Power Sys Dynamics And Stability
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Power System Dynamics and Stability
Power System Dynamics and Stability Dr. Federico Milano E-mail: [email protected] Tel.: +34 926 295 219
´ ´ ´ Departamento de Ingenier´ıa Electrica, Electronica, Automatica y Comunicaciones
´ Escuela Tecnica Superior de Ingenieros Industriales
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 1
2
Power System Dynamics and Stability
Programa de Doctorado
Technical and Economical Management of Generation, Transmission and Distribution Electric Energy Systems ´ ´ Area de Ingenier´ıa Electrica de la E.T.S. de Ingenieros Industriales de la Universidad de Castilla - La Mancha
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 2
3
Power System Dynamics and Stability
Note
➠ This course is partly based on the course ECE664 hold by Prof. Dr. C. ˜ Canizares at the University of Waterloo, Ontario, Canada. ˜ ➠ I wish to sincerely thank Prof. Dr. C. Canizares for his courtesy in sharing this material.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 3
4
Power System Dynamics and Stability
Objectives
➠ Understand the modeling and simulation of power systems from phasor analysis to electromagnetic transients.
➠ Discuss the basic definitions, concepts and tools for stability studies of power systems.
➠ Familiarize with basic concepts of computer modelling of electrical power systems.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 4
Power System Dynamics and Stability
5
Outlines: System Modeling
➠ Synchronous machine. ➠ Transformer. ➠ Transmission line. ➠ Cable. ➠ Loads.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 5
Power System Dynamics and Stability
6
Outlines: System Analysis
➠ Basic stability concepts: ➛ Nonlinear systems. ➛ Equilibrium points. ➛ Stability regions. ➠ Power Flow: ➛ System model. ➛ Equations and solution techniques. ➛ Contingency analysis.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 6
Power System Dynamics and Stability
7
Outlines: Voltage Stability
➠ Definitions. ➠ Basic concepts: ➛ Saddle-node bifurcation. ➛ Limit-induced bifurcation. ➠ Continuation Power Flow (CPF). ➠ Direct methods. ➠ Indices. ➠ Protections and controls. ➠ Real case example: August 2003 North American blackout. Universidad de Castilla-La Mancha June 26, 2008
Introduction - 7
Power System Dynamics and Stability
8
Outlines: Angle Stability
➠ Definitions. ➠ Small-disturbance: ➛ Hopf Bifurcations. ➛ Control and mitigation. ➛ Practical applications. ➠ Transient Stability (large-disturbance): ➛ Time domain. ➛ Direct Methods: ➳ Equal Area Criterion. ➳ Energy Functions. ➠ Real case example: May 1997 Chilean blackout. Universidad de Castilla-La Mancha June 26, 2008
Introduction - 8
Power System Dynamics and Stability
9
Outlines: Frequency Stability
➠ Definitions. ➠ Basic concepts. ➠ Protections and controls. ➠ Real case example: October 2003 Italian blackout. ➠ Real case example: November 2006 European blackout.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 9
Power System Dynamics and Stability
10
Outlines: Software Tools
➠ Outlines. ➠ UWPFLOW. ➠ Matlab. ➠ PSAT.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 10
11
Power System Dynamics and Stability
References
➠ P. Kundur, Power system stability and control, Mc Graw Hill, 1994. ➠ P. Sauer and M. Pai, Power system dynamics and stability, Prentice Hall, 1998.
➠ A. R. Bergen and V. Vittal, Power systems analysis, Second Edition, Prentice-Hall, 2000.
➠ C. A. Caizares, Editor, Voltage stability assessment: concepts, practices and tools, IEEE-PES Power System Stability Subcommittee Special Publication, SP101PSS, May 2003.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 11
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Power System Dynamics and Stability
References
➠ P. M. Anderson and A. A. Fouad, Power system control and stability, IEEE Press, 1994.
➠ J. Arrillaga and C. P. Arnold, Computer analysis of power systems, John Wiley, 1990.
➠ I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices, Oxford Science Publications, 1986.
➠ J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Second Edition, Springer-Verlag, 1993.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 12
13
Power System Dynamics and Stability
References
➠ M. Ili´c and J. Zaborszky, Dynamics and Control of Large Electric Power Systems, Wiley, New York, 2000. ˜ ➠ C. A. Canizares, UWPFLOW, available at www.power.uwaterloo.ca
➠ F. Milano, PSAT, Power System Analysis Toolbox, available at www.power.uwaterloo.ca
➠ Journal papers and technical reports. ➠ Course notes available on line.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 13
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Power System Dynamics and Stability
Evaluation
➠ Two projects are required. ➠ The projects concentrates in the various topics discussed in class. ➠ These will require the use of MATLAB, PSAT and UWPFLOW (the last two are free software for stability studies co-developed at the University of Waterloo, Canada).
➛ Reproducing examples presented in the slides using UWPFLOW and MATLAB.
➛ Stability analysis of the IEEE 14-bus test system using PSAT.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 14
15
Power System Dynamics and Stability
Evaluation
➠ Alternatively, the students can develop a “user defined model” in PSAT. ➠ Interested students are invited to contact Dr. Federico Milano.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 15
Power System Dynamics and Stability
16
Contents
➠ Introduction ➠ Generator Modeling ➠ Transmission System Modeling ➠ Load Modeling ➠ Power Flow Outlines ➠ Stability Concepts ➠ Voltage Stability ➠ Angle Stability ➠ Frequency Stability ➠ Software Tools ➠ Projects Universidad de Castilla-La Mancha June 26, 2008
Introduction - 16
Power System Dynamics and Stability
17
Generator Modeling
➠ Generator overview. ➠ Synchronous machine. ➠ Dynamic models of generators for stability analysis: ☞ Subtransient model ☞ Transient model ☞ Basic control models ➠ Steady-state model.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 1
18
Power System Dynamics and Stability
Generator Overview
➠ Generator components: Fuel
Steam at pressure, P
Torque
Power at voltage, V
Enthalpy, h
at speed, ω
Current, I
Turbine
Boiler
Generator
Firing
+
−
control
+
−
Ref ω
Power set-point
Excitation
Governor
+
−
system
Ref V
➠ Generator: ☞ Synchronous machine: AC stator and DC rotor. ☞ Excitation system: DC generator or static converter plus voltage regulator and stabilizer
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 2
19
Power System Dynamics and Stability
Generator Overview
➠ Generator with DC exciter and associated controls: Commutator
Field breaker
CT
dc Exc
Amplidyne field
ac Gen slip rings
Regulator transfer PT’s
M
Exciter field rheostat (manual control)
Magnetic amplifier
Stabilizer
Transistor amplifier
Reference and voltage sensing
Magnetic amplifier
Limiter sensing
Compensator
Other inputs
Other sensing 90−52
MG set PMG
M
Regulator power
Station auxiliary power
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 3
20
Power System Dynamics and Stability
Generator Overview
➠ Generator with static exciter and associated controls: Auxiliary power input start up
Voltage buildup element
Syn machine
FDR
Excitation power current transformer
CT
Excitation power potential transformer
slip rings
Power rectifier Excitation power Excitation breaker
PT’s Linear reactor Trinistat power amplifier
Gate circuitry
Base adjuster manual control Regulator transfer
Stabilizer
Reference & voltage sensing
Signal mixing amplifier
Limiter sensing
Rectifier current limit
Other sensing
Compensator
Voltage adjuster
Other inputs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 4
21
Power System Dynamics and Stability
Synchronous Machine a
ωr dr
Damper windings
c′
b′
Effects of induced currents in the rotor core
Q2
D
Q′2
b
Q1
F
Q′1
D′
θr ar
F′
DC field
a′
c qr
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 5
Power System Dynamics and Stability
22
Synchronous Machine
➠ Electrical (inductor) equations: v
dλ = ri + dt
dλ λ = L(θr )i + dt 1 T dL(θr ) Te = i i 2 dθr ➠ Mechanical (Newton’s) equations: J
dωr + Dωr dt dθr dt
= Tm − Te = ωr
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 6
23
Power System Dynamics and Stability
Synchronous Machine
➠ Stator equations: v0s rs vds = − vqs
rs
➠ Rotor equations: vF rF 0 = 0 0
i0s
0
λ0s
ids −ωr λqs − d λds dt λqs rs iqs −λds
rD rQ1 rQ2
iF
i D iQ1 iQ2
λF
d λ D + dt λQ1 λQ2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 7
24
Power System Dynamics and Stability
Synchronous Machine
➠ Magnetic flux equations:
λ0s λds λqs λF λD λQ1 λQ2
=
L0 Ld
Md
Md
Lq
Mq
Md
LF
Md
Md
Md
LD
Mq
Mq
LQ1
Mq
Mq
Mq
LQ2
i0s ids iqs iF iD iQ1 iQ2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 8
25
Power System Dynamics and Stability
Synchronous Machine
➠ Transformation equations: v0s vas vbs = P T vds vqs vcs P =
r
√
1/ 2
2 cos θr 3 sin θr
√
ias
i0s
ibs = P T ids iqs ics √
1/ 2 cos(θr − sin(θr −
1/ 2 2π 3 ) 2π 3 )
cos(θr + sin(θr +
2π 3 ) 2π 3 )
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 9
26
Power System Dynamics and Stability
Synchronous Machine
➠ Mechanical equations: 2 d 2 J ωr + Dωr p dt p d θr dt Te
= Tm − Te = ωr =
p (iqs λds − ids λqs ) 2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 10
27
Power System Dynamics and Stability
Synchronous Machine
➠ 3-phase short circuit at generator terminals: ia (t)
√
2|Ea (0)| x′′ d √
Subtransient component
2|Ea (0)| x′ d
Transient component
Steady state
t √
t
2|Ea (0)| xd
Ea (0) is the open-circuit RMS phase voltage.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 11
28
Power System Dynamics and Stability
Synchronous Machine
➠ Assuming balanced operation (null zero sequence), the detailed machine model can be reduced to phasor models useful for stability and steady-state analysis.
➠ Phasor models are based on the following assumptions: ➛ The rotor does not deviate “much” from the synchronous speed, i.e. ωr ≈ ωs = (2/p)2πf0 . ➛ The rate of change in rotor speed is ”small”, i.e. |dωr /dt| ≈ 0
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 12
29
Power System Dynamics and Stability
Synchronous Machine
➠ Subtransient models capture the full machine electrical dynamics, including the “few” initial cycles (ms) associated with the damper windings.
➠ Transient models capture the machine electrical dynamics starting with the field and induced rotor core current transient response.
➠ Damper windings transients are neglected. Steady-state models capture the machine electrical response when all transients have disappeared after “a few” seconds.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 13
Power System Dynamics and Stability
30
Subtransient Model
➠ External phase voltages and currents: 2 vas
θvas i2as θias
2 2 = vqs + vds vds +δ = tan−1 vqs
= i2qs + i2ds ids −1 = tan +δ iqs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 14
31
Power System Dynamics and Stability
Subtransient Model
➠ Subtransient “internal” voltages associated with the damper windings (D and Q1 ): d ′′ eq dt d ′′ e dt d e′′q − vqs
e′′d − vds
= =
1 ′ ′′ ′′ ′ )i − e − x + (x [e ds q] d d q ′′ Td0 1 ′ ′ ′′ ′′ [e − (x − x )i − e qs d q q d] ′′ Tq0
= rs iqs − x′′d ids
= rs ids + x′′q iqs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 15
32
Power System Dynamics and Stability
Subtransient Model
➠ Transient “internal” voltages associated with the field (F ) and rotor-core induced current windings (Q2 ): d ′ eq dt d ′ e dt d e′q − vqs
e′d − vds
= =
1 ′ ′ )i − e [e + (x − x ds f d q] d ′ Td0 1 ′ ′′ ′ [−(x − x )i − e qs q q d] ′ Tq0
= rs iqs − x′d ids = rs ids + x′q iqs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 16
33
Power System Dynamics and Stability
Subtransient Model
➠ Steady-state equations: ea − vqs
= rs iqs − xd ids
−vds
= rs ids + xq iqs
➠ Mechanical equations: d ∆ωr dt d δ dt
=
1 [Pm − vas ias cos(θvas − θias ) − D∆ωr ] M
= ∆ωr = ωr − ωs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 17
34
Power System Dynamics and Stability
Subtransient Model
➠ The subtransient reactances (x′′q , x′′d ) and open circuit time constants ′′ ′′ (Tq0 , Td0 ), as well as the transient reactances (x′q , x′d ) and open circuit ′ ′ time constants (Tq0 , Td0 ) are directly associated with the machine resistances and inductances:
xd
= ω0 Ld = xℓ + xM d
xq
= ω0 Lq = xℓ + xM q xM d xLF = xℓ + xLF + xM d xM q xLQ2 = xℓ + xLQ2 + xM q xM d xLF xLD = xℓ + xM d xLF + xM d xLD + xLF xLD xM q xLQ1 xLQ2 = xℓ + xM q xLQ1 + xM q xLQ2 + xLQ1 xLQ2
x′d x′q x′′d x′′q
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 18
Power System Dynamics and Stability
35
Subtransient Model
➠ Some definitions: xF
= xLF + xM d
xD
= xLD + xM d
xQ1
= xLQ1 + xM q
xQ2
= xLQ2 + xM q
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 19
36
Power System Dynamics and Stability
Subtransient Model
➠ Time constants: ′ Td0
=
′ Tq0
=
′′ Td0
=
′′ Tq0
=
xF ω 0 rF xQ2 ω0 rQ2 xM d xLD 1 xLD + ω 0 rD xD 1 xM q xLQ1 xLQ1 + ω0 rQ1 xQ1
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 20
37
Power System Dynamics and Stability
Subtransient Model
➠ Typical machine parameters: 2-pole
4-pole
Conventional
Conductor
Conventional
Conductor
Cooled
Cooled
Cooled
Cooled
xd x′d x′′ d
1.7-1.82
1.7-2.17
1.21-1.55
1.6-2.13
.18-.23
.264-.387
.25-.27
.35-.467
.11-.16
.23-.323
.184-.197
.269-.32
xq x′q x′′ q ′ Td0 ′′ Td0 ′ Tq0 ′′ Tq0
1.63-1.69
1.71-2.16
1.17-1.52
1.56-2.07
.245-1.12
.245-1.12
.47-1.27
.47-1.27
.116-.332
.116-.332
.12-.308
.12-.308
7.1-9.6
4.8-5.36
5.4-8.43
4.81-7.73
.032-.059
.032-.059
.031-.055
.031-.055
.3-1.5
.3-1.5
.38-1.5
.36-1.5
.042-.218
.042-.218
.055-.152
.055-.152
xℓ rs
.118-.21
.27-.42
.16-.27
.29-41
.00081-.00119
.00145-.00229
.00146-00147
.00167-00235
M
5-7
5-7
6-8
6-8
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 21
Power System Dynamics and Stability
38
Subtransient Model
➠ Typical machine parameters: Salient-pole
Combustion
Synchronous
Dampers
No dampers
Turbines
Compensator
xd x′d x′′ d
.6-1.5
.6-1.5
1.64-1.85
1.08-2.48
.25-.5
.25-.5
.159-.225
.244-.385
.13-.32
.2-.5
.102-.155
.141-.257
xq x′q x′′ q ′ Td0 ′′ Td0 ′ Tq0 ′′ Tq0
.4-.8
.4-.8
1.58-1.74
.72-1.18
= x′q
= x′q
.306
.57-1.18
.135-.402
.135-.402
.1
.17-.261
4-10
8-10
4.61-7.5
6-16
.029-.051
.029-.051
.054
.039-.058
−
−
1.5
.15
.033-.08
.033-.08
.107
.188-.235
xℓ rs
.17-.4
.17-.4
.113
.0987-.146
.003-.015
.003-.015
.034
.0017-006
M
6-14
6-14
18-24
2-4
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 22
39
Power System Dynamics and Stability
Subtransient Model
➠ In practice most of these constants are determined from short-circuit tests.
➠ All the e voltages are “internal” machine voltages directly assocated with the “internal” phase angle δ . ➠ The internal field voltage ef is directly proportional to the actual field dc voltage vF , and is typically controlled by the voltage regulator. ➠ The mechanical power Pm is controlled through the governor.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 23
40
Power System Dynamics and Stability
Subtransient Model
➠ A simple voltage regulator model (IEEE type 1): Se vr max vref +
Ka
+ − vm
−
Ta s + 1
−
vr +
1
vf
Te s + 1
vr min
1 Tr s + 1
Kf s Tf s + 1
V
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 24
41
Power System Dynamics and Stability
Subtransient Model
➠ A static voltage regulator model:
VT 1 1+sTR IT
VC
+ KIR
−
−
If +
0
Vuxl
+
Vmax
+
1+sTC 1+sTB
+ VS
If ref
vref
¯ | + (Rc + jXc )I¯ Vc = |V t t
−
Vmin
VA max 1+sTC1 1+sTB1
KA 1+sTA
−
Gate
+ ∗ Vef l
VA min
|VT |VR max − KC If Ef HV |VT |VR min
sKF 1+sTF
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 25
42
Power System Dynamics and Stability
Subtransient Model
➠ A simple governor model (hydraulic valve plus turbine): Torder +
ωref +
1/R − ω
∗ Tin
+ Tmin
Tmax Tin
1
T3 s + 1
T4 s + 1
Ts s + 1
Tc s + 1
T5 s + 1
Governor
Servo
Reheat
Tmech
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 26
43
Power System Dynamics and Stability
Subtransient Model
➠ A governor-steam turbine model: −
w_ref
+
P_ref
K1 1.0 1
1 + sT_1
1+ sT_R
1 + sT_2
+
1
1
1
T_C1
1 + sT_C2
sT_C3
− 0.0
+
P_GV
1
K_HP
F_HP
1 + sT_HP +
+
+
+ F_IP
−
1 − K_RH
P_mech P_MAX
+
1
1
1
sT_RH
1+sT_IP
1 + st_LP
F_LP
A
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 27
44
Power System Dynamics and Stability
Example
➠ For a 200 MVA, 13.8 kV, 60 Hz generator with the following p.u. data: rs
=
0.001096
xℓ
=
0.15
xd
=
1.7
xd
′
=
0.238324
LF = 0.00438
′′
⇒
xd
=
0.184690
⇒
LD = 0.00426
xq
=
1.64
Lq = 0.00435 Mq = 0.00395
xq
′
=
xq
xq
′′
=
0.185151
LQ1 = 0.00405
′
⇒
=
6.194876
rF = 0.000742
′′
⇒
Td0
=
0.028716
rD = 0.0131
′
⇒
Tq0
=
0
Tq0
′′
=
0.074960
p
=
2 M = 10 D = 0
Td0
Ld = 0.00451 Md = 0.00411
⇒
⇒
LQ2 = 0
⇒
⇒
rQ2 = 0 ⇒
rQ1 = 0.0540
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 28
45
Power System Dynamics and Stability
Example
➠ A three-phase fault from open circuit conditions, i.e. before the fault √ vas = 13.8/ 3 kV and ias = 0, and after the fault vas = 0, is simulated using the detailed machine equations:
vas , vbs ,vcs [kV]
2 1 0 −1 −2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 29
46
Power System Dynamics and Stability
Example 10 9 8
iF [p.u.]
7 6 5 4 3 2 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 30
47
Power System Dynamics and Stability
Example
ia [p.u.]
20 0 −20
0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ib [p.u.]
20 0 −20
ic [p.u.]
20
0
−20
t [s] Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 31
48
Power System Dynamics and Stability
Transient Model
➠ Obtained by eliminating electromagnetic differential equations and the damper winding dynamic equations.
➠ For this reason, the damping D in the mechanical equations, which is typically a small value, is assumed to be large to indirectly model the significant damping effect of these windings on ωr .
➠ It is the typical model used in stability studies.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 32
49
Power System Dynamics and Stability
Transient Model
➠ Neglecting the induced currents in the rotor (winding Q2 ): ′ =0 x′q = xq Tq0
⇒
e′d = 0
➠ This leads to the transient equations: d ′ e dt a
= =
e′a ∠δ
1 ′ ′ [e + (x − x )i − e f d ds d a] ′ Td0 1 [ef − ea ] ′ Td0
= vas ∠θvas + rs ias ∠θias +jx′d ids ∠(δ + π/2) + jxq iqs ∠δ
ea ∠δ
= vas ∠θvas + rs ias ∠θias +jxd ids ∠(δ + π/2) + jxq iqs ∠δ Universidad de Castilla-La Mancha
June 26, 2008
Generator Modeling - 33
50
Power System Dynamics and Stability
Transient Model
➠ The phasor diagram in this case is: e′a
= vas + rs ias + jx′d ids + jxq iqs
ea
= vas + rs ias + jxd ids + jxq iqs
ℑ
ea e′a
iqs
ids
jxq iqs
δ vas jxd ids rs ias
j(xd − x′d )ids ℜ
ias
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 34
51
Power System Dynamics and Stability
Transient Model
➠ For faults near the generator terminals, the q axis has little effect on the system response, i.e. iqs ≈ 0. ➠ This results in the classical voltage source and transient reactance generator model used in simple stability studies:
rs
jx′d
ias ∠θias +
e′a ∠δ
Pm
ωr
vas ∠θvas −
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 35
52
Power System Dynamics and Stability
Transient Model
➠ A further approximation in some cases is used by neglecting the field ′ dynamics, i.e. Td0 = 0. ➠ In this case, e′a is a “fixed” variable controlled directly through the voltage regulator via ef . ➠ The limits in the voltage regulator are used to represent limits in the field and armature currents.
➠ These limits can be “soft”, i.e. allowed to temporarily exceed the hard steady-state limits, to represent under- and over-excitation.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 36
53
Power System Dynamics and Stability
Steady-state model
➠ When all transient are neglected, the generator model becomes: ea ∠δ
= vas ∠θvas + rs ias ∠θias +jxd ids ∠(δ + π/2) + jxq iqs ∠δ
ea ias ∠θias
= ef = ids ∠δ + jiqs ∠δ
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 37
54
Power System Dynamics and Stability
Steady-state model
➠ For a round rotor machine (xd = x′d ), the steady-state model leads to the classical short circuit generator model:
rs
jxd
ias ∠θias +
ea ∠δ
Pm
ωr
vas ∠θvas −
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 38
55
Power System Dynamics and Stability
Steady-state model
➠ Based on this simple model, the field and armature current limits can be used to define the generator capability curves (for a given terminal voltage vas
= vt ): Q ef limit
Qmax
vt ea x
ia limit P
vt i ∗ a
Qmin Pmin −
Pmax
2 vt x
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 39
56
Power System Dynamics and Stability
Steady-state model
➠ Considering the voltage regulator effect, the generator can be modeled as a constant terminal voltage within the generator reactive power capability, delivering constant power (Pm ).
➠ This yields the PV generator model for power flow studies: P
Q
V
where P
Qmin
= Pm = constant, and V = vt = constant for ≤ Q ≤ Qmax ; otherwise, Q = Qmax,min and V is allowed to
change.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 40
57
Power System Dynamics and Stability
Example
➠ The generator data are: va
PG
v∞ = 1∠0 j0.1
θr (t) = ω0 t + π/2 + δ xd ′ Td0
M
= xq = 0.9, =
x′d = 0.2
2s
= large ⇒ δ = constant
The generator is operating in steady-state delivering PG
= 0.5 at
ea = 1.5. At t = 0 there is a fault and the line is disconnected. Find va (t) for t > 0. Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 41
58
Power System Dynamics and Stability
Example
➠ Steady-state conditions: ea ∠δ
⇒ ias PG
= vas ∠θvas + (rs + jxd )ias ∠θias = v∞ + j(xd + xL )ias 1.5∠δ − 1 = j(0.9 + 0.1) = 1.5 sin δ − j(1.5 cos δ − 1) = ℜ{v∞ i∗as } = ℜ{i∗as }
=
1.5 sin δ
⇒ δ
=
sin−1 (PG /1.5) = 19.47◦
ias
=
0.65∠ − 39.64◦
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 42
59
Power System Dynamics and Stability
Example
iqs ids ⇒ ea ∠δ
= ias cos(θias − δ)∠δ = 0.334∠19.47◦
= ias sin(θias − δ)∠(δ + π/2) = 0.558∠ − 70.53◦ = vas + rs ias + jx′d ids + jxq iqs = v∞ + j(xd + xL )ias + jx′d ids + jxq iqs
= 1 + j0.1(0.65∠ − 39.64◦ ) + j0.2(0.558∠ − 70.53◦ ) j0.9(0.334∠19.47◦ )
d ′ ea dt
= 1.110∠19.47◦ 1 ′ ′ = [e + (x − x )i − e a d ds d a] ′ Td0
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 43
60
Power System Dynamics and Stability
Example
➠ Transient: d ′ 1 ′ ′ ea (t) = [e + (x − x )i − e f d ds d a (t)] ′ dt Td0 ef = ea → steady-state ids
=
1.5
=
0 → open line
e′a (0) = d ′ ⇒ ea (t) = dt
1.110 1 [1.5 − e′a (t)] 2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 44
61
Power System Dynamics and Stability
Example
➠ Solution: d ′ ea (t) = −0.5e′a (t) + 0.75 dt ⇒ e′a (t) = Ae−0.5t + B e′a (0) = A + B = 1.110
e′a (∞) = B = 0.75/0.5 = 1.5 ⇒ e′a (t) = −0.390e−0.5t + 1.5
va (t) = e′a (t) → since ias = ids = iqs = 0 1.5
e′a
1.11 0
t
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 45
Power System Dynamics and Stability
62
Transmission System Modeling
➠ Transformers: ☞ Single phase: ➛ Detailed model ➛ Phasor model ☞ Three phase: ➛ Phase shifts ➛ Models
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 1
Power System Dynamics and Stability
63
Transmission System Modeling
➠ Transmission Lines: ☞ Single phase: ➛ Distributed parameter model ➛ Phasor lumped model ☞ Three phase: ➛ Distributed parameter model ➛ Reduced models ➠ Underground cables
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 2
Power System Dynamics and Stability
64
Single-phase Transformers
➠ The basic characteristics of this device are: ➛ Flux leakage around the transformer windings is represented by a leakage inductance Lℓ . ➛ The core is made of magnetic material and is represented by a magnetization inductance (Lm ≫ Lℓ ), but saturates.
➛ Losses in the windings (Cu wires) and core (hysteresis and induced currents) are represented with lumped resistances (r and Gm ). ➛ Steps up or down the voltage/current depending on the turn ratio a = N1 /N2 = V1 /V2 .
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 3
65
Power System Dynamics and Stability
Single-phase Transformers leakage core
i2
i1
+
+ v1
N1
v2
N2
λℓ1
λℓ2 λm
magnetizing
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 4
Power System Dynamics and Stability
66
Single-phase Transformers
➠ The equivalent circuit is: dλ di v = ri + = ri + L dt dt i1 r1 v1 = ⇒ i2 r2 v2 Lℓ1 + Lm Lm /a d i1 + dt i2 Lm /a Lℓ2 + Lm /a2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 5
67
Power System Dynamics and Stability
Single-phase Transformers
➠ Equivalent circuit: i1 + v1
r1
Lℓ1
i im
Gm
− Lm
Lℓ2
ai
r2
i2
+
+
+
e
e/a
v2
−
−
−
N1 : N2 a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 6
Power System Dynamics and Stability
68
Single-phase Transformers
➠ The phasor equivalent circuit is:
1 V1 = (Zℓ1 + Zm )I1 + Zm I2 a 1 aV2 = Zm I1 + (a2 Zℓ1 + Zm ) I2 a V1 Zℓ1 + Zm Zm /a I1 ⇒ = V2 Zm /a Zℓ2 + Zm /a2 I2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 7
69
Power System Dynamics and Stability
Single-phase Transformers
➠ Phasor equivalent circuit: I1 + V1 −
a2 Zℓ2
Zℓ1
+
Im Ym
I2 /a
I2 +
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 8
70
Power System Dynamics and Stability
Single-phase Transformers
➠ This can also be readily transformed into a ABCD input-output form based on the following approximation, since Zm ≫ Zℓ1 (Zℓ1 ≈ a2 Zℓ2 ):
V1 I1
=
Zℓ
a(1 + Zℓ Ym )
Zℓ /a
aYm 1/a A B V2 = C C −I2
V2 −I2
= r1 + jXℓ1 + a2 (r2 + jXℓ2 )
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 9
71
Power System Dynamics and Stability
Single-phase Transformers
➠ Phasor equivalent circuit with the approximation Zm ≫ Zℓ1 : I1 + V1 −
I2 /a
Zℓ
+
Im Ym
I2 +
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 10
Power System Dynamics and Stability
72
Single-phase Transformers
➠ Or Π form: Z =
Zℓ a 1 Zℓ
Y1
=
(1 − a)
Y2
=
(a2 Zℓ Ym + a2 + a)
1 Zℓ
1 for Ym ≈ 0 ≈ (a − a) Zℓ 2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 11
73
Power System Dynamics and Stability
Single-phase Transformers
➠ Π equivalent circuit: Z
I1
I2
+ V1
+ Y1
Y2
V2
−
−
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 12
Power System Dynamics and Stability
74
Single-phase Transformers
➠ Neglecting Zm (Ym is small given the core magnetic properties): 1. Time domain:
v1 i2
di1 + av2 = ri1 + L dt = −ai1
2. Phasor domain:
V1 I2
= Zℓ I1 + aV2 = −aI1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 13
75
Power System Dynamics and Stability
Single-phase Transformers
➠ Equivalent circuit neglecting Zm : I1
I2 = −I1 /a
Zℓ
+ V1
+
+
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 14
Power System Dynamics and Stability
76
Single-phase Transformers
➠ Certain transformers have built-in Under-Load Tap Changers (ULTC). ➠ This is either operated manually (locally or remote controlled) or automatically with a voltage regulator; the voltage control range is limited (≈
10%) and on discrete steps (≈ 1%).
➠ The time response is in the order of minutes, with 1-2 min. delays, due to ULTCs being implemented using electromechanical systems.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 15
Power System Dynamics and Stability
77
Single-phase Transformers
➠ These are typically used to control the load voltage side, and hence are used at subtransmission substations.
➠ Nowadays, power electronic switches are used, leading to Thyristor Controller Voltage Regulators (TCVR), which are faster voltage controllers and are considered Flexible AC Transmission systems (FACTS).
➠ These types of transformers are modelled using the same transformer models, but a may be assumed to be a discrete controlled variable through a voltage regulator with a dead-band.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 16
Power System Dynamics and Stability
78
Single-phase Transformers
➠ Transformers with special connections and under-load tap changers can also be used for phase shift control and are known as Phase Shifters.
➠ These control the phase shift difference between the two terminal voltages within approximately ±30◦ , thus increasing the power capacity of a transmission line (e.g. interconnection between Ontario and Michigan).
➠ Phase shifters are modeled using a similar model but the tap ratio is a phasor as opposed to a scalar:
a = a∠α ➠ A Π equivalent cannot be used in this case. Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 17
79
Power System Dynamics and Stability
Single-phase Transformers
➠ Phasor equivalent circuit with complex tap ratio a: I1 + V1 −
I2 /a∗
Zℓ
+
Im Ym
I2 +
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 18
Power System Dynamics and Stability
80
Single-phase Transformers (Example)
➠ Single phase 8/80 kV, 30 MVA transformer with Xℓ = 10% and Xm ≈ 10Xℓ . ➠ Detailed model parameters: a = Lℓ1
= =
Lℓ2
=
Lm
=
8/80 = 0.1 Xℓ1 Xℓ ≈ ω0 2ω0 0.1 (8 kV)2 = 0.283 mH 2 · 377 30 MVA Lℓ1 = 28.3 mH 2 a Xℓ 0.1 (8 kV)2 10 = 10 = 5.66 mH ω0 377 30 MVA Universidad de Castilla-La Mancha
June 26, 2008
Transmission System Modeling - 19
Power System Dynamics and Stability
81
Single-phase Transformers (Example)
➠ Per unit parameters: Zℓ
= j0.1
Zm
=
Ym
=
a =
10Zℓ = j1 1 = −j1 Zm 8/8 kV =1 80/80 kV
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 20
Power System Dynamics and Stability
82
Single-phase Transformers (Example)
➠ Per unit parameters:
A
=
B
=
C = D =
1(1 + j0.1(−j1)) = 1.1 j0.1 = j0.1 1 1(−j1) = −j1 1 =1 1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 21
Power System Dynamics and Stability
83
Single-phase Transformers (Example)
➠ Per unit parameters:
Z = Y1 Y2
= =
j0.1 = j0.1 1 1 =0 (1 − 1) j0.1 1 = −j1 (1 (j0.1)(−j1) + 1 − 1) j0.1 2
2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 22
Power System Dynamics and Stability
84
Three-phase Transformers a1
+
T1 +
aVa2 − b1
+
Va2 a:1
−
T2
c1
+
Vb2 a:1
−
T3
c2
+
aVc2 −
b2
+
aVb2 −
a2
Vc2 a:1
−
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 23
Power System Dynamics and Stability
85
Three-phase Transformers
➠ The 3 single-phase transformers form a 3-phase bank that induces a phase shift, depending on the connection:
Vab1 Vab2 ⇒ Vab1 a apu
= aVa2 √ 3∠30◦ Va2 = √ 3a∠30◦ Vab2 = √ 3a∠30◦ = √ 3apu ∠30◦ =
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 24
Power System Dynamics and Stability
86
Three-phase Transformers
∆−Y:
a=
√ √
3a∠30◦
3apu ∠30◦ √ a = 3a∠ − 30◦ √ apu = 3apu ∠ − 30◦
apu = Y−∆
:
Y−Y
:
a=a
∆−∆:
a=a
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 25
87
Power System Dynamics and Stability
Three-phase Transformers
➠ In balanced, “normal” systems, the net phase shift between the generation and load sides is zero, and hence is neglected during system analyses: Generator
Load
side
side
∆
Y
Y
∆
➠ In these systems, the p.u. per-phase models of the transformers are identical to the equivalent single-phase transformer models.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 26
88
Power System Dynamics and Stability
Three-phase Transformers
➠ For “smaller” transformers (e.g. load transformers), integral designs are preferred to transformers banks: a
b
c
core
3 φ windows
vabc1 vabc2
r13×3
L113×3
L123×3
L213×3
L223×3
=
r23×3
iabc1
+
iabc2 i d abc1 dt iabc 2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 27
Power System Dynamics and Stability
89
Saturation
➠ The magnetization inductance Lm changes with the magnetization current due to saturation of the magnetic core.
➠ Saturation occurs due to a reduction on the number of “free” magnetic dipoles in the enriched core.
➠ This results in the core behaving more like air than a magnet, i.e. magnetic “conductivity” decreases.
➠ It is typically represented using a piece-wise linear model.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 28
Power System Dynamics and Stability
90
Saturation
➠ The magnetization inductance Lm changes with the magnetization current due to saturation of the magnetic core: λm
Lm2 Lm1
ims
Lm (im ) =
L
m1
im
for
im ≤ ims
Lm2 for im > ims
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 29
Power System Dynamics and Stability
91
Single-phase Transmission Line
➠ Alossless line can be represented using a series of lumped elements: l
=
c = D
D µ0 ln ′ 2π R 2πǫ0 ln(D/R)
→
distance between wires
→
wire GMR
R
→
wire radius
µ0
=
ǫ0
=
R′
4π × 10−7 [H/m]
8.854 × 10−12 [F/m]
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 30
92
Power System Dynamics and Stability
Single-phase Transmission Line
i1
i
+
l [H/m] +
v1
dv
i2 −
c [F/m]
−
x
+
+
di v
v2
−
−
dx
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 31
Power System Dynamics and Stability
93
Single-phase Transmission Line
➠ The equations for this line are: dv di
di ∂v ∂i ⇒ = −l dt ∂x ∂t ∂i ∂v dv ⇒ = −c = −cdx dt ∂x ∂t
= −ldx
➠ These are D’Alambert equations with solution: 1 1 v1 (t) − i2 (t − τ ) + v2 (t − τ ) i1 (t) = Zc Zc | {z } I2 (t−τ )
i2 (t)
=
1 1 v2 (t) − i1 (t − τ ) + v1 (t − τ ) Zc Zc {z } | I1 (t−τ )
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 32
94
Power System Dynamics and Stability
Single-phase Transmission Line
➠ where: Zc
=
s = τ
=
p
l/c → chracteristic impedance 1 √ → wave speed lc d → travelling time for line length s
➠ Distributed parameter equivalent circuit: i1
r/2
+ v1
r/2
I2 (t − τ )
+
I1 (t − τ )
Zc
Zc
−
i2
v2 −
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 33
95
Power System Dynamics and Stability
Single-phase Transmission Line
➠ Example: t=0 +
E
i1 +
v1 −
i2 = 0 +
Trans. Line
v2
−
−
v1 (t) = E i2 (t) =
0
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 34
96
Power System Dynamics and Stability
Single-phase Transmission Line
➠ Example: t
I1 (t − τ )
I2 (t − τ )
i1
i2
0
0
0
E/Zc
0
τ
2E/Zc
0
E/Zc
2E
2τ
2E/Zc
2E/Zc
2E
3τ
0
2E/Zc
−E/Zc
4τ
0
0
5τ
2E/Zc
0
−E/Zc
0
E/Zc
0
E/Zc
2E
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 35
97
Power System Dynamics and Stability
Single-phase Transmission Line
➠ Example: i1 E/Zc τ
2τ
3τ
4τ
5τ
6τ
t
τ
2τ
3τ
4τ
5τ
6τ
t
−E/Zc v2 2E
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 36
Power System Dynamics and Stability
98
Single-phase Transmission Line
➠ Phasor model from the distributed line model: d V = −(r + jωl)I = −zI dx d I = −(jωc)V = −yI dx 0 −z V d V = ⇒ dx I −y 0 I
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 37
Power System Dynamics and Stability
99
Single-phase Transmission Line
➠ The solution to this set of linear dynamical equations is: cosh γd Zc sinh γd V2 d V1 = dx −I2 1/Zc sinh γd cosh γd I1 where
cosh γd = sinh γd = γ
=
Zc
=
eγd + e−γd 2 eγd − e−γd 2 √ zy → propagation constant p z/y → characteristic impedance
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 38
100
Power System Dynamics and Stability
Single-phase Transmission Line
➠ This can be converted into the Π equivalent circuit: Z′
I1
I2
+ V1
+ Y1′ /2
Y2′ /2
V2
−
−
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 39
Power System Dynamics and Stability
101
Single-phase Transmission Line
➠ Π equivalent circuit: ′
Z
=
sinh γd zd |{z} γd Z
Y
′
=
tanh γd yd |{z} γd Y
for
d < 250 km
⇒
for
d < 100 km
⇒
Z′ ≈ Z
Z′ ≈ Z
Y′ ≈ Y Y′ ≈ 0
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 40
102
Power System Dynamics and Stability
Three-phase Transmission Line dα n
n
dδ
2R a
b
c dβ
dγ
d×d 11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 41
103
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Typically the phase wires are bundled (e.g. 4 wires/phase) and the guard wires are grounded at every tower, i.e. correspond to the neutral.
➠ For a total of N wires, the per unit length equations are: 2
2
3
v1 6 7 6v 7 27 ∂ 6 6 7 . 7 ∂x 6 6 .. 7 4 5 vN
=
+
r11 6 6r 6 21 6 6 .. 6 . 4 rN 1 2 l11 6 6l 6 21 6 6 .. 6 . 4 lN 1
r12 r22 . . .
rN 2 l12 l22 . . .
lN 2
32
3
i1 76 7 6 7 . . . r2N 7 7 6 i2 7 76 7 . 76 . 7 .. . 76 . 7 . . 54 . 5 . . . rN N iN 3 2 3 i1 . . . l1N 7 6 7 6 7 . . . l2N 7 7 ∂ 6 i2 7 7 6 7 . 7 . 7 .. . 7 ∂t 6 . 7 6 . . 5 4 . 5 . . . lN N iN ...
r1N
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 42
104
Power System Dynamics and Stability
Three-phase Transmission Line 2
2
3
i1 6 7 6i 7 27 ∂ 6 7 6 − . 7 6 ∂x 6 . 7 4 . 5 iN
=
+
cN ×N
=
g11 6 6g 6 21 6 6 .. 6 . 4 gN 1 2 p11 6 6p 6 21 6 6 .. 6 . 4 pN 1
g12
...
g22
...
. . .
..
gN 2
...
p12
...
p22
...
. . .
..
pN 2
...
.
.
32
3
v1 76 7 6 7 g2N 7 7 6 v2 7 76 7 . 76 . 7 . 76 . 7 . 54 . 5 vN gN N 3−1 2 3 p1N v1 7 6 7 6v 7 p2N 7 7 6 27 ∂ 7 6 7 . 7 . 7 ∂t 6 . 7 . 7 6 . 5 4 . 5 pN N vN g1N
p−1 N ×N
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 43
105
Power System Dynamics and Stability
Three-phase Transmission Line
➠ In phasor form, these equations are: 2
2
3
V1 6 7 6V 7 27 d 6 6 7 . 7 6 dx 6 . 7 4 . 5 VN 2 3 I1 6 7 6I 7 27 d 6 6 7 . 7 dx 6 6 .. 7 4 5 IN
⇒
=
=
z11 6 6z 6 21 −6 6 .. 6 . 4 zN 1 2 y11 6 6y 6 21 −6 6 .. 6 . 4 yN 1
d V = −[z]I dx
z12
...
z22
...
. . .
..
zN 2
...
y12
...
y22
...
. . .
..
yN 2
...
.
.
32
3
I1 76 7 6 7 z2N 7 7 6 I2 7 76 7 . 76 . 7 . 76 . 7 . 54 . 5 zN N IN 32 3 y1N V1 76 7 6 7 y2N 7 6 V2 7 7 76 7 . 76 . 7 . 76 . 7 . 54 . 5 yN N vN z1N
d I = −[y]V dx
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 44
106
Power System Dynamics and Stability
Three-phase Transmission Line
➠ The image method is useful for computing line parameters: i
00 2Ri 11 00 11 00 11
dij j
hi
φij
Dij
111 000 2Rj 000 111 000 111 000 111
hj ρ [Ωm]
hj hi
Earth
j image
i image
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 45
Power System Dynamics and Stability
107
Three-phase Transmission Line
➠ The line parameters zij and yij can be computed using Carson’s formulas:
rii
= riint + ∆rii
riint → from tables
rij
=
∆rij
∆rij
=
4ω10−4 {π/8 − b1 a cos φ
+b2 [(c2 − ln a)a2 cos 2φ + φa2 sin 2φ]
+b3 a3 cos 3φ − d4 a4 cos 4φ − b5 a5 cos 5φ
+b6 [(c6 − ln a)a6 cos 6φ + φa6 sin 6φ] a =
+b7 a7 cos 7φ − d8 a8 cos 8φ − b9 a9 cos 9φ + . . .} p √ −4 4π 510 D f /ρ Universidad de Castilla-La Mancha
June 26, 2008
Transmission System Modeling - 46
Power System Dynamics and Stability
108
Three-phase Transmission Line
D=
2h
i
for i
=j
Dij for i 6= j √ 2 b1 = 6 s bk = bk−2 k(k + 2) c2 = 1.3659315 +1 k = 1, 2, 3, 4, 5, . . . s= −1 k = 5, 6, 7, 8, 13, . . .
0 for i = j φ= φij for i 6= j
1 b2 = 16 π d k = bk 4
1 1 ck = ck−2 + + k k+2
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
109
Three-phase Transmission Line
xii xij ∆xij
µ0 2hi = ω ln + xiint + ∆xii 2π Ri µ0 Dij = ω ln + ∆xij 2π dij =
xiint → from tables(≈ 0)
4ω10−4 {1/2(0.6159315 − ln a) − b1 a cos φ − d2 a2 cos 2φ
+b3 a3 cos 3φ − b4 [(c4 − ln a)a4 cos 4φ + φa4 sin 4φ] +b5 a5 cos 5φ − d6 a6 cos 6φ + b7 a7 cos 7φ
−b8 [(c8 − ln a)a8 cos 6φ + φa8 sin 8φ] + . . .}
gij pii
= gij ≈ 0 2hi 1 ln = 2πǫ0 Ri
1 2Dij pij = ln 2πǫ0 dij
Universidad de Castilla-La Mancha June 26, 2008
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110
Power System Dynamics and Stability
Three-phase Transmission Line f (Hz)
′ ′ Rac /Rdc
L′ac /L′dc
f (Hz)
′ ′ Rac /Rdc
L′ac /L′dc
2
1.0002
0.99992
4000
7.1876
0.15008
4
1.0007
0.99970
6000
8.7471
0.12258
6
1.0015
0.99932
8000
10.0622
0.10617
8
1.0026
0.99879
10000
11.2209
0.09497
10
1.0041
0.99812
20000
15.7678
0.06717
20
1.0164
0.99254
40000
22.1988
0.04750
40
1.0632
0.97125
60000
27.1337
0.03879
60
1.1347
0.93898
80000
31.2942
0.03359
80
1.2233
0.89946
100000
34.9597
0.03004
100
1.3213
0.85639
200000
49.3413
0.02124
200
1.7983
0.66232
400000
69.6802
0.01502
400
2.4554
0.47004
600000
85.2870
0.01227
600
2.9421
0.38503
800000
98.4441
0.01062
800
3.3559
0.33418
1000000
110.0357
0.00950
1000
3.7213
0.29924
2000000
155.5154
0.00672
2000
5.1561
0.21204
4000000
219.8336
0.00475
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
111
Three-phase Transmission Line
➠ The [z] parameters depend on the line frequqnecy ω , i.e. this is a frequency dependent model.
➠ These number of conductors, and hence equations, can be reduced based on the following observations:
➛ The voltage of all Nb conductors in a phase bundled are at the same voltage (e.g. v1 = v2 = · · · = vN b = va ). ➛ The current in each phase is shared approximately equally by each conductor in the bundle (e.g. i1 = i2 = · · · = iN b = ia /Nb ). ➛ The voltage in the guard wires is zero (e.g. vg = 0).
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
112
Three-phase Transmission Line
➠ This reduces the matrices to:
[z]N ×N
[y]N ×N
⇒
→
→
d Vabc dt d Iabc dt
zaa zab zac yaa yab yac
zab zbb zbc yab ybb ybc
zac
zbc zcc yac ybc ycc
= −[zabc ]Iabc = −[yabc ]Vabc
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 51
113
Power System Dynamics and Stability
Three-phase Transmission Line
➠ A line is transposed to balance the phases. ➠ The length of the barrel B must be much less than the wavelength (s/f ≈ 5000 km @ 60 Hz). B ≈ 50 km. B
a b c
zaa [[zabc ] = zab zac
zab zbb zbc
1 2 3
3 1 2
B/3
B/3
zac
zbc zcc
2 3 1
B/3
yaa [yabc ] = yab yac
yab ybb ybc
yac
ybc ycc
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
114
Three-phase Transmission Line
➠ The phasor equations can be diagonalized using eigenvalue (modal) analysis techniques:
[zm ] [ym ]
= TIT [z]N ×N TI (diagonal matrix)
= TVT [z]N ×N TV
Vm
= TV−1 V = TIT V
Im
= TI−1 I = TVT I
(diagonal matrix)
➠ There are N modes of propogation, one for each eigenvalue.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 53
115
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Diagonalization of a 3-phase transposed line through sequence transformation:
TS
=
a =
1 1 √ 1 3 1
V0pn
= TS−1 Vabc
I0pn
= TS−1 Iabc
1 2
a
a
1∠120◦
1
a a2
⇒
TS−1
1 1 = √ 1 3 1
1 a a2
1
a a 2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 54
116
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Transformation of zabc into z0pn : [z0pn ] = TS−1 [zabc ]TS zs + 2zm 0 = 0 zs − zm 0 0 zp = zn
0 z0 0 = 0 zs − zm 0
z0 ≈ 3zp
0 zp 0
0
0 zn
➠ Similar for [y0pn ] = TS−1 [yabc ]TS .
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 55
117
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Diagonalization of a 3-phase transposed line through 0αβ transformation:
V0αβ
= T −1 Vabc
= T −1 Iabc √ 2 0 1 p √ 1 T = √ 1 −1/ 2 ⇒ 3/2 3 p √ 1 −1/ 2 − 3/2 I0αβ
T −1 = T T
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 56
118
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Transformation of zabc into z0αβ : [z0αβ ]
= T T [zabc ]T zs + 2zm = 0 0
0 zs − zm 0
zα = zβ = zp
0 z0 0 = 0 zs − zm 0
z0 ≈ 3zα
0 zα 0
0
0 zβ
➠ Similar for [y0αβ ] = T T [yabc ]T .
