A novel heuristic optimisation algorithm for automated design of resonant compensators for shunt...

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Published in IET Power ElectronicsReceived on 2nd August 2012Revised on 10th March 2013Accepted on 11th May 2013doi: 10.1049/iet-pel.2012.0755

842The Institution of Engineering and Technology 2013

ISSN 1755-4535

A novel heuristic optimisation algorithmfor automated design of resonant compensatorsfor shunt active filtersWanchak Lenwari1, Nnamdi Okaeme2

1Department of Control System and Instrumentation Engineering, King Mongkut’s University of Technology Thonburi,

126 Prachauthid Road, Bangmod, Tungkru, Bangkok 10140, Thailand2Power Electronic Stafford (PES) Alstom Grid UK Ltd., St. Leonards Avenue, Stafford ST17 4LX, UK

E-mail: [email protected]

Abstract: This study presents the automated control design for shunt active filters using a novel heuristic optimisation algorithm.The hybrid bacterial foraging optimisation algorithm is specifically developed for the automated optimisation of the modifiedresonant compensator for the compensation of all main harmonics, 5th, 7th, 11th and 13th. The compensator is based on thesinusoidal internal model principle providing excellent improvement in control accuracy and performance for harmonicsignals. The compensator is implemented in a single rotating reference frame fixed to the supply voltage vector. Theautomated design procedure and a novel optimisation algorithm are presented in detail. Experimental results demonstrate theeffectiveness of the proposed control design and the accuracy of current tracking performance.

1 Introduction

The use of non-linear loads, particularly power electronicdevices in both industrial and domestic power distributionhas gradually degraded the quality of the electrical power[1]. Power electronic equipment acts as sources of voltageor current harmonics, and if these are of a sufficient size,system voltage distortion and even grid stability problemscan occur [2, 3]. Active power filter is one of the mostefficient solutions for power quality improvement inmodern electrical networks. It has been developed tomitigate the effects of harmonics. The shunt active filter(SAF) is designed to cancel the harmonic currents causedby disturbing non-linear loads by injecting currentharmonics into the distribution grid [3–5]. The effectivenessof the SAF can be determined by (i) the accurate control ofthe harmonic currents, and (ii) the ability to correctlyextract the polluting harmonics and generate referenceharmonics signals for the control system. With the maintarget of tracking the reference currents with the lowestpossible error, many researchers have developed andproposed a variety of control systems. These controlmethods are hysteresis control [5], multi-reference axiscontroller [6], fuzzy logic controller [7], sliding modecontrol [8], non-linear control [9], predictive control [10],repetitive control [11], robust controller [12] and resonantcontrollers [13, 14].Proportional plus resonant (PR) compensators have been

proposed and appear to be suitable for active power filtercontrol [13, 14]. The control systems based on thiscompensator have demonstrated excellent control properties

such as fast response, accurate control etc. However, toobtain effective parameters of the PR-type compensator, acomplicated design procedure is required. Therefore, themain aim of this paper is to propose a new optimisationprocedure based on the hybrid bacterial foragingoptimisation algorithm (HBFOA), which could automaticallydesign a high-performance PR-type compensator. Thecontrol scheme employs a combination of two resonantcompensators applied to a single dq reference frameconfiguration rotating at fundamental frequency, for thecompensation of all main harmonics. The design procedureand the principle of the proposed optimisation method arepresented. Experimental results confirm the effectiveness,accuracy and reliability of the proposed strategy.

2 Three-phase SAF

The SAF including overall control scheme is illustrated inFig. 1. The current compensator adjusts the inverter outputvoltage Vaf, with respect to the measured voltage at thepoint of common coupling, Vpcc, to force the inductorcurrent to match its demand value. The control employs asingle dq frame of reference, which is fixed to the measuredsupply voltage vector. The three-phase supply voltages aremeasured and a phase-locked loop is used to derive thevoltage angular position θ with respect to the ‘a’ phasewinding. There are three control loops as shown in theconfiguration of Fig. 1: the dc-link voltage control, and thed- and q-axis current compensators. For the current loop,zero steady-state error can be achieved with a simple

IET Power Electron., 2013, Vol. 6, Iss. 9, pp. 1842–1850doi: 10.1049/iet-pel.2012.0755

Fig. 1 SAF control structure

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proportional-integral (PI) or PID compensator forfundamental current control if the grid supply is ideal.However, these compensators do not control harmonicsaccurately because of the limitation of the designed controlbandwidth in practical. In this work, the current controlbased on resonant compensators is selected to track theharmonic currents with almost zero steady-state error.

