energy enigma-solution through genetic algorithm

7/25/2019 Energy Enigma-Solution Through Genetic Algorithm

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Project Report On

ENERGY

ENIGMA

Anshuman Das Mohapatra

113CS0133

Course Title: Optimization Methods in Engineering

Course ID: CE403


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With fossil fuels getting exhausted shortly and consequently less number of energy sources to

rely on, a huge energy crisis is expected. However, with a sustainable approach, energy can be

saved, which will reduce the intensity of the significant energy shortage in future. The main

objective of our project is to maximize the energy generation from a particular biogas plant by

locating a feasible region of its construction. The various proposed relationships between

population, economic development and efficiency are taken into account for generation of a

constrained non-linear optimization problem which is then solved through MATLAB. The

results are then compared with the population of various cities as per the Census of India, 2011.

The location of the Gas Pipeline Network of India plays a crucial role in the choosing a

solution among various cities in the feasible list. The approach in this project is however

gradual. We start with small linear optimization problem which can be solved graphically, then

add more variables to demonstrate the significance of other existing algorithms such asSimplex Algorithm, then go for a MATLAB solver to solve the main objective of this project

which cannot be solved by the Simplex approach. Therefore, certain other aspects of social

importance where this project can be used have also been covered.

ABSTRACT


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I am grateful to Prof. Sarat Kumar Das, without whose guidance and support this compilation

would not have been possible. I could not have a profound approach and solution to real life

problems through effective techniques without him. I am also very thankful to Prof. Anil

Kumar Bangia for inculcating and consolidating Linear Programming concepts possessed by

me.

ACKNOWLEDGMENT


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Contents

ABSTRACT ................................................................................................................................................... 1

ACKNOWLEDGMENT .................................................................................................................................. 2

INTRODUCTION .......................................................................................................................................... 4

LINEAR PROGRAMMING ............................................................................................................................ 5

GRAPHICAL SOLUTION.................................................................................................................... 5

UNDERSTANDING GRAPHICAL METHODOLOGY...................................................................... 5

FORMULATION OF EQUATIONS..................................................................................................... 6

SOLUTION TO THE PROBLEM......................................................................................................... 6

PROS & CONS TO GRAPHICAL METHOD...................................................................................... 7

PROBLEM STATEMENT.................................................................................................................... 7

ALGORITHM TO SOLVE THE STANDARD PROBLEM................................................................ 7

SIMPLEX METHOD OF SOLVING A LINEAR PROBLEM ............................................................ 8

DEALING WITH THE PROBLEM ................................................................................................................... 9

PROS & CONS TO SIMPLEX METHOD......................................................................................... 11

GENETIC ALGORITHM .............................................................................................................................. 11

FORMULATION OF NON-LINEAR MATHEMATICAL MODEL................................................ 12

GENERALIZED REDUCED GRADIENT ALGORITHM USING MICROSOFT EXCEL ......................................... 14

SOLVING THE MAIN PROBLEM.................................................................................................... 15

CONCLUSION ............................................................................................................................................ 16

BIBLIOGRAPHY ......................................................................................................................................... 17
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INTRODUCTION

Energy statistics of the contemporary period suggest that we rely heavily on fossil energy for

our energy demands. Fossil energy is a non-renewable source of energy. It takes centuries for

the bio forms deep inside the earths crust to be converted into petroleum energy. Recent

statistics suggest that the Fossil Energy Fuel Industry is now entering terminal deadline.Therefore the need of the hour is an efficient energy plan layout to preserve the fossil energy in

the environment and switch over to renewable sources of energy. In the present scenario,

population masses are growing rapidly. Every day the increasing population adds waste

elements in the form of pollutants to the surroundings, which in turn makes the situation worse

with the incident of a lot of communicable and non-communicable diseases. Also, unchecked

sewage treatment leads to pollution of the water bodies there by affecting the aquatic life. We

can use this bio-waste source as a source of energy, which will not only allow sustainable

development but also preserve the environment.

