A multi-objective robustness optimization method for bearingless permanent magnet synchronous motor

Abstract

The application discloses a kind of bearingless permanent magnet synchronous motor multi-objective robustness optimization method, belong to data processing technical field.Parameter and optimization target are obtained, significant parameter is filtered out by calculating sensitivity coefficient to realize dimension reduction;Gaussian process regression surrogate model is constructed based on significant parameter;Emperor penguin optimization algorithm is introduced to interact with surrogate model to obtain pareto front solution set, and entropy weight-ideal solution method is used to select certain optimization optimal solution;Based on Monte Carlo simulation, the failure rate is calculated, and if it does not meet the requirements, robustness optimization is carried out.Combining six sigma design theory in robustness optimization, an optimization model is established, the prediction variance output by the surrogate model is directly used as the estimated value of the standard deviation of the optimization target in robustness optimization, and the robustness optimal solution is selected by using the emperor penguin optimization algorithm again.The application avoids the huge amount of calculation of traditional double nested sampling, improves the global optimization ability and reduces the failure rate of the motor.

Description

Technical Field

[0001] This invention relates to the field of data processing technology, specifically to a multi-objective robustness optimization method for bearingless permanent magnet synchronous motors. Background Technology

[0002] Bearingless permanent magnet synchronous motors integrate torque drive and radial levitation force control, operating through the coordinated action of two sets of stator windings. A levitation force winding, with a pole pair count differing by one from the torque winding, generates a magnetic field in the stator air gap when a corresponding current is applied, leviting the rotor. This device eliminates mechanical contact and friction between the stator and rotor, significantly improving motor speed and lifespan. Therefore, this type of motor offers advantages such as frictionless operation, high precision, high speed, and low noise, making it suitable for high-speed, high-precision applications.

[0003] However, in practical engineering applications, the product qualification rate is crucial. Noise is unavoidable in the actual manufacturing of motors. Existing technologies, such as the Chinese patent application CN119940011A, "Multi-objective Optimization Method and System for Motors Based on Improved Dung Beetle Optimization Algorithm," aim to find the performance "peak" under ideal conditions, guaranteeing performance only under ideal models. When certain motor performance characteristics are sensitive to uncertainty, it may cause design violations, affecting motor reliability and manufacturing costs. Furthermore, the training cost of the surrogate model plays a significant role in optimization design. Existing technologies, such as the Chinese invention patent CN114398824A, "Multi-objective Robust Optimization Method for Motors Based on Local Surrogate Model," first establish a global surrogate model, then construct a local surrogate model, and the local surrogate model needs to be updated multiple times to meet requirements, significantly increasing the time cost of training the surrogate model during the optimization process. Summary of the Invention

[0004] The purpose of this invention is to provide a multi-objective robustness optimization method for bearingless permanent magnet synchronous motors to solve the problems mentioned in the background art.

[0005] To address the aforementioned technical problems, this invention provides a multi-objective robust optimization method for bearingless permanent magnet synchronous motors (BPMSMs), based on a Gaussian Process Regression (GPR) surrogate model for multi-objective robust optimization design of BPMSMs.

[0006] In mass production, unavoidable manufacturing tolerances (such as slight deviations in air gap dimensions) and batch variations in material properties exist. These factors can lead to inconsistent motor performance within the same batch, resulting in a decrease in yield. Robustness optimization takes these tolerance ranges into account during the design phase, thereby improving product yield.

[0007] By employing variance-based sensitivity analysis, the importance of all design parameters can be ranked, and four design parameters can be selected as significant parameters, thus solving the problem of excessive parameters leading to a significant increase in optimization time and cost.

[0008] GPR can build high-precision surrogate models with fewer training samples. It can provide not only the mean of its performance predictions but also the variance of the predictions. Under the Bayesian framework, it performs inference through priors, which can ensure accuracy while having a strong resistance to overfitting.

[0009] The technical solution adopted by the present invention to achieve the above-mentioned objective includes the following steps:

[0010] Obtain the various design parameters and their design ranges for the bearingless permanent magnet synchronous motor, and determine the optimization objectives. The optimization objectives include average torque, torque ripple, average levitation force, and levitation force ripple; the design parameters are the motor structural parameters.

[0011] Specifically:

[0012] First, deterministic optimization is performed. Based on the electromagnetic performance requirements of BPMSM, the design parameters required for global optimization are selected as optimization parameters, and the design range of the parameters is determined.

[0013] Define the optimization objective and its constraints. The optimization objective includes average torque. Torque pulsation Average levitation force Suspension force pulsation The requirements and constraints for the optimization objectives regarding motor performance indicators are as follows:

[0014] ;

[0015] ;

[0016] In the formula: - This represents the deterministic optimization objective function. Indicates constraints. ( =1,2,3,4) represent the constraint values ​​for each optimization objective. - The objective function representing the constraint.

