Modeling energy dissipation over stepped spillways using machine learning approaches
Introduction
Spillway is a major part of a dam for disposing the flood flows. Energy dissipation over dam’s spillway is usually achieved by (i) a standard stilling basin at downstream of the spillway to dissipate a large amount of flow energy with formatting a hydraulic jump, (ii) a high velocity water jet taking off from a flip bucket and impinging into a downstream plunge pool, and (iii) the construction of steps on the spillway to assist in energy dissipation.
The steps act as roughness elements to reduce flow acceleration and hence terminal velocity. A stepped spillway has a stepped ogee-profile spillway instead of the traditional smooth ogee-profile spillway, where a series of drops are made in the invert from the vicinity of the crest to the toe. Stepped spillway can reduce dimension construction of especial energy dissipaters because of its special shape, thus reduces construction cost and time of the project. Flow over a stepped spillway can be divided into three separate flow regimes, namely, nappe, skimming and transition flows. Nappe flow usually corresponds to relatively low discharges values and skimming flow corresponds to high discharges values while the intermediate discharges have transition flow regimes. In the nappe flow regime, water undergoes a succession of free-falling nappes. In the edge of each step, water becomes a jet of a free descent before it permeates the following step. Schematic representations of nappe and skimming flow regime are shown in Fig. 1. There are three types of nappe flows:
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Nappe flow with fully-developed hydraulic jump for low flow rate and small depth.
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Nappe flow with partially developed hydraulic jump.
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Nappe flow without hydraulic jump.
The skimming flow is characterized by a complete submersion of the totality of the steps which form the spillway. No diving was observed.
So far, considerable physical models have been developed for modeling stepped spillways. However, the obtained results are case-sensitive and could only be used as a preliminary guide for other similar cases. Young (1982) studied the feasibility of a stepped spillway for the upper Stillwater dam and managed a 75% energy reduction. Sorensen (1985) performed a physical model investigation for stepped spillways, where he found that adding a few steps to the face of the spillway eliminated the deflecting water jet. Christodoulou (1993) found that energy loss due to the steps depends primarily on the ratio of the critical depth to the height of the step, as well as on the number of steps. Sorensen (1985) studied the design of steps and their spacing on the spillway face in order to optimize the energy dissipation. Sorensen, 1985, Bayat, 1991, Diez-Cascon et al., 1991 Bindo et al. (1993), and Christodoulou (1993) measured the depth of the flow at the toe or along the length of stepped spillway models. Relative energy loss over stepped spillways, calculated based on these depths, ranged from 50% to 97%.
Christodoulou (1993) proposed an approximated method of estimating the energy loss including the number of steps. Rajaratnam (1990) used the idea of a fully developed region with a Reynolds shear stress between the skimming flow and the recirculating region on the steps, and developed an expression for the relative energy loss over a stepped spillway in terms of that on a smooth spillway and showed that significant energy losses could occur on a stepped spillway. Tozzi (1994) evaluated the variation of the friction factor. Based on experimental observations, the Darcy friction factor “f” was described by the equation:where Y is depth of flow, and k is roughness produced by the step.
Chanson (1994) has presented the following equations for the energy dissipation, for the nappe and skimming flow regimes, respectively:where is energy-loss ratio, ΔE is energy loss, Hdam is dam height (Hdam = Nh), Et is maximum head available, N is step number, h is height of step and yc is critical flow depth, f is friction factor, and α is spillway slope.
Chamani and Rajaratnam (1994) studied jet flow on stepped spillways and based on it presented an equation that is given below:where α is equal proportion of energy loss per steps. h, l and N are height, length and number of steps respectively, yc is equal critical depth, is energy-loss ratio, ΔE is energy loss, Et is equal specific energy at bottom of spillway.
Kells (1995) compared energy dissipation between nappe and skimming flow regimes on stepped chutes. Barani et al. (2005) obtained optimization of dimensions of stepped spillway and investigated flow energy dissipation over a physical model. Tabbora et al. (2005) analyzed flow over stepped spillway with finite element method. Kavianpour and Masoumi (2008) studied two physical models of the Siah Bisheh stepped spillway in Iran to determine the energy dissipation over stepped spillways and Naderi Rad and Teimouri (2010) studied simple stepped spillways to evaluate energy dissipation on them by numerical method.
In the recent years, application of Machine Learning (ML) [e.g. Artificial Neural Networks (ANNs), Neuro-Fuzzy models (NF) and Genetic Programming (GP)] in water resources engineering has become viable leading to numerous publications in this field. A complete review of all the applications is beyond the scope of this paper and only some literatures are mentioned here. Among others, ANNs have been applied for predicting the friction factor of open channel flow (Yuhong and Wenxin, 2009), determining flow discharge in straight compound channels (Zahiri and Dehghani, 2009), long term prediction of river discharge (Cheng et al., 2005), simulating groundwater levels (Taormina, 2012), and estimating daily pan evaporation values at different climatic zones (Kim et al., 2012).
Shiri et al. (2013) applied Genetic Expression Programming (GEP) for estimating daily evaporation through spatial and temporal data scanning. Kisi and Shiri (2012) estimated river suspended sediment by climatic variables implication by using of GEP and ANN. Shiri and Kisi (2012) estimated daily suspended sediment load using the hybrid wavelet-AI models. Shiri et al. (2012) compared various AI techniques for forecasting daily stream flow and found GEP as the superior model in this field. Kisi et al. (2013) introduced GEP as the best model of rainfall-runoff modeling among soft computing techniques.
