Modern Control Theory Applied to 200 MW Boiler-Turbine Unit Control

Copyright © IFAC Power Systems and Power Plant Control, Beijing, 1986

MODERN CONTROL THEORY APPLIED TO 200 MW BOILER-TURBINE UNIT CONTROL Wei-yong Xu and Hua-guang Zhang Department of Power Engineering, Northeast Electric Puwer Institute, Jilin, Chilla

Abstract. Four kinds of control design schemes for 200MW boiler-turbine unit are presented in this paper and they are (1) Smith-PID control, (2) Stochastic optimal control, (3) Optimal control using extended observor, and (4) The use of Linear Programming method. Analog-digital computer simulations are obtained using each of these control laws for comparison. Mathematical model, Smith-PID control, Stochastic optimal control, Extended observor, Linear Programming, Analog-digital computer simulation.

~eywords.

INTRODUCTION This paper deals with modern control theory applied to 200MW boiler-turbine unit control and an analog-digital simulation is discussed. Usually the control of boiler-turbine unit is based on the classical control the ory. As a result of successive construction of large capacity steam power plants, the steam power plants take over a large portion of rather quick load change. In this case, the application of modern control theory to find a new corttrol law for. ste.am po"er plant will become more important with each passing day.

Fig. 1 and the block diagram of input and output signals is shown in fig. 2.

In China more and more 200MW boiler-turbine units are installed and put into survice. Table 1 lists the principal specific ations of the boiler and turbine.

Fig. 1. The conceptual diagram of 200MW boiler-turbine unit in power plant 1. economizer, 2. downcomers, 3. waterwalls, 4. drum, 5. front platen, 6. pr imary spray desuperheater, 7. back platen, 8. steamsteam type reheater attemperater, 9. primary superheater, 10. secondary spray desuperheater, 11. secondary superheatec, 12. main steam pipeline, 1 3 . high pressure turbine, 14. primary reheater, 15. spray attemperator, 16. secondary reheater, 17. reheated steam pipeline, 18. intermediate and low pressure turbine, 19. condenser.

TABLE 1 Specifications of boiler and turbine

-----

Boiler Type MG 670/140-V drum, reheat type Full-load steam flow 670 Tons/Hour Superheater outlet steam Qressure 13720 KPa Superheater outlet steam temperature 0 540 C Reheater outlet steam temperature 540 0 C Fuel Pulverised fuel Manufacturer Haerbine Boiler Industry Turbine Type Tandem, 3 cylinders, 3 flows, reheat type steam condition 13720 KPa, 540'C at main stop valve 2500 KPa, 540·C at reheat stop valve Output 200MW R.P.M. 3000 Manufacturer Haerbine Turbine Industry

200MW unit is a typical example of multivariable system. Control loops within the boiler process show sign ificant mutual interaction. Under the conventional PID controller, it is not easy to fully compensate for these interactions and adjust the plant to fast and large load change with the controlled variables kept within the prescr ibed ranges. The diff iculty of controlling a mutually interact ing mul t ivariable system has been one of the principal factors that set the limit to the

The conceptual diagram of 200MW boilerturbine system in power plant is shown in

361

362

Wei-yong XU and Hua-guang Zhang of computer, the state equations of continuous-date system must be transited to the discrete form. From the point of view of boiler control, the turbine governor valve's position is regarded as reference disturbance, so the mathematical model can be devided into two parts-the main model and the disturbed model. The discrete state equations of main model are given by

Fig . 2. Block diagram of mul t ivar iable boiler-turbine system ~l--governor valve's position, W,--spray flow of primary desuperheater, W2--spray flow secondary desuperheater, Sr--reheated steam flow of reheater attemperator, B--fuel flow, Ps--superheated steam pressure, Ts--superheated steam temperature, Tp--steam temperature of platen, Tr--reheated steam temperature .

response of a steam power plant to the load changes required for the load-frequency control of an electric power system. The application of modern control theory presented in this paper is a step in the direction of solving this difficulty. Four kinds of control design schemes for 200MW boiler-turbine unit are discussed in this paper and they are (1) Smith-PlO control, (2) Stochastic optimal contrl, (3) Optimal control using extended observor, and(4)The application of Linear Programming method. For simulating and comparing these closedloop control systems an analog-digital computer simulation technique is adopted in which the analog computer is used for simulating the boiler-turbine unit and the digital computer is used for implementing one of these control laws. Fig. 3 shows the analog-digital computer simulation system .

