J. Cent. South Univ. (2019) 26: 2822-2832
DOI: https://doi.org/10.1007/s11771-019-4216-2
Improved hyper-spherical search algorithm for voltage total harmonic distortion minimization in 27-level inverter
A A KHODADOOST ARANI, H KARAMI, B VAHIDI, G B GHAREHPETIAN
Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran
Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: Multi-level inverters (MLIs) have become popular in different applications such as industrial power control systems and distributed generations. There are different forms of MLIs. The cascaded MLIs (CMLIs) have some special advantages among them such as more different output voltage levels using the same number of components and higher power quality. In this paper, a 27-level inverter switching algorithm considering total harmonic distortion (THD) minimization is investigated. Switching angles of the inverter switches are achieved by minimizing a THD-based objective function. In order to minimize the THD-based objective function, the hyper-spherical search (HSS) algorithm, as a novel optimization algorithm, is improved and the results of improved HSS (IHSS) are compared with HSS algorithm and other five evolutionary algorithms to show the advantages of IHSS algorithm.
Key words: 27-level inverter; cascade multi-level inverter; improved hyper-spherical search (IHSS) algorithm; total harmonic distortion (THD) minimization
Cite this article as: A A KHODADOOST ARANI, H KARAMI, B VAHIDI, G B GHAREHPETIAN. Improved hyper-spherical search algorithm for voltage total harmonic distortion minimization in 27-level inverter [J]. Journal of Central South University, 2019, 26(10): 2822-2832. DOI: https://doi.org/10.1007/s11771-019-4216-2.
1 Introduction
Nowadays, multi-level inverters (MLIs) have been used in many applications especially in adjustable speed drives, flexible AC transmission system (FACTS) devices and distributed generation units such as photovoltaic (PV) systems [1-3]. MLIs have some advantages that make them more popular than traditional inverters. They have lower switching frequency. This leads to lower switching losses and longer lifetime for these inverters. Total harmonic distortion (THD) is an important index for system power quality. Because of stepped AC waveform of output voltage, they have lower THD than square wave inverters and so filters can be eliminated [4]. Although other switching techniques such as space vector pulse width modulation (SVPWM) and sinusoidal pulse width modulation (SPWM) have better harmonics spectrum, but these techniques have more complexity and higher switching frequency [5-7]. Higher reliability of the MLIs, due to their modular structure, is another advantage of them [8]. Some of MLI advantages are shown in Figure 1.
Generally, MLIs are categorized in three main types: a) flying capacitor, b) diode-clamped and c) cascaded MLI (CMLI). In addition to switches, the basic elements in three mentioned MLI are diodes, capacitors and H-Bridge inverters, respectively. A CMLI is composed of two or more H-Bridge inverters in each phase and therefore it has modular structure. In addition, the number of elements is reduced in CMLI compared with other structures. The mentioned reasons cause CMLI to be used more than other structures [11].
Figure 1 Advantages of MLI [8-10]
CMLIs may have two possible structures: a) symmetrical and b) asymmetrical. In a symmetrical MLI (SMLI), primary DC voltage sources have equal values. But in an asymmetrical MLI (AMLI), these values are unequal. In addition, the SMLIs have less output voltage levels than AMLIs. For example, an AMLI-based 27-level inverter has three unequal DC voltage sources and 27 levels of the output voltage, while in SMLI-based model, it has only 7 levels.
Different switching algorithms can be defined based on various purposes and considering topology of MLIs. One of the most popular algorithms is selected harmonic elimination (SHE). In this algorithm, the switching angles are chosen, in order to remove higher order harmonics such as 5th, 7th, 11th, etc. Several switching angles based mathematical objective functions have been defined and optimized using numerical methods such as Newton-Raphson or evolutionary algorithms such as genetic algorithm (GA) or particle swarm optimization (PSO) [5, 12]. The SHE is a systematic method to achieve switching angles, but sometimes other power quality indexes such as THD of the output voltage are more important [13-15].
The THD minimization can be an objective function to obtain switching angles. Unlike to the SHE, this method has no clear mathematical objective functions. It has a general switching algorithm objective function that should be minimized. It should be defined for each topology of inverters. In Ref. [16], a topology of 27-level inverter for PV applications has been described and its switching angles have been obtained by SHE. HUA et al [17] have used the SHE to eliminate low order harmonics.
