  State estimation in water distribution networks (WDN), the problem of
estimating all unknown network heads and flows given select measurements, is
challenging due to the nonconvexity of hydraulic models and significant
uncertainty from water demands, network parameters, and measurements. To this
end, a probabilistic modeling for state estimation (PSE) in WDNs is proposed.
After linearizing the nonlinear hydraulic WDN model, the proposed PSE shows
that the covariance matrix of unknown system states (unmeasured heads and
flows) can be linearly expressed by the covariance matrix of three uncertainty
sources (i.e., measurement noise, network parameters, and water demands).
Instead of providing deterministic results for unknown states, the proposed PSE
approach (i) regards the system states and uncertainty sources as random
variables and yields variances of individual unknown states, (ii) considers
thorough modeling of various types of valves and measurement scenarios in WDNs,
and (iii) is also useful for uncertainty quantification, extended period
simulations, and confidence limit analysis. The effectiveness and scalability
of the proposed approach is tested using several WDN case studies.
