A robust optimization method for electromagnetic shielding structure of wireless power transmission system

Abstract

This invention, applicable to the field of wireless power transmission technology, provides a robust optimization method for the electromagnetic shielding structure of a wireless power transmission system, comprising: constructing a WPT system; designing an electromagnetic shielding structure to improve energy transmission efficiency and weaken the electromagnetic field in the non-operating area; building a K-Trans proxy model to integrate the uncertainty factors of the WPT system and the shielding structure design parameters, and modeling the data using an improved Transformer model combined with the Kolmogorov-Arnold representation theorem; and applying the MOEDO algorithm to perform multi-objective robust optimization based on the K-Trans proxy model to generate a Pareto optimal solution set that balances energy transmission efficiency and electromagnetic shielding performance. This invention proposes an electromagnetic shielding structure, optimized using the K-Trans proxy model and the MOEDO algorithm, to address the offset problem of the WPT system's coupling mechanism, providing new ideas for electromagnetic shielding and safety protection.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless power transmission, and in particular relates to a robust optimization method for an electromagnetic shielding structure of a wireless power transmission system. Background Art

[0002] With rapid socioeconomic development, traditional industrial production capacity has increased dramatically, and the resulting greenhouse effect has had a significant impact on human life. Traditional energy sources largely rely on oil consumption for energy production, resulting in significant carbon dioxide emissions that are deteriorating the global environment. Currently, the global energy landscape is gradually transitioning towards renewable, green, and environmentally friendly energy sources, with the shift towards electricity being particularly significant. Electricity utilizes energy storage components such as lithium batteries and does not generate greenhouse gas emissions during use. However, batteries have lower energy density than oil, and compared to traditional recharge methods, plug-in charging has the disadvantage of long charging times. Fast charging methods are not only inconvenient but also pose safety risks. To address the shortcomings of plug-in charging, wireless power transfer (WPT) technology has become a highly sought-after development direction. This technology enables safe and convenient charging and has been widely used in a variety of applications, including medical devices, electric bicycles, drones, and electric vehicles. Compared to traditional plug-in charging, it offers instant charging, significantly improving charging convenience. In the practical use of WPT systems, the coupling mechanism is the core component, and the coupling performance and structural characteristics between the transmitting and receiving coils directly determine the system's operational state. Maintaining the coupling mechanism in an ideal operating state is essential for achieving optimal system performance. However, in practice, coupling mechanism offset is unavoidable, leading to leakage electromagnetic fields during energy transmission. This not only increases the risk of exposure to the leakage electromagnetic fields surrounding the electric vehicle WPT system but also reduces energy transmission efficiency. While shielding structures can shield the WPT system's leakage magnetic field, protecting users from electromagnetic exposure, when the coupling mechanism is offset, the leakage magnetic field dose increases, further exacerbating the electromagnetic environment and threatening the user's health and safety. Therefore, given the uncertainty of coupling mechanism offset, how to reduce the leakage magnetic field while improving the system's energy transmission efficiency and enhancing the system's offset tolerance and electromagnetic protection performance has become an urgent issue to be addressed.

[0003] Furthermore, due to the numerous uncertainties inherent in the practical application of WPT systems, their operating conditions are also uncertain. To ensure the safe use of WPT products, the International Commission on Non-Ionizing Radiation Protection (ICNIRP) and the Institute of Electrical and Electronics Engineers (IEEE) have established standard LMF limits for WPT systems to guide electromagnetic safety assessments. ICNIRP uses the operating frequency of WPT systems as a basis for classification, establishing limit guidance for different frequency bands. Given that the operating frequency of WPT systems is typically 85kHz, magnetic flux density (B) is often used as an evaluation metric for the electromagnetic safety of WPT systems. Therefore, designing a reasonable electromagnetic shielding structure to limit the leakage electromagnetic field generated by the uncertain operating conditions of the WPT system is crucial to improving the system's electromagnetic safety and energy transmission efficiency.

[0004] Currently, various electromagnetic shielding measures have been proposed to address the leakage electromagnetic fields (EMFs) in WPT systems. The most common approach is to attach aluminum plates and ferrite materials to the transmitting or receiving coils, utilizing the eddy current effect generated by highly conductive metals to block LMFs. Some researchers have proposed new ferrite composite structures that can effectively control magnetic flux and improve efficiency. Others have studied the shielding performance of stacking ferrite and aluminum plates. However, ferrites are limited by weight, fragility, and significant temperature variations in magnetic properties. Covering large areas with ferrite poses challenges in designing compact WPT systems. Other researchers have derived an analytical formula for the mutual inductance between two circular coils at arbitrary relative positions on a double-layer electromagnetic shield. Nanocrystals are considered an alternative to ferrite cores due to their superior magnetic properties (magnetic saturation, permeability, Curie temperature) and mechanical properties (ductility and robustness). Designing magnetically conductive structures using nanocrystal cores can achieve higher efficiency, power density, and lower leakage electromagnetic fields than ferrite cores. However, the electromagnetic shielding structures in these studies are typically optimized under deterministic conditions, resulting in incomplete analysis of the results.

[0005] Although there have been many studies on the design of electromagnetic shielding structures for WPT systems, few studies have conducted robust design optimization (RDO) while considering the uncertainty factors in actual working conditions. The core of RDO is to reduce the impact of uncertainty on system performance by considering the influence of factors such as design parameters, design variables, and system analysis models. The surrogate model method can quantify the impact of relevant uncertainty parameters on the shielding effect of WPT systems, solving the problems of many variables and high optimization costs in the design of electromagnetic shielding structures. As an encoder-decoder structure, the Transformer architecture achieves relationship mapping through the self-attention mechanism and is widely used in various nonlinear problems. It can build high-precision surrogate models with fewer training samples and training time. However, existing studies have only completed uncertainty quantification and have not incorporated uncertainty factors into robust optimization design.

[0006] In view of the above-mentioned shortcomings of the prior art, the present invention proposes a robust optimization method for the electromagnetic shielding structure of a wireless power transmission system. Summary of the Invention

[0007] The object of the present invention is to provide a robust optimization method for the electromagnetic shielding structure of a wireless power transmission system, aiming to solve the problems raised in the above background technology.

[0008] The purpose of the present invention is achieved through the following technical solutions:

[0009] A robust optimization method for an electromagnetic shielding structure of a wireless power transmission system comprises the following steps:

[0010] Step 1: Construct a WPT system based on the principle of magnetic coupling resonance. The system uses a bilateral SS compensation circuit to ensure that the transmitter and receiver loops are in a resonant state to improve energy transmission efficiency.

