Fast harmonic power flow calculation using parallel processing

Z.A. Mariiios J.L.R. Pereira S. Carneiro, Jr.

Zndexing terms: Parallel processing, Power system harmonics, Powerflow

Abstract: The paper presents a novel approach to the study of power system harmonics, in which advantage is taken of the fact that the coupling between harmonics of different orders is negligible. Parallel processing techniques are adopted for the first time to solve this problem and it is shown that considerable savings in computing timings can be achieved. The results of simulations per- formed with the program are compared with measurements on a real system to validate the approach. The system is modelled using phase co- ordinates and frequency dependence of the param- eters and this allows the program to deal with difficult problems such as the unbalanced flow of harmonic currents, electromagnetic coupling between phase quantities etc. The solution process utilises sparse techniques applied to 3 x 3 blocks which ensures better numerical stability.

1 Introduction

The flow of harmonics in power systems has been the subject of growing concern among managers, engineers and researchers in electric utilities, research institutes and universities [l-71.

If one looks at the different kinds of loads which the power system has to supply, ranging from households to shops and to industries, the trend towards the intro- duction of more nonlinear devices is almost invariably present. Therefore, analyses of harmonics are likely to become a routine task for the engineer involved with the operation and planning of power systems. Harmonic analyses may be required to assess one or more of the following aspects:

(a) computation of additional losses produced by the flow of harmonics

(b) evaluation of the possible levels of harmonic volt- ages

(c) evaluation of the impact of new loads and new har- monic sources to be connected to the power system (d) estimation of resonance and overvoltages arising

from the interaction between the network and capacitor banks.

0 IEE, 1994 Paper 9742C (Pll), first received 14th July 1992 and in revised form 30th April 1993 The authors are at the Federal University of Rio de Janeiro, COPPE- EE/lJFRJ Caixa Postal 68504, Rio de Janeiro, Brazil

IEE Proc.-Gem. Transm. Distrib., Vol. 141, No. I , January I994

In the present work the power system variables are rep- resented in the frequency domain with the help of the well-known Fourier series expansion

where

f(t) = power system variable (voltage or current) a. = D C level which may be present in the variable f = fundamental frequency h = harmonic order in relation to fundamental

4h = phase angle of hth order harmonic ch = peak value for hth order harmonic

The network can be assumed to be passive and linear, provided that all nonlinear devices such as static VAR compensators (SVCs), convertor stations etc. are rep- resented as ideal current sources [ l ] or using Norton equivalents [SI. Therefore, each harmonic component in eqn. 1 can be. computed independently, and the overall performance is obtained by adding the contribution of every component to be considered, normally up to the 50th order.

The solution under steady-state conditions is obtained by direct solution of the well-known linear system of equations,

[4b'] = [Y;bc][v;bc] (2) where

[ I 3 = vector of hth order phase co-ordinate nodal

[V;"] = vector of hth order phase co-ordinate nodal

[Y;b'] = hth order phase co-ordinate admittance

Once eqn. 2 is solved for each harmonic order, the har- monic flow is computed from the nodal voltages. The overall power flow is then obtained by superposition with the conventional fundamental frequency load flow. Since the nodal network equations are formulated in phase quantities, unbalance effects caused by harmonics on impedances and sources will be readily calculated.

current injections

voltages

matrix

The authors are indebted to CENTROMIN Peru for providing financial assistance to the first author and for supplying the experimental data. Thanks are also due to the Ministry of Education

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The basic assumption that is commonly adopted [5] in the foregoing process is that the coupling effects that may exist between harmonics of different orders are neg- ligible. The separate solution for each harmonic is thus well suited for the introduction of parallel processing techniques.

2 Modelling techniques

The power system components will be modelled using the a, b, c phase frame of reference and actual quantities, that is, per-unit values are not used. Since the adopted models have been well studied in the literature, this Section will review the main assumptions and some modifications introduced for the present work.

