# Virtual Constant Signal Injection-Based MTPA Control for IPMSM Considering Partial Derivative Term of Motor Inductance Parameters

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## Abstract

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## 1. Introduction

_{e}/dβ). As the value of dq-axis inductance will change with the change in current amplitude, temperature and other factors, with the increase in current amplitude, the dq-axis inductance value changes more seriously. This indicates that ignoring the partial conductance value of dq-axis inductance to dq-axis current will lead to the deviation of dT

_{e}/dβ extracted by VSCIM from the actual value, and the greater the current amplitude, the greater the deviation. Therefore, when the current amplitude is large, the dq-axis current reference value obtained by VSCIM deviates from the MTPA point, and the MTPA control precision becomes worse.

## 2. MTPA Control Based on Virtual Constant Signal Injection Method

#### 2.1. Mathematical Model of dq-Axis of IPMSM

_{d}and u

_{q}are the dq-axis components of the motor stator voltage, respectively; i

_{d}and i

_{q}are the dq-axis components of the motor stator current, respectively; R is the motor stator resistance; L

_{d}and L

_{q}are the equivalent inductances of the dq-axis of the motor, respectively; ψ

_{f}is the permanent magnet flux linkage; ω

_{e}is the electrical angular velocity of the motor; T

_{e}is the torque of the motor; p is the number of pole pairs of the motor.

#### 2.2. The Principle of MTPA Control

_{s}is the magnitude of the current vector; β is the angle between the current vector and the q-axis.

_{s}is constant, there is an optimal current angle to maximize the torque, which is called the MTPA angle β

_{MTPA}.

_{d_MTPA}and i

_{q_MTPA}as the dq-axis current reference value, respectively.

#### 2.3. Virtual Constant Signal Injection Method

_{e}calculated by Equation (3) and the actual torque, and then the β

_{MTPA}obtained by dT

_{e}/dβ = 0 will have a deviation from the actual MTPA angle, so that the torque generated at this angle is not the maximum value. Fortunately, VCSIM has strong robustness to motor parameter changes in the process of realizing MTPA control.

_{e}/dβ can be expressed as

_{e}/dβ can be further expressed as

_{e}/dβ, we must first accurately obtain ∂T

_{e}/∂i

_{d}and ∂T

_{e}/∂i

_{q}. Defining the constant value signal A, and injecting A into i

_{d}and i

_{q}in Equation (3), respectively, the torque can be expressed as

_{d}, i

_{q}) can be expressed as

_{d}term is not included in ∂T

_{e}/∂i

_{d}, and the i

_{q}term is not included in ∂T

_{e}/∂i

_{q}, so the second-order and above partial derivatives in Equation (11) are all equal to 0. Equation (11) can be expressed as

_{e}/∂i

_{d}and ∂T

_{e}/∂i

_{q}obtained by Equations (14), (16) and (17) depends on the accuracy of the dq-axis current, dq-axis voltage and rotational speed. Due to the change in rotor position caused by the delay in the control cycle, there will also be a deviation between the voltage output by the current controller and the voltage actually applied to the motor terminal, and the deviation increases with the increase in the rotation speed. Therefore, the output voltage of the current controller needs to be corrected as follows before using it in Equations (15)–(17) [21].

_{do}and u

_{qo}are the dq-axis voltage output by the current controller, respectively; T

_{s}is the control period; k = 2sin(0.5ω

_{e}T

_{s})/(ω

_{e}T

_{s}).

_{e}/dβ can be obtained. The dq-axis current reference value at the MTPA point is obtained in the following manner:

- (a)
- D-axis reference current i
_{d_ref}

_{e}/dβ to generate i

_{d_ref}. When dT

_{e}/dβ ≠ 0, i

_{d_ref}is adjusted under the action of the integrator until dT

_{e}/d

_{β}= 0. At this time, i

_{d_ref}is the d-axis current at the MTPA point.

