Optimal stochastic scheduling of power generation systems with scheduling delays and large cost differentials

The optimal scheduling or unit commitment of power generation systems to meet a random demand involves the solution of a class of dynamic programming inequalities for the optimal cost and control law. We study the behavior of this optimality system in terms of two parameters: (i) a scheduling delay, e.g., the startup time of a generation unit; and (ii) the relative magnitudes of the costs (operating or starting) of different units. In the first case we show that under reasonable assumptions the optimality system has a solution for all values of the delay, and, as the delay approaches zero, that the solutions converge uniformly to those of the corresponding system with no delays. In the second case we show that as the cost of operating or starting a given machine increases relative to the costs of the other machines, there is a point beyond which the expensive machine is not used, except in extreme situations. We give a formula for the relative costs that characterize this point. Moreover, we show that as the relative cost of the expensive machine goes to infinity the optimal cost of the system including the expensive machine approaches the optimal cost of the system without the machine.