  Optimal sensor placement (SP) usually minimizes an impact measure, such as
the amount of contaminated water or the number of inhabitants affected before
detection. The common choice is to minimize the minimum detection time (MDT)
averaged over a set of contamination events, with contaminant injected at a
different location. Given a SP, propagation is simulated through a hydraulic
software model of the network to obtain spatio-temporal concentrations and the
average MDT. Searching for an optimal SP is NP-hard: even for mid-size
networks, efficient search methods are required, among which evolutionary
approaches are often used. A bi-objective formalization is proposed: minimizing
the average MDT and its standard deviation, that is the risk to detect some
contamination event too late than the average MDT. We propose a data structure
(sort of spatio-temporal heatmap) collecting simulation outcomes for every SP
and particularly suitable for evolutionary optimization. Indeed, the proposed
data structure enabled a convergence analysis of a population-based algorithm,
leading to the identification of indicators for detecting problem-specific
converge issues which could be generalized to other similar problems. We used
Pymoo, a recent Python framework flexible enough to incorporate our problem
specific termination criterion. Results on a benchmark and a real-world network
are presented.
