A CASE STUDY OF FUNDAMENTAL GREY-BOX MODELING

Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

www.elsevier.com/locate/ifac

A CASE STUDY OF FUNDAMENTAL GREY-BOX MODELING Kang Li, Steve Thompson, Gareth-Guan R. Duan, Jian-xun Peng School of Mechanical & Manufacturing Engineering Queen’s University Belfast Ashby Building, Stranmillis Rd., Belfast BT9 5AH, UK

Abstract: Fundamental grey-box modeling is an approach that can be used to model complex nonlinear dynamic systems for which the underlying mechanisms are either too complex or only partially known a priori. In this paper, further discussion is made on the motivations behind this method and the framework of this method is also extended. As a case study, it is used to model the NOx emission in a coal-fired power generation plant. Copyright © 2002 IFAC Keyword: nonlinear systems, modeling, power plants, power generation, pollution.

1.

The pragmatic approaches using both a priori and a posteriori information in system modeling sometimes are referred to as grey-box modeling techniques (Bohlin, 1994; Pearson and Pottmann, 2000; Tulleken, 1993, etc). In grey box approaches, physical modeling and system identification form two interacting paths. Depending on how much and, in what form priori information is used, various greybox modeling methods can be categorized. Fundamental grey-box modeling is one of such approaches (Li and Thompson, 2000 and 2001), in which it is assumed that the underlying mechanisms of the systems to be modeled are either too complex or only partially known a priori. So called “fundamental” knowledge can be relevant physical and chemical laws, or based on empirical experimental result. The use of term “fundamental” conforms to its definition, “forming or serving as an essential component of a system or structure”. Given that fundamental knowledge is available or partially known a priori a novel grey-box modeling technique is introduced and referred to as fundamental grey-box modeling.

INTRODUCTION

It is well established that in modeling complex engineering systems, both a priori and a posteriori information might have to be used, rather than solely relying on physical modeling or some data-dependent identification approaches. The reasons for the “pragmatic” approach are (Li and Thompson, 2001): • For physical modeling, the underlying physical and chemical laws of an engineering system (which are generally formulated as a set of PDEs and ODEs) some times can be too complex to build a simplified system model; or the underlying mechanism of the system is unknown or such knowledge is incomplete. Also the system under study may exhibit properties that change in an unpredictable manner, etc. • Data-dependent identification models may not be able to nest the “true” system structure therefore their prediction capacity cannot be guaranteed. In addition, for operational plant, safety and quality considerations often indicate that the duration of field experiments and the intensity of test-signal perturbation must be kept to a minimum. As a result, most popular and theoretic understood estimators may fail to produce physically consistent models from finite, noise corrupted data.

The following context is organized as follows. In section 2, further discussion is made on the

127

motivations behind the proposed fundamental greybox modeling approach, and an extended framework follows. In section 3, this modeling technique is applied in power generation plant NOx emission modeling. Section 4 is the conclusion.

2. A FURTHER DISCUSSION ON THE FUNDAMENTAL GREY-BOX MODELING Within the context of non-linear system modeling using linear-in-parameter polynomial model structure, neural networks, fuzzy systems or other “intelligent” methods, the underlying principle is to use a set of simple linear/nonlinear functions as the basis to approximate a complex function. It is true that every engineering system has its particular characteristics and may exhibits particular behavior, and how much success a modeling method can achieve therefore largely depends on the form of the approximation basis or functions that has been chosen. Hence is it possible to select the approximation basis, i.e. the activation function in the neural networks or the nonlinear terms in a linear-in-parameter generalized polynomial model structure, according to a priori knowledge of the engineering system? For engineering systems where a priori fundamental knowledge exists (though it may not be in a form that is suitable for system modeling) it is possible to extract basic information in the form of simplest expressions. For example reaction rates will be of the form exp(x;c) or oscillatory behavior might take the form sin(x;c), etc, for which x is a vector of variables, c is a vector of parameters. These simple functions acquired from the fundamental a priori knowledge of the studied system (these will be referred to as fundamental elements (FEs) in the sequel, since these simple functions are regarded as essential and basic functions), are associated with the behavior of the system. Such FEs may appear in the system model in a variety of forms, and can also undergo change (mutation). For example in the function sin(x;c), the value of the parameters in vector c can be different in different situations, while sin(x;c) may also undergo mutation and become cos(x;c). Once the FEs are collected, a system model which reflects the dynamics of the system may be produced by appropriately combining these FEs. At this model construction stage experimental or online operational data is required. Therefore, the proposed modeling technique involves a search for the fundamental elements (FEs) of the system, and then constructing the system model using appropriate combinations of these FEs, hence the title of the paper, fundamental grey-box modeling. This process can be briefly formulated as follows.

