EMTP simul(21)

7/27/2019 EMTP simul(21)

1/14

254 Power systems electromagnetic transients simulation

Build nodaladmittance

matrices

DFT

Component data

Simulationoutput(steady-state)

Simulationoutput

(transient)

Frequencydomain data

Time domaindata

Rationalfunction

coefficients ineither s or z

domain

Rational functionfitting

Frequency domainidentification

Time domainidentification

Figure 10.1 Curve-tting options

Z 13

Z 23

Z 33

Z 12

Z 22

Z 32

Z 11

Z 2 1

Z 3 1

V a

va

va

vc

V b

vb

vb

V c

vc

I a

I b

I c

=

Z 11

Z 2 1

Z 3 1

V a

V b

V c

I a

I h

I h

I h

0

0

=

. .

. .

. .

Z 22

Z 32

V a

V b

V c

I b

0

0

=

...

.

.

..

Z 33

V a

V b

V c I c

0

0=

. ..

. ..

..

Figure 10.2 Current injection

7/27/2019 EMTP simul(21)

2/14

Frequency dependent network equivalents 255

Y 13

Y 23

Y 33

Y 12

Y 22

Y 32

Y 11

Y 2 1

Y 3 1

I a

I a

I b

I b

I c

I c

I b

I c

I c

V a

V b

V c

=

Y 2 1

Y 3 1

I b

I c

V h

V h

V h

0

0

=

. .

. .

. .

Y 32

I a

I c

0

0

.. .

..

Y 33

I a

I b

I c V c

0

0=

. ..

. .

.

.

.

Y 11 I a V a

Y 22 I b V b= . .

Figure 10.3 Voltage injection

out some of the circuit parameters that need to be identied. The use of currentinjections, shown in Figure 10.2, is simpler in this respect.

10.5.1.1 Time domain analysis

Figure 10.4 displays a schematic of a system drawn in DRAFT (PSCAD/EMTDC),where a multi-sine current injection is applied. In this case a range of sine waves isinjected from 5 Hz up to 2500 Hz with 5 Hz spacing; all the magnitudes are 1.0 and theangles 0.0, hence the voltage is essentially the impedance. As the lowest frequencyinjected is 5 Hz all the sine waves add constructively every 0.2 seconds, resulting ina large peak. After the steady state is achieved, one 0.2 sec period is extracted fromthe time domain waveforms, as shown in Figure 10.5, and a DFT performed to obtainthe required frequency response. This frequency response is shown in Figure 10.6.As has been shown in Figure 10.2 the current injection gives the impedances for the

7/27/2019 EMTP simul(21)

3/14

7/27/2019 EMTP simul(21)

4/14


10 3

V c

3 0

10

10

3 0

50

0.8 0.85 0. 9 0.9 5 1

Figure 10.5 Voltage waveform from time domain simulation

10.5.1.2 Frequency domain analysis

Figure 10.7 depicts the process of generating the frequency response of an external

network as seen from its ports. A complete nodal admittance matrix of the network to be equivalenced is formed with the connection ports ordered last, i.e.

[Y f ]V f = I f (10.1)

where[Y f ] is the admittance matrix at frequency f

Vf is the vector of nodal voltages at frequency f I f is the vector of nodal currents at frequency f .

The nodal admittance matrix is of the form:

[Y f ] =

y11 y12 . . . y 1i . . . y 1k . . . y 1N y21 y22 . . . y 2i . . . y 2k . . . y 2N

......

. . ....

. . ....

. . ....

yi 1 yi 2 . . . y ii . . . y ik . . . y iN ...

.... . .

.... . .

.... . .

...yk1 yk2 . . . y ki . . . y kk . . . y kN

......

. . ....

. . ....

. . ....

yN 1 yN 2 . . . y Ni . . . y Nk . . . y NN

(10.2)

whereyki is the mutual admittance between busbars k and iyii is the self-admittance of busbar i .

