# Water Distribution in a Socio-Technical System: Resilience Assessment for Critical Events Causing Demand Relocation

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## Abstract

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## 1. Introduction

#### 1.1. Research Questions

- (i)
- What effect does the size of evacuated area have?
- (ii)
- With which sheltering scheme can demand be best fulfilled?
- (iii)
- Which part of the city is most vulnerable?
- (iv)
- Which areas are best suited for accommodating relocated demand?
- (v)
- How do the results of the resilience assessment compare to graph-theoretical resilience metrics?

#### 1.2. Related Work

“A resilient technical system guarantees a predetermined minimum of functional performance even in the event of disturbances or failure of system components, and a subsequent possibility of recovering at least the setpoint function. Resilience can be increased by adjusting the system state via monitoring, responding, learning and/or anticipating, as well as by systematically designing the system topology.”[14]

## 2. Materials and Methods

#### 2.1. Modelling WDSs

#### 2.2. Modelling Critical Events

#### 2.3. Sheltering Schemes

- centralized shelteringIn the centralized sheltering scheme, the set of shelter nodes is a singleton consisting of precisely one node, n, see Figure 1:$${S}_{\mathrm{cen}}=\{\exists n:n\in N\backslash E\}$$The required demand from the evacuation area E is summed for each time step t and transferred onto a single pre-defined central shelter node from the set of shelter nodes, $c\in {S}_{\mathrm{cen}}$:$${D}_{c,t}=\sum _{n\in E}{D}_{n,t}.$$While in reality, multiple sheltering locations are often dedicated to accommodate evacuated population, selecting a single shelter node represents the most critical case. Hence, centralized schemes with more than one shelter node have not been considered in the scope of this work but might be the subject of future research.
- near-at-hand shelteringIn the near-at-hand sheltering, demand from the evacuated area is distributed onto the immediate annulus around the evacuation area, with the width $0.5\phantom{\rule{0.166667em}{0ex}}r$ (Figure 1). Shelter nodes can then be expressed as a set of nodes ${S}_{\mathrm{nah}}$:$${S}_{\mathrm{nah}}=\{n\in N:r<|n\phantom{\rule{0.222222em}{0ex}}o|\le 3r/2\}.$$A probability p for each node in ${S}_{\mathrm{nah}}$ is generated using the truncated normal distribution function from the Python package scipy so that$$\sum _{n\in {S}_{\mathrm{nah}}}{p}_{n}=1.$$The required demand on a shelter node after demand relocation is then$${D}_{n,t}={p}_{n}\sum _{{n}^{\prime}\in E}{D}_{{n}^{\prime},t}\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{1.em}{0ex}}n\in {S}_{\mathrm{nah}}.$$
- diffuse shelteringDuring the diffuse sheltering, demand from the evacuated area is distributed onto the remaining nodes in the city (Figure 1). The set of shelter nodes, ${S}_{\mathrm{dif}}$, can thus be expressed as$${S}_{\mathrm{dif}}=N\backslash E$$

#### 2.4. Resilience Assessment

#### 2.5. Evaluation of Under-Serviced Nodes

#### 2.6. Basic Graph-Based Resilience Metrics

`single_source_dijkstra`from the Network X package [44]. The weighing was performed with the head loss $\Delta H$ averaged over the time of the simulation on each link in the WDS simulated under normal operating conditions (without demand relocation).