[0017] The sensitivity coefficients of the design parameters to the optimization objective are calculated. A weighted summation is used to calculate the comprehensive sensitivity coefficient. Based on the magnitude of the comprehensive sensitivity coefficient, significant and insignificant parameters are selected. The specific process is as follows:

[0018] A variance-based global sensitivity analysis method was used to evaluate the impact of each motor parameter on the optimization objective. Since there are numerous and interconnected motor parameters, sensitivity analysis can identify those with a significant impact on motor performance, thus reducing the difficulty of subsequent optimization.

[0019] The design range of each design parameter is divided into multiple horizontal intervals. With other design parameters keeping their initial values ​​unchanged, the simulation values ​​of the optimization target are obtained when a single design parameter takes different horizontal intervals.

[0020] First, calculate the sensitivity coefficients of the optimization targets separately. Then, calculate the comprehensive sensitivity coefficient using a weighted summation method. Finally, select the targets based on the magnitude of the comprehensive sensitivity coefficient. One significant parameter.

[0021] According to the formula: Calculate the sensitivity coefficient of each design parameter to each optimization objective. .

[0022] in, This indicates that, assuming other design parameters remain unchanged at their initial values, the design parameters... Optimization objective when taking different level intervals The conditional expected simulation value, This represents the variance of the expected simulated value under this condition. Indicate the optimization objective The total variance.

[0023] Formula used: Calculate the overall sensitivity coefficient .

[0024] in, These are the sensitivity coefficients for average torque sensitivity, torque ripple sensitivity, average suspension force sensitivity, and suspension force ripple sensitivity, respectively. The preset weighting coefficients are all set to 0.25.

[0025] The design parameters are sorted in descending order of comprehensive sensitivity coefficient. The design parameters with the highest number of preset positions are considered significant parameters, while the remaining design parameters are considered non-significant parameters. Finite element scanning optimization is then performed on the non-significant parameters.

[0026] Training samples are constructed using significance parameters as input variables, and the optimization objective is predicted based on Gaussian process regression, resulting in a Gaussian process regression surrogate model. Specifically:

[0027] Construct a surrogate model for predicting motor performance based on Gaussian process regression (GPR).

[0028] The core advantage of the GPR model lies in its probabilistic output capability, which simultaneously outputs the mean (performance prediction value) and variance (prediction uncertainty) for each prediction point.

[0029] This characteristic is relevant to the need to minimize performance fluctuations (standard deviation) in subsequent robustness optimization. The goal of GPR is naturally aligned. However, GPR models face two main challenges in high-dimensional parameter spaces:

[0030] First, the number of training samples required increases exponentially with the parameter dimension. Second, in high-dimensional space, the covariance function is difficult to effectively characterize the complex correlation between parameters, which can easily lead to model overfitting.

[0031] To address the aforementioned issues, variance-based sensitivity analysis dimensionality reduction techniques are deeply coupled with GPR modeling:

[0032] Instead of all design parameters, a select few significant parameters are used as input variables for the GPR model.

[0033] This dimensionality reduction process significantly reduces the size and condition number of the GPR covariance matrix, enabling the covariance function to more accurately capture the mapping relationship between parameters and performance with the same number of training samples, while reducing training time costs.

[0034] Specifically, different combinations of design parameters are taken as inputs, the optimization objective is taken as outputs, and sample points are uniformly sampled in a low-dimensional sample space composed of significant parameters to obtain training samples.

[0035] The GPR surrogate model is trained, and its accuracy is tested. The statistical characteristics of GPR are determined by its mean and covariance: ;

[0036] in, Let be any random parameter, and all of them be... 3D input vector, Represents the covariance matrix. This represents the mean.

[0037] Let the prior distribution of the observed target value be Training sample output With predicted sample output The joint Gaussian prior distribution is:

[0038] ;

[0039] In the formula, For training set Its own covariance matrix, It is a predicted sample With training set The covariance matrix, To predict the covariance of the sample itself, It is the identity matrix. This represents the noise variance.

[0040] The mean is 0, and the covariance matrix is... The Gaussian distribution.

[0041] In the GPR model, based on Bayes' theorem, the probability prediction distribution is continuously updated using observed data to infer the predicted sample output. Maximum Probable Predicted Posterior Distribution Predict the posterior distribution:

[0042] ;

[0043] test points The corresponding expected value and variance are as follows:

[0044] ;

[0045] .

[0046] The training samples are divided into training and validation sets to train and verify the accuracy of the Gaussian process regression surrogate model. Specifically:

[0047] In the training samples, 90% of the samples are used as the training set and 10% of the samples are used as the validation set to verify the fit and prediction accuracy of the surrogate model.

[0048] The GPR surrogate model obtained from the training set has good accuracy. Under these circumstances, the fitting risk assessment and accuracy verification are then performed on the validation set.

