A modified quantum particle swarm optimizer applied to optimization design of electromagnetic devices

Abstract

Quantum particle swarm optimization (QPSO) is a swarm intelligence method that has been successfully applied to solve a wide scope of electromagnetic inverse problems. The method encounters into local optima and insufficient diversity at the later phase of optimization. To address this type of issue, a new methodology is used to select the fittest particle, and a novel mutation mechanism is introduced, in which a mutation technique is applied on the global best particle to avoid the population from assembling and facilitating the individual to avoid the local optimum easily. In addition, a parameter updating strategy is proposed, which facilitates the optimizer to maintain a good balance between local and global searches. To demonstrate the merit and efficiency of the proposed methodology, the evaluated results from the case studies are presented.

Keywords

Electromagnetic design problem global optimization mutation quantum mechanics

1. Introduction

The design of electromagnetic devices has been of particular importance in the area of electromagnetic optimization, especially with the development of modern technology and increased growth of numerical examining machinery.

The optimization design of electromagnetic devices usually includes the characteristics of the optimization process, e.g. the limitation of constraint condition, the uncertainty of the outcomes, and the optimization method. Hence, when the optimization problem cannot be statistically characterized as continuous and differentiable functions, it is unsuitable to use the deterministic numerical approach.

In the last few years, evolutionary algorithms such as the tabu search method, ant colony optimization, genetic algorithm, and differential evolution method, have become prevalent in the field of optimization and have been efficiently applied to a wide scope of electromagnetic design optimizations. As, compared to traditional single point optimization techniques, evolutionary algorithms are population based stochastic methods which are characterized by their capability to determine the global optimal point in a short period, particularly when the objective functions are stochastic. There is no global stochastic method that can be applied successfully to all electromagnetic optimization problems. Consequently, it is important to research and develop new universal optimization methods to overcome electromagnetic design problems.

The particle swarm optimization (PSO) is a swarm intelligence optimization technique that was originated in 1995 [1]. The particle swarm optimizer is a simplistic concept that is easy to implement. Nevertheless, as compared to other well-developed stochastic methods, the PSO algorithm is an evolving methodology. E.g., the PSO optimizer is restricted by premature convergence while finding the global optima of a complex optimization problems and trapped to local minima. To address such issues in PSO, a quantum-based particle swarm optimizer (QPSO) was developed in [2]. In Quantum particle swarm optimizer, the activities of the individuals follow the principle of quantum mechanics, as compared to the principle of Newtonian mechanics that supposed in particle swarm. Hence, instead of the Newtonian method, a quantum-based motion is enforced in the evolution method of QPSO to guarantee that the individuals can be seems in any position and keep a balance between local and global searches. Though, there are still many concerns with QPSO that need to be solved.

In this regard, much effort has been done to improve the performance of QPSO and has successfully applied to many electromagnetic problems as reported. In [3], particle swarm optimization is combined with none-uniform steady state genetic algorithm. Cheng et al, have combine particle swarm with differential evolution for the imaging of a periodic conductor [4]. A novel particle swarm method with least square for the identification of axial magnetic bearing system was presented in [5]. Gan et al, have concatenate self-adaptive particle swarm (SAPSO) with least square to optimize the piezoelectric actuator [6]. A modified PSO is applied to the rectangular shape piezoelectric energy harvesting cantilever beam [7]. A new stable deviation based on quantum particle swarm optimization was presented in [8] to optimize the piezoelectric actuator for active vibration control.

Nevertheless, according to no free lunch theorem, there is no any optimization technique that solve all electromagnetic optimization problems. However, most of these techniques are problem oriented. Thus, there is a need to search and develop a new technique for the optimization of electromagnetic devices. In this context, an improved version of QPSO is proposed in this work, the improved version includes a novel methodology for the selection of the personal best particle and a new mutation method is applied to the global best particle. In addition, a new parameter updating rule is proposed that will improves QPSO, this enhances the convergence process and intensify the performance of the QPSO method. The proposed method is named as modified quantum particle swarm optimizer (MQPSO).

2. Quantum particle swarm optimization

Particle swarm optimizer as comparable to several evolutionary algorithms, it works with population, the population is known as a swarm and each individual is referred as a particle. In the particle swarm algorithm, each individual move around the search area looking for suitable space by its own experience and those of the group. As, a result information will share that will benefit the particles from the origination and previous experiences of all other particles in a large area during the evolutionary stage.

