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Structure of the PSCAD/EMTDC program 339

The main component models used in EMTDC, i.e. transmission lines, syn-

chronous generators and transformers, as well as control and switching modelling

techniques, have already been discussed in previous chapters.

Due to the popularity of the WINDOWS operating system on personal comput-

ers, a complete rewrite of the successful UNIX version was performed, resulting in

PSCAD version 3. New features include:

The function of DRAFT and RUNTIME has been combined so that plots are put

on the circuit schematic (as shown in Figure A.8).

The new graphical user interface also supports: hierarchical design of circuit pages

and localised data generation only for modified pages, single-line diagram data

entry, direct plotting of all simulation voltages, currents and control signals, without

writing to output files and more flexible multiple-run control.

A MATLAB to PSCAD/EMTDC interface has been developed. The interfaceenables controls or devices to be developed in MATLAB, and then connected in

any sequence to EMTDC components. Full access to the MATLAB toolboxes will

be supported, as well as the full range of MATLAB 2D and 3D plotting commands.

EMTDC V3 includes ideal switches with zero resistance, ideal voltage sources,

improved storage methods and faster switching operations. Fortran 90/95 will be

given greater support.

A new solution algorithm (the root-matching technique) is implemented for control

circuits which eliminates the errors due to trapezoidal integration but which is still

numerically stable.

Figure A.8 PSCAD Version 3 interface

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340 Power systems electromagnetic transients simulation

New transmission-line and cable models using the phase domain (as opposed to

modal domain) techniques coupled with more efficient curve fitting algorithms

have been implemented, although the old models are available for compatibility

purposes.

To date an equivalent for the very powerful MULTIPLOT post-processing program

is not available, necessitating exporting to MATLAB for processing and plotting.

PSCAD version 2 had many branch quantities that were accessed using the node

numbers of its terminals (e.g. CDC, EDC, GDC, CCDC, etc.). These have been

replaced by arrays (GEQ, CBR, EBR, CCBR, etc.) that are indexed by branch num-

bers. For example CBR(10,2) is the 10th branch in subsystem 2. This allows an

infinite number of branches in parallel whereas version 2 only allowed three switched

branches in parallel. Version 2 had a time delay in the plotting of current through indi-

vidual parallel switches (only in plotting but not in calculations). This was because themain algorithm only computed the current through all the switches in parallel, and the

allocation of current in individual switches was calculated from a subroutine called

from DSDYN. Old version 2 code can still run on version 3, as interface functions

have been developed that scan through all branches until a branch with the correct

sending and receiving nodes is located. Version 2 code that modifies the conductance

matrix GDC directly needs to be manually changed to GEQ.

Version 4 of PSCAD/EMTDC is at present being developed. In version 3 a circuit

can be split into subpages using page components on the main page. If there are

ten page components on the main page connected by transmission lines or cables,

then there will be ten subsystems regardless of the number of subpages branching

off other pages. Version 4 has a new single line diagram capability as well as a new

transmission line and cable interface consisting of one object, instead of the three

currently used (sending end, receiving end and line constants information page). The

main page will show multiple pages with transmission lines directly connected to

electrical connections on the subpage components. PSCAD will optimally determine

the subsystem splitting and will form subsystems wherever possible.

A.1 References

1 WOODFORD, D. A., INO, T., MATHUR, R. M., GOLE, A. M. and WIERCKX, R.:

Validation of digital simulation of HVdc transients by field tests, IEE Conference

Publication on AC and DC power transmission, 1985, 255, pp. 37781

2 WOODFORD, D. A., GOLE, A. M. and MENZIES, R. W.: Digital simulation of

DC links and AC machines, IEEE Transactions on Power Apparatus and Systems,

1983, 102 (6), pp. 161623

3 KUFFEL, P., KENT, K. and IRWIN, G. D.: The implementation and effectiveness

of linear interpolation within digital simulation, Electrical Power and EnergySystems, 1997, 19 (4), pp. 2214

4 Manitoba HVdc Research Centre: PSCAD/EMTDC power systems simulation

software tutorial manual, 1994

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AppendixB

System identification techniques

B.1 s-domain identification (frequency domain)

The rational function in s to be fitted to the frequency domain data is:

H(s) =a0 + a1s

1 + a2s2 + + ans

N

1 + b1s1 + b2s2 + + bnsn(B.1)

where N n.

