Figure 1 Types of applications that use machine learning
metamodels for Water Distribution Systems (WDS) and Urban Drainage
Systems (UDS)
Optimisation usually employs population-based algorithms (e.g., genetic
algorithms, particle swarm, ant colony optimisation, among others) which
require multiple runs. These algorithms create an initial population,
and they improve the obtained solutions through continuous iteration.
Usually, these algorithms employ mechanisms inspired on genetics, such
as crossover and mutation for finding (near) optimal solutions.
Evolutionary algorithms are the most well-established metaheuristic for
solving water resources problems (Maier et al., 2014); nonetheless, they
tend to be highly computationally intensive.
Optimisation can be used to formulate and solve multiple UWN problems.
This explains the high number of metamodeling publications dedicated to
this topic. A popular use of MLSMs for optimisation in UWNs is for the
(re)design of the networks. For example, applications that use MLSMs
include changes in pipe diameters and chlorine dosing rates (Andrade et
al., 2016; Bi & Dandy, 2014; Broad et al., 2005a; Sayers et al., 2019)
or operation of storage tanks and pumps (Broad et al., 2010; Martínez et
al., 2007; Salomons et al., 2007). The goal for design is to select
which new system components to install or identify existing ones to
substitute. For operation, the aim is to find an optimal policy on how
to operate the existing components. Regardless of the task, the goal is
to maximize the performance of the system described by the objective
function(s) and a number of constraints (e.g., physical, regulatory,
economic, among others). In addition, other problems such as water
quality model calibration (Dini & Tabesh, 2017), renovation planning
(Dini & Tabesh, 2019), and sensor placement (Behzadian et al., 2009)
have resorted to metamodels.
Although MLSMs accelerate optimisation algorithms, they come with a
series of drawbacks. First of all, these models need training data
(simulation examples) to calibrate their internal parameters (e.g., the
weights and biases in a neural network) to replicate the RS. Generating
a sufficiently large training dataset can be a time-consuming process,
and data sufficiency depends on the complexity of the input-output
mapping and it can not be known a priori. Secondly, the training process
is another optimisation process in itself, with its own hyperparameters
(e.g., learning rate, number of training epochs, parameter
initialization, among others depending on the optimiser) and its
convergence to a desired performance is not guaranteed. Furthermore,
errors of approximation in the RS can mislead the optimisation to
suboptimal or unfeasible solutions as noted by Broad et al. (2005b),
especially in zones near the boundaries or outside the training range.
When comparing water distribution with drainage systems, it is clear
that the applications of optimisation in UDSs are less diverse. The
reviewed papers focus on the optimisation of stormwater sewers’ design
with Low Impact Development (LID) management (Latifi et al., 2019; Raei
et al., 2019; Seyedashraf et al., 2021) or for flood mitigation (Huang
et al., 2015; W. Zhang et al., 2019). Meanwhile, WDS optimisation is
more varied, with applications to operation, calibration, sensor
placement, and long-term planning. This difference partially depends on
the stochastic nature of the rainfall events driving the functioning of
combined and stormwater sewers, which in turn favour real-time control
over the optimisation of the operations, typical of WDS. Also, the
research done on MLSMs for optimisation in UDSs is rather recent (2015
or later) compared to WDS (from 2005). Applications in UDSs that
typically do not use metamodels can benefit from the experience of
tackling similar problems in the context of WDSs. Examples include
sensor placement (Sambito et al., 2020), calibration (Tscheikner-Gratl
et al., 2016), and optimisation of operation (van Bijnen et al., 2017).
In contrast to off-line optimisation, real-time applications require
accurate answers with limited computational time. Real-time operation
uses the current state of the system to modify its behaviour and improve
its functioning in future time steps. In the case of UDSs, they are
usually designed to retain stormwater for a certain period, to avoid
combined sewer and stormwater outflows (Rosin et al., 2021; She & You,
2019) or to reduce flooding (Berkhahn et al., 2019; Chiang et al., 2010;
Keum et al., 2020; Kim et al., 2019; Kim & Han, 2020). Whereas, in
WDSs, the objective is to deliver high-quality drinking water while
minimizing pumping costs (Pasha & Lansey, 2014; Rao & Alvarruiz, 2007;
Rao & Salomons, 2007).
