Improved Method for Distributed Parameter Model of Solenoid Valve Based on Kriging Basis Function Predictive Identification Program

Abstract

In this paper, a method for the improvement of the calculation accuracy of the distributed parameter model (DPM) of electromagnetic devices is proposed based on the kriging basis function predictive identification program (PIP). Kriging is mainly an optimal interpolation method which uses spatial self-covariance, and takes a polynomial as the basis function. The accuracy of the kriging-based surrogate model can be improved by adjusting the related functions and hyperparameters. Based on the DPM of a solenoid valve, there exist certain errors in the estimation. They can be summarized as follows: Firstly, the estimation error of magnetic flux leakage (MFL) permeance is caused directly by the deviation of the magnetic flux tube due to the segmented magnetic field line. Secondly, the estimation error of soft magnetic resistance because of the nonlinearity of the permeability of soft magnetic material leads to the change of soft magnetic resistance alongside the magnetic flux. In this paper, an improved kriging error correction method is applied to modify the leak permeance and soft magnetic resistance calculation. The kriging basis function is adjusted to adapt to the data curve of the MFL permeance error data. The calculated MFL permeance data are compared with the error variation data to select the appropriate basis function. To improve the computational efficiency, the PIP is proposed to select the appropriate basis function. The modified MFL permeance data and soft magnetic resistance are substituted into the DPM for improving the computational accuracy and efficiency of the solenoid valve.

1. Introduction

The optimization design of modern solenoid valves faces more and more challenges with increasing performance requirements to realize higher latching force, higher output force, and more extensive robustness [1,2,3,4,5,6,7,8,9]. As a means of simulation, the FEM method has the absolute advantage of accuracy, and the disadvantage of efficiency. The distributed parameter model (DPM) method can effectively speed up the calculation, and kriging models can decrease error. The DPM and kriging models have become important factors affecting the design efficiency and accuracy of solenoid valves. Therefore, evaluating the applicability of the kriging model has an important theoretical significance and a practical engineering value for product design optimization [10,11,12,13,14,15].

With the increasing levels of computational technology and theoretical analysis for the DPM, the kriging model, whose inputs affect its outputs, plays an important role in engineering design and error correction [16,17,18,19,20,21]. The calculation accuracy of air leakage permeance and the soft magnetic resistance has an effect on the electromagnetic force calculation accuracy of solenoid valves. To reduce the relative error and improve the accuracy of target prediction based on sample data, various data fitting methods based on the kriging model have been proposed. In the design and optimization of products, kriging has a good performance in error correction [22,23,24,25,26,27,28]. An adaptive finite difference weighted essentially non-oscillatory (WENO) method with Gauss-kriging reconstruction is proposed to reduce dissipation in smooth regions of flow, while preserving high resolution around discontinuities for hyperbolic systems of conservation laws [29]. The work in [30] proposed a kriging model-based method for the reliability design and optimization of planetary gear, using a genetic algorithm. The kriging model was used to establish the gear reliability model to simplify the reliability calculation in this method [30]. Shin et al. performed an optimization study of response time to improve the dynamic performance of a direct-acting solenoid valve based on a kriging model [31]. A kriging metamodel-based multi-objective optimization strategy has been employed to optimize the valve-plate shape of the axial piston pump [32]. The kriging method was also used to build a surrogate model, which presents the relationship between dynamic responses and dynamic simulation of the valve train [33]. Many of the works where the kriging model has been adopted for error calculation mainly focus on the calculation errors associated with replacing the original analytical model with the kriging-based surrogate model.

In this paper, a method for the improvement of the calculation efficiency and accuracy of the DPM of electromagnetic devices is proposed based on the kriging basis function predictive identification program (PIP). To obtain an improved DPM performance, an appropriate basis function is selected by contrasting various basis functions with error item curves of air leakage permeance. To further improve the calculation efficiency, the PIP is introduced to prejudge the error item curves against standard functions. The modified leakage permeance and the soft magnetic resistance data are considered in the DPM of the electromagnetic device to calculate the electromagnetic force, thereby verifying the method proposed in this paper.

2. The Error Correction Method of DPM Based on Kriging Model

The kriging model is a global surrogate model derived from geostatistics, and it can mathematically provide an optimal linear unbiased estimation. This makes it an appropriate method to achieve the primary goal of this paper. To estimate the linear, unbiased and minimum estimated variance using kriging, the geometric features of information sample shape, size and space on samples are taken into account. Nonetheless, kriging is still a smooth interpolation method [34,35]. The electromagnetic calculation software is Matlab in this paper.

