Abstract
This study aimed to examine the fractal properties of the optimal hydraulic gradient surface (OHGS), a geometrical body that describes the way in which the available energy should be spent within a water distribution network to ensure the calculation of a minimum capital cost design. For this purpose, multiple benchmark and Colombian systems were optimized and then analyzed to compute the fractal dimension of the OHGS and of the underlying structure of each network, which included the examination of randomly generated nonoptimal designs to recognize the differences in the fractal behavior of a least-cost and a more expensive solution. The results showed a dependency between the fractal properties of the OHGS and those of the topological structure, flow, and energy distribution inside the corresponding optimized network. Moreover, it was found that the degree of irregularity of the OHGS tended to be higher compared to a nonoptimal energy dissipation pattern. This suggests the applicability of the fractal analysis in optimization and operational improvement procedures.
Practical Applications
The design of a least-cost water distribution network, which refers to the process of calculating appropriate pipe diameters that ensure the fulfilment of water demands and reduce the required investment to a minimum, has been of interest due to its complexity and importance for the viability of water supply infrastructure projects. In this study, the inherent characteristics of a least-cost design were explored by involving fractal analysis, a novel technique that has been used to solve several problems in the management and operation of water supply systems. The results show that the hydraulic features of a minimum-cost water distribution network, analyzed by considering the behavior of pressure and water flow throughout the system, exhibit certain fractal properties that can be used to distinguish it from more expensive alternatives. These findings lead to the recommendation to use fractal analysis in prospective water distribution network planning and management applications, such as new design and operational improvement strategies. Furthermore, fractal analysis may be useful in identifying the proper hydraulic behavior of a water distribution network so as to maximize its reliability, water quality, and other desirable features that improve its performance and level of service.
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Data Availability Statement
Some or all data, models, and code generated or used during the study are available in a repository online in accordance with funder data retention policies. The available materials are EPANET data files, Excel data files, and Python scripts to generate figures and tables. The materials can be found in the following repository: https://doi.org/10.5281/zenodo.6634751.
Reproducible Results
Figs. 1–3, 6–12, and S1 –S40 and Tables 2, 3, and S15 –S19 can be reproduced by accessing the uploaded material in Jaramillo (2022). Arturo Rodriguez (Research Assistant, Master’s Student in Structural Engineering, Universidad de los Andes) ran all scripts and reproduced all tables and figures.
Acknowledgments
The authors would like to thank PAVCO Wavin for providing essential information for the development of the research work as well as for their constant support in investigation activities concerning water supply.
References
Abdalfattah, S. F., Y. M. Makeen, I. Abdulbariu, and H. A. Ayinla. 2019. “Basic gravity geophysical processing and display in MATLAB: Application on NE African Sudanese Red Sea Province.” J. Afr. Earth. Sci. 150 (Nov): 346–354. https://doi.org/10.1016/j.jafrearsci.2018.10.018.
Alperovits, E., and U. Shamir. 1977. “Design of optimal water distribution systems.” Water Resour. Res. 13 (6): 885–900. https://doi.org/10.1029/WR013i006p00885.
Barber, C. B., D. P. Dobkin, and H. Huhdanpaa. 1996. “The quickhull algorithm for convex hulls.” ACM Trans. Math. Software 22 (4): 469–483. https://doi.org/10.1145/235815.235821.
Bi, W., and G. C. Dandy. 2014. “Optimization of water distribution systems using online retrained metamodels.” J. Water Resour. Plan. Manage. 140 (11): 04014032. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000419.
Bishop-Taylor, R., S. Sagar, L. Lymburner, and R. J. Beaman. 2019. “Between the tides: Modelling the elevation of Australia´s exposed intertidal zone at continental scale.” Estuarine Coastal Shelf Sci. 223 (31): 115–128. https://doi.org/10.1016/j.ecss.2019.03.006.
Bisoi, A. K., and J. Mishra. 2001. “On calculation of fractal dimension of images.” Pattern Recognit. Lett. 22 (6–7): 631–637. https://doi.org/10.1016/S0167-8655(00)00132-X.
