Motor stator auxiliary slot multi-objective optimization method based on information coefficient driving

Abstract

The invention discloses a motor stator auxiliary slot multi-objective optimization method based on information coefficient driving, and the method comprises the steps: obtaining an initial sample, and building an agent model; constructing a double-strategy component based on expectation and uncertainty; candidate points are uniformly generated in a design space, the expected utility of each point and four sub-acquisition function values are calculated, and the sub-function values are fused in respective components to form two component values; respectively calculating two corresponding information coefficients IC based on the two component values and the expected utility, and obtaining a basic weight through dynamic smoothing and normalization; the two basic weights are fused with the model confidence coefficient, the space coverage rate and the optimization stage indexes to form final weights, and a self-adaptive collection function is constructed; and adopting a self-adaptive acquisition function to evaluate all candidate points and performing batch sampling. According to the motor stator auxiliary slot multi-objective optimization method based on information coefficient driving, the effectiveness of the acquisition function is dynamically evaluated, the weight of the acquisition function is adaptively adjusted, and the performance of the motor is remarkably improved.

Description

Technical Field

[0001] This invention belongs to the field of motor design technology, specifically relating to a multi-objective optimization method for auxiliary slots of motor stator driven by information coefficients. Background Technology

[0002] Motor design optimization is a core issue in electrical engineering, with stator auxiliary slot size optimization directly impacting the motor's electromagnetic performance and operating efficiency. Traditional optimization methods, such as genetic algorithms and particle swarm optimization, suffer from high computational costs and slow convergence speeds, particularly when evaluating expensive functions like electromagnetic field finite element analysis.

[0003] Bayesian optimization (BO), as an efficient global optimization method, is particularly suitable for black-box optimization problems where objective function computation is expensive and derivative information is difficult to obtain. Its core idea is to approximate the objective function using a surrogate model (usually a Gaussian process) and to use a sampling function to guide the selection of sampling points. In multi-objective optimization scenarios, the ParEGO algorithm transforms the multi-objective problem into a single-objective problem through weighted scalarization, providing an effective framework for Bayesian optimization.

[0004] However, traditional Bayesian optimization methods face the following challenges in optimizing the size of stator auxiliary slots:

[0005] (1) A single acquisition function is insufficient to balance the contradiction between exploration and utilization;

[0006] (2) Fixed-weight strategies cannot adapt to the needs of different optimization stages;

[0007] (3) The model uncertainty was not adequately addressed;

[0008] (4) It is easy to get trapped in local optima in multi-objective trade-offs. Summary of the Invention

[0009] This invention provides a multi-objective optimization method for auxiliary slots of motor stators based on information coefficients to solve the aforementioned technical problems, specifically employing the following technical solution:

[0010] A multi-objective optimization method for auxiliary slots in motor stators based on information coefficients includes the following steps:

[0011] Define the stator auxiliary slot geometric parameters as design variables, and set a conflict optimization objective that weakens electromagnetic force harmonics while retaining electromagnetic torque;

[0012] Initial samples were obtained using the SLE continuous enumeration method, and a surrogate model for simultaneously predicting electromagnetic force and torque was established using Gaussian process regression.

[0013] Construct a dual-strategy component based on expectation and uncertainty. The expectation component integrates expectation improvement and robust expectation improvement, while the uncertainty component integrates uncertainty acquisition and information gain acquisition.

[0014] Candidate points are generated uniformly in the design space, and the expected utility of each point and the values ​​of four sub-collection functions are calculated in parallel. The sub-function values ​​are then merged within their respective components to form two component values.

[0015] Based on the values ​​of the two components and the expected utility, the corresponding two information coefficients IC are calculated, and the basic weights are obtained through dynamic smoothing and normalization.

[0016] The two basic weights are respectively fused with model confidence, spatial coverage and optimization stage indicators to form two final weights. The two final weights are then weighted and merged to construct an adaptive acquisition function.

[0017] The adaptive acquisition function is used to evaluate all candidate points and sample them in batches. The surrogate model is iteratively updated until the Pareto front converges, and the auxiliary slot structure scheme that balances low electromagnetic force and high torque is output.

[0018] Furthermore, the SLE continuous enumeration method uniformly divides the design space into equidistant square grids. According to the maximum-minimum distance criterion, that is, maximizing the minimum distance between any two sample points, 10 initial sample points are selected from them, so that the sample set has optimal space filling performance and projection characteristics, providing a comprehensive and balanced initial dataset for the surrogate model.

[0019] Furthermore, the Gaussian process regression uses the Matern 5 / 2 kernel function to construct a joint probability distribution. Through the covariance matrix operation of the training set and the test set, the predicted mean and variance of electromagnetic force and torque at the candidate points are output simultaneously. The variance is used to quantify the uncertainty of the model and provide a basis for exploring the subsequent acquisition function.

[0020] Furthermore, the four sub-function values ​​are obtained through multi-strategy parallel computation: expectation improvement measures the potential of a candidate point to surpass the current optimal solution; robust expectation improvement reduces optimistic estimation bias by setting stricter objectives; uncertainty acquisition integrates multi-objective prediction standard deviation to quantify information gaps; and information gain acquisition introduces spatial distance attenuation penalty to avoid repeated exploration in already sampled dense areas. Together, these four constitute a comprehensive evaluation of the value of the candidate point.

[0021] The values ​​of the four sub-functions are calculated as follows:

[0022] The desired improvement is to use a numerically stable improved function for calculation;

[0023] α EI (x)=max(f min -f(x), 0)

[0024] Among them, f min This is the current optimal value;

[0025] Robust expectations are improved by using the following formula to set more stringent targets and reduce the risk of optimistic estimations;

[0026]

[0027] Among them, f min It is the smallest function value currently observed, σ y It is the standard deviation of historical observations, which reflects the dispersion of the observed data. β is a non-negative robustness factor used to adjust the degree of conservatism for model uncertainty.

[0028] Uncertainty collection is weighted and summed using the following formula to calculate the standard deviation of multi-objective predictions;

[0029] α UC (x)=σ 2f (x)+σ 4f (x)+σ torque (x)

[0030] Where, σ 2f σ 4f and σ torque (x) represents the prediction standard deviation of each objective function model at point x;

[0031] Information gain acquisition is penalized by introducing a distance attenuation coefficient to the dense sampling area according to the following formula;

[0032]

[0033] Where, d min (x) represents the minimum distance to historical samples, and λ is the distance decay coefficient.

[0034] Furthermore, the information gain acquisition function applies an exponential penalty to the minimum distance from the candidate point to the historical sample point through a distance attenuation coefficient λ. The closer the distance, the greater the penalty, which enables the algorithm to prioritize exploring candidate points that are far from the sampled area and have high uncertainty, thereby improving the global exploration efficiency and enhancing sample diversity.

[0035] Furthermore, the expected utility of each candidate point is calculated by weighted summation of normalized electromagnetic force and torque, and the two information coefficients are calculated according to the following formulas.

[0036]

[0037] Among them, IC EI IC is the expected information coefficient. UC For uncertainty information coefficient, Candidate point x iCorresponding to expectations, Let U(x) be the mean of the expected value of the sample set. i ) is a candidate point x i Expected utility, U mean Candidate point x i The mean expected utility Candidate point x i Uncertainty Candidate point x i The mean of uncertainty.

[0038] The Pearson correlation coefficient is used to quantify the consistency between the expected component value and the uncertain component value and the expected utility. The higher the correlation, the stronger the guiding role of the strategy on the real optimization objective.

