# Real-Time Estimation of PMSM Rotor Flux Linkage for EV Application under Steady State and Free-Running Conditions

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## Abstract

**:**

## 1. Introduction

## 2. PMSM Modeling Including VSI Nonlinearity

_{dead}and DqV

_{dead}are the dq-axis distorted voltages caused by VSI nonlinearity; V

_{dead}is a constant when the PMSM is at steady state, and Dd and Dq can be denoted as [30]:

## 3. Rotor Flux Linkage Estimation

#### 3.1. Rotor Flux Linkage Estimation at Steady State Condition

_{s}), the corresponding q-axis equation can be expressed as (5) [23]. In other words, N cycles of PWM switching can be regarded as one control cycle, which consists of N − 1 cycles of vector control and one cycle for the injection of q-axis zero-voltage.

_{dead})

_{vc}are the q-axis command voltage, the dq-axis currents, and the q-axis distorted voltage due to VSI nonlinearity, respectively. ${i}_{d\_inj}$, ${i}_{q\_inj}$, and (DqV

_{dead})

_{inj}represent the dq-axis currents and the q-axis distorted voltage due to VSI nonlinearity when there is an injection of zero-voltage perturbation.

_{d}= 0 control and defining ${\eta}_{VSI}$= (N − 1)(DqV

_{dead}

_{)vc}+ (DqV

_{dead})

_{inj}, Equation (9) can be simplified to (10):

#### 3.2. Rotor Flux Linkage Estimation at Free-Running Condition

_{dead}as a constant. Hence, $\sum {}_{c=1}^{M}({V}_{dead}(c))$=$\sum {}_{d=1}^{M}({V}_{dead}(d))$ while $\sum {}_{c=1}^{M}(Dq(c))$=$\sum {}_{d=1}^{M}(Dq(d))$, and $\sum {}_{c=1}^{M}({\lambda}_{f}(c))$=$\sum {}_{d=1}^{M}({\lambda}_{f}(d))$=$M{\lambda}_{f}$. Then, (16) can be further simplified to (19):

## 4. Experimental Results

#### 4.1. Test Rig and Prototype PMSM

#### 4.2. Experimental Tests of Zero-Voltage Injection-Based Approach

_{s}= 2.2 Ω, and the test result after the addition of resistors with injection of zero-voltage perturbation at N = 5 is shown in Figure 8. It can be seen that the relationship between the q-axis command voltage and rotor speed is also approximately linear after the addition of resistors, and the two curves of q-axis command voltage with respect to rotor speed are almost parallel. Furthermore, the estimated rotor flux linkages are 0.2419 Wb and 0.2412 Wb before and after the addition of resistors, which is shown in Table 3. Thus, the variation of winding resistance has almost no influence on the estimation of rotor flux linkage.

- The PMSM is drawn by a induction machine and then is heated by connecting the external three-phase winding.
- The rotor speed of PMSM will be fixed to 600 rpm by the induction motor, and the corresponding line back EMF of PMSM under no-load condition will be measured. The rotor flux linkage under no-load condition can be calculated from these measured line back EMF.
- The PMSM will be loaded and the rotor flux linkage under loaded condition will be estimated by the proposed method N = 5.
- Repeat 1–3 under different PM temperatures.

