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arxiv: 2605.01707 · v1 · submitted 2026-05-03 · 📡 eess.SY · cs.SY

Nonsmooth Hydraulics, Smooth Control: System Theory Framework for Analyzing Water Networks

Ahmad F. Taha , Mohamad H. Kazma This is my paper

Pith reviewed 2026-05-09 17:12 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords water distribution networksdifferential algebraic equationsregularizationstability analysiscontrollabilityhydraulicsnetwork uncertainty
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The pith

Regularized DAE models of water networks yield error bounds on stability and controllability while matching EPANET trajectories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a control-theoretic framework for water distribution networks modeled by nonlinear differential algebraic equations that incorporate arbitrary topology along with pumps and valves. It first characterizes local well-posedness and the loss of differentiability caused by switching, then introduces regularization methods that smooth flow and pressure trajectories. Error bounds are established for DAE linearization, local stability, and finite-horizon controllability, together with a quantification of how network-induced parametric uncertainty affects these quantities. The smoothed models are shown to reproduce EPANET outputs closely on benchmark networks, while also exposing energy dissipation via a weighted Laplacian and ranking pipes by sensitivity to operating points.

Core claim

Regularization methods smooth flow and pressure trajectories under changing controls in a general nonlinear DAE model of water distribution networks, yielding explicit error bounds for linearization, local stability, and finite-horizon controllability that remain valid under network-induced parametric uncertainty; the resulting trajectories closely match those produced by EPANET on benchmark networks, and demand variation alters stability margins without removing local stability or pump authority.

What carries the argument

Regularization methods applied to the nonsmooth DAE model of WDN hydraulics that produce differentiable trajectories while preserving controllability and stability properties for error-bound analysis.

Load-bearing premise

The chosen regularization methods preserve the essential controllability and stability properties of the original nonsmooth DAE without introducing artifacts that invalidate the derived error bounds or local well-posedness results.

What would settle it

A side-by-side run of the smoothed DAE against EPANET on a benchmark network that produces trajectory deviations larger than the stated error bounds, or a specific demand-variation scenario in which the claimed local stability or pump authority is lost.

Figures

Figures reproduced from arXiv: 2605.01707 by Ahmad F. Taha, Mohamad H. Kazma.

Figure 1
Figure 1. Figure 1: Simple network describing (1). The red arrows denote positive withdrawals, and the blue arrows denote the chosen positive flow directions. regulate flow or pressure when active and require mode dependent residuals. Related complementarity-based optimization formulations appear for PRV localization and pressure management [36], [37]. All of these valves can be considered within the framework in this paper. … view at source ↗
Figure 2
Figure 2. Figure 2: A fixed mode hydraulic DAE can be represented locally by an algebraic view at source ↗
Figure 3
Figure 3. Figure 3: Effect of the smoothing width τs on the quintic transition. The top figure shows control trajectories for several τs values and the bottom figure shows the corresponding control rate du/dt inside the window. over a narrow time window. Let 0=t0<t1<···<tK be the control update times, and let u¯k denote the control vector prescribed on [tk,tk+1). Choose a smoothing width τs such that 0<τs<tk+1−tk for every k.… view at source ↗
Figure 4
Figure 4. Figure 4: High-level description of Theorem 4. linearized algebraic constraints, provided every hydraulically active subnetwork is anchored to a reservoir. Theorem 4 (Laplacian-parameterized local stability analysis). Under Assumption 2, define the local slope matrix K=diag(κe). Then K ≻0 and the inverse conductance matrix W =K−1 ≻0 forms the weighted graph Laplacian Lw =NWN⊤ (13) representing the linearized hydraul… view at source ↗
Figure 5
Figure 5. Figure 5: Summary of the paper’s technical contributions. view at source ↗
Figure 6
Figure 6. Figure 6: DAE-EPANET comparison for the Net3 one hour fixed control run (Theorem 1). (a) Trajectory overlay for a representative junction head and supported link flow (DAE solid and EPANET dashed are nearly identical). (b) Component-wise relative error envelope across heads (blue) and supported flows (red). The solid line is the median, and the shaded band is the 10–90% percentile range. (c) Total state error norms … view at source ↗
Figure 8
Figure 8. Figure 8: DAE-EPANET comparison for the 24-hour smoothed control run for view at source ↗
Figure 10
Figure 10. Figure 10: Numerical assessment of Procedures 1 and 2, overlaying Net3 (blue, circle markers) and Anytown (red, square markers). (a) Residual from the left hand side of (18), namely ∥Ah(1 + δ) −Ah(1) − (∂Ah/∂θ)|θ=1δ∥2 (solid), compared with the true matrix shift ∥∆Ah∥2 =∥Ah(1+δ)−Ah(1)∥2 (dotted). The residual stays well below the true shift across the swept range, confirming the bound in Theorem 5. (b) Top 6 pipe ro… view at source ↗
Figure 11
Figure 11. Figure 11: Aggressive operating point margin study for view at source ↗
Figure 12
Figure 12. Figure 12: Finite-horizon pump authority for the same view at source ↗
Figure 13
Figure 13. Figure 13: Toy network based on three nodes water system. The GPV is shown for view at source ↗
Figure 14
Figure 14. Figure 14: Topologies of the five networks used in the case studies. The layouts are topology-preserving but not to scale. Below each network the size tuple view at source ↗
read the original abstract

