Received: 22 May 2017
Revised: 16 October 2017
Accepted: 1 December 2017
DOI: 10.1002/etep.2522
RESEARCH ARTICLE
Application of ABC algorithm for selective harmonic elimination switching pattern of cascade multilevel inverter with unequal DC sources Seyyed Yousef Mousazadeh Mousavi1 | Mohammad Zabihi Laharami2
|
Alireza Niknam Kumle3 | S. Hamid Fathi3 1
School of Electrical Engineering, Iran University of Science and Technology (IUST), Tehran, Iran
Summary In this paper, Artificial Bee Colony (ABC) optimization algorithm is proposed
2
to solve nonlinear equation of Selective Harmonic Elimination pattern
Electrical Engineering Department, Babol University of Technology, Babol, Iran 3
Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran Correspondence Mohammad Zabihi Laharami, Department of Electrical Engineering, Babol University of Technology, Babol, Iran. Email:
[email protected]
considering unequal direct current sources. Satisfying fundamental component and eliminating low‐order harmonics are the objects of Selective Harmonic Elimination switching pattern. A 7‐level inverter fed by isolated unequal direct current sources is chosen for this study. The performance of ABC algorithm, which is 1 of the powerful and new evolutionary optimization patterns, is compared with particle swarm optimization, genetic algorithm, and bee algorithm used by other researchers. The comparison of results shows that ABC algorithm finds optimized switching angles with more accuracy and higher convergence compared with others. All aforementioned algorithms were run 30 times, and the statistical analyses show that the algorithm finds switching angles with higher accuracy and lower speed deviation than other algorithms. These comparisons are implemented in MATLAB software; furthermore, the experimental results are presented for the purpose of verification. KEYWORDS Artificial Bee Colony, cascade multilevel inverters, optimization methods, selective harmonic elimination
1 | INTRODUCTION In recent years, the advantages of multilevel inverters (MLIs) such as higher power quality, low voltage stress of components, and lower switching loss caused them to be the subject of ongoing studies1,2; furthermore, the application of these
Abbreviations: ABC, Artificial Bee Colony; AC, alternating current; BA, bee algorithm; CMLI, cascade multilevel inverter; DC, direct current; GA, genetic algorithm; MLI, multilevel inverter; PSO, particle swarm optimization; PWM, pulse width modulation; SD, standard deviation; SHE, selective harmonic elimination; SHEPWM, selective harmonic elimination pulse width modulation Symbols: Fitness(i), the fitness value of the ith solution; f(i), the value of the ith solution; m, number of phase output voltage level; M, Modulation index; Maxcycle, number of maximum cycles for foraging; Pi, probability of the ith food source; S, number of isolated DC sources; yi, ith data of data set; y, average value of data set; V *1 , desired value of fundamental component of output voltage; VDCi, ith voltage of isolated DC voltages; θi, ith angle of switching angles
Int Trans Electr Energ Syst. 2018;e2522. https://doi.org/10.1002/etep.2522
wileyonlinelibrary.com/journal/etep
Copyright © 2018 John Wiley & Sons, Ltd.
