- Email: [email protected]
Continuous Aspect in Petri Nets
copyright © !FAC Management and Control of Production and Logistics, Campmas, SP, BrazIl, 1997
PRODUCTION LINES MODELING BY DISCRETE I CONTINUOUS ASPECT IN PETRI NETS Isabel Demongodin·. Fran~ois Prunet"
*Depanment of Automatic Control and Production Systems - Ecole de Mines de Nantes La Chantrerie - 4, rue Alfred Kastler- BP 20722 - 44307 Nantes - FRANCE Tel: +33 (0)2.51.85.83.27 - Fax: +33 (0)2.51.85.83.49 - E-mail: [email protected] **Laboratoire d'1nformatique, de Robotique et de Microelectronique de Montpellier 161, rue Ada - 34392 Montpellier - FRANCE Tel: +33 (0)4.67.41.85.14 - Fax: +33 (0)4.67.41 .85.00 - E-mail: [email protected]
Abstract: To analyse and evaluate performances of high throughput production lines, Hybrid Petri nets are a powerful mathematical model. However, they are not accurate enough to represent, the dynamic behavior of the high speed accumulated system and the interaction between the physical part and the supervisory part. Thus, Batches Petri nets have been proposed as an extension of them. In this paper, a comparison between the Hybrid PN, and Batches PN are shown on a single production line. At the end of this study, an extension of Batches Petri nets for the control I supervision is presented by including the captor model. Copyright © 1998 IFAC ' Keywords: Discrete Petri nets, Hybrid Petri nets, Batches Petri nets, modeling, analysis, simulation, mixed production systems, flow of parts, supervision.
of Hybrid Petri nets, are defined. By the notion of internal coherent batch, in other words by a set of parts with the same density of distribution, they introduce a mathematical formalism of the parts flow circulation. This formalism is associated with a new kind of nodes, named batch nodes.
1. INTRODUCTION
A major step in the effort to enlarge the modeling power of Petri nets has been the extension known as the «Continuous Petri Nets». The motivation was the inability for successful modeling of discrete event systems with a large number of events. Thus, a continuous approximation was proposed to replace the counting of discrete variables with a flow approximation. Continuous Petri nets are thus approximations of discrete event systems allowing, basically, faster simulation of the latter without sacrificing the accuracy. The combination of Continuous with Discrete Petri nets leads to the «Hybrid Petri Nets» concept. ~n production domain, to optimize line throughput, it IS necessary to study the parts flow circulating between machines. This parts flow depends on the machines state, variations of element speed, and especially on the transient behavior of elements which assume moving and storing of products (accumulation conveyors, for instance). In order to represent transient behavior of transfer elements, Batches Petri nets, which are an extension
For the understanding of this paper, we assume that Petri nets theory and their concepts are known. First, high throughput production lines and an example are presented with a mixed point of view. This leads us into the presentation of hybrid Petri nets and batches Petri nets models. Some contributions of these Petri nets in the modeling of mixed systems, such as multibelts conveyors, are examined. Through the study of the example model, the applicability and the principles of modeling of these nets are illustrated. Quantities of products and their dynamic variations are compared for the hybrid Petri net and batches Petri net models. In the last section, we introduce the modeling of sensors in batches Petri net, which illustrates the interaction with the control part. Before concluding, extensions of batches Petri nets and
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decision, scheduling and so on, of manufacturing systems (Desrochers, et aI., 1994). They are powerful enough to generate analytical and theoretical results, and then properties are more and more studied. In the Discrete PN, two nodes are defined : the discrete place and the discrete transition. The net marking, a vector built up with the marking of the places (positive or zero integer) describes at a certain moment the Petri net state or more exactly the state of the system modeled by the Petri net. In order to describe not only what happens but also when it happens in the system, non autonomous PN have been developed; they enable to take time events into account. Then performance evaluations, conception and control, can be optimized. Usually, for a production line, machines and buffers with time dependent behavior, are very easy to be modeled via timed discrete PN. However, the multibelts conveyor is not so more studied on the accumulation length variation point of view. It could be modeled with a timed Petri nets where this transfer zone is split into cells (Bourrieres, et aI., 1988) (fig. 2). Then a product can move from one cell to the next one, if it is empty. This dedicated model is also called a FIFO system (First In - First Out).
generic models are introduced. They permit to obtain an uniform representation for high throughput production systems and to make design issues and procedures more clear.
2. PETRI NETS FOR HIGH THROUGHPUT PRODUCTION LINE MODELING IN TRANSIENT BERAVIOR WITH DISCRETE / CONTINUOUS EVENTS The high throughput production lines are composed of machines in series and conveyance systems which link the machines between them. The considered system is liable to a high number of short failures which lower the productivity of the whole system. The objective of the supervisor is to mask the short failures by using the possibilities of accumulation allowed on conveyors and that enlarges the productivity of the line. For our purposes, mixed systems are considered as combination of continuous/discrete event plant (such a bottling line) with a discrete-event supervisor which reacts to external events (planned or unforeseen). Then, the allocation decisions are discrete in nature while the technological process is hybrid. To illustrate different mixed approaches, a single physical process is modeled through the different Petri nets.
2.1.
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Fig. 2. Conveyor system with N cells
We consider a high throughput production line composed by accumulation systems. These systems are mono-driving speed, unidirectional, high capacity belt conveyors which move identical parts by friction. The manufacturing system, shown in fig. 1, consists of two machines M 1 and M2 with no failures linked by an accumulation belts conveyor. The first machine Ml has a throughput of VI = 70 parts/ min., and a buffer storage SI feeds it. In this stock, 40 identical products are placed every 4 minutes. Characteristics of the accumulation conveyor are: length = 15 m. ; maximal capacity = 225 parts ; speed = 5 mlmin ; maximal density = maximal capacity / length: d max = 15parts/m. This conveyor supplies the machine M2. This last one has a throughput V2 30 parts / min. The output of the second machine supplies an infinite buffer storage S2. Initially, all products, 175 parts, are in the first buffer SI and all the others elements are empty.
But, due to the large number of states in the dynamic behavior, a compromise between simulation speed and result accuracy must be found for this kind of discrete representation. Consequently, only a transfer delay is associated to the conveyor modeling, and the information linked to the product's position is lost.
2.2.
