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