Sequential Framework for HENS @ IITM

HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY

T HE S EQUENTIAL F RAMEWORK FOR H EAT
E XCHANGER N ETWORK S YNTHESIS

Rahul Anantharaman
rahul.anantharaman@ntnu.no

Department of Energy & Process Engineering
Norwegian University of Science and Technology

IIT Madras
Chennai, 18.12.2009


O UTLINE

1 HENS
Background
HENS in the 21st century
2 S EQUENTIAL F RAMEWORK
Introduction
Examples
Challenges
3 M IN U NITS SUB - PROBLEM
Background
Challenges
Model modiﬁcation
Model reformulation
Further work
4 S UMMARY


H EAT E XCHANGER N ETWORK S YNTHESIS

For a given set of hot and cold process streams as well as
external utilities, design a heat exchanger network that
minimizes Total Annualized Cost (TAC).
TAC = Capital Cost + Energy Cost

Sequential Framework Engine


S OLUTION METHODS

Evolutionary methods such as Pinch Design Method
Sequential synthesis methods
Simultaneous synthesis methods
Stochastic optimization methods


T IMELINE


HENS IN THE 21 ST CENTURY
R EVIEW

224 references published from 2000-2008
216 journal papers
48 jounals
43 countries
4 conference proceedings
10 Ph.D. theses
4 texts


R EVIEW

45

40

35

30

25

20

15

10

5

0
2000 2001 2002 2003 2004 2005 2006 2007 2008


R EVIEW

Computers & Chem Eng
43
77 Applied Thermal Eng

Industrial & Eng Chem
Research
Chem Eng Research & Design

39 Latin American Appl Research

Heat Transfer Eng
7
7 Chem Eng Science
7 23
7 7 10 Chem Eng and Processing
Ch E dP i
7


R EVIEW

50

45

40

35

30

25

20

15

10

5

0


R EVIEW


R EVIEW

HENS still an active area of research interest
Over 25% of references devoted to case studies
Pinch Analysis based evolutionary methods dominate
Sustained interest in simultaneous MINLP methods
Yee and Grossmann (1990) superstructure
Pressure drop and detailed HX design considerations
Small test problems
Number of references related to genetic programming and
other meta-heuristic methods increasing in frequency

Though there has been signiﬁcant developments in HENS
using mathematical programming methods, synthesis of large
scale HENS problems without simpliﬁcations and heuristics
have been lacking. This is an area that requires more research
for mathematical programming based approaches to be used in
the industry


M OTIVATION FOR THE S EQUENTIAL F RAMEWORK

Pinch based 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


M OTIVATION FOR THE S EQUENTIAL F RAMEWORK

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 included


U LTIMATE G OAL

Solve Industrial Size Problems
Defined to involve 30 or more streams
Include Industrial Realism
Multiple and ``Complex´Útilities
Constraints in Heat Utilization (Forbidden matches)
Heat exchanger models beyond pure countercurrent
Avoid Heuristics and Simplifications
No global or fixed ∆ Tmin
No Pinch Decomposition
Develop a Semi-Automatic Design Tool
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


O UR ENGINE

3 way trade-off

Compromise between Pinch Design and MINLP methods


E XAMPLE 1

Stream Tin Tout mCp ∆H h
K K kW/K kW 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 1.69 427.57 3.20
ST 650 650 - - 3.50
CW 293 308 - - 3.50
Exchanger cost ($) = 8,600 + 670A0.83 (A is in m2 )


E XAMPLE 1
L OOPING TO THE SOLUTION

HRAT ﬁxed at 20K (Qh,min = 244.1 kW & Qc,min = 172.6)
Umin = 8 units

Soln. No U EMAT (K) HLD TAC ($)
1 8 2.5 A 199,914
2 8 5 A 199,914
3 8 7.5 - No Soln
4 9 2.5 A 147,861
5 9 2.5 B 151,477
6 9 5 A 147,867
7 9 5 B 151,508
8 9 7.5 A 149,025
9 9 7.5 B 149,224
10 10 2.5 A 164,381
11 10 5 A 167,111
12 10 7.5 A 164,764


