An improved quantum particle swarm optimizer for electromagnetic design problem

Abstract

The expansion of global optimum methods for electromagnetic design optimization has been successful in the last few years. However, there is no any universal algorithm to be equally successful for all engineering inverse problems. In this regard, inspired from the classical particle swarm optimization (PSO) method and quantum mechanics, this paper presents an improved quantum particle swarm optimizer (MQPSO) by using a tournament selection strategy. Also, a new index, called the torment best (tbest), is incorporated into the QPSO to further enrich its performance. In addition, a new parameter updating strategy is proposed to tradeoff between the exploration and exploitation searches. The feasibility and merit of the proposed approach are verified by the numerical results on mathematic test functions and an electromagnetic inverse problem, namely the TEAM workshop problem 22.

Keywords

Global optimization electromagnetic design problem particle swarm optimization quantum mechanics

1. Introduction

Global optimization has been a dynamic and modern research area in the last few decades. However, many real world optimization problems are becoming complex that need robust optimization algorithms to work. To facilitate the explanation and description, an unconstrained optimization problem is formulated as an $N$ dimensional minimization problem as follows:

$\displaystyle\min f(x)$ $\displaystyle x=[x_{1},x_{2},...,x_{N}]$

where $N$ represents the number of parameters to be optimized.

Similarly, an electromagnetic design task includes a complex optimization problem, bounded by various competing design specifications and constrains, that can only be solved by using an efficient and robust optimization technique. Moreover, in the electromagnetic design optimization area, most problems can be defined by nonlinear relationship. Thus, to verify the performance and robustness of different optimization methods, a significant benchmark problem is the TEAM problem 22 [1].

Team benchmark workshop problem 22 contains in finding the optimal design of a superconducting magnetic energy storage (SMES) device to store a substantial amount of energy in the magnetic field using a very simple and uncostly coil arrangement that can be simply scaled up in size. Furthermore, the literature includes many references for different optimization methods in solving TEAM problem 22 [2, 3, 4, 5, 6, 7]. As a result, this problem is an ideal model to test different global optimizers.

The particle swarm optimization (PSO) [8] is an addition to the optimization world. It shares several similarities with the evolutionary methods. Moreover, PSO is a global optimization technique based on population in which each member is seen as a particle and each particle is a candidate solution to the optimization problem. In PSO each individual moves within the $N$ dimensional space associated with a velocity. However, the global search ability of a PSO is relatively low. Thus, as the concept of quantum mechanics is incorporated into PSO, it makes PSO a robust global optimizer because the quantum model of PSO can overcome the deficiencies of the traditional PSO [9].

Much efforts have been made to improve the performance of PSO and QPSO as reported. In [10], particle swarm optimization was combined with a none-uniform steady state genetic algorithm. Cheng et al., combined particle swarm with self-adaptive differential evolution for the imaging of a periodic conductor [11]. Particle swarm optimization with the least square method for the identification of axial magnetic bearing systems was presented in [12]. Gan and Zhang, combined a self-adaptive particle swarm (SAPSO) with the least square method to optimize the piezoelectric actuator [13]. A modified PSO was applied to the rectangular shape piezoelectric energy harvesting cantilever beam [14]. In [15], a novel stable deviation based on quantum behaved particle swarm optimization was presented to optimize piezoelectric actuator and sensor location for active vibration control. An improved quantum based particle swarm optimization was proposed based on a novel adaptive hybrid rule network [16]. An automatic image annotation was optimized using an improved quantum particle swarm optimization in [17]. An elitist breeding strategy for unconstrained optimization problems was presented using an improved QPSO method [18]. In [19], an improved QPSO algorithm with the chaotic search method was proposed. Yan et al., proposed a multi-objective quantum particle swarm optimization for electronic nose in wound infection [20]. A modified Quantum-inspired Particle Swarm Optimization (QPSO) algorithm for global optimizations of inverse problems was proposed in [21]. In the proposed algorithm, a new mutation strategy is applied on the personal best particle to improve its global searching ability, also an improved Factor (iF) is incorporated into the position update equation of QPSO to further enhance its convergence speed. In addition, a new parameter updating strategy is proposed to tradeoff between the exploration and exploitation searches.