[0049] Mainly based on correlation coefficient Using the normalized root mean square error (NRMSE) as the basic criterion, the accuracy of the Gaussian process regression surrogate model is tested. This can effectively identify overfitting models and avoid their adverse effects.

[0050] An interaction between the Emperor Penguin optimization algorithm and a Gaussian process regression surrogate model is introduced to obtain the Pareto front solution set. Specifically:

[0051] The compromise solution is obtained by using the Emperor Penguin Optimization (EPO) algorithm. The EPO algorithm is then combined with multi-objective optimization design to obtain the Pareto front solution set.

[0052] The EPO algorithm does not directly call the high-cost finite element simulation for each group of design parameters; instead, it inputs the parameters into the trained GPR model.

[0053] The GPR model quickly outputs the mean performance prediction for each individual. etc.) and prediction variance ( wait).

[0054] Within the multi-objective optimization framework, the EPO algorithm performs non-dominated sorting and crowding distance calculation based on the mean and variance of the GPR output, thereby guiding the population to evolve towards the Pareto front, which optimizes both mean and variance.

[0055] Furthermore, the prediction variance provided by GPR can serve as an adaptive exploration guide:

[0056] In the later stages of iteration, if the prediction variance is high in a certain region (low model confidence), the EPO algorithm will adaptively increase the population density in that region to achieve targeted sampling of uncertain regions and avoid getting trapped in local optima. The specific process of the interaction logic between EPO and the GPR model is as follows:

[0057] Assuming in In the 3D search space, the number of emperor penguins is In each iteration, the radius of the emperor penguin colony formed by the current population... ,in, .

[0058] This indicates the position of the emperor penguin with the best fitness in the current iteration. This represents the average location of all Emperor Penguins at present.

[0059] The temperature within the cluster is uneven, with the highest temperature at the center and the lowest temperature at the edges. for:

[0060] ;

[0061] In the formula T max Let t be the initial temperature. max This represents the maximum number of iterations.

[0062] Calculate the relationship between each emperor penguin and the current best individual. The relative distance between them affected by temperature :

[0063] ;

[0064] In the formula, For the first Only Emperor Penguins in the first Position at the next iteration As a social power factor, and It is a random number, and The range of values ​​is , The range of values ​​is .

[0065] Update the location of each emperor penguin .

[0066] in, , The position update operator, which is usually temperature-dependent, is used to fine-tune the step size, ensuring a larger step size at high temperatures (early iterations) for exploration and a smaller step size at low temperatures (late iterations) for development.

[0067] The entropy-weighted-ideal solution superiority-inferiority distance method is used to select a compromise solution from the Pareto front solution set as the optimal solution for deterministic optimization. The specific process is as follows:

[0068] After obtaining the Pareto front solution set, an optimized compromise solution is selected from the Pareto front solution set according to the weight assigned by the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS), and this compromise solution is taken as the optimal solution for deterministic optimization.

[0069] The Entropy Weight-TOPSIS method combines the entropy weight method and the TOPSIS method. The absolute value of the contribution of the BPMSM design parameters to the optimization objective is defined as the decision matrix of the TOPSIS method. , can be represented as:

[0070] ;

[0071] In the formula ; , Indicates the first The optimization parameter affects the first... The contribution of each optimization objective; To optimize the number of parameters (the total number of candidate solutions in the Pareto front solution set). To optimize the number of targets.

[0072] For easier comparison, the decision matrix is ​​regularized using the following formula:

[0073] ;

[0074] In the formula, yes The result after regularization.

[0075] Because the optimization objectives differ in importance, each objective carries a different weight. The degree of disorder in a system can be described by information entropy; the smaller the information entropy of an optimization objective, the less information it contains, and vice versa.

[0076] Calculate information entropy ;in, , For the first The candidate solution at the th... The formula for calculating the proportion under each optimization objective is as follows: , It is a constant and takes the value of .

[0077] The weight coefficients of each optimization objective are calculated using the entropy weight method. and satisfy .

[0078] in, .

[0079] The TOPSIS method ranks each candidate solution by calculating its distance to the ideal solution and the negative ideal solution. The ideal solution under the optimization objective and negative ideal solution It can be defined as:

[0080] ;

[0081] ;

[0082] The distance between each candidate solution and the ideal solution and the negative ideal solution can be calculated using the Euclidean distance formula. The comprehensive contribution coefficient of each candidate solution to the optimization objective can be defined as its relative proximity to the negative ideal solution, i.e.:

[0083] ;

[0084] ;

[0085] In the formula, and They represent the first The distances between each candidate solution and the ideal solution and the negative ideal solution; Indicates the first The comprehensive contribution coefficient of each candidate solution to the optimization objective is determined; the parameter combination with the largest comprehensive contribution coefficient is selected as the compromise solution.

