Using Bond Graphs to Model Production Systems with Machine Failure

Copyright © IFAC Control of Industrial Systems, Belfort, France, 1997

USING BOND GRAPHS TO MODEL PRODUCTION SYSTEMS WITH MACHINE FAILURE

P.-O. Lair- M. Femey- N. Zerhouni

Laboratoire de A1ecanique et Productique Belfort Technopole - Espace Bartholdi B.P.525 F-90016 BELFORT Cedex email: [email protected]

Abstract : During these last years, the problem of modelling production systems has received considerable attention. The standard tools generally used are queuing networks, Petri nets or Markov chains. The aim of this paper is to show that it is possible to use the . bond graph formalism to model such systems, and especially machine failures which can occur during their functioning. This tool allows a clear graphical representation, a study of the properties of the system from the bond graph structure, and finally a dynamic description by a state equation. Keywords : Bond graph - Production systems - Machine failures - Continuous variables Non-linear equations.

Resume : Ces dernieres annees, le probleme pose par la modelisation des systemes de production a suscite un vif interet. Les outils classiques generalement utilises sont les reseaux de files d'attentes, les reseaux de Petri ou les chaines de Markov. Le but de cet article est de montrer qu'i! est possible d'utiliser le formalisme bond graph pour modeliser les systemes de production, et notamment les pannes qui peuvent survenir sur les machines durant leur fonctionnement. Cet outil permet une representation graphique claire, une etude des proprietes du systeme cl partir de la structure bond graph, et enfin une description dynamique sous forme d'equation d'etat. Mots
1. INTRODUCTION

That is why this paper presents a bond graph model able to represent the dynamic behaviour of production systems, taking machine failures into account. This contribution is based on the use of the standard elements of the bond graph formalism. So, the model e:\.l'resses, on the one hand, the functioning of the stations of the system -a station is a set composed by a machine and an upstream stockand on the other hand, the effects of breakdO\ms

When production systems are functioning. they are bound by risks. Among them, there are machine failures, and a breakdo\\n results in a more or less consequent sloning do\m of the production. Then, it is interesting to simulate machine failures in order to understand the consequences in the whole system. And this allows to optimise some parameters of the system, especially the size of the stocks.

151

capacities between the machines. Between the moment when M; breaks down and the moment when this machine is repaired, the machine M;-l can unload its parts in the stock between M;_l and M;, while the machine M;+l can find parts to be processed in the stock between M; and M;+l. There is neither jamming nor starving, and the mean production of the workshop is equivalent to the production of the less productive machine. A real workshop is situated between these two borderline cases. If it is possible to evaluate the production of the workshop according to the characteristics of the machines (processing rate, failure frequencies) and the size of the stocks, then it is possible to solve some problems such as find the optimal size of the stocks. These processes must take into account the costs to realise a storage area and the production objectives. Machine failures can be either operation-dependent or time-<1ependent. Operation-dependent failures occur only while the machine is processing a part, while time-dependent failures can occur at any time. This work restricts the attention to time-dependent failure models.

which can disrupt the good functioning of the machines. Then, the bond graph model of the system results in a state equation, allowing to simulate the system with the obtained equations (Thomas, 1990), and consider a study to control the system.

2. PROBLEM STAlEMENT The bond graph formalism concerns systems with continuous evolution (Breedveld, 1990; Karnopp, et al., 1990; Breedveld and Dauphin-Tanguy, 1992), and the literature dealing with this tool is important (Breedveld, et al., 1991; Cellier, 1997). The context of this paper concerns continuous and approximated discrete system. First, the hypothesis is that the output flow of the machine is sufficiently high to compare with a fluid the flow of parts travelling the system. Then, continuous variables are used to study it. Besides, using a continuous model to describe the output flow of a machine or a stock level is generally a satisfactory approximation. A lot of papers deal with the description of production systems, explaining the different entities and parameters they bring into play (Cernault, 1988; Rodde, 1993). Some authors have been interested in modelling production system with the bond graph formalism (Dembele, 1993; Besombes and Marcon, 1993), but they do not introduce the concept of machine failures in their works. In a previous paper (Lair, et al., 1996), an analogy has been made between the flow of parts in the system and the flow variable of the bond graph. Similarly, storage phenomena in production system can be compared with bond graph storage elements. A bond graph model has been defined for a station composed by a machine and an upstream stock. Then, from the main bond graph model, it is possible to build a complex production system with several stations, divergent and convergent ways. The aim of the present work consists in introducing the concept of machine failure. An other hypothesis considers that a machine out of order implies that its processing rate is nil. In a workshop, the machine failure can have repercussions on all or part of the system, and this influence is dependent on the stocks between the machines. Let us consider the simple case of a production line constituted by the machines M l , M 2 , ... , on which the process of a part require to pass through the machines M j , then M], ..., in order. Supposing that the machine M; breaks do\\n. If there is no stock between the machines, the machine Ai;_l can not produce anymore: it is jammed. The machine Al;+l do not have work an)nlore: it is starved. The jamming echoes on all the upstream stations, and the starving on all the do\\nstream ones. Thus, the breakdo\\n of one machine entails the stopping of the whole of the workshop. Supposing now that there are stocks \\ith unlimited

