Multi-objective optimization of parameters for magnetically coupled resonant wireless energy transmission systems

Wen, Chunming; Chen, Minbo; Xu, Qing; He, Qiuli; Wu, Jiarong; Zhao, Xiaohui; Liang, Yuanxiong; Liang, Kairong

doi:10.1063/5.0131124

To improve the transmission efficiency and output power of a magnetically coupled resonant wireless energy transmission system, this paper proposes a multi-objective optimization method based on the analytical expression and the improved NSGA-II algorithm. First, the analytical expressions of the electrical and performance parameters in the magnetically coupled resonant mechanism are derived. On this basis, the mathematical model of the multi-objective optimization of the magnetically coupled mechanism is established, with the size, many turns, and wire diameter of the transmitting and receiving coils in the magnetically coupled mechanism as design variables; the resonant frequency of the system, the distance between the coils, and the load size as constraints; and the transmission efficiency and transmission power as optimization objectives. Then the mathematical model is solved by the improved NSGA-II algorithm to obtain the Pareto optimal solution. Finally, the physical system is built to verify the correctness of the simulation experiment and the reliability of the experiment. This study is instructive for optimizing the transmission efficiency and output power of a magnetically coupled resonant wireless energy transmission system.

I. INTRODUCTION

With the globalization of economic development, all countries in the world are experiencing huge pressure of energy shortage and environmental pollution, and for countries with large populations such as China, the environmental pollution problems brought about by economic growth are particularly prominent. Wireless energy transmission technology refers to the electric energy transmission technology that directly converts electric energy into electromagnetic waves and acoustic waves without wires or other physical contacts and transmits energy to the load through space. This technology realizes complete electrical isolation between power and load, which will greatly facilitate charging and guarantee safety, and it is conducive to promoting the development of the new energy vehicle industry. Therefore, the world’s major developed countries have attached great importance to the research of electric vehicle radio energy transmission technology.

Electric vehicle electric energy transmission technology is mainly divided into traditional wired charging and wireless charging methods. Traditional wired charging mainly uses the power cord plug-in charging, where the transmission process needs a wire connection; long time use will cause line wear leakage, electric sparks, and other safety hazards; and the charging pile occupies a lot of land, resulting in urban land resource tension. However, the wireless charging method can effectively avoid the disadvantages of the traditional way. Wireless power transmission is free from the dependence of traditional power transmission on wires, avoiding the frequent unplugging of power wires in the traditional wired charging method; solving the problems of being unsafe, unattractive, and unreliable and easy wear and tear; and making the system simple and at the same time conducive to later operation and maintenance. At the same time, the new wireless charging method has the advantages of high output power, high transmission efficiency, no additional installation site, etc. For users, it can be charged immediately after stopping, which greatly simplifies the operation process of car charging and has a good market prospect. Along with the development of smart grids and energy internet, to reduce energy consumption and carbon emission in transportation and guide and support the healthy development of new energy vehicle industry chains, electrochemical energy storage, and battery materials, various countries have proposed to vigorously develop electric vehicle wireless energy transmission technology.

Magnetically coupled resonant radio energy transmission charging efficiency is high and is currently more widely used in the field of electric vehicle wireless charging. Its principle is the use of high-frequency AC power in the primary coil to generate a high-density alternating magnetic field while the receiving coil couples the energy in the magnetic field. This coupled energy will be delivered to the electric vehicle receiving system, resulting in the magnetic resonance phenomenon. It is because of the many advantages of magnetically coupled radio energy transmission technology that it has become a key research area for electric vehicle wireless charging technology.

There are numerous studies on magnetically coupled resonant systems at home and abroad. Chan and Chau (2001)¹ proposed the concept of electric vehicle development to develop wireless electric vehicles from different angles and directions to promote the level of electric vehicle technology so that the commercialization of electric vehicles can be realized. Qiu et al. (2014)² proposed a quantitative comparison of magnetic couplers for electric vehicle wireless charging applications. Using finite element analysis, the performance of each type of coupler was evaluated based on magnetic flux distribution and effective charging area. Huang et al. (2014)³ proposed a radio energy transmission system with magnetic resonance coupling and applied the coupled mode theory to obtain an expression for the system efficiency. The system efficiency is improved by optimizing the key parameters of the system. The design was passed. The effectiveness of the proposed optimization method is verified by simulation and experimental results. Zhang et al. (2015)⁴ proposed an adaptive wireless power transfer technique to compensate for the effects of misalignment on electrical performance. The key to the proposed wireless power transfer technique is that the power supply unit uses a gapless alternating winding topology to generate a uniformly distributed electromagnetic field and a vertical and horizontal coil design to enhance energy acquisition. Wang et al. (2011)⁵ proposed the use of metamaterials to enhance coupling and improve the transmission efficiency of a wireless power transfer system based on coupled resonators. The measurement results show that the metamaterials can significantly improve the power transfer efficiency of the system and the fabricated system can be used to wirelessly transmit power to a 40 W light bulb. El-Shahat and Ayisire (2021)⁶ proposed a 3D model of transmitter and receiver coils for electric vehicle charging using magnetic resonance wireless power developed with ANSYS Maxwell. The model was integrated into the physical design of the magnetic resonance coupling using ANSYS Simplorer to optimize power consumption. The estimated efficiency is about 92.1%. Liu et al. (2018)⁷ proposed a shape-reconfigurable modular magnetically coupled resonant wireless power transfer (MCR-WPT) array system for multipurpose wireless charging applications. The proposed system achieves shape reconfiguration by combining different positions of resonant modules and structures without any control circuitry or any variation in resonant and matching capacitance. Cheng et al. (2018)⁸ proposed a novel wireless power transfer (WPT) system with repeater coils for multiple loads. The proposed proper resonance condition allows obtaining a constant load current for all loads when the resonance resistance of the coil is neglected. Guo et al. (2022)⁹ proposed a multi-parameter joint identification method at the receiver side of wireless power transfer (WPT), which enables the joint identification of several important parameters, such as battery voltage, charging current, rectifier bridge equivalent input impedance, and voltage and current at the receiver side, using only one measured voltage at the WPT transmitter side without the need to know the coil coupling state in advance. Tang et al. (2017)¹⁰ proposed the concept of optimal resonant frequency and efficacy synchronization factor for magnetically coupled resonant wireless energy transmission systems to derive the conditions under which the system transmission efficiency and power maximum occur synchronously. By comparing the experimental data and simulation results, the correctness of the theoretical analysis is proved, as well as for further optimization of the system efficacy synchronization performance. Xu et al. (2020)¹¹ proposed a method using split frequency to solve the problem of load power reduction in the over-coupled region while ensuring a higher efficiency of the system, effectively increasing the received power of the load. Jiang and Han (2019)¹² proposed a study of the transmission efficacy problem of magnetically coupled resonant wireless energy transmission systems to investigate the relationship between transferred power, transmission distance, transmission efficiency, and input frequency and load. Discussions and simulations are conducted to find the conditions that maximize the transmission efficacy. The conditions to maximize the transmission power are discussed and simulated.

