The environmental benefits of combining traditional infrastructure solutions for urban drainage (increasing storage volume) with real time control (RTC) strategies were investigated in the Lundofte catchment in Denmark, where an expensive traditional infrastructure expansion is planned to comply with environmental requirements. A coordinating, rule-based RTC strategy and a global, system-wide risk-based dynamic optimization strategy (model predictive control), were compared using a detailed hydrodynamic model. RTC allowed a reduction of the planned storage volume by 21% while improving the system performance in terms of combined sewer overflow (CSO) volumes, environmental impacts, and utility costs, which were reduced by up to 10%. The risk-based optimization strategy provided slightly better performance in terms of reducing CSO volumes, with evident improvements in environmental impacts and utility costs, due to its ability to prioritize among the environmental sensitivity of different recipients. A method for extrapolating annual statistics from a limited number of events over a time interval was developed and applied to estimate yearly performance, based on the simulation of 46 events over a five-year period. This study illustrates that including RTC during the planning stages reduces the infrastructural costs while offering better environmental protection, and that dynamic risk-based optimisation allows prioritising environmental impact reduction for particularly sensitive locations.

High quality on-line flow forecasts are useful for real-time operation of urban drainage systems and wastewater treatment plants. This requires computationally efficient models, which are continuously updated with observed data to provide good initial conditions for the forecasts. This paper presents a way of updating conceptual rainfall-runoff models using Maximum a Posteriori estimation to determine the most likely parameter constellation at the current point in time. This is done by combining information from prior parameter distributions and the model goodness of fit over a predefined period of time that precedes the forecast. The method is illustrated for an urban catchment, where flow forecasts of 0–4 h are generated by applying a lumped linear reservoir model with three cascading reservoirs. Radar rainfall observations are used as input to the model. The effects of different prior standard deviations and lengths of the auto-calibration period on the resulting flow forecast performance are evaluated. We were able to demonstrate that, if properly tuned, the method leads to a significant increase in forecasting performance compared to a model without continuous auto-calibration. Delayed responses and erratic behaviour in the parameter variations are, however, observed and the choice of prior distributions and length of auto-calibration period is not straightforward.

To prevent online models diverging from reality they need to be updated to current conditions using observations and data assimilation techniques. A way of doing this for distributed hydrodynamic urban drainage models is to use the Ensemble Kalman Filter (EnKF), but this requires running an ensemble of models online, which is computationally demanding. This can be circumvented by calculating the Kalman gain, which is the governing matrix of the updating, offline if the gain is approximately constant in time. Here, we show in a synthetic experiment that the Kalman gain can vary by several orders of magnitude in a non-uniform and time-dynamic manner during surcharge conditions caused by backwater when updating a hydrodynamic model of a simple sewer system with the EnKF. This implies that constant gain updating is not suitable for distributed hydrodynamic urban drainage models and that the full EnKF is in fact required.

Hydrodynamic models (numerical solutions of the Saint Venant equations) are at the core of simulating water movements in natural streams and drainage systems. They enable realistic simulations of water movement and are directly linked to physical system characteristics such as channel slope and diameter. This feature is important for man-made drainage structures as it enables straightforward testing of the effects of varying channel designs. In cities, models with hundreds up to tens of thousands of pipes are commonly used for drainage infrastructure. Their computational expense remains high and they are not suited for a systematic screening of design options, discussing water management options in workshops, as well as many real-time applications such as data assimilation.Hydrologists have developed many approaches to enable faster simulations. All of these do, however, compromise on the physical detail of the simulated processes (for example, by simulating only flows using linear reservoirs), and usually also on the spatial and temporal resolution of the models (for example, by simulating only flows between key points in the system). The link to physical system characteristics is thus lost. Therefore, it is challenging to incorporate such approaches into planning workflows where changing city plans require a constant revision of water management options.Recent advances in scientific machine learning enable the creation of fast machine learning surrogates for complex systems that preserve a high spatio-temporal detail and a physically accurate simulation. We present such an approach that employs generalized residue networks for the simulation of hydraulics in drainage systems. The key concept is to train neural networks that learn how hydraulic states (level, flow and surcharge volume) at all nodes and pipes in the drainage network evolve from one time step to another, given a set of boundary conditions (surface runoff). Training is performed against the output of a hydrodynamic model for a short time series.Once trained, the surrogates generate the same results as a hydrodynamic model in the same level of detail, and they can be used to quickly simulate the effect of many different rain events and climate scenarios. Considering pipe networks with 50 to 100 pipes, our approach achieves NSE values in the order of 0.95 for the testing dataset. Simulations are performed 10 to 50 times faster than the hydrodynamic model. Training times are in the order of 25 minutes on a single CPU. The surrogates are system specific and need to be retrained when the physical system changes. To minimize this overhead, we train surrogates for small subsystems which can subsequently be linked into a model for a large drainage network.Our approach is an initial application of scientific machine learning for the simulation of hydraulics that is readily combined with other recent developments. Future research should, in particular, explore the application of physics-informed loss functions for bypassing the generation of training data from hydrodynamic simulations, and of graph neural networks to exploit spatial correlation structures in the pipe network.