A cost model with several hydraulic constraints for optimizing in practice a trapezoidal cross section

A

flow area of channel (m²)

b

bed width of channel (m)

C

cost per unit length of canal ($/m)

C_e

cost of earthwork per unit length of channel ($/m)

C_L

cost of lining per unit length of channel ($/m)

C_w

capitalized cost of water lost per unit length of channel ($/m)

c_e

cost per unit volume of earthwork at ground level ($/m³)

c_L

cost per unit surface area of lining ($/m²)

c_r

increase in unit excavation cost per unit depth ($/m⁴)

c_w

cost per unit volume of water ($/m³)

E

evaporation discharge per unit free surface area (m/s)

Fr

Froude number

Fs

seepage function (dimensionless)

g

gravitational acceleration (m/s²)

k

hydraulic conductivity (m/s)

m

side slope of channel (dimensionless)

P

flow perimeter of channel (m)

p

penalty parameter (dimensionless)

Q

discharge (m³/s)

R

hydraulic radius (m)

S₀

bed slope of channel (dimensionless)

T_w

top width (m)

V

average velocity (m/s)

y_n

normal depth of flow in channel (m)

ɛ

average roughness height of canal lining (m)

λ

length scale (m)

ν

kinematic viscosity (m²/s)

ϕ

equality constraint (dimensionless)

Ψ

augmented function (dimensionless)

$

monetary unit

Irrigation channel networks are very substantial and are utilized as water supply and conveyance facilities, flood control and for use in other fields. Due to the high cost of channel construction, investigating the optimal and economic cross-sectional designs of these channels is a necessity. A channel in the network may be subjected to lining. The supported cost of a lined channel is less in comparison to unlined channels, since the lining protects against bed and bank erosion. Channels in alluvium are typically lined and capable of decreasing seepage. The seepage loss from channels has been estimated for different sets of special conditions. The analytical form of these solutions, which contain complex integrals and unknown implicit state variables, is not convenient in designing or estimating seepage from the existent channels (Morel-Seytoux 1964; Garg & Chawla 1970; Subramanya et al. 1973; Sharma & Chawla 1979).

Swamee et al. (2000) considered seepage loss in the objective functions and applied nonlinear optimization techniques to design explicit equations. Wachyan & Rushton (1987) used empirical measurements to reveal that main losses occurring from channels are related to whether they are lined or unlined. Kacimov (1992) developed a complex-variable method to optimize the shape design problems of channel beds. In order to minimize the cost function, the seepage losses and cost of lining was constrained by specified hydraulic characteristics. Additionally, Aksoy & Altan-Sakarya (2006) studied the optimum values of section variables for different channel types by minimizing their cost.

Usually, evaporation from a channel is only a small proportion of the total loss, but it becomes substantial for long channels running through arid regions. Warnaka & Pochop (1988) and Ikebuchi et al. (1988) compared the capabilities of different evaporative sorts of estimation models and found that the mass transfer-based models presented the most accurate results (Fulford & Sturm 1984).

The primary factor affecting the channel design is the channel surface forming material which determines the roughness coefficient, the minimum permissible velocity to avoid deposition of silt or debris, the constrained velocity to prevent erosion of the channel surface, and the topography of the channel route which indicates how much the section is hydraulically and/or economically efficient (Chow 1973). The choice of hydraulic parameters is a vital task in the hydraulic design of channels since they are entitled to high uncertainty. Dimitriadis et al. (2016) employed extended sensitivity analysis by simultaneously varying the input discharge, longitudinal and lateral gradients and roughness coefficient.

Monadjemi (1994) and Froehlich (1994) modeled optimized channel design using a Lagrangian undetermined multiplier method. Alternatively, for non-linear, non-convex optimization problems, the compound and implicit construction of the cost function and/or constraints makes the employment of customary gradient-based techniques very difficult, so that the optimization process cannot be applied in many locally optimized processes. This has caused the wide use of heuristic approaches, e.g., genetic algorithm (GA) (Goldberg 1989), particle swarm optimization (PSO) (Kennedy & Eberhart 1995), genetic programming (GP) (Koza 1992), gene expression programming (GEP) (Azamathulla 2012), and charged system search (Kaveh & Talatahari 2010), among many others. Different optimization algorithms are applied to open channel section design problems.

