A vector wind driven optimization algorithm for multi-objective optimizations of electromagnetic devices

Abstract

Wind driven optimization is a new entrant to evolutionary algorithms, and has been applied successfully to solve different engineering single objective (scalar) design problems. However, lukewarm efforts have been devoted to develop a vector optimizer using a wind driven optimization algorithm. This paper explores the potential of this new optimal method in solving multi-objective design problems in electromagnetics. In this regard, a vector wind driven optimization algorithm is proposed and validated using two inverse problems with promising numerical results.

Keywords

Wind driven optimization multi-objective evolutionary algorithm design optimization

1. Introduction

In engineering design problems, it is a common practice to ask a designer to reach a compromise between several seemingly conflicting criteria/objectives. Generally, the final solutions of such a problem are a set of tradeoffs, called Pareto optimal solutions, of different objectives or criteria. Consequently, an acceptable vector (multi-objective) optimizer should have the ability to find as many Pareto optimal solutions as possible, and these solutions should also be as uniformly distributed as possible. To achieve these two ultimate goals, a huge amount of efforts have been devoted to the advancement of evolutionary and stochastic algorithms, because of their suitability in producing a proper approximation of the Pareto optimal solutions in a single run [1,2].

Nevertheless, the situation in views of the searching ability, the numerical complexity, and the convergent speed of existing vector evolutionary and stochastic algorithms is still unsatisfactory. Moreover, there is no universal vector algorithm which can be applied equally and successfully to all multi-objective inverse problems according to the “no free lunch theorem” [3]. As a result, it is essential to retain and ensure there are sufficient diversity in the vector optimizers when studying multi-objective inverse and optimal problems in computational electromagnetics. In this regard, a vector wind driven optimization (WDO) algorithm is proposed due to its simpleness and easiness in numerical implementations.

Nature phenomenon is a perfect inspiration to develop evolutionary algorithms. The WDO algorithm is inspired from the modeling of the climate [4,5]. In our living environment, wind blows from the high pressure zone to the low pressure zone at various speeds to equalize the imbalances in air pressure. Based on Newton’s second law of motion and some simplifications, the velocity vector, v, and the position vector, x, of the WDO algorithm are updated using $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle v(k+1)=(1-{\alpha})v(k)-gx(k)+\left(RT\left|1-{\displaystyle \frac{1}{i}}\right|(x_{opt}-x(k))\right)+[cv^{\text{othr}\_\text{dim}}(k)]/i\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle x(k+1)=x(k)+{\Delta}t\times v(k+1)\end{eqnarray}$ (2) where, 𝛼 is a friction coefficient, g is the Gravitational constant, R is the universal gas constant, T is the temperature, c is a constant, i is the ranking among all air parcels, x _opt is the best parcel so far searched, $\Delta t$ is the step length.

Obviously, compared with other nature-inspired optimal algorithms, WDO is very simple and relatively easy to implement numerically. However, as a new stochastic algorithm, WDO may be prone to premature convergences when solving complex optimization problems [5], due to the possible diversity loss of the parcels and the corresponding imbalance between exploration and exploitation searches. Moreover, lukewarm efforts have been devoted to extend this new stochastic algorithm to multi-objective optimization problems [6], especially in the studies of inverse and optimization problems in electromagnetics. In this regard, this paper explores the potential of WDOs to solve a multi-objective design optimization or inverse problem of electromagnetic devices.

2. A vector wind driven optimization algorithm

In directions both to ensure enough diversities in the later searching stages of a single objective WDO and to extend the scalar WDO to a vector one, some improvements are proposed in this paper.

2.1. Improvements on scalar WDO algorithm

Physically, the second term, −gx (k), in (1) represents the gravitational force, which is in fact an attractive force, that pulls the parcels towards the absolute origin of the coordinate system. Since every parcel is always attracted towards this absolute origin under the influence of this gravitational force, the diversity of the parcels is reduced. To address this problem, this term is removed from the proposed WDO method. Moreover, to ensure a good balance between exploration and exploitation searches, a new parameter, w, is introduced in the position updating equation of (2). Consequently, (1) and (2) are improved to $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle v(k+1)=(1-{\alpha})v(k)+\left(RT\left|1-{\displaystyle \frac{1}{i}}\right|(x_{opt}-x(k))\right)+\left(\frac{cv^{\text{othr}\_\text{dim}}(k)}{i}\right)\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle x(k+1)=w\times x(k)+(1-w)\times {\Delta}t\times v(k+1)\end{eqnarray}$ (4) where w is a control parameter which decreases linearly from its maximum value to its minimal one in the iterative process of the proposed algorithm.