Bragalli, C., C. D’Ambrosio, J. Lee, A. Lodi, and P. Toth. 2012. “On the optimal design of water distribution networks: A practical MINLP approach.” Optim. Eng. 13 (2): 219–246. https://doi.org/10.1007/s11081-011-9141-7.
Clarke, K. C. 1986. “Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method.” Comput. Geosci. 12 (5): 713–722. https://doi.org/10.1016/0098-3004(86)90047-6.
Cunha, M., J. Marques, E. Creaco, and D. Savic. 2019. “A dynamic adaptive approach for water distribution network design.” J. Water Resour. Plan. Manage. 145 (7): 04019026. https://doi.org/10.1061/(ASCE)WR.1943-5452.0001085.
Cunha, M. C., and L. Ribeiro. 2004. “Tabu search algorithms for water network optimization.” Eur. J. Oper. Res. 157 (3): 746–758. https://doi.org/10.1016/S0377-2217(03)00242-X.
Cunha, M. C., and J. Sousa. 1999. “Water distribution network design optimization: Simulated annealing approach.” J. Water Resour. Plan. Manage. 125 (4): 215–221. https://doi.org/10.1061/(ASCE)0733-9496(1999)125:4(215).
Dandy, G. C., A. R. Simpson, and L. J. Murphy. 1996. “An improved genetic algorithm for pipe network optimization.” Water Resour. Res. 32 (2): 449–458. https://doi.org/10.1029/95WR02917.
Diao, K., D. Butler, and B. Ulanicki. 2017. “Fractality in water distribution networks.” In Proc., 15th Int. Computing and Control for the Water Industry (CCWI). Sheffield, England: Univ. of Sheffield.
Di Nardo, A., M. Di Natale, C. Giudicianni, R. Greco, and G. F. Santonastaso. 2017. “Complex network and fractal theory for the assessment of water distribution network resilience to pipe failures.” Water Sci. Technol. Water Supply 18 (3): 767–777. https://doi.org/10.2166/ws.2017.124.
Edgar, G. 2008. Measure, topology and fractal geometry. New York: Springer.
Erkal, B. G., and J. F. Hajjar. 2017. “Laser-based surface damage detection and quantification using predicted surface properties.” Autom. Constr. 83 (Oct): 285–302. https://doi.org/10.1016/j.autcon.2017.08.004.
Ezzeldin, R., B. Djebedjian, and T. Saafan. 2013. “Integer discrete particle swarm optimization of water distribution networks.” J. Pipeline Syst. Eng. 5 (1): 04013013. https://doi.org/10.1061/(ASCE)PS.1949-1204.0000154.
Falconer, K. 1990. Fractal geometry: Mathematical foundations and applications. West Sussex, England: Wiley.
Falconer, K. 1997. Techniques in fractal geometry. West Sussex, England: Wiley.
Farmani, R., R. Abadia, and D. Savic. 2007. “Optimum design and management of pressurized branched irrigation networks.” J. Irrig. Drain. Eng. 133 (6): 528–537. https://doi.org/10.1061/(ASCE)0733-9437(2007)133:6(528).
Featherstone, R. E., and K. K. El-Jumaily. 1983. “Optimal diameter selection for pipe networks.” J. Hydraul. Eng. 109 (2): 221–234. https://doi.org/10.1061/(ASCE)0733-9429(1983)109:2(221).
Fujiwara, O., and D. Khang. 1990. “A two-phase decomposition methods for optimal design of looped water distribution networks.” Water Resour. Res. 26 (4): 539–549. https://doi.org/10.1029/WR026i004p00539.
Geem, Z. W. 2008. “Particle-swarm harmony search for water network design.” Eng. Optim. 41 (4): 297–311. https://doi.org/10.1080/03052150802449227.
Gessler, J. 1985. “Pipe network optimization by enumeration.” In Proc., Computer Applications in Water Resources, 572–581. New York: ASCE.
