modelling by petri nets of the hybrid systems in ... · pdf filemodelling by petri nets of the...

Modelling by Petri nets of the hybrid systems in electrical engineering

BELKACEM SAIT, HASSANE ALLA

Department of Electrical Engineering

University of Ferhat ABBES Sétif, ALGERIA

[email protected], http://www.univ-setif

[email protected], http://www.lag.ensieg.inpg

Abstract: - Petri nets enable a discrete event system of any kind whatsoever to be modeled. They present two

interesting characteristics. Firstly they make it possible to model and visualize behaviors comprising

concurrency, synchronization and resource sharing. Secondly the theoretical results concerning them are

plentiful. The aim of this paper is to present a new approach of modelling of the Hybrid Dynamic Systems HDS

in the field of the electrical engineering systems. The modelling tool is the Hybrid Petri net HPN, an extension

of the Petri nets PN. The HPN unify within the same formalism the modelling of the continuous and

discontinuous phenomena basically non dissociable in an electrical engineering unit. It integrates two types of

PN: The continuous Petri net CPN models the part continues HDS; traditional PN models the discrete part of

the SDH. We represent in this article in the first time the tool of modelling the hybrid Petri net. Then the

methodology of representation by HPN of the static inverters, these constituting of the real examples of the

hybrid dynamic systems the discrete aspects of these systems are due to operations and the order of the

semiconductors (on/off) and the aspects continuous are due to the electrical and mechanical variables of

system. The designed model can be uses at the same time for simulation, the control and the monitoring of the

system.

Key-Words: - Hybrid dynamic systems, hybrid Petri nets, converter, modelling, simulation, monitoring

1 Introduction The Dynamic Continuous Systems DCS have

variables which have a continuous behavior in time

(voltage, current, speed, torque). They are often

model by differential or difference equations or

transfer functions. The systems based on the

principles of physics are continuous dynamic

systems. For the Dynamics Discrete systems DDS,

the space of the variables of the exits is a discrete

whole of value (states opening/closing of a switch,

numbers simultaneous switches input/output in a

static inverter, pulse repetition frequency for the

order of the interrupters). The traditional models

used for DDS are: Petri nets, automats, Grafcet.

The systems including the two characteristics

continuous and discrete are called the hybrid

dynamic systems. If the techniques of modeling of

the continuous and discrete systems are known, for

the hybrid dynamic systems a unified tool of

modeling is necessary:

- To understand and ensure the consistency of the interaction of the two parts, continuous

and discrete of the hybrid system;

- To contribute with the design, the supervision and the development of the order;

- To simplify simulation and to carry out more

precise formal analyses.

The modern electrical engineering sets constitute a

class of fast processes who’s at the same time

structural and functional complexity all is growing

These sets generally incorporate electromechanical

parts, static inverters concern the power electronics

and the element of control often treated on a

hierarchical basis providing the functions of brought

closer order, control, management of and the

reliability operating modes [1] and [2]

The Petri nets PN can thus be made profitable for

the study of the converters [3] in particular hybrid

Petri net HPN which associated within the same

formalism of the discrete representation and

continuous.

In this article we are interested in modeling of the

hybrid dynamic systems in electrical engineering; it

is composed of 3 parts. After the introduction, the

second left described the tool hybrid PN modeling,

in third parts applications of HPN to the modeling

of the static inverters.

2 Discrete, continuous and hybrid

Petri net We will represent in this section the discrete,

continuous and hybrid Petri net

2.1 Timed discrete Petri net In timed discrete PN, the temporizations can be

constant or variable, they are associated are in the

Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp151-156)

places, are with the transitions and one can pass

easily from a model to the other. We consider the

model here having time-lag constants associated

with the transitions. On the figure 1, d1=2 is

associated with T1 and d2 =3 is associated with T2

A firing draws occurs all the dj units of time and the

firing of each transition and limited to only one

firing at the same time.

The figure.1 represents the Petri net at the initial

moment (t=0). The transition T1 is validated a mark

is then reserved in the place P1 for firing from this

transition (the reserved marks are represented by a

small empty circle). After one d1 duration the

transition T1 is firing. This consists in withdrawing

the mark reserved in P1 and to add a mark not

reserved in P2 Then T1 and T2 is validated the marks

in P1 and P2 is reserved for the firing of these

transitions.

The evolution of marking for places P1 and P2 are

indicated on the figure 2.

P1

T1 d1=2

P2

T2 d2=3

P3

P4

Fig.1. Timed discrete Petri net

0 3 6 90

1

2marked of P1

time

marke

0 3 6 90

1

2marked of P2

time

marke

Fig. 2. Evolution of marking for P1 and P2

2.2 Timed continuous Petri net The temporizations associated with timed

continuous Petri net TCPN are usually expressed in

the form of speeds associated with the transitions.

