Using nonlinear programming method and gray wolf algorithm for estimating parameters of nonlinear Muskingum model

Document Type : Research Article


1 Department of Water Engineering, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

2 School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran.


Background and objectives: Flood routing is an important issue in river engineering. The flood routing methods are categorized into two groups of hydraulic and hydrologic methods. Hydraulic routing methods require considerable input data and time-consuming calculations. But hydrological methods need less input data and are less complicated in comparison with hydraulic routing methods. The hydrological routing methods are based on the continuity equation and a relationship between inflow/outflow values and flood storage. Linear Muskingum is a hydrological routing method commonly used in rivers routing. However, as the relation between channel storage and the inflow/outflow is nonlinear, the nonlinear form of this model is developed which has received special attention in recent years and several types of it have been proposed. Using the Muskingum method, while saving time, valuable information about the flood depth and hydrograph is obtained. However, the performance of these models is highly dependent on the optimal estimation of their parameters considering the study area characteristics.
Materials and methods: Although the nonlinear Muskingum models have special advantages over the linear Muskingum model. The hydrologists avoid from the nonlinear Muskingum models, because of the difficulties in estimation of their parameters. Therefore, researchers have attempted to estimate these parameters using the optimization algorithms. In this research, the nonlinear Muskingum model type 5 (NL5) is considered for flood routing and Nonlinear programing (NLP) is used for estimation of the optimal values of model parameters. The results are compared with the metaheuristic optimization algorithms of genetic algorithm (GA), particle swarm optimization algorithm (PSO) and Gray wolf optimizer (GWO). The objective function of the optimization algorithms was set to minimize the sum of squares of the difference between the measured and simulated values of flows (SSR). Wilson flood hydrograph (first case study), Wy River flood hydrograph in England (second case study) and the hydrograph presented by Vatankhah (2014) (third case study) were used as the case studies of this research.
Results: The performance of NL5 model was very good in the all considered cases. In the first case study, the maximum absolute error is less than seven percent. Also, in the second and third case studies, the maximum absolute errors are less than 20 percent and 10 percent, respectively. MARE, NSE, CC, DPO and DPOT measures were used to further evaluate the model performance. The closer values of MARE, DPO and DPOT to zero and the closer the values of the NSE and CC measures to one, show the better the performance of the model (Kult et al., 2014). In the first case study, the MARE values for NLP, GWO, PSO and GA algorithms are 0.011, 0.011, 0.012 and 0.012 m3/s, respectively. For the second case study, the MARE values are 0.104, 0.105, 0.103 and 0.104 m3/s, respectively; The values of this measure in the third case study for the mentioned optimization methods are 0.0301, 0.0301, 0.0301 and 0.0303 m3/s, respectively. The values of this measure show the perfect performance of NLP, GWO, PSO and GA techniques in estimation of NL5 parameters. DPO, DPOT, NSE and CC indices also show the same finding. SSR values in the first case study for NLP, GWO, PSO and GA optimization methods are 5.44, 5.44, 5.47 and 5.88, respectively. Also, SSR values for the second case study are 30837.6, 30848.2, 30880.1 and 30929.1, respectively. For the third case study, these values are 7356.7, 7432.1, 7391 and 7412.3. The simulation times for NLP, GWO, PSO and GA methods show that the processing time in the NLP method is much less than the other methods. The optimization methods are ranked based on their results accuracy and simulation time. NLP method is ranked first in the regard while is followed by GWO, PSO and GA in the next ranks, respectively. The comparison of the obtained SSR values in the current study and the previous studies which used the cases one and two, show that the NLP optimization method has better performance in estimation of NL5 model parameters. In this study, for the third case study (Vatankhah, 2014 data), which has not been routed by the Muskingum method previously, the results of routing with NL5 are compared with the results obtained with the Rang Kota method (Vatankhah, 2014). The SSR value when using NLP as the optimization tool for estimation of NL5 model parameters is 7356.8 m6/s2, while in the Rang Kota method it is 14441.3 m6/s2. Therefore, in this case study, the NL5 model has performed better than the Rang Kota method.
Conclusion: In the present study, NLP technique and powerful GWO algorithm were used to estimate the optimal values of NL5 model parameters and the results were compared with GA and PSO algorithms. The performance evaluation results indicate that the NLP method, in addition to being more accurate, also requires less time to estimate the optimal value of the parameters. The values of the objective function for the first case study for NLP, GWO, GA and PSO methods are 5.44, 5.44, 5.88 and 5.47 m6/s2, respectively, while these values for the second case study are 30837.6, 30848, 30929.1 and 3088.1 m6/s2 and for the third case study are 7356.7, 7432.1 7412.3 and 7391 m6/s2. NLP processing time is at least 10% less than the other considered optimization methods. Therefore, the NLP method is the best choice for estimating the optimal parameters of Muskingum type five by considering two factors of accuracy and speed of simulation even though all of the methods showed a very good performance. Also, in the third case study, which was not routed previously, by the Muskingum method, the results were compared with the Rang-Kota method, and the results showed that the NL5 model, which was solved by the NLP method, performed better. After NLP, GWO, PSO and GA methods had the better performance in estimating the NL5 parameters, respectively.


