Robust nonlinear slab temperature control design for an industrial reheating furnace

European Symposium on Computer-Aided Process Engineering - 14 A. Barbosa-P6voa and H. Matos (Editors) 9 2004 Elsevier B.V. All rights reserved.

811

Robust Nonlinear Slab Temperature Control Design for an Industrial Reheating Furnace Hatopan Sibarani and Yudi Samyudia* Department of Chemical Engineering McMaster University, Hamilton, Ontario L8S 4L7 Canada

Abstract In this paper, we develop a nonlinear model-based controller for optimally tracking a prespecified slab temperature trajectory of a finite time reheating process. The slab temperature controller is guaranteed to have a robustness property to cope with the uncertainty. We validate, and then compare, the nonlinear controller with the current industrial practice by means of simulation studies utilizing sets of plant measurement data. Simulation results are discussed to gain insights for an improved operation of the reheating furnace. Keywords: nonlinear control, reheating furnace, slab temperature

1. Introduction A walking beam reheating furnace (see Fig. la) represents one of the energy intensive units in the steel industries. For this type of furnace, a sequence of slabs walks through all heating zones with a dynamic pace rate before each slab is extracted from the furnace for further processing in the mill section. A properly heated slab is therefore necessary for achieving a desired final product as well as optimising the fuel consumptions and production rate.

[................................. 1 [ ~ ;,con~o;1T~. [_iL;ve':'::t_j~;;,;;-t -~ .! t i [i= c~176 F1 Furn~c~tr~

[

t

.i

th.

i I Temperature Estimator t~j jl .... ! I

I

Fig. la. A reheating furnace

Level -2 control systems | [. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J

Fig.lb. Industrial furnace control system

In the operation of reheating furnaces, one of the important variables to monitor and control is slab temperature; however there are no reliable sensors to measure it directly for feedback control. Moreover, the slab temperature control problem is complicated by the variations in the production line (e.g. size, steel grade), which create some uncertainty in the furnace operation. An effective slab temperature control should therefore be robust to such variations. A typical industrial control system for the reheating furnace is shown in Fig. l b (e.g. Schurko, et al (1987), Hollander (1982)). A distributed PI controller, together with a model-based estimation of slab temperature, has been applied as a supervisory controller in the Level 2. Most recent work of Norberg (1997), Pedersen (1999) and Evans, et al (2000) could also be considered within this control scheme. The model-based estimation strategy * Author to whom all correspondence should be adressed: [email protected]

812 has been shown to be an effective way of achieving a consistent reheating process (i.e. a small variability in the slab extract temperature). In this paper, we propose a constrained multivariable nonlinear control strategy for optimally tracking a pre-specified slab temperature trajectory. The nonlinear nature of the reheating process is not only due to heat transfer phenomena, but also due to the finite time nature of the control tasks (i.e. to track a pre-specified temperature profile for each slab in a finite time, starting from the charge-in time to the extract time) so that there is no steady state points to obtain a linearized model for a linear control design. Additionally, because of the combustion limitations, constraints in the rate of the change of manipulated variables (i.e. furnace temperature set points of all zones) should be explicitly considered. Fig. 2a illustrates the proposed slab temperature control system. [ ...... "Corlstr'ai-'nts ..................

i i

,

l

/

r ...........................................................

NoController .... ,TFZ L"-- Lc~llgH e ea. I

I dy

K~' I

Yv

I

8

i

! I Temperature Estimatol" r-i

II

fi

I

l

.._1 Nonlinear

"-I OPTIM

PLANT

T'-

I Nonlinear controller t ..................................................

Fig. 2b. Control design framework

Fig. 2a. Slab temperature controller

2. Nonlinear Slab Temperature Control Design The design of nonlinear slab temperature controller will follow the method of Samyudia and Lee (2002). Our control design objective is to optimally track a pre-specified slab temperature trajectory Ts* by employing a reference systems controller combined with nonlinear modelbased optimisation (see Fig 2b). Thermal models are used for slab temperature estimation and control design. A 2-D thermal model M(0) is used to estimate the slab temperature from the thermocouple measurements located at the side-wall of the furnace T~. For the jth slab, the 2-D thermal model is formulated as:

~)T(x, y,t) = k ~2T(x' y,t) Mj(0)" pCp

~)t

~)x2

+k

a2T(x, y,t) aY2

(1)

The boundary and initial conditions are set as:

k aT(x,n,t)=hc[Tft(zj aX

aT(0, y,t) = 0; ay

t)-T(x,H t)]+o'e,[T~(zj t)-Ta(x,n,t)] '

~T(L, y,t) /)y

~

= 0;

T(x, y,0) = Tinit

(la)

In the thermal model, we accommodate the cooling system in the furnace by imposing two different ranges of boundary conditions at the bottom of the slab as" 9 For 0.055 < x < 0.955 & 1.065 < x < 1.515"

k aT(x,O,t.._..__~)=hc[Tfb(zj,t)_T(x,O,t)] + cre[T~b (zj,t)_T 4 (x,0, t)] ax

(lb)

