A Comparison on PSO Variantsin Calculating HEPWM Switching Anglesfor a Cascaded H-Bridge MultilevelInverter

Norkharziana Mohd Nayan, Naziha Ahmad Azliand Shahrin Md. Ayob

Abstract This paper presents a comparison study on the application of variousPSO algorithms in calculating the Pulse Width Modulation (PWM) switchingangles of a cascaded H-bridge multilevel inverter. It aims to investigate thebehavior of the PSO algorithm in three different variants. The results are comparedto determine the most suitable algorithm for optimizing the multilevel inverteroutput voltage through Harmonic Elimination PWM (HEPWM) switchingtechnique.

Keywords Particle Swarm Optimization � Harmonic Elimination PWM � Inverter

1 Introduction

Multilevel inverter plays an important role in medium to high power conversionapplication. The applications include active power filters, static var compensator,unified power flow controller (UPFC), electric vehicles, and industrial motordrives [1–4]. Harmonic Elimination Pulse Width Modulation (HEPWM) offers

N. M. Nayan (&)School of Electrical System Engineering, Universiti Malaysia Perlis,02000 Kuala Perlis, Perlis, Malaysiae-mail: iana@ieee.org

N. A. Azli � S. Md. AyobDepartment of Energy Conversion (ENCON), Faculty of Electrical Engineering,Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysiae-mail: naziha@fke.utm.my

S. Md. Ayobe-mail: shahrin@fke.utm.my

James J. (Jong Hyuk) Park et al. (eds.), Computer Science and Convergence,Lecture Notes in Electrical Engineering 114, DOI: 10.1007/978-94-007-2792-2_30,� Springer Science+Business Media B.V. 2012

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elimination of dominant lower order harmonics and filters higher residualfrequencies in a multilevel inverter output voltage. In general, this method pro-duces a good harmonic performance for the inverter output. The elimination of theunwanted harmonics is obtained via the substitution of the nonlinear transcen-dental equations with a set of systems of linear equations.

Recent research on power converter control using PSO shows that the opti-mization algorithm can be successfully used to control a multilevel inverter [5, 6].Moreover, PSO algorithm is easy to implement and has been empirically shown toperform well on many optimization problems. Involvement of the PSO algorithmhas improved the overall system performance, especially on the aspect of speedprocessing and the quality of the output power. Multilevel inverters with HEPWMswitching controls are known to inherent characteristics such as nonlinearity,unavailability of a precise model and excessive complexity that make them suitedfor PSO control. Thus, it is expected to be able to reduce the computational burdenassociated with the solutions of the non-linear transcendental equations relevant tothe harmonic elimination problem.

The known problem of HEPWM switching technique for inverters in general iscomputing the switching angles as it involves the solving of transcendentalequations. The problem gets worse when the technique is applied to a multilevelinverter topology. In order to solve this problem, a solution using PSO method isproposed in this work. The advantage of this method is its capability in solving thenon-linear equations associated with HEPWM with fast generation of theswitching angles due to faster convergence when searching for the solutions. Thecalculated switching angles are also accurate to any desired value of modulationindex, which makes the angle resolution higher. The produced output voltage isexpected to be of high-quality with precision in the PWM switching angles whilethe unwanted harmonics are eliminated. Furthermore, via PSO method, thememory capacity can be reduced due to its online computational capability, whichmakes the system more flexible and interactive. In addition, using this method theelimination of selected harmonics could minimize the THD and motor losses [7].Thus, the filtering process can be omitted from the system. Since the inception ofPSO in 1995 [8], several researchers have modified the algorithm in order to makeit adaptable to their research application and perform faster than the originalversion. Common variants of the PSO algorithm are standard PSO, Inertia WeightPSO, Constriction Factor PSO and also Mutative PSO [9, 10].

This work aims to investigate the behavior of the PSO algorithm in threedifferent variants. The algorithms are used to compute the multilevel inverterswitching angles. The results are compared to determine an optimized algorithmfor optimizing switching control performance of multilevel inverter implementa-tion. It is expected that a suitable PSO algorithm that will produce a high reso-lution of optimized PWM switching angles and low THD voltage for a multilevelinverter can be achieved.

