Modeling and Mode1 Updating in the Real-Time Optimization of Gasoline Blending

Aseema Singh

A thesis submittea in conformity with the requirements for the degree of 4Iaster of Applied

Science

Department of Chernical Engineering and Applied Chemistry

University of Toronto

@ Copyright by Aseerna Singh (1997)

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MODELING AND MODEL UPDATING IN THE REAL-TIME OPTIMIZATION

OF GASOLINE BLENDING

Master of Applied Science, 1997, by Aseema Singh,

Graduate Department of Chernical Engineering and Applied Chemistry,

University of Toronto

Abstract

This thesis provides a new approach to gasoline blender control that can out-perform con-

ventional blend controllers. The proposed blend controuer includes appropriate noniinear

bleiiding modcls and adopts concepts froni mode1 predictive control theory in order to allow

it to liandlc stochastic dis t urbances in feeds tock quali t ies. The resulting Real-Time Opti-

mization (RTO) system is similar to model predictive control in that it predicts disturbances

over one time horizon and optimizes the blender control problem over another. It then irnple-

nieiits coiitrol action for tlic current tiinc-step aiid repeats the process in a recediiig horizon

fasiiiori.

r\iiotli(:r irriportaiit coritribiitiori is tlint of parnrrictcr observability for stcady-statc RTO sys-

tciiis. Tlic ciirrcritly nviiiablc mctliod providcs only n iiecessary biit riot sufficient coiiditioii.

A iicw ;rpproncli lias 11ce1i prescntcd wliicii uses fundainentai statistical priiiciples nxid is

~ippli~~:il~li! t.o ;uiy stcady-stntc systciii wlicrc secoiidary measiirements arc used to est iiiintc

uiiiiic;\siii-(*cl (liiaiitities. Obscrval>ility is tlicii cxtciided to a. iiicasiirc of degrce of observability.

Acknowledgment s

I would like to thank my supervisor, Dr. J. F. Forbes, for his valuable ,gidance and great

patience during this work. 1 also wish to express my deep appreciation to the Department of

Chernical and Materials Engineering, particularly the Cornputer Process Controi group, at the

University of Alberta. Findy , a special word of thanks goes to the members of my family for

their support and encouragement t hroughout t his project.

iii

Contents

Abstract

Acknowledgement s

List of Figures

List of Thbles

. . 11

iii

vii

vi ii

1 INTRODUCTION 1

9 1.1 RTOSTRUCTT,'RE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 1.2 RTO DESIGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 GASOLI3T BLEhDISG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 B L E n COXTROLLERS . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 THESIS OBJECTNES & SCOPE . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 THESIS CO>TTSTIOSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 GASOLINE BLENDING MODELS 15

2.1 OCT.L\T hT'rlBER 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1 BLESDISG OCT.L,S 72 SGlIBER METHOD . . . . . . . . . . . . . . . . 16

2.1.2 E T m l RT-70 SIETHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.3 STE\tTt91yT METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.4 ISTER4CTIOS 1. IETHOD . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 . 1 TReVSFOR\L\nOX SIETHOD . . . . . . . . . . . . . . . . . . . . . . 20

2.1.6 EXCESS JIETHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.7 ZMED METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.8 SUMPvIARY OF OCTANE BLENDGVG MODELS . . . . . . . . . . . . . 25

2.2 REID VAPOUR PRESSURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 THEORETICAL APPROACHES . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 INTERACTION METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.3 BLENDING INDEX METHOD . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.4 OTHER .ME THODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.5 SUMMARY OF RVP BLEXDNG MODELS . . . . . . . . . . . . . . . . 31

2.3 ASTM DISTILLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 INTERACTION METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 EMPIRICAL MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 BLENDING SIhIULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 PARAMETER OBSERVABILITY IN RTO SYSTEMS 35

3.1 BACKGROUND 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 PARAMETER OBSERVABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 DEGREE O F OBSERVABILITY . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 C O V . C E M.4TRIX APPROmlATION . . . . . . . . . . . . . . . . . . . 50

3.3.1 ESTIMATIOx EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 BLENDING CONTROL TECHNOLOGY 61

4.1 INTRODUCTlOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 GASOLI-NE BLEXDIXG BENCHMARK PROBLEM . . . . . . . . . . . . . . . 62

4.2.1 BLENDING PROBLEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2 STOCHASTIC D1STURBAi';CES . . . . . . . . . . . . . . . . . . . . . . 64

4.2.3 CONTROLLER PERFOm.IARiCE . . . . . . . . . . . . . . . . . . . . . . 65

4.3 IDEAL BLEND CONTROLLER . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.1 OPTIMAL BLEXD TRAJECTORY FOR BENCHMARK PROBLEM . . 68

4.4 CONVENTIONAL CONTROLLER . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4.1 CONVENTIONAL CONTROLLER PERFORMANCE STUDY . . . . . 71

4.5 PARAMETRIC MISMATCH CASE . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5.1 PARAMETRIC MISMATCH PERFORMAWCE STUDY . . . . . . . . . 74

4.6 TIMEHORIZON CONTROLLER . . . . . . . . . . . . . . . . . . . . . . . - . . 76

4.6.1 TIME-HORIZON CONTROLLER PERFORMAVCE STUDY . . . . . . 80

4.7 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 SUMhlARY, CONCLUSIONS, & RECOMMENDATIONS 84

5.1 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . 84

5.2 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 RECOMMENDATIONS . . . . . . . . . . . . . . . . . . . . - . . . . . . , . . . . 86

Nomenclature 88

Bibliography

Appendix A 101

Appendix B

Appendix C

Appendix D 108

Appendix E

Appendix F

Appendix G

Appendix H

List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 RTO Structure 3

1-2 Refinery Flow Diagram (blender feedstocks indicated in boId) . . . . . . . . . . . 6 CT 1-3 Gasoline Blending Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1-4 Blending Control Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3-1 Steady-State System and Estimator Considered by Stanley and Mah [1981] . . . . 38

3-2 Parameter Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Confidence Contours 44

3-4 Probability Contours for Example 3.2 . . . . . . . . . . . . . . . . . . . . . . . . 50

4 1 Blending Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 2 Reformate qualites during 24 hour blend . . . . . . . . . . . . . . . . . . . . . . . 66

4-3 Blended Octane using LP + Bias Controller . . . . . . . . . . . . . . . . . . . . . 72

4 4 BIended RVP using LP i- Bias Controller . . . . . . . . . . . . . . . . . . . . . . 73

4-5 Uncaptured Profit for 24 hour Blend . . . . . . . . . . . . . . . . . . . . . . . . . 83

vii

List of Tables

2.1 Cornparison of Octane Blending Models . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Prwtictive Accuracy of Octane Blending Models . . . . . . . . . . . . . . . . . 27

2.3 Predictive Accuracy of RVP Blending Models . . . . . . . . . . . . . . . . . . . . 31

4.1 Production Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Feedstock Economic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Feedstock Qualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Feedstock Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Deviation Erom Specification at End of Blend . . . . . . . . . . . . . . . . . . . . 75

4.6 Cornparison of Time-Horizon Controllers . . . . . . . . . . . . . . . . . . . . . . . 82

viii

Chapter 1

INTRODUCTION

The econornic benefits of process control are weli recognized in the chernicai and petrochemical

industries. These benefits have led to the widespread use of advanceci automation technologies

in such plants. The control system in these plants are designed to operate the process using

some pre-determineci policy, which may or may not be optimal. Additional benefits, which

can be crucial in providing a cornpetitive edge, can be reaiized by operating the process in

an economicdy optimal manner. This is usually achieved by the optimization of steady-state

operations policy [Forbes, 19941.

The major categories of steady-state optimization methods are direct search and model-based

optimization methods. Direct search methods 1e.g. McFarlane and Bacon, 19891 work on

the actual process and involve plant experimentation to obtain new steady-state operations.

The direction in which process economics are improved can be deduceà £iom the experimental

results and the process optimum is found by iteratively rnoving the process in this direction

after each plant experiment. However. the amount of experimentation required, especiaiiy for

large dimensional problems, can be prohibitive. Moreover , for complex and integrated plants

a i th slow dynamics, it can take a long time For the process to reach steady-state after each

perturbation. Using direct search methods on such plants would require the process to operate

outside the normal region of operation for long periods of time, which may not be acceptable.

Model-based optimization methods, on the ot her hand, work on mat hematical models t hat

represent the plant, rather than the process itself. Optimization algorithms are used to find the

optimum of the model, the results of which, after verification, are implemented on the plant.

A Real-Time Optimization (RTO) system is considered to be any closed-Loop control system

that is used to determine the optimal process operations and to implement this policy on the

plant through the underlying control system. On-Iine optimization is required when plants are

subject to disturbances which cause the procgs behaviour and the optimal process operation

to vary substantially with tirne. In such situations, the econornic optimum process operation

is not a point but rather a trajectory in time and the objective of the RTO system is to cause

the process to track this optimal trajectory.

One of the key petroleum refining processes which has long benefitted from RTO technology

is gasoline blending [Leung, 19851. This thesis considers RTO systems for gasoline blender

control and offers solutions for improving blender efficiency. In doing so, obsenability issues in

conventional steady-state RTO systems are also addressed.

1.1 RTO STRUCTURE

The structure of a typical RTO system is shown schematicdy in Figure 1-1 [Seferlis, 19951.

The constituent subsystems are briefly described below:

1. Process measurement: relevant measurements are taken which are used for updating

modek in the optimizer.

2. Data validation: at this stage, process measurements are examined to ensure that the

process has indeed reached steady-state. Also, Cross Error Detection and Data Reconcil-

iation are carried out. Gros Error Detection involves detecting errors in measurements

resulting from sensor failliles, leaks, etc., so as to eiirninate errornous data from being used

in mode1 updating. Data Reconciliation tries to adjust the data such that the material

and energy balances are satisfied [Seferlis, 1995).

3. Model updating: validated data is used to update the model in some way. Model updat-

ing involves obtaining new estimates of sorne of the model parameters using estimation

techniques such as regression analysis.

4. Optimization: model-based, steady-state optirnization is used to calculate controiler set-

points. The optimization algorithm can be a linear program (LP) or a nonlinear program

Validation 7 Controllen -_i,

I Sensors Process

figure 1-1: RTO Structure

(NLP) and, may require mixed integer programming (MIP) depending on the nature of

the mode1 equat ions.

5. Cornmand conditioning (also called post-optirnality or results analysis): at this stage, the

outputs from the optimizer are examined in order to ensure that only acceptable setpoints

which wili result in a significantly improved plant operation, are passed to the controllers

[Seferlis, 19951.

6. Control: the acceptable setpoints are used by the controllers to implement the operations

policy on the plant. The controller structure can range from simple PID loops to advanced

control schemes such as mode1 predictive control (MPC).

More detailed treatments of the RTO structure can be found in Seferlis [1995] and Forbes [1994].

1.2 RTO DESIGN

The performance of any RTO system will depend on the characteristics of each of the constituent

subsystems and their interactions with each other [Forbes, 19941. Al1 the components in the

RTO system depicted in Figure 1-1 have been studied to some extent by various researchers.

Krishnan [1990] has studied the problem of selecting appropriate measurements for model u p

dating. A measurement selection strategy based on minirnizing error in estimated parameters

has b e n provided. Issues in Gross Error Detection and Data Reconciliation have been ad-

dressed by Crowe [19881, Rosenberg e t al. [1987], and Tjoa and Biegler [1990].

Crucial to the success of the RTO systern is the quality of models used on the optimizer. Biegler

et al. [1985] have found that a model is adequate for use in RTO if model parameter \dues

can be found such that the gradients of the model with respect to the decision variables can

be forced to match those of the plant. Mode1 selection has also been studied by Forbes et al.

(19941 who have defined model adequacy in ternis of the model's ability to possess an optimum

at the true plant optimum. That is, a model is defined to be point-wise adequate if there exist

a set of values for the adjustable model pararneters such that the model-based optimization

possesses an optimum at the true plant optimum.

The model adequacy approach taken by Forbes e t al. (19941 d o w s for the selection of adjustable

model parameters. That is, model adequacy can be used as a criterion for selecting adjustable

model parameters. Sirnilarly, Roberts [1979], [1995] has stated t hat models and parameter

estimation strategies should not be selected in isolation; the characteristics of the process mode1

should be taken into account in order to allow the system to effectively handle plant-mode1

mismatch. Krishnan [1990] has also looked at the problem of selecting the adjustable parameters

in the model to be updated on-line and has suggested updating those parameters that have the

most effect on the optimal value of the objective function or alter the active constraint set.

Observability of the adjustable parameters from the available measurements has been studied

by Krishnan [1990].

Optimization algorithms have b e n developed for several classes of optimization problems such

as LP, XLP, and MIP, and are being continually improved for increased computational el-

ficiency. Optimization theory and algorithms c m be found in Fletcher (19871, Avriel [1976],

Balakrishnan and Boyd [1992], Duran and Grossman [1986], and Edgar and Himmelblau [1988].

Post optimality analysis has been studied by Seferlis [1995] and Miletic [19961. Tests have been

provided to determine whether or not setpoints generateà by the optimizer are to be passed to

the cont rollers.

While most of the work has focused on individual components of the W O system, the in-

teractions among the subsystems and their effect on the whole structure have received Little

attention. Forbes and Marlin [1996] have provided the first approach to making FLT0 decisions

based on the whole system. The approach is based on designhg RTO components such that

the loss of RTO performance is minimized. The measure of RTO performance used by Forbes

and Marlin [1996], the Design Cost Criteria' looks a t the loss of econornic opportunity resdting

frorn a given RTO design. This mesure considers the deviation of the manipulated variables

from the true plant optimum. An alternate approach due to Loeblin and Perkins [19961 uses

the average deviation of the objective function rather than the setpoints.

The past sections have focused on RTO system without application to any particular process.

Since this thesis is concerned nrith RTO of gasoline blending, gasoline blending will be discussed

next .

1.3 GASOLINE BLENDING

Gasoline is one of the most important refinery products since it can yield 60 to ?O percent of a

typical refinery's total revenue [DeWitt et ai., 19891. Fractional distillation of crude oil produces

several "cuts". Only a small portion of the distillation products, the light virgin naphtha or

light straight run (LSR) naphtha, is suitable for blending directly into gasoline. Other fractions

have to be further processed. A simpiified flow diagram of the refinery processes is given in

Figure 1-2. The flow diagram shows the main units and the refinery processes required for

producing gasoline feedstocks in a typical refinery. Some of the refining processes are briefly

described below [Givens, 19851:

1. Catalytic Reforming: this is an upgrading process that increases the octane nurnber of

the heavy virgin napht ha by catalytically convert ing low octane hydrocarbonç into high

octane ones. The most important reactions taking place include the conversion of rnethyl

cyclohexane (research octane number of 75) and n-heptane (zero octane by definition) to

toluene which has a research octane of 120.

2. Catalytic Cracking: this is a co~version p r o c w that breaks large, low volatility mole-

cules into smaller, more volatile ones. The product is then sent to the kactionator, a

Mixcd Butanes* B a t m a

Butyl- Alkylik +

Light Virgin Naphh . U R Niphth.

Virgin DiniIlau

n

u Distillation

b Virgin Distillate I Light CycIe Oit

c d oil

T L Huw Cycle 0il1

Vacuum Gas Oil Cracking Caral ytic bcaulytic Cu. A 1

Rcducui Cnidc Coke

Reductd Cnide Asphah

Figure 1-2: Refinery Flow Diagram (blender feedstocks indicated in bold)

distillation colurnn, that separates the stream according to boiling range. About 50% of

the catalyticdy cracked stream can be blended into gasoline without hrther upgrading.

Another 20% contains light gases from which n-butane and isebutane are separated and

used later for blending in gasoline.

3. Alkylation: down stream from the catalytic cracker, the alkylation unit further processes

light olefins produced during catalytic cracking. Although the light olefin stream has high

octane quality, its boiling range is too low for gasoline. Alkylation involves catalytically

reacting these olefins Rrith iso-butane to form high quality hydrocarbons such as i s e

octane.

4. Hydrotreating: feedstocks (including straight run naphtha frorn the crude distillation

unit) are treated with hydrogen in the presence of catalysts to reduce the amount of

sulphur, nitrogen, and oxygen.

Some of these refinery products are further separated into light and heavy fractions in order to

provide more ffexibility in blending. A large refinery can have more than 20 blender feedstocks

that are blended into several grades of gasoline [Civens, 19851. Gasoiine blending is the process

of combining these blender feedstocks, together with s m d amounts of additives (such as an-

tioxidants, corrosion inhibit ors, met al deactivators, detergents, and dyes) , to make a mixture

meeting certain quali ty specificat ions. S pecificat ions on gasohe qualities such as octane num-

ber, volatiiity, sulphur content, aromatics content, and viscosity are placed in order to ensure

acceptable engine performance as well as for environmental reasons. Feedstocks can be blended

out of storage tanks, intermediate tanks into which upstream processes are being fed. or directly

kom upstream processes (Agrawal, 19951. The gasoline blending process of blending feedstocks

frorn storage tanks, is shown in Figure 1-3.

LSR Naphtha

Reformate Regular

Additives

Figure 1-3: Gasoline Blending Process

Octane numbers are measures of a fuel's antiknock properties. Knocking occurs when a fuel/air

mixture ignites prematurely in the cylinder, producing a *'knocking" sound in the engine.

Knocking not only reduces the engine's power by working against the piston, but it also exerts

mechanical stress on the engine parts. The octane number of a fuel is defined as the percent of

iso-octane (2,2.4-trimethyl pentane) in a blend with n-heptane that exhibits the sarne resistance

to knocking as the test fuel under standard conditions in a standard engine [Palmer and Smith,

1985). On the octane scale. iseoctane is assigned an octane number of 100 and n-heptane a

number of O. Two standard test procedures are used to characterize the antiknock properties of

fuels for spark engines - the ASTM D-908 test gives the research octane number ( R O N ) and the

ASTM D357 gives the motor octane number (MON). The RON represents antilcnock proper-

ties under conditions of low speed and frequent accelerations while the h1ON represents engine

performance under more severe high speed conditions. The octane number posted at gasoline

stations is the arithmetic mean of the RON and MON [Gary and Handwerk.

19941. The main difference berneen automotive gasoline grades is their antiknock properties.

For example. regular and prernium gasolines are specified to have posted octane numbers of 87

and 92, respectiveiy.

Other important properties that afTect engine performance are volatility and boiling range.

