Optimal FOC-PID Parameters of BLDC Motor System Control Using Parallel PM-PSO Optimization Technique

Nguyen Tien Dat; Cao Van Kien; Ho Pham Huy Anh

doi:10.2991/ijcis.d.210319.001

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Volume 14, Issue 1, 2021, Pages 1142 - 1154

Optimal FOC-PID Parameters of BLDC Motor System Control Using Parallel PM-PSO Optimization Technique

Authors

Nguyen Tien Dat¹^{, 2}, Cao Van Kien¹^{, 2}, Ho Pham Huy Anh¹^{, 2}^{, *}^,

¹Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Viet Nam

²Vietnam National University Ho Chi Minh City (VNU-HCM), Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam

^*Corresponding author. Email: hphanh@hcmut.edu.vn

Corresponding Author

Ho Pham Huy Anh

Received 24 August 2020, Accepted 16 January 2021, Available Online 25 March 2021.

DOI: 10.2991/ijcis.d.210319.001 How to use a DOI?
Keywords: Particle swarm optimization (PSO); Parallel PSO technique; Parallel multi-population technique; Brushless DC electric motor; BLDC motor system; Field-oriented control (FOC-PID); Nonlinear Benchmark tests; Parallel multi-population particle swarm optimization (PM-PSO)
Abstract: This paper proposes a parallelization method for meta-heuristic particle swarm optimization algorithm to obtain a convincingly fast execution and stable global solution result. Applied the proposed method, the searching region is efficiently separated into sub-regions which are simultaneously searched using optimization algorithm. The structure of meta-heuristic algorithm is rebuilt as to execute in parallel multi-population mode. The closed loop system of brushless DC electric motor position control is used to verify the proposed method. The simulation and experiment results show that the proposed parallel multi-population technique obtains a competitive performance compared to the standard ones in both of precision and stability criteria. Especially, meta-heuristic algorithms running in parallel multi-population mode execute quite faster than standard ones. In particular, it shows an efficient improvement of the proposed method applied to identify of nonlinear Benchmark tests and to optimize proportional integral derivative parameters for field-oriented control scheme of the brushless DC electric motor system.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

ACRONYMS

BLDC	Brushless DC Electric Motor
FOC	Field-Oriented Control
PID	Proportional Integral Derivative
PSO	Particle Swarm Optimization
SPSO	Standard Particle Swarm Optimization
PM-PSO	Parallel Multi-Population Particle Swarm Optimization

1. INTRODUCTION

Particle swarm optimization (PSO) algorithm introduced by Eberhart and Kennedy [1,2] is a powerful basic algorithm for resolving different types of optimization applications. Its principle relies on the social-sharing information among particles aim to reach an evolutionary benefit. Thus those algorithms have been widely used in searching for the best optimum solution within a specified range in which random population is created.

Recently, many research related to PSO algorithms have been applied in various areas such as optimal problem [3], modification of landslide susceptibility mapping [4], estimating α ratio in driven piles [5], path planning [6], color image segmentation [7]. Regarding to brushless DC electric motor (BLDC) motor control there have been several advanced control techniques, such as optimal control, nonlinear control, and adaptive control, being applied for velocity and position control of BLDC [8,9]. Three elements of proportional integral derivative (PID) block can efficiently handle both of transient and steady control stages and then shows a simple but effective solution to numerous control tasks. Nevertheless, optimal updated PID parameters seem really hard. Recently evolutionary optimization algorithms, such as PSO, have been improved it in many ways to solve optimal problem [10–13]. Moreover, it is interested in applying PSO in tuning, optimizatio,n and identification BLDC motor. Those researches show that the PSO can solve lots of optimized problems, but it takes a lot of time to find out the best solution [14–19].

