graphical representation of fuzzy state space of a … representation of fuzzy state space of a...

Graphical Representation of Fuzzy State Space of a Boiler System

1NOOR AINY HARISH, 2RAZIDAH ISMAIL, 3TAHIR AHMAD 1,2 Faculty of Computer and Mathematical Sciences

University Teknologi MARA, 40450 Shah Alam, Selangor, MALAYSIA 3Department of Mathematics, Faculty of Science,

Universiti Teknologi Malaysia,81310 Skudai, Johor, MALAYSIA [email protected], [email protected], [email protected]

Abstract: - Graphical representation is useful to illustrate complex structure in a direct or intuitive way. A graph is a symbolic of network and its connectivity can be simplified as a set of linked notes. In recent studies of complex control system, directed graph has been introduced to define and interpret the interconnection structure underlying the dynamics of the interacting subsystem. Generally, the main components of a power plant are a gas turbine, a boiler and a steam turbine. Our interest in this study is to represent the boiler system using graphical representation. The boiler system consists of subsystems namely furnace, superheater, drum, riser and reheater. These subsystems will be transformed into vertices and interconnection between subsystems will be associated with edges of the graph. Initially, the input-output variables are identified using state space approach. The graphical representation of Boiler system acts as an initial stage for further exploration in application of Fuzzy Graph. Key-Words: - Fuzzy Graph, Combined Cycle Power Plant, State Space Model, Multi-connected System.

1 Introduction Graphical representation is extremely useful to illustrate complex structure in a direct and intuitive way, and such that they are widely used in many fields. Graph theory provides a mathematical modeling for studying interconnection among elements in natural and man-made systems [1]. In recent studies of complex control system, directed graphs have been introduced to define and interpret the interconnection structure underlying the dynamics of the interacting subsystems. Subsystems were associated with vertices while interconnection with edges of the graph [2]. Theorems and algorithm of graph theory represent the behavioral properties of the system as the properties of the vertices or edges of the graph [3]. Generally, the main components of a combined cycle of power plant are a gas turbine, a boiler and a steam turbine [4]. Our interest in this study is the boiler system. Like most real life processes, boiler dynamics are highly non-linear and finding an accurate model is almost impossible due to uncertainties in the system. The state space approach is based on time domain analysis and synthesis using state variables. It is a unified method for modeling, analyzing and designing a wide range of systems [5]. This approach is well studied and it provides a good approximation in modeling engineering and biological systems [6-9]. In addition, the mathematical equations are represented in matrix

algebra and first order differential equations. Therefore, only the size of matrix changes according to the complexities of the system.

2 A Boiler System Boiler system consists of five main subsystems namely furnace, riser, drum, reheater and superheater [4]. Based on the typical drum-type boiler, the feedwater is supplied to the drum where the water is evaporated. The water flows into downcomers, then enters the risers. In the risers, the heat from the furnace is used to increase the water temperature and eventually to cause evaporation. Thus the circulation of water, steam, and water and steam mixture takes place in the drum, the downcomers and the risers. Steam generated in the risers is separated in the drum where it flows through the superheater on to the high pressure turbines. It may be recycled to the boiler in the reheater where its energy content is increased.

Desuperheating spray water is introduced in the superheater for control of main steam temperature. As for the combustion process path, the risers absorb radiant heat in the furnace. The hot gases leaving the furnace transfer the heat by radiation and convection to the superheater. The heat then transferred by convection to the reheater and the economizer, before exiting the boiler via the stack. The burner tilt is used to change radiation heat distribution between the

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ISSN: 1790-5109 99 ISBN: 978-960-474-195-3

risers and the superheater. Schematic diagram of a boiler system is shown in Fig. 1[4].

