Nonlinear Muskingum Research Articles

The Flood Simulation of the Modified Muskingum Model with a Variable Exponent Based on the Artificial Rabbit Optimization Algorithm

In order to further improve the accuracy of flood routing, this article uses the Variable Exponential Nonlinear Muskingum Model (VEP-NMM), combined with the Artificial Rabbit Optimization (ARO) algorithm for parameter calibration, to construct the ARO-VEP-NMM flood routing model. Taking Wilson’s (1974) flood as an example, the model calculation results were compared and analyzed with the Muskingum model constructed with seven optimization algorithms. At the same time, six measured floods in the Zishui Basin were selected for model applicability testing. The results show that the ARO algorithm exhibits stronger robustness and search ability compared with other optimization algorithms and can better solve the parameter optimization problem of the Muskingum model. The use of the ARO-VEP-NMM model for flood routing accurately reflects the movement patterns of floods. The Nash coefficient of the Wilson section reached 0.9983, and the average Nash coefficient during the flood validation period in the Zishui Basin was 0.9, further verifying the adaptability and feasibility of the ARO-VEP-NMM model in flood routing. The research results can provide certain references and a theoretical basis for improving the accuracy of flood forecasting.

Development of a distributed nonlinear Muskingum model by considering snowmelt effects for flood routing in the Red River

This research paper presents the development of a nonlinear Muskingum model which achieves precise flood routing through river reaches while considering lateral inflow conditions. Fourteen pairs of flood hydrograph found at two specific United States Geological Survey (USGS) stations located along the Red River of the North, namely Grand Forks and Drayton, are used for the calibrations and validations of the Muskingum model. To enhance the accuracy of the procedure, a reach is divided into multiple sub-reaches, and the Muskingum model calculations are performed individually for each interval using the distributed Muskingum method. Notably, the model development process incorporates the use of the Salp Swarm algorithm. The obtained results demonstrate the effectiveness of the developed nonlinear Muskingum model in accurately routing floods through the very gentle river with a bed slope of (0.0002–0.0003). The events were categorized into three groups based on their dominant drivers: Group A (Snowmelt-driven floods), Group B (Rain-on-snow-induced floods), and Group C (Mixed floods influenced by both snowmelt and rainfall). For the sub-reaches in Group A, single sub-reach (NR = 1), the Performance Evaluation Criteria (PEC) yielded the highest value for SSE, amounting to 404.9 × 106. In Group B, when NR = 2, PEC results the highest value were SSE = 730.2 × 106. The number of sub-reaches in a model has a significant influence on parameter estimates and model performance, as demonstrated by the analysis of hydrologic parameters and performance evaluation criteria. Optimal performance varied across case studies, emphasizing the importance of selecting the appropriate number of sub-reaches for peak discharge predictions.

Beluga Whale Optimization Algorithm for Estimating Nonlinear Muskingum Model in Flood Routing

Beluga Whale Optimization Algorithm for Estimating Nonlinear Muskingum Model in Flood Routing

Distributed Muskingum model with a Whale Optimization Algorithm for river flood routing

Abstract This research introduces a novel nonlinear Muskingum model for river flood routing, aiming to enhance accuracy in modeling. It integrates lateral inflows using the Whale Optimization Algorithm (WOA) and employs a distributed Muskingum model, dividing river reaches into smaller intervals for precise calculations. The primary goal is to minimize the Sum of Square Errors (SSE) between the observed and modeled outflows. Our methodology is applied to six distinct flood hydrographs, revealing its versatility and efficacy. For Lawler's and Dinavar's flood data, the single-reach Muskingum model outperforms multi-reach versions, demonstrating its effectiveness in handling lateral inflows. For Lawler's data, the single-reach model (NR = 1) yields optimal parameters of K = 0.392, x = 0.027, m = 1.511, and β = 0.010, delivering superior results. Conversely, when fitting flood data from Wilson, Wye, Linsley, and Viessman and Lewis, the multi-reach Muskingum model exhibits better overall performance. Remarkably, the model excels with the Viessman and Lewis flood data, especially with two reaches (NR = 2), achieving a 21.6% SSE improvement while employing the same parameter set. This research represents a significant advancement in flood modeling, offering heightened accuracy and adaptability in river flood routing.

