## 5 Case Study  
The evaluation of the proposed methodology is carried out by means of a case study on the generation of the optimum operating procedure of a drum boiler. The problem consists of finding an operating procedure that takes the system from a given pressure and steam-flowrate values to the desired pressure and steam mass-flowrate values in the shortest time possible while avoiding excessive thermal stresses in the metal of the wall of the drum boiler. The simulator component was developed using the OpenModelica environment [25; 26]. OpenModelica has algorithms for solving differential equation systems, making it possible to observe changes in variables over time. The drum boiler simulation model in OpenModelica is based on the model reported by [PERSON] et al. [10]. The integration between the OpenModelica simulator and the optimization algorithm was carried out by implementing a two-way interface. First, the interface receives the operating procedure generated by the optimization algorithm. Then, it translates this procedure to a set of parameters for the simulation model. Finally, it evaluates and executes the simulation model. Conversely, the interface receives the results of the simulation and translates them into a format suited to the optimization algorithm. The optimization problem is formulated based on the work by [PERSON] et al. [17]. The goal is to reach given values of pressure and steam outlet flowrate by manipulating the heat inlet valve and the steam outlet valve. Accordingly, the optimization problem is formulated as:  
\[Min((A*S)+\alpha(P_{sat}-P_{goal})^{2}+\beta(q_{s}-q_{goal})^{2}) \tag{5}\]  
where  
\[S=\sum_{t_{0}}^{t_{f}}dt \tag{6}\]  
Subject to:  
\[0\leq V_{pos}\leq 1 \tag{7}\]  
\[0\,\frac{\text{MW}}{\text{min}}\leq\frac{dQ}{dt}\leq 25\,\frac{\text{MW}}{\text{ min}} \tag{8}\]  
\[0\leq Q\leq 500\,\text{MW} \tag{9}\]  
\[-10\,\text{MPa}\leq\sigma_{D}\leq 10\,\text{MPa} \tag{10}\]  
Equation (5) seeks to minimize the time it takes for the drum boiler to reach the goal. The parameters \(\alpha\) and \(\beta\) from Equation (5) are the ones specified by [PERSON] et al. [17]. When \(A=0\) the problem is reduced to finding a sequence of operations that is feasible but not necessarily optimal [9]. A feasible solution is a solution that reaches the goal state without violating any of the constraints. In this optimization problem \(P_{goal}\) is the desired internal pressure, \(q_{goal}\) is the desired steam mass-flow rate, \(\alpha\) and \(\beta\) are weights, \(P_{sat}\) is the steam pressure at the drum boiler, and \(q_{s}\) is the steam flow rate at the drum boiler outlet. Equation (6) represents the time that the system takes to get from the initial state to the final state. Equation (7) constrains the opening of the steam outlet valve (\(V_{pos}\)) to values between \(0\) (totally closed) and \(1\) (fully open). Equation (8) ensures that the heat rate does not exceed \(25\,\text{MW}/\text{min}\). It is a nonlinear constraint because it implies that there could be different heat ramp rates during the process. Equation (9) is a constraint of the accumulated heat limit of the drum boiler which must not exceed \(500\,\text{MW}\) and Equation (10) is a constraint that avoids excessive thermal stress in the drum boiler that must be less than \(10\,\text{MPa}\). The water is supplied by a control system and the steam flow is controlled by a valve. The interaction between the opening of the steam outlet valve and the heat rate in the drum boiler generates steam at pressure \(P_{sat}\) which exits at flow rate \(q_{s}\). The steam can later be sent to a superheater or directly to a steam turbine [9]. For this problem, the goal was set to \(P_{goal}=9\,\text{MPa}\), \(q_{goal}=180\,\text{kg}/\text{s}\). The weights were set to \(\alpha=10^{-4}\) and \(\beta=10^{-4}\). Parameter \(A\) is set to \(0\) during the initial solution generation and then changed to \(A=1\) for the rest of the algorithm.