Optimum distribution system harmonic filter design using a genetic algorithm

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Electric Power Systems Research 30 (1994) 263-267

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Optimum distribution system harmonic filter design using a genetic algorithm s Hanqing Yang, Gill G. Richards Electrical Engineering Department, University of New Orleans, New Orleans, LA 70148, USA

Abstract

This paper presents a technique for the anticipation of harmonic distortion as well as selection and placement of filters on distribution systems which have multiple harmonic sources and constantly changing characteristics caused by compensating capacitor switching and load impedance changes. The purpose is to predict and reduce harmonic voltage distortion for all buses in the system to comply with IEEE Standard 519. A genetic algorithm is used in this technique and shows itself an effective tool in optimizing a multivariable objective function.

Keywords: Harmonics; Filters; Genetic algorithms

1. Introduction

The increasing use of nonlinear loads, such as highpower semiconductor switch devices, variable-speed motor drives and fluorescent lighting, which inject harmonic currents into the distribution system, may cause severe power supply harmonic distortion. These harmonics, if disregarded or undetected, may present system operating problems resulting in complaints from customers and reduced life of power equipment as well as degraded efficiency and performance. IEEE Standard 519 [1] suggests a series of limitations for both customer harmonic current injection and utility harmonic voltage distortion levels. However, due to varying distribution network responses, even if customer harmonic current injection is under the IEEE 519 limitations, excessive harmonic voltages may still appear. To find an effective way to predict excessive harmonic voltages on a distribution system with shifting topology, a series of methods has been suggested. Techniques for predicting and filtering in cases with harmonic sources on several buses are discussed in Refs. [2-4], but the solution approach is qualitative and * Presented at the Third Biennial Symposium on Industrial Electric Power Applications, New Orleans, LA, USA, November 1992. 0378-7796/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSDI 0378-7796(94) 00865 -2

intuitive, rather than quantitative. In another case [5], the problem of harmonic reduction with shifting network capacitor banks and multiple distributed harmonic sources was solved by simply searching all possible capacitor bank combinations, a time-consuming calculation. In Ref. [5], it was possible because the only shifting system parameters considered were two switched capacitor banks. Other capacitors, loads, and harmonic sources were considered constant at their assumed values. This paper concerns the use of a genetic algorithm to search for the worst combination of switched capacitors, load impedances, and a given set of harmonic sources in a distribution system. The worst combination is the one which yields the worst total of the total harmonic distortion ( T T H D ) , which is related to the sum of all the T H D s of the individual buses. When the worst T T H D is found, all the individual bus T H D s are examined. If any one of them exceeds the IEEE 519 limit, a filter is placed in the 'worst-case' system, at a point that minimizes the T T H D . Repeated use of this procedure will lead to a selection and placement of filters which adequately reduce the harmonic distortion within the distribution system. A genetic algorithm is used to search the switching space because other directed search methods may be inadequate. Distribution system harmonics may be ex-

H. }rang, G.G. Richards /Electric Power Systems Research 30 (1994) 263 267

264

pected to be multimodal with respect to system parameters because of the high order of system equations, so that local maxima will defeat a gradient search. The genetic algorithm, described in Ref. [6], will find the global maximum, and is robust enough to accommodate a variety of responses and constraints.

Fig. 2. Crossover step.