A vapour pressure which is too high for the given ambient temperature wili resuit in vapour

stalling ana motor locking, while a vapour pressure which is too iow will lead to difficulties

in engine startup [Palmer and Smith, 19851. The boiling range also affects the engine during

startup and driving, and is particularly important for good performance during quick accel-

eration and high speed operatiom. Like vapour pressure, the boiling range has to be tailored

for a given geographicd region and season. There should be a fairly reasonable distribution

of light (more volatile), intermediate, and heavy (relatively non-volatile) components: however,

gasolines blended for use in cold climates need more iight components (front end volatility) than

t hose intended for use in w m e r climates [Palmer and Smith, 19851. In addition, governrnent

regulations place maximum restrictions on the vapour pressure to limit the emission of volatile

organic compounds into the atmosphere [Lo. 1994j. Other environmental restrictions include

maximum compositions of arornatics, olefin, sulphur, and oxygenates.

The gasoline blending challenge is to produce blends in such a way as to maximize profit while

meeting al1 quality specifications on a11 blends, in addition to satisfying any product dernands

and feedstock availability limits. Thus the blending problem is naturally posed as a constrained

optimization problem and the blender control is traditionally based on optimization technology

[Leffler, 19851.

1.3.1 BLEND CONTROLLERS

Blend controliers are used for making gasolines that meet quaiity specifications whiie satisfjring

demand and availability limits. In addition, the controller shodd work to rnaxirnize profit by

using the least expensive feedstocks to make the most valuable products, minimizing "quality

giveaway", and minimizing reblends. "Quaiity giveaway' refers to making gasoline with quaii-

ties that either exceed the minimum required or fa11 below the maximum allowed. thus reducing

the profit margin. It is estimated that consistent octane giveaway of 0.1 can cost a refinery

several millions of dollars a year [Givens, 19851. Reblending can also lead to significant cwts to

a refinery by taking up valuable tank space and blending time, reducing overd capacity Thus.

the controller shauld be able to blend the products optimally and get the blends on specification

the first time.

There is a wide range of sophistication ic blending control technology presently employed by

refiners. The typical state-of-the-art blending technology is built on t h e e levels: off-line opti-

rnizer or scheduier, on-line optimizer, and regulatory control [Sullivan, 19901, [ Agrawal, 19951.

The leveis of control strategy are shown schematicaily in Figure 1-4.

At the top of the hierarchy lies the planner or scfieduler which uses off-Iine optimization to

plan retinery operations for the long range (months), intermediate range (weeks). and short

range (days). Long-range planning can be performed at either the refinery or corporate level.

Long range forecasts of product dernands, crude oil prices, and process unit performances are

used to plan refinery operations several months to a few years into the future [Su l l i~n . 19901,

[Agrawal, 19%). Medium-range planning provides scheduling of refinery operations on a shorter

time frarne (a few weeks) and is used to improve on the long-range plan. In addition to econornic

forecasts, forecasts of refinery feedstock qualities and quantities, and tank availability are used

to check feasibility of the long range plan and to revise production targets set by the long-range

planner. Here, the refinery schedule is also adjusted for major unit upsets and unscheduled

shutdowns. Significant changes in expected crude quality and changes in operating flexibility

(such as additional constraints) are used at t his stage to revise refinery plans as well [Sullivan,

19901. Short-range planning works at the unit level, rather than at the refinery level, and plans

blender operations over a shorter time frame (1 to 2 days) [Agrawal, 19951. At this stage,

assumed feedstock qualities (or measured qualities, if available) are used to produce blend

f Econornics inventory

On-Iine op=t / setpoints I

I , -k i

Controller . , . l

Figure 1-4: Blending Cont rol Hierarchy

recipes which are then downloaded to the on-line optimizer. Usuall . all levels of planning

using an off-line optimizer are conducted using linear programming [Sulli\xn. 199O]. [Rarnsey

Jr. and Truesdale. 1990]. (DeWitt e t al.. 19891.

The on-line optimizer uses on-line blended quaiity information to rnodify the initial recipe

during the blend and provides ha1 blend recipes to be executed by the controller. On-line

optirnizers are designed to minimize deviation from the initial blend recipe or optimize some

economic performance (such as profit mauimization) while satisfjnng al1 constraints [Michaiek

e t aL. 19941. However, the move has been towards optimizing steady-state process economics.

Thus the on-line optimizer is usually a conventional RTO system. Vermeer et al. [1996] have

expanded on the conventionai steady-state RTO to perform dynarnic blend optimization. This

is achieved by including blending dynamics into the controller to calculate expected blended

qualities and using these expected qualities as feedback to the LP based on-line optimizer.

Incorporating blending dynamics in such a way allows the RTO layer to operate at a higher

frequency.

The blend recipes generated by the on-line optimizer are then implemented at the control

level. Final blend control is achieved through conventional distributed control systems (DCS) .

The DCS-resident controiler (usually PID loops) maintains flow ratio to match the recipe.

maintains a target flowrate for the sum of all feedstodc flows, and stops when the required

amount of product has been blended [Ajgawal, 19951. The control layer is aiso used to ensure

that quality specifications are met. It uses analyzer feedback and can deviate from the blend

recipe determined by the on-line optimizer in order to keep the products on specification.

The different subsystems in Fiewe 1-4 have to be integrated to provide information Aow (such

as tank and feedstock inventory) necessary for the success of the whole system. Databases are

used to keep track of feedstock inventory, tank availabiiity, feedstock qualities, and blended

quaiities. The subsystems mi t e to and/or retrieve information £rom these databases, thus

linking the subsysterns into an integrated network [Agrawal? 1995j, [Ramsey Jr. and Truesdale.

19901, [Palmer et al., 19951. [Ifichalek et al., 19941.

At the heart of the entire blending control system shown in Figure 1-4 is the RTO Iayer which

determines the final blend recipe and so the performance of the RTO system is crucial to the

performance of the whole gasoline blender control effort. The design and performance of the

gasoline blender RTO system is the focils of this t hesis.

The on-line optimizer is usually based on linear programming (e.g. Benefield and Broadway,

[1985], Diaz and Barsamian, [1996]? White and Hall, (19921. McDonald et al-, [l992], Serpemen

et al.' [1992]: Slichalek et al.. [1994]) even though most important gasoline properties have

been shown to blend in a nonlinear fashion [Rusin, 19751. -4s a result, blender control systems

must deal with structural mismatch. Some refiners have tri& addressing this problem by using

nonlinear models and employing nodinear programming (hTP) in the on-line optimizer (e .g .

Ramsey Jr. and Truesdaie? [1990]): however, linear programming remains as the predorninant

technolog-. oaing its popularity to reliability and ease of use [Magoulas et al., 19881. Another

approach that has been ernployed is to linearize the nonlinear models and use sequential linear

programming (SLP) (e.g. Diaz and Barsamian, j19961).

In an attempt to maintain accuracy of the rnodels in the optimizer, the feedback ernployed is

usually "bias updating" [e-g. Diaz and Barsamian, [1996], Bain et a l , [1993]). Bias updating

involves comparing measured blended qualities with those predicted by the models in the on-

Iine optimizer; the difference between the two is then added as an error term to the appropriate

blending models. Thus the RT0 system is most often formulateci as iinear programming ~ 4 t h

bias updating.

This approach of iinear programming with bias updating has proven quite successfui in practice

and is the basis of most commercially available biend control systems. Its success was largely

due to the practice of blending from well-mixed storage tanks. The current trends of in-line

biending and blending from "running" tanks presents a problem that such a control structure

cannot adequately address. This is because upst rearn proces variations result ing fkom cat al-yst

deactivation, heat exchanger fouling, etc.. will cause variations in the feedstock qualities. The

LP (or XLP) plus bias update formdation assumes constant feedstock qualities and so can

not adequately handle such time mrying feedstock qualities. Forbes and Mariin j1994j have

shown that this traditional blend controlier approach can lead to a substantial loss in blender

profitability given eLren a srnall cariation in feedstock qualities.

Xote that in the case of tirne-varying feedstocks, the bias updating approach can still be used

to make products because the lou-er level feedback controllers in the RTO system dl dek-iate

from the blend recipe determinecl by the on-line optimizer in order to meet specifications as

discussed above. Although products can be made in this manner to meet specifications, the

resulting blend u-ill not necessarily be optimal in that the best cornbination of feedstocks may

not have been used.

Some refiners have tried addressing this problem by feedback of laboratory anal-mes of feedstock

qualities and then using multiperiod optimization (e.g. McDonald et al., [1992] and Rigby et

al.. [1995]). The in tend between optirnizations is usually a day and feedstock qualities are

assumed to remain unchanged during these penods. Thus this approach cannot effectively deal

with higher fiequency disturbances arising Erom upstream process operation changes. Vermeer

et al. [1996] have used blended quality measurements to predict disturbances in t hese qualities.

The disturbances, however, are assurneci to he step disturbances that remain constant during

the optimization intenal. Therefore. this approach s d e n from the same drawback as the

multiperiod opt imization strategy.

This t hesis wiU address the probiems associated with structural mismatch and inability of

the bias update approach to adequately handle fluctuations in feedstock qualitis. A new

control approach that can incorporate nonlinear blending models and can deal with t ime -wing

qualities will be developed.

1.4 THESIS OBJECTIVE% & SCOPE

This thesis is concerned with the design of the blend controiler shown in Figure 1-4. and the

FE0 layer in particuiar. A s has been discussed in the previouç section. the W O iayer is

most often formulated as Iinear programrning with bias updating. The LP plus bias updating

formulation suffers &orn plant /mode1 mismat ch since most important gasoline quaiit ies blend

nonlinearly. Generaliy, the controiler is not capable of adequately handling disturbances in

feedstock qualities [Forbes and Mariin. N94!. The objective of this thesis is to proiide a blend

controller t hat incorporates models which capture the inherent nonlinearities of the blending

process. and one that can effectively deal with predictable kariations in feedstock qualities.

Gasoline blending models are examineci in Chapter 2 with respect to predictive accuracy and

their ability to be implemented for control purposes. Once scitable models have been selected.

it is crucial that their predictive accuracy be maintainecl by updating some parameters on-line.

Therefore it is important to ensure t hat the adjustable parameters in these rnodels be obsen-

able £rom the process measurements and the updating scheme. Chapter 3 look at parameter

obsenability for steady-state systems. A measure of observability for steady-sr ate systems is

dewloped and the concept is evtended to degree of obserrability. The observability measure

is based on fundamental statistical principles and can be applied to the blending rnodels. The

observability measure presented in Chapter 3 is not limited to the blending process or to RTO

systems, but is applicable to any steady-state system where unrneasured quantities are to be

est imated using available measurements.

Chapter 4 presents a new approach to blender control that can incorporate nonlinear models

and can effectivel~ deal with feedstock quality variations. The control approach is analogous

to 3lPC in that a time-horizon approach is taken. A gasoline blending case-study is used to

dernonstrate the problems associated with the conventional bias updating approach and the

benefits of adopting the new blending strategy.

Although the tirne-horizon strategy has been applied to the RT0 layer in this thesis, it codd

also be adopted into the off-line optimization iayer and used to improve the effectiveneçs of the

planning and scheduiing layer. Moreover, it codd be applied to on-line or off-line optirnization

of other processes that are subject to stochastic disturbances.

1.5 THESIS CONVENTIONS

In accordance with the petroleum industry terminolog-, light and heavy components refer to

relatively volatile and nonvolatile cornponents, respectively. Ali terrns are defined on first use

and used thereafter. The Nomenclature section lists all syrnbols employed in this thesis while

the acronynis used are summarized in the Glossary of Terms section.

Chapter 2

GASOLINE BLENDING MODELS

There are several properties that are important in characterizing automotive gasoline such as

octane number (ON), Reid vapour pressure (RVP), ASTM distillation points, viscosity, flash

point, and aniline point. Ideal mixing refers to a quality blending as its volumetric average

[Barrow, 19611; however, most gasoline properties blend in a non-ideal and nonlinear fashion,

necessitating the use of more complex blending models to predict these properties [Rusin, 19751.

In this section. blending models for key properties - octane, RVP. and M T M distillation points

- are presented and euamined. Blending models for other qualities can be found in Gary and

Handwerk [1994] or McLellan (19811 and wiil not be discussed here.

There are several characteristics that are desirable in a blending model. In this thesis the most

important of these characteristics are predictive accuracy and parsirnony. Also, since feedstock

qualities are dependent on upstream processes operations. they change over time. Therefore,

blending models should remain accurate over a range of qualities beyond the data where the

parameters are evaiuated.

2.1 OCTANE NUMBER

Octane numbers indicate the antiknock characteristics of gasoline or the ability of the gasoline

to resist detonation during combustion in the combustion chamber. The two most comrnon

types of octane nurnbers for spark engines are the research octane number (RON) and the

motor octane nurnber (MON) . The RON, defined by the Arnerican Society for Testing and

Materials under the designation ASTM D-908, represents the engine performance under city

driving conditions (fiequent acceleration) while the MON (ASTM D-357) represents engine

performance on the highway [Gary and Handwerk, 19941. Since RON and MON of a gasoline

blend in nonünear fashion, complex blending models are needed for accurate prediction of

blended octanes [Rusin, 1975].

2.1.1 BLENDING OCTANIE: NUMBER METHOD

In this method, fictitious blending octane numbers (BON's) are used for RON and M0.V.

The BON'S blend linearly on a volumetric basis to give the octane number of the blend [Gary

where: y is the volume fiaction of component i, (ON)Md is the octane number (RON or

MON) of the blend, (BON)i is the blending octane number (RON or II/fON) of the component.

n is the number of cornponents in the blend.

Blending octane numbers are u s u d y obtained from regression analysis on small data sets and

may be user-tuned. Thus their application is often based on user experience and judgment

[Rusin et al., 19811. BON's for some typical components can be found in Gary and Handwerk

(19941. The limitations of using BON's has long being recognized which h a . lead to the devel-

oprnent of more reliable methods that do not depend so heavily on user judgrnent [Rusin et a l ,

19811.

2.1.2 ETHYL RT-70 METHOD

The Ethyl method is one of the oldest models available in literature. This method has been used

as a benchmark against which newer models have been compared. Here, blending nonlinearity

is modeied explicitly as a function of the component sensitivity (RON - MON), olefin content,

and the aromatic content of the cornponents as given below [Heaiy et al., 19591:

where: r is RON, m is MON, s is sensitivity (RON - MON), O is olefin content (% by

volume), A is aromatic content (% by volume), and al , a2, a3, aa, as, atj are correlation

coefficients. Quantities accented with an overbar represent volumetric average.

Equations (2.2) and (2.3) contain a total of six parameters (a l , an, a3, a4, as, a6) for fiON and

i W N blending. In order to estimate these parameters, the following data would be needed:

RON, MON, olefin content, and aromatics content of each pure feedstock,

RON and MON of each blend.

One of the advantages of using the Ethyl method is that it c m be readily expandeci to include

the effects of other factors (such as sulphur content) on octane blending. These factors can

be included in the model in a way-similar to olefin and arornatics content by using additional

nonlinear t erms [Morris, 19751. Blending char acteristics can change considerably wi t h changes

in octane level and other feedstock qualities. Therefore, as properties of feedstocks change over

time with changes in upstream process operations, blending model parameters may change

as well [Morris, 19861. In the Ethyl models. the pararneters rnay be estimated on-line using

historical blend data. Thus! the mode1 can be made adaptive so that its accuracy does not

decrease over time. However, the models require knowledge of the olefin and aromatics content

which may not be readily available.

Some indication of the predictive accuracy of the models has been provided by Healy et al.

[l959]. The standard deviations of prediction errors for unleaded blends sirnilar to the ones the

coefficients were determined from are reporteci to be 0.66 and 0.85 octane numbers for RON

and MON, respectively. For independent blends, the standard deviations of prediction errors

were found to be 0.92 for RON and 0.61 for MON.

2.1.3 STEWART M33THOD

This method was proposed by Stewart [1959a] about the sarne time as ihe Ethyl method was

published and is siMlar to the Ethyl method in that blending nonlinearity is attributed to the

olefin content of the components. The blending models are:

where: is the volume of cornponent i in the blend, D, = *1, ONd is the per-

cent o l e h content of the blend and is the volumetric average of the cornponent olefin contents,

and c' and are constants.

Stewart [1959a] has detennined the constants à and C using least-squares analysis on 102 blends:

the value of ii was determined to be 0.01414 for RON and 0.01994 for M O N , the value of C

was determined to be 0.130 for RON and 0.097 for MON. However, oidy 10 of the blends used

were multicornponent; the rest were binary blends.

The standard deviation of prediction error reported by Stewart [1959a] is 0.77 octane numbers

for RON and 0.64 numbers for MON. However, this prediction error reported is for prediction

within the data set used for evaluating the correlation constants. No indication of the predictive

accuracy outside the development data set is provided. Also, any indication of the level of

industrial acceptance of this method could not be found.

2.1.4 INTERACTION METHOD

This method is based on the two-factor rnodels where the overall effect is attributed to the main

effects of each of the two factors and the nonlinearity is captured by an interaction terrn. The

interaction term accounts for the effect of one factor on the other and can be determined using

the design of experiments (Montgomery, 1991]. Similady, in the Interaction method, octane

blending nonlinearity is attributed t O the twefactor interactions arnong the components in the

blend and is accounted for by adding an interaction term to the volumetric average octane

nurnber. Only the twsway interactions between pairs of components are considered; the t hree-

way and higher interaction terrns are generally ignored [Morris, 19'751, [Morris, 19861, [Morris

et al.? 1994]. For an n component system, the octane number (RON or M N ) of the blend

can be calculated as:

where: I i , t is the interaction coefficient between components i and k.

The interaction coefficient, l&, between a pair of components is obtained by using the octane

numbers of the pure components and that of a 50:50 blend of the two components as given

below [Morris et al., 19941:

where is the octane number of a 50:50 blend of i and t.

n n 1 For an n component çystern, the number of two-way interaction coefficients are &$ for

RON and the same for MON, giving a total of n(n - 1) pararneters to be determined. The

experimental data required for estimating these parameters are:

a RON and MON for each pure component,

a RON and MON for 5050 blend of ail component combinations.

Thus, for either MON or RON in an n component system, the total number of fuels to be n(n 1 tested are n pure cornponents plus 5050 blends giving a total of v. The total

number of tests required for RON and MON is n(n + 1).

