The document discusses power flow analysis and solutions using the Gauss-Seidel method. It describes setting up the bus admittance matrix and node-voltage equations based on impedance values between nodes. The Gauss-Seidel method is then used to iteratively solve the nonlinear power flow equations to determine bus voltages and power flows by updating the solution for one variable at a time. Instructions are provided on applying the method to different bus types including slack, PQ and PV buses.
1. Energy Conversion Lab
POWER FLOW ANALYSIS
Power flow analysis assumption
steady-state
balanced single-phase network
network may contain hundreds of nodes and
branches with impedance X specified in per unit on
MVA base
Power flow equations
bus admittance matrix of node-voltage equation is
formulated
currents can be expressed in terms of voltages
resulting equation can be in terms of power in MW
2. Energy Conversion Lab
BUS ADMITTANCE MATRIX
Nodal solution
nodal solution is based on
the Kirchhoff’s current law
impedance is converted to
admittance
Bus admittance equations
the impedance diagram:
see Fig.6.1
ijijij
ij
jxrZ
y
+
==
11
3. Energy Conversion Lab
BUS ADMITTANCE MATRIX
Bus admittance
equations
the admittance is
based on bus-to-
bus: see Fig.6.2
if no connection
between bus-to-
bus, leave as zero
node voltage
equation is in the
form
busbusbus VYI =
4. Energy Conversion Lab
BUS ADMITTANCE MATRIX
Node-voltage matrix
Ibus=YbusVbus
Ibus is the vector of injected currents
Vbus is the vector of the bus voltage from reference
node
Ybus is the bus admittance matrix
=
n
i
n
i
V
V
V
V
I
I
I
I
2
1
2
1
nnnin2n1
iniii2i1
2n2i2221
1n1i1211
YYYY
YYYY
YYYY
YYYY
5. Energy Conversion Lab
BUS ADMITTANCE MATRIX
Node-voltage matrix
diagonal element Yii: sum of admittance connected to bus i
off-diagonal matrix Yij: negative of admittance between nodes I
and j
when the bus currents are known, bus voltages are unknown,
bus voltage can be solved as
inverse of bus admittance matrix is known as impedance matrix
Zbus
if matrix of Ybus is invertible, Ybus should be non-singular
ij
0
≠= ∑=
n
j
ijii yY
ijjiij yYY −==
busbusbus IYV 1−
=
1−
= busbus YZ
6. Energy Conversion Lab
BUS ADMITTANCE MATRIX
Node-voltage matrix
admittance matrix is symmetric along the leading diagonal,
which result in an upper diagonal nodal admittance matrix
a typical power system network, each bus is connected by a few
nearby bus, which cause many off-diagonal elements are zero
many zero off-diagonal matrix is called sparse matrix
the bus admittance matrix in Fig.(6.2) by inspection is
−
−
−
=
5.125.1200
5.125.220.50.5
00.575.85.2
00.55.25.8
jj
j-jjj
jjj
jjj
Ybus
7. Energy Conversion Lab
SOLUTION OF NONLINEAR ALGEBRA EQUATIONS
Techniques for iterative solution of non-linear
equations
Gauss-Seidal
Newton-Raphson
Quasi-Newton
Gause-Seidal method
consider a nonlinear equation f(x)=0
rearrange f(x) so that x=g(x), f(x)=x-g(x) or
f(x)=g(x)-x
guess an initial estimate of x = x(k)
use iteration, obtain next x value as x(k+1) = g(x(k))
criteria for stop iteration: |x(k+1)-x(k)| ≤ ε
ε is the desired accuracy
8. Energy Conversion Lab
GAUSE-SEIDAL METHOD
Nature of Gause-Seidal method
see Ex.(6.2) and Fig.(6.3)