To the best of the authors’ knowledge, only few studies have been carried out for evaluation of energy dissipation using machine learning approaches. Musavi and salmasi (2008) applied neuro-fuzzy (NF) technique to simulate the experimental data of energy dissipation over stepped spillways. The present work aims at application of GP technique for modeling energy dissipation over stepped spillways and comparing the results with those of ANN models. Consequently, GP (i.e. GEP) and ANN models are developed and tested using the experimental data set from laboratory and their performances are compared.
Machine learning, a branch of artificial intelligence, deals with the representation and generalization using data learning technique. Representation of data instances and functions evaluated on these instances are part of all machine learning systems. Generalization is the property that the system will perform well on unseen data instances; the conditions under which this can be guaranteed are a key object of study in the subfield of computational learning theory. There is a wide variety of machine learning tasks and successful applications (Mitchell, 1997). In general, the task of a ML algorithm can be described as follows: Given a set of input variables and the associated output variable(s), the objective is learning a functional relationship for the input–output variables set. Each target vector z is an unknown function f of the input vector x:
Genetic Expression Programming (GEP) was developed by Ferreira (2001) using fundamental principles of the Genetic Algorithms (GA) and Genetic Programming (GP). GEP is a procedure that mimics biological evolution to create a computer program to model some phenomenon. The problems are encoded in linear chromosomes of fixed-length as a computer program. In other words, a mathematical function is described as a chromosome with multi genes and developed using the data presented to it (Ferreira, 2001). GEP performs the symbolic regression using the most of the genetic operators of Genetic Algorithm (GA). However, there are some differences between GEP and GA. Any mathematical expression defined as symbolic strings of fixed-length (chromosomes) in GA is represented to be nonlinear entities of different size and shapes (parse trees). But in GEP, it is encoded as simple strings of fixed-length which are subsequently expressed as expression trees of different size and shape (Muñoz, 2005, Cevik, 2007). GEP algorithm begins selecting the five elements such as the function set, terminal set, fitness function, control parameters and stop condition.
The advantages of a system like Gene Expression Programming (GEP) are clear from nature, but the most important are (Ferreira, 2001): (1) the chromosomes are simple entities: linear, compact, relatively small, easy to manipulate genetically (replicate, mutate, recombine, etc.); (2) the expression trees are exclusively the expression of their respective chromosomes; they are entities upon which selection acts, and according to fitness, they are selected to reproduce with modification. There are also some problems regarding the GP (GEP) application. For instance, in some cases, it is usually observed that the program size (depth of parse tree) starts growing which leads to producing nested functions (i.e., the Bloat Phenomena) and is not accompanied by any corresponding increase in model fitness. It has some practical effects, because the large programs are computationally expensive to evolve and later use can be hard to interpret. The nested functions give no sense about the physical basis of studied phenomena (Poli and McPhee, 2008). To overcome this weakness, one should employ some penalization of complex models (limitation of the depth of the parse tree). The Parsimony Pressure tool may be considered as a powerful way for removing un-necessary nesting in the programs. Therefore, this method will be applied in the present study (Shiri and Kisi, 2011).
The procedure to model energy dissipation (as dependent variable) by using input variables is as follows:
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Selection of fitness function.
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Choosing the set of terminals T and the set of functions F to create the chromosomes.
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Choosing the chromosomal architecture.
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Choosing the linking function.
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Choosing the genetic operators.
In the present work the GeneXpro program was applied. Fig. 2 illustrates the general structure of a GEP modeling procedure. Detailed information about the aforementioned procedure is given in coming sections.
Artificial Neural Network (ANN) is learning systems that have solved a large amount of complex problems related to different areas (classification, clustering, regression, etc.) (Haykin and Cybenko, 1999), and is a system loosely modeled on the human brain. The field goes by many names, such as connectionism, parallel distributed processing, neuro-computing, neural intelligent system, machine learning algorithms and Artificial Neural Networks. It is an attempt to simulate within specialized hardware or sophisticated software. This simulation is achieved through multiple layers of simple processing elements called neurons. Each neuron is linked to certain of its neighbors with varying coefficients of connectivity that represent the strengths of these connections. A three-layers feed-forward is shown in Fig. 3. Hidden layer contains 3 neurons and output layer has 1 neuron. Input of this network contains 4 entries and output has 1. Learning is accomplished by adjusting these strengths to cause the overall network to output appropriate results. the parameters to be found by training are the weight vectors connecting the different nodes of the input, hidden, and output layers of the network by the so-called error-back-propagation method (a specialized version of the gradient-based optimization algorithm) (Haykin and Cybenko, 1999). During training the values of the parameters (weights) are varied so that the ANN output becomes similar to the measured output on a known data set.
Section snippets
Experimental setup
In present study, several experiments were done in the hydraulic laboratory of Tabriz University, Iran. The stepped section was installed in the flume with Plexiglas sides so that flow could be observed. Water was pumped from the sump to the stilling tank from which water entered the stepped spillway through an approach flume of 10 m length. The discharge was measured by means of magnetic flow meter installed in the supply line whit an accuracy of 5%. Velocities were measured with a
Result and discussion
In this study GEP and ANN models applied for modeling energy dissipation. A trial and error procedure is used to obtain the best percent of data blocks for training and testing phases. The aim of this procedure is to determine the best performance criteria. So three modes were considered that includes 30–70 (i.e. 30% of data for testing and 70% of them for training data) 35–65 and 40–60 modes. Between the three modes, state of 35–65 had best result in the criteria performance. So, in the both
Conclusions
The aim of this study is assessment of capability of machine learning (ML) approaches in predicting of stepped spillway energy dissipation. Consequently, GEP and ANN are applied with 12 input configurations. The data correspond to three kinds of flow regimes, namely, napped, skimming and combination of them (total data) were considered for modeling issue.
For total data (with unknown flow regime), M4 model comprising as input variables, is selected as best model. According to this
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