## 2 Dual Problem Formulation  
In its most analytical form, the objective of the unit commitment task is formulated to minimize the total generation cost (\(TGC\)), considering three main expenses. First, the fuel consumption cost \(F(S,P)\) is estimated based on the dynamic state (\(S_{i}^{t}\)) and actual power output (\(P_{i}^{t}\)) of each committed generator \(i\) during the time interval \(t\). This expenditure relies on the cost coefficients of each independent generator, which in turn accounts for the heat rate coefficients (\(a_{i}\), \(b_{i}\), \(c_{i}\)) and fuel specific cost \(f_{c}\). The following quadratic equation is generally used to define these expenses. \[F\big{(}S_{i}^{t},P_{i}^{t}\big{)}=S_{i}^{t}\Big{\{}f_{c}\Big{[}a_{i}\big{(}P_ {i}^{t}\big{)}^{2}+b_{i}P_{i}^{t}+c_{i}\Big{]}\Big{\}} \tag{1}\]  
The second aspect regards the emission cost \(E(S,P)\) that evaluates the gaseous emissions from different pollutants, including carbon dioxide (CO\({}_{2}\)), carbon oxide (CO), nitrogen oxides (NO\({}_{\rm x}\)), sulfur oxides (SO\({}_{\rm x}\)) and other hydrocarbon byproducts (C\({}_{\rm x}\)H\({}_{\rm y}\)). Assuming the emission cost coefficients (\(e_{i1}\), \(e_{i2}\), \(e_{i3}\)) of each generator, the following equation can be utilized to account for this expenditure [29]. \[E\big{(}S_{i}^{t},P_{i}^{t}\big{)}=S_{i}^{t}\Big{[}e_{i1}\big{(}P_{i}^{t}\big{)} ^{2}+e_{i2}P_{i}^{t}+e_{i3}\Big{]} \tag{2}\]  
The last portion of costs refers to the penalties (\(\Pi_{t}\) of Equation (3)) and deteriorates reliability. This relates the energy not-served (\(P_{ENS}\)), translated as load shedding, the spinning reserve not-served (\(P_{SRNS}\)) and the curtailed power from RES (\(P_{cut-RES}\)) with their respective penalty costs of \(\pi_{E}\), \(\pi_{SR}\) and \(\pi_{RES}\)[4]. \[\Pi^{t}=\pi_{E}P_{ENS}^{t}+\pi_{SR}P_{SRNS}^{t}+\pi_{RES}P_{cut-RES}^{t} \tag{3}\]  
Finally, a factor is added to both unit-specific costs to include the start-up cost with respect to fuel consumption (\(FSU_{i}\)) and associated emissions (\(ESU_{i}\)). The comprehensive objective can now be represented with the aid of Equation (4), considering a time horizon \(T\) and participating generators \(N\). \[TGC=\sum_{t=1}^{T}\left\{\sum_{i=1}^{N}\Big{[}F\big{(}S_{i}^{t},P_{i}^{t} \big{)}+E\big{(}S_{i}^{t},P_{i}^{t}\big{)}+S_{i}^{t}(FSU_{i}+ESU_{i})S_{i}^{t- 1}\Big{]}+\Pi^{t}\right\} \tag{4}\]  
The complicating constraints of the UC objective give priority to every action towards emissions inclination, efficiency improvement and RES integration. To this end, the power equilibrium expresses the equality constraints such that:  
\[\sum_{i=1}^{N}S_{i}^{t}P_{i}^{t}=P_{net}^{t}\,\ \forall t\in T \tag{5}\]  
\[P_{net}^{t}=P_{load}^{t}-P_{VRES}^{t} \tag{6}\]  
\[P_{VRES}^{t}=P_{PV}^{t}+P_{wind}^{t}+P_{ biomass}^{t}+P_{water}^{t}+P_{geo}^{t}+P_{CHP}^{t}+P_{FC}^{t} \tag{7}\]  
While the net load (\(P_{net}^{t}\)) equals the residual demand after the variable RES (\(P_{VRES}^{t}\)) involvement, the contribution of solar PVs (\(P_{PV}^{t}\)), wind (\(P_{wind}^{t}\)), biomass (\(P_{ biomass}^{t}\)), hydro and water alternatives such as wave and tidal (\(P_{water}^{t}\)), geothermal (\(P_{geo}^{t}\)), combined heat and power (\(P_{CHP}^{t}\)) and fuel cells using renewable fuels (\(P_{FC}^{t}\)) for electricity demand satisfaction requires increased spinning reserves (\(SR^{t}\)) to conserve the system's reliability.