rolling mill process based on extended coloured petri nets

ControlEng. Practice,Vol. 3, No. 10, pp. 1359-1371, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0967-0661/95 $9.50 + 0.00

Pergamon 0967-0661(95)00139-5

MODELLING AND SIMULATION OF A SOAKING PIT/ROLLING MILL PROCESS BASED ON EXTENDED COLOURED PETRI NETS 1 Y.Y. Yang, D.A. Linkens and N. Mort Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin St., SheffieM, $1 4DU, UK

(Received October 1994; in final form June 1995)

Abstract: In this paper a soaking pit/rolling mill process model is developed for a real

induslrial steel mill based on extended coloured Petri nets (ECPN). Both the discrete events and the continuous dynamics involved in the process are modelled in a unified framework, allowing better investigation of the interactions between these two parts. Corresponding ECPN diagrams are given for the critical components involved in the soaking pit/rolling mill process, including the arrival of hot ingots, the charging and discharging of soaking pits, the soaking process, the preheat furnace operation, the mill operation,etc. The ECPN diagrams are implemented in the Design/CPN environment in a Sun SPARC workstation. Computer simulations are carded out and typical results are shown, which are helpful for the improvement of soaking pit/rolling mill operations. Keyword: Steel manufacturing, mixed-mode systems, Petri nets, modelling, simulation.

the discrete events of the various unit operations for the sake of scheduling and operation control. However, research on mixed-mode modelling and integration of control and scheduling seems absent, although the continuous dynamics and the discrete events coexist and interact strongly in the soaking pit/rolling mill process.

I.I N T R O D U C T I O N The soaking pit/rolling mill process is still an important part in the iron and steel industry, although uow__a~__ys more and more modem steelmaking processes employ continuous casting, which does not utilize soaking pits. There are a considerable number of hot steel mills where soaking pits are used to produce hot ingots for the subsequent rolling mills. Energy eonsmnption by the soaking pits is huge and it absorbs a significant part of the total operational costs. Furthermore, the quality of soaking pits operation has a pronounced effect on the downstream mill operation as well as on the quality of the final steel product. Due to the importance of this process, a lot of research was carried out during the 70s and 80s for the modelling, op "tnnization, and control of the soaking pit/rolling mill process (Ashour and Bindingnavle, 1972; Patel et al., 1976; Lumelskey, 1983; Yang et al,, 1988; Rao et al., 1984; Lu and Williams, 1983a, 1983b). Some of the research concentrated on the continuous behaviour of the thermal dynamics of the soaking pits and related processes, while others focused on

The aim of this paper is to establish a soaking pit/rolling mill model for a real industrial steel mill complex with a combined discrete event/continuous dynamic in mind. It is believed that by considering the mutual effects of these two different parts the resulting mixed-mode model will be able to represent the physical process more faithfully. It is also natural to conjecture that control strategies and scheduling policies derived from the enhanced mixed-mode model will be superior to those based on more conventional models. The paper is organized as follows. In Section 2 a full description of the soaking pit/rolling mill process is given based on a practical steel mill complex. The development of a mixed-mode model for the soaking pit/rolling mill process is derived in Section 3, using extended

1 Thisresearchis suppcxt~byan EPSRCGrantGR/H/73585. 1359

Y.Y. Yang et al.

1360

coloured Petri nets (ECPNs). ECPN diagrams for the main components, such as the arrival of hot ingots, the charging and discharging of soaking pits, the rolling mill operation, the soaking process, etc., are also given in this section. The implementation of the mixed-mode model by Design/CPN (Meta, 1993) and typical simulation results are shown in Section 4. Finally, concluding remarks and future research topics are outlined in Section 5.

H' Fig. 2. The structural layout of a soaking pit/rolling mill complex

2. DESCRIPTION OF THE SOAKING PIT/ROLLING MII J. PROCF_.SS The following description is based on a practical industrial soaking pit/rolling mill complex within a steelworks located near Sheffield, UK. The soaking pit/rolling mill complex consists of eighteen soaking pits arranged in the soaking pit bay, and four mills with different capabilities to produce a variety of shapes and sizes of steel products. Two overhead travelling cranes, which span the pit-mill area, are responsible for charging the incoming ingots into the pits and drawing the ingots from the pits to the slabbing mill area for subsequent roll processing. Transfer of ingots in the mill area is carried out by a track conveyor system, which connects the related mills and other auxiliary facilities in the mill area. Hot ingots are produced by the process upstream where steel (in its liquid phase) is fu'st made from its raw materials by a particular steelmaking process (e.g., basic oxygen furnace, open hearth furnace, electric arc furnace, etc.), and then the steel liquid is tapped and poured into ingot moulds. Hot ingots are stripped off their moulds when they are partially solidified and then they are transported to the soaking pit area by the internal trains. Two preheat furnaces are also provided in order to supply hot ingots from a cold ingot bank when the number of hot ingots from the upstream steelworks is not sufficient to meet the demand. Figs. (1-2) show the ingot flow and the layout of the soaking pit/rolling mill complex.

