Chance Constrained Optimal Design of Composite ... - Semantic Scholar

Water Resour Manage DOI 10.1007/s11269-009-9548-5

Chance Constrained Optimal Design of Composite Channels Using Meta-Heuristic Techniques M. Janga Reddy · S. Adarsh

Received: 21 August 2009 / Accepted: 6 December 2009 © Springer Science+Business Media B.V. 2009

Abstract Optimal design of irrigation channels has an important role in planning and management of irrigation projects. The input parameters used in design of irrigation channels are prone to uncertainty and may result in failure of channels. To improve the overall reliability and cost effectiveness, optimal design of composite channels is performed as a chance constrained problem in this study. The models are developed to minimize the total cost, while satisfying the specified probability of the channel capacity being greater than the design flow. The formulated model leads to a highly non-linear and non-convex optimization problem having multimodal behavior. In this paper, the usefulness of two meta-heuristic search algorithms such as Genetic Algorithms (GA) and Particle Swarm Optimization (PSO) are investigated to obtain the optimal solutions. Two site specific cases of restricted top width and restricted flow depth are also analyzed. It is found that both the algorithms performing quite well in giving optimal solutions and handling the additional constraints. Keywords Optimal design · Composite channels · Chance constraint optimization · Genetic algorithms · Particle swarm optimization

1 Introduction Irrigation canals are one of the essential infrastructures required for sustainable development of agriculture sector of a nation. In the past, the optimal dimensions of different types of irrigation canals were determined by many researchers following

M. Janga Reddy (B) · S. Adarsh Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India e-mail: [email protected] S. Adarsh T K M College of Engineering, Kollam, 691005, India

M. Janga Reddy, S. Adarsh

the principles of calculus, considering uniform roughness along the perimeter (Guo and Hughes 1984; Loganathan 1991; Froehlich 1994; Monadjemi 1994). The cost effectiveness of irrigation projects can be achieved by selecting different lining materials along the perimeter of the channel considering seepage protection offered by a particular lining material, cost of construction, and effective life of the lining material, local availability, aesthetic requirements etc. Such channels with distinctly different lining materials along the perimeter are called as ‘composite channels’ (Chow 1959). The channel roughness is a major factor which influences the conveyance capacity of canals. For composite channels the varied surface roughness is accounted by defining an equivalent roughness coefficient for discharge computation. The concept of equivalent roughness was addressed first by Trout (1982); but this formulation considers only minimization of lining cost and neglect the excavation cost. Das (2000) determined the optimal trapezoidal channel cross-section with composite roughness using classical optimization technique involving Lagrange Multipliers (LM). Jain et al. (2004) used Genetic Algorithms (GA) for obtaining optimal channel dimensions and found that the method of equivalent roughness estimation significantly affect the optimal cost. Bhattacharjya (2006) used Sequential Quadratic Programming (SQP) for the optimal design of trapezoidal channel incorporating the critical flow constraint. Most of the above studies consider deterministic design of composite channels for specified lining materials, longitudinal slope and design flow. The design parameters are prone to uncertainties resulting from changes in roughness due to aging, imperfections in fabrication or inaccuracy in control of flow. This may result in failure of channels. Uncertainty modeling in design of open channels is addressed in different contexts, viz., the hydraulic uncertainty associated with flood levee capacity by Lee and Mays (1986); risk analysis in hydraulic design by Tung and Mays (1980) and Chow et al. (1988); probabilistic and reliability based design of open channels by Easa (1992, 1994), Das (2007, 2008), and Bhattacharjya and Satish (2008). Das (2007) presented a methodology for optimal design of composite channels with a goal of minimizing the total cost after satisfying the flooding probability constraint and solved using Projected Augmented Lagrangian multiplier (PAL) method. The optimization method requires derivative information on the objective function and simplification of functions in its final model construction. Bhattacharjya and Satish (2008) considered freeboard as an additional design variable in the channel design model and solved using a non-dominated sorting genetic algorithm technique. The study suggested that a considerable reduction in optimal cost can be achieved by this modification for lower values of flooding probabilities. However, the method may result in unsuitable designs for practical applications especially for lower values of flooding probabilities. Later a modified model is presented in Das (2008). However this study has not considered any site specific restrictions in its model formulation, so it may result in very wider channels with smaller flow depths or vice versa, which may not be acceptable for implementing in the field. It is also noticed that the conventional methods are inadequate to solve the highly nonlinear optimization problems. For example, for solving channel design problems researchers have used different techniques, for example Lagrangian multiplier (LM) method by Froehlich (1994) and Das (2000); Sequential quadratic programming method by Bhattacharjya (2006); Projected Augmented Lagrangian multiplier (PAL) method by Das (2007, 2008). The LM technique, by starting from any arbitrary initial point, the arrival to a final optimal solution could not be ensured.

