Nonlinear Exponential CARMA Model-Based

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002

A Nonlinear Exponential ARX Model Based Multivariable Generalized Predictive Control Strategy for Thermal Power Plants

Hui Peng*, Toru Ozaki**, Valerie Haggan-Ozaki† and Yukihiro Toyoda† †

*

College of Information Engineering, Central South University, Changsha 410083, P. R. China. He is currently a visiting researcher at the Institute of Statistical Mathematics, 4-6-7 Minami Azabu, Minato-ku, Tokyo 106-8569, Japan. E-mail: [email protected]

** The Institute of Statistical Mathematics, 4-6-7 Minami Azabu, Minato-ku, Tokyo 106-8569, Japan. E-mail: [email protected] † Sophia University, 4, Yonbancho, Chiyodaku, Tokyo 102-0081, Japan. E-mail: [email protected]. † † Bailey Japan Co. Ltd., 511 Baraki, Nirayama-cho, Tagata-gun, Shizuoka 410-2193, Japan.

E-mail:

[email protected].

Abstract: This paper presents a modeling and control method for thermal power plants having nonlinear dynamics varying with load. First a load-dependent exponential ARX (Exp-ARX) model that can effectively describe the plant nonlinear properties and requires only off-line identification is presented. The model is then used to establish a constrained multivariate multi-step predictive control (ExpMPC) strategy whose effectiveness is illustrated by a simulation study of a 600 megawatt (MW) thermal power plant. Although the predictive control algorithm may be used without resorting to online parameter estimation, it is much more reliable, and displays much better control performance than the usual GPC algorithm. Keywords: power systems, nonlinear systems, ARX models, system identification, constraints, predictive control. Nomenclature:

MWD: Megawatt demand or load demand, i.e. M (t ) ; STE:

Superheater outlet steam temperature error, i.e. y1 (t ) = K MSB x33 ;

RTE:

Reheater outlet steam temperature error, i.e. y2 (t ) = K RHB x15 ;

TPE: Main steam pressure error, i.e. y3 (t ) = x5 ; SP1: Primary spray attemperator; SP2:

Secondary spray attemperator;

1

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 FF:

Fuel flow rate;

FW:

Feedwater flow rate;

RGD: Opening of the reheater flue gas damper.

1. Introduction

parameter estimation to deal with plant non-linearity. In

In normal use, thermal power plants display daily load

real-time, the convergence and computational cost of

fluctuations which can be anything from 30% to 100% of

on-line

full load. The plant must be capable of operating as a

problems.

parameter

estimation

poses

considerable

controlled system across a wide range of working conditions. Since the relationships governing the main

Toyoda [7] introduced an algorithm based on an

controlled variables (main steam temperature, reheated

amplitude-dependent auto-regressive model [1], which

steam temperature, and main steam pressure) depend to a

uses an off-line method to identify the load-dependent

large extent on load, the behavior of the plant becomes

non-linearity of thermal power plants. However, in

quite non-linear. However, at present, the multi-loop PID

Toyota's method, the effects of stochastic process

controller remains the most widely used method in the

disturbances are not taken into account. Furthermore, the

process control.

cost function of the controller is not very appropriate,

Over the last 10 years, several advanced control methods

since it does not involve the outputs or the state errors,

for the control of thermal power plants have been

and on-line optimization has to be used to compute

reported. Examples are the generalized predictive control

control signals based on a nonlinear model and non-linear

(GPC) method based on local model networks [5], the

programming techniques. In this case, sometimes it is

self-tuning minimum variance control method for steam

impossible to obtain a feasible solution, and control may

temperature control [3], the self-tuning multi-loop

become divergent.

generalized predictive control (GPC) method [2], and the multivariable dynamic matrix control (DMC) method

In this paper, we also follow Toyota's method [7] of using

based on linear models [6]. Of these, the local model

an (amplitude dependent) exponential auto-regressive

network based GPC uses a set of local linear models to

model [1], thus obviating the problems resulting from

describe the non-linearity of the plant. However, in order

on-line parameter estimation. Like the model used by

to represent plant non-linearity satisfactorily, the required

Toyota [7], our model is an Exp-ARX model depending

number of local linear models often becomes too large to

on load as an exogenous variable. However, noting that

be implemented easily. The self-tuning GPC, DMC and

the dynamics of the thermal plant system are different

minimum variance control methods all require on-line

under situations of increasing or decreasing load, our

2

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 model also depends on the incremental change in load. As

