## 1 Introduction  
Permanent Magnet Synchronous Motor (PMSM) is one of the preferable motors due to its high efficiency, low cost, and simple structure [1]. However, PMSM performance may be disturbed considerably due to its chaotic behavior due to the disturbance of load, or the system parameters change [2, 3]. This behavior is an important problem in the operation of PMSM because of torque inconstancy, low-frequency fluctuations in current, and speed oscillations, which may lead, in turn, to motor collapse. The whole system stability, safety as well as the economic operation of the industrial process may be influenced by this unwanted behavior [4]. Therefore, there is a persistent need for efficient and accurate modeling of the motor's dynamic behavior to improve the chaotic performance control and prevent the ravage of the motor [2, 3]. Moreover, it is crucial to introduce an accurate and simple optimization technique to efficiently extract the parameters of these models with minimum execution time to restore the system's normal operation as soon as possible and prevent its damage [5]. Recently, fractional modeling becomes a new avenue in dynamic modeling. The fractional-order models provide a proper emulation of the system's physical response by adding new degrees of freedom for the mathematical model of the systems [6, 7, 8]. Two types of the fractional modeling of PMSM defined as an equal order and variable-order fractional models are currently published and validated to introduce more flexible and accurate modeling over the integer one [2, 9]. Therefore, they are selected to be tested in this work. In literature, two approaches are introduced to define the parameters of PMSM irregular behavior. The first one is the numerical methods that have been proposed for determining the PMSM model's parameters at a specific operating condition [2; 9]. The numerical techniques employed some simplifications and assumptions while handling such a nonlinear problem. Thus, the second approach has been proposed to identify the models' parameters using the meta-heuristic optimization techniques [10; 11]. [PERSON] et al. [12] proposed two novel meta-heuristic optimization algorithms named Chaotic Grasshopper Optimizer (CGOA) and Chaotic Grey wolf optimizer (CGWO) in addition to their basic versions (GWO and GOA) for parameters estimation of both equal and variable-order fractional models. Nevertheless, seeking better accuracy and lower time consumption, which are significant factors in predicting and controlling the motor's unpredictable performance as fast as possible, reliable developed algorithms should be proposed for optimal parameters estimation. The Particle Swarm Optimization Algorithm (PSO) is considered as the simplest one to be implemented and it has performed well on several fields [13]. While, there are some disadvantages in the PSO technique such as trapping in the local minimum where its searchability is insufficient and its convergence speed is slow because each particle is learned from both of the personal best as well as the global best [14]. To minimize these drawbacks, several variants of PSO have been reported in the literature to compromise between the exploration and exploitation processes. One of these variants is the Self-organizing Hierarchical PSO (HPSO-TVAC) that has been developed to reinforce the global exploration in addition to the local exploitation and thereby avoiding earlier convergence as well as reaching the global optimum during the latter stages of the search [15]. another variant known as Comprehensive Learning PSO (CLPSO) has been proposed where each particle learns from the best experiences of the other particles over various dimensions [16]. Additionally, a variant named Fitness Distance Ratio based PSO (FDR-PSO) has been introduced to enhance the capability of the local search [17]. Moreover, the variant named Distance-based locally informed PSO (LIPS) has been employed to enhance searching for multiple local optima in multi-modal problems [18].