Chance Constrained Optimal Design of Composite Channels

So the LM based methodology may perform unsatisfactorily for the highly nonlinear channel design problems. In the projected augmented Lagrangian (PAL) technique, starting from an initial guess solution, the PAL technique performs a sequence of linearly constrained subproblem solving many times (iterations). The SQP methods that employ convex quasi-Newton approximations can be slow when solving large or badly scaled problems, whereas methods that employ the exact Hessian of the Lagrangian are confronted with the difficult task of computing global solutions of indefinite quadratic programs (Gould et al. 2005). Thus the conventional methods may require several approximations or simplifications or derivative information on functions of the model, and then they may converge to local optimal solutions. Thus, there is a greater necessity to explore and apply new optimization methods to get optimal solutions. In this paper an effective approach is presented for optimal design of composite channels. In specific, it aims (1) to present a procedure for optimal design of composite trapezoidal irrigation channels by considering the uncertainty of the various design parameters, and the site specific features; (2) to apply and evaluate the performance of the two meta-heuristic techniques namely Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) methods in solving the model.

2 Meta-Heuristic Techniques for Optimization Meta-heuristic methods are population based random search techniques, which uses some heuristics to guide the search, and helps in quicker convergence to global optimal solutions. These methods start with a randomly generated initial population, and uses randomized decisions while searching for optimal solutions to a given problem, with aim of improved solutions over the generations. The optimal solution obtained from these methods does not depend on the initial solution supplied; they are derivative free optimization techniques; and they can overcome the problems of trapping at local optima, which are common in some classical gradient based optimization methods. Over the past few years there has been considerable research done in developing stochastic search algorithms and their applications for solving complex real life problems. Recent studies revealed that the stochastic search techniques have good potential for solving complex practical problems in engineering. In this context, the performance of two stochastic search algorithms GA and PSO are evaluated to solve the problem of chance constrained optimal design of trapezoidal channels. Brief details of these algorithms are given below. 2.1 GA The GA is a search technique based on the concept of natural selection inherent in natural genetics, which combines the survival of the fittest concept with genetic operators abstracted from nature (Holland 1975). The theory behind GA was proposed by Holland (1975) and further developed by Goldberg (1989) and others in 1980s. GA starts with random population of trial solutions called ‘chromosomes’. The fitness value (objective function value) associated with each chromosome is evaluated; and the new population for next generation is formed mainly through three genetic operations, namely selection, crossover and mutation. The selection operator