3. PROBLEM SOLUTION

3.i Building a model for one station The bond graph model proposed to model a station uses a C-element and a non-linear R-element (Lair, et al., 1996). The following fact have to be reminded: the flow bound to the storage C-element allows to observe the stocks evolution, whereas the flow bound to the R -element represents the output flow of the machine. The R-element is variable, it is a function of the maximum capacity of stocks, the machine processing rates, and the number of parts in stocks. Effectively, the output flow depends on the level of upstream and downstream stocks (Zerhouni and Alla, 1992). When a machine failure occurs, the input flow of parts is moved towards the stock. In the same time, the output flow becomes equal to zero. This phenomenon can be exploited by introducing a MTFelement between the 0- and i-junctions, in order to induce a discontinuity (Dauphin-Tangu)', et a/., 1989). The proposed model is presented figure 1.

Fig. 1. Bond graph model for one station.

152

The modulus of this element is Boolean, timedependent, and determines the two states of the machine : in order and out of order. When the machine is in order, m(t) = 1 and the MTF-element is transparent for the flow between the R-element and the 1-junction. When the machine is out of order, m(t) = 0, and the MTF-element involves a flow It = 0. Then, because of the presence of the 1junction, 16 = 0, this is to say that the output flow of the station is nil. In the same time, because jj = 0, h = fi, this is to say that the flow of parts entering the station is wholly moved towards the stock, represented by the C-element. Then, from the model presented figure 1, it is possible to build and study the behaviour of any production systems where machine failures can occur.

characteristic matrix appear data fitted to the system, such as stock capacities and machine processing rates. The obtained state equation is non-linear and variable. The variation depends on the stocks level, since it must take into account non-linear phenomena such as stocks saturation, blocking and starvation. But it also depends on the time variable, which appears through moduli m(t), present in the characteristic matrix. By observing simulations and studying the state equation, it can be interesting to analyse the effects of failures to make an appropriate choice of the stock capacities, establish prevention rules or maintenance policies.

4. APPLICATION: EXAMPLE

3.2 Environment Two types of interfaces appear in the system. First, the source interfaces, especially flow sources, have an imposed flow variable, independent of system characteristics. They represent the supply, that is to say the inputs of the system. Then, it appears the dynamic interfaces, which have the two power variables coupled to the system characteristics. So, the studied system appears as a subsystem of a bigger one. These output interfaces are identified by storage elements called Cs, placed downstream from final stations (figure 2). These elements are considered as a well. In other words, they have to collect a flow of parts at the output of the system without having any influence in the upstream elements. They must loose their dynamic feature, that is to say it is necessary to avoid the reaction (returned effort) caused by the Cs-element. This can be easily done by setting to infinity the capacity of the receiving stocks. This hypothesis is justified because, in a real production system, these stocks correspond to the place where the finished parts are collected together and removed. So, the expression e6 = q6 / Cs allows to free oneself of the reaction (effort e6). Effectively, if Cs tends towards infinity, e6 tends towards zero.

The simulated example is a production system with five stations (figure 3), results can be observed with an appropriate tool of simulation. The simulation of the bond graph model (figure 4) has been realised from the state equation. By way of comparison, the same system have been simulated with aT-timed Petri net model. The numerical values used in the following example are mentioned below. The machine frequencies are : UJ = 4 S·1 U2 = 2 S·1 U3 = 3 S·1 U4 = 2 S·1 Us = 3.5 S·I. The maximum capacities of the stocks are : Cmax J = 20 parts Cmax2 = 20 parts Cmax3 = 10 parts Cmax4 = 5 parts Cmaxs = 10 parts. The initial values of stocks are: q J (0) = 20 parts q2(0) = 10 parts q3(0) = Q4(0) = Qs(O) = 0.

