On the Control of Silicon Ratio in Ferrosilicon Production

Copyright © IFAC Automation in the Steel Industry, K yongju, Korea, 1997

ON THE CONTROL OF SILICON RATIO IN FERROSILICON PRODUCTION

Helgi Thor Ingason

Gudmundur R. Jonsson

IceLandic Alloys Ltd.

Dep. of Mech. and Ind. Engineering University of Iceland

Abstract: A Generalized Predictive Control , GPC, has been implemented at Icelandic Alloys Ltd. two ferrosilicon furnaces , for controlling the silicon ratio in tapped metal. The control is based on an ARX model of the furnace . The model is used to predict the silicon ratio one and two time steps ahead. The use of GPC has resulted in decreased variations in the chemical analysis of tapped metal and better results are also achieved in terms of keeping the silicon ratio near the desired value . Copyright© 1998 IFAC Keywords: Predictive Control, Parameter estimation, Modelling, Implementation.

1. INTRODUCTION

batch . Such a change can be made once per day . This paper descri bes the General i zed Predictive Control (GPC) which has been implemented on both furnaces at Icelandic Alloys Ltd. at Grundartangi, Iceland. A short description of the underlying theoretical principles is given with a discussion about the practical implementation of the control. Finally , an analysis of actual furnace data is presented, gIvIng indication about the performance of the control.

Ferrosilicon (FeSi75) is a two-phase mixture of the chemical compound FeSi2 and the element silicon . FeSi75 contains about 76 wt% silicon, 22 wt% iron and 2 wt% aluminum, calcium and other trace elements. Icelandic Alloys Ltd. produce 70000 tons annually FeSi75 in two 48 MV A submerged arc furnaces. The balance between silicon and iron in the tapped metal is regulated by adjusting the input of raw-materials to the furnace. Raw materials are charged batchwise to the top of the furnace. Each batch consists of a fixed amount of quartz (Si02) and a variable quantity of coal/coke (C) and iron oxide (Fe203). The quantity of coal/coke in each batch is adjusted to regulate the carbon balance of the raw material charge in the furnace and this control is beyond the scope of this paper.

2. THEORY 2.1 The chemical baLance of the charge

A simple model of the relation between silicon ratio (Si%) in tapped metal and the quantity of iron oxide per batch (kgFe) is as follows , (Halfdanarson, \995) : Si%

The silicon ratio is regulated by increasing or decreasing the quantity of iron oxide in each

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=0.98/ (1 + IS · kgFe / (700 · n»

( I)

The variable n represents the relative utilization of the quartz that is input to the furnace. The constant 0.98 expresses that Si and Fe are 98 wt% of the tapped metal, and the constant 700 represents that there are 700 kg quartz in each batch of raw materials. The constant 1.5 is based on experience at Icelandic Alloys Ltd. where 89% utilization of quartz is typically achieved with 120 kg of iron oxide per batch, when the Si% is 76%. The chemical analysis of the tapped metal varies from day to day . These changes may be attributed partly to variations in the internal environment of the furnace, e.g. temperature, and partly to fluctuations in the chemical composition of the raw materials. It is thus necessary to vary the quantity of iron oxide input in order to control the silicon ratio and keep it as close as possible to the desired value, 76% in the case of Icelandic Alloys Ltd.

2.3 Generalized Predictive Control (GPC)

The predicted value of Si% is compared to the set-point or desired value of Si% and the control calculates what kg Fe must be in order for these to be equal. In practice, too high variations from day to day in the iron oxide input to the furnace are unacceptable. The control therefore includes limitations as to how much it allows kgFe to change from one day to another. Following is a short description of GPC algorithm, see e.g . (Clarke et aI., 1987) and (Palsson et aI., 1994) for more details. The cost function to be minimized was chosen to be

J =

J-l

J-NI

(5)

with NI as the minimum costing horizon, N, as the maximum costing horizon, N. is the control horizon and wr + j as the future reference

2.2 ARX model of the furnace

The Generalized Predictive Control uses a mathematical model. of the furnace for predicting the silicon ratio. The model is of the type Auto Regressive, auXiliary input (ARX) and can be written as, see e.g . (Ljung, 1987): A(q) yet) = B(q )x(t - nk) + e(t)

E[IN~ (Yl+j _w1+j)2 + IN~ Aj (b.u1+j-1r]

output. Eg . (5) shows that besides penalizing deviations in the output from the reference value, the penalty is on changes in the input, Ut' i.e . Dour. The parameter A is typically found by trial and error.