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inverters in many cases such as flexible alternating current transmission system,3 high‐voltage direct current (DC),4 and power conditioning applications5-7 has risen. Several topologies and configurations have been suggested for MLIs. Diode clamped,8 cascaded H‐bridge, and flying capacitor are common configurations of MLIs.9,10 Besides these configurations, the hybrid and modified configurations of the MLIs are also presented.11-13 Cascaded H‐bridges have a modular and simple structure; furthermore, in this configuration, the number of components such as diodes and capacitors are fewer than diode clamped and flying capacitor MLIs.9 Additionally, unlike the diode clamp and flying capacitor MLIs, the control of the capacitor voltage fluctuation is not required.14 Different switching patterns have been proposed for controlling the harmonic content of MLI output voltage.15 Some major switching patterns can be listed as sinusoidal pulse width modulation, space vector modulation,16,17 selective harmonic elimination pulse width modulation (SHEPWM),11,18-20 and optimal minimization of the total harmonic distortion strategies.21-24 The optimal selection of DC source value for total harmonic distortion minimization is investigated in Fathi et al.25 A switching strategy that equalized the conduction losses of switches in cascade MLIs is also presented in Aghdam et al.26 The online calculation of switching angles for harmonic elimination is done in Salam,27 which is not an accurate method despite being fast. Moreover, decoupled current control of cascade MLI is introduced in Nademi et al.28 In SHEPWM, by using proper switching angles, low‐order harmonics of output voltage are eliminated while fundamental component of output voltage is regulated. The cascade H‐bridge MLIs are made of S number of H‐bridge units supplied by S number of isolated DC sources that are connected in series form. In conventional SHEPWM, up to S‐1 low‐order harmonics can be eliminated while the fundamental component is kept at its desired value. To that end, proper switching angles should be calculated from a set of transcendental equations. The problem is that the S equations are nonlinear. Hence, different methods have been suggested to solve the nonlinear equation; these methods are divided into 2 major categories. The first category is related to mathematics and iterative methods. Newton‐Raphson is 1 of the incorporated methods of this group18,29 in which the result is dependent on the initial guess; furthermore, this method is not suitable for inverters with a higher number of levels.18 Another mathematical method applied to SHEPWM is a resultant theory, which is a systematic method.30,31 In this approach, transcendental equations of SHEPWM are converted into an equivalent set of polynomial equations; then, the resultant theory is applied to solve them. The drawback of this method is that the algorithm would be complicated and time‐consuming in nonequal DC source application. Moreover, the best solution cannot be obtained in some particular modulation indexes.32 The second category is based on soft computing and evolutionary optimization algorithm that has been reported recently.33 The limitations related to increasing the number of inverter level, which has been a problem in the first category, especially in nonequal DC source case, are overcome in this category. Genetic algorithm (GA), which has been widely used, is applied to SHE in Ozpineci et al.32 Particle swarm optimization (PSO) algorithm is also presented in Taghizadeh and Hagh and Al‐Othman and Abdelhamid34,35 for solving SHE optimization in equal and nonequal DC source optimization problem. In Taghizadeh and Hagh,34 a comparison of PSO and GA is also presented. In Kavousi et al,36 the bee optimization algorithm, which is based on natural foraging behavior of honey, is applied to find switching angles of SHEPWM algorithm considering equal DC sources. Furthermore, GA and bee algorithm (BA) are also compared, and the results show that the BA finds the solution with more convergence and accuracy. Other optimization algorithms such as memetic algorithm and firefly algorithm are also applied in SHE optimization problem.37,38 In this paper, Artificial Bee Colony (ABC) optimization algorithm is applied for solving the SHEPWM of a 7‐level multilevel invert with nonequal DC sources. Furthermore, the performance of this method is compared with PSO, genetic, and BAs. The main advantages of the ABC algorithm include high convergence rate, few setting parameters (does not require external parameters such as cross‐over rate as in case of GA), simple implementation, high flexibility, and strong robustness against initialization.39-43 Both ABC and BA algorithms are inspired by foraging behavior of honey bee swarms. Bee algorithm executes primarily neighborhood searches to obtain better solutions, while ABC performs 1‐dimensional search according to the positions of the bees. In fact, this difference led to ABC's improved functionality compared with BA, especially in general search. Therefore, in BA algorithm, a higher number of bees are required than in the ABC algorithm to achieve the same accuracy. The main contributions of this paper are listed as follows: applying ABC algorithm to SHE optimization algorithm in a 7‐level application with unequal DC sources; comprehensive
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comparison of ABC algorithm with to BE, GA, and PSO algorithm; this comparison includes accuracy, convergence, and time consuming of this algorithm by statistical analyses; and experimental evaluation. The remaining part of paper is organized as follows: Section 2 is devoted to the topological description of cascaded MLI. Additionally, the mathematical equations are presented to optimize the switching angles to eliminate the fifth and seventh harmonics and satisfy the fundamental component. Then, the ABC algorithm is introduced in Section 3. In Section 4, the implementation steps of applying ABC algorithm to SHEPWM optimization are discussed. In Section 5, simulation results are presented and they are compared with other algorithms to show the effectiveness of the ABC algorithm; furthermore, experimental results are reported and discussed in this section. Finally, the paper is concluded in Section 6.
