# Differential Evolution Applied to a Multilevel Inverter--A Case Study
## 1 Introduction  
Metaheuristic algorithms are optimization algorithms attempt to improve the search for solutions in the search space iteratively. The algorithms may be inspired by biological phenomena or by the social behavior of species. Metaheuristic algorithms are usually problem-independent. Due to their specific advantages over classical methods, metaheuristics are used in the solution of different problems in the fields of engineering [1].  
Among the metaheuristic methods that have been used for the solution of these problems are differential evolution, particle optimization (PSO), genetic algorithm (GA), firefly algorithm (FA), and others. However, the differential evolution algorithm has distinguished itself for its efficiency and belongs to the evolutionary algorithms. Among the outstanding features of the differential evolution algorithm are its efficient global search capability, versatility to the characteristics of the problem, and not requiring a suitable initial solution [2].  
Differential evolution has been used in various engineering fields, such as acoustics, aerodynamics, aerospace, agriculture, automotive, biology, chemistry, defense, and others. The differential evolution algorithm has been widely used in electricals and electronics, in various communication systems (multi-input and multi-output systems), power allocation, system modeling, controller design for complex dynamic systems, radio network design, ultra-wideband radio systems, and power converters [3].  
Multilevel inverters (MLIs) can create a single output waveform by synthesizing multiple voltage levels, producing a stepped waveform approximating a sinusoidal signal. As the number of sources at the input to the inverter increases, so does the number of levels of the output waveform [4, 5]. Due to the stepped nature of the multilevel inverter output waveform, multiple sets of switching angles can be found for one modulation index (Mi) value [6].  
For applications related to power converters, the differential evolution algorithm has been used to minimize THD in multilevel inverters by calculating the optimal switching angles.  
The solution of the transcendental nonlinear equations is one of the main problems in calculating the optimal switching angles to minimize THD. Therefore, different optimization methods have been used in the literature to eliminate harmonics by solving the nonlinear transcendental equations that generate the Fourier series.  
Several papers mention performing a sweep of the Mi and obtaining results of a minimum THD using a Mi > 1 [6; 7; 8], which can produce an overmodulation, and they only considered using the minimum %THD found. In other papers [9; 10; 11; 12], \"coincidentally\", the minimum %THD found had a Mi > 0.6 and <1. Reviewing the angles found in these papers that coincide in the number of levels (7), they obtain a set of switching angles different from each other. Additionally, other articles do not mention which Mi they used. Moreover, they have different levels, loads, and phases implemented in the MLIs.  
This paper focuses on the differential evolution (DE) algorithm application to minimize the THD value in the output voltage waveform of a cascaded multilevel inverter and to analyze the obtained sets of switching angles calculated to achieve a required modular index according to the inverter application.