Numerical Estimation of Switched Reluctance Motor Excitation Parameters Based on a Simplified Structure Average Torque Control Strategy for Electric Vehicles
Abstract
:1. Introduction
2. The Proposed Simple ATC Strategy
3. Methodology
3.1. The Optimization Problem
3.2. Solution Method
3.2.1. Machine Modeling
3.2.2. The Limits of Switching Angles (θon, θoff)
- (1)
- Fitting of phase inductance L(i, θ) against the rotor position (θ) in the minimum inductance zone. In this zone, the inductance is only a function of θ. Hence, a simple exponential function is enough for the fitting . The coefficients a, b, and c are the fitting coefficients,
- (2)
- Calculation of parameter (kb) as the derivative of inductance ,
- (3)
- Determination of the initial turn-on angle ,
- (4)
- Estimation of the effective values Leff(i, θ) and kb-eff(i, θ) as the average values of L(i, θ) and kb, respectively, over the interval [θon-initial, θm],
- (5)
- Using Equation (14) to determine the best analytical solution for the turn-on angle.
3.2.3. Estimation of the Rated Torque for a Given Current Level
3.2.4. Estimation of Base Values
3.2.5. Estimation of Weight Factors
3.2.6. The Optimum Solution for Switching Angles
- (1)
- Define the operating point by the desired speed (ω) and reference current (iref),
- (2)
- Use Equation (14) to determine the best analytical turn-on angle (θon-analy),
- (3)
- Choose the margins ΔθA and ΔθB. Then, define the minimum and maximum limits (θA and θB) for the turn-on angle (θon),
- (4)
- Define the minimum and maximum limits ( and ) for the turn-off angle. For the tested 8/6 SRM, = θon + 15°, and = 25°,
- (5)
- Set θon = θA (the starting point) and θoff = (the starting point),
- (6)
- Run the simulation model of SRM, estimate the required indices (torque ripples and efficiency), and save the data,
- (7)
- Increase θoff by the desired resolution (Δθoff). Hence, θoff = θoff + Δθoff. Δθoff is set to 0.2°,
- (8)
- Repeat step 6 till θoff = ,
- (9)
- Increase θon by the desired resolution (Δθon). Hence, θon = θon + Δθon. Δθon is set to 0.2°,
- (10)
- Repeat steps 7 to 9 till θon = θB,
- (11)
- Estimate the base values (Trb and ηb) from the saved data using Equations (17) and (18),
- (12)
- Choose the weight factors (wr and wη) according to the desired optimization level,
- (13)
- Calculate the objective function (Fobj) using Equation (1),
- (14)
- Define the optimum switching angles that correspond to the minimum objective function.
4. Simulation Results and Discussion
4.1. The Steady-State Performance
4.2. The Dynamic Performance
4.2.1. Sudden Change in Reference Speed and Load Torque
4.2.2. Acceleration and Deceleration with Electric Vehicle Loading
5. Experimental Verification
5.1. Model Verification
5.2. Quantitative Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Symbol | Definition | Unit |
B | The combined rotor and load viscous friction coefficient | Kg.m2/s |
Fobj | The objective function | |
Iav | The average supply current | A |
ik | The phase current of kth phase | A |
iref | The reference current | A |
is | The instantaneous supply current | A |
J | The combined rotor and load inertia coefficient | Kg.m2 |
Lk | The phase inductance of kth phase | H |
Lu | The unaligned/minimum inductance | H |
n | The number of sampling point under calculation | |
q | The number of motor phases | |
R | The phase resistance | Ω |
Tav | The average torque | N.m |
Te | The total electromagnetic torque | N.m |
Tk | The phase torque of kth phase | N.m |