The frequency response of equation B.1 is:

H(j) =

Nk=0(ak (j)

k )nk=0(bk (j)

k )(B.2)

where b0 = 1.

Letting the sample data be c(j)+ jd(j), and equating it to equation B.2 yields

c(j) + jd(j) =a0 a2

2k + a4

4k + ja1k a3

3k a5

5k

1 b22k + b44k

+ j

b1k b33k b5

5k

(B.3)

or

(c(j) + jd(j))

1 b22k + b4

4k

+ j

b1k b3

3k b5

5k

=

a0 a22k + a4

4k

+ j

a1k a3

3k a5

5k

(B.4)

Splitting into real and imaginary parts yields:

dk

(j) b1

k

b3

3

k b

55

k + c(j) b

22

k+ b

44

k

a0 a22k + a4

4k

= c(j)

dk (j)

b22k + b4

4k

+ c(j)

b1k b3

3k b5

5k

a1k a33k a5

5k

= d(j)

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This must hold for each sample point and therefore assembling into a matrix equation

gives

d(j1)1 c(j1)21 d(j1)

31 t1 1 0

21 0

41 t2

d(j2)2 c(j2)22 d(j2)32 t1 1 0

22 0

42 t2

......

.... . .

......

......

......

. . ....

d(jk)k c(jk )2k d(jk )

3k t1 1 0

2k 0

4k t2

c(j1)1 d(j1)21 c(j1)

31 t3 0 1 0

31 0 t4

c(j2)2 d(j2)22 c(j2)

32 t3 0 2 0

32 0 t4

......

.... . .

......

......

......

. . ....

c(jk )k d(jk )2k c(jk )

3k t3 0 k 0

3k 0 t4

b1b2...

bna0a1...

an

=

c(j1)

c(j2)...

c(jk )d(j1)d(j2)

...d(jk )

(B.5)

where the terms t1, t2, t3, and t4 are

t1 =

sin

l

2

lk d(jk ) + cos

l

2

lk c (j k )

t2 = cos

l

2

lk

t3 =

cos

l

2

lk d(j k ) + sin

l

2

lk c (j k )

t4 = sin

l

2

lk

l = column number

k = row or sample number.

Equation B.5 is of the form:

[A11] [A12]

[A21] [A22]

a

b

=

C

D

(B.6)

where

aT = (a0, a1, a2, a3, . . . , an)

bT = (b1, b2, b3, . . . , bn)CT = (c(j1), c(j2), c(j3) , . . . , c(jk ))

DT = (d(j1), d(j2), d(j3) , . . . , d(jk ))

Equation B.6 is solved for the required coefficients (as and bs).

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System identification techniques 343

B.2 z-domain identification (frequency domain)

The rational function in the z-domain to be fitted is:

H(z) =a0 + a1z1 + a2z2 + + anzn

1 + b1z1 + b2z2 + + bnzn(B.7)

Evaluating the frequency response of the rational function in the z-domain and

equating it to the sample data (c(j) + jd(j)) yields

c(j) + jd(j) =

nk=0

ak e

kjt

1 +n

k=1

bk ekjt

(B.8)

Multiplying both sides by the denominator and rearranging gives:

c(j) jd(j) =

nk=1

(bk (c(j) + jd(j)) ak )e

kjt

+ a0

Splitting into real and imaginary components gives:

nk=1

((bk c(j) ak ) cos(kt) bk d(j) sin(kt)) a0 = c(j) (B.9)

for the real part and

nk=1

((bk c(j) ak ) sin(kt) + bk d(j) cos(kt)) = d(j) (B.10)

for the imaginary part.

