Genetic algorithms for optimizing stepped spillways to maximize energy dissipation Open Access

The following symbols are used in this paper:

Flow energy downstream of the smooth spillway (m);

Flow energy downstream is a stepped spillway (m);

E_res

Energy dissipation at the toe of spillway (m);

E_max

Maximum energy upstream of the spillway (m);

H_dam

Spillway height (m);

y_c

Critical depth over spillway crest (m);

ΔE/E_max

Relative energy dissipation (dimensionless);

α

Spillway slope (degrees);

w

Spillway width (m);

h

Step height (m);

H_dam

Spillway height (m);

H_d

Head over the spillway crest (m);

A

Functional relationship between variables (dimensionless);

f

Friction coefficient at the spillway surface;

f_max

Maximum coefficient of friction (dimensionless);

q

Flow rate per unit width of spillway (m²/s);

n

Number of chromosomes in a population (dimensionless);

f(x)

Fitness function (dimensionless);

d

Water depth in downstream spillway (m);

V

Velocity in downstream spillway (m/s);

F_r

Froude number in the downstream spillway (dimensionless);

L_stilling

Length of stilling basin (m);

H_stilling

Height of stilling basin wall (m);

EDR

Energy dissipation rate (dimensionless)

A stepped overflow spillway consists of steps that start near the crest of the spillway and continue downstream of the spillway. High-energy dissipation is one of the most important responsibility for this type of spillway, so that a large part of the kinetic energy of the flow is eliminated as the water passes successive steps (Roushangar et al. 2014). With stepped spillways, there is no need to use energy dissipation structures, which in turn reduces construction and operating costs (Khatibi et al. 2014).

Due to the increasing importance of stepped spillways, many research studies have been carried out to identify factors that affect energy dissipation. These studies have mainly utilized scaled laboratory tests with physical models (e.g. Pegram et al. 1999; Roushangar et al. 2018). In addition, a few studies have relied upon numerical simulation (e.g. Eghbalzadeh & Javan 2012; Salmasi & Samadi 2018).

There are three types of flow over stepped spillways which will now be discussed (Rajaratnam 1990; Chanson 1994):

• 1.

  Nappe flow: This type of flow occurs for low discharge and typically with large steps. Energy dissipation is caused by the mixing of the water jet with air, turbulence, and the formation of a complete or partial hydraulic jump on each step.

• 2.

  Skimming flow: In this type of flow, a pseudo-bottom is formed which connects the end lips of consecutive steps to each other. Eddies are formed so that much of the energy is captured by rotating currents. Stepped spillways are designed based on skimming flow regimes (Salmasi et al. 2020).

• 3.

  Transition flow: This type of flow is intermediate between nappe and skimming flow regimes and is not very important in terms of design. Oscillating flow over a stepped chute is a flow characteristic.

In Figure 1, sketches of the three flow regimes are provided.

Sorensen (1985) conducted a series of experiments on stepped spillways with ogee profiles and showed that steps can dissipate more energy than smooth spillways. Christodoulou (1993) found that the most important parameter affecting energy loss in stepped spillways is the ratio of critical depth (y_c) to step height (h) and the number of steps (N). Christodoulou (1993) showed that for a given discharge, the relative energy dissipation decreases with increasing N×h (which is approximately equal to the total height of the spillway). Chamani & Rajaratnam (1999) showed that the relative energy loss for stepped spillways is in the range 48–63%. They also found that for a certain discharge and constant values of h/l (h is the step height and l is the horizontal step length), a larger h leads to a greater energy loss. Fratino et al. (2000) performed experiments on stepped spillways in Zurich, Switzerland. They concluded that by reducing the downstream slope of the spillway, more energy is dissipated when the flow regime is in transition or skimming. Chaturabul (2002) carried out experiments on physical models to investigate the energy loss and flow velocity over stepped channels with end sills. They concluded that the existence of an end sill can increase the relative energy loss by ∼8%.

