consulta2 -sergio haffner

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Sistemas de Energia I – Folha de Consulta 2 – SHaffner Versão: 10/9/2007 Página 1 de 3 
 
Expressões gerais dos fluxos de corrente e potência nos ramos 
( ) ( ) mkmjkmkshkmkmkmkm VyeaVjbyaI kmϕ−−++= 2 ( ) ( ) mshkmkmkkmjkmmk VjbyVyeaI km ++−= ϕ 
( ) ( ) ( ) ( )[ ]kmkmkmkmkmkmmkkmkmkkmkm bgVVagVaP ϕθϕθ +++−= sencos2 
( ) ( ) ( ) ( ) ( )[ ]kmkmkmkmkmkmmkkmshkmkmkkmkm bgVVabbVaQ ϕθϕθ +−+−+−= cossen2 
 
Injeções de corrente e potência nas barras 
( ) ( )∑∑
Ω∈
−
Ω∈
−+








++=
k
km
k m
mkm
j
kmk
m
sh
kmkmkm
sh
kk VyeaVjbyajbI ϕ2 
VYI = (formulação matricial) 
sendo: 
 
I
 
– Vetor das injeções de corrente, cujas componentes são os fasores kI , NBk ,,2,1 L= ; 
V
 
– Vetor das tensões nodais, cujas componentes são os fasores kkk VV θ= , 
NBk ,,2,1 L= ; 
jBGY +=
 
– Matriz admitância nodal, cujos elementos são: 
km
j
kmkm yeaY km
ϕ−
−=
 
km
j
kmkm
j
kmmk yeayeaY kmmk
ϕϕ
−=−=
−
 
( )∑
Ω∈
++=
km
kmkm
sh
km
sh
kkk yajbjbY 2 
kΩ – Conjunto de barras vizinhas da barra k; 
K – Conjunto de todas as barras adjacentes à barra k, incluindo a própria barra k 
{ }( )kkK Ω∪= . 
 
∑∑
∈Ω∈
=+=
Km
mkm
m
mkmkkkk VYVYVYI
k
 
( )∑
∈
+=
Km
mmkmkmk VjBGI θ 
( )( )∑
∈
+−=
Km
kmkmkmkmmkk jjBGVVS θθ sencos 
( )∑
∈
+=
Km
kmkmkmkmmkk BGVVP θθ sencos 
( )∑
∈
−=
Km
kmkmkmkmmkk BGVVQ θθ cossen 
 
Fluxo de carga 
( )
( ) { }
( ) { }




∈=−−
∈=+−
∑
∑
∈
∈
PQ barras0cossen
PV e PQ barras0sencos
S1
esp
esp
kBGVVQ
kBGVVP
Km
kmkmkmkmmkk
Km
kmkmkmkmmkk
θθ
θθ
 
( )
( ) { }
( ) { }




∈−=
∈+=
∑
∑
∈
∈
Vθ e PV barrascossen
Vθ barrasencos
S2
kBGVVQ
kBGVVP
Km
kmkmkmkmmkk
Km
kmkmkmkmmkk
θθ
θθ
 
 
( ) ( )( )
( ) { }
( ) { }

∈=−=∆
∈=−=∆
⇒




=−=∆
=−=∆
=
PQ barras0,
PV e PQ barras0,
0,
0,
S1
esp
esp
esp
esp
kVQQQ
kVPPP
VQQQ
VPPP
kkk
kkk
θ
θ
θ
θ
 
Sistemas de Energia I – Folha de Consulta 2 – SHaffner Versão: 10/9/2007 Página 2 de 3 
 
 
 
 
Newton-Raphson 
PQ
PV PQ
←
+←








=
υ
υ
υ θ
V
x 
PQ
PV PQ
←
+←








∆
∆
=∆
υ
υ
υ θ
V
x 
( )
PQ
PV PQ
←
+←








∆
∆
= υ
υ
υ
Q
P
xg 
( ) ( ) ννν xxJxg ∆−= ( )[ ] ( )ννν xgxJx 1−−=∆ 
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
V
VQ
L
VQ
M
V
VPNVPH
V
VQVQ
V
VPVP
xJ
∂
∂
=
∂
∂
=
∂
∂
=
∂
∂
=
←
+←
↑
+
↑












∂
∂
∂
∂
∂
∂
∂
∂
−=
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ υ
υ
,,
,,
PQ
PV PQ
PQPVPQ
,,
,,
 








∆
∆






=








∆
∆
υ
υυ
υ
υ θ
VLM
NH
Q
P
 
( )
( )
( )