PARAMETER ESTIMATION FOR NON-LINEAR MUSKINGUM FLOOD ROUTING MODEL: NATURAL INSPIRED ALGORITHM

In any watershed the surface runoff is routed through the main course of the river to estimate the flow hydrograph at the outlet of the basin. One of the most widely used flood routing model in a river course is Muskingum model. The Muskingum model has three parameters, namely; k, x, and m, which are estimated based on the known input and outflow hydrograph of the channel. The accuracy of routing depends on the preciousness in estimating the model parameters. Many research works had been reported on improvising the parameter estimation by employing various advanced computational techniques. In this paper, we introduce a bat optimization algorithm for estimating the three parameters of the Muskingum model. Bat algorithm finds the global optimum for the parameters with random fly within the domain space. The results of the proposed model have been compared with various parameter estimation techniques reported for the Muskingum flood routing model. The comparative evaluation has been done based on four performance indicators, namely; a) Root Mean Square Error (RMSE), b) Mean absolute error (MAE), c) Coefficient of Correlation (R) , and d) Nash- Sutcliffe efficiency (E). From performance indicators it is observed that the proposed parameter estimation using a bat optimization algorithm outperforms in capturing the observed flow compared to other reported techniques. Key Words: Non-linear Muskingum model, Flood routing, parameter estimation, Bat optimization

A new technique for flood routing by nonlinear Muskingum model and artificial gorilla troops algorithm

Considering cost and time, the Muskingum method is the most efficient flood routing technique. The existing Muskingum models are only different in the storage equation and their efficiency depends on the model type and the estimation of different parameters. In this paper, the nonlinear Muskingum model is combined with a new lateral flow equation. Although the new lateral flow equation includes five decision variables, flood routing is done more accurately than previous studies. The new hybrid Muskingum model have 12 decision variables. To approximate the model decision variables, the artificial gorilla troops optimizer is utilized. The new Muskingum is examined for six case studies. The results of the new proposed method for these studies indicates the significant improvement of the model compared to previous research. Moreover, the sixth case study is the Dinavar River flood, which has not been used by researchers so far. Another significant point is the outstanding performance of the powerful artificial gorilla troops algorithm in minimizing the target function.

Parameter evaluation of a nonlinear Muskingum model using a constrained self-adaptive differential evolution algorithm

Abstract The precise evaluation of the Muskingum model (MM) parameters is quite critical for routing flood waves for achieving flood control in open channels. The MM is one of the popular techniques adopted for flood routing. Estimation of the MM parameters so as to provide the best fit for the observed and computed flow values is a global optimization problem. Several optimization techniques have been adopted in the past to serve this purpose, but efficient optimization algorithms are needed to overcome the local optima issues and improvement of accuracy. In this paper, the efficiency of three optimization algorithms, namely Jaya, Covariance Matrix Adaption-Evolution Strategy (CMAES) and self-adaptive differential evolution (SaDE), has been assessed in the evaluation of the Muskingum parameters. The sum of the square deviation of the observed outflow and computed outflow (SSQ) is considered an objective in this MM optimization problem. Also, a constraint is proposed in this paper to help the optimization algorithms in finding the optimal global solutions. The simulation results show that the sum of the square deviation of the observed outflow and computed outflow (SSQ) was the least for SaDE, followed by CMAES. HIGHLIGHTS Precise evaluation of Muskingum model (MM) parameters is quite critical for routing flood waves. Efficient optimization algorithms are needed to overcome local optima issues in the estimation of the Muskingum parameters. Jaya, Covariance Matrix Adaption-Evolution Strategy (CMAES) and self-adaptive differential evolution (SaDE) have been assessed. SaDE shows the best performance followed by CMAES.

