A robust methodology for design optimizations of electromagnetic devices under uncertainties

Abstract

To find the robust optimal solution of an engineering design problem under uncertainties, a novel robust optimization methodology is proposed. In the proposed methodology, the robust performances are treated as additional constraint functions, and the performance parameter is kept as the driving force to evolve the iteration procedures to find the exactly global robust optimal solution. An effective yet simple robust performance checking mechanism to check the robust performance constraints only to the potential robust “optimal” solutions to reduce the computational cost is introduced and its implemental procedure is developed. The proposed methodology can be used to search both the robust and the global optimal solutions of an electromagnetic design problem. Finally, the proposed methodology is applied to solve a well defined inverse problem, and its performance is compared to that of an existing robust optimizer.

Keywords

Robust optimization uncertainty design optimization inverse problem

1. Introduction

In a performance-based design optimization of electromagnetic devices, the goal is to find the global optimal solution merely according to the performance (objective) quality. Since imprecision and uncertainty are often inevitable and unavoidable in an engineering design problem, it is possible that slight variations or perturbations in the optimized variables obtained using such a method could result in either a significant performance degradation or an infeasible solution due to the violation of the design constraint functions if the optimized solution is very sensitive to the optimized decision parameters. In this regard, it is equally important to explore robust optimal techniques in the studies of design optimizations of electromagnetic devices under conditions of uncertainties in computational electromagnetics [1–3].

Robustness means some degree of insensitivity to small perturbations in the design variables and working conditions. To quantify the uncertainty in robust design optimizations, an expectancy metric (mean) is used in this paper [4]. For example, for a constrained minimization problem: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\min \quad f(x,{\delta})\\[3.0pt] \text{subject to}∼g_{i}(x,{\delta})\leq 0\quad (i=1,2,\ldots ,k),\end{array} & & \displaystyle\end{eqnarray}$ (1) the expectancy metric is generally defined as $\begin{eqnarray}\displaystyle h_{\text{exp}}(x)=\int _{-\infty }^{\infty }h(x,{\delta})p({\delta})d{\delta} & & \displaystyle\end{eqnarray}$ (2) where, x is the vector of design (decision) parameters (variables), h = f or g _i, δ is the vector of the uncertainty variables, p(δ) is the distribution probability of the uncertainties or disturbances.

As there is no close-form expression for the probability function p(δ) of the uncertainties in a real-world engineering problem, the robust performance parameter as defined in (2) is thus determined by using a numerical sampling approach from $\begin{eqnarray}\displaystyle h_{\text{exp}}(x)=\frac{\displaystyle \mathop{\sum }_{i=1}^{N_{s}}h(x_{i})}{N_{s}} & & \displaystyle\end{eqnarray}$ (3) where N _s is the number of the total sampling points generated in a small neighborhood of the specific point x in issue. Obviously, a large number of additional points in a small neighborhood of a given solution are sampled and their function values are used to approximate the robust metric of the specific solution. Since numerical approaches such as a finite element analysis are generally required to compute the function values of these sampling points, the salient deficiency of a robust optimizer is that its computational burden is extremely high compared to that for its global counterpart.

Moreover, in most of the existing robust optimization methodologies for robust optimization [2], the objective and the constraint functions are simply replaced by their robust counterparts, (2) or (3), to find the robust optimal solution of the original design problem. In this point of view, the robust performance is used as a predominant parameter in determining the searching direction of an optimizer in the optimization process. However, under some mild condition that the objective function is continuously differential, especially in cases that the expected fitness function is used as the robust parameter, only the (local) optimal and boundary solution has the potential to be a robust optimal solution [5]. In this regard, the original objective function should be selected as the driving force of the optimizer to stimulate enough competitive pressure to evolve the iterative procedures to find the exact optimal solution. In this regard, a different formulation, using additional constraint functions to ensure robust performances while the performance parameter being kept as the driving force to evolve the iteration procedures to find the exactly global robust optimal solution, for robust optimizations of electromagnetic devices is proposed. Moreover, an effective yet simple robust performance checking mechanism to check the robust performance constraints only to the potential robust “optimal” solutions to reduce the computational cost is introduced and its implemental procedure is developed.