[0086] The failure rate of the deterministic optimization optimal solution is calculated based on Monte Carlo simulation. If the failure rate does not meet the preset requirements, robust optimization is performed. Specifically:

[0087] The Physics of Failure (POF) of deterministic optimization results is tested. POF can be defined as follows: Simulations were performed using the Monte Carlo method (MCS). The probability that each optimization objective exceeds the constraints under these conditions.

[0088] The failure rates for each optimization objective were weighted and summed to calculate the overall failure rate. The weights for all four objectives were 0.25. The POF results indicate whether further robustness optimization is needed.

[0089] The process of calculating the failure rate based on Monte Carlo simulation is as follows:

[0090] The design parameters are set to follow a normal distribution. ,in This represents the optimal solution. , This represents manufacturing tolerances.

[0091] By using Monte Carlo simulation and random sampling, the probability of each optimization objective exceeding the constraints is obtained.

[0092] The overall failure rate is obtained by weighted summation of the failure rates for each optimization objective.

[0093] Significance parameters are used as design parameters for robust optimization. A robust optimization model and constraints are established by combining Six Sigma design theory and Monte Carlo simulation. The predicted variance output by the Gaussian process regression surrogate model is used as an estimate of the standard deviation of the optimization objective in robust optimization. The Emperor Penguin optimization algorithm is employed for optimization to obtain the Pareto front solution set for robust optimization. The optimal solution for robust optimization is selected using the entropy weight-ideal solution superiority distance method. Specifically:

[0094] The significance parameters selected based on the variance-based sensitivity method are used as design parameters, and the optimization objective is... , , , mean ( ) and standard deviation ( ).

[0095] Combining Design for Six Sigma (DFSS) theory with MCS, a sigma level of n=6 was chosen. The probabilistic output capability of the GPR surrogate model was directly used to construct a robust optimization objective:

[0096] The optimization objective constraint is: and The design parameter constraints are as follows: and .

[0097] in, and To optimize the mean and standard deviation of the target, To optimize the lower bound of the target, To optimize the upper bound of the target, and These are the lower and upper bounds of the design parameters, respectively. and These are the mean and standard deviation of the design parameters, respectively.

[0098] The fitness function of the Emperor Penguin optimization algorithm is defined as the mean of the output of the Gaussian process regression surrogate model. and standard deviation .

[0099] GPR avoids the huge computational overhead of traditional methods that require multiple MCS samplings to calculate the standard deviation by outputting the predicted variance for each set of design parameters.

[0100] This collaborative mechanism of "GPR probabilistic prediction + DFSS" makes the cost of a single iteration of robust optimization comparable to that of deterministic optimization, but the final solution has significantly lower manufacturing sensitivity.

[0101] To address the conflict between design parameters and optimization objectives in robust optimization, the EPO algorithm is used to optimize the objective through a surrogate model, obtaining the Pareto front solution set. The entropy-weighted TOPSIS method can select a compromise solution from the solution set and use it as the optimal solution for robust optimization.

[0102] During the optimization process, the fitness function of the EPO algorithm is directly defined as the mean of the GPR output. and standard deviation The algorithm iteratively optimizes according to the interaction logic of the EPO and GPR models.

[0103] The probability distribution of the optimal solution of robust optimization is obtained based on Monte Carlo simulation, and the failure rate distribution of deterministic optimization and robust optimization are compared.

[0104] According to DFSS, design parameters are considered to follow a normal distribution. , This represents the optimal solution obtained from the two designs. , This represents manufacturing tolerances, which are 1% of the optimization results in deterministic design.

[0105] Using MCS simulation, obtain , , , The overall probability distribution is shown. By comparing the POF distribution of deterministic optimization with that of robust optimization, the optimization effect of robust optimization can be verified.

[0106] Compared with the prior art, the beneficial effects achieved by the present invention are:

[0107] This invention compresses the original high-dimensional design space into a low-dimensional significant parameter space through sensitivity analysis, enabling the GPR surrogate model to maintain high prediction accuracy even with a limited number of training samples. This effectively overcomes the technical bottlenecks of high-dimensional GPR modeling being prone to overfitting and having excessively high training costs.

[0108] In this invention, the predicted output of the GPR surrogate model is not just a single value, but a complete probability distribution. For any new design point, GPR can not only provide the predicted mean of its performance (used to maximize performance), but also the variance of the prediction (directly used to quantify performance fluctuations). This variance value can be directly used as an estimate of the standard deviation of the optimization objective in robust optimization, avoiding the huge computational burden of traditional double-layer nested sampling (inner MCS + outer optimization).

[0109] In this invention, the entropy-weighted TOPSIS method is derived by combining the entropy-weighted method and the TOPSIS method. The TOPSIS method minimizes the distance between the proposed solution and the ideal optimal solution while maximizing the distance from the ideal worst solution. The entropy-weighted method determines the weights based on the amount of "information" provided by each indicator data. Considering the performance differences of each objective function, the weights are based on the degree of variation in the data itself, avoiding subjective weight assignment and making the optimization results more objective.

