Low Order Harmonics Minimization in Multilevel Inverters using Firefly Algorithm

Faiza Nawaz, Muhammad Yaqoob, Zhou Ming, Muhammad Taha Ali School of Electrical and Electronic Engineering

North China Electric Power University Beijing, China

{faiza.nawaz786, muhammad.yaqoob385, taha.meee}@gmail.com, zhouming@ncepu.edu.cn

Abstract— A solution to harmonics minimization problem using firefly algorithm (FFA) is proposed and results have been applied on multilevel inverter. 5th, 7th, 11th and 13th harmonics are minimized using selective harmonic elimination pulse-width modulation (SHE-PWM). The firefly algorithm (FFA) is used to generate optimized firing angles for multilevel inverter to generate desired outputs having minimized harmonics and lower total harmonics distortion. The output phase and line voltages are plotted in visual charts and their harmonic spectra is analyzed through fast fourier transform (FFT) analysis. The final results from simulations indicate minimization of the low order harmonics as desired.

Index Terms-- Multilevel inverter, H-bridge inverter , selective harmonic elimination pulse width modulation (SHEPWM), firefly algorithm (FFA), modulation index, total harmonics distortion, fitness function.

I. INTRODUCTION

Currently, power conversion technology through multilevel converters has been developing in the area of renewable energies and power electronics with excellent opportunities for further improvements in future [1]. One of the most important uses of this technology is in the medium to high voltage ranges. The use of renewable energy sources is also enabled by multilevel converters in addition to allowing high power ratings. For high power applications, multilevel converters system can also be integrated with renewable energy sources such as fuel, photovoltaic, and wind cells [2]. Thus in the hot research area of renewable energy, multilevel inverters find a great use.

Harmonics reduction is one of the most challenging problems related to multilevel inverters. In accordance with modulation and control for multilevel converters a number of techniques have been developed, some of which are selective harmonic elimination (SHE-PWM), space vector modulation (SVM), and sinusoidal pulse width modulation (SPWM) [3]. The selective harmonics elimination involves the solution of transcendental equations which are nonlinear; these equations have many possible solutions. Currently, the researchers are most widely using SHE-PWM due to its benefits over other

modulation techniques. A good initial guess is needed to acquire the result of transcendental equations using Newton-Raphson method. The next leap forward was to use the Resultant theory method for harmonics reduction. In this method the equations are converted to polynomials in order to obtain required solution but it is a very cumbersome process when degrees of polynomials increase. It was then suggested that these equations can only be solved using optimization techniques, in the past many heuristic techniques have been implemented to get the solution of transcendental equations. So, genetic algorithm (GA) and particle swarm optimization (PSO) [4], [5] is used. Ant colony optimization (ANT), neural networks (NN) and fuzzy logic also provided good solution for these equations. Moreover many derivatives of above mentioned algorithms have also been used.

Each algorithm has its own pros and cons. Some algorithms take much computational time but provide better result and vice-versa. Firefly algorithm (FFA) is proposed for solving nonlinear transcendental equations from SHE-PWM and to minimize the low-order harmonics in multilevel inverters.

This paper consists of five sections. Section II presents a theoretical overview of design and case setup. Implementation of firefly algorithm is discussed in Section III. Experiments and their analysis using MATLAB/Simulink and comparison of firefly algorithm with other optimization techniques is presented in Section IV. Lastly, discussions and concluding remarks are presented in Section V.

II. THEORATICAL OVERVIEW

A multilevel inverter, a power electronic device, is capable of providing desired alternating voltage level at the output using multiple lower level DC voltages as input. H-Bridge inverter shown in Fig. 1 has been selected for the implementation of firefly algorithm (FFA) because of its modularity, simplicity of control, requires less number of components, overall less weight and price as compared to the other types of inverters [6]. To provide a sinusoidal output, several H-bridge inverters are connected in series. Each cell contains one H-bridge and output voltage generated by this

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Figure 1. One phase of a cascaded H-bridge multilevel inverter

multilevel inverter is actually the sum of all the voltages generated by each cell i.e. if there are k cells in H-bridge multilevel inverter then the numbers of output voltage levels would be 2k+1 [7]. 11-level H-Bridge inverter is selected for implementation of firefly algorithm (FFA).

A. Selective Harmonic Elimination Pulse width Modulation

SHEPWM is used to eliminate harmonics in the outputs using angle method. Here switching angles are selected in such a way that the required output has the fundamental frequency and the unwanted frequencies are reduced to zero. Amplitude of odd harmonics in output can be calculated with applying fourier series expansion to the waveform [8].

