ELSEVIER

Electrical Power & Energy Systems, Vol. 18, No. 4, pp. 239-250, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0142-0615(95)00067-4 0142-0615/96/$15.00 + 0.00

LQG/LTR robust controller of cogeneration plant for disturbance rejection in electric frequency and steam pressure

B H Kim Plant Division 2, Hyundai Engineering Co. Ltd, 45. Pangi-dong, Songpa-ku, Seoul, Korea 1 38-1 50

Y - M Park, M - S Ch,oi and J -W Lee Department of Electrical Engineering. Seoul National University, 56-1. Sinrim-dong. Kwanak-ku, Seoul, Korea 1 51 -742

A robust controller design is considered for reducing the sensitiveness of electric .frequency and extraction steam pressure to the disturbances imposed on a cogeneration plant by applying LQG/LTR methodology. A sample cogeneration plant that includes the dynamics of the extraction condensing turbine and generator, actuators and valves, governor, and PID controllers is modelled as the 15th-order different&l equation for controller design. As one of the important steps in the LQG/LTR application, the analysis of the natural frequency of the generator swing is introduced to estimate the frequency band of the disturbances due to .the variation in loads, and to set up the design specification of the LQG/LTR controller. In the robust MIMO controller design procedure, the singular value Bode diagram of the sensitivity transfer function matrix is shaped to reduce disturbance effects on the power system frequency and steam pressure, according to the design specification. Case studies are carried out by adding a reduced llth-order controller to the nonlinear model of the cogeneration plant. Simulation results demonstrate the preferable performance of the LQG/LTR controller in disturbance rejection. Copyright © 1996 Elsevier Science Ltd,

Keywords: cogeneration plant dynamics, LQG/LTR, natural frequency oscillation analysis, disturbance rejection

Received 23 March 1995; revised 25 May 1995; accepted 6 July 1995

I. I n t r o d u c t i o n The ultimate goal of a control system designer is to build a system that will work in the real environment. As the real environment may change with time, and because the operating condition may vary due to load changes or other disturbances, the control system must be able to withstand these changes. Even if the environment does not change, the model uncertainty is another issue. The uncertainties of a design plant usually arise because of the linearization of a nonlinear system about an operating point, unmodelled dynamics, sensory actuator noise, and undesired external disturbances on different parts of the system. Accordingly the designer aims to obtain a robust controller that will work with the actual plant in a real situation to achieve the design objectives.

Among many approaches developed for the robust control problem, the linear quadratic Gaussian with loop transfer recovery (LQG/LTR) is preferred, owing to its systematic design procedure in resolving design issues such as stability robustness and the trade-off between performance and allowable control power. Since the methodology was developed by Doyle et al. 1-4, various applications of the methodology to the energy systems 5'6 and aerospace systems 7'8 have been considered.

Engineers operating a cogeneration plant as part of an industrial process plant for simultaneous supply of elec- tricity and steam often have a hard time keeping the power system frequency and extraction steam pressure within acceptable limits because of external disturbances such as abrupt changes in electrical and steam loads. In order to prevent extended operation at frequencies lower

239

240 LQG/L TR robust controller of cogeneration plant." B. H. Kim et al

than normal frequencies during system disturbances, load shedding schemes are commonly employed in the industrial plant 9. Although load shedding can secure the system frequency to normal, it interrupts the continuous operation of process plant loads, resulting in a large loss of production and economy. In a similar manner, the off- nominal steam pressure causes unnecessary steam blow- out at high pressure conditions and shortage of thermal energy supply at low pressure conditions. The engineers involved have made every endeavour to resolve these problems by empirical and physical improvement of the power system protection logic but have not tried to resolve the problems by applying modern optimal control theory. This paper presents the design procedure of a robust multi-input multi-output (MIMO) controller using the LQG/LTR to reduce the sensitivity of the electric frequency and the steam pressure to the external disturbances imposed on the cogeneration plant, and demonstrates the effective performance of the LQG/LTR controller by comparing it with a classical PID controller.

One of the important steps in the application of LQG/ LTR methodology is to determine the frequency band of the disturbance as the design specification of a robust controller. One of the objectives of this paper is to point out that the frequency band is estimated with reference to the natural frequency of the generator swing in the light of the undamped low frequency oscillation of external disturbances, in other words, variation in electrical loads. Another objective is to follow through each design step in detail so as to provide design engineers with an available reference to assist in more complicated applications.

