3.1. Case Studies
The application of this work concerned two case studies (
Figure 1a,b, respectively). The first is the skeletonized WDN of Santa Maria di Licodia, a small town in Sicily, Italy [
24]. The use of a skeletonized WDN is not expected to undermine the validity of the results since the pipes with larger diameter, which compose the WDN skeleton, are typically the best candidates for the installation of control valves.
The network layout is made up of
nn = 34 nodes (of which
n = 32 with unknown head and
n0 = 2 source nodes with fixed head, i.e., Nodes 33 and 34) and of
np = 41 pipes. The features of the WDN demanding nodes in terms of ground elevation
z, mean daily demand
d, and desired pressure head
hdes for full demand satisfaction, are reported in
Table 1. At the generic demanding node, the latter variable was set at the minimum value between 15 m and the daily lowest pressure head under uncontrolled conditions. This was done to avoid any pressure deficit increase at nodes that were pressure deficient ab initio.
Table 2 reports the features of the pipes in terms of length
L, diameter
D and Strickler roughness coefficient
k. The two source nodes, i.e., Nodes 33 and 34, have a ground elevation of 477.5 m a.s.l.
= 12 2-h-long time slots were used to represent the WDN daily operation. The patterns of the heads at the source nodes and of the hourly multiplying coefficient
Ch for nodal demands are reported in
Table 3.
As to leakage, two conditions were considered. In the former, values of the coefficient CL = 2.79 × 10−8 m0.82/s and of the exponent nleak = 1.18 were assumed for all the pipes of the network. The values of CL and nleak lead to a daily leakage volume WL = 1242.6 m3, which is about 44% of the water volume leaving the source nodes, as evaluated in the real network. In the latter, the coefficient CL was reduced to 0.85 × 10−8 m0.82/s, obtaining a daily leakage volume WL = 397.7 m3 (about 20% of the water volume leaving the source nodes) without control valves.
The second case study is a district of the pipe network model used as benchmark in the Battles of Water Networks of the last WDSA conferences [
19,
20]. This district is made up of
nn = 46 nodes (of which
n = 45 with unknown head and
n0 = 1 source node with assigned head, i.e., tank Node 46) and
np = 52 pipes. The features of the district demanding nodes and pipes are provided by Creaco and Pezzinga [
16], who also reported the values of the leakage exponent
nleak = 0.9 and of the leakage coefficient
CL ranging from 0.2 to 1 m
1.1/s in the various pipes. In this district, the leakage volume adds up to 118.7 m
3, which is 6.1% of the water volume entering the district, in the no valve scenario.
In the application, values equal to 105.8 m a.s.l. and 2.4 m, respectively, were assigned to the elevation and to the average water level of the source node. An inflow (that is a negative demand) takes place in correspondence to Node 1, at certain hours of the day, due to the activation of a pumping system that takes water from another district of the network. The detailed operation of this system, in terms of relationship between water discharge and head gain across the pump, is neglected in this work.
The district operation can be summarized in three time slots, for each of which the head of the source node, the inflow from the pump and the hourly demand coefficient for the demanding nodes were specified by Creaco and Pezzinga [
18]. No valve installation is allowed in the main line (
Figure 1b) that connects the pump with the tank node, to prevent any interferences with the filling/emptying process of the tank.
3.2. Results
In the application with SA [
15] to the first case study, the optimal locations of up to
Nmax = 10 control valves were searched for. This required 365 objective function evaluations.
In the application with GA [
17,
18], a population of 50 individuals and a total number of 50 generations, corresponding to total number of 2500 objective function evaluations, were used. A healthy initial population was generated in GA to guarantee high computation efficiency at the end of the optimization.
The applications of this work were performed in the Matlab(R) 2011b environment by making use of a single processor of an Intel(R) Core(TM) i7-7700 3.60 GHz unit.