813

~T(x,O,t) k~=Qskid (lc) ~x The spatial symbols x and y denote the thickness and length directions, zj corresponds to the jth slab at a particular location in the furnace, T represents the slab temperature at a particular (x,y) point, t is time, and p, c p , h c , k , a , e are the parameters 0 , associated with the slab 9

For 0 < x < 0 . 0 5 5 & 0 . 9 5 5 < x < l . 0 6 5 "

density, heat capacity, convection coefficient, thermal conductivity, Steven-Boltzman constant, and radiation heat transfer coefficient, respectively. Some of these parameters are known, but others are calibrated against the measurement data (Sibarani (2003)). The slab dimensions are: height H = 0.216 m and length L = 1.515 m. We assume a constant cooling

rateQski d = 1 . 7 3 x 1 0 6 j / m in, and the charge in temperature

Tinit

-

18~

Furthermore, we

define the slab temperature T~ as the x-average temperature at the 50% and 55% of its V" ,.,.,

"1

I~ft l" For the controller design, we employ the reduced 1-D thickness, and the inputs u j = ~Tfb ..1

thermal model after neglecting the temperature variation in the y-direction (length) of Mj(0), where it can be represented into the following DAE form:

dVs,j = F(Ts , T4, T 7 ) dt

~

0 = G(T i , u j )

(2)

i = 1..... 4,7 ..... 11

The nonlinear functions F and G are obtained from a finite difference approximation of the 1D thermal model by dividing the spatial x-direction into 10 equal sections with 11 node points, and by assuming the steady state models of T~ for i=1 ..... 4,7 ..... 11, and after substituting its boundary conditions of (la-lc). The temperature Ti denotes the slab temperature at the ith node point in the thickness direction, where Ts is associated with i = 5 and 6. By following the method of Samyudia and Lee (2002), we minimize the difference between

desired rate of change

dt

'

to generate optimal inputs u j

dTs,j dt

and the

by solving the following

constrained optimisation problem at each sampling time:

opt uj =arg min uJeffu

dTs

- F ( T s , T 4 , T 7)

j = l ..... N

(3)

j

where the desired rate of change is defined as" *

dt

= K opt (Ts,j (t) - es,j (t))

(4)

The constraints ~u is associated with the limits on the rate of change in the furnace zone temperature.

A

slab

temperature

trajectory Ts~j (t) is

pre-specified,

while

the

slab

temperature Ts,j (t) is estimated from T~ using (1). A systematic way of designing Kop t has been addressed in Samyudia and Lee (2002), where a robustness level bop t is guaranteed in

814 terms of gap metric (Vinnicombe, (2001)). For the slab temperature control problem with the nonlinear model of (2), Kop t can be designed using the loop shaping technique of McFarlane and Glover, (1992) with a design model G = 1 / s , which is shaped by a weighting function W to accommodate the performance specification. To solve (4), we arrange (2) such that two sets of algebraic equations are produced, where the first set is to obtain Ti (for i=1 ..... 11) by back substitution, and the other set, which is associated with (la-lc)), is solved for u~ pt by a constrained optimisation. Note that for N

opt

moving slabs in the furnace, there will be a set of N optimal u j

. So, from these optimal

inputs, we determine T/~,k (k=1,2,3,4) by employing a weighted averaging technique, where the pace rate v(t) is needed to locate the slab position. The procedure is as follows:

T~,k =

N 2 k opt 1 Y~ E wj (zj(t))uj (i) N k(zj(t)) j=l i=1 2Y~Wj

j = 1..... N

j=l I(Z ( t ) ' = l - - z

Wj

(t, / f 0 <

j

j

3 (01 wj(zj(t)) =

(t)
--Zj

-zj(t)

if if

-

2(Z ( t ) ) = ( ~

Wj

J

0 < z j (t) < 0.587 0.587
~ --zj(t)

if if

O
(5)

4 (t)) = { O i f O < z j ( t ) < 0 . 7 1 9 Wj(Zj

1 for

0.719
t z j (t) = I v(t).dt tc,j The charging time of the jth slab tc, j is given. Note that the weighted averaging technique requires us to select a weight distribution function, wjk that represents the correlations of the furnace temperatures at all zones with the to be determined set point T~, k at the kth zone. This means that the weighting function is selected to capture the nature interactions of the furnace temperatures at all zones. This implies that we consider the slab temperature control problem as a multivariable control problem. Table 1. Average furnace temperature set points of each zone

~~ Zone

Current Industrial Practice

1295 1328 1340 1269

One Two Three Four .

.

.

.

.