Ohtsu et al. (2004) proposed a method for designing stepped spillways, such that for constant values of slope, width, spillway height, step height, and flow rate, the amount of residual energy, depth and downstream water velocity can be calculated. They presented a step-by-step design method for stepped spillways. Tabbara et al. (2005) used the finite element method (FEM) to calculate the water surface profile on a stepped spillway using ADINA-f software. The calculated water surface profile was qualitatively consistent with the observed flow characteristics and was similar to the physical models. In addition, the relative energy dissipation was calculated and compared with the values obtained from the experiment. The comparison showed good agreement between the numerical and laboratory values.

Parsaie & Haghiabi (2019) used the M5 algorithm and multilayer perceptron neural networks (MLPNN) to estimate energy dissipation of flow over stepped spillways. They used the dataset published by Salmasi & Özger (2014). They indicated that the drop number and Froude number are the two important dimensionless parameters for energy dissipation. In addition, the accuracy of M5 algorithm showed an improvement compared to the MLPNN. Hassanvand et al. (2019) implemented the meta-heuristic harmony search algorithm (HSA) to select the best spillway type for Qeshlagh Dam, Iran. They considered four spillway types (free-flow, stepped, semicircular and cylindrical) in their study. The results showed that a free-flow spillway was the best choice in terms of cost and construction time. Jazayeri & Moeini (2020) refer to the high cost for dam spillway construction. To reduce this high cost, they considered the Tehri dam stepped spillway in India. The authors used the improved artificial bee colony (IABC) approach and improved particle swarm optimization (IPSO) for the optimal design of spillway. Ghorbani Mooselu et al. (2019) studied the Jarreh dam, Iran to optimize its stepped spillway dimensions. They used a Fuzzy Transformation Method (FTM) for optimization algorithm. In addition, they applied the FLOW3D software to complete the numerical simulations of the stepped spillway. Denga et al. (2020) used the Adleman–Lipton model as a bio-inspired algorithm in their study. Wang et al. (2021) used the optimized grey and Markov models for predicting domestic water consumption.

Genetic Algorithm (GA) is an evolutionary method. The advantages of GA compared to other search methods include:

• The nature of random search of this algorithm in the problem space is a type of parallel search. Because each of the random chromosomes generated by the algorithm is a new starting point for searching part of the problem state space, searching all of them simultaneously provides a significant advantage.

• Due to the breadth of the points being searched, GA achieves good results in a large search space.

• It is a kind of targeted random search that reaches different answers from different paths. In addition, there are no restrictions on the search and selection of random answers.

• The GA approach, through the use of competition, will select the optimum results from the population.

• It is simple to implement and does not require complex problem-solving procedures.

Evolutionary methods like GA have some shortcomings, which are mentioned below.

First, GA is a slow analysis method. Some problems with this method may require days or even weeks. Especially when issues are intricately intertwined. The main problem with GA is its high-operating costs. To solve problems, it is usually necessary to store several hundred chromosomes in memory and run the algorithm for several thousand generations. In addition, GA does not offer a guaranteed solution unless an appropriate interval is defined. This means that GA must has an objective function in order to be implemented.

The purpose of this study is to provide a method for optimizing stepped overflow spillways using a GA. A goal is to maximize energy dissipation on the stepped spillway. The Sarogh Dam smooth spillway was considered for the case study. The goal is to provide an optimized stepped spillway as an alternative to the existing smooth spillway. Maximizing energy dissipation downstream of a spillway helps reduce stilling basin dimensions. The decision variables include the width and height of the steps as well as the slope and height of the overflow spillway. Sarogh Dam spillway was designed and built as a smooth spillway. Inserting steps on this spillway and then finding the appropriate dimensions of the steps has not been performed.

This spillway is a smooth chute spillway having a 28 m height. The Sarogh Dam is located in West Azerbaijan province, 5.17 km northwest of Tekab city, Iran. Qaraqeh River is one of the tributaries of Sarogh River and the Sarogh River originates from the mountains of Tekab city and is one of the tributaries of the Zarrineh River. The purposes of this dam are:

• 1. Provide an annual supply of 42 million cubic meters (MCM) of agricultural water to the downstream regions.

• 2. Result in an annual supply of 3.10 MCM of water supply for Tekab city.

• 3. Serve as a modern irrigation and drainage network for 5,550 downstream hectares.

