cuckoo search a new optimization algorithm for harmonic elimination in multilevel inverter

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Copyright © 2012 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofBioinformatics and Intelligent Control

Vol. 1, 1–6, 2012

Cuckoo Search: A New Optimization Algorithm forHarmonic Elimination in Multilevel Inverter

Suman Debnath∗ and Dr. Rup Narayan RayDepartment of Electrical Engineering, National Institute of Technology, Agartala 799055, India

In this paper, a new optimization algorithm, called the cuckoo search (CS) algorithm, is introducedfor solving nonlinear transcendental equation derived from SHE-PWM. This research is the firstapplication of CS algorithm to the optimization of Total Harmonic Distortion (THD) of MultilevelInverter. In order to demonstrate the effectiveness of the CS algorithm, a THD optimization problemwas solved and the results were compared with those obtained using other well-known optimizationtechnique particle swarm (PSO) algorithm. The results demonstrate that the CS algorithm is a veryeffective and robust approach for the optimization of THD optimization problems.

Keywords: SHE-PWM, Cuckoo Search Algorithm, Multilevel Inverter.

1. INTRODUCTION

Minimum Total Harmonic Distortion (THD) is one of themost important requirements from multilevel inverter con-cerning good Power Quality.1 The switching strategies ofMultilevel Inverter are mainly classified in two categories.(1) High switching strategy (Sine Triangle Carrier PulseWidth Modulation (SPWM),2�3 Space Vector Pulse WidthModulation (SVPWM)4 etc.) and(2) Low Switching strategy (Selective Harmonic Elimi-nation Pulse Width Modulation (SHE-PWM),5–7 OptimalMinimization of Total Harmonic Distortion (OMTHD),Optimized Harmonic Stepped Waveform (OHSW)8 etc.).

Among these two strategies low switching strategy hasbeen widely used due to low switching losses. In OMTHDwithout any emphasis on special harmonics, all harmonicsin the same weight (i.e., THD) are minimized.9�10 SHE-PWM technique offers a tight control of the harmonicspectrum of a given voltage waveform generated by apower electronic converter along with a low number ofswitching transitions.7�11 The selected lower order harmon-ics are either zero (in SHE-PWM) or controlled withinallowable limit (in Selective Harmonic Controlled PulseWidth Modulation (SHC-PWM)). It involves the solutionof non-linear transcendental equation sets representing therelation between the amplitude of the fundamental wave,harmonic components and the switching angles.11

Many optimization techniques have been reported inliterature for the solution of non-linear transcendentalequation derived form SHE-PWM and minimize THD.

∗Author to whom correspondence should be addressed.

Particle Swarm Optimization (PSO),5�6�12 Genetic algo-rithm (GA),1�11�13 Harmony Search Algorithm (HSA),14

Bacterial Forgoing Algorithm (BFA)15 etc. has beenapplied for the minimization of THD in MultilevelInverter.Population based algorithms are working with a set of

solutions and trying to improve them. By the nature ofphenomenon simulated by the algorithm, population basedalgorithms can be divided into two groups: evolutionaryalgorithms (EA) and swarm intelligence based algorithms.The most prominent representative of the first group isgenetic algorithms (GA). GA is a method for moving apopulation of candidate solutions through fitness landscapeusing nature inspired operators: selection, crossover andmutation. But, second group of algorithms is of our par-ticular interest in this paper.A conceptual comparison of the cuckoo search with

PSO, Differential Evolution (DE) and Artificial BeeColony (ABC) suggest that CS and DE algorithms pro-vide more robust results than PSO and ABC.16�17 Theapplications of Cuckoo Search into engineering optimiza-tion problems have shown there promising efficiency. Forexample, for both spring design and welded beam designproblems, CS obtained better solutions than existing solu-tions in literature. A promising discrete cuckoo searchalgorithm is recently proposed to solve nurse schedulingproblem,18 data fusion in wireless sensor networks19 andstructural software testing.20

In this paper, a new optimization algorithm, called thecuckoo search algorithm (CS) algorithm21 is used to min-imize the overall THD of the output voltage of a multi-level inverter. The objective function derived from the SHE

J. Bioinf. Intell. Control 2012, Vol. 1, No. 1 xxxx-xxxx/2012/1/001/006 doi:10.1166/jbic.2012.1013 1

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Cuckoo Search: A New Optimization Algorithm for Harmonic Elimination in Multilevel Inverter Debnath and Ray

problem is minimized, to compute the switching angleswhile lower order harmonics are controlled within allow-able limits.This paper is organized as follows. The proposed

scheme is described in Section 2. Simulation results andcomparison are presented in Section 3 and finally conclu-sion in Section 4.

