2014 International Conference on Circuit, Power and Computing Technologies [ICCPCT]
Harmonic Elimination in Cascade Multilevel Inverters using Firefly Algorithm N.Karthik
R.Arul
Department of Electrical & Electronics Engg., Hindustan University, Padur Chennai, Tamilnadu, India
[email protected]
Department of Electrical & Electronics Engg., ARM College of Engineering & Technology Maraimalainagar, Tamilnadu, India angles with analytical proof have been utilized in [6] to minimize the value of THD. This method can be used to eliminate all the triplen harmonics, but not the harmonics of other orders. Selective harmonic elimination for multilevel inverters by a genetic algorithm (GA) approach is presented in [7]. However, this method is only applied to multilevel inverter with equal dc sources and needs considerable computational time. Reference [8] presented contemporary stochastic search techniques based on particle swarm optimization (PSO) to deal with the problem for equal dc sources. PSO and GA algorithms have been utilized by selection of the optimal switching angle for minimizing THD of the multilevel inverter in [9]. Switching is done by PWM method in it and switching angles are chosen to eliminate specific harmonics. This paper is used to eliminate the selected harmonics generated by multilevel inverter using proposed FFA algorithm by choosing the optimal switching angles. The proposed FFA algorithm is implemented to solve the SHE problem with equal and unequal dc sources. The proposed FFA algorithm is developed to deal with the problem where the number of switching angles is increased and the determination of these angles using conventional iterative methods as well as the resultant theory is not possible.
Abstract—In this paper a new method has been proposed to select optimal switching angles based on Firefly algorithm. The resultant equations for the computation of output voltage total harmonic distortion (THD) of a multilevel inverter are used as the objective function. This objective function is used to minimize the THD in the output voltage of a multilevel inverter. While minimizing the objective function, the selective harmonics such as the 5th, 7th, 11th and 13th can be controlled by using the Firefly algorithm. The simulations are performed for an 11 level cascaded multilevel inverter with equal and nonequal dc sources to show the validity of the proposed method. The results show that the proposed firefly algorithm can eliminate selective harmonics in the output voltage of a multilevel inverter. Keywords—Cascaded multilevel inverter selective harmonic elimination (SHE), Firefly Algorithm (FFA).
I. INTRODUCTION Multilevel inverters have attracted a great deal of attention in medium-voltage and high-power applications due to their lower switching losses, higher efficiency and more electromagnetic compatibility than those of conventional two-level inverters [1]. The multilevel inverters were introduced for the first time by Nabaei in 1981 to reduce amplitude of the harmonics and switching frequency of the three phase inverters. The general purpose of the multilevel inverter is to synthesize a desired voltage from several separate dc sources (SDCSs) such as solar cells, fuel cells, ultra capacitors, etc. [2]. The well-known multilevel topologies are cascaded H-bridge, neutral-point clamp or diode-clamped and flying capacitor. To control the output voltage and reduce the undesirable harmonics, different sinusoidal pulse width modulation (PWM) and space-vector PWM schemes are recommended for multilevel inverters; however, PWM techniques are not able to eliminate low-order harmonics completely [3]-[4]. Another approach is to select switching angles so that specific lower order dominant harmonics are suppressed. This technique is known as selective harmonic elimination (SHE) technique [5]. Real-time calculations of the switching
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II. POWER TOPOLOGY OF CASCADE MULTILEVEL INVERTERS Fig. 1(a) shows the structure of a single-phase Hbridge cascaded multilevel converter topology that is used to synthesize a staircase output waveform. Fig. 1(b) also shows the staircase voltage waveform generated by multilevel converters. A cascaded H-bridge multilevel converter consists of some SDCS. Each individual DC source is connected to one half bridge inverter and one full bridge inverter. The number of output-phase-voltage levels in a cascade multilevel inverter is 2s +1, where s is the number of SDCSs. To obtain the three-phase configuration, the outputs of three single-phase cascaded inverters can be connected in Y or Δ.
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2014 International Conference on Circuit, Power and Computing Technologies [ICCPCT]
III. HARMONIC ELIMINATION PROBLEM Case I: Multilevel Inverters with Nonequal DC Sources In this section, it is assumed that the levels of the dc sources are nonequal and can be measured.