3 Control system design

3.1 Modified PR compensator

The PR compensator was developed for application in thestationary reference frame [15], to accurately control asignal at fundamental frequency, while rejecting all otherfrequencies. It can be derived mathematically bytransforming an ideal PI compensator employed in asynchronous rotating frame rotating at the frequency ofinterest, to the stationary reference frame. It is based on acosine transfer function derived from the internal modelprinciple, as in (1)

Ccos(s) = Kp +Krv0s

s2 + v20

(1)

where Kp is the proportional gain, Kr is the gain of resonantterm and ω0 is the resonant frequency. The gain at resonantfrequency can theoretically be infinite. A high gain can beproblematic in this control system because it tends to reducethe system phase margin and robustness. However, the gainof resonant term must be high enough to achieve zero orquasi-zero steady-state error. A new term, the quality factor

Table 1 Harmonic sequences appeared in a dq frame ofreference

Harmonic in abcstationary frame

Sequence Harmonic order in a dqframe of reference

5 negative 67 positive 611 negative 1213 positive 1217 negative 1819 positive 18


Q, is therefore introduced in (2) to represent the ratio of thecentre or resonant frequency, f0, to the − 3 dB bandwidthwhich is fH− fL. The upper and lower frequencies, fH andfL, are defined as the frequencies where the gain hasdropped to 0.707 of the gain at centre frequency.

Q = f0fH − fL

(2)

The modified transfer function is given as in (3).

C(s) = Kp +Krv0s

s2 + (v0/Q)s+ v20

(3)

Lower values of Q will result in lower gain hence somesteady-state error will be present. However, a consequenceof the high gain at the resonant frequency is that thecompensator not only follows reference demands at thisfrequency, but is also able to reject external disturbances atthis frequency. For the application chosen – the SAF – thisis an important characteristic, as it allows the currentcontrol to perform correctly, even in the presence ofdisturbances at fifth and seventh harmonic frequencies, forexample, background harmonics on the system voltage, orharmonics introduced because of non-linearities in theinverter and pulse-width modulation (PWM) scheme.It should be noted that the resonant term provides only a

very small gain outside its band-pass region. Therefore, toobtain a desired transient behaviour, an additionalproportional gain Kp is required. The current compensatorintroduced in this paper is given in (4). The resonantfrequency corresponds to the appearance of two sets ofharmonic currents in rotating dq reference frame. The 5thand 7th harmonic currents are both seen as 300 Hz, whereasthe 11th and 13th harmonic currents are both seen as600 Hz. Harmonic sequences appeared in a dq frame ofreference is illustrated in Table 1.

C(s) = Kp +Kr1v1s− K1v

21

s2 + (v1/Q1)s+ v21

+ Kr2v2s− K2v22

s2 + (v2/Q2)s+ v22

(4)

1843& The Institution of Engineering and Technology 2013

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Each of the two pairs of harmonics appears at the same
frequency; but they have a difference in the sign of theq-axis current. Therefore one resonant compensator is ableto compensate two harmonic currents. This is a benefit ofdesigning the control in the dq reference frame. In (4), theposition of the zeros is indicated by the resonant gain Kr1

and Kr2. Thus, to precisely set the zeros positions, K1 andK2 are introduced in the compensator structure. They can beconsidered as ‘fine-tuning’ of the compensator and alsoallow the improvement in the characteristic of thecompensator. The stability of the current control systemshould be considered according to the configuration of tworesonant terms having two complex conjugated poles andzeros. From the root locus analysis, the system is initiallyunstable as one closed-loop pole is located outside thestability region thus it requires the dynamic compensation.Among a variety of possible compensation methods, thelead and lag compensation has been found to be effectiveand simple. A lead–lag compensator has a transfer functionof the form in (5).

Gc(s) =(s− z1)(s− z2)

(s− p1)(s− p2)(5)

They were added to the controller structure to improve anundesirable frequency response and the system stability. Bythe forward rectangular rule approximation, the discritisedcompensator is derived as given in (6).

D(z) = Kp +Az+ B

z2 + Cz+ D+ Ez+ F

z2 + Gz+ H

( )

× (z− a)(z− c)

(z− b)(z− d)

(6)

Since the two resonant frequencies, ω1 and ω2, are fixed to300 and 600 Hz, respectively, it is noted that there areeleven parameters (Kp, Kr1, K1, Q1, Kr2, K2, Q2, a, b, c andd ) that need to be optimised. In order to alleviate the designdifficulty, the new automated optimisation procedure andthe basic criteria for choosing these parameters will bepresented in the next section.

