EMTP simul(22)

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268 Power systems electromagnetic transients simulation

Frequency (Hz)

M a g n i t u d e

P h a s e ( d e g s )

0.4

0.5

0.6

0.7

0.8

0

10

20

30

200 400 600 800 1000 1200

Frequency (Hz)

200 400 600 800 1000 1200

Figure 10.14 Magnitude and phase response of a rational function

Table 10.1 Numerator and denominator coefficients

Numerator Denominator

s0 7.69230769e−001 1.00000000e+000

s1 7.47692308e−002 2.00769231e−001

s2 1.26538462e−004 1.57692308e−004

s

3

3.84615385e−

007 7.69230769e−

007

Table 10.2 Poles and zeros

Zero Pole

−1.59266199e+002 + 4.07062658e+002 ∗ j −1.00000000e+002 + 5.00000000e+002 ∗ j

−1.59266199e+002 − 4.07062658e+002 ∗ j −1.00000000e+002 − 5.00000000e+002 ∗ j−1.04676019e+001 −5.00000000e+000

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Frequency dependent network equivalents 269

Original frequency responseLeast squares fittingVector fitting Non-linear optimisation

200 400 600 800 1000 1200Frequency (Hz)

Original frequency responseLeast squares fittingVector fitting Non-linear optimisation

0.5

0.6

0.7

0.8

0.4

0

10

20

30

M a g n i t u d e ( o h m s )


200 400 600 800 1000 1200Frequency (Hz)

Figure 10.15 Comparison of methods for the fitting of a rational function

200 400 600 800 1000 1200

Frequency (Hz)

200 400 600 800 1000 1200

Frequency (Hz)

R e a l e r r o r ( % )

Least Squares Fitting

Non- linear optimisation

I m a g e r r o r ( % )

Least squares fittingVector fitting Non-linear optimisation

– 3.5

– 3

– 2.5

– 2

–1.5

–1

–0.5

0

–14

–12

–10

–8

–6

–4

– 2

0

Vector fitting

Figure 10.16 Error for various fitted methods

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Load

Figure 10.17 Small passive network

Table 10.3 Coefficients of z−1 (no weighting factors)

Term Denominator Numerator

z−0 1 0.00187981208257

z−1 −5.09271275503264 −0.00942678842550

z−2 12.88062106081476 0.02312960416135

z−3 −21.58018890110835 −0.03674152374824z−4 26.73613316059277 0.04159398757818

z−5 −25.81247589702268 −0.03448198061263

z−6 19.89428694917709 0.02039138329319

z−7 −12.26856666212080 −0.00756861064417

z−8 5.88983411589258 0.00077750907595

z−9 −2.00963299687702 0.00074985289424

z−10 0.36276901898885 −0.00029244729760

and cables complicates the fitting task, as their related hyperbolic function responses

are difficult to fit. This is illustrated with reference to the simple system shown in

Figure 10.17, consisting of a transmission line and a resistive load. A z-domain fit

is performed with the parameters of Table 10.3 and the fit is shown in Figure 10.18.

As is usually the case, the fit is good at higher frequencies but deteriorates at lower

frequencies. As an error at the fundamental frequency is undesirable, a weighting

factor must be applied to ensure a good fit at this frequency; however this is achieved

at the expense of other frequencies. The coefficients obtained using the weighting

factor are given in Table 10.4. Finally Figure 10.19 shows the comparison between

the full system and FDNE for an energisation transient.