Development of big data technologies have also initiated the generation of complex optimization problems with large size. The high calculation cost of these problems has accounted for the development of optimization algorithms with parallelization. PSO algorithm is one of the most popular swarm intelligence-based algorithms, which shows its robustness, simplicity, and global search proficiency. However, one of the major PSO obstacles is frequency of getting entrapped in local optima and being deteriorated as soon as the dimension of the problem increases. Hence, numerous efforts are made to reinforce its performance that includes the parallelization of PSO. The basic architecture of PSO inherits a natural parallelism, and receptiveness of fast processing machines has made this task pretty convenient. Therefore, parallelized PSO (PPSO) has emerged as a well-accepted algorithm by the research community. Several studies have been performed on parallelizing PSO algorithm so far [20–26]. To investigate and model the batch culture of glycerol to 1,3-propanediol, Yuan et al. [21] presented nonlinear switched system which is enzyme-catalytic, time-delayed and contains switching times, unknown state delays, and system parameters. The process contains state inequality constraints and parameter constraints accompanied by calibrated biological robustness as a cost function. To extract and estimate the parameters of the PV cell model, Ting et al. [22] proposed parallel swarm algorithm (PSA) that minimizes root-mean-square error that seeks the difference between determined and computed current value. They coded the fitness functions in OpenCL kernel and executed it on multi-core CPU and GPUs. Further, Liao et al. [23] proposed multi-core PPSO (MPPSO) for improving the computational efficiency of system operations of long-term optimal hydropower for solving the issue of speedily growing size and complexity of hydropower systems. The algorithm claims to achieve high-quality schedules for pertinent operations of the hydropower system. Li et al. [24] used parallel multi-population PSO with a constriction factor (PPSO) algorithm to optimize the design for the excavator working device. They authenticated kinetic and dynamic analysis models for the hydraulic excavator. Luu et al. [25] proposed a competitive PSO (CPSO) to improve algorithm performance with respect to stagnation problem and diversity. Arqub et al. [27,28] introduced the continuous genetic algorithm (GA) as a solver for boundary value problem of second-order systems and also proposed a new approach of fuzzy fractional derivative which is efficient and conformable to solve numerical solution.

Improved from above referred applications of PSO, this paper proposed the method splitting searching region into pieces of sub-region, in which the independent sub-population is created, aimed to greatly reduce computing time. This result is ameliorated in utilizing the power of multi-core processor through which all sub-regions of PSO structure are simultaneously searching by sub-population. Consequently, a best vector that contains global best stage of each sub-population is returned and the global best solution of main-population is successfully selected. Recently, new types of hardware that deliver massive numbers of parallel processing power so running parallel multi-population is possible. Moreover, recent CPU architectures have significant modifications leading to high-performance computing capabilities. One of those high-performance computing capabilities is Parallel Computing Toolbox that can solve intensive computing and data problems using multi-core processors. High-level constructions, such as parallel for loops, special types of arrays and parallelized digital algorithms, are supported. The toolbox allows programmer to use the functions supporting parallel calculation with MATLAB and other toolboxes. It is possible to use the toolbox with Simulink® to run several simulations of a model in parallel. The toolbox allows you to use all the processing power of multi-core computers by running applications on workers (MATLAB computing engines) that run locally so each sub-population can be setting up independently on each available worker. So, parallel multi-population PM-PSO algorithm is innovatively proposed for optimizing the PID parameters of BLDC controller. It shows the competitive performance compared to the standard ones.

The rest of this paper is structured as follows. Section 2 introduced the implementation of FOC algorithm to control BLDC motor system using MATLAB/Simulink platform. Section 3 presents the standard SPSO algorithm used to optimize PID parameters of BLDC FOC control. Section 4 presents the newly proposed parallel multi-population PM-PSO technique. Simulation and experiment results of proposed algorithm applied in four Benchmark tests and used to optimize PID parameters of BLDC FOC position control are fully presented in Section 5. Eventually Section 6 includes the conclusions.

2. BLDC FOC CONTROL ALGORITHM

The advantage of BLDC motors is not use mechanical components and brushes to commutate, in which the stator represents the coil and rotor includes permanent magnets. In this research, a 2-pole BLDC will be investigated, through which not only gathers the benefits of DC machine for example better speed ability with not mechanical commutator but includes the superiority of AC motor related to simple, high reliable, and no-maintenance. Furthermore, BLDC also possesses followed benefits: compact volume and powerful torque. Then it is often used in applications required highly driving quality. The FOC algorithm is applied to control a 3-phase synchronous motor in the dq0 domain. The advantage of this method is that it can simultaneously control the angle-speed and torque of the BLDC. The main structure of the FOC control algorithm can be divided into two main control layers. The first control layer is used to control two values d and q in order to generate the vector of Stator magnetic as desired. Meanwhile, the 2nd control layer is used to control the angular depend on controlling q value as presented in Figure 1.

Figure 2 shows that the BLDC rotor angle value is determined by using location sensor. The values id and iq can be determined via ia and ib using Clarke & Park transform equations. It is important to note that id=0 is preset value (Figure 1) and it needs to design a position control loop and iq-based current closed loop. The scheme of BLDC control system is fully illustrated in Figure 2. This BLDC scheme is with load, sine SPWM, VSI inverter, field-oriented control (FOC) implemented in 3 controllers. A cascading control loop is employed which includes 1 position and 2 current-loop control blocks. The role of PI control blocks which are designed here to adaptively modify the d-current component of the FOC control drive.

