## 5 Experiment
### The Performance Comparison of PMOEVO with Other Algorithms  
To evaluate the performance of the PMOEVO for solving the energy-production-scheduling problem, a comparison will be made between PMOEVO and traditional EVO, as well as other existing high-performance algorithms, such as NSGA-II [36], NSGA-III [37], MOPSO [38], MOEA/D [39], and MOGWO [40]. The parameter settings for the other algorithms, in addition to EVO and PMOEVO, are shown in Table 7. Each algorithm will be run 20 times, with the maximum runtime as the stopping condition for the iterations. Each algorithm is run separately based on the experimental setup parameters. The Pareto front from each algorithm run is calculated, and the resulting Pareto front graph is shown in Figure 11.  
In Figure 11, each point of the algorithms represents the Pareto front solution set produced after iterations (indicating the best-performing solutions found by the algorithm). Figure 11 plots the Pareto solution sets generated by each algorithm together, where in each subplot, points closer to the lower-left corner represent better performance (with lower costs and CO\({}_{2}\) emissions). Over the 20 experiments, the solutions generated by the PMOEVO proposed in this paper dominate the majority of the Pareto front. On one hand, the performance of the improved PMOEVO significantly outperforms the basic EVO algorithm, while on the other hand, most of the PMOEVO solutions dominate those generated by the other algorithms. Figure 10: Robust optimization effect diagram. Furthermore, the solutions produced by PMOEVO are uniformly distributed along the Pareto front. To further assess the algorithm's performance, the Pareto front ratio (PFR) and Inverted Generational Distance (IGD) are introduced as evaluation metrics. The selection of Pareto front ratio (PFR) and Inverted Generational Distance (IGD) as evaluation metrics for PMOEVO stems from their complementary strengths in assessing multi-objective optimization performance. PFR quantifies the quality of the solution set by measuring the proportion of solutions generated by the algorithm that match the benchmark front. Essentially, it reflects the algorithm's ability to identify practically deployable solutions in energy scheduling scenarios. This metric aligns with engineering requirements, where the number of high-quality solutions directly impacts the flexibility of operations and maintenance. IGD innovatively integrates convergence and diversity assessment through normalized distance computation. By averaging the minimum distances from solutions to the algorithm's output set \(X_{BS}\), it simultaneously captures proximity to the true Pareto front and distribution uniformity. The calculation method for PFR is as follows:  
\[PFR_{i}=\frac{card(X_{i}\cap X_{BS})}{card(X_{BS})} \tag{61}\]  
where \(PFR_{i}\) refers to the Pareto front ratio of algorithm \(i\). \(X_{BS}\) is the set of Pareto front solutions obtained from all algorithms participating in the comparison. \(X_{i}\) represents the solution set obtained by algorithm \(i\). \(card(X_{i}\cap X_{BS})\) refers to the number of solutions generated by algorithm \(i\) that are present in \(X_{BS}\). A larger value of \(PFR_{i}\) signifies better performance of the solution set obtained by algorithm \(i\). Before calculating IGD, first normalize the objective values of all solutions obtained by the algorithms using the maximum and minimum values. The calculation method for IGD is as follows:  
\[IGD_{i}=\frac{\sum_{l=1}^{card(X_{BS})}d_{x,i}}{card(X_{BS})} \tag{62}\]  
where \(d_{x,i}\) refers to the distance from the \(x\)-th solution on the Pareto front to the solution produced by the nearest algorithm \(i\). \(IGD\) refers to the convergence and diversity of the non-dominated solution set. The smaller the \(IGD_{i}\), the better the objective values of the solutions produced by algorithm \(i\) and the more uniform the distribution along the Pareto front.