Section II presents the modelling of the cogeneration plant. Section III presents the design specification of the LQG/LTR controller, including the estimation of the frequency band of the disturbances. In Section IV the controller is designed using the LQG/LTR procedure and its performance for disturbance rejection is evaluated. Time domain simulation results and discussions are also given, applying the designed controller to the nonlinear model of a sample cogeneration plant. Conclusions are given in Section V.

conceptional block diagram of the controlled cogenera- tion plant. The mathematical model of the extraction steam turbine and its parameters are physically calculated by fluid dynamics with reference to the manufacturer's design data u . The governor employed is the Woodward 505E type 12. The governor has to reference input adjusters to remove steady state errors of turbine generator speed and extraction steam pressure respectively at the steady state condition. Ultimately the overall cogeneration plant model is represented as the 15th-order differential equation. Figure 2 shows the cogeneration plant dynamics in the form of a transfer function block diagram. The gains and time constants of the PID controllers were empirically given by Ziegler-Nichols tuning 13. The robustness of the closed loop system based on this uncertain model has been demonstrated by dynamic simulations 14.

As Reference 5 suggests, the State Feedback Assisted Control (SFAC) configuration has the advantage of robust- ness in modifying the demand signal for an embedded classical output feedback PID controller. The purpose of this paper is to facilitate the implementation of the LQG/ LTR as an advanced model-based control technique within the established classical control structure ofcogeneration plant and at the same time methodically to achieve performance and robustness objectives 15.

A linearized 15th order cogeneration plant model is represented as (1) and used for controller design and analysis, while demonstrations are given by simulation applying the designed controller to the nonlinear system.

:~p(t) = Apxp(t) + Opup(t) (1)

yp(t) = Cpxp(t) + Dpup(t)

Gp(s) = Cp(sI - Ap)-lBp + Op (2)

yp = [A~ Ap] T

Up = [. lU2] T

where Aw, Ap are the frequency and steam pressure variation in per unit, and ul and u2 are the HP and LP valve inputs.

II. Cogenerat ion plant model The dynamic model includes generator rotor dynamics as well as single extraction steam turbine dynamics, time delay of actuators and valves, ratioers and limiters of high pressure (HP) and low pressure (LP) turbine valves, and speed and pressure PID controllers. Figure 1 shows the

II1. Design specification of robust LQG/LTR controller

II1.1 Specification of LQG/L TR controller An LQG/LTR based auxiliary controller, K(s), is incor- porated into the cogeneration plant dynamics, G(s), to

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , .

Electrical Load Variation

> ' ~ Valve ~ Turbine > Dynamics Dynamics

Auxiliaw Cxmtml Input Steam Load Vadation ~ j

Governer Dynamics

Figure 1. Conceptual block diagram of cogeneration plant dynamics

Frequency Steam Pressure

LQG/L TR robust controller of cogeneration plant: B. H. Kim e t a l 241

, ,

Z

- t1~1- >

>

+

~=

" t O 0 ~ ""

c o

6 6 o 0o Z t -

o . > .

x m ' ~ _ e-

. . . . ~ o

EE ~

• " ' ~ " IE ~. '~ ..~.J5 I ~-15 ' i , '

-~ " • E

( ~ • < ~ . -~

~ . 0 x O 0 0 wo(.~ . - -

Z Ila 4" w~ ~ 0

~; x~.~,,, .,. '-

~4

._~ U -

242 LQG/L TR robust controller of cogeneration plant." B. H. Kim et al

Disturbance d(s)

Figure 3. Block diagram of a M I M O feedback control system

establish a MIMO feedback control system, which is shown in Figure 3. The performance and stability of the MIMO feedback control system can be evaluated by the analysis of the transfer function matrix (TFM) in the frequency domain.

T(s) = G(s )K(s )

S (s ) = ( I + T(s)) -1

C(s) = ( I + T( s ) ) -~ T(s)

Loop TFM (3)

Sensitivity TFM (4)

Closed loop TFM (5)

The performance of command following is achieved when the minimum singular value of loop TFM O-min(T(j~)) >> 1, where w E ~r (~r is the frequency domain where the reference command input has energy and is usually at a low frequency band). The performance of disturbance rejection is attained when the maximum singular value of sensitivity TFM, amax(S(jw)) << 1, where w C f~d (~d is the frequency domain where the disturb- ance has energy and is usually at a low frequency band), and that of sensor noise insensitivity is achieved when the maximum singular value of closed loop TFM Crmax(C(jw)) << l, where w C ~n (~n is the frequency domain where the sensor noise has energy and is usually at a high frequency band).

The specification of the LQG/LTR controller is designed to meet the following objectives.