2.1 Synchronous machines and generators Since synchronous generators and machines are designed to produce balanced EMFs, for harmonic calculations these machines are represented using a shunt admittance matrix of the form

LP %-, 5-,

y h - m %-rn % - p

(3)

(4)

(5) The zero-sequence harmonic admittance YOh is obtained from the zero-sequence fundamental frequency machine reactance X , as

where

5 - p (yOh + 2yZh)/3 % - m = (yOh - y7.&/3

= l / ( jhX, + 3ZhJ (6) where Zh-" is the impedance which might be present in the neural of the machine, computed for each harmonic order. The negative-sequence harmonic admittance YZh is obtained from the negative-sequence fundamental fre- quency machine reactance X , as

yZh = l / W X z ) (7)

' I h = &h

where X, can be computed in several ways. Mahmoud [9] and Grady [lo] have adopted the simplest form,

L; + L4" 2

x , = 2nf

where L; and Li are, respectively, the direct- and quadrature-axis subtransient inductances of the machine. Densem et al. [5] have proposed that a negative sequence impedance 2, should be used in eqn. 7 instead of X z ; the value of Z , would be calculated from eqn. 8 for a 0.2 power factor. More sophisticated methods have been proposed, e.g. Semlyen et al. [11] but these may require data that are not normally available. For the present work, it was decided to use eqns. 7 and 8 to obtain the negative sequence admittance.

2.2 Transformers, shunt capacitors and reactors The transformer models adopted are discussed in detail in References 1, 5 and 12. Two- and three-winding trans- formers are represented, respectively, by 6 x 6 and 9 x 9 matrices in phase co-ordinates. The nodal admittance matrix of transformers, shunt reactors and capacitors are obtained from the primitive and incidence connection

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matrices using the well-known linear transformation techniques.

2.3 Transmission lines A multi-phase transmission line model was developed using modal analysis [3]. Routines were programmed to obtain the modal transformation matrix which is com- puted for each harmonic frequency. Once the modal vari- ables are calculated, inverse transformation to phase quantities is performed, and a n-equivalent representa- tion is constructed in the a, b, c frame of reference. The modal transformation is computed by solving the eigenvalue/eigenvector problem on the primitive line series impedance and shunt admittance matrices, the cal- culation of which is briefly discussed below.

The primitive series impedances of the line are written as

2," = 2:. + z:, z,, = z:, + z:, (9)

where the superscript 0 indicates that the parameter has been computed assuming a perfect earth return and "

refers to frequency-dependent nonideal earth correction terms. To compute for the later terms, the complex earth return plan concept, introduced by Deri et al. [6], was adopted, instead of the more traditional, but much more cumbersome Carson's series [4]. Given the importance attributed to the losses in the present studies, special con- sideration was dedicated to the calculation of the internal impedance of the conductors. The conductor self- impedance is given by

z:, = z:, + z:, (10)

where Z:, is the external and Za, the internal imped- ance. The parameter Z;, is a complex number where the real part is the resistance and the imaginary part is the internal reactance of the conductor. The internal react- ance is very small when compared with the external impedance Z:, and it was considered to have a constant value, independent of frequency. The internal resistance, however, is strongly influenced by the so-called skin effect within the conductor and its calculation demanded some special considerations. Expressions in terms of Bessel functions have been derived [13] but the cost to perform all calculations required for all the harmonics may become prohibitive if simplifications are not introduced. It was thus decided to use the following approximate expression which is based on the asymptotic behaviour of the Bessel series for higher frequencies [14]:

P\ ; P; = riJ(PU2nf h) = &-j=/4

where

R,, = DC resistance of the conductor

radius of the conductor 1 = ro/rl = ratio of the internal to the external

p = magnetic permeability of the conductor U = electrical conductivity of the conductor

