## 4 Results
### Analysis and Discussion  
The Hypervolume metric is used as an evaluation criterion for multi-objective optimization algorithms. The table below presents a comparison between the NSGA-II-MC algorithm designed in this paper and traditional multi-objective optimization algorithms. Nine different constrained multi-objective optimization algorithms, namely AGEMOEA [28], ANSGA-III [29], ARMOEA [30], CCMO [31], CTAEA [32], DCNSGA-III [33], NSGA-II [34], NSGA-III [35], and RVEA [36], which are considered to be the top performers in the field, are selected as benchmark algorithms. These algorithms were applied to solve the introduced energy management problem using the standard PlatEMO platform [37]. The general configurations of NSGA-II-MC and the compared algorithms are kept the same, and are the default set of parameters in PlatEMO:  
1. The probability of crossover, the distribution index of simulated binary crossover, the expectation of the number of mutated variables, and the distribution index of polynomial mutation are set to 1, 20, 1, and 20, respectively;
2. The maximum number of function evaluations was 100,000, i.e., the population size was 100 and the maximum number of iterations was set to 1000. Other algorithm-specific configurations were chosen to be exactly the same as those in PlatEMO. NaN values represent cases where the algorithm could not find any feasible solutions. Table 6 provides the average Hypervolume values obtained by each algorithm over 21 runs on various scales of energy management problems, as well as the variance in the metric values across the 21 runs for each algorithm. Moreover, we use the Wilcoxon rank-sum test \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Controllable Device**} & \multicolumn{2}{c}{**Minimum/Maximum**} & \multirow{2}{*}{**Earliest Start Time**} & \multirow{2}{*}{**Latest End Time**} & \multirow{2}{*}{**Operating Duration (h)**} & \multicolumn{1}{c}{**Energy Consumption (kWh)**} \\  & & & & & \\ \hline Device 1 & 35/105 & 5 & 21 & 6 & 420 \\ Device 2 & 40/120 & 8 & 22 & 3 & 240 \\ Device 3 & 50/150 & 9 & 18 & 4 & 400 \\ Device 4 & 65/195 & 9 & 18 & 2 & 260 \\ Device 5 & 55/180 & 8 & 20 & 5 & 300 \\ Device 4 & 20/300 & 12 & 23 & 7 & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters for controllable loads. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Device**} & \multicolumn{2}{c}{**Maximum/Minimum**} & \multirow{2}{*}{**Ramp-Up Power**} & \multirow{2}{*}{**Minimum Operation/Shutdown**} & \multirow{2}{*}{**Fuel Cost**} & \multirow{2}{*}{**Startup/Shutdown**} \\  & & & & **Duration (h)** & **Constant** \\ \hline
1 & 600/10 & 500 & 2/1 & 0.00044/0.48/3.2 & 3.1/3.36 \\
2 & 760/20 & 550 & 2/1.5 & 0.00054/0.55/3.6 & 3.52/4.2 \\
3 & 40/0.5 & 40 & 0.5/0.5 & 0.0015/0.74/2.5 & 1.1/1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for distributed controllable generation devices. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Device**} & \multicolumn{2}{c}{**Maximum/Minimum**} & \multirow{2}{*}{**Ramp-Up Power**} & \multirow{2}{*}{**Minimum Operation/Shutdown**} & \multirow{2}{*}{**Fuel Cost**} & \multirow{2}{*}{**Startup/Shutdown**} \\  & & & & **Constant** & **Costs (S)** \\ \hline
1 & 600/10 & 500 & 2/1 & 0.00044/0.48/3.2 & 3.1/3.36 \\
2 & 760/20 & 550 & 2/1.5 & 0.00054/0.55/3.6 & 3.52/4.2 \\
3 & 40/0.5 & 40 & 0.5/0.5 & 0.0015/0.74/2.5 & 1.1/1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for distributed controllable generation devices. with \(p<0.05\) to compare each algorithm with NSGA-II-MC. In the last column of the table, the symbols \"+\" and \"-\" indicate the number of test problems in which the compared algorithm shows significantly better performance or worse performance, respectively, than NSGA-II-MC. In addition, the symbol \"=\" indicates the number of test problems in which there is no significant difference between NSGA-II-MC and the compared algorithms.