However, QPSO may encounter a premature convergence as its ancestor of PSO faces when it is used to solve complex multimodal engineering inverse problems. Hence, to improve its performance and avoid from being trapped into local optima, this paper presents an improved quantum particle swarm optimization (MQPSO) using a tournament selection strategy. Also, some strategy to updating the algorithm parameters is proposed.

2. Quantum particle swarm optimizer

Particle swarm optimization imitates the group behavior of bird flocking and fish schooling, and the particles represent points in an $N$ dimensional problem space. A particle represents a potential solution. Consider $N$ dimensional space, the following position vector denotes the time evolution of a set of particles.

$\displaystyle x(t)=[x_{1}(t),x_{2}(t),.....,x_{N}(t)]^{T}$ (1)

The velocity vector is represented by

$\displaystyle v(t)=[v_{1}(t),v_{2}(t),.....,v_{N}(t)]^{T}$ (2)

where $T$ represents the transpose operator. The main idea of the standard PSO is the information exchange about the information for both local and global best particles, which can be done as follows.

$\displaystyle v(t+\Delta t)=wv(t)+c_{1}[\Phi_{1}](p_{\textit{best}}-x(t))+c_{2% }[\Phi_{2}](P_{g}-x(t))$ (3) $\displaystyle x(t+\Delta t)=x(t)+\Delta tv(t)$ (4)

where $\Delta t$ is the time step, $p_{\textit{best}}$ represent the local best vector and $P_{g}$ is the global best vector. The $\Phi_{i}$ represents the elements of the two diagonal matrices which is given as $[\Phi_{i}]=\textit{diag}(\varphi_{1}^{i},\varphi_{2}^{i},......,\varphi_{S}^{i})$ , where $i=$ 1, 2, represent a number of random variables with a uniform distribution between interval 0 and 1, $c_{1}$ and $c_{2}$ are the acceleration constants reflecting the weighting of the stochastic acceleration terms that attract each other towards local best and global best positions respectively, and $w$ is an inertia factor.

The Eq. (3) can be further simplified to

$\displaystyle v(t+\Delta t)=wv(t)+[\Phi_{]}(P-x(t))$ (5)

where,

$\displaystyle P=c_{1}\textit{diag}\left({\frac{\varphi_{1}^{1}}{c_{1}\varphi_{% 1}^{1}+c_{2}\varphi_{1}^{2}},\frac{\varphi_{2}^{1}}{c_{1}\varphi_{2}^{1}+c_{2}% \varphi_{2}^{2}},......,\frac{\varphi_{S}^{1}}{c_{1}\varphi_{S}^{1}+c_{2}% \varphi_{S}^{2}}}\right)p_{\textit{best}}+\,c_{2}\textit{diag}\left({\frac{% \varphi_{1}^{2}}{c_{1}\varphi_{1}^{1}+c_{2}\varphi_{1}^{2}},\frac{\varphi_{2}^% {2}}{c_{1}\varphi_{2}^{1}+c_{2}\varphi_{2}^{2}},......,\frac{\varphi_{S}^{2}}{% c_{1}\varphi_{S}^{1}+c_{2}\varphi_{S}^{2}}}\right)P_{g}$ (6)

and also,

$\displaystyle[\Phi]=\textit{diag}(c_{1}\varphi_{1}^{1}+c_{2}\varphi_{1}^{2},c_% {1}\varphi_{2}^{1}+c_{2}\varphi_{2}^{2},......,c_{1}\varphi_{S}^{1}+c_{2}% \varphi_{S}^{2}).$ (7)

In order for the PSO method to converge, all particles must approach the location $P$ as given by Eq. (6) and the convergence can be obtained by a proper tuning of the two acceleration coefficients of the PSO as shown in [22].

In a typical mechanics standard, a particle is described by its position vector $x$ and velocity vector $v$ that find the trajectory of the particle. In Newtonian method the particle moves along a determined trajectory, but the situation is different in quantum mechanics. In quantum mechanics, the word trajectory is meaningless, because of the uncertainty principle the position vector $x$ and velocity vector $v$ cannot be determined simultaneously. Thus, the PSO method is bound to work in a separate style if an individual particle in a PSO has the quantum behavior [9].

