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

Abstract
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.