E XAMPLE 1
B EST SOLUTION


E XAMPLE 1
C OMPARISON

No. of units Area (m2 ) Cost ($)
Colberg and Morari (1990) 22 173.6
Colberg and Morari (1990) 12 188.9 177,385
Yee and Grossmann (1990) 9 217.8 150,998
Sequential Framework 9 189.7 147, 861


E XAMPLE 2

Stream Tin Tout mCp ∆H h
(°C) (°C) (kW/°C) (kW) (kW/m2 °C)
H1 180 75 30 3150 2
H2 280 120 60 9600 1
H3 180 75 30 3150 2
H4 140 40 30 3000 1
H5 220 120 50 5000 1
H6 180 55 35 4375 2
H7 200 60 30 4200 0.4
H8 120 40 100 8000 0.5
C1 40 230 20 3800 1
C2 100 220 60 7200 1
C3 40 290 35 8750 2
C4 50 290 30 7200 2
C5 50 250 60 12000 2
C6 90 190 50 5000 1
C7 160 250 60 5400 3
ST 325 325 1
CW 25 40 2
Exchanger cost ($) = 8,000 + 500A0.75 (A is in m2 )


E XAMPLE 2
L OOPING TO THE SOLUTION

HRAT ﬁxed at 20.35°C (Qh,min = 11539.25 kW & Qc,min = 9164.25 kW)
Umin = 14 units

Soln. No U EMAT (C) 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 A 1,529,968
5 15 5 B 1,533,261
6 16 2.5 A 1,547,353


E XAMPLE 2
B EST SOLUTION


E XAMPLE 2
C OMPARISON

The solution given here with a TAC of $1,529,968, about
the same cost as the solution presented in the original
paper by Björk and Nordman (2005) (TAC $1,530,063)
When only one match was allowed between a pair of
streams the TAC reported by Björk & Nordman (2005) was
$1,568,745
The Sequential Framework allows only 1 match between a
pair of streams
Unable to compare the solutions apart from cost as the
paper did not present the networks in their work


C HALLENGES

C OMBINATORIAL E XPLOSION
Reason: Binary Variables in MILP models
Physical and engineering insights will mitigate, not remove, the problem
MILP models are the bottlenecks that limit problem size due to
computational time

L OCAL OPTIMA
Reason: Non-convexities in the NLP model
Convex estimators developed for MINLP models are computationally
intensive
Time to solve the basic NLP is not a problem

Sequence of MILP and NLP problems considerably easier to solve than
MINLP formulations


M INIMUM NUMBER OF UNITS SUB - PROBLEM
D EFINITION

Given:
a set H of hot process streams to be cooled,
a set C of cold process streams to be heated,
start and target temperatures, heat capacities and ﬂow rates of the hot
and cold process streams,
a set of utilities,
temperatures or temperature ranges and minimum requirement of the
utilities and
Exchanger Minimum Approach Temperature (EMAT) set to zero,
calculate the minimum number of matches between hot process streams and
utilities and cold process streams and utilities such that the heating and
cooling requirements for each stream are met.


F ORMULATION

X X
min z = yij (P1)
i∈H j∈C

s.t.

H
X
Ri,k − Ri,k −1 + Qijk = Qik ∀ i ∈ Hk , k ∈ TI
j∈Ck

C
X
Qijk = Qjk ∀ j ∈ Ck , k ∈ TI
i∈Hk
X
Qijk − Uij yij ≤ 0 ∀ i ∈ H, j ∈ C
k ∈TI

Rik ≥ 0 ∀ i ∈ Hk , k ∈ TI
Ri0 = RiK = 0 ∀i ∈H
Qijk ≥ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI
yij = {0, 1} ∀ i ∈ H, j ∈ C


C HALLENGES

Combinatorial explosion
Problem is proved to be
N P-hard in the strong sense


A LLEVIATING COMBINATORIAL EXPLOSION

The three major ways to improve the model solution time are:
1 Pre-processing to reduce model size using insight and
heuristics
2 Model modiﬁcation/reformulation
3 Improving efﬁciency of the B&B method


S HARPENING LP RELAXATION BY DECREASING BIG M

The gap, i.e. the difference between the LP relaxation and the
actual binary solution, is dependent on the value of Uij , the
upper limit on the amount of heat transfer between streams i
and j.
A smaller value of Uij corresponds to a smaller gap and thus
reduced computing times.



