03 Y Bus Matrix

Network

u

u

u

Nodes represent substation bus bars Branches represent transmission lines and transformers Injected currents are the flows from generator and loads

Used to form the network model of an interconnected power system

Iinj = Ybus ⋅ Vnode

Ik Vk

The matrix equation for relating the nodal voltages to the currents that flow into and out of a network using the admittance values of circuit branches

Power Systems I

l

l

The Bus Admittance Matrix

u

u

1 1 yij = = zij rij + j xij

impedances are converted to admittances

I k −inj = yk 0 Vk + yk 1 (Vk − V1 ) + yk 2 (Vk − V2 ) + K + ykn (Vk − Vn )

form the nodal solution based upon Kirchhoff’s current law

Constructing the Bus Admittance Matrix (or the Y bus matrix)

Power Systems I

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The Bus Admittance Matrix

line 23 z = j0.2 line 34 z = j0.08

line 12 z = j0.4

Power Systems I

2

generator 2 z = j0.8

4 Network Diagram

3

line 13 z = j0.2

1

generator 1 z = j1.0

j0.08

2 j0.2

4 Impedance Diagram

3

j0.2

1

j0.8

j1.0

j0.4

V2

V1

Matrix Formation Example

y34 = -j12.5 4

3

y12 = -j2.5 y13= -j5 y23= -j5

y20= -j1.25

y10= -j1.0

Power Systems I

Admittance Diagram

1

I1

KCL Equations

0 = y43 (V4 − V3 )

0 = y31 (V3 − V1 ) + y32 (V3 − V2 ) + y34 (V3 − V4 )

I 2 = y20V2 + y21 (V2 − V1 ) + y23 (V2 − V3 )

I1 = y10V1 + y12 (V1 − V2 ) + y13 (V1 − V3 )

2

I2

Matrix Formation Example

Power Systems I

 I1  ( y10 + y12 + y13 ) I   − y21  2 =  − y31 0     0 0 

− y43

− y32 0

( y31 + y32 + y34 )

− y13 − y23

− y12 ( y20 + y21 + y23 )

Matrix Formation of the Equations

0 = − y43V3 + y43V4

0 = − y31V1 − y32V2 + ( y31 + y32 + y34 )V3 − y34V4

I 2 = − y21V1 + ( y20 + y21 + y23 )V2 − y23V3

Rearranging the KCL Equations I1 = ( y10 + y12 + y13 )V1 − y12V2 − y13V3

Matrix Formation Example

0  V1  0  V2  ⋅  − y34  V3     y43  V4 

Power Systems I

0  V1  j5.00  I1  − j8.50 j 2.50  I   j 2.50 − j8.75  V  5 . 00 0 j  2 =   ⋅  2 j5.00 − j 22.50 j12.50  V3   0   j5.00       − 0 0 12 . 50 12 . 50 0 j j     V4 

Completed Matrix Equation Y11 = ( y10 + y12 + y13 ) = − j8.50 Y23 = Y32 = − y23 = j5.00 Y12 = Y21 = − y12 = j 2.50 Y33 = ( y31 + y32 + y34 ) = − j 22.50 Y13 = Y31 = − y13 = j5.00 Y34 = Y43 = − y34 = j12.50 Y22 = ( y20 + y21 + y23 ) = − j8.75 Y44 = y34 = − j12.50

Matrix Formation Example

Power Systems I

Matrix is symmetrical along the leading diagonal

l

Yij = Y ji = − yij

j≠i

Off-diagonal elements:

j =0

Yii = ∑ yij

Square matrix with dimensions equal to the number of buses Convert all network impedances into admittances n Diagonal elements:

l

l

l

l

Y-Bus Matrix Building Rules

Power Systems I

Line g1 g2 L1 L2 L3 L4 L5 L6

System Data Start End X value 1 0 1.00 5 0 1.25 1 2 0.40 1 3 0.50 2 3 0.25 2 5 0.20 3 4 0.125 4 5 0.50

Example

u

u

u

bus i

1:a

bus j complex number

a can be a

The flow of real power along a network branch is controlled by the angular difference of the terminal voltages The flow of reactive power along a network branch is controlled by the magnitude difference of the terminal voltages Real and reactive powers can be adjusted by voltage-regulating transformers and by phase-shifting transformers

The tap-changing transform gives some control of the power network by changing the voltages and current magnitudes and angles by small amounts

Power Systems I

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Tap-Changing Transformers

Ii

yt

Vx = 1a V j

Vx

I i = −a* ⋅ I j

basic circuit equations:

Vi

I i = yt (Vi − Vx )

1:a

Ij

Vj

the off-nominal tap ratio is given as 1:a the nominal turns-ratio (N1/N2) was addressed with the conversion of the network to per unit the transformer is modeled as two elements joined together at a fictitious bus x

Power Systems I

u

u

u

u

Modeling of Tap-Changers

I i = yt (Vi − Vx )

yt yt yt 1 I j = − * (Vi − a V j ) = − * Vi + 2 V j a a a

I j = − a1* I i

I i = −a* ⋅ I j

I i = yt (Vi − 1a V j )

Vx = 1a V j

Making substitutions

Power Systems I

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Modeling of Tap-Changers

 I i   yt  I  = − y a *  j  t

− yt a  Vi  2⋅ yt a  V j 

 y   yt  I j = − * Vi +  t2 V j  a   a 

 yt  I i = {yt }Vi + − V j  a

Matrix formation

Power Systems I

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YBus Formation of Tap-Changers

u

u

(a - 1) yt / a

i

non-tap side

yt / a

(1 - a) yt / a2

j

tap side

the off-diagonal element represent the impedance across the two buses the remainder form the shunt element

Valid for real values of a Taking the y-bus formation, break the diagonal elements into two components

Power Systems I

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l

Pi-Circuit Model of Tap-Changers