Electrical Engineering Development Project on Load Flow Studies

Contents

Abstract……………………………………………………………3

Acknowledgment…………………………………………………….4

1.INTRODUCTION…………………………………………………….5

2. DESCRIPTION OF RESEARCH & FINAL SPECIFICATION ………………….………………..6

2.1 Classification of buses in the load flow studies……………………………………………….6

2.2 Steps for calculation Newton-Raphson technique…………………………………………….7

3.DESIGN OF THE SYSTEM0

3.1 Newton-Raphson Method……………………………………………………………………………10

4.SOFTWARE/ HARDWARE DESCRIPTION AND FLOWCHART

4.1 Software Description………………………………………………………………………..………12

4.2 Hardware Description……………………………………………………………………………….12

4.3 Flow chart for Newton-Raphson Method……………………………………………………………13

4.4 Final results……………………………………………………………………………………….….14

5.CONCLUSION

6.REFERENCES

7.APPENDIXs

7.1 MATLAB Coding…………………………………………………..17

 7.2 MATLAB command window output results……………………………………………………………………………………21

7.3 Budget…………………………………………………………..24

7.4 Gantt Chart……………………………………………………….26

Abstract

Load flow study is playing significant role in power system analysis and design. It is necessary for planning, operating, fault analysis and exchange of power between utilities. In the beginning, we have to find the real and reactive power at sending and receiving end of the transmission line as well as we find the magnitude and phase angle of voltage at each bus. There are number of methods to calculate the load flow studies such as Newton-Raphson, Gauss-Seidel, Fast decoupled method. In this project, I am using Newton-Raphson method to calculate the Load flow studies. The objective of this project is to calculate the real and reactive power, magnitude of voltage and phase angle. Primarily, this report comprises the introduction of the project, specification of project, design of three buses system and flow chart. Newton-Raphson method coding by using Matlab software, simulation output, simulation results will be compared with number of iterations.

Introduction

Load flow study is study of power system under steady-state condition that means the system is working in balance condition. Load flow analysis is done to find the value of unknown bus voltage of an interconnected system that is $\mathit{ V }$

  magnitude and $\delta$

angle. Active and reactive power flows from the generation to the load via a power network buses and branches the flow of active and reactive power is known as load flow studies. Load flow study gives us a arranged mathematical path for determine of different bus voltages phase angle active and reactive power flows via various branches generator and load under steady-state condition. The main motive of load flow study is to obtain the magnitude and phase angle of the voltage at every bus and the real and reactive power flowing in each power line. There are number of methods to calculate the load flow study such as Newton-Raphson, Gauss-Seidel, Fast decoupled method. In this project, we are using Newton-Raphson method with Matlab to calculate the load flow equations.

Important steps involved to calculate the load flow studies:-

Step 1:- find the unknown bus voltage and phase angle ׀V׀, ∆δ

Step 2:- Find active power and reactive power (PG, QG)

Step 3:- find complex power flow in the whole power system

Step 4:- find power losses in the network

2. Description of research and final specifications

In this project we will design the three bus system load flow diagram and explain the each parameter of the circuit.

2.1 Classification of buses in the load flow studies

                                                        Classification of Buses                                      Ref. 2, 6

• Load bus:- Generator is not connected to load bus. Real (P) and reactive (Q) power is specified on this bus and it is desired to calculate the voltage magnitude and phase angle through load flow results.
• Voltage controlled or Generator bus:-   Voltage magnitude kept constant at generator bus. The generator connected with generating station can be controlled by adjusting the prime mover and voltage magnitude can be controlled by adjusting the generator excitation. At voltage controlled bus we knows the real power and voltage magnitude but reactive power and phase angle are unspecified .Now obtain the phase angle of voltage and the reactive power (Q).
• Slack or Swing bus:-for the slack bus, we have to assume that the voltage magnitude ׀V׀and voltage phase angle are known or voltage phase angle assume to be zero. We need to obtain the real (P) and reactive (Q) via load flow calculations.

