IEEE Robotics & Automation Magazine - June 2015 - 69
tokens that place p can hold at a time. With colors being introduced, a colored PN is formed. One can find the transition enabling and firing rules in [26] and [34].
We use a hybrid colored-timed PN (CTPN) to model the
system, which is defined as CTPN = (PD , PC, TD , TT , TC,
I, O, U, M 0), where PD and PC denote finite sets of discrete
and continuous places, respectively; TD, TT , and TC denote finite sets of discrete, timed, and continuous transitions, respectively; I and O denote input and output relations, respectively;
U ( p) and U (t) represent the color sets in P and T, respectively; and M 0 is the initial marking. Figure 2 shows the icons
used in our model.
A token in p ! PD has the same meaning as a basic PN. A
token p ! PC indicates that there is oil in the tank modeled by
p and its volume is given in a real number (called token volume for short). A t ! TD or t ! TT acts like the one in a basic
PN. When a t ! TC fires, oil is moved from one place to another, and its flow rate is decided by an OD. The firing duration of transition t ! TC determines if the token is completely
removed from its input place. Next, we present the PN modules for tanks and a pipeline.
Modeling the Devices
Due to a charging-tank-switch-overlap constraint, the behavior
of a storage tank is different from that of a charging tank; therefore, they have different models. The model for the former is
shown in Figure 3(a), in which we have K ( p) = 1,
6p ! PD , PC . Two places p s ! PC and p c ! PC are used to
model a tank's state. When M ( p s) = 1 or M ( p c) = 1, it means
that the tank has oil in it. M ( p s) = 1 indicates that the oil in the
tank is not ready to be discharged. Only when both M ( p c) = 1
and M ( p s) = 0 is the oil in a tank ready to be discharged.
Transitions t f ! TC and t ! TC are used to model the charging
and discharging of a tank, respectively. Transition t d ! TT models the oil residency time constraint. By t d ! TT , it guarantees
that the oil must stay in p s for a time associated with t d after
charging the tank ends. After firing t d ends, the token in p s goes
into p c, leading to the enabling of t, which means that the oil in
the tank can be discharged. Since M 0 ( p 3) = 1, p 3 ! PD, at any
time, at most one of transitions t f , t d, and t can fire, which guarantees that charging and discharging a tank at the same time
cannot occur. The inhibitor arc ( p s, t) and self-loop between p 3
and t d further ensures that the oil residency time constraint is
satisfied. The token volume associated with the token in
p 1 ! PC is used to model the available capacity of the tank at a
marking M. With this model, the dynamic behavior of a storage
tank is well modeled.
Now let us augment the model in Figure 3(a) to obtain the
PN model for a charging tank by adding a charging-tankswitch-overlap constraint. Assume that a distiller is being fed by
charging tank (CTK) CTK1, and it is followed by CTK2. Using
the charging-tank-switch-overlap constraint during period x ovr,
the distiller should be fed by both CTK1 and CTK2 simultaneously with oil amount c 1 and c 2, respectively. Assume that
the oil residency time is W. Let D = c 1 / (c 1 + c 2),
K = c 2 / (c 1 + c 2), and X = W + D. To model this constraint,
Discrete Transition
Timed Transition
Continuous Transition
Continuous Place
Discrete Place
Figure 2. The icons used in the PN model.
the process is treated not just as a constraint, but the process is
scheduled so that the oil in CTK2 that is fed into the distiller is
ready for discharging at least D time units earlier than W, and,
after being emptied, CTK1 is recharged at least K time units
later. Therefore, this constraint can be met when an appropriate
schedule is implemented. To model this mechanism, p 2 ! PC,
t h ! TT , inhibitor arc ( p 1, t f ), and a self-loop between p 3 and
t h are added to Figure 3(a) to form the PN model shown in Figure 3(b). At the same time, X and D time units are associated
with t d and t h, respectively.
The PN model for the pipeline with three different oil segments is shown in Figure 4. Places p 1 ! PC, p 2 ! PC, and
p 3 ! PC model the three segments, and t 1 ! TD and t 2 ! TD
are used to connect these places in a serial way. Transitions
t I 1 ! TC to t I k ! TC and t O 1 ! TC to t Ok ! TC are used to
model oil flowing into and out of the pipeline. Let
TI = {t I 1, f, t Ik} and TO = {t O 1, f, t Ok} be the sets of input
and output transitions of the pipeline. The firing of
t Oi, i ! {1, 2, f, k}, continuously consumes the token volume
in p 1, until M ( p 1) = 0. According to the transition enabling
and firing rules, then t 1 fires immediately, resulting in the
token in p 2 and its whole volume being moved into p 1 . This
way the transition t Oi ! TO can continuously fire. When a
transition t Ii, i ! {1, 2, f, k} fires, a token goes into p 3 immediately. The continuous firing of t Ii results in the continuous
tf
tf
ps
p3
ps
p1
td
p2
th
td
pc
p3
pc
p1
t
t
(a)
(b)
Figure 3. The PN modules for tanks.
JUNE 2015
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IEEE ROBOTICS & AUTOMATION MAGAZINE
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69
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