POWER SYSTEM STABILITY RESPONSE AND CONTROL USING …rx915z833/... · POWER SYSTEM STABILITY RESPONSE AND CONTROL USING SMALL SIGNAL ANALYSIS A Thesis Presented By Mark Germanos to

POWER SYSTEM STABILITY RESPONSE AND CONTROL USING SMALL SIGNAL ANALYSIS

A Thesis Presented

By

Mark Germanos

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements for the degree of

Master of Science

in the field of

Electrical and Computer Engineering

Northeastern University Boston, Massachusetts

August 2015

1

Abstract

Small signal stability analysis involves the investigation of small perturbation impacts in the

operating point of power systems on system stability. Small signal stability is typically related to

the generator and load properties. The purpose of this investigation is to examine the effects of

small disturbances on system stability and determine the best locations to place power system

stabilizers in order to avoid instability due to these disturbances. This will be accomplished by

evaluating the eigenvalues and eigenvectors of the linearized differential and algebraic

equations of the system. These calculations facilitate determination of the so called

participation factors, which will then be used to identify the relevant modes of the system that

are pushing the system towards instability. Finally, best locations for power system stabilizers

to minimize the effects of these dominant modes and to stabilize the system will be identified.

IEEE 39 Bus test system will be used to illustrate the stability analysis as well as the effective

stabilization of the system by proper placement of power system stabilizers for various power

system contingencies.

Dedication

I would like to dedicate the work of this thesis to my father, Germanos Germanos, who has

devoted himself to me as a father by encouraging me during my career and motivating me

towards the field of engineering by finding his success even at the most difficult challenges.

Key Words: Admittance matrix, Swing Equation, Modes, States, Participation factors, Stability,

Power system stabilizers, Rotor angle, Rotor speed, Damping, Disturbance.

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Table of Contents

Abstract ............................................................................................................................................ 1

Dedication ........................................................................................................................................ 1

Chapter 1.......................................................................................................................................... 6

1.1 Introduction ........................................................................................................................... 6

1.1.1 Historical Review – Power Systems ................................................................................ 6

1.1.2 Power System Stability – Small Signal Stability ............................................................... 8

1.2 Thesis Contribution .............................................................................................................. 10

Chapter 2........................................................................................................................................ 12

2.1 Electric System Network Model ........................................................................................... 12

2.2 Small Signal Stability ............................................................................................................ 17

2.3 Participation Factor Analysis ................................................................................................ 20

2.4 Power System Stabilizer ....................................................................................................... 25

Chapter 3........................................................................................................................................ 27

3.1 System Model – IEEE 39 Bus System ................................................................................... 27

3.1.1 General Outline ............................................................................................................. 27

3.1.2 Network ........................................................................................................................ 28

3.1.3 Generator Data ............................................................................................................. 28

3.1.4 Branch Data – Lines and Transformers ......................................................................... 28

3.1.5 Power and Voltage Data ............................................................................................... 29

3.2 Simulation Software ............................................................................................................. 29

Chapter 4........................................................................................................................................ 37

4.1 Simulation Results and Analysis ........................................................................................... 37

4.1.1 Methodology ................................................................................................................. 37

4.1.2 Base Case Modal Analysis ............................................................................................. 37

4.1.3 Stability Response to System Changes and System Stability Analysis .......................... 40

4.1.4 System Control and Power System Stabilizer Effect on System ................................... 56

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Chapter 5........................................................................................................................................ 85

5.1 Conclusion ............................................................................................................................ 85

5.2 Future work .......................................................................................................................... 86

Bibliography ................................................................................................................................... 88

Appendix A ..................................................................................................................................... 90

Appendix B ..................................................................................................................................... 98

Figure 1: Graphical Representation of Electrical System ............................................................... 13

Figure 2: Block Diagram Model Used in Stability Analysis ............................................................. 26

Figure 3: Load Flow data Reproduced in PowerWorld .................................................................. 31

Figure 4: Transient Stability Module .............................................................................................. 31

Figure 5: Pre-Defined Power System Stabilizer Model .................................................................. 32

Figure 6: Gen. Rotor Angle Plot Sample ......................................................................................... 33

Figure 7: Gen. Rotor Speed Plot Sample ........................................................................................ 34

Figure 8: Gen. Real Accelerating Power Plot Sample ..................................................................... 35

Figure 9: Mode (Eigenvalue) Graphic Representation ................................................................... 36

Figure 10: Graphical Display of Modes (Eigenvalues) .................................................................... 40

Figure 11: Test Case 1a Graphical Display of Modes (Eigenvalues) ............................................... 49

Figure 12: Test Case 1b Graphical Display of Modes (Eigenvalues)............................................... 51

Figure 13: Test Case 1c Graphical Display of Modes (Eigenvalues) ............................................... 53

Figure 14: Test Case 1d Graphical Display of Modes (Eigenvalues)............................................... 55

Figure 15: Gen. Rotor Angle Plot Prior to PSS Insertion ................................................................ 58

Figure 16: Gen. Rotor Speed Plot Prior to PSS Insertion................................................................ 59

Figure 17: Gen. Accel. Power Plot Prior to PSS Insertion ............................................................... 60

Figure 18: Modes (Eigenvalues) Plot Prior to PSS Insertion .......................................................... 62

Figure 19: Gen. Rotor Angle Plot During Fault ............................................................................... 63

Figure 20: Gen. Rotor Speed Plot During Fault .............................................................................. 64

Figure 21: Gen. Accel. Power Plot During Fault ............................................................................. 65

Figure 22: Modes (EIgenvalues) Plot During Fault ......................................................................... 67

Figure 23: Gen. Rotor Angle Plot During Contingency, Applied Fault, and Prior to PSS Insertion 68

Figure 24: Gen. Rotor Speed Plot During Contingency, Applied Fault, and Prior to PSS Insertion 69

Figure 25: Gen. Accel. Power Plot During Contingency, Applied Fault, and Prior to PSS Insertion

....................................................................................................................................................... 70

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Figure 26: Modes (Eigenvalues) Plot During Contingency, Applied Fault, and Prior to PSS

Insertion ......................................................................................................................................... 72

Figure 27: Gen. Rotor Angle Plot after PSS Insertion ..................................................................... 74

Figure 28: Gen. Rotor Speed Plot after PSS Insertion .................................................................... 75

Figure 29: Gen. Accel. Power Plot after PSS Insertion ................................................................... 76

Figure 30: Modes (Eigenvalues) Plot after PSS Insertion ............................................................... 78

Figure 31: Gen. Rotor Angle Plot after PSS Insertion ..................................................................... 79

Figure 32: Gen. Rotor Speed Plot after PSS Insertion .................................................................... 80

Figure 33: Gen. Accel. Power Plot after PSS Insertion ................................................................... 81

Figure 34: Modes (Eigenvalues) Plot after PSS Insertion ............................................................... 83

Figure 35: IEEE 39 Bus System ....................................................................................................... 90

Table 1: Base Case Load Flow Results ............................................................................................ 38

Table 2: Modal Analysis Summary Table ....................................................................................... 39

Table 3: List of Test Cases .............................................................................................................. 41

Table 4: Test Case 1a Load Flow Results ........................................................................................ 42

Table 5: Test Case 1b Load Flow Results ........................................................................................ 43

Table 6: Test Case 1c Load Flow Results ........................................................................................ 45

Table 7: Test Case 1d Load Flow Results ........................................................................................ 46

Table 8: Test Case 1a Modal Analysis Summary Table .................................................................. 48

Table 9: Test Case 1b Modal Analysis Summary Table .................................................................. 50

Table 10: Test Case 1c Modal Analysis Summary Table ................................................................ 52

Table 11: Test Case 1d Modal Analysis Summary Table ................................................................ 54

Table 12: Modes (Eigenvalues) Prior to PSS Insertion ................................................................... 61

Table 13: Modes (EIgenvalues) During Fault ................................................................................. 66

Table 14: Modes (Eigenvalues) During Contingency, Applied Fault, and Prior to PSS Insertion ... 71

Table 15: Modes (Eigenvalues) after PSS Insertion ....................................................................... 77

Table 16: Modes (Eigenvalues) after PSS Insertion ....................................................................... 82

Table 17: Generator Data .............................................................................................................. 91

Table 18: Branch Data .................................................................................................................... 94

Table 19: Power and Voltage Data ................................................................................................ 97

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V Pre-Transient Bus Voltage Magnitude

I Current

θ Pre-Transient Bus Voltage Angle

P Real Power

Q Reactive Power

𝑌 Bus Admittance Matrix

𝑌𝑖𝑗 Bus Admittance Matrix Element

𝐺𝑖𝑗 Real-Part of Bus Admittance Matrix Element

𝐵𝑖𝑗 Imaginary-Part of Bus Admittance Matrix Element

𝑥𝑑′ Generator Internal Reactance

E Generator Terminal (Internal) Voltage

𝛿′ Generator Terminal (Internal) Angle

δ0 Initial generator angle

𝜔𝑠 System Radian Frequency

𝑇𝑎 Accelerating Torque

𝑇𝑀 Mechanical Torque

𝑇𝑒 Electrical Torque

J Moment of Inertia

H Generator Inertia

𝛿 Generator Rotor Angle

𝜔 Generator Rotor Speed

d Linear Damping Coefficient

𝑃𝑎 Generator Accelerating Power

𝑃𝑀 Generator Mechanical Power

𝑃𝑒 Generator Electrical Power Output

𝐴𝜔 Generator Electrical power Output Equations Matrix

𝐷 Damping Factor Matrix

𝜆 Eigenvalue

𝑤 Left Eigenvector

ν Right Eigenvector

𝑝 Participation Factor

𝑉 Right Eigenvector Matrix

𝑊 Left Eigenvector Matrix

𝑃 Participation Factor Matrix

𝛾 Damping Ratio

𝜎 Real-Prt of Eigenvalue

𝛼 Imaginary-Part of Eigenvalue

Table of Symbols

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Chapter 1

1.1 Introduction

1.1.1 Historical Review – Power Systems

Power systems typically have three major components, generation, transmission and

distribution. . Power is produced by generators which deliver their output to the transmission

grid at various points. Transmission grid acts like a pool which accepts the power delivered by

the generators and delivers it to distribution systems connected to the transmission grid at

different points via distribution substations. The main function of all power systems is to

provide a constant and continuous supply of energy to all types of consumers ranging from large

industrial plants and commercial organizations to individual households. Because the reliability

of a power system is of utmost importance, a general understanding of the entire process is a

necessary prerequisite for power systems analysis.

Power can be generated by using fossil fuel, coal, water, nuclear sources, and wind. Fossil fuel

can be burned to produce steam, which in turn powers a steam turbine to produce energy. It

can also be burned in an engine or turbine to produce energy or a combination of both

mechanical and thermal (combined cycle). The byproducts of the combustion of coal are

utilized to provide steam which, in turn, powers a steam turbine to produce energy.

Hydroelectric power generation utilizes descending water kinetics to power turbines that turn

mechanical energy into electrical energy. Wind power works in much the same way with the

power source being wind. Nuclear fission produces extreme heat which, when used in

conjunction with cooling water, produces steam that drives a turbine in much the same way as

fossil fuel plants. Other methods of generation are solar, tidal, bio-fuel, etc. In conclusion,

power generation operates [or say all sources of power generation operate...] under the

principal of converting one type of energy into electrical energy, regardless of the means,

method, efficiency, or system used. Throughout the history of the industry, power generation

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has evolved based on economics, the availability of prime energy, efficiency and reliability,

safety and environmental issues.

A transmission system is the means to transport high voltage power generated by power plants

across long distances to its final destination where it is transformed to medium and low voltage.

This system includes equipment such as substations, transformers, insulated conductors,

capacitor banks, disconnect switches, and protection relays and breakers. When planning to

install a transmission system the following must be taken into consideration: location of

population and industry, terrain, size of the area to be covered, and future expansion.

