Optimization for Harmonic Multilevel Inverters

8/16/2019 Optimization for Harmonic Multilevel Inverters

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Electr Eng (2009) 91:221–228

DOI 10.1007/s00202-009-0135-9

ORI GI NAL P AP E R

Particle swarm optimization for harmonic eliminationin multilevel inverters

S. Barkat · E. M. Berkouk · M. S. Boucherit

Received: 28 November 2007 / Accepted: 22 October 2009 / Published online: 13 November 2009

© Springer-Verlag 2 009

Abstract In this paper, harmonic elimination problem in

multilevel inverters with any number of levels is redrafted asan optimization task. A new method based on particle swarm

optimization is proposed to identify the best switching angles

with the dual objectives of harmonic suppression and output

voltage regulation. The advantages of fundamental frequency

harmonic elimination and swarm intelligence are combined

to improve the quality of output voltage of multilevel invert-

ers. The validity of the proposed method is proved through

various simulation results.

Keywords Multilevel converter · Diode-clamped

multilevel inverter · Harmonic elimination · Particle swarm

optimization

1 Introduction

In recent years, static power converters have received more

and more attention because their usefulness for a wide range

of industrial and utility systems applications. These con-

verters produce current and voltage distorted waveforms.

The resulted harmonic pollution causes losses in power

equipment,poor power factor, and electromagnetic inference.

S. Barkat (B)Laboratoire d’Analyse des Signaux et Systèmes (LASS),

M’sila University, Ichbillia Road, M’sila 28000, Algeria

e-mail: [email protected]

E. M. Berkouk · M. S. Boucherit

Laboratoire de Commande des Processus (LCP),

Ecole Nationale Supérieure Polytechnique,

10 Hassen Badi Avenue, 16200 El Harrach, Algiers, Algeria


M. S. Boucherit


For mitigating the aforementioned problems, multilevel

power conversion, first proposed by Nabae [1], is one of themore promising techniques for reduced harmonic distortion

in the output waveform. Multilevel inverters incorporate a

topological structurethat allows a wanted outputvoltageto be

synthesized from among set of dc voltages sources. Various

multilevel topologies have been proposed. Diode-clamp, fly-

ing capacitor and cascade inverters are some of the examples.

Compared with the traditional two-level voltage inverter, the

primary advantage of multilevel inverters is their smaller out-

put voltage step, which results in high power quality, lower

harmonic components, better electromagnetic compatibility,

and lower switching losses [2]. Today, multilevel inverters

are extensively used in high-power applications with medium

voltage levels such as active power filters, static var compen-

sators, unified power flow controllers, electrical vehicles, and

industrial motor drives areas [3–5].

Several modulation and control strategies have been

adopted for multilevel inverters with a primary goal to shape

the harmonic spectrum of the output voltage waveform. The

proposed control strategies include among others multilevel

sinusoidalpulse widthmodulation (SPWM) and space-vector

modulation (SVM) [6,7]. However, switching losses and

voltage total harmonic distortion (THD) are still relatively

high for these proposed strategies [8]. Multilevel selective

harmonic elimination provides the opportunity to eliminate

the lower dominant harmonics and filter the higher residual

frequencies. Typically, this method yields good harmonic

performance with fundamental frequency switching which

reduce switching losses significantly. The main difficulty

for selective harmonic elimination method is to compute

the switching angles. Numerous approaches are available in

searching the optimal switching angles. Traditional Newton–

Raphson method is widely used in this area but can not be

applicable for a large number of switching angles if good

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222 Electr Eng (2009) 91:221–228

initial guesses are not available [8]. A second approach uses

block-pulse functions [9]. Harmonic elimination is achieved

via the replacement of nonlinear transcendental equations

with a set of systems of linear equations.

Another method based on symmetric polynomials and

results theory have been used to solve nonlinear transcenden-

tal harmonic elimination equations [10,11]. However, this

method reaches its practical limitations when the number of switching angle increases. An alternative technique based

on genetic algorithm (GA) optimization for harmonic elim-

ination problem has been reported in [12,13]. In references

[14–16], a hybrid method based on genetic algorithm and

direct search optimization technique is proposed in order to

reduce the computational burden.

This paper proposes to use particle swarm optimization

(PSO) to compute the optimal switching anglesfor multilevel

inverters. The diode-clamped multilevel inverter (DCMI) is

chosen as an example.