The kriging model can be summarized in two parts, which are represented by

$$y(x)={\displaystyle \sum _{j=1}^p{\beta }_j{f}_j(x)+Z(x)}$$

(1)

where f_j(x) is the basis, β_j is the regression coefficient, Z(x) is the stochastic distribution function.

A solenoid valve with a fan-shaped permanent magnet (PM) is selected as the example in this study, and its electromagnetic system is shown in Figure 1. The armature is a moving part, the angle of the fan-shaped PM is 20°, its inner radius is 8.5 mm, its outer radius is 17.5 mm and its height is 8 mm. The solenoid valve rated coil ampere-turns equal 1800 AT. The PM is divided into seven segments. The magnetic flux leakage (MFL) permeances of the flux tube from the outer radius side to the inner radius side are G₁, G₂, G₃, G₄, G₅, G₆ and G₇ [36,37,38,39,40].

The solenoid valve is excited in 0 AT, and its finite element method (FEM) responses are compared with the unmodified DPM, as shown in Figure 2. There are 20 sample points for every curve.

The FEM and DPM data are subtracted to obtain the MFL permeance error terms (i.e., ΔG₁, ΔG₂, ΔG₃, ΔG₄, ΔG₅, ΔG₆ and ΔG₇), which are then stored in the error matrix Er. The MFL permeance data are shown in Figure 3 which reveal a non-monotonic variation.

To further improve the error correction efficiency, multi-basis functions are applied to the adaptive error curve. The kriging basis functions are represented as follows:

Fourier Function

$$f(x)={\displaystyle \sum _i^n[a_i\sin (d_{i}wi)+b_i\sin (d_{i}wi)]}.$$

(2)

Polynomial Function

$$f(x)=1+{\displaystyle \sum _{i=1}^n{d}_i}+{\displaystyle \sum _{i=1}^n{d}_i^2}+{\displaystyle \sum {\displaystyle \sum _{i<j}{d}_i{d}_j}}.$$

(3)

Exponential Function

$$f(x)=e^{{\displaystyle \sum _{i=1}^n(-{\theta }_i,\left|d_j\right|)}}.$$

(4)

Gaussian Function

$$f(x)=e^{{\displaystyle \sum _{i=1}^n(-{\theta }_i,{\left|d_j\right|}^2)}},$$

(5)

where |d_i| is the distance between the known and predicted quantities.

The basis function output characteristics can be determined using the key nodes and the output of the model. Given a function k(x₁, x₂… x_n) and an input parameter x₁, x₂… x_n, the domain of the function can be expressed as follows:

$$\{\begin{array}{c}{X}_1\in [x_1^0-\Delta x_1,x_1^0+\Delta x_1]\\ X_2\in [x_2^0-\Delta x_2,x_2^0+\Delta x_2]\\ ⋮\\ X_n\in [x_n^0-\Delta x_n,x_n^0+\Delta x_n]\end{array},$$

(6)

where x₁, x₂… x_n are the coordinate positions of magnetic lines of flux with armature displacement, x₁₀, x₂₀… x_n0 are the center of fluctuations, Δx₁, Δx₂… Δx_n are the ranges of fluctuations. The basis function can be simplified to:

$$\Delta F\approx {\displaystyle \sum _{i=1}^n\Delta F_i}+{\Delta }^2(\Delta x),$$

(7)

where ΔF is the function output, Δ²(Δx) is the higher derivative of the input parameter fluctuations. Given that a function relation exists between k(x₁, x₂… x_n) and Δx₁, Δx₂… Δx_n, k(x₁, x₂… x_n), this function relation can be described as follows:

$$k\left(x_1^0,x_2^0\dots x_n^0\right)={\displaystyle \sum _{x_1^0-\Delta x_1,x_2^0-\Delta x_2\dots x_n^0-\Delta x_n}^{x_1^0+\Delta x_1,x_2^0+\Delta x_2\dots x_n^0+\Delta x_n}k\left(x_1,x_2\dots x_n\right)w|{}_{x_1,x_2\dots x_n}},$$