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
119
Three-phase Transmission Line
➠ These simplifications lead to the following per-phase (positive sequence), per-unit length formulas:
r
=
l
=
c =
rtables Nb µ0 Dm ln ′ 2π Rb 2πǫ0 ln DRmb
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
120
Three-phase Transmission Line
➠ where: ➛ Dm is the GMD of the 3 phases: p Dm = 3 dab dac dbc
➛ Rb′ is the GMR of the bundled and wires: p ′ Nb R′ d12 d13 · · · d1Nb Rb = ➛ Rb is the GMR of the bundled: p Nb Rd12 d13 · · · d1Nb Rb = Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 59
121
Power System Dynamics and Stability
Example 1
➠ 24.14 km transposed distributed line: 70.68 mH
0.741 Ω
Bus 1
15 miles
Bus 2
Bus 3
70.16 mH
1 ms
219.1 mH
√ 325/ 3 kV
251.2 Ω
R0′ = 0.3167 Ω/km
R1′ = 0.0243 Ω/km
L′0 = 3.222 mH/km
L′1 = 0.9238 mH/km
C0′ = 0.00787 µF/km
C1′ = 0.0126 µF/km
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
122
Example 1 2
3
187.79 cos(377t) 6 7 6187.79 cos(377t − 2π/3)7 4 5 187.79 cos(377t + 2π/3)
=
+ 2
3
vBU S10 6 7 6vBU S1 7 α5 4 vBU S1β 2
=
3
i1 6 07 6i1 7 4 α5 i1β
=
2 3 2 32 3 vBU S1a 0.714 0 0 i1 6 7 6 7 6 a7 6 vBU S1 7 + 6 0 6 7 0.714 0 7 4 4 5 4 i1b 5 b5 vBU S1c 0 0 0.714 i1c 2 32 3 0.07068 0 0 i1 7 6 a7 d 6 6 7 6 i1 7 0 0.07068 0 4 54 b5 dt 0 0 0.07068 i1c 32 3 2 vBU S1a 1 1 1 7 √ √ 76 √ 1 6 7 6 6 √ 4 2 −1/ 2 −1/ 2 5 4 vBU S1b 7 5 p p 3 0 3/2 − 3/2 vBU S1c | {z }
3 2 i1a 7 6 −1 6 7 T 4 i1b 5 i1c
T −1
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
123
Example 1
i r0 1 h vBU S10 − i10 (t) − I20 (t − τ0 ) i10 (t) = Zc0 2 i 1 h r0 i20 (t) = vBU S20 − i20 (t) − I10 (t − τ0 ) Zc0 2 r0 = 0.3167 × 24.14 = 7.6451 Ω r 3.222 × 10−3 = 639.85 Ω ZC0 = 0.00787 × 10−6 p τ0 = 24.14 3.222 × 10−3 0.00787 × 10−6 = 0.12156 ms
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
124
Example 1
i1α (t) = i2α (t) = rα
=
ZCα
=
τα
=
i rα 1 h vBU S1α − i1α (t) − I2α (t − τα ) Zcα 2 i 1 h rα vBU S2α − i2α (t) − I1α (t − τα ) Zcα 2 0.0243 × 24.14 = 0.5866 Ω r 0.9238 × 10−3 = 270.77 Ω 0.0126 × 10−6 p 24.14 0.9238 × 10−3 0.0126 × 10−6 = 0.08236 ms
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
125
Example 1
rβ
1 = Zcβ 1 = Zcβ = rα
ZCβ
= ZCα
i1β (t) i2β (t)
τβ
h h
vBU S1β vBU S2β
i rβ − i1β (t) − I2β (t − τβ ) 2 i rβ − i2β (t) − I1β (t − τβ ) 2
= τα
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 64
126
Power System Dynamics and Stability
Example 1 2 3 vBU S2a 6 7 6 vBU S2 7 4 b5 vBU S2c 2 3 i1 6 a7 6 i1 7 4 b5 i1c 3
2 vBU S2a 6 7 6 vBU S2 7 4 b5 vBU S2c
=
2 1 1 6 √ 6 1 34 1 | 2
=
=
√
2 √ −1/ 2 √ −1/ 2 {z T
3
32
3
vBU S20 76 7 7 6 3/2 5 4vBU S2α 7 5 p vBU S2β − 3/2 } 0
p
i2 6 07 7 T6 4i2α 5 i2β 2 32 3 251.2 0 0 i2 6 7 6 a7 6 7 −6 251.2 0 7 4 0 5 4 i2b 5 0 0 251.2 i2c 2 3 32 0.28926 0 0 i2 7 6 a7 d 6 6 7 6 i2 7 − 0 0.28926 0 4 54 b5 dt 0 0 0.28926 i2c
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 65
127
Power System Dynamics and Stability
Example 1 300
vaBUS2 vasource
200
[kV]
100
0
−100
−200
−300
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 66
128
Power System Dynamics and Stability
Example 1 300
200
100
[kV]
0
−100
−200
vaBUS2 vbBUS2 vcBUS2
−300
−400
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 67
129
Power System Dynamics and Stability
Example 2 50′ n
n
28′ 8′′
1′ a
b
c 45′
110′
40′ × 40′
11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
130
Example 2
➠ Line data: ➛ 4 Drake wires per phase (Nb = 4) ➛ 927 kcmil = 469.8 mm2 ➛ ACSR, 24 Al/13 steel, 3 layers ➛ R = 0.554 in = 1.407 cm ➛ R′ = 0.425 in = 1.080 cm ➛ rdc = 0.1032 Ω/mile = 0.0645 Ω/km ➛ rac@25◦ C = 0.1061 Ω/mile = 0.0663 Ω/km ➛ rac@100◦ C = 0.1361 Ω/mile = 0.0851 Ω/km ➛ d = 200 km ➛ f = 60 Hz → ω = 377 rad/s Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
131
Example 2
r
=
Dm
=
Rb′
=
Rb
=
l
=
c
=
rac@100◦ C = 0.02134 Ω/km Nb √ 3 45′ 45′ 90′ = 17.27 m √ 4 1.080 30.48 30.48 43.11 cm = 0.1442 m √ 4 1.407 30.48 30.48 43.11 cm = 0.1541 m 17.27 −7 = 0.9573 × 10−3 H/m 2 × 10 ln 0.1442 2π8.854 × 10−12 = 0.01179 µH/m 17.27 ln 0.1541
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
132
Example 2
z = r + jωl = y
= jωc =
γ
0.3615∠86.62◦ Ω/km
=
4.444 × 10−6 ∠90◦ S/km √ zy
0.00217∠88.31◦ km−1 r z = y =
Zc
=
285.21∠ − 1.69◦ Ω
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 71
Power System Dynamics and Stability
133
Example 2
A
= D = cosh γd eγd + e−γd = 2 = 0.9686∠0.11◦
B
= Zc sinh γd = =
C = =
eγd − e−γd Zc 2 71.46∠86.65◦ Ω 1 sinh γd Zc 8.787 × 10−4 ∠90.03◦ S
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
134
Example 2
Z = zd = Y
72.30∠86.62◦ Ω
= yd 8.89 × 10−4 ∠90◦ S sinh γd = Z γd = 71.33∠86.65◦ Ω ≈ Z tanh γd/2 = Y γd/2 =
Z′
Y′
=
9.032 × 10−4 ∠89.95◦ S ≈ Y
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
135
Cables
Steel pipe (filled with insulating oil) Skid wires Metallic tapes Paper/oil insulation Screen Conductor (stranded copper)
11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 000 111 00000000000 11111111111 000 111 00000000000 11111111111 000 111 00000000000 11111111111 00000000000 11111111111 0000 1111 00000000000 11111111111 0000 1111 00000000000 11111111111 0000 1111 0000000 1111111 00000000000 11111111111 0000000 1111111 00000000000 11111111111 000 111 0000000 1111111 00000000000 11111111111 000 111 0000000 1111111 00000000000 11111111111 000 111 0000000 1111111 00000000000 11111111111
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
136
Cables
➠ Single-phase and three-phase cables:
111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111
conductor
sheath
SF6 gas
111111111 000000000 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000000000 111111111 00 11 11 00 000000000 111111111 00 11 00 11 000000000 111111111 00 11 00 11 000000000 111111111 00 11 00 11 000000000 111111111 00 11 00 11 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 75
137
Power System Dynamics and Stability
Cables
➠ Model for a single-phase cable:
VC
1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 0000000 1111111 00000000 11111111 00000 11111 0000000 1111111 0000000 1111111 00000000 11111111 00000 11111 0000000 1111111 00000000 11111111 0000000 1111111 00000 11111 0000000 1111111 00000000 11111111 00000 11111 0 1 0000000 1111111 00000000 11111111 00000 11111 0 1 00000000 11111111 0000000 1111111 00000 11111 00000000 11111111 00000 11111 0000000 1111111 00000000 11111111 00000 11111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 V
11111111 00000000 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000 111 000000 111111 00000000 11111111 000 111 000000 111111 00000000 11111111 000 111 000000 111111 00000000 11111111 000000 111111 00000000 11111111
VS
I1 I2 I3
A
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
138
Cables
➠ Model for a single-phase cable: V1
= Vcore − Vsheath = VC − VS
V2
= Vsheath − Varmour = VS − VA
V3
= VA
IC
= I1
IS
= I1 − I2
IA
= I2 − I3
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 77
139
Power System Dynamics and Stability
Cables
➠ Loop equations:
V1
z11
z12
0
I1
d V2 = − z12 z22 z23 I2 dx V3 I3 0 z23 z33 V1 y1 0 0 I1 d I2 = 0 y2 0 V2 dx V3 I3 0 0 y3
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
140
Cables
➠ The elements of the impedance matrix z arecomputed as follows: z11
= zC ext + zCS + zS int
z12
= −zS mut
z22
= −zS ext + zSA + zAint
z23
= −zAmut
z33
= −zAext + zAE + zE
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
141
Cables
➠ These impedances are calculated using the formulas: 11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 q 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 r 00000000 11111111 00000000 11111111
zext zint zmut zins
ρm [I0 (mr)K1 (mq) + K0 (mr)I1 (mq)] 2πrD ρm [I0 (mq)K1 (mr) + K0 (mq)I1 (mr)] = 2πqD ρ = 2πqrD µ0 r −1 = y = jω 2π q =
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
142
Cables
D
= I1 (mr)K1 (mq) + K1 (mr)I1 (mq) s jωµ m = ρ ➠ Where In and Kn are modified Bessel functions, as follows: In (x)
=
∞ X
k=0
Kn (x)
=
x n+2k 1 k!Γ(n + k + 1) 2
π I−n (x) − In (x) 2 sin(nπ)
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
143
Cables
➠ The Gamma function Γ(n) is defined as follows: Γ(n) =
Z
∞
e−x xn−1 dx
0
Γ(n + 1) = nΓ(n) Γ(n + 1) = n! for n = 1, 2, 3, . . . ➠ For asymmetric cables a finite element method is needed to compute these impedances.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 82
144
Power System Dynamics and Stability
Cables
➠ From the relations between loop voltages/currents and node voltage/currents, the node equations are:
VC
zCC
d VS = − zCS dx VA zCA e.g.
d dx
zCS
zCA
zSS
zSA
zSA
zAA
IC
IS IA
zCC = z11 + 2z12 + z22 + 2z23 + z33 IC y1 −y1 0 VC IS = −y1 y1 + y2 −y2 VS IA 0 −y2 y2 + y3 VA
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 83
145
Power System Dynamics and Stability
Cables
➠ For a three-phase cable made of 3 single-phase cables: d [V] = −[z]9×9 [I] dx d [I] = −[y]9×9 [V] dx [zaa ]3×3 [z] = [zab ]3×3 [zac ]3×3 [yaa ]3×3 [y] = 0 0
[zab ]3×3
[zac ]3×3
[zbb ]3×3
[zbc ]3×3
[zbc ]3×3
[zcc ]3×3
0
0
[ybb ]3×3
0
0
[ycc ]3×3
Universidad de Castilla-La Mancha June 26, 2008
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146
Load Modeling
➠ RLC models. ➠ Induction motors. ➛ Detailed models. ➛ Phasor models. ➠ Aggregated models: ➛ Impedance models. ➛ Power models. ➛ Induction motor power models.
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 1
147
Power System Dynamics and Stability
Load Classification
➠ By demand level: ➛ Residential: lighting and heating (RL + controls); AC (motor + controls); appliances (small motors + controls). ➛ Commercial: similar types of devices as residential. ➛ Industrial: motor drives (induction and dc motor-based mostly); arc furnaces; lighting; heating; others (e.g. special motor drives).
➠ By type: ➛ RLC + controls. ➛ Drives: ac/dc motors + electronic controls. ➛ Special (e.g. arc furnace). Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 2
148
Power System Dynamics and Stability
Remarks on Load Classification
➠ Most controls are implemented using a variety of power electronic converters.
➠ Aggregate load models are necessary at the transmission system modeling level.
➠ Only large loads can be represented with their actual models.
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 3
Power System Dynamics and Stability
149
RLC Loads, Resistor
➠ Ideal, linear resistors (R), inductors (L) and capacitors (C ). ➠ Resistor: R
i +
v
−
➛ Time domain: v = Ri ➛ Phasor domain: V = RI
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 4
Power System Dynamics and Stability
150
RLC Loads, Inductor
➠ Inductor: L
i +
v
−
➛ Time domain: di v=L dt ➛ Phasor domain: V
= jωLI = ZL I
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 5
Power System Dynamics and Stability
151
RLC Loads, Inductor
➠ The inductor time domain model can be discretized using the trapezoidal integration method as follows:
i = ⇒
ik+1
1 L
Z
vdt
∆t = ik + (vk+1 + vk ) 2L ∆t ∆t = vk+1 + (ik + vk ) 2L 2L {z } | hk
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 6
Power System Dynamics and Stability
152
RLC Loads, Capacitor
➠ This yields the equivalent resistive circuit: vk+1 +
−
ik+1 hk
2L ∆t
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
153
RLC Loads, Capacitor
➠ Capacitor: C
i +
v
−
➛ Time domain: dv i=C dt ➛ Phasor domain: V
1 I ωC = ZC I = −j
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154
Power System Dynamics and Stability
RLC Loads, Capacitor
➠ The capacitor time domain model can be discretized using the trapezoidal integration method as follows:
v ⇒
vk+1 ik+1
=
1 C
Z
idt
∆t = vk + (ik+1 + ik ) 2C 2C 2C = vk+1 − (ik + vk ) ∆t ∆t {z } | hk
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
155
RLC Loads, Capacitor
➠ This yields the equivalent resistive circuit: vk+1 +
−
ik+1 hk
∆t 2C
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156
Power System Dynamics and Stability
Induction Motor
➠ Arbitrary 0dq reference: as
d
ω
q
ωr
c′s
r ar
θr
b′s
θ 0
s
cr bs
br cs
a′s
d
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157
Power System Dynamics and Stability
Induction Motor
➠ Electrical (inductor) equations: d [λabcs ] dt d [vabcr ] = [rabcr ][iabcr ] + [λabcr ] dt λ i Lsr(θr ) L abcs abcs = abcs Lsr (θr ) Labcr λabcr iabcr | {z } | {z } [vabcs ] =
[rabcs ][iabcs ] +
L(θr )
Te
=
[i]
1 T d[L(θr )] [i] [i] 2 dθr
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158
Power System Dynamics and Stability
Induction Motor
➠ Mechanical (Newton’s) equations: d J ωr + dωr dt d θr dt ➠ Stator transformation equations: vas v0s vds = Ks vbs vcs vqs
= Tm − Te = ωr
ias i0s ids = Ks ibs ics iqs
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159
Power System Dynamics and Stability
Induction Motor
➠ Where the transformation matrix Ks is as follows: 1/2 1/2 1/2 2 Ks = sin θ sin(θ − 2π/3) sin(θ + 2π/3) 3 cos θ cos(θ − 2π/3) cos(θ + 2π/3) 1 sin θ cos θ −1 Ks = 1 sin(θ − 2π/3) cos(θ − 2π/3) 1 sin(θ + 2π/3) cos(θ + 2π/3)
Universidad de Castilla-La Mancha June 26, 2008
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160
Power System Dynamics and Stability
Induction Motor
➠ Rotor transformation equations: var v0r vdr = Kr vbr vcr vqr
iar i0r idr = Kr ibr icr iqr
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161
Power System Dynamics and Stability
Induction Motor
➠ Where the transformation matrix Kr is as follows: 1/2 1/2 1/2 2 Kr = sin β sin(β − 2π/3) sin(β + 2π/3) 3 cos β cos(β − 2π/3) cos(β + 2π/3) 1 sin β cos β −1 Kr = 1 sin(β − 2π/3) cos(β − 2π/3) 1 sin(β + 2π/3) cos(β + 2π/3) β
= θ − θr
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Load Modeling - 16
162
Power System Dynamics and Stability
Induction Motor
➠ Stator equations: v0s rs vds = 0 vqs 0
0 rs 0
0
i0s
0
λ0s
d λ 0 ids + ω −λqs + ds dt λqs rs iqs λds
➠ Rotor equations referred to the stator: 2 a rr 0 0 ′ 0 λ′0r 0 i0r |{z} rr′ i′ + (ω − ωr ) −λ′ + d λ′ 0 = dr 0 qr dt dr rr′ 0 ′ ′ ′ 0 i λ λ qr qr dr 0 0 rr′ a=
Ns Nr
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 17
163
Power System Dynamics and Stability
Induction Motor
➠ Magnetic flux equations: Lls 0 0 λ0s 0 ls + M 0 L {z } | λ ds Ls λ 0 0 Ls qs = λ′ 0 0 0 0r ′ λdr 0 M 0 λ′qr 0 0 M
0 0 0 L′lr 0 0
0
0
i M 0 0s i ds 0 M iqs i′ 0 0 0r L′lr + M 0 i′dr | {z } ′ L′r iqr 0 L′r
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 18
164
Power System Dynamics and Stability
Induction Motor
➠ Mechanical equations: 2 d 2 J ωr + Dωr p dt p d θr dt d θ dt Te
= Tm − Te = ωr = ω =
3p (iqs λds − ids λqs ) 22
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165
Power System Dynamics and Stability
Induction Motor
➠ Equivalent circuit representation of these equations: iqs
rs
+ vqs
+
−
Lls
L′lr
−
+
rr′
Ns : Nr
iqr
(ω − ωr )λ′dr
ωλds M
vqr
−
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166
Power System Dynamics and Stability
Induction Motor ids
rs
+
−
+
Lls
L′lr
−
rr′
Ns : Nr
idr
(ω − ωr )λ′qr
ωλqs
vds
+
vdr
M
−
i0s
rs
Lls
L′lr
rr′
Ns : Nr
i0r
+ v0r
v0s − Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 21
167
Power System Dynamics and Stability
Induction Motor
➠ Assuming a balanced, fundamental frequency (ω = ω0 ) system, the model can be reduced to a p.u. “transient” model (3rd order model): Vas
= VasR + jVasI
Ias
= IasR + jIasI 1 ′ + (xS − x′ )IasI ] = ω0 σEI′ − ′ [ER T0 1 ′ ′ = −ω0 σER − ′ [EI + (xS − x′ )IasR ] T0
d ′ ER dt d ′ E dt I
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Load Modeling - 22
168
Power System Dynamics and Stability
Induction Motor ′ VasR − ER
VasI − EI′ d σ dt Te TL
= rs IasR − x′ IasI
= rs IasI + x′ IasR 1 = (TL − Te + Dω0 − Dσ) H 1 ′ (ER IasR + EI′ IasR ) = ω0 = f (σ)
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169
Power System Dynamics and Stability
Induction Motor
➠ where: σ xs x′ T0′
ω0 − ωr → slip = ω0 = xls + xm → stator reactance x′lr xm = xls + ′ → transient reactance xlr + xm =
x′lr + xm ω0 rr′
→
open circuit transient time constant
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170
Power System Dynamics and Stability
Induction Motor
➠ If the electromagnetic transients are neglected, the system can be reduced to the following quasi-steady state equivalent circuit model: ′ jx′lr jxls rs Iar rr′ Ias + Vas
jxm
rr′ 1−σ σ
−
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171
Power System Dynamics and Stability
Induction Motor
➠ Or equivalently: Ias
jx′lr
jxls
rs
′ Iar
+ jxm
Vas
rr′ σ1
−
➠ Plus mechanical equations.
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172
Power System Dynamics and Stability
Induction Motor
➠ This simplification can be justified from the point of view of the torque-speed characteristic: Te Te max Steady state characteristic
Transient start up
0
1
ωr ωc
=1−σ
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173
Power System Dynamics and Stability
Induction Motor
➠ If the mechanical dynamics are ignored, the slip σ becomes a fixed value, and hence the equivalent circuit can be reduced to a simple equivalent reactive impedance, i.e. a Z load.
➠ For loads with multiple IMs, a equivalent motor model can be used to represent these motors.
➠ Neglecting the motor dynamic equations in large or equivalent aggregate motors can lead to significant modeling errors, as the transient model and mechanical time constants can be on the same range as the generator time constants.
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174
Power System Dynamics and Stability
Induction Motor
➠ Double-cage IMs can be modeled by introducing an additional inductance on the rotor side, which leads to a “subtransient” model.
➠ Since rotor cores are laminated, eddy currents do not play a significant role on the system dynamics.
➠ Controls can also be modeled in detail by representing the converters with ideal electronic switches plus their control systems.
➠ For balanced, fundamental frequency systems, approximate equivalent models of the IM and its controls can be readily implemented.
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175
Power System Dynamics and Stability
Induction Motor
➠ For example, an IM with voltage-fed field oriented control can be represented by:
dωr dt
=
P (Te − TL ) 2J 2 L′ r rs
M 2 rr′
+ L′r σ
!
M rr′ ′ iqs − ωids + ′ λqr Lr σ
diqs dt
= −
dids dt
M L′r ′ − ωr λdr + vqs σ σ ! 2 L′ r rs + M 2 rr′ M rr′ ′ = − ids + ωiqs + ′ λdr ′ Lr σ Lr σ M L′r ′ + ωr λqr + vds σ σ Universidad de Castilla-La Mancha
June 26, 2008
Load Modeling - 30
176
Power System Dynamics and Stability
Induction Motor
dλ′qr dt dλ′dr dt Te σ dTref dt
′ ′ r r M r r ′ ′ = −(ω − ωr )λdr − ′ λqr + ′ iqs Lr Lr rr′ ′ rr′ M ′ = (ω − ωr )λqr − ′ λdr + ′ ids Lr Lr 3pM ′ ′ = (i λ − i λ qs ds dr qr ) 4L′r
= Ls L′r − M 2 2 pTL 3p M ′ = kq3 − i λ qs dr + kq4 (ωref − ωr ) 2J 8JL′r
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 31
177
Power System Dynamics and Stability
Induction Motor
duqs dt
rr′ M 3pM kq1 rr′ M ′2 ′ − = [ i λ i i + λ dr ωr qs dr ds qs ′ ′ ′ 4Lr Lr Lr σ ! ′2 2 ′ L′r ′ L r rs + M rr ′ iqs λdr − λ uqs ] + L′r σ σ dr 2 pTL 3p M ′ +kq1 kq3 − i λ qs dr 2J 8JL′r +kq2 Tref− 2pM′ iqs λ′dr aLr
duds dt
+kq1 kq4 (ωref − ωr ) ′ ′ M rr rr ′ ′ + k (λ − λ = kd1 λ i − d2 ref ds dr ) dr ′ ′ Lr Lr Universidad de Castilla-La Mancha
June 26, 2008
Load Modeling - 32
178
Power System Dynamics and Stability
Example
➠ A 21 MW load at 4 kV and 60 Hz is made of: ➠ An inductive impedance load with G = 0.06047, B = −0.03530. ➠ An aggregated induction motor model with rs = 0.07825, xls = 0.8320, rr′ = 0.1055, x′lr = 0.8320, xm = 16.48. ➠ This data is all in p.u. on a 100 MVA, 4 kV base.
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
179
Example
➠ The Z load model is then: PL PZ ⇒
PIM
21 MW = 0.21 = 100 MVA = PZ + PIM = VL2 G = 0.06 = 0.15
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180
Power System Dynamics and Stability
Example
ZIM
jxM (rr′ /σ + jx′lr ) = rs + jxls + ′ rr /σ + j(xM + x′lr ) −13.7114 + j1.7386/σ = 0.07825 + j0.832 + 0.1055/σ + j17.312 28.652/σ = 0.07825 + 0.01113σ 2 + 299.71 2 0.18342/σ + 237.37 +j 0.832 + 0.01113/σ 2 + 299.71
(0.00087/σ 2 + 28.652/σ + 23.452) + j(0.19628/σ 2 + 486.73) = 0.01113/σ 2 + 299.71
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Power System Dynamics and Stability
181
Example
PIM
= VL2 GIM = GIM
0.15 = ⇒σ
=
⇒ ZIM
=
YL
=
ZL
=
(0.01113/σ 2 + 299.71)(0.00087/σ 2 + 28.652/σ + 23.452) (0.00087/σ 2 + 28.652/σ + 23.452)2 + (0.19628/σ 2 + 486.73)2 0.0191 (by trial-and-error) 4.6221 + j3.0742 1 (G + jB) + ZIM 1 = 3.3654 + j2.1597 YL
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182
Power System Dynamics and Stability
Impedance Models
➠ Ignoring “fast” and “slow” transients, certain loads can be represented using an equivalent impedance.
➠ A Z load model is typically used for a variety of dynamic analysis of power systems.
➠ Using these load models, an equivalent impedance can be readily obtained for all loads connected at a particular bus at the transmission system level.
➠ ULTCs are used to connect distribution systems (subtransmission and LV systems and the loads connetced to these) to the transmission system to control the steady state voltage on the load side.
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183
Power System Dynamics and Stability
Power Models
➠ Hence, for “slow” dynamic analysis of balanced, fundamental frequency system models:
PL QL VL
2
VL = = PL0 VL0 2 VL 2 = VL BL = QL0 VL0 P L ≈ PL0 ≈ VL0 ⇒ QL ≈ QL0 VL2 GL
➠ This is typically referred as a constant P Q model.
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184
Power System Dynamics and Stability
Power Models
➠ Load recovery of certain loads (e.g. thermostatic) with respect to voltage changes can be modeled as:
= −
PL (t)
=
Nqs
y(t) VL (t) + QL0 Tq VL0 Nqt y(t) VL (t) + QL0 Tq VL0
= −
QL (t) =
Nps
VL (t) x(t) + PL0 Tp VL0 Npt VL (t) x(t) + PL0 Tp VL0
dx(t) dt
dy(t) dt
− PL0
− QL0
VL (t) VL0
Npt
VL (t) VL0
Nqt
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185
Power System Dynamics and Stability
Power Models
➠ This results in the following time response: P PL0
tf
t
tf
t
V VL0
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Load Modeling - 40
186
Power System Dynamics and Stability
Example
➠ Identification of LD1 -LD4 loads at paper mill in Sweden: Net
894 MVA, 30 kV
1 2
LD1 LD2 3
5 LD3 LD4
4
6
G1
G2
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
187
Example
➠ Measurements vs. model:
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 42
188
Power System Dynamics and Stability
Power Models
➠ Certain loads have been shown to behave in steady state as follows: αP VL PL = KP VLαP fLβP ≈ PL0 VL0 αQ VL α β QL = KQ VL Q fLQ ≈ PL0 VL0 ➠ If f ≈ f0 , then the following approximation holds: αP VL PL ≈ PL0 VL0 αQ VL QL ≈ PL0 VL0 Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 43
189
Power System Dynamics and Stability
Power Models Load
αP
αQ
βP
βQ
Filament lamp
1.6
0
0
0
Fluorescent lamp
1.2
3.0
-0.1
2.8
Heater
2.0
0
0
0
Induction motor (half load)
0.2
1.6
1.5
-0.3
Induction motor (full load)
0.1
0.6
2.8
1.8
Reduction furnace
1.9
2.1
-0.5
0
Aluminum plant
1.8
2.2
-0.3
0.6
Universidad de Castilla-La Mancha June 26, 2008
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190
Power System Dynamics and Stability
Power Models
➠ ZIP model: PL
= P LZ
QL
= QLZ
2
VL VL + P LI + P LP VL0 VL0 2 VL VL + QLI + QLP VL0 VL0
➠ Jimma’s model: PL
= P LZ
QL
= QLZ
2
VL VL + P LI + PL P VL0 VL0 2 VL dVL VL + QLI + QLP + KV VL0 VL0 dt Universidad de Castilla-La Mancha
June 26, 2008
Load Modeling - 45
191
Power System Dynamics and Stability
Induction Motor Power Models
➠ Walve’s model: PL
QL
dVL = Kpf f + Kpv VL + T dt dVL ≈ PL0 + Kpv (VL − VL0 ) + T dt = Kqf f + Kqv VL ≈ QL0 + Kqv (VL − VL0 )
Universidad de Castilla-La Mancha June 26, 2008
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192
Power System Dynamics and Stability
Induction Motor Power Models
➠ Mixed model: PL
= ≈
QL
= ≈
dVL Kpf f + Kpv VLα + Tpv dt dVL PL0 + Kpv (VL − VL0 )α + Tpv dt dVL Kqf f + Kqv VLβ + Tqv dt dV L QL0 + Kqv (VL − VL0 )β + Tqv dt
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 47
Power System Dynamics and Stability
193
Power Flow Outlines
➠ Power Flow: ➛ System model. ➛ Equations. ➛ Solution techniques: ➳ Newton-Raphson. ➳ Fast decoupled.
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 1
194
Power System Dynamics and Stability
Power Flow Model
➠ The steady-state operating point of a power system is obtained by solving the “power flow” equations.
➠ Power flow system model corresponds to the steady state model. ➠ Generator: ➛ Generates and injects power P in the system while keeping the output voltage V constant within active and reactive power limits (capability curve):
Pmin ≤ P ≤ Pmax Qmin ≤ Q ≤ Qmax Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 2
Power System Dynamics and Stability
195
Power Flow Model
➠ Thus, it is modeled as a P V bus: P = constant Q = unknown
V = constant δ = unknown
➠ When Q reaches a limit it becomes a P Q bus: P = constant Q = constant
V = unknown δ = unknown
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 3
Power System Dynamics and Stability
196
Power Flow Model
➠ Slack bus: ➛ The phasor model needs a reference bus. ➛ A “large” generator is typically chosen as the the reference bus, as it should be able to take the power “slack”: P = unknown Q = unknown
V = constant δ =0
Pslack =
X L
PL + Plosses −
X
PG
G
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 4
Power System Dynamics and Stability
197
Power Flow Model
➠ Hence, if Q reaches a limit: P = unknown Q = constant
V = unknown δ =0
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Power Flow Outlines - 5
198
Power System Dynamics and Stability
Power Flow Model
➠ Load: ➛ Loads are typically connected to the transmission system through ULTC transformers.
➛ Thus, most loads in steady state represent a constant power demand in the system, and hence are modeled as a P Q bus: P = constant Q = constant
V = unknown δ = unknown
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 6
199
Power System Dynamics and Stability
Power Flow Model
➠ Transmission system: ➛ AC transmission lines and transformers in steady state are basically modeled using the following model: Sk
Si
Vi
Z
Vk
Ik /a∠α
Ii
Ik Y1
Y2
a∠αVk
a∠α : 1
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 7
200
Power System Dynamics and Stability
Power Flow Model
➠ Hence: Si
= Vi I∗i
∗
1 +Vi Y1 = Vi (Vi − a∠αVk ) Z | {z } Y
= Vi (Y + Yi )∗ Vi∗ + (−a∠αY)∗ Vk∗ {z } | | {z } ∗ Yii
∗ Yik
➠ Similarly for Sk . Universidad de Castilla-La Mancha June 26, 2008
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201
Power System Dynamics and Stability
Power Flow Model
➠ Thus, for an N bus system interconnected through an ac transmission system, an N × N bus admittance matrix can be defined:
I Y 1 11 I Y 2 21 . . . . . . = I i Yi1 .. .. . . IN YN 1
I |{z}
node injections
= Ybus
Y12 Y22 .. .
Yi2 .. .
YN 2
···
··· ..
.
··· ..
.
···
Y1i Y2i .. .
Yii .. .
YN i
···
··· ..
.
··· ..
.
···
V |{z}
Y1N
V 1 Y2N V2 .. .. . . YiN Vi .. .. . . YN N VN
node voltages
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 9
202
Power System Dynamics and Stability
Power Flow Model
➠ where:
Ybus
P PN 1 Yii = k=1 Zik + Yi = sum of all the Y’s connected to node i = 1 Y = − ij Zij = negative of the Y between nodes i and j
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203
Power System Dynamics and Stability
Power Flow Equations
➠ In steady state, a system with n generators G and m loads L can be modeled as: SG1
SL1 VG1
VL1 Transmission System
Ybus (N × N ) N =n+m
SGn
SLm VGn
VLm
Universidad de Castilla-La Mancha June 26, 2008
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204
Power System Dynamics and Stability
Power Flow Equations
➠ Hence the power injections at each node are defined by: Si
= Pi + jQi = Vi I∗i = Vi
N X
∗ Yik Vk∗
k=1
= Vi ∠δi
N X
k=1
Si
=
S
Gi
−SL i
(Gij − jBij )Vk ∠ − δk for generator buses for load buses
Universidad de Castilla-La Mancha June 26, 2008
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205
Power System Dynamics and Stability
Power Flow Equations
➠ This yields two equations per node or bus: ∆Pi (δ, V, Pi ) = Pi −
N X
Vi Vk [Gik cos(δi − δk ) + Bik sin(δi − δk )] = 0
N X
Vi Vk [Gik sin(δi − δk ) − Bik cos(δi − δk )] = 0
k=1
∆Qi (δ, V, Qi ) = Qi −
k=1
➠ And two variables per node: ➛ PQ buses → Vi and δi
➛ PV buses → δi and Qi
➛ Slack buses → Pi and Qi Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 13
206
Power System Dynamics and Stability
Power Flow Equations
➠ These equations are referred to as the power mismatch equations. ➠ The equations are typically subjected to inequality constraints representing control limits:
0.95 Qmini
≤ Vi ≤
1.05 for all buses
≤ Qi ≤ Qmaxi for generator buses
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 14
Power System Dynamics and Stability
207
Power Flow Equations
➠ In summary, typical power flow data are as follows: Bus
Parameters
Variables
PQ
P, Q
V,θ
PV
P, V
Q, θ
slack
V,θ
P, Q
Universidad de Castilla-La Mancha June 26, 2008
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208
Power System Dynamics and Stability
Power Flow Equations
➠ In “constrained” power flow analysis, standard buses can degenerate if some limits is reached: Bus
Parameters
Variables
PQ
P , Vmax,min
Q, θ
PV
P , Qmax,min
V,θ
slack
Qmax,min , θ
P, V
➠ Continuation power flow techniques are by far more accurate and robust than standard power flow analysis if limits are taken into account.
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209
Power System Dynamics and Stability
Power Flow Equations
➠ The slack bus can be single or distributed. ➠ This refers to losses. ➛ For single slack bus model, all system losses are “cleared” by the slack bus.
➛ For distributed slack bus model losses are shared (equally or proportionally) among all or part of the generators:
➠ Continuation power flow techniques are by far more accurate and robust than standard power flow analysis if limits are taken into account.
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210
Power System Dynamics and Stability
Power Flow Equations
➠ The distributed slack bus model is based on a generalized power center concept.
➠ This is practically obtained by including in power flow equations a variable kG and rewriting the system active power balance as follows: nG X i
(1 + kG γi )PGi −
nP X i
PLi − Plosses = 0
➠ The parameters γi allow tuning the weight of the participation of each generator to the losses.
➠ For single slack bus model, γi = 0 for all generators but one. Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 18
211
Power System Dynamics and Stability
Power Flow Solution
➠ The power flow equations can be represented as F (z) = 0 ➠ There are 2 equations per bus, with 2 known variables and 2 unknown variables per bus; the problem is of dimension 2N . ➠ Since these equations are highly nonlinear due to the sine and cosine terms, Newton-Raphson (NR) based numerical techniques are used to solve them.
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212
Power System Dynamics and Stability
Power Flow Solution
➠ A “robust” NR technique may be used to solve these equations: 1. Start with an initial guess, typically Vi0
= 1, δi0 = 0, Q0Gi = 0,
0 Pslack = 0 (flat start).
2. At each iteration k(k
= 0, 1, 2, . . .), compute the “sparse” Jacobian
matrix:
∂F k = Jk = ∂z z
∂F1 ∂z1 |z k
...
.. .
..
∂FN ∂z1 |z k
...
.
∂F1 ∂zN |z k
.. .
∂FN ∂zN |z k
2N ×2N
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Power Flow Solution 3. Find ∆z k by solving the following linear set of equations (the sparse matrix Jk is factorized and not inverted to speed up the solution process):
Jk ∆z k = −F (z k ) 4. Computes the new guess for the next iteration, where α is a step control constant to guarantee convergence (0
< α < 1):
z k+1 = z k + α∆z k
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Power Flow Solution 5. Stop when:
kF (z k+1 )k = max |Fi (z k+1 )| ≤ ǫ ➠ This is basically the technique used in M ATLAB’s fsolve() routine, based on either numerical or actual Jacobians.
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Power Flow Solution
➠ If only the unknown bus voltage angles and magnitudes are calculated using NR’s method (the generator reactive powers and active slack power are evaluated later):
⇒
H N
Jk ∆z k = −F (z k ) M ∆P (δ k , V k ) ∆δ k = − L ∆V k /V k ∆Q(δ k , V k )
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Power Flow Solution
➠ Where: H
=
M
=
N
=
L =
∂∆P ∂δ (δk ,V k ) ∂∆P V ∂V (δk ,V k ) ∂∆Q ∂δ (δk ,V k ) ∂∆Q V ∂V (δk ,V k )
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Power Flow Solution
➠ Assuming: ➛ (δi − δk ) < 10◦ , then cos(δi − δk )
≈ 1
sin(δi − δk )
≈ δi − δk
➛ The resistance in the transmission system are small, i.e. R ≪ X , then Gij ≪ Bik . ➠ The M and N matrices may be neglected, and: H M
≈ V k B′V k
≈ V k B ′′ V k
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Power Flow Solution
➠ Thus the linear step equations may be decoupled and reduced to: B ′ ∆δ k B ′′ ∆V k
= −∆P (δ k , V k )/V k
= −∆Q(δ k , V k )/V k
where B ′′ is the imaginary part of the Ybus matrix, and B ′ is the imaginary part of the admittance matrix obtained by ignoring the system resistances, i.e.
B ′′ 6= B ′ .
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Power Flow Solution
➠ The fast Decoupled iterative method is then defined as follows: 1. Start with an initial guess, typically δi0 2. Solve for ∆δ k
→ B ′ ∆δ k = −∆P (δ k , V k )/V k .
3. Update → δ k+1 4. Solve for ∆V k
5. Update →
= 0, Vi0 = 1.
= δ k + ∆δ k .
→ B ′′ ∆V k = −∆Q(δ k , V k )/V k .
V k+1 = V k + ∆V k .
6. Compute unknown generator powers and check for limits. 7. Repeat process for k
= 1, 2, . . ., until convergence.
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Power Flow Solution
➠ This technique requires a relatively large number of iterations as compared to the robust NR method.
➠ It is significantly faster, as there is no need to re-compute and re-factorize the Jacobian matrix every iteration.
➠ It is sensitive to initial guesses and there is no guarantee of convergence, particularly for systems with large transmission system resistances.
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Example
➠ For the following system: 1
2
All lines: 200 MVA 138 kV
P2
P1
X = 0.1 p.u. B = 0.2 p.u.
3
Q
200 MVA 0.9 p.f. lagging
V1 = 1, δ1 = 0, V2 = 1, V3 = 1, and P2 = P1 /2. ➠ Determine the voltage phasor angles δ2 and δ3 and the shunt Q by solving the PF equations.
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Example
➠ The Ybus matrix is:
2 B +2 j = −j19.8 jX 2 1 Y12 = Y13 = Y23 = − = j10 jX −19.8 10 10 B11 B12 B13 Ybus = j 10 −19.8 10 = j B21 B22 B23 10 10 −19.8 B31 B32 B33 Y11
⇒
= Y22 = Y33 =
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Example
➠ Mismatch equation ∆P1 : ∆P1 0
= P1 − = P1 −
3 X
k=1 3 X
V1 Vk [G1k cos(δ1 − δk ) + B1k sin(δ1 − δk )] B1k sin(−δk )
k=1
= P1 + B12 sin(δ2 ) + B13 sin(δ3 ) = P1 + 10 sin(δ2 ) + 10 sin(δ3 )
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Example
➠ Mismatch equation ∆Q1 : ∆Q1 0
= Q1 − = Q1 +
3 X
k=1 3 X
V1 Vk [G1k sin(δ1 − δk ) − B1k cos(δ1 − δk )] B1k cos(−δk )
k=1
= Q1 − 19.8 + 10 cos(δ2 ) + 10 cos(δ3 )
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Power System Dynamics and Stability
Example
➠ Mismatch equation ∆P2 : ∆P2 0
= P2 −
3 X
k=1
= P1 /2 −
V2 Vk [G2k cos(δ2 − δk ) + B2k sin(δ2 − δk )]
3 X
k=1
B2k sin(δ2 − δk )
= 0.5P1 + 10 sin(δ2 ) + 10 sin(δ2 − δ3 )
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Example
➠ Mismatch equation ∆Q2 : ∆Q2 0
= Q2 − = Q2 +
3 X
k=1 3 X
k=1
V2 Vk [G2k sin(δ2 − δk ) − B2k cos(δ2 − δk )] B2k cos(δ2 − δk )
= Q1 + 10 cos(δ2 ) − 19.8 + 10 cos(δ2 − δ3 )
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Example
➠ Mismatch equation ∆P3 : ∆P3 0
= P3 −
3 X
k=1
= −0.9 −
V3 Vk [G3k cos(δ3 − δk ) + B3k sin(δ3 − δk )]
3 X
k=1
B3k sin(δ3 − δk )
= −0.9 + 10 sin(δ3 ) + 10 sin(δ3 − δ2 )
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Example
➠ Mismatch equation ∆Q3 : ∆Q3 0
= Q3 − = Q−
3 X
k=1
p
V3 Vk [G3k sin(δ3 − δk ) − B3k cos(δ3 − δk )]
1 − 0.92 +
3 X
k=1
B3k cos(δ3 − δk )
= Q − 0.436 + 10 cos(δ3 ) + 10 cos(δ3 − δ2 ) − 19.8
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Example
➠ Thus 6 equations and 6 unknowns, i.e. δ2 , δ3 , P1 , Q1 , Q2 , and Q, can be solved using M ATLAB’s fsolve() routine: >> global lambda >> lambda = 1; >> z0 = fsolve(@pf_eqs,[0 0 0 0 0 0], optimset(’Display’,’iter’)) Norm of step
First-order Trust-region Iteration Func-count f(x) optimality radius 0 7 0.945696 18 1 1 14 0.000661419 0.705901 0.0205 1 2 21 9.98637e-18 0.0257331 2.52e-09 1.76 Optimization terminated: first-order optimality is less than options.TolFun. z0 = -0.0100
-0.0500
0.6000
-0.1870
-0.1915
0.2565
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230
Example
➠ Where lambda is used to simulate constant power factor load changes and pf eqs.m is: function F = pf_eqs(z) global lambda d2 d3 P1 Q1 Q2 Q
= = = = = =
z(1); z(2); z(3); z(4); z(5); z(6);
F(1,1) F(2,1) F(3,1) F(4,1) F(5,1) F(6,1)
= = = = = =
P1 + 10*sin(d2) + 10*sin(d3); Q1 - 19.8 + 10*cos(d2) + 10*cos(d3); 0.5*P1 - 10*sin(d2) - 10*sin(d2-d3); Q2 + 10*cos(d2) - 19.8 + 10*cos(d2-d3); -0.9*lambda - 10*sin(d3) - 10*sin(d3-d2); Q - 0.436*lambda + 10*cos(d3) + 10*cos(d3-d2) - 19.8; Universidad de Castilla-La Mancha
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Example
➠ Observe that as the load (lambda) is increased, convergence problems show up: lambda = 20; z0 = fsolve(@pf_eqs,[0 0 0 0 0 0], optimset(’Display’,’iter’)) Norm of step
First-order optimality 360 143
Trust-region radius 1 1
Iteration Func-count f(x) 0 7 396.67 1 14 246.008 1 . . . 9 70 2.68094e-07 0.179729 0.000311 25.9 10 77 7.43614e-16 0.00127159 1.64e-08 25.9 Optimization terminated: first-order optimality is less than options.TolFun. z0 = -0.2501
-1.2613
12.0000
7.0658
4.8031
20.1667
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Example
➠ Convergence problems develop until the equations fail to converge: lambda = 22; z0 = fsolve(@pf_eqs,[0 0 0 0 0 0], optimset(’Display’,’iter’)) Norm of step
First-order optimality 396 168
Trust-region radius 1 1
Iteration Func-count f(x) 0 7 480.33 1 14 310.93 1 . . . 90 595 0.00622095 0.0211144 0.0322 0.0211 91 602 0.00621755 0.0211144 0.0155 0.0211 Maximum number of function evaluations reached: increase options.MaxFunEvals. z0 = -0.3322
-1.7225
13.1600
11.8633
8.5479
29.1072
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Stability Concepts Outlines
➠ Basic stability concepts ➛ Nonlinear systems: ➳ Ordinary differential equations (ODE) ➳ Differential algebraic equations (DAE) ➛ Equilibrium points: ➳ Definitions. ➳ Stability: ☞ Linearization. ☞ Eigenvalue analysis. ➛ Stability regions.
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ODE Systems
➠ Nonlinear systems are represented by a nonlinear set of differential equations:
x˙ = s(x, p, λ) where
➳ x → n state variables (e.g. generator angles)
➳ p → k controllable parameters (e.g. compensation) ➳ λ → ℓ uncontrollable parameters (e.g. loads)
➳ s(·) → n nonlinear functions (e.g. generator equations)
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ODE Systems
➠ For example, for a simple generator-system model: PG + jQG
Generator
jx′G
jxL
V1 ∠δ1
E ′ ∠δ
PL + jQL
System
jxth
V2 ∠δ2
V ∠0 Infinite bus (M
= ∞)
AVR
The generator is modeled as a simple d axis transient voltage behind transient reactance.
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ODE Systems
➠ The generator has only the mechanical dynamics: δ˙ ω˙
= ω = ωr − ω0 1 (PL − PG − Dω) = M
where
PG
= =
E′V sin δ x′G + xL + xth V1 V sin δ1 xL + xth
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ODE Systems
➠ If the AVR is modeled, V1 may be assumed to be kept constant by varying E ′ , with the generator’s reactive power within limits: V12 V1 V QG = − cos δ1 xL + xth xL + xth ⇒
QG min ≤ QG ≤ QG max
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ODE Systems
➠ If the AVR is not modeled: ➛ x = [δ, ω]T → state variables
➛ p = [E ′ , V ]T → controlled parameters ➛ λ = PL → uncontrolled parameters
➛ Hence, assuming p = [1.5, 1]T , M = D = 0.1, and x = 0.75: x˙ = x 1 2 s(x, p, λ) = x˙ 2 = 10λ = 20 sin x1 − x2
➠ These are basically the pendulum equations.
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DAE Systems
➠ Differential Algebraic Equation (DAE) models are defined as: x˙ = f (x, y, p, λ) 0
= g(x, y, p, λ)
where:
➛ y → m algebraic variables (e.g. load voltages)
➛ f (·) → n nonlinear differential equations (e.g. generator equations) ➛ g(·) → m nonlinear algebraic equations (e.g. reactive power equations)
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DAE Systems
➠ For example, for the generator-infinite bus example with AVR, for QG min ≤ QG ≤ QG max : δ˙ ω˙ 0 0 0
= ω 1 E′V = PL − sin δ − Dω M x V1 V E′V sin δ1 sin δ − = x xL + xth V1 E ′ V12 = −QG − ′ + ′ cos(δ1 − δ) xG xG V12 V1 V = QG − + cos δ1 xL + xth xL + xth Universidad de Castilla-La Mancha
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DAE Systems
➠ Thus, for: ➛ x = [δ, ω]T ➛ y = [E ′ , δ1 , QG ]T ➛ p = [V1 , V ]T = [1, 1]T ➛ λ = PL ➛ M = D = 0.1, x = 0.75, x′G = 0.25, xL + xth = 0.5 x˙ = x 1 2 f (x, y, p, λ) = x˙ 2 = 10λ − 13.33y1 sin x1 − x2 0 = 1.333y1 sin x1 − 2 sin y2 g(x, y, p, λ) = 0 = −y3 − 4 + 4y1 cos(y2 − x1 ) 0 = y3 − 2 + 2 cos y2 Universidad de Castilla-La Mancha
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DAE Systems
➠ If QG = QG max or QG = QG min : ➛ x = [δ, ω]T ➛ y = [E ′ , δ1 , V1 ]T ➛ p = [QG , V ]T = [±0.5, 1]T ➛ λ = PL ➛ M = D = 0.1, x = 0.75, x′G = 0.25, xL + xth = 0.5 x˙ = x 1 2 f (x, y, p, λ) = x˙ 2 = 10λ − 13.33y1 y3 sin x1 − x2 0 = 1.333y1 sin x1 − 2y3 sin y2 g(x, y, p, λ) = 0 = ∓0.5 − 4y32 + 4y1 y3 cos(y2 − x1 ) 0 = ±0.5 − 2y32 + 2y3 cos y2 Universidad de Castilla-La Mancha
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DAE Systems
➠ If the Jacobian matrix Dy g = [∂gi /∂yi ]m×m is nonsingular, i.e. invertible, along the trajectory solutions, the system can be transformed into an ODE system (Implicit Function Theorem):
y ⇒
= h(x, p, λ)
x˙ = f (x, h(x, p, λ), p, λ) = s(x, p, λ)
➠ In practice, this is a purely “theoretical” exercise that is not carry out due to its complexity.
➠ The system model should be revised when Dy g is singular.
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Equilibria
➠ For the ODE system, equilibria are defined as the solution x0 for given parameter values p0 and λ0 of the set of equations s(x0 , p0 , λ0 ) = 0 ➠ There are multiple solutions to this problem, i.e. multiple equilibrium points.
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Equilibria
➠ The “stability” of these equilibria is defined by linearizing the nonlinear system around x0 , i.e. ∂si (x0 , p0 , λ0 ) x − x0 ∆x˙ = | {z } ∂xj n×n ∆x
= Dx s|0 ∆x
➠ where Dx s|0 = Dx s(x0 , p0 , λ0 ) = ∂s/∂x|0 is the system Jacobian matrix evaluated at the equilibrium point.
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Equilibria
➠ From linear system theory, the linear system stability is defined by the eigenvalues µi of the Jacobian matrix Dx s|0 , which are defined as the solutions of the equation:
⇒
Dx s|0 v
= µv
right e-vector
Dx s|T0 w
→
= µw
→
left e-vector
det(Dx s|0 − µIn ) = 0
an µn + an−1 µn−1 + . . . + a1 µ + a0 = 0
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Equilibria
➠ There are n complex eigenvalues, left and right eigenvectors associated with the system Jacobian Dx s|0 . ➠ In practice, these eigenvalues are not computed using characteristic polynomial but other more efficient numerical techniques, as the costs associated with computing these values is rather large in realistic power systems.
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Equilibria
➠ These eigenvalues define the small perturbation stability of the ODE system, i.e. the “local” stability of the nonlinear system near the equilibrium points:
➛ Stable equilibrium point (s.e.p.): The system is locally stable about x0 if all the eigenvalues µi (Dx s|0 ) are on the left-half (LH) of the complex plane.
➛ Unstable equilibrium point (u.e.p.): The system is locally unstable about x0 if at least one eigenvalue µi (Dx s|0 ) is on the right-half (RH) of the complex plane.
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Equilibria
➠ The equilibrium point x0 is a bifurcation point if at least one eigenvalue µi (Dx s|0 ) is on the imaginary axis of the complex plane. ➠ Some systems have equilibria with eigenvalues on the imaginary axis without these being bifurcation points; for example, a lossless generator-infinite bus system with no damping.
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Equilibria
➠ For example, for the simple generator-infinite bus example with no AVR: x˙ 1
= x2
x˙ 2
= 10λ − 20 sin x1 − x2
the equilibrium points can be found from the steady-state (power flow) equations:
0
= x20
0
= 10λ − 20 sin x10 − x20
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Equilibria
➠ which leads to the solutions: sin−1 (λ/2) x10 = 0 x20
⇒
sin−1 (PL /2) δ0 = 0 ω0
➠ This yields basically three equilibrium points (other solutions are just “repetitions” of these three):
➛ s.e.p → −π/2 < x1s < π/2 ➛ u.e.p.1 → x1u1 = x1s + π ➛ u.e.p.2 → x1u2 = x1s − π
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Equilibria
➠ This corresponds to the intersections of PG = E ′ V /x sin δ with PL : Stable
Bifurcation
PG
EV /x
Unstable
Unstable
PL
δµ2 (xµ2 )
−π
π δs (xs )
π/2
δµ1 (xµ1 )
δ, (x1 )
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Equilibria
➠ The stability of these equilibria is determined using the system Jacobian: ∂s1 /∂x1 |0 ∂s1 /∂x2 |0 Dx s|0 = ∂s2 /∂x1 |0 ∂s2 /∂x2 |0 0 1 = −20 cos x10 −1
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Equilibria
⇒
det(Dx s|0 − µI2 )
−µ = det −20 cos x10
1
−1 − µ
= µ2 + µ + 20 cos x10 = 0 ⇒ PL = 1
µ1,2 ⇒
1 1√ 1 − 80 cos x10 =− ± 2 2 µ1,2 (Dx s|xs ) = −0.5 ± j4.132 3.192 µ1,2 (Dx s|xu1 /u2 ) = −5.192
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Equilibria
➠ For this system, the equilibria are: ➛ stable if ∂PG >0 ∂δ ➛ unstable if ∂PG <0 ∂δ ➛ bifurcation point for ∂PG =0 ∂δ
⇒
δ = π/2
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Equilibria
➠ For DAE systems, the equilibria z0 = (x0 , y0 ) for parameter values p0 and λ0 are defined as the solution to the nonlinear, steady state problem: f (x , y , p , λ ) = 0 0 0 0 0 F (x0 , y0 , p0 , λ0 ) = 0 g(x0 , y0 , p0 , λ0 ) = 0 ➠ In this case, the linearization about (x0 , y0 ) yields:
∆x˙ = Dx f |0 ∆x + Dy f |0 ∆y ∆0
=
Dx g|0 ∆x + Dy g|0 ∆y
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Equilibria
➠ Hence, by eliminating ∆y from these equations, one obtains: Dx s|0 = Dx f |0 − Dy f |0 Dy g|−1 0 Dx g|0 ➠ Observe that, as mentioned before, the nonsingularity of the Jacobian Dy g|0 in this case is required. ➠ The same local stability conditions apply in this case based on the eigenvalues of Dx s|0 .
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Equilibria
➠ For the generator-infinite bus example with AVR example (within QG limits), the steady-state solutions are obtained from solving the steady-state or power flow equations:
0
= x20
0
= 10λ − 13.33y10 sin x10 − x20
0
= 1.333y10 sin x10 − 2 sin y20
0
= −y30 − 4 + 4y10 cos(y20 − x10 )
0
= y30 − 2 + 2 cos y20
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Equilibria
➠ Which in M ATLAB format are: function f = dae_eqs(z) global lambda x10 x20 y10 y20 y30
= = = = =
z(1); z(2); z(3); z(4); z(5);
f(1,1) f(1,2) f(1,3) f(1,4) f(1,5)
= = = = =
x20; 10*lambda - 13.33 * y10 * sin(x10) - x20; 1.333 * y10 * sin (x10) - 2 * sin(y20); -y30 - 4 + 4 * y10 * cos(y20 - x10); y30 - 2 + 2 * cos(y20);
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Equilibria
➠ This generates the equilibrium point for λ = PL = 1: >> lambda = 1; >> z0 = fsolve(@dae_eqs,[0 0 1 0 1],optimset(’Display’,’iter’)) Norm of step
Iteration Func-count f(x) 0 6 102 1 12 1.52385 1 2 18 0.0050436 0.298687 3 24 2.72816e-05 0.0607967 4 30 2.0931e-13 0.000413599 5 36 4.98474e-28 5.13313e-08 Optimization terminated: first-order optimality
First-order Trust-region optimality radius 133 1 10.2 1 0.219 2.5 0.0534 2.5 4.89e-06 2.5 2.23e-13 2.5 is less than options.TolFun.
z0 = 0.7539
0
1.0959
0.5236
0.2679
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Equilibria
➠ And the symbolic Jacobian matrices: syms x1 x2 y1 y2 y3 real z = [x1 x2 y1 y2 y3]; F = dae_eqs(z); f = F(1:2); g = F(3:5); Dxf Dyf Dxg Dyg
= = = =
jacobian(f,[x1,x2]) jacobian(f,[y1,y2,y3]) jacobian(g,[x1,x2]) jacobian(g,[y1,y2,y3])
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➠ And the symbolic Jacobian matrices: Dxf = [ 0, [ -1333/100*y1*cos(x1), Dyf = [ 0, [ -1333/100*sin(x1),
1] -1]
0, 0,
Dxg = [ 1333/1000*y1*cos(x1), [ -4*y1*sin(-y2+x1), [ 0, Dyg = [ 1333/1000*sin(x1), [ 4*cos(-y2+x1), [ 0,
-2*cos(y2), 4*y1*sin(-y2+x1), -2*sin(y2),
0] 0]
0] 0] 0]
0] -1] 1]
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➠ These generate the following eigenvalues at the equilibrium point: Dxf = [ 0, [ -1333/100*y1*cos(x1), Dyf = [ 0, [ -1333/100*sin(x1),
1] -1]
0, 0,
Dxg = [ 1333/1000*y1*cos(x1), [ -4*y1*sin(-y2+x1), [ 0, Dyg = [ 1333/1000*sin(x1), [ 4*cos(-y2+x1), [ 0,
-2*cos(y2), 4*y1*sin(-y2+x1), -2*sin(y2),
0] 0]
0] 0] 0]
0] -1] 1]
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➠ These generate the following eigenvalues at the equilibrium point: x1 = z0(1); x2 = z0(2); y1 = z0(3); y2 = z0(4); y3 = z0(5); A = vpa(subs(Dxf),5); B = vpa(subs(Dyf),5); C = vpa(subs(Dxg),5); D = vpa(subs(Dyg),5); Dxs = A - B * inv(D) * C; ev = vpa(eig(Dxs),5) ev = -.50000+3.5698*i -.50000-3.5698*i
➠ Hence, this is a s.e.p. Universidad de Castilla-La Mancha June 26, 2008
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➠ A u.e.p. can be computed as well from these equations (neglecting QG limits): >> z0 = fsolve(@dae_eqs,[3 0 0.1 3 3],optimset(’Display’,’iter’)); z0 = 2.7465 >> >> >> >> >> >> >>
0
1.9491
2.6180
3.7321
x1 = z0(1); x2 = z0(2); y1 = z0(3); y2 = z0(4); y3 = z0(5); A = vpa(subs(Dxf),5); B = vpa(subs(Dyf),5); C = vpa(subs(Dxg),5); D = vpa(subs(Dyg),5); Dxs = A - B * inv(D) * C; ev = vpa(eig(Dxs),5)
ev = 4.2892 -5.2892 Universidad de Castilla-La Mancha June 26, 2008
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➠ Associated with every s.e.p. xs there is a stability region A(xs ), which basically corresponds to the region of system variables that are all attracted to xs , i.e.
x(t → ∞) → xs : x1 (0) A(xs ) xs
stable
x1 (t) ∂A(xs ) (stability region
x2 (0)
unstable
boundary)
x2 (t)
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Equilibria
➠ if A(xs ) is known, the stability of a system for large perturbations can be readily evaluated.
