Optimal Design of Energy Systems

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Optimal Design of Energy Systems

Chapter 11 Geometric Programming

Min Soo KIM

Department of Mechanical and Aerospace Engineering

Seoul National University

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11.1 Introduction

- Sum of polynomials

- Find optimum value of the function first

Chapter 11. Geometric Programming

objective function

constraints

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11.3 Degree of difficulty

Chapter 11. Geometric Programming

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D O D  T  N 

# of terms in objective

function + constraints # of variables

in this chapter DOD=0

6 3 1 0 1/ 2

y   x  x ex)

0 .5 2 1 / 5

1 2 1 2

1 2

5 2

5 0

y x x x x

x x

  



T=2 ( ) N=1 ( )

DOD=0

3 , 1 0x x1/ 2

 x

T=4 (3 from objective function + 1 from constraint) N=2DOD=1

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11.4 Solution for one independent variable, unconstrained

1 2

a a

y  c x  c x

optimized

u

1

u

₂

1 2

* 1 2

1 2

w w

a a

c x c x

y w w

   

    

   

1 2

w w

c c

w w

   

    

   

1 2

1 1 2 2

1

0 w w

a w a w

 

(at optimum)

1 2

1 1 2 1

2 1

1 2

2 1 2 1

* *

* 1 2

1 2

* *

* * * 1 2

1 2

(1 ) 0

,

( ) ( )

w w

a w a w

a a

w w

a a a a

u u

y w w

u u

y u u

w w

  

  

 

   

    

   

  

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11.4 Solution for one independent variable, unconstrained Ex)

1 1

2

y  x  x

^

 x x 

^

1 2

* 1/ 2 1/ 2

1 0 1 / 2

( 2 ) ( 2 ) 2 w w

w w

y

 

 

 

  

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11.5 How Geometric Programming works?

Chapter 11. Geometric Programming

1 2

1

w  w 

1 2

1 2 1 2

a a

y  c x  c x  u  u

1 2 1 2

1 2

1 2 1 2

w w a w a w

u u c x c x

g w w w w

       

         

       

optimize ln g  w

1

(ln u

1

 ln w

1

)  w

2

(ln u

2

 ln w

2

) subject to   w

₁

 w

₂

  1 0

(maximum of = maximum of )

g ln g

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Lagrange multiplier method w

₁

:

w

₂

:

(ln g )   0

   

1 1 1

1

ln u ln w w 1 0

w



    

2 2 2

2

ln u ln w w 1 0

w^



   

1 1

1 2 1 2

2 2

1 2

ln ln ln ln

1

u w

u u w w

u w

w w

    

 

1

2 2

2

1 u

w w

 u 

2 1

1 2 1 2

u , u

w w

u u u u

 

 

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1 2

1 2 1 2

1 2

1

/ (

1 2

)

2

/ (

1 2

)

u u

u u u u

u u

g u u

u u u u u u

 

   

            

At optimum ¹ ²

1 2

*

1 1

* *

1 1 2 2

* * * * *

1 1 2 2 1 1 2 2

( )

0

( ) 0

a a

d y x

c a x c a x d x

x c a x c a x a u a u

 

  

     

*

1 2

1

* 1 * 2 1

1 1

2 1

2

1 2

u a

w a a a

u u

a w a

a a

 



 

  

 

 

1 1 2 2

0

a w  a w 

2 1

1 2

1 2 1 2 1 2

* 1 2 1 2 *

1 2 1 2

a a

w w

a a a a a a

c x c x c c

g y

w w w w

 

       

          

       

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11.7 unconstrained, multivariable optimization

<Example 11.4> The pump and piping of example 11.1 are actually part of a waste-treatment complex, as shown in Fig. 11-1. The system accomplishes the treatment by a combination of dilution and chemical action so that the effluent meets code requirements. The size of the reactor can decrease as the dilution increases. The cost of the reactor is 150/Q, where Q is the flow rate in cubic meters per second. The equation for the pumping cost with Q broken out of the combined constant is (220 X 10¹⁵Q²)/D⁵ and the cost of pipe is 160D where D is diameter of pipe in milimiters. Use geometric programming to optimize the total system.

Chapter 11. Geometric Programming

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1 5 2 5

2 2 0 1 0 1 5 0

1 6 0 Q

y D

D Q

   

( 1)

3 2

D O D  T  N 

3

1 2

* 1 2 3

1 2 3

w w w

u

u u

y w w w

 

   

      

     

1 2 3

1

w  w  w 

1 2

2 3

5 0

2 0

w w

 

D:

Q:

1 1

1

2 1

5 5

w w

w   

₁

5

₂

1

₃

2

, ,

8 8 8

w  w  w 

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3

1 1 5 2

*

1 2 3

1 6 0 2 2 0 1 0 1 5 0

$ 3 0, 2 2 4

w

w w

y w w w

 

    

       

     

