Techref Overhead Line Models[1]

DIgSILENT Technical Documentation

Overhead Line Models

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Overhead Line Models Published by DIgSILENT GmbH, Germany Copyright 2007. All rights reserved. Unauthorised copying or publishing of this or any part of this document is prohibited. TechRef LineModels V1 Build 246 30.03.2007

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Table of Contents

Table of Contents 1 OVERHEAD LINE MODELS IN DIGSILENT POWERFACTORY ................................................................................. 4 1.1 GENERAL EQUATIONS FOR TRANSMISSION LINES.......................................................................................................... 4 1.2 LUMPED PARAMETER TRANSMISSION LINE MODEL ........................................................................................................ 6 1.3 DISTRIBUTED PARAMETER TRANSMISSION LINE............................................................................................................ 7 1.3.1 Bergeron´s Method. General Approach......................................................................................................................... 7 1.3.2 Line Model with Constant Parameters........................................................................................................................... 8 1.3.3 Line Model with frequency Dependent Parameters ...................................................................................................... 10 2 REFERENCES ....................................................................................................................................................... 11

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1 Overhead Line Models in DIgSILENT PowerFactory For steady-state solutions, such as load flow, short circuit or frequency sweep calculations among others, overhead lines can be modelled with sufficient accuracy as two-port networks with lumped parameters or PI-Circuits. For transient (RMS- and EMT-) simulations however, more sophisticated models are required. Depending on line length and involved frequencies, lines can usually any longer be represented with lumped parameters; distributed ones are required instead. Even more, constants or frequency-dependent distributed parameters can be considered. All these models are available in PowerFactory 13.1. In the following sections the basic principles behind these models will be explain, so that the user be able to select the most suitable one depending on each particularly study case. Models are presented first for single-phase lines and then generalized to multiple-phase lines by means of the modal transform.

1.1 General Equations for Transmission Lines Equations (1) and (2) describe the frequency behaviour of an incremental transmission line model of elemental

∆x

length

as depicted in Figure 1.

∆V I(x,t)

R´

I(x+∆x,t)

L´ ∆I

V(x,t) V(x+∆x,t) G´

C´

∆x x

x ∆x

Figure 1: Incremental model for a line of elemental length

Z′

∂V = I ( x) ⋅ Z ′ ∂x

(1)

∂I = V ( x) ⋅ Y ′ ∂x

(2)

is the series impedance per-unit length corresponding to the line voltage drop and

Y′

the shunt admittance

representing the current drawn to earth per-unit length as defined in (3) and (4) respectively.

Z ′ = R′ + jω ⋅ L′

(3)

Y ′ = G ′ + jω ⋅ C ′

(4)

The second derivatives of (1) and (2) with respect to x let us separate the voltage from the current and build the system of differential equations (5) and (6).

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∂ 2V = Z ′ ⋅ Y ′ ⋅ V ( x) ∂x 2

(5)

∂2I = Z ′ ⋅ Y ′ ⋅ I ( x) ∂x 2

(6)

The general solution is of the form:

U ( x) = K1 ⋅ eγ ⋅ x + K 2 ⋅ e −γ ⋅ x

(7)

Z C ⋅ I ( x) = − K1 ⋅ eγ ⋅ x + K 2 ⋅ e −γ ⋅ x

(8)

where

ZC

ZC

and

γ

is the characteristic impedance of the line as defined in (9) and

ZC =

γ

the propagation constant (10). Both

are frequency-dependent and uniquely characterize the behaviour of the transmission line.