The important parameters of BLDC control system are chosen as follows:

L_qL_d	H	8.5e-3	J	kg.m²	0.0008
R	Ohm	2.87	F	N	1.349e-05
λ	V.s	0.175	T_f	N.m	0
P		2

To simulate this control model, we need to perform the following below steps. In Simulink blocks of MATLAB, select the block “Permanent Magnet Synchronous Motor” with Trapezoidal Back-EMF waveform presented in Figure 3.

The mathematical model formulas (1–5) are presented for the nonlinear BLDC simulation model shown in Figure 2:

ddt.id=1Ld.vd−RLd.id+LqLd.p.wm.iq(1)

ddt.iq=1Lq.vq−RLq.iq+LqLd.p.wm.iq−λ.p.wmLq(2)

Te=1,5 pλ.iq+Ld−Lq.id.iq(3)

ddt.wr=1JTe−Tl−F.wm(4)

dθdt=wm(5)

with vd and vq represent the stator voltages in direct axis (d-axis) and quadrature axis (q-axis), respectively; id and iq are similar stator current components, respectively; Ld and Lq represent inductances of the direct- and quadrature-axis; R is the stator resistance; wm denotes the rotor mechanical speed. Te is electromagnetic torque. Tl, p, F and J represent the torque load, the pole pair, the viscous friction coefficient, and the inertia moment of the rotor, respectively.

The main control blocks are shown in Figure 1. The BLDC control system consist of transforming dq to abc block, SPWM block, Encoder signal transforming block and PI block used to control BLDC system.

“dq0 to abc” block was used in BLDC control system simulation (Figure 4). The main purpose of using this function is calculating abc value based on dq value provided from input. Input and output of this function block is shown in (6) and (7):

uaubuc=cos(wt)−sin(wt)1coswt−2π3−sinwt−2π31coswt+2π3−sinwt+2π31uduq0(6)

uduq0=23.cos(wt)coswt−2π3coswt+2π3−sin(wt)−sinwt−2π3−sinwt+2π3121212uaubuc(7)

The transformations of “dq0 to abc” and “abc to dq0 are shown in (6) and (7), respectively.

PID control schemes are now widely applied in numerous industrial control processes. It is important to note that the PID scheme is easy to be performed badly and even becoming unstable in case inappropriate values of PID controller which were chosen. Thus the fact is that it is crucially necessary to tune the PID coefficients to obtain the best performance within the bounded range.

The three coefficients of PID block must be carefully selected as to satisfy preset quality requirements, with respect to rising and steady time, overshoot percentage, etc. In this study parallel multi-population PM-PSO algorithm is innovatively applied to optimally identify the optimum coefficients of three PI control block for BLDC FOC-PID position control scheme.

3. OPTIMIZING FOC-PID PARAMETER BASED ON STANDARD PSO ALGORITHM

The standard PSO algorithm initializes a random population of solutions. At each generation, all of fitness function values are calculated to determine best value of those members in this population. In standard PSO, the member of population, called particle, move within a n-dimension of searching region with a velocity that is automatically adjusted base on its own experience and sharing information of its neighbors.

In standard PSO algorithm, a population included NP member are created for searching best stage within main searching range called X≜X1_lim,X2_lim,…,Xn_lim, where Xi_limit is a limited range vector of each dimension Xi_lim≜Xi_min,Xi_max and i=1,…,n is the dimension of searching area. The particle is denoted as xi→=xi1,xi2,…,xin, with xi∈X, are the border of searching region of the n^th dimension. The speed of each particle is denoted as vi→=vi1,vi2,..,viD and vmax is setting up manually. The best position (pbest) is calculated at first step of each iterations and stored as pi=pi1,pi2,…,piD which is compared to best position of all iterations (Pbest) called as global best position and denoted as Pi=Pi1,Pi2,…,PiD. In each iteration the particles' speed and position will be updated using (8) and (9):

vi=w*vi+R1*c1Pi−xi+R2*c2*pi−xi(8)

xi=xi+vi(9)

with w represents inertia value; c1, c2represent acceleration constant of particle to global and local best position in respectively; R₁, R₂ denote random vectors with uniform elements lied in [0, 1]. In Equation (8), the first part of velocity is adjusted by previous experience of itself. The second part is shown that velocity of particle influenced by the global and local best position according to c1, c2, R₁, R₂ parameters. New velocity vector is updated after found out local best position. This process is repeated until reaching the maximum iteration that is set earlier. Figure 5 shows the full flow-chart of standard PSO optimization algorithm.

