Theorem 1, p. 11 — without ramping constraints, the two-stage robust UC (2) and the multistage robust UC (5) are equivalent

Consider the unit commitment problem with budget uncertainty set $\mathcal D=\prod_t\mathcal D^t$ of (3), with $\Gamma\ge0$ and $\hat d_j^t>0$. Delete the ramping constraints, (1i) in the two-stage robust UC (2) and (5c) in the multistage robust UC (5). Then the two models are equivalent:

1. at every commitment $(x,u,v)\in X$ they have the same objective value,

where the minimum on the right is over non-anticipative dispatch policies $p^t(d^{[t]})$, depending only on the net loads $d^1,\dots,d^t$ realized up to period $t$, that satisfy (5b), (5d), (5e) for every $d\in\mathcal D$; 2. hence their optimal values over $X$ coincide.

The two-stage model lets the dispatch see the whole day's net loads; the multistage model does not. The theorem says that this extra information is worthless when the ramping constraints are absent, so the multistage model differs from the two-stage model only through ramping; Propositions 1 and 2 show that with ramping the two models can differ.

Formalization Note Periods are Fin T with Lean period $t$ being the paper's period $t+1$. Net-load trajectories are $d:\{0,\dots,T-1\}\to\mathbb R^{N_d}$. Values are extended reals: a minimum over an empty set is $+\infty$ (an infeasible dispatch problem costs $+\infty$). The standing assumptions $\Gamma\ge 0$ and $\hat d^t_j>0$ of the budget set (3) are carried as hypotheses; they make $\bar d\in\mathcal D$, so every $\mathcal D^t$ is nonempty. "Equivalent" is formalized as equality of the two objectives at every commitment in $X$ together with equality of the optimal values; equal objectives on $X$ also give the same optimal solutions.