Power System Analysis: Load flow studies:Gauss siedel,newton Raphson,Fast decoupled,dc power flow equations,problems

1. LOAD FLOW STUDIES

3.  In load flow analysis:
1) Voltage at various buses are found out.
2) Power injection into the transmission system is
obtained To ensure the reliable and economic
operation of the power system.
3) The likely difficulties including overloading of
one or more transmission lines, overloading of
generators, loss of one or more transmission
lines, shut down of generators etc. are studied.
 For normal operation of the power system
The voltages and powers are kept within certain
limits.

4.  The load flow studies generally require the following
steps:
1. Representation of the system by single line diagram.
2. Forming the impedance diagram.
3. Formation of network equations.
4. Solution of network equations.

7.  Load Buses : In these buses no generators are
connected and hence the generated real
power PGi and reactive power QGi are taken as
zero. The load drawn by these buses are defined
by real power -PLi and reactive power -QLi in which
the negative sign accommodates for the power
flowing out of the bus. This is why these buses are
sometimes referred to as P-Q bus. The objective
of the load flow is to find the bus voltage
magnitude |Vi| and its angle δi.

8. Voltage Controlled Buses : These are the buses where
generators are connected. Therefore the power
generation in such buses is controlled through a prime
mover while the terminal voltage is controlled through the
generator excitation.
Keeping the input power constant through turbine-
governor control and keeping the bus voltage constant
using automatic voltage regulator, we can specify
constant PGi and | Vi | for these buses.
This is why such buses are also referred to as P-V
buses. It is to be noted that the reactive power supplied
by the generator QGi depends on the system
configuration and cannot be specified in advance.
Furthermore we have to find the unknown angle δi of the
bus voltage.

9.  Slack or Swing Bus : Usually this bus is
numbered 1 for the load flow studies. This bus sets
the angular reference for all the other buses. Since
it is the angle difference between two voltage
sources that dictates the real and reactive power
flow between them, the particular angle of the slack
bus is not important. However it sets the reference
against which angles of all the other bus voltages
are measured. For this reason the angle of this bus
is usually chosen as 0° . Furthermore it is assumed
that the magnitude of the voltage of this bus is
known.

10.  Slack bus or Swing bus or Reference bus.
 Slack bus is a generator bus.
 Generator at slack bus supplies:
The specified real power injected into the system
at other buses- the total system output + Losses.

16. yp
q

17. To compute (k+1)th
iteration value of bus voltage, Vp
For buses less than p, (k+1)th
iteration values are
used.
For buses greater than p, kth
iteration values are
used.
For load buses – change in magnitude and phase angle
of voltage is allowed in each iteration.
For generator buses – change in voltage magnitude is
not allowed.
For slack bus – voltage magnitude and phase angle are
not modified.

18.  Reactive power is not specified.
 For using in the load flow equation, reactive power is
to be calculated.

20.  Qmax and Qmin will be specified.
 If limits violated, then Q is fixed at limit violated.
 Then treated as load bus.

21.  Acceleration factor – speeds up convergence.
 The accelerated value of the voltage in (k+1)th
iteration
Vp
k+1
= Vp
k
+ α(Vp
k+1
- Vp
k
)

22.  Advantages -
1. Simple technique.
2. Computer memory requirement is less.
3. Less computational time per iteration.
 Disadvantages -
1. Slow rate of convergence.
2. Therefore, large number of iterations.
3. As the number of buses increases, the number of iterations
increase.

23. Determine the voltage at the end of first iteration

27.  Slack bus power – calculated after computing the bus
voltage up to the specified accuracy.
 Pp-jQp=Vp* YƩ pqVq
k

31.  Powerful method of solving nonlinear algebraic
equations.
 Iterative method-Taylor series expansion.
 Faster, more accurate, reliable.
 Advantages:
◦ Convergence is better.
◦ Number of iterations required is less.
◦ Number of iterations independent of system size.
◦ Load flow solution of large power systems.

32.  Disadvantages:
◦ Solution technique is difficult.
◦ Calculation in each iteration is more.
◦ Computational time is more.
◦ Computer memory requirement is large.