Let us consider an N dimensional optimization problem, the time evolution of a population consists of Ms individuals that is referred by a position vector, x _m = (x _m,1, x _m,2, …, x _{m, N}), and a velocity vector, v _m = (v _m,1, v _m,2, …, v _{m, N}), at each generation ki, the position and velocity of a particle m are updated, defined as, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle v_{m}(ki+1)=wi\cdot v_{m}(ki)+a_{1}\cdot rd_{1}\cdot (p_{best}^{m}-x_{m}(ki))+a_{2}\cdot rd_{2}\cdot (p_{g}^{m}-x_{m}(ki))\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle x_{m}(ki+1)=x_{m}(ki)+v_{m}(ki+1)\end{eqnarray}$ (2) where m = 1,2, …, Ms represents the number of individuals of population, $p_{best}^{m}=(p_{best,1}^{m},p_{best,2}^{m},\ldots ,p_{best,N}^{m})$ is the local best position that particle m has ever achieved, $p_{g}^{m}=(p_{g,1}^{m},p_{g,2}^{m},\ldots ,p_{g,N}^{m})$ is the global best particle that particle m and its neighbor have ever found, a ₁ and a ₂ are the two acceleration coefficients respectively, rd ₁ and rd ₂ are the random numbers within range [0,1], wi is the inertia weight and ki represents the number of iteration.

To ensure the convergence of the particle swarm optimizer, trajectory analysis [9] exhibits, that each individual will converge to its center point P _m = (P _m,1, P _m,2, …, P _{m, N}), of which the coordinates are denoted as, $\begin{eqnarray}\displaystyle p_{m}=(c_{r1}\cdot p_{best}+c_{r_{2}}\cdot p_{g})/(c_{r1}+c_{r2}) & & \displaystyle\end{eqnarray}$ (3) or $\begin{eqnarray}\displaystyle p_{m}={\varphi}\cdot p_{best}+(1-{\varphi})\cdot p_{g} & & \displaystyle\end{eqnarray}$ (4) where 𝜑 is a random number within interval [0,1].

According to Monte Carlo technique, the position of an individual is denoted as, $\begin{eqnarray}\displaystyle x_{m}=p_{m}\pm \frac{L_{m}}{2}\ln (1/ra) & & \displaystyle\end{eqnarray}$ (5) where ra is a uniform random number within range [0,1]. The parameter L _m is determined by, $\begin{eqnarray}\displaystyle L_{m}=2{\alpha}\cdot |Mbest-x_{m}(ki)| & & \displaystyle\end{eqnarray}$ (6) where Mbest is called the mean best position, that is the average of the personal best positions of all individuals (p _best). Denoted by, $\begin{eqnarray}\displaystyle Mbest=\frac{1}{Ms}\mathop{\sum }_{m=1}^{Ms}p_{best}, & & \displaystyle\end{eqnarray}$ (7) where Ms represent the population size. Thus, the particles position will be updated accordingly, $\begin{eqnarray}\displaystyle x_{m}(ki+1)=p_{m}\pm {\alpha}\cdot |Mbest-x_{m}(ki)|\ln (1/ra) & & \displaystyle\end{eqnarray}$ (8) where 𝛼 is a contraction expansion parameter, that is used to control the convergence behavior of the optimization technique generally represented as, $\begin{eqnarray}\displaystyle {\alpha}=(1.0-0.5)\times \frac{(Mx-ki)}{Mx}+(0.5) & & \displaystyle\end{eqnarray}$ (9) where Mx is the maximum iteration and ki is the current iteration.

Hence, the PSO method combined with equation (8) is called the (QPSO).

3. The proposed MQPSO method

3.1. Selection of the fittest particle

In the QPSO optimizer, the diversity of the swarm is high at the initial phase of the optimization process. However, at the later phase of the search process, the convergence of the particles makes the diversity reduced rapidly, this increases the local search capability but decline the global search capability of the optimization technique. When the diversity of the individual becomes low during the evolution process, individuals may converge into a small area making further searches impossible. At this instance, if the individual has not attained the global best position and is at local minima then premature convergence happens.

Thus, to escape from such issues and to expand the algorithms efficiency, modification is made to the QPSO as follows.

Firstly, a new individual is produced in the current search domain by the following methodology: $\begin{eqnarray}\displaystyle p_{b2}(ki)=x_{m}(ki)-f(p_{g})/f(p_{best}) & & \displaystyle\end{eqnarray}$ (10) where f represents the fitness of the individual in issue, p _best and p _g are the particles with the personal and global best positions respectively, and x _m(ki) is the position vector of the current generation.

The p _b2 individual is then compared with the personal best particle (p _best) in the current swarm. If the p _b2 particle has a better fitness than the p _best particle, then p _b2 is replaced by p _best; otherwise, the p _best particle remains in the same position for the next generation cycle in the updating process.

