static and dynamic approaches for analyzing voltage stability

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2006; 16:277–296Published online 2 February 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/etep.87

Static and dynamic approaches for analyzing voltage stability

X. Lei*,y and D. Retzmann

Bijing XuJi Co. Xinxi Road 5, Shagdi, Beijing, 100085, China

SUMMARY

Voltage stability analysis always involves a large number of devices with various dynamic characteristics in powersystem. For properly capturing static and dynamic behaviors providing insights into characteristics of the voltagestability of the system, different system models are established for different time-scale studies. In this paper,relevant system models are discussed taking into account load characteristics. Based on these models, differentapproaches such as static analysis, time-domain simulation and quasi-steady state (QSS) approximation areimplemented into a simulation program, which is used as a comprehensive tool for analyzing voltage stability. Bycomplementary use of these approaches with the simulation program, voltage collapse stresses of the systemunder a critical operating condition can be analysed in steady state, and impacts of special events on the voltagestability can also be simulated with time-domain simulations. For time-domain simulations, a full system modeland a QSS approximation can be adopted for capturing different forms of instability mechanisms. A test system isused to demonstrate the feasibility and effectiveness of the approaches implemented. Copyright # 2006 JohnWiley & Sons, Ltd.

key words: voltage stability; static analysis; short-term and long-term dynamics approximation; quasi-steadystate approximation (QSS); time-domain simulation and modelling of load characteristics

1. INTRODUCTION

Dynamic phenomena causing voltage instability, occurring in electric power system subjected to

strong load demands, lead to a progressive decrease or, sometimes, fast drop of the voltage magnitude

at one or more busses, resulting sometimes in network islanding, thus leading to local or global

blackout.

Since environmental and economic constraints limit the construction of new generation and

transmission systems, and power demands are predicted to increase, the voltage instability problem

appears to be more and more topical and concerns about voltages instability risks are also rapidly

growing on.

Typical slowness of the voltage instability processes suggests treating the problem as steady-state

one. However, dynamic aspects cannot be neglected in the framework for voltage collapse analysis and

accurate mathematical model of the network components is required, if large perturbations occur and

Copyright # 2006 John Wiley & Sons, Ltd.

*Correspondence to: Xianzhang Lei, Bijing XuJi Co. Xinxi Road 5, Shagdi, Beijing, 100085, China.yE-mail: [email protected]

short-medium term transients become topical. As a consequence, to accurately simulate the system

voltage dynamics, overall system should be modeled by a set of differential and algebraic equations, in

which load flow equations are included as constraints.

Some new tools especially for voltage stability analysis have been developed. While developing

news tools, it is possible to exploit and extend conventional tools into this area. In this paper, the

authors introduce a comprehensive simulation program, and attempt to use this program for analyzing

voltage stability both in static and dynamic manner. With the program, components, which can

significantly affect voltage stability, can be modeled in detail. This also includes induction motors,

load with different dynamics and control systems, even such as AVR or over-excitation limiters

(OXLs) on synchronous generators, etc. By establishing adequate system models, voltage collapse

stresses of the system under a critical operating condition can be analyzed in steady state, and impacts

of special events (e.g., faults, motor starting, transformer’s tap changing, load varying) on the voltage

stability can also be simulated with time-domain simulations. For time-domain simulations, a full

system model and a quasi steady-state (QSS) approximation can be adopted for capturing different

forms of instability mechanisms.

This paper is organized as follows: in Section 2, static and dynamic models of a power system are

discussed, while modeling characteristics of some load and generator that appropriately represent the

system for the subsequent voltage collapse studies are described in Section 3. In Section 4, a

comprehensive simulation program, where static and dynamic approaches for analyzing voltage

stability are implemented, is presented. With this program, case studies on a test power system are

carried out with static analysis, time-domain simulation and QSS analysis, respectively for demon-

strating the usability of the simulation program.

2. POWER SYSTEM MODELS

Voltage collapse studies and their related tools are typically based on the following general

mathematical description of the system consisting of a set of algebraic and differential equations [1]:

_x ¼ f x; y; zð Þ_z ¼ h x; y; zð Þ0 ¼ g x; y; zð Þ

ð1Þ

where x 2 Rm represents the short-term state variables corresponding to fast dynamic states of

generators, induction motor loads, FACTS and HVDC controllers, etc; y 2 Rk corresponds to the

algebraic variables, usually associated to the transmission system and steady-state element models,

such as voltage magnitudes and phases at nodes, some generating sources and loads in the network;

z 2 Rn represents the long-term dynamic state variables of slow acting devices including well known

devices such as under-load tap-change (ULTC) transformers, OXLs and secondary voltage controls (if

any), etc. The differential equations represent the dynamic behavior of the system, while algebraic

equations represent the interaction of dynamic elements.

