PV- Curve Analysis of 3 Bus Power System using …...PV- Curve Analysis of 3 Bus Power System using MATLAB Manish Parihar1, M.K. Bhaskar2, Deepak Bohra3, Digvijay Sarvate4 1,3,4M.B.M

International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: [email protected]

Volume 6, Issue 5, September- October 2017 ISSN 2278-6856

Volume 6, Issue 5, September – October 2017 Page 201

Abstract: P-V curve analysis is use to determine voltage stability of power system network. Newton-Raphson method is used for load flow analysis. In this paper P-V curve is drawn for different load power factor condition. Newton-Raphson method along with determinant of power flow Jacobian matrix helps in calculating loadability limit and critical voltage collapse point at selected load bus. Results were obtained using MATLAB applications software. Keywords: PV Curve, Voltage Stability, Singularity of Jacobian, Power Transfer.

1. Introduction

Transmission systems are more becoming more stressed due to increased loads and inter-utility power transfers. With growing size, along with economic and environmental pressures, the efficient operation of the power system is becoming increasingly threatened due to problem of voltage instability and collapse. PV curves are most widely used voltage stability analysis tools and formed by increasing power at a particular area in steps and voltage (V) is observed at some critical load buses and then curves for those particular buses will be plotted The PV curve can provide real power and voltage margins using the knee of the curves as reference point. PV curves at constant power factor are used to get maximum power transfer at critical voltage.

Voltage corresponding to “maximum loading point” is called as critical voltage. If load is further increased, power flow equation doesn't have a solution i.e., Newton Raphson algorithm has not converged. Hence, the P-V curve can be used to determine the system’s critical voltage point and collapse margin. For a power system network, load buses (PQ buses) are identified to plot the P-V curves. Here at load side real power P is incremented at constant power factor 휙 and reactive power is given by equation 푄 = 푃 tan휙. Steps in P-V curve analysis:

1. Select a load bus, vary the load real power. Keep the power factor as constant.

2. Compute the power flow solution for the present load condition and record the voltage of the load bus.

3. Increase the load real power by small amount let say .5 푝.푢. and repeat step 2 until power flow does not have convergence.

4. P-V curve is plotted using the calculated load bus voltages for increased load values.

Above methodology used for drawing PV curve may lead to certain problems, as if load real power increment or step size is too low then power flow program takes much time to obtain solution and if load real power increment is high or too high then power flow program may fail to converge as it may skip voltage collapse point .It is evident that magnitude of ∆[퐽] decreases from high value to zero. The point of voltage stability limit being coincident with the singularity of the Jacobian ∆[퐽] = 0. by using above concept initially step size selected is high, checking ∆[ 퐽] at each step, if ∆[ 퐽] is below specified value then change step size to low value. The process is continued till ∆[ 퐽] reaches near zero at this point power is maximum and point is known as maximum loading point or point of voltage collapse.

2. Theoretical Background Newton Raphson Method is an iterative technique for

solving a set of various nonlinear equations with an equal number of unknowns. In this paper polar coordinate form is used. As shown in figure the current entering at bus 푖 is given by equation 퐼 = 푉 ∑ 푦 − ∑ 푦 푉 푗 ≠ 푖 This equation can be rewritten in terms of the bus admittance matrix as 퐼 = ∑ 푌 푉 , expressing in polar form we have 퐼 = ∑ 푌 푉 ∠ 휃 + 훿

Complex power at 푖 bus is 푃 − 푗 푄 = 푉 ∗ 퐼

퐼 = 푉 ∑ 푦 − ∑ 푦 푉 푗 ≠ 푖

This equation can be rewritten in terms of the bus admittance matrix as 퐼 = ∑ 푌 푉 , expressing in polar form we have

퐼 = ∑ 푌 푉 ∠ 휃 + 훿

Complex power at 푖 bus is 푃 − 푗 푄 = 푉 ∗ 퐼

PV- Curve Analysis of 3 Bus Power System using MATLAB

Manish Parihar1, M.K. Bhaskar2, Deepak Bohra3, Digvijay Sarvate4

1,3,4M.B.M Engineering College, Electrical Engineering Department, Jodhpur, Rajasthan, INDIA

2M.B.M Engineering College, Electrical Engineering department, Jodhpur, Rajasthan, INDIA

http://www.ijettcs.org

mailto:[email protected]




Figure 1 Bus 푖 of the power system

Substituting the value of current in complex power equation we get

푃 − 푗 푄 = 푉 ∠−훿 ∑ 푌 푉 ∠ 휃 + 훿 , simplify

and separating real and imaginary parts,

푃 = ∑ 푉 푌 푉 cos( 휃 −훿 + 훿 )

푄 = −∑ 푉 푌 푉 sin( 휃 −훿 + 훿 )

Elements of Jacobian matrix is obtained by taking partial derivatives of above equations with respect to magnitude and phase angle of voltages i.e., 푉

and 훿. the jacobian matrix gives the linearised relationship between small changes in magnitude and phase angle of voltages i.e., ∆ 푉

and ∆훿 with the small changes in real and reactive power ∆푃 and ∆푄.

