Reducing Design Time of Permanent Magnet Volume Minimization for IPMSM for Automotive Applications Using Machine Learning Osaka Prefecture University ◎Yuki Shimizu, Shigeo Morimoto, Masayuki Sanada, and Yukinori Inoue 2021/3/11 SAMCON2021

2 Agenda ⚫ Research background and purpose ⚫ Generation of training data ⚫ Prediction results ⚫ PM volume minimization design ⚫ Conclusion

3 ✓ Motors are used in a variety of products that run on electricity ⚫ Electric Vehicles ⚫ Drones ⚫ Industrial robots ⚫ HVAC ✓ IPMSMs have been widely adopted for such applications *IPMSM: Interior Permanent Magnet Synchronous Motor Stator core Rotor core Permanent magnet About IPMSM

4 Issue with IPMSMs for Automotive Applications ✓ IPMSMs for automotive applications face the problem of a long development period Finite Element Analysis (FEA) Because characteristics computations are performed for each element, characteristics analysis is highly time-intensive Characteristics in a Wide Speed Range To obtain driving characteristics within a speed-torque region, FEA must be performed repeatedly under various current conditions Torque Speed The speed-torque characteristics under various current conditions

5 Presentation Contents ✓ Propose a surrogate model construction method that can accurately predict the speed-torque characteristics of an IPMSM for automotive applications ✓ Use the trained surrogate models and real-coded genetic algorithm to minimize the permanent magnet (PM) volumes ✓ Show that our surrogate models can reduce the design time significantly Structure Predict by a surrogate model Speed Torque Driving characteristics with machine learning

6 Agenda ⚫ Research background and purpose ⚫ Generation of training data ⚫ Prediction results ⚫ PM volume minimization design ⚫ Conclusion

7 Example Fig. Settings for geometrical parameters d 9 d 8 (r 1 ,θ 1 ) d 2 *Polar coordinate with the axis center as the origin Settings for Geometrical Parameters ✓ Set geometrical parameters based on the rotor geometry of the double-layered IPMSM[1] ✓ Generate random numbers within the range of the upper and lower limit values of the geometry, and generate the shapes [1] Y. Shimizu et al., IEEJ Trans. Ind. Appl., Vol. 6, No. 6, pp. 401-408 (2017)

8 Target of Surrogate Model The maximum output control is performed with following equations 𝑇𝑎𝑣𝑔 = 𝑃 𝑛 𝛹𝑎 𝐼𝑎 cos 𝛽 + 1 2 𝐿𝑞 − 𝐿𝑑 𝐼𝑎 2 sin 2𝛽 𝑁𝑙𝑖𝑚 = 𝑉𝑜𝑚 𝛹𝑎 − 𝐿𝑑 𝐼𝑎 sin 𝛽 2 + 𝐿𝑞 𝐼𝑎 cos 𝛽 2 ✓ Predict the motor parameters under each current vector condition with surrogate models N-T characteristics Rotor geometry Predicting N T Motor parameters L d, L q β I a Ψ a I a PM flux linkage d-, q-axis inductance 𝛹𝑎 = 𝑓 𝐼𝑎 , 𝐱𝑔𝑒𝑜𝑚 𝐿𝑑 = 𝑔 𝐼𝑎 , 𝛽, 𝐱𝑔𝑒𝑜𝑚 𝐿𝑞 = ℎ 𝐼𝑎 , 𝛽, 𝐱𝑔𝑒𝑜𝑚 Learn the relationship between rotor geometry, current conditions, and motor parameters 𝐼𝑎 : Armature current 𝛽: Current phase angle 𝐱𝑔𝑒𝑜𝑚 : Geometrical parameter vector 𝑃𝑛 : Number of pole pairs, 𝑅𝑎 : Winding resistance, 𝑉𝑜𝑚 : Induced voltage limit Independent of current phase angle Computing

