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114 Power systems electromagnetic transients simulation

Input type (d):

y(z)

u(z)

=b/a(eat + 1/(at)(1 eat))z1 + b/a(1 1/(at)(1 eat))

(1 z1

eat

)

Table E.3 (Appendix E) summarises the resulting difference equation for the

family of exponential forms developed using root-matching techniques. The table

also contains the difference equations derived from trapezoidal integrator substitution.

The difference equations are then converted to the form:

(a0 + a1 z1)/(b0 + b1 z

1) or (a0 + a1z

1)/(1 + b1z1) if b0 is non-zero

Tables E.1 and E.2 give the coefficients of a rational function in z1 that repre-

sents each difference equation for the family of exponential forms, for admittanceand impedance respectively. It can be shown that the difference equation obtained

assuming type input (d) is identical to that obtained from the recursive convolution

technique developed by Semlyen and Dabuleanu [7].

5.5.1 Step response

A comparison of step responses is made here with reference to the simple switching

of a series RL branch, shown in Figure 5.2. Figure 5.15 shows the current magnitude

using the difference equations generated by Dommels method, root-matching forinput types (a), (b), (c) and (d) and the theoretical result for t = ( = 50s).

Figures 5.16 and 5.17 show the same comparison for t = 5 and t = 10, respec-

tively. Note that in the latter cases Dommels method exhibits numerical oscillation.

Root-matching type (a) gives the exact answer at each time point as its discretisation of

the input is exact. Root-matching type (b) gives the exact values but one time step late

as its discretisation of the input is a step occurring one time step later. Root-matching

type (c) is an average between the previous two root-matching techniques.

Current(amps)

Time (s)

Dommel Theoretical curve

0

20

40

60

80

100

0.00095 0.0012 0.00145

RM type A RM type B RM type C RM type D

Figure 5.15 Comparison of step response of switching test system fort =

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The root-matching method 115

Current(amps)

Time (s)

0

20

40

60

80

100

120

0.0006 0.0012 0.0018 0.0024 0.003



Figure 5.16 Comparison of step response of switching test system fort = 5

C

urrent(amps)

Time (s)

0

20

40

60

80

100

120

0.0005 0.001 0.0015 0.002 0.0025 0.003



Figure 5.17 Comparison of step response of switching test system fort = 10

Although from Figure 5.13 it would seem that root-matching type (d) should

provide the best approximation to an arbitrary waveform, this input resulted in sig-

nificant inaccuracies. The reason is that this discretisation is unable to model a pure

step, i.e. there will always be a slope, which is a function of t, as depicted in

Figure 5.18. However ift is sufficiently small then this method will provide a good

approximation to a step response.

Root-matching type (b) results in terms from the previous time step only, that is

only a current source but no parallel conductance. This can cause simulation problems

if a non-ideal switch model is used. If a switch is modelled by swapping between

high and low resistance states then even when it is OFF, a very small current flow

is calculated. This current is then multiplied by et/ and injected into the high

impedance switch and source, which results in a voltage appearing at the terminals. If

an ideal switch cannot be modelled, judicious selection of the switch parameters can

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t

U(t)

Time (ms)

Figure 5.18 Root-matching type (d) approximation to a step

Voltage(volts)

Time (s)

4

2

0

2

4

0.0046 0.0047 0.0048 0.0049 0.005



Figure 5.19 Comparison with a.c. excitation (5 kHz) (t = )

remove the problem, however a better solution is to use a controlled voltage source

when applying the step in voltage.

5.5.2 Steady-state response

The second test system, shown in Figure 5.6, consists of an RL branch, excited by a

5 kHz current source. Figure 5.19 shows the voltage response using a 10s step length

for each of the difference equations. The theoretical answer is 1.86 sin(t), where

= 57.52. Root-matching types (a), (b) and (d) give good answers; however,

root-matching type (c) gives results indistinguishable from Dommels method.

