This document discusses a lecture on power flow analysis in a power systems engineering course. It provides announcements about homework assignments and an upcoming midterm exam. It also discusses examples of the Gauss two bus power flow method and the inclusion of PV generator buses in the Gauss-Seidel iteration method. Key concepts covered include slack buses, generator reactive power limits, and methods to accelerate Gauss-Seidel convergence.
Load flow solution is the solution of the network under steady state conditions subjected to certain inequality constraints under which the system operates.
The modern power system around the world has grown in complexity of interconnection and
power demand. The focus has shifted towards enhanced performance, increased customer focus,
low cost, reliable and clean power. In this changed perspective, scarcity of energy resources,
increasing power generation cost, environmental concern necessitates optimal economic dispatch.
In reality power stations neither are at equal distances from load nor have similar fuel cost
functions. Hence for providing cheaper power, load has to be distributed among various power
stations in a way which results in lowest cost for generation. Practical economic dispatch (ED)
problems have highly non-linear objective function with rigid equality and inequality constraints.
Particle swarm optimization (PSO) is applied to allot the active power among the generating
stations satisfying the system constraints and minimizing the cost of power generated. The
viability of the method is analyzed for its accuracy and rate of convergence. The economic load
dispatch problem is solved for three and six unit system using PSO and conventional method for
both cases of neglecting and including transmission losses. The results of PSO method were
compared with conventional method and were found to be superior. The conventional
optimization methods are unable to solve such problems due to local optimum solution
convergence. Particle Swarm Optimization (PSO) since its initiation in the last 15 years has been
a potential solution to the practical constrained economic load dispatch (ELD) problem. The
optimization technique is constantly evolving to provide better and faster results.
While writing the report on our project seminar, we were wondering that Science and smart
technology are as ever expanding field and the engineers working hard day and night and make
the life a gift for us
Load flow solution is the solution of the network under steady state conditions subjected to certain inequality constraints under which the system operates.
The modern power system around the world has grown in complexity of interconnection and
power demand. The focus has shifted towards enhanced performance, increased customer focus,
low cost, reliable and clean power. In this changed perspective, scarcity of energy resources,
increasing power generation cost, environmental concern necessitates optimal economic dispatch.
In reality power stations neither are at equal distances from load nor have similar fuel cost
functions. Hence for providing cheaper power, load has to be distributed among various power
stations in a way which results in lowest cost for generation. Practical economic dispatch (ED)
problems have highly non-linear objective function with rigid equality and inequality constraints.
Particle swarm optimization (PSO) is applied to allot the active power among the generating
stations satisfying the system constraints and minimizing the cost of power generated. The
viability of the method is analyzed for its accuracy and rate of convergence. The economic load
dispatch problem is solved for three and six unit system using PSO and conventional method for
both cases of neglecting and including transmission losses. The results of PSO method were
compared with conventional method and were found to be superior. The conventional
optimization methods are unable to solve such problems due to local optimum solution
convergence. Particle Swarm Optimization (PSO) since its initiation in the last 15 years has been
a potential solution to the practical constrained economic load dispatch (ELD) problem. The
optimization technique is constantly evolving to provide better and faster results.
While writing the report on our project seminar, we were wondering that Science and smart
technology are as ever expanding field and the engineers working hard day and night and make
the life a gift for us
A brief introduction on the principles of particle swarm optimizaton by Rajorshi Mukherjee. This presentation has been compiled from various sources (not my own work) and proper references have been made in the bibliography section for further reading. This presentation was made as a presentation for submission for our college subject Soft Computing.
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Tutorial: Modelling and Simulations: Renewable Resources and Storage
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A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
TOP 10 B TECH COLLEGES IN JAIPUR 2024.pptxnikitacareer3
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6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
Tutorial for 16S rRNA Gene Analysis with QIIME2.pdf
Lecture 11
1. EE 369
POWER SYSTEM ANALYSIS
Lecture 11
Power Flow
Tom Overbye and Ross Baldick
1
2. Announcements
• Start reading Chapter 6 for lectures 11 and 12.
• Homework 8 is 3.1, 3.3, 3.4, 3.7, 3.8, 3.9, 3.10,
3.12, 3.13, 3.14, 3.16, 3.18; due 10/29.
• Homework 9 is 3.20, 3.23, 3.25, 3.27, 3.28,
3.29, 3.35, 3.38, 3.39, 3.41, 3.44, 3.47; due
11/5.
• Midterm 2, Thursday, November 12, covering
up to and including material in HW9.