Grouping in terms of the coefficients that are to be solved for (ak and bk ) yields:

nk=1

(bk (c(j) cos(kt) + d(j) sin(kt)) ak cos(kt)) a0 = c(j)

(B.11)

nk=1

(bk (d(j) cos(kt) c(j) sin(kt)) + ak sin(kt)) = d(j)

(B.12)

This must hold for all sample points. Combining these equations for each sample

point in matrix form gives the following matrix equation to be solved:[A11] [A12]

[A21] [A22]

a

b

=

C

D

(B.13)

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where

aT = (a0, a1, a2, a3, . . . , an)

bT = (b1, b2, b3, . . . , bn)

C

T =(

c(j1),

c(j2),

c(j3) , . . . ,

c(jm))DT = (d(j1), d(j2), d(j3) , . . . , d(jm))

This is the same equation as for the s-domain fitting, except that the matrix [A]

(called the design matrix) comprises four different submatrices [A11], [A12], [A21],

and [A22], i.e.

A11 =

1 cos(1t ) cos(n1t)1 cos(2t) cos(n2t)

......

. . ....

1 cos(mt) cos(nmt )

A21 =

0 sin(1t) sin(n1t)

0 sin(2t ) sin(n2t)...

.... . .

...

0 sin(mt) sin(nmt)

A12 =

R11 R12 R1nR21 R22 R2n

......

. . ....

Rm1 Rm2 Rmn

A22 =

S11 S12 S1nS21 S22 S2n

......

. . ....

Sm1 Sm2 Smn

where

Rik = c(ki ) cos(ki t) + d(ki ) sin(ki t)

Sik = d(ki ) cos(ki t ) c(ki ) sin(ki t)

n = order of the rational function

m = number of frequency sample points.

As the number of sample points exceeds the number of unknown coefficients

singular value decomposition is used to solve equation B.13.

The least squares approach smears out the fitting error over the frequency spec-

trum. This is undesirable as it is important to obtain accurately the steady-state

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condition. Adding weighting factors allows this to be achieved. The power frequency

is typically given a weighting of 100 (the other frequencies are weighted 1.0).

Adding weighting factors results in equations B.11 and B.12 becoming:

c(j)w(j) =

nk=1

(w(j)bk (c(j) cos(kt) + d(j) sin(kt))

ak w(j) sin(kt)) w(j)a0 (B.14)

d(j)w(j) =

nk=1

(w(j)bk (d(j) cos(kt) c(j) sin(kt))

ak w(j) sin(kt)) (B.15)

B.3 z-domain identification (time domain)

When the sampled data consists of samples in time rather than frequency, a rational

function in the z-domain can be identified, provided the system has been excited by a

waveform that contains the frequency components at which the matching is required.

This is achieved by a multi-sine injection.Given a rational function of the form of equation B.7, if admittance is being fitted

then

I(z)

1 + b1z1 + b2z

2 + + bnzn

=

a0 + a1z

1 + a2z2 + + anz

n

V (z) (B.16)

or

I(z) = I(z)

b1z1 + b2z

2 + + bnzn

+

a0 + a1z

1 + a2z2 + + anz

n

V (z) (B.17)

Taking the inverse z-transform gives:

i(t) =

nk=1

bk i(t kt) +

nk=0

ak v(t kt) (B.18)

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and in matrix form

v(t1) v(t1 t) v(t1 2t) v(t1 nt) i(t1 t) i(t1 2t ) i(t1 nt)

v(t2) v(t2 t) v(t2 2t) v(t2 nt) i(t2 t) i(t2 2t) i(t2 nt)

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

v(tk ) v(tk t) v(tk 2t) v(tk nt) i(tk t) i(tk 2t ) i(tk nt)

a0a1...

anb1b2...

bn

=

i(t1)

i(t2)

..

