evolutionary multi-objective optimisation of a complex steady … · 2017. 2. 7. · evolutionary...

Evolutionary Multi-Objective Optimisation of a

Complex Steady-State Process Flowsheet – The

‘Surrogate-Assisted’ Approach

Submitted in partial fulfilment of the requirements

of the degree of

Doctor of Philosophy

of the

Indian Institute of Technology Bombay, India

and

Monash University, Australia

by

Ishan Sharma

Supervisors:

Prof. Sanjay Mahajani (IIT Bombay)

Dr. Anuradda Ganesh (IIT Bombay)

Prof. Andrew Hoadley (Monash University)

The course of study for this award was developed jointly by

Monash University, Australia and the Indian Institute of Technology, Bombay

and was given academic recognition by each of them.

The programme was administrated by The IITB-Monash Research Academy

2016

Abstract

Decision making, in case of multiple objectives, involves analysing the trade-offs. Evolutionary

Multi-Objective Optimisation (MOO) is a derivative-free search method, which tries to mimic

the natural evolutionary process. It has the advantage of yielding a set of Pareto-optimal or

equally-good solutions in a single run. This eliminates the need for the Decision Maker (DM) to

a-priori articulate his/her preferences among the different objectives. Generating multiple

solutions may be considered unnecessary, as ultimately the DM is going to implement only one

of the multiple non-dominated solutions generated by evolutionary MOO. A large number of the

potential, or candidate, solutions just ‘die-off’ during the course of optimization. This is

specifically detrimental in case of high fidelity objective functions/models, which are often

computationally expensive and thus, generally require a significantly high computation time.

However, making an informed decision about the preferences a-priori is not always possible. In a

large number of instances, a set of non-dominated solutions can aid the DM to make an informed

decision. It is in these situations that the computation effort could be used as a ‘resource’ to fit

relatively lower fidelity surrogate approximations, which can be solved in a fraction of the time

needed to solve the high fidelity function/model. In the field of process design, some example of

high fidelity models include; 3D, 2D and 1D Computational Fluid Dynamics (CFD) based

models, cyclic or batch process models which need to be integrated with steady-state models.

The surrogate models can then be subsequently used, if expected to be of sufficient accuracy.

The surrogate may also be needed to be updated periodically. This is done with the dual

objective of ensuring better surrogate model accuracy/fidelity in the promising subdomain and to

use the additional information that is now available due to the high fidelity model evaluations

done since the last surrogate fitting step. The fidelity of the surrogate model is expected to only

selectively improve because the evolutionary algorithm is more likely to generate candidate

solutions from the promising subdomain, during the course of optimisation. When the surrogates

are to be used for optimisation, improving the global fidelity of the surrogates makes little sense

as this would inevitably require the high fidelity model to be evaluated for non-promising data

points. The aim of surrogate-assisted evolutionary MOO should thus be to converge as close to

the global optimum, as possible; while evaluating the high fidelity model as few number of

times, as possible.

This thesis includes a review of the recent application of surrogate-assisted evolutionary MOO in

the field of chemical engineering, with a specific interest in process design applications. The

Multiple Adaptive Spatially Distributed Surrogates (MASDS) algorithm has been modified in

order to better suit practical chemical engineering process design problems, where the final

solution space is often a small subset of the initial search space. In such a scenario, periodic

evolution of search space could also be done to ensure that the data points lying in the non-

promising regions do not contribute to surrogate model fitting, thereby, potentially improving the

surrogate accuracy (or fidelity) in the promising regions. A preliminary investigation has been

done to assess this hypothesis by applying the modified MASDS or mMASDS algorithm to two

numerical test problems. The results, thus obtained, are compared with those obtained from

MASDS algorithm by performing two separate runs, when starting from the same initial

population, while keeping all the other parameters the same.

The mMASDS algorithm is then demonstrated for a chemical engineering process design and

optimisation problem, involving simultaneous optimisation of economic and environmental

objectives in a coal to ammonia process with carbon capture. Two CO2 capture mechanisms have

been compared by performing two separate surrogate-assisted evolutionary MOO runs. Physical

absorption in chilled methanol and Activated Carbon based Pressure Swing Adsorption (PSA)

that have been investigated for CO2 capture. Both the chilled methanol and PSA models are

computationally expensive. For the chilled methanol case, the simulation model has a recycle;

this requires the simulation to be solved iteratively. While the PSA model needs to be solved

dynamically for a finite number of cycles, until a Cyclic Steady State (CSS) is achieved. This

makes the entire exercise computationally prohibitive. The CO2 capture unit models are thus

replaced with a set of surrogate models, predicting the CSS outputs. For the chilled methanol

case, the results from the surrogate-assisted run have been compared to those obtained from

Business-As-Usual (BAU) approach, where only the actual flowsheet model was used for

functional evaluation. Results show significant savings, measured in terms of the hypervolume

spanned by the Pareto-fronts obtained from the two approaches, for a fixed computational

budget.

The surrogate-assisted strategy thus allows for better integration of computationally complex

units into large-scale plant simulations. It also yields an array of surrogates, which can be used

for any future prediction of the objective function values.

To decide whether to update the surrogates or not, the use of rank correlation coefficient between

the surrogate and the actual models has been suggested for future implementation. This avoids

the extra computational effort wasted in unnecessarily updating the surrogates.

Both the MASDS and mMASDS algorithms involve comparing the surrogate model based

outputs with those obtained from the high fidelity models, during domination score computation.

This may result in the accurately evaluated promising, high fidelity data points dying-off during

the optimisation, due to erroneous surrogate predictions. It is thus suggested to maintain a

separate Actual Evaluated Pareto (AEP) which contains only the solutions obtained from high

fidelity model evaluations.

The surrogate-assisted evolutionary MOO is a powerful tool to be used for determining the trade-

offs, when the mathematical model/simulation is computationally expensive to be used with

conventional evolutionary algorithms. It allows the user to quickly hone in on the solution,

without spending too much time in evaluating the model/simulation. It has multiple applications

in chemical engineering and in particular, process design. As demonstrated in this work, it allows

the user to better integrate and optimise a computationally expensive subsection with the rest of

the plant.

i

Table of Contents

List of Figures ................................................................................................................................ iii

List of Tables ................................................................................................................................ vii

Nomenclature ............................................................................................................................... viii

Chapter 1 Introduction ................................................................................................................ 1

1.1 Context ............................................................................................................................. 1

1.2 Multi-Objective Optimisation (MOO) ............................................................................. 2

1.3 MOO methods .................................................................................................................. 4

1.4 Motivation for the present work ..................................................................................... 12

1.5 Goals and scope of the present work .............................................................................. 13

1.6 Outline ............................................................................................................................ 13

Chapter 2 Surrogate-assisted MOO and process design ........................................................... 15

2.1 Model management or evolution control ....................................................................... 17

2.2 Surrogate-assisted MOO and process design ................................................................. 19

2.3 Insights from the review and surrogate-assisted strategy selection ............................... 28

Chapter 3 The mMASDS algorithm and mathematical test problems ..................................... 29

3.1 The mMASDS algorithm ............................................................................................... 29

3.2 Differences between MASDS and mMASDS ............................................................... 33

3.3 Performance metric ........................................................................................................ 34

3.4 Mathematical test problems ........................................................................................... 35

3.4.1 Problem definition .................................................................................................. 35

3.4.2 Results ..................................................................................................................... 36

3.5 Conclusions .................................................................................................................... 41

Chapter 4 Process design and Optimisation case study: coal to NH3 process with carbon

capture ........................................................................................................................................... 42

4.1 Context ........................................................................................................................... 42

4.2 Coal to NH3 process details ............................................................................................ 43

4.2.1 Processing Options.................................................................................................. 44

4.3 Coal to NH3 flowsheets .................................................................................................. 49

ii

4.3.1 Coal to NH3 flowsheet with CO2 capture via physical absorption in chilled

methanol (RectisolTM

process)............................................................................................... 49

4.3.2 Coal to NH3 flowsheet with CO2 capture via physical adsorption on Activated

Carbon (PSA process) ........................................................................................................... 54

Chapter 5 Surrogate-assisted global MOO of the coal to NH3 flowsheet with CO2 capture via

physical absorption in chilled methanol (RectisolTM

process) ...................................................... 61

5.1 MOO problem formulation ............................................................................................ 61

5.2 Surrogate modelling for the RectisolTM

section ............................................................. 63

5.2.1 Dimensionality ........................................................................................................ 63

5.2.2 Surrogate models .................................................................................................... 63

5.2.3 Normalisation .......................................................................................................... 64

5.3 Results and discussion .................................................................................................... 65

Chapter 6 Surrogate-assisted global MOO of coal to NH3 flowsheet with CO2 capture via

physical adsorption on Activated Carbon (PSA process) ............................................................. 71

6.1 MOO problem formulation ............................................................................................ 71

6.2 Surrogate modelling for the PSA section ....................................................................... 72

6.2.1 Dimensionality ........................................................................................................ 72

6.2.2 Surrogate models .................................................................................................... 72

6.2.3 Normalisation .......................................................................................................... 73

6.3 Results and discussion .................................................................................................... 73

Chapter 7 Conclusions and prospects for further research ....................................................... 78

Future prospects for research ................................................................................................. 81

Appendix A Standalone chilled methanol based capture plant optimisation ........................... 83

Appendix B PSA modelling details .......................................................................................... 92

Appendix C Economic Assumptions ........................................................................................ 98

Appendix D Standalone PSA capture plant optimisation ....................................................... 100

Appendix E Two stage refrigeration system optimisation ...................................................... 105

References ................................................................................................................................... 113

Acknowledgments....................................................................................................................... 119

List of Publications……………………………………………………………………………..120

iii

List of Figures

Figure 1.1: Example of multi-objective optimisation in chemical engineering design. ................. 1

Figure 1.2: Dominated and nondominated solutions in objective variable space for the sample

problem, given by Equations 1.2, 1.3 and 1.4................................................................................. 3

Figure 1.3: Pareto plot for a hypothetical, two-objective optimisation problem ............................ 4

Figure 1.4: Graphical representation of the weighted sum method. ............................................... 5

Figure 1.5: Failure of the weighted sum method to find solutions in the non-convex region of the

Pareto plot ....................................................................................................................................... 6

Figure 1.6: Graphical representation of the ԑ-constraint method .................................................... 7

Figure 1.7: Crowding distance ) estimation (Deb et al., 2002) ................................. 10

Figure 1.8: Generation of geneneration from generation (reprinted from Deb et al.,

2002) ............................................................................................................................................. 11

Figure 2.1: An illustration explaining the model fidelity and computational effort trade-off ...... 15

Figure 2.2: Example of false minimum with the surrogate model (Jin, 2005) ............................. 16

Figure 2.3: Updating the surrogate by re-evaluating the optimum ............................................... 16

Figure 2.4: EC classification, as per Jin (2005) ............................................................................ 17

Figure 3.1: The mMASDS algorithm. Dashed rectangle highlights the modifications made to the

MASDS algorithm. ....................................................................................................................... 29

Figure 3.2: An illustration of the S-metric (represented by the shaded area). .............................. 34

Figure 3.3: Nondominated points obtained from the two runs for SCH problem after a fixed

budget of 600 original model evaluations ..................................................................................... 37

Figure 3.4: Parity plots of final set of surrogate models for (a) and (b) (SCH problem) .... 37

Figure 3.5: Evolution of the search space during surrogate-assisted MOO of SCH problem ...... 38

Figure 3.6: Nondominated points obtained from the two runs for ZDT2 problem after a fixed

budget of 1000 original model evaluations ................................................................................... 38

Figure 3.7: Parity plots of final set of surrogate models for (a) and (b) (ZDT2 problem) .. 39

Figure 3.8: Evolution of the search space during surrogate-assisted MOO of ZDT2 problem for

variables (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , (i) and (j) ............ 40

Figure 3.9: S values as a function of number of original model evaluation for the two runs ....... 41

Figure 4.1: Sour and clean shift configurations ............................................................................ 46

Figure 4.2 (a) and (b): The two possible configurations, in case the capture technology is unable

to separate H2S .............................................................................................................................. 48

Figure 4.3: Block diagram of the coal to ammonia process with RectisolTM

for CO2 capture (the

highlighted rectangular portion represents the RectisolTM

process and the surrogate model’s

boundary) ...................................................................................................................................... 50

Figure 4.4: (a) Flowsheet for coal gasification, (b) AGR and (c) NH3 synthesis sections ........... 52

Figure 4.5: Block diagram of the coal to ammonia process with PSA for CO2 capture (the

highlighted rectangular portion represents the PSA process and the surrogate model’s boundary)

....................................................................................................................................................... 55

iv

Figure 4.6: Flowsheet for (a) coal gasification and sulphur removal, (b) shift reactor, (c) PSA and

(d) NH3 synthesis sections ............................................................................................................ 58

Figure 4.7: Time chart for the PSA cycle ..................................................................................... 59

Figure 5.1: Information flow in the MOO framework .................................................................. 63

Figure 5.2: Surrogate model mapping for RectisolTM

case ........................................................... 64

Figure 5.3: Surrogate-assisted and BAU approach after (a) 250, (b) 350, (c) 450, (d) 550, (e) 650

and (f) 750 original model evaluations ......................................................................................... 66

Figure 5.4: S value as a function of number of original model evaluation for the two runs ........ 67

Figure 5.5: (a) , (b) , (c) , (d) , (e) , (f) and (g) values

corresponding to nondominated points, for surrogate-assisted run, shown in Figure 5.3(f) ........ 69

Figure 6.1: Surrogate model mapping for PSA case ..................................................................... 73

Figure 6.2: Pareto progression for the coal to NH3 case with CO2 capture via physical adsorption

on Activated Carbon (PSA process) ............................................................................................. 74

Figure 6.3: S value as a function of number of original model evaluation for the coal to NH3 case

with CO2 capture via physical adsorption on Activated Carbon (PSA process) .......................... 75

Figure 6.4: (a) , (b) and (c) values corresponding to nondominated points

obtained after 1425 original model evaluations ............................................................................ 76

Figure 6.5: Global vs. standalone PSA optimisation .................................................................... 76

Figure 6.6: Results obtained for the two flowsheets involving different carbon capture

technologies .................................................................................................................................. 77

Figure 7.1: The suggested surrogate-assisted evolutionary MOO algorithm ............................... 82

Figure A.1: Process flowsheet ...................................................................................................... 84

Figure A.2: Pareto front obtained for the standalone optimisation problem ................................ 87

Figure A.3: (a) Fourth/last stage flash pressure ( ), (b) Solvent (methanol) flow rate to the

absorber ( ) and (c) Temperature change in the bottoms heater ( ) corresponding to

the optimum objective function values ......................................................................................... 88

Figure A.4: Minimum specific energy penalty ( ) for different CO2 capture rates

( ) ........................................................................................................................................ 89

Figure B.1: A composite adsorbent (reprinted from Ruthven et al., 1994) .................................. 93

Figure D.1: The 4 bed PSA system ............................................................................................. 100

Figure D.2: Time chart for the PSA cycle. ................................................................................. 101

Figure D.3: Pareto plot for PSA standalone MOO ..................................................................... 103

Figures D.4: (a) Valve coefficient for Purge Valves (VPurge ) ( ), (b) H2 product flow

rate in adsorption step ( , (c) Adsorption step time ( ) and (d) Blow down pressure

( corresponding to the optimum objective function values ............................................... 104

Figure E.1: Single stage vapour compression refrigeration cycle .............................................. 105

Figure E.2: A hypothetical process GCC showing a two-stage refrigeration system ................. 106

Figure E.3: (a) GCC for CO2 pressurisation via compression and condensation (b) GCC for CO2

pressurisation via compression (Where; W1 and W2 are the two shaftwork and, and

are the two condensing duties) ...................................................................................... 108

v

Figure E.4: Extra refrigeration level at the CO2 condensation temperature ............................... 110

Figure E.5: Two stage refrigeration system optimisation algorithm .......................................... 110

vi

List of Tables

Table 1.1: Candidate solution sets .................................................................................................. 8

Table 1.2: Summary of the two approaches .................................................................................. 12

Table 2.1: Summary of works falling under the one-shot approach. ............................................ 23

Table 2.2: Summary of works involving periodic updating of surrogates. .................................. 27

Table 3.1: Surrogate-assisted MOO parameters ........................................................................... 36

Table 4.1: A comparison of different coal gasification technologies (Cortés et al., 2009) .......... 44

Table 4.2: Characteristics of the coal, used in this work .............................................................. 45

Table 4.3: Comparative assessment of Clean and Sour shift ........................................................ 46

Table 4.4: Performance of commercial WGS (DOE, 2007) ......................................................... 47

Table 4.5: Binary interaction parameters proposed by Smith and Sun (2013) ............................. 53

Table 4.6: Adsorbent and adsorption bed characteristics ............................................................. 60

Table 5.1: Decision variable ranges for the global optimisation (RectisolTM

case) ..................... 62

Table 5.2: Constraints for the global optimisation (RectisolTM

case) .......................................... 62

Table 5.3: Effect of purge fraction in NH3 synthesis loop on the objective functions ................. 70

Table 6.1: Decision variable ranges for the global optimisation (PSA case) .............................. 72

Table 6.2: Constraints for the global optimisation (PSA case) .................................................... 72

Table A.1: Feed gas properties ..................................................................................................... 85

Table A.2: Decision variable range for optimisation ................................................................... 86

Table A.3: CO2 compression and refrigeration requirement for four characteristic points, plotted

in Figure A.4 ................................................................................................................................. 90

Table A.4: Effect of pocket exploitation (Reference point on Pareto front– : 93.3 % and

: 3385.28 kWe) ......................................................................................................................... 90

Table B.1: LDF model parameter value for the system under consideration (Jee et al, 2001) ..... 95

Table B.2: Extended Langmuir Freundlich model parameter values for the system under

consideration (Jee et al., 2001) ..................................................................................................... 96

Table C.1: Basic economic assumptions ...................................................................................... 98

Table C.2: Economic parameters .................................................................................................. 99

Table C.3: Assumptions for COM prediction ............................................................................... 99

Table D.1: Feed gas properties ................................................................................................... 101

Table D.2: Adsorbent and adsorption bed characteristics .......................................................... 101

Table D.3: Decision variable range for optimisation ................................................................. 102

Table E.1: Results from detailed simulations to verify the assumption related to using a factor of

0.6 to account for deviations from an ideal isentropic operation ................................................ 109

vii

Nomenclature

Symbols

Annualized fixed capital investment

Molar flow rate of Ar in treated gas from absorber (kmol/h)

Blow down pressure (bar)

Number of constraints

Gas phase concentration of component (mol/m3)

Crossover probability for genetic algorithm

GRC at scale (USD)

The offspring set from Carbon footprint of the overall process (kg CO2e emitted/kg NH3 produced)

Molar flow rate of CH4 in treated gas from absorber (kmol/h)

Cost of operating labour (USD/y)

Heat capacity of the component (J/mol/K)

Heat capacity of adsorbent (J/kg/K)

Cost of raw materials (USD/y)

Molar flow rate of CO in treated gas from absorber (kmol/h)

Molar flow rate of CO2 in treated gas from absorber (kmol/h)

Molar flow rate of CO2 emitted in the stripper distillate (kmol/h)

Cost of manufacture (USD/y)

CO2 capture rate (%)

Cost of utilities (USD/y)

Valve coefficient for Purge Valves (VPurgei) (kmol/s/bar)

Cost of waste treatment (USD/y)

Axial dispersion coefficient (m2/s)

Molecular diffusivity (m2/s)

( ) Minimum shaftwork for pocket above/below the pinch assuming ideal isentropic

compression (MWe)

Shaftwork corresponding to the two-stage refrigeration system with one stage at

CO2 condensation temperature (MWe)

Minimum shaftwork for the two-stage refrigeration system assuming ideal

isentropic compression (MWe)

The external archive storing the information from all the original model

evaluation

Electrical power consumed by compressor (kWe)

Total electrical power required by the global process (kWe)

Optimum refrigeration electrical duty after adjusting for non-ideal isentropic

compression (kWe)

viii

Electrical power consumed by solvent recycle pump (kWe)

Electrical power consumed by solvent pump (kWe)

Approximate electrical power sacrificed by using LP steam in stripper

reboiler (kWe)

Scale factor for different components

Purge fraction in the NH3 synthesis loop

Fixed Capital Investment (USD)

H2 product flow rate in adsorption step (kmol/s)

Grass roots cost (USD)

Grass roots cost for the PSA process (USD)

Grass roots cost for the AGR section (USD)

Total gas phase enthalpy (J/m3)

Enthalpy of saturated water at the inlet pressure of the turbine (kJ/kg)

Enthalpy of saturated water at the outlet pressure of the turbine (kJ/kg)

Molar flow rate of H2 in treated gas from absorber (kmol/h)

HP steam to HTS reactor

The initial number of generations, for which the computationally expensive

detailed model is used for candidate solution evaluation

Maximum number of partitions allowed for

Set comprising of the minimum bounds for all decision variables, after

generation worth of evolution

Set comprising of the minimum bounds for all decision variables, at the start of

the search

Set comprising of the minimum bounds for all decision variables, at the start of

the surrogate-assisted set of generations

Rate of CO2 equivalent emissions (kg CO2e/y)

Rate of NH3 production (kg NH3/y)

Number of objective functions

‘Default’ mutation rate for genetic algorithm

‘High’ mutation rate for genetic algorithm

Mass flow rate of steam (kg/s)

Molecular weight (kg/kmol)

The number of equally sized trains for different components

The maximum number of generations

Domination count for individual p

Population size for genetic algorithm runs

Molar flow rate of N2 in treated gas from absorber (kmol/h)

Oxygen (95% pure) flow to the gasifier

Population size

ix

Partial pressure of ith

component (bar)

Pressure (bar)

The parent population set for the ith

generation

1st stage flash pressure

4th

stage flash pressure

Production cost of NH3 (USD/mt NH3)

Pressure at which CO2 is liquefied (bar)

Total power penalty associated with CO2 capture (kWe)

H2 Purity in the H2 product stream (%)

Equilibrium average solid-phase loading for the i

th component (mol/kg)

Average solid-phase loading for the ith

component (mol/kg)

Cooling water target (MWt)

Heat being given to the pocket (MWt)

Duty corresponding to (MWt)

Reboiler duty of stripper (Gcal/h)

Total cooling utility (MWt)

Adsorbent particle radius (m)

Universal gas constant (J/mol/K)

CO recovery across the absorber (%)

Percentage of the H2 in feed being recovered in the H2 product stream (%)

H2 recovery across the absorber (%)

The number of surrogate-assisted intermediate generations

The normalised hypervolume spanned by the Pareto front w.r.t. a reference point

Set of solutions dominated by individual p

The seed value supplied to the random number generator

Desired scale for different components

Specific energy penalty (kWh/kmol CO2 captured)

Chilled methanol to absorber (kmol/h)

Fraction of H2 in feed going with CO2 product stream

Fraction of CO2 in feed going with CO2 product stream

The surrogate model for the objective or constraint

The surrogate model for the objective or constraint

The scaling ratio for the PSA unit

Time (s)

Adsorption step time (s)

Pressurisation and Depressurisation step time (s)

Pressure equalisation time (s)

Average cross-sectional gas phase temperature in the adsorber bed (K)

x

Temperature (shifted) on the grand composite curve corresponding to (°C)

Intermediate temperature (shifted) level (°C)

Minimum temperature (shifted) on grand composite curve (°C)

Cooling water temperature (shifted) (°C)

Training set corresponding to the objective/constraint and partition

Temperature of treated gas from absorber (°C)

Wall temperature (ambient temperature) (K)

Effective heat transfer coefficient (J/m3/s/K)

Set comprising of the maximum bounds for all decision variables, after nmax


Set comprising of the maximum bounds for all decision variables, at the start of

the search

Set comprising of the maximum bounds for all decision variables, at the

start of the surrogate-assisted set of generations

Superficial gas velocity (m/s)

Mole fraction of CO2 in the treated syngas

Greek Symbols

The fraction of points in the EA, used to train and validate the surrogates

Number of valid surrogates for K partitions

Isosteric heat of adsorption (J/mol)

Temperature change in the bottoms heater (°C)

Minimum temperature difference for the heat exchanger network (°C)

Bed void fraction

Limiting NMSE on test data

Bulk density (kg/m3)

Gas phase molar density (kmol/m3)

Adsorbent particle density (kg/m3)

Shape factor

Valid partition ratio

Acronyms

AEP Actual Evaluation Pareto

AN Ammonium Nitrate

ANN Artificial Neural Network

ASU Air Separation Unit

AUD Australian Dollar

BAU Business As Usual

xi

BM Boston-Mathias

CEPCI Chemical Engineering Plant Cost Index

CFD Computational Fluid Dynamics

CCS Carbon Capture and Sequestration

COP Coefficient Of Performance

COM Cost Of Manufacture

DM Decision Maker

EI Expected Improvement

EA External Archive

EC Evolution Control

ECC Exergy Composite Curves

EGCC Exergy Grand Composite Curve

EoS Equation of State

FCI Fixed Capital Investment

GAs Genetic Algorithms

GCC Grand Composite Curve

GRC Grass Roots Costs

HHV High Heating Value

HP High Pressure

HTS High Temperature Shift

LAC Linde Ammonia Concept

LDF Linear Driving Force

LHD Latin Hypercube Design

LHS Latin Hypercube Sampling

LHV Lower Heating Value

LP Low Pressure

LTS Low Temperature Shift

mMASDS modified-MASDS

MASDS Multiple Adaptive Spatially Distributed Surrogates

MAF Moisture Ash Free

MOC Material Of Construction

MOO Multi-Objective Optimisation

MSDS Multiple Spatially Distributed Surrogates

NMSE Normalized Mean Square Error

NSGA Nondominated Sorting Genetic Algorithm

ParEGO Pareto Efficient Global Optimisation

PC-SAFT Perturbed Chain – Statistical Associated Fluid Theory

PFR Plug Flow Reactor

PDAEs Partial Differential Algebraic Equations

PSA Pressure Swing Adsorption

xii

RKS Redlich-Kwong-Soave

SOO Single-Objective Optimisation

SAEA Surrogate Assisted Evolutionary Algorithm

SUMO SUrrogate Modelling

USD US Dollar

VBA Visual Basic for Applications

VPSA Vacuum Pressure Swing Adsorption

WSG Water Gas Shift

1

Chapter 1

Introduction

1.1 Context

Everyday life involves individuals making decisions in circumstances where they have more than

one objective. These individuals, referred to as Decision Maker (DM) henceforth, either

knowingly or unknowingly make numerous such decisions. Under certain circumstances, the

nature of the problem may allow the DM to simultaneously optimise all of the objectives. This is

a ‘win-win’ situation where the multiple objectives are non-conflicting in nature. However, it is

often the case that an improvement in a particular objective is not possible, beyond a particular

limit, without worsening at least one of the others (‘win-lose’).

