**4. Case Study**

In this section, we apply the proposed approach to a moderate-size real-world water network, to check its effectiveness and applicability.

The proposed methodology is applied to the JYN network [40]. The network is composed by 300 nodes and two reservoirs. The mean inlet flow is around 2800 L/s. Figure 4 presents the topology of the network and the mean pressure at the nodes.

**Figure 4.** Topology and nodal mean pressure of JYN network.

During the day, the nodal demand varies. This fact makes the calculation of the sensitivity matrix hard. This complexity does not come from the computational effort, but because of the choice of the time step used. To avoid this decision, an extended period simulation is conducted, and the sensitivity time series is represented as a fuzzy number in the format (*μ* − *σ*, *μ*, *μ* + *σ*), where *μ* is the mean value of *si*,*<sup>j</sup>* and *σ* is the standard deviation. For leakage simulations, an emitter coefficient equal to 1 is adopted as discussed on the numerical example.

For a better understanding of the conditional entropy effect, the sensitivity and the conditional entropy matrices are presented in Figure 5a,b. The sensitivity parameter concentrates the nodes with highest scores around the node 300 (north-west side of the network), while the conditional entropy allows concentrations to be more scattered, giving higher scores to other nodes. A comparison between both can be drawn from this joint Figure 5.

Using solely the sensitivity matrix for sensor placement can lead to a concentration of nodes in zones of greater sensitivity. Despite the network sensitivity being maximised in this situation, there is no guarantee of good coverage. In contrast, the conditional entropy matrix, where the probabilities are calculated using the sensitivity matrix, as explained in Section 2.2, distributes the information along other sensitive zones, thereby guaranteeing improved scattering of sensors.

This can be achieved with the use of the fuzzy DEMATEL algorithm applied to the conditional entropy matrix. Using this algorithm enables us to obtain the rankings of nodes and identify those with lower influence on the others. This ranking is used to select a set of nodes to be monitored. The main interest of the methodology comes from combining the selection of high sensitivity nodes, based on the sensitivity matrix, with the lowest influencing nodes; namely, those minimising redundancy.

(**a**) Sensitivity index for JYN's network

(**b**) Conditional entropy for JYN's network

**Figure 5.** Comparison between sensitivity and conditional entropy for JYN's network. (**a**) Sensitivity index for JYN's network; (**b**) conditional entropy for JYN's network.

Considering the solution ranking obtained from fuzzy DEMATEL, Figure 6 presents, for the sake of simplicity, a scenario for just four sensors and their location in the water network.

An interesting property related to the hydraulic conditions of the network is that all the sensors are installed in low pressure zones. Usually, low pressure zones are more sensitive to changes in the network, so they have greater sensitivity than other zones. To check the improvement obtained from the use of the conditional entropy with respect to the performance using just the sensitivity matrix, Figure 7 presents the application of the fuzzy DEMATEL method using just the fuzzy sensitivity matrix, for the same scenario with four sensors. The concentration of sensors in the lowest pressure

zone of this network can be observed. This zone is also the one identified as the most sensitive by Equation (1). The use of the conditional entropy provides a more widespread distribution of sensors (see Figure 6). Of course, the low pressure zone (excessively) identified by just the sensitivity matrix is not missed when the conditional entropy is used. In addition, this methodology avoids the redundancy of information derived from the concentration of sensors in that low pressure zone, as illustrated in Figure 7, with a more widespread distribution of the four sensors.

**Figure 6.** Layout of a sensor network with four sensors using the conditional entropy as input for fuzzy DEMATEL.

**Figure 7.** Layout of a sensor network with four sensors using only the sensitivity as input for Fuzzy DEMATEL.

To further evaluate the difference between both solutions, the global sensitivity, which means the sum of the greatest sensitivity of each sensor, and the total entropy of the system, are calculated. Global values for these two metrics are calculated using only the columns of the sensitivity matrix corresponding to the sensors' positions. Using the conditional entropy as input for the fuzzy DEMATEL method, the global sensitivity equals 4.1985, while using as input the sensitivity matrix, the global sensitivity equals 5.4381. In terms of the total entropy, the conditional entropy leads to a value of 5.2990, while just the sensitivity matrix results in 4.7560. In both cases, the conditional entropy-based approach outperforms the results of the sensitivity matrix alone.

To compare the proposed methodology with two classical optimisation approaches, agent swarm optimisation [41] is applied as an optimisation engine.

Among the various published papers presenting optimisation-based approaches for sensor placement, we consider here [7], which applies a bi-objective optimisation, maximising the sensitivity of the sensors' network and the entropy of the system. In [7], the sensitivity is defined as in Equation (1), and the entropy as in Equation (3). Given a solution for the sensor placement problem, *X* = (*x*1...*xk*), it is possible to take the corresponding *k*-columns of the sensitivity matrix *S*. Using this new sensitivity submatrix *Sk*, it is possible to identify, for each simulated leak, the most sensitive sensor, as in Equation (5). Then the sensors' network sensitivity and entropy objective functions are calculated as:

$$F\_1 = \sum\_{\mathbf{x} \in X} a(\mathbf{x}).\tag{15}$$

$$F\_2 = \sum\_{\mathbf{x} \in \mathcal{X}} p(\mathbf{x}) \cdot \ln \frac{1}{p(\mathbf{x})}.\tag{16}$$

Finally, to apply single objective optimisation, the authors of [7] combine normalised values of *F*<sup>1</sup> and *F*2.

IngeniousWare®, in a modification of the methodology in [7], created a plugin for WaterIng©, software for optimising pressure sensor placement in water systems. In this case, the entropy based function is modified to guarantee a better spread of sensors in real networks (unpublished results).

The methodology of [7] and the commercial software WaterIng© are used, fixing a number of four sensors, as in the case study solved with the fuzzy DEMATEL approach. With the methodology of [7], the total sensitivity of the sensors' network results in 4.2261 and the total entropy in 5.5912, which are similar results to those obtained with the fuzzy DEMATEL approach. This is because, in general, the sensors are placed in the same region in both cases. WaterIng© produces a sensitivity value of 5.4920 and a total entropy of 5.3195. This approach produces better values for both sensitivity and entropy. Comparing the four approaches, the modified entropy approach proposed by IngeniousWare® manages to get the best values for both objective functions. Nevertheless, the fuzzy DEMATEL approach of this paper gives results which are comparable with [7], but without the need of running any optimisation process. Table 5 summarises the results obtained in this research.

**Table 5.** Comparison table for the four considered approaches.


Let us finally emphasise that the relation between the number of sensors and the global sensitivity is a good indicator for decision makers about the number of sensors to be installed. Figure 8 presents the increase of sensitivity with the increase of the number of sensors in the network. The bigger increases for small numbers of sensors, and an asymptotic trend for larger numbers of sensors may be

verified. The graph in Figure 8 also shows how by increasing the number of sensors beyond a certain point does not entail a corresponding relevant increase in monitoring.

**Figure 8.** Sensitivity vs. number of sensors for the network under study.
