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Adaptive Methods for Multiobjective Unit Commitment

Ece Tevruez, Aswin Kannan

TL;DR

The paper addresses multiobjective unit commitment by balancing generation cost, CO2 emissions, and renewable penetration. It advances scalable solution methods by combining Adaptive Weighted Sum (AWS) and epsilon-constraints with McCormick relaxations to linearize quadratic terms, enabling efficient exploration of the Pareto frontier on real hydro-thermal data. Empirical results show substantial reductions in computational time compared with direct Quadratic Constrained solves (QuadCon) while preserving front quality as measured by hypervolume HV. The work demonstrates practical impact by enabling faster Pareto frontier estimation for MOUC, and provides a framework extendable to larger networks and additional objectives. These contributions pave the way for integrating mathematical programming enhancements with MOOP in power systems and related domains.

Abstract

This work considers a multiobjective version of the unit commitment problem that deals with finding the optimal generation schedule of a firm, over a period of time and a given electrical network. With growing importance of environmental impact, some objectives of interest include CO2 emission levels and renewable energy penetration, in addition to the standard generation costs. Some typical constraints include limits on generation levels and up/down times on generation units. This further entails solving a multiobjective mixed integer optimization problem. The related literature has predominantly focused on heuristics (like Genetic Algorithms) for solving larger problem instances. Our major intent in this work is to propose scalable versions of mathematical optimization based approaches that help in speeding up the process of estimating the underlying Pareto frontier. Our contributions are computational and rest on two key embodiments. First, we use the notion of both epsilon constraints and adaptive weights to solve a sequence of single objective optimization problems. Second, to ease the computational burden, we propose a Mccormick-type relaxation for quadratic type constraints that arise due to the resulting formulation types. We test the proposed computational framework on real network data from [1,50] and compare the same with standard solvers like Gurobi. Results show a significant reduction in complexity (computational time) when deploying the proposed framework.

Adaptive Methods for Multiobjective Unit Commitment

TL;DR

The paper addresses multiobjective unit commitment by balancing generation cost, CO2 emissions, and renewable penetration. It advances scalable solution methods by combining Adaptive Weighted Sum (AWS) and epsilon-constraints with McCormick relaxations to linearize quadratic terms, enabling efficient exploration of the Pareto frontier on real hydro-thermal data. Empirical results show substantial reductions in computational time compared with direct Quadratic Constrained solves (QuadCon) while preserving front quality as measured by hypervolume HV. The work demonstrates practical impact by enabling faster Pareto frontier estimation for MOUC, and provides a framework extendable to larger networks and additional objectives. These contributions pave the way for integrating mathematical programming enhancements with MOOP in power systems and related domains.

Abstract

This work considers a multiobjective version of the unit commitment problem that deals with finding the optimal generation schedule of a firm, over a period of time and a given electrical network. With growing importance of environmental impact, some objectives of interest include CO2 emission levels and renewable energy penetration, in addition to the standard generation costs. Some typical constraints include limits on generation levels and up/down times on generation units. This further entails solving a multiobjective mixed integer optimization problem. The related literature has predominantly focused on heuristics (like Genetic Algorithms) for solving larger problem instances. Our major intent in this work is to propose scalable versions of mathematical optimization based approaches that help in speeding up the process of estimating the underlying Pareto frontier. Our contributions are computational and rest on two key embodiments. First, we use the notion of both epsilon constraints and adaptive weights to solve a sequence of single objective optimization problems. Second, to ease the computational burden, we propose a Mccormick-type relaxation for quadratic type constraints that arise due to the resulting formulation types. We test the proposed computational framework on real network data from [1,50] and compare the same with standard solvers like Gurobi. Results show a significant reduction in complexity (computational time) when deploying the proposed framework.
Paper Structure (51 sections, 34 equations, 9 figures, 30 tables, 1 algorithm)

This paper contains 51 sections, 34 equations, 9 figures, 30 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basics and Formulation
  3. Pareto Optimality
  4. Hypervolume Indicator:
  5. MOO Algorithms
  6. Weighted Sum Method
  7. Epsilon Constraints Method
  8. Problem Formulation
  9. Setup and Notation:
  10. Objective Functions
  11. Generation Costs:
  12. Emissions:
  13. Renewable Energy Penetration:
  14. Constraints
  15. Startup and Shutdown Constraints:
  16. ...and 36 more sections

Figures (9)

  • Figure 1: Feasible objective region and Pareto optimal set (Left) and the hypervolume indicator for a bi-objective problem. Plots are self generated.
  • Figure 2: The offset distances, $\delta_1$ and $\delta_2$
  • Figure 3: McCormick Envelopes with Underestimators and Overestimators
  • Figure 4: Total and Gurobi Runtimes for 10 weight pairs
  • Figure 5: Comparison of Gurobi Runtimes for Adaptive Weights using QuadCon, 1-layer and 2-layer McCormick
  • ...and 4 more figures

Theorems & Definitions (1)

  • Definition 1