# A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty
###### Abstract
Several strategies have been proposed to reduce the conservatism of the ROUC approach. For instance, min-max regret optimization [38], a unifying approach combining the elements and advantages of the SOUC and the ROUC [39], and a so-called non-conservative robust (security constrained) UC [38]. Remarkably, adaptive robust optimization of the UC (AROUC) produces a less conservative and more stable solution than ROUC approaches [40; 41; 42; 43]. This result is obtained due to adjustable decision variables (also known as _wait-and-see_ variables) that are calculated when some of the uncertain data are known at a given period (by considering information of past realizations of the random variables) [44; 45]. However, a known drawback of adaptive robust optimization (ARO) is the intractability that may occur for large scale problems given the NP-hardness of the problems [45]. The solution of this drawback results in the use of affinely adjustable robust optimization (AARO) approaches where adjustable variables are constrained to affine policies (AP), also known as affine arithmetic (AA) [45]. [PERSON] and [PERSON] [46] pioneered the combination of AA and ARO for the solution of the energy and reserve dispatch problem. The effectiveness of AA stimulates the use of this approach to tackle the uncertainty in optimal power flow problems [47; 48; 49; 50; 51; 52; 53], and UCs in power systems [54] and isolated micro-grids [55]. In recent years, some researchers have proposed including the affinely adjustable robust optimization within the UC (AAROUC), considering different sources of uncertainty. It was found that the AAROUC provides better results compared to conventional optimization methods when dealing with uncertainty in power systems. [PERSON] et al. [56] presented a distributionally robust optimization (DRO) model for UC in coordinated electricity and district heating networks. Simplified affine policies were used to improve the computational tractability of the model. [PERSON] et al. [57] introduced a model based on AARO to solve the UC problem and dispatch of generation units participating in electricity and heat markets. [PERSON] et al. [58] propose a DRO model to solve the UC (DROUC) considering only the volatility of wind power generation. DROUC produces less conservative solutions than ROUC. They applied formulations based on linear decision rules for the dispatch variables to improve computation time. Finally, [PERSON] et al. [42] presented a multistage adaptive robust optimization model for the UC only considering the uncertainty of the load. Simplified affine policies based on linear decision rules were explored to deal with large-scale power systems. The interested reader may refer to [45; 59] for the theoretic background regarding AARO. Researchers have concluded that AAROUC models (that incorporate the realization of the uncertain parameters in previous periods) significantly reduce the operational cost of the system when compared with UC models based on conventional deterministic, stochastic, and robust optimization models [42; 56]. Although AARO models have been proposed for the UC problem in the past [42; 46; 56; 58]. Previous works examined only one uncertainty source at a time when non-conventional renewable sources are considered. For instance, Ref. [42] only contemplates demand uncertainty, whereas Ref. [58] only includes wind generation uncertainty. In Ref. [56], the authors include wind and solar generation uncertainty but not demand uncertainty. Moreover, to the best of our knowledge, \(N-1\) security constraints are seldom studied in this context. Therefore, the contributions of this paper are threefold. First, we introduce a new AAROUC model where uncertainties in load, wind, and solar generation are simultaneously considered. This model relies on forecasts and forecast errors of the different sources of uncertainty to restrict the wait-and-see or adjustable variables of the model.