∆푃∆푄 = 퐽 퐽 퐽 퐽

∆훿∆ 푉

The term ∆푃 and ∆푄 known as power residue or mismatch, given by ∆푃 = 푃 − 푃 ∆푄 = 푄 −푄

∆훿∆ 푉

= 퐽 퐽퐽 퐽 ∆푃∆푄

The new estimate for bus voltages are 훿 = 훿 + ∆훿 and 푉 = 푉 + ∆|푉| . 퐽 = 퐽 퐽

퐽 퐽 , known as Jacobian matrix

3. Case Study Consider a three bus power system as shown in figure

generator at buses 1 and 3. The magnitude of voltage at slack bus 1 is 1.05 푝푢 and voltage magnitude at bus 3 is 1.04 푝푢 and real power generation at bus 3 is 200 MW. Bus 2 is a load bus consisting of 400 MW and 250 푀푣푎푟. Line impedances are marked in per unit on a 100 MVA base, and line charging suspectances are neglected.

Bus 1 Bus 2

Bus 3

.02 + j .04

.01 + j .03 .0125 + j .025Slack Bus

V1 = 1.05 0̥

PLOAD

MW

QLOAD

Mvar

200MW

V3 = 1.04 0̥

0̥

V2 = 1.0

Figure 2 Three bus system network

Load real power ( 푃 \ 푃 ) is incremented and load reactive power is ( 푄 \ 푄 ) is given by 푄 = 푃 tan휙 and constant generator real power 푃 =200 푀푊.

Ii

Vi y i1

yi2

... yin

yi0

V1

V2

Vn






3.1 CASE STUDY I

By varying load power 푃 and 푄 = 푃 tan휙 and constant generator real power 푃 = 200 푀푊 and power factor 휙 = 20° 푙푎푔. Results are obtained using Newton Raphson program is : Determinant of Jacobian = 511.45 Critical voltage (V ) = .567 p. u. 푅푒푎푙 푃표푤푒푟 푐푟푖푡푖푐푎푙(P ) = 16.76 푝.푢. 훿 푐푟푖푡푖푐푎푙(δ ) = −33.316° 푅푒푎푐푡푖푣푒 푃표푤푒푟 푐푟푖푡푖푐푎푙(Q ) = 6.10 푝.푢. 퐺푒푛푒푟푎푡표푟 푅푒푎푐푡푖푣푒 푃표푤푒푟 ( 푄 ) = 19.36 푝.푢. 푆푙푎푐푘 퐵푢푠 푅푒푎푙 푃표푤푒푟 ( 푃 ) = 24.42푝.푢. 푆푙푎푐푘 퐵푢푠 푅푒푎푐푡푖푣푒 푃표푤푒푟 ( 푄 ) = 7.37 푝.푢. 훿 푎푡 푏푙푎푐푘표푢푡 푐표푛푑푖푡푖표푛 = −19.78° And curves obtained shown below:

Figure 3 Iteration bar graph at 20 degree lag

Figure 4 PV curve at 20 degree lag

3.2 CASE STUDY II

By varying load power 푃 and 푄 = 푃 tan휙 and constant generator real power 푃 = 200 푀푊 and power factor 휙 = 0° .Results obtained using Newton Raphson program is:

Determinant of Jacobian = 519.85

critical voltage = .6035 p. u.

푅푒푎푙 푃표푤푒푟 푐푟푖푡푖푐푎푙 = 20.8780 푝. 푢. 훿 푐푟푖푡푖푐푎푙 = −47.3744°

푅푒푎푐푡푖푣푒 푃표푤푒푟 푐푟푖푡푖푐푎푙 = 0 푝. 푢.

퐺푒푛푒푟푎푡표푟 푅푒푎푐푡푖푣푒 푃표푤푒푟 ( 푄 ) = 18.864 푝.푢.

푆푙푎푐푘 퐵푢푠 푅푒푎푙 푃표푤푒푟 ( 푃 ) = 30.76푝.푢.

푆푙푎푐푘 퐵푢푠 푅푒푎푐푡푖푣푒 푃표푤푒푟 ( 푄 ) = 6.96 푝. 푢.