is used to determine which members of population are to be selected as ‘parents’ to produce ‘children’ (offspring). There are different types of selection operators’ available in literature (Deb 2001). Among them, the tournament selection method is one of the most popular and effective approach; and is adopted in the present study. In tournament selection scheme with tourney size two, two chromosomes are chosen randomly from the population and the one with higher fitness is selected. For crossover operation, some selected chromosomes are randomly assigned a mating partner, a random crossover location is selected and the genetic information is interchanged (called mating) to form ‘children’ with a certain probability known as ‘crossover probability’ (Pc ). Based on the number of location of information interchange, the crossover can be single point, two point or uniform. Studies have recommended that a crossover probability between 0.5–0.9 is an appropriate choice for solving most of the problems. The crossover operation is to be repeated among different pairs to form the new population. In order to maintain the diversity in a population (to explore new regions) and to avoid the chances of local trapping, an operation called mutation is applied. Mutation is achieved through a random local perturbation of bits (also called as genes) of a selected chromosome with a probability of mutation, Pm . A too low value of mutation probability will not result in creating useful chromosomes while too high value may result in generation of offspring with no resemblance with the parents. Thus a probability of mutation between 0.005–0.1 or 1/n (n is number of decision variables of the given problem) may be an appropriate choice (Deb 2001). The alteration of genes results in the formation of new chromosomes. More detailed description on the working of GA can be found elsewhere (Goldberg 1989; Deb 2001). Genetic algorithms have been applied successfully to solve many water resources problems, for example for rainfallrunoff modeling (Wang 1991); for hydrologic parameter estimation (Mohan and Vijayalakshmi 2008); for design of water distribution systems (Savic and Walters 1997); for reservoir operation (Oliveira and Loucks 1997; Wardlaw and Sharif 1999; Janga Reddy and Nagesh Kumar 2006); for design of open channels (Jain et al. 2004; Bhattacharjya and Satish 2008) etc. 2.2 PSO The PSO is a population based stochastic search optimization method inspired by the social behavior of bird flocking. The PSO was originally proposed by Kennedy and Eberhart (1995). Similar to other stochastic search algorithms such as GA, the PSO algorithm is also initialized with random solutions. But unlike GA, in basic PSO no operators inspired by natural evolution are used to produce a new generation of candidate solutions. The social sharing of information among individuals is the fundamental hypothesis of PSO. Particles fly through search space with velocities which are dynamically adjusted according to their own historical behaviors (cognitive learning experience) and social learning experiences. While using PSO, it can handle non-linearity and non-convexity of the problem domain easily; the search does not depend on initial population; and so the method can overcome the problem of trapping to local optima. The PSO is gaining popularity for solving many real world optimization problems in areas of biomedical, communication networks, control, electronics and electromagnetism, metallurgy, power systems, signal processing etc. (Poli 2008). Recently,


PSO also has been successfully applied for solving water resources management problems, for example, in water supply and related hydraulic design problems (Jung and Karney 2006; Suribabu and Neelakantan 2006; Montalvo et al. 2008); for reservoir operation and related water resources problems (Nagesh Kumar and Janga Reddy 2007; Janga Reddy and Nagesh Kumar 2007, 2009; Balter and Fontane 2008). From literature it was observed that there is no application of PSO method for channel design problems. 2.2.1 PSO Algorithm The random individual solution within a search space is called ‘particle’ and the population is called as the ‘swarm’. The dimension of the problem will be the number of decision variables involved. In a D dimensional search space, the i-th particle can be represented as a D dimensional vector Xi = (xi1 , xi2 .....xiD )T . The velocity (position change) of this particle is designated as Vi = (vi1 , vi2 ....viD )T . The best previously visited position of the ith particle is given by Pi = ( pi1 , pi2 , ..... piD )T . Defining g as the index of the best particle in the swarm and superscripts denoting the iteration number, the population dynamics can be described using the following equations. Velocity updating rule: ⎤ ⎡ n n Png − Xin − X P i ⎦ Vin+1 = χ ⎣ωVin + a1 r1 i + a2 r 2 (1) t t

Start Initialize the swarm with random position (X) and velocity (V) vectors

For each particle

Evaluate the fitness

Next particle

If fitness(X) is better than fitness(pbest), then update pbest = X

Update position of the particle using equation (2)

If fitness (pbest) is better than ‘gbest’, update gbest = pbest

No

Is termination criterion satisfied? Yes Stop & print the results

Fig. 1 The flow chart of PSO method

Update the velocity using equation (1)


Position updating rule: Xin+1 = Xin + (t) Vin+1

(2)

where d = 1,2,...,D, the index for decision variables; i = 1,2,...,N, the index for swarm population; N is the size of the swarm; a1 and a2 are called acceleration coefficients; r1 and r2 are uniformly generated random numbers between zero and 1; ω is the inertia weight ; χ is the constriction coefficient; t is time step (which is considered as unity). The control parameters of the algorithm play a vital role in getting good quality global optimal solutions. The flow chart of PSO method is shown in Fig. 1.

