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Method of Combined Static and Dynamic

Analysis of Voltage Collapse in Voltage

Stability AssessmentM.Hasani andM.Parniani

Abstract--Different analysis methods have been used for

voltage stability assessment. In comparison with static analysis

methods, little work has been done on dynamic analysis of large

interconnected power systems. Voltage instability can be studied

effectively with a combination of static approaches and time

simulations. This paper discusses voltage stability assessment

using mixed static and dynamic techniques. Using static methods,

a voltage stability based ranking is carried out to specify faint

buses, generators and links in power system. The system isanalyzed for most severe conditions. Then, time domain

simulation is performed for the conditions determined by voltage

instability ranking. The mixed approach benefits from

advantages of both static and dynamic analyses. The New

England (IEEE 39 bus) system was used as a test system.

Index Terms-- Voltage stability, Voltage collapse, Static

methods, Dynamic analysis, Contingency ranking

I. INTRODUCTION

N

im

recent years voltage stability is considered as an

portant concern to electric power industry. Voltageinstability and voltage collapse threaten power system

reliability and security. The voltage problems are often

associated with contingencies like unexpected line and

generator outages, insufficient local reactive power supply and

increased loading of transmission lines. Voltage collapse is

usually characterized by an initial slow and progressive

decrease and a final rapid decline in voltage magnitude at

different buses.

During the past years, there has been a continually

increasing attention to voltage stability assessment using

several analysis methods. Liu and Vu [1] presented a dynamic

description of voltage collapse by characterizing the voltage

stability regions in terms of the continuous tap changer model.

Lee et al [2] introduced a criterion for static voltage stability

enhancement and used accurate models for excitation systems,

tap changer and other equipment for analysis of dynamic

voltage stability. Voltage stability analysis using static and

M.Hasani is with the Department of Electrical Engineering, Sharif

University of Technology, Tehran, Iran (e-mail: hasani@ee.sharif.edu)

M.Parniani is an assistant professor in Department of Electrical

Engineering, Sharif University of Technology, Tehran, Iran (e-mail:

parniani@sharif.edu)

dynamic methods in small radial network was performed by

Morison et al [3]. Some advantages of dynamic simulation of

this phenomenon were shown by Deuse et al [4]. Taylor [5]

and Kundur [6] proposed different static methods and

dynamic simulation with appropriate models for voltage

stability assessments.

Although different approaches have been proposed and

employed for voltage collapse analysis till now, few literature

have dealt with dynamics of this phenomenon in largeinterconnected power systems. Most of the methods that are

applied to these networks are of static type. Little work has

been published on dynamic voltage stability analysis of these

systems, and the differences between the results of two

approaches have been rarely analyzed. The reason elaborate

modeling requirements and time-consuming computations of

dynamic simulation. This paper investigates the discrepancies

between static and dynamic techniques and combines these

techniques to exploit the advantages of both approaches.

In this paper, the New England (IEEE 39 bus) power system

shown in Fig.1 is used as the test system. First, a severity

ranking is carried out on the study system to specify faint buses, generators and links in terms of voltage instability.

Based on this ranking, the most severe conditions for

generator and branch outages and load increment are defined.

For each of these contingencies static methods are employed

to examine the network conditions and stability margins with

relevant curves and factors. Then, time domain simulation is

performed to scrutinize the same conditions. Results obtained

from these methods will be compared with each other. Then

capabilities and limitations of various methods are discussed.

II. VOLTAGE STABILITY ANALYSIS METHODS

A. Static Analysis

Many aspects of voltage stability problems can be

effectively analyzed by using static methods. These methods

examine the viability of the equilibrium point represented by a

specified operating condition of the power system. Static

I

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Fig.1 Schematic of New England (IEEE 39 bus) power system

approaches like sensitivity analysis, modal analysis and P-V

and Q-V methods for voltage stability assessment use a

system condition or snapshot for voltage stability evaluation.