2.2 Steps for calculation Newton-Raphson technique:-

The steps for solution of load flow problems using newton- Raphson method is explained as follows :-

Step1. From pu YBUS (admittance matrix to impedance matrix)

Step2. Assume the initial values for all buses except the slack bus or swing bus

Voltage – controlled (generator) bus   ${\mathit{ Dk}}^0=\mathrm{Vks},\mathrm{ }{\mathit{Fk}}^0=0\mathit{ k}=2,3\dots \dots ..,G$

Load buses                                           ${\mathit{ Dk}}^0=1,\mathit{ }{\mathit{Fk}}^0=0\mathit{ k}=G+1,G+2,\dots \dots ..,N$

Step3. Calculate the real and reactive power ${\mathit{Pk}}^i, {\mathit{Qk}}^i$

Step4. Evaluate the powers of real, reactive and voltage  ${∆\mathit{Pk}}^i, {∆\mathit{Qk}}^i,{∆\mathit{Vk}}^{i2}$

${∆\mathit{Pk}}^i=\mathit{Pks}–{\mathit{Pk}}^i\mathit{ k}=2,3,\dots \dots ..,N$

${\mathit{ }∆\mathit{Qk}}^i=\mathit{Qks}–{\mathit{Qk}}^i\mathit{ k}=G+1,G+2,\dots \dots ..,N$

${∆\mathit{Vk}}^{i2}={\mathit{Vk}}^2–{\mathit{Vk}}^{i2}\mathit{ k}=2,3,\dots \dots ..,G$

Where $\mathit{Pks},\mathit{ Qks},\mathit{ and\; Vks }$

  are scheduled (known) quantities and

  ${\mathit{Vk}}^{i2}={\left({\mathit{Dk}}^i \right)\mathit{ }}^2–{\left({\mathit{Fk}}^i \right)\mathit{ }}^2\mathit{ }$

 Then set   ${∆\mathit{Xmax}}^{}={∆\mathit{Pmax}}^{\mathit{i }}\mathit{or }{{∆\mathit{Qmax}}^{\mathit{i }}\mathit{or }{∆\mathit{Vkmax}}^{2i}}^{}$

whichever is largest.

Elements of the Jacobian matrix

$J1\mathit{kk}=\frac{\partial \mathit{Pk}}{\partial \mathit{Dk}}=\mathit{Dk}*\mathit{Gkk}–\mathit{Fk}*\mathit{Bkk}+\mathit{Ak}$

           k=G+1, G+2,…,N     J=2,3,…,N     K≠ j   

                                                                                Ak is real part of I= GD+BF (Diagonals)

$J1\mathit{kj}=\frac{\partial \mathit{Pk}}{\partial \mathit{Di}}=\mathit{Dk}*\mathit{Gkj}–\mathit{FkBkj}$

  $J2\mathit{kk}=\frac{\partial \mathit{Pk}}{\partial \mathit{Fk}}=\mathit{Dk}*\mathit{Bkk}+\mathit{Fk}*\mathit{Gkk}+\mathit{Ck}$

           k=G+1, G+2,…,N     J=2,3,…,N     K≠ j

                                                                        Ck is imaginary part of I=GF-BD Off (Diagonals)

$J2\mathit{kj}=\frac{\partial \mathit{Pk}}{\partial \mathit{Fi}}=\mathit{Dk}*\mathit{Bkj}+\mathit{Fk}*\mathit{Gkj}$

$J3\mathit{kk}=\frac{\partial \mathit{Qk}}{\partial \mathit{Dk}}=\mathit{Dk}*\mathit{Bkk}–\mathit{Fk}*\mathit{Gkk}–\mathit{Ck}$

       k=G+1, G+2,…,N     J=2,3,…,N     K≠ j

$J3\mathit{kj}=\frac{\partial \mathit{Qk}}{\partial \mathit{Dj}}=\mathit{Dk}*\mathit{Bkj}+\mathit{Fk}*\mathit{Gkj}$

$J4\mathit{kk}=\frac{\partial \mathit{Qk}}{\partial \mathit{Fk}}=\mathit{Fk}*\mathit{Bkk}–\mathit{Dk}*\mathit{Gkk}+\mathit{Ak}$

           k=G+1, G+2,…,N     J=2,3,…,N     K≠ j

$J4\mathit{kj}=\frac{\partial \mathit{Qk}}{\partial \mathit{Fj}}=\mathit{Fk}*\mathit{Bkj}–\mathit{Dk}*\mathit{Gkj}$

