  This paper studies an electricity market consisting of an independent system
operator (ISO) and a group of generators. The goal is to solve the DC optimal
power flow (DC-OPF) problem: have the generators collectively meet the power
demand while minimizing the aggregate generation cost and respecting line flow
limits in the network. The ISO by itself cannot solve the DC-OPF problem as
generators are strategic and do not share their cost functions. Instead, each
generator submits to the ISO a bid, consisting of the price per unit of
electricity at which it is willing to provide power. Based on the bids, the ISO
decides how much production to allocate to each generator to minimize the total
payment while meeting the load and satisfying the line limits. We provide a
provably correct, decentralized iterative scheme, termed BID ADJUSTMENT
ALGORITHM, for the resulting Bertrand competition game. Regarding convergence,
we show that the algorithm takes the generators' bids to any desired
neighborhood of the efficient Nash equilibrium at a linear convergence rate. As
a consequence, the optimal production of the generators converges to the
optimizer of the DC-OPF problem. Regarding robustness, we show that the
algorithm is robust to affine perturbations in the bid adjustment scheme and
that there is no incentive for any individual generator to deviate from the
algorithm by using an alternative bid update scheme. We also establish the
algorithm robustness to collusion, i.e., we show that, as long as each bus with
generation has a generator following the strategy, there is no incentive for
any group of generators to share information with the intent of tricking the
system to obtain a higher payoff. Simulations illustrate our results.