#### 4.3. Estimation under Free-Running Condition

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

L_{d}, L_{q} | dq-axis inductances (H) |

R | Stator winding resistance (Ω) |

λ_{f} | Rotor flux linkage (Wb) |

ω | Electrical angular speed (rad/s) |

θ | Electrical angle (rad) |

γ | Angles between current vector and q-axis (rad) |

${i}_{{}_{d}}$, ${i}_{{}_{q}}$ | Actual dq-axis currents (A) |

${i}_{d}^{\ast}$, ${i}_{q}^{\ast}$ | Command dq-axis currents (A) |

${u}_{d}$, ${u}_{q}$ | Actual dq-axis voltages (V) |

${u}_{d}^{\ast}$, ${u}_{q}^{\ast}$ | Command dq-axis voltages (V) |

Dd, Dq | Functions of θ and γ |

V_{dead} | Distorted voltage due to inverter nonlinearity (V) |

T_{s} | PWM switching period (S) |

N | Total number of PWM switching cycles included in one control cycle |

## References

- Zhu, Z.Q.; Howe, D. Electrical machines and drives for electric, hybrid, and fuel cell vehicles. Proc. IEEE.
**2007**, 95, 746–765. [Google Scholar] [CrossRef] - Sarlioglu, B.; Morris, C.T.; Han, D.; Li, S. Driving toward accessibility: A review of technological improvements for electric machines, power electronics, and batteries for electric and hybrid vehicles. IEEE Ind. Appl. Mag.
**2017**, 23, 14–25. [Google Scholar] [CrossRef] - Reigosa, D.; Fernandez, D.; Yoshida, H.; Kato, T.; Briz, F. Permanent-magnet temperature estimation in PMSMs using pulsating high-frequency current injection. IEEE Trans. Ind. Appl.
**2015**, 51, 3159–3168. [Google Scholar] [CrossRef] - Feng, G.; Lai, C.; Kar, N. Expectation-maximization particle-filter- and Kalman-filter-based permanent magnet temperature estimation for PMSM condition monitoring using high-frequency signal injection. IEEE Trans. Ind. Informat.
**2017**, 13, 1261–1270. [Google Scholar] [CrossRef] - Li, S.; Sarlioglu, B.; Jurkovic, S.; Patel, N.R.; Savagian, P. Analysis of temperature effects on performance of interior permanent magnet machines for high variable temperature applications. IEEE Trans. Ind. Appl.
**2017**, 53, 4923–4933. [Google Scholar] [CrossRef] - Chen, X.; Wang, J.; Griffo, A. A high-fidelity and computationally efficient electro-thermally coupled model for interior permanent magnet machines in electric vehicle traction applications. IEEE Trans. Transp. Electrific.
**2015**, 1, 336–347. [Google Scholar] [CrossRef] - Lai, C.; Feng, G.; Mukherjee, K.; Loukanov, V.; Kar, N.C. Torque ripple modeling and minimization for interior PMSM considering magnetic saturation. IEEE Trans. Power Electron.
**2018**, 33, 2417–2429. [Google Scholar] [CrossRef] - Morimoto, S.; Sanada, M.; Takeda, Y. Mechanical sensorless drives of IPMSM with online parameter identification. IEEE Trans. Ind. Appl.
**2006**, 42, 1241–1248. [Google Scholar] [CrossRef] - Inoue, Y.; Kawaguchi, Y.; Morimoto, S.; Sanada, M. Performance improvement of sensorless IPMSM drives in a low-speed region using online parameter identification. IEEE Trans. Ind. Appl.
**2011**, 47, 798–804. [Google Scholar] [CrossRef] - Liu, K.; Zhang, Q.; Chen, J.; Zhu, Z.Q.; Zhang, J. Online multiparameter estimation of nonsalient-pole PM Synchronous machines with temperature variation tracking. IEEE Trans. Ind. Electron.
**2011**, 58, 1776–1788. [Google Scholar] [CrossRef] [Green Version] - Kim, H.; Youn, M.; Cho, K.; Kim, H. Nonlinearity estimation and compensation of PWM VSI for PMSM under resistance and flux linkage uncertainty. IEEE Trans. Control. Syst. Technol.
**2006**, 14, 589–601. [Google Scholar] - Boileau, T.; Leboeuf, N.; Nahid, B.; Meibody, F. Online identification of PMSM parameters: Parameter identifiability and estimator comparative study. IEEE Trans. Ind. Appl.
**2011**, 47, 1944–1957. [Google Scholar] [CrossRef] - Feng, G.; Lai, C.; Kar, N.C. A novel current injection-based online parameter estimation method for PMSMs considering magnetic saturation. IEEE Trans. Magn.
**2016**, 52, 1–4. [Google Scholar] [CrossRef] - Underwood, S.; Husain, I. On-line parameter estimation and adaptive control of permanent magnet synchronous machines. IEEE Trans. Ind. Electron.
