Gradient methods for bilevel electricity grid expansion planning

Although electricity markets are optimized to operate at minimal cost, planners making long-term investments in transmission, generation, and battery storage capacity may also consider a wide array of other objectives, such as carbon emissions, consumer surplus, or generator profitability. Optimizing these objectives in a cost-based market environment is an example of bilevel capacity expansion planning (BEP), a nonconvex optimization problem that is hard to solve in general. Inspired by the success of optimization techniques in machine learning, we propose a stochastic gradient descent algorithm based on implicit differentiation for finding locally optimal solutions to large-scale BEP problems. We demonstrate that this approach scales well to high-resolution grid models with many scenarios, outperforming standard reformulation-based approaches by at least 25x and solving a case with more than 50 million variables in just a few hours. Additionally, while gradient descent only guarantees locally optimal solutions, we consistently observe optimality gaps of 5–10% empirically. Finally, we illustrate how planners can use warm starts to rapidly study expansion outcomes for a wide variety of technology cost, weather, and electrification assumptions.