Optimal Design of Energy Systems
Chapter 7 Optimization
Min Soo KIM
Department of Mechanical and Aerospace Engineering
Seoul National University
7.1 Introduction
• role of engineers = optimization
= finding the conditions of maximum or minimum
• difficult to optimize ← complex
• criterion required
• component simulation + system simulation
Chapter 7. Optimization
initial cost total cost
7.2 Levels of optimization
comparison of alternate concepts optimization within a concept
Chapter 7. Optimization
7.3 Mathematical Representation of Optimization Problems
• object(ive) function y : function to be optimized
• constraints
Chapter 7. Optimization
Independent variable↑
( 1, , n ) y y x x
equality constraint inequality constraint
( 1, , ) 0
i i x xn
( 1, , )
i i x xn Lj
( 1, , n )
y a Y x x m in y a m in Y m ax y m in ( y)
7.4 Water-chilling system
Chapter 7. Optimization
( co m p, co n d , eva p, C T , p u m p)
y x x x x x minimize
( co m p, co n d , eva p, C T , p u m p) p
Q x x x x x m c T
( , , , , ) 0 o
eva p co m p co n d eva p C T p u m p
t x x x x x C
( , , , , ) 1 0 0 o
c o n d c o m p c o n d e v a p C T p u m p
t x x x x x C
cost size
7.5 Optimization procedures
- Calculus method (Chap.8) : Lagrange Multipliers
- Search method (Chap.9) → multivariable optimization
exact optimum is approached not reached - Dynamic programming (Chap.10) → optimum function
- Geometric programming (Chap.11)
- Linear programming (Chap.12)
Chapter 7. Optimization
7.11
Setting up the Mathematical Statement of optimization Problem- To translate physical situation into mathematical statement - objective function – trivial
- constraints – difficult to establish
Chapter 7. Optimization
Strategy
(1) Specify all direct constraints
(2) Component characteristics + properties (3) Mass / energy balance
7.11
Setting up the Mathematical Statement of optimization Problem<Example 7.1> Intercooling of Air
Chapter 7. Optimization
t4
q3
, w p w
m c
t2
, a p a
m c
ta1
q2 q1 w1 c q4 1
t1
t3 ta2
: known
7.11
Setting up the Mathematical Statement of optimization Problem<Example 7.1>
Minimize first cost Refrigeration unit Precooler
Cooling tower
Chapter 7. Optimization
1 1 1
x c q
2 2 2
3 1
x c q
t t
3 3 3
x c q
7.11
Setting up the Mathematical Statement of optimization Problem<Example 7.1>
Total cost (objective function) Constraints (1)
(2) component
Chapter 7. Optimization
1 2 3
y x x x
1 , 3 2
2 , 1 3
( )
( )
a p a a
a p a a
q m c t t
q m c t t
1 1 1
x c q
2 2 2
3 1
x c q
t t
3 3 3
x c q
1 4 1
w c q (work)
1 1 w p w, ( 1 4) q w m c t t
7.11
Setting up the Mathematical Statement of optimization Problem<Example 7.1>
Constraints (3) balance
Chapter 7. Optimization
, ( 1 3 ) , ( 2 1)
a p a a w p w
m c t t m c t t
, ( 2 4) 3
w p w
m c t t q Refrigeration unit
Precooler
Cooling tower
7.11
Setting up the Mathematical Statement of optimization Problem<Example 7.1>
# of equations 9 → 2
# of unknowns
Minimize Subject to
Chapter 7. Optimization
1 2 3
1 2 3
1 2 3
1
, ,
, ,
, ,
q q q
t t t
x x x
w
10 → 3
1 2 3
y x x x
1 1 2 3
2 1 2 3
( , , ) 0
( , , ) 0
x x x
x x x
1, 2, 3
x x x