훿 푎푡 푏푙푎푐푘표푢푡 푐표푛푑푖푡푖표푛 = −24.96°

And curves obtained shown below:

Figure 5 Iteration bar graph at 0 degree

Figure 6 PV curve at 0 degree

3.3 CASE STUDY III

By varying load power 푃 and 푄 = 푃 tan휙 and constant generator real power 푃 = 200 푀푊 and power






factor 휙 = 20° 퐿푒푎푑 . Results obtained using Newton Raphson program is: Determinant of Jacobian = 1434 critical voltage( V ) = .7094 p. u. 푅푒푎푙 푃표푤푒푟 푐푟푖푡푖푐푎푙 ( 푃 ) = 24.75 푝.푢. 훿 푐푟푖푡푖푐푎푙 ( 훿 ) = −58.2864° 푅푒푎푐푡푖푣푒 푃표푤푒푟 푐푟푖푡푖푐푎푙( 푄 ) = −9.0112 푝. 푢. 퐺푒푛푒푟푎푡표푟 푅푒푎푐푡푖푣푒 푃표푤푒푟 ( 푄 ) = 18.05 푝.푢. 푆푙푎푐푘 퐵푢푠 푅푒푎푙 푃표푤푒푟 ( 푃 ) = 38.27푝.푢 푆푙푎푐푘 퐵푢푠 푅푒푎푐푡푖푣푒 푃표푤푒푟 ( 푄 ) = 7.100푝. 푢. 훿 푎푡 푏푙푎푐푘표푢푡 푐표푛푑푖푡푖표푛 = −31.01° And curves obtained shown below:

Figure 7 Iteration bar graph at 20 degree lead

Figure 8 PV curve at 20 degree lead

From graphs it is observed that as the load side power

factor varies from lagging to leading condition then the loading of the power system increases according to voltage stability. Thus, the monitoring of power system security becomes more complicated because the critical voltage might be close to voltages of normal operating range. Combining all curves for different power factor we get:

Figure 9 PV Curves at different power factors

Figure 10 Determinant of Jacobian matrix at different

power factors

4. CONCLUSION In this paper PV curve analysis is done for voltage

stability analysis. Power flow program is developed using Matlab software to analyze power system network. It is observed that power factor at load side has significant effect on loadability, critical voltage point of the system. Thus reactive power compensation is necessary for stability purpose and to reduce drop in bus voltage. REFERENCES

[1]. Abhijit Chakrabarti, de Abhinandan, Mukhopadhyay A.K.,Kothari D.P., 2010, An






Introduction to Reactive Power Control and Voltage Stability in Power Transmission Systems, PHI Learning Pvt. Ltd.

[2]. Basu, K. P., “Power Transfer Capability of Transmission Line Limited by Voltage Stability: Simple Analytical Expressions” IEEE Power Engineering Review, September 2000, pp 46-47.

[3]. Chemikala Madhava Reddy, Power System Voltage Stability Analysis, Department of Electrical Engineering June 2011

[4]. Khan, Asfar Ali, “PV Curves for Radial Transmission Lines”. Accepted for Publication in The Proceedings of National Systems Conference 2007 to be Held at Manipal Institute of Technology from 14-15 Dec.2007.

[5]. P. Kundur. Power System Stability and Control. McGraw Hill, New York, 1994

[6]. Peter W. Sauer, University of Illinois at Urbana-Champaign, Reactive Power and Voltage Control Issues in Electric Power System

[7]. Hadi Sadat, 2002.Power System Analysis, 2nd ed., New York: McGraw-Hill.

[8]. M. Parihar, M.K. Bhaskar,“ Radial Transmission Line Voltage Stability Analysis”, IJRERD (ISSN: 2455-8761), VOLUME02-Issue 09, September 2017, PP. 01-07.

AUTHOR Manish Parihar holds B.Tech Degree in Electrical & Electronics Engineering VIET, Rajasthan. Currently he is pursuing M.E. in Power Systems from MBM Engineering College, Jodhpur, Rajasthan. Also he is

associate member of The Institute of Engineers (AM163285-2).

Dr. M.K. Bhaskar received B.E. from Malviya Regional Enginnering College (MREC) known as MNIT, Jaipur and he completed his master of engineering and Phd. From M.B.M. engineering. college. He

is currently working as an professor in Electrical Engineering, M.B.M. Engineering College, J.N.V.University, Jodhpur, Rajasthan. Deepak Bohra is PG Scholar of Department of Electrical Engineering, M.B.M. Engineering College, JNV University, Jodhpur, Rajasthan, India. Digvijay Sarvate is PG Scholar of Department of Electrical Engineering, M.B.M. Engineering College, JNV University, Jodhpur, Rajasthan, India and he completed B.Tech from MERC, Jodhpur.



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PV- Curve Analysis of 3 Bus Power System using …...PV- Curve Analysis of 3 Bus Power System using MATLAB Manish Parihar1, M.K. Bhaskar2, Deepak Bohra3, Digvijay Sarvate4 1,3,4M.B.M