I Fig. 1. Flow of ingots within the soaking pit/rolling mill complex The time required to transport the hot ingots from the steelworks to the soaking pits (defined as track time) is subject to stochastic variations caused by congestion and equipment failure. As a result, sometimes hot ingots must wait on the tracks in the soakiqg pit area before being charged into one of the soaking pits, and at other times there are no hot

ingots available for the empty soaking pits. If there are too many hot ingots waiting in the pit area, some of them are removed to the cold ingot bank. When the supply of hot ingots from the upstream steelworks is not sufficient, preheated ingots are taken out of the preheat furnaces and then are charged into the soaking pit. At the same time, cold ingots are put into the preheat furnaces through the furnace entry to keep the preheat furnaces fully charged. Because the arrival of hot ingots from the steelworks is not completely predictable, the residing time of ingots in the preheat furnaces varies considerably. The charging of ingots to the preheat furnace is arranged four at a time by a bogie. During the preheating period, the bogie moves along the furnace, controlled by the pace of the additional hot ingot requirement, and when the bogie has reached the furnace exit the four ingots placed on it are removed to the destination soaking pit by an overhead crane. Occasionally, cold ingots are directly placed into a soaking pit, although this operation is mostly reserved for special cases. The total number of ingots charged into a soaking pit depends on its size, the average number being ten for a soaking pit in this case. The charging time depends on the destination pit, the location of input ingots, and the interaction between the cranes. Once loaded into a pit, the ingot soaking process begins. The initial ingot temperature is determined based on the track time and the characteristics of the steel ingots, if they come directly from the upstream steelworks. If the ingots come from the preheat furnaces, a constant initial temperature is assumed. Hot ingots with an average weight of around 4 tonnes are heated in the soaking pit until the ingot temperature is raised to a uniform value of around 1300°C. Then the ingots are ready to be removed from the pit for the subsequent rolling process. Fuel is regolated by a PID-type controller based on the given pit temperature set-point. Because direct measurement of the ingot temperature distribution within the production environment is impossible, the required heating time as well as the soaking pit temperature set point are determined on the basis of operating experience, heat balance, and the characteristics of the steel ingots in the soaking pit.

Modelling and Simulation of a Soaking Pit/Rolling Mill When a soaking pit is ready to discharge, the ingots are removed, one at a time, by a crane and placed on an ingot chariot. The ingot chariot, moving on the track connecting the pit area and the slabbing mill, sends the ingot to the slabbing mill in accordance with the pace of the mill's rolling rate. After being processed in the slabbing mill, the ingot (in a semiproduct form) will go through one or two of the remaining three mills to continue its rolling sequences according to the size and shape of the final product specified by the production schedule, before being finally sent to the product yard. As only a single ingot may be processed at any time in a mill, and each pit, on average, holds ten ingots, processing a batch of ten ingots takes around half an hour. If, during this time, another soaking pit becomes ready for discharge, it must wait until the first batch has been processed. Obviously, any waiting time for a soaking pit will result in extra energy consumption and such waiting should be kept as short as possible. 3. MODEL DEVELOPMENT OF A SOAKING PIT/ROLLING MILL PROCT_~S The soaking pit/rolling mill process is a typical mixed-mode system, where the operation of cranes, the charging and discharging activities of the soaking pits, the start/stop operations of the rolling mill, etc., can be described as discrete events, while the ingot soaking process in the soaking pits and the heating process in the preheat furnaces have typically continuous dynamics. Of course, the above classification is related to a certain level of abstraction and is not absolutely true under all circumstances. For example, the mill operation might also include many continuous dynamics at a more detailed level, such as the roll gap in each rolling pass, the roll force behaviour, and the edge speed of the roller, etc. No facility for breakdowns is explicitly modelled in the soaking pit/rolling mill process. All of the equipment is assumed fault-free and in normal working condition when available. The processing time for an equipment to fulfil a specific task is assumed to be a random variable in order that the effect of equipment faults can be approximately modelled by the selection of a probability distribution of this random variable. Pre-postcondition descriptions are adopted for the modelling exercise, since any of the processes (events) can be described by what are its necessary conditions, when will it happen (pre-conditions), how long will it last, and what are its results (post-conditions), in some way. After these pre-post conditions have been established, ECPNs (Yang and Linkens, 1994) and a mixed-mode modelling technique (Yang et al.,

1361

1994) are employed to derive the Petri net based model. Coloured Petri nets (Jensen, 1992) were developed from the basic Petri nets. The aim of introducing a colour is to simplify the Petri net structure by enfolding the original Petri net for those parts which have the same subnet-structure. Thus, with coloured Petri nets, we are able to model more complicated systems with a better visualisation because of a relatively small network structure, although it adds no extra power to the modelling ability. ECPNs are developed based on an extension of coloured Petri nets to provide facilities for modelling continuous dynamics. Dynamic colours, dynamic places and dynamic transitions are important components introduced in the ECPNs. A dynamic colour can have a set of continuous attributes attached on it, which can be used to represent a continuous dynamic. A dynamic place can hold tokens with dynamic colours, while a dynamic transition (Dtransition) can have functions to access and update the dynamic colours of the tokens residing in its input places and/or output places. The enabling and fn'ing rules are similar to those for ordinary coloured Petri nets except those of D-transitions. For a Dtransition, extra calculations are executed to update the value of each coloar state belonging to the dynamic colours of the related output places. For more details on ECPNs, readers are referred to (Yang and Linkens, 1994).

3.1. The Arrival of Hot Ingots This sub-process can be divided into two parts: the production of hot ingots at the upstream steelworks and the transportation of the hot ingots from the steelworks to the soaking pit area. Hot ingots are produced in batch mode (Kca~t ingots per batch) according to the production plan of the steelworks. Each batch of hot ingots takes a random production time X~,m, with the expectation denoted as "~c~t. The transportation of hot ingots from the steelworks to the soaking pit area is carried out by internal trains. It is also considered as a random time delay (~t,~l) process with Ktra,~l ingots in a batch, supposing that all of the preconditions for the transportation are satisfied. The hot ingots used in Stocksbridge Engineering Steel are of special engineering steel with relatively small size (average weight of 5 tons), and most of the ingots, when they arrive in the soaking pit area, are completely solidified with an average ingot temperature of 400800°C. An ingot is only partially solidified, however, when it is stripped from the ingot mould. The transfer of hot ingots to the cold ingot bank takes place when there are too many hot ingots waiting in the pit area, i.e., when the total number of hot ingots, K~sot,, exceeds the capacity of the pit area

Y.Y. Yang et al.