[0039] Furthermore, the dynamic smoothing is performed using the following formula:

[0040] IC EI_smooth =α(t)·IC EI +(1-α(t))·IC EI_previous

[0041] IC UC_smooth =α(t)·IC UC +(1-α(t))·IC UC_previous

[0042] Among them, IC EI_previous and IC UC_previous It is the smoothed IC value corresponding to the previous cycle, and the smoothing factor α(t) takes values ​​in segments as the number of iterations increases.

[0043] Furthermore, the model confidence level is calculated using the cross-validation mean absolute error formula:

[0044]

[0045] in, and The coefficients of determination for the test set are the three proxy models built on the training set. When the confidence level is >0.8, the collection function chooses to trust the model's prediction and will place more emphasis on the expected search strategy. When the confidence level is <0.3, the collection function chooses to question the model's prediction and will place more emphasis on the uncertain search strategy.

[0046] Furthermore, the spatial coverage rate is calculated using the following formula:

[0047]

[0048] Where K is the partitioning precision, N coveredThe number of grid cells covered by at least one sample point is calculated using the following formula:

[0049]

[0050] Where I is an indicator function, which takes the value 1 when the condition is met and 0 otherwise; D is the current set of sample points.

[0051] Furthermore, the optimization phase is divided into three stages according to the following formula: initial exploration, accelerated search, and convergence refinement.

[0052]

[0053] The advantages of this invention lie in the provision of a multi-objective optimization method for stator auxiliary slots driven by information coefficients. The improved adaptive acquisition function, driven by IC weight factors, employs multi-strategy fusion and a state-aware mechanism, effectively solving the exploration-exploitation balance problem in Bayesian optimization. This method significantly improves surrogate model accuracy, sampling efficiency, multi-objective optimization performance, and prediction reliability. By dynamically evaluating the effectiveness of the acquisition function and adaptively adjusting its weights, it significantly enhances the performance of Bayesian optimization in multi-objective engineering optimization problems. This method has achieved good results in the stator auxiliary slot size optimization problem, effectively balancing the conflicting objectives of minimizing electromagnetic force and maximizing torque, accelerating convergence, and avoiding local optima. Attached Figure Description

[0054] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0055] Figure 1 This is a schematic diagram of the multi-objective optimization method for auxiliary slots of motor stator based on information coefficients proposed in this application;

[0056] Figure 2 This is a schematic diagram of the convergence curves of the average relative error iteration process for different methods;

[0057] Figure 3 These are schematic diagrams illustrating the convergence curves of the RMSE iteration process using different methods;

[0058] Figure 4 These are schematic diagrams illustrating the convergence curves of the coefficient of determination iteration process for different methods;

[0059] Figure 5 It is a graph of the adaptive sampling function after multiple iterations of sampling;

[0060] Figure 6 This is a graph of the PI acquisition function after multiple iterations of sampling;

[0061] Figure 7 This is a graph of the EI acquisition function after multiple iterations of sampling;

[0062] Figure 8 This is a graph of the LCB acquisition function after multiple iterations of sampling;

[0063] Figure 9 This is a graph of the Thompson sampling function after multiple iterations of sampling;

[0064] Figure 10 This is a schematic diagram of the Pareto front solution of the surrogate model constructed by the adaptive acquisition function;

[0065] Figure 11 This is a schematic diagram of the Pareto front solution of the surrogate model constructed by the PI acquisition function;

[0066] Figure 12 This is a schematic diagram of the Pareto front solution of the surrogate model constructed by the EI acquisition function;

[0067] Figure 13 This is a schematic diagram of the Pareto front solution of the surrogate model constructed by the LCB acquisition function;

[0068] Figure 14 This is a schematic diagram of the Pareto front solution of the surrogate model constructed using the Thompson acquisition function;

[0069] Figure 15 This is a box plot of the prediction error distribution on the test set for the method of this application;

[0070] Figure 16 This is a box plot of the prediction error distribution on the test set for the PI method.

[0071] Figure 17 This is a box plot of the prediction error distribution on the test set for the EI method.

[0072] Figure 18 This is a box plot of the prediction error distribution on the test set for the LCB method.

[0073] Figure 19 This is a box plot of the prediction error distribution on the test set for the Thompson method;

[0074] Figure 20 It is the electromagnetic force FFT before optimization;

[0075] Figure 21 This is the electromagnetic torque before optimization;

[0076] Figure 22 It is the electromagnetic force FFT optimized using the method of this application;

[0077] Figure 23 The electromagnetic torque is optimized using the method described in this application;

[0078] Figure 24 This is a three-dimensional spatial-frequency distribution diagram of the electromagnetic force of the motor before optimization;

[0079] Figure 25 This is a three-dimensional spatial-frequency distribution diagram of the electromagnetic force of the motor after optimization using the method of this application;

[0080] Figure 26 This is a schematic diagram showing the motor's power consumption and stability before optimization;

[0081] Figure 27 This is a schematic diagram showing the power consumption and stability of the motor after optimization using the method described in this application;

[0082] Figure 28 This is a schematic diagram of the motor cogging torque waveform before optimization;

[0083] Figure 29 This is a schematic diagram of the motor cogging torque waveform after optimization using the method of this application;

[0084] Figure 30 It is a precise quantitative analysis diagram of the radial magnetic flux density waveform and spectrum of the motor under no-load air gap before optimization;

[0085] Figure 31 This is a precise quantitative analysis diagram of the radial magnetic flux density waveform and spectrum of the motor under no-load air gap after optimization using the method of this application;

[0086] Figure 32 It optimizes the harmonic spectrum characteristics of the no-load back EMF of the motor.

[0087] Figure 33 The harmonic spectrum characteristics of the motor's no-load back EMF after optimization using the method described in this application are shown. Detailed Implementation

[0088] Embodiments of the present invention are described in detail below. Examples of these embodiments are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0089] The accompanying drawings are merely illustrative of the invention and are not necessarily drawn to scale. Some of the block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0090] The flowchart shown in the attached diagram is merely an illustrative example and does not necessarily include all steps. For example, some steps may be broken down, while others may be combined or partially combined; therefore, the actual execution order may change depending on the specific circumstances.

[0091] In a specific embodiment of the present invention, the innovability and practicality of the invention are demonstrated by describing an exemplary embodiment in detail. This embodiment, in conjunction with the system architecture and flowcharts in the accompanying drawings, clearly illustrates the various key modules of the invention and their interactions. The embodiments of the present invention aim to provide those skilled in the art with a technical solution that is easy to understand and implement, while demonstrating the advantages and effects of the present invention in practical applications.

