# Comparative Performance Evaluation of Swarm Intelligence-Based FOPID Controllers for PMSM Speed Control

<figure><img src="image_1.png" /></figure>

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Hussain Ali Azzawi<fnref n="1" />*, Nihad M. Ameen<fnref n="2" /> ID, Sabah A. Gitaffa<fnref n="1" /> ID

Corresponding Author Email: eee.21.12@grad.uotechnology.edu.iq

https://doi.org/10.18280/jesa.560315

Received: 10 May 2023

Accepted: 11 June 2023

**Keywords:**

fractional-order proportional-integral-derivative (FOPID), proportional-integral-derivative (PID), permanent magnet synchronous motors (PMSMs), mGrey wolf optimization (GWO), ant colony optimization (ACO), particle swarm optimization (PSO)

## ABSTRACT

This paper presents a study focused on the design and performance evaluation of Fractional-Order Proportional-Integral-Derivative (FOPID) controllers for the speed control of Permanent Magnet Synchronous Motors (PMSMs). Effective speed control of PMSMs is of great importance in various applications such as robotics, electric vehicles, and industrial automation. However, achieving precise and efficient speed control poses several challenges due to the nonlinear and time-varying nature of PMSMs. To address these challenges, the study proposes the utilization of FOPID controllers, which offer advantages over traditional PID controllers, including improved robustness and greater flexibility in handling complex system dynamics. Additionally, the study explores the use of Swarm Intelligence (S.I.) algorithms for the design and tuning of FOPID controllers. Swarm Intelligence algorithms, such as Particle Swarm Optimization (PSO), ant colony optimization (ACO), and Grey Wolf Optimization (GWO), are known for their ability to effectively search and optimize complex parameter spaces. The main contribution of this work is the comparison and evaluation of PSO, GWO, and ACO algorithms for the design of FOPID controllers in PMSM speed control applications. The controllers are assessed through both simulations and experimental tests to analyze their performance in terms of speed-tracking accuracy, overshoot, and settling time. The key finding of the study is that the ACO-FOPID controller exhibits the best performance in terms of transient response. It achieves a rise time of 0.008978 s, a settling time of 0.01 s, and zero absolute time error (ITAE). These results indicate that the ACO-FOPID controller provides precise and fast speed control for PMSMs, making it a promising solution for practical applications. In summary, this study highlights the importance of PMSM speed control and the challenges associated with it. It introduces the FOPID controller as a potential solution and motivates the utilization of Swarm Intelligence algorithms for its design. The comparison of PSO, GWO, and ACO algorithms for FOPID controller design demonstrates the superiority of the ACO-FOPID controller in terms of transient response. This research contributes to the advancement of control systems for PMSMs and showcases the potential of Swarm Intelligence algorithms in optimizing complex control parameters.

## 1. INTRODUCTION

The Permanent Magnet Synchronous Motor (PMSM) is a widely used electric motor in various industrial applications due to its high efficiency, reliability, and precise speed control capabilities [1]. Achieving accurate speed control is essential for optimal motor performance and system efficiency. To address this requirement, control methods such as Proportional-Integral-Derivative (PID) controllers have been commonly employed [2]. However, the inherent limitations of PID controllers in dealing with nonlinear and time-varying systems have led to the development of more advanced control techniques, such as Fractional-Order Proportional-Integral-Derivative (FOPID) controllers, which offer higher precision and robustness [3, 4]. FOPID controllers have emerged as a promising solution for achieving accurate speed control in PMSMs. By incorporating fractional-order calculus into the controller design, FOPID controllers can effectively handle the complex dynamics of PMSMs, making them well-suited for industrial applications. Swarm Intelligence (SI) algorithms

provide a powerful optimization approach for tuning the control parameters of FOPID controllers. Inspired by the collective behavior of natural systems such as birds flocking, ant colonies, or schools of fish, SI algorithms mimic these behaviors to search for optimal solutions in complex optimization problems. The use of SI algorithms in FOPID controller design for PMSM speed control offers the potential to enhance system performance and robustness.

