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).