## Abstract

The significant role of open channels in agriculture include supplying drinking water, industry, irrigation and flood control, making these hydraulic structures an integral part of the water conveyance system. Determination of optimum dimensions with minimum construction costs is considered as the primary concern when designing artificial open channels. To achieve this, the compound channels were evaluated with the following constraints, viz. composite roughness, velocity, Froude number and channel stability. Grey Wolf Optimization (GWO) was used to determine the optimal geometry of the channel. Optimization results clearly showed that the variation of roughness coefficient and the increase of factor of safety increased costs by 60 and 20% respectively. The optimum suitable cross-section for the compound channels was obtained by conducting various model scenarios.

## HIGHLIGHTS

Grey wolf optimization (GWO) is proposed as an evolutionary method to achieve the optimum dimension of compound channels.

Two types of compound channel sections are investigated.

The constraints of roughness coefficient, velocity, Froude number and channel stability are evaluated.

Variation of roughness coefficient significantly impacts the construction cost.

## INTRODUCTION

Channels are the most important and major elements of any irrigation system. The building of the channels is an expensive part of the channel construction process; therefore, an economic design of the channel sections is a major concern of civil engineers. An economic channel section may be defined as the channel section that has the least total construction cost when considering all hydraulic parameters. The cost of the channel construction usually includes the cost of excavation, surface lining and maintenance. Thus, the first step of designing the channel is to determine the optimal dimensions to convey the required discharge with the minimum construction cost.

Generally, compound channels consist of a main channel and several floodplains with different geometry and roughness coefficients. Their cross-sectional design is generally based on a one-dimensional (1D) analysis of steady flow in which the composite roughness is conventionally expressed in the equivalent form. A number of mathematical models have been developed over the years in order to optimize the cost of construction. Among them, Trout (1982) offered a direct algebraic method to design an optimum channel with minimum lining material cost. Kacimov (1992) used the method of complex variables and series expansion to design an optimized channel section in which the cost function takes seepage losses and channel lining into account. Furthermore, successful application of the Lagrange multiplier method (LMM) in the optimization process of channel cross-sections has been reported in the literature (Froehlich 1994; Monadjemi 1994; Babaeyan-Koopaei *et al.* 2000; Das 2000; Anwar & Clarke 2005; Blackler & Guo 2009; Han *et al.* 2017). Furthermore, Swamee *et al.* (2000) made another attempt to design an optimized channel with different sections by considering the seepage losses. Ghazaw (2010) prepared a set of design charts in order to facilitate an easy design of the optimal channel dimensions and ensure minimum water loss from the channel section.

In recent years, various metaheuristic methods have been implemented for solving complicated nonlinear hydraulic problems (Bakhtyar & Barry 2009; Suribabu 2010; Ferdowsi *et al.* 2020). There are numerous published studies reporting the role of metaheuristic methods in the optimization process of open channels, one of which is the research carried out by Bhattacharjya & Satish (2007). He applied sequential quadratic programming (SQP) to derive a cost-effective open channel section incorporating critical flow conditions in the channel. Swamee *et al.* (2002) used a non-linear optimization technique for obtaining an explicit design equation with optimized roughness coefficients for triangular, rectangular and trapezoidal cross-sections. Jain *et al.* (2004) used a genetic algorithm (GA) and conducted research on optimization of channel dimensions and found that estimation of equivalent roughness led to a considerable cost reduction. Nourani *et al.* (2009) found that ant colony optimization (ACO) surpasses the GA in terms of the optimal designing of trapezoidal channel cross-sections. Reddy & Adarsh (2010) used the particle swarm optimization (PSO) method to optimize dimensions with different geometries and hydraulic constraints and presented the figures of optimized design for the trapezoidal compound channels. Sankaran & Manne (2013) also reported the effectiveness of the PSO algorithm in the optimization of a composite channel with the cross-sectional shape of a horizontal bottom and parabolic side. Orouji *et al.* (2016) confirmed the high potential of the shuffled frog-learning algorithm (SFLA) as a memetic metaheuristic method in determining the optimum design of open channels for increasing economic benefits. Liu *et al.* (2016) adopted cat swarm optimization (CSO) to find an optimal solution for designing the channel trapezoidal section and concluded that this method excels the GA and PSO methods that are being used for optimization of channel cross-section. Roushangar *et al.* (2017) explored the effect of several hydraulic constraints in designing the optimal trapezoidal channel cross-section through the GA algorithm. More recently, El-Ghandour *et al.* (2020) used PSO in order to achieve an optimal solution for designing irrigation canals. Farzin & Valikhan Anaraki (2020) used a combination of PSO and a bat algorithm (BA) for the design of trapezoidal open-channel cross-sections with minimum construction cost. Their hybrid model demonstrated higher potential for the optimal design of open channels. Niazkar (2020) proposed an artificial neural network (ANN) and GP as artificial intelligence models to design optimum trapezoidal-family lined channels. Some of the aforementioned methods showed poor performance in different field conditions. Some algorithms perform local exploitation at the mature stage of the search and a global exploratory search at the early stages of the evolutionary process. Few of the aforementioned methods demonstrate excellent global search capabilities but, they have some restrictions in their local search ability. Some of the techniques discussed above resulted in premature convergence. Therefore, the application of a more powerful heuristic method seems to be necessary in order to prevent premature convergence and accelerate the search process. The grey wolf optimization (GWO) algorithm was designed not to stay with local optimum points in the complex multimodal optimization problems. The nature of this algorithm provides a more varied search of the solution space for solving complex problems. Improved optimum solutions with reduced computation burden can be accessed in GWO in comparison to the existing stochastic search methods. The GWO is superior to these methods because: (i) The GWO has better capability for information sharing and the benefits of an improved conveying mechanism; (ii) it utilizes random function and considers three candidate solutions for getting the best performance and converges quickly from local minima towards global minima. The main advantage of the GWO algorithm over most of the widely known meta-heuristic algorithms is that the GWO algorithm operation requires no specific input parameters. Additionally, it is straightforward and free from computational complexity.

