## 3 Problem Formulation
### Objective Function  
In the development of this research the objective functions defined in [29; 30; 31] are taken as a guide. The direct objective is to maximize the system's overall generation capacity \(Egen_{sys}\) during a restoration period. Further, the indirect objective is to minimize the unserved load energy \(ENS_{sys}\) of the power system. In Equation (1), \(Egen_{sys}\) is the sum of MW generation capabilities over all units in the power system minus the start-up power requirements. \(E_{igen}\) and \(E_{jstart}\) are the MW capability of generator \(i\) and start-up requirement of NBS generator \(j\), respectively. Further, N and M are the number of BS and NBS units, respectively.  
\[Egen_{sys}=\sum_{i=1}^{N}E_{igen}-\sum_{j=1}^{M}E_{jstart} \tag{1}\]  
Figure 2: Simplified scenario of NCRES participation in a restoration process.  
Equation (1) can be expressed in terms of BS and NBS generators as indicated in Equation (2). In this case, k and j are the indexes of BS and NBS generators, respectively; while N and M are the total number of generators and NBS generators, respectively.  
\[\begin{split} Egen_{sys}=\left[\sum_{k=1}^{N-M}E_{kgen}+\sum_{j=1} ^{M}E_{jgen}\right]-\sum_{j=1}^{M}E_{jstart}\end{split} \tag{2}\]  
Figure 3 characterizes the capacity of BS and NBS units and the starting requirements of an NBS generator. The MW capability of each generator, over the system restoration horizon, can be calculated using the area under the curve of Figure 3a. Similarly, the start-up requirement for each NBS generator is represented using the area under the curve of Figure 3b. The MW capacity is obtained using Equation (3). In this case, \(P_{max}\) in [MW] is the maximum generator active power output, \(P_{start}\) in [MW] is the start-up power requirement for generator \(j\), \(R_{ri}\) in [MW/h] is the generator ramping rate, \(t_{start}\) in [h] and \(t_{start}\) in [h] are the restoration times of the ith and jth generators, respectively, and \(T\) is the total restoration time in [h].  
\[\begin{split} Egen_{sys}&=\sum_{i=1}^{N}E_{igen}- \sum_{j=1}^{M}E_{jstart}\\ &=\sum_{i=1}^{N}\biggl{\{}\frac{1}{2}P_{max}\frac{P_{max}}{R_{ri} }+P_{max}\biggl{[}T-\biggl{(}t_{start}+T_{ictp}+\frac{P_{max}}{R_{ri}}\biggr{)} \biggr{]}\biggr{\}}-\sum_{j=1}^{M}P_{jstart}\bigl{(}T-t_{jstart}\bigr{)}\\ &=\left\{\sum_{i=1}^{N}\biggl{[}\frac{(P_{max})^{2}}{2R_{ri}}+P_{ imax}\biggl{(}T-T_{ictp}-\frac{P_{imax}}{R_{ri}}\biggr{)}\biggr{]}-\sum_{j=1}^{M}P_{ jstart}T\right\}-\\ &\quad\left(\sum_{i=1}^{N}P_{imax}t_{start}-\sum_{j=1}^{M}P_{ jstart}t_{jstart}\right)\end{split} \tag{3}\]  
Simplifying Equation (3) as in [29] leads to Equation (4) where \(P_{jmax}\) is the maximum capability of NBS units, \(P_{jstart}\) is the start-up power requirement constraints of NBS units, and \(t_{jstart}\) is their corresponding start times.  
\[\begin{split} E_{sys}=\sum_{j=1}^{M}\bigl{(}P_{jmax}-P_{jstart} \bigr{)}t_{jstart}\end{split} \tag{4}\]  
The proposed objective function can be expressed as indicated in Equation (5), which also considers the minimization of the unserved load energy. In this case, the factor \((P_{Lyss}-P_{Lcl})\Delta t\) is the unserved load energy; where \(P_{Lyss}\) in [MW] is the total power demand of the system and \(P_{Lcl}\) in [MW] is the total restored load at time \(\Delta t\) in [h].  
\[\begin{split} minimize\sum_{j=1}^{M}\bigl{(}P_{jmax}-P_{jstart} \bigr{)}t_{jstart}+\sum_{t=1}^{T}\sum_{l=1}^{L}\bigl{(}P_{Lyss}-P_{Lcl}\bigr{)} \Delta t\end{split} \tag{5}\]