## 3 DR Resource Allocation Strategy Based on Multiple Response Modes
### DR Resource Allocation Model Based on Multiple Response Modes  
Considering the multiple response characteristics of DR resources, a day-head dispatch model of power systems is established. Here, we consider the startup/shutdown schedule of thermal units. Thus, the dispatch model in this paper is a mixed-integer nonlinear programming (MINLP) model to reduce the computational complexity of the proposed optimization with optimality guaranteed [28]. (1) Objective function  
After DR resources are involved in the dispatch process, the system combining thermal unit and wind power is optimized for the dispatch operation considering the uncertainty of wind power output. The objective function is to minimize the comprehensive expected system cost, which includes the startup and shutdown costs of thermal units. It can be established as:  
\[\min f=f_{1}+f_{2}+f_{3} \tag{13}\]  
where \(f\) is the total cost. \(f_{1}\), \(f_{2}\) and \(f_{3}\) are the thermal unit operation cost, DR dispatch cost and wind curtailment penalty cost, respectively. (i) Thermal unit operation cost  
Usually, the shutdown cost of thermal units is set as a smaller constant independent of the continuous operation period, while the startup cost is set as an exponential function of the time constant for the already shutdown time. Here, the startup and shutdown costs of the units are simplified as fixed parameters. \[STC_{i,t}=y_{i,t}\bullet C_{i} \tag{14}\]  
\[SDC_{i,t}=z_{i,t}\bullet Sd_{i} \tag{15}\]  
where \(y_{i,t}\) and \(z_{i,t}\) represent the startup and shutdown status of the \(i\)th unit, respectively. \(CS_{i}\) and \(Sd_{i}\) are the fixed startup and shutdown costs of the \(i\)th unit, respectively. \(STC_{i,t}\) and \(SDC_{i,t}\) represent the startup and shutdown costs of the \(i\)th unit, respectively. The coal cost of a unit is usually a binomial of its output, expressed as:  
\[f_{C}(p_{i}(t))=a_{i}\bullet p_{i}(t)^{2}+b_{i}\bullet p_{i}(t)+c_{i}\bullet u _{i}(t) \tag{16}\]  
where \(a_{i}\), \(b_{i}\) and \(c_{i}\) are the coal cost factors of the \(i\)th unit. To improve the computational efficiency, we use the piecewise linear function to linearize Equation (16) [29]. Then, the thermal unit operation cost can be expressed as:  
\[f_{1}=\sum_{t=1}^{T}\sum_{i=1}^{N_{C}}\left[f_{C}(u_{i}(t),p_{i}(t))+STC_{i,t} +SDC_{i,t}\right] \tag{17}\]  
where \(N_{C}\) represents the number of thermal units, \(T\) represents a dispatch cycle, \(u_{i}(t)\) and \(p_{i}(t)\) represent the operating state variable and the output of the \(i\)th thermal unit at time \(t\). (ii) Multiple DR resource dispatch cost  
The multiple DR resource dispatch cost can be expressed as:  
\[f_{2}=\sum_{t=1}^{T}[\sum_{m=1}^{N_{m}}C_{m}(t)+\sum_{j=1}^{N_{j}}C_{j}(t)+ \sum_{z=1}^{N_{Z}}C_{Z}(t)] \tag{18}\]  
where \(N_{m}\), \(N_{j}\) and \(N_{z}\) represent the number of IL, TL and PDR resources, respectively. (iii) Wind curtailment penalty cost  
The wind curtailment penalty cost can be expressed as:  
\[f_{3}=\min[\sum_{t=1}^{T}\sum_{w=1}^{N_{W}}W_{w}*P_{w,curt}^{s}] \tag{19}\]  
where \(W_{W}\) represents the price of the wind curtailment penalty. (2) Constraint conditions  
The constraints regarding this model include individual constraints and system constraints. The individual constraints contain DR resource multivariate response feature constraints and conventional thermal power characteristic constraints. System constraints contain system reserve constraints, power balance constraints and security constraints. (i) System constraints \(\bullet\) The spinning reserve constraint: \[\left\{ \begin{array}{l}\sum\limits_{i=1}^{N_{C}}u_{i}(t)\bullet p_{szi}(t)+ \sum\limits_{j=1}^{k}I_{j}(t)\bullet Q_{j-}(t)\geq\gamma\bullet L(t)+p_{w}(t) \bullet w_{u}\%\\ \sum\limits_{i=1}^{N_{C}}u_{i}(t)\bullet p_{sfi}(t)+\sum\limits_{j=1}^{k}I_{ j}(t)\bullet Q_{j+}(t)\geq(W^{\max}-P_{w}^{t})\bullet w_{d}\%\end{array} \right.\] (20) where \(p_{szi}(t)\) and \(p_{sfi}(t)\) are the positive and negative spinning reserve provided by the \(i\)th unit at time \(t\), respectively.