[0110] This invention combines Design Principles and Standardization (DFSS) with Manufacturing Components and Standardization (MCS) to address uncertainties involving manufacturing tolerances. DFSS can balance multiple conflicting objectives in the early stages of design and can take into account long-term production processes. Drifting allows future production yields to be considered during the design phase. MCS, based on the law of large numbers and the central limit theorem, simulates a probability distribution close to reality by repeatedly sampling within a range of design parameters, and can then calculate the corresponding... as well as This significantly improves the accuracy of multi-objective robustness optimization for motors.

[0111] This invention uses the GPR surrogate model, which differs from neural network models and response surface models. Neural network models require a very large number of training samples, while response surface models use a high-order polynomial to approximate the relationship between optimization parameters and the optimization objective, but the relationship between motor parameters and the optimization objective is highly nonlinear. GPR requires relatively fewer training samples, significantly reducing the time cost of training the surrogate model, and can sample in regions of high uncertainty, avoiding getting trapped in local optima. Attached Figure Description

[0112] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings:

[0113] Figure 1 This is a flowchart illustrating the implementation of deterministic optimization and robustness optimization in this invention;

[0114] Figure 2 This is a structural diagram of the stator and rotor of the motor of the present invention;

[0115] Figure 3 This is a sensitivity coefficient diagram obtained by the variance-based sensitivity analysis method of this invention;

[0116] Figure 4 This is the GPR model of the present invention for predicting average torque;

[0117] Figure 5 This is a Pareto solution set distribution diagram composed of levitation force and levitation force fluctuation in the robust design of this invention;

[0118] Figure 6 This is a distribution diagram of the average levitation force (POF) determined by the deterministic optimization of this invention;

[0119] Figure 7 This is a distribution diagram of the average levitation force (POF) for robustness optimization according to the present invention. Detailed Implementation

[0120] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0121] Example 1: Please refer to Figure 1 The overall implementation process of this invention is shown in the figure.

[0122] First, the design parameters, optimization objectives, and constraints are determined. A variance-based sensitivity analysis is performed on the design parameters. Significant parameters are optimized using multi-objective methods, while non-significant parameters are optimized using a scanning method.

[0123] Secondly, a GPR surrogate model is established based on the training samples, and the model is optimized using the EPO algorithm to obtain deterministic optimization results. If the results do not meet the reliability requirements, robust optimization needs to be performed. Robust optimization first determines the optimization objective, then establishes an optimization model and constraints based on DFSS, and then uses the EPO algorithm to optimize and obtain the optimal parameter combination. Finally, MCS is used to simulate the distribution of the objective and determine whether the POF of the robust optimization meets the requirements.

[0124] Please see Figure 2 The stator and rotor structure diagram of a bearingless permanent magnet synchronous motor.

[0125] In the deterministic optimization design described in the invention, combined with the electromagnetic performance requirements of BIPMSM, the optimization parameters are the magnetic rib height (R). i ), Magnetic bridge height (O2), Finite diameter of permanent magnet (D1), Magnetic bridge bottom width (O1), Pole spacing of permanent magnet (D) m ), permanent magnet thickness (T) m ), permanent magnet width (W) m Stator slot depth (H) s ), Maximum width of the slot wedge (B) s1 ) and the bottom width of the tank (B s2 ), optimize objective T a T r F a F r These are average torque, torque pulsation, average suspension force, and suspension force pulsation, respectively.

[0126] Based on the design requirements, the reasonable design range for each optimization parameter was determined, and the results are shown in Table 1. The optimization objectives are as follows:

[0127] ;

[0128] The optimization objective constraints are as follows:

[0129] ;

[0130] Based on the motor design requirements, the constraints are X1=1, X2=14, X3=200, X4=4.

[0131] Table 1

[0132]

[0133] Please see Figure 3Sensitivity coefficient diagram obtained by variance-based sensitivity analysis; perform variance-based sensitivity analysis on the optimized parameters, and calculate the sensitivity coefficients of the four optimization objectives for each parameter.

[0134] The design range for each design parameter is divided into 5 levels on average, R i Divided into five parameter levels: 2, 3, 4, 5, and 6, when R... i Set the parameter to 2, and keep all other parameters at their initial values. Use this as a set of parameters, R. i When the value is 3, the other parameters remain at their initial values. By doing this, 50 sets of parameters Y can be obtained. i (i=1,2,…,50). After finite element simulation of each set of optimization parameters, the simulation values ​​of the four optimization objectives are obtained, resulting in 50 sets of simulation data K. i (i=1,2,…,50), according to the sensitivity coefficient formula, each parameter is divided into 5 levels, resulting in 5 sets of simulation values. The sensitivity coefficients for the four optimization objectives of this parameter are then calculated based on these simulation values. The formula for obtaining the sensitivity coefficients is:

[0135] ;

[0136] In the formula, Indicates design parameters, Indicate design parameters The optimization objective for (i=1,2,3,4,5) (i=1,2,3,4) Simulation values, express variance Indicate design parameters The optimization objective for (i=1,2,3,4,5) The sum of the variances of (i=1,2,3,4).