The literature study mentioned above is of great importance for the improvement of the working characteristics of wireless energy transmission. However, the contents are mainly studied for a single influencing factor or a single target without considering the variation in system transmission characteristics in real situations with multiple factors and targets. In current wireless energy transmission systems, the maximum values of load power and transmission efficiency generally do not increase simultaneously at the same operating frequency. When a small electric vehicle is charged, as time increases, the output power decreases when the transmission efficiency increases. When the transmission efficiency decreases, the output power increases. Therefore, this paper explores and studies the optimization state of magnetically coupled resonant wireless charging technology, analyzes the circuit model affecting the transmission system, and analyzes the effects of unoptimized parameters, single optimized parameters, and simultaneous optimization on the system. In addition, some corresponding optimization methods are proposed to discuss the relationship between system transmission characteristics and load parameters and the design at different resonant frequencies to maximize the transmission efficiency and output power. Therefore, it is of great practical importance to study magnetically coupled wireless energy transfer systems.

This paper is organized in the following way. To address the problem that multi-objective optimization is easy to fall into local optimal solutions and to combine the variation in system transmission characteristics with multiple factors and multiple objectives in real situations, this paper establishes a mathematical model of radio energy transmission in magnetically coupled structures based on analytical expressions and derives expressions for the relationship between electrical energy transmission efficiency and circuit parameters based on circuit theory. The power and efficiency of the mathematical model is solved by using the improved Non-dominated Sorting Genetic Algorithm II (NSGA-II), with transmission distance, load resistance, and resonant frequency as key design parameters, and the fast non-dominated sorting strategy, congestion strategy, elite selection strategy, and binary tournament strategy are used to obtain the Pareto optimal solution set and analyze the effect of changing the parameter variation on the system power and efficiency. Experiments show that this multi-objective algorithm can greatly reduce the computation time, find the optimal solution quickly, and improve convergence and distributivity. Finally, a prototype physical system is established to prove the feasibility of the improved NSGA-II and to verify the effectiveness and reliability of the system, which is a guideline for optimizing magnetically coupled resonant wireless energy transmission systems.

II. ANALYSIS OF THE MAGNETICALLY COUPLED RESONANT TRANSMISSION MODEL

The radio energy transmission method is based on resonance and relies on the resonance generated by the transmitting resonant coil and the receiving resonant coil to realize radio energy transmission. When the system is in ideal resonance, the inductive and capacitive loads cancel each other, and the system is in a resistive state. At this time, the transmitting and receiving coils have the lowest impedance and the highest load current, and the energy generated by the magnetic field is transmitted with the lowest loss. When the system is in the detuned state, the transmitting coil cannot provide energy to the receiving coil normally, or it cannot even transmit energy. Therefore, the magnetically coupled resonant wireless energy transmission system needs to consider the parameter matching state. Figure 1 shows the equivalent circuit of a magnetically coupled wireless energy transmission network. In Fig. 1, U_m is the output voltage of the power supply, R_s is the internal resistance of the power supply, M is the mutual inductance between the transmitting and receiving coils, L₁ and L₂ are the equivalent mutual inductance of the transmitting and receiving coils, R₁ and R₂ are the equivalent loss resistance of the transmitting and receiving coils, C₁ and C₂ are the external series tuning capacitors of the transmitting and receiving coils, respectively, and R_L is the load resistance.

FIG. 1.

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Equivalent circuit of the magnetically coupled wireless energy transmission network.

The relationship between voltage and current can be derived from Kirchhoff’s law as follows:

(\begin{matrix} (R_{1} + R_{s} + j ω L_{1} + 1 / j ω C_{1}) {\dot{I}}_{1} - j ω M {\dot{I}}_{2} = {\dot{U}}_{s}, \\ - j ω M {\dot{I}}_{1} + (R_{2} + R_{L} + j ω L_{2} + 1 / j ω C_{2}) {\dot{I}}_{2} = 0 . \end{matrix}\}

(1)

When the transmitting coil resonates with the receiving coil,

(\begin{matrix} {\dot{I}}_{1} = \frac{{\dot{U}}_{S} (R_{L} + R_{2})}{(R_{S} + R_{1}) (R_{L} + R_{2}) + ω^{2} M^{2}}, \\ {\dot{I}}_{2} = \frac{j ω M {\dot{U}}_{S}}{(R_{S} + R_{1}) (R_{L} + R_{2}) + ω^{2} M^{2}} . \end{matrix}\}

(2)