.

i(tk )

(B.19)

where

k = time sample number

n = order of the rational function (k > p, i.e. over-sampled).

The time step must be chosen sufficiently small to avoid aliasing, i.e. t =

1/ (K1fmax), where K1 > 2 (Nyquist criteria) and fmax is the highest frequency

injected. For instance if K1 = 10 and t = 50s there will be 4000 samples

points per cycle (20 ms for 50 Hz). This equivalent is easily extended to multi-port

equivalents [1]. For an m-port equivalent there will be m(m + 1)/2 rational functions

to be fitted.

B.4 Prony analysis

Prony analysis identifies a rational function that will have a prescribed time-domain

response [2]. Given the rational function:

H(z) =Y(z)

U(z)=

a0 + a1z1 + a2z

2 + + anzN

1 + b1z1 + b2z2 + + bdzn(B.20)

the impulse response ofh(k) is related to H(z) by the z-transform, i.e.

H(z) =

k=0

h(k)z1 (B.21)

which can be written as

Y(z)

1 + b1z1 + b2z

2 + + bdzn

= U(z)

a0 + a1z1 + a2z

2 + + anzN

(B.22)

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This is the z-domain equivalent of a convolution in the time domain. Using the first

L terms of the impulse response the convolution can be expressed as:

h0 0 0 0h1 h0 0 0

h2 h1 h0 0...

......

. . ....

hn hn1 hn2 h0hn+1 hn hn1 h1

......

.... . .

...

hL hL1 hL2 hLn

1

b1b2...

bn

=

a0a1

...

aN0

0...

0

(B.23)

Partitioning gives: a

0

=

[H1]

[h1] | [H2]

1

b

(B.24)

The dimensions of the vectors and matrices are:

a (N + 1) vector

b (n + 1) vector

[H1] (N + 1) (n + 1) matrix

[h1] vector of last (L N ) terms of impulse response

[H2] (L N ) (n) matrix.

The b coefficients are determined by using the sample points more than n time

steps after the input has been removed. When this occurs the output is no longer

a function of the input (equation B.22) but only depends on the b coefficients and

previous output values (lower partition of equation B.24), i.e.

0 = [h1] + [H2] b

or

[h1] = [H2]b (B.25)

Once the b coefficients are determined the a coefficients are obtained from the

upper partition of equation B.24, i.e.

b = [H1]a

When L = N + n then H2 is square and, if non-singular, a unique solution for b is

obtained. IfH2 is singular many solutions exist, in which case h(k) can be generated

by a lower order system.

When m > n + N the system is over-determined and the task is to find a and bcoefficients that give the best fit (minimise the error). This can be obtained solving

equation B.25 using the SVD or normal equation approach, i.e.

[H2]T[h1] = [H2]

T[H2]b

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B.5 Recursive least-squares curve-fitting algorithm

A least-squares curve fitting algorithm is described here to extract the fundamental

frequency data based on a least squared error technique. We assume a sinusoidal signal

with a frequency of radians/sec and a phase shift of relative to some arbitrary

time T0, i.e.

y(t) = A sin(t ) (B.26)

where = T0.

This can be rewritten as

y(t) = A sin(t) cos(T0) A cos(t) sin(T0) (B.27)

Letting C1

= A cos(T0

) and C2

= A sin(T0

) and representing sin(t) and cos(t)

by functions F1(t) and F2(t) respectively, then:

y(t) = C1F1(t) + C2F2(t ) (B.28)

F1(t) and F2(t ) are known if the fundamental frequency is known. However, the

amplitude and phase of the fundamental frequency need to be found, so equation B.28

has to be solved for C1 and C2. If the signal y(t) is distorted, then its deviation from

a sinusoid can be described by an error function E, i.e.

