Journal of Advanced Informatics in Water, Soil, and Structure

Journal of Advanced Informatics in Water, Soil, and Structure

Performance Evaluation of Different Numerical Schemes for Muskingum Flood Routing (Case study: Karoon River, Iran)

Document Type : Research Article

Authors
Meisam Badfar 1
Reza Barati 2
Emrah Dogan 1
Gokmen Tayfur 3
1 Department of Civil Engineering, Sakarya University, Turkey,
2 Khorasan Razavi Water Authority, Mashhad, Iran,
3 Department of Civil Engineering, Izmir Institute of Technology, Turkey
10.22048/wss.2024.427805.1004
Abstract
The Muskingum model, widely used in flood routing, is the first order differential equation. The accuracy of the routing storage equation of the model depends on correct parameter estimation and a numerical method employed for its solution. In the present study, different explicit numerical methods are used for solving the equation. These methods include Euler, modified Euler, Runge-Kutta 4th order and Runge-Kutta-Fehlberg. For optimal parameter estimation, Shuffled Complex Evolution (SCE) algorithm is adopted. The performance of different numerical schemes are studied by appropriate evaluation criteria. The methods are tested against three historical and well-known flood data from literature and a field data from Karoon River, Iran as a natural river. Results indicated a good performance of the SCE algorithm and the Runge-Kutta 4th order method in the flood hydrograph numerical simulations. Regarding the sum of squared deviations and for the flood data of Karoon River, the Runge-Kutta 4th order method yielded 18% better results than traditional Euler method in the field condition of a natural river. Similar results can be observed for the first case study, where the Runge-Kutta 4th order method yielded 178% better results than traditional Euler method. However, for second and third case studies the results of all considered numerical methods nearly are in a same level of accuracy. Therefore, it can be said that an appropriate numerical scheme for a hydrological flood routing problem can be adopted by considering the relationship between storage volume and weighted flow.
Keywords
Hydrological Flood Routing
Numerical Methods
Optimization
Karoon River
Subjects
  1. Akbari, R., and Hessami-Kermani, M. R. (2021). Parameter estimation of Muskingum model using grey wolf optimizer algorithm. MethodsX, 8, 101589.
  2. Akbari, G. H., Nezhad, A. H., & Barati, R. (2012). Developing a model for analysis of uncertainties in prediction of floods. Journal of Advanced Research, 3(1), 73-79.
  3. Akbari, G. H., & Barati, R. (2012, April). Comprehensive analysis of flooding in unmanaged catchments. In Proceedings of the Institution of Civil Engineers-Water Management (Vol. 165, No. 4, pp. 229-238). Thomas Telford Ltd.
  4. Atashi, V., Barati, R., & Lim, Y. H. (2023). Distributed Muskingum model with a Whale Optimization Algorithm for river flood routing. Journal of Hydroinformatics, 25(6), 2210-2222.
  5. Atashi, V., Barati, R., & Lim, Y. H. (2023). Improved river flood routing with spatially variable exponent Muskingum model and sine cosine optimization algorithm. Environmental Processes, 10(3), 42.
  6. Atashi, V., Barati, R., & Lim, Y. H. (2023). Development of a distributed nonlinear Muskingum model by considering snowmelt effects for flood routing in the Red River. Scientific Reports, 13(1), 21356.
  7. Badfar, M., Barati, R., Dogan, E., & Tayfur, G. (2021). Reverse Flood Routing in Rivers Using Linear and Nonlinear Muskingum Models. Journal of Hydrologic Engineering, 26(6), 04021018.
  8. Barati, R. 2011. Parameter estimation of nonlinear Muskingum models using Nelder-Mead simplex algorithm, Journal of Hydrologic Engineering, 16(11), 946-954.
  9. Barati, R. 2013. Application of excel solver for parameter estimation of the nonlinear Muskingum models. KSCE Journal of Civil Engineering, 17(5), 1139-1148.
