Optimizing Parameters of the LinDistFlow Power Flow Approximation for Distribution Systems

TL;DR

This paper tackles the accuracy gap of LinDistFlow for distribution networks by introducing Optimized LinDistFlow (OLDF), which tunes diagonal line-parameter matrices $\mathbf{D}_{r}$ and $\mathbf{D}_{x}$ and introduces bias vectors $\boldsymbol{\gamma}$, $\boldsymbol{\rho}$, and $\boldsymbol{\varrho}$ to better emulate nonlinear DistFlow. The offline training uses a loss based on mean-squared voltage differences and analytic sensitivities, optimized via Truncated Newton Conjugate-Gradient (TNC); the resulting parameters are then deployed in online linear programs. Results show substantial reductions in $L_1$ and $L_\infty$ error (up to 92% and 88% respectively) across balanced and unbalanced test feeders and across varying loading and topology conditions, with effective application to hosting-capacity problems. OLDF thus provides a tractable, highly accurate linearized power-flow model that remains compatible with standard distribution-system optimization workflows and supports topology changes and active-control settings.

Abstract

The DistFlow model accurately represents power flows in distribution systems, but the model's nonlinearities result in computational challenges for many applications. Accordingly, a linear approximation known as \mbox{LinDistFlow} (and its three-phase extension LinDist3Flow) is commonly employed. This paper introduces a parameter optimization algorithm for enhancing the accuracy of this approximation for both balanced single-phase equivalent and unbalanced three-phase distribution network models, with the goal of aligning the outputs more closely with those from the nonlinear DistFlow model. Using sensitivity information, our algorithm optimizes the LinDistFlow approximation's coefficient and bias parameters to minimize discrepancies in predictions of voltage magnitudes relative to the nonlinear DistFlow model. The parameter optimization algorithm employs the Truncated Newton Conjugate-Gradient (TNC) method to fine-tune coefficients and bias parameters during an offline training phase to improve the LinDistFlow approximation's accuracy. % in optimization applications. Numerical results underscore the algorithm's efficacy, showcasing accuracy improvements in $L_{1}$-norm and $L_{\infty}$-norm losses of up to $92\%$ and $88\%$, respectively, relative to the traditional LinDistFlow model. We also assess how the optimized parameters perform under changes in the network topology and demonstrate the optimized LinDistFlow approximation's efficacy in a hosting capacity optimization problem.

Figures (11)

• Figure 1: A 2-bus system with line $(\pi_n,n)$ feeding bus $n$ from its parent $\pi_{n}$.
• Figure 2: Flowchart depicting the proposed parameter optimization algorithm.
• Figure 3: Average error ($\epsilon_{\text{avg}}^{OLDF}$) as a function of the number of training scenarios for the IEEE 33-bus test system.
• Figure 4: (a) Boxplots showing the distributions of the $\mathbf{D}_r$ parameter values for multiple test cases. Each test case is represented by two boxplots indicating the traditional and optimal $\mathbf{D}_r$ parameter values. (b) Scatter plots comparing the coefficient values $\mathbf{D}_r^{LDF}$ and $\mathbf{D}_r^{opt}$ for various test cases.
• Figure 5: (a) Boxplots showing the distributions of the $\mathbf{D}_x$ parameter values for multiple test cases. Each test case is represented by two boxplots indicating the traditional and optimal $\mathbf{D}_x$ parameter values. (b) Scatter plots comparing the coefficient values $\mathbf{D}_x^{LDF}$ and $\mathbf{D}_x^{opt}$ for various test cases.