[0011] Step 2: Design an electromagnetic shielding structure, which includes electrical shielding materials and magnetic materials to improve energy transmission efficiency by concentrating the electromagnetic field and weaken the electromagnetic field in non-working areas;

[0012] Step 3: Build a K-Trans proxy model, which integrates the uncertainty factors of the WPT system and the shielding structure design parameters, and uses the improved Transformer model combined with the Kolmogorov-Arnold representation theorem to model the data;

[0013] Step 4: Using the MOEDO algorithm, a multi-objective robust optimization is performed based on the K-Trans agent model to generate a Pareto optimal solution set to achieve a balance between the energy transmission efficiency and electromagnetic shielding performance of the WPT system.

[0014] Furthermore, in step 1, the bilateral SS compensation circuit satisfies the resonance condition:

[0015] ;

[0016] ;

[0017] in: ω is the resonant angular frequency, ω =2π f , f is the resonant frequency, take 85kHz; is the self-inductance of the transmitting coil, is the self-inductance of the receiving coil; and They are the transmitter compensation capacitor and the receiver compensation capacitor respectively.

[0018] Furthermore, in step 2, the shielding structure includes a nanocrystalline shielding layer, a ferrite unit, and an aluminum plate, and the nanocrystalline shielding layer is divided into inner nanocrystalline and outer nanocrystalline; at room temperature and 85kHz, the relative magnetic permeability of the nanocrystalline shielding layer material is 22000, and the relative magnetic permeability of the ferrite unit is 3300;

[0019] The shielding structure optimizes the magnetic flux path and reduces magnetic leakage based on the following formula:

[0020] ;

[0021] ;

[0022] in: represents magnetic flux; F m represents magnetomotive force; R m Reluctance; magnetic field The angle with the normal is ;magnetic field The angle with the normal is ; μ 1 and μ 2 are the relative magnetic permeabilities of nanocrystals and ferrites respectively.

[0023] Furthermore, in step 3, the input of the K-Trans proxy model includes shielding structure design parameters and uncertainty factors, and the shielding structure design parameters include the thickness of the ferrite unit t 1 , Distance from ferrite to WPT center d 1 , aluminum plate thickness t 2 Relative angles between the upper and lower ferrite layersα , the uncertainty factors include lateral offset , longitudinal offset and coil spacing ;

[0024] The output of the K-Trans proxy model is the maximum magnetic field intensity at the observation point B max and WPT system energy transmission efficiency η , process sequence data through position encoding and self-attention mechanism, and use KAN layer instead of MLP layer to improve model accuracy.

[0025] Furthermore, the position encoding operation in the network structure of the K-Trans agent model is expressed as:

[0026] ;

[0027] ;

[0028] in: It's location POS In the 2nd i Dimensional location information; It's location POS In the 2nd i+ Position information in 1 dimension; POS is the relative position of the element in the current sequence, i The total number of representative sequences is d The first i dimension; d MODEL The representative model dimension is d dimension.

[0029] Furthermore, in step 4, the optimization process of the MOEDO algorithm is as follows:

[0030] (1) Initialize the population and randomly generate a set of solutions;

[0031] (2) Mutation operation, which explores the solution space by introducing randomness and avoids the algorithm from falling into local optimality; the mutation operation includes memory-based mutation, in which for each solution x i , if the individual belongs to the memory solution, the following mutation operation is performed:

[0032] ;

[0033] in: is the generated solution; a , b It is a dynamic parameter generated by random factors; variance According to the expected rate of the current solution 1 / μ Calculated; Memoryless i It is an array that stores individual positions, including the position of each individual in the current population;

[0034] If the individual x i If it is not a memorized solution, the following mutation operation is performed:

[0035] ;

[0036] in: is a random number; is the current solution;

[0037] For boundary processing, to ensure the generated solution V i Within the allowable range of the solution space, the following correction is made:

[0038] ;

[0039] in: V ij It is the solution j The value of the dimension; max and min The functions are maximum function and minimum function respectively; lb j and ub j They are j The lower and upper bounds of the dimension;

[0040] (3) Non-dominated sorting and congestion sorting are used to maintain the quality and diversity of the solution set. The non-dominated sorting is used to select the Pareto optimal solution and classify the solutions according to the dominance relationship. The congestion sorting is used to ensure the diversity of the solution set and avoid the aggregation of solutions. In each frontier, the congestion distance is used to calculate the diversity of the solution. The formula is as follows:

[0041] ;

[0042] in: C i Is the solution i the degree of congestion; m is the dimension of the solution; j It is j The index of the objective function; and In the j On the objective function, the solution i The function values ​​of the adjacent solutions of ; f max, j and fmin, j They are j The maximum and minimum values ​​of the dimensional objective function value;

[0043] (4) Update operation, updating the solution set in the archive according to fitness and congestion; in the update operation, a solution set is maintained after each iteration, all historical optimal solutions are stored, and the optimal solution is selected according to non-dominated sorting to update the archive.

[0044] Compared with the prior art, the present invention has the following beneficial effects:

[0045] To address the offset uncertainty issues inherent in the coupling mechanisms of WPT systems, this paper proposes a highly offset-tolerant electromagnetic shielding structure combining ferrite units, aluminum plates, and a nanocrystalline shielding layer. This approach utilizes the K-Trans proxy model and MOEDO optimization algorithm, a modified Transformer architecture based on the KAN (Kan-like analytic architecture), to achieve efficient and robust optimization design of WPT system shielding structures. This approach incorporates offset uncertainty into the optimization design by constructing proxy models for different design objectives, replacing simulation evaluation with the traditional MLP unit. This approach improves model fitting and self-learning capabilities, enabling accurate approximation of high-dimensional functions. Compared to traditional optimization methods, the robust optimization algorithm of this paper effectively enhances the offset tolerance of WPT systems and offsets the negative effects of offset. Experimental results show that the electromagnetic shielding structure designed using the K-Trans proxy model and the MOEDO multi-objective optimization algorithm performs excellently. Under the same uncertainty conditions, the WPT system's over-limit probability is reduced to 0%, the average energy transmission efficiency reaches 84.79%, and the evaluation time cost is reduced by 90.97% compared to the traditional MC method. This paper not only provides practical application for the design of electromagnetic shielding structures for WPT systems but also opens up new avenues for ensuring human electromagnetic exposure safety. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 Flow chart of the method of the present invention.

[0047] Figure 2 Schematic diagram of the WPT system.

