Newton raphson method

newton raphson method of load flow analysis is discussed with flowchart ,algorithmic steps with example problem

1. NEWTON-RAPHSON METHOD
Contents:
Introduction
Algorithm
Flowchart
Advantages & Disadvantages
Example
Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi
1

2. Introduction
The Newton-Raphson technique, converges
equally fast for large as well as small
systems.
Most widely used for solving simultaneous
non-linear algebraic equations.
It is a successive approximation procedure
based on an initial estimate of the unknown
and the use of Taylor’s series expansion
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Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

3. Algorithm
Formulate ??????
�???????????? matrix
Assume flat start for starting voltage solution
�
??????
0
= 0, for i=1,2,3 …..N for all buses except slack bus
??????
??????
0
= 1.0 for i=M+1, M+2, ……N(for all PQ buses)
??????
?????? =??????
????????????��� for all PV buses and slack bus.
For load buses calculate �
??????
���
and �
??????
���
.
For PV buses, check for Q-limit violation.
If �
??????(&#3627408474;??????&#3627408475;) <&#3627408452;
??????
&#3627408464;&#3627408462;&#3627408473;
<&#3627408452;
??????(max), the bus acts as P-V bus.
If &#3627408452;
??????
&#3627408464;&#3627408462;&#3627408473;
> &#3627408452;
??????(max) &#3627408452;
??????(spec) = &#3627408452;
??????(max)
If &#3627408452;
??????
&#3627408464;&#3627408462;&#3627408473;
<&#3627408452;
??????(&#3627408474;??????&#3627408475;) &#3627408452;
??????(spec) = &#3627408452;
??????(max), the P-V bus will act as PQ
bus.
Compute mismatch vector using,
∆&#3627408451;
?????? = &#3627408451;
??????(??????&#3627408477;&#3627408466;&#3627408464;) - &#3627408451;
??????
&#3627408464;&#3627408462;&#3627408473;
∆&#3627408452;
?????? =&#3627408452;
??????(??????&#3627408477;&#3627408466;&#3627408464;) - &#3627408452;
??????
&#3627408464;&#3627408462;&#3627408473;


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Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

4. Contd…
Compute,
∆&#3627408451;
??????(max) = max∆&#3627408451;
??????; i=1,2,3 …..N’

∆&#3627408452;
??????(max) = max∆&#3627408452;
??????; i=M+1, M+2, ……N

Compute Jacobian matrix using Form J=
??????&#3627408451;
????????????
??????
??????&#3627408451;
????????????
??????&#3627408452;
????????????
??????
??????&#3627408452;
????????????

Obtain state vector
∆&#3627409151;
∆V
??????
=??????
−1
∆&#3627408451;
∆&#3627408452;

Update state vector using,
??????
&#3627408475;&#3627408466;??????
= ??????
&#3627408476;&#3627408473;&#3627408465;+∆V =??????
&#3627408476;&#3627408473;&#3627408465; 1+
∆V
??????&#3627408476;&#3627408473;&#3627408465;

&#3627409151;
&#3627408475;&#3627408466;??????
= &#3627409151;
&#3627408476;&#3627408473;&#3627408465;+∆&#3627409151;
This procedure is continued until,
∆&#3627408451;
?????? < &#3627409152; ∆&#3627408452;
?????? < &#3627409152;, otherwise go to step 3.

4
Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

5. Flowchart
5
Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

6. 6
Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

7. 7
Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

8. Example… 1.Perform an iteration of Newton-Raphson load flow method and determine the
power flow solution for the given system take base MVA as 100.(15)
Line
Bus
Rp.u Xp.u
Half line
charging
admittance
Yp/2(p.u)
Bus PL QL
From To
1 90 20
1 1 1 0.0839 0.5183 0.0636 2 30 10
Step 1:
Form &#3627408512;
&#3627408515;???????????? matrix
&#3627408512;
&#3627408515;???????????? =
&#3627409359;.&#3627409366;&#3627409362;&#3627409360; −&#3627409359;.&#3627409362;&#3627409358;&#3627409363;&#3627409359;.&#3627409367;&#3627409358;&#3627409362; &#3627409359;.&#3627409365;&#3627409361;&#3627409359;&#3627409362;
&#3627409359;.&#3627409367;&#3627409358;&#3627409362; &#3627409359;.&#3627409365;&#3627409361;&#3627409359;&#3627409362;&#3627409359;.&#3627409366;&#3627409362;&#3627409360; −&#3627409359;.&#3627409362;&#3627409358;&#3627409363;

Step 2: Assume the initial value, ?????? = 0 V=1.0
&#3627408511; =
??????
&#3627409360;
??????
&#3627409360;
=
&#3627409358;
&#3627409359;.&#3627409358;

