## 3 Parameter Identification Method and Process  
### Particle Swarm Optimization Algorithm Based on Random Perturbation  
The PSO algorithm is a bionic algorithm used to solve optimization problems. \(D\) is the parameter dimension. The position of the \(i\)-th particle is described as \(x_{i}=(x_{i1},x_{i2},\cdots,x_{iD})\). The velocity of the \(i\)-th particle is described as \(v_{i}=(v_{i1},v_{i2},\cdots,v_{iD}),1\leq i\leq m\). The historical best point experienced by the \(i\)-th particle is denoted as \(p_{i}=(p_{i1},p_{i2},\cdots,p_{iD})\). The best point for all particles is denoted as \(p_{g}=(p_{g1},p_{g2},\cdots,p_{gD})\). The position and velocity of the particles needed to be constantly updated by the following equation:  
\[ \begin{split} v_{iD}^{k+1}&=v_{iD}^{k}+C_{1}\times \xi\times(p_{iD}^{k}-x_{iD}^{k})+C_{2}\times\eta\times(p_{gD}^{k}-x_{iD}^{k})\\ x_{iD}^{k+1}&=x_{iD}^{k}+v_{iD}^{k+1}\end{split}  \tag{12}\]  
where \(C_{1}\) and \(C_{2}\) are learning factors. \(\xi\) and \(\eta\) are uniformly distributed random numbers in [0,1]. \(k\) reflects the \(k\)-th number of iterations. The concept of inertia weight was introduced in the particle's velocity updating equation to adopt global search followed by local search for solution efficiency [11]. The position and velocity were written as the following:  
\[ \begin{split}& v_{iD}^{k+1}=\omega\times v_{iD}^{k}+C_{1}\times \xi\times(p_{iD}^{k}-x_{iD}^{k})+C_{2}\times\eta\times(p_{gD}^{k}-x_{iD}^{k}),V _{\rm min}\leq v_{iD}^{k+1}\leq V_{\rm max}\\ &\omega=\omega_{\rm max}-(\omega_{\rm max}-\omega_{\rm min}) \times k/M1,1\leq k\leq M1\end{split}  \tag{13}\]  
where \(\omega\) is the inertia weight. \(V_{\rm min}\) and \(V_{\rm max}\) are the minimum and maximum velocity values. \(M\)I is the largest iteration. Let \(C_{1}\) and \(C_{2}\) change linearly and the convergence factor is introduced by the following expression:  
\[ \begin{array}{l}C_{1}=C_{\text{1 max}}-(C_{\text{1 max}}-C_{\text{1 min}})\times k /MI\\ C_{2}=C_{\text{2 max}}-(C_{\text{2 max}}-C_{\text{2 min}})\times k/MI\\ x_{iD}^{k+1}=x_{iD}^{k}+\lambda\times v_{iD}^{k+1}\end{array}  \tag{14}\]  
where \(\lambda\) is the convergence factor. The velocity and position of the particles cannot exceed their upper and lower limits. To further enhance the optimization ability, the particle initial population was generated by chaos method to achieve better initial solution. The two-dimensional cat map was written as follows:  
\[\left\{ \begin{array}{l}x_{iD}^{k+1}=\text{mod}(x_{iD}^{k}+v_{iD}^{k},1)\\ v_{iD}^{k+1}=\text{mod}(x_{iD}^{k}+2v_{iD}^{k},1)\end{array} \right. \tag{15}\]  
where \"mod\" means modular operation. Furthermore, the random perturbation method was introduced to avoid the parameters staying at the boundary value of the given parameter scope and get the appropriate solution. The global optimal value of the parameter was perturbed as follows:  
\[\text{z}_{\text{besti}}=\text{z}_{\text{besti}}\times(1+K\times\text{randn}) \quad i\in[1,D] \tag{16}\]  
where \(\text{z}_{\text{besti}}\) is a certain identified parameter and \"randn\" stands for the standard normal distribution function. The formula \"\(\text{z}_{\text{besti}}\times K\times\text{randn}\)\" reflects the perturbation value, which can make \(\text{z}_{\text{besti}}\) larger or smaller, wherein \(K\) is the perturbation coefficient which is  
* Have preprocessing of noise reduction, data resampling and data normalization to a range of 0 to 1 for some variables (e.g., \(N_{\text{r}}\), \(D_{\text{sp}}\) and so on) before parameter identification. * Make the model initially stable through variable initialization based on the differential equations of each module and program setting method. * After obtaining the constraints of variables and parameters, calculate some parameters according to the operating characteristics for PWR primary loop system. Identify parameters of each module using the RP-PSO algorithm. * Conduct sensitivity analysis for model parameters.