Application of evolution strategy to determine parameters of the multi-branch Foster and Cauer equivalent circuit of system with eddy-currents

Abstract

The paper discusses the strategy of searching for the extreme of multi-modal functions with the number of decision variables n ≥ 6, in a multidimensional design space. The optimization algorithm combining an evolution strategy (ES) with operators of a genetic algorithm has been proposed. The method of selecting subspaces that allows for narrowing of the design space has been proposed. The effectiveness of the proposed approach has been demonstrated on two selected test systems with eddy currents. First, the equivalent multi-branch Foster circuit of a solenoid coil with a massive conducting core has been determined. Next, the proposed approach has been applied in the analysis of a high frequency transformer, working in a wireless power transmission (WPT) system. In the second case, the equivalent circuit of the transformer has been formed by means of multi-branch Cauer circuits. Selected results of the calculations have been presented and discussed. The obtained results have been compared with the results of the performed finite element analysis (FEA) of the considered test systems.

Keywords

Optimization genetic algorithm eddy current problem Foster circuit Cauer circuit

1. Introduction

Optimization methods for designing new systems and devices or redesigning existing ones are used in many fields of technology, nowadays. Among the currently used methods, the most popular are approaches based on genetics and evolution strategies, commonly referred to genetic algorithms (GAs) [1,2]. In parallel with the GA, other optimization techniques are developed based on the behavior of various species of animals or insects, i.e. the Grey Wolf Optimization (GWO) [3,4], Bat Algorithm (BA) [5] or Particle Swarm Optimization (PSO) [6,7]. Despite the large number and variety of methods used, their authors are very often limited to solving tasks with a small number of decision variables, usually not exceeding 3–5 variables. However, in the case of more complex tasks aimed at finding an extreme not only for a multimodal function but also for many design variables, the most commonly used approach combines elements of an ES with a GA [8]. In the generally available literature, two basic and most frequently cited evolutionary strategies are listed: i.e. (a) strategy (μ, 𝜆) - the ES, in which a new generation of individuals is formed on the basis of the population of descendants constituting 𝜆 individuals, omitting in the further calculation process μ parental individuals [9]; and (b) strategy (μ + 𝜆) - the ES, in which a new population of individuals is created by selecting the best individuals from the parent population with the number μ individuals and the child population with the number 𝜆 [9,10]. In the case of the first strategies, the process of forming a new population is based on a random selection of 𝜆 parental individuals subjected to mutation and the next subsequent crossover operator. Algorithms based on this strategy are very often used to solve optimization problems in “continuous” multidimensional space Rⁿ. In the second of the above-mentioned strategies, mutation and crossover operators are used interchangeably. It can be said that these operators occur at the same frequency. In the (μ + 𝜆) ES, a new generation of individuals is formed through a random selection from the parent population. A temporary population of 𝜆 individuals (where 𝜆 ≥ μ) is created, which then undergoes the process of crossing and/or mutation, thus obtaining a descendant population of size 𝜆. Then, μ, the best individuals from the parental and descendant populations are selected. The algorithms based on the (μ + 𝜆) - ES are often applied for seeking solutions of combinatorial optimization problems, i.e. when the optimization problem is defined over a discrete design space. It should be added that nowadays the above-mentioned methods have greatly evolved. Currently, many authors using the ES propose modified algorithms, where along with operations of mutation, crossover and selection, elitism algorithms, in which the best individuals from the parental population are cloned to the new population [8,10], are implemented. This approach was also applied by the authors of this paper, when determining the values of parameters of the multi-branch Foster Circuit (FC) for the considered system, consisting of a solenoid coil and a conductive (aluminum) core (see, Fig. 1) [11], as well as when calculating the values of parameters of multi-branch Cauer Circuits (CC) [12], representing the branches of the equivalent circuit of the transformer working in a WPT system. The structure of the considered transformer with its equivalent circuit containing the CC has been presented in Fig. 2. For finding the extreme of the objective function describing the parameters of the Foster and Cauer circuits, the authors developed their own software in which a three stage optimization algorithm was used. In the proposed algorithm, apart from the elements combining the ES with GA operators (Fig. 3), the author’s procedure was implemented allowing for a gradual narrowing of the search area. In order to improve convergence of the developed optimization procedure the additional operator has been introduced to the GA. This imposed operator can be understood as an inflow of “New blood” to the population, when the GA is close to the local optimum. The results of the work on the development and implementation of the discussed above modification of a GA have been presented in this paper.

Fig. 1.

