The theory of modelling reaction and transport of substances within a water distribution system is described in a number of books and articles. A good overview of the methods that are included in most of the commercially available simulation software tools can be found in Rossman and Boulos (1996). In the following a very brief presentation of the problem of contaminant injection with sharp quality fronts (discontinuity) is given. Reaction and diffusion are not considered here. In this case the transport is described by the so-called Riemann problem (p. 49 in Toro, 2009): (PDE)ct+vcx=0,-∞<x<∞,t>0,(IC)cx,0=c0x=cLifx<0,cRifx>0. The partial differential equation (1a, PDE) includes the partial derivatives of the concentration c after time (ct) and location within the pipe (cx); v denotes the flow velocity. The initial value problem (IVP) of Eq. (1) differs from conventional transport equations only in the initial condition (Eq. 1b, IC). Here, there are two constant water qualities cL and cR that are upstream and downstream of point x=0. For x=0 the initial condition has a discontinuity. For solution the method of characteristics (MOC) can be used. When time passes, the discontinuity is propagated along the characteristic lines with flow velocity v. Accordingly, the solution of this simplest case of Riemann problems (Eq. 1) is cx,t=c0x-vt=cLifx-vt<0,cRifx-vt>0. For the Riemann equation, the characteristic that passes through x=0 is of special interest. It separates the (concentration) surface above the x–t-plane into one part where the concentration is cL and one part where the concentration is cR. This particular characteristic line is the only one across which the concentration changes. For implementation, the IVP (1) is solved for all pipes using a MOC. For later processing of the results, the traces of a particle that travels along the special characteristic line that starts from x=0 is stored. In the easiest case the flow velocity is constant and only the slope has to be stored. However, in extended period simulations the flow velocity and even the flow direction may change from external time step to time step. Therefore, it proved to be convenient to store the [time, position, value] triples of all particles for every external time step. The term external time step refers to the time interval between updates of the boundary conditions of the underlying hydraulic simulation model. Since the flow velocities change only after such an external time step, the full information [value, time, position] can be re-established from this information for any time and location.

In order to move from the pipe level – with the initial assumption of infinite pipe length from above – to the network level, additional boundary conditions have to be considered for a finite pipe length. The IVP then is transferred to an IBVP (initial boundary value problem). In general, the combination of pipes at a junction requires the formulation of additional mixing conditions at network nodes. Since no critical concentration can be given for unknown substances, here, the binary information (contaminated yes/no) of the front is transferred unchanged to all pipes having outflows from the junction.

The proposed source identification (SI) module is distinguished from existing solutions by its real-time capabilities, its integration within the EMS and the permanent calculation of the monitoring state of the sensor network even in the case with no alarm. The latter is useful especially when flow directions change due to network operations. The continuous backtracking of negative sensor alarms allows the visualization of the current monitoring state of the system. For any location in the network, the last time of observation is calculated. Given the limited coverage of the distribution network by the sensor network, a contamination may not be detected by the sensors because it can be outside of the covered area. It is important to be aware of this issue if other detections, like customer complaints, indicate an event contradicting the negative alarm states of the sensors.

In the context of SAFEWATER, one basic requirement was that the source identification algorithm must deliver results almost in real time. For that reason, all kinds of optimization-based approaches (see the Introduction) are not suitable due to the huge number of simulations and thus the long computing time that are required for stochastic optimization models. Therefore, a more direct approach was chosen for the solution of the inverse problem. It has some similarities to the particle backtracking algorithm that was first presented by Shang et al. (2002) and the approach presented by De Sanctis et al. (2010). Its core component is an event-driven simplified transport algorithm that does not consider reaction and diffusion mechanisms as described in the previous section. Another key performance indicator is the usage of memory. In the presented approach, a special format was developed that is used for storing the characteristic lines of the transport equation MOC. This information can then be used for particle tracking (forward and backward) as well as reconstruction of time curves (at a certain location) or the concentration along a pipe at a certain time. For implementation in SAFEWATER, an event-driven method is used that strongly simplifies the common water quality equations: instead of concentrations, only binary quality states are calculated (contaminated/not contaminated), neglecting reaction and diffusion terms and using simplified mixing equations. An event is triggered every time a change in the (binary) boundary conditions occurs (for example, start of intrusion in the forward case or release of a sensor alarm in the backward case). The algorithm sends a separator front through the system following the flow velocity of the water in the pipes (forward) or working against it (backtracking case). The moving front separates the network into regions that have distinct values for the binary quality state (contaminated/not contaminated). In the reverse case, a change in the sensor alarm state is considered a binary signal (no alarm → alarm). Based on this simplifying assumption, memory requirements and calculation time are minimized in comparison with common water quality solvers. The fact that in the case of a real event the substance and input concentration of a contamination are presumably not known may serve as a justification for the simplifying assumptions.