➠ However, determining this region is a rather difficult task. ➠ This can realistically be accomplished only for 2- or 3-dimendional systems using “sophisticated” nonlinear system analysis techniques.
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Equilibria λ = 0.2
10
5
x
x
xs
u2
u1
x
2
0
−5 ∂A(x ) s
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 0.5
10
5
x
x
u2
x
s
u1
x
2
0
−5
−10
∂A(x ) s
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 1.0
10
5
x
x
u2
x
s
u1
x
2
0
−5 ∂A(x ) s
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 1.5
10
5
x
xs
u1
x
2
0
∂A(x ) s
−5
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 1.8
10
5
x
u1
0 x
2
x
−5
s
∂A(x ) s
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 1.9
10
5
x 0 x
2
x
u1
s
∂A(x ) s
−5
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria
➠ In real systems, trial-and-error techniques are usually used: ➛ A contingency yields a given initial condition x(0). ➛ For the post-contingency system, the time trajectories x(t) can be computed by numerical integration.
➛ If x(t) converges to the post-contingency equilibrium point xs , the system is stable, i.e. x(0) ∈ A(xs ). ➛ If it diverges, the system is unstable.
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Voltage Stability Outlines
➠ Definitions. ➠ Basic concepts. ➠ Continuation Power Flow. ➠ Direct Methods. ➠ Indices. ➠ Controls and protections. ➠ Practical applications. ➠ Examples.
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➠ IEEE-CIGRE classification (IEEE/CIGRE Joint Task Force on Stability) Terms and Definitions, “Definitions and Classification of Power System Stability”, IEEE Trans. Power Systems and CIGRE Technical Brochure 231, 2003:
Power System Stability
Rotor Angle
Frequency
Voltage
Stability
Stability
Stability
Small Disturbance
Transient
Angle Stability
Stability
Large Disturbance Voltage Stability
Short Term
Short Term Short Term
Small Disturbance Voltage Stability
Long Term
Long Term
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➠ “Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact.”
➠ “Voltage stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition.”
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➠ The inability of the trnasmission system to supply the demand leads to a “voltage collapse” problem.
➠ This problem is associated with the “disappearance” of a stable equilibrium point due to a saddle-node or limit-induced bifurcation point, typically due to contingencies in the system (e.g. August 14, 2003 Northeast Blackout).
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➠ The concept of saddle-node and limit induced bifurcations can be readily explained using the generator-load example: PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
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➠ Neglecting for simplicity losses, electromagnetic dynamics, and the transient impedance in the d-axis transient model, the generator can be simulated with:
δ˙1 ω˙ 1
= ω1 = ωr − ω0 1 = (Pm − PG − DG ω1 ) M
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➠ The load can be simulated using the mixed models. ➠ For P , neglecting voltage dynamics (Tpv = 0) and voltage dependence (α = 0):
⇒
PL
= Kpf f2 + Kpv [V2α + Tpv V˙ 2 ]
PL
= P d + DL ω 2 1 (PL − Pd ) = ω2 = DL
δ˙2
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➠ For Q, neglecting frequency dependence (Kqf = 0) and voltage dependence (β = 0) QL ⇒
QL V˙ 2
= Kqf f2 + Kqv [V2β + Tqv V˙ 2 ] = Qd + τ V˙ 2 =
1 (QL − Qd ) τ
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Voltage Stability Concepts
➠ The transmission system yields the power flow equations: PL PG QL QG
V1 V2 sin(δ2 − δ1 ) XL V1 V2 sin(δ1 − δ2 ) = XL V1 V2 V22 + cos(δ2 − δ1 ) = − XL XL V12 V1 V2 = − cos(δ1 − δ2 ) XL XL = −
➠ Finally, since the system is lossless: P m = Pd Universidad de Castilla-La Mancha June 26, 2008
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➠ Hence, defining: δ ω ⇒ δ˙
= δ1 − δ2 = ω1 = ω − ω2
the system equations are:
ω˙ δ˙ V˙ 2
1 M
V1 V2 sin δ − DG ω Pd − XL V1 V2 1 sin δ − Pd = ω− DL XL 2 1 V1 V2 V = cos δ − Qd − 2 + τ XL XL =
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Voltage Stability Concepts
➠ Observe that these equations also represent a generator-dynamic load ′ , where V1 would stand system with no AVR and with XL including XG ′ . for EG ➠ The steady-state load demand may be assumed to have a constant power factor, i.e.
Qd = kPd ′ , ➠ If generator reactive power limits are considered, and neglecting XG
one has a DAE system.
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Voltage Stability Concepts
➠ For QGmin ≤ QG ≤ QGmax : 1 V10 V2 ω˙ = sin δ − DG ω Pd − M XL V10 V2 1 sin δ − Pd δ˙ = ω − DL XL 2 V V V 1 10 2 − 2 + cos δ − Qd V˙ 2 = τ XL XL 2 V10 V2 V10 + cos δ 0 = QG − − XL XL
with
x = [ω, δ, V2 ]T
y = QG
p = V10
λ = Pd
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Voltage Stability Concepts
➠ For QG = QGmin,max : ω˙ δ˙ V˙ 2 0
V1 V2 sin δ − DG ω Pd − = XL V1 V2 1 sin δ − Pd = ω− DL XL 2 V V1 V2 1 − 2 + cos δ − Qd = τ XL XL 1 M
= QGmin,max
V12 V1 V2 − + cos δ XL XL
with
x = [ω, δ, V2 ]T
y = V1
p = QGmin,max
λ = Pd
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Voltage Stability Concepts
➠ All the equilibrium point for the system with and without limits can be obtained solving the power flow equations:
0 0 0
V10 V20 sin δ0 XL 2 V20 V10 V20 − cos δ0 = kPd + XL XL 2 V10 V10 V20 = QG0 − + cos δ0 XL XL = Pd −
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➠ And the stability of these equilibrium points come from the state matrix: ➠ Without limits, or for QGmin ≤ QG ≤ QGmax : DG V10 V20 V10 − M XL sin δ0 − M − M XL cos δ0 V V V 10 20 10 Dx s|0 = 1 − DL XL cos δ0 − DL XL sin δ0 0 − Vτ10XVL20 sin δ0 −2 τVX20L + τVX10L cos δ0
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Voltage Stability Concepts
➠ For QG = QGmin,max : Dx s|0 = DG V10 V20 V10 − M − M XL cos δ0 − M XL sin δ0 V10 V20 V10 1 − cos δ − sin δ 0 0 DL XL DL XL 0 − Vτ10XVL20 sin δ0 −2 τVX20L + τVX10L cos δ0 T V10 0 − M XL sin δ0 −1 V10 V20 V V V 10 20 10 + cos δ0 − − D X sin δ0 −2 − sin δ 0 XL L L τ XL τ XL V20 V10 cos δ 0 τ XL XL cos δ0 Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ Assume XL = 0.5, M = 1, DG = 0.01, DL = 0.1, τ = 0.01, k = 0.25. ➠ With the help of M ATLAB and the continuation power flow routine of PSAT, for the system without limits and V1 = V10 = 1, the power flow solutions yield a “PV” or “nose” curve (bifurcation diagram):
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Voltage Stability Concepts 1
s.e.p.
0.9 0.8
x3 = V2
0.7
SNB
0.6 0.5 0.4 0.3 0.2 0.1 0
u.e.p. 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
λ = Pd
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Voltage Stability Concepts
➠ The saddle-node bifurcation point SNB corresponds to a point where the state matrix Dx s|0 is singular (one zero eigenvalue). ➠ This is typically associated with a power flow solution with a singular PF Jacobian Dz F |0 . ➠ This is not always the case, as for more complex dynamic models, the singularity of the state matrix does not necessarily correspond to a singularity o the PF Jacobian, and vice versa.
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➠ Observe that the SNB point corresponds to a maximum value λmax = Ps max ≈ 0.78, which is why is also referred to as the maximum loading or loadability point.
➠ For a load greater than Pd max , there are no PF solutions. ➠ This point is also referred to as the voltage collapse point.
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Voltage Stability Concepts
➠ Modeling a “contingency” at λ0 = Pd0 = 0.7 by increasing XL = 0.5 → 0.6: 1
operating point
0.9 0.8
x3 = V2
0.7 0.6
contingency
0.5 0.4 0.3 0.2
XL = 0.5 XL = 0.6
0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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➠ The dynamic solution yields a voltage collapse: 6
δ ω V2 V1
5
4
3 Contingency
Operating point
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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Voltage Stability Concepts
➠ For the system with limits and QGmax,min = ±0.5, the “PV” or “nose” curve is:
1 0.9 0.8
LIB
s.e.p.
x3 = V2
0.7 0.6
u.e.p.
0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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Voltage Stability Concepts
➠ In this case, the maximum loading or loadability point λmax = Pd max ≈ 0.65 corresponds to the point where the generator reaches its maximum reactive power limit QG = QG max = 0.5, and hence losses control of V1 . ➠ This is referred to a limit-induced bifurcation or LIB point. ➠ “Beyond” the LIB point, there are no more power flow solutions, due to the limit recovery mechanism of the AVR.
➠ In this case the LIB point is also a voltage collapse point.
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Voltage Stability Concepts
➠ Modeling a “contingency” at λ0 = Pd0 = 0.6 by increasing XL = 0.5 → 0.6: 1 0.9
operating point
0.8
contingency
x3 = V2
0.7 0.6 0.5 0.4 0.3 0.2
XL = 0.5 XL = 0.6
0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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➠ The dynamic solution also yields a voltage collapse: 6
δ ω V2 V1
5
4
3 Operating point
Contingency
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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Voltage Stability Concepts
➠ Collapse problems can be avoided using shunt or series compensation: PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
−jxC
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Voltage Stability Concepts
➠ In this case, the system equations are: ➠ For QGmin ≤ QG ≤ QGmax : ω˙ δ˙ V˙ 2 0
1 (Pd − V10 V2 BL sin δ − DG ω) M 1 (V10 V2 BL sin δ − Pd ) = ω− DL 1 = [−V22 (BL − BC ) + V10 V2 BL cos δ − kPd ] τ 2 = QG − V10 BL + V10 V2 BL cos δ =
where
BL =
1 XL
BC =
1 XC
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Voltage Stability Concepts
➠ For QG = QGmin,max : ω˙ δ˙ V˙ 2 0
1 (Pd − V10 V2 BL sin δ − DG ω) M 1 (V10 V2 BL sin δ − Pd ) = ω− DL 1 = [−V22 (BL − BC ) + V10 V2 BL cos δ − kPd ] τ 2 BL + V10 V2 BL cos δ = QGmin,max − V10 =
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Voltage Stability Concepts
➠ These equations generate the PV curves: ➠ Without limits: 1.5
XL = 0.6
XL = 0.5
V2
1
XL = 0.6 0.5
Bc = 0 Bc = 0 Bc = 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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Voltage Stability Concepts
➠ With limits: 1.5
XL = 0.6 XL = 0.5
V2
1
XL = 0.6 0.5
Bc = 0 Bc = 0 Bc = 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ = Pd
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Voltage Stability Concepts
➠ Applying compensation at t = 1.3 s without limits:
1.2 1
0.8
δ ω V2 V1
Apply compensation
0.6 Contingency 0.4
0.2 Operating point 0
−0.2
0
1
2
3
4
5
6
7
8
9
10
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ Applying compensation at t = 1.3 s with limits:
1.2 1
0.8
δ ω V2 V1
Apply compensation
0.6 Contingency 0.4
0.2 Operating point 0
−0.2
0
1
2
3
4
5
6
7
8
9
10
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Power System Dynamics and Stability
Continuation Power Flow
➠ These PV or nose curves are obtained using a continuation power flow. ➠ This technique “traces” the solutions of the power equations F (z, p, λ) = 0 as λ changes.
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Power System Dynamics and Stability
Continuation Power Flow
➠ The algorithm steps are: 1. Predictor
(∆z1 , ∆λ1 )
3. Corrector
(z0 , λ0 ) λ2
2. Parametrization
λ1
z2
λ z1
λ
➠ These methods “guarantee” convergence.
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Power System Dynamics and Stability
CPF Predictor
➠ Tangent vector method: ∂F dz − Dz F |1 dλ ∂λ 1 |{z}
⇒
∆z1 =
t1
k t1 kt1 k |{z} ∆λ1
(z2 , λ2 ) (∆z1 , ∆λ1 )
(z1 , λ1 )
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CPF Predictor
➠ Good method to follow “closely” the PV curves, but relatively slow. ➠ The tangent vector t defines “sensitivities” at any power flow solution point.
➠ This vector can also be used as an index to predict proximity to a saddle-node bifurcation, as opposed to using the smallest eigenvalue, which changes in highly non linear fashion.
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312
CPF Predictor
➠ Secant method: ∆z1
= k(z1a − z1b )
∆λ1
= k(λ1a − λ1b ) (z2 , λ2 )
(∆z1 , ∆λ1 ) (z1b , λ1b ) (z1a , λ1a )
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Power System Dynamics and Stability
CPF Predictor
➠ This method is faster but can have convergence problems with “sharp corners” (e.g. limits):
(z2 , λ2 ) (z1b , λ1b ) (z1a , λ1a )
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Power System Dynamics and Stability
CPF Parametrization
➠ Used to avoid singularities during the predictor step. ➠ Methods: ➛ Local: interchange a zi ∈ z with λ, i.e. “rotate” the PV curve.
➛ Arc length (s): assume z1 (s) and λ1 (s); thus solve for ∆z1 and ∆λ1 : ∂F Dz F |1 ∆z1 + + ∆λ1 = 0 ∂λ 1 ∆z1T ∆z1 + ∆λ21
= k
➠ No real need for parametrization in practice if step cutting is needed.
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CPF Corrector
➠ The idea is to add an equation to the equilibrium equations, i.e. solve for (z, λ) at a given point p: F (z, p, λ) = 0 ρ(z, λ) = 0 ➠ These equations are nonsingular for the appropriate choice of ρ(·).
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CPF Corrector
➠ Perpendicular intersection technique: (z1 , λ1 ) (∆z1 , ∆λ1 ) z (z2 , λ2 )
λ
F (z, p, λ) = 0 (z1 + ∆z1 − z)T ∆z1 + (λ1 + ∆λ1 − λ)∆λ1 = 0 Universidad de Castilla-La Mancha June 26, 2008
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CPF Corrector
➠ Local parametrization technique: (z1 , λ1 ) (∆z1 , ∆λ1 ) z
(z2 , λ2 )
λ
F (z, p, λ) = 0 λ = λ1 + ∆λ1 or zi = zi1 + ∆zi1 Universidad de Castilla-La Mancha June 26, 2008
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Direct Methods
➠ Allow to directly find the maximum loading point (saddle-node or limit induced bifurcation).
➠ This problem can be set up as an optimization problem:
Max. s.t.
λ F (z, p, λ) = 0 zmin ≤ z ≤ zmax pmin ≤ p ≤ pmax
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Power System Dynamics and Stability
Direct Methods
➠ If limits are ignored, the solution for given values of the control paramters (p = p0 ) to the associated optimization problem, based on the Lagrangian and KKT conditions, is given by:
F (z, p0 , λ)
0
→
DzT F (z, p0 , λ)w
= 0
DλT F (z, p0 , λ)w
= −1
steady stae solution
→ →
singularity condition
w 6= 0 condition
➠ This yields a saddle-node bifurcation point, and corresponds to the “left eigenvector” (w ) saddle-node equations.
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Direct Methods
➠ The solution to these nonlinear equations do not converge if the maximum loading point is a limit-induced bifurcation.
➠ In this case, the otpimization problem Max. s.t.
λ F (z, p0 , λ) = 0 zmin ≤ z ≤ zmax
must be solved using “standard” optimization techniques (e.g. Interior Point Method).
➠ The solution to this optimization problem may be a saddle-node or limit-induced bifurcation. Universidad de Castilla-La Mancha June 26, 2008
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Direct Methods
➠ Observe that if the control parameters p are allowed to change, the problem is transformed into a maximization of the maximum loading margin.
➠ Other optimization problems can be set up to not only maximize/guarantee loading margins but at the same time minimize costs (e.g. maximize social welfare).
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Direct Methods
➠ For example, for an OPF-based auction in electricity markets, the following multi-objective optimization problem can be posed: Min. s.t.
T G = − ω1 (CD PD − CST PS ) − ω2 λc
f (δ, V, QG , PS , PD ) = 0
→ PF equations
f (δc , Vc , QGc , λc , PS , PD ) = 0 → Max load PF eqs. λcmin ≤ λc ≤ λcmax
→ loading margin
0 ≤ PS ≤ PSmax
→ Sup. bid blocks
0 ≤ PD ≤ PDmax
→ Dem. bid blocks
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Power System Dynamics and Stability
Direct Methods
➠ With the phyiscal and security limits: Iij (δ, V ) ≤ Iijmax
→ Thermal limits
Iji (δ, V ) ≤ Ijimax Iij (δc , Vc ) ≤ Iijmax Iji (δc , Vc ) ≤ Ijimax QGmin ≤ QG ≤ QGmax → Gen. Q limits QGmin ≤ QGc ≤ QGmax Vmin ≤ V ≤ Vmax
→ V “security” lim.
Vmin ≤ Vc ≤ Vmax Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Direct Methods
➠ More information about this probem can be found in: ˜ F. Milano, C. A. Canizares and M. Invernizzi, “Multi-objective Optimization for PRicing System Security in Electricity Marktes”, IEEE Trans. on Power Systems, Vol. 18, No. 2, May 2003, pp. 596-604.
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Power System Dynamics and Stability
Indices
➠ Indices have been developed with the aim of monitoring proximity to voltage collapse for on-line applications.
➠ For example, the minimum real eigenvalue of the system Jacobian can be used to “measure” proximity to a saddle-node bifurcation, since this matrix becomes singular at that point.
➠ Many indices have been proposed, but the most “popular/useful” are: ➛ Singular value. ➛ Reactive power reserves.
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Indices
➠ The singular value index consists of simply monitoring the singular value of the Jacobian of the power flow equations as λ changes, e.g.
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327
Indices
➠ Observations: ➛ Computationally inexpensive. ➛ Highly nonlinear behavior, especially when control limits are reached. ➛ Cannot really be used to detect proximity to limit-induced bifurcation. ➛ Useful in some OPF-applications to help represent voltage stability as a constraints.
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328
Indices
➠ The reactive-power-reserve index consists of monitoring the “difference” between the reactive power generator output and its maximum limit, e.g. for a generator bus system:
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Power System Dynamics and Stability
Indices
➠ Observations: ➛ Computationally inexpensive. ➛ Highly nonlinear behavior. ➛ Only works for the “right” generators, i.e. the generators associated with the limit-induced bifurcation.
➛ Cannot really be used to detect proximity to a saddle-node bifurcation.
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Power System Dynamics and Stability
Voltage Stability Applications
➠ These concepts and associated techniques are applied to real power systems through the computation of PV curves.
➠ These curves are obtained either through a simple series of “continuous” power flows or using actual CPFs.
➠ In both cases, these “nose” curves are computed with respect to load changes, which are defined as follows:
PL
= PL0 + λ∆PL
QL
= QL0 + λ∆QL
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Power System Dynamics and Stability
Voltage Stability Applications
➠ Generator power changes are then defined, with the exception of the slack bus, as:
PG = PG0 + λ∆PG ➠ For a “distributed” slack bus model: PG = PG0 + (λ + kG )∆PG for all generators, where kG is a variable in the power flow equations replacing the variable power in the slack bus.
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Power System Dynamics and Stability
Voltage Stability Applications
➠ The Available Transfer Capability (ATC) of the tranmsission system is typically obtained from (NERC definition): ATC
= TTC + ETC + TRM
➠ The ATC can be associated to the maximum loading margin λmax of the system if N-1 contingency criteria are taken into account.
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Power System Dynamics and Stability
Voltage Stability Applications
➠ TTC or Total Transfer Capability is the maximum loading level of the system considering N-1 contingency criteria, i.e. the λmax for the worst realistic single contingency.
➠ ETC or Existent Transmission Commitments represents the “current” loading level plus any reserved transmission commitments.
➠ TRM or Transmission Reliability Margin which is an additional margin defined to represent other contingencies.
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Voltage Stability Applications
➠ For example, for WECC, systems should be operated a minimum 5% “distance” away from the maximum loadibility point when contigencies are considered:
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Power System Dynamics and Stability
Control and Protections
➠ To improve system loadability, i.e. avoid voltage stability problems, the most common control and protections techniques are:
➛ Increase reactive power output from generators, especially in the “critical” area (the area most “sensitive” to voltage problems).
➛ Introduce shunt compesation through the use of MSC, SVC or STATCOM (see slides 304 and 305).
➛ Use undervoltage relays.
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Power System Dynamics and Stability
Secondary Voltage Regulation
➠ One way of improving reactive power support is to coordinate the reactive power outputs of generators.
➠ The French and Italians typically referred to as “secondary voltage regulation” or control.
➠ The basic structure and controls are:
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Secondary Voltage Regulation
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Secondary Voltage Regulation
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Power System Dynamics and Stability
Secondary Voltage Regulation
➠ Observations: ➛ Control areas and associated pilot buses and controlled generators must be “properly” identified.
➛ Control “hierarchy” is important, i.e. PQR is “slower” than AVR, and RVR is “slower” than PQR.
➛ It is relatively inexpensive compared to shunt compensation solutions.
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Power System Dynamics and Stability
Undervoltage Relays
➠ These relays are installed on sub-transmission substations (loads) to shed during “long duration” voltage dips.
➠ The idea is to reduce load demand on the system to increase the loadability margin (see slide 334).
➠ Operation is somewhat similarly to taps in a LTC: ➛ Discrete load shedding steps (e.g. 1-2% of total load). ➛ Activated with a time delay (e.g. 1-2 mins.) after the voltage dips below certain values (e.g. 0.8-0.9 p.u.)
➛ The lower the voltage, the faster and larger the load shed.
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Power System Dynamics and Stability
Example
➠ For a 3-area sample system: 100 MW 150 MW 150 MW 60 MVAr
v3
150 MW 56 MVAr
Bus 2
Area 1
1.02∠0
50 MVAr R = 0.01 p.u. X = 0.15 p.u.
V2 ∠δ2
Bus 3 50 MVAr
50 MW 40 MVAr 100 MW
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Example
➠ For a 3-area sample system: Bus
∆PG
∆PL
∆QL
Name
(p.u.)
(p.u.)
(p.u.)
Area 1
1.5
0
0
Area 2
0
1.5
0.56
Area 3
0.5
0.5
0.40
➠ Using UWPFLOW to obtain the system PV curves, for a distributed slack bus model:
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Power System Dynamics and Stability
Example
➠ Data file in EPRI format (3area.wsc): HDG UWPFLOW data file, WSCC format 3-area example April 2000 BAS C C AC BUSES C C | SHUNT | C |Ow|Name |kV |Z|PL |QL |MW |Mva|PM |PG |QM |Qm |Vpu BE 1 Area 1 138 1 150 60 0 0 0 150 0 0 1.02 B 1 Area 2 138 2 150 56 0 50 0 100 0 0 1.00 B 1 Area 3 138 3 50 40 0 50 0 100 0 0 1.00
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Example
➠ Data file in EPRI format (3area.wsc): C C AC LINES C C M CS N C |Ow|Name_1 |kV1||Name_2 |kV2|||In || R | X | G/2 | B/2 |Mil| L 1 Area 1 138 Area 2 1381 15001 .01 .15 L 1 Area 1 138 Area 3 1381 15001 .01 .15 L 1 Area 2 138 Area 3 1381 15001 .01 .15 C C SOLUTION CONTROL CARD C C |Max| |SLACK BUS | C |Itr| |Name |kV| |Angle | SOL 50 Area 1 138 0. END
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Power System Dynamics and Stability
Example
➠ Generator and load change file (3area.k): C C UWPFLOW load and generation "direction" file C for 3-area example C C BusNumber BusName DPg Pnl Qnl PgMax [ Smax 1 0 1.5 0.0 0.0 0 0 2 0 0.0 1.5 0.56 0 0 3 0 0.5 0.5 0.40 0 0
Vmax Vmin ] 1.05 0.95 1.05 0.95 1.05 0.95
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Power System Dynamics and Stability
Example
➠ Batch file for UNIX (run3area): echo -1- Run base case power flow uwpflow 3area.wsc -K3area.k echo -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s echo - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 echo - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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Power System Dynamics and Stability
Example
➠ Batch file for Windows (run3area.bat): rem -1- Run base case power flow uwpflow 3area.wsc -K3area.k rem -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s rem - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 rem - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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Power System Dynamics and Stability
Example
➠ PV curves (threearea.m): Profiles 150
100
50
0
⇒
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
kVArea 3
138
kVArea 2
138
kVArea 1
138
1.4
1.6
λmax ≈ 1.6 p.u. Universidad de Castilla-La Mancha
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Power System Dynamics and Stability
Example
➠ Hence, if contingencies are not considered: X X ∆PL TTC = PL0 + λmax =
670 MW X ETC = PL0
TRM
⇒
ATC
=
350 MW
=
0.05TTC
=
33.5 MW
=
286.5 MW
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Power System Dynamics and Stability
Example
➠ The singular value index obtained with UWPFLOW is as follows: 0.7
Full matrix sing. value index
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
1.4
1.6
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ This information is extracted from the final report of the US-Canada Joint Task Force.
➠ The full details of the final report can be found on the Internet at: https://reports.energy.gov/B-F-Web-Part1.pdf
➠ The report is titled: U.S.-Canada Power System Outage Task Force. “Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations.” Washington DC: USGPO, April 2004.
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352
August 14 2003 Blackout
➠ 3 Interconnections, 10 Regional Reliability Councils:
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Regions and control areas of the North American Electric Reliability Council (NERC):
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Reliability coordinators (some are also system and market operators, such as the IMO and PJM):
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August 14 2003 Blackout
➠ Reliability coordinator and system/market operators in the Ohio area, where the system collapse started:
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356
August 14 2003 Blackout
➠ Four causes: ➛ Inadequate system understanding: FirstEnergy, ECAR ➛ Inadequate situational awareness: FirstEnergy ➛ Inadequate tree-trimming: FirstEnergy ➛ Inadequate diagnostic support: MISO, PJM
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August 14 2003 Blackout
➠ At 12:15 EDT, MISO began having problems with its state estimator; it did not return to full functionality until 16:04.
➠ Sometime after 14:14, FirstEnergy began losing its energy management system (EMS) alarms but did not know it.
➠ At 14:20, parts of FE’s EMS began to fail first remote sites, then core servers but FE system operators did not know this and FE IT support staff did not tell them.
➠ Without a functioning EMS, FE operators did not know their system was losing lines and voltage until about 15:45.
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ FirstEnergy did not do sufficient system planning to know the Cleveland-Akron area was seriously deficient in reactive power supply needed for voltage support.
➠ At 13:31 EDT, FE lost its Eastlake 5 unit, a critical source of real and reactive power for the Cleveland-Akron area.
➠ FE did not perform contingency analysis after this event.
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Lost of the Eastlake 5 unit and beginning of reactive power problems.
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Between 15:05 and 15:41 EDT, FE lost 3 345 kV lines in the Cleveland-Akron area under normal loading due to contact with too-tall trees but did not know it due to EMS problems.
➠ Line loadings and reactive power demands increased with each line loss. ➠ Between 15:39 and 16:08 EDT, FE lost 16 138kV lines in the Cleveland-Akron area due to overloads and ground faults.
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361
August 14 2003 Blackout
➠ Line outages:
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ At 16:05.57 EDT, FE lost its Sammis-Star 345 kV line due to overload. ➠ This closed a major path for power imports into the Cleveland-Akron area and initiated the cascade phase of the blackout.
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363
August 14 2003 Blackout
➠ The tipping point in Ohio:
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364
August 14 2003 Blackout
➠ Effect of a line trip: increasing loading on other lines.
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Effect of a line trip: decreasing voltages leading to voltage collapse.
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Cascade: ➛ Definition: A cascade on an electric system is a dynamic, unplanned sequence of events that, once started, cannot be stopped by human intervention.
➛ Power swings, voltage fluctuations and frequency fluctuations cause sequential tripping of transmission lines, generators, and automatic load-shedding in a widening geographic area.
➛ The fluctuations diminish in amplitude as the cascade spreads. Eventually equilibrium is restored and the cascade stops.
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367
August 14 2003 Blackout
➠ The system goes “haywire”:
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Higher voltage lines are better able to absorb large voltage and current swings, buffering some areas against the cascade (AEP, Pennsylvania).
➠ Areas with high voltage profiles and ample reactive power were not swamped by the sudden voltage and power drain (PJM and New England).
➠ After islanding began, some areas were able to balance generation with load and reach equilibrium without collapsing (upstate New York and southern Ontario).
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August 14 2003 Blackout
➠ Sequence of events:
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370
August 14 2003 Blackout
➠ Sequence of events:
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371
August 14 2003 Blackout
➠ Sequence of events:
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372
August 14 2003 Blackout
➠ Sequence of events:
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ When the cascade was over at 4:13 pm, as many as 50 million people in the northeast U.S. and the province of Ontario had no power.
➠ This is a “good” example of a voltage instability problem triggered by a series of contingencies.
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374
August 14 2003 Blackout
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Had the system properly monitored and NERC recommended operating guidelines followed, the system might have been saved.
➠ An operating rule regarding max. system loadability margins similar to WECC’s might have given the operators a better picture of the situation, but without proper monitoring, these would have probably not worked either.
➠ In the absence of these, under-voltage relays with ULTC blocking might have saved the system, as these would have automatically shed load when the voltages started to collapse in First Energy’s (FE) region in Ohio.
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Power System Dynamics and Stability
Voltage Stability Report
➠ Much more information regarding the issue voltage stability can be found in: ˜ ➠ C. A. Canizares, editor, “Voltage Stability Assessment: Concepts, Practices and Tools,” IEEE-PES Power System Stability Subcommittee Special Publication, SP101PSS, August 2002. IEEE-PES WG Report Award 2005.
➠ This is a 283-page report, published after 5 years in preparation, and coauthored by several voltage stability from around the world.
➠ More details about the report can be found at: http://thunderbox.uwaterloo.ca/˜claudio/claudio.html/#VSWG
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Angle Stability Outlines
➠ Definitions. ➠ Small disturbance: ➛ Hopf Bifurcations. ➛ Control and mitigation. ➛ Practical example. ➠ Transient Stability ➛ Time Domain. ➛ Direct Methods. ☞ Equal Area Criterion. ☞ Energy Functions. ➛ Practical applications. Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Angle Stability Definitions
➠ IEEE-CIGRE classification (IEEE/CIGRE Joint Task Force on Stability) Terms and Definitions, “Definitions and Classification of Power System Stability”, IEEE Trans. Power Systems and CIGRE Technical Brochure 231, 2003:
Power System Stability
Rotor Angle
Frequency
Voltage
Stability
Stability
Stability
Small Disturbance
Transient
Angle Stability
Stability
Large Disturbance Voltage Stability
Short Term
Short Term Short Term
Small Disturbance Voltage Stability
Long Term
Long Term
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Power System Dynamics and Stability
Angle Stability Definitions
➠ “Rotor angle stability refers to the ability of synchronous machines of an interconnected power system to remain in synchronism after being subjected to a disturbance. It depends on the ability to maintain/restore equilibrium between electromagnetic torque and mechanical torque of each synchronous machine in the system.”
➠ In this case, the problem becomes apparent through angular/frequency swings in some generators which may lead to their loss of synchronism with other generators.
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Power System Dynamics and Stability
Small Disturbance
➠ “Small disturbance (or small signal) rotor angle stability is concerned with the ability of the power system to maintain synchronism under small disturbances. The disturbances are considered to be sufficiently small that linearization of system equations is permissible for purposes of analysis.”
➠ This problem is usually associated with the appearance of undamped oscillations in the system due to a lack of sufficient damping torque.
➠ Theoretically, this phenomenon may be associated with a s.e.p. becoming unstable through a Hopf bifurcation point, typically due to contingencies in the system (e.g. August 1996 West Coast Blackout).
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381
Hopf Bifurcation I
➠ For the generator-load example, with AVR but no QG limits: PG + jQG
Generator
jx′G
PL + jQL
jxL
V1 ∠δ1
E ′ ∠δ
Kv s
−
V2 ∠δ2
V10 +
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Power System Dynamics and Stability
Hopf Bifurcation I
➠ The DAE model is: ω˙ δ˙ E˙ ′
E ′ V2 sin δ − DG ω Pm = X ′ E V2 1 sin δ − Pd = ω− DL X 1 M
= Kv (V10 − V1 ) ′ 2 1 V E V2 V˙ 2 = − 2 + cos δ − kPd τ X X ′ E V2 V1 V2 ′ sin δ − sin δ 0 = XL X 1 1 E ′ V2 V1 V2 2 0 = V2 − cos δ ′ + cos δ − XL X X XL Universidad de Castilla-La Mancha
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Power System Dynamics and Stability
Hopf Bifurcation I
➠ Observe that the algebraic constraint can be eliminated, since:
Thus:
V1r
= V1 cos δ ′
V1i
= V1 sin δ ′
V1i V2 E ′ V2 E ′ XL 0= − sin δ ⇒ V1i = sin δ XL X X
and
0 ⇒
V1r
1 E ′ V2 V1r V2 1 − + cos δ − = XL X X XL E ′ XL 1 + cos δ = V2 1 − XL X V22
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 7
384
Power System Dynamics and Stability
Hopf Bifurcation I
➠ This yields the following equations, which are better for numerical time domain simulations:
ω˙
=
δ˙
=
E˙ ′
=
V˙ 2
=
1 M
′
E V2 sin δ − DG ω Pm X ′ E V2 1 sin δ − Pd ω− DL X q 2 +V2 Kv V10 − V1r 1i 2 ′ 1 V E V2 − 2 + cos δ − kPd τ X X
➠ Observe that in this case, Pm = Pd , i.e. generation and load are assumed to be balanced. Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 8
385
Power System Dynamics and Stability
Hopf Bifurcation I
➠ The PV curves for M = 0.1, DG = 0.01, DL = 0.1, τ = 0.01, ′ = 0.5, V10 = 1, k = 0.25 are: Kv = 10, XG 1
OP
HB
0.9 0.8
HB
0.7
V2
0.6 0.5
xL = 0.6
0.4 0.3 0.2
xL = 0.5
0.1 0
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 9
386
Power System Dynamics and Stability
Hopf Bifurcation I
➠ The eigenvalues for the system with respect to changes in Pd for xL = 0.5: 3
2
Imag
1
0
−1
−2
−3 −5
−4
−3
−2
−1 Real
0
1
2
3
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 10
387
Power System Dynamics and Stability
Hopf Bifurcation I
➠ There is a Hopf bifurcation for Pd = 0.65, xL = 0.5: 6
4
HB 2
ℜ(µ)
0
−2
−4
−6
−8
−10
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 11
388
Power System Dynamics and Stability
Hopf Bifurcation I
➠ A Hopf bifurcation with eigenvalues µ = ±jβ yields a periodic oscillation of period:
T =
2π β
➠ Hence, for the example: µ ≈ ⇒
T
±j3
≈ 2s
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 12
389
Power System Dynamics and Stability
Hopf Bifurcation I
➠ The contingency xL = 0.5 → 0.6 yields: 2.5
ω δ E′ V1 V2
2
1.5
1
0.5
0
−0.5
0
0.2
0.4
0.6
0.8
t [s]
1
1.2
1.4
1.6
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 13
390
Power System Dynamics and Stability
Hopf Bifurcation I
➠ Notice that in this case no oscillations are observed, which is a “trademark” of Hopf bifurcations and small-disturbance angle instabilities.
➠ The reason for this is that the oscillation period is 2 s (typical in practice where these kinds of oscillations are in the 0.1-1 Hz range), but the bus voltage collapses well before the oscillations appear, which is atypical and is probably due to the chosen impedances and time constants.
➠ This example stresses the point that angle instabilities do lead to voltage collapse, and vice versa, voltage instabilities lead to angle/frequency oscillations, even though the reason behind each stability problem are fairly different.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 14
Power System Dynamics and Stability
391
Hopf Bifurcation II
➠ Single-machine dynamic-load system with SVC: PG + jQG
PL + jQT jX
E∠δ V
V ∠0 jBC
Vref
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 15
392
Power System Dynamics and Stability
Hopf Bifurcation II
➠ The total reactive power absorbed by the load and the SVC is as follows: EV V2 + cos(δ) + V 2 BC QT (V, δ) = − X X ➠ The SVC controller is modeled as a first order pure integrator. Vref
+
BC 1/sT − V
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 16
393
Power System Dynamics and Stability
Hopf Bifurcation II
➠ The resulting differential equations of the SMDL system with SVC are as follows:
δ˙ ω˙ V˙ B˙ C
= ω EV 1 [Pd − sin(δ) − Dω] = M X 1 1 EV = [−kPd + V 2 (BC − ) + cos(δ)] τ X X 1 (Vref − V ) = T
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 17
394
Power System Dynamics and Stability
Hopf Bifurcation II
➠ BC is the equivalent susceptance of the SVC; T and Vref are the SVC time constant and reference voltage, respectively.
➠ In the following, it is assumed that T = 0.01 s and Vref = 1.0 p.u. ➠ Observe that also in this case it is possible to deduce the set of ODE, i.e. the algebraic variables can be explicitly expressed as a function of the state variables and the parameters.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 18
Power System Dynamics and Stability
395
Hopf Bifurcation II
➠ The state matrix of the system is as follows:
0 EV − M X cos(δ) A= − EV sin(δ) τX 0
|
1
D | −M
|
0
|
0
|
0
| 0 E | − M X sin(δ) | 0 1 V2 | τ X [E cos(δ) − 2V + 2V BC X] | τ | − T1 | 0
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 19
396
Power System Dynamics and Stability
Hopf Bifurcation II
➠ Eigenvalue loci: 400
300
200
Imaginary
100
Hopf Bifurcation P =1.4143
0
d
−100
−200
−300
−400 −0.4
−0.2
0
0.2
0.4
0.6
0.8
Real
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 20
397
Power System Dynamics and Stability
Hopf Bifurcation II
➠ A complex conjugate pair of eigenvalues crosses the imaginary axis for Pd = 1.4143, thus leading to a Hopf bifurcation. ➠ The HB point is: (δ0 , ω0 , V0 , BC0 , Pd0 ) = (0.7855, 0, 1, 1.2930, 1.4143)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 21
398
Power System Dynamics and Stability
Hopf Bifurcation II
➠ Bifurcation diagram Pd -δ :
3
2.5
δ (rad)
2 max d
P 1.5
Hopf Bifurcation 1
0.5
0
0
0.2
0.4
0.6
0.8
1
P (p.u.)
1.2
1.4
1.6
1.8
2
d
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 22
399
Power System Dynamics and Stability
Hopf Bifurcation II
➠ We simulate a step change in Pd from 1.41 p.u. to 1.42 p.u. for t = 2 s. ➠ For t > 2 s the system does not present a stable equilibrium point and shows undamped oscillations (likely an unstable limit cycle), as expected from the P -δ curve.
➠ For t = 2.57 s, the load voltage collapses. ➠ Note that, in this case, the generator angle shows an unstable trajectory only after the occurrence of the voltage collapse at the load bus.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 23
400
Power System Dynamics and Stability
Hopf Bifurcation II
➠ Time domain simulation results:
δ (rad)
0.85
0.8
0.75 1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.3
2.4
2.5
2.6
t (s)
V (p.u.)
1.5 1 0.5 0 1.8
1.9
2
2.1
2.2
t (s)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 24
401
Power System Dynamics and Stability
Hopf Bifurcation II
➠ The use of the SVC device gives a birth to a new bifurcation, namely a Hopf bifurcation.
➠ This Hopf bifurcation cannot be removed by simply adjusting system parameters.
➠ However SVC and load dynamics can be coordinated so that the loadability of the system can be increased.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 25
Power System Dynamics and Stability
402
Control and Mitigation
➠ For the IEEE 14-bus test system: Bus 13
Bus 14 Bus 10 Bus 12
Bus 09
Bus 11
Bus 07
Bus 06 Bus 04
Bus 08
Bus 05 Bus 01
Breaker
Bus 02
Bus 03
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 26
403
Power System Dynamics and Stability
Control and Mitigation
➠ Generator speeds for the line 2-4 outage and 40% overloading: 1.001
1.0005
1
0.9995
0.999
0.9985
0.998
0
5
10
15 time (s)
20
25
30
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 27
404
Power System Dynamics and Stability
Control and Mitigation
➠ This has been typically solved by adding Power System Stabilizers (PSS) to the voltage controllers in “certain” generators, so that equilibriun point is made stable, i.e. the Hopf is removed. vs max vSI
Tw s Kw Tw s + 1
T1 s + 1
T3 s + 1
1
T2 s + 1
T4 s + 1
Tǫ s + 1
vs
vs min
➠ FACTS can also be used to address this problem.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 28
Power System Dynamics and Stability
405
Control and Mitigation
➠ For the IEEE 14-bus test system with PSS at bus 1: Bus 13
Bus 14 Bus 10 Bus 12
Bus 09
Bus 11
Bus 07
Bus 06 Bus 04
Bus 08
Bus 05 Bus 01
Breaker
Bus 02
Bus 03
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 29
406
Power System Dynamics and Stability
Control and Mitigation
➠ Generator speeds for the line 2-4 outage and 40% overloading: 1.001
1.0005
1
0.9995
0.999
0.9985
0.998
0
5
10
15 time (s)
20
25
30
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 30
407
Power System Dynamics and Stability
Control and Mitigation
➠ Data regarding this system are available at: http://thunderbox.uwaterloo.ca/∼claudio /papers/IEEEBenchmarkTFreport.pdf
➠ More details regarding this example can be found in: F. Milano, “An Open Source Power System Analysis Toolbox”, accpeted for publication on IEEE Trans. On Power Systems, March 2004.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 31
408
Power System Dynamics and Stability
Control and Mitigation
➠ For the IEEE 145-bus, 50-machine test system: 1
99
36
93
AREA 2 87
43 33 34
5 4
35 38
3
45 41
116
42
115
44
39
114
142
141 140
85
50
117 139 118 137
102
52
55
129 61
53 56 101
113
143
46
51 2
144
49
47 84
145
48
37
40 49
AREA 1
88
110
136
86
54
127 135 138
6 57 7
104
126
134
62
128
119
120 122
133 132 121
66
95 67
63
111
64
65
125 68
97
8 9
92
90
124
130
69
80 112
11 12
131
123
10
107
32
79
94 72
60 71 21
20
19
18
83
78
30
89
23
96
77
76
59 98 13
17
70 100
103
15 58
22 16
74
105
26
75
81 14
91
106 73 31
24 29
28
82 25
108
109
27
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 32
409
Power System Dynamics and Stability
Control and Mitigation
➠ For an impedance load model, the PV curves yield: (a) 0.94 Voltage (p.u.)
HB 0.92
Base Case Line 79−90 Outage
Operating point
Constant impedance load line 0.9
HB
0.88 0.86 0.84 0.82
0
0.002
0.004
0.006
0.008
0.01
0.012
(b) Base Case Line 79−90 Outage
L.F. (p.u.)
Voltage (p.u.)
0.94
Operating point 0.93
HB
HB
0.92
Constant impedance load line
0.91 0.9
2
2.5
3
3.5
4
4.5
5
λ
5.5
6 −3
x 10
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 33
410
Power System Dynamics and Stability
Control and Mitigation
➠ Indices based on the singular values have been proposed to predict Hopf bifucations: 0.015 HBI2 Base Case HBI Base Case 1 HBI Line 79−90 Outage 2 HBI1 Line 79−90 Outage
Hopf Indices
0.01
0.005
HB 0
0
0.002
0.004
0.006 λ
HB
0.008
0.01
0.012
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 34
411
Power System Dynamics and Stability
Control and Mitigation
➠ More details regarding this example can be found in: ˜ C. A. Canizares, N. Mithulananthan, F. Milano, and J. Reeve, “Linear Performance Indices to Predict Oscillatory Stability Problems in Power Systems”, IEEE Trans. On Power Systems, Vol. 19, No. 2, May 2004, pp. 1104-1114.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 35
412
Power System Dynamics and Stability
Small Disturbance Applications
➠ In practice, some contingencies trigger plant or inter-area frequency oscillations in a “heavily” loaded system, which may be directly associated with Hopf bifurcations.
➠ This is a “classical” problem in power systems and there are many examples of this phenomenon in practice, such as the August 10, 1996 blackout of the WSCC (now WECC) system.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 36
413
Power System Dynamics and Stability
Small Disturbance Applications
➠ Observe that the maximum loadability of the system is reduced by the presence of the Hopf bifurcation.
➠ This leads to the definition of a “dynamic” ATC value. 1 HB
HB
0.9 OP
0.8 0.7
ATC
ETC
TRM
V2
0.6 0.5 TTC
0.4 0.3 Worst Contingency
0.2 0.1 0
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 37
Power System Dynamics and Stability
414
August 10, 1996 WSCC Blackout
➠ This information was extracted from a presentation by Dr. Prabha Kundur, President and CEO of PowerTech Labs Inc.
➠ The material is availbale at: toronto.ieee.ca/events/oct0303/prabha.ppt
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 38
415
Power System Dynamics and Stability
August 10, 1996 WSCC Blackout
➠ System conditions: ➛ High ambient temperatures in Northwest, and hence high power transfers from Canada to California.
➛ Prior to main outage, three 500 kV line sections from lower Columbia River to loads in Oregon were out of service due to tree faults.
➛ California-Oregon interties loaded to 4330 MW north to south. ➛ Pacific DC intertie loaded at 2680 MW north to south. ➛ 2300 MW flow from Britsh Columbia.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 39
Power System Dynamics and Stability
416
August 10, 1996 WSCC Blackout
➠ Growing 0.23 Hz oscillations caused tripping of lines:
3000 2900
5
4
3
2800
4
2700
1
2600
2
2500 2400 2300 0 3
6 12 16 19 22
25 28 31 40 43 47 50 53 56 59 62 65 68 71 74
Time in seconds
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 40
Power System Dynamics and Stability
417
August 10, 1996 WSCC Blackout
➠ Event 1: 14:06:39 → Big Eddy-Ostrander 500 kV LG fault - flashed to tree ➠ Event 2: 14:52:37 → John Day-Marion 500 kV LG -flashed to tree ➠ Event 3: 15:42:03 → Keeler-Alliston 500 KV - LG - flashed to tree ➠ Event 4: 15:47:36 → Ross-Lexington 500 kV - flashed to tree ➠ Event 5: 15:47:36-15:48:09 → 8 McMary Units trip Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 41
Power System Dynamics and Stability
418
August 10, 1996 WSCC Blackout
➠ As a results of the undamped oscillations, the system split into four large islands.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 42
Power System Dynamics and Stability
419
August 10, 1996 WSCC Blackout
➠ PSS solution:
San Onofre (addition)
Palo Verde (tune existing)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 43
420
Power System Dynamics and Stability
August 10, 1996 WSCC Blackout
➠ PSS solution: 3000
With existing controls
2800
2600
Eigenvalue =
0.0597 + j1.771
Frequency =
0.2818 Hz
2400
Damping = 2200
0
15
30
45
61
−0.0337
72
Time in seconds 3000
With PSS modifications 2800
Eigenvalue =
−0.0717 + j1.673
Frequency =
0.2664 Hz
2600 2400
Damping = 2200 0
18
50 32 Time in seconds
68
−0.0429
75
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 44
421
Power System Dynamics and Stability
August 10, 1996 WSCC Blackout
➠ 7.5 million customers experienced outages from a few minutes to nine hours.
➠ Total load loss: 35,500 MW.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 45
422
Power System Dynamics and Stability
Transient Stability
➠ “Large disturbance rotor angle stability or transient stability, as it is commonly referred to, is concerned with the ability of the power system to maintain synchronism when subjected to a severe disturbance, such as a short circuit on a transmission line. The resulting system response involves large excursions of generator rotor angles and is influenced by the nonlinear power-angle relationship”.
➠ The system nonlinearities determine the system response; hence, linearization does not work in this case.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 46
423
Power System Dynamics and Stability
Transient Stability
➠ For small disturbances, the problem is to determine if the resulting steady state condition is stable or unstable (eigenvalue analysis) or a bifurcation point (e.g. Hopf bifurcation).
➠ For large disturbances, the steady state condition after the disturbance can exist and be stable, but it is possible that the system cannot reach that steady state condition.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 47
424
Power System Dynamics and Stability
Transient Stability
➠ The basic idea and analysis procedures are: ➛ Pre-contingency (initial conditions): the system is operating in “normal” conditions associated with a s.e.p.
➛ Contingency (fault trajectory): a large disturbance, such as a short circuit or a line trip forces the system to move away from its initial operating point.
➛ Post contingency (fault clearance): the contingency usually forces system protections to try to “clear” the fault; the issue is then to determine whether the resulting system is stable, i.e. whether the system remains relatively intact and the associated time trajectories converge to a “reasonable” operating point.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 48
425
Power System Dynamics and Stability
Transient Stability
➠ Based on non linear theory, this analysis can be basically viewed as determining wheter the fault trajectory at the “clearance” point is outside or inside of the stability region of the post-contingency s.e.p.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 49
426
Power System Dynamics and Stability
Time domain analysis
➠ Given the complexity of power system models, the most reliable analysis tool for these types of studies is full time domain simulations.
➠ For example, for the generator-load example: PG + jQG
Generator
jx′G
E ′ ∠δ
PL + jQL
jxL
V1 ∠δ1
V2 ∠δ2
−jxC
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 50
427
Power System Dynamics and Stability
Time domain analysis
➠ The ODE for the simplest generator d-axis transient model and neglecting AVR and generator limits is:
ω˙ δ˙ V˙ 2
1 (Pd − E ′ V2 B sin δ − DG ω) M 1 = ω− (E ′ V2 B sin δ − Pd ) DL 1 [−V22 (B − BC ) + E ′ V2 B cos δ − kPd ] = τ =
where
B=
1 1 = ′ X XG + XL
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 51
428
Power System Dynamics and Stability
Time domain analysis
➠ The objective is to determine how much time an operator would have to connect the capacitor bank BC after a severe contingency, simulated here as a sudden increase in the value of the reactance X , so that the system recovers.
➠ In this case, and as previously discussed in the voltage stability section, the contingency is severe, as the s.e.p. disappears.
➠ Full time domain simulations are carried out to study this problem for the parameter values M = 0.1, DG = 0.01, DL = 0.1, τ = 0.01, E ′ = 1, Pd = 0.7, k = 0.25, BC = 0.5.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 52
429
Power System Dynamics and Stability
Time domain analysis
➠ A contingency X = 0.5 → 0.6 at tf = 1 s, with BC connection at tc = 1.4 s yields a stable system: 1.4
ω δ V2
1.2
E′
1
0.8
0.6
0.4
tf
0.2
tc
0
−0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t [s] Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 53
430
Power System Dynamics and Stability
Time domain analysis
➠ If BC is connected at tc = 1.5 s, the system is unstable: 6
ω δ V2
5
E′
4
tc tf
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t [s] Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 54
431
Power System Dynamics and Stability
Direct Methods
➠ Time domain analysis is expensive, so direct stability analysis technique have been proposed based on Lyapounov’s stability theory.
➠ The idea is to define an “energy” or Lyapounov function ϑ(x, xs ) with certain characteristics to obtain a direct “measure” of the stability region
A(xs ) associated with the post-contingency s.e.p. xs . ➠ A system’s energy is usually a good Lyapounov function, as it yields a stability “measure”.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 55
432
Power System Dynamics and Stability
Direct Methods
➠ The rolling ball example can used to explain the basic behind these techniques: u.e.p.2
~ v m u.e.p.1
h
s.e.p.
➠ There are 3 equilibrium points: one stable (“valley” bottom), two unstable (“hill” tops).
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 56
433
Power System Dynamics and Stability
Direct Methods
➠ The energy of the ball is a good Lyapounov or Transient Energy Function (TEF):
W
= Wkinetic + Wpotential = WK + WP 1 = mv 2 + mgh 2 = ϑ([v, h]T , 0)
➠ The potential energy at the s.e.p. is zero, and presents local maxima at the u.e.p.s (WP 1 and WP 2 ). ➠ The “closest” u.e.p. is u.e.p.1 since WP 1 < WP 2 . Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 57
434
Power System Dynamics and Stability
Direct Methods
➠ The stability of this system can then be evaluated using this energy: ➛ if W < WP 1 , the ball remains in the “valley”, i.e. the system is stable, and will converge to the s.e.p. as t → ∞.
➛ If W > WP 1 , the ball might or might not converge to the s.e.p., depending on friction (inconclusive test).
➛ When the ball’s potential energy WP (t) reaches a maximum with respect to time t, the system leaves the “valley”, i.e. unstable condition.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 58
435
Power System Dynamics and Stability
Direct Methods
➠ The “valley” would correspond to the stability region when friction is “large”.
➠ In this case, the stability boundary ∂A(xs ) corresponds to the “ridge” where the u.e.p.s are located and WP has a local max. value. ➠ The smaller the friction in the system, the larger the difference between the ridge and ∂A(xs ). ➠ For zero friction, ∂A(xs ) is defined by WP 1 .