The main feature of Hybrid Petri nets is that they combine the advantages of aT-timed discrete PN for the representation of events (failures, decisions, ... ) and those of a continuous PN for the representation of flows (Alia, et aI., 1991). Continuous PN model considers constant firing maximal speed, and the notion of positive or zero real numbers representing marks in places, can describe, for example, tanks filled by liquid. It is the simplest continuous PN model because it is very easy to be simulated. Its extensions (asymptotic, controlled (Dubois, et aI., 1994)) and itself, allow modeling of some continuous linear systems and provides a good approximation of the behavior of many discrete-event systems modeled by regular discrete PN. In the context of this model, few events have to be considered, even when it approximates a timed Discrete PN with many reachable markings. These events are obtained from linear equations.
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Fig. 1. Example - production In the field of production lines, discrete Petri nets and their extensions (controlled, colored, object, stochastic, ... ) are used to model and analyze the behavior, properties, parallelism, synchronization, 200
While using Hybrid PN, the conveyor belt is no more considered as a discrete element but as a continuous element with input and output flows (Brinkman and Blaauboer, 1990). The real markings represent the quantity of « liquid » contained in places describing the conveyor. Then, the maximal firing speed, indicates how much the « valve » is opened (fig. 3).
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systems, consisting of discrete event or logical part cooperating with a linear continuous part is possible through a single place. Intrinsic characteristics are associated to this type of node, performing characteristics of mixed production elements. For a conveyor belt or a portion of a conveyor belt, a length, ( positive real constant), a maximum density, a driving speed, ( dis-continuous variables), are associated to the batch place, and describe the characteristics of conveyor. Marking value in this Petri net is an integer for a discrete place, a positive or null real number for a continuous place, and a set of internal coherent batches for a batch place. An internal coherent batch (LCIc ) represents a set of parts with the same characteristics of distribution on a continuous element C during a time interval (fig. 5). LCI c is characterized by three continuous variables (length, density, coordinate) in the batch place. This concept has proven to have an accurate representation of the real system (Demongodin and Prunet, 1993a). Associated delays to discrete places and associated firing flows to continuous and batch transitions, permit to take the notion of time into account in Batches PN. Conditions of enabling depend on the marking state of discrete and continuous places (reserved marks and non reserved marks). Notions of weakly and strongly enabled transitions, introduced in Hybrid PN, are extended for a batch transition. The conditions of enabling depend essentially on internal coherent batches characteristics, composing the marking of the batch place. For the firing of discrete and continuous transitions, marks (positive real or integer) are removed and added, respectively at their pre and post discrete and continuous places. For the firing of a batch transition, internal coherent batches are created or destroyed inside a batch place. Evolution algorithm of a Batches PN, carries out the evolution of batches inside a batch place in the first step, and then carries out the evolution of the places state, according to enabling transitions in the second step. Then linear variations and events inside a single place, allow a mixed modeling. A conveyor is characterized by the 5-tuple C=(L,V,dmax,e,s)' i.e. by its length, its driving speed, its maximum density, its input flow and its output flow (fig. 5).
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Fig. 3. Continuous representation of the conveyor It is obvious that the markings variations are proportional to the conveyance speed V, but also on the maximal density. Then the maximal firing speed is done by V.d max indicating the maximum valve flow. As the evolution in time of high throughput production line can be approximated by a continuous linear function, when the quantity of tokens is very large, we can establish the hybrid Petri net model describing the example (fig. 4). Stocks SI and S2 are TI modeled by continuous dl =4 places PI and P5. and the discrete tranSl!!on TIP I represents the supply of 40 parts every 4 min. T2 Continuous transitions T2 v2 =70 and T4 are associated with machines MI and M2 where the firing speeds are P2 V2 and V4 respectively. The conveyor is represented T3 by P2 and P3. A continuous v3=75 transition T3 models the transpon speed. its maximal P3 value equals V3 = 75 p/min .. in other words V3 = T4 V.d max =5 x 15. Place P4 V4 =30 limits the capacity of the conveyorC. Fig. 4. Hybrid Petri net model for the example
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Batches Petri net model
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Batches Petri net (Demongodin, et al., 1993b), a new member of the PN models family, is an extension of the Hybrid one, developed for discrete-event systems with a large number of parts. The Batches Petri net, defined through the introduction of a new kind of place and transition, namely the batch place and the batch transition. permit the modeling of discrete Continuous-time dynamic processes represented by systems of ordinary linear equations and discrete models. In this way, a unified representation of mixed
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while it needs 24 events for the B-PN. It is a good rapidity, for the high accuracy. The difference between the number of products circulating on the conveyor (fIg.7c), comes from the fIring speed and fIring flow calculation, on one hand. Thanks to the firing variables associated with continuous or batch transitions, systems variables are approximated by piece-wise constant functions (fIg. 7b, 7d). Higher the fIring variable is, worse the linear slop of system variables is (fig.7c). When the initial phase (date=lOmin.) is overstepped, the conveyor behavior is cyclic. The fIring flow of the upstream C-transition and B-transition of the conveyor models are equivalent after this functioning mode. In this phase, these transitions are saturated and their fIring speed and fIring flow are maximal.
the maximum output flow is the throughput of the machine M2. P4 is a batch place with accumulation. It contains the internal coherent batches of the conveyor, and represents the conveyance of the parts. P3 is a continuous place representing the maximum capacity of pans that the conveyor may convey. It acts like a semaphore. If the conveyor is empty, the marking of P3 is equal to the maximum capacity (as in the initial moment), it decreases as the conveyor receives parts, and increases when parts go out of the conveyor. Then, when the conveyor is full , this continuous place is empty and the transition T2 is not enabled (or weakly enabled).
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Comparison of Petri net models
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A simulator for Batches and Hybrid Petri nets, Simuleau developed at the LIRMM, permits to generate the evolution graphs of the Petri net. In this work, the fIrst aim of simulation is to test and estimate the accuracy of the approximation, by comparing markings behaviors of the mixed PN models. Therefore, for a pure discrete approach, a very complete study, comparing the discrete and continuous PN models of manufactured lines are explained in (Zerhouni, et aI., 1990). Thus, only mixed Petri net models are analyzed in this paper. Then for each place of the Hybrid PN and Batches PN, the markings values for buffers SI and S2, and for the conveyor are compared.