In which xit is current position of particle; xit+1 is current position of particle in next step. The new position of particle are calculated based on particle's velocity in previous generation w*vi, the distance from the local optimal R2*c2*pi−xi and the experience from the swarm R1*c1Pi−xi. The inertia weight w determines the inertial moving of the particle. The cognitive learning factor, presented with c1, is the influence of distance between current position of particle and local optimal to the velocity of particle while moving to new position in the next generation. The last parameter is c2 presented for social learning factor of particle which is affected to the velocity of current particle.

The PSO approach is used to optimally identify the parameters for PID block. This procedure can guide the swarm's particles shift to space with fitness value more available and eventually reach the global optimum solution. In Standard PSO method, each particle contains six members as P1,P2,P3,I1,I2,I3. It is explained that the searching region includes six-sized and particles have to hover in a six-dimension area. In detail, each particle in population must try to modify its position based on followed ingredients: the current position xi→=P1,P2,P3,I1,I2,I3; the current velocities vi→=vP1,vP2,vP3,vI1,vI2,vI3; the distance from the current position to pbest value; and the distance from the current position to Pbest.

Using (8) and (9), particle's speed eventually moves to pbest and Pbest, will be updated each step in processing iteration with its current location is updated using (9) as figured in Figure 5.

In this case, fitness function is determined as

J=∫02C1e2+C2e˙2dt.(10)

with e is the error between reference theta and actual theta of BLDC mentioned in Figure 1 and C1,C2 represent the weights of position error and the derivation of it.

Fitness function J with C1=10,C2=5 is used to evaluate the performance of step. And, update function show as Equation (8). After NG generation, the best stage, where value of member earned the lowest value of fitness function, is found by NP member. Figure 6 shows the principle of SPSO algorithm executed on single core.

4. PROPOSED PARALLEL MULTI-POPULATION TECHNIQUE APPLIED ON PM-PSO ALGORITHM

This paper proposes the parallel multi population technique called PM technique, the searching region is split into many sub-searching region according to classified parameter such as Xi…Xn dimension from original searching region X mentioned in Section 3. Searching range X is divided into ki parts on Xi-dimension where i=1,..,n. After all, there are k sub-searching region where k is calculated as k=∏i=1nki.

Firstly, the searching region is separated into k=4 pieces according to Xi−dimension shown as Figure 6. The method used to divide the population into k population on those sub-regions is each sub-region is sought for best local stage by NP/k members in NG generations. Each sub-region is processed on its respective core. After all, k population in k sub-region return a vector containing local best stage of k sub-region represented as pbest_local=pbest_1,…,pbest_k. Finally, the best global stage Pbest is selected by minimum selection strategy. Figure 7 illustrates the principle of parallel multi-population technique executed on multi cores proposed in this paper.

Pseudo codePbest=pbest_1;for i=1:k if fpbest_i<fpbest_global Pbest=pbest_i; endend

The parallel multi-population technique expected to perform for fast execution and stable results of PMPSO optimization algorithms. Firstly, splitting up searching region decreased local extremes rate of the results because there is k value. Secondly, the execution of upgraded algorithm takes less time than the original ones that are shown in experiment results. Finally, the standard deviation and average of results when using parallel multi-population technique are competitive compared to standard algorithms.

The performance and effectiveness of parallel multi-population are tested on four benchmark-test functions detailed below in Subsection 5.1. Then, it is compared to the standard SPSO algorithm. All experiments are run with Matlab 2016b on the Intel Core i5 8th Gen 1.6 Mhz and 8Gb RAM.

The control parameter of all algorithms used to optimize are listed. It is included inertia weight (w), particle's best weight (c1), swarm's best weight (c2) for PSO algorithm. The parallel multi-population PM-PSO algorithm mentions that the searching region is divided into four part according to X -dimension. This parameter is chosen to make sure that the number of computation carried out by processor in each algorithm is equal. All experiment was realized on Intel Core i5 computers shown as Table 1 below.

