## 3 Problem Formulation  
### Objective Function  
Stochastic programming simulates abundant scenarios to feature variability of random variables. It is an efficient method of solving uncertainties in power system dispatch issues. Most of the current research usually employs a two-stage SUC model to deal with wind power uncertainties. In the two-stage SUC model, the first-stage decision variable is the state of the generation start-up and shutdown. The commitment states of generators are resolved a day ahead on the basis of wind power forecast data. The second-stage decision variables are real-time generation and wind outputs. At the second-stage, numerous wind scenarios are generated as representatives of wind power uncertainties. Subsequently, the real-time economic dispatch in every scenario is solved in this stage on the basis of day-ahead UC from the first-stage. The object is to minimize the sum of the generators' start-up cost and the generators' expected operation cost. However, our paper introduces ACLs into the SUC problem and develops a two-stage SUC model considering the DR resource from ACLs and wind power. Thus, the ACL reduction as a decision variable should be considered in the second-stage, and the objective cost should include the expected compensation cost because of the ACL reduction. The objective function is described as follows:  
\[\begin{array}{rl}\min f=&\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N_{C}}C_{ UGC,i,t}(1-S_{G,i,t-1})S_{G,i,t}\\ &+\sum\limits_{s=1}^{N_{S}}g_{s}\sum\limits_{t=1}^{T}\left[\sum\limits_{i=1}^ {N_{C}}C_{G,i,t,s}S_{G,i,t}+\sum\limits_{n=1}^{N_{AC}}C_{AC,n,t,s}\right]\\ \\ &C_{G,i,t,s}=a_{i}P_{G,i,t,s}^{2}+b_{i}P_{G,i,t,s}+c_{i}\\ \\ &C_{AC,n,t,s}=K_{1}\cdot P_{AC,n,t,s}^{2}+(K_{2}+\lambda)\cdot P_{AC,n,t,s}\\ \end{array} \tag{13}\]  
where \(C_{UGC,i,t}\) refers to the start-up cost of the \(i\)th generation unit at time \(t\); \(C_{G,i,t,s}\) is the operation cost of the \(i\)th generation unit at time \(t\) in scenario \(s\); \(C_{AC,n,t,s}\) is the cost for the DR of the ACL; \(g_{s}\) is the probability of scenario \(s\); \(N_{G}\) is the total number of generation units; \(S_{G,i,t}\) is the switching state of the \(i\)th generation unit at time \(t\), with \(S_{G,i,t}=1\) for the generation unit being on and \(S_{G,i,t}=0\) for the generation unit being off; \(P_{G,i,t,s}\) is the active power of the \(i\)th generation unit at time \(t\) in scenario \(s\); \(a_{i}\), \(b_{i}\) and \(c_{i}\) are the cost coefficients of the \(i\)th generation unit, and \(P_{AC,n,t,s}\) is the load reduction of the \(n\)th-type ACL at time \(t\) in scenario \(s\). Equation (14) provides the calculation formula of the operation cost \(C_{G,i,t,s}\) of \(i\)-th generation unit at time \(t\) in scenario \(s\). The quadratic function is typically used to calculate the generators' operation cost [32]. Equation (15) represents the cost of using the \(n\)th-type ACLs to cut down the peak load at time \(t\) in scenario \(s\). It follows the same calculation pattern of Equation (6).  
Generally, the start-up cost of generators is described by an exponential function. In order to quickly calculate the start-up cost, this can be asymptotically approximated by a stairwise function [32] and formulated as follows:  
\[C_{UG,i,t}\geq K_{G,i,t}\left(S_{G,i,t}-\sum\limits_{n}^{l}S_{G,i,t-n}\right),\ l=1,\ldots,N_{Di} \tag{16}\]  
\[C_{UG,i,t}\geq 0 \tag{17}\]  
where \(K_{G,i,l}\) refers to the start-up cost of the \(i\)th generation unit at the interval \(l\), and \(N_{Di}\) is the total number of intervals in the start-up of the \(i\)th generation unit.