 Open Access
 Authors : Atef A. Elemary , M.M.R Ahmed , Ebtisam M. Saied , Mohamed A. Hamdy
 Paper ID : IJERTV11IS090010
 Volume & Issue : Volume 11, Issue 09 (September 2022)
 Published (First Online): 12092022
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Power System Voltage Stability Index
Atef A. Elemary1, M.M.R Ahmed2 Department of Electrical Engineering Jazan University
Jazan, Saudi Arabia
Ebtisam M. Saied3, Mohamed A. Hamdy4
Department of Electrical Engineering Benha University
Cairo, Egypt
Abstract After some damaging blackouts, voltage instability and collapse have become worldwide concern problems. Voltage instability is a nonlinear local phenomenon, which occurs due to insufficient reactive power production or as the inability to transfer it to consumers. In most dangerous cases, it can result in voltage collapse. Therefore, early identification of the process is essential. A power system is decomposed into selected radial paths regarding reactive power flow. Next, two bus equivalents
its simple algorithm and clear physical meaning. Banasila [9] used it in the optimal reactive power dispatch and expert system.
power system stability
for these radial transmission paths are determined. The analytical proof of maximal stable power transfer for two bus equivalents is available. However, each candidate critical load bus may be fed via more transmission paths. It is essential that a set of power sources, feeding the particular candidates critical reactive power drain bus, has sufficient reactive power reverse. Therefore, the proposed voltage collapse assessment procedure combines the voltage collapse proximity generators index and an available reactive power reserve of the involved generators. The methods properties and numerical results are presented.
Keywords Power System, voltage stability, indicator
rotor angle
stability
transient stability
frequency
stability
short term
short term
long term
large disturbance
long term
voltage
small disturbance
stability
short term
small disturbance

INTRODUCTION
In recent years, economical and environmental reasons have forced power systems to be operated closer to their security limits. Traditionally, their limits are associated with thermal and stability limitations have become less restrictive as nowadays most of electric utilities use faster relays to reduce the fault clearing time, fast response excitation systems and FAC,TS devices to reduce the amplitude of the disturbance. As a result, the power systems world is increasingly concerned with voltage stability and collapse problems, especially after blackouts of some world (New York, London,,), some of which have been reported to be associated with voltage collapse [1,3, 1115].
Recently IEEE/CIGRE task recommended numerous definitions in regarding to power system stability including voltage stability. Figure 1 summarizes these explanations. In general terms voltage index is applied to study voltage stability [1].
In order to prevent voltage collapse, fast and accurate voltage stability index is needed to help in monitoring the operating condition. In past articles, some useful voltage stability indices have been presented, such methods of testing the Jacobian, the voltage stability assessment using optimization, the point of collapse method ( POC), the voltage proximity indicator (VCPI), fast voltage stability index (FVSI) [6]. Methods of Thevennins impedance indicator, identifying the weak elements of a power system and the multiple load flow solutions (MLF)as in which, the voltage collapse indicators (VCPI) or (Lindex) [2, 1621] has raised especial regards for
Fig. 1. Classifications of power system stability.

VOLTAGE STABILITY
The voltage stability can be described with different definitions according to the points of views. Its the system ability to keep its voltage within assured specified limits at all loading conditions all over the hour. From the load point of view it can be defined as the load ability or (PV curve) to give more power as the load point of view it can be described as the load ability to supply more power as the loading is increased without voltage dips outside the limits as shown in Figure (2).
Fig. 2. PV curve for the simple
The most common technique of analyzing the transfer capacity of a system against voltage stability is to create a so called PV curve of the bus bar with load or production. PV curves are often respecting to as nose curves due to their shape [2225]. The nose point (generally raised to as the critical point) of the curve provides the maximum power which can be supplied to the load.
An alternative common technique of analyzing the transfer capacity of a system respecting voltage is to gsocalled so called QV curve for a bus bar [4, 9]. Figure 3, shows the reactive power which must be inserted into a bus bar to keep a specific voltage. These curves are plots of the critical bus voltage against the reactive power of the same bus. To obtain QV curve of a bus a fictitious synchronous capacitor is represented at that precise bus and the bus is supposed to be a voltagecontrolled bus without reactive bounds.
where:
Lj: voltage collapse proximity indicators
l: set of load buses
G: set of generators buses
Cji: elements of matrix C determined by:
L indicator varies in the range between 0 (no load system) and 1 (voltage collapse).
B. Fast Voltage Stability Index (VVSI)
The stability index of this method is defined by the following formula [7]
where:
Z: line impedance
X: line reactance
Qj : receiving end reactive power at load bus j Vi : complex voltage at generator bus i
Such that the stability index range is defined by:
(2)
Fig. 3. Q curve and reactive power margin

VOLTAGE INSTABILITY
As the disturbance increases in load or system changes this causes quick drop in voltage or drift downward, and operates and automatic system controls fail to halt the decay.
Causes of voltage instability can be summarized in the following:

Stressed power system (high active power loading Pr, and/ or also high reactive power loading Qr, in the system).