Akbari, R., Hessami-Kermani, M.-R. and Shojaee, S. (2020). Flood Routing: Improving Outflow Using a New Non-linear Muskingum Model with Four Variable Parameters Coupled with PSO-GA Algorithm. Water Resources Management. 34, 3291-3316.
Akbarifard, S., Qaderi, K. and Alinnejad, M. (2018). Parameter estimation of the nonlinear muskingum flood-routing model using water cycle algorithm. Journal of Watershed Management Research. 8, 34-43. (in Persian)
Barati, R. (2013). Application of excel solver for parameter estimation of the nonlinear Muskingum models. KSCE Journal of Civil Engineering. 17, 1139-1148.
Bozorg-Haddad, O., Abdi-Dehkordi, M., Hamedi, F., Pazoki, M. and Loáiciga, H.A. (2019). Generalized storage equations for flood routing with nonlinear Muskingum models. Water Resources Management. 33, 2677-2691.
Bozorg-Haddad, O., Sarzaeim, P. and Loáiciga, H. A. (2021). Developing a Novel Parameter-free Optimization Algorithm for Flood Routing, Scientific Reports. 11, 16183.
Chow, V.T. (1959). Open channel hydraulics. McGraw-Hill Civil Engineering Series McGraw-Hill: Toronto.
Chow, V.T., Maidment, D.R. and Larry, W. (1988). Mays. Applied Hydrology. International edition, MacGraw-Hill, Inc, 149 p.
Das, A. (2004). Parameter estimation for Muskingum models. Journal of Irrigation and Drainage Engineering. 130, 140-147.
Easa, S.M. (2014). New and improved four-parameter non-linear Muskingum model.  Proceedings of the Institution of Civil Engineers-Water Management. 167(5), 288-298.
Ehteram, M., Karami, H., Mousavi, S.-F., Farzin, S. and Sarkamaryan, S. (2017). Evaluation of the performance of bat algorithm in optimization of nonlinear Muskingum model parameters for flood routing. Iranian journal of Ecohydrology. 4, 1025-1032. (in Persian)
El Harraki, W., Ouazar, D., Bouziane, A. and Hasnaoui, D. (2021). Optimization of reservoir operating curves and hedging rules using genetic algorithm with a new objective function and smoothing constraint: application to a multipurpose dam in Morocco. Environmental Monitoring and Assessment. 193, 1-17.
Farahani, N., Karami, H., Farzin, S., Ehteram, M., Kisi, O. and El Shafie, A. (2019). A new method for flood routing utilizing four-parameter nonlinear Muskingum and shark algorithm. Water Resources Management. 33, 4879-4893.
Geem, Z.W. (2006). Parameter estimation for the nonlinear Muskingum model using the BFGS technique. Journal of Irrigation and Drainage Engineering. 132, 474-478.
Gill, M.A. (1978). Flood routing by the Muskingum method. Journal of Hydrology. 36, 353-363.
Haddad, O.B., Hamedi, F., Orouji, H., Pazoki, M. and Loáiciga, H.A. (2015). A re-parameterized and improved nonlinear Muskingum model for flood routing. Water Resources Management. 29, 3419-3440.
Hamedi, F., Bozorg-Haddad, O., Pazoki, M., Asgari, H.-R., Parsa, M. and Loáiciga, H.A. (2016). Parameter estimation of extended nonlinear Muskingum models with the weed optimization algorithm. Journal of Irrigation and Drainage Engineering. 142, 04016059.
Khalifeh, S., Esmaili, K., Khodashenas, S. and Akbarifard, S. (2020a). Data on optimization of the non-linear Muskingum flood routing in Kardeh River using Goa algorithm. Data in brief. 30, 105398.