Biogas typically refers to a mixture of differentgasesproduced by the breakdown oforganic

matter in the absence ofoxygen.Biogas can be produced from raw materials such as

agricultural waste,manure,municipal waste,plant material,sewage,green waste orfood

waste.It is a renewable energy source and in many cases exerts a very small carbon footprint.

Biogas can be produced byanaerobic digestion withanaerobic bacteria,which digest material

inside a closed system, orfermentation of biodegradable materials.

Biogas is primarilymethane (CH4) andcarbon dioxide (CO2) and may have small amounts of

hydrogen sulfide (H2S), moisture andsiloxanes.The gasesmethane,hydrogen,andcarbon

monoxide (CO) can be combusted or oxidized with oxygen. This energy release allows biogas

to be used as a fuel; it can be used for any heating purpose, such as cooking. It can also be used

in a gas engine to convert the energy in the gas into electricity and heat.

In this project, we aim to find the optimal location for setting up of a biogas plant in India, that

is, to find a feasible city with the requisite population size for an acceptable profit with

maximum efficiency of the plant as 95%.
https://en.wikipedia.org/wiki/Gashttps://en.wikipedia.org/wiki/Organic_matterhttps://en.wikipedia.org/wiki/Organic_matterhttps://en.wikipedia.org/wiki/Oxygenhttps://en.wikipedia.org/wiki/Manurehttps://en.wikipedia.org/wiki/Municipal_wastehttps://en.wikipedia.org/wiki/Plant_materialhttps://en.wikipedia.org/wiki/Sewagehttps://en.wikipedia.org/wiki/Green_wastehttps://en.wikipedia.org/wiki/Food_wastehttps://en.wikipedia.org/wiki/Food_wastehttps://en.wikipedia.org/wiki/Anaerobic_digestionhttps://en.wikipedia.org/wiki/Anaerobic_organismhttps://en.wikipedia.org/wiki/Fermentation_(biochemistry)https://en.wikipedia.org/wiki/Methanehttps://en.wikipedia.org/wiki/Carbon_dioxidehttps://en.wikipedia.org/wiki/Hydrogen_sulfidehttps://en.wikipedia.org/wiki/Siloxanehttps://en.wikipedia.org/wiki/Methanehttps://en.wikipedia.org/wiki/Hydrogenhttps://en.wikipedia.org/wiki/Carbon_monoxidehttps://en.wikipedia.org/wiki/Carbon_monoxidehttps://en.wikipedia.org/wiki/Carbon_monoxidehttps://en.wikipedia.org/wiki/Carbon_monoxidehttps://en.wikipedia.org/wiki/Hydrogenhttps://en.wikipedia.org/wiki/Methanehttps://en.wikipedia.org/wiki/Siloxanehttps://en.wikipedia.org/wiki/Hydrogen_sulfidehttps://en.wikipedia.org/wiki/Carbon_dioxidehttps://en.wikipedia.org/wiki/Methanehttps://en.wikipedia.org/wiki/Fermentation_(biochemistry)https://en.wikipedia.org/wiki/Anaerobic_organismhttps://en.wikipedia.org/wiki/Anaerobic_digestionhttps://en.wikipedia.org/wiki/Food_wastehttps://en.wikipedia.org/wiki/Food_wastehttps://en.wikipedia.org/wiki/Green_wastehttps://en.wikipedia.org/wiki/Sewagehttps://en.wikipedia.org/wiki/Plant_materialhttps://en.wikipedia.org/wiki/Municipal_wastehttps://en.wikipedia.org/wiki/Manurehttps://en.wikipedia.org/wiki/Oxygenhttps://en.wikipedia.org/wiki/Organic_matterhttps://en.wikipedia.org/wiki/Organic_matterhttps://en.wikipedia.org/wiki/Gas


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This is a mathematical technique for determining a way to achieve the best outcome

(such as maximum profit or minimum loss) in a given condition for some requirements

represented as linear relationships.

