Transcript Ingen lysbildetittel

ESCAPE-16 & PSE 2006
Garmisch-Partenkirchen, Germany
Developments in the
Sequential Framework
for Heat Exchanger Network
Synthesis of industrial
size problems
Rahul Anantharaman and Truls Gundersen
Dept of Energy and Process Engineering
Norwegian University of Science and Technology
Trondheim, Norway
Anantharaman & Gundersen, PSE/ESCAPE ’06
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Overview
1.
Introducing the Sequential Framework
1.
2.
3.
Motivation
Our Goal
Our Engine
1.
2.
2.
Challenges
1.
Combinatorial Explosion – MILP
1.
2.
2.
Automated starting values
Modal trimming method
Examples
1.
2.
4.
Temperature Intervals
EMAT as an area variable
Non-convexities - NLP
1.
2.
3.
Subproblems
Loops
7 stream problem
15 stream problem
Concluding remarks
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Motivation for the
Sequential Framework
 Pinch Methods for Network Design
 Improper trade-off handling
 Time consuming
 Several topological traps
 MINLP Methods for Network Design
 Severe numerical problems
 Difficult user interaction
 Fail to solve large scale problems
 Stochastic Optimization Methods for Network Design
 Non-rigorous algorithms
 Quality of solution depends on time spent on search
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Motivation for the
Sequential Framework
 HENS techniques decompose the main problem
 Pinch Design Method is sequential and evolutionary
 Simultaneous MINLP methods let math considerations define the
decomposition
 The Sequential Framework decomposes the problem into subproblems
based on knowledge of the HENS problem
 Engineer acts as optimizer at the top level
 Quantitative and qualitative considerations
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Our Ultimate Goal
 Solve Industrial Size Problems
 Defined to involve 30 or more streams
 Include Industrial Realism
 Multiple Utilities
 Constraints in Heat Utilization (Forbidden matches)
 Heat exchanger models beyond pure countercurrent
 Avoid Heuristics and Simplifications
 No global or fixed ΔTmin
 No Pinch Decomposition
 Develop Semi-Automatic Design Tool
 A tool SeqHENS is under development
 EXCEL/VBA (preprocessing and front end)
 MATLAB (mathematical processing)
 GAMS (core optimization engine)
 Allow significant user interaction and control
 Identify near optimal and practical networks
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Our Engine –
A Sequential Framework
Adjust EMAT
New HLD
Preoptim.
HRAT
LP
QH
MILP
U
QC
(EMAT=0)
EMAT
Vertical
MILP
HLD
1
Final
NLP
Adjust Units
Adjust HRAT
2
Network
3
4
Compromise between Pinch Design and MINLP Methods
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Challenges
 Combinatorial explosion (binary variables)
 Problem proved to be NP -complete in the strong sense
 Any algorithm may take exponential number of steps to reach optimality
 Use physical/engineering insights based on understanding of the problem
 Will not remove the problem but help mitigate it
 MILP and VMILP are currently the bottlenecks w.r.t. time (and thus size)
 Local optima (non-convexities in the NLP model)
 Convex estimators developed for MINLP models are computationally intensive
 Only very small problems have been solved
 Explore other options
 Time to solve the NLP is not a problem
 Relatively easier to solve than MINLP formulations
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Temperature Intervals (TIs)
in the VertMILP model
 Objective function is minimizing pseudo area


Qim, jn
min  

i
j
m
n 
U ij TLM ,mn 
H1
m-1
n-1
C1
 VertMILP model works best when the pseudo
area accurately reflects the actual HX area
i
m
n
j
 This happens when the number of TIs approaches infinity
HH
m+1
n+1
CC
 Size of the VertMILP model increases exponentially with the
number of temperature intervals
 The transportation model has a polynomial time algorithm
→ Keep number of TIs to a minimum while
ensuring the model achieves its objective
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Temperature Intervals (TIs) in
the VertMILP model
 Original philosophy of the VertMILP model
 Minimum area is achieved by vertical heat transfer
 Temperature intervals must facilitate