[0137] After calculating each sensitivity coefficient, a weighted sum is used to obtain the overall sensitivity coefficient, as shown in the formula:

[0138] ;

[0139] In the formula, , , , The weighting coefficients for the average torque sensitivity, torque pulsation sensitivity, average suspension force sensitivity, and suspension force pulsation sensitivity coefficients are all set to 0.25.

[0140] The overall sensitivity coefficients were calculated, and the overall sensitivity values ​​were sorted in descending order. The first four parameters were significance parameters, namely O2, D1, O1, and W. mAll other parameters are non-significant. Uniform sampling is performed within the sample space consisting of the design ranges of the four parameters. To ensure the accuracy of the surrogate model, the sample size is set to 200, which serves as the training sample for the GPR surrogate model. 90% of the samples are used as the training set, and 10% as the validation set to test the fit and accuracy of the surrogate model. R² values ​​ranging from 0 to 1 are used. 2 NRMSE and R are two fundamental criteria. 2 The closer the value is to 1, the better the predictive ability; the closer the NRMSE is to 0, the smaller the error. Accuracy results are shown in Table 2.

[0141] ;

[0142] ;

[0143] In the formula y i These are simulation values. Here, y represents the model's predicted value, N is the sample size, and y is the value predicted by the model. max y represents the maximum value of the simulation. min This represents the minimum simulated value.

[0144] Table 2

[0145]

[0146] Based on the deterministic optimization objective, the EPO algorithm is introduced to find the Pareto front solution set by finding the compromise solution. In 4-dimensional space, an initial population of 50 is used for iterative optimization. The entropy-weighted TOPSIS method can select the optimal solution for deterministic optimization from the solution set. First, the weights of each index are calculated using the entropy-weighted method. Then, the weights are multiplied by the standardized data, and the resulting data is used as the raw data for TOPSIS. Finally, TOPSIS is used to calculate the closeness of each evaluation object to the optimal solution, i.e., the comprehensive evaluation index of each candidate solution.

[0147] Please see Figure 4 The GPR model predicts the average torque. After optimization, the average torque increased from 0.85 Nm to 1.36 Nm, a 60% improvement; torque ripple decreased from 14.8% to 8.52%, a 42.4% reduction; average suspension force increased from 166.66 N to 224.35 N, a 34.6% improvement; and suspension force ripple decreased from 7.23% to 3.92%, a 45.8% reduction. The motor performance was significantly improved.

[0148] Please see Figure 5 The Pareto solution set distribution diagram consisting of the levitation force and levitation force fluctuations in the robust design is used; optimization is performed using a single-parameter finite element scan method for insignificant parameters. For R... iThe design range is 2mm to 6mm, with a step size of 0.5mm. Finite element scanning is performed, and the final R... i The target is optimal when the diameter is 5mm. Similarly, D... m The design range is 4mm to 5mm, with a step size of 0.1mm, and the final D is determined. m The optimal value is 4mm. T m The design range is 2mm to 3mm, with a step size of 0.1mm, and the optimal value is determined to be 3mm. H s The design range is 12mm to 15mm, with a step size of 0.2mm, and the optimal value is determined to be 12mm. B s1 The design range is 2.5mm to 3.5mm, with a step size of 0.1mm, and the optimal value is determined to be 2.5mm. B s2 The design range was from 3.5mm to 4.5mm, with a step size of 0.1mm, and the optimal value was finally determined to be 3.5mm. The parameter optimization results are shown in Table 3. The final deterministic optimization results show that the average torque increased from 0.85Nm to 1.44Nm, an improvement of 69.4%, the torque ripple decreased from 14.8% to 7.43%, a reduction of 49.8%, the average suspension force increased from 166.66N to 226.41N, an improvement of 35.9%, and the suspension force ripple decreased from 7.23% to 3.78%, a reduction of 47.7%.

[0149] To verify the reliability of the optimization, the Proof-of-Flight (POF) of the deterministic optimization results is tested. POF can be defined as:

[0150] ;

[0151] Based on the probability of each optimization objective exceeding the constraints in 10,000 scenarios simulated by MCS, the design parameters are assumed to follow a normal distribution N(x). i ,σ 2 xi ), x i σ represents the optimal solution obtained in deterministic optimization. xi =Δx i / 3, Δx i This represents manufacturing tolerance, which is 1% of the optimal solution of the deterministic optimization parameters. Where T... a The POF was 72.42%, T r The POF was 18.1%, F a The POF was 63.37%, F r The POF is 81.47%. The overall failure rate is calculated by weighting the failure rates of each optimization objective. The weight of each of the four optimization objectives is 0.25, and the calculated overall failure rate is 58.84%, which is not allowed in engineering. Therefore, robustness optimization needs to be carried out.

