Optimal Parameter Design by NSGA-II and Taguchi Method for RCD Snubber Circuit

A genetic algorithm and Taguchi method were used to optimize parameters for the residual-current device (RCD) snubber circuit of a DC-DC flyback converter. The most suitable algorithm was determined by using test functions to compare performance in three multi-objective optimization methods: non-dominated sort genetic algorithm-II (NSGA-II), multi-objective particle swarm optimization (MOPSO), and multi-objective differential evolution algorithm (MODE). Comparisons of coverage rate, distance between non-dominant solutions, and maximum walking distance showed that NSGA-II was superior to both MOPSO and MODE. Therefore, NSGA-II was used to obtain parameter values for the RCD snubber circuit. However, practical application of the parameter values was limited because the values could not meet the specifications required for real-world circuits. Thus, the parameter values obtained by NSGA-II were used in further factor-level experiments performed by Taguchi method. The experimental results indicated that, compared to previous design methods, the proposed NSGA-II and Taguchi method obtains better parameter values for the RCD snubber circuit.


I. INTRODUCTION
The residual-current device (RCD) snubber is usually used in flyback converter, in order to limit the voltage spikes caused by leakage inductance of the transformer [1]- [3]. This converter is an essential power electronics product that is widely used in many industries, including manufacturing, medicine, aerospace, and the defense. The stable, high quality power supplied from this converter directly enhances performance and safety in many electronic products. Therefore, how to select the proper snubber circuit and obtain the optimal design is an issue worth investigating [4], [5]. Sun et al. [4] proposed an improved genetic algorithm to optimize the design of AC-DC converter with snubber in switching The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney. power amplifier. Huang et al. [5] gave a single-objective optimal design method of DC-DC converter with an RCD snubber by using a genetic algorithm and the Taguchi method. An economical, practical, and efficient circuit design procedure is provided by Huang et al. [5], [6]. However, Huang et al. [5] only considered the single-objective problem of reducing the spike voltage. Reducing energy loss on the RCD snubber circuit is also an issue worth investigating. However, to the authors' best knowledge, there are no literatures to studying the optimal problem of reducing both spike voltage and energy loss on the RCD snubber circuit.
As the intelligent manufacturing industry evolves, singleobjective optimization methods have increasingly revealed limitations in meeting industrial demands. Therefore, multiobjective optimization has been widely used in the product design and production processes of various industries in recent years. The main purpose of multi-objective optimization is to resolve conflicts among various objectives [7]- [10]. Multi-objective optimization methods currently used in the industry are often derived from single-objective optimization methods such as genetic algorithm (GA), particle swarm optimization (PSO) and differential evolution algorithm (DEA).
The first multi-objective optimization algorithm, vector evaluated genetic algorithm (VEGA), was proposed by Schaffer [11]. The VEGA is easy to apply but can only find the best solution for a single objective. Improvements in VEGA proposed by Hajela and Lin [12] and by Fonseca and Fleming [13] resulted in the weight-based genetic algorithm and the multi-objective genetic algorithm (MOGA), respectively. The non-dominated sorting genetic algorithm (NSGA) was introduced by Deb and Srinivas from MOGA [14] and further refined by Deb et al. [15]. The original NSGA had two shortcomings: it was incapable of finding the best solution, and it tended to fall into the regional solution. The second generation NSGA (NSGA-II) addressed these shortcomings by applying an elite dominated strategy [15]. The many applications of NSGA-II have included pipe routing for aviation engines [16] and electricity price prediction [17]. Coello and Lechunga [18] proposed grid-based multi-objective particle swarm optimization (MOPSO) based on grid selection. A modification of MOPSO in 2004 enhanced its search capability by considering domination and non-domination relationships and by applying perturb and observe method [19]. A notable application of MOPSO is in cloud computing [20] to maximize broker profit while minimizing time cost. Abbass et al. [21] proposed the Pareto differential evolution algorithm (PDEA), which applied the dominance relation in DEA. Babu and Jehan [22] proposed the use of DEA in multi-objective optimization (MODE). Kukkonen and Lampinen [23] introduced the third evolution of DEA, which increased diversity of understanding. Huang et al. [24] proposed the self-adaptive differential evolution algorithm and Mirjalili [25] proposed a multi-objective optimization method based on dragonfly algorithm [26]. The MODE has been used to optimize the operating point of an auxiliary power unit to reduce its emissions [25].
Nowadays, one of popular solutions for the multiobjective optimal design problem is combined experimental design method with multi-objective optimization algorithm. Le Chau et al. [27] combined Taguchi method with DEA to solve the multi-objective optimization problem for a leaf compliant joint for micro-positioning systems. Dao and Huang [28] integrated Taguchi method with fuzzy logic to optimize a broad self-amplified 2-DOF monolithic mechanism. Dao et al. [29] utilized Taguchi method to obtain ranges for parameters, and then cuckoo search algorithm find the optimal parameter values according to the ranges obtained by Taguchi method. Huang and Dao [30], and Huynh et al. [31] exploited Taguchi method with grey relational analysis to optimize design of a 2-DOF flexure-based mechanism and the compliant mechanism flexure hinge, respectively. The validation results from those methods dovetailed nicely with the simulations.
Since follow-up studies have not adequately addressed the multi-objective optimal design problem of RCD snubber, this study proposes a multi-objective method for simultaneous optimization of reducing both spike voltage and energy loss on the RCD snubber circuit. The proposed multi-objective optimization method has many potential applications in circuit design. The experiment in this study used a flyback converter circuit as an example to verify the proposed approach.
Because the best solution found by the meta-heuristic optimization method cannot be directly used in practical applications, to meet the real-world component configuration, the best solution found by the meta-heuristic optimization method corresponds to several numerically approximated components, and then use the Taguchi method to find the most suitable configuration of real components and produce an approximate best solution that fits the real world.
This paper is organized as follows. Section 2 introduces the experimental design of the system. Section 3 introduces the design and implementation of three common multiobjective optimization methods. Section 4 presents the procedure for using Taguchi method to optimize the parameters and the design of RCD snubber circuit hardware. Finally, Section 5 concludes the study.