2. A robust methodology for design optimizations

2.1. Formulation of a robust optimization

As explained previously, the objective function should be selected as the driving force of a robust optimizer to stimulate enough competitive pressure to evolve the iterative procedures to find the exact robust optimal solution since, under some mild condition that the objective function is continuously differential, especially in cases that the expected fitness function is used as the robust parameter, only the local/global optimal and boundary solution has the potential to be a robust optimal solution. To eliminate the shortcomings of using the robust performance as the driving force in the existing robust optimal methodologies, and moreover, to incorporate a priori knowledge of a domain expert into a robust design optimization, some acceptable tolerance for performance degradations is introduced. Consequently, the proposed robust optimization formulation is given as $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\min \quad f(x)\\[3.0pt] \text{subject to}∼g_{i}(x)\leq 0\quad (i=1,2,\ldots ,k),\end{array} & & \displaystyle\end{eqnarray}$ (4) together with $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}f_{\text{exp}}(x)-f(x)\leq ({\rm\Delta}f)_{tolerance}\\[3.0pt] (g_{i})_{\text{exp}}(x)-g_{i}(x)\leq ({\rm\Delta}g_{i})_{tolerance}\end{array}\right. & & \displaystyle\end{eqnarray}$ (5) where (Δh)_tolerance (h = f or g _i) is the acceptable degradation of h(x).

Obviously, the robust performances are imposed as constraints in the proposed formulation, and the biasing force for driving the optimizer is still the quality of the objective performance. As explained previously, only a local/global optimal or a boundary solution has the potential to be a robust optimal solution, using the original objective function as the driving force will equip the algorithm to have the ability to find the globally robust optimal solution due the strong global searching ability of the tabu search algorithm as detailed in the next section. Moreover, a promising byproduct of the proposed methodology is that the global optimal solution, together with the robust optimal one, could be found in the same run.

2.2. A tabu search algorithm for robust optimizations

Even though any evolutionary algorithm can be used readily to find the robust optimal solution of a design problem as formulated (4) and (5), a simple and efficient robust oriented tabu search algorithm is proposed in this paper.

As explained previously, under some mild condition that the objective function is continuously differential, especially in cases that the expected fitness function is used as the robust parameter, a local/global optimum of the objective function or a solution distributed on the boundaries of the feasible parameter space has the potential to be the robust optimal solution of a constrained optimal design problem. In this point of view, it is required to check the robust performance constraints of (5) for only these potentially robust solutions, but not every intermediate individual, to reduce the heavily computational cost without any compromise on the solution quality, of a robust optimizer. However, it is not an easy task to identify if an intermediate solution is a potential robust one in the optimization process.

On the other hand, from a mathematical point of view, at a stationary point (a local or global optimal point), the partial derivatives are zero. However, using this criteria to identify if an intermediate point is a potential robust optimal point is also computationally expensive if the partial derivatives for every intermediate solution are computed using a numerical approach such as a finite element analysis. Nevertheless, from the iterative procedures of a tabu search algorithm [6], it is obvious that only the best neighborhood solution generated in the current state and the boundary solutions have the potential to be a global/local optimum. This salient feature lends that the proposed tabu search algorithm is convenient for developing an effective and simple robust performance checking mechanism and procedure to check the robust performance constraints only to those potential “optimal” solutions.

More specially, the proposed robust performance checking mechanism and procedure is based on a local reconstruction of the objective functions using a response surface model, and are implemented in the following 6 steps:

Step 1: Identify the best neighborhood solution generated in the current state and the boundary condition, and expressed them as x ^∗s;

Step 2: Construct a local response surface model of the objective function in a small neighborhood of every x ^∗ using the radial basis function;

Step 3: Compute the partial derivatives of every x ^∗ using the reconstructed local response surface model. If the partial derivatives $\Vert {\nabla}f(x^{\ast })\Vert _{2}^{0}$ is greater than a predefined value, go to Step 6;

Step 4: Compute directly $\Vert {\nabla}f(x^{\ast })\Vert _{2}^{1}$ using a finite element analysis. If the partial derivatives $\Vert {\nabla}f(x^{\ast })\Vert _{2}^{1}$ is greater than a predefined value, go to Step 6;

Step 5: Active the robust performance checking of (5) for x ^∗;

Step 6: Go to the next cycle of iterations.

