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Multifiquency Analysis with Time-Domain Simulation J. Usaola, J. G. Mayordomo

Abstract

This paper presents a novel method for harmonic analysis that uses techniques both in the time and the fre- quency domain. The linearparts of a network are modelled in the frequency domain, while the non-linear ones (converters, saturable transformers, etc.) can be modelled more easily in the time domain. Harmonic balance equations in the multi-frequency domain are set, and solved by a Newton-Raphson iterative procedure, with which it is easier to study cases with problems of convergence.

1 Introduction

Harmonic analysis methods can be classified in two general categories: frequency-domain and time-domain methods. Time-domain techniques involve numerical in- tegration of the non-linear system of differential equa- tions, from some assumed initial conditions until steady state is reached. Then, a Fourier analysis is performed to the voltage andcurrent waveforms of interest. This meth- od may need long computation times, and it is not pnc- tical on some occasions. For this reason fast steady-state techniques were developed [I ] , with which the steady state is found without having to calculate the whole tran- sient. These techniques have been frequently used [2-41.

Frequency-domain analysis methods, such as har- monic power flows like [ 161, can deal more efficiently with the linear parts of a network, and the size of the problem can be reduced using equivalents. However, to deal with non-linear elements is more difficult than in time domain. Analysis of non-linear networks in the fre- quency domain is accurately performed with the har- monic balance technique. For instance, in [5 ] converters in unbalanced systems are analyzed with a Gauss itera- tive procedure. In [6] the load-flow algorithm is used to solve this problem, but only balanced systems are con- sidered. In [7] and [8] a Newton-Raphson iterative algo- rithm is used in unbalanced systems. From these works, it seems that the powerful Newton-Raphson method is more useful when strong non-linearities or resonances are present, since the Gauss algorithm usually fails to converge under these conditions. Nevertheless, the men- tioned techniques can only take into account some kinds of non-linearities and even these, only under certain sim- plifying assumptions to avoid unmanageable equations. For instance, it is rather cumbersome to express analyt- ically the relationship between harmonic voltages and currents in converters in unbalanced networks if ripple in their DC side is going to be considered, as shown in [9] and [ 171. These equations must be found, for the la- cobian matrix of a Newton-Raphson algorithm, and they would not be useful for any other non-linear load.

In this paper a hybrid approach is described. It has been devised for three-phase networks with several non- linear systems that must be defined by the user. This method has several advantages - as the linear part of the network is modelled in the frequency domain, equiva- lents can be found, so that the computational effort de- creases. Besides, elements such as distributed parame- ter lines, or with frequency-dependent parameters could be modelled. The non-linear parts of the network of any structure (converters, non-linear inductances, etc.) are modelled in the time domain, with the only limitation of the required computation time. Finally, the use of New- ton-Raphson method improves the convergence. The al- gorithm has been implemented in a computer program called HYBRIS.

2 Description of the Method

In harmonic balance methods, the system is usually divided into its linear and non-linear part, and the har- monic voltages in the node, where a non-linear load has been connected, must be found solving the non-linear eq. (1)

Is - Y U = f (0, (1) where U is the vector of harmonic phase voltages, Y the three-phase harmonic admittance mamx, Is the vector of harmonic current sources and f (U) the current demand- ed by the non-linear device, dependent on the harmonic voltages. Eq. (1) must be solved with an iterative meth- od, such as Gauss or Newton-Raphson. If the Gauss method is chosen, the iterative process is

Y(k) U(k)"+' =I&) -Z(k)( i , (2) where Z(k)( i is the array of phase harmonic currents of order k demanded by the non-linear system at iteration 'i', namely I ( k ) = [ILl(k) IL2(k) IU(k)lr. In each itera- tion, the non-linear device is replaced by a current source. U(k) is also the array of phase voltages, U(k) = [ULl(k) U L Z W VL3(k)lT.