9 Training Data Generation and Analysis ✓ Generate training dataset for the PM flux linkage and train the surrogate model 1. Generate FEA conditions for Ψ a 2. Train the surrogate model for Ψ a 3. Generate FEA conditions for L d , L q 4. Predict Ψ a and Compute L d 4. Remove the outliers of L d 4. Compute L q 5. Train the surrogate models for L d ,L q Fig. 3. Generation and analysis flowchart for the training data. 1. Generate FEA conditions (Ψ a ) 2. Train a model for Ψ a ( ) ( ) ( ) ~ (0,140) (Arms) ~ ( , ) ( 1,...,11) e j j j geom lwr upr I U x U x x j    =   Surrogate model a geom I    =     x x Features Targets Randomly generate 2,000 cases, where β = 0° ( , ) U a b ：uniform distribution on interval (a,b) cos a o    = a b Prob. Phase current

10 Training Data Generation and Analysis ✓ Generate training dataset for inductances ✓ Compute q-axis inductances 3. Generate FEA conditions (L d , L q ) 4. Compute L q Randomly generate 6,000 cases ( ) ( ) ( ) ~140 (0,1) (Arms) ~ (0,90) (°) ~ ( , ) ( 1,...,11) e j j j geom lwr upr I U U x U x x j      =  i d i q sin cos o q a L I    = Compute L q from FEA results i d i q ~140 (0,1) e I U ~ (0,140) e I U NOT uniform Uniform Inverse transform method 1. Generate FEA conditions for Ψ a 2. Train the surrogate model for Ψ a 3. Generate FEA conditions for L d , L q 4. Predict Ψ a and Compute L d 4. Remove the outliers of L d 4. Compute L q 5. Train the surrogate models for L d ,L q Fig. 3. Generation and analysis flowchart for the training data.

11 Training Data Generation and Analysis 4. Predict Ψ a and compute L d Predict Ψ a using the surrogate model in Step 2 and compute L d Computed L d has the prediction error in Ψ a Computation error of L d becomes large in the case of small i d ( ) ˆ cos cos a a o a d d o a d d d L i L i i           − = − + = = − 　　 Ψ a prediction True Ψ a Prediction error Error of L d ✓ Predict PM flux linkages of each case and compute d- axis inductances using prediction and FEA results 1. Generate FEA conditions for Ψ a 2. Train the surrogate model for Ψ a 3. Generate FEA conditions for L d , L q 4. Predict Ψ a and Compute L d 4. Remove the outliers of L d 4. Compute L q 5. Train the surrogate models for L d ,L q Fig. 3. Generation and analysis flowchart for the training data.

12 Training Data Generation and Analysis ✓ Remove the outliers of the computed d-axis inductances and train the surrogate models for inductances 4. Remove the outliers of L d 5. Train models for L d ,L q Remove the outliers of L d using the Hampel identifier method d L q geom i i     =       x x d L q L ( ) ( ) ( ) ( ) ( ) ( ) med 3 1.4826 med med i i i d d d i i i L L L    − Data No. (1~6000) L d Thre- sholds Remove 1. Generate FEA conditions for Ψ a 2. Train the surrogate model for Ψ a 3. Generate FEA conditions for L d , L q 4. Predict Ψ a and Compute L d 4. Remove the outliers of L d 4. Compute L q 5. Train the surrogate models for L d ,L q Fig. 3. Generation and analysis flowchart for the training data. Features Targets Surrogate model Surrogate model Remove

13 Agenda ⚫ Research background and purpose ⚫ Generation of training data ⚫ Prediction results ⚫ PM volume minimization design ⚫ Conclusion

14 Prediction Accuracies of Surrogate Models ✓ Use Ridge regression, Support vector regression, and XGBoost Ridge SVR XGBoost Predicted (mH) d-axis inductance Analyzed (mH) Ridge SVR XGBoost q-axis inductance Predicted (Wb) Ridge SVR XGBoost Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) PM flux linkage

