Optimal Design of IPMSM for FCEV Using Novel Immune Algorithm Combined with Steepest Descent Method

Abstract

In this paper, the Novel Immune Algorithm (NIA) is proposed for an optimal design of electrical machines. By coupling the conventional Immune Algorithm and Steepest Descent Method, the NIA can perform fast and exact convergence to both global solutions and local solutions. Specifically, the concept of an antibody radius is newly introduced to improve the ability to navigate full areas effectively and to find new peaks by excluding already searched areas. The validity of the NIA is confirmed by mathematical test functions with complex objective function regions. The NIA is applied to an optimal design of an interior permanent magnet synchronous motor for fuel cell electric vehicles and to derive an optimum design with diminished torque ripple.

1. Introduction

Recently, as the problems of energy crisis and environmental pollution have become more serious, electric vehicles, such as fuel cell electric vehicles (FCEVs), are an increasing portion of the market, as they are cleaner than conventional internal combustion engine vehicles [1,2]. As FCEVs require high-power density and high efficiency, the interior permanent magnet synchronous motor (IPMSM) has been widely used for motors for FCEVs [3,4,5]. IPMSM is a motor in which permanent magnets are embedded in the rotor core. Owing to such structural characteristics, the IPMSM has either higher average torque or superior performance at high speed compared with a surface-mounted permanent magnet synchronous motor [6,7].

An engineer should design a motor for FCEV to have lower torque ripple for the purpose of comfortable driving and improved control stability, because torque ripple causes vibration and noise in the motor [8,9]. Therefore, this paper aims at the optimal design of an IPMSM for FCEVs to have lower torque ripple and satisfying rated torque.

The performance of an IPMSM can be improved by optimizing the design of the permanent magnet structure, and the objective function of torque ripple has a complex shape with many local minima. To analyze the load condition of the IPMSM accurately, the finite element method (FEM) is needed, as the IPMSM has nonlinear magnetic saturation [10]. However, the FEM requires an enormous amount of time and effort, especially when applied to search multiple peaks at multi-modal problems.

To address this problem, the conventional Immune Algorithm (IA), which mimics the human immune system, was devised [11]. However, like other stochastic algorithms, the IA needs many function calls and iterations to find the accurate peak. To resolve problems regarding computational cost, this paper proposes a Novel Immune Algorithm (NIA). The NIA is a hybrid algorithm that consists of the conventional IA and the Steepest Descent Method (SDM), which is a kind of deterministic method. In addition, the concept of an antibody radius is newly adopted for conventional IA, enhancing the ability to search entire areas efficiently. As a result, the NIA can significantly reduce the number of function calls to find multiple peaks accurately.

The NIA, conventional IA, and Niching Genetic Algorithm (NGA) were applied to two mathematical test functions with many local peaks, and the outstanding performance of the NIA was validated by comparison with the IA and NGA results. Finally, the proposed algorithm was applied to an optimal design of an IPMSM for FCEVs and to derive an optimum design that minimizes torque ripple and satisfies the rated torque and low cogging torque.

2. Conventional Immune Algorithm

The human immune system detects invading antigens such as viruses or germs that enter the human body and remove antigens by generating antibodies corresponding to the antigens. The IA is an algorithm that mimics this system, and it is applied for the optimization of the objective function. Among the many stochastic algorithms, the IA shows superior performance when applied to complex optimization problems with multiple local solutions [12]. Hence, this paper proposes an algorithm that is based on the IA. The IA accomplishes optimization using concepts such as antibody, memory cell, affinity, and expectation. The flow chart of the IA is shown in Figure 1, and the flow of the IA is described below.

• Problem definition: Objective function and restrictions are defined.
• Generation of antibody: Initial antibodies are generated randomly in the problem region and initial memory cells are selected according to antibody-antigen affinity. When the algorithm is applied to find peaks, the antibody-antigen affinity of an antibody (v) is defined as follows:

  $$af{f}_v=cu{r}_v$$

  (1)

  If the aim of the algorithm is minimization, antibody-antigen affinity is defined as:

  $$af{f}_v=-cu{r}_v$$

  (2)

  where cur_v is the objective function value.
• Expectation calculation: The expectation of each antibody is calculated and antibodies with low expectations are removed. The expectation of antibody (i) is calculated as follows:

  $$e_i=af{f}_i/c_i$$

  (3)

  where c_i is the concentration, which is defined as the number of antibodies having the similar value as antibody (i), divided by the total number of antibodies. Figure 2 and

  Table 1. Calculation result of concentration and expectation.