The gain-scheduling PID controllers and the constrained

will be shown in section 3, the model effectively

multivariable generalized predictive controller (ExpMPC)

represents the nonlinear dynamics of the system over a

presented in this paper will operate in parallel in the

wide range of operating conditions.

proposed control system. This is because the actual plant will commonly already be working with PID controllers,

In order to deal with the controller cost function problems

so parallel operation will obviate the need to drastically

noted in the use of Toyota's method [7], we have also

change the structure of the actual control system, and will

built a new type of multivariate multi-step-ahead

also guarantee operational safety.

predictor. Using this nonlinear predictor, a constrained

be

multivariate long-range predictive control strategy has

turbine-boiler system and the multi-loop PID controllers.

been designed. In section 5, the modeling and control

The outputs and the inputs of the controlled plant are

methods proposed in this paper are implemented in a

given as follows

simulation study of a 600MW fossil-fired thermal power

controlled

by the

Therefore the plant to

ExpMPC

consists

of

the

 y1 (t ) : STE (main steam temperature error / ℃)    Y(t ) =  y2 (t ) : RTE (reheat steam temperature error / ℃)   y3 (t ) : TPE (main steam pressure error / bar)

plant using MATLAB SIMULINK (The MathWorks, Inc.).

This paper will use a thermal power plant simulation of a

u1 (t ) : FFI (fuel flow increment)    ( ) : 1SPI (1st stage spray flow increment) u t 2    U(t ) = u3 (t ) : 2SPI (2st stage spray flow increment)   u4 (t ) : GRI (flue gas recirculation flow increment) u (t ) : TCI (turbine control increment)   5 

600MW sliding pressure type boiler-turbine system [8-9]

here, the unit of U(t ) is the open width of the damper

created in MATLAB SIMULINK as the plant to be

(expressed as a percentage), and the sampling interval is

controlled. This simulation model is based on a set of

30 seconds. For safe and economical operation of the

physical models designed in accordance with operational

plant, the errors of the output variables Y(t ) should be

and physical data of an actual plant and includes a total of

controlled to be as near to zero as possible, whatever the

40 variables. The model is not linearized near equilibrium,

load level. The non-linearity of the controlled plant may

and can imitate the strong nonlinear dynamic properties

be seen from its step responses. Figs. 1 and 2 are unit-step

of power systems under large load variations. In the

responses to u 3 (t ) and u 4 (t ) when the load demands

2. Thermal power plant non-linearity

simulation model, sets of conventional multi-loop

are 25%, 50%, 75% and 90%MWD (Megawatt Demand)

gain-scheduling PID regulators are also used to control

of the full load respectively.

the power plant as would be common in practice. The details of the simulation model may be described by a set

Figs. 1 and 2 show that the dominant time constants are

of differential equations given in Appendices A and B [8].

over 10 minutes, so a sampling interval of 30 seconds is

3

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 quite appropriate for digital control. The steady-state gain

3. Modeling of the controlled system

and step-response sequences are varying acutely with

In

load. The non-linear characteristics are mainly due to

auto-regressive model [1] may be written as follows

variation in plant static gains with loads. This provides a

the

single

p

variable

case,

the

2

x t = ∑ (ϕ i + π i e −γxt −1 ) x t −i + et

exponential

(1)

i =1

justification for using a load-dependent Exp-ARX model to describe the nonlinear dynamics of thermal power

where x t is the state, ϕ i , π i are constants, γ is a

plants.

scaling parameter, and et is white noise. The idea of the model is that it makes the instantaneous characteristic roots of the AR model depend on state amplitude. The model displays certain well-known features of nonlinear vibration theory, such as amplitude-dependent frequency, jump phenomena and limit cycle behavior.