Sarogh Dam is a rock-filled dam with a vertical clay core. The height from the riverbed is 66.5 m, the crest length is 418 m, body volume is 174 million m³, total reservoir volume is 40 MCM, and the useful reservoir volume is 35 MCM. The height of the chute is 25 m, the spillway width is 23 m, and a diversion system includes a tunnel length of 296 m. A photograph of Sarogh Dam is provided in Figure 2.

The discharge rates were 560, 776.9, and 422 m³/s, which are the design discharges of the spillway with a return period of 10,000 years, the probable maximum flood (PMF), and the discharge with a return period of 1,000 years, respectively. The location of Sarogh Dam is shown in Figure 3.

A spillway design discharge is usually a maximum flood discharge with a specified return period (10,000 years in this study). The design discharge was considered to be a known parameter. Since the flow is skimming, the optimal design of the spillway is determined using the skimming flow regime.

The term H_e/y_c is a dimensionless parameter that is used to separate quasi-uniform and non-uniform flows in stepped spillways. If ⁠, the flow is quasi-uniform and if ⁠, the flow is non-uniform.

For these two types of flows, the methods of calculating E_res for quasi-uniform and non-uniform flows are as follows (Ohtsu et al. 2004):

Constraints consist of a set of limitations that can be assigned to any of the parameters of the objective function. In this study, the constraints of the problem are:

• 1.
  The flow generated on the spillway must be of the skimming type. Accordingly, the following relation should be established.

  

  (13)

The value of P is 4.8 m.

• 4.
  The space allocated for the construction of the spillway can also be a limitation. Thus, the horizontal length of the spillway area cannot exceed a certain value. Considering that the horizontal length of the spillway is directly related to its slope and height, if the height of the spillway is indicated by H_chute and its slope is indicated by α, the following relation holds:

  

  

  (20)

Genetic algorithms are an effective search method for very large spaces that ultimately lead towards an optimal answer. This algorithm simulates the processes of survival in biological sciences to find the most appropriate answer to a problem. In a natural system, by choosing the right offspring, they have a higher ability to survive and reproduce, and after several generations, they reach a higher degree of competence. The natural process of selection is simulated by combining genetic operators to mimic the natural processes of selection. The steps of implementing the GA are as follows:

• 1.

  The problem is expressed in genetic language. Problem variables are encoded and are usually expressed as genes consisting of zero and ones (Sivanandam & Deepa 2008).

• 2.

  A population consisting of n chromosomes is formed randomly.

• 3.

  The fitness function f(x) is calculated for chromosome x.

• 4.

  A new population is generated in the following steps:

• • • Selection: Two chromosomes are selected based on parent competence. Chromosomes with higher competence are more likely to be selected.

  • • Crossover: The connection or non-connection is determined by a random number. If a transplant takes place, the parents become the new children, and if a transplant does not take place, the children will be exactly a copy of their parents.

  • • Mutation: whether or not a mutation takes place, is determined by a random number. The mutation changes some of the components of the new offspring.

  • • Acceptance: New children are placed in the new population.

• 5.

  The newly produced population is used to carry out the following steps.

• 6.

  If the final condition is met, the process is stopped and the best population response is provided.

• 7.

  If the condition is not satisfied, the second step is repeated.

In Figure 4, GP is described using a flowchart.

After specifying the objective function and its computational process and the specification of the problem constraints, a program m-file was written using Matlab software (Version R 2020b). The problem decision variables were entered into the program as functions. After determining the number of variables, the GA toolbox in Matlab software was used to determine the optimal combination of these variables so that the relative energy dissipation was a maximum.

During the optimization process, the objective function is multiplied by (−1). After entering the required information in the GA toolbox, the algorithm performs a search and the calculation process continues until one of the conditions related to stopping the algorithm are met. The selection of different options in the toolbox that determine the population and the probability of mutations and transplants by trial and error. The values of the parameters and the types of functions used to optimize the step are presented in Table 1.

The optimal design for a stepped spillway was based on the characteristics of the Sarogh Dam spillway. The schematic diagram of the Sarogh spillway and the decision variables are shown in Figure 5. Figure 6 shows the Sarogh spillway during its construction. The acceptable limits for design variables in the optimization process of this spillway are presented in Table 2.