15 ✓ SVR was most accurate Ridge SVR XGBoost Predicted (mH) d-axis inductance Analyzed (mH) Ridge SVR XGBoost q-axis inductance Predicted (Wb) Ridge SVR XGBoost PM flux linkage Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Ridge; the lowest accuracy because of linear regression SVR; the highest accuracy XGBoost; tendency to overfit the training data r2 (higher is better) train: 0.961 test: 0.963 r2 (higher is better) train: 0.999 test: 0.997 r2 (higher is better) train: 1.000 test: 0.983 SVR: Support Vector Regression r2: the coefficient of determination Prediction Accuracies of Surrogate Models

16 ✓ SVR was most accurate Ridge SVR XGBoost Predicted (mH) d-axis inductance Analyzed (mH) Ridge SVR XGBoost q-axis inductance Predicted (Wb) Ridge SVR XGBoost Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) PM flux linkage SVR: Support Vector Regression r2: the coefficient of determination r2 (higher is better) train: 0.864 test: 0.890 r2 (higher is better) train: 0.961 test: 0.973 r2 (higher is better) train: 0.996 test: 0.963 There are some plots where the predicted values deviate significantly because of the prediction error in Ψ a Ridge; the lowest accuracy because of linear regression SVR; high accuracy even for unknown test data XGBoost; learned perfectly even the prediction errors Prediction Accuracies of Surrogate Models

17 ✓ XGBoost was the most accurate Ridge SVR XGBoost Predicted (mH) d-axis inductance Analyzed (mH) Ridge SVR XGBoost q-axis inductance Predicted (Wb) Ridge SVR XGBoost Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) PM flux linkage Analyzed (mH) SVR: Support Vector Regression r2: the coefficient of determination r2 (higher is better) train: 0.918 test: 0.918 r2 (higher is better) train: 0.998 test: 0.994 r2 (higher is better) train: 1.000 test: 0.998 Ridge; the lowest accuracy because of linear regression SVR; not able to accurately predict the inductance variation XGBoost; the highest accuracy for both the training and test data Prediction Accuracies of Surrogate Models

18 ✓ SVR and XGBoost that can represent nonlinearity are accurate Ridge SVR XGBoost Predicted (mH) d-axis inductance Analyzed (mH) Ridge SVR XGBoost q-axis inductance Predicted (Wb) Ridge SVR XGBoost Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (Wb) Analyzed (Wb) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Predicted (mH) Analyzed (mH) Prediction Accuracies of Surrogate Models PM flux linkage

19 ✓ Proposed method reduced the RMSEs by 68.1% and 67.7% Results of Speed-Torque Characteristics Fig. 5. Prediction results for speed-torque characteristics (a) Motor parameter prediction I em = 134A I em = 30A Torque (Nm) Speed (min-1) Proposed method I em = 134A I em = 30A Torque (Nm) Speed (min-1) Conventional method (b) Direct torque-and-speed prediction *RMSE: Root Mean Square Error Initial shape RMSE (lower is better) 134A: 3.88 Nm 30A: 1.82 Nm RMSE (lower is better) 134A: 12.14 Nm 30A: 5.65 Nm

20 Agenda ⚫ Research background and purpose ⚫ Generation of training data ⚫ Prediction results ⚫ PM volume minimization design ⚫ Conclusion