This paper presents a comprehensive control-theoretic analysis of water distribution network (WDN) hydraulics. Starting from a general nonlinear differential algebraic equation (DAE) model of WDNs with arbitrary topology and network components (valves and pumps), we investigate three main questions. First, we study local well-posedness of the network dynamics and characterize the loss of differentiability introduced by pump and valve switching. Second, we introduce regularization methods that smooth flow and pressure trajectories under changing controls. Third, we establish error bounds for DAE linearization, local stability, and finite-horizon controllability, and quantify how network-induced parametric uncertainty impacts these properties. We demonstrate that the developed smoothed DAE models produce trajectories closely matching EPANET, a widely used WDN simulator, for various benchmark networks. The case studies also show that the WDN DAE exposes energy dissipation through a weighted Laplacian, ranks pipes by operating point sensitivity, and reveals that aggressive demand variation changes stability and controllability margins without eliminating local stability or pump authority. The developed theoretical foundations enable network analysis, mitigation strategies, and system design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a system-theoretic framework for water distribution networks modeled as nonsmooth DAEs with arbitrary topology and components. It analyzes local well-posedness and loss of differentiability at pump/valve switches, introduces smoothing regularization to restore differentiability, derives error bounds for DAE linearization, local stability, and finite-horizon controllability, quantifies the effect of network-induced parametric uncertainty on these properties, and validates that the smoothed models produce trajectories closely matching EPANET on benchmark networks. Additional insights include energy dissipation via a weighted Laplacian, pipe sensitivity ranking, and the impact of aggressive demand variation on stability/controllability margins.

Significance. If the error bounds and stability/controllability results transfer from the smoothed DAE to the original nonsmooth system, the work supplies a rigorous foundation for stability analysis and control design in WDNs under uncertainty and demand fluctuations. The pairing of theoretical derivations with external EPANET comparisons and the absence of free parameters in the core bounds are strengths that enhance applicability to infrastructure systems.

major comments (2)
  1. [error bounds and regularization sections] In the sections deriving error bounds for linearization, stability, and controllability (following the regularization): the bounds are established for the smoothed DAE, yet no explicit uniform estimate (in appropriate norms) is given for the trajectory or Jacobian distance between the smoothed and original nonsmooth DAE near switching surfaces. Without this, it is unclear whether the regularization perturbation dominates the linearization error term when assessing controllability Gramian or stability margins, undermining transfer of the quantitative claims to the physical network.
  2. [case studies and validation] Validation against EPANET: the claim of 'closely matching' trajectories for various benchmark networks is stated without reported quantitative error metrics (e.g., maximum deviation, L2 norms over time, or protocol details such as time-step alignment and switching event handling). This weakens support for the assertion that the smoothed models preserve essential properties of the original system.
minor comments (2)
  1. [analysis of energy dissipation] The definition and properties of the weighted Laplacian used to expose energy dissipation should be stated explicitly with an equation reference rather than assumed from context.
  2. [regularization methods] Notation for the smoothing parameter and its relation to switching thresholds could be made more uniform across the well-posedness and error-bound derivations to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and positive review, which highlights both the strengths and areas for improvement in our manuscript. We address each major comment below and will make revisions to strengthen the transfer of theoretical results and the validation evidence.

read point-by-point responses

  1. Referee: [error bounds and regularization sections] In the sections deriving error bounds for linearization, stability, and controllability (following the regularization): the bounds are established for the smoothed DAE, yet no explicit uniform estimate (in appropriate norms) is given for the trajectory or Jacobian distance between the smoothed and original nonsmooth DAE near switching surfaces. Without this, it is unclear whether the regularization perturbation dominates the linearization error term when assessing controllability Gramian or stability margins, undermining transfer of the quantitative claims to the physical network.