It should be noted that as the excitation is a current source and root-matching

type (b) is also a pure current source, there are two current sources connected to one

node. Hence, in order to get answers for this system a parallel conductance must

be added to enable Kirchhoffs current law to be satisfied. The conductance value

must be large enough so as not to influence the solution significantly but not too

large otherwise instability will occur. However, from a stability viewpoint the poles

in the z-plane for the complete solution fall outside the unit circle when parallel

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Voltage(volts)

0.0048 0.00485 0.0049 0.00495 0.005

Time (s)

8

4

0

4

8Dommel Theoretical curve


Figure 5.20 Comparison with a.c. excitation (10 kHz) (t = )

resistance is increased. Using a voltage source rather than current source excitation

would eliminate the need for a parallel resistor in the root-matching type (b).

The same conclusions are found from a simulation using 10 kHz as the excitation

frequency and a step length of 10 s. The theoretical answer is 3.30 sin(t ),

where = 72.43. In this case root-matching types (a), (b) and type (d) give

good answers, and again, root-matching type (c) gives results indistinguishable from

Dommels method (this is shown in Figure 5.20).

5.5.3 Frequency response

The frequency response of each difference equation can be reconstructed from the

rational function by using the following equation:

Z(f) =

ni=0 ai e

jitni=0 bi e

jit(5.18)

which for root-matching, simplifies to:

1

Gequ

et/ejt

Gequ= ((1 cos(t)et/) + j sin(t))/Gequ (5.19)

The magnitude and phase components are:

|Z(f)| =

(1 cos(t)et/)2 + sin(t)2/Gequ

and

Z(f) = tan1 sin(t)

1 cos(t)et/The corresponding equation for an s-domain function is:

h(f) =

nk=0 ak (j)

k

1 +n

k=1 bk (j)k

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RM type a

RM type b

RM type c

RM type d

Dommel

Theoretical

1

1.5

2

2.5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Frequency (Hz)

Magnitude

Phase(degs)

Figure 5.21 Frequency response for various simulation methods

The results of calculations performed using MATLAB (the code is given in

Appendix F.4), are displayed in Figure 5.21. These results were verified by performing

injection tests into the appropriate difference equation using PSCAD/EMTDC sim-

ulation software. As expected, the root-matching methods (a) and (b) provide the

closest match to the theoretical magnitude response, while root-matching methods

(c) and (d) are similar to the trapezoidal rule. The phase response clearly shows the

phase advance and phase lag inherent in the various discretisations used in the vari-

ous root-matching methods. Root-matching method (a) shows the phase advance and

method (b) the phase lag expected. Root-matching methods (c) and (d) and trapezoidalintegration show a considerably better phase response.

The trapezoidal rule assumes a linear variation between time points. An expo-

nential form of the difference equation can also be derived assuming constant input

between solution points. Hence the exponential form of a circuit or transfer function

is not unique but depends on the assumed variation in input between time points.

5.6 Example

For the test system shown in Figure 5.2, if the switch is closed at t = 1.0 s the exactsolution is:

i(t) =VDC

R(1 e(t1.0)/)

for t 1.0 where = L/R.

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For this example the parameters of the circuit are: R = 100 , L = 0.05 mH,

VDC = 100 V. If t = = 50s the difference equation obtained using the

trapezoidal rule is:

i(t + t) = 13

i(t + t) + 13

(v(t + t) + v(t))

For root-matching the difference equation is:

i(t + t) = i(t)e1 + v(t + t)(1 e1)

and the results are summarised in Table 5.3.

For t = 5 = 250 s the difference equations are:

i(t + t ) = 3

7i(t + t) + 5

7(v(t + t ) + v(t)) for the trapezoidal rule

i(t + t ) = i(t)e5 + v(t + t)(1 e5) for the root-matching method

and the corresponding results are summarised in Table 5.4.

Finally for t = 10 = 500s the difference equations are:

i(t + t ) = 23

i(t + t) + 56

(v(t + t) + v(t)) for the trapezoidal rule

i(t + t ) = i(t)e10 + v(t + t)(1 e10) for the root-matching method

and the results are summarised in Table 5.5.

