Solving Inverse Problem in Reheater System Modeling

Razidah Ismail and Noor Ainy Harish Faculty of Computer and Mathematical Sciences

Universiti Teknologi MARA Shah Alam, Selangor, Malaysia

razidah@tmsk.uitm.edu.my, ainy@tmsk.uitm.edu.my

Human perception of uncertainties plays an important role in handling parameter problems in multivariable system. Often, human experts describe the decision parameters of the systems through vague and uncertain statements. As a result, achieving a precise model of these systems become increasingly difficult and challenging especially in solving inverse problems in system modeling. The inverse problem is one type of ill-conditioned problems whereby the desired responses are given and a model is used to estimate the input parameters. Thus, this paper discusses a fuzzy state space approach for solving the inverse problem of a reheater system of the combined cycle power plant. A state space model of a reheater system with two input parameters and two output parameters is considered. Here, the uncertainties in the parameters are represented by fuzzy number, with its membership obtained from the human experts. The optimal combination of the input parameters is determined by using the Modified Optimized Defuzzified Value Theorem. These values are compared with those obtained using the forward simulation approach, highlighting some distinguish features of the fuzzy state space approach.

Keywords— Fuzzy State Space Model; Dynamic system; Fuzzy State Space Algorithm; Parameters Estimation

I. INTRODUCTION Almost all engineering analysis is based on models that

have agreeably smooth mathematical properties. On the other hand, any measurement is viewed as having some noise or uncertainties. However, combining measurements with engineering models almost always results in an ill-conditioned problem. The inverse problem, or more precisely inverse modeling, is one type of ill-conditioned or ill-posed problem. Pure mathematicians like to refer to Hadamard’s definition of “ill-posed problems”. A problem is ill-posed if the solution is not unique or if it is a continuous function of the data [1]. In Hadamard’s opinion, ill-posed problems do not have physical sense. Today, general agreement exists that ill-posed problems have “well-posed extensions” which are very meaningful. These well-posed extensions introduce a priori assumptions as to the unknowns. The techniques used today for solving inverse problems are as multivariate as the problems themselves. Some of these techniques are discussed in [2 – 4]. Thus, the interaction between the analysis of the inverse mathematical problem and the measurement of the real system used to

solve the problem is very useful for constructing a good model. In many practical applications, there can be disagreement between the data and the model due to inconsistency, even if the data is perfect [2]. Thus, inverse analysis requires us to study undetermined and over determined systems described with inaccurate and uncertain data.

Inverse problems come paired with direct or forward problems and of course the choice of which problem is called direct and which is called inverse is, strictly speaking, arbitrary. Nevertheless, considering only the direct problems would result in not looking at the problems from all sides and would fail to see the whole picture of the phenomenon [5]. In a typical direct problem, one is given a model, the initial conditions of the state variables, the forcing or input terms and is asked to produce a solution. The state space analysis of the dynamic multivariable system is an illustration of a direct problem [6 – 7]. In a general inverse problem, one is given a model and measurements of some state variables and is required to estimate the initial conditions, the forcing terms or the rest of the state variables. This situation occurs in most real-world systems where only desired states or outputs are specified. Thus, the task of the system analyst or engineer is to “design” the input parameters. Traditionally such inverse problems have been addressed by repeated simulation of forward problems, which requires excessive computer time and thus can be very costly. Such examples can be seen in [8 – 10].

Human perception of uncertainties plays an important role in handling parameters in multivariable system. These uncertainties are mostly reflected in uncertain model, uncertain system input or uncertain initial or boundary condition. Often, human experts describe the decision parameters of the system through vague and uncertain statements. Since uncertainty may contain useful information, several techniques and approaches that integrate uncertainties in system modeling had been studied by many researchers. The concept of fuzzy sets was introduced as a general model of uncertainty encountered in engineering system [11]. His approach provide a tool for handling ill-conditioned problem that exist as a result of combining measurement with engineering models.

