Session: Algorithms for Optimization and Equilibrium Models in Energy
Chair: Dominic Flocco
Cluster: Optimization Applications (Communication, Energy, Health, ML, ...)
Talk 1: Using spectral stepsizes and total variation in FWI
Speaker: Paulo Silva
Abstract: Full Wave Inversion is a computational technique that can unveil Earth's subsurface properties by solving large-scale inverse problems that minimize the difference between a wave propagation simulation based on a differential equation and actual observed data. In this talk, we will use the spectral proximal gradient (SPG) method to solve regularized FWI and compare the results to the direct application of L-BFGS-B, widely used by the geophysics community. SPG employs the spectral step introduced by Barzilai and Borwein and popularized by Raydan, with a non-monotone line search to achieve performance similar to quasi-Newton methods. At the same time, as it can cope with proximal terms, it can deal with different regularizations to induce desired properties in the model. Then, we used a total variation with box constraints regularization to recover high-quality models even from poor initial solutions. The number of iterations SPG requires is low, showing it might be useful to solve huge-scale 3D problems.
Talk 2: Modelling unit commitment constraints in a Cournot equilibrium model for a electricity market
Speaker: Mel Devine
Abstract: Electricity market modelling that captures the technical constraints of power system operation, such as start and shut down costs, require the use of binary programming so as to model the “on-off” condition of conventional generation units (Bothwell & Hobbs, 2017). Accurate modelling of power systems with higher levels of renewable generation requires such integer variables (Shortt, et al., 2012) and also high time resolution (Merrick, 2016). These high resolution integer models determine the least-cost schedule for generation technologies, which correlates to a strategy of profit-maximisation in a perfectly competitive market. However, electricity markets are better characterized by an oligopoly, where at least some firms have market power, giving them the ability to influence prices by varying their output and/or prices. Modelling this price-making ability requires equilibrium problems such as Mixed Complementarity Problems (MCPs), Mathematical Programs with Equilibrium Constraints (MPECs) or Equilibrium Problem with Equilibrium Constraints (EPECs). However, these problems cannot model a market where all market participants have integer decision variables. Despite notable exceptions (Gabriel, 2017; Weinhold and Gabriel, 2020), the literature on incorporating integer decision variables into equilibrium models is only emerging. In this work, we develop a Game Theory Optimisation model of a wholesale electricity market where all generators maximise profits through price-making, in addition to having integer decision variables. To solve our model, we employ the Gauss-Seidel diagonalisation algorithm, which is typically associated with EPEC models. The optimisation problem of each generator is solved individually, holding all other generators’ decision variables fixed. The algorithm iteratively solves each generator’s optimisation problem by fixing all other generators’ decisions, until it converges to a point where no generator has an optimal deviation. To improve computational efficiency, we also employ a rolling-horizon algorithm. In this talk, results will be presented on based on real-world data from the Irish power system. Thus we consider 16 generating firms that vary in size from large scale integrated firms with several generating technologies to small stand-alone firms. Furthermore, computational analysis, issues surrounding our methodological approach, and future directions will be discussed.
Talk 3: Exact Penalty Techniques via Difference-of-Convex Function Programming for General Bilinear Programs
Speaker: Dominic Flocco
Abstract: We present a new difference of convex functions algorithm (DCA) for solving general bilinear programs. The approach is based on the reformulation of bilinear constraints as difference of convex (DC) functions, more specifically, the difference of scalar, convex quadratic terms. This reformulation gives rise to a DC program, which is solved via sequential linear approximations of the concave term using standard DCA techniques. We explore variations on the classical DCA approach by considering linear- and Newton-proximal DCA and DC bundle methods. We apply this novel algorithmic framework to a variety of linear complementarity problems (LCP), mathematical programs with equilibrium constraints (MPEC) and bilevel optimization problems that arise in energy and infrastructure models.