128

Suppose a nonlinear unknown function f ( x ) has the following form: f ( x ) = f ( ϕ1( c1 ; x ),...,ϕ m ( cm ; x ))

(1)

where x is variable vector, and ϕ1( c1; x ),...,ϕ m ( cm ; x ) are fundamental elements (FEs) that are mutually linear independent, c i , i = 1,2 ,...,m are parameters in those FEs. Then instead of approximating f ( x ) by a set of linear/nonlinear functions that are irrelevant to f ( x ) itself, f ( x ) can be approximated as follows: f ( x ) = f ( ϕ 1 ( c1 ; x ),...,ϕ m ( c m ; x )) m

≈ f ( x ) x = x0 + ∑ bi ( ϕ i ( c i ; x0 + σx ) −ϕ i ( ci ; x0 ))

(2)

i =1

where bi =

f ′( x ) x = x0 . ϕ i′ ( c i ; x )

One issue in (2) is how to identify those fundamental elements (FEs) and particularly their forms. The choice of FEs indeed can be difficult. The following remarks on selecting FEs are made. Remarks: 1. FEs are the simplest form of expressions. If a priori information regarding the engineering system exists (based on first principle laws, chemical reactions, for example), one can then propose as many FEs as necessary. It is highly possible that these FEs may also be strongly correlated with each other. Therefore as to which FEs will be finally used to construct the model will be determined by a posteriori information through an identification method including model structure selection and parameter identification. 2. Some of the FEs may be the same type of function, and only the parameters in these functions are different. In this case, the derived model partially resembles the neural networks where the activation functions in the hidden layers are of the same type and only the weights and bias are different. However, it should be pointed out that, the FEs are derived from a priori information regarding the system mechanism. Therefore, it is highly possible that not all FEs are the same type of function. 3. If unknown a priori the parameters in these FEs may also have to be determined from experimental data. Fundamental grey-box modeling procedures Step 1. Establish the fundamental mechanisms of the system. Typically these will consist of a set of PDEs and ODEs.

Step 2. From these equations select a set of fundamental elements FEs in which each element describes a basic relationship between system variables.

ϕ0 ( t ) = 1  ϕ ( t ) = ϕ ( y( t − k ), u ( t − d − k )), i = 1,..., p i y j uj  i and y and uj j=1,2,..,m are output and inputs, θ is the parameter, ε ( t ) is a white noise series, ϕ i ( t ) denotes fundamental elements (FEs), and ky and kuj are time delays for inputs and output. Now if N samples are used for model training, (3) becomes:

Step 3 (optional). Construct a set of derived terms that are the production of two or more FEs, which are again based the mechanism governing the system behavior. The derived terms are used to reflect the couplings among system variables.

Y = ΦΘ + Ξ

Step 4. Those FEs and their derived terms constitute a term pool. Use plant data together with the term pool to establish a suitable linear-in-parameter model (generalized polynomial model) through model structure selection and parameter identification.

where: T T Y T = [ y( 1 ), y( 2 ),..., y( N )] Φ = [ ϕ0 ,ϕ1 ,ϕ 2 ,..,ϕ p ]

ϕ i = [ ϕ i ( 1 ), ϕ i ( 2 ),...,ϕ i ( N )] T , i = 0 ,1,2,.., p

Step 5. Validate the model.

