I. Introduction
Water networks appear in a wide spectrum of applications ranging from domestic infrastructure to industrial plants. The task of monitoring and regulation of the water states within such networks is shared among most of these applications. The work at hand proposes and investigates a possible framework for the realization of a holistic tool for model-based analysis, monitoring and control of fluid networks of different scales all while considering the spatial variation of the controlled quantity. A schematic of the proposed tool is shown in Fig. 1. The general idea is to construct multiple models of the network considering several aspects, e.g. thermal, chemical/disinfectant concentration, biological/microbial development. This allows the spatio-temporal monitoring of theses several water aspects for the modeled network. Moreover, the different actuation and sensing points within the network can be mapped to the model. Consequently, several model-based analysis and synthesis tasks can be performed, e.g. optimal sensor placement, controller design. One important requirement for the models used is their real-time capability for the monitoring and control tasks. On the other hand, the local elements constituting the network should preserve their accuracy regarding the spatial variation of the fluid property under consideration. That is to say that the distributed parameter character of the models is essential and can not be neglected. This emphasizes the two contradicting incentives of high fidelity modeling and computational complexity reduction required for real-time purposes. Open source as well as commercial tools representing water networks efficiently widely used for network analysis and monitoring (specially hydraulic) tasks exist, see [1] for a review. The most popular tool, which most of the tools are based on, can be claimed to be EPANET [2] developed by the U.S. Environmental Protection Agency which also include extensions integrating other water aspects such as water quality [3] and extensions allowing for real-time tasks, e.g. [4], [5]. Tools preserving the spatial resolution while pursuing real-time capability for the given tasks are to the authors knowledge still generally absent in literature. Most of the existing tools either dismiss the accurate representation of spatial variation of the modeled water properties or the model requirements needed for real-time model based monitoring and control, i.e. control oriented models. Hence, the dominating real-time control methods for water networks have been classical model free methods or methods relying on spatially lumped static models [6]. Recent efforts for considering full spatial resolution of the networks (considering water quality aspect) within a control oriented model can be also observed [7], where also a complexity reduction method for the constructed monolithic models are proposed [8]. However, the general strategy used in [7], [8] of formulating a monolithic model (emerging from direct spatial and temporal discretization of the whole network) and reducing the network’s model as a whole is unsuitable for the goals of the proposed tool here. As the reduced model in such case loses its structural interpretability, where it is impossible to trace the network local elements to certain part of the reduced order model. Moreover, the network parameters are absorbed within the reduced model indicating that the model could not be used for tasks requiring variation of the parameters, e.g. parameter identification. Hence, the paradigm shift of using reduced order meta-models for each local part of the network is adopted here. It should be clear to this point that investigating and developing reduction methods allowing for preserving the model accuracy with a reduced computational cost is essential to guarantee the practicability of the proposed tool. Moreover, the reduced models should be optimized reaching the lowest possible computational complexity enhancing the capability of the tool to construct networks of a different scales. Model Order Reduction (MOR) techniques offer solutions to the problem allowing for the reduction of computational complexity. Most importantly, in order to utilize the Reduced Order Models (ROMs) for identification and meta-modeling purposes, the model’s parametric dependencies must be preserved throughout the reduction process. This allows ROMs representing the network local element to be complemented within and employed by the other analysis and control modules of the tool, e.g. for parameter identification. Literature is rich with projection based MOR methods for finite dimensional systems. However, most of the methods developed for nonlinear systems have an empirical nature heavily relying on training data, and hence with no guarantee for parametric dependency and structure preservation. For example, in Proper Orthogonal Decomposition (POD) [9], [10], Full Order Model (FOM) simulations are directly used for the computation of the ROM basis by applying principal component analysis on the so-called snapshots matrix containing indexed spatio-temporal evolution of the FOM. Hence finding the spatial modes with the highest contribution to the dynamics and neglecting the rest. Alternatively in Trajectory Piecewise Linear Approximation (TPWL) methods [11], the ROM is formulated as a collection (weighted sum) of local linear models derived from FOM simulation data. On the other hand methods developed for linear systems overcome such reliance on training data establishing rigor regarding error bounds, structure preservation and even ROM error optimality (see [12], [13], [14]). Some of these methods such as balanced truncation or moment matching methods with their advantages are being recently transferred for systems with weak/structural nonlinearity such as bilinear [15], quadratic [16] and polynomial [17] systems. Developing globally optimal reduction methods for general nonlinear systems can be claimed to be an intractable task. Therefore, the main strategy adopted in this work for addressing ROM optimality and structure preservation is by focusing on the model formulation. That is to allow a weak nonlinear form for which optimal reduction methods are feasible.