Cost-Benefit Analysis for PMU Placement in Power Grids

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.

• The paper introduces a linear cost function for PMU placement, $\mathrm{C}(G;S,\beta)$, that balances the cost of installing PMUs and the operational cost of unobserved vertices, using $\beta$ to parameterize the trade-off.
• The framework yields specific, actionable guidelines for PMU deployment, as demonstrated with the Nordic32 test system, where the optimal number of PMUs is determined by $\beta$.
• For any given graph, the paper details realizability of useful sizes - a measure of how many sensors are optimally placed in a given context - allowing every possible useful size in its construction.

The cost function and its motivation

Power domination models the observance of a power grid by phasor measurement units (PMUs): a sensor placed at a vertex observes its closed neighborhood, and Kirchhoff's-law propagation (the zero-forcing rule) then observes any vertex with exactly one unobserved neighbor. The classical objective is the power domination number $\gamma_P(G)$, the minimum number of PMUs needed to observe every vertex. This paper departs from full observability: since PMU installation is expensive (the IESO estimated $50,000–$300,000 CAD per unit in 2021) while unobserved vertices carry their own operational cost, the authors introduce a linear cost function

$$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$

where $\beta$ is the Observance Cost Ratio—the cost of leaving a single vertex unobserved relative to the price of one PMU. Rather than estimating $\beta$, they treat it as an unknown parameter and characterize optimal placements ("$\beta$-best" sets) across all $\beta \in \mathbb{R}_{\geq 0}$. Two simplifying assumptions are made explicitly: uniform PMU placement cost and uniform per-vertex non-observance cost.

Two structural objects organize the analysis. The maximum observance $\mathrm{maxObs}(G;k)$ is the largest number of vertices observable with $k$ sensors; it is strictly increasing for $0 \le k \le \gamma_P(G)$. A size $50,000–$0 is useful if some $50,000–$1-sensor set is $50,000–$2-best on a non-degenerate interval of $50,000–$3. Trivially, $50,000–$4 is best when $50,000–$5 and any minimum power dominating set is best when $50,000–$6, so attention focuses on $50,000–$7, where $50,000–$8 is piecewise linear.

The framework is illustrated on the Nordic32-derived test system (60 vertices, $50,000–$9). Only sizes $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$0 are useful there; in particular, deploying exactly 5 PMUs is never cost-optimal, and if $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$1 the analysis prescribes 6 sensors. This demonstrates that the model yields concrete, actionable deployment guidance once $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$2 is estimated.

Which sizes can be useful

The first main result shows the notion of useful sizes is essentially unconstrained by graph structure:

Theorem (realizability). For any $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$3 with $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$4, there exists a graph whose set of useful sizes is exactly $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$5.

The proof uses cylindrical-grid gadgets $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$6 built from $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$7 with layered leaf structures ($$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$8) connecting to affix vertices on a clique $$\mathrm{C}(G;S,\beta)=|S|+\beta\,(|V(G)|-|\mathrm{Obs}(G;S)|),$$9. The key technical lemma establishes that if fewer than the required affix vertices carry sensors, at most $\beta$0 gadget vertices are observed—a bound independent of $\beta$1, hence of gadget size. This relies on a bipartite decomposition argument bounding zero forcing closure via star closures, together with the fact (from prior work) that star closure from fewer than $\beta$2 vertices in one partite class touches few columns of the cylindrical grid. Gadget sizes $\beta$3 are chosen recursively so that each $\beta$4 satisfies the interval condition of Lemma (characterization): size $\beta$5 is useful iff

$\beta$6

where $\beta$7. Sizes outside $\beta$8 violate this inequality by construction. One consequence worth noting: because gadgets grow superlinearly, the constructed graphs are large, so the theorem is an existence result rather than a statement about typical grid topologies.

Marginal observance and fort structure

The paper defines marginal observance $\beta$9 and marginal cost $\beta$0, giving telescoping identities for both $\beta$1 and $\beta$2. Two clean consequences follow: adding a $\beta$3-th sensor reduces cost exactly when $\beta$4, and if marginal observance is non-decreasing at $\beta$5, then size $\beta$6 cannot be useful. Conversely, strictly decreasing marginal observance forces every size to be useful.

