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Notes on Power System Load Flow Analysis using an Excel Workbook
Abstract
These notes describe the features of an MS-Excel Workbook which illustrates four
methods of power system load flow analysis
...
The Workbook also includes two
search algorithms: genetic algorithms and simulated annealing
...
Introduction
Load flow studies [1,2] are used to ensure that electrical power transfer from
generators to consumers through the grid system is stable, reliable and economic
...
Recently, however, there has been
much interest in the application of stochastic search methods, such as Genetic
Algorithms [3,4,5], to solving power system problems
...

The principles of power system load flow studies are taught within elective modules
in the later years of undergraduate electrical engineering courses, or as essential
components of specialist masters programmes in electrical power engineering
...
Pre-requisites include fundamental concepts
from a
...
circuit analysis, such as phasor notation, impedance and admittance, power
and reactive power, three-phase and per-unit systems, all of which are regarded as
‘difficult’ by many students
...

To assist in the teaching of load flow analysis techniques the Excel Workbook
illustrates four different methods of solving a simple load flow problem, which
nevertheless has all of the features to be found in a larger-scale system
...
These notes present an overview

1

of the general power system load flow problem and describe its solution using four
techniques: Newton-Raphson, Gauss-Seidel, Genetic Algorithm and Simulated
Annealing
...

2
...
As an
example, consider the very simple system represented by the single-line diagram in
Fig
...
Here two generators (1 and 2) are interconnected by one transmission line
and are separately connected to a load (3) by two other lines
...
2
...
1 Single-line diagram of a simple
example power system

Fig
...
1

For the circuit in Fig
...
For
example, at node 1:
I 1 = (y 12 + y 13 )V1 − y 12V 2 − y 13V 3

(1)

In general, for a system with r nodes, then at node n:
r

I n = Yn1V1 + Yn 2V2 +
...
+ Ynr V r = ∑ Ynk V k

(2)

k =1

where: Ynn = sum of all admittances connected to node n
Ynk = - (sum of all admittances connected between nodes n and k) = Ykn
In = current injected at node n
For the complete system of r nodes:

2

 I 1  Y11
...
Ynn
  
:
:   :
 I r  Yr 1
...
Y1r  V1 
:  : 
 

...
V n  or
 
:  : 

...


(3)

where [Y] is the nodal admittance matrix
...
The Excel Workbook (Sheet 2) allows
users to enter series impedance data for the three lines and then automatically
calculates the terms in the nodal admittance matrix
...
However, the
load flow problem is complicated by the lack of uniformity in the data about electrical
conditions at the nodes
...
Such nodes may also include links to other systems
...

b) Generator nodes, where the injected power, Pns, and the magnitude of the nodal
voltage |Vn| are specified
...
At the generator nodes
the voltage phase angle δn must be calculated
c) At least one node, termed the ‘floating bus’ or ‘slack bus’, where the nodal voltage
magnitude |Vn| and phase angle δn are specified
...
The
power and reactive power delivered at this node are not specified
...
1, which is analysed in the Excel Workbook, each
type of node is represented with node 1 being a floating bus, node 2 being a
generator node and node 3 being a load node
...
All three of these power values may be changed by
the user, though default values are provided (P2s = 1
...
5; Q3s = -0
...
The magnitude of the voltage at node 1 can be specified, with the default
value being 1
...
0∠0 0 )
...
1 pu ( V2 = 1
...
At the load node (node 3) the voltage magnitude and phase
angle have to be calculated ( V 3 = V3 ∠δ 3 )
...

3
...
1
...
x n ,
...
x n ,
...
x n ,
...
fn
...
xn ,
...
Kn
...
Applied to the load flow problem, the variables are the nodal
voltage magnitudes and phase angles, the functions are the relationships between
power, reactive power and node voltages, while the constants are the specified
values of power and reactive power at the generator and load nodes
...
2) at node n:
r

I n = ∑Ynk V k
k =1

so the complex power input to the system at node n is:
*
Sn = Vn I n

(5)

4

where the superscript * denotes the complex conjugate
...

0
The first step in the solution is to make initial estimates of all the variables: Vn0 , δ n

where the superscript

0

indicates the number of iterative cycles completed
...
These values are compared with the specified values to give a
power and reactive power error
...
g
...
 − − − 
−
0


:  ∆ Vn −1 
0

:   ∆ Vn 


 ∆ V 0 
:   n +1 


(11)

5

where the matrix of partial differentials is called the Jacobian matrix, [J]
...

