Optimization of water distribution systems design parameters using genetic algorithm
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• Suppose we have a problem!
• We don’t know how to solve it!
• What can we do?• Can we use a computer to
somehow find a solution for us?
• This would be nice! BUT Can it be done?
Start with a Dream
…
A Dumb Solution
!
A “blind generate and test” algorithm:
*RepeatGenerate a random
possible solutionTest the solution and see
how good it is
*Until solution is good enough
YES WHY NOT!!!
Can we use this dumb idea?• Erm... Sometimes – Yes:
– If there are only a few possible solutions– & you have enough time– then such a method could be used
• For most problems - No:– many possible solutions???– with no time to try them all !!!– My ‘Boss’ will kick me out !
*Generate a set of random solutions
#Repeat
Test each solution in the set (rank them)
Remove some bad solutions from set
Duplicate some good solutions
Make small changes to some of them
*Until best solution is good enough!
A “Less-Dumb” Idea (GA)
Conduit flow optimization of Design Parameters using GAs.
Seminar presentation by Tanay Kulkarni
Guided byDr. (Mrs.)K. C. Khare
Date: 02 April 2013, Dept. of Civil Engineering, SCOE, Pune.
continue dreaming…
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24.90m3/s
[100.00 m]
2.9
6
3.27.2
5.6
[95.00]
[94.00]
[92.00]
[90.00][94.00]
(300)
(400)
(350)
(300)
(250)
(200)
(200)
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Real Life Problem!
Optimize for cost & Diameter !
Single source, 7- links WDS with 5 Demand Nodes
Search Space in WDN = NX Where N= no. of commercial available pipes, X= no. of pipes !
Head Loss Equation:h = ѠLQα
CαHWDβ
Where, α & β are exponents and are taken as 1.85 & 4.87 resp. & w= 2.234 x 1012
Hazen-William Eq. but once can use Darcy-Weisbach also.
The critical path: 1-2-4
S Pj = H o – Hj min ; j= 1,…, (M – S)
L Pj
Max. Available friction Slope is defined.
L Pj = Minimum(L Pi + Lij); j = 1,…, (M - S)
NOW WE NEED TO DEFINE THE OBJECTIVE!
The Objective Function:
Minimize: f(D1,….,Dx) = ∑ u(Dx) X Lx; x=1,….,X Subjected to constraints
∑ Qx + qj = 0; j=1,…,(M - S)
∑ hx +∑ Ep = 0; y = 1,…,Y
Hj ≥ Hj min; j=1,…,(M - S)
Dx ϵ {Dmin,…,Dmax}; x=1,…,X
X incident on j
x ϵ y
Unit cost Length
Nodal demand
Total no. of nodes
Total no. of Source nodes
Energy added to water by pump
Total no. of Loops
Min. Permissible head at nodes
Total no. of links
Min. & Max. dia of commercially available pipes
Mathematical expression.
Time for penalty!Function defined to penalize the “infeasible solutions” to reduce their fitness.
THUS THE WEAKER SOLUTIONS CAN BE KILLED!
Penalty
Death
Static
Dynamic
Annealing
Niched
Self Organizing
Careful analysis in WDN revealed that:optimal solution lie at the boundary of feasible &
infeasible solutions.
Here, Self- organizing penalty function was used and thus the OBJECTIVE FUNCTION became
unconstrained!
The Objective Function Revised:
Minimize: f(D1,….,Dx) = ∑ u(Dx) X Lx + ∑ p X qj X {max (Hjmin – Hj , 0)}
x=1,….,X
Penalty Multiplier!Max violation of each pressure constraint at node j
Penalty? For what?
GRA-NET soft-tool was used! Population size = 60, Crossover % = 0.95, Mutation % = 0.02-0.05,
number of generations = 25!
And the Genetic Algorithm was RUN!
• With 14 commercially available pipes, the possible solutions are 147 (=105,413,504).• GA reduces the search space to 57
(=78,125).• With repetition of smallest available sizes it further reduces to 18,000 !!!
Rs. 44,48,250/-1500 evaluations
P3/166 MHz/ 32MB RAM.
1. (400mm)2. (300mm)3. (200mm)4. (300mm)5. (200mm)6. (150mm)7. (150mm)
RESULT IN 9 SECONDS!!!
i.e. 0.074 % of entire search space!i.e. 0.017 % of entire search space!
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24.90m3/s
[100.00 m]
6
3.27.2
5.6
[95.00]
[94.00]
[92.00]
[94.00]
(300)
(400)
(350)
(300)
(250)
(200)
(200)
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DedicationI dedicate this seminar of mine to Mr. Pramod Bhave for his beautiful contribution
to the field of Hydraulics Engineering.