Case		PSO (Serial Implementation)	w=0.8	PM-PSO (Parallel Implementation)	w=0.8
			c1=0.4		c1=0.4
			c2=0.3		c2=0.3
Case1010	NG	10		10
	NP	24		6 × 4
	No. core	1		4
Case1048	NG	10		10
	NP	48		12 × 4
	No. core	1		4
Case1060	NG	10		10
	NP	60		15 × 4
	No. core	1		4
Case1072	NG	10		10
	NP	72		18 × 4
	No. core	1		4
Details of multi-core processor
Specification		Intel Core i5 8250U CPU @ 1.60 MHz
Cores		4
Threads		8

PSO, particle swarm optimization.

Table 1

PSO parameter of each case and parameter of Intel Core i5 computers.

5. RESULTS AND DISSCUSSION

5.1. Four Benchmark Test Functions

The performance and effectiveness of parallel multi-population PMPSO are tested on four benchmark functions detailed below. Then, it is compared to the standard algorithm SPSO. All experiments are simulated by Matlab 2016b on Intel Core i5 8th Gen 1.6 Mhz and 8Gb RAM.

The principal parameters of two optimization algorithms used are listed in Table 2. It is included inertia weight (w), particle's best weight (c1), swarm's best weight (c2) which shows the same for both PM-PSO and SPSO algorithm. After few trials, the population and generation of standard PSO algorithm are chosen equal 50 and 2000 for all benchmark functions. The parallel multi-population PM-PSO algorithm shown in Figure 7 in which the searching region is divided into four parts according to X1-dimension, then the population and generation of proposed PM-PSO will be 50 and 500, respectively. Those parameters are chosen to make sure that the equality carried out by processor in each algorithm is equal. Table 2 illustrates the principal parameters of standard SPSO and proposed PM-PSO algorithms.

	SPSO	w=0.8	PM-PSO	w=0.8
		c1=0.4		c1=0.4
		c2=0.3		c2=0.3
NP	50		50
NG	2000		500
k	1		4

SPSO, standard particle swarm optimization; PSO, particle swarm optimization

Table 2

The parameters of SPSO, PM-PSO.

The four benchmark functions used to test the performance and effectiveness of proposed parallel multi-population PM-PSO are described as follows:

Rosen Brock: Valley-Shaped

f(x,y)=∑i=1nbxi+1−xi22+a−xi2,where a=1,b=100

Global minimum: fx*=f(1,…,1)=0

Dimension: 10, whereas X1_limit,X2_limit,…,X10_limit

Searching region: −30;30,−30;30,…,−30;30

Griewank: Many Local Minima

f(x,y)=1+∑i=1nxi24000−∏i=1ncosxii

Global minimum: f(x*)=f(0,…,0)=0

Dimension: 20, where as X1_limit,X2_limit,…,X20_limit

Searching region: −600;600,−600;600,…,−600;600

Ackley: Many Local Minima

f(x)=f(x1,…,xn)=−a.exp−b1n∑i=1nxi2−exp−b1n∑i=1ncoscxi+a+exp(1)

Global minimum: f(x*)=f(0,…,0)=0

Dimension: 20, whereas X1_limit,X2_limit,…,X20_limit

Searching region: −30;30,−30;30,…,−30;30

Michalewicz: Steep Ridges/Drops

f(x)=fx1,…,xn=−∑i=1nsinxisin2mixi2π

Global minimum: fx*=−9.66015

Dimension: 10, where as X1_limit,X2_limit,…,X10_limit

Searching region: 0;π,0;π,…,0;π

For each Benchmark Test, 50 independent runs are carried out and the statistical results of the best, worst, mean, and standard deviation for four optimization algorithms are shown in Table 3. The bold values show the best value of each case test, which is compared results between classical SPSO and parallel multi-technique PM-PSO, fully presented in the table.

Function		SPSO	PM-PSO
Rosenbrock	Best	4.0068	0.0017
	Worst	3.3315e+4	78.9591
	Mean	215.1817	7.2705
	StdDev	556.9688	12.0427
	Time (s/run)	0.2148	0.1697
Griewank	Best	0.0951	0.2916
	Worst	1.2870	3.5631
	Mean	0.7840	1.7050
	StdDev	0.3560	0.7924
	Time (s/run)	0.4165	0.1905
Ackley	Best	2.7652	3.0496
	Worst	10.2113	8.8752
	Mean	5.7396	6.3287
	StdDev	1.5309	1.3282
	Time (s/run)	0.6841	0.3033
Michalewicz	Best	−8.7674	−9.1561
	Worst	−4.2685	−5.7805
	Mean	−6.3755	−7.3387
	StdDev	1.0549	0.7246
	Time (s/run)	0.4733	0.1767

SPSO, standard particle swarm optimization; PSO, particle swarm optimization

Table 3

The experiment results of testing benchmark functions.