2.1. Static analysis based on load flow with full system model

Conventional load flow calculation is normally used to evaluate operation at a specific load level

specified by a given set of load and generation, where only algebraic part of the power system model is

278 X. LEI AND D. RETZMANN

Copyright # 2006 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2006; 16:277–296

considered and dynamic characteristics of the system is completely ignored. Concerning voltage

stability phenomena, especially when the system operating under stressed conditions, some dynamic

elements in the system such as generators, induction motors, load consumed with certain dynamic

characteristics and related controllers (e.g., AVR or OXLs, etc.) can significantly impact on steady-

state operations of the system. In this point of view, dynamic characteristics of the system elements

must be taken into account during the load flow calculation, in order to achieve such a solution that

provide insights into characteristics of the voltage stability of the system. Thus, to calculate operating

points in respect to the generator output and load consumed, the full system model Equation (1) should

be adapted, where parameters being relevant for dynamic behaviors of the system are considered.

To solve the load flow with the full system model Equation (1), conventional methods of Newton

family can be used. The procedure is similar to the conventional load flow calculation, except that the

Jacobian matrix of the full system model is stated in the form of

J ¼

@f@x

@f@y

@f@z

@h@x

@h@y

@h@z

@g@x

@g@y

@g@z

26664

37775 ¼

fx fy fz

hx hy hz

gx gy gz

264

375 ð2Þ

where only gy represents the Jacobian matrix of the conventional load power equation, others describe

interactions among the dynamic and algebraic variables in the system, respectively. By ignoring very

slow dynamics with an assumption of zi ¼ constant ði ¼ 1; 2; . . . nÞ, the Jacobian matrix Equation (2)

can be simplified as

Js ¼fx fy

gx gy

� �ð3Þ

The application of the Schur formula to the matrix Js in Equation (3) with the assumption that fx is not

singular, leads to establish the relationship

det Jsð Þ ¼ det fxð Þ det gy � gx f�1x fy

� �¼ det fxð Þ det Jdð Þ ð4Þ

where Jd is defined as a dynamic load flow matrix given in the form of

Jd ¼ gy � gx f�1x fy ð5Þ

In comparison with conventional load flow, the main advantage offered by the use of the full system

model is that it ensures a link with system differential equations, thus permitting interaction dynamics

to be taken into account.

The most common use of load power in static voltage stability analysis is to provide the P-U and U-

Q curves at selected load buses [3,5,6]. P-U curve shows the relationship between the critical bus

voltage variations and the system load changes which indicates system loadability, while U-Q

characteristics indicate reactive power margin under a given voltage condition and implies the

weak bus which needs more reactive power support. However, one of the major limitations of static

analysis is that it can not properly capture the instability caused by dynamic reasons.

STATIC AND DYNAMIC APPROACHES 279


2.2. Dynamic analysis based on time-domain simulation

Generally, dynamic analysis based on the time-domain simulation using the full system model

Equation (1) can capture all forms of instability mechanisms and thus to replicate the entire voltage

instability process with high accuracy. With further development of modern computer technology, it

has become quite feasible to deal with large-scale power system with a full system model Equation (1).

However, in engineering application, this time-domain simulation is inevitable very CPU-time

consuming when simulating a huge system with thousands of buses. Usually, a way to escape from

this situation without loosing calculation accuracy is to use multi-time scale simulation [3] with the

full system model Equation (1), where the simulation step can be changed while ensuring the

integration error is controlled in a reasonable limit.

For purpose of understanding voltage instability mechanisms, as well as devising faster analysis

methods, it is, however, advantageous to reduce the full system model for exploiting the time

separation, which exists between the short- and long-term phenomena [2]. A basic idea of this concept

consists in assuming that fast subsystem is infinite fast and can be replaced by its equilibrium

equations when dealing with the slow subsystem. Conversely, fast dynamics can be approximated by

considering the slow variables as practically constant during fast transients. This leads to a

significantly simple analysis of both subsystems.

