## 2 Materials and Methods
### Description of the Proposed Control Strategies
#### 2.1.2 Economic-Oriented Model Predictive Controllers  
The economic-oriented model predictive controllers follow a similar strategy to the standard MPC controllers, but in this case, the cost function includes a term representing the economic performance, the energy savings, or a measure of process performance associated to the economic profit. There are several types of economic-oriented controllers with these characteristics [17], and in this work, the economic optimizing controller (pure-EMPC) and the economics-oriented tracking controller (Hybrid-EMPC or HEMPC) have been considered. In the pure EMPC, the control actions are obtained considering a cost function that only includes this economic term, without any other tracking or control-movement terms. On the other hand, the economics-oriented tracking controller (HEMPC) cost function includes the economic term as well as the other terms explained in the previous section [25; 26]. Note that the HEMPC is more appropriated in the cases where some output variables must be maintained around some reference values to ensure the desired process performance, while also allowing for some deviation from the set point. This deviation is tolerable when economic performance and the tracking of the reference are conflicting objectives or in contrast, the simultaneous optimization of economics and control performance is possible because additional degrees of freedom are available [26]. The tuning parameters have to be carefully selected to find an appropriated compromise solution. A generalized mathematical description that includes both alternatives is presented here. \[\min_{\mathbf{u}}\int\limits_{0}^{\tau_{N}}(w_{E}\cdot J_{ECO}(\hat{\mathbf{x}}(t ),\mathbf{u}(t))+w_{C}\cdot J_{NMPC}(\hat{\mathbf{x}}(t),\mathbf{u}(t)))dt \tag{2}\]  
subject to:  
\[\dot{\hat{\mathbf{x}}}(t)=f(\hat{\mathbf{x}}(t),\mathbf{u}(t),\mathbf{v}(t)) \tag{2a}\] \[\hat{\mathbf{x}}(0)=\mathbf{x}(\tau_{k})\] (2b) \[g(\hat{\mathbf{x}}(t),\;\mathbf{u}(t))\leq 0,\;\forall t\in\;[0,\; \tau_{N}) \tag{2c}\]  
\(J_{ECO}\) in (Equation (2)) is an economic performance index measuring the operating profits or expenses, and \(J_{NMPC}\) is the MPC cost function described in the previous section. The relative importance of the objectives in the cost function is given by the weights \(w_{C}\) and \(w_{E}\). They have to be carefully selected to represent the desired compromise between economics and control performance. The constraints and initial conditions are defined by Equations (2a)-(2c). Since both objectives are represented as separate terms, the problem described results in a conventional MPC formulation when the weight \(w_{E}\) is set to zero (\(w_{E}=0,w_{C}\
eq 0\)), and in an economic-optimizing controller or pure EMPC formulation, when \(w_{C}\) is set to zero (\(w_{E}\
eq 0,\,w_{C}=0\)). If \(w_{E}\
eq 0,\,w_{C}\
eq 0\), then it is the case of an economic-oriented tracking controller or HEMPC, where the two terms have to be suitably weighted to represent the desired compromise between economics and control performance. The decision variables in Equation (2) are the manipulated inputs \(\mathbf{u}\), which are generally defined as a trajectory along the prediction horizon \(\mathbf{u}(t\,|\,\tau_{k})\). For each input trajectory \(\mathbf{u}\) that the optimizer proposes (note that the optimization is solved iteratively), the predicted states \(\hat{\mathbf{x}}\) and outputs \(\mathbf{y}\) are obtained using the non-linear model (1). However, in the implementation of the controller in this work, the trajectory \(\mathbf{u}(t\,|\,\tau_{k})\) is considered to be a constant value \(\mathbf{u}\) for simplicity, and this will be the control action applied to the plant. Therefore, there is only one decision value in Equation (2), and the problem can be solved easily by Sequential Quadratic Programming methods (SQP).