(8)

where w is the weight coefficient. For every node in the count, w = 1. However, if the sample points are not key nodes, then 0 < w < 1. Given that, the boundary condition of the input parameter is as follows:

$$\{\begin{array}{c}{x}_{1l}\le x_1\le x_{1h}\\ x_{2l}\le x_2\le x_{2h}\\ ⋮\\ x_{nl}\le x_n\le x_{nh}\end{array},$$

(9)

where x_il, x_ih are the top and bottom limitations of the ith input value. Combining Equations (7)–(9), ΔF_i can be defined as:

$$\Delta F_i={\displaystyle \sum _{(x_{lh},x_{lh}\dots x_{lh})}^{(x_{1h},x_{2h}\dots x_{nh})}w\Delta F(\Delta x_i)}|{}_{x_1,x_2\dots x_n}G(x_1,x_2\dots x_n)|{}_{x_1^0,x_2^0\dots x_n^0},$$

(10)

where G(x₁, x₂… x_n) is the interpolating function of x₁, x₂… x_n. Substituting G(x₁, x₂… x_n) into Equations (7) and (8), the following equations are obtained:

$${G(x_1,x_2\dots x_n)|}_{\left(x_1^0,x_2^0\dots x_n^0\right)}=e^{{\displaystyle \sum _{i=1}^n(-{\theta }_i,\left|d_i\right|)}},$$

(11)

$${G(x_1,x_2\dots x_n)|}_{\left(x_1^0,x_2^0\dots x_n^0\right)}=e^{{\displaystyle \sum _{i=1}^n(-{\theta }_i,{\left|d_i\right|}^2)}},$$

(12)

where $d_i=\left|x-x_i\right|$, x is the dependent variable, θ_i is the PM segment positions with respect to the armature displacement. The armature stroke is 2 mm, and θ_i can be obtained by particle swarm optimization (PSO) based on 20 nodes.

Based on the MFL permeance error data, and adopting a periodic function (i.e., Fourier series) for ΔG₃ fitting, it can be written as follows:

$$f(x)=a_0+{\displaystyle \sum _{i=1}^4[a_i\cos (ixw)+b_i\sin (ixw)]},$$

(13)

where x is the armature stroke, f(x) is the dependent variable of the MFL permeance. a₀ = −857, a₁ = 1384, b₁ = −45.96, a₂ = −71.14, b₂ = 43.84, a₃ = 212.3, b₃ = −17.28, a₄ = −28.1, b₄ = 2.529, w = 0.1202. Taking the derivative of Equation (13), df(x)/dx has four solutions defined in the range 0 to 2.

By adopting a polynomial function for ΔG₃ fitting, it can be written as follows:

$$f(x)=p_1{x}^7+p_2{x}^6+p_3{x}^5+p_4{x}^4+p_5{x}^3+p_6{x}^2+p_{7}x+p_8,$$

(14)

where p₁ = −5.34 × 10⁻⁶, p₂ = 3.26 × 10⁻⁵, p₃ = −7.49 × 10⁻⁵, p₄ = 8.1 × 10⁻⁵, p₅ = −4.26 × 10⁻⁵, p₆ = 1.08 × 10⁻⁶, p₇ = −1.33 × 10⁻⁶, p₈ = −3.92 × 10⁻⁸. Taking the derivative of Equation (14), i.e., df(x)/dx, yields four solutions.

By adopting an exponential function for ΔG₃ fitting, it can be written as follows:

$$f(x)=a{e}^{bx}+c{e}^{dx},$$

(15)

where a = −7.591 × 10⁻⁷, b = 4.862, c = 7.583 × 10⁻⁷, d = 4.863. Taking the derivative of Equation (15), i.e., df(x)/dx, has only one solution. It is obvious that the exponent is unfit for ΔG₃ fitting.

By adopting a Gaussian function for ΔG₃ fitting, it can be written as follows:

$$f(x)=a_1{e}^{-\frac{{(x-b_1)}^2}{2c_1^2}}+a_2{e}^{-\frac{{(x-b_2)}^2}{2c_2^2}}+a_3{e}^{-\frac{{(x-b_3)}^2}{2c_3^2}}+a_4{e}^{-\frac{{(x-b_4)}^2}{2c_4^2}}+a_5{e}^{-\frac{{(x-b_5)}^2}{2c_5^2}},$$

(16)

where a₁ = 1.28 × 10⁻⁶, b₁ = 1.906, c₁ = 0.2655, a₂ = 0, b₂ = 10.97, c₂ = 0.03011, a₃ = 2.89 × 10⁻⁵, b₃ = 1.29, c₃ = 0.6738, a₄ = −2.87 × 10⁻⁵, b₄ = 1.302, c₄ = 0.6887, a₅ = −0.6322 × 10⁻⁷, b₅ = 0.5452, c₅ = 0.1176. Taking the derivative of Equation (16), i.e., df(x)/dx, yields four solutions.