The functions p 1 and pi are nondimensional frequency- dependent qupntities. It was found in the present studies that the values for Z- obtained with eqn. 11 were less than 25% smaller than those computed with the classic Bessel series formulae [13], provided that the value of p', was higher than 3.5. For smaller values of p; the corres-

I E E Prof.-Gener. T r a m Dism'b., Vol. 141, No. I , January I994

pondence was not so good, and the error increased to almost 50% for p', = 0, that is, for DC or zero frequency. In order to correct this discrepancy, the following empiri- cal expressions were developed [12]:

0.5a Pi < 1 f (p; , q) = p;[O.11 - 0.2~3 + 0.7a - 0.11 1 < p i < 3.5

a = 1 - q (12) If the value off@;, q) computed from eqn. 12 is added to the real part on the right-hand side of eqn. 11, the resulting internal impedance ZL,,, will be within 2% of the value calculated with the classic Bessel formula, as long as q < 0.4. Since this condition applies to most practical conductors, the empirical approach was considered ade- quate for the harmonic calculations. Thus eqns. 11 and 12 present a good compromise between simplicity and accuracy.

The primitive shunt capacitances are computed in the usual way from the well-known potential (or Maxwell) coefficient expressions and are assumed constant with the frequency. The conductance is not considered in the cal- culations.

2.4 Series elements Three-phase series elements such as resistors, series com- pensation capacitors etc., connected between two nodes, are represented in terms of a rr-equivalent in which the admittances connected to ground are zero quantities.

2.5 Loads Several models have been proposed to represent loads during unbalanced operation and under the effect of har- monics. The models proposed in References 5 and 1 have been implemented in such a way that the user can select a single model or a combined representation of the load at a given busbar.

[0.26 p; > 3.5

3

The computation of additional losses caused by the flow of harmonics in the transmission system and in devices such as generators, transformers, reactors etc. is an important aspect in harmonic studies and can be per- formed in a straightforward way, from the equation

Computation of losses and harmonic distortion

where the symbol 9 designates the real part operator, n is the number of nodes of the device or subsystem being considered, and V f , If. i = a, b, c, are the harmonic volt- ages and currents as defined under eqn. 2.

The measure of voltage distortion is defined for each harmonic as [2]

(14) 4 VI

d h =- 100 (%)

and the total harmonic distortion as

T H D = pj An indication of the importance of the above calculations in system studies can be obtained considering the simple

IEE Proc.-Gener. Transm. Distrib., Vol. 141, No. I , January 1994

system of Fig. 1, which has a short-circuit level of 1100 MVA at 220 kV, 60 Hz.

Fig. 1 One-line diagram ofsimple system

The overall frequency response is shown in Fig. 2 and Table 1 gives some computed results when a 50A 5th order harmonic current is injected in node PCC. In Table 1, and represent, respectively, the system and

- 6 0 0 ~ i ' ' ' ' ' ' ' ' ~ 10 20 30 40 50 harmonic

Fig. 2 Ouerall system frequency response

Table 1 : Results for 5th harmonic

Phase a b C

/ 5 , , (A) 159L-93" 188~170" 235L32" (A) 169L70" 178L-26" 283L-160"

V 5 , c (kV) 35.8L-20" 37.7L-116" 50.6Ll lO" THD ("A) 28.19 29.68 39.81 Losses (kW) Total 6070

capacitor currents for the 5th order harmonic as shown in Fig. 1, and V5,c the voltage at bus PCC. It is seen that the 5th harmonic is unbalanced and attains very high values. It is also seen in Fig. 2 that the system has a natural resonance frequency which is very close to the 5th harmonic, and this originates an extremely high over- voltage. The total harmonic distortion is in excess of 28% in all phases. The total additional losses are 6.07 MW, which is less than 1% of the system capacity. They are, however, substantial if one considers that some simple procedures could be adopted to mitigate them. This aspect is not normally analysed in most system steady- state studies.