The last term, cv ^othr_dim(k)∕i in (3), introduces some perturbations to the velocity vector to retain its diversity. Obviously, it functions as a mutation operator as that is used in an evolutionary algorithm. However, towards the end of the searching stages of the algorithm in solving complex optimal design problems, the velocity vector might possibly converge to zero, giving rise to the so called stagnation phenomenon and the algorithm will then be trapped in a local optimum. In other word, in the early searching stage of a WDO algorithm, the diversity of the parcels is relatively high since the initial velocity vector is randomly and independently generated; and it will be continuously decreased with the evolution of the algorithm. In these points of views, the perturbed increment implemented by the mutation operation should be large when the diversity of the population is smaller and vice versa. Moreover, the standard deviation of the parcels in the current population is a proper metric to gauge the diversity of the population. In this regard, an adaptive mutation is designed, and (3) is modified to $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle v(k+1)=(1-{\alpha})v(k)+\left(RT\left|1-\displaystyle \frac{1}{i}\right|(x_{opt}-x(k))\right)+(2r-1){\Delta}_{\max }e^{-{\sigma}}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \begin{array}{@{}l@{}}{\sigma}=\sqrt{\left.\left(\displaystyle \mathop{\sum }_{i=1}^{N}(x_{i}(k)-{\mu})^{2}\right)\right/N}\\ {\mu}=\left.\displaystyle \mathop{\sum }_{i=1}^{N}x_{i}(k)\right/N\end{array}\end{eqnarray}$ (6) where, r is a random number uniformly generated in [0 1], Δ_max is the maximum perturbation increment, N is the size of the population.

2.2. Approaches oriented for multi-objective optimizations

In updating the position of a parcel in a scalar WDO algorithm, the “best” parcel so far searched is used to guide the search direction toward promising solutions. Nevertheless, the optimal solutions of a multi-objective design problem are not unique but a set of tradeoffs among different objectives. This feature will give rise to a dilemma in selecting the “best” parcel when updating the current parcel since the fitness values of all Pareto optimal solutions are the same. Moreover, the dominant concept is generally employed in a vector optimizer to determine the fitness value of an individual in a multi-objective optimal problem. However, such approach may only determine qualitatively the relationship of dominances and may not measure, quantitatively, the number of the improved objectives [7]. The amount of improvements in a specified objective may not be quantified either.

To address the aforementioned two issues, a metric gauging the improvements in the aforementioned two aspects is introduced. For a parcel, x _m, the proposed metric is precisely computed by using $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\varepsilon}_{improve}(x_{m})=\displaystyle \mathop{\sum }_{\stackrel{j=1}{j\neq k}}^{N}\displaystyle \mathop{\sum }_{l=1}^{k}sign[f_{l}(x_{j})-f_{l}(x_{m})]\left\{\displaystyle \frac{1}{k}+\displaystyle \frac{f_{l}(x_{j})-f_{l}(x_{m})}{(f_{l})_{\max }-(f_{l})_{\min }}\right\}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle sign(x)=\left\{\begin{array}{@{}ll@{}}1\quad & \quad (x>0)\\ 0\quad & \quad (\text{otherwise})\end{array}\right.\end{eqnarray}$ (8) where k is the number of the objectives.

Using this accumulation factor to ‘penalize’ the fitness value of all parcels in the current population, the fitness of x _m is given by $\begin{eqnarray}\displaystyle f_{fit}(x_{m})=rf_{fit}^{nor}(x_{m})+(1-r){\varepsilon}_{\text{improve}}(x_{m}) & & \displaystyle\end{eqnarray}$ (9) where, $f_{fit}^{nor}(x_{m})$ is the normally defined fitness of x _m using a dominance relationship, r is a random parameter uniformly generated in [0, 1].