By applying Monte Carlo method, the particle position can be updated by using the following equation:

$\displaystyle x(t+1)=P(t)\pm 0.5\times L(t)\times\ln(1/u)$ (8)

where $u$ represents a random number generated using a uniform distribution within the interval (0, 1).

According to some hypothesis, it is shown [22], that each particle will converge to the following position,

$\displaystyle P=\frac{\varphi_{1}.p_{\textit{best}}+\varphi_{2}.P_{g}}{\varphi% _{1}+\varphi_{2}}$ (9)

where $\varphi_{1}$ and $\varphi_{2}$ are two random numbers with uniform distribution between (0, 1).

The $L(t)$ parameter in Eq. (8) can be calculated from,

$\displaystyle L(t)=2\alpha\left|{P-x(t)}\right|$ (10)

where $\alpha$ is a control parameter known as the contraction expansion coefficient (CE) that can be tuned to control the convergence speed of the algorithm. Then, we will get the position updated equation as follows:

$\displaystyle x(t+1)=P\pm\alpha.\left|{P-x(t)}\right|.\ln(1/u)$ (11)

The global point which is called Mainstream or Mean best $(\textit{Mbest})$ is defined as the average personal best position of all particles in the swarm and is given by

$\displaystyle\left|{\textit{Mbest}}\right|=\frac{1}{M}\sum\limits_{i=1}^{M}{p_% {\textit{best}}}$ (12)

where $M$ is the population size. Thus, the particle will be updated by using the following equation:

$\displaystyle x(t+1)=P\pm\alpha.\left|{\textit{Mbest}-x(t)}\right|.\ln(1/u)$ (13)

The PSO algorithm based on Eq. (13) is generally known as a quantum behaved particle swarm optimization (QPSO).

3. An improved QPSO method

3.1 Introduction of tournament selection strategy

The QPSO algorithm has a strong global searching ability as compared to PSOs, however, still the QPSO algorithm may encounter a premature convergence when it is used to solve complex engineering design problems at the later stages of the optimization process. This is because at the initial evolution process the diversity of the population is high but later on it reduces rapidly. One of the facts is an improper balance between exploration and exploitation searches. To avoid premature convergence and to bring a good balance between exploration and exploitation searches, some modifications are proposed on the QPSO, and explained in the following paragraphs.

In a QPSO, the $p_{\textit{best}}$ , which has a better objective value, is always updated and then used in the updated equation of the position. However, the others ( $p_{\textit{best}}$ ) are overlapped (neglected). It means that all other $p_{\textit{best}}$ , previously achieved could not be referred back. Therefore, there is no information sharing from these best individuals that have been achieved at the preceding iterations. Nevertheless, by using the term of a “history best” concept, the previous knowledge and information about what has happened previously is the only guide for an accurate search. In this context, the $p_{\textit{best}}$ , is randomly selected from all personal best particles of the current population and is used to update the particle position. This new index terms “tournament best”, tbest, and is incorporated into the position updated equation as:

$\displaystyle p_{i}=\frac{\varphi_{1}\cdot P_{L}^{+}\varphi_{2}\cdot P_{g}+% \textit{tbest}}{2\cdot\varphi_{2}}$ (14)

where $\varphi_{1}$ and $\varphi_{2}$ are two uniform random numbers within the range of [0,1] and tbest is the new particle chosen by using a tournament selection strategy. The tbest is selected based on the fitness values of $p_{\textit{best}}$ . While $P_{L}$ and $P_{g}$ represent the particles with personal and global best positions respectively. The tournament selection is a method of selecting an individual from a population of individuals in a genetic algorithm. Tournament selection involves running several “tournaments” among a few individuals (particles) chosen at random from the population. The winner of each tournament (the one with the best fitness) is selected. Thus, for a minimization problem, the tournament selection procedure is explained as:

First choose two particles randomly from the history of the personal best particles $p_{\textit{best}}$ of the current population.

Compare the fitness values of these two particles and select the better one as tbest.

Thus, the proposed selection methodology will guarantee that the diversity of the population is conserved to avoid being trapped into local optima.

3.2 Parameter updating rule

In a QPSO, the contraction expansion coefficient, $\alpha$ , is an important parameter for the performance of QPSO algorithm, and has a significant impact on the global search ability and convergence of particles.