Several models of timed continuous Petri net were

defined: The continuous Petri net at constant speed

(CCPN); the continuous Petri net at variable speed

(VCPN), the speed can be a function of time or

function of marking of the places upstream, or

constant by interval; the asymptotic Petri net

(ACPN).

The difference between these models corresponds to

the type of approximation used in the calculation

instantaneous speeds of firing of the transitions.

Other authors also associated temporizations with

the places of CPN. We are considered here that first

CCPN basic tool which considers speeds maximum

of crossing of the transitions are constant

The figure 3 represents a timed continuous Petri net.

The maximum speeds of firing are V1=1/d1=0.5 and

V2=1/d2=0.33 is associated with the transitions T1

and T2. As long as P1 is not empty, the transition T1

is passable at the V1(t)=V1. Speed i.e. that the

quantity V1dt= 0.5dt is withdrawn from P1 and to

add to P2 between the moment T and t+dt. T2 can

also be crossed at the speed V2(t)=V2 bus V2<V1.

One can deduce markings from them from P1 and P2

by the following relations.

=−+=

−=+−=

6)()()0()(

62)()()0()(

2122

2111

ttVtVmtm

ttVtVmtm

(1)

These relations remain true until t=12.

For t>12, the speed of firing of T1 is limited by the

feed rate of P1 which is worth V2(t)=V2 thus V1(t)=2.

For t>12, m1(t)=0 and m2(t)=2. The evolution of

marking is illustrated on the figure.4.

P12

P2

0

T1 u1=0.5

T2 u2=0.33

Fig.3. The model CPN


0 6 12 180

1

2marked of P1

time

marke

0 6 12 180

1

2Marked of P2

Time

marke


2.3 Hybrid Petri net The timed hybrid Petri net (THPN) is composed of

two discrete and continuous timed Petri nets.

Figure 5 represents a timed hybrid PN in which the

set of places is P = P1, P2, P3, P4: the sets of

discrete places (PD) and continuous places (P

c) are

PD=P1, P2 and P

C =P3, P4, respectively.

Similarly, the sets of transition are T = T1, T2, T3,

T4, TD = T1, T2 and T

C = T3, T4. The initial

marking is: M0=(M0D, M0

C) where M0

D=(1, 0) and

M0C=(180,0).The timing associated with the discrete

transitions are d1=90 and the d2=60 and the maximal

speed associated with the continuous transitions are

v3=3 and v4= 2 (another notation will be introduced

in section 2.4.1).

Note that there are two marking invariants:

m1+m2=1 and m3+m4=180. The marking of the PN

can thus represented by (m1, m3) instead of (m1, m2,

m3, m4) which is redundant. This enables us to

represent the reachable space in the plane see

figure.5 where A represents the initial making.

P1

P2

T1 d1=90 T2 d2=60

P3180

T3 u3=3

P4

0

T4 u4=2

Fig.5. Model of HPN

Fig .5. Reachable spaces

The evolution marking of places P2 and P4 are given

by figure 6 following.

0 50 100 150 200 250 3000

0.3

0.6

0.9

Marked of P2

time

marke

0 50 100 150 200 250 3000

60

120

180Marked of P4

time

marke


2.4 Representation formal the hybrid

Petri net A timed hybrid Petri net THPN is a couple

<H,Tempo>.

- H is a marked Petri net H=<P, T, Pre, Post, h, M0>

P=P1,P2,…..Pn is a finite , non-empty, set of

places;

T=T, T, …Tm is a finite, non-empty, set of

transition =∩TP Ø. i.e. P and T are disjoint.

Pre: PxT → 0, 1 is the input incidence mapping;

Post: PxT → 0, 1 is the output incidence mapping

H : P∪ T → D, C called hybrid function

indicates for every node if it is a discrete node or

continuous one node.

Mo is the initial marking.

Pre and Post mapping must meet the following

criterion: If Pi and Tj are such that h(Pi)=D and

h(Tj)=C, then Pre(Pi,Tj)=Post(Pi,Tj) must be

verified.

- Tempo is the mapping that associates a positive

real number with each transition:

- For a D-transitions Tj, Tempo (Tj)=dj

where di is time which corresponds to firing

duration.


- For a C-transition the maximum firing

speed associated with a Tj is worth Vj=1/dj.

A D-transition has the priority over a C-transition.

