A Water Distribution Network (WDN) described by a graph G = (N , L) is given, where the nodes N represent junctions of pipes and water demand locations and the links L represent pipes and pumps. In this network there exists a subset of nodes, indicated by , where stationary water contamination sensors are installed. Each node i in N is associated with a water demand at the node location, indicated by di(k) ∈ R.
For this initial work it is assumed that there is a time period T for which water demands are constant, thus di(k) = di, k ∈ T . This is a typical case of night flow conditions. The status of the pumps is also assumed known and constant. Thus for a given network configuration and for the time period T, the water flows qj in all pipes j ∈ L are constant. We assume all pipes in L have valves installed that can remotely and instantly be closed or opened. The status of each pipe j, i.e. the status of the valve on each pipe, is indicated by uj(k) ∈ {0, 1}, where uj(k) = 0 if the valve is closed at the discrete time step k and uj(k) = 1 otherwise. By changing the valve settings, the incidence matrix of graph G is modified.