210311_samcon

Transcript

1. 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
   

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

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3. 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
   

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4. 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
   

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5. 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
   

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

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7. 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)
   

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8. 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
   

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9. 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
   

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10. 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.
    

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11. 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.
    

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12. 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
    

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

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14. 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
    

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15. 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
    

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16. 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
    

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17. 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
    

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18. 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
    

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19. 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
    

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

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21. 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
    

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22. 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
    

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23. 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.
    

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24. 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
    

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

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26. 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
    

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