[0048] Figure 3 This is the topology diagram of the SS compensation circuit.

[0049] Figure 4 It is a magnetic flux path map.

[0050] Figure 5 It is the WPT system simulation model.

[0051] Figure 6 Magnetic flux density distribution of the WPT system; (a) is unshielded; (b) is shielded by a combination of ferrite unit and aluminum plate; (c) is shielded by a combination of ferrite unit, aluminum plate and nanocrystalline shielding layer.

[0052] Figure 7 It is the magnetic field distribution inside the shielding structure.

[0053] Figure 8 Schematic diagram of the observation point; (a) is a top view; (b) is a side view.

[0054] Figure 9 Comparison of magnetic field strength and energy transmission efficiency curves for three combinations.

[0055] Figure 10 is the system energy transmission efficiency of the original structure and the maximum magnetic field intensity at a fixed observation point B max The probability distribution function (PDF) of the original structure is: (a) the maximum magnetic field intensity at a fixed observation point B max (b) is the PDF of the system energy transfer efficiency of the original structure.

[0056] Figure 11 To determine the performance PDF comparison between the optimal and robust optimal; where (a) is the maximum magnetic field intensity at a fixed observation point of the optimized structure B max (b) is the PDF of the system energy transfer efficiency of the optimized structure.

[0057] Figure 12 Figure 2 is a diagram of the experimental equipment.

[0058] Figure 13 Detailed diagram of the device; (a) is the transmitting coil; (b) is the receiving coil; (c) is the nanocrystalline shielding layer; (d) is the aluminum plate; (e) is the ferrite unit; and (f) is the acrylic pad.

[0059] Figure 14 For observation points B max Response surface diagram of ; where (a) is the observation point under the optimal structure B max Response surface diagram; (b) is the observation point under the robust optimal structure B max Response surface plot of .

[0060] Figure 15 Energy transfer efficiency η PDF of the optimal structure; (a) is the energy transfer efficiency under the optimal structure η PDF; (b) is the energy transfer efficiency under the robust optimal structure η PDF. DETAILED DESCRIPTION

[0061] In order to have a clearer understanding of the technical features, objectives and beneficial effects of the present invention, the technical solution of the present invention is now described in detail below, but it should not be understood as limiting the scope of implementation of the present invention.

[0062] The present invention provides a robust optimization method for the electromagnetic shielding structure of a wireless power transmission system, the flow chart of which is as follows: Figure 1 As shown, the specific content of the method is as follows:

[0063] 1.WPT system;

[0064] The WPT system is built based on the principle of magnetic coupling resonance, and its overall structural diagram is as follows: Figure 2 In this system, a high-frequency current flows through the coupling coil. Since the transmitting coil and the receiving coil have the same resonant frequency, a high-frequency resonant electromagnetic field is generated. This electromagnetic field serves as a medium for efficient wireless transmission of electrical energy. Figure 2 The meanings of the parameters are as follows: U s is the grid voltage, U DC1 is the voltage after rectification at the transmitter end, U 1 is the transmitting coil voltage, U 2 is the receiving coil voltage, C S1 is the transmitter compensation capacitor, C S2 For the receiving end compensation capacitor, M 12 is the mutual inductance between the transmitting coil and the receiving coil, L 1 is the self-inductance of the transmitting coil, L 2 is the self-inductance of the receiving coil. U DC2 is the rectified voltage at the receiving end, U BAT is the voltage across the battery load.

[0065] The working process of the WPT system is as follows: first, the industrial frequency AC power input from the power grid is converted into DC power by the transmitter rectifier circuit; then, the DC power is inverted into AC power by the high-frequency inverter circuit, and the power frequency output by the inverter circuit is consistent with the resonant frequency of the transmitter resonant circuit. At this time, under the action of the transmitter compensation circuit, the transmitter circuit works in a resonant state, and the circuit is purely resistive; according to Faraday's law of electromagnetic induction, the transmitter coil Tx The high-frequency electromagnetic field of the resonant frequency is excited into space, and the receiving coil RxAt this time, under the action of the receiving end compensation circuit, the receiving end loop also works in a resonant state and is purely resistive; finally, the receiving end rectifier circuit and chopper circuit convert the received high-frequency electric energy into the DC power required by the load.

[0066] To ensure the maximum efficiency of the WPT system during normal operation, the present invention adopts a bilateral SS compensation circuit, such as Figure 3 shown. Figure 3 middle, U S is the grid voltage, R S1 for Tx The equivalent resistance, C S1 , C S2 Are compensation capacitors, L 1 is the self-inductance of the transmitting coil, L 2 is the receiving coil self-inductance, M 12 is the mutual inductance between the transmitting coil and the receiving coil, R S2 for Rx The equivalent resistance, R L is the load resistance. To ensure the energy transmission efficiency of the WPT system, the SS compensation circuit should be in a resonant state, and the circuit parameters of the transmitter and receiver should meet the following requirements:

[0067] Formula 1: ;

[0068] Formula 2: ;

[0069] in: ω is the resonant angular frequency, ω =2π f , f As the resonant frequency, the present invention selects the WPT system f 85kHz, which is the most likely candidate frequency for most WPT systems; and They are the transmitter compensation capacitor and the receiver compensation capacitor respectively.

[0070] As the above formula shows, the self-inductance of the two coils affects the operating state of the entire WPT system. The offset uncertainty between the coupling mechanisms inevitably causes variations in the self-inductance of the two coils, which in turn reduces operating efficiency and increases the leakage magnetic field. Therefore, it is essential to design electromagnetic shielding structures to mitigate the negative impact of offset uncertainty on WPT systems.

[0071] 2.Electromagnetic shielding structure;

[0072] In WPT systems, the high-frequency coupled electromagnetic field surrounding the coupling coil is a key element in achieving energy transmission. Shielding structures concentrate the electromagnetic field, improving energy transmission efficiency while simultaneously weakening the electromagnetic field in non-operating areas. For WPT systems, the transmission of electromagnetic waves within the shielding structure directly determines its shielding effectiveness.

