## 5 Case Study
The nonlinearity of Equations (7) and (8) add complexity to the problem, which justifies the use of metaheuristic methods to find a solution. To ensure that the optimization algorithm converges, the stopping condition of the algorithm is met when \(1000\) iterations are performed. It is worth mentioning that several tests with more iteration limits were performed, but the algorithm always converged before 1000 iterations. Each operating procedure is represented according to the encoding scheme explained in Section 4.2.1, in which each operation is formed by combining discrete values of the heat flow and the valve position of the steam outlet valve. This results in the nine operations shown in Table 1.  
The execution time per operation is set to 60, 120, or 180 s. The repetition parameter is set to vary between 0 to 9. The length of the sequence is fixed to nine elements. Figure 7 shows an example of an operating procedure. The first element in the sequence represents operation 8 (\(dQ/dt=24\) MW/min, \(V_{pos}=0.6\)) being executed for 60 s and repeated three times. The feasibility function \(f(x)\) is calculated with Equation (11). An extra penalty can be applied to \(f(x)\) in case the total time of the generated sequence is less than 1200 s, a value too low to be feasible. \[ \begin{split} if\{(t<1200,f(x)=V_{1}+100;\,t>1200,f(x)=V_{1})\} \\ V_{1}=(400-Q)+f_{s}+G\\ G=\alpha(P_{sat}-P_{goal})^{2}+\beta(q_{s}-q_{goal})^{2}\end{split}  \tag{11}\]  
where \(Q\) is the accumulated heat that must reach 400 MW, \(f_{s}\) represents the number of times the thermal stress exceeded the 10 MPa limit throughout the process and \(G\) represents how far is the sequence of approaching the steam pressure and steam outflow goal. During the optimization process, new solutions are generated using the NOP. The NOP takes an existing solution and makes a mutation by randomly change one of the sequence elements (operation, time, and repetition). A neighborhood operator can be applied multiple times as a solution-diversification strategy. Figure 8 shows an example of NOP applied four times. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation** & **Heat Flow \(dQ/dt\)** & \(V_{pos}\) \\ \hline
1 & 8 & 0.0 \\
2 & 8 & 0.6 \\
3 & 8 & 1.0 \\
4 & 16 & 0.0 \\
5 & 16 & 0.6 \\
6 & 16 & 1.0 \\
7 & 24 & 0.0 \\
8 & 24 & 0.6 \\
9 & 24 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operations table. Figure 7: Solution representation of an operating procedure for the drum boiler optimization problem. The number of times that NOP is applied is set based on the value of \(f(x)\) as shown in Table 2. The number of times NOP is applied depends on the length of the solution for the given problem. In this case study, the maximum number of mutations is four because it changes half the values of the previous solution in the worst scenario for \(f(x)\). For the metaheuristic hybrid algorithm and the simulated annealing algorithm, it is necessary to specify the acceptance probability function. This function is represented by Equation (12):  
\[P=e^{-\frac{((t(x)-t(x))\cdot\tau)}{t}} \tag{12}\]  
where \(P\) is the probability of selecting a worse neighbor solution, \(T\) is a parameter that gradually decreases as the algorithm proceeds (\(T\) is also known as the annealing temperature), \(t(y)\) is the final time of the neighbor solution and \(t(x)\) the final time of the actual solution because the time is the value that is sought to optimize in this problem. \(\gamma\) is a parameter that magnifies the difference between two solutions. To select the value of \(\gamma\), the value of the Boltzman distribution was analyzed with a difference of 50 s between two solutions, varying the value of \(\gamma\) between 1 and 15. The value of \(\gamma=10\) was selected as it starts with a probability slightly larger than 50% but steadily decreasing through the iterations. Figure 9 shows the probability of selecting a worse solution (50 s worse) using different \(\gamma\) values in a run of 1000 iterations.