Dynamic modeling and control of coal fired fluidized bed boilers

DYNAMIC MODELING AND CONTROL

OF COAL FIRED

FLUIDIZED BED BOILERS

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DYNAMIC MODELING AND CONTROL

OF COAL FIRED

DYNAMIC MODELING AND CONTROL

OF COAL FIRED

FLUIDIZED BED BOILERS

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft, op gezag

varfcj ë~RëctoTM agrri f i cü svp r o f :d r s r PrAr S c h e nek-—

in het openbaar te verdedigen ten overstaan van een

commissie door het College van Dekanen

daartoe aangewezen,

op dinsdag 1 november 1988 te 14.00 uur

door

JOHANNES MICHAEL PETER VAN DER LOOIJ

geboren te Bergeyk

werktuigkundig

<&

Dit proefschrift is goedgekeurd door de promotoren

PROF.IR. D.G.H. LATZKO en PROF.IR. O.H. BOSGRA

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Looij, Johannes Michael Peter van der

Dynamic modeling and control of coal fired fluidized bed boilers/Johannes Michael Peter van der Looij. -Delft :

Faculty of Mechanical Engineering and Marine Engineering, Delft University of Technology Proefschrift Delft. - Met lit. opg. - Met samenvatting in het Nederlands.

ISBN 90-370-0017-7

SISO 652 UDC 621.3.036.5(043.3) Trefw.: wervelbedketels.

Copyright © 1988, Faculteit der Werktuigbouwkunde en Maritieme Techniek Technische Universiteit Delft

Alle rechten voorbehouden.

Niets uit dit rapport mag op enigerlei wijze worden verveelvoudigd of openbaar gemaakt zonder schriftelijke toestemming van de auteur.

Gebruik of toepassing van de gegevens, methoden en/of resultaten enz., die in het rapport voorkomen, geschiedt geheel op eigen risico. De Technische Universiteit Delft, Faculteit der Werktuigbouwkunde en Maritieme Techniek, aanvaardt geen enkele aansprakelijkheid voor schade, welke uit gebruik of toepassing mocht voortvloeien.

Any use or application of data, methods and/or results etc., occuring in this report will be at user's own risk. The Delft University of Technology, Faculty of Mechanical Engineering and Marine Engineering, accepts no liability for damages suffered from the use or application.

ACKNOWLEDGEMENTS

The work reported in this thesis was performed at the Laboratory for Thermal Power Engineering of Delft University of Technology under the scientific responsibility of prof.ir. D.G.H. Latzko and prof.ir. O.H. Bosgra. The author wishes to express his sincere appreciation for their constant encouragement and for many valuable discussions throughout this study.

The author gratefully acknowledges the various contributions made to the project.

Special thanks are directed to:

- Mr C.M. Verloop for his constant support regarding software development and maintenance,

- Mr D. Steen for his support regarding the development and adaption of freeboard and flue gas duct models, even after his graduation,

- Dr. B. van Leer for his enthusiastic support regarding the use and development of discretization schemes,

- Prof.ir. C.J. Hoogendoorn for his advice on the modeling of radiative heat transfer in the flue gas duct,

- Stork Boilers (ir. H. Anneveld and ir. F. Verhoeff) for their support and supply of information,

- the students Messrs H. Leentjes, J.A. Klelbergen, J.H. Hilke, P. Schoen, E.W.P den Bakker, M.J.E. Verschoor and J.C. Teeuwen for contributing to this research in the course of their graduate studies,

- Messrs J. Kerkhof, A. Korving, F.C.J. Krooss, W. Landlust, W.A.J. Middelkoop, C. Snel, A.J.J. Vincenten, C.J. Vis, from TUD, A. van

Haasteren and H. Temmink from TNO, B. Knops from Stork Boilers for their help in preparing and performing the experiments.

This project is partly supported by the Management Office for Energy Research PEO (research contract no. 4.351-3.10). Their support is gratefully acknowledged.

SUMMARY

This thesis deals with the dynamics and the associated control constraints of coal fired fluidized bed boilers. The treatment is restricted to

operating conditions,, i.e. startup and shutdown behaviour are not considered, nor are accidents such as e.g. pipe rupture.

Emphasis is laid on the modeling of fluidized bed combustor dynamics. The processes involved being too complex to rely on a purely theoretical treatment, an integral approach of theoretical modeling based on physical laws and experimental modeling based on system identification is

attempted. Experiments are performed under atmospheric and pressurized conditions with bituminous coals and anthracite on two different fluidized bed combustors ranging in thermal output from 1 to 4 MWth. Using a

prediction error method the order and the parameters of a state-space model are estimated. The resulting second order model is compared with a theoretical model where physical uncertainties are represented by

parameters. A sensitivity analysis quantifies these parameters in the form of the statistic distribution functions. The model is then extended to describe the bed dynamics of an industrial fluidized bed boiler featuring fly-ash recirculation and a multi-cell bed geometry.

Modules modeling the freeboard, flue gas duct, drum, evaporator,

economiser, air heater, superheater and desuperheater are then added. The former two are quasi-steady state, i.e. heat and mass storage is

neglected. In the drum model, which is a lumped model, special attention was paid to internal evaporation and condensation processes. In the evaporator model the influence of slip between the steam and water phases was emphasized. This spatially distributed model served as a basis for the derivation of the economiser, air heater, superheater and desuperheater modules. For the non-linear partial differential equation describing energy transport in the evaporator a difference scheme was derived which emphasizes the monotonocity of the solution. It is shown that a careful selection of explicit and implicit difference schemes leads to an efficient solution of the set of equations describing the fluidized bed boiler dynamics.

A sensitivity analysis showed that fluidized bed boiler dynamics are primarily gouverned by the heat capacity of water and steam next to char and energy storage in the fluidized bed. This which offers the possibility of reduced order simplified modeling of fluidized bed boilers without sacrificing accuracy. Furthermore it is shown that the uncertainty in char combustion modeling has a large impact on model predictions.

Replacement of the model's bed and freeboard modules by a radiative furnace module enabled its use for the dynamic simulation of pulverized coal fired industrial boilers. Comparison of typical transient responses of two industrial boilers of the same rating and steam conditions and identical geometry, except that one was equipped with an atmospheric fluidized bed and the other with a pulverized coal fired furnace, offered an insight into the differences in load following behaviour between these two types of coal fired industrial boilers.

Steps are taken towards the application of dynamic modeling capability of fluidized bed boilers to the control design, with emphasis on load

control. The control constraints are indicated and~ftf_iT~füfthërmo"rë_shown" that control strategy should be based on multivariable design methods, as interactions between the inputs and outputs of the system considered cannot be neglected.

SAMENVATTING

Dit proefschrift behandelt ,het dynamisch gedrag en de daaruit voortvloeiende randvoorwaarden voor het regelaarontwerp van kolen gestookte wervelbedketels-. Start- en stop-gedrag, alsmede

ongevalssituaties zoals b.v. pijp-breuk worden niet in beschouwing genomen.

Een belangrijk deel van de aandacht gaat uit naar de modelvorming van kolen gestookte wervelbedvuurhaarden (Engels: FBC). Aangezien dit systeem te complex is om te vertrouwen op zuiver theoretische modelvorming, wordt een geïntegreerde weg van theoretische (dwz. op fysische wetten

gebaseerde) en experimentele (dwz. op systeem identificatie berustende) modelvorming gevolgd. Hiertoe zijn experimenten uitgevoerd onder

atmosferische en boven-atmosferische omstandigheden met bitumineuze kolen en anthraciet in twee verschillende wervelbed vuurhaarden, variërend in vermogen van 1 to 4 MWth. Gebruik makend van een "prediction error" methode worden zowel de orde van als de parameters in een toestands model geschat. Het hieruit volgende tweede orde model is vergeleken met een tweede orde model waarbij onzekerheden in de fysische modelvorming gerepresenteerd zijn door parameters. Een volledig doorgevoerde

gevoeligheidsanalyse leidt tot kwantificering van deze parameters in de vorm van statistische verdelingsfuncties. Het op deze manier gevalideerde model is aangepast voor de beschrijving van de bed-dynamica van een

industriële wervelbedketel voorzien van vliegas-recirculatie en een uit meerdere secties bestaand bed.