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 59
436
Power System Dynamics and Stability
Direct Methods
➠ The direct stability test is only a necessary but not sufficient condition: ϑ(x, xs ) < c ⇒
x ∈ A(xs )
ϑ(x, xs ) > c ⇒
Inconclusive!
where the value of c is usually associated with a local maximum of a “potential energy” function.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 60
437
Power System Dynamics and Stability
Direct Methods
➠ For the simple generator-infinite bus example, neglecting limits and AVR: PG + jQG
Generator
jx′G
jxL
V1 ∠δ1
E ′ ∠δ
δ˙ ω˙ X
PL + jQL
System
jxth
V2 ∠δ2
V ∠0
= ω = ωr − ω0 ′ 1 EV = PL − sin δ − Dω M X ′ = XG + XL + Xth
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 61
438
Power System Dynamics and Stability
Direct Methods
➠ The kinetic energy in this system is defined as: WK =
1 M ω2 2
➠ And the potential energy is: Z WP = (Tc − Tm )dδ Z ≈ (Pc − Pm )dδ → in p.u. for ωr ≈ ω0 ≈
Z
δ
Z
δ
E′V ( (PG − PL )dδ ≈ − PL )dδ X δs δs
≈ −E ′ V B(cos δ − cos δs ) − PL (δ − δs ) where δs is the s.e.p. for this system. Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 62
439
Power System Dynamics and Stability
Direct Methods
➠ With WP presenting a very similar profile as the rolling ball example: PG unstable
stable
E′ V X
unstable
δu2
δs
δu1
max
WF max
WF 2 WF 1 min
δu2
δs
δu1
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 63
440
Power System Dynamics and Stability
Direct Methods
➠ Hence, the system Lyapounov function of TEF is: T EF
= ϑ(x, xs ) = ϑ([δ, ω]T , [δs , 0]T ) 1 = M ω 2 − E ′ V B(cos δ − cos δs ) 2 −PL (δ − δs )
➠ Thus, using similar criteria as in the case of the rolling ball: ➛ If T EF < WP 1 ⇒ system is stable.
➛ If T EF > WP 1 ⇒ inconclusive for D > 0 (“friction”). ➛ If T EF > WP 1 ⇒ unstable for D = 0 (unrealistic). Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 64
Power System Dynamics and Stability
441
Direct Methods
➠ This is equivalent to compare “areas” in the PG vs. δ graph (Equal Area Criterion or EAC): pre-contingency
PG
post-contingency
PL contingency (fault)
δ(0) = δspre
δ(tc ) δspost
δu1 post
δ
fault clearing time
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 65
442
Power System Dynamics and Stability
Direct Methods
➠ Thus, comparing the “acceleration” area: Z δ(tc ) (PL − PGf ault )dδ Aa = δspre
=
Z
δ(tc )
δspre
′
EV PL − Xf ault
dδ
➠ versus the “deceleration” area: Z δspost Ad = (PGpost − PL )dδ δ(tc )
=
Z
δspost
δ(tc )
E′V − PL dδ Xpost
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 66
Power System Dynamics and Stability
443
Direct Methods
➠ In conclusion: ➛ If Aa < Ad ⇒ system is stable at tc .
➛ If Aa > Ad ⇒ inconclusive for D > 0.
➛ If Aa > Ad ⇒ unstable for D = 0 (unrealistic).
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 67
444
Power System Dynamics and Stability
Direct Methods: Example 1
➠ A 60 Hz generator with a 15% transient reactance is connected to an infinite bus of 1 p.u. voltage through two identical parallel transmission lines of 20% reactance and negligible resistance. The generator is delivering 300 MW at a 0.9 leading power factor when a 3-phase solid fault occurs in the middle of one of the lines; the fault is then cleared by opening the breakers of the faulted line.
➠ Assuming a 100 MVA base, determine the critical clearing time for this generator if the damping is neglected and its inertia is assumed to be
H = 5 s. ➠ Assuming D = 0.1 s, determine the actual critical clearing time.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 68
Power System Dynamics and Stability
445
Direct Methods: Example 1
➠ Pre-contingency or initial conditions: PGpre QL
E′V sin δspre = PL = Xpre V2 E′V = − + cos δspre Xpre Xpre
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 69
Power System Dynamics and Stability
446
Direct Methods: Example 1
➠ Where: Xpre
=
0.15 +
0.2 = 0.25 2
300 MW PL = 100 MVA E′ 3 = sin δspre 0.25 QL = 3 tan(cos−1 0.9) E′ 1 + cos δspre 1.4530 = − 0.25 0.25
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 70
Power System Dynamics and Stability
447
Direct Methods: Example 1
⇒
Ei′pre
= E ′ sin δspre =
Er′ pre
0.75
= E ′ cos δspre =
E′
1.3633 q = E ′ 2rpre + E ′ 2ipre
= δspre
1.5559
Ei′pre Er′ pre
!
=
tan−1
=
28.82◦ = 0.5030 rad
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 71
448
Power System Dynamics and Stability
Direct Methods: Example 1
➠ Fault conditions: PGf ault
E′V sin δ Xf ault 1.5559 sin δ Xf ault
= =
where, using a Y-∆ circuit transformation due to the fault being in the middle of one of the parallel lines: jXf ault j0.15 j0.2 E ′ ∠δ
V ∠0 j0.1
j0.1
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 72
Power System Dynamics and Stability
449
Direct Methods: Example 1
Xf ault ⇒
PGf ault Aa
0.15 × 0.2 + 0.1 × 0.2 + 0.15 × 0.1 = 0.1 = 2.394 sin δ Z δ(tcc ) (PL − PGf ault )dδ = δspre
=
Z
δ(tcc )
0.503
= =
(3 − 2.394 sin δ)dδ
3(δ(tcc ) − 0.503) + 2.394(cos δ(tcc ) − cos(0.503)) 3δ(tcc ) + 2.394 cos δ(tcc ) − 3.6065
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 73
Power System Dynamics and Stability
450
Direct Methods: Example 1
➠ Post contingency conditions:
⇒
Xpost
=
=
0.15 + 0.2 = 0.35 E′V sin δ Xpost 4.446 sin δ
PGpost
=
3
=
4.446 sin δspost
δspost
=
42.44◦
=
0.7407 rad
⇒
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 74
451
Power System Dynamics and Stability
Direct Methods: Example 1
⇒ Ad
=
Z
π−δspost
δ(tcc )
=
Z
(PGpost − PL )dδ
2.4
δ(tcc )
(4.446 sin δ − 3)dδ
= −4.446(cos 2.4 − cos δ(tcc )) − 3(2.4 − δ(tcc )) = 3δ(tcc ) + 4.446 cos δ(tcc ) − 3.9215
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 75
452
Power System Dynamics and Stability
Direct Methods: Example 1
Aa
= Ad = 3δ(tcc ) + 2.394 cos δ(tcc ) − 3.6065 = 3δ(tcc ) + 4.446 cos δ(tcc ) − 3.9215
⇒
δ(tcc )
= 81.17◦ = 1.4167 rad
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 76
Power System Dynamics and Stability
453
Direct Methods: Example 1
➠ During the fault: δ˙ ω˙ M
⇒
= ω ′ 1 EV = sin δ PL − M Xf ault H = πf 5s = π60 Hz = 0.0265 s2
δ˙
= ω
ω˙
=
37.70(3 − 2.394 sin δ)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 77
454
Power System Dynamics and Stability
Direct Methods: Example 1
➠ Integrating these equations numerically for δ(0) = δspre = 28.82◦ : 220 200 180 160
δ [deg]
140 120 100 80 60 40 20
0
0.05
0.1
0.15
t [s]
0.2
0.25
0.3
0.35
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 78
455
Power System Dynamics and Stability
Direct Methods: Example 1
➠ For D = 0.1 and a clearing time of tc = 0.27 s, the system is stable:
δ [deg]
150
100
50
0
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
1.2
1.4
1.6
1.8
2
t [s]
10
ω [deg]
5
0
−5
−10
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 79
456
Power System Dynamics and Stability
Direct Methods: Example 1
➠ For a clearing time of tc = 0.28 s, the system is unstable; hence tcc ≈ 0.275 s: 2500
δ [deg]
2000 1500 1000 500 0
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
1.2
1.4
1.6
1.8
2
t [s]
40
ω [deg]
30
20
10
0
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 80
457
Power System Dynamics and Stability
Direct Methods: Example 2
➠ Generator-motor, i.e. system-system, cases may also be studied using the EAC method based on an equivalent inertia
M = M1 M2 /(M1 M2 ), and damping D = M D1 /M1 = M D2 /M2 . ➠ For the generator-load example neglecting the internal generator impedance and assuming an “instantaneous” AVR: PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 81
458
Power System Dynamics and Stability
Direct Methods: Example 2
➠ The “energy” functions, with or without generator limits, can be shown to be:
WK WP
1 M ω2 2 = −B(V1 V2 cos δ − V10 V20 cos δ0 ) 1 1 2 2 2 + B(V2 − V20 ) + B(V12 − V10 ) 2 2 V1 V2 − QG ln −Pd (δ − δ0 ) + Qd ln V20 V10 =
➠ The stability of this system can then be studied using the same “energy” evaluation previously explained for T EF = ϑ(x, x0 ) = WK + WP .
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 82
459
Power System Dynamics and Stability
Direct Methods: Example 2
➠ Thus for V1 = 1, XL = 0.5, Pd = 0.1, and Qd = 0.25Pd , the potential energy WP (δ, V2 ) that defines the stability region withr espect to the s.e.p. is:
8 7 6
WP
5 4 3 2 1 0 400
s.e.p. node
200
u.e.p. saddle
0
2 1.5 1
−200
δ
0.5 −400
0
V2
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 83
460
Power System Dynamics and Stability
Direct Methods: Example 2
➠ Simulating the critical contingency XL = 0.5 → 0.6 for Pd = 0.7 and neglecting limits, the “energy” profiles are: 0.2
Wp Wk+Wp
0.1
T EF
0
−0.1
−0.2
−0.3
−0.4 0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 84
461
Power System Dynamics and Stability
Direct Methods: Example 2
➠ The “exit” point on ∂A(xs ) is approximately at the maximum potential energy point.
➠ Thus, the critical clearing time is: tcc ≈ 1.42 s ➠ A similar value can be obtained through trial-and-error.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 85
462
Power System Dynamics and Stability
Direct Methods: Conclusions
➠ The advantages of using Lyapounov functions are: ➛ Allows reduced stability analysis. ➛ Can be used as an stability index. ➠ The problems are: ➛ Lyapounov functions are model dependent; in practice, only approximate “energy” functions can be found.
➛ Inconclusive if test fails. ➛ The post-perturbation system state must be known ahead of time, as the energy function is defined with respect to the corresponding s.e.p.
➠ Can only be used as an “approximate” stability analysis tool. Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 86
463
Power System Dynamics and Stability
Transient Stability Applications
➠ In practice, transient stability studies are carried out using time-domain trial-and-error techniques.
➠ These types of studies can now be done on-line even for large systems. ➠ The idea is to determine whether a set of “realistic” contingencies make the system unstable or not (contingency ranking), and thus determine maximum transfer limits or ATC in certain transmission corridors for given operating conditions.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 87
464
Power System Dynamics and Stability
Transient Stability Applications
➠ Thus, the maximum loadability of the system may be affected by the “size” of the stability region, leading to the definition of a “true” ATC value. 1 HB
HB
0.9 OP
0.8 0.7
ATC
ETC
TRM
V2
0.6 0.5 TTC
0.4 0.3 Worst Contingency
0.2 0.1 0
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 88
465
Power System Dynamics and Stability
Transient Stability Applications
➠ Critical clearing times are not really an issue with current fast acting protections.
➠ Simplified direct methods such as the “Extended Equal Area Criterion” (Y. Xue et al., “Extended Equal Area Criterion Revisited”, IEEE Transaction on Power Systems, Vol. 7, No. 3, Aug. 1992, pp. 1012-1022) have been proposed and tested for on-line contingency pre-ranking, and are being implemented for practical applications through an E.U. project.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 89
466
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ An example of an application of transient analysis techniques can be ˜ found in L.S. Vargas and C. A. Canizares, “Time Dependence of Controls to Avoid Voltage Collapse”, IEEE Transaction son Power Systems, Vol. 15, No. 4, November 2000, pp. 1367-1375.
➠ This paper discusses the May 1997 voltage collapse event of the main power system in Chile.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 90
467
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ Most of Chile lost power in a major blackout on Friday evening, snarling rush hour traffic in the capital. The blackout began at about 7:20 pm, and power was back in about a third of the affected areas at 9:00 pm.
➠ At 9:00 pm lights were gradually coming on in parts of the capital, home to 5 million people or one third of the country’s population. Television reports said power went out as far as Puerto Montt, a city some 600 miles south of Santiago, and in areas the same distance to the north.
➠ Source Reuters
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 91
468
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ Main system characteristics: ➛ Extension: 756626 km2 . ➛ Inhabitants: 14.5 mil. ➛ National consumptions: 33531 GWh. ➛ National peak load: 5800 MW. ➛ Installed capacity: 8000 MW. ➛ Frequency: 50 Hz. ➛ Trans. Level: 66/110/154/220/500 kV. ➛ Four interconnected systems: SING, SIC, AISEN, MAGALLANES
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 92
Power System Dynamics and Stability
469
Chilean Blackout (11/07/2003)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 93
470
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ Initial state of SIC system: ➛ 2500 MW load. ➛ Power flow south-north near 1000 MW (900 MW through 500 kV lines and 100 MW through 154 kV lines).
➠ Events: ➛ Line 154 kV trips. ➛ Major generator in the south hits reactive power limits and losses voltage control.
➛ Operator tries to recover falling voltages by connecting a capacitor bank near Santiago.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 94
Power System Dynamics and Stability
471
Chilean Blackout (11/07/2003)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 95
Power System Dynamics and Stability
472
Chilean Blackout (11/07/2003)
➠ The line trip and generator limits yield a voltage collapse associated with a limit-induced bifurcation problem:
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 96
Power System Dynamics and Stability
473
Chilean Blackout (11/07/2003)
➠ PV curves:
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 97
474
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ The connection of the capacitor bank after the generator limits are reached did not save the system, as the “faulted” system trajectories had “left” the stability region of the post contingency operating point.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 98
475
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ If the capacitor bank is connected before the generator limits are reached, the system would have been saved, as the “faulted” system trajectories were still within the stability of the post-contingency operating point.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 99
Power System Dynamics and Stability
476
Frequency Stability Outlines
➠ Definitions. ➠ Basic Concepts. ➠ Practical applications, controls and protections. ➛ Italian Blackout (28/10/2003) ➛ European Blackout (4/11/2006)
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 1
477
Power System Dynamics and Stability
Frequency Stability Definitions
➠ IEEE-CIGRE classification (IEEE/CIGRE Joint Task Force on Stability) Terms and Definitions, “Definitions and Classification of Power System Stability”, IEEE Trans. Power Systems and CIGRE Technical Brochure 231, 2003:
Power System Stability
Rotor Angle
Frequency
Voltage
Stability
Stability
Stability
Small Disturbance
Transient
Angle Stability
Stability
Large Disturbance Voltage Stability
Short Term
Short Term Short Term
Small Disturbance Voltage Stability
Long Term
Long Term
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 2
478
Power System Dynamics and Stability
Frequency Stability Definitions
➠ “Frequency Stability refers to the ability of a power system to maintain steady frequency following a severe system upset resulting in a significant imbalance between generation and load.”
➠ Thus, frequency stability analysis concentrates on studying the overall system stability for sudden changes in the generation-load balance.
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 3
479
Power System Dynamics and Stability
Frequency Stability Concepts
➠ For the generator-load example, with AVR but no QG limits: PG + jQG
Generator
jx′G
jxL
V2 ∠δ2
V1 ∠δ1
E ′ ∠δ
Kv s
PL + jQL
−
V10 +
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 4
480
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Neglecting losses and electromagnetic dynamics, the generator with a very simple AVR and no limits can be modeled using a d-axis transient model:
δ˙G ω˙ G E˙ ′
= ωG = ωr − ω0 1 (Pm − PG − DG ωG ) = M = Kv (V10 − V1 )
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 5
Power System Dynamics and Stability
481
Frequency Stability Concepts
➠ The load can be simulated using “simplified” mixed models and constant power factor:
δ˙2 V˙ 2
= ω2 = =
1 (PL − Pd ) DL
1 (QL − kPD ) |{z} τ Qd
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 6
482
Power System Dynamics and Stability
Frequency Stability Concepts
➠ The transmission system yields the power flow equations ′ ): (X = X L = X G PG
= =
QL
= =
QG
=
E ′ V2 PL = sin(δG − δ2 ) X V1 V2 sin(δ1 − δ2 ) XL V22 E ′ v2 − + cos(δ2 − δG ) X X V22 V1 V2 − + cos(δ2 − δ1 ) XL XL V12 V1 V2 − cos(δ1 − δ2 ) XL XL
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 7
Power System Dynamics and Stability
483
Frequency Stability Concepts
➠ Define: δ δ′ ω ⇒ δ˙
= δG − δ2 = δ1 − δ2 = ωG = ω − ω2
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 8
484
Power System Dynamics and Stability
Frequency Stability Concepts
➠ This yields the DAE model: ω˙ δ˙ E˙ ′
′
E V2 sin δ − DG ω X ′ E V2 1 sin δ − Pd = ω− DL X
=
1 M
Pm
= Kv (V10 − V1 ) 2 ′ 1 V2 E V2 ˙ V2 = − + cos δ − kPd τ X X E ′ V2 V1 V2 ′ sin δ − sin δ 0 = XL X 1 1 E ′ V2 V1 V2 2 0 = V2 − cos δ ′ + cos δ − XL X X XL Universidad de Castilla-La Mancha
June 26, 2008
Frequency Stability - 9
485
Power System Dynamics and Stability
Frequency Stability Concepts
➠ And the equilibrium equations: 0
=
0
=
0
=
0
=
0
=
0
=
E ′ V2 Pm sin δ − DG ω X E ′ V2 sin δ − Pd DL ω − X V10 − V1 V22 E ′ V2 − + cos δ − kPd X X E ′ V2 V1 V2 ′ sin δ − sin δ XL X 1 E ′ V2 V1 V2 1 2 V2 − cos δ ′ + cos δ − XL X X XL Universidad de Castilla-La Mancha
June 26, 2008
Frequency Stability - 10
486
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Hence: x = [ω, δ, E ′ , v2 ]T y
= [V1 , δ ′ ]T
p
= [Pm , V10 ]
λ
= Pd
➠ Observe that in this analysis, Pm 6= Pd to study the effect of generator-load imbalances in the system.
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 11
487
Power System Dynamics and Stability
Frequency Stability Concepts
➠ In normal operating conditions Pm = Pd ⇒ ω = 0, from the first 2 equilibrium equations.
➠ Hence, these equations can be replaced by the following 4 power flow equations, with 4 unknowns (E ′ , V2 , δ, δ ′ ): 0
=
0
=
0
=
0
=
E ′ V2 sin δ Pd − X V22 E ′ V2 −kPd − + cos δ X X V10 V2 sin δ ′ Pd − XL V22 V10 V2 kPd + − cos δ ′ XL XL
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 12
Power System Dynamics and Stability
488
Frequency Stability Concepts
➠ Simulating a 50% generation and load reduction, respectively, for M = 0.1, DG = 0.01, DL = 0.1, τ = 0.01, Kv = 10, XL = 0.5, ′ = 0.5, V10 = 1, Pd0 = 0.7, k = 0.25. XG ➠ Initial solution: ω
=
0
δ
=
0.7266
E′
=
1.3463
V2
=
0.7826
V1
=
1.0000
δ′
=
0.4636
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 13
489
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Time domain simulation: 1.5 1 0.5 0 −0.5 50% generator drop
−1 −1.5 −2 50% load drop
ω δ E′ V2 V1
−2.5 −3 −3.5
0
1
2
3
4
5
t [s]
6
7
8
9
10
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 14
490
Power System Dynamics and Stability
Frequency Stability Concepts
➠ The generation reduction yields a ≈0.5 Hz frequency drop and a load voltage increase.
➠ The load reduction yields a ≈0.5 Hz frequency increase, and an even larger load voltage increase, as the reactive power demand drops.
➠ The system reaches new s.e.p.s in both cases, as expected. ➠ Observe that the AVR keeps the generator terminal fairly stable and close to its set value V10 .
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 15
491
Power System Dynamics and Stability
Frequency Stability Concepts
➠ These frequency excursions due to generation-load imbalances are typical.
➠ It might lead to unstable conditions due to device protections such as frequency relays in generators and loads.
➠ Frequency problems may be solved manually by operators or automatically through controls and/or protections.
➠ Generator governors automatically regulate local frequency excursions.
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 16
492
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Centralized frequency regulators, such as automatic Area Control Error (ACE) regulators, may be used to regulate power exchanges among control areas by controlling the frequency deviations on the interties.
➠ Examples of frequency instabilities are: ➛ The Italian blackout of Tuesday, October 28, 2003 (Material courtesy of Prof. Alberto Berizzi, Politecnico di Milano, Italy).
➛ The European blackout of Saturday, November 4, 2006 (Material courtesy of Prof. Edmund Handschin, University of Technology, Dortmund, Germany).
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 17
Power System Dynamics and Stability
493
Italian Blackout (28/10/2003)
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 18
Power System Dynamics and Stability
494
Italian Blackout (28/10/2003)
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 19
495
Power System Dynamics and Stability
Italian Blackout (28/10/2003) 3:10.47 ETrans (operator of interties between Switzerland and rest of Europe) lets the GRTN (Italian operator) know of the Mettlen-Lavorgo line trip (1320 MW) and the overloading of the Sils-Soazza line (1650 MW), and requests a 300 MW demand reduction to relief the overload. 3:18.40 ETrans contacts EGL (Switzerland operator) requesting the tripping of a transformer in Soazza. 3:21.00 GRTN reduces the power imports by 300 MW.
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 20
496
Power System Dynamics and Stability
Italian Blackout (28/10/2003) 3:22.03 ATEL (Swiss energy company) changes the connection of the transformer at Lavorgo. 3:25.22 Protections trip the Sils-Soazza line (1783 MW). This is basically the beginning of the cascading events that follow, severing Italy from the rest of Europe and leading to the collapse of the Italian system.
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 21
Power System Dynamics and Stability
497
Italian Blackout (28/10/2003)
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Frequency Stability - 22
Power System Dynamics and Stability
498
Italian Blackout (28/10/2003)
➠ Imports from France:
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Frequency Stability - 23
Power System Dynamics and Stability
499
Italian Blackout (28/10/2003)
➠ Frequency in Italy:
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Power System Dynamics and Stability
500
Italian Blackout (28/10/2003)
➠ Frequency and voltages in Europe:
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Frequency Stability - 25
Power System Dynamics and Stability
501
Italian Blackout (28/10/2003)
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Frequency Stability - 26
Power System Dynamics and Stability
502
Italian Blackout (28/10/2003)
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 27
Power System Dynamics and Stability
503
Italian Blackout (28/10/2003)
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Frequency Stability - 28
504
Power System Dynamics and Stability
European Interconnected Systems
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Frequency Stability - 29
505
Power System Dynamics and Stability
European Interconnected Systems System
Capacity
Peak Load
Energy
Population
GW
GW
TWh
Mio.
Nordel
94
66
405
24
UPS/IPS
337
215
1285
280
UKTSOA
85
66
400
65
Nordel
600
390
2530
450
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506
Power System Dynamics and Stability
European Interconnected Systems
➠ 450 million people served ➠ 2530 TWh used ➠ 600 GW installed capacity at 500 e/kW = 300 Ge ➠ 230.000 km HV network at 400 000 e/km = 90 Ge ➠ Approx. 5.000.000 km MV+LV network ➠ 1500 e investment per EU citizen ➠ Largest man-made system
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 31
Power System Dynamics and Stability
507
European Blackout (4/11/2006)
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Frequency Stability - 32
Power System Dynamics and Stability
508
European Blackout (4/11/2006)
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509
Power System Dynamics and Stability
European Blackout (4/11/2006)
➠ Area 1: The frequency drops to 49 Hz, which causes an automatic load schedding.
➠ Area 2: Real power surplus of 6000 MW.
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510
Power System Dynamics and Stability
European Blackout (4/11/2006) 50.1 f [Hz] 50.0
frequency
49.9 49.8 49.7 49.6 49.5 49.4 49.3 49.2 49.1 49.0 48.9 22:10:00
22:10:10
22:10:20
22:10:30
22:10:40
22:10:50
22:11:00
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Power System Dynamics and Stability
511
European Blackout (4/11/2006)
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Frequency Stability - 36
Power System Dynamics and Stability
512
Contents
➠ Overview ➠ UWPFLOW ➠ Matlab ➠ PSAT
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Power System Dynamics and Stability
Overview
➠ Software packages for power system analysis can be basically divided into two classes of tools:
➛ Commercial softwares. ➛ Educational/research-aimed softwares.
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514
Overview
➠ Commercial softwares: ➛ PSS/E ➛ EuroStag ➛ Simpow ➛ CYME ➛ PowerWorld ➛ Neplan
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515
Power System Dynamics and Stability
Overview
➠ Commercial software packages follows an “all-in-one” philosophy and are typically well-tested and computationally efficient.
➠ Despite their completeness, these softwares can result cumbersome for educational and research purposes.
➠ commercial softwares are “closed”, i.e. do not allow changing the source code or adding new algorithms.
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516
Power System Dynamics and Stability
Overview
➠ For research purposes, the flexibility and the ability of easy prototyping are often more crucial aspects than computational efficiency.
➠ At this aim, there is a variety of open source research tools, which are typically aimed to a specific aspect of power system analysis.
➠ An example is UWPFLOW which provides an extremely robust algorithm for continuation power flow analysis.
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517
Power System Dynamics and Stability
Overview
➠ C anf FORTRAN are very fast but requires keen programming skills and are not suitable for fast prototyping.
➠ Several high level scientific languages, such as Matlab, Mathematica and Modelica, have become more and more popular for both research and educational purposes.
➠ At this aim, there is a variety of open source research tools, which are typically aimed to a specific aspect of power system analysis.
➠ Matlab proved to be the best user choice.
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518
Overview
➠ Matlab-based power system analysis tools: ➛ Power System Toolbox (PST) ➛ MatPower ➛ Voltage Stability Toolbox (VST) ➛ Power Analysis Toolbox (PAT) ➛ Educational Simulation Tool (EST) ➛ Power system Analysis Toolbox (PSAT)
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519
Power System Dynamics and Stability
Overview
➠ Comparison of Matlab-based power system analysis softwares: Package
PF
EST
X
CPF
OPF
SSA
TD
X
X X
MatEMTP MatPower
X
PAT
X
PSAT
X
X
PST
X
X
SPS
X
VST
X
EMT
GUI
GNE
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
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520
Power System Dynamics and Stability
Overview
➠ The features illustrated in the table are: ➛ power flow (PF) ➛ continuation power flow and/or voltage stability analysis (CPF-VS) ➛ optimal power flow (OPF) ➛ small signal stability analysis (SSA) ➛ time domain simulation (TD) ➛ graphical user interface (GUI) ➛ graphical network editor (GNE).
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521
Power System Dynamics and Stability
Overview
➠ An important but often missed issue is that the Matlab environment is a commercial and “closed” product, thus Matlab kernel and libraries cannot be modified nor freely distributed.
➠ To allow exchanging ideas and effectively improving scientific research, both the toolbox and the platform on which the toolbox runs should be free (Richard Stallman).
➠ An alternative to Matlab is the free GNU/Octave project.
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522
Power System Dynamics and Stability
UWPFLOW
➠ UWPFLOW is a research tool that has been designed to calculate local bifurcations related to system limits or singularities in the system Jacobian.
➠ The program also generates a series of output files that allow further analyses, such as tangent vectors, left and right eigenvectors at a singular bifurcation point, Jacobians, power flow solutions at different loading levels, voltage stability indices, etc.
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523
Power System Dynamics and Stability
UWPFLOW Features
➠ Adequate handling of generators limits, with generators being able to recover from a variety of limits, including S limits.
➠ Steady state models of generators and their control limits (AVR and Primemover limits) are included.
➠ Voltage dependent load models for voltage stability analysis are also included.
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524
Power System Dynamics and Stability
UWPFLOW Features
➠ Either BPA/WSCC ac-dc (HVDC systems) input data formats (and variations) or IEEE common format may be used.
➠ Detailed and reliable steady state models of SVC, TCSC and STATCOM models, and their controls with the corresponding limits are included.
➠ Secondary voltage control, as defined by ENEL (elecetricity company of Italy), can be modeled and simulated in the program.
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525
Power System Dynamics and Stability
UWPFLOW Features
➠ The program is able to compute the minimum real eigenvalue and the related right and left eigenvectors and several voltage stability indices.
➠ The program generates a wide variety of output ASCII and MATLAB (.m) files as well as IEEE common format data files.
➠ The program has being designed to automatically run script files.
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526
Power System Dynamics and Stability
UWPFLOW Usage
➠ Like any other UNIX program, i.e., command-line options (-option) with redirection of output (>) from screen into files: uwpflow [-options] input_file [[>]output_file]
➠ For example, to generate the program help: uwpflow -h
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527
Power System Dynamics and Stability
UWPFLOW Example
➠ 3-area sample system: 100 MW 150 MW 150 MW 60 MVAr
v3
150 MW 56 MVAr
Bus 2
Area 1
1.02∠0
50 MVAr R = 0.01 p.u. X = 0.15 p.u.
V2 ∠δ2
Bus 3 50 MVAr
50 MW 40 MVAr 100 MW
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UWPFLOW Example
➠ For a 3-area sample system: Bus
∆PG
∆PL
∆QL
Name
(p.u.)
(p.u.)
(p.u.)
Area 1
1.5
0
0
Area 2
0
1.5
0.56
Area 3
0.5
0.5
0.40
➠ Using UWPFLOW to obtain the system PV curves, for a distributed slack bus model:
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Power System Dynamics and Stability
UWPFLOW Example
➠ Data file in EPRI format (3area.wsc): HDG UWPFLOW data file, WSCC format 3-area example April 2000 BAS C C AC BUSES C C | SHUNT | C |Ow|Name |kV |Z|PL |QL |MW |Mva|PM |PG |QM |Qm |Vpu BE 1 Area 1 138 1 150 60 0 0 0 150 0 0 1.02 B 1 Area 2 138 2 150 56 0 50 0 100 0 0 1.00 B 1 Area 3 138 3 50 40 0 50 0 100 0 0 1.00
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530
UWPFLOW Example
➠ Data file in EPRI format (3area.wsc): C C AC LINES C C M CS N C |Ow|Name_1 |kV1||Name_2 |kV2|||In || R | X | G/2 | B/2 |Mil| L 1 Area 1 138 Area 2 1381 15001 .01 .15 L 1 Area 1 138 Area 3 1381 15001 .01 .15 L 1 Area 2 138 Area 3 1381 15001 .01 .15 C C SOLUTION CONTROL CARD C C |Max| |SLACK BUS | C |Itr| |Name |kV| |Angle | SOL 50 Area 1 138 0. END
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Power System Dynamics and Stability
UWPFLOW Example
➠ Generator and load change file (3area.k): C C UWPFLOW load and generation "direction" file C for 3-area example C C BusNumber BusName DPg Pnl Qnl PgMax [ Smax 1 0 1.5 0.0 0.0 0 0 2 0 0.0 1.5 0.56 0 0 3 0 0.5 0.5 0.40 0 0
Vmax Vmin ] 1.05 0.95 1.05 0.95 1.05 0.95
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532
Power System Dynamics and Stability
UWPFLOW Example
➠ Batch file for UNIX (run3area): echo -1- Run base case power flow uwpflow 3area.wsc -K3area.k echo -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s echo - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 echo - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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Power System Dynamics and Stability
UWPFLOW Example
➠ Batch file for Windows (run3area.bat): rem -1- Run base case power flow uwpflow 3area.wsc -K3area.k rem -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s rem - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 rem - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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534
Power System Dynamics and Stability
UWPFLOW Example
➠ PV curves (threearea.m): Profiles 150
100
50
0
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
kVArea 3
138
kVArea 2
138
kVArea 1
138
1.4
1.6
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Power System Dynamics and Stability
UWPFLOW Example
➠ The singular value index obtained with UWPFLOW is as follows (-0 option): 0.7
Full matrix sing. value index
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
1.4
1.6
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536
Power System Dynamics and Stability
Matlab Overview
➠ Matlab is a general purpose environment for mathematical and engineering analysis.
➠ Is vector/matrix based. A variable is by default a matrix. ➠ Is an “interpreted” language, thus can be slow for heavy applications. ➠ Is not open source. The GNU-Octave project provides a good alternative: http://www.octave.org
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Power System Dynamics and Stability
Matlab Example
➠ Generator-load example (see the introductory example in the Voltage Stability section starting from slide 279): : PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
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538
Power System Dynamics and Stability
Matlab Example
➠ For QGmin ≤ QG ≤ QGmax : V10 V2 1 sin δ − DG ω Pd − ω˙ = M XL 1 V10 V2 δ˙ = ω − sin δ − Pd DL XL 2 1 V V V 10 2 V˙ 2 = cos δ − Qd − 2 + τ XL XL 2 V10 V10 V2 0 = QG − − + cos δ XL XL
with
x = [ω, δ, V2 ]T
y = QG
p = V10
λ = Pd
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Power System Dynamics and Stability
Matlab Example
➠ For QG = QGmin,max : ω˙ δ˙ V˙ 2 0
1 M
V1 V2 = sin δ − DG ω Pd − XL V1 V2 1 sin δ − Pd = ω− DL XL 2 1 V1 V2 V = cos δ − Qd − 2 + τ XL XL = QGmin,max
V1 V2 V12 + cos δ − XL XL
with
x = [ω, δ, V2 ]T
y = V1
p = QGmin,max
λ = Pd
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Power System Dynamics and Stability
Matlab Example
➠ Assume XL = 0.5, M = 1, DG = 0.01, DL = 0.1, τ = 0.01, k = 0.25. ➠ The time domain integration can be solved with the help of M ATLAB.
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Power System Dynamics and Stability
541
Matlab Example
➠ Differential equations without limits: function dx = example(t,x) global M DL DG tau k Pd V10 if t <= 1 XL = 0.5; else XL = 0.6; end delta = x(1); omega = x(2); V2 = max(x(3),0); dx(1,1) = omega - (V10*V2*sin(delta)/XL-Pd)/DL; dx(2,1) = (Pd-V10*V2*sin(delta)/XL-DG*omega)/M; dx(3,1) = (-V2*V2/XL+V10*V2*cos(delta)/XL-k*Pd)/tau; Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 30
Power System Dynamics and Stability
542
Matlab Example
➠ Initialization and main routine: clear all global M DL DG tau k Pd V10 XL = 0.5; M = 0.1; DL = 0.1; DG = 0.01; tau = 0.01; k = 0.25; Pd = 0.7; V10 = 1; x0 = [0.4636; 0.0000; 0.7826]; tmax = 2; [t,x] = ode23(’example’,[0 tmax],x0);
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543
Matlab Example
➠ Graphical commands: figure plot(t,x(:,1),’b-’) hold on plot(t,x(:,2),’g--’) plot(t,max(x(:,3),0),’c-.’) plot([0 tmax],[V10 V10],’r:’) legend(’delta’,’omega’,’V2’,’V1’) xlabel(’t [s]’) ylim([-1 6])
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Power System Dynamics and Stability
Matlab Example
➠ The dynamic solution without limits: 6
δ ω V2 V1
5
4
3 Contingency
Operating point
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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Power System Dynamics and Stability
545
Matlab Example
➠ Differential equations without limits: function dx = example(t,x) global M DL DG tau k Pd V10 if t <= 1, XL = 0.5; else, XL = 0.6; end delta = x(1); omega = x(2); V2 = max(x(3),0); V10 = 1; Q = V10*V10/XL - V10*V2*cos(delta)/XL; if Q > 0.5 a = 1/XL; b = -V2*cos(delta)/XL; c = -0.5; V10 = (-b + sqrt(b*b - 4*a*c))/2/a; end dx(1,1) = omega - (V10*V2*sin(delta)/XL-Pd)/DL; dx(2,1) = (Pd-V10*V2*sin(delta)/XL-DG*omega)/M; dx(3,1) = (-V2*V2/XL+V10*V2*cos(delta)/XL-k*Pd)/tau; Universidad de Castilla-La Mancha June 26, 2008
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546
Power System Dynamics and Stability
Matlab Example
➠ The dynamic solution with limits: 6
δ ω V2 V1
5
4
3 Operating point
Contingency
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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547
PSAT Features
➠ PSAT has been thought to be portable and open source. ➠ PSAT runs on the commonest operating systems ➠ PSAT can perform several power system analysis: 1. Continuation Power Flow (CPF); 2. Optimal Power Flow (OPF); 3. Small signal stability analysis; 4. Time domain simulations.
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548
Power System Dynamics and Stability
PSAT Features
➠ PSAT deeply exploits Matlab vectorized computations and sparse matrix functions in order to optimize performances.
➠ PSAT also contains interfaces to UWPFLOW and GAMS which highly extend PSAT ability to solve CPF and OPF problems, respectively.
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549
Synoptic Scheme Simulink Models
Input
Other Data Format
Saved Results
Data Files Simulink Library
Simulink Model Conversion
Conversion Utilities
Power Flow & State Variable Initialization
User Defined Models
Settings
Interfaces GAMS
UWpflow
PSAT
Static Analysis
Dynamic Analysis
Optimal Power Flow
Small Signal Stability
Continuation Power Flow
Time Domain Simulation
Command History
Output
Plotting Utilities
Text Output
Save Results
Graphic Output
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550
Power System Dynamics and Stability
PSAT Features
➠ In order to perform accurate and complete power system analyses, PSAT supports a variety of static and dynamic models.
➠ Dynamic models include non conventional loads, synchronous machines and controls, regulating transformers, FACTS, wind turbines, and fuel cells.
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Power System Dynamics and Stability
551
PSAT Features
➠ Besides mathematical algorithms and models, PSAT includes a variety of additional tools, as follows: 1. User-friendly graphical user interfaces; 2. Simulink library for one-line network diagrams; 3. Data file conversion to and from other formats; 4. User defined model editor and installer; 5. Command line usage.
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Power System Dynamics and Stability
PSAT Features
➠ Not all features are available on GNU-Octave: Function
Matlab
GNU/Octave
Continuation power flow
yes
yes
Optimal power flow
yes
yes
Small signal stability analysis
yes
yes
Time domain simulation
yes
yes
GUIs and Simulink library
yes
no
Data format conversion
yes
yes
User defined models
yes
no
Command line usage
yes
yes
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Power System Dynamics and Stability
Getting Started
➠ PSAT is launched by typing at the Matlab prompt: >> psat which will create all structures required by the toolbox and open the main GUI.
➠ All procedures implemented in PSAT can be launched from this window by means of menus, buttons and/or short cuts.
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Power System Dynamics and Stability
554
Getting Started
➠ Main PSAT GUI:
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555
Power System Dynamics and Stability
Simulink Library
➠ PSAT allows drawing electrical schemes by means of pictorial blocks. ➠ The PSAT computational engine is purely Matlab-based and the Simulink environment is used only as graphical tool.
➠ A byproduct of this approach is that PSAT can run on GNU/Octave, which is currently not providing a Simulink clone.
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556
Simulink Library
➠ PSAT-Simulink Library:
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Power System Dynamics and Stability
Other Features
➠ To ensure portability and promote contributions, PSAT is provided with a variety of tools, such as a set of Data Format Conversion (DFC) functions and the capability of defining User Defined Models (UDMs).
➠ The set of DFC functions allows converting data files to and from formats commonly in use in power system analysis. These include: IEEE, EPRI, PTI, PSAP, PSS/E, CYME, MatPower and PST formats. On Matlab platforms, an easy-to-use GUI handles the DFC.
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558
Data Format Conversion
➠ GUI for data format conversion:
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559
Power System Dynamics and Stability
User Defined Models
➠ The UDM tools allow extending the capabilities of PSAT and help end-users to quickly set up their own models.
➠ Once the user has introduced the variables and defined the DAE of the new model in the UDM GUI, PSAT automatically compiles equations, computes symbolic expression of Jacobians matrices and writes a Matlab function of the new component.
➠ Then the user can save the model definition and/or install the model in PSAT.
➠ If the component is not needed any longer it can be uninstalled using the UDM installer as well.
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560
User Defined Models
➠ GUI for user defined models:
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561
Power System Dynamics and Stability
Command Line Usage
➠ PSAT is provided with a command line version. This feature allows using PSAT in the following conditions: 1) If it is not possible or very slow to visualize the graphical environment (e.g. Matlab is running on a remote server). 2) If one wants to write scripting of computations or include calls to PSAT functions within user defined programs. 3) If PSAT runs on the GNU/Octave platform, which currently neither provides GUI tools nor a Simulink-like environment.
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562
Power System Dynamics and Stability
Power System Model
➠ The standard power system model is basically a set of nonlinear differential algebraic equations, as follows:
x˙ = f (x, y, p) 0
= g(x, y, p)
∈ Rn ; y are the algebraic variables y ∈ Rm ; p are the independent variables p ∈ Rℓ ; f are the differential equations f : Rn × Rm × Rℓ 7→ Rn ; and g are the algebraic equations g : Rm × Rm × Rℓ 7→ Rm . where x are the state variables x
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Power System Dynamics and Stability
Power System Model
➠ PSAT uses these equations in all algorithms, namely power flow, CPF, OPF, small signal stability analysis and time domain simulation.
➠ The algebraic equations g are obtained as the sum of all active and reactive power injections at buses:
X gpc gp gpm − ∀m ∈ M g(x, y, p) = = gq gqm c∈Cm gqc
where gpm and gqm are the power flows in transmission lines, M is the T T T ] are the set and the power , gqc set of network buses, Cm and [gpc
injections of components connected at bus m, respectively.
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Power System Dynamics and Stability
Component Models
➠ PSAT is component-oriented, i.e. any component is defined independently of the rest of the program as a set of nonlinear differential-algebraic equations, as follows:
x˙ c
= fc (xc , yc , pc )
Pc
= gpc (xc , yc , pc )
Qc
= gqc (xc , yc , pc )
where xc are the component state variables, yc the algebraic variables
V and θ at the buses to which the component is connected) and pc are independent variables. Then differential equations f are built concatenating fc of all components.
(i.e.
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Power System Dynamics and Stability
Component Models
➠ These equations along with Jacobians matrices are defined in a function which is used for both static and dynamic analyses.
➠ In addition to this function, a component is defined by means of a structure, which contains data, parameters and the interconnection to the grid.
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Software Tools - 54
566
Power System Dynamics and Stability
Component Models: Example
➠ Let’s consider the exponential recovery load (ERL). ➠ The set of differential-algebraic equations are as follows: x˙ c1 x˙ c2
= −xc1 /TP + P0 (V /V0 )αs − P0 (V /V0 )αt
= −xc2 /TQ + Q0 (V /V0 )βs − Q0 (V /V0 )βt
Pc
= xc1 /TP + P0 (V /V0 )αt
Qc
= xc2 /TQ + Q0 (V /V0 )βt
where and P0 , Q0 and V0 are initial powers and voltages, respectively, as given by the power flow solution.
➠ Observe that a constant PQ load must be connected at the same bus as the ERL to determine the values of P0 , Q0 and V0 . Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 55
567
Power System Dynamics and Stability
Component Models: Example
➠ Component Data: Column
Variable
Description
Unit
1
-
Bus number
int
2
Sn
Power rating
MVA
3
Vn
Active power voltage coefficient
kV
4
fn
Active power frequency coefficient
Hz
5
TP
Real power time constant
s
6
TQ
Reactive power time constant
s
7
αs
Static real power exponent
-
8
αt
Dynamic real power exponent
-
9
βs
Static reactive power exponent
-
10
βt
Dynamic reactive power exponent
-
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 56
568
Power System Dynamics and Stability
Component Models: Example
➠ Exponential recovery loads are defined in the structure Erload, whose fields are as follows: 1.
con: exponential recovery load data.
2.
bus: Indexes of buses to which the ERLs are connected.
3.
dat: Initial powers and voltages (P0 , Q0 and V0 ).
4.
n: Total number of ERLs.
5.
xp: Indexes of the state variable xc1 .
6.
xq: Indexes of the state variable xc2 .
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 57
569
Power System Dynamics and Stability
PSAT Example
➠ This section illustrates some PSAT features for static and dynamic stability analysis by means of the IEEE 14-bus test system.
➠ All data can be retrieved from the PSAT web site: http://www.power.uwaterloo.ca/∼fmilano/
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 58
Power System Dynamics and Stability
570
PSAT Example
➠ IEEE 14-bus test system:
Bus 13 |V| = 1.047 p.u.
Bus 14 |V| = 1.0207 p.u.
Bus 10 |V| = 1.0318 p.u.
Bus 12 |V| = 1.0534 p.u.
Bus 09
Bus 11
|V| = 1.0328 p.u.
|V| = 1.0471 p.u.
Bus 07 |V| = 1.0493 p.u.
Bus 06 |V| = 1.07 p.u.
Bus 04
Bus 08
|V| = 1.09 p.u. |V| = 1.012 p.u.
Bus 05 Bus 01
|V| = 1.016 p.u.
|V| = 1.06 p.u.
Breaker
Bus 02 |V| = 1.045 p.u.
Breaker Bus 03 |V| = 1.01 p.u.
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 59
Power System Dynamics and Stability
571
PSAT Example
➠ Power flow report:
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 60
Power System Dynamics and Stability
572
PSAT Example
➠ Continuation power flow analysis (GUI):
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 61
Power System Dynamics and Stability
573
PSAT Example
➠ Continuation power flow analysis (plots):
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 62
574
Power System Dynamics and Stability
PSAT Example
➠ Nose curves at bus 14 for different contingencies for the IEEE 14-bus test system:
Base Case
1
Line 2-4 Outage Line 2-3 Outage
Voltage [p.u.]
0.8 0.6 0.4 0.2 0
1
1.1
1.2
1.3
1.4 λc
1.5
1.6
1.7
1.8
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 63
Power System Dynamics and Stability
575
PSAT Example
➠ Optimal power flow analysis (GUI):
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 64
576
Power System Dynamics and Stability
PSAT Example
➠ Comparison between OPF and CPF analysis: BCP
λ∗
M LC
ALC
[MW]
[p.u.]
[MW]
[MW]
None
259
0.7211
445.8
186.8
Line 2-4 Outage
259
0.5427
399.5
148.6
Line 2-3 Outage
259
0.2852
332.8
73.85
Contingency
➠ Because of the definitions of generator and load powers PG and PL , one has λc = λ∗ + 1.
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 65
577
Power System Dynamics and Stability
PSAT Example
➠ Time domain simulation: ➛ It has been used a 40% load increase with respect to the base case loading, and no PSS at bus 1. A Hopf bifurcation occurs for the line 2-4 outage resulting in undamped oscillations of generator angles.
➛ A similar analysis can be carried on the same system with a 40% load increase but considering the PSS of the generator connected at bus 1. In this case the system is stable.
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 66
578
Power System Dynamics and Stability
PSAT Example
➠ Time domain simulation (without PSS):
Generator Speeds [p.u.]
1.002 1.0015
ω1 - Bus 1 ω2 - Bus 2
1.001
ω3 - Bus 3 ω4 - Bus 6
1.0005
ω5 - Bus 8
1 0.9995 0.999 0.9985 0.998
0
5
10
15
20
25
30
Time [s]
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 67
579
Power System Dynamics and Stability
PSAT Example
➠ Eigenvalue analysis (with PSS): 10 8 6 4
Imag
2 0 −2 −4 −6 −8 −10
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Real
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 68
Power System Dynamics and Stability
580
Project 1
➠ Reproduce the examples illustrated in the following slides: ➛ Voltage Stability: Slides 341-350 (using UWPFLOW) ➛ Angle Stability: Slides 444-456 (using Matlab) ➛ Frequency Stability: 479-489 (using Matlab) ➠ The software UWPFLOW is freely available at: http://thunderbox.uwaterloo.ca/∼claudio/software/pflow.html
Universidad de Castilla-La Mancha June 26, 2008
Projects - 1
581
Power System Dynamics and Stability
Project 1
➠ Voltage stability (slides 341-350): ➛ Write the 3area.wsc data file in the EPRI data format (the format is fully described in the UWPFLOW documetation).
➛ Write the generator load change file 3area.k using the format described in the UWPFLOW documentation.
➛ Write a batch file for running the simulations as described in the slides.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 2
582
Power System Dynamics and Stability
Project 1
➠ Voltage stability (slides 341-350): ➛ Run a base case power flow, a continuation power flow with and without enforcing voltage and current limits. Compute also the singular value index.
➛ Report power flow results as given by UWPFLOW and plots PV curves and the singular value index using Matlab.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 3
583
Power System Dynamics and Stability
Project 1
➠ Angle stability (slides 444-456): ➛ Write a Matlab function with the system differential equations. ➛ Use a Matlab script file to initialize and solve the time domain simulation (function ode23). ➛ Using a trial-and-error technique find the clearing time tc of the system for D = 0.1, D = 0.05 and D = 0.2. ➛ For each value of the damping D, provide plots of the rotor angle δ and the rotor speed ω .
Universidad de Castilla-La Mancha June 26, 2008
Projects - 4
Power System Dynamics and Stability
584
Project 1
➠ Frequency stability (slides 479-489): ➛ Write a Matlab function with the system differential equations. ➛ Use a Matlab script file to initialize and solve the time domain simulation (function ode23). ➛ Run the time domain integration for a 25%, 50% and 60% generation drop at t = 1 followed by a 25%, 50% and 60% load drop at t = 5, respectively.
➛ For each value of the generation and load drop, provide plots of ω , δ , E ′ , V2 and V1 .
Universidad de Castilla-La Mancha June 26, 2008
Projects - 5
Power System Dynamics and Stability
585
Project 2
➠ Reproduce the results for the IEEE 14-bus tests system illustrated in the paper: F. Milano, “An Open Source Power System Analysis Toolbox”, accepted for publication on the IEEE Transactions on Power Systems, March 2005, 8 pages.
➠ The full paper as well as the software PSAT is available at: http://www.power.uwaterloo.ca/∼fmilano/
Universidad de Castilla-La Mancha June 26, 2008
Projects - 6
586
Power System Dynamics and Stability
Project 2
➠ The IEEE 14-bus test system is provided wintin the PSAT main distribution (folder tests). ➠ For the base case power flow, the continuation power flow and the optimal power flow routines, use the file:
d 014 dyn l10.mdl
➠ For the time domain simulations without PSS, use the file: d 014 dyn l14.mdl
➠ For the time domain simulations with PSS, use the file: d 014 pss l14.mdl
Universidad de Castilla-La Mancha June 26, 2008
Projects - 7
587
Power System Dynamics and Stability
Project 2
➠ Hints: ➛ For static analyses (PF, CPF, OPF), disable loading dynamic components by checking the box “Discard dynamic components” in the GUI Settings (within the menu “Edit” in the main window).
➛ To simulate a line outage in static analyses, uncheck the box “initially close” box (within the Simulink block mask) of the breaker of the line that one wants to keep out.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 8
Power System Dynamics and Stability
588
Project 2
➠ Hints: ➛ For OPF analysis, disable the “base case” powers in the GUI for OPF Settings and set the weighting factor to 1 (maximization of the ditance to voltage collapse).
➛ For time domain simulations, remember to uncheck “Discard dynamic components” box and do not allow the conversion to PQ buses when the program asks for.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 9
Power System Dynamics and Stability
Power System Dynamics and Stability Dr. Federico Milano E-mail: [email protected] Tel.: +34 926 295 219
´ ´ ´ Departamento de Ingenier´ıa Electrica, Electronica, Automatica y Comunicaciones
´ Escuela Tecnica Superior de Ingenieros Industriales
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 1
2
Power System Dynamics and Stability
Programa de Doctorado
Technical and Economical Management of Generation, Transmission and Distribution Electric Energy Systems ´ ´ Area de Ingenier´ıa Electrica de la E.T.S. de Ingenieros Industriales de la Universidad de Castilla - La Mancha
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 2
3
Power System Dynamics and Stability
Note
➠ This course is partly based on the course ECE664 hold by Prof. Dr. C. ˜ Canizares at the University of Waterloo, Ontario, Canada. ˜ ➠ I wish to sincerely thank Prof. Dr. C. Canizares for his courtesy in sharing this material.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 3
4
Power System Dynamics and Stability
Objectives
➠ Understand the modeling and simulation of power systems from phasor analysis to electromagnetic transients.
➠ Discuss the basic definitions, concepts and tools for stability studies of power systems.
➠ Familiarize with basic concepts of computer modelling of electrical power systems.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 4
Power System Dynamics and Stability
5
Outlines: System Modeling
➠ Synchronous machine. ➠ Transformer. ➠ Transmission line. ➠ Cable. ➠ Loads.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 5
Power System Dynamics and Stability
6
Outlines: System Analysis
➠ Basic stability concepts: ➛ Nonlinear systems. ➛ Equilibrium points. ➛ Stability regions. ➠ Power Flow: ➛ System model. ➛ Equations and solution techniques. ➛ Contingency analysis.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 6
Power System Dynamics and Stability
7
Outlines: Voltage Stability
➠ Definitions. ➠ Basic concepts: ➛ Saddle-node bifurcation. ➛ Limit-induced bifurcation. ➠ Continuation Power Flow (CPF). ➠ Direct methods. ➠ Indices. ➠ Protections and controls. ➠ Real case example: August 2003 North American blackout. Universidad de Castilla-La Mancha June 26, 2008
Introduction - 7
Power System Dynamics and Stability
8
Outlines: Angle Stability
➠ Definitions. ➠ Small-disturbance: ➛ Hopf Bifurcations. ➛ Control and mitigation. ➛ Practical applications. ➠ Transient Stability (large-disturbance): ➛ Time domain. ➛ Direct Methods: ➳ Equal Area Criterion. ➳ Energy Functions. ➠ Real case example: May 1997 Chilean blackout. Universidad de Castilla-La Mancha June 26, 2008
Introduction - 8
Power System Dynamics and Stability
9
Outlines: Frequency Stability
➠ Definitions. ➠ Basic concepts. ➠ Protections and controls. ➠ Real case example: October 2003 Italian blackout. ➠ Real case example: November 2006 European blackout.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 9
Power System Dynamics and Stability
10
Outlines: Software Tools
➠ Outlines. ➠ UWPFLOW. ➠ Matlab. ➠ PSAT.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 10
11
Power System Dynamics and Stability
References
➠ P. Kundur, Power system stability and control, Mc Graw Hill, 1994. ➠ P. Sauer and M. Pai, Power system dynamics and stability, Prentice Hall, 1998.
➠ A. R. Bergen and V. Vittal, Power systems analysis, Second Edition, Prentice-Hall, 2000.