They usually solve power flow equations of the network with

specific load increments until the point of voltage collapse is

reached. These techniques allow examination of a wide rangeof system conditions and can provide much insight into the

nature of this phenomenon by computation of the contributing

factors.

Sensitivity Analysis: This method calculates the relationship

between voltage changes and reactive power changes at

different buses using reduced Jacobian matrix [6]. Positive

sensitivities represent stable operation and as stability

decreases, the magnitude of the sensitivity increases becoming

infinite at the maximum loadability limit.

Modal Analysis: This analysis approach has the added

advantage that it provides information regarding the

mechanism of instability. Voltage stability characteristics ofthe system can be identified by computing the eigenvalues and

eigenvectors of the reduced Jacobian matrix [6]. Positive

eigenvalues represent voltage stability of system and the

smaller the magnitude, the closer the relevant modal voltage is

to being unstable. The magnitude of the eigenvalues can

provide a relative measure of the proximity to instability.

In this paper, bus and branch and generator participation

factors defined in [6] are widely used for identifying

appropriate measures to relieve voltage stability problems and

for contingency selections.

P-V Curve Method: These curves are produced by running a

series of load flow cases and relate bus voltages to load within

a special region. This methodology has the benefit of

providing an indication of proximity to voltage collapse

throughout the range of load levels. With power transfer

increase in a special region, its voltage profile will becomelower until a point of collapse is reached.

Q-V Curve Method: These curves are also produced by

running a series of load flow cases. They show the necessary

amount of reactive power to achieve a specified voltage level.

This method was developed due to difficulties in power flow

program convergence of stressed cases close to the maximum

loadability limit. To draw the Q-V Curves stretching from

unstable to stable region, special techniques are used to

achieve convergence [5]. Examples are using voltage sensitive

loads; artificially increasing reactive power sources limits and

fixing voltages at critical buses.

B. Dynamic Simulation

Time domain simulation with appropriate models for

devices clarifies this phenomenon more precisely. It shows the

time events and their chronology leading the system to final

phases of voltage collapse. The computer program solves the

differential-algebraic equations describing the power system.

It has the features of modeling important dynamics that are

influential in voltage instability such as dynamic load models,

OLTC, excitation system limiters and various other controllers

in the system. Dynamic analysis is useful for detailed study of

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specific voltage collapse situations and coordination of

protection and time dependent action of controls.

In recent years, capabilities of midterm and long-term time

domain simulation programs have been greatly improved. Full

dynamic simulation considering transient stability models for

these purposes seems to be time consuming. New programs

usually employ fast dynamic simulation techniques, which

have a good compromise between speed and accuracy [7]. The

DigSILENT software, which is suitable for long-termdynamic assessment, is used in the simulation presented in this

paper.

III. VOLTAGE STABILITY IN TEST SYSTEM

A. Modeling for Static and Dynamic Simulation

We will present the idea of mixed static and dynamic

analyses through an example. Then, we make general

conclusions and summarize the proposed approach. For static

power flow simulation, we followed the general methodology

used for planning and operation decision making. The loads,

which were located at the secondary of OLTC transformers,

were represented as voltage sensitive (both in active and

reactive power). Reference [5] describes the general

considerations for static analysis.

For dynamic simulation, we added OLTC transformer

models. Transformers were initialized at their neutral tap and

had 1% tap size (Except for T11-12 which was 2.5%). Other

data for tap changing transformers were 10% tap range with

20 steps of 1 or 2.5% and initial time delay was 5 second.

Based on typical measurements and report of similar

networks, we assumed active load to be 20% resistance and

80% dynamic (voltage and time dependent). Load represented

in bus 12 -which is of concern according to the followingstudies- has active and reactive power voltage dependence of

8.1= and 2.1= and time constants for active and reactive

power voltage dependence were 20 and 10 second. The loads

were represented on the low voltage side of the OLTC

transformers.