To strike a good balance between exploration and exploitation searches, the iterative procedures of the proposed algorithm are divided into two phases: an exploration search phase and an exploiting search phase. The goal of the exploration search phase is to guarantee the diversity of the parcels as well as the corresponding final solutions while that of the exploiting search phase is to find improved solutions efficiently and reliably. As a result, different searching strategies are used in the two searching phases. To steer the searches of the proposed vector WDO algorithm in the exploiting search phase to find improved high performance solutions, the metric parameter as defined in (7) is used as both the driving force to evaluate a parcel and the performance metric to select the “best” parcel in updating the velocity of a parcel while that in (9) is selected as the diving force to facilitate exploration searches of the whole feasible space. The algorithm starts from the exploration searching phase. Once a minimal number of “Pareto optimal” solutions are searched, the algorithm will enter into the exploiting searching phase until it has either successfully found an improved solution in every parcel or the number of consecutive iterations without finding any new “Pareto optimal” solution exceeds a predefined value. The two phases are continuously alternated until a global stop criterion is satisfied.

3. Numerical applications

To test the proposed WDO based vector optimizer, it is numerically experimented on different case studies and its performances are compared to those of existing ones.

3.1. Antenna array optimization

In this case study, a desired field pattern of a shaped beam with a cosecant variation is reconstructed using a nonuniformly spaced antenna array. The desired pattern is defined as [8]: the field will vary following a cosecant function in the interval cosθ ∈ [0.1,0.5] having a Maximum SideLobe Level (MSLL) smaller than −22 dB in the residual intervals. Mathematically, the desired pattern is expressed as $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}F_{desired}(\cos {\theta})=\left\{\begin{array}{@{}ll@{}}\text{cosecant}(\cos {\theta})\quad & \quad (0.1\leq \cos {\theta}\leq 0.5)\\ 0\quad & \quad (elsewhere)\end{array}\right.\\ MSLL_{Desired}\leq -22∼\text{dB}.\end{array} & & \displaystyle\end{eqnarray}$ (10)

To produce a close approximation to this desired field pattern, a completely non-uniform antenna array, as shown in Fig. 1, is optimized with a minimal number of elements with respect to the following objective function $\begin{eqnarray}\displaystyle \min f_{1}=\sqrt{\frac{\displaystyle \mathop{\sum }_{i=1}^{N}[f_{desired}^{norm}({\theta}_{i})-f_{designed}^{norm}({\theta}_{i})]^{2}}{\displaystyle \mathop{\sum }_{i=1}^{N}[f_{desired}^{norm}({\theta}_{i})]^{2}}} & & \displaystyle\end{eqnarray}$ (11) where, $f_{desired}^{norm}({\theta}_{i})$ is the value of the normalized desired radiation pattern at sampling point θ_i, $f_{designed}^{norm}({\theta}_{i})$ is the value of the normalized radiation pattern produced by a designed array of M elements, N is the number of total sampling points.

To compromise the requirement on the MSLL and the field pattern, the second objective is to minimize the MSLL. Consequently, the two-objective synthesis problem is formulated as $\begin{eqnarray}\displaystyle \min \left\{\begin{array}{@{}l@{}}f_{1}\quad \\ f_{2}=MSLLdesired.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (12) The computation of the field pattern is reffered to [8]. To quantitatively evaluate the performances of a vector optimal algorithm, the convergence metric 𝛾 is used to measure the convergence of its final solution set to a known set of the Pareto solutions, while the displacement metric is employed to gauge its uniformity or diversity performance of the attained solutions over the non-dominated front [7]. If the set consisting of the uniformly spaced true Pareto solutions, which are obtained using an exhaustive approach in this case study, is P ^∗; and the set of the final solution of the vector optimizer is Q, the aforementioned two metrics are then defined as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}=\displaystyle \mathop{\sum }_{i=1}^{|Q|}\left.\left(\min _{j\in [1,|P^{\ast }|]}(d(i,j))\right)\right/|Q|\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle displacement=\displaystyle \mathop{\sum }_{i=1}^{|P^{\ast }|}\left.\left(\min _{j\in [1,|Q|]}(d(i,j))\right)\right/|P^{\ast }|\end{eqnarray}$ (14) where d (⋅, ⋅) is the Euclidean distance of the two set solutions.

