# Synchronous Machine Winding Modeling Method Based on Broadband Characteristics
## 2 Theory Introduction
### Parameter Identification Based on Optimization Algorithm  
When identifying the model's parameters, the theoretical model with the experimental data are usually combined to identify the parameter value [19; 20; 21; 22; 23; 24; 25; 26]. As an essential method of parameter identification, the optimization algorithm has a strong search performance for a specific parameter range, and it has high adaptability for practical problems [19; 21; 23; 24]. In addition, it can visualize the performance of identification results through the value of the fitness function. Therefore, this paper chooses an optimization algorithm as a tool for parameter identification.  
First of all, the particle swarm optimization (PSO) algorithm is selected as the identification algorithm. The circuit constructed in this paper is relatively simple, and it does not need an algorithm with a strong search ability and complicated calculations. Second, because the PSO algorithm has few adjustable hyperparameters, the algorithm structure is relatively simple compared to other intelligent algorithms (simulated annealing algorithm, genetic algorithm, immune algorithm, etc.). It is also easy to implement in engineering and has a faster convergence speed [20; 22; 23]. Third, the algorithm structure of PSO is simple and it has high universality. Fourth, there are a large number of identification parameters in this study. The number of solutions of the problem can be directly taken as the dimension of the particle in the PSO algorithm. Thus, compared with other algorithms, the PSO algorithm is less affected by the dimension of the problem [20; 22; 23].  
In the particle swarm optimization algorithm, each particle is the random solution of the target to be optimized, and the objective function to be optimized determines the individual fitness of each particle. Each particle has its own speed and direction in the search process, and the particle will follow the current optimal particle to search in the solution space. The search method of the particles in the solution space is shown in Figure 3, \(v\) is the search speed of particles in the solution space, \(x\) is the example position, \(p_{best}\) is the best position in history, \(g_{best}\) is the best position at present, and \(d\) is the number of iterations. A flow chart of the particle swarm optimization algorithm is shown in Figure 4.  
Figure 3: Solution space of particle swarm optimization.  
The impedance is sampled when the frequency is equal to the frequency of each point of the actual measurement curve. Then a least-square optimization problem is formed between the impedance and the actual measurement value. The least-square optimization problem is taken as a fitness function. The smaller the value of the fitness function is, the closer the simulation curve is to the actual value. For the constructed model, a fitness function is constructed between the result of Equation (10) and the measurement curve:  
\[J=\sum_{f=1}^{N}\left(\frac{Z_{cm}(f)-Z_{cm1}(f)}{Z_{cm}(f)}\right)^{2}+\sum_{ f=1}^{N}\left(\frac{Z_{dm}(f)-Z_{dm1}(f)}{Z_{dm}(f)}\right)^{2} \tag{11}\]  
where \(f\) is the frequency, \(Z_{cm}\), \(Z_{cm1}\), \(Z_{dm}\), and \(Z_{dm1}\) are the common mode impedance measurement value and the identification value, and the differential mode impedance measurement value and the identification value, respectively.