II. EXPERIMENTAL DESIGN A. INTRODUCTION OF EXPERIMENTAL OBJECTS
The flyback converter circuit has several advantages, including its low cost and simple structure. Because it can implement multiple outputs, this circuit is widely used in auxiliary power systems that supply power to the whole system. To enhance power and to meet safety standards, practical applications of converter circuit designs must isolate input and output. The flyback buck-boost converter uses a coupled inductor for energy conversion. The storage and release of magnetic energy must be considered in the overall circuit design. Since the circuit structure of the flyback converter has no vibration characteristics, electrical isolation of the circuit is not required. Figure 1 shows the basic flyback converter equivalent circuit, where V in is input voltage; Q is an N channel MOSFET power switch; L m and L e are the transformer primary side excitation inductor and transformer primary leakage inductance, respectively; n is the transformer turn ratio; and D o , C o , and V o are the output diode, output capacitor, and output voltage, respectively.
A major limitation of the basic flyback converter is that magnetic inductor L m and leakage inductance L e in the transformer itself. To reduce voltage spikes and ringing noise, Hren added a basic flyback converter to an RCD snubber in Hren et al. [32]. Adding the flyback converter not only reduces noise and ringing phenomena, it also provides an energy path for the release of leakage inductance on two sides, which is essential for sharing the cross voltage for the power switch. Figure 2 shows the loop circuit, which is also known as a soft-switched flyback converter.  Energy loss in the RCD snubber circuit can be divided into two stages. This study considered both stages. Stage 1 is t 1 to t. When the dipole is directed, the energy stored leakage inductance L e is released through dipole D s to capacitor C s . According to the law of conservation of energy, energy loss in the circuit (W 1 ) can be indexed by subtracting the energy stored in the leakage inductance of transformer L e and the energy on capacitor C s from the energy on the transformer as shown in (2.1) below: where I Q denotes the peak voltage of the power switch, V cp is the initial voltage of capacitor C s ; V cH is the conduction state of the dipole. The final voltage value for capacitance C s is defined as follows: Stage 2 is t to t 1 . When the energy stored in C s is released at R s , energy loss W 2 from resistance R s at this time can be defined as The total energy loss P loss on the RCD snubber circuit can be defined as This study considered the optimal design of the smaller the better of two conflicting targets in the RCD snubber circuit: (1) Reducing the peak voltage on the power switch; (2) Reducing the energy loss on the RCD snubber circuit. The objective functions are as follows: Objective function 1 (peak voltage): Objective function 2 (total energy loss in snubber circuit):