A classic, chemical engineering example would be to simultaneously minimise both operating

and capital costs of the process shown in Figure 1.1.

T=130 °C

T=10 °C T=170 °C

T=240 °C

T=190 °C T=70 °CT=220 °C

T=30 °C

Figure 1.1: A hypothetical example of multi-objective optimisation in chemical engineering

design.

The hypothetical process shown in Figure 1.1 has two hot streams and two cold streams. Without

any heat integration, cold and hot utilities would be used in order to meet the cooling and heating

requirements, respectively. However, there is a significant scope of savings in utility costs by

thermally integrating the available hot and cold streams. In such a case, the minimum approach

temperature for the heat integrated exchanger network would be one of the decision variables

over which the optimisation needs to be performed. A lower value of the minimum approach

temperature implies lower utility (thus, operating) costs, due to a lower amount of utility flow

2

required. However, a lower value of minimum approach temperature may also require a higher

heat transfer area (and thus, capital costs) due to a decrease in the heat transfer driving force.

Apart from this, this cost optimisation problem could also have some constraints; for example, a

maximum limit imposed on the physical volume that a particular heat exchanger could occupy.

1.2 Multi-Objective Optimisation (MOO)

MOO refers to simultaneous optimisation of two or more objective functions. For multiple non-

conflicting objectives (win-win), there is only one, unique solution to the MOO problem.

However, the win-lose situation involving a conflict between the objective functions, is more

commonly encountered. The solution to such problems is non-trivial. The solution to such a

MOO optimisation comprises of multiple solutions, representing the trade-offs between the

objectives. These solutions are named after the Italian economist Vilfredo Pareto as Pareto-

optimal solutions.

Consider a general MOO problem as given below:

M

Subject to:

Where, , and

Rangaiah (2009) defines Pareto-optimal solutions as:

“The set: , and

is said to be a Pareto-optimal solution for the two-objective

problem in Equation 1.1, if and only if, no other feasible exists such that and

, with strict inequality valid for at least one objective.”

If, however, strict inequality is not valid for any of the objectives, the set is only weakly

Pareto-optimal. The Pareto-optimal solutions are also called nondominated or equally-good

solutions. This concept of dominance is best understood with the help of an example. Consider

the two-objective functions given by equation:

(1.2)

(1.3)

Suppose the aim is to minimise both and d by varying . The optimisation problem in

mathematical form is as follows:

Subject to:

Where,

(1.1)

(1.4)

3

To better understand the problem, the domain for variable needs to be broken down into three

subdomains; i.e. .The MOO problem needs to be analysed

separately in these three subdomains as follows:

In the subdomain , both the functions can be simultaneously minimised by

increasing . In other words, the two functions are non-conflicting in the subdomain

.

In the subdomain , both the functions can be simultaneously minimised by

decreasing . The two functions are also non-conflicting in the subdomain ,

In the subdomain , a decrease in is accompanied by a corresponding increase

in . The two functions are conflicting in the subdomain , such that an

improvement in cannot be achieved without worsening , and vice versa.

Figure 1.2 shows a few of the candidate solution sets in the objective variable search space. The

three points, marked by circles, are termed as nondominated or Pareto-optimal solutions. The

solutions marked by triangles have been dominated by at least one of the nondominated

solutions.

Figure 1.2: Dominated and nondominated solutions in objective variable space for the sample

problem, given by Equations 1.2, 1.3 and 1.4.

The three green points represent the trade-off between the two objectives in the subdomain

. The points (4, 0) and (0, 4), in the objective variable search space, represent the

results from the optimisation of individual functions and , respectively. The point (1, 1)

represents one of the infinite possible trade-off solutions, when both the objectives are

simultaneously optimised. The DM can then select either one of these solutions based upon an

appropriate selection criterion. The decision criterion may or may not be known a-priori, at the

start of the run. For example, in the cost optimisation example, the DM may not be interested in

a fixed discount rate a-priori but rather he/she may want to see the solutions for a range of

4

possible discount rates. Having a range of solutions also enables the DM to visualise the results

in both the objective and decision variable space, which in turn aids the DM to select a particular

solution. For the hypothetical, two-objective minimisation Pareto plot shown in Figure 1.3, the

DM would typically like to operate at point A, as even a small improvement in any one of the

objectives comes at a significantly greater worsening of the other. It is difficult to a-priori define

the optimisation problem such that it only gives this particular solution.

Figure 1.3: Pareto plot for a hypothetical, two-objective optimisation problem

The next section briefly discusses the various methods available to solve a MOO problem.

1.3 MOO methods

There are various ways to classify the different MOO methods as explained in detail by

Rangaiah (2009). In this work, the focus is on population based evolutionary methods. However,

to provide the reader a basic idea of the advantages and disadvantages of evolutionary

techniques, they have been discussed here along with another popular class of methods, called

the scalarisation methods (Rangaiah, 2009). In scalarisation methods, the MOO problem is

converted into a single or a series of Single-Objective Optimisation (SOO) problem(s). Given

below is a brief introduction to the scalarisation and evolutionary methods. The other MOO

methods commonly used are the multi-objective simulated annealing algorithm, particle swarm

optimisation, global criterion, value function and goal programing methods. The reader is

referred to Rangaiah (2009) for further reading of MOO and its applications in chemical

engineering.

Scalarisation methods:

As discussed in the previous paragraph, the scalarisation methods involve converting the

MOO problem into one or a series of SOO problem(s). The SOO problem(s) is (are) then

solved using either one of the derivative based SOO methods. Since, this type of methods are

dependent on derivative information, they can't guarantee convergence to global optimum.

5

The two most common scalarisation methods, i.e. the weighted sum and the ԑ-constraint

methods, have been discussed in this text. The reader is referred to Rangaiah (2009) for

further details regarding the other methods that fall under the scalarisation approach.

o The weighted sum method:

The weighted sum method involves the scalarisation of the MOO problem by assigning

‘weights’ to each one of the objective functions and minimizing the, thus formed,

weighted sum of objectives. The weighted sum method can be mathematically defined

as follows:

∑

Where, , with strict inequality valid for at least one of the objective functions

The SOO problem is then solved using a derivative based SOO optimisation method to

get one of the Pareto-optimal solutions, corresponding to the provided set of weights.

The weighted sum method can be graphically understood for a two-objective

minimisation problem by considering the negative of the weight ratios, i.e. ⁄ as

the slope of the line in Figure 1.4. The optimisation problem then reduces to finding the

intercept value; such the line just touches the feasible region at point . Different

solutions on the Pareto plot, represented in Figure 1.4 by the thick and bold boundary of

the feasible region, can hence be obtained by varying the slope ( ) of the line.

Figure 1.4: Graphical representation of the weighted sum method.

The graphical interpretation of the weighted sum method highlights one of its

shortcomings. The weighted sum method fails to find a Pareto-optimal solution lying in

the non-convex region of the Pareto plot, as depicted in Figure 1.5. No matter what the

(1.5)

6

slope of the line (i.e. the values of the weights for and ), the line won’t be tangential

to the Pareto plot.

Figure 1.5: Failure of the weighted sum method to find solutions in the non-convex region of the

Pareto plot

An even spread in the Pareto-optimal solution set is generally desired. In weighted sum

method, it is difficult to decide upon a set of weights a-priori, to ensure an equally spaced

Pareto plot. An equal spread of weights does not necessarily imply an equal spread in the

Pareto-optimal set (Das and Dennis, 1997).

o The ԑ-constraint method:

The ԑ-constraint method involves optimising either one of the objectives while treating

the others as constraints. Mathematically, the ԑ-constraint method can be represented by

Equation 1.6.

for all

Where, and being the upper bound fixed for the objective

The solution to Equation 1.6 is weakly Pareto-optimal but can be Pareto-optimal, if it is

unique.

The ԑ-constraint method can be graphically understood, for a two-objective minimisation

problem as shown in Figure 1.6.

(1.6)

7

Figure 1.6: Graphical representation of the ԑ-constraint method

Figure 1.6 also shows how the ԑ-constraint method is able to find Pareto-optimal

solutions, even in the non-convex region of the Pareto plot, in contrast to the weighted

sum method.

The problem with this method is related to the selection of appropriate values for the

bounds ( ) in order to ensure feasible solutions.

Evolutionary methods

These methods, in contrast to scalarisation methods, work with a population of solutions,

rather than a single solution. Evolutionary MOO methods try to generate an approximation of

the true Pareto plot by trying to mimic the biological evolution process. The evolutionary

algorithms are derivative-free search algorithms. Since they do not use derivative

information, they can be applied to problems that are discontinuous and multi-modal in

nature (Thibault, 2009). Genetic Algorithms (GAs) are an important class of evolutionary

algorithms and typically follow the following basic steps:

An initial population of decision variables is randomly generated based upon the range of

the decision variables specified by the user and the seed value. These sets of decision

variables are often referred to as chromosomes.

Objective function values are then computed for each set of decision variables or

chromosomes.

Each chromosome is then assigned a fitness value (or rank), depending upon the

corresponding objective function values, and/or a diversity parameter value, to ensure a

good spread in the solutions.

Based upon a particular selection algorithm, parents with favourable attributes (better

objective function values) are selected to undergo crossover and mutation in order to

8

produce offsprings which can be included in the population. It is expected that the

favourable attributes are passed on to the offsprings, thereby refining the pool of solution

over a number of generations.

The problem with these methods is the high computational effort associated with evaluating a

large number of candidate solutions before reaching the final Pareto-optimal set. This

problem is aggravated in cases where the objective functions are computationally complex

and hence, take a substantial amount of time to evaluate the objective function values.

The concept of dominance is often employed by methods falling under the evolutionary

approach and is best understood with the help of the following example.

Subject to

Where, ,

and

Consider three sets of and along with the corresponding objective function values as

shown in Table 1.1.

Table 1.1: Candidate solution sets

Solution set

A -2 -2 2 2

B 2 3 1/2 1

C 2 4 2/5 4/3

A comparison between candidate solution sets A and B yields that B dominates A, as the

value of both the objective functions for B is less than the corresponding values for A. A

similar comparison between candidate solution sets A and C shows that C also dominates A.

However, nothing can be inferred by making a comparison between B and C. Though B has a

better value of , but at the same time, it has a worse value for

. The candidate solution sets B and C are thus nondominated with respect to each

other. Based on these pairwise comparisons, a domination score is assigned to each candidate

solution set. A domination score is defined as the number of times a particular solution was

dominated by other solutions in pairwise comparisons. Hence, the domination scores for

solution sets A, B and C are 2, 0 and 0 respectively.

Nondominated Sorting Genetic Algorithm (NSGA)-II, proposed by Deb et al. (2002), is one

of the most widely used GA and hence, has been used as the basic evolutionary algorithm in

this research. Given below is a brief description of the NSGA-II algorithm.

(1.7)

9

o NSGA-II:

The two basic concepts at the heart of NSGA-II are as follows:

1. Fast nondominated sorting approach:

The earlier version of NSGA-II, i.e. NSGA, proposed by Srinivas and Deb (1995) was

criticised for being computationally expensive. A faster nondominated sorting

approach was thus incorporated into NSGA-II.

As explained earlier, an evolutionary algorithm starts by generating an initial

population of individuals or chromosomes. This is followed by the evaluation of the

objective function for these individuals. The next step is to sort these individuals

depending upon these objective function values.

The fast nondominated sorting algorithm proposed in NSGA-II proceeds by calculating

two entities for every individual of the population, namely,

Domination count ( ): Domination count refers to the number of solutions that

dominate the individual p.

: represents the set of solutions that were dominated by p.

The next step is to allot these individuals to different nondominated fronts. All the

individuals having a domination count of zero are assigned to the first nondominated

front. For every individual p, assigned to the first dominated front, the domination

count of every individual in the corresponding set is reduced by one. If, after this,

the domination count of any individual in becomes zero, it is assigned to the second

dominated front. Same procedure is then repeated for each and every individual of the

second nondominated front to generate the third nondominated front and so on.

2. Crowding comparison:

The two basic aims of any evolutionary algorithm are as follows:

Converge as close to the true optimal solution as possible.

Produce an even spread of Pareto points in the final solution.

To achieve the second aim, NSGA-II uses an entity named crowding

distance ). Deb et al. (2002) explained the concept of with the help

of a two-objective minimization, optimisation problem. The front represented by solid

circles, in Figure 1.7, is the first nondominated front. Crowding distance for point i is

the average side length of the rectangle represented by dashed lines.

10

Figure 1.7: Crowding distance ) estimation (Deb et al., 2002)

The basic steps involved in NSGA-II are as follows:

I. In the first step, a random initial population ( is generated, comprising of

(population size) individuals. This is followed by evaluation of objective function

values for these individuals.

II. The initial population is then sorted according to the fast nondominated sorting

algorithm. For unconstrained optimisation problems the concept of domination is

the same as what was explained earlier. The concept of dominance is a bit

different in case of optimisation problems involving constraints. In constrained

optimisation, a feasible solution is always given preference over an infeasible one.

A solution is considered feasible if it satisfies the specified constraint. A solution

is said to dominate a solution , in case any of the following conditions is found

to be true (Deb et al., 2002):-

1. Solution is feasible and solution is not.

2. Solutions and are both infeasible, but solution has a smaller overall

constraint violation.

3. Solutions and j are feasible and solution dominates solution .

An offspring population of size is then generated using binary tournament

selection, crossover and mutation.

III. The objective function value for these offsprings is then evaluated and the

combined parent and off-spring population of size is then sorted based upon

their nondomination score.

IV. The individuals corresponding to the first nondominated front are first selected to

make up the next generation. In case the number of individuals in the first

nondominated front is less than population size , individuals from second

nondominated front are included in the selection. This procedure is repeated until

11

individuals are selected. In case the number of individuals selected becomes

greater than , the individuals in the last included nondominated front are sorted

based on the crowding distance value and the excess individuals are rejected. This

step is pictorially explained in Figure 1.8.

V. The new population thus generated, goes through steps III, IV and V again until

some convergence criterion is met.

Figure 1.8: Generation of geneneration from generation (reprinted from Deb et al.,

2002)

Table 1.2 summarises the merits and demerits of the two broad approaches for solving MOO

problems.

12

Table 1.2: Summary of the two approaches

Approach Merits Demerits

Scalarisation Converts problem into one

or a series of SOO

problems, thereby, making

it possible to use the wide

array of derivative based

optimisation methods.

Since these methods typically rely on derivative

information, the methods falling under this

approach can’t guarantee convergence to global

optimum. However, evolutionary methods can

also be used to solve the resulting SOO

problem(s).

Multiple runs are required to generate a set of

Pareto-optimal solutions.

Specifying the weights and the upper bounds to

ensure equally spaced and feasible Pareto-optimal

solutions, respectively, is not straightforward.

Evolutionary Evolutionary methods,

being derivative-free, can be

applied to problems that are

discontinuous and multi-

modal in nature.

Typically, an evolutionary

MOO algorithm aims to

produce an equally spread

of Pareto-optimal solutions,

in a single run.

The major limitation of evolutionary methods is

the huge computational effort required to

repeatedly evaluate the objective and constraint

functions for each one of the candidate solutions.

Evolutionary methods may also be considered as

a waste of computational time (Rangaiah, 2009).

This is because a DM is usually interested in just

one Pareto-optimal solution. Hence, the extra

computational effort required to generate the

other solutions goes to waste. However, as

mentioned earlier, generating multiple Pareto-

optimal solutions also enables the DM to make an

informed decision regarding the selection of a

particular solution.

In practice, it is often difficult to detect whether a

particular evolutionary run has converged or not.

1.4 Motivation for the present work

As explained in the previous section, evolutionary methods like the NSGA-II algorithm can be

applied to a wider range of MOO problems, involving discontinuous and multi-modal functions.

In the context of MOO, the evolutionary methods are expected to produce a set of equally spaced

Pareto-optimal solutions, in a single run. Generating multiple solutions may be considered

unnecessary, as ultimately the DM is going to implement only one of multiple non-dominated

solutions generated by evolutionary MOO. However, multiple Pareto-optimal points are also

sometimes required to make an informed decision. Evolutionary methods are often criticised for

being extremely time consuming. To a large extent, the solution time for evolutionary MOO

depends on the computational complexity of the objective and constraint functions; the higher

13

the computational complexity, the greater would be the time required to get to the final Pareto

optimal solution set.

A significant amount of computational time could however be saved by replacing the

computationally expensive objective function evaluations with their cheaper approximations,

known as surrogate or response surface models. ‘Surrogate-assisted’ MOO refers to the use of

surrogate models, either partially or completely, during the course of a MOO run with an aim to

speed up the search. Generating a surrogate model ‘accurate enough’ in the entire decision

variable search space is a challenge. There is, thus, a need to continuously evolve or update the

surrogates with evolving generations. This is to ensure that the surrogates are accurate enough in

the evolved decision variable search space and the evolutionary algorithm does not converge to a

false optimum. To update the surrogates, the computationally expensive objective and constraint

functions also need to be selectively solved in order to collect additional input-output data during

the MOO run. The surrogates can then be updated by using these newly generated data in

addition to the existing input-output data.

In practice, it is often difficult to detect whether a GA run has converged or not.

1.5 Goals and scope of the present work

To review existing surrogate-assisted MOO strategies, with application in chemical

engineering process design.

To suggest improvements to the existing strategies to better suit practical chemical

engineering MOO problems.

To demonstrate the surrogate-assisted MOO strategy for a complex flowsheet optimisation

by applying it to a coal to ammonia process with carbon capture.

To interpret the results and suggest any improvements/modifications, if needed.

1.6 Outline

Chapter 2 reports the different surrogate-assisted MOO strategies, with applications in the

chemical engineering domain.

Chapter 3 suggests some modifications to an existing state-of-the-art surrogate-assisted MOO

strategy. The chapter also compares the performance of the surrogate-assisted strategy before

and after the modifications on two mathematical test problems.

Chapter 4 details the chemical engineering case study involving conversion of coal into ammonia

while capturing the CO2 formed during the process.

Chapter 5 and 6 report the surrogate-assisted MOO of the two coal to ammonia flowsheets,

involved CO2 capture by physical absorption in chilled methanol and physical adsorption on

activated carbon, respectively.

14

Chapter 7 includes the conclusions drawn from the research work and suggests future prospects

of further research.

15

Chapter 2

Surrogate-assisted MOO and process design

"Essentially, all models are wrong, but some are useful“ -George Box

One of the most reliable means to measure the relative effect of different factors/decision

variables on the objectives is to carry out a finite number of physical experiments. Physical

experiments are usually both time consuming and economically expensive. Hence, a

mathematical/computer model of a chemical process is often used as a substitute for the physical

experiment. Every mathematical/computer model represents an attempt by the modeller to

mimic the physical phenomenon. The fidelity of a mathematical/computer model represents the

extent to which it is able to mimic the physical phenomenon. There is usually a trade-off

between the accuracy of the model and the corresponding computational effort required to solve

the model. Figure 2.1 shows an example of such a trade-off involved while modelling a reactor

in the commercial process simulation software, Aspen PlusTM

. A Computational Fluid Dynamics

(CFD) simulation is also shown in the figure for comparison.

Figure 2.1: An illustration explaining the model fidelity and computational effort trade-off

The stoichiometric reactor is typically the reactor model having the least fidelity. A reaction

extent is specified for the model, on the basis of which, Aspen PlusTM

calculates the outlet stream

composition. An equilibrium reactor uses thermodynamic data to estimate the equilibrium

compositions, at the specified reaction temperature and pressure. The equilibrium model, thus,

has a higher fidelity, as compared to the stoichiometric reactor. However, the equilibrium model

may fail to represent the experimental observations, in case the reaction is kinetically controlled.

In such situations, a kinetic reactor model (for example, a plug flow reactor), which takes into

account both the relevant kinetics and thermodynamic data would have an even higher fidelity.

The different reactor models, arranged in order of increasing computational complexity, are the

Stoichiometric model ˂ Equilibrium model ˂ Kinetic reactor models. To further improve the

16

model fidelity, detailed CFD simulation for the reactor may also be carried out. However, the

cost of repeatedly evaluating the CFD simulation, for every candidate solution of the

evolutionary algorithm, would make the problem computationally prohibitive. It is under

situations like these that replacing the computationally complex model/simulation with a

relatively simpler but accurate enough substitute (surrogate) becomes a necessity for

evolutionary MOO. The lower fidelity model/models has/have to be accurate enough, in order to

ensure convergence to global optimum. Though surrogate-accuracy is not the sole criterion to

ensure convergence to global optimum, it is nevertheless desirable (Jin, 2011).