2. PROPOSED SCHEME

A higher level ac voltage can be synthesized by cascad-ing several lower level inverters supplied from equal orunequal dc sources. This configuration is known as mul-tilevel configuration of inverters. A single-phase structureof a cascaded multilevel inverter is shown in Figure 1.In case of equal dc sources. Vdc1 = Vdc2 � � � = Vdcn = Vdc.The synthesized ac output voltage waveform is the sumof all the individual inverter outputs. The number of out-put phase voltage levels of cascade multilevel inverter is,where N is the number of dc source. An output voltagewaveform of an 11-level cascade multilevel inverter withthree dc sources is shown in Figure 2.

2.1. Conventional Method

The output voltage waveform V �t� of the multilevelinverter as shown in Figure 2 can be represented by (1)

V �t�=�∑n−1

�an sinn�n+bn cosn�n� (1)

The even harmonics are absent (bn = 0) due to quarterwave symmetry of the output voltage.4 The n-th harmonican is expressed with the first quadrant switching angles�1��2� � � � �m.

an =(4Vdc

n�

) m∑k=1

cos�n�k� (2)

and0< �1 < �2 < � � ��k < ��/2� (3)

For any odd harmonics, (2) can be expanded up to the k-thterm where m is the number of variables corresponding toswitching angles �1 through �m of the first quadrant. Inselected harmonic elimination, an is assigned the desiredvalue for fundamental component and equated to zero forthe harmonics to be eliminated.7

a1=(4Vdc

�

) m∑k=1

cos��k�=M

a5=(4Vdc

5�

) m∑k=1

cos�5�k�= 0 (4)

��

an=(4Vdc

n�

) m∑k=1

cos�n�k�= 0

Fig. 1. Single-phase configuration of a multilevel inverter.

where M is the amplitude of the fundamental component.Nonlinear transcendental equations are thus formed andafter solving these equations, �1 through �k are computed.Triplen harmonics are eliminated in three-phase balancedsystem and these are not considered in (4). It is evidentthat �m−1� harmonics can be eliminated with m numberof switching angles. These nonlinear equations show mul-tiple solutions and the main difficulty is its discontinuityat certain points where no set of solution is available.5�6

This limitation is addressed in the present method to easethe online application at these points of discontinuity.

2.2. Proposed CS Method

Cuckoo Search (CS) was in its day a reasonably new meta-heuristic that imitates the breeding behavior of the cuckoobirds.21 When the breeding time has come, the cuckoobirds tend to lay their eggs in the nest of other birds. Thehost birds would either throw the eggs left by the cuckooout of the nest or decide to leave the nest and build anew home at another place. Further to confuse the hostbirds, some cuckoo birds were able to produce eggs that

Fig. 2. Output voltage waveform of a 11-level multilevel inverter.

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Debnath and Ray Cuckoo Search: A New Optimization Algorithm for Harmonic Elimination in Multilevel Inverter

look similar to the eggs of chosen host birds. This imita-tion would ensure that these eggs would be cared for bythe host birds and thus increase the cuckoo’s productivity.Once the cuckoo eggs hatched, the cuckoo chick wouldthrow out the host birds’ eggs of the nest, which werehatched slightly later than cuckoo eggs. Consequently, thecuckoo birds got more chance to be fed as the number ofchicks in the nest became less.The CS adopted three rules from the breeding behavior

of cuckoo birds, which were:• Each cuckoo laid one egg at a time. The eggs were leftin random nests.• The nest with a high quality of eggs (solutions) wouldbe carried to the next iterations.• The number of potential hosts was fixed, and the cuckooeggs could be found with a probability P a

a [0, 1]. If thehost bird discovered that the eggs were not hers, the alieneggs could be thrown away or the host birds simply aban-don the nest.

The capability of CS was verified by testing it on a setof mathematical benchmarks and a few engineering designproblems. In spite of using only two parameters, whichwere the population size of the cuckoo and Pa, the exper-imental results showed that the CS produced better solu-tions compared to the GA and PSO.At a first glance, it seems that there are some similari-

ties between CS and hill-climbing22 in respect with somelarge scale randomization. But, these two algorithms arein essence very different. Firstly, CS is population-basedalgorithm in a way similar to GA and PSO, but it usessome sort of elitism and/or selection similar to that used inharmony search. Secondly, the randomization is more effi-cient as the step length is heavy-tailed, and any large stepis possible. And finally, the number of tuning parametersis less than in GA and PSO, and thus CS can be mucheasier adapted to a wider class of optimization problems.The conventional SHE technique for multilevel inverter

has the disadvantage of complexity to solve the nonlineartranscendental equations that have multiple solutions.5�6

Moreover, at certain points, no solutions are available tosatisfy these equations. In the proposed PSO method, thecomplexity of finding the solution of these nonlinear equa-tions is avoided by converting the SHE problem to an opti-mization problem. The %THD of the output voltage canbe computed using.5

