A motor design method based on the Kriging surrogate model

Abstract

This invention discloses a motor design method based on a Kriging surrogate model. Specifically, it involves: collecting M combinations of motor size parameters as input sample data S, and using the corresponding motor performance as output sample data Y; forming a sample set from S and Y; training a Kriging surrogate model based on this sample set; constructing an optimization model with minimization as the objective function and the range of motor size parameter variations in the samples from which the Kriging surrogate model is trained as the feasible region; and solving the optimization model to obtain the motor parameters corresponding to the given motor performance. The method proposed in this invention improves the speed of solving the inverse mapping of the multi-performance Kriging surrogate model of a motor based on the coordinate descent method, and has good practicality and economy.

Description

Technical Field

[0001] This invention belongs to the field of motor technology, and in particular relates to a motor design method based on the Kriging surrogate model. Background Technology

[0002] In motor design, surrogate models are often combined with motor simulation models to optimize motor dimensional parameters, resulting in significant performance improvements even under certain conditions. Surrogate models can learn the mapping relationship between motor dimensional design parameters and motor performance from a small number of samples. However, in most applications of surrogate models, the dimensionality of the input variables is often much higher than the dimensionality of the output response. In such cases, the surrogate model can only achieve a unidirectional mapping; it can only use motor dimensional design parameters as input variables and motor performance as the output response, and cannot reverse the process, using motor performance as input variables and motor dimensional design parameters as the output response.

[0003] In his book *Principles of Geostatistics*, Georges Matheron systematically describes the Kriging interpolation method. This method boasts high prediction accuracy and provides a variance estimate when predicting the output response. Its principles are simple and easy to understand, making it commonly used in motor size optimization design. Furthermore, the Kriging interpolation model expression is differentiable with respect to the independent variable, possesses a clear structure, and is easily combined with other optimization models, allowing for flexible application in various scenarios. However, despite these advantages, the Kriging interpolation model can only predict the output response unidirectionally from a given input variable; it cannot predict the values ​​of the input variables inversely from a given output response.

[0004] After transforming the inverse mapping problem of the multi-performance Kriging surrogate model of a motor into an optimization model, the objective function of this optimization model is differentiable within the feasible region, and the constraints are linear. Rosen's projection gradient method, proposed in "The Gradient Projection Method for Nonlinear Programming," is well-suited for quickly solving this optimization model, ensuring that the optimization variables obtained in each iteration remain within the feasible region and rapidly approach the optimal value. However, in practical applications, due to the large number of optimization variables and the complexity of the model, it is very easy for the optimization model to get trapped in local optima.

[0005] Paul Tseng's coordinate gradient descent method, proposed in "A coordinate gradient descent method for nonsmoothseparable minimization," is simple and efficient, and has been applied to high-dimensional optimization problems in various fields such as machine learning, data mining, and power system optimization. However, as the number of motor performance parameters, i.e., the dimension of the output response of the Kriging surrogate model, increases, the number of extreme points of the objective function of the optimization model after inverse mapping also increases. This slows down the solution speed during iterative solving and makes the method highly susceptible to the influence of the order in which the components are solved in the coordinate gradient descent method. Summary of the Invention

[0006] Purpose of the invention: In order to solve the problems existing in the prior art, the present invention provides a motor design method based on the Kriging proxy model.

[0007] Technical solution: This invention discloses a motor design method based on the Kriging surrogate model, specifically as follows:

[0008] M combinations of motor size parameters are collected as input sample data S, and the corresponding motor performance is used as output sample data Y. S and Y form a sample set, and a Kriging surrogate model is trained based on the sample set. X is the input variable of the Kriging surrogate model, representing the motor parameters, X = x1, x2, x3, ..., x k ,…,x n ;x k Let k represent the k-th type of motor parameter, k = 1, 2, 3, ..., n, where n represents the total number of types of motor parameters;

[0009] To minimize Let y be the objective function, and construct an optimization model with the range of motor size parameters in the samples from which the Kriging surrogate model is trained as the feasible region; where y c For a given motor performance, y c =(y1,y2,…,y q ,…,y Q ) T T represents transpose, y q Let represent the q-th motor performance type, where q = 1, 2, ..., Q, and Q represents the total number of motor performance types; Λ is the coefficient matrix, Λ = diag{λ1, λ2, ..., λ...} q ,…,λ Q}, diag{.} is a diagonal function, λ q This represents the q-th weight coefficient;

[0010] Solve the optimization model to obtain the motor parameters based on the given motor performance.