[0092] In current common motor noise reduction designs, adding auxiliary slots to the motor teeth is a widely used method. However, to achieve the desired effect, this method requires determining reasonable dimensions for the auxiliary slots to avoid excessively weakening the motor's torque performance in exchange for asymmetric noise reduction. Therefore, this process necessitates optimization design, but current optimization methods require numerous experiments, and the convergence accuracy fluctuates with increasing nonlinearity. This application proposes a multi-objective Bayesian optimization method based on an IC numerically driven adaptive acquisition function to address this problem. This method significantly reduces the number of experiments, lowering experimental costs and computational burden. First, an initial sample set is constructed using the SLE (Simplified Leap) experimental method. This method greatly reduces the number of experiments and provides a more convergent sample point set for model building in the initial stage, as it possesses excellent space filling and superior detection and modeling capabilities for complex nonlinear relationships. Then, two sub-acquisition functions are constructed using expectation and uncertainty strategies. These two acquisition function components are responsible for exploration and development, respectively. The expectation-based component is constructed using a hybrid of expectation improvement and robust expectation improvement, while the uncertainty-based component achieves joint control through uncertainty acquisition and information gain acquisition. With two acquisition function components, the information coefficient (IC) is used to measure the correlation between the acquisition function value and the expected utility. Spearman's rank correlation coefficient is used to calculate the IC because it captures monotonic relationships and is insensitive to outliers. Specifically, a set of candidate points is uniformly sampled from the entire design space (to avoid historical sampling bias), and then the acquisition function value (e.g., EI, uncertainty acquisition, etc.) and expected utility are calculated for each candidate point. Expected utility is measured by a comprehensive quality scoring function that considers the performance of multiple objectives. Then, the correlation coefficient between the acquisition function value and the expected utility is calculated as the IC value for that acquisition function. A higher IC value indicates that the acquisition function is better able to identify high-performance points. The weights of each acquisition function are then dynamically adjusted based on the calculated IC values. Finally, the IC value, optimization stage, model confidence, and spatial coverage are integrated into a single weight calculation model to obtain the adaptive weights for each acquisition function. Traditional Bayesian optimization uses fixed acquisition functions (such as EI, PI, LCB, etc.) to balance exploration and exploitation, which often performs poorly when dealing with complex multi-objective problems. Fixed-weight acquisition functions cannot adaptively adjust according to the optimization process and model uncertainties, resulting in slow convergence and a tendency to get trapped in local optima. The adaptive acquisition function method based on information coefficients proposed in this application significantly improves the performance of Bayesian optimization in multi-objective engineering optimization problems by dynamically evaluating the effectiveness of the acquisition function and adaptively adjusting its weights.This method has achieved good results in the problem of optimizing the size of stator auxiliary slots. It effectively balances the conflicting objectives of minimizing electromagnetic force and maximizing torque, accelerates the convergence speed, and avoids local optima.

[0093] Specifically, such as Figure 1 As shown, this application discloses a multi-objective optimization method for auxiliary slots of a motor stator based on information coefficients, comprising the following steps:

[0094] S1: Define the stator auxiliary slot geometry parameters as design variables, and set a conflict optimization objective that weakens electromagnetic force harmonics while retaining electromagnetic torque.

[0095] To better demonstrate the superiority and innovation of this method, this application performs stator auxiliary slot optimization design on a 12-slot, 14-pole motor. Specifically, the stator auxiliary slots are divided into two variables: slot depth and slot width. Radial force wave analysis of the motor reveals that the motor exhibits more pronounced performance at 2x and 4x electromagnetic forces, thus contributing significantly to motor noise. Therefore, 2x and 4x electromagnetic forces are selected as optimization targets. Furthermore, to maximize the preservation of the motor's electromagnetic torque, electromagnetic torque is selected as a third optimization target.

[0096] S2: Initial samples are obtained using the SLE continuous enumeration method, and a surrogate model that simultaneously predicts electromagnetic force and torque is established using Gaussian process regression.

[0097] The SLE continuous enumeration method divides the design space into equidistant square grids. According to the maximum-minimum distance criterion, that is, maximizing the minimum distance between any two sample points, a number of initial sample points are selected from the grids to make the sample set have optimal space filling performance and projection characteristics, thus providing a comprehensive and balanced initial dataset for the surrogate model.

[0098] Specifically, the optimization variables for the motor in this application are slot width b and slot depth h, and the optimization objectives are twice the electromagnetic force f, four times the electromagnetic force f, and torque T. Therefore, when using the SLE method to generate sample points, the dimension of the sample space is 2, and the number of sample points is defined as n. This divides the design space into n... 2 Given a grid space of n squares, n squares will be selected to hold n sample points. These sample points will be distributed according to the minimum-maximum distance criterion, maximizing the distance between all sample points. The specific operation is as follows:

[0099] Process 1: The design space is equally divided into n 2 There are n two-dimensional subspaces, and then according to the minimum-maximum distance principle, each point from the first point to the last point will be determined to be in n. 2The first point P1 is selected in a random subspace between (i1,j1,…,g1,1) and (i1,j1,…,g1∈{1,2,…,n}). Then, the dataset can be represented as: P={P1,P2}.

[0100] Step 2: After the location of the first point is determined, the second sample point P2 will be placed in a spatial cell selected from (i2,j2,…,g2,2). At this time, the location of P2 needs to be determined by calculating the maximum distance between P1 and P2 as follows:

[0101] max(d((i2,j2,…,g2,2),P1))

[0102] in,

[0103]

[0104] Then P2 is placed into the sample set P = {P1, P2}.

[0105] Process 3: Similarly, when generating the t-th sample point, the original sample set P = {P1, P2, ..., P...} t-1}, P k The position is limited to (i t j t , ..., g t In t), and also need to calculate P. t All feasible positions are related to {P1, P2, ..., P} t-1 The distance to} is used to obtain the eigenvalue of the minimum distance as the basis for determining P. t The location is based on the following formula:

[0106] max(d((i2,j2,…,g2,2),P1))

[0107] in,

[0108]

[0109] Thus, the t-th sample point P t The location can then be determined, and the dataset can be updated to P = {P1, P2, ..., P}. t};

[0110] Process 4: Repeat process 3 continuously until the (n-1)th sample point is generated, and the last point P is generated. n It will be placed in the last cell space, ending the loop process, thus generating a complete dataset P = {P1, P2, ..., P}. n}

[0111] Here, the initial design points for the stator auxiliary slots are selected as 10, i.e., n=10, slot width b=0-4mm, and slot depth h=0-1.5mm. Choosing 10 points provides a dataset for the initial surrogate model. Later, through continuous exploration and development of the data acquisition function, supplementary data will be sought to improve the accuracy of the surrogate model and ultimately solve the optimization problem.

[0112] In the embodiments of this application, Gaussian process regression uses the Matern 5 / 2 kernel function to construct a joint probability distribution. Through the covariance matrix operation of the training set and the test set, the predicted mean and variance of electromagnetic force and torque at the candidate points are output simultaneously. The variance is used to quantify the uncertainty of the model and provide an exploratory basis for the subsequent acquisition function.

[0113] Specifically, based on the preliminary LSE experiments above, an original sample dataset P = {P1, P2, ..., P} was obtained. n Now, we construct a joint probability distribution for this dataset, which is a Gaussian process:

[0114] f 2f (x)~GP(μ 2f (x),κ 2f (x,x′));

[0115] f 4f (x)~GP(μ 4f (x),κ 4f (x,x′));

[0116] f T (x)~GP(μ T (x),κ T (x,x′)).

[0117]

[0118] In the above equation, x and x' are any two points in the input space. μ 2f (x), μ 4f (x) and μ T (x) are the mean functions of a Gaussian process, and are usually assumed to be 0. 2f (x,x'), k 4f (x,x') and k T (x,x') is the Matern 5 / 2 kernel function.

[0119] In the original dataset P = {P1, P2, ..., P...} n Based on this, it can be divided into two datasets: a training set and a training set. Here, the training set data is defined as {(X... i ,y i)|i=1,2,…,n}, where y i =[y i , 2f y i，4f y i，T ] T Here, the input and output can be defined as the following three functions.

[0120] y i，2f =f 2f (X i )+ε1;

[0121] y i，4f =f 4f (X i )+ε2;

[0122] y i，T =f T (X i )+ε3.

[0123] Among them, y i For observations with noise; f 2f (X i ), f 4f (X i ) and f T (X i ) respectively correspond to X i The observed values ​​of twice the electromagnetic force, 4 times the electromagnetic force, and torque. ε1-ε3 are noise components, satisfying ε i : N(0, σ 2 ).