[0152] Table 3

[0153]

[0154] In the robust optimization design described in the invention, the significance parameter is first used as the optimization parameter for robust optimization. The requirements and constraints of the robust optimization objective are as follows:

[0155] ;

[0156] ;

[0157] In the formula, μ Ta μ Tr μ Fa μ Fr T represents respectively a T r F a F r The mean of μ, where LSL and USL are the lower and upper bounds of the optimization objective, respectively. f and σ f To optimize the mean and standard deviation of the objective, x l x u μ xi and σ xi These represent the lower bound, upper bound, mean, and standard deviation of the design parameters, respectively; n is the sigma level, and in this invention, n=6.

[0158] A compromise solution was obtained using the EPO algorithm, and the entropy-weighted TOPSIS method was used to select the optimal solution for robustness optimization. Compared to the initial design, after robustness optimization, the average torque increased from 0.85 Nm to 1.13 Nm, an improvement of 32.9%, the torque ripple decreased from 14.8% to 6.74%, a reduction of 54.45%, the average suspension force increased from 166.66 N to 209.75 N, an improvement of 25.9%, and the suspension force ripple decreased from 7.23% to 3.47%, a reduction of 52%.

[0159] To verify the reliability of robust optimization, the point of failure (POF) of robust optimization is calculated based on Monte Carlo simulation, where the design parameters are assumed to follow a normal distribution N(x). i ,σ 2 xi ), x i σ represents the optimal solution obtained in robust design. xi =Δx i / 3, Δx i This represents manufacturing tolerance. To ensure consistency with the manufacturing tolerances of the same equipment, it is set to 1% of the optimal value of the deterministic optimization parameter. Please refer to the comparison results. Figure 6 and Figure 7As shown in the figure, after adopting the optimization method proposed in this invention, the overall failure rate of robust optimization was reduced from 58.84% to 10.35%, and the optimization effect was significantly improved. The results of deterministic optimization and robust optimization are shown in Table 4.

[0160] Table 4

[0161]

[0162] The above embodiments are merely for clearly illustrating the design method of the present invention, and are not intended to limit the implementation method. Those skilled in the art can make other variations, and it is not necessary to exhaustively list all implementation methods here.

[0163] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0164] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A multi-objective robustness optimization method for bearingless permanent magnet synchronous motors, characterized in that: The method includes: Step 1: Obtain the various design parameters and their design ranges for the bearingless permanent magnet synchronous motor, and determine the optimization objectives; the optimization objectives include average torque, torque pulsation, average levitation force, and levitation force pulsation; the design parameters are the motor structural parameters; Step 2: Calculate the sensitivity coefficient of the design parameters to the optimization objective, calculate the comprehensive sensitivity coefficient using weighted summation, and select significant parameters based on the magnitude of the comprehensive sensitivity coefficient; Step 3: Construct training samples using the significance parameters as input variables, and predict the optimization objective based on Gaussian process regression to obtain a Gaussian process regression surrogate model; Step 4: Divide the training samples into a training set and a validation set, and train and verify the accuracy of the Gaussian process regression surrogate model. Step 5: Introduce the Emperor Penguin optimization algorithm to interact with the Gaussian process regression surrogate model to obtain the Pareto front solution set; Step 6: Select a compromise solution from the Pareto front solution set using the entropy weight-ideal solution superiority distance method as the optimal solution for deterministic optimization; Step 7: Calculate the failure rate of the deterministic optimal solution based on Monte Carlo simulation. If the failure rate does not meet the preset requirements, perform robust optimization. Step 8: Use the significance parameter as the design parameter for robust optimization, establish a robust optimization model and constraints by combining Six Sigma design theory and Monte Carlo simulation, use the predicted variance output by the Gaussian process regression surrogate model as the estimated value of the standard deviation of the optimization objective in robust optimization, use the Emperor Penguin optimization algorithm for optimization, obtain the Pareto front solution set of robust optimization, and use the entropy weight-ideal solution superiority distance method to select the optimal solution of robust optimization; Step 9: Obtain the probability distribution of the optimal solution of robust optimization based on Monte Carlo simulation, and compare the failure rate distribution of deterministic optimization and robust optimization.

2. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 1, characterized in that: The specific process of step 2 is as follows: The design range of each design parameter is divided into multiple horizontal intervals. With other design parameters keeping their initial values ​​unchanged, the simulation values ​​of the optimization target are obtained when a single design parameter takes different horizontal intervals. According to the formula: Calculate the sensitivity coefficient of each design parameter to each optimization objective. ;in, This indicates that, assuming other design parameters remain unchanged at their initial values, the design parameters... Optimization objective when taking different level intervals The conditional expected simulation value, This represents the variance of the expected simulated value under this condition. Indicate the optimization objective The total variance; Formula used: Calculate the overall sensitivity coefficient ;in, These are the sensitivity coefficients for average torque sensitivity, torque ripple sensitivity, average suspension force sensitivity, and suspension force ripple sensitivity, respectively. Preset weighting coefficients; The parameters are sorted in descending order according to the comprehensive sensitivity coefficient, and the design parameters with the first preset number of positions are the significance parameters.

3. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 1, characterized in that: The specific process of step 3 is as follows: Different combinations of design parameters are used as inputs, and the optimization objective is used as the output. Sample points are uniformly sampled in a low-dimensional sample space composed of significant parameters to obtain training samples. Let the prior distribution of the observed target value be Training sample output With predicted sample output The joint Gaussian prior distribution is: ； In the formula, For training set Its own covariance matrix, It is a predicted sample With training set The covariance matrix, To predict the covariance of the sample itself, It is the identity matrix. For noise variance; The mean is 0, and the covariance matrix is... Gaussian distribution; Based on Bayes' theorem, the probability prediction distribution is updated using observed data to obtain the predicted sample output. The expected value and variance of the predicted posterior distribution.

4. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 1, characterized in that: The specific process of step 4 is as follows: The training samples are divided into a training set and a validation set according to a preset ratio; Using correlation coefficient The accuracy of the Gaussian process regression surrogate model is verified by normalized root mean square error (NRMSE).

5. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 1, characterized in that: In step 5, the interaction process between the Emperor Penguin optimization algorithm and the Gaussian process regression surrogate model is as follows: The population of Emperor Penguin optimization algorithm is input into the trained Gaussian process regression surrogate model; the Gaussian process regression surrogate model outputs the mean and variance of the performance prediction for each population individual. The Emperor Penguin optimization algorithm performs non-dominated sorting and crowding distance calculation based on the performance prediction mean and prediction variance. During the algorithm iteration process, for uncertain regions where the prediction variance is greater than a preset threshold, the number of emperor penguin individuals in that region is increased.

6. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 5, characterized in that: In step 5, the position update process of the Emperor Penguin optimization algorithm is as follows: Calculate the radius of the emperor penguin colony formed by the current population. ;in, ; This indicates the position of the emperor penguin with the best fitness in the current iteration. This represents the average location of all emperor penguins at present. Calculate the relationship between each emperor penguin and the current best individual. relative distance between : ； In the formula, For the first Only Emperor Penguins in the first Position at the next iteration As a social power factor, and It is a random number, and The range of values ​​is , The range of values ​​is ; Update the location of each emperor penguin ;in, , Position update operator.

7. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 1, characterized in that: In step 6, the process of selecting a compromise solution using the entropy weight-ideal solution superiority distance method is as follows: The decision matrix is ​​defined as each candidate solution in the Pareto front solution set and its corresponding value of the optimization objective. And perform regularization to obtain ; Calculate information entropy ;in, , For the first The candidate solution at the th... The formula for calculating the proportion under each optimization objective is as follows: , It is a constant and takes the value of ; This represents the total number of candidate solutions in the Pareto front solution set; The weight coefficients of each optimization objective are calculated using the entropy weight method. ;in, ; Define the ideal solution and negative ideal solution Calculate the distance between each candidate solution and the ideal solution. The distance between the sum and the negative ideal solution ; Calculate the overall contribution coefficient of each candidate solution to the optimization objective. ;in, The parameter combination with the largest comprehensive contribution coefficient is selected as the compromise solution.

8. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 1, characterized in that: Step 8, the process of establishing the robust optimization model and constraints, is as follows: The average values ​​of average torque, torque pulsation, average suspension force, and suspension force pulsation. and standard deviation To optimize the objective; Introducing Six Sigma design theory and setting preset parameters level ; The optimization objective constraint is: and The design parameter constraints are as follows: and ; in, and To optimize the mean and standard deviation of the target, To optimize the lower bound of the target, To optimize the upper bound of the target, and These are the lower and upper bounds of the design parameters, respectively. and These are the mean and standard deviation of the design parameters, respectively. The fitness function of the Emperor Penguin optimization algorithm is defined as the mean of the output of the Gaussian process regression surrogate model. and standard deviation .

9. The multi-objective robustness optimization method for a bearingless permanent magnet synchronous motor according to claim 1, characterized in that: The process of calculating the failure rate based on Monte Carlo simulation in steps 7 and 9 is as follows: The design parameters are set to follow a normal distribution. ,in This represents the optimal solution. , Represents manufacturing tolerances; The probability of each optimization objective exceeding the constraints is obtained by random sampling through Monte Carlo simulation; The overall failure rate is obtained by weighted summation of the failure rates for each optimization objective.

Images