At the next stage of the Newton-Raphson solution, the Jacobian is inverted
...
This requirement for matrix inversion is a
major drawback of the Newton-Raphson method of load flow analysis for large-scale
power systems
...
− − −
  : 
 
0 
 ∆Q n 
 : 



(12)

The approximate errors from (12) are added to the initial estimates to produce new
estimated values of node voltage magnitude and angle
...
However, they can be used
in another iterative cycle, involving the solution of Equations (9-14)
...

The description above relates specifically to a load node, where there are two
unknowns (the voltage magnitude and angle) and two equations relating to the
specified power and reactive power
...
The order of the
calculation can be reduced by 1
...
For the floating bus, both
voltage magnitude and angle are specified, so there is no need to calculate these
quantities
...
2
...
1 and analysed in the Excel Workbook, there
are three unknowns ( V3 ,δ 2 ,δ 3 ) and three set values of power and reactive power
(P2s, P3s, Q3s)
...
The terms in the Jacobian are obtained by partial
differentiation of (15-17)
...

(12) can be used to derive the approximate errors in the three variables and new
estimates formed from (13,14)
...

3
...
Sample Results for the Newton-Raphson Method

For the default system inputs, defined in Section 2, the Newton-Raphson method,
implemented in the 3rd sheet of the Excel Workbook, produces the results shown in
Table 1
...
When making comparisons with other
methods of solution, it is important to realise that each iterative cycle of the Newton7

Raphson method involves considerable computational effort, notably to invert the
Jacobian
...
Gauss-Seidel Method
4
...
General Approach

The Gauss-Seidel Method is another iterative technique for solving the load flow
problem, by successive estimation of the node voltages
...
However, this equation is used in the Gauss-Seidel method
as the basis for an iterative solution
...
In the p+1th iteration cycle when performing the calculation for
node n, the voltages at the nodes k=1…n-1 are available, but for the other node
voltages the values from the previous (pth) cycle have to be used
...
At the floating bus the voltage
V n is known and so does not need to be calculated
...
At these nodes the power Pn and voltage
magnitude Vn are specified, so in (23) Vnp +1 cannot be calculated because Qn (the

8

imaginary part of Sn) is unknown
...
The calculated value of Qn is
substituted back into (25) and the new estimate of generator node voltage is found
...
Therefore, it is common practice to
accelerate the iterative process, by adding to the newly-calculated value of each
variable an extra term proportional to the difference between the new and previous
values
...
V n*(p +1) − V n*(p)

}

(27)

where α is an ‘acceleration factor’, which has a typical value of 0
...

4
...
Application of the Gauss-Seidel Method to the Specific Problem

For the particular 3-node problem, introduced in Section 2, Equation (23) can be
used directly to calculate the voltage at the load node (node 3) every iterative cycle
...
3
...

Sample results for the default input data and using an acceleration factor α = 0
...

Comparing the results from Table 2 with those from Table 1, the slower convergence
of the Gauss-Seidel Method is evident: even after 10 iteration cycles the values of
phase angle are stabilised only to a single decimal place
...

Nevertheless both iterative methods require some complicated mathematical
operations: inversion of the large Jacobian matrix in the Newton-Raphson Method
and intermediate calculation of reactive power input at generator nodes in the GaussSeidel Method
...
Stochastic Search Techniques
5
...
General Principles

Recent developments in load flow analysis have moved attention away from the
iterative methods and towards so-called stochastic search methods
...
Both approaches use a series of trial
solutions to the problem and develop better solutions in the light of experience
gained from these trials
...

For the example problem being considered throughout this paper, there are three
variables: the voltage magnitude V3 and the phase angles δ 2 ,δ 3
...
The choice may be an
entirely random selection across the entire possible range of values (termed the
‘search space’) or the choice may be informed by previous experience
...
Therefore the
10

currents injected at each node can be evaluated directly using (2) and the
corresponding complex power input is calculated using (5)
...
The trial
values of node voltage lead to values of input power and reactive power (P, Q) that
do not exactly match the pre-defined values (Ps, Qs)
...
It is this selection process
which is defined by the particular search technique
...
2
...
A genetic algorithm [3,4] starts with a
random population of potential individuals, or chromosomes, each representing one
possible solution to a problem
...
The chromosomes are then evolved
through successive generations
...
The least fit chromosomes of each
population are then replaced by the offspring so that the population size remains
constant
...

A further refinement of the evolution process, again mirroring nature, is that any
chromosome in any generation has a finite probability of suffering mutation, in which
some of the genes are randomly perturbed
...