The GA has been successfully utilized for optimizing the design of open channels (Jain et al. 2004; Bhattacharjya & Satish 2007), as well as irrigation scheduling with flow of water through channel networks (Nixon et al. 2000), along with other hydraulic problems (Wu & Simpson 2002; Roushangar & Koosheh 2015). The GP has been applied in different engineering optimization problems (Sharifi et al. 2011). The GEP approach has also been used to solve engineering problems by deriving a new predictive model (Azamathulla & Ahmad 2013; Azamathulla 2013). PSO also has been successfully applied for solving water resources management problems (Janga Reddy & Nagesh Kumar 2009). Also Nourani et al. (2009) optimized composite channels using ant colony optimization.

To the authors’ knowledge, despite considerable investigations into providing a protocol for minimizing cross-sectional area of channels, there are no investigations dealing with constrained hydraulic parameters as well as the role of seepage cost on optimal channel sections. Therefore, in this study, six different models of optimum design for open channels were established. The first model was evaluated with no additional constraint equation, while in the second model, the channel top width was considered as an additional constraint. The third and fourth models had additional constraints of different values of velocity and Froude numbers, respectively. The fifth model utilized additional constraint of different depth values. The sixth model was determined for no seepage cost state. Finally, the applications section shows the effect of discharge (seventh model) and longitudinal bed slopes (eighth model) on optimum design of channels. We applied all models to a real case scenario and showed that the constrained hydraulic parameters have a great influence in the design of the trapezoidal channel section.

Selection of the geometric variables, e.g., side slope, bottom width, and flow depth for open channel sections varies according to the designer's perspective. Also, the longitudinal bed slope of the channel is influenced by topography that is considered as a constant amount in seven models. One of the important objectives is minimizing the total cost of a channel section, that it has the capability of passing the channel distance safely, which must be considered. Generally, the cost per unit length of a lined open channel section is defined as the summation of three terms, namely, the depth-dependent earthwork cost, the cost of lining, and the cost of water lost as seepage and evaporation. These terms are explained in the following sections.

where = equality restriction function.

Analysis of the optimal channel sections for a number of input variables led to the following generalized empirical equations in clear form for trapezoidal channel section (Swamee et al. 2000).

For a given set of data, a channel can be designed by minimizing Equation (10) and subjected to constraint given by Equation (11). Swamee's method is used to verify the results.

The GA is employed to solve the formulated nonlinear models. The GA is a search technique based on the genetics concept of natural selection, which combines an artificial survival of the fittest with genetic operators abstracted from nature (Holland 1975). The important difference between GA and classical optimization search techniques is that the GA generates a population of possible solutions, whereas the classical optimization techniques lead to a single solution. An individual solution in a population of solutions is analogous to a biological chromosome. While a natural chromosome specifies genetic characteristics of a human being, an artificial chromosome in GA indicates the values of varied decision variables representing a decision or a solution. For most GAs, candidate solutions are represented by chromosomes coded using a binary number system (Goldberg 1989). The GA that employs binary strings as its chromosomes is named binary-coded GA. The binary-coded GA contains three basic operators: selection, crossover or mating, and mutation. The selection function chooses parents for the next generation based on their scaled values from the fittest scaling function. In this study, the stochastic uniform selection is used in conjunction with elitism. The stochastic uniform selection lays out a line in which each parent corresponds to a section of the line of length proportional to its expectation.

This model is determined by Equations (10) and (11), representing no additional constraint when consisting of three variables.

For the designed section, the average flow velocity V_av could be achieved by Equation (8). However, in order to safely convey the required discharge through a channel, it is necessary to ensure that the velocity of the channel does not exceed the corresponding maximum velocity which is related to the roughness coefficient of that segment.

Let us suppose that the channel should be designed to transport a discharge of 100 on a longitudinal bed slope of 0.001. The channel passes through a stratum of typical soil, in which and (Table 2). Further, it is proposed to supply concrete lining with ⁠. The climatic condition of the channel area satisfies the ⁠. For the purpose of design, it is assumed that ⁠, (water at 20 °C), and ɛ = 1 mm (concrete lining) (Table 3).

Table 3

Modeling parameters of the trapezoidal channel

Flow factors .				Average roughness of height .	Cost of terms .
				ɛ (mm)
100	0.001	9.79		1	2	12	7	10

Flow factors .				Average roughness of height .	Cost of terms .
				ɛ (mm)
100	0.001	9.79		1	2	12	7	10

Note: For water at 20 °C.

