Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Revisiting the Optimal PMU Placement Problem in Multi-Machine Power Networks

Mohamad H. Kazma, Ahmad F. Taha

Abstract

To provide real-time visibility of physics-based states, phasor measurement units (PMUs) are deployed throughout power networks. PMU data enable real-time grid monitoring and control -- and are essential in transitioning to smarter grids. Various considerations are taken into account when determining the geographic, optimal PMU placements (OPP). This paper focuses on the control-theoretic, observability aspect of OPP. A myriad of studies have investigated observability-based formulations to determine the OPP within a transmission network. However, they have mostly adopted a simplified representation of system dynamics, ignored basic algebraic equations that model power flows, disregarded including renewables such as solar and wind, and did not model their uncertainty. Consequently, this paper revisits the observability-based OPP problem by addressing the literature's limitations. A nonlinear differential algebraic representation (NDAE) of the power system is considered. The system is discretized using various discretization approaches while explicitly accounting for uncertainty. A moving horizon estimation approach is explored to reconstruct the joint differential and algebraic initial states of the system, as a gateway to the OPP problem which is then formulated as a computationally tractable integer program (IP). Comprehensive numerical simulations on standard power networks are conducted to validate the different aspects of this approach and test its robustness to various dynamical conditions.

Revisiting the Optimal PMU Placement Problem in Multi-Machine Power Networks

Abstract

To provide real-time visibility of physics-based states, phasor measurement units (PMUs) are deployed throughout power networks. PMU data enable real-time grid monitoring and control -- and are essential in transitioning to smarter grids. Various considerations are taken into account when determining the geographic, optimal PMU placements (OPP). This paper focuses on the control-theoretic, observability aspect of OPP. A myriad of studies have investigated observability-based formulations to determine the OPP within a transmission network. However, they have mostly adopted a simplified representation of system dynamics, ignored basic algebraic equations that model power flows, disregarded including renewables such as solar and wind, and did not model their uncertainty. Consequently, this paper revisits the observability-based OPP problem by addressing the literature's limitations. A nonlinear differential algebraic representation (NDAE) of the power system is considered. The system is discretized using various discretization approaches while explicitly accounting for uncertainty. A moving horizon estimation approach is explored to reconstruct the joint differential and algebraic initial states of the system, as a gateway to the OPP problem which is then formulated as a computationally tractable integer program (IP). Comprehensive numerical simulations on standard power networks are conducted to validate the different aspects of this approach and test its robustness to various dynamical conditions.
Paper Structure (31 sections, 1 theorem, 58 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 31 sections, 1 theorem, 58 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction and Paper Contributions
  2. Introduction and Literature Review
  3. Paper Contributions and Organization
  4. Nonlinear Power System DAE Model
  5. Discrete-time Modeling of Power Systems
  6. Discrete-time Representation of NDAEs
  7. Structure-preserving $\mu$-NDAE Representation
  8. NDAE Numerical Solvability: Newton-Raphson Method
  9. Initial State Estimation: A MHE Approach
  10. Gauss-Newton for MHE
  11. Observability-based OPP in Power Networks
  12. Case Studies: Validation and Results
  13. OPP Implementation
  14. Simulating the Discretized Power System Dynamics
  15. Validating the Discrete $\mu$-NDAE Model
  16. ...and 16 more sections

Key Result

Proposition 1

The observability matrix $\boldsymbol W_o(\cdot)$ can be written as a linear combination of individual observability matrices computed from each individual PMU contribution as follows where $\mathcal{Z}_{i}$ corresponds to the selected i-th sensor that is encoded in matrix ${\tilde{\boldsymbol C}}$. That is, $\mathcal{Z}_{i}$ is a binary set that has a value of 1 at the i-th selected sensor locat

Figures (6)

  • Figure 1: Implementation of OPP framework for an NDAE power systems.
  • Figure 2: RMSE on both dynamic and algebraic states between the NDAE and $\mu$-NDAE discrete-time representations of the power systems: (a) $\mathrm{case}$-$\mathrm{9}$ ($\alpha_{L} = 2\%$), (b) $\mathrm{case}$-$\mathrm{39}$ ($\alpha_{L} = 5\%$), and (c) $\mathrm{case}$-$\mathrm{200}$ ($\alpha_{L} = 20\%$).
  • Figure 3: Transient differential $(\omega_{i})$ and algebraic $(\theta_{i})$ state trajectories under load and renewables disturbance: (a) $\mathrm{case}$-$\mathrm{9}$ ($\alpha_{L} = 2\%$), (b) $\mathrm{case}$-$\mathrm{39}$ ($\alpha_{L} = 5\%$), and (c) $\mathrm{case}$-$\mathrm{200}$ ($\alpha_{L} = 20\%$). The figures represent the power system's response to load/renewables perturbations; the system returns to steady-states conditions after a certain time-span.
  • Figure 4: Estimation error $\varepsilon$ resulting from the optimal PMU placements for each case and discretization methods.
  • Figure 5: Comparing optimal PMU placements resulting from $(a)$ the proposed OPP problem P5 and $(b)$ the OPP problem developed in Qi2015.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition 1
  • Remark 1
  • Definition 2: Hanba2009
  • Remark 2
  • Proposition 1
  • proof