Optimizing Parameters of the DC Power Flow

TL;DR

This work addresses the accuracy gap of DC power flow by optimizing its linear parameters to better match AC solutions. An offline training phase collects AC solutions across scenarios and uses gradient-based methods to minimize the loss $\mathcal{L}(\mathbf{b},\boldsymbol{\gamma},\boldsymbol{\rho})=\frac{1}{|\mathcal{E}|}\sum_{m\in\mathcal{M}} \|\mathbf{p}_m^{DC}-\mathbf{p}_m^{AC}\|_2^2$, adjusting $\mathbf{b}$ and biases to align DC flows with AC flows. The approach preserves the DC-PTDF structure and demonstrates orders-of-magnitude improvements in DC-AC accuracy across test systems, while maintaining online solution times similar to traditional DC methods. It also extends to $N-1$ contingency analysis, showing generalizable gains, and suggests clustering contingencies as a scalable path for future refinements.

Abstract

Many power system operation and planning problems use the DC power flow approximation to address computational challenges from the nonlinearity of the AC power flow equations. The DC power flow simplifies the AC power flow equations to a linear form that relates active power flows to phase angle differences across branches, parameterized by coefficients based on the branches' susceptances. Inspired by techniques for training machine learning models, this paper proposes an algorithm that seeks optimal coefficient and bias parameters to improve the DC power flow approximation's accuracy. Specifically, the proposed algorithm selects the coefficient and bias parameter values that minimize the discrepancy, across a specified set of operational scenarios, between the power flows given by the DC approximation and the power flows from the AC equations. Gradient-based optimization methods like Broyden-Fletcher-Goldfarb-Shanno (BFGS), Limited-Memory BFGS (L-BFGS), and Truncated Newton Conjugate-Gradient (TNC) enable solution of the proposed algorithm for large systems. After an off-line training phase, the optimized parameters are used to improve the accuracy of the DC power flow during on-line computations. Numerical results show several orders of magnitude improvements in accuracy relative to a hot-start DC power flow approximation across a range of test cases.

Figures (7)

• Figure 1: Flowchart describing the proposed algorithm.
• Figure 2: Training losses and times for the L-BFGS, TNC, BFGS, CG, and Newton-CG methods for the IEEE 300-bus system.
• Figure 3: (a) Boxplots showing the distributions of the $\mathbf{b}$ parameter values for multiple test cases. Each test case is represented by four boxplots indicating the cold-start, hot-start, and the optimal $\mathbf{b}$ parameter values. (b) Scatter plots comparing the coefficient values $\mathbf{b}^{hot}$ and $\mathbf{b}^{opt}$ for various test cases.
• Figure 4: a) Boxplots showing the distributions of the hot-start and optimal injection bias values, $\boldsymbol{\gamma}^{hot}$ and $\boldsymbol{\gamma}^{opt}$, across multiple test cases. b) Scatter plots comparing the bias values $\boldsymbol{\gamma}^{hot}$ and $\boldsymbol{\gamma}^{opt}$ for various test cases.
• Figure 5: (a) Boxplots showing the distributions of hot-start and optimal flow bias values, $\boldsymbol{\rho}^{opt}$ and $\boldsymbol{\rho}^{hot}$, across multiple test cases. (b) Scatter plots comparing the loss values $\boldsymbol{\rho}^{hot}$ and $\boldsymbol{\rho}^{opt}$ for various test cases.