vertical heat transfer
 Use Enthalpy Intervals to develop the vertical TIs
 The Normal and Enthalpy based (vertical)
TIs are the basis for the VertMILP model
EMAT
 Elaborate testing show that the VertMILP model achieves its
objective with this set of TIs
 Size of the model is reduced, on an average, by 10% (more
for larger models)
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EMAT as an Area Variable
 Choosing EMAT is not straightforward
 EMAT set too low (close to zero)
 non-vertical heat transfer (m=n) will have very small ΔTLM,mn and very
large penalties in the objective function
 EMAT set too high (close to HRAT)
 Potentially good HLDs will be excluded from the feasible set of solutions
 HLDs are affected by the choice of EMAT
 EMAT comes into play only when there is an extra
degree of freedom in the system : U > Umin
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Automated Starting Values and
Bounds for the NLP subproblem
 Multiple starting values for the NLP subproblem
 Ensure a feasible solution
 Explore different local optima
 Use physical insight to ensure `good´ local optima
 Heat Capacity Flowrates (mCps) identified to be the
decision variables
 Lower Bounds for Area were found to be crucial in getting a
feasible solution
 Information from the VertMILP subproblem is utilized
 4 different strategies for starting values were explored
 Ref.: Hilmersen S. E. and Stokke A., M.Sc Thesis , NTNU 2006
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Serial/Parallel mCp Generator
 Simple & flexible method
 Little physical insight needed
 Parallel arrangement gives
feasible solution to most
problems (90%)
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Clever Serial mCp Generator
 Serial configuration assumed for all streams
 Assigns demanding exchangers at the supply end
 Only stream temperatures are considered
 Heat exchanger duties & stream mCp values are not considered
 Assumed sequence of heat exchangers
 Hot supply end matched with ranked set of cold targets & vice versa
 Similar to the Ponton/Donaldson heuristic synthesis approach
 Only serial configuration is limiting in many cases
 Feasible solution in 50% of cases tested
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Combinatorial mCp Generator
 Utilizes heat loads, temperatures and
overall mCp values to assign stream flows
 Uses physical insight to determine flows
 Provides a feasible solution to the NLP
subproblem in all cases tested
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Modal Trimming Method for Global
Optimization of NLP subproblem
Search for local optimal solution
Objective function f(x)
Search for feasible solutions
f*0, x*0
f*1, x*1
f*2, x*2
x0
x2
x1
Variables x
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Modal Trimming Method for Global
Optimization of NLP subproblem
 Search for feasible solutions is the most
important step
Testing showed the Modal Trimming method to be inefficient
and computationally expensive for solving the NLP model
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Illustrating Example 1
Stream
Tin
(K)
Tout
(K)
mCp
(kW/K)
ΔH
(kW)
h
(kW/m2 K)
H1
626
586
9.802
392.08
1.25
H2
620
519
2.931
296.03
0.05
H3
528
353
6.161
1078.18
3.20
C1
497
613
7.179
832.76
0.65
C2
389
576
0.641
119.87
0.25
C3
326
386
7.627
457.62
0.33
C4
313
566
1.690
427.57
3.20
ST
650
650
-
-
3.50
CW
293
308
-
-
3.50
Exchanger cost ($) = 8,600 + 670A0.83 (A is in m2)
References:
Example 3 in Colberg, R. D. and Morari M., Area and Capital Cost Targets for Heat Exchanger Network Synthesis with Constrained
Matches and Unequal Heat Transfer Coefficients, Computers chem. Engng. Vol. 14, No. 1, 1990
Example 4 in Yee, T. F. and Grossmann I. E., Simulataneous Optimization Models for Heat Integration - II. Heat Exchanger Network
Synthesis, Computers chem. Engng. Vol. 14, No. 10, 1990
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Example 1 – Initial Values
Adjust EMAT
2
New HLD
HRAT
Q
MILP
LP H (EMAT=0)
QC
U
EMAT
Vertical
MILP
HLD
1
Final
NLP
Network
3
Adjust Units
HRAT fixed at 20K
QH = 244.1 kW
QC = 172.6 kW
Absolute Minimum
Number of Units = 8
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Example 1 – Looping to Solution
