## 3 Mathematical Model
### ENO Optimization Sub-Problem  
(1) Objective function  
As shown in (9), the objective function of ENO sub-optimization problem is determined by the hourly dispatch. It is to minimize the operating cost over the complete scheduling horizon with security-constraints. The first term of the objective function is the generation costs and startup cost of NGFUs, the second term is the generation costs and startup cost of non-NGFUs. The shutdown cost of units has been converted into the startup cost. \[\text{Min}\ obj\_E=\sum_{t=1}^{N_{T}}\left\{ \begin{array}{l}\sum\limits_{i \in GU}\left(G_{i,t}^{g}\eta_{i}+SUl_{i,t}\right)\\ +\sum\limits_{i\in NGU}\left(G_{i,t}^{c}+SUl_{i,t}\right)\end{array} \right\} \tag{9}\]  
(2) Units and network constraints  
Equations (10)-(21) are physical constraints of generating unit. Generation scheduling of each thermal unit is constrained by maximum and minimum output of the unit as presented in (10). The renewable power dispatch at individual hour is constrained by the renewable power forecast given in (11). Thermal units operating ramping up/down constraints are given in (12) and (13). Minimum on/off-limits are enforced by (14) and (15). The relationships between unit states and startup/shutdown indicators are given in (16). The logic between startup indicators and shutdown indicators are given in (17). Constraint (18) represents the generation costs of non-NGFUs. Constraint (19) represents the fuel consumption NGFUs. Constraint (20) shows the startup costs of thermal units. \[P_{i}^{\min}u_{i,t}\leq P_{i,t}\leq P_{i}^{\max}u_{i,t} \tag{10}\]  
\[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]  
\[P_{i,t}-P_{i,t-1}\leq UR_{i}(1-y_{i,t})+P_{i}^{\min}y_{i,t} \tag{12}\]  
\[P_{i,t-1}-P_{i,t}\leq DR_{i}(1-z_{i,t})+P_{i}^{\min}z_{i,t} \tag{13}\]  
\[\left[X_{i,t-1}^{\min}-T_{i}^{\min}\right][u_{i,t-1}-u_{i,t}]\geq 0 \tag{14}\]  
\[\left[X_{i,t-1}^{off}-T_{i}^{off}\right][u_{i,t}-u_{i,t-1}]\geq 0 \tag{15}\]  
\[y_{i,t}-z_{i,t}=u_{i,t}-u_{i,t-1} \tag{16}\]  
\[y_{i,t}+z_{i,t}\leq 1 \tag{17}\]  
\[G_{i,t}^{c}=\alpha_{i}^{c}u_{i,t}+\beta_{i}^{c}P_{i,t}+\gamma_{i}^{2}P_{i,t}^ {2},\forall i\in NGU \tag{18}\]  
\[G_{i,t}^{g}=\alpha_{i}^{g}u_{i,t}+\beta_{i}^{g}P_{i,t}+\gamma_{i}^{g}P_{i,t}^ {2},\forall i\in GU \tag{19}\]  
\[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]0  
Constraint (21) limits the gas consumption of NGFUs, the information of this constraint is from GNO optimization sub-problem-A. \[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]1  
For simplicity, the electric power transmission is modeled by (22)-(26) in DC power flow form. Constraint (22) shows the system power balance at each hour. Constraint (23)is the power balance for individual bus. Constraint (24) represents the DC power flow in branch \(br\) with first and last buses at \(j\) and \(k\). Constraint (25) means line capacity limit, \(pf^{\text{max}}\) is power flow limits. Constraint (26) is the phase angle for reference bus. \[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]2  
\[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]3  
\[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]4  
\[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]5  
\[P_{r,t}\leq P_{r,t}^{W} \tag{11}\]6  
(3) Demand response constraints  
The price-elastic load in the electricity network is based on the model reported in References [54; 55]. Equations (27)-(30) represent electrical load deviation and electricity price deviation in the demand response program. In this paper, the time-of-use electricity price is adopted to describe the price change in one day. In this demand response strategy, the satisfaction of customers should be considered. At the same time, customers' satisfaction cannot break the limit as shown in (31). Constraint (32) represents the load is shiftable and the sum of electrical load cannot be changed in one day.