1362

buffer, which is denoted as Kb,¢. The transfer to the cold ingot bank is completed in a hatch of Kt,~2

px tSteelvTofl~ tet~ ~

Pl: Production~

ingots with a random time interval 'U~,,,2. Table 1 shows the Pre-Post conditions for the events related to the arrival of hot ingots sub-process.

..-"

-.f-

I®,=.,

,

T~bl~ 1. Pr~-Post Conditions for the Arrival of Hot Ingots ~e-emditlem

Pmt-eemtitlom

Hotlsgot Produt-don: Productionplaais not m ~ ;

Kam hot ingots produced;

Cast machine is available.

Initialize insot atu-ilmtes; Release the cast machine;

Trtmsfcr Ingot to Pit Arm: K~

hot ingots or more

P6: Cr~meAvailable

O

: A Fusl°nPlace ~

are

available in the steeiwm'ks; Train is available~

Batch ~ Kvwl hot insots is moved to pit area; Release the Wain.

O

: A Normal Place

l~/mm~UOattq

CD:A .T-- I I

: A NormalTransition

EtwaC~llttm Clmek

Transfer Ingot to Cold Imfot Bank: Tl~ number ot hot ingom in

Batch ~f Kuaa hot ingots is

the pit ezea exceeds the capacity elf pit area buffer,

moved from pit area to the cold ingot bank; Release the crane.

Crane is available.

It is quite straightforward to derive the Petri net model when all the pre-post conditions of the related processes (events) have been estabfished: simply code the pre-conditions as the input places and the post-conditions as the output places, and add the related input arcs and output arcs to connect the prepost conditions to a transition which is used to represent the event concerned. Fig. 3. shows the resulting ECPN model, where specific graphic notation (symbols) developed in ECPNs (Yang and Linkens, 1994) are used with no further explanation.

3.2. Charging and Discharging of Soaking Pits Charging ingots into a soaking pit is assumed to be a simple time-delay process when all the preconditions for the charging activity are satisfied. The delay time is denoted as x ~ s which is a random variable with expectation ~ , , ~ .

The pre-conditions for the

soaking pit charging activity are: free space is available in the soaking pit in order to hold the ingot; at least one of the cranes is available for charging, and there are enough hot ingots in the pit input urea. One more additional condition for the charging activity is that only one soaking pit is allowed to enter the charging status. This means that a soaking pit cannot start its charging operation if there is already another soaking pit which is in its charging status. The post-conditions for the charging activity are: an ingot is charged into the soaking pit; the number of free spaces available in the soaking pit

Fig. 3. ECPN model for the ingot arrival sub-process is decreased by one; the soaking pit is set to the charging status; the crane is released after the charging activity; and the initial ingot temperature is calculated, along with other ingot attributes being updated. If the soaking pit is fully charged, then clear (reset) the pit's charging status. Discharging ingots from a soaking pit to the slabbing mill can be viewed roughly as an inverse operation of the ingot charging activity. When the ingot temperature in the soaking pit satisfies the temperature requirement (specification) of the rolling process, the ingots are ready for discharging. Other pre-conditions of the discharging operation include: a crane is available for discharging, the slabbing mill is ready to roll the discharged ingot, and no other soaking pit is in the discharging status. The related post-conditions are: an ingot is discharged from the soaking pit; the soaking pit is set to the discharging status; the crane is released from the discharging activity. If the soaking pit is empty after the discharging process, then clear the pit's discharging status. This will enable other soaking pits to enter the discharging operation. When those pre-conditions are satisfied, the discharging operation is treated as a random time-

delay process with the ,lelay time (discharging time)

Based on these pre-post conditions, the resulting ECPN model for the ingot charging and discharging sub-process is shown in Fig. 4, where transition Tsi is a high-level transition used to represent the ingot soaking process, which is to be discussed in Section 3.4.

1363

Modelling and Simulation of a Soaking Pit/Rolling Mill PS: P~t l ~ l u

~IsFutAm

~t~

~ isis d m l ~ l l rams

pI 2: SIM~IIB)~1Rd~l7

3.4. The Ingots Soaking Sub-process

* I ffi1,2,-~llt ISII~ ld ¢1I1~So~bl8 l~ts

Fig. 4. ECPN model for the charging and discharging sub-process

3.3. Rolling of Ingots in the Slabbing Mill After an ingot is discharged from a soaking pit, it is transferred to the slabbing mill for the rolling operation. According to the final specification of the product (given by the production plan), the ingot will go through some of the mills for processing before it is finally transferred to the product yard. This rolling sequence (the number of the mills to be used and the order to be rolled) is determined by process constraints based on the product specification. In the current work, only the slabbing mill is considered. It is assumed that all of the ingots, after they are processed in the slabbing mill, will be sent to the product yard. The detailed rolling sequences in the slabbing mill are ignored so that the rolling process can be treated as a random time delay 'C~ou] when the preconditions of rolling activity are satisfied. The rolling process will take place when there is a prepared hot ingot in the mill input area and the slabbing mill is available for rolling. The transfer of rolled steel products is also modelled as a simple random time-delay procedure with a timedelay of 'l;,nns3 • Based on these simplifications, the pre-post conditions for the rolling sub-process are listed in Table 2, and the resulting ECPN model for this simplified rolling sub-process is shown in Fig. 5. Table 2. Pr¢-Post Conditions for Rolling Sub-~ocess Pre-eondl~ts