7/27/2019 EMTP simul(21)

5/14

7/27/2019 EMTP simul(21)

6/14


y11

y11 y12 y1n y2 1

yn1 ykk ykN

y NN y Nk

y22

y2 n

y2 n

ynn

y11

y2 1 y3 1

y12

y12 y

1 N

y22

y22

y32

y13

y23

y33

y33

y1 N y2 N y3 N

y NN y

N 3 y

N 2 y

N 1

...

..

....

...

...

...

......

...

......

...

. . .

. . .. . .

. . . . . . . . . . . . . . .

... ... ...

0

0

0

0

0

0

0

. . .

. . .

. . .

. . .

. . .

...

. . .

. . .

. . .

. . .

. . . . . .

...

...

...

0

0

Port n

Port 1

Port 2

n Number of ports

k = N n + 1

N Number of nodes

Figure 10.7 Reduction of admittance matrices

Gaussian elimination is performed on the matrix shown in 10.2, up to, but notincluding the connection ports i.e.

y11 y12 . . . . . . y 1k . . . y 1N 0 y22 y23 . . . y 2k y2N 0 0 y33 y34 y3k y3N ...

.... . .

. . ....

...0 0 . . . 0 ykk . . . y kN ...

... . . . 0...

. . ....

0 0 . . . 0 yNk. . . yNN

(10.4)

The matrix equation based on the admittance matrix 10.4 is of the form:

[yA ] [yB ]0 [yD ]

V internalV terminal

=0

I terminal(10.5)

7/27/2019 EMTP simul(21)

7/14


2 500 Hz

5 Hz10 Hz

15 Hz

y11 y12 y1n y2 1

yn1

y22

y2 n

y2 n

ynn

......

......

. . .

. . .

. . .

y11 y12 y1n y2 1

yn1

y22

y2 n

y2 n

ynn

......

......

. . .

. . .

. . .

Figure 10.8 Multifrequency admittance matrix

The submatrix [yD ] represents the network as seen from the terminal busbars. If thereare n terminal busbars then renumbering to include only the terminal busbars gives:

y11 y1n...

. . ....

yn1 ynn

V 1...

V n

=

I 1...

I n

(10.6)

This is performed for all the frequencies of interest, giving a set of submatricesas depicted in Figure 10.8.

The frequency response is then obtained by selecting the same element from eachof the submatrices. The mutual terms are the negative of the off-diagonal terms of these reduced admittance matrices. The self-terms are the sum of all terms of a row(or column as the admittance matrix is symmetrical), i.e.

yself k =n

i = 1

yki (10.7)

The frequency response of the self and mutual elements, depicted in Figure 10.9,are matched and a FDNE such as in Figure 10.10 implemented. This is an admit-tance representation which is the most straightforward. An impedance based FDNEis achieved by inverting the submatrix of the reduced admittance matrices and match-ing each of the elements as functions of frequency. This implementation, shown inFigure 10.11 for three ports, is suitable for a state variable analysis, as an iterativeprocedure at each time point is required. Its advantages are that it is more intuitive,can overcome the topology restrictions of some programs and often results in morestable models. The frequency response is then tted with a rational function or RLC network.

Transient analysis can also be performed on the system to obtain the FDNE byrst using the steady-state time domain signals and then applying the discrete Fouriertransform.

7/27/2019 EMTP simul(21)

8/14


Frequency (Hz)

A d m i t t a n c e m a g n i

t u d e y11

y12 y13 y11

Frequency (Hz)

A d m i t t a n c e p h a s e a n g l e

Line styles

M arkers

0

0.05

0.1

0.15

0.2

0.2 5

0 500 1000 1500 2 000 2 500

100

0

100

2 00

2 000 500 1000 1500 2 000 2 500

y11 y12 y13

y

11

Figure 10.9 Frequency response

1 2

V oc1

y12

yself 2 yself 1

V oc2

Figure 10.10 Two-port frequency dependent network equivalent (admittanceimplementation)

The advantage of forming the system nodal admittance matrix at each frequencyis the simplicity by which the arbitrary frequency response of any given powersystem component can be represented. The transmission line is considered as themost frequency-dependent component and its dependence can be evaluated to great

7/27/2019 EMTP simul(21)

9/14


Z 12 I 2 + Z 13 I 3

Z 2 1 I 1 + Z 23 I 3

Z 3 1 I 1 + Z 32 I 2

I 1

I 2 Z 11

Z 22

Z 33

I 3

Figure 10.11 Three-phase frequency dependent network equivalent (impedanceimplementation)

accuracy. Other power system components are not modelled to the same accuracy atpresent due to lack of detailed data.