ETEP Vol. 6, No. 1, JanuaryFebruary 1996 53

.ETEP The simplicity of this method is its greatest advan-

tage, although its convergence may easily fail, especial- ly when strong resonances are present. In these cases a Newton-Raphson algorithm could be a better choice. The iterative equation providing the voltage increments in iteration 'm + 1' is:

(Y+ W(i)AU(i'' =Zs-YU"-f(u"). (3) W is asensitivity matrix between different harmonic var- iables:

(4)

The matrix w" is composed of sub-matrices kim. This term is the sensitivity sub-matrix of the harmonic 'k' of the current with respect to the harmonic 'm' of the voltage at iteration 'T. All the harmonics are coupled in this equation, and the separation between harmonics performed in eq. (2) is no longer possible. Therefore, the following tasks should be performed:

Decomposition of the network into its linear and non- linear parts. Modelling of the linear network in the frequency do- main and computation of an equivalent by Gaussian elimination. Loads are modelled as impedances as- suming rated voltages. Computation of harmonic currents injected by the non-linear devices. This is performed by means of a time-domain simulation for each device, in which the voltage U is known from the previous iteration. Formulation and solution of the non-linear algebraic eq. (1) with a Newton-Raphson iterative procedure. To do this, it is used the sensitivity matrix W, obtained from time-domain simulation data in each iteration. This is the most important part of the proposed algo- rithm, because it allows to use of this powerful itera- tive method in a multi-frequency algorithm.

lime-domain simulation can be Derformed bv a state-variables approach or by any otheimethod, such as nodal analysis with Norton equivalents associated to each dynamic element, like EMTP [ 101. Although time- domain simulation may require large computation times to reach steady state, the systems simulated in the time domain are of moderate size, because they are only small parts of the whole network. Besides, fast steady-state techniques could be applied. These techniques search the initial conditions from which steady state directly be- gins. The Newton method proposed by Aprille [ 11 seem to be the most effective of those algorithms, and its ap- plication to power systems was described in [4].

3 Sensitivity Matrix

The sensitivity matrix w" of eq. (3) was obtained in [ 111 for non-linear resistors in linear dynamic networks and was extended in [ 121 and [ 151 to non-linear induc- tances. However, these methods can not be applied to converters or other complex non-linear devices in three- phase networks, while the method described here is gen-

eral and can deal with any non-linear load. From now on we shall drop the superscript 'i' that indicates the itera- tion, since all the operations done belong to the same one.

Let us consider a known non-linear device connect- ed to a set of periodic voltages u(r) which demands a set of currents i(r) of period To. N samples of the currents and voltages equally spaced along To are known. We will find the sensitivity between the harmonic 'k' of the cur- rents and the harmonic '1' of the voltages. The extension to a system with several nonlinear devices will be shown later on. We will also drop the superscript 'i', because we will work within an iteration.

In such a system, the k-th harmonic of the currents could be written as

In this equation r,, could be written as r,, = (n/N)To. Many devices cm be modelled as piecewise linear, for instance, a converter. In this case there will be Q inter- vals with different topologies and expressions of cur- rents. In the interval q, the currents will have transient and steady-state components, and their expression for the instant r,, is

I (6) j (2%/ N)mn N-l

jq(r , , ) = iP(r,,) + C Yq(m)U(m)e m=O

where ip(rn) is the array of transient component of the three-phase currents and Yq(rn) is the harmonic admit- tance matrix of order m of the circuit. An example is shown in Appendix 1. If we include expression (6) in eq. (5 ) , we obtain:

(7)

nq is the sample from where the interval q begins. Ma- nipulating eq. (7) we obtain:

.U(rn).