A review of the literature shows that despite considerable investigations into finding a reliable solution for minimizing the cross-sectional area of channels, the combined effect of different Manning roughness coefficients, hydraulic constraints and channel stability have not been considered. In this study, GWO was employed with two scenarios for the optimization process of two types of channels, including a composite trapezoidal channel and a compound channel. In the first scenario, different roughness coefficients were considered following the Manning equation and moreover, the maximum velocity and Froude number were taken into account as hydraulic constraints in order to convey the required discharge safely. In the second scenario, a set of two different models were used to evaluate the stability of channels.

## OPTIMAL DESIGN FORMULATION

### The costs of earthwork, lining and water loss

*A*= flow area (m

^{2}); = depth of the centroid of the area of excavation from the ground surface (m).

*r*

*=*rate of interest ($/$/year) and = cost per unit volume of water ($/m

^{3}). The volumetric cost of water may differ for evaporation and seepage losses, depending upon the side effects caused by the seepage loss.

*k*= coefficient of permeability (m/s); = normal depth of flow in the channel (m) and

*Fs*= seepage function (dimensionless), which depends on channel geometry.

*T*= width of free surface (m) and

*E*= evaporation discharge per unit surface area (m/s).

*R*= hydraulic radius (m);

*S*

_{0}= longitudinal bed slope (dimensionless) and = equivalent roughness coefficient. Utilizing the continuity equation, the discharge

*Q*(m

^{3}/s) is obtained as:

The subscript * denotes the corresponding dimensionless parameters of each hydraulic parameter.

### Grey wolf optimization (GWO) algorithm

GWO is a newly developed intelligent algorithm inspired by the hierarchy and social behavior of grey wolves trying to hunt in nature and was proposed by Mirjalili *et al.* (2014). In general, the pack of wolves can be split into four groups; Alpha (*α*), Beta (*β*), Delta (*δ*) and the rest of the wolves are known as Omega (*ω*). The Alpha is the leader of the pack and can be considered as the most dominant wolf as the group follows his/her instructions. The domination level decreases from alpha to omega as depicted in Figure 1(a).

The GWO mechanism is conducted based on splitting a set of solutions to the given optimization problem into four groups. The first three solutions are regarded as *α*, *β*, *δ* and the remaining solutions refer to *ω* wolves. For the implementation of this mechanism, the hierarchy in each iteration is updated in accordance with the three best solutions. The illustration of the update location is shown in Figure 2.

*t*represents the current location, represents the matrix location and represents the distance separating the grey wolf and the prey, which can be calculated as follows:where and are randomly generated from (0 to 1). The previous equations allow a solution to relocate around the prey in a hyper-sphere form (Figure 2). However, this is not adequate to simulate the social intelligence of grey wolves. For simulation of the prey, the best solution obtained so far is considered as the alpha wolf is closer to the prey position, but the global optimal solution is unknown, so it is supposed that the top three solutions have a good idea of their location, therefore other wolves should be obligated to update their locations by using the following equations:where , and are calculated through the following equations:where , and can be determined by:

The prey encircling and attacking are repeated until an optimum solution is obtained or it reaches the maximum number of iterations.

The solution procedures for the channel cross-section are shown in Figure 3. Design of minimum cost irrigation channels involves minimization of the total earthwork cost, which varies with channel depth, cost of lining, and cost of water lost as seepage and evaporation, subject to uniform flow conditions in the channel.

## OPTIMAL DESIGN OF CHANNEL

### Scenario 1

The first type of compound channel, as depicted in Figure 4(a), was considered with different Manning roughness coefficients for the bed and sides, as well as various side slopes (*m*_{1} and *m*_{2}).

The first type consists of four models:

**Model I:** The roughness coefficient *n*_{1} changes on the bed, as any other coefficient is constant.

**Model II:** The roughness coefficient *n*_{2} changes on the side slope, as any other coefficient is constant.

**Model III:**The additional constraint of the Froude number. The constraint of Froude number is necessary for several cases. Then, an additional restriction can be imposed on the nonlinear optimization problem as represented by Equations (10) and (11). Fhe formulation of optimization remains the same, except the following additional restriction is imposed to restrict the Froude number to

*F*

_{max}:where = maximum permissible Froude number.

For the designed sections, the average flow velocity *V _{av}* could be achieved by Equation (24).

*n*is the roughness coefficient,

*P*is the wetted perimeter and

*i*shows the divided parts (main channel and floodplains). Developed models for the second type compound channel are listed in Table 1.