Theoretically, a lower fidelity approximate model (surrogate) for any kind of unit operation or

set of unit operations (i.e. a sub-problem) could be constructed using input-output data gathered

from solving the higher fidelity model a finite number of times.

Relying on a single set of surrogates, throughout the MOO run, is not recommended (Jin, 2011).

There is a risk that the surrogate-assisted approach may converge to a false optimum, due to the

introduction of false optimum by the surrogate models. Figure 2.2 illustrates one such possible

scenario, using a SOO minimisation problem. If the same surrogate model is used throughout the

optimisation run, it will converge to the false optimum shown in Figure 2.2. The problem with

this particular approach is that it lacks any feedback mechanism. For example, in this case, the

optimum solution could be re-evaluated with the help of original model, in order to check

whether the solution predicted by the surrogate-assisted approach is actually the optimum or not.

This newly gathered input-output datum can then be added to the initial input-output data, to fit a

new surrogate model, to ensure that the surrogate is better able to mimic the actual/original

function in the probable region of the search space. The updating of the surrogates is further

explained in Figure 2.3.

Figure 2.2: Example of false minimum with the surrogate model (Jin, 2005)

Figure 2.3: Updating the surrogate by re-evaluating the optimum

Original model

Surrogate model

Original model

Updated surrogate model New input-output

datum

False

optimum

17

Building globally accurate surrogates at the start of the evolutionary MOO is often difficult. As

explained in the next section, the surrogates could rather be updated periodically, after a set of

surrogate-assisted generations, in an evolutionary MOO run. The periodic updating, or

retraining, is extremely important in cases where the initial search space may significantly differ

from the final solution space. To update the surrogates, the computationally expensive objective

functions also need to be solved, along with the surrogates, in order to collect additional input-

output data during the MOO run. The surrogates can then be updated by using these newly

generated data in addition to the existing input-output data.

2.1 Model management or evolution control

An ‘ideal’ surrogate-assisted MOO strategy would converge to the global optimum while using

the original, computationally expensive original model the minimum number of times. This

necessitates the need for a ‘model management’ or ‘evolution control’ strategy, by means of

which the algorithm can decide whether to use the surrogate, or the original model for a

particular candidate solution evaluation. The aim of Evolution Control (EC) is to prevent

convergence to a false optimum while only selectively evaluating the original model. Jin (2005)

classified EC as either fixed EC or adaptive EC, as depicted in Figure 2.4.

Figure 2.4: EC classification, as per Jin (2005)

The basic EC approaches are as follows:

1. Fixed EC: Fixed EC involves a fixed frequency of original model evaluations. Fixed EC

can be implemented at the individual level (individual based fixed EC) and/or at the

generation level (generation based fixed EC).

Adaptive EC- The decision to carry out an

original evaluation at a particular point is

based on some or the other criterion.

EC

Fixed EC- The frequency

of original model

evaluations is fixed

Generation Based

Fixed EC

Individual Based

Fixed EC Generation Based

Adaptive EC

Individual Based

Adaptive EC

18

Individual based fixed EC: Individual based fixed EC implies that in each generation,

some fixed number of the candidate solutions are solved using the original function,

while others are evaluated with surrogate models.

One of the examples include; re-evaluating some fixed number of best solutions as

predicted by the surrogates, with the original model, before selecting candidate solutions

for the next pool of parents (Jin, 2011).

Another closely related model management strategy, which is not classified as an

individual based fixed EC, is the ‘pre-selection’ strategy proposed by Emmerich et al.

(2002). The pre-selection strategy is different from the ‘best’ solution strategy, explained

in the previous paragraph. While there is a chance of candidate solutions evaluated with

surrogate-models to be selected as a parent for the next generation in the best solution

strategy, the pre-selection strategy pre-screens all the candidate solutions before they are

selected to be included in the parent pool of solutions for the next generation.

Generation based fixed EC: Fixed EC can also be generation-based, such that the

original model is used to re-evaluating all the solutions obtained by using the surrogate

models. It may or may not be succeeded by surrogate re-training/updating.

One of the problems with fixed EC is that, the frequency of original model evaluations is

fixed, irrespective of the accuracy of the surrogates.

2. Adaptive EC: Adaptive EC involves a variable frequency which depends, either directly

or indirectly, upon the accuracy of the surrogates. Adaptive EC has also been

implemented at both individual and generational level.

Individual based adaptive EC: The decision regarding whether to use the surrogate or

the original function should also be dependent on the ability of the surrogate to give

accurate enough prediction for points ‘similar’ to the candidate solution under

consideration. The similarity can be assessed, for example, by calculating the Euclidian

distance of the candidate solution, in decision variable space, from the data points that

were used to fit/train the particular surrogate.

A kriging model, in contrast to other deterministic response surface methods, gives a

probability distribution as an output. The distribution has a mean value which is the

estimate for the dependent variable. A kriging model also provides a normally distributed

prediction error estimate. The prediction error is dependent on the distance of the

candidate solution to the nearest data point used to fit/train the surrogate. A data point in

close vicinity implies greater confidence in prediction, implying lower prediction errors,

and vice versa. The basic concept here too, is to gauge the prediction/extrapolative ability

of the surrogates beyond the training data points.

Generation based adaptive EC: Nair and Keane (1998) used an adaptive, generation

based EC strategy where they borrowed the ideas of expanding or contracting the trust

region (Dennis and Torczon, 1997). In their approach, Nair and Keane (1998) either

increased, or decreased the frequency of updating the surrogate models, based upon the

19

error in the prediction value at some of the best solutions. For this purpose the best

solutions have to be re-evaluated with the original model.

Jin (2011) also reports certain population based model management approaches. Population

based approaches involve multiple sub-population co-evolving, while each using its unique set of

surrogate models.

The next section details the two broad surrogate-assisted MOO approaches available in process

design and optimisation literature.

2.2 Surrogate-assisted MOO and process design

As far as chemical engineering design and optimisation literature is concerned, there appear to be

two broad approaches to surrogate-assisted MOO. Though this work is focussed on surrogate-

assisted evolutionary MOO, some of the relevant works dealing with surrogate-assisted

derivative–based optimisation have also been included here.

1. ‘One-shot’ approximation approach (Ray et al., 2009) :

The ‘one-shot’ approach involves fitting a unique set of surrogates at the start of the MOO

run and using the same set of surrogates throughout the MOO run. As illustrated in the

introductory section of this chapter, this approach is likely to converge to a false optimum, if

the surrogates are not accurate enough. Under the one-shot approach, there are two

sub-approaches, depending upon whether the surrogates are fitted in a single go, or by

adaptively sampling the training data. Given below is a brief description of the recent

research works following the one-shot approach:

Sub-approach I:

Eslick and Miller (2011) replaced the power plant and the cooling tower models by

surrogates in their study on MOO of a pulverised coal power plant, retrofitted with a CO2

capture and compression unit. Eslick and Miller (2011) used Latin Hypercube Design

(LHD), a type of space-filling approach, to sample the search space at the start of the

NSGA-II based MOO run.

Khatir et al. (2013) replaced CFD based complex models with surrogates, fitted at the

start of a GA enabled optimisation run. Khatir et al. (2013) first converted the MOO

problem into a SOO by combining all the objectives after applying the weighing method,

already discussed in Chapter 1.

Liu and Sun (2013) have used Support Vector Machines to approximate the complex

Pressure Swing Adsorption (PSA) process model involving the solution of a number of

Partial Differential Algebraic Equations (PDAEs). Liu and Sun (2013) used these

surrogates to simultaneously maximise O2 recovery and purity from a PSA unit

separating air into N2 and O2.

20

Tock and Marechal (2014) looked at the possibility of predicting the optimum for the

global optimisation of Natural Gas Combined Cycle (NGCC) with carbon capture

process while proposing the hypothesis that the optimum of a sub-section of the plant

coincides with the optimum of the entire plant. The sub-section chosen was the mono-

ethanolamine based CO2 capture unit. Tock and Marechal (2014) fitted multiple types of

surrogate models to predict the trade-off for the sub-problem as a function of a reduced

number of independent variables. The primary benefit of such an approach is that by

predicting the optimum of the sub-problem directly, the global optimisation problem need

not be explicitly optimised with respect to the decision variables inherent to the sub-

problem which were originally, eight in number. However, generating the training data

required several NSGA-II based MOO runs using the computationally expensive Aspen

PlusTM

based CO2 capture model.

Lambert et al. (2015) used Artificial Neural Network (ANN) models to replace a

potassium carbonate solvent based CO2 capture model in Aspen PlusTM

.

Sub-approach II:

Some of the other works have focussed on increasing the accuracy of the surrogates by

following adaptive sampling strategies.

In the work of Fahmi and Cremaschi (2012), the initial training set was generated

randomly from a uniform distribution. Different number of hidden layers and number of

neurons were tried to select the best network architectures for the fitted ANN models.

Nuchitprasittichai and Cremaschi (2013) tried to improve the global accuracy of the

surrogates by using the ‘incremental’ Latin Hypercube Sampling (LHS) (or iLHS). The

iLHS strategy has an initial sample size which is ten times the number of decision

variables, generated via LHS. This is then followed by evaluating the samples and

training the surrogates. The extrapolative or generalisation ability of the surrogate is then

gauged by estimating the cross validation errors. If the surrogates are found to be valid,

they are used with derivative based SOO method. However, if the surrogates are found to

be invalid, a new set of surrogates is trained by generating a new Latin Hypercube

Design (LHD) by increasing the sample size by 33%. The new surrogates are again tested

for their generalisation ability, followed by increasing the sample size even further, if

required. The problem with the iLHS strategy is that every time the sample size is

increased, the previously evaluated sample points need to be discarded, as new input sets

are generated by the LHS strategy. All the previously evaluated points thus become

redundant In the early stages of the search, only a general trend of the objective

functions, as a function of independent variables, is desired. The prediction accuracy of

the surrogates can then be selectively improved in the promising regions of the search

space, to ensure an efficient utilisation of the limited computational budget available.

21

Eason and Cremaschi (2014) proposed two sampling strategies to select the initial

training data. The first sampling strategy starts with generating an initial sample set with

an appropriate space-filling technique like LHS. This is then followed by dividing these

data-points into ‘K´ sub-sets and fitting an equal number of surrogate models. A set of

newly proposed sample points are then randomly generated within the search space to

test the prediction uncertainty of the surrogates. The sample point giving the maximum

prediction uncertainty is then added to the existing sample points used to fit the

surrogates, in an attempt to improve the prediction quality of the surrogates. The second

sampling strategy, proposed by Eason and Cremaschi (2014), focuses on considering the

weighted average of the normalised Euclidian distance of the candidate solution from

existing sample points, along with the normalised prediction uncertainty at that point.

Eason and Cremaschi (2014) tested the proposed sampling techniques by testing it

against mathematical test problems, along with a CO2 absorption case study.

These two strategies, as well as the iLHS strategy, are focussed on selectively exploring

the search space, in order to improve the generalisation ability of the surrogates. The

authors however conclude that if the surrogates are to be used for optimisation, the

sampling strategy should favour the ‘best’ performance regions. This can be done by

carrying out optimisation and sample selection in tandem. This kind of strategy which

favours promising regions in sample selection is exploitative in nature in contrast to the

explorative nature of selection strategies proposed by Eason and Cremaschi (2014).

Mogilicharla et al. (2015) used the Expected Improvement (EI) criterion to sequentially,

and selectively sample the search space. The criterion of EI can however be applied just

to those kind of surrogate models which provide a measure of uncertainty in their

prediction. Only two types of models are able to satisfy the above criterion, viz. kriging

and polynomial response surfaces. The EI criterion aims to strike a balance between

exploitation and exploration strategies. The EI criterion for kriging models is defined as

per Equation 2.1.

( ) (

) (

)

Where,

The present best value from optimisation

: The model prediction at

: Cumulative density function of standard normal distribution

: Probability density function of standard normal distribution

: Prediction standard deviation

Kriging models are interpolating in nature; that is to say that they necessarily have to pass

through the sample points. The basic assumption at the core of kriging model

construction is that the value of the function at any point is correlated to the values at

(2.1)

22

neighbouring sample points, based upon their separation in different directions. The

correlation is assumed to be strong with nearby points and weak with far-away points. In

Equation 2.1 the first term on the right hand side contributes to the exploitative nature of

EI by considering the difference between the present best value ( ) and the

expected/predicted value ( ) provided by the kriging model. The second term in

Equation 2.1 promotes exploration of the search space in case of a high value of

prediction variance at the prospective sample point. The prediction variance of a kriging

model is dependent upon the distance of the prospective sample point to already sampled

data points; greater the distance, higher the uncertainty in prediction, and vice versa. The

problem with EI based sampling approach is that its application is limited to surrogate

models which provide some measure of uncertainty in their prediction, such as kriging

models. In principle, however, some other distance based metric can also be used for

other types of surrogate models. Additionally, the sampling algorithm and optimisation

can be carried out in tandem, as the main aim of constructing the surrogates is global

optimisation and not to construct a globally accurate surrogate.

A summary of the works falling under this approach is given in Table 2.1.

23

Table 2.1: Summary of works falling under the one-shot approach O

ne-

shot

or

No E

C a

pp

roach

Salient attributes Literature Remarks

Sub-approach I:

Surrogate training in a single go.

Surrogates once fitted are not altered.

Initial sample points can be selected randomly or via

sophisticated space filling techniques (like LHD or full factorial

design).

Initial sample size decided on the basis of heuristics.

Eslick and Miller (2011); ;

Khatir et al. (2013), Lambert

et al. (2015)

Such approaches are likely to

converge to a false optimum.

It’s difficult to fit a globally

accurate surrogate model in a

single go.

Such an approach lacks any

feedback mechanism.

Sub-approach II:

Globally accurate surrogate models.

Initial sampling done via space filling techniques.

Typically, the sample size is increased by adding more input-

output data points.

Selection of new data points can either be done in any of the

following ways:

Exploration of the search space by either choosing a higher

number of data points generated by any of the space filling

techniques. The prediction uncertainty can also be used as a

decision criterion to promote exploration of the search space.

Exploiting the information about the promising regions of the

search space to sample points which are expected to give the

greatest improvement in the objective values.

A combination of exploration and exploitation strategies.

The sample size can be

increased, on the basis of the

prediction errors to produce a

new space filling design as in

Nuchitprasittichai and

Cremaschi (2013).

New sample points can also

be selected based upon the

prediction uncertainty of the

surrogate estimates, as in

Eason and Cremaschi (2014).

Mogilicharla et al. (2015)

used the EI to strike a balance

between exploitation and

exploration strategies.

This approach is most suited

for cases where a globally

accurate model is to be

trained.

However, if the surrogates

are to be used for

optimisation, the model

fidelity should be selectively

improved to limit the

original, computationally

complex model evaluations.

Miscellaneous Tock and Marechal (2014). Restricted applicability.

24

2. Approaches involving some form of EC or periodic update of the surrogates:

Caballero and Grossmann (2008) along with Henao and Maravelias (2011) had a ‘domain

contraction’ step in their works dealing with surrogate-assisted optimisation. In the

domain contraction approach, the domain of the search space is updated based upon the

results from the optimiser. This is to increase the local accuracy of the surrogates which

in turn increases the probability of converging to the global optimum (Henao and

Maravelias, 2011).

The Surrogate Assisted Evolutionary Algorithm (SAEA), proposed by Ray et al. (2009),

makes the use of a combination of fixed generation based EC and adaptive EC. In

SAEA, all the input-output data gathered from the original model evaluations is stored in

an ‘External Archive’ (EA). The EA is divided into two subsets, namely, training and test

subset. The surrogate models are fitted and/or validated with the training data. The

extrapolation ability of the fitted surrogate models is checked based on their predictions

with test data. If the prediction error for a surrogate model is less than a user provided

threshold limit; the surrogate is termed to be ‘valid’ and can be used instead of the

original model. However, if the surrogate model is found to be ‘invalid’; the original

model is used to evaluate the objective functions. SAEA includes fixed generation based

EC and adaptive individual based EC. The original model is used to re-evaluate the

results obtained after a fixed number of surrogate-assisted generations. The uncertainty in

the surrogate model prediction is related to the distance of the candidate solution from the

sample/training set. SAEA uses the minimum normalised Euclidean distance of the

candidate solution from any of the training data points. A maximum threshold limit for

this distance is specified, reflecting the user confidence in the surrogate predictions. Ray

et al. (2009) applied the SAEA algorithm to an alkylation process optimisation problem.

Isaacs et al. (2009) proposed another set of surrogate-assisted MOO algorithms, namely,

Multiple Spatially Distributed Surrogates (MSDS) and Multiple Adaptive Spatially

Distributed Surrogates (MASDS). The primary difference between SAEA and these two

algorithms is that while SAEA considers just one surrogate model to predict an

output/constraint over the entire search space; MSDS and MASDS use multiple, spatially

distributed surrogate models to predict a dependent variable. To train the surrogate

models, the search space is partitioned into clusters. As a result, multiple, locally

accurate surrogates can be trained. In MSDS, contrary to MASDS, the number of

partitions of search space is decided a-priori. In MASDS different number of surrogates

are fitted for every output/constraint, followed by selecting the particular set of surrogates

corresponding to the maximum ‘valid partition ratio’ ( ), as defined by Equation 2.2.

(2.2)

25

Where,

Number of valid surrogates for K partitions

: Valid partition ratio

Number of partitions of the search space

Isaacs et al. (2009) found MASDS outperforming both MSDS and SAEA while

achieving a better solution set in fewer or similar number of original model evaluations.

Husain and Kim (2010) proposed a surrogate-assisted evolutionary MOO procedure to

optimise a microchannel heat sink. In this approach, initial sample points are selected on

the basis of a three-level full factorial design. The first set of surrogates is then fitted

based on the initial sample set. The surrogates are subsequently used for GA based MOO.

A fixed number of predicted nondominated points are then re-evaluated and added to the

existing sample points. A new set of surrogates is then fitted based on the updated

training set, to be used again for GA based MOO. This sequential re-evaluation, surrogate

update and GA based MOO are performed until a termination criterion is met. Husain

and Kim (2010) note that such a selective enhancement in the surrogate model’s fidelity

requires a lower expense and computation time, as compared to strategies which focus on

improving the global fidelity of the surrogate models.

In Mitra and Majumder (2011), the original, computationally complex model and

surrogate model are alternately used for functional evaluations. Mitra and

Majumder (2011) also note that the time required to train the surrogate models may

become so great, especially in the later part of the run, that the computation time saved by

using the surrogates would be wasted in surrogate training. Thus, the surrogate model is

only updated if the existing surrogate model is found not to be accurate enough. In

SAEA, MSDS and MASDS there is a limit to the maximum number of training samples,

to limit the surrogate training time.

Mitra (2013) also follows a similar approach as that followed by Mitra and

Majumder (2011), apart from using the original model for functional evaluations, after

every two successive surrogate assisted generations. Meanwhile, the prediction accuracy

of the surrogates is also checked after every surrogate assisted generation, to decide

whether to use the original or the surrogate model for functional evaluation.

The EI based approach was first applied to MOO by Knowles (2005), who named this

approach Pareto Efficient Global Optimisation (ParEGO) after the EGO procedure for

SOO proposed by Jones et al. (1998). Beck et al. (2015) used the design expression

shown in Equation 2.3 for NSGA-II based optimisation. The expression switches

between optimising the kriging predictions ( (i.e. exploitation) and the prediction

variances ( (i.e. exploration) from one iteration to another.

26

( )

( )

Where, is the surrogate-based design criteria and represents the iteration number

Beck et al. (2015) used this strategy to simultaneously optimise the purity and recovery

of CO2 from a post-combustion, Vacuum Pressure Swing Adsorption (VPSA) process.

The applicability of such an approach is limited to kriging models as it requires the

estimate of uncertainty in prediction. However, as noted earlier, the normalised Euclidean

distance of the point from already sampled points can also be used to qualitatively

estimate the uncertainty in prediction.

A summary of the works falling under this approach is given in Table 2.2.

(2.3)

27

Table 2.2: Summary of works involving periodic updating of surrogates A

pp

roach

es u

sin

g E

C o

r p

erio

dic

up

dati

ng o

f su

rrogate

s

Salient attributes Literature Remarks

The original, computationally

complex model is used along with

the surrogate model during the

optimisation run to collect

additional input-output data points.

These additional data points are

used to regularly update the

surrogate models to selectively

improve the fidelity of the

surrogates

Typically, such approaches also

rely on the concepts of exploration

and exploitation of the search

space

SAEA, proposed by Ray et al. (2009), uses a

combination of fixed generation based EC and

adaptive individual based EC.

MSDS and MASDS, proposed by Isaacs et al.

(2009), involve training multiple surrogates for

each objective and constraint.

The approach adopted by Husain and Kim (2010)

involved a fixed number of nondominated solutions

obtained from using the GA with surrogate models,

being re-evaluated in order to update the surrogates.

Mitra and Majumder (2011) used the original and

the surrogate model for every alternate generation,

with an aim not to rely on surrogate models for too

long.

The work by Mitra (2013) involved the use of

original model after two successive surrogate-

assisted generations. To limit the time required for

surrogate fitting, the prediction error of the

surrogates was also checked. The surrogates were

updated only after being found sufficiently

inaccurate.

Beck et al. (2015) adapted the EI based EGO

approach to MOO in their work on VPSA process

optimisation.

By selective improvement in the

fidelity of the surrogate models, such

approaches aim to avoid the extra

computational effort required to

produce globally accurate

surrogates.

The domain contraction or periodic

evolution of the search space can

also be implemented to increase the

local accuracy of the surrogates.

Prediction uncertainties, in case of

models other than kriging, can be

related to the separation between the

candidate solution and the

sample/training set.

Multiple, locally accurate, surrogates

can be trained in different sub-

regions of the search space by

clustering the available data set.

28

2.3 Insights from the review and surrogate-assisted strategy selection

In the present case, since the end use of surrogates is in process optimisation, the optimisation

and selective sampling should be carried out in tandem. The surrogates need to be periodically

updated to have better predictions in the promising sub-set of the search space. In the present

work, the author did not want to be restricted to a particular type of surrogate. This is the reason

why the EI based techniques have not been investigated in this work. However, this approach

seems to be effective, as evident from the body of works discussed in the previous section. In

principle, both MASDS and EI based approaches aim to selectively exploit and explore the

search space.

This work suggests periodic evolution of the decision variable search space to be included into

the MASDS framework. This is to increase the local accuracy of the surrogates which in turn

increases the probability of converging to the global optimum. A similar domain contraction

steps have been applied by Caballero and Grossmann (2008) and Henao and Maravelias (2011).

The modified MASDS algorithm is henceforth referred to as modified-MASDS or mMASDS.

29

Chapter 3

The mMASDS algorithm and mathematical test problems

3.1 The mMASDS algorithm

The broad framework of the mMASDS algorithm is reported in Figure 3.1. The modifications

made to the MASDS algorithm are shown within the dashed rectangle.

Start

Total number

of generations = nmax

Stop

Start the GA run with original model for the

first few generations



Train the surrogate models for objectives and

constraints and test their generalisation

abilities against test data.




If any of the surrogates is found to be invalid,

multiple surrogates are fitted for that particular

dependent variable


multiple surrogates are fitted for that particular

dependent variable

Continue the GA run with surrogate models

for a fixed number of generations

Continue the GA run with surrogate models

for a fixed number of generations

Re-evaluate the intermediate results of GA

using original models

Re-evaluate the intermediate results of GA

using original models

Run the Genetic Algorithm using original models for

a single generation but with a high mutation rate in

the entire decision variable search space

Run the Genetic Algorithm using original models for

a single generation but with a high mutation rate in

the entire decision variable search space

Evolve/Update the search space Evolve/Update the search space

Yes

No

Figure 3.1: The mMASDS algorithm. Dashed rectangle highlights the modifications made to the

MASDS algorithm.