Then the complex power of the system is

\{\begin{cases} S_{1} = U_{s} I_{1}^{*} = (R_{s} + R_{1}) I_{1} I_{1}^{*} + j ω M I_{2} I_{1}^{*}, \\ S_{2} = j (R_{L} + R_{2}) I_{2} I_{2}^{*} + j ω M I_{1} I_{2}^{*} . \end{cases})

(3)

In the above-mentioned equation, $I_{1}^{*}$ and $I_{2}^{*}$ are the complex power currents, and the load power is

\begin{align} P_{L} = {|I_{2}|}^{2} R_{L} & = \frac{{(ω M U_{s})}^{2} R_{L}}{{[{(ω M)}^{2} + (R_{s} + R_{1}) (R_{L} + R_{2})]}^{2}} \\ = \frac{U_{s}^{2}}{R_{s} + R_{1}} \cdot \frac{R_{L}}{R_{L} + R_{2}} \cdot \frac{k^{2} Q_{1} Q_{2}}{{(1 + k^{2} Q_{1} Q_{1})}^{2}}, \end{align}

(4)

where k is the coupling coefficient. It is known that

k = \frac{M}{\sqrt{L_{1} L_{2}}}, Q_{1} = \frac{ω L_{1}}{R_{1} + R_{s}}, Q_{2} = \frac{ω L_{2}}{R_{2} + R_{L}} .

(5)

Then the system transmission efficiency is

\begin{align} η = \frac{P_{L}}{P_{s}} = \frac{{|i_{2}|}^{2} R_{L}}{U_{s} |i_{1}|} & = \frac{R_{L}}{R_{L} + R_{2}} \cdot \frac{ω M}{({[ω M)}^{2} + (R_{s} + R_{1}) (R_{L} + R_{2})]} \\ = \frac{R_{L}}{R_{L} + R_{2}} \cdot \frac{k^{2} Q_{1} Q_{2}}{1 + k^{2} Q_{1} Q_{1}} . \end{align}

(6)

The formula for calculating the resistance of the transmitting coil and the receiving coil is

R_{i} = \sqrt{\frac{ω μ_{0}}{2 δ}} \frac{n_{i} r_{i}}{a},

(7)

where a is the coil copper wire diameter, μ₀ is the vacuum permeability, δ is the coil copper wire conductivity, and n_i and r_i are the number of turns and the radius of the i coil, respectively. Then the coil mutual inductance is

M = \frac{π μ_{0} {(r_{1} r_{2})}^{2} {(n_{1} n_{2})}^{0.5}}{2 d^{3}} .

(8)

Hence, the maximum value of the load power is

P_{Lmax} = \frac{U_{s}^{2}}{4 (R_{s} + R_{1})} \cdot \frac{R_{L}}{R_{L} + R_{2}} .

(9)

From the above-mentioned analysis, it can be seen that in the magnetically coupled resonant wireless energy transmission system, the maximum transmission power of the load can be achieved by increasing the supply voltage or by reducing the resistance of the series connection or the output impedance of the power supply. The coil mutual inductance is mainly regulated by the transmission distance, and an increase in the transmission distance can be achieved by increasing the power supply frequency, increasing the coil inductance, and reducing the impedance.

In magnetically coupled resonant wireless energy transmission mechanism indicators, the output power and transmission efficiency are the most basic and important indicators. The less the output power is, the better. However, the transmission efficiency and output power are higher under the premise of satisfying the demand of the actual electric vehicle. To sum up, the optimization goal of the magnetically coupled mechanism is transmission efficiency and output power. The resonant frequency of the system, the distance between coils, and the load size are used as constraints. Sec. III will investigate and analyze this mathematical model.

III. OPTIMIZATION OF THE DESIGN OF THE MODEL

In real life and scientific research, there are many optimization problems. Researchers have found that these problems are often large and complex optimization problems and only very few of them are single-objective optimization problems; most real-life optimization problems are multi-dimensional and belong to multi-objective optimization problems (MOPs). Solving multi-objective optimization problems and obtaining a more satisfactory optimal solution are difficult. For the optimization problem of magnetically coupled transmission systems, many intelligent optimization algorithms have been used at home and abroad, among which the NSGA-II has been applied in many engineering fields. Zhang et al. (2020)¹³ proposed a novel three-coil magnetically coupled resonant radio energy transmission system. A chaotic particle swarm optimization algorithm is applied to optimize the system and obtain the optimal parameters corresponding to the highest transmission efficiency. It was found that the new three-coil magnetically coupled resonant system has higher transmission efficiency. Chen and Liu (2022)¹⁴ proposed a numerical optimization method that can simultaneously achieve power distribution and efficiency maximization. The maximum efficiency is achieved by the particle swarm optimization (PSO) algorithm. Finally, the superiority of the control method is verified by building a simulation model. Su et al. (2022)¹⁵ proposed an electric coupled wireless power transmission (EC-WPT) system with an underwater rotating coupling mechanism and gave a multi-constrained multi-objective optimization method based on the second-generation Non-Dominated Sorting Genetic Algorithm Ⅱ (NSGA-II), and the prototype EC-WPT system with an underwater rotating coupling mechanism was built experimentally to achieve 311 W power transmission with 87% efficiency. Yang and Chen (2021)¹⁶ proposed an improved scalar artificial bee colony algorithm to search for all the poles in a magnetically coupled resonant wireless energy transmission system. The analysis of the system model illustrates the effect of the frequency splitting phenomenon and the necessity of multi-peak search, which can improve and stabilize the output power of the system in the over-coupled state while ensuring transmission efficiency. Feng et al. (2022)¹⁷ proposed Non-Dominated Sorting Genetic Algorithm Ⅱ (NSGA-II) selection to study the civil aircraft customization option meritocracy model. The system optimization resulted in a 20.71% reduction in fleet maintenance cost and a 2.576% increase in availability. Liang et al. (2022)¹⁸ proposed the Non-dominated Sorting Genetic Algorithm (NSGA-II) to study wireless energy transmission systems. By analyzing the two objective functions of mutual inductance and equivalent coupler loss impedance (ECLI), 20 sets of Pareto optimal points were evaluated using the Technique for Ordering Preferences by Similarity to Ideal Solution (TOPSIS) method, and five sets of design points with different weights were selected. The validity of the optimization process was verified by comparing the optimization results with the initial results. Liu et al. (2021)¹⁹ proposed an improved NSGA-II based on the spatial density (SD) operator and combined it with a computer graphics-based surface parameterization method and computational fluid dynamics (CFD) simulation. The optimization results show a 1.29% increase in rated operating condition efficiency, an 8.8% increase in the mass flow rate, a 0.74% increase in the pressure ratio, and a 6.2% increase in overall profitability. The normal operating condition efficiency increased by 1.2%, the mass flow rate increased by 9.1%, the pressure ratio increased by 0.24%, and the overall profitability increased by 10%.