(1) The classical PID controllers combined with an electronic governor are currently removing the steady state errors of the turbine generator speed and extraction steam pressure in such a manner as to carry out the command following. Therefore the auxiliary LQG/LTR controller must carry out dis- turbance rejection rather than the command follow- ing. Accordingly O'max(S(j~))<< 1, in other words, amax(S(jw)) << 0 in dB in the disturbance's frequency band of interest ~5.

(2) All control loops shall properly maintain reasonable stability margins to guarantee stability in the pres- ence of simplified or unmodelled dynamics and changes in model parameters.

(3) Excessive control input commands shall be avoided so as to prevent nonlinearlities due to control input command.

(4) The LQG/LTR controller shall be formed with an adequately low order model that is sufficient to realize it.

111.2 Estimation of frequency band of disturbances The disturbances imposed on the cogeneration plant usually arise from abrupt variations in electrical and steam loads, both of which cause a variation in electric

> output y(s)

~:: Sensor Noise n(s)

frequency and extraction steam pressure mutually. As the variation in steam load remains at a lower frequency band than the electric load, the disturbance in steam load can be rejected if the disturbance in electric load is rejected. The variation in generator output results from either variations in domestic in-plant electric loads per- tinent to the cogeneration plant or variation in the tie- line power flow between the cogeneration plant and the external power system. The generator output is deter- mined by the torque angle, 6 (in electro radians), and it is varied by the change in power system frequency. There- fore the frequency domain characteristics of disturbances due to the variation in electric load can be understood with reference to the natural frequency of generator system. The natural frequency oscillation of the angle, 6, is analysed by the representation of the 1-generator infinite-bus model, shown in Figure 4.

The second-order synchronous generator model includes the torque relation as follows 16.

= (Vm - :re- TD) (6)

-- ~Vb(W - 1) (7)

where !

Te ~ Pe ---- e Vt sin 6

! e = V t +jX~d i

Pe = Vti cos O

Tm, Te, TD: mechanical, electric, and damping torque in p.u. M~b: accelerating torque in p.u. 6: torque angle in electric radians ~ b = 27rf f : system frequency

The linearization of the equations (6) and (7) results in

Genera tor

Z = 1 .862 + j 5 . 6 6 3

Y = 0.636 - j0.313

Vt

I_LI %? ~ =G+jB

Figure 4. One-generator inf in i te-bus model of a sample cogenerat ion plant

LQG/L TR robust controller of cogeneration plant: B. H. Kim et al 243

equations (8) to (11).

MA~b = ATm - ATe - A T o (8)

AT M = --DMAW , AT E = DEAW (9)

ATm - ATM = -DMAW, ATe = K1A6 + DEAW,

AT n = DAw (10)

( m s 2 - (D M + D E + D)s + WbK1)A6 = 0 (11)

where D M and DE are extra mechanical and electrical damping coefficients respectively. The solution of equation (11) gives

S = [--~n + Jr/-( 1 -- (2)]'~n (12)

Wn -~- V~(WbK1/M), ~n = (DM + DE + O)/2WbM

(13)

where w n is the undamped natural frequency of the mechanical mode in radians per second and ~n is the damping coefficient in p.u. Here the constant K1 is obtained as explained below.

From Figure 4, the current injected from the synchro- nous generator is represented as equations (14) and (15) 16 .

Z i = (1 + Z Y ) V t - V 0 (14)

[ R - R X ] [ I ~ I = [ C I . - C 2 Vd [s in6]

(15) C I = R e ( I + Y Z ) = I + R G - X B

C2 = Im(1 + YZ) = XG + RB

According to the phase diagram for the low-order synchronous generator, the magnitudes of direct axis voltage Vd and quadrature axis voltage Vd are:

[0], [0 (,6, Vq = 1 e q - x~ 0 iq

Substituting (16) into (151) and solving id and iq gives

[ ] [ . -X2 - X ' l L c o s 6 J ' ' id Y d l , V o R 2 [s in6]

(17) iq = ynJeq--Z-~e 2 R,

where

Yd ~- (C1X1 - C2R2)/(RIR2 + X1X2)

Yq ~- (C1R1 + C2X2)/(RIR2 + XIX2)

By linearization of equation (17), the sensitivities of the generator injection currents ia, iq to the torque angle, 6, are represented as

(18)

R 1 "~= R - C2Xtd, X 1 _~ X + C l X q

X 2 "~= X -~- ClX/d, R 2 ~- R - C 2 x q

where 6o is the initial angle. The electric torque of the synchronous machine near

the synchronous speed can be approximated by

Te ~- Pe = idVd + idVq (19)