Modellen voor vrijboord, rookgaskanaal en drum, verdamper, voedingwater-voorwarmer, luchtverhitter, oververhitter en inspuitkoeler zijn

gepresenteerd. De eerste twee beperken zich tot de quasi-statische

toestand: opslag van massa en energie is verwaarloosd. Bij de modelvorming van de drum, waarin de grootheden onafhankelijk van de ruimtelijke

coördinaat zijn verondersteld, is extra aandacht geschonken aan interne verdamping en condensatie. De invloed van slip tussen de water en stoom faze in de verdamper is nader onderzocht. Het ruimtelijk verdeelde

verdamper model dient als basis voor de modellen van economiser, luchtverhitter, oververhitter en inspuitkoeler.

Bij de afleiding van het differentie-schema voor de niet-lineaire partiele differentiaal-vergelijking die het transport van energie in het

verdampermodel beschrijft, is de nadruk gelegd op de monotoniciteit van de oplossing. Aangetoond wordt dat een selectieve combinatie van expliciete en impliciete differentie schema's tot een efficiënte oplossing leidt van het stelsel van vergelijkingen dat het dynamisch gedrag van wervelbed ketels beschrijft.

Vervanging van de bed en vrijboord modules door een stralings-warmteoverdracht vuurhaard model maakte het mogelijk het model te gebruiken voor het voorspellen van het dynamisch gedrag van poederkoolgestookte industrie ketels. Vergelijking van een aantal responsies van twee industrie ketels met hetzelfde vermogen en

stoomcondities, de ene voorzien van een atmosferisch wervelbedvuurhaard, de ander van een poederkoolvuurhaard, verdiepte het inzicht in het dynamisch gedrag van deze twee typen kolen gestookte industrie ketels.

Een gevoeligheidsanalyse laat zien dat de het dynamisch gedrag van de wervelbedketel in belangrijke mate bepaald wordt door de warmtecapaciteit van het water en de stoom, en de char en energie opslag in het bed

materiaal. Dit biedt de mogelijkheid om met behoud van nauwkeurigheid lagere orde-modellen af te leiden. Daarnaast wordt aangetoond dat de nog aanwezige onzekerheden in de modelvorming van de verbranding van

ontvluchtigde kolen grote invloed hebben op de voorspellingen van het ketelmodel.

Een eerste stap in de richting van regelaar ontwerp voor een wervelbed ketel is gezet, waarbij de nadruk ligt op de belastingregeling. De randvoorwaarden waaraan een dergelijke regeling zou moeten voldoen zijn aangegeven. Daarnaast is aangetoond dat een dergelijk ontwerp gebaseerd dient te zijn op multivariabele ontwerp methodieken, daar de interacties tussen de diverse in- en uitgangen niet verwaarloosbaar zijn.

CONTENTS page ACKOWLEDGEMENTS 0.3 SUMMARY 0.4 SAMENVATTING 0.6 CONTENTS 0.8 LIST OF SYMBOLS 0.13 CHAPTER 1 INTRODUCTION 1.1 1.1. Fluidized bed coal combustion. 1.1

1.2. Djrnamics and control of fluidized bed boilers. 1.4

1.3. Aims and scope of this study. 1.5 1.4. Organisation of the text. 1.6

CHAPTER 2 FLUIDIZED BED MODELING. 2.1

2.1. Introduction. 2.1 2.2. Outline of the reference model. 2.2

2.2.1. Char mass balance. 2.3 2.2.2. Energy balance. 2.6 2.2.3. Fluidization/gas mixing. 2.6

2.3. Principal uncertainties. 2.9 2.4. Experimental modeling: the black box approach. 2.11

2.4.1. Overview of experiments. 2.11

2.4.2. Model set. 2.14 2.4.3. Uniqueness of the parametrization. 2.16

2.4.4. Identification method. 2.16 2.4.5. Model validation. 2.18 2.4.6. Model order. 2.19 2.4.7. Accuracy of the estimated models. 2.25

2.5.1. Parametric description of uncertainties in the theoretical

model. 2.27 2.5.2. Estimation procedure. 2.30

2.5.3. Results. 2.34 2.6. White box modeling. 2.41

2.6.1. Combustion rate coefficient. 2.41 2.6.2. Heat transfer coefficient. 2.47 2.7. Conclusions and recommendations. 2.48 2.8. Modeling bed hydrodynamics. 2.50 2.9. Modeling of a multi-cell atmospheric fluidized bed. 2.51

2.9.1. Lateral mixing. 2.52 2.9.1.1. Introduction. 2.52 2.9.1.2. Dispersion model. 2.53 2.9.1.3. Dispersion coefficient. 2.55 2.972. Recirculation. ~' 2756 2.9.3. Combustion rate coefficient 2.59

CHAPTER 3 FREEBOARD MODEL 3.1

3.1. Introduction. 3.1 3.2. Model description and selection. 3.2

3.3. Characteristics of reference model. 3.3

3.3.1. Particle velocity. 3.4 3.3.2. Entrainment. 3.4 3.3.3. Combustion. 3.5 3.3.4. Energy balance of a section. 3.6

3.3.4.1. Radiative heat exchange. 3.7 3.3.4.2. Convective heat exchange. 3.8

CHAPTER 4 FLUE GAS DUCT MODEL. 4.1

4.1. Introduction. 4.1 4.2. Assumptions. 4.2 4.3. Balance equations. 4.4 4.4. Radiative heat transfer. 4.6

4.4.1. Absorption coefficients. 4.7 4.4.2. Direct exchange areas. 4.8 4.4.3. Total exchange areas. 4.10 4.4.4. Directed flux areas. 4.13 4.5. Convective heat transfer. 4.13

4.5.1. Heat transfer to smooth tube bundles. 4.13 4.5.2. Heat transfer to finned tube bundles. 4.14

CHAPTER 5 BALANCE OF BOILER MODEL. 5.1

5.1. Introduction. 5.1 5.2. Drum model. 5.2 5.2.1. General. 5.2 5.2.2. Model description. 5.4 5.2.2.1. Assumptions. 5.4 5.2.2.2. Balance equations. 5.5 5.2.2.3. Internal flows. 5.6

5.2.2.3.1. Surface condensation on and evaporation from the

water volume (processes 6 and 7). 5.6 5.2.2.3.2. Condensation on metal surfaces (processes 8, 9 and