21 Minimizing PM Volume by Real Number GA ✓ Minimize the permanent magnet volumes with a combination of the surrogate models and real number genetic algorithm Start Create initial population Unsatisfied End Satisfied Select parents Generate children by UNDX-m Modify children to satisfy geometric constraint Calculate fitness function Select top children Termination conditions Fitness function 𝑓𝑖𝑡𝑛𝑒𝑠𝑠 = 𝑉(𝐱 𝑔𝑒𝑜𝑚 ) 𝑉 𝑖𝑛𝑖𝑡 + 𝑃𝑇(𝐱𝑔𝑒𝑜𝑚 ) + 𝑃𝐴𝐷(𝐱𝑔𝑒𝑜𝑚 ) 𝑉(𝐱𝑔𝑒𝑜𝑚 ): PM volume 𝑉𝑖𝑛𝑖𝑡 : PM volume of initial shape (100cm3) 𝑃𝑇 𝐱𝑔𝑒𝑜𝑚 , 𝑃𝐴𝐷 (𝐱𝑔𝑒𝑜𝑚 ): Penalty functions Torque constraints 𝑃 𝐱geom = max 0,197 × 1.03 − 𝑇𝑝𝑟𝑒𝑑1 + max 0,40 × 1.03 − 𝑇𝑝𝑟𝑒𝑑2 𝑇𝑝𝑟𝑒𝑑1,2 : Torque prediction N T P B 11000min-1 P A 40Nm 197Nm 3000min-1 Penalty when not satisfied Prediction T pred1 T pred2 Penalty function

22 Applicability Domain Constraints ✓ Set the applicability domain constraints with OCSVM Applicability domain constraints ・Set the applicability domain using OCSVM ・Use geometrical parameters of training dataset x geom (1) x geom (2) ：training data Decision boundary Applicability domain Penalty function ( ) ( ) ( ) max 0, AD geom OCSVM geom P f = − x x 𝑓𝑂𝐶𝑆𝑉𝑀 : output of OCSVM （negative when outside AD） *OCSVM: One-Class Support Vector Machine *Applicability domain: Area where the accuracy of a model is guaranteed Start Create initial population Unsatisfied End Satisfied Select parents Generate children by UNDX-m Modify children to satisfy geometric constraint Calculate fitness function Select top children Termination conditions

23 Prediction Accuracy for Best Individuals ✓ Ran the optimization design five times ✓ Prediction error rates of all models are less than 5% ✓ All the best individuals satisfied the constraints Fig. Torque prediction results of the best individuals at required drive points. 0 50 100 150 200 250 1st 2nd 3rd 4th 5th 1st 2nd 3rd 4th 5th PA PB Torque (Nm) 予測 FEA @3000 min-1 (P A ) @11000 min-1 (P B ) Reqd: 197Nm Reqd: 40Nm N T P B 11000 min-1 P A 40Nm 197Nm 3000 min-1 Pred.

24 0 500 1000 1500 2000 2500 3000 FEAのみ 提案法 FEAのみ 提案法 FEAのみ 提案法 FEAのみ 提案法 FEAのみ 提案法 1st 2nd 3rd 4th 5th Computation time (hour) 1762 2744 1789 1898 1121 78.5 79.1 78.6 78.5 78.0 Comparison of Optimization Design Time ✓ The total computation time of the proposed method can be reduced to less than 1/14th to 1/35th Fig. Computation time for optimization design Convergent generation 128 gen. 200 gen. 130 gen. 138 gen. 81 gen. × 1 22 × 1 35 × 1 23 × 1 24 × 1 14 1st 2nd 3rd 4th 5th Only FEA Proposed method Only FEA Proposed method Only FEA Proposed method Only FEA Proposed method Only FEA Proposed method

25 Agenda ⚫ Research background and purpose ⚫ Generation of training data ⚫ Prediction results ⚫ PM volume minimization design ⚫ Conclusion

26 Conclusion ✓ Proposed a surrogate model construction method that can accurately predict the speed-torque characteristics ✓ Following methods were the most accurate for each parameter ⚫ Permanent magnet flux density; SVR ⚫ d-axis inductance; SVR ⚫ q-axis inductance; XGBoost ✓ Proposed shapes that reduced the PM volumes while satisfying the required torques using surrogate models and real-coded genetic algorithm ✓ The proposed method took less than 1/14th to 1/35th of the optimization design time compared to FEA-only design