In the field of PMSM speed control, several studies have explored the use of different optimization algorithms and controller designs. These studies have focused on techniques such as Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), FireFly Algorithm (F.A.), Grey Wolf Optimization (GWO), Genetic Algorithm (G.A.), and Nelder-Mead (NM). Here is a summary of the main findings from the cited studies:

(1) Belkhadir et al. [5] proposed the use of the ACO algorithm for self-tuning controller parameters, aiming to improve system efficiency and stability.

<footnote n="1">¹Electrical Engineering Department, University of Technology, Baghdad 10066, Iraq</footnote>

<footnote n="2">²Communication Engineering Department, University of Technology, Baghdad 10066, Iraq</footnote>
(2) Agarwal et al. [6] utilized ACO to calculate gain coefficients for a PID speed controller, showing improved dynamic performance compared to traditional PID tuning methods.

(3) Chiranjeevi et al. [7] compared the performance of FPA-based PID and FOPID controllers with FireFly Algorithm (F.A.) and PSO methods. FOPID controllers outperformed PID controllers in terms of setup time and oscillation amplitude, with FPA exhibiting better performance than F.A. and PSO in terms of ISE-based responses.

(4) Yadav and Verma [8] compared the Ziegler-Nichols (Z-N) method with the PSO technique for controlling PMSM speed. They found that the PSO-based PID speed controller improved motor performance across different operating conditions.

(5) El-Saadawi et al. [9] introduced the Gray Wolf Optimization (GWO/FOPID) method for speed control of a D.C. motor. The Nelder-Mead (NM) algorithm-based FOPID approach yielded the best results, achieving zero overshoot with low settling and rising times.

(6) Szczepanski et al. [10] investigated the Model Reference Adaptation System (MRAS) for PMSM speed control. Their results showed that the adaptive ABC-based SFC (State Feedback Control) improved performance compared to non-adaptive conditions.

(7) Ibrahim et al. [11] compared the effectiveness of FOPID controllers with other techniques, such as PSO and ACO algorithms. The PSO-FOPID approach outperformed ACO-FOPID and ACO-PID controllers in terms of transient and frequency response, although it exhibited a larger maximum overshoot.

The motivation behind this study is to investigate and compare the performance of different SI algorithms in tuning FOPID controllers for accurate speed control of PMSMs. The goal is to determine the best controller design that can overcome the limitations of traditional PID controllers and achieve optimal speed control in PMSMs.

By evaluating the performance of FOPID controllers with SI algorithms such as Particle Swarm Optimization (PSO), ant colony optimization (ACO), and Grey Wolf Optimization (GWO), we can identify the most effective approach for achieving accurate speed control in PMSMs. The study aims to improve system efficiency, reliability, and overall motor performance, thereby contributing to advancements in industrial automation and energy efficiency.

The format of this essay is as follows: The fractional order controller is shown in Section 2. The Mathematical Modeling of the PMSM Motor is the subject of Section 3. A summary of optimization methods and various metaheuristic PSO, GWO and ACO algorithms are provided in Section 4. Results from the tests are applied in Section 5. then highlights the conclusions of the proposed system.

## 2. FRACTION ORDER CONTROLLER

A Fractional Order Controller (FOC) is a type of controller used in control systems that incorporates fractional calculus principles in its design. Unlike traditional controllers, which utilize integer order differential and integral operations, FOCs employ fractional order derivatives and integrals.

In mathematical terms, a fractional order controller utilizes non-integer orders, such as fractional values to define the control action. These fractional orders allow the controller to

capture the dynamics and memory effects of the controlled system more accurately.

The key advantage of using FOCs lies in their ability to handle systems with non-linearities, time delays, and complex dynamics more effectively. By incorporating fractional calculus, FOCs can better capture the behavior of these systems and provide improved control performance.

FOCs offer several benefits over traditional integer order controllers. They exhibit increased flexibility, adaptability, and robustness, making them suitable for controlling complex processes and systems. FOCs can provide better tracking of setpoints, rejection of disturbances, and stability in the presence of uncertainties.

The design and implementation of FOCs require specialized mathematical tools and algorithms that deal with fractional calculus operations. These tools enable the analysis and synthesis of FOCs for specific control applications.

FOCs find applications in various fields, including robotics, chemical processes, power systems, biomedical systems, and many others. They offer an alternative approach to control system design, allowing for improved control performance in challenging and dynamic environments.