[0011] Furthermore, specifically: the Kriging proxy model The expression is as follows:

[0012]

[0013] Among them, f b (X) represents the basis functions of the regression model. p represents the number of basis functions, and r(X) is the correlation vector between X and S. θ is a hyperparameter. The function expression is:

[0014]

[0015] s m,k This represents the k-th motor parameter in the m-th input sample data;

[0016] β * and γ * The expression is:

[0017] β * =(F T R -1 F) -1 F T R -1 Y

[0018] γ * =R -1 (Y-Fβ * )

[0019] The expressions for F and R are:

[0020]

[0021] Furthermore, a genetic algorithm is used to calculate the hyperparameter θ.

[0022] Furthermore, the expression for optimizing the model is shown below:

[0023]

[0024] in, This represents the q-th performance of the motor predicted by the Kriging surrogate model. s m,k This represents the k-th motor parameter in the m-th input sample data.

[0025] Furthermore, set λ q The expression is: and These are the minimum and maximum values ​​of the performance of the q-th type of motor, respectively.

[0026] Furthermore, when solving the optimization model, the objective function is repeatedly descented using the coordinate descent method, with each descent direction corresponding to the coordinate axis direction of a random single component in the input variable, so that the objective function eventually converges.

[0027] Furthermore, the descent iteration specifically includes:

[0028] After arranging the parameters in the input variable X in a preset order, a descent iteration is performed.

[0029] Let the input variable at the j-th iteration be X. j If X needs to be changed in the (j+1)th iteration j The i-th component will then Updated to:

[0030]

[0031] in, The iteration step size, Where m j The smallest non-negative positive integer that satisfies the following inequality:

[0032]

[0033] Where β∈(0,1), σ∈(0,0.5), the expression for f(.) is:

[0034] s m,i This represents the i-th motor parameter in the m-th input sample data;

[0035] Then the input variable for the (j+1)th iteration is

[0036] Furthermore, the Kriging surrogate model requires the sample data to be normalized before training.

[0037] Beneficial effects:

[0038] 1. The method of the present invention is simple and easy to understand, and the objective function is differentiable with respect to the optimization variables in the feasible region. The constraints are linear, and the motor size parameters that meet the performance requirements of multiple motors can be easily obtained through known algorithms. Furthermore, the response speed of the Kriging surrogate model is much faster than that of the finite element model, which can save a lot of time consumed by the finite element model calculation.

[0039] 2. When solving the optimization model after the inverse mapping transformation of the multi-performance Kriging surrogate model of a motor, this invention uses an improved coordinate gradient descent method to make the optimization variables gradually approach the optimal solution. That is, only after each random selection of a component, a descent iteration is performed on that component. In other words, the number of iterations to solve for the optimal value of the component in each stage is limited to 1. Compared with the general case, i.e. the unimproved coordinate gradient descent algorithm, the degree of descent by changing the component after one iteration is greater than the degree of descent by not changing the component without limiting the number of iterations. The improved coordinate gradient descent method can solve complex high-dimensional optimization models faster. Moreover, because it frequently changes the component for descent iteration, the iteration order of the component has little impact on the speed of the algorithm. This improvement is simple and easy to understand, and can efficiently solve complex high-dimensional optimization models. Attached Figure Description

[0040] Figure 1 This is a flowchart of the present invention.

[0041] Figure 2 This is a structural diagram of an 8-pole δ-type permanent magnet flat wire motor with a 48-slot stator and an 8-pole rotor, as shown in the embodiment. Detailed Implementation

[0042] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0043] In the field of motor body parameter design, it is often necessary to consider multiple motor performance application requirements to determine motor size parameters. It is also necessary to predict the motor performance corresponding to different combinations of motor size parameters using a finite element model with a high confidence level. This involves establishing a surrogate model with motor size parameters as input variables and multiple motor performance characteristics as output responses. The process of transforming the inverse mapping problem of the surrogate model into the solution of an optimization model can use multiple motor performance characteristics as input variables and motor size parameters as output responses. The surrogate model's response speed is much faster than the finite element model; introducing a surrogate model to replace the finite element model can save a significant amount of computation time. The technical solution adopted in this invention limits the number of descent iterations after randomly selecting components in each stage during the solution of the motor multi-performance optimization model, ultimately enabling a fast and stable solution for the motor parameter combinations corresponding to the multiple motor performance requirements.