The short-term time scale is the time scale of synchronous generators and their regulators (AVRs

and governors), induction motors, HVDC and FACTS devices. The corresponding dynamics last

typically for several seconds following a disturbance. For a short-term approximation with an

assumption of the slow variables remaining constant, the full system model Equation (1) is reduced

to short-term dynamic model given the form of

_x ¼ f x; y; zð Þ0 ¼ g x; y; zð Þ

ð6Þ

The long-term time scale is to deal with slow subsystems regarding phenomena, controllers and

protecting devices, which act over several minutes following a disturbance. It is also assumed that

components of slow variables are so designed to act after the short-term transients have died out, to

avoid unnecessary actions or even unstable interactions with short-term dynamics. This indicates that

the following equilibrium is already established when simulating long-term dynamics of the system

0 ¼ f x; y; zð Þ ð7Þ

In particular, the QSS approximation is used for dealing with long-term dynamics dominated by a

slow subsystem. Replacing the full system model Equation (1) with the condition described in

Equation (7), the QSS can be then stated in the form of

_z ¼ h x; y; zð Þ0 ¼ f x; y; zð Þ0 ¼ g x; y; zð Þ

ð8Þ

The QSS method gives a series equilibrium of the fast subsystem driven by the slow subsystem. This

replacement greatly decreases the order of the system equations. Furthermore, because the remained



differential equations are only those relevant to slow subsystem, longer integration step can be used.

Depending on the adopted model, integration step can be up to 10s, while keeping the accuracy of the

simulation [3]. As a result, QSS approximation is dramatically fast compared with full model

simulation. For instance, on the 1200-bus system described in Reference [4], the simulation of system

behavior over 15 minutes following a major contingency takes only about 15 seconds.

3. MODELING OF COMPONENT CHARACTERISTICS

The term of load model refers to the active and reactive load equations used to represent the

characteristics of the load in voltage stability studies. Due to the fact that loads are generally voltage

dependent, load modeling is a critical aspect of voltage stability analysis. This is much unlike in the

angle stability studies, where loads are commonly modeled as constant power and constant

impedance. For capturing the characteristics of different loads, various approaches for load modeling

have been addressed in literature. A comprehensive summery can be found in References [7] and [8].

3.1. Static load models

Basically, there are two kinds of the load models: static model and dynamic model. A static model

expresses the active and reactive powers at any instant of time as functions of the bus voltage

magnitude and frequency at the same instant. Static load model is used both for essentially static load

components (e.g., resistive and lighting load), and as an approximation for dynamic load components

(e.g., motor-driven loads), and it includes constant impedance load model, constant current load

model, and constant power load model. Other two commonly used static load models that represent the

power relationship to voltage magnitude are as a polynomial equation and an exponential equation,

given in the following form respectively:

Exponential load model

P ¼ P0

U

U0

� ��

;Q ¼ Q0

U

U0

� ��

ð9Þ

Polynomial load model

P ¼ P0 a1

U

U0

� �2

þ a2

U

U0

� �þ a3

" #

Q ¼ Q0 a4

U

U0

� �2

þ a5

U

U0

� �þ a6

" # ð10Þ

where P and P0 are actual and initial active power; Q and Q0 are actual and initial reactive power and U

and U0 are actual and initial voltage; � and � indicate exponential factors; aiði ¼ 1; 2; . . . 6Þ stand for

multiplying factors. Discussions on the selection of the exponential and multiplying parameters can be

found in Reference [9]. Note that restoration dynamics of the load described in Equations (9) and (10)

can significantly influence the voltage recovery after a disturbance, and thus these dynamics shall be

considered for voltage stability studies too. A detailed description of this load restoration dynamics is

given in Reference [13].



3.2. Dynamic load models

A dynamic load model expresses, on the other hand, the active and reactive powers at any instant of

time as functions of voltage magnitude and frequency at past instants of time and, usually,

including the present instant. Difference or differential equations can be used to represent such

models. As a typical dynamic load, models of the electrical machines have been well established

and widely implemented into simulation programs as standard models for voltage collapse

analysis. In addition to that, several References [10–12] have recently proposed simplified dynamic

load models intending to capture the essential behavior of loads with different transient and steady-

state characteristics, such as thermostatically-controlled loads and (with considerable care) some

motor-driven loads. While the form in which these models are presented appears quite different, it

can be shown that all, except for Reference [12], can be generalized to the block diagram shown in

Figure 1. The only difference in the model proposed in Reference [12] is that the final summation is

replaced by a multiplication.

In this model, the steady-state load-voltage characteristic is represented by the function g(U), which

can be either an exponential or polynomial function in U. For a thermostatic load, this would normally

be represented as constant power. The transient characteristic is represented by the function f(U),

which is often constant impedance. Frequency sensitivity can also be included in both of these

functions. Note that in this model, Pnom, Ps, and Pt indicate the load power consumed at nominal

conditions, i.e., 1.0 p.u. voltage and the rated frequency, the active power in a steady-state and at a

transient condition, respectively.

3.3. Modeling of ULTC

The ULTC tends to bring the load voltages back to their base point when voltage dropping due

to lack of power infed. Since system loads are generally voltage dependent, a voltage recovery

will generally result in an increase of power consumed toward their pre-contingency level.

This will cause further stress on the system and can eventually lead to a voltage collapse

condition. Thus, proper modeling of the ULTC is one of the basic requirements for voltage

collapse studies.