The mean best position is the average position of the personal best particles. So, when the diversity is low then this methodology will select a fittest personal best particle that will affect the Mean best position (Mbest) and will reset it. The main cause for resetting the Mean best position (Mbest) is that to increase the gap between mean best position and current particle, and in this way, will avoid from local minima and intensify the algorithm stability.

3.2. Mutation mechanism

Secondly, a new mutation mechanism is introduced by applying the following mutation mechanism on the individual having the global best position (p _g), defined as, $\begin{eqnarray}\displaystyle G_{best}=z\times p_{g}\times (1+E_{1}) & & \displaystyle\end{eqnarray}$ (11) where G _best is a new global best particle, E ₁ is an exponential random number, z is a user defined constant that is set to be 0.1 by a trial and error method, having undertaken a series of comprehensive experiments and emulated numerous case studies.

When the proposed mutation strategy is applied, the displacement of the new best particle (G _best) will expand the gap between the personal best particles (p _best) and the mean best position (Mbest), and hence, increase the diversity. In this context, the displaced new global best particle (G _best) will replaced the mean best from its original position, which increases the distance between the mean best position (M _best) and the current particles. This enhances the mechanisms search limit, the algorithm will avoid being trapped into local minima, and consequently enhances the algorithm performance.

3.3. Parameter updating strategy

The contraction expansion coefficient 𝛼 is an important coefficient for the performance of the QPSO method and has a significant impact on the global searching capability and convergence behavior of the optimizer. Hence, if the value of 𝛼 is constant, the balance between local and global searches will not be maintained. As a direct result, the particles would not achieve a global optimum solution.

However, it is also stated that the exploitation and exploration searches require a minimum and maximum value as a control parameter. It means that a constant value of 𝛼 parameter will prevent particles from being trapped into local optima, when dealing with hard optimization problems. Consequently, without proper adjustment of the 𝛼 parameter, premature convergence will occur.

Many researchers have proposed different strategies to adjust the value of 𝛼 parameter and increase the convergence speed of the optimizer [9,10]. The most general value for 𝛼 parameter in the literature is to set it initially to 1 and then reduce linearly to 0.5.

In this work, to bring stability between the exploitation and exploration searches, and to intensify the convergence speed of the algorithm, a new parameter updating formula is proposed which will be further incorporated into QPSO updated equation, defined as, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}=(0.5)\times (1-ki/Mx)^{q}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle q=\exp (ki-rand)\end{eqnarray}$ (13) where rand is a random number within range [0,1], ki is the current iteration, and Mx is the maximum iterations.

On the bases of the parameter updating equation (12), the values of 𝛼 vary dynamically during the search process to ensure a tradeoff between the local and global searches.

4. Numerical validation

4.1. Mathematical test

To validate the usefulness and the global search capability of the MQPSO, firstly it has been applied to several shifted version of the mathematical test functions [11]. The detail of these functions is presented in the following paragraph. $\begin{eqnarray}f_{1}(y)=\mathop{\sum }_{m=1}^{N}\left(\mathop{\prod }_{j=1}^{m}s_{j}\right)^{2}+b_{1},s=y-o,y=[y_{1},y_{2},\ldots ,y_{N}],y\in [-100,100]^{N}\end{eqnarray}$ global optimum: y ^∗ = o, f ₁(y ^∗) = b ₁ = −450. $\begin{eqnarray}\displaystyle f_{2}(y) & = & \displaystyle \mathop{\sum }_{m=1}^{N}(100.(s_{m+1}-s_{m}^{2})^{2}+(s_{m}-1)^{2})+b_{2},s=y-o+1,y=[y_{1},y_{2},\ldots ,y_{N}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle y\in [-100,100]^{N},\nonumber\end{eqnarray}$

Global optimum: y ^∗ = o, f ₂(y ^∗) = b ₂ = 390 $\begin{eqnarray}\displaystyle f_{3}(y) & = & \displaystyle \frac{1}{4000}\mathop{\sum }_{m=1}^{N}s_{m}^{2}-\mathop{\prod }_{m=1}^{N}\cos \left(\frac{s_{m}}{\sqrt{m}}\right)+1+b_{3},s=y-o,y=[y_{1},y_{2},\cdots ∼,y_{N}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle y\in [-600,600]^{N},\nonumber\end{eqnarray}$

Global optimum: y ^∗ = o, f ₃(y ^∗) = b ₃ = 390.