Gause-Seidal method needs many iterations to
achieve desired accuracy
no guarantee for the convergence, depend on the
location of initial x estimate
9. Energy Conversion Lab
GAUSE-SEIDAL METHOD
Nature of Gause-Seidal method
solution: if initial estimate x is within convergent
region, solution will converge in zigzag fashion to one
of the roots
no solution: if initial estimate x is outside convergent
region, process will diverge, no solution found
in some case, an acceleration factor α is added to
improve the rate of convergence:
x(k+1) = x(k) +α[g(x(k))-x(k)], where α>1
acceleration factor should not too large to produce
overshoot
see Ex.(6.3) for the acceleration factor used
10. Energy Conversion Lab
GAUSE-SEIDAL METHOD
Extend one variable to n variable equations using Gause-
Seidal method
consider the system of n equations in n variables and solving for
one variable from each equation in one time of iteration
the updated variable x1
(k+1) calculated in first equation in
Eq.(6.12) is used in the calculation of x2
(k+1) in the second
equation
Ex: in the 2nd iteration x2
(k+1) = c2+g2(x1
(k+1)+x2
(k)+x3
(k)+…+xn
(k))
at n iteration to complete n variables, the x1
(k+1),…,xn
(k+1) is
tested against x1
(k),…,xn
(k) for accuracy criterion
nnn
n
n
cxxxf
cxxxf
cxxxf
=
=
=
),,,(
),,,(
),,,(
21
2212
1211
),,,(
),,,(
),,,(
21
21222
21111
nnnn
n
n
xxxgcx
xxxgcx
xxxgcx
+=
+=
+=
11. Energy Conversion Lab
POWER FLOW SOLUTION
Power Flow (Load Flow)
operating condition: balanced, single phase model
quantities used in power flow equation are: voltage
magnitude |V|, phase angle δ, real power P, and
reactive power Q
system bus classification:
slack bus (swing bus): taken as reference where |V| and
∠V are specified. It makes up the loss between generated
power and scheduled loads
load bus (PQ bus): P and Q are specified, |V| and ∠V are
unknown
regulated bus (PV bus): P and |V| are specified, ∠V and Q
are unknown
12. Energy Conversion Lab
POWER FLOW EQUATION
Power flow formulation
consider bus case in Fig.(6.7)
current flow into bus i:
express Ii in terms of P,Q:
the power flow equation becomes
the power flow problem results in algebraic nonlinear equations
which must be solved by iteration methods
ij
10
≠−= ∑∑ ==
j
n
j
ij
n
j
ijii VyyVI
*
i
ii
i
V
jQP
I
−
=
ij
10
*
≠−=
−
∑∑ ==
j
n
j
ij
n
j
iji
i
ii
VyyV
V
jQP
13. Energy Conversion Lab
GAUSS-SEIDEL POWER FLOW EQUATION
Gauss-Seidel power flow solution
solving Vi: for PQ bus, assume P,Q are known
solving Pi: for slack bus, assume V is known
solving Qi: for PV bus, assume |V| is known
ij
)(
)(*
)1(
≠
+
−
=
∑
∑+
ij
k
kijk
i
sch
i
sch
i
k
i
y
Vy
V
jQP
V
ijRe )(
10
)()(*)1(
≠
−= ∑∑
≠
==
+ k
j
n
ij
j
ij
n
j
ij
k
i
k
i
k
i VyyVVP
ijIm )(
10
)()(*)1(
≠
−−= ∑∑
≠
==
+ k
j
n
ij
j
ij
n
j
ij
k
i
k
i
k
i VyyVVQ
14. Energy Conversion Lab
GAUSS-SEIDEL POWER FLOW EQUATION
Instructions for Gauss-Seidel solution
there are 2(n-1) equations to be solved for n bus
voltage magnitude of the buses are close to 1pu or
close to the magnitude of the slack bus
voltage magnitude at load buses is lower than the slack
bus value
voltage magnitude at generator buses is higher than
the slack bus value
phase angle of load buses are below the reference