Universiti Teknologi MARA Malaysia and Ministry of Higher Education, Malaysia

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Inverse problem may be formulated as a problem of combination of information: the experimental information about data, a priori information about parameters and the theoretical information [12]. In line with this, Fuzzy State Space Modeling for solving inverse problems in multivariable dynamic systems was developed [13-14]. The Fuzzy State Space algorithm (FSSA) is the main feature in this modeling approach, where the uncertain value parameters of the system to be controlled are represented by fuzzy numbers [15], with their membership function derived from expert knowledge. Thus, the objective of this paper is to illustrate the fuzzy state space approach for solving the inverse problem of a reheater system of the combined cycle power plant. For this purpose, a state space model of the reheater system with two input and two output parameters is considered. The effectiveness of this modeling approach was proven by implementing it to the state space model of a furnace and superheater system. The results demonstrate that the proposed approach is reasonable and provides an innovative tool for decision-makers [13-14]. In order to facilitate the implementation of FSSA to other multivariable systems, such as reheater system, an efficient computational tool together with the user’s interface was developed.

II. STATE SPACE MODEL OF REHEATER SYSTEM Any multivariable systems with n inputs, m outputs and p

state variable can be describe mathematically as sets of first order differential equations, which can be represented in a vector matrix form [16]. The state space representation is a convenient model structure which does not requires solving higher order differential equation and only the size of matrix changes according to the number of input and output. The mathematical representation of the systems can be described as follow:

)()( tuBtxAdt

xd += (1)

y C x ( t ) D u ( t )= + (2)

where A is the state matrix of order p, B is the input matrix of order p n× , C the output matrix of order m p× and D is the direct transmission matrix of order m n× . For time–invariant system, A, B, C and D are constant matrices. The vector

dtxd is the time derivatives of )(tx . If D = 0, this

implies that there is no direct connection between the input )(tu and the output y (t). Equation (1) is the state equation

whereas equation (2) is the output equation.

Reheater system is one of the subsystems in a boiler of a combined cycle in power plant [8]. Modeling of reheater system based on forward problems are discussed in [9-10]. Here, the linear state space model of a reheater system developed in [17] is adopted. The model is based on first principal physical and the thermodynamics laws with appropriate simplifications. The variables or parameters used in the development of the state space model are the input parameters, output parameters and the state

parameters. The state equation and the output equation of the reheater system can be represented by (3) and (4) respectively.

[ ]

[ ]+=

ri

rs

rh

rh

rh

rh

hQ

BxT

Ax

dtd

Tdtd

(3)

=rh

rh

r

ro

xT

CTP (4)

where

−=

rh

ro

rh

rirh

rhr

rirh

wv

wkCMwk

A

ρ

0

=

rh

ri

rhr

vw

CMB0

01

+−=

01

0pr

refprrefror c

TchhRC

the state vector, =rh

rh

xT

tx )( , the input vector, =ri

rs

hQ

tu )(

and the output vector =r

ro

TP

ty )( .

Therefore, the linear state space model of the reheater system is given by the state equation, (3) and the output equation, (4) with two state variables, two input and two output parameters. The state variables consists of rhT (metal tube temperature in K) and rhx (product of reheated steam density and outlet steam specific enthalpy in Jm-3) whereas the input parameters are rsQ (heat supplied to reheater from

the furnace model in Js-1) and rih (inlet steam specific enthalpy in Jkg-1). The output parameters from the state space model of a reheater system are roP (outlet steam pressure in Pa) and Tr (reheater steam temperature in K). The nomenclature for the other symbols is listed in the Appendix.

III. INVERSE PROBLEM: FUZZY STATE SPACE APPROACH Fuzzy state space approach is based on Fuzzy State Space

Algorithm developed in [13] for solving inverse problem in multivariable system. Fuzzy State Space Model of multivariable dynamic system is defined as follows:

)()(

)()()(:~

~

tCxty

tuBtAxtxSgF

=

+=•

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where ~u denotes the fuzzified input vector [ ]T

nuuu ,...., 21

and ~y denotes the fuzzified output vector [ ]T

myyy ,...., 21 with initial conditions as 00 =t and 0)( 00 == txx . The elements of state matrix ppA × , input matrix npB × , and

output matrix pmC × are known to specified accuracy. The uncertain value parameters of the system are represented by a triangular fuzzy number with their membership function derived from expert knowledge.

