Pso Switching Optimisation

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Optimized switching scheme of cascaded H-bridge multilevel inverter

using PSO

Vivek Kumar Gupta, R. Mahanty

Department of Electrical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India

a r t i c l e i n f o

Article history:

Received 17 October 2013Received in revised form 14 July 2014

Accepted 23 July 2014

Available online 23 August 2014

Keywords:

Particle swarm optimization (PSO)

Cascaded H-bridge multilevel inverter

Harmonics

Sinusoidal pulse width modulation (SPWM)

a b s t r a c t

The paper uses particle swarm optimization (PSO) to determine the optimum switching angles of

cascaded H-bridge multilevel inverter (CH-MLI) so as to produce the required fundamental voltage and

reduce the harmonic content. This is done by solving the transcendental equations characterizing the

harmonic content. The validity of the proposed method is verified through simulation studies for

three-phase, five-level CH-MLI. To compare the results obtained using PSO, the simulation studies have

been extended for three-phase, five-level CH-MLI using sinusoidal pulse width modulation (SPWM). The

results obtained using PSO are found superior as compared to SPWM in terms of total harmonic distortion

at different modulation indices.

2014 Elsevier Ltd. All rights reserved.

Introduction

Multilevel inverters (MLIs) produce a desired output voltage

from several levels of DC voltages as inputs. By taking sufficient

number of DC sources, a nearly sinusoidal voltage waveform can

be obtained. MLIs have been receiving increasing attention in

recent years for high voltage and high power applications [16].

To control the output voltage and reduce undesired harmonics,

sinusoidal pulse width modulation (SPWM) and space vector mod-

ulation techniques have been conventionally used in MLIs [7,8].

Methods such as selective harmonic elimination (SHE) or pro-

grammed pulse width modulation (PWM) techniques have also

been used extensively, wherein specific higher order harmonics

such as 5th, 7th, 11th and 13th are suppressed in the output volt-

age of the inverter[913]. The major intricacy associated with such

methods is to solve the nonlinear transcendental equations charac-

terizing the harmonics, which can be solved by iterative tech-

niques such as NewtonRaphson method. However, this methodis not suitable in cases involving a large number of switching

angles if good initial guess is not available. Another approach based

on mathematical theory of resultant, wherein transcendental

equations that describe the SHE problem are converted into an

equivalent set of polynomial equations and then mathematical

theory of resultant is utilized to find all possible sets of solutions

for the equivalent problem has also been reported[14]. However,

as the number of harmonics to be eliminated increases (up to five

harmonics), the degrees of the polynomials in the equations

become so large that solving them becomes very difficult. Recently,

this problem has been solved by stochastic optimization methods

based on genetic algorithm (GA) approach in a simpler manner

[15].The GA has been successfully applied to MLIs to find all pos-

sible set of solutions for switching angles. However, the quality of

solution deteriorates using GA as the level of inverter increases in

MLI. Moreover, the convergence speed of the GA algorithm is low

and each step to find the switching angles is a time consuming pro-

cess. In this paper particle swarm optimization (PSO) [1620]has

been applied to solve the SHE problem of cascaded H-bridge MLI

(CH-MLI).

Cascade H-bridge MLI

The basic circuit of a single-phase m-level CH-MLI is shown in

Fig. 1[4,5]. It consists of (m1)/2 cells connected in series in each

phase. Each cell consists of single-phase H-bridge inverter with

separate DC source. There are four active devices in each cell and

a cell can produce three voltage levels 0, Vdc/2 and Vdc/2. When

switches S1 andS2 of one H-bridge inverter are closed, the output

voltage isVdc/2 and when switches S3 andS4 are closed, the out-

put voltage is +Vdc/2. When either the switches S1 and S3 or the

switches S4 and S2 are closed, the output voltage is 0. Higher

output voltage levels can be obtained by connecting these cells

in cascade. The phase voltageVanin a CH-MLI is the sum of voltages

of individual cells.

Van V1V2V3 Vm 1

http://dx.doi.org/10.1016/j.ijepes.2014.07.072

0142-0615/2014 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +91 542 2575388.

E-mail address:[email protected](R. Mahanty).

Electrical Power and Energy Systems 64 (2015) 699707

Contents lists available at ScienceDirect

Electrical Power and Energy Systems

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j e p e s
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whereV1,V2,V3,. . .,Vm are the output voltages of each cell.

The output voltage of a CH-MLI is shown in Fig. 2. This can be

represented as

Vt X1n1

an sin nanbn cos nan 2

where

an 4Vdc

np

Xmk1

cos nak

and all switching angles must satisfy the condition

0ha1ha2 hamhp

2: 3

The even harmonics are zero due to quarter wave symmetry of the

output voltage (bn= 0).

Particle swarm optimization

PSO is an intelligent algorithm which relies on exchanging

information through social interaction among particles [1620].

