Information Technology Reference
In-Depth Information
It follows from the definition that if substitution
θ
is
and elemen-
tary condition
C
is “
r
0
is in
A
(
σ, ω
)
since at least duration
2” then
h,
3
{
σ/s
2
,ω/o
2
}
=
θ
C
,
where
h
is the timed history defined by table 8. As well, if substitution
θ
is
{
|
and elementary conditions
C
and
C
are “
r
3
is in
A
(
σ, ω
)
since at least duration
3” and “
r
4
is in
A
(
σ
,ω
)
since at least duration
3” then
h,
7
σ/s
0
,σ
/s
1
,ω/o
2
}
=
θ
C
,where
h
is the timed history defined by table 8. The
concept of timed history is used to model the behaviour of timed protection sys-
tems. We shall say that timed history
h
is a model for timed protection system
Π
,insymbols
h |
=
Π
, iff there exists a timed sequence (
v
0
,∆
0
), (
v
1
,∆
1
),
...
for
h
such that for all non-negative integers
n
, there exists a substitution
θ
n
and a
timed command
α
n
∈ Π
with elementary conditions
C
n
,
...
,
C
i
n
and primitive
operations
π
n
,
...
,
π
j
n
=
θ
C
and
h,
7
|
|
such that:
=
θ
n
C
n
,
...
,
h, v
n
+1
|
=
θ
n
C
i
n
,
-
h, v
n
+1
|
θ
n
θ
n
π
j
n
-
∆
n
−→
π
n
◦ ...◦−→
∆
n
+1
.
We shall say that the sequence (
v
0
,∆
0
,θ
0
,α
0
), (
v
1
,∆
1
,θ
1
,α
1
),
...
is a dynamic
timed sequence for
h
. It is obvious from the definition that the timed history
h
shown in table 8 is a model for the timed protection system
Π
of table 7.
It is now time to get more precise concerning the question of safety in timed
protection systems. Let
Π
be a timed protection system and
∆
be a protection
state. If
d
is a positive real number then
Π
is said to be
d
-unsafe for
r
with
respect to
∆
iff there exists a timed history
h
with dynamic timed sequence
(
v
0
,∆
0
,θ
0
,α
0
), (
v
1
,∆
1
,θ
1
,α
1
),
...
such that:
-
h
=
Π
,
-
the following conditions are satisfied for some individual
s
of type subject
and some individual
o
of type object:
•
|
if
s
is in
S
n
and
o
is in
O
n
then
r
is not in
A
n
(
s, o
),
s
is in
S
n
+1
,
o
is in
O
n
+1
,and
r
is in
A
n
+1
(
s, o
),
-
∆
0
=
∆
,
•
for some non-negative integer
n
such that
v
n
+1
≤
d
. We shall also say that
d
is
a waiting period of
Π
for
r
with respect to
∆
. For example, with respect to
∆
,
the timed protection system
Π
defined in table 7 is 1-unsafe for
r
0
, 1-unsafe for
r
1
, 1-unsafe for
r
2
, 3-unsafe for
r
3
, 4-unsafe for
r
4
, and 7-unsafe for
r
5
,where
∆
is the protection state defined by table 1.
Table 8.
Timed history
h
v
h
(
v
)
v ∈
[0
,
1[
∆
∆
v ∈
[1
,
3[
∆
v ∈
[3
,
4[
∆
v ∈
[4
,
7[
∆
(4)
v ∈
[7
, ∞
[