10). 5.7 5.2.2.3.3. Condensation within the steam volume (process 11). 5.8

5.2.2.4. Final set of equations. 5.8

5.3. Steam generator model. 5.10

5.3.1. General. 5.10 5.3.2. Model description. 5.11

5.3.2.1. Assumptions. 5.11 5.3.2.2. Balance equations. 5.12

5.3.2.3. Boundary conditions. 5.16 5.3.2.4. Heat transfer correlations. 5.17

5.3.2.5. Slip correlation. 5-1 8

5.3.2.6. Friction correlations. 5.18 5.4. Single phase heat exchanger model. 5.20

5.4.1. General. 5■2 0

5.4.2. Model description. 5-2 0

5.4.2.1. Assumptions. 5-2 0

5.4.2.2. Balance equations. 5.21

5.4.2.3. Correlations. 5.21

5.5. Tube wall model. 5-2 2

5.6. Miscellaneous. 5.23 5.6.1. Turbine valve and turbine model. 5.23

5.6.1.1. Turbine valve. 5.23

5.6.1.2. Turbine. 5-2 4

5.6.2. Pump model. 5-2 4

5.6.3. Water-steam properties. 5.25

CHAPTER 6 DISCRETIZATION AND SOLUTION METHOD. 6-1

6.1 Introduction. 6-l

6.2. Discretization of the advection equations (energy balances). 6.2

6.2.1. Explicit versus implicit discretization. 6.3

6.2.2. Explicit discretization scheme. 6.5 6.3. Discretization and solution of the mass and momentum balances 6.9

6.3.1. Discretization. 6-1 0

6.3.2. Solution. 6•U

6.4. Ordinary differential equations in time. t>.11

6.5. Choice of integration step. 6.12 6.5.1. Choice of space integration step. 6-1 3

6.5.2, Choice of time integration step. 6.1k

6 . 7 . F i n a l r e m a r k s . 6 . 2 1

CHAPTER 7 SIMULATION RESULTS. 7 . 1

7.1. Introduction. 7.1 7.2. Pulverized coal fired furnace model. 7.2

7.3. Comparison of transients of the DYFBEB and the DYPFIB model. 7.6

7.4. Sensitivity analysis. 7.12 7.5. Model order reduction. 7.20 7.6. Concluding remark. 7.22

CHAPTER 8 FIRST STEPS TOWARDS ATMOSPHERIC FLUIDIZED BED BOILER

CONTROL 8.1 8.1. Introduction. 8.1 8.2. Part load operation: a steady state approach. 8.1

8.3. Control scheme setup. 8.4 8.3.1. SISO versus MIMO control design. 8.4

8.3.2. Selection of inputs and outputs for MIMO control design. 8.8

8.3.3. Constraints on the control scheme. 8.10

CHAPTER 9 REVIEW AND PROSPECT. 9.1 9.1. On modeling approaches. 9.1 9.2. On dynamic modeling and control of coal fired fluidized bed

boilers. 9.3

REFERENCES 10.1

APPENDIX A Experimental facilities. A.l APPENDIX B Time series of identification experiments. B.l

APPENDIX C Step responses of the experimental models. C.l APPENDIX D Results of the estimation of the parameters 8. D.l

LIST OF SYMBOLS A A A Ar a a a B b C C C D c c D D D' d d E E e F Fr f

i

J

system matrix, state space model

pre-exponential factor in the reaction rate constant Archimedes number

amplitude of PRBS signal coefficient in system matrix thermal diffusivity

distribution matrix, state space model coefficient in distribution matrix output matrix, state space model molar concentration

concentration concentration

flux correction factor, defined by (6-14)

-drag-coefficient . coefficient in output matrix

specific heat

throughput matrix state space model dispersion coefficient

reduced firing density defined by (7-5) coefficient in throughput matrix

diameter

activition energy of the reaction emission of a black radiator error vector, defined by equation system matric, state space model Froude number

valve opening characteristic defined by (5-67) friction coefficient

direct area for radiative exchange between area i and area j

total area for radiative exchange between area i and area j m s K 2 _i m s kmol m % _3 kg m kJ kg" K" 2 .1 m s kJ kg" 2 kW m

FiFj

V j

V j

V j

^ J

directed flux area for radiative exchange from area i to area j

directed flux area for radiative exchange from area j to area i

Kalman gain matrix gravitation constant

direct area for radiative exchange between volume i and area j

total area for radiative exchange between volume i and area j

directed flux area for radiative exchange from volume i to area j

directed flux area for radiative exchange from area j to volume i

output matrix, state space model heat of reaction

height

specific enthalpy cost function

mass interchange rate absorption coefficient coefficient

specific absorption coefficient specific absorption coefficient overall heat transfer coefficient combustion rate constant per particle combustion rate constant for all particles rate constant

cycle length of PRBS signal length

mass

monotonicity factor mass flow

two-phase mass flow

fin-number defined by (4-48) kJ kg m kJ kg" 2 . 1 m kg l . 1 m Pa 2 kJ m K . l m s 3 . 1 . 1 m kg s . l m s m k g kg s kg s . l m

N number of samples Nu Nusselt number n number of sections n order of the reaction P power

P covariance of parameter estimates Pr Prandtl number

p pressure

t pitch of the tubes perpendicular on the flow direction kW

kPa

p partial pressure k P a

Q heat flow kW Q reduced furnace efficiency defined by (7-4)

Q weight matrix q heat flux k w m .2 .1 .1 1 . 1 entropy kJ kg K R gas constant kJ k"101 K R thermal resistance m s kJ . l R amount of component released kmol s

Re Reynolds number ~ Sh Sherwood number

S perimeter m

S slip s

s entropy of two-phase mixture kJ kg K

s- fin thickness m T mean temperature K T transformation matrix -T temperature K t time s m At sample time s At time step l

At clock frequency of PRBS signal s U superficial velocity m s s . l u, input vector volume flow volume m 3 .1 V volume flow m s 3

v specific volume

W normalized particle distribution W total radiated energy flux We Weber number

w mass fraction x steam quality x, state vector

x normalized valve lift v

X cross flow factor, defined by eq. (2-15)

Y amount of oxygen required for combustion of the volatiles °2 y, output vector z space z, state vector Az space step Greek symbols

a heat transfer coefficient

a absorption coefficient

a void fraction

P factor defined by (2-1) P factor defined by (5-33) P factor defined by (5-34)

A diagonal system matrix

A differences, defined by (6-4) and (6-5) A gas temperature drop over the furnace

A ratio of gas temperature drop A to the adiabatic flame temperature defined by (7-6)

5 bubble fraction

5 relative tube wall roughness £ vector of innovations e emissivity

£3 third exponential integral, defined by (4-10)

T coefficient defined by (2-21) 7 coefficient defined by (2-20)

7

<z

n

z

angle with vertical polytropic exponent thermal conductivity factor defined by (2-64) coefficient defined by (5-57)

hollow space fraction defined by (4-12) dynamic viscosity

coefficient defined by (5-29) reflection coefficient

density

density of two-phase mixture covariance of the innovations

inaccuracy of the operating conditions surface tension standard deviation Stephan-Boltzmann constant -Courant-number,_defined-by_(6^2) row dominance column dominance residence time time constant .1 -i kW m K kg m s 3 .1 m kg kg m" kg m" N m 2 .4 kW m K

ratio of mean wall temperature to adiabatic flame temperature

parameter vector two-phase multiplier

amount of component consumed due to volatile combustion kmol s mechanism factor

coefficient

Subscripts af as aw B BJ B b C02 c eg ch CO conv crit d d d des dr e e el ex exch est

FR

f

f

gw

h

H

adiabatic flue gas wall Bankoff

Bankoff-Jones bed

bubble

carbon dioxide component combustion flue gas char coal convection critical mass transfer particle downcomer desuperheater drum equivalent emulsion elutriated on exit

exchange due to lateral mix estimated freeboard surface fin fast bubble fixed carbon gas

from gas to wall water

H20 water component i inner in inert in on entrance inc incidented j j-th space step lam laminair m mean mf minimal fluidization mod modified n n-th gray gas 02 oxygen component o outer p pressure r riser rad radiative ree r e_c_i rculation s solid s reaction kinetic

s saturated or superheated steam sas saturated steam

saw saturated water slow slow bubble t tube t terminal tu turbine turb turbulent tv turbine valve tw cooled-wall v volatile matter w wall

Superscripts

k k-th time step T transpose -1 inverse

Abbreviations

AFBC Atmospheric Fluidized Bed Combustion AFB Atmospheric Fluidized Bed

AIC Average Information Criterion

ARX Auto Regressive with eXogeneous input FBC Fluidized Bed Combustion

LHV Lower Heating Value MIMO Multi Input Multi Output

PFBC Pressurized Fluidized Bed Combustion PRBS Pseudo Random Binary Sequence

SISO Single Input Single Output SVD Singular Value Decomposition

CHAPTER 1 INTRODUCTION

This thesis is concerned with the dynamic behaviour of coal-fired

fluidized bed boilers. Accordingly the studies reported herein bear upon two engineering areas: fluidized bed combustion and boiler dynamics and control. The treatment is restricted to operating conditions, i.e. startup and shutdown behaviour are not considered, nor are accidents such as e.g. pipe rupture.