Fig. 1. Schematic Diagram of a Boiler System

3 Graphical Representation of Boiler System

Graph theory is a useful representation because the elements of the graph can be defined so as to have a one to one correspondence with the elements of

many kinds of engineering systems [3]. In addition, a graph is a symbolic of network and of its connectivity which implies an abstraction of the reality that can be simplified as set of linked nodes [10]. A directed graph E,VG , often referred as a

graph, is defined by a set V of ‘nodes’ and E of ‘links’ (or ‘arcs’). The set of nodes and edges can be conveniently labeled by jv,......,v,v,vV 321 and

ne,.....,e,e,eE 321 respectively. The boiler system can be visualized in the form of a graph as shown in Fig. 2. Initially, the state space model is used to determine the input and output parameters for each subsystem as shown in Table 1 [8], [12]. The complete nomenclature can be found in [4].

Table 1. State Space Representation of a Boiler System

Sub-system

Matrices A, B and C in state space equations State Vector

Input-Output Vector

Super-heater

refpsrefspsp

R

Qf

pk

Qf

pk

vQf

pk

Qf

pk

v

refpsrefspsp

Rvsv

sCM

kvsv

sCM

k

A

Tchhc

Tchh)c(

ppf

ppf

s

s

ss

ssv

ss

vsv

ss

ssv

ss

vsv

s

s

s

sts

s

sts

s

2

2

2

2

2

1

2

1

11

11

s

sts

v

CMB

10

01

refpsrefscp

R

Qf

pk

Qf

pk

Qf

pk

Qf

pk

Tchhcp

R

Tchh

cT

Tchh

h

c

hR

C

Tchhpss

s

ss

ssv

ss

vsv

ss

ssv

ss

vsv

refpsrefspss

s

refpsrefs

psref

refpsrefs

ref

ps

ss

2

2

22

0

0

sstT

)t(x

s

gswQ

)t(u

vss

wTp

ty

Riser

rr

r

rr

rrtr

r

V

w

V

JkcM

Jk

A

0 ,

rr

wrtr

V

hcM

B

0

01

wrv hhC

10

rrt

hT

)t(x

dir

wQ

)t(u

xty

Reheater

rh

ro

rh

rirhrhrc

rirh

w

V

wkCM

wk

A

0 ,

rh

ri

rhrc

V

wCM

B0

01

01

0pr

refprrefror c

TchhR

C

rh

rhxT

)t(x

rirs

hQ

)t(u

rro

TP

ty

Drum

L

dowL

dow

V

vV

v

A0

0 ,

wre hxhxB 1

11

0

10

L

dowwL

V

vV

C

1DdL

xm

)t(x

r

eww

)t(u

dw

wh

ty

Furnace

F

gEGF

V

TRkA

,

000000

111

FFF VVVB

,

gEG

glgegsgEGf

gegrgsgEGf

grgsEGgsEG

fEGg

grgsEGgsEG

fEGg

TR)TT(cTRk)TT(cTRk

TcwQw

kRT

TcwQw)(

kRT

C

1 EGx

G

A

F

www

)t(u

Gesrsisir

pQQQQ

ty


ISSN: 1790-5109 100 ISBN: 978-960-474-195-3

Fig. 2. Graphical Representation of Boiler System The construction of the graphical representation of boiler system can be formalized by the following proposition. Proposition The boiler system of a combined cycle power plant in Ordys et. al, 1994 [4] can be presented by a graph

:E,VG where a set )G(V of vertices corresponds to the subsystems related to the boiler and a set )G(E of edges corresponds to the input-

output variables associated with the components. However, the graphical representation of a boiler system in Fig. 2 is static. Therefore, a more comprehensive graphical representation of the system is needed which is outlined in the following section

4 Fuzzy State Space Model Fuzzy State Space Model (FSSM) of a furnace was introduced and followed by Fuzzy State Space algorithm with the purpose of solving the inverse problem where the input parameters that achieve the desired outcome can be deduced [12]. In this algorithm, the uncertainties in the parameters are presented by fuzzy numbers [13] which are integrated in the state space model of the system. The fuzzy algorithm has been shown to give good parameter estimation for a superheater system [14].The definition of FSSM of a multivariable dynamic system is given as follows: Definition A Fuzzy State Space Model of a multivariable dynamic system is defined as

gFS : tu~BtxAtx.

txCty~

where u~ denotes the fuzzified input vector Tnu,....u,u 21

and y~ denotes the fuzzified output vector

Tmy,....y,y 21 with initial conditions as 00 t and

000 txx . The elements of state matrix p pA ,

input matrix p nB , and output matrix m pC are

known to specified accuracy. The multi-connected systems of FSSM can be viewed as a system of FSSM, gFS which is a collection of subsystems 1gFS ,

2gFS ,…, gFjS .j,....,,21 For each of gFS in multi-

connected system, it can be transformed into a point in the Euclidean n-space where the elements of A, B and C matrices can be written as coordinates of a point in a finite dimensional space. Thus, FSSM can be embedded in Euclidean space by using the following transformation.