One modeling technique used to capture the effect of the finite range of ULTC taps is to enforce

constant MVA behavior for a range of voltages around nominal. Outside of this range load voltage

sensitivity is included. As with the generation protection, the ULTC must be modeled with their actual

T

1

(U)fPnom

sP

nomP

tP

+

+-+

U

nomP

(U)gPnom

Figure 1. Simplified dynamic load model.



tap range and size, voltage controls and dead-bands, and settings for tap delay and tap motion time. So

that, it can be modeled as

rkþ1 ¼rk þ�r if U2 > U0

2 þ d and rk < rmax and delay > Tk

rk ��r if U2 < U02 � d and rk > rmin and delay > Tk

rk else

8>><>>: ð11Þ

where rk is the ratio of the step-down transformer at tap position k, rmax and rmin are the upper and the

lower limits of the ratio, respectively, U2 is the voltage magnitude at the secondary bus of the

transformer, d is the dead-band to prevent an ULTC from acting too frequently, �r is the size of each

tap step and Tk is delay time of the tap change action when the voltage is out of dead-band. Tk is not

necessarily constant and can be variable with the voltage so that an ULTC can have an inverse-time

delay. Obviously, this model will be implemented as a discrete model for voltage collapse studies.

3.4. Modeling of AVR and OXL

At a post-contingency (after tripping a faulted transmission line) steady-state operating condition,

where the generator field current is above its rated value, the reactive output may be above the value

obtained from the generator reactive capability curve. After several minutes of operation at this

condition, the OXL control resets the excitation voltage to its rated value. This typically causes a small

oscillation, and brings the field current down to rated. The reset action will cause a reduction in the

reactive output and terminal voltage of the machine. Under a sufficiently stressed state, the loss of the

transmission line and subsequent OXL action can cause other machines to reach excitation limits. This

action, along with other control actions and the characteristics of the system loads, can drive the

system into a voltage collapse. Thus, field current limit enforced by the generator OXL control

function is of particular concern when studying voltage stability.

Many types of OXLs are encountered in practice for voltage stability studies. The model proposed

by Reference [3] is as an example shown in Figure 2. This model limits the field current Ifd under Ilimfd

by injecting a signal xoxl into the AVR main summing junction and has an inverse-time characteristic

of overload capability allowing smaller over-excitations to last longer.

0xt

0x t

limfdI

fdIoxlx

+ 1s

Ks

0

0x t ≥

0x t

-Kr

tx

RsT11

AK

refU

tUfdE

+_

Over excitation limiter

_

_ <

Figure 2. AVR and OXL model.



Since the action of OXLs may differ greatly from plant to plant, and if detailed models can not be

assembled and implemented, the model illustrated in Figure 2 can used as a general AVR and OXL

model for voltage stability studies.

4. A COMPREHENSIVE SIMULATION PROGRAM

Network torsion machine control (NETOMAC) is a simulation program used widely for the

simulation of electromechanical and electromagnetic transient phenomena, as well as long-term

time scale and steady-state behaviors of a power system [14,15]. Figure 2 shows the capabilities of the

program.

As shown in Figure 3, the load flow calculation mode is not only used as an independent function for

static analysis. It is also used as a precondition for the subsequent simulation by solving system

differential equations. This program uses a current iteration process with node admittance matrix for

determining the system operating point. Thus, it is robust in critical load flow calculations. Voltage

dependence of loads and actions of controllers as well as automatic setting of transformer tap changers

can be included in load flow calculations so that a true equilibrium can be obtained.

In time-domain, the program provides an instantaneous value mode similar to the EMTDC/EMTP

program and a stability mode like the PSS/E program. In stability mode, the networks are in the form

of complex impedance instead of differential equations. Machines are modeled by differential

equations. A number of standard models of machines and controllers such as synchronous machine,

induction motor, AVR, HVDC, and FACTS devices, etc., are implemented in a standard library and

can be used in simulation directly. Users can also define their own models for particular objectives

very conveniently, especially e.g. models of loads and network elements with voltage dependent

characteristics described in the previous section. In addition, the program provides also various time-

domain calculation modes used for voltage stability studies that are shaded in Figure 3, so that the

Simulation Models for System Components, Machines, Controllers and Control Units

Loadflow Initial Conditions Only Loadflow Loadflow Operating Point

Network in a-b-cAdmittances

differential equations

Single-line network complex admittancesalgebraic equations

System componentslinearization

Time range (ns~s)Instantaneous values

Time range (s~min)RMS values

Time range (min~h)RMS values

Frequencydomain analysis

Eigenvalueanalysis

Electromechanicalphenomena

for full system modelin eq (1) or

short-term dynamicmodel in eq. (6)

Electromechanicalphenomena

QSS analysisfor long-term dynamic

modelin eq. (7)

Loadflow for specialrequirements

Calculation withJacobian matrix

in eq. (2) orin eq. (3)

Small-signalcharacteristics

System oscillationand damping, Net

reduction,Controller layout

Electromagneticphenomena

for short-termdynamic

using modelin eq. (1) or in eq. (6)

Figure 3. General view of the capability of NETOMAC.



voltage collapse can be analyzed with a detailed model or a QSS approximation. In Figure 4,

procedures of the program with different methodologies for voltage stability studies are illustrated.