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Fig. 7. Comparison between H-PN and B-PN The variation of batches densities, constant over the intervals of time, characterizes the situation, and a delay is added to the accumulation time of products (fIg.7c). This delay permits to represent more accuracy the transient behavior. We can also remark that the marking tends towards its stationary value at each period. Obviously, the influences of SI and Conveyor on the buffer S2, are correlated, since both influence the place marking and the fIring flow/speed calculation. In the H-PN, if a continuous place is empty, the fIring speed of its downstream . transitions depends on '" /./•. ..... ./ ...... ..the fIring speed of its '" upstream tranSItIons . '" ....the Consequently, ../ instantaneous parts' l ..··: flow through machine " M2 is not null. For the Fig. 8. Variation of S2 B-PN the respective batch transition has a null fIring flow. Then parts arrive at the output of the conveyor after 3 minutes, corresponding to the minimum transfer delay to pass from M1 to M2.
Dynamic behavior : evolution of the quantities of parts: In order to study the dynamic behavior of the example and the corresponding mixed Petri net models (H-PN and B-PN), different simulations with a duration equals to 20 min. have been executed. A solid line, represents the B-PN behavior, while a dash line represents the C-PN functioning. The buffer SI is represented by the place PI in each PN. The dynamic of this stock is shown in fIg.7a. The marking, representing the quantity of products inside SI, has a cyclical behavior after the initial functioning mode, at date 3 min. After this moment, its period is equal to 4 minutes, corresponding to the supplying delay. We see (fIg.7a) that variables have exactly the same behavior for H-PN and B-PN besides their accuracy is different, but in both models, this buffer is represented with a continuous place. Then, this variation illustrates the principle of interaction between the discrete part net and continuous part net.. In fIgures 7b, 7c, 7d, the difference between H-PN and B-PN leads to the number of events happening during 20 min. : for the H-PN we get 14 events to carry out the behavior,
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For the performance evaluation point of view, Batches Petri net is a good model as the state and the quantity of products circulating inside the production system can be known at every time. The average occupation of stocking, of busy times, and so on, can be carried out. Also, from the evolution graph, it is possible to analyze cyclic behavior, performances, etc. On the other hand, continuous flow models for design optimization of transfer lines are more and more studied (Suri, et aI., 1993; Dallery, et aI. , 1996) by analytical models. Modeling and simulation of a production line with many conveyors may be easier since Batches Petri nets allow to defined only one place by mixed model without sacrificing the accuracy.
Results analysis: The modeling with H-PN constitutes an approximation of the system mode led by timed discrete transition but batch model allows to fix the accuracy of the results; it doesn't depend on the cells concept. Concerning simulation time, the approximation becomes more efficient when the place markings increases for H-PN and D-PN if the conveyor is represented with n cells. Actually, with a modeling by discrete timed transition Petri nets, more events occur, since every time a token is transfer from a place to another one, an event happens. In a discrete event model, we get an explosion of the states number, because the time delay on the conveyor must be split in small intervals to get a sufficient accuracy. An other reason concerns the operations of the machines where only one part could be manufacturing at a time. As a consequence, the conveyor and machines models introduce the implicit limitation on the D-PN. So, the number of discrete events and consequently of states, is very large. In the Continuous PN model, the number of state changes to be handled during the dynamic behavior is not only smaller than that of the corresponding discrete model, but is also the minimum achievable. Consequently, Hybrid PN model contains the necessary minimum amount of information to describe the state of the system. A modeling of conveyor belts based on this tool has be established and studied. But the results show that this model considers the conveyor belt as a transfer (pure delay) only and does not take account of delay and accumulation phenomena in transient behavior. In fact, in the H-PN model, it is assumed that as soon as a part is introduced in the conveyor, it is available to go out. This is not true on the real conveyor because of the conveying delay of the parts. As the authors said, this model over-estimates the throughput when the system is not saturated. More accurate model, an extension named Batches PN has been introduced to take this possibility into account. In the Batches PN, the batch place performs an accurate modeling of the belt conveyor due to its mixed aspect. The conveyor belt is characterized by its length, driving speed, maximum capacity, input flow and output flow through the batch place and its upstream and downstream batch transitions. From the above concept, we have formalized the state of the batches as a function of time, switching between different continuous models. In this way, batches defined inside this batch place correspond to a discrete linear variation of continuous variables corresponding to products. The global model takes into account the state of the transfer element, in accumulation behavior or in free handle behavior, the state of the driving speed variations which can occur at any time and, the state of upstream and downstream connections with other physical ~Iements. Simulation of high throughput production hnes modeled by B-PN (Jive very eO'ood results. A . SImple model of a bottling line, of the Perrier company, can be found in the thesis of Demongodin.
3. EXTENSION OF BATCHES PETRI NETS FOR CONTROL I SUPERVISION Batches Petri nets are a natural model for manufacturing systems where discrete logic and continuous systems intermix through physical phenomena. This model introduces identified tokens which evolve discontinuously. Within supervisory control, there are a number of classes of functionalities that are necessary for the supervisor to carry out its tasks (Prunet, et aI. , 1996). In this context, an important class is the monitoring of the system' operation and the issuance of alarms or reactions to specific events. Such events may be the simple activation of sensors which detect an accumulation length or a product detection in a transfer zone. Modeling such actions can be a substantial task in the study of the supervisory system behavior and performance. Thus it is important to validate the power of Batches PN in this setting.
3.1.