Thus, if the value of $\alpha$ is constant, the balance between local and global searches will be disturbed and the particles could not find the global optimum when dealing with complex multimodal and engineering inverse problems. However, it is also stated that the local and global searches need a minimum and maximum values of $\alpha$ . It means that a constant contraction coefficient parameter will result the algorithm to be trapped to local optima while dealing with complex optimization and inverse problems. Thus, it is clear that without proper adjustment of $\alpha$ will result in premature convergence.

Therefore, different researchers have proposed different strategies to adjust the contraction coefficient parameter and control the convergence speed of the algorithm [23]. The most common value for $\alpha$ in the literature is to set it initially to 1.0 and then reduced it linearly to 0.5.

Thus, in order to tradeoff between exploration and exploitation searches, and to speed up the convergence process of the proposed approach, a new parameter updating formula is proposed as:

$\displaystyle\alpha=\cos\left({\frac{t}{2}-\textit{rand}}\right)\times(\textit% {MaxIter}-t)/\textit{MaxIter}$ (15)

where rand is a uniform random number within the interval of [0, 1], MaxIter is the maximum number of iterations and $t$ is the current iteration.

Obviously, on the bases of cosine function values, the contraction coefficient $\alpha$ also varies dynamically during the optimization process to guarantee a good balance between the exploration and exploitation searches.

4. Numerical validations

4.1 Mathematical tests

To evaluate the performance of the proposed MQPSO along with standard QPSO [9], the original PSO [8], the Gaussian Quantum Behaved Particle Swarm Optimization approaches for constrained engineering design problems (GQPSO) [24] and an improved Quantum Behaved Particle Swarm Optimization algorithm with an weighted mean best position (WQPSO) [25], seven well known benchmark functions, as shown in Table 1, are used.

In the experimental studies, each experiment is performed 30 independent trials and the final outcomes are recorded in Table 2. In this case study, the population size is 80 for 30 dimension problems with a corresponding number of generations of 2000. The minimum objective function value for all functions is zero. Moreover, Figs 1 $\sim$ 3 give the convergence comparison of the proposed method with other optimal approaches (30 independent runs) in logarithmic scale of the best objective function on some well-known functions using a population size of 80 with number of generations being 2000 for the corresponding dimension of 30.

Table 1
Standard benchmark functions

Function	Expression	Search domain
Sphere	$f_{1}(x)=\sum\limits_{i=1}^{D}{x_{i}^{2}}$	( $-$ 100, 100)
Rosenbrock	$f_{2}(x)=\sum\limits_{i=1}^{D}(100.(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2})$	( $-$ 100, 100)
Griewank	$f_{3}(x)=\frac{1}{4000}\sum\limits_{i=1}^{D}{x_{i}^{2}-\prod\limits_{i=1}^{D}{% \cos(\frac{xi}{\sqrt{i}}})+1}$	( $-$ 600, 600)
DeJong’s	$f_{4}(x)=\sum\limits_{i=1}^{D}{i\cdot x_{i}^{4}}$	( $-$ 100, 100)
Schwefel22	$f_{5}(x)=\sum\limits_{i=1}^{D}{\left\|{x_{i}}\right\|}+\prod\limits_{i=1}^{D}{% \left\|{x_{i}}\right\|}$	( $-$ 100, 100)
Rastrigin	$f_{6}(x)=\sum\limits_{i=1}^{D}{[x_{i}^{2}-10\cos(2\pi x_{i})+10]}$	( $-$ 5.12, 5.12)
Ackley	$f_{7}(x)=-a\exp\left({-b\sqrt{\frac{1}{D}}\sum\limits_{i=1}^{D}{x_{i}^{2}}}% \right)-\exp\left({\frac{1}{D}\sum\limits_{i=1}^{D}{\cos(c\cdot x_{i})}}\right% )+a+\exp(1)$	( $-$ 32, 32)

Table 2

Mean (first row) and Variance (second row) of different optimal algorithms for 30 dimensional problems