2. Genetic algorithm The main idea of the genetic algorithm (GA) comes from the mechanism of natural selection and natural genetics, that is, 'the fittest member of a population has the highest probability for survival and reproduction'. By exploiting this philosophy, a GA can be used to approximate global maxima or minima even in a function with a large number of local maxima or minima. The genetic maximization process can be described as follows: it is desired to find a set of parameters which will maximize a 'fitness' function (similar to an objective function in an optimization problem). Several possible parameter sets with random values (the 'population') are chosen. Each parameter set takes the form of a string of binary bits representing parameter values. Each bit string (member of the population) is tested to find its fitness by substitution into the fitness function. The population is then reproduced, with the same total number of bit strings. However, the new generation contains the same members as the old, but now in numbers proportional to their fitness. Thus the parameter sets tending to maximize the fitness function occur in greater numbers. This is the first step of the GA, called the reproduction step. Fig. 1 shows a reproduction step example using ,V2 as a fitness function, where each bit string (parameter set) denotes one X value (taken from Ref. [6]). In this step, string 2, with high fitness, is reproduced twice. On the other hand, string 3, with low fitness, is entirely eliminated from the population. Next, to produce variety, the members are randomly paired. Each paired string exchanges a randomly chosen portion of its bits with its mate. This produces a new set of members which maintain many of the characteristics of their predecessors. This is the crossover step, shown in Fig. 2. After crossover, the population undergoes mutation, the third operation in the genetic algorithm. In this

No

String

1 2 3 4

01101 Ii000 01000 i0011 Total

Fitness

% of T o t a l

169 576 64 361

14.4 49.2 5.5 30.9

1170

i00.0

Fig. 1. Reproduction step.

New Population

,

Fig, 3. Mutation step.

step, some portion of the bits in the total population is randomly altered (Fig. 3). Typically, an average of one in one thousand bits is changed. This function prevents the algorithm from losing some potentially useful information. For example, if the entire population of bit strings has zero for its second bit, this condition cannot be altered by the reproduction and crossover steps. Eventually mutation will change this bit, which might have prevented an optimal bit string from forming. The whole process is then reiterated until convergence is obtained.

3. Objective function To search for the worst combination of capacitors, loads, and harmonic sources to find the 'worst-case' values as suggested in Refs. [2-4], two kinds of objective function can be used: 1. Total of total harmonic distortion (TTHD) [7] TTHD = C ~ ' THDi2) 1/2 where N is the number of buses. 2. Maximum total harmonic distortion ( M T H D ) M T H D = max{THD~ }

i = 1, 2 . . . . . N

In this paper we choose the TTHD as the objective (fitness) function, because it is better behaved under shifting filter positioning than the MTHD. For example, if there is a bus whose harmonic resonance results in severe distortion at several other buses, the TTHD will weight this bus. Selecting such a bus for filtering may be the optimal strategy. The following example illustrates this approach.

I II000 [Ii000

11OOlI

4. Example distribution system A 21-bus distribution network revised from Ref. [4] is shown in Fig. 4. Table 1 gives the network impedances.

265

H. Yang, G.G. Richards /Electric Power Systems Research 30 (1994)263-267

Table 1 Network impedances Source impedance: Zs = (0.126 +jl.3863) x 1 0 - 3 p.u. Transformer impedance: Ztr = (8.9477 +j134.72) x 10.3 p.u.

i9

230/13.8KV

l

10 1200KVAR each

Line impedances

(p.u.)

2800

21

600KVAR each

From

To

RL( x 10-3)

XL( x 10-3)

Xc( × 10-3)

1 1 1 2 2 3 4 5 6 7 8 10 11 11 12 13 13 14 15 16

2 21 12 3 10 4 5 6 7 8 9 11 19 20 13 14 18 15 16 17

52.174 20.794 1.008 16.383 13.611 32.262 2.773 20.92 22.684 10.208 9.074 9.578 2.647 24.323 52.552 53.182 39.067 32.892 7.940 14.745

92.376 445.872 1.638 29.238 24.197 57.467 5.041 49.401 40.328 18.147 16.131 17.139 4.033 43.352 93.384 107.120 93.888 78.891 19.030 35.412

3.712 5.740 8.920 34.104 41.210 17.351 197.807 26.730 24.726 54.946 61.815 58.178 3.570 23.000 10.867 10.527 14.330 17.100 70.650 38.039

12

Load impedances (p.u.) From 2 3 4 5 7 8 9 10 11 13 14 15 17 18 19 20

To 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

RL

XL

117.56 12.00 91.43 234.15 218.17 24.00 7.424 118.03 14.084 423.64 24.302 124.13 97.658 164.56 87.28 59.368

3.53 1.80 13.71 7.02 6.54 3.60 1.11 3.54 2.12 12.70 3.64 3.73 14.64 4.94 2.62 8.91

Capacitor reactances (p.u.) From 6 7 8 9 11 16 17 20 21

To 0 0 0 0 0 0 0 0 0

Xc

20.00 20.00 20.00 20.00 40.00 40.00 20.00 40.00 8.57

13

14

15

17

16

I

6OO

1200KV~

Fig. 4. 21-bus distribution network.

