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.

is nonlinear; therefore, the original linear form of the Muskingum flood-routing model has been superseded by the following continuity and nonlinear Muskingum model, respectively 8,9,45 : where S t , I , and O t denote the channel storage (with dimension of L 3 ) of a river reach, rate of inflow with dimension of L 3 /T to a river reach, and rate of outflow (with dimension of L 3 /T) to a river reach, respectively, at time t; K , χ , and m = storage-time constant parameter, dimensionless weighting factor, and dimensionless parameter related to the nonlinearity of the flood wave, respectively. The following Eq. in which Δt denotes the time step of hydrograph simulation. Therefore, the outflow O t is calculated with Eq. (6) as follows: The values of the parameters K,χ , and m must be calibrated to achieve accurate outflow predictions with Eq. (6). Parameter calibration and hydraulic prediction can be achieved efficiently and accurately with EMOAs. The next section describes TLBO for the estimation of the Muskingum parameters with this parameter-free metaheuristic optimization algorithm in detail.
The teaching-learning-based optimization (TLBO) algorithm. Teaching-learning-based optimization (TLBO) is a population-based, meta-heuristic, optimization algorithm inspired by the swarm intelligence of a population seeking to change from a current situation to an optimal situation by overall knowledge improvement (i.e. grades) of students in a classroom 39 . The peculiar feature of TLBO is that it does not require algorithmic parameters for its implementation other than general parameters ubiquitous to all evolutionary optimization algorithms such as population size and the number of iterations. Recall the GA features crossover and mutation rates, whose values affect the optimization performance and the accuracy of results 38 .
TLBO starts searching for the optimal solution of a well-posed problem with an initial population whose members' scores are the values of the decision variables, such as grades earned by students. TLBO strives to improve the population's quality by means of a "Teacher Phase" and a "Learner Phase" to achieve a solution that is very near to the globally optimal solution. In the following the "Teacher Phase" and "Learner Phase" are discussed.  www.nature.com/scientificreports/ In a classroom the teacher has the highest-level knowledge of a particular topic and endeavors to teach the students to advance their overall knowledge, and, consequently, raise their individual and average grades in exams. In the "Teacher Phase" the person with the best grade is considered as a teacher making efforts to transfer knowledge to the other learners (i.e. students) in a class. I Suppose there are n students in a class and that they have an average grade M i in exam i. The most successful student with the best grade X T,i in exam i among the n students plays the teacher's role. The difference between the teacher's level and the average level of knowledge ( Diff i ) in exam i is expressed as follows: in which, r i = random number in [0, 1] , and T F = a random number that accounts for the teacher factor that depends on teaching quality, and equals either 1 or 2.
By teaching and transferring knowledge to students their new, improved, grades in the next exam are defined by Eq. (9): where, X ′ j,i and X j,i = new and old grades of student j in exam i, respectively.X j,i is transferred to the "Learner Phase" if the old grade is better than the new one, otherwise X ′ j,i is transferred. In the "Learner Phase" the top students help their peers to improve their knowledge through, for example, team work in assignments, leading to new grade improvement ( X ′′ ). This helping interaction between two students A and B in each exam i is defined as follow: This teacher-students interaction is the fundamental inspiration for TLBO, in which the number of students in the class is the algorithm population size and the number of exams is the number of iterations. The TLBO steps are defined as follow: (1) Define the population size, the number of iterations, and the objective function.
(3) Evaluate the objective function for n students in exam i. (4) Select the student with the best grade as teacher and calculate Diff i for exam i. (5) To calculate X ′ j,i for n students in exam i. (6) Compare X j,i and X ′ j,i and select the better one for transferring to the next step. (7) Select randomly each pair of students and calculate X ′′ j,i . (8) Compare X ′ j,i and X ′′ j,i and select the better one for transferring to the next step. (9) Evaluate the objective function for all students, check whether the stop criterion is satisfied (the optimal solution is achieved), otherwise the algorithm will iterate from step (4).
A more in-depth description of TLBO can be found in Refs. 38,39,46 .