2 | CASCADED MULTILEVEL INVERTER 2.1 | Cascaded multilevel inverter topology Figure 1 shows a cascaded 7‐level inverter topology. As depicted in this figure, a CMLI consists of some single phase full bridge (H‐bridge) units that are connected in series form; hence, the structure is modular and simple. Each full bridge (cell) unit can convert the isolated DC voltage (ie, VDCi) to 3‐level voltages −VDCi, 0, and +VDCi. The number of phase output voltage level (m) of a CMLI inverter, which is fed by S number of isolated DC sources, can be written as m ¼ 2S þ 1
(1)
The output voltage will be more similar to sinusoidal wave when the number of cells is increased; hence, through fundamental frequency switching, the output voltage harmonic content will be desirable.
2.2 | Selective harmonic elimination pulse width modulation Figure 2 shows a typical half‐cycle of phase voltage waveform of the inverter with unequal DC sources. Due to half‐wave symmetry of output voltages of the inverter, the Fourier series expansion of the output voltage will be written as ∞
4V DC ðK 1 cosnθ1 þ K 2 cosnθ2 þ K 3 cosnθ3 þ … þ K s cosnθs Þ sinnωt n¼1;3;5 nπ
V ðωt Þ ¼ ∑
VDC1
VDC2
VDC3
FIGURE 1
Seven‐level cascaded multilevel inverter (CMI).
VOUT
(2)
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VDC1+ VDC2+ VDC3 VDC1+ VDC2 VDC1
θ1
θ2
θ
3
2
π −θ 3
π −θ 2 π −θ1 FIGURE 2
Output voltage of 7‐level cascaded multilevel inverter
where VDC.Ki and VDC are DC voltages of ith cell and nominal DC voltage, respectively. S represents the number of DC sources. Furthermore, the following conditional constraint must be satisfied: θ1 ≤ θ 2 ≤ θ3 ≤ … ≤ θ n ≤
π 2
(3)
In selective harmonic elimination method, the objective is eliminating low‐order harmonics of output voltage. S‐1 number of harmonics can be eliminated by the proper selection of α1‐αs switching angles. In addition, the fundamental component of output voltage should be satisfied. The amplitude of fundamental and nth voltage harmonics are written as 8 4 > > V 1 ¼ V DC ðK 1 cosθ1 þ K 2 cosθ2 þ K 3 cosθ3 Þ × > > π < V 5 ¼ V DC ðK 1 cos5θ1 þ K 2 cos5θ2 þ K 3 cos5θ3 Þ × 4=ð5π Þ > > > > : V n ¼ V DC ðK 1 cosnθ1 þ K 2 cosnθ2 þ K 3 cos nθ3 Þ × 4 nπ
(4)
A 7‐level inverter fed by 3 isolated DC sources is chosen in this paper as a case study (S = 3 in 1). The multithird harmonic will be removed in output line voltage of 3 phase 3 wire inverters. Hence, the switching angles should satisfy the following equations to eliminate the fifth and seventh harmonics and satisfy the fundamental harmonic: 8 > < K 1 cosθ1 þ K 2 cosθ2 þ K 3 cosθ3 ¼ ðπ=2ÞM K 1 cos5θ1 þ K 2 cos5θ2 þ K 3 cos5θ3 ¼ 0 > : K 1 cos7θ1 þ K 2 cos7θ2 þ K 3 cos7θ3 ¼ 0
(5)
where M is the modulation index defined as M = V1/3VDC, while V1 represents the desired value of fundamental component of output voltage.