39.  The process is repeated till
i)the largest ( in magnitude)element in the left column
of the equation , B matrix is less than a prespecified
value
or
ii) the largest element in the column vector C matrix is
less than a prespecified value

40.  In the formulation – quantities in rectangular form.
Vp = | Vp|<δp=ep+jfp
Pp=u1(e,f)
Qp=u2(e,f)

45. 

56.  Having calculated the Jaobian matrix and the
residual column vector corresponding to the initial
solution , the desired increment voltage vector
can be calculated by using any standard technique
 The next better solution will be
ep
1
=ep
0
+Δep
0
fp
1
=fp
0
+ Δfp
0

57.  These values of voltages will be used in the next
iteration .The process will be repeated and the bus
voltages will be
ep
k+1
=ep
k
+ Δep
k
and fp
k+1
= fp
k
+ Δfp
k
 The process is repeated till the magnitude of the largest
element in the residual vector is less than the
prespecified value

58. 1. Assume a suitable solution for all buses except slack
bus.Vp=1+j0 for p=2,3,4..n
2. Set convergence criterion =ϵ
3. Set iteration count , k=0 and bus count p=1
4. Calculate the real & reactive power Pp & Qp
5. Evaluate ΔPp
k
=Psp - Pp
k
6. Check for generator bus. If yes , compare the Qp
k
with the limits .

59. If the limit is violated , fix the reactive power generation
to the corresponding limit and treat the bus as a load
bus for that iteration
7. If limit is not violated evaluate the voltage residue |
ΔVp|2
= |Vp|2
spec - |Vk
p|2
8.Evaluate ΔQp
k
=Qsp-Qp
k
9.Advance bus count by 1 ie p=p+1
10.Determine the largest of the absolute values of the
residue .If that value is less than ϵ go to step 16

60. 11.Evaluate elements of Jacobian matrix
12.Calculate the voltage increments Δep
k
and Δfp
k
13.Calculate new bus voltages ep
k+1
=ep
k
+ Δep
k
and fp
k+1
=
fp
k
+ Δfp
k
14.Evaluate cosδ and sin δ for all voltages
15.Advance iteration count k=k+1 and repeat the
process
16.Evaluate bus and line powers

62.  Powerful method of solving nonlinear algebraic
equations.
 Iterative method-Taylor series expansion.
 Faster, more accurate, reliable.
 Advantages:
◦ Convergence is better.
◦ Number of iterations required is less.
◦ Number of iterations independent of system size.
◦ Load flow solution of large power systems.

63.  Disadvantages:
◦ Solution technique is difficult.
◦ Calculation in each iteration is more.
◦ Computational time is more.
◦ Computer memory requirement is large.

64. Fast Decoupled Load Flow
 Extension of Newton Raphson method formulated in
polar coordinates
 makes certain approximations resulting into fast
algorithm for LF solution
 Pp-jQp = V*
pIp
 Ip = n
Σq=1Ypq.Vq
 Pp-jQp=Vp
*n
Σq=1Ypq.Vq

65. )(
)4.....(*
)3........(
)2......(
)1......(*
)(
1
qpj
n
q
pqqPPP
J
pqpq
qqq
PPP
PPP
pq
pq
eYVVjQP
eYY
VV
VV
IVjQP
δδθ
θ
δ
δ
+−
=
∑=−
=
〈=
〈−=
=−

66.  Vp=|Vp|exp(jδp) and Ypq = |Ypq|exp(-jθpq)
Pp-jQp= |Vp| exp(-jδp)n
Σq=1 |Ypq|exp(-jθpq)|Vq|exp(jδq)
= n
Σq=1 |Vp||Vq||Ypq|[+j(θpq- δp+ δq)]
Pp= n
Σq=1|VpVqYpq|cos(θpq- δp+ δq)
Qp=-1* n
Σq=1|VpVqYpq|sin(θpq-δp+ δq)
where p= 1,2,3,….,n

67. Pp= |VpVpYpp|cosθpp + n
Σq=1 |VpVqYpq|cos(θpq- δp+ δq)
q≠p
Qp=-( |VpVpYpp|sinθpp + n
Σq=1 |VpVqYpq|sin(θpq- δp+ δq) )
q≠p
After linearization can be written in matrix form as
Where H ,N ,J and L are elements of Jacobian matrix