O(k) comes from iP(r,,). If now we take the derivative of currents with respect to voltages:

Along one period, the different values of the admit- tance could have a shape such as Fig. 1. As these values follow a periodic pattern, at each time instant r,, the har- monic admittance of order 'm', Y,,(m), could be written in the following way:

(9)

54 ETEP Vol. 6, No. 1, JanuarylFebruary 1996

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Data input Form linear network equivalent

Set initial conditions

Steady state found from

admittances

Non-linear equations in frequency domain

New harmonic voltages 4 Calculation of mismatch E

Harmonic computation in linear network

I End 1 rnllo9.2i

Fig. 2. Flow chart of HYBRIS (N-R: Newton-Raphson)

If this mismatch is smaller than a tolerance, the pro- cess has finished. Otherwise, return to Step II. The flow- chart of Fig. 2 represents the computational procedure.

5 Examples

In this section some examples are provided to show the capabilities of the algorithm implemented in pro- gram HYBRIS. Equivalence between its results and

6'2 MvA 4.8 Mvar 6.3 MVA a=l.o , + A = I . o

ET1109.3A A =0"5 h7.25 MW

Fig 3. Example 1 (S- =I0 MVA)

ETI 109.4A

Fig 4. Non-linear part of the network in Fig. 3

those obtained from more conventional techniques is demonstrated through two examples that study the be- haviour of a six-pulse diode rectifier connected to a three-phase network. This is shown in Fig. 3 and 4. Data about this network are given in [8].

- Example 1

Two cases have been studied. In case a) there are no capacitors in bus 8. The harmonic voltages in this bus and the harmonic currents injected by the converter have been found for three different values of the smoothing reactance, and E had to change to keep the demanded power unchanged. Results are shown in Tab. 1. Tab. 2 displays the same variables for case b), when capacitors of 4.8 Mvar has been connected to bus 8. The capacitors of case b) produce a strong resonance that raises the value of the 5th harmonic voltage. This makes the con- vergence worse and the number of iterations increases. The solution could not be reached using Gauss algorithm in any of the cases. Results were obtained with a mis- match smaller than and have been compared with the harmonic analysis program in the frequency domain

Tab. 3 shows a comparison between the sensitiv- ities in the last iteration of case b) with Xd = 5 p. u. between some harmonic voltages and currents, found

ARMO-E [8].

& = 5 & = O S &=0.05 E=2.1837 . E=2.1783 E=2.1689

6 it. 7 it. 6 it. U, mod. 0.9893 0.9886 0.9874

II mod. 0.7326 0.7357 0.7422 arg. - 18.86' - 19.55" -20.73"

U5 mod. 0.0523 0.0571 0.0686 - 10.98" - 18.43" -28.01"

I, mod. 0.1289 0.1410 0.1693 84.02" 76.56" 67.00"

U7 mod. -0.0477 0.0426 0.0342

I7 mod. 0.0774 0.0691 0.0556 arg. . 47.27" 54.67" 72.56"

UII mod. 0.0203 0.0195 0.0 1 89 arg. -47.92" 42.5 1 " 29.91"

Ill mod. 0.03 18 0.0305 0.0296 arg- 140.45" 135.03" 122.36

arg. -5.45" -5.45" -5.45"

arg. - 59.89" -52.48" -34.58"

Tab. 1. Example 1, case a) (values in p. u.)

56 ETEP Vol. 6, No. I, JanuaryFebruary 1996

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x d = 5 X, = 0.5 X d = 0.05 E = 2.2423 E = 2.2378 E = 2.3 128

10 it. 7 it. 8 it. UI mod. 1.034 I 1.03 12 I 1.0370

-5.50" -5.5 I" -5.47" I, mod. 0.6953 0.7055 0.6824

w. - 12.04" - 14.%" -9.17" U, mod. 0.2664 0.2542 0.1069

arg. -39.22" -35.82" -37.85" Is mod. 0.059 I 0.0564 0.0238

125.79" 129.03" I 27.04" (17 mod. 0.0034 0.0565 0.1083

17 mod. 0.0064 0.1058 0.2026 77.28" 56.06" 116.47"

UII mod. 0.0098 0.0044 0.0200 arg . 128.13" - 147.98 - 12.66"

III mod. 0.0365 0.0165 0.0741 a. 37.0 I O 123.93" 256.28"

arg. - 178.46" 156.09" - 138.72

Tab. 2. Example 1, case b) (values in p. u.)