The PSO conducts search using swarm of particles randomly gener-

ated initially. Each particle i (i= 1 to swarm size) possesses a cur-

rent position xi= [xi1xi2. . .xid] and a velocity ui= [ui1ui2. . .uid],

where d is the dimension of search space. The position of the par-ticle represents 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) which corre-

sponds to the best fitness value in the searched space. Each particle

can also access the global best position (gbest) that is the overall

best place found by one member of the swarm. The particles profit

from their own experiences and previous experience of other par-

ticles during the exploration, to adjust their velocity, in direction

and amount. The concept of a moving particle is illustrated in

Fig. 3. The basic concept behind the PSO technique is to change

the velocity of each particle towards its pbestand gbestpositions

at each time step. This means that each particle tries to modify

its current position and velocity according to the distance between

its current position andpbestand the distance between its current

position andgbest. In its canonical form, PSO is modeled as follows:

Let x and u denote the coordinates (position) and flight speed

(velocity) of a particle respectively in a search space. The position

of theith particle is represented as

xi xi1xi2 . . .xid in the d-dimensional space:

The best previous position of the ith particle is recorded and repre-

sented as

pbesti pbesti1pbesti2 . . .pbestid

The index of the best particle among all the particles in the group is

represented as

Gbest gbest1gbest2 . . .gbestd

The velocity of the ith particle is represented as

ui ui1ui2 . . . uid

The modified velocity and position of each particle can be calcu-

lated using the current velocity and the distance from pbestid and

gbestdas given in(4) and (5).

uk11d ukidc1r1pbestidx

kid c2r2gbestdx

kid 4

xk11d xkidu

k1id i 1; 2; 3;. . . m 5

wheremis the number of particles in group, kis the number of iter-

ations (generation), d is the number of dimensions corresponds to

number of members of each particle, uk1id

is the velocity of member

d of particle iat iteration k + 1, ukidis the velocity of member d of par-

S1

S2

Vdc2

+

-

S1 S3

Vdc1+

-

S4

Vdcn

+

-

S1

S3

S2

S2

S3

S4

S4

Fig. 1. Single-phase m-level CH-MLI.

Fig. 2. Output voltage waveform of CH-MLI. Fig. 3. Concept of modification of searching points.

700 V.K. Gupta, R. Mahanty/ Electrical Power and Energy Systems 64 (2015) 699707


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ticlei at iterationk,umind 6 uid 6 umaxd ,x

k1id is the position of member

d of particle i at iteration k +1, xk

id is the position of member d ofparticle i at iteration k, c1 is the constant weighing factor corre-

sponding topbest,c2is the constant weighing factor corresponding

togbest,r1and r2are the random numbers between 0 and 1, pbestidis the local best position of member d of particle i.

Fig. 4. (a) Simulink model of three-phase five-level CH-MLI, (b) sub-circuit of three-phase five-level CH-MLI using PSO and (c) sub-circuit of RL load.

V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699707 701
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umax determines the fitness of the regions to be searched

between the present position and target position. If umax is too

high, particles may fly past good solutions. Ifumax is too small, par-

ticles may not explore sufficiently beyond local solutions. In many

experiences with PSO, it is often set at 1020% of the dynamic

range of the variable in each dimension. The constants c1 and c2represent the weighing of the stochastic acceleration terms that

pull each particle towards thepbestandgbestposition. Low values

of pbest and gbest allow particles to roam far from the target

regions before being tugged back. On the other hand, high values

Fig. 5. Sub-circuit of pulse generation block using PSO (for phase A).

Fig. 6. Switching pulses of three-phase CH-MLI (for phase A).



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of pbest and gbest result in abrupt movement towards target

regions. The acceleration constants c1 and c2 are often set to 1.8

according to past experiences. The velocity and position update

are described by (4) and (5). A new velocity for each particle is

based on the particles previous velocity, the particles location at

which the best fitness has been achieved so far and the population

global location at which the best fitness has been achieved so far.

The constantsc1andc2provide the correct balance between explo-

ration and exploitation (individuality and sociality). Acceleration is

weighted by a random term, with separate random numbers being

generated for acceleration toward pbest and gbest locations. The

random numbers provide stochastic for the particles velocities inorder to simulate the real behavior of the birds in a flock. Fig. 3

shows the concept of modification of searching points described

by(4) and (5).

An inertia weight parameter w is introduced in order to

improve the performance of the original PSO model. This parame-

ter plays the role of balancing the global search and local search

capability of PSO. A better chance of finding the global optimum

within reasonable number of iterations can be achieved by incor-

porating this parameter into the velocity update given in (4), as

follows:

uk1id wukidc1r1pbestidx

kid c2r2gbestdx

kid 6

Suitable selection of inertia weight provides a balance between glo-bal and local exploration abilities and thus require less iterations to

find the optimal solution. The inertia weight often decreases line-

arly from about 0.9 to 0.4 during a run. In general, the inertia weight

is set according to the following equation:

w wmax wmaxwmin

itermax

iter 7

where wmax and wmin are the maximum and minimum values of

inertia weight, anditeranditermax are current and maximum values

of iteration.