[0073] Electrical shielding materials mainly rely on their high electrical conductivity to attract electromagnetic fields, thereby reducing the possibility of electric fields entering non-working areas. When electromagnetic fields enter the electrical shielding structure, eddy currents are generated. Eddy currents have two functions: on the one hand, they generate a magnetic field on the shielding body that is opposite to the direction of energy transmission, reducing the possibility of these magnetic fields escaping to other areas; on the other hand, eddy currents generate heat, consuming the energy of the electric field entering the shielding body, thereby achieving shielding of the electric and magnetic fields. The eddy current induced electromotive force generated by the alternating electromagnetic field on the outside of the shielding material E The calculation formula is as follows:

[0074] Formula 3: ;

[0075] in: is the gradient operator symbol; B is the magnetic induction intensity; t represents a certain moment. The eddy current generated in the shielding structure can be expressed by volume integral, as shown in Equation 4:

[0076] Formula 4: ;

[0077] in: I for vortex; V is the integrated volume; Represents the total magnetic flux that interlinks with the shield; j is an imaginary unit; ω is the system resonant angular frequency; A is the magnetic vector; σ is the conductivity of the shielding material; ε is the dielectric constant. From this formula, we can see that the eddy current is affected by the system working angular frequency. ω , conductivity of shielding materials σ and dielectric constant εThe eddy current phenomenon generates a field in the opposite direction of energy transmission, which can offset any leakage magnetic flux that may escape and achieve shielding of the leakage magnetic field. Although materials with high electrical conductivity are effective in shielding electric fields, they also have some disadvantages. For example, eddy currents may be generated in time-varying electromagnetic fields, which will affect the magnetic field state around the receiving coil. The heat generated by eddy currents not only increases the loss of transmitted energy, but also poses certain safety hazards. Therefore, the current WPT shielding structure usually uses a combination of electrical and magnetic materials to reduce losses, improve the energy transmission efficiency of the WPT system, and make the shielding structure have better shielding performance.

[0078] When designing a shielding structure, the position, shape, and size of the shielding material can be modified based on the actual application. When analyzing the magnetic field's positional state, magnetic lines of force can be used to represent it. Because magnetic lines of force are closed and cannot be interrupted, shielding the magnetic field can only be achieved by guiding them to reduce the magnetic field that escapes outside the transmission mechanism. In a magnetic field, the relationship exists:

[0079] Formula 5: ;

[0080] in: represents magnetic flux; F m represents magnetomotive force; R m Represents magnetic resistance. By analyzing the relationship between the various quantities in this formula, it can be seen that the magnetic flux mainly passes through the shielding material with low magnetic resistance and high magnetic permeability, and only a small part passes through the air, which effectively reduces the leakage magnetic flux entering the air. When constructing a magnetic field shielding structure, the low magnetic resistance path is as follows: Figure 4 shown.

[0081] The shielding performance of the shielding structure reaches its best state when the magnetic flux in the magnetic material is not saturated, and it can also improve the coupling between the transmitting and receiving coils. When designing a magnetic shielding structure, in addition to considering the shielding effect, it is also necessary to comprehensively consider the requirements of the use environment and adjust the shape, weight, etc. of the shielding structure. The operating frequency of the WPT system of the present invention is 85kHz. The working magnetic field belongs to the low-frequency magnetic field. It mainly uses a high-permeability shielding body to shunt the magnetic flux to achieve shielding. The following relationship exists on the boundary surface of the two magnetic media:

[0082] Formula 6: ;

[0083] Where: magnetic field The angle with the normal is ;magnetic field The angle with the normal is ; μ 1 and μ2 are the relative permeabilities of the two magnetic media. This shows that the magnetic induction intensity inside the high-relative-permeability shield is much greater than that outside. Meanwhile, the magnetic lines of force in the low-relative-permeability material are nearly perpendicular to those in the high-relative-permeability material. This confines most low-frequency magnetic field energy to the shield's interior, achieving a shielding effect.

[0084] 3.K-Trans agent model;

[0085] During the actual operation of WPT systems, there is offset uncertainty between coupling mechanisms. The key to solving this problem is to incorporate offset uncertainty into the optimization design to achieve a robust optimization and mitigate the impact of offset uncertainty. However, existing processing methods typically require a large number of samples and are often driven by probability, which can increase computational time and lead to large errors in the results. Considering the complexity and high simulation cost of WPT systems, this paper adopts a K-Trans-based method to incorporate the effects of uncertainty and achieve robust optimization.

[0086] The Transformer model is more sensitive to changes in the data set and has better adaptability for regression tasks due to the long-range dependencies of its multi-head self-attention mechanism. In addition, the Transformer model uses distributed GPU parallel training, which is a significant advantage over LSTM (Long Short-Term Memory) and GRU (gated recurrent unit). The traditional Transformer model has an MLP (Multi-Layer Perceptron) layer after the self-attention layer. The connection between neurons in the MLP is usually a real value, representing the weight, and the neuron itself is equipped with a nonlinear activation function, usually a sigmoid function, so the calculation process of the MLP is to first weight the weight input and then introduce nonlinearity through the activation function. KAN is based on the Kolmogorov–Arnold (KA) representation theorem. The KA representation theorem states that any continuous function f ( x 1 ,..., x n ) can be expressed as a nested combination of a finite number of single-variable functions, and the formula is:

[0087] Formula 7: ;

[0088] in: is a multivariate continuous function, x is a ndimensional vector, i.e., the set of input variables; n Indicates the number of input variables; p and q They are indicators representing the number of functions and the variables that each function acts on; is the radial basis function, Φ q is the activation function of the output layer. KAN utilizes the KA representation theorem to represent weight parameters as a B-spline function. A B-spline function directly connects two neurons and is essentially a piecewise polynomial function. It achieves high smoothness at intersections of polynomial blocks, particularly near boundaries. The KA theorem states through the above equation that a high-dimensional function can be reduced to one-dimensional functions of the order of magnitude of learned polynomials. However, these one-dimensional functions are not necessarily smooth and easy to learn. Therefore, high-dimensional decomposition is achieved by parameterizing each one-dimensional function as a B-spline function. Replacing the MLP with a KAN allows the weights to have learnable activation functions. This not only leverages KAN's improved interpretability, but also achieves self-learning by moving the activation function to the "edges" and further smoothing the data by parameterizing it as a B-spline function. This allows the model to both learn features and accurately optimize these learned features, resulting in a smooth function that approximates the data. This improves the model's representational capabilities and ensures accurate approximation of high-dimensional functions. It also decomposes multidimensional functions into a combination of single-variable functions, reducing computational complexity.

[0089] The K-Trans proxy model consists of four parts: model input, encoder, decoder, and model output. In the WPT system of the present invention, the implementation steps of the K-Trans proxy model are as follows:

[0090] Input data: input data of the model The uncertainty factors in WPT application scenarios and shielding structure design parameters are integrated. The maximum magnetic field intensity at four fixed observation points around the WPT system is used. B max and energy transmission efficiency η For model output, realize the construction of proxy model under uncertainty scenario.