As can be seen from the results shown in Table 3 and Figure 8, the parallel multi-technique applied on PM-PSO yields superior results compared to the classical ones in running time criteria. It is clear that the running time of the heuristic algorithms applied PM Technique is significantly faster than the classical ones. In detail, Rosenbrock function optimized by PSO take 0.21s per run meanwhile it take 0.17s in PM-PSO technique. Same results as Rosenbrock function, Griewank, Ackley, and Michalewicz tested by classical PSO take 0.42s, 0.68s, and 0.47s worse than approximately three times when running by PM-PSO that is 0.19s, 0.30s, and 0.18s, respectively. Abovementioned principle of both algorithms, PSO have to use two “for loop,” one for calculating all member value and one for moving member after that. As indicated in Figure 8, The parallel multi-technique is strongly improved processing time of classical PSO algorithm and the returned results are completely finer which shown in Table 3.

In the case of the optimization of the Griewank using PM-PSO, the searching region [−600; 600] is divided into four sub-regions [[−600; −300]; [−300; 0]; [0; 300]; [300; 600]] according to X1-dimension. Then the global minimum of those benchmark functions is placed on the line that divides original area so this fact means that it is hard to find minimum by standard PSO algorithm. The best value of Griewank found after 50 runs by SPSO worse than running by PM-PSO, nearly three times (0.4165 [sec] compared to 0.1905 [sec]).

5.2. Proposed PM-PSO Method Applied in Optimization of FOC-PID Parameters

For each case, 10 independent classical PSO training tests are investigated and the statistical results related to the J value, training time are completely shown in Tables 4–7.

Number of Calculation Loop	SPSO		Proposed PM-PSO
Number of Calculation Loop	Total Generation	PopSize	Total Generation	PopSize
200	10	24	10	6 × 4
Runtime	J value	Training time (sec)	J value	Training time (sec)
1	6.2170e+03	982.6	5.7058e+03	349.5
2	6.3356e+03	988.6	*4.9523e+03()**	313.2
3	5.7399e+03	995.3	6.4139e+03	303.5
4	5.2306e+03	986.3	5.1691e+03	300.9
5	5.3676e+03	970.5	6.5410e+03	299.9
6	6.0665e+03	*967.5()**	5.7916e+03	*298.7()**
7	5.2380e+03	984.4	7.1795e+03	304.3
8	4.7115e+03	972.0	5.3953e+03	303.1
9	*4.4546e+03()**	981.8	5.0330e+03	304.6
10	5.0771e+03	970.3	5.7747e+03	309.1
Mean	5.4400e+03	979.9	5.8000e+03	308.7

SPSO, standard particle swarm optimization; PSO, particle swarm optimization

Table 4

Case 1 – 10gen24particle.

Number of Calculation Loop	SPSO		Proposed PM-PSO
Number of Calculation Loop	Total Generation	PopSize	Total Generation	PopSize
300	10	48	10	12 × 4
Runtime	J value	Training time (sec)	J value	Training time (sec)
1	5.1414e+03	1969.6	4.9245e+03	604.3
2	5.3568e+03	2051.3	4.6878e+03	600.2
3	4.7614e+03	2081.1	5.1898e+03	601.0
4	5.1346e+03	1974.0	4.5614e+03	604.4
5	4.6158e+03	1966.9	4.4830e+03	603.8
6	5.8932e+03	1965.5	*4.4384e+03()**	*597.7()**
7	*4.4208e+03()**	1966.5	6.5609e+03	603.2
8	4.4246e+03	1999.8	5.6027e+03	601.2
9	6.6357e+03	1971.0	5.2908e+03	601.1
10	4.6925e+03	*1962.4()**	5.1543e+03	599.8
Mean	5.1100e+03	1990.8	5.0900e+03	601.7

SPSO, standard particle swarm optimization; PSO, particle swarm optimization

Table 5

Case 2 – 10gen48particle.

