This presentation is about power system voltage stability.
What is voltage stability?
How voltage instability occurs?
How to improve voltage stability of the system?
This presentation is about power system voltage stability.
What is voltage stability?
How voltage instability occurs?
How to improve voltage stability of the system?
Classification Of Power System StabilityAravind Shaji
The Slide Deals With Power System Stability. Contents Include
Power System Stability Overview
Power System Stability: A Proposed Definition
Need of Stability Classification
Classification of stability
Power System Stability Classification
Rotor Angle Stability
Voltage Stability
Frequency Stability
Rotor Angle Stability vs. Voltage Stability
References
he main purpose of transient stability studies is to determineThe main purpose of transient stability studies is to determine
whether a system will remain in synchronism following major
disturbances such as transmission system faults, sudden load
changes, loss of generating units, or line switching.
Random events & other changes such as random variation in loads continually excite low frequency electromechanical oscillations within the power system which are continually observable in the power system measurements such as those obtained from PMUs. These oscillations also happen to characterize the Small Signal or Oscillatory Stability which is a measure of the power system’s ability to maintain synchronism under small disturbances, and is usually associated with power system’s inability to dampen such oscillations. Damping of these oscillations (typically expressed in %) is therefore a measure of the grid’s robustness to withstand small events or gradual changes in the system. For example, 10% damping implies that the oscillations will decay out in 10 oscillatory cycles while 20 % damping implies that these oscillations will die out in 5 oscillatory cycles (thus higher damping suggests greater robustness and a better ability to mitigate oscillations).
Transient Stability Assessment and Enhancement in Power SystemIJMER
Power system is subjected to sudden changes in load levels. Stability is an important concept
which determines the stable operation of power system. For the improvement of transient stability the
general methods adopted are fast acting exciters, circuit breakers and reduction in system transfer
reactance. The modern trend is to employ FACTS devices in the existing system for effective utilization
of existing transmission resources. The critical clearing time is a measure to assess transient instability.
Using PSAT, the critical clearing time (CCT) corresponding to various faults are calculated. The most
critical faults were identified using this calculation. The CCT for the critical faults were found to change
with change in operating point. The CCT values are predicted using Artificial Neural Network (ANN) to
study the training effects of ANN. TCSC is selected as the FACTS device for transient stability
enhancement. Particle Swarm Optimization method is used to find the optimal position of TCSC using
the objective function real power loss minimization. The result shows that the technique effectively
increases the transient stability of the system
Transient stability analysis on a multi machine system in psateSAT Journals
Abstract
Modern power system are subject to large disturbances such as three phase short circuit faults. When a fault occurs on a system
the generators rotor angle becomes unstable and thus it losses synchronism with the system and it becomes unstable. Thus
transient stability analysis can be performed on a system in order to understand the generators performance when subjected to a
short circuit fault. When the system is subjected to a fault the generator experiences transient oscillations in rotor speed and
angle which can be effectively suppressed with the incorporation of Automatic Voltage Regulator (AVR) and Power System
Stabilizer (PSS). The simulations have been performed using the MATLAB/PSAT software.
Keywords—Transient stability, Three phase fault Faults, AVR, PSS.
Classification Of Power System StabilityAravind Shaji
The Slide Deals With Power System Stability. Contents Include
Power System Stability Overview
Power System Stability: A Proposed Definition
Need of Stability Classification
Classification of stability
Power System Stability Classification
Rotor Angle Stability
Voltage Stability
Frequency Stability
Rotor Angle Stability vs. Voltage Stability
References
he main purpose of transient stability studies is to determineThe main purpose of transient stability studies is to determine
whether a system will remain in synchronism following major
disturbances such as transmission system faults, sudden load
changes, loss of generating units, or line switching.