Models . | The roughness coefficient changes . | |||
---|---|---|---|---|

n_{1} (on the bed of main channel)
. | n_{2} (on the side slope of main channel)
. | n_{3} (on the bed of bed floodplain)
. | n_{4} (on the side slope of the floodplain)
. | |

I | 0.012–0.03 | Constant | Constant | Constant |

II | Constant | 0.018–0.063 | Constant | Constant |

III | Constant | Constant | 0.025–0.07 | Constant |

IV | Constant | Constant | Constant | 0.028–0.082 |

Models . | The roughness coefficient changes . | |||
---|---|---|---|---|

n_{1} (on the bed of main channel)
. | n_{2} (on the side slope of main channel)
. | n_{3} (on the bed of bed floodplain)
. | n_{4} (on the side slope of the floodplain)
. | |

I | 0.012–0.03 | Constant | Constant | Constant |

II | Constant | 0.018–0.063 | Constant | Constant |

III | Constant | Constant | 0.025–0.07 | Constant |

IV | Constant | Constant | Constant | 0.028–0.082 |

### Scenario 2

Trapezoidal channels have slopes at the sides. The aim of slope stability analyses is to contribute to the safe and economical design of excavation, embankment, earth dams and earth channels. In this situation, nonclassical algorithms such as the GWO algorithm may provide better performance. The stability of an earthen slope can be determined by calculating the factor of safety (FS) for the most critical slip circle, which generally is dependent on the soil parameters. The soil parameters are cohesion *C*, friction angle *φ* and unit weight of the soil *γ*. By increasing the slope degree, the probability of the earth channel collapse increases and if the slope gradient is less than a specific value, the channel construction cost will increase. Thus, the slope of the sides should be determined in a way that meets the stability and economic criteria at the same time. Figure 5(a) shows a homogeneous earthen slope with a trial slip circle. Let *BC* be the trial slip circle with radius *r*. The center of the slip circle is O (*p, q*) and B (0, 0) is the toe of the slope. The stability of the channel slope can be ascertained by calculating the safety factor for the most critical slip circle (Easa *et al.* 2011).

The minimum *FS* required for the stability of the channel side slope is considered as 1.5. The limit equilibrium method of the slice method was employed to calculate the critical *FS*. Fellenius method of slices is one of the most frequently used iterative procedures to calculate the *FS* for an earthen slope (Fellenius 1936). In this scenario, two models were analyzed.

#### Model I

*FS*(Equation (26)). The

*FS*for the critical slip circle is calculated using the developed GWO algorithm based on the optimization model:where and are the upper limits and and

*q*are the lower limits for the coordinate of the center of the slip circle with a radius of

_{l}*r*. The coordinates of the slip circle are determined as X-Y coordinates (Figure 5(b)) (Sengupta & Upadhyay 2009).

#### Model II

*FS*was applied as an additional constraint function (Equation (27)) to the Manning constraint on the objective function (cost function):The

*FS*of the side slopes should be less than the permissible

*FS*(Fp).

*L*is the length of the failure arc. The parameters

*C*(effective cohesion), and (effective internal friction angle) are known constants.

*T*is the tangential slice of each slice and N is the normal force.

This limitation on the side slopes' *FS* is incorporated with an inequality constraint in the optimization model. This process continued until the following function was established

## RESULTS AND DISCUSSIONS

### First scenario

The compound channel was designed to transport a discharge of 100,200 m^{3}/sec on a longitudinal bed slope of 0.0001. It passes through a stratum of typical soil in which , and were calculated for the roughness In this scenario, different values of roughness coefficients were considered for different bed covers and sides.

#### First type (constraint of manning equation with different roughness coefficients)

In the first model, different values from 0.012 to 0.03 were considered for *n*_{1}, which is the roughness coefficient of the channel bed (*n*_{2} and *n*_{3} were considered as 0.013 and 0.014, respectively). In this model, the discharge was considered as 100 m^{3}/s. In the second model, *n*_{2} was considered as a variable over the range 0.018–0.063. The results obtained using different values of *n*_{1} and *n*_{2} for composite trapezoidal are tabulated in Table 2.

Model . | n_{1}
. | n_{2}
. | n_{3}
. | Fr . | V (m/s) . | n
. _{e} | b
. | h
. | m_{1}
. | m_{2}
. | C($/m) . |
---|---|---|---|---|---|---|---|---|---|---|---|

Model I | 0.012 | 0.013 | 0.014 | 0.247 | 1.582 | 0.013 | 5.410 | 6.177 | 0.716 | 0.847 | 554.012 |

0.016 | 0.013 | 0.014 | 0.226 | 1.485 | 0.014 | 6.182 | 6.285 | 0.502 | 0.942 | 574.200 | |

0.020 | 0.013 | 0.014 | 0.216 | 1.404 | 0.015 | 6.055 | 6.320 | 0.957 | 0.694 | 595.581 | |

0.024 | 0.013 | 0.014 | 0.197 | 1.329 | 0.017 | 6.042 | 6.751 | 0.705 | 0.806 | 616.559 | |

0.028 | 0.013 | 0.014 | 0.183 | 1.241 | 0.018 | 6.741 | 6.750 | 0.896 | 0.645 | 641.865 | |

0.030 | 0.013 | 0.014 | 0.183 | 1.215 | 0.018 | 6.238 | 6.712 | 0.559 | 1.237 | 655.661 | |

Model II | 0.011 | 0.018 | 0.014 | 0.230 | 1.438 | 0.015 | 6.1845 | 5.899 | 0.974 | 0.927 | 588.030 |

0.011 | 0.028 | 0.014 | 0.179 | 1.213 | 0.019 | 6.641 | 6.797 | 0.693 | 0.921 | 652.063 | |

0.011 | 0.038 | 0.014 | 0.146 | 1.036 | 0.023 | 7.164 | 7.427 | 0.695 | 0.877 | 721.518 | |