O RIGINAL U
8 9
<X =
H C
X
Uij = min Qik , Qjk
: ;
k k

M ODIFIED U
8 9
<X h “ ” “ ” i=
H C H C H C
X
Uij = min Qik , Qjk , max min mCpi , mCpj · Tsi − Tsj − EMAT , 0
: ;
k k


L OCAL U

L OCAL U: Deﬁne maximum amount of heat exchanged between streams on a
tempertaure interval basis

80 1 9
< X =
HA C
Uijk = min @ Qi k , Qjk
¯
: ;
¯
k ≤k

Logical constraint utilizing the local U

Qijk − Uijk yij ≤ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI

This constraint will reduce the feasible region as compared to the earlier one - thus
leading to a tighter formulation.


I NTEGER C UTS

The gap between the LP relaxation based lower bound and
the optimal integer solution. This gap can be reduced by
employing integer cuts to the model.
A potential drawback of adding such cuts is the increase in
model size and hence computation time.
2 types of integer cuts applied to the model
1 Compulsory matches
2 Minimum number of matches per stream


M ODEL MODIFICATION
T EST PROBLEM 22TP1

ROOT NODE LP RELAXATION VALUES FOR TEST PROBLEM 22TP1 WITH IP SOLUTION 23

U deﬁnition No integer Compulsory Minimum matches Both cuts
cuts matches per stream
Eq 1 12.21 - - -
Global Eq 2 15.17 16.27 16.82 18.62
Local Eqs 3,4 15.78 16.56 17.63 18.62


M ODEL MODIFICATION
T EST PROBLEM 21TP1


Eq 1 11.93 - - -
Global Eq 2 14.30 15.14 14.49 15.21
Local Eqs 3,4 14.87 15.39 14.95 15.40


M ODEL MODIFICATION
T EST PROBLEM 21TP2

ROOT NODE LP RELAXATION VALUES AND TOTAL SOLUTION TIMES FOR TEST PROBLEM
21TP2 WITH IP SOLUTION 22

Eq 1 16 - - -
20 s - - -
Global Eq 2 17.67 18.80 18.13 18.83
19 s 27 s 23 s 21 s
Local Eqs 3,4 18.80 18.80 18.81 18.83
36 s 46 s 45 s 42 s


M ODEL MODIFICATION
D ISCUSSION

Modified U and Local U definitions give tighter lower bounds than
original U definition
Compulsory matches integer cuts always improve the lower bound
Minimum number of matches integer cuts have varying results
21TP1 and 22TP1 still do not solve in less than 12 hours

Gap is not the only measure of the complexity of an integer problem. Two of
the main issues in this particular integer problem are the number of feasible
solutions and the number of multiple optima.


S ET PARTITIONING FORMULATION

SSis be deﬁned to represent all feasible sets of matches, s ∈ Si , between a hot stream
i and all cold streams:

SSis = { j | j ∈ C if stream j is in set of matches s for stream i}

s 1 2 3 4 5 6 7
C1 1 0 0 1 1 0 1
C2 0 1 0 1 0 1 1
CW 0 0 1 0 1 1 1
(
1, if set of matches s is chosen for hot process stream i
λis =
0, otherwise

S ET PARTITIONING CONSTRAINT
X
λis = 1 ∀i∈H
s∈Si



Maximum number of set of matches for a hot stream =
2nc − 1
Potentially large number of binary variables will be
introduced
Use thermodynamics and physical insight to reduced the
number of binary variables
The set partitioning constraint is expected to have more
efﬁcient branching characteristics



The constraints for defining the set of feasible matches are:
1 At least one cold process stream or utility in the set of matches must have a
supply temperature lower than or equal to the hot process stream’s target
temperature and satisfy the hot process stream’s duty in this temperature range.
2 The total heat demand for the set of cold process streams or utilities below the
hot process stream’s supply temperature must be greater than or equal to the
hot process stream’s total heat duty.
3 The total number of cold process streams and utilities in a set of matches should
not exceed a user-specified maximum value.
4 For cases with streams having large duties, the number of streams in a set of
matches for a utility must be larger than one.

Constraints 1 and 2 are thermodynamically based while constraints 3 and 4 are
heuristics based on insights gained from testing various problems.

Constraint 3 is user specified to allow the user to impart their knowledge about the
problem on hand to reduce the problem size.