Z′ Y′

(9)

γ = Z′⋅Y′ K1

and

(10)

K 2 are adjusted to verify the border conditions at node s. According to the references in Figure 2:

⎧U ( x = 0) = U s x=0⇒⎨ ⎩ I ( x = 0) = I s

and

⎧U ( x = l ) = U r x=l⇒⎨ ⎩I ( x = l ) = − I r

⎛V + I ⋅Z ⎞ ⎛V − I ⋅Z ⎞ Vr = ⎜ s s C ⎟ ⋅ eγ ⋅l + ⎜ s s C ⎟ ⋅ e −γ ⋅l 2 2 ⎠ ⎝ ⎠ ⎝

(11)

⎞ − γ ⋅l ⎟⋅e ⎟ ⎠

(12)

⎛V − I ⋅Z I r = ⎜⎜ s s C ⎝ 2 ⋅ ZC

Is

⎞ γ ⋅l ⎟⋅e ⎟ ⎠

Zc , γ

Vs

⎛V + I ⋅Z − ⎜⎜ s s C ⎝ 2 ⋅ ZC

Ir Vr

Figure 2:Border conditions Using matrix representation, (11) and (12) can be rewritten in terms of hyperbolic functions as (13), which is usually known as ABCD -parameter representation of two-port networks.

⎡ cosh γ ⋅ l ⎡Vr ⎤ ⎢ = ⎢ I ⎥ ⎢ 1 ⋅ sinh γ ⋅ l ⎣ r⎦ ⎣ ZC

− Z C ⋅ sinh γ ⋅ l ⎤ ⎥ = ⎡ A B ⎤ ⋅ ⎡Vs ⎤ − cosh γ ⋅ l ⎥ ⎢C D ⎥ ⎢ I s ⎥ ⎣ ⎦ ⎣ ⎦ ⎦

(13)

Equation (13) describes accurately the input-output relationship of the transmission line in the frequency domain. Further explanations about the derivation of these equations can be found in [1,2].

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1.2 Lumped Parameter Transmission Line Model The ABCD parameters deduced before can be used now to calculate the circuit elements of an equivalent Π circuit of the line, as depicted in Figure 3.

Zπ

Is Vs

Yπ

Zπ= −Β

Ir Vr

Yπ

Yπ = (1−Α)/Β

Figure 3: Elements of equivalent π circuit where

Zπ = Z C ⋅ sinh γ ⋅ l = Z ′ ⋅ l ⋅

sinh γ ⋅ l γ ⋅l

cosh γ ⋅ l − 1 1 = ⋅Y ′ ⋅ l ⋅ Yπ = Z C ⋅ sinh γ ⋅ l 2

⎛ γ ⋅l ⎞ thg ⎜ ⎟ ⎝ 2 ⎠ γ ⋅l 2

(14)

(15)

Zπ and Yπ in Figure 3 are frequency-dependent parameters. This circuit exactly represents the transmission line for any frequency and hence it is used for modelling long transmission lines in steady-states calculations (for example load flow or harmonics). However, it can not be used for transient simulations, where many frequencies are involved at the same time. Let consider however the series expansion of the hyperbolic functions defining the ABCD parameters.

1 1 1 cosh ϑ = 1 + ⋅ ϑ 2 + ⋅ ϑ 4 + ⋅ϑ 6 + K 2 24 720 sinh ϑ

ϑ Using

1 1 1 = 1 + ⋅ϑ 2 + ⋅ϑ 4 + ⋅ϑ 6 + K 6 120 5040

ϑ = γ ⋅l = Z′ ⋅Y′ ⋅l

(16)

(17)

A and B can be expanded as follows:

1 1 2 A = cosh γ ⋅ l = 1 + ⋅ Z ′ ⋅ Y ′ ⋅ l 2 + ⋅ (Z ′ ⋅ Y ′) ⋅ l 4 + K 2 24

(18)

⎛ sinh γ ⋅ l ⎞ 1 ⎡ 1 ⎤ 2 ⎟⎟ = Z ′ ⋅ l ⋅ ⎢1 + ⋅ Z ′ ⋅ Y ′ ⋅ l 2 + B = Z ′ ⋅ l ⋅ ⎜⎜ ⋅ (Z ′ ⋅ Y ′) ⋅ l 4 + K⎥ 120 ⎣ 6 ⎦ ⎝ γ ⋅l ⎠

