# Grid Monitoring Using Phasor Measurement Units

a combinatorial problem related to the monitoring of high-voltage electrical grid## A “phasor measurement unit”, you say?

Monitoring the links of an electrical network can be carried out by means of Phasor Measurement Units (PMUs). Placed on a network node, a PMU can locally performs electrical measurements on incident links. If a node is not monitored by a PMU but all it incident links are monitored excepted one, Using Kirchoff’s current law and Ohm law one can deduce the voltage of the remaining unmonitored link – this is what we call *a propagation rule-*, and then of other links by consecutive applications of this rule. The problem of placing an optimal number of PMUs on the nodes for complete network monitoring has been well studied.

We study here a recent variant of the problem consisting to have the PMUs on the network links rather than the nodes. Here a PMU monitors only one edge, but the same propagagion rule exists. Here again, the problem is to find a minimum number of PMU to place for complete network monitoring.

## The challenge

Finfing the minimum set of edges to place PMUs is a very difficult combinatorial problem. Sometimes, only a few number of PMU are sufficient to cover a large network, and sometimes we require amost one PMU per link. The propagation rule plays an essential role but is very difficult to apprehend, i.e. adding only one more PMU can trigger a propagation rule, that can then trigger another one and so on.

The aim of our research is to have a better understanding of this problem : what makes it difficult to solve? Can we solve it easily on restricted networks topologies and how? We try among other to find sufficient conditions on networks topologies allowing to apply polynomial-time exact algorithms, or to describle what minimal conditions make the problem difficult to solve. We aim also to design efficient polynomial-time algorithms for some restricted network topologies, or constant factor approximation algorithms on hard-to-solve network topologies.

# Our results so far

#### Publications

- 2 international conference in combinatorics
- 1 national conference (french) in networks
- 1 pending paper in an international journal

#### People involved in the project

- Annie Chateau
- Benoit Darties
- Rodolphe Giroudeau
- Matthias Weller