# Dynamic Coal Flow-Based Energy Consumption Optimization of Scraper Conveyor
## 5 Case Study and Validation
### Optimization Simulation and Results
#### 5.2.2 Simulation Results and Analysis  
The optimization of the scraper conveyor energy consumption model was performed, and the Pareto front solutions are shown in Figure 26. As seen in the figure, the optimization objectives typically exhibit conflict and incommensurability, meaning that there is no single feasible solution that satisfies all objectives simultaneously (such as minimizing energy consumption and minimizing the coal block area). However, in practical applications, decision-makers often need to choose the optimal operating conditions based on specific goals. Therefore, this study introduces the Ideal Point Method to evaluate the Pareto front. In each set of optimal solutions, there is typically a trade-off relationship between the two conflicting objective functions. The research suggests that the closer the design point is to the ideal point, the better its overall performance [39]. Since the objective values in the figure are unevenly distributed, direct numerical comparison is not appropriate. [PERSON] and [PERSON] [40] proposed that the actual values can be replaced by a normalized Pareto front for analysis, which makes the objectives comparable. Therefore, the optimization \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Values** & **Parameter** & **Values** \\ \hline Longwall length (L/m) & 340 & Cutting depth (H/m) & 0.8 \\ Triangle coal length (L\({}_{3}\)/m) & 50 & Coal seam dip angle (\(a_{3}\)) & 6 \\ Tecovery rate (c/\({}^{\circ}\)) & 0.98 & Mining height (B/m) & 3.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Working face parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Values** \\ \hline Population size & 100 \\ Number of generations & 200 \\ Crossover probability & 0.9 \\ Mutation probability & 0.1 \\ Probability parameter & 0.5 \\ Crossover distribution index & 10 \\ Mutation distribution index & 20 \\ Number of iterations & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: NSGAII-ARSBX parameters. objectives were normalized to make energy consumption and coal block area comparable. The normalized expressions for the objective functions E and Sa are as follows:  
\[E=\frac{E-E_{min}}{E_{max}-E_{min}} \tag{78}\]  
\[S_{a}=\frac{S_{a}-S_{amin}}{S_{amax}-S_{amin}} \tag{79}\]  
\(E_{max}\) and \(E_{min}\) denote the maximum and minimum values of the objective function along the horizontal axis of the Pareto front, respectively. Similarly, \(S_{amax}\) and \(S_{amin}\) represent the maximum and minimum values of the objective function along the vertical axis of the Pareto front. The goal of this study is to achieve a balance between energy consumption and lump coal area. Therefore, the ideal point aims to minimize both energy consumption and lump coal area simultaneously. However, it is important to note that this ideal point does not lie on the Pareto boundary. As shown in Figure 27, the ideal point is marked as point P. Point B, which is closest to the ideal point on the Pareto front, is defined as the optimal compromise solution on the Pareto boundary. Point B represents a solution that strikes a reasonable balance among multiple optimization objectives. After normalizing the Pareto front, an ideal optimal point \(p(E_{i},S_{i})\) can be determined based on specific optimization targets. In this study, the coordinates of the ideal optimal point are (\(E_{min}\)-\(S_{amin}\)). Figure 26: Pareto front solutions. Figure 27: Definition of different ideal states in the Pareto front. The Euclidean distance from a Pareto front point to the ideal optimal point is defined as follows:  
\[d=\sqrt{(E-E_{i})^{2}+(S_{a}-S_{ai})^{2}} \tag{80}\]  
Figure 27 presents the Pareto front scatter plot, illustrating the relationship between coal block area and energy consumption. According to the Ideal Optimal Point Method, Point B is identified as the closest solution on the Pareto front to the ideal point. At this point, the coal block area is 4912 mm\({}^{2}\) and energy consumption is 1.2617 \(\times\) 10\({}^{9}\) KJ.