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Accelerating Chance-constrained SCED via Scenario Compression

Qian Zhang, Le Xie

TL;DR

This paper shows that by exclusively employing the vertices after convex hull compression, an equivalent solution can be obtained compared to utilizing the entire scenario set by presenting the compression risk validation scheme to assess the risk arising from the sample space.

Abstract

This paper studies some compression methods to accelerate the scenario-based chance-constrained security-constrained economic dispatch (SCED) problem. In particular, we show that by exclusively employing the vertices after convex hull compression, an equivalent solution can be obtained compared to utilizing the entire scenario set. For other compression methods that might relax the original solution, such as box compression, this paper presents the compression risk validation scheme to assess the risk arising from the sample space. By quantifying the risk associated with compression, decision-makers are empowered to select either solution risk or compression risk as the risk metric, depending on the complexity of specific problems. Numerical examples based on the 118-bus system and synthetic Texas grids compare these two risk metrics. The results also demonstrate the efficiency of compression methods in both problem formulation and solving processes.

Accelerating Chance-constrained SCED via Scenario Compression

TL;DR

This paper shows that by exclusively employing the vertices after convex hull compression, an equivalent solution can be obtained compared to utilizing the entire scenario set by presenting the compression risk validation scheme to assess the risk arising from the sample space.

Abstract

This paper studies some compression methods to accelerate the scenario-based chance-constrained security-constrained economic dispatch (SCED) problem. In particular, we show that by exclusively employing the vertices after convex hull compression, an equivalent solution can be obtained compared to utilizing the entire scenario set. For other compression methods that might relax the original solution, such as box compression, this paper presents the compression risk validation scheme to assess the risk arising from the sample space. By quantifying the risk associated with compression, decision-makers are empowered to select either solution risk or compression risk as the risk metric, depending on the complexity of specific problems. Numerical examples based on the 118-bus system and synthetic Texas grids compare these two risk metrics. The results also demonstrate the efficiency of compression methods in both problem formulation and solving processes.
Paper Structure (14 sections, 6 theorems, 22 equations, 9 figures, 5 tables)

This paper contains 14 sections, 6 theorems, 22 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Chance-Constrained SCED
  4. The Scenario Approach
  5. Scenario Compression
  6. Convex Hull Compression
  7. Box Compression
  8. Compression Risk
  9. Time Complexity
  10. Case Study
  11. 118-bus System
  12. Synthetic Texas Grid
  13. Conclusion
  14. Acknowledgment

Key Result

Theorem 1

Consider the number of scenarios $N$ is larger than the number of decision variables $n$. Given a confidence parameter $\beta \in (0,1)$, for any $k = 0,1,\dots,n$ consider the polynomial equation in the $t$ variable This equation has exactly two solutions in $[0,+\infty)$, which are named as $\underline{t}(k) <$. Supposing $\underline\epsilon(k):= max\{0,1-\bar{t}(k)\}$ and $\bar{\epsilon}(k):=

Figures (9)

  • Figure 1: The comparison of risk upper bound from different papers
  • Figure 2: The comparison of convex hull compression (black dotted lines) and box compression (red solid lines)
  • Figure 3: The comparison of (convex hull) compression risk and solution risk
  • Figure 4: The convex hull compression time under different dimensionality
  • Figure 5: The simple illustration of affine transformation from black points to red points, where the black random variables are uniformly distributed in [0, 1] of each coordinate direction but with the same value in X-direction and Z-direction.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Definition 1: Support Constraint
  • Definition 2: Solution Complexity
  • Definition 3: Solution Risk
  • Theorem 1: Solution Risk Bounds garatti2022risk
  • Definition 4: Vertex of Convex Hull
  • Theorem 2: Invariant Solution
  • proof
  • Corollary 1: Invariant Solution Complexity
  • proof
  • Definition 5: Compression Risk campi2023compression
  • ...and 7 more