Ξ T = [ ε ( 1 ),ε ( 2 ),...,ε ( N )]

Two technical problems stand out: 1. How to identify the parameters unknown in the FEs and their derived terms? 2. How to select the model structure (the term pool can be very large therefore there is a combination problem, also the correlation between terms can be strong)?

The loss function is defined: E( Θ ) = Ξ T Ξ

Y TY = 1.

Then the minimal cost function can be constructed in a recursive way:  ( Y T Kqϕ q +1 )2 Eq +1( Θˆ q+1 ) − Eq ( Θˆ q ) = −  ϕ qT+1K qϕ q +1   K qϕ q +1ϕ qT+1K qT = − K K (6)  q +1 q ϕ qT+1K qϕ q+1   K0 = I N × N , E0 = 1    where Θˆ q is the estimated parameter vector with q terms in the model. According to (6), if a new term, e.g. ϕ q+1 , in the term pool, will be selected into the model, its contribution to decreasing the cost function will solely depends on the value of ( Y T K qϕ q +1 )2

ϕ Tq +1Kqϕ q +1

where

K q does not depend on

ϕ q+1 . Therefore, the computation in the selection process is greatly reduced.

Consider a generalized nonlinear polynomial NARX model (nonlinear auto-regressive with exogenous input) that can be used to represent a non-linear dynamic system: y( t ) = ∑θ iϕi ( t ) + ε ( t )

i.e. ϕiT ϕi = 1, i = 1,2,..., p ,

normalized,

With respect to the first approach, various analytic methods might be applied. However for the second approach, some intelligent methods, such genetic algorithms (Peng, et al, 2001), may have to be used. In this paper, the first approach is applied to construct the fundamental grey-box model. That is, the parameters in the FEs and their derived terms are first identified, then a step-wise forward algorithm is used to construct the model (Thompson and Li, 2000). In the step-wise forward model construction process, at each time only the term in the term pool that contribute most to decreasing the cost function will be selected.

p

(5)

It is supposed that all regressors in (4) are

Two general approaches can be used to resolve the above problems: 1. The identification of parameters for FEs and their derived terms, as well as the identification of model structure, are taken as two separate and sequential processes. 2. The identification of parameters for FEs and derived terms, as well as the identification of model structure, are taken as an integrated process.

Model :

(4)

Algorithm modeling

1

Step-wise

fundamental

grey-box

Step 1. To identify the fundamental elements (FEs) and construct the derived terms based on a priori knowledge about the system mechanism. Those FEs and derived terms will constitute a term pool.

(3)

i =0

where,

129

Step 2. Build an auxiliary NARX model Ì au comprising of all terms in the term pool. Select N training samples. Build another NARX model Ì pri initially containing no terms. Let Tau be the

1.

2.

terms set comprising all terms in the term pool, T pri be the term set comprising all terms in Ì pri . Now initialise the intermediate variables in (6). Set the minimal acceptable decrease in the cost function δ E when a new term is added into Ì pri .

3.

Step 3. Term selection phase. The best term in Tau satisfying the following criterion will be included:

4.

ϕ q+1 : max ϕr

( Y T K qϕ r )2

ϕ rT K qϕ r

, ϕ r ∈ Tau

(7)

The mechanism governing thermal NO formation is believed to be expressed as:

Step 4. Check phase. Check whether

∆E =

( Y T K qϕ q +1 )2

ϕ qT+1K qϕ q +1

≤ δE

Raw particles are mixed together with preheated air and fired into the furnace by burners normally located within corners of the furnace. At first devolatilisation takes place and volatile combustion (mainly combustion of volatile hydrocarbons) then take place in the core of the fireball forming in the furnace. In the devolatilisation and volatile combustion process, fuel NOx formation and reduction are taking place, also SOx, COx, and H2O are formed in the process. Then chars combustion takes place in the outer layer of the fireball in the furnace. CO2, char NOx, thermal NOx, as well as ashes, etc, are formed in this process.

d [ NO ]T =2 k1 [N2] [O2] × dt 1 − [ NO ] 2 / k [ O 2 ][ N 2] 1 + k −1[ NO ] /( k 2[ O ] + K 3[ OH ])

(8)

is satisfied. If (8) is true, stop the model selection process, otherwise, go on.