For thermal power plants, an unsophisticated method of describing the dynamics could be by using different linear ARX models at certain fixed load levels or operating points, where non-linearity will result from the power load-cycling operations. Extending this idea, a more Fig. 1

Unit step responses to u 3 (t ) .

flexible description of the change in dynamics could be provided by a time-varying ARX model, such as the Exp-ARX model below, whose parameters change with load demand. Such a model can represent the dynamic behavior of the system over the whole operating range, and may be estimated off-line with only slightly more computational effort than fitting a single linear ARX model. This load-dependent multi-variable Exp-ARX model may be constructed as follows

A t ( q −1 )Y(t ) = Bt ( q−1 )U(t − 1) + Dt ( q −1 )∆M (t − 1) + T( q−1 )ξ (t ) (2) Fig. 2

Unit step responses to u 4 (t ) .

where Y(t ) ∈ ℜ n , U(t − 1) ∈ ℜ m , M (t − 1) is the load

4

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 demand, and ∆M (t − 1) is the load demand increment,

operating point form which the parameters of model (2)

{ ξ( t ) ∈ ℜ }

track the load variations.

n

denotes the noise sequence which is

assumed to satisfy

{

The estimation procedure for the Exp-ARX model is

}

E { ξ(t ) | Ft −1 } = 0 , E ξ(t ) ξ(t ) T = Ω

computationally simple, involving only the use of the

where Ft denotes the σ -algebra generated by the data

least squares methods to estimate the coefficients

up to and including time t, and Ω is a positive definite −1

−1

Φ xi ( j )( x = a, b, d ; j = 0,1) after first fixing the model

−1

matrix. In (2), At (q ) , Bt (q ) , and D t (q ) are the n×n ,

n×m

orders

and n × 1 load-dependent polynomial

matrices in the unit delay operator q

−1

ka, kb, kd , kt

and

the

parameters

λ a , λ b , λ d , Q a , Qb , Q d , M 0 , ∆M 0 . The Akaike Information

respectively,

Criterion

(AIC) may be used to choose the model

−1

T(q ) is the n × n Hurwitz design polynomial matrix,

orders [4]. Generally, λ a , λ b , λ d , Q a , Qb and Q d may

which is used to represent prior knowledge about the

be chosen as follows

process noise, because successful identification of the M max − M 0 ∆M max − ∆M 0

2

actual noise model structure which is multi-source and

λi = − log(α i ) /

time-varying is unlikely. In (2), take

Qi = diag ( β i1 , β i 2 ) , i = a, b, d ,

β i1 = 1 , β i 2 = 100 ~ 500 , α i ∈ [0.1 ~ 0.00001].

 A t ( q ) = I + A t,1q + " + A t,ka q  −1 −1 − kb  Bt ( q ) = Bt,0 + Bt,1q + " + Bt,kb q  −1 −1 − kd  Dt ( q ) = Dt,0 + Dt,1q + " + Dt,kd q  −1 −1 − kt  T( q ) = I + T1q + " + Tkt q −1

−1

, Qi

− ka

In this case, the model orders and other parameters used (3)

were ka = 8, kb = 14, kd = 14, T(q −1 ) = I ,

and 2 M ( t −1) − M 0  − λa  A = Φ (0) + Φ (1)e ∆M ( t −1) −∆M 0 Qa ai ai  t ,i 2  M ( t −1) − M 0 − λb ∆M ( t −1) −∆M 0 Q  b Bt ,i = Φbi (0) + Φbi (1)e  2 M ( t −1) − M 0 − λd  ∆M ( t −1) −∆M 0 Q d Dt ,i = Φdi (0) + Φdi (1)e  

Qa = Qb = Qd = diag (1 , 300) , M 0 = 20(%) , ∆M 0 = 0, λ a = 0.0001 , λ b = λ d = 0.1 .