**2010**, 57, 2435–2443. [Google Scholar] [CrossRef] - Shi, Y.; Sun, K.; Huang, L.; Li, Y. Online identification of permanent magnet flux based on extended Kalman filter for IPMSM drive with position sensorless control. IEEE Trans. Ind. Electron.
**2012**, 59, 4169–4178. [Google Scholar] [CrossRef] - Dang, D.Q.; Rafaq, M.S.; Choi, H.H.; Jung, J.W. Online parameter estimation technique for adaptive control applications of interior PM synchronous motor drives. IEEE Trans. Ind. Electron.
**2016**, 63, 1438–1449. [Google Scholar] [CrossRef] - Rafaq, M.S.; Mwasilu, F.; Kim, J.; Choi, H.H.; Jung, J.W. Online parameter identification for model-based sensorless control of interior permanent magnet synchronous machine. IEEE Trans. Power Electron.
**2017**, 32, 4631–4643. [Google Scholar] [CrossRef] - Xiao, X.; Chen, C.M.; Zhang, M. Dynamic permanent magnet flux estimation of permanent magnet synchronous machines. IEEE Trans. Appl. Supercond.
**2010**, 20, 1085–1088. [Google Scholar] [CrossRef] - Hamida, M.A.; Leon, J.D.; Glumineau, A.; Boisliveau, R. An adaptive interconnected observer for sensorless control of PM synchronous motors with online parameter identification. IEEE Trans. Ind. Electron.
**2013**, 60, 739–748. [Google Scholar] [CrossRef] - Gatto, G.; Marongiu, I.; Serpi, A. Discrete-time parameters identifi-cation of a surface-mounted permanent magnet synchronous machine. IEEE Trans. Ind. Electron.
**2013**, 60, 4869–4880. [Google Scholar] [CrossRef] - Wilson, S.D.; Stewart, P.G.; Taylor, B.P. Methods of resistance estimation in permanent magnet synchronous motors for real-time thermal management. IEEE Trans. Energy Convers.
**2010**, 25, 698–707. [Google Scholar] [CrossRef] [Green Version] - Feng, G.; Lai, C.; Mukherjee, K.; Kar, N.C. Current injection-based online parameter and VSI nonlinearity estimation for PMSM drives using current and voltage dc components. IEEE Trans. Transp. Electrific.
**2016**, 2, 119–128. [Google Scholar] [CrossRef] - Xie, G.; Lu, K.; Dwivedi, S.K.; Riber, R.J.; Wu, W. Permanent magnet flux online estimation based on zero-voltage vector injection method. IEEE Trans. Power Electron.
**2015**, 30, 6506–6509. [Google Scholar] [CrossRef] - Feng, G.; Lai, C.; Mukherjee, K.; Kar, N.C. Online PMSM magnet flux-linkage estimation for rotor magnet condition monitoring using measured speed harmonics. IEEE Trans. Ind. Appl.
**2017**, 53, 2786–2794. [Google Scholar] [CrossRef] - Liu, K.; Zhu, Z.Q. Position-offset-based parameter estimation using the adaline NN for condition monitoring of permanent magnet synchronous machines. IEEE Trans. Ind. Electron.
**2015**, 62, 2372–2383. [Google Scholar] [CrossRef] - Liu, K.; Zhu, Z.Q. Online estimation of the rotor flux linkage and voltage source inverter nonlinearity in permanent magnet synchronous machine drives. IEEE Trans. Power Electron.
**2014**, 29, 418–427. [Google Scholar] [CrossRef] [Green Version] - Zhou, S.; Liu, K.; Hu, W.; Chen, Y.; Zhang, D.; Huang, Q.; Tong, Q.; Zhang, Q. Harmonic-separation-based direct extraction and compensation of inverter nonlinearity for state observation control of PMSM. IEEE Access.
**2021**, 9, 142028–142045. [Google Scholar] [CrossRef] - Park, D.; Kim, K. Parameter-independent online compensation scheme for dead time and inverter nonlinearity in IPMSM drive through waveform analysis. IEEE Trans. Ind. Electron.
**2014**, 61, 701–707. [Google Scholar] [CrossRef] - Deng, W.; Xia, C.; Yan, Y.; Geng, Q.; Shi, T. Online multiparameter identification of surface-mounted PMSM considering inverter disturbance voltage. IEEE Trans. Energy Convers.
**2017**, 32, 202–212. [Google Scholar] [CrossRef] - Feng, G.; Lai, C.; Tjong, J.; Kar, N.C. Noninvasive Kalman filter based permanent magnet temperature estimation for permanent magnet synchronous machines. IEEE Trans. Power Electron.
**2018**, 33, 10673–10682. [Google Scholar] [CrossRef]