Number of Calculation Loop	SPSO		Proposed PM-PSO
Number of Calculation Loop	Total Generation	PopSize	Total Generation	PopSize
400	10	60	10	15 × 4
Runtime	J value	Training time (sec)	J value	Training time (sec)
1	5.4947e+03	2440.3	4.4346e+03	752.3
2	4.4797e+03	2455.1	*4.4370e+03()**	750.9
3	4.4787e+03	2820.5	5.3184e+03	*743.7()**
4	*4.4246e+03()**	2496.6	4.5327e+03	748.1
5	6.2065e+03	2435.6	5.1911e+03	753.1
6	6.0658e+03	2435.6	5.6657e+03	770.5
7	5.8426e+03	2321.4	4.8502e+03	779.0
8	4.4703e+03	2302.9	4.4719e+03	778.0
9	4.4347e+03	*2291.5()**	5.3862e+03	764.5
10	4.4544e+03	2297.4	4.5680e+03	756.0
Mean	5.0400e+03	2429.8	4.8900e+03	759.6

SPSO, standard particle swarm optimization; PSO, particle swarm optimization

Table 6

Case 3 – 10gen60particle.

Number of Calculation Loop	SPSO		Proposed PM-PSO
Number of Calculation Loop	Total Generation	PopSize	Total Generation	PopSize
400	10	72	10	18 × 4
Runtime	J value	Training time (sec)	J value	Training time (sec)
1	4.8721e+03	2767.6	4.9562e+03	898.3
2	4.5403e+03	2751.7	4.6085e+03	893.9
3	5.1795e+03	2774.0	5.5953e+03	890.1
4	6.0641e+03	2775.9	4.7629e+03	887.6
5	4.5016e+03	2765.3	5.1834e+03	888.0
6	4.4239e+03	2765.8	4.7745e+03	889.7
7	4.9346e+03	2765.6	5.0763e+03	889.2
8	*4.4156e+03()**	*2751.2()**	*4.6017e+03()**	*884.6()**
9	4.4768e+03	2756.5	5.0967e+03	888.7
10	4.4788e+03	2764.3	5.0502e+03	892.1
Mean	4.7900e+03	2763.79	4.9700e+03	890.2

Bold(*) is the best result of each case in SPSO and PM-PSO experiment.

Bold(**) is the best result of J value and training-time in each method.

SPSO, standard particle swarm optimization; PSO, particle swarm optimization

Table 7

Case 4 – 10gen72particle.

The best results of four testing cases shown in Tables 4–7 are concisely tabulated in Table 8 as follows,

	SPSO		PM-PSO		Ratio SPSOrtPMPSOrt
Case	J	Training Time (sec)	J	Training Time (sec)	Ratio SPSOrtPMPSOrt
1	5.4400e+03	979.9	5.8000e+03	308.7	3.174
2	5.1100e+03	1990.8	5.0900e+03	601.7	3.309
3	5.0400e+03	2429.8	4.8900e+03	759.6	3.199
4	4.7900e+03	2763.8	4.9700e+03	890.2	3.105

SPSOrt Training time of SPSO

PMPSOrt Training time of PM-PSO

SPSO, standard particle swarm optimization; PSO, particle swarm optimization

Table 8

Best results of four testing cases.

Table 8 shows that the training time of SPSO implemented on single core takes 979.9s in total by 10 generations and 24 particles meanwhile it takes 1990.8s by 10 generations and 48 particles. It also takes 2429.8s in total by 10 generations and 60 particles. Eventually in the last case it takes 2763.79s by 10 generations and 72 particles. We see that the fitness value decrease significantly when the experiment carried out more number of generations and particles. As presented in Table 8, the proposed PM-PSO method not only generate outperforming results compared to SPSO but also found more fitting solution for optimizing PID parameter due to J value quickly converging to minimum. The experiment results show that PM-PSO providing more precise and stable outcome than SPSO.

In Table 8, the training time of PM-PSO, implemented on 4 cores, is excellently ameliorated and reduced to only 308.7s in case of 10 generations and 6 particles meanwhile it takes 601.7s in case of 10 generations and 12 particles, 759.6s in total in case of 10 generations and 15 particles and eventually the case of 10 generations and 18 particles it takes 890.2s. Using the last “ratio” column of Table 8, which is clear to note that the training time of the optimized PID parameters using SPSO technique on single core are quite worse than the training time of proposed PM-PSO executing on multi-core over 310%.

The results are shown in Tables 4–7. Those experiments show that the mean value of J value reduces gradually when the number of particle increased from 24 to 72 particles. It means the proposed algorithm returned the stable results from independence cases when the number of particles is increasing.

In detail, the resulted optimized PID parameters of three PI blocks, implemented in BLDC FOC-PID controller, using PM-PSO and SPSO are adequately shown in Tables 9 and 10, respectively.