[0044] like Figure 1 As shown, the specific process of this embodiment is as follows:

[0045] This embodiment uses a permanent magnet flat wire motor with a 48-slot stator and an 8-pole delta-type rotor as an example to describe the implementation process of the invention in detail. Figure 2 This is a structural diagram of the motor.

[0046] First, M = 100 combinations of motor size parameters are collected as input sample data S using a Latin hypercube sampling experiment design and combined with a finite element model. In this embodiment, the total number of motor performance parameters is Q = 2. The average torque and average loss performance of the motors corresponding to the input sample data are used as output sample data Y. Based on the sample set, a Kriging surrogate model is trained with multiple motor size parameters as input variables and multiple motor performance parameters as output responses. The parameter tuning process during training is solved using a genetic algorithm and k-fold cross-validation. Then, given the motor performance y... c =(y1,y2) T ,by To minimize the objective function, an optimization model is constructed using the range of motor size parameters from the samples trained on the Kriging surrogate model as the feasible region. Finally, the objective function is iteratively descentd using coordinate descent, with each descent direction corresponding to a random single component of the input variable on the coordinate axis, until the objective function converges to 0, thus obtaining the result corresponding to the given motor performance y. c Several corresponding combinations of motor size parameters

[0047] In this embodiment, the sampling process selected 11 motor parameters as input variables X = (x1, x2, ..., x...). 11 = (Rsi, Wslot, Hslot, Wslot_open, Wtip, ..., Lpm2), and its initial parameter values ​​are shown in Table 1.

[0048] Table 1

[0049]

[0050]

[0051] Any sample s in the input sample data S m =(s m,1 ,s m,2 ,...,s m,k ,...,s m,n ) is an n-dimensional vector, corresponding to n motor size parameters (n=11), s m,k This refers to the motor size parameter of the kth motor in the mth sample.

[0052] Given motor performance requirements y c =(y1,y2) T =(120,2500) TLet y1 represent the motor torque performance requirement, y1 = 120 Nm, and y2 represent the motor loss performance requirement, y2 = 2500 W. The objective function of the optimization model is to minimize the coefficient matrix Λ = diag{λ1, λ2} that keeps all performance values ​​on the same scale, and the difference vector between the given performance values ​​and the Kriging surrogate model. The square of the product's L2 norm, i.e. Where λ1:λ2=1 / (Y1) max -Y1 min ) 2 :1 / (Y2 max -Y2 min ) 2 ,λ1+λ2=1,Y1 min and Y1 max Y2 represents the minimum and maximum values ​​of the motor torque performance in the sample dataset Y, respectively. min and Y2 max These represent the minimum and maximum values ​​of the motor loss performance in the sample dataset Y, respectively. The feasible region is determined by the range of values ​​of the samples in the sample dataset S used to construct the Kriging surrogate model, i.e., the k-th component x of variable X. k The feasible range is [x k min ,x k max ],in The expression for the optimization model constructed in this invention is:

[0053]

[0054] in, This represents the motor torque performance predicted by the Kriging surrogate model. This represents the motor loss performance predicted by the Kriging surrogate model.