Since octane blending characteristics may change with changes in octane level and other feed-

stock qualitieso the interaction coefficients may become outdated if significant changes occur

in the octane pool [blorris, 19861. Thus the predictive accuracy of the Interaction model may

decrease with time as the feedstock qualities change with changes in upstream operations or

crude srnitches. However, unlike the Ethyl RT-70 method, the Interaction model camot easily

be made adaptive since the interaction coefficients (IiVk) are not correlation coefficients but

are determined experimentaily. Instead of determining the interaction coefficients frorn binary

blends, the interaction coefficients could be obtained using regression analysis on blend data

(data on al1 feedstock qualities as well as blended qualities). The interaction coefficients could

then be updated using historical blend data and made adaptive. However, Interaction model

requires a large number of parameters to be determined. For example, a 7 component blend

would require determination of 42 interaction coefficients while the Ethyl RT-70 method will

require estimation of only 6 pararneters. Therefore, a large amount of blend data would be

required to update the interaction coefficients. &O, d the model parameters would have to

be re-evaluated as new feedstocks are entered into the blend.

Some indication of the predictive accuracy of the blending models have been provided in litera-

ture. Morris [1975] has studied the accuracy of the Interaction method and has compared it to

the Ethyl RT-70 approach. It has been reported that whiie the Interaction models fit data more

closely than the Ethyl modek (this is to be expected since the interaction model has many more

parameters that the Ethyl one), the predictive accuracy for a generai untested data set using

predetermined parameters is about the sa-ne for the two methods. The standard dellation of

prediction error for both the methods were reported by Morris Il9751 to be between 0.6 to 1.0

octane nurnbers.

While there has been some industrial acceptance of the Interaction models, they have not been

widely used. This codd be due to the large number of parameters that need to be determined.

The transformation method involves transfonning the octane number, which blends nonlinearly,

to a quality that does blend linearly and ideally. This new quantity is then blended ideally and

the result is transformed back to octane number [Husin et al.. 19811. This method is similar to

the Blending Octane Nurnber (BON) method in t hat transformed qualities are blended ideally;

however. systemat ic methods of transforrning the octane nurnbers to the transformed qualities

and the blended transformed qualities badc to blended octane numbers are provided. While the

Transformation method is applicable to leaded as well as unleaded gasolines, only the unleaded

case will be presented here. Discussions on the application of the transformation method to

leaded gasolines can be found in Rusin et al. [1981]. For unleaded gasolines, t his method takes

into account the interaction of major classes of hydrocarbons and the change in conditions

(severity levels) inside the engine when measuring different hiels. The effect of severity levels

are accounted for by relating component octanes and sensitivities to reference levels. The steps

required for unleaded gasoline are outlined below.

Step 1: Transform properties for each cornponent:

(a) evaluate sensitivity at reference engine severity levek,

R0Ni - MON* Si =

1 - ci(RON, - RONypf) + c2(MONi - MONref)

(b) adjust octanes to reference levels,

(c) det errnine hydrocarbon type adjustment ,

(d) adjust octanes to "remove" hydrocarbon type interaction effects.

Step II: Blend transformed properties linearly:

Step III: transform the blended properties

(a) hydrocarbon type adjustments,

t=1

to blended octane numbers:

(b) adjust t ransformed octanes to include effects of hydrocarbon type interactions,

ROiv, , ,_ , = RONtam,bl,nd - H R O N ~ ~ . , ~

(c ) cdcda t e sensit ivity at reference octane severity levels.

(d) adjust £rom reference to actual severity levels.

where: ci, cz, c3, y, CS, cg, C T , cg, kl, and k2 are constants to be determined, s is sensitivity

(RON - MON), RON,.! and MOiV,,j are the reference RON and MON arbitrarily set at 90

[Rusin et al. 19811. H is effect of hydrocarbon type interaction, Ai, Pz, 0, are volume fractions

of aromat ics, p a r a h , and olefins for component i , respect ively.

These blending models for RON and MON contain at least ten parameters (RON,. f , illON., j ,

ci, c2, c3. q, CS, ~ g , c7, cg, kl , and k2) which can be determined From the following experi-

mental blending data:

RON, MON, aromatics, par&, and olefin content of each component,

RON and hiON of each blend.

Predictive accuracy of the Transformation rnodels was tested by Rusin et al. [1981] who used

data £rom 564 gasoiines of varying compositions £rom severai independent blending studies.

They split the data from each source roughly into halves and used the first half for determining

the model parameters. The second haif of the data set, together with data lrom two additional

complete studies, was used to test the predictive accuracy of the rnodels. The standard de+

ations of prediction errors were found to be around 0.4 to 0.5 of an octane number for both

RON and MON blending.

As with the Ethyl method, the Transformation rnethod requires RON, MON: olefin content.

and aromatics content. The Transformation method also requires paraffin co~tent . Moreover,

the Transformation method requires the determination of ten regression coefficients compared

to only six in the Ethyl models (three each for RON and MON). Furthemore, the Ethyl

approach is much superior to the Transformation approach in terms of both mode1 simplicity

and ease of use.

2.1.6 EXCESS METHOD

This method de& with deviations hom ideality by applying an "excess" term. These excess

values are added to the volurnetric average of the component octane numbers to predict the

blended octane number (RON or MON) [Muller, 19921 as:

where: ( O N ) j is the octane nurnber of blend j , (ON): is the excess octane number associated

with component i in blend j, and uij is the volume fraction of component i in blend j.

In this linear model, the excess octane number, (ON):, gives an indication of the contribution

to nonideality in blend j by component i. These parameters for each blend are determined

by preparing a series of "est-blends". The "rest-blends" are obtained by blending as many

sarnples of the fuel as there are components but omitting one component after another. Kote

that in preparing the "rest-blends" the volume of each component (except the omitted one)

remains the same so the total volume and volurnetric composition vary. Detaiis of determining

the excess numbers can be found in Muller [19921.

The number of (ON)$ values to be determined for each fiel grade (blend) is n and the exper-

imental data needed to estimate these parameters are:

a octane number (RON and MON) of each component,

0 octane number (RON and MON) of the blend,

octane number (Rollr and fi1 ON) of the n 'test-blends" .

The number of fuels that need to be tested are (n + d(n t 1)). Thus, a total of 2(n + d(n t 1))

tests would be needed for MON and RON combined, where d is the number of gasoline grades.

The Excess met hod Ieads to linear blending models which has the adkantage of alloaing the use

of linear programming (a-hich is widely used in industry [Magodas et aL. 19881) in determining

the optimal blend recipe. However, the "excess" values are calculated for a given blend at its

nominal composition. Therefore. the accuracy of this method will decrease as the composition

deviates Erom this nominal point and nerv blending studies would have to be comrnissioned

periodically to recalculate "excess numbers". Also, the parameters determined are specific to

each blend. That is. "excess numbers" for different gades of gasoline (such as regular. prernium.

etc.) have to be determined separately whereas for other models. the same set of parameters is

applicable to ail gasoline grades.

S o indication of the predictive accuracy or the level of industrial acceptance of this method

codd be found.

This method correlates feedstock octane numbers to predict the blended octane number. The

parameters of the equation are estimated using regression analysis on experimental data [Zahed

et al.. 19931. The blending mode1 for ROM is:

An analogous rnethod c m be used for MON:

blend.

Thus the parameters that need to be determincd are -Ido, dZl, ...' Ji,, and k3 for ROS. and

Qa, QI, ..-, Qn: and k4 for ?JlO:V. The data needed for estimating t h e pararneters are RO-\:

and MON of each blend and of the n components.

-4s with other modeis, the accuracy of this method is limited to the data set that is used in

estimaring the parameters since blendhg characteristics and therefore mode1 pararneters may

change as feedstock qualities changes over t h e . The rnodei parameters may be updated on-line

using historical blend data. Zahed et <IL j1993! have eompared the predictive accuracy of the

rnodels in Equations (2.9) and (2.10) ~ l t h that of the Stewart method and have shmn it to

te better at predicting blended ROX for a set of expenmental data. Standard del-iations of

prediction errcr were not prorided.

2.1.8 S l l M M A R Y OF OCTANE BLENDING MODELS

Table 2.1 sumarizes blending models dixussed above and Table 2.2 compares the preàictive

a c c u r q of the difhent models. Accu- for interpolation (prediction aithin the development

data set j as well as extrapolation (prediction outside the development data set j are provided.

Surprisingly for the Transformation method. Rusin et al : 1981: have reported a higher predic-

tion accu- for extrapolation than for interpolation.

The Excess method. being lineu. is valid only u-ithin the vicinity of the nominal blend. .Uso.

there is little information a~ailable with regards to the predictive a c c u r q of the Excess. Stewart

or Z&ed methods. The Interaction method has been shown to be no more accurate than the

Ethyl method in predicting blended octar.es on general. untesteci data sets even though it has

many more parameters than the Ethyl rnodels. Xlthough the Transformation models seem to

provide slightly better predictive accu- than the Ethyl rnodels. they require more feedstock

# of Parameters

6

Mode1 1 Parameters

Interaction

Transformation

n for ,lfOhr 272 totai for each made

Ethyl

kl, kz 1

Table 2.1: Cornparison of Octane Blending Modeis

al:a2,a3 Q4,aa.M

Stewart

L , k

Cl. c2. c3. c4

CS. q. cï cg

n for RO-V

quality data and contain more parameters (les parsimonious). .\loreorer. the Ethyl models are

much more simple and easy to use than the Transformation models.

"("Lbr L \ f ~ ~ 2 n(n - 1) total

IO

2.2 REID VAPOURPRESSURE

I "'"-'i for ROX I

.

The Reid Vapour Pressure (RVP). defineci by the -4merican Society for Testing and Ilarerial.

under the designation ASTN D-323-56. giws an indication of the volatility of a gasoiine blend

and is approximately the vapour pressure of the gasoline at 100°F (3B°C). The RVP of a

gasoline blend &ects the gasoline performance in r e m of ease of starting* engine n-arm-up.

and rate of acceleration [Gan and Handwerk. 19941. R V P is &O important because mavimurn

specifications on R V P Limit the amount of n-butane. a relatively cheap source of octane. that

can be added to a blend.

a. c

2.2.1 THEORETICAL APPROACHES

2

One of the first theoretical approaches alailable is that by Stewart [1959b] who presented a

met hod for predict ing blended RV P's by mat hemat ically simulat ing the processes occurring

during the Reid test (which measures the equilibrium pressure for a constant-volume partial

vapourizarion of the hiel in air). The Stewart approach uses component data (such as feedstock

1 Mode1 1 RON 1 MON 1

Ethyl Stewart

I 1

&frorn Heaiy e t al [1359] &€rom Stewart (1953a]

' €rom Morris [1975! 'frorn Rusin et aL [1981]

S / A not available

Interpolation Extrapolation 1 Interpolation Extrapolation

Int eract ion Transformation

I I

Table 2.2: Predictive Accuracy of Onane Blending Modeis

O-8e 0.99 O. 77@ Y / A

Excess Zahed

composition and component volatility ) and t hermodynamic relationships in simulat ing the test.

Several simplikng asjurnptions are made including: the presence of air and water vapour in

the test chamber are ignored. the ahsolute pressure is taken as the RVP. volatile components

are assumeci to have the density of butanes. and the nonvolatile components are assurneci to

have the thermal expansion characteristics of n-octane [Stewart. lgjgbj.

0.79~ 0.61~ 0 . 6 4 ~ W.4

0.28' 0.8' 0.51 A 0.43A

The equations for estirnating the blended R V P provided by Stewart '1959bi are:

0.33' 0.64' 0.60' O . S A

Reponed are standard deviacions of prediction error in octane number

S/,4 S/A S/ -4 X/A

a-here: d, is the mole fraction of component i in the liquid phase. 7, is the acth-ity coefficient

of component i. u, is the volume fraction of component i in the blend (feed) at 1 atm and 60

O F . is the rnolar density of component i (pound-moles per barrel as liquid at 1 atm and

60 OF). pblmd is the density of the blend as saturated liquid at 60 OF (pound-moles per barrel).

and k is the number of components in the b l e d (components over al1 feedstocks).

S/A S/A S/A Y/ -4

The modeis given in Equation set (2.1 1) do not proxide (RVP)b(d explicitly as a function

of component data: therefore. an iterative approach would have to be taken in çolving for the

blended RVP. The data required in using the Stewart method are RVP's of each feedsrock.

volume fractions of ail components in each feedstock. molar densities of all cornponents in each

feedstock, and estimates of activity coefficients of all components.

Vazques et ai. [1992] have presented an algonthm for calcdating the blended RVP's based

on the work of Stewart [1959b]. The mode1 assumes additivity of Iiquid and gas volumes and.

as in the Stewart case, the algorithm is based on an air and water-Eree model. This approach

differs £rom the Stewart method in the equation of state used and it requires that the molar

composition of the feedstocks be knoun. An iterative procedure for calcdating the vapour-

liquid equiiibri~m for the flashing of the gasoline feed is provided. The steps required by this

method are outlined below:

Step 1: Calculate the molecular weight of the blend:

Step 2: Evaluate density of blend at 35 OF. 60 O F o and 100 "F. Compute liquid

expansion of blend as:

Step 3: Periorm "flash calculation" at 100 OF using the Gas Procesors Association

Soave-Redlich-Kwong equation of state [Vazques et al. l g E ! . It is sugested that

4 of 0.97 be used initially.

Step 4: Calculate new $ using :

Step 5: Use $ from Step 4 to re-do the "flash calculation" in Step 3 and repeat i until F converges to an acceptable degree.

Step 6: The pressure obtained irom the equation of state in Step 3 using the

converged $ is the blended RVP.

where: F is the m a s of the feed used in the Reid test, Y is the mass of the vapour phase

obtained by flashing F, and is the mass of the Iiquid phase; (Ad W ) L , ( M W ) V , ( M W ) i , and

are the molecular weights of the liquid phase, the vapour phase, component 2 , and

the blend, respectively; py , p ~ , and PLF. are the densities of the liquid phase, vapour phase,

and liquid feed (at 100 OF); p35, pcO, and plm are the densities of the Liquid blend at 35 OF, 60

OF, and 100 OF, respectively; di is the mole fraction of component i, and uo is the expanded

liquid volume.

Details of how the blend densities at the different temperatures and the densities of the liquid

and vapour phases can be obtained have not been provideci. Also, it is not clear how d u e s

needed in Step 4 can be obtained Erom the equation of state in Step 3. The equation of state

used is the Soav+Rediich-Kwong equation [Vazques e t al. 19921:

&?' 6T p z - - v - b v

(V + 6 )

where: P is the pressure, l? is the gas constant. v is gas volume. and ii and b are constants.

The parameters ü and b are determined according to the Gas Processors Association [Vazques

et al. 19921 rules as:

where: rn is the number of components over al1 feedstocks, 4 and aj are the Soave-Redlich-

Kwong constants for the pure component i and j , 6, is the Soave-Redlich-Kwong constant for

pure component i, and Ko and Ki are the binary interaction parameters used to represent the

vapour-liquid equilibria of binary mixtures..

This method requires that the molar composition of al1 the feedstocks be known which may not

be readily amilable. Also, correlations for estimating densities would be required. In addition,

experimental data may be needed to determine the parameters in the Soave-Redlich-Kwong

[Vazques et al. 19921 equation of state.

The major drawback of the theoretical approaches is that they are not easy to use on-line in

that they require cornplex calculations and iterative solutions. h o , they require data su& as

feedstock molar compositions and component activity coefficients which may not 'De available.

2.2.2 INTERACTION METHOD

The interaction approach has &O been appüed to R V P blending by Morris [1975]. The blending

mode1 is of the same f o m as in Equarions (2.6) and (2.7) evcept RVP is blended instead of

ON (octane numbers). Here also, only interactions between pairs of components is considered:

the three-way and higher interaction te- are ignored. The Interaction mode1 is:

where: n is the number of feedstocks, ÏiYk is the interaction coefficient between feedstock i and

k, and (RVP)i,k is the R V P of a 50:50 blend of i and k.

The number of interaction coefficients required wauld be 9. Some indication of the accu-

racy of the Interaction method c m be found in Morris [1975] where the standard deviation of

prediction error on an 8 cornponent. 42 blend study is reported to be 0.18 psi.

2.2.3 BLENDING INDEX METHOD

-4.n easy to use empiricai method developed by the Chevron Research Company is the Blending

Index method [Gary and Handwerk. 19941 . In this approach, blended RVP's are predicted

using the Reid Vapour Pressure Blending Indices (RVPBI) which blend linearly. The RVPBI's

are given in Gary and Handwerk [1994] and are of the form shown below [Gary and Handwerk,

19941:

RVPBI = (RVP)'.~' (2.12)

n

(RVPBI)M,nd = C (RVPBI),

1 Method 1 Standard Deviation 1

Stewart

I r ~ o l a r average 1.17 1

(psi) 0.76

Ideal blending* Haskel and Beavon 119421

*ldeel blending is volurnctric average R V P

1.30 1.01

Al1 data from Stewart [1959b]

Table 2.3: Predictive Accuracy of RVP Blending Models

2.2.4 O T m R METHODS

Haskell and Beavon [1942] presented a method which involves the volumetnc averaging of the

component RVPs except those of butanes which were treated differently; the butanes are

assigned variable "blending pressure values" which were calculated based on the RVP of the

butane and that of the debutanized blend (blend with al1 components except the butanes).

Another simple approach that bas been used involves molar averaging (not volumetric) of

the component RVPs based on the blend composition [Stewart, 1959b1. However, molar

composition of feedstocks may not be readily available.

2.2.5 SUMR/LARY OF RVP BLENDING MODELS

Cornparisons of predictive accuracy of some of the methods can be found in Stewart [1959b]

looked at the standard deviation of prediction error for 67 blends wing different models.

The reported standard deviations are presented in Table 2.3. It is not surprising that the

method based on modeling the actual process provides more accurate predict ions. However,

the thwretical rnethods by Stewart [1959bI and Vazques e t al. [1992] are not suitable for use

in on-line control systems due to their computational requirements. The interaction method is

also not suitable for on-line implementation since it requires that a lot of parameters be updated

on-line as discussed in the octane blending case. Although not as accurate as the theoretical

methoàs. simpler methods such as the Blending Index method can easily be incorporated in

on-line controllers.

ASTM DISTILLATION

Gasolines, since they are mixtures of hydrocarbons with different boiling points, vapourize over

temperature ranges. For example, the light virgin naphtha stream vapourizes throughout a

temperature range of 30°C - 90 O C . The volatility characteristics of gasoline is measured by

the ASTM D-86 distillation which is a standardized laboratory batch distillation that is carried

out at atmospheric pressure wit hout fractionat ion. It gives amount (percent volume) dist illed

to a given temperature. Several temperatures dong the distillation curve are used as reference

points in comparing the volatility properties of gasolines. Equivalently, distillation properties

can be specified as temperatures at a given percent evaporated. Some of the methods for

predicting blended distillation points available in Literature are briefiy discussed next.