Distribution is simply the transport of medium and low voltage electricity to the end user. The

equipment involved in this phase of a power system includes switchgear, motor control centers,

transformers, breakers, and distribution panels.

The transmission and distribution systems vary in size and configuration but all have limitations,

related to safety and reliability which are important issues to consider in operating these

systems.

In the 1920’s, when power system stability emerged as a field of study, power systems were

relatively simple being small and having limited connections. Therefore stability was not a major

concern by reason of a simple system design and operation. A larger issue was synchronization

which was attended to by the design of a high quality transmission system. In addition, because

of limited technical understanding and the absence of detailed models along with an inadequate

computational ability, efficiency and reliability were low.

As power systems grew and became interconnected in the 1930’s, improvements were made,

such as the development of scaled physical models which enhanced system analysis. However,

the inability to perform complicated calculations impeded further development and restricted

the analysis of the network only rather than to the individual components, such as generators,

8

transformers, etc. During this time stability of systems was increased by faster protection

devices and better regulating components within the system [1].

With the advent of the computer in the 1950’s, the focus of analysis was no longer limited to the

network, but equipment performance could be analyzed and evaluated as well. Later as

interconnection of systems became a focus, transient stability became a greater concern [2].

Today the field of power system stability involves ongoing development in a variety of areas

including new analytical methods, computer generated calculations, more detailed models,

more complex testing, and higher performance levels of equipment.

1.1.2 Power System Stability – Small Signal Stability

Power System Stability is defined as that property of a power system that enables it to remain in

a state of operating equilibrium under normal operating conditions and to regain an acceptable

state of equilibrium after being subjected to a disturbance. This definition shapes the aspects of

power systems that each technique and methodology must address. This mentioned equilibrium

to be maintained both before and after disturbances incorporates several aspects and is

dependent upon specific system parameters. Synchronization of the machines within the system

and voltage stability is of primary concern. The components that govern these characteristics

are necessary to form the building blocks of stability analysis. One of the chief components of

synchronization is machine rotor angle. Rotor angle is the angle between a rotor axis of the

machine and its stator axis. In order for the system to remain stable, all of the synchronous

machines within a power system must maintain a synchronized state of equilibrium between

their rotor angles. Thus, rotor angle and system stability are one of the fundamental concerns of

analysis. Another important factor of system stability that involves the rotors of the machines is

the non-linear relationship between the machine electric power output and its rotor angle. It

accounts for both the amount of power transferred and the system’s ability to return to

equilibrium i.e. stable after a certain disturbance. Finally, the third major component of power

system stability is rotor speed. The change in rotor speed after a certain disturbance can

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significantly affect the damping characteristics of the system. Therefore, oscillatory stability

becomes a vital point of emphasis. Having explained stability and its three major components,

the two designations of disturbances involved in stability analysis will be defined. These two

categories are small-signal stability analysis of small disturbances and transient stability analysis

of large disturbances. Small signal stability, covered in this analysis, is the study of system

response to small disturbances. These disturbances result in the system having to cope with

certain issues from rotor angle un-synchronism or oscillation problems due to insufficient

damping. Synchronism issues often involve application of voltage regulation but oscillation

problems form the majority of instability events and are more complex due to the fact that

there are many different oscillation modes. Small signal stability involves the analysis of power

systems around an equilibrium point ‘0’. This allows the complex dynamics of the system to be

transformed to a system of linear equations. In other words, stability is examined for small

disturbances using a simplified linear system model. The idea of small signal stability analysis is

to compose characteristics of the system with minimal error while significantly decreasing both

the complexity and computation effort involved. As the size and complexity of the power

systems increases, the computational effort involved can exceed the reasonable limits even

when using today’s capable and powerful computers. Therefore, techniques can be utilized to

simplify the process and still have a precise and useful outcome of a power system which has a

system response and behavior with minimal error and effort involved. The term “small signal” is

associated to minor disturbances on the system which are used to assess the simplified system

model. Small disturbances within a system are a very common type of disturbance and could be

used to represent the system as a linearized model. These disturbances could be represented as

load changes within a power system. In summary, small signal stability transforms a complex

non-linear power system to a linear system model which could be analyzed as a stable system

around an equilibrium point ‘0’ while it experiences small disturbances eliminating complex

dynamics so that an assumption could be made to make the computation effort more feasible.

However, linear approximations are only made at a certain point in time and these

approximations will change for other time instances. Transient stability analysis attributes to the

10

analysis of power systems including the complex non-linear dynamics. Larger disturbances are

analyzed at different operating points since the behavior of the system changes during a certain

period of time. Transient stability analysis differs in the nature and scale of a disturbance but

often leads to a state similar to small signal instability. Regarding transient instability, first swing

issues or long term end state instability resulting from growing oscillations could lead to

synchronization issues [3].

1.2 Thesis Contribution

The purpose of this study is to introduce power systems including the multi-machine classical

model, the simplified electrical network, and small signal stability including the swing equation

and its linear transformation. Then the system is analyzed using modal analysis in order to

determine the different modes of oscillation and the mode most likely to be driven to instability.

This will facilitate our analysis to identify how the system operates under certain conditions.

Moreover, the techniques specified above will enable us to determine a location to control the

system thus restoring its stability. The remainder of the thesis is devoted to:

Chapter 2: The second chapter of this literary work is first devoted to understanding the

derivation of the electric network model including the classical machine model and its inclusion

to the transmission system admittance network. Second, the chapter establishes a

mathematical foundation behind small signal stability and modal analysis techniques. Third,

introduce the power system stabilizer power system stabilizer and its role in controlling the

system. A literature review is conducted to explain and explore the derivations of the numerical

methods used by the simulation software to formulate the model, analyze the power system,

and explain the impact of the power system stabilizer on a corresponding machine by using

numerical methods and block diagrams to illustrate those facts. The small signal stability areas

addressed include linearization of non-linear differential equations i.e. the swing equation which

models a power system, modal analysis via state space representation, eigenvalue analysis,

eigenvectors, and participation factor analysis. Upon completion of Chapter 2, a thorough

11

understanding of the electric network model, small signal stability analysis process, and the

power system stabilizer should be obtained.

Chapter 3: The third chapter is devoted to introducing both the power system that is used in this

research and the simulation software used to conduct that analysis. The IEEE 39 Bus system is

presented along with all of the system data inputs that are necessary to model and analyze

system operation. These inputs include voltage magnitude and angle, real and reactive power,

impedance, admittance, and component data for the system buses, generators, loads,

transformers and transmission lines. In addition, the simulation software MATLAB and

PowerWorld are present. Software information that is detailed includes the capabilities and

calculative processes involved in system analysis. Examples of results yielded by MATLAB and

PowerWorld are shown, and explanations are given as to the interpretation and significance of

the tables, charts, and graphs.

Chapter 4: The fourth chapter is devoted to presenting the simulation results and their

associated analysis. A methodology is also outlined as to the approach taken during the

research. The results published in this chapter include load flow results, system eigenvalues,

mode distribution, and participation analysis using the modal analysis results. In addition to

these, machine rotor angle, machine rotor speed, and generator accelerating power plots

before and after controlling the system through power system stabilizers at determined

locations are presented.

Chapter 5: The fifth chapter is devoted to discussing the conclusions that are to be drawn from

the research results and future work. These conclusions include the system’s stability, the

system component that participates most in modal stability, and the location for controlling the

system to restore stability.

Following Chapter 5, Appendix A contains all the IEEE 39 Bus system data, while Appendix B

contains all the modal analysis simulation results that were not directly addressed in Chapter 4

and scripts used in programming simulations. This information can be drawn upon when a more

12

detailed account of the modal stability of the various system configurations is needed. Finally, a

list of references completes the thesis work.

Chapter 2

2.1 Electric System Network Model

The machine model used is the classical model of a multi-machine system. This model is based

on the fact that the mechanical power input is constant, damping or asynchronous power is

negligible, constant voltage behind the transient reactance model for the synchronous machines

is valid, mechanical rotor angle of a machine coincides with the angle of the voltage behind the

transient reactance, and loads are represented by passive impedances [4].

This model is useful for small signal stability analysis i.e. transient stability for short time

periods. The assumption that the damping factor is negligible is improved upon by assuming a

linear damping characteristic i.e. a damping torque diωi is added to the inertial torque in the

swing equation. The damping coefficient di includes various damping torque components, both

mechanical and electrical. Values of the damping coefficient usually used in stability studies are

in the range of 1-3 pu. This represents turbine damping, generator electrical damping, and the

damping effect of electrical loads [4].

The electrical network obtained for an m-machine system is as shown in Figure 1. Node 0 is the

reference node (neutral). Nodes 1, 2… m are the internal machine buses, or the buses to which

the voltages behind transient reactances are applied. Passive impedances connect the various

nodes and connect the nodes to the reference at load buses. The initial values of machine

voltage are determined from the pre-transient conditions. The machine voltage magnitudes are

held constant during the transient in this study [4].

13

Xd’1

IL1

E1<δ1

Transmission System

Admittance Matrix

m-machine system

m generation

r constant impedance loads

Xd’m

Em<δm

ILr

Node 0

Im

I1

Figure 1: Graphical Representation of Electrical System

The loads are converted to equivalent admittances. The needed data for this step are obtained

from the load flow study. Thus, if a certain load bus has a voltage ��𝐿, real power 𝑃𝐿 , reactive

power 𝑄𝐿 , and current 𝐼�� flowing into a load admittance ��𝐿 = 𝐺𝐿 + 𝑗𝐵𝐿, then

𝑃𝐿 + 𝑗𝑄𝐿 = ��𝐿𝐼��∗ = 𝑉𝐿

2(𝐺𝐿 − 𝑗𝐵𝐿)

(2.1)

The equivalent shunt admittance ��𝐿 at that bus is given by

��𝐿 = −(𝐺𝐿 + 𝑗𝐵𝐿) = −(𝑃𝐿 𝑉𝐿2⁄ − 𝑗(𝑄𝐿 𝑉𝐿

2⁄ ))

14

There is a negative sign for ��𝐿, since loads are assumed as injected quantities [2].

(2.2)

The internal voltages of the generators are also calculated from the load-flow data. These

internal angles are computed from the pre-transient terminal voltages.

𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑤𝑖𝑡ℎ

𝐼 = 𝐼1 + 𝑗𝐼2

(2.3)

𝑇ℎ𝑒𝑛 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑃 + 𝑗𝑄 = ��𝐼∗ 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑑𝑒𝑟𝑖𝑣𝑒𝑑

𝐼1 + 𝑗𝐼2 = (𝑃 − 𝑗𝑄) 𝑉⁄

(2.4)

𝑎𝑛𝑑 𝑠𝑖𝑛𝑐𝑒 𝐸∠𝛿′ = �� + 𝑗𝑥𝑑′ 𝐼 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑑𝑒𝑟𝑖𝑣𝑒𝑑

𝐸∠𝛿′ = (𝑉 + 𝑄𝑥𝑑′ 𝑉⁄ ) + 𝑗(𝑃𝑥𝑑

′ /𝑉)

(2.5)

The initial generator angle δ0 is obtained by adding the pre-transient voltage angle θ to δ’.

In order to combine the system admittance matrix with the load admittances and generator

impedances, the following procedure is taken: The system admittance matrix is modified by

adding each load admittance to its corresponding bus (node). Then the system admittance

matrix is augmented to include the generator terminal buses (nodes) by creating a diagonal

matrix of the generator admittances 𝑦 which is simply the inverse of its impedance and then

adding in the fashion shown in equation (2.6).