C. SELECTION FOR CONTROL PARAMETERS
Based on the above objective functions analysis, the peak value of power switch V ds_pk and total energy loss P loss are affect by electricity leakage L e , snubber capacitor C s and resistance R s . The control parameters considered in this paper are the three circuit components R s , C s and L e . The range of resistance R s and capacitance C s is the main consideration in ready-made products on the market, where R s is [1 -10] (unit: K ), C s is [1 -100] (unit: nF), and L e is [32, 43.5, 58] (unit: µH ) from three self-winding transformers according to the parameters calculated in this experiment. Figures 3 and 4 show that two objectives change with snubber resistance R s and that two objectives change with snubber capacity C s , respectively. Figure 3 shows that, as R s increases, peak voltage increases but loss decreases. Figure 4 further shows that, as C s increases, peak voltage decreases, but loss increases, however, the increase is not obvious. Figures 5-6 show how the efficiency of the converter increase with R s and C s , respectively.
The above discussion reveals that the optimal design must resolve the conflict between the two objectives. Therefore, this study used a multi-objective optimization method to   design the optimal circuit given the conflict between two targets. The effect of reducing voltage surge V ds_pk and snubber loss P loss on power switch Q were then determined.

D. PARAMETER SETTINGS FOR OPTIMIZATION
This study compared three multi-objective optimization methods: NSGA-II, MOPSO and MODE. Table 1 shows the parameter settings used in the comparisons. In the experiment, the three multi-objective optimization methods were used to calculate the optimal solution space when using the parameters shown in Table 1. The number of function evaluation is a critical criterion to compare optimization methods. The number of function evaluation is the number of evaluating main objective functions during iteration. In the  study, the greatest number of function evaluations is 6 × 10 4 . NSGA-II does not adopt an external storage (archive) for the Pareto optimality. Instead, each generation first performs genetic operations on the population to obtain subpopulations and then merges the two populations to perform non-inferior ranking and crowding distance ranking to obtain a new generation of populations until the end of the calculation. Therefore, there is no need to set the number of archives in NSGA-II.

A. NON-DOMINATED SORTING GENETIC ALGORITHM BASED ON ELITE STRATEGY (NSGA-II)
The NSGA-II was proposed by Deb and Srinivas [14] and Deb et al. [15]. The novel feature of NSGA-II is that nondominated sorting and clustering are used to form several different levels of Pareto front solutions in the solution set. The NSGA-II adds the concept of crowding distance and competitive choice. The algorithm not only reduces computational complexity, it also overcomes the limitation of the original NSGA, which is its tendency to fall into regional solutions. The NSGA-II also reduces computational complexity to O(mN 2 ). The procedure of the NSGA-II is roughly similar to that of the GA. That is, it generates offspring groups by selection, crossover, and mutation, and then performs union actions with the parents. Then, the chromosomes of offspring groups and parents are sorted by the non-dominated solution, and the crowding degree of each non-dominated level parameter is calculated. Finally, according to non-dominated relationship and crowding degree, the best chromosome is chosen to enter the next generation until the termination condition is met. Figure 7 shows a flowchart of the NSGA-II.
The crossover and mutation operations of the NSGA-II were performed by simulated binary crossover [33]   and polynomial mutation operator [34], respectively. These two operators are used to deal with the upper and lower bound. The settings for NSGA-II controllable parameters would be strongly affected the responses. The authors have many experiences using the Taguchi experimental method to set parameters to improve system performances [35]- [37]. Therefore, in the study, the Taguchi method was used to arrange the settings for controllable parameters. The settings for NSGA-II were crossover rate = 0.9, mutation rate = 0.1, η c = 20, and η m = 20. Figure 8 shows the best solution space obtained. Table 2 shows the 10 sets of representative parameters chosen from 200 sets in the solution space.