Theorem Given a fuzzy state space systems

pmnpppgFj CBAC,B,AS

such that

ppp

p

aa

aaA

1

111

,

pnp

n

bb

bbB

1

111

mpm

p

cc

ccC

1

111

.

The transformation

mppnppgFj

mppnppmnppp

c,...,c,b,...,b,a,...,aS

ECBA:

111111

2

is a bijective map. Proof

(i) Let *gFgF SS

'

mp''

pn''

pp'

mppnpp

'mpmp

'

'pnpn

''pppp

'

'mp

'mp

'pn

'pn

'pp

'pp

'''

'''

c,.....,c,.....,b,....,b,a,.....,ac,.....,c,.....,b,....,b,a,.....,a

cc,........,cc,bb,........,bb,aa,.......,aa

c,......,cc,.....,c,b,.....,bb,.....,b,a,.....,aa,.....,a

CC,BB,AAC,B,AC,B,A

111111

111111

1111

11111111

1111

1111

1111

In other words,

*gFgF SS where is a function

3e

10e

2e

8e 12e2v

6v

5v

4v

14e

13e

11e

6e 9e

7e

3v

1e

4e 5e


ISSN: 1790-5109 101 ISBN: 978-960-474-195-3

(ii) Let *gFgF SS ,

therefore,

'mp

''np

''pp

'mpnppp

c,......,c,b.......b,a,.......,ac,.....,c,b,......b,a,.....,a

111111

111111

,

and

,aa.,,.........aa 'pppp

' 1111

,bb.,,.........bb 'npnp

' 1111 'mpmp

' cc.,,.........cc 1111

ppp

p

aa

aa

1

111

,aa

aa

'pp

'p

'p

'

1

111

npn

p

bb

bb

1

111

,bb

bb

'np

'n

'p

'

1

111

npn

p

cc

cc

1

111

'mp

'm

'p

'

cc

cc

1

111

(iii) Pick any point, mpnpppr Then , mpnppp c,.....,c,b,......b,a,.....,ar 111111 where Rc,b,a ijijij .

Consequently, pmpnppgF MBAS such that

mpnpppgF c,.....,c,b,......b,a,.....,aS 111111

with

ppp

p

aa

aaA

1

111

,

mpn

p

bb

bbB

1

111

mpm

p

cc

ccC

1

111

Since r is arbitrary thus is onto.

'''''' C,B,AC,B,ACC,BB,AA Therefore, *

gFgF SS is one to one

Table 2. FSSM to Euclidean n-space

Sub- system

Order of matrices

k Vertex *iS in Euclidean n-space

Super-heater

23

2222

C

,B,A

14

000

10

100000222112113 ,,,

v,,

CM,,,,,,a,a,a,aS

ssts

*

Riser

21

2222

C

,B,A

10

0000

100000222112112 ,,,

V

h,,

CM,,,,,,a,a,a,aS

rr

w

rts

*

Reheater

22

2222

C

,B,A

12

0000

100000222112114 ,,,

V

w,,

CM,,,,,,a,a,a,aS

rh

ri

rhs

*

Drum

22

2222

C

,B,A

12

0011100000005 ,,hx,h,x,,,,,,,

V

v,,,

V

vS wre

L

dow

L

dow*

Furnace 1591

11

C,B

,A

15

000

111000000001 ,,,

V,

V,

V,,,,,,,,,

V

TRkS

FFFF

GEGF*


ISSN: 1790-5109 102 ISBN: 978-960-474-195-3

By using this transformation, the Fuzzy State Space Model of the boiler system can be embedded in Euclidean space. The detailed is shown in Table 2. The Euclidean k-space for each vertex is determined by the total size of A, B and C matrices in each subsystem of FSSM. These vertices will be embedded into the same Euclidean n-space. 15 is the maximum value of k which will represent the size of the n-space. The mapping of FSSM to Euclidean space will be the initial stage for further exploration view of FSSM.