Figure 4 shows the contrasting condition between basis functions and MFL permeance error ΔG₃. It is obvious that there is a large error between the exponential function-based fitted curve and ΔG₃. The mean error for this particular case is over 100%. The polynomial function-based fitted curve, Fourier function-based fitted curve and Gaussian function-based fitted curve have mean errors reaching 38.4%, 30.1% and 14.2%, respectively.

To further improve computational efficiency, the reliability index of samples is calculated by four algorithms to achieve optimal basis function. The algorithms are:

• Particle Swarm Optimization(PSO)
• Response Surface (RS)
• Linear Second order moment (LS)
• Monte Carlo (MC)

The relative error of the basis functions can be obtained by taking MC as the base. Firstly, the failure probability of the samples is calculated, then the reliability index can be obtained using the calculation results. The reliability index is calculated by the MC method and it is used as the standard with a termination condition of 10⁻⁷.

Table 1. The basis function model reliability index of ΔG3.

Function	Algorithm	Iteration	Count	Time (s)
Gaussian	PSO	132	660	987
	RS	189	945	1229
	LS	147	588	1143
	MC	-	105	2961
Fourier	PSO	162	810	1328
	RS	198	990	1843
	LS	204	816	1687
	MC	-	105	3063
Polynomial	PSO	153	765	1125
	RS	207	1035	2063
	LS	192	768	1763
	MC	-	105	3012

shows the reliability index of ΔG₃.

Table 1 shows that Gaussian function with PSO has the smallest count for ΔG₃, while the Fourier function with PSO has the smallest count for ΔG₁.

ΔG₃ is presented as an example. The polynomial function is adopted for the armature stroke, Gaussian error data and ΔG₃. This function can be written as follows:

$$\Delta G_3=p_{00}+p_{10}x+p_{01}{y}^3+p_{20}x+p_{11}xy,$$

(17)

where p₀₀ = −2.057 × 10⁻⁷, p₁₀ = 7.069 × 10⁻⁷, p₀₁ = −0.02261, p₂₀ = −4.141 × 10⁻⁷, p₁₁ = 0.3128. x is the armature stroke, y is the Gaussian error data. Figure 5 shows Gaussian error data fitting.

The MFL permeance of the solenoid valve G₃ can be improved. Now, assuming that P is the FEM data matrix and Q is the DPM data matrix, Er can be written as follows:

Er = P − Q

(18)

If ΔG depicts the permeance error of the sampling point, Er can be derived as follows:

$$Er=\left[\begin{array}{llll}\Delta G_{11}& \Delta G_{12}& \cdots & \Delta G_{1m}\\ \Delta G_{21}& \ddots & & \Delta G_{2m}\\ ⋮& & \ddots & ⋮\\ \Delta G_{n1}& \Delta G_{n2}& \cdots & \Delta G_{nm}\end{array}\right],$$

(19)

where n is the PM segment number, m is the sampling point position. i = 1, 2, …n. j = 1, 2, …m.

The basis function can be defined as f_k(x₁, x₂…x_q), k = 1, 2, 3, …n, where q is the dimension. f_k(x₁, x₂…x_q). The function can be written as:

$$f\left(x_1,x_2,\dots ,x_q\right)=\min \left\{f_k\left(x_1,x_2,\dots ,x_q\right)-Er\left[k;\right]\right\}.$$

(20)

Based on Equation (20), the polynomial function is adopted for the armature stroke, l, ΔG and f(x₁, x₂…x_q). This function is written as follows:

$$\Delta G=p_{00}+p_{10}l+p_{01}{f}^3(x_1,x_2\dots x_q)+p_{20}l+p_{11}lf(x_1,x_2\dots x_q).$$

(21)

Equation (21) can also be written as follows:

$$f_{er}\left(l,x_1,x_2,\dots ,x_q\right)=X\left[\Delta G,l,f\left(x_1,x_2,\dots ,x_q\right)\right]$$

(22)

where f_er(l, x₁, x₂…x_q) is the penalty function for the error data of the sampling point with armature displacement. The ith MFL permeance or soft magnetic resistance can be defined as q_i(x), where x is the unknown quantity. Therefore, w_i(x) can be obtained as follows:

$$w_i(x)=f_{eri}(x)+q_i(x)$$

(23)

where w_i(x) is the improved DPM data, and it is very close to the FEM data.