2
3. Wind Blade Failure
Photo source: Peoria Journal Star
Several years ago, a 140 foot,
6.5 ton blade broke off from a
Suzlon Energy wind turbine.
The wind turbine is located
in Illinois. Suzlon Energy is
one of the world’s largest
wind turbine manufacturers;
its shares fell 39% following
the accident. No one was hurt
and wind turbines failures
are extremely rare events.
(Vestas and Siemens turbines
have also failed.) 3
6. Gauss Two Bus Power Flow Example
•A 100 MW, 50 MVAr load is connected to a
generator through a line with z = 0.02 + j0.06
p.u. and line charging of 5 MVAr on each end
(100 MVA base).
•Also, there is a 25 MVAr capacitor at bus 2.
•If the generator voltage is 1.0 p.u., what is V2?
SLoad = 1.0 + j0.5 p.u. 6
j0.05 j0.05
7. Gauss Two Bus Example, cont’d
2
2 bus
The unknown is the complex load voltage, .
To determine we need to know the ,
which is a 2 2 matrix. The capacitors have
susceptances specified by the reactive power
at the rated voltage.
Line
V
V
×
Y
bus
11 22
1 1
series admittance = 5 15.
0.02 0.06
5 14.95 5 15
Hence .
5 15 5 14.70
( Note: 15 0.05; 15 0.05 0.25).
j
Z j
j j
j j
B B
= = −
+
− − +
= − + −
= − + = − + +
Y
7
8. Gauss Two Bus Example, cont’d
1 1
2
*
2
2 2 2*
22 1, 22
( 1)
2
2
Note that =1.0 0 is specified, so we do not update .
We only consider one entry of ( ), namely ( ).
1 S
Equation to solve: ( ).
1 1 0.5
Update:
5 14.70 (
n
k k
k k
V V
h V h V
V Y V h V
Y V
j
V
j V
ν
= ≠
+
∠
= − = ÷
− +
=
−
∑
( )
(0)
2
( ) ( )
2 2
( 5 15)(1.0 0)
)*
Guess 1.0 0 (this is known as a flat start)
0 1.000 0.000 3 0.9622 0.0556
1 0.9671 0.0568 4 0.9622 0.0556
2 0.9624 0.0553
v v
j
V
v V v V
j j
j j
j
ν
− − + ∠ ÷
= ∠
+ −
− −
− 8
9. Gauss Two Bus Example, cont’d
2
* *
1 1 11 1 12 2 1 1
ˆFixed point: 0.9622 0.0556 0.9638 3.3
Once the voltages are known all other values can
be determined, including the generator powers and
the line flows.
ˆ( ) 1.023 0.239 ,
V j
S V Y V Y V j P jQ
= − = ∠ − °
= + = − = −
1 1
2
2
In actual units 102.3 MW, 23.9 MVAr
The capacitor is supplying 25 23.2 MVAr
P Q
V
= =
=
9
10. Slack Bus
In previous example we specified S2 and V1
and then solved for S1 and V2.
We can not arbitrarily specify S at all buses
because total generation must equal total
load + total losses.
We also need an angle reference bus.
To solve these problems we define one bus
as the “slack” bus. This bus has a fixed
voltage magnitude and angle, and a varying
real/reactive power injection. In the
previous example, this was bus 1. 10
11. Gauss for Systems with Many Buses
*
( 1) ( )
( )*
1,
( ) ( ) ( )
1 2
( 1)
With multiple bus systems we could calculate
new values of the voltages as follows:
S1
( , ,..., )
But after we've determined , it is
i
i
n
v vi
i ik kv
ii k k i
v v v
i n
v
i
V
V Y V
Y V
h V V V
V
+
= ≠
+
= − ÷
÷
=
∑
( )
a better estimate
of the voltage at bus than , so it makes sense to use
this new value. Using the latest values is known as the
Gauss-Seidel iteration.
v
ii V
11
12. Gauss-Seidel Iteration
( 1) ( ) ( ) ( )
2 12 2 3
( 1) ( 1) ( ) ( )
3 13 2 3
( 1) ( 1) ( 1) ( ) ( )
4 14 2 3 4
( 1) ( 1) (
1 2 3
Immediately use the new voltage estimates:
( , , , , ) (bus 1 is slack),
( , , , , )
( , , , , )
( , ,
v v v v
n
v v v v
n
v v v v v
n
v v v
n n
V h V V V V
V h V V V V
V h V V V V V
V h V V V
+
+ +
+ + +
+ +
= …
= …
= …
=
M
1) ( 1) ( )
4, , )
Gauss-Seidel usually works better than the Gauss, and
is actually easier to implement.