1.1. Fluidized bed coal combustion.

The combined effets of renewed interest in coal as a primary energy

carrier, spawned by the oil price hikes in the seventies, and of increased awareness of the environmental damage associated with some of its

combustion products has led to increased emphasis on the development of coal conversion techniques for energy production, such as fluidized bed combustion (FBC).

Coal fired fluidized bed combustors possess the following advantages over pulverized coal fired furnaces:

- effective reduction of SO emissions by addition of limestone or

x J

dolomite to the bed material

- lower NO emissions due to the relatively low bed temperature (1050 - 1200 K)

- high volumetric energy production due to increased heat transfer rates from furnace (i.e. the bed) to working fluid (i.e.

water/steam flowing through tubes immersed in the bed)

This advantage is particularly pronounced for pressurized fluidized bed combustors, which also generally yield higher combustion efficiencies than the atmosspheric alternative.

The technique of fluidization of solid particles by an upward flow of gas (cf. figure 1-1) has long been applied in chemical industry, where the intensive contact between particles and fluid is used to enhance chemical reactions. The superficial air velocity at which the particles start to float is generally referred to as minimum fluidization velocity. Gas flow in excesss of that required for minimum fluidization percolates the bed in nearly particle-free cavities, like bubbles in a boiling liquid, and contributes to an enhanced mixing of the solids: the bubbling bed regime. Further increase of gas flow causes the fluidization velocity to exceed the free fall velocity and the particles to be ejected from the

bed: the circulating bed regime.

The present study is limited to bubbling fluidized beds.

a A n

t t t

t t !

!

I

_| ,

. |_

_| ,

FIGURE 1-1. Illustration of the bubbling fluidized bed concept.

Coal fired fluidized beds essentially consist of inert particles,

typically of the order of 1 mm equivalent diameter, carrying approximately 0.1 to IX (by weight) of coal particles of up to 10 mm equivalent

diameter.

By contrast, particle sizes in other chemical reactors are generally well below 0.1 mm. This difference in the solid component of the bed,

illustrated in the diagram of figure 1-2 proposed by Geldart, is mainly responsible for the limited applicability of the vast amount of chemical reactor data of FBC and for the resulting need for research specific to

the latter application. Overviews of such research are e.g. presented by OLOFSSON (1980) and SMOOT (1984).

sooo

SO 100 SCO IOO0

FIGURE 1-2. Classification of particles. Taken from GELDART (1986).

Bubbles bursting at the bed level cause the ejection of particles into the zone above the bed, the so-called freeboard. The freeboard is usually operated at temperatures approximating that of the bed in order to enhance chemical reactions to achieve higher combustion efficiencies and lower pollutant emissions.

The height of the freeboard, determined by the pursuit of high once-through combustion efficiencies 1) is usually in the order of 3 m. This compares with bed heights in the order of 1 and 5 m for commercial FBC's of the atmospheric and pressurized type, respectively. Carbon containing fly-ash, caught in cyclones or bag filter, is recirculated to the fluidized bed in order to increase overall combustion efficiency.

1.2. Dynamics and control of fluidized bed boilers.

The advent of the high speed digital computer as a simulation tool has spawned a vast amount of literature on dynamic behaviour and control of industrial and utility boilers (DOLEZAHL and BERNDT (1986), DIBELIUS and GEBHARDT (1983), SCRUTON et al. (1987), GORNER (1988), FRANKE (1974)). The objects of these studies, including those of pulverized coal fired design, differ essentially from coal fired fluidized bed boilers by the relative magnitudes of the furnace and water/steam side time constants. Contrary to pulverized coal fired boilers the dynamic behaviour of coal fired fluidized bed boilers is governed to a large extend by the dynamics of the combustor itself: e.g. the time constant of the bed is typically in

2 1

the order of 0(10 ) , as compared to 0(10 ) for the time constants of the water/steam side. This is not only due to the heat and char stored in the bed but also to the large size (up to 10 mm) of the coal particles and the

1 2

resulting large combustion time constants (0(10 to 10 ) ) .

By contrast, time constants on the furnace side of convential boilers are o

of the order 0(10 ) , causing their dynamics to be largely governed by water/steam side behaviour and permitting quasi steady state modeling of

the furnace part.

A further characteristic of fluidized bed boilers are the severe

limitations imposed on the load following capability by the fairly narrow temperature range of the fluidized bed, 1050 to 1200 K, maintained in order to minimize SO and NO emissions and to prevent ashes from fusing. The following measures serve to improve load following capability:

* division of the bed into sections which may be slumped for load decrease, thereby decreasing heat transfer almost instantaneously to roughly one third of the value before slumping, NOWOTNY (1986). Modularization, where a plant consists of several separate small boilers can be considered an extreme of this approach

* mounting part of the heat exchanger tubes in the lower part of the freeboard just above the bed. An increase in bed height due to e.g. an increase in fluidization velocity or bed mass will then result in an increase in heat transfer rate, as the heat transfer coefficient

in the bed is approximately 5 times higher than that in the freeboard.

1.3. Aims and scope of this study.

In the recent past KOOL (1985) explored the dynamic behaviour of a

pressurized coal fired fluidized bed combustor. The present study extends this work to the freeboard and water/steam side of industrial coal-fired fluidized bed boilers, with the aim to contribute to their improved (i.e. smoother and faster) load following capability.

Attention is paid to:

- experimental modeling (i.e. system identification based on

experimental data) and theoretical modeling (i.e based on a priori knowledge in terms of physical laws) of fluidized bed combustors of both atmospheric and pressurized design

- theoretical modeling of the freeboard, the flue gas duct and the water/steam side~orf~FBCrdFiven boilers

In addition, exploratory studies were undertaken on:

- the design of control strategies for a fluidized bed boiler with special reference to the 90 MWth atmospheric fluidized bed boiler of AKZO Salt and Basic Chemicals in Hengelo, the Netherlands will be given described in appendix E.

It should be emphasized that the present thesis is confined to phenomena which are directly related to the load following characteristics of the

fluidized bed boiler, to the complete exclusion of all aspects concerning pollutant emissions. This latter topic, while of paramount importance to the viability of coal fired fluidized bed boilers, hardly affects their load following characteristics.

1.4. Organisation of the text.

Chapter 2 covers the modeling of fluidized bed combustors. In search of the optimal model complexity and of quantitative data for several model parameters, an integral treatment of experimental and theoretical modeling is attempted, using the simplified second order model of KOOL (1985) as our point of departure. A prediction error method is used to derive

experimental models from data collected from two fluidized bed combustors. The resulting model is extended to the AKZO boiler geometry and process conditions. The latter design differs from the two test facilities mainly by the recirculation of the fly-ash and by a multi-cell bed geometry. Modeling of the freeboard is covered in chapter 3. Three models of

increasing complexity with respect to the interactions between combustion, particle trajectories and entrainment, and arguments are presented for the selection of one of these models for inclusion in the overall boiler model.

The flue gas duct model, treated in chapter 4, describes radiative and convective heat transfer from the flue gas to the superheaters, economiser and air heater tubes. Modeling of the secondary side of these components and of the evaporators and drum is described in chapter 5.

The discretization of the various mass, momentum and energy balances and the subsequent solution of the resulting set of algebraic equations is discussed in chapter 6 from the viewpoint of a user in search of an approach combining accuracy with computational efficiency.