[0091] The input data first passes through the embedding layer (Input Embedding), a learnable parameter matrix that maps each input into a fixed-size vector, converting the input data into a vector representation. During the actual operation of WPT equipment, uncertainty factors can significantly impact the overall operation of the system, causing charging times to increase or even prevent charging. The shielding structure design parameters and the allowable ranges of uncertainty factors involved in the model of the present invention are shown in Tables 1 and 2, respectively. In engineering applications, since the offset between the coils of a WPT system is equally likely to occur in all directions, it is usually considered to be uniformly distributed, denoted by U.

[0092] Table 1 Design variables and their distribution parameters

[0093]

[0094] Table 2 Uncertainty factors and their distribution parameters

[0095]

[0096] Positional encoding: Since the Transformer architecture is completely based on the attention mechanism and does not contain loop or convolutional structures, it cannot understand the order of the input sequence. To solve this problem, the Transformer manually adds positional encoding. PE The position encoding operation can be expressed as follows:

[0097] Formula 8: ;

[0098] Formula 9: ;

[0099] in: It's location POS In the 2nd i Dimensional location information; It's location POS In the 2nd i+ Position information in 1 dimension; POS is the relative position of the element in the current sequence, i The total number of representative sequences is d The first i dimension; d MODEL The representative model dimension is d dimension. No matter how long the sequence is, PE Both can help the model learn information based on relative positions.

[0100] Encoder: The core of the improved Transformer encoder is the self-attention mechanism and KAN layer inside the encoder. It is composed of 6 encoders stacked together, each encoder consists of two sub-layers. Given a key ( K ) and the corresponding value ( P For each query ( Q ), the purpose of the self-attention mechanism is to calculate its attention to each value ( P i ) should be given attention (weight), emphasizing the most relevant values. The calculation formula is:

[0101] Formula 10: ;

[0102] in: softmax The function projects the weights into the range (0,1) and divides by To avoid very small gradients, h is a matrix K Dimensions of self-attention Attention is to calculate the attention within each sequence, i.e. Q = K = P= Process sequences and perform calculations on both the input and output sides; T Represents transpose.

[0103] Decoder: The Transformer decoder plays an important role in the autoregressive (AT) mode. In AT mode, each element generated by the decoder depends on the previously generated elements and is generated step by step. This allows the decoder to better capture dependencies between sequences.

[0104] By training this model, a proxy model for the electromagnetic shielding structure design of WPT systems that accounts for offset uncertainty can be obtained. Using the MC method, this proxy model generates corresponding outputs, which can be used to assist in optimization. Calculations based on these samples can replace simulations, saving time and effort in the optimization process.

[0105] 4.MOEDO optimization algorithm;

[0106] In the optimization design of WPT systems, the K-Trans agent model has been used to quantify the impact of uncertainty in the design process. During the optimization phase, only the design parameters are optimized, and uncertainty is embedded as a background factor in the optimization process, thus achieving robust optimization.

[0107] This paper uses the Multi-Objective Exponential Distribution Optimizer (MOEDO), an intelligent optimization algorithm, for calculations. MOEDO is a heuristic method that integrates exponential distribution theory and dynamic optimization concepts. It uses an enhanced non-dominated sorting and crowding distance mechanism, integrated with an information feedback mechanism (IFM), to transform multi-objective tasks into single-objective subtasks, thereby improving the algorithm's convergence and efficiency while overcoming local optimality.

[0108] The optimization process of the MOEDO algorithm is as follows:

[0109] (1) Initialization of population: In the early stage of MOEDO, the population needs to be initialized, just like other heuristic algorithms. X winners operation, randomly generating a set of solutions X 0={ x 1, x 2,..., x N}, each solution x i =( x i1 , x i2 ,..., x id ) is a d dimensional design vector, and satisfy the value range of the decision space, that is, x ij ∈[ lb j , ub j ],in lb j and ub j They are j In the main evolution process, the core of MOEDO is to simulate the process of natural selection, including three main operations: mutation, selection and update.

[0110] (2) Mutation operation: The goal of the mutation operation is to explore the solution space and avoid the algorithm from falling into the local optimum by introducing randomness. MOEDO adopts a memory-based mutation operation. First, the mean of the top several optimal solutions in the population is selected as the guide vector X guide , , represents the weighted average of the first three optimal solutions, where represents the optimal solution found in the iteration, Indicates the first i All columns of the row.

[0111] For each solution xi , the mutation process must first undergo conditional judgment. If the individual x i If it is a memory solution, the following mutation operation is performed:

[0112] Formula 11: ;

[0113] in: is the generated solution; a , b It is a dynamic parameter generated by random factors; variance According to the expected rate of the current solution 1 / μ Calculated; memoryless i It is an array that stores individual positions, containing the position of each individual in the current population.

[0114] If the individual x i If it is not a memorized solution, the following mutation operation is performed:

[0115] Formula 12: ;

[0116] in: is a random number; is the current solution.

[0117] For boundary processing, MOEDO ensures that the generated solution V i Within the allowable range of the solution space, the following correction is made:

[0118] Formula 13: ;

[0119] in: V ij It is the solution j The value of the dimension; max and min The functions are maximum function and minimum function respectively.

[0120] (3) Non-dominated sorting and congestion sorting. MOEDO uses two sorting strategies, non-dominated sorting and congestion sorting, to maintain the quality and diversity of the solution set.

[0121] Non-dominated sorting: used to select Pareto optimal solutions. In non-dominated sorting, the solutions in the population are divided into different Pareto frontiers. The individuals on the frontier are non-dominated, that is, no solution dominates other solutions. The solutions are divided into multiple levels according to the dominance relationship. The dominance relationship is expressed as: x i Dominant solution x j, the condition is that for all objective functions f i ( x i )≤ f i ( x j )and f i ( x i )≠ f i ( x j ).

[0122] Crowding sorting: used to ensure the diversity of the solution set and avoid clustering of solutions. In each frontier, the crowding distance is used to calculate the diversity of solutions. The crowding distance is an indicator to measure whether the solutions are clustered in the target space. The formula is as follows:

[0123] Equation 14: ;

[0124] in: C i Is the solution i the degree of congestion; m is the number of objective functions; j is the index of the objective function; and They are j The objective function solution i The adjacent solution function value of ; f max, j and f min, j They are j The maximum and minimum values ​​of the dimensional objective function value.

