J-A hysteresis model parameters estimation using GA

Advances in Electrical and Electronic Engineering 174 J-A HYSTERESIS MODEL PARAMETERS ESTIMATION USING GA Bogomir Zidari 1 , Damijan Miljavec 2 1TECES, Gosposvetska cesta 84, 2000 Maribor, Slovenia, e-mail: zidari[email protected]i 2Univerza v Ljubljani, Fakulteta za elektrotehniko, Tržaška 25, 1000 Ljubljana, Slovenia, e-mail: mi[email protected]i-lj.si Summary This paper presents the Jiles and Atherton (J-A) hysteresis model parameter estimation for soft magnetic composite (SMC) material. The calculation of Jiles and Atherton hysteresis model parameters is based on experimental data and genetic algorithms (GA). Genetic algorithms operate in a given area of possible solutions. Finding the best solution of a problem in wide area of possible solutions is uncertain. A new approach in use of genetic algorithms is proposed to overcome this uncertainty. The basis of this approach is in genetic algorithm built in another genetic algorithm. 1. INTRODUCTION J-A hysteresis model is one of the most investigated hysteresis model in recent years focusing on estimation of five model parameters using different optimization techniques. One of these techniques are well known genetic algorithm. To determine five J-A model parameters it is adequate having only one measured hysteresis loop [1]. For this purpose the measured DC hysteresis loops were obtained. Fig. 1 shows the measured DC hysteresis loop for soft magnetic composite material Somaloy 500 TM [2] at different induction levels. - 7200 -5200 -3200 -1200 1200 3200 5200 7200 1.5 1 0.5 0 -0.5 -1 -1.5 Applied field (A/m) Induction ( T ) 1.4 T 0.9 T 0.5 T Fig. 1. Measured DC curves at different induction levels. The DC hysteresis loop measured values from Fig. 1 at induction level 1.4 T are going to be further used to optimize five J-A model parameters . 2. FEROMAGNETIC J-A HYSTERESIS MODEL Mathematical model for ferromagnetic hysteresis introduced by Jiles and Atherton (J-A model) is based on physical principles of ferromagnetic material and not strictly on mathematical or experimental curve fitting [3]. The J-A model assumes that total magnetization M of ferromagnetic material is decomposed into an irreversible M irr and a reversible M rev magnetization component [3], [4]. The form of total differential magnetization susceptibility dM/dH is: ( ) ( ) dH dM c MMk MM c dH dM an irran irran + −− − −= αδ 1 , (1) where M an is anhysteretic magnetization curve represented by Langevin function:       + −      + =MH a a MH MM San α α coth . (2) Five material – dependent parameters appearing in (eq. 1) and (eq. 2) are: Ms – magnetization saturation, a – shape parameter, α – main field parameter, k – domain wall pinning constant, proportional to hysteresis losses, c – domain flexing constant. The parameter δ has no physical meaning and is equal to +1 for dH/dt > 0 and to –1 otherwise. The induction B can be calculated using: ( ) . 0 MHB += µ (3) At first the optimization of five parameters included in J-A model is done using measured results of hysteresis loops and by genetic algorithms. 3. GENETIC ALGORITHMS FOR J-A MODEL PARAMETERS IDENTIFICATION To get the best fitted modelled hysteresis (J-A hysteresis model parameters values) the genetic algorithms (GAs) are used. The genetic algorithms are optimization technique based on the concept of natural selection and genetics [5]. The algorithm starts with a set of possible solutions – individuals called population J-A hysteresis model parameters estimation using GA 175 [5], [1]. Solutions from a population are used to form a new population by hope that the new population will be better that the old one. Solutions that will form a new set of solutions are selected according to their fitness. This is repeated until stop criteria (for example, number of generations or improvement of the best solution) is satisfied. The GAs can optimize with continuous or discrete parameters and can handle numerically generated data, experimental data or analytical functions [1]. The construction of population for J-A models parameters used in GAs for optimization process is shown on Figure 2 [1].               = nnnnnn fitnesscMska fitnesscMska pop α α ΜΜΜΜΜΜ ΜΜΜΜΜΜ 111111 Fig. 2. Construction of population . n – number of individuals included in population. Each row of population represents one individual (chromosome), composed of parameters of J-A model called genes (column). The last column represents the estimation of the individual – fitness according to chosen evaluation of individual (for example, sum square error - SSE). In this case the fitness is obtained by difference between measured and computed induction B: ( )  = −−= mBcBmabsfitness 1i ii (4) Bm – measured induction , Bc – calculated induction using J-A model, m – number of measurements. GAs searches for the best solution in an area of possible solutions individually determined for each parameter (Table 1). The area of possible solutions (searching area) is limited with upper and lower limit values. It is uncertain, if the global optimum exists in this limited area. With selection, crossover and mutation methods as GAs operators in gaot Matlab toolbox, the new better population is formed. Selection function is based on the normalized geometric distribution. Crossover is arithmetic function that takes two individuals and performs an interpolation along the line formed by the two individuals to get the new individual. Some individual genes are chosen to mutate using non-uniform probability distribution of mutation. Using selected population (Fig. 2), estimation of