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Basic Optimization Theory. LP (Linear Programming) NLP (Non-Linear Programming) IP (Integer Programming) MIP (Mixed Integer Programming) MINLP (Mixed Integer Non-Linear Programming). Types Of Optimization. Parameter Optimization Configuration Optimization - PowerPoint PPT Presentation
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Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Basic Optimization Theory
• LP (Linear Programming)
• NLP (Non-Linear Programming)
• IP (Integer Programming)
• MIP (Mixed Integer Programming)
• MINLP (Mixed Integer Non-Linear Programming)
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Types Of Optimization
• Parameter Optimization
• Configuration Optimization
• Operational Optimization
• Topology Optimization
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Topology Optimization
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
The General Non-Linear Problem
b
bb
subject to:min. max.
m21m
2212
1211
21
)(
)(
)(
)( or
xxxg
xxxgxxxg
xxxf
n
n
n
n
,..., ,
,..., ,
,..., ,
,..., ,
Objective Function
Constraints
Design Space
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Extrema: Ordering Situation
0 10 20 30 40 500
200
400
600
800
1000
$
Order Quantity
Total Cost
Transport Cost
Ordering Cost
OPT.
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Extrema: Heat Exchanger
0 100 200 300 400 5000
200
400
600
800
1000
$
Heat Exchanger Area
Total Cost
Material Cost
Energy Cost
OPT.
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Local vs. Global Extrema
-1,2
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0 2 4 6 8 10 12 14
f(x)=sin(30x)/x
f(x)
x
localmin
globalmax stationary
point
localmin
localmax
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Convexityf(x) f(x)
x x
f(x) is a convex function if and only if for any given two points x1 and x2 in the function domain and for any constant 0 1
f(x1 +(1- )x2) f(x1)+(1- )f(x2)
x1 x2
f(x1)
f(x2)
x1+(1-)x2
f(x1)+(1- )f(x2)
f(x1 +(1- )x2)
Convex Function Concave Function
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
The Hessian • The gradient vector of f at x
n
2
1
...
)(
x
f
x
f
x
f
xf
nn
2
n1
2
1n
2
11
2
2 ...
...
...
...
...)(
xx
f
xx
f
xx
f
xx
f
xf
• The Hessian Matrix of f at x
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Conditions for convexity
• H(x) is positive semi-definite if and only if xTHx ≥ 0 for all x and there exists and x 0 such that xTHx = 0. => Convexity
• H(x) is positive definite if and only if xTHx > 0 for all x 0.
• H(x) is indefinite if and only if xTHx > 0 for some x, and xTHx < 0 for some other x.
How can we use Hessian to determine whether or not f(x) is convex?
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Possible Solutions To Convex Problemsobjective function
level curve
optimal solution
Feasible Region
linear objective,nonlinear constraints
objective function level curve
optimal solution
Feasible Region
nonlinear objective,nonlinear constraints
objective function level curve
optimal solution
Feasible Region
nonlinear objective,linear constraints
objective function level curves
optimal solution
Feasible Region
nonlinear objective,linear constraints
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Line SearchLine search techniques are in essence optimization algorithms for one
dimensional minimization problems.
They are often regarded as the backbones of nonlinear optimization algorithms.
Typically, these techniques search a bracketed interval.
Often, unimodality is assumed.
Line search techniques are in essence optimization algorithms for one dimensional minimization problems.
They are often regarded as the backbones of nonlinear optimization algorithms.
Typically, these techniques search a bracketed interval.
Often, unimodality is assumed.
Exhaustive search requires N = (b-a)/ + 1 calculations to search the above interval, where is the resolution.
a bx*
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Bracketing Algorithm
Two point search (dichotomous search) for finding the solution to minimizing ƒ(x):
0) assume an interval [a,b]
1) Find x1 = a + (b-a)/2 - /2 and x2 = a+(b-a)/2 + /2 where is the resolution.
2) Compare ƒ(x1) and ƒ(x2)
3) If ƒ(x1) < ƒ(x2) then eliminate x > x2 and set b = x2
If ƒ(x1) > ƒ(x2) then eliminate x < x1 and set a = x1
If ƒ(x1) = ƒ(x2) then pick another pair of points
4) Continue placing point pairs until interval < 2
a bx1 x2
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Golden Section Search
a
a
b
b
a - b
In Golden Section, you try to have b/(a-b) = a/b
which implies b*b = a*a - ab
Solving this gives a = (b ± b* sqrt(5)) / 2
a/b = -0.618 or 1.618 (Golden Section ratio)
Note that 1/1.618 = 0.618
Discard
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Golden Section Search Algorithm
a bx1 x2
{Initialize}
x1 = a + (b-a)*0.382
x2 = a + (b-a)*0.618
f1 = ƒ(x1)
f2 = ƒ(x2)
{Loop}
if f1 > f2 then
a = x1; x1 = x2; f1 = f2
x2 = a + (b-a)*0.618
f2 = ƒ(x2)
else
b = x2; x2 = x1; f2 = f1
x1 = a + (b-a)*0.382
f1 = ƒ(x1)
end
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
The 2D case
A
C
B
Local optimal solution
Feasible Region
D
EF
G
Local and global optimal solution
X1
X2
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
The 2D case
(From John Rasmussen, 1999)
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Example: Heat ExchangerProblem: Find the radius of tubes in a heat exchanger tomaximize the total surface area.