3 | ARTIFICIAL BEE COLONY ALGORITHM Artificial Bee Colony algorithm proposed by Dervis Karaboga in 2005 is inspired by foraging behavior of honey bee swarms.39 Artificial Bee Colony algorithms provide a population based on metaheuristic procedure and are applied to different optimization purposes such as multidimensional and multimodal optimization problems. This algorithm performs a kind of neighborhood search combined with random search.40 Searching food sources is obtained by 3 bee groups: employed, onlookers, and scout bees. Onlookers wait in the dance area to decide and choose a food source. Employed bees fly to the food source and collect the nectar. A scout is an employed bee whose food resource is abandoned or exhausted by onlookers or employed bees. A scout finds the new food source randomly.41 Their purpose is to discover the places of food sources with a large amount of nectar. Onlookers and scouts perform global search, while local search is carried out by employed bees. Half of the hive population is allocated to onlookers, and the other half is assigned to employed bees. For every food source, only 1 employed bee is assigned. They translate the information of their determined food sources by dancing in the dance room of hive. This waggle dance contains 3 pieces of information. First, it determines the nectar amount corresponding to the food source quality by the duration of the waggle run.43 Second, the distance between the hive
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and food source is indicated by the intensity of the waggles. Finally, the direction of bees indicates the direction of food source in relation to the sun. The angle between the vertical and waggle run indicates the angle between the sun and food source. The food source location represents a possible solution to the optimization problem.42,43 Bees evaluate different food sources according to nectar quality and energy usage. According to the dance, onlookers choose 1 of the better and more attractive food sources. Employed bees and onlookers modify the initial solutions. An employed bee changes to a scout if its food source is rejected by employed and onlookers because of low quality. Scouts start searching the area surrounding the hive to find new food sources randomly and replacing them with abandoned sources.44,45
4 | IMPLEMENTING A RTIFICIAL BEE COLONY OPTIMIZATION A LG O R I T H M S The steps to determine θ = [θ1θ2θ3] are listed below. Furthermore, the flowchart of this algorithm is depicted in Figure 3. Step 1. In the first step, the required parameters of the ABC algorithm such as the number of colony size (the number of employed bees plus the number of onlookers), the value of limit (the number of iterations for releasing a food source), constraints, and the number of maximum cycles for foraging that is represented by Maxcycle are determined. Step 2. The initial conditions of each employed bee are generated in the second level of the algorithm (θ = [θ1θ2θ3]); in other words, each bee in the population is randomly initialized with the following constraints: π 0 < θi < ; θ 1 < θ2 < θ 3 2
(6)
Step 3. In the third step of optimization, first, the cost function value is determined by Equation 7 after that the fitness function is derived from Equation 836:
FIGURE 3
Flowchart of Artificial Bee Colony (ABC) algorithm
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f ði Þ ¼
V * −V 1 100 1 * V1 *
4
2 2 1 1 * V5 * V7 þ 50 þ 50 V1 V1 5 7
Fitness ðiÞ ¼
AL.