0.011 | 0.048 | 0.014 | 0.120 | 0.881 | 0.029 | 8.025 | 7.944 | 0.893 | 0.683 | 800.745 | |

0.011 | 0.058 | 0.014 | 0.108 | 0.819 | 0.032 | 8.538 | 8.289 | 0.629 | 0.865 | 839.845 | |

0.011 | 0.063 | 0.014 | 0.098 | 0.752 | 0.035 | 0.814 | 0.999 | 0.710 | 0.932 | 899.568 |

Model . | n_{1}
. | n_{2}
. | n_{3}
. | Fr . | V (m/s) . | n
. _{e} | b
. | h
. | m_{1}
. | m_{2}
. | C($/m) . |
---|---|---|---|---|---|---|---|---|---|---|---|

Model I | 0.012 | 0.013 | 0.014 | 0.247 | 1.582 | 0.013 | 5.410 | 6.177 | 0.716 | 0.847 | 554.012 |

0.016 | 0.013 | 0.014 | 0.226 | 1.485 | 0.014 | 6.182 | 6.285 | 0.502 | 0.942 | 574.200 | |

0.020 | 0.013 | 0.014 | 0.216 | 1.404 | 0.015 | 6.055 | 6.320 | 0.957 | 0.694 | 595.581 | |

0.024 | 0.013 | 0.014 | 0.197 | 1.329 | 0.017 | 6.042 | 6.751 | 0.705 | 0.806 | 616.559 | |

0.028 | 0.013 | 0.014 | 0.183 | 1.241 | 0.018 | 6.741 | 6.750 | 0.896 | 0.645 | 641.865 | |

0.030 | 0.013 | 0.014 | 0.183 | 1.215 | 0.018 | 6.238 | 6.712 | 0.559 | 1.237 | 655.661 | |

Model II | 0.011 | 0.018 | 0.014 | 0.230 | 1.438 | 0.015 | 6.1845 | 5.899 | 0.974 | 0.927 | 588.030 |

0.011 | 0.028 | 0.014 | 0.179 | 1.213 | 0.019 | 6.641 | 6.797 | 0.693 | 0.921 | 652.063 | |

0.011 | 0.038 | 0.014 | 0.146 | 1.036 | 0.023 | 7.164 | 7.427 | 0.695 | 0.877 | 721.518 | |

0.011 | 0.048 | 0.014 | 0.120 | 0.881 | 0.029 | 8.025 | 7.944 | 0.893 | 0.683 | 800.745 | |

0.011 | 0.058 | 0.014 | 0.108 | 0.819 | 0.032 | 8.538 | 8.289 | 0.629 | 0.865 | 839.845 | |

0.011 | 0.063 | 0.014 | 0.098 | 0.752 | 0.035 | 0.814 | 0.999 | 0.710 | 0.932 | 899.568 |

The results (Table 2 and Figure 6) show that the cost has an ascending trend but velocity has a descending trend with an increase in the roughness of the channel's bed (*n*_{1}) and sides. The cost of roughness coefficients *n*_{1}, *n*_{2} and *n*_{3} (0.011, 0.063, and 0.014) is 62% more than the roughness coefficients (0.012, 0.013 and 0.014).

The constraints of the Froude number for all the models of subcritical, critical and supercritical were studied in the range of 0.2–1.2. In these models (III, IV and V), the values of Manning's roughness coefficients of *n*_{1}, *n*_{2}, and *n*_{3} were 0.020, 0.018, and 0.015, respectively. As a predecessor, it is clear that for better stability of the designed channel the flow regime must be subcritical, for which the Froude number should be less than unity (Fr < 1). Figure 7 depicts the variation of the Froude number values versus velocity, cost, hydraulic depth and total area. As can be observed, velocity has an ascending trend unlike area, cost and hydraulic depth, which have a descending trend by increasing Froude number.

A detailed investigation on the obtained results indicates that the construction cost decreases from 16.600 to 14.045 by increasing the Froude number from 0.203 to 0.298. However, when the Froude number increases from 1.016 to 1.176, the cost only decreases from 8.969 to 8.424. A comparison between Fr = 0.8 and Fr = 0.2 shows that the total cost of construction decreases by approximately 41%. Furthermore, it can be seen that the total cost of channel construction in supercritical flow regimes is much less than subcritical flow regimes.

In order to safely convey the required discharge, it is necessary to ensure that the velocity along the channel does not exceed the corresponding maximum velocity. Model III was evaluated for different values of *V _{av}* ranging from 1.5 to 3 m/s. The optimization results in Table 3 show that total area and cost of the channel construction increase as the velocity decreases. Furthermore, as shown in Table 3, the values

*A*

_{2}and

*A*

_{3}are almost close to each other because similar numbers were obtained for the side slopes of the channel and most of the changes were observed in

*A*

_{1}which was due to an increase in the width of the channel bed.

