authors detailed in these studies. They found that, generally speaking, PSO works
faster than the GA (being very effective for small networks or few sensors) but, when
the problem complexity increases (e.g. when more sensors are considered), the GA
tends to find placements with higher efficiency. In this regard, the authors observed
that PSO may tend to be trapped in a local suboptimum, probably because it has
memory of past successes and therefore tends to explore around those recorded
configurations; whereas when it is necessary to leap from one region of the search
space to a distant other region, crossover operations like those in a GA are probably
more effective. Finally, the authors stressed that, although relatively small networks
were used in their studies, trying to find optimal placements for larger numbers of
sensors than those detailed in their papers would be prohibitive in terms of computational time required to obtain a solution.
Steffelbauer and Fuchs-Hanusch [78] extended the work by Steffelbauer et al.
[79] in which the effect of demand uncertainty on modelled predictions of pressure
was incorporated in the optimal sensor placement problem (solved by using the
method proposed by Casillas et al. [9, 10] but adapted in order to penalise potential
sensor locations with high uncertainties) by using Monte Carlo simulation to calculate pressures for multiple realisations of nodal demands. In Steffelbauer and FuchsHanusch [78], the authors solved the problem for different numbers of sensors
ranging from two to ten (the study by Steffelbauer et al. [79] was limited to four
sensors) taking into account different strengths of uncertainties. One of the main
findings was that incorporating uncertainties leads to very different optimal placements than without uncertainties. Indeed, without uncertainties the algorithm tended
to place sensors in regions with high demand uncertainties spread over the whole
system. With high strength uncertainty, on the other hand, the sensors tended to be
clustered “too much” in regions with low demand uncertainties, thus indicating that
points which are sensitive to leaks/bursts are also likely to be points which are most
sensitive to demand variations and, hence, not ideal locations to place sensors
at. Worth of note in this study is also the fact that the authors derived a costbenefit function to describe the relation between the number of sensors and the
leak/burst localisation quality. The main reason for this was to provide water
companies with a methodology to answer the question of how many sensors are
needed to identify a specific number of leak/burst scenarios correctly. They found
that the simple cost-benefit function they derived follows a power law. That is to say,
for a linear improvement of the localisation quality, the number of sensors has to
double. Furthermore they observed that the power law behaviour still applies even if
demand uncertainties are accounted for. The only difference to simulations without
uncertainties is that the localisation quality for a placement with a particular number
of sensors decreases as the strength of the uncertainty increases.
A further interesting investigation into the issue of demand uncertainty can be
found in Puleo et al. [80]. In this study, the authors proposed an “identifiability
analysis” [81] method that makes use of the Fisher information matrix to select
points that are sensitive to leaks/bursts and also provide less correlated measurements under uncertain demands. They performed Monte Carlo simulations whereby
demand was randomly drawn from a normal distribution and, through limited tests
Review of Techniques for Optimal Placement of Pressure and Flow Sensors. . .
41
faster than the GA (being very effective for small networks or few sensors) but, when
the problem complexity increases (e.g. when more sensors are considered), the GA
tends to find placements with higher efficiency. In this regard, the authors observed
that PSO may tend to be trapped in a local suboptimum, probably because it has
memory of past successes and therefore tends to explore around those recorded
configurations; whereas when it is necessary to leap from one region of the search
space to a distant other region, crossover operations like those in a GA are probably
more effective. Finally, the authors stressed that, although relatively small networks
were used in their studies, trying to find optimal placements for larger numbers of
sensors than those detailed in their papers would be prohibitive in terms of computational time required to obtain a solution.
Steffelbauer and Fuchs-Hanusch [78] extended the work by Steffelbauer et al.
[79] in which the effect of demand uncertainty on modelled predictions of pressure
was incorporated in the optimal sensor placement problem (solved by using the
method proposed by Casillas et al. [9, 10] but adapted in order to penalise potential
sensor locations with high uncertainties) by using Monte Carlo simulation to calculate pressures for multiple realisations of nodal demands. In Steffelbauer and FuchsHanusch [78], the authors solved the problem for different numbers of sensors
ranging from two to ten (the study by Steffelbauer et al. [79] was limited to four
sensors) taking into account different strengths of uncertainties. One of the main
findings was that incorporating uncertainties leads to very different optimal placements than without uncertainties. Indeed, without uncertainties the algorithm tended
to place sensors in regions with high demand uncertainties spread over the whole
system. With high strength uncertainty, on the other hand, the sensors tended to be
clustered “too much” in regions with low demand uncertainties, thus indicating that
points which are sensitive to leaks/bursts are also likely to be points which are most
sensitive to demand variations and, hence, not ideal locations to place sensors
at. Worth of note in this study is also the fact that the authors derived a costbenefit function to describe the relation between the number of sensors and the
leak/burst localisation quality. The main reason for this was to provide water
companies with a methodology to answer the question of how many sensors are
needed to identify a specific number of leak/burst scenarios correctly. They found
that the simple cost-benefit function they derived follows a power law. That is to say,
for a linear improvement of the localisation quality, the number of sensors has to
double. Furthermore they observed that the power law behaviour still applies even if
demand uncertainties are accounted for. The only difference to simulations without
uncertainties is that the localisation quality for a placement with a particular number
of sensors decreases as the strength of the uncertainty increases.
A further interesting investigation into the issue of demand uncertainty can be
found in Puleo et al. [80]. In this study, the authors proposed an “identifiability
analysis” [81] method that makes use of the Fisher information matrix to select
points that are sensitive to leaks/bursts and also provide less correlated measurements under uncertain demands. They performed Monte Carlo simulations whereby
demand was randomly drawn from a normal distribution and, through limited tests
Review of Techniques for Optimal Placement of Pressure and Flow Sensors. . .
41