Post-conditions

This sub-process is quite different from those subprocesses described previously due to the substantial involvement of continuous thermal dynamics. When a soaking pit is fully charged, the ingot soaking process begins in order to raise the ingot temperature to a specified level which is required for rolling. In the soaking pit, heat is fast transferred to the ingot surface from the high temperature pit wall and pit gases by means of radiation and convection. Then, this heat energy is transferred further to the interior of the ingot by heat conduction. After a certain soaking time, the average ingot temperature reaches a level of around 1300°C, which is one of the preconditions for rolling. The soaking process is then terminated and the soaked ingots are discharged from the soaking pit one by one for subsequent rolling. It is obvious that two interacting dynamics coexist in the soaking process: the start and ending of the soaking process (discrete events) and the thermal behaviour of the ingot temperature (continuous dynamics). Modelling and control of soaking pits on their continuous behaviour has attracted a lot of research effort and many different models exist in the literature (Lu and Williams, 1983a; Yang et al., 1988; Rao et al., 1984; Lumelsky 1983). These models can be divided into two classes: empirical (recursive) and analytical models. The former are established from experimental data collected from the production line or experimentation, by using some identification or model-fitting techniques. The latter are derived by using the energy balance equations relating to the heating process occurring in the soaking pits. Because the heat transfer mechanism and the corresponding boundary conditions are very complicated in a practical soaking pit, a methodology incorporating the analytical model with identification is favoured. The heat transfer between ingots and the pit environment in a soaking pit is carried out primarily by radiation and convection. All ingots in a single soaking pit are assumed to have the same temperature distribution, provided that the ingots are of the same grade and geometrical size. Most of P12: Mill Ready

m: ~ in~ t ~ Am

~

p 1~,. [ ~ a

~',~sc~

"" ~ - O ~ - ~ -

Roi~ng Ingot in the slabbing milL" Hot ingot exists in the mill

An ingot is rolled to final product;

area; Slabbing mill is available.

Release the slabbing mill; Set mill-ready status.

Trtmsfer flm~l ingot products: Ingot product is available fat storing,

The product is moved to product yard; Update statistics.

P16: Ingot P t c d l ~ ill lhx~laet Ys~rd ITS:Trms l ~ ° t I~ | to Prodaa Y ~ I

I

e'c~

I-"

Fig. 5. ECPN model for the rolling sub-process

Y.Y. Yang et al.

1364

the ingots are completely solidified when they arrive at the soaking pit area, i.e., there is no molten core in the ingot. According to Lu and Williams (1983a), the temperature distribution within the ingot is assumed to be one-dimensional under the cylindrical co-ordination system as shown in Fig. 6. Based on the heat balance principle and some basic heat transfer equations, the ingot temperature can be represented by a partial differential equation together with the related boundary conditions and initial conditions: OT,(r,t)= 1 ~

f.. OT,(r,t)~. r < R; t > t. cpr "~r [ "r ~ :

~t

where ews is the equivalent radiative heat transfer coefficient between the soaking pit and the ingots, (p ,s is the view factor between the pit gases and the ingots, ¢p~ is the view factor between the pit wall and the ingots, e, is the emi~ivity of the ingots, o is the Stefan-Boltzmann constant, f,s is the convective heat transfer coefficient between the pit gas and the ingots. If the ingot comes directly from the upstream steelworks, the initial ingot temperature is determined based on the track time using the following equation:

(3)

T,o(r) = 1". + (To(r)- T,)e -'~'~-'~'r" ~T'(r't)I 3r

ir-s

=q(T,(R,t),T,(t))IK

(1)

T,(r, t,) = T,o(r) where T, (r, t) is the temperature distribution of the ingot; Tp(t) is the soaking pit temperature; K, p, and c are the specific conductivity, density and specific heat coefficients of the ingot, respectively; R is the equivalent radius of the ingot; q(T~ Te) is the heat flow density from the pit to the ingot surface; and To(r) is the initial ingot temperature distribution. The soaking pit temperature Tp(t) is mainly determined by the fuel supply and can be obtained through on-line measurements.

where To is the ambient temperature, T,~(r) is the ingot temperature associated with the minimum track time t,,t, tt,,~t is the actual track time, and Ts is the time constant for an ingot with radius R. If the ingot comes from the preheat furnaces, then the initial ingot temperature is given by: T. (r) = T,,.~

(3a)

where Tp,,~,~ which is a constant, is the ingot temperature when drawn from the preheat furnaces. In order to facilitate the computation, equation (1) is converted into a disaete-time version using a finite difference approximation. Suppose the discrete-time step is At, and the discrete spatial step is Ar, then by using the following approximation:

R - (L! * L2MOLI+L~

t = kAt, k = 0,L2..... ; r = iAr; i = 1,2..... N; n = int(R / Ar);

~T,(r,t) = T,(iAr,(k + l)At)- T,(iAr,/rAt). ~t At ~Tf(r,t) = T,((i+ l)Ar,kAt)- Tr(/Ar,kAt) ;

Fig. 6. Ingot temperature distribution with a cylindrical co-ordination

9

There are many factors, such as the view factors between the soaking pit and the ingots, the emissivity of the ingots, the shape of the soaking pit and the ingots, the relative position of the ingots in the pit, the pit gas temperature, etc., which will affect the heat flow density from a soaking pit to an ingot surface. So the heat flow density q(T,, Tp) is the most difficult term to determine in equation (1). The pit wall temperature and the pit gas temperature are assumed to be identical and the following equation has been adopted to calculate the heat flow density in this paper:

e~ = q~,,e. + % . e .

