Design of irrigation canals with minimum overall cost using particle swarm optimization – case study: El-Sheikh Gaber canal, north Sinai Peninsula, Egypt

Nowadays, the scarcity of freshwater sources, climate change and the deterioration of freshwater quality have a great impact on the lives of human beings. As such, improving the design of irrigation canals will reduce water losses through evaporation and seepage. In this paper, particle swarm optimization (PSO) is used to determine the optimum design of irrigation canals’ cross-sections with the objective to minimize the overall costs. The overall costs include the costs of earthwork, lining, and water loss by both seepage and evaporation. The velocity constraints for sedimentation and erosion have been taken into consideration in the proposed design method. The proposed PSO is compared with both the Probabilistic Global Search Lausanne (PGSL) and classical optimization methods to verify its usefulness in optimal design of canals’ cross-sections. The proposed PSO is then used to design El-Sheikh Gaber canal, north Sinai Peninsula, Egypt and the obtained dimensions are compared with the existing canal dimensions. To facilitate the use of the developed model, optimal design graphs are presented. The results show that the reduction of overall cost ranged from 28 to 41% and consequently, the proposed PSO algorithm can be reliably used for the design of irrigation open canals without going through the conventional and cumbersome trial and error methods.


GRAPHICAL ABSTRACT INTRODUCTION
Water canals are used to convey water from its source to its final destination for different purposes. One of the main objectives of these canals is to convey water for irrigation with minimal cost. As such, the design of water canals was investigated by many researchers. For example, Lindley (), Lacey (, ), and other researchers developed procedures for the design of stable water canals by the tractive force method originally developed by Lane (, ) and others. Recently, several optimization techniques have been applied to determine the optimal cross-section dimensions such as the non-linear method (Swamee et al. a, b, c, ); particle swarm optimization (PSO) (Reddy & Adarsh ); differential evolution algorithm (Turan & Yurdusev ) The optimization models used in the literature can be divided into three categories. The first category minimizes water losses due to both evaporation and seepage. Swamee & Chahar () reported that about 50% of water supplied at the head of the canal may be lost in transit to the field. Seepage losses are mainly dependent on subsoil hydraulic conductivity, geometry of the canal, and water table location relative to the canal. Evaporation losses are directly proportional to the canal's surface area, particularly for long canals carrying small discharges in arid regions (Swamee & Chahar ). In the literature, much effort has been put into the optimal design of canal cross-sections. Swamee et al. (b), for example, applied non-linear optimization for the optimal design of three canal shapes with the objective of minimizing seepage losses. They found that trapezoidal sections have the least seepage losses and cross-sectional area. Ghazaw () adopted Lagrange's method of undetermined multipliers to determine the optimal canal dimensions with the objective of minimizing both seepage and evaporation water losses.
They presented a number of design charts to facilitate optimal canal design. Kentli & Mercan () compared both GA and sequential quadratic programming methods for optimal canal sections design considering seepage and evaporation losses. They applied the two methods on several shapes of cross-sections. In a recent effort, Dong et al. () presented an improved cat swarm optimization algorithm to design canals' cross-sections with low water losses in irrigation areas. They enhanced the efficiency of the conventional cat swarm optimization by adding exponential inertia weight coefficient and mutation. The application of the improved technique on a study area shows a 20% water loss reduction compared to the original design.
The second category of optimization models considers the minimization of both earthwork and lining costs considering canal uniform flow condition. Earthwork costs depend on the flow area and vary with canal depth while the lining cost varies with the wetted perimeter length. Swamee et al.
(c) applied a non-linear optimization technique for the minimum cost design of lining canals with different shapes. They concluded that the minimum area crosssection is the one that minimizes both costs of lining and earthworks. However, when costs of excavation with canal depth are taken into consideration, the optimal section is wider and shallower. This conclusion is also reached by costs, and water losses costs due to seepage and evaporation using non-linear optimization. They concluded that the optimal section is wider and shallower than the minimum area section due to the increased cost when the canal depth increases, while the wider canals increase the cost due to the evaporation water losses. Adarsh & Sahana () used PGSL to determine the optimal dimensions of triangular, rectangular, and trapezoidal canal shapes by minimizing the costs of excavation, lining, water losses (due to seepage and evaporation) and land acquisition. They added two site-specific constraints to suit the design of real-field trapezoidal canals.
In the present study, the third category of optimization models is adopted in order to minimize the cost of excavation, lining, water losses due to seepage and evaporation. For triangular cross-section canals: where m is the canal side slope.
For rectangular cross-section canals: where b is the canal bed width in meters.
For trapezoidal cross-section canals: in which Q is the flow rate of the canal in (m 3 /s), g is the acceleration due to gravity (m/s 2 ); R is the hydraulic radius (m) defined (¼A/P), s is the canal bed slope, ε is the average roughness height for the canal lining (m), and ν is the water kinematic viscosity (m 2 /s).
The following constraints are considered: where ξ is a small positive number.
where V is the average flow velocity (m/s) and V max and V min are the maximum and minimum permissible velocities, respectively. the best position reached before for each particle i, (P i ); the moving velocity (V i ) of each particle i are identified as follows: Also, during each cycle of the evolution process, the position of the particle (g) having the best fitness (best objective function value) in the whole solution space (P g ) is determined. Accordingly, the velocity V i of each particle is updated to reach the position of the best particle g, as follows (Shi & Eberhart ): Then, particles update their positions (new solutions generated) using their new velocities V i , as follows: where c 1 and c 2 are positive constants (usually, c 1 ¼ c 2 ¼ 2) named learning factors; RN 1 and RN 2 are random numbers generated in the range [0, 1]; V max is the maximum particle velocity change (Kennedy & Eberhart ); and ω is an operator proposed by Shi & Eberhart ()  Once a particle's new position is determined (i.e. new solution is formed) using Equation (11), the particle then moves towards it (Shi & Eberhart ). The main parameters of the PSO optimization are the population size (number of particles), number of evolution cycles, the maximum change of a particle velocity V max and ω. Figure 2 shows the flow chart of the PSO algorithm.

