## 3 Modeling Details and Case Studies  
The general formulation of our SHTUC is presented in (8)-(19). \[f_{\star}=\min\!\sum\limits_{\mathsf{g}\in\mathcal{G}}\!\!\left[\!\sum\limits_ {\mathsf{t}\in\mathcal{T}}\!\!\left(\!\mathsf{CS}_{\mathsf{g}}\!\cdot\!a_{ \mathsf{gt}}+\sum\limits_{\mathsf{s}\in\mathcal{S}}\mathbf{C}_{\mathsf{g}}\! \cdot\!\!tg_{\mathsf{gt}}\!\right)\!\right]+\!\sum\limits_{\mathsf{b}\in \mathcal{B}}\sum\limits_{\mathsf{t}\in\mathcal{T}}\mathbf{CL}\!\cdot\!(\delta_{ \mathsf{bt}}^{+}+\delta_{\mathsf{bt}}^{-})+\sum\limits_{\mathsf{s}\in\mathcal{ S}}f_{\mathsf{s}}^{\mathsf{w}}(v) \tag{8}\]\[\mathrm{s.t}:\sum_{\mathrm{o=t-TD}_{g}+1}^{\mathrm{t}}a_{\mathrm{g} 0}\leq I_{\mathrm{g}t},\sum_{\mathrm{o=t-TD}_{g}+1}^{\mathrm{t}}b_{\mathrm{g}0} \leq 1-I_{\mathrm{g}t} \tag{9}\] \[a_{\mathrm{g}t}-b_{\mathrm{g}t}=I_{\mathrm{g}t}-I_{\mathrm{g}t-1},z_{\mathrm{h}t}-u_{\mathrm{h}t}=w_{\mathrm{h}t}-w_{\mathrm{h}t-1}\] (10) \[z_{\mathrm{h}t},u_{\mathrm{h}t},u_{\mathrm{g}t},a_{\mathrm{g}t}, b_{\mathrm{g}t}\in\{0,1\}\] (11) \[I_{\mathrm{g}t}\mathbf{\cdot}\mathbf{\underline{g}}_{g}\leq t_{ \mathrm{g}t}\mathbf{\leq}I\mathbf{\cdot}\mathbf{\overline{p}}_{g}\] (12) \[tg_{\mathrm{g}t}-tg_{\mathrm{g}t-1}\leq I_{\mathrm{g}t-1}\mathbf{ \overline{R}}_{g}+(1-I_{\mathrm{g}t-1})\mathbf{\cdot}\mathbf{SU}_{g}\] (13) \[tg_{\mathrm{g}t-1s}-tg_{\mathrm{g}t}\leq I_{\mathrm{g}t}\mathbf{ \cdot}\mathbf{\overline{R}}_{g}+(1-I_{\mathrm{g}t})\mathbf{\cdot}\mathbf{SD}_ {g}\] (14) \[v_{\mathrm{h}t}-v_{\mathrm{h}t-1s}+f^{\mathrm{v}}_{\mathrm{h}t}( q,s)+\mathbf{A}_{\mathrm{h}ts}=0\] (15) \[\mathbf{\underline{V}}_{\mathrm{h}}\leq v_{\mathrm{h}ts}\leq \mathbf{\overline{V}}_{\mathrm{h}},w_{\mathrm{h}t}\mathbf{\underline{Q}}_{ \mathrm{h}}\leq q_{\mathrm{h}ts}\leq w_{\mathrm{h}t}\mathbf{\overline{Q}}_{ \mathrm{h}},0\leq s_{\mathrm{h}ts}\leq\mathbf{\overline{S}}_{\mathrm{h}}\] (16) \[0\leq hg_{\mathrm{h}ts}\leq f^{\mathrm{hg}}_{\mathrm{h}ts}(q,s)\] (17) \[\mathrm{s.t}:\sum_{\mathrm{o=t-TD}_{g}+1}^{\mathrm{t}}a_{\mathrm{g} 0}\leq I_{\mathrm{g}t},\sum_{\mathrm{o=t-TD}_{g}+1}^{\mathrm{t}}b_{\mathrm{g}0} \leq 1-I_{\mathrm{g}t} \tag{9}\]0 (18) \[\mathrm{s.t}:\sum_{\mathrm{o=t-TD}_{g}+1}^{\mathrm{t}}a_{\mathrm{g} 0}\leq I_{\mathrm{g}t},\sum_{\mathrm{o=t-TD}_{g}+1}^{\mathrm{t}}b_{\mathrm{g}0} \leq 1-I_{\mathrm{g}t} \tag{9}\]1  
In our model, the indices and respective sets containing them are \(\mathrm{g}\in\mathcal{G}\) for thermal generators, \(\mathrm{h}\in\mathcal{H}\) for hydro plants, \(\mathrm{b}\in\mathcal{B}\) for buses, \(\mathrm{l}\in\mathcal{L}\) for transmission lines, and \(\mathrm{t}\) and \(\mathrm{o}\in\mathcal{T}\) for periods. In (5), thermal generators' start-up costs are \(\mathbf{CS}\), and we assume that the shutdown cost is null. The thermal-generation costs are \(\mathbf{C}\); \(\mathbf{CL}\) is the per-unit cost of load shedding (\(\delta^{+}\)) and generation surplus (\(\delta^{-}\)). Expected future-operation cost for scenario \(\mathrm{s}\) is represented by the piecewise-affine function, \(f^{\omega}_{\mathrm{s}}(v^{\mathrm{s}}):\mathbb{R}^{|\mathcal{H}|}\to \mathbb{R}\), where \(v^{\mathrm{s}}\in\mathbb{R}^{|\mathcal{H}|}\) are the reservoir volumes in the last period of scenario \(\mathrm{s}\). The first-stage decisions are thermal generators' commitment, start-up, and shutdown, respectively, \(I\), \(a\), and \(b\), and their hydro counterparts (\(w\), \(z\), and \(u\)). Set \(\mathcal{X}\) in (1) contains the feasible commitments of thermal and hydro generators in our SHTUC, and it is defined by Constraints (9)-(11). In this work, we model the statuses of hydro plants with associated binary variables only in the first 48 h, to reduce the computational burden. For the remaining periods, the hydro plants are modeled only with continuous variables. The minimum up-time Constraint (9) ensures that, once turned on, thermal generator \(\mathrm{g}\) remains on for at least \(\mathbf{TU}_{\mathrm{g}}\) periods.