## 3. Results

#### 3.1. Effect of Evacuation Radius

#### 3.2. Effect of Sheltering Scheme

#### 3.3. Vulnerable Areas

#### 3.4. Identifying Shelter Nodes

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

WDS | Water Distribution System |

UN | United Nations |

USN | under-serviced nodes |

WNTR | Water Network Tool for Resilience [35] |

WSA | Water Service Availability |

## References

- Schott, D. Europäische Urbanisierung (1000–2000): Eine umwelthistorische Einführung; Böhlau: Köln, Germany, 2014. [Google Scholar]
- Melosi, M. The Sanitary City: Urban Infrastructure in America from Colonial Times to the Present; Creating the North American Landscape; Johns Hopkins University Press: Baltimore, MD, USA, 2000. [Google Scholar]
- Engels, J.I.; Schenk, G.J. Macht der Infrastrukturen—Infrastrukturen der Macht. Überlegungen zu einem Forschungsfeld. In Wasserinfrastrukturen und Macht von der Antike bis zur Gegenwart; Historische Zeitschrift, Beiheft; Förster, B., Bauch, M., Eds.; Oldenbourg: München, Germany, 2014; Volume 63, pp. 22–58. [Google Scholar]
- Moss, T. Remaking Berlin: A History of the City through Infrastructure, 1920–2020; Infrastructures, MIT Press: Cambridge, MA, USA; London, UK, 2020. [Google Scholar]
- Stör, C.; Rosendorff, K. Frankfurt: Weltkriegsbombe im Nordend Gesprengt—Absperrungen Aufgehoben | Frankfurt. Available online: https://www.fr.de/frankfurt/frankfurt-am-main-nordend-bombe-evakuierung-stadtteil-news-90652435.html (accessed on 8 June 2021).
- Beirut Port Explosion: UNICEF Swiftly Restores Water to Damaged Homes, Schools and Hospitals | UNICEF Lebanon. 2020. Available online: https://www.unicef.org/lebanon/stories/beirut-port-explosion-unicef-swiftly-restores-water-damaged-homes-schools-and-hospitals (accessed on 12 June 2021).
- Texas Weather: President Biden Declares Major Disaster—BBC News. Available online: https://www.bbc.com/news/world-us-canada-56129833 (accessed on 22 July 2021).
- Engels, J.I. (Ed.) Key Concepts for Critical Infrastructure Research; Springer Fachmedien Wiesbaden: Wiesbaden, Germany, 2018. [Google Scholar] [CrossRef]
- United Nations International Strategy for Disaster Reduction. 2009 UNISDR Terminology on Disaster Risk Reduction. Available online: https://www.unisdr.org/files/7817_UNISDRTerminologyEnglish.pdf (accessed on 27 July 2021).
- Holling, C.S. Resilience and Stability of Ecological Systems. Annu. Rev. Ecol. Syst.
**1973**, 4, 1–23. [Google Scholar] [CrossRef][Green Version] - Hollnagel, E. (Ed.) Resilience Engineering: Concepts and Precepts; Ashgate: Farnham, UK, 2010. [Google Scholar]
- Hollnagel, E. Resilience Engineering in Practice: A Guidebook; Ashgate Studies in Resilience Engineering, Ashgate: Farnham, UK; Burlington, VT, USA, 2011. [Google Scholar]
- Scharte, B. Educating engineers for resilience. CSS Policy Perspect.
**2019**, 7, 63. [Google Scholar] [CrossRef] - Altherr, L.C.; Brötz, N.; Dietrich, I.; Gally, T.; Geßner, F.; Kloberdanz, H.; Leise, P.; Pelz, P.F.; Schlemmer, P.D.; Schmitt, A. Resilience in Mechanical Engineering—A Concept for Controlling Uncertainty during Design, Production and Usage Phase of Load-Carrying Structures. Appl. Mech. Mater.
**2018**, 885, 187–198. [Google Scholar] [CrossRef][Green Version] - Shin, S.; Lee, S.; Judi, D.; Parvania, M.; Goharian, E.; McPherson, T.; Burian, S. A Systematic Review of Quantitative Resilience Measures for Water Infrastructure Systems. Water
**2018**, 10, 164. [Google Scholar] [CrossRef][Green Version] - Ulusoy, A.J.; Stoianov, I.; Chazerain, A. Hydraulically informed graph theoretic measure of link criticality for the resilience analysis of water distribution networks. Appl. Netw. Sci.