[0125] (4) Update operation: After each iteration, MOEDO needs to maintain a solution set to store all historical optimal solutions and select the optimal solution based on non-dominated sorting to update the archive. For each frontier, individuals are selected and sorted by congestion to maintain the diversity of solutions. Finally, the solution set in the archive is updated based on the fitness and congestion of the solution to ensure that the optimal solution is retained in the archive, thus achieving the update archive.

[0126] The algorithm pseudo code is as follows:

[0127]

[0128] In the algorithm pseudocode, X guide is the guide vector; are three optimal solutions respectively, and the average is taken to find a comprehensive optimal solution; For the time+ In the 1st iteration i The generated value of each individual; is the variance; is the maximum number of iterations; are random numbers respectively.

[0129] In operations where offset uncertainty is embedded, the design variables remain the same, whether in robust optimization or traditional deterministic optimization. The offset uncertainty variables serve only as random inputs to the proxy model. The Pareto front sought by MOEDO is always the optimization result for design factors. By combining the MOEDO optimization method with the K-Trans proxy model, both deterministic optimization and robust optimization of WPT systems with offset uncertainty can be achieved.

[0130] The specific implementation of the present invention is described in detail below with reference to specific embodiments.

[0131] Example 1: This example conducts multi-faceted research on the WPT system, aiming to optimize system performance. The specific contents are as follows:

[0132] 1.WPT system simulation model;

[0133] The WPT system simulation model constructed by the present invention adopts a composite shielding structure (see Figure 5 ), based on the combination of aluminum plates and ferrite units, an ultra-thin single layer of nanocrystalline material with high relative permeability is added. At room temperature and 85kHz, the relative permeability of the nanocrystalline shielding material reaches 22,000, far exceeding the 3,300 of the ferrite unit.

[0134] In this composite shielding structure, the inner small strip-shaped nanocrystals guide the leakage magnetic field at the center of the coil into the coupling path, the outer strip-shaped nanocrystals guide the leakage magnetic field in the non-working area back to the working area, the ferrite units compensate for the shielding gaps of the nanocrystal shielding layer material, the aluminum plate performs electrical shielding, and the three materials are stacked. Compared with the traditional large-area ferrite-covered shielding structure, the use of nanocrystals to replace part of the ferrite can not only reduce the overall weight of the WPT system and save the space occupied by the shielding structure, but also improve the robustness of the overall structure. However, because the ferrite units of the present invention are not placed symmetrically, the magnetic field distribution around the system is asymmetric. In addition, the offset uncertainty between the coupling mechanisms will also affect the overall shielding effect. Therefore, designing a suitable combination and placement of shielding materials is crucial to ensuring that the WPT system achieves optimal performance under the influence of offset uncertainty.

[0135] The WPT system consists of a transmitting coil Tx , receiving coil Rx , shielding layers at both ends and conversion circuit. Among them, Txand Rx The dimensions are all 200cm×200cm, and the cross-sectional area of ​​the coils is 1.3×10 -5 mm 2 , are composed of 15 turns of coil winding, and the coil wires are made of metal copper wrapped with an insulating layer. Tx and Rx When the alignment and coil spacing are 5 cm, the system works under ideal conditions, the system input power of WPT is 300 W, and the operating frequency is 85 kHz.

[0136] Magnetic flux density around the WPT system before and after adding the shielding structure B The distribution of Figure 6 As shown in Figure 2, it can be seen that when the WPT system is working normally, the magnetic field distribution on the cross section tends to diverge outward, and the magnetic field density at the center of the coil is relatively large (see Figure 6 (a)); After adding the shielding structure, the magnetic field around the system is obviously confined to the working area, and the magnetic fields on the upper and lower surfaces of the WPT system are significantly reduced due to the shielding structure (see Figure 6 (b) and (c) in the figure), which can improve the safety of the system. The magnetic field intensity distribution inside the shielding structure of the WPT system is shown in the figure. Figure 7 As shown in the figure, the magnetic field strength of the receiving nanocrystal shield is greater than that of the five ferrite units. This is due to the large difference in relative permeability between the nanocrystal shield and the ferrite units, which increases the tendency of the magnetic field to propagate toward the nanocrystal shield, and the ferrite units play a supporting role in shielding. Furthermore, the magnetic field strength of the inner nanocrystals is greater than that of the outer nanocrystals. This is because during coil energy transfer, the magnetic field at the coil center is guided through the inner nanocrystals, thereby improving magnetic field coupling efficiency and reducing magnetic leakage.

[0137] According to the ICNIRP 2010 guidelines, the WPT system of the present invention operates at a frequency of 85kHz and uses 27μT as the electromagnetic environment protection standard. Since the magnetic flux density is higher at the vertical center between the coils, fixed observation points are set up in four directions: the vertical center between the coupling mechanisms and 20cm from the coil center. V 1. V 2. V 3 and V 4 (see Figure 8 (a) and (b) in the figure are used to evaluate the shielding effect. The magnetic field intensity distribution at the four observation points and the energy transmission efficiency of the WPT system with and without the shielding structure are shown in the figure. η like Figure 9 As shown in Table 3, Group 1 represents no shielding, Group 2 represents combined shielding of aluminum plate and ferrite unit, and Group 3 represents combined shielding of aluminum plate, ferrite unit and nanocrystalline shielding layer.

[0138] Table 3 Four observation points B value

[0139]

[0140] pass Figure 9 As shown in Table 3, compared with Group 1, after adding Group 2 shielding, the observation point B The maximum value is reduced by 85.19%. At the same time, the energy transmission efficiency of the WPT system is reduced by 7.2%. This is because after adding the shielding layer, although the leakage magnetic field of the environment is reduced, the eddy current effect of the aluminum plate causes part of the electrical energy to be converted into heat energy, resulting in a decrease in efficiency. Since the shielding structure is asymmetric, the maximum value of the magnetic field strength at the four observation points is calculated. B max As an indicator of shielding effect. After adding the nanocrystalline shielding layer, compared with the Group 2 combination, B max The reduction was 30.05%, while the energy transmission efficiency of the WPT system increased by 8.58%. This shows that the nanocrystalline shielding layer material, with its high relative permeability, can improve the system coupling degree, thereby improving the energy transmission efficiency of the WPT system. Therefore, when optimizing the shielding structure design, it is necessary to determine the optimal thickness of the aluminum plate and ferrite unit, as well as the optimal placement of the ferrite unit, to achieve the best combination with the nanocrystalline shielding layer.