    Authors: We agree that an explicit uniform estimate (in suitable norms) for the trajectory and Jacobian distance between the smoothed and original nonsmooth DAE near switching surfaces is needed to rigorously justify transferring the quantitative bounds. In the revised manuscript we will add a dedicated subsection deriving such an estimate. Under the standing local Lipschitz continuity of the DAE right-hand side away from switches and the O(ε) approximation property of the chosen regularization (with ε the smoothing parameter), we will prove that the trajectory deviation remains bounded by Cε (C independent of the linearization) in a neighborhood of any switching surface. This separates the regularization perturbation from the linearization error and allows the stability and controllability margins to be transferred to the original system for sufficiently small ε. revision: yes

  2. Referee: [case studies and validation] Validation against EPANET: the claim of 'closely matching' trajectories for various benchmark networks is stated without reported quantitative error metrics (e.g., maximum deviation, L2 norms over time, or protocol details such as time-step alignment and switching event handling). This weakens support for the assertion that the smoothed models preserve essential properties of the original system.

    Authors: We concur that quantitative error metrics and protocol details are required to substantiate the 'closely matching' claim. In the revised manuscript we will expand the case-studies section with explicit quantitative comparisons: tables reporting maximum absolute deviations and time-integrated L2 norms for nodal pressures and pipe flows on each benchmark network; specification of the time-step sizes employed by the smoothed DAE integrator and by EPANET; description of time-grid alignment; and explicit handling of switching events (e.g., detection thresholds and interpolation). These additions will provide concrete, reproducible evidence that the smoothed models preserve the essential dynamics of the original system. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivations remain independent of inputs

full rationale

The paper begins with a general nonlinear DAE model of arbitrary WDN topology, characterizes loss of differentiability at switches, introduces regularization to obtain a smoothed DAE, derives error bounds and stability/controllability results for that smoothed system, and validates trajectories against the external EPANET simulator on benchmark networks. No quoted step reduces a claimed prediction, bound, or property to a fitted parameter or self-referential definition by construction; external simulator matching supplies independent evidence rather than internal fitting. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only access yields no identifiable free parameters, axioms, or invented entities; full text would be required to audit the DAE formulation, regularization functions, or uncertainty models.

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Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    Environmental Protection Agency,Drinking W ater Infrastructure Needs Survey and Assessment: Seventh Report to Congress, U.S

    U.S. Environmental Protection Agency,Drinking W ater Infrastructure Needs Survey and Assessment: Seventh Report to Congress, U.S. Environmental Protection Agency, Washington, DC, 2023, ePA 810-R-23-001

    work page 2023

  2. [2]

    Water distribution systems analysis,

    U. Shamir and C. D. D. Howard, “Water distribution systems analysis,”Journal of the Hydraulics Division, ASCE, vol. 94, no. 1, pp. 219–234, 1968

    work page 1968

  3. [3]

    A gradient algorithm for the analysis of pipe networks,

    E. Todini and S. Pilati, “A gradient algorithm for the analysis of pipe networks,” inComputer Applications in W ater Supply: V olume 1 – Systems Analysis and Simulation. London, UK: John Wiley & Sons, 1988, pp. 1–20

    work page 1988

  4. [4]

    T. M. Walski, D. V . Chase, D. A. Savic, W. Grayman, S. Beckwith, and E. Koelle,Advanced W ater Distribution Modeling and Management. Waterbury, CT: Haestad Press, 2003

    work page 2003

  5. [5]

    Rossman, H

    L. Rossman, H. Woo, M. Tryby, F. Shang, R. Janke, and T. Haxton,EP ANET 2.2 User Manual, U.S. Environmental Protection Agency, Washington, DC, 2020, ePA/600/R-20/133

    work page 2020

  6. [6]

    Epyt: An EPANET-Python toolkit for smart water network simulations,

    M. S. Kyriakou, M. Demetriades, S. G. Vrachimis, D. G. Eliades, and M. M. Polycarpou, “Epyt: An EPANET-Python toolkit for smart water network simulations,”Journal of Open Source Software, vol. 8, no. 92, p. 5947, 2023

    work page 2023

  7. [7]

