Post on 17-Aug-2015
#+TITLE: Optimisation problems andmethods in ChemicalEngineering: a personal survey
#+AUTHOR: Eric S FragaCentre for Process Systems EngineeringDepartment of Chemical Engineering
#+INSTITUTE: University College London (UCL)#+DATE: 6 June 2014
One Day Workshop on Applied and NumericalMathematicsUniversity of Greenwich
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Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
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What is Chemical Engineering?
Chemical Engineering about changing raw materials intouseful products.
Other engineering disciplines deal with things;Chemical Engineering deals with stuff.
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Modelling
Mathematical models ofsystems that work with stuffare complex:
- nonlinear and nonsmooth,e.g.
∆Hk = w(Qk
CHW
)βLk∑m
d −γm ykm
- combinatorial- significant
uncertainties inparameters and models
350
375
400
425
6 8 10
Annual
ized
cost
(10
3 $
/y)
Operating pressure, unit 4 (atm)
f(X)
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Outline
Presentation will be personal survey of problems andoptimisation methods used over the past 20 years,ranging from off-the-shelf optimisation software throughto custom programs for specific applications.
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Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
–:–- talk.tex 14% 6/42 [Process synthesis] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Process design
Objective is to determineunit operations and theirinterconnections so as toachieve a specific task,usually defined by a productspecification.
?1
2
3
4
������������������������������
������������������������������
F
F
F
P
P
P
P
P
P
H SO2 4
Reactor D1Fluorspar
M
D2
Vapour effluent
HF
Excess
Makeup
M
HF +
A1Makeup
SolidEffluent
H SO2
H SO2
4
4
H SO2 4
ABC
B−−C
−−BA
A
B
C
–:–- talk.tex 16% 7/42 [Process synthesis] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Discrete Mathematical Approach
- Convert MINLP to discreteproblem: component flows,stream enthalpies, unitoperations.
⇒ Search a large, but finite,graph.
- Combine dynamic programmingwith implicit enumeration:
f (s ) = minu
{cu (s ) +
np∑i =1
f (pi )}
ABD
AB/D A → C
AB
A→C
BC
BCD
BC/D
B/C
A B C D
ESF & K I M McKinnon (2004), Ind Eng Chem Res 43(1):144-160.
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Example model
A short-cut distillation column model:
Nmin =log
[(xd
1−xd
) (1−xb
xb
)]logαavg
Rmin =∑i
αi xD ,i
αi − φ− 1
where 1− q =∑i
αi zF ,i
αi − φN − Nmin
N + 1= 0.75
[1−
(R − Rmin
R + 1
)0.5668]
to estimate N stages and R reflux ratio necessary forcosting, and where α are thermophysical properties.
–:–- talk.tex 21% 9/42 [Process synthesis] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
N -best solutions
ESF (1996), in State of the Art in Global Optimisation, Kluwer, 627-651.
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Multiple objectives
-1.8
-1.5
-1.2 2e+07
4e+07
6e+07
0
2000
4000
Annual cost (M$) SPI (m
2/year)
CT
WM
(kg w
ate
r/year)
10 best flowsheets according to cost
Minimum cost flowsheet
#2 & #3 flowsheets
by cost
#1 SPI
B
A
#1 CTWM
M A Steffens, ESF & I D L Bogle (1999), Comp Chem Eng 23(10):1455-1467.
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Uncertainty
A R F O’Grady, ESF, I D L Bogle (2001), Chemical Papers 55:376-381.
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Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
–:–- talk.tex 30% 13/42 [Heat integration] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Process heat integration
Matchappropriateheating andcooling needs toreduce costs andenvironmentalimpact.
P=
8.1
, R
=1
1.0
, S
=7
9
P=
5.6
, R
=0
.9,
S=
24
P=
7.8
, R
=5
.9,
S=
62
P=
¼.3
, R
=2
.3,
S=
21
H1
H4
C4
C3C2
H2 H3
C1
Unit 3
Unit 1
Unit 2
Unit 4
n−
Pen
tan
e
iso
−P
enta
ne
C3 C1 C2
Fee
d
Pro
pan
eis
o−
Bu
tan
e
H4
H4
n−
Bu
tan
eH4
–:–- talk.tex 33% 14/42 [Heat integration] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Simultaneous design
- Integrated process design is a nonlinear &combinatorial problem:
minx ,y
f (x , y ) = g (x ) + h (x , y )
g (x ) base process designh (x , y ) process heat integration
- Solve with an embedded hybrid method:
minx
f (x ) = g (x ) + miny
h (x , y )
with gradient or direct search for x and GA for y .
ESF & A Zilinskas (2003), Adv Eng Software 34:73-86.