A comparison of simulation results of the atmospheric fluidized bed boiler model with a pulverized coal fired boiler model, the impact of some of the key hypotheses on model predictions and some comments on low order

modeling may be found in chapter 7.

The interactions between the various inputs and outputs, the principal control design constraints together with an outline of a possible strategy for load control, are presented in chapter 8.

The main conclusions to be drawn from the present study and some recommendations for further work are given in chapter 9.

CHAPTER 2 FLUIDIZED BED MODELING.

2.1. Introduction.

Taking the second order non linear dynamic PFBC model developed by KOOL (1985) as our point of departure, the present chapter has, in order to be able to accurately model the dynamic behaviour of the AKZO fluidized bed, the following aims:

* to use experimental evidence obtained by the present author for - verifying the adequacy of the model order proposed by Kool

to decrease the margins of uncertainty of those parameters found most relevant for predicting the combustor's dynamic behaviour

* to extend the model by adding a description of horizontal mixing of solids in the bed, and recirculation of fly-ash to the bed

The chapter starts with a summary outline of Kool's model (section 2.2), followed by an overview of the principal uncertainties in this reference model (section 2.3). The experiments preformed by the author are

summarized in section 2.A (more detailed information being supplied in Appendix A ) . Here a state space model is introduced and the experimental evidence obtained is used to validate the model and to select a proper model order. This part of the modeling effort may be termed "black box modeling", no attempt being made at this stage to quantify physical model parameters. By contrast, the contents of section 2.5 might be called "grey box modeling" as the uncertainties pointed out in section 2.3 are

introduced in the form of parameters whose values are estimated by a comparison with the experimental model. In section 2.6 correlations from literature predicting char reaction kinetics and bed heat transfer are compared with the experimental evidence obtained in order to facilitate a choice between them. The results of the modeling effort are summarized in section 2.7. Un unsuccesful attempt at experimentally obtaining values of the dispersion coefficient characterizing the bed flow regime is briefly reported in section 2.8. Section 2.9 represents the extensions of Kool's model describing horizontal mixing of solids and recirculation of fly-ash.

2.2. Outline of the reference model.

We use the second order non linear dynamic PFBC model, KOOL (1985) as our reference model. The relevant phenomena are schematically shown in figure 2-1.

AIR MASS FLGH FLUIOIZED BED BOUNDARY

OUTLET VALVE OPENING (PFBC ONLY)

COAL KASS FLOH

GAS FLOW DYNAMICS

ATTRITION / ELUTRIATION

energy f l a w ! ; energy flaw COOLANT DYNAMICS

BED TEMPERATURE

OXYGEN CONCENTRATION

FIGURE 2-1. Phenomena considered in the theoretical model.

Bed coolant dynamics are not described by the model, which is only justified for the test facilities used and described in appendix A. Chapter 5 covers a description of bed coolant dynamics for the AKZO facility.

Considering that the time constants for combustion and heat transfer exceed those for gas dynamics, for fluidization and gas mixing and for devolatilization by an order of magnitude, a steady state description is used for the latter phenomena.

The description of gas dynamics is straightforward and will not be discussed here.

2.2.1. Char mass balance.

The following assumptions underly the reference model: the solids are perfectly mixed

the char particle diameter distribution is not strongly dependent on the operating conditions

the char combustion rate may be evaluated using the mean oxygen concentration in the emulsion phase

the combustion of char particles can be described by a shrinking particle model i.e. combustion is concentrated on the outer surface of the particles and the formed ash layer immediately rubbs off the reaction kinetics are of order one,

and defining: w . + w

fi = -£c as

fc — — - . the char mass b a l a n c e may be formulated a s :

dM . ch

m . (1 - w - w, ) - m , - m , , (2-2)

v v h ' c h , c c h . e l '

d t c o , i n v v h ' c h , c c h . e l

where t h e char combustion r a t e i s given by:

c h , c c ch 02, e

with the overall combustion rate coefficient k for a first order reaction c

in oxygen defined by: 7? a " Vd ) c "ch o d B

with W„(d) the normalized char p a r t i c l e diameter d i s t r i b u t i o n i n the bed. ft

so far unkown and coal type depending extend, the normalized char particle diameter distribution is an input to the model.

The rate coefficient for a fixed particle diameter d is given by:

(2-5)

kc( d ) ■

ks( d ) ks( d )

x T

<t>

V

i.e. assumed to depend upon:

a) the mass transfer coefficient:

D

k

(d) - Sh

-f (2-6)

b) the first order reaction rate constant (written in Arrhenius form):

k (d) - A T exp(- E/R T ) (2-7)

c) the mechanism factor describing the heterogeneous reaction at the particle (internal and external) surface, viz.:

C + 7 02 - (2 - h CO + (7 - 1) C02 (2-8)

The temperature of a burning char particle T is calculated from a quasi steady state energy balance around the char particle and the surrounding gas:

a (T - TD) + o £ (T *- Tn4) - 12 k (d) H C. ^ (2-9)

*) k and k (d) though using the same symbol, differ in fact in dimensions (respectively m3 kg-1 s~' and m s—*) where k refers to the combustion rate

coefficient for all particles and k (d) to the combustion rate coefficient per particle.

where a denotes the heat transfer coefficient for a single particle, e the particle emissivity and a the Stephan Boltzmann constant. The particle

emissivity has been taken as 0.85, ROSS et al. (1981).

The mechanism factor <j>, indicating whether CO or C02 is transported from

the particle surface, is calculated according to WEN and DUTTA (1978) with the ratio of CO to C 02 production at the surface according to ARTHUR

(1951).

Assuming a dominating influence of a t t r i t i o n on elutriation, correlated to the surface area of the particles, the char elutriation rate can be

written as follows, KOOL (1985):

mu - k . (U - U _) M . (2-10)

ch.el el v mf' ch

where the minimum fluidization velocity U f will be calculated according to WEN and YU (1966). Assume that the diameter dependance of Sh and kg may

be represented by one characteristic diameter d, the char elutriation rate constant may be written as:

-^-Sh-D-1

1

where the coal type depending attrition rate constant k is defined by:

_ Igt'a ( 2.1 2 )

a U - U „ mf

and Qx, Q2 are the first and second moment of the char particle diameter

2.2.2. Energy balance.

Assuming:

a uniform bed temperature,

complete oxidation in the bed of CO, formed by char combustion, to C02, and

complete combustion of the volatiles in the bed,

the energy balance may be written as: dc dT <Cs Ms + Ms TB Si*) d T " p Hfc *ch,c + m . [w H + (T. - Tn) c ] + co,in v v x in B' co m , (T. c - TD c ) - k A„ (I. ■ I ) (2-13) g,in v in p B p ' t v B w' 6 g.in *cg 2.2.3. Fluidization/pas mixing,

Fluidization and the related gas mixing phenomena are characterized by a two phase description, i.e. an emulsion phase consisting of both gas and particles is thought to be percolated by a bubble phase. Gas flow through the emulsion phase is equal to or greater than the flow needed for minimum fluidization. Gas may be exchanged between the bubble and emulsion phase.

The distribution of gas flow over the two phases is defined by:

i

-

r

<

U

-

U

mf)

(2

"

U)

where T is a parameter whose value seems to range between 0.5 and 0.75 ( for r-1 gas flow through the emulsion phase would correspond to the minimum flow needed for fluidization). Gas mixing in the bed is affected by the foregoing, by bubble properties and by the rate of gas exchange between the bubble and emulsion phases. In general this gas interchange is

characterized by a dimensionless parameter, the so called cross flow factor: K*. S AR HR „ _ _be B _ g (2-15) Vb where

K, = mass exchange rate coefficient [ s- 1]

H ~

6 = bubble fraction = 1 - - [-] HB

For a discussion on the diffusive and convective contributions to the crucial coefficient K, the reader is referred to KOOL (1985).