2.4.1 Flow rate and maximal speed

A flow rate denoted by Uj, [7] corresponds to the

maximal speed provided by server associated with

transition Tj. In figure 5, the flow rate associated

with T3 is 3; since the number of servers associated

with T1 always one (one token in P1), the maximal

speed is V3= U3. m1= U3=3. In a timed Hybrid PN

we have:

- The flow rate Uj corresponds to its maximal speed

if its D-enabling degree is 1.

- The maximal speed of transition Tj is the product

its flow rate by its D-enabling degree

- The D-enabling degree denoted D(Tj,, m) = min mi

for Dji

PTP ∩∈ 0 .

2.5 Hybrid Petri nets with speed

depending on the C-marking Informally, the main idea is as follows: the firing

speed of a C-transition is proportional to the

minimum (maximum) marking of its input places.

This model is called variable speed hybrid PN,

various model are explained. We have interested in

this section a Differential Hybrid Petri Nets DHPN.

2.5.1 Differential Hybrid Petri Nets

In [9], these authors define differential places, and

differential transition, resembling our C-places and

C-transitions, with some difference: the marking of

a differential place may de negative and the weights

of arcs to or from a differential place may be

negative. Enabling of a differential transition is

similar to D-enabling of a C-transition. If all the

marking and the arc weights were non-negative, the

behaviour of DHPN could be modelled by hybrid

PN. If the marking of every place has either an

upper bound or lower bound, it can be replaced by

non-negative variable thanks to a simple change of

variable.

3 Application To illustrate the method we considered two

examples, in first example the electric quantities are

positive and negative, whereas in the second

examples are positive:

1) Diode circuit with LC load,

2) The DC chopper.

The simulation of these examples was carried out

under simulator SIRPHYCO (SImulator for RdP

HYbrids and COntinues).

3.1 Diode with LC load A diode circuit with an LC load is shown in

figure.7, D is perfect Diode, L =7 mH is a pure

inductance and C=5µF is a perfect condenser having

an initial voltage V0=60V.

Fig.7. Diode circuit with LC load

When switch S is closed at t=0, the discharging

current of the capacitor is expressed as.

dt

dvCi

dt

diLv

c

c

c

c

−=

=− 0

(2)

02

2

=+td

vdLCv c

c (3)

The solution of the equation is:

tBtAcv ωω sincos += , With LC

1=ω (4)

A and B are constants obtained from the initial

conditions:

- 0

)0( vvc

= , then Av =0

- 0)0( =i , then B=0

By replacement in the equation one obtains:

tLC

vi

tvv

c

c

ω

ω

sin

cos

0

0

=

=

(5)

Can be written as follows:

=

=

tLCL

Cvi

tLC

vv

c

c

1sin

1cos

0

0

With

L

Cvi

vvv

c

c

0

00

0 ≤≤

+≤≤−

↑=

↓−=

cc

c

c

c

vsii

vsidt

dvci

LL0

(6)

The variation of vc and ic are to give as follows:

L

V

dt

di

C

I

dt

dv

cc

cc

=

−=

(7)

The representation by hybrid Petri nets HPN of the

system is given by the following figure 8.


Fig 8. The model HPN of circuit LC

The physical sizes are modelled by a place

continuous and a transition continues the continuous

places P5 indeed and P6 respectively represent the

evolution of the current and the expression voltage

of the condenser. The continuous transitions T2, T3

and T4 are associated firing speeds from which

depends on the marking of the continuous places.

The maximal firing speed is defined as follows:

iiimUV .= , with Ui is the flow rate associated with

transition Tj and mi is the marked of Pi.

The maximal firing speeds associated with

continuous transition are:

T4 : C

mU

6

4= ; T2 : )0,

5max(

0

2 L

vmU

−= , and

T3: )0,max(50

3 L

mvU

−= .

The initial marking for HPN is )000

,( CD MMM = ,

the initial making P5=120, The evolution of the

marking of the places P5 and P6 in HPN are

represented by the figures 9 and 10.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (s)

m(P5)

Fig 9. Marking of the place P5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

20

40

60

80

100

120

time (s)

marke

Fig 10. Marking of the place P6

The Waveforms of the current and the tension of the

circuit are given by marking the places P5 and P6

Such as: The current ic=m(P5) and the tension Vc=

m(P6)-v0

3.2 DC Chopper A DC chopper converts directly from DC to DC and

also known as DC-to-DC converter. A chopper can

be considered as DC equivalent to a transformer

with a continuously variable turn’s ratio. Like a

transformer, it could be used to step-down as shown

the figure 10 or step-up a DC voltage source.

