Small signal stability analysis

PRESENTED BY:-
Bhupendra Kumar
Integrated M.Tech.

Contents
 Basic concepts and definitions
 Classification of power system stability
 Rotor angle stability
 Transient stability analysis
 Voltage stability
 Voltage collapse
 Factors affecting voltage stability
 Classification of voltage stability
 Small signal stability

 Stability of dynamic systems
 Dynamic stability analysis
 Effect of excitation system
 Power system stabilizers
 Control of PSS
 Applications of PSS
 AESOPS algorithm
 MAM method
 Characteristics of small signal stability problem
 References and bibliography

BASIC CONCEPTS AND DEFINITIONS
 Power system stability may be broadly defined as
that property of a power system that enables it to
remain in a state operating equilibirium under normal
operating conditions and to regain an acceptable
state of equilibirium after being subjected to a
disturbance.
 Instability in a power system may be manifested in
many different ways depending on the system
configuration and operating mode.
 Traditionally ,the stability problem has been one of
maintaining Synchronus operation.

BASIC CONCEPTS AND
DEFINITIONS(cont..)
 Since power system rely on synchronus machines for
electrical power generation, a necessary condition for
satisfactory system is that all syn. machines should
remain in synchronism.
 This aspect of stability is influenced by the dynamics
of generators rotor angles and power-angle
relationships.
 Instability may also encountered without loss of
synchronism. For example a system consisting of a
synchronus generator feeding an induction motor
load through a transmission line can become
unstable because of the collapse of load voltage.

BASIC CONCEPTS AND
DEFINITIONS(cont..)
 In the evaluation of stability the concern is the behavior of
the power system when subjected to a transient
disturbance. The disturbance may be small or large.
 Small disturbances in the form of load changes take place
continually, and the system adjusts itself to the changing
conditions.
 The system must be able to operate under these changes
and also be capable of surviving numerous disturbances of
severe nature, such as short- ckt. on transmission line, loss
of a large generator or load , or loss of a tie between two
subsystems.

Classification Of Power
System Stability

Small signal stability analysis

ROTOR ANGLE STABILITY
 Rotor angle stability is the ability of synchronous
machines of a power system to remain in
synchronism.
 In other words, rotor angle or load angle stability
denotes the angular displacement between stator
and rotor speeds.
 It is directly proportional to the speed of the m/c i.e.
the load connected to the generator.
 If the Angle is beyond to liable limit, the system will
come out of synchronism.

Transient Stability Analysis
 For transient stability analysis we need to consider
three systems
1. Prefault - before the fault occurs the system is
assumed to be at an equilibrium point
2. Faulted - the fault changes the system equations,
moving the system away from its equilibrium point
3. Post fault - after fault is cleared the system
hopefully returns to a new operating point

VOLTAGE STABILITY & VOLTAGE
COLLAPSE
 Voltage Stability-It refers to the ability of the
system to maintain a steady frequency, following a
system drastic change resulting in a significant
imbalance between generated and demand power
 Voltage stability margins-

Factors affecting voltage stability
 Voltage stability is a problem in power systems which are
heavily loaded, faulted or have a shortage of reactive
power.
 The nature of voltage stability can be analyzed by examining
the production, transmission and consumption of reactive
power.
 The reactive characteristics of AC transmission lines,
transformers and loads restrict the maximum of power
system transfers.
 The power system lacks the capability to transfer power
over long distances or through high reactance due to the
requirement of a large amount of reactive power at some
critical value of power or distance.

Scenario of classic voltage collapse
 The large disturbance causes the network characteristics to
shrink dramatically. The characteristics of the network and
load do not intersect at the instability point. A load increase
beyond the voltage collapse point results in loss of
equilibrium, and the power system can no longer operate. This
will typically lead to cascading outages.
 The load voltage decreases, which in turn decreases the load
demand and the loading of EHV transmission lines. The
voltage control of the system, however, quickly restores
generator terminal voltages by increasing excitation. The
additional reactive power flow at the transformers and
transmission lines causes additional voltage drop at these
components.

PV-curve
1. Power systems are operated in the upper part of the PV-curve. This
part of the PV-curve is statically and dynamically stable.
 The head of the curve is called the maximum loading point. The
critical point where the solutions unite is the voltage collapse point.
The maximum loading point is more interesting from the practical
point of view than the true voltage collapse point, because the
maximum of power system loading is achieved at this point. The
maximum loading point is the voltage collapse point when constant
power loads are considered, but in general they are different.
 The voltage dependence of loads affects the point of voltage collapse.
The power system becomes voltage unstable at the voltage collapse
point. Voltages decrease rapidly due to the requirement for an infinite
amount of reactive power.