UI UJ Ul Re Im Re Irn Re Im

11 HYBRIS Re -2.044 1.221 -0.116 -0.432 -0.213 -0.170 Im -12.504 2.285 0.974 -1.104 0.212 -0.490

ARMO-E Re -2.054 1.184 -0.117 -0.435 -0.213 -0.172 Irn -12.146 2.057 0.970 - 1 . 1 1 1 0.212 -0.497

IS HYBRIS Re 5.449 -2.292 0.1 13 1.318 0.450 0.689 Irn 4.480 0.482 -1.356 -0.097 -0.883 0.141

ARMO-E Re 5.555 -2.175 0.102 1.310 0.447 0.682 Irn 4.849 0.583 -1.356 -0.102 -0.884 0.141

f7 HYBRIS Re 3.412 -1.283 -0.187 0.964 0.119 0.801 Irn 1.548 1.463 -1.242 -0.626 -1.109 -0.117

ARMO-E Re 3.480 -1.203 -0.198 0.955 0.1 14 0.795 Irn 1.471 1.492 -1.237 -0.625 -1.109 -0.114

Tab. 3. Comphson between sensitivities from HYBRIS and ARMO-A

with HYBRIS and with the program ARMO-E, that uses Xia Cpr Heydt's formulation [6] . Differences are due to time discretization, that only allows discrete variations of the overlap angles.

- Example 2

In this example one of the phases of the capacitors (star-connected and isolated from ground) in bus 8 has changed from 1.6 Mvar to 0.2 Mvar. This imbalance in- creases the distortion of the waveforms, as shown in Fig. 5 to 7, obtained with the program HYBRIS. An interruption in the commutation process may be appre- ciated in Fig. 5, and a detail of this wave-form is given in Fig. 7. This phenomenon cannot be easily considered in the frequency domain, especially when ripple in the DC side must be modelled, but it does not bring special problems to time domain simulation. The high value of 15th harmonic voltage (more than 0.16 p.u. in phase 'Ll') is one of the reasons of this interruption. In this case, the structural imbalances in the network are isolat- ed from ground, and therefore there are neither zero se-

t UL2LI lL2

ET1109.SA

Fig. 5. Voltage u ~ L I and current iL2 at bus 8

f uL2L3 'L3

€TI 109.6A t - Fig. 6. Voltage uWu and current iw at bus 8

t UL2 L I 'L3

l31109.7A t -

Fig. 7. Detail of Fig. 5 showing the intemption of the com- mutation process

quence voltages nor currents. Harmonic magnitudes of orders 3, 9, 15, etc., have only direct and inverse se- quences components. Comparisons of these results with those obtained by EMTP are shown in Fig. 8.

0.20 p.u. 0.16

0.12

U 0.08

0.04

0

ET1109.8A 1 3 5 7 9 11 13 15 17 19

V -

Fig. 8. Harmonic voltages (phase LI) in bus 8

ETEP Vol. 6, No. 1, JanuarylFebmary 1996 57

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6 Conclusions

A new harmonic analysis method has been de- scribed. It uses time-domain simulation for the non-lin- ear part of the network while the linear part of the net- work is modelled as an harmonic equivalent. Newton al- gorithm has been used to reach convergence. This meth- od allows the study of complex networks without the computation times required by EMTP and other time- simulation programs. This method has been implement- ed in a computer program called HYESRIS. Numerical results have been compared with EMTP.

