An Efficient Method for THD Minimization in Multilevel Inverter's

Abstract

In this paper, minimization of total harmonic distortion (THD) is discussed for the output voltage of multilevel inverter. THD minimization is an efficient method in reduction of the harmonic components of the inverter's output voltage. In multilevel inverters, the switching angles can be selected in a way that the output voltage THD will be minimized. Given that THD minimization is an optimization problem, intelligent algorithm is found to be an appropriate alternative in this regard. Shuffled Frog Leaping Algorithm and Harmony Search Algorithm are employed to find optimum switching angles to generate desired voltage value in the possible minimum THD. The obtained results of two algorithms are compared with each other to determine that which algorithm is more efficient in this regard. Also, both simulation and experimental results indicate superiority of this approach over the published work using GA in this concept. The experiments are conducted on a seven-level inverter to validate the feasibility of presented approach.

Headnote

Abstract: In this paper, minimization of total harmonic distortion (THD) is discussed for the output voltage of multilevel inverter. THD minimization is an efficient method in reduction of the harmonic components of the inverter's output voltage. In multilevel inverters, the switching angles can be selected in a way that the output voltage THD will be minimized. Given that THD minimization is an optimization problem, intelligent algorithm is found to be an appropriate alternative in this regard. Shuffled Frog Leaping Algorithm and Harmony Search Algorithm are employed to find optimum switching angles to generate desired voltage value in the possible minimum THD. The obtained results of two algorithms are compared with each other to determine that which algorithm is more efficient in this regard. Also, both simulation and experimental results indicate superiority of this approach over the published work using GA in this concept. The experiments are conducted on a seven-level inverter to validate the feasibility of presented approach.

Keywords: Shuffled Frog Leaping Algorithm (SFLA), Harmony Search Algorithm (HS), Multilevel Inverter, Minimization of Total Harmonic Distortion, Phase Voltage

(ProQuest: ... denotes formulae omitted.)

1. Introduction

In the recent years, power electronics engineers have centralized their attention to multilevel inverters. Compared to conventional two level inverters, stepwise output voltage is the basic advantage of multilevel inverters. This advantage will have some results such as better Electromagnetic Capability (EMC), lower switching losses, higher power quality, reduction of dv/dt stresses, lower total harmonic distortion (THD), needlessness of a transformer at distribution voltage and lower rating on power semiconductor switches [1-5]. Due to the advantages mentioned above for multilevel inverters, they are employed in many applications such as: distributed generation [6], micro grids [7-8], FACTs devices [9], High Voltage Direct Current (HVDC) [10], and electrical vehicles [11-12]. Multilevel inverters are mainly classified into three configurations which are the flying capacitor [13], diode clamped [14-15] and cascaded multilevel inverters [16]. Multilevel inverters are divided into two categories from the aspect of DC source value, which are called symmetric and asymmetric topologies. For DC voltage sources, capacitors, batteries and renewable energy sources can be implemented. In symmetric topology all DC sources have the same value but in asymmetric topology they have different values. Asymmetric multilevel inverters generate higher number of voltage steps for a definite number of switches compared to symmetric ones. The stepwise output voltage of multilevel inverters composed by a number of DC voltage sources [17]. An increase in the number of steps, will lead to generation of a near sinusoidal output voltage of multilevel inverter. This results in a considerable reduction in output voltage THD. Nevertheless, problems such as voltage unbalance, circuit layout and voltage clamping, limit the possible number of levels. Consequently, reducing the THD of output voltage waveform is a vital issue in designing useful and efficient multilevel inverters. Hence, improving the output waveform quality and minimizing THD has become the subject of many recent papers. The performance of multilevel converters depends on its switching strategy. To qualify the output voltage waveform of multilevel inverters, different modulation strategies have been proposed.