Random events & other changes such as random variation in loads continually excite low frequency electromechanical oscillations within the power system which are continually observable in the power system measurements such as those obtained from PMUs. These oscillations also happen to characterize the Small Signal or Oscillatory Stability which is a measure of the power system’s ability to maintain synchronism under small disturbances, and is usually associated with power system’s inability to dampen such oscillations. Damping of these oscillations (typically expressed in %) is therefore a measure of the grid’s robustness to withstand small events or gradual changes in the system. For example, 10% damping implies that the oscillations will decay out in 10 oscillatory cycles while 20 % damping implies that these oscillations will die out in 5 oscillatory cycles (thus higher damping suggests greater robustness and a better ability to mitigate oscillations).
Transient Stability Assessment and Enhancement in Power SystemIJMER
Power system is subjected to sudden changes in load levels. Stability is an important concept
which determines the stable operation of power system. For the improvement of transient stability the
general methods adopted are fast acting exciters, circuit breakers and reduction in system transfer
reactance. The modern trend is to employ FACTS devices in the existing system for effective utilization
of existing transmission resources. The critical clearing time is a measure to assess transient instability.
Using PSAT, the critical clearing time (CCT) corresponding to various faults are calculated. The most
critical faults were identified using this calculation. The CCT for the critical faults were found to change
with change in operating point. The CCT values are predicted using Artificial Neural Network (ANN) to
study the training effects of ANN. TCSC is selected as the FACTS device for transient stability
enhancement. Particle Swarm Optimization method is used to find the optimal position of TCSC using
the objective function real power loss minimization. The result shows that the technique effectively
increases the transient stability of the system
Transient stability analysis on a multi machine system in psateSAT Journals
Abstract
Modern power system are subject to large disturbances such as three phase short circuit faults. When a fault occurs on a system
the generators rotor angle becomes unstable and thus it losses synchronism with the system and it becomes unstable. Thus
transient stability analysis can be performed on a system in order to understand the generators performance when subjected to a
short circuit fault. When the system is subjected to a fault the generator experiences transient oscillations in rotor speed and
angle which can be effectively suppressed with the incorporation of Automatic Voltage Regulator (AVR) and Power System
Stabilizer (PSS). The simulations have been performed using the MATLAB/PSAT software.
Keywords—Transient stability, Three phase fault Faults, AVR, PSS.
Tutorial: Introduction to Transient Analysis with PowerFactory. This tutorial is a simple introduction to transient simulation using DIgSILENT PowerFactory
The calculation of a Triangle Voltage Stability Index (TVSI) for monitored alternating-current circuits using voltage data from a PSSE load flow study. The analysis provides TVSI values for monitored transmission circuits in the Bulk Electric System under varying power transfers and contingencies.
TVSI provides an indication of the closeness of the load voltage to potential voltage collapse. To provide situational awareness to system operators, AEP proposes monitoring the phase angle across a low loss EHV overhead circuit operating in a system environment and comparing the angle to an established phase angle loci as a proxy for TVSI.
This monitoring could be independent of the line loading or the associated line impedance.
Up/Down Converter Linear Model with Feed Forward and Feedback Stability AnalysisCSCJournals
After knowing most power electronics circuits, we can say that the operation that we expected are under ideal conditions. The goal of power conversion was achieved by proper circuit configuration and proper switching. Our electronics circuits are treated under ideal conditions and operate in a nominal way. The steady state in most power electronics circuit is periodic steady state. Usually the focus is in operation were the behavior is the same from cycle to cycle.
In this paper we will deal with disturbances that will cause the power electronics circuit to deviate from nominal. This includes unexpected changes in the input or load. Also we look at the transient due to start. This deviation from nominal is called dynamic behavior. If this dynamic behavior do not change the desired output significantly we do not do any corrective action. This is rarely the case, however.
We have to design the system in order not to deviate from the desired nominal conditions. We need a control system recover to desired specifications. The compensator must operate to restore nominal conditions. second, it must maintain the circuit and guide it to nominal conditions by advancing or delaying the switching time. In this paper we are going to analyze the dynamic behavior due to disturbances or fault and how to control and guide the system to normal.
