Port-Hamiltonian based control of water distribution networks

Port-Hamiltonian Modeling of Hydraulics in 4th Generation District Heating Networks

“…Thus, the open-loop of any i ∈ L valve is equivalent to that of a control valve in series with a pipe element, i.e., to (6). By closing the loop with u v,i as in (23), we obtain a closed-loop system equivalent to (24), which is clearly of the form (27). Moreover, EIP and qi = q * i directly follow from Proposition 4 and Assumption 3.…”

“…with appropriate vectors and matrices. Moreover, for any i ∈ D ∪ L ∪ P boost ∪ M ∪ H, (27) is EIP with supply rate (z i − zi )(d i − di ) and positive definite storage function…”

“…Lemma 2: The model of any i ∈ P \ P boost ∪ C, i.e., any pipe edge without booster pump and any capacitive node, can be written as (27) with an appropriate choice of the defining vectors and matrices. Furthermore, these models are EIP with some positive definite storage function in the form of (28).…”

“…For any i ∈ D VSP , we combine the open-loop DGU model (5) with u P,i as in (3) and fix u v,i = ūv,i > 0 (due to s v,i = 1) to write the closed-loop as in (27)…”

“…The same restrictions underlie the DHN models used in [13], [22], [23], [24], where, additionally, only single producer DHNs are considered. Further noteworthy works that contribute towards a passivity-based control design and analysis of DHNs by introducing port-Hamiltonian system (PHS) models are [25], [26]; see also [27], where water distribution networks are modeled as PHSs. However, [25] exhibits the same setup restrictions as [13], [22], [23], [24], while [26] overcomes most of these restrictions with the exception of modeling valves as non-actuated components and considering pumps installed at every producer and consumer.…”