Although PSO shares many similarities with GA, the clas-

sical PSO does not have genetic operators such as cross-over and mutation which leads to easy implementation of

this method. The particle swarm optimization (PSO) is a

relatively new optimization algorithm proposed firstly by

Kennedy and Eberhart [17]. The core idea behind PSO is

to emulate the social behavior of a flock of birds seeking

food. This stochastic optimization procedure is based on

the movement and intelligence of swarms, which are able

to solve the optimization problems by social interactions.

The most attractive feature of the PSO is the fact that no

gradient information of the objective function is required.

Successful applications of PSO to several optimization prob-

lems, like PID controller optimization [18] and feed for-

ward neutral network design [19] have demonstrated its

potential.

The paper is arranged as follows. A general description of

an n-level DCMI is established in Sect. 2. The Fourier anal-

ysis of output voltage is presented in Sect. 3. The design of

objective function is formulated in Sect. 4. In Sect. 5, the pro-

posed minimization technique based on PSO is introduced.

The adopted optimization algorithm is detailed in Sect. 6. To

prove the feasibility of the proposed method, Sect. 7 provides

simulations for 5, 7 and 11-level DCMI. Finally, in Sect. 8

concluding remarks are given.

2 Diode-clamped multilevel inverter structure

Figure 1 illustrates the basic power circuit of one phase

leg of DCMI. Normally, one leg of an n-level DCMI has

2(n − 1) main switches (T ki, T

ki with i = 1, . . . , n − 1)

and 2(n− 1) main diodes ( Dki, Dki with i = 1, . . . ,n− 1).

In addition, this topology needs 2(n − 2) clamping diodes

( Dcki, Dcki with i = 1, . . . ,n − 2). k denotes leg

number.

k2D

k(n 2)D

−

k(n 1)D

−

k1Dk1T

k2T

k(n 2)T −

k(n 1)T −

ck1D

ck2D

ck(n 3)D

−

ck(n 2)D −

k1D′

k2D′

k(n 2)D −′

k(n 1)D −′

k1T′

k2T′

k(n 2)T

−′

k(n 1)T

−′

ck2D′

ck(n 3)D −′

ck1D′

ck(n 2)D −′

dcV

n 1−

dcV

n 1−

dcV

n 1−

dcV

n 1−

dcV O a

ai

Fig. 1 One leg of an n-level diode-clamped multilevel inverter

Table 1 Switching table of an n-level DCMI

Output voltage Vao Switch state

T k 1 T k 2 · · · T k (n−2) T k (n−1)

V dc/2 1 1 · · · 1 1

V dc(n − 3)/2(n − 1) 1 1 · · · 1 0

.

.

....

.

.

. · · ·...

.

.

.

−V dc(n − 3)/2(n − 1) 1 0 · · · 0 0

−V dc/2 0 0 · · · 0 0

If theneutral point O is considered asthe outputphase volt-

age reference point, then the circuit generates n output volt-

age levels, where n is assumed an odd number greater than

three. This can be possible by connecting in series (n−1) dc

sources to ac side via (n−1) power switches. The maximum

resulting output voltage Vao swings from V dc/2 to −V dc/2

[20,21].

Assuming that all dc sources have the same voltage

V dc/(n − 1), different switching states provide different out-

put voltages. The lower group switches requires the comple-

mentary gating pulsesof theupper group of the same number.

That means if T ki is On, T

k (n−i) is Off. Table 1 lists the volt-age output levels possible for one phase of an n-level DCMI.

State condition 1 means that the switch is On, and 0 means

that the switch is Off.

3 Fourier analysis

The DCMI can produce a general quarter-wave symmetric

stepped voltage waveform synthesized by (n − 1) equal dc

voltage sources such as the one depicted in Fig. 2.

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Electr Eng (2009) 91:221–228 223

ω t

aoV

2π23π

dcV

n−1

0

1θ

2θ n−12

θ π2π

dcV

2

dcV

2−

dcV

n−1−

dc

n 3V

2(n−1)

−

dc

n−2V

2(n−1)−

Fig. 2 Quarter-symmetric stepped-voltage waveform of an n-level

DCMI

By applying Fourier seriesanalysis, theoutput voltage can

be expressed as

V ao(t ) =

∞k =0

V 2k +1 sin(2k + 1)ωt (1)

Where V 2k +1 is the amplitude of the (2k+1)th harmonic volt-

age given by

V 2k +1 =4V dc

(2k + 1) (n − 1) π

n−12

i=1

cos (2k + 1) θ i (2)

θ i (i = 1, . . . , (n − 1)/2) are switching timing angles. They

indicate the On or Off instant of power switches. Not that

only odd harmonics are considered. The even harmonics are

zero due to the symmetry of the output voltage.