15

𝑌𝑎𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑 =

[

𝑦 | −𝑦 | 0

− | − − − − −−𝑦−0

| 𝑌𝑚𝑜𝑑𝑖𝑓𝑖𝑒𝑑]

(2.6)

Finally, all the nodes are eliminated except for the internal generator nodes thus obtaining the

reduced network. The reduction can be achieved by matrix operation since all the nodes have

zero injection currents except for the internal generator nodes. The reduction is shown in

equation (2.7).

[𝐼𝑚−0

] = [𝑌𝑚𝑚 | 𝑌𝑚𝑟

− − −𝑌𝑟𝑚 | 𝑌𝑟𝑟

] [𝑉𝑚−𝑉𝑟

]

Where m defines a generator node and r a remaining node.

𝐴𝑛𝑑 𝑏𝑦 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝐼𝑚 = (𝑌𝑚𝑚 − 𝑌𝑚𝑟𝑌𝑟𝑟−1𝑌𝑟𝑚)𝑉𝑚

(2.7)

𝑡ℎ𝑢𝑠 𝑌𝑟𝑒𝑑𝑢𝑐𝑒𝑑 = (𝑌𝑚𝑚 − 𝑌𝑚𝑟𝑌𝑟𝑟−1𝑌𝑟𝑚)𝑉𝑚 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑎𝑛 𝑚 × 𝑚 𝑚𝑎𝑡𝑟𝑖𝑥.

(2.8)

Note that this reduction can be done if the loads are assumed to be constant admittances.

Since the network buses have been eliminated, the internal nodes may be renumbered from 1,

…, m [2].

Where

𝐼�� = ∑𝑌𝑖𝑗 𝐸��

𝑚

𝑗=1

𝑖 = 1,… ,𝑚

16

(2.9)

The real electric power into the network at node 𝑖, which is the electrical power output 𝑃𝑒𝑖 of

machine 𝑖 in an m – machine system is given by

𝑃𝑒𝑖 = 𝑅𝑒[𝐸��𝐼��∗]

Replacing 𝐼��

𝑃𝑒𝑖 = 𝑅𝑒 [𝐸𝑖𝑒𝑗𝛿𝑖 ∑𝑌𝑖𝑗

∗𝐸��

∗𝑚

𝑗=1

]

𝑃𝑒𝑖 = 𝑅𝑒 [𝐸𝑖𝑒𝑗𝛿𝑖 ∑(𝐺𝑖𝑗 − 𝑗𝐵𝑖𝑗)𝐸𝑗𝑒

−𝑗𝛿𝑖

𝑚

𝑗=1

]

𝑃𝑒𝑖 = 𝑅𝑒 [∑(𝐺𝑖𝑗 − 𝑗𝐵𝑖𝑗)𝐸𝑖𝐸𝑗

𝑚

𝑗=1

[cos(𝛿𝑖 − 𝛿𝑗) + 𝑗 sin(𝛿𝑖 − 𝛿𝑗)]]

𝐸𝑖 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑏𝑒ℎ𝑖𝑛𝑑 𝑡𝑟𝑎𝑛𝑠𝑖𝑒𝑛𝑡 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑐𝑒 𝑓𝑜𝑟 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑖

𝐼𝑖 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑜𝑢𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑛𝑜𝑑𝑒 𝑖

𝑌𝑖𝑖 = 𝐺𝑖𝑖 + 𝑗𝐵𝑖𝑖 𝑖𝑠 𝑎 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑒𝑡𝑤𝑜𝑟𝑘 𝑎𝑑𝑚𝑖𝑡𝑡𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑌𝑟𝑒𝑑𝑢𝑐𝑒𝑑

𝑌𝑖𝑗 = 𝐺𝑖𝑗 + 𝑗𝐵𝑖𝑗 𝑖𝑠 𝑎𝑛 𝑜𝑓𝑓 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑒𝑡𝑤𝑜𝑟𝑘 𝑎𝑑𝑚𝑖𝑡𝑡𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑌𝑟𝑒𝑑𝑢𝑐𝑒𝑑

𝑑𝑒𝑓𝑖𝑛𝑒:

𝛿𝑖𝑗 = 𝛿𝑖 − 𝛿𝑗

𝑃𝑒𝑖 = ∑𝐸𝑖𝐸𝑗(𝐵𝑖𝑗 sin 𝛿𝑖𝑗 + 𝐺𝑖𝑗 cos 𝛿𝑖𝑗)

𝑚

𝑗=1

17

(2.10)

2.2 Small Signal Stability

The process begins with modeling the system in order to mathematically quantify the system’s

response to whatever stimulus is injected. Necessary assumptions are made to achieve minimal

error and complexity. The system behavior is then presented and analyzed based on the design

methodology used in this study [5]. Small signal stability analysis is the transformation of the

complex dynamic system to a linearized model. In other words, the non-linear system equations

are linearized around an equilibrium point. Most often state-space representation is used. This

representation format quantifies the system into the simplest of terms while still holding the

true characteristics needed to outline the behavior of the system with minimal error. The

components that make up this representation are the state variables of the power system itself,

and it is the combination of these variables along with system inputs that determine how the

system performs. For any given system, there exists a singular behavior pattern for that system

given a set of inputs [6]. Nevertheless, there are often nonexclusive state space representations

for that system, as differing state variables can often be chosen to accurately represent the

system. The equilibrium points around which the analysis takes place basically takes a snapshot

of the response of the system to a given input at a specific instant of time i.e. setting all of the

derivatives to zero and then solving the equations graphically representing zero velocity [6].

Linear systems only have one equilibrium point and satisfy the equation f(x0) =0 [7] and,

therefore, consist of information about the stability of the system. The nature of linear systems

is that they are able to depict stability or instability independent of the input. The nature of the

system depends only on the system itself. Any stable system will return to a stable state

assuming zero input.

18

The state space representation and its corresponding linear system model of a power system

begin with transforming the nonlinear differential equations. These differential equations are

those, such as the popular swing equation, that numerically model the operation of the

different components of the system. In the instance of the generator models, the angular

dynamics between the rotor and stator axis depend on the angular difference [5].

The swing equation governs the motion of the machine rotor relating the inertia torque to the

resultant of the mechanical and electrical torques on the rotor as shown in equation (3.1) [4]

𝑑𝛿𝑖

𝑑𝑡= 𝜔𝑖 − 𝜔𝑠 𝑖 = 1,… ,𝑚

(3.1)

𝐽𝑖𝑑𝜔𝑖

𝑑𝑡= 𝑇𝑎𝑖 = 𝑇𝑀𝑖 − 𝑇𝑒𝑖

𝑤ℎ𝑒𝑟𝑒 𝐽𝑖 =2𝐻𝑖

𝜔𝑠

Due to the fact that the angular speed 𝜔𝑖 is nearly constant the accelerating power 𝑃𝑎𝑖 is

numerically nearly equal to the accelerating torque 𝑇𝑎𝑖 [4] and substituting 𝐽𝑖

2𝐻𝑖

𝜔𝑠

𝑑𝜔𝑖

𝑑𝑡= 𝑃𝑎𝑖 = 𝑃𝑀𝑖 − 𝑃𝑒𝑖 − 𝑑𝑖𝜔𝑖

𝑊ℎ𝑒𝑟𝑒 𝑑𝑖𝜔𝑖 𝑖𝑠 𝑎 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 𝑡𝑜𝑟𝑞𝑢𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑑𝑑𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛

(3.2)

𝐿𝑖𝑛𝑒𝑎𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑎𝑟𝑜𝑢𝑛𝑑 𝑎𝑛 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡 0 𝑔𝑖𝑣𝑒𝑠

𝑑

𝑑𝑡𝛥𝛿𝑖 = 𝛥𝜔𝑖

19

𝑑

𝑑𝑡𝛥𝜔𝑖 =

𝜔𝑠

2𝐻𝑖

[𝛥𝑃𝑀𝑖 − 𝛥𝑃𝑒𝑖 − 𝑑𝑖𝛥𝜔𝑖]

(3.3)

𝐵𝑒𝑐𝑎𝑢𝑠𝑒 𝑃𝑀𝑖 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑛𝑑 𝛥𝑃𝑒𝑖 = ∑𝜕𝑃𝑒𝑖

𝜕𝛿𝑗𝛥𝛿𝑗

𝑚

𝑗=1𝑗≠𝑖

𝑑

𝑑𝑡𝛥𝛿𝑖 = 𝛥𝜔𝑖

𝑑

𝑑𝑡𝛥𝜔𝑖 = −

𝜔𝑠

2𝐻𝑖∑(

𝜕𝑃𝑒𝑖

𝜕𝛿𝑗𝛥𝛿𝑗)

𝑚

𝑗=1

−𝜔𝑠

2𝐻𝑖𝑑𝑖𝛥𝜔𝑖

(3.4)

𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑛𝑔 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (2.10)

−𝜕𝑃𝑒𝑖

𝜕𝛿𝑗 𝑖𝑠 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑎𝑠 𝑠ℎ𝑜𝑤𝑛 𝑖𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (3.5)

= −∑𝐸𝑖𝐸𝑗(𝐵𝑖𝑗 cos 𝛿𝑖𝑗0 − 𝐺𝑖𝑗 sin𝛿𝑖𝑗0)

𝑚

𝑗=1𝑗≠𝑖

𝑓𝑜𝑟 𝑗 = 𝑖

= 𝐸𝑖𝐸𝑗(𝐵𝑖𝑗 cos𝛿𝑖𝑗0 − 𝐺𝑖𝑗 sin 𝛿𝑖𝑗0) 𝑓𝑜𝑟 𝑗 ≠ 𝑖

(3.5)

𝐼𝑛 𝑚𝑎𝑥𝑡𝑟𝑖𝑥 𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠

20

[ 𝛥𝛿1

⋮𝛥𝛿𝑚

−

𝛥𝜔1

⋮𝛥𝜔𝑚

]

= [0 | 𝐼− − −𝐴𝜔 | 𝐷

]

[ 𝛥𝛿1

⋮𝛥𝛿𝑚−

𝛥𝜔1

⋮𝛥𝜔𝑚]

(3.6)

𝑇ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓𝐴𝜔𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 −𝜔𝑠

2𝐻𝑖

𝜕𝑃𝑒𝑖

𝜕𝛿𝑗. 𝑇ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴𝜔 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠

𝐴𝜔 =

[ −

𝜔𝑠

2𝐻1

𝜕𝑃𝑒1

𝜕𝛿1⋯ −

𝜔𝑠

2𝐻1

𝜕𝑃𝑒1

𝜕𝛿𝑚

⋮ ⋱ ⋮

−𝜔𝑠

2𝐻𝑚

𝜕𝑃𝑒𝑚

𝜕𝛿1⋯ −

𝜔𝑠

2𝐻𝑚

𝜕𝑃𝑒𝑚

𝜕𝛿𝑚 ]

0

𝑤ℎ𝑒𝑟𝑒 [𝐴𝜔]0 𝑖𝑠 𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑧𝑒𝑑 𝑎𝑡 𝑎 𝑝𝑜𝑖𝑛𝑡 ′0′ 𝑖𝑛 𝑡𝑖𝑚𝑒

(3.7)

𝑇ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓𝐷 𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 −𝜔𝑠

2𝐻𝑖𝑑𝑖. 𝑇ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝐷 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠

𝐷 =

[ −

𝜔𝑠

2𝐻1𝑑1 ⋯ 0

⋮ ⋱ ⋮

0 ⋯ −𝜔𝑠

2𝐻𝑚𝑑𝑚]

0

𝑤ℎ𝑒𝑟𝑒 [𝐷]0 𝑖𝑠 𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑧𝑒𝑑 𝑎𝑡 𝑎 𝑝𝑜𝑖𝑛𝑡 ′0′ 𝑖𝑛 𝑡𝑖𝑚𝑒

(3.8)

2.3 Participation Factor Analysis

Due to the large size of the power system, it is often necessary to construct reduced order

models for dynamic stability studies by retaining only a few modes [2]. The appropriate

definition and determination as to which state variables significantly participate in the selected

21

modes become very important. This requires a tool for identifying the state variables that have

significant state variables for an eigenvalue λi are those that correspond to large entries in the

corresponding eigenvector νi. However, the entries in the eigenvector are dependent on the

dimensions of the state variables which in this study are the generator rotor angle δ and speed

ω. Thus, participation factor analysis is used to measure state variable participation. For linear

systems, participation factor analysis is used in the identification of how each dynamic variable

affects a given system mode which corresponds to an eigenvalue [8].

