Formula Sheets: Load Flow Studies | Power Systems - Electrical Engineering (EE) PDF Download

 Page 1


Power S ystems F ormula Sheet: Load Flow
Studies
1. Load Flow Basics
• Objective : Determine bus voltages, l ine flows, and losses in a power system
under steady-state conditions.
• Bus Types :
– Slack Bus: KnownV (magnitude and angle), unknownP andQ .
– PV Bus (Gener ator): KnownP and|V| , unknownQ andd .
– PQ Bus (Load): KnownP andQ , un known|V| andd .
• Power Flow Equations (for busi ):
P
i
=
n
?
k=1
|V
i
||V
k
||Y
ik
| cos(?
ik
+d
k
-d
i
)
Q
i
=
n
?
k=1
|V
i
||V
k
||Y
ik
| sin(?
ik
+d
k
-d
i
)
whereV
i
=|V
i
|?d
i
is bus voltage,Y
ik
=|Y
ik
|??
ik
is admittance matrix element,n
is number of buses.
2. Bus A dmittance Matrix (Y-bus)
• Y-bus Definition :
I = Y
bus
V
whereI is bus current vector ,V is bus voltage vector .
• Y-bus Elements :
Y
ii
=
?
(admittances connected to busi )
Y
ik
=- (admittance between busi andk ), i?= k
• Line A dmittance : F or a line between busesi andk :
y
ik
=
1
R
ik
+jX
ik
1
Page 2


Power S ystems F ormula Sheet: Load Flow
Studies
1. Load Flow Basics
• Objective : Determine bus voltages, l ine flows, and losses in a power system
under steady-state conditions.
• Bus Types :
– Slack Bus: KnownV (magnitude and angle), unknownP andQ .
– PV Bus (Gener ator): KnownP and|V| , unknownQ andd .
– PQ Bus (Load): KnownP andQ , un known|V| andd .
• Power Flow Equations (for busi ):
P
i
=
n
?
k=1
|V
i
||V
k
||Y
ik
| cos(?
ik
+d
k
-d
i
)
Q
i
=
n
?
k=1
|V
i
||V
k
||Y
ik
| sin(?
ik
+d
k
-d
i
)
whereV
i
=|V
i
|?d
i
is bus voltage,Y
ik
=|Y
ik
|??
ik
is admittance matrix element,n
is number of buses.
2. Bus A dmittance Matrix (Y-bus)
• Y-bus Definition :
I = Y
bus
V
whereI is bus current vector ,V is bus voltage vector .
• Y-bus Elements :
Y
ii
=
?
(admittances connected to busi )
Y
ik
=- (admittance between busi andk ), i?= k
• Line A dmittance : F or a line between busesi andk :
y
ik
=
1
R
ik
+jX
ik
1
• Shunt A dmittance : Includes line charging and shunt capacitors:
Y
ii
= Y
ii
+
y
ik, shunt
2
3. Gauss-Seidel Method
• V oltage Update (for PQ busi ):
V
(k+1)
i
=
1
Y
ii
[
P
i
-jQ
i
V
(k)*
i
-
?
m?=i
Y
im
V
(k)
m
]
wherek is iter ation number ,V
*
i
is complex con jugate of voltage.
• F or PV Bus : UpdateQ
i
, then adjust|V
i
| to specified value, compute d
i
.
• Convergence : Stop when|V
(k+1)
i
-V
(k)
i
| < ? (small toler ance).
4. Newton-Raphson Method
• Power Mismatch Equations :
?P
i
= P
i, spec
-P
i, calc
?Q
i
= Q
i, spec
-Q
i, calc
• Jacobian Matrix :
[
?P
?Q
]
=
[
J
1
J
2
J
3
J
4
][
?d
?|V|
]
where:
J
1
=
?P
i
?d
j
, J
2
=
?P
i
?|V
j
|
, J
3
=
?Q
i
?d
j
, J
4
=
?Q
i
?|V
j
|
• Jacobian Elements :
?P
i
?d
i
=
?
k?=i
|V
i
||V
k
||Y
ik
| sin(?
ik
+d
k
-d
i
)
?P
i
?|V
i
|
=2|V
i
||Y
ii
| cos?
ii
+
?
k?=i
|V
k
||Y
ik
| cos(?
ik
+d
k
-d
i
)
?Q
i
?d
i
=-
?
k?=i
|V
i
||V
k
||Y
ik
| cos(?
ik
+d
k
-d
i
)
?Q
i
?|V
i
|
=2|V
i
||Y
ii
| sin?
ii
+
?
k?=i
|V
k
||Y
ik
| sin(?
ik
+d
k
-d
i
)
• Update Rule :
[
d
|V|
]
(k+1)
=
[
d
|V|
]
(k)
+
[
?d
?|V|
]
2
Page 3