4 Harmonic source representation

Some possible starting points to determine harmonic injections in the system are to examine existing records of harmonic behaviour in the network, or to obtain data from the manufacturer of new harmonic-producing equipment which is to be installed in the system. In the present work, harmonic injections are represented using a probabilistic approach, which is more representative of the true system operating conditions, particularly when more than one harmonic source is connected to the network [lS, 121. The method consists of the five steps described below, which are to be repeated for any given node, subjected to a range of predetermined operating conditions, and for each phase and each harmonic order:

Step I ; obtain (by measurements) harmonic current injections

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Step 2: identify the variables’ maxima and minima in order to construct a frequency distribution table

Step 3: find an approximate normal (and discrete) dis- tribution function; compute its medium and standard deviation

Step 4 : select an harmonic injection (modulus and phase) and test for its probability of occurrence

Step 5 : introduce phase angle correction as related to the fundamental componeiit of the voltage.

5 Computational aspects

The electrical network is represented in terms of nodal equations which lead to a set of linear equations given by eqn. 2. This set of equations is sparse and the solution is obtained using the bifactorisation algorithm presented in Reference 16.

The algorithm of Reference 16 which was designed for single-phase systems was modified for three-phase systems by converting the numerical values (the admit- tance coefficients), which were originally scalar, into 3 x 3 blocks and by making the numerical operations with these matrices throughout the algorithm. It is important to notice that the normalisation of the equa- tions in the bifactorisation process requires the inversion of a 3 x 3 matrix and pivoting is required for harmonic studies. To demonstrate the effectiveness of the 3 x 3 block approach, comparisons were made with the con- ventional (scalar) technique. Fig. 3 shows that the pro- posed technique is very stable numerically and gives identical results for the impedances in the three phases. The conventional approach is clearly unstable in this case, and produces large errors below 120 Hz. However, there are cases in which the two methods produce the same results throughout the frequency spectrum [12].

frequency. HZ

Fig. 3 Frequency response ofnodal impedance at busbar CAR50

With the sparse programming technique, computer time is saved and memory requirements are reduced. However, the complete solution of an harmonic problem requires a substantial amount of computer time, because the whole solution process must be repeated for various frequencies to determine the system behaviour. Since in the present work the flow of harmonics is typically calcu- lated for frequencies up to 3 kHz in 15 Hz steps, each study will require 200 solutions. To reduce computer time, it is proposed to use a parallel processing approach in which the sets of instructions for different frequencies are computed simultaneously. This technique is imple- mented on an IBM PC compatible with a transputer board in one of the slots of the mother board, as shown in Fig. 5.

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The sequential algorithm for the numerical solution of the electrical network and computation of the harmonic flows, voltage distortion and additional losses is pre- sented in Fig. 4.

Read system data f = 1 5 while ( f d 3000) do

Compute the network parameters for frequency f Compute the admittance matrix in a sparse format Solve the set of linear equations Y”’cV’bc = PC Compute the harmonic flows Compute the additional losses throughout the system Evaluate harmonic voltage distortion at every node Store the results f = f + 1 5

end while Compute the total voltage distortion at every node Compute the total additional loses

Fig. 4 Sequential algorithm

The parallel machine used to test the proposed algo- rithm is an order 2 hypercube, having in each node a 32 bit INMOS transputer, T800 RISC processor, with 2 Mb of local memory, connected to a host microcomputer using an Intel 80286 microprocessor, as shown in Fig. 5. The potential throughput of the machine is 40 mips/6 megaflops. The language used to develop the program was the 3L Parallel Fortran [17].

Fig. 6 Order 2 hypercube connected IO on IBM-PC compatible

In Fig. 5 the host computer is connected to the clients T1, T2, T3 and T4, and T1 is called the root. The inter- process communication between host and clients must be done through T1. The server task AF is responsible for the interprocess communication and for loading the exe- cutable tasks throughout the clients. The task F is exe- cuted in the client T1 to ensure compatibility between the different communication protocols. The 3L Parallel Fortran [17] provides the main routines for the data exchanges through the communication channels.