Table 5.3 Response fort = = 50 s

Exact solution Trapezoidal rule Root-matching

1.0 0.0 0.0 0.0

1.0 + t 63.212056 33.333333 63.212056

1.0 + 2t 86.466472 77.777777 86.466472

1.0 + 3t 95.021293 92.2530864 95.021293

1.0 + 4t 98.168436 97.530864 98.168436

1.0 + 5t 99.326205 99.176955 99.326205

Table 5.4 Response fort = 5 = 250 s


1.0 0.0 0.0 0.0

1.0 + t 99.326205 71.428571 99.326205

1.0 + 2t 99.995460 112.244898 99.9954601.0 + 3t 99.999969 94.752187 99.999969

1.0 + 4t 100.000000 102.249063 100.000000

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Table 5.5 Response fort = 10 = 500 s


1.0 0.0 0.0 0.0

1.0 + t 99.995460 83.333333 99.995460

1.0 + 2t 100.000000 111.111111 100.000000

1.0 + 3t 100.000000 92.592593 100.000000

To demonstrate why root-matching is so good let us consider the exact response

at a discrete time tk , i.e.

i(tk ) =Vdc

R(1 e(tk 1.0)/) (5.20)

which, expressed as a function of a previous time point at tk t, becomes:

i(tk ) =Vdc

R(1 e(tk 1.0)/) =

Vdc

R(1 et/e(tk t1.0)/) (5.21)

Now the same must be true for the previous time point, hence from equation 5.20:

i(tk t) =Vdc

R(1 e(tk t1.0)/) (5.22)

Hence

e(tk t1.0)/ = 1 R

Vdci(tk t ) (5.23)

Substituting equation 5.23 in equation 5.21 gives:

i(tk ) =

Vdc

R

1 et/

1 R

Vdc i(tk t)

= et/i(tk t) +

Vdc

R

1 et/

(5.24)

which is exactly the difference equation for the root-matching method.

5.7 Summary

An alternative to the difference equation using the trapezoidal integration developed

in Chapter 4 for the solution of the differential equations has been described in this

chapter. It involves the exponential form of the difference equation and has been devel-

oped using the root-matching technique. The exponential form offers the following

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advantages:

Eliminates truncation errors, and hence numerical oscillations, regardless of the

step length used.

Can be applied to both electrical networks and control blocks. Can be viewed as a Norton equivalent in exactly the same way as the difference

equation developed by the numerical integration substitution (NIS) method.

It is perfectly compatible with NIS and the matrix solution technique remains

unchanged.

Provides highly efficient and accurate time domain simulation.

The exponential form can be implemented for all series and parallel RL, RC , LC

and RLC combinations, but not arbitrary components and hence is not a replacement

for NIS but a supplement.

5.8 References

1 WATSON, N. R. and IRWIN, G. D.: Electromagnetic transient simulation of power

systems using root-matching techniques, Proceedings IEE, Part C, 1998, 145 (5),

pp. 4816

2 ANGELIDIS, G. and SEMLYEN, A.: Direct phase-domain calculation of trans-

mission line transients using two-sided recursions, IEEE Transactions on Power

Delivery, 1995, 10 (2), pp. 94173 WATSON, N. R. and IRWIN, G. D.: Accurate and stable electromagnetic transient

simulation using root-matching techniques, International Journal of Electrical

Power & Energy Systems, Elsevier Science Ltd, 1999, 21 (3), pp. 22534

4 CAMPOS-BARROS, J. G. and RANGEL, R. D.: Computer simulation of modern

power systems: the elimination of numerical oscillation caused by valve action,

Proceedings of 4th International Conference on AC and DC Power Transmission,

London, 1985, Vol. IEE Conf. Publ., 255, pp. 2549

5 WATSON, N. R. and IRWIN, G. D.: Comparison of root-matching techniques

for electromagnetic transient simulation, IEEE Transactions on Power Delivery,2000, 15 (2), pp. 62934

6 WATSON, N. R., IRWIN, G. D. and NAYAK, O.: Control modelling in electro-

magnetic transient simulations, Proceedings of International Conference on Power

System Transients (IPST99), June 1999, pp. 5448

7 SEMLYEN, A. and DABULEANU, A.: Fast and accurate switching transient

calculations on transmission lines with ground return using recursive convolutions,

IEEE Transactions on Power Apparatus and Systems, 1975, 94 (2), pp. 56171

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Chapter 6

Transmission lines and cables

6.1 Introduction

Approximate nominal PI section models are often used for short transmission lines

(of the order of 15 km), where the travel time is less than the solution time-step, but

such models are unsuitable for transmission distances. Instead, travelling wave theory

is used in the development of more realistic models.