∑ (1)

where Vn is the harmonic amplitude. The angles are restricted between 0-90 degrees. Due to odd quarter wave symmetry, the harmonics of even order become zero so Vn can be expressed as:

∑ cos :0 : (2)

Now for an 11-level inverter transcendental equations are:

/5 (3) 5 5 5 55 (4) 7 7 7 77 (5) 11 11 1111 11

(6)

13 13 1313 13 (7)

where V1 also termed as modulation index represents the fundamental harmonic which we have to maximize. V5, V7, V11 and V13 are odd harmonics to be eliminated or minimized. Modulation index is presented as:

(0≤M≤1) (8)

In order to eliminate the harmonics 5,7,11 and 13, five switching angles , , , and is needed to find.

B. Firefly Algorithm

The firefly algorithm (FFA) is inspired by the flashing behavior of fireflies and a metaheuristic algorithm. Firefly has three factors of interest [9].

1) Attractiveness: Considering two fireflies as i and j, brightness I as a function of x where x is a certain point in a space i.e. I(x). They are at a distance r from each other and intensity I(r) varies monotonically with the distance as:

2^)( ro eIrI γ−= (9)

where Io is original intensity of light, r is the distance between two fireflies and γ is light absorption coefficient and. As intensity is also a function of attractiveness, we can write the relationship as:

r

oe γββ −= (10) where βo is the attractiveness at r=0. In most of cases βo=1. β also varies monotonically with r. So, equation (3) can be rephrased as:

)1(,)( ^ ≥= mer mro

γββ (11)

Characteristic length Ѓ for a fixed γ is:

∞→→=Γ − mm ,1/1γ (12)

2) Distance: Considering two fireflies i and j at points xi and xj respectively, then the distance r can be mathematically defined as:

rij = || xi-xj || ∑ , , (13)

where d is the number of dimensions; xi,k is the kth component of coordinate xi and xj,k is the kth component of coordinate xj.

3) Movements: While considering two fireflies i and j, movement of i is attracted toward j due to extra brighteness of j can be summarized as: Є (14) where xi is current position of fireflies; second term is attractiveness as seen by other fireflies; third term of the equation is due to random movement of fireflies with the

random variable Єi, which can be replaced by rand-1/2 in MATLAB/Simulink to generate a random number for a given range and α is randomization parameter.

C. The Pseudo Code

By keeping the above three factors in mind, pseudo code of the firefly algorithm can be implemented [10] as in Fig. 2.

Figure 2. Pseudo Code of firefly algorithm.

III. IMPLEMENTATION OF FIRFLY ALGORITHM

The basic principle of firefly algorithm is that to initialize a random swarm of bees and after that the transcendental equations with respect to current position of bees is computed. The transcendental equations are then represented in the form of a single quantity which is called the fitness function.

The fitness function is the measure of the brightness of a firefly. The algorithm tends to find the brightest bee in each alteration and it is the best solution for a current iteration. It does that by sorting and ranking the fireflies based on their light intensity. Fireflies tend to move towards the brighter fireflies by computing the attraction parameter. The algorithm finds the optima when the bee glowing brightest is obtained and all the fireflies gather around the optima closely.

A. Fitness Function Implementation

The fitness function is shown below: 100 ∑ 50 (15)

where i = 1, 2,…, S, subject to 0≤ θi ≤ π/2; M is the required modulation index; S is the number of switching angles; V1 is obtained value of fundamental harmonic and hs is the order of the Sth variable harmonic at the output of a multi-level inverter, e.g. h2 =5 and h4=11.

The top priority is to maximize first harmonic i.e. the fundamental harmonic then second priority is to reduce the remaining harmonics, 5,7,11 and 13. In line to line output voltage the triplen harmonics are already eliminated in Y connection. Fitness function also termed as cost is calculated based on position of each firefly. So better is the position of firefly, lower will be the values of low order harmonics and maximum will be the amplitude of fundamental harmonic. This bee will be glowing brightly and the best solution of algorithm will be the bee with the best value of fitness function in certain iteration. In each iteration, the position of firefly is updated. Cost or the value of fitness function with respect to new position of firefly is calculated again. Sorting is performed. Bee with least cost and maximum brightness is best solution for current iteration. The loop is run for 50 times after which best solution is saved. This is performed for each value of Modulation index over the range of 0 to 1.

IV. SIMULATION RESULTS AND THEIR ANALYSIS

The cascaded H-bridge inverter’s model is developed in MATLAB/Simulink using built-in blocks and components.