30

The pseudo-code of the mMASDS algorithm is reported below.

Inputs (need to be specified a-priori):

: The initial number of generations, for which the computationally expensive. original model

is used for candidate solution evaluation

: The number of surrogate-assisted intermediate generations

: The maximum number of generations for the evolutionary MOO run

: The fraction of points in the EA, used to train and validate the surrogates

: ‘Default’ mutation rate for GA

: ‘High’ mutation rate for GA

: Crossover probability for GA

: Population size

: Limiting Normalised Mean Square Error (NMSE) on test data

: The seed value supplied to the random number generator

: Set comprising of the minimum bounds for all decision variables, at the start of the

search

: Set comprising of the maximum bounds for all decision variables, at the start of the

search

: Set comprising of the minimum bounds for all decision variables, at the start of


: Set comprising of the maximum bounds for all decision variables, at the start of


: Set comprising of the minimum bounds for all decision variables, after generation

worth of evolution

: Set comprising of the maximum bounds for all decision variables, after


: Number of objective functions

: Number of constraints

1: {}

2: 3: 4: Add to

5: 6:

7: 8: Add to

9: 10:

11:

12: 13: Add to

14: 15:

31

16: 17: 18: 19: 20:

21: )

22: )

23:

24: 25: 26: 27: 28: 29:

30: )

31: ( )

32: 33: 34: 35: 36:

37: 38: Add and to

39: 40: 41:

42:

43:

44:

: The initial population is created using a random number generator from the seed

value provided by the user. The initial population can also be generated using sophisticated

sampling techniques like LHS.

: The objective and constraint values are evaluated using the computationally

expensive, original models.

: The candidate/prospective solutions are ranked based upon the objective function values

and the constraint-handling approach, proposed in Deb et al. (2002).

: This step involves a combination of binary tournament selection, crossover and mutation

(with ‘default’ mutation probability), in order to produce off-springs from the parent

generation.

32

: This step introduces elitism and contributes towards improving the

convergence properties of NSGA-II (Deb et al. 2002). The combined parent and off-spring

sets are then reduced, by selecting the best solutions, to form the parent set for the next

iteration.

: Only a fraction, α, of the data in is used for training and validation

of the surrogates. The remaining data is used to test the generalization/extrapolative abilities of

the surrogates. function involves the partitioning of into and

, while involves further partitioning of in case a single

surrogate model is found to be inaccurate in describing the function behaviour throughout the

search space. The k-means clustering algorithm is used for partitioning purposes.

: This step involves training and validation of the surrogates. The surrogates are fitted only

in the promising subset of the initial decision variable search space, obtained from periodic

evolution of search space.

: This step involves testing the generalization ability of the surrogates by using them to

predict the objective and constraint values corresponding to the ‘un-seen’, test data and

comparing them against the values from the original model evaluation. A surrogate is termed to

be valid, if the error it gives for test data is below a particular threshold value, i.e. , specified

by the user.

: Initially only one surrogate model is fitted for each objective function and constraint. If,

however, any one of them is found to be invalid, multiple surrogates are fitted to better represent

the original model behaviour. The number of surrogates to be fitted is decided adaptively. The

valid partition ratio (Isaacs et al., 2009), , is calculated for each value of . If for a particular

value of , all the surrogates come out to be valid, further partitioning, for higher values of , is

not done. This is done in order to limit the computational effort required in fitting the additional

surrogates. If, however, there does not exist a value of , for which all the surrogates are valid;

the value of which has the highest is chosen, along with the corresponding surrogate

models for objectives and constraints. The algorithm for the function is as follows:

Input:

: Valid partition ratio

1:

2:

3: [ ]

4:

5:

33

6:

7: 8:

9: 10:

11:

12:

13:

14: 15: 16:

: The objective and constraints values are evaluated using the surrogate models, given

that the candidate solution is ‘similar-enough’ to the data used in order to fit the surrogates. This

‘similarity’ is measured by calculating the normalized Euclidean distance between the candidate

solution and all the points used in order to fit the surrogates. If the minimum distance, among all

of these distances, is less than a particular threshold value, the surrogate is expected to have

satisfactory extrapolative ability to predict the outputs for the candidate solution. However, if the

minimum distance is greater than the threshold value, the computationally expensive original

models is used to predict the outputs for the candidate solution.

: This step involves a combination of binary tournament selection, crossover and mutation

(with ‘high’ mutation probability), in order to produce off-springs from the parent generation.

A high mutation probability is selected to give the GA a chance to reconsider the discarded

regions for the search space by searching in whole of the initial decision variable search space.

: This step uses the recent set of nondominated solutions in order to evolve/update the

search space.

3.2 Differences between MASDS and mMASDS

In contrast to the MASDS algorithm, the mMASDS algorithm involves the decision variable

search space (or the domain) also evolving during the course of the evolutionary search. This is

done by adapting the search space every time the surrogate models are re-trained, after the initial

training. The search space is either ‘truncated’ or ‘expanded’ based on the most recent generation

results. The periodic evolution enables the user to discard the ‘un-promising’ datasets in the EA

and prevent them from contributing to the surrogate models fitting. It is expected that the

accuracy of the surrogate models thus fitted, would be better in the promising regions. Every

surrogate-assisted set of generations is followed by re-evaluating all the solutions by the original

model. This is then followed by high mutation evolution for one generation using the original

model. This is done to give the GA an opportunity to relook in the discarded region, so as to

ensure that any promising region does not get erroneously discarded. It must however be noted

34

that prediction accuracy is not the only criterion that ensures convergence to true optima, but

nevertheless, it is desirable (Jin, 2011).

In Section 3.4, the algorithm is demonstrated with two mathematical test problems in order to

identify the advantages, if any, it offers in comparison to MASDS.

3.3 Performance metric

In this work, the ‘S’-metric (Zitzler and Thiele, 1999), or the hypervolume metric, has been used

to quantitatively compare two sets of nondominated, Pareto-optimal solution sets. The S-metric

is defined as the hypervolume spanned by the Pareto plot and some reference point. The

reference point is chosen such that all the points on the Pareto fronts dominate it. Figure 3.2

gives an illustration of the S-metric. Variables ϕ1 and ϕ2 represent the two objectives that are

being minimised. For a two-dimensional Pareto plot, S-metric is simply the area enclosed by the

nondominated front and the reference point, depicted in Figure 3.2 by the green point. The S-

metric for the case shown in Figure 3.2 is calculated as shown in Equation 3.1. Each

nondominated point makes a rectangular boundary, and dominated points such as shown in

Figure 3.2 which lie inside the boundary are not included. The higher the S-metric value, the

better the solution set is.

∑

Figure 3.2: An illustration of the S-metric (represented by the shaded area).

(3.1)

35

3.4 Mathematical test problems

In this chapter, the MASDS and mMASDS algorithms are applied to two chosen mathematical

test problems. These test problems have been chosen, from a suite of problems typically used by

researchers in the field of evolutionary MOO. The two test problems, i.e. the SCH (convex

Pareto) and ZDT2 (concave Pareto) have been specifically chosen because they typify the case

where the solution space is a very small subset of the initial search space. This is often the case

with many practical chemical engineering design problems where not much is known about the

optimal solutions a-priori and hence, the final solution space may be significantly different from

the initial domain of the problem.

3.4.1 Problem definition

The two test problems considered in this work are as follows:

1. SCH problem:

The SCH problem is a two-objective, single variable, minimisation problem. The Pareto

front is convex.

Objectives:

(3.2a)

(3.2b)

The final solution is a small subset of the search space, i.e. .

2. ZDT2 problem:

The ZDT2 problem is a two-objective, ten-variable, minimisation problem. The Pareto

front is concave.

Objectives:

(3.3a)

; Where, ⁄ and ∑

(3.3b)

The final solution is a small subset of the search space, i.e. and

Table 3.1 reports the different parameter values chosen for the MOO runs.

36

Table 3.1: Surrogate-assisted MOO parameters

Parameter Parameter description Problem

SCH ZDT2

init The initial number of generations using original model 1 1

Number of intermediate surrogate assisted generations 10 3

Maximum number of generations 50 12

Fraction of EA used for training 0.9 0.9

Maximum number of partitions allowed for training data 5 5

‘Default’ mutation rate for GA 0.01 0.005

‘High’ mutation rate for GA 0.02 0.01

Crossover probability for GA 0.85 0.85

Population size 100 100

Limiting NMSE on test data 0.5 0.5

The periodic evolution of the search space results in periodic localisation of the search in the

promising region. As a result, it is expected that the surrogate models would exhibit better

accuracy in this localised (promising) region of the search space. Search localisation coupled

with better accuracy of the surrogates in the promising region is expected to speed up the search.

To test this hypothesis, the mMASDS algorithm has been applied to the two test problems. The

results have been compared against the results obtained with MASDS algorithm for the same

number of original model evaluations. In order to single out the effect of periodic evolution of

search space, all the parameters, listed in Table 3 have been kept the same for the two runs, to

facilitate comparison.

For these mathematical test problems, adaptive EC has not been considered as it has been

assumed that all the surrogates are accurate enough and have good generalisation abilities in the

search space. However, if this is not the case; poor accuracy of surrogates could result in a

relatively poor Pareto-optimal solution set. ANN surrogate models are fitted for the test

problems. The distance threshold ( ), for both the cases, was kept as 0.05 times the length of the

solid diagonal length.

3.4.2 Results

SCH results

Figure 3.3 shows the nondominated solutions obtained for the SCH test problem, in the two

surrogate-assisted MOO runs, within a fixed budget of 600 original model evaluations. The

reference point chosen for the S value calculation was (3.97, 4.08).

37

Figure 3.3: Nondominated points obtained from the two runs for SCH problem after a fixed

budget of 600 original model evaluations

The periodic evolution of search space seems to have accelerated the search, as evident from the

S values. Figures 3.4 (a) and (b) show the parity plots for the final set of surrogate models

obtained from the two runs. The plotted points are the nondominated points obtained after the

final set of surrogate-assisted generations.

As evident from Figures 3.4 (a) and (b) the final set of surrogate models exhibit better accuracy,

in the promising subset of the initial decision variable space, for the mMASDS run. It is

interesting to note here that though the final set of surrogate models obtained from the MASDS

algorithm were inaccurate, it converged to the same optimum, approximately. This is due to the

error in the surrogates being systematic in nature.

Figure 3.4: Parity plots of final set of surrogate models for (a) and (b) (SCH problem)

38

Figure 3.5 reports how the search space (in terms of range of the decision variable x) was

evolved during the surrogate-assisted MOO run. As evident from Figure 3.5, the GA was able to

significantly hone in on the solution, i.e. .

Figure 3.5: Evolution of the search space during surrogate-assisted MOO of SCH problem

ZDT2 results

Figure 3.6 shows the nondominated solutions obtained for the ZDT2 test problem, in the two

surrogate-assisted MOO runs, within a fixed budget of 1,000 original model evaluations. The

reference point chosen for S value calculation was (1.00, 1.18).

Figure 3.6: Nondominated points obtained from the two runs for ZDT2 problem after a fixed

budget of 1000 original model evaluations

The mMASDS appears to have outperformed MASDS for ZDT2 problem as well, as is evident

from the respective S values. However, the surrogate model accuracy did not differ much in the

promising subset of the search space across the two optimisation runs as evident in Figures 3.7(a)

39

and (b). Figure 3.8 reports how the search space was evolved during the surrogate-assisted MOO

run.

Figure 3.7: Parity plots of final set of surrogate models for (a) and (b) (ZDT2 problem)

40

Figure 3.8: Evolution of the search space during surrogate-assisted MOO of ZDT2 problem for

variables (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , (i) and (j)

In case of variable , the search range was erroneously truncated at the end of 600 original

evaluations. However, the range was later expanded to include the wrongfully discarded region.

The high mutation generation provides the GA an opportunity to include any region of the search

space that may have been erroneously discarded during periodic evolution. Figure 3.9 shows

how the ‘normalised’ S value, for the two runs, varies as a function of the number of original

model evaluations. The normalization is with respect to the S value obtained for a manually

generated Pareto front. The Pareto front is generated by considering 100 equally spaced values

for , between 0 and 1 (the final solution of ZDT2 problem) and calculating the

corresponding values for . Same reference point (i.e. (1.00, 1.18) was chosen to calculate the S

value. As shown in Figures 3.8 (a) and 3.9, the faulty truncation of the range for appears to

have had an impact on the corresponding S value. However, as the search progresses, the

benefits of periodic evolution of search space are reflected in higher S values as compared to the

MASDS case.

41

Figure 3.9: S values as a function of number of original model evaluation for the ZDT2 run

3.5 Conclusions

For both of the mathematical test problems investigated, periodic evolution of search space as

proposed in mMASDS resulted in an increase in rate of convergence. For SCH problem, periodic

evolution of search space resulted in locally accurate surrogates. However, for ZDT2 problem,

the accuracy of the final set of surrogate models was similar, irrespective of search space

evolution.

It can hence be concluded, that periodic evolution of search space does exhibit some valuable

advantages, for the type of problems considered in this section where the initial search space may

be significantly different to the final solution space. The advantages include faster overall

convergence and potential for fitting locally accurate surrogate models, in the context of

surrogate-assisted MOO. As seen in the case of ZDT2 problem, there may be situations in which

periodic evolution may have some negative effects. It is hence suggested not to use periodic

evolution for the first few surrogate-assisted set of generations. This reduces the probability of

the algorithm discarding any promising region without sufficiently searching the search space.

It must be noted that the conclusions drawn here are only based on a single run of the stochastic

algorithms; that too for problems where the solution space is just a small subset of the search

space. Therefore, any generalisation of these conclusions would require further problem-specific

investigations, involving multiple runs of the stochastic algorithms.

The following chapter illustrates the process design application of the mMASDS algorithm.

42

Chapter 4

Process design and Optimisation case study: coal to ammonia process with

carbon capture

4.1 Context

Amid concerns regarding climate change caused by anthropogenic greenhouse gas emissions,

Carbon Capture and Sequestration (CCS) is expected to be a ‘critical component’ in the portfolio

of low-carbon technologies for the future (IEA, 2013). Though CCS implementation in coal

based power plants has received most of the research interest, implementing CCS in coal based

bulk chemical plants (for example; ammonia, methanol etc.) is likely to be cheaper and easier.

This is primarily due to the fact that most of these plants already have a carbon dioxide (CO2)

capture/removal unit and hence, implementing CCS has a smaller economic penalty.

Ammonia (NH3) is one such bulk chemical which is a precursor for nitrogen containing

fertilizers and industrial explosives, and hence is produced globally.

The financial assistance for this work has been provided by Orica Ltd. through IITB-Monash

Research Academy (project number: IMURA0221 (B). Orica Mining Services is the largest

supplier of mining explosives in the world (Orica, 2012). Orica operates a Natural Gas to

Ammonium Nitrate (AN) facility at their Kooragang Island site, New South Wales, Australia.

The aim of this case study is to assess the techno-economic feasibility of a relatively small scale,

low carbon footprint method of ammonia production from black coal in eastern Australia. This

ammonia would be used to produce bulk explosives to be used at nearby mining sites. Since the

plant is only supplying explosives to nearby mines; it only needs sufficient capacity for these

mines, which explains its relatively small scale.

The black coal (specified by Orica Ltd.) would be sourced locally from the remotely-distributed

coal mining locations in Australia. To limit the carbon footprint of the coal based ammonia

process, CCS needs to be incorporated. Ammonia (NH3) production implicitly involves CO2

removal, because any oxide presence is detrimental to the NH3 synthesis catalyst. However, there

is an explicit need to pressurise the CO2 stream to its supercritical state and ensuring that the CO2

stream is at least 95 % pure. Historically, natural gas has been the preferred feedstock for NH3

production due to its relatively low price and general availability. The C:H ratio for coal is

significantly higher than that for natural gas; as a result, coal to ammonia plants have a

significantly higher carbon footprint as compared with natural gas to ammonia plants. CCS is

also expected to increase the cost of ammonia production from coal.

Ammonia production from coal is a well-established process and its cost has been predicted in

the past (Appl, 1999). However, optimising the global coal to ammonia with CO2 capture is still

43

unexplored. The mass and energy interactions between the capture and CO2 pressurisation units

and rest of the process plant also need to be taken into account.

4.2 Coal to ammonia process details

Coal accounts for 27 % of the global NH3 production capacity, but this capacity is mainly

localised in China (IFA, 2008), which lacks natural gas reserves. However, compared to natural

gas, the higher C:H ratio in coal results in higher CO2 emissions per unit of NH3 produced,

thereby making carbon capture important as a means of reducing CO2 emissions.

The coal to NH3 conversion process broadly involves these processing steps:

1. Gasification of the coal to form a gaseous mixture primarily consisting of CO, CO2, H2,

H2O, H2S, COS and CH4. The gaseous mixture is called syngas. The sulphur present in the

coal forms either H2S or COS upon gasification.

2. The syngas is typically produced at a high temperature. There is thus, a need to cool the

syngas before sending it for further processing, which is usually at a lower temperature.

3. The syngas then undergoes shift reaction and acid gases (CO2 and H2S) removal, not

necessarily in this order. There are different processing options for each of the operation,

as explained later in this section. The shift reaction involves most of the CO content in

syngas reacts with steam and gets converted to CO2 and H2, as per the Equation 4.1.

CO + H2O ↔ CO2 + H2 (4.1)

The shift reaction catalyst may or may not be prone to sulphur poisoning. The COS

hydrolysis reaction, given by Equation 4.2, can be carried out in the shift reactor itself, in

case a sulphur tolerant shift catalyst is used.

COS + H2O ↔ CO2 + H2S (4.2)

The acid gases (CO2 and H2S) present in the syngas, also need to be removed to get a H2

stream, devoid of any oxide and sulphur. Sulphur needs to be removed as it also acts as a

poison to the ammonia synthesis catalyst (Maxwell, 2004). COS is not readily removed in

most acid gas removal systems (Cortés et al., 2009). Hence, the COS may need to be

hydrolysed to form CO2 and H2S, before the syngas can be de-sulfurized, if the shift

catalyst is prone to sulphur poisoning.

4. This is then followed by further treatment of the H2 stream, in a methanator to remove any

remaining carbon oxides. This is done because any oxide presence is detrimental to the

downstream NH3 synthesis catalyst (Appl, 1999). In the methanator, a small fraction of the

H2 present in the syngas reacts with carbon oxides on a Nickel Oxide catalyst, in order to

form CH4, which acts as an inert in the synthesis reaction.

CO + 3H2 ↔ CH4 + H2O (4.3)

44

CO2 + 4H2 ↔ CH4 + 2H2O (4.4)

5. The H2 and N2 are then reacted over a fused, promoted Magnetite (Maxwell, 2004) in the

ammonia synthesis loop. The per-pass conversion can vary between 20-30 %, at the usual

commercial operating conditions (Maxwell, 2004). To prevent inert (primarily, CH4 and

Ar) build-up, a purge also needs to be taken out from the recycle loop.

There are a number of different processing options available for each stage of coal to H2

conversion process. These are summarised next, along with the chosen processing steps.

4.2.1 Processing Options

Gasification technology

Available technologies:

The feedstock and scale of operation for the coal to NH3 process in this study have been

specified by Orica Ltd. Based upon the comparison between various gasification technologies, as

summarised in Table 4.1, entrained flow gasification technology was chosen to be capable of

processing the high rank, less reactive, black coal specified by Orica Ltd. For a detailed analysis

of different gasification technologies and various types of gasifiers, the reader is referred to the

work by Cortés et al. (2009).

Table 4.1: A comparison of different coal gasification technologies (Cortés et al., 2009)

Fixed Bed Fluidized Bed Entrained Flow

Operating

Temperature (°C) 300-1100 650-1100 1300-1900

Operating Pressure

(bar) 10-100 10-40 25-80

Product Gas

Syngas contains tar,

phenols

Lower tar and phenol

content

Higher content of ash

and char

No tars and phenols

Sensible heat needs to be

recovered

Gas Outlet

Temperature (°C) 400-600 700-900 900-1600

Residence Time (s) 900-3600 5-100 1-10

Coal Rank Low Low Any

Capacity Low Moderate High

Chosen technology:

A slurry-fed entrained flow gasifier has been chosen based on the feedstock, i.e. eastern

Australian black coal. The proximate and ultimate analysis of coal used in this work is listed in

Table 4.2.

45

Table 4.2: Characteristics of the coal, used in this work

Proximate analysis (weight %)

Total moisture (as received) 13.90

Fixed carbon (dry basis) 43.66

Volatile matter (dry basis) 45.38

Ash (dry basis) 10.86

Ultimate analysis (weight %) (dry basis)

Carbon 69.07

Hydrogen 5.34

Nitrogen 0.89

Chlorine 0

Sulphur 0.50

Oxygen 13.34

Ash 10.86

Sulphur analysis (weight % of original dry coal) (Sum is equal to 0.5 %, as in ultimate

analysis)

Pyritic 0.022

Sulphate 0.011

Organic 0.467

Heat of combustion (dry basis)

HHV 7020 kcal/kg

Choice of shift catalyst:

Available technologies:

Water Gas Shift (WGS) involves reaction between carbon monoxide and steam to produce

hydrogen and carbon dioxide. WGS reaction is a mildly exothermic reaction (Equation 4.1).

Reactor temperatures vary from 200 °C to 500 °C (Cortés et al., 2009). The syngas can be shifted

either before or after desulphurization. If the syngas is shifted before the desulphurization it is

termed as sour gas shift. The catalyst used in such a reactor is resistant to sulphur poisoning.

There is no need for a separate COS hydrolysis step as the COS hydrolysis occurs in the shift

rector itself. If the syngas is shifted after desulfurization it is termed as clean gas shift. The

catalyst used in such a reactor is prone to sulphur poisoning.

Figure 4.1 shows the block flow diagrams for sour and clean shift configurations.

46

Sour Shift Configuration

Clean Shift Configuration

Figure 4.1: Sour and clean shift configurations

In the sour shift configuration, the raw syngas, after solids removal and water scrubbing, is

reheated and fed to the shift reactors. The COS in the syngas is also hydrolysed to H2S (IEA,

2003). The shifted gas is then fed to the acid gas removal unit for H2S and CO2 removal.

The clean shift configuration is relatively complex because of the large number of unit

operations involved. It also involves repetitive cooling and heating of the syngas. Clean shift

configuration is considered to be infeasible for Texaco gasifier with water quenching (IEA,

2003) because of the need to condense all the steam produced during cooling. Clean shift catalyst

is cheaper than sour shift catalyst, even after considering the additional cost of COS hydrolysis

catalyst (IEA, 2003). In a clean shift configuration, the molar ratio of H2S and CO2 is higher in

the sulphur recovery section which makes it easier to produce a concentrated stream of H2S to be

sent to Claus unit for sulphur recovery (Cortés et al., 2009). Table 4.4 reports the performance of

commercial clean and sour shift catalysts.

