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Stochastic Security Constrained AC Optimal Power Flow Using General Polynomial Chaos Expansion

Ghulam Mohy-ud-din, Yunqi Wang, Rahmat Heidari, Frederik Geth

TL;DR

This work develops a general polynomial chaos expansion–based, chance-constrained AC security-constrained optimal power flow (gPCE-CC-SCOPF) to address uncertainty from renewable generation and discrete contingencies. It adopts a two-stage framework in rectangular coordinates: a base-case gPCE-based stochastic problem and linked contingency cases, with an objective to minimize expected generation cost while penalizing constraint violations. The approach yields improved uncertainty capture and reveals a broader set of insecure contingencies than deterministic SCOPF, at modest increases in cost and computation time, through deterministic reformulations of probabilistic constraints. Overall, it enables reliable, risk-aware operation under non-Gaussian randomness without resorting to sampling or heavy relaxation, and is validated on multiple IEEE test systems with degree-2 accuracy.

Abstract

Addressing the uncertainty introduced by increasing renewable integration is crucial for secure power system operation, yet capturing it while preserving the full nonlinear physics of the grid remains a significant challenge. This paper presents a stochastic security constrained optimal power flow model with chance constraints supporting nonlinear AC power flow equations and non Gaussian uncertainties. We use general polynomial chaos expansion to model arbitrary uncertainties of finite variance, enabling accurate moment computations and robust prediction of system states across diverse operating scenarios. The chance constraints probabilistically limit inequality violations, providing a more flexible representation of controllable variables and the consequent power system operation. Case studies validate the proposed models effectiveness in satisfying operational constraints and capturing uncertainty with high fidelity. Compared to the deterministic formulation, it also uncovers a wider set of unsecure contingencies, highlighting improved uncertainty capture and operational insight.

Stochastic Security Constrained AC Optimal Power Flow Using General Polynomial Chaos Expansion

TL;DR

This work develops a general polynomial chaos expansion–based, chance-constrained AC security-constrained optimal power flow (gPCE-CC-SCOPF) to address uncertainty from renewable generation and discrete contingencies. It adopts a two-stage framework in rectangular coordinates: a base-case gPCE-based stochastic problem and linked contingency cases, with an objective to minimize expected generation cost while penalizing constraint violations. The approach yields improved uncertainty capture and reveals a broader set of insecure contingencies than deterministic SCOPF, at modest increases in cost and computation time, through deterministic reformulations of probabilistic constraints. Overall, it enables reliable, risk-aware operation under non-Gaussian randomness without resorting to sampling or heavy relaxation, and is validated on multiple IEEE test systems with degree-2 accuracy.

Abstract

Addressing the uncertainty introduced by increasing renewable integration is crucial for secure power system operation, yet capturing it while preserving the full nonlinear physics of the grid remains a significant challenge. This paper presents a stochastic security constrained optimal power flow model with chance constraints supporting nonlinear AC power flow equations and non Gaussian uncertainties. We use general polynomial chaos expansion to model arbitrary uncertainties of finite variance, enabling accurate moment computations and robust prediction of system states across diverse operating scenarios. The chance constraints probabilistically limit inequality violations, providing a more flexible representation of controllable variables and the consequent power system operation. Case studies validate the proposed models effectiveness in satisfying operational constraints and capturing uncertainty with high fidelity. Compared to the deterministic formulation, it also uncovers a wider set of unsecure contingencies, highlighting improved uncertainty capture and operational insight.

Paper Structure

This paper contains 22 sections, 17 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. gPCE for Non-convex Stochastic SCOPF
  3. gPCE Foundations
  4. Uncertainty Propagation and Chance Constraints
  5. Proposed Stochastic SCOPF Framework
  6. Objective Function
  7. Base Case Constraints
  8. Constraints for reference bus $i, \forall i \in \mathcal{I}^{\mathsf{ref}}$
  9. Constraints for generator $g, \forall g \in \mathcal{G}$
  10. Constraints for bus $i$, $\forall i \in \mathcal{I}$
  11. Constraints for ac branch $l, \forall lij \in \mathcal{T}$
  12. Contingency Case Constraints
  13. Constraints for reference bus $i, \forall i \in \mathcal{I}^{\mathsf{ref}}$
  14. Constraints for bus $i, \forall i \in \mathcal{I}, \forall k \in \mathcal{K}$
  15. Constraints for ac branches, $\forall lij \in \mathcal{T}, \forall k \in \mathcal{K}$
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Conceptual uncertainty propagation in stochastic SCOPF. White: deterministic feasible set; black: operating point; gray: variability induced by uncertain inputs. The sketch is qualitative; nonconvex AC constraints can yield irregular feasible regions and paths.
  • Figure 2: Generator active power bound violation profile under contingency set.
  • Figure 3: Branch current limit violation profile under contingency set.