Some of popular switching strategies are space vector modulation, minimization of the total harmonic distortion, pulse width modulation (PWM), selective harmonic elimination (SHE), and sinusoidal pulse width modulation (SPWM) [18-23]. Also in [24] a technique has been proposed based on optimal minimization of THD (OMTHD). Selective harmonic elimination approach minimizes or eliminates some low-order harmonics and achieves to the fundamental component. But in THD minimization approach all of the harmonic components are considered together to be minimized while having the fundamental component. Commonly an optimization algorithm is employed to search for the best switching angles by which the minimum possible THD will be achieved. In this paper THD minimization is applied to the phase voltage. Shuffled Frog Leaping Algorithm (SFLA) and Harmony Search (HS) Algorithm which are two new approaches of optimization algorithms are developed and employed to solve the problem. Then the simulation and experimental results are compared together and with other studies. Provided comparison validate that the obtained results are better in this work. The rest of the paper is organized as follows. Section II gives a brief description to the staircase output voltage of a seven-level inverter. Section III describes THD minimization in more details. In section IV and section V, SFLA and HS Algorithms are described, respectively. In section VI and VII simulation and experimental result are given and compared, respectively. Finally section VIII is devoted to conclusion.

2. Output voltage of multilevel inverter

Figure 1 shows a typical stepped waveform of the phase voltage of a seven-level inverter with equal DC sources.

Assuming the waveform, only three angles α1, α2, and α3 are required to determine the mentioned waveform. Regarding to switching angles, such a waveform can be demonstrated in terms of step function ( t). For positive half cycle of the waveform, we have:

...

For negative half cycle in which is lagging in radians we have:

...

Considering (1) and (2) the whole waveform can be obtained from following:

...

Applying Fourier analysis to , such a waveform yields the following expression for harmonic components of the phase voltage and so the rms value of fundamental calculation (for odd n):

...

This is an odd function and it only contains odd harmonics, i.e. even harmonics are eliminated. From the following equation, it is concluded that the maximum possible value of the fundamental component is obtained when α1, α2, and α3 are all equal to zero:

...

The phase voltage rms value can be easily calculated by considering the waveform in figure 1:

...

The ratio of the sum of the RMS value of power of all harmonic components to the RMS value of power of the fundamental frequency component is defined as THD. So the following equation gives the THD value.

...

Finally, for the phase-voltage THD the following analytical expression is obtained:

...

3. THD Minimization

THD Minimization technique aims to determine the optimum switching angles while generating the fundamental component with the possible minimum THD value. This is a problem that an optimization algorithm is used to solve it. In this paper SFLA and HS algorithms are used which are general purpose stochastic global search algorithms. These algorithms do not need functional derivative information to search for the solutions to solve the problem.

The objective function is defined as follows to solve the problem:

...

V is the required fundamental component in per unit and substitute from equation (5). It varies from zero to one. The first term of the objective function, * 1 1 |V -V | , is the absolute value of error, which is required for adjusting the fundamental harmonic. A weighting factor of ten has been applied to terms of error to increase the importance of fundamental component. The value of errors factor can be eristic and is tentative. The fundamental component reaches a nearly to the desired as well as the possible minimum THD in the output with the presented weighting factor. The following basic constraint must be satisfied with switching angles which are the solutions of aforementioned objective function:

...

The obtained results from both SFLA and HS are compared together to determine which one is more efficient to solve THD minimization problem. And also, a comparison is provided with the published paper which uses GA in this regard. The comparison will finally confirm that the attained results in this work cause lower phase voltage THD value.

4. Shuffled Frog Leaping Algorithm (SFLA)

In the SFLA, there are several frogs with the same structure but different adaptabilities and each of them represents a possible solution to an optimization problem [25]. The population of frogs is shared into some frog memeplexes according to particular rules and each memeplex represents a type of meme. Frogs in the memeplex conduct local survey of solution space according to particular strategies which permit the meme exchange between local individuals. The initial population is made of F frogs generated randomly and the fitness value f (i) for the ith frog can be evaluated and sorted in descending order to form memeplexes. The entire population of F frogs is divided into m memeplexes and each memeplex consists of n frogs. To avoid local optima, a sub memeplex is carried out in each memeplex, which contains frogs chosen on the basis of their respective fitness value. If the fitness is better, it is chosen easier. The worst solutions are updated and replaced with a better one. The calculations carry on for a definite number of iterations. Then, for global information exchange all frogs are shuffled. The local survey and the shuffling procedure continue until a pre-defined convergence criterion is met.