    A software framework for assessing the resilience of drinking water systems to disasters with an example earthquake case study,

    K. A. Klise, M. L. Bynum, D. Moriarty, and R. Murray, “A software framework for assessing the resilience of drinking water systems to disasters with an example earthquake case study,”Environmental Modelling & Software, vol. 95, pp. 420–431, 2017

    work page 2017

  8. [8]

    WaterModels.jl: An open source Julia/JuMP package for steady state water network optimization,

    B. Tasseff, R. Bent, and C. Coffrin, “WaterModels.jl: An open source Julia/JuMP package for steady state water network optimization,” https://github.com/lanl-ansi/WaterModels.jl, 2019

    work page 2019

  9. [9]

    A new derivative-free linear approximation for solving the network water flow problem with convergence guarantees,

    S. Wang, A. F. Taha, L. Sela, M. H. Giacomoni, and N. Gatsis, “A new derivative-free linear approximation for solving the network water flow problem with convergence guarantees,”W ater Resources Research, vol. 56, no. 3, pp. no–no, 2020

    work page 2020

  10. [10]

    M. H. Chaudhry,Applied Hydraulic Transients, 3rd ed. New Y ork, NY: Springer, 2014

    work page 2014

  11. [11]

    E. B. Wylie, V . L. Streeter, and L. Suo,Fluid Transients in Systems. Englewood Cliffs, NJ: Prentice Hall, 1993

    work page 1993

  12. [12]

    A review of water hammer theory and practice,

    M. S. Ghidaoui, M. Zhao, D. A. McInnis, and D. H. Axworthy, “A review of water hammer theory and practice,”Applied Mechanics Reviews, vol. 58, no. 1–6, pp. 49–75, 2005

    work page 2005

  13. [13]

    Improved rigid water column formulation for simulating slow transients and controlled operations,

    J. Nault and B. Karney, “Improved rigid water column formulation for simulating slow transients and controlled operations,”Journal of Hydraulic Engineering, vol. 142, no. 4, p. 04015082, 2016

    work page 2016

  14. [14]

    Parameterization of offline and online hydraulic simulation models,

    J. Deuerlein, O. Piller, and A. Simpson, “Parameterization of offline and online hydraulic simulation models,” inProc. Procedia Engineering, vol. 119. Elsevier, 2015, pp. 559–568

    work page 2015

  15. [15]

    A unified framework for pressure driven network analysis,

    O. Piller and J. van Zyl, “A unified framework for pressure driven network analysis,” inProc. CCWI, 2007

    work page 2007

  16. [16]

    Port hamiltonian based control of wa- ter distribution networks,

    R. Perryman, C. Taylor, and A. Ramsey, “Port hamiltonian based control of wa- ter distribution networks,”Systems & Control Letters, vol. 162, p. 105197, 2022

    work page 2022

  17. [17]

    Port hamiltonian models for flow of incompressible fluids in rigid pipelines with faults,

    L. Torres and G. Besancon, “Port hamiltonian models for flow of incompressible fluids in rigid pipelines with faults,” inProc. IEEE Conf. Decision and Control (CDC). IEEE, 2019, pp. 8244–8249

    work page 2019

  18. [18]

    Applications of graph spectral techniques to water distribution network management,

    A. Di Nardo, M. Di Natale, G. Santonastaso, V . Tzatchkov, and V . Alcocer- Y amanaka, “Applications of graph spectral techniques to water distribution network management,”W ater, vol. 10, no. 1, p. 45, 2018

    work page 2018

  19. [19]

    Complex network analysis of water distribution systems,

    A. Y azdani and P . Jeffrey, “Complex network analysis of water distribution systems,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 21, no. 1, p. 016111, 2011, also available as arXiv:1008.1770

    work page arXiv 2011

  20. [20]

    Application of predictive control strategies to the management of complex networks in the urban water cycle,

    C. Ocampo-Mart´ınez, V . Puig, G. Cembrano, and J. Quevedo, “Application of predictive control strategies to the management of complex networks in the urban water cycle,”IEEE Control Systems Magazine, vol. 33, no. 1, pp. 15–41, 2013

    work page 2013

  21. [21]