[0124] The joint distribution of the test and training sets, which then conform to a Gaussian distribution, can be represented as follows:

[0125]

[0126] The mean in the above formula is

[0127]

[0128] The variance is:

[0129]

[0130] In the above formula, X is the training set, X * This is the test set. κ 2f (X * X * ), κ 4f (X * X * ), κ T (X * X *), κ 2f (X * ,X)κ 4f (X * ,X)κ T (X * ,X)κ 2f (X, X), κ 4f (X, X) and κ T (X, X) are both Matern 5 / 2 kernel functions.

[0131] At this point, the Gaussian process regression model for the three optimization objectives has been established. Based on the current model, a rough performance prediction can be made for the new design point. However, since the initial sample size and quality cannot fully meet the requirements of high-precision modeling, the data acquisition function needs to continuously add new sample points to the original dataset to enrich the diverse data required for model construction. Of course, the Gaussian process regression itself will provide the mean and variance of the predicted points, which can serve as the direction and basis for the data acquisition function to find the optimal point.

[0132] S3: Construct a dual-strategy component based on expectation and uncertainty. The expectation component integrates expectation improvement and robust expectation improvement, while the uncertainty component integrates uncertainty acquisition and information gain acquisition.

[0133] S4: Generate candidate points uniformly in the design space, calculate the expected utility of each point and the values ​​of four sub-collection functions in parallel, and merge the sub-function values ​​within their respective components to form two component values.

[0134] S5: Calculate the two information coefficients IC corresponding to the values ​​of the two components and the expected utility, and obtain the basic weights through dynamic smoothing and normalization.

[0135] For the above three steps, the four sub-function values ​​are obtained through multi-strategy parallel computation: expectation improvement measures the potential of candidate points to surpass the current optimal solution; robust expectation improvement reduces optimistic estimation bias by setting stricter objectives; uncertainty collection integrates multi-objective prediction standard deviation to quantify information gaps; and information gain collection introduces spatial distance attenuation penalty to avoid repeated exploration in already sampled dense areas. Together, these four constitute a comprehensive evaluation of the value of candidate points.

[0136] Due to the small number of initial sample points and the randomness of sampling, the constructed surrogate model still has some errors compared to the model that meets the accuracy requirements. Therefore, it is necessary to continuously add new sample points to the dataset to continuously update the accuracy of the surrogate model.

[0137] Currently, Bayesian optimization (BO), as a sequential optimization strategy, exhibits unique advantages in handling computationally expensive black-box function optimization problems. Its core idea is to approximate the objective function by constructing a surrogate model (such as a Gaussian process) and using an acquisition function to guide the selection of the next sampling point. However, traditional Bayesian optimization methods face severe challenges in complex multi-objective optimization problems. Fixed acquisition function strategies cannot adapt to the characteristic changes at different stages of the optimization process, leading to an imbalance between exploration and exploitation. Some existing acquisition functions either favor exploration, easily get trapped in low-quality local optima, or have low sampling efficiency, resulting in slower model convergence. Considering these issues, this application proposes an adaptive acquisition function that can simultaneously meet the requirements of exploration and exploitation while ensuring the quality and efficiency of the recommended new sample points, thus enabling faster and better construction of the surrogate model.

[0138] In the optimization problem of auxiliary slots in motor stators, the initial number of sampling points is limited (usually 10-20), making it difficult to fully cover the design space. To construct an effective surrogate model and guide subsequent optimization searches, a sufficient number and diversity of candidate points are needed. These candidate points are generated by uniformly sampling from the design space to avoid historical sample selection bias. Specifically, a uniformly distributed grid point set is generated within the boundaries of the design variables, and then a certain number of candidate points are randomly selected from it to calculate the information coefficient.

[0139] For this problem, the dimensions of the motor stator structure are determined, therefore the size range of the auxiliary slots can also be determined. Here, the design space is defined by the lower bound of the variable lb. i and the upper world ub i The candidate point set C is defined as follows:

[0140] C={x∈R d |x i =lb i +(ub i -lb i )·g i g i ∈G i}

[0141] Among them, G i These are grid points uniformly distributed along the i-th dimension. To ensure computational efficiency, N points are randomly selected from the initial grid. c N points constitute the candidate point set, where N c In this application, it is 2000, here, ub i =[4, 1.5] T , lb i =[0,0]T .

[0142] By sampling uniformly from the entire design space, biases introduced by historical sample points are avoided, thus enabling a more accurate evaluation of the performance of the acquisition function across the entire space.

[0143] Based on obtaining the candidate point set, an effective acquisition function needs to be designed to guide the optimization search. Traditional single acquisition functions struggle to balance the conflict between exploration and utilization; therefore, this paper proposes an adaptive combined acquisition function method. This method is jointly determined by the expected improvement and uncertainty, with each having its own weighting factor. This weighting factor is based on the dynamic calculation of the Information Coefficient (IC). The framework of the acquisition function is as follows, which is also the adaptive acquisition function constructed in subsequent step S7. The difference is that step S7 constructs a definite adaptive acquisition function after the final weights have been calculated.

[0144] A(x i )=ω EI ·(0.7·α EI +0.3·α REI )+ω Unc ·(0.6·α UC +0.4·α IG )

[0145] Here, ω EI and ω Unc The weighting factors representing expected improvement and uncertainty exploration are dynamically adjusted through an adaptive mechanism, ω. EI +ω Unc＝1 α EI This provides a basic assessment of the optimal solution improvement potential for the desired improvement function; α REI To improve the robust expectation function and enhance its adaptability to model uncertainties; α UC For uncertain acquisition functions, it promotes sampling of unexplored regions; α IG This is the information gain acquisition function, avoiding repeated exploration in already sampled regions. In the component where improvement is desired, α... EI This accounts for 70%, ensuring continuous improvement of the current optimal solution; α REI It accounts for 30%, providing robustness guarantees against model uncertainties. Among the uncertainty components, α... UC Acquiring data based on fundamental uncertainties, primarily to advance the exploration of high-variance regions; α IG For information gain acquisition, a modulating factor is used, and spatial diversity is considered. With this acquisition function as the fundamental basis, A(x) is compared by traversing the entire candidate point space. iThe numerical values ​​can be used to filter out candidate points that can improve the accuracy of the proxy model. The following is the process of setting up the parameters of the acquisition function:

[0146] (1) For each candidate point x i For ∈C, an improved expected improvement function is used to enhance numerical stability:

[0147] α EI (x)=max(f min -f(x), 0)

[0148] Here f min This represents the current minimum function value. In actual calculations, a numerically stable implementation is used to handle extreme value cases to ensure computational robustness.

[0149] Having established the basic expected improvement function, we can use the currently observed minimum function value f. min To evaluate the improvement. However, due to model uncertainty, the qualitative estimate may not be accurate. Therefore, the robust expectation improvement function introduces a robustness factor for f. min Adjustments are made to obtain a more conservative optimal estimate. The robust expectation improvement function is defined as:

[0150]

[0151] in, f min It is the smallest function value currently observed, σ y This is the standard deviation of historical observations, reflecting the dispersion of the observed data. β is a non-negative robustness factor used to adjust for the degree of conservatism regarding model uncertainty. Thus, Compared to f min Smaller. In minimization problems, this means setting a more stringent objective. Therefore, greater improvement is needed before a point is considered to have potential. This helps avoid overly optimistic estimates of improvement, thus increasing the robustness of the exploration.

[0152] After having the expected improvement component, for the uncertainty corresponding to each point, the uncertainty acquisition function integrates multiple objective functions to predict the uncertainty:

[0153] α UC (x)=σ 2f (x)+σ 4f (x)+σ torque (x)

[0154] Where, σ 2f σ 4f and σ torque (x) represents the prediction standard deviation of each objective function model at point x.

