# Evaluation of energy dissipation on stepped spillway using evolutionary computing

**Part of the following topical collections:**

## Abstract

In this study, using the M5 algorithm and multilayer perceptron neural network (MLPNN), the capability of stepped spillways regarding energy dissipation (ED) was approximated. For this purpose, relevant data was collected from valid sources. The study of the developed model based on the M5 algorithm showed that the Drop and Froude numbers play important roles in modeling and approximating the ED. The error indices of M5 algorithm in training were *R*^{2} = 0.99 and RMSE = 2.48 and in testing were *R*^{2} = 0.99 and RMSE = 2.23. The study of developed MLPNN revealed that this model has one hidden layer which includes five neurons. Among the tested transfer functions, the great efficiency was related to the Tansing function. The error indices of MLPNN in training were *R*^{2} = 0.97 and RMSE = 3.73 and in testing stages were *R*^{2} = 0.97 and RMSE = 3.98. Evaluation of the results of both applied methods shows that the accuracy of the MLPNN is a bit less than the M5 algorithm.

## Keywords

Energy dissipation Soft computing Drop number Spillways M5 algorithm## Abbreviations

- ANFIS
Adaptive neuro-fuzzy inference system

- CFD
Computational fluid dynamic

- DN
Drop number

- ED
Energy dissipation

*Fr*Froude number

*g*Gravity acceleration

- GEP
Genetic expression programming

- GMDH
Group method of data handling

*H*Total head of flow

*h*_{s}Height of steps

*L*_{c}Length of crest

*L*_{s}Length of steps

- MARS
Multivariate adaptive regression splines

- Max
Maximum

- Min
Minimum

- MLPNN
Multilayer perceptron neural network

*N*Number of steps

*q*Discharge per width

*y*_{0}Flow depth over the crest

*y*_{1}and*y*_{2}Conjugated depths of hydraulic jump

## Introduction

Spillways are structures that are extensively used to evacuate surplus flow over reservoir capacity of dams. One of the most important hydraulic problems of spillways is the high flow velocity, which causes cavitation and scouring at its downstream. Hence, dissipation of the energy of flow is the primary issue of spillways. In this way, using the baffles along the chute of the spillway, stepped spillway, ski jump buckets and stilling basin at the toe of spillways have been suggested (Heller et al. 2005; Movahedi et al. 2019; Xiao et al. 2015). Baffles usually are used in the small dam projects. The use of other mentioned structures is common in large dams projects (Erfanain-Azmoudeh and Kamanbedast 2013). An economic examination of three options including stilling basin, flip bucket and stepped spillway for designing the energy dissipater in large dams indicated that the use of stepped spillway is a logical decision (Christodoulou 1993). The advantage of stepped spillway in comparison with other energy dissipater structures is related to reduce or remove the probability of cavitation occurrence on the spillway (Frizell et al. 2013; Pfister and Hager 2011). The flow pattern over the stepped spillways was classified into three as napped, transition and skimming flows. The napped flow appears in the low discharge, and in the skimming flow, there is a virtual boundary between the jet stream and the steps. The transition regime is a status between the napped and skimming flow (Shahheydari et al. 2014). Although the energy lost in the nape regime is more than the skimming flow, but due economic reasons, the stepped spillways are designed under skimming flow condition. By advancing the computer technology, the investigators have tried to study the properties of flow over the spillway using the numerical methods (Parsaie et al. 2016a, 2018b). Numerical modeling is divided into two main groups as computational fluid dynamic (CFD) and soft computing. In the field of CFD, the governing equations which are usually Navier–Stokes equations are solved along the turbulence models such as K-epsilon and RNG . (Kim and Park 2005). Fortunately, nowadays, a number of user-friendly CFD packages have proposed to easily apply the CFD techniques along the physical modeling to reduce the cost of experiments (Salmasi and Samadi 2018). Along the physical and CFD modeling, another field of numerical modeling, i.e., soft computing techniques have been implemented for accurately present the results of experimental studies which are based on the defining the depended desired parameter with correspond to measuring the independent variables (Azamathulla et al. 2016; Maghsoodi et al. 2012; Najafzadeh et al. 2017; Najafzadeh and Zeinolabedini 2019; Sihag et al. 2019; Wu 2011). Among the soft computing models using the ANFIS by Salmasi and Özger (2014) and the GEP by Roushangar et al. (2014), MARS and GMDH methods by Parsaie et al. (2018a, c) have been reported to predict the energy dissipation over the stepped spillway. Reviewing the literature shows that the M5 algorithm for modeling the capabilities of stepped spillway has not been test; hence, in this a formula based on the M5 algorithm for modeling and predicting the performance of stepped spillways regarding energy dissipation is proposed.

## Materials and methods

### Energy dissipation involved parameters

*h*

_{s}and

*L*

_{s}, respectively.

*H*

_{w}is the height of dam,

*y*

_{0}is the depth of flow over the crest,

*L*

_{c}is the length of crest, and

*y*

_{1}and

*y*

_{2}are the conjugated depths of hydraulic jump, respectively. The energy dissipation over the stepped spillway is estimated using the Bernoulli equation in the upstream and downstream of the spillway. As given in Eq. (1), the total upstream energy is cleared with \(H_{0}\) and downstream total energy as presented in Eq. (2) is calculated with \(H_{1}\).