····1

And the source is :

S = 0.5 part/sec (step)

Fig. 2. Bond graph model for a final station nith receiving stock.

3.3 State equation The systematic methods to analyse the graph lead to the elaboration of a continuous state equation. In the

Fig. 3. Considered system

153

As for the R-elements, they can be expressed by :

9..L_:l1RI

=

.CI C3 u\ .mzn(l,q!,C3 -q3)

(11)

:J:L_:l1~=

.C2 C3 u2 .mzn(I,%, C3 -q3)

(12)

:l1-_!k_!hR

3

Fig. 4. Bond graph model of the system. The state vector corresponding to the bond graph model (figure 4) is:

X

= [ql q2 q3 q4 q3 q6 q7t

=

.

C3 C4 Cs u3 .mzn(l,q3,C4 -q4' CS-qs)

(13)

R4 = q4. C4 ,u4 .mzn(l.q4)

(14)

(15)

(1)

The state equation can be expressed as :

X = A(X)'X +B.U

Figure 5 represents the evolution of stocks for the considered system, from bond graph and Petri net models without machine failures. This diagram shows the difference between the continuous and the discrete evolution. A zoom on the stationary part of figure 5 is shown figure 6. At t = 25 s, the flows fi bound to C.--elements remain equal to zero, that means that the stationary state is reached. The constant flow source Sf goes on supplying the system, except the station 2, and so, it gives the stationary level at the stocks (variables qj). The obtained values at t = 25 s are the following: qJ(25) = Sf/ u J= 0.125 P

(2)

with the output equation written as :

= C(X)'X

y

(3)

in fact, it is a part of the equation (2), corresponding to the elements q6 et q7 . Effectively, they represent the flow of parts entering the stocks Cs1 and CS2. The input vector U represents the flow sources of the system. The elements of the matrix A are variable, because they depend on the state vector. The hypothesis is that there is always a space to collect the parts downstream from the final stations 4 and 5, in other words, the stocks Cs1 and CS2 have unlimited capacities. The obtained equations are presented below. The moduli mj appear in the matrix A of the state equation, so they have a direct consequence on the functioning of the machines. 2

R1Cj

.

h.

=

2

_...!.!!i- q2 + -.!!!L q3

(5)

R2~

R2C2

m;

(m;

mi)

1 q3=!J=--q\- - +mi - + - -q3 R\C1 RI ~ R3 C3

mi

,,~

discrete contin uous

(4)

R1C3

2

ti2 =

Level of stocks 1 to 5

2

_-.!!!L q\ +-!!!L q3

ti\ =J; =J;3

ql25) =0 qi25) = Sf/ u3 = 0.167 P q4(25) = Sf/ u4 = 0.25 P qi25) = Sf/ Us = 0.143 p.

,,~

(6)

+ R C q2 - R C q4 - R C q5 22

. _. I'

q4

_,,~

-)4 -

RC q333

'=I'=mi

q5

JS

RC q3 33

(

34

m;) RC + RC q4,,~

34

44

35

,,~ RC 35

5 q5(7)

m;) ,,~ RC + C qs - R C q4

_(,,~

35

11

"SS

2

(8)

time (s)

34

2

. _ I' _ m4 m4 q6 - J6 - RC q4 - R C q6 4

o ~!:tftjfffi~~El!It"~~ o 5 10 15 20 25

4

4

s\

Fig. 5. Comparison of simulation results for BG and PN models (\\ithout failures).

(9)

154

fIO\N(f)

3] -

------------------.------------.-----. ~ 0.5

3. r

l

2.~

Oltput nows stations 2 and 3

.

.

.

- ---

\ • - -- --- . -

23

---.------ --

- -'-2

1.~ -- - --

Fig. 6. Zoom on the stationary part

Q

O

time (s)

The continuous approximation gives mean values inferior to one in stationary state. TIlls means that the real level of stocks, in other words, during the discrete evolution, periodically oscillates between zero and one part.

I

-- ----- -'- --- - ---. -- --- --

--------

. L C-

Q.5 -------------

jJ

--

L - - -----~ - -

1 ----- -

25

24

1

.--? --.. ----.. --- - _

2~---

.

.

...,----------------------~------------------~-----------~----------1

\

.

$

4.

10

1;3 l

18

15

1

20

25

~

*%\%i

Fig. 8. Corresponding output flows of the 2 failing machines

Figure 7 represents the evolution of the same system, where a breakdown occurs on machine 2 between t = 4 s and t = 8 s, and another one on machine 3 between t = 13 s and t = 18 s. The corresponding output flow of the two failing machines is shown in figure 8.