Substituting Vd and Vq from (16) into (19) yields

T e = iqeq + (Xq - X'd)idi q (20)

Substituting the Aid and Aiq given by linearization of (17) into the linearized result of (20), the coefficient K1 is given by

OTe ( X q - t , . Oid t _ t • K 1 - - ~ - - Xd)lqo - - ~ "4- {eq0 + (Xq Xd)ld0 } Oiq

(21)

The natural frequency wn of the generator for the sample cogeneration plant is calculated to be 4.43 rad/s at the steady state and has a low frequency oscillation mode in the frequency band of 3-5 rad/s. The calculated natural frequency is referred to the estimation of the frequency band of disturbances in setting up the design specification of the LQG/LTR controller as one of the important steps in the LQG/LTR application.

IV. LQG/LTR controller design

IV.1 Design plant model (DPM) The linear model of the cogeneration plant has trans- mission zeros at the left half s-plane and it is a minimum phase plant. However one of the transmission zeros stays at s -- 0. Because of this the singular values of the open loop TFM of the cogeneration plant are very small at the low frequency band. So it is desirable to increase the singular values at the low frequency band by adding integrators to the cogeneration plant 15. Figure 5 shows the comparison of singular values of the plant with and without integrators. The integrator prevents high fre- quency oscillation of the control input to the turbine valve owing to its character of a low pass filter. The integral control action is provided by appending two integrators to the 15th-order cogeneration plant model, one in each control input channel. This results in the augmented 17th-order design plant model (DPM) as follows.

k = Ax + Bu, y = Cx (22)

H where the state variable x = , y = yp. LXpJ

The system matrices A, B, and C are given by:

A = Ap ; B = ; C = [ 0 Cp] (23)

where Ap is a 15 x 15 matrix, Bp is a 15 x 2 matrix, Cp is a 2 x 15 matrix and I is a 2 x 2 matrix.

IV.2 Scaling of design plant model The singular value based stability robustness of the MIMO control system is sensitive to scaling. The cogener- ation plant in normal operation has a range of variation in electric frequency of about ]Aw I = 0.01 p.u. and varia- tion in steam pressure of about lAP1 = 0.1 p.u. While a unity change in IAPI will be insignificant, a unity change in IAwl will be considered a large perturbation. The importance of scaling is demonstrated in Reference 6. As scaling prevents a wide spread in the open loop singular value raised from the physical units being used for the system input and output description, it can make it easier to design the target filter loop (TFL).

244 LQG/L TR robust controller of cogeneration plant. B. H. Kim et al

50

-50

-100

-150 10 .2

SV of Open Loop Transfer Function Matrix GOw)

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

. !iii 2i i iiiiiii ! i !i i :i ii

! i ! i i:,ili i i i i i i li{i2:Plantwithintegrator i ~ i i i ! ~

'. : : ', : : : : : : : ', ,. , , : : , . : ,. : : : ' . : : : ' . : : : : : : : ' , { : = = : 1 : : : : : 1 I l : ~ 1 ~ l I { : I ' , I : = : : : ~ 1 :

i :: :~ i;iiii :: :: iii::iii ! i i!i::i'.i i i i:::~i:::: i :: :: i : , '~i i i i :: i i i : ~ i i l i i i i i : , i : , i :, i i :~ii:,i

10 -~ 100 101 102 Frequency - Rad/Sec

Figure 5. Singular value of open loop transfer function matrix

IV.3 Target filter loop ( TFL ) design The LQG/LTR procedure consists of two steps: first designing the Kalman filter in such a manner that the filter loop satisfies the performance and stability robust- ness requirements, and second recovering this loop asymp- totically by tuning the regulator.

The target filter loop transfer function matrix, GF(S), has the form:

OF(S) = Ccb(s)H (24)

where ,I)(s) = (sI - A) -1 and H(s) is the filter gain matrix. The TFL design problem is to obtain the filter gain

matrix H to meet the design objective of stability robust- ness.

First consider the Kalman filter model as

2 = A x + Bu + L~ ( 2 5 )

y = C x + O

where ( and 0 are zero mean Gaussian white noise processes at the input and output with covariance Q and R(=#) respectively, y are the controlled plant out- puts. The intensity of noise, ~, is unity. The disturbance input matrix, L, and the intensity, #, of sensor noise, 0, are used as design parameters to determine the singular value shape of the target filter loop (TFL) that would meet the desired performance and robustness specifi- cation 15. Equation (26) is the Kalman filter frequency domain equivalent (KFDE). By iteration, the design parameter, L, is varied and determined to obtain the desired loop shape of the TFL, and the design parameter, #, is selected so that Crmax(S(j0.,')) < 0 in dB at the frequency band of disturbances of interest, to satisfy the desired cross-

over frequency as required by the design specification.