When the magnitudes of the Fourier coefficients are nor-

malized with respect to V dc/(n − 1), we obtain:

V 2k +1 =4

(2k + 1) π

n−12

i=1

cos (2k + 1) θ i (3)

All switching angles must satisfy the condition

0 < θ 1 < θ 2 < · · · < θ (n−1)/2 <π

2(4)

4 Objective function design

The task here is to choosethe switching angles θ i (i = 1, . . . ,

(n − 1)/2) such that the relative fundamental component V 1is equal to the desired normalized voltage V re f /V dc and the

(n − 3)/2 low-order harmonics of V ao(t ) are equal to zero.

Harmonic elimination problem is converted in optimiza-

tion problem and can be stated formally as follows:

Let Fitness(θ i ) the objective function, which can be

written as:

Minimize

Fitness (θ i ; i = 1, . . . , (n − 1)/2) = w1 V 1− (n − 1) M /2

+

(n−1)/2 j=2

w j

V j (5)

Where M is the modulation index defined as follows:

M =2V ref

(n − 1)V dc(6)

and wi (i = 1, . . . , (n − 1)/2) are positive weights which

can give more importance to impose the fundamental over

harmonic elimination.

With the objective function (5), the PSO technique is usedto find the optimal θ i (i = 1, . . . , (n − 1)/2).

5 Particle swarm optimization

Particle swarm optimization is an intelligent algorithm which

relies on exchanging information through social interaction

among particles. The PSO conducts searches using a swarm

of particles randomly generated initially. Each particle i

(i = 1 to swarm size) possesses a current position pi =

pi1 pi2 . . . pi N and a velocity vi = vi1 vi2 · · · vi N , N is the dimension of search space. The position of the particlerepresents a possible solution of the problem. The velocity

indicates the change in the position from one step to the next.

Each particle memorizes its personal best position (pbest i )

which corresponds to the best fitness value in the searched

places. Each particle can also access to the global best posi-

tion (gbest) that is theoverallbest place found by onemember

of the swarm. Namely, particles profit from their own expe-

riences and previous experience of other particles during the

exploration, to adjust their velocity, in direction and amount

[22,23]. The concept of a moving particle is illustrated in

Fig. 3.

The velocity of each particle can be updated iterativelyaccording to the following rule:

vi (k + 1) = wvi (k ) + c1r 1 ( pbes t i − pi (k ))

+ c2r 2 (gbest − pi (k )) (7)

Where

vi (k ) is the current velocity of particle i at iteration k ;

pi (k ) is the current position of particle i at iteration k .

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224 Electr Eng (2009) 91:221–228

Current velocity

Personnel bestperformance

Global best

performance

Current

position

New position

Fig. 3 Concept of modification of searching points

The inertia weight w governs how much of previous

velocity should be retained from the previous time step. The

acceleration coefficients c1 > 0 and c2 > 0 influence the

maximum size of the step that a particle can take in a single

iteration. r 1 and r 2 are two independent random sequencesuniformly distributed in [0,1]. These sequences are used to

affect the stochastic nature of the algorithm.

The first term of right-hand side of the velocity update

equation is the inertia velocity of particle, which reflects the

memory behavior of particle. The second term in the veloc-

ity update equation is associated with cognition since it only

takes into account the private thinking and own experiences

of particles. This component is a linear attraction toward the

local best position ever found by the given particle. But the

third term in the same equation represents the social collabo-

ration and interaction between the particles. This component

is a linear attraction toward the global best position found byany particle.

Each particle investigates the search space from its new

local position using the following equation:

pi (k + 1) = pi (k ) + vi (k ) (8)

After a number of iterations, the particles will eventually

cluster around the area where fittest solutions are.

6 Implementation of PSO for harmonic elimination

problem

In order to describe the implementation of the PSO in har-

monic elimination problem of multilevel inverters, the fol-

lowing pseudo-code is adopted.

Step 1: Initialization

For each particle:

– Initialize the position θ i (0)=

θ i1(0) θ i2(0) · · · θ i n−12

(0)

of each particle with random angles that respect the con-

straints (4);

– Initialize the velocity vθ i (0)=

vθ i1(0)vθ i2(0) · · · vθ in−1

2 (0)

of each particle to random values;

– Initialize the best fitness Fit ness_ pbes t i of particle i.

End for

– Initialization of the best fitness Fit ness_gbest of the

swarm.

