Genetic algorithm and extended analysis optimisation techniques for switched capacitor active filters - Comparative study

M.Welsh, PMehta and M.K.Datwish

Abstract: The proliferation of nonlinear loads has led to considerable interest in the development of active filters designed to eliminate current harmonics from power supply networks. The switched capacitor family of active filters developed at Brunel University is examined, hghlighting the complexity of the control technique which limits the effectiveness of these filters. Two different approaches are investigated with the aim of reducing the computational requirements associated with the control technique, and hence increasing the applicability of the switched capacitor active filters. The approach presented eases the somewhat laborious but unavoidable computational requirements. Genetic algorithms (GAS) and extended analysis (EA) are investigated, and provide an alternative to the calculus-based techniques which historically have dominated optimisation problems.

1 Introduction

The principle of filtering current harmonics generated by nonlinear loads is shown in Fig. 1 [I]. Due to the nonlinear nature of the load, the load current iL consists of a funda- mental component il, as well as harmonic components iH, whch in the absence of the active filter are present within the supply current is, polluting the power system. The par- allel connected active filter generates a compensation signal iF, whch then supplies the harmonic components required by the load, shown below:

is = i L - ZF + i f f - iF (1) Therefore, if iF = iH, the supply current is will only contain the fundamental component.

'S

nonlinear load

1 active 1 filter -

Fig. 1 Harmonic filtering

Most active filters designed for this purpose are voltage (or current) fed inverter (or converter) configurations, using various PWM techniques to control the compensation sig- nal. This configuration relies on the availability of a con- stant DC voltage or current source in order to generate the required compensation signal. This is accomplished by the use of large reservoir capacitors or inductors, which are not

0 IEE, 2000 IEE Proceedings onlie no. 20000006 DO1 1 0 . 1 0 4 9 / i p - e p a : 2 ~ Paper first received 4th June and in r e d form 8th September 1999 The authon are with the Department of Electrid and Electronic Engineering, Brunel University, UK

only expensive but are also physically very large. The energy required to maintain this constant DC level is obtained from the supply via the inverter, and this limits the performance of the fiter.

The switched capacitor active filter was first introduced in 1990 [2], and is fundamentally different from inverter- based active filters. The switched capacitor configuration removes the requirement for a large current or voltage source, which leads not only to a reduction in cost but also in physical size. The switching frequency of the fdter is also reduced, i.e. 24kHz compared to a relatively high switch- ing frequency of 20-40kHz for inverter configurations. A typical switched capacitor circuit with nonlinear load is shown in Fig. 2 [2, 31. The nonlinear load shown in Fig. 2 is representative of most computing and electronic equip- ment power supplies. The switching pattern for S, is evalu- ated in the next Section and is shown in Fig. 3. The switching pattern for S, is the complement of SI. Full details of the operation of ths circuit are given in [3] .

26.5Q

II= c=80pF Fig. 2 Switched capcrcitorjilter with rwnlineav loud

The basic operating principle of switched capacitor filters is summarised in the following Section followed by an anal- ysis of the conventional optimisation technique. The genetic algorithms (GAS) are then discussed, followed by the extended analysis (EA) method. The simulation and practi- cal results are presented at the end of the paper with the comparison of the convergence times for each of the three techniques.

21 IEE Proc-Electr. Power Appl.. Vol. 147. No. I , Junuury 2000

filter /current

required current

switching pattern for S,

required current

switching pattern for S,

'i-1 'i 'i+i

Fig.3 filter current control

2 Filter analysis

The switched capacitor active fiter generates the required compensation signal by controlling the first and second derivatives of the fdter current by appropriate choice of switching pattern. Using Fig. 2 as an example, the rate of change of current within the inductor is controlled as fol- lows: Switch S2 closed SI open, inductor connected to ground:

Assuming constant input voltage for the duration of the switching period, the second derivative of the filter current with this circuit configuration is zero. Switch S, open SI closed, inductor in series with the capaci- tor:

(3)

The second derivative of the fiter current is now given as:

Eqns. 2 4 show that the polarity and rate of change of the filter current can be controlled by varying the circuit con- figuration, as shown in Fig. 3.