5 Conclusion The static graphical representation of the boiler system has been successfully modeled using the basic graph theoretical concept. We finally outlined general procedure in construction of graphical presentation for Fuzzy State Space Model of a system. A sample construction of a boiler system is presented. This new approach will simplify the schematic diagram of interconnection of subsystems in a boiler. Thus, the graphical representation will lead to the development of fuzzy graphical representation of a boiler where the technique to determine the membership values of each subsystems will be discussed.

References: [1] F.Harary, Graph Theory. Addison Wesley

Publishing Company. California, USA, 1967. [2] D.D.Siljak, Dynamic Graphs, Nonlinear

Analysis: Hibrid System. Article in Press. http:// www.sciencedirect.com. Retrieved 2 August, 2007.

[3] O.Shai, & K.Preiss, Graph theory representations of engineering systems and their embedded knowledge. Artificial Intelligence in Engineering, 13, 1999, pp. 273-285.

[4] A.W.Ordys, A.W.Pike, M.A.Johnson, R.M. Katebi, & M.J.Grimble, Modelling and Simulation of Power Generation Plants. Springer-Verlag, London, 1994.

[5] N.S. Nise, Control Systems Engineering. Addison Wesley. Menlo Park, California, 1995.

[6] I.D.J.Onsen, R.A.M.Yers, & J.M.Flemming, Meta-Analysis of Animal Movement using State-Space Models. Ecology. 84(11), 2003, pp. 3055-3063.

[7] W.L.Berendrecht, A.W.Heemink, F.C.Van Geer, & J.C.Behrels, State Space Modeling of Water Tables Fluctuations in Swiching Regimes. Journal of Hydrology. 292, 2004, pp. 249-261.

[8] R.Ismail, Furnace Modelling using State Space Representation. Scientific Research Journal, 3(1), 2006, pp. 37 – 52.

[9] W.L.Berendrecht, A.W.Heemink, F.C.Van Geer, & J.C.Behrels, A non-linear state space approach to model groundwater fluctuations. Journal of Hydrology. Advances in Water Resources, 2006, pp. 959-973.

[10] S.Abu Bakar, S.Baharum, & T.Ahmad, Eigenvalue of Fuzzy Graph Type-3. National Seminar on Fuzzy Theory and Applications, Shah Alam, Malaysia, 2008.

[11] N.A.Harish, R.Ismail, & T.Ahmad, Modeling of Superheater System using a State Space Representation, Proc. of Int. Conference on Science & Technology: Application in Industry & Education, Penang,12-13 Dec, 2008.

[12] R.Ismail, T.Ahmad, R.S.Ahmad, & S.Ahmad, Optimal Parameter Estimation of MISO System based on Fuzzy Numbers, Jurnal Teknologi Maklumat and Sains Kuantitatif. 8(1), 2006, pp. 1-10.

[13] A.Kaufman, & M.M.Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1985.

[14] N.A.Harish, R.Ismail, & T.Ahmad, A fuzzy algorithmic approach for estimating the input parameter of a superheater system. National Seminar on Fuzzy Theory and Applications. Shah Alam, Malaysia, 2008.

Nomenclature

9e - irh

10e - dw w,h

11e - rw,x

12e - vw

1v - Furnace 1gFS

2v - Superheater 2gFS

3v - Reheater 3gFS

4v - Riser 4gFS

5v - Drum 5sFS

1e - irQ

2e - rsQ

3e - gsQ

4e - GA,F w,ww

5e - esisG Q,Q,P

6e - sw

7e - ss T,P

8e - ror P,T

13e - re w,w

14e - dw


ISSN: 1790-5109 103 ISBN: 978-960-474-195-3

graphical representation of fuzzy state space of a … representation of fuzzy state space of a...

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