The Representation of Hydrological Dynamical Systems Using Extended Petri Nets (EPN)

Abstract

This work presents a new graphical system to represent hydrological dynamical models and their interactions. We propose an extended version of the Petri Nets mathematical modeling language, the Extended Petri Nets (EPN), which allows for an immediate translation from the graphics of the model to its mathematical representation in a clear way. We introduce the principal objects of the EPN representation (i.e., places, transitions, arcs, controllers, and splitters) and their use in hydrological systems. We show how to cast hydrological models in EPN and how to complete their mathematical description using a dictionary for the symbols and an expression table for the flux equations. Thanks to the compositional property of EPN, we show how it is possible to represent either a single hydrological response unit or a complex catchment where multiple systems of equations are solved simultaneously. Finally, EPN can be used to describe complex Earth system models that include feedback between the water, energy, and carbon budgets. The representation of hydrological dynamical systems with EPN provides a clear visualization of the relations and feedback between subsystems, which can be studied with techniques introduced in nonlinear systems theory and control theory.

On the relations between the hydrological dynamical systems of water budget, travel time, response time and tracer concentrations

“…The dictionary of the dynamical system shown in Table 1 gives a meaning to the variables used. In a change with respect to Bancheri et al (2019), here we number the dictionary rows for easier use in the sections that follow.…”

“…On the top left (DynWB) we have the EPN representation of the water budget of the model used in Kirchner (2016Kirchner ( , 2019; on the top right (TTD) we have the representation of travel time distributions; on the bottom left (DynRTD) we have the representation of life expectancy distributions; and on the bottom right (DynC) we have the representation of the concentrations of a tracer (Gabrielli et al, 2018), for instance, could be such a case. As already mentioned, in EPN nomenclature (Bancheri et al, 2019) the reservoirs (represented by circles) are called places and the fluxes (represented by squares) are called transitions. With EPN there is a topological part, called topology of the model, which resolves the system behaviour by means of a group of places that receive and exchange transitions (see as an example Figure 1 at top left).…”

“…Applying the rules introduced, it is easy to cast any model into the EPN representation, as is clearly shown in Bancheri et al (2019) where three examples are reported.…”

“…Any HDSys can be seen to be composed of topology and semantics: the former describing the connections between the compartments of the budget and the latter associating symbols to topology, meanings to symbols, and expressions to fluxes, as presented in Bancheri et al (2019) and outlined in Appendix . For a complete understanding of EPN, Bancheri et al (2019) is a mandatory reading, where it is shown that they have the capabilities to represent the more complex coupled Earth Systems. However, a short video has also been created to communicate the main EPN concepts and it can be found via Data S1.…”

The GEOframe-NewAge Modelling System Applied in a Data Scarce Environment

“…The sERM is represented by extended Petri nets (EPNs) [57] in Figure 2. The model is composed of four integrated reservoirs: canopy (green reservoir), root zone (orange reservoir), runoff (purple reservoir), and groundwater (yellow reservoir).…”