This study presents selective harmonic elimination pulse width modulation technique-based hybrid asynchronous PSO-Newton-Raphson (APSO-NR) algorithm for the elimination of undesired harmonics in cascaded H-bridge multilevel inverter. The proposed algorithm is applicable to all levels of MLI having equal and non-equal DC sources. In the proposed method, ring topology-based APSO algorithm is hybrid with NR method. APSO worked as a global search technique and NR is used for the refinement of best solutions. APSO-NR is applied to the seven-level inverter to eliminate fifth and seventh harmonics. In simulations, the performance of the proposed algorithm is compared with genetic algorithm, bee algorithm and particle swarm optimisation. The results proved that the proposed algorithm is efficient, and gives more precise firing angles in less number of iterations with high capability of tackling local optima. For the 48% of modulation index range, APSO-NR minimised the fitness function value lower than (10−25). The proposed algorithm is validated through the experimental implementation of the three-phase seven-level inverter.