Giudicianni, C., A. Di Nardo, R. Greco, and A. Scala. 2021. “A community-structure-based method for estimating the fractal dimension, and its application to water networks for the assessment of vulnerability to disasters.” Water Resour. Manage. 35 (4): 1197–1210. https://doi.org/10.1007/s11269-021-02773-y.
Harris, C. R., K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, and D. Cournapeau. 2020. “Array programming with NumPy.” Nature 585 (Feb): 357–362. https://doi.org/10.1038/s41586-020-2649-2.
Hunter, J. D. 2007. “Matplotlib: A 2D graphics environment.” Comput. Sci. Eng. 9 (3): 90–95. https://doi.org/10.1109/MCSE.2007.55.
Iwanek, M., D. Kowalski, B. Kowalska, and P. Suchorab. 2020. “Fractal geometry in designing and operating water networks.” J. Ecol. Eng. 21 (6): 229–236. https://doi.org/10.12911/22998993/123501.
Jaramillo, A. 2020. Análisis de la geometría fractal de la superficie óptima de presiones en el diseño optimizado de redes de distribución de agua potable [Analysis of the fractal geometry of the Optimal Pressure Surface in the optimized design of water distribution networks]. [In Spanish.] Bogotá, Colombia: Universidad de los Andes.
Jaramillo, A. 2022. FractalAnalysis_OHGS_v3: v.4. Bogotá, Colombia: Universidad de los Andes. https://doi.org/10.5281/zenodo.6634751.
Jian-Hua, L., Y. Bo-Ming, and Z. Ming-Qing. 2009. “A model for fractal dimension of rough surfaces.” Chin. Phys. Lett. 26 (11): 116101. https://doi.org/10.1088/0256-307X/26/11/116101.
Ju, W., and N. S.-N. Lam. 2009. “An improved algorithm for computing local fractal dimension using the triangular prism method.” Comput. Geosci. 35 (6): 1224–1233. https://doi.org/10.1016/j.cageo.2008.09.008.
Kadu, M. S., R. Gupta, and P. R. Bhave. 2008. “Optimal design of water distribution networks using a modified genetic algorithm with reduction in search space.” J. Water Resour. Plan. Manage. 134 (2): 147–160. https://doi.org/10.1061/(ASCE)0733-9496(2008)134:2(147).
Kim, J. S., K. I. Goh, G. Salvi, E. Oh, B. Kahng, and D. Kim. 2007. “Fractality in complex networks: Critical and supercritical skeletons.” Phys. Rev. E 75 (1): 016110. https://doi.org/10.1103/PhysRevE.75.016110.
Kowalska, B., D. Kowalski, and H. Ewa. 2020. “Fractal-heuristic method of water quality sensor locations in water supply network.” Water 12 (3): 832. https://doi.org/10.3390/w12030832.
Kowalski, D., B. Kowalska, T. Blawucki, P. Suchorab, and K. Gaska. 2019. “Impact assessment of distribution network layout on the reliability of water delivery.” Water 11 (3): 480. https://doi.org/10.3390/w11030480.
Kyriakou, M. S., and S. G. Vrachimis. 2019. “EPANET-benchmarks.” Accessed July 25, 2020. https://github.com/KIOS-Research/EPANET-Benchmarks.
Lam, N. S. N., H. Qium, D. A. Quattrochi, and C. W. Emerson. 2002. “An evaluation of fractal methods for characterizing image complexity.” Cartogr. Geogr. Inf. Sci. 29 (1): 25–35. https://doi.org/10.1559/152304002782064600.
Lansey, K. E., and L. W. Mays. 1989. “Optimization model for water distribution system design.” J. Hydraul. Eng. 115 (10): 1401–1418. https://doi.org/10.1061/(ASCE)0733-9429(1989)115:10(1401).
Lee, S. C., and S. I. Lee. 2001. “Genetic algorithms for optimal augmentation of water distribution networks.” J. Korea Water Resour. Assoc. 34 (5): 567–575.