(7)
1 f ðiÞ þ 1
(8)
where V1* is the desired value of fundamental component of output voltage. In Equation 7, the first term is related to the fundamental component. Using power 4, this part will be negligible under the 1% of the desire value. Its second and third terms refer to the elimination of the fifth‐ and seventh‐order harmonics. These parts will be ignored when their amount is less than 2% of the fundamental harmonics. If the harmonics exceed the limitation, the cost function will be affected to the power of 2. It should be mentioned that the weighing factor of each term is the inverse of the harmonic order because eliminating the low‐order harmonics is more important. In Equation 8, Fitness(i) represents the fitness value of the ith solution and f(i) is earned by placing the ith solution in the cost function. Step 4. After initialization, the population is subject to repetition of the search process cycles of the employed, onlooker, and scout bees, respectively. Employed bees search the neighborhood of food source, and if fitness of the new food source is better than the old source, they replace it with the new one and save it in their memory. Then, employed bees share their food source information with onlookers waiting in the hive. Step 5. Each onlooker selects a food source after watching the dance of employed bees and evaluates the nectar information taken from all employed bees. The food source that has higher quality will have more chance to be selected by onlookers compared with the one with lower quality. Then, onlookers go to that source, and after choosing a neighbor around that site, they evaluate its nectar amount and compare its performance with that of its old one. If the nectar of the new food source is better than the old one, it is replaced with the old one in the memory. The ith food source, which is selected with the probability Pi by onlookers, is calculated by using the following expression: P ði Þ ¼
a × FitnessðiÞ maxðFitnessÞ þ b
(9)
where a and b are constants that are limited between 0 to 1 and max(Fitness) represents the maximum amount of fitness function until solution i. Step 6. Food sources are abandoned while they are not improved for a preselected number of iteration, and employed bees of these food sources become scouts. Scouts find new food sources, and abandoned food sources are replaced with new food sources. Step 7. The algorithm is terminated if the iteration counter(iter) reaches Maxcycle; otherwise, the iteration counter is incremented (iter = iter + 1) and jumps to step 4.
5 | RESULTS AND DISCUSSIONS Artificial Bee Colony optimization algorithm is implemented for SHE optimization problem of cascaded 7‐level inverter by using MATLAB software. The DC voltage sources are assumed to be nonequal, and the parameter of Ki that determines their magnitude values is in accordance to Table 1. To verify the effectiveness of the algorithm for solving SHEPWM optimization problem, 4 following case studies are considered as follows: • Case I: Applying ABC algorithm to SHEPWM and its evaluation. • Case II: Statistic comparison of the performance of ABC to BA, GA, and PSO optimization algorithms for solving SHEPWM problem. • Case III: Changing the iteration number from 100 to 300 and 1000 and repeating comparison in Case 2. • Case IV: Experimental validation. TABLE 1 The parameter of direct current (DC) sources K1
K2
K3
1.1
1
0.8
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5.1 | Case I: Applying Artificial Bee Colony algorithm to selective harmonic elimination pulse width modulation and its evaluation In the first case study, ABC population size is assumed 500, the number of iteration is considered 200, and the ABC algorithm is applied to solve the SHEPWM problem. The amount of cost function and the optimized angles are plotted in Figures 4 and 5, respectively. Figure 4 shows that, in most of the modulation indexes, the amount of cost function is between 10−12 and 10−14; hence, the effectiveness of this algorithm for solving SHE problem is evident; furthermore, Figure 6 shows the fundamental and low‐order harmonic voltage versus modulation index (M). As depicted in this figure, the fundamental component of output voltage is satisfied, while the low‐order harmonic voltages are minimized especially for the ranges of M in which the cost function has lower values. It is evident from Equation 5 that finding the optimized switching angles depends on the amount of modulation index. In other words, the amount of modulation index should be updated and the switching pattern has to be recalculated; otherwise, considerable harmonics will be generated in output voltage of the inverter. So, the optimized angles are completely independent of the previous response for each modulation index. For instance, Figure 7 shows the percentage of fifth‐ and seventh‐order harmonics with this assumption that the same switching pattern calculated for the equal DC source is applied to unequal DC sources with Ki listed in Table 1. As depicted in this figure, the amounts of fifth‐ and seventh‐order harmonics are significant in some modulation indexes. In Figure 7, the amounts of fifth‐ and seventh‐order harmonics depend on the switching angles (Equation 3). So the disturbances are normal in Figures 5 and 7.
5.2 | Case II: Statistic comparison of the performance of Artificial Bee Colony to bee algorithm, genetic algorithm, and particle swarm optimization algorithm In the second case study, as a comparison, PSO, GA, and BA optimization algorithms are also applied to find the solution to the SHE problem in 3 part, convergence, accuracy, and time.