Low power factor of loads of loads, which results higher
values of Q.

Load characteristics at low voltage magnitudes different from their designed values.


STEADY STATE VOLTAGE STABILITY ASSESSMENT
There are different methods used to steady state voltage stability assessment. These methods are basically depends on one of the following approaches:
A. Voltage Collapse Proximity Indicators (VCPI)
The criteria of this indicator is defined as follows [7,9].
(1)

Thevennins Impedance Indicator
To calculate this indicator it is required to calculate the equivalent Thevennins impedance (Zth) for each load bus in the system. The ratio of this impedance to the equivalent load impedance (ZL) at the same bus is taken as the indicator. Zth of a load bus is calculated by removing the load at this bus, replacing all loads and generators in the system by an equivalent impedances with appropriate signs, forming Ybus and inverting it to obtain Zbus , and ZTh is equal to the diagonal elements of Zbus corresponding to the bus required. These steps repeated at each bus as well as load flow is to be carried out for each case to determine the equivalent Thevennins voltage. The certerion for voltage stability by this indicator is given by:


MATHEMATICAL DERVIATION
In this section a proposed method is made to calculate the indicator based on the change of reactive and actie powers. The L indicator [7,9] depend on the calculation of load flow in each once. To overcome this problem a proposed method is investigated. Figure 4 shows two buses connected by transmission line to derive a new mathematical formula of change voltage and its angle based on change of active and reactive power angle. Figure 5, shows phasor diagram of the system shown in figure 4.
where:
(12)
Fig. 4. Two bus radial system
For incremental change of , we get Therefore, equation 12 can be put in the form
Vs
I' X
Vs
Rewrite equation 13 gives:
(13)
2
1
Vr
' I R
I
V
Vx
(14)
From equations 9 and 14, we can get the following relation
Fig. 5. Phasor diagram
From figure 5 the following relation are calculated [10]
(3)
(4)
(15)
We can assume that therefore equation 15 yields to:
(16)
Rewrite equation 6 yields to:
(5)
(6)
Equation 16 gives a new formula to calculate the change of voltage and its angle based on change of active and reactive power of loads. Also, this equation depends on the base case load flow and network data. Therefore, the repeated load flow in each case of change active and reactive power is not required.
(7)
For small change of angle , the following assumption are made:
1, and Equation 7, can be put in the form
(8)
Equation 8 can be put in the form:

CASE STUDY
To test the capability of the proposed method the following applications are made.

Two bus system
The pu data of the system under studying are given in pu:
Vs = 1.05
P = 0.3
Q = 0.2
R = 0.083
X = 0.31
In this system the following applications are made
such that
Equation 11 can be put in the form:
(9)
(10)
(11)

System under Condition of Power Factor4
In this case the active and reactive power are change to get constant power factor. The test results are demonstrated as follows

L indicator (VCPI)

From Fig. 6 it is noted that the system under study is stable when the active power P and reactive power Q reach the values 1.38 pu and 0.92 pu. Respectively. Otherwise the system collapse to unstable region when the active power P and reactive power Q greater than the above values.
Fig. 6 Active and reactive power in pu relation with Linductor at constant pf

FVSI indicator
It can be noted that from figure 7 the system under study is stable when the active power P is vary up to the value of.2 pu and the reactive power Q is vary up to the value of 0.8 pu. The system behavior is changed to unstable when the active power P and reactive power Q greater than 1.2and 0.8 pu
respectively.
Fig 7 Active and reactive power in pu relation with FVSIindicator at constant pf

Thevennins impedance indicator
In this case it is noted that from figure 8 the system under study is stable when the active P is varying up to the value of
1.2 pu and the reactive power Q is varying up to the value of
0.8 pu, otherwise the system is unstable. The system behavior is changed to unstable when the active power P reaches 1.21 pu and the reactive power Q reaches 0.8067pu.
Fig. 8 Active and reactive power in pu relation with ZThinductor at constant pf
2). System under Condition of Constant Active Power P
In this case P = 0, and the applications are made as shown below.