316 N. M. Nayan et al.

2 Harmonic Elimination PWM

A typical output voltage waveform of a HEPWM is shown in Fig. 1 containsharmonics and the output function f(t) can be expressed in Fourier’s series as:

f ðtÞ ¼X1

n¼1

ðan sin nan þ bn cos nan Þ ð1Þ

Due to the quarter wave symmetry of the output voltage, the even harmonics areabsent and only odd harmonic are present [11]. The amplitude of the nth harmonican is expressed only with the first quadrant switching angles a1, a2,…, aN

an ¼4

np

� �1þ 2

XN

k¼1

�1ð Þkcos nak Þ" #

ð2Þ

and the solution must satisfy:

0\a1\a2\ � � �\aN\p2

� �ð3Þ

For any odd harmonics, Equation (2) can be expanded up to the kth term.Where N is the number of variables corresponding to switching angles a1 throughaN of the first quadrant.

In order to eliminate the selected harmonic, an is assigned the desired value forfundamental component and equated to zero for the harmonics to be eliminated.

Nonlinear transcendental equations are formed. a1 through aN are calculated bysolving this equation. It is evident that (N-1) harmonic can be eliminated withN number of switching angles.

a1 ¼4

np

� �1þ 2

XN

k¼1

�1ð Þkcos ak Þ" #

¼ M

a5 ¼4

5p

� �1þ 2

XN

k¼1

�1ð Þkcos 5ak Þ" #

¼ 0

an ¼4

np

� �1þ 2

XN

k¼1

�1ð Þkcos nak Þ" #

¼ 0

ð4Þ

α1α2α3 αN

1

0 180360

ωt

Fig. 1 Harmonic elimination PWM waveform

A Comparison on PSO Variants 317

where M is the amplitude of the fundamental component.The main issue with regard to the HEPWM method is computation the

switching angles. Several methods for computing optimal PWM switching anglesincludes Newton–Raphson, Walsh function, Resultant Theory, Genetic Algorithm(GA), curve fitting techniques and PSO [4, 9, 16–18]. Newton–Raphson method iswidely used in this area, but it is not appropriate for a large number of switchingangles if good initial guesses are not available and may end in local optima.A second approach is Walsh’s functions commonly used to solve a linear equation.An alternative technique based on GA is very effective in solving optimal PWMswitching angle. Unfortunately this method is complicated and has a mutationprobability. The Resultant Theory’s method has been used to solve non-lineartranscendental harmonic elimination equations by converting them into polyno-mial equations. However, this method is unattractive and has a limitation when thenumber of inverter level increases as the degrees of the polynomial will alsoincrease that may lead to numerical difficulty and substantial computationalburden. Curve fitting techniques present a solution for the optimal PWM switchingangle’s trajectories with polynomial equations. However, the problem with curvefitting techniques arises when a large look-up table needs to be stored on thecontrol system memory. In addition the sampling interval for the PWM switchingangle is low when digitally implemented. Neural network control, fuzzy logic andPSO are parts of the latest trends on computing optimal PWM switching angles.Neural network and fuzzy logic control are quite complicated since the parametersneed repetitive training before being implemented on the system and this maybecome a time-consuming task. PSO has similar characteristics with GA, but it iseasy to implement as it does not have genetic operators such as cross over andmutation which may drag to longer processing time.

3 PSO: The Theory and its Variants

PSO is relatively another type of an optimization algorithm originated fromKennedy and Eberhart [12]. Theory of this stochastic optimization procedure isbased on the movement and intelligence of swarms, which has the ability to solveoptimization problems with social interactions. Research conducted by Al-Othman[4] on the harmonic elimination problem of a cascaded H-bridge inverter with non-equal DC sources proves that PSO have reduced the computational burden asso-ciated with the solution of the non-linear transcendental equations. The mainreason for choosing PSO in this work is that PSO is very simple in concept, easy toimplement and computationally efficient. In power converter areas, PSO mainlyinvolves inverter control, AC/AC choppers, photovoltaic controller and motordrives systems [9, 11, 16, 21]. It is proven that PSO algorithm is very efficientrelative to other evolutionary computation techniques in order to solve optimiza-tion problems.