Optimum Value of Cost Function
10
10
10
10
10
10
10
FIGURE 4
Cost function versus modulation index
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Modulation Index
Optimum switching Angles (Degree)
90
θ
80
θ
70
θ
60 50 40 30 20 10 0 0.4
FIGURE 5
Optimized switching angles
0.5
0.6
0.7
0.8
Modulation Index
0.9
1
1.1
1.2
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100
Amplitude of Harmonic ( )
80 -3
60
40
20
0 0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
FIGURE 6
0.9
1
1.1
1.2
FIGURE 7 Percentage of the fifth‐ and seventh‐harmonic generated if the switching pattern calculated
Modulation Index
The percentage of fifth and seventh harmonics
3
Harmonic Amplitudes ( )
2.5
2
1.5
1
0.5
0 0.4
0.5
0.6
0.7
0.8
Modulation Index
5.2.1 | Convergence For evaluating their convergence characteristics, these algorithms are run 30 times. The population size of these algorithms and ABC algorithm is assumed to be the same and equal to 100. Because the convergence depends on the initial value, for a better comparison, random initial values are considered the same in each run. The number of iteration required for reaching the cost function equal to desired value (in this simulation: 10−6) or lower than that is saved and plotted for 30 times run in Figure 8. Some criteria such as maximum, minimum, average number, and standard deviation (SD) are used to compare these algorithms numerically; these criteria are listed in Table 2. It can be denoted from this table that the average number of required iteration to achieve desired accuracy in ABC algorithm is 9, while it is 1.5, 2.5, and 3.5 times in BA, PSO, and GA, respectively. The maximum and minimum numbers of required iteration are 18 and 1 in ABC, 36 and 8 in PSO, 33 and 2 in BA, and 3 in GA in 30 runs, while sometimes, GA is unable to achieve the desired accuracy at its maximum (100 iteration); this inability is due to entrapment in local optimum. The SD is a statistical measurement of the spread of the numbers in a set of data (yi) from its average value (y) defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n ∑ ðy −yÞ2 SD ¼ n−1 i¼1 i
(10)
When a data set has a high SD, it means the values are widely scattered over a broad range (less reliable), while a low SD means that they are tightly bunched up on either side of the mean value (more reliable).46,47 It is the most robust and
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Number of Required Iteration
40 35 30 25 20 15 10 5 0
0
5
10
15
20
25
30
Run
FIGURE 8
Comparison among Artificial Bee Colony (ABC) and particle swarm optimization (PSO) convergence characteristic
TABLE 2 Comparison among bee algorithm (BA), particle swarm optimization (PSO), genetic algorithm (GA), and Artificial Bee Colony (ABC) convergence characteristic in 30 times run Average Number of Required Iterations
Standard Deviation
Minimum Number of Required Iterations
Maximum Number of Required Iterations
GA
30
35.785
3
100
ABC
9
4.4329
1
18
PSO
22
7.851
8
36
BA
14
8.7727
2
33
Algorithm
widely used criterion of dispersion because unlike the range (minimum and maximum), it takes into account every variable in the dataset. Standard deviation is usually presented in conjunction with the mean, and 1 of its useful properties is that, unlike the variance, it is expressed in the same units as the data and without it; comparisons among several data sets cannot be made effectively. The SD of ABC is 4.4329, where the value is almost double in BA and PSO and 8 times in GA. So, the results of ABC are closer to the average value than other algorithms; therefore, it generates more reliable results. The results of ABC present the lowest among other methods to reach the desired value of fitness function; hence, this algorithm has better convergence characteristics.
