# A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty
## 4 AARO Model for Security Constrained Unit Commitment (SCUC)
### AARO Formulation to Solve the SCUC Problem  
In this formulation, the non-adjustable variables are those binary variables belonging to Equations (9)-(13) (first stage decisions), while the adjustable variables belong to the dispatch Equations (14)-(19). Dispatch decisions, which are being taken every hour, depend on the knowledge of the uncertainty parameters up to the current operating time. Therefore, dispatch decisions are non-anticipative variables. The AAROUC perfectly models this condition, since it allows considering Automatic Generation Control (AGC) actions where hydro-thermal generators can correct errors only at the current time. In Equation (21), \(g_{i,t}^{affine}\) represents the affine form of the power dispatch variable for each generator \(g_{i,t}\), \(g_{i,t}^{0}\) being its central value, and \(\tau\leq t\) determines the non-anticipativity condition. \(\gamma_{i,t,\omega,\tau}\), \(\gamma_{i,t,\eta,\tau}\), \(\gamma_{i,t,\tau}\), \(\varepsilon_{\text{w},\tau}\), \(\varepsilon_{\text{g},\tau}\), \(\varepsilon_{\text{g},\tau}\), represent the deviation variables and forecast errors related to the uncertainty of wind generation, solar generation, and demand, respectively. Forecast errors \(\varepsilon_{\text{w},\tau}\), \(\varepsilon_{\text{p},\tau}\), \(\varepsilon_{\text{g},\tau}\) belong to the uncertainty set bounded by the a maximum level of uncertainty as shown in Section 4.2:  
\[g_{i,t}^{affine}=g_{i,t}^{0}+\sum_{w\in W}\sum_{\tau\leq t}\gamma_{i,t,w,\tau} \cdot\varepsilon_{w,\tau}+\sum_{p\in P}\sum_{\tau\leq t}\gamma_{i,t,p,\tau} \cdot\varepsilon_{p,\tau}+\sum_{s\in S}\sum_{\tau\leq t}\gamma_{i,t,\tau} \cdot\varepsilon_{s,\tau}\quad\forall i\in I,t\in T \tag{21}\]  
Finally, in consideration of the theoretical background provided in Section 2, the deterministic model presented in Section 3, the uncertainty of demand and that of renewable energy sources (i.e., wind and solar generation), a new AARO model to solve the SCUC is presented below:  
\[\min_{x,y,z,\lambda}\quad\sum_{t=1}^{T}\sum_{i=1}^{I}[FC_{i}\cdot x_{i,t}+Csu_{ i}\cdot y_{i,t}+Csd_{i}\cdot z_{i,t}]+\lambda \tag{22}\]  
Subject to:  
Binary constraints of the UC. On-off status and minimum up and down time constraints given in Equations (9)-(13). _Dispatching cost_. The constraint given by (23) represents the cost of dispatch. The additional variable \(\lambda\) sets an upper bond by denoting the highest cost for the dispatch problem. Constraints (24)-(26) are the absolute values of the deviations (related to the objective function) generated by the uncertainties of wind and photovoltaic generation, as well as the load, respectively:  
\[\sum_{t=1}^{T}\sum_{i=1}^{I}C_{i}\cdot g_{i,t}^{0}+\sum_{w\in W}\sum_{\tau\leq t }\eta_{w,\tau}^{1}\cdot\varepsilon_{w,\tau}^{max}+\sum_{p\in P}\sum_{\tau\leq t }\eta_{p,\tau}^{1}\cdot\varepsilon_{p,\tau}^{max}+\sum_{s\in S}\sum_{\tau\leq t }\eta_{\tau}^{1}\cdot\varepsilon_{s,\tau}^{max}\leq\lambda \tag{23}\]  
\[-\eta_{w,\tau}^{1}\leq\sum_{t=1}^{T}\sum_{i=1}^{I}C_{i}\cdot\gamma_{i,t,w, \tau}\leq\eta_{w,\tau}^{1}\qquad\forall w\in W,\tau\leq t \tag{24}\]  
\[-\eta_{p,\tau}^{1}\leq\sum_{t=1}^{T}\sum_{i=1}^{I}C_{i}\cdot\gamma_{i,t,p,\tau }\leq\eta_{p,\tau}^{1}\qquad\forall p\in P,\tau\leq t \tag{25}\]  
\[-\eta_{\tau}^{1}\leq\sum_{t=1}^{T}\sum_{i=1}^{I}C_{i}\cdot\gamma_{i,t,\tau} \leq\eta_{\tau}^{1}\qquad\forall\tau\leq t \tag{26}\]  
Variables \(\gamma_{i,t,w,\tau},\gamma_{i,t,p,\tau},\gamma_{i,t,\tau}\) represent the adjustment of the hydro-thermal generator (\(i\)) at time \(t\) and past period \(\tau\), given the deviation of forecast errors of the wind generator (\(w\)), solar generator (\(p\)) and total demand, respectively. Likewise, auxiliary linear variables \(\eta_{w,t}^{1},\eta_{p,t}^{1},\eta_{t}^{1}\) ensure the linearity of the decision rule for affine policies of the objective function, given the deviation of forecast errors of the wind generator (\(w\)), solar generator (\(p\)) and total demand, correspondingly, at time \(t\) and past period \(\tau\). _Generation limits_.