The focus in this paper is dynamics and control and an appropriate model for the non linear circuit to apply this control. From our experience we conceder faults that the system might fall into and we consider treatment from these situations. In our treatment we consider more parameters and more elements to have more flexibility to control the system. The cost of falling in one of these faults might be too high so we have to think of a solution.
Application of SVC on IEEE 6 Bus System for Optimization of Voltage Stabilityijeei-iaes
The problem of voltage or current unbalance is gaining more attention recently with the increasing awareness on power quality. Excessive unbalance among the phase voltages or currents of a three phase power system has always been a concern to expert power engineers. The study of shunt connected FACTS devices is an associated field with the problem of reactive power compensation related problems in today’s world. In this study an IEEE-6 bus system has been studied & utilized in order to study the shunt operation of FACTS controller to optimize the voltage stability
In the modern power system the reactive power compensation is one of the main issues, the transmission of active power requires a difference in angular phase between voltages at the sending and receiving points (which is feasible within wide limits), whereas the transmission of reactive power requires a difference in magnitude of these same voltages (which is feasible only within very narrow limits). The reactive power is consumed not only by most of the network elements, but also by most of the consumer loads, so it must be supplied somewhere. If we can't transmit it very easily, then it ought to be generated where it is needed." (Reference Edited by T. J. E. Miller, Forward Page ix).Thus we need to work on the efficient methods by which VAR compensation can be applied easily and we can optimize the modern power system. VAR control technique can provides appropriate placement of compensation devices by which a desirable voltage profile can be achieved and at the same time minimizing the power losses in the system. This report discusses the transmission line requirements for reactive power compensation. In this report thyristor switched capacitor is explained which is a static VAR compensator used for reactive power management in electrical systems.
Seminar Topic For Electrical and Electronics Engineering (EEE)
Ekeeda Provides Online Video Lectures, Tutorials & Engineering Courses Available for Top-Tier Universities in India. Lectures from Highly Trained & Experienced Faculty!
A detailed review of technology of hybrid electric vehicleDHEERAJ DHAKAR
This paper presents the development of hybrid electric
vehicles, classifications of hybrid electric vehicles based on the arrangement of the internal combustion engine and the
electric motor for traction.
Facts controllers for power flow control a brief reviewDHEERAJ DHAKAR
This paper provides a review of FACTS devices. The value of these FACTS is the improvement of security and efficiency of power transmission networks. Fast controllability in emergency situation provides increased flexibility and therefore stability and security advantages. The flexibility in control allows operating closer to stability limits and improve the efficiency of existing networks
Power factor correction using bridgeless boost topologyDHEERAJ DHAKAR
Power quality is becoming a major concern for
many electrical users. The high power non linear loads
(such as adjustable speed drives, arc furnace, static power
converter etc) and low power loads (such as computer, fax
machine etc) produce voltage fluctuations, harmonic
currents and an inequality in network system which results
into low power factor operation of the power system. The
devices commonly used in industrial, commercial and
residential applications need to go through rectification for
their proper functioning and operation. Due to the
increasing demand of these devices, the line current
harmonics create a major problem by degrading the power
factor of the system thus affecting the performance of the
devices. Hence there is a need to reduce the input line
current harmonics so as to improve the power factor of the
system. This has led to designing of Power Factor
Correction circuits. Power Factor Correction (PFC)
involves two techniques, Active PFC and Passive PFC. An
active power factor circuit using Boost Converter is used for
improving the power factor. This thesis work analyzes the
procedural approach and benefits of applying Bridgeless
Boost Topology for improving the power factor over Boost
Converter Topology. A traditional design methodology
Boost Converter Topology is initially analyzed and
compared with the Bridgeless Boost topology and the
overall Power Factor (PF) can be improved to the
expectation. Method of re-shaping the input current
waveform to be similar pattern as the sinusoidal input
voltage is done by the Boost converter and the related
controls that act as a Power Factor Correction (PFC)
circuit. Higher efficiency can be achieved by using the
Bridgeless Boost Topology. In this paper simulation of Boost
Converter topology and Bridgeless PFC boost Converter is
presented. Performance comparisons between the
conventional PFC boost Converter and the Bridgeless PFC
Boost Converter is done.