Recently, some new algorithms for problem optimization have been proposed which have a fast convergence rate [18-20]. Some researchers have used optimization algorithms in engineering problems [21-24]. The minimization of the THD has been carried out by optimization algorithms such as GA, PSO, harmony search algorithm (HSA), simulated annealing (SA) and imperialist competitive algorithm (ICA) [25-30]. In Ref. [31], hyper-spherical search (HSS) algorithm, as a novel method for optimization of non-linear functions, has been introduced and applied to mathematical problems. This is the first time that the HSS algorithm is used for the THD minimization of a 27-level inverter as an engineering problem. In Ref. [31], it was shown that the HSS converges faster toward a better solution compared to GA, HSA and particle swarm optimization (PSO) in some mathematical test problems. However, the application of HSS in an engineering problem, especially in the THD minimization as one of the most important issues in the multi-level inverters, has not been investigated in previous studies. On the other hand, HSS has been compared with HSA, GA and PSO in Ref. [28], but has not been compared with SA previously. Thus, in this paper, the HSS algorithm is applied to the problem and the results show the power and effectiveness of HSS over PSO, HSA, GA, SA and ICA. In addition, an improvement is proposed in HSS and the improved HSS (IHSS) is compared with HSS to show the effectiveness of this improvement.
In the next section, the detailed structure of the 27-level inverter, as an AMLI, is discussed and switching pattern of different switches is expressed. Then, the objective function for the THD optimization is introduced in Section 3. The explanation of the HSS algorithm is presented and the IHSS algorithm is proposed in Section 4. A 27-level inverter is simulated by MATLAB/ Simulink and the results of evolutionary algorithms are shown in Section 5. Finally, the conclusion of paper is drawn in the last section.
2 27-level inverter
A 27-level inverter is an AMLI and has unequal DC voltage sources in its input. Figure 2 shows the detailed diagram of a 27-level inverter. As can be observed, the 27-level inverter includes three H-Bridge inverters that their output voltages are unparalleled and therefore summed in each time as follows:
(1)
where Voi is the output voltage of each H-bridge and i is the number of the H-bridge inverter. Each H-bridge can generate three output voltages of +V, 0 and –V , where V is the magnitude of DC voltage source. Therefore, it can be concluded that 27 states are available. These levels are: (+13, +12, +11, …, +1, 0, -1, …, -11, -12, -13)×Vdc. By defining switching angles (α1 to α13) output voltage waveform in a half cycle, can be shown in Figure 3. In addition, the switching pattern for this half cycle is given in Table 1. A similar pattern is used for the negative half cycle.
Figure 2 Detailed structure of 27-level inverter
Figure 3 Output voltage of 27-level inverter in half cycle
Table 1 Switching pattern of 27-level inverter
3 Objective function
As can be seen in Figure 3, the output voltage cannot be completely a sinusoidal waveform and has several harmonics. Using Fourier analysis, different harmonics are expressed based on switching angles (α1 to α13) as follows:
(2)
where Vdc is the amount of the smallest DC source and n is the number of harmonics. In Eq. (2), it can be seen that the output voltage can be expressed by n sets of equation that are dependent of α1 to α13. Since the waveform has half-wave symmetry, even harmonics are eliminated. The THD can be defined as a measurement of the harmonic distortion in a signal and is generally formulated as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency. Therefore, it can be expressed as follows:
(3)
where Vo1rms is obtained as follows:
(4)
4 IHSS algorithm
4.1 Original HSS
The HSS algorithm has been introduced in Ref. [26] for the first time. This procedure can be modelled as follows [32]:
Min{f(x)|x∈X}
subject to: g(x)≥0 and h(x)=0 (5)
In the optimization procedure, f(x) as objective function (OF) is minimized. There are some inequality and equality constraints in the most real problems, g(x) and h(x), respectively. Decision variables, x, have their allowable ranges as Xi,min≤xi≤Xi,max, which the OF value depends on.
The procedure of HSS algorithm is defined into the following steps:
Step 1: Initialization of particles
The HSS algorithm is initialized in the four sub-step as follows:
a) Parameters initialization: The user assigns the algorithm parameters such as Npop (number of initial population), NSC (number of hyper-sphere centers), Prangle (the probability of changing an angle), rmin, rmax and Nnewpar (number of newly generated particles in each iteration).
b) Generation of initial population: In order to start the HSS algorithm, Npop solutions, called particles, are generated randomly with a uniform probability from [Xi,min, Xi,max]. The OF is evaluated for each particle. The dimensions of the optimization problem can be determined by knowing the number of variables. As described in Section 3, the THD of the inverter is minimized by choosing optimal switching angles. Therefore, as shown in Figure 3 and Table 1, there are 13 variables for the optimization procedure as switching angles. Figure 4 shows that how the variables (switching angles) form the particles.