,.~ 1 C ( s I - A ) 1 L (26) GF(s)

After satisfactory values of L and # have been obtained, the Kalman filter gain matrix, H, is calculated from the filter algebraic Riccati equation (FARE):

AP + PA T + LL T __1 p c T c P = 0 (27) #

H = 1 p c T (28) #

IV.4 Loop transfer recovery (L TR)

After the filter gain matrix, H, has been obtained, the control gain matrix, G, is calculated through the LTR. This is to calculate the full state feedback regulator gain matrix, G, via the optimal control technique of the cheap control linear quadratic regulator (LQR) problem. The optimal performance measure is given as

J = [x'Qx + utRu]dt (29) o

where Q(Q _> 0) and R(R>0) are a state weight and control weight matrix respectively. The optimal control input is given by

u = - G x (30)

with G = R-lBTK. The control gain matrix, G, is calculated from equation

(31), the control algebraic Riccati equation (CARE), where the weight matrix A = c T c and R = pI the control

r(s) ,~, H > I ~ ( S ) ;:> C Y(: - - - - -~

Figure 6. Structure of target filter loop

LQG/L TR robust controller of cogeneration plant." B. H. Kim et al 245

140

120

m 100 " 0

¢ 8o

-~ so 0 >

h 4o

c

"~ 20

0

TARGET FILTER LOOP

iniinii

-20 I I I l l l 0,01 0.1

Figure 7. Singular value of target filter loop

i

!

i

1 frequency band, rad/sec

" H~,,, \ \

i i , l l

10 100

weight parameter r --* 0. ]'he control gain, G, is iteratively determined by adjusting the design parameter p to recover the singular value of loop TFM to the TFL, so that G(s)K(s) ~- Ge(s ).

KA + AXK + c T c - 1 KBBT K = 0 (31) P

G = _1 Bz K (32) P

The transfer function matrix, K(s), of the LQG/LTR controller is then given by

K(s) = G(sI - A + H C + BG)- lH (33)

and the overall loop transfer function matrix T(s)= G(s)K(s), where G(s) is tlae transfer function matrix of the augmented scaled plant, e.g. C(sI - A)-IB.

During the LTR process, as p comes close to zero, the control inputs, u, increase so much. The control inputs given by the LQG/LTR controller are furnished to the steam turbine valves' actuators through the integrator. There are limits of the valves' openings and closings because of the valves' physical sizes, which impose limits in increase of the control inputs required. During the simulation of the plant compensated by the designed controller, the controller's inputs are managed so that they have relatively low values compared with the governor inputs. On this condition a trade-off should be made to determine the lowest possible p, which is selected to be 0.01 after reviewing the time simulation results of the compensated plant.

IV.5 LQG/L TR controller's order reduction and evaluation The LQG/LTR controller designed in the above section has the form of the 17th-order, the same as the integrator augmented cogeneration plant model. It is desirable to reduce the order of the controller to keep fewer com- ponents or computing resources in its implementation. Various methods for model reduction of the large scale linear systems have been studied and proposed 17. These are, for example, Pade approximations, modal approxi- mation, or continued fraction expansions. In this study the method of optimal Hankel-norm approximations is used 1°. This technique makes it possible to calculate the

achievable error between the frequency responses of the full-order model and any reduced-order model. The technique has an easier computation algorithm than other methods and requires less computational demand. The object function is to minimize the Hankel-norm [[K(s) - K r ( S ) [ [ H where/(r(S) is the reduced r order con- troller. The 17th-order controller is reduced to the 11 th- order model within the extent of keeping the desirable performance and stability robustness of the controller connected plant, G(s)K(s).

The Bode diagrams of singular values c~[T(jw)], a[S(jw)], a[C(jw)] are reviewed and compared for the full-order and reduced-order models, whose Bode dia- grams closely match each other in the frequency band of interest. As shown in Figure 8, the command following performance is evaluated by means of the shape of a[T(ja~)] and it is achieved in the frequency band of interest, to say, O'min[T(ja~)] >0dB at co<5.62rad/s for the full 17th-order controller and at a; < 6.34 rad/s for the reduced 1 lth-order controller.