[0055] In this embodiment, the Kriging surrogate model requires maximum / minimum normalization of the sample data before training, that is, scaling the data to between 0 and 1 to eliminate the influence of different units. The basis functions of the Kriging surrogate model are set as constant basis functions, and the model expression can be given by the following formula:

[0056]

[0057] Where the input variable X is an n-dimensional vector, f b (X) represents the basis functions of the regression model:

[0058] f b (X)=[f1 b (X),f2 b (X),…,fp b (X)] T

[0059] In this example, f b (X) is taken as a constant basis function, i.e., p = 1, f1 b (X) = 1, r(X) is the correlation vector between X and S:

[0060]

[0061] θ is a hyperparameter. The function expression is:

[0062]

[0063] Where θ, w, x1 are all The input parameters of the function. Then we can obtain:

[0064]

[0065] β * and γ * The weight vector for the corresponding function:

[0066] β * =(F T R -1 F) -1 F T R -1 Y

[0067] γ * =R -1 (Y-Fβ * )

[0068] Where F and R are:

[0069]

[0070] Given randomized, multidimensional initial agent model input variables In this example, assuming the sequence is as follows in Table 1: "1(x1,Rsi)→2(x2,Wslot)→3(x3,Hslot)→…→11(x…) 11 The component order of "2(x2,Wslot)→1(x1,Rsi)→3(x3,Hslot)→…→11(x1,Rsi)" is used for cyclic iteration to obtain the order 1, following the sequence "2(x2,Wslot)→1(x1,Rsi)→3(x3,Hslot)→…→11(x1,Rsi ... 11 The components of ",Lpm2)" are iterated in sequence to sequence 2, following the order "3(x3,Hslot)→1(x1,Rsi)→2(x2,Hslot)→…→11(x 11The components of ",Lpm2)" are iterated in sequence to sequence 3, and so on, according to "11(x 11 ,Hslot)→1(x1,Rsi)→2(x2,Hslot)→…→10(x 10 The components of ",Wpm2)" are iterated sequentially to sequence 11.

[0071] Randomly select an initial value X for the input variable X within the feasible region. 0 The components are iteratively decreased in a cyclical order of 1, 2, 3, ..., 11, and this step is repeated continuously. The input variable for the j-th iteration is X. j =(x1) j x2 j ,...,x n j Assume that the (j+1)th iteration requires changing X. j The i-th component Then for components Update it to The iterative formula follows the one-dimensional projective gradient method based on the Armjio criterion, where The iteration step size, The direction is downward, and the input variable for the (j+1)th iteration is... That is, other components remain unchanged, only component 2 remains unchanged. A one-step descent iteration is performed following the one-dimensional projective gradient method based on the Armjio criterion. The components... Descent iteration step size and descent direction The search process follows the Armjio criterion, β∈(0,1), σ∈(0,0.5). In this embodiment, β=0.5, σ=0.2. Where m k The smallest non-negative positive integer that satisfies the following inequality:

[0072]

[0073] The expression for f(.) is: That is, f(x) i x in ) i As variables, Only the corresponding x is in i Modify, everything else remains the same;

[0074] The above process is repeated until the objective function of the optimization model converges to the global minimum of 0 or the total number of iterations is greater than or equal to the maximum number of iterations, thus obtaining the motor performance setpoint y. cThe corresponding combination of motor size parameters as input variables of the surrogate model The termination condition is set to convergence of the objective function to 0 when the objective function is less than ε (in this example, ε is set to 1e-5), at which point the algorithm stops.

[0075] Comparing the improved coordinate gradient descent method with the original coordinate gradient descent method, the improved method performs only one iteration of descent when updating variables, thus reducing the number of components... Updated to After one iteration of descent, the components are replaced according to the order of the cyclic iterations. Continue the iterative descent; while the original coordinate descent method does not limit the number of iterative descents, performing m iterative descents on the same component in each stage, with the iterative formula remaining the same. After each stage of the solution is completed, the components are updated. That is, do not change the component until the value of the component that minimizes the objective function is found.

[0076] Initial value X for input variable X 0 Using the original coordinate gradient descent method and the improved coordinate gradient descent method respectively, according to 1, 2, ...

[0077] The components were iteratively decreased in the cyclic sequence of 3, ..., 11, and the number of iterations required to terminate the iteration was recorded. The results are shown in Table 2.

