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Economic Dispatch of a Single Micro-Gas Turbine Under CHP Operation with Uncertain Demands

Miel Sharf, Iliya Romm, Michael Palman, Daniel Zelazo, Beni Cukurel

TL;DR

This paper tackles robust economic dispatch for a single micro gas turbine operating in CHP mode under uncertain power and heat demands. It recasts the known- and unknown-demand ED problem as robust shortest-path problems on a discrete-time graph, and develops two uncertainty-set frameworks: an $\ell^\infty$-based box and a mixed $\mathcal{L}_1/\mathcal{L}_\infty$ budgeted set, with recourse to utility costs. The main contributions are exact linear-time solutions for the $\mathcal{L}_\infty$ case and exact quadratic-time, plus an approximate linear-time, algorithm for the mixed-norm case, validated via a detailed case study using real demand profiles and tariffs. The results show that the robust approaches outperform a nominal forecast-based ED, with the mixed-norm method achieving benchmark-like performance on several days, highlighting practical value for robust microgrid operation. This work lays the foundation for robust CHP-ED with discrete turbine models and points to scalable extensions to small turbine arrays and rolling-horizon schedules.

Abstract

This work considers the economic dispatch problem for a single micro-gas turbine, governed by a discrete state-space model, under combined heat and power (CHP) operation and coupled with a utility. If the exact power and heat demands are given, existing algorithms can be used to give a quick optimal solution to the economic dispatch problem. However, in practice, the power and heat demands can not be known deterministically, but are rather predicted, resulting in an estimate and a bound on the estimation error. We consider the case in which the power and heat demands are unknown, and present a robust optimization-based approach for scheduling the turbine's heat and power generation, in which the demand is assumed to be inside an uncertainty set. We consider two different choices of the uncertainty set relying on the $\ell^\infty$- and the $\ell^1$-norms, each with different advantages, and consider the associated robust economic dispatch problems. We recast these as robust shortest-path problems on appropriately defined graphs. For the first choice, we provide an exact linear-time algorithm for the solution of the robust shortest-path problem, and for the second, we provide an exact quadratic-time algorithm and an approximate linear-time algorithm. The efficiency and usefulness of the algorithms are demonstrated using a detailed case study that employs real data on energy demand profiles and electricity tariffs.

Economic Dispatch of a Single Micro-Gas Turbine Under CHP Operation with Uncertain Demands

TL;DR

This paper tackles robust economic dispatch for a single micro gas turbine operating in CHP mode under uncertain power and heat demands. It recasts the known- and unknown-demand ED problem as robust shortest-path problems on a discrete-time graph, and develops two uncertainty-set frameworks: an -based box and a mixed budgeted set, with recourse to utility costs. The main contributions are exact linear-time solutions for the case and exact quadratic-time, plus an approximate linear-time, algorithm for the mixed-norm case, validated via a detailed case study using real demand profiles and tariffs. The results show that the robust approaches outperform a nominal forecast-based ED, with the mixed-norm method achieving benchmark-like performance on several days, highlighting practical value for robust microgrid operation. This work lays the foundation for robust CHP-ED with discrete turbine models and points to scalable extensions to small turbine arrays and rolling-horizon schedules.

Abstract

This work considers the economic dispatch problem for a single micro-gas turbine, governed by a discrete state-space model, under combined heat and power (CHP) operation and coupled with a utility. If the exact power and heat demands are given, existing algorithms can be used to give a quick optimal solution to the economic dispatch problem. However, in practice, the power and heat demands can not be known deterministically, but are rather predicted, resulting in an estimate and a bound on the estimation error. We consider the case in which the power and heat demands are unknown, and present a robust optimization-based approach for scheduling the turbine's heat and power generation, in which the demand is assumed to be inside an uncertainty set. We consider two different choices of the uncertainty set relying on the - and the -norms, each with different advantages, and consider the associated robust economic dispatch problems. We recast these as robust shortest-path problems on appropriately defined graphs. For the first choice, we provide an exact linear-time algorithm for the solution of the robust shortest-path problem, and for the second, we provide an exact quadratic-time algorithm and an approximate linear-time algorithm. The efficiency and usefulness of the algorithms are demonstrated using a detailed case study that employs real data on energy demand profiles and electricity tariffs.

Paper Structure

This paper contains 20 sections, 8 theorems, 28 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Micro Gas Turbines
  3. Economic Analysis of Power Generation
  4. Shortest-Path Algorithms and Uncertain Demand
  5. Contributions
  6. Turbine Models, ED, and Robust Optimization
  7. Discrete State-Space Models for Turbines
  8. Economic Dispatch and the Shortest-Path Problem
  9. Robust Optimization
  10. Robust Shortest-Path Problems
  11. Robust Economic Dispatch with Uncertain Demands
  12. Positively-Extreme Profiles and $\mathcal{L}_\infty$-based Uncertainty
  13. Mixed $\mathcal{L}_1\setminus\mathcal{L}_\infty$ Uncertainty
  14. Discussion about Uncertainty Sets and Algorithms
  15. Case Studies
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathcal{W}$ be any bounded closed subset of $\mathbb{R}^{2T}$, and assume all functions $w_e$ satisfy eq.Monotonicity. The problem eq.RSPP for $\mathcal{W}$ is equivalent to the problem eq.RSPP for ${\rm pe}(\mathcal{W})$, i.e.,

Figures (8)

  • Figure 1: The state transition graph corresponding to the ED problem for the turbine in Example \ref{['eq.StartUpMinimum']} with time horizon $T=5$.
  • Figure 2: Normed ball in $\mathbb{R}^2$. The black set is an $\ell^\infty$-normed ball, and the green set is an $\ell^1$-normed ball. The yellow set is a larger $\ell^\infty$-normed ball, which is the analogue of \ref{['eq.Linfty']}. The blue set is the Minkowski sum of the black and purple sets, which is the analogue of \ref{['eq.MixedNorm']}. The blue set is a subset of the yellow set which does not include points in which both entries are large in absolute value.
  • Figure 3: Engine schematic cycle.
  • Figure 4: Solution grid over the states of the gas turbine model.
  • Figure 5: Schedules produced by the algorithms for February 5$^{\rm th}$ (winter).
  • ...and 3 more figures

Theorems & Definitions (27)

  • Remark 1
  • Example 1
  • Example 2
  • Definition 1
  • Example 3
  • Example 4
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • ...and 17 more