Table 4.3: Comparative assessment of Clean and Sour shift

Clean Shift Sour Shift

Catalyst prone to Sulphur poisoning Sulphur Tolerant Catalyst

Selective H2S capture is easier (Lower

CO2:H2S ratio in H2S Absorber)

No need for separate COS hydrolysis

Cheaper catalyst and smaller reactor Lower number of unit operations

Involves repetitive cooling and heating of

feed gas

Costly Catalyst

Requires separate COS hydrolysis step Retains the steam from gasifier (particularly

important for slurry fed gasifiers)

Coal Gasifier Shift

Reaction

Cooling/Water

Scrubbing

H2S and CO

2

Removal

CO2 H2S

H2 Rich

Gas

Coal Gasifier

Shift Reacti

on

Cooling/

Water

Scrubbing

CO2

Removal

CO2 H2S

H2 Rich

Gas

COS

Hydrolysis

and H2S

Removal

47

Table 4.4: Performance of commercial WGS (DOE, 2007)

Clean Shift

Sour Shift

Attribute

Low/Medium

Temperature

Clean Shift

High

Temperature

Clean Shift

Catalyst Form Pellets Pellets pellets

Active Metals Cu/Zn Fe/Cr Co/Mo

Reactor Type

multiple fixed

beds

multiple fixed

beds

multiple fixed

beds

Temperaturea

(°C) 200-270 300-500 250-550

CO in Feed Low moderate to high high

Residual CO (% Volume) 0.1-0.3 3.2-8.0 0.8-1.6

Approach to Equilibrium (°C) 8-10 8-10 8-10

Minimum Steam/CO Ratio

(molar) 2.6 2.8 2.8

Sulphur Tolerance (ppmv) <0.1 <100 >100b

COS Conversion Low Moderate moderate

Durability (years) 3-5 5-7 2-7

a – Lower Temperature Limit is set by water dew point at the operating pressure and b –

Sulphur is required in the syngas to maintain catalyst activity

Chosen technology:

The choice of shift catalysis is mainly dependent on the chosen CO2 capture technology. Since

the gasifier is slurry fed, entrained flow gasifier; sour shift is the preferred configuration in case

the capture technology is able to separate H2S along with CO2. As noted in Table 4.4, the sour

shift configuration, avoids the need for an additional COS hydrolysis step, in addition to

avoiding the repetitive cooling and heating of the syngas.

However, the activated carbon based PSA unit is not able to reversibly separate the H2S. A

separate sulphur removal unit would need to be added before the PSA unit. In such a case, the

sulphur removal unit also needs to be preceded by a COS hydrolysis unit. Two possible

arrangements of different processing steps have been shown in Figure 4.2.

During selective H2S removal, some amount of CO2 would also be lost along with the H2S

stream. For the capture technologies considered in this work, higher the mole fraction of CO2 in

the syngas stream, higher is the co-removal of CO2, along with H2S. Configuration ‘b’ is thus

preferred in case of PSA process, based on physical adsorption of CO2 on Activated Carbon, as

the capture technology. This is because of a lower mole fraction, and thus lower co-absorption,

of CO2 during sulphur removal.

48

Figure 4.2 (a) and (b): The two possible configurations, in case the capture technology is unable

to separate H2S

Choice of capture technology:

As of 2002, there were about 30 AGR processes commercially available (DOE-National Energy

Technology Laboratory, 2012). Only four (RectisolTM

, SelexolTM

, SulfinolTM

and MDEA) of

these processes had been demonstrated or implemented in the 18 commercial-size coal or coke

gasification based plants worldwide as of 2002 (DOE-National Energy Technology Laboratory,

2012).

Out of these 18 plants, nine plants manufactured chemicals and the other nine were IGCC

applications (DOE-National Energy Technology Laboratory, 2012). Out of the nine plants meant

for chemical production, eight used RectisolTM

process for AGR while, only one used the

SelexolTM

process (DOE-National Energy Technology Laboratory, 2012). As per DOE-National

Energy Technology Laboratory (2012), “While Rectisol is more costly, it is preferred for treating

coal-based syngas because it allows for very deep sulphur removal (<0.1 ppmv H2S plus COS),

and also because it can remove HCN, NH3, and many metallic trace contaminants (including

iron- and nickel-carbonyls, and mercury) to provide additional catalyst protection”. Out of the

nine IGCC plants, six plants used MDEA while the remaining three used RecisolTM

, SelexolTM

and SulfinolTM

processes (DOE-National Energy Technology Laboratory, 2012).

Chosen Technologies:

The RectisolTM

process is typically used in coal to NH3 plants to remove hydrogen

sulphide (H2S) and CO2 from the shifted syngas. The RectisolTM

process uses chilled methanol at

temperatures between -20 and -70 °C (Sun and Smith, 2013) to absorb the acid gas components

present in the syngas. The RectisolTM

process is typically used to produce an extremely pure

hydrogen stream, with deep sulphur removal (Trop et al., 2014), for production of chemicals

such as ammonia and methanol. Atsonios et al. (2015) and Olajire (2010) list various advantages

offered by the RectisolTM

process which include; its non-corrosive nature, high thermal as well

as chemical stability of methanol, high selectivity towards CO2 and H2S, commercial availability

of large scale units and more suitability for CO2 capture at high partial pressures as compared to

Selective H2S

Removal

CO2

Removal H2 Sour Shift

Cooled

Syngas

(a)

Clean

Shift CO2

Removal H2

COS

Hydrolysis

Selective

H2S

Removal

Cooled

Syngas

(b)

49

other chemical solvents. The sour shift configuration, as depicted in Figure 4.1, is considered

with bulk removal of H2S and CO2 with the RectisolTM

process.

The other CO2 capture technology considered in this work is the PSA process, relying on

periodic physical adsorption and desorption of CO2 on activated carbon. The PSA is another

capture process which is used in some modern natural gas to NH3 plants, such as those based on

the Linde Ammonia Concept (LAC) technology (Maxwell, 2004). The N2 needed for NH3

synthesis is added after the CO2 removal via a PSA unit, in the LAC technology, which is similar

to coal based ammonia production. The PSA process is increasingly being studied for pre-

combustion capture (For example, Schell et al., 2013; Riboldi and Bolland, 2015 and Riboldi et

al., 2014). The PSA process considered in this work involves multiple packed beds of activated

carbon, operating dynamically, in a cyclic manner. The clean shift configuration, as depicted in

Figure 4.2 (b), is considered for CO2 capture with the PSA process. A small RectisolTM

unit has

also been used ahead of the shift catalysis to selectively remove H2S.

The next section briefly discusses the two flow-sheets considering the two CO2 capture

technologies.

4.3 Coal to NH3 flowsheets

The feed basis has been fixed at 36,365.5 and 51,950 kg/h of eastern Australian black coal for

the two capture technologies, considered in this work. Different coal flow rates have been chosen

to ensure a similar NH3 production rates for the two cases.

4.3.1 Coal to NH3 flowsheet with CO2 capture via physical absorption in chilled methanol

(RectisolTM

process)

Implementing CCS is expected to increase the cost of NH3 production, apart from reducing the

carbon footprint. Hence, optimising the coal to NH3 plant (with CCS) from both environmental

and economic perspectives allows the designer to pick the desired operating point from the best

points considering both objectives.

A standalone flowsheet of the chilled methanol based capture plant was first optimised to get a

better understanding of the system. The MOO was performed using the original Aspen PlusTM

model for the RectisolTM

process. The results thus obtained, have been reported in Appendix A.

It must be noted here that an optimally operating, standalone capture unit does not necessarily

imply an optimally operating, coal to NH3 plant. Optimising the CO2 capture unit, within the

context of NH3 production (i.e. ‘global’ optimisation) should provide the best overall results.

Process details

The block diagram for the coal to NH3 process with chilled methanol based CO2 capture unit is

reported in Figure 4.3. The simulation model is computationally expensive as it involves a

50

recycle around the absorber. The simulation thus needs to be solved iteratively. The surrogate

boundary has been highlighted with the dashed rectangle in Figure 4.3. Figures 4.4(a), (b) and (c)

depict the process flow sheet for NH3 plant.

Figure 4.3: Block diagram of the coal to ammonia process with RectisolTM

for CO2 capture (the

highlighted rectangular portion represents the RectisolTM

process and the surrogate model’s

boundary)

Super-critical CO2

@ 100bar

Produced

NH3

CO2 emission

source (2)

Air

Compression

N2

Compression NH

3 synthesis

reactor

NH3

Condensation

Purge gas burning and heat

recovery

Drying

Methanation

CO2 emission

source (1)

Absorber

Water

HP Steam

Slag

Gasification High

temperature

shift

Low

temperature

shift

Water removal

and drying

O2 (95% pure)

Coal

Water

Stripper

CO2

compression

Solvent

flashing

51

Entrained Flow Gasifier

Coal Slurry

(68 % w/w

solids)

O2

(95 % pure)Slag

Syngas Cooler

HTS Reactor

Inter Cooler 1

LTS Reactor

Inter Cooler 2

Flash Drum 1

Water

Molecular Sieves

Dried Syngas to

Chilled Methanol

Absorber

Water

(a)

Absorber

Make-up

Methanol

Dried Syngas

Treated Hydrogen to

NH3 synthesis Section

Bottoms

Heater

1st Stage

Flashing

2nd Stage

Flashing

3rd Stage

Flashing 4th Stage

Flashing

Stripper

H2S and CO2

Mixture

Solvent Recycle

Pump

Methanol

Recycle

(b)

CO2

PressurisationCaptured CO2

@ 100 bar

52

Methanator

Treated H2

Molecular Sieves

Water

N2

Inter

Cooler 3

Inter

Cooler 4

Inter

Cooler 5

Ammonia

Reactor

Ammonia

Condenser

Produced

NH3

Recycle Heater

Purge Gas

Combustor

Flue

Gases

(c)

Air

Figure 4.4: (a) Flowsheet for coal gasification, (b) AGR and (c) NH3 synthesis sections

The coal-water slurry (68 % w/w solids) and O2 (95% pure) stream are fed to the entrained flow

gasifier, operating at a pressure of 41 bar. Redlich-Kwong-Soave (RKS) cubic equation of state

with Boston-Mathias (BM) alpha function has been used as the property method as per Akhlas et

al. (2015) and Aspen Tech (2008a), except for the acid gas removal section of the flowsheet.

The gasifier is modelled as a combination of RYield reactor and RGibbs reactor models, as per

Aspen Tech (2010). After slag removal, the raw syngas is cooled and fed to the high and low

temperature sour shift reactors. The two shift reactors have been modelled as adiabatic

equilibrium reactors with intercooling. High Pressure (HP) steam at 38 bar and 252 °C is fed to

the High Temperature Shift (HTS) reactor. The shifted syngas coming out of the Low

Temperature Shift (LTS) reactor is cooled before being de-hydrated. The dehydrated syngas is

further cooled to -21 °C, by means of refrigeration, before being fed to the chilled methanol

absorber. In the absorber, the acid gas components, primarily H2S and CO2, are physically

absorbed in chilled methanol, entering the absorber at -42 °C. The absorber is equipped with

side-coolers to remove the heat of CO2 absorption. The spent methanol solvent from the absorber

bottoms is first heated and then flashed to recover the co-absorbed CO and H2 in the methanol

solvent. The solvent is then fed to a series of flash drums, whose primary purpose is to recover

CO2 at different pressures. The liquid solvent coming out of the last flash stage is fed to the

stripper for thermal regeneration of the solvent so that it can be recycled back to the absorber

after being cooled to -42 °C. The CO2 contained in the stripper top product (which also contains

H2S) is assumed not to be captured and hence is emitted to atmosphere from the Claus Plant

(outside of the scope of this study). The Perturbed Chain – Statistical Associated Fluid Theory

(PC-SAFT) Equation of State (EoS) has been implemented to predict the phase equilibrium. The

53

binary interaction parameters proposed by Smith and Sun (2013), and reproduced in Table 4.5,

are used in the simulation. Smith and Sun (2013) validated these binary interaction parameters

against industrial data. Other pure component properties were taken from the methanol wash

model published by Aspen Tech (2008b).

Table 4.5: Binary interaction parameters proposed by Smith and Sun (2013)

i CO2 H2S

j Methanol Methanol

aij 0 0

bij 0.02456573 -0.04015458

cij -0.00449643 -0.05477222

dij 0 0

eij 0 0

Tref 298.15 298.15

The absorber and stripper have been modelled as RadFrac columns with 200 and 25 equilibrium

stages, respectively. The flashed CO2 at different pressures is fed to a compression train, where it

is pressurised to supercritical conditions at 100 bar, so that it can be sent for sequestration. The

treated hydrogen from the absorber is fed to a methanator to convert the remaining carbon oxides

into methane. The methanator is modelled as an isothermal equilibrium reactor in Aspen PlusTM

.

The hydrogen stream is then mixed with the N2 stream from the Air Separation Unit (ASU) and

the mixture is compressed to a pressure of 92 bar. The recycle stream from the NH3 synthesis

loop is then added to the mixture and further compressed to 143 bar, before being fed to the NH3

synthesis reactor. The NH3 synthesis reactor is modelled as a combination of three adiabatic

equilibrium reactors with intercoolers. In most of the NH3 plants, conversion values close to

equilibrium ones can be attained (Appl., 2006). The outlet gas from the ammonia reactors is then

cooled to -30 °C and flashed to 1 bar in order to obtain pure ammonia as liquid. The gas from the

flash drum is recycled except for a small fraction taken out as purge. The purge stream is burned

to recover energy for generating HP steam. The ASU simulation is outside the scope of this

study.

The refrigeration for chilled methanol and ammonia synthesis sections are assumed to be

provided by two separate, two-stage refrigeration systems with propylene and NH3 as

refrigerants, respectively. The two-stage refrigeration systems have been optimised as per the

methodology reported in Appendix E.

The specific energy consumption (kJ⁄kg O2) for the ASU is taken from Hu et al. (2010). Process

Grand Composite Curve (GCC) is used to predict the steam generation potentials. The electrical

power generation potential from different varieties of steam is estimated with the help of

Salisbury approximation (Salisbury, 1942). Scope 2 emission factor of 0.82, for electricity

purchased from grid in Queensland (Department of Industry, Innovation, Climate Change,

54

Science, research and Tertiary Education, 2013), is used to estimate the CO2 equivalent

emissions from any net electricity consumption.

Relatively low fidelity but accurate enough, equilibrium calculation based, reactor models have

been used in this work, since the primary focus is on optimisation of the carbon capture process.

4.3.2 Coal to NH3 flowsheet with CO2 capture via physical adsorption on Activated

Carbon (PSA process)

Similar to the RectisolTM

case, a standalone flowsheet of the PSA based capture plant was first

optimised, for a fixed inlet composition of a binary mixture of H2 and CO2. The results thus

obtained, have been reported in Appendix D.

Process details

The block diagram for the coal to NH3 process with PSA based CO2 capture unit is reported in

Figure 4.5. The PSA model is solved dynamically for a finite number of cycles, until a Cyclic

Steady State (CSS) is achieved. The surrogate boundary has been highlighted with the dashed

rectangle in Figure 4.5. Figures 4.6 (a), (b), (c) and (d) depict the process flow sheet for NH3

plant with PSA based carbon capture.

55

Figure 4.5: Block diagram of the coal to ammonia process with PSA for CO2 capture (the

highlighted rectangular portion represents the PSA process and the surrogate model’s boundary)

H2S + CO2

(Emission

Source 1)

CO2

(Emission

Source 2)

CO2 Compression

and Condensing

Super-critical CO2

@ 100bar

Purge gas

burning and heat

recovery

CO2 (Emission

Source 3)

Air

NH3

Condensation

Purge gas

burning and heat

recovery

Produced

NH3

Compression NH

3

Synthesis

Reactor

HP

Steam Slag

Gasification

O2 (95% pure)

Coal

Water

Cooling

and Water

Removal

Water

COS

Hydrolysis

H2S

Removal

by chilled

methanol

High

Temperature

Shift

Low

Temperature

Shift

Water

removal PSA Methanation

Compression

N2

Drier

Water

56

Entrained Flow Gasifier

Coal Slurry

(68 % w/w solids)

O2 (95 % pure)

Slag

Syngas

Cooler 1

Flash

Drum 1

Water

Syngas

Cooler 2

Molecular Sieves

Water

Absorber

Make-up Chilled

Methanol @ -42 ⁰C

Sulfur free Raw Syngas

Stripper

H2S and CO2

Mixture

Solvent Recycle

Pump

Chilled Methanol

Recycle(a)

HTS Reactor

Inter Cooler 1

LTS Reactor

Inter Cooler 2

Water

Molecular Sieves

Dried, shifted syngas to PSA unit

Water

HP Steam

Sulfur free Raw Syngas

(b)

57

CO2

Condenser

FD

HX1

HX2

Bed 1 Bed 2 Bed 3 Bed 4

VF1 VF2 VF3 VF4

VP1

VP2

VP3

VP4

H2 Product Tank

H2 Product to

Methanator

VPurge1

VPurge2

VPurge3

VPurge4

VW1 VW2 VW3 VW4

CO2 Tank

CO2 Product

VPEQ12 VPEQ23 VPEQ34

VPEQ13

VPEQ14

VPEQ24

Pump

Captured

CO2 @ 100 bar

(c)

Combustor

Air

Flue Gases

CO2

compression

(5 stages)

Dried, shifted syngas

from shift reactors

58

Methanator

H2 product from PSA

Molecular Sieves

Water

N2

Inter Cooler 3

InterCooler 4

Inter Cooler 5 Ammonia

Reactor

Ammonia Condenser

ProducedNH3

Purge Gas Combustor

Air

Flue Gases

(d)

Figure 4.6: Flowsheet for (a) coal gasification and sulphur removal, (b) shift reactor, (c) PSA and

(d) NH3 synthesis sections

Coal is fed to the gasifier, operating at 41 bar, in the form of slurry (68% w/w coal). Chilled

methanol (@ -42 °C) is used to remove H2S. It has been assumed that the sulphur stream is sent

to a Claus plant. Any CO2 going with the H2S is considered as an emission. The sulphur free

syngas is then shifted in two stages before being dried and fed to the PSA unit. The PSA unit

separates the shifted syngas into two streams, one rich in H2 and the other, rich in CO2. A 4 bed

PSA unit has been taken as an example. The 4 bed PSA unit is operated in a 12 step cycle. The

PSA cycle schedule is shown in Figure 4.7. The cycle consists of five basic processing steps

namely; adsorption, blow down, purge, pressurization and pressure equalisation. Activated

carbon is used as the adsorbent. The CO2 stream is pressurised to an intermediate pressure,

before being condensed at -42 °C, in order to preferentially condense and separate CO2. The

liquid CO2, thus produced, is pumped to supercritical conditions. The gases that remain are

combusted in order to raise steam to be used for power generation.

Further modelling details of the PSA unit are reported in Appendix B.

59

PRES AD PED1 PED3 BD PG PEP1 PEP3

PEP3 PRES AD PED1 PED3 BD PG PEP1

BD PG PEP1 PEP3 PRES AD PED1 PED3

PED3 BD PG PEP1 PEP3 PRES AD PED1PED2 PEP2

PED2 PEP2

PEP2 PED2

PEP2 PED2

Figure 4.7: Time chart for the PSA cycle. Steps are denoted as: PRES: Pressurisation; AD:

Adsorption; PED1: First Pressure Equalisation (depressurisation); PED2: Second Pressure

Equalisation (depressurisation); PED3: Third Pressure Equalisation (depressurisation); BD: Blow

down; PG: Purging; PEP1: First Pressure Equalisation (pressurisation); PEP2: Second Pressure

Equalisation (pressurisation); PEP3: Third Pressure Equalisation (pressurisation)

The H2 product from PSA is fed to the methanator, in order to convert any residual carbon oxides

into methane. The product gases are cooled, dried and mixed with N2, before being compressed

to 143 bar, and fed to NH3 synthesis reactor. The product gases are cooled down and NH3 is

separated as a liquid. A fraction of the remaining gases is purged, so as to prevent inert build-up.

The purge gases are burned to produce steam, which is used for power generation. The

remaining gases are recycled back to the synthesis reactor. In all, there are three CO2 emission

sources, viz. the H2S removal unit and the two purge gas burners.

The PSA unit has been modelled in Aspen AdsorptionTM

(version 8.4), while the remaining

segments have been modelled in Aspen PlusTM

(version 8.4). The PSA model assumes non-

isothermal operation of the adsorption beds, which is closer to the actual operating conditions

than if the beds are assumed to operate isothermally. The feed gas to the PSA has been assumed

to be a binary mixture of H2 and CO2, the major components of the shifted syngas. The other

component present in a significant quantity is CO, which does not have much affinity towards

activated carbon. Moreover, the amount of CO left after shift reaction is relatively low and does

not have a significant impact on the objective function values. It is hence assumed that the entire

CO goes with the H2 product stream. The adsorbent beds in the PSA model have fixed

dimensions. As a result, there is a need to scale the PSA unit so that it is able to process the

amount of syngas being fed from the shift reactors. The criterion on the basis of which scaling is

performed is that the residence time for the feed gas, during the adsorption step, should be the

same at both the scales. All the other unit operations have been modelled in the same way as the

RectisolTM

case.

The refrigeration for CO2 condensation and ammonia synthesis sections are assumed to be

provided by two separate, two-stage refrigeration systems with propylene and NH3 as

refrigerants, respectively. The two-stage refrigeration systems have been optimised as per the

methodology reported in Appendix E.

The adsorbent properties, along with adsorption bed characteristics have been summarised in

Table 4.6.

60

Table 4.6: Adsorbent and adsorption bed characteristics

Diameter of adsorption beds 3.5 m

Length of adsorption beds 3 m

Average adsorbent particle radius 0.00115 m (Jee et al., 2001)

Adsorbent particle density 850 kg/m3 (Jee et al., 2001)

Adsorption bed void fraction 0.433 (Jee et al., 2001)

61

Chapter 5

Surrogate-assisted global MOO of the coal to NH3 flowsheet with CO2 capture

via physical absorption in chilled methanol (RectisolTM

process)

5.1 MOO problem formulation

The standalone MOO problem solved in Appendix A was aimed at maximising the CO2 capture

rate and minimising the corresponding electrical power penalty. The standalone problem was

solved for a specific input concentration profile, which in reality; it is a function of decision

variables related to the gasification and WGS section. In this sense, the standalone optimisation

did not properly integrate the CO2 capture unit with the upstream and downstream processing

units. Tock and Maréchal (2014) provided a solution to this problem but with restricted

applicability. For their approach to be valid, the optimum of the sub-section (the capture unit in

our case) of the plant should coincide with the optimum of the entire plant. There are usually

mass and heat integration opportunities between the sub-section (capture unit) and rest of the

plant. Such integration is bound to shift the optimum of the sub-section (capture unit) when

optimised along with rest of the plant, i.e. global optimisation.

The chilled methanol based CO2 capture model may take anywhere between 5 – 30 s in order to

converge on an Intel Core 2 Quad processor (3 GHz), depending upon the initial estimates

supplied to the Aspen PlusTM

(version 7.3) simulation. As a result, a 50 population size evolved

over 50 generations can take as much as 21 h just to solve the absorber and stripper system.

The global optimisation problem was formulated as a MOO problem. The costing methodology,

adopted in this work, has been summarised in Appendix C. For a fixed flow rate of coal, i.e.

36,365.5 kg/h, the Production Cost of NH3 and the Carbon Footprint (CF) of the overall

plant were considered as the two objectives to be minimised. The two objectives are defined as

per Equations 5.1 and 5.2.

⁄ (5.1)

⁄ (5.2)

Where,

: Production Cost of NH3 (USD/mt NH3)

: Annualised Fixed Capital Investment (FCI) (USD/y)

: Cost of manufacture (USD/y)

: Rate of NH3 production (kg NH3/y)

𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒

62

: Carbon footprint of the overall process (kg CO2 equivalent emissions/kg NH3

produced)

: Rate of CO2 equivalent emissions (kg CO2/y)

The decision variables and their initial ranges are given in Table 5.1; while Table 5.2 lists the

constraints specified for the MOO run.