The program can provide block-oriented language to establish corresponding models of the

controllers. Using standard control blocks, e.g., PI, PID, etc., which are implemented in a library of

the program, controllers commonly used in power system such as exciters, governors can be easily

modeled. In addition to the standard blocks, mathematical or logical statements can also be used to

describe some special functions such as discrete actions (e.g., ULTCs). Using this language, load

characteristics, e.g., equations given in Equations (9,10) or even functions described in Figure 1 can be

implemented as a variable-power unit, where voltage (U ) and frequency (f ) are defined as inputs (or

any other system variables affecting outputs) and active and reactive power as outputs. By use of such

a variable-power unit, various load characteristics and even simplified FACTS controllers and

generators can be modeled. By use of such a variable-power unit, various load characteristics and

even simplified FACTS controllers and generators can be modeled such that these elements can be

involved in both of power flow calculations and time-domain simulations, where load voltage-

dependency, controller limits and discrete functions, etc., are taken into account. The implementation

of these special units in the program is illustrated in Figure 5.

Due to the modular structure with the complex of algorithms implemented, this program offers

many advantages for voltage stability studies that are summarized as follows:

* Establishing various models depending on the requirement of studies, including full systemmodel, short-term time scale model, and QSS approximation model;

Establishing system modelstaking into account e.g.: Load characteristics Controls e.g. AVR & OXL Discrete functions e.g. ULTC

Steady state calculationPre-contingency

P-U and U-Q curvesfor detecting voltage collapse

Voltage collapse atPcritical or Qcritical

Time domain calculationPre- and post-contingency

Full system model in eq. (1)

Short-termdynamics

modelin eq. (6)

Long-termdynamics

QSS modelin eq. (8)

z remainingconstant

equilibrium ofshort-term dynamics

Identifying voltage collapse andAssessing voltage security

.

.

.

Figure 4. Studies for voltage collapse with the program.

Load flowcalculation

U, f, etc.

P+jQ

Variablepower Block

Time-domainsimulation

U, f, etc.

P+jQ

Figure 5. Variable-power unit participating in load flow calculation and time-domain simulation.



* Modeling system components flexibly, including various generators, induction motors, and loadcharacteristics, devices with transient actions (e.g., exciters, governors and FACTS as wellHVDC), devices with slow actions (e.g., OXLs) and discrete actions (ULTCs);

* Static studies for determining P-U and Q-U curves with full system model or reduced model;* Time-domain simulations with full system model by changing integration’s steps or with QSS

approximation using electromechanical mode;* Time-domain simulation with short-term time-scale model using electromechanical or

electromagnetic mode;* Implementing voltage collapse indexes and further carrying out studies on voltage security

assessment flexibly;* Providing useful tools such as optimization and eigenvalue modes for voltage stability studies.

In the following section, case studies with different modes provided by the program are discussed.

5. CASE STUDIES

In this section, a test system shown in Figure 6 is used to illustrate the applications of NETOMAC in

voltage stability analysis in different time scales introduced in Section 2. The system represents a load

center supplied mostly by remote generators G1 and G2 and partly by a local generator G3. An ULTC

is equipped to automatically regulate the secondary voltage magnitude (Ubus11) of the step-down

transformer (T6) at Bus 11. Parameters of the system elements can be found in Reference [16].

5.1. Case study with static analysis

In this section, voltage characteristics in steady state are studied based on conventional P-U and U-Q

curves generated by a series of calculations of the system equilibrium. The case study is carried out

with two different models of L8, i.e., an exponential load model given in Equation (9) and a motor-load

shown in Figure 7. The goal of the study is to demonstrate impacts of the load voltage-dependency on

the voltage stability of the system. In this study, three sets of the �-� value in Equation (9) are used

�¼ �¼ 0 (constant power)

�¼ �¼ 1 (constant current)

�¼ �¼ 2 (constant impedance).

G1 T1 G3

G2

T5

T4

T3

T2

ULTC

1

Ubus9Ubus10

7

6

5

Ubus3

2

Ubus1T6

L8

L11

Ubus8

Ubus7

8

10

3

9

Figure 6. Test system.



