## 6 Experiments and Results
### Experiment A  
The benchmark solution is a representative startup profile reported by [PERSON] et al. [17]. In this solution, the operating procedure consists of keeping the heat input at a constant value of 8 MW/min and the steam output valve fully open since the very beginning. After executing the operating procedure with the OpenModelica simulation model, the startup completed in 3000 s.  
Figure 10 shows the plot of function \(f(P_{sat},q_{s})=\alpha(P_{stem}(t)-P_{goal})^{2}+\beta(q_{steam}-q_{goal})^ {2}\) which measures the distance to the goal state over time. The [PERSON] operating procedure was a feasible solution as it never exceeded the limits imposed by the thermal stress constraint, maintaining a stress value between \(-10\) MPa and 10 MPa. The goal state was reached after 3000 s. The stress profile is shown in Figure 11.  
The micro genetic algorithm (mGA) implemented the same encoding scheme of this paper. The probabilities used in all the experiments were 10% for mutation and 20% for crossover. The population of the mGA consisted of 5 individuals, and the termination criteria were set to a maximum of 40 generations and 20 epochs, respectively. The metaheuristic hybrid algorithm was initiated with a randomly generated solution and stopped after 1000 iterations. Table 3 shows the solution obtained as a result of the optimization. Table 4 shows the decoded operating procedure based on the operations shown in Table 1. After reaching the desired goal, the heat inlet valve and the steam outlet valve took the values of 0 MW/min and full open respectively. Figure 9: Probability of selecting a better or a worse solution (50 s worse), using different \(\gamma\) values, depending on the progress of a run of 1000 iterations. Figure 11: Changes of thermal stress in the drum boiler over time with the benchmark solution. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time** & **Heat Inlet Valve** & **Steam Outlet Valve** & **Accumulated Time** \\ \hline
60 & 24 & 0.6 & 60 \\
840 & 8 & 1 & 900 \\
660 & 24 & 1 & 1560 \\ - & 0 & 1 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Operating procedure decoded from the solution shown in Table 3.  
Figure 10: Change of the distance to the goal state over time with the benchmark solution. The operating procedure obtained with the metaheuristic hybrid algorithm produced a feasible thermal stress profile of the thick-walled component. This thermal stress profile had a similar shape pattern and magnitude as the one obtained with either the benchmark solution or the solution obtained with the micro genetic algorithm, which means that the drum boiler integrity was not affected. Figure 13 shows the behavior of the thermal stress, which is related to the structural integrity of the drum boiler. The feed water flow had to be controlled so that the water level inside the drum was kept at its set point. A PI controller was used for this purpose. Figure 14 shows the behavior of the water level during the entire drum boiler startup process, where the metaheuristic hybrid algorithm achieved a more stable pattern compared to the micro genetic algorithm and the benchmark solution. Figure 12: Change of the distance to the goal state over time over time for the benchmark solution (red line), the micro genetic algorithm (black line) and the proposed metaheuristic hybrid algorithm (green line). Figure 13: Comparison of the the thick-walled thermal stresses generated with the benchmark solution (red line), the micro genetic algorithm solution (black line) and the metaheuristic hybrid algorithm solution (green line). Figure 14 shows that the water level changed abruptly during the drum boiler startup process. The instability of the liquid level could be reduced with the use of a non-linear controller that can adapt to the non-linearity of the model or the use of the gain scheduling approach which involved the application of different controller tuning parameters as a process transitions from one operating range to another [27].