## 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

The total cost function of the channel per unit length C ($/m) is calculated as follows: (1) The earthwork cost (monetary unit per unit length, e.g.$/m) is given as:
(2)
where = cost per unit volume of earthwork at ground level ; = the additional cost per unit volume of excavation per unit depth ; A = flow area (m2); = depth of the centroid of the area of excavation from the ground surface (m).
The cost of lining (monetary unit per unit length, e.g. $.m−1) is expressed as: (3) where = cost of unit lining (monetary unit per unit area of lining, e.g.$.m−2 and P = flow perimeter (m).
The capitalized cost of water lost ($/m) can be expressed as: (4) (5) (6) where r= rate of interest ($/$/year) and = cost per unit volume of water ($/m3). 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).
The objective function of this research is the reduction of cost. The uniform flow equation is treated as a constraint and is inserted into the optimization models. The most commonly used uniform flow resistance formula is the Manning equation (Chow 1973):
(7)
where R = hydraulic radius (m); S0 = longitudinal bed slope (dimensionless) and = equivalent roughness coefficient. Utilizing the continuity equation, the discharge Q (m3/s) is obtained as:
(8)
Combining Equations (1) and (8) forms the general optimization algorithm for a minimum cost section of open channel. The terms of these models are all in dimensional forms. In order to facilitate the detection of the effects of variables on the models, the aforementioned equations are transformed to dimensionless forms, through defining a length scale, as follows:
(9)
Using Equations (1), (8) and (9), the problem of determining the optimal channel section formed in dimensionless form is reduced to:
(10)
(11)

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).

Figure 1

The social hierarchy of grey wolves.

Figure 1

The social hierarchy of grey wolves.

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.

Figure 2

Illustration of position updating mechanism of ω wolves according to positions of α, β and δ wolves (Al Shorman et al. 2020).

Figure 2

Illustration of position updating mechanism of ω wolves according to positions of α, β and δ wolves (Al Shorman et al. 2020).

The main principal in the GWO algorithm is searching, encircling, hunting, and attacking the prey. Before the hunting process, the grey wolves are encircling the prey. The encircling behavior of grey wolves can be expressed as:
(12)
where represents the next location of any wolf, represents the position vector of the grey wolf, 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:
(13)
(14)
(15)
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:
(16)
where , and are calculated through the following equations:
(17)
(18)
(19)
where , and can be determined by:
(20)
(21)
(22)

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.

Figure 3

Solution procedures.

Figure 3

Solution procedures.

## 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 (m1 and m2).

Figure 4

(a) Composite trapezoidal channel geometry, (b) Compound channel geometry.

Figure 4

(a) Composite trapezoidal channel geometry, (b) Compound channel geometry.

The first type consists of four models:

Model I: The roughness coefficient n1 changes on the bed, as any other coefficient is constant.

Model II: The roughness coefficient n2 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 Fmax:
(23)
where = maximum permissible Froude number.
Model IV: The additional constraint of velocity (maximum velocity). If the average velocity of the compound channel is more comparable to the permissible velocity of the channel, the average velocity can be restricted utilizing the following restriction:
(24)
where is average velocity of the channel.

For the designed sections, the average flow velocity Vav could be achieved by Equation (24).

In the second type, the design was conducted for the compound channel section consisting of a main channel with floodplain, as shown in Figure 4(b). These channels are composed of two parts. The main channel with lower bed level usually has a main section close to a rectangular or trapezius shape. This part transports base discharge that flows mostly in the rivers. The bed and walls of the main channel in natural rivers are formed from sand with poor or no vegetation. Thus, these parts have less roughness than the floodplain in terms of hydraulic aspect. To determine the hydraulic parameters of such streams, it is necessary to consider an equivalent roughness for the whole wetted perimeter of the channel. To calculate the roughness coefficient known as the combined roughness coefficient, different methods with specific hypotheses were presented by researchers. The equivalent roughness for the compound channel was presented by using different methods such as Horton (1933), Einstein (1934), etc. In this study, the Horton's method was used to calculate the equivalent roughness. Horton presented the following equation by assuming equal flow in all parts:
(25)
in which is the equivalent roughness coefficient, 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.
Table 1

Different models for second type design

ModelsThe roughness coefficient changes
n1 (on the bed of main channel)n2 (on the side slope of main channel)n3 (on the bed of bed floodplain)n4 (on the side slope of the floodplain)
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
ModelsThe roughness coefficient changes
n1 (on the bed of main channel)n2 (on the side slope of main channel)n3 (on the bed of bed floodplain)n4 (on the side slope of the floodplain)
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).