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Using Equation (9), the was determined to be 15.9 m, so the following parameters were identified using Equation (10): ɛ_* = 6.3 × 10⁻⁵, ϑ_* = 1.75 × 10⁻⁷, C_l* = 0.75, C_r* = 2.27, C_ws* = 0.63, and C_wE* = 0.125.

Table 4 compares the Swamee method (Swamee et al. 2000) with the proposed GA-based Model I of the present study (no additional constraint model). The sum total construction costs in both methods for Model I (no additional restriction) are 429.27 and 417.26, respectively. It can be noted from Table 4 that the proposed GA-based model shows better results compared to the Swamee method, with relatively less expensive values.

Table 8 sums up the optimization results for Model V, where y_max is considered a constraint in the event that unfavorable strata depth of channel should not go beyond a certain limit, because the excavation may not be economical or due to some other problem, such as the presence of shallow ground water table. This may require restriction on the maximum permissible flow depth. Therefore, the bottom width of the channel should be significantly increased to provide the necessary cross-sectional area for conveying the required flow discharge, which leads to increasing section area as well as the total cost and evaporation loss.

Table 8

Optimization result for different values of depths as additional restriction (Model V)

Parameter .	When depth is: .
	y ≤ 1.5 .	y ≤ 2 .	y ≤ 2.5 .	y ≤ 3 .	y ≤ 3.5 .
b (m)	18.60	13.22	9.77	7.55	5.91
y (m)	1.49	1.98	2.49	2.98	3.49
m	0.55	0.54	0.53	0.5	0.53
A (m2)	29.05	28.41	27.72	25.27	27.25
Cost of evaporation	40.29	30.55	24.72	21.35	19.21
Cost	597.87	417.59	449.01	425.91	417.59

Parameter .	When depth is: .
	y ≤ 1.5 .	y ≤ 2 .	y ≤ 2.5 .	y ≤ 3 .	y ≤ 3.5 .
b (m)	18.60	13.22	9.77	7.55	5.91
y (m)	1.49	1.98	2.49	2.98	3.49
m	0.55	0.54	0.53	0.5	0.53
A (m2)	29.05	28.41	27.72	25.27	27.25
Cost of evaporation	40.29	30.55	24.72	21.35	19.21
Cost	597.87	417.59	449.01	425.91	417.59

Note: Cost equivalent ⁠; ⁠, ⁠.

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According to Table 9, Model VI, which eliminates seepage from the objective function, caused a decrease in total cost (47% in the problem presented in this paper). It clearly shows the significant effect of seepage on the optimal parameter of a channel.

The best state and minimum cost of each model are listed in Table 10. It can be observed that Models I, III, and V produce bottom widths of 5.159, 5.399 and 5.917 m, while Models II, IV, and VI produce 2.575, 4.927, and 4.77 m, respectively. The depths of flow are the same, but the side slopes given by Models II and IV have smaller magnitudes, in comparison with the other models. The total cross-sectional area obtained by Models II and IV (13.374 m² and 22.568 m², respectively) are approximately 50% and 17%, less than I and V models.

In the present research, the effect of parameter restrictions on optimal design of trapezoidal channels was investigated. The proposed non-linear optimization formulation consists of minimizing the cost of lining, the depth-dependent unit volume earthwork, water lost by seepage, and evaporative losses of the open channel that are constrained by uniform flow conditions and the resistance equation. For the aforementioned optimization issue, six different models were proposed. These non-linear models were evaluated as including no restriction, constrained normal depth, constrained velocity of flow, constrained Froude number, and constrained top width. The optimization formulations corresponding to all of the models are investigated in the present research and solved using GA. The results indicate that for Models III and V (with constrained velocity and normal depth), the total cost of construction is high and for Models II and IV (with constrained top width and Froude number (Fr > 0.7)), the cost of construction in the open channels is smaller. Also, the results indicate that a model which disregards seepage in the objective function causes a considerable decrease in the cost. The obtained result suggests that seepage cost plays an important role on optimal design of open channels. Also, the different applications in the last two models for different discharge with constant bed slope (S₀ = 0.001) and varying longitudinal bed slopes with constant discharge (Q = 100 m³/s, Q = 250 m³/s) was evaluated. It was concluded that with increasing discharge and bed slope, cost of channel construction increases and decreases, respectively.

It is to be noted that the present study is conducted for a specified set of input values and it can be easily extended to any other combination of input design parameters. Also the proposed models for design of open channels are simpler to implement and effective for practical applications, thus it can be used for reliable design of irrigation channels.