## 2 Fatigue Characteristics of High-Power IGBT Model
### Fault Model Fitted by Weighted LSM  
However, it is not enough to analyze the experimental data of bi-directional aging under a single current. In order to obtain a more consistent distribution function with the practical trend of failure rate, this paper will analyze the bi-directional aging under different current values. Take the bi-directional 900A experiment as an example. The data collected from the experiment are plotted in Figure 10. In Figure 10, the horizontal axis is the number of cycle times and the vertical axis is cumulative failure rate. It should be noted that when the test current decreases, the aging speed decreases sharply [25]. It is too difficult to obtain complete aging data through experiments. Therefore, only the part with a failure rate of 0.4 is used for analysis. Through the previous analysis, Weibull distribution is selected to fit the failure rate distribution. The least squares method can be used to estimate the parameters of non-linear regression analysis. The traditional methods of least squares are direct search method, lattice search method, Gauss-Newton method, Newton-Raphson method, etc. [26]. Direct search method and lattice search method are inefficient and seldom used in practical engineering [27]. Gauss-Newton method is also called linearization method. Newton-Raphson method is an improvement of Gauss-Newton method. However, in some cases, these two methods may require much iteration to converge, and in a few cases, they may not converge at all [26]. To a certain extent, the problem of the above method can be avoided by using a genetic algorithm for least squares. GA (genetic algorithm) is an adaptive stochastic search method that is based on natural selection and biogenetic theory [28]. Firstly, the solution of the problem to be solved is regarded as a number of the individuals in the population. The individual in the population is coded and expressed as a string. Then, according to the fitness of the individuals, the selection, crossing and variation operations between the individuals are simulated to generate new solutions. In the process of population cyclic evolution, individuals gradually evolve to the state of the approximate optimal solution; therefore, the optimal solution of the problem can be obtained. The flow chart is shown in Figure 11.  
Figure 10: Curve of bi-directional aging under 900A. The most important part of GA is fitness function. When GA simulates the selection of nature, the adaptation of each generation of population to conditions corresponds to a necessary parameter, which determines how much probability the individuals in this population participate in the heredity. The function of evaluating this probability is called fitness function. The suitability of fitness function selection will affect the whole GA. Selecting a different fitness function will result in different convergence and accuracy of the whole algorithm [29]. Aiming at the fitting of Weibull distribution in this paper, the purpose of our fitting is to find the appropriate values of \(\lambda\) and \(k\), so we take \(\lambda\) and \(k\) as variables in the initial population. The fitness function selected is:  
\[fitness=\sum_{i=1}^{n}{(1-e^{-(\frac{r_{i}}{\lambda})^{k}}-Y_{i})^{2}} \tag{16}\]  
In Equation (16), \(x_{i}\) is the number of experimental cycles and \(Y_{i}\) is the failure rate corresponding to the number i experimental cycle in the actual experimental data. It should be noted that GA is to solve the fitness function to a minimum value, and the corresponding variables of the minimum value as the output of the optimal solution. The concept of LSM is also used in the construction of fitness function. The purpose of the fitness function is to minimize the absolute value of the difference between the output value of the function to be fitted and the actual data. After summing the squares of the difference between each output value and the practical data, we can calculate a function that is closest to the curve that was formed by the practical data.