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Model of heat exchanger
Cost model
C = α + βA γ
Area of exchanger
A = QU ∆TLMTD
Driving force for heat exchange
∆TLMTD = ∆Tin −∆Tout
log ∆Tin − log ∆Tout
and the various temperatures are the result ofthermophysical property predictions, functions of T andP design variables.
–:–- talk.tex 38% 16/42 [Heat integration] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Outer methods for process structure
Key Code MethodPG gradproj project gradientsNP projbfgs quasi-Newton projectedND method by ShorNM fmins Nelder & Mead simplexHJ hooke Hooke & JeevesIF imfil Implicit filteringCS Coordinate search
using penalty functions for methods designed forunconstrained optimisation (ND, NM, HJ, IF).
–:–- talk.tex 40% 17/42 [Heat integration] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Genetic algorithm for heat exchanges
Two implementations:GA1 nl × nc ordered pairs (j , r ) where nl is number
of levels, nc number of cold streams,j ∈ [0, nh ] index for hot stream and r fractionof available heat to exchange.
GA2 vector of ne values i ∈ [0, nc × nh ] where ne
represents the number of exchanges to allowand i the actual exchange to consider.
GA1 implements a larger and more comprehensive searchspace; GA2 however is constant in length.
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Impact of embedded stochastic method
350
375
400
425
6 8 10
An
nu
aliz
ed c
ost
(1
03 $
/y)
Operating pressure, unit 4 (atm)
f(X,Y*)
f(X)
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Zoomed view
342
344
7.6 8 8.4
An
nu
aliz
ed c
ost
(1
03 $
/y)
Operating pressure, unit 4 (atm)
f(X,Y*)
f(X)
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Some performance results
Alg best ave std nf ninf time(106 $) (106 $) (106 $) (s)
NM 8.40 9.35 0.813 906 13 1 472HJ 8.39 8.39 0.001 810 78 1 594IF 8.61 10.15 1.762 554 42 1 170CS 8.81 10.06 1.584 170 44 244GA-2L 8.53 8.75 0.192 814 58 1 703GA-1L 8.39 8.51 0.122 107 399 2 239 1 239
- GA-1L solves combined problem, f (x , y ), directly forbenchmarking.
- GA-2L uses a GA for outer method.- Hooke & Jeeves direct search algorithm is most
consistent and achieves best solution.
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Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
–:–- talk.tex 52% 22/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Efficient carbon capture
Identification
of solvent or
nanoporous
material
Detailed
process modelling
Try
again
Process
integration
Molecular modelling
and/or experiments
Done
Yes
Yes
Yes
No
No
No
Evaluation
- reduce efficiency lossdue to carbon capture.
- combined materials andprocess design.
- evaluation based onexperiments, detailedmodelling and processsimulation andoptimisation.
EPSRC EP/G062129/1
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Modelling I
Component mass balances (axial dispersed plug flowmodel):
dc i
dt+ 1− εb
εb
dQi
dt+ ∂(uci )
∂z+ ∂Ji
∂z= 0
dQi
dt= εp
dc mi
dt+ (1− εp )dqi
dt= k p
iAp
Vp(ci − c m
i )
Energy balance for the adsorbate in the gas phase:
εbdUf
dt= −(1− εb )∂Up
∂t− εb
∂(Hf u )∂z
− ∂JT
∂z
−Nc∑
i =1
∂(Ji Hi )∂z
− hwAc
Vc(Tf − Tw )
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Modelling IIEnergy balance for the adsorbate in the solid phase:
∂Up
∂t= εp
dUp ,f
dt+ (1− εp )dUp ,s
dt= hp
Ap
Vp(Tf − Tp )
Energy balance in the bed wall:
ρw Cp ,w∂Tw
∂t= −hw
Ac
Vw(Tw − Tf )− Uαwl (Tw − T∞)
and so on.
As simulation must reach cyclic steady state,⇒ computational effort is significant.
–:–- talk.tex 59% 25/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Behaviour of objective function
Obje
ctiv
e f
unct
ion v
alu
e
Along a line in design space
⇒ motivates use of surrogate modelling (responsesurface modelling, meta-modelling, ...).G Fiandaca, ESF & S Brandani (2009), Engineering Optimization 41(9):833-854.
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Surrogate model
- a fast approximation of model’s responsey (x ) : Rp → R where X ⊂ Rp is the space with pdesign variables.
- suitable for black box optimisation models as thesurrogate model is non-intrusive.
- based on training data: a set of known designpoints.
Most surrogates have form
y (x ) =q∑
k =1βk hk (x ) + ε(x )
with regressors hi (·) and a residual random process, ε(·).
–:–- talk.tex 64% 27/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Kriging
A statistical interpolating approach used forapproximating deterministic models.
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
y(x)
x
–:–- talk.tex 66% 28/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Optimisation
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Optimiser
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1
Pur
ity (
%)
λ
We use evolutionary stochastic methods to cater formulti-modality of objective function.