B. MULTI-OBJECTIVE PARTICLE SWARM OPTIMIZATION (MOPSO)
The main architecture of the evolution process in this study was MOPSO, which was first proposed by Coello et al. [19]. This optimization method uses external temporary data to  retain the non-dominated solution set in each iteration. The particles are then guided to the best solution by the nondominated solution in the external temporary data. This optimization method increases the capability to search for the best solution. Figure 9 is a flow chart of the MOPSO used in this study.
In MOPSO, c 1 , c 2 , and χ are set to 2.05, 2.05, and 0.73, respectively, obtained by the Taguchi method. Figure 10 shows the best solution space obtained by MOPSO. Table 3 shows the ten representative parameter sets chosen from the 200 parameter sets in the solution space.

C. MULTI-OBJECTIVE DIFFERENTIAL EVOLUTION ALGORITHM (MODE)
Many different methods of applying MODE have been proposed in recent years [21]. This study used the MODE   architecture proposed by Huang et al. as the main architecture of the evolutionary process [24]. Figure 11 is a flowchart showing how MODE applies the concept of external temporary data. The crossover rate of the MODE in this study was set to 0.6 and F was set to [0, 2], obtained by the Taguchi method. Figure 12 shows the best solution space obtained by MODE, and Table 4 shows the 10 representative parameter sets chosen from 200 sets in the solution space. Figure 13 compares the performance of the three multiobjective optimization methods. Figure 13 shows that NSGA-II outperformed MODE and MOPSO. The MODE had the worst diversity. Since human judgement of optimization performance is unreliable, this study applied FIGURE 11. Flowchart of multi-objective differential evolution algorithm. and discussed optimization performance evaluation methods described in the literature.

1) COVERAGE METRICS
For a clear performance comparison of different algorithms, the best non-dominated solutions obtained by multi-objective optimization were analyzed and compared. Here, coverage metrics (C) were calculated by two non-dominated solutions [38]. The calculation was the ratio of all solutions in non-dominated solution set V to all solutions dominated at U. The function was as follows: where C (U , V ) = 1 means that all the non-dominated solution sets in V are dominated by U . Otherwise, VOLUME 8, 2020

2) INTERVAL OF NON-DOMINATED SOLUTIONS (S)
The objective function values must be normalized before calculating the distance between the non-dominated solutions and the maximum dispersion distance. The distance between non-dominated solutions is calculated mainly to evaluate the distribution density of non-dominated solutions [39].
where f m is the objective function value of the non-dominated solution.

3) MAXIMUM DISPERSAL DISTANCE
The maximum dispersal distance (D) is defined as farthest distance between two non-dominated solutions in the set of non-dominated solutions. A large value for D indicates a wide distribution of optimization results. The calculation is as follows.
To evaluate optimization performance, the interval between non-dominated solutions and the maximum dispersal distance is calculated for each of the three optimization methods. The coverage metrics between the optimizations is then calculated. For each of the three optimization methods, Table 5 shows the interval between the non-dominated solutions and the maximum dispersal distance. Table 6 further compares the coverage metrics between NSGA-II and MOPSO, between NSGA-II and MODE, and between MODE and MOPSO.

