## 4 Chaotic Ensemble Particle Swarm Optimizer (C.EPSO)
### Ensemble Particle Swarm Optimizer (EPSO)  
EPSO is the most recent variant of PSO that combine different variants of PSO such as CLPSO, inertia weight PSO, LIPS, HPSO-TVAC and FDR-PSO variants to create more robust algorithm that is able to solve various optimization problems of different applications [19]. The population size in EPSO is consisting of two groups of subpopulations, a small group and a large one. The small group utilizes CLPSO algorithm while the large one uses the other predetermined variants. In the large group, for updating a particle, one of PSO strategies is adaptively selected depending on the ratio of success of every methodology in  
Figure 1: The block diagram for estimating the parameters of PMSM fractional models. the latest iterations [19]. Mathematical control equation of PSO variants that employed in EPSO are listed as follows:  
Control equations of inertia weight PSO  
basic PSO is combined with a control parameter named inertia weight \(w\) to compromise between the local as well as the global searches. \(w\) has value deceased linearly with time. The velocity of a particle is computed as follows  
\[ \begin{split}& V_{i}^{d}=w*V_{i}^{d}+c_{1}*rand1_{i}^{d}*(pbest_{i}^ {d}-Z_{i}^{d})+\\ & c_{2}*rand2_{i}^{d}*(gbest^{d}-Z_{i}^{d})\end{split}  \tag{6}\]  
where \(i\) indicates the agents (\(i=1,2,\ldots,N\)) while \(d\) indicates the dimension (\(d=1,2,\ldots,D\)). \(Z_{i}^{d}\) is the location of \(i^{th}\) particle and \(V_{i}^{d}\) is the agent velocity in the population. \(pbest_{i}^{d}\) is the best location of the particle \(i\) at \(d\) dimension. The \(gbest^{d}\) is the best location found by population of the swarm so far. The \(w\) is decreasing linearly in range of 0.9-0.2 during the run time [19] to enhance the balance between the global and local search. \(c_{1}\) and \(c_{2}\) are the acceleration coefficients varied with time, where \(c_{1}=2\) and \(c_{2}=2\). \(rand1_{i}^{d}\) and \(rand2_{i}^{d}\) are random numbers in range of \([0,1]\). Control equations of modified CLPSO  
CLPSO has been proposed for tackling the problem of trapping the original PSO in the local minimum and far from the global one for multi-model optimization problem [16]. In CLPSO, each particle has been learned from pests of other particles at different dimensions. The new velocity employed in EPSO variant is as follows [16]:  
\[ \begin{split}& V_{i}^{d}=w*V_{i}^{d}+c_{1}*rand1_{i}^{d}*(pbest_{f_{ i}(d)}^{d}-Z_{i}^{d})+c_{2}*rand2_{i}^{d}*\\ &(gbest^{d}-Z_{i}^{d})\end{split}  \tag{7}\]  
where \(pbest_{f_{i}(d)}^{d}\) is the best place of a particle \(i\), \(f_{i}(d)=[fi(1),fi(2),...,fi(D)]\) shows that ith particle moves to its own or to the other's \(pbest_{i}^{d}\) for each dimension \(d\). \(c_{1}\) and \(c_{2}\) are time varying acceleration coefficients, \(c_{1}\) is used in the range of 2.5-0.5 while \(c_{2}\) is used in the range of 0.5-2.5. \(rand1_{i}^{d}\) and \(rand2_{i}^{d}\) are randomly generated numbers in the range of \([0,1]\). Control equations of FDR-PSO  
FDR-PSO has been introduced to addressed the convergence propblem in PSO. In FDR-PSO, each particle learns from the neighboring particle's experience (\(nbest\)) that have a better fitness than itself. Consequently, the \(i^{th}\) particle's velocity component in \(d^{th}\) dimension is upgraded through the following equation [17]:  
\[ \begin{split}& V_{i}^{d}=w*V_{i}^{d}+c_{1}*rand1_{i}^{d}*(pbest_{i}^ {d}-Z_{i}^{d})+c_{2}*rand2_{i}^{d}*\\ &(gbest^{d}-Z_{i}^{d})+c_{3}*(nbest_{j}^{d}-Z_{i}^{d})\end{split}  \tag{8}\]  
where \(c_{1}\), \(c_{2}\) and \(c_{3}\) are time varying acceleration coefficients, they equaled to 1, 1 and 2, respectively. \(nbest\) is the experience of neighboring particle. Control equations of HPSO-TVAC  
HPSO-TVAC has been introduced to avoid premature convergence in the early stages and enhance the convergence to the global optimum solution [15].