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Evaluating Power Flow Manifold from Local Data around a Single Operating Point via Geodesics

Qirui Zheng, Dan Wu, Franz-Erich Wolter, Sijia Geng

Abstract

The widespread adoption of renewable energy poses a challenge in maintaining a feasible operating point in highly variable scenarios. This paper demonstrates that, within a feasible region of a power system that meets practical stability requirements, the power flow equations define a smooth bijection between nodal voltage phasors (angle and magnitude) and nodal active/reactive power injections. Based on this theoretical foundation, this paper proposes a data-based power flow evaluation method that can imply the associated power flow manifold from a limited number of data points around a single operating point. Using techniques from differential geometry and analytic functions, we represent geodesic curves in the associated power flow manifold as analytic functions at the initial point. Then, a special algebraic structure of the power flow problem is revealed and applied to reduce the computation of all higher-order partial derivatives to that of the first-order ones. Integrating these techniques yields the proposed data-based evaluation method, suggesting that a small number of local measurements around a single operating point is sufficient to imply the entire associated power flow manifold. Numerical cases with arbitrary directional variations are tested, certifying the efficacy of the proposed method.

Evaluating Power Flow Manifold from Local Data around a Single Operating Point via Geodesics

Abstract

The widespread adoption of renewable energy poses a challenge in maintaining a feasible operating point in highly variable scenarios. This paper demonstrates that, within a feasible region of a power system that meets practical stability requirements, the power flow equations define a smooth bijection between nodal voltage phasors (angle and magnitude) and nodal active/reactive power injections. Based on this theoretical foundation, this paper proposes a data-based power flow evaluation method that can imply the associated power flow manifold from a limited number of data points around a single operating point. Using techniques from differential geometry and analytic functions, we represent geodesic curves in the associated power flow manifold as analytic functions at the initial point. Then, a special algebraic structure of the power flow problem is revealed and applied to reduce the computation of all higher-order partial derivatives to that of the first-order ones. Integrating these techniques yields the proposed data-based evaluation method, suggesting that a small number of local measurements around a single operating point is sufficient to imply the entire associated power flow manifold. Numerical cases with arbitrary directional variations are tested, certifying the efficacy of the proposed method.
Paper Structure (17 sections, 10 theorems, 35 equations, 11 figures, 4 tables, 1 algorithm)

This paper contains 17 sections, 10 theorems, 35 equations, 11 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Analytic Representation of Geodesic Curves
  3. Basic Quantities in Differential Geometry
  4. The Geodesic Equation
  5. Analytic Representation of Geodesic Curves
  6. Rational Representation of Geodesic Curves
  7. Shortest Path on Riemannian Manifold
  8. Power Flow Map in Differential Geometry
  9. The Power Flow Map and The Associated Manifold
  10. The Full Power Flow Map and Its Partial Derivatives
  11. Evaluating Power Flow Manifold from Local Measurements
  12. Numerical Simulations
  13. Manifold Estimation from Random Local Data
  14. Impact of Data Ranges on Approximation Errors
  15. Conclusion
  16. ...and 2 more sections

Key Result

Lemma 1

Consider the smooth manifold $Y$ in Definition def:map. $X$ is the smooth parameterization of $Y$. The solution to eq:geodesic is a smooth function $x^m$ with respect to $\lambda$, which, thus, admits a Taylor series expansion. The $k$-th order coefficient $\Omega_k$ of the Taylor series expansion where $\dot{x}^m = dx^m/d\lambda$; $\Omega_k$ is the $k$-th order coefficient of the Taylor series;

Figures (11)

  • Figure 1: Topologies of the studied systems.
  • Figure 2: Manifold constructions for the 4-bus system (Orange dots: true manifold, green curves: proposed estimation, blue stars: data points used).
  • Figure 3: Data points used in the 4-bus system (blue stars).
  • Figure 4: Voltage stability boundary of the 4-bus system.
  • Figure 5: Manifold constructions for the 9-bus system (Orange dots: true manifold, green curves: proposed estimation, blue stars: data points used).
  • ...and 6 more figures

Theorems & Definitions (26)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 16 more