PV-curve
 The lower part of the PV-curve (to the left of the voltage collapse point) is
statically stable, but dynamically unstable. The power system can only
operate in stable equilibrium so that the system dynamics act to restore
the state to equilibrium when it is perturbed.
V1=400 kV and X=100 Ohm

Classification of Voltage stability
Small-disturbance Voltage Stability- this category
considers small perturbations such as an incremental
change in system load.
 It is the load characteristics and voltage control
devices that determine the system capability to
maintain its steady-state bus voltages.
 This problem is usually studied using power-flow-based
tools (steady state analysis).
 In that case the power system can be linearised
around an operating point and the analysis is typically
based on eigenvalue and eigenvector techniques

Large-disturbance Voltage Stability
 Here, the concern is to maintain a steady bus voltages
following a large disturbance such as system faults,
switching or loss of load, or loss of generation.
 This ability is determined by the system and load
characteristics, and the interactions between the different
voltage control devices in the system.
 Large disturbance voltage stability can be studied by using
non-linear time domain simulations in the short-term time
frame and load-flow analysis in the long-term time frame
(steady-state dynamic analysis)
 The voltage stability is. however, a single problem on which
a combination of both linear and non-linear tools can be
used.

Short-term voltage stability
 Short-term voltage stability is characterized by
components such as induction motors, excitation of
synchronous generators, and electronically controlled
devices such as HVDC and static var compensator.
The time scale of short-term

Long-term voltage stability
 The analysis of long-term voltage stability requires
detailed modeling of long-term dynamics
 Two types of stability problems emerge in the long-term
time scale:
1. Frequency problems may appear after a major
disturbance resulting in power system islanding.
Frequency instability is related to the active power
imbalance between generators and loads. An island
may be either under or over-generated when the
system frequency either declines or rises.
2. Voltage problems

Small Signal Stability
 Small signal stability refers to the system’s ability to
maintain steady voltages when subjected to small
perturbations such as incremental changes in system load.
 This form of stability is influenced by the characteristics of
loads, continuous controls, and discrete controls at a given
instant of time.
 This concept is useful in determining, at any instant,
how the system voltages will respond to small system
changes.

Forms of Instability
Two forms of Instability
occur under these
conditions:
 Steady Increase in Rotor
Angle due to lack of
sufficient Synchronising
Torque
 Rotor oscillations of
increasing amplitude due
to lack of sufficient
damping torque

Small-Signal Stability of Multi-machine
Systems
 Analysis of practical power systems involves the
simultaneous solution of equations representing the
following:
 Synchronous machines, and the associated excitation
systems and prime movers.
 Interconnecting transmission network.
 Static and dynamic (motor) loads
 Other devices such as HVDC converters, static var
compensators

• For system stability studies it is appropriate to
neglect the transmission network and machine stator
transients.
• The dynamics of machine rotor circuits, excitation
systems, prime mover and other devices are
represented by differential equations.
• The result is that the complete system model
consists of a large number of ordinary differential
and algebraic equations.

Stability of a Dynamic System
 The stability of a linear system is entirely independent
of the input, and the state of the stable system with
zero input will always will return to the origin of the
state space, independent of the finite initial state.
 In contrast stability of the non linear system depends
on the type and magnitude of input, and the initial
state.

 In control system theory, it is common to classify
nonlinear stability into following categories:-
 Local Stability
 Finite Stability
 Global Stability

Local problems
 Associated with rotor angle oscillations of a single
generator or a single plant against the rest of the power
system. Such oscillations are called local plant mode
oscillations.
 Most commonly encountered small-signal stability
problems are of this category.
 Local problems may also be associated with oscillations
between the rotors of a few generators close to each
other.

 Such oscillations are called inter-machine or inter-plant
mode oscillations.
 The local plant mode and interplant mode oscillations
have frequencies in the range of 0.7 to 2.0 Hz.
 Analysis of local small-signal stability problems requires
a detailed representation of a small portion of the
complete interconnected power system.

Finite Stability
 If state of a system remains within a finite region R, it
is said to be stable within R. if further, the state of the
system returns to the equilibrium point from any
point within R, it is asymptotically stable within finite
region R.

Global Stability Problems
 Global small-signal stability problems are caused by
interactions among large groups of generators and
have widespread effects.
 They involve oscillations of a group of generators in
one area swinging against a group of generators in
another area. Such oscillations are called inter-area
mode oscillations.

Eigen value And Stability
• Stability of the linearized system is described by the
eigenvalues of the state matrix
• A real eigen value, or a pair of complex eigenvalues, is
usually referred to as a mode

• For a complex mode =j, two quantities are of main
interest:
 Frequency (in Hz) :f=ω/2π
 Damping ratio (in %):ξ=100×σ/√(σ^2+ω^2)
• The system is unstable if  is negative
• To ensure the acceptable performance, a damping margin
in the range of 3%-5% is normally required

Model Characteristics
• While an eigenvalue indicates the stability, its right
and left eigenvectors give much more information on
the characteristics of the mode
• The right eigenvector shows the mode shape, i.e., the
observability of the mode
 A mode should be observable from generator
rotor oscillations if the generator is high in its
mode shape

• A weighted left eigenvector shows the participation
factors, i.e., the controllability of the mode
 A mode should be controllable from generator if the
generator is high in its participation factors
• A generator which is high in the mode shape of a
mode is not necessarily high in the participation
factor of the same mode

DYNAMIC STABILITY ANALYSIS
 The analysis of dynamic stability can be performed by
deriving a linearized state space model of the system
in the following form
p X = A X + B u
 Where the matrices A and B depend on the system
parameters and the operating conditions.
 The Eigen values of the system matrix A determine
the stability of the operating point.

 The Eigen value analysis can be used not only for the
determination of the stability regions, but also for the
design of the controllers in the system.

Small Signal Stability of Single M/C
Infinite Bus System
 In this section we study the small signal performance of a
single machine connected to a large system through
transmission lines.
 A general system configuration is shown as-
 Fig. a reduced as fig. b using Thevenin’s Equivalent such
as virtually there is no change in voltage & frequency of
Thevenin’s voltage E.
 Such a voltage source of constt. Voltage & frequency is
referred as Infinite Bus.

 For any given system condition, the magnitude of the
infinite bus voltage E remains constt. when the
machine is perturbed. However as the steady state
system conditions change, magnitude of E may
change, representing a changed operating condition of
external network.

Effects of Excitation System
 In this section we extend the state space model & block
diagram to include the excitation system.
 Fast excitation-systems are usually acknowledged to be
beneficial to transient stability.
 These fast excitation changes are not necessarily
beneficial in damping the oscillations that follow the first
swing.

 They sometimes contribute growing oscillations several
seconds after the occurrence of a large disturbance.
 With proper design and compensation, a fast exciter can
be an effective means of enhancing stability in the
dynamic range as well as in the first few cycles after a
disturbance.

Some General Comments on the
Effect of Excitation on Stability
 For less severe transients, the effect of modern fast
excitation systems on first swing transients is marginal.
 For more severe transients or for transients initiated by
faults of longer duration, these modern exciters can have
a more pronounced effect.

 Their effects on damping torques are small; but in the
cases where the system exhibits negative damping
characteristics, the voltage regulator usually
aggravates the situation by increasing the negative
damping.
 Supplementary signals to introduce artificial
damping torques and to reduce inter machine and
intersystem oscillations have been used with great
success.

 Large interconnected power systems experience
negative damping at very low frequencies of
oscillations. The parameters of the PSS for a
particular generator must be adjusted after careful
study of the power system dynamic performance.

POWER SYSTEM STABILIZERS
 The dynamic stability of a system can be improved by
providing suitably tuned power system stabilizers on
selected generators. to provide damping to critical
oscillatory modes.
 Suitably tuned Power System Stabilizers (PSS), will
introduce a component of electrical torque in phase
with generator rotor speed deviations resulting in
damping of low frequency power oscillations in which
the generators are participating.

 The input to stabilizer signal may be one of the locally
available signal such as changes in rotor speed, rotor
frequency, accelerating power or any other suitable
signal.

CONTROL OF PSS
A Typical Control Schematic Diagram of
Power System Stabilizer

STRUCTURE OF COMPLETE POWER
SYSTEM MODEL
 For system stability it is appropriate to neglect the
transmission network and machine stator transients.
 The dynamics of machine rotor ckts., excitation
systems, prime movers and other devices
represented by differential equations.
 Result in that the complete system model consists of
a large no. of ordinary differential and algebric
equation.

SPECIAL TECNIQUES FOR ANALYSIS
OF VERY LARGE SYSTEMS
 Two methods have been found to be efficient, and,
they complement each other in meeting the
requirements of small-signal stability-
 1. the AESOPS algorithm
 2. the Modified Arnoldi Method(MAM)

THE AESOPS ALGORITHM
 The AESOPS algorithm is a type of selective eigen
value analysis method and it is found very effective in
computing modes.
 This allows the efficient studies of local modes without
the need to reduce the system model

CHARACTERISTICS OF SMALL
SIGNAL STABILITY PROBLEM
In large power systems, small signal
stability problem may be either local or
global in nature.

LOCAL PROBLEMS
 Local problems involve a small party of the system.
 They may be associated with rotor angle oscillations
of a single generator or a single plant against the rest
of power system.
 Such oscillations are called local plant mode
oscillations.
 the stability problems related to such oscillations
are similar to infinite bus system.

 Local problems may also associated with oscillations
between the rotors of a few generators close to each
other.
 Such oscillation are called inter-machine or interplant
mode oscillations.
 Other possible local; problems include instability of
modes associated with controls of equipments such as
generator excitation system, HVDC converters, and
static var compensators.

is in order of 0.1 to 0.3 Hz.
Higher frequency modes involving subgroups of
generators swinging against each other.
The frequency of these oscillations is typically in the
range of 0.4 to 0.7 Hz.

References and Bibliography
 Prabha Kundur , Power System Stability and Control , TMH
Publication,2008.
 Kimbark E W, Power System Stability, Volume I, III, Wiley
publication.
 C. Radhakrishna : “Stability Studies of AC/DC Power Systems” ,
Ph. D. Thesis , submitted to Indian Institute of Technology
Kanpur, India, 1980.
 Presentation on Voltage collapse- M.H.Sadegi
 Small Signal Stability Analysis Study:
study prepared by Powertech Labs Inc. for ERCOT
 www.Google.com

Small signal stability analysis

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