(9)

where k1, k-1 are forward and backward reaction rate of N2 and O2 to NO and N, k2 and k-2 is the forward and backward reaction rate of N and O2 to NO and O, k3 is the forward reaction rate of N and OH to NO and H, k = ( k1 / k−1 )( k2 / k −2 ) is the equilibrium constant for the reaction between N2 and O2. The value for [O] (the concentration of the oxygen atom) and [OH] was obtained from the predicted concentration of major species using the partial equilibrium assumption. Through some extensive experimental study prompt NO formation mechanism is believed to have the following form:

Step 5. Update phase. Add ϕ q+1 into T pri and delete ϕ q+1 from Tau , update intermediate variables in (6). Step 6. Go to step 2.

3. APPLICATION The fundamental grey-box modelling approach has previously used to build a NOx emission model for a coal-fired power generation plant in Northern Ireland (Li and Thompson 2001). In this paper, this technique is used to model NOx emission in another coal-fired power generation plant. Because these two plants operate differently this study tests the technique.

d [ NO ] p dt

= fT β Apr [ O2 ] a [ N2 ][ fuel ] b exp( Ea / RT )

(10)

where f is a correction factor applicable for all aliphatic Alkane hydrocarbon fuels. T β represents the non-Arrhenius behaviour of the equation at conditions where the maximum flame temperature is exceptionally high or low. APr is the pre-exponential factor, a and b are reaction order constants for oxygen and fuel respectively and vary between 0 and 1, depending on the rate of consumption of fuel and oxidiser. E a is the activation energy. For the fuel NO, its formation and decay are correlated with a pair of competitive parallel reactions, each first order in HCN, which represents the pool of nitrogen containing species:

In coal-fired power generation plant, the major part of NOx emission in power generation plant has been found to be NO. According to De Soete (1975), there are three main sources of NO in combustion, namely thermal NO, Prompt NO and Fuel NO. Thermal NO results from the reaction of atmospheric nitrogen and oxygen at high temperature, prompt NO is formed by the reaction of nitrogen with hydrogen derived radicals in the fuel-rich zone of combustion, while fuel NO results when nitrogen compounds present in the fuel are released and react with oxygen. In almost all coal-fired power plant, fuel NO is the major contribution to NOx emission, and some of the fuel NO can be released from the devolitisation of the fuel while some from the oxidation of the char. This NOx emission mechanism is summarized as follows:

d [ NO ] f dt

b exp( −33700 / T ) |HCN → NO = 1010 ρX CN X O 2

d [ N2 ] |NO → N 2 = 3 × 1012 ρX CN X bNO exp( −30000 / T ) dt

130

(11)

where, X is the mole fractions of the chemical species, b is the reaction order for molecular oxygen (that is the function of oxygen concentration), ρ is the density. The two reaction rates are included in the transport equations for HCN and NO and form the basis for the fuel NO post processor, which allows the calculation of NO formation for a pulverised coal flame.

also let 9 7 5 u10 = ∑ ui ( t ) , u11 = ∑ ui ( t ) , u12 = ∑ ui ( t ) (13) i =8 i =6 i =1

Based on (9) to (11), the following types of function are taken as fundamental elements (FEs): Fi = ( u i ( t )) ci , i = 1,2,...,12 (c / u ( t )) Fi = e j i , i = 13,14 ,...,18 , j = 1,2 ,3,4 ,5 ,10 (14)

