Post on 05-Feb-2018
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
Sait_belkacem19@yahoo.fr, http://www.univ-setif
Hassane.Alla@inpg.fr, 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
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp151-156)
0 6 12 180
1
2marked of P1
time
marke
0 6 12 180
1
2Marked of P2
Time
marke
Fig. 4. Evolution of marking for P1 and P2
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
Fig. 6. Evolution of marking for P2 and P4
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.
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp151-156)
- 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.
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp151-156)
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.
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp151-156)
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.
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Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp151-156)