Obviously, in this iterative procedure, the most computational heavy task is the computation of $\Vert {\nabla}f(x^{\ast })\Vert _{2}^{1}$ using a finite element analysis. To determine the partial gradients, $\Vert {\nabla}f(x^{\ast })\Vert _{2}^{1}$ , the finite-difference approach is readily applicable. Since such approach will compute only one partial derivative in one direction in one computation, it is computationally expensive or prohibitive for problems with large decision parameters. To overcome this disadvantage of computing only one partial derivative in one computation of the finite-difference approach, the stochastic approximation (SA) method [7] is used.

In the SA method, a n-dimensional random vector (n is the number of the decision parameters), $[{\rm\Delta}_{1}^{-1}∼{\rm\Delta}_{2}^{-1}\cdots {\rm\Delta}_{n}^{-1}]^{T}$ , consisting of independent and symmetrically distributed entries with finite inverse moment expectations, is firstly generated using the symmetric Bernoulli (±1) distributions; and the partial gradient information of f(x ^∗) for all n-dimensions is approximated at one computation using $\begin{eqnarray}\displaystyle {\nabla}f(x^{\ast })=\frac{f(x^{\ast }+c{\rm\Delta})-f(x^{\ast }-c{\rm\Delta})}{2c}[{\rm\Delta}_{1}^{-1}∼{\rm\Delta}_{2}^{-1}\cdots {\rm\Delta}_{n}^{-1}]^{T}. & & \displaystyle\end{eqnarray}$ (6)

Obviously, the proposed stochastic approximation method realizes the computational savings of n times relative to the finite difference approximation in determining the gradient information. It should be pointed out that the proposed stochastic approximation certainly brings in errors in estimating the gradient information. The error in the gradient information may result in an unnecessary activation or a lost of checking procedures for the robust performance, resulting in more iterations to converge to the global optimal solution. However, the quality of the final solution will not be degraded since the stochastic nature of the tabu search algorithm.

Moreover, the radial basis function used in this paper is the multiquadrics function [8]. Given a series of sampling points x _j (j = 1,2, …, N) and the corresponding function values f _j, the approximation of function f using a multiquadrics radial basis function is given as $\begin{eqnarray}\displaystyle f(x)=\mathop{\sum }_{j=1}^{N}c_{j}H_{j}(\Vert x-x_{j}\Vert ) & & \displaystyle\end{eqnarray}$ (7) where c _j is the interpolation coefficient and is determined from $\begin{eqnarray}\displaystyle XC=F & & \displaystyle\end{eqnarray}$ (8) where C = [c ₁ c ₂ ⋯ c _N]^T, F = [f ₁ f ₂ ⋯ f _N]^T, X is a N × N matrix, and x _ij is given by $\begin{eqnarray}\displaystyle x_{ij}=H_{j}\Vert x_{i}-x_{j}\Vert , & & \displaystyle\end{eqnarray}$ (9) H(r) is the radial basis function and the multiquadrics radial basis function is used in this paper. The multiquadrics radial basis function is defined as $\begin{eqnarray}\displaystyle H(r)=(r^{2}+h)^{{\beta}}\quad (0\lt {\beta}\lt 1) & & \displaystyle\end{eqnarray}$ (10)

In this paper, the neighborhood solutions of the current state are used to reconstruct the local response surface model of the objective function. In other word, no additional sampling points are required.

3. Numerical application

To validate the proposed robust optimal methodology, it is used to solve different case studies and compared with other existing robust optimal approaches. Due to space limitations, only the numerical results on the robust optimal counterpart of the Team Workshop problem 22 of a superconducting magnetic energy storage (SMES) configuration with three free parameters [9] are reported.

As shown in Fig. 1, the system consists of two concentric coils which are the inner main solenoid and the outer shielding solenoid for reducing the stray field. The current directions of the two coils are opposite to each other. The SMES design is to obtain the desired stored energy with minimal stray fields:

Fig. 1.

The schematic diagram of a SMES.

(1) The energy stored in the system should be 180MJ;

(2) The generated magnetic field inside the solenoids must not violate the specific physical conditions in order to guarantee the super-conductivity of the coils $\begin{eqnarray}\displaystyle J_{i}\leq (-6.4|(B_{\text{max}})_{i}|+54)(\text{A}/\text{mm}^{2})\quad (i=1,2) & & \displaystyle\end{eqnarray}$ (11) where J _i and |B _max|_i are the current density and the maximal magnetic flux density in the i ^th coils, respectively.