A triangular fuzzy number (TFN) is a special type of fuzzy number with three parameters: smallest possible value (a1), most desirable value (a2) and the largest possible value (a3). It can be represented by their breaking points

( )321 a;a;aA = , which is defined as

( )

[ ]

[ ]

otherwise

aax

ax

a,ax

aaxa

aaax

x

,

A

32

2

21

23

3

32

1

0

1

∈

=

∈

−−

−−

=μ

The α -cuts of TFN define a set of closed intervals. The

intervals are ( ) ( )[ ] ( ]1.0,332121 ∈∀+−+− ααα aaaaaa

The graph of typically TFN is illustrated in Fig. 1. Thus, triangular membership function is used to represent imprecise, vague or uncertain parameters because they are mathematically easy to implement due to its simple formula, computational efficiency and easy to manipulate [18].

Fig. 1: A triangular fuzzy number

The development of the algorithm is based on three phases of a fuzzy system, i.e fuzzification, fuzzy environment and defuzzification. For fuzzification, the crisp values are converted to fuzzy values. These values are processed in the fuzzy environment. Finally the fuzzy value is converted to crisp value for interpretation in the defuzzification phase. The framework shown in Fig. 2, gives a general procedure that leads to the development of Fuzzy State Space Algorithm for multivariable system.

IV. PARAMETER ESTIMATION OF THE REHEATER SYSTEM Multivariable systems can be represented in a

decomposed form as a set of coupled multiple-input single-output (MISO) systems [19]. Thus, the global modeling problem of the reheater system can be reduced to MISO system. The implementation of the fuzzy state space approach to the reheater system is discussed according to three phases of fuzzy system as mentioned earlier. An interactive approach using Interactive Parameter Estimator [20] is used for the computations involved in this algorithm.

A. Phase 1: Fuzzification Each of the input parameters of the reheater system is

represented by a TFN, ( )321 a;a;aA = , as shown in Table 1. The α -cuts of all the fuzzified input parameters, with increment of 0.2 are listed in Table 2. These values are used to calculate indF , the fuzzy values of induced output or performance parameter gFS . indF is determined by taking the maximum and minimum value of each performance parameter. These values are used to plot the graph of indF .

Each output parameter can be expressed as linear combination of the input parameters [21]. Using the steady state operating data [8], the input-output equations are determined as

rirsro hQP 1756.01542.0 +=

rirsr hQT 33 101002.0101081.0 ×+×=

x

1

3a 2a 1a

A

( )xAμ

Fig. 2: Framework for fuzzy algorithm

Fuzzy value

Fuzzy value

Crisp value

PHASE 1

• For each input and output parameters, specify the preferred values and its domain

• Set the alpha cut, [ ]10,∈α • Generate combination of the end points

interval of the parameters • Substitute in the performance parameter • Obtain the induced and preferred

performance parameter

PHASE 2

• Plot the graphs. • Determine the intersection of two graphs. • Obtain the fuzzy number.

PHASE 3

• Use back substitution • Generate combination of the end points

intervals • Apply the Modified Optimized Deffuzzified

Value Theorem

Crisp value

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Each of the input values in Table 2 is substituted in input-output equations so as to obtain the corresponding performance parameter, indF for the output parameters.

Table 1: Input parameters specifications

Input parameters ( )321 a;a;aA =

Qrs ( )666 105.3;1017.3;105.2 ×××

hri ( )666 105.3;1003.3;105.2 ×××

Table 2: α -cuts of Input parameters and induced

performance parameter for Pro

α -cuts 0 0.2 0.4

rsQ 610× [2.50, 3.50] [2.634, 3.434] [2.768, 3.368]

rih 610× [2.50, 3.50] [2.606, 3.406] [2.712, 3.312]

indF (Pro) [0.824, 1.154] [0.863. 1.127] [0.903, 1.100]

indF (Tr) [520.3, 728.3] [545.3, 711.8] [570.4, 695.2]

α -cuts 0.6 0.8 1.0

rsQ 610× [2.902, 3.302] [3.036, 3.236] [3.170, 3.170]

rih 610× [2.818, 3.218] [2.924, 3.124] [3.030, 3.030]

indF (Pro) [0.942, 1.074] [0.981, 1.047] [1.021, 1.021]

indF (Tr) [595.5, 678.7] [620.5, 662.2] [645.6, 645.6]

The domain and the desired value for the output

parameters are specified in Table 3. α -cuts with increment of 0.2 are used to calculate SgFF , the fuzzy values of preferred or desired output parameters. Combination of the endpoints of intervals for all output parameters with respect to each particular α -cuts are determined. These values are used to plot the graph of SgFF .