TL | The load torque | N.m |
Tmax | The maximum value of instantaneous motor torque | N.m |
Tmin | The minimum value of instantaneous motor torque | N.m |
Tr | The torque ripple | N.m |
Trated | The rated motor torque for a given value of reference current and speed | N.m |
Trb | The base value of torque ripple | N.m |
Tref | The reference torque | N.m |
Ts | The sampling period | s |
VDC | The DC supply voltage | V |
vk | The phase voltage of kth phase | V |
wr | The weight factor of torque ripple | |
wη | The weight factor of efficiency | |
θ | The rotor position | Deg. |
θA | The minimum limit of the turn-on angle | Deg. |
θB | The maximum limit of the turn-on angle | Deg. |
θm | The angle where rotor poles start to overlap with stator poles | Deg. |
θon | The turn-on angle | Deg. |
The analytically obtained turn-on angle | Deg. | |
θoff | The turn-off angle | Deg. |
The minimum limit of the turn-off angle | Deg. | |
The maximum limit of the turn-off angle | Deg. | |
λk | The phase flux linkage of kth phase | Wb |
λ(0) | The initial flux-linkage | Wb |
η | The efficiency | |
ηb | The base value of efficiency | |
ω | The angular velocity of rotor | Rad/s |
τ | The time of one electric cycle | s |
ΔθA | The lower margin of the turn-on angle | Deg. |
ΔθB | The higher margin of the turn-on angle | Deg. |
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Geometry Parameter | Value |
---|---|
Output power (kW) | 4.0 |
Rated voltage (v) | 600 |
Rated speed (r/min) | 1500 |
Phase resistance (Ω) | 0.642 |
Air-gap length | 0.4 |
Height of rotor/stator pole | 18.1/29.3 |
Stator outside diameter | 179.5 |
Shaft/Bore diameters | 36/96.7 |
Rotor/stator pole arc | 21.5°/20.45° |
Stack length | 151 |
Turns per pole | 88 |
Parameter | Experimental | Simulation |
---|---|---|
Tav (N.m) | 4.1519 | 3.9661 |
Tr (%) | 72.9793 | 71.2504 |
η (%) | 80.12 | 82.78 |
RMSE of current (A) | 0.2007 | |
RMSE of torque (N.m) | 0.3609 | |
Maximum current error (A) | 0.8018 | |
Maximum torque error (N.m) | 1.1114 | |
Turn-on angle θon (°) | 3 | |
Turn-off angle θoff (°) | 19 | |
Reference current (A) | 6 | |
Hysteresis current band (A) | 0.3 |
Parameter | At 627 r/min | At 806 r/min | ||
---|---|---|---|---|
Conventional | Proposed | Conventional | Proposed | |
Tav (N.m) | 6.459 | 6.388 | 4.834 | 4.827 |
IRMS (A) | 5.488 | 4.585 | 4.706 | 4.097 |
Tr (%) | 68.763 | 30.822 | 72.821 | 36.254 |
η (%) | 78.911 | 77.575 | 79.224 | 78.540 |
Tav / IRMS (N.m/A) | 1.177 | 1.393 | 1.027 | 1.178 |
θon (°) | 4.65 | 6.8 | 5.1 | 5.6 |
θoff (°) | 19.28 | 23 | 19.55 | 22 |
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Hamouda, M.; Abdel Menaem, A.; Rezk, H.; Ibrahim, M.N.; Számel, L. Numerical Estimation of Switched Reluctance Motor Excitation Parameters Based on a Simplified Structure Average Torque Control Strategy for Electric Vehicles. Mathematics 2020, 8, 1213. https://doi.org/10.3390/math8081213
Hamouda M, Abdel Menaem A, Rezk H, Ibrahim MN, Számel L. Numerical Estimation of Switched Reluctance Motor Excitation Parameters Based on a Simplified Structure Average Torque Control Strategy for Electric Vehicles. Mathematics. 2020; 8(8):1213. https://doi.org/10.3390/math8081213
Chicago/Turabian StyleHamouda, Mahmoud, Amir Abdel Menaem, Hegazy Rezk, Mohamed N. Ibrahim, and László Számel. 2020. "Numerical Estimation of Switched Reluctance Motor Excitation Parameters Based on a Simplified Structure Average Torque Control Strategy for Electric Vehicles" Mathematics 8, no. 8: 1213. https://doi.org/10.3390/math8081213