angle
phase angle of generator buses are above the
reference angle
15. Energy Conversion Lab
INSTRUCTIONS FOR G-S SOLUTION
Instructions for PQ bus solution
real and reactive power Pi
sch, Qi
sch are known
starting with an initial estimate of voltage using Vi
equation
Instructions for PV bus solution
Pi
sch, |Vi| are specified
assume Vi = |Vi|∠0o, solve the Qi equation as below
ij
)(
)(*
)1(
≠
+
−
=
∑
∑+
ij
k
jijk
i
sch
i
sch
i
k
i
y
Vy
V
jQP
V
ijIm )(
10
)()(*)1(
≠
−−= ∑∑
≠
==
+ k
j
n
ij
j
ij
n
j
ij
k
i
k
i
k
i VyyVVQ
16. Energy Conversion Lab
INSTRUCTIONS FOR G-S SOLUTION
Instructions for PV bus solution
when Qi
(k+1) is available, solve Vi using equation below
since |Vi| is specified, keep imaginary part of Vi,
calculate real part of Vi
solve Vi
stopping criteria
ij
)(
)(*
)(
)1(
≠
+
−
=
∑
∑+
ij
k
kijk
i
k
i
sch
i
k
i
y
Vy
V
jQP
V
{ } { }( )2)1(2)1(
Re ++
−= k
ii
k
i VimagVV
{ } { })1()1()1(
ImRe +++
+= k
i
k
i
k
i VjVV
{ } { } { } { } εε ≤−≤− ++ )()1()()1(
ImIm,ReRe k
i
k
i
k
i
k
i VVVV
17. Energy Conversion Lab
INSTRUCTIONS FOR G-S SOLUTION
Instructions for PV bus solution
to accelerate the convergence, using the following
approximation after new Vi is obtained
α is in the range between 1.3 to 1.7
voltage accuracy in |Vi| and ∠δ is in the range between
0.00001 to 0.00005
( ))()()()1( k
i
k
cali
k
i
k
i VVVV −+=+
α
18. Energy Conversion Lab
INSTRUCTIONS FOR G-S SOLUTION
Instructions for V, ∠δ slack bus solution
solve Pi
solve Qi
accuracy: the largest ΔPΔQ is less than the
specified value, typically is about 0.001pu
ijRe )(
10
)()(*)1(
≠
−= ∑∑
≠
==
+ k
j
n
ij
j
ij
n
j
ij
k
i
k
i
k
i VyyVVP
ijIm )(
10
)()(*)1(
≠
−−= ∑∑
≠
==
+ k
j
n
ij
j
ij
n
j
ij
k
i
k
i
k
i VyyVVQ
19. G-S Power flow Homework
For the one-line diagram shown below, using the G-S method
to determine all bus voltages (magnitude and phase) and
show the power flow solution between the buses assume the
regulated bus (#2) reactive power limits are between 0 and
600Mvar.
20. Energy Conversion Lab
NEWTON RAPHSON METHOD
Newton Raphson method for solving one variable
consider the solution of one-dimensional equation f(x)=c
assume x = x(0)+∆x(0)
f(x)=f(x(0)+∆x(0))=c
use Taylor’s series expansion
assume ∆x(0) is very small, higher order terms of expansion can
be neglected, Taylor series becomes
assume f(x(0))=c-∆c(0), the equation becomes ∆c(0)≅(df/dx)(0)∆x(0)
the new approximation of x
( ) cx
dx
fd
x
dx
df
xfxxf =+∆
+∆
+=∆+
2)0(
)0(
2
2
)0(
)0(
)0()0()0(
!2
1
)()(
cx
dx
df
xfxxf =∆
+=∆+ )0(
)0(
)0()0()0(
)()(
)0(
)0(
)0()1(
∆
+=
dx
df
c
xx
21. Energy Conversion Lab
NEWTON RAPHSON METHOD
Newton Raphson method for solving one variable
the new approximation of x
Newton Raphson algorithm
for more information, see Ex.(6.4)
Newton’s method converges faster than Gauss-Seidal, the root
may converge to a root different from the expected one or
diverge if the starting value is not close enough to the root
)0(
)0(
)0()1(
∆
+=
dx
df
c
xx
)()()1(
)(
)(
)(
)()(
)(
kkk
k
k
k
kk
xxx
dx
df
c
x
xfcc
∆+=
∆
=∆
−=∆
+
22. Energy Conversion Lab
NEWTON RAPHSON METHOD FOR n VARIABLES
Newton Raphson method for solving n variables
nn
n
nnn
nn
n
n
n
n
cx
x
f
x
x
f
x
x
f
xfxxf
cx
x
f
x
x
f
x
x
f