As motor-load, a steady-state model is used, which is represented by the well-known equivalent

circuit of a motor with a constant mechanical load Tm as shown in Figure 7.

For simplicity, Bus 1 in the test system shown in Figure 6 is defined as a slack bus, while Bus 2 and

Bus 3 in the system serve as P-U buses. To consider impacts of the field current limit of G2 and G3

voltage behaviors of the system, an upper limit of the reactive power output for each generator is

defined. In this study, generator G2 and G3 are modeled with a variable-power unit, respectively, in

which the relationship among the active and reactive power as well as voltage magnitude is magnitude

is described according to real situations. Since here effects of the load models are concerned, the active

power output and the terminal voltage magnitude of G2 and G3 are considered as constant,

respectively.

To produce P-U and U-Q curves, series of load flow calculations are performed until each load flow

fails to converge. The process runs fully automatically with the program NETOMAC. Results shown

in Figures 8 and 9 demonstrate that the load voltage-dependency largely affects the system voltage

characteristics. Under low voltage conditions, the constant impedance-load provides a load relief,

which is favorable to a stable operation of the system, whereas the constant power-load always keeps

its load requirement, so making the system prone to instability. The most serious situation is with the

motor-load. When highly loaded, motor absorbs more reactive power under low voltage conditions

than constant power-load, thus, it causes firstly voltage collapse of the system.

Rs Xs

Xm

Xr

U I

Rr

P=Tm

Figure 7. Steady-state equivalent circuit of motor with constant mechanical torque.

Figure 8. P-U curves with different types of the load (L8) at Bus 8.



5.2. Case study with time-domain simulations

In this section, time-domain simulation is applied to analyze system dynamics following a loss of one

transmission line between Bus 6 and Bus 7 (without a fault). The initial operating condition is shown

in Table I.

Generators are modeled by Park equations with stator transient neglected. The model of the AVR

with the OXL used is shown in Figure 2 and the ULTC model applied is described in Equation (11),

respectively. Load L11 consists of a mixture with a half constant-current and a half constant-

impedance load for both active and reactive power. Load L8 is of two different types of the load: in

case 1, the type of L8 is the same as that of L11, while in case 2, L8 is of an equivalent induction motor

modeled as shown in Figure 7.

5.2.1. Case 1: Long-term instability. Simulation results are shown in Figures 10–13. In correspon-

dence with dynamic behaviors of the system, changes in ULTC ratios are given in Figure 14. The

evolution of the sequence of the events leading the instability is clearly reproduced.

The outage of one transmission line results in an increased reactive loss on the transmission line and

thus, largely reduces the transmission capability. As a consequence, voltages of the system drop. To

keep the terminal voltage magnitude constant, the AVR of G2 and G3 boost their field currents to

increase their reactive power outputs. With increasing the reactive power, the system becomes

transient stable and enters into a steady state by 40 seconds. During this time, the ULTC starts to act to

regulate voltage at Bus 11 (Ubus11) by lowering the tap ratio. After several tap changes, it succeeds in

Figure 9. U-Q curves with different types of the load (L8) at Bus 8.

Table I. An initial condition for time-domain simulations.

Bus P (MW) Q (MVar)

1 4085 12562 1734 7213 1154 10.908 3320 1030

11 3435 985



0.85

0.9

0.95

1

1.05

1.1

0 50 100 150 200 250

U (p.u.)

Time (sec)

Bus 11

Bus 7

Bus 3

Figure 10. Voltages at buses.

2.8

3

3.2

3.4

3.6

3.8

4

0 50 100 150 200 250

I field,G3 (p.u.)

Time(sec)

Figure 11. G3 field current.

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250

Q G3 (p.u.)

Time (sec)

Figure 12. G3 reactive power output.



bringing the voltage back into dead-band. Because of the limited transmission capability of the

transmission lines between Bus 6 and Bus 7, the reactive power needed for bringing Ubus11 back is

mainly contributed from G3 by overshooting the field current. At about 101 seconds after that, the

system enters into a steady state again. However, at this moment, the field current of G3 is still over its

limit which is not allowed for a long-term operation due to generator protections. Thus, after 80

seconds, the OXL is activated to force the field current back to the limit’s value. This leads to a

reduction of reactive power generation and subsequently a decrease of the system voltage. To avoid a

further fall of the system voltage, the ULTC acts again and tries to regulate Ubus11 back to its pre-

contingency level. However, without reactive support from generators, every step change has an

adverse effect, i.e., reduces the voltage. Therefore, voltages of the system continuously decrease until

the ULTC reaches its limit. In this case, the system voltage is already at a very low level that can not be

tolerant for a long time. To maintain a stable operation of the system, further measures must be taken

which will not be discussed in this paper. As additional information, behaviors of the rotor angle of the

generators G2 and G3 are also illustrated in Figure 13.