[0141] 2.K-Trans agent model;

[0142] Taking into account the actual application of the WPT system, according to the uncertainty factors and their distribution in Table 2, a random sampling method is used to calculate the system energy transmission efficiency of the original structure and the maximum magnetic field intensity at a fixed observation point. B max The probability distribution function (PDF) of the improved Transformer proxy model is compared with the calculation results of the MC method (considering the cost of simulation calculation, the number of simulations of the MC method is 10,000). The comparison results are as follows Figure 10 As shown in (a) and (b), it can be seen that the calculation results of the improved Transformer (K-Trans) are basically consistent with those of MC, and the calculation accuracy is slightly improved compared with MC and the Transformer before the improvement. Under the condition of uncertainty, the average energy transmission efficiency of the WPT system predicted by the proxy model is 78.83%, and the magnetic field strength is B maxThe average value is 22.64 μT. Due to the influence of offset uncertainty, the original WPT structure has the possibility of exceeding the limit, with a probability of 27.34%. Therefore, the shielding structure needs to be optimized to reduce the probability of exceeding the limit. The calculation time of the three methods for a single structure is shown in Table 4:

[0143] Table 4 Calculation time of three methods

[0144]

[0145] The data in Table 4 shows that, under the same WPT system structure, the computational cost of the MC method is very high when running 10,000 iterations, while the K-Trans method saves 90.97% of the computational time. This demonstrates that K-Trans significantly improves computational efficiency while maintaining accuracy.

[0146] 3. Shielding structure optimization;

[0147] In the optimization process, there are two methods: deterministic optimum and robust optimum. Deterministic optimum is to directly calculate by optimization without considering other factors; while robust optimization needs to consider the offset uncertainty between the coupling mechanisms of the WPT system. Therefore, the present invention embeds the uncertainty influence as the data background into the optimization process to achieve the purpose of finding the robust optimum. For the WPT system, the system has high working efficiency and the observation point is B max Achieving the optimal state is the most ideal. The design variables corresponding to the target points in the Parato solution are shown in Table 5:

[0148] Table 5 Optimal design variable combinations in two cases

[0149]

[0150] In actual use, WPT systems rarely maintain ideal operating conditions, as offset uncertainty between coupling mechanisms can significantly alter their operating conditions. To verify the effects of offset uncertainty on the WPT system's energy transmission efficiency and electromagnetic shielding, we used a surrogate model to calculate the probability distribution curves for energy transmission efficiency and magnetic field strength, based on the parameter settings in Table 2 and the optimized parameters in Table 5. These curves were then compared with the pre-optimization probability distribution functions.

[0151] Table 6 Statistical parameter values ​​obtained from PDF

[0152]

[0153] Figure 11The data in (a) and (b) and Table 6 show that the robustly optimized system achieved an average efficiency of 86.60% and an average magnetic field strength of 13.42 μT. However, the determined optimal structure was significantly affected, with an average power of 80.73% and an average magnetic field strength of 20.63 μT. Furthermore, from the perspective of limit-exceeding probability, the determined optimal structure achieved a limit-exceeding probability of 19.09%, while the robust optimal structure WPT system's magnetic flux leakage exceeded the limit probability by 0, eliminating the possibility of limit-exceeding, thus ensuring the user's electromagnetic exposure safety. Therefore, the robustly optimized WPT system has stronger anti-drift capabilities. The robust optimization method, assisted by the K-Trans agent model, can effectively eliminate the negative impact of uncertainty on the WPT system, improving the stability of the system's electromagnetic shielding, energy transmission efficiency, and user safety.

[0154] 4. Experimental verification;

[0155] In order to verify the effectiveness of the K-Trans agent model-assisted robust optimization method, a 300W, 85kHz WPT experimental system was constructed (see Figure 12 and Figure 13 ). The system consists of a transmitting coil, a receiving coil, a conversion circuit on both sides, a shielding layer and a resistive heating load. The present invention adjusts the coil spacing by using acrylic material pads that have no effect on the electromagnetic field. In the experiment, the magnetic field strength measurement device is based on the ELT-400 series probes of the German Narda company used for safety assessment of radiation exposure to magnetic fields on the human body. This type of probe complies with the measurement standards of ICNIRP 2010. The following is a verification based on the established experimental system. In the process of verifying the effect of the robust optimal shielding structure, this experiment will include three uncertainty factors: 、 and The steps are all set to 1cm, and the coil spacing is different ∆Z The numerical plane is used as a distinction to obtain different coil spacings In the case The corresponding efficiency and magnetic field strength data were used to plot a four-dimensional response surface diagram, further verifying the shielding performance of the two structures under the influence of uncertainty factors. The uncertainty factor distribution parameters in Table 2 show that, with a step of 1 cm, there are 363 possible combinations of the three uncertainty factors for each shielding structure during the experiment. These 363 points are used to plot the response surface diagram.

[0156] Since the shielding structure of the WPT system of the present invention is asymmetric, and The values ​​of are the same, but their effects on the whole system are different. The response surface diagrams of the two structures under different combinations are shown in Figure 2. Figure 14 and Figure 15As shown, the three coordinate axes represent three uncertainty factors, and the color legend on the right represents B max and energy transmission efficiency η The numerical value of . Figure 14 and Figure 15 It can be seen that in different On the numerical plane, the magnetic field shielding capability of the robust optimal structure is better than that of the determined optimal structure. B max Distribution and energy transfer efficiency η Distribution, all in =4cm. This is because when the coil spacing becomes smaller, the coupling effect of the two coils is better, so the energy transmission efficiency is higher; at the same time, the magnetic field is more concentrated in the working area, so the leakage magnetic field in the non-working area is less. Tx and Rx In the presence of offset, the robust optimal structure can better suppress the leakage of magnetic field. In addition, in terms of energy transmission efficiency η In terms of improvement, the robust optimal structure can also maintain high efficiency when the coil is offset. In summary, in terms of the performance of the two shielding performance indicators, the robust optimal structure can make the system work more stable and safe under the influence of uncertain factors. η Mean, B max Mean and system out-of-limit probability under the influence of uncertain factors P The data are shown in Table 7:

[0157] Table 7 Data after shielding the two under the influence of offset uncertainty

[0158]

[0159] The data in Table 7 show that under the influence of uncertain factors, the overall shielding performance of the robust optimal structure is better than that of the determined optimal structure. η It increased by 4.95%. B max The mean value dropped by 5.17 μT compared with the latter, and the probability of exceeding the limit was reduced most significantly. Under the robust optimal structure, the probability of exceeding the limit of the WPT system leakage magnetic field was reduced to 0%, effectively protecting the electromagnetic exposure safety of users.

