## 3 Problem Formulation
### Constraints
#### 3.2.2 Start-Up Power Requirement  
This constraint is represented by Equation (7), where \(P_{igen(t)}\) in [MW] is the generation capability function of generator \(i\), \(P_{start(t)}\) in [MW] is the start-up power requirement function of NBS generator \(i\), RV is the number of NCRES generators, and \(P_{restart(t)}\) in [MW] is the start-up requirement function of the NCRES generators. \[\sum_{i=1}^{N}P_{igen}(t)-\sum_{j=1}^{M}P_{istart}(t)-\sum_{rv=1}^{RV}P_{restart }(t)\geq 0,\ \ \ t=1,2,\ldots,T \tag{7}\]  
Equations (1)-(7) represent a nonlinear combinatorial optimization problem. The proposed formulation that corresponds to a MILP problem relies on a series of transformations that are described in the following paragraphs. Initially, it is necessary to introduce new decision variables for defining the generator capability function of Figure 2(a), which corresponds to a piecewise linear function. Note that Figure 2(a) shows the point \((t_{start}+t_{ictp},0)\) at which the generator begins to ramp up. The point \((t_{start}+t_{ictp}+P_{max}\,/\,Rri,P_{max})\), at which the generator reaches its maximum generation capability separates the curve into three segments: \(t_{i1}^{t}\), \(t_{i2}^{t}\), and \(t_{i3}^{t}\). Here, \(w_{i1}^{t}\) and \(w_{i2}^{t}\) are binary decision variables that restrict these three variables within the corresponding range. Equation (8) represents the piecewise generator capability function. Figure 3: Characterization of conventional generators: (**a**) capacity curves for BS and NBS units; (**b**) starting curve of an NBS generator. \[ \begin{split} P_{igen}(t)&=R_{ri}\cdot t_{l2}^{t}\\ t&=t_{11}^{t}+t_{12}^{t}+t_{i3}^{t}\\ w_{l1}^{t}\big{(}t_{istart}+T_{ictp}\big{)}&\leq t_{i1 }^{t}\leq t_{istart}+T_{ictp}\\ w_{l2}^{t}\frac{P_{\textit{imax}}}{R_{ri}}&\leq t_{i2 }^{t}\leq w_{l1}^{t}\frac{P_{\textit{imax}}}{R_{ri}}\\ 0\leq t_{i3}^{t}&\leq w_{l2}^{t}\left(T-t_{istart}-T _{ictp}-\frac{P_{\textit{imax}}}{R_{ri}}\right)\\ w_{l2}^{t}&\leq w_{l1}^{t}\\ w_{l2}^{t},w_{i1}^{t}&\in\{0,1\}\\ t_{i1}^{t},t_{i2}^{t},t_{i3}^{t}&\in(0,1,2,\ldots,T )\end{split}  \tag{8}\]  
In Equation (8), the start-up time of BS units is zero (\(t_{\textit{{{{{{kistart}}}}}}}=0\)). In consequence, Equation (8) can be reformulated as in Equation (9). \[ \begin{split} P_{igen}(t)&=R_{ri}\cdot t_{l2}^{t}\\ t&=t_{i1}^{t}+t_{i2}^{t}+t_{i3}^{t}\\ w_{k1}^{t}T_{ictp}&\leq t_{k1}^{t}\leq T_{\textit{{{ {{kctp}}}}}}\\ w_{j1}^{t}\big{(}t_{istart}+T_{jctp}\big{)}&\leq t_{j1 }^{t}\leq t_{jstart}+T_{jctp}\\ w_{l2}^{t}\frac{P_{\textit{imax}}}{R_{ri}}&\leq t_{i2 }^{t}\leq w_{i1}^{t}\frac{P_{\textit{imax}}}{R_{ri}}\\ 0\leq t_{k3}^{t}&\leq w_{k2}^{t}\left(T-T_{\textit{{{ {{{kctp}}}}}}}-\frac{P_{\textit{{{{{{ktx}}}}}}}}{R_{rk}}\right)\\ 0\leq t_{j3}^{t}&\leq w_{j2}^{t}\left(T-t_{istart}-T _{jctp}-\frac{P_{\textit{imax}}}{R_{rj}}\right)\\ w_{l2}^{t}&\leq w_{i1}^{t}\\ w_{l2}^{t},w_{l1}^{t}&\in\{0,1\}\\ t_{i1}^{t},t_{i2}^{t},t_{i3}^{t}&\in(0,1,2,\ldots,T )\end{split}  \tag{9}\]  
Equation (9) corresponds to a non-linear constraint. Note that there is a product of one binary decision variable (\(w_{j1}^{t}\)) and one integer decision variable (\(t_{\textit{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{ } }                                                                }}}}}}}}}}}}\)). Equation (9) can be transformed into linear expressions by using Equation (10). In this case, \(BigM\) is an upper positive bound of \(y_{jh}^{t}\) and \(w_{jh}^{t}\) is its associated binary variable. If \(w_{jh}^{t}=1\), then the constraint is relaxed and is met by default \(y_{jh}^{t}\leq BigM\). Otherwise; if \(w_{jh}^{t}=0\), then \(y_{jh}^{t}\leq 0\). So this constraint allows modeling the proposition \(w_{jh}^{t}=0\Longrightarrow y_{jh}^{t}\leq 0\). Furthermore, if \(y_{jh}^{t}>0\) then \(w_{jh}^{t}=1\). If \(y_{jh}^{t}\leq 0\) the constraint does not imply anything \(y_{jh}^{t}>0\Longrightarrow w_{jh}^{t}=1\).