A_{*2}
. | A_{*3}
. | A_{*1}
. | b_{*}
. | y_{*}
. | m_{1}
. | m_{2}
. | Fr . | V(m/s) ≤ . | T_{*}
. | C($/m) . |
---|---|---|---|---|---|---|---|---|---|---|

0.20 | 0.10 | 0.97 | 0.99 | 0.98 | 0.42 | 0.20 | 0.22 | 1.63 | 1.59 | 776.93 |

0.11 | 0.11 | 0.98 | 0.95 | 1.03 | 0.20 | 0.20 | 0.22 | 1.72 | 1.36 | 770.96 |

0.09 | 0.08 | 0.86 | 0.93 | 0.92 | 0.20 | 0.20 | 0.27 | 2.00 | 1.30 | 700.50 |

0.07 | 0.07 | 0.69 | 0.83 | 0.83 | 0.20 | 0.20 | 0.36 | 2.50 | 1.16 | 624.00 |

0.06 | 0.06 | 0.57 | 0.77 | 0.75 | 0.20 | 0.20 | 0.45 | 3.00 | 1.07 | 572.07 |

A_{*2}
. | A_{*3}
. | A_{*1}
. | b_{*}
. | y_{*}
. | m_{1}
. | m_{2}
. | Fr . | V(m/s) ≤ . | T_{*}
. | C($/m) . |
---|---|---|---|---|---|---|---|---|---|---|

0.20 | 0.10 | 0.97 | 0.99 | 0.98 | 0.42 | 0.20 | 0.22 | 1.63 | 1.59 | 776.93 |

0.11 | 0.11 | 0.98 | 0.95 | 1.03 | 0.20 | 0.20 | 0.22 | 1.72 | 1.36 | 770.96 |

0.09 | 0.08 | 0.86 | 0.93 | 0.92 | 0.20 | 0.20 | 0.27 | 2.00 | 1.30 | 700.50 |

0.07 | 0.07 | 0.69 | 0.83 | 0.83 | 0.20 | 0.20 | 0.36 | 2.50 | 1.16 | 624.00 |

0.06 | 0.06 | 0.57 | 0.77 | 0.75 | 0.20 | 0.20 | 0.45 | 3.00 | 1.07 | 572.07 |

#### Second type (main channel with floodplain)

In this type, the discharge was assigned as 200 m^{3}/s for roughness coefficients of *n*_{1} to *n*_{4} (Figure 2(b)) and the results are presented in Table 4. The results indicate that the velocity and construction cost have an ascending trend with the increase of the roughness coefficient. In model I, *n*_{1} varies from 0.012 to 0.03. The comparison between two types of channels shows that the costs increased for the second type of channel (with floodplain). For example, the total construction cost for the roughness coefficient of 0.012 is increased by around 22% in comparison to the first type. However, the depth, bottom width and side slopes for the main part of the channel (trapezoidal) reduce in the second type. Roughness coefficient (*n*_{2}) varies from 0.018 to 0.043 in model II. Considering the presented results for the roughness coefficient of 0.043 and the roughness coefficient of 0.018, it can be clearly seen that increasing the roughness coefficient would increase the total cost by 23%. In the models III the coefficient (*n*_{3}) varies from 0.025 to 0.07. Comparison between the roughness coefficient of 0.025 and roughness coefficient of 0.07 shows that the cost is raised by approximately 55%, however the velocity has decreased by 45%. In the last model (IV) the coefficient (*n*_{4}) varies from 0.028 to 0.082. In this model, a comparative result between *n*_{4} = 0.028 and *n*_{4} = 0.082 demonstrated a 60% increase over the optimized cost. Generally, as can be observed from the results, increasing roughness coefficient yields an increasing trend in the optimal cost for models I–IV. However, considering model IV, the construction cost is more sensitive to variation of the roughness coefficient of *n*_{4}. It is also noticed that the side slopes of *m*_{1} and *m*_{2} with coefficients of *n*_{1} = 0.01, *n*_{2} = 0.013, *n*_{3} = 0.02 and *n*_{4} = 0.082 were optimized to the highest values of 0.533 and 0.832 respectively, and consequently, it leads to the maximum cost of construction of 1,153.193 C($/m). The optimized hydraulic parameters for developed models obtained by GWO are very distinctive. For example, the depths of flow values were different for all models ranging from 9.649 m (obtained from model III) to 14.353 m (obtained from model IV). Furthermore, the width of channel at the water surface (B) was optimized in the range of 21.761 m (obtained from model I) to 37.714 m (obtained from model IV).

Model . | n_{1}
. | n_{2}
. | n_{3}
. | n_{4}
. | Fr . | V (m/s) . | m_{1}
. | m_{2}
. | C ($/m) . | B . | H . |
---|---|---|---|---|---|---|---|---|---|---|---|

Model I | 0.012 | 0.013 | 0.02 | 0.022 | 0.175 | 1.394 | 0.400 | 0.407 | 679.512 | 22.036 | 10.368 |

0.016 | 0.013 | 0.02 | 0.022 | 0.174 | 1.377 | 0.465 | 0.409 | 687.376 | 22.651 | 10.209 | |

0.020 | 0.013 | 0.02 | 0.022 | 0.163 | 1.339 | 0.371 | 0.360 | 698.880 | 21.761 | 10.765 | |

0.024 | 0.013 | 0.02 | 0.022 | 0.161 | 1.312 | 0.379 | 0.393 | 709.465 | 22.428 | 10.781 | |

0.028 | 0.013 | 0.02 | 0.022 | 0.166 | 1.310 | 0.613 | 0.369 | 718.465 | 23.881 | 10.041 | |

0.030 | 0.013 | 0.02 | 0.022 | 0.161 | 1.288 | 0.549 | 0.389 | 724.491 | 23.800 | 10.308 | |

Model II | 0.01 | 0.018 | 0.02 | 0.022 | 0.168 | 1.345 | 0.471 | 0.384 | 699.238 | 22.698 | 10.345 |

0.01 | 0.023 | 0.02 | 0.022 | 0.158 | 1.277 | 0.543 | 0.349 | 728.469 | 23.454 | 10.425 | |