To generate the required compensation signal a suitable switchmg pattern must be determined. Assuming half wave symmetry, 1 + n/2 differential equations with variable co- efficients, filter current and capacitor voltage must be solved, where n is the number of switchmg instants. Since the switching pattern cannot therefore be obtained by direct substitution using an objective function, a process of optimisation is. used, the optimisation variables being the switchmg instants.

3 Conventional optimisation technique

A measure of the effectiveness of any fdter is the total har- monic distortion (THD) after filtering for a given load, denoted as the cost function J defined below:

n=3

where A, and B, are the load harmonic coefficients, X, and Y, are the filter harmonic coefficients and n is the harmonic index. Therefore, by minimising the cost function J, the switching instants are optimised and the required compen- sation signal is generated, resulting in the load harmonics being fitered.

The present approach calculates the cost function using a fourth order Runge-Kutta procedure to obtain discrete samples of the filter current, a simple discrete Fourier trans- form then evaluates the harmonic coefficients and hence

the cost function J. The optimisation procedure directs the search for an appropriate switching pattern using a hill climbing technique developed by Fletcher and Reeves [4], as follows: Step 1: Given xo (initial values of switching instants), com- pute go = V'xo)' and do = -go.

Step 2: Fork = 0, 1, ..., n - 1 (a) Set xk+l = xk + a&, where a k minimises

f l X k + Ordk)

Compute &+I = vfxk+l)'

(b) Unless k = n - 1 set dk+l = %k+l + p d k

where p k = &+I gk+lkkgk) Step 3: Replace xo by x, and go back to step 1. where step 2(a) involves a linear search for the value of a k which minimises xk in the direction dk (a single-dimensional search).

Two constraints restrict the optimisation of the switching parameters to ensure a feasible solution is always gener- ated:

22-1 I zi I Zi+l (6)

z, 5 0.02 (7) Hill climbing techniques predominantly use the partial derivative of the cost function or objective function with respect to each of the optimisation variables, switching instants, to direct the search, step 2(b). However, in t h s case the partial derivatives are not directly obtainable, and therefore a numerical approximation is made. This of course degrades the accuracy of the optimisation proce- dure, but far more importantly the computational time requirements of calculating n/2 partial derivatives are con- siderable and prevent the filter from being able to adapt to changes in the nonlinear load current within an acceptable time period.

Two new methods are investigated in order to reduce the time required for the determination of a suitable switching pattern to generate the compensation signal. The first method eliminates the need for auxiliary information regarding the optimisation surface, i.e. partial derivatives, by using a genetic algorithm [5] to direct the search for a suitable switchmg pattern. The second method extends the analysis of the filter circuit of Fig. 2, reducing the computa- tional requirements of calculating the cost function and its derivatives.

4 Genetic algorithm (GA)

Based on models of biological evolution, GAS have proven to be highly effective in a range of optimisation problems, including vast multimodal functions. They are highly suited to search spaces which are not well defined or have a high number of local minima, which plague more traditional calculus-based search methods. By removing the need for auxiliary information regarding the optimisation surface the computational requirements are reduced as will be shown.

The GA necessitates the need for the optimisation varia- bles to be coded as symbols, in this case as a binary string, the symbols are known as genes and the string generated from these genes is termed a chromosome. To ensure that the constraint (eqn. 6) defined above is not broken by the optimisation procedure, the variables were coded as Axj, where i = 1, 2, 3, ..., N, rather than explicit switching times,

IEE Proc.-Electr. Power Appl., Vol. 147, No. 1, January 2000 22

as follows: 20 = 0 z1 = 2 0 + Ax1 = az, 2 2 = 20 + Ax1 + Ax2 = Ax1 + Ax2

z5 = zo + Ax1 + Az2 + 3 * . + A22-1+ Az5 ( 8 )

The number of bits per Axi defines the resolution of the optimisation. In this case a resolution of lps is required. Hence the number of bits required is defined as follows:

0.01 0.000001

max integer = ~ = 10000 (9)

no. of bits = log'' 'OooO = 13.28 (10) log10 2

Therefore, the number of bits per switching instant is 14, giving a resolution of 610ns per switching instant.

Due to the way in which the variables are coded, the final switching time is a summation of each of the optimi- sation variables. Ths could cause the second of the con- straints defined above to be violated if the sum of the switchmg deltas is greater than half the cycle period, i.e.