Gauss-Seidel is used in practice instead of Gauss.
v v
nV V+ +
…
12
13. Three Types of Power Flow Buses
There are three main types of buses:
– Load (PQ), at which P and Q are fixed; goal is to
solve for unknown voltage magnitude and angle at
the bus.
– Slack at which the voltage magnitude and angle
are fixed; iteration solves for unknown P and Q
injections at the slack bus
– Generator (PV) at which P and |V| are fixed;
iteration solves for unknown voltage angle and Q
injection at bus:
special coding is needed to include PV buses in the
Gauss-Seidel iteration. 13
14. Inclusion of PV Buses in G-S
* *
1
( ) ( )* ( )
1
To solve for at a PV bus we must first make a
guess of using the power flow equation:
Hence Im is an
estimate of the reactive power injectio
k
i
i
n
i i ik k i i
k
n
v v v
i i ik
k
V
Q
S V Y V P jQ
Q V Y V
=
=
= = −
= −
∑
∑
( ) ( )
n.
For the Gauss iteration we use the known value
of real power and the estimate of the reactive power:
v v
i i iS P jQ= + 14
15. Inclusion of PV Buses, cont'd
( 1)
( )*
( 1) ( )
( )*
1,
( 1) ( 1)
( 1)
( 1)
Tentatively solve for
1
In update, set .
But since is specified, replace by .
That is, set
i
v
i
v n
v vi
i ik kv
ii k k i
v
i i
v
i i i
i i
V
S
V Y V
Y V
V V
V V V
V V
ν
ν
+
+
= ≠
+ +
+
+
= − ÷
÷
∠ = ∠
=
∑%
%
%
15
16. Two Bus PV Example
Bus 1
(slack bus)
Bus 2
V1 = 1.0 V2 = 1.05
P2 = 0 MW
z = 0 .0 2 + j 0 .0 6
Consider the same two bus system from the previous
example, except the load is replaced by a generator
16
j0.05j0.05
17. Two Bus PV Example, cont'd
( ) ( )* ( )
22 2
1
( ) ( )* ( ) ( )*
21 221 2 2 2
( )* ( )*
( 1) ( ) ( )2 2
2 212 1( )* ( )*
22 221, 22 2
(0)
2
( ) ( 1) ( 1)
2 2 2
Im ,
Im[ ]
1 1
Guess 1.05 0
0 0 0.457
k
n
v v
k
k
n
k k
k k
v v v
Q V Y V
Y V V Y V V
S S
V Y V Y V
Y YV V
V
v S V V
j
ν
ν ν ν ν
ν ν
ν ν ν
ν ν
=
+
= ≠
+ +
= −
= − +
= − = − ÷ ÷
= ∠ °
+
∑
∑%
%
1.045 0.83 1.050 0.83
1 0 0.535 1.049 0.93 1.050 0.93
2 0 0.545 1.050 0.96 1.050 0.96
j
j
∠ − ° ∠ − °
+ ∠ − ° ∠ − °
+ ∠ − ° ∠ − ° 17
18. Generator Reactive Power Limits
The reactive power output of generators
varies to maintain the terminal voltage; on
a real generator this is done by the exciter.
To maintain higher voltages requires more
reactive power.
Generators have reactive power limits,
which are dependent upon the generator's
MW output.
These limits must be considered during the
power flow solution. 18
19. Generator Reactive Limits, cont'd
During power flow once a solution is
obtained, need to check if the generator
reactive power output is within its limits
If the reactive power is outside of the limits,
then fix Q at the max or min value, and re-
solve treating the generator as a PQ bus
– this is know as "type-switching"
– also need to check if a PQ generator can again
regulate
Rule of thumb: to raise system voltage we
need to supply more VArs. 19
20. Accelerated G-S Convergence
( 1) ( )
( 1) ( ) ( ) ( )
(
Previously in the Gauss-Seidel method we were
calculating each value as
( )
To accelerate convergence we can rewrite this as
( )
Now introduce "acceleration parameter"
v v
v v v v
x
x h x
x x h x x
x
α
+
+
=
= + −
1) ( ) ( ) ( )
( ( ) )
With = 1 this is identical to standard Gauss-Seidel.