  10. Barati, R., Rahimi, S., & Akbari, G. H. (2012). Analysis of dynamic wave model for flood routing in natural rivers. Water Science and Engineering, 5(3), 243-258.
  11. Barati, R., Akbari, G. H., & Rahimi, S. (2013). Flood Routing of an Unmanaged River Basin Using Muskingum--Cunge Model; Field Application and Numerical Experiments. Caspian Journal of Applied Sciences Research, 2(6). 8-20.
  12. Barati, R., Neyshabouri, S. S., & Ahmadi, G. (2014). Sphere drag revisited using shuffled complex evolution algorithm. River flow, 978, 345-353.
  13. Barati, R. (2018). Discussion of “application of genetic programming to flow routing in simple and compound channels” by Elahe Fallah-Mehdipour, Omid Bozorg-Haddad, Hossein Orouji, and Miguel A. Mariño. Journal of Irrigation and Drainage Engineering, 144(5), 07018015.
  14. Brutsaert, W. (2023). Hydrology. Cambridge university press.
  15. Chu, H. J., and Chang, L. C. (2009). “Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model.” Journal of Hydrologic Engineering, 14(9), 1024-1027.
  16. Chu, H. J. (2010) “Closure to ‘Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model” by H.-J. Chu and L.-C. Chang’ ” Journal of Hydrologic Engineering, 15(11), 952–954.
  17. Das, A. (2004). “Parameter estimation for Muskingum models.” Journal of Irrigation and Drainage Engineering, 130(2):140–147.
  18. Das, A. (2007). “Chance-constrained optimization-based parameter estimation for Muskingum models.” Journal of Irrigation and Drainage Engineering, 133(5):487–494.
  19. Das, A. (2010). “Discussion of ‘Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model by H.-J. Chu and L.-C. Chang.’ ” Journal of Hydrologic Engineering, 15(11), 946–948.
  20. Duan, Q. (2003). Global optimization for watershed model calibration. Calibration of watershed models, 6, 89-104.
  21. Duan, Q. Y., Gupta, V. K., and Sorooshian, S. (1992). “Effective and efficient global optimization for conceptual rainfall-runoff models.” Water Resources Research, 28(4), 1015–1031.
  22. Duan, Q. Y., Gupta, V. K., and Sorooshian, S. (1993). “Shuffled complex evolution approach for effective and efficient global minimization.” Journal of Optimization Theory and Applications, 76(3), 501–521.
  23. Duan, Q. Y., Sorooshian, S., and Gupta, V. K. (1994). “Optimal use of the SCE-UA global optimization method for calibrating watershed models.” Journal of Hydrology, 158, 265–284.
  24. Easa, S. M., Barati, R., Shahheydari, H., Nodoshan, E. J., & Barati, T. (2014). Discussion: New and improved four-parameter non-linear Muskingum model. In Proceedings of the Institution of Civil Engineers-Water Management (Vol. 167, No. 10, pp. 612-615). Thomas Telford Ltd.
  25. Ehteram, M., Binti Othman, F., Mundher Yaseen, Z., Abdulmohsin Afan, H., Falah Allawi, M., Najah Ahmed, A., ... and El-Shafie, A. (2018). Improving the Muskingum flood routing method using a hybrid of particle swarm optimization and bat algorithm. Water, 10(6), 807.
  26. Farahani, N., Farzin, S., and Karami, H. (2018). Flood routing by Kidney algorithm and Muskingum model. Natural Hazards, 1-19.
  27. Farzin, S., Singh, V. P., Karami, H., Farahani, N., Ehteram, M., Kisi, O., ... and El-Shafie, A. (2018). Flood routing in river reaches using a three-parameter Muskingum model coupled with an improved bat algorithm. Water, 10(9), 1130.
  28. Geem, Z. W. (2006). “Parameter estimation for the nonlinear Muskingum model using the BFGS technique.” Journal of Irrigation and Drainage Engineering, 132(5), 474–478.
  29. Geem, Z. W. (2011). “Parameter Estimation of the Nonlinear Muskingum Model Using Parameter-Setting-Free Harmony Search.” Journal of Hydrologic Engineering, 16(8), 684–688.
  30. Gerald, C. F., and Wheatley, P. O. 2004. Applied numerical analysis, 7th Ed., Addison-Wesley, Boston.