Khalifeh, S., Esmaili, K., Khodashenas, S.R. and Khalifeh, V. (2020b). Estimation of nonlinear parameters of the type 5 Muskingum model using SOS algorithm. MethodsX. 7, 101040.
Khalifeh, S., Esmaili, K., Khodashenas, S.R. and Modaresi, F. (2021a). Estimation of Nonlinear Parameters of Type 6 Hydrological Method in Flood Routing with the Spotted Hyena Optimizer Algorithm (SHO), Research Square.
Khalifeh, S., Khodashenas, S.R., Esmaili, K. and Khalifeh, V. (2021b). Estimation of the type 5 Muskingum nonlinear model parameters in the flood routing with The Harris Hawks Optimization Algorithm (HHO). Irrigation and Water Engineering. 11, 128-141. (in Persian)
Kult, J., Choi, W. and Choi, J. (2014). Sensitivity of the Snowmelt Runoff Model to snow covered area and temperature inputs. Applied Geography. 55, 30-38
Mccarthy, G. (1938). The unit hydrograph and flood routing, Conference of North Atlantic Division. US Army Corps of Engineers, New London, CT. US Engineering.
Mohammad Rezapour Tabari, M. and Emami Dehcheshmeh, S.A. (2018). Development of Nonlinear Muskingum Model Using Evolutionary Algorithms Hybrid. Iran-Water Resources Research. 14, 160-167. (in Persian)
Mohammadi Ghaleni, M. and Ebrahimi, K. (2013). Evaluation of direct search and genetic algorithms in optimization of muskingum nonlinear model parameters-a flooding of Karoun river, Iran. Water and Irrigation Management. 2, 1-12. (in Persian)
Mohammadi Ghaleni, M., Bozorg Hadad, O. and Ebrahimi, K. (2010). Parameter Estimation of the Nonlinear Muskingum Model Using Simulated Annealing. Water and Soil, 24(5), 908-919. (in Persian)
Mohan, S. (1997). Parameter estimation of nonlinear Muskingum models using genetic algorithm. Journal of Hydraulic Engineering. 123, 137-142.
Norouzi, H. and Bazargan, J. (2019). Using the Linear Muskingum Method and the Particle Swarm Optimization (PSO) algorithm for calculating the depth of the rivers flood. Iran-Water Resources Research. 15, 344-347. (in Persian)
O'donnell, T. (1985). A direct three-parameter Muskingum procedure incorporating lateral inflow. Hydrological Sciences Journal. 30, 479-496.
Ouyang, A., Liu, L.-B., Sheng, Z. and Wu, F. (2015). A class of parameter estimation methods for nonlinear muskingum model using hybrid invasive weed optimization algorithm. Mathematical Problems in Engineering, 2015, 1-15.
Rajabi, D., Karami, H., Hosseini, K., Mousavi, S.F. and Hashemi, S.A. (2015). Estimating Optimum Parameters of Non-Linear Muskingum Model of Routing using Imperialist Competition Algorithm (ICA). Journal of Water and Soil Science. 19, 321-334. (in Persian)
Vatankhah, A.R. (2014). Evaluation of explicit numerical solution methods of the Muskingum model. Journal of Hydrologic Engineering, 19, 06014001.
Wilson, E. (1974). Hydrograph Analysis. Engineering Hydrology. Springer.
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, 94-106. (in Persian)
Volume 17, Issue 3 - Serial Number 173
September 2022
Pages 31-46
  • Receive Date: 21 September 2021
  • Revise Date: 14 February 2022
  • Accept Date: 21 February 2022
  • First Publish Date: 21 February 2022