XU Bowei，LI Junjun

(1.Science ＆ Technology Division，Shanghai Maritime Univ.，Shanghai 201306，China;2.College of Engineering Science ＆ Technology，Shanghai Ocean Univ.，Shanghai 201306，China)

0 Introduction

In recent years，multilevel inverters have been drawing growing attention in high power application at higher voltage level.［1］At present，staircase modulation is one of the common control methods to control cascaded H-bridge.When staircase modulation is used，the value of total harmonic distortion(THD)of the output voltage is low without any filter.Another major advantage of staircase modulation is that the switching frequency is lower than that of other control methods，which means switching losses are reduced.［2］

The primary step is to calculate a set of switching angles in staircase modulation.Recently，many method，such as Newton’s method，polynomial composition method，homotopy algorithm，genetic algorithm(GA)and particle swarm optimization(PSO)algorithm，have been applied to optimization of switching angles.［1-4］Most researchers ignored the triple harmonics.

PSO algorithm is a method of evolutionary computation.［5-7］It was first introduced by Doctor Eberhart and Doctor Kennedy［5］.PSO has strong ability to find the most optimistic result.Meanwhile，it has a disadvantage of local minimum.To enhance the searching ability of PSO，many modified algorithms have been developed.Some modified algorithms bring higher computational complexity despite of improving searching results.In this paper，a sort of modified PSO algorithm is presented.This method can intensify the global exploration ability during the former period and the local exploitation ability during the latter period.This method is applied to optimization of switching angles in 11-level inverter while the triple harmonics are taken into account.And better optimization results are obtained.

1 Harmonic reduction for cascaded multilevel inverters

The cascaded multilevel inverter is formed of several H-bridge inverter units which are connected together.If the output voltage’s four quarter cycles are symmetrical，there is only one switching angle needed to be determined for an inverter unit.And even harmonics are eliminated automatically.

In three-phase circuits，the triple harmonics of phase voltage are counteracted in line voltage.［8］Therefore，many researchers have not taken the triple harmonics into account when they studied the harmonic optimization of cascaded multilevel inverters.But if we study the single phase circuits or three-phase four-wire circuits，the triple harmonics will be considered.

The objective function is the minimum value of THD，which is defined as follows:

Subject to:

where θ=［θ1，θ2，…，θm］，θ1，θ2，…，θmare switching angles of inverter units;UmL(2k+1)is the amplitude of the 2k +1th harmonic of output voltage;UmL1is the amplitude of fundamental wave.

2 SPSO algorithm

Standard particle swarm optimization (SPSO)［9］first initializes a group of random particles.These particles find the best solution by iteration.The iteration formulations are as follows:

Besides，the velocity Vidis limited by maximum velocity Vmax，d.

3 MPSO

A kind of modified particle swarm optimization(MPSO)，in which the inertia weight and acceleration constants change along with iteration step，is proposed to improve the searching ability of PSO.

3.1 Parameter setting

For any optimization search algorithm，generally，it is a good idea for the algorithm to possess higher exploitation ability at the beginning to find a suitable seed，and higher exploration ability to quicken the convergence rate.［9］Therefore，some methods are raised to make the inertia weight become a decreasing function of time.［9-11］

If the inertia weight is always high at the beginning and always low sequentially，the global exploration ability during the former period and the local exploitation ability during the latter period will be enhanced.The negative arc tangent function y=-arctan x (Fig.1)has this character.

Fig.1 negative arc tangent function

Then ω changes with negative arc tangent function in this paper:

where ωmidis the middle value of ω(t);A1determines the change extent of the amplitude of ω(t)，A1＞0;B1is the magnitude of the changing rate of ω(t)，B1＞0.

ω1denotes ω(1)，then

While ω declines with negative arc tangent function，c1and c2change with arc tangent function:

where c1midand c2midare the middle value of c1(t)and c2(t)，respectively.c11and c21denote c1(1)and c2(1)respectively，then

where A2，B2，A3and B3are similar with A1and B1，A2，B2，A3，B3＞0.For the sake of convenience we assume that B1=B2=B3＞0.