Fig 11. The DC chopper

3.2 Principle of operation and modelling In this structure with two switches (thyristor and

diode), four combinations of configuration are a

priori are possible. For a continuous and stationary

operation there are two following configurations:

- Configuration 1

-H conduct and D blocked, for 0<t< αT, the load current is:

=

−−+==

0

)1)(()()(

D

nHc

i

t

R

EI

R

Etiti

τ (8)

- Configuration 2

-H blocked and D conduct, for αT<t< T, the load current is:

−−==

=

))(

1()()(

0

τ

αTtItiti

i

mDc

h

(9)

With, Im, In, and ∆I=Im-In are respectively,

maximum, minimal and peak to peak load current, α is the duty cycle of chopper, T is the chopping

period, f is the chopping frequency, and τ =L/R is the time-constant. With E=220V, R=8Ω, L=7mH,

α=0.4 we have obtained Im=15A, In=10A.

The hybrid model is represented on the figure 12 the

two places P1 and P2 are associated with the two

configurations of the operation of the system. Its,

respectively model the behaviour of the chopper and

the diode. A mark in P1 corresponds has the

conduction of H and the current which it conduit is

indicated by the marking of the place.


The Waveform of the current is represented by the

marking of P3 in figure 13.

15

15

10

10

T3 v3=4

T4 v4=2.5

P1

P310

T1 d1=0

P2

P415 T2 d2=2

Configuration 1

Configuration 2 Fig. 12. Behaviour Model for chopper by HPN

0 3 6 9 12 15 180

5

10

15marked of P3

time

marke

0 3 6 9 12 15 180

0.4

0.8

marked of P1

time

marke

Fig 13. Marking of the place P1 and P3.

4 Conclusion A variety of models have been presented, each one

with its own particular.

In this paper, we are interested in the modeling of

the static inverters electric constituting real hybrid

dynamic systems. We proposed a methodology

based on a graphic and formal tool to represent the

continuous and discrete parts system and their

interactions.

We used the discrete Petri nets to model the discrete

part and the continuous Petri net to modeling the

part continuous in dynamic hybrid system, the

model is hybrid Petri net. The hybrid Petri nets may

by used when some part can be modeled by a

continuous PN, Wile another need a discrete

modeling. We presented in the first time the tool of

modeling, a made progressive study at summer,

starting with PN traditional, then continuous PN and

finally hybrids PN, then the modeling of the static

inverters. The evolution of marking in the model

corresponds well to the description of the behavior

and variations of the parameters of the systems. In

this article we showed that the Petri nets lend

themselves very well to model the hybrid dynamic

systems example the static inverters. Moreover, in a

lot of industrial systems, there are continuous sub-

systems connected to discrete ones. Using a single

simulation tool could prove very powerful to

coordinate the simulation of both.

References:

[1] H. ALLA, les réseaux de Petri: un outil particulièrement adapté à la modélisation des

système hybrides, ADPM’94, Bruxelles,

Novembre 1994.

[2] G W BRAMS, Réseau de Petri: théorique et

pratique, Edition Masson Paris 1983.

[3] J LE BAIL, H ALLA, R DAVID, Réseaux de Petri hybride, TSI, vol 11, n°5/1992 pp 95 à

120

[4] E DEBOIS, H ALLA. R DAVID, Les Réseaux de Petri à vitesse fonction du temps, APPI, vol,

28, n°5, pp 425 à 443, juin 1994.

[5] M SECHILARIU, J BERRUE, J LFERRIER,

Technique de commande des convertisseurs

statiques utilisant les réseaux de Petri,

Symposium ADPM’94, Bruxelles, novembre

1994.

[6] R David, H Alla, Du Grafcet aux Réseaux de Petri, Edition Hermès Paris 1992, 2

ième édition.

[7] R David, H Alla, Discrete, Continuous, and Hybrid Peti Nets, Springer-Verlag Berlin 2004.

[8] R David, H Alla, Petri Nets for Modelling of Dynamics Systems-A Survey, Automatica, Vol

30, N°2, pp175-202, 1995

[9] I Demongodin, N T Koussoulas, Differential

Petri Nets: Representing Continuous systems in

a Discrete-Event World, IEEE transactions on

Automatic Control, Special Issue on Hybrid

Systems Vol.43 pp. 573-579, 1998

[10] I Demongodin, F Prunet, Extension of hybrid

Petri nets for accumulation systems, IMACS

Decembre 1992

[11] E Dubois, H. Alla, R. David, continuous Petri Net with Maximal speeds depending on time,

CIMAT94, 10-12 October1994, Troy (New

York).


modelling by petri nets of the hybrid systems in ... · pdf filemodelling by petri nets of the...

Documents