The gas mixing characteristics in turn determine oxygen supply and oxygen consumption, i.e. the combustion rate, throughout the bed.

Under the assumption of negligible CO production oxygen consumption is defined by:

* - -1- rii (2-16)

02 120- chTc

as regards char combustion, and by:

*„ - R Y_ m . (2-17) 02 v 02 co,in

as regards volatile combustion, where R denotes the volatiles released [kmol volatiles/kg coal] and Y the oxygen required for combustion of the

U2

volatiles [kmol 02/kmol volatiles]. The composition of the volatiles released is determined using the relations of LOISON and CHAUVIN (1964).

Two alternative flow patterns are considered, each resulting in a

i) a fast bubble flow regime i.e. upward velocity of the bubbles exceeding that of the gas in the emulsion phase, with a perfectly mixed emulsion phase. The mean oxygen concentration may be

approximated by: * + * C02,e,fast " C02,in " tl U ƒ ( 2"1 8 ) z B mod with: - . - . , .„ c „, (2-19) mod x mf '

where 7 accounts for the combined effects of flow division and gas exchange between bubble and emulsion phase.'

7 - r exp(- X) (2-20)

with:

b) a slow bubble flow regime, where the gas flow in plug flow through bubble and emulsion phases. By taking the influence of char and volatile combustion separately the oxygen concentration may be approximated by:

C = - - I ^ + - $ ) (2-22)

02,slow ^ <■ *n 2 02 ; U z z ; In (1

V C02, i n

Compared to a combined treatment, the mathematically easier to handle equation (2-22) will lead to oxygen concentrations up till 10% higher for a high volatile coal. This however is of no interest in what follows.

The oxygen concentration above the fluidized bed is given by: R m . - TTTT m Z . = 100 09 ,ex .28.966 g.in 0-, v co. in 120 ch.c. (2-23) 28.966 mg,in 2.3. Principal uncertainties.

The principal uncertainties, i.e. those that might cause the largest quantitative deviations from the model described above, derive from the assumptions underlying eq. (2-3) for the char combustion rate, viz:

time-invariant char particle diameter distribution. As seen from eq. (2-4), this assumption underlies the use of the time-invariant overall combustion rate coefficient k . In general the char particle diameter distribution is taken as time-dependant, see e.g. LOUIS and TUNG (1982).

a mass transfer coefficient k,(d) obtained from the definition of the

a

Sherwood number eq. (2-6) and the value of the Sherwood number used therein. The compilation of correlations presently in use given by LA NAUZE (1985b) shows values of Sh differing by as much as a factor of 3.

- a first order reaction in oxygen, whereas LAURENDEAU (1978) argues that under typical FBC conditions the reaction order is approximately 0.5. Only at temperatures above 1500 K the order approaches one. It should be noted however that apart from bedtemperature the reaction order depends on the type of coal used and on the oxygen

concentration.

- Extensive review articles i.a. by LAURENDEAU (1978) and SMITH (1982) deal with reaction kinetics, intrinsic as well as extrinsic. Most of them are written in the Arrhenius form. Most researchers when modeling char combustion use the extrinsic kinetic data of FIELD et al. (1967) without questioning although his data shows a

substantially higher activation energy than e.g. SMITH (1970). - Elutriation rate. ARENA et al. (1983) concluded that elutriation

feed coal or formed during primary fragmentation, 2) fines resulting as elutriable residues after combustion and attrition and 3) fines generated by attrition. According to DONSI et al. (1980) the

contribution of attrition dominates over that of elutriable residues. KOOL (1985) found the elutriation rate in his experiments to be proportional to the char feed rate rather than to the accumulated char mass, but attributed this deviation from expression (2-10) to insufficient freeboard height in his experimental combustor. The location where the volatiles burn: uniform in the bed, as implicitly assumed in the foregoing, in the lower or upper part of the bed, or (partly) above it.

The flow regime, which combined with the assumption regarding the volatile combustion determines the oxygen concentration in the emulsion phase.

The heat transfer coefficient, and for a vertical heat exchanger even the heat exchanging area determined by bed expansion is not

accurately known.

Although BELLGARDT (1985) showed that lateral mixing is far from perfect, we assume perfect solids mixing for the test facilities, whose dimensions are described below. For the AKZO fluidized bed however we will take a limited lateral solids mixing into account, see section 2.9.

The primary purpose of the experimental effort described in the following sections is to quantify the above mentioned uncertainties.

We will start the treatment with experimental modeling, discussed in the next section. Linear black box models, whose structures are determined by mathematics instead of physics, will be estimated from dynamic

experiments, in which perturbations of i.a. coal and air mass flow are sufficiently small to permit linear modeling. Our attention is primarily focused on determination of model complexity, i.e. model order. In section 2.5 the experimental models determined at various operating conditions are compared with a linearized version of the non-linear theoretical model described in section 2.2, yielding a quantification of the above mentioned uncertainties.

2.4. Experimental modeling: the black box approach.

2.4.1. Overview of experiments.

Experiments have been performed in two facilities, subsequently referred to as THD-3 and TNO. The main design data are listed in table 2-1. Further details can be found in appendix A.

bed dimension bed height freeboard height operating pressure thermal power

heat transfer surface

bed bulk material

mass mean particle coal feeding system

m m m MPa MW size THD-3 </> 0.485 0.3 - 1.5 2.6 - 3.8 0.1 - 1.0 0.05 - 1.0 vertical U-tubes, water cooled silica sand,

narrow sieve fraction, = 0.78 mm

over bed feeding

TNO 2.25*1. 1.05 4.7 0.1 1.0 - 4.0 horizontal tubes, steam raising

coal ash and limestone, wide size distribution, ~ 1.5 mm

under bed feeding

TABLE 2-1. Experimental facilities: main design data.

Five types of coal differing both in chemical composition and in particle size were used in the experiments:

- anthracite, known i.a. for its tendency to fragment

- a bituminous coal with relatively high ash content, size range 4 to 7 mm, subsequently referred to as bituminous I

- a bituminous coal with relatively low ash content, size range 1 to 6 mm, subsequently referred to as bituminous II

- a bituminous coal with relatively low ash content, size range 0 to 10 mm, subsequently referred to as Polish 5

- a bituminous coal with relatively low ash content, size range 0 to 6 mm, subsequently referred to as Polish 7

The first three types of coal were used in the THD-3 facility, the latter two in the TNO facility.

The composition of the coal types is listed in table 2-2.

Proximate (%) moisture volatiles ash Ultimate (%) carbon hydrogen nitrogen sulphur ash oxygen (remainder) LHV 1) inthracite 1.9 8.5 3.6 88.7 3.2 1.3 0.9 3.6 2.3 34400 bituminous I 1.4 21.9 28.6 59.9 3.6 1.3 1.4 28.6 5.2 24160 bituminous II 1.9 31.0 5.7 79.8 4.8 1.1 0.7 5.7 7.9 33020 Polish 5

Pseudo Random Binary Sequences (PRBS), see figure 2-2, were used as test signals. These signals were simultaneously applied to coal feeder and air inlet valve. Additional test signals were applied to the flue gas outlet valve of the THD-3 facility under pressurized conditions, and to the feed water valve and the steam pressure controller of the TNO facility.

At

The selection of amplitudes a, cycle lengths L and clock frequencies

.1

At was based on preliminary step response experiments and on the results of KOOL (1985) and VERHEY (1985). The amplitudes were designed for a maximum excitation of approximately 20 K in bed temperature. A compilation of the chosen amplitudes, cycle lengths and clock frequencies is given in appendix A.

