## 3 DR Resource Allocation Strategy Based on Multiple Response Modes
### DR Resource Allocation Model Based on Multiple Response Modes
\(w_{u}\%\) and \(w_{d}\%\) are the percentage for positive and negative spinning reserves owing to the wind power output forecast error, respectively. \(Q_{j-}(t)\) and \(Q_{j+}(t)\) represent the shedding and adding load at time \(t\). \(\bullet\) System network security constraint We use the DC current model to characterize the system security constraint in the day-ahead scheduling model to ensure the convergence of the solution and computational efficiency. To reduce the solution complexity, only the maximum transmission capacity constraint of the line is considered as: \[PF_{l}(t)=B_{l}\bullet\theta(t)\] (21) where \(PF_{l}(t)\) represents the active power of the \(l\)th branch. \(B_{l}\) and \(\theta(t)\) represent the conductance matrix and voltage phase angle, respectively. \(\bullet\) Power balance constraint \[\sum\limits_{i=1}^{N_{C}}p_{i}^{s}(t)+\sum\limits_{m=1}^{Nm}D_{m}^{s}(t)+\sum \limits_{j=1}^{N_{j}}Q_{j}^{s}(t)+\sum\limits_{z=1}^{N_{z}}Q_{z}^{s}(t)+\sum \limits_{w=1}^{N_{w}}\left[P_{w}^{s}(t)-P_{w,curt}^{s}(t)\right]=L(t)\] (22) where \(P_{w,curt}^{s}(t)\) and \(L(t)\) represent the amount of wind power curtailment and the forecast load at time \(t\), respectively. \(\bullet\) Stability constraint As the scale of wind power integrated into the system increases, the stability of the system decreases. Thus, the stability of the system requires that the output provided by the thermal units cannot be lower than the minimum level, expressed as: \[\sum\limits_{i=1}^{N_{C}}p_{i}^{s}(t)\geq L(t)\bullet\beta\] (23) where \(\beta\) is the minimum demand factor to meet the system stability requirements. (ii) Individual constraints \(\bullet\) The upper and lower limits of the output constraint \(u_{i,t}\bullet p_{i,\min}\leq p_{i}(t)\leq u_{i,t}\bullet p_{i,\max}\) (24) where \(p_{i,\min}\) and \(p_{i,\max}\) represent the minimum and maximum output of the \(i\)th thermal unit, respectively. \(\bullet\) The minimum startup/shutdown time constraint \[\left\{ \begin{array}{l}[u_{i}(t-1)-u_{i}(t)][T_{i,on}(t-1)-UT_{i}]\geq 0 \\ [u_{i}(t)-u_{i}(t-1)][T_{i,off}(t-1)-DT_{i}]\geq 0\end{array} \right.\] (25) where \(UT_{i}\) and \(DT_{i}\) are the minimum online and offline hours of the \(i\)th thermal unit, respectively, \(T_{i,off}(t-1)\) and \(T_{i,on}(t-1)\) are the accumulated offline and online time of the \(i\)th thermal unit, respectively. * The ramping constraint \[\left\{ \begin{array}{l}p_{i}(t)-p_{i}(t-1)\leq R_{up,j}\\ p_{i}(t-1)-p_{i}(t)\leq R_{down,i}\end{array} \right.\] (26) where \(R_{up,i}\) and \(R_{down,i}\) are the upward and downward ramping rate of the \(i\)th thermal unit, respectively. * The maximum startup and shutdown power constraint \[\left\{ \begin{array}{l}\overline{p}_{i}(t)\leq SD_{i}\bullet Z_{i}(t+1)+p_{ i}^{\max}\bullet\left[\mathbf{u}_{i}(t)-z_{i}(t+1)\right]\quad\text{shutdowns at time}t+1\\ p_{i}(t)\geq p_{i}(t-1)-RD_{i}\bullet\mathbf{u}_{i}(t)-SD_{i}\bullet z_{i}(t) \quad\text{shutdowns at time}t\\ \overline{p}_{i}(t)\leq p_{i}(t-1)+RU_{i}\bullet\mathbf{u}_{i}(t-1)+SU_{i}\bullet y _{i}(t)\quad\text{starts up at time}t\end{array} \right.\] (27) where \(SD_{i}\) and \(SU_{i}\) are the maximum shutdown and startup power of the \(i\)th unit, respectively. * Constraints of DR resources in Equations (6)-(9).