2.3.1 INTERACTION METHOD

The interaction method, as described for octane blending, c m bo used to predict the ASTN

distillation points as percent evaporated at a given temperature [Morris, 19831, [Morris. 19751.

Again, only the two-way interactions are considered and the interaction coefficients between

al1 the component pairs would have to be determined for each temperature on the distillation

curve. As in the octane and RVP blending cases, determination of the interaction coefficients

require the quality (percent evaporated at that temperature) of the individual components as

well as those of the 50:50 blends and are calculated as described for the octane blending case.

This method is only applicable to the intermediate points. not to end points of gasoline blends.

The biggest drawback to using the Interaction mode1 is the large number of parameters that n n 1) wouid have to be determined. For each temperature on the distillation curve, &f-- parameters

are needed a-here n is the number of components. Thus for a 7 cornponent blend with 5 points

on the distillation curve, 105 interaction coefficients would be required.

2.3.2 EMPIRICAL MODELS

Empirical models are quite often used in predicting the ASTM distillation points of blends

and owe their popularity to their ease of use. Stanley and Pingrey [1954] have developed

correlations for predicting the end points of distillation curves when only the blend composition

and component ASTM dist ilIat ions are known. Correlation analysis on experiment al data was

used (details of whidi were not provided) to produce tables of values that can be used for

distillation end point estimation. This method does not provide estimates for intermediate

points on the distillation cuve.

Dhulesia [1984] has reported using models of the form:

where: al and a* are constants, ü is the percent volume evaporated, TI and T, are the final

and initial boiling points, respectively, of the blend.

Another empirical mode1 reported is of the form [DeWitt et al., 19891 :

where: a3 and a4 are constants, Ùk is the percent evaporated at temperature k for the blend,

and is the percent evaporated at temperature k for component i in the blend.

Indications of predictive accuracy of the models could not be found; thus sufficient information

is not amilable to ailow cornparisons to be drarvn arnong the diEerent models.

2.4 BLENDING SIMULATION

In order to conduct blending control studies by simulation, models that can closely represent the

blending process are required. Blending models are also needed for off-iine blend optimization,

as well as on-line blend optimization and control. For the purposes of the present research, only

octane number (RON and MON) and RVP blending will be simulated. Ot her qualities will

not be considered. The blending simulation has been restricted to the three most important

gasoline properties so that the important results can be highlighted wit hout adding unnecessary

complications.

A good blending mode1 should provide acceptable predictive accuracy, it should be parsimonious

and easy to use, and it should not require a large data set to be trained on. Also, for on-line

optimization, a mode1 that can be readily updated on-line would be desirable. This is because

models are not perfect and their predictive accuracy usudy degrades over tirne (as feeùstock

properties and t huç blending characteristics change wit h tirne) unless they are appropriately

updated periodicaily.

Octane blending rnodels have been discussed and cornpared in Section 2.1. .4mong the models

found in published literature, the Ethyl RT-70 [Healy et al., 19591 modek were found to exhibit

the best combination of predictive accuracy and parsimony. Also, the Ethyl models c m readily

be made adaptive for on-iine implementation. Healy et al. [1959] have w d data from 135

blends to determine the correlation coefficients. Thus the Et hyl correlations wit h the published

coefficient values were chosen over the other octane blending models.

RVP blending models have b e n discussed in Section 2.2. Very iittle information is currently

amilable in Literature that can be used to choose the best RVP blending model. .4iso, blending

data is not available that can allow the different models to be compared for predictive accuracy.

Thus' based on ease of use and a high degree of industrial acceptance, the Biending Index

method was chosen for modeling RVP blending. The selected models are presented again for

completeness in the following equation set and &il1 henceforth be used to model the gasoline

blending process:

The parameters used for octane nurnber blending are [Healy et al., 19591: ai = 0.03224, a2 =

0.00101. a3 = O, a4 = 0.04450, as = 0.00081. and as = -0.00645.

These equations and parameters will be considered to adequately represent the actual blending

process for the purposes of the studies contained in this thesis.

Chapter 3

PARAMETER OBSERVABILITY

IN RTO SYSTEMS

In Chapter 2, gasoline blending models were examinai for use in the gasoline blending RTO

system s h o w in Fibwe 1-4. Since the selected blending models are to be uçed on-line, the

adjustable parameters in the models would have to be updated on-iine in order to maintain

mode1 accuracy. Therefore. it is important that al1 adjustable models pararneters be observable

for the available measurements. That is, the measurements should d o w unique estimates of the

adjustable parameters to be obtainod t hrough some estimator. The success of any RTO system

is dependent on al1 the adjustable parameters being observable from the process measurements

and given estimation scheme .

This chapter investigates parameter observability and begins with a survey of the currently

available methods of determining observability. A new measure of parameter observability is

t hen developed based on hndamental stat istical principles and the observability concept is

extended to degree of observability. The chapter concludes with examples to illustrate bow the

new observability measure can be applied to diEerent estimation schemes.

3.1 BACKGROUND

Observability was originally defined and studied for linear dynarnic systems by Kalman [1960]

as a dual to controllability. Kalrnan observability is used to determine whether a given set

of measurements contain sdficient information from which the states of the system can be

determined (Brogan [1991], Friedland, [l986]). It will only be briefly discussed here since the

focus of the present work is RTO of steady-state operations policy. Consider the standard

state-space representation of a dynarnic linear process:

where: y is a vector of state \&ables: u is a vector of control moves, z is a vector of rneasure-

ments or outputs, and c(t ) and v(t) are noise terms.

The system given in Equation set (3.1) is said to be completely obsenable if every initial state

y(to) can be detennined through knowledge of the system control, u(t), and output. y ( t ) , over

some b i t e time i n t e r d to 5 t 5 ti [Ray, 19813. For systerns where the matrices A and C do

not vary with time, observability can be checked by constructing the observability matrix M:

where: q is the number of state variables.

The linear dynamic system with constant matrices A and C is completeiy observable if and

only if the rank of the observability matrix M is q, where q is the number of state variables

[Ray, 1981]. Here complete observability means that every state variable is observable. If only

a subset of the state variables are obsenable, the system is said to be partially observable.

Conditions for other classes of dynamic systems can be found in Bryson and Ho [1976] and

Astrom [1970].

So te t hat checking for observability by test ing the rank of the observability mat rix can yield only

two possible outcornes - observable or unobservable. No indication of degree of observability is

provided by the test. Gudi [1995] has looked a t the singular values of the observability matrix

to characterize degree of observability. Gudi (19951 has used the condition number, the ratio

of the largest singular value of M to the smallest, as a measure of degree of observability.

A smdl condition number indicat es t hat the state variables are "strongly observable" while

a large condition number indicates "weak observability" . Since condition numbers are scale

dependent [Golub and Van Loan, 19891, such a measure of degree of observability will also be

scale de pendent.

The concept of obsenability was extendeci £rom linear dynarnic systems to nonlinear steady-

state systems by Stanley and Mah (19811 while retaining the fundamentai idea behind observ-

ability. That is, it is used to check if changes in all the state variables c m be determined

through knowledge of the process outputs (measurements) and the steady-state models that

describe the process. Stanley and Mah [1981] considered a steady-state system of dimension q

with m measurements shown in Figure 3-1. They defined the system as the triplet (S, C, V)

where S is the feasible set of States that always satisfy equality, inequaiity and strict inequality

constraints, C is the vector of equations that relate the measurements to the state variables,

and V is the set from which measurement noise realizations are obtained.

The measurements are given by the measurement functions as:

where: y is a vector of state variables, v is the noise vector, z is the measurement vector,

and S is the closure of set S consisting of S and dl its limit points. In the case of perfect

measurements, V = {O}, v = 0, and z = C(y).

The authors define the steady-state system (S, C, V) as being in "standard forrn?' if S =

{y : w ( y ) = O, y E N} where vector w contains plant mode1 equations such as material and

energy balances, thermodynamic relations, etc., N is the union of a finite number of convex

sets, N, c W, that defines the plant operating region. Since equations in vectors C and/or w

may be noniinear. observability is defined as a local property.

Stanley and hlah [1981] have provided some definitions regarding observability that are useful

for the present work. These are given below.

Definition 1 Local Unobservabd2ty: in a system (S, Cl V ) let y, E S and I be an index

Measurement Noise

Figure 3-1: Steady-State System and Estimator Considered by Stanley and Mah (19811.

1 . Feasible Set , Y C 5 6 9 & l i Y^

set. y' is locally unobsentable a t y, if there exists a sequence { y k } h , such that:

S

That is? the states included in the index set I are defined to be locally unobsemble if there are

changes within this set of states that do not result in a corresponding change in measurements.

For the situation described in the defuùtion of local unobservability. even though y, differs frorn

y,, they both result in the same measurement value. As a result. one cannot be distinguished

frorn the other and the system is said to be locally unobservable at y,. Observability is then

defineci by Stanley and hl& [1981] in ternis of unobservability:

Definition 2 Locd Observability: in a system {S, C, V ) let y, E S'. Y' zs locally observable

ut y, if it is nat unobsentable.

5

Stanley and ~Mah [1981] have also provideci a method of testing local observability which is

+

Steady-State System

1 4- Estimator

given in the following theorem:

Theorem 3 For a system gzven in the standard fonn S = { y : #(y) = O, y E N}? y, E S. let

w and C be continuously dzfirentzable in some nezghbourhood of 9p wntaining y,. The system r f

is calculable on some set S, C S containing y, if rank V,w(yo) / vyC(ya) / = q d e n q i-s the numhi

L J of states in the steady-state system. Vyw(y.) and VyC(yo) are the Jacobian m a h c e s ofw and

('? respectively, evaluated at y,.

Thus. instead of checking the rank of the observability matrix M. as in the case of linear dynamic f 7

systems. the rank of the augmenteci Jacobian rnatrix 1 Vyw(YO) 1 should be checked. As in 1 VYC(YO) J

the linear dynamic case. if the rank is found to be l e s than q: the state variables are not

completely observable. The rationale behind the rank requirement for local obsenability can

be illustratecl using first-order Taylor series approximation [Cornin and Szczarba. 1995i to the

proces model. w(y): and measurement equations. z = ('(y), about y,:

which can be written as:

In order to ensure that changes in states (Ay) are uniquely obsenable from the changes in r 1

the measurements (Az)? the rnatrix must have rank q. That is. the augmenteci

Jacobian matrix m u t not be column rank deficient.

The concepts of state observability for steady-stat e s-wtems was extended to parameter observ-

ability by Krishnan [1990] a-ho considered a generai nonlinear system where the stead-state

model of the process and the measurement equations are defined as:

where: B E W' is the adjustable parameter vector to be determined.

As an extension of the rank condition in Theorem 3 for local state obwrvability, for al1 the

pararneters and states of the systems to be obsenable at y., the observability criterion presented

by Krishnan [1990] requires that:

rank 1 V Y ~ Y O )

where: q is the number of state variables and p is the number of parameters to be determined.

The need for the above rank requirement can be iilustrated usinp the Implicit Function Theorem

[Corwin and Szczarba, 19951 :

du =

dC =

These can be rearranged to give:

r

Thus for this Linear system of equations to have a unique solution with respect to dy and dfl

at y,, the rank of the matriu:

must be the number of parameters plus number of states.

As illustrated in Figure 3-1, the objective behind an observability study is to determine if

estimates of the system states (or the pararneters as in the case of parameter observability)

can be obtained korn the measurernents and the estimator. While the parameter observabil-

ity criterion in Equation (3.2) contains information about the steady-state process and the

measurernents, no information about the estimator, Qi, is included. The characteristics of the

estimation procedure can have a very significant effect on whether or not the unrneasurable

quantities (pararneters and states) can be determineci from the measurements. Even when

these quantities are observable through the mode1 and measurement equations, a poorly de-

signed estimation procedure may not allow the measurements to be uniquely mapped ont0

est imates, rendering the unmeasurable quant it ies uno bservable. Therefore, the characterist ics

of the estimator should be considered-

3.2 PARAMETER OBSERVABILITY

The obsemability criteria presented by Stanley and Mah [1981j and Krishna [1990] do not con-

sider the effect of the estimation procedure. Their methods for determining obsenability have

ben based only on the models that describe the steady-state mode1 equations and process mea-

surements. If the system is unobservable according to the rank cnterion in Equation (3.2). then

the variables will cert ainly be unobservable through the whole updating procedure. However . cornpliance with Equation (3.2) does not guarantee observability through the entire updating

process. That is, Equation (3.2) provides a necessary but not sufficient condition for local

obsenability. For a sufficient condition, properties of the estimator would have to be included

in formulat ing a measure of observability. In t his section pararneter obsembili t y for nonlinear

stead-state systems wili be considered and a new measure of obsenability will be developed

based on statistical principles. The concepts udi then be extended to provide measures of

degree of observability. First, a definition of pararneter observability is provided:

Definition 4 Parameters are dejïned to be obsentable fmm a given set of pmcess rneasurernents

if a unique set of pammeter estimates c m be determined wing a speci,5ed updating scheme.

In order to gain insight into how pararneters may become unobservable, consider a steady-state

system for which the adjustable parameters ( f i ) have to be estimated using the measurements

(z) as shon-n in Figure 3.2.

For the purpases of this work? measurements are assurneci to be corrupted with noise and so

have a randorn component. These measurements get mapped to the parameter estirnates ( f i ) . Thus. the parameter estimates (b) also have a random component. i;hile measurement error

may be normally dist ributed. the same is not necessarily true for the resulting est imates. Figure

3-2 shows the propagation of measurement noise through what may be a nonlinear updating

Fi-we 3-2: Parameter Updating

z

scheme, a-hich will distort the distribution characteristics The developments of this chapter

assume that the parameter estimates may be locally approximated in some small neighbourhood

of the nominal estimates using multikw-iate normal distribution (ie.. 8 - -~p(& Qd) j. The

probability density hinction of for positiue definite QB can then be arpressed as [Chathrld

and Coilins. 19801:

For a bivariate case. this funcrion represents a beU-shaped surface whose probzbiiity contours

4

Updator

on the pararneter plane are concentric ellipses around p. In general. the surface contours in the

P b

parameter space are pdimensionai efipsoids. Thus. the function in Equation (3.4) exhibits a

single extremum which is a maxima located at P .

The surface would not possess a unique maxima if the surface were to become Bat in which case

the updator would not be able to use the rneasurements to obtain a unique set of parameter

estirnates. The conditions under which such a situation couid arise can be examined by looking

at the distribution of fi more closelc Xote that the inverse of the co=?uiance matrix appears in

the exponential t e m in Equation !3.4) and the determinant of Qo appears in the denorninator

of the pre-exponential terrn. One definition of the determinant of Qg is [Chatfield and Collins.

where Xi. .. .. Xp are the eigemdues of Qa .

The eigenvalues of 4s'. q, are reciprocals of the eigenvalucs of Qg (2. e., qi = k, i = 1, .... p ) B

[Chatfield and Collins. 1980j. Soa-. as an eigendue q, decreases. lQbl increases and the prob

ability density function decreases in height. At the same time, the hvper-ellipsoid elongates in

the direction of the corresponding eigenvector of QI', T j , as will be discussed later. In the B limit as rlj - 0, the surface becomes Bat and ahib i t s no extremum; a t this point a unique set

of estimates does not exkt and the parameters are unobservable. Then observability can be

dehed in term of the properties of QB.

Conditions for observability can &O be seen by examining the shape of the joint confidence

interval of the parameter estirnates. For a prescnbed probability f ( B ) and a given covariance

m a t h Qg , the probability density hinction can be written as [Chatfield and Collins. 19801:

Taking logarithms of both sides and multiplying by -1 gives the following quadratic equation:

The random variable c2 follows a chi-squared distribution with p degrees of fieedom [Chatfield

and Collins. 19801. For a prescribed d u e of 2, this equation describes the surface on which the

probability density h c t i o n has a constant value. Since Qg is required to be positive definite

for validity of Equation (3.4): the surface in Equation (3.7) is a pdirnensional ellipsoid that

describes the joint confidence region of the parameter estimates [Sorenson. 1980]. The lengths of

the semi-axes of t hese hyper-eUipsoids indicate the uncertainty in the comesponding eigenvector

directions in the space of the parameters [Sorenson. 19801. [Krishnan. 19901. The semi-axes

2 of the hyper-eilipsoid have magnitudes ?; and have directions defined by the eigenvectors T,

[Sorenson, 19801. For good estimates then. large eigenvalues of Q-' are desireci. D

Alternatively? conditioning of the joint confidence region can be studied by exarnining Q-' B which is the Hessian of 3. -4s an eigenvaiue of Q;'. q,, approaches zero. the Hessian becomes

very ill-conditioned. As 11, approaches zero. the confidence contours in the parameter space

stretch out in the direction of r, and in the Iimit become unbounded as depicted for a bivariate

case in Figures 3-3(a) to 3-3(d).

In Figure 3-3(a). the contours are concentric circles which inàicate a perfectly conditioned Q;'

(ail eigenvalues are qua i ) . As one of the eigenvalues decreases. the contours elongate as shown

Figure 3-3: Confidence Contours

in Figures 3-3(b) and (c). Figure 3-3(d) shows the limit case when one of the eigenvahes of Q: l B

reaches zero and the contours become unbounded. At th% point, a unique B does not exist and

the parameters become unobservable. Hence an alternate definition of parameter observability

can be provided in terms of the eigenvalues of the inverse of the conriance matrix:

Definition 5 For any updating pmcedure luhere parameter estimates (p) (assumed to be locally

normally distributed) are to be determined using measurements (z), the parameters are obseni-

able zf al1 the eigenvalues of Q:' are strictly positive (positive and nonrem) where Qp is the P

covariance mat& of 6.

In order to demonstrate how parameter obsenability relates to the covariance rnatrix of the

estimates, a simple linear example is used below. The exarnple will illustrate how parameters

can become unobservable and how unobservability of parameters can be detected.

Consider a simple problem of estirnating parameters (P) using measurements (2)

which are related by the model:

where: z is a vector of measurernents or observations, p is the vector of parameters

to be estimated, x is a matrix of operating conditions, and v is a vector of errors

in the observations distributed as N(@, Q,), Qv = a21. Note that the operating

conditions in x are variables but not random variables.

The least squares estirnate: P , is given by the normal equations [Draper and Smith.