Figure 4 Variables forming particles
c) Nomination of hyper-sphere centers: Hyper-sphere centers (SCs) are the best NSC particles according to their evaluated OFs. A particle is a 1×N vector, [p1, p2, …, pN], in the N-dimensional problem, where pi is the decision variable for the ith dimension and f(p1, p2, …, pN) is the OF value of a particle.
d) Assign particles to hyper-spheres: After the nomination of SCs, there are Npopds-NSC particles which should be assigned to the SCs to make hyper-spheres. This assignment is carried out considering the dominance of SCs, called DSC, which is defined as follows:
(6)
where
(7)
Each SC will possess round{DSC×(Npop-NSC)} number of particles at the initial step.
Step 2: Searching process
The space is searched by particles to find better positions which have better OF value in it. In this step, there are two parameters, r and θ. The θ is the angle of particle in the spherical coordinates, if the SC is the origin of coordinates.
The searching space for each particle is a sphere whose center is corresponding SC and the radius r. If R is the distance between the particle and corresponding SC, the parameter r is a random value between [rmin×R, rmax×R]. In N-dimensional problem, each particle has N-1 angles which are changed by randomly values between (0, 2π) with the probability of Prangle.
During searching procedure, some particles may find positions which have lower OF value than their corresponding SCs. In this situation, the particle and SC exchange their labels. In other words, a particle named as new SC and the previous SC is changed to particle of this new SC. At the end of Step 2, this situation should be checked.
Step 3: Recovery of dummy particles
Each SC and its particles make a set. Dummy particles are ones that have the largest OF values in their sets and seem that cannot find optimal position in this condition. Therefore, it is better to change their searching spaces and corresponding SCs as shown in Figure 5. The worst set, which has the dummy particle, has the worst objective function set (SOF), which is defined as follows:
(8)
Figure 5 Step 3 procedure
In general, the value of γ is set to 0.1 [33, 34]. The dummy particles from the set which has the worst SOF, should be assigned to another SC. This aim is achieved based on the assigning probability (AP) to divide the dummy particle among SCs as calculated in the following:
(9)
where difference of objective function set is:
(10)
It should be noted that after some iterations, the algorithm may have an SC with no particles. In this situation, this SC is labeled as particle and assigned to other SCs.
Step 4: Generating new particles
In this step, Nnewpar new particles are generated and replaced by the worst particles to make the algorithm more flexible. It is expected that the worst particles among all of them cannot reach the global position and so, should be replaced with some new particles, which maybe search the space better. In this paper, the Nnewpar is set to 5% of the initial population number and will be assign according to the fourth sub-step of Step 1.
Step 5: Convergence of algorithm
The HSS algorithm will be terminated if the maximum of iteration number is reached or the difference between the OF value of the best SCs in two consequential iterations becomes lower than a predetermined threshold (e.g., in this paper is set to 10-6).
Figure 6 shows the flowchart of the HSS algorithm based on the steps expressed above.
Figure 6 Flowchart of HSS algorithm
4.2 Improved HSS
As mentioned before, in searching process there are two parameters, r and θ. Changing these parameters leads to search the space by particles to find better positions. This process can be improved by a modification in particle movement process. It is proposed that after the mentioned search process in the original HSS algorithm and changing the positions of particles by changing their r and θ, the Nsubst number of decision variables of each particle maybe substituted directly by decision variables of its corresponding SC. For example, assumed that in 5-dimentional problem, after search process of original HSS algorithm, a particle is [p1, p2, p3, p4, p5] and an SC is [sc1, sc2, sc3, sc4, sc5]. By considering the improvement process, the mentioned particle may substitute Nsubst (for example Nsubst=2) number of its decision variables in the probability of Psubst by the decision variables of the SC and makes a new particle like [sc1, p2, p3, sc4, p5], which the first and fourth decision variables are substituted by the first and fourth decision variables of its corresponding SC. The effects of this improvement are investigated in this paper.
5 Simulations and results
5.1 Comparison of IHSS with HSS in three benchmark functions
In this section, the IHSS and HSS algorithms are used for three benchmark functions [31]. Similar to papers about introducing new optimization algorithm, at first the performance of IHSS should be investigated in some popular benchmark functions. In this paper, three popular benchmark functions in this area are used and shown on Figure 7.
In other words, these problems (benchmark functions) should be solved by HSS and IHSS. As can be seen, these have different local minimum and the algorithm should find the global minimum. In Figure 8, these problems have been solved by HSS and IHSS and so, the performance of HSS and IHSS has been compared by solving these problems.