Table 5.1: Decision variable ranges for the global optimisation (RectisolTM

case)

Decision Variable Initially specified range

Oxygen (95% pure) flow to the gasifier 31,300 – 36,700 kg/h

HP steam to HTS reactor 20,000 – 70,000 kg/h

First stage flash pressure 10 – 30 bar

Fourth/Last stage flash pressure 0.1 – 10 bar

Chilled methanol to absorber 4,500 – 5,600 kmol/h

Temperature change in the bottoms heater 0 – 40 ℃

Purge fraction in the NH3 synthesis loop 0.01 – 0.1

Table 5.2 lists the constraints specified for the MOO run.

Table 5.2: Constraints for the global optimisation (RectisolTM

case)

Variable ≥ Or ≤ Value

Amount of NH3 produced ≥ 1,250 kmol/h

The NSGA-II algorithm has been implemented via a binary coded version, which is embedded as

a macro in Excel Visual Basic for Applications (VBA). The choice of a binary coded over a real

coded version of NSGA-II was purely based on logistical reasons. Hence, a real coded algorithm

could also have been used for all practical purposes. The evolutionary MOO framework was

developed by Sharma et al. (2012) particularly to facilitate the interface between the

Optimisation algorithm and a flowsheeting software such as Aspen PlusTM

. The heat integration,

Excel-based module developed by Harkin et al. (2012) is used to carry out heat pinch analysis,

which is used in the coal to ammonia case study as part of a two stage refrigeration Optimisation.

Figure 5.2 depicts the overall flow of information in the complete framework. The GA produces

a generation of individuals to be sequentially evaluated by Aspen PlusTM

. The output and

constraints values thus gathered are sent to the Optimisation algorithm to generate the

individuals constituting the next generation.

63

Decision variablesDecision variables Decision variablesDecision variables

Aspen PlusTM

Simulation

Aspen PlusTM

Simulation

Two stage

refrigeration system

optimisation

Two stage

refrigeration system

optimisationValue for objective

functions and

constraints

Value for objective

functions and

constraints

Value for objective

functions and

constraints

Value for objective

functions and

constraints

Optimisation

algorithm

Optimisation

algorithm NSGA-II Excel-Visual Basic

Interface

NSGA-II Excel-Visual Basic

Interface

Figure 5.1: Information flow in the MOO framework

5.2 Surrogate modelling for the RectisolTM

section

5.2.1 Dimensionality

Ideally the number of independent variables in the surrogate models should be kept as low as

possible. This is to avoid the ‘curse of dimensionality’, which refers to the fact that as the

dimensionality or the number of independent variable increases, the amount of input-output data

required to predict a statistically significant relationship between the dependent and independent

variables, increases exponentially. In the present case, the performance of the RectisolTM

process,

apart from depending on the decision variables specific to RectisolTM

process, also depends on

the mole fraction or molar flow rates of different components entering the RectisolTM

process. In

such a case, the dimensionality of the surrogate models would be, , where represents

the number of components. However, the mole fractions or molar flow rates of individual

components are not mutually independent. A combination of and values is

responsible for producing a fixed set of input profile to the RectisolTM

process. Hence, in the

present case, the dimensionality of the surrogate models can be brought down to 6. If however,

the number of variables (like, and ), affecting the performance of the RectisolTM

process, had been greater than the number of components, the dimensionality of the problem

could only be brought down to .

5.2.2 Surrogate models

An appropriate set of surrogate models should be able to provide all the information that is

needed to calculate the objective function values and constraints. In the present work, eleven

dependent variables have been chosen, as shown in Figure 5.2.

64

Where,

: Total electrical power required by the global process (kWe)

: Molar flow rate of CO in treated gas from absorber (kmol⁄h)

: Molar flow rate of CO2 in treated gas from absorber (kmol⁄h)

: Molar flow rate of H2 in treated gas from absorber (kmol⁄h)

: Molar flow rate of N2 in treated gas from absorber (kmol⁄h)

: Molar flow rate of CH4 in treated gas from absorber (kmol⁄h)

: Molar flow rate of Ar in treated gas from absorber (kmol⁄h)

: Molar flow rate of CO2 emitted in the stripper distillate (kmol⁄h)

: Reboiler duty of stripper (Gcal⁄h)

: Temperature of treated gas from absorber (℃)

: Grass roots cost for the RectisolTM

process (USD)

Figure 5.2: Surrogate model mapping for RectisolTM

case

The surrogate models in this work are fitted by using SUMO toolbox developed by

Gorissen et al. (2010). The fitted models are feed-forward neural networks where MATLAB’s

gads toolbox selects network parameters using a GA. The ANN models are trained via the back-

propagation algorithm. In order to avoid ‘overfitting’, Bayesian regularisation has also been

considered.

5.2.3 Normalisation

Normalising the input values is critical to the success of ANN model fitting (Chaturvedi, 2007).

If the independent variables differ significantly in scale the weights of the synaptic connection

from the two inputs will be significantly different. If however, the inputs were first normalised,

the magnitude of the weight would reflect the strength of the synaptic connection or in other

words, the degree to which the dependent variable, depends on the respective independent

variable.

𝑂 𝐺

𝐻𝑃𝑆𝐻𝑇𝑆

𝑃

𝑃

𝑆𝐹𝐴𝑏𝑠

𝑇𝐵𝐻

𝑓𝑖(𝑂 𝐺 𝐻𝑃𝑆𝐻𝑇𝑆 𝑃 𝑃 𝑆𝐹𝐴𝑏𝑠 𝑇𝐵𝐻)

Surrogate models for RectisolTM

process

𝐸𝑃𝑟𝑒𝑞 𝐶𝑂𝑜𝑢𝑡

𝐶𝑂 𝑂𝑢𝑡 𝐻 𝑂𝑢𝑡 𝑁 𝑂𝑢𝑡

𝐶𝐻 𝑂𝑢𝑡

𝐴𝑟𝑂𝑢𝑡

𝐶𝑂 𝑒𝑚𝑖𝑡

𝑄𝑟𝑒𝑏𝑆 𝑇𝑇𝐺

𝐺𝑅𝐶𝑅

65

In this work, all the independent variables have been normalised between -1 and 1. The

output values have also been normalised, as it resulted in a decrease in the cross-validation error

for the respective model.

5.3 Results and discussion

The following parameters are chosen for the chilled methanol case, as per the framework:

3

0.9

0.1 (to check for validity)

5

6

Distance threshold,

= 55

0.05

0.1

0.9

50

Figures 5.3 (a), (b), (c), (d), (e) and (f) depict the nondominated fronts obtained from the

surrogate-assisted and Business As Usual (BAU) approach, for same number of original model

evaluations. The BAU approach does not involve any surrogate-assistance and relies only on the

Aspen PlusTM

model for functional evaluations. Both runs started with the same initial

population due to the same seed value. All the S-metric values have been normalised with

respect to the S-metric value for the surrogate-assisted approach after 750 original model

evaluations. Hence, the normalised S-metric values represent the approach to the final solution,

assumed to be achieved after 750 original model evaluations. The maximum value for the two

objectives, 789.656 and 2.855, found across all the nondominated fronts, is chosen as the

reference point for S-metric calculation.

66

Figure 5.3: Surrogate-assisted and BAU approach after (a) 250, (b) 350, (c) 450, (d) 550, (e) 650

and (f) 750 original model evaluations

The S value for surrogate-assisted run is consistently higher than that for BAU run as shown in

Figure 5.4.

67

Figure 5.4: S value as a function of number of original model evaluation for the two runs

With the help of surrogate-models, the GA is quickly able to hone into the promising subset of

the initial decision variable search space, reflected by high S values. As a result, a significant

amount of computation time need not be invested in evaluating the computationally complex

original model. This advantage is even more significant because only a fraction of all the

evaluated points are going to be included in the final solution. However, an effective evolution,

assisted by surrogate models, can only be ensured when the surrogate models are accurate

enough to be used as substitutes to the original model. As evident from Figures 5.3(a) through

(f), features such as multiple, spatially distributed surrogate fitting (adapted from Isaacs et al.,

2009) and periodic search space evolution have been able to ensure faster convergence for a

given computational budget. Periodic evolution of search space also ensures that the

computational effort can be effectively apportioned and more stress can be laid on exploring in

the promising regions of the search space. For both the surrogate-assisted MOO strategies, i.e.

MASDS and mMASDS, the EA (repository containing all the input-output data) keeps getting

enriched in promising data. This enrichment is more pronounced in the latter case where periodic

evolution of the search space is also incorporated. As a result of this enrichment, the becomes

sparse with respect to un-promising data. Therefore, the surrogate models are prone to suffer

from prediction inaccuracies in the un-promising regions, which could lead to false optima. This

is another way in which periodic evolution of search space may help to avoid convergence to

false optimum.

Figures 5.5 (a) through (g) show the decision variable values for the Pareto-optimal points,

depicted in Figure 5.3(f), for the surrogate-assisted case.

69

Figure 5.5: (a) , (b) , (c) , (d) , (e) , (f) and (g) values

corresponding to nondominated points, for surrogate-assisted run, shown in Figure 5.3(f)

The scale of the horizontal axis, in Figures 5.5 (a) through (g), has been adjusted to represent the

initial range in which the respective variable has been varied. The convergence with respect to

decision variables is a bit slow. This is mainly because of the number of bits, used to express

every variable, being kept constant; as a result, the actual resolution of the solution with respect

to every variable keeps on changing, as the search space evolves. However, the general trend for

every variable can easily be gauged from Figures 5.5 (a) through (g).

The oxygen flow to gasifier ( ) prefers to take the lowest value possible, which has been

fixed, so as to ensure a minimum operating temperature for the slagging gasifier. HP steam flow

to the shift reactor is also limited to a small range as compared to the initial search space. The

first stage flash pressure doesn’t seem to portray any particular correlation to the objectives.

However, fourth/last stage flash pressure and the optimum objective function values appear to be

strongly correlated. This is because the fourth/last stage flash pressure directly regulates the

amount of CO2 captured. The solvent flow rate variation is also restricted within a small subset

of the initial range. The importance of heat integration is highlighted by the high values for

temperature change in the bottoms heater, as shown in Figure 5.5 (f). The purge fraction for NH3

synthesis loop and objective values show a strong negative correlation, as shown in

Figure 5.5 (g).

All the decision variables reported in Figures 5.5 (a) through (g) have been obtained by the GA

after a rigorous search, considering the effect of all the decision variables on the objective

function values. This makes interpreting the results, reported in Figures 5.5 (a) through (g), a bit

difficult. Nevertheless, some insights can be gathered for the purge fraction in ammonia

synthesis loop, i.e. , to understand the negative correlation between the purge fraction and the

carbon footprint of the plant, i.e. . Table 5.3 shows how a variation in purge fraction affects

70

the objective function values. Two different values for purge fraction have been considered,

while keeping all the other variables the same.

Table 5.3: Effect of purge fraction in NH3 synthesis loop on the objective functions

(USD⁄mt NH3) (mt CO2 equivalent emissions/mt

NH3 produced)

NH3 production rate

(kmol NH3/h)

0.03 710.82

0.59 1375.17

0.0986 736.72 0.47 1294.73

An increase in purge fraction allows for more hydrogen being burnt in the purge gas combustor

which decreases the net amount of electricity required. Hence the CO2 emissions associated with

electricity consumption are reduced. The NH3 production rate, however, is not reduced

significantly in comparison.

For all of the decision variables, except the two flash pressures and purge fraction, the range of

optimum values is significantly different than the initial search space. This is the kind of

situation where periodic evolution of search space is expected to yield high dividends. In fact,

such a situation, where the optimum decision variable space is not known beforehand, is quite

common in real life Optimisation problems. In such situations, fitting surrogates in the entire

search space may cause poor surrogate generalization abilities in the promising region.

Since the present strategy involves a comparison of actual and approximate functional values,

there is a possibility of actual values getting discarded from the nondominated front, due to

possible inaccuracies in the surrogate predictions. This significantly hampers the search process.

To overcome this deficiency, it is recommended to maintain a separate ‘actual evaluation Pareto’

or AEP, that would only have the best ranking individuals as evaluated via the actual model.

Every time the surrogate predicted solutions are re-evaluated with the actual model, they would

be compared to the solutions in the AEP set, to decide the new Pareto front. The new Pareto set

would then be used as the starting point for the next set of surrogate-assisted generations.

71

Chapter 6

Surrogate-assisted global MOO of coal to NH3 flowsheet with CO2 capture via

physical adsorption on Activated Carbon (PSA process)

6.1 MOO problem formulation

The global optimisation problem was formulated as a MOO problem. The recovery and purity of

H2 from PSA is expected to be lower than that from RectisolTM

. Hence, an additional amount of

coal is required in order to produce a similar amount of NH3, as in RectisolTM

case.

The dynamic PSA Aspen Adsorption (version 8.4) model may take anywhere between 15 – 30

min in order to achieve CSS on an Intel Core i7 processor (3.40 GHz) for a single calculation.

For a fixed flow rate of coal, i.e. 51950 , the Production Cost of NH3 and Carbon

Footprint (CF) of the overall plant were considered as the two objectives to be minimised. The

two objectives are defined as per Equations 6.1 and 6.2.

⁄ (6.1)

⁄ (6.2)

Where,

: Production Cost of NH3 (USD/mt NH3)

: Annualised FCI (USD/y)

: Cost of manufacture (USD/y)

: Rate of NH3 production (kg NH3/y)

: Carbon footprint of the overall process (kg CO2 equivalent emissions/kg NH3

produced)

: Rate of CO2 equivalent emissions (kg CO2/y)

The decision variables and their initial ranges are given in Table 6.1.

72

Table 6.1: Decision variable ranges for the global optimisation (PSA case)

Decision Variable Initially specified range

Oxygen (95% pure) flow to the gasifier 43,700 – 55,000 kg/h

HP steam to HTS reactor 48,000– 200,000 kg/h

Valve coefficient for Purge Valves (VPurge ) 0.000766– 0.00457 kmol/bar/s

Adsorption step time 1 – 100 s

Blow down pressure 0.1 – 1 bar

Pressurisation and Depressurisation step time 10 – 300 s

Pressure equalisation time 5 – 200 s

H2 product flow rate in adsorption step 0.1 – 3 kmol/s

Pressure at which CO2 is liquefied 15 – 65 bar

Purge fraction in the NH3 synthesis loop 0.01 – 0.1

Table 6.2 lists the constraints specified for the MOO run.

Table 6.2: Constraints for the global optimisation (PSA case)

Variable ≥ Or ≤ Value

Purity of H2 product from PSA ≥ 0.8

Amount of NH3 produced ≥ 1,100 kmol/h

Purity of CO2 product from PSA ≥ 0.6

6.2 Surrogate modelling for the PSA section

6.2.1 Dimensionality

The dimensionality of the surrogate models used to replace the PSA model is 7. This includes the

six decision variables inherent to the PSA section and the inlet H2 mole fraction.

6.2.2 Surrogate models

To replace the computationally complex PSA model, four dependent variables are chosen as

shown in Figure 6.1. The maximum number of sample points used to fit the surrogates are 450.

All excess sample points are used for testing the generalisation abilities of the fitted surrogates.

ANN surrogate models are fitted for this case study as well.

73

Where,

: Fraction of H2 in feed going with CO2 product stream

: Fraction of CO2 in feed going with CO2 product stream

: The scaling ratio for the PSA unit

: Grass roots cost for the PSA process (USD)

Figure 6.1: Surrogate model mapping for PSA case

6.2.3 Normalisation

The independent variables, along with the output, have been normalised before fitting

the surrogate models.

6.3 Results and discussion

The following parameters are chosen for the PSA case, as per the framework:

3

0.9

0.1 (to check for validity)

5

4

Distance threshold,

= 43

0.01

0.02

0.85

75

The Pareto progression with increasing number of Aspen Adsorption model evaluations is

reported in Figure 6.2.

𝐻 𝑀𝐹

𝐶𝑣𝑃𝑢𝑟𝑔𝑒

𝑡𝐴𝑑𝑠

𝐵𝐷𝑃

𝑡𝑃𝐷𝑃

𝑡𝑃𝐸𝑄

𝑓𝑖(𝐻 𝑀𝐹 𝐶𝑣𝑃𝑢𝑟𝑔𝑒 𝑡𝐴𝑑𝑠 𝐵𝐷𝑃 𝑡𝑃𝐷𝑃 𝑡𝑃𝐸𝑄 𝐹𝐻 𝑃𝐿𝑖𝑞 )

Surrogate models for PSA process

𝑆𝐹𝐻

𝑆𝐹𝐶𝑂

𝑆𝑅𝑃𝑆𝐴

𝐺𝑅𝐶𝑃𝑆𝐴 𝐹𝐻

74

Figure 6.2: Pareto progression for the coal to NH3 case with CO2 capture via physical adsorption

on Activated Carbon (PSA process)

The surrogate-assisted, evolutionary MOO run was stopped as soon as the maximum number of

generations ( = 43) was reached. This problem appears to be multi-modal in nature, with the

evolutionary algorithm getting stuck at different local optimums during the course of

optimisation. In this case, the surrogate-assisted results have not been compared with the BAU

results because of the exorbitant time required, in order to do so. The issue discussed in the last

paragraph of the last chapter related to non-dominated solutions, evaluated via the original

model, not getting retained due to erroneous surrogate model predictions may explain the reason

why the circled point (1307.32, 0.96), in Figure 6.2, did not get retained between 825 and 975

actual model evaluations.

Figure 6.3 reports how the S-metric evolved with increasing number of original model

evaluations. All the S-metric values have been normalised with respect to the S-metric value for

the surrogate-assisted approach after 1425 original model evaluations. Hence, the normalised S-

metric values represent the approach to the solution achieved after 1425 original model

evaluations. The maximum value for the two objectives, 1555.95 and 2.04, found across all the

nondominated fronts, is chosen as the reference point for S-metric calculation.

75

Figure 6.3: S value as a function of number of original model evaluation for the coal to NH3 case

with CO2 capture via physical adsorption on Activated Carbon (PSA process)

Among the decision variables; , and were found to strongly correlated to the

optimum objective function values reported in Figure 6.2. Figures 6.4 (a), (b) and (c) report ,

and values corresponding to the nondominated solutions obtained after 1425

original model evaluations. The scale of the horizontal axis, in Figures 6.4 (a) through (c), has

been adjusted to represent the initial range in which the respective variable has been varied.

76

Figure 6.4: (a) , (b) and (c) values corresponding to nondominated points

obtained after 1425 original model evaluations

Figure 6.5 reports the results from the standalone optimisation (reported in Appendix D)

alongside the results when the PSA unit is used in context of NH3 production (global

optimisation), as obtained after 1425 original model evaluations.

Where,

: H2 Purity in the H2 product stream (%)

: Percentage of the H2 in feed being recovered in the H2 product stream (%)

Figure 6.5: Global vs. standalone PSA optimisation

77

The objectives for the two optimisation runs were different. For the global optimisation problem,

almost all of the carbon oxides, still left in the H2 stream, are converted to CH4 in the

methanator. For every mole of CO and CO2, there is a penalty of three and four moles of H2,

respectively. As a result, there are lower limits to both and in the global

optimisation problem. The standalone optimisation also did not take into account the productivity

(represented by the amount of feed the PSA could process in unit time) of the PSA unit. The

productivity of the PSA unit is dependent on the individual step times and hence, greatly affects

the capital cost of the PSA unit.

Figure 6.6 reports a comparison of the two flowsheets considered for coal to ammonia

conversion. The chilled methanol based flowsheet produced cheaper and lower carbon footprint

ammonia using this particular eastern Australian black coal, under the same set of economic

assumption. The 4-bed, 12-Step PSA process, taken as the case study, has a significantly lower

H2 purity and recovery as compared to the chilled methanol based process; the chilled methanol

process had both H2 purity and recovery > 99%. The methanol refrigeration penalty was more

than compensated by the low H2 purity and recovery in case of PSA. To improve the H2 purity

and recovery, relatively more complex PSA processes like the Gemini process, developed by Air

Products and Chemicals Inc. (Sircar and Golden, 2010), can be employed. The four basic steps,

considered in this work, need to be augmented by steps such as co-current CO2 rinse and

additional pressure equalisation steps (as in Gemini process).

Figure 6.6: Results obtained for the two flowsheets involving different carbon capture

technologies

78

Chapter 7

Conclusions and prospects for further research

The utility of a mathematical model is dependent upon both its ability to mimic accurately the

physical phenomenon as well as the computational demand. There is, typically, a trade-off

involved between the fidelity of a process model/simulation and the computation time.

Evolutionary MOO is a search method with an aim to mimic the process of natural evolution to

produce a set of Pareto-optimal solutions for problems involving multiple conflicting objectives.

Evolutionary MOO is a derivative free Optimisation method which requires solving the

simulation/model a number of times. For problems where a high fidelity model is used, the time

required to execute an evolutionary MOO run is often unacceptable.

Surrogate-assisted evolutionary MOO can be used in such situations by replacing the high

fidelity model, either partially or completely, with a cheaper approximate, often known as the

surrogate model. The straight forward surrogate-assisted evolutionary MOO approach is to fit a

surrogate model at the start of the evolutionary MOO run by sampling data points using a space-

filling technique, such as the LHD. However, for evolutionary MOO applications, the surrogate

need not be globally-accurate; rather the fidelity/accuracy of the surrogate only needs to be

selectively improved by sampling additional data points during the course of Optimisation. The

aim of surrogate-assisted evolutionary MOO is to converge as close as possible to the global

optimum, while evaluating the high fidelity/actual/original model as few times as possible.

Generally, an additional data point is sampled and evaluated via the original model if the

surrogate is expected to be significantly inaccurate. The nondominated solutions are also

periodically re-evaluated via the original models, to keep the search on track. The surrogates can

also be regularly updated to ensure better accuracy in the promising subset of the search space.

As a part of this research the recent applications of surrogate-assisted evolutionary MOO, in the

chemical engineering design field were reviewed. There are broadly two surrogate-assisted

evolutionary MOO approaches reported by researchers in the field of process design and

Optimisation. The first approach uses a fixed set of surrogate models throughout the evolutionary

MOO run. The aim of this approach is to build a globally accurate set of surrogate models which

are to be used during the course of evolutionary MOO, without any update. Typically, this

approach is likely to converge to a false optimum. The second approach, on the other hand,

involves periodic updating of surrogates by selectively improving the fidelity of surrogates. EC

or model management is often used to decide whether to use the surrogate or the original model

for evaluating a particular candidate solution. For cases where the end use of surrogates is in

optimisation, the one-shot surrogate-assisted approach appears to be unnecessary, as

evolutionary approach to optimisation provides an opportunity to selectively improve the

surrogate fidelity, thereby saving precious computation time.

79

Though surrogate-accuracy is not the sole criterion affecting the surrogate-assisted approaches’

ability to converge to the global optimal, it is nevertheless desirable. For this research, the

MASDS algorithm was modified to better suit practical chemical engineering design and MOO

problems, where the final solution space is often significantly smaller than the search space. The

modified MASDS or mMASDS algorithm involves periodic evolution of the search space; this is

expected to potentially improve the surrogate fidelity in the promising subset of the search space.

This hypothesis was tested for two numerical test problems. For the SCH problem, periodic

evolution of search space resulted in a locally more accurate final set of surrogates. the However,

for ZDT2 problem, the accuracy of the final set of surrogate models was just marginally better,

irrespective of search space evolution. Nevertheless, based on the single set of evolutionary

MOO runs executed, it was observed that rate of convergence to the global optimum was higher

for mMASDS, as compared to the MASDS algorithm. These test problems were specifically

chosen, as they represented the situation where the final solution space was just a small subset of

the initial search space. Additionally, it has also been suggested not to use periodic evolution for

the first few surrogate-assisted set of generations. This reduces the probability of the algorithm

erroneously discarding any promising region. It is expected that by the time the algorithm

finishes the first few surrogate-assisted set of generations, it would have sufficiently searched the

original solution space.