When considering the optimization of power and efficiency parameters from the perspective of product design for wireless charging mode operation, the highest transmission efficiency and maximum output power are targeted while satisfying the transmission distance, load resistance, and system resonance frequency. The coil size, number of turns, and turn spacing of the transmitting and receiving coils are determined, where the size of the mutual inductance of the coil structure is adjusted by changing the axial distance between the coils. When the design of a real-life magnetically coupled mechanism considers the power and efficiency optimization problem, the problem becomes a multi-parameter, multi-objective optimization problem. Since the objectives are often constrained by each other, it is difficult to obtain an optimal answer.

In this paper, we propose an improved NSGA-II and establish a multi-objective optimization model to apply the improved NSGA-II to the multi-objective optimization solution to address the shortcomings of the multi-objective optimization algorithm in solving practical problems. There have been related research on the NSGA-II, among which the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) proposed by Deb, an American professor,²⁰ uses fast non-dominated sorting, elite retention strategy, and crowded distance sorting. The algorithm can find the global optimal solution quickly and rapidly, and the Pareto optimal solution will be uniformly distributed over the whole Pareto front. Compared with the traditional NSGA, the fast non-dominated sorting genetic algorithm proposed by the NSGA-Ⅱ reduces the complexity of the algorithm by one dimension, and the introduction of the elite retention strategy improves the accuracy of the optimization results and ensures the diversity of good individuals in the population.

In this section, we focus on the multi-objective version of the genetic algorithm proposed in this paper: the non-dominated ranking genetic algorithm (NSGA-II). To implement the NSGA-II, the fast non-dominated ranking strategy and the crowding degree strategy are mainly utilized. The purpose of utilizing these two important strategies is twofold: first, to be able to find non-dominated solutions by dividing all individuals into non-dominated ranks. Second, because the different hierarchies in the NSGA-II are established by individual fitness to differentiate the multi-objective problem, it is not possible to judge an individual based on the fitness of one objective alone, so it is possible to judge which group the individual belongs to by ranking the non-dominated sequence and crowding degree together. The elite selection strategy and the binary tournament selection strategy also have two purposes: first is to give preference to individuals with high ranking in the random selection of individuals, and second is to give preference to those with high crowding if the ranking is the same. Rather, the bidding tournament strategy selects parents suitable for reproduction as a way to generate offspring by crossover variation, using binary crossover and polynomial variation with real number coding, and selects individuals preferred in the merged population to form a new population. The following is an improved strategy for the NSGA-II.

A. Fast non-dominated sorting strategy

The fast non-dominated ranking sets two parameters for each particle in the population, the number of non-dominated and dominated sets. The highest-ranked non-dominated solution set is found first, and then the ranking of other individuals is performed. The main operation is shown as follows:

Calculate n_i and S_i of all particles. For example, if the calculation knows that particle i is dominated by particle j, the n_i of particle i is self-increasing by 1, and the index of particle i is put into the S_j set of particle j.
Put the particles with n_i = 0 into F1, which is the Pareto optimal solution set at this time.
Access the dominating set S_i of all particles in F1, and put the n_i of the members in it by self-subtracting 1.
Put the particles with n_i = 0 into the corresponding rank, visit the dominating set S_i of the corresponding rank, decrease the n_i of its members by 1, and repeat this operation.

Figure 2 shows the correspondence of ranks, where the first rank is the Pareto optimal solution, which is not dominated by any particle. All particles in the second rank are dominated by at least one particle in the first rank. The other ranks are in the same order. The pseudo-code for fast non-dominated ranking is shown in Algorithm 1.

FIG. 2.

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Fast non-dominated sorting chart.

Algorithm 1.

Fast non-dominated sort.

Initialization pop, n_q, n_p, S_p, S_q, F_k

For p in pop do

For q in pop do (q ≠ p)

If p ≺ q then

n_q + = 1;

S_p = S_p ∪ {q};

End

If p ≺ q then

n_p + = 1;

S_q = S_q ∪ {p};

End

If n_p = = 0 then F₁ = F₁ ∪ {p}; end

End

k = 1

While F_k ≠ ∅ do

Rank = ∅

For p in F_k do

For q in S_p do

n_q−= 1

If n_q = = 0 then Rank = Rank ∪ {q}; end

End

k + = 1

F_k = Rank

End

B. Congestion strategy

According to the above-mentioned fast non-dominated ranking method, the particles can be accurately classified into several ranks, where F1 is the Pareto optimal solution and the particle quality in this rank is the best. As the i in Fi increases, the quality of the particles in the corresponding rank becomes worse. Although it is possible to sort the particles according to their quality using fast non-dominated sorting, there is a high probability that multiple particles will appear in the same rank. For these particles, we cannot distinguish their good or bad quality again, which leads to the NSGA-Ⅱ not being able to assign the rank to each particle smoothly. To solve this problem, a crowding degree strategy is used in this paper. This strategy sets a predefined distance for the particles in different ranks and then calculates the number of neighboring solutions within that distance as their crowding degree. The formula for calculating the crowding degree is