- 2 AX% 5 0.01 (11) i=l

To force the above constraint to hold, the sum of the switchmg deltas must be reduced, and this is accomplished as follows:

thus ensuring that a feasible solution is generated. This cre- ates the need for a penalty function: the fitness of the indi- vidual whose optimisation variables required adjustment is decreased in line with the degree to which the constraint was violated. A simple implementation of this is as follows:

The optimisation variables now coded as individual binary strings are concatenated together to form a single string (chromosome) as shown in Fig. 4.

a 01 00 b 1101 coo11 d 0101

\/ \/ chromosome 1 01 001 101 chromosome 2 00110101

Fig.4 Optunrsatwn c o d q

Each chromosome represents a preliminary solution to the optimisation. A set of genes that make up a preliminary solution is called a schema. During each iteration (genera- tion) the individuals in the current population are evaluated using a measure of fitness, in ths case the THD after filter- ing. The selection procedure is best described by the use of an example. Using a simple roulette wheel based selection procedure, each individual is allocated a sector of the

IEE ProL -Electr Power A p p l , Vu1 147 No I , Junuary 2000

wheel. The size of the sector is determined by the fitness of the individual: fitter individuals will have a large sector area, whereas a less fit individual will be allocated a smaller sector size. The pointer then spins; the sector the pointer is pointing to once it ceases rotation is the selected parent and is placed in the mating pool, thus creating the required probability-driven selection process, as shown in Fig. 5.

population selected parents

A 01001 101

pointerl B 001 1001 1 ------+ 01001 101 c 0001 001 I / oollolol D 00101 100

E 00110101 FOOOOI 01 1

Fig. 5 sekction mchi .un

This process is simulated in software by first adjusting the fitness values of each of the individuals within the pop- ulation, so that the fittest individuals have the highest fit- ness values. Since the selection procedure described above requires the function to be a maximisation rather than a minimisation, we must therefore adjust the fitness values of the population. In order to generate the random nature of the selection procedure, the sum of the population fitness values is calculated and multiplied by a random number between 0 and 1. Then, starting at the top of the popula- tion each of the fitnesses are added together; as soon as the total is greater than the random fitness value generated that individual is selected, and this process is repeated until suf- ficient parents have been selected. Thus the individuals with the higher (lower) adjusted fitness are selected more fre- quently. The genetic operator's crossover, mutation and reproduction are then applied to the selected parents to produce two offspring within the next generation (Fig. 6). This process is repeated until the new population is com- plete.

crossover point

parent 1 child 1 01001 001

parent 2 o l o o ~ ~ ~ l o l o l } - { child 2 10110101

Fig. 6 Crossover and reprodiction

One of the most sigmfkant properties of GAS is the con- cept of implicit parallelism [6], in which the population simultaneously searches many dfferent regions of the objective surface. Therefore, by the processes of reproduc- tion, crossover and mutation individuals compete to increase or decrease the number of their offspring in each population.

Several GAS have been developed which reproduce vari- ous aspects of the evolution process, such as the crowding factor model, the generalised crossover model, the elitist expected value model and many others. Each model attempts to improve some aspect of the reproduction proc- ess. Only by an investigation of the applicability of the var- ious models can a suitable selection be made. In this paper the crowding factor model was selected, due to its improved performance on multimodal functions. The crowding factor model increases the competition for limited resources for &e individuals, which has the effect of decreasing life expectancy and birth rates. Less crowded regions acheve hgher life expectancy and birth rates since the competition for resources is decreased.

23

5 Extended analysis

Analysing the filter current waveform as ( N + 1) time inter- vals, where N is the number of switching instants per cycle (N even), the circuit configuration is fixed within each time interval as shown in Fig. 7. Therefore, a representative equation describing the filter current at all points within each switching period can be obtained.

case 1 Fig. 7 Circuit separation

case 2

By splitting the filter into two 'separate' linear circuits, the analysis is simplified using forced and natural responses of the linear circuits during the time interval k (where k = 0 to N), as follows:

i F , = i f o r c e d b + ' h a t u r a l h .

