(14h) Efficient Economic Model Predictive Control of Water Treatment Process Using Learning-Enabled Koopman Operator

The treatment of used water is a systematic process that removes impurities from the influent feed and converts it into safer and clear effluent that can meet specific requirements. The water treatment facilities have been playing a critical role in the preservation of the ecosystems and the protection of human health [1][2]. Modern water treatment facilities typically incorporate multiple tightly interconnected physical units. Their dynamic operations involve a variety of physical, biological, and chemical phenomena, and typically exhibit complex nonlinear behaviors that are affected by significant external disturbances, including fluctuations in the flow rates and variations in the compositions of the inlet flows [3][4]. As the environmental regulations continue to tighten, the industry is demanding advanced control solutions that can be used to manage the real-time dynamic operations of water treatment facilities to ensure the production of consistent high-quality treated water while maintaining the energy and chemical consumption and operating costs at a satisfactory level [5][6].

Over the past decades, research attention has been dedicated to the development of control strategies for wastewater treatment plants (WWTP). Classical proportional-integral (PI) control has been applied for ammonia removal in WWTPs [7]. Another commonly adopted control paradigm for WWTPs is the model predictive control (MPC). As compared to conventional control, MPC is capable of explicitly addressing constraints on quality variables and optimality considerations. Nonetheless, MPC does not explicitly take into account the associated operating costs, and this limitation restricts its capability of optimizing the overall process operation performance. Economic model predictive control (EMPC) offers a promising alternative in the field of optimal control [8]. By seamlessly integrating economic process optimization and dynamic control into one framework, EMPC has the potential to simultaneously optimize the effluent quality and reduce the operating costs for WWTPs [9].

However, these EMPC designs face constraints in three key aspects that limit their broader applicability. First, these EMPC methods rely on an accurate first-principles model of the underlying process, which can be restrictive when accurate parameters are partially or entirely unavailable, or when the mechanistic knowledge of the nonlinear process is not completely known. Second, the implementation of these EMPC designs requires access to full-state information, which requires that all the states are measured online or a nonlinear state estimator is designed [10]. Lastly, when applied to nonlinear systems, the existing EMPC designs require solving non-convex nonlinear optimization problems, which can be computationally expensive and time-consuming.

In recent years, the Koopman operator theory has gained substantial research attention, owing to its capability to represent the dynamics of complex nonlinear processes in a linear manner [11]. Several cost-effective algorithms, including dynamic mode decomposition (DMD) [12] and extended dynamic mode decomposition (EDMD) [13][14][15], have emerged for the construction of approximated Koopman operators and Koopman linear models using temporal data of a system/process. To streamline the design of the observable functions, researchers have proposed deep learning Koopman modeling methods [16][17]. Koopman operator provides a promising alternative approach to address complex nonlinear control problems efficiently by leveraging linear control theories. This motivates us to explore whether the Koopman modeling framework may be exploited for efficient EMPC.

In this work, we aim to bridge this gap by proposing a Koopman-based EMPC framework, which is designed to alleviate the prohibitive computational burden associated with the existing non-convex EMPC frameworks to achieve the efficient and economic operation of WWTPs. Specifically, we propose a data-driven modeling framework, referred to as the Deep Input-output Koopman Operator (DIOKO) model, to predict the future economic operational cost and safety-related states. By using only input data and economic cost data, the proposed modeling approach learns a computationally efficient latent space, where future key performance indices related to WWTP operations are accurately predicted. Based on the established DIOKO model, a convex computationally efficient EMPC scheme is formulated. The constraints on safety/water quality-related states are incorporated into the formulated EMPC scheme. This data-driven EMPC scheme is used for managing the real-time operation of the nonlinear WWTP, resulting significantly improved overall operational performance under various operational conditions.

References

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