In summary, a Fractional Order Controller (FOC) is a controller that utilizes fractional calculus principles and non-integer orders to achieve improved control performance, flexibility, and adaptability in controlling complex systems [12, 13]. The Laplace transform of a PID controller can be written as in Eq. (1).

$$G_c(s) = \frac{u(s)}{e(s)} = K_p + \frac{K_i}{s} + K_d s \quad (1)$$

where, $K_p$: proportional gain; $G_c$: transfer function of PID controller; $K_i$: integral gain; $K_d$: derivative gain.

The FOPID Eq. (2) describes the controller's generalized transfer function [14].

$$G_c(s) = \frac{u(s)}{e(s)} = K_p + K_i(s^{-\lambda}) + K_d(s^{\mu}) \quad (2)$$

where, $G_c(s)$: FOPID transfer function; $u(s)$: Controller output. $e(s)$: An error has been produced; $K_p$, $K_i$, and $K_d$: are the earnings equitably distributed, integral and derivative terms, respectively; $\mu$: Fractional portion of the derivative part; $\lambda$: fractional component of the integral term.

Figure 1 shows the block diagram of the PID controller. Figure 2 shows the block diagram of the FOPID controller.

<figure><img src="image_2.png" /><figcaption>Figure 1. PID controller block diagram</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2. FOPID controller block diagram</figcaption></figure>

### 3. MATHEMATICAL MODELING OF PMSM MOTOR

A Permanent Magnet Synchronous Motor (PMSM) is a type of electrical motor in which a permanent magnet serves as the magnetic field source for the stator. These motors offer a high torque/inertia ratio, a high-power density, a high efficiency, are durable, and are simple to maintain [14]. Following is a derivation of the modelling of PMSM that was done to establish the transfer function between output and input [15].

$$V_d = R_s i_d + p \lambda_d - w_e \lambda_q \quad (3)$$

$$V_q = R_s i_q + p \lambda q + w_e \lambda_d \quad (4)$$

$$T_e = \left(\frac{3}{2}\right) P(\lambda_{af} i_q + (L_d - L_q) i_d i_q) \quad (5)$$

$$T_e = T_L + B w_r + J p w_r \quad (6)$$

$$w_e = P w_r \quad (7)$$

where,

$$\lambda q = L_q i_q \quad (8)$$

$$\lambda d = L_d i_d + \lambda_{af} \quad (9)$$

Currents along the d and q axes are denoted by $i_d$ and $i_q$, whereas voltages along those axes are written as $V_d$ and $V_q$. The pole pair count is represented by $P$; The inductances $L_q$, $L_d$ are on the q and d axes; $p$ is the derivative operator; $T_L$, $T_e$ are the load and electric torques; The damping coefficient is denoted by $B$; The inertial moment is characterised by $J$, and the mutual flux or airgap flux is represented by $\lambda_{af}$.

Assume that every one of the Eqs. (4)-(5) is nonlinear and that the id variable is forced to zero by the vector-controlled behaviour of the PMSM. Eqs. (4)-(5) can be solved, as illustrated in the following paragraphs [16].

$$V_q = R_s i_q + p L_q i_q + w_e \lambda_{af} \quad (10)$$

$$V_d = -w_e L_q i_q \quad (11)$$

$$T_e = (3/2) P \lambda_{af} i_q - k_t i_q \quad (12)$$

Transfer Function of PMSM:

$$\frac{\mathrm{Wr}(S)}{\mathrm{Vq}(S)} = \frac{k_t}{(R + \mathrm{LqS})(JS + B) + k_t P \lambda_{af}} \quad (13)$$

The entire T.F is written in numerical value below [16].

$$\frac{wr(s)}{v_q(s)} = \frac{4.705S + 2.219}{S^3 + 7.504S^2 + 3.36S + 2.702} \quad (14)$$

### 4. INTELLIGENT SWARMS

The phrase "swarm intelligente" was first used by Gerardo Beni and Jing Wang in a 1989 article. Swarm intelligence techniques are population-based stochastic methods applied to combinatorial optimization issues, which should not be mistaken for intelligent swarms. Due to their local interactions with their environment, these challenges require the creation of functional global patterns from the aggregate conduct of straightforward individuals. A meta-heuristic method for solving complicated issues is swarm intelligence [17].