𝐺𝑖𝑣𝑒𝑛 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑦𝑠𝑡𝑒𝑚 �� = 𝐴𝑥

A participation factor is a sensitivity measure of an eigenvalue to a diagonal entry of the system

which is defined as

𝑝𝑘𝑖 =𝜕𝜆𝑖

𝜕𝑎𝑘𝑘

(3.9)

Where λi is the ith system eigenvalue, akk is a diagonal entry in the system A matrix, and pki is the

participation factor relating the kth state variable to the ith eigenvalue. pki can also be defines as

𝑝𝑘𝑖 =𝑤𝑘𝑖ν𝑘𝑖

𝑤𝑖𝑡ν𝑖

(3.10)

Where wki and νki are the kth entries in the left and right eigenvector associated with the ith

eigenvalue. The right eigenvector νi and the left eigenvector wi associated with the ith eigenvalue

λi satisfy the following equations

𝐴ν𝑖 = 𝜆𝑖ν𝑖

(3.11)

22

𝑤𝑖𝑡𝐴 = 𝑤𝑖

𝑡𝜆𝑖

(3.12)

To establish the equivalence the following system is considered

[𝐴 − 𝜆𝑖𝐼]ν𝑖 = 0

(3.13)

𝑤𝑖𝑡[𝐴 − 𝜆𝑖𝐼] = 0

(3.14)

Assuming that the eigenvalues and eigenvectors vary continuously with respect to the elements

of the matrix A, the following equation is derived as shown

(𝐴 + 𝛥𝐴)(ν𝑖 + 𝛥ν𝑖) = (𝜆𝑖 + 𝛥𝜆𝑖)(ν𝑖 + 𝛥ν𝑖)

(3.15)

[𝐴ν𝑖] + [𝛥𝐴ν𝑖 + 𝐴𝛥ν𝑖] + [𝛥𝐴𝛥ν𝑖] = [𝜆𝑖ν𝑖] + [𝛥𝜆𝑖ν𝑖 + 𝜆𝑖𝛥ν𝑖] + [𝛥𝜆𝑖𝛥ν𝑖]

(3.16)

[𝐴 − 𝜆𝑖𝐼]𝛥ν𝑖 + 𝛥𝐴ν𝑖 = 𝛥𝜆𝑖ν𝑖

(3.17)

𝑤𝑖𝑡[𝐴 − 𝜆𝑖𝐼]𝛥ν𝑖 + 𝑤𝑖

𝑡𝛥𝐴ν𝑖 = 𝑤𝑖𝑡𝛥𝜆𝑖ν𝑖

(3.18)

𝑤𝑖𝑡𝛥𝐴ν𝑖 = 𝑤𝑖

𝑡𝛥𝜆𝑖ν𝑖

23

(3.19)

The sensitivities of λi with respect to diagonal entries of A are related to the participation

factors. Assuming that the kth diagonal entry of A akk is perturbed then it will result in

𝑤𝑖𝑡𝛥𝐴ν𝑖 = 𝑤𝑖

𝑡𝛥a𝑘𝑘ν𝑖 = 𝑤𝑖𝑡𝛥𝜆𝑖ν𝑖

(3.20)

𝑠𝑜𝑙𝑣𝑖𝑛𝑔 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 𝑔𝑖𝑣𝑒𝑠 𝑡ℎ𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝑎𝑠 𝑝𝑘𝑖 =𝑤𝑘𝑖ν𝑘𝑖

𝑤𝑖𝑡ν𝑖

Participation factor analysis is a type of analysis that indicates the influence (participation) of a xi

on its corresponding mode λk. This process is explained as shown in equation (3.27).

𝑉: 𝑟𝑖𝑔ℎ𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑚𝑎𝑡𝑟𝑖𝑥

𝑊: 𝑙𝑒𝑓𝑡 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 𝑚𝑎𝑡𝑟𝑖𝑥

𝑃 = 𝑝𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝑚𝑎𝑡𝑟𝑖𝑥

𝑉 × 𝐿 = 𝐼 𝑤ℎ𝑒𝑟𝑒 𝐼 𝑖𝑠 𝑡ℎ𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑚𝑎𝑡𝑟𝑖𝑥

𝑉 = ([𝜈11

𝜈1𝑘

⋮]… [

𝜈𝑖1

𝜈𝑖𝑘

⋮]… [

𝜈𝑛1

𝜈𝑛𝑘

⋮])

𝑊 =

(

[𝑤11 𝑤1𝑘 ⋯]⋮

[𝑤𝑖1 𝑤𝑖𝑘 ⋯]⋮

[𝑤𝑛1 𝑤𝑛𝑘 ⋯])

state

mode

mode

state

24

�� = 𝐴𝑥 → [𝑥1

𝑥��

⋮] = ([

𝑎11

𝑎1𝑘

⋮]… [

𝑎𝑖1

𝑎𝑖𝑘

⋮]… [

𝑎𝑛1

𝑎𝑛𝑘

⋮]) [

𝑥1

𝑥𝑘

⋮]

𝑃 = [

𝑝11 ⋯⋮ ⋯

𝑝1𝑖 ⋯⋮ ⋯

𝑝11 ⋯⋮ ⋯

𝑝𝑘𝑖 ⋯⋮ ⋯

]

𝑃(𝑘, 𝑖) = |𝑊(𝑖, 𝑘) · 𝑉(𝑘, 𝑖)|

(3.21)

After identifying an unstable mode represented by eigenvalue λk=σk±jαk where either σk >0 or

|σk|<ε and using participation factor analysis, it can be determined which state xi has most

influence on that particular mode λk by choosing the largest value pki of column matrix Pk. By

using this analysis, it can be deduced that returning the system back to stability i.e. letting σk <0

and |σk|>ε of the unstable mode λk is primarily dependent on controlling state xi. Moreover, the

state xi is associated with a specific generator which provides a starting point to control the

system.

Another factor which provides a good idea of the behavior of the system is the damping ratio γ

given by the following equation.

mode

mode

state

state

25

𝛾 =−𝜎

√𝜎2 + 𝛼2

(3.22)

2.4 Power System Stabilizer

During the 1950’s new generator units were equipped with voltage regulators (VR). However, as

the new units with their associated VR became a large percentage of the generating capability of

the system, the system began to experience instability during small magnitude oscillations and

low frequencies which persisted for a long period of time. Power System Stabilizers were

introduced to provide damping to the machine oscillations by producing an electrical torque on

the rotor which is in phase with speed variations [5].

A basic power system stabilizers design is based on the concept of classical control theory. This

design has been widely used in actual systems. It is simply a phase feedback control of a

generator’s compensated signal derived from speed, power, or frequency which then generates

a supplementary rotor torque to damp the oscillations [2].

Power system oscillations are a characteristic of the system and they are inevitable. Power

system stabilizers can help the power system oscillations damping. This controller has become

an accepted solution for oscillatory instability problems and thus improves system stability [2].

Power System Stabilizers are generator control equipment which are used in feedback to

enhance the damping of rotor oscillations caused due to small disturbances [2].

The power system stabilizer model as shown in Figure 28 is provided by the stability analysis

software tool PowerWorld. Once the power system stabilizer(s) is inserted in the system the

constants A1 through A6 and T1 through T6 are tuned using trial and error to achieve acceptable

damping on the rotor angle oscillations caused by a disturbance.

26

The power system stabilizers provided additional damping torque to the system usually a few

power swings in the rotor angle oscillations. If a disturbance is likely to render the system

unstable, when no control is provided, the additional damping assists the system to return to a

stable state.

Figure 2: Block Diagram Model Used in Stability Analysis

27

Chapter 3

3.1 System Model – IEEE 39 Bus System

3.1.1 General Outline

In order to succeed in the interpretation and analysis of our data using the analytical methods

outlined in Chapter 2, a test system must be used as a starting point for our analysis. In other

words, system data will be required and then applied to the equations derived in Chapter 2.

Thus, the IEEE 39 bus system is chosen as the representative test system.

This system is also well known as the 10-machine New-England Power System. Generators are

represented as the aggregation of a large number of generators. In this test system, one

generator is selected as the swing bus. For this test system, bus 31 was selected as the swing

bus and its characteristics will display as such. The presentation of system characteristics and

general descriptions for the various parameters needed by MATLAB and PowerWorld to

properly model the test system can be found in Appendix A. All parameters are from the book

titled 'Energy Function Analysis for Power System Stability.

28

3.1.2 Network

The network configuration and connectivity is depicted as shown in Figure 35 in Appendix A.

3.1.3 Generator Data

Regarding the modeling of the generators’ dynamic characteristics in both MATLAB and

PowerWorld, there are a number of parameters that must be inputted. These include:

Machine inertia

Machine sub-transient reactance

Machine armature resistance

All generator data can be found in Table 17 in Appendix A.

3.1.4 Branch Data – Lines and Transformers

Branch data is used to form the network admittance matrix using MATLAB

Concerning the line data, the data needed for the equivalent pi line model includes the

following:

The connected buses on both sides

The line impedance (resistance and reactance) in per unit

Total charging susceptance in per unit.

Concerning the transformer data, all transformers are modeled as two winding transformers

and their data includes the following:

The connected buses on both sides

The internal transformer winding impedance in per unit

29

The turns ratio and phase angle difference with respect to the ‘from bus’

All branch data can be found in Table 18 in Appendix A.

3.1.5 Power and Voltage Data

All data concerned with the power and voltage that are inputted into MATLAB to conduct load

flow analysis are as follows:

Type of bus varying from load bus to generator bus

Voltage set point of the bus in per unit

Real Power that is consumed by the load in MW

Reactive Power that is consumed by the load in Mvar

Real Power that is supplied by the generator in MW

Reactive Power that is supplied by the generator in Mvar

Generator unit number connected to the bus

There are three different types of buses when conducting a load flow listed below:

A swing/slack bus where both the bus voltage V and angle θ are kept constant

A generator or PV bus where the real power P and voltage V are kept constant

A load or PQ bus where the real and imaginary power P and Q are kept constant while

its corresponding voltage V and angle θ are variable

All power and voltage data can be found in Table 19 in Appendix A.

3.2 Simulation Software

In order to model the power system, the system data as described above is inputted in both

MATLAB and PowerWorld. The description, capabilities and functionality of these systems as

they pertain to the system analysis involved in this study are described as follows.

30

Using MATLAB scripts, the admittance matrix is calculated and stored. The load flow calculation

is performed using the Newton-Raphson method. Load flow data in addition to machine data is

used to model the dynamic performance of the system as a system of linear equations using

small signal stability. Small disturbances represented as load changes are implemented

iteratively on the system by modifying the load data and performing a load flow calculation. This

iterative process is repeated until the system renders unstable by modal analysis i.e. eigenvalue

and eigenvector calculation. Finally, participation factors are calculated which will indicate the

state’s influence on its corresponding modes; thus determining the most influential mode. All

results will be shown in tabular format.

The power system steady state and dynamic model is reproduced in PowerWorld as shown in

Figure 3. Then by utilizing the transient stability module which is presented in Figure 4, plots for

generator rotor angles, rotor speeds, and accelerating power as shown in Figures 6, 7, and 8 are

used to display generator transient performance. Predefined power system stabilizer models

with their corresponding parameter inputs as shown in Figure 5 are used to determine locations

by participation factor analysis to stabilize the system. The plots as shown in Figure 6, 7, and 8

will also be used to depict the influence of the power system stabilizers on controlling the

system and returning it to a stable one.

