## 2 Problem Formulation
### Risk Averse Stochastic Unit Commitment with Transmission Reserves
The constraints of the aforementioned SUC model are given as follows:  
* The Unit Constraints: \[x_{g}^{t},u_{g}^{t},v_{g}^{t}\in\{0,1\},\;t\in\mathcal{T},g\in\mathcal{G}\] (5) \[x_{g}^{t}-x_{g}^{t-1}=u_{g}^{t}-v_{g}^{t},\;t\in\mathcal{T},g\in\mathcal{G}\] (6) \[\sum\limits_{\tau=t-Ut_{Tg+1}}^{t}u_{g}^{\tau}\leq x_{g}^{t},\;t\in \mathcal{T},g\in\mathcal{G}\] (7) \[\sum\limits_{\tau=t-DTg+1}^{t}v_{g}^{\tau}\leq 1-x_{g}^{t},\;t\in \mathcal{T},g\in\mathcal{G}\] (8) \[P_{g}^{\min}x_{g}^{t}\leq P_{g}^{t}\leq\Big{(}P_{g}^{\max}-r_{g}^{t}\Big{)}x_ {g}^{t},\;t\in\mathcal{T},g\in\mathcal{G}\] (9) \[-R_{g}^{D}\leq P_{g}^{t}-P_{g}^{t-1}\leq R_{g}^{U},\;t\in \mathcal{T},g\in\mathcal{G}\] (10) \[r_{g}^{t}\leq R_{g}^{10}x_{g}^{t},\;t\in\mathcal{T},g\in\mathcal{G}\] (11)
* The System Constraints: \[\sum\
olimits_{g\in\mathcal{G}}P_{g}^{t}=\sum\
olimits_{n\in\mathcal{N}} \big{(}D_{n}^{t}-\hat{W}_{n}^{t}\big{)},\;t\in\mathcal{T}\] (12) \[\big{|}P_{l}^{t}\big{|}=\Big{|}\sum\
olimits_{n\in\mathcal{N}}S_{l,n}\Big{(}D_{ n}^{t}-\hat{W}_{n}^{t}-\sum\
olimits_{g\in\mathcal{G}_{n}}P_{g}^{t}\Big{)}\Big{|}\leq P_{l}^{ \max},\;t\in\mathcal{T},l\in\mathcal{L}\] (13)\[\sum_{g\in G}r_{g}^{t}\geq\sum_{n\in\mathcal{N}}\bigl{(}\gamma_{D}D_{n}^{t}+\gamma_ {W}\hat{W}_{n}^{t}\bigr{)},\;t\in\mathcal{T} \tag{14}\]  
where \(G\), \(N\), \(L\), \(T\) are the sets of generators, nodes, transmission lines and time periods, respectively; \(G_{n}\subseteq G\) is the subset of generators at bus \(n\); \(v_{g}^{t}\) is the shut-down binary variable of generator \(g\) at time \(t\); \(P_{g}^{\min}\) and \(P_{g}^{\max}\) are the minimum and maximum output levels of generator \(g\); \(R_{g}^{D}\) and \(R_{g}^{U}\) are the ramp-down and ramp-up rates of generator \(g\), and \(R_{g}^{10}\) is the 10-min ramp rate. \(DT_{g}\) and \(UT_{g}\) are the minimum-down/ -up times of generator \(g\). \(P_{l}^{\max}\) is the power flow limit for line \(l\). \(D_{n}^{t}\) and \(\hat{W}_{n}^{t}\) denote the net load and day-ahead wind power forecast at bus \(n\) and time \(t\), respectively, with the reserve capacity coefficients \(\gamma_{D}\) and \(\gamma_{W}\). \(S_{l,n}\) is the shift factor which characterizes the contribution from bus \(n\) to power flow on line \(l\). Equations (5) and (6) correspond to units on/off, start-up and shut-down constraints. Equations (7) and (8) are the constraints of the generator's minimum up and downtime. Equation (9) enforces the maximum and minimum generation capacity limits. Equation (10) enforces ramping up and down limits. Equation (11) confines the reserve capacity to the 10-min generation ramping capability. As for the system constraints, Equation (12) is the power balance constraint, which ensures that the power provided by generators must be equal to the net-demand (load minus the wind power) at all times. Equation (13) applies the DC power flow model to impose transmission line limits. Equation (14) establishes the minimum system-wide reserve requirements. The aforementioned constraints (Equations (5)-(14)) are common in conventional UC formulations. In this work, we extend to quantify the transmission reserve requirements in the UC by imposing a limit on the risk of congestion in real-time dispatch. According to [PERSON]'s inequality [PERSON] (1993), the upper bound of congestion probability on line \(l\) is formed as  
\[x_{g}^{t}-x_{g}^{t-1}=u_{g}^{t}-v_{g}^{t},\;t\in\mathcal{T},g\in\mathcal{G}\]0  
where  
\[x_{g}^{t}-x_{g}^{t-1}=u_{g}^{t}-v_{g}^{t},\;t\in\mathcal{T},g\in\mathcal{G}\]1  
where \(\alpha_{l}\) is a user-defined parameter, called _Conserativeness Factor_, which is based on the degree of conservativeness in assessing the risk of congestion. This parameter is similar to the confidence level defined in the chance-constraint programming [PERSON] (1993) that enforces the inequality to a certain level of probability; thus, \(0<\alpha_{l}<1\).