Consider a discretized time horizon and a set of possible states S = {0, ..., n} for a given system. The system must be in exactly one state at a time.
We introduce generalized min-up/min-down constraints as follows. If the system switches to state i at time t, then it must remain in state i for at least L(i) time periods. These constraints generalize minimum up and down time constraints from the literature in the sense that the system has anarbitrary number n of possible states, instead of only two states (up and down).

Generalized min-up/min-down constraints appear in practical Unit Commitment Problems where nuclear and hydro production units have discrete production levels.

In this paper, we study the generalized min-up/min-down polytope. We also study a variant featuring precedence constraints between states.