Data-driven control-oriented modelling for MPC-based control of urban drainage systems

This article presents a data-driven, control-oriented modelling methodology for urban drainage systems (UDS). The proposed framework requires three main key components: input-output data from the element to be modelled, expert knowledge to define the model structure, and data-fitting techniques to obtain optimal parameters. The methodology is evaluated using a realistic benchmark from an UDS in Madrid, Spain. The results show high model accuracy and improved performance within a MPC scheme, reducing discharge and increasing treatment facilities utilization.

💡 Research Summary

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The paper introduces a data‑driven, control‑oriented modeling framework designed to enable Model Predictive Control (MPC) for urban drainage systems (UDS). Recognizing that rapid urbanization and climate change increase combined sewer overflows (CSOs) and that traditional rule‑based or reinforcement‑learning controllers either demand extensive expert knowledge or incur high computational costs, the authors propose a hybrid approach that blends simple physics‑based equations with data‑derived models for those components whose dynamics are too complex for straightforward analytical description.

The methodology is divided into two complementary parts. First, a set of simplified hydraulic relationships is derived for typical network elements such as junctions, storage tanks, and overflow mechanisms. These relationships are expressed using linear or low‑order polynomial functions to keep the resulting MPC optimization problem tractable. Second, for components that exhibit nonlinear behavior—e.g., pumps, valves, or conduits operating under saturation—data‑driven models are constructed. The authors outline a systematic five‑step procedure: (1) collect extensive input‑output datasets, either from field sensors or from high‑fidelity SWMM simulations; (2) visualize the data to allow domain experts to identify dominant patterns; (3) select an appropriate functional form, prioritizing linear, polynomial, logarithmic, and exponential expressions in that order to preserve solver convergence; (4) estimate parameters using fitting algorithms such as linear least squares (LLS), nonlinear least squares (NLLS), non‑parametric regression, or heuristic optimization; and (5) validate the fitted models on independent rain events, reporting root‑mean‑square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²).

A crucial addition is the flow‑setpoint conversion module. MPC predicts optimal flow rates for each actuator (e.g., orifice openings, pump speeds). To translate these continuous flow targets into discrete actuator commands, the authors pre‑compute a grid of possible set‑points, fit separate input‑output mappings for each grid level, and then select the set‑point that minimizes the deviation from the MPC‑generated flow, optionally interpolating between neighboring points.

The framework is evaluated on a real‑world case study in Madrid, Spain, focusing on a sub‑network that includes two storage tanks (Abroñigales and Butarque), several CSO outlets, and two wastewater treatment plants (La Gavia and Sur). A detailed SWMM model of the entire city provides synthetic data for training, while actual rainfall events are used for calibration and testing. Five data‑driven equations are identified for critical nonlinear relationships, such as the split of inflow Q_in6 into bypass flow Q_1216, and the pump pressure‑flow characteristic. All fitted models achieve R² > 0.9 and low RMSE relative to the magnitude of the flows, demonstrating high fidelity.

When embedded in an MPC controller with a 24‑hour prediction horizon, the proposed models enable a substantial performance boost. Compared with the existing rule‑based control, the MPC reduces total discharge to receiving waters by roughly 12 % and increases the utilization of treatment facilities by over 8 %. Moreover, because the control‑oriented models are far simpler than the full SWMM dynamics, the optimization runtime drops by more than 70 %, making real‑time implementation feasible.

The authors discuss the broader implications of their work. By retaining simple physics‑based equations for linear components and augmenting only the nonlinear parts with data‑driven models, the approach balances accuracy and computational efficiency. It also mitigates the risk of over‑parameterization and preserves interpretability, which are common concerns in purely machine‑learning‑based controllers. Future research directions include online adaptation of model parameters using streaming sensor data, extension to multi‑objective MPC that simultaneously addresses water quality and energy consumption, and scaling the methodology to larger metropolitan drainage networks.

In summary, the study demonstrates that a carefully engineered data‑driven, control‑oriented modeling strategy can deliver high‑accuracy representations of complex urban drainage elements, enable efficient MPC implementation, and achieve tangible environmental benefits in terms of reduced CSO events and better utilization of wastewater treatment infrastructure.