3 Model Formulation for Chance Constrained Composite Channel Design The optimal design of composite trapezoidal channels can be performed by solving a non-linear model with an objective function of minimization of unit cost of construction subjected to Manning’s equation for discharge computation as an equality constraint. Here the construction cost includes cost of excavation and cost of lining. The design of deterministic model may not be reliable always because the input design parameters are prone to uncertainties. To improve overall reliability and cost effectiveness, optimal design of trapezoidal open channels should take into account the uncertainties associated with cost parameters, channel surface roughness, dimensions, bed slope and designed flow. For real life applications, earlier studies have suggested use of mean dimensions with specified tolerances and a chance constrained optimal design methodology. A chance constrained design thus allows a margin of errors for construction and safety measures. In this section, a chance constrained model is presented. Figure 2 shows the definition sketch of a trapezoidal channel. The notations used in this figure denote, n1 , n2 , n3 be the roughness parameters for the sides and the bed of the channel; Tt be the top width; Tw be the flow top width and f be the freeboard. The development of chance constrained optimization model is based on the following assumptions: (1) the side slopes of the freeboard region are also lined with the same lining materials used in the flow region; (2) Horton’s (1933) formula is valid for the computation of equivalent roughness coefficient; (3) the Manning’s n, longitudinal slope (S) and design flow (Q) are mutually independent random

Tt Tw f 1

n2

n1

y n3

z2 b Fig. 2 Definition sketch of a trapezoidal canal

1 z1


variables and flow depth is dependent upon these variables; (4) all random variables follow normal distribution; (5) cost parameters are probabilistic and (6) first order uncertainty analysis is valid. By applying the principles of first order uncertainty analysis (Chow et al. 1988), the probabilistic cost function (C1 (x)) can be expressed as, ⎛ ⎞1/2 2 2 2 ∂C ∂C 2 ∂C 2 2 ∂C 2 2 ⎜ Sb ∂b + S y ∂ y + Sz1 ∂z1 + Sz2 ∂z2 ⎟ C1 (x) = k1 C + k2 ⎝ 2 2 2 ⎠ ∂C ∂C ∂C ∂C 2 2 2 2 +Sc0 ∂co + Sc1 ∂c1 + Sc2 ∂c2 + Sc3 ∂c3 (3) where, C = c0 At + c1 P1 + c2 P2 + c3 P3 is the deterministic cost per unit length of the channel; k1 and k2 are weighing coefficients; c0 is the excavation cost per unit area per m length of the channel; c1 , c2 and c3 are lining cost for the perimeter segments 1, 2, 3 respectively per m length of the channel. P1 , P2 , P3 are the perimeters of the two side slopes and the bottom of the channel respectively; b = base width of the canal, in m; y = flow depth, in m; z1 :1 and z2 :1 are the side slopes of the channel (as shown in Fig. 2); Sb , S y , Sz1 , Sz2 , Sco , Sc1 , Sc2 , Sc3 are the standard deviations of above described design parameters and cost parameters. In the chance constrained design the probability of the channel capacity being greater than the design flow can be described as: ⎛ ⎞ √ 5/3 SAw ⎜ ⎟ P ⎝ (4) 2/3 > Q⎠ ≥ α 3/2 ni Pwi where Q = expected design flow, m3 /s; Aw = expected flow area, m2 ; Pwi = expected wetted perimeter of individual sides, m; S = expected longitudinal slope of the channel; α is the flow exceedence probability; i = 1,2,..., j (here j = 3 for trapezoidal channels). The probabilistic optimization model for channel design can be expressed as:

Subject to

⎛

Minimize : C1 (x)

(5)

Maximize : α

(6)

√

⎜ P ⎝

⎞ 5/3 SAw

3/2

⎟ 2/3 > Q⎠ ≥ α

(7)

ni Pwi

In this model, minimization of cost and maximization of probability of flow exceedence are two conflicting goals. The Pareto optimal solutions can be generated by solving a transformed scalar problem. √ 5/3 SAw 2/3 , Eq. 7 can be rewritten as By taking Q1 = 3/2 ni

Pwi

P (Q1 > Q) ≥ α where, Q1 is the expected channel capacity.

(8)


The above probabilistic constraint can be written as an equality constraint in terms of standard normal variable as follows: (Q1 − Q) = Zα S Q1

(9)

where, Z a is the standard normal variable and S Q1 is the standard deviation of expected channel capacity. For solving in a meta-heuristic optimization framework, the chance constrained optimization model can be represented as follows: 2 1/2 2 ∂C Minimize : C1 (x) = k1 C + k2 Sx (10) ∂x Subjected to:

g1 (b , y, z1 , z2 ) = ε − (Q1 − Q) − S Q1 Z α ≥ 0

(11)

where, Sx is the standard deviation of x, the vector of all design variables; ε is a very mall positive real number.

