Sample-Based Piecewise Linear Power Flow Approximations Using Second-Order Sensitivities

TL;DR

This work tackles the nonlinearity of AC power flow by introducing Conservative Piecewise Linear Approximations (CPLA) that selectively target dominant nonlinear directions identified via second-order sensitivities. By rotating coordinates with SVD and applying piecewise linear segments only along high-curvature directions while retaining conservative linear bounds elsewhere, the method achieves higher fidelity than standard CLA without prohibitive complexity. The authors formulate a regression-based CPLA with continuity constraints, reduce decision variables through continuity, and exploit parallelism for scalability. Numerical results across multiple test cases demonstrate substantial error reductions (up to ~90% in some cases) with a manageable computational footprint, highlighting CPLA’s potential for MILP-based planning and optimization in power systems.

Abstract

The inherent nonlinearity of the power flow equations poses significant challenges in accurately modeling power systems, particularly when employing linearized approximations. Although power flow linearizations provide computational efficiency, they can fail to fully capture nonlinear behavior across diverse operating conditions. To improve approximation accuracy, we propose conservative piecewise linear approximations (CPLA) of the power flow equations, which are designed to consistently over- or under-estimate the quantity of interest, ensuring conservative behavior in optimization. The flexibility provided by piecewise linear functions can yield improved accuracy relative to standard linear approximations. However, applying CPLA across all dimensions of the power flow equations could introduce significant computational complexity, especially for large-scale optimization problems. In this paper, we propose a strategy that selectively targets dimensions exhibiting significant nonlinearities. Using a second-order sensitivity analysis, we identify the directions where the power flow equations exhibit the most significant curvature and tailor the CPLAs to improve accuracy in these specific directions. This approach reduces the computational burden while maintaining high accuracy, making it particularly well-suited for mixed-integer programming problems involving the power flow equations.

Figures (6)

• Figure 1: The figure presents a visual comparison between a standard linear approximation (depicted on the left) and CLAs (depicted on the right). The solid line represents the nonlinear function being analyzed. In the left illustration, the dotted line portrays a conventional first-order Taylor approximation centered at point $\times$, whereas in the right illustration, the dotted line above (below) signifies an over- (under-)estimating approximation.
• Figure 2: Flowchart illustrating the computation processes for the CPLA method. Red-dashed boxes highlight the steps involving the second-order sensitivity matrix. Steps marked with $*$ indicate parallelizable processes.
• Figure 3: An example of a CPLA is shown, where the yellow and red planes (each referred to as a "region" in this paper) underestimate the quadratic function $y = -4x_1^2 + x_2$ (blue manifold). The breakpoint is at $x_1 = -2$, with $x_1 \leq -2$ and $x_1 > -2$ referred to as two segments. The yellow (red) plane corresponds to the function $y = 12x_1 + x_2 + 16$ ($y = 4x_1 + x_2$).
• Figure 4: Percentage error reduction of voltage magnitudes vs. number of breakpoints in the 141-bus system at bus 80 when considered (a) one singular vector and (b) two singular vectors.
• Figure 5: Error histograms from the underestimating CLA (blue) and the underestimating CPLA (red) for voltage magnitudes at bus 20 in the IEEE 30-bus system.