MODEL VERIFICATION
The  respectively. Although the obtained solution is inferior with respect to the classical optimization method used, it is noted that a violation of nearly 5 m 3 /s occurred in the flow constraint, thus meaning that the obtained solution using the classical optimization method is infeasible. The proposed method was able to determine the optimal solution, which satisfied the required discharge for the different shapes of the canal cross-sections. As such, the cost obtained by the current method is larger as the discharge is higher than the classical optimization. Consequently, the proposed PSO algorithm is applicable for the optimal design of real-life canal sections.

MODEL APPLICATION ON A REAL-LIFE CASE STUDY
The proposed PSO algorithm is applied to design three cross-sections of trapezoidal shape for the El-Sheikh    (5)) (m 3 /s) C (Equation (1)

OPTIMAL DESIGN CHARTS
The PSO algorithm is utilized in this section to prepare and present optimal design charts that achieve minimum total cost (Figures 4-7). The land acquisition cost is considered as one of the cost elements that should be incorporated to achieve minimum total costs (Adarsh & Sahana ). Consequently, the objective function, Equation (1), is modified as follows:  Gabr (2018).     the third design chart, Figure 6, can be used to determine the earthwork and lining costs; while the fourth design chart, Figure 7, can be used to determine the seepage and evaporation water losses for a given design discharge and bed slope of the trapezoidal cross-section.
It is noticed from the developed design charts that, for a given discharge, b, y, cost and Q l decrease as the value of the bed slope (s) increases. Also, both the values of bottom width and flow depth increase as the design discharge increases but the rate of increase of the bottom width is larger than the flow depth. It can be seen that for a given small design discharge the costs of both earthwork and lining decrease as the value of the bed slope (s) increases with a small rate compared with a large relative rate in larger design discharges.
The procedure of using these charts is given as follows: (1) identify the canal design discharge and bed slope; (2)  To verify the design charts with the results obtained before (Table 3), both Q and s for sections 2 and 3 in El-Sheikh Gaber canal are adopted. The results obtained from Figures 4-7 for b, y, Q, and Q l are 7.3 m, 6.2 m, 16,000 LE/ m, and 1190 m 3 /year, respectively, for section 2 and 6.6 m, 5.9 m, 15,000 LE/m and 1,100 m 3 /year, respectively for section 3. It is noticed that the results of section 2 are the same as those listed in Table 3 while the results of section 3 are close to those listed in Table 3 because m is considered equal to 1 and the corresponding value in Table 3 is 1.217.

CONCLUSIONS
In this paper, the PSO algorithm is adopted to determine the optimal geometric design of rectangular, triangular, and trapezoidal canals' cross-sections for minimum overall cost.
The proposed PSO is compared with both the PGSL and classical optimization methods, given in the literature, to verify its usefulness in canals' cross-sections optimization.