In Figure 9, cr[S(ja;)] are compared to examine the performance of disturbance rejection. O'max[S(j~)] < 0 dB, say at ~ < 5.62 rad/s for the 17th-order controller and at a; < 6.15 rad/s for the 11 th-order controller. The perform- ance of disturbance rejection is attained in the interested frequency range of 3-5rad/s, where the disturbances caused by load variation are imposed on the cogeneration plant.

In Figure 10, cr[C(ja;)] are compared to check that the performance of noise insensitivity. Omax[C(jco)] is close to zero up to a; -- 10 rad/s for the full-order controller and up to w -- 14 rad/s for the reduced 1 lth-order controller. amin[C(ja;)] is also close to zero up to 13 rad/s for the both orders. Bode plots of g[C(j~)] go down to negative mag- nitudes steeply as the frequency band goes to 10 or more rad/s, which illustrates that the designed LQG/LTR con- trollers of both 17th- and 1 lth-orders preserve immunity to sensor noise with a high frequency bandwidth.

IV.6 Eigenvalue analysis of full and reduced order controller The eigenvalue analysis is carried out from the full 17th- order to the 11 th-order during the order reduction process of the designed LQG/LTR controller. Table 1 indicates the comparison of eigenvalues among the 17th-, 13th-,

246 LQG/L TR robust controller of cogeneration plant. B. H. Kim et al

100

a0 ~ ",,...

i \ m 6 0

¢ 40 ! 03

--~ 20 0

g .-. : o~ -20 I

-40

-6o I ).01

i i i i

II I i

i

OPEN LOOP TFM

I I

i

0,1

!

i

I

i

1

f requency bond, r a d / s e c

\

10

\ \

100

I -- SVmox-full ord. -- SVmin-fult ord. -- SVmax-reduced ord. - - SVmin-reduced ord.

Figure 8. Singular value of loop TFM

~0 ¸

0

- I 0 ra "o - 2 0

>~ - 3 0

--~ -40 >

-50

~ - 6 0

-70 /

-80 /

- 9 0 0.01

/

SENSITIVITY TFM

1

f requency bond, r a d / s e c

i

1111

0.1

t I

LI . TI . I i l . t Ilf

II [ ti it! f Ill

10 I00

[ ~ SVmox-full ord. - - SVmin-full ord. - - SVmox-reduced ord. - - SVmin-reduced ord. ]

Figure 9. Singular value of sensitivity

and 1 lth-order models. The dominant damping coeffi- cients Al, ~2 remain until the 13th-order, however they disappear in the 1 lth-order. Though the 1 lth-order model has less damping effect than the larger-order models, it is beneficial from the viewpoint of actual implementation. So, in this paper, the l lth-order model is chosen to demonstrate the time domain simulation result.

IV.7 Time domain simulation

The above LQG/LTR controller design relies on the linear models taking advantage of its well proven

theory of linear systems. The linear controller design allows quick initial evaluation of the stability robustness of the closed loop system. However the initial evaluation is not sufficient for confirming whether the designed controller will behave satisfactorily when implemented in the real plant that is actually nonlinear and uncertain. The reduced 11th-order model of the LQG/LTR con- troller is applied to the 15th-order nonlinear model of the cogeneration plant and tested in the time domain for nonlinear verification using the numerical analysis technique 18.

LQG/L TR robust controller of cogeneration plant: B. H. Kim et al 247

-10 °

i:> -20

o

> - 5 0

3

c -40

-50

-60 0.01 0.1

10

CLOSED LOOP TFM

I 10 1oo

frequency band, rod/see

l ~5Vrnox-fu! ord. ~SVrnin-ful ord. - - 5Vrnox-reduced ord. - - SVmin--reduced ord.

Figure 10. Singular value of closed loop TFM

Table 1. Eigenvalues of fidl and reduced order controllers

Eigenvalues Full 17th order 13th order 1 l th order Ai (multiply 100) (multiply 100) (multiply 100)