3.2 Hybrid bacterial foraging optimisationalgorithms

The HBFOA was specifically developed for the automatedoptimisation of the PR compensators as given in (6) to filterout the 5th, 7th, 11th and 13th harmonic component. TheHBFOA is the result of the selective combination of certainfavourable functions of two of the more widely employedevolutionary algorithms: the genetic algorithm (GA) and thebacterial foraging optimisation algorithm (BFOA) [16–25].The HBFOAwas formed by selectively combining specific

functions of the GA and BFOA in a bid to create an algorithmthat inherently possessed excellent local and global searchability, and hence a faster convergence time to the bestpossible solution. The novel HBFOA proposed in this paperis formed through the combination of specific features ofthe parent algorithms, GA and BFOA, with a specific focuson the parameter optimisation of the PR compensator forthe prototype SAF. Fig. 2 shows a flow chart for theHBFOA and, in the subsequent paragraphs, the differentfunctions necessary for its implementation will be describedin detail in the following sections.


3.2.1 Initialisation: This involves the random generationof the positions for each bacterium within the colony (eachbacterium represents a control solution – a combination ofvalues for the parameters of the compensator). The otherparameters defined at this point include the number ofbacterium in the colony, the number of chemotactic steps,the number of reproductive steps, the number andprobability of elimination and dispersal events; theseparameters are inherited from the BFOA. The possiblevalues for each of the controller parameters are generatedand stored within the matrix θi ( j, k, l ): i represents theindex for each bacterium, whose position holds the possiblevalue for the best solution for the controller parameters,within the colony, j represents the index for the number ofchemotactic steps taken by each bacterium, k is the indexfor the number of reproductive steps and l is the index forthe number of elimination and dispersal events. The nutrientconcentration at the different positions occupied by thebacteria colony is calculated and stored in J (i, j, k, l ).

3.2.2 Chemotaxis: This defines the motion of eachbacterium in response to areas that have the best nutrientconcentrations (or solutions to the optimisation problem).A nutrient concentration function is used to quantify theconcentration of nutrients in any bacterium location – thisrepresents positions within the search space that provide thebest sets of solutions for the controller parameters. In thesearch for areas with the best nutrient concentrations, eachbacterium takes a specified number of chemotactic steps;each step may be a tumble, which is movement in a randomdirection, or a swim/run, which is movement in a specifieddirection with the bacterium always alternating betweenthese two modes during the search and foraging procedure.The chemotactic function gives the BFOA its inherentexcellent local search procedure; but, in order to enablewithin the HBFOA a global search and foraging function,the mutation function of the GA is evoked every time asuccessful chemotactic step is made.On initialisation, the nutrient concentration (the quality of

the control response produced) at each bacterial position iscalculated and made equal to Jlast. After the bacteriumtumbles in a random direction, the nutrient concentration iscalculated at this new bacterial position and made to equalJtumble. Jlast is then compared with Jtumble: if Jtumble is lessthan Jlast (this is called a successful tumble), Jtumble

becomes equal to Jlast and a swim will occur in thedirection specified by the tumble; if Jtumble is greater than orequal to Jlast (this is called an unsuccessful tumble), thevalue of Jlast remains unchanged and the algorithm eitherimplements another chemotactic step (tumble) for thecurrent bacterium or a new chemotactic step (tumble) forthe next bacterium within the colony. In the case that aswim occurs, the nutrient concentration at the new positionis calculated and made equal to Jswim; and then comparedwith Jlast: if Jswim is less than Jlast (a successful swim),Jswim becomes equal to Jlast and another swim will occur inthe same direction as the previous swim; if Jswim is greaterthan or equal to Jlast (an unsuccessful swim), the value ofJlast is unchanged. At this juncture, the current position ofthe bacterium after the unsuccessful swim is mutated andthe new nutrient concentration of the bacterium at themutated position is calculated as Jswim and compared withJlast. If a successful swim occurs after mutation, anotherswim (and possibly a mutation) is performed by thebacterium; if the swim is unsuccessful or the number ofpermissible swims, Ns, is reached, the algorithm implements


Fig. 2 HBFOA flow chart

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a new chemotactic step (tumble) for the next bacterium in thepopulation. Once the total number of chemotactic steps, Nc,has been taken by each bacterium in the colony (that hastotal number indicated by Nb) the algorithm moves on toimplement its reproduction function.