Z I ( S ) = RL + S L

(14)

(15)

(16)

For case I, shown in Fig. 7:

where the single root is R L L

S I == Qi = --

Hence

Consequently

giving the instantaneous variations of the current iik (tk-J for case I during the time interval k

W L and 71 =tan- ' ( n,)

This equation can then be used to determine the filter cur- rent during the time interval k while the circuit remains in this state. At each switchmg transition the initial conditions change. Therefore the preliminary calculations must be reevaluated.

The above analysis is repeated to determine an equiva- lent equation for case 11, giving:

2;:. ( t ) = K I I sin(wt + 7 1 1 ) + ,;'eat cos($t + 0,") (21)

where V,WC

K I I = J ( W R L C ) 2 + (1 - W 2 L C ) Z

and 7r

((1 "&) 711 = - - tan-' 2

Three assumptions are made throughout the above analy- sis: (i) The capacitor voltage remains constant while switched out of the circuit. (ii) The switches SI and S2 act in anti-phase, such that they carry the input current alternately. (iii) The transition period between the two branches is neg- ligible.

This leads to a further improvement regarding the evalu- ation of the cost function. By dividing the filter current waveform into N + 1 distinct sections, and obtaining N + 1 equations describing the filter current for each section, the Fourier coefficients of the filter current can now be calcu- lated using analytical methods rather than the discrete transform used previously. The Fourier analysis of the above two cases is presented in the Appendix (Section 9).

6 Simulation and practical results

The harmonics generated by the nonlinear load in Fig. 2 were used to evaluate the performance of the optimisation software, implemented using a 450MHz Pentium I1 PC. Fig. 8 shows the convergence properties of each of the methods outlined above. Comparing with the original pro- cedure, the x-axis shows the time required to generate a suitable switching pattern is reduced by a factor of seven for the GA method and 40 for the extended analysis method. The GA method does, however, fail to converge to a stable minimum quickly, unlike the extended analysis method, and ths is due to the crowding factor model forc- ing the algorithm to explore new search areas when attempting to find the global minima.

100 904.

8 d F

Fig.8

conventional

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

J 60 50

....... ......................................... 10

0 200 400 600 800 time, s

Convergence gruph for vurbus opthisation technyues

___ ....... ..................................... ........ nt: ~ : . . . j . . ,

. . . . . . . . . . . . . . , . . . . . . . . ' . . . A:\r . . . . . . . . . . . . . . . . . . ...

-1oA i ....... :.__.____: ........ I ........ I ................ : ..................................................... 40 50 60 70 80 90 100

time, ms Fig.9 smwhted current waveformr

The filter and nonlinear load of Fig. 2 were simulated using PSPICE, after having determined a suitable switchmg pattern using any one of the above methods. As Fig. 9

IEE Proc -Electr. Power Appl. , Vol. 147. No. I . January 2000 24

shows, the switched capacitor filter successfully fdters the harmonics generated by the nonlinear load. The high fre- quency component superimposed upon the supply current waveform is due to the switching frequency of the fiter, in this case 6kHz.

Fig. 10 shows the block diagram of the experimental setup used for the circuits of Fig. 2. Fig. 11 shows the practical results of the system which is rated at 3.5kVA. As the waveforms show, the harmonics generated by the non- linear load are filtered out from the supply current leaving a predominantly sinusoidal waveform, again with the high frequency switching component from the fiter switching superimposed upon the supply current. The THD of the prototype system drops from approximately 102% in the uncompensated case to 5.5% for the compensated case.

nonlinear

sampled current signal

and timers

optimal switching I Process I angles

Fig. I O Block diugrmn of experhntdsetup

load current

supply current 20 A

. .

switching pulses

Fig. 1 1 Prototype w a ~ ~ ~ o r m s

It is worthwhile to note that due to the time delay taken by any of the above three optimisation techniques, the filter operates satisfactorily under steady state condtions. This is acceptable for medium power applications (such as medium voltage distribution networks) where the harmonic spectrum of the system does not change within the span of the fundamental 20ms cycle.

