# A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty
## 2 Theorical Background
### Affinely Adjustable Robust Optimization (AARO)  
There are two types of variables within AARO: adjustable and non-adjustable. In this case, the non-adjustable variables are those sided with the UC constraints, while the adjustable ones are those that belong to the dispatch constraints. One of the most important features when using AARO is that the adjustable variables can be tuned by themselves (considering the deviation produced in past periods), based on the known uncertainty data which is fundamental in multi-period problems [45].  
The first step in developing a formulation using AARO is to establish a linear decision rule (LDR) through affine rules. For example, allowing \(\pi_{t}\) be a linear function of the demand \(d_{\tau}\) where \(\tau=1,2,\cdots,t-1\)  
\[\pi_{t}=\pi_{t}^{0}+\sum_{\tau=1}^{t-1}\pi_{t}^{\tau}\cdot d_{\tau} \tag{3}\]  
where \(\pi_{t}^{0}\) and \(\pi_{t}^{\tau}\), \(\forall t,\tau\), are new decision variables introduced to define the LDR. In AARO, the optimal values of \(\pi_{t}^{0}\) and \(\pi_{t}^{\tau}\) are computed instead of computing \(\pi_{t}\)[45].  
The second step consists of defining an uncertainty set for the objective function, and for each constraint of the problem, which could be a polyhedral or ellipsoidal set. It is important to select a set of uncertainties that allows the AARO model to be solved efficiently. For example, it is not possible to know the future value of the demand variable \(d_{t}\), but it is possible to know that:  
\[d_{t}\in[d_{t}^{*}(1-\theta_{t}),d_{t}^{*}(1+\theta_{t})] \tag{4}\]  
where \(d_{t}^{*}\) is the nominal value of the demand for each period \(t\), and \(\theta\) is the uncertainty level. Thus, probability distributions are not necessary to define the uncertain parameters within the model.  
Without loss of generality, the following equivalences are used to reformulate the LDR (3), considering the uncertainty set (4):\[\sum_{t=1}^{T}\pi_{t}\cdot d_{t}\leq b,\forall d_{t}\in[d_{t}^{*}(1- \theta_{t}),d_{t}^{*}(1+\theta_{t})]\] \[\Updownarrow\] \[\sum_{t:\pi_{t}\leq 0}\pi_{t}\cdot d_{t}^{*}(1-\theta_{t})+\sum_{t: \pi_{t}\geq 0}\pi_{t}\cdot d_{t}^{*}(1+\theta_{t})\leq b\] \[\Updownarrow\] \[\sum_{t=1}^{T}\pi_{t}\cdot d_{t}^{*}+\theta_{t}\cdot\sum_{t=1}^{T }|\pi_{t}|\cdot d_{t}^{*}\leq b \tag{5}\]  
where \(b\) is a real number.  
Therefore, considering the equivalence (5), the decision rule (3) is redefined as follows:  
\[\pi_{t}=\pi_{t}^{0}+\sum_{\tau=1}^{t-1}\pi_{\tau}\cdot d_{\tau}^{*}\pm\theta_{ t}\cdot\sum_{\tau=1}^{t-1}|\pi_{\tau}|\cdot d_{\tau}^{*} \tag{6}\]  
Finally, in order to eliminate the absolute value in (6), it is necessary to introduce a linear decision variable \(\eta_{\tau}\), to guarantee the linearity of the decision rule. In this way, the AARO formulation is obtained for an adjustable variable limited by an uncertainty set:  
\[\pi_{t}=\pi_{t}^{0}+\sum_{\tau=1}^{t-1}\pi_{\tau}\cdot d_{\tau}^{*}\pm\sum_{ \tau=1}^{t-1}\eta_{\tau}\cdot\theta_{\tau}\cdot d_{\tau}^{*}\]  
\[-\eta_{\tau}\leq\pi_{\tau}\leq\eta_{\tau},\tau=1,\cdots,t-1 \tag{7}\]  
In this expression, signs of term \(\eta_{\tau}\cdot\theta_{\tau}\) depend on the side of the inequality where the adjustable variable is.