Larger values of may result in faster convergence.
v v v v
x h x xα
α
α
+
= + −
20
21. Accelerated Convergence, cont’d
( 1) ( ) ( ) ( )
Consider the previous example: 1 0
(1 )
Matlab code: alpha=1.2;x=x0;x=x+alpha*(1+sqrt(x)-x).
Comparison of results with different values of
1 1.2 1.5 2
0 1 1 1 1
1 2 2.20 2.5 3
2
v v v v
x x
x x x xα
α
ν α α α α
+
− − =
= + + −
= = = =
2.4142 2.5399 2.6217 2.464
3 2.5554 2.6045 2.6179 2.675
4 2.5981 2.6157 2.6180 2.596
5 2.6118 2.6176 2.6180 2.626 21
22. Gauss-Seidel Advantages
Each iteration is relatively fast (computational
order is proportional to number of branches +
number of buses in the system).
Relatively easy to program.
22
23. Gauss-Seidel Disadvantages
Tends to converge relatively slowly, although
this can be improved with acceleration.
Has tendency to fail to find solutions,
particularly on large systems.
Tends to diverge on cases with negative
branch reactances (common with
compensated lines)
Need to program using complex numbers.
23
24. Newton-Raphson Algorithm
The second major power flow solution
method is the Newton-Raphson algorithm
Key idea behind Newton-Raphson is to use
sequential linearization
General form of problem: Find an such that
( ) 0
x
f x =
24
25. Newton-Raphson Method (scalar)
( )
( )
( )
( ) ( ) ( ) ( ) ( )
2 2( ) ( )
2
1. Represent by a Taylor series about the
current guess . Write for the deviation
from :
( ) ( ) ( )
1
( )
2
higher order terms.
v v v v v
v v
f
x x
x
df
f x x f x x x
dx
d f
x x
dx
ν
ν
∆
+ ∆ = + ∆ +
+ ∆ +
25
26. Newton-Raphson Method, cont’d
( ) ( ) ( ) ( ) ( )
( )
1
( ) ( ) ( )
2. Approximate by neglecting all terms
except the first two
( ) ( ) ( )
3. Set linear approximation equal to zero
and solve for
( ) ( )
4. Sol
v v v
v
v v v
f
df
f x x f x x x
dx
x
df
x x f x
dx
ν ν
−
+ ∆ ≈ + ∆
∆
∆ = −
( 1) ( ) ( )
ve for a new estimate of solution:
v v v
x x x+
= + ∆ 26
27. Newton-Raphson Example
2
1
( ) ( ) ( )
( ) ( ) 2
( )
( 1) ( ) ( )
( 1) ( ) ( ) 2
( )
Use Newton-Raphson to solve ( ) 0,
where: ( )= 2.
The iterative update is:
( ) ( )
1
(( ) 2)
2
1
(( ) 2).
2
v v v
v v
v
v v v
v v v
v
f x
f x x
df
x x f x
dx
x x
x
x x x
x x x
x
−
+
+
=
−
∆ = −
∆ = − −
= + ∆
= − −
27
28. Newton-Raphson Example, cont’d
( 1) ( ) ( ) 2
( )
(0)
( ) ( ) ( )
3 3
6
1
(( ) 2)
2
Matlab code: x=x0; x = x-(1/(2*x))*(x^2-2).
Guess 1. Iteratiting, we get:
( )
0 1 1 0.5
1 1.5 0.25 0.08333
2 1.41667 6.953 10 2.454 10
3 1.41422 6.024 10
v v v
v
v v v
x x x
x
x
x f x xν
+
− −
−
= − −
=
∆
−
−
× − ×
×
28
29. Sequential Linear Approximations
Function is f(x) = x2
- 2.
Solutions to f(x) = 0 are points where
f(x) intersects x axis.
At each
iteration the
N-R method
uses a linear
approximation
to determine
the next value
for x
29
30. Newton-Raphson Comments
• When close to the solution the error
decreases quite quickly -- method has what is
known as “quadratic” convergence:
– number of correct significant figures roughly
doubles at each iteration.
• f(x(v)
) is known as the “mismatch,” which we
would like to drive to zero.
• Stopping criteria is when f(x(v)
) < ε
30
31. Newton-Raphson Comments
• Results are dependent upon the initial guess.
What if we had guessed x(0)
= 0, or x(0)
= -1?
• A solution’s region of attraction (ROA) is the
set of initial guesses that converge to the
particular solution.
• The ROA is often hard to determine.
