Aggregations of electric loads can provide reserves to power systems, but their available reserve capacities are time-varying and not perfectly known when the system operator computes the optimal generation and reserve schedule. In this paper, we formulate a chance constrained optimal power flow problem to procure minimum cost energy, generator reserves, and load reserves given uncertainty in renewable energy production, load consumption, and load reserve capacities. Assuming that uncertainty distributions are not perfectly known, we solve the problem with distributionally robust optimization, which ensures that chance constraints are satisfied for any distribution in an ambiguity set built upon the first two moments. We use two ambiguity sets to reformulate the model as a semidefinite program and a second-order cone program and run computational experiments on the IEEE 9-bus, 39-bus, and 118-bus systems. We compare the solutions to those given by two benchmark reformulations; the first assumes normally distributed uncertainty and the second uses large numbers of uncertainty samples. We find that the use of load reserves, even when load reserve capacities are uncertain, reduces operational costs. Also, the approach is able to meet reliability requirements, unlike the first benchmark approach and with lower computation times than the second benchmark approach.
- Chance-constrained optimal power flow (CC-OPF)
- convex optimization
- load control
- moment-based ambiguity set
- uncertain reserves