The overexcitation limiter was modeled as a part of AC2A

standard type of AVR. Armature current limit may be

encountered, requiring either severe reduction in field

current/reactive power or reduction in active power. Armature

current limit was modeled separately in simulations.

B. Contingency Ranking

The base case condition in IEEE 39 bus was used for

contingency selection. It is noteworthy that this contingency

selection is based on voltage instability in power system and is

different from conventional contingency ranking used for

system security and reliability studies. All of static analysis

methods introduced in part II were used to select the most

severe conditions corresponding to line, transformer and

generator outages and load increment in different buses.

The relationship between voltage and reactive power

changes was assessed for all of the buses using sensitivity

analysis. The results identified buses 12 and 2 as the most and

least sensitive buses with sensitivity factors 0.0334 and 0.0074

%/MVAR respectively. Modal analysis shows that the least

eigenvalues of jacobian matrix in the network has the

magnitude of 9.5609, which has sufficient margin to

instability. The sizes of bus, branch and generator

participation factors corresponding to this mode were

calculated. Bus 12 indicated the most effectiveness of

remedial actions in stabilizing that mode. Line8-9 (linebetween buses 8 & 9) consumes the most reactive power in

response to an incremental change in reactive load at bus 12,

and the generator G3 supplies the most reactive power in such

situation. The P-V and Q-V methods confirm these results.

Based on these observations, three scenarios were selected for

the rest of analyses:

1. Line 8-9 outage

2. Generator G3 outage

3. Load increment at bus 12

IV. STATIC ANALYSIS OF VOLTAGE STABILITY

For voltage stability assessment with static methods all three

cases defined in the previous section were analyzed. Fig.2

shows Q-V sensitivity results for different buses in network

after transmission line outage. Bus 12 shows to be the most

sensitive bus in response to reactive power changes. Using

modal analysis, different participation factors were calculated

for the lowest eigenvalue with magnitude of 6.633. Table.1

shows the bus participation factors indicating nearly the same

result as sensitivity analysis. Branches (line and transformers)

participation factors are shown in Table.2 and generator

participation factors are presented in Table.3.

Figures.3 and 4 show the P-V and Q-V curves for the most

and least sensitive buses in voltage stability assessment. In Q-

V curve method, bus 12 has the least stable condition with 585

MVAR reactive power margin to instability and bus 19 has

the greatest one. P-V curve method identified bus 7 as the

weakest bus in terms of voltage stability.

Similar analyses were performed for two other scenarios

defined in contingency selections. Approximately the same

results were obtained for these conditions. For instance, all

cases have sufficient reactive power margin, and hence are

considered as voltage stable. Results of these stable conditions

are directly comparable with the steady state values at the end

of stable dynamic simulation. Detailed results of the second

and third scenarios are not shown here for the sake of brevity.In the load increment scenario, active and reactive powers

were increased with a fixed pattern until the power flow

program did not converge. Then the network was analyzed in

the situation just before divergence. The critical load at bus 12

in this condition was 44 MW and 420 MVAR as compared to

the initial loading of 8.5 MW and 88 MVAR.

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Fig. 2. Q-V sensitivity results for different buses after line 8-9 outage

Fig. 3. P-V curves for the most and least stable buses after line 8-9 outage

Fig. 4. Q-V curves for the most and least stable buses after line 8-9 outage

TABLE.1

BUS PARTICIPATION FACTORS CORRESPONDING TO THE LEAST STABLE MODE

IN FIRST CONTINGENCY

Bus

Par.

Factor

Bus

No

Bus

Par.

