## 2 Power Supply Recovery Model
### Power Supply Recovery Model Based on the Whole Network Reconstruction
#### 2.3.1 Objective Function  
(1) Maximum load recovery amount. For the network reconstruction of a floating nuclear power plant, the load is divided into three classes according to the load priority. The objective function established to maximize the recovery amount of the load after fault is:  
\[\max f_{1}=a_{1}\sum_{i=1}^{k_{1}}x_{1i}P_{g1i}+a_{2}\sum_{j=1}^{k_{2}}x_{2j} P_{g2j}+a_{3}\sum_{q=1}^{k_{3}}x_{3q}P_{g3q} \tag{1}\]  
where \(P_{g1}\), \(P_{g2}\) and \(P_{g3}\) respectively represent the power of the first-, second- and third-class loads and \(x_{1}\), \(x_{2}\) and \(x_{3}\) represent the operating states, for which the value of 1 indicates that the load is put into operation, and the value of 0 indicates that the load is out of operation. \(k_{1}\), \(k_{2}\) and \(k_{3}\) are the total amount of loads for all classes, and \(a_{1}\), \(a_{2}\) and \(a_{3}\) are the weights of loads for different classes. The values of different classes of loads in the process of network reconstruction can be fully reflected by assigning larger weights to important loads.  
(2) Minimum switching operation cost. Switching operation cost is an important index for measuring the efficiency of network reconfiguration. For the internal system of the floating nuclear power plant, when the main generator supplies power, inush current suppression should be realized by pre-magnetization. The switching operation process is complex, and the operation cost is high. The other switches are common switches with low operation costs. Different weight coefficients are given for different switching operation costs. The objective function is:  
\[\min f_{2}=k_{P}X_{p}+k_{M}X_{M} \tag{2}\]  
where \(X_{p}\) and \(X_{M}\) respectively represent the numbers of pre-magnetization-related switches and common switches, for which the state should be changed in the process of network reconstruction. \(k_{p}\) and \(k_{M}\) are the corresponding weight coefficients.  
(3) Highest generator operating efficiency. The generator has the lowest energy consumption at rated power and is most beneficial for operation safety. Therefore, the mode with higher operation efficiency should be preferred during network reconstruction. The objective function is:  
\[\max f_{3}=\min_{i\in\tau}\left\{\begin{array}{c}P_{gi}\\ \end{array}\right\} \tag{3}\]  
where \(\tau\) is the generator set that is put into operation, and \(P_{gi}\) and \(P_{giN}\) respectively represent the actual and rated output power of the generator \(i\). The corresponding efficiency of the generator with the lowest operating efficiency among the four generators is selected as the objective function. When its calculated result is large, it indicates that the operating efficiency of each generator is high.  
Considering the above three objectives comprehensively, the normalization process is carried out to solve the problems of different dimensions. On this basis, according to the importance of each indicator, the weighted summation method is adopted for establishing the maximization objective function as:  
\[\left\{\begin{array}{l}\max F=w_{1}\mu_{1}+w_{2}\mu_{2}+w_{3}\mu_{3}\\ \mu_{1}=\frac{f_{1}}{\sum\limits_{i=1}^{k_{1}}P_{gi1}+a_{2}\sum\limits_{i=1}^{k _{2}}P_{g2j}+a_{3}\sum\limits_{q=1}^{k_{2}}P_{g3q}}\\ \mu_{2}=1-\frac{f_{2}}{k_{P}X_{p}+k_{M}X_{M}}\\ \mu_{3}=f_{3}\end{array}\right. \tag{4}\]where \(X_{PS}\) and \(X_{MS}\) represent the total number of pre-magnetization related switches and common switches, respectively. \(w_{1}\), \(w_{2}\) and \(w_{3}\) are the weights of each objective function, which can be determined by the analytic hierarchy process [29]. In the hierarchical analysis, it is considered that the load recovery value is the most important, the switching operation cost is the second and the generator operation efficiency is the last.