The results and speed of convergence for both algorithms are shown in Figure 8. As shown in the results, the IHSS algorithm has better results and converges faster, especially for G2 and G3 benchmark functions. Moreover, the best and mean results of different 20 runs are listed in Table 2. The IHSS algorithm has better mean for all three functions and in G2 achieves better minimum result.
5.2 THD minimization
The results and comparison of the THD minimization for 27-level inverter using several optimization algorithms are discussed in this subsection. All of the optimization algorithms including GA, PSO, HSA, SA, ICA, IHSS and HSS are run in the same computer.
Figure 7 Plot of three benchmark functions:
The initial population is set to 100 for all of them. The mean, minimum and maximum of each algorithm for 100 different runs are listed in Table 3.
As can be seen in Table 3, the SA has the worst result among mentioned algorithms in this problem. The IHSS, HSS and ICA have the best minimum. However, an advantage can be observed in mean and maximum THD, achieved by IHSS and HSS. It can be concluded that the IHSS has the best minimum, maximum and mean values than other optimization algorithms.
Figure 8 Convergence diagram of IHSS and HSS:
The switching angles of the best results are listed in Table 4. As can be seen, the achieved angles in different algorithms are unequal. While this difference in in smaller angles (α1 to α7) is less, in larger angles (α8 to α13), this difference is greater. The archived angles and calculated THD show that this problem presumably is similar to “G1” or (G3) in Figure 7 and has several local minimum.
Table 2 Best and mean results of algorithms for optimization problems
Table 3 THD of different algorithms
5.3 Simulation of 27-level inverter
The 27-level inverter with Vdc=30 V, is simulated by MATLAB/Simulink. Therefore, three DC voltage levels (30, 90 and 270 V) are used in simulations. The result of simulations for different sets of switching angles obtained from different optimization algorithms is shown in Figures 9 and 10. In Figure 9, the output voltage of 27-level inverter using achieved angles by IHSS has been shown. Moreover, the output voltage of 27-level inverter using achieved angles by other algorithms have been shown in Figure 10. Comparing these figure, it can be seen that the voltage waveform of obtained by the IHSS is more similar to a sinusoidal waveform. In addition, as expected, the results achieved by SA have the worst voltage waveform and lead to high amount of THD.
In order to study the results of the optimization, the THD and amplitude of fundamental component of output voltage are shown in Figures 11 and 12 using FFT analysis for different algorithms.
Table 4 Switching angles obtained by different optimization algorithms
Figure 9 Output voltage waveform of IHSS
Figure 10 Output voltage waveform of different algorithms
Figure 11 FFT analysis of output voltage waveforms using IHSS
6 Conclusions
A 27-level inverter was studied in this paper. In order to determine its switching angles, a THD-based objective function was defined. A novel optimization algorithm, called HSS was used for the minimization and an improvement in HSS was proposed. The result of IHSS algorithm was compared with HSS and five algorithms including GA, PSO, HSA, SA and ICA. The performance of IHSS compared with others in an engineering problem, THD minimization problem of a multi-level inverter has been investigated. The results of simulation for different algorithms show that IHSS has the best convergence speed and solution finding compared with HSS, GA, PSO, SA, HSA and ICA, not only in mathematical problems, but also in THD minimization of a multi-level inverter as the engineering problem. Finally, FFT analysis was carried out for the output voltage waveforms to show the supervisory of the IHSS algorithm. Similar to other meta-heuristic algorithms, the main drawback of IHSS is that it cannot guarantee to reach the global minimum.
Figure 12 FFT analysis of output voltage waveforms using other algorithms
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(Edited by HE Yun-bin)
中文导读
27级电平逆变器电压全谐波失真最小化的改进超球面搜索算法
摘要：多电平逆变器(MLIs)在工业电源控制系统和分布式电源等不同领域中得到了广泛的应用。MLIs有多种形式，级联多线阵(CMLIs)具有不同的输出电压等级，使用相同数量的元件和更高的电能质量等特殊优点。本文研究了一种考虑总谐波失真最小化的27级电平逆变器开关算法。逆变器开关的开关角通过最小化基于THD的目标函数来实现。为了使基于THD的目标函数最小化，对一种新的优化算法—超球面搜索算法进行了改进，并将改进后的超球面搜索算法与其他五种进化算法的结果进行了比较，说明了超球面搜索算法的优越性。
关键词：27级逆变器；级联多级逆变器；改进的超球面搜索算法；总谐波失真(THD)最小化
Received date: 2017-12-03; Accepted date: 2018-12-07
Corresponding author: B VAHIDI; Tel: +98-21-64543330, Fax: +98-21-66406469; E-mail: vahidi@aut.ac.ir