31

Figure 3: Load Flow data Reproduced in PowerWorld

Figure 4: Transient Stability Module

32

Figure 5: Pre-Defined Power System Stabilizer Model

33

Figure 6: Gen. Rotor Angle Plot Sample

34

Figure 7: Gen. Rotor Speed Plot Sample

35

Figure 8: Gen. Real Accelerating Power Plot Sample

It is easy to observe the system’s stability in that all the modes (eigenvalues) are located to the

left half of the imaginary plane. Had there been any unstable modes, they would have been

located to the right half of the imaginary plane. Degree of stability is also easy to detect. The

farther away each mode is from the imaginary plane the more stable the system is. And the

closer a mode is to the imaginary plane, the closer that mode it to being unstable. In addition,

damping information can be graphically determined. The farther away the mode is from the

36

imaginary axis the larger the damping of that mode will be and vice-versa [8]. An example of

that graphical analysis is shown in Figure 9.

Figure 9: Mode (Eigenvalue) Graphic Representation

Chapter 3 concludes the background information and system data that will be used in the power

system research. The IEEE 39 bust test system was identified as well as its inputs. The MATLAB

script and PowerWorld software was also introduced. All of the scripts calculative procedures

pertaining to small signal stability, modal analysis, and participation factor calculation are

described in the literature review. The research methodology will now be presented in Chapter

4.

37

Chapter 4

4.1 Simulation Results and Analysis

4.1.1 Methodology

After defining the electrical system that will be put under test and describing the simulation

software and scripts, a system analysis and simulation will follow. The analysis and simulation

process of the IEEE 39 Bus system (New England system) contains several steps.

4.1.2 Base Case Modal Analysis

The modes of an undisturbed system or the Base case are calculated to achieve a good

understanding of the system stability and response behavior. This is achieved by running a load

flow calculation on the Base case and calculating the eigenvalues using small signal stability

analysis as described in the previous chapters. Once the results are obtained, a good starting

point will be established in order to have a good comparison for later results.

Using the data provided as depicted in Chapter 3 an admittance matrix is calculated followed by

a load flow calculation. The Base case load flow results are shown in Table 1.

38

Table 1: Base Case Load Flow Results

Bus V (PU) Theta (Rad) P (PU) Q (PU)

1 1.0474 -0.1473 0 0

2 1.0487 -0.1004 0 0

3 1.0302 -0.1501 -3.22 -0.024

4 1.0039 -0.1677 -5 -1.84

5 1.0053 -0.1503 0 0

6 1.0077 -0.1387 0 0

7 0.997 -0.1767 -2.338 -0.84

8 0.996 -0.1853 -5.22 -1.76

9 1.0282 -0.1802 0 0

10 1.0171 -0.0947 0 0

11 1.0127 -0.1097 0 0

12 1.0001 -0.109 -0.075 -0.88

13 1.0143 -0.1064 0 0

14 1.0117 -0.1336 0 0

15 1.0154 -0.135 -3.2 -1.53

16 1.0318 -0.108 -3.29 -0.323

17 1.0335 -0.1274 0 0

18 1.0309 -0.1435 -1.58 -0.3

19 1.0499 -0.0178 0 0

20 0.9912 -0.0352 -6.28 -1.03

21 1.0317 -0.066 -2.74 -1.15

22 1.0498 0.0117 0 0

23 1.0448 0.0082 -2.475 -0.846

24 1.0373 -0.1059 -3.086 0.92

25 1.0576 -0.0762 -2.24 -0.472

26 1.0521 -0.0965 -1.39 -0.17

27 1.0377 -0.1308 -2.81 -0.755

28 1.0501 -0.0352 -2.06 -0.276

29 1.0499 0.013 -2.835 -0.269

30 1.0475 -0.0582 2.5 1.4617

31 0.982 0 5.1161 1.9366

32 0.9831 0.0448 6.5 2.0516

33 0.9972 0.0732 6.32 1.0994

34 1.0123 0.0554 5.08 1.6578

35 1.0493 0.0983 6.5 2.1246

36 1.0635 0.1453 5.6 1.0121

37 1.0278 0.0423 5.4 0.0045

38 1.0265 0.1363 8.3 0.2285

39 1.03 -0.1755 -1.04 -1.6171

39

Referring to Table 1, the voltages at the PQ buses are between 0.99p.u. and 1.06p.u. which

indicate that the system is voltage stable as expected of an undisturbed system.

Once the load flow is complete, the initial values are determined. In addition to that, all loads

are converted to constant impedances and with the internal generator impedance data the

reduced admittance matrix is formed. Using these results the system is linearized using small

signal analysis; the eigenvalues are calculated with their corresponding largest participation

factor and damping ratio determining the most influential state on that mode. A summary table

of the results in addition to a plot of the modes is depicted in Table 2 and Figure 10.

Table 2: Modal Analysis Summary Table

Eigenvalue/Mode Largest Participation factor State Gen_Index Damping ratio

1 -0.2959 0.5451 speed 5 1

2 0 0.1272 angle 2 1

3 -0.1891 + 4.1623i 0.2147 speed 5 0.0454

4 -0.1891 - 4.1623i 0.2147 speed 5 0.0454

5 -0.2973 + 5.9988i 0.3458 speed 9 0.0495

6 -0.2973 - 5.9988i 0.3458 speed 9 0.0495

7 -0.3002 + 6.4350i 0.1521 angle 2 0.0466

8 -0.3002 - 6.4350i 0.1521 angle 2 0.0466

9 -0.3073 + 7.2381i 0.1947 speed 8 0.0424

10 -0.3073 - 7.2381i 0.1947 speed 8 0.0424

11 -0.2737 + 8.2921i 0.2637 speed 6 0.033

12 -0.2737 - 8.2921i 0.2637 speed 6 0.033

13 -0.2758 + 8.5519i 0.2566 speed 10 0.0322

14 -0.2758 - 8.5519i 0.2566 speed 10 0.0322

15 -0.3238 + 9.7490i 0.306 angle 3 0.0332

16 -0.3238 - 9.7490i 0.306 angle 3 0.0332

17 -0.3512 + 9.1023i 0.3833 angle 4 0.0386

18 -0.3512 - 9.1023i 0.3833 angle 4 0.0386

19 -0.3321 + 9.2995i 0.4039 speed 7 0.0357

20 -0.3321 - 9.2995i 0.4039 speed 7 0.0357

40

Figure 10: Graphical Display of Modes (Eigenvalues)

Referring to Table 2 and Figure 10, it is observed that the system is stable as expected since all

the modes λ are located on the left half of the imaginary plane. In addition to that, all modes

have large absolute values of their real-part σ which display that the mode is far away from the

origin and damping factors ϒ greater than 3% which also displays that the mode’s slope is very

steep. These observed factors indicate that the system is stable.

4.1.3 Stability Response to System Changes and System Stability Analysis

In the second part of our analysis, small disturbances are simulated in two distinct ways applied

to the system; one way is concerned in load changes, while the other in topological changes .

Similar calculations will be executed and a comparison with the Base case will be made. This

41

comparison will portray the response of these small disturbances. Also, the results by analyzing

the modal behavior due to each disturbance are quantified. In addition to that, using

participation factor analysis, the influence of a specific state on certain modes is determined and

a general idea of system control can then be identified.

In order to simulate a small disturbance as a load change, additional real load is added

incrementally to a load bus and in the case of a topology change a line is disconnected as a

design contingency. Finally, a load flow calculation is obtained for each test case.

Each load increment and the design contingency are both represented by test cases as seen in

Table 3.

Table 3: List of Test Cases

Test Case Load Bus Load Increment (MW)

1a 3 1000

1b 3 2000

1c 3 3900

Test Case Design Contigency

1d Line 9-39 Disconnected

42

The load flow results for each test case are shown in Tables 4 through 7.

Table 4: Test Case 1a Load Flow Results


1 1.0433 -0.0699 0 0

2 1.0368 -0.1603 0 0

3 1.0084 -0.2747 -13.22 -0.024

4 0.9887 -0.2224 -5 -1.84

5 0.9945 -0.1711 0 0

6 0.9979 -0.1568 -0.0001 0

7 0.9869 -0.1849 -2.338 -0.84

8 0.9857 -0.1883 -5.22 -1.76

9 1.0212 -0.0839 0 0

10 1.0082 -0.1212 0 0

11 1.0035 -0.1335 0 0

12 0.9902 -0.1366 -0.075 -0.88

13 1.0041 -0.1378 0 0

14 0.999 -0.1778 0 0

15 1.0038 -0.1894 -3.2 -1.53

16 1.0208 -0.1661 -3.29 -0.323

17 1.019 -0.2022 0 0

18 1.013 -0.2374 -1.58 -0.3

19 1.0452 -0.0624 0 0

20 0.9884 -0.0753 -6.28 -1.03

21 1.0232 -0.1175 -2.74 -1.15

22 1.0448 -0.033 0 0

23 1.0395 -0.0363 -2.475 -0.846

24 1.027 -0.1623 -3.086 0.92

25 1.0494 -0.135 -2.24 -0.472

26 1.0419 -0.1567 -1.39 -0.17

27 1.0252 -0.1981 -2.81 -0.755

28 1.0438 -0.0853 -2.06 -0.276

29 1.0453 -0.0338 -2.835 -0.269

30 1.0475 -0.1084 3.0366 2.164

31 0.982 0 5.7199 2.39

32 0.9831 0.0296 6.9574 2.5342

33 0.9972 0.0342 6.6854 1.4202

34 1.0123 0.0214 5.4122 1.825

35 1.0493 0.06 6.9446 2.5263

36 1.0635 0.1098 5.9373 1.2628

37 1.0278 -0.009 5.7105 0.3897

38 1.0265 0.0964 8.7408 0.5587

39 1.03 -0.0138 5.3481 -1.2621

43

Referring to Table 4, with a load addition of 1000MW at Bus #3, the voltages at the PQ buses

decrease to approximately 99.5% of Base case levels.