When applied to the load flow problem, the genes are the nodal voltage magnitude
and phase angle values and each chromosome contains a complete set of the genes
needed to define uniquely a trial solution
...
For the example problem, a genetic algorithm
solution is implemented in Sheet 5 of the Excel Workbook
...
An
initial population of 10 chromosomes forms the first generation
...
The values
of the genes, voltage magnitude V3

and the phase angles δ 2 ,δ 3 , for each

chromosome are shown in rows 1-3 of the Table
...
The set values for the power inputs are (P2s = 1
...
5; Q3s = -0
...
The chromosomes are then ranked
(row 11) by error
...

Chromosomes for the second generation are derived from the previous generation’s
chromosomes according to their ranking
...
The first two chromosomes are copies of
the two highest-ranked chromosomes of the previous generation
...
Each
gene in chromosome 6 takes the average value of the genes in chromosomes 1 and
2
...
Finally, chromosome 10 of the
second generation has genes which are entirely randomly-selected
...


12

The cycle of chromosome evaluation and breeding continues through many
generations
...
A typical variation of error with
generation number is shown in Fig
...
The genetic algorithm quickly reduces the
error, but a very large number of generations is needed to bring the error close to
zero
...
The effect of this redefinition is apparent
in Fig
...


error

2
...
5

1
...
5

0
...
3 Typical error associated with the best chromosome as a function
of the number of generations

5
...
The
probability of Y being accepted for investigation depends on the proximity of Y to X
and the extent to which the solution has been developed, as represented by a
‘temperature’ parameter, T, which reduces throughout the annealing process
...

To apply this concept to load flow studies in general, it is assumed that the solutions
X and, Y, represent information about possible nodal voltage values
...
Table 4 presents
sample results from the simulated annealing technique, which is implemented in
Sheet 6 of the Excel Workbook
...

The range indicates the extent of the search space and is defined as:
range =

(δ 2MAX

(

− δ 2MIN ) + V3
2

− V3 MIN

MAX

) + (δ
2

− δ 3MIN )

2

3 MAX

2

(35)

A new set of voltage values, Y, are selected at random
...
PA is
compared to a random value r1 in the range [0 – 1]
...
If r1 < PA then Y is accepted for evaluation
...
For example, in the first row of Table 4, the displacement between X and Y is
small (0
...
Therefore the

14

calculated acceptance probability is high (0
...
The random number r1 = 0
...

When Y is accepted for evaluation the corresponding nodal voltages are used to
calculate the input currents {from (2)} and the complex power inputs {from (5)}
...
This error is compared to the
error EX obtained for the solution X in the swap probability function, PS:
PS =

1

 E − E X
1 + exp Y

 E X


  Ts 
 *  
 T 
  

(38)

PS is compared to a random value r2 in the range [0 – 1]
...
Substitution of Y for X is most likely to occur if the error Ey is small, in
which case the swap probability is high
...
In row 1 of Table 4 the values of EY and EX are almost
equal and the swap probability is 0
...
However the random number r2 is 0
...
Therefore in the second row of Table 4 the ‘current best
values’ are the Y values from row 1
...
19) which is substantially larger
than EX (1
...
33
...
68, so a swap does not occur
...
The probability of such an event happening
diminishes as the temperature reduces
...
The current and power calculations do not need to be made in
these circumstances
...
In
this work, the decrement function proposed by Xin [6] is used, with the temperature
being inversely proportional to the number of potential solutions investigated
...
Fig
...
However there are instances where the
15

error increases
...
In Fig
...

Comparing the two search methods, genetic algorithms and simulated annealing, the
number of calculations required to produce an acceptable error is similar: for the
default power system parameters the genetic algorithm operates over 200
generations, with 8 new chromosomes to be evaluated in each generation, giving a
total of 1600 evaluations
...
If a random search was conducted across the entire
search space then to obtain comparable resolution in the solution would require the
evaluation of approximately 300 voltage magnitude values and 200 values of each
phase angle, giving a total of 300x200x200 = 1
...


error

3
...
5
2
...
5
1
...
5
0
0

100

200

300

400

500

600

700

number of investigated solutions
Fig
...
Summary of Results and System Calculations

Sheet 7 of the Excel Workbook summarises the results of the nodal voltage
calculations from the four methods
...
If necessary, the calculation cycle can be
repeated by pressing the (F9) key
...
Therefore, included in the Sheet is the calculation
of the power and reactive power flow through each of the three lines and a
calculation of the total power and reactive power consumed by the transmission
system
...
)
7
...

8
...
E
...
D
...
2’, (Pergamon
Press, 2nd edition, 1977)
...
D
...
, ‘Elements of power system analysis’, (McGraw-Hill, 4th
edition, 1982)
...
F
...
S
...
Kwong, ‘Genetic algorithms: concepts and
applications’, IEEE Transactions on Industrial Electronics, 43 (1996), 5, pp
...

