A Lindenmayer system for heat exchanger network design with stream splitting

.

A Lindenmayer system for heat exchanger
network design with stream splitting

Eric S Fraga

Centre for Process Systems Engineering
Department of Chemical Engineering
UCL (University College London)

29 April – 1 May 2008

c 2008, All rights reserved.

Eric S Fraga (CPSE/UCL) Heat exchanger network design ACDM 2008 1 / 30

Heat exchanger networks.

Outline

Heat exchanger networks
1

An evolutionary strategy
2

Implementation
3

Results
4

Conclusions
5




Energy consumption is often the largest cost of a process.
One means of reducing
energy use is through process
integration:
Identify matches for
transfer of excess heat in
one part of the process to
another part.
Problem is highly combinatorial, discontinuous and non-convex.



Problem deﬁnition and costing
Example problem† Exchangers are costed using
˙
Stream Tin Tout Q
Ccapital = α + βAγ
(K) (K) (kW)
2
typically with γ ≈ 3 and the area,
H1 443 333 30
A, is a function of the log-mean
H2 423 303 15 temperature diﬀerence (LMTD):
C1 293 408 20 ∆Tin − ∆Tout
LMTD =
log ∆Tin − log ∆Tout
C2 353 413 40

†
Pariyani et al. (2006). Computers & Chemical Engineering 30:1046.



Combinatorial aspects

H1
Many possible
alternative matches ... H2
... with alternative
order of placement.
C1
Streams may be split
as well. Split
C2 Mix
Result is highly
combinatorial.


An evolutionary strategy.

Outline

1

2

Implementation
3

Results
4

Conclusions
5



Evolving a HEN structure

Biological analogue† :
Genotype: the plan
Phenotype: the instance
For HEN, the aim is to use a genotype to represent overall
structure: possible matches and stream splits.
The phenotype is an instantiation of the genotype with speciﬁc
matches.
Question becomes one of representation for the genotype and how
this can be manipulated in an evolutionary manner.

†
De Jong (2006). Proc. ACDM 2006, I C Parmee (ed.), 23-25.



Lindenmayer Systems
Prusinkiewicz & Lindenmayer† presented a Lindenmayer system, often
known as an L-system:
The central concept of
L-systems is that of rewriting.
In general, rewriting is a
technique for deﬁning complex
objects by successively replacing
parts of a simple initial object
using a set of rewriting rules or
Image courtesy Solkoll @ wikipedia.
productions.
We wish to apply this concept of rewriting to streams in heat
exchanger network synthesis problems to create potential integration
structures.
†
Prusinkiewicz & Lindenmayer (1990). “The algorithmic beauty of plants,” Springer-Verlag.



L-system definition
An L-system is defined by a tuple,

G = V , ω, P

where
V the alphabet or set of symbols which can be
replaced in a string by specific combinations
symbols from the same set.
ω the initial configuration (set of strings).
P (⊂ V × V ∗ ) is the set of replacement rules.
For the heat exchanger network synthesis problem, we define an
L-system, GHEN .



GHEN – 1. Symbols V
The alphabet includes:
+, - denote the heating and cooling requirements of each
stream;
S, E the start and end of each stream;
x indication of exchange;
s, m split and mix; and,
[, ] start and end of split stream segments.
The full alphabet, therefore, is

V ≡ {−, +, S, E, s, m, [, ]}



GHEN – 2. Starting representation, ω

The starting set of symbols is a set of strings, one for each
stream in the network problem deﬁnition.
Each cold stream is represented initially by the string S-E.
Each hot stream by E+S.
The hot and cold streams are written in opposite order to
indicate the use of counter-current heat exchangers.



GHEN – 2. Starting representation, ω

The starting set of symbols is a set of strings, one for each
stream in the network problem deﬁnition.
Each cold stream is represented initially by the string S-E.
Each hot stream by E+S.
The hot and cold streams are written in opposite order to
indicate the use of counter-current heat exchangers.
 