D^{i} = \sum_{j = 1}^{njob} \frac{a b s (f_{j}^{i + 1} - f_{j}^{i - 1})}{a b s (f_{j}^{\max} - f_{j}^{\min})},

(10)

where Dⁱ denotes the crowding degree of the ith particle in a certain rank, njob is the number of objective functions of the problem, $f_{j}^{i + 1}$ and $f_{j}^{i - 1}$ are the fitness values of the i + 1st and i − 1st particles in the jth objective function, respectively, and $f_{j}^{\max}$ and $f_{j}^{\min}$ are the maximum and minimum values of the jth objective function, respectively. The crowding degree policy is shown in Fig. 3, where f₁ and f₂ are the objective functions.

Fig. 3.

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Congestion strategy chart.

C. Elite selection strategy

After obtaining the parents and children of the NSGA-II in the multi-objective optimization process in Sec. III B, we need to select the best group as the next generation, and the elite selection strategy is used here. First, we select all the dominated solutions on the surface of the dominated front and then select the ranks in turn. While selecting the rank, we compare the number of populations to be added each time and determine whether the initial number of populations is redundant. If the number of populations remaining after selecting all ranks is not enough, the last rank is sorted from highest to lowest according to the crowding strategy, and the highest value is selected until the number matches the number of populations.

Initially, a random parent population P₀ is created, and the parent and offspring are combined to select a non-dominated sorting frontier solution to determine whether the population size matches. When the population size is not enough, the first N elements of the last rank P are created, and the crowding distance P_t+1 is selected using selection, hybridization, and mutation. The pseudo-code for the elite selection strategy is shown in Algorithm 2.

Algorithm 2.

Elite selection strategy.

Initialization pop, Q₀, N,

F

R_t = P_t ∪ Q_t

F = f a s t - n o n - d o m i n a t e d - s o r t (R_{t})

P_t+1 = ∅ and i = 1

u n t i l |P_{t + 1}| + |F_{i}| \leq N

c r o w d i n g - d i s t a n c e - a s s i g n m e n t (F_{i})

P_{t + 1} = P_{t + 1} \cup F

i = i + 1

S o r t (F_{i}, ≺_{n})

P_{t + 1} = P_{t + 1} \cup F_{i} [1 : (N - |P_{t + 1}|)]

Q_{t + 1} = m a k e - n e w - p o p (P_{t + 1})

t = t + 1

D. Binary tournament selection strategy

The binary tournament selection strategy in the NSGA-II removes a certain number of individuals from the population and then selects the best one to enter the offspring population. The operation is repeated until the size of the new population reaches the size of the original population. A binary tournament removes two individuals in the overall population at one time and then removes the best individual among these individuals to keep in the set of the next-generation population. The steps are as follows: the number of individuals is determined as two individuals, two individuals are randomly selected in the population, and each individual is selected with the same probability; according to the fitness value of each individual, the individual with the best fitness is selected to enter the next generation population. The final number of repetitions is the size of the population until the size of the new population reaches the size of the original population. Figure 4 shows the flow chart of the specific optimization process of the NSGA-Ⅱ.

FIG. 4.

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Execution process of the NSGA-II.

IV. EXPERIMENTAL ANALYSIS

In this paper, the NSGA-II algorithm is used to evaluate the optimization efficacy model with two optimization objectives as evaluation metrics. Based on the Pareto optimal solution set, the dominant ranking strategy, congestion strategy, elite selection strategy, and binary bidding race strategy are introduced to effectively attenuate the less reliable solutions at both ends of the Pareto optimal solution and reduce the adverse effects on the system.

The transmitting and receiving coils are set to have the same structural parameters, and to ensure that they work in a resonant state during transmission, the corresponding compensation capacitance of the coils is changed when the frequency changes. The simulation parameters in this paper are presented in Table I, where n₁ and n₂ are the number of turns of the coil, a is the wire diameter, and the coil spacing is set to 0.1 m. In the following, the effects of the magnetic coupling mechanism before and after the improvement will be analyzed by unoptimized, single-parameter optimization and multi-parameter optimization of the system.

TABLE I.

Simulation optimization parameters.

U_m (V)	r₁ (cm)	R_L (Ω)	R_s (Ω)	L₀ (μH)	C₁ (μF)	d (m)
220	9.5	1	50	1.96	33.3	0.1
L₁ (μH)	r₂ (cm)	n₁	C₂ (nF)	n₂	a (mm)	f (kHz)
218	9.5	10	20.8	10	2.5	50.5

U_m (V)	r₁ (cm)	R_L (Ω)	R_s (Ω)	L₀ (μH)	C₁ (μF)	d (m)
220	9.5	1	50	1.96	33.3	0.1
L₁ (μH)	r₂ (cm)	n₁	C₂ (nF)	n₂	a (mm)	f (kHz)
218	9.5	10	20.8	10	2.5	50.5

A. Simulation of the unoptimized magnetically coupled mechanism

When the system parameters change, the performance of the system also changes, and there is often a conflict between these performance parameters. This paper uses unoptimized system parameters to analyze the effects on magnetically coupled resonant wireless energy transmission systems.