[0078] Table 2

[0079] order 1 2 3 4 5 6 7 8 9 10 11 average value Range Number of iterations in the original method 109 97 86 60 81 74 45 154 67 70 61 82 109 Improved method iteration count 20 22 26 33 31 29 21 31 23 23 22 26 13

[0080] As shown in the table, the original coordinate gradient descent method requires 45 iterations when solving in sequence 7 and 154 iterations when solving in sequence 8, with a range of 109. In contrast, the improved coordinate gradient descent method requires 20 iterations when solving in sequence 1 and 33 iterations when solving in sequence 4, with a range of 13. This demonstrates that the iteration order of the components has a significantly greater impact on the original coordinate gradient descent method than on the improved method. Furthermore, the original coordinate gradient descent method requires an average of 82 iterations, while the improved method requires an average of 26 iterations, making the original method slower. Therefore, the improved coordinate gradient descent method not only reduces the impact of component order on the speed of iteratively solving the optimization model but also significantly improves the solution speed.

[0081] The above embodiments are merely illustrative examples of the present invention and do not limit its scope of protection. Those skilled in the art can make partial changes to it. For example, the type and performance of the motor can be selected according to the actual situation, and the type of proxy model used to fit the mapping relationship between the combination of motor size parameters and the motor performance can be selected according to the actual situation. Any equivalent substitution in any form that conforms to the spirit of the invention falls within the scope of protection of the present invention.

Claims

1. A motor design method based on the Kriging surrogate model, characterized in that, Specifically: M combinations of motor size parameters are collected as input sample data S, and the corresponding motor performance is used as output sample data Y. S and Y form a sample set, and a Kriging surrogate model is trained based on the sample set. X is the input variable of the Kriging surrogate model, representing the motor parameters, X = x1, x2, x3, ..., x k ,…,x n ;x k Let k represent the k-th type of motor parameter, k = 1, 2, 3, ..., n, where n represents the total number of types of motor parameters; To minimize Let y be the objective function, and construct an optimization model with the range of motor size parameters in the samples from which the Kriging surrogate model is trained as the feasible region; where y c For a given motor performance, y c =(y1,y2,…,y q ,…,y Q ) T T represents transpose, y q Let represent the q-th type of motor performance, where q = 1, 2, ..., Q, and Q represents the total number of motor performance types; Λ = diag{λ1, λ2, ..., λ...} q ,…,λ Q }, diag{.} is a diagonal function, Λ is the coefficient matrix, λ q This represents the q-th weight coefficient; Solve the optimization model to obtain the motor parameters based on the given motor performance; When solving the optimization model, the objective function is repeatedly descented once based on the coordinate descent method. Each descent direction is the coordinate axis direction corresponding to a random single component in the input variable, so that the objective function eventually converges. The descent iteration specifically refers to: After arranging the parameters in the input variable X in a preset order, a descent iteration is performed. Let the input variable be X in the j-th iteration. j If X needs to be changed in the (j+1)th iteration j The i-th component will then Updated to: in, The iteration step size, Where m j The smallest non-negative positive integer that satisfies the following inequality: Where β∈(0,1), σ∈(0,0.5), the expression for f(.) is: s m,i This represents the i-th motor parameter in the m-th input sample data; Then the input variable for the (j+1)th iteration is 2. The motor design method based on the Kriging surrogate model according to claim 1, characterized in that, Specifically, the Kriging proxy model The expression is as follows: Among them, f b (X) represents the basis functions of the regression model. p represents the number of basis functions, and r(X) is the correlation vector between X and S. θ is a hyperparameter. The function expression is: s m,k This represents the k-th motor parameter in the m-th input sample data; β * and γ * The expression is: β * =(F T R -1 F) -1 F T R -1 Y; c * =R -1 (Y-Fβ * ); The expressions for F and R are:

3. The motor design method based on the Kriging surrogate model according to claim 2, characterized in that, The hyperparameter θ is calculated using a genetic algorithm.

4. The motor design method based on the Kriging surrogate model according to claim 1, characterized in that, The expression for the optimization model is shown below: in, This represents the q-th performance of the motor predicted by the Kriging surrogate model. s m,k This represents the k-th motor parameter in the m-th input sample data.

5. The motor design method based on the Kriging surrogate model according to claim 1, characterized in that, Set λ q The expression is: and These are the minimum and maximum values ​​of the performance of the q-th type of motor, respectively.

6. The motor design method based on the Kriging surrogate model according to claim 1, characterized in that, Before training, the Kriging surrogate model requires normalization of the sample data to its maximum and minimum values.

Images