In this example, it is a s s u m e d that b a l a n c e d 5th an d 7th h a r m o n i c s only are g e n e r a t e d at the indicated buses, so only the positive sequence n e t w o r k is m o d e l l e d , with h a r m o n i c currents c o m i n g f r o m n o n l i n e a r loads such as m o t o r drives at eight o f the buses. A m o t o r drive consists o f a rectifier, which p r o d u c e s a square cu r r en t w a v e in the supply, an d an inverter, w h i ch has little effect on the supply current. T h e n o n l i n e a r loads are m o d e l l e d as harm o n i c cu r r en t sources, with the large series R - L c o m b i n a t i o n s listed in T a b l e 1 across t h e m to represent o t h e r local loads. N i n e o f the buses have switchable capacitors for v o l t a g e or p o w e r f a c t o r correction. Th e h a r m o n i c loads that a p p e a r on buses w i t h o u t h a r m o n i c cur r e nt sources are a s s u m e d to be largely i n d u c t i o n m o t o r s r ep r esen t ed by the R - L c o m b i n a t i o n s indicated in T a b l e 1. In o r d e r to e v a l u a t e the T T H D f o r the fitness function, Zbu s is repeatedly f o r m e d for the e x a m p l e network. It has been a s s u m e d that the h a r m o n i c loads on buses w i t h o u t m o t o r drives can vary in f o u r steps: 0, 1/3, 2/3, an d 1 times the full load. C a p a c i t o r s are applied in f o u r steps. T h e h a r m o n i c sources which represent the m o t o r drives have m a x i m u m values specified by I E E E S t a n d a r d 519. T h e m a g n i t u d e s o f the per unit currents o f these h a r m o n i c sources, which are a f u n c t i o n o f the S C R defined in I E E E 519, are sh o w n in Tab l e 2. All h a r m o n i c sources are a s s u m e d to be in phase.

5. Bit string T h e a r r a n g e m e n t o f bits for the bit strings representing m e m b e r s o f the genetic a l g o r i t h m p o p u l a t i o n is s h o w n Table 2 Harmonic current sources at the nonlinear load buses Harmonic source bus

2

5

7

10

13

15

18

19

SCRa 483 668 480 436 1705 270 470 296 Current 5th 9.0 4.5 4.9 9.0 3.3 8.5 6.4 1.2 injected 7th 9.0 4.5 4.9 9.0 3.3 8.5 6.4 1.2 ( x 0.0001 p.u.) a SCR is defined in Ref. [ 1].

H. Yang, G.G. Richards / Electric Power Systems Research 30 (1994) 263-267

266 CIC 2 . . . . . . . . . . . .

C9

"

~lI~ .........

L8

I I

v"

]

Bit i--18 for capacitors

v--/

~

Table 3 Worst-case configuration Worst-case T T H D = 0.4162 p.u. Bus

Load (p.u.)

Xc (p.u.)

Bit 19--34 for loads

Harmonic voltage (p.u.) 5th

7th

0.00663 0.00926 0.00949 0.00995 0.00999 0.01027 0.01050 0.01056 0.01061 0.00964 0.00984 0.00665 0.00785 0.00871 0.00933 0.00940 0.00940 0.00816 0.00986 0.00991 0.00982

0.0664 0.0805 0.0822 0.0855 0.0857 0.0885 0.0908 0.0918 0.0928 0.0827 0.0843 0.0666 0.0731 0.0804 0.0856 0.0869 0.0869 0.0731 0.0843 0.0854 0.1832

THD (%)

Fig. 5. Bit string arrangement.

in Fig. 5. Binary numbers are used in these strings for speed of manipulation.