“Dream Works!!!”- Steven Spielberg
References• Mahendra Kadu, Rajesh Gupta, Pramod Bhave,”Optimal Design of Water
Networks Using a Modified Genetic Algorithm with Reduction in Search Space”, JWRPM 134(2) © ASCE/ MARCH/APRIL 2008 / 147-159.
• S. N. Sivanandanam, S. N. Deepa, “Introduction to Genetic Algorithm”,ISBN 978-3-540-73189-4 ©Springer- Verlag Berlin Heidelberg 2008.
• Prashant Shinde, K. C. Khare, ”Optimization of Water Distribution Network”, Seminar Report ©SCOE-DCE 2011
Thank you!
Genetic Algorithms - History• Pioneered by John Holland in the 1970’s• Got popular in the late 1980’s• Based on ideas from Darwinian Evolution• Can be used to solve a variety of problems that are not easy to solve
using other techniques
BACK-UP!!!
Evolution in the real world• Each cell of a living thing contains chromosomes - strings of DNA• Each chromosome contains a set of genes - blocks of DNA• Each gene determines some aspect of the organism (like eye colour)• A collection of genes is sometimes called a genotype• A collection of aspects (like eye colour) is sometimes called a phenotype• Reproduction involves recombination of genes from parents and then
small amounts of mutation (errors) in copying • The fitness of an organism is how much it can reproduce before it dies• Evolution based on “survival of the fittest”
The MetaphorNature Genetic Algorithm
Environment Optimization problem
Individuals living in that environment
Feasible solutions
Individual’s degree of adaptation to its surrounding environment
Solutions quality (fitness function)
Nature Genetic Algorithm
A population of organisms (species)
A set of feasible solutions
Selection, recombination and mutation in nature’s evolutionary process
Stochastic operators
Evolution of populations to suit their environment
Iteratively applying a set of stochastic operators on a set of feasible solutions
Pipe Diameter (mm)
Unit Cost in Rupees
150 1115
200 1600
250 2154
300 2780
350 3475
400 4255
450 5172
500 6092
600 8189
700 10670
750 11874
800 13261
900 16151
1000 19395
Commercially available pipe sizes and their costs
Advantages :A GA has a number of advantages.
#It can quickly scan a vast solution set.
# Bad proposals do not effect the end solution negatively as they are simply discarded.
#The inductive nature of the GA means that it doesn't have to know any rules of the problem - it works by its own internal rules.
#This is very useful for complex or loosely defined problems.
Disadvantages :
• A practical disadvantage of the genetic algorithm involves longer running times on the computer. Fortunately, this disadvantage continues to be minimized by the ever-increasing processing speeds of today's computers.
• If we have a hammer, all problems looks like a nail!!!
GA in Classes of Search TechniquesSearch Techniqes
Calculus Base
Techniques
Guided random search techniques Enumerative Techniques
BFSDFS Dynamic Programming
Tabu SearchHill Climbing
Simulated Annealing
Evolutionary Algorithms
Genetic Programming Genetic Algorithms
Fibonacci Sort
Distribution tree or sub tree
Path/ sub-pathserial number
Path Length of path(m)
HGL at Source of Path (m)
HGL at End of Path (m)
Max. available friction loss (m)
Slope of path
(a) Distribution tree
1 1 1-2 300 100.00 95.00 5.00 0.01667
2 1-3 400 100.00 94.00 6.00 0.01500
3 1-2-4 500 100.00 94.00 6.00 0.01200
4 1-2-5 550 100.00 92.00 8.00 0.01454
5 1-2-5-6 750 100.00 90.00 10.00 0.01333
(b) Distribution sub-tree
1.1 2 2-5 250 96.40 92.00 4.40 0.01760
3 2-5-6 450 96.40 90.00 6.40 0.01422
S Pj = H o – Hj min ; j= 1,…, (M – S)
L Pj
Column 7:Max. available friction loss (m) = HGL @ Source of path – HGL @ End of Path
Determination of Critical Path & Critical Sub-paths
Link number
Length(m)
Head Loss(m)
Discharge (m3 / s)
Diameter (m) Candidate Diameters (mm)
1 300 3.60 12.366 348.37 250, 300, 350, 400, 450
2 400 6.00 6.934 267.11 150, 200, 250, 300, 350
3 200 2.40 2.900 200.81 150, 150, 200, 250, 300
4 250 3.56 8.472 291.41 200, 250, 300, 350, 400
5 200 2.85 3.200 201.30 150, 150, 200, 250, 300
6a 350 2.40 0.994 - 150, 150, 150, 200, 250
7a 300 1.16 0.728 - 150, 150, 150, 200, 250