Lindicator (VCPI)
Figure 9 show that the system under study is stable when the reactive power Q is vary up to the value of 1.1196 pu and the angle Ã˜ is vary up to 1.309 rad., the system behavior is changed to the unstable when the reactive Q reaches 1.7014 pu and the angle Ã˜ reaches 1.3963 rad.
Fig. 9 Reactive power relation with Linductor at constant P

FVSI indicator
It is noted that from figure 10, the system under study is stable when the reactive power Q is varying up to the value of 0.8242 pu and the angle Ã˜ is vary up to 1.2217. The system behavior is changed to the unstable when the reactive power Q reaches 1.1196 pu and the angle Ã˜ reaches 1.309 pu.
Fig. 10 (a) Variation of reactive power with FVSI inductor
(b) Variation pf angle with FVSI inductor

Thevennins impedance indicator
Figure 11 shows that the system under study is stable when the reactive power Q is vary up to the value of 0.8242 pu and the angle is vary up to 1.2217, the system behavior is changed to the unstable when the reactive power Q reaches 1.1196 pu and the angle Ã˜ reaches 1.309 rad.
Fig. 11(a) Change of reactive power with ZThinductor at constant power
(b) Change of Power Factor angle ZThinductor at constant power
3) System under Condition of Constant Reactive Power Q
In this case Q = 0, and the applications are made as shown below
a) Lindicator (VCPI)
The system under study shown in figure 12 is stable when the active power P is varying up to the value of 2.286 pu and the angle Ã˜ is varying up to 0.0873. the system behavior is changed to the unstable when the reactive power P reaches 5.7273 pu and the angle Ã˜ reaches 0.0349 rad.
Fig. 12 (a) Change of reactive power with ZThinductor at constant power
(b) Change of Power Factor angle ZThinductor at constant power
b) FVSI indicator
We can note that the function between active power and FVSI indicator is constant function because the FVSI indicator depends on the reactive power Q, such that Q = 0 and there is the same function between angle Ã˜ and FVSI indicator
c). Thevennins impedance indicator
From figure 13 it is noted that

The system under study is stable when the active power P is varying up to the value of 2.286 pu and the power factor angle is varying up to 0.0873 rad.

The system behavior is changed to the unstable when the active power P reaches 2.8601 pu and the power factor angle Ã˜ reaches to 0.698 rad.
Fig. 13 (a) Active power as a function with ZThinductor at constant Q
(b) Power Factor as a function with ZThinductor at constant Q


APPLICATION ON IEEE 6BUS POWER SYSTEM
NETWORK
Figure 14 shows the configuration of IEEE 6bus network power system. This figure includes base case load flow. In this section the stability index under base case condition is made and the effect of load change on the stability index is also investigated.
Fig. 14 IEEE 6=bus power system network

Base Case results
TABLE 1 SHOWS THE SYSTEM IS STABLE BASED ON THE BASE CASE
Bus number
Lindicator
ZTh indicator
4
0.093
0.0692
5
0.1143
0.0793
6
0.0956
0.0824

Change in Bus 4
The effect of load change on bus 4 at constant P, Q and power factor on the stability are studied and shown in Table 2. It is noted that the system is stable at all condition
TABLE 2: L AND ZTH INDICATORS AT CONSTANT P, Q AND POWER FACTOR WITH CHANGES AT BUS 4
Constant active power P
Constant reactive power Q
Constant power factor (pf)
Q
Lindicator
ZTh indicator
P
Lindicator
ZTh indicator
P
Q
Lindicator
ZTh indicator
buses
buses
buses
buses
buses
buses
4
5
6
4
5
6
4
5
6
4
5
6
4
5
6
4
5
6
0.8
0.108
0.075
0.8
0.087
0.099
0.8
0.496
0.055
0.107
1
0.138
0.088
1
0.077
0.091
1
0.619
0.064
0.138
1.2
0.168
0.108
1.2
0.066
0.082
1.2
0.744
0.073
0.17
1.4
0.198
0.123
1.4
0.055
0.073
1.4
0.868
0.082
0.204
1.6
0.228
0.142
1.6
0.044
0.063
1.6
0.991
0.091
0.24
1.8
0.261
0.169
1.8
0.033
0.055
1.8
1.11
0.099
0.278
2
0.295
0.209
2
0.021
0.047
2
1.24
0.109
0.319