Factor

Bus

No

0.0373170.084812

0.0261270.08228

0.0254160.07987

0.0228240.073840.0149210.06925

0.0126260.06773

0.0051230.06286

0.0047280.062514

0.0047220.057311

0.0038190.056813

0.0022290.051918

0.0013200.048810

0.0011250.039615

TABLE.2

BRANCH PARTICIPATION FACTORS IN FIRST CONTINGENCY

Branch

Par.Factor

Branch

Name

Branch

Par.Factor

Branch

Name

Branch

Par.Factor

Branch

Name

0.05245-60.132526-291T3

0.0483T90.138-50.6813T2

0.047313-100.1269T40.262716-19

0.03667-80.121614-40.2284T6

0.0337T13-120.1199T70.227227-26

0.030328-290.108128-260.191521-22

0.026724-160.097211-60.186226-25

0.026T11-120.0873T50.164315-16

0.02525-20.087121-160.16016-7

0.0225T10.08183-40.147324-23

0.018823-220.0810-110.141817-27

0.0145T80.0814-130.141218-3

0.0143T20-190.070616-170.140714-15

0.01191-390.0554-50.133318-17

TABLE.3

GENERATOR PARTICIPATION FACTORS IN FIRST CONTINGENCY

Gen Par.

Factor

Gen No

13

0.86372

0.48126

0.3994

0.2939

0.27367

0.19555

0.14268

0.12191

0.034710

V. DYNAMIC SIMULATION

Figs. 5-7 show the results of time domain simulation for

three cases identified in the previous part. In this figures bus

12 is the load side of the OLTC transformer and bus 11 and 13

are high voltage side of relevant transformers. In all cases, the

simulated voltages have reached their stable steady state

conditions. However, in Fig.7 which has the load increment of

36 MW and 330 MVAR (to receive the critical load defined in

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the previous section) stability can not be achieved because of

considerable decline in load side voltage which is likely to

activate protection devices.

Acceptability of stable conditions acquired by dynamic

simulation can be judged by the post-disturbance voltage

levels, the remaining reactive power reserves at generating

plants and the time available for operator action. We used

ANSI standard C84.1-1989 for reliability criteria [8], which

suggests 92% voltage for consumer service in firstcontingencies. Also, undervoltage load shedding may be

devised using dynamic simulation results in order to enhance

system reliability. (This was not considered here).

In the first and second scenarios which were stable in both

static and dynamic analyses, the result of static methods can

be directly compared with the results at the steady state of

dynamic simulation. Results for voltage profiles in different

buses and generators reactive power margin are presented in

Table 4 and 5. According to the results of Table 5, the static

method seems to be more conservative. Further investigations

not presented in this paper - are carried out with the aid of

dynamic simulation, to examine the effects of generator

armature and field current limiters, OLTC control parameters,

and load characteristics.

Fig. 5. Voltage for different buses after line 8-9 outage

Fig. 6. Voltage for different buses after generator G3 outage

Fig. 7. Voltage for different buses after load increment at bus 12

TABLE.4

COMPARISON OF VOLTAGES AT LOAD BUSES CALCULATED BY STATIC AND

DYNAMIC METHODS

Voltage

(Dynamic)

Voltage

(Static)

Bus

No

0.990.9730.950.954

0.930.947

0.930.938

0.990.9512

0.960.9815

0.981.0016

0.990.9918

0.920.9820

0.991.0121

1.011.0323

0.991.0124

1.041.0425

1.021.0226

1.001.0027

1.031.0328

1.031.0429

1.001.0031

TABLE.5

COMPARISON OF REACTIVE POWER RESERVES CALCULATED BY STATIC AND

DYNAMIC METHODS

MVAR Reactive

Margin (Dynamic)

MVAR Reactive

Margin (Static)

Gen No

12191

30372

52753

39294

84795

37466

55627

35498

20159

8011210

VI. GENERAL CONSIDERATIONS

Although the voltage instability is a dynamic phenomenon,

different static analysis methods have been proposed and

widely used in different networks. Static methods are

generally easier to implement and require less computing

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time. Some of these methods, e.g. P-V and Q-V curves,

provide valuable information about stability margin. Some

others like modal analysis are very useful to identify the

pattern of voltage instability and thus to devise appropriate

remedial actions. In contrast, dynamic simulation yields more

accurate results; and requires more elaborate models and

computing time. In time domain simulation all controllers,

protective relays, dynamic model of loads, tap-changer

controller and etc can be taken into consideration.And the main limitations in this analysis method are:

1. With dynamic simulation, stability margin for busvoltage is not directly computed.

2. In interconnected networks, because of existing severalcontrollers and protection devices with overlapped time

domain actions, distinction of main factor affecting

instability might be a problem.