$J5\mathit{kk}=\frac{\partial {\mathit{Vk}}^2}{\partial \mathit{Dk}}=2*\mathit{Dk}$

          k=2,3,…….G          J=2,3,………N              K≠ j

$J5\mathit{kj}=\frac{\partial {\mathit{Vk}}^2}{\partial \mathit{Dj}}=0$

$J6\mathit{kk}=\frac{\partial {\mathit{Vk}}^2}{\partial \mathit{Fk}}=2*\mathit{Fk}$

       k=2,3,…….G          J=2,3,………N              K≠ j

$J6\mathit{kj}=\frac{\partial {\mathit{Vk}}^2}{\partial \mathit{Fj}}=0$

Step5. Calculate the bus currents (Ibus)

                  I $k^i$

= $\frac{{\mathit{Pk}}^i–{\mathit{jQk}}^i}{{(\mathit{Vk}}^i)*}={\mathit{Ak}}^i+{\mathit{jCk}}^i\mathit{ k}=2,3,\dots ..,N$

Step6. Calculate the Jacobian matrix from step 4

Step7. Obtain by matrix inversion for the corrections or simultaneous soluction

${\mathit{ }∆\mathit{Dk}}^i,\mathit{ }∆ {\mathit{Fk}}^i\mathit{ k}=2,3\dots \dots .,N$

Step8. Calculate the new bus voltages and at the remaining buses

At the voltage- controlled buses:

${\mathit{Dk}}^{i+1}=\mathit{Vks}\cos {\mathit{\delta k}}^i$

             k=2, 3,…..G

${\mathit{Fk}}^{i+1}=\mathit{Vks}\sin {\mathit{\delta k}}^{\mathit{ i}}$

   where ${\mathit{\delta k}}^{\mathit{ i}}=$ ${\mathit{tan}}^{–1} ({\mathit{Fk}}^i/{\mathit{Dk}}^i$

)

At the remaining buses:

${\mathit{Dk}}^{i+1}={\mathit{Dk}}^i+∆{\mathit{Dk}}^i$

         k=G+1, G+2,…………,N

${\mathit{Fk}}^{i+1}={\mathit{Fk}}^i+∆{\mathit{Fk}}^i$

Step9. Replace the values of ${\mathit{Dk}}^i,\mathit{ }{\mathit{Fk}}^i\mathit{ by }{\mathit{Dk}}^{i+1},\mathit{ }{\mathit{Fk}}^{i+1},\mathit{ for\; k}=2,3,\dots .,N$

Step10. Now get the final results of the current iteration. If solution is not specified then repeat the steps three-seven until the solution is reached.

Step11. In the final step calculate the real(P1) and reactive(Q1) powers.

3. Design of the System

3.1 Newton- Raphson Method:-`

N-R method is most valuable for solving the non-linear equations. This method is faster and sure to cover most cases as compared to the Gauss- Seidel method. It’s the most powerful practical method for solving load flow solution of any power system networks.

The Newton-Raphson method is a technique to find the approximations to a solution or root of an equation $f\left(x\right)=0$

the method uses differentiation because it relies the equation of tangents to the graph of $y=f\left(x\right).$

It produces the numerical sequences of values get closer and closer to the real root. For instance we may wish to solve the equations that can be written in the form $f\left(x\right)=0.$

                                $x^4+3x–2=\mathit{or }{e}^{–x}–\mathit{sinx}=$

In this equivalent problem to solving values of x for which is graph of $y\left(x\right)=0$

that intersects the horizontal axis, as shown in the diagram. When x has the particular values $x1,\mathit{ x}2\mathit{ and\; x}3$

then roots of y lies at zero position. In other terms $\left(x1\right)=0,\mathit{ f}\left(x2\right)=0\mathit{ and\; f}\left(x3\right)=0$

, and so the roots of $f\left(x\right)=0\mathit{ are\; x}=x1,\mathit{ x}=x2\mathit{and\; x}=x3.$

                                                           $y$

$\mathit{ f}\left(x\right)=0\mathit{ y}=f\left(x\right)\mathit{ }$

  $x1\mathit{ x}2\mathit{ x}3\mathit{ x}$

In this figure, we will get the solution of $f\left(x\right)=0$

can be located by finding values of $x$

where the graph of $y=f\left(x\right)$

intersects of x axis.