The operating conditions of the thirteen experiments, selected for further analysis, are compiled in table 2-3

test facility 1 THD-3 2 THD-3 3 THD-3 4 THD-3 5 THD-3 6 THD-3 7 THD-3 8 THD-3 9 THD-3 10 TNO 11 TNO 12 TNO 13 TNO coal type anthracite anthracite bituminous I anthracite anthracite anthracite bituminous II anthracite anthracite polish 5 polish 7 polish 7 polish 7 PB(kPa) 100 100 100 100 400 400 400 400 400 100 100 100 100 TB(K) 1155 1140 1105 1120 1145 1145 1120 1165 1155 1115 1090 1140 1125 C02 )ex<% ) 4.2 4.3 9.7 6.0 5.8 5.8 6.0 5.7 2.3 3.2 3.2 3.2 3.2 Uf(m/s) 1.04 1.04 1.04 1.05 1.07 1.07 1.10 1.27 1.09 1.98 1.88 2.27 2.17 Ms(kg) 96 96 73 94 144 144 152 152 170 2300 2400 2400 2400 m . (kg/s) co, in 6/ 0.00391 0.00405 0.00394 0.00377 0.0155 0.0156 0.0174 0.0193 0.0224 0.107 0.109 0.125 0.125

TABLE 2-3. Operating conditions of the measurements.

We note the following:

- In most experiments the duration of the PRBS excitation was limited by the availability of the facilities, which was restricted to normal working hours.

- When burning the bituminous II coal in the THD-3 facility the bed height exhibited a continuous increase due to ash build up in the bed. With a bed content of approximately 95 kg, the ash build up resulted in strongly non-linear system behaviour essentially

This problem was overcome by operating with a lower bed content of 73 kg.

- Drift phenomena were eliminated using a first order high pass filter. - DEN BAKKER (1987) used spectral analysis to show that heat transfer

and combustion phenomena were not correlated with water/steam

dynamics at the TNO facility, thereby effectively reducing the number of inputs to two.

- The data was recorded with a sampling rate of 1/15 Hz at the THD-3 facility and of 1 Hz at the TNO facility. For the experimental modeling, described in the following of section 2.4, sampling rates of 1/30 and 1/40 Hz were used for the THD-3 and the TNO data, respectively. A digital low pass filter was used before data reduction to limit aliasing effects.

- Samples of the bed material, taken under a protective nitrogen atmosphere, revealed that the bituminous I coal burned according to a shrinking core model: a decreasing core of carbon and ash is

surrounded by a virtually carbon-free product layer largely unaffected by attrition. The combustion of the other coal types closely followed the shrinking particle model.

2.4.2. Model set.

We assume that the system under consideration can be described by the adjectives linear, time invariant, stochastic, discrete time,

multivariable, i.e. that the system can be described by figure 2-3, where u, e R , y e R and £, e R denote the input, output and noise vectors at time instant k, respectively. Here the input vector consists of coal mass flow, air mass flow and, under pressurized conditions, also of the outlet valve opening. Bed temperature and oxygen concentration in the flue gas above the bed are taken as output.

We furthermore assume that (e, ) forms a white sequence with covariance W. From the variety of models proposed in literature for the system of figure 2-3 we selected the state space model, as a linear model can conveniently be written in this form.

HU)

Six)

■ i - i i

FIGURE 2-3. The system under consideration.

This results in a parametrization of the deterministic system G(x) as follows:

*k+l " A Xk + B \

yk - c xk + D "k

(2-24)

(2-25)

We assume that the stochastic system H(z) may be parametrized by:

zk+l " F Zk + G £k

H Zk + £k

(2-26)

(2-27)

The order of the system H(z) is not necessarily equal to that of system G(x).

The total model can now be written as follows:

,Xk+l. .A 0 . xk. ^ ,B. ^ ,0. ( L ) - ( 0 F > (z > + <0> " k+ (G> £k k+1 k yk = (C H) ( ) + D ^ + ,k k (2-28)

which can also be written as:

(2-29) ^ -C # * k+ D u k+ « k

This representation is commonly referred to as the innovations representation form.

3. Uniqueness of the parametrization.

In system identification it is essential to know which parametrizations of a linear dynamical system are appropriate for identification, i.e. are such that the parametrization is uniquely identifiable.

A fundamental property of linear multivariable systems is that no unique model set is able to represent all systems of given order, say n. The set of all systems of order n can only be represented as a finite union of model sets each characterized by its structure indices.

Now there are basically two ways to let each system be represented by an identifiable model. One way is to let all the systems of order n be described by a union of discrete identifiable model sets, which are then called canonical. The alternative is to let the set of all systems be described by a union of identifiable but overlapping model sets, which are often called pseudo-canonical. The pseudo-canonical form possesses more parameters, but is able to represent almost any system for a given set of

structure indices, leaving only the order of the model to be determined. Here a transformation matrix based on a "nice selection" of the rows of the observability matrix is used to achieve a pseudo-canonical form, DENHAM (1974).

k. Identification method.

A single-step prediction error formulation is used to estimate the parameters in the model described by equation (2-29), GOODWIN and PAYNE

Jn

#

The estimation criterion to be minimized is defined as:

N T

J1 - £ £! qf i (2-30)

i=»l

T

where N is the number of samples and Q - Q > 0 is a weighting matrix. If

. l

Q is taken as S , where 2 is the covariance of the innovations e, then the smallest asymptotic parameter covariance is obtained. The results obtained with the J1 criterion are then equivalent to those obtained using the stationairy maximum likelihood criterion defined by, GOODWIN and PAYNE

(1977):

N

J2 = det ( 2 £ e ) (2-31)

i-1

VAN ZEE and BOSGRA (1982) derive easy to use expressions for the gradients of the criterion Jx, which greatly reduces the computational costs of the parameter optimization. We therefore adopt the criterion Jj for the parameter optimization.

Let S0 be the true parameter vector and assume e, to be a white sequence then the distribution of ,/N (S — 60) is asymptotically normal:

/N (f?N - 60) e As N(0,P) (2-32)

The covariance matrix of the parameters P may be estimated by use of the Hessian of Jx.

A first estimate of the deterministic state space model was obtained by using a least squares method with an autoregressive model with exogeneous input (ARX-model) followed by an approximate realization of the impulse response computed with the ARX model. We refer to KOOL (1985) for a detailed treatment of the latter two models.

2.4.5. Model validation.

In the validation phase we have to decide whether or not the model obtained from the identification procedure is acceptable.

In order to obtain the nice asymptotical properties of the model the prediction errors should be white and the specific choice of the model set, determined by the orders of the deterministic G(x) and the stochastic system H ( z ) , should be correct.

With increasing order an estimation criterion will decrease monotoneously. This leads to the risk of overfitting, i.e. of obtaining models which are too complex. There are several ways to avoid overfitting:

- fit models of different complexity and apply hypothesis testing - use criteria which take the model complexity into account, e.g. the

AIC criterion AKAIKE (1974)

- cross validation: the measurement set is divided into two sets: an estimation and a validation dataset. Models of varying complexity are estimated on the estimation dataset. With the model obtained a

criterion say J2 will be determined on the validation dataset. The model with the lowest criterion value on the validation dataset will be the optimal one. For an extensive treatment on cross-validation we refer to JANSSEN (1988).

Here we restrict ourselves to cross-validation and validate the model as follows:

- the autocorrelation function is used to test the whiteness of the prediction errors. On the estimation dataset it shows the quality of the fit and on the validation dataset it shows the quality of the estimated model: i.e. its capability to predict the output from a given input

- the stationary maximum likelihood criterion J2, equation (2-31), is used to determine the orders of the systems G(x) and H(z). Here it possesses the advantage of being independent upon any weighting.

2.4.6. Model order.

The measurements together with the sequence of the prediction errors, the so called residuals, are shown in appendix B. Figure 2-4 reveals that the prediction errors of the model estimated on measurement 7 are not white.