A l -- 1.3144 -- 1.4060 Disappear A2 - 1.0861 - 1.4060 Disappear A3 -0.9909 - 0.6961i -1.0230 - 0.6836i -1.1478 - 0.6122i /~4 -0.9909 + 0.6961i -1.0230 + 0.6836i -1.1478 + 0.6122i As -0.6949 - 0.3879i Disappear Disappear )% -0.6949 + 0.3879i Disappear Disappear A 7 -0.3780 - 0.4117i -0.3852 - 0.3735i -0.4390 - 0.3348i A8 -0.3780 -t- 0.4117i -0.3852 + 0.3735i -0.4390 + 0.3348i ~9 -0.3420 - 0.8625i -0.3426 - 0.8621i -0.3463 - 0.8619i A~ 0 -0.3420 + 0.8625i -0.3426 + 0.8621i -0.3463 + 0.8619i All -0.1081 Disappear Disappear A12 -0.0631 - 0.5361i -0.0631 - 0.5361i -0.0629 - 0.5362i A~3 -0.0631 + 0.5361i -0.0631 + 0.5361i -0,0629 + 0.5362i /~14 -0.0424 - 0.2877i -0.0423 - 0.2876i -0.0430 - 0.2870i At5 -0.0424 + 0.2877i -0.0423 + 0.2876i -0.0430 + 0.2870i LI 6 -0.0231 Disappear Disappear A17 -0.0001 -0.0001 -0.0001

(1) Case 1: unit step change in electric load This simulates the case that the stepwise change in electric load demand from 66.7% to 100% is applied at t = 0.5 s to the cogeneration plant including the conventional PID controller and to the same plant compensated by the designed auxiliary LQG/LTR controller. As indicated in Figure 11, the maximum variation in the electric fre- quency is reduced from -0.65 Hz at the 60 Hz system without the auxiliary controller to -0 .3 Hz with the auxiliary controller, which affords an acknowledgeable improvement of 0.35 Hz in frequency variation. It should be emphasized that the improvement of 0.35 Hz in the industrial power system can significantly remove the nui- sance load shedding of essential loads and unnecessary interruption of production in the process plant industries.

Meanwhile the maximum variation in extraction steam pressure is decreased from +9.6% to 0.9%, resulting in an improvement of 8.7% being gained.

(2) Case 2: unit step change in steam load This simulates the stepwise change in steam load from 56.4% to 100% being applied at t = 0.5 s. As shown in Figure 12, the maximum frequency deviations without and with the LQ G /LTR controller are -0.020 Hz and -0 .014 Hz respectively, while the extraction steam pres- sure deviations are - 3 . 8 % and - 1 . 0 % respectively. 2.8% is improved in the steam pressure, although the frequency improvement is negligible, 0.006 Hz.

(3) Case 3: sinusoidal variation in electrical and steam load In this test sinusoidal variations of 0.667 + 0.3 sin (4t) p.u.

248 LQG/L TR robust controller of cogeneration plant." B. H. Kim et al

0.2

0,1

0

-0.1

°~ -0.2 o"

& -0 .3

~ -0 .4

-0 .5

-0.6

FREQUENCY & PRESSURE VARIATION

-0.7

time in second

I - - freq. w/control - - freq. w/o control - - pres. w/control

Figure 11. Time response of stepwise disturbance in electrical load

0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0,02

-0.01

-0

--0.01 1'4 15

- - pres. w/o control !

.E

.=

0;

0.015

0.01

0 .005 3 :

~ o

g ID

-0 ,005

~o o. -0.01

-0 .015

- 0 , 0 2

FREQUENCY & PRESSURE VARIATION -0.005

- - 0 . 0 0 5

g --0.ol ~

Q.

- -0 ,015 .E

- 0 . 0 2 ~

- 0 , 0 2 5 E °

- - 0 .03

- -0 .035

' -0 .04 2 ; 1; 17 1'4 ,6

time in second

I - - freq. w/control - - freq. w/o control - - pres. w/control - - pres. w/o control 1 )

Figure 12. Time response of stepwise disturbance in steam load

in electric load and 0.564 + 0.3 sin (t)p.u. in steam load is imposed at t = 0.Ss on the cogeneration plant. As indicated in Figure 13, the LQG/LTR controller reduces the system frequency from - 0 . 5 9 H z ± 0 . 7 4 H z to - 0 . 2 8 H z + 0 . 3 2 H z and the steam pressure from -2 .5+2 .5% to -0.2% 4-0.2%. The frequency of 0.30Hz/0.42 Hz and steam pressure of 2.3%/2.3% are respectively improved in the negative/positive direction. This improvement will provide significant room on both

system frequency and steam pressure to the engineers who suffer from external disturbances.