(4)

+ r,((i- l)Ar, ~ t ) (~)' and applying the energy balance principle on the boundary sections (section 1 and section /9"), the following equivalent discrete-time model is obtained: T,(t,k+ I) -- (I- 2a)T,(l,t) + ~T,(2,k)

T,(i,k + 1) = aT,(i- Lk) + (1 - (2 +l)(x)T,(i,k)

q(T,, Tp) = e.tc [(Tr (t) + 273)' - (T, (R,t) + 273)'] + f~,[(T,(t)- T,(R,t)]

8r Ar ZT,(r,t) = T,((i + l)Ar,/cAt)-2T(iAr, kAt) ~r~ (At)2

l

(2)

+(I+I)ctT,(i +l,k);

i = 2,3..... N - 1;

1

(5)

Modelling and Simulationof a Soaking Pit/RollingMill

BT,(r.t)= 1 ~ r ( g r BT,(r,t)~, r,(t)< r < R;

T,(N.k+ I)= 2(N- I)ocT,(N_ l.k) (2N-I)

~t

+ (I- 2(N- I)cz)T,(N _ Lk) + (2N-I)

cpr

~r

)"

T,(r,t~,s~¢o= 1",;

dr,(t) (

1365

K ~T,(r,t)l,.~,,

dt = pH,

-)

~T, (r,t) 1,., where ~ = K a t / ( c p A r ' ) , and T,(i, k) is the abbreviation of T, (jAr, hat).

~r

~r

'|

(8)

=q(T,(R,t),Tp(t))l K

T,(r, to)=T,o(r) r, (o) = r,o

Choosing the state vector as x(k) = [T,(1.k).T,(2.k).....T,(N. k)]r, and the input vector as u(k)= T,(k), equation (4) can be written

as the following discretestate-spacemodel:

x(k + 1) = f, (x(k), u(k),k) x(O) = Xo. k = 0,1,2....

(6)

Similarly, equation (8) can be discretised and the resulting discrete time state space model takes the form: X(k + l) = f , (X(k),u(k),k) X(0) = X0.

(9)

where j~ is an N dimensional non-linear vector function, and xo is the initial condition of the state vector.

where X=[x',r,]' is the extended state vector containing both the ingot temperature distribution and the size of the steel liquid core.

Note that eqn~!!ons (1.6) are only valid for ingots without molten liquid core. Occasionally, there is the possibility that hot ingots coming directly from the upstream steelworks contain a liquid core, e.g. when the ingot is big and the track time is relatively short. An ingot with a liquid core can be divided into two parts: a solid part and a liquid part. The ingot temperature for the solid part still satisfies equations (1-6), while the thermal behaviour of the liquid part can be best described by the moving of the liquid/solid boundary because the temperature of the liquid part will be maintained at its melting point. Suppose that the liquid core occupies the centre of the ingot with a radius of rt, and the steel solidification takes place only at the boundary, then the radius of the steel liquid core will change with time as the solidification progresses. Applying heat balance at the liquid/solid boundary, the following dynamic equation is obtained, which is similar to Lu and Williams" result (1993a):

Equation (6) and equation (9) can be merged into the following model:

dr~(t) dt :

K ~T,(r,t)l,..~o> pn,

r, (0) = r,o

~

l

(7)

T,(r,t~ ,~,,,,, = I". where lit is the latent heat of the molten steel, T,m is the steel melting temperature, and rto is the initial radius of the steel liquid core. The overall thermal behaviour of the soaking process when a ingot contains a partial liquid core is described by the following equation:

x(k + I) = F(X(k),u(k),k)

X(0) = Xo,

(I0)

where F is a switch-able non-linear vector function defined by:

~'[f:(x(k),u(k),t),0]', /fr,(k)= 0 FO = [f~(X(k),u(k),k), if r~(k)> 0

(11)

Equation (10) describes the dynamic behaviour of the ingot soaking process, where the ingots can be completely solidified or contain a liquid core. The above continuous dynamics can also be modelled using ECPNs by simply treating the single calculation of equation (10) as an event, and implementing it as a dynamic transition (Yang et al., 1994). The whole soaking process can be divided into three parts: the start of the soaking process, soaking the ingot, and the ending of the soaking process. When the soaking pit is fully charged (i.e., a batch of K , o ~ ingots has been charged), the soaking process starts. During the soaking process the ingot temperature distribution and the size of the steel liquid core are governed by equation (10). After the average ingot temperature reaches the requited 1 N

level ( ' ~ ( k ) = - ~ T , ( i , k ) > T , ou) and

the ingot

temperattm~ difference is small enough (Ax = max [T,(1,k), T,(2,k), .... T,(N,k)] . rain [T,(I,k), T,(2,k), .... TIN, k)] < AT, ou), the soaking process ends and the ingots in the pit are ready for discharging. The

Y.Y. Yang et al.

1366

resulting ECPN model for the soaking process is shown in Fig. 7.

I

represented by the withdrawing activity alone, and the we-post conditions of withdrawing hot ingots out of the preheat furnaces are outlined in Table 3. The process of taking a hatch of Ker~,~ &hot ingots from the preheat furnace and putting the same number of cold ingots into the preheat furnace will take a time interval of xt,~,4, which is a random variable with a certain probability distribution. The corresponding ECPN model for the preheat furnace is shown in Fig. 8. £inSot < Ksoakb~8

Fig. 7. ECPN model for the soaking sub-process

x(O) ffi ~reheat

~, PT: Ingot In Cold Ingot-

. •

/,

'~

.