(19)

Taking into consideration up to the second order terms of the series expansion for A and B, equations (18) and (19) can be approximated as:

Z π = B = Z ′ ⋅ l = R ′ ⋅ l + j ω ⋅ L′ ⋅ l

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(20)

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A −1 Yπ = = B

1 1+ ⋅ Z ′ ⋅Y ′ ⋅ l 2 1 1 2 = ⋅ Y ′ ⋅ l = ⋅ (G′ ⋅ l + jω ⋅ C ′ ⋅ l ) 2 2 Z′⋅l

(21)

The resulting Π circuit, usually known as nominal Π, is shown in Figure 4. Different to the prior model, this circuit contains only lumped R, L, G and C elements. Hence the model is not only useful for steady-state calculations (short line model) but it can be use for transients simulations as well.

R´

G´/2

C´/2

L´

G´/2

C´/2

Figure 4: Nominal Π model for transmission lines. The model error depends on the weight of truncated terms in the series expansion, which on his part depends on the factor

f ⋅l

(frequency times length). For overhead lines less than 250 km and power frequency, this approximation

is very satisfactory and the error can be neglected. Longer lines can be modelled in PF by dividing them into routes, derived from the same line type. In this case the transmission line is divided into N transmission routes and then N mathematical Π models will be used in the calculations. The following general rule is to be regarded: the higher the involved frequencies are, the shorter is the line length, which can be represented with one nominal Π model and i.e. the larger the number of routes which are required. However, transient simulations involving higher frequencies require more accurate models for long transmission lines. Distributed parameter models should be preferred, which are described in the next section.

1.3 Distributed Parameter Transmission Line 1.3.1 Bergeron´s Method. General Approach. We refer again to (11) and (12), which represent the input-output relationships for the transmission line. Border conditions have been adjusted according to Figure 2. We rewrite the expressions for explicitness:

⎛V − I ⋅Z ⎞ ⎛V − I ⋅Z ⎞ Vr = ⎜ s s C ⎟ ⋅ eγ ⋅l + ⎜ s s C ⎟ ⋅ e −γ ⋅l 2 2 ⎠ ⎝ ⎠ ⎝ ⎛V − I ⋅Z ⎞ ⎛V + I ⋅Z ⎞ Z C ⋅ I r = ⎜ s s C ⎟ ⋅ eγ ⋅l − ⎜ s s C ⎟ ⋅ e −γ ⋅l 2 2 ⎝ ⎠ ⎝ ⎠ For the wave travelling from node s to node k we can obtain an expression of the form (V+I Zc ) by subtracting (12) from (11),

U r − Z C ⋅ I r = (U s + Z C ⋅ I s ) ⋅ e −γ ⋅l

(22)

or rewritten as

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Ir =

Ur ⎛ U − ⎜ Is + s Z C ⎜⎝ ZC

⎞ − γ ⋅l ⎟⋅e ⎟ ⎠

(23)

As it can be observed in (22), the expression

U s + Z C ⋅ I s for the border conditions encountered when leaving node e −γ ⋅l . The same

s, is the same when arriving at node r, after having been multiplied with the propagation factor applies for (23).

Repeating the same procedure but setting now the initial conditions at node r and then travelling with the wave from node r to node s, we can obtain,

U s − Z C ⋅ I s = (U r + Z C ⋅ I r ) ⋅ e −γ ⋅l

(24)

or rewritten as

Is =

Us ⎛ U − ⎜⎜ I r + r ZC ⎝ ZC

⎞ − γ ⋅l ⎟⋅e ⎟ ⎠

(25)

These are the Bergeron’s equations in the frequency domain. According to (22) to (25) the line behaviour at both I

ends can be represented then by controlled current sources ( W ) in parallel with an impedance controlled voltage source ( W

U

) in serie with an imedance

Z C or by Z C , as shown in Error! Reference source not

found.. To convert the equation set into the time domain we have to apply the inverse Fourier transform,