[0155] Since uncertainty-based sampling might lead to repeated sampling in already sampled areas (where model uncertainty may be temporarily high), the information gain sampling function, by introducing a distance penalty, avoids repeated exploration in already densely sampled areas, thus encouraging the algorithm to explore regions that have both high uncertainty and are far from existing sample points. Information gain sampling, taking into account both uncertainty and spatial distribution, can be defined as:

[0156]

[0157] Where, d min (x) represents the minimum distance to historical samples, and λ is the distance decay coefficient. This function avoids repeated sampling in densely sampled areas, improving exploration efficiency. The information gain acquisition function exponentially penalizes the minimum distance from candidate points to historical sample points through the distance decay coefficient λ. The closer the distance, the greater the penalty, causing the algorithm to prioritize exploring candidate points that are far from sampled areas and have high uncertainty, thereby improving global exploration efficiency and enhancing sample diversity.

[0158] After obtaining the normalized results of the expected value and uncertainty of the acquisition function for each candidate point, it is also necessary to construct the IC weighting factor based on the expected utility. The expected utility of each candidate point is calculated by weighted summation of normalized electromagnetic force and torque, preferably based on a surrogate model, to calculate the expected utility of the candidate points:

[0159]

[0160] in and α T Let be the weight coefficients for each objective, satisfying... and The normalized electromagnetic force value (minimizes the objective), and the normalized moment value (maximizes the objective). The normalization method is as follows:

[0161]

[0162] Once the metrics for assessing expected utility are available, the EI and the information coefficients between uncertainty and expected utility can be calculated. The Pearson correlation coefficient is used to quantify the consistency between the expected and uncertain component values ​​and expected utility; a higher correlation indicates a stronger guiding effect of the strategy on the true optimization objective.

[0163] Desired improvement in information coefficient:

[0164]

[0165] Information coefficients from uncertain data collection:

[0166]

[0167] in, Candidate point x i Corresponding to expectations, Let U(x) be the mean of the expected value of the sample set. i ) is a candidate point x i Expected utility, U mean Candidate point x i The mean expected utility Candidate point x i Uncertainty Candidate point x i The mean of uncertainty.

[0168] After obtaining information coefficients regarding expectation and uncertainty, and considering the iterative properties of sampling, smoothing is also necessary to reduce fluctuations in the improved IC value and enhance stability and reliability.

[0169] IC EI_smoth =α(t)·IC EI +(1-α(t))·IC E_previous

[0170] IC UC_smooth =α(t)·IC UC +(1-α(t))·IC UC_previous

[0171] Among them, IC EI_previous and IC UC_previous This is the smoothed IC value corresponding to the previous period. α(t) is the dynamic smoothing factor, which is determined by the number of calculations.

[0172] This was then further transformed into weighting factors based on expectation and uncertainty search strategies. The normalization results are as follows.

[0173]

[0174] With the weighted shadow built on the information coefficient, the search strategies based on expectation and uncertainty also have directionality and adaptability. This allows the acquisition function to play the dual role of correlation discovery and effectiveness evaluation that the information coefficient possesses. It can also measure the consistency between the search direction of the two guiding strategies and the actual optimization goal, reducing the blindness and complexity of manual parameter tuning and giving the system stronger adaptive performance.

[0175] S6: The two basic weights are fused with the model confidence, spatial coverage and optimization stage indicators to form two final weights. The two final weights are then weighted and merged to construct an adaptive acquisition function.

[0176] After establishing the information coefficient as the foundation for the two basic weights, other factors need to be incorporated into the weight construction to enhance the robustness and generalization ability of the data collection function. This application utilizes model confidence, spatial coverage, and optimization stage metrics. These factors, in conjunction with the information coefficient-based weights, ensure the accuracy of data-driven approaches, enhance the robustness and generalization ability of the method across multiple dimensions, and enable the construction of more robust decision-making strategies.

[0177] Model confidence evaluation assesses the current accuracy level of the model, informing the data acquisition function whether to focus more on exploration or development, thus serving a reinforcing function. Therefore, the model confidence used to evaluate cross-validation results is as follows:

[0178]

[0179] in, and These are the determination coefficients of three surrogate models built on the training set for the test set, respectively, when the confidence level C... modle When the confidence level is greater than 0.8, the acquisition function makes a choice based on the prediction of the trust model, placing greater emphasis on the expected search strategy; when the confidence level C... modle When the value is less than 0.3, the acquisition function makes choices that question the model's predictions, and places greater emphasis on search strategies that address uncertainty.

[0180] The spatial coverage factor is used to quantify the uniformity of sample point distribution in the design space, preventing excessive clustering of sampling points in certain areas and resulting in missing information in other areas. The design space is uniformly divided into K grids (K=5 recommended), and the proportion of grids containing sample points is calculated out of the total number of grids.

[0181]

[0182] Where K is the partitioning precision, N covered The number of grid cells covered by at least one sample point is calculated using the following formula:

[0183]

[0184] Where I is an indicator function, which takes the value 1 when the condition is met and 0 otherwise; D is the current set of sample points.

[0185] After combining the model confidence and spatial coverage, a composite index can be obtained:

[0186]

[0187] Optimization stage identification is a key component of the adaptive acquisition function. By analyzing the state characteristics of the process, it dynamically identifies the optimization stage, providing strategic guidance for weight adjustment. Therefore, the needs for exploration and utilization differ significantly at different optimization stages. Based on the current number of sample points, the stages are divided into:

[0188]

[0189] After obtaining the parameterized expressions of these systems, they are merged as follows:

[0190]

[0191] With two final weights ω EI and ω Unc The weighted merging of two component values ​​constructs an adaptive acquisition function, which is the final ω. EI and ω Unc Substitute into the following equation,

[0192] A(x i )=ω EI ·(0.7·α EI +0.3·α REI )+ω Unc ·(0.6·α UC +0.4·α IG )

[0193] This completes the adaptive acquisition function, allowing the surrogate model to continuously increase the number of sample points in batches according to the adaptive acquisition function, thereby improving accuracy more quickly.

[0194] S7: The adaptive acquisition function is used to evaluate all candidate points and sample them in batches. The surrogate model is iteratively updated until the Pareto front converges, and the auxiliary slot structure scheme that balances low electromagnetic force and high torque is output.

[0195] To further illustrate the significant advantages of the proposed method in building accurate surrogate models, this application also selects other traditional acquisition functions to continuously supplement the initial sample points with new sample points, thereby improving the accuracy of the surrogate model. The superiority of this application is comprehensively compared through the process and the final accuracy and quality performance of the surrogate model.

[0196] This application selects commonly used EI acquisition functions, PI acquisition functions, LCB acquisition functions, and Thompson acquisition functions to compare and illustrate the adaptive acquisition function of this application.

[0197] First, based on the SLE experimental method proposed in this application, 10 sample points are initially recommended. Using this sample point information, the corresponding 2x electromagnetic force, 4x electromagnetic force, and torque are simulated and calculated in Maxwell. After obtaining the initial sample set, a surrogate model for each optimization objective is established using Gaussian regression. These three surrogate models establish the functional relationship between the geometric dimensions of the stator auxiliary slot and the electromagnetic performance. The accuracy of these models determines the numerical accuracy and correctness of the final Pareto front solution and optimal solution. Therefore, it is necessary to continuously supplement new sample points using acquisition functions to improve the accuracy of the surrogate models. Here, 10 new sample points are recommended in each iteration, and each method iterates 4 times. Ultimately, each method supplements the original dataset with 40 sample points, which can more quickly improve optimization efficiency. Therefore, considering these requirements, this application proposes an adaptive optimization algorithm based on information coefficients to design the optimal stator auxiliary slot size. The following is a detailed analysis of the performance of various acquisition functions.

