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Differentiating Through Power Flow Solutions for Admittance and Topology Control

Samuel Talkington, Daniel Turizo, Sergio A. Dorado-Rojas, Rahul K. Gupta, Daniel K. Molzahn

Abstract

The power flow equations relate bus voltage phasors to power injections via the network admittance matrix. These equations are central to the key operational and protection functions of power systems (e.g., optimal power flow scheduling and control, state estimation, protection, and fault location, among others). As control, optimization, and estimation of network admittance parameters are central to multiple avenues of research in electric power systems, we propose a linearization of power flow solutions obtained by implicitly differentiating them with respect to the network admittance parameters. This is achieved by utilizing the implicit function theorem, in which we show that such a differentiation is guaranteed to exist under mild conditions and is applicable to generic power systems (radial or meshed). The proposed theory is applied to derive sensitivities of complex voltages, line currents, and power flows. The developed theory of linearizing the power flow equations around changes in the complex network admittance parameters has numerous applications. We demonstrate several of these applications, such as predicting the nodal voltages when the network topology changes without solving the power flow equations. We showcase the application for continuous admittance control, which is used to increase the hosting capacity of a given distribution network.

Differentiating Through Power Flow Solutions for Admittance and Topology Control

Abstract

The power flow equations relate bus voltage phasors to power injections via the network admittance matrix. These equations are central to the key operational and protection functions of power systems (e.g., optimal power flow scheduling and control, state estimation, protection, and fault location, among others). As control, optimization, and estimation of network admittance parameters are central to multiple avenues of research in electric power systems, we propose a linearization of power flow solutions obtained by implicitly differentiating them with respect to the network admittance parameters. This is achieved by utilizing the implicit function theorem, in which we show that such a differentiation is guaranteed to exist under mild conditions and is applicable to generic power systems (radial or meshed). The proposed theory is applied to derive sensitivities of complex voltages, line currents, and power flows. The developed theory of linearizing the power flow equations around changes in the complex network admittance parameters has numerous applications. We demonstrate several of these applications, such as predicting the nodal voltages when the network topology changes without solving the power flow equations. We showcase the application for continuous admittance control, which is used to increase the hosting capacity of a given distribution network.
Paper Structure (19 sections, 2 theorems, 44 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 2 theorems, 44 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Power flow models
  3. Admittance-parameterized power flow equations
  4. Differentiating power flow solutions with respect to admittance parameters
  5. Nodal voltages
  6. Line current flows
  7. Shuntless networks
  8. Generalization to networks with shunts
  9. Line power flows
  10. Polar form
  11. Rectangular form
  12. Generalization to networks with shunts
  13. Application of Admittance Sensitivities
  14. Numerical Experiments
  15. Power flow solution prediction as admittance changes
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\boldsymbol{x} \in \mathbb{C}^n$ be a nodal voltage state inducing power injections $\boldsymbol{s} \in \mathbb{C}^n$. There exists a matrix-valued function $\boldsymbol{F}: \mathbb{C}^n \to \mathbb{C}^{n \times n(n-1)/2}$ such that the complex conjugate of the power injections can be written a where $\boldsymbol{F}: \mathbb{C}^n \to \mathbb{C}^{n \times n(n-1)/2}$ is defined explicitly in eq

Figures (6)

  • Figure 1: Illustration of the $\pi$-line model and the line parametrization used in our work.
  • Figure 2: The simple 5-bus test case shown in its final configuration ($\gamma=1$).
  • Figure 3: Illustrative linearization of nodal voltage magnitudes (top) and branch current magnitudes (bottom) as a function of the network topology change for the simple test case shown in Fig. \ref{['fig:simple-network']}.
  • Figure 4: Relative error of approximating the AC power flow solution produced by the power set of switching configurations for the case33bw radial network.
  • Figure 5: Optimal solutions of the voltage regulation problem for the CIGRE low voltage test case CIGREREF. The plot compares the minimum curtailment solution using the full AC power flow equations and the linearized formulation \ref{['eq:linear-program-vreg']} with admittance bounds ranging between $\pm 0 \%$ and $\pm 50\%$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof