[Paper Review] Chance-Constrained AC Optimal Power Flow -- A Polynomial Chaos Approach

Tillmann Mühlpfordt, Line Roald+3 more|arXiv (Cornell University)|Mar 27, 2019

Probabilistic and Robust Engineering Design31 references4 citations

TL;DR

This paper proposes a polynomial chaos expansion (PCE)-based method for solving chance-constrained AC optimal power flow (AC-OPF) problems without sampling, linearization, or relaxation. By representing uncertain injections via PCE, the method accurately propagates uncertainty through nonlinear AC power flow equations and enables moment-based reformulation of chance constraints, achieving high accuracy with low-degree polynomials (N_d = 2–3).

ABSTRACT

As the share of renewables in the grid increases, the operation of power systems becomes more challenging. The present paper proposes a method to formulate and solve chance-constrained optimal power flow while explicitly considering the full nonlinear AC power flow equations and stochastic uncertainties. We use polynomial chaos expansion to model the effects of arbitrary uncertainties of finite variance, which enables to predict and optimize the system state for a range of operating conditions. We apply chance constraints to limit the probability of violations of inequality constraints. Our method incorporates a more detailed and a more flexible description of both the controllable variables and the resulting system state than previous methods. Two case studies highlight the efficacy of the method, with a focus on satisfaction of the AC power flow equations and on the accurate computation of moments of all random variables.

Motivation & Objective

• Address the challenge of incorporating stochastic uncertainties in AC optimal power flow while preserving the full nonlinear AC power flow equations.
• Overcome limitations of existing methods that rely on linearizations, relaxations, or sampling, which compromise feasibility or computational efficiency.
• Enable accurate and computationally tractable chance-constrained AC-OPF formulations using polynomial chaos expansion (PCE) for arbitrary uncertainties of finite variance.
• Ensure satisfaction of both equality (AC power flow) and inequality (chance) constraints under uncertainty, with minimal reliance on assumptions about distributional families.
• Demonstrate the method’s efficacy on standard test systems (5-bus and 30-bus) with focus on moment accuracy and constraint violation control.

Proposed method

• Use polynomial chaos expansion (PCE) to represent uncertain nodal injections as a series of orthogonal polynomials in random variables, enabling spectral representation of uncertainty.
• Propagate uncertainty through the full nonlinear AC power flow equations via PCE, avoiding linearization or sampling while preserving nonlinearity.
• Reformulate chance constraints using statistical moments (mean and variance) derived directly from the PCE representation, enabling deterministic optimization.
• Formulate the chance-constrained AC-OPF as a deterministic nonlinear program by expressing constraints in terms of PCE-derived moments.
• Employ a sparse, basis-adaptive PCE approach to reduce computational cost while maintaining accuracy, especially for low-degree expansions.
• Solve the resulting deterministic optimization problem using standard nonlinear programming solvers, with generation policies expressed as non-affine functions of uncertainty realizations.

Experimental results

Research questions

• RQ1 Can polynomial chaos expansion be used to accurately represent and propagate uncertainty through the full nonlinear AC power flow equations without linearization or sampling?
• RQ2 To what extent does the degree of the PCE basis affect the accuracy of AC power flow satisfaction and chance constraint violation probability?
• RQ3 How sensitive is the method’s performance to mismatches between the true and modeled probability distributions of uncertain injections?
• RQ4 Can PCE-based moment reformulations of chance constraints ensure reliable constraint satisfaction across diverse uncertainty realizations?
• RQ5 What is the trade-off between computational cost and solution accuracy when using low-degree PCE (e.g., N_d = 2 or 3) for AC-OPF under uncertainty?

Key findings

• The AC power flow equations are satisfied to high numerical accuracy (within 10^-3 to 10^-4 per bus) even with low-degree PCE (N_d = 2 or 3), demonstrating robustness to nonlinearity.
• For the 5-bus system, the maximum power flow violation is below 0.001 p.u. across all uncertainty realizations when using N_d = 2, indicating high accuracy.
• Empirical constraint satisfaction remains close to the target 85% level (1−ε=0.85) across different uncertainty distributions, with only minor deviations when the true distribution differs from the model.
• The method shows higher sensitivity to errors in the assumed distribution family than to errors in distribution parameters, indicating that distributional modeling is critical.
• Generation control policies derived from PCE are non-affine and exhibit significant curvature, especially around binding constraints, reflecting the nonlinear nature of reactive power control.
• The upper active power limit (p_g,3^max = 4.3) may be violated by the policy, but such violations occur with sufficiently low probability to satisfy the chance constraint, confirming feasibility under uncertainty.