  31. Gill, M. A. (1978). “Flood routing by Muskingum method.” Journal of hydrology, 36(3–4), 353–363.
  32. Gupta, H. V., Sorooshian, S., and Yapo, P. O. (1999). “Status of automatic calibration for hydrologic models: comparison with multilevel expert calibration.” Journal of Hydrologic Engineering, 4(2), 135-143.
  33. Hosseini, S. M. (2009). “Application of Spreadsheets in Developing Flexible Multiple-Reach and Multiple-Branch Methods of Muskingum Flood Routing.” Computer Applications in Engineering Education, 17(4), 448–454.
  34. Khalifeh, S., Esmaili, K., Khodashenas, S., and Akbarifard, S. (2020). Data on optimization of the non-linear Muskingum flood routing in Kardeh River using Goa algorithm. Data in Brief, 30, 105398.
  35. Kim, J. H., Geem, Z. W., and Kim, E. S. (2001). “Parameter estimation of the nonlinear Muskingum model using harmony search.” Journal of the American Water Resources Association, 37(5), 1131–1138.
  36. Luo, J., and Xie, J. (2010). “Parameter estimation for nonlinear Muskingum model based on immune clonal selection algorithm.” Journal of Hydrologic Engineering, 15(10), 844-851.
  37. Mohan, S. (1997). “Parameter estimation of nonlinear Muskingum models using genetic algorithm.” Journal of Hydraulic Engineering, 123(2), 137–142.
  38. Nelder, J. A., and Mead, R. (1965). “A simplex method for function minimization.” Computer Journal., 7, 308-313.
  39. Niazkar, M., and Afzali, S. H. (2015). Assessment of modified honey bee mating optimization for parameter estimation of nonlinear Muskingum models. Journal of Hydrologic Engineering, 20(4), 04014055.
  40. Niazkar, M., and Zakwan, M. (2022). Parameter estimation of a new four-parameter Muskingum flood routing model. In Computers in Earth and Environmental Sciences (pp. 337-349). Elsevier.
  41. Ramirez, J. A. (2010). Accessed October 4, 2012. http://www.engr.colostate.edu/∼ramirez/ce_old/
  42. Samani, H. M. V., and Shamsipour, G. A. (2004). “Hydrologic flood routing in branched river systems via nonlinear optimization.” Journal of Hydraulic Research, 42(1), 55-59.
  43. Tung, Y. K. (1985). “River flood routing by nonlinear Muskingum method.” Journal of Hydraulic Engineering, 111(12), 1447–1460.
  44. Vatankhah, A. (2010). “Discussion of ‘Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model by H.-J. Chu and L.-C. Chang.’ ” Journal of Hydrologic Engineering, 15(11), 949–952.
  45. Vatankhah, A. R. 2014. Evaluation of explicit numerical solution methods of the Muskingum model. Journal of Hydrologic Engineering, 19(8), 949-952.
  46. Vatankhah, A. R. (2021). The lumped Muskingum flood routing model revisited: the storage relationship. Hydrological Sciences Journal, 66(11), 1625-1637.
  47. Wilson, E. M. (1974). “Engineering hydrology.” MacMillan, Hampshire, U.K.
  48. Xu, D. M., Qiu, L., & Chen, S. Y. (2012). Estimation of nonlinear Muskingum model parameter using differential evolution. Journal of Hydrologic Engineering, 17(2), 348-353.
  49. Yang, T., Li, P., & Wang, X. (2020). Convergence analysis of an improved BFGS method and its application in the Muskingum model. Mathematical Problems in Engineering, 2020(1), 4519274.
  50. Yoon, J., and Padmanabhan, G. (1993). Parameter estimation of linear and nonlinear Muskingum models. Journal of Water Resources Planning and Management, 119(5), 600-610.
  51. Zeinali, M., and Pourreza-Bilondi, M. (2018). Estimation of optimal parameters of the nonlinear Muskingum model using continuous ant colony algorithm. Irrigation and Water Engineering, 8(3), 94-106.
Journal of Advanced Informatics in Water, Soil, and Structure
Volume 1, Issue 1 - Serial Number 1
January 2025
Pages 47-61

  • Receive Date 29 November 2023
  • Revise Date 22 February 2024
  • Accept Date 26 February 2024
  • First Publish Date 01 January 2025
  • Publish Date 01 January 2025