31
32. Multi-Variable Newton-Raphson
1 1
2 2
Next we generalize to the case where is an -
dimension vector, and ( ) is an -dimensional
vector function:
( )
( )
( )
( )
Again we seek a solution of ( ) 0.
n n
n
n
x f
x f
x f
= =
=
x
f x
x
x
x f x
x
f x
M M
32
33. Multi-Variable Case, cont’d
i
1 1
1 1 1 2
1 2
1
1 2
1 2
The Taylor series expansion is written for each f ( )
( ) ( ) ( ) ( )
( ) higher order terms
( ) ( ) ( ) ( )
( ) higher order terms
n
n
n n
n n
n
n
n
f f
f x x f x x x x x
x x
f
x x
x
f f
f x x f x x x x x
x x
f
x x
x
∂ ∂
+ ∆ = + ∆ + ∆ +…
∂ ∂
∂
∆ +
∂
∂ ∂
+ ∆ = + ∆ + ∆ +…
∂ ∂
∂
∆ +
∂
x
M
33
34. Multi-Variable Case, cont’d
1 1 1
1 2
1 1
2 2 2
2 2
1 2
1 2
This can be written more compactly in matrix form
( ) ( ) ( )
( )
( ) ( ) ( )( )
( )
( )
( ) ( ) ( )
n
n
n
n n n
n
f f f
x x x
f x
f f f
f x
x x x
f
f f f
x x x
∂ ∂ ∂
∂ ∂ ∂
∆
∂ ∂ ∂ ∆
∂ ∂ ∂= +
∂ ∂ ∂
∂ ∂ ∂
x x x
x
x x xx
f x +Δx
x
x x x
L
L
M
M O O M
L
higher order terms
nx
∆
+
M
34
35. Jacobian Matrix
1 1 1
1 2
2 2 2
1 2
1 2
The by matrix of partial derivatives is known
as the Jacobian matrix, ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
n
n
n n n
n
n n
f f f
x x x
f f f
x x x
f f f
x x x
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂
∂ ∂ ∂=
∂ ∂ ∂
∂ ∂ ∂
J x
x x x
x x x
J x
x x x
L
L
M O O M
L
35
36. Multi-Variable N-R Procedure
Derivation of N-R method is similar to the scalar case
( ) ( ) ( ) higher order terms
( ) ( ) ( )
To seek solution to ( ) 0, set linear
approximation equal to zero: 0 ( ) ( ) .
+ ∆ = + ∆ +
+ ∆ ≈ + ∆
+ ∆ =
= + ∆
∆ = −
f x x f x J x x
f x x f x J x x
f x x
f x J x x
x 1
( 1) ( ) ( )
( 1) ( ) ( ) 1 ( )
( )
( ) ( )
( ) ( )
Iterate until ( )
v v v
v v v v
v
ε
−
+
+ −
= + ∆
= −
<
J x f x
x x x
x x J x f x
f x
36
37. Multi-Variable Example
1
2
2 2
1 1 2
2 2
2 1 2 1 2
1 1
1 2
2 2
1 2
Solve for = such that ( ) 0 where
( ) 2 8
( ) 4
First symbolically determine the Jacobian
( ) ( )
( ) =
( ) ( )
x
x
f x x x
f x x x x x
f f
x x
x x
f f
x x
x x
=
= + −
= − + −
∂ ∂
∂ ∂
∂ ∂
∂ ∂
x f x
J x
37
38. Multi-variable Example, cont’d
1 2
1 2 1 2
1
1 1 2 1
2 1 2 1 2 2
4 2
( ) =
2 2
4 2 ( )
Then
2 2 ( )
Matlab code: x1=x10; x2=x20;
f1=2*x1^2+x2^2-8;
f2=x1^2-x2^2+x1*x2-4;
J = [4*x1 2*x2; 2*x1+x2 x1-2*x2];
[x1;x2] =
x x
x x x x
x x x f
x x x x x f
−
+ −
∆
= − ∆ + −
J x
x
x
[x1;x2]-inv(J)*[f1;f2].
38
39. Multi-variable Example, cont’d
(0)
1
(1)
1
(2)
( )
1
Initial guess
1
1 4 2 5 2.1
1 3 1 3 1.3
2.1 8.40 2.60 2.51 1.8284
1.3 5.50 0.50 1.45 1.2122
At each iteration we check ( ) to see if itν
−
−
=
−
= − = − −
= − = −
x
x
x
f x
(2)
is
0.1556
below our specified tolerance : ( )
0.0900
If = 0.2 then done. Otherwise continue iterating.
ε
ε
=
f x
39