[4] M
...
Cheng, ‘Genetic algorithms and engineering design’, (John Wiley
& Sons, Inc
...

[5] J
...
Xu, C
...
Chang, and X
...
Wang, ‘Constrained multiobjective global
optimisation of longitudinal interconnected power system by genetic algorithm’,
IEE Proceedings, Generation, Transmission & Distribution, 143 (1996), 5, pp 435446
...


17

Iteration

δ2o

|V3|

δ3o

0

10
...
900

-20
...
813

1
...
773

2

-0
...
093

-3
...
226

1
...
210

4

-0
...
050

-5
...
224

1
...
278

Table 1: Results from the Newton-Raphson method

Iteration

δ2o

|V3|

δ3o

0

10
...
900

-20
...
156

1
...
985

2

-3
...
046

-7
...
251

1
...
935

4

-0
...
050

-5
...
552

1
...
490

6

-0
...
050

-5
...
354

1
...
379

8

-0
...
050

-5
...
229

1
...
271

10

-0
...
049

-5
...

1
2
3

1

2

3

4

5

6

7

8

9

10

δ2 (radians)
|V3|

0
...
000

0
...
100

0
...
100

0
...
900

0
...
900

-0
...
100

-0
...
100

-0
...
900

-0
...
900

0
...
937

δ3 (radians)

0
...
0+
0
...
0
+0
...
0
+1
...
000
0
...
000
1
...
175
-1
...
75j
-1
...
75j
0
...
42j
0
...
955
0
...
603
3

-0
...
19
+0
...
19
+0
...
78
-0
...
902
-5
...
363
5
...
175
-1
...
24j
-1
...
24j
0
...
54j
0
...
781
-2
...
031
5

-0
...
017
-0
...
017
-0
...
25
+2
...
150
-4
...
585
4
...
175
-0
...
75j
-0
...
75j
4
...
42j
-4
...
093
1
...
986
10

-0
...
72
+0
...
72
+0
...
96
-0
...
764
-0
...
633
2
...
175
-0
...
24j
-0
...
24j
4
...
538j
-4
...
167
-1
...
782
9

-0
...
55
-0
...
55
-0
...
43
+2
...
764
-0
...
182
2
...
106
-0
...
034j
0
...
37j
1
...
57j
-1
...
115
-1
...
546
6

best

2nd best

mutate
best

mutate
best

mutate
best

random

1

2

3

4

5

6

7

8

9

10

δ2 (radians)
|V3|

0
...
000

-0
...
100

0
...
000

0
...
100

-0
...
000

-0
...
050

0
...
000

0
...
977

0
...
000

0
...
972

δ3 (radians)

0
...
175

-0
...
000

0
...
087

-0
...
000

0
...
075

4

I1

5

I2

6

I3

7
8
9
10
11

P2
P3
Q3
E
ranking

12
13
14
15
16

2nd Generation
Chromosome No
...
00

1
...
00

1
...
0

0
...
10

0
...
10

0
...
15

Y

0
...
43j

0
...
45j

0
...
06

0
...
93

0
...
49

Y

2

0
...
10

0
...
93

1
...
03

0
...
02

0
...
69

0
...
61
-2
...
60
+2
...
61

0
...
99

3
...
33

0
...
02

1
...
02

1
...
2

-0
...
98

-0
...
22

0
...
70

N

0
...
00

0
...
000

0
...
81
4

0
...
69
4

4

0
...
10

0
...
93

1
...
03

1
...
14

0
...
66

0
...
93
-1
...
76
+1
...
07

-1
...
83

0
...
68

0
...
03

1
...
14

0
...
3

0
...
92

0
...
32

0
...
14

Y

0
...
16j

0
...
23j

0
...
52

-2
...
74

0
...
13

N

6

-0
...
00

-0
...
75

1
...
03

1
...
08

0
...
86

0
...
351
...
90
+0
...
43

-0
...
86

1
...
38

0
...
03

1
...
14

0
...
5

-0
...
96

0
...
27

0
...
50

N

0
...
00

0
...
000

0
...
81
4

0
...
54
4

8

-0
...
00

-0
...
75

1
...
08

1
...
11

0
...
48

0
...
00

0
...
000

0
...
00
0

1
...
10
6

0
...
03

1
...
14

0
...
5

-0
...
92

0
...
21

0
...
46

Y

-0
...
21j

1
...
22j

-0
...
04

-1
...
62

0
...
35

R
E
J
R
E
J
N

Count

V3x

δ3Y (radians)

Swap

δ2Y (radians)

Acceptance

δ3x (radians)

New random values

δ2x (radians)

Current best values

Ts
/T

Table 4: Typical initial results from Simulated Annealing

20



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