The sequential algorithm was modified and tested on this parallel environment. Fig. 6 shows the client tasks and Fig. 7 shows the server task. The communication routines were introduced to provide the interprocess communication.

6

A 40 kV busbar, 200 MW installed capacity power system, having about 700 km of transmission lines (220,

Application to a real system

I

Get data h = l while (h d 50) do

f=60h - (60 - 15) Compute the first 7 instructions inside the

h = h + l ‘do while’ loop on Fig. 4

end while Send results to task T1

Fig. 6 Tasks Ti, i = 2,3,4

IEE Proc.-Gew. Transm. Distrib, Vol. 141, No. I , January 1994

138,69,50, 12 and 10 kV), was selected as a test case. The system is situated in the Central Region of Peru and supplies mainly industrial loads, including several metal smelters which alone require 90 MW installed capacity. These smelters are connected to buses ALAMlO and ZNll as shown in the one-line diagram of Fig. 10. Some- time in the past the network developed harmonic over- voltages and phase unbalance, particularly under light-load conditions, and this led to the decision to undertake extensive measurements on the system.

Send data

Compute the instructions inside the 'do while' on Fig. 6 Get results from T2, T3 and T4 Compute the total voltage distortion at every node Compute the total additional losses

Fig. 7 Task T I (root task)

The measurement techniques are described in detail in References 12 and 3 and for space reasons are not given here. Fig. 8 shows the calculated and measured spectra of the phase a voltage at bus ZN11. The harmonics are mainly due to an AC-DC convertor station supplying a smelter plant. The AC-DC convertor was represented using the methods of Section 4. The dominant fre- quencies which are present in the voltage at ZNll are the 5th, 7th, 11th and 13th harmonics and it is seen that good correspondence has been obtained between the experimental and computed results. The maximum devi- ation in this case is less than 20%. The THD for the same operating condition as that of Fig. 8, computed from eqn. 15, was 4.2% using the measured harmonics, as com- pared to 4.7% using the calculated values.

40 '0 4 8 12 16 20 2 4 28

harmonic order Fig. 8 Voltagefrequency spectrum - bus-ZNII -phase A

An interesting indication of the level of harmonic pen- etration can be obtained from Fig. 9, which shows the calculated and measured spectra of the phase a current, flowing in the line L 538 (connecting buses PACHSO and ORNVA, see Fig. 10). It is Seen that essentially the same dominant harmonics which were observed in phase a voltage (Fig. 8) are present in line L-538 phase a current, as was to be expected.

Q 4 5 2

OO 1 A 2 l k i o i4 218 hormonic order

Fig. 9 Currentfrequency spectrum - line L-538 - phase A

Fig. 11 shows the calculated frequency responses of the network as seen from buses ORNVASO and MOR050, with and without capacitor banks. This Figure clearly shows the resonance peaks introduced by the capacitor banks connected to the buses.

IEE Proc.-Gener. T r a m . Distrib., Vol. 141, No. I , January 1994

The real power system of Fig. 10 was used to test the algorithms proposed. The sequential algorithm of Fig. 4 was tested on the IBM 4381 and on the parallel machine (Fig. 5) in which the algorithm was executed in one trans- puter only, in order to compare computer systems' per- formance. The IBM 4381 required 420s of CPU time and approximately 2 h of clock time when the system was using 99% of its CPU time and 33% of its 15 Mb of RAM. The parallel machine took only 964s of clock time.

Z N l l

One-line diagram of test system Fig. 10

C l o o t ,.

frequency, HZ Fig. 11 Network frequency response at buses ORNVASO and MOROSO

ORNVA without capacitor hank . . . - -. ORNVA with capacitor hank ~ _ _ _ MOROM with capacitor bank __ MOROM without capacitor bank

. . . . . . .

The parallel machine was also used with the four transputers to run the parallel algorithm of Figs. 6 and 7 and it required 292 s of clock time.

