Proposition 6, pp. 18–19 — under the W_it-policy, robust generation and ramping limits reduce to the extreme total-load scenarios

Consider the affine multistage robust unit commitment model under the $W_{it}$-policy $p^t_i(\mathbf d)=w^t_i+W_{it}\sum_{j\in\mathcal N_d}d^t_j$, with arbitrary output limits $p^{\min}_i,p^{\max}_i$, ramp rates $RD^t_i,RU^t_i,SD^t_i,SU^t_i$, commitments $x^t_i,u^t_i,v^t_i$, initial state $x^0_i,p^0_i$, and policy coefficients $w^t_i,W_{it}$. Let $\mathcal D=\prod_t\mathcal D^t$ be the budget uncertainty set (3), and let $\mathbf d_{\max},\mathbf d_{\min}\in\mathcal D$ be trajectories whose total load is maximal, respectively minimal, over $\mathcal D$ in every period ((19), (20)). Let $\mathbf d_{minmax}(t)$ and $\mathbf d_{maxmin}(t)$ be the scenarios (21), (22).

Then for every generator $i$ and every period $t$:

1. the robust generation limits (8c) over $\mathcal D$ are equivalent to the same constraints over the finite set $\{\mathbf d_{\min},\mathbf d_{\max}\}$;
2. the robust ramping-down limits (8d) at $t$ over $\mathcal D$ are equivalent to the same constraints over $\{\mathbf d_{\min},\mathbf d_{\max},\mathbf d_{minmax}(t),\mathbf d_{maxmin}(t)\}$, and so are the robust ramping-up limits (8e).

In symbols, for (8c),

This is Proposition 6 of the paper. It means that, under the $W_{it}$-policy, the generation and ramping constraints can be imposed in advance through finitely many scenarios, so that only the worst-case cost constraint and the transmission constraints need constraint generation.

Formalization Note. The statement is for the $W_{it}$-policy; the $W_i$-policy (9) ("any simpler policy") is the instance $W_{it}=W_i$. Lean periods are 0-based. At the first period the ramping constraints compare with the initial dispatch $p^0_i$ and use $x^0_i$ in place of $x^{t-1}_i$. The commitments are arbitrary reals (the paper's $\{0,1\}$ restriction is not needed). The scenarios $\mathbf d_{\max},\mathbf d_{\min}$ are hypotheses: any members of $\mathcal D$ that maximize, respectively minimize, the total load of every period. The standing assumptions $\Gamma\ge0$, $\hat d^t_j>0$ of (3) are not needed once $\mathbf d_{\max},\mathbf d_{\min}\in\mathcal D$ is assumed, and are omitted.