X(K+l)=AX(K)+BU(K) Yl(K)=CX(K)

( 1) (2 )

where the control vector U(K) is (3 ) U(K)=(Wl,W2,Sr,B) and X is l2xl matrix, A is 12x12 matrix, B is 12x4 matrix, C is 4x12 matrix , Yl is 4xl vector.

The discrete state equations of the disturbed model are given by X(k+l)=AX(k)+Bu(k) Y2(k)=CX(k)

(4 ) (5 )

where the control vector u(k) is

u(k ) = ).LT

(

and X is 6xl matrix, A is 6x6 matrix, B is 6xl matrix, C is 4x6 matrix, Y2 is 4xl vector.

6)

The total output vector is Y(k)=Yl(k)+Y 2 (k) where Y(k)=(Ts,Tp,Tr , Ps)

(7 ) (8 )

PERFORMANCE INDEXES Under normal operating condition in case of a steam power plant , it is required that the pressure and temperature of superheated steam and the temperature of reheated steam be kept within specified values about their rated values. This is necessary to maintain plant efficiency and long life of equipment. The design of control scheme is to find an optimal control law such that the above output parameters are kept within the prescribed values as the load demand changes from power system. The optimal control design is aimed at obtaining a system that is the best possible with respect to a certain performance index or design criterion. Two performance indexes are chosen for comparing control schemes. One is the quadratic performance index for the linear regulator problem, J 1 = l' Y( K ) T Qk Y( k) + "[ U( k ) T Rk U( k) ( 9 )

k·'

Fig. 3. Analog-digital computer simulation system. MATHEMATICAL MODEL A linearized mathematical model of 200MW, drum-type, reheat boiler-turbine system is developed which is suitable for control system analysis and design by considing small perturbations around a steady state operating point . The thermodynamic equations are usually derived in continuous form . Since the application of modern control theory is based on the digital computer control, and for saving the CPU time

ko,

where Qk and Rk are the weighting matrixes We choose Qk=diag(5, 3, 5, 4) and Rk=I. Another is the absolute error criterion which is used for estimating the overshoot of output variables, J2= 0<1.<00 max ICkY(k)1 where Ck is the weighting vector,

(10)

Ck= ( 5, 3, 5, 4) . SMITH-PlO CONTROL As is well known, the Smith-predictor is suitable to time-delay process and easy to implement with computer control. A block diagram of Smith-PlO control is shown in

Modern Control Theory Fig. 4. Assume that the predictor compensates the time delay of process, the inner loop transfer function will be a constant Ki (Fig. 4a), so the transfer function of predictor Gpr(z) is GpdZ)=Ki-Gp(z)

(11 )

where Gp(z) is the transfer function of process. Usually the predictor is connected as feedback (Fig. 4b). In order to eliminate the steady-state error and obtain a better transient response, the regulator Gr(z) is still selected to be PID type. The following equations are written directly from Fig. 4b. El(Z)=R(z)-Y(z)

( 12)

E2(z)=(Ki- Gp(z»U(z)

(l3 )

E(z)=El(Z)-E2(Z)

( 14 )

KiT(z+l) U(z)=(K p +

Kd(z-l) +

2(z-1)

)E( z)

Tz

The parameters Kp , Ki, Kd is determined by means of method. Fig. 5 shows the oi Smith-PID control when turbine valve's position

(15)

of PID regulator optimum seaching process outputs the change of A.1!T=15%.

363 X(k+l)=AX(k)+BU(k)+V(k)

(16)

Y(k)=CX(k)+W(k)

(17)

where V(k) is the plant driving noise and W(k) is the measurement noise. Assume further that noise V(k) and W(k) are both white, Gaussian with zero mean and of known covariance, i.e. E[V(k)VT(j»)= OeOkj

( 18)

E[W(k)WT{j»)= Reb~

( 19)

We choose Oe=I,Re=I, Okj is Dirac-function. The block diagram of stochastic optimal control system is shown in Fig. 6. The control system design so called LOG problem is baSed on (1) deterministic perturbation control (linear quadratic control), (2) stochastic state estimation (Kalman filter), and(3) linearized stochastic control and it leads to an OVerall colsedloop optimal control system. According to the separation theorem, the design steps for time-invarient system are expreSSed by the following. Step 1. Optimal control problem calculation. Solve the following equations, U(k)=-K(k)X(k) (20) K(k)=Rk+BTS(k+l)R)-lBTS(k+l)A (21) S(k)=ATS(k+l)A-AT S(k+l)B(BTS(k+l)B+Rkl l *BTS(k+l)A+O K

(22)

where K(k) is the feedback gain matrix, S(k) is the Riccati gain matrix. When k=n. S(n)=O. (. )

(b)

v

Fig.

k)

W( k)

4. The blocks diagram of Smith-PID control system

T

n I(pa

'c

400

2.0

300

1.5

201)

1J )

100

O.?

o -lOO - 0 . ·,

(1<. )

. _~6~O~O~__~~__~~O \. t,Sec s

-200 _1. tl -300 -1. '-'

Fig.