6. Results and analysis The result of a genetic algorithm run using the T T H D as the fitness function is given in Fig. 6 and Table 3. Fig. 6 shows the convergence of the most 'fit' member of each generation to a stable maximum. The actual CPU time for this run was 6 minutes on an 80486 IBM-compatible PC equipped with a MicroWay Number Smasher 860 card. Table 3 shows the worst-case (most 'fit') configuration of the distribution network using the T T H D . Note that the 7th harmonic distortion of every bus exceeds 3%, the limit specified by IEEE Standard 519. To reduce this distortion, a 7th harmonic filter is used to minimize the T T H D by searching all buses for the best position, holding the loads and capacitors shown in Table 3 constant. The result gives bus 2 as the best position to put a 7th harmonic filter to reduce the TTHD. With the 7th filter at bus 2, the genetic algorithm is run again. The result given in Table 4 shows that the most distorted order is now the 5th, which still exceeds the IEEE 519 limit. To reduce all the harmonic voltages to less than the IEEE 519 levels (with this severe case of eight maximum harmonic source buses), two more filters were required

0.45 0.40 r~ 0.35 0,30

r~

~

/

/

f

0.25 0.20 0.15

" ; ' '~' 'lb"'1'3''¢6' 'ff '2'2'~ ' ~ ' '3'( ~: ~ ' 4 ' GENERATIONS

Fig. 6. Convergence of the algorithm,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 a

source off off source source off off source off

off off off off 64.00

source off source off source source off

64.00 off

160.00 34.28

6.67 8.10 8.28 8.61 8.62 8.91 9.14 9.24 9.34 8.33 8.49 6.69 7.35 8.09 8.61 8.74 8.74 7.36 8.49 8.60 18.3

a Worst harmonic distortion bus.

Table 4 Worst-case configuration with the 7th harmonic filter on bus 2 Worst-case T T H D = 0.2691 p.u. Bus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17a 18 19 20 21

Load (p.u.)

Xc (p.u.)

source off off source source off off source off

20.00 off off off 64.00

source off source off source source off

off 20.00

40.00 34.28

Harmonic voltage (p.u.) 5th

7th

0.0384 0.0526 0.0549 0.0596 0.0600 0.0639 0.0640 0.0640 0.0640 0.0540 0.0550 0.0385 0.0466 0.0557 0.0623 0.0639 0.0669 0.0467 0.0550 0.0565 0.0569

0.00635 2E - 10 0.00028 0.00082 0.00087 0.00118 0.00132 0.00132 0.00132 0.00039 0.00055 0.00650 0.01490 0.02444 0.03145 0.03314 0.03629 0.01492 0.00059 0.00058 0.01750

a Worst harmonic distortion bus.

THD (%)

3.89 5.26 5.49 5.96 6.00 6.39 6.40 6.40 6.40 5.40 5.50 3.90 4.89 6.08 6.98 7.20 7.61 4.90 5.50 5.65 5.95

H. Yang, G.G. Richards /Electric Power Systems Research 30 (1994)263-267 Table 6 Worst-case configuration with five filters in place Worst-case M T H D = 0.0209 p.u.

Table 5 Worst-case configuration with three filters in place Worst-case T T H D = 0.04383 p.u. Bus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 a

Load (p.u.)

Xc (p.u.)

source off off source source off off source off

80.00 20.00 off 20.00 40.00

source off source off source source off

40.00 20.00

40.00 13.712

Harmonic voltage (p.u.)

