## 2 Problem Description
### The Reserve Provision Model of ESS
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Situation**} & \multicolumn{4}{c}{**ESS Status**} & \multirow{2}{*}{**Upward Reserve**} & \multirow{2}{*}{**Downward Reserve**} \\ \cline{2-2} \cline{5-5}  & **Before Reserve Activation** & & & \\ \hline
1 & 1 & 1 & \(P_{ESS,RU2}^{l,s}\) & \(P_{ESS,RD1}^{l,s}\) \\ \hline
2 & 1 & 0 & infeasible & \(P_{ESS,RD1}^{l,s}+P_{ESS,RD2}^{l,s}\) \\ \hline
3 & 0 & 1 & \(P_{ESS,RU1}^{l,s}+P_{ESS,RU2}^{l,s}\) & infeasible \\ \hline
4 & 0 & 0 & \(P_{ESS,RU1}^{l,s}\) & \(P_{ESS,RD2}^{l,s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Four reserve provisioning situations of the ESS. Figure 5: The ESS status before and after reserve activation. limitations of these upward reserve portions are represented by Equations (5)-(8). In detail, we have the following:  
* Situation 1: In this situation, \(u_{ESS}^{t,s}\) and \(u_{ESS,gtr,gtr,gtr}^{t,s}\) are equal to 1 (Table 1). Consequently, Equation (7) is non-binding while Equation (8) forces \(P_{ESS,RUL2}^{t,s}\) not to exceed \(P_{ESSmax}-P_{ESS,disch}^{t,s}\). Besides, \(u_{ESS}^{t,s}=1\) means that \(P_{ESS,ch}^{t,s}=0\), so that Equation (6) forces \(P_{ESS,RUL1}^{t,s}\) to be equal to zero. * Situation 3: With \(u_{ESS}^{t,s}=0\) and \(u_{ESS,gtr,gtr}^{t,s}=1\), Equation (6) forces \(P_{ESS,RUL1}^{t,s}\) to be equal to \(P_{ESS,ch}^{t,s}\) while Equation (8) ensures \(P_{ESS,RUL2}^{t,s}\) is limited by \(P_{ESSmax}\). Additionally, it can be seen that Equations (5) and (7) do not affect Equations (6) and (8). * Situation 4: With \(u_{ESS}^{t,s}\) and \(u_{ESS,gtr,gtr}^{t,s}\) are equal to 0, Equations (6) and (8) become to be non-binding. Equation (5) limits \(P_{ESS,RUL1}^{t,s}\) by \(P_{ESS,ch}^{t,s}\) while Equation (7) forces \(P_{ESS,RUL2}^{t,s}\) to be equal to zero. \[0\leq P_{ESS,RUL1}^{t,s}\leq P_{ESS,ch}^{t,s}\] (5) \[P_{ESS,ch}^{t,s}-\Big{(}1-\Big{(}u_{ESS,gtr}^{t,s}-u_{ESS}^{t,s} \Big{)}\Big{)}bigM\leq P_{ESS,RUL1}^{t,s}\leq P_{ESS,ch}^{t,s}+\Big{(}1-\Big{(}u _{ESS,gtr}^{t,s}-u_{ESS}^{t,s}\Big{)}\Big{)}bigM\] (6) \[-u_{ESS,gtr}^{t,s}bigM\leq P_{ESS,RUL2}^{t,s}\leq u_{ESS,gtr}^{t, s}bigM\] (7) \[-\Big{(}1-u_{ESS,gtr}^{t,s}\Big{)}bigM\leq P_{ESS,RUL2}^{t,s}\leq P _{ESSmax}-P_{ESS,disch}^{t,s}+\Big{(}1-u_{ESS,gtr}^{t,s}\Big{)}bigM\] (8)  
Similarly, depending on each situation in Table 1, the amount of downward reserve provided by ESS can be \(P_{ESS,RUL1}^{t,s}\), \(P_{ESS,RDL2}^{t,s}\) or both, and limited by Equations (9)-(12). \[0\leq P_{ESS,RUL1}^{t,s}\leq P_{ESS,disch}^{t,s} \tag{9}\] \[P_{ESS,disch}^{t,s}-\Big{(}1-\Big{(}u_{ESS}^{t,s}-u_{ESS,gtr}^{ t,s}\Big{)}\Big{)}bigM\leq P_{ESS,RDL1}^{t,s}\leq P_{ESS,disch}^{t,s}+\Big{(}1-\Big{(}u _{ESS}^{t,s}-u_{ESS,gtr}^{t,s}\Big{)}\Big{)}bigM\] (10) \[-\Big{(}1-u_{ESS,gtr}^{t,s}\Big{)}bigM\leq P_{ESS,RDL2}^{t,s}\leq \Big{(}1-u_{ESS,gtr}^{t,s}\Big{)}bigM\] (11) \[-u_{ESS,gtr}^{t,s}bigM\leq P_{ESS,RD2}^{t,s}\leq P_{ESSmax}-P_{ESS,ch}^{t,s}+u_{ESS,gtr}^{t,s}bigM \tag{12}\]  
Assuming that with the pre-determined operating schedule in the energy market, the ESS energy level after time interval \(t\) is \(E_{ESS,DA}^{t,s}\). After providing upward/downward reserve, the ESS energy level will be changed by an amount of \(E_{ESS,SR}^{t,s}\). It can be seen that \(E_{ESS,SR}^{t,s}\) is positive in the case of downward reserve and vice versa, negative in the case of upward reserve. Equation (13) determines the adjustment \(E_{ESS,SR}^{t,s}\) with \(\eta\) is the ESS charge/discharge efficiency and \(\Delta t\) is the time interval duration. Equation (14) ensures that the ESS reserve energy provision does not affect its scheduling in the energy market. Besides, Equation (15) shows that the ESS energy level should be limited by its rated capacity at any time. Figure 5:0  
Figure 5:1  
Figure 5:2