318 N. M. Nayan et al.

3.1 PSO: The Theory

PSO is inspired by social behaviour of birds flocking or fish schooling. PSOconducts the searching process using a population of particles. Each particle is apotential solution to the problem under investigation. Each particle in a givenpopulation adjusts its position by flying in multi-dimensional search space until anunchanging position of the fittest particle is encountered. The concept; at each timestep changing the velocity and position of each particle toward its pbest and gbestis according to Equations (5) and (6)

vidðt þ 1Þ ¼ w� vidðtÞ½ � þ c1 � r1 � pidðtÞ � xidðtÞð Þ½ �þ c2 � r2 � pidðtÞ � xidðtÞð Þ½ � ð5Þ

xidðt þ 1Þ ¼ xidðtÞ þ vidðt þ 1Þ ð6Þ

For a target problem which has n-dimensions;

xi = (xi1, xi2, …, xin)T; position vector of the ith particleVi = (vi1, vi2, …, vin)T; velocity vector of the ith particlePi = (pi1, pi2, …, pin)T; best position of each particle/pbestPg = index of best particle among all particle in the population/gbestw = inertia weightc1 and c2 = acceleration constantr1 and r2 = random number in a range of 01

Pi is the best fitness so far that particles have achieved. The first part ofEquation (5) is the inertia weight part whereas w represents the degree of themomentum of the particles. The second part is the cognition part, which representsthe independent behaviour of the particle itself. The rest parts of the equationrepresent the social part, which represents the collaboration among the particles.c1 and c2 represent the weighting of the cognition and social parts that pull eachparticle towards pbest and gbest positions. Variable xi is the solution found byeach particle. The goodness of the solution is evaluated based on its fitness value.

3.2 The PSO Variants

Since its inception in 1995, several researchers have modified the algorithm inorder to make it adaptable to their research application and perform faster than theoriginal version. The common variants of the PSO algorithm are as follows:

i) The original PSO or standard PSO—An explorative PSO algorithm. Unfortu-nately, it may suffer from late convergence as it goes closer to the maximumiterations. The equation for the standard PSO is given by Equation (7);

A Comparison on PSO Variants 319

Vid ¼ Vid þ C1 pBest � Xidð Þ � rand1 þ C2 gBest � Xidð Þ � rand2 ð7Þ

ii) Inertia weight PSO—Good for focus but lacks in solution quality. The equa-tions for the standard PSO are given by Equations (5) and (6). The inertiaweight is usually decreased linearly from 0.9 to 0.4 to improve convergence.

iii) Constriction factor PSO—best for focus and solution quality, better perfor-mance in continuous value. Unfortunately lacks in discrete and binaryproblems.

Vid ¼ x Vid þ C1 pBest � Xidð Þ � rand1 þ C2 gBest � Xidð Þ � rand2ð Þ ð8Þ

v ¼ 2

2� u�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 � 4u

p������; u ¼ c1þ c2;u[ 4 ð9Þ

iv) Mutative PSO—Rapid convergence but may be premature. This algorithmkills off non-performing particles and replaces them with mutated version ofgbest.

4 Simulation

The simulation is done by implementing a PSO algorithm on calculating the PWMswitching angles of the power switches in a multilevel inverter to produce the itsoutput voltage. Through PSO algorithm, optimum PWM switching angles of themultilevel inverter can be determined. With that, a high processing speed andresolution of the generated switching angles can be established. A single-phasecascaded H-bridge multilevel inverter PSO algorithm is applied to this system. ThePSO control algorithms are designed and simulated using MATLAB/Simulink. Inaddition, the training of the PSO control algorithm on a representative set of inputand targets pairs have been done using the same simulation software.

4.1 The Software Development

The development and implementation stages of the PSO algorithm on the systemare shown in Fig. 2. It begins with the determination of the number of angles perquarter cycle, N for the HEPWM equations. With that, Equation (4) is used tocalculate the switching angles by adapting the equation into an m-file program-ming command with PSO algorithm. Subsequently, this program is trained todetermine the correct weighting parameter solutions for the HEPWM switchingangles through a training simulation. Then, the algorithm is integrated with themultilevel inverter simulation system to verify the control parameters. After

320 N. M. Nayan et al.

achieving the optimized PWM switching angles, another program in C is devel-oped to generate the pulse widths for the HEPWM. In order to drive the powerswitch on the inverter’s side, the pulse generation algorithm is programmed on aDSP to produce the PWM pulses.