Change in bus 5
The effect of load change on bus 5 at constant P, Q and power factor on the stability are studied and shown in Table 3. It is noted that the system is stable at all condition of studies.
TABLE 3: L AND ZTH INDICATORS AT CONSTANT P, Q AND POWER FACTOR WITH CHANGES AT BUS 5
Constant active power P
Constant reactive power Q
Constant power factor (pf)
Q
Lindicator
ZTh indicator
P
Lindicator
ZTh indicator
P
Q
Lindicator
ZTh indicator
buses
buses
buses
buses
buses
Buses
4
5
6
4
5
6
4
5
6
4
5
6
4
5
6
4
5
6
0.8
0.13
0.086
0.8
0.108
0.112
0.8
0.496
0.074
0.121
1
0.162
0.102
1
0.097
0.097
1
0.619
0.084
0.156
1.2
0.195
0.126
1.2
0.086
0.088
1.2
0.744
0.094
0.193
1.4
0.228
0.143
1.4
0.074
0.079
1.4
0.868
0.103
0.232
1.6
0.263
0.167
1.6
0.062
0.068
1.6
0.991
0.112
0.274
1.8
0.302
0.199
1.8
0.05
0.059
1.8
1.11
0.121
0.319
2
0.349
0.249
2
0.037
0.05
2
1.24
0.131
0.368

Change in Bus 6

The effect of load change on bus 6 at constant P,Q and power factor on the stability are studied and shown in Table 4. It is noted that the system is stable at all condition of studied
TABLE 4: L AND ZTH INDICATORS AT CONSTANT P, Q AND POWER FACTOR WITH CHANGES AT BUS 6
Constant active power P
Constant reactive power Q
Constant power factor (pf)
Q
Lindicator
ZTh indicator
P
Lindicator
ZTh indicator
P
Q
Lindicator
ZTh indicator
buses
buses
buses
Buses
buses
Buses
4
5
6
4
5
6
4
5
6
4
5
6
4
5
6
4
5
6
0.8
0.108
0.09
0.8
0.091
0.133
0.8
0.496
0.061
0.123
1
0.135
0.106
1
0.082
0.096
1
0.619
0.07
0.157
1.2
0.161
0.131
1.2
0.072
0.089
1.2
0.744
0.078
0.193
1.4
0.189
0.15
1.4
0.062
0.079
1.4
0.868
0.086
0.232
1.6
0.222
0.174
1.6
0.051
0.071
1.6
0.991
0.094
0.273
1.8
0.264
0.209
1.8
0.04
0.064
1.8
1.11
0.101
0.316
2
0.343
0.262
2
0.029
0.055
2
1.24
0.109
0.363
It is noted that from Table 2,3, and 4 the changes in active and reactive power at any conditions the voltage stability indicators Lindicator and ZTh indicator are increase and the stability of the system are decrease.