**2018**, 3, 31. [Google Scholar] [CrossRef] - Shuang, Q.; Liu, H.J.; Porse, E. Review of the Quantitative Resilience Methods in Water Distribution Networks. Water
**2019**, 11, 1189. [Google Scholar] [CrossRef][Green Version] - Liu, W.; Song, Z. Review of studies on the resilience of urban critical infrastructure networks. Reliab. Eng. Syst. Saf.
**2020**, 193, 106617. [Google Scholar] [CrossRef] - Lorenz, I.S.; Pelz, P.F. Optimal Resilience Enhancement of Water Distribution Systems. Water
**2020**, 12, 2602. [Google Scholar] [CrossRef] - Diao, K.; Sweetapple, C.; Farmani, R.; Fu, G.; Ward, S.; Butler, D. Global resilience analysis of water distribution systems. Water Res.
**2016**, 106, 383–393. [Google Scholar] [CrossRef][Green Version] - Lestakova, M.; Logan, K.; Lorenz, I.; Pietsch, M.; Friesen, J.; Steinke, F.; Pelz, P.F. Towards a Common Ontology for Investigating Resilience of Interdependent Urban Systems. In Proceedings of the Joint International Resilience Conference, JIRC2020, Online, 23–27 November 2020; Comes, T., Hölscher, C., Eds.; University of Twente: Twente, The Netherlands, 2020; pp. 101–104. [Google Scholar]
- Herrera, M.; Abraham, E.; Stoianov, I. A Graph-Theoretic Framework for Assessing the Resilience of Sectorised Water Distribution Networks. Water Resour. Manag.
**2016**, 30, 1685–1699. [Google Scholar] [CrossRef][Green Version] - Meng, F.; Fu, G.; Farmani, R.; Sweetapple, C.; Butler, D. Topological attributes of network resilience: A study in water distribution systems. Water Res.
**2018**, 143, 376–386. [Google Scholar] [CrossRef] - Cassottana, B.; Aydin, N.Y.; Tang, L.C. Quantitative Assessment of System Response during Disruptions: An Application to Water Distribution Systems. J. Water Resour. Plan. Manag.
**2021**, 147, 04021002. [Google Scholar] [CrossRef] - Todini, E. Looped water distribution networks design using a resilience index based heuristic approach. Urban Water
**2000**, 2, 115–122. [Google Scholar] [CrossRef] - Creaco, E.; Franchini, M.; Todini, E. Generalized Resilience and Failure Indices for Use with Pressure-Driven Modeling and Leakage. J. Water Resour. Plan. Manag.
**2016**, 142, 04016019. [Google Scholar] [CrossRef] - Prasad, T.D.; Park, N.S. Multiobjective Genetic Algorithms for Design of Water Distribution Networks. J. Water Resour. Plan. Manag.
**2004**, 130, 73–82. [Google Scholar] [CrossRef] - Hashimoto, T.; Stedinger, J.R.; Loucks, D.P. Reliability, resiliency, and vulnerability criteria for water resource system performance evaluation. Water Resour. Res.
**1982**, 18, 14–20. [Google Scholar] [CrossRef][Green Version] - Kjeldsen, T.R.; Rosbjerg, D. Choice of reliability, resilience and vulnerability estimators for risk assessments of water resources systems/Choix d’estimateurs de fiabilité, de résilience et de vulnérabilité pour les analyses de risque de systèmes de ressources en eau. Hydrol. Sci. J.
**2004**, 49, 755–767. [Google Scholar] [CrossRef] - Zhuang, B.; Lansey, K.; Kang, D. Resilience/Availability Analysis of Municipal Water Distribution System Incorporating Adaptive Pump Operation. J. Hydraul. Eng.
**2013**, 139, 527–537. [Google Scholar] [CrossRef] - Revelli, R.; Ridolfi, L. Fuzzy Approach for Analysis of Pipe Networks. J. Hydraul. Eng.
**2002**, 128, 93–101. [Google Scholar] [CrossRef] - Seifollahi-Aghmiuni, S.; Bozorg Haddad, O.; Mariño, M.A. Water Distribution Network Risk Analysis Under Simultaneous Consumption and Roughness Uncertainties. Water Resour. Manag.
**2013**, 27, 2595–2610. [Google Scholar] [CrossRef] - Baños, R.; Reca, J.; Martínez, J.; Gil, C.; Márquez, A.L. Resilience Indexes for Water Distribution Network Design: A Performance Analysis Under Demand Uncertainty. Water Resour. Manag.
**2011**, 25, 2351–2366. [Google Scholar] [CrossRef] - Lu, L.; Wang, X.; Ouyang, Y.; Roningen, J.; Myers, N.; Calfas, G. Vulnerability of Interdependent Urban Infrastructure Networks: Equilibrium after Failure Propagation and Cascading Impacts. Comput. Aided Civ. Infrastruct. Eng.