[0198] like Figure 2-4 The figure shows the mean relative error, RMSE, and coefficient of determination (R²) for different methods during the iteration process. 2The dynamic comparison of these indicators is shown. Based on the experimental chart data, it can be seen that under continuous sampling, the average relative error, RMSE, and coefficient of determination of the surrogate models obtained by all methods reach a reasonable level. Among them, the adaptive acquisition function based on the IC weight factor achieves average values ​​of 1.41%, 77.09, and 0.9883 for the average relative error, RMSE, and coefficient of determination for each optimization objective, which is significantly better than the other methods. The IC method has the lowest average relative error (2.3% for 2 times electromagnetic force, 2.1% for 4 times electromagnetic force, and 0.8% for torque), which is 39.5%-46.7% higher than the traditional method, and the RMSE index (e.g., 29.37 for the 2 times electromagnetic force model) is 73% better than the PI method (107.61). During the optimization iteration process, it can be found that although the PI acquisition function is initially superior to the other methods, the quality drops sharply after 4 times electromagnetic force, so the robustness of this method is poor. The LCB acquisition function, EI acquisition function, and Thompson acquisition function are significantly slower than the adaptive acquisition function in terms of optimization speed. The adaptive sampling function exhibits the fastest convergence speed across all three objective functions (2x electromagnetic force, 4x electromagnetic force, and torque), achieving a stable error level with only two samplings, compared to 3-4 samplings required by traditional methods. These results consistently demonstrate that the IC weighting factor method, through its adaptive weight adjustment mechanism, can dynamically balance exploration and utilization, prioritizing sampling points that contribute most to model improvement. This results in higher model accuracy with fewer samplings, significantly improving the reliability of the surrogate model and laying a solid foundation for subsequent optimization stages. It has significant application value in computationally expensive engineering optimization problems.

[0199] like Figure 5-9The figure shows the sampling function graphs after multiple iterations of various sampling functions. Through the sampling function distribution graphs, the sampling characteristics of different sampling functions and their differences in performance in the exploration-utilization balance can be clearly observed. The improved adaptive method proposed in this application shows a significant advantage in the distribution of recommended points. Its sampling points are evenly distributed in the design space, covering both potential areas with high EI values ​​(slot width 1.5-2.5 mm) and boundary areas with higher uncertainty (slot widths below 0.5 mm and above 3.5 mm), demonstrating a good exploration-utilization balance capability. In contrast, the EI sampling function is overly concentrated near the current optimal solution (slot width around 2.0 mm), lacking exploration of unknown areas; the PI sampling function, although relatively dispersed, lacks systematic correlation between points, resulting in low exploration efficiency; the LCB sampling function is overly biased towards boundary areas, focusing entirely on exploration while neglecting the fine development of potential areas; while Thompson sampling ensures diversity, it lacks a clear optimization direction. These distribution characteristics corroborate the results of previous model accuracy analysis, further verifying that the improved adaptive method, through multi-strategy fusion and dynamic weight adjustment, can achieve more intelligent sampling decisions, providing a reliable guarantee for efficient Bayesian optimization.

[0200] Figure 10-14 Pareto front solutions for surrogate models constructed using multiple methods. Based on the 3D Pareto front distribution plot, the significant differences in multi-objective optimization performance of different acquisition functions can be clearly observed. The adaptive acquisition function proposed in this application ( Figure 10 The system exhibits optimal Pareto front exploration capability, with its Pareto solution set widely distributed across the performance space comprised of the three objective functions. While maintaining a high torque level of 90-100 Nm, it successfully found multiple non-dominated solutions for 2 times the electromagnetic force (4000-6000 N) and 4 times the electromagnetic force (2000-3000 N), forming a complete and uniformly distributed Pareto front. In contrast, the PI acquisition function ( Figure 11 The solution set of the LCB acquisition function is relatively concentrated, mainly limited to local optima, and cannot effectively explore the entire Pareto front. Figure 13 Although the solution set is widely distributed, its quality is poor, and most solutions perform poorly on the torque target; while Thompson sampling ( Figure 14 The results show a random distribution and lack a clear optimization direction. These comparisons fully demonstrate that the improved adaptive method, through multi-strategy fusion and dynamic weight adjustment, can achieve comprehensive and effective exploration in the complex three-objective optimization space, providing decision-makers with a set of uniformly distributed and high-quality Pareto optimal design schemes, demonstrating significant advantages in multi-objective motor design optimization problems.

[0201] Figure 15-19Box plots of error distributions for test sets are constructed for multiple methods. Based on the box plots of predicted error distributions on the test sets, the performance of the improved adaptive acquisition function proposed in this application can be systematically evaluated. The box plot analysis clearly shows that the improved adaptive method ( Figure 15 The model exhibits the best performance in terms of prediction errors for all three objective functions, with its error distribution characterized by a low median, narrow interquartile range, and few outliers. Specifically, the median prediction errors for 2x electromagnetic force, 4x electromagnetic force, and torque are all maintained at a low level, and the box range is compact, indicating that the established surrogate model has high accuracy and strong stability. In contrast, the PI acquisition function ( Figure 16 ) and EI acquisition function ( Figure 17 While performing reasonably well on some targets, the error distribution exhibits significant dispersion, with numerous outliers, resulting in insufficient consistency in model predictions. LCB acquisition function ( Figure 18 The Thompson sampling method performed the worst, with significantly higher prediction errors for all three targets, particularly for the 2x electromagnetic force, indicating that its sampling strategy is insufficient to support the establishment of an effective surrogate model. Figure 19 The error distribution exhibits unstable characteristics. Although the median is acceptable, there are many outliers with a wide distribution range. These error distribution characteristics corroborate the aforementioned experimental results, jointly demonstrating that the surrogate model established by the improved adaptive method through the intelligent sampling strategy has optimal prediction accuracy and reliability, providing solid technical support for motor design optimization.

[0202] Depend on Figure 20-23 It can be seen that, based on the optimization of the adaptive acquisition function, the Pareto front solution is obtained. While ensuring that the electromagnetic torque decreases by less than 2%, the 2x and 4x electromagnetic forces are kept as low as possible. Here, a slot width of 1.5172 mm and a slot depth of 1.1379 mm are chosen. The optimized algorithm outputs 2x electromagnetic force, 4x electromagnetic force, and electromagnetic torque values ​​of 3639.9 N, 1864.7 N, and 108.08 N*m, respectively. Maxwell simulation results show that before optimization, the 2x electromagnetic force, 4x electromagnetic force, and electromagnetic torque values ​​were 8148.5 N, 4281.7 N, and 110.1 N*m, respectively; after optimization, the values ​​are 3593.4 N, 1838.6 N, and 108.1 N*m, respectively. It can be observed that the predicted values ​​of the optimized algorithm for the three indicators have errors of 1.2%, 1.4%, and 0.01% relative to the simulated values, which meets the accuracy requirements. After optimization, the values ​​of the three indicators decreased by 55.9%, 57.0%, and 1.8% respectively compared to before optimization, with the electromagnetic force being twice and the electromagnetic torque being four times that of the original. This greatly reduced the electromagnetic force and the electromagnetic torque being four times that of the original, while ensuring that the torque was not significantly lost.