Table 5: Test Case 1b Load Flow Results


1 1.0193 -0.0019 -0.0001 0.0001

2 1.0076 -0.2443 -0.0005 0.0005

3 0.9628 -0.4349 -23.2182 -0.0235

4 0.9521 -0.2988 -4.9997 -1.8396

5 0.9656 -0.2066 -0.0003 0.0002

6 0.9714 -0.1884 -0.001 0.0006

7 0.9582 -0.2076 -2.3378 -0.84

8 0.9561 -0.2058 -5.2197 -1.7598

9 0.9991 0.0062 0.0001 0.0001

10 0.9859 -0.1624 -0.0001 0.0001

11 0.9797 -0.1717 0 0

12 0.9652 -0.1794 -0.075 -0.88

13 0.9793 -0.1851 -0.0001 0.0001

14 0.9694 -0.2411 -0.0001 0.0001

15 0.979 -0.2668 -3.1998 -1.5299

16 0.9985 -0.248 -3.29 -0.3228

17 0.989 -0.3035 -0.0001 0.0001

18 0.9755 -0.3611 -1.5801 -0.2999

19 1.0363 -0.1292 0.0001 0.0001

20 0.9833 -0.1377 -6.28 -1.03

21 1.0065 -0.1917 -2.74 -1.1499

22 1.0353 -0.0992 0.0001 0.0001

23 1.0296 -0.1024 -2.475 -0.8459

24 1.0062 -0.2423 -3.086 0.92

25 1.029 -0.2204 -2.2397 -0.4721

26 1.0207 -0.243 -1.39 -0.1699

27 0.9994 -0.2921 -2.81 -0.755

28 1.0316 -0.1602 -2.06 -0.276

29 1.0366 -0.1048 -2.8349 -0.2689

30 1.0475 -0.1814 3.5732 3.8444

31 0.982 0 6.6805 3.5427

32 0.9831 0.0021 7.4147 3.6449

33 0.9972 -0.0268 7.0508 2.0257

34 1.0123 -0.0346 5.7444 2.1268

35 1.0493 0.0007 7.3892 3.2503

36 1.0635 0.0534 6.2746 1.6985

37 1.0278 -0.0854 6.0209 1.3076

38 1.0265 0.0327 9.1816 1.1533

39 1.03 0.1428 11.7363 1.0308

44




1 0.8888 0.0482 -0.0005 0.0001

2 0.8589 -0.5923 -0.0017 0.0001

3 0.7309 -1.0297 -37.1139 -0.0239

4 0.752 -0.5994 -5.0005 -1.8398

5 0.8078 -0.3701 -0.0005 0

6 0.8261 -0.3361 -0.0006 0.0003

7 0.8036 -0.3462 -2.3379 -0.84

8 0.7979 -0.3349 -5.2199 -1.76

9 0.8985 0.1209 -0.0002 0

10 0.8642 -0.3382 -0.0002 0

11 0.8496 -0.3388 0 0

12 0.8285 -0.3615 -0.075 -0.88

13 0.8445 -0.3817 -0.0002 0

14 0.8094 -0.4959 -0.0002 0.0001

15 0.8507 -0.5878 -3.1997 -1.53

16 0.8879 -0.5861 -3.2902 -0.323

17 0.838 -0.6986 -0.0005 0.0001

18 0.7849 -0.8267 -1.5806 -0.2999

19 0.994 -0.4321 -0.0001 0

20 0.9599 -0.4351 -6.28 -1.03

21 0.9262 -0.5092 -2.74 -1.15

22 0.9912 -0.3965 0 0

23 0.9837 -0.3996 -2.475 -0.846

24 0.9044 -0.5763 -3.086 0.92

25 0.9231 -0.5882 -2.2398 -0.4721

26 0.9172 -0.6096 -1.39 -0.17

27 0.872 -0.6767 -2.8099 -0.755

28 0.9756 -0.5007 -2.06 -0.276

29 0.9977 -0.4366 -2.835 -0.269

30 1.0475 -0.5031 4.319 12.3172

31 0.982 0 10.0025 9.9428

32 0.9831 -0.134 8.0505 9.4484

33 0.9972 -0.3196 7.5587 4.8293

34 1.0123 -0.3221 6.2061 3.4612

35 1.0493 -0.2834 8.0072 6.4909

36 1.0635 -0.2252 6.7434 3.5785

37 1.0278 -0.4297 6.4524 5.9685

38 1.0265 -0.2861 9.7942 3.7209

39 1.03 0.3612 20.6157 12.2172

45

Table 6: Test Case 1c Load Flow Results



46

Table 7: Test Case 1d Load Flow Results


1 1.0469 -0.1361 0 0

2 1.0465 -0.0967 0 0

3 1.0246 -0.1488 -3.22 -0.024

4 0.9923 -0.1693 -5 -1.84

5 0.9887 -0.153 0 0

6 0.9919 -0.1412 0 0

7 0.9752 -0.181 -2.338 -0.84

8 0.9713 -0.1902 -5.22 -1.76

9 0.9571 -0.1893 0 0

10 1.0071 -0.0958 0 0

11 1.0007 -0.1111 0 0

12 0.9888 -0.1102 -0.075 -0.88

13 1.0043 -0.1075 0 0

14 1.0021 -0.1347 0 0

15 1.0099 -0.1352 -3.2 -1.53

16 1.0282 -0.1075 -3.29 -0.323

17 1.0297 -0.1266 0 0

18 1.0264 -0.1426 -1.58 -0.3

19 1.0486 -0.0171 0 0

20 0.9905 -0.0345 -6.28 -1.03

21 1.0293 -0.0653 -2.74 -1.15

22 1.0485 0.0125 0 0

23 1.0434 0.0091 -2.475 -0.846

24 1.0341 -0.1054 -3.086 0.92

25 1.0557 -0.073 -2.24 -0.472

26 1.0498 -0.0945 -1.39 -0.17

27 1.0347 -0.1294 -2.81 -0.755

28 1.049 -0.0331 -2.06 -0.276

29 1.0491 0.0152 -2.835 -0.269

30 1.0475 -0.0543 2.5 1.5863

31 0.982 0 5.1229 2.522

32 0.9831 0.0452 6.5 2.5191

33 0.9972 0.074 6.32 1.1843

34 1.0123 0.0561 5.08 1.6971

35 1.0493 0.0992 6.5 2.2199

36 1.0635 0.1463 5.6 1.0661

37 1.0278 0.0455 5.4 0.0849

38 1.0265 0.1385 8.3 0.2798

39 1.03 -0.1596 -1.04 -0.4022

47

Referring to Table 7, with a system topology change by applying the contingency of

disconnecting Branch 9 – 39, it is noticed that the PQ bus voltages and angles change from that

of the Base case. For instance, the voltage and associated angle at Bus #1 which was

1.0474p.u.∠-0.1473rad decreased to 1.0469p.u.∠-0.1361rad. This will lead to the change in the

system’s initial conditions affecting the system’s stability.

Regarding the test cases 1a, 1b, and 1c, it can be concluded from the results that as the load is

increased at the load bus, the voltage levels decrease pushing the system into voltage instability.

This manipulation of the voltage levels i.e. that lead to voltage instability will give a good

understanding of how the modes behave due to this manipulation or disturbance.

Next using the test cases (1a, 1b, 1c, and 1d) as specified above, the eigenvalues are calculated

and a modal analysis is done to compare with that of the Base case; thus indicating the

difference in behavior after the disturbance. Participation factor analysis is also used to show

the states’ influence on modes providing a starting point of controlling the system if it should be

driven to instability.

48

Table 8: Test Case 1a Modal Analysis Summary Table


1 -0.282 0.5696 speed 1 1

2 0 0.1268 angle 5 1

3 -0.1981 + 4.2915i 0.2007 speed 1 0.0461

4 -0.1981 - 4.2915i 0.2007 speed 1 0.0461

5 -0.2964 + 6.0825i 0.344 speed 9 0.0487

6 -0.2964 - 6.0825i 0.344 speed 9 0.0487

7 -0.2987 + 6.4986i 0.1429 angle 5 0.0459

8 -0.2987 - 6.4986i 0.1429 angle 5 0.0459

9 -0.3075 + 7.2972i 0.1944 speed 6 0.0421

10 -0.3075 - 7.2972i 0.1944 speed 6 0.0421

11 -0.2714 + 8.4361i 0.2565 speed 3 0.0322

12 -0.2714 - 8.4361i 0.2565 speed 3 0.0322

13 -0.2745 + 8.7065i 0.252 speed 10 0.0315

14 -0.2745 - 8.7065i 0.252 speed 10 0.0315

15 -0.3510 + 9.2979i 0.3635 angle 8 0.0377

16 -0.3510 - 9.2979i 0.3635 angle 8 0.0377

17 -0.3366 + 9.4063i 0.3801 speed 4 0.0358

18 -0.3366 - 9.4063i 0.3801 speed 4 0.0358

19 -0.3232 + 9.8445i 0.3032 angle 7 0.0328

20 -0.3232 - 9.8445i 0.3032 angle 7 0.0328

49

Figure 11: Test Case 1a Graphical Display of Modes (Eigenvalues)

Referring to Table 8 and Figure 11, with load added as per Test Case 1a, the absolute value of

the real-part σ of modes λ5, λ6 and λ19, λ20 decreased to 0.2964 and 0.3232 or 99.6% and

97.3% of the Base case levels respectively. In addition to that, the damping ratio ϒ of modes λ5,

λ6 and λ19, λ20 have decreased to 4.87% and 3.28% or 98.4% and 92% of the Base case levels

respectively. These values still indicate a stable system; however, the degree of stability is less

than that of the Base case. Thus, the modes λ5, λ6 and λ19, λ20 are considered most likely to

become unstable and will be taken into account.

50

Table 9: Test Case 1b Modal Analysis Summary Table


1 -0.2607 0.6065 speed 1 1

2 0 0.1238 angle 5 -1

3 -0.2094 + 4.3385i 0.1819 speed 1 0.0482

4 -0.2094 - 4.3385i 0.1819 speed 1 0.0482

5 -0.2951 + 6.1719i 0.3447 speed 9 0.0478

6 -0.2951 - 6.1719i 0.3447 speed 9 0.0478

7 -0.2981 + 6.5939i 0.142 angle 5 0.0452

8 -0.2981 - 6.5939i 0.142 angle 5 0.0452

9 -0.3072 + 7.3843i 0.1926 speed 6 0.0416

10 -0.3072 - 7.3843i 0.1926 speed 6 0.0416

11 -0.2597 + 8.7021i 0.2211 angle 3 0.0298

12 -0.2597 - 8.7021i 0.2211 angle 3 0.0298

13 -0.2886 + 9.0008i 0.2867 angle 2 0.032

14 -0.2886 - 9.0008i 0.2867 angle 2 0.032

15 -0.3548 + 9.5843i 0.3121 angle 8 0.037

16 -0.3548 - 9.5843i 0.3121 angle 8 0.037

17 -0.3329 + 9.5189i 0.2927 speed 4 0.035

18 -0.3329 - 9.5189i 0.2927 speed 4 0.035

19 -0.3223 + 9.9823i 0.2983 angle 7 0.0323

20 -0.3223 - 9.9823i 0.2983 angle 7 0.0323

51

Figure 12: Test Case 1b Graphical Display of Modes (Eigenvalues)

Referring to Table 9 and Figure 12, with load added as per Test Case 1b, the absolute value of

the real-part σ of considered modes λ5, λ6 and λ19, λ20 decreased to 0.2951 and 0.3223 or

99.2% and 97% of the Base case levels respectively. In addition to that, the damping ratio ϒ of

modes λ5, λ6 and λ19, λ20 have decreased to 4.78% and 3.23% or 98.4% and 90.5% of the Base

case levels respectively. The additional decrease of the absolute value of the real-part σ of the

mode and the damping ratio ϒ illustrates that considered modes are becoming less and less

stable.

52

Table 10: Test Case 1c Modal Analysis Summary Table


1 -0.2409 + 3.9715i 0.1205 speed 1 0.0605

2 -0.2409 - 3.9715i 0.1205 speed 1 0.0605

3 -0.183 0.7397 speed 1 1

4 0 0.1528 angle 1 -1

5 -0.2916 + 6.2393i 0.3618 speed 9 0.0467

6 -0.2916 - 6.2393i 0.3618 speed 9 0.0467

7 -0.2983 + 6.7133i 0.1521 angle 5 0.0444

8 -0.2983 - 6.7133i 0.1521 angle 5 0.0444

9 -0.3121 + 7.5141i 0.1877 speed 6 0.0415

10 -0.3121 - 7.5141i 0.1877 speed 6 0.0415

11 -0.2737 + 9.1772i 0.2469 speed 10 0.0298

12 -0.2737 - 9.1772i 0.2469 speed 10 0.0298

13 -0.3317 + 9.9632i 0.4138 speed 4 0.0333

14 -0.3317 - 9.9632i 0.4138 speed 4 0.0333

15 -0.3448 +10.0861i 0.3883 angle 8 0.0342

16 -0.3448 -10.0861i 0.3883 angle 8 0.0342

17 -0.2951 +10.4831i 0.3871 angle 2 0.0281

18 -0.2951 -10.4831i 0.3871 angle 2 0.0281

19 -0.3187 +10.3846i 0.2784 angle 7 0.0307

20 -0.3187 -10.3846i 0.2784 angle 7 0.0307

53

Figure 13: Test Case 1c Graphical Display of Modes (Eigenvalues)

Referring to Table 10 and Figure 13, with load added as per Test Case 1c, the absolute value of

the real-part σ of modes λ5, λ6 and λ19, λ20 decreased to 0.2916 and 0.3187 or 98.1% and 96%

of the Base case levels respectively. In addition to that, the damping ratio ϒ of modes λ5, λ6 and

λ19, λ20 have decreased to 4.67% and 3.07% or 94.4% and 85.2% of the Base case levels

respectively.