The changes of ω，c1and c2can enhance the global exploration ability during the former period and the local exploitation ability during the latter period.

3.2 Parameter range

(1)ω(t)is a monotonic decreasing function，while c1(t)and c2(t)are monotonic increasing functions.Because ω(t)，c1(t)and c2(t)are normally plus，it is requested that:

(2)If the particle trajectories are convergent，ω，c1and c2satisfy the following equation［12］:

Then

Equations (14)and (19)are both considered，then

3.3 Convergence rate of particle trajectories

It is shown in Fig.2 (where φ=c1R1+c2R2)that the parameter range of SPSO can be divided into five different areas.［13］If ω1=0.729，c11=c21=1.494 45，T=30，A1=0.03，A2=A3=0.01，B=0.2，then ωmid，c1midand c2midcan be calculated，and equation (20)is satisfied.

Fig.2 parameters’areas of PSO

4 Application of MPSO

In this paper，an 11-level inverter which consists of five H-bridges is optimized by MPSO.The direct voltage of a H-bridge is E=20 V，and frequency of inverter’s output voltage is 50 Hz.

The equation (1)is used to be cost function.There are 10 particles，30 iterations in MPSO.The parameters setting of MPSO is the same as those in section 4.3.SPSO，PSO with the strategy of nonlinearly decreasing inertia weight［10］，LAPSO［14］are used to be compared with MPSO.The swarm scale and the number of iteration are the same in these four algorithms.In SPSO，ω=0.729，c1=c2=1.494 45.In PSO with the strategy of nonlinearly decreasing inertia weight，ω，c1and c2are the same as literature［10］.In LAPSO，ω=0.729，c1=c2=1.494 45，ρ=0.2，α=1，To=0.001.MATLAB 7.1 is used to write programs in a computer with Pentium Dual 2.00 GHz，2 GHz EMS memory，Windows XP.

Fig.3 Changes of ω，c1，c2，ρ(A)

These four algorithms run for 100 times respectively.The average value of THD，minimum value of THD，computing time of 100 times’running are shown in Table 1.We can see in Table 1 that the average value of THD and the minimum value of THD of MPSO are better than those of SPSO and PSO with the strategy of nonlinearly decreasing inertia weight，andthe computing time is even less than SPSO.Compared with MPSO，LAPSO’s average value of THD is better.The minimum value of THD of LAPSO is close to that of MPSO，but the computing time of LAPSO is much longer than that of MPSO.

Table 1 Comparison of optimization results

Define modulation index as M=UmL1/(m·E)［2］，the values of THD and switching angles calculated by MPSO are shown in Fig.4 and 5 (Because of only 100 times，a few modulation indices are covered).Most modulation indices are between 1.0 and 1.1，while all values of THD are larger than 6%.

Fig.6 Inverter’s output voltage and its fundamental wave when M=1.033 6，value of THD=6.097 1

When the minimum value of THD calculated by MPSO is 6.097 1，the modulation index is 1.033 6.In this case，the inverter’s output voltage and its fundamental wave are shown in Fig.6.The amplitude of fundamental wave and harmonics is shown in Fig.7.When the minimum value of THD calculated by LAPSO is 6.100 8，the modulation index is 1.035 7.In this case，the inverter’s output voltage and its fundamental wave are shown in Fig.8.The amplitude of fundamental wave and harmonics is shown in Fig.9.

Simulation results show that the optimization results calculated by MPSO are better than that of other methods.Harmonics of best result calculated by MPSO have been well controlled.Therefore MPSO is an efficient method for reducing harmonic for cascaded multilevel inverters.

5 Conclusion

A kind of modified PSO is designed to minimize the value of THD of cascaded multilevel inverters considering the triple harmonics.The inertia weight，which declines with negative arc tangent function，is always high at the beginning and always low sequentially.And the acceleration constants change with arc tangent function.Then the global exploration ability is always high during the former period，and the local exploitation ability is always high during the latter period.The efficiency of MPSO is verified by harmonic reduction for 11-level inverter.

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