➠ C. A. Caizares, Editor, Voltage stability assessment: concepts, practices and tools, IEEE-PES Power System Stability Subcommittee Special Publication, SP101PSS, May 2003.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 11
12
Power System Dynamics and Stability
References
➠ P. M. Anderson and A. A. Fouad, Power system control and stability, IEEE Press, 1994.
➠ J. Arrillaga and C. P. Arnold, Computer analysis of power systems, John Wiley, 1990.
➠ I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices, Oxford Science Publications, 1986.
➠ J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Second Edition, Springer-Verlag, 1993.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 12
13
Power System Dynamics and Stability
References
➠ M. Ili´c and J. Zaborszky, Dynamics and Control of Large Electric Power Systems, Wiley, New York, 2000. ˜ ➠ C. A. Canizares, UWPFLOW, available at www.power.uwaterloo.ca
➠ F. Milano, PSAT, Power System Analysis Toolbox, available at www.power.uwaterloo.ca
➠ Journal papers and technical reports. ➠ Course notes available on line.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 13
14
Power System Dynamics and Stability
Evaluation
➠ Two projects are required. ➠ The projects concentrates in the various topics discussed in class. ➠ These will require the use of MATLAB, PSAT and UWPFLOW (the last two are free software for stability studies co-developed at the University of Waterloo, Canada).
➛ Reproducing examples presented in the slides using UWPFLOW and MATLAB.
➛ Stability analysis of the IEEE 14-bus test system using PSAT.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 14
15
Power System Dynamics and Stability
Evaluation
➠ Alternatively, the students can develop a “user defined model” in PSAT. ➠ Interested students are invited to contact Dr. Federico Milano.
Universidad de Castilla-La Mancha June 26, 2008
Introduction - 15
Power System Dynamics and Stability
16
Contents
➠ Introduction ➠ Generator Modeling ➠ Transmission System Modeling ➠ Load Modeling ➠ Power Flow Outlines ➠ Stability Concepts ➠ Voltage Stability ➠ Angle Stability ➠ Frequency Stability ➠ Software Tools ➠ Projects Universidad de Castilla-La Mancha June 26, 2008
Introduction - 16
Power System Dynamics and Stability
17
Generator Modeling
➠ Generator overview. ➠ Synchronous machine. ➠ Dynamic models of generators for stability analysis: ☞ Subtransient model ☞ Transient model ☞ Basic control models ➠ Steady-state model.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 1
18
Power System Dynamics and Stability
Generator Overview
➠ Generator components: Fuel
Steam at pressure, P
Torque
Power at voltage, V
Enthalpy, h
at speed, ω
Current, I
Turbine
Boiler
Generator
Firing
+
−
control
+
−
Ref ω
Power set-point
Excitation
Governor
+
−
system
Ref V
➠ Generator: ☞ Synchronous machine: AC stator and DC rotor. ☞ Excitation system: DC generator or static converter plus voltage regulator and stabilizer
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 2
19
Power System Dynamics and Stability
Generator Overview
➠ Generator with DC exciter and associated controls: Commutator
Field breaker
CT
dc Exc
Amplidyne field
ac Gen slip rings
Regulator transfer PT’s
M
Exciter field rheostat (manual control)
Magnetic amplifier
Stabilizer
Transistor amplifier
Reference and voltage sensing
Magnetic amplifier
Limiter sensing
Compensator
Other inputs
Other sensing 90−52
MG set PMG
M
Regulator power
Station auxiliary power
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 3
20
Power System Dynamics and Stability
Generator Overview
➠ Generator with static exciter and associated controls: Auxiliary power input start up
Voltage buildup element
Syn machine
FDR
Excitation power current transformer
CT
Excitation power potential transformer
slip rings
Power rectifier Excitation power Excitation breaker
PT’s Linear reactor Trinistat power amplifier
Gate circuitry
Base adjuster manual control Regulator transfer
Stabilizer
Reference & voltage sensing
Signal mixing amplifier
Limiter sensing
Rectifier current limit
Other sensing
Compensator
Voltage adjuster
Other inputs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 4
21
Power System Dynamics and Stability
Synchronous Machine a
ωr dr
Damper windings
c′
b′
Effects of induced currents in the rotor core
Q2
D
Q′2
b
Q1
F
Q′1
D′
θr ar
F′
DC field
a′
c qr
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 5
Power System Dynamics and Stability
22
Synchronous Machine
➠ Electrical (inductor) equations: v
dλ = ri + dt
dλ λ = L(θr )i + dt 1 T dL(θr ) Te = i i 2 dθr ➠ Mechanical (Newton’s) equations: J
dωr + Dωr dt dθr dt
= Tm − Te = ωr
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 6
23
Power System Dynamics and Stability
Synchronous Machine
➠ Stator equations: v0s rs vds = − vqs
rs
➠ Rotor equations: vF rF 0 = 0 0
i0s
0
λ0s
ids −ωr λqs − d λds dt λqs rs iqs −λds
rD rQ1 rQ2
iF
i D iQ1 iQ2
λF
d λ D + dt λQ1 λQ2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 7
24
Power System Dynamics and Stability
Synchronous Machine
➠ Magnetic flux equations:
λ0s λds λqs λF λD λQ1 λQ2
=
L0 Ld
Md
Md
Lq
Mq
Md
LF
Md
Md
Md
LD
Mq
Mq
LQ1
Mq
Mq
Mq
LQ2
i0s ids iqs iF iD iQ1 iQ2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 8
25
Power System Dynamics and Stability
Synchronous Machine
➠ Transformation equations: v0s vas vbs = P T vds vqs vcs P =
r
√
1/ 2
2 cos θr 3 sin θr
√
ias
i0s
ibs = P T ids iqs ics √
1/ 2 cos(θr − sin(θr −
1/ 2 2π 3 ) 2π 3 )
cos(θr + sin(θr +
2π 3 ) 2π 3 )
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 9
26
Power System Dynamics and Stability
Synchronous Machine
➠ Mechanical equations: 2 d 2 J ωr + Dωr p dt p d θr dt Te
= Tm − Te = ωr =
p (iqs λds − ids λqs ) 2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 10
27
Power System Dynamics and Stability
Synchronous Machine
➠ 3-phase short circuit at generator terminals: ia (t)
√
2|Ea (0)| x′′ d √
Subtransient component
2|Ea (0)| x′ d
Transient component
Steady state
t √
t
2|Ea (0)| xd
Ea (0) is the open-circuit RMS phase voltage.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 11
28
Power System Dynamics and Stability
Synchronous Machine
➠ Assuming balanced operation (null zero sequence), the detailed machine model can be reduced to phasor models useful for stability and steady-state analysis.
➠ Phasor models are based on the following assumptions: ➛ The rotor does not deviate “much” from the synchronous speed, i.e. ωr ≈ ωs = (2/p)2πf0 . ➛ The rate of change in rotor speed is ”small”, i.e. |dωr /dt| ≈ 0
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 12
29
Power System Dynamics and Stability
Synchronous Machine
➠ Subtransient models capture the full machine electrical dynamics, including the “few” initial cycles (ms) associated with the damper windings.
➠ Transient models capture the machine electrical dynamics starting with the field and induced rotor core current transient response.
➠ Damper windings transients are neglected. Steady-state models capture the machine electrical response when all transients have disappeared after “a few” seconds.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 13
Power System Dynamics and Stability
30
Subtransient Model
➠ External phase voltages and currents: 2 vas
θvas i2as θias
2 2 = vqs + vds vds +δ = tan−1 vqs
= i2qs + i2ds ids −1 = tan +δ iqs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 14
31
Power System Dynamics and Stability
Subtransient Model
➠ Subtransient “internal” voltages associated with the damper windings (D and Q1 ): d ′′ eq dt d ′′ e dt d e′′q − vqs
e′′d − vds
= =
1 ′ ′′ ′′ ′ )i − e − x + (x [e ds q] d d q ′′ Td0 1 ′ ′ ′′ ′′ [e − (x − x )i − e qs d q q d] ′′ Tq0
= rs iqs − x′′d ids
= rs ids + x′′q iqs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 15
32
Power System Dynamics and Stability
Subtransient Model
➠ Transient “internal” voltages associated with the field (F ) and rotor-core induced current windings (Q2 ): d ′ eq dt d ′ e dt d e′q − vqs
e′d − vds
= =
1 ′ ′ )i − e [e + (x − x ds f d q] d ′ Td0 1 ′ ′′ ′ [−(x − x )i − e qs q q d] ′ Tq0
= rs iqs − x′d ids = rs ids + x′q iqs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 16
33
Power System Dynamics and Stability
Subtransient Model
➠ Steady-state equations: ea − vqs
= rs iqs − xd ids
−vds
= rs ids + xq iqs
➠ Mechanical equations: d ∆ωr dt d δ dt
=
1 [Pm − vas ias cos(θvas − θias ) − D∆ωr ] M
= ∆ωr = ωr − ωs
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 17
34
Power System Dynamics and Stability
Subtransient Model
➠ The subtransient reactances (x′′q , x′′d ) and open circuit time constants ′′ ′′ (Tq0 , Td0 ), as well as the transient reactances (x′q , x′d ) and open circuit ′ ′ time constants (Tq0 , Td0 ) are directly associated with the machine resistances and inductances:
xd
= ω0 Ld = xℓ + xM d
xq
= ω0 Lq = xℓ + xM q xM d xLF = xℓ + xLF + xM d xM q xLQ2 = xℓ + xLQ2 + xM q xM d xLF xLD = xℓ + xM d xLF + xM d xLD + xLF xLD xM q xLQ1 xLQ2 = xℓ + xM q xLQ1 + xM q xLQ2 + xLQ1 xLQ2
x′d x′q x′′d x′′q
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 18
Power System Dynamics and Stability
35
Subtransient Model
➠ Some definitions: xF
= xLF + xM d
xD
= xLD + xM d
xQ1
= xLQ1 + xM q
xQ2
= xLQ2 + xM q
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 19
36
Power System Dynamics and Stability
Subtransient Model
➠ Time constants: ′ Td0
=
′ Tq0
=
′′ Td0
=
′′ Tq0
=
xF ω 0 rF xQ2 ω0 rQ2 xM d xLD 1 xLD + ω 0 rD xD 1 xM q xLQ1 xLQ1 + ω0 rQ1 xQ1
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 20
37
Power System Dynamics and Stability
Subtransient Model
➠ Typical machine parameters: 2-pole
4-pole
Conventional
Conductor
Conventional
Conductor
Cooled
Cooled
Cooled
Cooled
xd x′d x′′ d
1.7-1.82
1.7-2.17
1.21-1.55
1.6-2.13
.18-.23
.264-.387
.25-.27
.35-.467
.11-.16
.23-.323
.184-.197
.269-.32
xq x′q x′′ q ′ Td0 ′′ Td0 ′ Tq0 ′′ Tq0
1.63-1.69
1.71-2.16
1.17-1.52
1.56-2.07
.245-1.12
.245-1.12
.47-1.27
.47-1.27
.116-.332
.116-.332
.12-.308
.12-.308
7.1-9.6
4.8-5.36
5.4-8.43
4.81-7.73
.032-.059
.032-.059
.031-.055
.031-.055
.3-1.5
.3-1.5
.38-1.5
.36-1.5
.042-.218
.042-.218
.055-.152
.055-.152
xℓ rs
.118-.21
.27-.42
.16-.27
.29-41
.00081-.00119
.00145-.00229
.00146-00147
.00167-00235
M
5-7
5-7
6-8
6-8
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 21
Power System Dynamics and Stability
38
Subtransient Model
➠ Typical machine parameters: Salient-pole
Combustion
Synchronous
Dampers
No dampers
Turbines
Compensator
xd x′d x′′ d
.6-1.5
.6-1.5
1.64-1.85
1.08-2.48
.25-.5
.25-.5
.159-.225
.244-.385
.13-.32
.2-.5
.102-.155
.141-.257
xq x′q x′′ q ′ Td0 ′′ Td0 ′ Tq0 ′′ Tq0
.4-.8
.4-.8
1.58-1.74
.72-1.18
= x′q
= x′q
.306
.57-1.18
.135-.402
.135-.402
.1
.17-.261
4-10
8-10
4.61-7.5
6-16
.029-.051
.029-.051
.054
.039-.058
−
−
1.5
.15
.033-.08
.033-.08
.107
.188-.235
xℓ rs
.17-.4
.17-.4
.113
.0987-.146
.003-.015
.003-.015
.034
.0017-006
M
6-14
6-14
18-24
2-4
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 22
39
Power System Dynamics and Stability
Subtransient Model
➠ In practice most of these constants are determined from short-circuit tests.
➠ All the e voltages are “internal” machine voltages directly assocated with the “internal” phase angle δ . ➠ The internal field voltage ef is directly proportional to the actual field dc voltage vF , and is typically controlled by the voltage regulator. ➠ The mechanical power Pm is controlled through the governor.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 23
40
Power System Dynamics and Stability
Subtransient Model
➠ A simple voltage regulator model (IEEE type 1): Se vr max vref +
Ka
+ − vm
−
Ta s + 1
−
vr +
1
vf
Te s + 1
vr min
1 Tr s + 1
Kf s Tf s + 1
V
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 24
41
Power System Dynamics and Stability
Subtransient Model
➠ A static voltage regulator model:
VT 1 1+sTR IT
VC
+ KIR
−
−
If +
0
Vuxl
+
Vmax
+
1+sTC 1+sTB
+ VS
If ref
vref
¯ | + (Rc + jXc )I¯ Vc = |V t t
−
Vmin
VA max 1+sTC1 1+sTB1
KA 1+sTA
−
Gate
+ ∗ Vef l
VA min
|VT |VR max − KC If Ef HV |VT |VR min
sKF 1+sTF
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 25
42
Power System Dynamics and Stability
Subtransient Model
➠ A simple governor model (hydraulic valve plus turbine): Torder +
ωref +
1/R − ω
∗ Tin
+ Tmin
Tmax Tin
1
T3 s + 1
T4 s + 1
Ts s + 1
Tc s + 1
T5 s + 1
Governor
Servo
Reheat
Tmech
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 26
43
Power System Dynamics and Stability
Subtransient Model
➠ A governor-steam turbine model: −
w_ref
+
P_ref
K1 1.0 1
1 + sT_1
1+ sT_R
1 + sT_2
+
1
1
1
T_C1
1 + sT_C2
sT_C3
− 0.0
+
P_GV
1
K_HP
F_HP
1 + sT_HP +
+
+
+ F_IP
−
1 − K_RH
P_mech P_MAX
+
1
1
1
sT_RH
1+sT_IP
1 + st_LP
F_LP
A
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 27
44
Power System Dynamics and Stability
Example
➠ For a 200 MVA, 13.8 kV, 60 Hz generator with the following p.u. data: rs
=
0.001096
xℓ
=
0.15
xd
=
1.7
xd
′
=
0.238324
LF = 0.00438
′′
⇒
xd
=
0.184690
⇒
LD = 0.00426
xq
=
1.64
Lq = 0.00435 Mq = 0.00395
xq
′
=
xq
xq
′′
=
0.185151
LQ1 = 0.00405
′
⇒
=
6.194876
rF = 0.000742
′′
⇒
Td0
=
0.028716
rD = 0.0131
′
⇒
Tq0
=
0
Tq0
′′
=
0.074960
p
=
2 M = 10 D = 0
Td0
Ld = 0.00451 Md = 0.00411
⇒
⇒
LQ2 = 0
⇒
⇒
rQ2 = 0 ⇒
rQ1 = 0.0540
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 28
45
Power System Dynamics and Stability
Example
➠ A three-phase fault from open circuit conditions, i.e. before the fault √ vas = 13.8/ 3 kV and ias = 0, and after the fault vas = 0, is simulated using the detailed machine equations:
vas , vbs ,vcs [kV]
2 1 0 −1 −2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 29
46
Power System Dynamics and Stability
Example 10 9 8
iF [p.u.]
7 6 5 4 3 2 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 30
47
Power System Dynamics and Stability
Example
ia [p.u.]
20 0 −20
0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ib [p.u.]
20 0 −20
ic [p.u.]
20
0
−20
t [s] Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 31
48
Power System Dynamics and Stability
Transient Model
➠ Obtained by eliminating electromagnetic differential equations and the damper winding dynamic equations.
➠ For this reason, the damping D in the mechanical equations, which is typically a small value, is assumed to be large to indirectly model the significant damping effect of these windings on ωr .
➠ It is the typical model used in stability studies.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 32
49
Power System Dynamics and Stability
Transient Model
➠ Neglecting the induced currents in the rotor (winding Q2 ): ′ =0 x′q = xq Tq0
⇒
e′d = 0
➠ This leads to the transient equations: d ′ e dt a
= =
e′a ∠δ
1 ′ ′ [e + (x − x )i − e f d ds d a] ′ Td0 1 [ef − ea ] ′ Td0
= vas ∠θvas + rs ias ∠θias +jx′d ids ∠(δ + π/2) + jxq iqs ∠δ
ea ∠δ
= vas ∠θvas + rs ias ∠θias +jxd ids ∠(δ + π/2) + jxq iqs ∠δ Universidad de Castilla-La Mancha
June 26, 2008
Generator Modeling - 33
50
Power System Dynamics and Stability
Transient Model
➠ The phasor diagram in this case is: e′a
= vas + rs ias + jx′d ids + jxq iqs
ea
= vas + rs ias + jxd ids + jxq iqs
ℑ
ea e′a
iqs
ids
jxq iqs
δ vas jxd ids rs ias
j(xd − x′d )ids ℜ
ias
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 34
51
Power System Dynamics and Stability
Transient Model
➠ For faults near the generator terminals, the q axis has little effect on the system response, i.e. iqs ≈ 0. ➠ This results in the classical voltage source and transient reactance generator model used in simple stability studies:
rs
jx′d
ias ∠θias +
e′a ∠δ
Pm
ωr
vas ∠θvas −
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 35
52
Power System Dynamics and Stability
Transient Model
➠ A further approximation in some cases is used by neglecting the field ′ dynamics, i.e. Td0 = 0. ➠ In this case, e′a is a “fixed” variable controlled directly through the voltage regulator via ef . ➠ The limits in the voltage regulator are used to represent limits in the field and armature currents.
➠ These limits can be “soft”, i.e. allowed to temporarily exceed the hard steady-state limits, to represent under- and over-excitation.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 36
53
Power System Dynamics and Stability
Steady-state model
➠ When all transient are neglected, the generator model becomes: ea ∠δ
= vas ∠θvas + rs ias ∠θias +jxd ids ∠(δ + π/2) + jxq iqs ∠δ
ea ias ∠θias
= ef = ids ∠δ + jiqs ∠δ
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 37
54
Power System Dynamics and Stability
Steady-state model
➠ For a round rotor machine (xd = x′d ), the steady-state model leads to the classical short circuit generator model:
rs
jxd
ias ∠θias +
ea ∠δ
Pm
ωr
vas ∠θvas −
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 38
55
Power System Dynamics and Stability
Steady-state model
➠ Based on this simple model, the field and armature current limits can be used to define the generator capability curves (for a given terminal voltage vas
= vt ): Q ef limit
Qmax
vt ea x
ia limit P
vt i ∗ a
Qmin Pmin −
Pmax
2 vt x
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 39
56
Power System Dynamics and Stability
Steady-state model
➠ Considering the voltage regulator effect, the generator can be modeled as a constant terminal voltage within the generator reactive power capability, delivering constant power (Pm ).
➠ This yields the PV generator model for power flow studies: P
Q
V
where P
Qmin
= Pm = constant, and V = vt = constant for ≤ Q ≤ Qmax ; otherwise, Q = Qmax,min and V is allowed to
change.
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 40
57
Power System Dynamics and Stability
Example
➠ The generator data are: va
PG
v∞ = 1∠0 j0.1
θr (t) = ω0 t + π/2 + δ xd ′ Td0
M
= xq = 0.9, =
x′d = 0.2
2s
= large ⇒ δ = constant
The generator is operating in steady-state delivering PG
= 0.5 at
ea = 1.5. At t = 0 there is a fault and the line is disconnected. Find va (t) for t > 0. Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 41
58
Power System Dynamics and Stability
Example
➠ Steady-state conditions: ea ∠δ
⇒ ias PG
= vas ∠θvas + (rs + jxd )ias ∠θias = v∞ + j(xd + xL )ias 1.5∠δ − 1 = j(0.9 + 0.1) = 1.5 sin δ − j(1.5 cos δ − 1) = ℜ{v∞ i∗as } = ℜ{i∗as }
=
1.5 sin δ
⇒ δ
=
sin−1 (PG /1.5) = 19.47◦
ias
=
0.65∠ − 39.64◦
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 42
59
Power System Dynamics and Stability
Example
iqs ids ⇒ ea ∠δ
= ias cos(θias − δ)∠δ = 0.334∠19.47◦
= ias sin(θias − δ)∠(δ + π/2) = 0.558∠ − 70.53◦ = vas + rs ias + jx′d ids + jxq iqs = v∞ + j(xd + xL )ias + jx′d ids + jxq iqs
= 1 + j0.1(0.65∠ − 39.64◦ ) + j0.2(0.558∠ − 70.53◦ ) j0.9(0.334∠19.47◦ )
d ′ ea dt
= 1.110∠19.47◦ 1 ′ ′ = [e + (x − x )i − e a d ds d a] ′ Td0
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 43
60
Power System Dynamics and Stability
Example
➠ Transient: d ′ 1 ′ ′ ea (t) = [e + (x − x )i − e f d ds d a (t)] ′ dt Td0 ef = ea → steady-state ids
=
1.5
=
0 → open line
e′a (0) = d ′ ⇒ ea (t) = dt
1.110 1 [1.5 − e′a (t)] 2
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 44
61
Power System Dynamics and Stability
Example
➠ Solution: d ′ ea (t) = −0.5e′a (t) + 0.75 dt ⇒ e′a (t) = Ae−0.5t + B e′a (0) = A + B = 1.110
e′a (∞) = B = 0.75/0.5 = 1.5 ⇒ e′a (t) = −0.390e−0.5t + 1.5
va (t) = e′a (t) → since ias = ids = iqs = 0 1.5
e′a
1.11 0
t
Universidad de Castilla-La Mancha June 26, 2008
Generator Modeling - 45
Power System Dynamics and Stability
62
Transmission System Modeling
➠ Transformers: ☞ Single phase: ➛ Detailed model ➛ Phasor model ☞ Three phase: ➛ Phase shifts ➛ Models
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 1
Power System Dynamics and Stability
63
Transmission System Modeling
➠ Transmission Lines: ☞ Single phase: ➛ Distributed parameter model ➛ Phasor lumped model ☞ Three phase: ➛ Distributed parameter model ➛ Reduced models ➠ Underground cables
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 2
Power System Dynamics and Stability
64
Single-phase Transformers
➠ The basic characteristics of this device are: ➛ Flux leakage around the transformer windings is represented by a leakage inductance Lℓ . ➛ The core is made of magnetic material and is represented by a magnetization inductance (Lm ≫ Lℓ ), but saturates.
➛ Losses in the windings (Cu wires) and core (hysteresis and induced currents) are represented with lumped resistances (r and Gm ). ➛ Steps up or down the voltage/current depending on the turn ratio a = N1 /N2 = V1 /V2 .
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 3
65
Power System Dynamics and Stability
Single-phase Transformers leakage core
i2
i1
+
+ v1
N1
v2
N2
λℓ1
λℓ2 λm
magnetizing
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 4
Power System Dynamics and Stability
66
Single-phase Transformers
➠ The equivalent circuit is: dλ di v = ri + = ri + L dt dt i1 r1 v1 = ⇒ i2 r2 v2 Lℓ1 + Lm Lm /a d i1 + dt i2 Lm /a Lℓ2 + Lm /a2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 5
67
Power System Dynamics and Stability
Single-phase Transformers
➠ Equivalent circuit: i1 + v1
r1
Lℓ1
i im
Gm
− Lm
Lℓ2
ai
r2
i2
+
+
+
e
e/a
v2
−
−
−
N1 : N2 a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 6
Power System Dynamics and Stability
68
Single-phase Transformers
➠ The phasor equivalent circuit is:
1 V1 = (Zℓ1 + Zm )I1 + Zm I2 a 1 aV2 = Zm I1 + (a2 Zℓ1 + Zm ) I2 a V1 Zℓ1 + Zm Zm /a I1 ⇒ = V2 Zm /a Zℓ2 + Zm /a2 I2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 7
69
Power System Dynamics and Stability
Single-phase Transformers
➠ Phasor equivalent circuit: I1 + V1 −
a2 Zℓ2
Zℓ1
+
Im Ym
I2 /a
I2 +
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 8
70
Power System Dynamics and Stability
Single-phase Transformers
➠ This can also be readily transformed into a ABCD input-output form based on the following approximation, since Zm ≫ Zℓ1 (Zℓ1 ≈ a2 Zℓ2 ):
V1 I1
=
Zℓ
a(1 + Zℓ Ym )
Zℓ /a
aYm 1/a A B V2 = C C −I2
V2 −I2
= r1 + jXℓ1 + a2 (r2 + jXℓ2 )
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 9
71
Power System Dynamics and Stability
Single-phase Transformers
➠ Phasor equivalent circuit with the approximation Zm ≫ Zℓ1 : I1 + V1 −
I2 /a
Zℓ
+
Im Ym
I2 +
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 10
Power System Dynamics and Stability
72
Single-phase Transformers
➠ Or Π form: Z =
Zℓ a 1 Zℓ
Y1
=
(1 − a)
Y2
=
(a2 Zℓ Ym + a2 + a)
1 Zℓ
1 for Ym ≈ 0 ≈ (a − a) Zℓ 2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 11
73
Power System Dynamics and Stability
Single-phase Transformers
➠ Π equivalent circuit: Z
I1
I2
+ V1
+ Y1
Y2
V2
−
−
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 12
Power System Dynamics and Stability
74
Single-phase Transformers
➠ Neglecting Zm (Ym is small given the core magnetic properties): 1. Time domain:
v1 i2
di1 + av2 = ri1 + L dt = −ai1
2. Phasor domain:
V1 I2
= Zℓ I1 + aV2 = −aI1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 13
75
Power System Dynamics and Stability
Single-phase Transformers
➠ Equivalent circuit neglecting Zm : I1
I2 = −I1 /a
Zℓ
+ V1
+
+
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 14
Power System Dynamics and Stability
76
Single-phase Transformers
➠ Certain transformers have built-in Under-Load Tap Changers (ULTC). ➠ This is either operated manually (locally or remote controlled) or automatically with a voltage regulator; the voltage control range is limited (≈
10%) and on discrete steps (≈ 1%).
➠ The time response is in the order of minutes, with 1-2 min. delays, due to ULTCs being implemented using electromechanical systems.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 15
Power System Dynamics and Stability
77
Single-phase Transformers
➠ These are typically used to control the load voltage side, and hence are used at subtransmission substations.
➠ Nowadays, power electronic switches are used, leading to Thyristor Controller Voltage Regulators (TCVR), which are faster voltage controllers and are considered Flexible AC Transmission systems (FACTS).
➠ These types of transformers are modelled using the same transformer models, but a may be assumed to be a discrete controlled variable through a voltage regulator with a dead-band.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 16
Power System Dynamics and Stability
78
Single-phase Transformers
➠ Transformers with special connections and under-load tap changers can also be used for phase shift control and are known as Phase Shifters.
➠ These control the phase shift difference between the two terminal voltages within approximately ±30◦ , thus increasing the power capacity of a transmission line (e.g. interconnection between Ontario and Michigan).
➠ Phase shifters are modeled using a similar model but the tap ratio is a phasor as opposed to a scalar:
a = a∠α ➠ A Π equivalent cannot be used in this case. Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 17
79
Power System Dynamics and Stability
Single-phase Transformers
➠ Phasor equivalent circuit with complex tap ratio a: I1 + V1 −
I2 /a∗
Zℓ
+
Im Ym
I2 +
aV2
V2 −
− a:1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 18
Power System Dynamics and Stability
80
Single-phase Transformers (Example)
➠ Single phase 8/80 kV, 30 MVA transformer with Xℓ = 10% and Xm ≈ 10Xℓ . ➠ Detailed model parameters: a = Lℓ1
= =
Lℓ2
=
Lm
=
8/80 = 0.1 Xℓ1 Xℓ ≈ ω0 2ω0 0.1 (8 kV)2 = 0.283 mH 2 · 377 30 MVA Lℓ1 = 28.3 mH 2 a Xℓ 0.1 (8 kV)2 10 = 10 = 5.66 mH ω0 377 30 MVA Universidad de Castilla-La Mancha
June 26, 2008
Transmission System Modeling - 19
Power System Dynamics and Stability
81
Single-phase Transformers (Example)
➠ Per unit parameters: Zℓ
= j0.1
Zm
=
Ym
=
a =
10Zℓ = j1 1 = −j1 Zm 8/8 kV =1 80/80 kV
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 20
Power System Dynamics and Stability
82
Single-phase Transformers (Example)
➠ Per unit parameters:
A
=
B
=
C = D =
1(1 + j0.1(−j1)) = 1.1 j0.1 = j0.1 1 1(−j1) = −j1 1 =1 1
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 21
Power System Dynamics and Stability
83
Single-phase Transformers (Example)
➠ Per unit parameters:
Z = Y1 Y2
= =
j0.1 = j0.1 1 1 =0 (1 − 1) j0.1 1 = −j1 (1 (j0.1)(−j1) + 1 − 1) j0.1 2
2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 22
Power System Dynamics and Stability
84
Three-phase Transformers a1
+
T1 +
aVa2 − b1
+
Va2 a:1
−
T2
c1
+
Vb2 a:1
−
T3
c2
+
aVc2 −
b2
+
aVb2 −
a2
Vc2 a:1
−
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 23
Power System Dynamics and Stability
85
Three-phase Transformers
➠ The 3 single-phase transformers form a 3-phase bank that induces a phase shift, depending on the connection:
Vab1 Vab2 ⇒ Vab1 a apu
= aVa2 √ 3∠30◦ Va2 = √ 3a∠30◦ Vab2 = √ 3a∠30◦ = √ 3apu ∠30◦ =
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 24
Power System Dynamics and Stability
86
Three-phase Transformers
∆−Y:
a=
√ √
3a∠30◦
3apu ∠30◦ √ a = 3a∠ − 30◦ √ apu = 3apu ∠ − 30◦
apu = Y−∆
:
Y−Y
:
a=a
∆−∆:
a=a
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 25
87
Power System Dynamics and Stability
Three-phase Transformers
➠ In balanced, “normal” systems, the net phase shift between the generation and load sides is zero, and hence is neglected during system analyses: Generator
Load
side
side
∆
Y
Y
∆
➠ In these systems, the p.u. per-phase models of the transformers are identical to the equivalent single-phase transformer models.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 26
88
Power System Dynamics and Stability
Three-phase Transformers
➠ For “smaller” transformers (e.g. load transformers), integral designs are preferred to transformers banks: a
b
c
core
3 φ windows
vabc1 vabc2
r13×3
L113×3
L123×3
L213×3
L223×3
=
r23×3
iabc1
+
iabc2 i d abc1 dt iabc 2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 27
Power System Dynamics and Stability
89
Saturation
➠ The magnetization inductance Lm changes with the magnetization current due to saturation of the magnetic core.
➠ Saturation occurs due to a reduction on the number of “free” magnetic dipoles in the enriched core.
➠ This results in the core behaving more like air than a magnet, i.e. magnetic “conductivity” decreases.
➠ It is typically represented using a piece-wise linear model.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 28
Power System Dynamics and Stability
90
Saturation
➠ The magnetization inductance Lm changes with the magnetization current due to saturation of the magnetic core: λm
Lm2 Lm1
ims
Lm (im ) =
L
m1
im
for
im ≤ ims
Lm2 for im > ims
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 29
Power System Dynamics and Stability
91
Single-phase Transmission Line
➠ Alossless line can be represented using a series of lumped elements: l
=
c = D
D µ0 ln ′ 2π R 2πǫ0 ln(D/R)
→
distance between wires
→
wire GMR
R
→
wire radius
µ0
=
ǫ0
=
R′
4π × 10−7 [H/m]
8.854 × 10−12 [F/m]
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 30
92
Power System Dynamics and Stability
Single-phase Transmission Line
i1
i
+
l [H/m] +
v1
dv
i2 −
c [F/m]
−
x
+
+
di v
v2
−
−
dx
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 31
Power System Dynamics and Stability
93
Single-phase Transmission Line
➠ The equations for this line are: dv di
di ∂v ∂i ⇒ = −l dt ∂x ∂t ∂i ∂v dv ⇒ = −c = −cdx dt ∂x ∂t
= −ldx
➠ These are D’Alambert equations with solution: 1 1 v1 (t) − i2 (t − τ ) + v2 (t − τ ) i1 (t) = Zc Zc | {z } I2 (t−τ )
i2 (t)
=
1 1 v2 (t) − i1 (t − τ ) + v1 (t − τ ) Zc Zc {z } | I1 (t−τ )
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 32
94
Power System Dynamics and Stability
Single-phase Transmission Line
➠ where: Zc
=
s = τ
=
p
l/c → chracteristic impedance 1 √ → wave speed lc d → travelling time for line length s
➠ Distributed parameter equivalent circuit: i1
r/2
+ v1
r/2
I2 (t − τ )
+
I1 (t − τ )
Zc
Zc
−
i2
v2 −
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Transmission System Modeling - 33
95
Power System Dynamics and Stability
Single-phase Transmission Line
➠ Example: t=0 +
E
i1 +
v1 −
i2 = 0 +
Trans. Line
v2
−
−
v1 (t) = E i2 (t) =
0
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Transmission System Modeling - 34
96
Power System Dynamics and Stability
Single-phase Transmission Line
➠ Example: t
I1 (t − τ )
I2 (t − τ )
i1
i2
0
0
0
E/Zc
0
τ
2E/Zc
0
E/Zc
2E
2τ
2E/Zc
2E/Zc
2E
3τ
0
2E/Zc
−E/Zc
4τ
0
0
5τ
2E/Zc
0
−E/Zc
0
E/Zc
0
E/Zc
2E
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Transmission System Modeling - 35
97
Power System Dynamics and Stability
Single-phase Transmission Line
➠ Example: i1 E/Zc τ
2τ
3τ
4τ
5τ
6τ
t
τ
2τ
3τ
4τ
5τ
6τ
t
−E/Zc v2 2E
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 36
Power System Dynamics and Stability
98
Single-phase Transmission Line
➠ Phasor model from the distributed line model: d V = −(r + jωl)I = −zI dx d I = −(jωc)V = −yI dx 0 −z V d V = ⇒ dx I −y 0 I
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 37
Power System Dynamics and Stability
99
Single-phase Transmission Line
➠ The solution to this set of linear dynamical equations is: cosh γd Zc sinh γd V2 d V1 = dx −I2 1/Zc sinh γd cosh γd I1 where
cosh γd = sinh γd = γ
=
Zc
=
eγd + e−γd 2 eγd − e−γd 2 √ zy → propagation constant p z/y → characteristic impedance
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 38
100
Power System Dynamics and Stability
Single-phase Transmission Line
➠ This can be converted into the Π equivalent circuit: Z′
I1
I2
+ V1
+ Y1′ /2
Y2′ /2
V2
−
−
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 39
Power System Dynamics and Stability
101
Single-phase Transmission Line
➠ Π equivalent circuit: ′
Z
=
sinh γd zd |{z} γd Z
Y
′
=
tanh γd yd |{z} γd Y
for
d < 250 km
⇒
for
d < 100 km
⇒
Z′ ≈ Z
Z′ ≈ Z
Y′ ≈ Y Y′ ≈ 0
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 40
102
Power System Dynamics and Stability
Three-phase Transmission Line dα n
n
dδ
2R a
b
c dβ
dγ
d×d 11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 41
103
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Typically the phase wires are bundled (e.g. 4 wires/phase) and the guard wires are grounded at every tower, i.e. correspond to the neutral.
➠ For a total of N wires, the per unit length equations are: 2
2
3
v1 6 7 6v 7 27 ∂ 6 6 7 . 7 ∂x 6 6 .. 7 4 5 vN
=
+
r11 6 6r 6 21 6 6 .. 6 . 4 rN 1 2 l11 6 6l 6 21 6 6 .. 6 . 4 lN 1
r12 r22 . . .
rN 2 l12 l22 . . .
lN 2
32
3
i1 76 7 6 7 . . . r2N 7 7 6 i2 7 76 7 . 76 . 7 .. . 76 . 7 . . 54 . 5 . . . rN N iN 3 2 3 i1 . . . l1N 7 6 7 6 7 . . . l2N 7 7 ∂ 6 i2 7 7 6 7 . 7 . 7 .. . 7 ∂t 6 . 7 6 . . 5 4 . 5 . . . lN N iN ...
r1N
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Transmission System Modeling - 42
104
Power System Dynamics and Stability
Three-phase Transmission Line 2
2
3
i1 6 7 6i 7 27 ∂ 6 7 6 − . 7 6 ∂x 6 . 7 4 . 5 iN
=
+
cN ×N
=
g11 6 6g 6 21 6 6 .. 6 . 4 gN 1 2 p11 6 6p 6 21 6 6 .. 6 . 4 pN 1
g12
...
g22
...
. . .
..
gN 2
...
p12
...
p22
...
. . .
..
pN 2
...
.
.
32
3
v1 76 7 6 7 g2N 7 7 6 v2 7 76 7 . 76 . 7 . 76 . 7 . 54 . 5 vN gN N 3−1 2 3 p1N v1 7 6 7 6v 7 p2N 7 7 6 27 ∂ 7 6 7 . 7 . 7 ∂t 6 . 7 . 7 6 . 5 4 . 5 pN N vN g1N
p−1 N ×N
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 43
105
Power System Dynamics and Stability
Three-phase Transmission Line
➠ In phasor form, these equations are: 2
2
3
V1 6 7 6V 7 27 d 6 6 7 . 7 6 dx 6 . 7 4 . 5 VN 2 3 I1 6 7 6I 7 27 d 6 6 7 . 7 dx 6 6 .. 7 4 5 IN
⇒
=
=
z11 6 6z 6 21 −6 6 .. 6 . 4 zN 1 2 y11 6 6y 6 21 −6 6 .. 6 . 4 yN 1
d V = −[z]I dx
z12
...
z22
...
. . .
..
zN 2
...
y12
...
y22
...
. . .
..
yN 2
...
.
.
32
3
I1 76 7 6 7 z2N 7 7 6 I2 7 76 7 . 76 . 7 . 76 . 7 . 54 . 5 zN N IN 32 3 y1N V1 76 7 6 7 y2N 7 6 V2 7 7 76 7 . 76 . 7 . 76 . 7 . 54 . 5 yN N vN z1N
d I = −[y]V dx
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Transmission System Modeling - 44
106
Power System Dynamics and Stability
Three-phase Transmission Line
➠ The image method is useful for computing line parameters: i
00 2Ri 11 00 11 00 11
dij j
hi
φij
Dij
111 000 2Rj 000 111 000 111 000 111
hj ρ [Ωm]
hj hi
Earth
j image
i image
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Transmission System Modeling - 45
Power System Dynamics and Stability
107
Three-phase Transmission Line
➠ The line parameters zij and yij can be computed using Carson’s formulas:
rii
= riint + ∆rii
riint → from tables
rij
=
∆rij
∆rij
=
4ω10−4 {π/8 − b1 a cos φ
+b2 [(c2 − ln a)a2 cos 2φ + φa2 sin 2φ]
+b3 a3 cos 3φ − d4 a4 cos 4φ − b5 a5 cos 5φ
+b6 [(c6 − ln a)a6 cos 6φ + φa6 sin 6φ] a =
+b7 a7 cos 7φ − d8 a8 cos 8φ − b9 a9 cos 9φ + . . .} p √ −4 4π 510 D f /ρ Universidad de Castilla-La Mancha
June 26, 2008
Transmission System Modeling - 46
Power System Dynamics and Stability
108
Three-phase Transmission Line
D=
2h
i
for i
=j
Dij for i 6= j √ 2 b1 = 6 s bk = bk−2 k(k + 2) c2 = 1.3659315 +1 k = 1, 2, 3, 4, 5, . . . s= −1 k = 5, 6, 7, 8, 13, . . .
0 for i = j φ= φij for i 6= j
1 b2 = 16 π d k = bk 4
1 1 ck = ck−2 + + k k+2
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Transmission System Modeling - 47
Power System Dynamics and Stability
109
Three-phase Transmission Line
xii xij ∆xij
µ0 2hi = ω ln + xiint + ∆xii 2π Ri µ0 Dij = ω ln + ∆xij 2π dij =
xiint → from tables(≈ 0)
4ω10−4 {1/2(0.6159315 − ln a) − b1 a cos φ − d2 a2 cos 2φ
+b3 a3 cos 3φ − b4 [(c4 − ln a)a4 cos 4φ + φa4 sin 4φ] +b5 a5 cos 5φ − d6 a6 cos 6φ + b7 a7 cos 7φ
−b8 [(c8 − ln a)a8 cos 6φ + φa8 sin 8φ] + . . .}
gij pii
= gij ≈ 0 2hi 1 ln = 2πǫ0 Ri
1 2Dij pij = ln 2πǫ0 dij
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 48
110
Power System Dynamics and Stability
Three-phase Transmission Line f (Hz)
′ ′ Rac /Rdc
L′ac /L′dc
f (Hz)
′ ′ Rac /Rdc
L′ac /L′dc
2
1.0002
0.99992
4000
7.1876
0.15008
4
1.0007
0.99970
6000
8.7471
0.12258
6
1.0015
0.99932
8000
10.0622
0.10617
8
1.0026
0.99879
10000
11.2209
0.09497
10
1.0041
0.99812
20000
15.7678
0.06717
20
1.0164
0.99254
40000
22.1988
0.04750
40
1.0632
0.97125
60000
27.1337
0.03879
60
1.1347
0.93898
80000
31.2942
0.03359
80
1.2233
0.89946
100000
34.9597
0.03004
100
1.3213
0.85639
200000
49.3413
0.02124
200
1.7983
0.66232
400000
69.6802
0.01502
400
2.4554
0.47004
600000
85.2870
0.01227
600
2.9421
0.38503
800000
98.4441
0.01062
800
3.3559
0.33418
1000000
110.0357
0.00950
1000
3.7213
0.29924
2000000
155.5154
0.00672
2000
5.1561
0.21204
4000000
219.8336
0.00475
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 49
Power System Dynamics and Stability
111
Three-phase Transmission Line
➠ The [z] parameters depend on the line frequqnecy ω , i.e. this is a frequency dependent model.
➠ These number of conductors, and hence equations, can be reduced based on the following observations:
➛ The voltage of all Nb conductors in a phase bundled are at the same voltage (e.g. v1 = v2 = · · · = vN b = va ). ➛ The current in each phase is shared approximately equally by each conductor in the bundle (e.g. i1 = i2 = · · · = iN b = ia /Nb ). ➛ The voltage in the guard wires is zero (e.g. vg = 0).
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 50
Power System Dynamics and Stability
112
Three-phase Transmission Line
➠ This reduces the matrices to:
[z]N ×N
[y]N ×N
⇒
→
→
d Vabc dt d Iabc dt
zaa zab zac yaa yab yac
zab zbb zbc yab ybb ybc
zac
zbc zcc yac ybc ycc
= −[zabc ]Iabc = −[yabc ]Vabc
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 51
113
Power System Dynamics and Stability
Three-phase Transmission Line
➠ A line is transposed to balance the phases. ➠ The length of the barrel B must be much less than the wavelength (s/f ≈ 5000 km @ 60 Hz). B ≈ 50 km. B
a b c
zaa [[zabc ] = zab zac
zab zbb zbc
1 2 3
3 1 2
B/3
B/3
zac
zbc zcc
2 3 1
B/3
yaa [yabc ] = yab yac
yab ybb ybc
yac
ybc ycc
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 52
Power System Dynamics and Stability
114
Three-phase Transmission Line
➠ The phasor equations can be diagonalized using eigenvalue (modal) analysis techniques:
[zm ] [ym ]
= TIT [z]N ×N TI (diagonal matrix)
= TVT [z]N ×N TV
Vm
= TV−1 V = TIT V
Im
= TI−1 I = TVT I
(diagonal matrix)
➠ There are N modes of propogation, one for each eigenvalue.
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 53
115
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Diagonalization of a 3-phase transposed line through sequence transformation:
TS
=
a =
1 1 √ 1 3 1
V0pn
= TS−1 Vabc
I0pn
= TS−1 Iabc
1 2
a
a
1∠120◦
1
a a2
⇒
TS−1
1 1 = √ 1 3 1
1 a a2
1
a a 2
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 54
116
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Transformation of zabc into z0pn : [z0pn ] = TS−1 [zabc ]TS zs + 2zm 0 = 0 zs − zm 0 0 zp = zn
0 z0 0 = 0 zs − zm 0
z0 ≈ 3zp
0 zp 0
0
0 zn
➠ Similar for [y0pn ] = TS−1 [yabc ]TS .
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 55
117
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Diagonalization of a 3-phase transposed line through 0αβ transformation:
V0αβ
= T −1 Vabc
= T −1 Iabc √ 2 0 1 p √ 1 T = √ 1 −1/ 2 ⇒ 3/2 3 p √ 1 −1/ 2 − 3/2 I0αβ
T −1 = T T
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Transmission System Modeling - 56
118
Power System Dynamics and Stability
Three-phase Transmission Line
➠ Transformation of zabc into z0αβ : [z0αβ ]
= T T [zabc ]T zs + 2zm = 0 0
0 zs − zm 0
zα = zβ = zp
0 z0 0 = 0 zs − zm 0
z0 ≈ 3zα
0 zα 0
0
0 zβ
➠ Similar for [y0αβ ] = T T [yabc ]T .
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 57
Power System Dynamics and Stability
119
Three-phase Transmission Line
➠ These simplifications lead to the following per-phase (positive sequence), per-unit length formulas:
r
=
l
=
c =
rtables Nb µ0 Dm ln ′ 2π Rb 2πǫ0 ln DRmb
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 58
Power System Dynamics and Stability
120
Three-phase Transmission Line
➠ where: ➛ Dm is the GMD of the 3 phases: p Dm = 3 dab dac dbc
➛ Rb′ is the GMR of the bundled and wires: p ′ Nb R′ d12 d13 · · · d1Nb Rb = ➛ Rb is the GMR of the bundled: p Nb Rd12 d13 · · · d1Nb Rb = Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 59
121
Power System Dynamics and Stability
Example 1
➠ 24.14 km transposed distributed line: 70.68 mH
0.741 Ω
Bus 1
15 miles
Bus 2
Bus 3
70.16 mH
1 ms
219.1 mH
√ 325/ 3 kV
251.2 Ω
R0′ = 0.3167 Ω/km
R1′ = 0.0243 Ω/km
L′0 = 3.222 mH/km
L′1 = 0.9238 mH/km
C0′ = 0.00787 µF/km
C1′ = 0.0126 µF/km
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
122
Example 1 2
3
187.79 cos(377t) 6 7 6187.79 cos(377t − 2π/3)7 4 5 187.79 cos(377t + 2π/3)
=
+ 2
3
vBU S10 6 7 6vBU S1 7 α5 4 vBU S1β 2
=
3
i1 6 07 6i1 7 4 α5 i1β
=
2 3 2 32 3 vBU S1a 0.714 0 0 i1 6 7 6 7 6 a7 6 vBU S1 7 + 6 0 6 7 0.714 0 7 4 4 5 4 i1b 5 b5 vBU S1c 0 0 0.714 i1c 2 32 3 0.07068 0 0 i1 7 6 a7 d 6 6 7 6 i1 7 0 0.07068 0 4 54 b5 dt 0 0 0.07068 i1c 32 3 2 vBU S1a 1 1 1 7 √ √ 76 √ 1 6 7 6 6 √ 4 2 −1/ 2 −1/ 2 5 4 vBU S1b 7 5 p p 3 0 3/2 − 3/2 vBU S1c | {z }
3 2 i1a 7 6 −1 6 7 T 4 i1b 5 i1c
T −1
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
123
Example 1
i r0 1 h vBU S10 − i10 (t) − I20 (t − τ0 ) i10 (t) = Zc0 2 i 1 h r0 i20 (t) = vBU S20 − i20 (t) − I10 (t − τ0 ) Zc0 2 r0 = 0.3167 × 24.14 = 7.6451 Ω r 3.222 × 10−3 = 639.85 Ω ZC0 = 0.00787 × 10−6 p τ0 = 24.14 3.222 × 10−3 0.00787 × 10−6 = 0.12156 ms
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
124
Example 1
i1α (t) = i2α (t) = rα
=
ZCα
=
τα
=
i rα 1 h vBU S1α − i1α (t) − I2α (t − τα ) Zcα 2 i 1 h rα vBU S2α − i2α (t) − I1α (t − τα ) Zcα 2 0.0243 × 24.14 = 0.5866 Ω r 0.9238 × 10−3 = 270.77 Ω 0.0126 × 10−6 p 24.14 0.9238 × 10−3 0.0126 × 10−6 = 0.08236 ms
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
125
Example 1
rβ
1 = Zcβ 1 = Zcβ = rα
ZCβ
= ZCα
i1β (t) i2β (t)
τβ
h h
vBU S1β vBU S2β
i rβ − i1β (t) − I2β (t − τβ ) 2 i rβ − i2β (t) − I1β (t − τβ ) 2
= τα
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Transmission System Modeling - 64
126
Power System Dynamics and Stability
Example 1 2 3 vBU S2a 6 7 6 vBU S2 7 4 b5 vBU S2c 2 3 i1 6 a7 6 i1 7 4 b5 i1c 3
2 vBU S2a 6 7 6 vBU S2 7 4 b5 vBU S2c
=
2 1 1 6 √ 6 1 34 1 | 2
=
=
√
2 √ −1/ 2 √ −1/ 2 {z T
3
32
3
vBU S20 76 7 7 6 3/2 5 4vBU S2α 7 5 p vBU S2β − 3/2 } 0
p
i2 6 07 7 T6 4i2α 5 i2β 2 32 3 251.2 0 0 i2 6 7 6 a7 6 7 −6 251.2 0 7 4 0 5 4 i2b 5 0 0 251.2 i2c 2 3 32 0.28926 0 0 i2 7 6 a7 d 6 6 7 6 i2 7 − 0 0.28926 0 4 54 b5 dt 0 0 0.28926 i2c
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 65
127
Power System Dynamics and Stability
Example 1 300
vaBUS2 vasource
200
[kV]
100
0
−100
−200
−300
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 66
128
Power System Dynamics and Stability
Example 1 300
200
100
[kV]
0
−100
−200
vaBUS2 vbBUS2 vcBUS2
−300
−400
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Transmission System Modeling - 67
129
Power System Dynamics and Stability
Example 2 50′ n
n
28′ 8′′
1′ a
b
c 45′
110′
40′ × 40′
11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
130
Example 2
➠ Line data: ➛ 4 Drake wires per phase (Nb = 4) ➛ 927 kcmil = 469.8 mm2 ➛ ACSR, 24 Al/13 steel, 3 layers ➛ R = 0.554 in = 1.407 cm ➛ R′ = 0.425 in = 1.080 cm ➛ rdc = 0.1032 Ω/mile = 0.0645 Ω/km ➛ rac@25◦ C = 0.1061 Ω/mile = 0.0663 Ω/km ➛ rac@100◦ C = 0.1361 Ω/mile = 0.0851 Ω/km ➛ d = 200 km ➛ f = 60 Hz → ω = 377 rad/s Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
131
Example 2
r
=
Dm
=
Rb′
=
Rb
=
l
=
c
=
rac@100◦ C = 0.02134 Ω/km Nb √ 3 45′ 45′ 90′ = 17.27 m √ 4 1.080 30.48 30.48 43.11 cm = 0.1442 m √ 4 1.407 30.48 30.48 43.11 cm = 0.1541 m 17.27 −7 = 0.9573 × 10−3 H/m 2 × 10 ln 0.1442 2π8.854 × 10−12 = 0.01179 µH/m 17.27 ln 0.1541
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
132
Example 2
z = r + jωl = y
= jωc =
γ
0.3615∠86.62◦ Ω/km
=
4.444 × 10−6 ∠90◦ S/km √ zy
0.00217∠88.31◦ km−1 r z = y =
Zc
=
285.21∠ − 1.69◦ Ω
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133
Example 2
A
= D = cosh γd eγd + e−γd = 2 = 0.9686∠0.11◦
B
= Zc sinh γd = =
C = =
eγd − e−γd Zc 2 71.46∠86.65◦ Ω 1 sinh γd Zc 8.787 × 10−4 ∠90.03◦ S
Universidad de Castilla-La Mancha June 26, 2008
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134
Example 2
Z = zd = Y
72.30∠86.62◦ Ω
= yd 8.89 × 10−4 ∠90◦ S sinh γd = Z γd = 71.33∠86.65◦ Ω ≈ Z tanh γd/2 = Y γd/2 =
Z′
Y′
=
9.032 × 10−4 ∠89.95◦ S ≈ Y
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
135
Cables
Steel pipe (filled with insulating oil) Skid wires Metallic tapes Paper/oil insulation Screen Conductor (stranded copper)
11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 000 111 00000000000 11111111111 000 111 00000000000 11111111111 000 111 00000000000 11111111111 00000000000 11111111111 0000 1111 00000000000 11111111111 0000 1111 00000000000 11111111111 0000 1111 0000000 1111111 00000000000 11111111111 0000000 1111111 00000000000 11111111111 000 111 0000000 1111111 00000000000 11111111111 000 111 0000000 1111111 00000000000 11111111111 000 111 0000000 1111111 00000000000 11111111111
Universidad de Castilla-La Mancha June 26, 2008
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136
Cables
➠ Single-phase and three-phase cables:
111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111
conductor
sheath
SF6 gas
111111111 000000000 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 000000000 111111111 00 11 11 00 000000000 111111111 00 11 00 11 000000000 111111111 00 11 00 11 000000000 111111111 00 11 00 11 000000000 111111111 00 11 00 11 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Cables
➠ Model for a single-phase cable:
VC
1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 0000000 1111111 00000000 11111111 00000 11111 0000000 1111111 0000000 1111111 00000000 11111111 00000 11111 0000000 1111111 00000000 11111111 0000000 1111111 00000 11111 0000000 1111111 00000000 11111111 00000 11111 0 1 0000000 1111111 00000000 11111111 00000 11111 0 1 00000000 11111111 0000000 1111111 00000 11111 00000000 11111111 00000 11111 0000000 1111111 00000000 11111111 00000 11111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 V
11111111 00000000 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000 111 000000 111111 00000000 11111111 000 111 000000 111111 00000000 11111111 000 111 000000 111111 00000000 11111111 000000 111111 00000000 11111111
VS
I1 I2 I3
A
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
138
Cables
➠ Model for a single-phase cable: V1
= Vcore − Vsheath = VC − VS
V2
= Vsheath − Varmour = VS − VA
V3
= VA
IC
= I1
IS
= I1 − I2
IA
= I2 − I3
Universidad de Castilla-La Mancha June 26, 2008
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139
Power System Dynamics and Stability
Cables
➠ Loop equations:
V1
z11
z12
0
I1
d V2 = − z12 z22 z23 I2 dx V3 I3 0 z23 z33 V1 y1 0 0 I1 d I2 = 0 y2 0 V2 dx V3 I3 0 0 y3
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
140
Cables
➠ The elements of the impedance matrix z arecomputed as follows: z11
= zC ext + zCS + zS int
z12
= −zS mut
z22
= −zS ext + zSA + zAint
z23
= −zAmut
z33
= −zAext + zAE + zE
Universidad de Castilla-La Mancha June 26, 2008
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141
Cables
➠ These impedances are calculated using the formulas: 11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 q 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 r 00000000 11111111 00000000 11111111
zext zint zmut zins
ρm [I0 (mr)K1 (mq) + K0 (mr)I1 (mq)] 2πrD ρm [I0 (mq)K1 (mr) + K0 (mq)I1 (mr)] = 2πqD ρ = 2πqrD µ0 r −1 = y = jω 2π q =
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
142
Cables
D
= I1 (mr)K1 (mq) + K1 (mr)I1 (mq) s jωµ m = ρ ➠ Where In and Kn are modified Bessel functions, as follows: In (x)
=
∞ X
k=0
Kn (x)
=
x n+2k 1 k!Γ(n + k + 1) 2
π I−n (x) − In (x) 2 sin(nπ)
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
143
Cables
➠ The Gamma function Γ(n) is defined as follows: Γ(n) =
Z
∞
e−x xn−1 dx
0
Γ(n + 1) = nΓ(n) Γ(n + 1) = n! for n = 1, 2, 3, . . . ➠ For asymmetric cables a finite element method is needed to compute these impedances.
