## 2 Problem Description  
In this section, we develop a chance-constrained multi-objective formulation to address uncertain wind power availability. The two objectives are to maximize the utilization of wind power \(E_{W}\) and to minimize the system operation cost \(C_{GL}\), which including the costs of thermal generator and the operation costs of SLs and HLs. HLs and SLs are adopted in the optimization. The formulation is to determine the regulate or shift operations of the responsive loads. The detailed formulation is shown as follows:  
\[\left\{ \begin{array}{c}\max_{EW}=\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N_{W }}P_{Wi}^{t}\Delta T\\ \min_{GL}=C_{gm}+C_{SL}+C_{HL}\end{array} \right. \tag{1}\]  
\(s.t.\)  
\[\sum\limits_{j=1}^{N_{G}}u_{Gj}^{t}P_{Gj}^{t}+\sum\limits_{i=1}^{N_{W}}P_{Wi}^{ t}=P_{L}^{t}+\sum\limits_{a=1}^{N_{S}}S_{Sa}^{t}P_{Sa}^{t}+\sum\limits_{b=1}^{N_{b}}S_{HB }^{t}P_{Hb^{\prime}}^{t},\ t\in T \tag{2}\]  
\[\sum\limits_{j=1}^{N_{G}}u_{Gj}^{t}(P_{Gj}^{\max}-P_{Gj}^{t})\geq R_{L}^{t}+ \gamma\sum\limits_{i=1}^{N_{W}}P_{Wi}^{t},\ t\in T \tag{3}\]  
\[u_{Gj}^{t}P_{Gj}^{\min}\leq P_{Gj}^{t}\leq u_{Gj}^{t}P_{Gj}^{\max},\ j\in G,\ t\in T \tag{4}\]  
Figure 1: Introduction of an HL and an SL. (**a**) The scheduling characteristic of an HL. (**b**) The scheduling characteristic of an SL. \[-u_{Gj}^{t-1}+u_{Gj}^{t}-u_{Gj}^{k}\leq 0;\] \[(1\leq k-(t-1)\leq T_{Gj,\mu r}^{\min}j\in G,\ t\in T) \tag{5}\]  
\[u_{Gj}^{t-1}-u_{Gj}^{t}+u_{Gj}^{k}\leq 1;\] \[(1\leq k-(t-1)\leq T_{Gj,\mu r}^{\min}j\in G,\ t\in T) \tag{6}\]  
\[P_{Gj}^{t}-P_{Gj}^{t-1}\leq(2-u_{Gj}^{t-1}-u_{Gj}^{t})P_{Gj}^{\max}+(1+u_{Gj}^ {t-1}-u_{Gj}^{t})R_{Gj}^{up} \tag{7}\]  
\[ \begin{array}{c}P_{Gj}^{t-1}-P_{Gj}^{t}\leq(2-u_{Gj}^{t-1}-u_{Gj}^{t})P_{Gj}^ {\max}+(1-u_{Gj}^{t-1}+u_{Gj}^{t})R_{Gj}^{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1}{dumn{1} {dumn{1}}}}}}}}}}}}}}}\\ (j\in G,\ t\in T)\end{array}  \tag{8}\]  
\[0\leq P_{Wi}^{t}\leq P_{Wi}^{\max}i\in W,\ t\in T \tag{9}\]  
\[\Pr\{P_{Wi}^{t}-\gamma P_{Wi}^{t}\leq\zeta_{wi}^{t}\}\geq 1-\varepsilon\ i\in W,\ t\in T \tag{10}\]  
\[-S_{Sa}^{t}P_{Smax}^{t}\leq P_{S,a}^{t}\leq S_{Sa}^{t}P_{Smax}^{t}a\in SL,\ t\in T \tag{11}\]  
\[\sum_{t=1}^{T}S_{Sa}^{t}P_{Sa}^{t}=0,\ a\in Sa \tag{12}\]  
\[0\leq P_{H,b}^{t}\leq S_{Hb}^{t}P_{Hbmax}^{t}\ \text{b}\in HL,\ t\in T \tag{13}\]  
\[ \begin{array}{c}-S_{Sa}^{t-1}+S_{Sa}^{t}-S_{Sa}^{k}\leq 0;\\ (1\leq k-(t-1)\leq T_{Sa,\mu r}^{\min},\ a\in SL)\end{array}  \tag{14}\]  
\[ \begin{array}{c}-S_{Hb}^{t-1}+S_{Hb}^{t}-S_{Hb}^{k}\leq 0\\ (1\leq k-(t-1)\leq T_{Hb,\mu r}^{\min},b\in HL)\end{array}  \tag{15}\]  
\[0\leq\sum_{t=1}^{T}[S_{Sa}^{t}-S_{Sa}^{t-1}\big{|}\leq M_{Sa},\ a\in SL \tag{16}\]  
\[0\leq\sum_{t=1}^{T}[S_{Hb}^{t}-S_{Hb}^{t-1}]\leq M_{Hb},b\in HL \tag{17}\]  
\[-P_{ij}^{\max}\leq b_{ij}\cdot(\theta_{i}-\theta_{j})\leq P_{ij}^{\max}\ (j\in i,j\
eq i,\forall i) \tag{18}\]  
\[P_{G}-P_{L}=-B^{\prime}\theta \tag{19}\]  
where  
\[C_{gen}=\sum_{t=1}^{T}\sum_{j=1}^{N_{G}}\{u_{Gj}^{t}[l_{j}+m_{j}P_{Gj}^{t}+n_{ j}(P_{Gj}^{t})^{2}]\Delta T+[u_{Gj}^{t}(1-u_{Gj}^{t-1})+(1-u_{Gj}^{t})u_{Gj}^{t-1}] \lambda_{Gj}\} \tag{20}\]  
\[C_{SL}=\sum_{t=1}^{T}\sum_{a=1}^{N_{S}}\lambda_{Sa}S_{Sa}^{t}P_{Sa}^{t}\Delta T \tag{21}\]  
\[C_{HL}=\sum_{t=1}^{T}\sum_{b=1}^{N_{H}}\lambda_{Hb}S_{Hb}^{t}P_{Hb}^{t}\Delta T \tag{22}\]  
In the above formulation, the functions (1) represents the two objectives. One is to maximize utilization of wind power; the other is to minimize the system operation cost, which is composed of the generation cost of thermal generators \(C_{gen}\), the operation cost of shiftable loads \(C_{SL}\) and the operation cost of high-energy loads \(C_{HL}\). Constraint (2) represents the power balance of the system. is the total amount of conventional system load over time period \(t\) which doesn't include SL and HL Constraint (3) describes that the reserve needed for loads forecast error and wind power forecast error should be satisfied.