Soft Computing

, Volume 22, Issue 4, pp 1263–1285 | Cite as

Self-adaptive differential evolution algorithm with hybrid mutation operator for parameters identification of PMSM

  • Chuan Wang
  • Yancheng Liu
  • Xiaoling Liang
  • Haohao Guo
  • Yang Chen
  • Youtao Zhao
Methodologies and Application


Parameters identification of permanent magnetic synchronous motor (PMSM), which significantly influences the control performance of the drive system, is an important and challenging task of power electronic system. The problem requires both high solution quality and fast convergence speed due to the constraints of hardware. This paper presents a self-adaptive differential evolution algorithm with hybrid mutation operator (SHDE) for parameters identification problem. In this method, a novel mutation operator, called “current-to-archive-best,” is developed by mixing the best solutions randomly selected from archive set and current population. Thus, the algorithm could use the best searching memories so far to generate promising solutions, yielding a faster evolving procedure. Besides, the corresponding control parameters of SHDE are also self-adapted without tedious trial-and-error progress to get appropriate values. Moreover, the parameters estimation program is inserted into the PMSM simulation that is solved by using Newton–Raphson method without any pre-assumption and simplification. This framework, which may be used under any working conditions with large disturbance, is different from other publications, resulting in wider applications. The proposed method applied to parameters identification of PMSM is evaluated on a PMSM drive system with two different operations. The comprehensive results and statistical analyses, compared with other state-of-the-art algorithms, show that SHDE could find high-quality solutions with higher convergence speed and probability.


Parameter identification Differential evolution (DE) Neutral selection operator Permanent magnet synchronous machine Ranking-based mutation operators 

List of symbols


Amplification of speed loop after PI controller


Amplification of current loop after PI controller


Crossover rate


Optimization problem dimension


Scaling factor


Generation index of DE algorithm


Inertia coefficient \((\hbox {kg}\,\hbox {m}^{2}\))


d-Axis inductance (H)


Lower limit of d-axis inductance (H)


Upper limit of d-axis inductance (H)


q-Axis inductance (H)


Lower limit of q-axis inductance (H)


Upper limit of q-axis inductance (H)


Stator resistance (\(\Omega )\)


Lower limit of stator resistance \((\Omega )\)


Upper limit of stator resistance \((\Omega )\)


Electromagnetic torque (\(\hbox {N}\,\hbox {m}\))


Load torque (\(\hbox {N}\,\hbox {m}\))


Trail vector for ith individual


Mutation vector for ith individual


Target vector, i.e., ith individual

\(i_{d}, i_{q}\)

dq-Axis current (A)

\(i_{d\_\mathrm{ref}}, i_{q\_\mathrm{ref}}\)

Reference dq-axis current (A)


Step size (s)


Iteration index of Newton–Raphson method

\(k_{\max }\)

Max iteration number of Newton–Raphson method


Number of pole pairs


Population size of DE algorithm


Sampling time index of PMSM simulation

\(v_{d}, v_{q}\)

dq-Axis voltage (V)

\(v_{a}, v_{b}, v_{c}\)

abc-Axis voltage (V)

\(\varepsilon \)

Small positive constant of termination criteria for Newton–Raphson method

\(\theta \)

Rotor position (rad)

\({\psi }_{\mathrm{r}}\)

Rotor flux linkage (Wb)

\({\psi }_{\mathrm{r}}^\mathrm{min}\)

Lower limit of rotor flux linkage (Wb)

\({\psi }_{\mathrm{r}}^\mathrm{max}\)

Upper limit of rotor flux linkage (Wb)

\({\omega }_{\mathrm{e}}\)

Electrical angular speed (rad/s)

\({\omega }_{\mathrm{m}}\)

Mechanical angular speed (rad/s)

\({\omega }_{m\_\mathrm{ref}}\)

Reference mechanical angular speed (rad/s)



The authors would like to thank Natural Science Foundation of Liaoning Province, China, under Contract No. 2014025006; Education Department General Project of Liaoning Province, China, under Contract No. L2014209; Fundamental Research Funds for the Central Universities under Contract No. 3132016011 for financially supporting this research.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Chuan Wang
    • 1
  • Yancheng Liu
    • 1
  • Xiaoling Liang
    • 1
  • Haohao Guo
    • 1
  • Yang Chen
    • 1
  • Youtao Zhao
    • 1
  1. 1.Marine Engineering CollegeDalian Maritime UniversityDalianPeople’s Republic of China

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