Linking TLBO to the Muskingum model. Figure 2 depicts the flowchart of the algorithmic "TLBO-Muskingum" method to estimate the Muskingum model parameters (i.e. K,χ , and m). The algorithm starts by defining the population size (number of students), number of iterations (number of exams), and the objective function. Next, the initial population of Muskingum's parameters is generated randomly. The flood hydrograph is then simulated with Eqs. (5) and (6). The values of the objective function for each sequence of scores earned by the students are calculated following the Muskingum simulation. The next step improves the current population of decision variables (i.e., the estimates of K,χ , and m) by calculating the mean value of the objective function and selecting the best solution as the teacher of the population. Afterward, the population of parameters is updated in the teaching phase (i.e. moving the population toward the teachers' sequence of simulated outflows) and the learning phase (i.e. updating the population based on the interaction between the students). In other words, the new population of parameters is generated with the modifier operator such that each student (or parameter estimate) starts moving towards the best solution in the population by means of the linear and random base equation (this is the Teacher Phase). Furthermore, the improvement of the population is guided by the interactions between students using a linear equation based on the difference between their positions (this is the so-called Learner Phase). The Muskingum simulation is repeated with the improved or updated population and the objective function is re-evaluated. The optimal solution is reported whenever the user-specified termination criterion is satisfied. Otherwise, the iterations involving improvement of the current population, Muskingum simulation, evaluation of objective functions, and assessment of the termination criterion proceed until convergence is achieved. The population size is set equal to 100, the number of iterations is 500, and the objective function is expressed as below: www.nature.com/scientificreports/ where, SSD is the sum of the squared deviation between the observed and simulated outflows at time interval t, O i is the observed outflow at time interval i, and O i is the simulated outflow at time interval i.

Results and discussion
The performance of the "TLBO-Muskingum" in estimating the three-parameters of nonlinear Muskingum model is evaluated with two well-known benchmark problems: (1) the Wilson cased study 47 , and (2) the River Wye in the United Kingdom 48 , both with one single peak flood event. The former one is a popular benchmark problem based on the data provided by Wilson (1974) and has been employed commonly in several studies 9,14,22,[50][51][52][53] to be linked to EMOAs for estimation of nonlinear Muskingum model's parameters. The minor lateral nature of the flow makes this case study ideal for flood routing studies 48 . The second case study is based on flood event of the River Wye in Dec 1960 in the United Kingdom 49 . The 69-75 km riverbed characteristics (without tributaries and small lateral inflow) make it ideal for flood-routing calibration purposes. www.nature.com/scientificreports/ Along with objective function evaluation (SSD), the Nash-Sutcliffe Efficiency (NSE) is herein calculated for more precise evaluation of flood routing and optimization performance in hydrological point of view. The NSE is a normalized index of error variance 53 that measures the predictive skill of hydrologic models. It takes value in the range of (− ∞, 1]. The closer NSE is to 1, the more accurately the hydrological model performs 54 . The NSE is defined as follows: where NSE is the Nash-Sutcliffe Efficiency index, O i is the observed outflow at time interval i, O i is the simulated outflow at time interval i, and O is the average of observed outflow. The "TLBO-Muskingum" has been implemented for 10 runs for each of the benchmark problems. The objective function values in all 500 iterations have been plotted in Fig. 3. It is shown in Fig. 3 that in both case studies the converges was achieved by iteration 500, the best objective function is 169.3 and 102,511.9 for the Wilson and Wye River benchmarks, respectively. For more precise insight, the zoom-in objective function values have been extracted up to iteration 50 in Fig. 3, which depicts the fast convergence of TLBO in reaching the optimal solution. Figure 3 also illustrates how all 10 runs reached the same objective function, which results in negligible difference between the 10 runs for each benchmark problem. The calculated average, minimum (min), maximum (max), and standard deviation (SD) statistics of all the runs are listed in Tables 1 and 2 for the Wilson and Wye River benchmarks, respectively. These