Range of dataset assigned to stages of prepared soft computing models (Salmasi and Özger 2014)

Stage | Range | |
| DN | | | EDR |
---|---|---|---|---|---|---|---|

Training | Min | 0.234 | 0.094 | 0.000 | 3.000 | 15.000 | 15.136 |

Max | 9.339 | 13.781 | 0.104 | 50.000 | 45.000 | 96.580 | |

AVEG | 4.128 | 2.556 | 0.012 | 18.309 | 33.943 | 59.292 | |

STDV | 1.401 | 2.566 | 0.022 | 14.202 | 13.465 | 23.133 | |

Testing | Min | 0.307 | 0.233 | 0.000 | 3.000 | 15.000 | 13.145 |

Max | 6.883 | 6.327 | 0.109 | 50.000 | 45.000 | 96.441 | |

AVEG | 4.438 | 2.344 | 0.014 | 17.839 | 34.032 | 56.412 | |

STDEV | 1.500 | 1.799 | 0.028 | 15.323 | 14.226 | 23.534 |

#### M5 tree model

*T*the dataset inputs into the tree branches, and

*T*

_{i}the dataset in leafs. SD is the standard deviation. With the growth and development of M5 tree model, it is feared that the performance of the model leads to have so local behavior so usually in the second stage of the model development the pruning the tree is considered. To this purpose, the Quinlan algorithm is used. In this algorithm allowed to the tree to have enough growing then the branched which has not influence effect on the precision improvement is pruned. Figure 2 shows a schematic shape of the M5 tree model development. In Fig. 2a, the

*X*1 and

*X*2 are the input variables (independent parameters) and

*Y*is the output data (dependent parameter), and Fig. 2b shows the tree model development for mapping the input and outputs data.

### Review on ANNs

- 1.
Gaussian: \(F(x) = a\exp \left( { - \frac{{\left( {x - b} \right)^{2} }}{{c^{2} }}} \right)\)

- 2.
Sigmoidal: \(F(x) = \frac{1}{1 + \exp ( - x)}\)

- 3.
Tansing: \(F(x) = \frac{2}{{\left( {1 + \exp ( - 2x)} \right)}} - 1\)

## Results and discussion

*R*

^{2}= 0.99 and in testing stages were RMSE = 2.23 and

*R*

^{2}= 0.99.

To examine the efficiency of the M5 algorithm, its performance was compared with MLPNN a common model of soft computing methods. The same dataset was used for training and testing of MLPNN. The MLPNN which was proposed by Haghiabi et al. (2018) was considered. They recommend that, in order to reduce the trial and error process in designing the structure of the MLPNN, first, a single-layer network, which contains a number of neurons equal to the number of input features, is considered. Then in next stage, the different type transfer functions can be tested to define the best of them. In this study, the structure of developed MLPNN model is shown in Fig. 3. As shown in this figure, the developed MLPNN has one hidden layer which includes five neurons. The best performance of transfer function is related to Tansing function. The performance of developed MLPNN in training and testing stages is shown in Fig. 4. The error indices of MLPNN in training were *R*^{2} = 0.97 and RMSE = 3.73 and in testing stages were *R*^{2} = 0.97 and RMSE = 3.98. Comparing the performance of M5 algorithm with MLPNN shows that the accuracy of M5 algorithm is a bit more than MLPNN. The performance of stepped spillways regarding energy dissipation has been predicted using group method of data handling (GMDH), genetic programming (GP), support vector machine (SVM) and multivariate adaptive regression splines (MARS) (Parsaie et al. 2016b, 2018a, c). According to the reports, the error indices of MARS technique in preparation stages were *R*^{2} = 0.99 and RMSE = 0.65. The error indices of GMDH in development stages (training and testing stages) were *R*^{2} = 0.95 and RMSE = 5.4. The performance of SVM and GP in training and testing was *R*^{2} = 0.98, RMSE = 2.61 and *R*^{2} = 0.96, RMSE = 4.94, respectively. Comparing the performance of developed M5 algorithm with previous applied models shows that the accuracy of M5 algorithm is a bit more than the GP and GMDH, and is a bit less than the MARS and SVM.

## Conclusion

Stepped spillways are hydraulic structures that are commonly used in water engineering and watershed projects. These structures have been very much considered due to the economics, easy in construction and proper operation of energy dissipation and elimination of probability of cavitation. In watershed projects, this type of spillways can also be constructed from local materials such as loose rock dams. In this paper, new formula based on the M5 algorithm was proposed for estimating the performance of stepped spillways regarding energy dissipation. To compare the performance of M5 algorithm with other type of soft computing methods, the MLPNN was chosen. Results of M5 algorithm showed that there is good agreement between observed data and M5 algorithm outputs. Reviewing the structure of formula derived from the M5 algorithm declared that the Drop and Froude numbers are the main parameters used for feature classification in M5 algorithm. Results of MLPNN model showed that this model also has good performance in predicting the performance of stepped spillways regarding energy dissipation of flow. However, the accuracy of M5 algorithm is a bit more than the MLPNN. The best performance of tested transfer function during the development of the MLPNN model is related to Tansing function.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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