Many wholesale electricity markets call on the independent system operator (ISO) to determine day-ahead schedules for generators based on a centralized unit commitment. Up until recently, the Lagrangian relaxation (LR) algorithm was the only practical means of solving an ISO-scale unit commitment problem, and it was the solution technique used by most ISOs. Johnson et al.  demonstrate, however, that equity, incentive, and efficiency issues will arise from use of LR solutions, because different commitments that are similar in terms of total system costs can result in different surpluses to individual units. Recent advances in computing capabilities and optimization algorithms now make solution of the mixed-integer programming (MIP) formulation by means of branch and bound (B&B) tractable, often with optimality gaps smaller than those of LR algorithms, which has led some ISOs to adopt B&B algorithms and others proposing to do so.
With the move towards B&B, one obvious question is whether the use of MIP will eliminate or reduce the issues with LR raised by Johnson et al. Using actual market data from an ISO, we demonstrate that both LR and MIP solutions will suffer the same equity issues, unless the ISO unit commitment problems can be solved to complete optimality within the allotted timeframe—which is beyond current computational capabilities. Our results further demonstrate that the size of the payoff deviations are not monotone in the size of the optimality gap, meaning smaller optimality gaps from B&B will not necessarily mitigate the issues Johnson et al. raise. We show that the use of “make-whole” payments, which ensure units recover any startup and no-load costs not recovered by inframarginal energy rents, can help to reduce surplus volatility and differences to some extent.