Integration a/sensors in Batches Petri nets
The interaction between physical model and supervisory part, for behavior and performance evaluation, is assumed by the introduction of sensors on the process. More precisely, accumulation sensors permit to detect a certain amount of accumulated products in a transfer zone. Some accumulation levels are so detected (fig. 9) by putting accumulation sensors on the conveyor. Each time sensor value 'u~
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changes, an event is produced, and the control laws are computed. More precisely, the test of accumulation sensors refers to the event of a quantity exceeding a level or to the event of a quantity crossing from above a lower level. Through the batch concept, sensor events consist to detect the moment when the output batch has a maximal density, representing an accumulation, and has its length inferior, superior or equal to the sensor' s position. This switching detection permits to inform the supervision part. With this information, it can regulate throughputs of machines and transfer zones, in order to optimize the production. Since we needed to know whether the accumulation length is inferior or superior to a value, we have introduced this new concept in the batch place. Deduced from the result of the analysis part, Batches Petri net is powerful enough to represent the accumulation length variation. In fact, the notion of output internal coherent batch permits to determine the accumulation length (fig. 10) at the output of the conveyor. For the example, the accumulation is created at date 3min. , -_,~",_,~, increases (date 3.278 min.) and then decreases until the second batch arrives (date 5.778 min.). During the spent time this batch is completely accumulated, the o , '-:.....:.-,.!-,. , • accumulation length increases linearly, until Fig. 10. Variation length the batch is merged with the output batch. From this date (5.816 min.) the accumulation goes continuously out from the conveyor, and decreases until the meeting with an other batch. This discrete-continuous variation shows that Batches PN is perfectly suitable in modeling and simulating mixed systems, due to the fact that it is able to model continuous variables with discontinuities. The position detection of the accumulation is implicit, in other words the evolution algorithm carries out the date of switching. As the physical flows circulating on the conveyor belts are not only function of the accumulation phenomena, but also of the driving speed, event dates are done by : * (J = to + [( IJ + Xn • L ) . dmax / t1Jsl : switching from occulted state to not occulted state,
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In Controlled Batches Petri nets, transitions are associated with firing flows, which evolve in time and indicate a change in the system. This change comes from internal events such as instantaneous variation of the driving speed when the conveyor is in failure, or external events, which correspond to a change in the external environment (like variation of the driving speed due to a decision from the supervisor). Then to
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The studied example is now extended with the introduction of two sensors on the conveyor: position of the first sensor at 5 meters from the conveyor entrance, and position of the second one at 13 meters, that we note Cap(5), Cap(13) respectively. By
204
both the theoretical results obtained and for its impact on tools which simulate systems that combine continuous and discrete phenomena. The fact that discrete and continuous variables may be used together especially in the same expressions represents the B-PN ability in handling hybrid systems. Furthermore, design of integrated systems can be put on a new basis for high throughput production lines.
ontrol the conveyance speed is to impose on the ~odel some firing flows andlor speeds done by the supervisor part. Therefore this kind of Petri nets are suitable for modeling systems subjected to external constraints, such as logical controllers or real time systems. A formal description is done in (Audry, et aI., 1994), and shows the possibility to build a behavior model which takes the normal and faulty running mode of each part of the productions lines into account. So, generic models have been defined, representing physical elements of a manufactured line. They model a particular behavior described in controlled B-PN. So, the complete modeling of a high throughput production line, is obtained by merging models representing the physical production elements. Introduced in Simuleau software, this approach has been applied to the bottling line of the Perrier company.
5. REFERENCES AlIa H. and R. David (1988). Modelling of production Systems by continuous Petri Nets. Conference on CAD/CAM -Detroit, USA. AlIa H., J. Lebail and R. David (1991). Hybrid Petri net. European Control Conference, Grenoble, France, pp. 1472-1477. Audry, N., I. Demongodin and F. Prunet (1994). Modeling of high throughpUl production lines by using generic models described in Batches Petri nets. Robotics and Automation, IEEE San Diego, USA, pp. 807-812. Bourrieres, J.P. and Chevillard (1988). Modelisation des con voyeurs Cl transfert libre a ['aide des reseaux de perri. Revue d'automatique et de productique appliquees, VoU, n0 4 -, pp. 87-100. Brinkrnan. P.L. and W.A. Blaauboer (1990). Timed continuous Petri nets - A tool for analysis and simulation of discrete event systems. European Simulation Symposium, Ghent, Belgium. pp.I64-168. Dallery, Y; and S.B. Gershwin (1992). Manufacturing flow line systems: a review of models and analytical results. Journal of Queueing Systems vol. 12, pp. 3-94. Demongodin, I., and F. Prunet (1993a). Simulation modelling for accumulation conveyors in transient behavior. COMPEURO, Seventh Annual Computer Conference, IEEE, Paris- Evry, France, pp. 29-37. Demongodin, I., N. Audry and F. Prunet (1993b). Batches Petri Nets. International Conference on Systems, Man and Cybernetics, IEEElSMC, Le Touquet, France, pp. 607- 617. Desrochers, A.A. and R.Y. AI-Jaar (1994). Applications of Petri nets in Manufacturing Systems : Modeling. Control, and Perfonnance Analysis. Book - IEEE PRESS Control Systems Society. Dubois, E., H. Alia and R. David (1991). Continuous Petri Net with Maximal Speeds Depending on Time. 15th International Conference on Application and theory of Petri nets, Zaragoza, Spain. Prunet, F., M. Caradec and N. Audry (1996). Running modes and faulty behavior conveyors modeling with extended Petri nets. Computational Engineering in Systems Applications, CESA'96 IMACS Multiconference -IEEElSMC, Lille, France, pp. 291-296. Silva, M. and E. Teruel (1996). Petri nets for the design and operation of manufacturing systems. 17th. International Conference on Application and Theory of Petri Nets. Osaka, Japan, pp. 31-61. Suri R. and B.-R. Fu (1993) Using continuous flow models to enabled rapid analysis and optimization of discrete production lines. Proceedings of the 19th Annual NSF Grantees Conference on Design and Manufacturing Systems Research, Charlotte. Ne. Zerhouni, N. and H. Alia (1990). Dynamic analysis of manufacturing systems using continuous Petri nets. IEEE International Conference on Robotics and automation, pp. 1070-1075
4. CONCLUSIONS Petri nets are mathematical models which represent real processes, describe event sequences and interpreted interactions between a system and its environment. As the model enables qualitative and quantitative analysis, applications have been developed swiftly. Continuously new models have been proposed (and are still being proposed), in order to describe more complex processes (Silva, et al., 1996). The first purpose of this work consisted in representing high throughput production line in Petri nets. As a result, it appears that the Batches Petri net fits perfectly well the mixed model, such as multi belts conveyor. Due to the ability of Batches Petri nets to handle with state variables, which are alternately real variables and constants, those models are efficiently suitable for dynamic behavior. The second aim of this study was to compare the different Petri nets for mixed systems. The number of tokens in discrete places may be large, therefore the use of discrete Petri net is limited, since there are many reachable markings and simulation time becomes too long. By modeling the number of tokens by real numbers, Hybrid Petri nets with positive real markings remove these problems. In addition, it turns out that such models can also model flow processes. Some simulations proved that Batches Petri nets are more advanced for the dynamic functioning of high throughput production systems. Their formalism leads to the mode ling of mixed system as production line with accumulation transfer elements. The high level of detail of these models allows to study the transient phenomena of these systems, while requiring the minimum amount of information and simulation time . It is very well suited for high speed conveyor control, and is compatible with discrete Petri nets. Thus it is possible to include in B-PN, stochastic, interpreted and many other kinds and extensions of Petri nets. This study is interesting for
205
PRODUCTION LINES MODELING BY DISCRETE I CONTINUOUS ASPECT IN PETRI NETS Isabel Demongodin·. Fran~ois Prunet"
*Depanment of Automatic Control and Production Systems - Ecole de Mines de Nantes La Chantrerie - 4, rue Alfred Kastler- BP 20722 - 44307 Nantes - FRANCE Tel: +33 (0)2.51.85.83.27 - Fax: +33 (0)2.51.85.83.49 - E-mail: [email protected] **Laboratoire d'1nformatique, de Robotique et de Microelectronique de Montpellier 161, rue Ada - 34392 Montpellier - FRANCE Tel: +33 (0)4.67.41.85.14 - Fax: +33 (0)4.67.41 .85.00 - E-mail: [email protected]
Abstract: To analyse and evaluate performances of high throughput production lines, Hybrid Petri nets are a powerful mathematical model. However, they are not accurate enough to represent, the dynamic behavior of the high speed accumulated system and the interaction between the physical part and the supervisory part. Thus, Batches Petri nets have been proposed as an extension of them. In this paper, a comparison between the Hybrid PN, and Batches PN are shown on a single production line. At the end of this study, an extension of Batches Petri nets for the control I supervision is presented by including the captor model. Copyright © 1998 IFAC ' Keywords: Discrete Petri nets, Hybrid Petri nets, Batches Petri nets, modeling, analysis, simulation, mixed production systems, flow of parts, supervision.