3. MFL Permeance Prejudge the Error Data Based on Kriging PIP

To improve the overall calculation efficiency of the program, the predictive identification program is applied to prejudge the scatter data before employing the kriging base function.

The basis function can be defined as f(x), $a=(x_1,x_2,\dots x_n)\in R^n$, where f(x) is continuous and differentiable in the field of definition.

$$\nabla f(x)(x-a)<0,$$

(24)

where f(x) is the maximum with x = x_k.

Similarly,

$$\nabla f(x)(x-a)>0,$$

(25)

where f(x) is the minimum with x = x_k.

The extreme function can be analyzed according to Equations (24) and (25), and the gradient of f(x) can be written as follows:

$$\nabla f(x)={\left[\frac{\partial f(x)}{\partial x_1},\frac{\partial f(x)}{\partial x_2},\dots ,\frac{\partial f(x)}{\partial x_n}\right]}^T.$$

(26)

Based on the characteristic of the first derivative, the adjacent points x_i and x_i+1 are the local maxima or minima. Therefore, the adjacent points are used as the criteria to judge the curve of the function. If the adjacent points of the function data are monotonically increasing, and the second derivative product is positive, this shows that the function is unimodal and it has a maximum. Therefore, it can be obtained as follows:

$$\{\begin{array}{l}\frac{\partial f(x)}{\partial x_1}=0,\frac{\partial f(x)}{\partial x_2}=0,\dots ,\frac{\partial f(x)}{\partial x_n}=0\\ f(x_i)<f(x_{i+2})\\ f(x_{i+1})<f(x_{i+3})\\ \frac{{\partial }^2(x){\partial }^{2}f(x)}{{\partial }^2{x}_i{\partial }^2{x}_{i+2}}>0\end{array},$$

(27)

Similarly,

$$\{\begin{array}{l}\frac{\partial f(x)}{\partial x_1}=0,\frac{\partial f(x)}{\partial x_2}=0,\dots ,\frac{\partial f(x)}{\partial x_n}=0\\ f(x_i)>f(x_{i+2})\\ f(x_{i+1})>f(x_{i+3})\\ \frac{{\partial }^2(x){\partial }^{2}f(x)}{{\partial }^2{x}_i{\partial }^2{x}_{i+2}}>0\end{array}.$$

(28)

From Equation (28), the function is unimodal and has a minimum.

Similarly,

$$\{\begin{array}{l}f(x_i)<f(x_{i+2}),f(x_i)>f(x_{i+2})\\ \frac{\partial f(x)}{\partial x_1}=0,\frac{\partial f(x)}{\partial x_2}=0,\dots ,\frac{\partial f(x)}{\partial x_n}=0\\ f(x_{i+1})>f(x_{i+3}),f(x_{i+1})<f(x_{i+3})\end{array}.$$

(29)

If the adjacent points of the function data are monotonically increasing or decreasing, then the function has a multi-crest. As such, the derivative can identify the extremum of the scatter data, and the shape of the multi-crest based on the number of extremum points.

To identify the more complicated scatter data curve, the standard function for numerical matching of the scatter data is applied. When the trend of the curve for the standard function is the same as the trend of the distribution for the scatter data, it can be said that the corresponding kriging basis function is practical for this type of standard function.

Based on the trend of the ΔG_i curve, the Schwefel function has similar characteristics. Figure 6 shows the standard Schwefel function.

The Schwefel function can be written as follows:

$$f(x)=x\sin \left(\sqrt{\left|x\right|}\right)$$

(30)

From Equation (30), the parameters can be modified to have the following:

$$f(x)=-1-2.6x\sin \left(-3.8+15\sqrt{\left|x-0.02\right|}\right)$$

(31)

The Schwefel functions are compared with ΔG₃ and ΔG₄, as shown in Figure 7.