4) CONCLUSIONS OF PERFORMANCE EVALUATION
The experimental results in Tables V-VI show that each of the three multi-objective optimization methods revealed advantages and disadvantages. The average values obtained for the 10 representative parameter sets were used for the performance comparisons. The performance comparison results were as follows: Coverage rate C (bigger the better): NSGA-II > MODE > MOPSO. Maximum dispersal distance D (bigger the better): NSGA-II > MOPSO > MODE. Non-dominated solution space S (smaller the better): NSGA-II < MOPSO < MODE. From above results and discussion, NSGA-II has better performance than MODE and MOPSO. Therefore, this paper used NSGA-II to optimize the design of an RCD buffer circuit.

E. TAGUCHI METHOD
The Taguchi method is a quality design method compiled and proposed by Taguchi [40]. Its main concept is to conduct experimental design in a systematic way to understand the causes of product function variation in the most efficient way and try to find the ideal quality. The Taguchi method can effectively reduce experimental costs and obtain highquality products. The use of orthogonal arrays for experimental planning is the most important step of the Taguchi method. Generally speaking, L a (b c ) is the symbol of the orthogonal array, a represents the number of experiments, b is the level number, and c is the number of factors that can be placed. Therefore, according to the needs, the appropriate orthogonal array can be selected for experimental configuration. Table 7 is a commonly used orthogonal array of L 8 (2 7 ). After the experiment configuration is completed and the experiment is executed, Table 8 is an example of L 8 (2 7 ). In Table 8, A 1 is the first level of factor A, A 2 is the second level of factor A, and the other factors and levels can be deduced by analogy; y is the experimental result, and η is the S/N ratio converted from the experimental result. Then, start to make a response table and draw a response chart. Table 9 and Figure 14 are examples of the response table and response chart, respectively. The response values of each level of each factor are averaged from the experimental results corresponding to the level. For example, the response value of A 1 in the response table is averaged from η 1 , η 2 , η 3 , and η 4 , as shown in Eq. (3.5).
The response values of other factors and their levels are also calculated in the same way. The response chart is drawn from each response value in the response table. Finally, the best combination of factors can be obtained through the response table and response chart.