#### 4.1 Particle swarm optimization (PSO)

Particle swarm optimization is an optimization technique inspired by the behaviour of animal swarms like birds or bees. It is a population-based optimizing approach that may be used to iteratively search through the space of possible answers to discover the best solution to a problem. Particle Swarm Optimization (PSO) is a stochastic and swarm intelligence-based optimization technique. Nevertheless, the classic PSO suffers from early convergence when discovering the global optimal [18]. A "swarm" of potential solutions is initiated in a PSO algorithm, and each answer is portrayed by a "particle" in a swarm. The particles travel around the solution space, and their positions are updated depending on the location of the top-performing particles in the hive (the "global best") and their individual personal best. A set of equations governs particle movement, considering the particle's present position, global or personal best. Eqs. (15) and (16) [19, 20] do this:

$$\begin{aligned} V_{jd}(k+1) = & w * V_{jd}(k) + C_1 * \mathrm{rand}(P_{jd}(K) - X_{jd}(k)) + C_2 * \mathrm{rand}(P_{jd}(K) - X_{jd}(k)) \end{aligned} \quad (15)$$

$$X_{jd}(k+1) = X_{jd}(k) + V_{jd}(k) \quad (16)$$

where, $C_1$ and $C_2$ are constant acceleration coefficients that have a positive value, and where the value of $C_1$ is greater than $C_2$.

The rand is based on either a random integer or a mathematical formula. function that produces a random number in the range of 0 to 1. The moment of inertia is denoted by the sign $w$, where $w$ is less than one.

#### 4.2 Grey wolf optimization (GWO)

Is a meta-heuristic algorithm inspired by the hunting behaviour of grey wolves in nature. In GWO, a pack of grey wolves cooperate to find suitable solutions to optimization problems by imitating the wolf pack's social hierarchy and hunting behaviour [21]. The algorithm works as follows:

Initialize the positions of the wolves randomly within the search space. Update the parts of the wolves based on their fitness values, which are evaluated using the objective function of the problem. Determine the leader, beta, and delta wolves, the three wolves with the highest fitness values.
The ACO Algorithm's characteristics were set up to match the information in Table 3. Figure 6 demonstrates the step response of PMSM Motor velocity when controlled by a standard PID-ACO.

<table><caption>**Table 3.** ACO values parameters</caption><thead><tr><td>Parameters' names</td><td>Values</td></tr></thead><tbody><tr><td>Evaporate rate</td><td>0.5</td></tr><tr><td>Scaling rate</td><td>2</td></tr><tr><td>NO. of iteration (N)</td><td>100</td></tr><tr><td>Size of sample</td><td>40</td></tr><tr><td>Pop. size</td><td>20</td></tr><tr><td>size of Step</td><td>1</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Figure 6.** Speed response of the PMSM motor using the control unit ACO - PID</figcaption></figure>

<table><caption>**Table 4.** GWO values parameters</caption><thead><tr><td>Parameters' names</td><td>Values</td></tr></thead><tbody><tr><td>Evaporate rate</td><td>0.5</td></tr><tr><td>Scaling rate</td><td>2</td></tr><tr><td>NO. of iteration (N)</td><td>100</td></tr><tr><td>Size of sample</td><td>40</td></tr><tr><td>size of Step</td><td>1</td></tr><tr><td>Pop. size</td><td>20</td></tr></tbody></table>

The response of the PMSM motor's speed stepping is shown when it is controlled by an ACO-PID controller in Figure 6.

The GWO Algorithm was configured using the parameters shown in the Table 4. Figure 7 illustrates the step response of

PMSM Motor speed when controlled by a typical PID controller using GWO.

<figure><img src="image_5.png" /><figcaption>**Figure 7.** GWO-PID controller step response of PMSM motor speed</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 8.** Block diagram for comparison between the PSO-PID, GWO-PID and ACO-PID Simulink in MATLAB</figcaption></figure>

In Figure 8, block diagram shows comparison of PSO-PID, GWO-PID and ACO-PID Simulink in MATLAB.