Table 3: Output parameters specification

Output parameters Domain Desired value

Pro [1 × 106, 1.5 × 106] 1.24 × 106

Tr [7.10 × 102 , 7.50× 102] 7.446 × 102

B. Phase 2: Fuzzy environment The intersection of fuzzy preferred output parameter

and the fuzzified performance parameter is determined by superimposing the two graphs in order to obtain the f*- value. Fig. 3 and Fig. 4 indicate the fuzzy value obtained for each of the output parameter. Since the membership function designates the degree of desirability, the largest fuzzy value f* = 0.4132 is chosen and used in the rest of the algorithm.

C. Phase 3: Defuzzification With f* = 0.4132, the steps in the defuzzification

process are carried out to calculate the best possible combination of the input parameters in order to accommodate all the constraints defined in the process of fuzzification. The selection of the optimal combination for the input parameters is determined by the Modified Optimized Defuzzified Value Theorem [13]. The optimized parameter estimation for the reheater system are shown in Table 4. The optimized input parameters of the reheater system are Qrs = 3.087 × 106 Js-1 and hri = 3.087 × 106 Jkg-1. The percentage difference from the desired values is 2.62% and 11.76% respectively. With these input values, the optimized output parameters are Pro = 1.06 × 106 Pa and Tr = 6.282 × 102 oK with a percentage difference of 14.51% and 15.63% respectively. In order to properly model the uncertainties and further improve the results, the parameters of fuzzy numbers need to be adjusted based on the historical data and human experience. For a better resolution, α -cuts with much smaller increment can be used.

Table 4: Optimized input and output parameters

f* = 0.4132 Calculated value Desired value Difference (%)

Qrs 3.087 × 106 3.17 × 106 2.62

hri 2.674 × 106 3.03 × 106 11.76

Pro 1.06 × 106 1.24 × 106 14.51

Tr 6.282 × 102 7.446 × 102 15.63

These values can be compared with the result obtained

from simulation carried out by [8]. The main purpose of this comparison is to highlight the difference between inverse modelling by fuzzy concept and a widely accepted simulation approach. The calculated values of input parameters using the fuzzy state space approach differ by

Fig 3. Reheater system (Pro) f* = 0.4132

Fig 4. Reheater system (Tr) f* = 0.2227

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2.7% for Qrs and 11.7% for hri. The good results obtained in this application show that this approach may become an interesting and innovative tool for decision-makers. It will provide broader and more useful information for power generation planning purposes. Besides, it is relatively easy to take into account experts’ knowledge and considerations for establishing the membership functions.

Table 5 Comparison of Optimized Input parameters

Input parameter Ismail’s Ref. [8] difference (%)

Qrs 3.087 × 106 3.174 × 106 2.7

hri 2.674 × 106 3.029 × 106 11.7

V. CONCLUSION AND RECOMMENDATION A fuzzy state space approach was presented for solving

the inverse problem in reheater system modeling, whereby the manipulation of imprecise, uncertain quantities is considered. The uncertain value parameters of the system are represented by fuzzy numbers with their membership function derived from expert knowledge. In many respects, fuzzy numbers depict the physical world more realistically than single-valued numbers, as the concept takes into account the fact that all phenomena have a degree of uncertainty. The ability of this method to address inverse problems in multivariable systems directly is an outstanding advantage especially in reducing computation time and cost.

To facilitate the implementation of this approach, an interactive interface was developed. Using this computational tool, the users can evaluate more alternatives in less time, and at the same time, the users can obtain more information on the performance of each of those alternatives. Since fuzzy state space approach is designed for solving inverse problem in any multivariable dynamic system, further study is undertaken to explore the possibility of integrating with other mathematical techniques in order to gain a better understanding of the system.

ACKNOWLEDGMENT The authors would like to thank Research Management

Institute, Universiti Teknologi MARA Malaysia and the Ministry of Higher Education of Malaysia for the financial support through the Fundamental Research Grant Scheme 600-RMI/ST/FRGS 5/3 Fst (23/2008). Their gratitude is also extended to anonymous reviewers for their valuable comments and suggestions.