The tradeoff curves of
WL as a function of
Nval obtained with SA and GA in the first leakage condition are reported in
Figure 2. As for GA, two Pareto fronts are shown, the final one (after 50 generations) and that after seven generations. The latter is associated with 350 objective function evaluations (very close to SA). Though GA explored solutions with
Nval values up to
np (see
Section 2.3), only the solutions with
Nval ≤ 10 were reported in the graph for a consistent comparison with SA. This graph shows that the curve of SA and the final curve of GA have identical trend up to
Nval = 3. To the right of
Nval = 3, the curve obtained with GA dominates that of SA, in that it provides lower values of
WL for each value of
Nval. Furthermore, the distance between the two curves increases as
Nval grew. This seems to suggest that, for low number of
Nval, SA can provide identical or close results to those of GA. Conversely, for high number of
Nval, the better performance of GA stands out, due to the wider exploration of the research space. However, this comes at an about seven times larger computational cost. As was expected, the curve of GA after seven generations features worse solutions than the final curve of GA to the right of
Nval = 3. The curve of GA after seven generations enables the results of GA to be compared with those of SA, given the same computational budget. Interestingly, neither curve prevails in absolute terms. In fact, identical solutions are observed up to
Nval = 3. GA after seven generations is slightly better for
Nval = 7, 9 and 10, while not offering any solutions for
Nval = 8. SA, instead, is better for
Nval = 5 and 6. Summing up the results in
Figure 2, the better performance of GA stands out only with a higher computational budget.
An insight into the different results obtainable with SA and GA is provided in
Table 4, which reports the optimal valve locations and the
WL values for the two algorithms. Whereas the valve positions suggested by SA and GA are the same up to
Nval = 4, the two algorithms behave differently starting from the optimal location of five valves. In fact, at Step 5, SA suggests adding valve in Pipe 33 to the optimal location of
Nval = 4 valves, in which the valve-fitted pipes are 3, 7, 14 and 27. Instead, for the sake of optimality, GA proposes valve elimination in Pipe 14 and valve insertion in Pipes 25 and 26, while moving from
Nval = 4 to
Nval = 5. The locations of SA and GA for
Nval = 5 valves are shown in
Figure 3. As
Table 4 shows, this yields a 4.36% benefit of GA compared to SA, in terms of
WL. The percentage benefit of GA tends to grow up to almost 10% for
Nval = 10.
Another example of the results obtained is provided in the graphs in
Figure 4, associated with the optimal location of three valves, installed in Pipes 3, 7 and 27, respectively, for both SA and GA.
Figure 4a shows the temporal pattern of
V, optimized through the iterated LP at each time slot.
Figure 4b shows the pressure head
hdown at the downstream node, which can be used as the time varying setting to be prescribed if the water utility chooses to perform the pressure regulation by means of pressure reducing valves (PRVs). In fact, PRVs has the downstream pressure head as setting. This variable is always equal to
hdes (that is 15 m) at the valve in Pipe 3 because the downstream node of Pipe 3, namely Node 32, is a terminal node of the network. It is always equal to
hdes also for the valve in Pipe 7, though the downstream node of this pipe, namely Node 3, is not terminal. This happens because the descending topography downstream of Pipe 7 facilitates the meeting of the pressure requirements. The case of the valve in Pipe 27, which has Node 9 as downstream node, is similar. However, this valve cannot reduce
hdown to
hdes at nighttime, even if it is almost fully closed (
V ≈ 0).
An analysis should also be carried out concerning the possibility to convert the control valves proposed by the optimizer to isolation valves. This could be done when the optimal settings V are very close to 0 throughout the day. Furthermore, isolation valves could be closed in some pipes in parallel to the installed control valves if water flow is remarked to bypass the latter. However, neither situation was remarked to occur in the present case study. Furthermore, it must be noted that closing an isolation valve may decrease WDN redundancy and reliability.
The tradeoff curves of
WL as a function of
Nval obtained with SA and GA in the second leakage condition are reported in
Figure 5. This figure leads to similar remarks as
Figure 2, as far as the comparison of the two algorithms is concerned.
Table 5 gives some insight into optimal valve locations,
WL values and benefits of GA compared to SA. The comparison of
Table 5 (low starting leakage) and
Table 4 (high starting leakage rate) points out that the reduction in the starting leakage has the following minor effects:
The optimal valve locations change for
Nval > 3 (e.g., for
Nval = 4, the optimal valve locations with GA are 3-7-24-27 and 3-7-14-27 in
Table 5 and in
Table 4, respectively).
The benefits of GA in
Table 4 tend to grow with
Nval increasing, whereas the values in
Table 5 tend to stabilize around 6.5%.
The applications to the second case study were carried out similarly to the first case study. The results are reported in
Figure 6 and
Table 6. The former reports the Pareto fronts obtained through SA and GA, whereas the latter reports optimal valve locations and
WL provided by the two algorithms, as well as the benefits of GA. Similar to the first case study, these benefits stand out only for
Nval ≥ 5. However, they are smaller (always below 4.32%), due to the different structure of the WDN. In fact, while the network of Santa Maria di Licodia is very interconnected, the network of the second case study is made up of quite independent branch structures fed by the main line, along which valves cannot be installed. This attenuates the non-linearities pertinent to optimal valve location.