10.5.2 Time domain identication

Model identication can also be performed directly from time domain data. However,in order to identify the admittance or impedance at a particular frequency there mustbe a source of that frequency component. This source may be a steady-state type asin a multi-sine injection [4], or transient such as the ring down that occurs after adisturbance. Prony analysis (described in Appendix B) is the identication techniqueused for the ring down alternative.

10.6 Fitting of model parameters

10.6.1 RLC networks

The main reason for realising an RLC network is the simplicity of its implementionin existing transient analysis programs without requiring extensive modications.

7/27/2019 EMTP simul(21)

10/14


R0 R1

L0 L1 L2

C 1 C 2

Rn 1

Ln 1 Ln

C n 1 C n

R2 Rn

Figure 10.12 Ladder circuit of Hingorani and B urbery

The RLC network topology, however, inuences the equations used for the ttingas well as the accuracy that can be achieved. The parallel form (Foster circuit) [1]represents reasonably well the transmission network response but cannot model an

arbitrary frequency response. Although the synthesis of this circuit is direct, themethod rst ignores the losses to determine the L and C values for the requiredresonant frequencies and then determines the R values to match the response atminimum points. In practice an iterative optimisation procedure is necessary afterthis, to improve the t [5][7].

Almost all proposed RLC equivalent networks are variations of the ladder circuitproposed by Hingorani and Burbery [1], as shown in Figure 10.12. Figure 10.13 showsthe equivalent used by Morched and Brandwajn [6], which is the same except for theaddition of an extra branch ( C and R ) to shape the response at high frequencies.

Do and Gavrilovic [8] used a series combination of parallel branches, which althoughlooks different, is the dual of the ladder network.The use of a limited number of RLC branches gives good matches at the selected

frequencies, but their response at other frequencies is less accurate. For a xed numberof branches, the errors increase with a larger frequency range. Therefore the accuracyof an FDE can always be improved by increasing the number of branches, though atthe cost of greater complexity.

The equivalent of multiphase circuits, with mutual coupling between the phases,requires the tting of admittance matrices instead of scalar admittances.

10.6.2 Rational function

An alternative approach to RLC network tting is to t a rational function to a responseand implement the rational function directly in the transient program. The tting can

7/27/2019 EMTP simul(21)

11/14


R0 R1 Rn 1 R2 Rn R

C

L0 L1 Ln 1 L2 Ln

C 1 C n 1C 2 C n

Figure 10.13 Ladder circuit of Morched and B randwajn

be performed either in the s -domain

H(s) = e s a 0 + a 1 s + a 2 s 2 + + aN s N

1 + b1 s + b2 s 2 + + bn s n(10.8)

or in the z-domain

H(z) = e l t a 0 + a 1 z + a2 z2 + + a n z n

1 + b1 z + b2 z2 + + bn z n(10.9)

where e s or e l t represent the transmission delay associated with the mutualcoupling terms.

The s -domain has the advantage that the tted parameters are independent of thetime step; there is however a hidden error in its implementation. Moreover the ttingshould be performed up to the Nyquist frequency for the smallest time step that isever likely to be used. This results in poles being present at frequencies higher thanthe Nyquist frequency for normal simulation step size, which have no inuence onthe simulation results but add complexity.

The z-domain tting gives Norton equivalents of simpler implementation andwithout introducing error. The tting is performed only on frequencies up to theNyquist frequency and, hence, all the poles are in the frequency range of interest.However the parameters are functions of the time step and hence the tting must beperformed again if the time step is altered.