A. THD and Modulation

Theoretical values of THD corresponding to the obtained optimum values of firing angles are presented in Fig. 3. The graph is depicts the modulation index versus total harmonic distortion based on theoretical values of THD calculated through the formula. The best theoretical results of THD lie in the range M=0.8 to 0.9. So, the desired THD is achieved from M=0.6 onwards. Fig. 4 shows the trend of the firing angles versus modulation index. At each value of M, optimum value of each firing angle obtained from firefly algorithm is plotted to obtain a graphical representation. Different colors show different angles θ1 to θ5. In Fig. 5, Y- axis shows cost function/fitness function values depicted in terms of logarithmic scale to the base 10. In the range of M varying from 0.6 to 0.8, it can be observed that regular peaks indicate abrupt rise and fall in value of our fitness function. During the interval 0.8 to 0.85, the solution is reaching global minima. In our case the minimum value of cost function is achieved in the same interval. It is because the harmonics are decreased the most so cost becomes least. For the purpose of depicting results and elaborating the achievement of our goal of low-order harmonics minimization, if M is chose to be

Figure 3. Modulation index Vs Total Harmonic distortion THD

0 0.2 0.4 0.6 0.8 10

5

10

15

20

THD

Modulation index

THD VS Modulation index

Figure 4. Modulation index Vs firing a

Figure 5. Modulation index vs. cost function/fit

0.81 and run the algorithm, the obtained angles come out to be θ1= 6.86o, θ2= 14θ4=36.59o and θ5=57.86o.

B. 11-Level H-Bridge Inverter

The cascaded H bridge inverter’s model isBuilt-in component of MATLAB/Simulink purpose.

1) Configuration of inverter: The internalinverter requires careful modeling. The modbridge inverter Phase A has 5 blocks in the and number of output voltage levels is 2k+1block within this inverter block generates +cycle and -100V for negative cycle. Total vby each cell thus giving rise to a +500V twhich is desired goal. Same configuration aand C. C. FFT Analysis

1) FFT Analysis of Phase Voltage: The haof each phase voltage is analyzed. The valuearound 10% and is achieved. It is becauharmonics in the system that phase THD higher. The FFT analysis of Phase A and B7 and Fig. 8. The results show that 5th, 7harmonics are reduced to minimum values.

0.6 0.65 0.7 0.75 0.8 0.850

20

40

60

80

100fir

ing

angl

es th

eta

Modulation index

0.6 0.65 0.7 0.75 0.8 0.850

1

2

3

4

cost

on

logr

ithm

ic b

ase

10 s

cale

Modulation index

angles

tness function.

values of firing 4.34o, θ3= 23.52o,

s shown in Fig. 6. are used for this

l configuration of del of cascaded H- model since k=5 1=11 levels. Each 100V for positive voltage generated to -500V outputs

applies to Phase B

armonic spectrum e of THD must be use of the triplen

comes out to be B is shown in Fig. 7th, 11th and 13th

Figure 6. Three phase and 11-level cas

It is because the switching angfirefly algorithm (FFA) minimize harmonic elimination (SHE) elimharmonics as desired. The FFT anacan be seen in Fig. 9.

Figure 7. FFT Analysis of Ph

0.9 0.95 1

0.9 0.95 1

scaded H-Bridge inverter.

gles theta generated from the THD and selective

minates the low order alysis of Phase C voltage

hase A Voltage

Figure 8. FFT Analysis of Phase B Vo

Figure 9. FFT Analysis of Phase C Vo

2) FFT Analysis of Line Voltages: Harmeach line voltage is also observed. THD mand it can be seen from the results that this vThe FFT analysis of line voltages AB and Fig. 10 and Fig. 11.

oltage

oltage

monic spectrum of must be within 5%

value is obtained. BC are shown in

The main goal of eliminating the been achieved. Further options on Mused to view the exact values of shows the harmonic spectrum of line

Figure 10. FFT Analysis of L

Figure 11. FFT Analysis of Li

5th, 7th, 11th and 13th has MATLAB/Simulink can be

each harmonic. Fig. 12 e voltages CA.

Line Voltage AB

ine Voltages BC

Figure 12. FFT Analysis of Line Volt

D. Comparison with other algorithms

In firefly algorithm fireflies can gather amore closely, so it is better than genetic algothe absorption coefficient and randomizatfirefly algorithm can be adjusted to surpasparticle swarm optimization (PSO) algorithmthe results of harmonics reduction are in sommodulation index values. Many forms of PSresults with various pros and cons. In casmore efficient, reliable and fast. The rangindex on which we get good results is moPSO. Also the 5th harmonic is minimized algorithm but it is not minimized in the other

V. CONCLUSION

Low-order harmonics in the H-bridge mare minimized using selective harmonic elim

tages CA

around the optima orithm (GA). Also ion parameter in s the genetic and

ms. In case of GA, me defined sets of SO have generated se of firefly, it is ge of modulation ore than GA and in case of firefly r algorithms.

multilevel inverter mination through

firefly algorithm. Optimized firinwhich are further used in the invoutputs having minimized harmonicthe results it can be seen that the fsuccessfully implemented utilizielimination technique.

The Simulink model of multilevthe results of firefly algorithm areinverter. The output phase and linvisual charts and their harmonic through fast fourier transform (FFTspectra indicate significant reducharmonics as desired.

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