%THD =[(

1

a21

) �∑n=5

�an�2

]1/2

×100

Where n = 6i±1�i = 1�2�3� � � �� (5)

In the developed CS algorithm, the same expression ofthe voltage THD is considered as the objective functionF �� and minimized with the constraints of individual har-monics limits and minimal variations of switching angles.The formulation of the problem will be as follows:

Minimize

F ��= F ��1��2� � � � �m� (6)

Subjected to:

0< �1 < �2 < � � ��k <

(�

2

)

�1 =M

�5 = �1

�7 = �2 (7)

��

an ≤ �n

where �1� �2� � � � �n are the allowable limits of individualharmonics. With a considerable number of Cuckoo andlarge number of iteration, the algorithm searches for allprobable set of solutions and finally computes the angles�1 through �m to contribute either the lowest THD or nextto the lowest one based on changes in the switching angles,keeping the individual harmonics within the limits as spec-ified by (7). Also at the modulation indices of disconti-nuity, the switching angles �1 through �m are computedbased on possible minimum voltage THD optimizing theindividual harmonics.Figure 3 shows a flowchart of the proposed algorithm.

Like other evolutionary algorithms, the proposed algorithmstarts with an initial population of cuckoos. These initialcuckoos have some eggs to lay in some host birds’ nests.Some of these eggs which are more similar to the hostbird’s eggs have the opportunity to grow up and becomea mature cuckoo. Other eggs are detected by host birdsand are killed. The grown eggs reveal the suitability of thenests in that area. The more eggs survive in an area, themore profit is gained in that area. So the position in whichmore eggs survive will be the term that CS algorithm isgoing to optimize.

2.3. PSO Method

The PSO methodology is a very powerful tool for opti-mization of nonlinear functions. The method was discov-ered through simulation of a simplified social model viz.bird flocking, fish schooling, etc.8 and presently beingused in many applications for optimization of nonlinearequations.Figure 4 shows a flowchart of the PSO algorithm. In the

proposed PSO method, the complexity of finding the solu-tion of these nonlinear equations is avoided by convertingthe SHE problem to an optimization problem. The %THDof the output voltage can be computed using.5

In the developed PSO algorithm, the same expressionof the voltage THD is considered as the objective function

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Fig. 3. Flowchart of proposed CS algorithm.

F �� and minimized with the constraints of individual har-monics limits and minimal variations of switching angles.With a considerable number of generations and large

number of population in each generation, the algorithmsearches for all probable set of solutions and finally com-pute the angles �1 through �m to contribute either thelowest THD or next to the lowest one based on changesin the switching angles, keeping the individual harmonicswithin the limits as specified by (7). Also at the mod-ulation indices of discontinuity, the switching angles �1

through �m are computed based on possible minimumvoltage THD optimizing the individual harmonics.

3. SIMULATION RESULTS ANDDISCUSSIONS

The proposed scheme has been simulated in MAT-LAB/Simulink environment. For three dc sources, the mul-tiple sets of angles present within the modulation indexrange of 0.45–1.10, eliminating harmonics are computed.Cost of fitness function versus Cuckoo Iteration at 1 Mod-ulation Index is shown in Figure 5. Switching angles ver-sus Modulation Index is shown in Figure 6. Harmonic

Fig. 4. Flowchart of proposed PSO algorithm.

Spectrum for output phase voltage at 1 Modulation Indexis shown in Figure 7. The voltage THD against modula-tion index is shown in Figure 7 using CS and PSO as anoptimization tool. Percentage magnitude of Fifth and Sev-enth Harmonics at particular Modulation Index is shown

Fig. 5. Cost of fitness function versus cuckoo iteration at modulationindex 1.


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Debnath and Ray Cuckoo Search: A New Optimization Algorithm for Harmonic Elimination in Multilevel Inverter

Fig. 6. Switching angle versus modulation index for 7-level multilevelinverter using CS considering lowest THD.

Fig. 7. Voltage THD versus modulation index for 7-level multilevelinverter using CS and PSO considering lowest THD.

in Figure 8. In the CS algorithm, 50 cuckoos are initializedwith a minimum of 2 eggs and maximum 4. Algorithm isconfigured with motion coefficient as 4, number of clus-ters as 2, control parameter of egg lying as 0.5, switchingangle range in between 0 to 900. In PSO algorithm theacceleration factors c1 and c2 are taken as 0.2 and 0.5 for40 numbers of particles. For 7-Level Multilevel Inverterthe THD is being optimized up to 49th order.Figure 5 shows the cost value minimization with num-

ber of iterations. Once the near global minimum is reachedthe fitness growth is saturated. The solution for a 3 vari-able system is obtained in less than 20 iterations wherethe maximum no of iterations is 101. Figure 7 shows acomparison of THD minimization applying CS algorithm

Fig. 8. Percentage of controlled voltage harmonics versus modulationindex for 7-level multilevel inverter using cuckoo search algorithm.