x(t) = y(t) + E (B.29)

For a least squares method of curve fitting, the size of the error function is measured

by the sum of the individual residual squared values, such that:

E =

ni=1

{xi yi }2 (B.30)

where xi = x(t0 + it) and yi = y(t0 + it). From equation B.28

E =

ni=1

{xi C1F1(ti ) C2F2(ti )}2 (B.31)

where the residual value r at each discrete step is defined as:

ri = xi C1F1(ti ) C2F2(ti ) (B.32)

In matrix form:

r1r2...

rn

=

x1x2...

xn

F1(t1) F2(t1)F1(t2) F2(t2)

......

F1(tn) F2(tn)

C1C2

(B.33)

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or r

=

X

F

C

(B.34)

The error component can be described in terms of the residual matrix as follows:

E = [r]T[r] = r21 + r22 + + r

2n

= [[X] [F][C]]T[[X] [F][C]]

= [X]T[X] [C]T[F]T[X] [X]T[F][C] + [C]T[F]T[F][C] (B.35)

This error then needs to be minimised, i.e.

E

C= 2[F]T[X] + 2[F]T[F][C] = 0

[F]T [F][C] = [F]T[X]

[C] = [[F]T[F]]1[F]T[X]

(B.36)

If[A] = [F]T[F] and [B] = [F]T[X] then:

[C] = [A]1[B] (B.37)

Therefore

[A] =

F1F2

F1 F2

=

F1F1(ti ) F1F2(ti )

F2F1(ti ) F2F2(ti )

=

a11 a12a21 a22

Elements of matrix [A] can then be derived as follows:

a11n =

F1(t1)...

F1(tn)

T

F1(t1)...

F1(tn)

=

n1i=1

F21 (ti ) + F2

1 (tn)

= a11n1 + F2

1 (tn) (B.38)

etc.

Similarly:

[B] =

F1(ti )x(ti )

F2(ti )x(ti )

=

b1b2

and

b1n

= b1n1

+ F1

(tn

)x(tn

) (B.39)

b2n = b2n1 + F2(tn)x(tn) (B.40)

From these matrix element equations, C1 and C2 can be calculated recursively using

sequential data.

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B.6 References

1 ABUR, A. and SINGH, H.: Time domain modeling of external systems for elec-

tromagnetic transients programs, IEEE Transactions on Power Systems, 1993,

8 (2), pp. 67177

2 PARK, T. W. and BURRUS, C. S.: Digital filter design (John Wiley Interscience,

New York, 1987)

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Appendix C

Numerical integration

C.1 Review of classical methods

Numerical integration is needed to calculate the solution x(t + t) at time t + t

from knowledge of previous time points. The local truncation error (LTE) is the error

introduced by the solution at x(t + t) assuming that the previous time points are

exact. Thus the total error in the solution x(t + t) is determined by LTE and the

build-up of error at previous time points (i.e. its propagation through the solution).

The stability characteristics of the integration algorithm are a function of how thiserror propagates.

A numerical integration algorithm is either explicit or implicit. In an explicit

integration algorithm the integral of a function f, from t to t + t, is obtained

without using f (t + t). An example of explicit integration is the forward Euler

method:

x(t + t) = x(t) + t f(x(t),t) (C.1)

In an implicit integration algorithm f (x(t+ t),t+ t) is required to calculate

the solution at x(t + t ). Examples are, the backward Euler method, i.e.

x(t + t) = x(t) + t f (x(t + t),t+ t) (C.2)

and the trapezoidal rule, i.e.

x(t + t) = x(t) +t

2[f(x(t),t) + f (x(t+ t),t+ t)] (C.3)

There are various ways of developing numerical integration algorithms, such as

manipulation of Taylor series expansions or the use of numerical solution by poly-

nomial approximation. Among the wealth of material from the literature, only a few

of the classical numerical integration algorithms have been selected for presentation

here [1][3].

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