Universidad de Castilla-La Mancha June 26, 2008
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144
Power System Dynamics and Stability
Cables
➠ From the relations between loop voltages/currents and node voltage/currents, the node equations are:
VC
zCC
d VS = − zCS dx VA zCA e.g.
d dx
zCS
zCA
zSS
zSA
zSA
zAA
IC
IS IA
zCC = z11 + 2z12 + z22 + 2z23 + z33 IC y1 −y1 0 VC IS = −y1 y1 + y2 −y2 VS IA 0 −y2 y2 + y3 VA
Universidad de Castilla-La Mancha June 26, 2008
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145
Power System Dynamics and Stability
Cables
➠ For a three-phase cable made of 3 single-phase cables: d [V] = −[z]9×9 [I] dx d [I] = −[y]9×9 [V] dx [zaa ]3×3 [z] = [zab ]3×3 [zac ]3×3 [yaa ]3×3 [y] = 0 0
[zab ]3×3
[zac ]3×3
[zbb ]3×3
[zbc ]3×3
[zbc ]3×3
[zcc ]3×3
0
0
[ybb ]3×3
0
0
[ycc ]3×3
Universidad de Castilla-La Mancha June 26, 2008
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146
Load Modeling
➠ RLC models. ➠ Induction motors. ➛ Detailed models. ➛ Phasor models. ➠ Aggregated models: ➛ Impedance models. ➛ Power models. ➛ Induction motor power models.
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 1
147
Power System Dynamics and Stability
Load Classification
➠ By demand level: ➛ Residential: lighting and heating (RL + controls); AC (motor + controls); appliances (small motors + controls). ➛ Commercial: similar types of devices as residential. ➛ Industrial: motor drives (induction and dc motor-based mostly); arc furnaces; lighting; heating; others (e.g. special motor drives).
➠ By type: ➛ RLC + controls. ➛ Drives: ac/dc motors + electronic controls. ➛ Special (e.g. arc furnace). Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Remarks on Load Classification
➠ Most controls are implemented using a variety of power electronic converters.
➠ Aggregate load models are necessary at the transmission system modeling level.
➠ Only large loads can be represented with their actual models.
Universidad de Castilla-La Mancha June 26, 2008
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149
RLC Loads, Resistor
➠ Ideal, linear resistors (R), inductors (L) and capacitors (C ). ➠ Resistor: R
i +
v
−
➛ Time domain: v = Ri ➛ Phasor domain: V = RI
Universidad de Castilla-La Mancha June 26, 2008
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150
RLC Loads, Inductor
➠ Inductor: L
i +
v
−
➛ Time domain: di v=L dt ➛ Phasor domain: V
= jωLI = ZL I
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
151
RLC Loads, Inductor
➠ The inductor time domain model can be discretized using the trapezoidal integration method as follows:
i = ⇒
ik+1
1 L
Z
vdt
∆t = ik + (vk+1 + vk ) 2L ∆t ∆t = vk+1 + (ik + vk ) 2L 2L {z } | hk
Universidad de Castilla-La Mancha June 26, 2008
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152
RLC Loads, Capacitor
➠ This yields the equivalent resistive circuit: vk+1 +
−
ik+1 hk
2L ∆t
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Power System Dynamics and Stability
153
RLC Loads, Capacitor
➠ Capacitor: C
i +
v
−
➛ Time domain: dv i=C dt ➛ Phasor domain: V
1 I ωC = ZC I = −j
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154
Power System Dynamics and Stability
RLC Loads, Capacitor
➠ The capacitor time domain model can be discretized using the trapezoidal integration method as follows:
v ⇒
vk+1 ik+1
=
1 C
Z
idt
∆t = vk + (ik+1 + ik ) 2C 2C 2C = vk+1 − (ik + vk ) ∆t ∆t {z } | hk
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 9
Power System Dynamics and Stability
155
RLC Loads, Capacitor
➠ This yields the equivalent resistive circuit: vk+1 +
−
ik+1 hk
∆t 2C
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Load Modeling - 10
156
Power System Dynamics and Stability
Induction Motor
➠ Arbitrary 0dq reference: as
d
ω
q
ωr
c′s
r ar
θr
b′s
θ 0
s
cr bs
br cs
a′s
d
Universidad de Castilla-La Mancha June 26, 2008
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157
Power System Dynamics and Stability
Induction Motor
➠ Electrical (inductor) equations: d [λabcs ] dt d [vabcr ] = [rabcr ][iabcr ] + [λabcr ] dt λ i Lsr(θr ) L abcs abcs = abcs Lsr (θr ) Labcr λabcr iabcr | {z } | {z } [vabcs ] =
[rabcs ][iabcs ] +
L(θr )
Te
=
[i]
1 T d[L(θr )] [i] [i] 2 dθr
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 12
158
Power System Dynamics and Stability
Induction Motor
➠ Mechanical (Newton’s) equations: d J ωr + dωr dt d θr dt ➠ Stator transformation equations: vas v0s vds = Ks vbs vcs vqs
= Tm − Te = ωr
ias i0s ids = Ks ibs ics iqs
Universidad de Castilla-La Mancha June 26, 2008
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159
Power System Dynamics and Stability
Induction Motor
➠ Where the transformation matrix Ks is as follows: 1/2 1/2 1/2 2 Ks = sin θ sin(θ − 2π/3) sin(θ + 2π/3) 3 cos θ cos(θ − 2π/3) cos(θ + 2π/3) 1 sin θ cos θ −1 Ks = 1 sin(θ − 2π/3) cos(θ − 2π/3) 1 sin(θ + 2π/3) cos(θ + 2π/3)
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 14
160
Power System Dynamics and Stability
Induction Motor
➠ Rotor transformation equations: var v0r vdr = Kr vbr vcr vqr
iar i0r idr = Kr ibr icr iqr
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 15
161
Power System Dynamics and Stability
Induction Motor
➠ Where the transformation matrix Kr is as follows: 1/2 1/2 1/2 2 Kr = sin β sin(β − 2π/3) sin(β + 2π/3) 3 cos β cos(β − 2π/3) cos(β + 2π/3) 1 sin β cos β −1 Kr = 1 sin(β − 2π/3) cos(β − 2π/3) 1 sin(β + 2π/3) cos(β + 2π/3) β
= θ − θr
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 16
162
Power System Dynamics and Stability
Induction Motor
➠ Stator equations: v0s rs vds = 0 vqs 0
0 rs 0
0
i0s
0
λ0s
d λ 0 ids + ω −λqs + ds dt λqs rs iqs λds
➠ Rotor equations referred to the stator: 2 a rr 0 0 ′ 0 λ′0r 0 i0r |{z} rr′ i′ + (ω − ωr ) −λ′ + d λ′ 0 = dr 0 qr dt dr rr′ 0 ′ ′ ′ 0 i λ λ qr qr dr 0 0 rr′ a=
Ns Nr
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 17
163
Power System Dynamics and Stability
Induction Motor
➠ Magnetic flux equations: Lls 0 0 λ0s 0 ls + M 0 L {z } | λ ds Ls λ 0 0 Ls qs = λ′ 0 0 0 0r ′ λdr 0 M 0 λ′qr 0 0 M
0 0 0 L′lr 0 0
0
0
i M 0 0s i ds 0 M iqs i′ 0 0 0r L′lr + M 0 i′dr | {z } ′ L′r iqr 0 L′r
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 18
164
Power System Dynamics and Stability
Induction Motor
➠ Mechanical equations: 2 d 2 J ωr + Dωr p dt p d θr dt d θ dt Te
= Tm − Te = ωr = ω =
3p (iqs λds − ids λqs ) 22
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 19
165
Power System Dynamics and Stability
Induction Motor
➠ Equivalent circuit representation of these equations: iqs
rs
+ vqs
+
−
Lls
L′lr
−
+
rr′
Ns : Nr
iqr
(ω − ωr )λ′dr
ωλds M
vqr
−
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 20
166
Power System Dynamics and Stability
Induction Motor ids
rs
+
−
+
Lls
L′lr
−
rr′
Ns : Nr
idr
(ω − ωr )λ′qr
ωλqs
vds
+
vdr
M
−
i0s
rs
Lls
L′lr
rr′
Ns : Nr
i0r
+ v0r
v0s − Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 21
167
Power System Dynamics and Stability
Induction Motor
➠ Assuming a balanced, fundamental frequency (ω = ω0 ) system, the model can be reduced to a p.u. “transient” model (3rd order model): Vas
= VasR + jVasI
Ias
= IasR + jIasI 1 ′ + (xS − x′ )IasI ] = ω0 σEI′ − ′ [ER T0 1 ′ ′ = −ω0 σER − ′ [EI + (xS − x′ )IasR ] T0
d ′ ER dt d ′ E dt I
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 22
168
Power System Dynamics and Stability
Induction Motor ′ VasR − ER
VasI − EI′ d σ dt Te TL
= rs IasR − x′ IasI
= rs IasI + x′ IasR 1 = (TL − Te + Dω0 − Dσ) H 1 ′ (ER IasR + EI′ IasR ) = ω0 = f (σ)
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169
Power System Dynamics and Stability
Induction Motor
➠ where: σ xs x′ T0′
ω0 − ωr → slip = ω0 = xls + xm → stator reactance x′lr xm = xls + ′ → transient reactance xlr + xm =
x′lr + xm ω0 rr′
→
open circuit transient time constant
Universidad de Castilla-La Mancha June 26, 2008
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170
Power System Dynamics and Stability
Induction Motor
➠ If the electromagnetic transients are neglected, the system can be reduced to the following quasi-steady state equivalent circuit model: ′ jx′lr jxls rs Iar rr′ Ias + Vas
jxm
rr′ 1−σ σ
−
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171
Power System Dynamics and Stability
Induction Motor
➠ Or equivalently: Ias
jx′lr
jxls
rs
′ Iar
+ jxm
Vas
rr′ σ1
−
➠ Plus mechanical equations.
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172
Power System Dynamics and Stability
Induction Motor
➠ This simplification can be justified from the point of view of the torque-speed characteristic: Te Te max Steady state characteristic
Transient start up
0
1
ωr ωc
=1−σ
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173
Power System Dynamics and Stability
Induction Motor
➠ If the mechanical dynamics are ignored, the slip σ becomes a fixed value, and hence the equivalent circuit can be reduced to a simple equivalent reactive impedance, i.e. a Z load.
➠ For loads with multiple IMs, a equivalent motor model can be used to represent these motors.
➠ Neglecting the motor dynamic equations in large or equivalent aggregate motors can lead to significant modeling errors, as the transient model and mechanical time constants can be on the same range as the generator time constants.
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174
Power System Dynamics and Stability
Induction Motor
➠ Double-cage IMs can be modeled by introducing an additional inductance on the rotor side, which leads to a “subtransient” model.
➠ Since rotor cores are laminated, eddy currents do not play a significant role on the system dynamics.
➠ Controls can also be modeled in detail by representing the converters with ideal electronic switches plus their control systems.
➠ For balanced, fundamental frequency systems, approximate equivalent models of the IM and its controls can be readily implemented.
Universidad de Castilla-La Mancha June 26, 2008
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175
Power System Dynamics and Stability
Induction Motor
➠ For example, an IM with voltage-fed field oriented control can be represented by:
dωr dt
=
P (Te − TL ) 2J 2 L′ r rs
M 2 rr′
+ L′r σ
!
M rr′ ′ iqs − ωids + ′ λqr Lr σ
diqs dt
= −
dids dt
M L′r ′ − ωr λdr + vqs σ σ ! 2 L′ r rs + M 2 rr′ M rr′ ′ = − ids + ωiqs + ′ λdr ′ Lr σ Lr σ M L′r ′ + ωr λqr + vds σ σ Universidad de Castilla-La Mancha
June 26, 2008
Load Modeling - 30
176
Power System Dynamics and Stability
Induction Motor
dλ′qr dt dλ′dr dt Te σ dTref dt
′ ′ r r M r r ′ ′ = −(ω − ωr )λdr − ′ λqr + ′ iqs Lr Lr rr′ ′ rr′ M ′ = (ω − ωr )λqr − ′ λdr + ′ ids Lr Lr 3pM ′ ′ = (i λ − i λ qs ds dr qr ) 4L′r
= Ls L′r − M 2 2 pTL 3p M ′ = kq3 − i λ qs dr + kq4 (ωref − ωr ) 2J 8JL′r
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 31
177
Power System Dynamics and Stability
Induction Motor
duqs dt
rr′ M 3pM kq1 rr′ M ′2 ′ − = [ i λ i i + λ dr ωr qs dr ds qs ′ ′ ′ 4Lr Lr Lr σ ! ′2 2 ′ L′r ′ L r rs + M rr ′ iqs λdr − λ uqs ] + L′r σ σ dr 2 pTL 3p M ′ +kq1 kq3 − i λ qs dr 2J 8JL′r +kq2 Tref− 2pM′ iqs λ′dr aLr
duds dt
+kq1 kq4 (ωref − ωr ) ′ ′ M rr rr ′ ′ + k (λ − λ = kd1 λ i − d2 ref ds dr ) dr ′ ′ Lr Lr Universidad de Castilla-La Mancha
June 26, 2008
Load Modeling - 32
178
Power System Dynamics and Stability
Example
➠ A 21 MW load at 4 kV and 60 Hz is made of: ➠ An inductive impedance load with G = 0.06047, B = −0.03530. ➠ An aggregated induction motor model with rs = 0.07825, xls = 0.8320, rr′ = 0.1055, x′lr = 0.8320, xm = 16.48. ➠ This data is all in p.u. on a 100 MVA, 4 kV base.
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 33
Power System Dynamics and Stability
179
Example
➠ The Z load model is then: PL PZ ⇒
PIM
21 MW = 0.21 = 100 MVA = PZ + PIM = VL2 G = 0.06 = 0.15
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Load Modeling - 34
180
Power System Dynamics and Stability
Example
ZIM
jxM (rr′ /σ + jx′lr ) = rs + jxls + ′ rr /σ + j(xM + x′lr ) −13.7114 + j1.7386/σ = 0.07825 + j0.832 + 0.1055/σ + j17.312 28.652/σ = 0.07825 + 0.01113σ 2 + 299.71 2 0.18342/σ + 237.37 +j 0.832 + 0.01113/σ 2 + 299.71
(0.00087/σ 2 + 28.652/σ + 23.452) + j(0.19628/σ 2 + 486.73) = 0.01113/σ 2 + 299.71
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 35
Power System Dynamics and Stability
181
Example
PIM
= VL2 GIM = GIM
0.15 = ⇒σ
=
⇒ ZIM
=
YL
=
ZL
=
(0.01113/σ 2 + 299.71)(0.00087/σ 2 + 28.652/σ + 23.452) (0.00087/σ 2 + 28.652/σ + 23.452)2 + (0.19628/σ 2 + 486.73)2 0.0191 (by trial-and-error) 4.6221 + j3.0742 1 (G + jB) + ZIM 1 = 3.3654 + j2.1597 YL
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182
Power System Dynamics and Stability
Impedance Models
➠ Ignoring “fast” and “slow” transients, certain loads can be represented using an equivalent impedance.
➠ A Z load model is typically used for a variety of dynamic analysis of power systems.
➠ Using these load models, an equivalent impedance can be readily obtained for all loads connected at a particular bus at the transmission system level.
➠ ULTCs are used to connect distribution systems (subtransmission and LV systems and the loads connetced to these) to the transmission system to control the steady state voltage on the load side.
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183
Power System Dynamics and Stability
Power Models
➠ Hence, for “slow” dynamic analysis of balanced, fundamental frequency system models:
PL QL VL
2
VL = = PL0 VL0 2 VL 2 = VL BL = QL0 VL0 P L ≈ PL0 ≈ VL0 ⇒ QL ≈ QL0 VL2 GL
➠ This is typically referred as a constant P Q model.
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184
Power System Dynamics and Stability
Power Models
➠ Load recovery of certain loads (e.g. thermostatic) with respect to voltage changes can be modeled as:
= −
PL (t)
=
Nqs
y(t) VL (t) + QL0 Tq VL0 Nqt y(t) VL (t) + QL0 Tq VL0
= −
QL (t) =
Nps
VL (t) x(t) + PL0 Tp VL0 Npt VL (t) x(t) + PL0 Tp VL0
dx(t) dt
dy(t) dt
− PL0
− QL0
VL (t) VL0
Npt
VL (t) VL0
Nqt
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185
Power System Dynamics and Stability
Power Models
➠ This results in the following time response: P PL0
tf
t
tf
t
V VL0
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 40
186
Power System Dynamics and Stability
Example
➠ Identification of LD1 -LD4 loads at paper mill in Sweden: Net
894 MVA, 30 kV
1 2
LD1 LD2 3
5 LD3 LD4
4
6
G1
G2
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 41
Power System Dynamics and Stability
187
Example
➠ Measurements vs. model:
Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 42
188
Power System Dynamics and Stability
Power Models
➠ Certain loads have been shown to behave in steady state as follows: αP VL PL = KP VLαP fLβP ≈ PL0 VL0 αQ VL α β QL = KQ VL Q fLQ ≈ PL0 VL0 ➠ If f ≈ f0 , then the following approximation holds: αP VL PL ≈ PL0 VL0 αQ VL QL ≈ PL0 VL0 Universidad de Castilla-La Mancha June 26, 2008
Load Modeling - 43
189
Power System Dynamics and Stability
Power Models Load
αP
αQ
βP
βQ
Filament lamp
1.6
0
0
0
Fluorescent lamp
1.2
3.0
-0.1
2.8
Heater
2.0
0
0
0
Induction motor (half load)
0.2
1.6
1.5
-0.3
Induction motor (full load)
0.1
0.6
2.8
1.8
Reduction furnace
1.9
2.1
-0.5
0
Aluminum plant
1.8
2.2
-0.3
0.6
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Power System Dynamics and Stability
Power Models
➠ ZIP model: PL
= P LZ
QL
= QLZ
2
VL VL + P LI + P LP VL0 VL0 2 VL VL + QLI + QLP VL0 VL0
➠ Jimma’s model: PL
= P LZ
QL
= QLZ
2
VL VL + P LI + PL P VL0 VL0 2 VL dVL VL + QLI + QLP + KV VL0 VL0 dt Universidad de Castilla-La Mancha
June 26, 2008
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191
Power System Dynamics and Stability
Induction Motor Power Models
➠ Walve’s model: PL
QL
dVL = Kpf f + Kpv VL + T dt dVL ≈ PL0 + Kpv (VL − VL0 ) + T dt = Kqf f + Kqv VL ≈ QL0 + Kqv (VL − VL0 )
Universidad de Castilla-La Mancha June 26, 2008
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192
Power System Dynamics and Stability
Induction Motor Power Models
➠ Mixed model: PL
= ≈
QL
= ≈
dVL Kpf f + Kpv VLα + Tpv dt dVL PL0 + Kpv (VL − VL0 )α + Tpv dt dVL Kqf f + Kqv VLβ + Tqv dt dV L QL0 + Kqv (VL − VL0 )β + Tqv dt
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Load Modeling - 47
Power System Dynamics and Stability
193
Power Flow Outlines
➠ Power Flow: ➛ System model. ➛ Equations. ➛ Solution techniques: ➳ Newton-Raphson. ➳ Fast decoupled.
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194
Power System Dynamics and Stability
Power Flow Model
➠ The steady-state operating point of a power system is obtained by solving the “power flow” equations.
➠ Power flow system model corresponds to the steady state model. ➠ Generator: ➛ Generates and injects power P in the system while keeping the output voltage V constant within active and reactive power limits (capability curve):
Pmin ≤ P ≤ Pmax Qmin ≤ Q ≤ Qmax Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 2
Power System Dynamics and Stability
195
Power Flow Model
➠ Thus, it is modeled as a P V bus: P = constant Q = unknown
V = constant δ = unknown
➠ When Q reaches a limit it becomes a P Q bus: P = constant Q = constant
V = unknown δ = unknown
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
196
Power Flow Model
➠ Slack bus: ➛ The phasor model needs a reference bus. ➛ A “large” generator is typically chosen as the the reference bus, as it should be able to take the power “slack”: P = unknown Q = unknown
V = constant δ =0
Pslack =
X L
PL + Plosses −
X
PG
G
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Power Flow Outlines - 4
Power System Dynamics and Stability
197
Power Flow Model
➠ Hence, if Q reaches a limit: P = unknown Q = constant
V = unknown δ =0
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198
Power System Dynamics and Stability
Power Flow Model
➠ Load: ➛ Loads are typically connected to the transmission system through ULTC transformers.
➛ Thus, most loads in steady state represent a constant power demand in the system, and hence are modeled as a P Q bus: P = constant Q = constant
V = unknown δ = unknown
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199
Power System Dynamics and Stability
Power Flow Model
➠ Transmission system: ➛ AC transmission lines and transformers in steady state are basically modeled using the following model: Sk
Si
Vi
Z
Vk
Ik /a∠α
Ii
Ik Y1
Y2
a∠αVk
a∠α : 1
Universidad de Castilla-La Mancha June 26, 2008
Power Flow Outlines - 7
200
Power System Dynamics and Stability
Power Flow Model
➠ Hence: Si
= Vi I∗i
∗
1 +Vi Y1 = Vi (Vi − a∠αVk ) Z | {z } Y
= Vi (Y + Yi )∗ Vi∗ + (−a∠αY)∗ Vk∗ {z } | | {z } ∗ Yii
∗ Yik
➠ Similarly for Sk . Universidad de Castilla-La Mancha June 26, 2008
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201
Power System Dynamics and Stability
Power Flow Model
➠ Thus, for an N bus system interconnected through an ac transmission system, an N × N bus admittance matrix can be defined:
I Y 1 11 I Y 2 21 . . . . . . = I i Yi1 .. .. . . IN YN 1
I |{z}
node injections
= Ybus
Y12 Y22 .. .
Yi2 .. .
YN 2
···
··· ..
.
··· ..
.
···
Y1i Y2i .. .
Yii .. .
YN i
···
··· ..
.
··· ..
.
···
V |{z}
Y1N
V 1 Y2N V2 .. .. . . YiN Vi .. .. . . YN N VN
node voltages
Universidad de Castilla-La Mancha June 26, 2008
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202
Power System Dynamics and Stability
Power Flow Model
➠ where:
Ybus
P PN 1 Yii = k=1 Zik + Yi = sum of all the Y’s connected to node i = 1 Y = − ij Zij = negative of the Y between nodes i and j
Universidad de Castilla-La Mancha June 26, 2008
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203
Power System Dynamics and Stability
Power Flow Equations
➠ In steady state, a system with n generators G and m loads L can be modeled as: SG1
SL1 VG1
VL1 Transmission System
Ybus (N × N ) N =n+m
SGn
SLm VGn
VLm
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204
Power System Dynamics and Stability
Power Flow Equations
➠ Hence the power injections at each node are defined by: Si
= Pi + jQi = Vi I∗i = Vi
N X
∗ Yik Vk∗
k=1
= Vi ∠δi
N X
k=1
Si
=
S
Gi
−SL i
(Gij − jBij )Vk ∠ − δk for generator buses for load buses
Universidad de Castilla-La Mancha June 26, 2008
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205
Power System Dynamics and Stability
Power Flow Equations
➠ This yields two equations per node or bus: ∆Pi (δ, V, Pi ) = Pi −
N X
Vi Vk [Gik cos(δi − δk ) + Bik sin(δi − δk )] = 0
N X
Vi Vk [Gik sin(δi − δk ) − Bik cos(δi − δk )] = 0
k=1
∆Qi (δ, V, Qi ) = Qi −
k=1
➠ And two variables per node: ➛ PQ buses → Vi and δi
➛ PV buses → δi and Qi
➛ Slack buses → Pi and Qi Universidad de Castilla-La Mancha June 26, 2008
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206
Power System Dynamics and Stability
Power Flow Equations
➠ These equations are referred to as the power mismatch equations. ➠ The equations are typically subjected to inequality constraints representing control limits:
0.95 Qmini
≤ Vi ≤
1.05 for all buses
≤ Qi ≤ Qmaxi for generator buses
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
207
Power Flow Equations
➠ In summary, typical power flow data are as follows: Bus
Parameters
Variables
PQ
P, Q
V,θ
PV
P, V
Q, θ
slack
V,θ
P, Q
Universidad de Castilla-La Mancha June 26, 2008
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208
Power System Dynamics and Stability
Power Flow Equations
➠ In “constrained” power flow analysis, standard buses can degenerate if some limits is reached: Bus
Parameters
Variables
PQ
P , Vmax,min
Q, θ
PV
P , Qmax,min
V,θ
slack
Qmax,min , θ
P, V
➠ Continuation power flow techniques are by far more accurate and robust than standard power flow analysis if limits are taken into account.
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209
Power System Dynamics and Stability
Power Flow Equations
➠ The slack bus can be single or distributed. ➠ This refers to losses. ➛ For single slack bus model, all system losses are “cleared” by the slack bus.
➛ For distributed slack bus model losses are shared (equally or proportionally) among all or part of the generators:
➠ Continuation power flow techniques are by far more accurate and robust than standard power flow analysis if limits are taken into account.
Universidad de Castilla-La Mancha June 26, 2008
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210
Power System Dynamics and Stability
Power Flow Equations
➠ The distributed slack bus model is based on a generalized power center concept.
➠ This is practically obtained by including in power flow equations a variable kG and rewriting the system active power balance as follows: nG X i
(1 + kG γi )PGi −
nP X i
PLi − Plosses = 0
➠ The parameters γi allow tuning the weight of the participation of each generator to the losses.
➠ For single slack bus model, γi = 0 for all generators but one. Universidad de Castilla-La Mancha June 26, 2008
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211
Power System Dynamics and Stability
Power Flow Solution
➠ The power flow equations can be represented as F (z) = 0 ➠ There are 2 equations per bus, with 2 known variables and 2 unknown variables per bus; the problem is of dimension 2N . ➠ Since these equations are highly nonlinear due to the sine and cosine terms, Newton-Raphson (NR) based numerical techniques are used to solve them.
Universidad de Castilla-La Mancha June 26, 2008
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212
Power System Dynamics and Stability
Power Flow Solution
➠ A “robust” NR technique may be used to solve these equations: 1. Start with an initial guess, typically Vi0
= 1, δi0 = 0, Q0Gi = 0,
0 Pslack = 0 (flat start).
2. At each iteration k(k
= 0, 1, 2, . . .), compute the “sparse” Jacobian
matrix:
∂F k = Jk = ∂z z
∂F1 ∂z1 |z k
...
.. .
..
∂FN ∂z1 |z k
...
.
∂F1 ∂zN |z k
.. .
∂FN ∂zN |z k
2N ×2N
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213
Power System Dynamics and Stability
Power Flow Solution 3. Find ∆z k by solving the following linear set of equations (the sparse matrix Jk is factorized and not inverted to speed up the solution process):
Jk ∆z k = −F (z k ) 4. Computes the new guess for the next iteration, where α is a step control constant to guarantee convergence (0
< α < 1):
z k+1 = z k + α∆z k
Universidad de Castilla-La Mancha June 26, 2008
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214
Power System Dynamics and Stability
Power Flow Solution 5. Stop when:
kF (z k+1 )k = max |Fi (z k+1 )| ≤ ǫ ➠ This is basically the technique used in M ATLAB’s fsolve() routine, based on either numerical or actual Jacobians.
Universidad de Castilla-La Mancha June 26, 2008
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215
Power System Dynamics and Stability
Power Flow Solution
➠ If only the unknown bus voltage angles and magnitudes are calculated using NR’s method (the generator reactive powers and active slack power are evaluated later):
⇒
H N
Jk ∆z k = −F (z k ) M ∆P (δ k , V k ) ∆δ k = − L ∆V k /V k ∆Q(δ k , V k )
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
216
Power Flow Solution
➠ Where: H
=
M
=
N
=
L =
∂∆P ∂δ (δk ,V k ) ∂∆P V ∂V (δk ,V k ) ∂∆Q ∂δ (δk ,V k ) ∂∆Q V ∂V (δk ,V k )
Universidad de Castilla-La Mancha June 26, 2008
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217
Power System Dynamics and Stability
Power Flow Solution
➠ Assuming: ➛ (δi − δk ) < 10◦ , then cos(δi − δk )
≈ 1
sin(δi − δk )
≈ δi − δk
➛ The resistance in the transmission system are small, i.e. R ≪ X , then Gij ≪ Bik . ➠ The M and N matrices may be neglected, and: H M
≈ V k B′V k
≈ V k B ′′ V k
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218
Power System Dynamics and Stability
Power Flow Solution
➠ Thus the linear step equations may be decoupled and reduced to: B ′ ∆δ k B ′′ ∆V k
= −∆P (δ k , V k )/V k
= −∆Q(δ k , V k )/V k
where B ′′ is the imaginary part of the Ybus matrix, and B ′ is the imaginary part of the admittance matrix obtained by ignoring the system resistances, i.e.
B ′′ 6= B ′ .
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Power System Dynamics and Stability
Power Flow Solution
➠ The fast Decoupled iterative method is then defined as follows: 1. Start with an initial guess, typically δi0 2. Solve for ∆δ k
→ B ′ ∆δ k = −∆P (δ k , V k )/V k .
3. Update → δ k+1 4. Solve for ∆V k
5. Update →
= 0, Vi0 = 1.
= δ k + ∆δ k .
→ B ′′ ∆V k = −∆Q(δ k , V k )/V k .
V k+1 = V k + ∆V k .
6. Compute unknown generator powers and check for limits. 7. Repeat process for k
= 1, 2, . . ., until convergence.
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220
Power System Dynamics and Stability
Power Flow Solution
➠ This technique requires a relatively large number of iterations as compared to the robust NR method.
➠ It is significantly faster, as there is no need to re-compute and re-factorize the Jacobian matrix every iteration.
➠ It is sensitive to initial guesses and there is no guarantee of convergence, particularly for systems with large transmission system resistances.
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221
Power System Dynamics and Stability
Example
➠ For the following system: 1
2
All lines: 200 MVA 138 kV
P2
P1
X = 0.1 p.u. B = 0.2 p.u.
3
Q
200 MVA 0.9 p.f. lagging
V1 = 1, δ1 = 0, V2 = 1, V3 = 1, and P2 = P1 /2. ➠ Determine the voltage phasor angles δ2 and δ3 and the shunt Q by solving the PF equations.
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
222
Example
➠ The Ybus matrix is:
2 B +2 j = −j19.8 jX 2 1 Y12 = Y13 = Y23 = − = j10 jX −19.8 10 10 B11 B12 B13 Ybus = j 10 −19.8 10 = j B21 B22 B23 10 10 −19.8 B31 B32 B33 Y11
⇒
= Y22 = Y33 =
Universidad de Castilla-La Mancha June 26, 2008
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223
Power System Dynamics and Stability
Example
➠ Mismatch equation ∆P1 : ∆P1 0
= P1 − = P1 −
3 X
k=1 3 X
V1 Vk [G1k cos(δ1 − δk ) + B1k sin(δ1 − δk )] B1k sin(−δk )
k=1
= P1 + B12 sin(δ2 ) + B13 sin(δ3 ) = P1 + 10 sin(δ2 ) + 10 sin(δ3 )
Universidad de Castilla-La Mancha June 26, 2008
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224
Power System Dynamics and Stability
Example
➠ Mismatch equation ∆Q1 : ∆Q1 0
= Q1 − = Q1 +
3 X
k=1 3 X
V1 Vk [G1k sin(δ1 − δk ) − B1k cos(δ1 − δk )] B1k cos(−δk )
k=1
= Q1 − 19.8 + 10 cos(δ2 ) + 10 cos(δ3 )
Universidad de Castilla-La Mancha June 26, 2008
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225
Power System Dynamics and Stability
Example
➠ Mismatch equation ∆P2 : ∆P2 0
= P2 −
3 X
k=1
= P1 /2 −
V2 Vk [G2k cos(δ2 − δk ) + B2k sin(δ2 − δk )]
3 X
k=1
B2k sin(δ2 − δk )
= 0.5P1 + 10 sin(δ2 ) + 10 sin(δ2 − δ3 )
Universidad de Castilla-La Mancha June 26, 2008
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226
Power System Dynamics and Stability
Example
➠ Mismatch equation ∆Q2 : ∆Q2 0
= Q2 − = Q2 +
3 X
k=1 3 X
k=1
V2 Vk [G2k sin(δ2 − δk ) − B2k cos(δ2 − δk )] B2k cos(δ2 − δk )
= Q1 + 10 cos(δ2 ) − 19.8 + 10 cos(δ2 − δ3 )
Universidad de Castilla-La Mancha June 26, 2008
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227
Power System Dynamics and Stability
Example
➠ Mismatch equation ∆P3 : ∆P3 0
= P3 −
3 X
k=1
= −0.9 −
V3 Vk [G3k cos(δ3 − δk ) + B3k sin(δ3 − δk )]
3 X
k=1
B3k sin(δ3 − δk )
= −0.9 + 10 sin(δ3 ) + 10 sin(δ3 − δ2 )
Universidad de Castilla-La Mancha June 26, 2008
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228
Power System Dynamics and Stability
Example
➠ Mismatch equation ∆Q3 : ∆Q3 0
= Q3 − = Q−
3 X
k=1
p
V3 Vk [G3k sin(δ3 − δk ) − B3k cos(δ3 − δk )]
1 − 0.92 +
3 X
k=1
B3k cos(δ3 − δk )
= Q − 0.436 + 10 cos(δ3 ) + 10 cos(δ3 − δ2 ) − 19.8
Universidad de Castilla-La Mancha June 26, 2008
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229
Power System Dynamics and Stability
Example
➠ Thus 6 equations and 6 unknowns, i.e. δ2 , δ3 , P1 , Q1 , Q2 , and Q, can be solved using M ATLAB’s fsolve() routine: >> global lambda >> lambda = 1; >> z0 = fsolve(@pf_eqs,[0 0 0 0 0 0], optimset(’Display’,’iter’)) Norm of step
First-order Trust-region Iteration Func-count f(x) optimality radius 0 7 0.945696 18 1 1 14 0.000661419 0.705901 0.0205 1 2 21 9.98637e-18 0.0257331 2.52e-09 1.76 Optimization terminated: first-order optimality is less than options.TolFun. z0 = -0.0100
-0.0500
0.6000
-0.1870
-0.1915
0.2565
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
230
Example
➠ Where lambda is used to simulate constant power factor load changes and pf eqs.m is: function F = pf_eqs(z) global lambda d2 d3 P1 Q1 Q2 Q
= = = = = =
z(1); z(2); z(3); z(4); z(5); z(6);
F(1,1) F(2,1) F(3,1) F(4,1) F(5,1) F(6,1)
= = = = = =
P1 + 10*sin(d2) + 10*sin(d3); Q1 - 19.8 + 10*cos(d2) + 10*cos(d3); 0.5*P1 - 10*sin(d2) - 10*sin(d2-d3); Q2 + 10*cos(d2) - 19.8 + 10*cos(d2-d3); -0.9*lambda - 10*sin(d3) - 10*sin(d3-d2); Q - 0.436*lambda + 10*cos(d3) + 10*cos(d3-d2) - 19.8; Universidad de Castilla-La Mancha
June 26, 2008
Power Flow Outlines - 38
231
Power System Dynamics and Stability
Example
➠ Observe that as the load (lambda) is increased, convergence problems show up: lambda = 20; z0 = fsolve(@pf_eqs,[0 0 0 0 0 0], optimset(’Display’,’iter’)) Norm of step
First-order optimality 360 143
Trust-region radius 1 1
Iteration Func-count f(x) 0 7 396.67 1 14 246.008 1 . . . 9 70 2.68094e-07 0.179729 0.000311 25.9 10 77 7.43614e-16 0.00127159 1.64e-08 25.9 Optimization terminated: first-order optimality is less than options.TolFun. z0 = -0.2501
-1.2613
12.0000
7.0658
4.8031
20.1667
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Example
➠ Convergence problems develop until the equations fail to converge: lambda = 22; z0 = fsolve(@pf_eqs,[0 0 0 0 0 0], optimset(’Display’,’iter’)) Norm of step
First-order optimality 396 168
Trust-region radius 1 1
Iteration Func-count f(x) 0 7 480.33 1 14 310.93 1 . . . 90 595 0.00622095 0.0211144 0.0322 0.0211 91 602 0.00621755 0.0211144 0.0155 0.0211 Maximum number of function evaluations reached: increase options.MaxFunEvals. z0 = -0.3322
-1.7225
13.1600
11.8633
8.5479
29.1072
Universidad de Castilla-La Mancha June 26, 2008
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233
Stability Concepts Outlines
➠ Basic stability concepts ➛ Nonlinear systems: ➳ Ordinary differential equations (ODE) ➳ Differential algebraic equations (DAE) ➛ Equilibrium points: ➳ Definitions. ➳ Stability: ☞ Linearization. ☞ Eigenvalue analysis. ➛ Stability regions.
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Power System Dynamics and Stability
234
ODE Systems
➠ Nonlinear systems are represented by a nonlinear set of differential equations:
x˙ = s(x, p, λ) where
➳ x → n state variables (e.g. generator angles)
➳ p → k controllable parameters (e.g. compensation) ➳ λ → ℓ uncontrollable parameters (e.g. loads)
➳ s(·) → n nonlinear functions (e.g. generator equations)
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Power System Dynamics and Stability
ODE Systems
➠ For example, for a simple generator-system model: PG + jQG
Generator
jx′G
jxL
V1 ∠δ1
E ′ ∠δ
PL + jQL
System
jxth
V2 ∠δ2
V ∠0 Infinite bus (M
= ∞)
AVR
The generator is modeled as a simple d axis transient voltage behind transient reactance.
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236
ODE Systems
➠ The generator has only the mechanical dynamics: δ˙ ω˙
= ω = ωr − ω0 1 (PL − PG − Dω) = M
where
PG
= =
E′V sin δ x′G + xL + xth V1 V sin δ1 xL + xth
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Power System Dynamics and Stability
237
ODE Systems
➠ If the AVR is modeled, V1 may be assumed to be kept constant by varying E ′ , with the generator’s reactive power within limits: V12 V1 V QG = − cos δ1 xL + xth xL + xth ⇒
QG min ≤ QG ≤ QG max
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238
ODE Systems
➠ If the AVR is not modeled: ➛ x = [δ, ω]T → state variables
➛ p = [E ′ , V ]T → controlled parameters ➛ λ = PL → uncontrolled parameters
➛ Hence, assuming p = [1.5, 1]T , M = D = 0.1, and x = 0.75: x˙ = x 1 2 s(x, p, λ) = x˙ 2 = 10λ = 20 sin x1 − x2
➠ These are basically the pendulum equations.
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Power System Dynamics and Stability
239
DAE Systems
➠ Differential Algebraic Equation (DAE) models are defined as: x˙ = f (x, y, p, λ) 0
= g(x, y, p, λ)
where:
➛ y → m algebraic variables (e.g. load voltages)
➛ f (·) → n nonlinear differential equations (e.g. generator equations) ➛ g(·) → m nonlinear algebraic equations (e.g. reactive power equations)
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Power System Dynamics and Stability
240
DAE Systems
➠ For example, for the generator-infinite bus example with AVR, for QG min ≤ QG ≤ QG max : δ˙ ω˙ 0 0 0
= ω 1 E′V = PL − sin δ − Dω M x V1 V E′V sin δ1 sin δ − = x xL + xth V1 E ′ V12 = −QG − ′ + ′ cos(δ1 − δ) xG xG V12 V1 V = QG − + cos δ1 xL + xth xL + xth Universidad de Castilla-La Mancha
June 26, 2008
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Power System Dynamics and Stability
241
DAE Systems
➠ Thus, for: ➛ x = [δ, ω]T ➛ y = [E ′ , δ1 , QG ]T ➛ p = [V1 , V ]T = [1, 1]T ➛ λ = PL ➛ M = D = 0.1, x = 0.75, x′G = 0.25, xL + xth = 0.5 x˙ = x 1 2 f (x, y, p, λ) = x˙ 2 = 10λ − 13.33y1 sin x1 − x2 0 = 1.333y1 sin x1 − 2 sin y2 g(x, y, p, λ) = 0 = −y3 − 4 + 4y1 cos(y2 − x1 ) 0 = y3 − 2 + 2 cos y2 Universidad de Castilla-La Mancha
June 26, 2008
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Power System Dynamics and Stability
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DAE Systems
➠ If QG = QG max or QG = QG min : ➛ x = [δ, ω]T ➛ y = [E ′ , δ1 , V1 ]T ➛ p = [QG , V ]T = [±0.5, 1]T ➛ λ = PL ➛ M = D = 0.1, x = 0.75, x′G = 0.25, xL + xth = 0.5 x˙ = x 1 2 f (x, y, p, λ) = x˙ 2 = 10λ − 13.33y1 y3 sin x1 − x2 0 = 1.333y1 sin x1 − 2y3 sin y2 g(x, y, p, λ) = 0 = ∓0.5 − 4y32 + 4y1 y3 cos(y2 − x1 ) 0 = ±0.5 − 2y32 + 2y3 cos y2 Universidad de Castilla-La Mancha
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➠ If the Jacobian matrix Dy g = [∂gi /∂yi ]m×m is nonsingular, i.e. invertible, along the trajectory solutions, the system can be transformed into an ODE system (Implicit Function Theorem):
y ⇒
= h(x, p, λ)
x˙ = f (x, h(x, p, λ), p, λ) = s(x, p, λ)
➠ In practice, this is a purely “theoretical” exercise that is not carry out due to its complexity.
➠ The system model should be revised when Dy g is singular.
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➠ For the ODE system, equilibria are defined as the solution x0 for given parameter values p0 and λ0 of the set of equations s(x0 , p0 , λ0 ) = 0 ➠ There are multiple solutions to this problem, i.e. multiple equilibrium points.
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➠ The “stability” of these equilibria is defined by linearizing the nonlinear system around x0 , i.e. ∂si (x0 , p0 , λ0 ) x − x0 ∆x˙ = | {z } ∂xj n×n ∆x
= Dx s|0 ∆x
➠ where Dx s|0 = Dx s(x0 , p0 , λ0 ) = ∂s/∂x|0 is the system Jacobian matrix evaluated at the equilibrium point.
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➠ From linear system theory, the linear system stability is defined by the eigenvalues µi of the Jacobian matrix Dx s|0 , which are defined as the solutions of the equation:
⇒
Dx s|0 v
= µv
right e-vector
Dx s|T0 w
→
= µw
→
left e-vector
det(Dx s|0 − µIn ) = 0
an µn + an−1 µn−1 + . . . + a1 µ + a0 = 0
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➠ There are n complex eigenvalues, left and right eigenvectors associated with the system Jacobian Dx s|0 . ➠ In practice, these eigenvalues are not computed using characteristic polynomial but other more efficient numerical techniques, as the costs associated with computing these values is rather large in realistic power systems.
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➠ These eigenvalues define the small perturbation stability of the ODE system, i.e. the “local” stability of the nonlinear system near the equilibrium points:
➛ Stable equilibrium point (s.e.p.): The system is locally stable about x0 if all the eigenvalues µi (Dx s|0 ) are on the left-half (LH) of the complex plane.
➛ Unstable equilibrium point (u.e.p.): The system is locally unstable about x0 if at least one eigenvalue µi (Dx s|0 ) is on the right-half (RH) of the complex plane.
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➠ The equilibrium point x0 is a bifurcation point if at least one eigenvalue µi (Dx s|0 ) is on the imaginary axis of the complex plane. ➠ Some systems have equilibria with eigenvalues on the imaginary axis without these being bifurcation points; for example, a lossless generator-infinite bus system with no damping.
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➠ For example, for the simple generator-infinite bus example with no AVR: x˙ 1
= x2
x˙ 2
= 10λ − 20 sin x1 − x2
the equilibrium points can be found from the steady-state (power flow) equations:
0
= x20
0
= 10λ − 20 sin x10 − x20
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➠ which leads to the solutions: sin−1 (λ/2) x10 = 0 x20
⇒
sin−1 (PL /2) δ0 = 0 ω0
➠ This yields basically three equilibrium points (other solutions are just “repetitions” of these three):
➛ s.e.p → −π/2 < x1s < π/2 ➛ u.e.p.1 → x1u1 = x1s + π ➛ u.e.p.2 → x1u2 = x1s − π
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➠ This corresponds to the intersections of PG = E ′ V /x sin δ with PL : Stable
Bifurcation
PG
EV /x
Unstable
Unstable
PL
δµ2 (xµ2 )
−π
π δs (xs )
π/2
δµ1 (xµ1 )
δ, (x1 )
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➠ The stability of these equilibria is determined using the system Jacobian: ∂s1 /∂x1 |0 ∂s1 /∂x2 |0 Dx s|0 = ∂s2 /∂x1 |0 ∂s2 /∂x2 |0 0 1 = −20 cos x10 −1
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⇒
det(Dx s|0 − µI2 )
−µ = det −20 cos x10
1
−1 − µ
= µ2 + µ + 20 cos x10 = 0 ⇒ PL = 1
µ1,2 ⇒
1 1√ 1 − 80 cos x10 =− ± 2 2 µ1,2 (Dx s|xs ) = −0.5 ± j4.132 3.192 µ1,2 (Dx s|xu1 /u2 ) = −5.192
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➠ For this system, the equilibria are: ➛ stable if ∂PG >0 ∂δ ➛ unstable if ∂PG <0 ∂δ ➛ bifurcation point for ∂PG =0 ∂δ
⇒
δ = π/2
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➠ For DAE systems, the equilibria z0 = (x0 , y0 ) for parameter values p0 and λ0 are defined as the solution to the nonlinear, steady state problem: f (x , y , p , λ ) = 0 0 0 0 0 F (x0 , y0 , p0 , λ0 ) = 0 g(x0 , y0 , p0 , λ0 ) = 0 ➠ In this case, the linearization about (x0 , y0 ) yields:
∆x˙ = Dx f |0 ∆x + Dy f |0 ∆y ∆0
=
Dx g|0 ∆x + Dy g|0 ∆y
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➠ Hence, by eliminating ∆y from these equations, one obtains: Dx s|0 = Dx f |0 − Dy f |0 Dy g|−1 0 Dx g|0 ➠ Observe that, as mentioned before, the nonsingularity of the Jacobian Dy g|0 in this case is required. ➠ The same local stability conditions apply in this case based on the eigenvalues of Dx s|0 .
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➠ For the generator-infinite bus example with AVR example (within QG limits), the steady-state solutions are obtained from solving the steady-state or power flow equations:
0
= x20
0
= 10λ − 13.33y10 sin x10 − x20
0
= 1.333y10 sin x10 − 2 sin y20
0
= −y30 − 4 + 4y10 cos(y20 − x10 )
0
= y30 − 2 + 2 cos y20
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➠ Which in M ATLAB format are: function f = dae_eqs(z) global lambda x10 x20 y10 y20 y30
= = = = =
z(1); z(2); z(3); z(4); z(5);
f(1,1) f(1,2) f(1,3) f(1,4) f(1,5)
= = = = =
x20; 10*lambda - 13.33 * y10 * sin(x10) - x20; 1.333 * y10 * sin (x10) - 2 * sin(y20); -y30 - 4 + 4 * y10 * cos(y20 - x10); y30 - 2 + 2 * cos(y20);
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➠ This generates the equilibrium point for λ = PL = 1: >> lambda = 1; >> z0 = fsolve(@dae_eqs,[0 0 1 0 1],optimset(’Display’,’iter’)) Norm of step
Iteration Func-count f(x) 0 6 102 1 12 1.52385 1 2 18 0.0050436 0.298687 3 24 2.72816e-05 0.0607967 4 30 2.0931e-13 0.000413599 5 36 4.98474e-28 5.13313e-08 Optimization terminated: first-order optimality
First-order Trust-region optimality radius 133 1 10.2 1 0.219 2.5 0.0534 2.5 4.89e-06 2.5 2.23e-13 2.5 is less than options.TolFun.
z0 = 0.7539
0
1.0959
0.5236
0.2679
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➠ And the symbolic Jacobian matrices: syms x1 x2 y1 y2 y3 real z = [x1 x2 y1 y2 y3]; F = dae_eqs(z); f = F(1:2); g = F(3:5); Dxf Dyf Dxg Dyg
= = = =
jacobian(f,[x1,x2]) jacobian(f,[y1,y2,y3]) jacobian(g,[x1,x2]) jacobian(g,[y1,y2,y3])
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➠ And the symbolic Jacobian matrices: Dxf = [ 0, [ -1333/100*y1*cos(x1), Dyf = [ 0, [ -1333/100*sin(x1),
1] -1]
0, 0,
Dxg = [ 1333/1000*y1*cos(x1), [ -4*y1*sin(-y2+x1), [ 0, Dyg = [ 1333/1000*sin(x1), [ 4*cos(-y2+x1), [ 0,
-2*cos(y2), 4*y1*sin(-y2+x1), -2*sin(y2),
0] 0]
0] 0] 0]
0] -1] 1]
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➠ These generate the following eigenvalues at the equilibrium point: Dxf = [ 0, [ -1333/100*y1*cos(x1), Dyf = [ 0, [ -1333/100*sin(x1),
1] -1]
0, 0,
Dxg = [ 1333/1000*y1*cos(x1), [ -4*y1*sin(-y2+x1), [ 0, Dyg = [ 1333/1000*sin(x1), [ 4*cos(-y2+x1), [ 0,
-2*cos(y2), 4*y1*sin(-y2+x1), -2*sin(y2),
0] 0]
0] 0] 0]
0] -1] 1]
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➠ These generate the following eigenvalues at the equilibrium point: x1 = z0(1); x2 = z0(2); y1 = z0(3); y2 = z0(4); y3 = z0(5); A = vpa(subs(Dxf),5); B = vpa(subs(Dyf),5); C = vpa(subs(Dxg),5); D = vpa(subs(Dyg),5); Dxs = A - B * inv(D) * C; ev = vpa(eig(Dxs),5) ev = -.50000+3.5698*i -.50000-3.5698*i
➠ Hence, this is a s.e.p. Universidad de Castilla-La Mancha June 26, 2008
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➠ A u.e.p. can be computed as well from these equations (neglecting QG limits): >> z0 = fsolve(@dae_eqs,[3 0 0.1 3 3],optimset(’Display’,’iter’)); z0 = 2.7465 >> >> >> >> >> >> >>
0
1.9491
2.6180
3.7321
x1 = z0(1); x2 = z0(2); y1 = z0(3); y2 = z0(4); y3 = z0(5); A = vpa(subs(Dxf),5); B = vpa(subs(Dyf),5); C = vpa(subs(Dxg),5); D = vpa(subs(Dyg),5); Dxs = A - B * inv(D) * C; ev = vpa(eig(Dxs),5)
ev = 4.2892 -5.2892 Universidad de Castilla-La Mancha June 26, 2008
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➠ Associated with every s.e.p. xs there is a stability region A(xs ), which basically corresponds to the region of system variables that are all attracted to xs , i.e.
x(t → ∞) → xs : x1 (0) A(xs ) xs
stable
x1 (t) ∂A(xs ) (stability region
x2 (0)
unstable
boundary)
x2 (t)
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➠ if A(xs ) is known, the stability of a system for large perturbations can be readily evaluated.
➠ However, determining this region is a rather difficult task. ➠ This can realistically be accomplished only for 2- or 3-dimendional systems using “sophisticated” nonlinear system analysis techniques.