H1:E+S,
 
 
 
H2:E+S,
 
For our example, ω =  .
 C1:S-E, 

 
C2:S-E;
 



GHEN – 3. Rule set, P

Target → Replacement
Rule Description
- → x-
R1 Add an exchanger to a cold stream
- → s[x-][x-]m-
R2 Split a cold stream
+ → x+
R3 Add an exchanger to a hot stream
+ → m]x+[]x+[s+
R4 Split a hot stream
S→S
R5 A do-nothing rule

Note: the rule for splitting a hot stream creates a structure that is
the reverse of that created by a cold stream split rule.



GHEN summary

The L-system for HEN design is context free.
It is non-deterministic, perfect for an evolutionary algorithm.
The majority of strings (words generated from V ∗ starting with
ω) represent a valid genotype for the HEN design problem.
The genotype describes a conﬁguration with
locations of splits and
locations for integrated exchangers.


Implementation.

Outline

1

2

Implementation
3

Results
4

Conclusions
5


Implementation.

The evolutionary algorithm


Implementation.

Given: population size, np ; number of generations, ng ; the L-system,
GHEN = V , ω, P .
Outputs: Best solution found.
p ← ω {Initialise population}
1:

return best solution in p
9:


Implementation.

GHEN = V , ω, P .
1:
for i = 1, . . . , ng do
2:

9:


Implementation.

GHEN = V , ω, P .
1:
for i = 1, . . . , ng do
2:
Select g from population p {Random or ﬁtness based}
3:
Choose rule, r ∈ P {Random}
4:

9:


Implementation.

GHEN = V , ω, P .
1:
for i = 1, . . . , ng do
2:
3:
4:
r
g → g {Apply rule}
5:

9:


Implementation.

GHEN = V , ω, P .
1:
for i = 1, . . . , ng do
2:
3:
4:
r
5:
Evaluate g {Instantiate phenotype from g }
6:

9:


Implementation.

GHEN = V , ω, P .
1:
for i = 1, . . . , ng do
2:
3:
4:
r
5:
Evaluate g {Instantiate phenotype from g }
6:
Insert g into population subject to diversity constraint
7:
Shrink p if necessary {so that |p| ≤ np }
8:
9: return best solution in p


Implementation.

Phenotype instantiation

A genotype describes the overall structure of a HEN.
To instantiate a phenotype from a genotype:
link the integrated exchangers and
1

deﬁne the appropriate optimisation problem to size exchangers
2

and determine split factors.
Linking exchangers is non-deterministic so a single genotype may
lead to diﬀerent phenotypes.
The do-nothing rule, R5, allows for multiple instances of the
same genotype in a population.


Implementation.

Exchanger matching for phenotype instantiation


Implementation.

Given: g, the genotype to instantiate.
Outputs: ϕ, a phenotype instantiation of the genotype


Implementation.

1: ϕ ← g {Phenotype is initially the genotype.}

return ϕ
10:


Implementation.

2: while ∃ unassigned exchange term in ϕ do

return ϕ
10:


Implementation.

x1 ← random unassigned exchange term in ϕ
3:
x2 ← complementary random unassigned exchange in ϕ
4:

return ϕ
10:


Implementation.

3:
4:
if x2 then
5:

return ϕ
10:


Implementation.

3:
4:
if x2 then
5:
x2 ← new random complementary exchange term
6:

return ϕ
10:


Implementation.

3:
4:
if x2 then
5:
6:
if x2 then {Match might not be possible}
7:
return φ {Infeasible instantiation}
8:

return ϕ
10:


Implementation.

3:
4:
if x2 then
5:
6:
if x2 then {Match might not be possible}
7:
return φ {Infeasible instantiation}
8:
create heat exchange link between x1 and x2
9:
10: return ϕ


Implementation.

Example evolution with phenotype instantiation
Initial: { H1:E+S, H2:E+S, C1:S-E, C2:S-E; }


Implementation.


H1

H2

C1

C2


Implementation.

+R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; }

H1

H2

C1

C2


Implementation.

+R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; }

H1

H2

C1

C2

H1:E+S, H2:Ex(C1)+S, C1:Sx(H2)-E, C2:S-E;


Implementation.