  Point	Concentration	Expectation
  A	1/4	2 × 4/1 = 8
  B	2/4	3 × 4/2 = 6
  C	1/4	5 × 4/1 = 20
  D	2/4	3 × 4/2 = 6

  show how expectation works in the algorithm. When the value of B and D is considered as similar, the concentration of A and C is low, and concentration of B and D is relatively high. Therefore, the expectation is calculated as listed in Table 1, and you can figure out that the local peak A is more likely to survive than B or D.
• Crossover, mutation: The antibody group is updated by replacing removed antibodies. Mutants are randomly generated in the entire problem area and crossover is done by the antibodies remaining from step 3.
• Affinity calculation: The antibody-antigen affinity of the new antibody group is calculated with (3). Additionally, the antibody-antibody affinity of the memory cell is calculated as follows:

  $$af{f}_{v,w}=1/(1+dis{t}_{v,w})$$

  (4)

  where dist_v,w is the distance between newly generated antibody (v) and antigen (w). Antigens are entities involved in the memory cells. With this step, antibodies near the memory cells are candidates of elimination, enhancing the ability to find unknown areas.
• Memory cell renewal: Cells with high antibody-antibody affinity in memory cells are removed, and antibodies with high antibody-antigen affinity are added to the memory cell.
• Convergence check: If the value of the memory cell group is not changed by N iteration, it is judged that the convergence condition is satisfied, and the algorithm terminates and the optimal solutions are saved to the memory cell. If the convergence condition is not satisfied, each step is repeated from step 4, and the algorithm proceeds.

3. Proposed Algorithm

Even though the conventional IA performs better in multi-modal optimization problems than other heuristic algorithms, the IA also has a problem that requires many function calls to converge to an accurate peak like other stochastic algorithms. To address the problem with computational cost, the NIA is proposed. The NIA improves the performance of the conventional IA using three concepts.

3.1. Multi-Jittered Sampling

As shown in Figure 3a, a conventional IA generates initial solutions randomly in the problem domain. The NIA adopts Multi-Jittered Sampling (MJS) to generate initial solutions. At first, the MJS divides the problem region into N × M subregions, and generates one sample for each subregion. In Figure 3b, the red lines divide the problem region into subregions and only one sample exists in each subregion, where N and M are both 4. The subregions are subdivided into N × M grid, marked with black lines, to place only one sample in one subregion. Additionally, there are no other samples in each row and column. In other words, initial samples are scattered throughout the entire region and have randomness too, enhancing the ability to find local minima. By adopting the MJS method, a more superior set of initial samples can be obtained, compared with the conventional method of generating samples on the entire region randomly.

3.2. Hybrid Steepest Descent Method

The IA, which is a kind of stochastic optimization method, can find global and local solutions. However, the IA requires a lot of function calls to converge to an accurate optimal solution. On the other hand, the SDM, one of the deterministic optimization methods, finds the exact optimal solution quickly using gradient information along the search direction. However, in the complex objective function domain with many local minima, the SDM has the problem of premature convergence to a local solution, instead of a global solution [13]. The NIA combines the SDM with the IA to compensate for the shortcomings and maximize the advantages. The SDM is applied to memory cells and new antibodies, allowing them to converge directly to local peaks nearby. As a result, the NIA can find accurate peaks with a remarkably reduced number of function calls and can search both global and local solutions.

3.3. Antibody Radius

The NIA utilizes the concept of an antibody radius to maximize its ability to navigate unexplored area. The antibody radius is the distance between a memory cell and the closest memory cell. New antibodies, or mutations, are generated outside the antibody radius and the radius is adjusted automatically when a new memory cell is added.

Figure 4 intuitively shows how the new antibody is added and how the antibody radius is adjusted. On the left, there is a memory cell with the antibody radius set beforehand, and the new antibody is generated outside the circle. Using the SDM, the new antibody converges to the peak nearby and the antibody radius is adjusted automatically.

3.4. Flow Chart of the NIA

The flow chart of the NIA is shown in Figure 5. First, the objective function and constraints are defined, and initial samples are created using MJS. The red box shows how the antibody group is updated as the algorithm progresses. Among the initial samples, antibodies with high value are added to the memory cell group. The antibody-antibody affinity of the antibody group is calculated with “(4)” and low affinity antibodies are removed. Then the expectation is calculated according to “(3)” and high expectation antibodies are differentiated into the memory cell group.

The blue box is the part that handles the memory cell group. Initial memory cells or newly differentiated memory cells find local peaks using the SDM. If a new peak is found, it is added to the memory cell. If it converges to a peak that has already been found, it is removed. The antibody radius is automatically adjusted to the distance from the nearest memory cell. If the convergence condition is not satisfied, the antibodies with high expectations are added to the memory cell, and the algorithm is repeated.