Fig.3 shows the root loci of the simulation plants (4)

represented by the fitted Exp-ARX model, and illustrates how the model dynamics vary with load. It also reveals that the model has an ability to represent the non-linear

where * denotes a vector norm, λ a , λb and λ d are

dynamics of the plant due to working point changes. Fig.

the scaling parameters, Q a , Qb and Q d

are 2 × 2

4 shows a comparison between the actual data and data

weighting matrices, M 0 and ∆M 0 are the datum load

generated by the fitted Exp-ARX model, where the plant inputs were a set of independent PRBS signals with unit

and its increment respectively. In the exponential factors

amplitude, from which it is clear that the Exp-ARX

of (4), M (t − 1) and ∆M (t − 1) respectively represent

model provides a very close fit to the actual data.

the messages of ‘position’ and ‘velocity’ of the system

5

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 4.1 Multi-step-ahead prediction

Theorem 1: Multivariable nonlinear predictor Consider the systems described by (2)-(4). If we can obtain {M (t + j − 1), ∆M (t + j − 1) | j = 1,2,", N } , and constrain { U (t + j − 1) | j = 1,2,", N } to be Ft -measurable, then j ( j = 1,2, " , N ) step ahead optimal predictive outputs are ˆ (t + j | t ) = E { Y(t + j ) | F } = G' ( q −1 )U(t + j − 1) Y t j + Y0 ( t + j ) (5) where

Y0 (t + j ) = T( q−1 )−1 F j ( q −1 )Y(t ) + T( q −1 )−1 H j ( q −1 )U(t − 1)

Fig. 3

+ T( q −1 )−1 E'j ( q −1 )Dt + j ( q −1 )∆M (t + j − 1)

Root loci with load change

(6) E j (q ), F j (q ), G j (q ) , and H j (q ) are the solutions '

−1

−1

'

−1

−1

of two Diophantine equations T(q −1 ) = E'j (q −1 ) At + j (q −1 ) + q − j F j (q −1 ) −1

'

−1

−1

−1

'

−1

(7) −1

E j (q )Bt + j (q ) = T( q )G j ( q ) + q H j ( q )

here

(8)

deg(E'j ) = j − 1 , deg(F j ) = max{ ka − 1, kt − j} , deg(G 'j ) = j − 1 , deg(H j ) = max{ kb − 1, kt − 1} .

Proof: Multiply (7) by Y (t + j ) and introduce (2) and (8) into the resulting expression. This yields Y(t + j ) = G'j ( q −1 )U(t + j − 1) + Y0 (t + j )

(9)

ˆ ( q −1 )ξ (t + j ) + V  ( q −1 )ξ (t ) +V j j here

Fig. 4 Comparison of actual outputs and model outputs while load changes through the range 40% to 90%

~ −1 −1 −1 ' −1 −1 j ˆ (q −1 )q j + V V j j ( q ) = T( q ) E j ( q ) T( q ) q ˆ (q −1 ) = V ˆ +V ˆ q −1 + " + V ˆ q − j +1 V j j1 j2 jj ~ −1 ~ ~ −1 V j ( q ) = V j1 + V j 2 q + "

.

require { U(t + j − 1) | j = 1,2,", N } to be Ft -measurable and ignore the random disturbance term ~ V j (q −1 )ξ (t ) , then (9) implies (5). □

4. Multi-step predictive control

If

The model based multi step predictive control method has various advantages, such as rolling

Theorem 2: The recursive solution of the Diophantine equation (7) may be achieved as follows

optimization, multi-step-ahead prediction and

E'j +1 (q −1 ) = E'j (q −1 ) + E j q − j , j = 1,2,", N

feedback correction etc. Here, an Exp-ARX model

−1

based multi-step predictive control algorithm (ExpMPC) will be presented for thermal power

−1

−1

F j +1 (q ) = q[F j ( q ) − E j A t + j (q )]

(11)

E j = F j (0)

(12) −1

−1

−1

E0 = I, F1 (q ) = q[T(q ) − A t +1 (q )]

plant control.

(10)

(13)

Proof: Subtracting (7) from the equation obtained by substituting j with j + 1 in (7), we get 6

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 E'j +1 ( q −1 )A t + j +1 ( q−1 ) − E'j ( q −1 )A t + j ( q −1 )

Yr (t ) = Y (t ), j = 0,1", N − 1  Yr (t + j + 1) = (1 − µ )Yr (t + j ) + µW* , µ ∈ (0,1]

(14)

+ q − j [q −1F j +1 ( q −1 ) − F j ( q −1 )] = 0

here W* is the set-point. From (5-6), thus yields

In general, the variance of load demand between two sampling points can be seen to be very small, and therefore At + j +1 (q −1 ) ≅ At + j ( q −1 ) .