After contrasting the PSO method with the ACO technique, we arrive at the results in Table 4.

<table><caption>**Table 5.** A comparison of the outcomes obtained by the various suggested algorithms (traditional PID controller)</caption><thead><tr><td>Method</td><td>Input</td><td>M.P. (%)</td><td>Rise Time (sec)</td><td>Settling Time (sec)</td><td>Steady State Error</td></tr></thead><tbody><tr><td rowspan="3">PSO-PID</td><td>K<sub>p</sub> = 194.3689</td><td rowspan="3">3.646</td><td rowspan="3">0.08855</td><td rowspan="3">0.0132</td><td rowspan="3">0.000067</td></tr><tr><td>K<sub>i</sub> = 139.8394</td></tr><tr><td>K<sub>d</sub> = 10.0119</td></tr><tr><td rowspan="3">ACO-PID</td><td>K<sub>p</sub> = 40.7362</td><td rowspan="3">1.484</td><td rowspan="3">0.0934</td><td rowspan="3">0.0521</td><td rowspan="3">0.000191</td></tr><tr><td>K<sub>i</sub> = 45.2896</td></tr><tr><td>K<sub>d</sub> = 6.3493</td></tr><tr><td rowspan="3">GWO-PID</td><td>K<sub>p</sub> = 75.5372</td><td rowspan="3">4.747</td><td rowspan="3">0.0309</td><td rowspan="3">0.0455</td><td rowspan="3">0.000261</td></tr><tr><td>K<sub>i</sub> = 61.8052</td></tr><tr><td>K<sub>d</sub> = 9.5482</td></tr></tbody></table>

In terms of performance, Table 5 shows that the PSO and GWO algorithms are more valuable than the ACO algorithm, especially with regard to maximum overrun. A simulation of a PID controller with the molecular organization of the proposed system is shown in Figure 9 below. This Figure also compares each PSO algorithm's maximum bypass, fixation time and rise time. GWO and ACO.

Figure 10 shows PMSM motor system diagram with FOPID Control Simulink in MATLAB. Figure 11 demonstrates that the PSO-FOPID controller provides improved dynamic properties (overshoot and settling time) of a system response. Figure 12. demonstrates that the GWO-FOPID controller provides improved dynamic properties (overshoot and settling time) of a system response.
<figure><img src="image_7.png" /><figcaption>**Figure 9.** Compares the performance of the ant colony optimization method, the grey wolf optimization system, and the particle swarm optimization technique.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 10.** The block diagram of the PMSM system incorporates the FOPID controller Simulink in MATLAB.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 11.** The step response of the PMSM motor speed when a PSO-FOPID controller controls it.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 12.** The step response of the PMSM motor speed when controlled by a GWO-FOPID controller.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 13.** The step response of the PMSM motor speed when controlled by an ACO-FOPID controller.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 14.** PSO-PID and PSO-FOPID step response of permanent magnet synchronous motor (PMSM) motor speed</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 15.** The step response of the PMSM motor speed when controlled by a GWO-PID or GWO-FOPID controller.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 16.** Demonstrates the step response of the PMSM motor speed when controlled by an ACO-FOPID controller.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 17.** Block diagram for comparison between the FOPSO, FOGWO and FOACO Simulink in MATLAB.</figcaption></figure>

Figure 13 demonstrates that the ACO-FOPID controller provides improved dynamic properties (overshoot and settling time) of a system response compared to those produced by the PID controller.

Figure 14 demonstrates how the PSO-FOPID controller outperforms the PSO-PID controller regarding reliability and route tracking.

Figure 15 demonstrates how the GWO-FOPID controller outperforms the GWO-PID controller regarding reliability and route tracking.
Figure 16 demonstrates that the ACO-FOPID controller provides improved dynamic properties (overshoot and settling time) of a system response compared to those produced by the PID controller.

Figure 17 shows block diagram showing comparison of FOPSO, FOGWO, and FOACO Simulink in MATLAB.

The comparison of the PSO-FOPID, ACO-FOPID, and GWO-FOPID is depicted in Figure 18. demonstrates that the ACO-FOPID.

Figure 18 Block diagram for compares the FOPSO, FOACO, and FOGWO. The controller produces better dynamic properties (settling time and excess) of the system response than the PSO and GWO controllers.