(2)

The cost function may also be expressed in vector form as

AR refers to the autoregressive part A(q)y(t) and X refers to the extra input B(q)Xt-nk. q-I is

a delay operator, y(t)q -I = yet -I) and nk is the time delay, i.e. the number of delays from input to output. e(t) is the prediction error, i.e. the difference between the actual output value and the predicted output value. Finally, A(q) and B(q) are polynomials in the delay operator:

with Yt =

[Yt+N

=

[Yt+N

Wr

DoUr

(3)

=

, •• • ,

I

I

r wl+N, r,

YI+N,

, ••• ,

[b.ut+N ,··· ,b.u +N, I

1

r

and

with na and nb as the orders of the respective pol ynomi als.

The output may be written as a sum of two independent parts, i.e . the predicted output and an error term or

The ARX model, upon which the GPC is based, is of the form (see section 3. 1):

(7)

(4)

with

i.e. the Si% value today (Sit) is based on Si% and kgFe yesterday and a constant d.

Yt containing the predictions

Yt

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=

[Yt+I'r,·· ·,Yr+N'tf

and

Yt containing the prediction errors , i.e .

In order to prevent offset , the control signal may be filtered and may be written as , see (Palsson et aI. , 1994) (12)

Inserting this into Eq. (6) yields If b.u r = Ur - u r- J , then gr =[-ur- 1 ,0, ... 1

The GPC assumes that the predictions can be expressed as a linear combination of present and future controls . Following the lines in (Palsson et aI., 1994), the predicted output is written as

0

-1

F=

0

0

0

0

0

-1

0

0

0

0

-1

,of and

j

Yt+ jit = L,hi.t+j Ul+j-i

(9)

+ Vj .t

Inserting Eq . (12) into Eq. (11) gives

1=1

J=E[(HtUt+Vr-w rf(H r Ut+Vt-Wt)+

where the former term contains unknown values until the control values are known, but the latter one contains known values at time t. Eq . (9 ) can be written on vector form as

(FUr +gt)T ArF(u r +gr)]+ E[YtTYr] (13)

The GPC is then obtained by differentiating the cost function with respect to u r ' i.e .

(10) where

yielding the solution:

Ut = [Ut "",Ul+N U _l]T ,

T Ut =(H:Ht +Ft ArFt

v t = [VNI.t , ... , VN: .t]T

r' [H:(w t - vJ-F TArgr ] ( 14)

and r-

Ht

=

hl. t+N,

0

~.t+NI+l

hl. t+NI

hNu.1

hNu -1.1 h Nu .l- 1

hNu +1.1-1

Since only the first element in u r is used, the control signal becomes

o o

14 =[1,0,... ,0

h1.l

](KHr +F;T~F;r'[K(Wt -vJ-F Ar~) T

(15)

~. J-l +h1.l-J

The are several advantages in using GPC in the present case instead of e.g. Minimum Variance Control.For instance, GPC is more robust than MVC with regard to changes in system time delays and to changes in the parameters of the system mode l.

N:-N I-N u +2 L,hi.t+N: ;=1

where I =t+N J +Nu -1 . Note that the matrix

Hr is shown time dependent to indicate that The following guidelines were used in chosing the design parameters, i.e . NI' Nz and Nu :

this version of the algorithm can cope with deterministic time varying system parameters, see (Palsson et aI. , 1994) . Inserting Eq . (l0) into Eq. (9) yields

NI: Naturally, N, $ k = system time delay . If k is unknown, then NI = 1.

J=4~ Uy+vt -Wt((Hr Uy +vr -Wr)+eo;J\t..J,]+

Ely;yrJ

Nz: Usually , NI ~ nb with nb the degree of the B polynomial in the ARX model.

(11 )

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N. : Often it is sufficient to put N. = 1 but for complicated systems, N. is chosen at least as large as the time constants corresponding to illconditioned poles of the system.

3.2 User interface A sample is taken from each tap for chemical analysis . All samples from each of the three shifts are crushed and mixed together to form one representative sample for that shift. The results from chemical analysis are available at 10.00 in the morning , six days a week (excluding Sundays).

It was found , see the following section, that an adequate ARX model is gi ven by

Cov(Si%)

where d is a constant. With NI' = 1 and Nz = N. =2, the control becomes , cf. Eq . (14 ):

0 . 2 2 . - - - - - - - - - - - - - - ---, 0.20 0.18

(17 ) 0,16 0.14 +---+---+----t--+--+----1

o Furthermore , w, =

r,

[w w

g, = [-Fe,_1

F, = [

~ I ~J

Or, A=,tl

10

20

30 40 Cov(kgFe)

50

60

Fig . 1 Variance of Si % as a function of the variance of kgFe for different values of A.. (18 ) The results are then input to an Excel based user interface, specially developed for thi s purpose . The program uses the average of the three analysis for its calculation . After the program calculation, results are presented as indicated in Fig . 2 .