3.2.3 Swarming: This function is modelled by thecell-to-cell attractant/repulsion function. It effectivelyaccounts for the way a bacterium secretes attractants whileforaging in areas with increasing nutrient concentration toenable convergence of the colony to areas of high nutrientconcentration. It also models the secretion of repellents by abacterium foraging within areas of decreasing nutrientconcentration to encourage divergence of the colony awayfrom such areas. This swarming function helps to direct the


search procedure in such a way that more control solutionswhich give better responses are found; wherever the searchencounters poor control solutions, there the procedureensures that such control solutions are discouraged fromfurther testing.

3.2.4 Reproduction: This function is implemented wheneach bacterium within the colony has taken the specifiednumber of chemotactic steps. The reproduction within theHBFOA happens in three stages [16–19]: the first stageinvolves ranking each bacterium or control solution withinthe colony according to the quality of their nutrientconcentration values (according to the quality of theresponses produced by the combination of controlparameter values). For the purpose of this work, the


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nutrient concentration is equal to the best nutrientconcentration value experienced by the bacterium over thetotal number of chemotactic steps. The second stageinvolves adopting the elitist selection function [16–18].In this stage, the positions of a number of the mostsuccessful foraging bacteria within the colony are passed onto the next colony of bacteria unaltered. In the third stage,there is enforced information exchange between theremaining portions of the colony to generate the newpositions for the remaining members of the new colonythrough the application of the following GA reproductionfunctions: crossover and mutation [19, 20]. The collectiveconvergent and divergent effects of crossover and mutation,respectively, ensure that through successive colonies, theHBFOA arrives at the most optimal solution to theoptimisation problem. The reproduction process isperformed until the number of permissible reproductivesteps, Nre, has been reached.
3.2.5 Elimination and dispersal: This function isincorporated within the algorithm to account for changeswithin the environment of the colony that could occurwithin the foraging environment of the colony as a result ofchanges in climatic conditions, the overall level of nutrientconcentration etc. Elimination involves randomly killing offsome poorly performing bacteria within the colony.Elimination events result in a reduction in the total size ofthe colony and the dispersal events compliment theseelimination events by replacing the dead bacteria with newones located in possibly better locations. These events serveto enable the global search and foraging technique of theHBFOA.

3.2.6 Termination: The HBFOA terminates when thepermissible number of elimination and dispersal events,Ned, has been reached. Table 2 shows the parameters usedfor the implementation of the hybrid bacterial foraging; itincludes all of the parameters of the BFOA and some of theparameters of the GA.

3.3 Control optimisation process

The process for validating the HBFOA takes place in twostages: the first stage involves optimising the digitalcompensators on the mathematical model of the active filtermodel using MATLAB/Simulink. The second stageinvolves testing the digital implementation of the optimisedcompensator in conjunction with the experimental

Table 2 Parameters for HBFOA

initialisation no. of bacterialdimensions:

10

no. of bacteria in colony: 50chemotaxis no. of tumbles 10

no. of swims 8mutations type decimal

probability 0.3reproduction crossover type multi-point

probability 0.8mutations type decimal

probability 0.2elimination anddispersal

no. of elimination/dispersal events

6

probability 0.2termination maximum number of

elimination and dispersalevents

6


prototype. The optimisation process is repetitive; the sameprocedure is performed until the termination criterion isreached. The implementation of the first stage involves thefollowing steps:

1. At the start of the algorithm, a specified number ofcompensators, whose parameters correspond to thepositions of the bacteria within the colony, are generated.2. Each compensator solution is tested on the mathematicalmodel of the active filter.3. After the tests are performed on the mathematical model,the output of the compensator is analysed to quantify thequality of the output. The nutrient concentration functionused is the integral of the absolute error [26] between thereference current demand and the measured current.

Once all the fitness values have been obtained, thealgorithm uses these values in ranking the compensatorsolutions in order to apply the appropriate function in theright sequence until the termination criterion is reached.The first stage of the optimisation process takes a total of∼ 0.6 s. The termination criterion is chosen in such a waythat, on average, the total time for the entire process takes∼ 4–5 h; this period can be set to vary depending on thespeed of the processing power used in the optimisationprocess. As proposed in [25] – the BFOA was shown to bethe least performing out of the GA, BFOA and HBFOA.This paper suggested that the results could be generalisedacross all other application and not one specific to drives.Hence, the focus for this paper was to compare theperformance of the GA (a popular and good performingalgorithm) with the HBFOA (a new and hybrid algorithm).Fig. 3 shows the progress made by HBFOA and GA intheir search for optimal parameters for the SAFcompensator. It can be observed that on an average of tensuccessive control optimisations using the HBFOA, withinthe first 6500 function evaluations, the HBFOA convergesto 20% of the final objective value and eventuallyconverges to a final objective value of 0.28052874 after31423 function evaluations. The compensator thatcorresponds to the final objective value has the equation asgiven in (7). The HBFOA converges to 10% of finalobjective value of 0.28052874 after 8300 functionevaluation. The GA achieves a final objective value of