The ACO-FOPID controller outperforms the PID controller. When all of the trials that resulted from the simulation are considered, it has been shown that the ACO-FOPID controller, which is superior to both the PSO-FOPID and the GWO-FOPID controllers, is the most effective way to attain

performance as well as durability. This leads us to conclude that the stability and resilience of the system have grown, as illustrated in Figure 18 and Table 6.

<figure><img src="image_16.png" /><figcaption>Figure 18. Compares the FOPSO, FOACO, and FOGWO</figcaption></figure>

<table><caption>**Table 6.** A comparison of the outcomes obtained by the various suggested algorithms (traditional FOPID controller)</caption><thead><tr><th>Method</th><th>Input</th><th>M.P. (%)</th><th>Settling Time (sec)</th><th>RiseTime (sec)</th><th>Steady State Error</th></tr></thead><tbody><tr><td>PSO-FOPID</td><td>K<sub>p</sub>=135.7952 K<sub>i</sub>=48.8753<br>Lambda=0.798<br>K<sub>d</sub>=20.8952<br>Mu=0.985<br>K<sub>p</sub>=231.5537<br>K<sub>i</sub>=228.8589</td><td>0.452</td><td>0.0326</td><td>0.0182</td><td>0.00179</td></tr><tr><td>ACO-FOPID</td><td>Lambda=0.987<br>K<sub>d</sub>=34.5458<br>Mu=0.998<br>K<sub>p</sub>=263.34<br>K<sub>i</sub>=174.73</td><td>0.024</td><td>0.01</td><td>0.008978</td><td>0.000058</td></tr><tr><td>GWO-FOPID</td><td>Lambda=0.676<br>K<sub>d</sub>=32.205<br>Mu=0.96</td><td>0.505</td><td>0.0139</td><td>0.0085</td><td>0.0014</td></tr></tbody></table>

## 6. CONCLUSIONS

The search suggested adjusting the FOPID controller for managing the speed of the PMSM motor using the GWO, PSO, and ACO algorithms with the main goal of minimizing the ITAE objective function, which assesses the controller's capability to monitor the intended speed while reducing error over time. The study looked at a number of measures, including transient response, frequency response, settling time, rising time, and ITAE, to compare the efficiency of the proposed ACO-FOPID controller with that of other controllers, including ACO-PID, FOPID, GWO-PID, FOPID, PSO-PID, and PSO-FOPID. According to the comparison data, the ACO-FOPID controller outperformed the other controllers, indicating that it has the potential to regulate the speed of PMSM motors. The study hypothesizes that future investigation could enhance the performance and resilience of the controller.

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## NOMENCLATURE

<table><tr><td>Rs</td><td>stator resistance [Ω]</td></tr><tr><td>Ø</td><td>rotor magnetic flux [Weber]</td></tr><tr><td>Lq, Ld</td><td>quadrature and direct axis inductance [H]</td></tr><tr><td>P</td><td>number of pole pairs</td></tr><tr><td>Iq, Id</td><td>quadrature and direct axis currents [A]</td></tr><tr><td>p</td><td>derivative concerning time</td></tr><tr><td>Eb</td><td>back emf [V]</td></tr><tr><td>Te</td><td>electromagnetic torque [Nm]</td></tr><tr><td>T</td><td>load torque [Nm]</td></tr><tr><td>B</td><td>friction coefficient</td></tr><tr><td>J</td><td>moment of inertia [Kg - m²]</td></tr><tr><td>Kt</td><td>torque constant</td></tr><tr><td>We</td><td>angular rotation [rad/sec]</td></tr><tr><td>λd, λq</td><td>flux linkages [weber]</td></tr><tr><td>λaf</td><td>mutual flux between magnet and stator.</td></tr><tr><td>Xt</td><td>Position of particle</td></tr><tr><td>Vt</td><td>Velocity of particle</td></tr><tr><td>pbest</td><td>Best position of a single particle</td></tr><tr><td>gbest</td><td>Best position of all particles</td></tr><tr><td>w</td><td>Inertial Weight factor</td></tr><tr><td>C2, C1</td><td>Acceleration Coefficients</td></tr></table>