## 3 Modeling Details and Case Studies
Likewise, the minimum downtime in (9) requires that once \(\mathrm{g}\) has been turned off, it must remain off for at least \(\mathbf{TD}_{\mathrm{g}}\) periods. Constraints (10) guarantee the satisfaction of logical relations of status, start-up, and shutdown for thermal and hydro plants. The sets \(\mathcal{Y}\)s are defined by (12)-(19). Constraints (12) are the usual limits on thermal generation \(tg\); (13) and (14) are the up and down ramp-rate limits, and the start-up and shutdown requirements of generators \(\mathrm{g}\). Equation (15) is the mass balance of the hydro plant \(\mathrm{h}\)'s reservoir. The \(\mathbf{A}_{\mathrm{h}ts}\) is the inflow to reservoir \(\mathrm{h}\) in period \(\mathrm{t}\) of scenario \(\mathrm{s}\). Moreover, the affine function \(f^{\mathrm{v}}_{\mathrm{h}ts}(q,s):\mathbb{R}^{2|\mathcal{H}|\mathcal{T}| \mathcal{X}|}\to\mathbb{R}\) maps the inflow to \(\mathrm{h}\)'s reservoir in period \(\mathrm{t}\) of scenario \(\mathrm{s}\) given the vectors of turbine discharge \(q\) and spillage \(s\). The constraints in (16) are the limits on reservoir volume, \(v\), turbine discharge, \(q\), and spillage, \(s\). In (17), the piecewise-affine function \(f^{\mathrm{hg}}_{\mathrm{h}ts}(q,s):\mathbb{R}^{2|\mathcal{H}|\mathcal{T}| \mathcal{X}|\mathcal{S}}\to\mathbb{R}\) bounds the hydropower generation \(hg_{\mathrm{h}ts}\) of plant \(\mathrm{h}\). We use the classical DC network model: Equation (18) is the bus power balance, where the linear function \(f^{\mathrm{P}}_{\mathrm{bts}}(tg,hg,\delta^{+},\delta^{-}):\mathbb{R}^{| \mathcal{T}|\mathcal{X}|\mathcal{S}|\cdot(|\mathcal{G}|+|\mathcal{H}|+2| \mathcal{B}|)}\to\mathbb{R}\) maps the controlled generation at each bus into the power injection at bus \(\mathrm{b}\), \(\mathbf{WG}_{\mathrm{bts}}\) is the wind generation at bus \(\mathrm{b}\), and \(\mathbf{L}_{\mathrm{b}t}\) is the corresponding load at \(\mathrm{b}\). Lastly, (19) are the limits on the flow of transmission line \(\mathrm{l}\) in period \(\mathrm{t}\) and scenario \(\mathrm{s}\), defined by the affine function \(f^{\mathrm{I}}_{\mathrm{bst}}(tg,hg,\delta^{+},\delta^{-}):\mathbb{R}^{| \mathcal{T}|\mathcal{X}|\cdot(|\mathcal{G}|+|\mathcal{H}|+2|\mathcal{B}|)}\to \mathbb{R}\). We assess our algorithm on a 46-bus system with 11 thermal plants, 16 hydro plants, 3 wind farms, and 95 transmission lines. The system's installed capacity is 18,600 MW, from which 18.9% is due to thermal plants, hydro plants represent 68.1%, and wind farms have a share of 13%. We consider a one-week-long planning horizon with hourly discretization. Thus, a one-scenario instance of our SHTUC would have 7848 binary variables and 5315 constraints at the first stage; and 36,457 continuous variables and 100,949 constraints for each scenario in the second stage. Furthermore, the weekly peak load in the baseline case is 11,204 MW--nearly 60.2% of the installed capacity. The hydro plants are distributed over two basins and include both run-of-river ones and plants with reservoirs capable of regularization. Further information about the system can be found in the multimedia files attached. The uncertainty comes from wind generation and the inflows. In all tests, we use a scenario set with 256 scenarios. To assess how our algorithm performs in distinct scenario sets, three sets (A, B, and C) are considered. Moreover, we use three initial useful-reservoir-volume levels: 40%, 50%, and 70%. The impact of different load levels on the performance of our algorithms is analyzed through three load levels: low (L), moderate (M), and high (H). Level H is our baseline case regarding load. Levels M and L have the same load profile as H's, but with all loads multiplied by factors of 0.9 and 0.8, respectively. Lastly, to investigate how our algorithm's convergence rate is affected by different choices of initial stability centers, we implement two strategies for obtaining the initial stability center. In both strategies, we solve an expected-value problem, as defined in Reference [5].