Flood Routing using Gravitational Search Algorithm and Investigation of Hydrological Parameters Uncertainty of Nonlinear Muskingum Model

Introduction: The occurrence of flood in the human’s history has always been one of the mankind concerns. The methods of confronting this destructive phenomenon are of utter importance between researchers. One of the categories against this issue is flood routing. The financial losses of flood to human societies have made it too important to predict the occurrence of floods, so that it is necessary to accurately predict flood. In order to predict the outflows, the extraction of flood hydrographs is required in the downstream. The routing methods are divided into two hydraulic and hydrological groups. Hydraulic methods require the historical data and the solution of equations mainly through complex hydraulic methods is time consuming. However hydrological methods are preferred because of simplicity of their relative concepts. They are easy to be implemented and economize the simulation time. It is believed to be popular with researchers and has always tried to improve the accuracy of the results of the hydrological methods, which has become a good alternative to hydraulic methods. The Muskingum method is the most widely used hydrological routing technique which is divided into two groups of linear Muskingum and nonlinear Muskingum, depending on the relationship between the amount of storage and the inflows and outflows. Methodology: Various methods for estimating the hydrological parameters of Muskingum model have been presented. Techniques for estimating the parameters of the Muskingum model can be classified into three categories: mathematical techniques, phenomenonmimicking techniques and hybrid algorithms. Among various methods of routing, three parameters non-linear Muskingum method are widely popular. Evolutionary algorithms are used to estimate the optimal parameters of the non-linear Muskingum method because of their convergence rate, no need to make highly accurate initial estimate of the hydrological parameters and their randomness nature. In this paper, a gravitational search algorithm which is based on the Kepler algorithm was first used to routing three different hydrographs. In fact, Kepler algorithm is inspired by the elliptical motion of planets around the sun. At different times, the planets are too close to the sun, which represent the stage of the exploration of the Akbari et al., 2019 Flood Routing using Gravitational Search ... Journal of Hydraulics 14 (3), 2019 18 algorithm, and at the other times, the planets are far away from the sun and express the stage of exploitation of the algorithm. Results and discussion: Using the combination of gravitational search algorithm and Kepler algorithm (GSA-Kepler), the parameters of the Muskingum model are calculated for routing three different hydrographs: Wilson (1974), Wye River and Veissman and Lewis (2003). The first example is a benchmark problem that was first considered by Wilson (1974) to estimate the parameters of the Muskingum model. This river has no branch to the Belmont and has little flow. The results of the GSA-Kepler and the Segmented Least Squares Method, BFGS, HJ + DFP, HJ + CG, Genetic Algorithm, Immune Clonal Selection Algorithm, Harmony Search Algorithm and Free Parameter Setting Harmony Search Algorithm are compared with each other. The second example is the flood hydrograph in the Wye River. It has no tributaries from Erwood to Belmont and has little lateral flow. The third model is a multi-peak flow hygrograph that was first studied by Veissman and Lewis (2003). For the second example, the results of the GSA-Kepler algorithm, COBSA, PSO, DE, GA, BFGS and WOA are showed and for the third example, the results of the GSA-Kepler are compared with the results of the WOA and MHBMO algorithms. After determining the optimal hydrologic parameters, their uncertainty is estimated using the possibility theory. Selecting an analysis of uncertainty depends on many factors such as knowledge of uncertainty sources and model complexities. There is no definite guideline for choosing the specific uncertainty analysis method that properly works. The principles of analyzing the possibility theory are based on fuzzy theory, which was first pronounced by Zadeh in 1965. To investigate the uncertainty of the non-linear Muskingum model parameters based on the possibility theory, the aforementioned algorithm and other algorithms include Least Squares Method, Gravitational Search Algorithm, BFGS algorithm, HJ + DFP, HJ + CG, Genetic Algorithm, Immune Clonal Selection Algorithm, Harmony Search Algorithm and Free Parameter Setting Harmony Search Algorithm were used. Then, three triangular membership functions were assigned to the hydrological variables and the uncertainty of these parameters was calculated using the fuzzy alpha cut method. Conclusion: Comparing the results of the GSA-Kepler with the results of the previous studies shows that the combined algorithm used in this study has an acceptable accuracy and high convergence rate. Based on the fuzzy alpha cut method, it is determined that for Wilson (1974) the uncertainty of parameter k is greater than the uncertainty of parameters x and m.