The proposed MQPSO is evaluated against the original QPSO [2], GQPSO [13], and LIQPSO [14].

Each of these functions used in this case study are minimization problems and the minimum value for each objective function is zero. Table 1 tabulate the average performance comparison of different optimal algorithms. Moreover, Figs 1–3 give the convergence comparison of the proposed MQPSO method with other comparable optimal approaches (30 independent runs) in logarithmic scale of the best objective function on some standard benchmark functions using a population size of 80 with number of generations being 2000 for corresponding dimension of 30.

Table 1
Mean (first row) and Variance (second row) of different stochastic methods

Algorithms f ₁(y) f ₂ (y) f ₃ (y)

QPSO 5.9342 × 10⁻⁶ 0.1376 3.0716 × 10⁻²

2.8627 × 10⁻¹⁰ 3.8629 × 10⁻² 4.8620 × 10⁻³

LIQPSO 1.23924 5.09190 2.9314 × 10⁻²

0.38211 0.52149 7.6824 × 10⁻⁴

GQPSO 1.9307 8.9316 1.8028 × 10⁻¹

0.2683 0.1308 1.2303 × 10⁻²

MQPSO 1.1339 × 10⁻⁷ 7.8273 × 10⁻² 5.8461 × 10⁻³

2.7820 × 10⁻¹³ 5.7432 × 10⁻⁵ 1.6560 × 10⁻⁷

Algorithms	f ₁(y)	f ₂ (y)	f ₃ (y)
QPSO	5.9342 × 10⁻⁶	0.1376	3.0716 × 10⁻²
	2.8627 × 10⁻¹⁰	3.8629 × 10⁻²	4.8620 × 10⁻³
LIQPSO	1.23924	5.09190	2.9314 × 10⁻²
	0.38211	0.52149	7.6824 × 10⁻⁴
GQPSO	1.9307	8.9316	1.8028 × 10⁻¹
	0.2683	0.1308	1.2303 × 10⁻²
MQPSO	1.1339 × 10⁻⁷	7.8273 × 10⁻²	5.8461 × 10⁻³
	2.7820 × 10⁻¹³	5.7432 × 10⁻⁵	1.6560 × 10⁻⁷

Fig. 1.

Convergence comparison of different optimal algorithms for solving f ₁ function.

Fig. 2.

Convergence comparison of different optimal algorithms for solving f ₂ function.

Fig. 3.

Convergence comparison of different optimal algorithms for solving f ₃ function.

On the three shifted problems, MQPSO and the original QPSO performs better. The original QPSO significantly improved its performance especially on f ₁ and f ₃ as compared to other tested optimal methods. The proposed MQPSO beats all the tested algorithms on f ₁ to f ₃. However, LIQPSO shows better performance on f ₃ as compared to QPSO and GQPSO. The GQPSO algorithm is completely fails and could not generated good results on the shifted problems and stuck into local optimum point.

Comparing the results and convergence plots among these four QPSO algorithms. In this context, the proposed MQPSO found an appropriate mean behavior in approximately initial generations on most of the tested functions during the search process while all other optimal methods stuck into local minima. Thus, the convergence plots also demonstrate that the convergence speed of the proposed MQPSO method is very fast and the proposed MQPSO is a global optimizer on many tested functions. LIQPSO converges faster than GQPSO and original QPSO. However, the original QPSO and GQPSO yield to a balanced performance between the local and global versions. Thus, among the four algorithms, MQPSO has perform significantly better.

4.2. Numerical application

To further validate the performance of the MQPSO optimizer, it has been applied to the TEAM 22 problem as shown in Fig. 4. The TEAM problem 22 is the optimization of SMES (superconducting magnetic energy storage) device as reported in [12,15–21]. The system contains two concentric superconducting coils, carrying currents in the opposite direction. The inner solenoid and outer shielding solenoid that is used to minimal the stray field. The optimum design of SMES is focused to achieve a favorable energy stored with negligible stray fields. Thus, the design should fulfil:

(1) The stored energy in the device should be 180 MJ.

(2) The magnetic field produced in the solenoids must not disrupt certain physical condition to ensure the superconductivity.

(3) The mean stray field in 22 measurement points along line A and line B at a distance of 10 m should be as small as possible.

To ensure the superconductivity of the conductors, the constraint equation between the current density of the two solenoids and their magnetic flux densities should fulfil: $\begin{eqnarray}\displaystyle J_{t}\leq (-6.4|(B_{\text{mx}})_{t}|+54)(A/mm^{2})\quad (t=1,2) & & \displaystyle\end{eqnarray}$ (14) where J _t and |B _mx|_t are respectively, the current density and maximum magnetic flux density in the tth coil.