- (b)
- Q-axis reference current i
_{q_ref}

_{e}(i

_{d},i

_{q})/i

_{q}, and T

_{e_ref}is the reference torque.

## 3. Error Analysis and Error Compensation Method of Virtual Constant Signal Injection Method

#### 3.1. Error Analysis

_{d}and L

_{q}are regarded as constants independent of i

_{d}and i

_{q}when VCSIM obtains ∂T

_{e/}∂i

_{d}and ∂T

_{e}/∂i

_{q}. However, in the actual operation of the IPMSM, the dq-axis inductance varies with the current.

_{d}| and i

_{q}, both L

_{d}and L

_{q}show a downward trend, and the larger |i

_{d}| and i

_{q}are, the greater the decline in L

_{d}and L

_{q}is. Representing L

_{d}and L

_{q}as implicit functions with i

_{d}and i

_{q}as independent variables, it can be expressed as

_{e}/∂i

_{d}and ∂T

_{e}/∂i

_{q}can be expressed as

_{e}/dβ can be expressed as

_{e}/dβ and Δ∂T

_{e}/∂i

_{q}will cause a deviation between the actual reference value of the current and the ideal reference value of the current, which will cause the motor output torque to fail to accurately track the reference torque.

#### 3.2. Error Compensation Method

_{e}/∂i

_{d}and Δ∂T

_{e}/∂i

_{q}.

_{e}/∂i

_{d}and Δ∂T

_{e}/∂i

_{q}, the dq-axis inductance is considered to be proportional to the dq-axis current. In other words, the second-order and above partial derivatives of dq-axis inductance to dq-axis current are zero. From Equations (22)–(24), Δ∂T

_{e}/∂i

_{d}and Δ∂T

_{e}/∂i

_{q}can be expressed as

_{dr}/∂i

_{d}− ∂L

_{qr}/∂i

_{d}and N = ∂L

_{dr}/∂i

_{q}− ∂L

_{qr}/∂i

_{q}.

_{e}/∂i

_{d}and Δ∂T

_{e}/∂i

_{q}lies in obtaining the values of M and N. If the measured motor has a current inductance relation table, ∂L

_{dr}/∂i

_{d}, ∂L

_{qr}/∂i

_{d}, ∂L

_{dr}/∂i

_{q}and ∂L

_{qr}/∂i

_{q}can be obtained by linear fitting. Then, M and N are calculated using the fitting results. In the actual operation of the motor, the inductance current relation table is not completely accurate, so the M and N values should be slightly adjusted according to the experimental results to obtain more accurate compensation effect. If the measured motor does not have a current inductance relation table, it is necessary to select different values of M and N to carry out the experiment. By observing the magnitude of the deviation between the dq-axis current and the MTPA point, the appropriate M and N values are selected.

## 4. Experimental Results and Analysis

^{−6}and 1.1 × 10

^{−6}, respectively.

## 5. Conclusions

_{e}/dβ, which causes the output current to deviate from the MTPA point, thus affecting the motor efficiency and torque output ability. In order to solve this problem, a method to compensate for the partial conductance error by solving the partial conductance value between the d-q axis inductance and the d-q axis current is proposed in this paper. The experimental results show that, compared with the existing VCSIM, the proposed method can realize MTPA control more accurately, and the motor efficiency and torque output capacity are improved significantly, especially under the condition of a large current amplitude.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Relationship between dq-axis inductance and dq-axis current: (