Figure 13. Rotor angle of G2 and G3.

0.85

0.875

0.9

0.925

0.95

0.975

1

1.025

1.05

0 50 100 150 200 250

r (p.u.)

Time (sec)

Figure 14. ULTC ratio.



5.2.2. Case 2: Short-term instability. In the case of L8 as an induction motor, the instability

mechanism is completely different from that demonstrated in case 1. Simulation results are shown

in Figures 15 and 16. Correspondingly, the procedure of tap changes of the ULTC is demonstrated in

Figure 17.

0.9

0.92

0.94

0.96

0.98

1

0 10 20 30 40

U bus7 (p.u.)

Time (sec)

Figure 15. Voltage at Bus 7.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40

Q motor (p.u.)

Time (sec)

Figure 16. Reactive power drawn by motor.

0.975

0.981

0.988

0.994

1.000

1.006

0 10 20 30 40

r (p.u.)

Time (sec)

Figure 17. ULTC ratio.



The tripping one of the transmission lines also triggers the action of the ULTC, which tries to bring

Ubus11 back. After the first action of the ULTC, a new stable equilibrium with increased motor slip and

reduced voltage at Bus 7 (Ubus7) is obtained. But the second tap change reduces Ubus7 further and

causes the motor loosing its equilibrium. During further drops of the voltage Ubus7, large amounts of

the reactive power are absorbed. This results in a voltage collapse in very short time. This effect can be

clearly observed in Figures 15 and 16.

5.3. Case study with QSS analysis

As described in Section 2, long-term voltage dynamics assume that inter-machine synchronizing

power oscillations have damped out, and focus mainly on actions of slow elements such as ULTCs

and OXLs, etc., where fast dynamics are replaced by their equilibrium equations and thus

further simplified to static models. In this section, the QSS approximation is adopted for

long-term voltage stability studies, where all dynamic models of generators and AVRs are replaced

by the corresponding static ones. With the assumption of the direct axis synchronous reactance Xd

being equal to the quadrature axis synchronous reactance Xq, the following equations are

established

PG ¼ EqUt

Xd

sinð� � �Þ

QG ¼ EqUt

Xd

cosð� � �Þ � U2

Xd

ð12Þ

where PG and QG represent the active and reactive power output of the generator respectively, Ut is the

generator terminal voltage magnitude, Eq is the q-axis voltage, � is rotor angle, and � is terminal

voltage phase angle. With assumption of active power output PG remaining constant and equal to PG0,

the static model of the generator can be simplified as

QG ¼ EqUt

Xd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � PG0Xd

EqUt

� �2s

� U2t

Xd

ð13Þ

and the static model of AVR as:

Eq ¼ KAðUref � UtÞ ð14Þ

where KA is the gain of the AVR shown in Figure 2 and Uref is the reference voltage magnitude.

Equations of generator and the AVR are now only algebraic ones, which are expressed in their compact

form given in Equation (8) and are implemented in variable-power units within the simulation

program. The OXL and the ULTC are modeled in the same way as that in the time-domain simulation

with the full system model.

Figures 18 and 19 compare some results obtained from QSS approximations and detailed

simulations described in the previous section. Although some differences are observed at critical

time, results achieved with the QSS still demonstrate a satisfactory coherence with that obtained from

time-domain simulations with the full system model.



6. CONCLUSION

For properly capturing static and dynamic behaviors providing insights into characteristics of the

voltage stability of a power system, various models for voltage stability studies are established in

multi-time-scale. In addition to P-U and U-Q analysis, full, long-term dynamic and short-term

dynamic simulations, modal analysis, as well as security-constrained optimal power flow analysis are

valuable tools for providing insight into the voltage instability and collapse phenomenon. This paper

has presented a comprehensive way for using different approaches that are implemented into the

simulation program. With this program adequate models can be established taking characteristics of

load and other elements of the system into account. Based on these models, voltage collapse stresses of

the system under a critical operating condition can be analyzed in steady state, and impacts of special

events on the voltage stability can also be simulated with time-domain simulations. For time-domain

simulations, a full system model and a QSS approximation can be adopted for capturing different

forms of instability mechanisms. Case studies validated the flexibility of the approaches implemented

for voltage stability studies.

Further possibilities of using the simulation program presented in this paper are such as studies on

voltage security assessment and studies on improvement of the voltage stability of a system by

0.86

0.9

0.94

0.98

1.02

1.06

1.1

0 50 100 150 200 250

U bus7 (p.u.)