+ k5 cos 5θ5 = 0
(3)
k1 cos 7θ1 + k2 cos 7θ2 + k3 cos 7θ3 + k4 cos 7θ4 + k5 cos 7θ5 = 0
(4)
k1cos 11θ1 + k2cos 11θ2 + k3cos 11θ3 + k4cos 11θ4 + k5cos 11θ5 = 0 (5) k1cos 13θ1 + k2cos 13θ2 + k3cos 13θ3 + k4cos 13θ4 (6) + k5cos 13θ5 = 0 The nominal dc voltage is considered to be 100 V, and the ki values are the same as those in Table I. TAB LE I. TYP IC AL VALUES OF K I k1
k2
k3
k4
k5
1.08
0.98
0.9
0.86
0.8
The magnitudes of the dc voltage levels in the multilevel inverter considered are given in Table II which corresponds to the ki coefficients given in Table I. TAB LE II. DC VOLTAGE LE VE LS VDC1
VDC2
VDC3
VDC4
VDC5
21.6
19.6
18
17.2
16
Case II: Multilevel Inverters with Equal DC Sources In this case, it is assumed that the levels of the dc sources of the cascade inverter cells are equal and constant, i.e., Vdc1 = Vdc2 = Vdc3 = Vdc4 = Vdc5 = Vdc. By applying Fourier series analysis, the output voltage of multilevel inverters with equal sources can be described as follows: ∑
VDC
cos nθ1 cos nθ2 cos nθ3 7 … cos nθn sin n ωt Where VDC is the nominal dc voltage and the switching angles must satisfy the following condition:
Fig. 1. (a) Topology of a single-phase cascaded inverter. (b) Staircase output phase voltage.
The output voltage of multilevel inverters with nonequal sources can be described as follows: 4VDC ∑50 V ωt k1 cos nθ1 +k2 cos nθ2+ … N=5,7,11,… nπ kn cos nθn sin n ωt 1
, ,
,…
π
π 8 2 The number of harmonics which can be eliminated from the output voltage of the multilevel inverter is s – 1 where s is the number of individual DC sources. The switching angles θ1, θ2, θ3, θ4 and θ5 can be chosen such that the voltage total harmonic distortion is a minimum. The switching angles can be found by solving the following equations 0
Where kiVDC is the ith dc voltage. The switching angles can be found by solving the following equations k1cos θ1 + k2cos θ2 + k3cos θ3 + k4cos θ4 (2) + k5cos θ5 = M k1 cos 5θ1 + k2 cos 5θ2 + k3 cos 5θ3 + k4 cos 5θ4
839
θ1
θ2
θ5
2014 International Conference on Circuit, Power and Computing Technologies [ICCPCT]
cos θ1 + cos θ2 + cos θ3 + cos θ4 cos θ5 M 9 cos 5θ1 + cos 5θ2 + cos 5θ3 + cos 5θ4 + cos 5θ5 = 0 (10) cos 7θ1 + cos 7θ2 + cos 7θ3 + cos 7θ4 + cos 7θ5 = 0 (11) cos 11θ1 + cos 11θ2 + cos 11θ3 + cos 11θ4 + cos 11θ5 = 0 (12) cos 13θ1 + cos 13θ2 + cos 13θ3 + cos 13θ4 + cos 13θ5 = 0 (13) In (9), modulation index M is defined as M = V1/sVdc and V1 is the fundamental of the required voltage.
Move firefly i towards j in all d dimensions end if Attractiveness varies with distance r via exp [−γr2] Evaluate new solutions and update light intensity end for j end for i Rank the fireflies and find the current best end while Post process results and visualization The switching angles θ1, θ2, θ3, θ4, θ5 can be chosen such that the THD of the output voltage is minimized. These angles are normally chosen so as to cancel some predominant lower frequency harmonics. To eliminate 5th, 7th, 11th and 13th harmonics assuming that the peak fundamental output voltage is the same as its maximum value, the following equations are solved for different modulation indices. The system has 5 transcendental equations with unknown θ1, θ2, θ3, θ4 and θ5. f(θ1,θ2,θ3,….., θs = [|M - |V1|/sVdc + (|V5| + |V7| + …. + |V3s-2 or 3s-1|)/sVdc] (14) To solve this set of non-linear transcendental equations, Firefly Algorithm (FFA) technique is used. Newton-Raphson method is derivative dependent and may end in local optima; however, a judicious choice of initial values alone guarantees convergence. So optimization techniques like Firefly Algorithm (FFA) is employed for minimization of harmonics in order to reduce the computational burden associated with the solution of the non-linear transcendental equation of the conventional SHE method. An accurate solution will be guaranteed with FFA even for a higher number of switching angles than other techniques would be able to calculate for a given computational effort. Hence FFA seems to be promising methods for applications when a large number of DC sources are sought in order to eliminate more low-order harmonics to further reduce the THD. For demonstration of the output voltage quality, one may define the harmonic distortion of the universal line voltage as below: 2 ∑ % 100 (15) n ) /H1
IV. FIREFLY ALGORITHM The firefly algorithm (FFA) is a meta-heuristic algorithm, inspired by the flashing behaviour of fireflies. The primary purpose for a firefly's flash is to act as a signal system to attract other fireflies. Now this can idealize some of the flashing characteristics of fireflies so as to consequently develop firefly inspired algorithms. For simplicity in describing our new Firefly Algorithm (FFA) [10], there are the following three idealized rules. On the first rule, each firefly attracts all the other fireflies with weaker flashes. All fireflies are unisex so that one firefly will be attracted to other fireflies regardless of their sex. Secondly, attractiveness is proportional to their brightness which is inversely proportional to their distances. For any two flashing fireflies, the less bright one will move towards the brighter one. The attractiveness is proportional to the brightness and they both decrease as their distance increases. If there is no brighter one than a particular firefly, it will move randomly. Finally, no firefly can attract the brightest firefly and it moves randomly. The brightness of a firefly is affected or determined by the landscape of the objective function. For a maximization problem, the brightness can simply be proportional to the value of the objective function. Other forms of brightness can be defined in a similar way to the fitness function in genetic algorithms. Based on these three rules, the basic steps of the firefly algorithm (FFA) can be summarized as the pseudo code shown below. PSEUDO CODE OF THE FFA:
THD is an appropriate tool for measurement of quality of the output signals harmonic. Thus, this function is selected as the objective function for optimization. The optimal value of THD is obtained using firefly optimization algorithm.