3.5. Verification of the Proposed Algorithm

In order to verify the reliability and superiority of the algorithm, conventional IA, NGA, and the NIA are applied to the optimization of two mathematical functions. Test function 1 is defined as follows:

$$f_1(x,y)={\displaystyle \sum _{i=1}^{n{p}_1}\frac{b_i}{1+[{(x-x_i)}^2+{(y-y_i)}^2]/a_i}}$$

(5)

and test function 2 is given as:

$$f_2(x,y)={\displaystyle \sum _{k=1}^{n{p}_2}\frac{60}{1+[{(x-x_k)}^2+{(y-y_k)}^2]/a_k}}$$

(6)

where (x_i, y_i) and (x_k, y_k) are the coordinates of the peaks of the test functions, and a_i, a_k, and b_i are the values related to the magnitude of the peaks. np₁ and np₂ are the values that determine the number of peaks of each function and the values are 11 and 36, respectively. As shown in Figure 6, the two test functions have complex objective domains with 11 and 36 peaks, respectively. The test was repeated 100 times in total and it was judged to be successful when the found peak rates were more than 95% of the actual peak value. The test results are listed in

Table 2. Performance comparison between the NIA, IA, and NGA.

Test Function 1 [11 Peaks]	Number of Function Call	Success Rate [%]
IA	2060.0	58.45
NGA	1650	89.09
NIA	1365.2	99.45
Test Function 2 [36 Peaks]	Number of Function Call	Success Rate [%]
IA	3390.0	84.0
NGA	3140	94.44
NIA	1878.1	99.64

. As the NIA combines with the deterministic method, SDM, it shows a tremendous improvement in the success rate compared to the conventional IA. In terms of the number of function calls, the NIA is superior to the IA, as the NIA generates samples effectively considering the antibody radius. Moreover, the NIA seems to be superior compared with the NGA, as the NIA adjusts the radius automatically. It is obvious that the NIA shows outstanding performance in both accuracy and computational cost, compared with a conventional IA and NGA.

4. Application to the Optimal Design of IPMSM for FCEV

4.1. Analysis Model and Design Variables

Electrical vehicles have been increasing as a portion of the commercial market over the past few decades for the purpose of reducing air pollution in cities [14]. For the commercialization of FCEVs, high torque density and efficiency are required to increase the power transmission and the saving of energy because the motors solely drag a car without the engine and only use the energy from a battery [15]. The IPMSM is selected as the model for the FCEV motor, because the IPMSM has the advantages of high torque density and efficiency [16,17]. However, due to saliency, the IPMSM suffers from high torque ripple, causing undesired noise and vibration during its operation and reducing driving impression and control stability [18]. As a motor for FCEVs should have lower torque ripple, the purpose of design is to diminish torque ripple [19].

Torque ripple of IPMSM can be reduced depending on the shape of the rotor or placement of the permanent magnets [20,21]. Therefore, as shown in Figure 7, the design variables are determined as the angle of each magnet and the length between two magnet layers to change the shape of the permanent magnets inside the rotor core. The values related to the rest of the shape are fixed and set the constraints that the magnets are not deviated from the rotor core and the magnets in each layer are not overlapped. The length of the bridge and center post are kept uniform in order to guarantee mechanical stability, even if design parameters are changed. Detailed specifications and requirements of an objective motor are listed in

Table 3. Specifications and requirements of an objective motor.

Parameter	Unit	Value
Pole number	-	6
Slot number	-	27
Rated torque	[Nm]	310
Rated power	[kW]	97.4
Rotation speed	[rev/min]	3000
Stator inner/outer diameter	[mm]	172/240
Rotor inner/outer diameter	[mm]	50/170
Air gap	[mm]	1
Stacking length	[mm]	240
Stator and rotor core material	-	JFE steel 35JN230
Permanent magnet material	-	NEOMAX-42

, and the available range of each variable is indicated in

Table 4. Range of design variables.

Variable	Unit	Range
θ1	[degree]	125–140
θ2	[degree]	100–150
length	[mm]	12–18.5

. In detail, the number of poles and slots of the initial model is 6 and 27, respectively. The rotor has a multipolar structure and a multi-layered magnet to satisfy the high-power density and high torque density requirements of FCEVs. For the stator, the number of slots was selected through cogging torque comparison of the other pole and slot combinations [22]. For the practical design, parameters that related to structure of the stator and rotor were selected according to [22].

The proposed NIA is applied for an optimal design of the IPMSM for FCEVs. Whenever new antibodies are generated in the algorithm, shape design of the IPMSM reflecting each parameter value is performed and the FEM, using an analysis tool, is executed to obtain results.