~ ~ ~ ~ Y(t ) = G t U (t ) + Y0 (t )

where

In this way, (14) becomes [E'j +1 ( q−1 ) − E'j ( q −1 )]A t + j ( q −1 ) + q− j [q−1F j +1 ( q−1 ) − F j ( q −1 )] = 0

(15)

 G0    G0  G1   #  # % ~ . Gt =  G G " G  N u −1 Nu −2 0   # # # #     G N −1 G N − 2 " G N − N u 

.

From the above equation, we can see that the first j-term coefficients of E'j +1 (q −1 ) and E'j (q −1 ) must be equal, so that (10)-(12) are established. If we let j = 1 in (7), the initial conditions (13) can be obtained. □

Let

Theorem 3: The solution of the Diophantine equation (8) may be obtained using the following recursion

the

predictive

Exp-ARX

model-based

multi-step

algorithm

(ExpMPC)

control

cost-function be defined as

G 'j +1 (q −1 ) = G 'j (q −1 ) + G j q − j ,

~ ~ J = Y(t ) − Yr (t )

H 'j +1 (q −1 ) = q[H j ( q −1 ) + E j Bt + j (q −1 ) − T(q −1 )G j ], G j = H j (0) + E j Bt + j (0), G 0 = E0Bt +1 (0), H1 (q −1 ) = q[E0Bt +1 (q −1 ) − T(q −1 )G 0 ].

where

The proof of theorem 3 is similar to the proof of theorem 2, and is omitted here.

~ + U (t )

2 I ( n× N )×( n × N )

2 R

.

(16)

R = diag{r1 ,", rN u } is the weighting matrix.

Introducing (15) into (16), and ignoring constant terms, we obtain a quadratic version of the ExpMPC

4.2 The constrained multi-variable generalized predictive

cost-function equivalent to (10)

control (ExpMPC) strategy

1 T  T  U  (t ) + [Y  (t ) − Y  (t )]T G  (t ) ( G t G t + R )U J = U 0 r t t . 2

Equations 5 and 6 may be used to obtain a vector-matrix

 * (t ) can be The future optimal control sequence U

form for the predicted output. We first define ~ ˆ (t + 1 | t )T , Y ˆ (t + 2 | t )T , " , Y ˆ (t + N | t )T ]T Y (t ) = [ Y

obtained by searching

~ Y0 (t ) = [Y0 (t + 1)T , Y0 (t + 2)T ,", Y0 (t + N )T ]T

 (t ) U

such that

~ J is

minimized under some constraints as follows

~ U(t ) = [U(t )T , U(t + 1)T ,", U(t + N u − 1)T ]T

 * (t ) = arg min J U  (t ) U

~ Yr (t ) = [Yr (t + 1)T , Yr (t + 2)T ,", Yr (t + N )T ]T

subject to

here N is the prediction horizon whereas N u is the

~  G ~ t  ~  U (t ) ≤ − G t 

control horizon after which controls are assumed to be

~ ~  Y max − Y0  ~ ~  , − Ymin + Y0 

zeros, i.e. U(t + j ) = 0, j ≥ N u . In the definitions above,

   U min ≤ U( t ) ≤ U max .

~ Yr (t ) is the output reference sequence that usually may

In a actual power plant control, the first vector U* (t ) of

be designed as

 * (t ) is used. the solution U

7

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 5. Simulation results In the following simulation, N = 20 , Nu = 4, R = 35I, µ = 0.4, and the Exp-ARX model (2) identified in Section 3 are used as internal models of the ExpMPC strategy. The comparisons of the control performance of ExpMPC and the gain-scheduling PID controls are shown in Fig.5.

It

is clear that over a wide load operating range, the control performance of ExpMPC based on the Exp-ARX models is far better than conventional PID control as shown in Fig.5. It may also be seen from Fig.6 that the ExpMPC

Fig.6 Control performance of ExpMPC and GPC based

based on the Exp-ARX models has demonstrably better

on a global linear ARX model

control performance than a GPC using a global linear ARX model as its internal model. (Here, the global linear

6. Conclusion

ARX model was identified using the same data and

In this paper, a multivariable nonlinear exponential ARX

model orders as for the Exp-ARX model).