Fig. 3 Objective value achieved after a given number ofevaluations of the objective function


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0.2810; although this objective value is similar to thatachieved by the HBFOA, the superiority of the HBFOA ishighlighted given that the GA only converges to 10% of its
Table 3 SAF system and component parameters forexperimental test

Parameters Values

AC supply RMS voltage andfrequency

120 V, 50 Hz

DC capacitor 1000 µFDC side voltage 400 Vactive filter inductance 7.54 mH,

resistance 0.3 Ωanti-aliasing filter fc = 1940 HzIGBT Semikron IGBT moduledeadtime 4 μsDC voltage compensator (0.3z− 0.295)/(z− 1)PWM switching frequency 5 kHz

Fig. 4 Steady-state response of fifth harmonic control

Fig. 5 Steady-state response of seventh harmonic control


final objective value after 20220 function evaluations.

D(z) = 4.0958+ 0.0595z− 0.3272

z2 − 1.863z+ 0.9999

(

+ 1.5338z− 0.1345

z2 − 1.479z+ 0.9697

)

× (z+ 0.4614)(z+ 0.1207)

(z− 0.4562)(z+ 0.7352)

( )(7)

4 Control implementation and experimentalresults

The experimental rig comprised a 10 kVA SAF controlledby a TMS320C6711 digital signal processor (DSP). Dataacquisition and pulse generation are coordinated by anfield-programmable gate array (FPGA). The SAF and


Fig. 6 Steady-state response of eleventh harmonic control

Fig. 8 Steady-state response of 5th + 7th + 11th + 13th harmonic control

Fig. 7 Steady-state response of thirteenth harmonic control

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1848 IET Power Electron., 2013, Vol. 6, Iss. 9, pp. 1842–1850& The Institution of Engineering and Technology 2013 doi: 10.1049/iet-pel.2012.0755

Fig. 9 Transient response of fifth harmonic control

Table 4 Spectrum analysis of simulation results in steady state

Test Magnitudeerror, %

Phase error,degree

fifth harmonic control 1.51 − 1.97seventh harmonic control 3.18 − 4.22eleventh harmonic control 6.79 − 12.55thirteenth harmonic control 8.34 − 9.565th + 7th + 11th + 13thharmonic control

5th 7.31 − 4.117th 9.44 − 10.3711th 28.39 − 23.8313th 32.31 − 22.63

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compensator parameters used for experimental test are shownin Table 3 with a balanced supply. The experimental datahave been captured using the transducers to the dataacquisition system and imported into MATLAB for plottingall experimental results.For the following tests, the harmonic references, ia*,

were derived from a sinusoidal reference generator in theDSP controller and made available via digital-to-analogueconverters. Therefore, the performance of the currentcompensator itself can be evaluated, independent of thedynamics of the harmonic identifier. The experimentalresults presented are as follows. Figs. 4–7 showsteady-state control of a 5th (3 A), 7th (3 A), 11th (3 A)and 13th (3 A) harmonics, respectively. Fig. 8 showssteady-state control of harmonics combined. It can clearlynote that the actual currents accurately followed thereference currents with very small error in amplitude andphase as presented in Table 4. To test the transientperformance, the response to a step change of fifthharmonic current is investigated as shown in Fig. 9,where the current amplitude was changed from 0–3 A. Asmentioned earlier, the transient response is mainlycontrolled by Kp. A larger Kp are likely to be favoured bycontrol designers since it indicates a faster response and alower damping or a larger overshoot, but the systemstability must be concerned. By contrast, a small Kp willresult in a slow response and high damping. However,one can observe that the actual current rapidly tracks thereference even with the low value of Kp [see (7)], which


is preferable in the controller design. This also shows theadvantage of introducing the modified compensatorstructure in this paper.

5 Conclusions

The novel HBFOA has been proposed in this paper with aspecific focus on the parameter optimisation of theproportional plus two resonant plus lead–lag compensatorsfor active filtering. The algorithm is formed through thecombination of specific features of the parent algorithms,

GA and BFOA, which maintains a general validity and itsoptimised features. The optimisation is performed on themathematical model of the SAF system and presents anexcellent rate of convergence. The experimental resultsshow excellent control of all the main lower order harmoniccurrents. From the results, HBFOA appears to be suitablefor the design of a complicated compensator structure.It can therefore be considered to be an excellent tool andcan be applied in automating the commissioning of manyother different applications.

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