Modern FACTS controllers are being used to control the power through the current power transmission system.
The power transfer can be controlled by using these devices in an efficient and effective manner in transmission
lines. FACTS controllers are having some downsides i.e. their bulky size, higher cost, reliability and break-in
the transmission line, which makes it obsolete to use in modern power system network. These downsides can be
fulfilled by a new compound which is scalable, light weighted and cost effective devices that are distributed-
FACTS (D-FACTS). D-FACTS controllers are distributed version of conventional lumped FACTS controllers
and their cost is low due to lower ratings of component and reliability also increases due to redundancy of
devices. Enhanced Power Flow Controller (EPFC) is all a D-FACTS device which is a distributed version of
thyristor controlled series controller. This paper discusses extensive review about the EPFC and its application.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
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• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Cosmetic shop management system project report.pdf
Chapter8
1. Chapter 8
VOLTAGE STABILITY
The small signal and transient angle stability was discussed in Chapter 6 and 7.
Another stability issue which is important, other than angle stability, is voltage
stability. Voltage stability can be defined as the ability of the system to retain system
voltages within acceptable limits when subjected to disturbance. If the disturbance is
large then it is called as large-disturbance voltage stability and if the disturbance is
small it is called as small-signal voltage stability [1]-[2]. Voltage stability can be a
local phenomenon where only a particular bus or buses in a particular region have
voltage stability issue and this may not affect the entire system. Voltage stability can
be a global phenomenon where many of the system buses experience voltage stability
problems which can also trigger angle stability problems and hence can affect the
entire system. Some of the voltage stability problems can start as a local problem and
escalate to global stability problem.
Voltage stability of a system can be analysed either by static analysis or
dynamic analysis. In static analysis the system is assumed to be in steady state and
hence instead of taking the DAE of the system only algebraic equations are
considered. This type of analysis is suitable for small-disturbances in the system. For
large disturbances the DAE are solved and the system response over a certain period
of time is observed. It is important that for voltage stability the loads should be
properly modelled as each type of load will affect the system voltage stability in a
different way. Similarly the tap changing transformers, shunt or series reactive power
compensators, generator reactive power limits, line charging capacitance should be
included in the system representation to get an accurate picture of voltage stability.
8.1 Basic Concept of Voltage Stability
To understand the concept of voltage stability let there be a two bus system with
one end being sending end and the other being receiving end as shown in Fig. 8.1. Let
the voltage of sending end be SV and that of receiving end be RV . The transmission
8.1
2. line is represented by a short line mode with impedance TZ T . If a load of
impedance LZ L is connected at the receiving end.
Fig. 8.1: Single line diagram of two bus system
The magnitude of current in the transmission line is given as, taking the sending end
voltage as reference,
2 2
0
T T LZ Z 2 cos
S S
L T L T L T L
V V
I
Z Z Z Z
(8.1)
The receiving end voltage magnitude is then defined as
2 2
T LZ Z 2 cos
S L
R
T L T L
V Z
V
Z Z
(8.2)
The real and reactive power consumed by the load is given as
2
2 2
S L
T L
V Z
Z Z
cos
2 cos
L
L
T L T L
P
Z Z
(8.3)
2
2 2
S L
T L
V Z
Z Z
sin
2 cos
L
L
T L T L
Q
Z Z
(8.4)
SV
TZ
RV
LZ
I
8.2
3. Let the transmission line impedance be taken as 0.25TZ pu , neglecting
resistance as usually the reactance to resistance ratio is quite high in a transmission
network. Let the sending end voltage be 1 pu . The real and reactive power of the load
can be increased by decreasing the load impedance LZ maintaining a constant power
factor say 0.8. The receiving end voltage is plotted in Fig. 8.2 with varying real power
consumed by the load, with the parameters mentioned above. This curve is called as
P-V curve.