5. Process outputs of Smith-PID control when A~T=15%

Fig. 6. The block diagram of stochastic optimal control system Step 2. Optimal filting problem calculation. Solve the following equations, X(k)=X(klk-l)+Ke[Y(k)-CX(klk-l»)

STOCHASTIC OPTIMAL CONTROL It is assumed that the unCertain disturbanCeS in boiler-turbine unit are of random nature and so it is considered to be white noise. Then this stochastic noise can be included in our state-variable model with inputs. Also, the measurement noise is considerd to be random and white. Define the linearized stochastic model by

(23 )

X(klk-l)=AX(k-l)+BU(k-l)

(24 )

Ke(k)=P(klk+l)CT(CP(klk-l)CT+Re)-l P(k+llk)=AP(klk-l)AT-AP(klk-l)C T *

(25)

*(CP(klk-l)CT+R e ) *cP(klk-l)AT +Oe (26) p(lIO)=APoAT+O e (27) where Ke(k) is the filter gain matrix, P(k) is the Riccati gain matrix. When k=O, P(O)=Var X(O).

364

Wei-yong XU and Hua-guang Zhang

The process outputs of the LQG control is shown in Fig. 7 when /l)J.T;15 %.

Let

X* (k);

[ Ee ( K )]

A* ;

b

Y( k)

Bl I

C * ;(C 0) T

'c

Kpa

?O

400 30()

E(k);C * X* (K)

1. 5

?OtJ

1. 0

100

0.5

(29 )

We obtain the state equation X* (k+1);A * X* (k)+B * U(k)

( 31)

The corresponding state regulator is U(k);K * X* (K) Where K* ;(K

(30 )

(32)

I)

0 f')')

_1 0')

-

) .

)

- .")')

-1 . 0

- 3D')

-1. 5

Fig.

l ,):):J t , .... f' C

7.

Process outputs of LQG control system when A.J!,r ;15 %

In the LQG c ontrol the filter estimates the state variables and the control law is based on the linear quadratic criteria . Since state feedback variables don't have the intergrat component, they do not guarantee in general that the steady-state errors can be eliminated. We c an imagine that any deterministic disturbance W(k) is generated by certainty initial values ViOl and r( O) which apply on a suppositional disturbance model (Fig. 8). Define a error vector Ee(k), i.e Ee(k) ; V(k)-X(k). Thus, the state equation of whole model is

r '-'- -

Authores propose a method for design the extended observor. Decompose the observor gain matrix

H*;[~~l

OPTIMAL CONTROL USING EXTENDED OBSERVOR

[E e (k+1)] ;[A Bl [Ee(k)j _[BjU(k) lr (k+1) 0 I i (k) 0

In practice, not all the state variables are accessible, it is necessary to observe the states from information contained in the output as well as the input v ariables. The next step is the design of extended observor that produces an approximation to the state vector.

(28 )

( 33)

The calculation of matrix Ht is in a manner similar to design common observor. But the characteristic equation is

IZI-A+BK+H~cl;o

(34)

In order the responce of Ee to be deadbeat, the characteristic equation roots are all at z;O, so the matrix H7 is obtained. The objective of design H2* is to reconstruct r(k) which is approaching to r(K) as fast as posible . However it is impossible in practic , because r(k) has to lag one beat behind r(k). Another reason is that the disturbance usaully includes the stochastic noise. Assume r(k);~r(k), the following characteristic equation is obtained

I ~Z-~I-H2 CBI;0 Let Z;O, we get the solution of Hi

-

H~;-C(CB)-l

(35) (36 )

Table 2 is recommended for choosing ()( which is obtained by means of several different process simulation. In this case Tmin is the smallest time constant of disturbed model. TABLE 2 Values of

Fig.

~

8 . The block diagram of optimal control system using extended observor.

Let's divide the control vector U( k) two parts,

into

U(k);U l (k)+U 2 (k) If U1(k);r(k), the influence of r(k ) to Y(k) can be compensated, so it is regard as forward control . U2 (k) is the feedback control, the feedback gain matrix K(k) is determined by solving the linear quadratic control problem discussed bef ore .