Case	Generation	PopSize	P1	P2	P3	I1
1	10	24	20	1.2460	1	0.9283
2	10	48	0.4108	0.6623	1	10
3	10	60	0	1.2771	1	10
4	10	72	0.5117	0.6619	1	0.4515

PID, proportional integral derivative; SPSO, standard particle swarm optimization.

Table 9

Resulted optimized PID parameters of SPSO.

Case	Generation	PopSize (×4)	P1	P2	P3	I1	I2	I3
1	10	6	0.1362	1.4686	0.4238	8.3042	4.5211	0
2	10	12	0	1.2857	1	0	0	0
3	10	15	0	1.2801	1	0	0	0
4	10	18	0	1.2824	1	0.6217	0	0.6816

PID, proportional integral derivative; SPSO, standard particle swarm optimization.

Table 10

Resulted optimized PID parameter of PM-PSO

Figure 9 shows the step response of the proposed PMPSO-PID and standard PSO-PID FOC BLDC controller for all cases in which the eventual resulted optimized PID parameter using proposed PMPSO and standard PSO shown in Table 11. The best response for each test-case is shown in Figure 9 where the recovery time reaches the reference mechanical angle of BLDC motor around 0.09s. The overshoot of step response is decreased significantly when the number of both generation and particle is increased. Figure 9 shows the step response of the closed loop BLDC system by using proposed parallel PMPSO algorithm that the recovery time reaches the reference mechanical angle of BLDC motor around 0.2s as same rate as one using SPSO algorithm.

		PM-PSO PID	Standard SPSO PID
PI angular velocity	P3	1	1
PI angular velocity	I3	0	0
PI i_q	P2	1.2801	1.2771
PI i_q	I2	0	0
PI i_d	P1	0	0
PI i_d	I1	0	10
Max overshoot	%	23.74	23.85
Steady state theta value	theta	1.049	1.053
Steady state error	e (%)	0.19	0.57

PID, proportional integral derivative; SPSO, standard particle swarm optimization.

Table 11

Eventual resulted optimized PID parameters.

Figure 9 and Table 11 comparatively show the best case of performance responses of closed loop BLDC system between proposed Parallel multi PM-PSO and standard SPSO algorithm. Those results confirm that the proposed parallel multi PM-PSO proves better than standard SPSO algorithm not only in steady-state error but also in transient-time response.

Eventually the comparative convergence curve of proposed parallel multi PM-PSO and standard SPSO algorithm are shown in Figure 10. Those results demonstrate that the proposed parallel multi PM-PSO obtains better fitness value than standard PSO algorithm not only in optimally minimum fitness value but also in number of converging iterations.

6. CONCLUSIONS

In this work, a comparison of using proposed PMPSO-PID and standard SPSO-PID for optimally identifying PID parameters of BLDC FOC-PID controller is satisfactorily shown. Moreover, the well-known benchmark functions such as Rosenbrock, Griewank, Ackley, and Michalewicz are used to test proposed technique. The results show that multi-population technique applied on PM-PSO algorithm achieves competitive performance compared to the standard SPSO one. The results show that the proposed PMPSO-PID controller performs an efficient tuning method for PID parameters. By comparing between proposed PMPSO-PID and standard SPSO-PID, it shows that both of PM-PSO and SPSO algorithm enhance the dynamic performance of the closed loop BLDC system. As a consequent the parallel multi-population (PM) technique applied on PM-PSO optimization algorithm which yields superior results compared to the classical ones in running-time requirement.

CONFLICTS OF INTEREST

The authors have declared that there is no conflicts of interest.

AUTHORS' CONTRIBUTIONS

Nguyen T.D.: original idea, edit the draft, prepare software and source code. Cao V.K.: improve idea, complete source code. Ho P.H.A.: complete the content and edit the final paper.

ACKNOWLEDGMENTS

We acknowledge the support of time and facilities from Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this study.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 1142 - 1154
Publication Date: 2021/03/25
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.210319.001 How to use a DOI?
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ris enw bib

TY  - JOUR
AU  - Nguyen Tien Dat
AU  - Cao Van Kien
AU  - Ho Pham Huy Anh
PY  - 2021
DA  - 2021/03/25
TI  - Optimal FOC-PID Parameters of BLDC Motor System Control Using Parallel PM-PSO Optimization Technique
JO  - International Journal of Computational Intelligence Systems
SP  - 1142
EP  - 1154
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210319.001
DO  - 10.2991/ijcis.d.210319.001
ID  - Dat2021
ER  -

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