Lin, M. D., Y. H. Liu, G. F. Liu, and C. W. Chu. 2007. “Scatter search heuristic for least-cost design of water distribution networks.” Eng. Optim. 39 (7): 857–876. https://doi.org/10.1080/03052150701503611.
Liu, H., C. A. Shoemaker, Y. Jiang, G. Fu, and C. Zhang. 2020. “Preconditioning water distribution network optimization with head loss-based design method.” J. Water Resour. Plan. Manage. 146 (12): 04020093. https://doi.org/10.1061/(ASCE)WR.1943-5452.0001299.
Lopez-Giraldo, S., and J. Saldarriaga. 2004. “Optimal cost potable water network distribution design with genetic algorithms.” In Proc., World Water and Environmental Resources Congress 2004. Reston, VA: ASCE.
Mandelbrot, B. 1977. Fractals: Form, chance and dimension. San Francisco: W.H. Freeman and Company.
Mandelbrot, B. 1983. The fractal geometry of nature. New York: W.H. Freeman and Company.
Marchi, A., G. Dandy, A. Wilkins, and H. Rohrlach. 2014. “Methodology for comparing evolutionary algorithms for optimization of water distribution systems.” J. Water Resour. Plan. Manage. 140 (1): 22–31. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000321.
Mitchell, M. 1999. An introduction to genetic algorithm. Cambridge, MA: Massachusetts Institute of Technology.
Montesinos, P., A. Garcia-Guzman, and J. L. Ayuso. 1999. “Water distribution network optimization using a modified genetic algorithm.” Water Resour. Res. 35 (11): 3467–3473. https://doi.org/10.1029/1999WR900167.
Moosavian, N., and B. J. Lence. 2020. “Flow-uniformity index for reliable-based optimal design of water-distribution networks.” J. Water Resour. Plan. Manage. 146 (3): 04020005. https://doi.org/10.1061/(ASCE)WR.1943-5452.0001161.
Ormsbee, L., and T. Walski. 2016. “Darcy-Weisbach versus Hazen-Williams: No Calm in West Palm.” In Proc., World Environmental and Water Resources Congress 2016. Reston, VA: ASCE.
Páez, D., D. Hernández, and J. Saldarriaga. 2013. “Optimal design of pipes in series with pressure driven demands.” In Proc., World Environmental and Water Resources Congress 2013: Showcasing the Future, 857–868. Reston, VA: ASCE.
Páez, D., C. Salcedo, A. Garzón, M. A. González, and J. Saldarriaga. 2020. “Use of energy-based domain knowledge as feedback to evolutionary algorithms for the optimization of water distribution networks.” Water 12 (11): 3101. https://doi.org/10.3390/w12113101.
Páez, D., J. Saldarriaga, L. López, and C. Salcedo. 2014. “Optimal design of water distribution systems with pressure driven demands.” Procedia Eng. 89 (Nov): 839–847. https://doi.org/10.1016/j.proeng.2014.11.515.
Persson, B. N. J. 2014. “On the fractal dimension of rough surfaces.” Tribol. Lett. 54 (1): 99–106. https://doi.org/10.1007/s11249-014-0313-4.
Prasad, T. D., and N. S. Park. 2004. “Multiobjective genetic algorithms for design of water distribution networks.” J. Water Resour. Plan. Manage. 130 (1): 73–82. https://doi.org/10.1061/(ASCE)0733-9496(2004)130:1(73).
Przestacki, D., R. Majchrowski, and L. Marciniak-Podsadna. 2016. “Experimental research of surface roughness and surface texture after laser cladding.” Appl. Surf. Sci. 388 (Apr): 420–423. https://doi.org/10.1016/j.apsusc.2015.12.093.
Qiu, M., M. Housh, and A. Ostfeld. 2021. “Analytical optimization approach for simultaneous design and operation of water distribution-systems optimization.” J. Water Resour. Plan. Manage. 147 (3): 06020014. https://doi.org/10.1061/(ASCE)WR.1943-5452.0001330.