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Mascow University Publishing, 1984. [2] E.Hensel. Inverse Theory and Applications for Engineers.

Eaglewood Cliffs, N.J.: Prentice Hall, 1991 [3] A.I.Prilepko, D.G.Orlovsky and I.A. Vasin. Methods for Solving

Inverse Problems in Math. Physics.New York:Marcel Dekker,Inc., 2000.

[4] J.A.Souza, J.V.C.Vargas, O.F.vonMeien, W.P.Martignoni and J.C.Ordonez, “The inverse methodology of parameter estimation for

model adjustment, design, simulation, control and optimizations of fluid catalytic cracking (FCC) risers”, J Chem Technol Biotechnol 2009, 84:343 – 355.

[5] C.W. Groetsch, Inverse Problems: Activities for undergraduate. Washington DC: Mathematical Association of America, 1999.

[6] R.Ismail (2006). “Furnace Modeling using State Space Representation”, Scientific Research Journal, 3(1), 37 – 52, 2006.

[7] N.A.Harish, R. Ismail and T. Ahmad. “Modeling of Superheater system using a state space representation”. Proc. of the Int. conf on Sc & Tech.: Application in industry & education. 11-13 Dec 2008. Malaysia: Penang

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

[9] K.Y.Lee, L.Ma, W.H. Jung and S.H.Kim. “Intelligent modified predictive optimal control of Reheater Steam Temperature in a Large-Scale Boiler Unit”, IEEE Power & Energy Soc Gen.Meeting, 2009.

[10] D.Zeng, J.Liu, J.Liu and X.Xie. “Model and simulation of Reheater based on Thermal System Dynamics”, Advanced Materials Research, 322 pp 322 – 327, 2011.

[11] L.A.Zadeh, “Fuzzy Sets”.Inform. and Control.8(3): 338 – 353, 1965. [12] A.Tarantola and B.Valette, “Inverse Problems = Quest for

Information”. Journal of Geophysics. 50:159 – 170, 1982. [13] R. Ismail, T. Ahmad, S. Ahmad and R.S. Ahmad. “An Inverse Fuzzy

State Space Algorithm for Optimization of Parameters in Furnace System”. Proc. of the Joint 2nd Int. Conf. on Soft Computing and Intelligent Syst and 5th Int. Conf. Symp on Adv Intelligent Syst, Yokohama, Japan., 2004.

[14] R.Ismail, K.Jusoff, T, Ahmad, S. Ahmad and R.S. Ahmad. “Fuzzy state space model of multivariable control systems”. Computer and Information Science. 2(2): pp. 19 - 25, May 2009

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

[16] K.Ogata, K. Modern System Engineering. Upper Saddle River: Prentice-Hall International, 1997.

[17] N.A.Harish, R. Ismail and T. Ahmad. “State Space Modeling of a Reheater system in the Power Generation Plant”. 2013 IEEE Symp on Computers & Informatics (ISCI2013). Malaysia: Langkawi.

[18] W.Pedrycz.” Why Triangular Membership Functions?. Fuzzy Sets and Systems. 64: 21 – 30, 1994.

[19] C.C.Lee. “Fuzzy logic in control systems: Fuzzy Logic controller – Part 1”, IEEE Trans. Syst. Cyber. 20: 404 – 418, 1990.

[20] R.Ismail, R.A.Halim and N.A.Harish. “Interactive Parameter Estimator of a Superheater System: A GUI for Industrial Applications”, IEEE Proc. of Int. Conf. on Business, Engineering and Ind. Appl, Kuala Lumpur, Malaysia. 227-231, 2011

[21] J.S.Bay. Fundamental of Linear State Space Systems. New York: WCB/McGraw-Hill, 1999.

APPENDIX

riw - reheater inlet steam mass flow (kg/s)

row - reheater outlet steam mass flow (kg/s)

rv - reheater volume (m 3 )

rhρ - reheated steam density (kg/m 3 )

rsQ - heat supplied to the reheater (from the furnace model) (J/s)

rM - mass of reheater tubes (kg)

rhT - metal tube temperature (K)

rhk - an experimental coefficient

roh - outlet steam specific enthalpy (J/kg)

refh - reference steam enthalpy condition (J/kg)

refT - reference steam temperature condition

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