Considered system consisting of solenoid coil with conductive core (a) and its equivalent Foster Circuit (b) [11].

Fig. 2.

Construction of considered transformer working in WPT system (a) and its equivalent circuit containing the Cauer Circuits (b).

Fig. 3.

Block scheme of the proposed evolution strategy (ES).

2. Evalution strategy algorythm

The block diagram of the optimization strategy discussed in the paper has been shown in Fig. 3. The proposed strategy distinguishes three levels/stages of calculations. The lowest level of the strategy, level III, includes classic GA operators, i.e. crossover, tournament selection and mutation, as well as the “random of new individuals” operator proposed by the authors. The operator’s task is to introduce to the given ith population new individuals that form in the genetics the so-called “New blood” supply. The implementation of this operator into the code is very simple and involves the introduction of newly selected at random individuals to the created descendant population. The participation of this operator in the optimization process can be noticed especially in the final phase of the calculations, that is, when most of the individuals have the same genes, and the value of the objective function is still far from the value of the sought extreme. In reference to the remaining GA operators, the following were used in the work: (a) the tournament selection operator, consisting in a random selection from the superior (parental) population of N - individuals (in the work, N = 20 is assumed) forming the so-called tournament group, from which the individual with the best adaptation is transferred to the newly created population of descendants; (b) one-point crossover, in which parental specimens were selected at random and then returned to the main set; (c) and the mutation operator, consisting in a random selection of individuals, which randomly made changes in the binary value of a single gene in a selected chromosome. The calculations using the above-mentioned 4 operators are performed in parallel. The best individuals obtained in each operation create a new ith population. The ith population includes also individuals from the previous (i −1)th population. The number of individuals from a given GA operation and the (i −1)th population depends on the values of the weight coefficients imposed during the optimization process. Based on the conducted research and number of test calculations the authors noticed that the simplest and effective approach is a uniform selection of the best individuals given by each operator for the current and also the previous population. It means that the algorithm discussed here uses the same weighting coefficients, equal to 0.2 for each of the operators used, as well as the same number of individuals from the parent population. The calculation process at level III ends after reaching the desired number of populations Nⁱ. Then, at level II, the information about the best individual obtained from the calculations is stored in the BI_j table. In the case, when the number of iterations of N^j at level II has not been reached, a new start population is created and the level III calculations are repeated. The last stage (level I) of the proposed strategy includes changing the ranges of decision variables S_n (design space), where S_n𝜖〈S_{n, min}, S_{n, max}〉. In the discussed algorithm the change concerns narrowing of the design space within the currently best individual X_n^opt. In the presented approach the range of S_n in the kth iteration was changed according to the following formula: $\begin{eqnarray}\displaystyle (\mathbf{S}_{n})^{k+1}\in \langle \mathbf{S}_{n,\min }^{k+1},\mathbf{S}_{n,\max }^{k+1}\rangle =\langle (\mathbf{X}_{n}^{opt})^{k}-r^{-{\beta}}(\mathbf{S}_{n,\max }-\mathbf{S}_{n,\min }),(\mathbf{X}_{n}^{opt})^{k}+r^{-{\beta}}(\mathbf{S}_{n,\max }-\mathbf{S}_{n,\min })\rangle \cap \mathbf{R}_{+} & & \displaystyle\end{eqnarray}$ (1) where: (S_n)^k+1 represents the range (size) of the decision variables in the (k +1)th iteration; (X_n^opt)^k is a vector of decision variables for the best individual from the kth iteration; S_{n, min} and S_{n, max} represent, respectively, the lower and upper initial values of the decision variables, while S_{n, min}^k+1 and S_{n, max}^k+1 describe the minimum and maximum values of the decision variables in the (k +1)th iteration; r^−𝛽 describes the speed coefficient of narrowing of the S_n space, where 𝛽 = k and r ∈ (1,2〉. In the work, the value of the narrowing speed coefficient r was assumed equal to 1.15. The optimization process ends after reaching the preset number of iterations N^k of the level I calculations.

It should be noted that the application of the relationship (1) in the discussed algorithm not only allows for changing (narrowing) the S_n design space in the subsequent iterations, which directly translates into increasing the “resolution” of the GA in the considered sub “areas” of the design space, but also gives the opportunity to search for the optimum of the goal function outside the initially predefined area. Such functionality of the proposed algorithm is irreplaceable when the design space is unknown and difficult to define at the very beginning of the optimization process, and thus, when it is uncertain, whether the optimum of the objective function is within the selected area of the design space. In addition, by using the intersection of sets, in this case the common part between the set 〈S_{n, min}^k+1, S_{n, max}^k+1 〉 and the set of real positive R₊ numbers, it was ensured that values of decision variables will always be positive. It should be pointed out that the parameters of the equivalent circuits sought by the authors of the article are values greater than zero.