+R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; }

H1

H2

C1

C2

H1:Ex(C1)+S, H2:E+S, C1:Sx(H1)-E, C2:S-E;


Implementation.

+R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; }
+R2: { H1:E+S, H2:E+S, C1:Sx-E, C2:Ss[x-][x-]m-E; }
H1

H2

C1

Split
C2 Mix


Implementation.

+R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; }
H1

H2

C1

Split
C2 Mix

H1:Ex(C2)+S, H2:Ex(C1)x(C2)+S, C1:Sx(H2)-E, C2:Ss[x(H2)-][x(H1)-]m-E;


Implementation.

+R1: { H1:E+S, H2:E+S, C1:Sx-E, C2:S-E; }
H1

H2

C1

Split
C2 Mix

H1:Ex(C2)+S, H2:Ex(C2)x(C1)+S, C1:Sx(H2)-E, C2:Ss[x(H2)-][x(H1)-]m-E;


Implementation.

Embedded optimisation problem
A network structure is
created from the
phenotype.
The structure deﬁnes a
nonlinear programme
(NLP).
The decision variables are
xi ∈ [0, 1], the fraction
to exchange, so that
Qi = xi Qi,max , and
xj ∈ [0, 1], the split
fraction.


Implementation.

A network structure is H1
created from the
phenotype. H2
nonlinear programme x1
(NLP). x2 x3
The decision variables are C1
to exchange, so that x4
C2 Mix
fraction.


Implementation.

A network structure is H1
created from the
phenotype. H2
nonlinear programme x1
(NLP). x2 x3
The decision variables are C1
to exchange, so that x4
C2 Mix
fraction.
The NLP represents a superstructure and is solved using a hybrid
stochastic & direct search procedure† .
† ˘
ESF (2006). In “Computer Aided Methods for Optimal Design and Operations”, Zilinskas & Bogle (ed.), 1-14.


Results.

Outline

1

2

Implementation
3

Results
4

Conclusions
5


Results.

Summary of performance
A wide range of case studies has been considered:

Objective function value (×103 $ y −1 )
Case study
Best Mean Worst σ
1. 4SP 83.5 84.5 89.1 1.07
2. 10SP1 44.9 45.1 45.4 0.171
3. Morton 1620. 1680. 1720. 35.
4. Lewin A 573. 575. 594. 6.61
5. Aromatics 2940. 2960. 2980. 13.

Best known solution obtained or bettered in all cases.


Results.

4SP Network viewer

genotype:[R3(1), R4(1), R3(4), R2(1), R1(1), R1(4)]

H1

H2

C1

C2

8.35E4


Results.

4SP. Evolution of objective function


Results.

4SP. Variation in ﬁnal solution obtained


Results.

10SP1 Network viewer

genotype:[R3(1), R3(4), R3(2), R3(5), R3(3), R4(3)]

H2

H3

H5

H1

H4

C4

C3

C5

C1

C2

4.49E4


Results.

Morton
Network viewer

genotype:[R3(2), R3(3), R3(3), R4(1), R3(4), R1(2), R3(1)]

H2

H3

H1

C3

C1

C2

1.62E6 6.19E5


Results.

Lewin A Network viewer

genotype:[R2(1), R2(1), R1(4)]

H3

H2

H5

H1

H4

C1

5.73E5


Results.

Aromatics Network viewer

genotype:[R4(4), R3(5), R3(3), R3(2), R3(1), R3(1), R3(1), R3(1), R2(3)]

H1

H2

H3

H4

C4

C3

C5

C1

C2

2.94E6


Conclusions.

Outline

1

2

Implementation
3

Results
4

Conclusions
5


Conclusions.

Summary
The challenging optimisation
problem of HEN design with Network viewer

genotype:[R3(1), R4(1), R2(1), R3(4), R1(4), R1(1), R3(3)]

stream splitting has been H2

addressed through a simple H1

two-level algorithm with a
Lindenmayer system for C1

evolving designs.
C2

The result is a robust and 8.35E4

eﬀective tool for heat
exchanger network design.

http://www.homepages.ucl.ac.uk/~ucecesf/research/


A Lindenmayer system for heat exchanger network design with stream splitting

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