Step 3: calculate &#3627408503;
&#3627409360;
&#3627408516;&#3627408514;??????
&#3627408504;
&#3627409360;
&#3627408516;&#3627408514;??????
and ∆&#3627408503;
&#3627409360; and ∆&#3627408504;
&#3627409360;
&#3627408503;
&#3627409360;
&#3627408516;&#3627408514;??????
= ??????
2
??????
1
??????
12 cos(??????
12+&#3627409151;
2 -&#3627409151;
1) + ??????
2
??????
2
??????
22 cos(??????
22+&#3627409151;
2 -&#3627409151;
2) = -0.015
p.u
&#3627408504;
&#3627409360;
&#3627408516;&#3627408514;??????
= - ??????
2
??????
1
??????
12 sin(??????
12+&#3627409151;
2 -&#3627409151;
1) - ??????
2
??????
2
??????
22 sin(??????
22+&#3627409151;
2 -&#3627409151;
2) = -0.157
p.u
∆&#3627408451;
2 = &#3627408451;
2(????????????????????????) - &#3627408451;
2
??????????????????
= -0.285
∆&#3627408452;
2 =&#3627408452;
2(????????????????????????) - &#3627408452;
2
??????????????????
= 0.057
Step 4: Form Jacobian matrix

??????&#3627408503;&#3627409360;
∆&#3627409151;2
??????
2
??????&#3627408503;&#3627409360;
????????????&#3627409360;
??????&#3627408504;&#3627409360;
??????&#3627409151;2
??????
2
??????&#3627408503;&#3627409360;
????????????&#3627409360;

∆&#3627409151;
2
∆ ??????
2
??????2
=
∆&#3627408503;
&#3627409360;
∆&#3627408504;
&#3627409360;
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Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

9. ??????&#3627408503;&#3627409360;
∆&#3627409151;2
= ??????
2
??????
1
??????
12 sin(??????
12+&#3627409151;
2 -&#3627409151;
1) = 1.973
??????
2
??????&#3627408503;&#3627409360;
????????????&#3627409360;
= ??????
2
??????
1
??????
12 cos(??????
12+&#3627409151;
1 -&#3627409151;
2) +2 ??????
2
??????
2
??????
22 cos(??????
22) =0.289

??????&#3627408504;&#3627409360;
??????&#3627409151;2
= ??????
2
??????
1
??????
12 cos(??????
12+&#3627409151;
1 -&#3627409151;
2) = - 0.3197
??????
2
??????&#3627408503;&#3627409360;
????????????&#3627409360;
= ??????
2
??????
1
??????
12 sin(??????
12+&#3627409151;
1 -&#3627409151;
2) +2 ??????
2
??????
2
??????
22 sin(??????
22) = 1.66
Step 5: Compute ∆??????,

∆&#3627409151;
2
∆ ??????
2
??????2
=
??????&#3627408503;&#3627409360;
∆&#3627409151;2
??????
2
??????&#3627408503;&#3627409360;
????????????&#3627409360;
??????&#3627408504;&#3627409360;
??????&#3627409151;2
??????
2
??????&#3627408503;&#3627409360;
????????????&#3627409360;

−1

∆&#3627408503;
&#3627409360;
∆&#3627408504;
&#3627409360;

=
1.9730.289
−0.31961.66

−1

−0.285
0.057

=
−0.145
0.064

&#3627409151;
2
1
= &#3627409151;
2
0
+∆&#3627409151;
2 = - 0.145rad
??????
2
1
= ??????
&#3627409360; +
∆ ??????
2
??????2
= 1.0064p.u 9
Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

10. Advantages and disadvantages of N-R method.
Advantages:
The N-R method is faster, more reliable and the results are
accurate.
Requires less number of iterations for convergence
The number of iterations are independent of the size of the
system
Suitable for large system
Disadvantages
The programming logic is more
Memory requirement is more
Number of calculations per iteration are high

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Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi

11. Conclusion:
The Newton-Raphson method is explained with
algorithm, flowchart and with an example.
References:

1. Hadi Saadat, ‘Power System Analysis’, Tata McGraw Hill Education Pvt. Ltd., New Delhi,
21st reprint, 2010.
2. Kundur P., ‘Power System Stability and Control, Tata McGraw Hill Education Pvt. Ltd., New
Delhi, 10th reprint, 2010.
3. Pai M A, ‘Computer Techniques in Power System Analysis’, Tata Mc Graw-Hill Publishing
Company Ltd., New Delhi, Second Edition, 2007.
4. J. Duncan Glover, Mulukutla S. Sarma, Thomas J. Overbye, ‘ Power System Analysis & Design’,
Cengage Learning, Fifth Edition, 2012.
5. Olle. I. Elgerd, ‘Electric Energy Systems Theory – An Introduction’, Tata McGraw Hill Publishing
Company Limited, New Delhi, Second Edition, 2012.
6. C.A.Gross, “Power System Analysis,” Wiley India, 2011.
7. M.Jeraldin Ahila “Power System Analysis”, Lakshmi Publications, Chennai, Eleventh Edition
2017.
8. Web Sources
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Kongunadu College of Engineering & Technology Newton-Raphson Method Prepared by,
Mrs.S.Revathi