7/27/2019 EMTP simul(21)

12/14


The two main classes of methods are:

1 Non-linearoptimisation (e.g. vector-tting and the LevenbergMarquardt method),which are iterative methods.

2 Linearised least squares or weighted least squares (WLS). These are direct fastmethods based on SVD or the normal equation approach for solving an over-determined linear system. To determine the coefcients the following equation issolved:

d 11 d 12 d 1, 2m+ 1d 21 d 22 d 1, 2m+ 1...

.

.... .

.

..d k1 d k2 d k, 2m+ 1

.

b1b2...

bm

a 0a 1...

a m

=

c(j 1) c(j 2)... c(j k )

d(j 1) d(j 2)... d(j k )

(10.10)

This equation is of the form [D ] x = b whereb is the vector of measurement points ( b i = H(j i ) = c(j i ) + jd(j i ))[D ] is the design matrixx is the vector of coefcients to be determined.

When using the linearised least squares method the tting can be carried out inthe s or z-domain, using the frequency or time domain by simply changing the designmatrix used. Details of this process are given in Appendix B and it should be notedthat the design matrix represents an over-sampled system.

10.6.2.1 Error and gure of merit

The percentage error is not a useful index, as often the function to be tted passesthrough zero. Instead, either the percentage of maximum value or the actual error canbe used.

Some of the gures of merit (FOM) that have been used to rate the goodness of t are:

Error RMS = ni = 1 y Fittedi y Datai2

n(10.11)

Error Normalised = ni = 1 y Fittedi y Datai2

ni = 1 y Datai 2(10.12)

Error Max = MAX y Fittedi yDatai (10.13)

The t must be stable for the simulation to be possible; of course the stability of thet can be easily tested after performing the t, the difculty being the incorporation

7/27/2019 EMTP simul(21)

13/14

7/27/2019 EMTP simul(21)

14/14


Transforming back to discrete time:

i(n t) = a 0v(n t) + a 1 v(n t t ) + a 2 v(n t 2 t )

+ +a m

v(n t

m t)

(b 1 i(n t

t )

+b2 i(n t

2 t )

+ + bm i(n t m t))

= G equiv v(n t) + I History (10.16)

where

G equiv = a 0

I History = a 1 v(n t t ) + a 2 v(n t 2 t ) + + a m v(n t m t)

(b 1 i(n t t ) + b2 i(n t 2 t ) + + bm i(n t m t))

As mentioned in Chapter 2 this is often referred to as an ARMA (autoregressivemoving average) model.

Hence any rational function in the z-domain is easily implemented without error,as it is simply a Norton equivalent with a conductance a 0 and a current source I History ,as depicted in Figure 2.3 (Chapter 2).

A rational function in s must be discretised in the same way as is done whensolving the main circuit or a control function. Thus, with the help of the root-matching

technique and partial fraction expansion, a high order rational function can be splitinto lower order rational functions (i.e. 1 st or 2 nd ). Each 1 st or 2 nd term is turned intoa Norton equivalent using the root-matching (or some other discretisation) techniqueand then the Norton current sources are added, as well as the conductances.

10.8 Examples

Figure 10.14 displays the frequency response of the following transfer function [11]:

f(s) = 1s + 5

+ 30 + j 40s ( 100 j 500 )

+ 30 j 40s ( 100 j 500 )

+ 0.5

The numerator and denominator coefcients are given in Table 10.1 while thepoles and zeros are shown in Table 10.2. In practice the order of the response is notknown and hence various orders are tried to determine the best.

Figure 10.15 shows a comparison of three different tting methods, i.e. leastsquares tting, vector tting and non-linear optimisation. Allgave acceptable ts withvector tting performing the best followed by least squares tting. The correspondingerrors for the three methods are shown in Figure 10.16. The vector-tting error is soclose to zero that it makes the zero error grid line look thicker, while the dotted leastsquares t is just above this.

Obtaining stable ts for well behaved frequency responses is straightforward,whatever the method chosen. However the frequency response of transmission lines

Documents

EMTP simul(21)