For the IPMSMs, the magnitude of the average torque changes with the current phase angle. Figure 8 shows the average torque of the initial model and the current phase angle is changed in 2° intervals from 0° to 90°. When the rotor shape of an IPMSM is changed, the current phase having maximum average torque changes. Therefore, the algorithm is set to find the current phase with maximum torque whenever the design changes.

4.2. Optimization Results

By applying FEM analysis to the NIA, three optimal models with diminished torque ripple are derived. Figure 9 shows the objective region, which is torque ripple, according to the variables and has nonlinear shape with multiple solutions. The final design is selected considering not only torque ripple but also average torque and cogging torque. In

Table 5. Variables of initial and optimal models.

Model	Initial Model	Model 1	Model 2	Model 3
θ1 [degree]	127.67	130.77	141.50	131.33
θ2 [degree]	122.22	111.33	124.00	130.08
length [mm]	13.0	18.5	12.0	17.0

, the obtained values of design variables for the initial and the candidate models are listed.

Table 6. Performance comparison of initial and optimal models.

Model	Initial Model	Model 1	Model 2	Model 3
Torque Ripple [%]	19.58	2.68	4.10	4.36
Average Torque [Nm]	322.39	304.97	323.71	315.49
Cogging Torque [Nm]	24.11	6.23	21.52	2.92
AC phase [degree]	43	45	39	44

numerically shows torque ripple, average torque, cogging torque, and current phase having maximum torque of each model.

In detail, in terms of torque ripple, model 1 is a better design than the other candidates and it also has a relatively low cogging torque. However, model 1 does not meet the rated torque requirement. Model 2 has a relatively small torque ripple and the highest average torque, but its cogging torque value is too high. Model 3 seems to be the most suitable design when considering torque ripple, average torque, and cogging torque. Therefore model 3 is decided as the optimum design.

4.3. Result Analysis of Optimum Design

Figure 10 shows the permanent magnet structure of initial and optimum design of the IPMSM. Magnet usage of the optimum model is 6.4% lower than the initial model, however, the average torque is only 2% smaller than the initial model. Figure 11a shows the torque waveform of the initial model and optimized model. The average torque of the initial model is 322.39 Nm, and of the optimized model is 315.49 Nm, which is reduced by 2.1% compared to the initial model. However, the torque ripple of the optimized model is decreased by 77.73% compared to the initial model, with the optimal model being 4.36% and the initial model being 19.58%. Figure 11b shows the cogging torque comparison between the initial model and optimum model. The cogging torque of the initial model is 24.11 Nm and of the optimized model is 2.92 Nm, which is 87.89% less than the initial model.

The magnetic flux density of the optimum models is displayed in Figure 12. The maximum values of the initial model and optimum model are 2.61 T and 2.47 T, respectively, and the optimum model shows 22.95% lower saturation than that of the initial model.

Table 7. Power characteristics of the initial model and optimum model.

Model	Initial Model	Optimum Model
Iron loss [W]	507.8	473.5
Copper loss [W]	2162.7	2162.7
Total loss [W]	2670.5	2636.1
Input power [kW]	103.9	101.7
Output power [kW]	101.3	99.1
Efficiency [%]	97.4	97.4

represents power characteristics of both models, and there is no difference for the efficiency.

For the IPMSM, bilateral and central bridges suffer maximum mechanical stress and are most vulnerable to breakage during high-speed rotation [23,24]. Therefore, analysis of mechanical stress at high speed should be considered when designing the IPMSMs.

Table 8. Setting for the mechanical stress analysis.

Parameter	Value
Young’s modulus (Core/PM)	210/120 [GPa]
Poisson’s ratio (Core/PM)	0.3/0.3
Density (Core/PM)	7850/8400 [kg/m3]
Rotation speed	3000/6000 [rpm]
Yield stress	250 [MPa]

shows parameters of mechanical stress analysis. Maximum rotational speed of the rotor is set as 6000 rpm, which is twice the rated rotation speed, to ensure stability against breakage at high speed. Figure 13 shows the result of mechanical stress analysis at the maximum speed. The maximum Von Mises stress of maximum speed and rated speed are 114.74 MPa and 28.5 MPa, respectively. The results show that the optimal design does not exceed the yield stress value of 230 MPa, and is safe from breakage even at high speed.

5. Conclusions

This paper suggests the NIA for multimodal optimization. By applying the proposed algorithm and the conventional IA to the test function, outstanding performance of the NIA was verified. The NIA is applied to the optimal design of the IPMSM for FCEVs to diminish torque ripple, considering average torque and cogging torque. By combining the FEM with the NIA, the optimal design result is derived within a significantly reduced computation time, and the torque ripple of the optimized model was decreased by 77.73% compared to the initial model. The proposed algorithm is expected to be widely used in the design of other electrical machines that need to consider various aspects.