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Robust Continuous-Time Generation Scheduling under Power Demand Uncertainty: An Affine Decision Rule Approach

Youngchae Cho, Insoon Yang, Takayuki Ishizaki

TL;DR

This work addresses robust continuous-time generation scheduling under demand uncertainty using a non-anticipative decision-rule approach. It constructs a tractable surrogate by bounding the demand set $ ext{Xi}$ with a superset $ ext{Omega}$ built from two piecewise-affine envelopes and restricting decisions to an affine rule in the current demand, specifically $\hat{x}(\xi,t)=oldsymbol{\\alpha}oldsymbol{\xi}(t)+(1-oldsymbol{mma}(t))oldsymbol{eta}_{k(t)}+oldsymbol{mma}(t)oldsymbol{eta}_{k(t)+1}$, yielding a finite-dimensional LP solvable by a cutting-plane method. The paper proves that this LP-based surrogate upper-bounds the original non-anticipative problem and provides an explicit LP reformulation whose constraints are generated via a systematic cutting-plane algorithm. Numerical demonstrations on a six-bus system show that the method delivers non-anticipative, ramp-feasible generator trajectories for all $oldsymbol{\xi} olinebreak[4] ext{ in } ext{Omega}$ and achieves competitive worst-case costs relative to scenario-based baselines. Overall, the approach offers a scalable, practical framework for robust continuous-time generation scheduling under uncertainty.

Abstract

Most existing generation scheduling models for power systems under demand uncertainty rely on energy-based formulations with a finite number of time periods, which may fail to ensure that power supply and demand are balanced continuously over time. To address this issue, we propose a robust generation scheduling model in a continuous-time framework, employing a decision rule approach. First, for a given set of demand trajectories, we formulate a general robust generation scheduling problem to determine a decision rule that maps these demand trajectories and time points to the power outputs of generators. Subsequently, we derive a surrogate of it as our model by carefully designing a class of decision rules that are affine in the current demand, with coefficients invariant over time and constant terms that are continuous piecewise affine functions of time. As a result, our model can be recast as a finite-dimensional linear program to determine the coefficients and the function values of the constant terms at each breakpoint, solvable via the cutting-plane method. Our model is non-anticipative unlike most existing continuous-time models, which use Bernstein polynomials, making it more practical. We also provide illustrative numerical examples.

Robust Continuous-Time Generation Scheduling under Power Demand Uncertainty: An Affine Decision Rule Approach

TL;DR

This work addresses robust continuous-time generation scheduling under demand uncertainty using a non-anticipative decision-rule approach. It constructs a tractable surrogate by bounding the demand set with a superset built from two piecewise-affine envelopes and restricting decisions to an affine rule in the current demand, specifically , yielding a finite-dimensional LP solvable by a cutting-plane method. The paper proves that this LP-based surrogate upper-bounds the original non-anticipative problem and provides an explicit LP reformulation whose constraints are generated via a systematic cutting-plane algorithm. Numerical demonstrations on a six-bus system show that the method delivers non-anticipative, ramp-feasible generator trajectories for all and achieves competitive worst-case costs relative to scenario-based baselines. Overall, the approach offers a scalable, practical framework for robust continuous-time generation scheduling under uncertainty.

Abstract

Most existing generation scheduling models for power systems under demand uncertainty rely on energy-based formulations with a finite number of time periods, which may fail to ensure that power supply and demand are balanced continuously over time. To address this issue, we propose a robust generation scheduling model in a continuous-time framework, employing a decision rule approach. First, for a given set of demand trajectories, we formulate a general robust generation scheduling problem to determine a decision rule that maps these demand trajectories and time points to the power outputs of generators. Subsequently, we derive a surrogate of it as our model by carefully designing a class of decision rules that are affine in the current demand, with coefficients invariant over time and constant terms that are continuous piecewise affine functions of time. As a result, our model can be recast as a finite-dimensional linear program to determine the coefficients and the function values of the constant terms at each breakpoint, solvable via the cutting-plane method. Our model is non-anticipative unlike most existing continuous-time models, which use Bernstein polynomials, making it more practical. We also provide illustrative numerical examples.

Paper Structure

This paper contains 15 sections, 4 theorems, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Proposed Approach
  4. Approximation of Demand Trajectory Sets
  5. Affine Decision Rule
  6. Proposed Model
  7. Solution Method
  8. LP Reformulation
  9. Cutting-Plane Method
  10. Illustrative Examples
  11. Conclusions
  12. Proof of Proposition \ref{['prop:cost']}
  13. Proof of Proposition \ref{['prop:ineq']}
  14. Proof of Proposition \ref{['prop:ramp']}
  15. Proof of Proposition \ref{['prop:bal']}

Key Result

Proposition 1

The objective function in (eq:prob_our_obj) can be rewritten as follows: where $\alpha_{d}\in{\mathbb R}^G$ denotes the $d$th column of $\alpha$ and

Figures (4)

  • Figure 1: Obtaining $\left(\overline{\omega}_d,\underline{\omega}_d\right)$. (a) $\overline{\omega}^{\prime+}_{d,j},\underline{\omega}^{\prime+}_{d,j}\ge0$. (b) $\overline{\omega}^{\prime+}_{d,j},\underline{\omega}^{\prime+}_{d,j}\le0$.
  • Figure 2: (a) Construction of $\Omega_1$. (b), (c) Enlarged views of (a).
  • Figure 3: (a)--(c) The test demand trajectories. (d)--(f) The power output trajectories obtained using our decision rule for the test demand trajectories.
  • Figure 4: (a) The total demand computed with the trajectories in $\hat{\Omega}^{\rm sc}$ (grey) and the test trajectories from $\Omega$ (green). (b)--(d) The power output trajectories obtained using $\hat{x}^{\rm sc}$ (red) and our decision rule (blue) for the test demand trajectories. (e), (f) Enlarged views of (d). The yellow segments in (d)--(f) indicate the ramp-rate limit violations of generator 1, caused by $\hat{x}^{\rm sc}$.

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4