Closure to “Reverse Flood Routing in Rivers Using Linear and Nonlinear Muskingum Models” by Meisam Badfar, Reza Barati, Emrah Dogan, and Gokmen Tayfur

Closure to “Reverse Flood Routing in Rivers Using Linear and Nonlinear Muskingum Models” by Meisam Badfar, Reza Barati, Emrah Dogan, and Gokmen Tayfur

Discussion of “Reverse Flood Routing in Rivers Using Linear and Nonlinear Muskingum Models” by Meisam Badfar, Reza Barati, Emrah Dogan, and Gokmen Tayfur

Discussion of “Reverse Flood Routing in Rivers Using Linear and Nonlinear Muskingum Models” by Meisam Badfar, Reza Barati, Emrah Dogan, and Gokmen Tayfur

A new method for dividing flood period in the variable-parameter Muskingum models

Abstract The Muskingum routing model is favored by water engineers owing to its simplicity and accuracy. A large amount of research is done to improve the accuracy of the model. One way to do so is to consider variable hydrological parameters during the flood routing period. In this study, the random selection (RS) method was proposed to divide the flood period of the nonlinear Muskingum model into three sub-periods. The proposed method was based on RS of members in each sub-region. It was applied to rout three flood hydrographs, and the objective function was the sum of squared errors. Comparing the results from the three variable-parameter nonlinear Muskingum model with those from the variable-parameter nonlinear Muskingum models in previous studies, the proposed model optimized the objective function in these hydrographs up to 61%. The uncertainty analysis of Muskingum parameters for Wilson's hydrograph was performed by the fuzzy alpha cut method, and it was found that the uncertainty of the parameter x is greater than the uncertainty of the parameters k and m.

Optimization of the Nonlinear Muskingum Model Parameters for the River Routing, Tigris River a Case Study

Flood forecasting and management are one of the most important strategies necessary for water resource and decision planners in combating flood problems. The Muskingum model is one of the most popular and widely used applications for the purpose of predicting flood routing. The particle swarm optimization (PSO) methodology was used to estimate the coefficients of the nonlinear Muskingum model in this study, comparing the results with the methods of genetic algorithm (GA), harmony search (HS), least-squares method (LSM), and Hook-Jeeves (HJ). The average monthly inflow for the Tigris River upstream at the Al-Mosul dam was selected as a case study for estimating the Muskingum model's parameters. The analytical and statistical results showed that the PSO method is the best application and corresponds to the results of the Muskingum model, followed by the genetic algorithm method, according to the following general descending sequence: PSO, GA, LSM, HJ, HS. The PSO method is characterized by its accurate results and does not require many assumptions and conditions for its application, which facilitates its use a lot in the subject of hydrology. Therefore, it is better to recommend further research in the use of this method in the implementation of future studies and applications.

Developing a novel parameter-free optimization framework for flood routing

The Muskingum model is a popular hydrologic flood routing technique; however, the accurate estimation of model parameters challenges the effective, precise, and rapid-response operation of flood routing. Evolutionary and metaheuristic optimization algorithms (EMOAs) are well suited for parameter estimation task associated with a wide range of complex models including the nonlinear Muskingum model. However, more proficient frameworks requiring less computational effort are substantially advantageous. Among the EMOAs teaching–learning-based optimization (TLBO) is a relatively new, parameter-free, and efficient metaheuristic optimization algorithm, inspired by the teacher-student interactions in a classroom to upgrade the overall knowledge of a topic through a teaching–learning procedure. The novelty of this study originates from (1) coupling TLBO and the nonlinear Muskingum routing model to estimate the Muskingum parameters by outflow predictability enhancement, and (2) evaluating a parameter-free algorithm’s functionality and accuracy involving complex Muskingum model’s parameter determination. TLBO, unlike previous EMOAs linked to the Muskingum model, is free of algorithmic parameters which makes it ideal for prediction without optimizing EMOAs parameters. The hypothesis herein entertained is that TLBO is effective in estimating the nonlinear Muskingum parameters efficiently and accurately. This hypothesis is evaluated with two popular benchmark examples, the Wilson and Wye River case studies. The results show the excellent performance of the “TLBO-Muskingum” for estimating accurately the Muskingum parameters based on the Nash–Sutcliffe Efficiency (NSE) to evaluate the TLBO’s predictive skill using benchmark problems. The NSE index is calculated 0.99 and 0.94 for the Wilson and Wye River benchmarks, respectively.