In this section, the optimal dimensions for stepped spillways with different horizontal lengths (L) are presented. The optimization program achieved the highest relative energy dissipation with a design flood discharge of 560.2 m³/s and a horizontal spillway length of L = 230 m. Iteration 55 resulted in the largest energy dissipation (Figure 7). Each iteration consists of 150 GA generations; with each generation, the fitness function moves towards the optimal value. The GA stops when the fitness function reaches a constant value for several consecutive generations.

Figure 7 shows the evolution of fitness values with total 150 GA generations. As seen in the figure, the algorithm stops when the fitness function has reached a constant value. Figure 7 shows that the fitness function converges rapidly for the first ∼30 iterations.

By implementing the optimization program, the optimal values of the spillway slope, the spillway width, the step height, the spillway height, and the water head on the spillway crest were determined. Using these results, other factors required for spillway design can also be obtained. The optimization program was also performed for horizontal lengths of 220, 210 and 200 m; for each of these lengths, after ∼50 iteration, optimal solutions were obtained.

Table 3 shows the results of the optimization program for different horizontal lengths and for a design discharge Q = 560.2 m³/s.

In order to investigate situations where the actual discharge differs from the design discharge (Q_d), optimizations were performed for the probable maximum flood (PMF) and floods with a return period of 1,000 years. For this part of the analysis, two other discharges, Q = 776.9 m³/s and Q = 422 m³/s, were selected as inputs to the optimization program. Then the optimal values for different lengths were determined. The fitness function and the optimal value for the desired horizontal lengths are shown in Figures 8 and 9. Results of stepped spillway optimization with discharge values of Q = 776.9 m³/s and Q = 422 m³/s are presented in Tables 4 and 5.

From Tables 3–5, it can be seen that the optimal height of the steps for different slopes of the spillway are similar. The optimization program was implemented to further investigate this issue for a wider range of the spillway slopes. In order to achieve more general results, the slope of the spillway varied from ⁠. The results of optimization and the changes in relative energy dissipation with spillway slope and y_c/h are provided in Figures 10 and 11.

Based on Figure 10, with increasing chute slope, the relative energy dissipation decreases from 68.96% to 58.5% for Q = 776.9 m³/s. For Q = 560.2 m³/s relative energy dissipation decreases from 73.97% to 63.19%. Finally, for Q = 422 m³/s, the relative energy dissipation decreases from 76.87% to 66.39%.

Figure 11 shows that with increasing discharge, the relative energy dissipation decreases. A physical explanation for this phenomenon is that as the flow rate increases, the steps on the spillway will sink below the water surface and their roughness will decrease. In this case, the skimming flow regime is formed on the spillway. As mentioned earlier, the rotation of water under the pseudo-bottom causes energy dissipation. This shows that the energy dissipation not only depends on the rough geometry, but also depends on the hydraulics of the flow (depth or discharge). This issue has already been discussed in Salmasi et al. (2020).

The highest relative energy dissipation relates to the lowest chute slope, i.e., h/l = 0.12. For this chute slope, the relative energy dissipation changes from 76.45% to 68.42% depending on the discharge.

The changes in the optimal height of the steps relative to the discharge are presented in Figure 12. Three discharges of 422, 560.2 and 776.9 m³/s were used. The height and width of the spillway were fixed for different discharges and slopes (H_dam = 26 m and w = 25 m). Figure 12 shows that with increasing discharge over the stepped spillway, the optimum step height increases, and the number of steps decreases. In fact, with increasing discharge, step roughness decreases due to submergence. Thus, for obtaining greater energy dissipation, GA suggests larger steps. The step heights increase from 1.64 m to 2.25 m as the discharge changes from 422 m³/s to 776.9 m³/s for h/l = 0.177. In addition, the chute slope (h/l) does not significantly affect step height.

Figure 13 provided a comparison of relative energy dissipation (ΔE/E_total) between stepped and smooth spillways. Relative energy dissipation for both spillways have been plotted against discharge (Q). Average relative energy dissipation for stepped spillway is 73.27% while for smooth spillway it is 35.52%. This shows that in optimized stepped spillway, 106.23% more energy dissipation occurs. Obviously, more energy dissipation on stepped spillway leads to a reduction of the downstream stilling basin dimensions. In addition, increased air entrainment in stepped spillways can prevent cavitation in the chute bottom. Usually high velocities occur downstream of the spillway/chute. The fast flow can cavitate and destroy a concrete chute. Stepped spillways have the positive effect of reducing velocity downstream of the spillway and thus some cavitation facilities like a chute bottom ramp or tunnels for chute aerators are unnecessary. Some techniques of chute aeratin can be found in Pfister et al. (2011).