In the case of WDSs, the reviewed real-time applications concern
optimisations, in which MLSMs are essential to reduce the computational
time for evaluating the fitness function used by an evolutionary
algorithm. Consequently, these applications suffer from the drawbacks
already mentioned for optimisation with MLSMs. Real-time applications
for UDS concern Real-Time Control (RTC), where operation and validation
relies on real data (Beeneken et al., 2013; Langeveld et al., 2013; Lund
et al., 2018). This is an issue since the usual targets are infrequent
events, i.e., outflows and flooding; therefore, the availability of
records may be scarce or non-existent.
The third application in order of frequency is uncertainty analysis of
the UWNs’ performance. These analyses are usually carried out via
multiple simulations to test the response of the system to multiple
possible scenarios or uncertain input conditions, leveraging the
computational efficiency of SMs. In WDSs, ANNs have been used to replace
computationally expensive models for accelerating Monte Carlo analyses.
For example, Yoon et al. (2020) performed a seismic risk assessment of a
water distribution network considering earthquakes of different
magnitudes and epicentres. In UDSs, Beh et al., (2017) used metamodels
to directly estimate reliability and vulnerability metrics. In this
case, resorting to MLSMs was crucial for the feasibility of the study.
Otherwise, the explicit robustness assessment would have been impossible
in practice. Creating a metamodel for uncertainty analysis entails
having a model with explicit robustness as output, or generating a
training dataset with multiple runs per example. However, the former is
rarely the case and the latter consumes a large quantity of
computational budget.
Other works tested the ability of ANNs to estimate the state of the
system at ungauged points with measurements from a limited amount of
sensors. Lima et al. (2018) and Meirelles et al. (2017) used recorded
pressure at strategically located sensors and an ANN to estimate the
pressure of all the nodes in a WDS. SMs for state estimation not only
decreases the degrees of freedom for the addressed calibration problem
but, according to the authors, they could also be used to detect
anomalies and predict the current state of the network in real-time.
Nevertheless, in these studies, the pressure in all the nodes is known
since the MLSM is trained on simulations. For applications depending on
sensor data, only a few nodes would be known and it would not be
possible to estimate the error for the ungauged nodes. One alternative
to handle this issue is to use some sensors for training and others for
testing. This way, it is possible to estimate the error at the unseen
nodes. However, this process reduces the training data available, and it
is not clear how representative the testing sensors are with respect to
the remaining ungauged nodes. This may lead to unjustified trust in the
model and consequent errors.
Metamodels for UDSs have also been used to complement LFPB surrogates,
either to approximate some parts of the model (e.g., the most
time-consuming) or to correct the predictions produced by a model. Wolfs
& Willems (2017) created a modular approach in which they replaced the
hydraulic simulation of drainage flow between subcatchments with an ANN,
this was part of a bigger framework in which the goal was to simulate
outgoing discharges for a given rainfall event. Similarly, Bermúdez et
al. (2018) employed an ensemble of ANNs to accelerate a component of an
LFPB model, used to estimate the occurrence and magnitude of flooding.
On the other hand, Vojinovic et al. (2003) used MOUSE (MOdel for Urban
Sewers), a hydrodynamic process model, to estimate flows within
wastewater pipes during wet weather periods and trained a neural network
to compensate for the output errors (residuals), leading to an overall
increase in accuracy. Even though this hybrid approach bridges both
metamodeling practices, the LPFB metamodel inherits the RS problems,
e.g., database creation and training difficulties.
In summary, SMs in water networks have been primarily used for
optimisation and real-time applications due to their ability to quickly
evaluate outputs while remaining sufficiently accurate. This avoids
running computationally expensive hydrodynamic models. Nevertheless, the
use of these metamodels is not bound to these two applications. They can
replace the original model for uncertainty analyses and state
estimation, or help the original model by correcting outputs or
approximating computationally expensive components.
3.2 Case studies
Figure 2 shows the number of case studies analysed in the reviewed
literature. In WDSs, each paper usually presents two or more networks.
Since the papers introduce new problem formulations or methodologies,
the authors apply them to different networks to prove that the methods
work independently of the choice of the system. Studies in optimisation
usually follow a common pattern where preliminary trials are done on
small benchmark networks before proceeding with implementation in bigger
real case scenarios. This pattern is repeated in all the cases, whether
it is on the same paper or in sequential papers, as in the case of the
POWADIMA project by Martínez et al., 2007; Rao & Alvarruiz, 2007; and
Salomons et al., 2007. In the cases of real-time applications, the
networks were usually modified benchmarks of medium size. For
applications in uncertainty analysis and state estimation, the networks
were real cases of large size. The reviewed papers for UDSs, in contrast
to WDS, present only applications with real networks, some of them with
modifications (e.g., Berkhahn et al., 2019; She & You, 2019).