#+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

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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

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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.

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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

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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

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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.

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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).

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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

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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.

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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(·).

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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

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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.

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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

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Pareto front: n = 64

0

20

40

60

80

100

0 20 40 60 80 100

Pur

ity (

%)

Recovery (%)

NSGA-IISbNSGA-IISbNSGA-II ALM

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Pareto front: n = 96

0

20

40

60

80

100

0 20 40 60 80 100

Pur

ity (

%)

Recovery (%)

NSGA-IISbNSGA-IISbNSGA-II ALM

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Pareto front: n = 176

0

20

40

60

80

100

0 20 40 60 80 100

Pur

ity (

%)

Recovery (%)

NSGA-IISbNSGA-IISbNSGA-II ALM

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Pareto front: n = 256

0

20

40

60

80

100

0 20 40 60 80 100

Pur

ity (

%)

Recovery (%)

NSGA-IISbNSGA-IISbNSGA-II ALM

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Topic

* Introduction

* Process synthesis

* Heat integration

* Carbon capture

* Power generation

* Conclusions

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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

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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

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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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