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Towards Optimality Preserving Aggregation for Scheduling Distributed Energy Resources

Riccardo Remo Appino, Veit Hagenmeyer, Timm Faulwasser

TL;DR

This paper investigates aggregation of heterogeneous (storage) devices with time-varying power and energy constraints and proposes mild conditions on the constraints of each device guaranteeing the applicability of an aggregated model in scheduling without any loss of optimality in comparison to the complete problem.

Abstract

Scheduling the power exchange between a population of heterogeneous distributed energy resources and the corresponding upper-level system is an important control problem in power systems. A key challenge is the large number of (partially uncertain) parameters and decision variables that increase the computational burden and that complicate the structured consideration of uncertainties. Reducing the number of decision variables by means of aggregation can alleviate these issues. However, despite the frequent use of aggregation for storage, few works in the literature provide formal justification. In the present paper, we investigate aggregation of heterogeneous (storage) devices with time-varying power and energy constraints. In particular, we propose mild conditions on the constraints of each device guaranteeing the applicability of an aggregated model in scheduling without any loss of optimality in comparison to the complete problem. We conclude with a discussion of limitations and possible extensions of our findings.

Towards Optimality Preserving Aggregation for Scheduling Distributed Energy Resources

TL;DR

This paper investigates aggregation of heterogeneous (storage) devices with time-varying power and energy constraints and proposes mild conditions on the constraints of each device guaranteeing the applicability of an aggregated model in scheduling without any loss of optimality in comparison to the complete problem.

Abstract

Scheduling the power exchange between a population of heterogeneous distributed energy resources and the corresponding upper-level system is an important control problem in power systems. A key challenge is the large number of (partially uncertain) parameters and decision variables that increase the computational burden and that complicate the structured consideration of uncertainties. Reducing the number of decision variables by means of aggregation can alleviate these issues. However, despite the frequent use of aggregation for storage, few works in the literature provide formal justification. In the present paper, we investigate aggregation of heterogeneous (storage) devices with time-varying power and energy constraints. In particular, we propose mild conditions on the constraints of each device guaranteeing the applicability of an aggregated model in scheduling without any loss of optimality in comparison to the complete problem. We conclude with a discussion of limitations and possible extensions of our findings.

Paper Structure

This paper contains 13 sections, 5 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Scheduling for Integrated Energy Systems
  3. Energy-Constrained Distributed Energy Resources
  4. Scheduling with Individual Device Models
  5. Scheduling and Aggregated Models
  6. Abstract Problem Statement
  7. Calculating Bounds for the Aggregated Model
  8. Ensuring Schedule Feasibility
  9. Existence and Computation of Consistent Dispersions
  10. Proofs of Theorem \ref{['thm:theorem_1']} & \ref{['thm:existence_of_ideal_states']}
  11. Discussion
  12. Conversion Losses
  13. Conclusion

Key Result

Lemma 1

Consider Problems eq:abstract_opt_problem and eq:abstract_opt_problem_aggregated, let $\bold{x}^\star\in\mathbb{R}^{N_x\cdot K}$ and $\bold{y}^\star\in\mathbb{R}^{N_y\cdot K}$ denote the respective optimal solutions. The identity holds for arbitrary continuous choices of $\ell:\mathbb{R}^{N_y} \to \mathbb{R}$ if and only if for all $k \in \mathcal{K}$. $\blacksquare$

Figures (5)

  • Figure 1: Schematic representation of an integrated energy system.
  • Figure 2: Ilustration of Example 1.
  • Figure 3: Illustration of the energy constraints \ref{['eq:complete_power_limit_energy_focus']} and \ref{['eq:complete_power_limit_energy_focus_agg']} in Example \ref{['ex:inf_states']}.
  • Figure 4: Illustration of Example 2.
  • Figure 5: Graphical representation of a consistent dispersion of $E(k)$.

Theorems & Definitions (13)

  • Lemma 1: Optimality preserving aggregation
  • Example 1: Non-dispersable aggregation
  • Example 2
  • Definition 1: Consistent dispersion of $E(k)$
  • Remark 1: Dispersions and two-stage scheduling
  • Theorem 1: Existence of a consistent dispersion
  • Theorem 2: Recursive existence of consistent dispersions
  • proof : Proof of Theorem \ref{['thm:theorem_1']}
  • Lemma 2: Case (i)
  • proof
  • ...and 3 more