Variables in (1) to (3) are intermediate variables and they are not measurable or observable. In the fundamental grey-box modelling approach, (9) to (11) are basic knowledge relating to the NOx formation and used to derive the fundamental elements. However it is also essential to identify the manipulated variables and how they are incorporated into (9) to (11). Hence an understanding of the plant operational process is required.

and similar to a previous paper (Li and Thompson, 2001), the following derived terms are used: cp

Di = ( F j )

c

( Fk ) q , j ∈ { 1,2 ,...,12 }, k ∈ { 1,2 ,...,18 }, j ≠ k

(15) The final number of terms in the term pool, which includes the FEs and their derived terms, is 102.

The coal-fired power generation plant studied in this paper has the capacity of 500 MWe. This unit is installed with a Low NOx concentric firing system (LNCFS) where the burners are provided with overfire air, and a proportion of the air in the combustion chamber is offset from the walls. The coal delivery system for each unit comprises of four subsystems: coal feeder, coal mill, separator and PF pipe-work to supply fuel to the burners. There are five coal mills in the coal delivery system. The furnace of this boiler is separated into two sides, namely A and B-side, by a wall. Each side has 4 burners per level and there are 5 levels of burners with each level linked to one of the five mills. The fuel flow (feeder speed) to each of the mills can be measured. Each mill feeds eight burners on a level (4 on the A-side and 4 on the B). Damper settings on each side of the boiler tend to be ganged together. Based on the unit operation process, the following variables are identified to be related with NOx formation:

7000 samples from plant filed operation are used to identify the parameters in the fundamental elements and their derived terms. This data set is also used to construct the NARX model based on algorithm 1. δ E in algorithm 1 is chosen to be 0.00002. Finally 10 FEs and derived terms are selected to construct the model, and the resulted model takes the form: y( t ) + a1 y( t − 1 ) + a2 y( t − 2 ) + a3 y( t − 3 ) + a4 y( t − 4 ) = b0 + b1q −1F2 ( t )F10 ( t ) + b2 q −1F2 ( t )F14 ( t ) + b3q −1F11( t )F18 ( t ) + b4 q −4 F2 ( t )F18 ( t )

(16)

+ b5q −3 F4 ( t ) F10 ( t )ε ( t ) + b6 q −1F3( t )F9 ( t ) + b7 q −1F3( t )F15 ( t ) + b8 q −4 F4 ( t )F10 ( t ) + b9q −3 F9 ( t )F18 ( t ) + b10 q −3 F7 ( t ) + ε ( t )

The model is used to predict NOx emission in the power plant over another 3 time periods. Figures 1 to 3 list the long-term prediction performance over these 3 periods, i.e. the long-term prediction model takes the following form:

1. Speed of the conveyor belt feeding the coal for each of the five mills (rpm): ν 1( t ) , ν 2 ( t ) , ν 3( t ) ,

ν 4 ( t ) , ν 5( t ) . 2. O2 in A and B-side of the furnace that are measured at economiser (percentage in wet): O 21 ( t ) , O 22 ( t ) . 3. Tilting position of burners in A and B-side of the furnace (degree, relative to horizontal): θ1( t ) , θ 2( t ) .

ˆy( t ) + a1 ˆy( t − 1 ) + a2 yˆ( t − 2 ) + a3 yˆ( t − 3 ) + a4 yˆ( t − 4 ) = b0 + b1q −1F2 ( t )F10 ( t ) + b2 q −1F2 ( t )F14 ( t ) + b3q −1F11( t )F18 ( t ) + b4 q −4 F2 ( t )F18 ( t )

(17)

+ b5q − 3 F4 ( t )F10 ( t )ε ( t ) + b6 q −1F3 ( t )F9 ( t ) + b7 q −1F3 ( t )F15 ( t ) + b8 q − 4 F4 ( t )F10 ( t ) + b9 q − 3 F9 ( t )F18 ( t ) + b10 q −3 F7 ( t )