of Hybrid Petri nets, are defined. By the notion of internal coherent batch, in other words by a set of parts with the same density of distribution, they introduce a mathematical formalism of the parts flow circulation. This formalism is associated with a new kind of nodes, named batch nodes.
1. INTRODUCTION
A major step in the effort to enlarge the modeling power of Petri nets has been the extension known as the «Continuous Petri Nets». The motivation was the inability for successful modeling of discrete event systems with a large number of events. Thus, a continuous approximation was proposed to replace the counting of discrete variables with a flow approximation. Continuous Petri nets are thus approximations of discrete event systems allowing, basically, faster simulation of the latter without sacrificing the accuracy. The combination of Continuous with Discrete Petri nets leads to the «Hybrid Petri Nets» concept. ~n production domain, to optimize line throughput, it IS necessary to study the parts flow circulating between machines. This parts flow depends on the machines state, variations of element speed, and especially on the transient behavior of elements which assume moving and storing of products (accumulation conveyors, for instance). In order to represent transient behavior of transfer elements, Batches Petri nets, which are an extension
For the understanding of this paper, we assume that Petri nets theory and their concepts are known. First, high throughput production lines and an example are presented with a mixed point of view. This leads us into the presentation of hybrid Petri nets and batches Petri nets models. Some contributions of these Petri nets in the modeling of mixed systems, such as multibelts conveyors, are examined. Through the study of the example model, the applicability and the principles of modeling of these nets are illustrated. Quantities of products and their dynamic variations are compared for the hybrid Petri net and batches Petri net models. In the last section, we introduce the modeling of sensors in batches Petri net, which illustrates the interaction with the control part. Before concluding, extensions of batches Petri nets and
199
decision, scheduling and so on, of manufacturing systems (Desrochers, et aI., 1994). They are powerful enough to generate analytical and theoretical results, and then properties are more and more studied. In the Discrete PN, two nodes are defined : the discrete place and the discrete transition. The net marking, a vector built up with the marking of the places (positive or zero integer) describes at a certain moment the Petri net state or more exactly the state of the system modeled by the Petri net. In order to describe not only what happens but also when it happens in the system, non autonomous PN have been developed; they enable to take time events into account. Then performance evaluations, conception and control, can be optimized. Usually, for a production line, machines and buffers with time dependent behavior, are very easy to be modeled via timed discrete PN. However, the multibelts conveyor is not so more studied on the accumulation length variation point of view. It could be modeled with a timed Petri nets where this transfer zone is split into cells (Bourrieres, et aI., 1988) (fig. 2). Then a product can move from one cell to the next one, if it is empty. This dedicated model is also called a FIFO system (First In - First Out).
generic models are introduced. They permit to obtain an uniform representation for high throughput production systems and to make design issues and procedures more clear.
2. PETRI NETS FOR HIGH THROUGHPUT PRODUCTION LINE MODELING IN TRANSIENT BERAVIOR WITH DISCRETE / CONTINUOUS EVENTS The high throughput production lines are composed of machines in series and conveyance systems which link the machines between them. The considered system is liable to a high number of short failures which lower the productivity of the whole system. The objective of the supervisor is to mask the short failures by using the possibilities of accumulation allowed on conveyors and that enlarges the productivity of the line. For our purposes, mixed systems are considered as combination of continuous/discrete event plant (such a bottling line) with a discrete-event supervisor which reacts to external events (planned or unforeseen). Then, the allocation decisions are discrete in nature while the technological process is hybrid. To illustrate different mixed approaches, a single physical process is modeled through the different Petri nets.
2.1.
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Fig. 2. Conveyor system with N cells
We consider a high throughput production line composed by accumulation systems. These systems are mono-driving speed, unidirectional, high capacity belt conveyors which move identical parts by friction. The manufacturing system, shown in fig. 1, consists of two machines M 1 and M2 with no failures linked by an accumulation belts conveyor. The first machine Ml has a throughput of VI = 70 parts/ min., and a buffer storage SI feeds it. In this stock, 40 identical products are placed every 4 minutes. Characteristics of the accumulation conveyor are: length = 15 m. ; maximal capacity = 225 parts ; speed = 5 mlmin ; maximal density = maximal capacity / length: d max = 15parts/m. This conveyor supplies the machine M2. This last one has a throughput V2 30 parts / min. The output of the second machine supplies an infinite buffer storage S2. Initially, all products, 175 parts, are in the first buffer SI and all the others elements are empty.
But, due to the large number of states in the dynamic behavior, a compromise between simulation speed and result accuracy must be found for this kind of discrete representation. Consequently, only a transfer delay is associated to the conveyor modeling, and the information linked to the product's position is lost.
2.2.