Chapter 4 details the process design and evolutionary MOO case study, involving production of

ammonia from coal, while capturing and pressurizing the CO2 produced to supercritical

conditions. The case study was specified by Orica Ltd. with the aim of assessing the techno-

economic feasibility of a relatively small scale, low carbon footprint method of ammonia

production from black coal in eastern Australia. This ammonia would be locally used to produce

bulk explosives to be used at nearby mining sites. Since the remote plant is only meant to supply

explosives to nearby mines, this explains its relatively small scale. Two CO2 capture mechanisms

have been compared by performing two separate surrogate-assisted evolutionary MOO runs.

Physical absorption in chilled methanol and Activated Carbon based PSA were shortlisted for

capturing the CO2. Both processes typify complex process flowsheet simulations, which takes a

considerable time to simulate and compute objective functions for one set of decision variables.

For the chilled methanol case, the simulation model is computationally expensive as it involves a

recycle around the absorber. The simulation thus needs to be solved iteratively, until it

converges. The PSA model is solved dynamically for a finite number of cycles, until a Cyclic

Steady State (CSS) is achieved. The optimisation of coal to ammonia process with PSA based

CO2 capture, thus requires the dynamic model to be solved every time a candidate solution is

generated by the evolutionary MOO algorithm.

The results from the surrogate-assisted run of the chilled methanol case have been compared to

those obtained from BAU approach, where only the original flowsheet model was used for

functional evaluations. The surrogate-assisted run, consistently achieved a better solution set in

an equal number of original model evaluations, as measured in terms of the hypervolume

80

spanned by their respective Pareto fronts. The optimal range of decision variables was observed

to be significantly different than the initial search range. The fourth/last stage flash pressure and

the purge fraction for NH3 synthesis loop were found to be strongly correlated to the optimum

carbon footprint of the coal to ammonia process. The fourth/last stage flash pressure had a strong

positive correlation with the carbon footprint of the process, as the total amount of CO2 captured

is directly regulated by the extent to which the solvent is flashed. An increase in purge fraction,

on the other hand, allows for more hydrogen being burnt in the purge gas combustor which

decreases the net amount of electricity required. Hence the CO2 emissions associated with

electricity consumption are reduced.

The 4-bed, 12-Step PSA capture process produced ammonia at a higher cost, and carbon

footprint, as compared to the chilled methanol based process. The primary reason for this was the

significantly lower H2 purities and recoveries of the 12-step PSA process, as compared to

RectisolTM

. Though there is no theoretical proof of the results being converged, the final results

are not expected to change the conclusions significantly. The results were also compared to those

obtained from standalone optimisation of the same PSA unit. The comparison shows how the

optimum for the standalone processing sections may differ when considered in the context of

their application. The PSA process used in this case study had a basic configuration, involving a

lower number of beds as those typically used in commercial PSA based CO2 capture processes,

like the Gemini process. This work also used a limited variety of processing steps. However, the

optimisation strategy would largely be independent of the process model/configuration used.

Hence, the PSA MOO case study demonstrated the benefits of the surrogated assisted strategy,

because such a flowsheet could not have been optimised without surrogate assistance.

The surrogate-assisted evolutionary MOO is a useful tool for complex process flowsheet

analyses. It can be applied when one or more sub-process models are computationally expensive

and hence can’t be repeatedly evaluated. The surrogate-assisted approach thus allows the user to

better integrate and optimise these sub-processes with the rest of the plant. The research in the

field of surrogate-assisted evolutionary MOO is mainly targeted at minimising the number of

times the computationally complex model is solved, while ensuring convergence to the global

optimum. The success of surrogate-assisted evolutionary MOO is, to a significant extent, also

depends on how the problem is formulated and executed. Some of the significant steps involved

during problem formulation and execution are as follows:

Identification of appropriate variables, both independent and dependent, needed to predict the

sub-processes’ performance.

Normalising the dependent variables, along with independent variables, as and when

required. A dependent variable values are typically normalized when they differ by over an

order of magnitude.

81

Deciding the frequency of generation based EC, surrogate updating and search space

evolution. Both the MASDS and mMASDS algorithms are, in a way, rigid as they do not

allow these frequencies to vary.

The first two bullet points have been demonstrated for the two CO2 capture flowsheet

optimisation problems.

Since surrogate accuracy is not the sole criterion for convergence to global optimum, the

generation EC and surrogate updating frequencies should rather be based on whether the ranks of

the nondominated solutions based on the original and the surrogate models are significantly

correlated or not. This is further explained in the next section while listing some future avenues

of research.

In this work, the frequency of search space evolution has been linked to the frequency of

generation based EC and surrogate updating. It is advisable not to evolve the search space during

the initial stages of the search. There is a greater chance of a promising region in the decision

variable search space getting rejected, before the evolutionary algorithm has had a chance to

sufficiently explore the search space.

Another common issue with MASDS and mMASDS is that they consider the surrogate based

objective values at par with their original model based counterparts. This leads to some ‘high-

confidence’ nondominated solutions evaluated via the original models dying-off, due to

unrealistically better surrogate predictions. This issue is also discussed in detail in the following

section.

In this work, the surrogate model fitting is carried out using the SUMO toolbox (Gorissen et al.,

2010) in MATLAB. The GA, coded in Excel VBA, is able to interact with MATLAB and

commercial process simulators, both steady-state and dynamic. The surrogate-assisted MOO

approach can theoretically be automated in the future and may involve information transfer

between commercial process simulators, MATLAB and Excel VBA.

A by-product of such a surrogate-assisted MOO run is the vast array of surrogate models that

were originally built during the MOO run. If need be, the most accurate surrogate model can be

chosen to make any future functional evaluation.

Future prospects for research

One possible improvement could be to use the rank correlation coefficient between the

surrogate and the original models to decide whether to update the surrogate or not, thereby

avoiding the extra computational effort wasted in unnecessarily updating the surrogates, as

suggested by Jin (2011). The objective is to ensure that the ranks of the nondominated

solutions based on the original and the surrogate models are significantly correlated.

The issue with both MASDS and mMASDS algorithms is that they treat the surrogate based

outputs at par with the actual outputs, while making comparisons. As a result, there may be

82

situations where nondominated solutions evaluated via the original models die-off due to

erroneous surrogate output prediction. A separate ‘actual evaluation Pareto’ or AEP is

suggested to be maintained. The AEP would only contain the best ranking individuals as

evaluated via the original model. The approach is depicted in Figure 7.1. The intention here

is to avoid any promising point, as predicted by the original model, being discarded. In this

approach the ‘active’ Pareto set alternates between the AEP and the intermediate mixed

evaluation Pareto or MEP.

Start

Total number

of generations = nmax

Stop







multiple surrogates are fitted for that

particular dependent variable

Continue the GA run by creating a separate copy of the AEP Pareto set, called the Surrogate Evaluation Pareto (SEP)

The SEP is then used as the ‘active’ Pareto set for this set of surrogate-assisted generations

Meanwhile, any actual model outputs would also be fed to the AEP set fot comparison to avoid their ‘death’ due to potentially inaccurate surrogate model outputs

Re-evaluate the solutions in the SEP set via actual models.

An updated AEP set would then be the active Pareto set after combining it with the

reevaluated SEP set

Run the Genetic Algorithm using original models for a single

generation but with a high mutation rate in the entire decision

variable search space

Evolve/Update the search space

Yes

No

Figure 7.1: The suggested surrogate-assisted evolutionary MOO algorithm

A theoretical proof of convergence of any such surrogate-assisted evolutionary algorithm

must also be developed so as to ensure that they converge to the same optimum as achieved

with the BAU approach.

83

Appendix A

Standalone chilled methanol based capture plant optimisation

Process description

Figure A.1 shows the process flowsheet for the chilled methanol unit, considered in this work.

Methanol at a temperature of −42 °C is fed to the absorber, in which the acid gases (that is, CO2

and H2S) are absorbed. The feed gas composition and the flow rates are given in Table A.1. The

absorber is equipped with side coolers in order to remove the heat of absorption. The acid gas-

laden solvent is then heated and flashed in order to release the co-absorbed CO and H2. The

released gases are recycled back to the absorber. The solvent is then flashed in a sequence of

flash drums, in order to release CO2 at different pressures. The flashed solvent is then fed to the

stripper to undergo thermal regeneration with the help of LP steam. The flowsheet has been

simulated in the Aspen PlusTM

software (version 7.3). PC-SAFT EoS was implemented to predict

the phase equilibrium. The binary interaction parameters proposed by Smith and Sun (2013)

were used in the simulation. A two-stage propylene based cycle is used to meet the refrigeration

demands, which has been optimised using the algorithm, summarised in Appendix E.

84

Absorber

Methanol

@ -42oC

Treated Gas

H2S/CO2

Rich Solvent

Bottoms

Heater

T 101

C6

First Stage

(T 102)

Flash Drum

Second Stage

(T 103)

Flash DrumThird Stage

(T 104)

Flash Drum

Fourth Stage

(T 105)

Flash Drum

Solvent

Pump

Stripper

Heater

C1Intercooler 1

C2

Intercooler 2C3

Intercooler 3C4

Intercooler 4C5

Condenser

T 106

H2S+CO2

Reboiler

Solvent Recycle

Pump

Solvent

CoolerMethanol recycle

Gas Feed

Captured CO2

@ 100 bar

Figure A.1: Process flowsheet

85

Table A.1: Feed gas properties

Pressure 41 bar

Temperature -21 °C

Total molar flow rate 1899.434 kmol/h

Mole fractions

CO2 0.4646

H2 0.4932

H2S 0.0018

CO 0.0161

N2 0.0072

COS 2.8 ppm

CH4 0.0047

Ar 0.0124

Problem definition

The overall optimisation problem is formulated as given by Equation A.1.

Maximise: & Minimise: ∑ (kWe)

w.r.t.: , , and

Subject to: ≤ 10 ppm, ≥ 98 % and ≥ 99.8 %

Where,

: CO2 capture rate (%)

: Total power penalty associated with CO2 capture (kWe)

: Electrical power consumed by compressor iC (kWe)

Optimum refrigeration electrical duty after adjusting for non-ideal isentropic

compression (kWe)

: Electrical power consumed by solvent recycle pump (kWe)

: Electrical power consumed by solvent pump (kWe)

: Approximate electrical power sacrificed by using Low Pressure (LP) steam in

stripper reboiler (kWe)

: First stage flash pressure (bar)

: Fourth/last stage flash pressure (bar)

: Solvent (methanol) flow rate to the absorber (kmol/h)

: Temperature change in the bottoms heater (°C)

: Mole fraction of CO2 in the treated syngas

: CO recovery across the absorber (%)

(B.1)

86

: H2 recovery across the absorber (%)

The electrical power sacrificed by using LP steam in the stripper reboiler has been estimated

using the Salisbury approximation (Salisbury, 1942), as given by Equation A.2.

(A.2)

Where,

: Enthalpy of saturated water at the inlet pressure of the turbine (kJ/kg)

: Enthalpy of saturated water at the outlet pressure of the turbine (kJ/kg)

: Mass flow rate of steam (kg/s)

The explored decision variable space is given in Table A.2.

Table A.2: Decision variable range for optimisation

Decision Variable Range

10–38 (bar)

0.1–10 (bar)

2,000–2,900 (kmol/h)

0–50 (°C)

The basic steps involved in solving the present problem by NSGA-II algorithm are as follows:

I. As the first step, a random initial population is generated, comprising of (=30)

individual chromosomes, consisting of 4 genes, each corresponding to one decision

variable. The value for the population size is chosen based on the suggestions given

by Reeves (2003).

II. This is followed by evaluation of objective function values (i.e. and ) for these

individual chromosomes. The initial population is then sorted according to the fast

nondominated sorting algorithm, suggested by Deb et al. (2002).

III. An offspring population of size is then generated using binary tournament selection,

crossover (with a probability of 0.9) and mutation (with a probability of 0.005).

IV. The objective function value for these off-springs is also evaluated and the combined

parent and offspring population of size ‘2 ’ is then sorted based upon their

nondomination score.

V. The individuals corresponding to the first nondominated front are then selected to make

up the next generation. In case the number of individuals in the first nondominated front

is less than population size , individuals from second nondominated front are included

in the selection. This procedure is repeated until individuals are selected. In case the

87

number of individuals selected becomes greater than , the individuals in the last

included nondominated front are sorted based on the crowding distance value and the

excess individuals are rejected.

VI. The new population thus generated, goes through steps III, IV and V again until the

maximum number of generations (i.e. 50) is reached.

Results and Discussion

The final Pareto front (CO2 capture rate vs. Power penalty) is shown in Figure A.2.

Fourth/last stage flash pressure, solvent flow rate and the extent of heating in the bottoms heater

were found to have the most significant impact on the Pareto front. The values of the decision

variables are reported in Figures A.3 (a), (b) and (c). Qualitatively, the results seem to be in

agreement with those reported by Liu et al. (2015), who also reported the pressure of the

low-pressure flash as an important factor affecting CO2 capture rate.

Figure A.2: Pareto front obtained for the standalone optimisation problem

88

Figure A.3: (a) Fourth/last stage flash pressure ( ), (b) Solvent (methanol) flow rate to the

absorber ( ) and (c) Temperature change in the bottoms heater ( ) corresponding to the

optimum objective function values

The extent of heating in the bottoms heater (that is, ) controls the amount of the co-

absorbed H2 and CO being recycled back to the absorber. Low values of imply a lower

solvent temperature at the exit of the fourth/last stage of flashing, thereby increasing the extent of

heat integration among the heater just before the inlet of the stripper and solvent cooler. The

extent of heating in the bottoms heater also regulates the amount of CO2 that can be captured for

a given flash pressure level. Hence, there are two options available to increase the CO2 capture

level:

Flashing of the solvent to a lower pressure

Increasing the extent of heating in the bottoms heater

As can be seen from Figures A.3 (a) and A.3 (c), the preferable option in is to have lower values

of the fourth/last stage flash pressure. The recovery constraints dictate that the solvent needs to

be heated to a certain extent, as shown in Figure A.3 (c). This increased heating in the bottoms

heater limits the extent of heat integration in the low temperature region. In order to achieve high

capture rates, the solvent needs to be heated to a higher extent, as evident in Figure A.3 (c).

The solvent flow rate was fixed at a particular level, as is evident in Figure A.3 (b). This

particular level of solvent flow corresponds to the minimum amount of solvent needed in order

to meet the minimum quality constraint for the treated gas ( ≤ 10 ppm).

Another interesting trend is observed when the optimum specific energy penalty (i.e. ),

given by Equation A.3, for different points on the Pareto front are plotted against the

corresponding CO2 capture levels.

(A.3)

89

As can be seen in Figure A.4, the plot exhibits a minima. The results seem to be in agreement

with those reported by Liu et al. (2015), who predicted a minimum at ~80% capture rate. The

trends observed in Figure A.4 can be understood, keeping in mind the different contributors to

the total power penalty. The total power penalty associated with CO2 capture is mainly

composed of the compression penalty, the refrigeration penalty, and the approximate amount of

electrical power sacrificed by using LP steam in the stripper reboiler ( ). The refrigeration

penalty is composed of solvent refrigeration. The solvent refrigeration requirements largely

remains the same, with varying capture rates. This is due to the constant solvent flow rate

required in order to achieve the quality constraint ( ≤ 10 ppm) for the treated syngas. It is

only at high capture rates, that the solvent flow rate and consequently solvent refrigeration duty

increase drastically. The CO2 compression penalty, on the other hand, decreases with decreasing

capture rates. Figure A.4, can hence be understood as the cumulative result of these two different

factors. It should be noted that when the capital cost is also included, this minima is likely to be

shifted to the right.

Figure A.4: Minimum specific energy penalty ( ) for different CO2 capture rates ( )

To illustrate the point made in the previous paragraph, Table A.3 lists four points on the plot in

Figure A.4. The four points are picked so that the transition from low capture rates (~69%) to

high capture rates (~99%) can be analysed. As can be seen in Table A.3, the specific

compression work increases with increasing CO2 capture rate whereas, the specific refrigeration

work first decreases and then increases with increasing capture rate.

90

Table A.3: CO2 compression and refrigeration requirement for four characteristic points, plotted

in Figure A.4

Point CO2

compression

work (kWe)

Shaftwork

for

refrigeration

system (kWe)

Other

electrical

work

requirements

(kWe)

Specific

compression

work

(kWeh/kmol CO2)

Specific

refrigeration

work

(kWeh/kmol CO2)

CO2

capture

rate

(%)

1 1015.86 1121.13 138.99 1.65 1.83 69.57

2 1286.42 1146.57 145.11 1.83 1.63 79.52

3 1362.38 1151.25 146.45 1.88 1.59 81.87

4 2500.42 1781.33 187.89 2.84 2.02 99.71

Pocket exploitation (explained in detail in Appendix E) was the preferred refrigeration design

option for all the points on the Pareto front. The shaftwork savings achieved by having pocket

exploitation instead of a single stage refrigeration system are reported in Table A.4.

Table A.4: Effect of pocket exploitation (Reference point on Pareto front– : 93.3 % and

: 3385.28 kWe)

Single stage refrigeration Pocket Exploitation Shaftwork savings

1445.95 kWe 1095.76 kWe 24.22 %

The detailed analysis reported in the preceding paragraphs was based on the results obtained both

in the objective function and decision variable space. Such an interpretation of results would not

be possible, had the two objective functions been combined into a single objective function. For

example, if minimising the specific energy penalty ( ) had been the only objective, the

result would have been a single point corresponding to the minimum values of in

Figure A.4. However, in that case the results could not have been interpreted the way they have

been done in this work. Keeping the two objectives separate allows the DM to visualise and

analyse the solution set, which in turn facilitates better understanding of the complex trade-offs

involved between the objectives and the effect of various decision variables on the objective

functions.

Considering an average computation time per simulation as 17.5 s, the total time required for the

standalone optimisation was at least 8 h on an Intel Core 2 Quad processor (3 GHz).

Conclusions

A heat integrated, non-selective RectisolTM

process is optimised to obtain the minimum energy

penalty associated with different CO2 capture rates. The analysis helps in the selection of

optimum operating conditions for the unit. Pocket exploitation was the preferred design option

91

for the refrigeration system. The specific energy consumption for different capture rates shows a

minimum for the investigated case.

92

Appendix B

PSA modelling details

Introduction

PSA involves preferential adsorption of some components at high pressures and subsequent

desorption of the same components at low pressures. The solid on which adsorption takes place

is called adsorbent and the species getting adsorbed on the adsorbent are called adsorbate. A

fixed amount of adsorbent can only store a finite amount of adsorbate. Hence, there is a need to

regenerate the adsorbent so that it can be reused.

An industrial pressure swing adsorption process has several fixed beds (filled with adsorbent)

operating in parallel. An adsorption bed in a PSA process typically undergoes the following

series of steps in a cyclic manner:

1. Pressurization with feed: In this step the newly regenerated bed is pressurized up to the

adsorption pressure by feed stream.

2. Adsorption: This step involves preferential adsorption of one or more components in feed

gas on the adsorbent, thus increasing the concentration of other components in gaseous

phase.

3. Pressure equalisation (pressurisation/de-pressurisation): A pressure equalization step is often

employed to increase the recovery rate of the less adsorbed species. In pressure equalization

step an exhausted bed (after adsorption step) is used to pressurize a newly regenerated bed

(after purge step). A PSA cycle could have any number of pressure equalization steps,

depending upon the recovery rates required. An increase in number of pressure equalization

steps requires an increase in number of beds thereby increasing the capital cost.

4. Blow down: This step typically involves a reduction in adsorption bed’s pressure thereby

releasing the components that were preferentially adsorbed in the adsorption step, at a higher

pressure.

5. Purge: In purge step a product stream is redirected to the bed under regeneration, thus

decreasing the partial pressure of the preferentially adsorbed component in the gas phase.

This step further enhances the extent of desorption.

Commercial adsorbents are typically made from microporous material in order to ensure a large

adsorbent area and consequently a larger adsorption capacity (Agarwal, 2010). Some adsorbents

may also have macropores formed due to aggregation of fine particles into pellets (Agarwal,

2010). Figure B.1 depicts a typical adsorbent particle having both micro and macropores

(Ruthven et al., 1994).

93

Figure B.1: A composite adsorbent (reprinted from Ruthven et al., 1994)

PSA Modelling

The PSA models used in this work are based on following assumptions:

1. Non-isothermal operation.

2. Plug flow with axial dispersion flow pattern.

3. Negligible concentration gradients in the radial direction.

4. The gas phase has been assumed to behave as an idea gas mixture.

5. The overall mass transfer rate is assumed to be described by an overall lumped resistance.

Modelling equations

1. Material balance: The component material balance in the bulk phase is given by

Equation B.1.

(B.1)

Accumulation of component in the bulk phase

Accumulation of component on the adsorbent

Convective mass transfer

Axial dispersion term

Where,

Gas phase concentration of component (mol/m3)

: Bed void fraction

94

: Time (s)

: Average solid-phase loading for the ith

component (mol/kg)

Superficial gas velocity (m/s)

Adsorbent particle density (kg/m3)

: Axial dispersion coefficient (m2/s)

The axial dispersion coefficient is estimated by using Equation B.2 proposed by Edwards

and Richardson (1968):

(

) (B.2)

Where,

: Molecular diffusivity (m2/s)

: Particle radius (m)

The rate of accumulation of component

on the adsorbent mainly depends on three

types of mass transfer resistances, namely:

Mass transfer resistance between bulk fluid phase and the external surface of the

adsorbent particle.

Mass transfer resistance due to the macropores structure of the adsorbent. This mass

transfer resistance is often the rate determining resistance (Agarwal, 2010).

Mass transfer resistance offered in the micropores of the adsorbent particle.

In this work, a Linear Driving Force (LDF) model has been used. In the LDF model, all the

mass transfer resistances are lumped into a single overall resistance. The LDF model is

represented by Equation B.3.

(B.3)

Where,

: Equilibrium average solid-phase loading for the i

th component (mol/kg)

The LDF model has successfully been applied by Jee et al, 2001 on a similar five

component (H2, CO2, CO, CH4 and N2) system with activated carbon and zeolite 5A as

adsorbents. The LDF model parameter ( ) value for different components for an activated

carbon system as reported by Jee et al., 2001 are given in Table B.1. The present work uses

the same LDF parameters.

95

Table B.1: LDF model parameter value for the system under consideration (Jee et al., 2001)

Component LDF coefficient ( ) (s-1

)

H2 0.700

CO2 0.036

2. Momentum balance: Flow through a packed bed is often associated with pressure drop.

Ergun equation has been used in this work to relate superficial velocity to pressure drop.

Ergun equation is given by Equation B.4.

(

) (B.4)

Where,

: pressure (bar)

: Shape factor

: Molecular weight (kg/kmol)

: Gas phase molar density (kmol/m3)

Ergun equation is valid for both laminar and turbulent flow conditions (Ruthven, 1984)

3. Adsorption isotherm: Adsorption isotherms are used to predict the equilibrium average

loading, , at a particular temperature. The general form of adsorption isotherms is

represented by Equation B.5.

(B.5)

In this work, extended Langmuir Freundlich model has been used to predict the multi-

component adsorption equilibrium. The extended Langmuir Freundlich model is given by

Equation B.6. Jee et al., 2001 had proposed the value of parameters to be used in extended

Langmuir Freundlich adsorption equation for the system under consideration. The parameter

values proposed by Jee et al., 2001 are reported in Table B.2.