Reverse Flood Routing in Rivers Using Linear and Nonlinear Muskingum Models

AbstractOne of the key factors for flood modeling and control is the flood hydrograph, which is not always available due to lack of flood discharge observations. In reverse flow routing, hydraulic ...

تخمین پارامترهای مدل غیرخطی نوع پنجم ماسکینگام در روندیابی سیلاب با الگوریتم نوین شاهین هریس

One of the major issues in the hydrology is the prediction of flooding and subsidence, or the rise and fall of river hydrographs at a certain point. This can be analyzed by flood routing. The Muskingum method is one of the hydrological methods that can be used to save time and money with simple operation and proper accuracy of flood routing. The use of meta-heuristic methods has shown satisfactory results so far. Therefore, in this study, the performance of Harris hawks optimization (HHO) algorithm in estimating the optimal parameters of the non-linear Muskingum model has been evaluated. In this paper, the fifth type of Muskingum nonlinear model (NL5) was first used to evaluate the HHO algorithm in the Wilson River (Applied Research) and the Karun River (Case Study). In order to evaluate the desirability of the research findings, the results were compared to the results of the genetic algorithms (GA) and harmonic search (HS). The results of the HHO algorithm for both the Wilson and Karun rivers indicate the minimization of the sum of squares (SSQ) as the objective function, which is 11.64 for the Wilson River and 143050.02 for the Karun River. Based on the results, the HHO has better performance than the GA and HS algorithms, so the proposed algorithm can be used with good confidence to estimate the optimal values of nonlinear Muskingum model parameters.

The modified PRP conjugate gradient algorithm under a non-descent line search and its application in the Muskingum model and image restoration problems

In this paper, a modified Polak–Ribiere–Polyak (PRP) method, which possesses the following desired properties for unconstrained optimization problems, is presented. (i) The search direction of the given method has the gradient value and the function value. (ii) A non-descent backtracking-type line search technique is proposed to obtain the step size $$\alpha _k$$ and construct a point. (iii) The method inherits an important property of the classical PRP method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. (iv) The strongly global convergence and R-linear convergence of the modified PRP method for nonconvex optimization are established under some suitable assumptions. (v) The numerical results show that the modified PRP method not only is interesting in practical computation but also has better performance than the normal PRP method in estimating the parameters of the nonlinear Muskingum model and performing image restoration.

Parameter estimation of Muskingum model using grey wolf optimizer algorithm

Flood routing plays a crucial role in prevention of major economic and human losses, which, in this study, has been conducted via both three- and four-constant parameter non-linear Muskingum models for four hydrographs, along with the Grey Wolf Optimizer (GWO) algorithm. Three benchmark examples and a real example (Karun river) were investigated. The routing results of the Karun River revealed that in the estimation of the hydrological parameters using the GWO technique, SSQ became 59294 cms in the three-parameter model, compared to the genetic, artificial bee colony (ABC), simulated annealing (SA) and shuffled frog leaping (SFLA) algorithms, decreasing by 68%, 67%, 56% and 55% in comparison with the best modelings performed. As for the four-parameter model, the amount of reduction was 18% with respect to the particle swarm optimization algorithm.•The flood routing is carried out by two non-linear Muskingum model.•The main purpose of this work is to make a comprehensive study between models optimized by AGWO, GWO and other meta-heuristic algorithms.•In order to compare the results of the GWO algorithm to those of more recent algorithms, the flood routing was performed by using the Augmented Grey Wolf Optimizer algorithm as well.

The Projection Technique for Two Open Problems of Unconstrained Optimization Problems

There are two problems for nonconvex functions under the weak Wolfe–Powell line search in unconstrained optimization problems. The first one is the global convergence of the Polak–Ribiere–Polyak conjugate gradient algorithm and the second is the global convergence of the Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method. Many scholars have proven that the two problems do not converge, even under an exact line search. Two circle counterexamples were proposed to generate the nonconvergence of the Polak–Ribiere–Polyak algorithm for the nonconvex functions under the exact line search, which inspired us to define a new technique to jump out of the circle point and obtain the global convergence. Thus, a new Polak–Ribiere–Polyak algorithm is designed by the following steps. (i) Given the current point and a parabolic surface is designed; (ii) An assistant point is defined based on the current point; (iii) The assistant point is projected onto the surface to generate the next point; (iv) The presented algorithm has the global convergence for nonconvex functions with the weak Wolfe–Powell line search. A similar technique is used for the quasi-Newton method to get a new quasi-Newton algorithm and to establish its global convergence. Numerical results show that the given algorithms are more competitive than other similar algorithms. Meanwhile, the well-known hydrologic engineering application problem, called parameter estimation problem of nonlinear Muskingum model, is also done by the proposed algorithms.