Function	PSO	QPSO	GQPSO	WQPSO	MQPSO
$f_{1}(x)$	4.542 $\times$ 10 ${}^{-3}$	8.1044 $\times$ 10 ${}^{-51}$	3.3864 $\times$ 10 ${}^{-30}$	3.201 $\times$ 10 ${}^{-54}$	4.955 $\times$ 10 ${}^{-274}$
	3.937 $\times$ 10 ${}^{-5}$	2.9486 $\times$ 10 ${}^{-87}$	8.7833 $\times$ 10 ${}^{-56}$	2.014 $\times$ 10 ${}^{-97}$	4.3827 $\times$ 10 ${}^{-373}$
$f_{2}(x)$	70.293	40.927	26.527	50.0236	27.328
	47.682	31.443	0.2895	0.5231	4.9770
$f_{3}(x)$	3.531 $\times$ 10 ${}^{-2}$	8.1286 $\times$ 10 ${}^{-3}$	1.1084 $\times$ 10 ${}^{-10}$	4.3621 $\times$ 10 ${}^{-4}$	6.492 $\times$ 10 ${}^{-12}$
	2.768 $\times$ 10 ${}^{-3}$	9.5644 $\times$ 10 ${}^{-5}$	6.0711 $\times$ 10 ${}^{-16}$	1.2351 $\times$ 10 ${}^{-7}$	7.59254 $\times$ 10 ${}^{-21}$
$f_{4}(x)$	9.631 $\times$ 10 ${}^{-14}$	6.3227 $\times$ 10 ${}^{-65}$	1.6402 $\times$ 10 ${}^{-40}$	7.215 $\times$ 10 ${}^{-67}$	5.6728 $\times$ 10 ${}^{-291}$
	7.519 $\times$ 10 ${}^{-19}$	2.6123 $\times$ 10 ${}^{-87}$	8.9837 $\times$ 10 ${}^{-73}$	8.023 $\times$ 10 ${}^{-108}$	2.7962 $\times$ 10 ${}^{-394}$
$f_{5}(x)$	6.872 $\times$ 10 ${}^{-2}$	1.2731 $\times$ 10 ${}^{-14}$	2.6645 $\times$ 10 ${}^{-15}$	4.180 $\times$ 10 ${}^{-27}$	7.7275 $\times$ 10 ${}^{-217}$
	8.946 $\times$ 10 ${}^{-4}$	3.2432 $\times$ 10 ${}^{-23}$	2.2853 $\times$ 10 ${}^{-27}$	8.934 $\times$ 10 ${}^{-41}$	4.9823 $\times$ 10 ${}^{-376}$
$f_{6}(x)$	27.382	19.491	3.91 $\times$ 10 ${}^{-8}$	7.54 $\times$ 10 ${}^{-11}$	9.53 $\times$ 10 ${}^{-15}$
	14.936	4.977	6.78 $\times$ 10 ${}^{-14}$	6.92 $\times$ 10 ${}^{-19}$	7.29 $\times$ 10 ${}^{-27}$
$f_{7}(x)$	5.26 $\times$ 10 ${}^{-4}$	3.97 $\times$ 10 ${}^{-14}$	1.09 $\times$ 10 ${}^{-12}$	7.62 $\times$ 10 ${}^{-9}$	3.84 $\times$ 10 ${}^{-17}$
	8.51 $\times$ 10 ${}^{-7}$	4.38 $\times$ 10 ${}^{-25}$	2.30 $\times$ 10 ${}^{-19}$	5.63 $\times$ 10 ${}^{-16}$	9.54 $\times$ 10 ${}^{-29}$

Figure 1.

Convergence comparison of different optimal algorithms for solving $f_{1}$ .

Figure 2.

Convergence comparison of different optimal algorithms for solving $f_{3}$ .

Figure 3.

Convergence comparison of different optimal algorithms for solving $f_{5}$ .

Moreover, to facilitate the performance comparisons of different optimal methods, we categorize the algorithms into three categories:

Category A: an algorithm is called category A algorithm if its final solution is significantly improved as compared to other tested variants; Category B: an algorithm is called a category B algorithm if its final solution has slight improvements compared to other tested variants; Category C: an algorithm is called a category C algorithm if its final solution is poor as compared to that of its counterpart.