    Control of tree water networks: A geometric programming approach,

    L. Sela Perelman and S. Amin, “Control of tree water networks: A geometric programming approach,”W ater Resources Research, vol. 51, no. 10, pp. 8409–8430, 2015

    work page 2015

  22. [22]

    Receding horizon con- trol for drinking water networks: The case for geometric programming,

    S. Wang, A. F. Taha, N. Gatsis, and M. H. Giacomoni, “Receding horizon con- trol for drinking water networks: The case for geometric programming,”IEEE Transactions on Control of Network Systems, vol. 7, no. 3, pp. 1151–1163, 2020

    work page 2020

  23. [23]

    Non-linear economic model predictive control of water distribution networks,

    Y . Wang, V . Puig, and G. Cembrano, “Non-linear economic model predictive control of water distribution networks,”Journal of Process Control, vol. 56, pp. 23–34, 2017

    work page 2017

  24. [24]

    Geometric programming-based control for nonlinear, DAE-constrained water distribution networks,

    S. Wang, A. F. Taha, N. Gatsis, and M. H. Giacomoni, “Geometric programming-based control for nonlinear, DAE-constrained water distribution networks,” inProceedings of the American Control Conference, Philadelphia, PA, USA, 2019, pp. 1470–1475

    work page 2019

  25. [25]

    Optimal scheduling of water distribution systems,

    M. K. Singh and V . Kekatos, “Optimal scheduling of water distribution systems,”IEEE Transactions on Control of Network Systems, vol. 7, no. 2, pp. 711–723, 2020

    work page 2020

  26. [26]

    GPU- accelerated stochastic predictive control of drinking water networks,

    A. K. Sampathirao, P . Sopasakis, A. Bemporad, and P . P . Patrinos, “GPU- accelerated stochastic predictive control of drinking water networks,”IEEE Transactions on Control Systems T echnology, vol. 26, no. 2, pp. 551–562, 2018

    work page 2018

  27. [27]

    Optimal pump control for water distribution networks via data-based distributional robustness,

    Y . Guo, S. Wang, A. F. Taha, and T. H. Summers, “Optimal pump control for water distribution networks via data-based distributional robustness,”IEEE Transactions on Control Systems T echnology, vol. 31, no. 1, pp. 114–129, 2023

    work page 2023

  28. [28]

    Real-time pressure control in water distribution networks: Stability guarantees via gain-scheduled internal model control,

    G. Galuppini, E. Creaco, and L. Magni, “Real-time pressure control in water distribution networks: Stability guarantees via gain-scheduled internal model control,”IEEE Transactions on Control Systems T echnology, vol. 32, no. 3, pp. 731–743, 2024

    work page 2024

  29. [29]

    Exploiting structural observability and graph colorability for optimal sensor placement in water distribution networks,

    J. J. V an Gemert, V . Breschi, D. R. Yntema, K. J. Keesman, and M. Lazar, “Exploiting structural observability and graph colorability for optimal sensor placement in water distribution networks,” in2025 European Control Conference (ECC). IEEE, 2025, pp. 546–551

    work page 2025

  30. [30]

    Local sensitivity of pressure-driven modeling and demand-driven modeling steady-state solutions to variations in parameters,

    O. Piller, S. Elhay, J. Deuerlein, and A. Simpson, “Local sensitivity of pressure-driven modeling and demand-driven modeling steady-state solutions to variations in parameters,”Journal of W ater Resources Planning and Management, vol. 143, no. 2, p. 04016074, 2017

    work page 2017

  31. [31]

    P . R. Bhave and R. Gupta,Analysis of water distribution networks. Alpha Science Int’l Ltd., 2006

    work page 2006

  32. [32]

    T. M. Walski, D. V . Chase, and D. A. Savic,W ater Distribution Modeling. Waterbury, CT: Haestad Press, 2001

    work page 2001

  33. [33]

    Modelling techniques in dynamic control of water distribution systems,

    B. Coulbeck and M. J. H. Sterling, “Modelling techniques in dynamic control of water distribution systems,”Measurement and Control, vol. 11, no. 10, pp. 385–389, 1978

    work page 1978

  34. [34]

    Dynamic simulation of water distribution systems,

    B. Coulbeck, “Dynamic simulation of water distribution systems,”Mathematics and Computers in Simulation, vol. 22, no. 3, pp. 222–230, 1980

    work page 1980

  35. [35]