As can be seen from Fig. 5(a), the transmission efficiency of the magnetically coupled mechanism with unoptimized parameters decreases with the increase in the transmission distance. When the transmission efficiency is small, the energy between the magnetic fields cannot be transferred uniformly because the coupling transmission distance of the two coils is too far, which leads to an increase in the difficulty of magnetic coupling. Because the output voltage is constant, the transmission efficiency also decreases as the load resistance decreases. When the load resistance decreases, the current increases, and more power is consumed by the internal resistance of the power supply, so the transmission efficiency decreases. As can be seen from Fig. 5(b), in the case of unoptimized load resistance, when the transmission efficiency and output power are selected to be optimal at the same time, the load resistance is 0.28, the transmission efficiency is 54.64%, and the output power is 425 W, which is shown in the red box in the figure. One can see that the transmission efficiency is very low, and the reason is that the load resistance does not match the transmission characteristics of the system. In addition, after the real-life system is formed, the corresponding impedance values are determined one after another, and the corresponding load resistance value when the transmission efficiency of the system reaches its maximum, which is the impedance matching value of the system. Figure 5(c) shows that the transmission efficiency and output power are not optimized and the transmission efficiency is only 19.45% when the transmission distance is set at 0.1 m. The output power is 283 W. From the above-mentioned analysis, it can be seen that the output characteristics of the magnetically coupled structure without optimized design parameters are not stable.

FIG. 5.

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(a) Unoptimized transmission efficiency. (b) Unoptimized load resistance. (c) Unoptimized transmission efficiency and output power.

B. Single-objective optimization simulation of the magnetically coupled mechanism

In the practical application of wireless energy transmission systems, the transmission efficiency, load resistance, and output power are dynamically changing in a certain range, so in the study of the magnetically coupled resonant wireless transmission system, to ensure that the magnetically coupled resonant structure can be flexible with high transmission efficiency and output power, the relationship between the optimization algorithm and transmission efficiency and transmission power will be studied in the following.

From Fig. 6(a), it can be seen that when the transmission efficiency is optimized with a single target, the transmission efficiency is 78.36% and the output power is 184 W when the transmission distance is set to 0.1 m. However, as shown in Fig. 6(b), the output power is 400 W and the transmission efficiency is only 39.28% when the working condition is to the set transmission distance. The reason for the above is that when the single target optimization magnetic coupling system is used, the actual reality is not taken into account, which often results in the situation that the transmission efficiency increases while the output power is relatively low, or the output power increases while the transmission efficiency is relatively low.

FIG. 6.

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(a) Schematic diagram of single objective optimized transmission efficiency. (b) Schematic diagram of single objective optimized output power.

Therefore, the design of magnetically coupled structural systems is a multi-parameter, multi-objective optimization problem, and multiple objectives are often coordinated with each other. A comprehensive constraint function and optimization objective function are needed to establish a multi-objective mathematical model of the system. In this paper, the improved NSGA-II will be used to optimize the magnetically coupled resonant system, combining the practical application requirements, assigning different weights to multiple objective functions, and then calculating a set of optimal solutions for design reference, providing a basis for the comprehensive performance of the system to reach the optimum.

C. Multi-objective optimization solution for the magnetically coupled mechanism

In this paper, an improved non-dominated ranking genetic algorithm NSGA-II is used for parameter design optimization, and the optimization of design parameters is carried out based on the model established in the previous paper. The maximum individual vector, the minimum individual vector, and the transmission distance are used as design variables, and the maximum transmission efficiency and output power are used as optimization objectives. Before conducting the NSGA-II iteration, the range of the optimization parameters needs to be constrained, and its upper and lower limits are given as the generated matrix vectors.

The population size is set to 50, and the number of evolutionary iterations is 500, i.e., the algorithm terminates after 500 evolutionary iterations, and the other parameters are kept as default. Table II lists the constrained parameter values.

TABLE II.

Constraint parameter values.

Binding variables	Lower limit	Upper limit
min_range	0	10
max_range	1	100
d (m)	0	0.4

In the improved NSGA-II, the magnetically coupled radio transmission system utilizes constraints for better optimality-seeking efficiency and power. At each iteration, the global optimum position is perturbed using the bid-race strategy, which increases the probability of the algorithm jumping out of the local optimum. Second, the adaptive step size is introduced to improve the algorithm’s optimality-seeking accuracy. In the case of finding the optimal efficiency, the update mechanism of the position avoids the blindness of the current individual position update and can better seek the optimal solution for the transmission efficiency.

The NSGA-II is used to perform a multi-objective optimization search for the established magnetic coupling efficacy optimization model, and the frontier solutions of Pareto for transmission efficiency and output power are obtained. As shown in Fig. 7(a), different optimal Pareto frontier solutions are obtained by the improved NSGA-II and the conventional NSGA-II. The frontier solutions obtained by the conventional NSGA-II are scattered, weakly stable, and not strongly convergent, and it can be known that the diversity and distribution of the conventional NSGA-II are not good. The improved NSGA-II not only has excellent convergence but also has excellent diversity and distribution of the resulting Pareto optimal solutions. As shown in Fig. 7(b), after the magnetic coupling mechanism uses the improved NSGA-Ⅱ to optimize the output power and transmission efficiency simultaneously, the transmission efficiency is 87.61% and the output power is 440 W when the transmission distance is set to 0.1 m, and the system is more realistically close to the real situation, the multi-objective optimized transmission efficiency is 11% higher than that when the transmission efficiency is optimized by a single objective, and the multi-objective output power is 40 W higher than that of the single objective. The optimized output power is improved by 40 W, which proves that the performance of the magnetically coupled resonant wireless energy transmission system is optimized by using the improved NSGA-II. It is hoped that it has some reference value for the application of a magnetically coupled resonant wireless transmission system.

FIG. 7.

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(a) Schematic diagram of the Pareto front solution. (b) Schematic diagram of simultaneous optimized output power and transmission efficiency.