Knee
Point
Fig. 8.2: P-V curve
It can be observed from P-V curve that as the load increases from no load
condition the receiving voltage starts dropping form initial no load voltage of 1 pu .
The increase in the current I due to decrease in the load impedance LZ is more than
the drop in the receiving end voltage magnitude and hence the real power
consumed by the load will increase. The maximum real power, 1
RV
pu , will be delivered
to the load when L TZ Z . Once operating point reaches the maximum power point or
critical point or knee point of the curve any further decrease in load impedance, to
8.3
4. increase the load, will result in further drop in the receiving end voltage and
simultaneously the real power consumed by the load will also decrease. This is
because after knee point the drop in the voltage is more than the increase in the line
current leading to a decrease in the real power consumed by the load.
8.1.1 Effect of load type on voltage stability
Fig. 8.3: P-V curve with stable and unstable operating points
Consider the P-V curve shown in Fig. 8.3. For a given real power consumed by
the load there are two possible receiving end voltages. As can be seen from operating
points A and B from Fig. 8.3, for a real power of 0.6 pu there are two receiving end
voltages of 0.85 pu and 0.20 pu , approximately. If the load is a constant MVA load
then the operating point A is stable where as the operating point B is unstable. In fact
for a constant MVA load any operating point below the knee point is unstable. This
phenomenon can be understood from the Fig. 8.3. Suppose due to some disturbance
the operating point moves from point A to C. At the operating point C the real power
is more than the real power at operating point A. Since, the load is constant MVA
8.4
5. load the current will decrease and therefore the operating point move back from C to
A. The operating point may not stop immediately at operating point A but may
oscillate around it and will settle down with enough damping. Hence, the operating
point A is a stable operating point. Now consider the operating point B. Again let the
operating point move to point D due to disturbance. At operating point D the real
power is less than that at operating point B. Since the load is a constant MVA load at
operating point D the current increases, as compared to operating point B, which leads
to a further voltage drop and further increase in the line current. This phenomenon
continues till the voltage at the receiving end becomes zero and hence called as
voltage collapse. Therefore, the operating point B is unstable operating point. It can
be observed from P-V curve that the curve above knee point has negative
that is the change in real power and voltage are in opposite direction, for a increase in
power voltage decreases and for a decrease in power voltage increases. The curve
below the knee point has positive and at the knee point is zero.
/RdV dP
/R LdV dP
L
L/RdV dP
Fig. 8.4: P-V curve for different power factor loads
8.5
6. Figure 8.4 show the P-V curves for loads with different power factors. It can be
observed from the figure that as the load power factor moves from lagging to leading
the knee point is shifted towards higher real power and higher voltage. This shows
that the voltage stability improves as the power factor moves from lagging to leading
loads.
Fig. 8.5: Q-V curve for different load real powers
Just as the load real power was plotted with respect to the receiving end voltage
for varying load impedances, the load reactive power variation with respect to the
receiving end voltage, with load real power being constant, can also be plotted.
Figure 8.5 shows the variation of receiving end voltage with variation in load reactive
power for three different real power loads. Again the locus of knee point is marked.
On the curve left of the locus of the knee point the reactive power and the voltage
move in opposite directions that is is negative and this region is unstable.
The curve on the right of the locus of the knee point the reactive power and the
voltage move in the same direction and hence is positive and this region is
stable. In the unstable region even if reactive power compensation is done through
shunt capacitance at the receiving end bus, the voltage of the receiving end bus will
/ RdQ dV
/ RdQ dV
8.6
7. not improve. One way of finding the voltage stability is to check the sensitivity of
each bus voltage with respect to the reactive power injected at that bus and if the
sensitivity is positive then it means the operating point is stable and is on the right
side of the locus of the knee points in Fig. 8.4 and if it is negative then it is on the left
side. This stable region and unstable region are only applicable to constant MVA
loads. In case of constant impedance and constant current loads the loads interact with
the system and settle at a new operating point as there is no requirement of constant
MVA.