Time(sec)

QSS approximation

Detailed model

Figure 18. Voltage at Bus 7.

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250

Q G3 (p.u.)

Time (sec)

QSS approximation

Detailed model

Figure 19. G3 reactive power output.



selecting adequate measures, e.g., FACTS or HVDC, etc. Other functions available in the program

such as optimization and eigenvalue calculation provide also useful tools for voltage stability studies.

7. LIST OF SYMBOLS AND ABBREVIATIONS

7.1. Symbols

x; z short-term and long-term state variable

_x; _z derivatives of short-term and long-term state variable

y algebraic variable

f, h short-term, long-term dynamics

g functions of algebraic variables

Rn n-dimension euler domain

J, Js Jacobian matrix of a full system and a reduced system

Jd, f(�), g(�), h(�), partition of Jacobian matrix

U, U0 voltage and nominal voltage

Ut generator terminal voltage

Uref reference voltage

P, Q active and reactive power consumption at U

P0, Q0 active and reactive power consumption at U0

PG, QG active and reactive power output of generator

PG0 initial active power output of generator

Pnom active power at normal condition

Ps, Pt steady-state and transient active power

f(U), g(U) function of transient and steady-state load-voltage characteristic

�, � exponential factor

ai multiplying factor

Efd, Ifd, Ilimfd field voltage, field current, and its limit

xt internal state variable of OXL

xoxl output signal of OXL

r, rk, rmin, rmax ratio, k-th ratio, minimum and maximum ratio of ULTC

d voltage difference dead band of ULTC

�r size of tap step of ULTC

Tk time delay of tap change of ULTC

U; I, complex voltage and current

Rs, Rr stator and rotor resistances

Xs, Xr stator and rotor leakage reactance

Xm rotor magnetizing reactance

Tm mechanical torque

Xd, Xq direct and quadrature axis synchronous reactance

Eq the q-axis voltage

�, ! rotor angle and rotor speed

� terminal voltage phase angle

Kr, KA gain constant

T, TR time constant

s Laplace operator



@ partial derivative

(�)�1 matrix inversion

7.2. Abbreviations

AVR automatic voltage regulator

OXL over excitation limiter

ULTC under load tap changer

QSS quasi steady-state

FACTS flexible AC transmission systems

HVDC high voltage DC systems

NETOMAC network torsion machine control

EMTDC electromagnetic transients program for DC-application

EMTP electromagnetic transients program

PSS/E power-system simulation for engineering

PI proportional and integral block

PID proportional and integral and differential block

REFERENCES

1. Cutsem TV, Vournas CD. Voltage stability analysis in transient and mid-term time scales. IEEE Transactions on PowerSystems 1996; 1:146–154.

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6. IEEE Working Group on Voltage Stability: Suggested techniques for voltage stability analysis. IEEE Publication93TH0620-5PWR, USA, 1993.

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AUTHORS’ BIOGRAPHIES

Xianzhang Lei (1958) received the B.S. degree from Zhejiang University, China, and hisM.Sc. and Ph.D. degrees in Electrical Engineering from the Technical University of BerlinGermany, in 1982, 1987, and 1992, respectively. From 1987 to 1993 he worked as a researchfellow in the Department of Electrical Engineering at the Technical University of BerlinGermany. After that, he was as Post-Doctor fellow at the Yale University, USA. Since 1994 hehas been a senior manager in the Power Transmission and Distribution Group at Siemens inErlangen Germany. Now he is working with XJ Group in China as Vice President and ChiefTechnology Officer. He is junction-professor at several universities in the People’s Republicof China. (XJ Group, 100065 Beijing China, Phone: þ86 10 62982288, Fax: þ86 10

62984582, E-mail: [email protected])

Dietmar Retzmann (1947), VDE, graduated in Electrical Engineering (Dipl.-Ing.) atTechnical University of Darmstadt Germany, in 1974. He received Dr.-Ing. degree in 1983from the University of Erlangen-Nuremberg Germany. He is with Siemens ErlangenGermany, since 1982 and currently head of the Power Electronic Applications Departmentin the Power Transmission and Distribution Services Division. He is involved in developmentand testing of Advanced Technology Applications for HVDC, FACTS, Power Quality, andProtection. He co-operates with universities in Asia, Europe, and America. Since 1998 heis guest-professor of Tsinghua University in Beijing China. (Director Power ElectronicApplications, Network Analysis & Consulting, Power Transmission and Distribution Ser-

vices, Siemens AG, EV SE NC4, P. O. Box 3220, D-91050 Erlangen/Germany, Phone: þ49 9131 7-3 47 39,Fax: þ49 9131 7-3 44 45, E-mail: [email protected])



static and dynamic approaches for analyzing voltage stability

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