In this case study, a 19-element nonuniform antenna array is optimized to find the Pareto optimal solutions of the two conflicting objectives of (12). For performance comparisons, this case study is solved by using the proposed algorithm and the improved vector tabu search algorithm [9]. Moreover, to evaluate the average performance of an algorithm, each algorithm is run randomly and independently 10 times. The average performances of the 10 runs for the two algorithms are summarized as:

(1) the numbers of iterations for the proposed and the vector tabu search algorithms are, respectively, 26794 and 29465;

(2) the parameters 𝛾 and displacement of the final solution for the proposed algorithm are, respectively, 0.000016 and 0.00028; while the metrics 𝛾 and displacement of the final solution for the vector tabu search method are, respectively, 0.000018 and 0.00029; evidencing the advantages of the proposed vector WDO algorithm in solving multi-objective optimal design problems.

3.2. Design optimizations of a multi-sectional pole arcs of a large hydrogenerator

The design optimization of the multi-sectional pole arcs of a large hydrogenerator [10] is then solved. The problem is mathematically formulated as $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\max ∼∼\quad \quad \quad B_{f1}(X)\\ \min ∼∼\quad \quad \quad \quad e_{v}\\ \text{s.t}\quad \quad SCR-SCR_{0}\geq 0\\ \quad \quad \quad \quad \quad X_{d}^{\prime }-X_{d0}^{\prime }\leq 0\\ \quad \quad ∼∼∼THF-THF_{0}\leq 0\end{array} & & \displaystyle\end{eqnarray}$ (15) where, B _f1 is the amplitude of the fundamental component of the flux density in the air gap, e _v is the distortion factor of a sinusoidal voltage of the machine under no-load condition, THF is the abbreviation of the Telephone Harmonic Factor, $X_{d}^{^{\prime }}$ is the direct axis transient reactance of the generator, SCR is the abbreviation of the Short Circuit Ratio, SCR ₀ ( $X_{d0}^{\prime }$ and THF ₀) is the limit of the corresponding performance parameter.

The geometrical parameters, the center positions and radii of the multi-sectional arcs of the pole shoe, as shown in Fig. 2, are optimized to obtain a series of trade-off solutions of this case study. For performance comparisons, the multi-sectional pole arcs of a 300 MW, 20-pole hydro-generator are optimized by using the proposed vector WDO and the improved vector tabu search optimizers [9]. In the numerical implementations, the magnetic fields of the hydrogeneraor generator at no-load conditions are computed using a finite element method, and the performance parameters as required in (15) are then determined based on these finite element solutions. To give a fair comparison, the number of the final (Pareto) solutions is fixed for the two algorithms. Moreover, to give the average performances of the two algorithms, they are independently and randomly run 10 times. Since the true Pareto solutions, P ^∗, of this case study are unavailable, they are approximated by using the bested ones searched by the two vector optimizers in the 20 runs. The average parameters 𝛾 and displacement of the final solutions for different algorithms are given in Table 1. Also, the corresponding iterative numbers used by the two algorithms are presented in Table 1.

From the average performances of the three performance parameters, it is concluded that:

(1) the quality of the final solutions of the proposed WDO algorithm is better than that of the improved vector tabu search algorithm, because the average value of the convergence metric of the final solutions for the proposed algorithm is smaller than that for the improved tabu search method.

(2) the diversity of the final solutions of the proposed WDO algorithm is identical to that of the improved vector tabu search method, because the average values of the displacement metric of the final solutions for the two algorithms are almost the same.

(3) the proposed WDO based vector algorithm is a little faster than the improved vector tabu search method.

Table 1

Performance comparison of different algorithms on case study two

Algorithm	𝛾	Displacement	No. Iterations
The proposed WDO algorithm	0.000026	0.00045	1235
The improved Tabu search method	0.000028	0.00045	1248

4. Conclusions

This paper strives to develop a WDO based vector optimizer for multi-objective inverse and optimization problems. The numerical results on two case studies being reported have demonstrated that: (1) compared with an improved vector tabu search algorithm, the proposed algorithm can use relatively fewer iterations to find relatively higher quality final solutions; (2) the proposed WDO based vector optimizer can easily keep a good balance between minimizing the distance from the found solutions to the true Pareto front and maximizing the diversity among the found Pareto solutions.

Footnotes

Acknowledgements

This work was supported by the Research Grant Council of the Hong Kong SAR Government under project PolyU 152254/16E.

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