Linear programming can be solved either through graphical method or by using simplex

method.

Solution is achieved from the following steps:

I.

Formulation of objective equation and constraint equations from the problem

II.

Representing the objective equation and the constraint equations in a graph

III. Finding out the feasible region from the graph

Let a person wants to start a furniture business of tables and chairs. The cost price of each table

is Rs. 100 and that of each chair is Rs. 20. The selling price of the same is Rs. 120 and Rs. 35respectively. The capacity of the shop is 30 articles. The man can invest a maximum of Rs.

2000 for the business. He wants to have chairs more than or equal

to twice the number of tables. We need to find the number of chairs and tables the man should

buy so as to have a maximum profit.

Figure 1 Figure 2

LINEAR PROGRAMMING

GRAPHICAL SOLUTION

UNDERSTANDING GRAPHICAL METHODOLOGY


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Let the man needs to buy x number of chairs and y number of tables. Our main objective function

is

Maximize p=15x+20y, where p is the profit obtained from the business.

Subject to constraints:

x>=0; y>=0

x+y


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SOLUTION OBTAINED:

X = 18

Y = 9

P = 450

It is convenient for problems involving two variables. So for the main objective of this project

which is to determine an optimal location for the set up of a bio gas plant this method is not

suitable. Hence, we go

for the simplex method which facilitates the use of two or more variables to easily optimize a

linear programming problem.

To find out: The Grade point (p) of Population (x), Profit (y) and Efficiency (z) of Bio-Gas

Plant to be set up in a city by solving the following linear problem.

P-Grade point scale assigned to respective cities

Problem: To maximize p = x + y + z

Subject to constraints:

x + y + z 200.(i) (Maximum grade point is 200)

2x + yz 180(ii) (Twice the population with profit half of that and reduced

efficiency must yield a maximum grade point of 180)

x + 2y +2z 220(iii) (Reduced population with twice the profit and increased efficiency

by a factor of 2 must yield a maximum gradepoint of 220)

ALGORITHM TO SOLVE THE STANDARD PROBLEM

To generalize the main optimization problem, all the variables x, y & z are brought to left side

i.e.,

xyz + p = 0

Slack variables s, t, & u are introduced in the inequalities (i), (ii) & (iii) above to equate them

with zero as under.

x + y + z + s = 200

2x + yz + t = 180

-x + 2y + 2z + u = 220

Then a spread sheet is generated using Simplex Method.

PROS & CONS TO GRAPHICAL METHOD

PROBLEM STATEMENT


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SIMPLEX METHOD OF SOLVING A LINEAR PROBLEM

Tableau - 1

x y z s t u p Ans. Ratio

s 1 1 1 1 0 0 0 200 200t 2 1 -1 0 1 0 0 180 90

u -1 2 2 0 0 1 0 220

p -1 -1 -1 0 0 0 1 0

Tableau - 2

x y z s t u p Ans.

s 0 1 3 2 -1 0 0 220 2R1-R2 220

x 2 1 -1 0 1 0 0 180 Pivot Row 180

u 0 5 3 0 1 2 0 620 2R3+R2 124

p 0 -1 -3 0 1 0 2 180 2R4+R2

Tableau - 3

x y z s t u p Ans.

s 0 0 12 10 -6 -2 0 480 5R1-R3 40

x 10 0 -8 0 4 -2 0 280 5R2-R3

y 0 5 3 0 1 2 0 620 Pivot Row 206.67

p 0 0 -12 0 6 2 10 1520 5R4+R3

Tableau - 4

x y z s t u p Ans.

z 0 0 12 10 -6 -2 0 480 Pivot Row

x 30 0 0 20 0 -10 0 1800 2R1+3R2

y 0 20 0 -10 10 10 0 2000 4R3-R1

p 0 0 0 10 0 0 10 2000 R4+R1

Conclusion:

x = 60

y = 100

z = 40

p = 200


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The above smaller values are taken for a small hypothetical village just to establish the

linear relationships.