Loop

{For each particle

Step 2: Objective function evaluation

– Compute the Fitenessi value of each particle i of the

swarm using the cost function given by (5);

Step 3: Personal best position updating

If Fitenessi < Fit ness_ pbes t iThen Fiteness_ pbes t i = Fitenessi and θ pbest i = θ iEnd if

Step 4: Global best position updating

If Fitenessi < Fit ness_gbest

Then Fiteness_gbest = Fitenessi and θ gbest = θ i

End if

End for

For each particle

Step 5: Position and velocity updating

vθ i = wvθ i + c1r 1(θ pbes ti − θ i ) + c2r 2(θ gbest − θ i )

θ i = θ i + vθ iEnd for

} Until a sufficiently good fitness value is reached.

7 Simulation results

The proposed PSO based method has been successfully

applied to a number of levels of diode-clamped inverter to

illustrate its feasibility. Our aim is to generate an optimal

control of multilevel inverter for a given value of the modu-

lation index M . The parameter M is incremented in step of

0.001.

The proposed method offers the advantage that does not

require severe parameters tuning. To expedite the search for

an optimal solution, c1 and c2 are set to 1.8, the coefficient

was set to 0.75. The weighted factors: w1 is set to 10 and

wi (i = 2, . . . , (n − 1)/2) are set to 1. The number of parti-

cles for PSO is 20. The dc source of each multilevel is givenby V dc = 100(n − 1)/2.

To indicate the quality of output voltage, the total line

voltage harmonic distortion is defined as follows:

THD(%) = 100

100k =1 V

26k ±1

V 1(9)

The even and third harmonic and its multiple are not com-

puted in THD because do not appear in the line voltage.

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Electr Eng (2009) 91:221–228 225

Fig. 4 PSO-harmonic

elimination technique for 5-level

DCMI a Switching angles.

b Cost function c Output

voltage relatively to the middle

point O for M = 0.85 d Lowest

line voltage THD e Line output

voltage for M = 0.85

f Harmonic spectrum of line


The optimal switching angles for 5,7 and 11-level DCMI

are shown in Figs. 4a, 5a and 6a. It is important to note that

the proposed minimization method finds all sets of solutions.

According to the simulation results, the solution is not con-

tinuous for some modulation index and there are several sets

of solutions for some other modulation index.

As seen on Figs. 4b, 5b and 6b, any solution that yields a

cost function less than 0.001 is accepted. We clearly notice

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226 Electr Eng (2009) 91:221–228

Fig. 5 PSO-harmonics

elimination technique for 7-level

DCMI (a) Switching angles

(b) Cost function (c) Output

voltage relatively to the middle

point O for M = 0.85

(d) Lowest line voltage THD

(e) Line output voltage for

M = 0.85 (f ) Harmonic

spectrum of line voltagefor M = 0.85

that thenumber of solutionsfor each M increases or decreases

in according to precision constraint value by which solutions

are calculated.

Using the optimal switching angles calculated above, sim-

ulations have been conducted to verify that the fundamental

frequency switching can achieve high control performance.

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Electr Eng (2009) 91:221–228 227

Fig. 6 PSO-harmonic

elimination technique for

11-level DCMI (a) Switching

angles (b) Cost function

(c) Output voltage relatively to

the middle point O for

M = 0.85 (d) Lowest line

voltage THD (e) Line output


(f ) Harmonic spectrum of linevoltage for M = 0.85

The optimized staircase voltages are depicted in Figs. 4c,

5cand 6c. From theabove simulation results, it canbe derived

that the increased number of levels results in a better approx-

imation to a sinusoidal wave form and provides the opportu-

nity to eliminate more harmonics content.

The THD is different for different solution sets; there-

fore, the lowest THD are shown in Figs. 4d, 5d and 6d.

It can be seen that the THD is high for the low modula-

tion index range and decreases when the number of levels

increases.

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228 Electr Eng (2009) 91:221–228

Figures 4e, 5e and 6e illustrate the line voltages wave-

forms when modulation index is M = 0.85. Figures 4f, 5f

and 6f show the first 100 harmonics (FFT) of line voltages.

From the FFT analysis of line voltages, it is seen that all har-

monics chosen to be eliminated and the third harmonic and

its multiple have been strongly eliminated as expected.

8 Conclusion

In this paper, a novel strategy to eliminate harmonics in mul-

tilevel inverters has been described which exploits the swarm

intelligence. Particle swarm optimization is used to improve

the harmonic elimination technique for multilevel inverters,

which exhibits clear advantages in term of low switching fre-

quency and high output quality. This study hasshown that the

particle swarm optimization is more suitable for multilevel

invertersoptimal control design. This optimization algorithm

is simple to implement, effective and inexpensive in term of

memory and time required.

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Optimization for Harmonic Multilevel Inverters