Begin FFA Procedure; Initialize algorithm parameters: Objective function f(x), x = (x1, x2, . . . , xd)T Initialize a population of fireflies xi(i = 1, 2, . . ., n) Define light absorption coefficient γ while (t < MaxGeneration) for i = 1: n all n fireflies for j = 1: i all n fireflies Light intensity Ii at xi is determined by f(xi) if (Ij > Ii)
V. SIMULATION RESULTS The proposed algorithm has been simulated in Matlab/Simulink software for an 11-level cascaded H-bridge inverter with equal and nonequal dc sources. The proposed firefly algorithm is one of the
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2014 International Conference on Circuit, Power and Computing Technologies [ICCPCT]
switching angles, which is itself similar to the number of cascade half bridge multilevel inverters. For every firefly, the optimal objective is determined for m variables. (2) The main aim is to minimize the specified harmonics. Therefore, a relation must exist between the objective function and these harmonics. THD is considered as the objective function in this problem.
most efficient optimization algorithms. Simulation results are presented for different values of modulation indices for multilevel inverters with equal and nonequal dc sources. Table III compares the optimized values of switching angles and THDs evaluated using PSO and proposed FFA algorithm for multilevel inverters with nonequal dc sources. Table IV compares the optimized values of switching angles and THDs evaluated using proposed FFA and evolutionary computation algorithms for multilevel inverters with equal dc sources. It can be seen that from Table III and Table IV, the line voltage THD increases slightly when the modulation index decreases. The following steps express how these algorithms are implemented for a cascade H-bridge 11-level inverter. (1) Population of the fireflies is initiated by random positions between 0 to π/2 and speed in m-dimensional space of the problem, such that the dimensions of each firefly are equal to the controllable
The switching angles are varied within the range of 0 to 90° only. The proposed firefly algorithm gives the lower % THD compared to the other iterative methods. The lowest THD value is occurred at 1.075 modulation index value for multilevel inverters with nonequal dc sources. In this paper lower order harmonics have been eliminated by using the equation for THD as the objective function and have given better results in minimization of THD for up to the 50th order of harmonics.
TABLE III. OUTPUT SWITCHING ANGLES OBTAINED USING THE PROPOSED FFA AND PSO FOR MULTILEVEL INVERTER WITH NONEQUAL DC SOURCES Modulation Index(M)
0.47 0.7 1.075
Optimization Technique
Switching Angles θ1
θ2
θ3
THD θ4
θ5
%
PSO[11]
37.714
52.811
68.196
86.250
89.396
9.822
FFA
1.939
51.031
55.330
64.647
68.991
6.683
PSO[11]
16.734
36.028
56.241
62.346
88.367
5.416
FFA
51.033
55.606
59.187
63.018
68.091
3.775
PSO[11]
6.850
8.490
21.883
28.011
42.950
4.717
FFA
18.615
23.601
27.472
30.885
35.835
2.426
TABLE IV. OUTPUT SWITCHING ANGLES OBTAINED USING THE PROPOSED FFA AND PSO FOR MULTILEVEL INVERTER WITH EQUAL DC SOURCES Modulation Index(M)
Optimization Technique
Switching Angles θ1
0.5
0.6
θ2
θ3
THD θ4
θ5
%
Newton Raphson Method[12]
35.52
45.49
57.20
69.20
84.92
8.77
Continuous GA[12]
4.671
30.83
37.44
48.55
89.42
11.12
Modified PSO[12]
35.55
45.36
57.12
69.97
84.68
8.60
FFA
31.66
34.26
38.83
80.94
86.32
6.91
Newton Raphson Method[12]
26.64
43.93
51.53
62.39
72.50
7.24
Continuous GA[12]
11.64
33.58
33.74
56.79
78.11
10.67
Modified PSO[12]
10.79
29.88
46.18
63.11
87.56
8.14
FFA
51.27
56.09
59.80
63.50
68.55
5.96
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2014 International Conference on Circuit, Power and Computing Technologies [ICCPCT]
0.8
1.0
Newton Raphson Method[12]
6.569
18.94
27.18
45.13
62.24
5.55
Continuous GA[12]
6.62
23.10
44.42
54.37
63.57
8.59
Modified PSO[12]
6.884
19.01