{

}

(26)

{

}

(27)

u s (t ) = F −1 Z C ⋅ I s + (U r + Z C ⋅ I r ) ⋅ e −γ ⋅l

ur (t ) = F −1 Z C ⋅ I r + (U s + Z C ⋅ I s ) ⋅ e −γ ⋅l or rewritten

⎫ ⎧U ⎛ U ⎞ is (t ) = F −1 ⎨ s − ⎜ I r + r ⎟ ⋅ e −γ ⋅l ⎬ Zc Zc ⎝ ⎠ ⎭ ⎩ ⎧U ⎛ ⎫ U ⎞ ir (t ) = F −1 ⎨ r − ⎜ I s + s ⎟ ⋅ e −γ ⋅l ⎬ Zc ⎠ ⎩ Zc ⎝ ⎭

(28)

(29)

where both, the characteristic impedance

Z C = Z C (ω ) =

R ′ + j ω ⋅ L′ G ′ + jω ⋅ C ′

(30)

and the propagation constant

γ = γ (ω ) = α (ω ) + jβ (ω ) =

(R′ + jω ⋅ L′) ⋅ (G′ + jω ⋅ C ′)

(31)

are frequency dependent, even for constant distributed parameters R’, L’, G’ and C’.

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1.3.2 Line Model with Constant Parameters A simple model for constant distributed parameters can be obtained for the case of lossless and distortionless lines. In that case, (30) reduces to

ZC =

L′ C′

(32)

being real and frequency independent. The dumping coefficient

α =0

and hence from (31)

γ = jβ = jω L′ ⋅ C ′

(33)

Regardless of frequency, all waves travel with the same propagation velocity, given by

v=

ω = β

1 L′ ⋅ C ′

(34)

so that we can define a travel time (frequency independent) as

τ=

l = l ⋅ L′ ⋅ C ′ v

(35)

In terms of travel time, the propagation constant can be rewritten as

γ ⋅ l = j β ⋅ l = j ω L′ ⋅ C ′ ⋅ l = j ω ⋅ τ

(36)

and after simplifications we can rewrite (24) as

U s = Z C ⋅ I s + WrU

(37)

U r = Z C ⋅ I r + WsU

(38)

with

WrU = (U r + Z C ⋅ I r ) ⋅ e

− jω ⋅τ

WsU = (U s + Z C ⋅ I s ) ⋅ e

− jω ⋅τ

The inverse Fourier transform of the phase shift

(39)

(40)

e

− jω ⋅τ

in the frequency domain becomes a time delay

τ

in the time

domain and (37) to (40) therefore transform to

u s (t ) = Z C ⋅ is (t ) + wUr

(41)

ur (t ) = Z C ⋅ ir (t ) + wUs

(42)

wUr = ur (t − τ 0 ) + Z C ⋅ ir (t − τ )

(43)

wUs = u s (t − τ 0 ) + Z C ⋅ is (t − τ )

(44)

Idem in term of current sources

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is (t ) =

u s (t ) + wrI (t ) ZC

(45)

ir (t ) =

ur (t ) + wsI (t ) ZC

(46)

wrI (t ) = ir (t − τ ) +

ur (t − τ ) ZC

(47)

wsI (t ) = is (t − τ ) +

us (t − τ ) ZC

(48)

The equivalent circuits are shown in Error! Reference source not found., either with voltage or current sources. For the transient simulation, the voltages and currents at one side of the line are calculated upon the voltage and current at the other side one time delay back in time (picked up from the history vector), which is the Bergeron’s method.