statistics are not significantly different at each time step and the SD values are small (the average standard deviation between the simulated outflows in all time steps for all 10 runs is 6 × 10 −6 and 7 × 10 −5 for the Wilson and Wye River examples, respectively). The small SD values stem from the high convergence capability of the TLBO in outflow simulations, which confirms the high reliability and robustness of "random-based" TLBO method in flow prediction. The average values of the simulated outflow were used for further analysis. It is important to notice that the results from the 10 runs were calculated without calibration of any algorithmic parameters. The observed and average simulated outflows timeseries calculated with the optimal values of the Muskingum parameters are listed in Tables 3 and 4 for the Wilson and Wye River case studies, respectively. The SSD as objective function from calculated outflows along with the NSE index as a standard hydrological performance metric are listed in Tables 3 and 4, as well. The calculated SSD is shown as function of the iteration number in Fig. 3. The NSE values clearly show how accurately the "TLBO-Muskingum" simulates the outflows, and consequently how the three parameters of Muskingum are estimated precisely. The NSE value for the Wilson case study is 0.99, demonstrating the excellent performance of the optimization process. The NSE equals 0.94 regarding the Wye River, also showing high algorithmic accuracy. Recall that the closer the NSE to 1, the more accurate the model prediction is.  www.nature.com/scientificreports/ Figure 4 depicts the timeseries of the observed and simulated hydrographs for case studies 1 and 2, along with the inflows. The simulated outflow hydrographs correspond to the average estimates of the Muskingum parameters. It is seen in Fig. 4 the overall good fit between observations and predictions. The accuracy of the "TLBO-Muskingum" algorithm's predictions is relatively higher for the first benchmark problem (NSE = 0.99 and R 2 = 0.99, see Table 5). The accuracy is nevertheless high in the second case study (NSE = 0.94 and R 2 = 0.97, see Table 5), except for the peak outflow: which is underestimated. This is evident in Fig. 5 illustrating the scatterplots of the observed and simulated outflows. Clearly in the case of the Wilson example the accuracy is high for the entire range of the outflows (R 2 = 0.99), while as discussed earlier, the predictive skill of the TLBO-Muskingum method for estimating lower outflows is superior to that associated with the peak outflows. The prediction of high flows may be improved by using a longer time series in the training phase of the Muskingum model. Furthermore, applying TLBO to the four-parameter Muskingum model may lead to better performance of the flood routing model with respect to peak flows, which may be addressed in future work. Table 5 lists the optimal values of the Muskingum's parameters calculated by TLBO ( K,χ , and m), the NSE, and the average run time obtained with 10 runs. The average run time for the Wilson and Wye examples equal 2.531 and 3.488 s, respectively. This shows the rapid convergence of the TLBO-Muskingum method.
For comparison purposes, the results from coupling GA 18 and PSO 22 to three-parameter Muskingum flood routing for Wilson example with the same objective function have been extracted and presented in Fig. 6. The sensitivity analysis has been implemented in the application of both GA and PSO to increase the accuracy of optimization-simulation Muskingum flood routing 18,22 which is computationally expensive and time-consuming, additionally the optimization perform sensitive to the algorithmic parameters to reach the global optima. The SSD value for fine-tuned GA and PSO are 23.0 and 36.9, respectively. In addition, deviation of peak flood (DPO)

Concluding remarks
There are many evolutionary optimization algorithms such as the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) among well-known ones, and Cat Swarm Optimization (CSO) and Flower Pollination Algorithm (FPA) among recently developed ones. These algorithms calculate near global optimal solutions for complex problems. Yet, their performance relies on the pre-calibration of algorithmic parameters, for there is no deterministic method for their assignment. The specification of evolutionary algorithmic parameters is commonly guided by experienced gained with similar optimization problems, if available. This paper implemented TLBO to estimate the three parameters of the nonlinear Muskingum models. TLBO was coupled with a nonlinear Muskingum flood routing model to make optimal predictions of outflow hydrographs by means of K,χ , and m calibration, which is the major challenge in Muskingum application for flood routing purposes.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.