From the above figures one can see that multiproces- sor machines, which are cheaper when compared to mainframes, are becoming competitive, particularly to solve the class of problems involving large amounts of number crunching with independent tasks, when minimal interprocess communication is required.

7 Conclusions

Considerable attention has been given in the present work to the adequate modelling of power system com- ponents and devices under the combined effect of harmonics and unbalanced operation.

The modelling techniques adopted were validated by comparisons of simulation results and field tests on a real power system. A novel formulation of the skin effect has shown good compromise between computing time and accuracy.

The importance of taking into consideration the addi- tional losses introduced by higher order harmonics in power systems was also demonstrated.

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Numerical stability of the solution process was obtained for the complete frequency spectrum with the introduction of the 3 x 3 block formulation.

The introduction of the parallel processing technique was very effective in reducing computer timings. The use of mainframe computers with a medium load would typically require almost 2 h of clock time as compared to only about 5 min when the parallel machine is used. Since harmonic studies require a large number of case studies, the cost in terms of manpower and computing could render such studies prohibitive. The availability of a parallel machine would clearly be much more eco- nomical.

8 References

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2 CIGRE WORKING GROUP 36-05: Transformers and instru- ments for measuring harmonics’, Electra, 1989, 124, pp. 91-97

3 ARRILLAGA, J., BRADLEY, D.A., and BODGER, P.S.: ‘Power system harmonics’(John Wiley, 1985)

4 CARSON, J.R.: ‘Wave propagation in overhead wires with ground return’, Bell Syst. Tech. J., 1926,s pp. 534-554

5 DENSEM, T.J., BODGER, P.S., and ARRILLAGA, J.: ‘Three- phase transmission system modelling for harmonic penetration studies’, IEEE Trans., 1984, PAS-103, (2), pp. 310-317

6 DERI, A., TEVAN, G., SEMLYEN, A., and CASTANHEIRA, J.: ‘The complex ground return plane, a simplified model for homo- geneous and multi-layer earth return’, IEEE Trans., 1981, PAS-100, (a), pp. 3686-3692

7 DOUGLAS, J.: The future of transmission switching to silicon’, IEEE Power Engineering Review, 1989,9, (lo), pp. 23-25

8 XU, W., MARTI, J.R., and DOMMEL, H.W.: ‘A multiphase har- monic load flow solution technique’. PWRS/IEEE/PES, Paper 90 WM 098-4, Atlanta, USA, 1990

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10 GRADY, W.M., and HEYDT, G.T.: ‘Prediction of power system harmonics due to gaseous discharge lightning’, IEEE Trans., 1985, PAS-104, (3). pp. 554-561

11 SEMLYEN, A., EGGLESTON, J.F., and ARRILLAGA, J.: ‘Admit- tance matrix model of a synchronous machine for harmonic analysis’, IEEE Trans., 1987, PW-2, (4), pp. 833-840

12 MARIROS, Z.A.: ‘Fluxo de harmhicos em sistemas de pot2ncia utilisando processamento paralelo’. Master’s thesis, Universidade Federal do Rio de Janeiro, COPPE/UFRJ, 1991

13 STEVENSON. W.D.: ‘Elements of wwer systems analvsis’ (McGraw-Hill,’ 1962)

COPPENFRJ. 1983 14 PORTELA, C.M.: ‘Regimes transitbrios’. Portuguese edition,

15 YAMAYEE, Z:A., and KAZIBWE, W.E.: ‘Probabilistic modelling of distribution harmonics: data collection and analysis’, Electrical Power & E w g y Systems, 1987.9, (3). pp. 189-192

16 ZOLLENKOPF, K.: ‘Conference on large sparse sets of linear equations’ (Academic Press, 1971)

17 EDINBURGH PORTABLE COMPILERS LTD.: ’Parallel Fortran, user guide’. 3L Ltd. and EPCL, 1988

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