Objective function formulation

The objective function has been chosen to get the optimized

switching angles ai i 1; 2;. . . ;n1

2

so that the relative funda-

mental componentV01is equal to the desired voltage and the lower

order harmonics are equal to zero, wheren is the number of levels

of the inverter.

The harmonic elimination problem is converted into optimiza-

tion problem and is rewritten as

Fitness pbest ai ; i 1; 2;. . . ;n 1

2

w1 V01

n 1

2 M

X13

k2m1

w2m1

V0

2m1 8

(V)

(V)

(V)

(s)

Fig. 7. Line voltages three-phase CH-MLI using PSO at modulation index 0.85: (a) Vab, (b) Vbcand (c)Vca.

Fig. 8. Harmonic spectrum of line voltage Vab using PSO at modulation index 0.85.

V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699707 703
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Fig. 9. Sub-circuit of three-phase five-level CH-MLI using SPWM.

Fig. 10. Sub-circuit of pulse generation block using SPWM.



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shows the line voltages Vab, Vbcand Vca of three-phase, five-level

cascaded CH-MLI andFig. 8shows the harmonic spectrum of one

of the line voltages Vab using PSO at modulation index 0.85. It is

evident from the harmonic spectrum that the lower order harmon-

ics reduce significantly. The total harmonic distortion (THD) ofVabat this modulation index is found to be 7.21%. The THD further

reduces at increased values of modulation indices.

Three-phase CH-MLI using SPWM

To compare the results of three-phase, five-level CH-MLI using

PSO, the three-phase, five-level CH-MLI has been simulated usingone of the widely used PWM techniques, SPWM. Fig. 9shows the

sub-circuit of three-phase, five-level CH-MLI using SPWM. This

consists of six modules of H-bridges. The sub-circuit of pulse gen-

eration block using SPWM is shown inFig. 10. The line voltagesVab,

VbcandVcaof three-phase, five-level cascaded CH-MLI are shown in

Fig. 11and the harmonic spectrum of one of the line voltages Vabusing SPWM at modulation index 0.85 is shown in Fig. 12. The

THD ofVab at this modulation index is 13.21%.

Comparison between three-phase CH-MLI using PSO and SPWM

Table 1gives the comparison of THDs using PSO and SPWM at

different values of modulation indices.Fig. 13shows the graphical

representation of THD versus modulation index for the three-

phase, five-level CH-MLI using PSO and SPWM. It is observed that

as the modulation index increases, THD of output voltage of three-

phase CH-MLI decreases for both PSO and SPWM, however, the

THD of output voltage using PSO is less as compared to SPWM at

all values of modulation indices. Hence the quality of the output

voltage is better in PSO as compared to SPWM.

Conclusion

In this paper, PSO has been used to eliminate some selectedlower order harmonics of CH-MLI. In PSO, there is only one fitness

value which move towards the global optimal point in each itera-

tion. This makes the PSO method computationally faster. The con-

vergence of the PSO is better than the other evolutionary methods.

The PSO converges to the global or near global point with respect

to the last function. The voltage harmonics minimization of five-

level CH-MLI inverter using PSO includes velocity equations, equal-

ity and inequality constraints and creation of initial position. The

application of velocity calculation in PSO is a powerful strategy

to improve the global searching ability. Simulations have been car-

ried out for three-phase five-level CH-MLI using PSO and SPWM

and their FFT analysis has been carried out for different modulation

indices. The simulation results show that the THD is less in the out-

put voltage using PSO as compared to SPWM.Hence PSO can be used for improving the harmonic elimination

problem of MLIs. The proposed method exhibits advantages in

terms of switching frequency and high output voltage quality.

The present study shows that PSO is suitable for MLIs optimal

design. However, practical implementation of the proposed

scheme requires further study specially under varying load

conditions.

References

[1] Lai JS, Peng FZ. Multilevel convertersa new breed of power converters. IEEETrans Ind Appl 1996;32(3):50917.

[2] Rodrguez J, Lai J, Peng FZ. Multilevel inverters: a survey of topologies, controlsand applications. IEEE Trans Ind Electron 2002;49(4):72438.

[3] Tolbert LM, Peng FZ, Habetler TG. Multilevel converter for large electric drives.IEEE Trans Ind Appl 1999;35(1):3644.

Fig. 12. Harmonic spectrum of line voltage Vab using SPWM at modulation index 0.85.

Table 1

THDs (%) for three-phase five-level CH-MLI using PSO and SPWM.

Modula tion inde x THD (%) using PSO THD ( %) using S PW M

0.85 7.21 13.21

0.875 6.34 13.04

0.9 6.17 12.76

0.925 5.91 12.48

0.95 5.68 12.16

0.975 5.13 11.84

1 4.84 11.411.025 4.69 10.94

1.05 4.18 10.69

1.075 4.34 10.38

1.1 4.04 9.61

1.125 3.76 9.15

1.15 3.48 8.64

1.175 3.16 8.27

1.2 2.81 7.56

Fig. 13. THD versus modulation index for three-phase five-level CH-MLI using PSOand SPWM.

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