xfxxf
cx
x
f
x
x
f
x
x
f
xfxxf
=∆
∂
∂
++∆
∂
∂
+∆
∂
∂
+=∆+
=∆
∂
∂
++∆
∂
∂
+∆
∂
∂
+=∆+
=∆
∂
∂
++∆
∂
∂
+∆
∂
∂
+=∆+
)0(
)0(
)0(
2
)0(
2
)0(
1
)0(
1
)0()0()0(
2
)0(
)0(
2)0(
2
)0(
2
2)0(
1
)0(
1
2)0(
2
)0()0(
2
1
)0(
)0(
1)0(
2
)0(
2
1)0(
1
)0(
1
1)0(
1
)0()0(
1
)()(
)()(
)()(
23. Energy Conversion Lab
NEWTON RAPHSON METHOD FOR n VARIABLES
Rearrange in matrix form
The matrix can be written as
ΔC(k) = J(k) ΔX(k)
∆
∆
∆
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
−
−
−
)0(
)0(
2
)0(
1
)0()0(
2
)0(
1
)0(
2
)0(
2
2
)0(
1
2
)0(
1
)0(
2
1
)0(
1
1
)0(
)0(
22
)0(
11
n
n
nnn
n
n
nn x
x
x
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
fc
fc
fc
24. Energy Conversion Lab
NEWTON RAPHSON METHOD FOR n VARIABLES
The Newton-Raphson algorithm for n-
dimensional case is
X(k+1) = X(k) +ΔX(k) = X(k) + [J(k)]-1ΔC(k)
where
−
−
−
=∆
)(
)(
22
)(
11
)(
k
nn
k
k
k
fc
fc
fc
C
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
)()(
2
)(
1
)(
2
)(
2
2
)(
1
2
)(
1
)(
2
1
)(
1
1
)(
k
n
n
k
n
k
n
k
n
kk
k
n
kk
k
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
J
∆
∆
∆
=∆
)(
)(
2
)(
1
)(
k
n
k
k
k
x
x
x
X
25. Energy Conversion Lab
NEWTON RAPHSON METHOD FOR n VARIABLES
The Newton-Raphson algorithm
J(k) is called the Jacobian matrix
solution to X(k+1) is inefficient because it involves
inverse of J(k) , a triangular factorization is used to
facilitate the computation
in MATLAB, the operator “” (i.e., ΔX=JΔC) is used to
apply the triangular factorization
Newton-Raphson method converge to solution
quadratically when near a root
The limitation is that it does not generally converge to
a solution from an arbitrary starting point
26. Energy Conversion Lab
LINE FLOWS AND LOSSES
Complex power flow between bus i,j
for line model, see Fig. 6.8
current flow from bus i to bus j
current flow from bus j to bus i
complex power Sij from bus i to j and Sji from j to i
power loss in the line i-j
for more Gauss-Seidel method examples, see Ex. (6.7)
and Ex. (6.8)
iijiijilij VyVVyIII 00 )( +−=+=
jjijijjlji VyVVyIII 00 )( +−=+−=
**
jijjiijiij IVSIVS ==
jiijjiL SSS +=− )(
27. Energy Conversion Lab
NEWTON-RAPHSON POWER FLOW
Real power flow in terms of Vi , ∠δ, and Yij
Reactive power flow
Newton-Raphson matrix form: ΔC(k) = J(k) ΔX(k)
diagonal and off-diagonal elements of J1
( )∑=
+−=
n
j
jiijijjii YVVP
1
cos δδθ
( )∑=
+−−=
n
j
jiijijjii YVVQ
1
sin δδθ
∆
∆
=
∆
∆
VJJ
JJ
Q
P δ
43
21
( )
( ) ijsin
sin
≠+−−=
∂
∂
+−=
∂
∂
∑≠
jiijijji
j
i
ij
jiijijji
i
i
YVV
P
YVV
P
δδθ
δ
δδθ
δ
28. Energy Conversion Lab
NEWTON-RAPHSON POWER FLOW
Newton-Raphson matrix form: ΔC(k) = J(k) ΔX(k)
diagonal and off-diagonal elements of J2
diagonal and off-diagonal elements of J3
∆
∆
=
∆
∆
VJJ
JJ
Q
P δ
43
21
( )
( ) ijcos
coscos2
≠+−=
∂
∂
+−+=
∂
∂
∑≠
jiijiji
j
i
ij
jiijijjiiiii
i
i
YV
V
P
YVYV
V
P
δδθ
δδθθ
( )
( ) ijcos
cos
≠+−−=
∂
∂
+−=
∂
∂
∑≠
jiijijji
j
i
ij
jiijijji
i
i
YVV
Q
YVV
Q
δδθ
δ
δδθ
δ
29. Energy Conversion Lab
NEWTON-RAPHSON POWER FLOW
Newton-Raphson matrix form: ΔC(k) = J(k) ΔX(k)
diagonal and off-diagonal elements of J4
power residuals ΔPi
(k) ΔQi
(k)
new estimates for bus voltages
∆
∆
=
∆
∆
VJJ
JJ
Q
P δ
43
21
( )
( ) ijsin
sinsin2
≠+−−=
∂
∂
+−−−=
∂
∂
∑≠
jiijiji
j
i
ij
jiijijjiiiii
i
i
YV
V
Q
YVYV