Figure 2 shows the flowchart of SFLA.

5. Harmony Search Algorithm

Harmony search is a recently developed music-based metaheuristic optimization algorithm [26]. It was roused by the observation that the goal of music is to search for a impeccable state of harmony. Specifically, the process by which the musicians (who have never played together before) quickly refine their individual improvisation through adjusting the pitches of their instruments resulting in a pleasing harmony. This harmony in music is similar to find the optimality in an optimization process. Simple concept, few parameters to adjust, and easy implementation make HS as the major competitor of other evolutionary algorithms. In music improvisation, each player sounds any pitch within the possible range, together making one harmony vector. If all the pitches make a good harmony, that experience is stored in each player's memory, and the possibility to make a good harmony is increased next time. Figure 3 represents the optimization procedure of the HS algorithm.

6. Simulation Results

To confirm the attained results, a simulation is carried out for a 7-level inverter with equal DC sources according to the waveform shown in figure 1. SFLA and HSA are applied to minimize the Eq. (9), for finding the optimal switching angles. The optimum switching angles (degree), simulated values for phase voltage THD and line voltage THD versus modulation index for SFLA and HS algorithms are plotted in Figure 4 and Figure 5, respectively.

As explained before, different optimum switching angles are found by different algorithms. Figure 6(a) shows the obtained values from simulation for normalized amplitude of fundamental component for both algorithms versus the modulation index. Phase Voltage THD for both algorithms versus the modulation index is plotted in Figure 6(b).

Figure 6(a) shows that SFLA is more powerful and mostly maintained fundamental component value close to the desired, despite that HS seems that is not able to find the switching angles generating the desired value. To corroborate the efficiency of obtained switching angles the phase voltage THD from both algorithms are shown in Figure 6 (b). The comparison shows that values of phase voltage THD for SFLA are always lower than value of phase THD voltage using HSA. The switching angles, related to obtained minimum value of phase voltage THD for SFLA and HSA are given in Table 1 and Table 2, respectively.

From the above tables, it is inferred that the minimum phase voltage THD value attained from using SFLA is lower than the THD value achieved by HSA. Also, the least phase voltage THD value obtained from using SFLA is lower than the phase THD value given in [27] using GA to solve THD minimization problem. So, for THD minimization of phase voltage, SFLA seems to be more effective. As a case study the related phase voltage and Line voltage for the V values given in Table I and II, and also harmonic content of them are shown in Figure 7 and Figure 8 for both SFLA and HS algorithms, respectively.

Figure 7 and 8, validate the ability of obtained switching angles in generation of phase voltage with minimum possible THD, for both algorithms. Compared to Figure 8 (b), at Figure 7 (b) the amplitude of harmonics, mostly are lower. So, SFLA is more efficient than HSA to minimize the phase voltage THD.

7. Experimental Results

The output phase voltage and the output line-to-line voltage of the seven-level inverter prototype are shown in Figure 9(a) and (b), respectively. This is related to the case that switching angles are adjusted using SFLA algorithm. Figure 9 is related to table 1.

In Figure 10(a) and (b) the output phase voltage and the output line voltage of the seven-level inverter case study are presented. The switching angles are obtained from employing HS algorithm. Figure 10 relates to table 2.

The waveform exhibited in Figure 9 and 10 corroborate the simulation results, which validates the feasibility of obtained results and accuracy of the study. Form this paper, it is concluded that SFLA is more efficient to solve the phase THD minimization problem. Because, the minimum phase voltage THD achieved by SFLA is the least in comparison with HS and GA algorithms.

8. Conclusion

Minimization of Total Harmonic Distortion (THD) for phase voltage of multilevel inverter was studied in this work. SFLA and HSA were the two optimization algorithms which were applied to solve the problem. Simulation and experimental results for both of the algorithms were presented for a seven-level inverter. The results of the two aforementioned algorithms were compared together and also with a work published using GA. It is shown that both HSA and SFLA algorithms provide better switching angles to minimize the phase voltage THD. From the attained results it is inferred that SFLA has more efficiency to minimize phase voltage THD while providing the fundamental component.

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