V. C o n c l u s i o n s In this paper a multi-input multi-output (MIMO) robust controller using the LQG/LTR methodology is designed for the cogeneration plant, with the objective of improv- ing the electric frequency and extraction steam pressure

LQG/L TR robust controller of cogeneration plant: B. H. Kim et al 249

0.8

0.6

0.4 -r

0.z

g

-0 .2

-0.4

-0.6

FREQUENCY & PRESSURE VARIATION

i

/ t / /A/1/ / t /

J

12 14 flme in second

0.03

-0.02

0.01

"0

"-0.01

- 0 , 0 2

-0.03 16

I - ~ freq, w/control - - freq. w /o control ~ pres. w/control - - pres. w /o control I

Figure 13. Time response of sinusoidal disturbances in both electrical and steam load

response which is the prime concern of engineers who are involved in the design anti operation of the cogeneration plant. The application of the methodology to the cogener- ation system control is discussed in detail to provide a reference for design engineers who are involved in similar problems.

As one of the important steps in the L Q G / L T R application, the analysis of the natural frequency of the generator output is introduced in order to estimate the frequency band of the disturbances due to variations in loads and to set up the design specification of the LQG/ LTR controller.

A state space mathematical model is calculated at the 15th-order, taking an example of a real cogeneration plant. A 17th-order controller is designed for the 17th- order design plant model augmented with two integra- tors, and then reduced to an 1 l th-order controller using the Hankel-norm approximation method to enable its practical implementation. For nonlinear verification, time domain simulations are performed combining the 1 lth- order controller as an auxiliary controller to the 15th-order nonlinear cogeneration plant model including embedded classical PID controllers. Simulation results demonstrate that the L Q G / L T R controller significantly improves the performance of electric frequency and extraction steam pressure in the cogeneration plant against abrupt changes in loads.

VI. References 1 Doyle, J C and Stein, G 'Robustness with observers' IEEE

Trans. Aurora. Control Vol AC-24 (1976) pp 607-611

2 Doyle, J C and Stein, G 'Multivariable feedback design; concepts for a classical/modern synthesis' IEEE Trans. Aurora. Control Vol AC-26 (1981) pp 4-16

3 Athans, M 'A tutorial on the LQG/LTR method' Proc. ACC (1986)

4 Athans, M Lecture notes on multivariable control systems MIT (1986)

5 Ben-Abdennour, A, Edwards, R M and Lee, K Y 'LQG/LTR robust control of nuclear reactors with improved tempera- ture performance' IEEE Trans. Nuclear Science Vo139 No 6 (December 1992)

6 Garg, S 'Turbofan engine control system using LQG/LTR methodology' Proc. ACC Vol 1 (June 1989)pp 134-141

7 Athans, M, Kapassouris, P, Kappos, E and Spang III, H A 'Linear quadratic Gaussian with loop recovery method- ology for the F-100 engine' AIAA Journal of Guidance, Control and Dynamics Vol 9 (January-February 1986) pp 45-52

8 Ridgley, B D, Banda, S S, McQuade, T E and Lynch, P J 'Linear quadratic Gaussian with loop recovery method- ology for an unmanned aircraft' AIAA Journal of Guidance, Control and Dynamics Vol 10 (January-February 1987) pp 82-89

9 Kundur, P Power system stability and control McGraw-HiU (1994) pp 444-448, pp 623-626

10 Glover, K 'All optimal Hankel-norm approximations of linear multivariable systems and their L~ error bounds' Int. J. ControlVol 39 No 6 (1984) pp 1115-1193

11 Kawasaki Heavy Industries Ltd Mathematical model of single extraction turbine Hyundai Oil Company, Refinery Construction Project (June 1986)

12 Woodward Governor Company Woodward 50510B governor manual, Application note 43027 Electronic controls for steam turbine applications (1978)

13 Zeigler, J G and Nichols, N B 'Optimum setting for auto- matic controllers' Trans. A S M E Vol 64 (1942) pp 759-768

14 Park, Y M, Kim, B H and Cho, B W 'A dynamic modeling and simulation of cogeneration plant' Proceeding of Inter- national Symposium on Electric Power Business for Korea Electricity Centennial, Korea Electric Power Corp. (November 1987) pp 226-238

250 LQG/L TR robust controller of cogeneration plant." B. 14. Kim et al

15 Maciejowski, J M Multivariable feedback design Addison- Wesley (1989) pp 222-264

16 Yu, Y N Electric power system dynamics Academic Press (1983) pp 65-94

17 Jamshidi, M Large scale systems modeling and control North-Holland series in System Science and Engineering (1982) pp 66-102

18 Shahian, B and Hassul, M Control system design using Matlab Prentice-Hall Intl. Ed. (1993) pp 366-394

Appendix Constants and parameters of the study system are avail- able upon request, which include system matrices of the cogeneration plant model, the designed controller and the reduced controller, the scaling matrices, and the design parameters (#, p, H).