Notice that the suppositional disturbance model is nut necessary for the application of control scheme. All we need do are design extended observor and controller . The process outputs of the control system with extended observor is shown in Fig. 9 when A)J.r; 15%.

365

Modern Control Theory

As is well known, the control variables treated in LP problem must be positive, while those in the actual system can not Rlways be kept positive. To solve this problem we express the control variables by the following equation

T

P Kpa

C

400

?o

300

1. 5

200

1.0

100

0.5

o

0

U(k);U+(k)-U-(k)

(44)

where U+(k)~O, U-(k)~O and the object function of LP problem is expressed by (C:U+(j)+C~U-(j»~ min J J

1000

t,sec

-100 -0.5

(45)

Then , the problem of LP will be expressed

-200 - 1 . 0

[N - N I] -300 - 1.5

[~o:]=(L)-(M)JLT

(46)

where I is unit matrix. Fig. 9 Process outputs of optimal control system with extended o bser v or

The specified limits of U(k) are defind O$U+(k)~U+

(k)

(47)

O~U-(k)~U~ax(k)

(48 )

max

APPLICATION OF LINEAR PROGRAMMING The problem of finding optimal control can be formulated by Linear Programming ( LP ) in the following way. Ths state equations of disturbed model

P

T C 2.0

X(K+l)=AX(K)+B~

(4 )

Kg" 4 ()

Y (K)= CX(K) 2

(5 )

300

1. 5

200

1.0

1()()

0.5

0

0

- 100

-" . :,;

or ( 37 )

The state equations of main model X(K+l)=AX(K)+BU(K) Y1 (K)= CX(K)

or

Fig. 10. shows the process outputs of optimal control system by applying LP method when 61lT= 15 %.

( 1) (2 )

800

1000 t , Sec

- 200 -1. '1

n-'

1 k

Y1 ( n ) = L CA n - - BU(k) k
(38 )

- 300 _ 1 . ;

The total output

Y(n)=Yl(n)+Y2(n) =( 'rCAn-l-kln.LtT+

r""'

CAn-l-kBU(k) ( 39 ) The equations from time point n=O to n=m are collectively expressed as koO

koo

Y (1) Y( 2) Y( 3)

CB C(AB+B)

Y(m)

Cr'Am-1-k

CB CAB CB E(A2~+AB+B) V+ CA2B CAB

CB . ... .. .. . ...

...... . . . .

B

U( 0) U( 1) U(2)

Clt'B CA·'B· ·CB U(m-l)

1<--0

or stated in matrix form Y=M*V+N*U

(40)

...,

The equation (39) can be Y(m)=M(m)JiT+ rN(m-k)U(k) (41) m k:."o where M(m)=!'CA"'"",-kg and N(m-k)=CA",+k B

"".

If we want to keep the output deviations at t=m within L(m), then the last equation must be satisfied or

-L(m)~ M(m )V+ '1:' N(m-K )U( k)H( m) •• 0

10-,

Fig . 10 . The process outputs of optimal control system by applying LP method when AllT z 15 %.

M(m)V+ t,N(m-k)U(k)+tJ o(m)=L(m)

(42)

where l'1 o(m) is a 4x1 slack variable vector , providing the permissible output deviations. t!o(m) is defined by O ~LlJm) ~ 2L(m) (43)

CONCLUSION Four different optimal control laws are designed for 200MW boiler-turbine unit. The results of the comparison are summarizeJ i~ Fig. 11 and Table 3 . Fig . 11 shows t:le superheated steam temperature curves while A.lLT= 15 %. Table 3 lists the values of performance indexes Jl and J2' It can be concluded that the system response in case of optimal control with ex tended observor is better than others. The performance of LQG control is not as good as we will. The reason is that the noise covariances a re choosen unsatisfiedly. Evidently the performance indexes of conventional PID control is the worst . The next step we are planning to do is to implement the foregoing control schemes at a 200MW boiler-turbine unit of Northeas t Flectricity Generating Board . The conventional analog PID control is already mounted and used for the boiler-turbine unit. In this case for implementin g optima l cont r o l we want to adopt a system name d ADC (Anajog Digital Control) in whi ch a d igital comp uter works in conjuc -

366

Wei-yong XU and Hua-guang Zhang

TABLE 3 Comparison b Etween indexes Control law

The following eq uati on s are written dirEctly from Fig. 8.