3. METHODOLOGY 3.1 Implementing GPC

Itr:larxlic Al~ Lld

A thorough analysis , involving e .g. cro ss validation , was carried out in order to find a suitable system mode l. The result was the following first order ARX model (parameter standard deviation in parenthesis) :

Si, - 0.44Si,_1 = -O.0028Fel-l + 46.1 + er (0.07) (0.001) (5.6)

GrC"r ~1iu'lt""111

Fumace2 t.N rami. July 25.

Si'" OS 2A.

7Sj(l'J.

Si..· ES PI. 7 7~.I S'" Iron 0J00e per Inch 122 Ic!: SiIH.;s 25. 7 7S.!I3"· ..

INewram! I IDeL last ram! I IFn
(19 )

The parameter estimation was based on daily data from furnace 1 in the year 1995 . The numbers in parenthesis are the standard deviations of the corresponding coefficients . Fig. 2 . Result window for the GPC program (Gudmundsson , 1996).

It was found through simulations that a reasonable value for A. in Eq. (18) is 0.0007 , see Fig . 1 which shows how the variance of output Si % as a function of the variance of input kgFe for various values of A.. If A. = 0 , i.e. no penalty on the input , the variance of the output is about 0. 147 but the variance of the input is much higher compared to when A. = 0 .0007 . A variance of 20 kgFe was found acceptable by Icelandic Alloys Ltd .

The operator can delete the last record , i f it is discovered that an error has been made in the input. The operator can view the Si% (average values) and kgFe for the last 60 days. Finally, the operator can plot the prediction error , er ' for the ARX model. The error chart gives the operator a change to monitor the process as a

76

part of the Statistical Process Control system at Icelandic Alloys Ltd.

71.0

-r-----------------------------------

76.~

4. RESULTS 76.0

4.1 Time series The silicon ratio for the period 1.1.95 31. 7 .96 is shown in Fig. 3 . The dotted line shows the actual values but the solid line shows a running average, based on the iast 7 values .

7S.5

+-.........__+_+_+_

14 ,1.9'

0.'

IIJ.?S

6 .S,9'

__+_-+---I-+-+--<~__+_+_+_+___+__+__1

a .• .."

1.7 .95

I .b ?"

:7 .1 ."6

..

77

, . , .

..

"

:: :

0.'

...~..

:.

~ ~

:

,:

::

• •

::

,': ~ . ' • " '.

.

'.,'

~.

t',

.. :. .

.' , ;

. ~.'-

"

(j

PC

_ _ ~GPC

.. ~

+-.........--+-+-+__+_-+---I-+-+--
1" .1.H Il.l ,"S

76

#

t'

,"

Bdore 0.2

/ \

o' • . •

0 .3

6 ,S.'"

1.7 .9S

16 .' .9S 21 .10 ,9516 .1: .95 10 .:.961'1 ... .96

:7 ,7 ,'1('1

Fig.4 a) Average values and b) standard deviation, for the silicon ratio , based on the period 1.1.95-31.7.96. The period is subdivided into 14 days periods and the parameters are calculated for each 14 day period.

75

~~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~~~~~~~

-~~~~~~~~O~N~NM~~~~ -~N~~~OO~~---~~~~~~~ ~~~NN~~~NNNNNNN

N

6 ... .?6

-r-----------------------------------

~r-------------------------------------'

N

10 .::: .""

0.7 0 .•

~

11.10 .')516 . 1: .'"

N

I~r-----------------------------------~

The distribution of changes in the iron oxide input from day to day is analyzed by making a frequency plot of two equally long periods. before and after the implementation of GPe. This is shown in Fig. 5.

13U

1211

122

ilK II~

110

Req.

Uk'1 ~

o,

~

;:; '"0

-

;:;

~

~

~

'" '"(!;- '-" ~

~

0

0

N

~

o, ~

0

~

~

'" ...o,0

,.; 0

;:; g so;

~

o,

~

o,

~

o,

~ ~ ~ ~ '" N ~

~

o,

~

o,

....

:::; :::;

'C

'~" 0

-c o, '" '" N

~

0

0

~

N

~

'C

'"(!; ~

N

o, '"o, '" ~

0

~

N

~ ~

'C

...'0"

.

11 'M thout GPC

N

Fig . 3 a) The silicon ratio and b) the iron oxide per raw material batch during the period 1.1.95-31.7.96.

