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Discrete Shortest Paths in Optimal Power Flow Feasible Regions

Daniel Turizo, Diego Cifuentes, Anton Leykin, Daniel K. Molzahn

TL;DR

This work reframes transitions between OPF operating points as a discretized shortest-path problem inside the ACOPF feasible space to minimize the amplitude of control actions. It develops a specialized log-barrier interior-point method that exploits the resulting sparse block-tridiagonal structure, and augments the model with per-node power-flow states to maintain tractable Jacobians. A homotopy-based procedure initializes the path by gradually enforcing constraints, ensuring a feasible start for the interior-point solver. Numerical experiments across diverse test cases demonstrate the method can produce locally shortest, discretized paths with a fixed number of intermediate points, offering a scalable and controllable approach to safe OPF transitions. The framework highlights practical gains for real-time operations and suggests future work on guaranteeing full continuous-feasibility of path segments and adaptive breakpoint placement.

Abstract

Optimal power flow (OPF) is a critical optimization problem for power systems to operate at points where cost or other operational objectives are optimized. Due to the non-convexity of the set of feasible OPF operating points, it is non-trivial to transition the power system from its current operating point to the optimal one without violating constraints. On top of that, practical considerations dictate that the transition should be achieved using a small number of small-magnitude control actions. To solve this problem, this paper proposes an algorithm for computing a transition path by framing it as a shortest path problem. This problem is formulated in terms of a discretized piece-wise linear path, where the number of pieces is fixed a priori in order to limit the number of control actions. This formulation yields a nonlinear optimization problem (NLP) with a sparse block tridiagonal structure, which we leverage by utilizing a specialized interior point method. An initial feasible path for our method is generated by solving a sequence of relaxations which are then tightened in a homotopy-like procedure. Numerical experiments illustrate the effectiveness of the algorithm.

Discrete Shortest Paths in Optimal Power Flow Feasible Regions

TL;DR

This work reframes transitions between OPF operating points as a discretized shortest-path problem inside the ACOPF feasible space to minimize the amplitude of control actions. It develops a specialized log-barrier interior-point method that exploits the resulting sparse block-tridiagonal structure, and augments the model with per-node power-flow states to maintain tractable Jacobians. A homotopy-based procedure initializes the path by gradually enforcing constraints, ensuring a feasible start for the interior-point solver. Numerical experiments across diverse test cases demonstrate the method can produce locally shortest, discretized paths with a fixed number of intermediate points, offering a scalable and controllable approach to safe OPF transitions. The framework highlights practical gains for real-time operations and suggests future work on guaranteeing full continuous-feasibility of path segments and adaptive breakpoint placement.

Abstract

Optimal power flow (OPF) is a critical optimization problem for power systems to operate at points where cost or other operational objectives are optimized. Due to the non-convexity of the set of feasible OPF operating points, it is non-trivial to transition the power system from its current operating point to the optimal one without violating constraints. On top of that, practical considerations dictate that the transition should be achieved using a small number of small-magnitude control actions. To solve this problem, this paper proposes an algorithm for computing a transition path by framing it as a shortest path problem. This problem is formulated in terms of a discretized piece-wise linear path, where the number of pieces is fixed a priori in order to limit the number of control actions. This formulation yields a nonlinear optimization problem (NLP) with a sparse block tridiagonal structure, which we leverage by utilizing a specialized interior point method. An initial feasible path for our method is generated by solving a sequence of relaxations which are then tightened in a homotopy-like procedure. Numerical experiments illustrate the effectiveness of the algorithm.
Paper Structure (20 sections, 2 theorems, 106 equations, 4 figures, 3 tables, 2 algorithms)

This paper contains 20 sections, 2 theorems, 106 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Shortest Path OPF Problem Formulation
  3. Optimal Control Problem
  4. Piece-wise Linear Path Approximation
  5. Log-Barrier Newton Method Implementation
  6. Interior Point Iteration
  7. Newton Step Correction
  8. Newton Step Permutation
  9. Newton Iteration Algorithm
  10. Initial Feasible Path Generation
  11. Numerical Experiments
  12. Implementation
  13. Example: Two Variants of the 9-Bus Case
  14. Multiple scale OPF cases
  15. Tests on the number of control actions
  16. ...and 5 more sections

Key Result

Theorem 1

For any $j=1,\ldots,K+1$ define $b_j(p) = w_j d_j$ and assume that $b_j (p)\neq 0$ for all $j=1,\ldots,K+1$ (this is true if and only if $d_j \neq 0$). Let $q_j$ be If $\sum^{K+1}_{j=1} q_j(p) \neq 0$, then $D_{\mathcal{E}}$ is full rank.

Figures (4)

  • Figure 1: Variables $u$ and $x$ in a neighborhood of $u_0$ and $x_0$ are related by the power flow mapping $\varphi$. Feasible sets generated by inequalities in $x$ can be mapped back to feasible sets in $u$ and vice-versa. As the power flow mapping $\varphi$ is nonlinear, the geometry of the mapped feasible sets will be altered.
  • Figure 2: Variant 1 of the 9-bus case. The straight line path is not feasible, but the algorithm deforms the path to achieve feasibility.
  • Figure 3: Variant 2 of the 9-bus case. The endpoints are disconnected, so the algorithm fails to find a feasible path.
  • Figure 4: Variant 1 of the 9-bus case. For smaller barrier parameters, the path length decreases until stabilizing at the shortest path. However, very small barrier parameters introduce numerical artifacts.

Theorems & Definitions (6)

  • Theorem 1
  • proof : Proof
  • Theorem 2
  • proof : Proof
  • proof : Proof
  • proof : Proof