**1. Introduction**

The Selective Harmonic Elimination (SHE) and Selective Harmonic Mitigation (SHM) [1,2] has been described for the first time in the 1960s in [3] and disseminated by Patel and Hoft [4,5]. Since that time SHE has been introduced in a number of industrial applications where power electronics was proposed [6]. The challenge is progress in the development of techniques for solving SHE/SHM non-linear transcendental equations

Since the early days, iterative techniques such as Newton–Raphson (N–R) [4,5], Gauss–Newton have been employed to solve these equations. The convergence of these methods depends on the initial guess, which is a complex problem and in many cases is not successful. This disadvantage encourages researchers to develop more effective techniques. Thus, Chaison et al. in [7] proposed the method based on the conversion of transcendental equations into an equivalent set of polynomials. The high degree of polynomial requires specialized software to compute it. The combination of Groebner's bases and symmetric polynomials was applied to solve the mentioned polynomials [8]. However, it generates ambiguous solutions which make it less useful. The main disadvantage of iterative techniques is they do not find an optimum solution.

The development of evolutionary algorithms opens new opportunities in the field of solving SHE equations [9]. These algorithms present numerous benefits such as independence from an initial guess, utilization of simple algebra, lower computational costs, formulation of multi-constrained problems. One of the most popular evolutionary algorithms is Particle Swarm Optimization (PSO) proposed for finding switching angles for PWM VSI inverter to eliminate low order voltage harmonic [10] and to optimize dc-link current harmonics [11]. The application of numerous algorithms are proposed for SHE in the literature: Imperial Colonial Algorithm (ICA) [12], Genetic Algorithm [13], Ant Colony Algorithm [14], bee optimization technique (BA) [15], Bacterial Foraging Algorithm [16], Firefly Algorithm (FA) [17], Shuffled Frog Leaping (SFL) algorithm [18], Backtracking Search Algorithm (BSA) and Differential Search Algorithm (DSA) [9], Whale Optimization Algorithm (WOA) [19]. The use of the Grasshopper Optimization Algorithm (GOA) for SHE has not been studied so far.

The GOA is an algorithm recently developed and introduced by Saremi et al. [20] in 2017. In recent two years, the GOA gained great attention in many research fields due to its high efficiency of solving a different kind of optimization problems. It was tested for constrained and unconstrained test functions with promising results [21]. The GOA was proposed for solving multi-objective optimization problems [22] modified by the application of Opposition-Based Learning (OBL) [23]. Modifications of GOA to improve its performance are has been developed and studied: Adaptive GOA (AGOA), Grey Wolf Optimizer (GWO) and Natural Selection (NS) [24], Gaussian mutation, and Leavy- flight strategy [25].

The efficiency of GOA has been compared with existing evolutionary algorithms utilized for different optimization problems. In [26] GOA adaptation for energy loss reduction and voltage stability factor was proposed and compared with PSO, Gravitational Search Algorithm (GSA), and Artificial Bee Colony (BA) algorithms. In [27], the comparison of GOA with PSO and WOA (Wale Optimization Algorithm) was used to optimize the PI controller parameters in the microgrid. Since the first presentation, GOA has found its implementation in numerous industrial applications such as optimization of the parameters of proton membrane fuel cells (PEMFC) [28], the stability of microgrid applications [29] and energy management [30], medicine [31], the technology of image processing [32], and financial issues [25] as well.

In this paper, the recently developed GOA is applied to eliminate low-order voltage harmonics (5th, 7th, 11th, and 13th) in low-frequency VSI based drive. The hypothesis to prove is that GOA represents a higher probability of convergence than PSO applied for SHE problem with similar computation effort. Results for GOA and modified GOA are compared with PSO. The main criterion of comparison is the probability of convergence. The following modification of GOA are examined: Natural Selection (NS), Adaptive GOA (AGOA), Opposite Based Learning (OBL), and Grey Wolf Optimizer (GWO). Experimental results are presented to validate simulation analysis.

The rapid development of controllers for high and medium power converters provides an opportunity for the application of modulation techniques; a decade ago, they used to be known as difficult to use. This type of modulation is SHE-PWM. When it was invented its applicability was very low and nowadays it competes with the most advanced and popular modulations [33]. Its application is studied for grid connectors [34] and railway vehicles [35] as well. Moreover, the separation of a modulator from the controller brings the possibility of implementation of the SHE-PWM with space vector modulation (SVPWM) [36]. Authors of this paper claim that for railway vehicles the most efficient is hybrid modulation studied in [37] where SHE-PWM and SVPWM are used interchangeably and the choice depends on the operating conditions. This solution is the most reasonable and allows to utilize the advantages of both techniques: dynamics of SVPWM and harmonics control of SHE-PWM.

The attention focused on SHE-PWM stimulates research towards increment of efficiency in the calculation of switching angles. In [38] the comprehensive review of SHE-PWM focused on various aspects, is presented. One of the mentioned aspects is the utilization of optimization-based techniques for solving SHE equations and they were divided into four groups: Genetic Algorithms (GA) Particle Swarm Optimization (PSO) Differential Evolution (DE) and Hybrid. According to the "no free lunch" theorem applied to the bio-inspired optimization algorithms [39], there is the most suitable solver for a specific optimization task. Solving SHE equations developed for voltage source inverter is a task of variable complexity that depends on assumptions like the number of switching angles, modulation

index, dead times between switching, switching frequency, and others. Thus, there is a possibility that for different optimization region different algorithm is most suitable. Thus, every recently developed algorithm should be evaluated towards application for solving SHE-PWM equations. In this paper, the authors present the study for the application of the GOA algorithm. The novelty of this study is proof that there is a range of SHE problems where the GOA algorithm gives a higher possibility of convergence with lower computational effort than widely used and appreciated PSO. Results presented in this paper encourage further research to discover the full potential of the GOA algorithm regarding the presented problem by comparing it with a wider representation of bio-inspired algorithms.

The problem undertaken in the study is considered as a single criteria optimization problem. SHE problem could be considered as a multi-objective optimization problem as each harmonic value as a function of optimization variables could be considered as a separate objective function. However, all considered harmonics in the problem of SHE should be eliminated for the same optimization variables, therefore the desirable optimization solution is a utopian solution from the point of view of the multi-optimization approach [40]. Accordingly, the optimization functions have been aggregated to a single optimization variable by means of the dedicated relationship proposed in the article.