27.80
45.77
62.52
4.69
FFA
51.24
55.98
59.64
63.14
68.20
4.50
Continuous GA[12]
3.24
9.496
17.17
28.03
28.14
6.85
Modified PSO[12]
6.129
8.00
12.68
21.67
29.30
4.10
FFA
3.13
9.71
16.23
22.31
27.91
3.774
Fig.2. Flow chart for solving the problem using firefly algorithm
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2014 International Conference on Circuit, Power and Computing Technologies [ICCPCT]
[3] D. G. Holmes and T. A. Lipo, “Pulse Width Modulation for Power Converters,” Piscataway, NJ: IEEE Press, 2003. [4] S. Kouro, J. Rebolledo, and J. Rodriguez, “ Reduced switchingfrequency modulation algorithm for high-power multilevel inverters ,” IEEE Transactions Industrial Electronics, vol. 54, no. 5, pp. 2894–2901, Oct. 2007. [5] W.Fei, X.Du and B.Wu, “A generalized half-wave symmetry SHE-PWM formulation for multilevel voltage inverters,” IEEE Transactions Industrial Electronics, vol. 57, no. 9, pp.3030-3038, Sep. 2010. [6] Liu, Y., Hong, H and Huang, A. Q. (2009), “Real-time calculation of switching angles minimizing THD for multilevel inverters with step modulation,” IEEE Transactions Industrial Electronics, 56(2), 285-293. [7] Ozpineci, B., Tolbert, L. M., and Chiasson, J. N. (2005), “Harmonic optimization of multilevel converters using genetic algorithms,” Power Electronics Letters, IEEE,3(3), 92-95. [8] Rup Narayan Ray, Debashis Chatterjee and Swapan Kumar Goswami(2010), “A PSO based optimal switching technique for voltage harmonic reduction of multilevel inverter, Expert Systems with Applications,” Elsevier on 37(2010),7796-7801. [9] Sarvi, M., & Salimian and M. R. (2010, August), “Optimization of specific harmonics in multilevel converters by GA & PSO,” In Universities Power Engineering Conference (UPEC), 2010 45th International (pp. 1-4). IEEE. [10] Hassanzadeh, T. Meybodi and M,R.Mahmoudi, “An improved Firefly Algorithm for optimization in static environment,” Fifth Iran Data Mining Conference / IDMC 2011. [11] Taghizadeh, H and Hagh, M. T. (2010), “Harmonic elimination of cascade multilevel inverters with nonequal DC sources using particle swarm optimization,” IEEE Transactions Industrial Electronics, 57(11), 3678-3684. [12] Joshi Manohar Vesapogu, Sujatha Peddakotla and Seetha Rama Anjaneyulu Kuppa, “Harmonic analysis and FPGA implementation of SHE controlled three phase CHB 11-level inverter in MV drives using deterministic and stochastic optimization techniques,” Vesapogu et al.SpringerPlus2013,2:370.
VI. CONCLUSION The FFA based optimization technique has been proposed to minimize the overall THD of the output voltage of 11-level cascaded H-bridge inverter with equal and nonequal dc sources. A FFA based algorithm is developed to compute the switching angles for minimization of overall voltage THD while the individual selected harmonics are optimized within the allowable limits. Simulation results are provided for an 11-level cascaded Hbridge inverter to validate the accuracy of computational results. The proposed firefly algorithm searches for all possible set of solutions to contribute the minimum THD. The result of the proposed algorithm have been compared with the results available in the literature and found to be better. ACKNOWLEDGEMENT The author wish to thank the Management of Hindustan University, Padur Chennai for their support and encouragement to carry out this work. REFERENCES [1] J. Rodríguez, J. Lai, and F. Peng, “Multilevel inverters: A survey of topologies, controls and applications,” IEEE Transactions Industrial Electronics, vol. 49, no. 4, pp. 724–738, Aug. 2002. [2] J. M. Carrasco, L. G. Franquelo, J. T. Bialasiewicz, E. Galvan, R. C. PortilloGuisado, M. A. M. Prats, J. I. Leon, and N.MorenoAlfonso, “Power-electronic systems for the grid integration of renewable energy sources: A survey,” IEEE Transactions Industrial Electronics., vol. 53, no. 4, pp. 1002– 1016, Jun. 2006.
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