**2018**, 33, 300–315. [Google Scholar] [CrossRef] - Klise, K.A.; Bynum, M.; Moriarty, D.; Murray, R. A software framework for assessing the resilience of drinking water systems to disasters with an example earthquake case study. Environ. Model. Softw.
**2017**, 95, 420–431. [Google Scholar] [CrossRef] - Weiler, H.C. Trinkwasser im Katastrophengebiet. Aufbereitung und Verteilung. ZB Ziviler Bevölkerungsschutz
**1965**, 3, 16–23. [Google Scholar] - Weiler, H.C. Notrufe aus der Wasserwüste: „Schickt Trinkwasser!“: Wie funktioniert die Versorgung in Notstandsgebieten? ZB Ziviler Bevölkerungsschutz
**1966**, 4, 2–6. [Google Scholar] - Gehrmann, K.H. Trinkwassernotversorgung—Netzunabhängig. Bericht über die Erprobung eines Tiefsaugverfahrens zur Förderung von Trinkwasser aus Brunnen zur Trinkwassernotversorgung. Zs-Magazin
**1974**, 6, 18–21. [Google Scholar] - Such, W.; Keil, R. Aktueller Stand der Vorsorgemaßnahmen zur Trinkwasser-Notversorgung nach dem Wassersicherstellungsgesetz. Aufbau einer von den friedensmäßig genutzten Anlagen der zentralen Wasserversor-gung unabhängigen Trinkwasser-Notversorgung steht im Vordergrund. Zivilschutz Mag.
**1987**, 7/8, 35–40. [Google Scholar] - World Bank Group; European Union; United Nations. Beirut Rapid Damage and Needs Assessment. Available online: https://openknowledge.worldbank.org/handle/10986/34401 (accessed on 27 July 2021).
- Overview—WNTR 0.3.1 Documentation. 2019. Available online: https://wntr.readthedocs.io/en/latest/overview.html (accessed on 12 June 2021).
- Hagberg, A.A.; Schult, D.A.; Swart, P.J. Exploring Network Structure, Dynamics, and Function using NetworkX. In Proceedings of the 7th Python in Science Conference, Pasadena, CA, USA, 19–24 August 2008; Varoquaux, G., Vaught, T., Millman, J., Eds.; SciPy: Austin, TX, USA, 2008; pp. 11–15. [Google Scholar]
- Networkx.Algorithms.Centrality.betweenness_CENTRALITY—NetworkX 2.5 Documentation. 2020. Available online: https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.centrality.betweenness_centrality.html?highlight=betweenness#networkx.algorithms.centrality.betweenness_centrality (accessed on 12 June 2021).
- Networkx.Algorithms.shortest_paths.weighted.single_source_dijkstra—NetworkX 2.5 Documentation. 2020. Available online: https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.single_source_dijkstra.html#networkx.algorithms.shortest_paths.weighted.single_source_dijkstra (accessed on 12 June 2021).
- Weiner, U. Wir alle dürfen nicht nur nur Zuschauer sein: Obdachlose warten auf Hilfe. Hambg. Abendblatt
**1962**, 43, 3. Available online: https://www.abendblatt.de/archive/1962/pdf/19620220.pdf/ASV_HAB_19620220_HA_003.pdf (accessed on 28 July 2021). - Gatermann, F. Eine Welle der Fürsorge in allen Massenquartieren: Privatleute holten Tausende Obdachlose/ Kein Mangel an Helfern. Hambg. Abendblatt
**1962**, 42, 5. Available online: https://www.abendblatt.de/archive/1962/pdf/19620219.pdf/ASV_HAB_19620219_HA_005.pdf (accessed on 28 July 2021). - Reese, C. Katastrophengebiet Süderelbe jetzt für Evakuierte gesperrt: Scharfe Kontrollen an allen Zufahrtsstraßen. Hambg. Abendblatt
**1962**, 49, 9. Available online: https://www.abendblatt.de/archive/1962/pdf/19620226.pdf/ASV_HAB_19620226_HA_009.pdf (accessed on 28 July 2021). - Hambuger, bitte helft! Nehmt Obdachlose auf! Bisher 119 Tote geborgen. Hambg. Abendblatt
**1962**, 42, 1. Available online: https://www.abendblatt.de/archive/1962/pdf/19620219.pdf/ASV_HAB_19620219_HA_001.pdf (accessed on 28 July 2021). - Ley, R. (Ed.) Gespräche mit Zeitzeugen. In Die Nacht der Großen Flut; Ellert & Richter: Hamburg, Germany, 2006; pp. 72–133. [Google Scholar]
- Noch 1887 Evakuierte in Notlagern. Hambg. Abendblatt
**1962**, 56, 3. Available online: https://www.abendblatt.de/archive/1962/pdf/19620307.pdf/ASV_HAB_19620307_HA_003.pdf (accessed on 28 July 2021).