Thus, the performances of different stochastic algorithms are compared in Table 2. It is obvious from Table 2, that: Our proposed MQPSO is a category A algorithm for test functions $f_{1},f_{3},f_{4},f_{5},f_{6}$ and $f_{7}$ , because for all these test functions the MQPSO demonstrates better performance as compared to other stochastic methods; WQPSO is a category B algorithm for test functions $f_{1}$ , $f_{4}$ , $f_{5}$ and $f_{6}$ as compared to other tested optimal methods and category C on test function $f_{2}$ , $f_{3}$ and $f_{7}$ . However, it does not display better performance comparable to MQPSO; also the GQPSO is a category A algorithm for test functions $f_{2}$ and category B on $f_{3}$ , while on remaining test functions $f_{1}$ , $f_{4}$ , $f_{5},f_{6}$ and $f_{7}$ , it does not shows any improvement and placed in category C. The proposed MQPSO is a Category B algorithm only on test function $f_{2}$ . Similarly, the original QPSO algorithm is a category B one on test function $f_{7}$ and category C on remaining test functions $f_{1}$ , $f_{2},f_{3},f_{4},f_{5}$ and $f_{6}$ . However, the PSO algorithm is a category C algorithm on all tested functions $f_{1}$ to $f_{7}$ , because it does not illustrate any improvement on its counterpart.

Thus, one can conclude from the performances of different optimal algorithms in Table 2 on 30 dimensional problem, that:

The proposed MQPSO is a category A algorithm for test functions $f_{1}$ , $f_{3}$ , $f_{4}$ , $f_{5}$ , $f_{6}$ and $f_{7}$ , among all algorithms;

The WQPSO is a category A algorithm for test functions $f_{1}$ , $f_{4}$ , $f_{5}$ and $f_{6}$ as compared to the GQPSO, QPSO and PSO but excluding MQPSO.

The GQPSO is a category A algorithm on test function $f_{2}$ , category B on $f_{3}$ and category C on remaining test functions i.e. $f_{1}$ , $f_{4}$ , $f_{5}$ , $f_{6}$ and $f_{7}$ as compared to other stochastic algorithms.

The original QPSO is a category A algorithm on test function $f_{7}$ as compared to PSO, WQPSO and GQPSO but excluding MQPSO.

The QPSO and GQPSO algorithms fall in category C on test function $f_{1}$ , $f_{4},f_{5}$ and $f_{6}$ .

The PSO algorithm is a category C on all tested functions $f_{1}$ to $f_{7}$ .

Thus, from the above discussion and statistical analysis, it is observed that: 1) the proposed MQPSO method found an appropriate mean behavior in approximately the initial generations on most tested functions during the evolution process while all other well designed optimal methods stuck in local optima and 2) the quality of the final solution searched by the proposed MQPSO method is significantly higher than those of other four optimal stochastic algorithms. Nevertheless, the proposed MQPSO is a global optimizer on most of the tested functions.

4.2 Numerical application

To validate further the feasibility of the proposed MQPSO approach, and to show its merit and efficiency for solving an engineering inverse problem in electromagnetics, the proposed approach is then applied to a well-known benchmark problem, TEAM workshop problem 22.

As explained previously, TEAM workshop problem 22 is a benchmark problem for testing different optimization techniques. The problem is a SMES (superconducting magnetic energy storage system) design optimization as shown in Fig. 4 and reported in [1, 2, 3, 4, 5, 6, 7, 21].

Figure 4.

Schematic diagram of SMES.

The system consists of two concentric coils that are the inner main solenoid and outer shielding solenoid to reduce the stray field. The directions of currents in the two coils are opposite to each other. The goal of the design is to achieve a desired store energy with a minimal stray field. Hence, the design should fulfil: 1) the energy stored in the device should be 180 MJ; 2) the magnetic field produced inside the solenoid must not violate certain physical conditions to ensure superconductivity; and 3) the mean stray field at 21 measurement points along lines $A$ and $B$ at a distance of 10 meters should be as small as possible [26].

To guarantee the superconductivity of the conductors, the constraint equation stipulates the current density of two solenoids and the magnetic flux density as follows:

$\displaystyle J_{i}\leqslant(-6.4\left|{(B_{\max})_{i}}\right|+54,(i=1,2)$ (16)

where $J_{i}$ and $(B_{\max})_{i}$ are the current density and maximum magnetic flux density in the $i^{\text{th}}$ coil, respectively.