* * * *

1 1

* *

2 2

* * * * 3

3 3

1 6 0 = 1 1 8

1 5 0 / 0 .0 1 9 8 /

u w y D D m m

u w y

u w y Q Q m s

  



   

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0

 y

1 5 2

3

6 1 5

1 5

5 2

1 5 3

1 5

5 6

1 5

( 5) 2 2 0 1 0 1 6 0

1 6 0 0

5 2 2 0 1 0

2 2 0 1 0 2 1 5 0

0 1 6 0 2 2 0 1 0 2

1 5 0

5 2 2 0 1 0 0

1 6 0 5 2 2 0 1 0

y Q

Q D

D D

y Q

Q D Q

D

D D

       

  

      



    

  

 

Lagrange Multiplier Method

3

= 1 1 8

0 .0 1 9 8 /

D m m

Q  m s

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11.8 Constrained optimization with DOD=0

objective function

1 2 3

y  u  u  u

x₁, x₂, x₃, x₄ subject to

4 5 1

u  u 

geometric programming

3

1 2

3

1 2

1 2 3

w

w w

u

u u

y g

w w w

 

   

       

     

1 3

1 1 1 2 1 4

1 1 2 3 4

a

a a a

c x x x x

1 2 3

1

w  w  w 

( 1) 0

5 4

D O D  T  N  

variables

should be 1

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11.8 Constrained optimization with DOD=0

1 1

1 2 3

w u

u u u

  

² ₁ ₂² ₃

w u

u u u

  

³ ₁ ₂³ ₃

w u

u u u

  

5 4

4 5

1

w w

u u u u

w w

 

      

   

4 5 1

w  w 

4

4 4

4 5

w u u

u u

 



⁵ ₄ ⁵ ₅ ⁵

w u u

u u

 



5 4

4 5

1

M w

u

w w

 

    

   

3 5

1 2 4

3 5

1 2 4

1 2 3 4 5

w M w

w w M w

u u

u u u

y g

w w w w w

   

     

           

         

M : arbitrary constant

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1 2 3 4 5

( u u u )  ( u u ) 0

      

4 5

1

u  u 

* * * * *

1 1 1 2 1 2 3 1 3 4 1 4 5 1 5

1 2 1 3 1 4

0

a u a u a u a u a u

a a a

 

    



divide these equations by

y

^*

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* *

* * *

3 5

1 2 4

1 1 * 2 1 * 3 1 * 4 1 * 5 1 *

1 2 1 3 1 4

u u 0

u u u

a a a a a

y y y y y

a a a

 

    



w₁ w₂ w₃ Mw₄ Mw₅

1 2 3

1

w  w  w 

4 5

M w  M w  M

3 5

1 2 4

* 1 2 3 4 5

1 2 3 4 5

w M w

w w M w

u u

u u u

y w w w w w

   

     

          

         

6 unknowns : w

_i

,M

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Chapter 11. Geometric Programming

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<solution> total cost y

1 .2 1 .5

( 2 5 0 0 0 .0 0 0 3 2 ) 2, 5 6 0, 0 0 0

y  n   p  D

3 0, 0 0 0 m

n  L

2 2

3

2 5

0 .1 6 1

(0 .0 2 ) (1 0 0 0 / ) 0 .4 1 5 0

2 / 4 2

L V L L

p f kg m

D  D D D



 

       

1 .2

7 5, 0 0 0, 0 0 0 9 .6

1 .5

2, 5 6 0, 0 0 0

y p D

L L

   

2 .4 1 0

5

p D 1 L

 

optimize

subject to

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3

1 2 4

*

1 2 3 4

7 5, 0 0 0, 0 0 0

^w

9 .6

^w

2, 5 6 0, 0 0 0

^w

2 .4 1

^{M w}

y w w w w

 

     

        

       

1 2 4

2 4

3 4

1 2 3

4

0

1 .2 0

1 .5 5 0

1

w w M w

w M w

w w w

M w M

   

 

  

 L :

Δp : D :

1 0.0385, 2 0.1923, 3 0.7692, 0.2308, 4 0.2308

w  w  w  M   M w  

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0 .0 3 8 5 0 .1 9 2 3 0 .7 6 9 2 0 .2 3 0 8

*

7 5, 0 0 0, 0 0 0 9 .6 2, 5 6 0, 0 0 0 2 .4 1

0 .0 3 8 5 0 .1 9 2 3 0 .7 6 9 2 1

y

       



                

*

$ 4 1 0,1 5 0 y 

* 1

7 5, 0 0 0, 0 0 0 ( 4 1 0,1 5 0 )(0 .0 3 8 5)

u   L

*

4 7 5 0

L  m

* 1 .5

3

( 4 1 0,1 5 0 )(0 .7 6 9 ) 2, 5 6 0, 0 0 0

u   D

*

0 .2 4 6

D  m

*

*5

2,1 8 8, 0 0 0 2 1 8 8 2 .4 1 0

p L P a kP a

  D  