19661 :

The estimates f i are normally distributed with mean and covariance matrix Qg ;

the covariance mat* is given by [Draper and Smit hl 19661:

Case A:

22.944

The eigenvalues of 48' are 0.48,53.02, and 2.49. Since di the eigenvalues of Q:' are P

nonzero, the parameters are observable. .&O, the least squares estimates obtained

( B ) are quite close to the true value of the parameters (P).

Case B: If rank[x] < 3, the m a t h ( X T X ) would not be invertible and the para-

rneters would not be observable. Consider the case where the third row is dependent

on the first two:

Here Q-' = and its eigenvalues are 2.489, 0, and 73-51 1. The zero eigenvalue P

of Q F ~ indicates that the parameters are unobsernble and the least squares estimate

cannot be obtained.

Case C: In this case x is not singular but is poorly conditioned (the condition

number Q, the ratio of the largest eigenvalue of x to the smallest? is 285.3):

660.350 329.875 -445-000

329.875 165.313 -222.500

-445.000 -222.500 300.000 1 The Ieast squares estimate, P? is a very poor estirnate of P (in that the estimates are

quite far £rom the true values of the parameters) because the parameters are prac-

tically unobservable. This can be seen by examining the eigenvalues of QY' which B are 8.9*10-~. 2.43, and 72.3. Although al1 the eigenvalues are nonzero, one of the

eigenvalues is very small, causing the parameters to become practically u n o b ~ n ; -

able. Also, the joint confidence region is very ill-conditioned - the condit ion number,

Q = 3n.U Iirnin

, is 8.1*104 and so the joint confidence i n t e d is very elongated in the

direction corresponding to vmin That is, the uncertainty in parameter estimates in

the direction corresponding to qmi, is almost 105 times larger than in the direction

corresponding to q,, .

In this rather simple linear example it is easy to recognize the estimation problem as being

pooriy posed by exarnining the condition number of X . Nonlinear estimation problems can be

checked in a similar manner by linearizing the nodinea. problem about some nominal operating

point and then checking the condition number of the resulting matrix. Such an approach ni11

only locally test the estimation probIern and wili not be able to detect celineuity of the

nonlinear mode1 equations at other operating points. That is, c~lineari ty of equations at

certain points, which can cause the parameters to become practicdy unobservable, may not

be easily detectable. In such situations, test ing for local observability by exarnining the inverse

of the covariance matrix of pararneter estimates as proposed here. will identiS the obtained

estimates as being poor. Note that the least squares estimates obtained in Case C are very

poor (B is not close to ,6) even though ail the eigenvalues of QT' are nonzero and so the B

updating problem passes the observability test. This highlights the need to consider degree of

observability rather than a binary observability test.

3.2.1 DEGREE OF OBSERVABILZTY

The observability criterion for pararneter observability avaiiable in literature by Krishnan [1990]

given in Equation (3.2) determines whet her or not quantit ies (states/parameters) are observ-

able. Thus there are only two possible outcornes from the observability test: observable or

unobservable. This test provides no information about the quality of estimates. For dynarnic

systems. Gudi (19951 has used the condition nurnber of the obsenmbility matrk M to differen-

tiate between "strong" versus "weak observability. The approach taken by Gudi [1995] could

be applied to pararneter observability of steady-state systems by examining the condition num-

ber of the augmented Jacobian matrix in Equation (3.2) to get an indication of the degree

of obsembility. However. as has been pointed out earlier, the augmented matr~u contains no

information about the estimator. In this section, it will be shoan how the proposed parameter

observability measure naturally leads to a measure of degree of observability that is based on

the statistical properties of the pararneter estirnates.

Degree of observability, in this context, refers to the quality of estimates that can be obtained

from a given set of measurements via the updator. In designing experiments for precise para-

meter estimation, the joint confidence region of the estimates has been used to obtain measures

of estimate quality [Sutton and MacGregor, 19771, [Box and Lucas, 19591' [Box and Hunter,

19651. Quality of parameter estimates is related to the characteristics, both size (volume) and

shape, of the joint confidence region of parameter estimates [Sutton and MacGregor, lW7]. In

general, updators yielding confidence regions with s m d volumes provide better estimates than

those resulting in large volumes (of similar shape). Confidence regions that take the shape of

a hyper-sphere are more desirable than very elongated regions since the more elongated the

hyper-eliipsoid gets, the more correlated the estimates become. Box and Lucas [1959] and Box

and Hunter 119651 have used the determinant of Qg as measure of the volume of the joint

confidence region. The rationale behind using the determùiant is that IQDJ is the product of

the eigenvalues of Qp which represent magnitudes of the semi-axes of the hyper-ellipsoid (joint

confidence region). Therefore. the larger the determinant , the greater the volume.

However, the volume criterion gives no indication of the shape of the confidence region. An

indication of the shape or conditioning of the region can be obtained by looking at the condition

number of the Hessian of the function c2, QT' : P

The condition number, ~ 2 , takes on its minimum value of unity when the joint confidence region

is a hyper-sphere and increases as the region becomes elongated. Another measure of shape

provided by W n g (19481 is:

This measure. R, also takes on a minimum value of 1 and increases as the region becomes

more elongated or ill-conditioned. Wote that R is a function of al1 the eigenvalues of Qil and so incorporates information about uncertainty in al1 the p independent directions in the

parameter space whereas ~2 considers only the best (rnost certain) and worst (least certain)

directions. However, r;2 places an upper bound on any measure of conditioning and so will be

adop ted here.

Thus both the size and shape of the joint confidence region have to be considered in determin-

ing the degree of parameter observability. For "strong" observability, small lQBl and nz are

required. Among regions wit h similar volumes, the one yielding the smallest value for r ; ~ wodd

be preferred, while among regions with similar conditioning the one encompassing the smallest

volume (for a given confidence level) shodd be chosen.

4 simple example is used below to illustrate the use of the shape criterion in selecting among

estimators that yield confidence regions of s i d a volumes.

EXAMPLE 3.2

T T Measurernents z are used to update parameters a = [ p, ] = [ j 5 ] s i n g

two different estimators A and B. The inverse of the covariance rnatrk of the

parameter estimates resdt ing from the two estimators are:

The eigenvalues

and 0.43. Since

are 1 and 5 and the eigenvalues of Q- are 11.57 ( P 1 ) B

1 ( Q ~ ) 1 = 0.20, the joint confidence regions frorn both

the estimators enclose the same volume for a given confidence level. However. the

condition numbers of (4; ' ) A and (Q; ' ) , are vastly different :

The confidence contours resulting horn the two estimators are shown in Figures 3-4

(4 and (b).

The confidence regions resulting hom estimator B are much more elongated than

t hose from estimator A. Therefore est imator A provides much stronger parameter

observability and so shouid be chosen over B.

In this section a criterion for determining whether or not parameters are observable from a

given updating scheme and process measurernents has been provideci. The criterion is based on

evarnining the inverse of the covariance matrix of estimates (QT' ). The idea was then extended P

to obtaining an indication of the degree of obsenmbility which again looks at the characteristics

of ~i'. The rest of the chapter ail1 focus on obtaining estimates of Qg .

Figure 3-4: Probability Contours for Example 3.2

3.3 COVARTANCE MAT- APPROXIMATION

.ei approximation to Qp can be obtained by representing the updating procedure in Figure 3-2

as a nonlinear map:

which produces parameter estimates £rom process measurements. When O is at least once

continuously differentiable in z, changes in the adjustable parameters (AB) in response to

srnall changes in the rneasurements (Az) can be approximated using a first-order Taylor series

as:

which can be expressed as:

where: 2 is a sensitivity rnatrix (sensitivity of parameters to measurements).

Thus small changes in z, including measurement noise, are approxirnately mapped to pertur-

bations in B through - As such, the propagation of measurement noise through the updator

to produce a variance in the parameter estimates, Qg, can also be locally approximated using

E. Forbes [1994] has provided an expression for approximating Q6:

where: Q, is the covariance matrix of measurement noise.

Thus measurement noise propagates through the updator, the properties of which are captured

in g, to produce a variance in the parameter estimates. Determination of will be examined

for some estimation techniques in the next section.

It has already been estabiished that any measure of observabiiity should include information

about al1 components of the parameter updating systern. Qd depends on 2 (in addition to

Q z ) which in turn depends on 0, the updator. Thus Qg contains information about the whole

updating subsystem. How relates to the cornponents of the updator will also be discussed

for some estimation techniques in the next section.

3.3.1 ESTIMATION EXAMPLES

Consider a steady-state process described by:

for which the pararneters 0 have to be estimated using the measurements z. There are many

alternative estimation techniques which may be used in updating the adjust able pararneters.

This section will focus on t hree common met hods (back cdculat ion, optimizat ion- based esti-

mation, and prediction filter based estimation) and illustrate how c m be obtained for each

case. The covariance rnatrix of estimates, Q8, can then be approxirnated using g. Note that since t here are no output equations or state variables involved for steady-state systems

described by w ( z , P ) = O, Krishnan's [1990] obsenability criterion in Equation (3.2) will reduce

rank [V& = p

BACK CALCULATION

The parameters can be back-calculateci using aven values for measurements (z) by solving the

following sirnultaneous set of (nonlinear) equations for 8:

The mode1 equations ( w ) are solvable for B if rank of V p w is equal to the number of parameters

as required by the rank criterion provided by IOishnan [1990] aven in Equation (3.10).

The sensitivity matm can be determined by taking the total differential of Equation (3.11):

h m which can be caiculated assuming that rank of % (or Vow) is p, the number of a0

parameters. QB can then be approxirnated using Equation (3.9). Sote that calcdation of

requires that rank criterion by Krishnan j19901 in Equation (3.10) be satisfied.

Since in back-calculation al1 the characteristics of the estimator are included in the mode1

equations ( w ) thernselves. the approach provided by Krishnan [1990] includes information about

dl cornponents of the estimator. Also. from Equation (3.12) it can be seen that (and therefore

Q B ) contains al1 the characteristics of the estimator as well. Thus for back-calculation. both

approaches d l provide necessary and sufficient conditions for parameter obsemability.

OPTIMIZATION-BASED ESTIMATION

Parameters are often updated using optimization based methods such as nonlinear regression.

-4 generd optimization based estimation can be formulateci as:

O P.€

subject to :

where: O is the objective hinction for parameter-estimation optimization problem. f is a vector

containing the appropriate mode1 equations. and E is a vector of residuals.

The optimization variables in Problem (3.13) are B. and E. Eliminating the residuals kom the

problem transforrns it into its reduced space [Fletcher, 19871 where it becornes an unconstrained

optirnizat ion problem:

mjn o(D:z) (3.14) P

Sufficiency conditions for a minimum require t hat the reduced gradient (Oro) ~an i sh at the

optimum and that the reduced Hessian be positive definite for a unique minimum:

Forbes j19941

at ives e i s t ) :

~f O be positive definite

has provided an expression for the reduced gradient (nhere the appropriate deriv-

The sensithlty matrix can be obtained by taking the total differential of the reduced gradient

and rearranging:

Thus depends on the inverse of the reduced Hessian and the effect of perturbations in

the measurements on the reduced gradient. An expression for the reduced Hessian has b e n

provided by Forbes 119941:

From Equation (3.15) it can be seen that contains information about all components of the

parameter updating procedure including the characteristies of the objective function as n-el1 ai at.-O the steady-stare modeis that descnbe the procev (contained in the vector f ) . Similarl- 7

n-iU &O contain information about the objective function as weli as the mode1 equations.

-4ithough Equation (3.15) semes WU to illustrate the information contained in S. this formu-

lation cannot be readily used to cornpute g. .An alternate approach to determinhg is ro use

local sensitivity analqsis [Seferlis. 1995: which is based on the Implicit Function Theorem :Cor-

uin and Sznarba. 19951 and describes the fim-order changes to the optimum subject t o small

perturbations about a nominal point. The approach to sensitivity analysis taken by Ganesh cf3 and Biegler il9871 rields a linear system of equations that allous for easier computation of z.

The aut hors develop sensiti\-ity analgis under the assumption that the fol1oning condit ions are

satisfied at the local minimum. e': 1. The functions in Problem 3.13 are at least tnice continuously differential in B and

at l e s t once in z for a neighbourhood of (B'. 2'). 2. The constraint gradients are linearly independent at b' and strict complementary

slachess [Edgar and Himmelblau. 1988j holds for the Functions in Problem 3.13 at

a' with unique Lagrange multipiiers. p.

3. The second-order sufficiency condirions are met [Ganesh and Biqler. l98ïj.

From the Karush-Kuhn-Tucker (KKT) conditions !Edgar and Hirnmelblau. 1988::

a-here: L = 0 - pTf is the Lagrangian of the optimization problem. p is a \=or of Lagrange

mult ipiiers.

Applying the Implicit Function Theorem [Cornin and Szaarba. 1995i and rearran,$ng Dives:

£rom a-hch the sensirivin matrix. v&' !or 2). can be calculateci. U M e analytical solutions

can be obtained for maIl optimization problems. numerical methods =-ould have to be utilized

for larger ones. The cm.ariance rnatnx of parameter esrimares !Q8 j c m then be apprmimated

using Equation 13.9).

From this definition of 2 as ad. it can be seen that contains information about al1 corn-

ponents of the updating procedure. The Lwangian L contains the objective function O as nell

as the model equations (contained in f ) . Thus the characreristics of O are capturd in TjZLœ.

T, . ,LS. V: L'. Tg, L'. and T,, L'. Information on the model equations are contained in these

te- as well as in T,fœ. Tâfœ. and Vcfs.

.b an example of optimizarion-baseci estimation. consider weighted leact-squares regession:

subjed to :

where: A is a weighting matrix.

In this case. f = w@.zj - 6. and so rjf =Tau. Therefore. Equation (3.16) becomes:

where the Lagrangïan is:

L = E'AE t pT[o(,&z) - E]

Rom Equation (3.18) it can be seen that 2 contains information about ali components of the

estimator - the model equations in the constraints as u d as the objective function. Character-

istics of the objective function are containeci in the ternis VB,Lal Vr,cL', v~L'. VB,L? and

V,,L*? while mode1 information is included in these terms as well as in V,w'. 'VjwT*. V,w'.

and V.ok. The method presented by I(rishnan jl99O! examines ody the r e m V D w and so looks

only at the constraints in the estimator in Probiem (3.17) and neglects the objective function

ent irely. providing a necessary but not sdicient condition for parameter obsemability. Test ing

for obsemibility using Qg on the other hand. provides a sufficient condition since i t includes

al1 aspects of the estimator.

ESTIMATION USING PREDICTION FILTERS

Filtering is comrnonly used in control s-ystems and involva using current measurements to u p

date previous estimates of the unmeasured quantities to protide the most up teda te estimates

in a sequential fashion [Ray. 19811. Filters include Katman and extended Kaiman filters :Chu.

19873 shich incorporate -tem model information. The Kalrnan family of filters have been

applied very successfuily over the past three decades for state estimation. Prediction filters

based on Box and Jenkins :19T6i methods are also widely used. Other prediction filter design

rnethods include those baseci on Discounted Least Squares approach [llontgomeq- and Johnson.

1976; and the Bayesian approach :Ilontgomey and Johnson. 1976:.

In filtering. the updating s t r a t e e is t o estimate the pararneters at the current time-step t k

(&t, ) using their predictions from the prediction time s e p ( B t k i t 4 - , j and a correction term

a-hich involves the current measurernents (z t , ) and their predictions from the previous time step

( i tkIt4- ,) [Ray. 19813. Thus the expression for updating the adjustable pararneters is given by:

a-here: P t k i t C is the vector of parameter estimates at time t , using data up to time tt ' ,Btki,- ,

are predicted estimates for time t k using data up to time t t - l z t , are measurements obtained

at time t t , itkl,,-, are predicted measurements at time tk using data u p to time t t - l ' and K is

the £ilter matrix.

Step ahead predictions of parameters (Pt, (,,-, ) and measurernents (î,, (,,- I ) can be obt ained

using appropriate prediction models. When such information is not amilable, one approach is

to assume that the measurements and the parameter estimates wiii remain unchangeci over the

next t ime-step:

The updating expression can then be mit ten as:

and rearranged to give:

Comparing Equation (3.19) with Equation (3.8): the sensitivity of the curent estirnates to the

measurements. d4tk'tk. is sirnply the fiiter matriu: &tk

Thus the sensitivity matrix ail1 depend on filter design. The co~ariance matrix of parameter

estimates Qq can be obtained using via Equation (3.9) as: ( -1

The met hod for determining parameter obsembility presented by Krishnan [1990] ws steady-

state models ( w ) that describe the process as given in Equation (3.10). These models are

commonly used in the design of the Kalman farnily of filters [Chiu, 19871 (in addition to other

information such as noise characteristics). However. certain other classes of filters such as

prediction filters bas4 on tirne-series andysis [Box and Jenkins, 19761 may not directly contain

these steady-state models. Moreover, the filter matrix may be user-tuned in which case the

filter matrix (K) will not contain any mode1 information. Thus for filter based-estimation.

determination of parameter observability using the approach of Knshnan [1990] uses only partial

6iter information (as in the case of the Kaiman f d y of filters) or no information at dl.

Therefore, Knshnan's approach has only lirnited application to some fiiter designs and no

application to ot her designs. The st atist ical approach presented in t his chapter, however. can

be applied to all filten and will provide a sufficient condition for parameter observability since

it uses the filter matrix directly in calcdating Q p as shown in Equation (3.20).

3.4 DISCUSSION

Estimation of unmeasured quantities fiom secondary measurements for steady-state systerns

is an important problem in the chernical industry. For example, concentrations in distillation

colwrins are often estimated using temperature measurements. However? care should be taken

to ensure that unique estimates of these quantities can be obtained from the available mea-

surements and estimator. Moreover, the estimates must have an acceptable level of statistical

certainty.

The approach to determining parameter obsenability for steady-state systems provided by

Krishnan [1990] considers only the steady-state models that represent the plant. It does not

take into account the characteristics of the estimator. It has been shown in this chapter that

t his approach cannot provide a sufficient condition for pararneter obsenabili ty. A statistical

approach to determining observability of parameters from the measurements and estimator has

been presented which takes into account d l components of parameter estimation to provide a

sufficient condition. The ideas behind parameter observability have been extended to degree of

observability which look at quaiity of estimates.

The proposed approach to observability and degree of observability involves examining the

eigenvalues of the inverse of the covariance matrix of pararneter estimates (QT'). Since eigen- B

values of a matrix are dependent on the scaling applied to the elements of the matriw, degree of

observability will also be scale dependent. Gudi (19951 detemines degree of state observability

in dynamic systems by looking at the singular values of the obsembility matriw (M) which

are also s a l e dependent. Therefore, the matrices Q:' and M would have to be pre-scaled P (by scaling ail elements to the same order of magnitude) before degree of observability can

be determined. In the steady-state parameter obsemability case, Q-' should be based on al1 P parameters (p) being scaled to the same order of magnitude.