**Figure 1.**Zero-voltage injection-based approach under variable speed control. (

**a**) Schematic diagram of zero-voltage injection-based approach; (

**b**) processes of data measurement and parameter estimation of the proposed method.

**Figure 2.**Proposed method under free-running condition. (

**a**) Current control diagram; (

**b**) process of data measurement.

**Figure 4.**Waveforms of PMSM with and without injection of zero-voltage perturbation (N = 5, i

_{d}= 0, i

_{q}=3 A, 300 rpm). (

**a**) Dq-axis currents; (

**b**) rotor speed; (

**c**) Dq-axis command voltages within 2 s; (

**d**) Dq-axis command voltages within 10 cycles of PWM switching. −15.

**Figure 5.**PMSM waveforms under variable speed control (i

_{d}= 0, i

_{q}= 3 A). (

**a**) Dq−axis currents; (

**b**) rotor speed.

**Figure 7.**Nominal and estimated rotor flux linkage of the proposed and POPE method at different values of N.

**Figure 8.**Measured q-axis command voltage with respect to rotor speed (N = 5, i

_{d}= 0, i

_{q}= 3 A).

**Figure 9.**PM temperature measurement. (

**a**) Observation hole in end cover of PMSM; (

**b**) thermal image captured by a high-precision infrared thermal imager.

**Figure 10.**Comparison between estimated and measured rotor flux linkage under different PM temperatures. (

**a**) Estimated/measured rotor flux linkage; (

**b**) variation in the rotor flux linkage.

**Figure 11.**PMSM waveforms at free-running condition with different initial speeds. (

**a**) 100 rpm. (

**b**) 200 rpm. (

**c**) 400 rpm.

**Figure 12.**Estimated rotor flux linkage and its estimation error with respect to different initial rotor speeds of free-running. (

**a**) Nominal and estimated rotor flux linkage; (

**b**) estimation error of rotor flux linkage.

Parameter | Value |
---|---|

Rated power | 3 kW |

Rated current | 6 A |

Rated speed | 3000 rpm |

Number of pole pairs | 3 |

Nominal d-axis inductance | 13.8 mH |

Nominal q-axis inductance | 22.6 mH |

Nominal rotor flux linkage (T = 25 C) | 0.2458 Wb |

Stator winding resistance (T = 25 C) | 0.98 Ω |

Parameter | Typical Value |
---|---|

Turn-on delay (Ton) | 0.10 μs |

Turn-off delay (Toff) | 0.60 μs |

PWM switching period (Ts) | 100 μs |

Dead time (Tdead) | 2 μs |

Voltage drop of the switching tube (Vsat) | 1.45 V |

Voltage drop of the freewheeling diode (Vd) | 1.55 V |

Winding Resistances R (Ω) | Estimated λ_{f} (Wb) |
---|---|

0.98 | 0.2419 |

3.18 | 0.2412 |

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## Share and Cite

**MDPI and ACS Style**

Wen, B.; Liu, K.; Zhou, J.; Zhou, S.; Hu, W.; Chen, Y.; Huang, C.; Huang, Q.
Real-Time Estimation of PMSM Rotor Flux Linkage for EV Application under Steady State and Free-Running Conditions. *World Electr. Veh. J.* **2022**, *13*, 83.
https://doi.org/10.3390/wevj13050083

**AMA Style**

Wen B, Liu K, Zhou J, Zhou S, Hu W, Chen Y, Huang C, Huang Q.
Real-Time Estimation of PMSM Rotor Flux Linkage for EV Application under Steady State and Free-Running Conditions. *World Electric Vehicle Journal*. 2022; 13(5):83.
https://doi.org/10.3390/wevj13050083

**Chicago/Turabian Style**

Wen, Bisheng, Kan Liu, Jing Zhou, Shichao Zhou, Wei Hu, Yongdan Chen, Chao Huang, and Qing Huang.
2022. "Real-Time Estimation of PMSM Rotor Flux Linkage for EV Application under Steady State and Free-Running Conditions" *World Electric Vehicle Journal* 13, no. 5: 83.
https://doi.org/10.3390/wevj13050083