The study is carried out for the wireless charging system of small electric vehicles, and the variation in transmission efficiency and output power under different conditions can be seen in Fig. 8. When the optimization method is not used, the transmission efficiency and output power are very low and do not meet the practical application of the wireless charging system. When the common single-objective optimization is used, it is easy to fall into the local optimum in the process of finding the best solution, and the solution of the objective function is easy to reach the best solution, but it does not meet the real-life scenarios. For co-optimization, a multi-objective function solution is used to jointly find a set of suitable matching values of system parameters to achieve co-optimization of system transmission efficiency and output power to meet the requirements of wireless charging for small electric vehicles.

FIG. 8.

View large Download slide

Comparison of transmission efficiency and output power characteristics.

V. PHYSICAL VERIFICATION

To verify the effectiveness of the improved optimization algorithm for the improved magnetic coupling model, a magnetically coupled resonant coil system with a frequency tracking structure is designed for verification.

A. Experimental platform

The parameters of the experimental platform are selected to choose the system parameters obtained through algorithm optimization. The coil diameter of 9.46 cm is surrounded by copper wire of 2.5 mm wire diameter, the number of turns is 10, the resonant frequency set by the spectrum analyzer is 50 kHz, the output voltage of the power supply is 220 V, and the experimental platform is built as shown in Fig. 9. The values of the main components and coil circuit parameters of the prototype are shown in Table III.

FIG. 9.

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Magnetic coupled resonant system experimental platform (1 DC power supply, 2 digital oscilloscope, 3 signal generator, 4 inverter circuit, 5 power amplifier, 6 load, 7 spectrum analyzer, 8 digital frequency meter, 9 filter circuit, 10 voltage regulator circuit module, 11 transmit resonant coil, and 12 receive resonant coil).

TABLE III.

Prototype main components and coil circuit parameter values.

Parameters	Value/model	Parameters	Value/model	Parameters	Value/model
Digital oscilloscope	SDS1102X-E	U_s	220	Coupling coefficient	0.33
DC power supply	MN-3205D	L₀	2.2 μH	Resonant frequency	50 kHz
Signal generator	SDG1025	C₁	33 µF	Load resistance	360 mΩ
Spectrum analyzer	GA40XX	L₁	220 μH	Digital frequency meter	OPA615
Voltage regulator module	LM2596	Coil spacing	10 cm	C₂	22 nF
Power amplifier	LM1875	Load	Light source	Number of turns of coils	10

Parameters	Value/model	Parameters	Value/model	Parameters	Value/model
Digital oscilloscope	SDS1102X-E	U_s	220	Coupling coefficient	0.33
DC power supply	MN-3205D	L₀	2.2 μH	Resonant frequency	50 kHz
Signal generator	SDG1025	C₁	33 µF	Load resistance	360 mΩ
Spectrum analyzer	GA40XX	L₁	220 μH	Digital frequency meter	OPA615
Voltage regulator module	LM2596	Coil spacing	10 cm	C₂	22 nF
Power amplifier	LM1875	Load	Light source	Number of turns of coils	10

B. Validation results

To verify the effectiveness of the optimized magnetic coupling model, the experimental platform uses eight bulbs of the same power, two in series with a group, and then the switch control will be in four parallel groups to access the magnetically coupled resonant system. The load is adjusted by replacing the bulbs with different powers of 25, 38, 75, and 100 W. The experimental results are shown in Fig. 10. When the receiving coil and transmitting coil are at a set spacing of 10 cm, with the increase in load resistance, as indicated by the red box in the figure, when the resistance value is optimized close to 365 mΩ, the transmission efficiency is 82.36%, and the output power is 433 W, at which time the output power and transmission efficiency stability are the strongest and the system performance is optimal. Therefore, the physical verification experiment selects 360 mΩ close to the optimized resistance value to allow the output power and transmission efficiency of the magnetically coupled resonant system to be improved simultaneously.

FIG. 10.

View large Download slide

Graph of experimental results of load parameter optimization.

Figure 11 shows the graph of the physical verification results after optimization using the NSGA-II. Table III shows the best set of optimal solutions in the multi-objective optimization for design guidance, and the prototype physical experiments will verify the correctness of the previous simulation. In the magnetically coupled resonant wireless energy transmission system, the system resonant frequency, load resistance, and transmission distance are the key factors affecting the system transmission efficiency and output power. In the physical experiment, the optimal matching parameters of the improved NSGA-II will be used to improve the transmission efficiency and output power, and the load resistance is 360 mΩ, the coil diameter is 9.46 cm, the number of turns is 10, and the transmission distance is 0.1 m. Therefore, under a certain transmission distance, the appropriate load resistance and resonant frequency can make the output power and transmission efficiency reach the maximum. The experiment shows that the closer the parameters are to the optimized algorithm, the more the transmission efficiency and output power of the magnetically coupled resonant system are improved simultaneously, which is consistent with the simulation experiment after the multi-objective optimization in the previous paper and provides the basis for the optimal comprehensive performance of the system.

FIG. 11.

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Physical verification experimental result graph.

C. Error analysis

From the experimental verification results mentioned above, the trend of the experimental measurements and the simulation experiments is basically the same, but not exactly the same, because errors exist. In the following, the comparison of transmission efficiency and output power obtained from simulation and experiment at 0.1 m will be analyzed, and the reasons for the generated errors will be analyzed and elaborated. A comparison of the simulated and experimental values is shown in Table IV.

TABLE IV.

Comparison of simulated and experimental values.