[0203] according to Figure 24-25The three-dimensional distribution comparison diagram of electromagnetic force space-frequency can be used to make a key evaluation of the engineering effectiveness of the optimization method of this application. Figure 24-25 This visually demonstrates the changes in the distribution of electromagnetic force harmonic components inside the motor before and after the optimization of the stator auxiliary slot dimensions. This is a core indicator for measuring the effectiveness of electromagnetic vibration and noise reduction. The spectrum diagram before optimization (…) Figure 24 In this study, electromagnetic energy is highly concentrated in a few low-order space-frequency modes, with the DC component (0,0) having an amplitude as high as 1.83 × 10⁻⁶. 4 The amplitude of the second-order spatial and second-order frequency force wave (2,2) also reaches 2.65 × 10⁻⁶. 3 This intense energy concentration is the main cause of electromagnetic vibration and noise in motors. After adjusting the slot shape parameters using the adaptive optimization method proposed in this paper, the optimized spectrum ( Figure 25 Significant improvements were observed: the overall harmonic energy distribution became more dispersed and flattened, and the amplitude of key low-order force waves was effectively suppressed. The amplitude of the (2,2)th order force wave was reduced to 2.55 × 10⁻⁶. 3 The reduction was approximately 3.8%; meanwhile, although the amplitude of some individual harmonics, such as the (6,-6) order, increased to 2.41 × 10⁻⁶ after post-optimization. 2 However, this is a normal phenomenon of spectrum reshaping, which is that vibration energy is transferred from low-order modes that are sensitive to noise to high-order modes that contribute less to noise through groove design. This can achieve active transfer of energy distribution.

[0204] Figure 26-27 The comparison graphs showing the motor efficiency before and after optimization allow for a crucial assessment of the overall engineering benefits of the optimization scheme in this application. The two graphs clearly demonstrate the comparative changes in motor energy distribution and operating efficiency before and after optimization. In the state before optimization ( Figure 26 In this study, the average operating efficiency of the motor was 73.27%, and the output power was 19.70W. Its main losses consisted of copper loss of 4.11W (15.3%) and iron loss of 2.04W (7.6%). After global optimization of the stator auxiliary slot dimensions using the adaptive optimization method proposed in this application, the optimized state ( Figure 27The results show that the average efficiency of the motor slightly decreased to 72.62%, while the output power slightly increased to 19.74W. Simultaneously, copper loss increased to 4.27W and iron loss increased to 2.13W. It can be seen that the stability of the efficiency curve is significantly improved after optimization, with the fluctuation range narrowing dramatically from approximately 1.12 percentage points (72.67%-73.79%) before optimization to approximately 0.43 percentage points (72.44%-72.87%), indicating enhanced motor operational stability. This series of analyses ultimately proves that the method proposed in this application successfully found a comprehensive optimal engineering trade-off: at a small efficiency cost (a reduction of approximately 0.65 percentage points), it achieved a significant improvement in electromagnetic vibration and noise characteristics, while ensuring stable output power and improved operational stability. This perfectly embodies the core value of multi-objective optimization in balancing conflicting objectives and achieving optimal overall performance in practical engineering problems.

[0205] Figures 28-29 The comparison results of the motor cogging torque waveform before and after optimization are presented. Before optimization, the cogging torque waveform exhibits periodic pulsations with large amplitude and sharp peaks, indicating strong harmonic magnetic field energy generated by the interaction between the permanent magnet and the stator slots. After optimization, the peak value of the cogging torque is significantly reduced, the waveform amplitude is greatly attenuated, and the smoothness of the pulsation is significantly improved. This result directly corresponds to the aforementioned analysis conclusion on the harmonic reconstruction of the no-load air gap magnetic field: by adjusting the size of the auxiliary slot through the Bayesian optimization algorithm, the harmonic distribution of the air gap permeability is changed, effectively suppressing specific harmonics corresponding to the cogging period (especially the harmonic components affecting the cogging torque), thereby weakening the excitation source of the cogging torque at its source. This proves that the multi-objective optimization method can not only optimize the electromagnetic force and back EMF characteristics, but also effectively suppress cogging torque, which has clear engineering value for improving the low-speed stability of the motor and reducing vibration and noise.

[0206] according to Figures 30-31The precise quantitative analysis of the unloaded air gap radial magnetic flux density waveform and spectrum before and after optimization clearly demonstrates the optimization effect of the magnetic field harmonic structure. Before optimization, the fundamental amplitude in the spectrum was 0.2843 mT, the third harmonic amplitude was 0.0765 mT, and the amplitudes of the fifth, seventh, ninth, and eleventh harmonics were 0.0359 mT, 0.0258 mT, 0.0137 mT, and 0.0110 mT, respectively. After optimization, the spectrum data shows that the fundamental amplitude decreased slightly to 0.2813 mT, while all harmonics were effectively suppressed: the third harmonic decreased slightly to 0.0750 mT, and the fifth, seventh, ninth, and eleventh harmonics decreased to 0.0350 mT, 0.0195 mT, 0.0118 mT, and 0.0049 mT, respectively, with the attenuation of the seventh and eleventh harmonics being the most significant. This harmonic reconstruction phenomenon indicates that the auxiliary slot size obtained through the algorithm finely controls the harmonic distribution of the air gap magnetic permeability. While basically maintaining the intensity of the fundamental wave and the main lower harmonics (3rd), it significantly weakens the amplitude of higher harmonics (especially the 7th and 11th). This targeted harmonic suppression reduces the key harmonic excitations that cause cogging torque pulsation and specific electromagnetic force waves at the source of the magnetic field. This is directly related to the aforementioned reduction in cogging torque amplitude and improvement in the quality of the back EMF waveform. Thus, while maintaining the main magnetic flux capability of the motor, it achieves a comprehensive improvement in vibration noise characteristics and operational stability by purifying the harmonic components of the magnetic field.

[0207] Figures 32-33 The harmonic spectrum characteristics of the motor's no-load back EMF before and after optimization were compared. Before optimization, the 3rd, 5th, 7th, 9th, and 11th harmonics dominated the back EMF, contributing most significantly to the total harmonic distortion (THD). After optimization, the harmonic structure was reconstructed: lower-order harmonics generally decreased, with the 3rd harmonic decreasing by 0.5%, the 5th harmonic by 5.2%, and the 9th harmonic by 0.2%; higher-order harmonic amplitudes generally increased, with the 7th harmonic increasing by 3.1% and the 11th harmonic by 9.4%. The contribution ratio of each harmonic to THD also changed significantly. This harmonic reconstruction phenomenon corresponds to the optimization results of the no-load air gap magnetic field, further verifying that adjusting the auxiliary slot size can optimize the harmonic distribution while maintaining the fundamental magnetic field strength, thereby improving the overall electromagnetic performance of the motor.

[0208] Based on the experimental verification and analysis of this system, the improved adaptive acquisition function proposed in this application demonstrates comprehensive and excellent performance in the multi-objective optimization problem of stator auxiliary slot size. Through comparative analysis with traditional acquisition functions (PI, EI, LCB) and advanced sampling methods (Thompson sampling), this application draws the following important conclusions:

[0209] In terms of surrogate model construction, the improved adaptive method achieves rapid improvement in model accuracy and stable convergence through its unique IC weighting factor mechanism. Experimental data show that this method requires only 2 samples to reach a stable error level on the three objective functions (2 times electromagnetic force, 4 times electromagnetic force, and torque), which is about 50% faster than the 3-4 samples required by traditional methods. The final surrogate model exhibits excellent prediction accuracy on the test set, with the average relative error controlled below 2.3% and the coefficient of determination R0. 2 With a value of over 0.98, it provides a reliable model basis for the optimization process.

[0210] Regarding sampling strategies, the improved adaptive method achieves an intelligent balance between exploration and utilization through a multi-strategy fusion mechanism. The sampling function distribution diagram clearly shows that the sampling points recommended by this method cover both high-potential optimal solution regions and boundary regions with higher uncertainty, forming a systematic spatial exploration pattern. In contrast, EI sampling is overly concentrated near the current optimal solution, LCB sampling is excessively biased towards boundary exploration, and Thompson sampling lacks a clear optimization direction, all failing to achieve an effective balance.

