## 5 Simulation Results
### Comparison with Existing Methods  
To verify that the MPGA method can quickly and reliably search for the optimal solution of network reconstruction, it is compared with the commonly used optimization algorithms. The traditional single-population genetic algorithm (SGA) and the double sub-swarms particle swarm optimization algorithm (DSPSO) [33] are selected as the comparison algorithms. Take fault scenario 2 as an example. When SGA is adopted, 200 individuals are set, the iteration number is 200, the crossover probability is 0.7 and the mutation probability is 0.014. The algorithm is also run continuously five times. The convergence results are shown in Figure 15.  
For fault scenario 2, when SGA is used, the optimal fitness values corresponding to the convergence results of five runs are 0.91765, 0.93295, 0.9217, 0.93064 and 0.90096. The convergence results are not all the same, only one-time convergences to the optimal solution, with an average time of 4.92 s.  
Figure 14: The revised system topology under fault scenario 3. Figure 13: Convergence procedure of the MPGA under fault scenario 3. This is because the SGA only relies on the genetic evolution of a single population in the genetic process; it cannot provide the optimal recovery strategy. The corresponding switch-coding state with the fitness of 0.9217 is: 10101 11110 111111 11100 11110 01111 11000 10011 11101 11101 11100 111100 11111 111100 11111 111100 11111 11111. Decode the network reconstruction result. The corresponding network reconstruction scheme is: turn on switches S\({}_{5}\), S\({}_{9}\), S\({}_{40}\), S\({}_{57}\), S\({}_{58}\) and turn off switches S\({}_{2}\), S\({}_{37}\), S\({}_{44}\), S\({}_{49}\), S\({}_{50}\). Comparing with the optimal reconstruction scheme obtained by MPGA, the differences lie in that load L\({}_{20}\) exits, which increases the off-power load. Close switch S\({}_{5}\) and S\({}_{9}\) but not S\({}_{4}\) and S\({}_{10}\), so the tie lines are not effectively connected. This is equivalent to invalid switch closing, which increases the number of switch operation costs. The outputs of each generator are 7.75 p.u., 8 p.u., 1 p.u. and 0, and the lowest operating efficiency of the generator is 96.87%. The operating efficiency is reduced. When DSPSO is adopted, the size of the main group is set as 200, and the size of the auxiliary group is set as 100. The dimensions of both the main group and the auxiliary group are set as 20, the number of iterations is 200, the acceleration factor is 2 and the initial inertia weight is 0.9. The algorithm is also run continuously five times. The convergence results are shown in Figure 16.  
When DSPSO is used, the optimal fitness values corresponding to convergence results of five runs is 0.93295, 0.9191, 0.93295, 0.93295 and 0.93295. In most cases, it could converge to the optimal solution, only once failing to do so. The average searching time is 4.13 s. The optimal solution is searched through the joint search of the main group and the auxiliary group. Although it is helpful for finding the optimal solution, the search time is long due to the large sizes of the main group and the auxiliary group. By comparing with different algorithms, the maximum fitness based on MPGA, SGA and DSPSO algorithm is 0.93295. This solution was verified by multiple simulation results, and it can be viewed as the optimal fitness. Moreover, the corresp  
Figure 16: Convergence procedure of the DSPSO under fault scenario 2. Figure 15: Convergence procedure of the SGA under fault scenario 2.  
is considered the optimal solution. Under the condition of five consecutive simulations, MPGA achieves the optimal solution all five times, and the reliability is 100%. SGA achieves the optimal solution only once, and the reliability is 20%. DSPSO achieves the optimal solution four times, and the reliability is 80%. In the searching process, the average times for MPGA, SGA, DSPSO are 3.12 s, 4.92 s and 4.13 s, respectively.