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Equilibria λ = 0.2
10
5
x
x
xs
u2
u1
x
2
0
−5 ∂A(x ) s
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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10
5
x
x
u2
x
s
u1
x
2
0
−5
−10
∂A(x ) s
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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10
5
x
x
u2
x
s
u1
x
2
0
−5 ∂A(x ) s
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 1.5
10
5
x
xs
u1
x
2
0
∂A(x ) s
−5
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 1.8
10
5
x
u1
0 x
2
x
−5
s
∂A(x ) s
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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Equilibria λ = 1.9
10
5
x 0 x
2
x
u1
s
∂A(x ) s
−5
−10
−15 −10
−8
−6
−4
−2
0 x1
2
4
6
8
10
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➠ In real systems, trial-and-error techniques are usually used: ➛ A contingency yields a given initial condition x(0). ➛ For the post-contingency system, the time trajectories x(t) can be computed by numerical integration.
➛ If x(t) converges to the post-contingency equilibrium point xs , the system is stable, i.e. x(0) ∈ A(xs ). ➛ If it diverges, the system is unstable.
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Voltage Stability Outlines
➠ Definitions. ➠ Basic concepts. ➠ Continuation Power Flow. ➠ Direct Methods. ➠ Indices. ➠ Controls and protections. ➠ Practical applications. ➠ Examples.
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➠ IEEE-CIGRE classification (IEEE/CIGRE Joint Task Force on Stability) Terms and Definitions, “Definitions and Classification of Power System Stability”, IEEE Trans. Power Systems and CIGRE Technical Brochure 231, 2003:
Power System Stability
Rotor Angle
Frequency
Voltage
Stability
Stability
Stability
Small Disturbance
Transient
Angle Stability
Stability
Large Disturbance Voltage Stability
Short Term
Short Term Short Term
Small Disturbance Voltage Stability
Long Term
Long Term
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➠ “Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact.”
➠ “Voltage stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition.”
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Voltage Stability Definitions
➠ The inability of the trnasmission system to supply the demand leads to a “voltage collapse” problem.
➠ This problem is associated with the “disappearance” of a stable equilibrium point due to a saddle-node or limit-induced bifurcation point, typically due to contingencies in the system (e.g. August 14, 2003 Northeast Blackout).
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Voltage Stability Concepts
➠ The concept of saddle-node and limit induced bifurcations can be readily explained using the generator-load example: PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
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➠ Neglecting for simplicity losses, electromagnetic dynamics, and the transient impedance in the d-axis transient model, the generator can be simulated with:
δ˙1 ω˙ 1
= ω1 = ωr − ω0 1 = (Pm − PG − DG ω1 ) M
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➠ The load can be simulated using the mixed models. ➠ For P , neglecting voltage dynamics (Tpv = 0) and voltage dependence (α = 0):
⇒
PL
= Kpf f2 + Kpv [V2α + Tpv V˙ 2 ]
PL
= P d + DL ω 2 1 (PL − Pd ) = ω2 = DL
δ˙2
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➠ For Q, neglecting frequency dependence (Kqf = 0) and voltage dependence (β = 0) QL ⇒
QL V˙ 2
= Kqf f2 + Kqv [V2β + Tqv V˙ 2 ] = Qd + τ V˙ 2 =
1 (QL − Qd ) τ
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➠ The transmission system yields the power flow equations: PL PG QL QG
V1 V2 sin(δ2 − δ1 ) XL V1 V2 sin(δ1 − δ2 ) = XL V1 V2 V22 + cos(δ2 − δ1 ) = − XL XL V12 V1 V2 = − cos(δ1 − δ2 ) XL XL = −
➠ Finally, since the system is lossless: P m = Pd Universidad de Castilla-La Mancha June 26, 2008
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➠ Hence, defining: δ ω ⇒ δ˙
= δ1 − δ2 = ω1 = ω − ω2
the system equations are:
ω˙ δ˙ V˙ 2
1 M
V1 V2 sin δ − DG ω Pd − XL V1 V2 1 sin δ − Pd = ω− DL XL 2 1 V1 V2 V = cos δ − Qd − 2 + τ XL XL =
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Voltage Stability Concepts
➠ Observe that these equations also represent a generator-dynamic load ′ , where V1 would stand system with no AVR and with XL including XG ′ . for EG ➠ The steady-state load demand may be assumed to have a constant power factor, i.e.
Qd = kPd ′ , ➠ If generator reactive power limits are considered, and neglecting XG
one has a DAE system.
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➠ For QGmin ≤ QG ≤ QGmax : 1 V10 V2 ω˙ = sin δ − DG ω Pd − M XL V10 V2 1 sin δ − Pd δ˙ = ω − DL XL 2 V V V 1 10 2 − 2 + cos δ − Qd V˙ 2 = τ XL XL 2 V10 V2 V10 + cos δ 0 = QG − − XL XL
with
x = [ω, δ, V2 ]T
y = QG
p = V10
λ = Pd
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➠ For QG = QGmin,max : ω˙ δ˙ V˙ 2 0
V1 V2 sin δ − DG ω Pd − = XL V1 V2 1 sin δ − Pd = ω− DL XL 2 V V1 V2 1 − 2 + cos δ − Qd = τ XL XL 1 M
= QGmin,max
V12 V1 V2 − + cos δ XL XL
with
x = [ω, δ, V2 ]T
y = V1
p = QGmin,max
λ = Pd
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➠ All the equilibrium point for the system with and without limits can be obtained solving the power flow equations:
0 0 0
V10 V20 sin δ0 XL 2 V20 V10 V20 − cos δ0 = kPd + XL XL 2 V10 V10 V20 = QG0 − + cos δ0 XL XL = Pd −
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➠ And the stability of these equilibrium points come from the state matrix: ➠ Without limits, or for QGmin ≤ QG ≤ QGmax : DG V10 V20 V10 − M XL sin δ0 − M − M XL cos δ0 V V V 10 20 10 Dx s|0 = 1 − DL XL cos δ0 − DL XL sin δ0 0 − Vτ10XVL20 sin δ0 −2 τVX20L + τVX10L cos δ0
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Voltage Stability Concepts
➠ For QG = QGmin,max : Dx s|0 = DG V10 V20 V10 − M − M XL cos δ0 − M XL sin δ0 V10 V20 V10 1 − cos δ − sin δ 0 0 DL XL DL XL 0 − Vτ10XVL20 sin δ0 −2 τVX20L + τVX10L cos δ0 T V10 0 − M XL sin δ0 −1 V10 V20 V V V 10 20 10 + cos δ0 − − D X sin δ0 −2 − sin δ 0 XL L L τ XL τ XL V20 V10 cos δ 0 τ XL XL cos δ0 Universidad de Castilla-La Mancha June 26, 2008
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Voltage Stability Concepts
➠ Assume XL = 0.5, M = 1, DG = 0.01, DL = 0.1, τ = 0.01, k = 0.25. ➠ With the help of M ATLAB and the continuation power flow routine of PSAT, for the system without limits and V1 = V10 = 1, the power flow solutions yield a “PV” or “nose” curve (bifurcation diagram):
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s.e.p.
0.9 0.8
x3 = V2
0.7
SNB
0.6 0.5 0.4 0.3 0.2 0.1 0
u.e.p. 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
λ = Pd
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Voltage Stability Concepts
➠ The saddle-node bifurcation point SNB corresponds to a point where the state matrix Dx s|0 is singular (one zero eigenvalue). ➠ This is typically associated with a power flow solution with a singular PF Jacobian Dz F |0 . ➠ This is not always the case, as for more complex dynamic models, the singularity of the state matrix does not necessarily correspond to a singularity o the PF Jacobian, and vice versa.
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ Observe that the SNB point corresponds to a maximum value λmax = Ps max ≈ 0.78, which is why is also referred to as the maximum loading or loadability point.
➠ For a load greater than Pd max , there are no PF solutions. ➠ This point is also referred to as the voltage collapse point.
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ Modeling a “contingency” at λ0 = Pd0 = 0.7 by increasing XL = 0.5 → 0.6: 1
operating point
0.9 0.8
x3 = V2
0.7 0.6
contingency
0.5 0.4 0.3 0.2
XL = 0.5 XL = 0.6
0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ The dynamic solution yields a voltage collapse: 6
δ ω V2 V1
5
4
3 Contingency
Operating point
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ For the system with limits and QGmax,min = ±0.5, the “PV” or “nose” curve is:
1 0.9 0.8
LIB
s.e.p.
x3 = V2
0.7 0.6
u.e.p.
0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ In this case, the maximum loading or loadability point λmax = Pd max ≈ 0.65 corresponds to the point where the generator reaches its maximum reactive power limit QG = QG max = 0.5, and hence losses control of V1 . ➠ This is referred to a limit-induced bifurcation or LIB point. ➠ “Beyond” the LIB point, there are no more power flow solutions, due to the limit recovery mechanism of the AVR.
➠ In this case the LIB point is also a voltage collapse point.
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ Modeling a “contingency” at λ0 = Pd0 = 0.6 by increasing XL = 0.5 → 0.6: 1 0.9
operating point
0.8
contingency
x3 = V2
0.7 0.6 0.5 0.4 0.3 0.2
XL = 0.5 XL = 0.6
0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ The dynamic solution also yields a voltage collapse: 6
δ ω V2 V1
5
4
3 Operating point
Contingency
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ Collapse problems can be avoided using shunt or series compensation: PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
−jxC
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ In this case, the system equations are: ➠ For QGmin ≤ QG ≤ QGmax : ω˙ δ˙ V˙ 2 0
1 (Pd − V10 V2 BL sin δ − DG ω) M 1 (V10 V2 BL sin δ − Pd ) = ω− DL 1 = [−V22 (BL − BC ) + V10 V2 BL cos δ − kPd ] τ 2 = QG − V10 BL + V10 V2 BL cos δ =
where
BL =
1 XL
BC =
1 XC
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ For QG = QGmin,max : ω˙ δ˙ V˙ 2 0
1 (Pd − V10 V2 BL sin δ − DG ω) M 1 (V10 V2 BL sin δ − Pd ) = ω− DL 1 = [−V22 (BL − BC ) + V10 V2 BL cos δ − kPd ] τ 2 BL + V10 V2 BL cos δ = QGmin,max − V10 =
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304
Power System Dynamics and Stability
Voltage Stability Concepts
➠ These equations generate the PV curves: ➠ Without limits: 1.5
XL = 0.6
XL = 0.5
V2
1
XL = 0.6 0.5
Bc = 0 Bc = 0 Bc = 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ = Pd Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ With limits: 1.5
XL = 0.6 XL = 0.5
V2
1
XL = 0.6 0.5
Bc = 0 Bc = 0 Bc = 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ = Pd
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Power System Dynamics and Stability
Voltage Stability Concepts
➠ Applying compensation at t = 1.3 s without limits:
1.2 1
0.8
δ ω V2 V1
Apply compensation
0.6 Contingency 0.4
0.2 Operating point 0
−0.2
0
1
2
3
4
5
6
7
8
9
10
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307
Power System Dynamics and Stability
Voltage Stability Concepts
➠ Applying compensation at t = 1.3 s with limits:
1.2 1
0.8
δ ω V2 V1
Apply compensation
0.6 Contingency 0.4
0.2 Operating point 0
−0.2
0
1
2
3
4
5
6
7
8
9
10
t [s] Universidad de Castilla-La Mancha June 26, 2008
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308
Power System Dynamics and Stability
Continuation Power Flow
➠ These PV or nose curves are obtained using a continuation power flow. ➠ This technique “traces” the solutions of the power equations F (z, p, λ) = 0 as λ changes.
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Power System Dynamics and Stability
Continuation Power Flow
➠ The algorithm steps are: 1. Predictor
(∆z1 , ∆λ1 )
3. Corrector
(z0 , λ0 ) λ2
2. Parametrization
λ1
z2
λ z1
λ
➠ These methods “guarantee” convergence.
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310
Power System Dynamics and Stability
CPF Predictor
➠ Tangent vector method: ∂F dz − Dz F |1 dλ ∂λ 1 |{z}
⇒
∆z1 =
t1
k t1 kt1 k |{z} ∆λ1
(z2 , λ2 ) (∆z1 , ∆λ1 )
(z1 , λ1 )
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Power System Dynamics and Stability
CPF Predictor
➠ Good method to follow “closely” the PV curves, but relatively slow. ➠ The tangent vector t defines “sensitivities” at any power flow solution point.
➠ This vector can also be used as an index to predict proximity to a saddle-node bifurcation, as opposed to using the smallest eigenvalue, which changes in highly non linear fashion.
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312
CPF Predictor
➠ Secant method: ∆z1
= k(z1a − z1b )
∆λ1
= k(λ1a − λ1b ) (z2 , λ2 )
(∆z1 , ∆λ1 ) (z1b , λ1b ) (z1a , λ1a )
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Power System Dynamics and Stability
CPF Predictor
➠ This method is faster but can have convergence problems with “sharp corners” (e.g. limits):
(z2 , λ2 ) (z1b , λ1b ) (z1a , λ1a )
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Power System Dynamics and Stability
CPF Parametrization
➠ Used to avoid singularities during the predictor step. ➠ Methods: ➛ Local: interchange a zi ∈ z with λ, i.e. “rotate” the PV curve.
➛ Arc length (s): assume z1 (s) and λ1 (s); thus solve for ∆z1 and ∆λ1 : ∂F Dz F |1 ∆z1 + + ∆λ1 = 0 ∂λ 1 ∆z1T ∆z1 + ∆λ21
= k
➠ No real need for parametrization in practice if step cutting is needed.
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Power System Dynamics and Stability
CPF Corrector
➠ The idea is to add an equation to the equilibrium equations, i.e. solve for (z, λ) at a given point p: F (z, p, λ) = 0 ρ(z, λ) = 0 ➠ These equations are nonsingular for the appropriate choice of ρ(·).
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Power System Dynamics and Stability
CPF Corrector
➠ Perpendicular intersection technique: (z1 , λ1 ) (∆z1 , ∆λ1 ) z (z2 , λ2 )
λ
F (z, p, λ) = 0 (z1 + ∆z1 − z)T ∆z1 + (λ1 + ∆λ1 − λ)∆λ1 = 0 Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
317
CPF Corrector
➠ Local parametrization technique: (z1 , λ1 ) (∆z1 , ∆λ1 ) z
(z2 , λ2 )
λ
F (z, p, λ) = 0 λ = λ1 + ∆λ1 or zi = zi1 + ∆zi1 Universidad de Castilla-La Mancha June 26, 2008
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318
Power System Dynamics and Stability
Direct Methods
➠ Allow to directly find the maximum loading point (saddle-node or limit induced bifurcation).
➠ This problem can be set up as an optimization problem:
Max. s.t.
λ F (z, p, λ) = 0 zmin ≤ z ≤ zmax pmin ≤ p ≤ pmax
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Power System Dynamics and Stability
Direct Methods
➠ If limits are ignored, the solution for given values of the control paramters (p = p0 ) to the associated optimization problem, based on the Lagrangian and KKT conditions, is given by:
F (z, p0 , λ)
0
→
DzT F (z, p0 , λ)w
= 0
DλT F (z, p0 , λ)w
= −1
steady stae solution
→ →
singularity condition
w 6= 0 condition
➠ This yields a saddle-node bifurcation point, and corresponds to the “left eigenvector” (w ) saddle-node equations.
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Power System Dynamics and Stability
Direct Methods
➠ The solution to these nonlinear equations do not converge if the maximum loading point is a limit-induced bifurcation.
➠ In this case, the otpimization problem Max. s.t.
λ F (z, p0 , λ) = 0 zmin ≤ z ≤ zmax
must be solved using “standard” optimization techniques (e.g. Interior Point Method).
➠ The solution to this optimization problem may be a saddle-node or limit-induced bifurcation. Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Direct Methods
➠ Observe that if the control parameters p are allowed to change, the problem is transformed into a maximization of the maximum loading margin.
➠ Other optimization problems can be set up to not only maximize/guarantee loading margins but at the same time minimize costs (e.g. maximize social welfare).
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Power System Dynamics and Stability
Direct Methods
➠ For example, for an OPF-based auction in electricity markets, the following multi-objective optimization problem can be posed: Min. s.t.
T G = − ω1 (CD PD − CST PS ) − ω2 λc
f (δ, V, QG , PS , PD ) = 0
→ PF equations
f (δc , Vc , QGc , λc , PS , PD ) = 0 → Max load PF eqs. λcmin ≤ λc ≤ λcmax
→ loading margin
0 ≤ PS ≤ PSmax
→ Sup. bid blocks
0 ≤ PD ≤ PDmax
→ Dem. bid blocks
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Power System Dynamics and Stability
Direct Methods
➠ With the phyiscal and security limits: Iij (δ, V ) ≤ Iijmax
→ Thermal limits
Iji (δ, V ) ≤ Ijimax Iij (δc , Vc ) ≤ Iijmax Iji (δc , Vc ) ≤ Ijimax QGmin ≤ QG ≤ QGmax → Gen. Q limits QGmin ≤ QGc ≤ QGmax Vmin ≤ V ≤ Vmax
→ V “security” lim.
Vmin ≤ Vc ≤ Vmax Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
Direct Methods
➠ More information about this probem can be found in: ˜ F. Milano, C. A. Canizares and M. Invernizzi, “Multi-objective Optimization for PRicing System Security in Electricity Marktes”, IEEE Trans. on Power Systems, Vol. 18, No. 2, May 2003, pp. 596-604.
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325
Power System Dynamics and Stability
Indices
➠ Indices have been developed with the aim of monitoring proximity to voltage collapse for on-line applications.
➠ For example, the minimum real eigenvalue of the system Jacobian can be used to “measure” proximity to a saddle-node bifurcation, since this matrix becomes singular at that point.
➠ Many indices have been proposed, but the most “popular/useful” are: ➛ Singular value. ➛ Reactive power reserves.
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326
Indices
➠ The singular value index consists of simply monitoring the singular value of the Jacobian of the power flow equations as λ changes, e.g.
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Power System Dynamics and Stability
327
Indices
➠ Observations: ➛ Computationally inexpensive. ➛ Highly nonlinear behavior, especially when control limits are reached. ➛ Cannot really be used to detect proximity to limit-induced bifurcation. ➛ Useful in some OPF-applications to help represent voltage stability as a constraints.
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328
Indices
➠ The reactive-power-reserve index consists of monitoring the “difference” between the reactive power generator output and its maximum limit, e.g. for a generator bus system:
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Power System Dynamics and Stability
Indices
➠ Observations: ➛ Computationally inexpensive. ➛ Highly nonlinear behavior. ➛ Only works for the “right” generators, i.e. the generators associated with the limit-induced bifurcation.
➛ Cannot really be used to detect proximity to a saddle-node bifurcation.
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Power System Dynamics and Stability
Voltage Stability Applications
➠ These concepts and associated techniques are applied to real power systems through the computation of PV curves.
➠ These curves are obtained either through a simple series of “continuous” power flows or using actual CPFs.
➠ In both cases, these “nose” curves are computed with respect to load changes, which are defined as follows:
PL
= PL0 + λ∆PL
QL
= QL0 + λ∆QL
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331
Power System Dynamics and Stability
Voltage Stability Applications
➠ Generator power changes are then defined, with the exception of the slack bus, as:
PG = PG0 + λ∆PG ➠ For a “distributed” slack bus model: PG = PG0 + (λ + kG )∆PG for all generators, where kG is a variable in the power flow equations replacing the variable power in the slack bus.
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332
Power System Dynamics and Stability
Voltage Stability Applications
➠ The Available Transfer Capability (ATC) of the tranmsission system is typically obtained from (NERC definition): ATC
= TTC + ETC + TRM
➠ The ATC can be associated to the maximum loading margin λmax of the system if N-1 contingency criteria are taken into account.
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Power System Dynamics and Stability
Voltage Stability Applications
➠ TTC or Total Transfer Capability is the maximum loading level of the system considering N-1 contingency criteria, i.e. the λmax for the worst realistic single contingency.
➠ ETC or Existent Transmission Commitments represents the “current” loading level plus any reserved transmission commitments.
➠ TRM or Transmission Reliability Margin which is an additional margin defined to represent other contingencies.
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Power System Dynamics and Stability
Voltage Stability Applications
➠ For example, for WECC, systems should be operated a minimum 5% “distance” away from the maximum loadibility point when contigencies are considered:
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335
Power System Dynamics and Stability
Control and Protections
➠ To improve system loadability, i.e. avoid voltage stability problems, the most common control and protections techniques are:
➛ Increase reactive power output from generators, especially in the “critical” area (the area most “sensitive” to voltage problems).
➛ Introduce shunt compesation through the use of MSC, SVC or STATCOM (see slides 304 and 305).
➛ Use undervoltage relays.
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Power System Dynamics and Stability
Secondary Voltage Regulation
➠ One way of improving reactive power support is to coordinate the reactive power outputs of generators.
➠ The French and Italians typically referred to as “secondary voltage regulation” or control.
➠ The basic structure and controls are:
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337
Secondary Voltage Regulation
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Power System Dynamics and Stability
338
Secondary Voltage Regulation
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Power System Dynamics and Stability
Secondary Voltage Regulation
➠ Observations: ➛ Control areas and associated pilot buses and controlled generators must be “properly” identified.
➛ Control “hierarchy” is important, i.e. PQR is “slower” than AVR, and RVR is “slower” than PQR.
➛ It is relatively inexpensive compared to shunt compensation solutions.
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Power System Dynamics and Stability
Undervoltage Relays
➠ These relays are installed on sub-transmission substations (loads) to shed during “long duration” voltage dips.
➠ The idea is to reduce load demand on the system to increase the loadability margin (see slide 334).
➠ Operation is somewhat similarly to taps in a LTC: ➛ Discrete load shedding steps (e.g. 1-2% of total load). ➛ Activated with a time delay (e.g. 1-2 mins.) after the voltage dips below certain values (e.g. 0.8-0.9 p.u.)
➛ The lower the voltage, the faster and larger the load shed.
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Power System Dynamics and Stability
Example
➠ For a 3-area sample system: 100 MW 150 MW 150 MW 60 MVAr
v3
150 MW 56 MVAr
Bus 2
Area 1
1.02∠0
50 MVAr R = 0.01 p.u. X = 0.15 p.u.
V2 ∠δ2
Bus 3 50 MVAr
50 MW 40 MVAr 100 MW
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342
Example
➠ For a 3-area sample system: Bus
∆PG
∆PL
∆QL
Name
(p.u.)
(p.u.)
(p.u.)
Area 1
1.5
0
0
Area 2
0
1.5
0.56
Area 3
0.5
0.5
0.40
➠ Using UWPFLOW to obtain the system PV curves, for a distributed slack bus model:
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Power System Dynamics and Stability
Example
➠ Data file in EPRI format (3area.wsc): HDG UWPFLOW data file, WSCC format 3-area example April 2000 BAS C C AC BUSES C C | SHUNT | C |Ow|Name |kV |Z|PL |QL |MW |Mva|PM |PG |QM |Qm |Vpu BE 1 Area 1 138 1 150 60 0 0 0 150 0 0 1.02 B 1 Area 2 138 2 150 56 0 50 0 100 0 0 1.00 B 1 Area 3 138 3 50 40 0 50 0 100 0 0 1.00
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344
Example
➠ Data file in EPRI format (3area.wsc): C C AC LINES C C M CS N C |Ow|Name_1 |kV1||Name_2 |kV2|||In || R | X | G/2 | B/2 |Mil| L 1 Area 1 138 Area 2 1381 15001 .01 .15 L 1 Area 1 138 Area 3 1381 15001 .01 .15 L 1 Area 2 138 Area 3 1381 15001 .01 .15 C C SOLUTION CONTROL CARD C C |Max| |SLACK BUS | C |Itr| |Name |kV| |Angle | SOL 50 Area 1 138 0. END
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Power System Dynamics and Stability
Example
➠ Generator and load change file (3area.k): C C UWPFLOW load and generation "direction" file C for 3-area example C C BusNumber BusName DPg Pnl Qnl PgMax [ Smax 1 0 1.5 0.0 0.0 0 0 2 0 0.0 1.5 0.56 0 0 3 0 0.5 0.5 0.40 0 0
Vmax Vmin ] 1.05 0.95 1.05 0.95 1.05 0.95
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Power System Dynamics and Stability
Example
➠ Batch file for UNIX (run3area): echo -1- Run base case power flow uwpflow 3area.wsc -K3area.k echo -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s echo - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 echo - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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Power System Dynamics and Stability
Example
➠ Batch file for Windows (run3area.bat): rem -1- Run base case power flow uwpflow 3area.wsc -K3area.k rem -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s rem - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 rem - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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Power System Dynamics and Stability
Example
➠ PV curves (threearea.m): Profiles 150
100
50
0
⇒
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
kVArea 3
138
kVArea 2
138
kVArea 1
138
1.4
1.6
λmax ≈ 1.6 p.u. Universidad de Castilla-La Mancha
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Power System Dynamics and Stability
Example
➠ Hence, if contingencies are not considered: X X ∆PL TTC = PL0 + λmax =
670 MW X ETC = PL0
TRM
⇒
ATC
=
350 MW
=
0.05TTC
=
33.5 MW
=
286.5 MW
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Power System Dynamics and Stability
Example
➠ The singular value index obtained with UWPFLOW is as follows: 0.7
Full matrix sing. value index
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
1.4
1.6
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ This information is extracted from the final report of the US-Canada Joint Task Force.
➠ The full details of the final report can be found on the Internet at: https://reports.energy.gov/B-F-Web-Part1.pdf
➠ The report is titled: U.S.-Canada Power System Outage Task Force. “Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations.” Washington DC: USGPO, April 2004.
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352
August 14 2003 Blackout
➠ 3 Interconnections, 10 Regional Reliability Councils:
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353
Power System Dynamics and Stability
August 14 2003 Blackout
➠ Regions and control areas of the North American Electric Reliability Council (NERC):
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Reliability coordinators (some are also system and market operators, such as the IMO and PJM):
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
August 14 2003 Blackout
➠ Reliability coordinator and system/market operators in the Ohio area, where the system collapse started:
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
356
August 14 2003 Blackout
➠ Four causes: ➛ Inadequate system understanding: FirstEnergy, ECAR ➛ Inadequate situational awareness: FirstEnergy ➛ Inadequate tree-trimming: FirstEnergy ➛ Inadequate diagnostic support: MISO, PJM
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 82
357
Power System Dynamics and Stability
August 14 2003 Blackout
➠ At 12:15 EDT, MISO began having problems with its state estimator; it did not return to full functionality until 16:04.
➠ Sometime after 14:14, FirstEnergy began losing its energy management system (EMS) alarms but did not know it.
➠ At 14:20, parts of FE’s EMS began to fail first remote sites, then core servers but FE system operators did not know this and FE IT support staff did not tell them.
➠ Without a functioning EMS, FE operators did not know their system was losing lines and voltage until about 15:45.
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 83
358
Power System Dynamics and Stability
August 14 2003 Blackout
➠ FirstEnergy did not do sufficient system planning to know the Cleveland-Akron area was seriously deficient in reactive power supply needed for voltage support.
➠ At 13:31 EDT, FE lost its Eastlake 5 unit, a critical source of real and reactive power for the Cleveland-Akron area.
➠ FE did not perform contingency analysis after this event.
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 84
359
Power System Dynamics and Stability
August 14 2003 Blackout
➠ Lost of the Eastlake 5 unit and beginning of reactive power problems.
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 85
360
Power System Dynamics and Stability
August 14 2003 Blackout
➠ Between 15:05 and 15:41 EDT, FE lost 3 345 kV lines in the Cleveland-Akron area under normal loading due to contact with too-tall trees but did not know it due to EMS problems.
➠ Line loadings and reactive power demands increased with each line loss. ➠ Between 15:39 and 16:08 EDT, FE lost 16 138kV lines in the Cleveland-Akron area due to overloads and ground faults.
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 86
Power System Dynamics and Stability
361
August 14 2003 Blackout
➠ Line outages:
Universidad de Castilla-La Mancha June 26, 2008
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362
Power System Dynamics and Stability
August 14 2003 Blackout
➠ At 16:05.57 EDT, FE lost its Sammis-Star 345 kV line due to overload. ➠ This closed a major path for power imports into the Cleveland-Akron area and initiated the cascade phase of the blackout.
Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
363
August 14 2003 Blackout
➠ The tipping point in Ohio:
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 89
Power System Dynamics and Stability
364
August 14 2003 Blackout
➠ Effect of a line trip: increasing loading on other lines.
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365
Power System Dynamics and Stability
August 14 2003 Blackout
➠ Effect of a line trip: decreasing voltages leading to voltage collapse.
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366
Power System Dynamics and Stability
August 14 2003 Blackout
➠ Cascade: ➛ Definition: A cascade on an electric system is a dynamic, unplanned sequence of events that, once started, cannot be stopped by human intervention.
➛ Power swings, voltage fluctuations and frequency fluctuations cause sequential tripping of transmission lines, generators, and automatic load-shedding in a widening geographic area.
➛ The fluctuations diminish in amplitude as the cascade spreads. Eventually equilibrium is restored and the cascade stops.
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 92
Power System Dynamics and Stability
367
August 14 2003 Blackout
➠ The system goes “haywire”:
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Voltage Stability - 93
368
Power System Dynamics and Stability
August 14 2003 Blackout
➠ Higher voltage lines are better able to absorb large voltage and current swings, buffering some areas against the cascade (AEP, Pennsylvania).
➠ Areas with high voltage profiles and ample reactive power were not swamped by the sudden voltage and power drain (PJM and New England).
➠ After islanding began, some areas were able to balance generation with load and reach equilibrium without collapsing (upstate New York and southern Ontario).
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 94
Power System Dynamics and Stability
369
August 14 2003 Blackout
➠ Sequence of events:
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 95
Power System Dynamics and Stability
370
August 14 2003 Blackout
➠ Sequence of events:
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 96
Power System Dynamics and Stability
371
August 14 2003 Blackout
➠ Sequence of events:
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 97
Power System Dynamics and Stability
372
August 14 2003 Blackout
➠ Sequence of events:
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Voltage Stability - 98
373
Power System Dynamics and Stability
August 14 2003 Blackout
➠ When the cascade was over at 4:13 pm, as many as 50 million people in the northeast U.S. and the province of Ontario had no power.
➠ This is a “good” example of a voltage instability problem triggered by a series of contingencies.
Universidad de Castilla-La Mancha June 26, 2008
Voltage Stability - 99
Power System Dynamics and Stability
374
August 14 2003 Blackout
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Voltage Stability - 100
375
Power System Dynamics and Stability
August 14 2003 Blackout
➠ Had the system properly monitored and NERC recommended operating guidelines followed, the system might have been saved.
➠ An operating rule regarding max. system loadability margins similar to WECC’s might have given the operators a better picture of the situation, but without proper monitoring, these would have probably not worked either.
➠ In the absence of these, under-voltage relays with ULTC blocking might have saved the system, as these would have automatically shed load when the voltages started to collapse in First Energy’s (FE) region in Ohio.
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Voltage Stability - 101
376
Power System Dynamics and Stability
Voltage Stability Report
➠ Much more information regarding the issue voltage stability can be found in: ˜ ➠ C. A. Canizares, editor, “Voltage Stability Assessment: Concepts, Practices and Tools,” IEEE-PES Power System Stability Subcommittee Special Publication, SP101PSS, August 2002. IEEE-PES WG Report Award 2005.
➠ This is a 283-page report, published after 5 years in preparation, and coauthored by several voltage stability from around the world.
➠ More details about the report can be found at: http://thunderbox.uwaterloo.ca/˜claudio/claudio.html/#VSWG
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Power System Dynamics and Stability
377
Angle Stability Outlines
➠ Definitions. ➠ Small disturbance: ➛ Hopf Bifurcations. ➛ Control and mitigation. ➛ Practical example. ➠ Transient Stability ➛ Time Domain. ➛ Direct Methods. ☞ Equal Area Criterion. ☞ Energy Functions. ➛ Practical applications. Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 1
378
Power System Dynamics and Stability
Angle Stability Definitions
➠ IEEE-CIGRE classification (IEEE/CIGRE Joint Task Force on Stability) Terms and Definitions, “Definitions and Classification of Power System Stability”, IEEE Trans. Power Systems and CIGRE Technical Brochure 231, 2003:
Power System Stability
Rotor Angle
Frequency
Voltage
Stability
Stability
Stability
Small Disturbance
Transient
Angle Stability
Stability
Large Disturbance Voltage Stability
Short Term
Short Term Short Term
Small Disturbance Voltage Stability
Long Term
Long Term
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 2
379
Power System Dynamics and Stability
Angle Stability Definitions
➠ “Rotor angle stability refers to the ability of synchronous machines of an interconnected power system to remain in synchronism after being subjected to a disturbance. It depends on the ability to maintain/restore equilibrium between electromagnetic torque and mechanical torque of each synchronous machine in the system.”
➠ In this case, the problem becomes apparent through angular/frequency swings in some generators which may lead to their loss of synchronism with other generators.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 3
380
Power System Dynamics and Stability
Small Disturbance
➠ “Small disturbance (or small signal) rotor angle stability is concerned with the ability of the power system to maintain synchronism under small disturbances. The disturbances are considered to be sufficiently small that linearization of system equations is permissible for purposes of analysis.”
➠ This problem is usually associated with the appearance of undamped oscillations in the system due to a lack of sufficient damping torque.
➠ Theoretically, this phenomenon may be associated with a s.e.p. becoming unstable through a Hopf bifurcation point, typically due to contingencies in the system (e.g. August 1996 West Coast Blackout).
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 4
Power System Dynamics and Stability
381
Hopf Bifurcation I
➠ For the generator-load example, with AVR but no QG limits: PG + jQG
Generator
jx′G
PL + jQL
jxL
V1 ∠δ1
E ′ ∠δ
Kv s
−
V2 ∠δ2
V10 +
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 5
382
Power System Dynamics and Stability
Hopf Bifurcation I
➠ The DAE model is: ω˙ δ˙ E˙ ′
E ′ V2 sin δ − DG ω Pm = X ′ E V2 1 sin δ − Pd = ω− DL X 1 M
= Kv (V10 − V1 ) ′ 2 1 V E V2 V˙ 2 = − 2 + cos δ − kPd τ X X ′ E V2 V1 V2 ′ sin δ − sin δ 0 = XL X 1 1 E ′ V2 V1 V2 2 0 = V2 − cos δ ′ + cos δ − XL X X XL Universidad de Castilla-La Mancha
June 26, 2008
Angle Stability - 6
383
Power System Dynamics and Stability
Hopf Bifurcation I
➠ Observe that the algebraic constraint can be eliminated, since:
Thus:
V1r
= V1 cos δ ′
V1i
= V1 sin δ ′
V1i V2 E ′ V2 E ′ XL 0= − sin δ ⇒ V1i = sin δ XL X X
and
0 ⇒
V1r
1 E ′ V2 V1r V2 1 − + cos δ − = XL X X XL E ′ XL 1 + cos δ = V2 1 − XL X V22
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 7
384
Power System Dynamics and Stability
Hopf Bifurcation I
➠ This yields the following equations, which are better for numerical time domain simulations:
ω˙
=
δ˙
=
E˙ ′
=
V˙ 2
=
1 M
′
E V2 sin δ − DG ω Pm X ′ E V2 1 sin δ − Pd ω− DL X q 2 +V2 Kv V10 − V1r 1i 2 ′ 1 V E V2 − 2 + cos δ − kPd τ X X
➠ Observe that in this case, Pm = Pd , i.e. generation and load are assumed to be balanced. Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 8
385
Power System Dynamics and Stability
Hopf Bifurcation I
➠ The PV curves for M = 0.1, DG = 0.01, DL = 0.1, τ = 0.01, ′ = 0.5, V10 = 1, k = 0.25 are: Kv = 10, XG 1
OP
HB
0.9 0.8
HB
0.7
V2
0.6 0.5
xL = 0.6
0.4 0.3 0.2
xL = 0.5
0.1 0
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 9
386
Power System Dynamics and Stability
Hopf Bifurcation I
➠ The eigenvalues for the system with respect to changes in Pd for xL = 0.5: 3
2
Imag
1
0
−1
−2
−3 −5
−4
−3
−2
−1 Real
0
1
2
3
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 10
387
Power System Dynamics and Stability
Hopf Bifurcation I
➠ There is a Hopf bifurcation for Pd = 0.65, xL = 0.5: 6
4
HB 2
ℜ(µ)
0
−2
−4
−6
−8
−10
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 11
388
Power System Dynamics and Stability
Hopf Bifurcation I
➠ A Hopf bifurcation with eigenvalues µ = ±jβ yields a periodic oscillation of period:
T =
2π β
➠ Hence, for the example: µ ≈ ⇒
T
±j3
≈ 2s
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 12
389
Power System Dynamics and Stability
Hopf Bifurcation I
➠ The contingency xL = 0.5 → 0.6 yields: 2.5
ω δ E′ V1 V2
2
1.5
1
0.5
0
−0.5
0
0.2
0.4
0.6
0.8
t [s]
1
1.2
1.4
1.6
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 13
390
Power System Dynamics and Stability
Hopf Bifurcation I
➠ Notice that in this case no oscillations are observed, which is a “trademark” of Hopf bifurcations and small-disturbance angle instabilities.
➠ The reason for this is that the oscillation period is 2 s (typical in practice where these kinds of oscillations are in the 0.1-1 Hz range), but the bus voltage collapses well before the oscillations appear, which is atypical and is probably due to the chosen impedances and time constants.
➠ This example stresses the point that angle instabilities do lead to voltage collapse, and vice versa, voltage instabilities lead to angle/frequency oscillations, even though the reason behind each stability problem are fairly different.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 14
Power System Dynamics and Stability
391
Hopf Bifurcation II
➠ Single-machine dynamic-load system with SVC: PG + jQG
PL + jQT jX
E∠δ V
V ∠0 jBC
Vref
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 15
392
Power System Dynamics and Stability
Hopf Bifurcation II
➠ The total reactive power absorbed by the load and the SVC is as follows: EV V2 + cos(δ) + V 2 BC QT (V, δ) = − X X ➠ The SVC controller is modeled as a first order pure integrator. Vref
+
BC 1/sT − V
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 16
393
Power System Dynamics and Stability
Hopf Bifurcation II
➠ The resulting differential equations of the SMDL system with SVC are as follows:
δ˙ ω˙ V˙ B˙ C
= ω EV 1 [Pd − sin(δ) − Dω] = M X 1 1 EV = [−kPd + V 2 (BC − ) + cos(δ)] τ X X 1 (Vref − V ) = T
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 17
394
Power System Dynamics and Stability
Hopf Bifurcation II
➠ BC is the equivalent susceptance of the SVC; T and Vref are the SVC time constant and reference voltage, respectively.
➠ In the following, it is assumed that T = 0.01 s and Vref = 1.0 p.u. ➠ Observe that also in this case it is possible to deduce the set of ODE, i.e. the algebraic variables can be explicitly expressed as a function of the state variables and the parameters.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 18
Power System Dynamics and Stability
395
Hopf Bifurcation II
➠ The state matrix of the system is as follows:
0 EV − M X cos(δ) A= − EV sin(δ) τX 0
|
1
D | −M
|
0
|
0
|
0
| 0 E | − M X sin(δ) | 0 1 V2 | τ X [E cos(δ) − 2V + 2V BC X] | τ | − T1 | 0
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 19
396
Power System Dynamics and Stability
Hopf Bifurcation II
➠ Eigenvalue loci: 400
300
200
Imaginary
100
Hopf Bifurcation P =1.4143
0
d
−100
−200
−300
−400 −0.4
−0.2
0
0.2
0.4
0.6
0.8
Real
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 20
397
Power System Dynamics and Stability
Hopf Bifurcation II
➠ A complex conjugate pair of eigenvalues crosses the imaginary axis for Pd = 1.4143, thus leading to a Hopf bifurcation. ➠ The HB point is: (δ0 , ω0 , V0 , BC0 , Pd0 ) = (0.7855, 0, 1, 1.2930, 1.4143)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 21
398
Power System Dynamics and Stability
Hopf Bifurcation II
➠ Bifurcation diagram Pd -δ :
3
2.5
δ (rad)
2 max d
P 1.5
Hopf Bifurcation 1
0.5
0
0
0.2
0.4
0.6
0.8
1
P (p.u.)
1.2
1.4
1.6
1.8
2
d
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 22
399
Power System Dynamics and Stability
Hopf Bifurcation II
➠ We simulate a step change in Pd from 1.41 p.u. to 1.42 p.u. for t = 2 s. ➠ For t > 2 s the system does not present a stable equilibrium point and shows undamped oscillations (likely an unstable limit cycle), as expected from the P -δ curve.
➠ For t = 2.57 s, the load voltage collapses. ➠ Note that, in this case, the generator angle shows an unstable trajectory only after the occurrence of the voltage collapse at the load bus.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 23
400
Power System Dynamics and Stability
Hopf Bifurcation II
➠ Time domain simulation results:
δ (rad)
0.85
0.8
0.75 1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.3
2.4
2.5
2.6
t (s)
V (p.u.)
1.5 1 0.5 0 1.8
1.9
2
2.1
2.2
t (s)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 24
401
Power System Dynamics and Stability
Hopf Bifurcation II
➠ The use of the SVC device gives a birth to a new bifurcation, namely a Hopf bifurcation.
➠ This Hopf bifurcation cannot be removed by simply adjusting system parameters.
➠ However SVC and load dynamics can be coordinated so that the loadability of the system can be increased.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 25
Power System Dynamics and Stability
402
Control and Mitigation
➠ For the IEEE 14-bus test system: Bus 13
Bus 14 Bus 10 Bus 12
Bus 09
Bus 11
Bus 07
Bus 06 Bus 04
Bus 08
Bus 05 Bus 01
Breaker
Bus 02
Bus 03
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 26
403
Power System Dynamics and Stability
Control and Mitigation
➠ Generator speeds for the line 2-4 outage and 40% overloading: 1.001
1.0005
1
0.9995
0.999
0.9985
0.998
0
5
10
15 time (s)
20
25
30
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 27
404
Power System Dynamics and Stability
Control and Mitigation
➠ This has been typically solved by adding Power System Stabilizers (PSS) to the voltage controllers in “certain” generators, so that equilibriun point is made stable, i.e. the Hopf is removed. vs max vSI
Tw s Kw Tw s + 1
T1 s + 1
T3 s + 1
1
T2 s + 1
T4 s + 1
Tǫ s + 1
vs
vs min
➠ FACTS can also be used to address this problem.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 28
Power System Dynamics and Stability
405
Control and Mitigation
➠ For the IEEE 14-bus test system with PSS at bus 1: Bus 13
Bus 14 Bus 10 Bus 12
Bus 09
Bus 11
Bus 07
Bus 06 Bus 04
Bus 08
Bus 05 Bus 01
Breaker
Bus 02
Bus 03
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 29
406
Power System Dynamics and Stability
Control and Mitigation
➠ Generator speeds for the line 2-4 outage and 40% overloading: 1.001
1.0005
1
0.9995
0.999
0.9985
0.998
0
5
10
15 time (s)
20
25
30
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 30
407
Power System Dynamics and Stability
Control and Mitigation
➠ Data regarding this system are available at: http://thunderbox.uwaterloo.ca/∼claudio /papers/IEEEBenchmarkTFreport.pdf
➠ More details regarding this example can be found in: F. Milano, “An Open Source Power System Analysis Toolbox”, accpeted for publication on IEEE Trans. On Power Systems, March 2004.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 31
408
Power System Dynamics and Stability
Control and Mitigation
➠ For the IEEE 145-bus, 50-machine test system: 1
99
36
93
AREA 2 87
43 33 34
5 4
35 38
3
45 41
116
42
115
44
39
114
142
141 140
85
50
117 139 118 137
102
52
55
129 61
53 56 101
113
143
46
51 2
144
49
47 84
145
48
37
40 49
AREA 1
88
110
136
86
54
127 135 138
6 57 7
104
126
134
62
128
119
120 122
133 132 121
66
95 67
63
111
64
65
125 68
97
8 9
92
90
124
130
69
80 112
11 12
131
123
10
107
32
79
94 72
60 71 21
20
19
18
83
78
30
89
23
96
77
76
59 98 13
17
70 100
103
15 58
22 16
74
105
26
75
81 14
91
106 73 31
24 29
28
82 25
108
109
27
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 32
409
Power System Dynamics and Stability
Control and Mitigation
➠ For an impedance load model, the PV curves yield: (a) 0.94 Voltage (p.u.)
HB 0.92
Base Case Line 79−90 Outage
Operating point
Constant impedance load line 0.9
HB
0.88 0.86 0.84 0.82
0
0.002
0.004
0.006
0.008
0.01
0.012
(b) Base Case Line 79−90 Outage
L.F. (p.u.)
Voltage (p.u.)
0.94
Operating point 0.93
HB
HB
0.92
Constant impedance load line
0.91 0.9
2
2.5
3
3.5
4
4.5
5
λ
5.5
6 −3
x 10
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 33
410
Power System Dynamics and Stability
Control and Mitigation
➠ Indices based on the singular values have been proposed to predict Hopf bifucations: 0.015 HBI2 Base Case HBI Base Case 1 HBI Line 79−90 Outage 2 HBI1 Line 79−90 Outage
Hopf Indices
0.01
0.005
HB 0
0
0.002
0.004
0.006 λ
HB
0.008
0.01
0.012
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 34
411
Power System Dynamics and Stability
Control and Mitigation
➠ More details regarding this example can be found in: ˜ C. A. Canizares, N. Mithulananthan, F. Milano, and J. Reeve, “Linear Performance Indices to Predict Oscillatory Stability Problems in Power Systems”, IEEE Trans. On Power Systems, Vol. 19, No. 2, May 2004, pp. 1104-1114.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 35
412
Power System Dynamics and Stability
Small Disturbance Applications
➠ In practice, some contingencies trigger plant or inter-area frequency oscillations in a “heavily” loaded system, which may be directly associated with Hopf bifurcations.
➠ This is a “classical” problem in power systems and there are many examples of this phenomenon in practice, such as the August 10, 1996 blackout of the WSCC (now WECC) system.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 36
413
Power System Dynamics and Stability
Small Disturbance Applications
➠ Observe that the maximum loadability of the system is reduced by the presence of the Hopf bifurcation.
➠ This leads to the definition of a “dynamic” ATC value. 1 HB
HB
0.9 OP
0.8 0.7
ATC
ETC
TRM
V2
0.6 0.5 TTC
0.4 0.3 Worst Contingency
0.2 0.1 0
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 37
Power System Dynamics and Stability
414
August 10, 1996 WSCC Blackout
➠ This information was extracted from a presentation by Dr. Prabha Kundur, President and CEO of PowerTech Labs Inc.
➠ The material is availbale at: toronto.ieee.ca/events/oct0303/prabha.ppt
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 38
415
Power System Dynamics and Stability
August 10, 1996 WSCC Blackout
➠ System conditions: ➛ High ambient temperatures in Northwest, and hence high power transfers from Canada to California.
➛ Prior to main outage, three 500 kV line sections from lower Columbia River to loads in Oregon were out of service due to tree faults.
➛ California-Oregon interties loaded to 4330 MW north to south. ➛ Pacific DC intertie loaded at 2680 MW north to south. ➛ 2300 MW flow from Britsh Columbia.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 39
Power System Dynamics and Stability
416
August 10, 1996 WSCC Blackout
➠ Growing 0.23 Hz oscillations caused tripping of lines:
3000 2900
5
4
3
2800
4
2700
1
2600
2
2500 2400 2300 0 3
6 12 16 19 22
25 28 31 40 43 47 50 53 56 59 62 65 68 71 74
Time in seconds
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 40
Power System Dynamics and Stability
417
August 10, 1996 WSCC Blackout
➠ Event 1: 14:06:39 → Big Eddy-Ostrander 500 kV LG fault - flashed to tree ➠ Event 2: 14:52:37 → John Day-Marion 500 kV LG -flashed to tree ➠ Event 3: 15:42:03 → Keeler-Alliston 500 KV - LG - flashed to tree ➠ Event 4: 15:47:36 → Ross-Lexington 500 kV - flashed to tree ➠ Event 5: 15:47:36-15:48:09 → 8 McMary Units trip Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 41
Power System Dynamics and Stability
418
August 10, 1996 WSCC Blackout
➠ As a results of the undamped oscillations, the system split into four large islands.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 42
Power System Dynamics and Stability
419
August 10, 1996 WSCC Blackout
➠ PSS solution:
San Onofre (addition)
Palo Verde (tune existing)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 43
420
Power System Dynamics and Stability
August 10, 1996 WSCC Blackout
➠ PSS solution: 3000
With existing controls
2800
2600
Eigenvalue =
0.0597 + j1.771
Frequency =
0.2818 Hz
2400
Damping = 2200
0
15
30
45
61
−0.0337
72
Time in seconds 3000
With PSS modifications 2800
Eigenvalue =
−0.0717 + j1.673
Frequency =
0.2664 Hz
2600 2400
Damping = 2200 0
18
50 32 Time in seconds
68
−0.0429
75
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 44
421
Power System Dynamics and Stability
August 10, 1996 WSCC Blackout
➠ 7.5 million customers experienced outages from a few minutes to nine hours.
➠ Total load loss: 35,500 MW.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 45
422
Power System Dynamics and Stability
Transient Stability
➠ “Large disturbance rotor angle stability or transient stability, as it is commonly referred to, is concerned with the ability of the power system to maintain synchronism when subjected to a severe disturbance, such as a short circuit on a transmission line. The resulting system response involves large excursions of generator rotor angles and is influenced by the nonlinear power-angle relationship”.
➠ The system nonlinearities determine the system response; hence, linearization does not work in this case.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 46
423
Power System Dynamics and Stability
Transient Stability
➠ For small disturbances, the problem is to determine if the resulting steady state condition is stable or unstable (eigenvalue analysis) or a bifurcation point (e.g. Hopf bifurcation).
➠ For large disturbances, the steady state condition after the disturbance can exist and be stable, but it is possible that the system cannot reach that steady state condition.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 47
424
Power System Dynamics and Stability
Transient Stability
➠ The basic idea and analysis procedures are: ➛ Pre-contingency (initial conditions): the system is operating in “normal” conditions associated with a s.e.p.
➛ Contingency (fault trajectory): a large disturbance, such as a short circuit or a line trip forces the system to move away from its initial operating point.
➛ Post contingency (fault clearance): the contingency usually forces system protections to try to “clear” the fault; the issue is then to determine whether the resulting system is stable, i.e. whether the system remains relatively intact and the associated time trajectories converge to a “reasonable” operating point.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 48
425
Power System Dynamics and Stability
Transient Stability
➠ Based on non linear theory, this analysis can be basically viewed as determining wheter the fault trajectory at the “clearance” point is outside or inside of the stability region of the post-contingency s.e.p.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 49
426
Power System Dynamics and Stability
Time domain analysis
➠ Given the complexity of power system models, the most reliable analysis tool for these types of studies is full time domain simulations.
➠ For example, for the generator-load example: PG + jQG
Generator
jx′G
E ′ ∠δ
PL + jQL
jxL
V1 ∠δ1
V2 ∠δ2
−jxC
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 50
427
Power System Dynamics and Stability
Time domain analysis
➠ The ODE for the simplest generator d-axis transient model and neglecting AVR and generator limits is:
ω˙ δ˙ V˙ 2
1 (Pd − E ′ V2 B sin δ − DG ω) M 1 = ω− (E ′ V2 B sin δ − Pd ) DL 1 [−V22 (B − BC ) + E ′ V2 B cos δ − kPd ] = τ =
where
B=
1 1 = ′ X XG + XL
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 51
428
Power System Dynamics and Stability
Time domain analysis
➠ The objective is to determine how much time an operator would have to connect the capacitor bank BC after a severe contingency, simulated here as a sudden increase in the value of the reactance X , so that the system recovers.
➠ In this case, and as previously discussed in the voltage stability section, the contingency is severe, as the s.e.p. disappears.
➠ Full time domain simulations are carried out to study this problem for the parameter values M = 0.1, DG = 0.01, DL = 0.1, τ = 0.01, E ′ = 1, Pd = 0.7, k = 0.25, BC = 0.5.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 52
429
Power System Dynamics and Stability
Time domain analysis
➠ A contingency X = 0.5 → 0.6 at tf = 1 s, with BC connection at tc = 1.4 s yields a stable system: 1.4
ω δ V2
1.2
E′
1
0.8
0.6
0.4
tf
0.2
tc
0
−0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t [s] Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 53
430
Power System Dynamics and Stability
Time domain analysis
➠ If BC is connected at tc = 1.5 s, the system is unstable: 6
ω δ V2
5
E′
4
tc tf
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t [s] Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 54
431
Power System Dynamics and Stability
Direct Methods
➠ Time domain analysis is expensive, so direct stability analysis technique have been proposed based on Lyapounov’s stability theory.
➠ The idea is to define an “energy” or Lyapounov function ϑ(x, xs ) with certain characteristics to obtain a direct “measure” of the stability region
A(xs ) associated with the post-contingency s.e.p. xs . ➠ A system’s energy is usually a good Lyapounov function, as it yields a stability “measure”.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 55
432
Power System Dynamics and Stability
Direct Methods
➠ The rolling ball example can used to explain the basic behind these techniques: u.e.p.2
~ v m u.e.p.1
h
s.e.p.
➠ There are 3 equilibrium points: one stable (“valley” bottom), two unstable (“hill” tops).
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 56
433
Power System Dynamics and Stability
Direct Methods
➠ The energy of the ball is a good Lyapounov or Transient Energy Function (TEF):
W
= Wkinetic + Wpotential = WK + WP 1 = mv 2 + mgh 2 = ϑ([v, h]T , 0)
➠ The potential energy at the s.e.p. is zero, and presents local maxima at the u.e.p.s (WP 1 and WP 2 ). ➠ The “closest” u.e.p. is u.e.p.1 since WP 1 < WP 2 . Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 57
434
Power System Dynamics and Stability
Direct Methods
➠ The stability of this system can then be evaluated using this energy: ➛ if W < WP 1 , the ball remains in the “valley”, i.e. the system is stable, and will converge to the s.e.p. as t → ∞.
➛ If W > WP 1 , the ball might or might not converge to the s.e.p., depending on friction (inconclusive test).
➛ When the ball’s potential energy WP (t) reaches a maximum with respect to time t, the system leaves the “valley”, i.e. unstable condition.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 58
435
Power System Dynamics and Stability
Direct Methods
➠ The “valley” would correspond to the stability region when friction is “large”.