Reca, J., and J. Martínez. 2006. “Genetic algorithms for the design of looped irrigation water distribution networks.” Water Resour. Res. 42 (5): W05416. https://doi.org/10.1029/2005WR004383.
Reca, J., J. Martínez, C. Gil, and R. Baños. 2008. “Application of several meta-heuristic techniques to the optimization of real looped water distribution networks.” Water Resour. Manag. 22 (10): 1367–1379. https://doi.org/10.1007/s11269-007-9230-8.
Saldarriaga, J. 2016. Hidráulica de Tuberías: Abastecimiento de Agua, Redes y Riegos [Pipeline Hydraulics: Water Supply, Networks and Irrigation]. [In Spanish.] Bogotá, Colombia: Alfaomega.
Saldarriaga, J., D. Páez, P. Cuero, and N. León. 2013. “Optimal design of water distribution networks using mock open tree topology.” In Proc., World Environmental and Water Resources Congress 2013: Showcasing the Future, 869–880. Reston, VA: ASCE.
Saldarriaga, J., D. Páez, N. León, L. López, and P. Cuero. 2014. “Historical development of power used methods for WDS design and their evolution towards optimization metaheuristics.” Procedia Eng. 70 (Feb): 1470–1484. https://doi.org/10.1016/j.proeng.2014.02.162.
Saldarriaga, J., D. Páez, N. León, L. López, and P. Cuero. 2015. “Power use methods for optimal design of WDS: History and their use as post-optimization warm starts.” J. Hydroinfor. 17 (3): 404–421. https://doi.org/10.2166/hydro.2014.013.
Saldarriaga, J., D. Páez, C. Salcedo, P. Cuero, L. L. López, N. León, and D. Celeita. 2020. “A direct approach for the near-optimal design of water distribution networks based on power use.” Water 12 (4): 1037. https://doi.org/10.3390/w12041037.
Saldarriaga, J., L. Pulgarín, P. Cuero, and N. Duque. 2017. “Pipeline hydraulics academic software.” [In Spanish.] In Proc., IBERO-American Seminar on Water and Drainage Networks (SEREA 2017). Bogotá, Colombia: Universidad de los Andes.
Saldarriaga, J., S. Takahashi, F. Hernández, D. M. Díaz, and S. Ochoa. 2010. “An energy methodology for the design of water distribution systems.” In Proc., World Environmental and Water Resources Congress 2010: Challenges of Change. Reston, VA: ASCE.
Saldarriaga, J., G. Villalba, and S. Takahashi. 2005. “Algorithms of combinatorial optimization applied to water distribution networks design.” In Proc., World Water and Environmental Resources Congress 2005. Reston, VA: ASCE.
Sarkar, N., and B. B. Chaudhuri. 1992. “An efficient approach to estimate fractal dimension of textural images.” Pattern Recognit. 25 (9): 1035–1041. https://doi.org/10.1016/0031-3203(92)90066-R.
Savic, D. A., and G. A. Walters. 1997. “Genetic algorithms for least-cost design of water distribution networks.” J. Water Resour. Plan. Manage. 123 (2): 67–77. https://doi.org/10.1061/(ASCE)0733-9496(1997)123:2(67).
Schaake, J., and D. Lai. 1969. Linear programming and dynamic programming application to water distribution network design, 69. Cambridge, MA: Massachusetts Institute of Technology.
Sherali, H. D., S. Subramanian, and G. V. Loganathan. 2001. “Effective relaxations and partitioning schemes for solving water distribution network design problems to global optimality.” J. Glob. Optim. 19 (1): 1–26. https://doi.org/10.1023/A:1008368330827.
Sirsant, S., and M. J. Reddy. 2020. “Assessing the performance of surrogate measures for water distribution network reliability.” J. Water Resour. Plan. Manage. 146 (7): 04020048. https://doi.org/10.1061/(ASCE)WR.1943-5452.0001244.
Sivanandam, S. N., and S. N. Deepa. 2008. Introduction to genetic algorithms. New York: Springer.