'

PS: Hot lagot In

' "l']'r91:HeatIl~olff

PltIl~At'ea

3.5. The Heating of lngots in the Preheat Furnaces Two preheat furnaces in the soaking pit/rolling mill process act as an additional hot ingot source When there are insufficient hot ingots in the pit area to charge an empty soaking pit, hot ingots in batches of Kp,,h~,s are drawn out at the hot end (exit) of the preheat furnaces, and the same number of cold ingots are put into the preheat furnace through the cold end (entry) to keep the furnace fully filled. These hot ingots from the preheat furnaces are then charged into the soaking pit with a high priority. Cold ingots are heated in the preheat furnace via radiation, convection, and conduction while they are gradually moved from the cold end to the hot end in the preheat furnace. The temperature of the preheat furnaces is controlled in such a way that when the ingots reach the hot end, their temperature is approximately 700 ° C, similar to those coming directly from the upstream steelworks. The ~ m e method as described in Section 3.4 can be used to establish the preheat furnace model by considering both the continuous dynamics and discrete events. However, a simplified version has been used here since the preheat furnaces act as an auxiliary equipment in the soaking pit/rolling mill process. Assume that the temperature of the ingots is constant (denoted as Tpr,~,,t) when they arrive at the hot end of the preheat furnace, despite their residing time and heating history in the preheat furnaces. The behaviour of preheating furnace operation can be Table 3. Pre-Post Conditions for Preheat Furnaces Pre-comations

Post-coaOltions

Withdraw I n g o t o u t o f a P r e h e a t Furnace

Hot ingots in the pit area are insufficient for charging a

soakingpit; Kw,~en~ ingots are available in the cold ingot bank; A bogie is available for putting cold ingots into the furnace.

ingots is &awn from the

furnaceto the pit area; Releasethebogie: Updatethe ingotattributes.

I

I

,

W - -

~-- : l gpreheatln 8

,

p19:. Bogie Avaiinble ,

r~mu~n

L] @ "f trtm4

•ilpl,

r

I

I Kprekeatin8

Fig. 8. ECPN model for the preheating sub-process 4. SIMULATION OF THE SOAKING PIT/ROLLING MII J. PROCESS The ECPN model of the soaking pit/rolling mill complex developed in Section 3 is implemetit&l in the Design/CPN software environment (Meta, 1993). Design/CPN is an interactive tool for performing modelling and simulation with colonred Petri nets. It provides: an editor for creating and manipulating colonred Petri nets (CPNs), a simulator for executing CPNs, interactive monitoring and debugging capabilities, and facilities for displaying simulation results. In Design/CPN, tokens are CPN data objects. Every token used in a CPN must be one of the colour sets declared for the CPN. A place is a location that can contain tokens of a particular colour set. All tokens in a place must have the same colour set. A place may optionally have a name and an initial token set, with the default initial token set of empty. The transitions in Design/CPN are more powerful than those m conventional Petri nets (Jensen, 1992; Murata, 1989). A transition can have a name field, a guard field, a code field, and a time delay field. The guard field is used to set up the additional conditions for the enablement of the transition, while the code field is used to specify what extra functions should be carried out when the transition is fired. An arc in Design/CPN is a connection between a place and a transition. Every arc has a direction, either from a place to a transition (called an input arc) or from a transition to a place (called an output arc). An arc has an arc inscription attached, which is used to specify the pre-conditions (for input arc) or the postconditions (for output arc) for the related transition.

Modelling and Simulation of a Soaking Pit/Rolling Mill In Design/CPN, the criteria for the enablement of a transition are: each of the input places of the transition contains the tokens specified by the transition's input are inscription, and the Wansition's guard field evaluates to true. When a transition is enabled it can be fired; otherwise the transition cannot be fwed. When a transition is fired, the following actions will be taken. First, it will bind the tokens in the input places to their input arcs according to the arc inscriptions. These bound tokens will be removed from the input places. Then, it will evaluate each output arc inscription and put the resulting tokens specified by the output arc inscriptions in the corresponding output place. The functions specified in the code field will also be executed if the transition's code field is not empty. For those transitions with a time delay field, the tokens produced by the transition in the output places will not be available for the following transition enablement until the specified time delay has passed. The soaking pit/rolling mill model has been implemented hierarchically in Design/CPN as shown in Fig. 9. The top level is an overview of the whole process, while the bottom level is the detail CPN model of the associated event (component). The middle level is devoted to the key individual subprocesses, such as the arrival of hot ingots, the ingot soaking, the ingot rolling, etc. Due to the flexibility of the hierarchical structure, it is possible to extend the model to any desired level of detail by expanding the relevant hierarchies. lhe $cskl

Pit-

Fig. 9. Hierarchy of the soaking pit/rolling mill model In order to carry out the simulation effectively, a man/machine interface has been designed to facilitate the interaction and intervention for various computer simulations. An input file is also designated to provide all the simulation environment variables as well as the model parameters which can be varied from one simulation to another. Output fries are created for the purpose of post-simulation analysis due to the limited analysis capability within Design/CPN environment. Critical performance and system behaviour can be displayed on-line while the simulation is being carried out. Fig. 10 shows the flowchart of the soaking pit/rolling mill model.

1367

ECPN Model ] Initialization

t Variables L.

["

fOutput

n~

r m - - - ] ofECpN Sinmlatice

I

I M°da

Fig. 10. Flowchart of the simulation model To study the behaviour and characteristics of the soaking pit/rolling mill complex, a series of computer simulations have been carried out for different working conditions and system configurations. The default parameters and environments of the soaking pit/rolling mill process are listed in Table 4. Table 4. Default Model Parameters and Simulation Environments Para-

I~nlt

Para-

meters

Vakns

meters

De/salt Valmss

'~.,z

6000 s

T,~

1250 °C

%,,~

60 s

AT,~,a

50 °C

"ra~,~,~

90 s

T,~

30 Oc

x,~/ql

180 s

T~o.,~

700 °C

't~,ul

1260 s

At

60 s

X~2

260 s

c

470.0 J/Kg-°C

1:t,a~

180 s

p

7500.0 Kg/m 3

1;t,~

300 s

K

27.9 W/m-°C

K~

150

Csw

0.4

30

tpq

0.3

K,,~..

30

~

0.7

K.ffi,a

30

fz.