Other factors also affect the NOx emission. But they will be treated as model noise/disturbance. In conclusion, the above 9 variables will be used as the input variables in the model to be built:

where ˆy( t ) is the model prediction, and others are real system inputs. If model performance is defined as:

u1( t ) = ν 1( t ) , u2( t ) = ν 2 ( t ) , u3( t ) = ν 3( t ) u4 ( t ) = ν 4 ( t ) , u5 ( t ) = ν 5 ( t ) , u6 ( t ) = O 21( t )

MP =

u8 ( t ) = θ1( t ) , u9 ( t ) = θ 2 ( t ) (12)

131

N

N

i =1

i =1

2 2 ∑ ei / ∑ yi

(18)

where N is the number of samples, ei , i = 1,2,..., N is the modeling prediction error, y i , i = 1,2 ,..., N is the NOx emission. Then the long term prediction performance of the grey-box model is 0.103. Linear ARX model and no-linear ARX model with up to two-order terms are also built. For the linear ARX model, the long-term prediction performance is 0.171 and for the nonlinear ARX model, the long-term prediction performance is 0.113. The number of terms in grey-box model is 14 (not include the DC factor), whereas the linear ARX model and the nonlinear ARX model have 22 terms. Obviously the fundamental grey-box model is able to give better long-term prediction performance with much less terms in the model.

models. Journal of Process Control, 4, pp. 301-315. Thompson, S., K. Li (2000). Fundamental grey-box modelling and identification of non-linear dynamic systems, QUB Report Number 2842. Tulleken, H. J.A.F. (1993). Grey-box modelling and indentification using physical knowledege and Bayesian techniques. Automatica, 29, pp. 285308. Model prediction and NOx emission 600

500

NOx emission (ppm)

400

4. CONCLUSION

-. NOx emission . Model prediction

300

200 -. Error

100

In this paper, a further discussion has been made on the motivation behind the fundamental grey-box modeling approach, and an extended framework has also been proposed. This modeling method has been used to model the NOx emission in a coal-fired power generation plant, and simulation results using data from the power plant show the merits of this new method.

0

-100 0

500

1000

1500

Time (minute)

Fig. 1 Long term prediction over period 1 Model prediction and NOx emission 600

ACKNOWLEDGEMENTS

500

Acknowledgement is made to the British Coal Utilisation Research Association and the UK Department of Trade and Industry for a grant in aid of this research but the views expressed are those of the authors, and not necessarily those of BCURA or the Department of Trade and Industry.

NOx emission (ppm)

400 -. NOx emission 300

. Model prediction

200

-. Error

100

0

REFERENCES

-100 0

500

1000

1500

Time (minute)

Bohlin, T., (1994). Case study of grey box identification. Automatica, 30, pp. 307-318. De Soete, G.G., (1975). Overall reaction rates of NO and N2 formation from fuel nitrogen. 15th Symposium (international) on Combustion, The Combustion Institute, 1093-1102. ETSU (1997). Coal R&D programme. Technology status report: NOx control for puliverised coalfired power plant, ETSU, Harwell. Li, K. and Thompson, S. (2001). Fundamental Greybox Modelling. European Control Conference, Porto, Portugal, pp.3648-3653. Peng, J., K. Li, S. Thompson (2001). GA based software for power generation plant NOx emission modelling. 6th International Conference on Technologies and Combustion for A Clean Environment, Vol. 2, 881-887, Porto, Portugal. Pearson RK, Pottmann M (2000). Gray-box identification of block-oriented nonlinear

Fig. 2 Long term prediction over period 2 Model prediction and NOx emission 500

400

NOx emission (ppm)

300

-. NOx emission . Model prediction

200

-. Error 100

0

-100 0

500

1000 Time (minute)

Fig. 3 Long term prediction over period 3

132

1500