The main feature of Hybrid Petri nets is that they combine the advantages of aT-timed discrete PN for the representation of events (failures, decisions, ... ) and those of a continuous PN for the representation of flows (Alia, et aI., 1991). Continuous PN model considers constant firing maximal speed, and the notion of positive or zero real numbers representing marks in places, can describe, for example, tanks filled by liquid. It is the simplest continuous PN model because it is very easy to be simulated. Its extensions (asymptotic, controlled (Dubois, et aI., 1994)) and itself, allow modeling of some continuous linear systems and provides a good approximation of the behavior of many discrete-event systems modeled by regular discrete PN. In the context of this model, few events have to be considered, even when it approximates a timed Discrete PN with many reachable markings. These events are obtained from linear equations.
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While using Hybrid PN, the conveyor belt is no more considered as a discrete element but as a continuous element with input and output flows (Brinkman and Blaauboer, 1990). The real markings represent the quantity of « liquid » contained in places describing the conveyor. Then, the maximal firing speed, indicates how much the « valve » is opened (fig. 3).
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systems, consisting of discrete event or logical part cooperating with a linear continuous part is possible through a single place. Intrinsic characteristics are associated to this type of node, performing characteristics of mixed production elements. For a conveyor belt or a portion of a conveyor belt, a length, ( positive real constant), a maximum density, a driving speed, ( dis-continuous variables), are associated to the batch place, and describe the characteristics of conveyor. Marking value in this Petri net is an integer for a discrete place, a positive or null real number for a continuous place, and a set of internal coherent batches for a batch place. An internal coherent batch (LCIc ) represents a set of parts with the same characteristics of distribution on a continuous element C during a time interval (fig. 5). LCI c is characterized by three continuous variables (length, density, coordinate) in the batch place. This concept has proven to have an accurate representation of the real system (Demongodin and Prunet, 1993a). Associated delays to discrete places and associated firing flows to continuous and batch transitions, permit to take the notion of time into account in Batches PN. Conditions of enabling depend on the marking state of discrete and continuous places (reserved marks and non reserved marks). Notions of weakly and strongly enabled transitions, introduced in Hybrid PN, are extended for a batch transition. The conditions of enabling depend essentially on internal coherent batches characteristics, composing the marking of the batch place. For the firing of discrete and continuous transitions, marks (positive real or integer) are removed and added, respectively at their pre and post discrete and continuous places. For the firing of a batch transition, internal coherent batches are created or destroyed inside a batch place. Evolution algorithm of a Batches PN, carries out the evolution of batches inside a batch place in the first step, and then carries out the evolution of the places state, according to enabling transitions in the second step. Then linear variations and events inside a single place, allow a mixed modeling. A conveyor is characterized by the 5-tuple C=(L,V,dmax,
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Fig. 3. Continuous representation of the conveyor It is obvious that the markings variations are proportional to the conveyance speed V, but also on the maximal density. Then the maximal firing speed is done by V.d max indicating the maximum valve flow. As the evolution in time of high throughput production line can be approximated by a continuous linear function, when the quantity of tokens is very large, we can establish the hybrid Petri net model describing the example (fig. 4). Stocks SI and S2 are TI modeled by continuous dl =4 places PI and P5. and the discrete tranSl!!on TIP I represents the supply of 40 parts every 4 min. T2 Continuous transitions T2 v2 =70 and T4 are associated with machines MI and M2 where the firing speeds are P2 V2 and V4 respectively. The conveyor is represented T3 by P2 and P3. A continuous v3=75 transition T3 models the transpon speed. its maximal P3 value equals V3 = 75 p/min .. in other words V3 = T4 V.d max =5 x 15. Place P4 V4 =30 limits the capacity of the conveyorC. Fig. 4. Hybrid Petri net model for the example
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Batches Petri net (Demongodin, et al., 1993b), a new member of the PN models family, is an extension of the Hybrid one, developed for discrete-event systems with a large number of parts. The Batches Petri net, defined through the introduction of a new kind of place and transition, namely the batch place and the batch transition. permit the modeling of discrete Continuous-time dynamic processes represented by systems of ordinary linear equations and discrete models. In this way, a unified representation of mixed
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while it needs 24 events for the B-PN. It is a good rapidity, for the high accuracy. The difference between the number of products circulating on the conveyor (fIg.7c), comes from the fIring speed and fIring flow calculation, on one hand. Thanks to the firing variables associated with continuous or batch transitions, systems variables are approximated by piece-wise constant functions (fIg. 7b, 7d). Higher the fIring variable is, worse the linear slop of system variables is (fig.7c). When the initial phase (date=lOmin.) is overstepped, the conveyor behavior is cyclic. The fIring flow of the upstream C-transition and B-transition of the conveyor models are equivalent after this functioning mode. In this phase, these transitions are saturated and their fIring speed and fIring flow are maximal.
the maximum output flow is the throughput of the machine M2. P4 is a batch place with accumulation. It contains the internal coherent batches of the conveyor, and represents the conveyance of the parts. P3 is a continuous place representing the maximum capacity of pans that the conveyor may convey. It acts like a semaphore. If the conveyor is empty, the marking of P3 is equal to the maximum capacity (as in the initial moment), it decreases as the conveyor receives parts, and increases when parts go out of the conveyor. Then, when the conveyor is full , this continuous place is empty and the transition T2 is not enabled (or weakly enabled).
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.. 2.4.
Comparison of Petri net models
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A simulator for Batches and Hybrid Petri nets, Simuleau developed at the LIRMM, permits to generate the evolution graphs of the Petri net. In this work, the fIrst aim of simulation is to test and estimate the accuracy of the approximation, by comparing markings behaviors of the mixed PN models. Therefore, for a pure discrete approach, a very complete study, comparing the discrete and continuous PN models of manufactured lines are explained in (Zerhouni, et aI., 1990). Thus, only mixed Petri net models are analyzed in this paper. Then for each place of the Hybrid PN and Batches PN, the markings values for buffers SI and S2, and for the conveyor are compared.