In order to use the same fitted network for an active FDNE, the same transmission

line is used with a source impedance of 1 ohm. Figure 10.20 displays the test system,

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0 1000 2000 3000 4000 5000

|Y ( f )| & | N ( z)/ D ( z)|

Frequency (Hz)

angle (Y ( f )) & angle ( N ( z)/ D ( z))

0

0.05

0.1

0.15

–100

–50

0

50

100

M a g n i t u

d e


0 1000 2000 3000 4000 5000

Figure 10.18 Magnitude and phase fit for the test system

Table 10.4 Coefficients of z−1 (weighting-factor)

Term Denominator Numerator

z−0 1 1.8753222e−003

z−1 −5.1223634e+000 −9.4562048e−003

z−2 1.3002665e+001 2.3269772e−002

z−3 −2.1840662e+001 −3.7014495e−002

z−4 2.7116238e+001 4.1906856e−002

z−5 −2.6233100e+001 −3.4689620e−002

z−6 2.0264580e+001 2.0419347e−002

z−7 −1.2531812e+001 −7.4643948e−003

z−8 6.0380835e+000 6.4923773e−004

z−9 −2.0707968e+000 8.2779560e−004

z−10 3.7723943e−001 −3.1544461e−004

which involves energisation, fault inception and fault removal. The response using the

FDNE with weighting factor is shown in Figure 10.21 and, as expected, no steady-

state error can be observed. Using the fit without weighting factor gives a better

representation during the transient but introduces a steady-state error. Figures 10.22

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–150

–100

–50

0

50

100

150

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

– 2

–1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

V o l t a g e ( k V )

C u r r e n t ( k A )

Figure 10.19 Comparison of full and a passive FDNE for an energisation transient

Frequency

dependent

network

equivalentLoad

Fault

Circuit

breaker

t close=0.096 s

t fault=0.025 s

Duration= 0.35 s

Figure 10.20 Active FDNE

and10.23 show a detailed comparison forthe latter case (i.e. without weightingfactor).

Slight differences are noticeable in the fault removal time, due to the requirement

to remove the fault at current zero. Finally, when allowing current chopping the

comparison in Figure 10.24 results.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Time (s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time (s)

ActualFDNE

– 200

–100

0

100

200

–1

0

1

2

3


C u r r e n t ( k

A )

Figure 10.21 Comparison of active FDNE response

Time (s)

ActualFDNE

–150

–100

–50

0

50

100

150

0.09 0.092 0.094 0.096 0.098 0.1 0.102 0.104 0.106 0.108

Time (s)

0.09 0.092 0.094 0.096 0.098 0.1 0.102 0.104 0.106 0.108 –0.2

–0.1

0

0.1

0.2



Figure 10.22 Energisation

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0.32 0.34 0.36 0.38 0.4 0.42 0.44

V o l t a g

e ( k V )

Time (s)

0.32 0.34 0.36 0.38 0.4 0.42 0.44

Time (s)

C u r r e n t ( k

A )

ActualFDNE

– 200

–100

0

100

200

–1

0

1

2

3

Figure 10.23 Fault inception and removal

0.37 0.375 0.38 0.385 0.39 0.395 0.4Time (s)

0.37 0.375 0.38 0.385 0.39 0.395 0.4

Time (s)

ActualFDNE

–500

0

500

1000

–1

0

1

2

3



Figure 10.24 Fault inception and removal with current chopping

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6 MORCHED, A. S. and BRANDWAJN, V.: ‘Transmission network equivalents

for electromagnetic transient studies’, IEEE Transactions on Power Apparatus

and Systems, 1983, 102 (9), pp. 2984–94

7 MORCHED, A. S., OTTEVANGERS, J. H. and MARTI, L.: ‘Multi port fre-

quency dependent network equivalents for the EMTP’, IEEE Transactions on

Power Delivery, Seattle, Washington, 1993, 8 (3), pp. 1402–12

8 DO, V. Q. and GAVRILOVIC, M. M.: ‘An interactive pole-removal method for

synthesis of power system equivalent networks’, IEEE Transactions on Power

Apparatus and Systems, 1984, 103 (8), pp. 2065–70

9 JURY, E. I.: ‘Theory and application of the z-transform method’ (John Wiley,

New York, 1964)

10 OGATA, K.: ‘Modern control engineering’ (Prentice Hall International, Upper

Saddle River, N. J., 3rd edition, 1997)

11 GUSTAVSEN, B. and SEMLYEN, A.: ‘Rational approximation of frequencydomain response by vector fitting’, IEEE Transaction on Power Delivery, 1999,

14 (3), pp. 1052–61

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Chapter 11

Steady state applications

11.1 Introduction

Knowledge of the initial conditions is critical to the solution of most power system

transients. The electromagnetic transient packages usually include some type of fre-

quency domain initialisation program [1]–[5] to try and simplify the user’s task.