Using the same logic a program is compiled in C++ to compute the larger values which

are the integral multiples of the already established relationships.

The above smaller values are taken for a small hypothetical village just to establish the

linear relationships.

Using the QSopt softwareto compute for the larger constraint values which are the

close to the population census data.

MAIN PROBLEM:

Maximize

Objective function: x+y+z

Subject

c1:x+y-z


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SOLUTION:

Objective:

10000190.00000

Primal Solution Values:

x = 6660095.00000

y = 3340000.00000

z = 95.00000

Both Ahmedabad and Chennai have population close to the above obtained value.

However, since, Ahmedabad is much closer to the Gas Pipeline Network of India, it should

be chosen as the optimal location to facilitate maximum efficiency and profit. The Gas

Pipeline Network would enhance the energy transportation at least cost.

Figure 4

Figure 5


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The Simplex method can be used to solve Linear Programming Problems quite easily.

However, the real life problems are non-linear in nature. To arrive at a more accurate result we

use the APMonitor Interface in MATLAB and solve the non-linear programming problem as

formed below. The interface uses the Genetic Algorithm approach to arrive at the optimalsolution.

GENETIC ALGORITHM

The genetic algorithm is a method for solving both constrained and unconstrained optimization

problems that is based on natural selection, the process that drives biological evolution. The

genetic algorithm repeatedly modifies a population of individual solutions. At each step, the

genetic algorithm selects individuals at random from the current population to be parents and

uses them to produce the children for the next generation. Over successive generations, the

population "evolves" toward an optimal solution. You can apply the genetic algorithm to solvea variety of optimization problems that are not well suited for standard optimization

algorithms, including problems in which the objective function is discontinuous, non-

differentiable, stochastic, or highly nonlinear. The genetic algorithm can address problems

of mixed integer programming, where some components are restricted to be integer-valued.

The genetic algorithm uses three main types of rules at each step to create the next generation

from the current population:

Selection rulesselect the individuals, calledparents, that contribute to the population at the

next generation.

Crossover rulescombine two parents to form children for the next generation.

Mutation rulesapply random changes to individual parents to form children.

The genetic algorithm differs from a classical, derivative-based, optimization algorithm in two

main ways, as summarized in the following table.

Tableau - 5

Classical Algorithm Genetic Algorithm

Generates a single point at each iteration. The

sequence of points approaches an optimal

solution.

Generates a population of points at each

iteration. The best point in the population

approaches an optimal solution.

Selects the next point in the sequence by a

deterministic computation.

Selects the next population by computation

which uses random number generators.

In order to arrive at a better optimal solution the results of the APMonitor Interface Solver(using Genetic Algorithm) were compared with the Generalised Reduced Gradient Algorithm

of Microsoft Excel.

PROS & CONS TO SIMPLEX METHOD


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In order to formulate the objective function, we need to consider the hypothesis that we want

squared profit with respect to the consuming population. With the population growing steadily,

the yielded plant efficiency is acceptable. So, the objective function can be given as

maximize p = x1+x22+x3

where, p is the Grade Point of the plant.

Figure 6

Source: U.S. Census Bureau, International Data Base, June 2011 Update

The figure above suggests that GDP growth of the country decreases in exponentially as the

population growth increases every year (this can happen in most of the developing countries

where the population may be a liability). So we use Data Science here to predict one of the

constraints as

-2*x1+x22+x3


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rate of population growth do not necessarily imply low rates of per capita income if the

population is an asset. This leads us to the third constraint as

x1+x2-x3=0, x1=0

x3>=0, x3


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The MATLAB solver computed the results in a variable y which had the following values:

Figure 9

Figure 10

The values suggest that, we can construct a biogas plant with 95% efficiency and a profit per

year of Rs. 3.1649e+03where the population is currently 99,96,930.

