## 3 Model Parameter Identification and Comparison
### Model Parameter Identification Using PSO  
For the _n_RC and E_n_RC, the model parameters that need to be identified and optimized can be expresses as:  
\[\left\{ \begin{array}{l}V_{j}=\left[\begin{array}{ccc}R_{0}^{+}&R_{0}^{-}\\ V_{j}=\left[\begin{array}{ccc}R_{0}^{+}&R_{0}^{-}\\ R_{0}^{+}&R_{0}^{-}\\ \end{array}\right.&\underbrace{\tau_{1}\;R_{1}}_{1 st\;RC}&\underbrace{\tau_{2}\;R _{2}}_{2 nd\;RC}\\ V_{j}=\left[\begin{array}{ccc}R_{0}^{+}&R_{0}^{-}\\ \end{array}\right.&\underbrace{\tau_{1}\;R_{1}}_{1 st\;RC}&\underbrace{\tau_{2}\;R _{2}}_{2 nd\;RC}\\ \end{array}\right.&\cdots&\underbrace{\tau_{n}\;R_{n}}_{n-th\;RC}\\ \end{array} \right]\right.\left(\text{if the model is }n\text{RC},\;n=1,2,3\right) \tag{8}\]  
From Equation (8), it can be easily found that the identification parameters of the E_n_RC model are only two more than that of the _n_RC model. In the process of identification and optimization, the closer the model terminal voltage to the measured terminal voltage, the more accurate are the model parameters. Therefore, the root-mean-square error (RMSE) between the model terminal voltage and the measured terminal voltage can be employed as the fitness value to assess the model parameters and acquire the optimal model parameters that make the model voltage closest to the measured voltage. The objective function for the optimization can be expressed as:  
\[\min:g(V)=\sqrt{\frac{1}{n}\sum_{k=1}^{n}(u_{i,k}(V)-\hat{u}_{i,k})^{2}}, \tag{9}\]  
where \(g(V)\) is the objective function, \(u_{i,k}\) is the model voltage, and \(\hat{u}_{i,k}\) is the measured voltage. During parameter optimization, the upper and lower bounds of the parameters can be obtained by experimental results. The bounds of the same parameters for different models are maintained the same for fair comparison. In this study, the PSO algorithm is used for global optimization for parameter identification of the above model. The PSO is a typical swarm intelligence algorithm, inspired by flocks of birds in search for food; it has been successfully applied for artificial neural-network training, function optimization, and pattern classification [26]. Because of the emergence of several variants over time, certain researchers have attempted to define a standard PSO version, with updates to incorporate the latest advances. The most recent standard PSO version was defined in 2011 and is referred to as the Standard PSO2011 (SPSO2011). The following is a brief description of the basic principles of PSO algorithm, and detailed descriptions of the PSO can be found in Reference [27]. Each particle in the algorithm represents a potential solution to the problem. The state of each particle is represented by its position \(x\), velocity \(v\), and fitness. In the iteration process, particle states are continuously updated until the termination criteria are met. Assume that in a D-dimensional search space, there is a swarm consisting of \(n\) particles, \(\mathbf{X}\) = (\(\text{X}_{1}\), \(\text{X}_{2}\),..., \(\text{X}_{n}\)), where the velocity of the _i_th particle is expressed as a D-dimensional vector \(\mathbf{X}_{i}\) = (\(\text{X}_{i1}\), \(\text{X}_{i2}\),..., \(\text{X}_{nD}\))\({}^{\text{T}}\), the individual extremum is expressed as \(\mathbf{P}_{i}\) = (\(\text{P}_{i1}\), \(\text{P}_{i2}\),..., \(\text{P}_{nD}\))\({}^{\text{T}}\), and the swarm extremum is expressed as \(\mathbf{P}_{best}\) = (\(\text{P}_{g1}\), \(\text{P}_{g2}\),..., \(\text{P}_{gD}\))\({}^{\text{T}}\).