Fig. 9. Harmonic spectrum of output phase voltage for a 7-level multi-level inverter at 1 Md up to 49th order having THD 5.1969% using CS.

and PSO algorithm at each Modulation Index (Md). Theresults show that CS algorithm gives better performance inTHD minimization than PSO. Figure 8 shows the selectedharmonics (fifth and seventh) are within allowable limits3 percents of fundamental. At the same time other order ofharmonics up to 49th order are minimized using proposedoptimization technique to meet the IEEE 519 harmonicstandard.

4. CONCLUSION

A new optimization algorithm, called the cuckoo searchalgorithm (CS) algorithm for THD minimization in Cas-caded H-Bridge Multilevel Inverter is proposed. Selectedlower order of harmonics are controlled within allowablelimits while the fundamental output voltage is maintainedat desired level, thus resulting in the minimum THD andthe corresponding switching angles are computed. Fromthe presented case studies it is observed that CS techniqueprovides superior performance compared to PSO as far asthe minimization of THD is concerned. This method wasapplied for 7 level Cascaded Multilevel inverter with equalinput dc voltage sources. This method can be extended toany number of levels of Multilevel Inverter.

References and Notes

1. S. Debnath and R. N. Roy, Int. J. Eng. Res. Appl. 2, 385 (2012).2. N. Mohan, T. M. Undeland, and W. P. Robbins, Power Electronics:

Converters, Applications, and Design 2nd edn., Wiley, New York(1995).

3. D. G. Holmes and T. A. Lipo, Pulse Width Modulation for PowerConverters: Principles and Practice, John Wiley (2003).

4. N. Celanovic, Space Vector Modulation and Control of MultilevelConverters, [dissertation], Virginia Polytechnic Institute and StateUniversity (2000).

5. R. Ray, D. Chatterjee, and S. K. Goswami, Applied Soft. Comput.l9, 1315 (2009).

6. R. N. Ray, D. Chatterjee, and S. K. Goswami, IET Power Electron.2, 646 (2009).

7. H. S. Patel and R. G. Hoft, IEEE Trans. Ind. Appl. 3, 310 (1973).8. S. Sirisukprasert, Optimized Harmonic Stepped-Waveform for Mul-

tilevel Inverter, [dissertation], Virginia Polytechnic Institute and StateUniversity (1999).

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9. N. Yousefpoor, N. Farokhnia, S. H. Fathi, and J. S. Moghani, Pro-ceedings of International Conference on Electric Power and EnergyConversion Systems (2009), p. 1.

10. N. Yousefpoor, S. H. Fathi, N. Farokhnia, and H. A. Abyaneh, Pro-ceedings of IEEE Conference on Industrial Electronics and Applica-tions (2010), p. 498.

11. S. Kamaraj and V. S. Thamizharasan, Proceedings of 2011 1st Inter-national Conference on Electrical Energy Systems, India, January(2011).

12. J. Kennedy and R. Eberhart, Proceedings of IEEE International Con-ference on Neural Networks, Perth, Australia (1995).

13. C. B. Venkatramanan, K. S. Jayakumar, and B. Yuvarani, Eur. J. Sci.Res. 69, 449 (2012).

14. Z. W. Geem, Music-Inspired Harmony Search Algorithm: Theoryand Applications, Springer (2009).

15. H. R. Baghaee and M. Mirsalim, Proceedings of 25th InternationalPower System Conference (2010), p. 442.

16. A. H. Gandomi, X. S. Yang, and A. H. Alavi, Eng. Comput. 27, 1(2011).

17. R. Rajabioun, Applied Soft. Comp. 11, 5508 (2011).18. M. Dhivya, M. Sundarambal, and L. N. Anand, Int. J. Commun.

Network Syst. Sci. 4, 249 (2011).19. M. Dhivya and M. Sundarambal, Int. J. Mobile Commun. 9, 642

(2011).20. K. Perumal, J. M. Ungati, G. Kumar, N. Jain, R. Gaurav, and P. R.

Srivastava, Lecture Notes in Computer Sciences 7077, 46 (2011).21. X. S. Yang and S. Deb, Proceedings of World Congress on Nature

and Biologically Inspired Computing (NaBIC 2009), India (2009),p. 210.

22. M. M. Nesheli, C. Othman, and R. A. Moradkhani, WSEAS Trans.Advances Eng. Educ. 6, 203 (2009).

Received: 16 April 2012. Accepted: 21 August 2012.


cuckoo search a new optimization algorithm for harmonic elimination in multilevel inverter

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