[0160] In summary, the proposed optimization design strategy based on the K-Trans proxy model combined with the MOEDO method achieves multi-objective robust optimization of the WPT system shielding structure while effectively reducing computational costs. This approach not only improves the system's energy transmission efficiency and shielding performance under the influence of coupling offset uncertainty, but also significantly enhances the system's electromagnetic safety and offset tolerance, showing promising prospects for engineering applications.

[0161] The above are only preferred embodiments of the present invention. It should be pointed out that for those skilled in the art, several variations and improvements can be made without departing from the concept of the present invention. These should also be regarded as the scope of protection of the present invention. These will not affect the effect of the implementation of the present invention and the practicality of the patent.

Claims

1. A robust optimization method for the electromagnetic shielding structure of a wireless power transmission system, characterized in that: The following steps are involved: Step 1: Construct a WPT system based on the principle of magnetic coupling resonance, i.e., a wireless power transmission system. The system uses a bilateral SS compensation circuit, which is used to ensure that the transmitter and receiver loops are in a resonant state to improve energy transmission efficiency. Step 2: Design an electromagnetic shielding structure, which includes electrical shielding materials and magnetic materials to improve energy transmission efficiency by concentrating the electromagnetic field and weaken the electromagnetic field in non-working areas; Step 3: Build a K-Trans proxy model, which integrates the uncertainty factors of the WPT system and the shielding structure design parameters, and uses the improved Transformer model combined with the Kolmogorov-Arnold representation theorem to model the data; Step 4: Using the MOEDO algorithm, a multi-objective robust optimization is performed based on the K-Trans agent model to generate a Pareto optimal solution set to achieve a balance between the energy transmission efficiency and electromagnetic shielding performance of the WPT system; In step 2, the shielding structure includes a nanocrystalline shielding layer, a ferrite unit and an aluminum plate, and the three materials are stacked; The nanocrystal shielding layer consists of inner and outer nanocrystals. The inner nanocrystals direct the leakage magnetic field from the center of the coil into the coupling path, while the outer nanocrystals direct the leakage magnetic field from the non-working area back to the working area. The ferrite units compensate for the shielding gaps in the nanocrystal shielding material, and the aluminum plate provides electrical shielding. At room temperature and 85kHz, the relative magnetic permeability of the nanocrystal shielding material is 22,000, while the relative magnetic permeability of the ferrite units is 3,300. The shielding structure optimizes the magnetic flux path and reduces magnetic leakage based on the following formula: Where: Φ represents magnetic flux; F m represents magnetomotive force; R m Reluctance; magnetic field The angle with the normal is θ1; the magnetic field The angle with the normal is θ2; μ1 and μ2 are the relative magnetic permeabilities of the two magnetic media, nanocrystal and ferrite, respectively; In step 3, the inputs of the K-Trans proxy model include shielding structure design parameters and uncertainty factors. The shielding structure design parameters include the ferrite unit thickness t1, the distance d1 from the ferrite to the WPT center, the aluminum plate thickness t2, and the relative angle α between the upper and lower ferrite layers. The uncertainty factors include the lateral offset ΔX, the longitudinal offset ΔY, and the coil spacing ΔZ. The output of the K-Trans proxy model is the maximum magnetic field intensity B at the observation point. max and the energy transmission efficiency η of the WPT system, process sequence data through position encoding and self-attention mechanism, and use KAN layer to replace MLP layer to improve model accuracy.

2. The method for robust optimization of electromagnetic shielding structure of wireless power transmission system according to claim 1, characterized in that: In step 1, the bilateral SS compensation circuit satisfies the resonance condition: Where: ω is the resonant angular frequency, ω = 2πf, f is the resonant frequency, which is 85kHz; L1 is the self-inductance of the transmitting coil, L2 is the self-inductance of the receiving coil; C S1 and C S2 They are the transmitter compensation capacitor and the receiver compensation capacitor respectively.

3. The method for robust optimization of electromagnetic shielding structure of wireless power transmission system according to claim 1, characterized in that: The position encoding operation in the network structure of the K-Trans agent model is expressed as: Among them: PE ( pos,2i ) is the position information of position pos in the 2i dimension; PE ( pos,2i+1 ) It is the position information of position pos in the 2i+1th dimension; pos is the relative position of the element in the current sequence, i represents the total number of sequences is the i-th dimension in the d-dimensional; d MODEL The representative model dimension is d dimension.

4. The method for robust optimization of electromagnetic shielding structure of wireless power transmission system according to claim 1, characterized in that: In step 4, the optimization process of the MOEDO algorithm is as follows: (1) Initialize the population and randomly generate a set of solutions; (2) Mutation operation, which explores the solution space by introducing randomness and avoids the algorithm from falling into local optimality; the mutation operation includes memory-based mutation, where for each solution x i , if the individual belongs to the memory solution, the following mutation operation is performed: V i =a·(Memoryless i -variance)+b·X guide ; Where: V i is the generated solution; a, b are dynamic parameters generated by random factors; variance is calculated based on the expected rate 1 / μ of the current solution; Memoryless i It is an array that stores individual positions, including the position of each individual in the current population; If individual x i If it is not a memorized solution, the following mutation operation is performed: V i =b·(Memoryless i -variance)+log(φ)·X i ; Where: φ is a random number; X i is the current solution; For the boundary processing, to ensure the generated solution V i Within the allowable range of the solution space, the following correction is made: V ij =max(min(V ij ,ub j ),lb j ); Where: V ij is the value of the jth dimension of the solution; max and min functions are the maximum value function and the minimum value function respectively; lb j With ub j are the lower and upper bounds of the j-th dimension respectively; (3) Non-dominated sorting and congestion sorting are used to maintain the quality and diversity of the solution set. The non-dominated sorting is used to select the Pareto optimal solution and classify the solutions according to the dominance relationship. The congestion sorting is used to ensure the diversity of the solution set and avoid the aggregation of solutions. In each frontier, the congestion distance is used to calculate the diversity of the solution. The formula is as follows: Where: C i is the congestion degree of solution i; m is the dimension of the solution; j is the index of the jth objective function; f i+1,j and f i-1,j are the function values ​​of the adjacent solutions of solution i on the jth objective function; f max,j and f min,j are the maximum and minimum values ​​of the objective function value of the j-th dimension respectively; (4) Update operation, updating the solution set in the archive according to fitness and congestion; in the update operation, a solution set is maintained after each iteration, all historical optimal solutions are stored, and the optimal solution is selected according to non-dominated sorting to update the archive.

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