Adjust EMAT
2
New HLD
HRAT
LP
QH MILP
QC (EMAT=0)
U
EMAT
Vertical
MILP
1
Final
HLD NLP
Network
3
Adjust Units
Soln. No
U
EMAT (K)
HLD#
INVESTMENT COST ($)
1
8
2.5
A
199,914
2
8
2.5
B
Not feasible
3
9
2.5
A
147,861
4
9
2.5
B
151,477
5
9
5.0
A
147,867
6
9
5.0
B
151,508
7
10
2.5
A
164,381
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Example 1 – `Best´ Solution
HRAT = 20, EMAT = 2.5, ΔTsmall= 3
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Example 1 – Solution Comparisons
No of Units
Area (m2)
Cost
Remarks
Colberg & Morari
(1990)
22
173.6
-
Colberg & Morari
(1990)
12
188.9
$177,385
Synthesized network by evolution
Yee and Grossmann
(1990)
9
217.8
$150,998
Optimized w.r.t. cost
$147,861
MILP optimized w.r.t ”area”
NLP optimized w.r.t cost
Our work
9
189.7
Optimized w.r.t area
Spaghetti design
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Illustrating Example 2
Stream
Tin
(°C)
Tout
(°C)
mCp
(kW/°C)
ΔH
(kW)
h
(kW/m2 °C)
H1
H2
H3
H4
H5
H6
H7
H8
C1
C2
C3
C4
C5
C6
C7
ST
CW
180
280
180
140
220
180
200
120
40
100
40
50
50
90
160
325
25
75
120
75
40
120
55
60
40
230
220
290
290
250
190
250
325
40
30
60
30
30
50
35
30
100
20
60
35
30
60
50
60
3150
9600
3150
3000
5000
4375
4200
8000
3800
7200
8750
7200
12000
5000
5400
2
1
2
1
1
2
0.4
0.5
1
1
2
2
2
1
3
1
2
Exchanger cost ($) = 8,000 + 500A0.75 (A is in m2)
Reference:
Björk K.M and Nordman R., Solving large-scale retrofit heat exchanger network synthesis problems
with mathematical optimization methods, Chemical Engineering and Processing. Vol. 44, 2005
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Example 2 – Initial Values
Adjust EMAT
2
New HLD
HRAT
LP
QH MILP
QC (EMAT=0)
U
EMAT
Vertical
MILP
HLD
1
Final
NLP
Network
3
Adjust Units
HRAT fixed at 20.35 C
QH = 11539.25 kW
QC = 9164.25 kW
Absolute Minimum
Number of Units = 14
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Example 2 – Looping to Solution
Adjust EMAT
2
New HLD
HRAT
LP
QH MILP
QC (EMAT=0)
U
EMAT
Vertical
MILP
1
Final
HLD NLP
Network
3
Adjust Units
Soln. No
U
EMAT (K)
HLD#
TAC ($)
1
14
2.5
A
1,545,375
2
15
2.5
A
1,532,148
3
15
2.5
B
1,536,900
4
15
5.0
A
1,529,968
5
15
5.0
B
1,533,261
6
16
2.5
A
1,547,353
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Example 2 – `Best´ Solution
HRAT = 20.35
EMAT = 5
ΔTsmall= 4.9
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Example – Solution Comparison
 The solution given here with a TAC of $1,529,968, about the
same cost as the solution presented in the original paper
(TAC $1,530,063)
 When only one match was allowed between a pair of
streams the TAC is reported as $1,568,745 - Björk &
Nordman (2005)
 The Sequential Framework allows only 1 match between a pair of streams
 Solution at Iteration 2 (TAC $ 1,532,148) provides a slightly
more expensive but slightly less compless network
 Unable to compare the solutions apart from cost as the
paper did not present the networks in their work
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Global vs Local Optimum
 Global optima in each of the subproblems does
not, by itself, ensure overall global optimum for
the HENS problem
 Inherent feature of any problem decomposition
 The emphasis has been on utilizing knowledge of
the problem and engineering insight to achieve a
network close to global optimum
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Concluding Remarks
 Sequential Framework has many advantages
 Automates the design process
 Allows significant User interaction
 Numerically much easier than MINLPs
 Progress




EMAT identified as an optimizing `area variable´
Improved HLDs from VertMILP subproblem
Algorithm for generating optimal TIs for the VertMILP
Significantly better and automated starting values for NLP subproblem
 Limiting elements
 NLP model for Network Generation and Optimization
 Enhanced convex estimators are required to ensure global optimum
 VertMILP Transportation Model for promising HLDs
 Significant improvements required to fight combinatorial explosion
 MILP Transhipment model for minimum number of units
 Similar combinatorial problems as the Transportation model
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