8.2 Static Analysis
In static analysis [3] of voltage stability the snapshots of the entire system at
different instants is considered and at each instant the system is assumed to be in
steady state that the rates of changes of the dynamic variables are zero. Hence, instead
of considering all the differential algebraic equations only algebraic power balance
equations are considering assuming that the system is in steady state. At each instant
whether the system is voltage stable or not and also how far the system is from
instability can be assessed. There are two methods for assessing whether system is
voltage stable or not. They are sensitivity analysis [4] and modal analysis [5].
8.2.1 Sensitivity analysis
The power balance equations are taken assuming that the system is in steady
state. For, , assuming bus number 1 is a slack bus, the real power balance
equation is given as
2,......,i n
1
( ) cos 0
n
Gi Di i i j ij i j ij
j
P P V VV Y
(8.5)
The reactive power balance equations are taken at the load buses, for
1, 2,......,g gi n n n
8.7
8. (8.6)
1
( ) sin 0
n
Di i i j ij i j ij
j
Q V VV Y
The linearized form of equations (8.5) and (8.6) is given as
2 2
1 1g g
n nP PV
n nQ QV
n n
P
P J J
Q VJ J
Q V
or
P
J
Q V
(8.7)
The matrix is the Jacobian matrix used in Newton-Raphson load flow analysis. For
constant real power is zero hence from equation (8.7)
J
P
1
P PVJ J
V
(8.8)
Substituting (8.8) in (8.7) an simplifying leads to
(8.9)
1
QV P PV QQ J J J J V
Let,
1
R QV P PV QJ J J J J
, then
1
RV J Q
(8.10)
The diagonal elements of the matrix 1
RJ
represents the sensitivity of the voltage
with respect to the reactive power injected at that bus. This is nothing but the slope of
8.8
9. Q-V curve given in Fig. 8.4. Hence, if the V-Q sensitivity of an ith
bus is positive then
the system is voltage stable and if it is negative then the system is voltage unstable. A
small positive value of sensitivity indicates that the system is more voltage stable and
if the sensitivity is small with negative value then the system is highly voltage
unstable.
8.2.2 Modal Analysis
Voltage stability can also be estimated by the eigen values and eigen vectors of
the matrix . The matrix can be expressed in terms of right and left eigen vector
matrices as
1
RJ
,
RJ
RJ (8.11)
1
RJ
1
(8.12)
Where, is a diagonal matrix with eigen values along its diagonal. Substituting
(8.12) in (8.10) lead to, noting that
I
1 1
RV J Q Q
(8.13)
1
V
Q
Q
(8.14)
Let, v V and , then or q 1
v q
1
i
i
v iq
(8.15)
Where, is the ith
bus modal voltage and is the modal reactive power. If the
eigen value is positive then the system is voltage stable. The larger the eigen value
with positive sign the better the stability and if the eigen value has very small positive
value then it is very close to
iv iq
0i , that is the critical point or knee point, and hence
8.9
10. very close to instability. If the eigen value is negative then the system is voltage
unstable.
8.3 Dynamic Analysis
The voltage stability of a system can also be assessed by the transient simulation
over a period of time [3]. The differential algebraic equations, given in (8.16), of the
system should be solved through numerical methods and the transient simulation
should be carried out for few minutes to completely observer the interaction of
generators, static loads, dynamic loads, tap changing transformer etc.
