A Hybrid Algorithm for Parameter Extraction of Energetic Hysteresis Model

Abstract

Magnetic materials are widely used in motors, reactors, power-frequency and high-frequency transformers and other electrical devices, which are of great significance for the safety, efficiency and stability of power system. Since the most distinguishing features of this kinds of materials is hysteresis characteristics, an accurate mathematical modelling of the hysteresis loop is a prerequisite for the design and optimization of magnetic cores in electrical devices. As far as now, various hysteresis models have been proposed in the related literatures [1]. Among them, the energetic hysteresis model developed by Hans Hauser is very promising for its simple formula and accurate representation of hysteresis loop [2]. Many factors, such as stress, temperature, frequency and magnetization direction can be incorporated in this model. However, determining the model parameters with reasonable accuracy and efficient computation is still a major challenge. The existing methods for parameter identification of energetic model have been divided into two categories. The first one is using the formulas proposed by Hans [3]. This method not only needs the data of original magnetization curve and major loop, but also involves the solving of transcendental and approximate equations, which could result in inefficient solution and low accuracy. In order to overcome this problem, a curve fitting method is proposed [4], which uses simulated annealing (SA) method to identify parameters based on the objective function of minimum error between measured and calculated hysteresis loops. However, it has the intrinsic drawback of stochastic optimization algorithm that the convergence rate is extremely slow near the global minimum, and there is no guarantee to reach the optimal point [5]. This paper proposes a hybrid algorithm which combines the most appealing features of two kinds of optimization methods viz. the stochastic approach of SA and the deterministic nature of Levengerg-Marquardt (L-M) algorithm. Since SA has an ability to avoid traps in local minima, it is used to approach to the area where global optimal solution is located in the initial iteration period of this hybrid algorithm. Then a normalization of the sensitivity function is built to improve the convergence of L-M algorithm, which can converge quickly towards global minimum. Based on the introduced commutation criteria, the current best solution of SA algorithm is transferred to normalized L-M algorithm as its initial parameter. The simulation and experimental results show that the proposed hybrid algorithm leads to a considerable reduction in computation time and higher accuracy, so it can be used to precisely and quickly extract the parameters of energetic hysteresis model. The test is carried out on the same benchmark model [4], and the root-mean-square error (RMSE) of calculated loop is taken as the objective function. Besides, the value of this parameters are already known, and the bounds of the parameters are taken in a wider range to verify the robustness of the proposed hybrid technique. Firstly, SA is used to identify the parameters. The reduction of RMSE varying with iteration number is shown as Fig. 1. It is evident that the RMSE almost does not change after 23th iteration. This result confirms that SA has a good global search capability, but its local optimization capability is poor, and the maximum error of optimized parameters is up to 27.8%. The final best calculated loop by SA is shown in Fig. 2. However, when it comes to hybrid algorithm, the L-M algorithm takes over the calculation from SA when iteration number reaches 29, and then it converges to the global optimal parameters in just 6 interation. Fig. 2 gives the calculated loop by the hybrid algorithm. The parameters match exactly with that of benchmark model curve.