The similarity between the Schwefel function, ΔG₃ and ΔG₄ can be obtained using the Tanimoto method. This method is stated as follows:

$$T(x_1,x_2)=\frac{{\displaystyle \sum x_{1i}{x}_{2i}}}{{\displaystyle \sum x_{1i}^2+{\displaystyle \sum x_{2i}^2}-{\displaystyle \sum x_{1i}{x}_{2i}}}}$$

(32)

By calculation, the similarity of ΔG₃ with the Schwefel function is 0.7654, and the similarity of ΔG₄ with the Schwefel function is 0.7775. The similarities of ΔG with the Schwefel function are further detailed in

Table 2. Schwefel function and ΔG similarity.

Schwefel Similarity	ΔG1	ΔG2	ΔG3	ΔG4	ΔG5	ΔG6	ΔG7
	0.005	0.0662	0.7654	0.7775	0.1674	0.5698	0.0013

.

In Table 2, the similarities of ΔG₃ and ΔG₄ with the Schwefel function are over 0.7, while the similarities of other ΔG₁, ΔG₂, ΔG₅, ΔG₆ and ΔG₇ are the same as the trigonometric function. The similarities of ΔG are compared with the trigonometric function, as shown in

Table 3. Trigonometric function and ΔG similarity.

Trigonometric Similarity	ΔG1	ΔG2	ΔG3	ΔG4	ΔG5	ΔG6	ΔG7
	0.8625	0.921	0.231	0.1922	0.8124	0.7958	0.7642

.

It can be concluded that the similarities which are over 0.7 conform to the corresponding standard function. Hence, the kriging basis function can be selected. When the similarity of the ΔG curve and the Schwefel function is over 0.7, ΔG can be modified by the Gaussian function. When the similarity of the ΔG curve and the trigonometric function is over 0.7, ΔG can be modified by the Fourier function.

4. The Error Correction of MFL Permeance and Soft Magnetic Resistance of Solenoid Valves

To verify the practicality of the proposed method, the kriging model based on the basis function adjustment is used to modify the error of the MFL permeance and soft magnetic resistance data of the solenoid valve magnetic system. Substituting modified data into the DPM, the electromagnetic force can be obtained.

From Figure 8, it can be observed that the unmodified DPM permeance mean error reaches 13.1% (see Figure 2), and the modified DPM mean error reaches 4.7%.

The soft magnetic resistance in la₁ and la₂ segments is taken as an example, without the presence of an excitation current in the coil and at different armature positions. Figure 9 shows the soft magnetic resistance of FEM and the unmodified and modified DPM. The unmodified DPM permeance mean error reaches 9.94%, while the modified DPM permeance mean error reaches 3.7%.

The modified MFL permeance and soft magnetic resistance data are substituted into the DPM. The FEM results are compared with the modified and unmodified DPM results of the force on the armature for a coil excitation of 1800 AT and −1800 AT, as shown in Figure 10. The unmodified DPM mean error is 10.2%, and the modified DPM mean error is 3.8%. The FEM and measurement results mean error is 3%.

Figure 11 shows the flow chart of the improved method for the DPM of solenoid valves based on the kriging basis function PIP. An appropriate function is selected by the PIP for error correction, and by substituting modified data into the DPM, the electromagnetic force can be obtained. Under the same calculation conditions, the output force of one armature displacement point cost 420 s in the FEM, and 56 s in the DPM.

5. Conclusions

To improve the calculation accuracy and efficiency of the solenoid valve DPM, an improved method for the DPM based on the kriging basis function PIP is proposed in this paper. From the experiments and the results, the following conclusions are drawn:

• Based on the characteristics of the kriging basis function curve, the relationship between the kriging basis function and the MFL permeance error data can be obtained, and an appropriate function is selected by contrasting various basis functions with error data curves. Then it is applied to gain error compensation between the FEM and DMP data. The PIP is introduced to prejudge the error data by comparing the standard function to the selected basis function. The modified MFL permeance and the soft magnetic resistance data are then substituted into the DPM of the electromagnetic device to calculate the attraction force.
• The proposed method can effectively improve the calculation accuracy of the solenoid valve electromagnetic system. Compared with the FEM data, the unmodified MFL permeance of the DPM mean error is 13.1%, and the modified MFL permeance of the DPM mean error is 4.7%. The unmodified MFL permeance of the DPM mean error is 9.94%, and the modified MFL permeance of the DPM mean error is 3.7%.
• The results of the DPM solenoid valve electromagnetic system in the case study showed a significant improvement. Particularly, the calculation accuracy improved by reducing the DPM mean error from 10.2% to 3.8%.