    Dynamical stability of water distribution networks,

    N. Masuda and F. Meng, “Dynamical stability of water distribution networks,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 475, no. 2230, p. 20190291, 2019

    work page 2019

  36. [36]

    Optimal localization of pressure reducing valves in water distribution systems by a reformulation approach,

    P . Dai and P . Li, “Optimal localization of pressure reducing valves in water distribution systems by a reformulation approach,”W ater Resources Management, vol. 28, no. 10, pp. 3057–3074, 2014

    work page 2014

  37. [37]

    A new mathematical program with com- plementarity constraints for optimal localization of pressure reducing valves in water distribution systems,

    L. Hien, B. De Schutter, and F. Logist, “A new mathematical program with com- plementarity constraints for optimal localization of pressure reducing valves in water distribution systems,”Applied W ater Science, vol. 11, no. 10, p. 166, 2021

    work page 2021

  38. [38]

    An efficient algorithm for solving piecewise- smooth dynamical systems,

    N. Guglielmi and E. Hairer, “An efficient algorithm for solving piecewise- smooth dynamical systems,”Numerical Algorithms, vol. 89, pp. 1311–1334, 2022

    work page 2022

  39. [39]

    Extended-period analysis with a transient model,

    Y . R. Filion and B. W. Karney, “Extended-period analysis with a transient model,”Journal of Hydraulic Engineering, vol. 128, no. 6, pp. 616–624, 2002

    work page 2002

  40. [40]

    Quadratic head loss approximations for optimisation problems in water supply networks,

    F. Pecci, E. Abraham, and I. Stoianov, “Quadratic head loss approximations for optimisation problems in water supply networks,”Journal of Hydroinformatics, vol. 19, no. 4, pp. 493–506, 2017

    work page 2017

  41. [41]

    Real-time quintic hermite interpolation for robot trajectory execution,

    M. Lind, “Real-time quintic hermite interpolation for robot trajectory execution,” P eerJ Computer Science, vol. 6, p. e304, 2020

    work page 2020

  42. [42]

    EP ANET -Benchmarks: A collection of several benchmark water networks,

    KIOS Research and Innovation Center of Excellence, “EP ANET -Benchmarks: A collection of several benchmark water networks,” https://github.com/ KIOS-Research/EPANET-Benchmarks, 2026, accessed April 30, 2026

    work page 2026

  43. [43]

    The impacts of spatially variable demand patterns on water distribution system design and operation,

    K. Diao, R. Sitzenfrei, and W. Rauch, “The impacts of spatially variable demand patterns on water distribution system design and operation,”W ater, vol. 11, no. 3, p. 567, 2019

    work page 2019

  44. [44]

    Calibration of design models for leakage management of water distribution networks,

    L. Berardi and O. Giustolisi, “Calibration of design models for leakage management of water distribution networks,”W ater Resources Management, vol. 35, pp. 2537–2551, 2021. 13

    work page 2021

  45. [45]

    Teschl,Ordinary Differential Equations and Dynamical Systems, ser

    G. Teschl,Ordinary Differential Equations and Dynamical Systems, ser. Graduate Studies in Mathematics. Providence, RI: American Mathematical Society, 2012, vol. 140

    work page 2012

  46. [46]

    Kunkel and V

    P . Kunkel and V . Mehrmann,Differential-Algebraic Equations: Analysis and Numerical Solution, ser. EMS Textbooks in Mathematics. Z ¨urich, Switzerland: European Mathematical Society, 2006, vol. 2

    work page 2006

  47. [47]

    Hairer and G

    E. Hairer and G. Wanner,Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd ed., ser. Springer Series in Computational Mathematics. Berlin, Germany: Springer, 1996, vol. 14

    work page 1996

  48. [48]

    Godsil and G

    C. Godsil and G. Royle,Algebraic Graph Theory, ser. Graduate Texts in Mathematics. New Y ork: Springer, 2001, vol. 207

    work page 2001

  49. [49]

    H. K. Khalil,Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002

    work page 2002

  50. [50]

    R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed. Cambridge, UK: Cambridge University Press, 2013. APPENDIXA NDAE MODEL FOR ASIMPLENETWORK This appendix writes the full DAE of Section III for the Threenodesbenchmark shown in Fig. 13. The goal is not to introduce a new model, but to make the general construction concrete by displaying the incidence blo...

    work page 2013