THD (%)

5th

7th

0.00425 3 E - 10 0.00024 0.00072 0.00076 0.00106 0.00129 0.00132 0.00135 0.00027 0.00038 0.00431 0.00768 0.01115 0.01352 0.01403 0.01468 0.00789 0.00041 0.00039 0.02271

0.00013 2 E - 10 0.00177 0.00525 0.00556 0.00854 0.01077 0.01129 0.01176 0.00039 0.00056 0.00013 0.00041 3 E - 11 0.00071 0.00076 0.00084 0.00083 0.00060 0.00059 0.00021

267

0.43 0.00 0.18 0.53 0.56 0.86 1.08 1.14 1.18 0.05 0.07 0.43 0.77 0.11 1.35 1.40 1.47 0.79 0.07 0.07 2.27

a Worst harmonic distortion bus.

by the procedure, a 5th harmonic filter at bus 2 and a second 7th harmonic filter at bus 14. With these filters in place, the worst-case harmonics are shown in Table 5. The most distortion is now 5th harmonic on bus 21, which is below the IEEE 519 limit. Table 6 shows the result of the same procedure using the M T H D as the fitness function. The same procedure as with the TTHD was followed, maximizing distortion by switching, then finding an optimum filter placement. In order to reach the result of Table 6, five filters were needed, 5th filters at buses 9 and 17, and 7th filters at buses 9, 17, and 21. This indicates that the T T H D approach may avoid excessive filtering in some complex distribution systems.

Bus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 a

Load (p.u.)

Xc (p.u.)

source off off source source off off source off

20.00 20.00 20.00 20.00 40.00

source off source off source source off

off off

40.00 13.712

Harmonic voltage (p.u.)

THD (%)

5th

7th

0.00202 0.00066 0.00056 0.00039 0.00037 0.00024 0.00013 0.00006 2E - 10 0.00065 0.00068 0.00200 0.00109 0.00051 0.00013 0.00008 1 E - 10 0.00084 0.00069 0.00070 0.02092

0.00205 0.00388 0.00348 0.00270 0.00263 0.00180 0.00094 0.00045 1E - 10 0.00454 0.00490 0.00205 0.00202 0.00126 0.00071 0.00046 6 E - 11 0.00245 0.00493 0.00517 3 E - 11

0.28 0.39 0.35 0.27 0.26 0.18 0.09 0.05 0.00 0.45 0.49 0.29 0.23 0.14 0.07 0.05 0.00 0.26 0.50 0.52 2.09

a Worst harmonic distortion bus.

Acknowledgements The work reported in this paper was supported by the Electric Power Research Institute, the Entergy Corporation, Louisiana Power and Light Co., and the Mobil Oil Chalmette Refinery.

References [1] IEEE Guide for Harmonic Control and Reactive Compensation of

Static Power Converters, IEEE Standard 519-1981. [2] M.F. McGranaghan, R.C. Dugan, J.A. King and W.T. Jewell, Distribution feeder harmonic study methodology, IEEE Trans. Power Appar. Syst., PAS-103 (1984) 3663-3671. [3] R.C. D u g a n and C.D. Ko, Analyzing and controlling harmonic distortion on distribution feeders, Proc. Int. Conf. Harmonics in

Power Systems, Worcester Polytechnic Institute, Worcester, MA, USA, 1984, pp. 38-44.

7. Conclusions The genetic algorithm was successfully applied to search for the worst-case combination under conditions of shifting system response caused by variable compensating capacitors and loads. A series of filters were properly selected and placed. The result shows this approach may be helpful for harmonic distortion anticipation and filtering in distribution systems.

[4] R.C. D u g a n and D.T. Rizy, Harmonic considerations for electrical distribution feeders, Rep. No. ORNL/Sub/81-95011/4, Oak Ridge National Laboratory, Oak Ridge, TN, Mar. 1988. [5] R.K. Hartana and G.G. Richards, O p t i m u m filter design for distribution feeders with multiple harmonic sources, Electr. Power Syst. Res., 23 (1992) 103-114. [6] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, M A , 1989. [7] G.G. Richards and H. Yang, Distribution system harmonic worst case design using a genetic algorithm, IEEE Trans. Power Delivery, 8 (1993) 1484-1489.