5.2.2 | Accuracy The accuracy of algorithms is also compared. Both the population size and the number of iteration are assumed to be 100 for each algorithm. Furthermore, each algorithm is run 30 times and the amount of cost function is saved and shown in Figure 9. It shows that in 30 runs, the amount of cost function in ABC algorithm is the lowest value among other algorithms in 29 runs; the defined criteria for comparing these methods are shown in Table 3 in which they are indicative of the superiority of ABC algorithm; hence, ABC algorithm can find the optimized angle with more accuracy.
5.2.3 | Time Another parameter compared in this paper is the consumption time in simulations. Because ABC has more complexity than PSO and GA, it is predictable that the ABC algorithm consumes more time. The average times consumed for achieving the desired accuracy are 0.16, 0.22, 0.77, and 0.32 seconds in PSO, GA, BA, and ABC algorithms, respectively. These results are implemented with the Intel Corei7 and 2.4‐GHz processor with 100 iterations. Although these results show that PSO algorithm consumes lower time than ABC at 100 iterations, it should be considered that the average required iterations for reaching the desired amount of accuracy is about 2.5 times in PSO according to Table 2; hence, ABC can reach the desired value of accuracy in short time and with more reliability (according to Table 3) in comparison with PSO and other methods.
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10
10
Cost Function
10
10
10
10
10
0
5
10
15
20
25
30
Run
FIGURE 9 Comparison the accuracy of particle swarm optimization (PSO), genetic algorithm (GA), bee algorithm (BA), and Artificial Bee Colony (ABC) algorithm
TABLE 3 Comparison the minimum cost function among bee algorithm (BA), particle swarm optimization (PSO), genetic algorithm (GA), and Artificial Bee Colony (ABC) in 30 times run with 100 Average of Minimum Cost Function
Algorithm
Standard Deviation
Minimum of Minimum Cost Function
Maximum of Minimum Cost Function
GA
5.3452 × 10−7
4.3805 × 10−7
3.4642 × 10−8
1.6418 × 10−6
ABC
4.8085 × 10−9
3.3078 × 10−9
7.2962 × 10−11
1.2304 × 10−8
PSO
1.5868 × 10−7
2.5331 × 10−7
6.3228 × 10−9
1.2844 × 10−6
BA
1.0979 × 10−7
7.6895 × 10−8
4.1119 × 10−9
3.4337 × 10−7
5.3 | Case III: Changing the iteration number from 100 to 300 and 1000 and repeating comparison in Case 2 Because the result of the algorithms depends on iteration, to confirm the validity of the results, the number of iterations is changed in 2 different values of 300 and 1000. According to the results listed in Table 4, ABC algorithm is superior in all criteria except 1. The only exception is the minimum cost function in 1000 iterations where the PSO algorithm is slightly better than the ABC algorithm; however, the proposed algorithm outperforms PSO according to the main criteria (mean) of minimum values in 30 runs because its SD is smaller than that of PSO algorithm. The obtained results confirm the previous results, which indicated the superiority of ABC.