3. Calculation results

During the first stage of the investigations, the proposed strategy of searching for the extreme of the multimodal function in a multidimensional design space was tested by the authors on the example of the Foster Circuit for the system with eddy currents (Fig. 1a). The Foster circuit with the number of branches n =5 has been considered. For this case, the objective function of the following form was minimized: $\begin{eqnarray}\displaystyle f(\mathbf{R},\mathbf{L})=\left(\mathop{\sum }_{i}^{M}\left|{\displaystyle \frac{\text{Re}\left(\text{}\underline{Z}_{FEM}({\omega}_{i})-\text{}\underline{Z}({\omega}_{i},\mathbf{R},\mathbf{L})\right)}{\text{Re}(\text{}\underline{Z}_{FEM}({\omega}_{i}))}}+\frac{\text{Im}(\text{}\underline{Z}_{FEM}({\omega}_{i})-\text{}\underline{Z}({\omega}_{i},\mathbf{R},\mathbf{L}))}{\text{Im}(\text{}\underline{Z}_{FEM}({\omega}_{i}))}\right|^{2}\right)^{\frac{1}{2}} & & \displaystyle\end{eqnarray}$ (2) where: R(R = [R₁, R₂, …, R_n]) and L(L = [L₁, L₂, …, L_n]) are the single-column matrices representing the desired resistance and inductance values of the FC branch [11,13], respectively, while R_n, L_n > 0; Z _FEM denotes the impedance of the considered system in a complex form determined on the basis of a field model of the studied system, while the symbol Z represents the impedance values calculated on the basis of the searched values of the R and L parameters; M describes the number of samples identifying the given Foster Circuit for ith pulsation of the ω_i supply source.

The calculations for the above described optimization task were carried out on a population of 1000 individuals with starting resistance and inductance values equal to R_n $\epsilon$ (0,1000〉[𝛺] and L_n $\epsilon$ (0,1000〉[mH], respectively. It was assumed that the number of internal iterations: (a) Nⁱ at level III is equal to 20, (b) N^j at level II equals 100, and (c) N^k of level I equals 25. The obtained values of the decision variables R_n and L_n for the best individual in a given level I iteration have been summarized in Table 1. The values of the equivalent resistance R_c and inductance L_c seen from the side of the terminals of the considered system with eddy currents at a given frequency, calculated for the determined parameters of the Foster Circuit, have been shown in Fig. 4. The obtained characteristics have been compared with the equivalent parameters of the system calculated by means of field model parameters (R_FEM and L_FEM).

Table 1

The values of decision variables and fitness function of the best individual for kth iteration obtained for the Foster circuit using the proposed ES

	kth
	1	10	20	25
R₁, L₁	174.28 Ω,	17.20 Ω,	11.29 Ω,	10.72 Ω,
	802.94 mH	20.13 mH	15.01 mH	14.36 mH
R₂, L₂	2.56 Ω,	2.73 Ω,	2.81 Ω,	2.81 Ω,
	8.56 mH	9.04 mH	10.65 mH	10.96 mH
R₃, L₃	909.29 Ω,	2404.05 Ω,	2808.79 Ω,	2918.22 Ω,
	275.32 mH	186.06 mH	207.82 mH	210.60 mH
R₄, L₄	259.01 Ω,	313.30 Ω,	367.55 Ω,	384.65 Ω,
	591.52 mH	125.61 mH	123.56 mH	125.74 mH
R₅, L₅	161.08 Ω,	131.51 Ω,	147.75 Ω,	149.96 Ω,
	89.91 mH	649.85 mH	740.04 mH	744.44 mH
f (R,L)	1.4662	0.0391	0.0305	0.0295

Fig. 4.

Comparison of the equivalent impedance components calculated by means of the Foster Circuit (n = 5) and field model.

The optimization task presented above has been compared by the authors with (a) the task in which the classic genetic algorithm (GA) [2] was used to search for the optimal values of Foster equivalent circuit parameters and (b) the approach based on the algorithm proposed in the work, in which the operator of “random of new individuals” and the design space narrowing technique, were excluded. It was assumed that the values of the initial parameters in relation to the resistance and inductance will be the same as in the case study task, in which the algorithm of Fig. 3 was used, i.e. it was agreed that R_n $\epsilon$ (0,1000〉[Ω] and L_n $\epsilon$ (0,1000〉[mH]. For both tasks considered here, the values of the obtained Foster circuit parameters for the GA as well as the ES without the “random of new individuals” operator and the design space narrowing technique, are given in Tables 2 and 3, respectively.

