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Cost-Benefit Analysis for PMU Placement in Power Grids

Beth Morrison, Sean English, Johnathan Koch

TL;DR

The paper develops a cost-benefit framework for placing phasor measurement units (PMUs) in power grids by modeling observance through the power domination process. It introduces the cost function $\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|)$, analyzes beta-best sensor sets via $\mathrm{maxObs}(G;k)$ and $\mathrm{minC}(G;\beta)$, and formalizes the notion of useful sizes. It proves that any prescribed set of useful sizes containing $0$ and $\gamma_P(G)$ can be realized on some graph, and provides fort-based bounds that guarantee minimum power-dominating sets are beta-best for broad ranges of $\beta$; it also develops a marginal observance/cost theory to guide incremental sensor decisions. The work uses grid gadgets and fort-related concepts to connect observability, sensor count, and cost, offering a constructive approach to cost-aware PMU deployment with implications for grid reliability and efficiency.

Abstract

Power domination is a graph-theoretic model for the observance of a power grid using phasor measurement units (PMUs). There are many costs associated with the installation of a PMU, but also costs associated with not observing the entire power grid. In this work, we propose and study a power domination cost function, which balances these two costs. Given a graph $G$, a set of sensor locations $S$, and a parameter $β$ (which is the ratio of the cost of a PMU to the cost of non-observance of any given vertex), we define the cost function \[ \mathrm{C}(G;S,β)=|S|+β\cdot (|V(G)|-|\mathrm{Obs}(G;S)|) \] where $|\mathrm{Obs}(G;S)|$ is the number of vertices observed by sensors placed at $S\subseteq V(G)$ in the power domination process. We explore the values of $k$ for which there is a set $S$ of size $k$ that minimizes this cost function, and explore which values of $β$ guarantee that it is optimal to observe the entire power grid to minimize cost. We also introduce notions of marginal cost and marginal observance, providing tools to analyze how many PMUs one should install on a given power grid.

Cost-Benefit Analysis for PMU Placement in Power Grids

TL;DR

The paper develops a cost-benefit framework for placing phasor measurement units (PMUs) in power grids by modeling observance through the power domination process. It introduces the cost function , analyzes beta-best sensor sets via and , and formalizes the notion of useful sizes. It proves that any prescribed set of useful sizes containing and can be realized on some graph, and provides fort-based bounds that guarantee minimum power-dominating sets are beta-best for broad ranges of ; it also develops a marginal observance/cost theory to guide incremental sensor decisions. The work uses grid gadgets and fort-related concepts to connect observability, sensor count, and cost, offering a constructive approach to cost-aware PMU deployment with implications for grid reliability and efficiency.

Abstract

Power domination is a graph-theoretic model for the observance of a power grid using phasor measurement units (PMUs). There are many costs associated with the installation of a PMU, but also costs associated with not observing the entire power grid. In this work, we propose and study a power domination cost function, which balances these two costs. Given a graph , a set of sensor locations , and a parameter (which is the ratio of the cost of a PMU to the cost of non-observance of any given vertex), we define the cost function where is the number of vertices observed by sensors placed at in the power domination process. We explore the values of for which there is a set of size that minimizes this cost function, and explore which values of guarantee that it is optimal to observe the entire power grid to minimize cost. We also introduce notions of marginal cost and marginal observance, providing tools to analyze how many PMUs one should install on a given power grid.
Paper Structure (15 sections, 16 theorems, 71 equations, 4 figures, 1 table)

This paper contains 15 sections, 16 theorems, 71 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The power domination process and cost function
  3. Example of useful sizes and beta-best sets
  4. Main results
  5. Definitions, notation, and organization
  6. Useful sizes
  7. Grid gadgets
  8. Bounding the observance on a grid gadget
  9. Proof of Theorem 2.7
  10. Determining how many PMUs are needed
  11. Marginal cost and marginal observance
  12. Marginal observance and minimum fort size
  13. Using the power domination number of sensors
  14. Concluding remarks
  15. Acknowledgements

Key Result

Lemma 2.1

Let $G$ be a bipartite graph with partite sets $V_1$ and $V_2$ and $\delta(G)\geq 2$. Given a set $T\subseteq V(G)$, we have

Figures (4)

  • Figure 1: The power grid described in C2020, which we will call $G$. This graph has 60 vertices and has power domination number equal to 11. The dark blue vertices constitute a minimum power dominating set. Table \ref{['tab:betabestsets']} shows the maximum observance for each size from $0$ to $11$, along with the relevant cost functions and intervals on which the given size is useful.
  • Figure 2: A plot of the cost functions from Table \ref{['tab:betabestsets']}. Segments of functions on intervals where they are $\beta$-best are indicated, and all other segments and functions are displayed in gray.
  • Figure 3: The graph ${K_4}\boxminus_{{\{u_1,u_2,u_3\}}}{\boxplus^{3}_{4,6}}$
  • Figure 4: The graph $G$ as described in Case 3 with $K = \{0,2,4,6 , 8, 10, 12, 14, 16\}$ indicated in red, and $X_1 = \{1,3,5\}$, $X_2 = \{7,9,11\}$, and $X_3 = \{13,15,17\}$ identified by the dashed curves.

Theorems & Definitions (35)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Lemma 2 from DH2006
  • Lemma 2.4
  • proof
  • Claim 2.5
  • proof
  • Lemma 2.6
  • ...and 25 more