In RTO systems, parameter observability can be used in selecting adjustable pararneters. Krish-

nan [1990] has provided a met hod for selecting adjustable paramet ers which invoives identifjing

"key" pararneters and then updating these parameters on-line. Key parameters are determined

by perturbing each parameter at a tirne and noting the resulting change in the objective funct ion

value and any active set changes. Parameters that cause the most change in objective function

value per unit change in parameter. or thase that cause the active con.stra.int set to change. are

selected as adjustable parameters. However, the objective function in RTO system is based on

process econornicç which is usually o d y poorly known. Therefore, the predicted profit could be

quite different than the actual profit. Thus selecting adjustable pararneters based on objective

function changes could lead to poor RTO design. Forbes and Marlin [1996] have provided an

alternative approach to adjustable parameter selection which is based on minimizing loss of

RTO performance. Loss of RTO performance is determined as the Design Cost which considers

performance loss due to: offset from true plant optimum, and variance in predicted optimal

setpoints. In the Design Cost approach. those pararneters which when updated on-line result

in the l e s t Design Cost. are selected as adjustable pararneters. However, this s t r a t e3 (as

weIl as that presented by Krishnan [1990j) does not consider parameter observability. Thus.

it could in some cases, lead to the selection of parameters that cannot be estimated from the

measurements and estimator and so cannot be updated on-line.

The first step in any parameter selection scheme is to ensure that the selected parameters

d l be observable frorn the a~xilable measurements and chosen estimation scheme. Parameter

observability can be tested in two steps. First, the method presented by Krishnan [19901 c m

be used which provides a necessary condition for observability. Parameters that fail to satisfy

the rank criterion in Equation (3.2) can be discarded from the set of candidate adjustable

pararneters at this point. Next. the observability approach presented in this chapter should

be used or1 the remaining pararneters to test if the parameters would be observable from the

measurements and the estimation scheme. Parameters that are not observable or those that are

ody weakly obsenxble can also be eliminated. Note that the proposed statistical approach to

determining parameter obsembility will recognize and eliminate paramet ers that fail to satisfy

the rank criterion in Equation (3.2) since it provides a necessary and sufficient condition for

pararneter observability. However, the second test is specific to a given estimation scheme. Thus

if obsenabiiity t hrough ot her estimation schemes have to be tested, the parameters eliminated

by the h s t step wodd not have to be tested again. The above procedure will narrow the set of

possible adjustable parameters to those t hat exhibit an acceptable degree of observability. The

Design Cost approach to adjustable parameter selection presented by Forbes and Marlin [1996!

can then be applied to the resulting srnder set.

Thus the observability measure developed in this Chapter can be applied to RTO systems

as well as to any steady-state system where parameters are to be estimated frorn secondary

measurements.

Chapter 4

BLENDING CONTROL

TECHNOLOGY

This chapter builds on the modeling and parameter observability work presented in Chapters 2

and 3 to provide an efficient blend controller. The chapter begins by presenting the perfect or

ideal blend controller which is then used as a benchmark for comparing controller performance.

In addition to providing a performance bendunark. the ideal/perfect controller provides insight

into better controller design. Xext , the nidely used control strategy and the problems associated

with it are examined. Finally, a new control approach that overcomes the shortcornings of the

current strategy is presented. Performance of diEerent blend controllers are demonstrated using

an automotive gasoline blending case-study t hroughout the chapter.

4.1 INTRODUCTION

As discussed in Chapter 1, gasoline blending involves combining refinery feedstocks to make

products meeting certain quality specifications. The control structure used for blending gasoline

is given in Figure 1-4. At the heart of the controller is an RTO layer that is usually based on

linear programming with bias updating. The success of this control strategy has been largely

due to the practice of blending from well-rnixed storage tanks. However, increased cornpetition

has forced refiners to reduce on-site tankage inventory by blending out of "running" tanks (or

direct in-line blending). "Running' tanks are those intermediate tanks into which feedstocks

from upstream processes are being pumped. Also, process strearns can be routed directly

to the blender without first being pumped into either an intermediate storage or "running"

tank [Agrawal, 19953. Since blending feedstock qualities are dependent on upstream process

operations and upstream process changes ( e.g. catalyst deactivation, heat exchanger fouling,

crude oil switches, etc.), they may not be constant and may vary significantly during blending.

This chapter focuses on the problem of controlling a gasoline blender which is subject to sto-

chastic disturbances in the feedstock qualities. Such disturbances nat uraily arise when the

blender is using "running" tanks as feeds and upstream process operations change. The objec-

tives of this chapter are to demonstrate the problems that existing blender control systems may

encounter when faced with feedstock quality disturbances, to provide a control strategy that

can effectively deal with such disturbances, and to illustrate the performance improvements

associated wit h the proposed cont rol approach.

4.2 GASOLINE BLENDING BENCHMARI( PROBLEM

An automotive gasoline brending problem is used to compare the performance of biend con-

trollers in this chapter. Different blend controllers will be applied to the benchmark problem.

4.2.1 BLENDLNG PROBLEM

The blending problem studied here is based on a case-study by Forbes and Marlin [1994] and

involves simultaneously blending two grades of automotive gasoline using five feedstocks as

shoan in Figure 4-1. For the purposes of this illustrative problem, the product specifications

for each grade of gasoline will be lirnited to: minimum limits on RON and MON? and a

maximum limit on the RVP. Constraints are also placed on maximum and minimum product

demands and feedstock availabilities. Product specifications and economic data are provided

in Table 4.1 and feedstock data are in Tables 4.2 and 4.3.

In this problem, gasoline is to be blended for 24 hours with an RTO interval of 2 hours. As

discussed in Chapter 2, the gasoline blending process is simulated using the Ethyl RT-70 [Healy

et al.! 19591 and the Blending Index [Gary and Handwerk, 19941 models. These models are

presented in Equation set (2.16).

value ($1 b bl) ma.. demand limit

+chosen for this study

Table 4.1: Production Requirements

(bbl/day min. demand fimit

(bbl/dav)

l ( Available 1 Cost 1

Regular 33.00 8000

Premium 3 7.00 10000

7000

Table 4.2: Feedstock Economic Data

10000

reformate LSR naphtha

n- but ane catalytic gas.

alkylate

pnrantlicscs i i~ ir ler fccdsrock indicntc t a b l a in Hcnly P Z of. [195!3]

+ + froni Forbcs aiid Xlarlin [1991]

from Forbcri anci MnrLn [11)91j

(bbllday) 12000 6500 3000 4500 7000

Feedstock

RON MON

Ole f in (%) Armatics (%) RVP (psi)

Table 4.3: Feedstock Qualities

($/bbl) 34.00 26.00 10.30 31.30 37.00

frotir Hccily et al. [11)59]

reformate (C14a)

94.1 80.5 1 .O

58.0 3.8

LSR napht ha (clh)

70.7 68.7 1.8 2.7 12.0

n-butane (C25a)

93.8 90.0

O O

138.0'~

catalytic gas. (C12a)

92.9 80.8 48.8 22.8 5.3

alkylate ('34

95.0 91.7

O O

6.6

LSR naphtha 1 m!

* Regular

Figure 4- 1: Blending Process

n- butane I *

catalytic gas.

4.2.2 STOCHASTIC DISTURBANCES

For the illustrative purposes of this case-stud- disturbances are to be added to RON. MON,

and RVP of the feedstocks. In order to sirnplib the benchmark blending problem, it was

assumed that disturbances were present oniy in the feedstock that affects the optimal operations

the most. In order to determine the feedstock whose qualities the optimal blend recipe is most

sensitive to? a local sensitivity analysis [Seferlis. 19951 was conducted around the nominal blend.

The nominal optimal blend was obtained using NLP with MINOS in GAMS [Brooke e t al., 19921

using data given in Tables 4.1, 4.2, and 4.3, and the blending models in Equation set (2.16). The

results are swnmarized in Appendk A. After scaling al1 the decision variables and parameters

to the same order of magnitude, the linear system of equations given in Ganesh and Biegler

[1987] was solved analytically using MAPLE [Char et ai., 19911 for the pararnetric sensitivity

matrix (VqxS) of the optimal process variables (x') to the parameters (p). Five submatrices

of Vgx*, each corresponding to a feedstock, were obtained and their norms (largest singular

values) computed. The results are summarized in Table 4.4.

C

*

.c

Premium

Table 4.4: Feedstock Sensitivity

Feedst ock reformate LSR naphtha n- but ane catalytic gas. alkylat e

The submat rut corresponding to the reformate qualities yielded the Iargest singular value. Thus.

stochastic disturbances were added to the reformate stream' for the purposes of this example.

as:

Subrnatrix il'orm 3.66 1.20

0.005 1.63 O

b

T where: %f.t = [ RON i f ~ . Y RVP ] is a vector of reformate qualities at time t : BEI

ref ,t is a vector of nominal reformate qudities, and dl is a scalar disturbance.

The scalar disturbance, d t , was modeled as steps of random height entering a well rnixed tank.

The tank was assurned to hold one third the availability of reformate (as given in Table 4.3)

per eight-hour shift, resulting in the following transfer function:

where: 2-' is the backward shift operator and st are steps of random height.

.S noise vector was generated for each 2 h o u time interval over the blend horizon and noise

was assurned to remain unchanged during each RTO intenal. Soise was then added to the

reformate qualities over the whole blend horizon. The resulting RON, 610N. and R V P of the

reformate Stream over the 24 hour period is shoam in Figure (4-2).

4.2.3 CONTROLLER PERFORMANCE

Controller performance is measured by the profit earned from applying the controller on the

benchmark problem posed in this section. When a controller produces blends that meet al1

Reformate Qualities

RTO Interval

Figure 4-2: Reformate qualites dunng 24 hour blend.

quality specifications. the profit level can easily be computed using the economic da ta provided

in Table 4.2. However, if the products at the end of the 24 hour blend horizon are found to be

off-specification. the- may not be suitable for sale as produced, and would likely be routed to

.slop' tanks and reblended. In such situations! profit calcdation is not as simple.

One option is to include the cost of reblending in determining profit level. Alternatively. the

off-specification portions of blends (products made during RTO intervals that are causing the

whole 24 h o u blend to be off-specification) can be "removed" from the blend. The profit can

then be calcuiated based on the rnodified srnaller blend that does meet quality specifications.

..Uthough blend portions cannot be 'kemoved" or unblended from a blended product in practice,

it can easily be done for simulation studies for the purposes of accessing controller performance.

Since the economic data required for determining loss of profit due to reblending is not available

for this study, the second alternative of "removing" off-specification products is adopted.

Small violations of quality specifications can often be tolerated since product quality measure-

ments are limited by analyzer sensitivity. For the purposes of this study, it is assumed that

violations of up to -0.03 octane number for RON and MON, and +O.005 psi for RVP cm

be tolerated. That is, blended quaiities that fall outside these tolerance limits are deemed

off-specification.

4.3 IDEAL BLEND CONTROLLER

An ideal blend controller is one that gives the maximum possible profit £rom the blending

process. while meeting all blended quality specificat ions, and sat isfying demand and araila bil-

ity limits. In order to achieve perfect control, the controller must have perfect knowledge of

the blending process. That is, it should employ blending rnodels t hat capture the t m e blending

behaviour. In addit ion, the controller must have perfect knowledge of al1 feedstock qualities at

d l times throughout the entire blending tirne horizon. For batch blending of gasoiine (blending

to a storage tank for a given length of time as in the benchmark problem) , the ideal blending

control problem c m be formulated as an optimization problem where the objective is to ma.-

irnize profit over the entire blending time horizon, subject to meeting quaiity constrainrs and

demand/availability limits at the end of the blend. This problem has the form:

sub ject to:

where: c is a constant vector containing process economics, x are the feedstock or component

flows. R is a matrix containing known feedstock qualities. g is a vector of blending equations

that represent the true blending behaviour. w is a weighting vector (usually containing only

ones and zeros to indicate the presence of a specific Stream in a blend). s are the blended quality

specifications, h contains product demand and feedstock availability constraints. and t,, and tf

are the initial and final blend times, respectively.

The controller works to maximize profit o\*er the whole blend by integrating blending economics

from the start of the blend (t,) to the end of the blend (tf), rather than at any point in time.

The first set of constraints:

ensure that quality specifications are met at the end of the blend. These constraints use perfect

knowledge of the feedstock qualities (R) over the entire blend horizon (hom t , to tf) as well as

perfect blending models in vector g. The second set of constraints:

integrate mass balance information over the blend horizon in order to satisfy d l product demand

and feedstock availability lirnits.

Problem (4.3) contains integrals in the objective h c t i o n as we11 as in the constraints and

so would have to be solved using dynamic programming tediniques [Bellman and Lee. 19841.

.4lternatively! the problem can be discretized and solved using standard XLP rnethods [.4niel.

19761. The solution to Problem (4.3) will provide optimal feedstock flom ( x ( t ) ) over the whole

blending horizon which is the optimal blend trajectory. The d u e of the objective function at

the optimum represents the theoretical maximum profit that can be eamed £rom the blending

process.

The ideal controller in Problem (4.3) requires future knowledge of feedstock qualities and so

cannot be realized in practice. However. it can serve as a benchmark against whkh performance

of other controilers can be evaluated. and so it can provide insight into better controller design.

4.3.1 OPTIMAL BLEND TRAJECTORY FOR BENCHMARK PROBLEM

The perfect controller u formulated in Problem (4.3) was applied to the benchmark blending

problem. Vector g contained blending models for RON, MON, and R V P for both gasoline

grades. The blending models and conçtraints used here were the same as those used for sim-

ulating the blending process given in Equation (2.16). The dernand/aiailability constraints in

vector h use m a s balances and assume no density changes upon blending. Feedstock flows xi

at time t. and the wighting vector w are given as:

where: X ~ J . .... xt.5 are B o a s at time t to regular gasoiine of reformate. LSR naphtha. n-butane.

catalytic gasoline. and alblate. respectively: 16. .... xlo are B o a s at time t to premium gasoline

of reformate. LSR naphtha. n-butane. catai'ic gasoline. and aikylate. respectively.

The demand/availability limits are of the form:

The perfect controller was applied to the blending problem by approximating the integais in

Problem (1-31 n-ith mrnat ions . and wlving using IIISOS/G;\.\IS YBrooke et al . 1992:. The

optimal blend trajecton and the active constraints during each KI0 i n t e n d are provideci in

Xppendix B. The theoretical maximum profit &-as found to be S66.276.55 for the 2 1 hour blend.

4.4 CONVENTIONAL CONTROLLER

As ha. been discussed in Chapter 1. at the h a r t of the blender control q n e m is an R f O Iayer

ïhat is wudy based on linear propamraing ai th b i s updaring :.\fagodas et aL. 1988;. The

objective of the on-line optirnizer is usually optimization of steady-state economic performance

'1-e. . maximization of profit flowj. Thus the blending control problem is m m ofren forrndated

as:

max X

cTx

subject to:

- where: c is a constant t-ector containing process economics. R is a constant matrLv contain-

ing the feedstock qudity blending indices. s are the blended quality specifications. x are the

f&to& or component flm. w is a weighting vector (usually containing o d y ones and zeros

to indicate the presence of a specific Stream in a blend), b are the biases wd to update the

blending model to maintain its accuracy. and h is a vector of linear equations for the maximum

and minimum product demands and feedstock availabilities.

The matrix R contains blending indices such as Blending Octane Sumbers (BO.\'-s) that

b l e d linearly as volumeçric averages. However. as discussed in Chapter 2. most important

gasoline qudities (such as octane number. Reid Vapour Pressure. and the -%S'Pd Distillation

points) blend nonlinearly. Since linear blending models are unable to capture the true blending

behariour. structural mismatch is introduced into the optimizer. Bias updating is used to

compensate for such piant/model mismatch as weU as other modeling inaccuracies.

The RTO layer works by:

1. t aking measurernents of blended qualities.

2. cdculating biases as the difference between measured blended qualities and those pre-

dicted by the linear model &cc-i. where xb-1 are the current feedstock Rom.

3. solrlng Problem (4.1) using the newly calculateci biases to obtain new blender feedstock

f l o ~ ( x ~ )

4. implementing blend recipe and wait ing for steady-state.

The process iterates over the whole blend horizon. That is. the above four steps are carried out

at each RTO in tend until end of blend is reached.

In the bias update formulation of Problern (4.4). the only adjustable pararneters are the bias

terms whle ail the feedstock qualities are treated as fixed pararneters. Forbes and 4Iarlin (19941

have shown that a bias update approach can lead to a substantial l o s in blender profitability

given even a s m d variation in the feedstock qualities. Thus the current blender control structure

cannot adequat ely address the blending pro blem where the cornponent qualit ies are varying .

The conventional LP with bias updating controller sufTers from two limitations: inability to

incorporate the inherent blending nonlinearities. and ineffectiveness in dealing with fluctuating

feedstock qualities. The effect of t hese limitations on cont roller performance is demonstrated

by applying the controller to the gasoline blending problem descnbed in Section 4.2.

4.4.1 CONVENTIONAL CONTROLLER PERFORMANCE STUDY

The traditional LP with bias updating blending controller given in Problem (4.4) was used on

the blending problem posed in Section 4.2. Blending Octane Nurnbers (BON'S) [Gary and

Handwerk, 19941 were used for RON and MUN and blending indices (RVPBl's) were used

for RVP blending [Gary and Handwerk. 19941. Hence, matrix R contained the BON'S and

RVPBl's for al1 the feedstocks. The linear models used in the RTO layer result in structural

mismatch in octane blending.

The demand/availability constraints use mass balances and assume no density changes upon

blending. Feedstock flows at time t. xt, and the weighting vector. w, are given as:

where: x t , ~ , ..., xt,s are flows at time t to regular gasoline of reformate, LSR napht ha. n-butane,

catalytic gasoline, and alkylate, respectively; xs, ..., xlo are flows at time t to premium gasoline

of reformate, LSR napht ha, n- butane, catalyt ic gasoline, and alkylate, respectively.

The demand/availability limits at time t are formulated as:

The following steps were carried out at the kth RTO interval:

1. simulate blending process using xk-1 and the blending rnodels in Equation (2.16) to get

"measured" blended quali t ies,

2. calculate biases for next step,

3. solve Problem (4.4) to calculate xk.