V
Q
δδθ
δδθθ
)()()()(
, k
i
sch
i
k
i
k
i
sch
i
k
i QQQPPP −=∆−=∆
)()()1()()()1(
, k
i
k
i
k
i
k
i
k
i
k
i VVV ∆+=∆+= ++
δδδ
30. Energy Conversion Lab
NEWTON-RAPHSON POWER FLOW
Procedure for Newton-Raphson method:
PQ bus: set |Vi
(0)|=1.0, δi
(0)=0.0
PV bus: set δi
(0)=0.0
set PQ bus equation for J matrix elements:
set PV bus equation for J matrix elements:
)()()()(
, k
i
sch
i
k
i
k
i
sch
i
k
i QQQPPP −=∆−=∆
( )∑=
+−=
n
j
jiijijjii YVVP
1
cos δδθ
( )∑=
+−−=
n
j
jiijijjii YVVQ
1
sin δδθ
( )∑=
+−=
n
j
jiijijjii YVVP
1
cos δδθ
)()( k
i
sch
i
k
i PPP −=∆
31. Energy Conversion Lab
NEWTON-RAPHSON POWER FLOW
Procedure for Newton-Raphson method:
use above equation to calculate Jacobian matrix (J1, J2,
J3, J4)
solve Δ|V| and Δδ using Newton-Raphson matrix
update Δ|V| and Δδ by
repeat the calculation until
for example: see Ex.(6.10)
εε ≤∆≤∆ )()(
, k
i
k
i QP
∆
∆
=
∆
∆
VJJ
JJ
Q
P δ
43
21
)()()1()()()1(
, k
i
k
i
k
i
k
i
k
i
k
i VVV ∆+=∆+= ++
δδδ
32. Energy Conversion Lab
FAST DECOUPLED POWER FLOW
Fast decoupled power flow solution:
the algorithm is based on Newton-Raphson method
when transmission lines has a high X/R ratio, the
Newton-Raphson method could be further simplified
Consider the Newton-Raphson power flow
equation
ΔP are less sensitive to |V| and most sensitive to Δδ
ΔQ is less sensitive to Δδ and most sensitive to |V|
we can reasonably eliminate J2 and J3 elements in
Jacobian matrix
∆
∆
=
∆
∆
VJJ
JJ
Q
P δ
43
21
33. Energy Conversion Lab
FAST DECOUPLED POWER FLOW
Consider the Newton-Raphson power flow equation
the power flow equation reduces to
ΔP = J1Δδ = [∂P/∂δ]Δδ, ΔQ = J4Δ|V| = [∂Q/∂|V|]Δ|V|
∂Pi/∂δi = -Qi - |Vi|2Bii, Bii = |Yii|sinθii is the imaginary part of the
diagonal elements
since Bii >> Qi, ∂Pi/∂δi (diagonal elements of J1) can be further
reduced to ∂Pi/∂δi = - |Vi|Bii (|Vi|2 ≈|Vi| )
off diagonal element of J1: ∂Pi/∂δi = - |Vi||Vj|Yijsin(θij-δi+δj), since δj-δi
is quite small, θij-δi+δj = θij, J1 = ∂Pi/∂δj = - |Vi||Vj|Bij
since |Vj|≈1, off diagonal elements of J1 = ∂Pi/∂δj = - |Vi|Bij
∆
∆
=
∆
∆
VJ
J
Q
P δ
4
1
0
0
34. Energy Conversion Lab
FAST DECOUPLED POWER FLOW
Consider the Newton-Raphson power flow
equation
similarly, diagonal elements of J4: ∂Qi/∂|Vi| = - |Vi|Bii
off diagonal elements of J4: ∂Qi/∂|Vj| = - |Vi|Bij
therefore, ΔP and ΔQ has the following forms
B’ and B” are the imaginary part of Ybus
the updated Δδ and Δ|V| can be obtained from
to calculate PQ bus, use simplified J1 and J4 to obtain
solution
to calculate PV bus, J4 can be further eliminated, only
J1 is used to obtain solution
VB
V
Q
B
V
P
ii
∆−=
∆
∆−=
∆ '''
,δ
[ ] [ ]
V
Q
BV
V
P
B
∆
−=∆
∆
−=∆
−− 11
",'δ
35. Energy Conversion Lab
FAST DECOUPLED POWER FLOW
Comparison between fast decouple power flow
solution and Newton Raphson power flow solution
fast decoupled solution requires more iterations than
Newton Raphson solution
fast decoupled solution requires less time per iteration
since decoupled solution needs less time for iteration,
the overall computation time may be less than using
the Newton Raphson method
fast decoupled solution often used in fast computation
of power flow, for example, contingency analysis or on-
line control of power flow
see Ex. 6.12