CONCLUSION
A method for computation static voltage collapse proximity indicator VCPI, FVSI and ZTh indicator are presented. Some case studies conducted on typical power networks have also been presented. These networks represented by two buses connected by transmission line and IEEE 6buses system. This paper presented a new formula to get change in voltage and its angles based on the change on active and reactive powers.
This formula is used to calculate Lindex, FVSI index and ZTh index directly without need to compute load flow in each time. Therefore, this formula depends only to compute base case load flow for one time to find the initial steady state condition. This study is important to prevent a possible voltage collapse with simple calculations of voltage stability indicator.
REFERENCES
[1] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. Van Cutsem and V. Vittal, "Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions," IEEE Transactions on Power Systems, vol. 19, no. 3, pp. 13871401, 2004. [2] V. Ajjarapu et al. The continuation power flowA tool for steady state voltage stability analysis, IEEE Trans Power Syst, 1992 [3] Prabha Kundur, Overview on Definition and Classification of Power System Stability, on Beharfof IEEE/CIGRE Joint Task Force on Stahilia Terms and Definition. [4] R.A. Shlueter, AVoltage Stability Security Assessment Method, IEEE Trans. Power System 13 (4), 1998, 14231438. [5] F.Gubina, A method for local Voltage Instability Assessment, University of Ljubljana. [6] S.M. Abdel Maksoud, Voltage stability Assessment with Particular References to optimal Operation of Power System, Ph.D. Thesis, Zagazig University, 2002. [7] D.Thukaram, K. Parthasarathy, H.P. khincha, A. Narendranath Udupa, Bansila, voltage Stability Improvement: Case Studies of Indian Power Networks, Elec. Power Syst. Res. 44, 1998, 3544. [8] Ismail Musirin, Titik Khawa Abdul Rahman, Member IEEE, On lone Voltage Stability bases Contingency Ranking using Fast Voltage Stability Index, IEEE, 2002. [9] D. Bansilal, Thukaram, K. Parthasarathy, Optimal Reactive Power Dispatch for voltage Stability Improvement, Elec. Power Energy System 18(7), 1996, 461468. [10] M.Z. ElSadek, Power System Voltage Stability and Power Quality, Book, Assiut Uinversity, Copy Right, 2002.
Generator 
P (Mw) 
Q (Mvar) 
1 
107.9 
16 
2 
60 
74.4 
3 
60 
89.6 
[11] M. A. Kabir, M. M. H. Sajeeb, M. N. Islam and A. H. Chowdhury, "Frequency transient analysis of countrywide blackout of Bangladesh Power System on 1st November 2014," in 2015 International Conference on Advances in Electrical Engineering (ICAEE), 2015
[12] P. T. Son and N. Voropai, "The Major Outage in South Vietnam in 2013: The Nature of Blackout, Security Measures and Strategy of National Power System Modernization," in International Conference on Problem of Critical Structures, 2015.
load 
P (Mw) 
Q (Mvar) 
4 
70 
70 
5 
70 
70 
6 
70 
70 
[13] O. P. Veloza and F. Santamaria, "Analysis of major blackouts from 2003 to 2015: Classification of incidents and review of main causes," The Electricity Journal, vol. 29, no. 7, pp. 4249, 2016
[14] H. Haes Alhelou, M. HamedaniGolshan, T. Njenda and. P. Siano, "A Survey on Power System Blackout and Cascading Events: Research Motivations and Challenges," Energies, vol. 4, no. 682, p. 12, 2019. [15] S. D. Anagnostatos, C. D. Halevidis, A. D. Polykrati, P. D. Bourkas and. C. G. Karagiannopoulos, "Examination of the 2006blackout in Kefallonia Island, Greece," International Journal of Electrical Power & Energy Systems, vol. 49, pp. 122127, 2013
[16] S. Ratra, R. Tiwari and K. Niazi, "Voltage stability assessment in power systems using line voltage stability index," Computers and Electrical Engineering, vol. Available: https://doi.org/10.1016/j.compeleceng.2017.12.046, 2018. [17] S. Yari and H. Khoshkhoo, "Assessment of Line Stability Indices In Detection of Voltage Stability Status," in IEEE International Conference on Environment and Electrical Engineering and IEEE Industrial and Commercial Power Systems Europe (EEEIC / I&CPS Europe), Milan, 2017. [18] Musirin and T. Abdul Rahman, "Novel fast voltage stability index (FVSI) for voltage stability analysis in power transmission system," in Student Conference on Research and Development, SCOReD, Shah Alam, Malaysia, 2002. [19] Chakrabarti and S. Halder, Power System Analysis: Operation and Control, 3rd ., New Delhi: PHI Learning Private Ltd, 2010. [20] S. Sahu, S. Reddy and S. Kumar, "New Voltage Stability Index (NVSI) for Voltage Stability Analysis in Power System," International Journal of Electrical and Electronics Engineering Research (IJEEER), vol. 2, pp. 1320, 2012. [21] M. Haruna, "An Overview on Voltage Stability Indices as Indicators of Voltage Stability for Networks with Distributed Generations Penetration," International Journal of Science, Technology and Society, vol. 3, no. 4, pp. 214219, 2015. [22] U. C. Ogbuefi, C. L. Ugwu and O. N. O., "Analysis of Nigeria Power System Voltage Collapse Incidences From 2000 To 2017," IOSR Journal of Electrical and Electronics Engineering (IOSRJEEE), vol. 13, no. 12, pp. 2834, 2018. [23] I. Samuel, J. Katende and F. Ibikunle, "Voltage Collapse and the Nigerian National Grid," Covenant University, Ota, Nigeria, 2012. [24] I. Samuel, "A New Voltage Stability Index For Predicting Voltage Collapse In Electrical Power System," Covenant University Repository, Ota, 2017. [25] I. Samuel, J. Katende, S. A. Daramola and A. Awelewa, "Review of System Collapse Incidences on the 330kV Nigerian National Grid," International Journal of Engineering Science Invention, vol. 3, no. 4, pp. 5559, 2014.
Bus 
V (kV) 
1 
o 241.5 0 
2 
o 241.6 3.1 
3 
246.1 3.3o 
4 
227.6 4.2o 
5 
o 226.7 5.3 
6 
o 231 5.9 
APPENDIX