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Alleviating the Curse of Dimensionality in Minkowski Sum Approximations of Storage Flexibility

Emrah Öztürk, Timm Faulwasser, Karl Worthmann, Markus Preißinger, Klaus Rheinberger

TL;DR

The paper tackles the dimensionality barrier in aggregating flexible energy devices by developing a vertex-based inner approximation for Minkowski sums of polytopes representing device flexibilities. It proves that the sums of per-device extreme actions yield $2^d$ vertices of the full Minkowski sum and that their convex hull $\mathcal{A} = \text{Conv}(\{v^j\})$ provides an efficient inner approximation with $\mathcal{A} \subseteq \mathcal{M}$. An energy-storage–specific, non-optimizing algorithm computes the extreme actions for polytopes $\mathcal{B}(S_0,S_f,p)$, with a correction step to meet final-energy constraints, plus a complete multi-device aggregation workflow. A complementary disaggregation method distributes aggregated profiles back to devices without optimization, and extensive benchmarks show the approach outperforms ten state-of-the-art inner approximations in speed and accuracy, enabling day-long, quarter-hourly planning with practical computational costs and potential microcontroller deployment.

Abstract

Many real-world applications require the joint optimization of a large number of flexible devices over time. The flexibility of, e.g., multiple batteries, thermostatically controlled loads, or electric vehicles can be used to support grid operation and to reduce operation costs. Using piecewise constant power values, the flexibility of each device over $d$ time periods can be described as a polytopic subset in power space. The aggregated flexibility is given by the Minkowski sum of these polytopes. As the computation of Minkowski sums is in general demanding, several approximations have been proposed in the literature. Yet, their application potential is often objective-dependent and limited by the curse of dimensionality. We show that up to $2^d$ vertices of each polytope can be computed efficiently and that the convex hull of their sums provides a computationally efficient inner approximation of the Minkowski sum. Via an extensive simulation study, we illustrate that our approach outperforms ten state-of-the-art inner approximations in terms of computational complexity and accuracy for different objectives. Moreover, we propose an efficient disaggregation method applicable to any vertex-based approximation. The proposed methods provide an efficient means to aggregate and to disaggregate energy storages in quarter-hourly periods over an entire day with reasonable accuracy for aggregated cost and for peak power optimization.

Alleviating the Curse of Dimensionality in Minkowski Sum Approximations of Storage Flexibility

TL;DR

The paper tackles the dimensionality barrier in aggregating flexible energy devices by developing a vertex-based inner approximation for Minkowski sums of polytopes representing device flexibilities. It proves that the sums of per-device extreme actions yield vertices of the full Minkowski sum and that their convex hull provides an efficient inner approximation with . An energy-storage–specific, non-optimizing algorithm computes the extreme actions for polytopes , with a correction step to meet final-energy constraints, plus a complete multi-device aggregation workflow. A complementary disaggregation method distributes aggregated profiles back to devices without optimization, and extensive benchmarks show the approach outperforms ten state-of-the-art inner approximations in speed and accuracy, enabling day-long, quarter-hourly planning with practical computational costs and potential microcontroller deployment.

Abstract

Many real-world applications require the joint optimization of a large number of flexible devices over time. The flexibility of, e.g., multiple batteries, thermostatically controlled loads, or electric vehicles can be used to support grid operation and to reduce operation costs. Using piecewise constant power values, the flexibility of each device over time periods can be described as a polytopic subset in power space. The aggregated flexibility is given by the Minkowski sum of these polytopes. As the computation of Minkowski sums is in general demanding, several approximations have been proposed in the literature. Yet, their application potential is often objective-dependent and limited by the curse of dimensionality. We show that up to vertices of each polytope can be computed efficiently and that the convex hull of their sums provides a computationally efficient inner approximation of the Minkowski sum. Via an extensive simulation study, we illustrate that our approach outperforms ten state-of-the-art inner approximations in terms of computational complexity and accuracy for different objectives. Moreover, we propose an efficient disaggregation method applicable to any vertex-based approximation. The proposed methods provide an efficient means to aggregate and to disaggregate energy storages in quarter-hourly periods over an entire day with reasonable accuracy for aggregated cost and for peak power optimization.
Paper Structure (13 sections, 7 theorems, 14 equations, 5 figures, 2 tables, 4 algorithms)

This paper contains 13 sections, 7 theorems, 14 equations, 5 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Application to Energy Storage
  5. Benchmark Results
  6. Disaggregation for Vertex-based Approximations
  7. Conclusions
  8. Proof of Lemma 1
  9. Proof of Lemma 2
  10. Proof of Proposition 1
  11. Proof of Proposition 2
  12. Proof of Lemma 3
  13. Proof of Theorem 2

Key Result

Lemma 1

Let polytopes $\mathcal{P}(A_i, b_i) \subset \mathbb{R}^{d}$, $i \in \{1,\ldots,{n}\}$, fulfill Assumptions assumption and assumption2. Further, let $v^j, v^k \in \mathbb{R}^{d}, j,k \in \{-1, 1\}^{d}$ satisfy eq:y. Then, the following holds:

Figures (5)

  • Figure 1: Illustration of Assumption \ref{['assumption2']}. While the polytope on the left satisfies the assumption, the polytope on the right does not.
  • Figure 2: Left: vectors $y^{(-1, 1)}_1$, $y^{(-1, 1)}_2$ within the polytopes shown in dashed blue and solid green, and the sum $v^{(-1, 1)}$ shown in the Minkowski sum $\mathcal{M}$ in dash-dotted black. Right: all possible vectors $v^j, j \in \{-1, 1\}^2$ in the Minkowski sum with the resulting set $\mathcal{A}$ in orange.
  • Figure 3: The set $\mathcal{B}(S_0, S_f, p)$ in solid green and the set $\mathcal{B}(S_0, \underline{S}, p)$ in dashed blue. The crosses on $\mathcal{B}(S_0, \underline{S}, p)$ indicate the $y^j$, and the arrow with dot visualizes the correction process.
  • Figure 4: Boxplot for UPR values with 100 batteries, $d=12, 14, \ldots, 96$ time periods, and $g=d^2$. For each time period, the approximation is calculated 50 times.
  • Figure 5: Results for experiments with tuples $(n,d) \in \{30\} \times \{4, 8, 12, 16, 24\}$ in the first column and experiments with tuples $(n,d) \in \{2, 6, 10, 20, 30\} \times \{24\}$ in the second column. The cost UPR values are shown in the first row, the peak UPR values in the second row and the calculation times in the third row.

Theorems & Definitions (16)

  • Definition 1: Extreme actions
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Theorem 1: Extreme actions define vertices
  • proof
  • Lemma 3
  • Theorem 2
  • proof
  • ...and 6 more