X X
min z = cis λis (P2)
i∈H s∈Si

s.t.
H
X
j∈Ck

C
X
i∈Hk
X X
Qijk − Uij λis ≤ 0 ∀ i ∈ H, j ∈ C
k ∈TI s∈Pij
X
λis = 1 ∀i ∈H
s∈Si
X X
λis ≥ 1 ∀j ∈C
i∈H s∈Pij
X X
λis ≤ max value ∀j ∈C
i∈H s∈Pij

cis = card (SSis ) ∀ i ∈ H, s ∈ Si
Rik ≥ 0 ∀ i ∈ Hk , k ∈ TI
Ri0 = RiK = 0 ∀i ∈H
Qijk ≥ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI
λis = {0, 1} ∀ i ∈ H, s ∈ Si


XX
min z = yij (P3)
i j

s.t.
H
X
j∈Ck

C
X
i∈Hk
X
Qijk − Uij yij ≤ 0 ∀ i ∈ H, j ∈ C
k ∈TI
X
λis = 1 ∀i ∈H
s∈Si
X X
λis ≥ 1 ∀j ∈C
i∈H s∈Pij
X X
λis ≤ max value ∀j ∈C
i∈H s∈Pij
X
λis = yij ∀ i ∈ H, j ∈ C
s∈Pij

Rik ≥ 0 ∀ i ∈ Hk , k ∈ TI
Ri0 = RiK = 0 ∀i ∈H
Qijk ≥ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI


M ODEL REFORMULATION
T EST PROBLEM 22TP1


Model Binary U model Additional LP relaxation
variables constraints value
P1 143 Global Eq-2 Eqs 5,6 18.62
Local Eqs 3,4 Eqs 5,6 18.62
P3 2500 Global Eq-2 Eqs 5,6 18.62
with λis Local Eqs 3,4 Eqs 5,6 18.62
P4 2625 Global Eq-2 Eqs 5,6 18.67
with µjt Local Eqs 3,4 Eqs 5,6 18.67
P5 4999 Global Eq-2 None 18.67
with λis & µjt Local Eqs 3,4 18.67


T EST PROBLEM 21TP1


Model Binary U model Additional LP relaxation
variables constraints value
P1 131 Global Eq-2 Eqs 5,6 15.21
Local Eqs 3,4 Eqs 5,6 15.40
P3 4645 Global Eq-2 Eqs 5,6 15.74
with λis Local Eqs 3,4 Eqs 5,6 15.82
P4 5435 Global Eq-2 Eqs 5,6 16.33
with µjt Local Eqs 3,4 Eqs 5,6 16.45
P5 9879 Global Eq-2 None 16.86
with λis & µjt Local Eqs 3,4 16.93


D ISCUSSION

Reformulated models may results in strengthening the LP
relaxation
Contain more information regarding thermodynamics of the
matches than the basic model
Reformulated models for 21TP1 and 22TP2 cannot be
solved within 12 hours
These models are larger and with more binary variables
counteracting any potential beneﬁt
Reduction in gap not signiﬁcant enough


F URTHER WORK

Optimum value is reached early in the solution process
and most of the effort is expended in proving optimality.
Develop heuristics to stop the search after an appropriate
solution time.
Identifying subnetworks by relaxing stream temperatures
and ﬂow rates it may be possible to get a good initial bound
on the minimum number of units using U = N + L − S.
This value could be used by CPLEX as the initial lower
bound thus tightening the gap.


F URTHER WORK

All NP-complete problems have a phase separation i.e.
both hard and easy regions, with a sharp boundary
between them. On crossing that frontier, the problem
undergoes a phase transition, analogous to the boiling or
freezing of water. Identifying the phase transition of the
minimum units problem would enable the user to decide on
using deterministic methods for solving it for other
non-deterministic methods depending on which phase the
problem happens to be in.


S UMMARY
Sequential Framework has many advantages
Automates the design process
Allows significant User interaction
Progress
EMAT identified as an optimizing ‘area variable´
Improved HLDs from Stream match generator subproblem
Significantly better and automated starting values for NLP
subproblem
Global optimum for NLP ensured using a modified GBD
scheme
Limiting elements
Stream match generator 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
Reliability of NLP solutions is no longer limiting

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