These programs, however, are not part of the electromagnetic transient simulation

discussed in this book. The starting point in the simulation of a system disturbance isthe steady-state operating condition of the system prior to the disturbance.

The steady-state condition is often derived from a symmetrical (positive sequence)

fundamental frequency power-flow program. If this information is read in to initialise

the transient solution, the user must ensure that the model components used in the

power-flow program represent adequately those of the electromagnetic transient pro-

gram. In practice, component asymmetries and non-linearities will add imbalance

and distortion to the steady-state waveforms.

Alternatively the steady-state solution can be achieved by the so-called ‘brute

force’ approach; the simulation is started without performing an initial calculationand is carried out long enough for the transient to settle down to a steady-state condi-

tion. Hence the electromagnetic transient programs themselves can be used to derive

steady-state waveforms. It is, thus, an interesting matter to speculate whether the cor-

rect approach is to provide an ‘exact’ steady state initialisation for the EMTP method

or to use the latter to derive the final steady-state waveforms. The latter alternative is

discussed in this chapter with reference to power quality application.

A good introduction to the variety of topics considered under ‘power quality’ can

be found in reference [6] and an in-depth description of the methods currently used

for its assessment is given in reference [7].

An important part of power quality is steady state (and quasi-steady state)

waveform distortion. The resulting information is sometimes presented in the time

domain (e.g. notching) and more often in the frequency domain (e.g. harmonics and

interharmonics).

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Steady state applications 279

The resulting converter currents were then expressed in terms of harmonic current

injections to be used in a new iteration of the a.c. system harmonic flow. This method,

based on the fixed point iteration (or Gauss) concept, had convergence problems

under weak a.c. system conditions. An alternative IHA based on Newton’s method [9]

provided higher reliability at the expense of greatly increased analytical complexity.

However the solution accuracy achieved with these early methods was very lim-

ited due to the oversimplified modelling of the converter (in particular the idealised

representation of the converter switching instants).

An important step in solution accuracy was made with the appearance of the

so-called harmonic domain [9], a full Newton solution that took into account the mod-

ulating effect of a.c. voltage and d.c. current distortion on the switching instants and

converter control functions. This method performs a linearisation around the operating

point that provides sufficient accuracy. In the present state of harmonic domain devel-

opment the Jacobian matrix equation combines the system fundamental frequencythree-phase load-flow and the system harmonic balance in the presence of multiple

a.c.–d.c. converters. Although in principle any other type of non-linear component

can be accommodated, the formulation of each new component requires consider-

able skill and effort. Accordingly a program for the calculation of the non-sinusoidal

periodic steady state of the system may be of very high dimension and complexity.

11.4 Phase-dependent impedance of non-linear device

Using perturbations the transient programs can help to determine the phase-dependent

impedance of a non-linear device. In the steady state any power system component

can be represented by a voltage controlled current source: I = F ( V ), where I and V

are arrays of frequency phasors. The function F may be non-linear and non-analytic.

If F is linear, it may include linear cross-coupling between frequencies, and may be

non-analytic, i.e. frequency cross-coupling and phase dependence do not imply non-

linearity in the frequency domain. The linearised response of F to a single applied

frequency may be calculated by:

I RI I

=

⎡⎢⎢⎣

∂F R∂V R

∂F R∂V I

∂F I

∂V R

∂F I

∂V I

⎤⎥⎥⎦

V RV I

(11.1)

where F has been expanded into its component parts. If the Cauchy–Riemann con-

ditions hold, then 11.1 can be written in complex form. In the periodic steady state,

all passive components (e.g. RLC components) yield partial derivatives which satisfy

the Cauchy–Riemann conditions. There is, additionally, no cross-coupling between

harmonics for passive devices or circuits. With power electronic devices the Cauchy–

Riemann conditions will not hold, and there will generally be cross-harmonic coupling

as well.