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Fig. 7. Comparison between H-PN and B-PN The variation of batches densities, constant over the intervals of time, characterizes the situation, and a delay is added to the accumulation time of products (fIg.7c). This delay permits to represent more accuracy the transient behavior. We can also remark that the marking tends towards its stationary value at each period. Obviously, the influences of SI and Conveyor on the buffer S2, are correlated, since both influence the place marking and the fIring flow/speed calculation. In the H-PN, if a continuous place is empty, the fIring speed of its downstream . transitions depends on '" /./•. ..... ./ ...... ..the fIring speed of its '" upstream tranSItIons . '" ....the Consequently, ../ instantaneous parts' l ..··: flow through machine " M2 is not null. For the Fig. 8. Variation of S2 B-PN the respective batch transition has a null fIring flow. Then parts arrive at the output of the conveyor after 3 minutes, corresponding to the minimum transfer delay to pass from M1 to M2.
Dynamic behavior : evolution of the quantities of parts: In order to study the dynamic behavior of the example and the corresponding mixed Petri net models (H-PN and B-PN), different simulations with a duration equals to 20 min. have been executed. A solid line, represents the B-PN behavior, while a dash line represents the C-PN functioning. The buffer SI is represented by the place PI in each PN. The dynamic of this stock is shown in fIg.7a. The marking, representing the quantity of products inside SI, has a cyclical behavior after the initial functioning mode, at date 3 min. After this moment, its period is equal to 4 minutes, corresponding to the supplying delay. We see (fIg.7a) that variables have exactly the same behavior for H-PN and B-PN besides their accuracy is different, but in both models, this buffer is represented with a continuous place. Then, this variation illustrates the principle of interaction between the discrete part net and continuous part net.. In fIgures 7b, 7c, 7d, the difference between H-PN and B-PN leads to the number of events happening during 20 min. : for the H-PN we get 14 events to carry out the behavior,
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For the performance evaluation point of view, Batches Petri net is a good model as the state and the quantity of products circulating inside the production system can be known at every time. The average occupation of stocking, of busy times, and so on, can be carried out. Also, from the evolution graph, it is possible to analyze cyclic behavior, performances, etc. On the other hand, continuous flow models for design optimization of transfer lines are more and more studied (Suri, et aI., 1993; Dallery, et aI. , 1996) by analytical models. Modeling and simulation of a production line with many conveyors may be easier since Batches Petri nets allow to defined only one place by mixed model without sacrificing the accuracy.
Results analysis: The modeling with H-PN constitutes an approximation of the system mode led by timed discrete transition but batch model allows to fix the accuracy of the results; it doesn't depend on the cells concept. Concerning simulation time, the approximation becomes more efficient when the place markings increases for H-PN and D-PN if the conveyor is represented with n cells. Actually, with a modeling by discrete timed transition Petri nets, more events occur, since every time a token is transfer from a place to another one, an event happens. In a discrete event model, we get an explosion of the states number, because the time delay on the conveyor must be split in small intervals to get a sufficient accuracy. An other reason concerns the operations of the machines where only one part could be manufacturing at a time. As a consequence, the conveyor and machines models introduce the implicit limitation on the D-PN. So, the number of discrete events and consequently of states, is very large. In the Continuous PN model, the number of state changes to be handled during the dynamic behavior is not only smaller than that of the corresponding discrete model, but is also the minimum achievable. Consequently, Hybrid PN model contains the necessary minimum amount of information to describe the state of the system. A modeling of conveyor belts based on this tool has be established and studied. But the results show that this model considers the conveyor belt as a transfer (pure delay) only and does not take account of delay and accumulation phenomena in transient behavior. In fact, in the H-PN model, it is assumed that as soon as a part is introduced in the conveyor, it is available to go out. This is not true on the real conveyor because of the conveying delay of the parts. As the authors said, this model over-estimates the throughput when the system is not saturated. More accurate model, an extension named Batches PN has been introduced to take this possibility into account. In the Batches PN, the batch place performs an accurate modeling of the belt conveyor due to its mixed aspect. The conveyor belt is characterized by its length, driving speed, maximum capacity, input flow and output flow through the batch place and its upstream and downstream batch transitions. From the above concept, we have formalized the state of the batches as a function of time, switching between different continuous models. In this way, batches defined inside this batch place correspond to a discrete linear variation of continuous variables corresponding to products. The global model takes into account the state of the transfer element, in accumulation behavior or in free handle behavior, the state of the driving speed variations which can occur at any time and, the state of upstream and downstream connections with other physical ~Iements. Simulation of high throughput production hnes modeled by B-PN (Jive very eO'ood results. A . SImple model of a bottling line, of the Perrier company, can be found in the thesis of Demongodin.
3. EXTENSION OF BATCHES PETRI NETS FOR CONTROL I SUPERVISION Batches Petri nets are a natural model for manufacturing systems where discrete logic and continuous systems intermix through physical phenomena. This model introduces identified tokens which evolve discontinuously. Within supervisory control, there are a number of classes of functionalities that are necessary for the supervisor to carry out its tasks (Prunet, et aI. , 1996). In this context, an important class is the monitoring of the system' operation and the issuance of alarms or reactions to specific events. Such events may be the simple activation of sensors which detect an accumulation length or a product detection in a transfer zone. Modeling such actions can be a substantial task in the study of the supervisory system behavior and performance. Thus it is important to validate the power of Batches PN in this setting.
3.1.
Integration a/sensors in Batches Petri nets
The interaction between physical model and supervisory part, for behavior and performance evaluation, is assumed by the introduction of sensors on the process. More precisely, accumulation sensors permit to detect a certain amount of accumulated products in a transfer zone. Some accumulation levels are so detected (fig. 9) by putting accumulation sensors on the conveyor. Each time sensor value 'u~
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changes, an event is produced, and the control laws are computed. More precisely, the test of accumulation sensors refers to the event of a quantity exceeding a level or to the event of a quantity crossing from above a lower level. Through the batch concept, sensor events consist to detect the moment when the output batch has a maximal density, representing an accumulation, and has its length inferior, superior or equal to the sensor' s position. This switching detection permits to inform the supervision part. With this information, it can regulate throughputs of machines and transfer zones, in order to optimize the production. Since we needed to know whether the accumulation length is inferior or superior to a value, we have introduced this new concept in the batch place. Deduced from the result of the analysis part, Batches Petri net is powerful enough to represent the accumulation length variation. In fact, the notion of output internal coherent batch permits to determine the accumulation length (fig. 10) at the output of the conveyor. For the example, the accumulation is created at date 3min. , -_,~",_,~, increases (date 3.278 min.) and then decreases until the second batch arrives (date 5.778 min.). During the spent time this batch is completely accumulated, the o , '-:.....:.-,.!-,. , • accumulation length increases linearly, until Fig. 10. Variation length the batch is merged with the output batch. From this date (5.816 min.) the accumulation goes continuously out from the conveyor, and decreases until the meeting with an other batch. This discrete-continuous variation shows that Batches PN is perfectly suitable in modeling and simulating mixed systems, due to the fact that it is able to model continuous variables with discontinuities. The position detection of the accumulation is implicit, in other words the evolution algorithm carries out the date of switching. As the physical flows circulating on the conveyor belts are not only function of the accumulation phenomena, but also of the driving speed, event dates are done by : * (J = to + [( IJ + Xn • L ) . dmax / t1Jsl : switching from occulted state to not occulted state,
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3.3.