Due to limitations of time domain simulation in analyzing

voltage stability in interconnected power systems, a new

method of combined static and dynamic analysis of voltage

collapse is introduced in this paper. In this method, first, a

contingency ranking for voltage stability is carried out on the

study system. Based on this ranking, the most severe

conditions including generators/lines outages and load

changes are identified. For each contingency condition, static

methods are employed again to examine the stability

conditions. Time domain simulation -with more detail models

for these parts- is then performed for the selected contingency

cases. With the result of dynamic analysis, appropriate

controllers and protection devices can be selected for the

system to overcome voltage instability. The method

considerably reduces computations of dynamic analysis with

no complex and detail models required for all equipment.

VII. CONCLUSION

In this paper, a new approach using combination of static

and dynamic methods was proposed for voltage stability

assessment. Using static methods, a voltage stability ranking

was performed to define faint buses, generators and links in

terms of voltage stability. Then, these parts are modeled with

more detail and dynamic analysis was used for most severe

conditions. Results from different static approaches were

compared with more accurate time domain simulations.

Although static methods based on power flow analysis is

suitable for screening, final decisions involving several

considerations both in planning and operation should be

confirmed by more accurate time domain simulation in which

different characteristics of multiple controllers, protectionrelays and coordination of them are taken into account.

VIII. REFERENCES

[1] C.C.Liu and K.T.Vu, "Analysis of tap-changer dynamic andconstruction of voltage stability regions," IEEE Trans. on Circuit and

Systems, Vol.36, No.4, pp.575-590, Apr 1989

[2] B.H.Lee and K.Y.Lee, "Dynamic and static voltage stabilityenhancement of power systems," IEEE Trans. on Power Systems, Vol.8,

pp.231-238, Feb. 1993

[3] G.K.Morison, B.Gao and P.Kundur, "Voltage Stability analysisusing static and dynamic approaches," IEEE Trans. on Power Systems,

Vol. PWRS8, No.3, pp.1159-1171, Aug.1993

[4] J.Deuse and M.Stubbe, "Dynamic simulation of voltagecollapses," IEEE Trans. on Power Systems, Vol.8 pp.894-900, Aug.1993

[5] C.W.Taylor, Power System Voltage Stability, New York:McGraw-Hill, 1994

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

[7] T.Van Cutsem, and C.Vournas, Voltage Stability of ElectricPower Systems, Kluwer Academic Publishers, 1998

[8] American National Standard Institute, "American NationalStandard for Electric Power Systems and Equipment Voltage Ratings,"

ANSI C84.1-1989.

Masoud Hasani was born in July 30, 1978. He received the B.Sc. and M.Sc.

degrees in Electrical Power Engineering from Sharif University of

Technology (SUT), Tehran, Iran, in 2001 and 2004 respectively.

He is currently a Ph.D. student in Electrical Power Engineering at Sharif

University of Technology.

Mostafa Parniani obtained his B.Sc. and M.Sc. degrees in Electrical Power

Engineering from Tehran Polytechnic and Sharif University of Technology

(SUT), in 1987 and 1990 respectively; and Ph.D. in Electrical Engineering

from the University of Toronto in 1995. Since then, he has been with the

Electrical Engineering Department of SUT as an assistant professor. He has

worked with Ghods Niroo Consulting Engineers Co., Electric Power ResearchCenter, and Niroo Research Institute. He has also been a member of IEEE

Task Force on Slow Transients, as well as national committees in his field. His

areas of interest are power system control and dynamics, reactive power

control, and applications of power electronics in power systems.

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