**Figure 1.**An overview of sheltering schemes (

**a**–

**c**) and a comparative depiction of the evacuation radii (

**d**). Nodes in the evacuated area (E) are portrayed in blue, shelter nodes are portrayed in yellow. Consumer nodes (black) are differentiated from non-consumer nodes (grey). (

**a**) centralized sheltering scheme, (

**b**) near-at-hand sheltering scheme, (

**c**) diffuse sheltering scheme.

**Figure 2.**Mean and min-max interval resilience index $\overline{\rho}$ for evacuation radii: (

**a**) centralized sheltering scheme, (

**b**) near-at-hand sheltering scheme, (

**c**) diffuse sheltering scheme.

**Figure 3.**Mean and standard deviation of number of under-serviced nodes (USN) $\overline{\eta}$ for all evacuation radii r: (

**a**) textitcentralized sheltering scheme, (

**b**) near-at-hand sheltering scheme, (

**c**) diffuse sheltering scheme.

**Figure 4.**Mean resilience index ${\overline{\rho}}^{\left(o\right)}$ for each node serving as the evacuation origin for all analyzed evacuation radii and sheltering schemes. (

**a**) centralized sheltering (

**b**) near-at-hand sheltering (

**c**) diffuse sheltering.

**Figure 5.**Mean resilience index ${\overline{\rho}}^{\left(s\right)}$ for each node serving as the shelter node in the centralized sheltering scheme for all analyzed evacuation radii.

**Figure 6.**Results of the graph-based resilience metrics. (

**a**) betweenness centrality, (

**b**) shortest hydraulic paths length.

Name | Magnitude |
---|---|

${r}_{1}$ | $1/4$ |

${r}_{2}$ | $1/2$ |

${r}_{3}$ | $3/4$ |

${r}_{4}$ | 1 |

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**MDPI and ACS Style**

Logan, K.T.; Leštáková, M.; Thiessen, N.; Engels, J.I.; Pelz, P.F.
Water Distribution in a Socio-Technical System: Resilience Assessment for Critical Events Causing Demand Relocation. *Water* **2021**, *13*, 2062.
https://doi.org/10.3390/w13152062

**AMA Style**

Logan KT, Leštáková M, Thiessen N, Engels JI, Pelz PF.
Water Distribution in a Socio-Technical System: Resilience Assessment for Critical Events Causing Demand Relocation. *Water*. 2021; 13(15):2062.
https://doi.org/10.3390/w13152062

**Chicago/Turabian Style**

Logan, Kevin T., Michaela Leštáková, Nadja Thiessen, Jens Ivo Engels, and Peter F. Pelz.
2021. "Water Distribution in a Socio-Technical System: Resilience Assessment for Critical Events Causing Demand Relocation" *Water* 13, no. 15: 2062.
https://doi.org/10.3390/w13152062