## 4 Description of Search Algorithms
### Overview of Moth-Flame Optimization Algorithm
Two variations were made in the FPA process: (a) The estimation of a set of global orientations for all members of the population towards global or local pollination; (b) The construction of a set of better solution vectors related to all generated global orientations. This set is compared with each iteration with a fixed number of real solution vectors to select the best one among them based on their fitness values. In MOD-FPA two matrices are calculated at each iteration \(t\); one contain \(N_{or}\) sets, where each set is represented by a column with a random number of orientations toward global pollination. The second matrix includes the same number of sets as the first, but has \(n_{f}-rand_{Glog}\) orientations toward local pollination. Please note that the filling of each set is computed using the switch FPA parameter \(p\). The computed solution vector \(x_{i}^{t+1}\), in \(t+1\) requires of \((g^{*}-x_{i}^{t})\) and \((x_{j}^{t}-x_{k}^{t})\) according to (14) and (16), respectively. The MOD-FPA has been proposed to use a solution vector \(x_{i}^{t+1}\) generated \(N_{p}\) pairs of pollen lists extracted from different flowers of the same plants species, each iteration either using global or local pollination. This presents significant improvements to the FPA basic version in terms of found solutions quality [31]. The \(N_{p}\)\((x_{j}^{t}-x_{k}^{t})\) terms can be generated using successively \(N_{p}\) pairs of pollen determined at each new iteration \(t\). However, it is important to note that each generated complete orientation is associated with a pair of pollen gametes lists. More specifically, each set number \(n_{o}\) is assorted to the pair number \(n_{p}\). This means that \(N_{or}=2N_{p}\). As a result, the global exploitation algorithm characteristic is significantly enhanced. It should be noted that MOD-FPA offers the possibility to have a unique best solution generated during one cycle, either through a local or global pollination. The flow chart of MOD-FPA algorithm is shown in Figure 2. A detailed description of these processes is presented in [31]. ## 5 Controllers Tuning Methodology

In this section, six different reactive, nature-inspired algorithms with the proposed objective function (12) are employed to search optimum values of three PI controllers for the velocity trajectory tracking controller. The block diagram of the control scheme is given in Figure 1 and PMSM parameters are given in Table 1. Six parameters in velocity trajectory tracking controller can be tuned simultaneously to attain an improvement in motor performance. The control scheme, together with optimized PI are implemented using Matlab/Simulink 9.1 on Window 8.1 Professional Intel(r) CoreTM i7-4712 HQ CPU @ 2.3 GHz 16 GB RAM. The parameter settings for each algorithm used in the experiment are shown in Table 2. The six algorithms use the same objective function (12) for the optimization of each algorithm. The performance of the optimized PI controller is evaluated with objective function (12) that considers rise time, settling time, steady state error, overshoot, RMSE, JISE, JIAE, JITSE, and JITAE. The search for optimal gains is done with nominal PMSM parameters and a nominal load torque of 2 Nm. For finding the optimum values of PI parameters through MOD-FPA, DA, WOA, FPA, MOA, and CSA, 25 independent runs of each algorithm were computed so that a statistical analysis could be performed on data, see Table 3.

Figure 2: Flowchart of MOD-FPA. The optimal gains value for the velocity trajectory tracking controller are tuned with six different nature-inspired algorithms are tabulated in Table 4. Table 3 lists the statistical comparison between six search algorithms in terms of the best, mean, worst fitness values, and computational (CPU) time. The best results of each study and comparative analysis are highlighted in bold in the corresponding table. The results presented in Table 3 show that MOD-FPA offered the best fitness values (\(11.6044\)) compared to other nature-inspired algorithms.