Figure 11

Although Bangalore has the population necessary for this purpose, however, Delhi must be

chosen as the feasible city, since it is closer to the Gas Pipeline Network of India.

GENERALIZED REDUCED GRADIENT ALGORITHM USING

MICROSOFT EXCEL

The GRG Solving method alonelike virtually all classical nonlinear optimization

algorithmscan find a locally optimalsolution to a reasonably well-scaled, non-convex model.

At times, Solver will stop before finding a locally optimal solution, when it is making very

slow progress (the objective function is changing very little from one trial solution to another)

or for other reasons.

When the message Solver found a solution appears, it means that the GRG method has founda locally optimal solutionthere is no other set of values for the decision variables close to the

current values that yields a better value for the objective function. Figuratively, this


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means that Solver has found a peak (if maximizing) or valley (if minimizing) but if the

model is non-convex, there may be other taller peaks or deeper valleys far away from the

current solution.

Mathematically, this message means that the Karush - Kuhn - Tucker (KKT) conditions for

local optimality have been satisfied (to within a certain tolerance, related to the Precisionsetting in the Solver Options dialog).

SOLVING THE MAIN PROBLEM

Using the same set of nonlinear constraints, and the objective function, formulae were

incorporated into the Excel spreadsheet to connect the constraints to respective cells. The Excel

Solver Add-in was made active under the File->Options menu and the solver button under the

Data tab was used to insert the Solver Parameters as follows.

Figure 12

The solver solved the nonlinear constraint optimization problem using the GRG nonlinear

method (which was selected under the Solver Parameters dialog) and the respective cells in the


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Excel spreadsheet were filled up with the optimal values.

Figure 13

Although the solver generates almost identical results to that of APMonitor Interface of

MATLAB, however, it can be easily noted that the GRG Nonlinear Algorithm doesnt

converge to the true values faster than the Genetic Algorithm. It stops with a population

requirement of 9996930.062 because of maximum iterations limit and fails to give the realistic

figure of 9996930 (which is obtained in case of Genetic Algorithm Method).

Three methods yielded two different results. The Simplex Algorithm suggested Ahmedabad as

the most feasible city for the setting up of the Biogas plant whereas the MATLAB solver and

the Excel solver brought Delhi as the feasible city. Since, the input to the solver in the second

case was a Non-Linear Optimization Problem which is more realistic than a LPP, therefore,

Delhi is the most feasible city for the given purpose. In other words, setting up a biogas plant at

Delhi would mean sustainable energy resource for years to come with boost to the GDP of the

country and a means for efficient waste usage. The Gas Pipeline Network in close proximity to

Delhi would mean easy access to the resource for nearby cities. Also we concluded that

Genetic Algorithm converges faster than GRG Nonlinear Algorithm with a maximum limit on

the number of iterations (under the Solver Parameters properties).

CONCLUSION


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BIBLIOGRAPHY

http://cleantechnica.com/2015/07/20/the-fossil-fuel-energy-industry-is-now-entering-

terminal-decline/

in.mathworks.com

www.apmonitor.com www.wikipedia.org

www.youtube.com/watch?v=Q2zgz0ag0L0 (Mathematical Optimization with

MATLAB)

http://www.excel-easy.com/data-analysis.html
http://www.apmonitor.com/http://www.apmonitor.com/http://www.wikipedia.org/http://www.wikipedia.org/http://www.youtube.com/watch?v=Q2zgz0ag0L0http://www.excel-easy.com/data-analysis.htmlhttp://www.excel-easy.com/data-analysis.htmlhttp://www.excel-easy.com/data-analysis.htmlhttp://www.youtube.com/watch?v=Q2zgz0ag0L0http://www.wikipedia.org/http://www.apmonitor.com/

energy enigma-solution through genetic algorithm

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