Asymmetric Supply Function Equilibrium with Constant Marginal Costs

In a real-time electric power auction, the bids of producers consist of committed supply as a function of price. The bids are submitted under uncertainty, before the demand by the Independent System Operator has been realized. In the Supply Function Equilibrium (SFE), every producer chooses the supply function maximizing his expected profit given his residual demand. I consider a uniform-price auction with a reservation price, where demand is inelastic and exceed the market capacity with a positive probability, and firms have identical constant marginal costs but asymmetric capacities. I show that under these conditions, there is a unique SFE, which is piece-wise symmetric.