–:–- talk.tex 71% 30/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Case study: 6 step, 2 bed PSA
Bed 1
Bed 2
Vaccum
Tank
V1
V2
V4
V5
V7
V3 V6
Vent tank
Vent
Feed tank
Feed
BH
BH
BH
BH
- 6 design variables.- 3 objective functions:
recovery, purity and power(but will illustrate 2).
- computational effort large:30-60 minutes per objectivefunction evaluation.
J Beck, D Friedrich, S Brandani & ESF (2012),
Proc 22nd ESCAPE, Elsevier, 1217-1221
–:–- talk.tex 73% 31/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Pareto front: n = 64
0
20
40
60
80
100
0 20 40 60 80 100
Pur
ity (
%)
Recovery (%)
NSGA-IISbNSGA-IISbNSGA-II ALM
–:–- talk.tex 76% 32/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Pareto front: n = 96
0
20
40
60
80
100
0 20 40 60 80 100
Pur
ity (
%)
Recovery (%)
NSGA-IISbNSGA-IISbNSGA-II ALM
–:–- talk.tex 78% 33/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Pareto front: n = 176
0
20
40
60
80
100
0 20 40 60 80 100
Pur
ity (
%)
Recovery (%)
NSGA-IISbNSGA-IISbNSGA-II ALM
–:–- talk.tex 80% 34/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Pareto front: n = 256
0
20
40
60
80
100
0 20 40 60 80 100
Pur
ity (
%)
Recovery (%)
NSGA-IISbNSGA-IISbNSGA-II ALM
–:–- talk.tex 83% 35/42 [Carbon capture] --------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
–:–- talk.tex 85% 36/42 [Power generation] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
The problem
Determine power schedulefor minimum fuel cost forset of online thermalunits subject to a numberof issues:
- prohibited operatingzones
- transmission losses- valve point loadings
using mathematicalprogramming toolbox.
minPi
z =∑i
Fi (Pi )
4000
6000
8000
10000
12000
150 200 250 300 350 400 450
F(P
1)
P1 [MW]
Fi (Pi ) = ai P 2i +bi Pi +ci +|ei sin (fi (Pi ,min − Pi ))|
L Yang, ESF & L G Papageorgiou (2013), Elec Power Sys Res 95:302-308.
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ResultsMean Cost Best Cost Method Year
CSOMA 121,415.05 121,414.70 Cultural algorithm 2010FAPSO-VDE 121,412.61 121,412.56 Particle swarm 2011DE 121,422.72 121,416.29 Differential Evolution 2008SOMA 121,449.88 121,418.79 Self-org migration 2000EP-SQP 122,379.63 122,323.97 Genetic Algorithm 2008CDEMD 121,526.73 121,423.40 Differential Evolution 2009BBO 121,512.06 121,418.27 Biogeography 2010HGA 121,784.04 121,418.27 Genetic Algorithm 2008HDE 122,304.30 121,698.51 Differential Evolution 2009MTS 121,798.51 121,532.10 Tabu search 2011UHGA 121,602.81 121,424.48 Genetic Algorithm 2008MDE 121,418.44 121,414.79 Differential Evolution 2010VLEMIQP 121,412.54 121,412.54 Mathematical Prog 2013
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Caveat: modelling
ChemicalEngineeringproblems arebased on modelsof thetransformationof stuff. Thesemodels aredifficult toobtain andsometimes theresults are notcorrect.
0
500
1000
1500
2000
2500
3000
20 60 100
0
F(P
7)
dF
/dP
P7 [MW]
valve effectsno effects
dF/dP
–:–- talk.tex 92% 39/42 [Power generation] ------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Restricted search
Problem Demand DVLMILP Excess DVLMILP Best Gapoutput output cost known
(MW) (MW) (%) cost (%)13 units 1800 1802 0.11 17964 17960 0.0213 units 2520 2525 0.20 24174 24164 0.0440 units 10500 10501 0.01 121986 121413 0.47
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Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
–:–- talk.tex 97% 41/42 [Conclusions] -----------------------------<[ ]> <[[]]> <**> < ** > «» < ? >
Summary
marketobjective
processstructure
bespoke software
optimisedoperation
directsearch
processscheduling
mathematical programming
heatintegration
DS+ GA
dynamicoperation
MOGA + surrogate
Thanks to Dr Joakim Beck,
Professor David Bogle, Dr Rob
O’Grady, Professor Lazaros
Papageorgiou, Dr Mark Steffens
and Dr Lingjiang Yang at UCL;
Professor Stefano Brandani
(Edinburgh), Dr Daniel Friedrich
(Edinburgh), Professor Ken
McKinnon (Edinburgh) and
Professor Antanas Zilinskas
(Lithuania).
http://www.ucl.ac.uk/~ucecesf/
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