**a**) d-axis inductance, (

**b**) q-axis inductance.

**Figure 6.**MTPA tracking experimental result of VCSIM and proposed method when the motor speed is 1000 r/min.

**Figure 7.**The comparison diagram of motor efficiency at speed of 1000 r/min and torque of 150 N·m. (

**a**) VCSIM, (

**b**) the proposed method.

**Figure 8.**The current and torque waveforms at speed of 1000 r/min and T

_{e_ref}= 50, 100, 150, 200, 250, 300, 320 N·m: (a) VCSIM; (

**b**) the proposed method.

**Figure 9.**The current and torque waveforms at speed of 3000 r/min and T

_{e_ref}= 50, 100, 150, 200, 250, 300, 320 N·m: (

**a**) VCSIM; (

**b**) the proposed method.

Parameters | Symbol | Value | Unit |
---|---|---|---|

Pole pairs | p | 4 | \ |

Flux linkage | ψ_{f} | 0.09398 | Wb |

Stator resistance | R_{s} | 0.032 | Ω |

d-axis inductance | L_{d} | 0.437 | mH |

q-axis inductance | L_{q} | 1.119 | mH |

Rated speed | n_{N} | 3820 | r/min |

Rated torque | T_{N} | 150 | N·m |

Peak torque | T_{P} | 320 | N·m |

Rated voltage | U_{N} | 540 | V |

Rated current | I_{N} | 135 | A |

Peak current | I_{P} | 275 | A |

**Table 2.**Output torque, current amplitude and efficiency when applying VCSIM and the proposed method.

T_{e_ref} (N·m) | VCSIM | Proposed Method | Error | |||||
---|---|---|---|---|---|---|---|---|

T_{e1} (N·m) | I_{s1} (A) | η_{1} | T_{e2} (N·m) | I_{s2}(A) | η_{2} | I_{s2} − I_{s1} (A) | η_{2} − η_{1} | |

30 | 30.23 | 50.27 | 96.77% | 30.11 | 50.50 | 96.78% | 0.25 | 0.01% |

60 | 59.77 | 86.26 | 95.63% | 59.67 | 85.81 | 95.75% | 0.48 | 0.12% |

90 | 89.21 | 119.40 | 94.62% | 89.23 | 118.67 | 94.78% | 0.13 | 0.16% |

120 | 118.94 | 150.92 | 93.76% | 118.91 | 152.03 | 93.98% | −1.46 | 0.22% |

150 | 148.67 | 186.66 | 92.74% | 148.78 | 185.54 | 93.15% | −2.54 | 0.41% |

180 | 178.53 | 222.81 | 91.63% | 178.89 | 219.27 | 92.19% | −3.98 | 0.57% |

210 | 208.15 | 260.21 | 90.32% | 208.83 | 253.24 | 91.23% | −7.11 | 0.92% |

240 | 237.83 | 303.56 | 88.88% | 238.67 | 286.58 | 90.29% | −14.88 | 1.42% |

270 | 266.72 | 345.27 | 87.07% | 268.61 | 321.37 | 89.17% | −25.51 | 2.09% |

300 | 292.35 | 385.79 | 84.97% | 297.99 | 357.91 | 87.92% | −34.86 | 2.96% |

320 | \ | \ | \ | 316.93 | 384.37 | 86.89% | \ | \ |

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**MDPI and ACS Style**

Miao, Q.; Li, Q.; Xu, Y.; Lin, Z.; Chen, W.; Li, X.
Virtual Constant Signal Injection-Based MTPA Control for IPMSM Considering Partial Derivative Term of Motor Inductance Parameters. *World Electr. Veh. J.* **2022**, *13*, 240.
https://doi.org/10.3390/wevj13120240

**AMA Style**

Miao Q, Li Q, Xu Y, Lin Z, Chen W, Li X.
Virtual Constant Signal Injection-Based MTPA Control for IPMSM Considering Partial Derivative Term of Motor Inductance Parameters. *World Electric Vehicle Journal*. 2022; 13(12):240.
https://doi.org/10.3390/wevj13120240

**Chicago/Turabian Style**

Miao, Qiang, Qiang Li, Yamei Xu, Zhichen Lin, Wei Chen, and Xinmin Li.
2022. "Virtual Constant Signal Injection-Based MTPA Control for IPMSM Considering Partial Derivative Term of Motor Inductance Parameters" *World Electric Vehicle Journal* 13, no. 12: 240.
https://doi.org/10.3390/wevj13120240