	Optimization parameters		Optimization results
	Coil spacing	Resonant frequency	Output power	Transmission efficiency (%)
Simulation value	0.1 m	50.5 kHz	440 W	87.61
Experimental value	0.1 m	50 kHz	424 W	82.36
Error value	0	0.5 kHz	16 W	4.64
Error rate	0	1%	3.772%	5.633

	Optimization parameters		Optimization results
	Coil spacing	Resonant frequency	Output power	Transmission efficiency (%)
Simulation value	0.1 m	50.5 kHz	440 W	87.61
Experimental value	0.1 m	50 kHz	424 W	82.36
Error value	0	0.5 kHz	16 W	4.64
Error rate	0	1%	3.772%	5.633

As can be seen from Table IV, the simulated value is the value obtained through the optimization of the NSGA-II, while the experimental value is the value obtained through the physical experimental measurement. The main reasons for the error between the simulated and experimental measured values in Table IV are as follows: when the experimental platform ensures that the system frequency is fixed, the frequency tracking control is used, and the change in the internal resistance of the power supply is changed within a certain range. In addition, the coil itself has losses, material problems, a vertical distance between the coils and the distance of the coil center point in the horizontal direction (the so-called degrees of freedom are not calibrated), a degree of heat generation of the whole magnetically coupled system, and ambient temperature. However, the improved algorithm does not have a large impact on the results obtained from the system optimization, and the error remains within acceptable limits, so this paper verifies the validity of the application in magnetically coupled resonant wireless energy transmission systems.

VI. CONCLUSION

In this paper, the system structure and multi-parameter matching problem are optimized to improve the transmission efficiency and output power of the magnetically coupled resonant wireless energy transmission system. In the paper, the analytical expressions of the coil parameters are first designed. The size, number of turns, and wire diameter of the transmitting and receiving coils are used as design variables; the resonant frequency, distance between coils and load size of the system are used as constraints; and the transmission efficiency and transmission power are used as optimization objectives. Then the multi-objective optimization mathematical model of the system is established by integrating the constraint function and optimization objective function. On this basis, the NSGA-II is introduced, and the optimization results under different conditions are compared to prove that the proposed improved NSGA-II applies to the wireless energy transmission system. The Pareto fronts of transmission efficiency and output power are obtained, and the optimal values are solved by simulation for design reference. Finally, the prototype physical system is established to verify the correctness of the theory and the reliability of the experiment. This study has certain advantages in the optimization of wireless energy transmission parameters, transmission efficiency, and power and provides the basis for the optimal comprehensive performance of the system.

This study has certain advantages in the optimization of wireless energy transmission parameters and improving the transmission efficiency and output power. Based on the literature review and the problems raised by the university-enterprise cooperation project, the study of a magnetically coupled resonant wireless energy transmission system is carried out. Based on the review of a large number of references, the idea, structure, and optimization method of this research paper are proposed, the theoretical structure is simulated and analyzed, and the experimental system is built to verify and conform to the scope of the literature review.

Most of the previous research is optimized for single objective parameters. The research in this paper is carried out for the new structure and multi-objective influencing factors by improving the strategy, thus realizing the improvement of the NSGA-II, getting the system to get the key parameters, and verifying the simulation experiment and the physical experiment together. In addition, the experiment proves that it can reach the requirements of small electric vehicles for charging transmission efficiency and power.

The present research only completes simulation experiments and prototype experiments in the laboratory without considering the influence of temperature on the system performance and without large-scale practical application. Future research will further improve the reliability of the system and ensure the stability of the experimental system under different temperatures. It is hoped that this paper will provide some guidance for the realization of magnetically coupled resonant radio energy transmission and optimization for electric vehicles.

SUPPLEMENTARY MATERIAL

See the supplementary material for the complete experimental data of the studied physical system.

ACKNOWLEDGMENTS

This work was sponsored in part by the National Natural Science Foundation of China, under Grant No. 61662005, by the 2021 Project of Scientific Research Basic Ability Improvement for Young and Middle-aged Teachers in Guangxi Universities (Grant No. 2021KY0164), and by the Guangxi Key Laboratory of Power System Optimization and Energy Technology.

This study was approved by the Guangxi University for Nationalities.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Chunming Wen: Conceptualization (equal); Project administration (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Minbo Chen: Conceptualization (equal); Project administration (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Qing Xu: Software (equal); Visualization (equal). Qiuli He: Resources (equal); Validation (equal). Jiarong Wu: Conceptualization (equal); Methodology (equal). Xiaohui Zhao: Resources (equal); Validation (equal). Yuanxiong Liang: Supervision (equal). Kairong Liang: Supervision (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

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Author(s)

Multi-objective optimization of parameters for magnetically coupled resonant wireless energy transmission systems

I. INTRODUCTION

II. ANALYSIS OF THE MAGNETICALLY COUPLED RESONANT TRANSMISSION MODEL

III. OPTIMIZATION OF THE DESIGN OF THE MODEL

A. Fast non-dominated sorting strategy

B. Congestion strategy

C. Elite selection strategy

D. Binary tournament selection strategy

IV. EXPERIMENTAL ANALYSIS

A. Simulation of the unoptimized magnetically coupled mechanism

B. Single-objective optimization simulation of the magnetically coupled mechanism

C. Multi-objective optimization solution for the magnetically coupled mechanism

V. PHYSICAL VERIFICATION

A. Experimental platform

B. Validation results

C. Error analysis

VI. CONCLUSION

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Supplementary Material

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Multi-objective optimization of parameters for magnetically coupled resonant wireless energy transmission systems Open Access

I. INTRODUCTION

II. ANALYSIS OF THE MAGNETICALLY COUPLED RESONANT TRANSMISSION MODEL

III. OPTIMIZATION OF THE DESIGN OF THE MODEL

A. Fast non-dominated sorting strategy

B. Congestion strategy

C. Elite selection strategy

D. Binary tournament selection strategy

IV. EXPERIMENTAL ANALYSIS

A. Simulation of the unoptimized magnetically coupled mechanism

B. Single-objective optimization simulation of the magnetically coupled mechanism

C. Multi-objective optimization solution for the magnetically coupled mechanism

V. PHYSICAL VERIFICATION

A. Experimental platform

B. Validation results

C. Error analysis

VI. CONCLUSION

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Supplementary Material

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Submit your article

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Multi-objective optimization of parameters for magnetically coupled resonant wireless energy transmission systems