Song, C., L. K. Gallos, S. Havlin, and H. A. Makse. 2007. “How to calculate the fractal dimension of a complex network: The box covering algorithm.” J. Stat. Mech: Theory Exp. 2007 (3): P03006. https://doi.org/10.1088/1742-5468/2007/03/P03006.
Spasic, S. 2014. “On 2D generalization of Higuchi’s fractal dimension.” Chaos Solitons Fractals 69 (Sep): 179–187. https://doi.org/10.1016/j.chaos.2014.09.015.
Sun, W., G. Xu, P. Gong, and S. Liang. 2006. “Fractal analysis of remotely sensed images: A review of methods and applications.” Int. J. Remote Sens. 27 (22): 4963–4990. https://doi.org/10.1080/01431160600676695.
Sung, Y. H., M. D. Lin, Y. H. Lin, and Y. L. Liu. 2007. “Tabu search solution of water distribution network optimization.” J. Environ. Manage. 17 (3): 177–187.
Székely, G. J., M. L. Rizzo, and N. K. Bakirov. 2007. “Measuring and testing dependence by correlation of distances.” Ann. Stat. 35 (6): 2769–2794. https://doi.org/10.1214/009053607000000505.
Todini, E., and S. Pilati. 1987. “A gradient algorithm for the analysis of pipe networks.” In Computer applications in water supply: Volume 1—Systems analysis and simulation, 1–20. New York: Wiley.
Tolle, C. R., T. R. McJunkin, and D. J. Gorsich. 2003. “Suboptimal minimum cluster volume cover-based method for measuring fractal dimension.” IEEE Trans. Pattern Anal. Mach. Intell. 25 (1): 32–41. https://doi.org/10.1109/TPAMI.2003.1159944.
Vargas, K., C. Salcedo, and J. Saldarriaga. 2019. “Dimensión fractal e identificación de potenciales sectores de servicio en redes de distribución de agua potable utilizando criterios hidráulicos. [Fractal dimension and identification of potential service sectors in water distribution networks using hydraulic criteria].” [In Spanish.] Revista Recursos Hídricos 40 (2): 27–38. https://doi.org/10.5894/rh40n2-cti3.
Vargas, K., and J. Saldarriaga. 2019. “Analysis of fractality in water distribution networks using hydraulic criteria.” In Proc., World Environmental and Water Resources Congress 2019. Reston, VA: ASCE.
Virtanen, P., et al.2020. “SciPy 1.0: Fundamental algorithms for scientific computing in Python.” Nat. Methods 17 (3): 261–272. https://doi.org/10.1038/s41592-019-0686-2.
Wang, Q., M. Guidolin, D. Savic, and Z. Kapelan. 2015. “Two-objective design of benchmark problems of a water distribution system via MOEAs: Towards the best-known approximation of the true Pareto front.” J. Water Resour. Plan. Manage. 141 (3): 04014060. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000460.
Wen, T., and K. H. Cheong. 2021. “The fractal dimension of complex networks: A review.” Inform. Fusion 73 (Jan): 87–102. https://doi.org/10.1016/j.inffus.2021.02.001.
Wu, I. 1975. “Design of drip irrigation main lines.” J. Irrig. Drain. Div. 101 (4): 265–278. https://doi.org/10.1061/JRCEA4.0001064.
Zheng, F., A. R. Simpson, and C. Zecchin. 2011. “A combined NLP-differential evolution algorithm approach for the optimization of looped water distribution systems.” Water Resour. Res. 47 (8): 10394. https://doi.org/10.1029/2011WR010394.
Zhou, G., and N. S.-N. Lam. 2005. “A comparison of fractal dimension estimators based on multiple surface generation algorithms.” Comput. Geosci. 31 (10): 1260–1269. https://doi.org/10.1016/j.cageo.2005.03.016.
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Journal of Water Resources Planning and Management
Volume 149 • Issue 1 • January 2023
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Received: Mar 4, 2021
Accepted: Jun 27, 2022
Published online: Nov 3, 2022
Published in print: Jan 1, 2023
Discussion open until: Apr 3, 2023
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