(Exp-ARX) model has been presented, which can very closely describe the nonlinear dynamic behaviors of thermal power plants over the whole operating range. Since Exp-ARX models are similar to linear ARX models fixed at certain load levels or operating points, actually this is a class of local linearization models which does not need to resort to on-line parameter estimation. They may be applied to describe a class of nonlinear systems satisfying certain smooth conditions. On the other hand, the Exp-ARX model can be also regarded as a special NARMAX (nonlinear ARMAX) model. Notice that there is no necessity to give up the idea of the

Fig. 5

original NARMAX model as a result of the birth of the

Control performance of ExpMPC and PID

Exp-ARX model. However, in actual applications, using Exp-ARX models is very convenient, because some identification approaches of linear models may be used to identify the models.

8

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 The MIMO constrained long-range predictive

1994.

control strategy based on the Exp-ARX models

[4] T. Ozaki and H. Oda, ‘’Nonlinear time series model

proposed in this paper has shown much better

identification by Akaike’s information criterion,’’ In

control

Information and Systems, B. Dubuission, Ed.

performance

than

conventional

Oxford: Pergamon Press, 1978, pp. 83-91.

gain-scheduling PID control and the usual GPC

[5] G. Prasad, E. Swidenbank and B. W. Hogg, ‘’A local

based on a global linear ARX model. Since our

model networks based multivariable long-range

suggested predictive controller does not resort to

predictive control strategy for thermal power

on-line parameter estimation, as is done in

plants,’’ Automatica, vol. 34, pp. 1185-1204, Oct.

general self-tuning control, it also has better

1998.

reliability than the usual self-adapting control

[6] J. A. Rovnak and R. Corlis, ‘’Dynamic matrix based

algorithms.

control of fossil power plants,’’ IEEE Transactions on Energy Conversion, vol. 6, pp. 320-326, June Acknowledgement

1991.

The authors are grateful to the Ministry of Education,

[7] Y. Toyoda, K. Oda and T. Ozaki, ‘’The nonlinear

Culture, Sports, Science and Technology, Japan, for

system identification method for advanced control of

supporting H. Peng’s visit to the Institute of Statistical

the fossil power plants,’’ in Proceedings of 11th IFAC

Mathematics, Tokyo, Japan.

Symposium on System Identification, 1997, pp. 8-11, Fukuoka, Japan.

References

[8] Y. Toyoda, ‘’ Identification of nonlinear systems,’’ Ph.

[1] V. Haggan and T. Ozaki, ‘’Modeling nonlinear

D Thesis, Kyushu University, Japan, Jan. 2000.

random vibrations using an amplitude-dependent

[9] J. Kusunoki, H. Furukoshi and A. Takami,

autoregressive time series model,’’ Biometrika, vol.

‘’Advanced

68, pp. 189-196, Jan. 1981.

process

simulation

system,’’

in

Proceedings of IFAC Workshop on Modelling and

[2] B. W. Hogg and N. M. El-Rabaie, ‘’ Multivariable

Control of Electric Power Plants, 1983, pp. 482-503,

generalized predictive control of a boiler system,’’

Como, Italy.

IEEE Transactions on Energy Conversion, vol. 6, pp. 282-288, June 1991. [3] S. Matsumura, K. Ogata, S. Fujii, H. Shioya and H. Nakamura, ‘’ Adaptive control for the steam temperature of thermal power plants,’’ Control Engineering Practice, vol. 2, pp. 567-575, Apr.

9

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 Appendix A. The plant model 1.

dx14 (t ) 2 [ K RHV x7 (t ) − x12 (t ) − x13 (t ) =− dt TRV 2 (u0 (t ))