7 List of Symbols, Sub- and Superscripts

7.1 Symbols

Z(k)

I L I L ~ L ~ ( ~ ) harmonic phase currents of order ‘k’ i(t)

vector of harmonic currents of order k = [ILI(~) I ~ 2 ( k ) Iu(k)I‘)

vector of instantaneous phase currents at time instant t ( i ( t ) = [ iL l (k) i ~ ~ ( k ) iU(k)lT) vector of harmonic current sources coming from the linear part of the system number of samples per period sample from where the interval ‘q’ begins harmonic of order ‘r’ of the evolution with time of the harmonic admittance of order ‘m’ of a non-linear system in its steady state number of intervals in a period in with un- changed topology of the non-linear system fundamental period vector of harmonic phase volta es of order k ( W k ) = [ ULl(k) UL2W UL3(k>l 1 sub-matrix of the sensitivities between the real part of the phase current harmonic k with respect to the imaginary part of the phase voltage harmonic m harmonic admittance matrix of order ‘m’

%

7.2 Sub- and Superscripts

i iteration k, I , m harmonic order n time instant 4

R, I T transpose

interval inside a period of a piecewise linear system real and imaginary part of complex number

Appendix 1: Example of Harmonic Admittance Matrix for a Converter

In a converter such as that shown in Fig. 4, when valves 1 and 2 are conducting, the matrix that relates har- monic phase currents with harmonic voltage sources in steady state is:

In the same figure, if valves 4,5 and 6 are conduct- ing, the relationships between the line voltages and the loop currents 11, 12, I3 are:

2, = j 2 0 4 , Z 2 = Z 3 = R + j k o ( 2 4 + 4 ) ,

Z I 2 = ZI3 = Z,, = Z31 = j o 4 , 2, = Z32 = R + j w ( 2 4 + Ld)-

Inverting the matrix:

If we consider that

the relationship between the line currents and the phase currents is:

Appendix 2: Sensitivities in Real and Imaginary Parts

Instantaneous currents and voltages are real vari- ables, with periodic values after N samples, so their har- monic values have the following property:

Z(k ) = Z*(N-k), (AW

U ( k ) = U’(N-k), (Alb)

where the asterisk (*) denotes the complex conjugate. Therefore, there are only 1 + ( N - 1)/2 independent har- monics. For one singIe-phase non-linear load, the sensi- tivity matrix P has (2N - 1) x (2N - 1) complex terms, where N is the number of harmonics considered (1 is subtracted because of LX). However, if the sensitivities of currents with respect to real and imaginary parts of voltages are found, the matrix W would have (2N - 1)

58 ETEP Vol. 6, No. I, JanuaryEebruary 1996

E TEP x ( 2 N - 1) real terms,'halving the memory requirements. In order to find these exmessions. let us write harmonic

References

currents in real and imaginary pa& namely: [ I 1

(N-1112

+ C (ZR(k) sin (21cf0r)+jZ1(k) cos(27tf0 r)). [31 k=l Therefore, the relationships between real and com-

( A m

(A2b)

plex currents are: [41 zR(k) = j(Z(k) - ~ * ( k ) ) ,

Z'(k) = Z ( k ) + Z*(k), ZR(0) = 2Z(O). (A2c) [5l

The same equations could be easily found for the voltages. Now, to find the expressions of the new sen- [6] sitivities, the current must be arranged in the follow- ing way:

~ ( k ) = Z R +jz'(k) = ~ ( k ) + pk (o)u(o) (71

if we make

(A4) [91

P ... (...I = PR( ...)+ jqf,( ...I, (A51

using eq. (Al ) and equalling real and imaginary parts of eq. (A3), the sensitivities remain as follows:

the

Aprille, Z J.; Trick, T. 14.: Steady-state analysis of non- linear circuits with periodic inputs. Proc. IEEE 60 (1972) no. 1 , pp. 108- 114 Rajagopalan, V : Computer-Aided Analysis of Power Electronic Systems. New York/USA: M. Dekker, 1987 Grotzbach, M.; von Lutz, R.: Evaluation of Converter Harmonics in Power Systems by a Direct and Analytical- ly Based State Variable Approach. EPE, Brussels/ Belgium 1985, Proc. vol. 1, pp. 2.153-2.158 Usaola, 1.; Mayordorno, J . G.: Fast Steady-State Tech- nique for Harmonic Analysis. IV. Int. Conf. on Har- monics in Power Syst., BudapestIHungaria 1990, Proc. pp. 336-342 Yacamini, R.; de Oliveira, J.C.: Harmonics in Multiple Converter Systems: a Generalized Approach. Proc. IEE 127(1980)no.2,pp.96-106 Xia. D.; Heydt, G. T: Harmonic Power Flow Studies. P. I a. 11. IEEE Trans. on Power Appar. a. Syst. PAS. PAS-