Connecting forts (sets $\beta$7 such that no outside vertex has exactly one neighbor in $\beta$8) to marginal observance yields the paper's second main theorem. Since the complement of any observed set under a non-dominating placement contains a fort, the minimum fort number $\beta$9 lower-bounds the final marginal observance; small-fort analyses give bounds on all marginals:

• If $\beta$0 (no isolated vertices), then $\beta$1 for all $\beta$2.
• If $\beta$3 (equivalently, no isolated vertices and no twins), then $\beta$4.

These feed into the central practical result:

Theorem (when full observability is optimal). Let $\beta$5 equal $\beta$6, $\beta$7, or $\beta$8 according as $\beta$9 or $\beta \in \mathbb{R}_{\geq 0}$0. If $\beta \in \mathbb{R}_{\geq 0}$1, then any minimum power dominating set is $\beta \in \mathbb{R}_{\geq 0}$2-best. Moreover, this threshold is best possible when $\beta \in \mathbb{R}_{\geq 0}$3.

Sharpness is witnessed by explicit constructions: an isolated vertex for $\beta \in \mathbb{R}_{\geq 0}$4, a matching for $\beta \in \mathbb{R}_{\geq 0}$5, and a clique-of-$\beta \in \mathbb{R}_{\geq 0}$6 construction with disjoint 3-vertex forts for $\beta \in \mathbb{R}_{\geq 0}$7, where a single well-placed sensor beats full domination whenever $\beta \in \mathbb{R}_{\geq 0}$8. For graphs without small forts, the threshold approaches $\beta \in \mathbb{R}_{\geq 0}$9 from below—so in reasonably robust grids, a PMU costing more than roughly three times a vertex's non-observance cost justifies complete observability.

A further structural result supports heuristic search: every connected graph on at least three vertices admits a minimum power dominating set contained entirely in the union of entrances of minimal forts. Via the fort-cover formulation of power domination, this licenses pruning any vertex not in such an entrance from branch-and-bound or ILP searches without losing optimality.

Limitations and open questions

The authors are explicit about several gaps. First, $\mathrm{maxObs}(G;k)$0 is never estimated from data; the model's practical value depends on obtaining reliable Observance Cost Ratios, which remains external to the paper. Second, the marginal-observance lower bounds do not extend cleanly: the natural conjecture $\mathrm{maxObs}(G;k)$1 is posed but unproven, and the hoped-for bound $\mathrm{maxObs}(G;k)$2 fails in general—the square grid $\mathrm{maxObs}(G;k)$3 has $\mathrm{maxObs}(G;k)$4 yet $\mathrm{maxObs}(G;k)$5. Third, complexity is unresolved: computing $\mathrm{maxObs}(G;k)$6 is NP-complete, and even for trees—where $\mathrm{maxObs}(G;k)$7 is linear-time computable—the greedy locally-optimal choices underlying those algorithms break down for $\mathrm{maxObs}(G;k)$8 with $\mathrm{maxObs}(G;k)$9; whether computing $k$0 is NP-complete on trees is left open. Finally, the realizability construction produces very large graphs, and the uniform-cost assumptions ignore real heterogeneity in both PMU siting costs and per-bus criticality.

Conclusion

This paper reframes PMU placement as an explicit cost–benefit trade-off parameterized by a single ratio $k$1, yielding a piecewise-linear minimum cost function whose breakpoints identify exactly which deployment sizes merit consideration. Its two principal contributions—a universality theorem showing any set of useful sizes containing $k$2 and $k$3 is realizable, and sharp thresholds guaranteeing that full observability minimizes cost based on minimum fort size—together delineate both the expressive range and the practically decisive regime of the model. The marginal observance machinery and the entrance-of-minimal-forts pruning result provide usable tools for computation, while the open complexity question on trees marks the most immediate barrier to algorithmic deployment.