4 Model Application and Results The algorithms of PSO and real coded GA are implemented in MATLAB programming and are used to solve the developed models. 4.1 Model Validation for Deterministic Design First the meta-heuristic techniques are applied to a deterministic channel design problem to verify the performance of PSO and GA methods with the conventional methods that were applied in the past. The deterministic model involves minimizing the total cost (excavation cost and lining cost per unit length of canal) subjected to Manning’s equation as the only equality constraint. The data used in the study: Q = 100 m3 /s; n1 = 0.02, n2 = 0.018, n3 = 0.015; S = 0.0016, c1 = 0.6, c2 = 0.2, c3 = 0.25, c4 = 0.3 and freeboard f = 0.5. To solve the deterministic model using PSO method, the following control parameters are used: swarm size (N) = 100; inertia weight (ω) = linearly varying from 1.2 to 0.4; acceleration coefficients a1 = 1 and a2 = 0.5; constriction coefficient (χ ) of 0.9. The model is solved for two cases fixed side slope condition and no restriction on side slope using the PSO method and the results compared with the earlier solutions of Lagrangian multiplier method (Das 2000), and GA method (Jain et al. 2004). The results of the LM, GA and PSO methods are presented in Table 1. It can be seen that there is not much significant difference in optimal costs of the PSO, GA and LM methods. However for complicated problems such as model that is presented in this study, (which accounts detailed uncertainty of various parameters), it may not be easy to apply and get optimal solutions with the conventional methods. Since, they may require several approximations or simplifications or derivative information on functions of the model. Hence, for solving highly nonlinear, nonconvex, multi-modal problems with complex functions, the traditional methods are not that attractive to

Chance Constrained Optimal Design of Composite Channels Table 1 Results of deterministic model for channel design, provides a comparison of optimal solutions obtained by meta-heuristic techniques PSO and GA with LM method Method

b(m)

Fixed side slope condition 3.776 LM methoda GAb 3760 3.776 PSOc No restriction on side slope 5.826 LM methoda 5.433 GAb 5.796 PSOc

y(m)

z1

z2

Cost (Rs 1000/m)

3.783 3.788 3.783

1.0 1.0 1.0

1.0 1.0 1.0

24.568 24.563 24.568

4.052 4.211 3.962

0.247 0.272 0.284

0.265 0.296 0.321

22.958 22.973 22.964

a Das

(2000) et al. (2004) c Present study b Jain

use them. Where as the meta-heuristic techniques such as PSO and GA can overcome these issues and can provide acceptable solutions to the problem in hand. In the following, the model formulated in Section 3 is applied for different cases of channel design. 4.2 Chance Constrained Model Without Any Site Specific Constraints 4.2.1 Channels with Composite Roughness The chance constrained design is performed for a composite trapezoidal channel for a specified weight combination of k1 = 1 and k2 = 1. The data for the representative mean values of discharge (Q), Manning’s roughness coefficient (n), bed slope (S) and cost parameters are considered as the same as mentioned in above section. The standard deviation of all design parameters is assumed to be 10% of the expected value. The chance constrained model is solved using both GA and PSO algorithms. The control parameters that are adopted for PSO method involves: a swarm size (N) of 100; inertia weight (ω) is linearly varying from 1.2 to 0.4; acceleration coefficients a1 = 1 and a2 = 0.5; constriction coefficient (χ ) of 0.9. The parameters used for GA model includes: a population size of 100, two point crossover operator with a crossover probability of 0.8, adaptive feasible mutation function with a mutation probability of 0.05, in conjunction with an elite count of 5. For selection scheme, a tournament selection scheme is adopted with a tourney size of two. To handle the constraints of the model a penalty function approach is adopted, i.e., if the generated member solution violates any of the constraints, such members are penalized by giving a suitable penalty to their fitness. To check the performance of the algorithms, the search process is attempted with different random populations and it is observed that the algorithms are consistently converging to the optimal or near optimal solutions. The model is solved for a set of flow exceedence probabilities ranging from 0.1 to 0.9 in steps of 0.1. The results obtained by both PSO and GA methods are shown in Fig. 3. It can be observed that the performance of both methods is very close and there is not much significant difference in their solutions. However, for few cases PSO solutions are slightly better than GA. Figure 4 depicts the detailed


