Dual first-order methods for efficient computation of convex hull prices

TL;DR

The paper tackles the computation of Convex Hull prices in realistic electricity markets, where non-convex unit-commitment constraints create duality gaps. It reframes CH pricing as a dual, nonsmooth convex optimization problem and benchmarks a broad suite of first-order methods—subgradient, parameter-free, and smoothing-based—on large real-world UC instances. The authors introduce and evaluate novel dual-method variants (including bundle-level and recent adaptive schemes) and provide a Julia toolbox ConvexHullPricing.jl for practitioners. Key findings show that bundle proximal level methods and carefully tuned subgradient variants offer strong performance, while smoothing-based approaches show promise in specific datasets; a practical hybrid solver is proposed to balance speed and accuracy in computing CH prices. The work advances scalable, implementable CH pricing in large-scale power systems, with clear guidance on method selection and practical acceleration techniques.

Abstract

Convex Hull (CH) pricing, used in US electricity markets and raising interest in Europe, is a pricing rule designed to handle markets with non-convexities such as startup costs and minimum up and down times. In such markets, the market operator makes side payments to generators to cover lost opportunity costs, and CH prices minimize the total "lost opportunity costs", which include both actual losses and missed profit opportunities. These prices can also be obtained by solving a (partial) Lagrangian dual of the original mixed-integer program, where power balance constraints are dualized. Computing CH prices then amounts to minimizing a sum of nonsmooth convex objective functions, where each term depends only on a single generator. The subgradient of each of those terms can be obtained independently by solving smaller mixed-integer programs. In this work, we benchmark a large panel of first-order methods to solve the above dual CH pricing problem. We test several dual methods, most of which not previously considered for CH pricing, namely a proximal variant of the bundle level method, subgradient methods with three different stepsize strategies, two recent parameter-free methods and an accelerated gradient method combined with smoothing. We compare those methods on two representative sets of real-world large-scale instances and complement the comparison with a (Dantzig-Wolfe) primal column generation method shown to be efficient at computing CH prices, for reference. Our numerical experiments show that the bundle proximal level method and two variants of the subgradient method perform the best among all dual methods and compare favorably with the Dantzig-Wolfe primal method.

Figures (2)

• Figure 1: Plot of the best objective gap ${\mathcal{L}}_* - {\mathcal{L}}_{best}^k$ achieved so far versus time. For each dataset (multiple instances), we aggregate the individual plots per instance by smoothing them over a same time interval and taking their arithmetic mean.
• Figure 2: Visual comparison of all methods on all instances. Methods in the bottom-left corner are the most competitive. Each method is associated with its mean time to reach $5 \times 10^{-6}$ relative error and mean relative error across all instances. Methods stopped after 900 seconds are penalized to 1000 seconds. All methods that should be outside the plot are represented either as a right pointing triangle if they exceeded the time budget or a up pointing triangle if their relative error is too big. A diamond marker means that the method exhibits both large relative error and exceeded budget.

Theorems & Definitions (5)

• Lemma 1
• Lemma 2: First-order oracle
• Lemma 3
• proof
• Lemma 4