For solution of the source identification problem at each external time step (change in the flow vector), the alarm states of the sensors (positive, negative, unknown) are sent through the system in reverse time. If at least one of the sensors is positive, the nodes and pipes upstream of the sensor are identified, which are possible candidates for the location of the contamination source. If more sensors have positive alarm states, normally released at different times, the results of the backtracking calculations are combined. Necessary conditions for source candidate locations are that (1) signals of all positive sensors passed the location during the backtracking, and (2) no signal of a negative sensor was observed. Since the conditions are necessary but not sufficient, the backtracking, in general, cannot give unique source locations. Based on a worst case assumption a unique source location is selected that serves as input for a simplified look-ahead transport calculation that gives an estimation over the future spread of contamination. As a worst case, the node with the biggest outflow volume is chosen. This second step is necessary since the backtracking algorithm in general does not give unique results. The region of possible source locations can be quite large depending on the efficiency of the sensor network.

It is important to mention that negative sensor alarms also deliver useful information. The backtracking of negative alarm states (under the assumption of perfect sensors) identifies the nodes upstream of these sensors and the times when the signal passes the nodes. A contamination starting from such a node before the calculated arrival time of the negative separator front is not possible since, in this case, the sensor alarm state must have been positive as well. Consequently, the nodes can be withdrawn from the list of source candidates. Another important application of backtracking of negative sensor alarms consists in the continuously updated monitoring state of the system. The backtracking algorithm calculates a kind of reverse water age for each location under the protection of the sensor network. That means that for any location the time of the last observation is known. In addition, the nodes and pipes that are not covered by the sensor network can be visualized and updated in real time.

For estimation of the future spread of contamination and the identification of isolation valves, the simplified forward transport calculation is used. Here, the assumption is again that more complicated water quality calculations such as incomplete mixing, reaction and diffusion are of secondary importance for this particular problem. It is expected that at the beginning of a real contamination event the kind and severity of the substance are unknown. In the SAFEWATER project a second water quality solver was implemented that deals with all of these issues. Once more information about the chemical properties of the agent is available, the enhanced water quality solver can be used for more detailed calculation of concentrations.

In the following section the SI algorithm is demonstrated using the example of the SAFEWATER test network in Zurich (Fig. 2). The total pipe length of the network is about 160 m. The system consists of transparent PVC pipes (diameter 100 mm) and has three major outlets that allow the operation with adjustable demands. In addition to flow and pressure measurements at the system inlet there are four additional flow measurements installed within the pipe network. The measurement devices are connected to a central OPC server. The hydraulic online simulation is connected with the OPC server through a bidirectional interface provided by the SirOPC online OPC client. The hydraulic simulation runs every 5 s with updated boundary values from the OPC server. The calculation results, namely the flow velocities, are stored in a native binary data format from where they can be accessed by the SI plugin and transferred to the transport solvers.

Water quality sensors are installed at five locations. Four sensors are distributed over the network (orange circles) and one is located directly downstream from the contamination source (red circle) in order to control the entrance water quality for the injection scenarios. For demonstration purposes, the sensor alarm times were calculated by running a forward transport calculation that starts at 09:02. The SI software tool is able to record the earliest time when the contamination passed the sensors. During the tests, a valve on the lower left-hand side was closed. Therefore, there is almost no flow in direction towards sensor E and the alarm time is beyond the scenario end time (09:30). The subsequent alarm times are also shown in Fig. 2. From Fig. 3 it can be seen that the more information is available (through subsequent additional alarms), the smaller the area of possible source candidates (red marked pipes). In Fig. 3a all pipes upstream of sensor B are possible locations. In Fig. 3b the combination of the upstream pipes of sensor B and of sensor D decreases the set of source candidates because the locations directly upstream of sensor B cannot explain the alarm at sensor location D. The area is further reduced after the third alarm (Fig. 3c). Possible source locations must also be upstream of sensor C. The orange and blue marked pipes show the estimated spread at the current time and in a predefined look-ahead time, respectively.

The sensors do not only deliver valuable information in case of an alarm. Backtracking of negative sensor alarm states also helps to reduce the area of possible source candidates (see for comparison De Sanctis et. al., 2010). Assuming perfect sensors, the upstream locations of the negative sensors cannot be the contamination source since otherwise the contamination had to be detected by the sensor. Negative sensors can clear parts of the source candidate locations. A node (or pipe) is a possible candidate for the contamination source if the sensor signals of all sensors with positive alarms are observed and no signal of negative sensors can be found for these locations. A particular weighting scheme has been implemented, considering also the duration and time delay between the different signals. The criterion for a node to be a source candidate location is that all signals that passed that node are positive signals. The set of source candidate locations consists of a more or less large area, depending on the design of the sensor network, the number of sensors and the topology of the network graph. For example, if forward calculations shall be carried out for the estimation of the current spread of contamination, a single source node has to be selected.