35

GA-comp

PSO-comp

GA-unif

0.2

0.4

0.6

PSO-unif

Cost (Rs 1000/m)

30 25 20 15 10 5 0 0.1

0.3

0.5

0.7

0.8

0.9

Exceedence probablity (α) Fig. 3 Comparison of the results of PSO and GA methods for chance constrained design of trapezoidal channel without any site specific restrictions. It shows optimal costs obtained for different exceedence probabilities (α) for both the cases of composite channel (GA-comp and PSO-comp) and for uniform roughness channel (GA-unif and PSO-unif)

Pareto optimal solutions obtained using PSO for composite trapezoidal channels, which clearly shows that minimization of cost and maximization of probability of flow exceedence are two conflicting goals. To minimize the expected cost, one has

32

0.4

0.5

8

6

z2

30

0.4

z1

5.5

b

5

4.5

Optimal bed width (b ) (m)

0.2

0.2

Optimal flow depth (y) (m)

26

0.3

Optimal side slope (z1)

28

Optimal side slope (z2)

Optimal cost (Rs 1000/m)

0.3 6

Cost

4

0.1 24

0.1

22

0

y 4

0

3.5

2 0

0.2

0.4

0.6

0.8

1

Flow exceedence probability

Fig. 4 Detailed results of optimal solutions obtained using PSO for chance constrained design of composite trapezoidal channels


to move towards the origin along the vertical axis and to maximize the probability of exceedence of channel capacity over the design flow, one has to move away from origin along the horizontal axis. With an increase in flow exceedence probability, an increasing trend is observed in optimal bed width and optimal flow depth; while a decreasing trend is observed in the optimal side slope. For the value of probabilities greater than 0.7, the magnitude of optimal side slope is very small (less than 0.1) and practically the channel becomes rectangular for probabilities closer to 0.9. For higher values of probabilities in order to maintain the stability of the channel, it may require fixing the side slope before making any decision on the channel design. 4.2.2 Channels with Uniform Roughness Channels with uniform roughness (n1 = n2 = n3 = n) is considered as a special case of composite channels. The values of design parameters are Q = 100 m3 /s, n = 0.02, S = 0.0016, c0 = 0.6 and c1 = c2 = c3 = 0.2 and freeboard = 0.5. The chance constrained model involving Eqs. 10 and 11 is solved using both PSO and GA methods for different flow exceedence probabilities after setting the same control parameters as that used for composite roughness case. The results of the model are shown in Fig. 3 itself. The detailed optimal solutions obtained using PSO for uniform roughness case is shown in Fig. 5. The behavior of the optimal design curves and Pareto front is similar to that of composite roughness case, but an increase in cost is noticed for different probabilities from the corresponding solutions of composite roughness case. It is also noticed that there is an increase in optimal flow depth and

32

6

0.5

7

z 0.4

b

0.2

5

4

26

Optimal bed width (b) (m)

0.3

Optimal flow depth (y) (m)

28

6

Optimal side slope (z)


30

Cost

y 5

4 0.1

24

0

3

3 0

0.2

0.4

0.6

0.8

1


Fig. 5 Detailed results of optimal solutions obtained using PSO for chance constrained design of trapezoidal channels for uniform roughness case


side slope values and a reduction in optimal bed width, when compared with the corresponding solutions of composite channels. 4.3 Solution of Chance Constrained Model with Site Specific Constraints The conventional chance constrained optimization formulation can be modified to make it suit for fulfilling geometrical restrictions imposed by field conditions. The maximum availability of right of way and maximum allowable flow depth are two such scenarios considered independently in this study. 4.3.1 Top Width Restricted Chance Constrained Design of Trapezoidal Channel The land acquisition costs will be significantly high if the top width is very large and land availability may also be an issue in certain site conditions. The restriction on top width of the channel can be accounted by imposing an additional constraint given by Eq. 12, along with the basic chance constrained model formulation consisting of Eqs. 10 and 11. g2 (b , y, z1 , z2 ) = (Tmax − Tt ) ≥ 0