, ,

Nonlinear controller

S e t 1- bop t =0.62 1193 1247 1297 1307

Set 2 - bop t = 0.70 1227 1237 1306 1297

815

3. Simulation Results and Discussion The nonlinear slab temperature algorithm of (3-5) is implemented for the validated reheating furnace simulator by utilizing the pace rate data gathered from our industrial partner. A portion of pace rate data from one-day operation of the furnace is presented in Fig. 3a, where under the one-day pace rate dynamics the performance of the current industrial slab temperature controller is shown in Fig 3b. Note that in this one-day furnace operation, the furnace temperature set points at zones 1-3 were set at their maximum values, while that of zone 4 was varying. This strategy was employed for maximizing the production rate.

1=oo

0.46

1~----/

0.4

t~

0.36

.o aoo

,9o)

0.3 eoo

0.26

/

0.2 0,18 0.10

20

40

60

80

100

120

| ~a f

140

....

time - mtnu~

IocaUon

Fig. 3a. A portion of pace rate data

*(•i41S

1 ...~~ or

8o0

/

i~

~

1200

1

1000

(1087 • 64)

719V

o,.~'*

2,;

~ (1113~:79) (1002+ 121)

~ 800

(959 + 97)

E

156)

600 (666 :I: 103)

4OO

(meter)

Fig. 3b. Industrial control performance

~ 400 200

1'o

1'~

~o

location

~5

~o

~

,'o

~

~

10

(meter)

20

30

40

location (meter)

(b)

(a)

Fig. 4. Nonlinear control performance: (a) bopt = 0.62 9 (b) bopt = 0.70 Two

sets

of

Kopt were

designed

to

represent

different

robustness

objectives" bop t = 0.62 and bop t = 0.70. Then, we utilize the pace rate data in Fig. 3a. The nonlinear controller performance is shown in Fig. 4a and 4b, respectively. The simulation results reveal that a reduction up to 50% in the variability of the slab extract temperature (from +59~ to +29~ and +21~ is achieved using the nonlinear controller as compared to the current industrial practice. However, the nonlinear controller tends to heat the slabs under the target at zones 1 to 3. This is due to the constraints we posed on the rate of change of the furnace temperature. Also, such a better performance is achieved by operating the furnace at lower furnace temperature set points of zones 1-3 (see Table 1). This implies that we are able to gain a better variability in the slab extract temperature without requiring the furnace to operate at

816 their maximum set points for zones 1 to 3, as the current industrial practice does, while attaining the same production rate (e.g. the same pace rate). Note that operating at lower furnace temperature set point would lead to a possibility of saving in the fuel consumptions. Further, by analyzing Figs. 4a and 4b, we observe that if we try to achieve a higher robustness (i.e. from bopt = 0.62 to bopt = 0.70 ), the variability in the slab extract temperature increases from +21~ to +29~ This result indicates that there should be a trade-off between achieving a small variability in the extract slab temperature and attaining a higher robustness level, where bopt can be used as a design indicator.

4. Conclusions In this paper, we have addressed the slab temperature control problem for a typical industrial reheating furnace and applied the Samyudia and Lee's method for designing a robust nonlinear slab temperature controller. The nonlinear controller has been shown to achieve a better variability in the slab extract temperature, while operating at lower furnace temperature set points for zones 1-3, than the current industrial practice. A trade-off between achieving a small variability in the extract slab temperature and attaining a higher robustness level has also been observed, where bopt could be used as an indicator for the design.

Acknowledgement The authors would like to acknowledge the research support provided by McMaster Steel Research Centre, and partly by McMaster Advanced Control Consortium. Also, we would like to thank DOFASCO, Inc (M.Dudzic, B. Nelson, A Cheung and B. Poole) for their supports in providing us with the plant measurement data.

References Evans, D.M., A. Belambe, M.R. Bovan and I.G. Whitley, 2000, Reheat furnace supervisory control systems at Ipsco, Inc., AISE Steel technology, July/August. Hollander, F., 1982, Iron and Steel Engineer (January): 44-52. McFarlane, D.C. and K. Glover, 1992, IEEE Trans. Auto Control, 37(6):759-769. Norberg, P.O., 1997, Scandinavian Journal of Metallurgy, 26:206-214. Pedersen, L.M., 1999, Modeling and control of plate mill processes, PhD Thesis, Department of Automatic Control, Lund Institute of Technology, Sweden. Schurko, R.J., C. Weistein, M.K. Hanne, and D.J. Pellecchia, 1987, Iron and Steel Engineer (May):37-42. Samyudia, Y. And P.L. Lee, 2002, Robust generic model control (GMC) design for uncertain nonlinear processes, Proc. Of the 41st IEEE Conf. On Decision and Control, Las Vegas, NV: 1046-1047. Sibarani, H., 2003, Nonlinear slab temperature estimation and control of an industrial reheating furnace, M.A.Sc. Thesis, McMaster University, Canada Vinnicombe, G., 2001, Uncertainty and Feedback, Imperial College Press.