1.3.3 Line Model with frequency Dependent Parameters Except for lossless and distortionless lines as seen above, the characteristic impedance (30) and propagation constant (31) are frequency dependent, even for lines with constant distributed parameters. The variation of

Zc

and

γ

is most pronounced in the zero sequence mode, and hence frequency-dependent models

should be preferred when zero sequence currents or voltages are involved, as for example in the case of single phase-to-ground faults. A line model, which takes into account frequency dependent parameters, can also be based on Bergeron´s method for distributed parameters. However different from the case with constant parameters, we can not make here any longer the assumption, which simplify the inverse Fourier transform mentioned in 1.3.2. In the following, only equations for the equivalent circuit with current sources are described. Similar equations can be rewritten however for the equivalent circuit with voltage sources as well. Explicitly writing the frequency-dependent parameters, the input current at node s is

Is =

Us

Z C (ω )

+ WrI (ω )

⎛ U WrI (ω ) = −⎜⎜ I r + r ZC ⎝ where

A(ω )

(49)

⎞ ⎟ ⋅ A(ω ) ⎟ ⎠

(50)

is the propagation factor defined as

A(ω ) = e −γ (ω )⋅l

(51)

Equation (49) must be transformed now from the frequency domain into the time domain. The inverse Fourier transform of the controlled current source

WrI can be evaluated by means of the convolution

integral and hence ∞ ⎡ i (t − u ) ⎤ wrI (t ) = − ∫ ⎢ir (t − u ) + r ⎥ ⋅ a(u ) ⋅ du Zc ⎦ 0⎣

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(52)

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a(t ) = F −1 {A(ω )} , τ min travel time of the fastest waves and τ max travel time of the slowest ones. Furthermore a (t ) is zero up to t = τ min and tends to zero for t → τ max . For this reason we only need to evaluate

with

τ min

the convolution integral between

and

τ max .

The still unresolved issue is the evaluation of the inverse Fourier transforms of

Z c (ω )

and

A(ω ) . Both parameters

can be approximated by rational functions directly in the frequency domain [3] and then expanded into partial fractions.

A(ω ) ,

For the propagation factor

Aapp (s ) = e − s ⋅τ with

s = jω

and

Aapp (s ) =

min

⋅k ⋅

(s + z1 )⋅ (s + z2 )K(s + zn ) (s + p1 )⋅ (s + p2 )K(s + pm )

(53)

n < m . Expanding into partial fractions k1

k2

+

(s + p1 ) (s + p2 )

+L+

km

(54)

(s + pm )

and then the inverse Fourier transform becomes a sum of exponentials:

for t < τ min ⎧0 aapp (t ) = ⎨ − p1 (t −τ min ) − p m (t −τ min ) − p 2 (t −τ min ) L kme for t ≥ τ min + k2e ⎩k1e

(55)

For the characteristic impedance follows

Z c − app (s ) = k ⋅ with

(s + z1 )⋅ (s + z2 )K(s + zn ) (s + p1 )⋅ (s + p2 )K(s + pn )

(56)

s = jω . Expanding into partial fractions Z c − app (s ) = k0 +

k1

+

k2

L

kn

(57)

(s + p1 ) (s + p2 ) (s + pn )

Equation (57) corresponds to a RC network as shown in Fig(), where:

R0 = k0

and

Ri =

ki pi

,

Ci =

1 ki

with

i = 1K n

The frequency-dependent line model seen from node s becomes a current source

(58)

aapp (t )

in parallel with a RC

network. The precision of the model depends on the quality of the rational function approximations of

Z c (ω )

and

A(ω ) .

Power Factory uses the Bode’s procedure proposed in [3]. The result of the approximations is shown in the element dialog (amplitude and phase error for each propagation mode). The user is encouraged to verify, if the approximations are accurate enough in the frequency range of interest for the transient simulation. If not the case, the frequency range for parameter fitting and the main travelling is to be readjusted.

2 References 1. 2.

H. Dommel, EMTP Theory Book. 1992. B. R. Oswald, Netzberechnung 2, Berechnung transienter Vorgänge in Elektroenergieversorgungsnetzen, VDE Verlag 1996.

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3.

J.R. Marti, The problem of frequency dependence in transmission line modelling. PhD. Thesis, The University of British Columbia, Vancouver, Canada, April 1981.

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