Blended Octane

0

Figure 4 3 : Blended Octane using LP + Bias Controller

The above steps were repeated at each RTO interval, and Problem (1.4) was solved using linear

prograrnming in GAMS [Brooke et ai., 19921. The optimization results (feedstock flowrates and

active constraints) for each ElTO intenal are given in Appendiv C. The product qualities a t

the end of the 24 hour blending period are s h o w in Figures 4 3 and 4 4 as deviations from

specification.

Whiie there is significant quaiity giveaway in RON and MON of reguiar and RON of pre-

Mum gasoline, RVPs of both grades of gasoline are above the maximum specified and the

ib.iON of prernium grade is below the minimum dowed. In addition, RVPof both grades is

off-specification throughout the biend. As a result, both grades of gasoline produced are off-

specification and may not be suitable for sale. Since both grades are off-specification t hroughout

the blend horizon, portions of blends cannot be removed to obtain blends that do meet speci-

fications. Hence, a profit level for the conventional controller could not be cornputed.

In this case-study, the conventionai controuer fails to make products meeting quaiity specifica-

tions. Poor controller performance can be attributed to a combination of structural plant/model

mismatch resulting from the use of linear blending models, and parametric mismatdi aiising

frorn fluctuations in reforrnat e quali t ies.

Blended RVP

RVP Regular RVP Premium

Figure 44 : Blended RVP using L P + Bias Controller

4.5 PARAR.IETFUC MtSMXTCH CASE

As discussed in Chapter 2, linear gasoline blending models are not able to capture the tme

blending behaviour because most important gasoline qualities blend nonlineariy. Since blending

nonlinearity is one of the major sources of blending inaccuracy, accurate modeling of proper-

ties such as octane number and boiling points is crucial to the success of the blend controiler

[Ramsey Jr. and Truesdale, 19901. Recognizing the problems associateci with blending non-

linearity, some refiners have tned incorporating nonlinear blending models in their controllers.

An approach to incorporating nonlinear blending models has been to linearize the models and

then use sequential iinear programming (SLP) which retains the controiler formulation given

in Problem (4.4) (e.9. Diaz and Barsamian, [1994]). Another approach is to incorporate the

nonlinear rnodels directly in the blender control problem by formulating the on-line optimizer as

a nonlinear programming (NLP) problem with bias updating (e-g. Ramsey Jr. and Truesdale.

19901:

subject to:

where: g(x) is a vector of nonlinear blending models.

Sote that Problem (4.5) differs from the LP + bias update controller in Problem (4.4) only in

the blending models used and can be solved using standard NLP techniques [Avriel, 19761.

4.5.1 PARAPViETRPC MISPVIATCH PEWORMANCE STUDY

The NLP -+ bias update controller given in Problem (4.5) was used on the benchmark autome

tive gasoline blending problem posed in Section 4.2. The controller was designed to eliminate al1

sources of plant/model mismatch by using the same blending models as used in simulating the

blending process (see Section 2.4). Since the feedstock qualities used in the RTO were fixed at

their nominal values for al1 feedstocks except reformate, bias updating was used to compensate

for fluctuations in reformate qualities. Product demand and feedstock availability limits were

( Quaüty 1 RON 1 RVP 1 MON 1 RVP

Table 4.5: Deviation £rom Specification at End of Blend

Deviation

the same as used in the LP + bias update case and the quaiity constraints were of the form:

where: gl and g2 are the RON blending mode1 applied to regular and premium grades, respec-

tively, g3 and g4 are the MON blending mode1 for regular and premium grades, and g~ and g6

are the R V P blending mode1 for regular and premium gasoline. The blending models are given

in Equations ( 2 . 1 6 ) .

(reguiar) -0.1

Simulation of blending control was conducted as descri bed for the LP+bias update case except

that the blend recipe at each RTO interval was obtained using ,ULP with MINOS in GAMS

[Brooke et al.? 19921. The cornplete blend recipe can be found in Appendix D. Upon completion

of the blend, four of the blended qualities in the product tanks were found to be off-specification

as summarized in Table 4.5.

For the purposes of this study, deviations of up to -0.03 octane nurnber for RON and MON

and f0.005 psi for RVP were deemed acceptable. These tolerance levels were satisfied for both

grades when gasoline produced during the l S t , g th , lo th, and 121h intervals were excluded form

the final products. The excluded gasoline was routed for reblending.

(regdar) 0.015 psi

Eliminating products from 4 of the 12 RTO intends resulted in much smaller batches and

the profit earned from the blends is $44,105.96, much Iess than the theoretical maximum of

( pr emiurn) -0.05

( prernium) 0.01 psi

$66,276.55. However, performance of the NLPtbias update controuer is far better than t hat

of the conventional LPfbias controller used in Section 4.4.1. Thus eliminating structural m i s

mat ch from the convent ional LP + bias cont rouer improves cont roller performance drarnat icdy.

However, the bias updating strategy is not able to adequately handle pararnetric mismatch

arising from stochastic feedstock disturbances, resulting in the production of off-specification

products.

4.6 TIME-HORIZON CONTROLLER

Cornparing the performance of the LPtbias controller of Section 4.4.1 and the XLPtbias con-

troller of section 4.5.1 clearly shows that the use of appropriate nonlinear blending models is

crucial to the success of the blend controller. By eliminating structural plant/model mismatch.

a very significant improvement in controller performance was obtained for the benchmark prob-

lem. However, as shown in the case-study, the resulting NLP+bias updating controller can still

lead to the production of off-specification blends when disturbances are present in feedstock

qualities. This is because in the bias updating structure, the feedstock qualities are treated as

Lied parameters in the controller while they vary in the actuai plant. The resulting parametric

mismatch cannot be compensated for by bias update [Forbes anci Marlin, 19941. Thus, such a

blender control strategy is not well suited for control application where feedstock qualities are

fluctuating.

Several control practitioners have reco,dzed the problems associated wit h Auctuating compo-

nent qualities and have offered some partial solutions by incorporating multi-period optirnization

capabilities into t heir blender control systems (e-g. SlcDonald et al. [l992], Rigby et al. [l9951).

These methods approach the problem of feedstock quality changes by infrequent feedback of

feedstock properties hom laboratory analyses, and predicting future control actions using these

laboratory based measurements. The update intervai for such systerns is typically a day and

component qualities are assumed to remain unchangeci in the future. Thus, they cannot effec-

t ively deal with the higher frequency dist urbances associated wi t h upstream process operat ion

changes. Another approach involves using mode1 predictive control technology where on-line

blended quality measurements are used to predict disturbances in t hese qualit ies [Vermeer et

al., 19961. However, in their work Vermeer e t al. assumed disturbances in the blended qualities

as step disturbances that persist over the b l e d horizon. Although a move in the right direction.

t his met hod does not utilize noise modeis that exploit feedstock quality disturbance dynamio

to predict feedstock qualities in the future. As a result, existing methods cannot adequately

address the issue of handling stochastic feedçtock dist urbances.

In this section, a new blender control approach wiU be presented that can effectively deal

wit h changes in feedstock qualities. Mode1 predictive control (MPC) schemes ( e.g. Cut ler and

Rarnaker [19791) have enjoyed considerable success both in academia and in industry in the last

15 years since their development . MPC systems have been widely appiied to systerns subject to

stochastic disturbances [Garcia et al., 19891, [Ogunnaike and Ray, 19941. The success of MPC

systems c m be attributed in large part to their ability to exploit knowledge of process dynarnics

to predict future plant behaviour using appropriate modeis. However, to date, such strategies

have not been appiied to RTO systems. In this work, the disturbance handling capabilities

of MPC systems are adopted in developing a new blend controller which is based on insights

provided by the ideai controller,

The ideal controller in Problem (4.3) cannot be implernented because it requires future knowl-

edge of feedstock qualit ies. Although perfect knowledge of feedstock qudit ies at al1 t imes during

the blend is not realizable' past feedstock qualities c m be used to predict these qualities into

the future. Thus at each RTO intend! predicting feedstock qualities during the remaining

blending time horizon ni11 allow future control actions to be based on knowledge of feedstock

behaviour over the whole blend horizon ( p s t and future). This requires the blending problem

to be posed as an optimization probIern over the whole blend horizon as in the case of the

ideal controller. This approach generates blend recipes for the rernaining blend horizon. The

recipe for the current time-step can be implemented and the process of prediction/optimization

repeated at each time-step in a receding horizon fashion until end of blend is reached.

Çuch a blender control strategy is very similar to MPC in that they both involve predicting

disturbances over a time horizon and basing control action by optimizing over a time horizon.

In both cases, only the current control action is implernented and the whole process is repeated

in receding horizon fashion. However, while the length of the prediction as well as the control

(optimization) horizons are tuning parameters in MPC, only the prediction horizon is w r

tuned in the blend controller. This is because the optimization horizon in the blend controller

is h e d at each RTO i n t e d as the remaining blend horizon.

The blender control strategy discussed above and the ideal controller work towards meeting

quality specifications only at the end of the blend. If blending has to be tenninated prior t o

reaching end of blend, the products made thus far could be far Erom being on specification. In

order to guard against such situations? an additionai constraint may be added to the opt irniza-

tion problem that would ensure that product specifications are met at ail times in the future.

This control strat egy can be mat hematically formdated as:

subject to:

where: R is a rnatrix containing past and present measured

containing the predicted feedstock qualities, g is a vector of

feedstock qualities.

blending equat ions

R is a matrix

which rnay be

linear or nonlinear. and t,. t , , tf are the initial. present, and final blend times, respectiveiy,

and h contains product demand and feedstock availability constraints.

x ~ t e that the Problem (4.6) is very similar to the perfect controller in Problem (4.3). The pro-

posed blend controller differs from the ideal controller in that the blend horizon is divided into

two parts: past (and present) and future. Also, Problem (4.6) contains an additional constraint

to ensure that product specifications are met during blending. The objective functions in P r o b

lems (4.6) and (4.3) are equ iden t even though Problem (4.6) integrates blending economics

only from t , to tl rat her than over the whole blend horizon. This is because the economics due

to the past blended products is a constant and will not affect decisions on future control action.

In other words, future control action cannot change economics due to past blended products.

The first set of constraints:

look a t predicted end of blend quaiities which are composai of past blended qualities (to to t p )

and future predicted blended qualities. The future predicted blended qualities are obtained by

integrating the blending models Erom t p to t f using predicted feedstock qualities. ~ ( t ) . Feedstock

qualities c m be predicted using conventional prediction filter techniques (e-g. Box and Jenkins.

[1976], Chiu, [1987]). These constraints emure that products meet quaiity specification at end

of the blend.

The second set of constraints:

look at future blended quaiities based on blending models and predicted feedstock qualities.

These constraints attempt to blend products such that each partial batch made during individ-

ual RTO intends meet specifications. Finail- the third set of constraints:

integrate m a s baiances fkom the start of the blend to the end of the blend

ail demand and availability limits over the whole blend.

in order to satisfy

Like the ideal controller. the time-horizon based controller in Problem (4.6) can be solved using

dmarnic programrning [Bellman and Lee. 19841 or the problem can be discretized and then

solved with SLP [Avriel. 19761.

The blender control problem in Problem (4.6) is unique compared to other blend controllers

in that . at each RTO inten-al. the controller look back in time to what has been bIended so

far while anticipating future trends in feedstock qualities. It then adaptively alters the blend

recipe in order to maxirnize profit while satisfymg al1 constraints. The advantages of the time

horizon formulation are that during each RTO interval the controller has an opportunity to:

1. cornpensate for past rneasured off-specification products:

2. pre-compensate for ant icipated trends in feedstock qualities,

3. re-capture any past quality ';giveawaf.

Therefore, the controller should be able to effectively deal with stochastic disturbances in feed-

stock qualities and provide better performwice than the bias update approach. The effectiveness

of the time-horizon controller will be demonstrated using the benchmark automot ix-e gasoline

blending case-study of Section 4.2.

The time-horizon based controller requires that some of the feedstock qualities be forecasted

into the future. First. though, the feedstock qualities that are to be updated on-line and fore-

casted have to be identified. It is desirable to keep the number of adjustable parameters Iow

since the more parameters that need to be updated on-line. the more measurements t hat would

be required. However. the selected adjustable feedstock qualities should be able to capture

the effects of major disturbances. Selection of adjustable parameters for RTO systenis is dis-

cussed in Section 3.4 where a selection strategy based on the Design Cost criterion (Forbes and

Slarlin. 19961 is recommended. The suggested adjustable parameter selection strategy incorpo-

rates parameter observability information. Parameter obsenability is discussed in Chapter 3

where a new mesure of obsembility is dewloped based on fundamental statisticd principles.

Even when a rigorous selection strategy is not applied. those parameters that exhibit strong

obsenability (have well condition4 joint confidence regions) from the anilable measurements

and aven estimation scheme should be selected orer those that only have a Ioa- degree of

observability.

4-6.1 TIMEHORIZON CONTROLLER PERFORMANCE STUDY

The proposed tirne-horizon based cont roller in Problem (4.6) was discretized (by approximating

integrals rvith summations) and used on the blending problem pmed in Section 4.2. The

blending models used in the vector g contained no structural mismatch with the simulated

blending process and are as giwn in Equations (2.16). Product and feedstock availability

constraints were as describeci for the ideal controller in Section 4.2.

In this simulation case-study. disturbances are known to be present only in reformate RON.

-%ION, and RVP. Therefore, feedstock qualities to be forecasted (adjustable parameters) did

not have to be selected. h o . it was assurneci for this case-study that the reformate qualities

(RON, MON, and RVP) are rneasured on-line. As a result. an observability test on these

parameters was not conducted. Generdy, however, feedst ock qualities are not rneasured and

have to be estimated from other rneasurements. In such situations, obserwbility (fiom the

available measuremeats and estimator) of the qualities to be forecststed wiIl have to be ensured

using the methods developed in Chapter 3.

The t hree reformate qualities were forecasteci in the time-horizon based controller using a predic-

tion filter. In this simulation study. the actud noise model was used in designing the prediction

füter. In general. the noise mode1 would not be h o a n and can be developed using identification

techniques [e.g. Ljung, 19871. The prediction filter made minimum mean squared error fore-

casts [Box and Jenkins, 19761 of the reformate qualities using the noise model given in Equation

(4.2). The prediction filter equations are provided in Appendix H.

Three time-horizon based controllers were designed which differed in their size of the prediction

horizon. 1-step. Zsteps. and Psteps ahead predictions were wd. The prediction filter used

the noise model given in Equation (4.2) and two samples of historical reformate quality data

to make 1-step. 2-steps. and 3-steps ahead minimum mean squared error forecasts [Box and

Jenkins, 19761 of the reformate qualities for each FU0 intend. The qualities were forecasted

at each RTO intenal £rom t , to tf-i and the predictions were held constant for al1 i n t ends

b e o n d the predict ion horizon.

The three controllers (with 1. 2. and 3 step ahead prediction horizons) were used on the blend-

ing problern with SLP usinp MISOS in GAMS [Brooke et aL. 19921. The results from the

controllers at each RTO instant are provided in Appendices El F. and G. While controllers

with prediction horizons of 2 and 3 steps yielded products meeting quality tolerance levels?

ROX and R V P in regular g a ~ l i n e produced using 1-step ahead prediction filter failed to meet

quali ty specificat ions wit hin the speci fied tolerances. Evcluding regular gasoline produced dur-

ing the last inten-al brought the ROX up and the R V P d o m to within tolerance a t the cost

of reducing the size of the blend. thus decreasing the profit level.

Profit earned kom the controllers for the 24 hour blend are given in Table 4.6.

Table 4.6: Cornparison of Tirne-Horizon Controllers

,U three controllers provide profit levels that are quite close to the theoretical maximum of

S66.3r6.55 but performance of tirne-horizon controllers increase slightly as the size of the pre-

diction horizon increases since longer prediction horizons are able to exploit more of the noise

dynarnics.

4.7 DISCUSSION

-4 gasoline blending case-study has been used in this chapter to compare the performance of

blend controllers. The convenriond LP - bias update controller U'~LS found to perfom quite

poorly on the benchmark problern and it has been thown that the conventional controller can

lead to a large performance los when stochastic disturbances are present in feedsrock qualitiec.

Se-=. ail structural p1ant;modeI mismatch u-as eliminated in formularing the XLP - bia. con-

rroller. In this cootroiler. the fdtock qualities were wumed ro be fised and b i s updating nas

u'ed ro compensate for parametric misrnarch resulting from fluctuations in reformate qudities.

The SLP - bias controller a-as found to proride si ,dcant performance improvements O\-er the

conventional LP - bias controller on the benchmark blen&ng problem. However. the controller

can stiii lead to the production of off-specification blends as illusrrated in the cass tudy.

Further performance improvements can be obtained for such -tem. a-here stochastic dis

turbances are present in the feedstock qualities. b~ using the proposeci the-horizon based

controlier. The p r o p o d controller diffen from the 572 - bias updzte controller in that it

optirnizes the blending problem over the a-hole blend horizon rather than at a point in time.

In addition. it updates the feedstock qualities directl- thus addressing the issue of parametric

mismatch directly. The performance of three tirne-horizon based controllers and the hT,P - bias controller are compared in Figure 4.5- Controikr performance is based on the amount

of theoretical maximum profit uncaptured by each controiier. The profit levels for the h;LP

+ bias update and the 1-step ahead prediction the-horizon based controller u-ere calcuiated

using modified blends that meet quali l specifications.

Controller Performance

Figure 4-5: Cncaptured Profit for 24 h o u Blend

-As is evident £rom Figure 4-5. the proposeci time-horizon based controllers are far more efficient

at handling stochastic disturbances than the conrroller based on the conventional bias updatinp

approach. This is because the time-horizon controllers update feedstock qualities directly and

work to rnavimize profit over the whole blend horizon rather than at a point in time.

Chapter 5

SUMMARY, CONCLUSIONS, &

RECOMMENDATIONS

This thesis has focused on the design of the gasoline blend controller. The widely used blend

controiler (LP + bias update) has to deal mith structural plant/model mismatch and also is

not capable of effectively handüng feedstock quaiity disturbances as shoan in Chapter 4. In

t his t hesis, a blend controller that incorporates suitable gasoline blending rnodels and one t hat

can effectively handle predictable changes in feedstock qualities has b e n presented and shown

to out-perfonn the conventional controllers.