Table 2

The values of decision variables and fitness function of the best individual for kth iteration obtained for the Foster circuit using the ES without operator “random of new individuals”

	kth
	1	10	20	25
R₁, L₁	354.17 Ω,	40.32 Ω,	90.59 Ω,	311.04 Ω,
	140.05 mH	143.64 mH	465.61 mH	471.24 mH
R₂, L₂	428.78 Ω,	631.38 Ω,	586.53 Ω,	585.87 Ω,
	814.97 mH	854.68 mH	459.73 mH	664.13 mH
R₃, L₃	378.56 Ω,	22.36 Ω,	27.46 Ω,	19.19 Ω,
	756.03 mH	173.11 mH	617. 97 mH	190.15 mH
R₄, L₄	793.60 Ω,	718.54 Ω,	792.29 Ω,	494.15 Ω,
	39.15 mH	159.89 mH	97.98 mH	86.19 mH
R₅, L₅	2.69 Ω,	3.74 Ω,	2.68 Ω,	2.92 Ω,
	4.29 mH	7.25 mH	6.52 mH	6.53 mH
f (R, L)	1.6811	0.5983	0.2923	0.1407

Table 3

The values of decision variables and fitness function of the best individual for kth iteration obtained for the Foster circuit using the classical GA [2]

	kth
	1	50	100	200
R₁, L₁	954.31 Ω,	454.81 Ω,	500.42 Ω,	375.62 Ω,
	464.45 mH	526.33 mH	500.75 mH	502.34 mH
R₂, L₂	226.73 Ω,	216.02 Ω,	71.51 Ω,	1.74 Ω,
	505.54 mH	506.55 mH	536.77 mH	612.51 mH
R₃, L₃	14.38 Ω,	14.41 Ω,	9.81 Ω,	8.81 Ω,
	366.86 mH	50.77 H	48.87 mH	47.83 mH
R₄, L₄	875.72 Ω,	376.22 Ω,	438.06 Ω,	492.69 Ω,
	118.39 mH	87.18 mH	86.85 mH	86.85 mH
R₅, L₅	11.31 Ω,	3.51 Ω,	3.54 Ω,	3.54 Ω,
	3.92 mH	7.71 mH	7.34 mH	7.34 mH
f (R, L)	9.8343	0.3282	0.1689	0.1403

Studying the summarized above results, it can be seen that the obtained values of the FC parameters are unsatisfactory in comparison with the results given in Table 1. Better concordance can be achieved by “manual” interference with the optimization task, i.e. changing the scope of decision variables. However, the problem to face is the answer to the question of how to define the space of the decision variables to ensure the desired accuracy of the FC parameters. One option is to increase the considered design space until satisfactory results will be obtained. Nevertheless, in such an approach repetitive calls of the algorithm are required and extending of the design space will lead to deterioration of “resolution” of the decision variable space under consideration. In this regard, the implementation of dependencies (1) in the code gives a significant advantage to algorithms in the selection of decision variable spaces.

Fig. 5.

Comparison of the equivalent impedance components calculated for the magnetizing branch of the transformer equivalent circuit after using the Cauer Circuit (n = 3) and field model.

Fig. 6.

Comparison of the equivalent impedance components obtained for the horizontal branches of the transformer equivalent circuit after using the Cauer Circuit (n = 3) and field model.