5 Results

In order to select the suitable PSO algorithm for the PWM switching anglesimplementation, several simulations using various types of PSO algorithm asdiscussed in the previous section have been done. Computation of the PWMswitching angles for eight angles per quarter cycle (N = 8 and) amplitudemodulation, ap1 M = 1 is made using Equation (4). The PSO equation param-eters are: Swarm size = 50, Dimension = 8, Maximum iteration = 1000.Table 1 presents the comparison made on the PSO algorithm variants based onthe computed angle value, best fitness and number of maximum iteration withthe switching angles obtained from Newton–Raphson’s method [13] used as thereference angle.

HEPWM PSO DSP Gate Driver

Power Switch

Fig. 2 The system block diagram

Table 1 Comparison of the PSO variants

Standard PSO Inertia weight PSO Constriction factor PSO

Best fitness 8.324 9 10-6 1.1102 9 10-16 0Last iteration 350 302 126Reference angle Calculated anglea1 = 15.2985� 15� 15� 15�a2 = 20.7779� 20.1� 21� 21�a3 = 30.8777� 30� 30� 30�a4 = 41.5097� 38� 42� 38�a5 = 47.0563� 57� 48� 49.2042�a6 = 62.2443� 63� 57� 57�a7 = 64.3129� 64� 64� 64�a8 = 89.8477� 90� 87.6362� 90�

A Comparison on PSO Variants 321

6 Discussions

The simulation result shows that implementation of the standard PSO to calculatethe PWM switching angles of the multilevel inverter suffers from late convergenceand immature value. It can be seen that the calculated angles do not converge tostable values and they are also inaccurate. Meanwhile, from the Inertia WeightPSO best fitness, there is a significant difference on maximum iteration comparedto the standard PSO. The Inertia Weight PSO converges faster than the StandardPSO. The computed angle values are much closer to the reference angles but notvery accurate. Constriction Factor PSO has the fastest convergence performancebased on its final iteration to achieve best fitness and have stable angle values. Thecalculated angles are closer to the reference angles compared to Inertia WeightPSO and standard PSO. Therefore, it is concluded that the Constriction Factor PSOis the most suitable algorithm for implementing in a multilevel inverter system.However, in order to achieve more optimal results further training should be doneby changing the PSO parameters. Apart from that, the objective function of thePSO algorithm can also be changed according to the programmer’s creativity as itmay give different way of iteration and best fitness value. Nevertheless, StandardPSO and Inertia weight PSO can as well be used to give an optimal result only ifthe processing time is not the work constraint.

7 Conclusions

In this paper, three types of PSO algorithms namely standard PSO, Inertia WeightPSO and Constriction Factor PSO have been used to calculate the PWM switchingangles of a multilevel inverter. A brief summary on HEPWM and PSO has beengiven. Based on the simulation result, PSO algorithm can be successfully used tocalculate the HEPWM switching angles of a multilevel inverter. At the same time,the selected harmonics in the multilevel inverter output voltage are eliminated withlow THD percentage. Implementation of the PSO based system on a multilevelinverter switching control has produced a high accuracy of switching anglesgeneration with high quality output voltage waveform. From the simulation withthree types of PSO algorithm, it is found that PSO implementation performancesaccording to the selected application give a distinct characteristic. Implementationof Standard PSO suffers from late convergence and immature value. While withInertia weight PSO the computed results are good for focus, but lacks in solutionquality. Whereas, with Constriction PSO algorithm the computed angle values isbest for focus and solution quality. Simulation results also show that ConstrictionFactor PSO is the potential method in producing an optimize pulse generation forthe multilevel inverter. It is expected that with the implementation of the PSOalgorithm, a high processing speed system with less complexity in the control partis produced.

322 N. M. Nayan et al.

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