J1 403

0.6

601

2.2

V(k+1)=AV(k)+Br(k+1)

35~

0.9

f(k+1) = r(k)

(59)

LP

431

2.1

W(k)=CV(k)

(60)

PID

100

2.5

Smith-PID LQG With ExtEndEd obsErvor

J2

f(K+1) = r(k)+Ar(k + 1 )

"*

A ~r(k+1)=H2(W(k)-CX(k ) -CE e(k))

(56) (57) (58 )

so r(k+1)=r(k) +H 2 ( W(k )-CX(k) -CE E (k))

A * n k) +H2( * CAV( k-1) -CX( k) =n k) +H2CB

p

-cEe(k))

'I'

Kpa

·c

400

;> . 0

30' )

1.~

?OO

1 . '1

10 "

'.5

0

0

* * =r
zr(z)=r(Z)+H2CBr(Z)+H~F2(Z) 600

00 l, f,ec

(~Z-~I-H~CB)r(Z)+HZF2(z)=0

(63)

The characteristic equation is

I~Z-O(I-H2CBI=0

extended observor

- ?O!) -1.0

Let Z=O, we get H2* from E~ (35 ) HZ=-(X(CB)

_ ·nf) -1 . 5

11.

(62)

Let r(k) = ~r(k), equation (62) becomes

-1 00 -0 . 5

Fig.

(61)

where F 2 (k)=CAV(k-1)-CX(k)-CE e (k) Taking the Z-transform on both side of equation (61), WE have

REsponsE curVES of superhEated stEam tEmpErature for each control law.

tion with thE convEntional analog PID controllers. A 16-bit microcomputer will be seleted to pErform the task of optimal control. Of course thE theoretical mathematical modEl should be modified according to the actual operating condition fo 200MW boiler-turbine. The weighting matrixes Qk and Rk must bE choosen carEfully and bE examinEd by observing the response characteristics of real boilEr-turbine unit. APPENDIX Proof the expression of calculatin~! The following equations are written dirEctly from Fig. 8.

AE

E (k+1)= HiAE(k) AE(k)= W(k)- Y(k)- CEe(k) r(k+1)= r(k)+ Or(k+1)

(49) (50) (51)

U(k)=U1(k)+U 2 (k)=KE e (k)+r(k)

(52)

Ee (k+1)= Aie(k)-BU(k)+AEe(k+1)+B~(kI53)

SO~

A

*

Ee(k+1)= (A-BK)E e (k)+H1(W(k)-Y(k)-

-CB (k) )=(A-BK-Hic)Ee(k)+HiF1(k) e (54) whEre F1(k)=W(k)-Y(k) Taking the Z-transform on both sidE of equation (54), we havE Ee(Z)=[zI-A+BK+HiCj-1HiF1(Z)

(55)

The charactEristic equation of this systEm is IZI-A+BK+HiCI=O Let z=O, we get Hi from Eq (34).

( 34 )

..

Proof the expression of calculating H,

(35 ) (36 )

REFFRENCES Athans, M. (1971). The role and use of the stochastic 1inear-quadratic-Gaussian problem in control system dEsign. IEEE Trans. Autom. Control, Ac-16, 529 -552. Isermann, R. (1977). DigitalE rEgElsystEms. Springer-Ver1ag, Berlin . Kuo, B. (1980). Digital control system . HRW, NEW York . Li, T. (1985). A simulation model for the power generating unit of thermal power stat ion. ProcEEdings of intErn at ional conference on industrial porcess mo del1ing and control, Vol. I, Hangzhow, pp. 206 212. Nakamura, H. and H. Akaike (1981). Statistical identification of optimal control of supercritica1 thErmal powEr plants, Automatica, Vol. 17, No.1, 143 -155 . Tysso, A. and J.C.BrEmbo (1978) . Installation and operation of a mu1tivariable ship boilEr control system . Automatica, Vo1.14, 213-221. Uchida, M. , H.Nakamura and K. Kawai (1981), Application of linEar programming to thermal power plant control. Proceeding of 8th IFAC world congrEss, Kyoto, pp. 3033 3038 Wa1lace, J. and R Clarck (1983). ThE application of Ka1man filtering Estimation techniques in power station control systems, IEEE Trans. Autom, Control, Ac-18, 416 427 . Xu, Wand Y.P Kakad (1985). RobustnESs of optimal control system for once-through bolier. Proceeding of international conferEnce on industrial prOCESS modelling and control, Vol. 11., Hangzhow, pp. 205 218. Xu,W and Y.P . Kakad (1983). Optimal control of once-through boiler. Proceeding of 15th southeastErn symposium on system theory. Huntsvi11e, pp. 205-218 .