0.01 | 0.028 | 0.02 | 0.022 | 0.144 | 1.205 | 0.389 | 0.381 | 755.961 | 23.357 | 11.235 | |

0.01 | 0.033 | 0.02 | 0.022 | 0.136 | 1.139 | 0.658 | 0.221 | 797.224 | 24.378 | 10.823 | |

0.01 | 0.038 | 0.02 | 0.022 | 0.128 | 1.075 | 0.688 | 0.266 | 833.068 | 25.862 | 10.970 | |

0.01 | 0.043 | 0.02 | 0.022 | 0.122 | 1.023 | 0.635 | 0.374 | 862.438 | 27.244 | 11.288 | |

Model III | 0.01 | 0.013 | 0.025 | 0.022 | 0.176 | 1.359 | 0.658 | 0.408 | 702.465 | 24.280 | 9.649 |

0.01 | 0.013 | 0.03 | 0.022 | 0.154 | 1.261 | 0.438 | 0.397 | 731.906 | 23.377 | 10.769 | |

0.01 | 0.013 | 0.035 | 0.022 | 0.140 | 1.185 | 0.390 | 0.342 | 765.015 | 23.301 | 11.320 | |

0.01 | 0.013 | 0.04 | 0.022 | 0.131 | 1.105 | 0.429 | 0.342 | 806.500 | 25.177 | 11.340 | |

0.01 | 0.013 | 0.045 | 0.022 | 0.119 | 1.013 | 0.405 | 0.310 | 859.746 | 26.965 | 11.611 | |

0.01 | 0.013 | 0.05 | 0.022 | 0.112 | 0.954 | 0.427 | 0.331 | 900.374 | 28.525 | 11.778 | |

0.01 | 0.013 | 0.055 | 0.022 | 0.104 | 0.904 | 0.412 | 0.291 | 937.856 | 28.751 | 12.196 | |

0.01 | 0.013 | 0.06 | 0.022 | 0.104 | 0.913 | 0.534 | 0.291 | 935.316 | 28.624 | 12.530 | |

0.01 | 0.013 | 0.065 | 0.022 | 0.091 | 0.802 | 0.422 | 0.305 | 1,028.174 | 31.736 | 12.643 | |

0.01 | 0.013 | 0.07 | 0.022 | 0.084 | 0.739 | 0.407 | 0.298 | 1,094.761 | 34.331 | 12.870 | |

Model IV | 0.01 | 0.013 | 0.02 | 0.028 | 0.164 | 1.294 | 0.503 | 0.491 | 720.993 | 24.443 | 10.295 |

0.01 | 0.013 | 0.02 | 0.034 | 0.142 | 1.160 | 0.371 | 0.549 | 777.229 | 25.531 | 11.244 | |

0.01 | 0.013 | 0.02 | 0.04 | 0.129 | 1.090 | 0.453 | 0.402 | 815.059 | 25.272 | 11.525 | |

0.01 | 0.013 | 0.02 | 0.046 | 0.122 | 0.992 | 0.446 | 0.703 | 876.317 | 29.841 | 11.655 | |

0.01 | 0.013 | 0.02 | 0.052 | 0.110 | 0.942 | 0.459 | 0.497 | 910.462 | 28.353 | 12.228 | |

0.01 | 0.013 | 0.02 | 0.058 | 0.094 | 0.833 | 0.232 | 0.630 | 999.081 | 30.138 | 13.702 | |

0.01 | 0.013 | 0.02 | 0.064 | 0.097 | 0.829 | 0.503 | 0.649 | 1,006.067 | 32.368 | 12.638 | |

0.01 | 0.013 | 0.02 | 0.07 | 0.088 | 0.766 | 0.426 | 0.724 | 1,068.638 | 34.025 | 13.321 | |

0.01 | 0.013 | 0.02 | 0.076 | 0.079 | 0.711 | 0.313 | 0.686 | 1,132.312 | 33.979 | 14.353 | |

0.01 | 0.013 | 0.02 | 0.082 | 0.081 | 0.699 | 0.533 | 0.832 | 1,153.193 | 37.714 | 13.397 |

Model . | n_{1}
. | n_{2}
. | n_{3}
. | n_{4}
. | Fr . | V (m/s) . | m_{1}
. | m_{2}
. | C ($/m) . | B . | H . |
---|---|---|---|---|---|---|---|---|---|---|---|

Model I | 0.012 | 0.013 | 0.02 | 0.022 | 0.175 | 1.394 | 0.400 | 0.407 | 679.512 | 22.036 | 10.368 |

0.016 | 0.013 | 0.02 | 0.022 | 0.174 | 1.377 | 0.465 | 0.409 | 687.376 | 22.651 | 10.209 | |

0.020 | 0.013 | 0.02 | 0.022 | 0.163 | 1.339 | 0.371 | 0.360 | 698.880 | 21.761 | 10.765 | |

0.024 | 0.013 | 0.02 | 0.022 | 0.161 | 1.312 | 0.379 | 0.393 | 709.465 | 22.428 | 10.781 | |

0.028 | 0.013 | 0.02 | 0.022 | 0.166 | 1.310 | 0.613 | 0.369 | 718.465 | 23.881 | 10.041 | |

0.030 | 0.013 | 0.02 | 0.022 | 0.161 | 1.288 | 0.549 | 0.389 | 724.491 | 23.800 | 10.308 | |

Model II | 0.01 | 0.018 | 0.02 | 0.022 | 0.168 | 1.345 | 0.471 | 0.384 | 699.238 | 22.698 | 10.345 |

