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Power System Steady-State Estimation Revisited

Pavel Rytir, Ales Wodecki, Martin Malachov, Pavel Baxant, Premysl Vorac, Miloslava Chladova, Jakub Marecek

Abstract

In power system steady-state estimation (PSSE), one needs to consider (1) the need for robust statistics, (2) the nonconvex transmission constraints, (3) the fast-varying nature of the inputs, and the corresponding need to track optimal trajectories as closely as possible. In combination, these challenges have not been considered, yet. In this paper, we address all three challenges. The need for robustness (1) is addressed by using an approach based on the so-called Huber model. The non-convexity (2) of the problem, which results in first order methods failing to find global minima, is dealt with by applying global methods. One of these methods is based on a mixed integer quadratic formulation, which provides results of several orders of magnitude better than conventional gradient descent. Lastly, the trajectory tracking (3) is discussed by showing under which conditions the trajectory tracking of the SDP relaxations has meaning.

Power System Steady-State Estimation Revisited

Abstract

In power system steady-state estimation (PSSE), one needs to consider (1) the need for robust statistics, (2) the nonconvex transmission constraints, (3) the fast-varying nature of the inputs, and the corresponding need to track optimal trajectories as closely as possible. In combination, these challenges have not been considered, yet. In this paper, we address all three challenges. The need for robustness (1) is addressed by using an approach based on the so-called Huber model. The non-convexity (2) of the problem, which results in first order methods failing to find global minima, is dealt with by applying global methods. One of these methods is based on a mixed integer quadratic formulation, which provides results of several orders of magnitude better than conventional gradient descent. Lastly, the trajectory tracking (3) is discussed by showing under which conditions the trajectory tracking of the SDP relaxations has meaning.
Paper Structure (33 sections, 3 theorems, 57 equations, 3 figures, 3 tables)

This paper contains 33 sections, 3 theorems, 57 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Sensors and the Measurement Chain
  4. SCADA Mesurements
  5. SCADA Voltage amplitude measurement
  6. SCADA Branch Power Measurements
  7. SCADA Injection Measurements
  8. PMU Measurements
  9. PMU Voltage Phasor Measurements and Phase Angle Measurement
  10. PMU Current Phasor Measurements
  11. Model
  12. Power System State
  13. Branch Model
  14. Branch and Bus Admittance matrices
  15. Power Equations
  16. ...and 18 more sections

Key Result

Lemma 1

Let $M_{1}\left(g_{p,k},y\right)$ be defined by eq_definition_of_moment_order_1 and suppose that which corresponds to imposing $g_k = 0$ in the original polynomial optimization problem. Then the matrix $M_{1}\left(g_{p,k},y\right)$ has only zero eigenvalues and thus for each $k \in \left\{ 1,2,\ldots,n\right\}$ and all multi-indices $\alpha$ and $\beta$.

Figures (3)

  • Figure 1: SCADA measurement chain
  • Figure 2: PMU measurement chain
  • Figure 3: Transmission branch model (line or transformer). $N=\tau e^{{\mathrm{\mathbf{i}}\mkern1mu}\theta_\textrm{shift}}$, where $\tau$ is a tap ratio and $\theta_\textrm{shift}$ is a phase shift of the transformer. If the branch is a line, then $N=1$. From side of the branch is on the left, to side is on the right.

Theorems & Definitions (10)

  • Definition 1: Moment matrix of order $d$
  • Definition 2: Localizing matrix of order $d$ with respect to polynomial $a(x)$
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 3: inner and outer limits
  • Definition 4: continuity of a set valued map
  • Definition 5: basic regularity conditions
  • Theorem 3: the classification of trajectories of time-varying SDPs