In many cases it is desirable to ignore the phase dependence and obtain a complex

impedance which is as near as possible to the average phase-dependent impedance.

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Since the phase-dependent impedance describes a circle in the complex plane as

a function of the phase angle of the applied voltage [10], the appropriate phase-

independent impedance lies at the centre of the phase-dependent locus. Describing

the phase-dependent impedance as

Z =

z11 z12

z21 z22

(11.2)

the phase independent component is given by:

Z =

R −X

X R

(11.3)

where

R = 12

(z11 + z22) (11.4)

X = 12

(z21 − z12) (11.5)

In complex form the impedance is then Z = R + j X.

In most cases an accurate analytic description of a power electronic device is

not available, so that the impedance must be obtained by perturbations of a steady-

state model. Ideally, the model being perturbed should not be embedded in a larger

system (e.g. a.c. or d.c. systems), and perturbations should be applied to control

inputs as well as electrical terminals. The outcome from such an exhaustive study

would be a harmonically cross-coupled admittance tensor completely describing the

linearisation.

The simplest method for obtaining theimpedance by perturbationis to sequentially

apply perturbations in the system source, one frequency at a time, and calculate

impedances from

Zk =V k

I k(11.6)

The Zk obtained by this method includes the effect of coupling to the source

impedance at frequencies coupled to k by the device, and the effect of phase depen-

dency. This last means that for some k, Zk will be located at some unknown position

on the circumference of the phase-dependent impedance locus. The impedance at fre-

quencies close to k will lie close to the centre of this locus, which can be obtained by

applying two perturbations in quadrature. With the two perturbations of the quadra-

ture method, enough information is available to resolve the impedance into two

components; phase dependent and phase independent.

The quadrature method proceeds by first solving a base case at the frequency

of interest to obtain the terminal voltage and total current: (V kb , I kb ). Next, two

perturbations are applied sequentially to obtain (V k1, I k1) and (V k2, I k2). If the source

was initially something like

Eb = E sin (ωt) (11.7)

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Steady state applications 281

then the two perturbations might be

E1 = E sin (ωt) + δ sin (kωt) (11.8)

E2 = E sin (ωt) + δ sin (kωt + π/2) (11.9)

where δ is small to avoid exciting any non-linearity. The impedance is obtained by

first forming the differences in terminal voltage and injected current:

V k1 = V k1 − V kb (11.10)

V k2 = V k2 − V kb (11.11)

I k1 = I k1 − I kb (11.12)

I k2 = I k2 − I kb (11.13)

Taking real components, the linear model to be fitted states that

V k1R

V k1I

=

zk11 zk12

zk21 zk22

I k1R

I k1I

(11.14)

and V k2R

V k2I

=

zk11 zk12

zk21 zk22

I k2R

I k2I

(11.15)

which permits a solution for the components zk11, etc:⎡⎢⎢⎣

V k1R

V k1I

V k2R

V k2I

⎤⎥⎥⎦ =

⎡⎢⎢⎣

I k1R I k1I 0 0

0 0 I k1R I k1I

I k2R I k2I 0 0

0 0 I k2R I k2I

⎤⎥⎥⎦

⎡⎢⎢⎣

zk11

zk12

zk21

zk22

⎤⎥⎥⎦ (11.16)

Finally the phase-independent impedance in complex form is:

Zk =12

(zk11 + zk22) + 12

j (zk21 − zk12) (11.17)

11.5 The time domain in an ancillary capacity

The next two sections review the increasing use of the time domain to try and find

a simpler alternative to the harmonic solution. In this respect the flexibility of the

EMTP method to represent complex non-linearities and control systems makes it an

attractive alternative for the solution of harmonic problems. Two different modelling

philosophies have been proposed. One, discussed in this section, is basically a fre-

quency domain solution with periodic excursions into the time domain to update thecontribution of the non-linear components. The alternative, discussed in section 11.6,

is basically a time domain solution to the steady state followed by FFT processing of

the resulting waveforms.

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