It is noticed that with each load increase from 1000MW to 2000MW and until 3900MW, the

absolute value of the real-part σ of the modes and the damping ratio decreases for considered

modes λ5, λ6 and λ19, λ20. In other words, the modes are approaching the origin and their

54

associated slope is becoming less steep. Therefore, they are approaching instability. Referring to

the participation factor column of Tables 8, 9, and 10, state 9 associated with Generator 9 has

the largest influence on modes λ5, λ6 with a participation factor value of 0.3440, 0.3447, and

0.3618 respectively and state 7 associated with Generator 7 has the largest influence on modes

λ19, λ20 with a participation factor value of 0.3032, 0.2983, and 2784 respectively.

Table 11: Test Case 1d Modal Analysis Summary Table


1 -0.2968 0.5436 speed 1 1

2 0 0.1271 angle 5 -1

3 -0.1933 + 4.1065i 0.2084 speed 1 0.047

4 -0.1933 - 4.1065i 0.2084 speed 1 0.047

5 -0.3236 + 9.7584i 0.3053 angle 7 0.0331

6 -0.3236 - 9.7584i 0.3053 angle 7 0.0331

7 -0.3326 + 9.3112i 0.4007 speed 4 0.0357

8 -0.3326 - 9.3112i 0.4007 speed 4 0.0357

9 -0.3754 + 9.0575i 0.4474 angle 8 0.0414

10 -0.3754 - 9.0575i 0.4474 angle 8 0.0414

11 -0.2911 + 8.4360i 0.2912 angle 2 0.0345

12 -0.2911 - 8.4360i 0.2912 angle 2 0.0345

13 -0.2935 + 5.9928i 0.3496 speed 9 0.0489

14 -0.2935 - 5.9928i 0.3496 speed 9 0.0489

15 -0.3048 + 6.4371i 0.1634 angle 5 0.0473

16 -0.3048 - 6.4371i 0.1634 angle 5 0.0473

17 -0.2281 + 6.9409i 0.4377 speed 10 0.0328

18 -0.2281 - 6.9409i 0.4377 speed 10 0.0328

19 -0.3076 + 7.2471i 0.1915 speed 6 0.0424

20 -0.3076 - 7.2471i 0.1915 speed 6 0.0424

55

Figure 14: Test Case 1d Graphical Display of Modes (Eigenvalues)

Referring to Table 11 and Figure 14, with the design contingency applied as per Test Case 1d, the

absolute value of the real-part σ of modes λ17, λ18 decreased to 0.2281 or 65% of the Base case

levels respectively. In addition to that, the damping ratio ϒ of modes λ17, λ18 have decreased to

3.28% and 85% of the Base case levels respectively. It is concluded that the modes are

approaching the origin and their associated slope is becoming less steep. Thus, the modes λ17,

λ18 are considered most likely to become unstable and will be taken into account.

Referring to the participation factor column of Tables 11, state 10 associated with Generator 10

has the largest influence on modes λ17, λ18 with a participation factor value of 0.4377.

56

From the modal analysis performed, it is noticed for the case of load incremental changes as the

voltage levels are continuously decreased, the modes of the system subsequently move closer

to the origin. It is also noticed that the slope of the eigenvalues for lower voltage levels is less

steep; thus, reducing the damping factor which is a major factor in system stability. Similar

modal behavior was noticed from the analysis of the design contingency (topology

change).These results prove the connection between a small disturbance either as a load change

at a bus or a change in the system’s topology and modal stability. The more severe the

disturbances, the more the modes of the system move towards instability.

From the participation factor analysis performed, it is noticed, for these specific disturbances,

that states associated with Generators 7 and 9 have a large influence on modes λ19, λ20 and λ5,

λ6 respectively for test cases 1a, 1b, and 1c while Generator 10 has a large influence on modes

λ17, λ18 for test case 1d. This is deduced due to the fact that for every load increment or design

contingency the modes are becoming more and more unstable and have a smaller and smaller

damping factor based on the modal plots comparing to the Base case. In summary, an effective

way of controlling the system could be done by controlling the identified generators.

4.1.4 System Control and Power System Stabilizer Effect on System

In order to depict system control and analyze the effect of power system stabilizers on the

system, the analyses for the load incremental changes and the topology change driven

instability must be handled differently.

For the first case of load change driven instability, the system is rendered into instability by

increasing the load at the load bus to a specific MW level.

Regarding the second case of topology change driven instability, a three-phase bolted fault,

which is a fault where all three phases and neutral are bolted together with zero impedance to

ground, was simulated at a specific bus to depict how the Base case system will ride through the

fault. Secondly, the design contingency as specified in Test Case 1d was applied to increase the

57

likelihood of the system becoming less stable as shown in the previous analysis. Then an

identical fault was applied to the system prior to the insertion of any power system stabilizers.

The system is then analyzed to illustrate its behavior including machine rotor angle, machine

rotor speed and generator accelerating power plots. Modes (eigenvalues) are calculated and

plotted to further show system instability. The following characteristics are used in order to

identify system instability.

Loss of synchronism between generator rotor angles

Drastic increase of generator rotor speeds

Unstable generator accelerating power

Modes located on the right half of the imaginary plane

Using the previous analysis, power system stabilizers are inserted into the determined locations

and a complementary system analysis is done during the disturbance to prove that stabilizing

the system by controlling it from the specified locations can be done effectively.

Results prior to the insertion of the power system stabilizers for the first case, where load Bus

#3 experiences a load increase of 5300 MW, are depicted in Figures 15 through 18 and Table 12.

58

Figure 15: Gen. Rotor Angle Plot Prior to PSS Insertion

Referring to Figure 15, after approximately 3.4 seconds, Generator 9 rotor angle separates from

the rest of the generators i.e. loses synchronism. At approximately 4 seconds all remaining

generators separate. Thus, all generators become unsynchronized with each other. The rotor

angles separated due insufficient damping torque.

59

Figure 16: Gen. Rotor Speed Plot Prior to PSS Insertion

Referring to Figure 16, after approximately 3.4 seconds, Generator 9 rotor speed increases

drastically beyond the stable system frequency of 60Hz. At approximately 4 seconds all

remaining generators except Generator 1 follow suit. During the 10 second time frame, the

rotor speeds of all generators excluding Generator 1 continue to increase beyond 65Hz and it is

safe to assume that they will not re-stabilize at system frequency (60Hz).

60

Figure 17: Gen. Accel. Power Plot Prior to PSS Insertion

Referring to Figure 17, at approximately 4 seconds all generators’ accelerating power oscillates

at an unstable frequency. Throughout the time frame of 10 seconds the generators’ accelerating

power do not stabilize but remain in this unstable condition due to insufficient damping.

61

Table 12: Modes (Eigenvalues) Prior to PSS Insertion

Eigenvalue/Mode

1 2.0764

2 -0.0236 + 3.8037i

3 -0.0249 + 3.7873i

4 -0.0346 + 6.6750i

5 -0.0302 + 5.5455i

6 -0.0290 + 6.4856i

7 -0.0330 + 7.0433i

8 5.8018

9 -0.0275 + 3.5229i

10 1.3349

11 -2.8064

12 -0.0236 - 3.8037i

13 -0.0249 - 3.7873i

14 -0.0346 - 6.6750i

15 -0.0302 - 5.5455i

16 -0.2900 - 6.4856i

17 -0.0330 - 7.0433i

18 -5.8567

19 -0.0275 - 3.5229i

20 -1.3612

62

Figure 18: Modes (Eigenvalues) Plot Prior to PSS Insertion

Referring to Table 12 and Figure 18, the values of modes λ1, λ8 and λ10 are 2.0764, 5.8018, and

1.3349 respectively which are real and positive; thus, located on the right half of the imaginary

plane. These unstable modes indicate system instability.

Results for the second case, which is applying a three-phase bolted fault at Bus #5 with a

clearing time of 0.24 seconds before and after applying the specified design contingency, are

shown in Figures 19 through 22 and Table 13. Both scenarios are prior to power system

stabilizer insertion.

63

Figure 19: Gen. Rotor Angle Plot During Fault

Referring to Figure 19, once the fault is applied at 1 second all the generator rotor angles

oscillate. However, during the time frame of 10 seconds, the rotor angle oscillations are

dampened indicating a stable system.

64

Figure 20: Gen. Rotor Speed Plot During Fault

Referring to Figure 20, once the fault is applied at 1 second, the generators’ rotor speeds

increase to a maximum of 60.6Hz and a minimum of 59.85Hz. However, during the time frame

of 10 seconds, the rotor speeds stabilize between 60Hz and 60.45Hz indicating stability.

65

Figure 21: Gen. Accel. Power Plot During Fault

Referring to Figure 21, once the fault is applied at 1 second all the generators’ accelerating

power outputs oscillate. However, during the time frame of 10 seconds, the oscillations are

dampened indicating a stable system.

66

Table 13: Modes (EIgenvalues) During Fault

Eigenvalue/Mode

1 -0.0049 + 2.7645i

2 -0.0424 + 6.9112i

3 -0.0448 + 6.8726i

4 -0.0549 + 8.5336i

5 -0.0487 + 6.3247i

6 -0.0464 + 7.9202i

7 -0.0527 + 8.5084i

8 -0.0550 + 8.7919i

9 -0.0599 + 5.7702i

10 -0.0147 + 8.4840i

11 -0.0049 - 2.7645i

12 -0.0424 - 6.9112i

13 -0.0448 - 6.8726i

14 -0.0549 - 8.5336i

15 -0.0487 - 6.3247i

16 -0.0464 - 7.9202i

17 -0.0527 - 8.5084i

18 -0.0550 - 8.7919i

19 -0.0599 - 5.7702i

20 -0.0147 - 8.4840i

67

Figure 22: Modes (EIgenvalues) Plot During Fault

Referring to Table 13 and Figure 21, all modes λ are complex with negative real-parts σ; thus,

located on the left half of the imaginary plane. In this case they are damped oscillatory modes

indicating stability.

68

Figure 23: Gen. Rotor Angle Plot During Contingency, Applied Fault, and Prior to PSS Insertion

Referring to Figure 23, at 1 second the fault is applied, and after approximately 0.24 seconds or

at the same time the fault is cleared, Generator 1 rotor angle separates from the rest of the

generators i.e. loses synchronism. At approximately 4 seconds all remaining generators

separate. Thus, all generators become unsynchronized with each other. The rotor angles

separated due to insufficient damping torque.

69

Figure 24: Gen. Rotor Speed Plot During Contingency, Applied Fault, and Prior to PSS Insertion

Referring to Figure 24, after the fault is applied (1 second) all generator rotor speeds except that

of Generator 1 increase beyond the stable system frequency of 60Hz. At approximately 4.8

seconds, well beyond the time the fault is cleared (1.24 seconds), the same generator rotor

angles continue to increase while Generator 9 rotor angle increases even more drastically.

During the 10 second time frame, the rotor speeds of all generators excluding Generator 1

continue to increase to more than 70Hz; therefore, it is safe to assume that they will not re-

stabilize at system frequency (60Hz).

70

Figure 25: Gen. Accel. Power Plot During Contingency, Applied Fault, and Prior to PSS Insertion

Referring to Figure 25, after the fault is applied (1 second), all generators’ accelerating power

oscillates at an unstable frequency. Throughout the time frame of 10 seconds, the generators’

accelerating power does not stabilize but remain in this unstable condition due to insufficient

damping.

