[Paper Review] An Integral Formulation and Convex Hull Pricing for Unit Commitment

Yanan Yu, Yongpei Guan+1 more|arXiv (Cornell University)|Jun 19, 2019

Electric Power System Optimization9 references4 citations

TL;DR

This paper proposes a mixed-integer linear programming formulation (MEUC) that enables exact convex hull pricing for unit commitment by solving a single linear program, significantly reducing uplift payments. The method achieves optimal convex hull prices without requiring complex mixed-integer optimization, as validated by computational experiments showing up to 91% reduction in uplift payments compared to traditional methods on a modified IEEE-118 bus system.

ABSTRACT

Reducing uplift payments has been a challenging problem for most wholesale markets in US. The main difficulty comes from the unit commitment discrete decision makings. Recently convex hull pricing has shown promises to reduce the uplift payments. However, it has been intractable to obtain the optimal convex hull price. In this paper, we describe an innovative approach to decide the optimal convex hull price by simply solving a linear program. We also provide an example to illustrate the calculation process. The final computational experiments on a revised IEEE-118 bus system verify the cost effectiveness by utilizing our proposed approach.

Motivation & Objective

• To address the challenge of high uplift payments in US wholesale electricity markets caused by non-convexity in unit commitment (UC) problems.
• To develop an exact method for computing optimal convex hull prices that minimize uplift payments without relying on heuristic or approximate approaches.
• To provide a computationally tractable formulation that achieves integral solutions for single-generator UC problems with complex constraints, including ramping, min-up/down times, and variable start-up costs.
• To demonstrate the effectiveness of the proposed method through computational experiments on a revised IEEE-118 bus system under varying load conditions.

Proposed method

• Develops a mixed-integer extended UC (MEUC) formulation that captures all relevant generator constraints: min-up/down times, ramping limits, variable start-up costs, and convex generation cost functions.
• Introduces an integral formulation that guarantees integer solutions for the single-generator UC problem by solving a single linear program, avoiding the need for mixed-integer programming.
• Uses dynamic programming to prove the correctness of the integral formulation and establishes its equivalence to the convex hull of the UC polytope.
• Derives the optimal convex hull price by solving a linear program that minimizes uplift payments, leveraging dual variables from the MEUC formulation.
• Applies the method to a modified IEEE-118 bus system with 54 generators and 24-hour horizon, using Gurobi 8.0.1 to solve the LPs.
• Compares uplift payments across three pricing methods: traditional LMP (TLMP), approximated CHP-Primal, and the proposed MEUC formulation.

Experimental results

Research questions

• RQ1 Can a linear programming formulation be designed to compute the optimal convex hull price for unit commitment without solving mixed-integer programs?
• RQ2 Does the proposed MEUC formulation achieve exact integral solutions for single-generator UC problems with complex operational constraints?
• RQ3 To what extent can the proposed method reduce uplift payments compared to traditional LMP and approximated convex hull pricing methods?
• RQ4 How does the performance of the MEUC formulation scale across different load scenarios in a realistic power system network?

Key findings

• The proposed MEUC formulation achieves the lowest uplift payments across all tested cases, with a maximum reduction of 91.0% compared to the traditional LMP (TLMP) method.
• On average, the MEUC formulation reduces uplift payments by 65.2% compared to TLMP and 45.0% compared to the approximated CHP-Primal method across all 11 test cases.
• In the nominal load case (Case 0), the MEUC formulation yields $46 in uplift payments, compared to $132 under TLMP and $71 under the approximated CHP-Primal method.
• The method consistently outperforms both benchmarks across all 10 load variation scenarios, with uplift payments under MEUC ranging from $90 to $671, significantly lower than the $356–$1064 range under the approximated CHP-Primal method.
• The results confirm that the MEUC formulation captures the convex hull of the UC polytope more accurately than prior approximations, especially by accounting for ramping and time-dependent start-up costs.
• Although uplift payments are minimized, they are not zero, indicating that the full integral formulation for the entire system remains an open research challenge.