0.01 | 0.023 | 0.02 | 0.022 | 0.158 | 1.277 | 0.543 | 0.349 | 728.469 | 23.454 | 10.425 | |

0.01 | 0.028 | 0.02 | 0.022 | 0.144 | 1.205 | 0.389 | 0.381 | 755.961 | 23.357 | 11.235 | |

0.01 | 0.033 | 0.02 | 0.022 | 0.136 | 1.139 | 0.658 | 0.221 | 797.224 | 24.378 | 10.823 | |

0.01 | 0.038 | 0.02 | 0.022 | 0.128 | 1.075 | 0.688 | 0.266 | 833.068 | 25.862 | 10.970 | |

0.01 | 0.043 | 0.02 | 0.022 | 0.122 | 1.023 | 0.635 | 0.374 | 862.438 | 27.244 | 11.288 | |

Model III | 0.01 | 0.013 | 0.025 | 0.022 | 0.176 | 1.359 | 0.658 | 0.408 | 702.465 | 24.280 | 9.649 |

0.01 | 0.013 | 0.03 | 0.022 | 0.154 | 1.261 | 0.438 | 0.397 | 731.906 | 23.377 | 10.769 | |

0.01 | 0.013 | 0.035 | 0.022 | 0.140 | 1.185 | 0.390 | 0.342 | 765.015 | 23.301 | 11.320 | |

0.01 | 0.013 | 0.04 | 0.022 | 0.131 | 1.105 | 0.429 | 0.342 | 806.500 | 25.177 | 11.340 | |

0.01 | 0.013 | 0.045 | 0.022 | 0.119 | 1.013 | 0.405 | 0.310 | 859.746 | 26.965 | 11.611 | |

0.01 | 0.013 | 0.05 | 0.022 | 0.112 | 0.954 | 0.427 | 0.331 | 900.374 | 28.525 | 11.778 | |

0.01 | 0.013 | 0.055 | 0.022 | 0.104 | 0.904 | 0.412 | 0.291 | 937.856 | 28.751 | 12.196 | |

0.01 | 0.013 | 0.06 | 0.022 | 0.104 | 0.913 | 0.534 | 0.291 | 935.316 | 28.624 | 12.530 | |

0.01 | 0.013 | 0.065 | 0.022 | 0.091 | 0.802 | 0.422 | 0.305 | 1,028.174 | 31.736 | 12.643 | |

0.01 | 0.013 | 0.07 | 0.022 | 0.084 | 0.739 | 0.407 | 0.298 | 1,094.761 | 34.331 | 12.870 | |

Model IV | 0.01 | 0.013 | 0.02 | 0.028 | 0.164 | 1.294 | 0.503 | 0.491 | 720.993 | 24.443 | 10.295 |

0.01 | 0.013 | 0.02 | 0.034 | 0.142 | 1.160 | 0.371 | 0.549 | 777.229 | 25.531 | 11.244 | |

0.01 | 0.013 | 0.02 | 0.04 | 0.129 | 1.090 | 0.453 | 0.402 | 815.059 | 25.272 | 11.525 | |

0.01 | 0.013 | 0.02 | 0.046 | 0.122 | 0.992 | 0.446 | 0.703 | 876.317 | 29.841 | 11.655 | |

0.01 | 0.013 | 0.02 | 0.052 | 0.110 | 0.942 | 0.459 | 0.497 | 910.462 | 28.353 | 12.228 | |

0.01 | 0.013 | 0.02 | 0.058 | 0.094 | 0.833 | 0.232 | 0.630 | 999.081 | 30.138 | 13.702 | |

0.01 | 0.013 | 0.02 | 0.064 | 0.097 | 0.829 | 0.503 | 0.649 | 1,006.067 | 32.368 | 12.638 | |

0.01 | 0.013 | 0.02 | 0.07 | 0.088 | 0.766 | 0.426 | 0.724 | 1,068.638 | 34.025 | 13.321 | |

0.01 | 0.013 | 0.02 | 0.076 | 0.079 | 0.711 | 0.313 | 0.686 | 1,132.312 | 33.979 | 14.353 | |

0.01 | 0.013 | 0.02 | 0.082 | 0.081 | 0.699 | 0.533 | 0.832 | 1,153.193 | 37.714 | 13.397 |

Figure 8 presents a comparison of the variation of the optimal cost based on the changes in roughness values. As can be seen the cost has an ascending trend both horizontally and vertically, indicating the parallel relationship of the cost and roughness. Accordingly, it can be deduced that an increase in the roughness yields to an increase in the cost.

### Second scenario: stability

In this scenario, two different models were studied to analyze the stability. In the first model, the factor of safety (FS) was considered as an objective function. The design assumptions are shown in Table 5 and the associated results are presented in Table 6. In this model, the coordinates of slip circle were obtained by minimizing the factor of safety.