Effectiveness of the proposed strategy has been evaluated based on a number of case study problems. The results for the second considered system, i.e. a WPT system transformer, are presented and discussed below. The parameters of two forms of Cauer circuits [14], i.e. (a) RL Cauer Form I and (b) RL Cauer Form II (see, Fig. 2) have been determined. The R, L parameters of the CC have been calculated by the minimization of the goal function ((2)), however in the case of the Cauer Circuit (b), instead of Z_FEM the admittance Y_FEM of the system has been employed. Due to sufficient accuracy of the impedance fitting, the CC of the number of branches, n = 3, has been considered. The determined parameters of the CC have been used in the developed software, dedicated to the analysis of operation of a high frequency transformer, working in a WPT system. The considered WPT system consists of two pancake coils with field concentrators (ferrite blocks), see Fig. 2. The resistances R_σ and leakage inductances L_σ of the transformer windings have been represented by Cauer Circuit Form I (a), while the components of the magnetizing branch, i.e. R_M and L_M of the transformer, have been represented by means of Cauer circuit Form II (b). Values of the parameters of the CC have been calculated using the proposed approach, referring to the full 3D FEM model of the system employing the complex vector potential formulation 𝛺 − T − T₀ [15]. The reference values of R_{σ_FEM} and L_{σ_FEM} and the magnetizing branches R_{M_FEM} and L_{M_FEM} have been determined by mean of a 3D FEM model as a function of supply frequency. Next, based on the determined R_{σ_FEM}, L_{σ_FEM} and R_{M_FEM}, L_{M_FEM} versus frequency characteristics, the resistances R_n, R_Mn and inductances L_n, M_n, for both forms of the CC have been determined. Each time, the calculations for the considered optimization task were carried out on a population of 2500 individuals for initial (starting) values of the resistance and the inductance equal to R_n $\epsilon$ (0,5000〉[Ω] and L_n $\epsilon$ (0,5000〉[mH], respectively. It was assumed that the number of internal iterations: (a) Nⁱ at level III is equal to 25, (b) N^j at level II equals 100, and (c) N^k of level I equals 30. The calculated values of the resistances R_n and inductances L_n for the best individual in the kth iteration (i.e. values of decision variables giving minimal value of the goal function) have been summarized in Table 4 and Table 5, respectively, for CC Form I (a) representing the magnetizing branch of the transformer, and CC Form II (b) representing the winding resistances and leakage inductances. The comparisons between frequency dependence of the equivalent circuit parameters seen from the terminals, i.e. R_{M_CC}, L_{M_CC} and R_{M_FEM}, L_{M_FEM} have been shown in Fig. 5, while the corresponding comparisons of R_{σ_CC}, L_{σ_CC} and R_{σ_FEM}, L_{σ_FEM} are shown in Fig. 6. It can be noted that satisfactory concordance between the 3D FEM and the proposed here equivalent circuit approach has been achieved in a whole range of supply frequency.

Table 4

The values of decision variables and fitness function of the best individual for kth iteration obtained for the Cauer Circuit representing the magnetizing branch of the transformer equivalent circuit after using the proposed ES

	kth
	1	10	20	30
R_M1, M₁	1012.91 Ω,	808.01 Ω,	126,78 Ω,	127.52 Ω,
	60.99 μH	25.43 μH	25.83 μH	26.24 μH
R_M2, M₂	1381.51 Ω,	4211.42 Ω,	2098.95 Ω,	1578.77 Ω,
	1535.6 mH	150.17 μH	106.72 μH	105.38 μH
R_M₃, M₃	3113.07 Ω,	11082.1 Ω,	9195.34 Ω,	9000.02 Ω,
	1677.5 mH	254.98 μH	142.13 μH	138.74 μH
f (R_M, M)	4.2039	0.5109	0.0558	0.0223

Table 5

The values of decision variables and fitness function of the best individual for kth iteration obtained for the Cauer Circuit representing the horizontal branches of the transformer equivalent circuit after using the proposed ES

	kth
	1	10	20	30
R₁, L₁	0.1922 Ω,	0.15652 Ω,	0.15227 Ω,	0.15237 Ω,
	72.573 μH	14.0618 μH	12.0654 μH	12.0655 μH
R₂, L₂	1137.57 Ω,	746.41 Ω,	857.59 Ω,	855.52 Ω,
	426.67 mH	56.83 mH	50.81 mH	50.94 mH
R₃, L₃	1829.22 Ω,	1659.48 Ω,	2060.55 Ω,	2131.54 Ω,
	89.85 mH	10.78 mH	6.957 mH	7.076 mH
f (R, L)	26.11	0.862	0.00091	0.00057

4. Conclusions

On the basis of the presented results, it can be concluded that the proposed optimization strategy is well suited to finding the extreme of the multimodal functions of several variables and allows for the effective determination of the Foster and Cauer Circuits parameters for the systems with eddy currents. The proposed by the authors’ three-level optimization strategies can be effectively employed for solving other types of optimization problems characterized not only by multimodal objective functions but also for the creation of multi-degree-of-freedom design spaces. The advantage of the proposed strategy is the ability to impose the procedure of narrowing/focusing design space around the best individuals in the sequence of levels of the proposed optimization strategy. By using this procedure, the ES algorithms are better adapted to search for the optimum of multimodal goal functions in multidimensional design spaces, simultaneously allow for a higher resolution of this space.

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