96

∑

( )

Where,

: Partial pressure of ith

component (bar)

Table B.2: Extended Langmuir Freundlich model parameter values for the system under

consideration (Jee et al., 2001)

Component (mmol/g)

(mmol g-1

K-1

)

(atm-1

)

(K) (-) (K)

H2 16.943 -2.100 0.625 1229 0.980 43.03

CO2 28.797 -7.000 100.0 1030 0.999 -37.04

4. Energy Balance: Adsorption process is accompanied by an evolution of energy, which is

referred to as heat of adsorption. In this work, the bulk gaseous phase and adsorbent

particles have been assumed to be at the same temperature (Agarwal, 2010). The thermal

conduction in the solid phase has been assumed to be negligible (Agarwal, 2010). Only fluid

to wall heat transfer has been considered. The energy balance equation has been reported in

Equation B.7. The isosteric heat of adsorption for H2 and CO2 are 8.42 and 24.8 kJ/mol,

respectively (Agarwal, 2010).

( ∑ )

∑

(B.7)

Where.

: Heat capacity of the component (J/mol/K)

: Universal gas constant (J/mol/K)

: Bulk density (kg/m3)

: Heat capacity of adsorbent (J/kg/K)

: Average cross-sectional gas phase temperature in the adsorber bed (K)

: Isosteric heat of adsorption (J/mol)

: Total gas phase enthalpy (J/m3)

: Effective heat transfer coefficient (J/m3/s/K)

: Wall temperature (ambient temperature) (K)

(B.6)

97

Numerical method

A central differencing scheme has been used for spatial discretization. The first order spatial

derivative (convection) term has been discretised by using central differencing scheme of fourth

order, given by Equation B.8.

(B.8)

The second order spatial derivative (dispersion) term has been discretised by using central

differing scheme of second order, given by Equation D.9.

(B.9)

98

Appendix C

Economic Assumptions

The NH3 production cost is estimated from the annualised capital cost and the annual

manufacturing costs. The basic economic assumptions taken in this work are listed in Table C.1.

Table C.1: Basic economic assumptions

Coal price 1.5 AUD/GJ LHV

Electricity price 150 AUD/MWh

Plant life 20 y

Discount rate 15 %

Carbon tax None

Plant availability 90 %

Operating labour required 37

Labour cost 90000 USD/y

AUD to USD conversion rate AUD = 0.77 USD

The grass roots cost or the for ASU; O2 compression; N2 compression; coal handling,

storage and preparation; slurry-fed gasifier and water gas shift reactors has been taken from the

literature and reported in Table C.2.

The grass roots cost for rest of the plant is calculated via the strategy suggested in

Turton et al. (2009). It involves estimating the bare module equipment costs (sum of direct and

indirect module costs) at base conditions (Material Of Construction (MOC) carbon steel and near

atmosphere operation). The bare module cost for non-base conditions is then estimated by using

pressure and MOC correction factors. The contingency, fee and auxiliary facility costs are then

added to the total module cost to get the grass roots cost. The cost correlations given in

Turton et al. (2009) have been used for this purpose.

For heat exchange capital costs, balanced composite curves are used to estimate the required heat

exchanger area, as per Smith (2005). The heat exchanger cost function used in Girardin et al.

(2009), based on the number of units anticipated and total area, is used to estimate the equipment

cost of the exchangers.

The Chemical Engineering Plant Cost Index (CEPCI) is used to take into account the effect of

time on purchased equipment costs. All the reported costs in this work are in terms of 2014 USD,

unless otherwise specified. The manufacturing costs or the Cost Of Manufacture ( ) have

also been estimated based on the methodology suggested in Turton et al. (2009). The total

manufacturing cost is the sum of direct, fixed costs and general expenses. The is estimated

from , cost of operating labour , utilities , waste treatment and raw

materials as per Equation C.1.

99

⁄ (C.1)

Equation C.1 is based on the assumptions listed in Table C.3, as per Turton et al. (2009).

Table C.2: Economic parameters

Component Scaling

parameter

(Million

USD) (2002)

ASU O2 produced 40.4 (Holt,

1998)

1839 mt/day 0.5 (Kreutz

et al., 2005)

2

O2 compression Compression

power

6.3 (Kreutz et

al., 2005) 10 MWe 0.67 (Kreutz

et al., 2005)

2

N2 compression Compression

power

4.7 (Kreutz et

al., 2005) 10 MWe 0.67 (Kreutz

et al., 2005)

2

Coal handling, storage

and preparation

Raw coal feed 29.1 (Holt,

1998)

2367 mt/day 0.67 (Kreutz

et al., 2005)

2

Gasifier, syngas cooler

and scrubber

Moisture Ash

Free (MAF) coal

input (LHV)

144.3 (Kreutz

et al., 2005) 697 MWth 0.67 (Kreutz

et al., 2005)

2

WGS reactors, heat

exchangers

MAF coal input 39.8 (Kreutz

et al., 2005) 1377 MWth 0.67 (Kreutz

et al., 2005)

2

Grass Roots Cost ( ) = ⁄ ; where, is the number of equally sized

trains, is the at scale , is the desired scale and is the scale factor (Kreutz et al.,

2005)

Table C.3: Assumptions for COM prediction

COM item Value used in this work (Turton et al., 2009)

Direct manufacturing costs

Direct supervisory and clerical labour costs

Maintenance and repair costs Operating supplies costs

Laboratory charges Patents and royalties

Fixed manufacturing costs

Local taxes and insurance Plant overhead costs

General manufacturing expenses

Administration costs Distribution and selling costs

Research and development

The of the plant is annualised and added to the annual , to estimate the cost of NH3

production.

100

Appendix D

Standalone PSA capture plant optimisation

Process description

Figure D.1 shows the process flowsheet for the 4-bed PSA unit. It follows a 12-step cycle, as

shown in Figure D.2. Activated Carbon has been used as the adsorbent to physically adsorb and

desorb CO2 in a cyclic manner. The feed gas has been assumed to be a binary mixture of H2 and

CO2. The feed gas specifications are given in Table D.1. The process has been simulated in

Aspen Adsorption (version 8.4). The model is described in detail in Appendix B.

Bed 1 Bed 2 Bed 3 Bed 4

VF1 VF2 VF3 VF4

VP1 VP2 VP3 VP4

Feed Tank

H2 Product Tank

H2 Product

VPurge1VPurge2 VPurge3 VPurge4

VW1 VW2 VW3 VW4

CO2 Tank

CO2 Product

VPEQ12 VPEQ23VPEQ34

VPEQ13

VPEQ14

VPEQ24

Dried, shifted syngas from shift reactors

Figure D.1: The 4 bed PSA system

101

PRES AD PED1 PED3 BD PG PEP1 PEP3

PEP3 PRES AD PED1 PED3 BD PG PEP1

BD PG PEP1 PEP3 PRES AD PED1 PED3

PED3 BD PG PEP1 PEP3 PRES AD PED1PED2 PEP2

PED2 PEP2

PEP2 PED2

PEP2 PED2

Figure D.2: Time chart for the PSA cycle. Steps are denoted as: PRES: Pressurisation; AD:

Adsorption; PED1: First Pressure Equalisation (depressurisation); PED2: Second Pressure

Equalisation (depressurisation); PED3: Third Pressure Equalisation (depressurisation); BD: Blow

down; PG: Purging; PEP1: First Pressure Equalisation (pressurisation); PEP2: Second Pressure

Equalisation (pressurisation); PEP3: Third Pressure Equalisation (pressurisation)

Table D.1: Feed gas properties

Pressure 41 bar

Temperature 40 °C

Mole fractions

CO2 0.49

H2 0.51

The adsorbent properties, along with adsorption bed characteristics have been summarised in

Table D.2.

Table D.2: Adsorbent and adsorption bed characteristics

Diameter of adsorption beds 3.5 m

Length of adsorption beds 3 m

Average adsorbent particle radius 0.00115 m (Jee et al., 2001)

Adsorbent particle density 850 kg/m3 (Jee et al., 2001)

Adsorption bed void fraction 0.433 (Jee et al., 2001)

Problem definition

The overall optimisation problem is formulated as given by Equation D.1.

Maximise: & Maximise:

w.r.t.:

Subject to: % 602HP

Where,

: H2 Purity in the H2 product stream (%)

: Percentage of the H2 in feed being recovered in the H2 product stream (%)

: Valve coefficient for Purge Valves (VPurge ) (kmol/bar/s)

(D.1)

102

: H2 product flow rate in adsorption step (kmol/s)

: Adsorption step time (s)

: Blow down pressure (bar)

The explored decision variable space is represented in Table D.3.

Table D.3: Decision variable range for optimisation

Decision Variable Range

0.000766 – 0.00257 kmol/bar/s

0.1–2 kmol/s

1– 60 s

0.1 – 4 bar

The basic steps involved in solving the present problem by NSGA-II algorithm are as follows:

I. As the first step, a random initial population is generated, comprising of (=25)

individual chromosomes, consisting of 4 genes, each corresponding to one decision

variable. The value for the size is chosen based on the suggestions given

by Reeves (2003).

II. This is followed by evaluation of objective function values (i.e. and) for these individual

chromosomes. The initial population is then sorted according to the fast nondominated

sorting algorithm, suggested by Deb et al. (2002).

III. An offspring population of size is then generated using binary tournament selection,

crossover (with a probability of 0.9) and mutation (with a probability of 0.0011).

IV. The objective function value for these off-springs is also evaluated and the combined

parent and offspring population of size ‘2 ’ is then sorted based upon their

nondomination score.

V. The individuals corresponding to the first nondominated front are then selected to make

up the next generation. In case the number of individuals in the first nondominated front

is less than population size , individuals from second nondominated front are included

in the selection. This procedure is repeated until individuals are selected. In case the

number of individuals selected becomes greater than , the individuals in the last

included nondominated front are sorted based on the crowding distance value and the

excess individuals are rejected.

VI. The new population thus generated, goes through steps III, IV and V again until the

maximum number of generations (i.e. 35) is reached.

103

Results and Discussion

The final Pareto front (H2 Purity vs. H2 Recovery), obtained after 35 generations is shown in

Figure D.3. Figures D.4 (a) through (d) show the decision variable values corresponding to the

nondominated solutions.

For low CO2 capture rates (or for ~<95%), the H2 product flow rate in adsorption step

( ) appears to have hit the upper bound. The blow down pressure (i.e. ) and the valve

coefficient for purge valves (i.e. ), however, seem to close to their respective lower

bounds. For low CO2 capture rates, i.e. for ~<95%, the adsorption step time ( ) was the

only decision variable that had a significant impact on the objectives. Increasing the purity of H2,

required a lowering of adsorption time.

For high CO2 capture rates (or for ~>95%), a decrease in purge flow rate, coupled with an

increase in blow down pressure and purge flow rate is also required. Such a relatively low scale

operation, indicted by low values, allows for an increase in , as shown in Figure D.4(c).

For coal to ammonia process, we would typically be interested in values ~>85%, to limit

the amount of H2 loss in the methanator.

Considering an average computation time per simulation as 22.5 min, the total time required for

the standalone optimisation was at least 13 days on an Intel Core i7 processor (3.40 GHz).

Figure D.3: Pareto plot for PSA standalone MOO

104

Figures D.4: (a) Valve coefficient for Purge Valves (VPurge ) ( ), (b) H2 product flow

rate in adsorption step ( , (c) Adsorption step time ( ) and (d) Blow down pressure (

corresponding to the optimum objective function values

Conclusions

A standalone, 4 bed PSA unit has been optimised to maximise both H2 recovery and purity for a

binary feed, having a fixed composition. For the global coal to ammonia optimisation problem,

other decision variables, affecting the productivity of the PSA unit also need to be considered. In

addition to this, variables affecting the amount of feed processed by the PSA unit per unit time

are the pressurisation and depressurisation time, along with the pressure equalisation step time.

105

Appendix E

Two stage refrigeration system optimisation

Multi-stage refrigeration system optimisation

Refrigeration systems are employed in process plants whenever there is a requirement of cooling

below the ambient temperature. Refrigeration systems are typically associated with high

operating and capital costs. It is for this reason that the optimisation of refrigeration systems is

very important when optimising any process plant that operates at a sub-ambient temperature.

Among the various types of refrigeration systems employed in the industry, vapour compression

cycles are the most common. Figure E.1 depicts a basic single stage vapour compression

refrigeration cycle.

Figure E.1: Single stage vapour compression refrigeration cycle

The saturated liquid refrigerant undergoes adiabatic expansion, which results in the refrigerant

getting cooled to the saturation temperature at the lower pressure. The pressure to which the

refrigerant is flashed is decided by the minimum temperature to which the process streams need

to be cooled less the minimum temperature difference . The expanded refrigerant

temperature then satisfies the refrigeration needs of the process by heat exchange in the

evaporator. The saturated refrigerant vapour that leaves the evaporator is then compressed to a

suitable pressure, so that it can again be condensed against cooling water. The shaftwork

required to compress the refrigerant vapour is known to dominate the operating cost of a vapour

compression refrigeration system (Lee, 2001).

The refrigeration demand of process plants is generally distributed over a range of temperatures

and can be seen from the process GCC. A GCC represents a plot of the difference between heat

supplied from all the hot streams and the heat needed by all the cold streams against shifted

temperatures. Shifted temperature refers to the temperature being adjusted according to the

minimum temperature difference assumed for the heat exchanger network. For more details on

Condenser

Evaporator

Compressor

Expansion Valve

Saturated LiquidRefrigerant

106

the GCC, the reader is referred to Kemp (2007). For example, in the hypothetical process GCC,

shown in Figure E.2, Q1 amount of refrigeration could be supplied at temperature, T1 while the

remaining Q2 amount could be provided at temperature level T2. When compared to the case

where all the refrigeration duty was provided at the minimum temperature level, the

configuration shown in Figure E.2 will have lower shaftwork consumption. However, the capital

cost of such systems is also expected to be higher. Such systems, with multiple refrigeration

temperature levels, are called multi-stage refrigeration systems.

Figure E.2: A hypothetical process GCC showing a two-stage refrigeration system

Another option available to reduce the shaftwork of vapour compression systems is to condense

a part of the refrigerant against a process stream cooler than the temperature of cooling water-

thereby reducing the extent to which the refrigerant needs to be compressed ,subsequently

reducing the compressor shaftwork. The region enclosed by the vertical axis and the GCC in

Figure E.2 is typically referred to as a ‘pocket’. The negatively and positively sloped segments of

a GCC represent ‘pseudo-hot’ and ‘pseudo-cold’ streams, respectively. A pocket represents

intra-process heat transfer, in the sense that the pseudo-hot stream heats the pseudo-cold stream.

The pseudo-cold stream could instead be used to condense a part of the compressed refrigerant.

In such a case, the pseudo-hot stream could be cooled with the help of cooling water. In addition

to the two alternatives discussed, Lee (2001) discussed a variety of other design options available

in order to optimise refrigeration systems.

Refrigeration system optimisation methodology

The different options for optimisation of vapour-compression refrigeration systems were

discussed by Lee (2001). These include design features such as: economisers, aftercoolers,

107

presaturators, desuperheaters, suction vapour-liquid heat exchangers, pocket exploitation, multi-

stage cycles and cascade refrigeration systems. The reader is referred to the referenced document

for further details.

Figures E.3 (a) and E.3 (b) depict the typical GCCs for CO2 capture processes involving CO2

pressurisation via compression and condensation and via just compression, respectively. In case

of RectisolTM

, the CO2 is only compressed whilst the PSA process involves CO2 pressurisation

via compression and condensation.

From a given GCC, one can directly extract the refrigeration heat duty to be supplied at different

temperature levels. To convert these thermal duties into an equivalent shaftwork, three

approaches could be used, namely, the Exergy Grand Composite Curve (EGCC), the Coefficient

of Performance (COP) or the detailed approach using a simulation/model of the refrigeration

system.

108

Figure E.3: (a) GCC for CO2 pressurisation via compression and condensation (b) GCC for CO2

pressurisation via compression (Where; W1 and W2 are the two shaftwork and, and

are the two condensing duties)

Linnhoff and Dhole (1989) proposed the usage of an exergy based method to estimate the

shaftwork requirements of a refrigeration system. This method is based on using the area

enclosed by the Exergy Composite Curves (ECC) and utilities to estimate the required shaft

work. It has been used by many researchers over the years. Hackl and Harvey (2013) used it in

their work that was based on an industrial case study. Gatti et al. (2013, 2014(a), 2014(b)) have

also used this approach to estimate shaftwork for a variety of refrigeration systems that they

considered. The second method is the COP-based method (Smith (2005)). This approach makes

use of process GCC to estimate the duty at different temperatures and to predict the

corresponding shaftwork with the help of the COP. The third approach is dependent on

performing detailed simulations for different design options and estimating the corresponding

shaftwork. This approach introduces an extra layer of complexity within the MOO framework.

In the present work, the relatively simpler and easy-to-implement COP-based method is used to

estimate the shaftwork at different temperature levels. An average factor of 0.6 has been used for

each stage to account for the deviations from an ideal isentropic operation. The value of 0.6 was

found to be a reasonable assumption for the refrigerant under consideration, i.e. propylene, by

carrying out separate detailed simulations of a two-stage refrigeration system. Table E.1

summaries the results obtained from these simulations.

109

Table E.1: Results from detailed simulations to verify the assumption related to using a factor of

0.6 to account for deviations from an ideal isentropic operation

Refrigeration Temperature

levels (°C)

Shaftwork from

simulation (MWe)

Shaftwork from COP

Method (assuming the

0.6 factor) (MWe)

Dataset 1 -5.69

1.52 1.56 -47

Dataset 2 -2.80

1.10 1.07 -47

For the GCC shown in Figure E.3 (a), the ‘pocket exploitation’ optimisation problem can be

formulated as follows:

minmin

minmin

min

minmin

min

minmin

,/

*5.015.273*

*5.015.273

*5.015.273

TT

TTT

TT

TTTQQQQ

TT

TTTQ

MinE

CWS

cond

cond

condcondCWtot

cond

cond

cond

idealiBPAP

S

S

S

S

(E.1)

condQ and ScondT are linearly related and the exact relation can be deduced by decomposition of

the GCC.

It is important to note that exploiting pocket 2 would require the pseudo-hot stream to be

supplied with additional refrigeration. All such pockets have been ignored in this work since they

would require an additional refrigeration stage for the pseudo-hot stream.

Alternatively, as can be seen from Figure E.3 (b), the total refrigeration duty could also be

supplied at two sub-ambient temperature levels. The minimum temperature difference between

the two refrigeration levels has been taken as 15 °C to ensure a practical minimum pressure ratio

between the two stages. The optimisation problem, in this case, can be mathematically

formulated as follows:

110

minmin

minmin

min

min

,

*5.015.273*

*5.015.273

TT

TTTQQ

TT

TTTQ

MinE

CWSevaptot

evap

evapCWS

evap

idealWP

S

S

(E.2)

For the case involving CO2 condensation, another promising option would be to have an

additional refrigeration stage at the CO2 condensation temperature. This option is explained in

Figure E.4. The shaftwork corresponding to this option is given by Equation E.3.

Figure E.4: Extra refrigeration level at the CO2 condensation temperature

minmin

minmin

min

min

,

*5.015.273*

*5.015.273

TT

TTTQQQ

TT

TTTQ

E

CWS

CWevaptot

evap

evapCWS

evap

idealT

S

S

(E.3)

After considering all design options, the minimum shaftwork among all these options is selected

as the optimum shaftwork and the particular design option as the optimum configuration.

111

Thus,

idealTidealWPidealiBPidealiAPoptref EEEEMinE ,,,,, ,,, (E.4)

The refrigeration system optimisation algorithm is shown in the form of a flowchart in

Figure E.5.

112

Figure E.5: Two stage refrigeration system optimisation algorithm

113

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Acknowledgments

I would like to express my gratitude and thanks to my supervisors; Prof. Andrew Hoadley, Prof.

Sanjay Mahajani and Prof. Anuradda Ganesh, for their constant guidance and supervision.

During the highs and lows of my PhD journey, they have always been there to provide the much

needed support. I sincerely acknowledge the constructive suggestions provided by Prof. Santanu

Bandyopadhyay, Prof. Anand B. Rao, Prof. Yogendra Shastri, Prof. Sankar Bhattacharya, Prof.

Akshat Tanksale during the annual research progress committee meetings. I also thank Prof.

Arun Sadashio Moharir (IITB), Prof. Paul Webley (University of Melbourne) and Prof. François

Maréchal (EPFL) for providing me guidance at different junctions of my PhD journey.

I would also like to acknowledge the financial and academic support provided by Orica Ltd.

through IITB-Monash research academy (Project ID: IMURA 0221 (B)). In particular, I would

like to acknowledge the support from Dr. Greg Rigby and Dr. Milinda Ranasinghe at Orica.

I am thankful to the staff at IITB-Monash research academy namely; Prof. Mohan

Krishnamoorthy, Dr. Murali Sastry, Anasuya ma’am, Kuheli ma’am, Mamta ma’am, Nancy

ma’am, Jayasree, Laya, Priyanka, Kiran, Rahul and Bharat, for their timely and much needed

assistance. I would like to thank Mrs. Jill Crisfield for helping me during my stay at Monash.

I would also like to specially acknowledge Yughabala for her constant support and motivation

throughout my PhD. I would also thank my friend Saurabh because of whom the last five years

just whisked away. I would specially thank him and his flat mate, Prateek, for lending me his

sofa to ‘occasionally’ crash. I am also thankful to my friends Ramil, Sourav, Vineet, Mohit

Prashant and Prabhav for the eventful time I had during my stay at Bombay. I am also thankful

to my lab mates; Pratham, Nilam, Bhoja, Sravan, Ganesh, Kapish, Sanchit, Sminu, Rohidas and

Detke; for all the help and support in my PhD journey.

Last but not the least; I would like to thank my parents for the never ending stream of

encouragement and love that I was fortunate enough to receive. The constant support of my

sister, Swati, and my fiancée, Meenakshi, is also deeply acknowledged.

Ishan Sharma

120

List of Publications

Journal Publications/Book Chapter

Sharma I., Hoadley A., Mahajani S.M., Ganesh A., Chapter 14: MOO of a Complex Process

– A Surrogate-Assisted Approach, G.P. Rangaiah (Ed.), Multi-objective Optimisation:

Techniques and Applications in chemical engineering (2nd Edition), Singapore: World

Scientific (In Press)

Sharma I., Hoadley A., Mahajani S.M., Ganesh A., Multi-Objective Optimisation of a

RectisolTM

process for CO2 capture, Journal of Cleaner Production, 119, 196-206, 2016

Sharma I., Hoadley A., Mahajani S.M., Ganesh A., Methodology for “surrogate-assisted"

Multi-Objective Optimisation (MOO) for computationally expensive process flowsheet

analysis, Chemical Engineering Transactions, 45, 349-354, 2015

Paper in Conference Proceedings

Sharma I., Arora P., Hoadley A., Mahajani S.M., Ganesh A., Remote, small-scale, ‘greener’

routes of ammonia production, Proceedings of Efficiency, Cost, Optimisation, Simulation

and Environmental Impact of Energy Systems (ECOS)-2016, Portoroz, Slovenia, 19th June to

23rd June, 2016

Sharma I., Hoadley A., Mahajani S.M., Ganesh A., “Surrogate-assisted” optimisation of

Pressure Swing Adsorption (PSA) Process, Proceedings of Asian Pacific Confederation of

Chemical Engineering (APCChE)-2015, Melbourne, Australia, 27th

September to 1st

October,

2015

Sharma I., Hoadley A., Mahajani S.M., Ganesh A., Optimisation of pressure swing

adsorption (PSA) process for producing high purity CO2 for sequestration purposes,

Chemical Engineering Transactions, 39, 1111-1116, 2014

Sharma I., Hoadley A., Mahajani S.M., Ganesh A., Automated optimisation of multi stage

refrigeration systems within a multi-objective optimisation framework, Chemical

Engineering Transactions, 39, 25-30, 2014

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