Adaptive Power Flow Approximations with Second-Order Sensitivity Insights

Abstract

The power flow equations are fundamental to power system planning, analysis, and control. However, the inherent non-linearity and non-convexity of these equations present formidable obstacles in problem-solving processes. To mitigate these challenges, recent research has proposed adaptive power flow linearizations that aim to achieve accuracy over wide operating ranges. The accuracy of these approximations inherently depends on the curvature of the power flow equations within these ranges, which necessitates considering second-order sensitivities. In this paper, we leverage second-order sensitivities to both analyze and improve power flow approximations. We evaluate the curvature across broad operational ranges and subsequently utilize this information to inform the computation of various sample-based power flow approximation techniques. Additionally, we leverage second-order sensitivities to guide the development of rational approximations that yield linear constraints in optimization problems. This approach is extended to enhance accuracy beyond the limitations of linear functions across varied operational scenarios.

Figures (8)

• Figure 1: A conceptual example of a traditional linear approximation (left) and a conservative linear approximation (right). The solid line represents the nonlinear function of interest. The dotted line in the left figure is a traditional first-order Taylor approximation around point $\times$ while the dotted top (bottom) line in the right figure is an over- (under-) estimating approximation.
• Figure 2: An example of an overestimating linear function $\mathbf{c}^T\mathbf{x} + d$ (blue plane) of a quadratic function $ax_1^2 + bx_2$ (red manifold) where $\mathbf{x}^T = [x_1, \ x_2], a = -2, b = 2, \mathbf{c}^T = [-8, \ 2]$, and $d = 8$.
• Figure 3: Flowchart depicting the computation processes for the importance sampling and CLA/CRA methods. Steps with $*$ are parallelizable. Red-dashed boxes highlight the computation of the second-order sensitivity matrix, while black-dashed boxes indicate sample drawing without importance sampling.
• Figure 4: The left plot shows a comparison between the first- (red crosses) and second-order Taylor approximations (blue asterisks) and the Padé approximant (black circles) for voltage magnitudes at bus 1 in the IEEE 24-bus system. The green line at $45^\circ$ represents zero approximation error. The smoothed histogram in the right plot displays the errors from the first- (red dashed line) and second-order Taylor approximations (blue solid line), along with the [1/1] multivariate generalization of the Padé approximant (thick black line).
• Figure 5: The left plot compares linear approximation (LA) in red crosses and rational approximation (RA) in blue circles for current flow on a branch connecting buses 15 and bus 21 in IEEE 24-bus system. The green line indicates the zero approximation error. The right plot shows the error histogram from the linear approximation (LA) and the rational approximation (RA).