Existence and Uniqueness of Physically Correct Hydraulic States in Water Distribution Systems -- A theoretical analysis on the solvability of non-linear systems of equations in the context of water distribution systems

Planning and extension of water distribution systems (WDSs) plays a key role in the development of smart cities, driven by challenges such as urbanization and climate change. In this context, the correct estimation of physically correct hydraulic states, i.e., pressure heads, water demands and water flows, is of high interest. Hydraulic emulators such as EPANET or more recently, physic-informed surrogate models are used to solve this task. They require a subset of observed states, such as heads at reservoirs and water demands, as inputs to estimate the whole hydraulic state. In order to obtain reliable results of such emulators, but also to be able to give theoretical guarantees of their estimations, an important question is whether theoretically, the subset of observed states that the emulator requires as an input suffices to derive the whole state, purely based on the physical properties, also called hydraulic principles, it obeys. This questions translates to solving linear and non-linear systems of equations. Previous articles investigate on the existence question under the term observability analysis, however, they rely on the approximation of the non-linear principles using Taylor approximation and on network-dependent numerical or algebraic algorithms. In this work, we provide purely theoretical guarantees on the existence and uniqueness of solutions to the non-linear hydraulic principles, and by this, the existence and uniqueness of physically correct states, given common subsets of them -- a result that seems to be common-sense in the water community but has never been rigorously proven. We show that previous existence results are special cases of our more general findings, and therefore lay the foundation for further analysis and theoretical guarantees of the before-mentioned hydraulic emulators.