Figure 5

(a) Channel slope with circular slip circle and (b) failure circle (in the method of slice). (X, Y): center of the failure circle; r: radius of the failure circle; H: height of the slope; b: width of a slice; W1: weight of slice; N1: normal component of forces in slice along failure circle; T1: tangential component of forces in slice along failure circle.

Figure 5

(a) Channel slope with circular slip circle and (b) failure circle (in the method of slice). (X, Y): center of the failure circle; r: radius of the failure circle; H: height of the slope; b: width of a slice; W1: weight of slice; N1: normal component of forces in slice along failure circle; T1: tangential component of forces in slice along failure circle.

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

In the first model, the objective function was considered as a minimization criterion for the FS (Equation (26)). The FS for the critical slip circle is calculated using the developed GWO algorithm based on the optimization model:
(26)
where and are the upper limits and and ql are the lower limits for the coordinate of the center of the slip circle with a radius of r. The coordinates of the slip circle are determined as X-Y coordinates (Figure 5(b)) (Sengupta & Upadhyay 2009).

#### Model II

In the second model, the allowed FS was applied as an additional constraint function (Equation (27)) to the Manning constraint on the objective function (cost function):
(27)
(28)
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 m3/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 n1, which is the roughness coefficient of the channel bed (n2 and n3 were considered as 0.013 and 0.014, respectively). In this model, the discharge was considered as 100 m3/s. In the second model, n2 was considered as a variable over the range 0.018–0.063. The results obtained using different values of n1 and n2 for composite trapezoidal are tabulated in Table 2.

Table 2

Optimization results for different values of n1 and n2 for composite trapezoidal (Scenario I, Model I)

Modeln1n2n3FrV (m/s)nebhm1m2C($/m) Model I 0.012 0.013 0.014 0.247 1.582 0.013 5.410 6.177 0.716 0.847 554.012 0.016 0.013 0.014 0.226 1.485 0.014 6.182 6.285 0.502 0.942 574.200 0.020 0.013 0.014 0.216 1.404 0.015 6.055 6.320 0.957 0.694 595.581 0.024 0.013 0.014 0.197 1.329 0.017 6.042 6.751 0.705 0.806 616.559 0.028 0.013 0.014 0.183 1.241 0.018 6.741 6.750 0.896 0.645 641.865 0.030 0.013 0.014 0.183 1.215 0.018 6.238 6.712 0.559 1.237 655.661 Model II 0.011 0.018 0.014 0.230 1.438 0.015 6.1845 5.899 0.974 0.927 588.030 0.011 0.028 0.014 0.179 1.213 0.019 6.641 6.797 0.693 0.921 652.063 0.011 0.038 0.014 0.146 1.036 0.023 7.164 7.427 0.695 0.877 721.518 0.011 0.048 0.014 0.120 0.881 0.029 8.025 7.944 0.893 0.683 800.745 0.011 0.058 0.014 0.108 0.819 0.032 8.538 8.289 0.629 0.865 839.845 0.011 0.063 0.014 0.098 0.752 0.035 0.814 0.999 0.710 0.932 899.568 Modeln1n2n3FrV (m/s)nebhm1m2C($/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 (n1) and sides. The cost of roughness coefficients n1, n2 and n3 (0.011, 0.063, and 0.014) is 62% more than the roughness coefficients (0.012, 0.013 and 0.014).

Figure 6

Effect of various roughness coefficient (n2) on velocity and construction cost of the channel.