Chapter 2 examines gasoline blending modeis for use in the blend controller. Slodek for octane

number. Reid \=pour pressure' and A S D I distillation points are discussed with respect to

predictive accuracy and ease with whch the model can be updated on-üne in order to maintain

its predictive accuracy. The Ethyl RT-70 [Heaiy et al.. 19591 model for octane number blending,

and the Blending Index method [Gary and Handwerk. 19941 for R V P blending were found to

be the most suitable for contrai purposes arnong published models.

Since some of the mode1 parameters would have to be updated on-line in the blend controller,

it is important to ensure that estimates of these parameters be can obtaineà using the available

measurements and chosen estimator. Chapter 3 develops a method of determining parameter

observability for such steady-state systems.

The design of the new blend controiler that incorporates appropriate nonlinear blending rnodels

is then presented in Chapter 4. The proposed controller adopts concepts bom MPC systems

which aliows it to deal with stochastic disturbances in feedstock qualities.

5.2 CONCLUSIONS

The use of LP with bias updating for the control of gasoline blending can lead to a large per-

formance loss when stochastic disturbances are present in feedstock properties as shown in the

case-study in Chapter 4. As a first step towards capturing more profit, plant/model mismatch

should be rninimized. The use of 'JLP with bias updating increases controller performance, but

can still lead to the production of off-specification blends as illustrateci in the case-study. This

is because in. the bias update approach, the feedstock qualities are treated as fixed parameters

even though they are changing in the real process.

The time-horizon b a s 4 blend controller proposed in this thesis directly addresses the issue

of feedstock quality fluctuations by treating these qualities as adjustable parameters that are

updated on-line. Moreover, the control strategy predicts feedstock quali t ies over the blend

horizon to plan blending operations over the entire blend horizon. At each RTO interval,

the controller predicts feedstock qualities over a time horizon. and optimizes the blending

problem over the remaining blend horizon. It then implements control action for the curent

time-step and repeats the process at the next time step in a receding horizon fashion until

end of blend. Thus the proposed time-horizon controller can be thought of as an economic

optirnizing model-predictive control system t hat is analogous to the process control technology

that petroleum refiners have widely deployed throughout t heir plants [e. g. Cut ler and Ramaker,

1979j. Although computationally more intensive, the time-horizon based controller offers a new

approach to handling feedstock quality disturbances, and c m provide efficient control of the

gasoline blender as iUustrated in Chapter 4.

The tirne-horizon RTO approach developed for the blend controller can be applied to on-line

optimization of any process that experiences stochast ic disturbances t hat c a w the optimal

operations to drift with time. The proposed RTO approach would predict trends in major

disturbances entering the plant and optimize the steady-state process over a time horizon so as

to include predicted future plant behaviour in determining optimal operatiom. In addition. the

time-horizon optimization approach can be applied to off-line optimization applications such

as refinery planning and scheduling where predict able trends exist . Here again, predict able

disturbances can be exploited in order to optimize operations over a period of time rather than

at a point in time.

The parameter observability test developed in Chapter 3 is based on fundamental statistical

principles. and has been shown to provide a necessary and sufficient condition for obsenability of

parameters hom available measurements and estimator for steady-state systems. The approach

naturally leads to a measure of degree of observability that looks at the joint confidence region

of the parameter estimates. The observability and degree of observability concepts developed

in this thesis can be used in the design of al1 RTO systems. The tests can be used for selecting

adjust able parameters such that t hey have an acceptable degree of observability. Moreover,

the observability concepts and tests developed in Chapter 3 are not limited to RTO systems.

Rat her, they are applicable to any steady-state system where secondary measurements are used

to obtain estimates of unrneasured quantities.

Future work stemming fiom this thesis include applications of the parameter observability

work of this thesis for RTO design. as well as further work on the design of the gasoline b l e d

cont rouer.

The approach to determining parameter observability developed in this thesis can be used for

developing RTO design practices. That is, it can be used as a basis for selecting adjustable

mode1 parameters as well as for selecting process measurements for updating these parame-

ters. In developing a met hod for select ing adj ustable parameters and/or measurements, the

test developed in this thesis can be used to ensure that the adjustable parameters exhibit an

acceptable degree of observability.

The time-horizon blend controller is based on forecasting some of the feedstock qualities into

the future using a prediction filter. The prediction filter only looks at the behaviour of the

feedstock qualities in the past in order to make future predictions. It does not directly include

any information on upstream process operations changes and how these changes affect blender

feedstock qudities. The proposed blend controller can be improved by developing models

between some upstream process variables (measured disturbances) and the feedstock quaiities

of interest. The models can then be used in a feed forward configuration in order to aUow the

contrder to handle feedstock disturbances (arising fkom measured changes in upstream process

operations) more effectively.

Also, the developments of this thesis have only been applied to the design of the RTO layer in

the blend controller. The ideas should be extended to the off-Iine optimization layer as well.

That is, blending models selected in Chapter 2 and the time-horizon optimization approach

developed in Chapter 4 shouid be adopted in the off-Iine scheduler and planner for increased

bIender efficiency.

Nornenclat ure

system matrix in Problem (3.1)

aromat ics content ( % volume)

correlation coeEcients in Ethyl RT-70 model, i=1,..,6

constant in the Stewart model for octane blending

constant in Soave-Redlich-Kwong equation of state

constant in Soave-Redlich-Kwong equation of state

matrix in Problem (3.1) that relates rate of change of state variables to controlled variables

bias vector

constant in Soave-Rediich-Kwong equation of st ate

matrix in Problem (3.1) that relates observations to states

cost vector

random variable defined in Problem (3.7)

random variable defined in Problem (3.6)

constant in the Stewart method for octane blending

parameters in Transformation method

variable for feedstock i defined in the Stewart method for octane blending

number of gasoline grades

scalar disturbance at time t

mass of feed to RVP test chamber

probability density function of parameter estirnates

vector of blending models

adjustment factor for hydrocarbon type in Transformation method

vector of demand and availability limits

interaction coefficient between feedstocks i and k in octane blending

interaction coefficient between feedstocks i and k in R V P blending

filter matrix

constant in Soave-Redlich-Kwong equation of st ate

number of components over all feedstocks

parameters in Transformation met hod

pararneters in Zahed rnethod

Lagrangian of optimization problem

mass of liquid phase

observability matrix for dynarnic systems

model parameter in Zahed method

model pararneters in Zahed method, i = 1, ..., n

motor octane nurnber

molecular weight of i

union of a finite number of convex sets that defines plant operation region

normal distribution

convex regions that define the plant operating region, Nt c W

number of feedstocks in blend

olefin content (% volume)

paraffin content (% volume)

pressure

number of adjustable pararneters

covariance mat rix of parame ter est imates

covariance matrix of process measurements

model parameter in Zahed method

model parameters in Zahed method, i = 1, ..., n

vector of reformate qudities

number of state variables in dynarnic system

matrix cont aining feedstock qualities

matrix cont aining feedst ock blending indices

measure of shape of joint confidence region

gas constant

research octane number

feasible set of states in steaciy-state system

closure of set S consisting of S and al1 its limit points

vector of quality specifications

octane sensitivity (research octane number - motor octane number)

steps of random height at time t

random impulses at tirne t

ternperat ure

vector of contolled variables

volume fraction of feedstock i in blend

set from which rneasurement noise realizations are obtained

m a s of vapour phase in RVP test chamber

volume of feedstock i in blend

measurement noise vector

gas volume

percent volume evaporated

expanded liquid volume

weighting vect or

decision variables in gasoline blending problerns (feedstock flowrates)

state variables E 3P

rate of change of y

initial states of process

process measurement vector

backward shift operat or

Pb1 end

o2

constants in blending mode1 in Equation (2.14)

constants in blending model in Equation (2.15)

adjustable parameters E !RP

adjustable parameter estimates E 9ip

mean of adjustable parameter estirnates, f i activity coefficient of component i

vector of steady-state equations that reiate measurement to States

ith eigenvalue of Q:' P

condition nurnber of a matrix based on L2 n o m

weighting matriv

ith eigenvalue of Q8

Lagrange mult i pliers

mole fkaction of cornponent i in the liquid phase

vector of residuals in Problem (3.13)

noise vector in Problem (3.1)

initial noise vector in Problem (3.1)

density of i at 100 OF (pound-moles per barrel)

molar density of component i (pound-moles per barrel as liquid at 1 atm and 60 O F )

density of blend as saturated liquid at 60 O F (pound-moles per barrel)

variance of measurement noise in Example 3.1

objective h c t i o n of optimization-based estimation problem in Problem (3.13)

map (possibly nonlinear) representing pararneter estimation probiem

matrix of operating conditions in Example 3.1

vector of steady-state plant model equations

OPERATORS

V gradient

V* Hessian

IAl determinant of matrix A

transpose of matrix A

SUPERSCRIPTS

* optimalvalue

predictedvalue

Glossary of Terms

ASTM

bias updating

BON

L P

LSR

1MIP

MON

MPC

NLP

ON

P D

RON

RTO

RVP

RVPBI

SLP

American Society for Testing and Materials

mode1 updating by applying an error tenn

Blending Octane Number

Linear Programming

Light Straight Run

Mixed Int eger Programming

Mot or Octane Xurnber

Mode1 Predict ive Control

Nonlinear Programming

Octane Number

Proportional-Integral-Derivative

Research Octane Xumber

Real-Time Optimization

Reid Vapour Pressure

Reid Vapour Pressure BIending Index

Sequent ial Linear Programming

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Appendix A: Nominal Blend

Results for the optimization problem for the nominal blend of Section 4.2 are provided in this

ap pendix.

OptimaI BIend Recipe

fcedstocks in bl>l/day

Feedst ock

reformate

LSR naphtha

n-butane

catalytic gas.

alkylat e

-1 RVP (psi) 10.8

Regular

5430

2570

1.99

O

O Blended Qualit ies

- - - -

maximum RVP in regular gasoline

maximum RVP in premium gasoline

minimum RON in regular gasoline

minimum MOX in premium gasoline

maximum demand of regular gasoline

maximum demand of premiurn gasline

maximum availability of catalytic gas.

al1 available cataiytic gas. to premium gas.

no alkylate to regular gasoline

no alkylate to premium gasoline

Lagrange multipiier

$1737 9 $2023 $ $3110.5 @ $5679.3 3 $1.57 " $4.58

$2.38

$0.15 bbl $4.36

$0.43

Active Const raints

101

Appendix B: Ideal controller on

benchmark problem

The optimization results for the ided controuer on the blenchmark problem of Section 4.2 are

provideci in this appendk.

LSR Naphtha Cataiytic Gas.

Ided Blend Recipe for Regular Gasoiine

feedstock flows in 10'

Reformate LSR Naphtha n-But ane Catalytic Gas.

Ideal Blend Recipe for Prernium Gasoline

feedstock flows in 10'

RON

MON

Regular

maximum demand of regdar gasoline 1

Premium

1

l

Constraint

maximum availability of catalytic gas.

RV P (psi) Fi i id Blendcd Qunlitics Activc coristraints ovcr wholc blciid

10.8 10.8 maximum demand of premium gasoline

C0NSTRA.INT 1 I N T E R V A L 1

Active constraints during each RTO interval

x indicates the constraint is active

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

maximum RV P in regdar g asoline ~ X X X X ~ X X ~ X ,Y x

maximum RVP in premium gasoline x x x x x x x x x x x x

minimum RON in regular gasohe X ~ X X X X X X ~ X x

minimum RON in premium gasoline x x x x x x x x x x x x

minimum M O N in premium gasoline x x x x x x x x x x x x

maximum cataiytic gas. to premium x x x x x x x x x x x x

minimum catalytic gas. to reodar x x x x x ~ x x x x x .Y

minimum alkylate to regular gasoline

maximum alkylate to premium gasoline

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

Appendix C: LP plus bias controller

on benchmark problem

This appendiv sumarizes results for the opt imization problem for the conventional LP + bias

controller on the benchmark problem in Section 4.2.

The blend recipe implemented by the controller at each RTO intenai is given below:

Reformate LSR Naphtha

O. 157

O. 175

0.204

0.306

0.191

0.186

O. 186

0.186

0.186

0.186

0.186

0.186

Cataiytic Gas. Alkylate

Blend Recipe for Regular Gasoline

feedstock flows in 104

1 RTO Interval 1

Reformate LSR Naphtha - - -

Catalytic Gas.

Blend Recipe for f remium Gasoline

feedstock flows in 104

Alkylate

O

O

O

ROrV

Regular

89.6

Premium

92.8

Active comraints during each RTO intenal

maximum RVP in regular gasoline

maximum RVP in premium gasoline

minimum ROIV* in regdar gasoline

minimum ROiV in premium gasoline

minimum :CIO&- in reguiar gasoline

minimum MON in premium gasoline

maximum demand reguiar gasoline

maximum demand premium gasoline

maximum cataiytic gas. to premium

minimum akylate to regular gasoline

maximum alkylate to premium gasohe

minimum butane to reguiar gasoline

minimum catallic pas. to premium

x indicates the constraint is active

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x x x x x x x x x x

x

x

x l x I

x

x

x

x

x

x

x

x

x

x

x

1 2 3 4 5 7 8 9 1 0 1 1 1 2

x

x

x

x

x

x

x

~ x ~ x x x x x x x

x

x

x

1

x

x

x

x I x

x

x

x

x

x

x

x

1

x ) x x

x

x

x

x

x x

x x

x

x , x

4

?

.Y

x

x

x

x

x

l x

x

x

x

x

x x x x

I

x

x

x

x

x

Appendix D: NLP plus bias

controller on benchmark problem

Results for the optimization problem for the NLPfbias controlier (Section 4.5.1) on the bench-

mark problem of Section 4.2 are provided in this appendix.

The blend recipe implemented by the controller at each RTO intervai is given below:

Reformate LSR Xaphtha n-But ane - - - -

Catalytic Gas.

Blend Recipe for Regular Gasoline

feeàstock flows in 104

1 RTO Interval I

Reformate LSR Saphtha

0.150

o. 220

0.1 15

0.111

0.114

0.113

O. 109

O. 109

0.110

o. 102

O .O90

0.087

n-But ane Catdytic Gas. Alkylat e

Blend Recipe for Premium Gasoline

feedstock floa-s in 104

The folloning constraints were found to be active during al1 RTO in tends oves the blend:

1. maximum RVP in re,oular gasoline

2. maximum RVP in premiurn gasoline

3. minimum RO-V in re*ar gasoline

4. minimum MO:\- in premium gasoline

5. maximum demand of re,dar gasoline

6. maximum demand of premium gasoline

7. maximum supply of cataiytic gas.

8. minimum alkylate to regdar gasoline

9. maximum *late to premium gasoiine.

Appendix E: 1-St ep ahead controller

on benchmark problem

The optimization results for the time-horizon based 1-step ahead prediction controuer on the

blenchmark problem of Section 4 . 2 are summarized in this appendk

Reformate LSR Xaphtha

BIend Recipe for Regular Gasotine

feedstock Boas in 104

Reformat e LSR Naphtha

Blend Recipe for Premium Gasoline

feedstock flows in 104

Regular Premium

77.5

RVP (psi) Final BIcndrvl Qiialitic~

Constraint 1

maximum availability of cata11ic gas.

1 maximum demand of premium gasoline 1 Active constraints ovcr wholcr blerid

maximum demand of regular gasoline

The following constraints were found to be active during each RTO intenal for the "current"

t i meet ep:

1

1. rnauiniurn RVP in regular gasoline

2. maximum R V P in premium gasoline

111

3. minimum RON in regular gasoline

4. minimum RON in premium gasoline

5. minimum MON in premium gasoline

6. minimum cat alyt ic gas. t o regular gasoline.

Appendix F: 2-step ahead cont roller

on benchmark problem

The optimization results for the tirnehorizon based Zstep ahead prediction cont roller on the

blenchmark problem of Section 4.2- are provided in this appendix.

Reformate - - ---

Catalytic Gas. Alkylate

Blend Recipe for Regular Gasoline

feedstock Bows in 10'

Reformate LSR Naphtha a-But ane Catalytic Gas.

Blend Recipe for Premium Gasoline

feedstock flows in 104

The following constraints were found to be active for the whole bknding problem:

1. maximum availability of catalytic gas.,

2. maximum demand of regular gasoline,

3. maximum dernand of prernium gasoline.

The foilowing constraints were found to be active during each RTO interval for the "current"

time-step:

1- maximum RVP in reguiar gasoline,

2. maximum R V P in prernium gasoline,

3. minimum RON in regular gasoline,

4. minimum RON in prsmium gasoline,

114

5. minimum MON in premium gasoline,

6 . minimum catalytic gas. to regular gasoline.

Appendix G: bstep ahead controller

on benchmark problem

The optimization results for the time-horizon based 3step ahead prediction controller on the

blenchmark problem of Section 4.2- are provided in this appendix.

RTO Interval -

Reformate LSR Napht ha

0.072 0.030

Blend Recipe for Regular Gasoline

feedçtock flows in 10' "

-

n-Butane

3.1*10-~

0.006

0.006

0.006

5.2*10-~

O

0.007

2.4*10-~

2.9*10-~

0.009

0.009

O

-- .-

Catalyt ic Gas.

O

O

O

O

O

O

O

O

O

O

O

O

Reformate LSR Naphtha

Blend Recipe for Premiurn Gasoline

feedstock flows in 104

The foilowing constraints were found to be active over the blending problem:

1. maximum availabili ty of catalyt ic gas..

2. maximum demand of regular gasoline,

3. maximum demand of premium gasoline.

The following constraints were found to be active during each RTO interval for the "current"

time-step:

1. maximum R V P in regular gasoline,

2. maximum RVP in premium gasoline,

3. minimum RON in regular gasoline,

4. minimum RON in premium gasoline,

5. minimum hf O N in premium gasoline,

6 . minimum catalytic gas. t o regular gasoiine.

Appendix H: Predict ion Filter

This appendix descnbes the prediction filter used in forecasting the feedstock qualities in Section

4.6.1.

The reformate qualities are given in Equation (4.1) as:

In order to predict the reformate qualities? q,.f,, , the scalar disturbances are first calculated

using the above equation (with known Be,) The scalar disturbances are then forecasted and

used to calculate predictions of q,,! using Equation (4.1).

The scalar disturbance given in Equation (4.2) can be wit ten as:

where: si are random impulses that are integrated to give steps of random height, st.

One, two. and three step ahead forecasts of dt are made using minimum mean squared error

forecasts [Box and Jenkins? 19761. The filter equations are given below:

where: 4 (l), &(2),and d t (3 ) are 1, 2, and 3 step ahead forecasts of dt , respectively

Forecasting was started using two samples of historicai data and by ini t idy setting si-, and

si to zero. The impulses were then calculateci using:

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