[0211] In terms of multi-objective optimization performance, the improved adaptive method demonstrates superior Pareto front exploration capabilities. The 3D Pareto front distribution shows that the solution set obtained by this method is widely and uniformly distributed in the design space composed of the three objective functions. It successfully found a high-quality design scheme that significantly reduces electromagnetic force (4000-6000N for twice the electromagnetic force, 2000-3000N for four times the electromagnetic force) while maintaining high torque (90-100Nm), forming a complete and high-quality Pareto front.

[0212] Regarding the reliability of model predictions, error analysis results further validated the advantages of the improved adaptive method. The surrogate model established exhibited minimal prediction error and optimal stability on the test set, with a concentrated and uniform error distribution, significantly outperforming other comparative methods. This high-precision model prediction provides a reliable basis for optimization decisions, ensuring the practical effectiveness of the final design scheme.

[0213] In summary, the improved adaptive acquisition function proposed in this application effectively solves the explore-exploitation balance problem in Bayesian optimization through IC weight factor driving, multi-strategy fusion, and state-aware mechanisms. It significantly outperforms traditional methods in terms of surrogate model accuracy, sampling efficiency, multi-objective optimization performance, and prediction reliability. This research not only provides an effective solution for stator auxiliary slot size optimization but also offers general methodological guidance for multi-objective optimization problems in complex engineering systems, possessing significant theoretical and engineering application value. Future research will focus on extending this method to higher-dimensional optimization problems and further exploring its application potential in other engineering fields.

[0214] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the above embodiments do not limit the present invention in any way, and all technical solutions obtained by equivalent substitution or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A multi-objective optimization method for auxiliary slots of a motor stator based on information coefficients, characterized in that, Includes the following steps: Define the stator auxiliary slot geometric parameters as design variables, and set a conflict optimization objective that weakens electromagnetic force harmonics while retaining electromagnetic torque; Initial samples were obtained using the SLE continuous enumeration method, and a surrogate model for simultaneously predicting electromagnetic force and torque was established using Gaussian process regression. Construct a dual-strategy component based on expectation and uncertainty. The expectation component integrates expectation improvement and robust expectation improvement, while the uncertainty component integrates uncertainty acquisition and information gain acquisition. Candidate points are generated uniformly in the design space, and the expected utility of each point and the values ​​of four sub-collection functions are calculated in parallel. The sub-function values ​​are then merged within their respective components to form two component values. Based on the values ​​of the two components and the expected utility, the corresponding two information coefficients IC are calculated, and the basic weights are obtained through dynamic smoothing and normalization. The two basic weights are respectively fused with model confidence, spatial coverage and optimization stage indicators to form two final weights. The two final weights are then weighted and merged to construct an adaptive acquisition function. The adaptive acquisition function is used to evaluate all candidate points and sample them in batches. The surrogate model is iteratively updated until the Pareto front converges, and the auxiliary slot structure scheme that balances low electromagnetic force and high torque is output.

2. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients as described in claim 1, characterized in that, The SLE continuous enumeration method divides the design space into equidistant square grids. According to the maximum-minimum distance criterion, that is, maximizing the minimum distance between any two sample points, a number of initial sample points are selected from the grids to make the sample set have optimal space filling performance and projection characteristics, thus providing a comprehensive and balanced initial dataset for the surrogate model.

3. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients as described in claim 1, characterized in that, The Gaussian process regression uses the Matern 5 / 2 kernel function to construct a joint probability distribution. Through the covariance matrix operation of the training set and the test set, the predicted mean and variance of electromagnetic force and torque at the candidate points are output simultaneously. The variance is used to quantify the uncertainty of the model and provide a basis for exploring the subsequent acquisition function.

4. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients as described in claim 1, characterized in that, The four sub-function values ​​are obtained through multi-strategy parallel computation: expectation improvement measures the potential of a candidate point to surpass the current optimal solution; robust expectation improvement reduces optimistic estimation bias by setting stricter objectives; uncertainty acquisition integrates multi-objective prediction standard deviation to quantify information gaps; and information gain acquisition introduces spatial distance attenuation penalty to avoid repeated exploration in already sampled dense areas. Together, these four constitute a comprehensive evaluation of the value of the candidate point. The values ​​of the four sub-functions are calculated as follows: The desired improvement is to use a numerically stable improved function for calculation; α EI (x)=max(f min -f(x),0) Among them, f min This is the current minimum function value; Robust expectations are improved by using the following formula to set more stringent targets and reduce the risk of optimistic estimations; Among them, f min It is the current minimum function value, σ y It is the standard deviation of historical observations, which reflects the dispersion of the observed data. β is a non-negative robustness factor used to adjust the degree of conservatism for model uncertainty. Uncertainty collection is weighted and summed using the following formula to calculate the standard deviation of multi-objective predictions; a UC (x)=σ 2f (x)+σ 4f (x)+σ torque (x) Where, σ 2f σ 4f and σ torque (x) represents the prediction standard deviation of each objective function model at point x; Information gain acquisition is penalized by introducing a distance attenuation coefficient to the dense sampling area according to the following formula; Where, d min (x) represents the minimum distance to historical samples, and λ is the distance decay coefficient.

5. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients according to claim 4, characterized in that, The information gain acquisition function applies an exponential penalty to the minimum distance from candidate points to historical sample points using a distance attenuation coefficient λ. The closer the distance, the greater the penalty, which enables the algorithm to prioritize exploring candidate points that are far from the sampled area and have high uncertainty, thereby improving global exploration efficiency and enhancing sample diversity.

6. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients according to claim 1, characterized in that, The expected utility of each candidate point is calculated by weighted summation of normalized electromagnetic force and torque, with the two information coefficients calculated using the following formulas. Among them, IC EI IC is the expected information coefficient. UC For uncertainty information coefficient, Candidate point x i Corresponding to expectations, Let U(x) be the mean of the expected value of the sample set. i ) is a candidate point x i Expected utility, U mean Candidate point x i The mean expected utility Candidate point x i Uncertainty Candidate point x i The mean of uncertainty. The Pearson correlation coefficient is used to quantify the consistency between the expected component value and the uncertain component value and the expected utility. The higher the correlation, the stronger the guiding role of the strategy on the real optimization objective.

7. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients as described in claim 6, characterized in that, The dynamic smoothing is executed as follows: IC EI_smooth =α(t)·IC EI +(1-α(t))·IC EI_previous IC UC_smooth =α(t)·IC UC +(1-α(t))·IC UC_previous Among them, IC EI_previous and IC UC_previous It is the smoothed IC value corresponding to the previous cycle, and the smoothing factor α(t) takes values ​​in segments as the number of iterations increases.

8. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients according to claim 1, characterized in that, The model confidence level is calculated using the cross-validation mean absolute error formula: in, and The coefficients of determination for the test set are the three proxy models built on the training set. When the confidence level is greater than 0.8, the collection function chooses to trust the model's prediction and will place more emphasis on the expected search strategy. When the confidence level is less than 0.3, the collection function chooses to question the model's prediction and will place more emphasis on the uncertain search strategy.

9. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients as described in claim 8, characterized in that, The spatial coverage rate is calculated using the following formula: Where K is the partitioning precision, N covered The number of grid cells covered by at least one sample point is calculated using the following formula: Where I is an indicator function, which takes the value 1 when the condition is met and 0 otherwise, and D is the current set of sample points.

10. The multi-objective optimization method for auxiliary slots of motor stator based on information coefficients according to claim 9, characterized in that, The optimization phase is divided into three stages according to the following formula: initial exploration, accelerated search, and convergence refinement.

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