Figure 6

Effect of various roughness coefficient (n2) on velocity and construction cost of the channel.

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 n1, n2, and n3 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.

Figure 7

Changes in velocity (1), the total cost of construction (2), the cross section area (3) and hydraulic depth (4) with different Froude number.

Figure 7

Changes in velocity (1), the total cost of construction (2), the cross section area (3) and hydraulic depth (4) with different 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 Vav 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 A2 and A3 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 A1 which was due to an increase in the width of the channel bed.

Table 3

Optimal values of parameters considering velocity constraint (Scenario I, Model IV)

A*2A*3A*1b*y*m1m2FrV(m/s) ≤T*C($/m) 0.20 0.10 0.97 0.99 0.98 0.42 0.20 0.22 1.63 1.59 776.93 0.11 0.11 0.98 0.95 1.03 0.20 0.20 0.22 1.72 1.36 770.96 0.09 0.08 0.86 0.93 0.92 0.20 0.20 0.27 2.00 1.30 700.50 0.07 0.07 0.69 0.83 0.83 0.20 0.20 0.36 2.50 1.16 624.00 0.06 0.06 0.57 0.77 0.75 0.20 0.20 0.45 3.00 1.07 572.07 A*2A*3A*1b*y*m1m2FrV(m/s) ≤T*C($/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 m3/s for roughness coefficients of n1 to n4 (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, n1 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 (n2) 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 (n3) 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 (n4) varies from 0.028 to 0.082. In this model, a comparative result between n4 = 0.028 and n4 = 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 n4. It is also noticed that the side slopes of m1 and m2 with coefficients of n1 = 0.01, n2 = 0.013, n3 = 0.02 and n4 = 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($/m). The optimized hydraulic parameters for developed models obtained by GWO are very distinctive. For example, the depths of flow values were different for all models ranging from 9.649 m (obtained from model III) to 14.353 m (obtained from model IV). Furthermore, the width of channel at the water surface (B) was optimized in the range of 21.761 m (obtained from model I) to 37.714 m (obtained from model IV). Table 4 Optimization results for different values of n1, n2, n3 and n4 for compound channel (Scenario I, second type) Modeln1n2n3n4FrV (m/s)m1m2C ($/m)BH
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
Modeln1n2n3n4FrV (m/s)m1m2C ($/m)BH 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. Figure 8 Comparison of the results obtained from models I, II, III and IV for design of compound channel without any site specific restrictions. Figure 8 Comparison of the results obtained from models I, II, III and IV for design of compound channel without any site specific restrictions. ### 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. Table 5 Soil parameters used for stability of slope Height of the slopeSoil cohesionNumber of slicesSlope inclination angleAngle of shearing resistance h (m) C (kpa) β° ϕ° 15 30 15 Height of the slopeSoil cohesionNumber of slicesSlope inclination angleAngle of shearing resistance h (m) C (kpa) β° ϕ° 15 30 15 Table 6 Parameters of the critical slip circle by minimizing SF of the slip circle (Scenario II, Model I) Parameters Safety of factor Fs 0.744294 0.659687 0.592622 0.53818 Coordinate center of circle 8.532285 8.588164 8.588396 8.608878 Coordinate center of circle 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 8.532285 8.588164 8.588396 8.608878 Coordinate center of circle 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 ($/m)). Table 7 Optimal values of parameters considering FS constraint (Scenario II, first type, Model II) FsC ($/m)FrV (m/s)Tbhm1m2
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
FsC ($/m)FrV (m/s)Tbhm1m2 736.711 0.204 1.202 3.405 4.315 6.305 1.612 1.653 1.2 788.024 0.204 1.177 3.607 3.048 7.197 1.956 1.745 1.5 843.72 0.165 0.932 4.779 11.548 4.232 2.438 1.948 Table 8 Optimal values of parameters considering FS constraint (Scenario II, second type, Model II) FsC ($/m)FrV (m/s)m1m2BH
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
FsC (\$/m)FrV (m/s)m1m2BH
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.

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