Ω∉=
Ω∈−=
∂
∂
=
+−=
∂
∂
=
∂
∂
=
∑
Ω∈
kkl
kklklklkllk
l
k
kl
m
kmkmkmkmmk
k
k
kk
lH
lBGVVPH
BGVVPH
VPH
k
0
cossen
cossen
, θθ
θ
θθ
θ
θ
θ
 
( )
( )
( )









Ω∉=
Ω∈+=
∂
∂
=
++=
∂
∂
=
∂
∂
=
∑
Ω∈
kkl
kklklklklk
l
k
kl
m
kmkmkmkmmkkk
k
k
kk
lN
lBGV
V
PN
BGVGV
V
PN
V
VPN
k
0
sencos
sencos2
, θθ
θθ
θ
 
( )
( )
( )









Ω∉=
Ω∈+−=
∂
∂
=
+=
∂
∂
=
∂
∂
=
∑
Ω∈
kkl
kklklklkllk
l
k
kl
m
kmkmkmkmmk
k
k
kk
lM
lBGVVQM
BGVVQM
VQ
M
k
0
sencos
sencos
,
θθ
θ
θθ
θ
θ
θ
 
( )
( )
( )









Ω∉=
Ω∈−=
∂
∂
=
−+−=
∂
∂
=
∂
∂
=
∑
Ω∈
kkl
kklklklklk
l
k
kl
m
kmkmkmkmmkkk
k
k
kk
lL
lBGV
V
Q
L
BGVBV
V
Q
L
V
VQ
L
k
0
cossen
cossen2
,
θθ
θθ
θ
 
Sistemas de Energia I – Folha de Consulta 2 – SHaffner Versão: 10/9/2007 Página 3 de 3 
 
 
 
Newton desacoplado 
( ) ( )
( ) ( )




∆+=
∆⋅′=∆⋅




∆+=
∆⋅′=∆⋅
+
++
−
+
−
PQ barras
PQ barras,,QV Iteração 
PV e PQ barras
PV e PQ barras,,Pθ Iteração 
1
111
2
1
1
1
2
1
ννν
νννννν
ννν
νννννν
θθ
θθθ
θθθ
VVV
VVLVQV
VHVPV
 
( )
( )









Ω∉=
Ω∈−=
∂
∂
=
−
−
=+−=
∂
∂
=
=′
∑
Ω∈
−
kkl
kklklklkll
l
k
k
kl
kkk
k
k
m
kmkmkmkmm
k
k
k
kk
lH
lBGVP
V
H
BVV
QBGVP
V
H
HVH
k
0
cossen
1
cossen
1
'
'
'
1 θθ
θ
θθ
θ
 
( )









Ω∉=
Ω∈−=
∂
∂
=
−=−+−=
∂
∂
=
=′
∑
Ω∈
−
kkl
kklklklkl
l
k
k
kl
kk
k
k
m
kmkmkmkmm
k
kk
k
k
k
kk
lL
lBG
V
Q
V
L
B
V
QBGV
V
B
V
Q
V
L
LVL
k
0
cossen
1
cossen
121
'
'
2
'
1 θθ
θθ
 
 
Desacoplado rápido 
( )
( )




∆+=
∆⋅′′=∆⋅




∆+=
∆⋅′=∆⋅
+
+
−
+
−
PQ barras
PQ barras,QV Iteração 
PV e PQ barras
PV e PQ barras,Pθ Iteração 
1
11
2
1
1
1
2
1
ννν
νννν
ννν
νννν
θ
θθθ
θθ
VVV
VBVQV
BVPV
 







Ω∉=
Ω∈−=
=
′
−
Ω∈
−∑
kkl
kklkl
m
kmkk
lB
lxB
xB
B
k
0'
1'
1'
 





Ω∉=
Ω∈−=
−−≈
′′
kkl
kklkl
sh
kkkkk
lB
lBB
BBB
B
0''
''
''
 
sendo shkB a soma de todas as susceptâncias que ligam o nó k à terra. 
 
Expressões gerais 
( ) ( ) jXRI
V
jYjZ +=== ωω
1
 
( ) ( ) jBGV
I
jZjY +=== ωω
1
 
 
jQPjVIVIIVIVS +=+==+−=⋅= θθθθφφ sencos* θ
φ
φ
φ cos
3
3
3 == Q
P
FP 
θθφφ cos3cos33 LLL IVIVP == θθφφ sen3sen33 LLL IVIVQ == LLL IVIVS 333 == φφ