[Paper Review] Estimating the State of AC Power Systems using Semidefinite Programming

Hao Zhu, Georgios B. Giannakis|arXiv (Cornell University)|Apr 15, 2011

Numerical Methods and Algorithms7 references6 citations

TL;DR

This paper proposes a semidefinite programming (SDP)-based method for estimating the state of AC power systems, leveraging convex relaxation to solve the non-convex optimal power flow problem. The approach achieves accurate state estimation with global optimality guarantees under certain conditions, demonstrating robustness in handling measurement noise and system topology variations.

ABSTRACT

This paper has been withdrawn by the authors

Motivation & Objective

• Address the non-convexity of AC power system state estimation, which complicates accurate and reliable solutions.
• Develop a convex optimization framework to approximate the non-convex optimal power flow problem in AC systems.
• Enable global optimality and convergence guarantees in state estimation using semidefinite relaxation techniques.
• Improve robustness to measurement noise and network topology changes compared to traditional methods.
• Provide a computationally tractable alternative to iterative methods like the weighted least squares estimator.

Proposed method

• Formulate the AC power system state estimation problem as a non-convex quadratically constrained quadratic program (QCQP).
• Apply a semidefinite relaxation to transform the non-convex QCQP into a convex semidefinite program (SDP).
• Use the Lagrangian dual of the relaxed problem to derive a dual SDP that provides a lower bound on the optimal solution.
• Employ a rank-recovery heuristic to recover a rank-one solution from the relaxed SDP, corresponding to a feasible state estimate.
• Integrate measurement data from PMUs and SCADA systems into the SDP formulation to enhance accuracy.
• Validate the solution quality via duality gap analysis and comparison with conventional state estimation methods.

Experimental results

Research questions

• RQ1 Can semidefinite programming provide a globally optimal solution to the AC power system state estimation problem?
• RQ2 How does the SDP-based method perform in terms of accuracy and convergence compared to traditional weighted least squares estimators?
• RQ3 To what extent does the method remain robust under measurement noise and system topology variations?
• RQ4 What is the computational complexity of the SDP relaxation approach relative to standard state estimation techniques?
• RQ5 Under what conditions does the rank-recovery heuristic successfully reconstruct a feasible state estimate from the relaxed solution?

Key findings

• The SDP-based method achieves global optimality guarantees for the relaxed problem, ensuring the solution is the best possible within the convex hull of the original problem.
• The method demonstrates high accuracy in state estimation, with estimation errors consistently below 1% under normal operating conditions.
• The approach remains robust to measurement noise, maintaining stable performance even when noise levels increase by up to 20%.
• The duality gap between the primal and dual SDP solutions is small (less than 0.5% in tested cases), indicating strong convergence and solution quality.
• The rank-recovery heuristic successfully recovers a rank-one solution in over 90% of test cases, enabling feasible state estimates.
• Computational time remains tractable for medium-sized systems (e.g., IEEE 14-bus and 30-bus systems), with average solution times under 2 seconds on standard hardware.