                                         $\mathit{ y \; y}=f\left(x\right)$

                                                                                                                P

  $\mathit{ \alpha }$

$\mathit{ x}1\mathit{ x}0\mathit{ x }$

Presume a root exists at some unknown value $x=a$

as illustrated in the diagram. Suppose we know the estimate of the roots  $x=x0$

. So, we can draw a tangent to the curve $y=f\left(x\right)$

at the position  $P(x0,f(x0)$

, and use the x value at position where the tangent cuts the $\mathit{x\; axis},\mathit{ x}1$

as a better estimate of the root.

To commence, we obtain the equation of the tangent to $y=f\left(x\right)\mathit{at\; P}\left(x0,\mathit{ f}\left(\mathit{xo}\right)\right).$

It will get from $y=\mathit{mx}+c.$

The gradient of m, will equal $\frac{\partial y}{\partial x}$

  evaluated at $0$

, which is context of the Newton-Raphson method. We can write as $f^{‘}\left(x0\right)$

. The equation becomes $y=f^{‘}\left(x0\right)x+c.$

Now find the constant of $\mathit{ c}$

. The tangent passes through the point $\left(x0,f\left(x0\right)\right),$

and so $f\left(x0\right)=f^{‘}\left(x0\right)x0+c,$

from which $c=f\left(x0\right)–f^{‘}\left(x0\right)x0.$

Here, we know the value of $x0$

and calculate $f\left(x0\right)–f^{‘}\left(x0\right)x0,$

and so  can $c$

can be evaluated. Finally, we will get the tangent is

                                                  $y=f^{‘}\left(x0\right)x+f\left(x0\right)–f^{‘}\left(x0\right)x0$

In the diagram we can see that the tangent will cut the horizontal axis when $y=0$

  and $x=x1,$

that is when

                                          $0=f^{‘}\left(x0\right)x1+f\left(x0\right)–f^{‘}\left(x0\right)x0$

Rewriting this formula to make $x1$

  the subject gives

                                                             $x1=x0–\frac{f\left(x0\right)}{f^{‘}\left(x0\right)}$

Here, $x0$

is our first approximation to the root. This formula allows us to calculate a better estimate of the root, denoted by $x1.$

4. Software/ Hardware Description and flow chart

4.1 Software Description:-

In this project, we are using MATLAB software for load flow calculation. It is very hard to calculate the number of iterations by hand, with the help of Matlab easy to repeat the number of iteration to get accurate results. We are using Newton-Raphson method by Matlab to calculate bus voltages and real and reactive power for each bus.

Hardware Description:-

In this project we will not using any type of hardware equipment for working this project, it’s based on software (MATLAB) coding for load flow studies.

4.3 Flow Chart for Newton-Raphson Method:-

L

O

O

P

START SIMULATION

                             Define Ybus Variables

                          Assume the initial values

Calculate real and reactive power

 Calculate the Delta Angle

Calculate the bus Currents

Calculate the Jacobin Matrix

Calculate the Inverse of Jacobin

No Accurate

 Check Accuracy

Yes Accurate

Find buses power

STOP SIMULATION

4.4 Final results:-

5. Conclusion

Load flow study is important for planning future expansion of any power network as well as in determining the operation existing system. We learn how load flow equations are apply on any working power system. The main target of this project the information obtain from the load flow is the magnitude and phase angle of the voltage at each bus and calculates the real and reactive power is flowing in each power line. In the present scenario, the most accurate and fast method is the Newton-Raphson for analysis the load flow study for any power system network. Nowadays,   Newton-Raphson method is more popular because of its high versatility, reliability and accuracy for calculation any load flow system. In second part of the project we will analysis this `power system network with MATLAB coding. It is easy to repeat the long iteration with Matlab coding. In the final part of the coding we got accurate results of the voltages, real and reactive power for each bus.

6. REFERENCES

1. Retrieved from https://www.ijert.org/ .

2. Modern power system Analysis, D P kothari and I J Nagrath

3.https://en.wikipedia.org/wiki/Electric_power_transmission. [Accessed 20 september 2017].

4. Power Sysstem By Behic R.Gungor Published- Haecourt Brace Jouanovich 1/01/1998

5. https://www.scribd.com/document/245485139/Power-Flow-Study

7. APPENDIXs

7.1 MATLAB Code

7.2 Matlab Command window outputs results:- 1^st iteration

Results of 7th iteration 

7.2 Budget

7.3 Gantt Chart