12.0

K,~am

10

o

5.67x108 W/m2 °K4

Kt,~,n ~

4

Tp(t)

1350 °(2

The simulation is supposed to end when a total of 30 cast ingot batches with the dimension of 600x600x1500 (nun) have been processed by the slabbing mill. All the ingots are completely sofidified when they arrive at the soaking pit area. A cold start initial condition is selected, i.e., all of the resources (equipmen0 are free and available at the initial time. The normal system configuration is: 18 soaking pits, 2 cranes, 2 preheat furnaces, 1 slabbing mill. Simulation results for seventeen soaking pits,

Y.Y. Yang et al.

1368

fourteen soaking pits, ten soaking pits, and eight soaking pits are shown in Figs. 11 and 12, with other process parameters specified in Table 4. Typical ingot temperature behaviour during the soaking period is shown in Fig. 13.

Ii'

Fig. 11 shows that when the total number of soaking pits in use exceeds fourteen, the slabbing mill is the bottleneck of the whole soaking pit/rolling mill process, and that effort should be focused on how to increase the rolling speed of the slabbing mill in order to achieve the maximum production throughput. No significant performance

8 Pits

+i+ i

1#

!

iii

4#

7#

+

~ii 10#

13#

~

17 Pits

3

j2

0 8 Pits

10 Pits

14 Pits

17 Pits

(b) Effects on the average mill waiting time

l

I!

16#

(a) 17 soaking pits

400 3OO 2oo loo 0

14 Pits

(a) Effects on the total production time

,¢

~i

10 Pits

SPits

10 Pits

14 Pits

17 Pits

i

(c) Effects on the average throughput Fig. 12. Effects of the total number of soaking pits 1#

4#

7#

1~

13#

16#

(b) 14 soaking pits

i =k......-...w+,i,l =IIIIIIII o

1#

4#

7#

10#

13#

improvement is obtained by increasing the soaking pit number beyond fourteen. However, when the number of the soaking pits is below fourteen, the soaking pits are becoming the crucial factor. In this case the improvement of soaking pit operation becomes vital to the production throughput, as is clearly indicated in Fig. 12. Fig. I l, also indicates that the average soaking time does not change significantly with the total number of soaking pits in

16#

(c) 10 soaking pits

1400 1300 ,.,. 1200

ImS " l

4#

7#

lO#

1100 1000 900

[ 1°3111--1#

~

801) 13#

16#

700 1

(d) 8 soaking pits Fig. 11. Average pits working/waiting time versus total pits in use

101

201

301

Time (Mill.)

Fig. 13. Ingot temperature profile during soaking

period

M o d e l l i n g a n d S i m u l a t i o n o f a S o a k i n g Pit/Rolling Mill

use. This is due to the fact that the average soaking time (or soaking pit working time) is determined by the time required to raise the ingot temperature to its pre-spectfied level, which is mainly determined by the soaking pit temperature profile and the initial ingot temperature. The change of total soaking pit number does not affect these factors directly. To study the influence of the mill rolling rate on the average throughput and the soaking/waiting time of the soaking pits, the total number of soaking pits is fixed at eighteen, and there is no preheat furnace available for preheating the cold ingots. The (hot) ingot buffer capacity Kt~¢is set to infinity because of the assumption of no preheat furnaces, other conditions for simulation being the same as described above unless otherwise stated. Various mill rolling rates are simulated by Design/CPN, and typical simulation results for the mill rolling rate of 1, 2, 3, 4 minute/per ingot (or, equivalently, 0.25, 0.33, 0.5, 1.0 ingot/minute) are shown in Figs. 1415. Fig. 14(b) shows that when the mill rolling rate is greater than 0.5 ingot/minute, the soaking pits are the critical equipment in the whole process. The average waiting time of the soaking pit, when the

600

1369

80 60 20 0 0.25

0.33

0.5

1.0

Ingot/Min Ingot/Min Ingot/Min Ingot/Min (a) Effects of the mill rate on the production time

'::to.:

A 0.25

0.33

0.5

1.0

Ingot/Min Ingot/Min Ingot/Min Ingot/Min (b) Effects of the mill rate on the mill waiting time

! 20 15 I0 5 ~, 0 "¢

0.25 0.33 0.5 1.0 Ingol/Min Ingot/Min Ingot/Min Ingot/Min

(c) Effects on the average throughput

Fig. 15. Effects of the different mill rolling rates

<

440

,. ,. ,. I ". : 3 5 7

:

"- : : : : "- : -" "- "9 11 13 15 17

Pit N u m b e r

(a) Effect of the mill rate on the pit soaking lime 400

~

0.25IngotfMin

300 2o0

loo ,< 0 3

5

7

9

11

13

15

17

Pit N u m b e r

(b) Effect of the mill rate on the pit waiting time Fig. 14. Average pit soaking/waiting time versus mill rate

mill rolling rate is greater than 0.5 ingot/minute, is very small, indicating the soaking pits are always busy soaking the ingots in order to provide enough ingots for the rolling mill. Since the soaking pits are the critical equipment in the whole process, increasing the mill rolling capability beyond 0.5 ingot/minute makes little contribution to the overall throughput. Fig. 15(c) shows that the average output increased only by 1.19 ingot/hour (from 18.79 ingot/hour to 19.96 ingot/hour) when the mill rolling rate was doubled from 0.5 to 1.0 ingot/minute, whereas the mill average waiting time increased significantly, i.e., from 0.67 minute to 1.49 minute. It is clear that the increased mill rolling capability is mainly wasted by idle waiting, and situations like this should be avoided to make the full use of all kinds of production equipment. This can be achieved by increasing the soaking pit capability via adding extra soaking pits or intensifying the soaking operation. When the rolling mill rate is below 0.5 ingot/minute, the rolling mill is the bottleneck of the process and efforts to raise the mill rolling capacity up to 0.5

Y.Y. Yang et al.