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

In the three parameters problem, the inner solenoid is fixed. The dimensional parameters of the outer solenoid are optimized under the following constraint conditions: 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. Moreover, the current densities for the two coils are set to be 22.5 A/mm². Also, for the convenience of the numerical implementation, (11) is simplified to |B _max| ≤ 4.92 T. Under such simplified conditions, this optimal design problem is formalized as: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle\min \quad f=w_{1}\frac{B_{stray}^{2}}{B_{norm}^{2}}+w_{2}\frac{|Energy-E_{ref}|}{E_{ref}}\\[12.0pt] \text{Subject to}\quad |B_{\max }|_{i}\leq 4.92∼\text{T}\end{array} & & \displaystyle\end{eqnarray}$ (12) where, Energy is the stored energy in the SMES device, E _ref = 180MJ, B _norm = 2 × 10⁻⁴ T. $B_{stray}^{2}$ is a measure of the stray fields evaluated along 22 equidistant points of Line A and Line B of Fig. 1 using $\begin{eqnarray}\displaystyle B_{stray}^{2}=\left.\mathop{\sum }_{i=1}^{22}(B_{stray})_{i}^{2}\right/22. & & \displaystyle\end{eqnarray}$ (13)

To determine the electromagnetic field and the performance parameters as required in (12) and (13), two dimensional finite element method is used in this paper.

In the numerical experiment, the uncertainty in the decision parameters is set to ±1% limits of the ranges of the corresponding decision variables. For performance comparison, this case study is solved by using the proposed, and the combined polynomial Chaos and PSO approach [10]. Under the same implementation and simplification conditions as reported in [10], the performances in a typical run of the proposed, and the combined polynomial chaos and PSO approaches for robust optimizations are summarized as:

(1) The final solutions (both global and robust optimal ones) obtained using the two robust optimal approaches are nearly the same. The stray field and the stored energy under the two different optimized solutions are: the stray field is 7.55 × 10⁻⁷ and the stored energy is 179.2141 MJ for the robust optimal solution; while the stray field is 7.75 × 10⁻⁷ and the stored energy is 179.9956 MJ for the global optimal solution.

(2) The global optimal solution searched by the two robust optimizers is nearly identical to that of the best one so far searched in literature [9].

(3) The numbers of iterations used by the proposed methodology, and the combined polynomial Chaos and PSO approach are, respectively, 2456 and 3018; i.e., the number of iterations used by the proposed methodology is about 85% of that of the combined polynomial chaos and PSO approach [10].

(4) To show the robustness of the two optimal solutions against small variations, some post-processing numerical experiments are conducted. In a more detail description, 50 random perturbations with ±1% limits of the bounds of the corresponding decision variables as the uncertainty are randomly applied to the two optimized decision variables, and the performance parameters of the totally 100 perturbed solutions are computed. The numerical results have revealed that:

(a) The averaged performance degradations of the robust optimal solution are that the stray field is degraded an incremental value of 0.35, and the averaged deviation of the stored energy is 0.0015, both in relative values. The base values for the stray field and the stored energy are, respectively, the ones as given in (1) for the robust optimal solution.

(b) The averaged performance degradations of the global optimal solution are that the stray field is degraded an incremental value of 0.82 and the averaged deviation of the stored energy is 0.02, both in relative values. The base values for the stray field and the stored energy are, respectively, the ones as given in (1) for the global optimal solution.

(c) These comparison results demonstrate that the robustness of the robust optimal solution is significantly stronger than that of the performance based global optimal solution.

From the previous numerical results and comparisons, it is natural to conclude that the feasibility and advantage of the proposed robust optimization methodology is positively confirmed.

4. Conclusions

To find the robust optimal solutions of an electromagnetic device design problem under uncertainties, a different formulation and the corresponding methodology, of the robust optimization, are proposed. The numerical results on a case study have demonstrated that the proposed formulation and optimization methodology can use relatively few iterations to find the same quality final solutions compared to an efficient combined polynomial chaos and evolutionary method. Moreover, a promising byproduct of the proposed robust search methodology is that it can find the global optimal solution of a design problem under uncertainties in a single run.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51377139 and by the Science and Technology Department of Zhejiang Province under Grant No. 2016C31037.

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