~

i

~

m2(t)

---1"- '..-..-..--

Figures 9 and 10 present the time-diagrams of the moduli m:z(t) and m3(t), indicating the moments of the breakdowns. The others moduli remain equal to one during the simulation.

8 10 15 tim e (s)

4

- . 20

25

18 20

25

Fig. 9. Time-diagram ofmlt) Levels of stocks 1 to 5

~

machine failures

. m3(t) ;

5

.

•

10 13 tim e (s)

Fig. 10. Time-diagram ofm/tJ 5

.

~%~~~~

time (s)

4

8

20

10 13

25

5. CONCLUSION

18

TIlls paper has sho\\n that it is possible to model machine failures in production systems, \\ith the bond graph formalism. Experimental results have proved that the proposed model )ields very accurate results. TIlls has been possible by comparing the movement of parts with a continuous flow, and introducing time-modulated MTFelements. The result is a mixing of continuous variables \\ith Boolean ones_ TIlls approach can

Fig_ 7. Simulation results for BG model ("ith failures). It is important to note that, on the graphs above, obtained results are very convincing as for the accuracy of the continuous approximation, compared \\ith the discrete evolution. TIlls remark is as true for the model including machine failures as the basic one.

155

possibly be used to model hybrid systems where continuous and discrete variables appear. This could be studied in next works. The stock of any station of the system has an influence on the upstream and downstream stations, so it is interesting to choose a non-linear R-element to model the behaviour of a station. For a given initial configuration, the non-linear state equation allows to know the behaviour of the system, and more precisely the stocks level of each station. Simulations are useful to easily observe effects of one or several machine failures, fixing the moduli m(t) to precise values, or bring in random values to interfere. That is why a simulation interface, built from the Matlab software (Moler, et al., 1987), has been developed in addition to this work. Besides, since the standard bond graph structure is conserved, the state representation is the beginning of the study of stability and elaboration of control (consisting of applying appropriate values to the input flows to reach a desired stock levels). This analysis is one of the aims of the next works.

Kamopp, D., D.L. Margolis, RC. Rosenberg (1990).

Systems dynamics : a unified approach.

2nd

edition. Lair, P.-o., N. Zerhouni, M. Ferney (1996). Using bond graphs to model production systems. CL'vfAT'96, Grenoble, France. Moler, C., J. Little, S. Bangert (1987). Matlab User's Guide, New York, Mathworks. Rodde, G. (1993). Les systemes de production : modelisation et performances. Hermes. Thoma, Ju. (1990). Simulation by bond graphs. Springer-Verlag. Zerhouni, N., H. Alla (1992). Sur l'analyse des lignes de fabrication par reseaux de Petri continus. RAJRO APII, Vol. 26, No 3, pp. 253276.

REFERENCES Besombes, B., E. Marcon (1993). Bond-graphs for modeling of manufacturing systems, pp. 256261, ICSMC, Le Touquet, France. Borne, P., G. Dauphin-Tanguy, J.-P. Richard, F. Rotella, I. Zambettakis (1992). Modelisation et identification de processus, pp. 25-80, tome 2, chap. 5, Editions Technip. Breedveld, P.e. (1990). Fundamentals of bond graphs. IMACS Annals on computing and applied mathematics. Vol. 3, pp. 7-14. Breedveld, P.C., G. Dauphin-Tanguy (1992). Bond graphs for engineers. North-Holland. Breedveld, P.C., RC. Rosenberg, T. Zhou (1991). Bibliography of bond graph theory and application. Journal of the Frank/in Institute, 328(5/6), pp. 1067-1109. Cellier, F.E. (1997). World wide web - the global library: a compendium of knowledge about bond graph research. Proceedings of ICBGM'97, 3rd International Conference on Bond Graph Modeling and Simulation, Phoenix, Arizona, January 12-15, SCS Publishing, San-Diego, California, Simulation Series, Vol. 29, No 1, pp. 187-191, ISBN 1-56555-050-1. Cernault, A. (1988). Simulation des systemes de

production: methodes, langages et applications. Cepadues Editions. Dauphin-Tangu)', G., C. Sueur, C. Rombaut (1989). approach of corrunutation Bond graph phenomena. AIPAC'89, Nancy, France, 3-5 juillet, IFAC Laxenburg, pp. 297-301. Dembele, S. (1993). Contribution a la modelisation

qualitative des flux dans I'usine manufacturiere. These, Besan90n, France.

156