Turbine/Main steam pressure system

+ x14 (t )]

dx1 (t ) 1 = − [ x1 (t ) − u2 (t )] dt Tf

dx15 (t ) 1 = [ K RHO (u2 (t ))TRTE (u0 (t )) x14 (t ) − x15 (t )] dt TRHB

dx2 (t ) 1 =− [ x2 (t ) − K FWD u1 (t )] dt TFW

dx16 (t ) 2 = [ K RHO (u2 (t ))TRTE (u0 (t )) x14 (t ) dt THR (u0 (t )) − x16 (t )]

dx3 (t ) 1 K u (t )u7 (t ) x5 (t ) =− x3 (t ) + LP 0 [ + 1] dt TLP TLP f 2 (u0 (t )) f1 (u0 (t ))

dx17 (t ) 2 = [ x16 (t ) − x17 (t )] dt THR (u0 (t ))

dx4 (t ) 1 K u (t )u7 (t ) x5 (t ) =− x4 (t ) + HP 0 [ + 1] dt THP THP f 2 (u0 (t )) f1 (u0 (t ))

dx18 (t ) 1 = [ x17 (t ) − x18 (t )] dt TRHT

dx5 (t ) 1 = [ x1 (t ) − x2 (t ) + K FWD u1 (t )] dt TB +

3.

u0 (t )u7 (t ) x (t ) [ 5 − 1] TB f 2 (u0 (t )) f1 (u0 (t ))

dx19 (t ) 2 =− [ x19 (t ) + CSH x3 (t ) + CSH x4 (t ) dt TPR1 (u0 (t )) − CSH (u2 (t ))]

dx6 (t ) 1 = [ K MT (u0 (t )) x1 (t ) − K MT (u0 (t )) x2 (t ) dt TMT (u0 (t ))

dx20 (t ) 1 =− [ x3 (t ) + x4 (t ) + x20 (t ) − u1 (t )] dt TS (u0 (t ))

− x6 (t ) + K FWD K MT (u0 (t ))u1 (t )] − 2.

Boiler/Superheater system

K MT (u0 (t ))u0 (t )u7 (t ) x5 (t ) + 1] [ TMT (u0 (t )) f 2 (u0 (t )) f1 (u0 (t ))

dx21 (t ) 1 =− [ x21 (t ) dt TSD (u0 (t ))

Boiler/Reheater system

− K SD (u0 (t ))TSGD (u0 (t ))u4 (t − LSH )]

dx7 (t ) 1 =− [ x3 (t ) + x4 (t ) + x7 (t ) − u1 (t )] dt TR (u0 (t ))

dx22 (t ) K R1 1 =− [ x3 (t ) + x4 (t ) + x22 (t ) − u2 (t )] dt TR1 (u2 (t )) K R1

dx8 (t ) 1 =− [ x8 (t ) + TRGD (u0 (t )) KGD (u0 (t ))u3 (t − LRH )] dt TGD dx9 (t ) 1 [ K RRH x3 (t ) + K RRH x4 (t ) + x9 (t ) =− dt TRRH (u0 (t ))

dx23 (t ) 2 = [ x19 (t ) − K S x20 (t ) + x21 (t ) dt TPR 2 (u0 (t )) + x22 (t ) − x23 (t )]

− K RRH u2 (t )]

dx24 (t ) 1 =− [ x24 (t ) − K S 1 (u8 (t ))T1SP (u0 (t ))u5 (t )] dt TS1 (u0 (t ))

dx10 (t ) 2 =− [ x10 (t ) + CRH x3 (t ) + CRH x4 (t ) dt TRH 1 (u0 (t ))

dx25 (t ) 2 = [ K PR (u2 (t )) x23 (t ) + x24 (t ) dt TPL1 (u0 (t ))

− CRH (u2 (t ))]

− x25 (t ) − K S 1 (u8 (t ))T1SP (u0 (t ))u5 (t )] dx26 (t ) K RP 1 =− [ x3 (t ) + x4 (t ) + x26 (t ) − u2 (t )] dt TRP (u0 (t )) K RP

dx11 (t ) 2 = [ K RRH x7 (t ) − x8 (t ) − x9 (t ) dt TRH 2 (u0 (t )) + x10 (t ) − x11 (t )]

dx27 (t ) 2 =− [ K1 x20 (t ) − x25 (t ) − x26 (t ) + x27 (t )] dt TPL 2 (u0 (t )) dx28 (t ) 1 =− [ x28 (t ) − K S 2 (u8 (t ))T2 SP (u0 (t ))u6 (t )] dt TS 2 (u0 (t ))

dx12 (t ) 2 = [ K RH (u2 (t )) x11 (t ) − x12 (t )] dt TRV 1 (u0 (t )) dx13 (t ) KTG 1 =− [ x3 (t ) + x4 (t ) + x13 (t ) − u2 (t )] dt TTG (u0 (t )) KTG

dx29 (t ) 2 =− [ x6 (t ) − K PL (u2 (t )) x27 (t ) − x28 (t ) dt TFS 1 (u0 (t )) + x29 (t ) + K S 2 (u8 (t ))T2 SP (u0 (t ))u6 (t )]