1370

ingot/minute will be properly rewarded. This can be seen from Fig. 14 and Fig. 15. For example, when the mill rolling rate is raised from 0.25, to 0.33, to 0.5 ingot/minute respectively, the average soaking pit waiting time is significantly reduced (see Fig. 14(b)) and the average throughput is increased from 13.34, to 16.47, to 18.79 ingot/hour, respectively. The average mill waiting time increases significantly when the mill rolling rate reaches 0.5 ingot/minute, indicating that now the rolling mill is becoming a critical part of the whole process. It is interesting to note in Fig. 14(a) that when the mill rolling rate increased from 0.25 to 0.50 ingot/minute, the average soaking time, i.e., the average time needed for a soaking pit to provide a batch of hot ingots for the rolling mill, decreased accordingly although the soaking pit capacity remained the same. This is caused by the reduced track time (the time period from when the ingot is stripped from the mould to when it is charged into a soaking pi0 due to the increased mill capacity, which affects the average throughput. The longer the track time, the cooler will be the initial ingot temperature when it is charged for soaking, and the more difficult it is to be soaked. However, there is no big difference in the soaking times for mill rolling rate of 0.5 and 1.0 ingot/minute, since the average throughput does not increase much, and neither does the track time. This phenomenon shows that in a hybrid system, the relationship among different factors is very complicated and often cannot be predicted by simple reasoning. Simulation is very useful for revealing these hidden relationships. Simulations can be designed easily to provide other useful information about the soaking pit/rolling mill process, such as those related to the cost behaviour, working element queue length, etc. They are very useful for providing guidelines for production improvement and operation planning through properly designed experiments. These aspects will not be discussed further in this paper. 5. CONCLUSIONS In this paper a mixed-mode model based on ECPNs for a soaking pit/rolling mill complex has been established. The unique characteristic of the ECPN model is that the discrete event part and the continuous variable part are modelled in a unified way which allows the full consideration of the interactions between these two different parts. The description of the soaking pit/rolling mill complex is based on a real industrial enterprise near to Sheffield, UK. Sub-models for the critical components in the complex, including the soaking pits, the rolling mill, the preheat furnaces, the

cranes, etc., are derived, with detail procedures and illustrations. Techniques for the model implementation using Design/CPN are also given. Various computer simulations have been conducted on a Sun SPARC workstation in order to study the characteristics and behaviours of the soaking pit/rolling mill complex. Effects of the operating patterns of different equipment and the number of available resomces are also studied by simulation. From these simulations, it is possible to determine the critical equipment (bottleneck) of the whole process and to identify the most suitable operation pattern which provides useful guidance on improving the soaking pit/rolling mill operation. The modelling approach established here is also applicable to the similar continuous coating/rolling process, provided that the relevant pre-post conditions and the continuous dynamics of the casting machines are available. There are other benefits to be gained from using this soaking pit/rolling mill model. One potentially useful study would be to perform operation optimization and production scheduling based on the developed model. Here, perturbation techniques and heuristic approaches are promising in order to find out the optimal working patterns for the individual equipment. Also, model-based production scheduling is likely to produce considerable performance improvement. These will be the subject of future research studies. 6. ACKNOWLEDGEMENTS The authors wish to thank Mr. Coldwell of the Stocksbridge Engineering Steel, Mr. I-Iibberd and Mr. Oates of the British Steel Swinden Lab. for providing the detail data and specifications of the soaking pit/rolling mill process and for their helpful discussions. REFERENCES Ashour, S., and Bindingnavle, S., G. (1972), "An optimal design of a soaking-pit rolling-mill system", Simulation, June, 207-214. Jensen, K. (1992), Coloured Petri Nets -- Basic Concepts, Analysis Methods and Practical Use, Vol. 1, Springer-Verlag. Lu, Y. Z., and Williams, T. J. (1983a), "Energy savings and productivity increases with computers -- A case study of the steel ingot handling process", Computers in Industry, 4(1), 1-18. Lu, Y. Z., and Williams, T. J. (1983b), Modelling, Estimation, and Control of Soaking Pit .- An E.rample of the Development and Application

Modelling and Simulationof a Soaking Pit/RollingMill

of Some Modem Control Techniques to Industrial Processes, Published by Instrument Society of America, New York, USA. Lumelskey, V. J. (1983), "Estimation and prediction of unmeasurable variables in the steel mill soaking pit control systems", IEEE Trans. on Automatic Control, AC-28(3), 388-400. Meta Software Corporation (1993), Design/CPN Tutorial for X-Windows, Version 2.0, 1993 Meta software. Murata, T. (1989), "Petri nets: Properties, analysis and applications", Proceedings of IEEE, 77 (4), 541-580. Patel, C., Ray, W. H., and Szekely, J. (1976), "Computer simulation and optimal scheduling of a soaking pit-slabbing mill system", Metallurgical Transactions, 71], 119-130. Rao, T. R. S., Upton, E. A., Rupar, D. L., and Ellis, R. (1984), "Optimization of ingot heating in

1371

soaking pits", Iron and Steel Engineer, 61(10), 34-38. Yang, D. R., Wang, F. Y., Chang, K. S., and Lee, W. K. (1988), "Optimal control of temperature and fuel consumption for soaking pits in a steel plant -- A distributed parameter approach", Optimal Control Applications and Methods, 9(4), 417-442. Yang, Y. Y., Linkens, D. A., and Banks, S. P. (1994), "An unified approach for modelling of hybrid systems", Proceedings of the European Simulation Multiconference, Barcelona, Spain, 1-3 June, 240-244. Yang, Y. Y., and Linkens, D. A. (1994), "Extended coloured Petri nets and its application in mixedmode system modelling", Submitted to the Workshop of Hybrid Systems and Autonomous Control, Cornell University, New York, USA, 28-30 October.