In the three parameters problem, the dimensions of the outer solenoid are optimized under 2.6 m $<r_{2}<$ 3.4 m, 0.204 m $<h_{2}/2<$ 1.1 m, 0.1 m $<d_{2}<$ 0.4 m while the inner solenoid is fixed at $r_{1}=$ 2 m, $h_{1}/2=$ 0.8 m, $d_{1}=$ 0.27 m. Moreover, the current densities of the coils are set to be 22.5 A/mm ${}^{2}$ Hence, for the convenience of mathematical implementation, Eq. (16) is simplified to $\left.{B_{\max}}\right|\leqslant$ 4.92 T. Under the previous simplifications, mathematically, the optimization problem is defined as:

$\displaystyle\min OF=\frac{B_{\textit{stray}}^{2}}{B_{\textit{normal}}^{2}}+% \frac{\left|{\textit{Energy}-E_{\textit{ref}}}\right|}{E_{\textit{ref}}}$ (17)

where the reference stored energy is $E_{\textit{ref}}=$ 180 MJ, $B_{\textit{normal}}=$ 3 $\mu$ T. $B_{\textit{stray}}^{2}$ is computed from:

$\displaystyle B_{\textit{stray}}^{2}={\sum\limits_{i=1}^{21}{\left|{B_{\textit% {stray},i}}\right|^{2}}}/21$ (18)

where $B_{\textit{stray},i}$ is the magnetic flux density at the $i^{\text{th}}$ equidistant pointe along lines $A$ and $B$ , as shown in Fig. 4.

In the numerical study, the performance parameters as required by Eqs (17) and (18) are determined based on two dimensional finite element computations.

To compare the performance of proposed MQPSO with other well designed stochastic approaches, this case study is solved using the original QPSO [9], PSO [8], GQPSO [24] and WQPSO [25]. Also, the results obtained by other three optimal methods DE2 [27] and ARDGDE2 [27] and a Modified QPSO [28] have been taken from the literature for comparisons. Moreover, the maximum number of generations is set to 50 with corresponding population size of 15 with 10 independent runs for each optimal algorithm. Table 3 compares the final results obtained by using different stochastic approaches.

Table 3

Performance comparison of different optimal algorithms on TEAM workshop problem 22

Algorithms	Min (best)	Mean	Max	SD	$r_{2}$	$h_{2}/2$	$d_{2}$
PSO	0.1487	0.1639	0.1823	0.1107	3.2530	0.2246	0.3804
WQPSO	0.1251	0.1371	0.1541	0.01221	3.1172	0.3214	0.2942
GQPSO	0.1222	0.1282	0.1472	0.01397	3.1723	0.2319	0.3892
QPSO	0.1077	0.1189	0.1348	0.01284	3.0786	0.2414	0.3795
DE2	0.3967	5.4716	19.551	6.4238	1.8000	1.1986	0.3801
ARDGDE2	0.2296	2.2967	8.1377	2.5668	1.8424	0.9577	0.4184
Modified QPSO	0.0903	0.0942	0.1293	0.01260	3.1198	0.3008	0.3079
MQPSO	0.0694	0.0819	0.0938	0.00126	3.0942	0.2404	0.3925

From the numerical results on this case study, it is obvious that the proposed method outperforms all other well designed stochastic approaches in terms of both convergence speed (iterations) and solution quality (objective function value).

Thus, one can reveal that the proposed MQPSO has formed better outcomes by using almost the same computational cost (number of generations).

5. Conclusion

An improved QPSO method for global optimizations of inverse problems is presented in this work. Experimental results on standard mathematical test functions and an engineering inverse problem have been reported to validate the effectiveness and merit of the proposed method. The proposed method has a significant advantage of having a faster convergence speed as compared to other tested optimal approaches. Moreover, the results of the proposed MQPSO approach are very satisfying and may represent an important contribution to enhance the QPSO performance for other electromagnetic problems. As for future studies, it is important to investigate other optimal approaches in order to solve electromagnetic design problems.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under Grants No. 51377139 and the Science Technology Department of Zhejiang Province under Grants No. 2016C31G2010049.

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An improved quantum particle swarm optimizer for electromagnetic design problem

Abstract

Keywords

1. Introduction

2. Quantum particle swarm optimizer

3.1 Introduction of tournament selection strategy

4.1 Mathematical tests

Table 1 Standard benchmark functions

Footnotes

Acknowledgments

References

Table 1
Standard benchmark functions