10

IEEE Trans. On Control Systems Technology, Vol. 10, No. 2, 256-262, March, 2002 SGD Demand:

dx30 (t ) 2 =− [ K 2 x20 (t ) − x29 (t ) + x30 (t ) − x31 (t )] dt TFS 2 (u0 (t )) dx31 (t ) K RF 1 =− [ x3 (t ) + x4 (t ) + x31 (t ) − u2 (t )] dt TRF (u0 (t )) K RF

u3 (t ) = K RHB K P 3 (u0 (t )) x15 (t ) + x40 (t ) + RGDD (t ) ; RGD Demand: u4 (t ) = −u3 (t ) ;

dx32 (t ) 1 = [ K PL (u2 (t )) x27 (t ) − x32 (t )] dt TPLT

SP1 Demand : u5 (t ) = fV 1 (0.01x32 (t ) + SP1DD(t )) ; SP2 Demand:

dx33 (t ) TSTE (u0 (t )) [ KTP x5 (t ) + K SHO (u2 (t )) x30 (t )] = dt TMSB −

1 TMSB

u6 (t ) = fV 2 (0.01K MSB x33 (t ) + SP 2 DD (t )) ;

x33 (t )

TCV Demand: u7 (t ) = fV 3 [ K P 2 (u0 (t ))( − x3 (t ) − x4 (t ) + u0 (t )) + x39 (t )]

dx34 (t ) 2TSTE (u0 (t )) [ KTP x5 (t ) + K SHO (u2 (t )) x30 (t )] = dt TMS (u0 (t )) −

+ f 2 (u0 (t ))

2 x34 (t ) TMS (u0 (t ))

Turbine outlet steam flow: u8 (t ) =

dx35 (t ) 2 = [ x34 (t ) − x35 (t )] dt TMS (u0 (t )) dx36 (t ) 2 = [ x35 (t ) − x36 (t )] dt TMST 4.

u0 (t )u7 (t ) x5 (t ) + 1] . [ f 2 (u0 (t )) f1 (u0 (t ))

Notice that the inputs defined above are the input signals in the simulation model, and are different to the outputs

PID controllers

used by the predictive controller proposed in this paper.

dx37 (t ) K (u (t )) = − P0 0 x5 (t ) dt TI 0 (u0 (t )) dx38 (t ) K K (u (t )) = − MSB P1 0 x33 (t ) dt TI 1 (u0 (t ))

CSH (⋅) : primary superheater inlet steam temperature; CRH (⋅) : reheater inlet steam temperature;

dx37 (t ) K (u (t )) = − P2 0 [ x3 (t ) + x4 (t ) − u0 (t )] dt TI 2 (u0 (t )) dx40 (t ) K RHB K P 3 (u0 (t )) = x15 (t ) dt TI 3 (u0 (t ))

f1 (⋅) :

relationship between MWD and TP set-point;

f 2 (⋅) :

relationship between MWD and TCV set-point.

Limiters:

Appendix B. Input signals and functions MW Demand : u0 (t ) = MWD(t ) + MWDD (t ) ; FW Demand :

u1 (t ) = u0 (t ) − K P 0 (u0 (t )) x5 (t ) + x37 (t ) ;

FF Demand : u2 (t ) = u0 (t ) − K P 0 (u0 (t )) x5 (t ) + x37 (t ) − K MSB K P1 (u0 (t )) x33 (t ) + x38 (t ) + FFDD(t )

;

;

 −2,  fV 1 ( x ) =  x,  4, 

x < −2 −2< x < 4 ; x>4

 −4,  fV 2 ( x ) =  x, 6, 

x < −4 −4< x 6

 −10,  fV 3 ( x ) =  x, 10, 

11

x < −10 − 10 < x < 10 . x > 10