Height of the slope . | Soil cohesion . | Number of slices . | Slope inclination angle . | Angle of shearing resistance . |
---|---|---|---|---|

h (m) | C (kpa) | n | β° | ϕ° |

5 | 15 | 4 | 30 | 15 |

Height of the slope . | Soil cohesion . | Number of slices . | Slope inclination angle . | Angle of shearing resistance . |
---|---|---|---|---|

h (m) | C (kpa) | n | β° | ϕ° |

5 | 15 | 4 | 30 | 15 |

Parameters . | . | . | . | . | . |
---|---|---|---|---|---|

Safety of factor | Fs | 0.744294 | 0.659687 | 0.592622 | 0.53818 |

Coordinate center of circle | X | 8.532285 | 8.588164 | 8.588396 | 8.608878 |

Coordinate center of circle | Y | 4.931274 | 4.959163 | 4.959276 | 4.969501 |

Parameters . | . | . | . | . | . |
---|---|---|---|---|---|

Safety of factor | Fs | 0.744294 | 0.659687 | 0.592622 | 0.53818 |

Coordinate center of circle | X | 8.532285 | 8.588164 | 8.588396 | 8.608878 |

Coordinate center of circle | Y | 4.931274 | 4.959163 | 4.959276 | 4.969501 |

As shown in Table 6, the minimum FS was obtained as 0.53. Thus, the coordinates of slip circle for the analysis of the second model were considered as 8.6 and 4.9.

In the next model of this scenario, the *FS* value was applied as an additional constraint to the objective function (cost function). The coordinates obtained from the first model (x,y) were entered into the calculations. The results are provided in Tables 7 and 8. The obtained results indicate that the side slopes of both first and second types of the compound channel increase to values of around 2 and 2.5, respectively, with an increase in the FS value. Table 8 shows that by imposing FS = 1.5 as a constraint, channel bottom width *b* increased considerably (30.530) compared with model 1 which imposed FS = 1. In this situation, the total construction cost of the composite trapezoidal channel increases to 843.72 ($/m). The results obtained from optimizing the compound channel with floodplain demonstrated that the determined variable *H* is less sensitive to variation of the factor of safety. On the other hand, for the factor of safety FS = 1.2, the optimal channel depth is required to be optimized as 44 m whereas the value for FS = 1.5 is optimized to 30.530 m with a 12.24% increase over the cost calculated for FS = 1.2 (874.047 C ($/m)).

Fs . | C ($/m) . | Fr . | V (m/s) . | T . | b . | h . | m_{1}
. | m_{2}
. |
---|---|---|---|---|---|---|---|---|

1 | 736.711 | 0.204 | 1.202 | 3.405 | 4.315 | 6.305 | 1.612 | 1.653 |

1.2 | 788.024 | 0.204 | 1.177 | 3.607 | 3.048 | 7.197 | 1.956 | 1.745 |

1.5 | 843.72 | 0.165 | 0.932 | 4.779 | 11.548 | 4.232 | 2.438 | 1.948 |

Fs . | C ($/m) . | Fr . | V (m/s) . | T . | b . | h . | m_{1}
. | m_{2}
. |
---|---|---|---|---|---|---|---|---|

1 | 736.711 | 0.204 | 1.202 | 3.405 | 4.315 | 6.305 | 1.612 | 1.653 |

1.2 | 788.024 | 0.204 | 1.177 | 3.607 | 3.048 | 7.197 | 1.956 | 1.745 |

1.5 | 843.72 | 0.165 | 0.932 | 4.779 | 11.548 | 4.232 | 2.438 | 1.948 |

Fs . | C ($/m) . | Fr . | V (m/s) . | m_{1}
. | m_{2}
. | B . | H . |
---|---|---|---|---|---|---|---|

1 | 812.691 | 0.120 | 0.615 | 0.863 | 2.499 | 21.754 | 8.495 |

1.2 | 874.047 | 0.108 | 0.570 | 1.327 | 2.5 | 44.00 | 8.204 |

1.5 | 981.06 | 0.090 | 0.494 | 1.872 | 2.499 | 30.530 | 8.138 |

Fs . | C ($/m) . | Fr . | V (m/s) . | m_{1}
. | m_{2}
. | B . | H . |
---|---|---|---|---|---|---|---|

1 | 812.691 | 0.120 | 0.615 | 0.863 | 2.499 | 21.754 | 8.495 |

1.2 | 874.047 | 0.108 | 0.570 | 1.327 | 2.5 | 44.00 | 8.204 |

1.5 | 981.06 | 0.090 | 0.494 | 1.872 | 2.499 | 30.530 | 8.138 |

## CONCLUSIONS

In this paper, the newly developed GWO algorithm was utilized for solving nonlinear problems of the optimum design of compound open channels. For this purpose, two scenarios with several models were presented to optimize two types of compound channel sections. In the first scenario, the objective function was based on the construction cost in which a compound channel with different roughness coefficients and side slopes was studied. The effect of variation in Manning roughness coefficient on the construction cost and optimized dimensions was also studied, which showed that the construction cost increased with the increase of roughness. The states of no restriction, a restricted Froude number, restricted velocity of flow and restricted top width were also imposed on the models. It was observed that by imposing these restrictions on velocity and Froude number, the construction cost increases. The design was implemented for optimizing the channel dimensions and costs considering different roughness coefficients. The total cost had an ascending trend with an increase in roughness coefficient. Besides, the obtained results can be used to obtain the optimized dimensions for different roughness coefficients. In the second scenario, the developed optimization model was used to calculate the optimal channel dimensions of a stable channel section. In this scenario, two models were studied to analyze stability. In the first model the FS value was considered as the objective function and in the next model, the factor of safety was applied as an additional constraint to the objective function (cost function). It was concluded that by increasing the factor of safety, side slopes and cost of channel construction increases. It is hoped that future research efforts will be aimed in this direction to improve the suggested models by incorporating other constraints. Furthermore, the real terrain profile with unsteady flows can also be used for evaluating excavation costs.

## DATA AVAILABILITY STATEMENT

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