Power S ystems F ormula Sheet: Load Flow
Studies
1. Load Flow Basics
• Objective : Determine bus voltages, l ine flows, and losses in a power system
under steady-state conditions.
• Bus Types :
– Slack Bus: KnownV (magnitude and angle), unknownP andQ .
– PV Bus (Gener ator): KnownP and|V| , unknownQ andd .
– PQ Bus (Load): KnownP andQ , un known|V| andd .
• Power Flow Equations (for busi ):
P
i
=
n
?
k=1
|V
i
||V
k
||Y
ik
| cos(?
ik
+d
k
-d
i
)
Q
i
=
n
?
k=1
|V
i
||V
k
||Y
ik
| sin(?
ik
+d
k
-d
i
)
whereV
i
=|V
i
|?d
i
is bus voltage,Y
ik
=|Y
ik
|??
ik
is admittance matrix element,n
is number of buses.
2. Bus A dmittance Matrix (Y-bus)
• Y-bus Definition :
I = Y
bus
V
whereI is bus current vector ,V is bus voltage vector .
• Y-bus Elements :
Y
ii
=
?
(admittances connected to busi )
Y
ik
=- (admittance between busi andk ), i?= k
• Line A dmittance : F or a line between busesi andk :
y
ik
=
1
R
ik
+jX
ik
1
• Shunt A dmittance : Includes line charging and shunt capacitors:
Y
ii
= Y
ii
+
y
ik, shunt
2
3. Gauss-Seidel Method
• V oltage Update (for PQ busi ):
V
(k+1)
i
=
1
Y
ii
[
P
i
-jQ
i
V
(k)*
i
-
?
m?=i
Y
im
V
(k)
m
]
wherek is iter ation number ,V
*
i
is complex con jugate of voltage.
• F or PV Bus : UpdateQ
i
, then adjust|V
i
| to specified value, compute d
i
.
• Convergence : Stop when|V
(k+1)
i
-V
(k)
i
| < ? (small toler ance).
4. Newton-Raphson Method
• Power Mismatch Equations :
?P
i
= P
i, spec
-P
i, calc
?Q
i
= Q
i, spec
-Q
i, calc
• Jacobian Matrix :
[
?P
?Q
]
=
[
J
1
J
2
J
3
J
4
][
?d
?|V|
]
where:
J
1
=
?P
i
?d
j
, J
2
=
?P
i
?|V
j
|
, J
3
=
?Q
i
?d
j
, J
4
=
?Q
i
?|V
j
|
• Jacobian Elements :
?P
i
?d
i
=
?
k?=i
|V
i
||V
k
||Y
ik
| sin(?
ik
+d
k
-d
i
)
?P
i
?|V
i
|
=2|V
i
||Y
ii
| cos?
ii
+
?
k?=i
|V
k
||Y
ik
| cos(?
ik
+d
k
-d
i
)
?Q
i
?d
i
=-
?
k?=i
|V
i
||V
k
||Y
ik
| cos(?
ik
+d
k
-d
i
)
?Q
i
?|V
i
|
=2|V
i
||Y
ii
| sin?
ii
+
?
k?=i
|V
k
||Y
ik
| sin(?
ik
+d
k
-d
i
)
• Update Rule :
[
d
|V|
]
(k+1)
=
[
d
|V|
]
(k)
+
[
?d
?|V|
]
2
5. Power Flow in Lines
• Power Flow (from busi to busk ):
S
ik
= P
ik
+jQ
ik
= V
i
(I
ik
)
*
where:
I
ik
=(V
i
-V
k
)y
ik
+V
i
y
ik, shunt
2
• Line Losses :
P
loss
= P
ik
+P
ki
, Q
loss
= Q
ik
+Q
ki
6. DC Load Flow
• Assumptions : |V
i
|˜1 p.u., sin(d
i
-d
k
)˜ d
i
-d
k
, neglect reactive power .
• Power Flow :
P
i
=
n
?
k=1
d
i
-d
k
X
ik
whereX
ik
is lin e reactance.
• Matrix F orm :
P = Bd
whereB is the susceptance matrix (B
ik
=-
1
X
ik
,B
ii
=
?
1
X
ik
).
3