Controlled Batches PN - Generic models
In Controlled Batches Petri nets, transitions are associated with firing flows, which evolve in time and indicate a change in the system. This change comes from internal events such as instantaneous variation of the driving speed when the conveyor is in failure, or external events, which correspond to a change in the external environment (like variation of the driving speed due to a decision from the supervisor). Then to
Example
The studied example is now extended with the introduction of two sensors on the conveyor: position of the first sensor at 5 meters from the conveyor entrance, and position of the second one at 13 meters, that we note Cap(5), Cap(13) respectively. By
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both the theoretical results obtained and for its impact on tools which simulate systems that combine continuous and discrete phenomena. The fact that discrete and continuous variables may be used together especially in the same expressions represents the B-PN ability in handling hybrid systems. Furthermore, design of integrated systems can be put on a new basis for high throughput production lines.
ontrol the conveyance speed is to impose on the ~odel some firing flows andlor speeds done by the supervisor part. Therefore this kind of Petri nets are suitable for modeling systems subjected to external constraints, such as logical controllers or real time systems. A formal description is done in (Audry, et aI., 1994), and shows the possibility to build a behavior model which takes the normal and faulty running mode of each part of the productions lines into account. So, generic models have been defined, representing physical elements of a manufactured line. They model a particular behavior described in controlled B-PN. So, the complete modeling of a high throughput production line, is obtained by merging models representing the physical production elements. Introduced in Simuleau software, this approach has been applied to the bottling line of the Perrier company.
5. REFERENCES AlIa H. and R. David (1988). Modelling of production Systems by continuous Petri Nets. Conference on CAD/CAM -Detroit, USA. AlIa H., J. Lebail and R. David (1991). Hybrid Petri net. European Control Conference, Grenoble, France, pp. 1472-1477. Audry, N., I. Demongodin and F. Prunet (1994). Modeling of high throughpUl production lines by using generic models described in Batches Petri nets. Robotics and Automation, IEEE San Diego, USA, pp. 807-812. Bourrieres, J.P. and Chevillard (1988). Modelisation des con voyeurs Cl transfert libre a ['aide des reseaux de perri. Revue d'automatique et de productique appliquees, VoU, n0 4 -, pp. 87-100. Brinkrnan. P.L. and W.A. Blaauboer (1990). Timed continuous Petri nets - A tool for analysis and simulation of discrete event systems. European Simulation Symposium, Ghent, Belgium. pp.I64-168. Dallery, Y; and S.B. Gershwin (1992). Manufacturing flow line systems: a review of models and analytical results. Journal of Queueing Systems vol. 12, pp. 3-94. Demongodin, I., and F. Prunet (1993a). Simulation modelling for accumulation conveyors in transient behavior. COMPEURO, Seventh Annual Computer Conference, IEEE, Paris- Evry, France, pp. 29-37. Demongodin, I., N. Audry and F. Prunet (1993b). Batches Petri Nets. International Conference on Systems, Man and Cybernetics, IEEElSMC, Le Touquet, France, pp. 607- 617. Desrochers, A.A. and R.Y. AI-Jaar (1994). Applications of Petri nets in Manufacturing Systems : Modeling. Control, and Perfonnance Analysis. Book - IEEE PRESS Control Systems Society. Dubois, E., H. Alia and R. David (1991). Continuous Petri Net with Maximal Speeds Depending on Time. 15th International Conference on Application and theory of Petri nets, Zaragoza, Spain. Prunet, F., M. Caradec and N. Audry (1996). Running modes and faulty behavior conveyors modeling with extended Petri nets. Computational Engineering in Systems Applications, CESA'96 IMACS Multiconference -IEEElSMC, Lille, France, pp. 291-296. Silva, M. and E. Teruel (1996). Petri nets for the design and operation of manufacturing systems. 17th. International Conference on Application and Theory of Petri Nets. Osaka, Japan, pp. 31-61. Suri R. and B.-R. Fu (1993) Using continuous flow models to enabled rapid analysis and optimization of discrete production lines. Proceedings of the 19th Annual NSF Grantees Conference on Design and Manufacturing Systems Research, Charlotte. Ne. Zerhouni, N. and H. Alia (1990). Dynamic analysis of manufacturing systems using continuous Petri nets. IEEE International Conference on Robotics and automation, pp. 1070-1075
4. CONCLUSIONS Petri nets are mathematical models which represent real processes, describe event sequences and interpreted interactions between a system and its environment. As the model enables qualitative and quantitative analysis, applications have been developed swiftly. Continuously new models have been proposed (and are still being proposed), in order to describe more complex processes (Silva, et al., 1996). The first purpose of this work consisted in representing high throughput production line in Petri nets. As a result, it appears that the Batches Petri net fits perfectly well the mixed model, such as multi belts conveyor. Due to the ability of Batches Petri nets to handle with state variables, which are alternately real variables and constants, those models are efficiently suitable for dynamic behavior. The second aim of this study was to compare the different Petri nets for mixed systems. The number of tokens in discrete places may be large, therefore the use of discrete Petri net is limited, since there are many reachable markings and simulation time becomes too long. By modeling the number of tokens by real numbers, Hybrid Petri nets with positive real markings remove these problems. In addition, it turns out that such models can also model flow processes. Some simulations proved that Batches Petri nets are more advanced for the dynamic functioning of high throughput production systems. Their formalism leads to the mode ling of mixed system as production line with accumulation transfer elements. The high level of detail of these models allows to study the transient phenomena of these systems, while requiring the minimum amount of information and simulation time . It is very well suited for high speed conveyor control, and is compatible with discrete Petri nets. Thus it is possible to include in B-PN, stochastic, interpreted and many other kinds and extensions of Petri nets. This study is interesting for
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