TABLE 4 Comparison the minimum cost function among particle swarm optimization (PSO), genetic algorithm (GA), bee algorithm (BA), and Artificial Bee Colony (ABC) algorithm in 300 and 1000 iterations and 30 times run Average of Minimum Cost Function
Iteration
Algorithm
300
GA ABC PSO BA
3.1812 1.1239 3.0879 1.2954
1000
GA ABC PSO BA
4.8552 × 1.9092 × 3.0669 × 9.1280 ×
× × × ×
Standard Deviation
10−7 10−9 10−8 10−7
3.6679 1.5832 1.3066 1.086
10−7 10−10 10−8 10−8
6.5014 × 1.7853 × 1.382 × 6.4019 ×
× × × ×
Minimum of Minimum Cost Function
Maximum of Minimum Cost Function
10−7 10−9 10−8 10−7
9.5224 × 10−9 6.7495 × 10−11 3.6519 × 10−10 1.1833 × 10−8
1.7874 7.7543 3.8591 5.0604
× × × ×
10−6 10−9 10−8 10−7
10−7 10−10 10−8 10−8
8.2285 × 10−12 3.6710 × 10−12 3.6555 × 10−12 1.0255 × 10−8
2.6724 6.9435 3.6907 2.2794
× × × ×
10−6 10−10 10−8 10−7
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5.4 | Case IV: Experimental validation A 3‐phase 2‐kW cascade 7‐level inverter is built and implemented for verifying the simulation results of SHE problem. Each phase of this inverter consists of 3 full bridge cells connected in series form. The DC voltages of sources are assumed adjustable, and their nominal voltage is 20 V. The frequency is 50 Hz. Due to low cost, simplicity, and high‐speed computations, the AT90CAN128 microcontroller is used to control the switching signals. The switching angle is loaded in this microcontroller by open‐source software “Code Vision.” The switching signal is transferred by an opto‐coupler 6N137 used to insulate the control board from IGBT switches. The structure of the implemented prototype is shown in Figure 10.The amplitude of DC source voltage is in accordance with the variables K1, K2, and K3, which are listed in Table 1, while the nominal DC voltage is considered 20 V. The first experimental test has been done for M = 0.8; the output phase and line voltage waveforms and their harmonic contents for this modulation index are shown in Figures 11 and 12, respectively. The switching angles are in accordance to the angles shown in Figure 5. The experimental result is obtained by using Tektronix TDS2002 digital oscilloscope. Using VoltechPM3000A universal power analyzer, the harmonic spectrum of the phase and line voltages is also depicted in Figures 11 and 12. As shown in this figure, the percentages of fifth and seventh harmonics are low and are equal to 0.19 and 0.38, respectively. Triple harmonics are also eliminated in line voltage. In this figure, the experimental and simulation results are also compared. In the second step, the experimental and simulation results for different amounts of modulation index are also compared and listed in Table 5, which shows the effectiveness of the pattern and also validates the simulation results.
FIGURE 10
The experimental prototype
(A) FIGURE 11
(B)
Experimental result for phase voltage, A waveform, and B, harmonic content
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FIGURE 12
TABLE 5
AL.
(B)
Experimental result for line voltage, A waveform, and B, harmonic content
Comparison of simulation and experimental
M
Firing Angles
0.92
θ1 = 18.5862 θ2 = 46.4102 θ3 = 65.4089
0.73
θ1 = 38.1080 θ2 = 57.2946 θ3 = 72.8344
V 1 =V *1
V5/V1
V7/V1
THDp%
THDL%
Simulation Experiment
1 0.9743
0 0.0007
0 0.0145
19.71 21.31
10.46 11.78
Simulation Experiment
1 0.9691
0 0.0187
0 0.0036
44.18 45.51
12.02 13.82
6 | CONCLUSION In this paper, harmonic elimination method is presented for eliminating low‐order harmonics in MLI with unequal DC sources. Artificial Bee Colony algorithm is proposed to determine the optimum switching angles of MLIs, and it is implemented by MATLAB software. The simulation results show that ABC is able to find the optimum switching angles to generate desirable voltage. Also, the proposed algorithm is compared with GA, PSO, and BA in 30 times run with the same initial values. Based on all the criteria for convergence and accuracy, ABC is superior over other algorithms. Furthermore, to confirm the validity of the results, the number of iteration is changed in 2 different values 300 and 1000. Changing the number of iterations also confirmed previous results. The aforementioned results are also corroborated by changing the number of iterations. Finally, experimental results verify the simulation results. ORCID Mohammad Zabihi Laharami
http://orcid.org/0000-0002-2733-0523
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How to cite this article: Mousazadeh Mousavi SY, Zabihi Laharami M, Niknam Kumle A, Fathi SH. Application of ABC algorithm for selective harmonic elimination switching pattern of cascade multilevel inverter with unequal DC sources. Int Trans Electr Energ Syst. 2018;e2522. https://doi.org/10.1002/etep.2522