(12)

where, Tmax is the maximum permissible top width and for a freeboard height of f , the top width for trapezoidal channels Tt = b + (z1 + z2 ) (y + f ). From the solutions of earlier model for composite roughness case, it is observed that the top width exceeds 8 m for probability values greater than 0.7 and for uniform roughness case it falls in the range of 7–8 m for probability values in the range 0.7–0.9. Thus Tmax is adopted as 7.0 m for the present study. The model is solved using PSO method for both composite and uniform roughness cases for different

32 Channels with uniform roughness Channels with composite roughness


30

28

26

24

22

20 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Fig. 6 Comparison of the trade-offs between optimal cost and flow exceedence probability for composite and uniform roughness channels, in the case of top width restricted chance constrained channel design


flow exceedence probabilities. A comparison of the trade-offs between optimal cost and flow exceedence probabilities for composite and uniform roughness channels are shown in Fig. 6. It was observed that there is an increase in cost for higher flow exceedence probabilities in both the cases. In the case of composite channels, the optimal cost varies between 24.08 and 29.10 Rs 1,000/m for different values of flow exceedence probabilities. For channels with uniform roughness, the optimal cost varies between 22.25 and 31.09 Rs 1,000/m for different values of flow exceedence probabilities. Thus the obtained Pareto optimal fronts clearly show that the optimal cost increases over that of no restriction case 4.3.2 Depth Restricted Chance Constrained Design of Trapezoidal Channel In the case of existence of unfavorable strata, depth of channel should not go beyond a certain limit, because the excavation may not be economical or due to some other problems such as the presence of shallow ground water table. This may require restriction on the maximum permissible flow depth. Thus, the limiting depth scenario can be addressed in channel design problem by imposing Eq. 13 as an additional constraint in the optimization formulation involving Eqs. 10 and 11. g3 (y) = (ymax − y) ≥ 0

(13)

The maximum permissible flow depth (ymax ) was selected as 3.5 m, and hence the overall depth of the channel can be limited to 4 m. On solving the depth restricted model using PSO method, Fig. 7 depicts a comparison of the trade-offs between optimal cost and flow exceedence probability for composite and uniform roughness channels. From these solutions, it is observed that for different values of flow exceedence probabilities, the optimal cost varies in between 23.90 and 29.60 Rs

34 Channels with uniform roughness Channels with composite roughness


32

30

28

26

24

22

20 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Fig. 7 Comparison of the trade-offs between optimal cost and flow exceedence probability for composite channel with uniform roughness case, in the case of depth restricted chance constrained channel design


1,000/m for composite channels; whereas for uniform roughness case it varies in between 25.36 and 32.52 Rs 1,000/m. Here also the trade-offs for both the cases show that the optimal cost increases over that of no restriction case.

5 Discussion The chance constrained optimal design of open channels enables the user to account the uncertainty associated with the design parameters. It allows specified tolerances for mean values of channel dimensions. 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. From the practical implementation perspective, the uniform roughness condition is more relevant and the optimal design can be made for the user specified probability values. It is also noticed from the results of chance constrained model for both composite and uniform roughness cases, that for higher values of probabilities in order to maintain the stability of the channel, the design is to be made after fixing the side slope to specified values considering the site conditions. The results of the study shows that the meta-heuristic techniques GA and PSO are capable of handling the non-linearity and non-convexity associated with the chance constrained optimization model; the methods are derivative free optimization techniques; and they are successfully handling the additionally imposed site specific constraints and giving optimal solutions.

6 Conclusions In this paper, to avoid the unsuitable designs for practical applications and to account the detailed uncertainty, an improved approach is presented for optimal design of composite channels considering the site-specific features (such as restriction on top width and restriction on flow depth), and accounting the uncertainty in cost functions, and then solved using meta-heuristic techniques, namely Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) methods. The meta-heuristic methods are found to be quite successful in solving the optimization models for chance constrained channel design in spite of additional complexities incurred from site specific constraints. The proposed methodology for probabilistic design of open channels is simpler to implement and effective for practical applications, thus it can be used for reliable design of irrigation canals.

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