Flood Routing: Improving Outflow Using a New Non-linear Muskingum Model with Four Variable Parameters Coupled with PSO-GA Algorithm

Flood is one of the most destructive natural disasters that damages people’s lives dramatically. Thus, it is crucial for researchers and politicians to research flood routing. The non-linear Muskingum model has been significantly considered by engineers and researchers in flood routing. In this study, in order to increase the accuracy of outflow prediction, the new non-linear Muskingum model, with four variable parameters, is proposed for the first time. In the proposed model, the inflows are divided into three sub-regions, and each of the four hydrologic parameters has a various value in each sub-region. How to select the sub-regions, as well as the values of the hydrologic parameters, is determined by combining both the Particle Swarm Optimization and Genetic Algorithm. The proposed model is studied in four case studies. Compared to the non-linear Muskingum model with three parameters, the amount of sum squared deviation (SSQ) decreased 52 and 6.9% for the first and second case studies, respectively. Compared to the best variable parameter model, the SSQ for the third and fourth case studies reduced 76 and 62%, respectively. The results showed that the SSQ was considerably decreased significantly in all of the four case studies, and the proposed model has superiority over other non-linear Muskingum models, which have been used by other researchers so far.

Locally tuned hybridized particle swarm optimization for the calibration of the nonlinear Muskingum flood routing model

Abstract In the literature about the parameter estimation of the nonlinear Muskingum (NL-MUSK) model, benchmark hydrographs have been subjected to various metaheuristics, and in these studies the minor improvements of the algorithms on objective functions are imposed as ‘state-of-the-art’. With the metaheuristics involving more control variables, the attempt to search global results in a restricted solution space is not actually practical. Although metaheuristics provide reasonable results compared with many derivative methods, they cannot guarantee the same global solution when they run under different initial conditions. In this study, one of the most practical of metaheuristics, the particle swarm optimization (PSO) algorithm, was chosen, and the aim was to develop its local search capability. In this context, the hybrid use of the PSO with the Levenberg–Marquardt (LM) algorithm was considered. It was detected that the hybrid PSO–LM gave stable global solutions as a result of each random experiment in the application for four different flood data. The PSO–LM, which stands out with its stable aspect, also achieved rapid convergence compared with the PSO and another hybrid variant called mutated PSO.

Identification of Parameters Influencing the Accuracy of the Solution of the Nonlinear Muskingum Equation

Two nonlinear versions of the Muskingum equation are considered. The difference between both equations relates to the exponent parameter. In the first version, commonly used in hydrology, this parameter is considered as free, while in the second version, it takes a value resulting from the kinematic wave theory. Consequently, the first version of the equation is dimensionally inconsistent, whereas the proposed second one is consistent. It is shown that the difference between the results provided by both versions of the nonlinear Muskingum equation depends on the longitudinal bed slope of a channel. For an increasing slope, when a propagating wave becomes more kinematic, the value of the exponent considered as the free parameter tends to its value resulting from the kinematic wave theory. Consequently, when the character of an open channel flow tends to a kinematic one, the dimensionally inconsistent version of the nonlinear Muskingum equation becomes a consistent one. The results of the numerical analysis suggest that apart from the parameters involved in the Muskingum equation, usually considered as free, the parameters of the numerical method of the solution (the number of reservoirs and the time step) should be considered also as free parameters. This conclusion results from the fundamental property of the Muskingum equation, relating to the numerical roots of the wave attenuation process. All numerical examples and tests relate to the solutions of the system of Saint Venant equations, considered as the benchmark.