7 Conclusions

In order to reduce the computation times of optimisation techniques for switched capacitor active filters, two new methods are presented and compared in this paper. In cer- tain filter applications a fast response is required and con- ventional optimisation techniques are not adequate due to the long computation times, which slows the fdter response to system harmonic current variations. The genetic algo- rithms are analysed and implemented. They are shown to improve the computation times considerably. The fdter response is improved further by applying the extended analysis to the optimisation process. The reduction in time is due to the crowding factors of GAS, which tend to search for better solutions in the neighbourhood of the best solu- tion found so far. Ths factor causes the GA to be slower than the extended analysis. Both the GA and the extended analysis reached the same set of switching patterns as the conventional techniques.

IEE Proc.-Electr. Power A p p l , Vol. 147, No. I , Junuury 2000

8 References

1 AKAGI, H.: 'New trends in active filters'. Proceedings of the EPE-95, Sevilla, 1995

2 MEHTA, P., DARWISH, M., and THOMSON, T.: 'Switched capac- itor filters', ZEEE Trans. Power Electron., 1990, 5, (3), pp. 331-336

3 KOOZEHKANI, Z.D., MEHTA, P., and DARWISH, M.K.: 'An active filter for retrofit applications'. PEVD-96, Nottingham, UK, 1996, pp. 50-55

4 LUENBERGER, D.G.: 'Introduction to linear and nonlinear pro- gramming' (Addison Wesley, 1973)

5 GOLDBERG, D.E.: 'Genetic algorithms in search, optimisation and machine leaming' (Addison Wesley, 1989)

6 DAVIES, L.: 'Handbook of genetic algorithms' (Van Nostrand Rein- hold, New York, 1991)

9 Appendix

Applying the Fourier series expansion to eqns. 20 and 21, the values of A, and B, can be calculated as follows. A set of harmonic coefficients, AA, Bnk, can be obtained for each of the switchmg instants. By summing the harmonic coeffi- cients for each of the switchng periods, the harmonic coef- ficients for the complete cycle are obtained, A,, B,.

k=l k=l

where

A = f ] i ~ , ( t ) cos(nwt)dt 7r

n k

t L - 1

and

B,, = 7r / ipk ( t ) sin(nwt)dt

AI = 5 [ j: KI sin(wt + T I ) cos(nwt)dt

(23) tli-1

Applying the above equations for both case I and case 11, the following equations can be obtained.

ALk terms for case I:

nh 7r th-1

+ / PLe"tcos(nwt)dt t k - 1

whch for y1 > 1 gives

AI - Kr [-cos((n + 1)wt + rr) ,lC 27r (n+ 1)

x [ecut [nu sin(nwt) + a cos(nwt)l]

(25) For n = 1 this gives

(26)

25

B$ terms for case I:

+ J PLeatsin(nwt)dt

tl. - 1

which for n > 1 gives - sin((n - 1)wt - 71)

(n - 1 ) w

s in ( (n + 1)wt + 71) " 1 tli-1

- (n + 1 ) w

1 t h

+ [ W"eQt [a s in (nwt ) - n w cos(nwt)l 7r((nw)2 + a 2 ) t b - 1

(28 ) For n = 1 this gives

[a sin(wt) + w cos(wt)]

(29) terms for case 11:

This generates very similar equations to those above:

which for n > 1 gives

A;; - W K I I [ - cos((n + 1)wt + YZZ)

COS((^ - 1)wt - 711)

27r (n + l ) w

+ (n - 1)w

x [($ - n w ) sin(($ - n w ) t + 6,")

+U cos(($ - n w ) t + e;')]

x [($ + n w ) sin(($ + n w ) t + 0,")

+a cos(($ + n w ) t + e;')] (31)

For n = 1 ths gives

[($ - U ) sin(($ - w)t + 0;')

t k

+acos(($ - w ) t + H ; ' ) ] ] t k - 1

[($ + w) sin(($ + w ) t + of ' )

+aces(($ + w)t + e;')] 1 tk t h . - l

terms for case 11:

[a s in ( ($ + w)t + 0,")

t k

-(!b + w) cos(($ + w)t + e ; q ] t k - 1

[a s in( ($ - w)t + 0,")

1 t k

-($ - w) cos(($ - w)t + 0 3 1 t k - 1

( 3 5 ) Each of the above equations represents the harmonic

coefticients for a given switching period, and therefore by a summation of the harmonic coefficients for each of the switching instants the harmonic coefficients for the com- plete cycle are obtained.

IEE Proc.-Electr. Power A p p l , Vol 147, No. I , January 2000 26