( , , , )
( , )
0 ( , , )
d q
d q
d q
x f x I V u
I h x V
g x I V
-
-
-
=
=
=
(8.16)
The transient simulation should be done for different fault scenarios and the
system behaviour should be observed. If the system voltages are restored to
acceptable values after fault clearing then the system is voltage stable if not the
system is voltage unstable. The simulations needs to be done for few minutes because
the time constant involved can be very small like generator exciter or very large like
induction motors or tap changing transformers. For voltage stability assessment load
modelling is important hence both static and dynamic loads should be modelled. The
reactive power compensating devices like series / shunt capacitor and static VAr
compensator should also be included in the system model.
8.3.1 Small-disturbance analysis
Small-disturbance (signal) analysis can also be used for voltage stability
analysis. The DAE given in equation (8.16) can be represented as
( , ( , ), , )
0 ( , ( , ), , )
x f x h x V V u
g x h x V V l
=
=
(8.17)
8.10
11. Where, in (8.17) is a constant representing the loading of the system, as given in
(8.18)
0
0
1
1
D D
D D
P P
Q Q
(8.18)
0 , 0D DP Q are the real and reactive power loads at all the load buses for base case.
The idea is to increase the loading factor in steps and at each step find the eigen
values of the linearized system given in (8.17). The linearized form of the system
given in (8.17) is given as
0
Vf f f f
x x u
x V u
Vg g g
x
x V
qq
qq
é ùDé ù¶ ¶ ¶ ¶
ê úê úD = D + + D
ê úê ú D¶ ¶ ¶ ¶ë û ë û
é ùDé ù¶ ¶ ¶
ê úê ú= D +
ê úê ú D¶ ¶ ¶ë û ë û
(8.19)
Here, 1 1 1
T T
nV V V V V n n hence the linearized form shown in
(8.19). On further simplification of (8.19) the following expression can be derived
1
sysJ
sys
f f f g g g f
x x u
x V V x u
f
J x u
u
q q
-é ùé ù é ù é ù é ù é ù¶ ¶ ¶ ¶ ¶ ¶ ¶ê úê ú ê ú ê ú ê ú ê úD = - D + Dê úê ú ê ú ê ú ê ú ê ú¶ ¶ ¶ ¶ ¶ ¶ ¶ë û ë û ë û ë û ë ûê úë û
é ù¶
ê ú= D + D
ê ú¶ë û
(8.20)
The matrix sysJ is called as the system Jacobian and is different from the load
flow Jacobian. The eigen value of the system Jacobian sysJ for different loading
conditions, obtained by varying the loading factor , are computed. If a one of the
eigen value becomes zero or a pair of complex conjugate eigen values are on the
imaginary axis for a particular loading factor then the load corresponding to the
8.11
12. loading factor is a critical load at which the system becomes voltage unstable. If a
complex conjugate pair of eigen values are on imaginary axis then that operating
point is called as Hopf-bifurcation point. If one of the eigen value is on the origin then
that operating point is called as saddle node point. The saddle node bifurcation point
represents the knee point. Hopf-bifurcation point occurs at a load which is less than
the load at which the saddle node bifurcation point occurs. Hence, due to the
dynamics involved the system can become voltage unstable even before the operating
point reaches the knee point.
8.12
13. 8.13
References
1. IEEE, Special publication 90TH0358-2-PWR, Voltage stability of power
systems: concepts, analytical tools, and industry experience, 1990.
2. CIGE task force 38-02-10, Modelling of voltage collapse including dynamic
phenomena, 1993.
3. G. K. Morison, B. Gao, and P. Kundur, “Voltage stability evaluation using
static and dynamic approaches,” IEEE PES summer meeting, July 2-16, 1992,
Seattle, Washington.
4. N. Flatabo, R. Ognedal and T. Carlsen, “Voltage stability conditions in a
power transmission system calculated by sensitivity methods,” IEEE Trans.,
Vol. PWRS-5, No. 4, pp. 1286-1293, November.
5. 1990G. K. Morison, B. Gao, and P. Kundur, “Voltage stability evaluation
using modal analysis,” IEEE Trans., Vol. PWRS-7, No. 4, pp. 1529-1542,
November 1992.