Although this study has not dealt with the economic aspects of stepped spillway construction, the results of this study show that by optimizing the geometric dimensions of the stepped spillway, the maximum energy loss can be achieved. Having the maximum energy loss will help reduce the size of the stilling basin and will be effective in leading to a cost-effective project.

In order to evaluate the validity of the results obtained from the optimization with the objective function presented by Ohtsu et al. (2004), Figures 14–16 are provided.

Figures 14–16 show that increasing the discharge leads to a decrease in the relative energy dissipation. For example, in a constant spillway slope of h/l = 0.577, increasing the discharge from 400 to 700 m³/s causes a decrease in the relative energy dissipation from 62% to 56%. With increasing step height, the relative energy dissipation initially shows an upward trend but thereafter reaches a constant value. Consequently, changing the height of the steps has little effect on relative energy dissipation. For a constant discharge, increasing the spillway slope leads to a decrease in relative energy dissipation.

GA has advantages that make it more powerful than other problem-solving methods. GA is flexible and can be used for a wide range of problems. An important advantage of GA is that it simplifies complex problems and solve problems with excessive limitations with faster corrections. Then, because this method is progressive, intermediate answers can be extracted at any time. GA provides a number of optimal answers in each stage and subsequent stages provide better quality answers than the previous stage. This provides relatively optimal solutions if operations are stopped in the middle of the solution.

In this study, energy dissipation was studied for the Sarogh Dam spillway. This dam is located in the province of West Azerbaijan in Iran. The spillway of this dam is a smooth design (without steps). The performance of a stepped alternative was also considered. The objective function maximizes energy dissipation downstream of the spillway. Due to the large number of variables and the non-linearity of the objective function and constraints, the GA was used to maximize the objective function.

This study showed that GA is successful in finding the optimal solution of the objective function. The energy loss for a stepped spillway is significantly greater that when the spillway is smooth. Although the construction cost was not calculated for the two types of spillways, it was hydraulically proven that increasing the energy dissipation in the stepped spillway would lead to a more economical project. Also, the application of GA for optimizing the dimensions of stepped spillway was discussed; the use of other optimization methods are beyond the scope of this study. Other optimization methods such as the meta-heuristic HSA, Improved Artificial Bee Colony (IABC), Improved Particle Swarm Optimization (IPSO), FTM can be considered as a suggestion for future studies. The main results are as follows:

• 1

  Application of GA determined the best combination of stepped spillway geometry that provide the greatest relative energy dissipation.

• 2

  Conversion of a smooth spillway to a stepped spillway significantly improved the reduction in the kinetic energy of the flow downstream of the spillway and thus reduced the dimensions of the stilling basins at the spillway toe. This, in turn, reduces the volume of concrete for stilling basin construction and reduced the costs associated with construction and operation.

• 3

  In this study, the EDR value for stepped spillway with a height of 28.66 m and with different discharges was in the range 42.14–60.42%, which according to the results obtained by Pegram et al. (1999), lay within acceptable limits. According to Pegram et al. (1999), for a stepped spillway with a height of 58 m and a step height of 2 m, the EDR value reduced from about 60% for y_c = 0.3 m to 50% for y_c = 3.5 m.

• 4

  Examining the changes in the optimal step height to the discharge for different slopes of the spillway, it was found that for a constant discharge, the optimal height of the steps is almost constant. In other words, the optimal height of the steps has no effect on energy dissipation and what leads to changes in energy dissipation is the change in the slope of the spillway. This was confirmed by sensitivity analysis the of relative energy dissipation with respect to the design parameters. It should also be noted that the increase in discharge is followed by an increase in the optimal height of the steps.

The authors declare that they have no conflict of interest.

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

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

This paper is the outcome of a research project supported by the University of Tabriz research affairs office.

All relevant data are included in the paper or its Supplementary Information.