68.  Assumptions made:
1. Real power changes(ΔP) are less sensitive to changes
in voltage magnitude and are mainly sensitive to
angle
2. Reactive power changes are less sensitive to change
in angle but mainly sensitive to change in voltage
magnitude
With these

69.  Similarly off-diagonal element of L is
Lpq= = |VpVqYpq|sin(θpq- δp+ δq)
= |VpVq|[Bpqcos(-δp+ δq)+Gpq( cos(-δp+δq))]
Hpq=Lpq= |VpVq|[Gpqcos(-δp+ δq) +Bpqcos(-δp+ δq)]
Diagonal elements of H are given as
Hpp= =- n
Σq=1 |VpVqYpq|sin(θpq- δp+ δq)
q≠p
= Qp + |VpVpYpp|sin θpp= +Qp+Vp
2
Bpp

70. Lpp= =|2Vp
2
Ypp|sin θpp +
n
Σq=1 |VpVqYpq|sin(θpq-δp+δq)
q≠p
=Qp+Vp
2
Bpp
In fast decoupled method following approximations are
further made for evaluating Jacobian element
Cos(-δp+δq)≈ 1
Gpqsin(-δp+δq) << Bpq(cos(-δp+δq) and
Qp<<BppVp
2

71.  The Jacobian elements now become
Lpq = Hpq=-|VpVq|Bpq for q≠p
and
Lpp= Hpp = Qp+Bpp|Vp|2
With these
where Bpq` and Bpq`` are the elements of [-Bpq]

76.  Non –iterative algorithm
 Dropping Q-V equation
 Vp=1per unit
)1.....(
;
.]'[
...
2
1
2
1
















∆
∆
∆
=












∆
∆
∆
n
B
Pn
P
P
θ
θ
θ

77.  DC load flow can also be applied to find the fairly good
approximation of the unknown voltages that can be
used as initial values in a Newton –Raphson/Decoupled
load flow solution or calculations.
 The DC power flow is only good for calculating MW flows
on transmission lines & transformers. It gives no
identification of what happens to voltage magnitudes, or
MVAR or MVA flows. The power flowing on each line
using the DC power flow in them

78. 
 Step 1 : Read the number of buses.
 Step 2 : Read the reactance values between the buses.
 Step 3 : Formulate the C matrix by inversing the reactance values.
 Step 4 : Read the reference bus as i.
 Step 5 : Formulate the B matrix by eliminating ith
row and column
of C matrix.
 Step 6 : Calculate
 Step 7 : Evaluate

 Step7:
[ ] [ ] [ ]PB *1−
=θ[ ] [ ] [ ]PB *1−
=θ
P1−
= Bθ
( )ki
ik
ik
X
P θθ −=
1
∑=
N
ki PP ∑=
N
ki PP
∑=
N
k
P
i
P

79. START
Read the no: of buses and power on each bus
Read the reactive value between buses
Formulate the C matrix with reactance value, C=1/x
Read the reference bus number as i
Delete the ith row and ith column of matrix C to form the
matrix
STOP
PB *1−
( )ki
ik
ik
X
P θθ −=
1
∑= ki PP
C
al
u
at
e
=

80.  Determine the line power flow and slack bus
power by dc load flow analysis

81. 









−−
−−
−−
5.645.2
495
5.255.7






−
−
95
55.7












−
−
=





2
1
2
1
95
55.7
θ
θ
P
P






−





=





1
65.0
1765.01177.0
1177.02118.0
2
1
θ
θ






−
=





1.0
02.0
2
1
θ
θ
Susceptance Matrix:
B’=
B =

82. ( )1
=−= KI
IK
IK
X
P θθ MW60
( )31
13
13
1
θθ −=
X
P 02.0*
4.0
1
= MW5=
( )32
23
23
1
θθ −=
X
P
( )1.0
25.0
1
−= MWP 4023 −=
133233 PPP +==
540 −= MW35=
Slack bus power