A simple and elegant travelling wave model of the lossless transmission line has

already been described in Chapter 4 in the form of a dual Norton equivalent. Themodel is equally applicable to overhead lines and cables; the main differences arise

from the procedures used in the calculation of the electrical parameters from their

respective physical geometries. Carsons solution [1] forms the basis of the overhead

line parameter calculation, either as a numerical integration of Carsons equation,

the use of a series approximation or in the form of a complex depth of penetration.

Underground cable parameters, on the other hand, are calculated using Pollacks

equations [2], [3].

Multiconductor lines have been traditionally accommodated in the EMTP by a

transformation to natural modes to diagonalise the matrices involved. Original sta-bility problems were thought to be caused by inaccuracies in the modal domain

representation, and thus much of the effort went into the development of more accu-

rate fitting techniques. More recently, Gustavsen and Semlyen [4] have shown that,

although the phase domain is inherently stable, its associated modal domain may be

inherently unstable regardless of the fitting. This revelation has encouraged a return

to the direct modelling of lines in the phase domain.

Figure 6.1 displays a decision tree for the selection of the appropriate transmission

line model. The minimum limit for travel time is Length/c where the c is the speed

of light, and this can be compared to the time step to see if a PI section or travelling

wave model is appropriate. Various PI section models exist, however the nominal

(or coupled) PI, displayed in Figure 6.2, is the preferred option for transient solutions.

The exact equivalent PI is only adequate for steady-state solution where only one

frequency is considered.

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Is travelling time greater

than time step?

Is physical

geometry of line available

(i.e. conductor radius and

positions)?

PI section BergeronFrequency-

dependent

No

No Yes

Yes

Use travelling wave model

Use R, X& B informationFrequency-dependent

transmission line model

Start

Figure 6.1 Decision tree for transmission line model selection

R

ia

ib

ic

va va

vb

vc

vb

vc

R

R

Figure 6.2 Nominal PI section

6.2 Bergerons model

Bergerons model [5] is a simple, constant frequency method based on travelling wave

theory. It is basically the model described in Chapter 4. Here, the line is still treated

as lossless but its distributed series resistance is added in lump form. Although the

lumped resistances can be inserted throughout the line by dividing its total length into

several sections, it makes little difference to do so and the use of just two sections at

the ends is perfectly adequate. This lumped resistance model, shown in Figure 6.3,

gives reasonable answers provided that R/4 ZC , where ZC is the characteristic

(or surge) impedance. However, for high frequency studies (e.g. power line carrier)

this lumped resistance model may not be adequate.

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Transmission lines and cables 125

4R 4R2R

ZC CZ

ikm(t)

vk(t)

in (t)imk(t)

io (t)

vm (t)

Ik(t/2 )Io (t/2 )

Im (t /2 )In (t/2 )

ZC ZC

Figure 6.3 Equivalent two-port network for line with lumped losses

ikm (t)

Ik(t /2)

Im (t /2)

vk(t) vm (t)

imk(t)

Z0+R

4Z0+

R

4

Figure 6.4 Equivalent two-port network for half-line section

By assigning half of the mid-point resistance to each line section, a model of half

the line is depicted in Figure 6.4, where:

ikm(t ) =1

ZC + R/4vk (t) + Ik (t /2) (6.1)

and

Ik (t /2) =1

ZC + R/4vm(t /2)

ZC R/4

ZC + R/4

im(t /2) (6.2)

Finally, by cascading two half-line sections and eliminating the mid-point vari-

ables, as only the terminals are of interest, the model depicted in Figure 6.5 is obtained.

