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On Degeneracy Issues in Multi-parametric Programming and Critical Region Exploration based Distributed Optimization in Smart Grid Operations

Haitian Liu, Ye Guo, Hao Liu

TL;DR

This work tackles degeneracy in multi-parametric LP/QP when optimizing smart-grid operations with renewable penetration. It introduces a unified degeneracy-handling framework that transforms mpLP/QP to a multi-parametric linear complementarity problem (mpLCP), solves via Lemke's method, and uses auxiliary LCPs and complementary-basis enumeration to recover all critical regions containing a given parameter. The authors embed this approach into an improved Critical Region Exploration (CRE) for distributed optimization, incorporating cutting planes and adaptive step sizes to ensure robust finite convergence even under degeneracy, and validate the method on eight multi-area tie-line scheduling benchmarks. The results show comparable or superior performance to state-of-the-art distributed methods, with strong robustness to initial conditions and problem scale, highlighting significant practical potential for coordinated transmission-distribution optimization in smart grids.

Abstract

Improving renewable energy resource utilization efficiency is crucial to reducing carbon emissions, and multi-parametric programming has provided a systematic perspective in conducting analysis and optimization toward this goal in smart grid operations. This paper focuses on two aspects of interest related to multi-parametric linear/quadratic programming (mpLP/QP). First, we study degeneracy issues of mpLP/QP. A novel approach to deal with degeneracies is proposed to find all critical regions containing the given parameter. Our method leverages properties of the multi-parametric linear complementary problem, vertex searching technique, and complementary basis enumeration. Second, an improved critical region exploration (CRE) method to solve distributed LP/QP is proposed under a general mpLP/QP-based formulation. The improved CRE incorporates the proposed approach to handle degeneracies. A cutting plane update and an adaptive stepsize scheme are also integrated to accelerate convergence under different problem settings. The computational efficiency is verified on multi-area tie-line scheduling problems with various testing benchmarks and initial states.

On Degeneracy Issues in Multi-parametric Programming and Critical Region Exploration based Distributed Optimization in Smart Grid Operations

TL;DR

This work tackles degeneracy in multi-parametric LP/QP when optimizing smart-grid operations with renewable penetration. It introduces a unified degeneracy-handling framework that transforms mpLP/QP to a multi-parametric linear complementarity problem (mpLCP), solves via Lemke's method, and uses auxiliary LCPs and complementary-basis enumeration to recover all critical regions containing a given parameter. The authors embed this approach into an improved Critical Region Exploration (CRE) for distributed optimization, incorporating cutting planes and adaptive step sizes to ensure robust finite convergence even under degeneracy, and validate the method on eight multi-area tie-line scheduling benchmarks. The results show comparable or superior performance to state-of-the-art distributed methods, with strong robustness to initial conditions and problem scale, highlighting significant practical potential for coordinated transmission-distribution optimization in smart grids.

Abstract

Improving renewable energy resource utilization efficiency is crucial to reducing carbon emissions, and multi-parametric programming has provided a systematic perspective in conducting analysis and optimization toward this goal in smart grid operations. This paper focuses on two aspects of interest related to multi-parametric linear/quadratic programming (mpLP/QP). First, we study degeneracy issues of mpLP/QP. A novel approach to deal with degeneracies is proposed to find all critical regions containing the given parameter. Our method leverages properties of the multi-parametric linear complementary problem, vertex searching technique, and complementary basis enumeration. Second, an improved critical region exploration (CRE) method to solve distributed LP/QP is proposed under a general mpLP/QP-based formulation. The improved CRE incorporates the proposed approach to handle degeneracies. A cutting plane update and an adaptive stepsize scheme are also integrated to accelerate convergence under different problem settings. The computational efficiency is verified on multi-area tie-line scheduling problems with various testing benchmarks and initial states.
Paper Structure (19 sections, 3 theorems, 40 equations, 12 figures, 1 table, 2 algorithms)

This paper contains 19 sections, 3 theorems, 40 equations, 12 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Dealing with degeneracies with multi-parametric linear complementary problems
  3. Muiti-parametric linear/quadratic programming problem
  4. Characterizing different types of degeneracies
  5. Find all critical regions containing the given parameter
  6. An Improved Critical Region Exploration
  7. Distributed problem setup
  8. General critical region exploration process
  9. An improved scheme
  10. Application to multi-area tie-line scheduling
  11. Case studies
  12. Benchmarks and simulation settings
  13. Computation time comparisons under a cold start
  14. Analysis of degeneracy handling of each benchmark
  15. Benchmark tests under random initial system states
  16. ...and 4 more sections

Key Result

Lemma 1

( cf.borrelli_bemporad_morari_2017): Consider the general convex mpQP eq:mpP, the set of feasible parameters ${\bm{\Theta}}^\star_i$ is polyhedral and ${\bf J}_i({\bm{\theta}})$ is convex and piecewise linear/quadratic over ${\bm{\Theta}}^\star_i$.

Figures (12)

  • Figure 1: Illustrations on the violation of (a) LICQ/SOSC, (b) SCS condition. Indices in the figures represent the active constraints set, and the degenerated region is shown in a green-shaded area, thick lines, and dots. Red shaded areas represent the non-degenerate region of the problems. Due to the singularity, the degenerate space is plotted by directly projecting the variable space onto the parameter space.
  • Figure 2: Illustrative process to generate basis candidates hercegEnumerationbasedApproachSolving2015a. When $p=3$, all possibles candidates are listed in level 3.
  • Figure 3: Generation and selection of basis candidates when $p=3$, ${\cal W}_j = 1$, ${\cal Z}_j = 2$, ${\cal D}_j = 3$.
  • Figure 4: The architecture of CRE. Red/green arrows indicate upward/downward communication links and blue arrows indicate physical connections.
  • Figure 5: An illustration for the multi-area power system.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Corollary 1
  • Corollary 2