• 'MthGPC

The GPe was implemented 18.12.95 which is indicated by a small solid triangle in Fig. 4. o~~~~~-r~~~~~r&~~

4.2 Analysis

<-6 -6 -5 4 -3 -2 -1 0 1 2 3 4 5 6 >6 Di fference bet ween days

In order to evaluate the performance of the GPe, a simple analysis is done. The period 1.1.95-31.7.96 is divided into periods of 14 days and the average and standard deviation of the silicon ratio is calculated for each period. Fig . 5 shows plots of the results.

Fig. 5 A frequency plot of the distribution of difference in kgFe between two adjacent days. Total umber of data points is 221. Furthermore, the use of reduced the correlation seen in Fig. 6 autocorrelation function periods is shown.

77

GPe has significantly in the Si% as can be where the sample of the Si% from both

shift samples. This means that once a week the orders of the GPC, regarding kgFe for that day, are not followed. This must have a negative effect on the control performance . Other problems have been encountered. Both Figs. 3a and 4a show that Si% decreases in February and March 1996 but increases again in April. This is explained by slow variations in the internal furnace environment. The model parameters were based on furnace data for 1995 . These constants are not final and must be re-calculated regularly as the furnace performance may differ from time to time, e.g. because of different raw materials . By recalculating the coefficient d it is possible to some extent account for those slow variations in the furnace. Given the average of the kgFe for a 60 days period of time, the coefficient is calculated by d=(1+at) ·76-btkgFewd ' The

0.8

0.6 0.4 0.2

o -0.2

5

10

Lag

15

20

25

Fig. 6 The autocorrelation functions of Si% before and after implementing GPC.

5. CONCLUSIONS The time series presented in Fig. 3 indicate that there is a change in the process be ha vi or when the GPC is implemented. Fig. 3a clearly shows that kgFe is much more dynamic after than before. This is explained by the fact that the control before GPC was much more rigid, it allowed Si% to vary freely within defined limits, and reacted only if those limits were crossed. Furthermore, changes in kgFe, from day to day, that exceeded 3 kg were not allowed. Fig. 3a indicates that the Si% is kept near 76%, and oscillations in the signal are less after the implementation of GPC. These observations are confirmed by the statistical analysis presented in Fig. 4 where oscillations in the 14 days average values for Si% seem to decrease somewhat with the implementation of the GPC. The impact of GPC on Si% is best seen in the standard deviation presented in Fig. 4b. The standard deviation drops significantly and is near 0.4 after the GPC is implemented. This is the expected variance of the signal (0.16), as discussed in the Theory section. The increased dynamics of kgFe, observed in Fig. 3b is confirmed in Fig. 5. Changes in kgFe, between two days, were none in the majority

coefficient d has been updated irregularly, using this equation, when a trend has been discovered in the data. In the future, the coefficient along with other model parameters will be calculated and updated on-line.

6. ACKNOWLEDGEMENTS The authors wish to thank Mr. Atli B. Gudmundsson for his work on designing the GPC and help regarding the statistical analysis of the performance of the control. Icelandic Alloys Ltd . is thanked for supporting this work.

REFERENCES Clarke, D.W., C. Montadi, and P.S. Tuffs (1987). Generalized Predictive Control Part II . The Basic Algorithm, Automatica , Vol. 23, No. 2. pp 137 - 148 . Gudmundsson, A.B (1995) . Control charts and statistical quality control at Icelandic Alloy Ltd. (in Icelandic), BSc thesis, Department of Mechanical and Industrial Engineering, University of Iceland. Gudmundsson, A.B (1996) . An Excel Based User Interface for GPC - Revised Version, Excel program, Icelandic Alloys Ltd. Halfdanarson, J.H. (1995). The relation between Si%, quarts utilization and kgFe , Internal report, Icelandic Alloys Ltd. Ljung, L. (1987). System Identification Theory for the User, Prentice-Hall , Englewood Cliffs, N.J . Palsson, O.P, H. Madsen and H.T. Sogaard (1994 ):Generalized Predicti ve Control for Non-stationary Systems, Automatica, Vol. 30, No. 12. pp 1991 - 1997.

of cases before GPC, and never more than ±3 kg. This diagram changes drastically after GPC and the changes are now smeared out to the limits of ±6 kg. Finally, the autocorrelation functions presented in Fig. 6, indicate that correlation in Si% has dropped significantly after the implementation of GPc. The GPC assumes that chemical analysis is done every day . This is not the case at Icelandic Alloys Ltd. where no chemical analysis is made on Sundays . In addition, chemical analysis is done irregularly around holidays, e .g. Christmas and Easter, but no more than 2 days pass between analysis of

78