➠ In this case, the stability boundary ∂A(xs ) corresponds to the “ridge” where the u.e.p.s are located and WP has a local max. value. ➠ The smaller the friction in the system, the larger the difference between the ridge and ∂A(xs ). ➠ For zero friction, ∂A(xs ) is defined by WP 1 .
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 59
436
Power System Dynamics and Stability
Direct Methods
➠ The direct stability test is only a necessary but not sufficient condition: ϑ(x, xs ) < c ⇒
x ∈ A(xs )
ϑ(x, xs ) > c ⇒
Inconclusive!
where the value of c is usually associated with a local maximum of a “potential energy” function.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 60
437
Power System Dynamics and Stability
Direct Methods
➠ For the simple generator-infinite bus example, neglecting limits and AVR: PG + jQG
Generator
jx′G
jxL
V1 ∠δ1
E ′ ∠δ
δ˙ ω˙ X
PL + jQL
System
jxth
V2 ∠δ2
V ∠0
= ω = ωr − ω0 ′ 1 EV = PL − sin δ − Dω M X ′ = XG + XL + Xth
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 61
438
Power System Dynamics and Stability
Direct Methods
➠ The kinetic energy in this system is defined as: WK =
1 M ω2 2
➠ And the potential energy is: Z WP = (Tc − Tm )dδ Z ≈ (Pc − Pm )dδ → in p.u. for ωr ≈ ω0 ≈
Z
δ
Z
δ
E′V ( (PG − PL )dδ ≈ − PL )dδ X δs δs
≈ −E ′ V B(cos δ − cos δs ) − PL (δ − δs ) where δs is the s.e.p. for this system. Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 62
439
Power System Dynamics and Stability
Direct Methods
➠ With WP presenting a very similar profile as the rolling ball example: PG unstable
stable
E′ V X
unstable
δu2
δs
δu1
max
WF max
WF 2 WF 1 min
δu2
δs
δu1
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 63
440
Power System Dynamics and Stability
Direct Methods
➠ Hence, the system Lyapounov function of TEF is: T EF
= ϑ(x, xs ) = ϑ([δ, ω]T , [δs , 0]T ) 1 = M ω 2 − E ′ V B(cos δ − cos δs ) 2 −PL (δ − δs )
➠ Thus, using similar criteria as in the case of the rolling ball: ➛ If T EF < WP 1 ⇒ system is stable.
➛ If T EF > WP 1 ⇒ inconclusive for D > 0 (“friction”). ➛ If T EF > WP 1 ⇒ unstable for D = 0 (unrealistic). Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 64
Power System Dynamics and Stability
441
Direct Methods
➠ This is equivalent to compare “areas” in the PG vs. δ graph (Equal Area Criterion or EAC): pre-contingency
PG
post-contingency
PL contingency (fault)
δ(0) = δspre
δ(tc ) δspost
δu1 post
δ
fault clearing time
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 65
442
Power System Dynamics and Stability
Direct Methods
➠ Thus, comparing the “acceleration” area: Z δ(tc ) (PL − PGf ault )dδ Aa = δspre
=
Z
δ(tc )
δspre
′
EV PL − Xf ault
dδ
➠ versus the “deceleration” area: Z δspost Ad = (PGpost − PL )dδ δ(tc )
=
Z
δspost
δ(tc )
E′V − PL dδ Xpost
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 66
Power System Dynamics and Stability
443
Direct Methods
➠ In conclusion: ➛ If Aa < Ad ⇒ system is stable at tc .
➛ If Aa > Ad ⇒ inconclusive for D > 0.
➛ If Aa > Ad ⇒ unstable for D = 0 (unrealistic).
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 67
444
Power System Dynamics and Stability
Direct Methods: Example 1
➠ A 60 Hz generator with a 15% transient reactance is connected to an infinite bus of 1 p.u. voltage through two identical parallel transmission lines of 20% reactance and negligible resistance. The generator is delivering 300 MW at a 0.9 leading power factor when a 3-phase solid fault occurs in the middle of one of the lines; the fault is then cleared by opening the breakers of the faulted line.
➠ Assuming a 100 MVA base, determine the critical clearing time for this generator if the damping is neglected and its inertia is assumed to be
H = 5 s. ➠ Assuming D = 0.1 s, determine the actual critical clearing time.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 68
Power System Dynamics and Stability
445
Direct Methods: Example 1
➠ Pre-contingency or initial conditions: PGpre QL
E′V sin δspre = PL = Xpre V2 E′V = − + cos δspre Xpre Xpre
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 69
Power System Dynamics and Stability
446
Direct Methods: Example 1
➠ Where: Xpre
=
0.15 +
0.2 = 0.25 2
300 MW PL = 100 MVA E′ 3 = sin δspre 0.25 QL = 3 tan(cos−1 0.9) E′ 1 + cos δspre 1.4530 = − 0.25 0.25
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 70
Power System Dynamics and Stability
447
Direct Methods: Example 1
⇒
Ei′pre
= E ′ sin δspre =
Er′ pre
0.75
= E ′ cos δspre =
E′
1.3633 q = E ′ 2rpre + E ′ 2ipre
= δspre
1.5559
Ei′pre Er′ pre
!
=
tan−1
=
28.82◦ = 0.5030 rad
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 71
448
Power System Dynamics and Stability
Direct Methods: Example 1
➠ Fault conditions: PGf ault
E′V sin δ Xf ault 1.5559 sin δ Xf ault
= =
where, using a Y-∆ circuit transformation due to the fault being in the middle of one of the parallel lines: jXf ault j0.15 j0.2 E ′ ∠δ
V ∠0 j0.1
j0.1
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 72
Power System Dynamics and Stability
449
Direct Methods: Example 1
Xf ault ⇒
PGf ault Aa
0.15 × 0.2 + 0.1 × 0.2 + 0.15 × 0.1 = 0.1 = 2.394 sin δ Z δ(tcc ) (PL − PGf ault )dδ = δspre
=
Z
δ(tcc )
0.503
= =
(3 − 2.394 sin δ)dδ
3(δ(tcc ) − 0.503) + 2.394(cos δ(tcc ) − cos(0.503)) 3δ(tcc ) + 2.394 cos δ(tcc ) − 3.6065
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 73
Power System Dynamics and Stability
450
Direct Methods: Example 1
➠ Post contingency conditions:
⇒
Xpost
=
=
0.15 + 0.2 = 0.35 E′V sin δ Xpost 4.446 sin δ
PGpost
=
3
=
4.446 sin δspost
δspost
=
42.44◦
=
0.7407 rad
⇒
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 74
451
Power System Dynamics and Stability
Direct Methods: Example 1
⇒ Ad
=
Z
π−δspost
δ(tcc )
=
Z
(PGpost − PL )dδ
2.4
δ(tcc )
(4.446 sin δ − 3)dδ
= −4.446(cos 2.4 − cos δ(tcc )) − 3(2.4 − δ(tcc )) = 3δ(tcc ) + 4.446 cos δ(tcc ) − 3.9215
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 75
452
Power System Dynamics and Stability
Direct Methods: Example 1
Aa
= Ad = 3δ(tcc ) + 2.394 cos δ(tcc ) − 3.6065 = 3δ(tcc ) + 4.446 cos δ(tcc ) − 3.9215
⇒
δ(tcc )
= 81.17◦ = 1.4167 rad
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 76
Power System Dynamics and Stability
453
Direct Methods: Example 1
➠ During the fault: δ˙ ω˙ M
⇒
= ω ′ 1 EV = sin δ PL − M Xf ault H = πf 5s = π60 Hz = 0.0265 s2
δ˙
= ω
ω˙
=
37.70(3 − 2.394 sin δ)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 77
454
Power System Dynamics and Stability
Direct Methods: Example 1
➠ Integrating these equations numerically for δ(0) = δspre = 28.82◦ : 220 200 180 160
δ [deg]
140 120 100 80 60 40 20
0
0.05
0.1
0.15
t [s]
0.2
0.25
0.3
0.35
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 78
455
Power System Dynamics and Stability
Direct Methods: Example 1
➠ For D = 0.1 and a clearing time of tc = 0.27 s, the system is stable:
δ [deg]
150
100
50
0
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
1.2
1.4
1.6
1.8
2
t [s]
10
ω [deg]
5
0
−5
−10
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 79
456
Power System Dynamics and Stability
Direct Methods: Example 1
➠ For a clearing time of tc = 0.28 s, the system is unstable; hence tcc ≈ 0.275 s: 2500
δ [deg]
2000 1500 1000 500 0
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
1.2
1.4
1.6
1.8
2
t [s]
40
ω [deg]
30
20
10
0
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 80
457
Power System Dynamics and Stability
Direct Methods: Example 2
➠ Generator-motor, i.e. system-system, cases may also be studied using the EAC method based on an equivalent inertia
M = M1 M2 /(M1 M2 ), and damping D = M D1 /M1 = M D2 /M2 . ➠ For the generator-load example neglecting the internal generator impedance and assuming an “instantaneous” AVR: PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 81
458
Power System Dynamics and Stability
Direct Methods: Example 2
➠ The “energy” functions, with or without generator limits, can be shown to be:
WK WP
1 M ω2 2 = −B(V1 V2 cos δ − V10 V20 cos δ0 ) 1 1 2 2 2 + B(V2 − V20 ) + B(V12 − V10 ) 2 2 V1 V2 − QG ln −Pd (δ − δ0 ) + Qd ln V20 V10 =
➠ The stability of this system can then be studied using the same “energy” evaluation previously explained for T EF = ϑ(x, x0 ) = WK + WP .
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 82
459
Power System Dynamics and Stability
Direct Methods: Example 2
➠ Thus for V1 = 1, XL = 0.5, Pd = 0.1, and Qd = 0.25Pd , the potential energy WP (δ, V2 ) that defines the stability region withr espect to the s.e.p. is:
8 7 6
WP
5 4 3 2 1 0 400
s.e.p. node
200
u.e.p. saddle
0
2 1.5 1
−200
δ
0.5 −400
0
V2
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 83
460
Power System Dynamics and Stability
Direct Methods: Example 2
➠ Simulating the critical contingency XL = 0.5 → 0.6 for Pd = 0.7 and neglecting limits, the “energy” profiles are: 0.2
Wp Wk+Wp
0.1
T EF
0
−0.1
−0.2
−0.3
−0.4 0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
t [s]
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 84
461
Power System Dynamics and Stability
Direct Methods: Example 2
➠ The “exit” point on ∂A(xs ) is approximately at the maximum potential energy point.
➠ Thus, the critical clearing time is: tcc ≈ 1.42 s ➠ A similar value can be obtained through trial-and-error.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 85
462
Power System Dynamics and Stability
Direct Methods: Conclusions
➠ The advantages of using Lyapounov functions are: ➛ Allows reduced stability analysis. ➛ Can be used as an stability index. ➠ The problems are: ➛ Lyapounov functions are model dependent; in practice, only approximate “energy” functions can be found.
➛ Inconclusive if test fails. ➛ The post-perturbation system state must be known ahead of time, as the energy function is defined with respect to the corresponding s.e.p.
➠ Can only be used as an “approximate” stability analysis tool. Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 86
463
Power System Dynamics and Stability
Transient Stability Applications
➠ In practice, transient stability studies are carried out using time-domain trial-and-error techniques.
➠ These types of studies can now be done on-line even for large systems. ➠ The idea is to determine whether a set of “realistic” contingencies make the system unstable or not (contingency ranking), and thus determine maximum transfer limits or ATC in certain transmission corridors for given operating conditions.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 87
464
Power System Dynamics and Stability
Transient Stability Applications
➠ Thus, the maximum loadability of the system may be affected by the “size” of the stability region, leading to the definition of a “true” ATC value. 1 HB
HB
0.9 OP
0.8 0.7
ATC
ETC
TRM
V2
0.6 0.5 TTC
0.4 0.3 Worst Contingency
0.2 0.1 0
0
0.1
0.2
0.3
0.4
Pd
0.5
0.6
0.7
0.8
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 88
465
Power System Dynamics and Stability
Transient Stability Applications
➠ Critical clearing times are not really an issue with current fast acting protections.
➠ Simplified direct methods such as the “Extended Equal Area Criterion” (Y. Xue et al., “Extended Equal Area Criterion Revisited”, IEEE Transaction on Power Systems, Vol. 7, No. 3, Aug. 1992, pp. 1012-1022) have been proposed and tested for on-line contingency pre-ranking, and are being implemented for practical applications through an E.U. project.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 89
466
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ An example of an application of transient analysis techniques can be ˜ found in L.S. Vargas and C. A. Canizares, “Time Dependence of Controls to Avoid Voltage Collapse”, IEEE Transaction son Power Systems, Vol. 15, No. 4, November 2000, pp. 1367-1375.
➠ This paper discusses the May 1997 voltage collapse event of the main power system in Chile.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 90
467
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ Most of Chile lost power in a major blackout on Friday evening, snarling rush hour traffic in the capital. The blackout began at about 7:20 pm, and power was back in about a third of the affected areas at 9:00 pm.
➠ At 9:00 pm lights were gradually coming on in parts of the capital, home to 5 million people or one third of the country’s population. Television reports said power went out as far as Puerto Montt, a city some 600 miles south of Santiago, and in areas the same distance to the north.
➠ Source Reuters
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 91
468
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ Main system characteristics: ➛ Extension: 756626 km2 . ➛ Inhabitants: 14.5 mil. ➛ National consumptions: 33531 GWh. ➛ National peak load: 5800 MW. ➛ Installed capacity: 8000 MW. ➛ Frequency: 50 Hz. ➛ Trans. Level: 66/110/154/220/500 kV. ➛ Four interconnected systems: SING, SIC, AISEN, MAGALLANES
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 92
Power System Dynamics and Stability
469
Chilean Blackout (11/07/2003)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 93
470
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ Initial state of SIC system: ➛ 2500 MW load. ➛ Power flow south-north near 1000 MW (900 MW through 500 kV lines and 100 MW through 154 kV lines).
➠ Events: ➛ Line 154 kV trips. ➛ Major generator in the south hits reactive power limits and losses voltage control.
➛ Operator tries to recover falling voltages by connecting a capacitor bank near Santiago.
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 94
Power System Dynamics and Stability
471
Chilean Blackout (11/07/2003)
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 95
Power System Dynamics and Stability
472
Chilean Blackout (11/07/2003)
➠ The line trip and generator limits yield a voltage collapse associated with a limit-induced bifurcation problem:
Universidad de Castilla-La Mancha June 26, 2008
Angle Stability - 96
Power System Dynamics and Stability
473
Chilean Blackout (11/07/2003)
➠ PV curves:
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Angle Stability - 97
474
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ The connection of the capacitor bank after the generator limits are reached did not save the system, as the “faulted” system trajectories had “left” the stability region of the post contingency operating point.
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Angle Stability - 98
475
Power System Dynamics and Stability
Chilean Blackout (11/07/2003)
➠ If the capacitor bank is connected before the generator limits are reached, the system would have been saved, as the “faulted” system trajectories were still within the stability of the post-contingency operating point.
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Angle Stability - 99
Power System Dynamics and Stability
476
Frequency Stability Outlines
➠ Definitions. ➠ Basic Concepts. ➠ Practical applications, controls and protections. ➛ Italian Blackout (28/10/2003) ➛ European Blackout (4/11/2006)
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Frequency Stability - 1
477
Power System Dynamics and Stability
Frequency Stability Definitions
➠ IEEE-CIGRE classification (IEEE/CIGRE Joint Task Force on Stability) Terms and Definitions, “Definitions and Classification of Power System Stability”, IEEE Trans. Power Systems and CIGRE Technical Brochure 231, 2003:
Power System Stability
Rotor Angle
Frequency
Voltage
Stability
Stability
Stability
Small Disturbance
Transient
Angle Stability
Stability
Large Disturbance Voltage Stability
Short Term
Short Term Short Term
Small Disturbance Voltage Stability
Long Term
Long Term
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478
Power System Dynamics and Stability
Frequency Stability Definitions
➠ “Frequency Stability refers to the ability of a power system to maintain steady frequency following a severe system upset resulting in a significant imbalance between generation and load.”
➠ Thus, frequency stability analysis concentrates on studying the overall system stability for sudden changes in the generation-load balance.
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Frequency Stability - 3
479
Power System Dynamics and Stability
Frequency Stability Concepts
➠ For the generator-load example, with AVR but no QG limits: PG + jQG
Generator
jx′G
jxL
V2 ∠δ2
V1 ∠δ1
E ′ ∠δ
Kv s
PL + jQL
−
V10 +
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Frequency Stability - 4
480
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Neglecting losses and electromagnetic dynamics, the generator with a very simple AVR and no limits can be modeled using a d-axis transient model:
δ˙G ω˙ G E˙ ′
= ωG = ωr − ω0 1 (Pm − PG − DG ωG ) = M = Kv (V10 − V1 )
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Frequency Stability - 5
Power System Dynamics and Stability
481
Frequency Stability Concepts
➠ The load can be simulated using “simplified” mixed models and constant power factor:
δ˙2 V˙ 2
= ω2 = =
1 (PL − Pd ) DL
1 (QL − kPD ) |{z} τ Qd
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Frequency Stability - 6
482
Power System Dynamics and Stability
Frequency Stability Concepts
➠ The transmission system yields the power flow equations ′ ): (X = X L = X G PG
= =
QL
= =
QG
=
E ′ V2 PL = sin(δG − δ2 ) X V1 V2 sin(δ1 − δ2 ) XL V22 E ′ v2 − + cos(δ2 − δG ) X X V22 V1 V2 − + cos(δ2 − δ1 ) XL XL V12 V1 V2 − cos(δ1 − δ2 ) XL XL
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Frequency Stability - 7
Power System Dynamics and Stability
483
Frequency Stability Concepts
➠ Define: δ δ′ ω ⇒ δ˙
= δG − δ2 = δ1 − δ2 = ωG = ω − ω2
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Frequency Stability - 8
484
Power System Dynamics and Stability
Frequency Stability Concepts
➠ This yields the DAE model: ω˙ δ˙ E˙ ′
′
E V2 sin δ − DG ω X ′ E V2 1 sin δ − Pd = ω− DL X
=
1 M
Pm
= Kv (V10 − V1 ) 2 ′ 1 V2 E V2 ˙ V2 = − + cos δ − kPd τ X X E ′ V2 V1 V2 ′ sin δ − sin δ 0 = XL X 1 1 E ′ V2 V1 V2 2 0 = V2 − cos δ ′ + cos δ − XL X X XL Universidad de Castilla-La Mancha
June 26, 2008
Frequency Stability - 9
485
Power System Dynamics and Stability
Frequency Stability Concepts
➠ And the equilibrium equations: 0
=
0
=
0
=
0
=
0
=
0
=
E ′ V2 Pm sin δ − DG ω X E ′ V2 sin δ − Pd DL ω − X V10 − V1 V22 E ′ V2 − + cos δ − kPd X X E ′ V2 V1 V2 ′ sin δ − sin δ XL X 1 E ′ V2 V1 V2 1 2 V2 − cos δ ′ + cos δ − XL X X XL Universidad de Castilla-La Mancha
June 26, 2008
Frequency Stability - 10
486
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Hence: x = [ω, δ, E ′ , v2 ]T y
= [V1 , δ ′ ]T
p
= [Pm , V10 ]
λ
= Pd
➠ Observe that in this analysis, Pm 6= Pd to study the effect of generator-load imbalances in the system.
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Frequency Stability - 11
487
Power System Dynamics and Stability
Frequency Stability Concepts
➠ In normal operating conditions Pm = Pd ⇒ ω = 0, from the first 2 equilibrium equations.
➠ Hence, these equations can be replaced by the following 4 power flow equations, with 4 unknowns (E ′ , V2 , δ, δ ′ ): 0
=
0
=
0
=
0
=
E ′ V2 sin δ Pd − X V22 E ′ V2 −kPd − + cos δ X X V10 V2 sin δ ′ Pd − XL V22 V10 V2 kPd + − cos δ ′ XL XL
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Frequency Stability - 12
Power System Dynamics and Stability
488
Frequency Stability Concepts
➠ Simulating a 50% generation and load reduction, respectively, for M = 0.1, DG = 0.01, DL = 0.1, τ = 0.01, Kv = 10, XL = 0.5, ′ = 0.5, V10 = 1, Pd0 = 0.7, k = 0.25. XG ➠ Initial solution: ω
=
0
δ
=
0.7266
E′
=
1.3463
V2
=
0.7826
V1
=
1.0000
δ′
=
0.4636
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Frequency Stability - 13
489
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Time domain simulation: 1.5 1 0.5 0 −0.5 50% generator drop
−1 −1.5 −2 50% load drop
ω δ E′ V2 V1
−2.5 −3 −3.5
0
1
2
3
4
5
t [s]
6
7
8
9
10
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Frequency Stability - 14
490
Power System Dynamics and Stability
Frequency Stability Concepts
➠ The generation reduction yields a ≈0.5 Hz frequency drop and a load voltage increase.
➠ The load reduction yields a ≈0.5 Hz frequency increase, and an even larger load voltage increase, as the reactive power demand drops.
➠ The system reaches new s.e.p.s in both cases, as expected. ➠ Observe that the AVR keeps the generator terminal fairly stable and close to its set value V10 .
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Frequency Stability - 15
491
Power System Dynamics and Stability
Frequency Stability Concepts
➠ These frequency excursions due to generation-load imbalances are typical.
➠ It might lead to unstable conditions due to device protections such as frequency relays in generators and loads.
➠ Frequency problems may be solved manually by operators or automatically through controls and/or protections.
➠ Generator governors automatically regulate local frequency excursions.
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Frequency Stability - 16
492
Power System Dynamics and Stability
Frequency Stability Concepts
➠ Centralized frequency regulators, such as automatic Area Control Error (ACE) regulators, may be used to regulate power exchanges among control areas by controlling the frequency deviations on the interties.
➠ Examples of frequency instabilities are: ➛ The Italian blackout of Tuesday, October 28, 2003 (Material courtesy of Prof. Alberto Berizzi, Politecnico di Milano, Italy).
➛ The European blackout of Saturday, November 4, 2006 (Material courtesy of Prof. Edmund Handschin, University of Technology, Dortmund, Germany).
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Frequency Stability - 17
Power System Dynamics and Stability
493
Italian Blackout (28/10/2003)
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Frequency Stability - 18
Power System Dynamics and Stability
494
Italian Blackout (28/10/2003)
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Frequency Stability - 19
495
Power System Dynamics and Stability
Italian Blackout (28/10/2003) 3:10.47 ETrans (operator of interties between Switzerland and rest of Europe) lets the GRTN (Italian operator) know of the Mettlen-Lavorgo line trip (1320 MW) and the overloading of the Sils-Soazza line (1650 MW), and requests a 300 MW demand reduction to relief the overload. 3:18.40 ETrans contacts EGL (Switzerland operator) requesting the tripping of a transformer in Soazza. 3:21.00 GRTN reduces the power imports by 300 MW.
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 20
496
Power System Dynamics and Stability
Italian Blackout (28/10/2003) 3:22.03 ATEL (Swiss energy company) changes the connection of the transformer at Lavorgo. 3:25.22 Protections trip the Sils-Soazza line (1783 MW). This is basically the beginning of the cascading events that follow, severing Italy from the rest of Europe and leading to the collapse of the Italian system.
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Frequency Stability - 21
Power System Dynamics and Stability
497
Italian Blackout (28/10/2003)
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Frequency Stability - 22
Power System Dynamics and Stability
498
Italian Blackout (28/10/2003)
➠ Imports from France:
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Frequency Stability - 23
Power System Dynamics and Stability
499
Italian Blackout (28/10/2003)
➠ Frequency in Italy:
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Frequency Stability - 24
Power System Dynamics and Stability
500
Italian Blackout (28/10/2003)
➠ Frequency and voltages in Europe:
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Frequency Stability - 25
Power System Dynamics and Stability
501
Italian Blackout (28/10/2003)
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Frequency Stability - 26
Power System Dynamics and Stability
502
Italian Blackout (28/10/2003)
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 27
Power System Dynamics and Stability
503
Italian Blackout (28/10/2003)
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Frequency Stability - 28
504
Power System Dynamics and Stability
European Interconnected Systems
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Frequency Stability - 29
505
Power System Dynamics and Stability
European Interconnected Systems System
Capacity
Peak Load
Energy
Population
GW
GW
TWh
Mio.
Nordel
94
66
405
24
UPS/IPS
337
215
1285
280
UKTSOA
85
66
400
65
Nordel
600
390
2530
450
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Frequency Stability - 30
506
Power System Dynamics and Stability
European Interconnected Systems
➠ 450 million people served ➠ 2530 TWh used ➠ 600 GW installed capacity at 500 e/kW = 300 Ge ➠ 230.000 km HV network at 400 000 e/km = 90 Ge ➠ Approx. 5.000.000 km MV+LV network ➠ 1500 e investment per EU citizen ➠ Largest man-made system
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 31
Power System Dynamics and Stability
507
European Blackout (4/11/2006)
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Frequency Stability - 32
Power System Dynamics and Stability
508
European Blackout (4/11/2006)
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Frequency Stability - 33
509
Power System Dynamics and Stability
European Blackout (4/11/2006)
➠ Area 1: The frequency drops to 49 Hz, which causes an automatic load schedding.
➠ Area 2: Real power surplus of 6000 MW.
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Frequency Stability - 34
510
Power System Dynamics and Stability
European Blackout (4/11/2006) 50.1 f [Hz] 50.0
frequency
49.9 49.8 49.7 49.6 49.5 49.4 49.3 49.2 49.1 49.0 48.9 22:10:00
22:10:10
22:10:20
22:10:30
22:10:40
22:10:50
22:11:00
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Frequency Stability - 35
Power System Dynamics and Stability
511
European Blackout (4/11/2006)
Universidad de Castilla-La Mancha June 26, 2008
Frequency Stability - 36
Power System Dynamics and Stability
512
Contents
➠ Overview ➠ UWPFLOW ➠ Matlab ➠ PSAT
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Software Tools - 1
513
Power System Dynamics and Stability
Overview
➠ Software packages for power system analysis can be basically divided into two classes of tools:
➛ Commercial softwares. ➛ Educational/research-aimed softwares.
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Software Tools - 2
Power System Dynamics and Stability
514
Overview
➠ Commercial softwares: ➛ PSS/E ➛ EuroStag ➛ Simpow ➛ CYME ➛ PowerWorld ➛ Neplan
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Software Tools - 3
515
Power System Dynamics and Stability
Overview
➠ Commercial software packages follows an “all-in-one” philosophy and are typically well-tested and computationally efficient.
➠ Despite their completeness, these softwares can result cumbersome for educational and research purposes.
➠ commercial softwares are “closed”, i.e. do not allow changing the source code or adding new algorithms.
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Software Tools - 4
516
Power System Dynamics and Stability
Overview
➠ For research purposes, the flexibility and the ability of easy prototyping are often more crucial aspects than computational efficiency.
➠ At this aim, there is a variety of open source research tools, which are typically aimed to a specific aspect of power system analysis.
➠ An example is UWPFLOW which provides an extremely robust algorithm for continuation power flow analysis.
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Software Tools - 5
517
Power System Dynamics and Stability
Overview
➠ C anf FORTRAN are very fast but requires keen programming skills and are not suitable for fast prototyping.
➠ Several high level scientific languages, such as Matlab, Mathematica and Modelica, have become more and more popular for both research and educational purposes.
➠ At this aim, there is a variety of open source research tools, which are typically aimed to a specific aspect of power system analysis.
➠ Matlab proved to be the best user choice.
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 6
Power System Dynamics and Stability
518
Overview
➠ Matlab-based power system analysis tools: ➛ Power System Toolbox (PST) ➛ MatPower ➛ Voltage Stability Toolbox (VST) ➛ Power Analysis Toolbox (PAT) ➛ Educational Simulation Tool (EST) ➛ Power system Analysis Toolbox (PSAT)
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Software Tools - 7
519
Power System Dynamics and Stability
Overview
➠ Comparison of Matlab-based power system analysis softwares: Package
PF
EST
X
CPF
OPF
SSA
TD
X
X X
MatEMTP MatPower
X
PAT
X
PSAT
X
X
PST
X
X
SPS
X
VST
X
EMT
GUI
GNE
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
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Software Tools - 8
520
Power System Dynamics and Stability
Overview
➠ The features illustrated in the table are: ➛ power flow (PF) ➛ continuation power flow and/or voltage stability analysis (CPF-VS) ➛ optimal power flow (OPF) ➛ small signal stability analysis (SSA) ➛ time domain simulation (TD) ➛ graphical user interface (GUI) ➛ graphical network editor (GNE).
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Software Tools - 9
521
Power System Dynamics and Stability
Overview
➠ An important but often missed issue is that the Matlab environment is a commercial and “closed” product, thus Matlab kernel and libraries cannot be modified nor freely distributed.
➠ To allow exchanging ideas and effectively improving scientific research, both the toolbox and the platform on which the toolbox runs should be free (Richard Stallman).
➠ An alternative to Matlab is the free GNU/Octave project.
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Software Tools - 10
522
Power System Dynamics and Stability
UWPFLOW
➠ UWPFLOW is a research tool that has been designed to calculate local bifurcations related to system limits or singularities in the system Jacobian.
➠ The program also generates a series of output files that allow further analyses, such as tangent vectors, left and right eigenvectors at a singular bifurcation point, Jacobians, power flow solutions at different loading levels, voltage stability indices, etc.
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Software Tools - 11
523
Power System Dynamics and Stability
UWPFLOW Features
➠ Adequate handling of generators limits, with generators being able to recover from a variety of limits, including S limits.
➠ Steady state models of generators and their control limits (AVR and Primemover limits) are included.
➠ Voltage dependent load models for voltage stability analysis are also included.
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Software Tools - 12
524
Power System Dynamics and Stability
UWPFLOW Features
➠ Either BPA/WSCC ac-dc (HVDC systems) input data formats (and variations) or IEEE common format may be used.
➠ Detailed and reliable steady state models of SVC, TCSC and STATCOM models, and their controls with the corresponding limits are included.
➠ Secondary voltage control, as defined by ENEL (elecetricity company of Italy), can be modeled and simulated in the program.
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Software Tools - 13
525
Power System Dynamics and Stability
UWPFLOW Features
➠ The program is able to compute the minimum real eigenvalue and the related right and left eigenvectors and several voltage stability indices.
➠ The program generates a wide variety of output ASCII and MATLAB (.m) files as well as IEEE common format data files.
➠ The program has being designed to automatically run script files.
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Software Tools - 14
526
Power System Dynamics and Stability
UWPFLOW Usage
➠ Like any other UNIX program, i.e., command-line options (-option) with redirection of output (>) from screen into files: uwpflow [-options] input_file [[>]output_file]
➠ For example, to generate the program help: uwpflow -h
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Software Tools - 15
527
Power System Dynamics and Stability
UWPFLOW Example
➠ 3-area sample system: 100 MW 150 MW 150 MW 60 MVAr
v3
150 MW 56 MVAr
Bus 2
Area 1
1.02∠0
50 MVAr R = 0.01 p.u. X = 0.15 p.u.
V2 ∠δ2
Bus 3 50 MVAr
50 MW 40 MVAr 100 MW
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Power System Dynamics and Stability
528
UWPFLOW Example
➠ For a 3-area sample system: Bus
∆PG
∆PL
∆QL
Name
(p.u.)
(p.u.)
(p.u.)
Area 1
1.5
0
0
Area 2
0
1.5
0.56
Area 3
0.5
0.5
0.40
➠ Using UWPFLOW to obtain the system PV curves, for a distributed slack bus model:
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Software Tools - 17
529
Power System Dynamics and Stability
UWPFLOW Example
➠ Data file in EPRI format (3area.wsc): HDG UWPFLOW data file, WSCC format 3-area example April 2000 BAS C C AC BUSES C C | SHUNT | C |Ow|Name |kV |Z|PL |QL |MW |Mva|PM |PG |QM |Qm |Vpu BE 1 Area 1 138 1 150 60 0 0 0 150 0 0 1.02 B 1 Area 2 138 2 150 56 0 50 0 100 0 0 1.00 B 1 Area 3 138 3 50 40 0 50 0 100 0 0 1.00
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 18
Power System Dynamics and Stability
530
UWPFLOW Example
➠ Data file in EPRI format (3area.wsc): C C AC LINES C C M CS N C |Ow|Name_1 |kV1||Name_2 |kV2|||In || R | X | G/2 | B/2 |Mil| L 1 Area 1 138 Area 2 1381 15001 .01 .15 L 1 Area 1 138 Area 3 1381 15001 .01 .15 L 1 Area 2 138 Area 3 1381 15001 .01 .15 C C SOLUTION CONTROL CARD C C |Max| |SLACK BUS | C |Itr| |Name |kV| |Angle | SOL 50 Area 1 138 0. END
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Software Tools - 19
531
Power System Dynamics and Stability
UWPFLOW Example
➠ Generator and load change file (3area.k): C C UWPFLOW load and generation "direction" file C for 3-area example C C BusNumber BusName DPg Pnl Qnl PgMax [ Smax 1 0 1.5 0.0 0.0 0 0 2 0 0.0 1.5 0.56 0 0 3 0 0.5 0.5 0.40 0 0
Vmax Vmin ] 1.05 0.95 1.05 0.95 1.05 0.95
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Software Tools - 20
532
Power System Dynamics and Stability
UWPFLOW Example
➠ Batch file for UNIX (run3area): echo -1- Run base case power flow uwpflow 3area.wsc -K3area.k echo -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s echo - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 echo - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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Software Tools - 21
533
Power System Dynamics and Stability
UWPFLOW Example
➠ Batch file for Windows (run3area.bat): rem -1- Run base case power flow uwpflow 3area.wsc -K3area.k rem -2- Obatin PV curves and maximum loading uwpflow 3area.wsc -K3area.k -cthreearea.m -m -ltmp.l -s rem - with bus voltage limits enforced uwpflow 3area.wsc -K3area.k -c -7 -k0.1 rem - with current limits enforced uwpflow 3area.wsc -K3area.k -c -ltmp.l -8 -k0.1
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Software Tools - 22
534
Power System Dynamics and Stability
UWPFLOW Example
➠ PV curves (threearea.m): Profiles 150
100
50
0
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
kVArea 3
138
kVArea 2
138
kVArea 1
138
1.4
1.6
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Software Tools - 23
535
Power System Dynamics and Stability
UWPFLOW Example
➠ The singular value index obtained with UWPFLOW is as follows (-0 option): 0.7
Full matrix sing. value index
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8 L.F. [p.u.]
1
1.2
1.4
1.6
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Software Tools - 24
536
Power System Dynamics and Stability
Matlab Overview
➠ Matlab is a general purpose environment for mathematical and engineering analysis.
➠ Is vector/matrix based. A variable is by default a matrix. ➠ Is an “interpreted” language, thus can be slow for heavy applications. ➠ Is not open source. The GNU-Octave project provides a good alternative: http://www.octave.org
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Software Tools - 25
537
Power System Dynamics and Stability
Matlab Example
➠ Generator-load example (see the introductory example in the Voltage Stability section starting from slide 279): : PG + jQG
PL + jQL jxL
V1 ∠δ1
V2 ∠δ2
Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 26
538
Power System Dynamics and Stability
Matlab Example
➠ For QGmin ≤ QG ≤ QGmax : V10 V2 1 sin δ − DG ω Pd − ω˙ = M XL 1 V10 V2 δ˙ = ω − sin δ − Pd DL XL 2 1 V V V 10 2 V˙ 2 = cos δ − Qd − 2 + τ XL XL 2 V10 V10 V2 0 = QG − − + cos δ XL XL
with
x = [ω, δ, V2 ]T
y = QG
p = V10
λ = Pd
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539
Power System Dynamics and Stability
Matlab Example
➠ For QG = QGmin,max : ω˙ δ˙ V˙ 2 0
1 M
V1 V2 = sin δ − DG ω Pd − XL V1 V2 1 sin δ − Pd = ω− DL XL 2 1 V1 V2 V = cos δ − Qd − 2 + τ XL XL = QGmin,max
V1 V2 V12 + cos δ − XL XL
with
x = [ω, δ, V2 ]T
y = V1
p = QGmin,max
λ = Pd
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540
Power System Dynamics and Stability
Matlab Example
➠ Assume XL = 0.5, M = 1, DG = 0.01, DL = 0.1, τ = 0.01, k = 0.25. ➠ The time domain integration can be solved with the help of M ATLAB.
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Software Tools - 29
Power System Dynamics and Stability
541
Matlab Example
➠ Differential equations without limits: function dx = example(t,x) global M DL DG tau k Pd V10 if t <= 1 XL = 0.5; else XL = 0.6; end delta = x(1); omega = x(2); V2 = max(x(3),0); dx(1,1) = omega - (V10*V2*sin(delta)/XL-Pd)/DL; dx(2,1) = (Pd-V10*V2*sin(delta)/XL-DG*omega)/M; dx(3,1) = (-V2*V2/XL+V10*V2*cos(delta)/XL-k*Pd)/tau; Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 30
Power System Dynamics and Stability
542
Matlab Example
➠ Initialization and main routine: clear all global M DL DG tau k Pd V10 XL = 0.5; M = 0.1; DL = 0.1; DG = 0.01; tau = 0.01; k = 0.25; Pd = 0.7; V10 = 1; x0 = [0.4636; 0.0000; 0.7826]; tmax = 2; [t,x] = ode23(’example’,[0 tmax],x0);
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Matlab Example
➠ Graphical commands: figure plot(t,x(:,1),’b-’) hold on plot(t,x(:,2),’g--’) plot(t,max(x(:,3),0),’c-.’) plot([0 tmax],[V10 V10],’r:’) legend(’delta’,’omega’,’V2’,’V1’) xlabel(’t [s]’) ylim([-1 6])
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Power System Dynamics and Stability
Matlab Example
➠ The dynamic solution without limits: 6
δ ω V2 V1
5
4
3 Contingency
Operating point
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t [s] Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
545
Matlab Example
➠ Differential equations without limits: function dx = example(t,x) global M DL DG tau k Pd V10 if t <= 1, XL = 0.5; else, XL = 0.6; end delta = x(1); omega = x(2); V2 = max(x(3),0); V10 = 1; Q = V10*V10/XL - V10*V2*cos(delta)/XL; if Q > 0.5 a = 1/XL; b = -V2*cos(delta)/XL; c = -0.5; V10 = (-b + sqrt(b*b - 4*a*c))/2/a; end dx(1,1) = omega - (V10*V2*sin(delta)/XL-Pd)/DL; dx(2,1) = (Pd-V10*V2*sin(delta)/XL-DG*omega)/M; dx(3,1) = (-V2*V2/XL+V10*V2*cos(delta)/XL-k*Pd)/tau; Universidad de Castilla-La Mancha June 26, 2008
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546
Power System Dynamics and Stability
Matlab Example
➠ The dynamic solution with limits: 6
δ ω V2 V1
5
4
3 Operating point
Contingency
2
1
0 Voltage collapse −1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t [s] Universidad de Castilla-La Mancha June 26, 2008
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Power System Dynamics and Stability
547
PSAT Features
➠ PSAT has been thought to be portable and open source. ➠ PSAT runs on the commonest operating systems ➠ PSAT can perform several power system analysis: 1. Continuation Power Flow (CPF); 2. Optimal Power Flow (OPF); 3. Small signal stability analysis; 4. Time domain simulations.
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548
Power System Dynamics and Stability
PSAT Features
➠ PSAT deeply exploits Matlab vectorized computations and sparse matrix functions in order to optimize performances.
➠ PSAT also contains interfaces to UWPFLOW and GAMS which highly extend PSAT ability to solve CPF and OPF problems, respectively.
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549
Synoptic Scheme Simulink Models
Input
Other Data Format
Saved Results
Data Files Simulink Library
Simulink Model Conversion
Conversion Utilities
Power Flow & State Variable Initialization
User Defined Models
Settings
Interfaces GAMS
UWpflow
PSAT
Static Analysis
Dynamic Analysis
Optimal Power Flow
Small Signal Stability
Continuation Power Flow
Time Domain Simulation
Command History
Output
Plotting Utilities
Text Output
Save Results
Graphic Output
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550
Power System Dynamics and Stability
PSAT Features
➠ In order to perform accurate and complete power system analyses, PSAT supports a variety of static and dynamic models.
➠ Dynamic models include non conventional loads, synchronous machines and controls, regulating transformers, FACTS, wind turbines, and fuel cells.
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Power System Dynamics and Stability
551
PSAT Features
➠ Besides mathematical algorithms and models, PSAT includes a variety of additional tools, as follows: 1. User-friendly graphical user interfaces; 2. Simulink library for one-line network diagrams; 3. Data file conversion to and from other formats; 4. User defined model editor and installer; 5. Command line usage.
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552
Power System Dynamics and Stability
PSAT Features
➠ Not all features are available on GNU-Octave: Function
Matlab
GNU/Octave
Continuation power flow
yes
yes
Optimal power flow
yes
yes
Small signal stability analysis
yes
yes
Time domain simulation
yes
yes
GUIs and Simulink library
yes
no
Data format conversion
yes
yes
User defined models
yes
no
Command line usage
yes
yes
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Power System Dynamics and Stability
Getting Started
➠ PSAT is launched by typing at the Matlab prompt: >> psat which will create all structures required by the toolbox and open the main GUI.
➠ All procedures implemented in PSAT can be launched from this window by means of menus, buttons and/or short cuts.
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Power System Dynamics and Stability
554
Getting Started
➠ Main PSAT GUI:
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555
Power System Dynamics and Stability
Simulink Library
➠ PSAT allows drawing electrical schemes by means of pictorial blocks. ➠ The PSAT computational engine is purely Matlab-based and the Simulink environment is used only as graphical tool.
➠ A byproduct of this approach is that PSAT can run on GNU/Octave, which is currently not providing a Simulink clone.
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Power System Dynamics and Stability
556
Simulink Library
➠ PSAT-Simulink Library:
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557
Power System Dynamics and Stability
Other Features
➠ To ensure portability and promote contributions, PSAT is provided with a variety of tools, such as a set of Data Format Conversion (DFC) functions and the capability of defining User Defined Models (UDMs).
➠ The set of DFC functions allows converting data files to and from formats commonly in use in power system analysis. These include: IEEE, EPRI, PTI, PSAP, PSS/E, CYME, MatPower and PST formats. On Matlab platforms, an easy-to-use GUI handles the DFC.
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Power System Dynamics and Stability
558
Data Format Conversion
➠ GUI for data format conversion:
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559
Power System Dynamics and Stability
User Defined Models
➠ The UDM tools allow extending the capabilities of PSAT and help end-users to quickly set up their own models.
➠ Once the user has introduced the variables and defined the DAE of the new model in the UDM GUI, PSAT automatically compiles equations, computes symbolic expression of Jacobians matrices and writes a Matlab function of the new component.
➠ Then the user can save the model definition and/or install the model in PSAT.
➠ If the component is not needed any longer it can be uninstalled using the UDM installer as well.
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Power System Dynamics and Stability
560
User Defined Models
➠ GUI for user defined models:
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Power System Dynamics and Stability
Command Line Usage
➠ PSAT is provided with a command line version. This feature allows using PSAT in the following conditions: 1) If it is not possible or very slow to visualize the graphical environment (e.g. Matlab is running on a remote server). 2) If one wants to write scripting of computations or include calls to PSAT functions within user defined programs. 3) If PSAT runs on the GNU/Octave platform, which currently neither provides GUI tools nor a Simulink-like environment.
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562
Power System Dynamics and Stability
Power System Model
➠ The standard power system model is basically a set of nonlinear differential algebraic equations, as follows:
x˙ = f (x, y, p) 0
= g(x, y, p)
∈ Rn ; y are the algebraic variables y ∈ Rm ; p are the independent variables p ∈ Rℓ ; f are the differential equations f : Rn × Rm × Rℓ 7→ Rn ; and g are the algebraic equations g : Rm × Rm × Rℓ 7→ Rm . where x are the state variables x
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563
Power System Dynamics and Stability
Power System Model
➠ PSAT uses these equations in all algorithms, namely power flow, CPF, OPF, small signal stability analysis and time domain simulation.
➠ The algebraic equations g are obtained as the sum of all active and reactive power injections at buses:
X gpc gp gpm − ∀m ∈ M g(x, y, p) = = gq gqm c∈Cm gqc
where gpm and gqm are the power flows in transmission lines, M is the T T T ] are the set and the power , gqc set of network buses, Cm and [gpc
injections of components connected at bus m, respectively.
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564
Power System Dynamics and Stability
Component Models
➠ PSAT is component-oriented, i.e. any component is defined independently of the rest of the program as a set of nonlinear differential-algebraic equations, as follows:
x˙ c
= fc (xc , yc , pc )
Pc
= gpc (xc , yc , pc )
Qc
= gqc (xc , yc , pc )
where xc are the component state variables, yc the algebraic variables
V and θ at the buses to which the component is connected) and pc are independent variables. Then differential equations f are built concatenating fc of all components.
(i.e.
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Power System Dynamics and Stability
Component Models
➠ These equations along with Jacobians matrices are defined in a function which is used for both static and dynamic analyses.
➠ In addition to this function, a component is defined by means of a structure, which contains data, parameters and the interconnection to the grid.
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566
Power System Dynamics and Stability
Component Models: Example
➠ Let’s consider the exponential recovery load (ERL). ➠ The set of differential-algebraic equations are as follows: x˙ c1 x˙ c2
= −xc1 /TP + P0 (V /V0 )αs − P0 (V /V0 )αt
= −xc2 /TQ + Q0 (V /V0 )βs − Q0 (V /V0 )βt
Pc
= xc1 /TP + P0 (V /V0 )αt
Qc
= xc2 /TQ + Q0 (V /V0 )βt
where and P0 , Q0 and V0 are initial powers and voltages, respectively, as given by the power flow solution.
➠ Observe that a constant PQ load must be connected at the same bus as the ERL to determine the values of P0 , Q0 and V0 . Universidad de Castilla-La Mancha June 26, 2008
Software Tools - 55
567
Power System Dynamics and Stability
Component Models: Example
➠ Component Data: Column
Variable
Description
Unit
1
-
Bus number
int
2
Sn
Power rating
MVA
3
Vn
Active power voltage coefficient
kV
4
fn
Active power frequency coefficient
Hz
5
TP
Real power time constant
s
6
TQ
Reactive power time constant
s
7
αs
Static real power exponent
-
8
αt
Dynamic real power exponent
-
9
βs
Static reactive power exponent
-
10
βt
Dynamic reactive power exponent
-
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568
Power System Dynamics and Stability
Component Models: Example
➠ Exponential recovery loads are defined in the structure Erload, whose fields are as follows: 1.
con: exponential recovery load data.
2.
bus: Indexes of buses to which the ERLs are connected.
3.
dat: Initial powers and voltages (P0 , Q0 and V0 ).
4.
n: Total number of ERLs.
5.
xp: Indexes of the state variable xc1 .
6.
xq: Indexes of the state variable xc2 .
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569
Power System Dynamics and Stability
PSAT Example
➠ This section illustrates some PSAT features for static and dynamic stability analysis by means of the IEEE 14-bus test system.
➠ All data can be retrieved from the PSAT web site: http://www.power.uwaterloo.ca/∼fmilano/
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Power System Dynamics and Stability
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PSAT Example
➠ IEEE 14-bus test system:
Bus 13 |V| = 1.047 p.u.
Bus 14 |V| = 1.0207 p.u.
Bus 10 |V| = 1.0318 p.u.
Bus 12 |V| = 1.0534 p.u.
Bus 09
Bus 11
|V| = 1.0328 p.u.
|V| = 1.0471 p.u.
Bus 07 |V| = 1.0493 p.u.
Bus 06 |V| = 1.07 p.u.
Bus 04
Bus 08
|V| = 1.09 p.u. |V| = 1.012 p.u.
Bus 05 Bus 01
|V| = 1.016 p.u.
|V| = 1.06 p.u.
Breaker
Bus 02 |V| = 1.045 p.u.
Breaker Bus 03 |V| = 1.01 p.u.
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Power System Dynamics and Stability
571
PSAT Example
➠ Power flow report:
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Power System Dynamics and Stability
572
PSAT Example
➠ Continuation power flow analysis (GUI):
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Power System Dynamics and Stability
573
PSAT Example
➠ Continuation power flow analysis (plots):
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574
Power System Dynamics and Stability
PSAT Example
➠ Nose curves at bus 14 for different contingencies for the IEEE 14-bus test system:
Base Case
1
Line 2-4 Outage Line 2-3 Outage
Voltage [p.u.]
0.8 0.6 0.4 0.2 0
1
1.1
1.2
1.3
1.4 λc
1.5
1.6
1.7
1.8
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Power System Dynamics and Stability
575
PSAT Example
➠ Optimal power flow analysis (GUI):
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576
Power System Dynamics and Stability
PSAT Example
➠ Comparison between OPF and CPF analysis: BCP
λ∗
M LC
ALC
[MW]
[p.u.]
[MW]
[MW]
None
259
0.7211
445.8
186.8
Line 2-4 Outage
259
0.5427
399.5
148.6
Line 2-3 Outage
259
0.2852
332.8
73.85
Contingency
➠ Because of the definitions of generator and load powers PG and PL , one has λc = λ∗ + 1.
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577
Power System Dynamics and Stability
PSAT Example
➠ Time domain simulation: ➛ It has been used a 40% load increase with respect to the base case loading, and no PSS at bus 1. A Hopf bifurcation occurs for the line 2-4 outage resulting in undamped oscillations of generator angles.
➛ A similar analysis can be carried on the same system with a 40% load increase but considering the PSS of the generator connected at bus 1. In this case the system is stable.
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578
Power System Dynamics and Stability
PSAT Example
➠ Time domain simulation (without PSS):
Generator Speeds [p.u.]
1.002 1.0015
ω1 - Bus 1 ω2 - Bus 2
1.001
ω3 - Bus 3 ω4 - Bus 6
1.0005
ω5 - Bus 8
1 0.9995 0.999 0.9985 0.998
0
5
10
15
20
25
30
Time [s]
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579
Power System Dynamics and Stability
PSAT Example
➠ Eigenvalue analysis (with PSS): 10 8 6 4
Imag
2 0 −2 −4 −6 −8 −10
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Real
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Power System Dynamics and Stability
580
Project 1
➠ Reproduce the examples illustrated in the following slides: ➛ Voltage Stability: Slides 341-350 (using UWPFLOW) ➛ Angle Stability: Slides 444-456 (using Matlab) ➛ Frequency Stability: 479-489 (using Matlab) ➠ The software UWPFLOW is freely available at: http://thunderbox.uwaterloo.ca/∼claudio/software/pflow.html
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581
Power System Dynamics and Stability
Project 1
➠ Voltage stability (slides 341-350): ➛ Write the 3area.wsc data file in the EPRI data format (the format is fully described in the UWPFLOW documetation).
➛ Write the generator load change file 3area.k using the format described in the UWPFLOW documentation.
➛ Write a batch file for running the simulations as described in the slides.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 2
582
Power System Dynamics and Stability
Project 1
➠ Voltage stability (slides 341-350): ➛ Run a base case power flow, a continuation power flow with and without enforcing voltage and current limits. Compute also the singular value index.
➛ Report power flow results as given by UWPFLOW and plots PV curves and the singular value index using Matlab.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 3
583
Power System Dynamics and Stability
Project 1
➠ Angle stability (slides 444-456): ➛ Write a Matlab function with the system differential equations. ➛ Use a Matlab script file to initialize and solve the time domain simulation (function ode23). ➛ Using a trial-and-error technique find the clearing time tc of the system for D = 0.1, D = 0.05 and D = 0.2. ➛ For each value of the damping D, provide plots of the rotor angle δ and the rotor speed ω .
Universidad de Castilla-La Mancha June 26, 2008
Projects - 4
Power System Dynamics and Stability
584
Project 1
➠ Frequency stability (slides 479-489): ➛ Write a Matlab function with the system differential equations. ➛ Use a Matlab script file to initialize and solve the time domain simulation (function ode23). ➛ Run the time domain integration for a 25%, 50% and 60% generation drop at t = 1 followed by a 25%, 50% and 60% load drop at t = 5, respectively.
➛ For each value of the generation and load drop, provide plots of ω , δ , E ′ , V2 and V1 .
Universidad de Castilla-La Mancha June 26, 2008
Projects - 5
Power System Dynamics and Stability
585
Project 2
➠ Reproduce the results for the IEEE 14-bus tests system illustrated in the paper: F. Milano, “An Open Source Power System Analysis Toolbox”, accepted for publication on the IEEE Transactions on Power Systems, March 2005, 8 pages.
➠ The full paper as well as the software PSAT is available at: http://www.power.uwaterloo.ca/∼fmilano/
Universidad de Castilla-La Mancha June 26, 2008
Projects - 6
586
Power System Dynamics and Stability
Project 2
➠ The IEEE 14-bus test system is provided wintin the PSAT main distribution (folder tests). ➠ For the base case power flow, the continuation power flow and the optimal power flow routines, use the file:
d 014 dyn l10.mdl
➠ For the time domain simulations without PSS, use the file: d 014 dyn l14.mdl
➠ For the time domain simulations with PSS, use the file: d 014 pss l14.mdl
Universidad de Castilla-La Mancha June 26, 2008
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587
Power System Dynamics and Stability
Project 2
➠ Hints: ➛ For static analyses (PF, CPF, OPF), disable loading dynamic components by checking the box “Discard dynamic components” in the GUI Settings (within the menu “Edit” in the main window).
➛ To simulate a line outage in static analyses, uncheck the box “initially close” box (within the Simulink block mask) of the breaker of the line that one wants to keep out.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 8
Power System Dynamics and Stability
588
Project 2
➠ Hints: ➛ For OPF analysis, disable the “base case” powers in the GUI for OPF Settings and set the weighting factor to 1 (maximization of the ditance to voltage collapse).
➛ For time domain simulations, remember to uncheck “Discard dynamic components” box and do not allow the conversion to PQ buses when the program asks for.
Universidad de Castilla-La Mancha June 26, 2008
Projects - 9
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