(The magnitude of pressure drops are not considered)
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Setting up the problem:
What is the best value of r ?What if we added a maximum allowable pressure drop?
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Ideal Rankine Cycle Balance• Assumptions: steady flow process, no generation, neglect KE and PE changes for all four devices,• First Law: 0 = (net heat transfer in) - (net work out) + (net energy flow in)
0 = (qin - qout) - (Wout - Win) + (hin - hout)
T
s
1
2
3
4
• 1-2: Pump (q=0) Wpump = h2 - h1 = v(P2-P1)
• 2-3: Boiler (W=0) qin = h3 - h2
• 3-4: Turbine (q=0) Wout = h3 - h4
• 4-1: Condenser (W=0) qout = h4 - h1Thermal efficiency = Wnet/qin = 1 - qout/qin = 1 - (h4-h1)/(h3-h2)
Wnet = Wout - Win = (h3-h4) - (h2-h1)
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
T
s
12
3
4
(a) lower pressure(temp)
s
T
1
2
(b) Superheating (c) increase pressure
s
T
1
2
3
4
Low qualityhigh moisture content
1
2
4
Red area = increase in W netBlue area = decrease in W net
Thermal efficiency can be improved by manipulating the temperatures and/or pressures in various components(a) Lowering the condensing pressure (lowers TL, but decreases quality, x4 )(b) Superheating the steam to a higher temperature (increases TH but requires higher temp materials)(c) Increasing the boiler pressure (increases TH but requires higher temp/press materials)
Possibility for improvement of thermal efficiency:
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Reheating• The optimal way of increasing the boiler pressure without increasing the moisture
content in the exiting vapor is to reheat the vapor after it exits from a first-stage turbine and redirect this reheated vapor into a second turbine.
boiler
high-Pturbine
Low-Pturbine
pump
condenser1
2
3
4
56
T
s
1
2
3 5
6
4
high-Pturbine
low-Pturbine
4
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Reheating
• Reheating allows one to increase the boiler pressure without increasing the moisture content in the vapor exiting from the turbine.
• By reheating, the average temperature of the vapor entering the turbine is increased, thus, it increases the thermal efficiency of the cycle.
• Multistage reheating is possible but not practical. One major reason is because the vapor exiting will be superheated vapor at higher temperature, thus, decrease the thermal efficiency.
• Energy analysis: Heat transfer and work output both change
•
qin = qprimary + qreheat = (h3-h2) + (h5-h4)
Wout = Wturbine1 + Wturbine2 = (h3-h4) + (h5-h6)
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Regeneration• From 2-2’, the average temperature is very low, therefore, the heat addition process is at
a lower temperature and therefore, the thermal efficiency is lower.• Use a regenerator to heat the liquid (feedwater) leaving the pump before sending it to
the boiler. This increases the average temperature during heat addition in the boiler, hence it increases efficiency.
T
s1
2
2’
3
4
Lower tempheat addition
T
s1
23
4
5
6
7
Use regenerator to heat up the feedwater
higher tempheat addition
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Regeneration Cycle• Improve efficiency by increasing feedwater temperature before it enters the boiler.• Two Options:
– Open feedwater : Mix steam with the feedwater in a mixing chamber.– Closed feedwater: No mixing.
Pump 24
Pump 1
OpenFWHboiler
condenser1
23
5
67
1
23
4
5
6
7
T
s
Open FWH
(y) (1-y)
(y)
(1-y)
Analysis, Modelling and Simulation of Energy Systems, SEE-T9Mads Pagh Nielsen
Analysis Of Regenerative Cycle
• Assume y percent of steam is extracted from the turbine and is directed into open feedwater heater.
• Energy analysis:
qin = h5-h4, qout = (1-y)(h7-h1),
Wturbine, out = (h5-h6) + (1-y)(h6-h7)
Wpump, in = (1-y)Wpump1 + Wpump2
= (1-y)(h2-h1) + (h4-h3)
= (1-y)v1(P2-P1) + v3(P4-P3)
• In general, more feedwater heaters result in higher cycle efficiencies.• However this does not mean that it is necessary a practical optimal solution!
(Note: Ideal pumps)