1.4 0.7

- 0 . 7

i.v

N0nMHLIZE0 nUTOCOflRELOTION BEDTEHPERflTURE

N0RMHUZE0 AUTOCORRELATION OXTGEN CONCENTRATION

1.4 0 . 7 0 -0.7 1.4 0.7 0 -0.7 <SEC> 1200 <SEC>

FIGURE 2-4. Autocorrelation function of the residuals of experiment 7 on: a. the estimation dataset, and

b. the validation dataset.

This measurement is therefore omitted from further analysis. The other experiments are very well estimated: the autocorrelation function shows

that the prediction errors on the validation dataset are almost as white as those on the estimation dataset, as can be seen in figure 2-5 for experiment 1.

Validation by means of the J2 criterion is illustrated in table 2-4, showing the value of J2 on validation dataset 2 for different model orders of the model estimated on estimation dataset 1.

0.7 0 - 0 . 7 1.1 0.7 0

NOflHUUZEO nurCCOnHELflTION BEOTEHPERRTURE

NORnfl'.IÏEO AUTOCORRELATION OXYGEN CONCENTRATION

- 1 . 1 - 3.7 - O - -0.7

V-*

FIGURE 2 - 5 . A u t o c o r r e l a t i o n f u n c t i o n of t h e r e s i d u a l s of experiment 1 on: a. the estimation dataset, and

b. the validation dataset.

stochastic order 4 1 2 . 2 8 . 2 6 8 . 6 4 7 . 0 1 8 . 3 7 8 . 5 7 6 . 9 7 9 . 2 8 8 . 8 6 6 . 6 5 7 . 0 9 7 . 7 4 1 1 . 0 1 1 . 8 1 2 deterministic order

TABLE 2-4. J2» 1 0s, determined for the validation dataset 2 with the model estimated on dataset 1.

Table 2-5 shows the construction of the estimation and validation datasets from the experiments, together with the resulting order of the

deterministic and stochastic systems.

It appeared that for experiment 11 the Hessian of the criterion JlF constructed during the optimization, became singular for the optimal model,with orders 5 and 2 for the deterministic and stochastic models respectively. In order to overcome this difficulty we selected the model with orders 4 and 2 as the optimal one.

estimation dataset 1 3-lst part 4-lst part 5 7-lst part 8-lst part 9-1st part 10-lst part 11-lst part 12-lst part 13-lst part validation dataset 2 3-2nd part 4-2nd part 6 7-2nd part 8-2nd part 9-2nd part 10-2nd part 11-2nd part 12-2nd part 13-2nd part deterministic order 5 3 4 4 4 5 5 5 4 4 5 stochastic order 4 4 3 3 2 2 4 2 2 1 1

TABLE 2-5. Division of the measurement sets into estimation and validation datasets and the estimated order of the model.

From table 2-5 no firm conclusion can be drawn with respect to the proper (deterministic) model order, though a fourth or fifth order model would appear appropriate.

The author attributes the differences in model order to the variation in quality of the experiments, influenced amongst others by the types of coal used.

In view of the limitations of sampling rate mentioned earlier and of the limited availability of analogue filters the data may also be tainted with aliasing effects.

Furthermore the single step prediction error method may also have been a factor in the inconclusive evidence on the model order.

The importance of the choice of the order of the estimated model is evidenced in figure 2-6 by the comparison of the unit step responses of the optimal model of experiment 8 with a third order model. The figure indicates considerable differences with respect to timeconstants (generally smaller for the lower order model) and static gain.

eullIT VMIVC ---■ BEOIEHPEMIUAE CMl NN» riOH — • BE01EMEMIUM

FIGURE 2-6. Unit step responses of the model of experiment 8 for two different orders.

D : 3th order deterministic model, o : 5th order deterministic model.

In general every model consists of a combination of parallel and serie processes. For each extra process in serie a delay of one time-step will be found in the impulse response of the whole system. In figure 2-7 the

first part of the impulse response of the model without its D-matrix shows that only one step delay can be found which implicates that the different orders can be treated as parallel.

A transformation of the estimated model with the modal matrix (a matrix consisting of the eigenvectors of the matrix A) decouples the model into its different modes. The system matrix A is then transformed into a diagonal matrix A. The diagonal consists of the eigenvalues of A. When the eigenvalues are complex, the first upper and lower diagonal of matrix A are partly filled with parameters, TAKAHASHI et al. (1972), and decoupling of the model is only partly possible.

AIR NflSS FU8H — * OEOTENfERRTURE

*

t

t

'

.

D D Q D 0 D 0 70 1110 210 280 3S0 fc» <5EC>

*

t

t

COHL KflSS rtOW — > tmHEH COMENTMTJON

m D

B a D o m n o o O 70 tUO 210 2BU 350

•> ' 5 E C >

FIGURE 2-7. The first 11 points of the impulse response of the deterministic model of experiment 1 without its D-matrix.

TOTAL RESPONSE

ORDER 4

ORDER 1 ( C O M P L E X )

TIME <SEC>« 10 3

FIGURE 2-8. Modal decomposition of the response of the bedtemperature for a unit step change in coal feed rate of the deterministic model for experiment 1.

This decomposition has been used to show the separate influence of each order. The unit step responses of the bed temperature to a perturbation in coal feed rate are given in figure 2-8 for experiment 1 (order 1 is formed with a complex eigenvalue) and figure 2-9 for experiment 10. These figures reveal that the system can be decoupled into a fast and a second order slow part with time constants of 10 to 80 and 200 to 1400 seconds respectively.

TOTAL RESPONSE

TIME <5fc'C>« 10■

FIGURE 2-9. Modal decomposition of the response of the bedtemperature for a unit step change in coal feed rate of the deterministic model for experiment 10.

The behaviour of the fast phenomena (i.e. their static gain and time constant) differed for each experiment. A close re-examination of the experimental conditions did not reveal any physical explanation. This forced us to see the fast phenomena as parasitic, and we will therefore reduce the order of the model to two.

Several ways to reduce the order of a model exist. A well known and in theoretical modeling often used method is treating the relatively fast phenomena as quasi steady-state. For the experimental state space model this would lead to adding the static gain of the fast phenomena to the D-matrix and deleting the fast parts of the A, B and C matrices. It will be clear that using this procedure the resulting lower order model can differ considerably from the original model in the high frequency part of the impulse response. More complex methods add to the foregoing a slight adjustment of the (slow part of the) A, B and C matrices of the model, thereby minimizing some measure of differences between the impulse

responses. Here we will use modal decomposition to reduce the order of the experimental model to two by adding the static gain of the fast phenomena to the D-matrix of the model and deleting the fast part of the A, B and C matrices.

2.4.7. Accuracy of the estimated models.

Assuming normally distributed parameters with a covariance matrix P as given by (2-32), Monte Carlo simulation is used to show the influence o~f the inaccuracy of the parameters, represented by their covariance matrix, on the unit-step responses. The results of these simulations are shown in appendix C. As an example figure 2-10 shows the unit step response of the bed temperature to a perturbation in air feed rate. The bundle of lines given in these figures represent the inaccuracy of the parameters of the model on input-output behaviour, i.e. the step response. It shows that the high frequency behaviour of the models is very well estimated. The low frequency behaviour is considerably less accurate. The experiments at the TNO facility are generally less accurate than those at the THD-3 facility.

o

-50

TIME <SEC»< 10 3

FIGURE 2-10. Monte Carlo simulation of the response of the bedtemperature to a unit step change of the coal mass flow of experiment 1.

20 -19.5 19.0 18.5 -+• <SEC>* 10 3 100 50 0 -50 -> TIME <SEC>x 10 3

FIGURE 2-11. The effect of the construction of the test signal on the accuracy of the estimated model.

A. Coal mass flow of experiment 8 consisting of a PRBS with a step superimposed.

B. Monte Carlo simulation of the response of the bedtemperature to a unit step change of the coal mass flow.