It has the same form as the previous models but the current source representing the

history terms is more complicated as it contains conditions from both ends on the line

at time (t /2). For example the expression for the current source at end k is:

Ik (t ) = ZC

(ZC + R/4)2(vm(t ) + (ZC R/4)imk(t ))

+R/4

(ZC + R/4)2(vk (t ) + (ZC R/4)ikm (t )) (6.3)

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ikm (t)

Ik (t )

Im (t )

vk(t) vm (t)

imk(t)

Z0+R

4Z0+

R

4

Figure 6.5 Bergeron transmission line model

In the EMTDC program the line model separates the propagation into low and high

frequency paths, so that the line can have a higher attenuation to higher frequencies.

This was an early attempt to provide frequency dependence, but newer models (in the

phase domain) are now preferred.

6.2.1 Multiconductor transmission lines

Equations 4.22 and 4.23 are also applicable to multiconductor lines by replacing the

scalar voltages and currents by vectors and using inductance and capacitance matrices.The wave propagation equations in the frequency domain are:

dVphase

dx

=

Zphase

[Iphase] (6.4)

dIphase

dx

=

Yphase

[Vphase] (6.5)

By differentiating a second time, one vector, either the voltage or current, may be

eliminated giving:

d2Vphase

dx2

=

Zphase

dIphasedx

=

Zphase

[Yphase][Vphase] (6.6)

d2Iphase

dx2

=

Yphase

dVphasedx

=

Yphase

Zphase

[Iphase] (6.7)

Traditionally the complication of having off-diagonal elements in the matrices of

equations 6.6 and 6.7 is overcome by transforming into natural modes. Eigenvalue

analysis is applied to produce diagonal matrices, thereby transforming from coupled

equations in the phase domain to decoupled equations in the modal domain. Each

equation in the modal domain is solved as for a single phase line by using modal

travelling time and modal surge impedance.

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Transmission lines and cables 127

The transformation matrices between phase and modal quantities are different for

voltage and current, i.e.

[Vphase] = [Tv][Vmode] (6.8)

[Iphase] = [Ti ][Imode] (6.9)

Substituting equation 6.8 in 6.6 gives:d2[Tv ]Vmode

dx2

=

Zphase

Yphase

[Tv][Vmode] (6.10)

Hence

d2Vmode

dx2

= [Tv]1

Zphase

Yphase

[Tv ][Vmode] = [][Vmode] (6.11)

To find the matrix [Tv ] that diagonalises [Zphase][Y

phase] its eigenvalues and eigen-

vectors must be found. However the eigenvectors are not unique as when multiplied

by a non-zero complex constant they are still valid eigenvectors, therefore some

normalisation is desirable to allow the output from different programs to be com-

pared. PSCAD/EMTDC uses the root squaring technique developed by Wedepohl for

eigenvalue analysis [6]. To enable us to generate frequency-dependent line models

the eigenvectors must be consistent from one frequency to the next, such that the

eigenvectors form a continuous function of frequency so that curve fitting can beapplied. A NewtonRaphson algorithm has been developed for this purpose [6].

Once the eigenvalue analysis has been completed then:

[Zmode] = [Tv ]1[Zphase][Ti ] (6.12)

[Ymode] = [Ti ]1[Yphase][Tv ] (6.13)

[Zsurge i ] =

Zmode(i,i)

Ymode(i,i)(6.14)

where [Zmode] and [Ymode] are diagonal matrices.

As the products [Zphase][Y

phase] and [Y

phase][Zphase] are different so are their

eigenvectors, even though their eigenvalues are identical. They are, however,

related, such that [Ti ] = ([Tv ]T)1 (assuming a normalised Euclidean norm, i.e.n

j=1 T2

ij = 1) and therefore only one of them needs to be calculated. Looking at

mode i, i.e. taking the ith equation from 6.11, gives:

d2Vmode idx2

= ii

Vmode i

(6.15)

and the general solution at point x in the line is:

Vmode i (x) = e ix VFmode i (k) + e

ix VBmode i (m) (6.16)

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