IV. OPTIMAL DESIGN OF RCD SNUBBER CIRCUIT
This study then used the NSGA-II to optimize the design of an RCD snubber circuit in a flyback converter. Figure 15 is a flowchart showing how the NSGA-II multi-objective optimization method searches within the wide range of the solution space and then converges the best non-dominated solutions in the best solution space. To obtain the optimal   design, the parameter combination most suitable for the user is then selected from the solution set and adjusted by the Taguchi method with actual passive components.
Although there are some softwares, like GAMS that is nonlinear optimizer, the GAMS is a solver for single-objective problems. However, this study is a multiple-objective problem including both objectives of V ds and P loss . The classical algorithms for multi-objective problems include NSGA-II, MOPSO, and MODE. Therefore, this study used the three algorithms to solve the RCD buffer circuit problem and   compared them in term of performance. From the results, NSGA-II has better performance among the three algorithms.
Comparisons of the three algorithms confirmed the superior performance of the NSGA-II. Therefore, NSGA-II was then used to find the optimal solution space. Since multiobjective optimization obtains elastic sets of solutions instead of a single optimal solution, users can choose suitable solutions from the solution set according to their needs. Since the two objectives in this experiment had equal importance, the trade-off parameters were selected as the snubber parameters of the flyback converter. Table 10 shows a set of parameters obtained by NSGA-II and previous design. Figure 16 shows the optimization results.
The results in Table 10 are continuous theoretical values. In a real-world application, components with these best-case parameters would probably be unavailable. Therefore, the orthogonal experiment in this study was performed using the actual values for three commercially available models   to approximate the theoretical values obtained by NSGA-II. Table 11 shows the control factors and levels.
Since the three-factor experiment with three levels requires 6 degrees of freedom, the orthogonal array selects L 9 (3 4 ). In this case, the S/N ratio of the smaller the better was adopted. The V ds and P loss were obtained by direct measurement. To prevent the use of different units from affecting the results, data for the two objectives were normalized before the experiment was executed. Table 12 shows the orthogonal array and data. Table 13 and Figure 17 are the response table and chart, respectively, obtained according to the S/N ratios in each experiment. The data in the table and figure show that the best parameter combination in the experiments was A 1 B 3 C 1 .
Finally, a validation experiment was performed using the best parameter combination. Table 14 shows the trade-off best results, obtained by the NSGA-II, MOPSO, and MODE, and the optimal results, achieved from the method obtained by NSGA-II and tuned by the Taguchi (NSGA-II-Taguchi) method, and comparison with the previous design method. Figure 18 shows the actual circuit with the   optimized parameters. Figure 19 shows the V ds waveform with V dsp = 162V . A comparison with the waveform before optimization (Figure 20 with V dsp = 190V ) reveals a substantial improvement. Reduction percentages, compared with the previous design result, for V ds and P loss by the NSGA-II-Taguchi method are 14.74% and 2.31%, respectively. Reduction percentages, compared with the previous design result, for V ds obtained by the NSGA-II, MOPSO, and MODE are 13.68%, 11.58%, and 13.16%, and for P loss obtained by the NSGA-II, MOPSO, and MODE are 1.87%, 1.60%, and 0.77%, respectively. From Table 14, the trade-off best result obtained the NSGA-II-Taguchi method has better performance.
Huang et al. [5], [6] used a GA-related algorithm and the Taguchi method to optimize the design of DC-DC converter for spike voltage reduction. In this paper, the authors attempt to just consider the influence of peak voltage. For comparing performances between the NSGA-II-Taguchi method and the method of Huang et al. [5], [6], the V ds is regarded as the objective. The V ds can be obtained 125V   with R s = 1 k , C s = 100 nF, and L e = 32 µH by the NSGA-II-Taguchi method. Table 15 shows the experimental results obtained by the NSGA-II-Taguchi method and the method of Huang et al. [5], [6] and comparison with the previous design method. Figure 21 shows the V ds waveform with V dsp = 125V , and it also indicates a more substantial improvement by contrast with Figure 20. The optimized result is also better than that with V dsp = 138V obtained by the method of Huang et al. [5], [6]. Reduction percentage, compared with the previous design result, for V ds obtained by the method Huang et al. [5], [6] is 27.37% and for V ds obtained by the NOTT is 34.21%. Therefore, the optimal method obtained by the NSGA-II-Taguchi method for designing a DC-DC converter with an RCD snubber is effective and feasible.

V. CONCLUSION
This study used NSGA-II and Taguchi method to optimize the parameter values for an RCD snubber circuit of a DC-DC flyback converter. The above discussion of the working principle of the RCD snubber circuit in the flyback converter described the circuit components that affect the power switch voltage loss and circuit energy loss, which were used as parameters. Three multi-objective optimization methods were implemented and analyzed: NSGA-II, MOPSO, and MODE. Comparisons confirmed the superior performance of NSGA-II. Additionally, the specific steps needed to implement a practical design after the optimal solution space is calculated by the NSGA-II algorithm. By combining NSGA-II algorithm with Taguchi method, the proposed method can obtain the optimal design without the need for practical experience or trial-and-error method. At the same time, the proposed method solves the problem of theoretical optimization results being restricted by the currently existing component specifications. The design results were realized in the fabrication of a printed circuit board. The experiment verified that performance with the proposed parameter optimization is superior to those without parameter optimization as well as with parameter optimization given by Huang et al. [5], [6]. Notably, the process required only nine experiments, which substantially reduced circuit design time and cost. For application in industries, the proposed NSGA-II-Taguchi method can quickly find design parameters and reduce the complexity of considering both spike voltage and energy loss on the RCD snubber circuit. From the real applied responses of Metal Industries Research and Development Centre (www.mirdc.org.tw) and cooperation manufacturers, it has been approved that the RCD snubber circuit designed by the proposed NSGA-II-Taguchi method can improve the effectiveness of the equipment in reducing spike voltage and energy loss. Thus, the proposed optimization method is both economical and practical.