Christoforidis, G. P.; Meliopoulos, A. P.: Effects of Mod- elling on the Accuracy of Harmonic Analysis. IEEE Trans. on Power Delivery PWRD-5 (1990) no. 3,

Mayordomo, J. G.: Pire: Coyto. A.: Computer Program for Analyzing Converter Harmonics in Power Systems. Application to Static VAR Compensation Analysis. IEEE Int. Workshop on Control Syst. in New Energy AppI., MadridSpain 1987, Proc. pp. 115- 124 Xia, D.; Shen, 2; Liao. Q.: Solution of non charac- teristic harmonics caused by multiple factor in HVDC transmission systems. 111. Int. Conf. on Har- monics in Power Systems, NashvilleAJSA 1988, Proc. DD. 222-228

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1131 Brigham, 0.: The Fast Fourier Transform and its Applications. Englewood Cliffs, NJNSA: Prentice- Hall, 1988

[ 141 Usaola, J.: Steady-State of Power Systems with Nonlin- ear Elements with an Hybrid Algorithm in the Time and Frequency Domains (in Spanish). Ph.D. Thesis, ETS de 11. Madridspain, 1990

[I51 Medina. A.; Arrillaga, J.: Generalised Modelling of Power.Transformers in the Harmonic Domain. IEEE Trans. on Power Delivery PWRD-7 (1992) no. 3,

[16] Valcdrcel, M.; Mayordorno, J. G.: Harmonic Power Flow for Unbalanced Systems. IEEE Trans. on Power Delivery PWRD-8 (1993) no. 4, pp. 2052-2059

[ 171 Carpinelii. G.; Gagliardi, E ; Russo, M.; Sturchio, A.: Steady-State Mathematical Models of Battery Storage Plants with Line-commutated Converters. V. Int. Conf. on Harmonics in Power Syst.. Atlanta/USA 1992,

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pp. 1090- 1098

pp. 1458-1465

ROC. pp. 24 I - 250

Manuscript received on April 11. I994

ETEP Vol. 6, No. I, JanuaryFebruary 1996 59

ETEP The Authors

Julio Usaola (1961) received his B.S. degree and his Ph.D. degree in Electri- cal Engineering from E.T.S. de Inge- nieros Industrides de MadridSpain in 1986and 1990,respectively.In 1988 he joined the Department of Electrical En- gineering in E.T.S. de Ingenikros Indus- triales de Madrid, supported by a grant of the Spanish Ministq of Education, where he remained until 1994. He is presently a Lechuer in the Department

of Engineering in the Universidad Carlos III de Madrid. His re- search interests include circuit theory and harmonic analysis in power systems. (Universidad Carlos 111 de Madrid, Escuela Po- litecnica Superior, Butarque 15, E-28911 Legants (Madrid)/ Spain, T +34 116249404, Fax +34 116249465)

Julio G. Mayordomo (1956) received his B.S. degree and his Ph.D. degree in Electrical Engineering from E.T.S. de Ingenieros Industriales de Madrid/ Spain in 1980 and 1986, respectively. In 1980 he joined the Department of Electrical Engineering where he is presently Lecturer of Electrical Engi- neering. His research interests include transient phenomena in networks and harmonic analysis in power systems.

(E.T.S.I.I., Dpto. Ingenieria Eltctrica. Jost Gutitrrez Abascal 2, E-28006 Madridspain, T + 34 1 13 36 3 1 74, Fax +34 113 363008)

60 ETEP Vol. 6, No. 1, JanuuylFebruary 1996