In the three parameters optimization problem of SMES design, the inner solenoid is fixed at, R ₁ = 2 m, H ₁∕2 = 0.8 m, D ₁ = 0.27 m. The dimensions of the outer solenoid are optimized following the constraints: 2.6 m < R ₂ < 3.4 m, 0.204 m < H ₂∕2 < 1.1 m and 0.1 m < D ₂ < 0.4 m. Furthermore, the current densities for the two coils are set to be 22.5 A/mm² in opposite directions. Hence, for the easiness of mathematical implementation, equation (14) is referred to as, B _mx | ≤ 4.92 T. Utilizing this adaptation, the optimization problem is expressed as, $\begin{eqnarray}\displaystyle Min∼F=\frac{B_{st}^{2}}{B_{n}^{2}}+\frac{|En-E_{r}|}{E_{r}}\text{ subject to }|B_{\text{mx}}\leq 4.92T, & & \displaystyle\end{eqnarray}$ (15) where, B _n = 3 × 10⁻³ T, En is the stored energy in the SMES device, (B _mx)_t (t = 1, 2) is the maximum field in the tth coil, $B_{st}^{2}$ is evaluated on 22 equidistance points along line A and B as shown in Fig. 4, using the following equation, $\begin{eqnarray}\displaystyle B_{st}^{2}=\mathop{\sum }_{t=1}^{22}B_{st,t}^{2}/22. & & \displaystyle\end{eqnarray}$ (16)

In this case study, the electromagnetic field and performance parameters such as magnetic energy as required in (15) and (16), are calculated using two-dimensional finite element technique.

To compare performances, this problem is solved respectively, using the MQPSO optimizer, basic PSO [1], QPSO [2] and GQPSO [13]. For this purpose, we set the maximum number of generations to be 100, the population size to be 20. Each algorithm runs 20 times independently. Also, the results obtained by using four other optimal algorithms DE1, DE2, ARDGDE1, ARDGDE2 [22] have been taken from the literature for comparisons. Table 2 and Table 3 compares the final optimal results of the different stochastic algorithms.

Fig. 4.

SMES configuration.

Table 2

Comparison of different stochastic methods on TEAM Workshop Problem 22

Optimizer	Min (Best)	Mean	SD	Max	R ₂	H ₂ ∕2	D ₂
PSO	0.148	0.1639	0.1107	0.182	3.253	0.224	0.380
GQPSO	0.122	0.1397	0.0113	0.154	3.172	0.231	0.389
QPSO	0.107	0.1124	0.0125	0.125	3.078	0.241	0.379
MQPSO	0.072	0.0874	0.0062	0.097	3.093	0.246	0.382

Table 3

Comparison of different stochastic methods on TEAM Workshop Problem 22

Optimizer	R ₂	H ₂ ∕2	D ₂	$B_{st}^{2}$	f _opt
DE1	3.5174	1.1600	0.2627	34.176 × 10⁻⁷	2.9292
DE2	1.8000	1.1986	0.3801	11.006 × 10⁻⁷	0.3967
ARDGDE1	1.8080	1.7944	0.1377	29.861 × 10⁻⁷	2.2359
ARDGDE2	1.8424	0.9577	0.4184	7.6107 × 10⁻⁷	0.2296
Improved CE	3.1098	0.3112	0.2979	7.7500 × 10⁻⁷	0.0909
MQPSO	3.0937	0.2461	0.3827	7.5712 × 10⁻⁷	0.0729

The outcomes of Tables 2–3, reveals the merit and advantages of the proposed MQPSO compared to other tested optimal algorithms. Thus, from the numerical outcomes and statistical analysis, one can conclude that the proposed MQPSO optimizer has significant performance than other tested optimal techniques in terms of both the final outcome searched and convergence speed (number of iterations).

5. Conclusion

This work has presented an improved quantum-based particle swarm optimizer for the purpose of optimizing electromagnetic devices. The characteristics of the proposed MQPSO method is its simple in nature and implementation. The experimental outcomes on the two case studies validate the usefulness of the proposed method and is highly applicable when compared to well-developed stochastic approaches. Nevertheless, the proposed MQPSO method is a favorable alternative for the optimization of electromagnetic design studies.

Footnotes

Acknowledgements

This work is supported in part by the National Key R&D Program of China (No. 2018YFB0803600), National Natural Science Foundation of China (No. 61801008), Beijing Natural Science Foundation National (No. L172049), Beijing Science and Technology Planning Project (No. Z171100004717001).

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