fitness (eq. 4) and selected searching area (Tab. 1) with GAs the identification of J-A model parameters is done. The optimization results with 40 individuals after 100 generations of GAs on measured hysteresis loop with induction of 1.4 T are presented in Tab. 2. Parameters Upper area limit Lower area limit a 2000 100 α 0.002 0.0001 k 2500 100 Ms 1.7*10 6 1*10 6 c 0.3 0.01 Tab. 1. GAs parameter area limits for measured DC hysteresis loop B = 1.4 T. J-A model parameters Optimized values a 694,684 α 0,000614 k 320,873 Ms 1149531,87 c 0,1469 Tab. 2. Optimized J-A model parameter values for measured DC hysteresis loop B = 1.4 T . The results were not satisfied in spite of repeated calculations with GAs. The GAs did not find the global optimum in defined searching area. If taking the wide searching not clear in convergence of the best solution. In [1] is proposed to calculate the suitable ranges for the J-A model parameters with a trial-and-error procedure using a small number of individuals with few generations by observing the error (fitness). From previous work with GAs the improve english with wide search area, is proposed. The main idea is to use genetic algorithm built in another genetic algorithm called GgaA. Flow diagram of GgaA is shown in Fig. 3. Advances in Electrical and Electronic Engineering 176 Fig. 3. Flow diagram of GgaA. The main property of GgaA is, that the outer GA finds the best upper and lower searching area limits and inner GA is used to estimate the fitness. In fact from wide searching area the region containing global optimum is extracted out. The outer GA individual of the population is composed with eleven genes, which represents upper and lower limits of search area plus fitness. For example, the first gene of individual represents the upper boundary area limit for parameter a, the second gene represents the lower boundary area limit for parameter a, and in the same manner further for the rest of parameters. The inner GA uses 40 individuals and 100 generations. This number of individuals and generations is selected because the searching area of inner GA is narrow and is expected to find global optimum for all searching areas (fast convergence) delivered from outer GA. To determine the fitness for outer GA the inner GA finds the J-A model not clear for current area limits. It uses measured values of DC hysteresis loops and fitness from (eq. 4). Inner GA is also used when GgaA stopping conditions are reached (number of generations of outer GA). In this case the inner GA optimizes J-A model parameters with upper and lower limits of parameters searching area optimized with GgaA. The J-A model parameter identification is done using new GgaA optimization way on measured hysteresis loop with induction of 1.4 T. The results are presented in Fig. 4, Tab. 3 and Tab. 4. Outer GA is set to 10 individuals and 20 generations. Parameters of GgaA Search area limits for GgaA a upper limit 4000 ÷ ÷÷ ÷ 1950 a lower limit 1950 ÷ ÷÷ ÷ 100 α upper limit 0,004 ÷ ÷÷ ÷ 0,00195 α lower limit 0,00195 ÷ ÷÷ ÷ 0,0001 k upper limit 4000 ÷ ÷÷ ÷ 1950 k lower limit 1950 ÷ ÷÷ ÷ 100 Ms upper limit 2,4*10 6 ÷ ÷÷ ÷ 10 6 Ms lower limit 10 6 ÷ ÷÷ ÷ 0,4*10 6 c upper limit 0,3 ÷ ÷÷ ÷ 0,1475 c lower limit 0,1475 ÷ ÷÷ ÷ 0,005 Tab. 3. GgaA parameter area limits for measured DC hysteresis loop B = 1.4 T. J-A model parameters Search area limits optimized with GgaA Optimized values with inner GA a 1974.577 ÷ 124.577 1188,541 α 0,00197 ÷ 0,000121 0,001633 k 1974.577 ÷ 137.744 322,618 Ms 1,6*10 6 ÷ 0,9*10 6 1,268*10 6 c 0,2479 ÷ 0,04657 0,2220997 Tab. 4. Optimized J-A model parameter values for measured DC hysteresis loop B = 1.4 T using GgaA . -6 -4 -2 0 0.5 1 1.5 -0.5 -1 -1.5 B (T) H (A/m) Section A x10 3 0246 Measured Modeled Fig. 4. Optimization result of GgaA for measured DC hysteresis loop B = 1.4 T . From Fig. 4 it can be seen that modelled hysteresis loop is very well fitted to the measured one. Discussion does not exist and work done with GgaA’s. Stop if conditions are met Genetic operators for Inner GA End Genetic operators for outer GA Stop if conditions are met Inner GA Definition of fitness for inner GA Fitness for inner GA new population Fitness for outer GA new population Outer GA Initial population outer GA Definition of fitness for Initial population Calculation of J-A hysteresis model parameters estimation using GA 177 4. CONCLUSION The identification of J-A hysteresis model parameters with genetic algorithms for SMC material is efficient method when searching area is pre defined with knowledge of possible solutions. Good possible solutions can be obtained from empirical equitation in characteristic points on hysteresis loop delivered from (eq. 1) and (eq. 2) [6]. If no assumption on solutions is present and wide searching area range is selected the use of GgaAs approach gives good results. This is clearly seen from Figure 4. Obtained results are improve English of J-A hysteresis model for SMC material. Bestfitted hysteresis model parameters could be used to calculate hysteresis losses in SMC material. REFERENCES [1] J. V. Leite, S. L. Avila, N. J. Batistela, W. P. Carpes, Jr., N Sadowski, P. Kuo-peng, and J. P. A. Bastos: Real Coded Genetic Algorithm for Jiles-Atherton Model Parameters Identification, IEEE Trans. Magn., vol. 40, no. 2, March 2004. [2] Patricia Jansson: Soft magnetic composites – a rapidly expanding materials group, Höganäs AB, Sweden, 1999. [3] D. Jiles: Introduction to Magnetism and Magnetic Materials, Encyclopedia Britannica, Fifteenth Edition 1989. [4] D. C. Jiles, J. B. 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