flexible ac transmission systems ( (2)
flexible ac transmission systems ( (2)
... are:
)sincos(
2
ijijijijjiiiiij
bgVVgVP
θθ
+−=
))sin()cos ((
seiijseiijsei
bgVV
θ
θ
θ
θ
−+−−
(2 .36)
)cossin(
2
ijijijijjiiiiij
bgVVbVQ
θθ
−−−=
))cos()sin ((
seiijseiijsei
bgVV
θ
θ
θ
θ
−−−−
(2 .37)
)sincos(
2
jiijjiijjijjjji
bgVVgVP
θθ
+−=
))sin()cos ((
sejijsejijsej
bgVV
θ
θ
θ
θ
−+−+
(2 .38)
Fig. ... ,
ijjj
bb = .
The operating constraint of the SSSC (active power exchange via...
flexible ac transmission systems ( (3)
... Analysis
))sin()cos ((
))sin()cos ((
2
innininnininn
ininininniinnni
sebsegVseV
bgVVgVP
θθθθ
θθθθ
−+−+
−+−−=
(3 .45)
innininnininn
ininininninnnni
sebsegVseV
bgVVbVQ
θθθθ
θθθθ
−−−+
−−−−−=
cos()sin ((
))cos()sin ((
2
(3 .46)
where
)/1Re(
in
Zse=
in
g
,
)/1Im(
in
Zse=
in
b
.
in
P ,
in
Q (n=j, k) are the active and
reactive power flows of two IPFC branches ... equations (3 .50)...
flexible ac transmission systems ( (4)
... FACTS in O PF Analysis 115
))sin()cos ((
2
ininininniinnni
bgVVgVP
θθθθ
−+−−=
))sin()cos ((
innininnininn
sebsegVseV
θ
θ
θ
θ
−+−+
(4 .61)
))cos()sin ((
2
ininininninnnni
bgVVbVQ
θθθθ
−−−−−=
))cos()sin ((
innininnininn
sebsegVseV
θ
θ
θ
θ
−−−+
(4 .62)
where
)/1Re(
inin
g ... OPF
))cos()sin ((
ijiijijiijiji
ij
i
sebsegVseV
se
P
θθθθ
θ
−−−=
∂
∂
(4 .123)
))sin()cos ((
ijiijiji...
flexible ac transmission systems ( (5)
... as:
°
°
¯
°
°
®
==+
≠≠−−
=
∂
∆∂
) ,() (
) ,() cossin(
2
pmijBVQ
pmijBGVV
P
pp
ii
p
i
p
i
pm
ij
pm
ij
pm
ij
pm
ij
m
j
p
i
m
j
p
i
θθ
θ
(5 .59)
°
°
¯
°
°
®
==−
≠≠+−
=
∂
∆∂
) ,(/
) ,() sincos(
pmijVPBV
pmijBGV
V
P
p
i
p
i
pp
ii
p
i
pm
ij
pm
ij
pm
ij
pm
ij
p
i
m
j
p
i
θθ
(5 .60)
152 ... parts:
0
2
3
2
1
)Re(
1201
=++=−=∆
b
se
b
se
a
se
jb
se
a
seRe
FEEeV
$
VV
(5 .79)
0
2
1...
flexible ac transmission systems ( (6)
... =−−
min
0000
)( (6 .42)
0hsuyh =−+
max
0000
)( (6 .43)
0yg = )(
ii
(i =1,2,…,Nc) (6 .44)
0hslyh =−−
min
)(
iiii
(i =1,2,…,Nc) (6 .45)
0hsuyh =−+
max
)(
iiii
(i =1,2,…,Nc) (6 .46)
0uu =−
i0
(i =1,2,…,Nc) (6 .47)
where
µ
>0,
0sl ... )(
00
µ
L
(6 .50)
) )((
min
0000
0
hslyh
ʌl
−−−=∇
µ
L
(6 .51)
) )((
max
0000
0
hsuyh
ʌu
−+−=∇
µ
L
(6 .52)
00
0...
flexible ac transmission systems ( (7)
... Systems
¦¦
−+−+
¦¦
+−
−=∆
==
==
cbapcbam
m
i
p
i
pm
i
m
i
p
i
pm
i
p
i
p
i
cbapcbam
pm
i
pm
i
pm
i
pm
i
m
i
p
i
ii
BgGgEV
BgGgVV
PgPg
,,,,
,,,,
))sin()cos (( [
)sincos([
δθδθ
θθ
(7 .6)
¦¦
−−−+
¦¦
−−
−=∆
==
==
cbapcbam
m
i
p
i
pm
i
m
i
p
i
pm
i
p
i
p
i
cbapcbam
pm
i
pm
i
pm
i
pm
i
m
i
p
i
ii
BgGgEV
BgGgVV
QgQg
,,,,
,,,,
))cos()sin (( [
)cossin([
δθδθ
θθ
(7 .7)
where ... given by
t
],...
flexible ac transmission systems ( (8)
... circuit
LFC
high capacity line
low capacity
line
transmission direction
LFC placed
crossways to
critical lines
LFC placed in
critical line with
high capacity
LFC placed in
critical line with
low capacity
Fig. ... 53
Conneforde
Maasbracht
Siersdorf
Rommerskirchen
Vigy
Uchtelfangen
Meeden
Diele
Hengelo
Gronau
Achene
Lonny
Aubange
Moulaine
Avelin
Avelgem
Germany
(D)
Netherlands
(NL)
Belgium
(B...
flexible ac transmission systems ( (9)
... function V(T) exists for time points T≥0 [9].
³
≥∀+≤
T
TudttutyVTV
0
00 (. ), )() ( )() (
(9 .3)
New System
with NISC
(stable)
Stable equilibrumStable equilibrum
Original
System
(stable)
x
1
(t)
x
2
(t)
„New“ ... sub-
SCADA
Coordination
y
Analysis
»
¼
º
«
¬
ª
ℑ
→ℑ
222
2211
') ,(
': )(
uux
uuGu
Analysis
»
¼
º
«
¬
ª
ℑ
→ℑ
222
2211
') ,(
': )(
uux
uuG...
flexible ac transmission systems ( (10)
... to compute the impact of the FACTS-devices on
lines on parallel paths. It computes the GSDF (generation shift distribution factors,
[5]) in order to quantify the impacts of FACTS-devices on all ... atic analysis of
network t opology with
backtracking algorithm
z
control path
z
all parallel paths
for every
FACTS-device
computing t he impact of the FACTS-
device on parallel paths w ith GSDF
det...
flexible ac transmission systems ( (11)
... filter known t o be ( 11.17), see e.g. [7].
p
T
T
m
T
QkKkukgkKkK
kgkkpkp
kykpkuk
QkukKku
kukK
kg
+−−−=
+−=
−−=
+−
−
=
) 1() ( )() 1() (
)() () 1() (
)() 1() ( )(
)() 1() (
)() 1(
)(
ε
ε
(1 1.17)
To ensure good ... tracking, a regularized constant trace algorithm is used with
4
21
10/ ≅cc .
300 11 Wide Area Control of FACTS
p
T
Qc
kKtr
kKc
kK
kKk...
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