[Paper Review] A Linear-Programming Approximation of AC Power Flows

Carleton Coffrin, Pascal Van Hentenryck|arXiv (Cornell University)|Jun 16, 2012

Optimal Power Flow Distribution35 references4 citations

TL;DR

This paper proposes LPAC models, a linear programming approximation of AC power flows that incorporate both active and reactive power, voltage magnitudes, and phase angles by using piecewise linear approximations of cosine terms and Taylor series for nonlinearities. The method achieves high accuracy in active power, reactive power, voltage magnitudes, and phase angles across IEEE and MATPOWER benchmarks, enabling reliable use in MIP-based optimization for applications like capacitor placement and power restoration.

ABSTRACT

Linear active-power-only DC power flow approximations are pervasive in the planning and control of power systems. However, these approximations fail to capture reactive power and voltage magnitudes, both of which are necessary in many applications to ensure voltage stability and AC power flow feasibility. This paper proposes linear-programming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation. The LPAC models are built on a convex approximation of the cosine terms in the AC equations, as well as Taylor approximations of the remaining nonlinear terms. Experimental comparisons with AC solutions on a variety of standard IEEE and MatPower benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes. The potential benefits of the LPAC models are illustrated on two "proof-of-concept" studies in power restoration and capacitor placement.

Motivation & Objective

• Address the limitations of traditional Linearized DC (LDC) models, which ignore reactive power and voltage magnitudes, by developing a more comprehensive linear approximation.
• Enable optimization applications requiring reactive power and voltage magnitude constraints—such as capacitor placement and power system restoration—by extending the LDC model framework.
• Improve accuracy and reliability of power system optimization under normal and contingent operating conditions compared to LDC models.
• Provide a computationally efficient alternative to nonlinear programming and metaheuristic methods that often require extensive tuning and lack global optimality guarantees.
• Demonstrate the feasibility of integrating the LPAC models into mixed-integer programming (MIP) frameworks for complex power system problems.

Proposed method

• Use a piecewise linear approximation of the cosine term in AC power flow equations to model the nonlinear relationship between voltage angles and active power.
• Apply Taylor series expansions to approximate the remaining nonlinear terms in the power flow equations, particularly those involving voltage magnitudes.
• Construct two variants: a cold-start model using Taylor approximations for all nonlinearities and a warm-start model using a piecewise-linear approximation for cosine and Taylor for other terms.
• Formulate the resulting approximations as linear programs (LPs) that can be efficiently solved using standard LP and MIP solvers.
• Ensure convexity and feasibility by restricting the domain of the cosine approximation to (−π/2, π/2), where cosine is convex.
• Integrate the LPAC models into MIP formulations for optimization problems involving voltage constraints and reactive power flows.

Experimental results

Research questions

• RQ1 Can a linear programming model accurately approximate both active and reactive power flows, voltage magnitudes, and phase angles in AC power systems?
• RQ2 How does the accuracy of the LPAC model compare to full AC power flow solutions across standard IEEE and MATPOWER test cases under normal and contingent conditions?
• RQ3 To what extent can the LPAC models be used as a reliable substitute for AC power flow in MIP-based optimization problems involving reactive power and voltage constraints?
• RQ4 Can the LPAC models effectively support complex applications such as power system restoration and capacitor placement while maintaining computational efficiency and solution quality?
• RQ5 What is the impact of using piecewise linear and Taylor-based approximations on the overall accuracy and scalability of the model compared to existing linear and nonlinear approaches?

Key findings

• The LPAC models achieve high accuracy in approximating active and reactive power, phase angles, and voltage magnitudes across a range of IEEE and MATPOWER test systems.
• Experimental comparisons show that the LPAC models produce results very close to full AC power flow solutions, with small absolute and percent differences across all key variables.
• The warm-start LPAC model, which uses a piecewise linear approximation for cosine and Taylor series for other terms, demonstrates superior accuracy compared to the cold-start version.
• The models are robust under various contingencies, maintaining accuracy even when system conditions deviate from normal operation.
• Proof-of-concept studies in capacitor placement and power restoration demonstrate that the LPAC models can be effectively embedded into MIP formulations to yield feasible and high-quality solutions.
• The LPAC models offer a viable, scalable, and computationally efficient alternative to nonlinear programming and metaheuristic methods, with better integration potential into existing MIP-based optimization pipelines.