71

Table 14: Modes (Eigenvalues) During Contingency, Applied Fault, and Prior to PSS Insertion

Eigenvalue/Mode

1 -0.0050 + 0.2392i

2 -0.0318 + 6.2489i

3 -0.0336 + 6.2700i

4 -0.0407 + 7.2569i

5 -0.0361 + 4.3146i

6 -0.0345 + 7.2405i

7 -0.0391 + 7.8415i

8 -0.0405 + 5.2856i

9 5.8869

10 -0.0110 + 5.4224i

11 -0.0050 - 0.2392i

12 -0.0318 - 6.2489i

13 -0.0336 - 6.2700i

14 -0.0407 - 7.2569i

15 -0.0361 - 4.3146i

16 -0.0345 - 7.2405i

17 -0.0391 - 7.8415i

18 -0.0405 - 5.2856i

19 -5.8869

20 -0.0110 - 5.4224i

72

Figure 26: Modes (Eigenvalues) Plot During Contingency, Applied Fault, and Prior to PSS

Insertion

Referring to Table 14 and Figure 26, the value of mode λ9 is 5.8869 which is real and positive;

thus, located on the right half of the imaginary plane. This unstable mode indicates system

instability.

When Figures 19 through 22 are compared with Figures 23 through 26, a difference in generator

behavior (including rotor angles, speeds and accelerating power) and in mode values is observed

even though the same fault is applied to the system (fault type: bolted three-phase, location:

Bus #5, clearing time: 0.24 seconds). This discrepancy arises since the applied contingency

73

(disconnecting Branch 9 – 39) changed the system’s conditions making the system’s stability

more vulnerable when a disturbance might occur.

It can be clearly seen that the system has been driven to an unstable condition by looking at the

generator behavior where the machine rotor angles and speeds lose synchronism while

experiencing drastic acceleration. The system’s modes also illustrate instability of the system

since some modes are now located at the right half of the imaginary plane i.e. σ > 0.

In order to control the system through stabilizing it, the unstable modes must be now shifted to

the left half of the imaginary plane σ < 0. This can be established by providing sufficient positive

damping to the rotor angles so they do not lose synchronism with each other.

Power system stabilizers are then inserted at Generators 7 & 9 for the first case and 10 for the

second cases as specified previously to provide this necessary damping in order to control the

system.

First case results after the insertion of the power system stabilizers at Generators 7 & 9 are

depicted in Figures 27 through 30 and Table 15.

74

Figure 27: Gen. Rotor Angle Plot after PSS Insertion

Referring to Figure 27, once the power system stabilizers are inserted at Generators 7 and 9 and

the disturbance of increasing the load at Bus #3 to 5300MW is applied, the necessary positive

damping torque is provided to the system after a few oscillations by the power system

stabilizers, and it is clear that the rotor angles no longer separate but remain in synchronism

with each other. Moreover, during the time frame of 10 seconds, the rotor angle oscillations are

continually dampened indicating a stable system.

75

Figure 28: Gen. Rotor Speed Plot after PSS Insertion

Referring to Figure 28, once the power system stabilizers are inserted, the generator rotor

speeds no longer increase beyond the system frequency of 60Hz. However, there is a

continuous decrease in rotor angle speeds since the load increase of 5300MW at Bus #3 is kept

beyond the time frame of 10 seconds.

76

Figure 29: Gen. Accel. Power Plot after PSS Insertion

Referring to Figure 29, generators’ accelerating power no longer oscillates at an unstable

frequency and are dampened indicating a stable system.

77

Table 15: Modes (Eigenvalues) after PSS Insertion

Eigenvalue/Mode

1 -0.0056 + 2.4630i

2 -0.0481 + 5.7488i

3 -0.0508 + 5.5380i

4 -0.0619 + 6.9791i

5 -0.0549 + 4.4030i

6 -0.0523 + 6.8534i

7 -0.6132 + 0.0142i

8 -0.0620 + 7.7896i

9 -0.6281 + 0.0088i

10 -0.0167 + 7.5755i

11 -0.0056 - 2.4630i

12 -0.0481 - 5.7488i

13 -0.0508 - 5.5380i

14 -0.0619 - 6.9791i

15 -0.0549 - 4.4030i

16 -0.0523 - 6.8534i

17 -0.6132 - 0.0142i

18 -0.0620 - 7.7896i

19 -0.6281 - 0.0088i

20 -0.0167 - 7.5755i

78

Figure 30: Modes (Eigenvalues) Plot after PSS Insertion

Referring to Table 15 and Figure 30, previously unstable modes λ1, λ8 and λ10 are now stable

with values of -0.0056+2.4630i, -0.062+7.7896i, and -0.0167+7.5755i respectively; thus, they

have been shifted to the left half of the imaginary plane. All modes are now damped oscillatory

modes indicating system stability.

Second case results, after the insertion of the power system stabilizer at Generator 10, are

depicted in Figures 31 through 34 and Table 16.

79

Figure 31: Gen. Rotor Angle Plot after PSS Insertion

Referring to Figure 31, once the power system stabilizer is inserted at Generator 10 and design

contingency (disconnecting Branch 9 – 39) and fault are applied the necessary positive damping

torque is provided to the system after a few oscillations by the power system stabilizer, and it is

clear that the rotor angles no longer separate but remain in synchronism with each other.

Moreover, during the time frame of 10 seconds, the rotor angle oscillations are continually

dampened indicating a stable system. For example, at the first swing the minimum for

Generator 1 rotor angle is approximately -105 degrees, at the second swing the minimum is

80

approximately -90 degrees, and at the last swing in the 10 second time frame the minimum is at

approximately -65 degrees.

Figure 32: Gen. Rotor Speed Plot after PSS Insertion

Referring to Figure 32, once the power system stabilizer is inserted, the generator speeds

increase to a maximum of 63Hz and a minimum of 59Hz due to the applied fault. There is no

further increase beyond unstable levels as was the case in the results shown in Figure 24. During

81

the time frame of 10 seconds, the rotor speeds stabilize between 60Hz and 61.8Hz indicating

stability.

Figure 33: Gen. Accel. Power Plot after PSS Insertion

Referring to Figure 33, the inserted power system stabilizer provide the necessary positive

damping torque to the system after a few oscillations, and it is clear that the generators’

accelerating power no longer oscillates at unstable frequencies but are continually dampened

throughout the time frame of 10 seconds indicating a stable system. For example, at the first

swing the minimum for Generator 1 accelerating power is approximately -1200 MW, at the

82

second swing the minimum is approximately -1000MW, and at the last swing in the 10 second

time frame the minimum is at approximately -800MW.

Table 16: Modes (Eigenvalues) after PSS Insertion

Eigenvalue/Mode

1 -0.0047 + 2.0959i

2 -0.0410 + 6.8546i

3 -0.0433 + 6.9256i

4 -0.0533 + 8.5675i

5 -0.0475 + 5.3651i

6 -0.6067 - 0.0157i

7 -0.0506 + 8.5510i

8 -0.6046 - 0.0150i

9 -0.0582 + 6.5662i

10 -0.0142 + 8.5926i

11 -0.0047 - 2.0959i

12 -0.0410 - 6.8546i

13 -0.0433 - 6.9256i

14 -0.0533 - 8.5675i

15 -0.0475 - 5.3651i

16 -0.6067 + 0.0157i

17 -0.0506 - 8.5510i

18 -0.6046 + 0.0150i

19 -0.0582 - 6.5662i

20 -0.0142 - 8.5926i

83

Figure 34: Modes (Eigenvalues) Plot after PSS Insertion

Referring to Table 16 and Figure 34, previously unstable mode λ9 is now stable with a value of

-0.0582+6.5662i; thus, it has been shifted to the left half of the imaginary plane. All modes are

now damped oscillatory modes indicating system stability.

It is concluded that the insertion of the power system stabilizers successfully stabilized the

system by relocating the unstable modes to the left half of the imaginary plane. In other words,

they have provided enough damping to keep the generators in synchronism throughout the

disturbance; thus, pushing the system back into stability.

84

In this chapter, the small signal stability analysis results of an IEEE 39 bus system were

presented. In defining various system inputs including bus data, generator data, line data, and

transformer data, simulation results were produced for the modal analysis of the system.

Furthermore, analysis was completed to study the behavior of the system during small

disturbances by increasing the load at a specific load bus in the system or by applying a design

contingency to the system which in this case was a disconnection of a branch. In summary, as

the stability margin decreased, so did modal stability. Participation analysis was also done to

deduce which state primarily influenced each mode thus determining a general location for

system control. Finally, the system was driven into instability and then power system stabilizers

were inserted at the specified locations illustrating that a system can be controlled or stabilized

by inserting the power system stabilizers at these specified locations.

Chapter 5 will review these results and draw conclusions based on the data. The conclusions will

include system stability, modal and participation factor analysis. In addition, future work

involved in continuing the research will be cited.

85

Chapter 5

5.1 Conclusion

Chapter 1 presented the foundational building blocks of power systems and power system

stability defining what each and how they evolved till present day. The major components of a

power system, generation, transmission and distribution were covered and examples of

different types of power systems were explained. Power system stability and its different

designations were then outlined. The approach of small signal stability is to simplify the analysis

and calculations of a system by transforming the complex dynamic system used in the time

varying differential methodology of transient analysis into a system of linear equations. Finally,

the contents of the thesis as pertaining to the expected contributions were introduced. Chapter

2 followed with a mathematical and literary review of the electrical system model, small signal

stability including the computational derivations to transform the dynamic differential equation

to a state space representation of a linearized modeled power system, state matrix calculation,

eigenvalue determination, left and right eigenvectors and participation matrix identification,

and an introduction to power system stabilizer highlighting its role in power system control.

Chapter 3, based on the foundation laid by the previous chapters, introduced the IEEE 39 Bus

system and simulation software that utilized the numerical analysis techniques that have been

established. The system inputs were presented and details are shown for exactly how the

simulation software analyzed the system. Chapter 4 bridged the background information

provided in the first three chapters and the research results by first outlining the approach and

methodology used in this effort. Modal analysis of an initial 39 bus system, that yields the

eigenvalues of the system was used to determine whether the system is stable or not at that

particular equilibrium point. Modal and participation factor analysis was performed during a

specified small disturbance either by load increments or applied design contingencies to identify

which modes are most likely to become unstable followed by locating the most influential states

on the identified modes i.e. how significant a role each generator (or group of generator) had on

the system during the small disturbance; thus, determining a location(s) for control i.e. system

86

stabilization. Finally, the system was pushed into instability and then control through power

system stabilizers was applied to the determined locations in order to re-stabilize the system

after the disturbance. Stability results prior and post power system stabilizer insertion were

compared to illustrate how the system response transformed from being unstable to stable by

combining modal and participation factor analysis to identify power system stabilizer locations

for control. Previous work was done using modal analysis to determine the degree of system

stability as well as power system stability response and control using power system stabilizers.

The relation between the severity of the disturbance and modal stability analyzed in this

research and in references [9] [10] [11] achieved similar conclusions. Furthermore, the results

obtained after control were applied to the system depicted stable behavior similar to references

[12] [13] [14] [15] [16] where rotor angle oscillations were damped, rotor speeds were stabilized

at system frequency, and unstable modes were moved to the stable region of the imaginary

plane. The research performed was a learning experience starting from the general concept of

small signal stability followed by the mathematical technique of modal and participation

analysis, power system modeling, to finally stability response analysis and control theory.

5.2 Future work

The research and methodology presented in this paper serve as a useful tool in solving real

world power system problems. In large power systems, such as industrial facilities, the modal

stability and participation factor analysis techniques can allow for a number of advantages.

These advantages include stability determination and monitoring as systems grow and expand

as well as achieving operational and stability improvement in the most efficient manner possible

by focusing engineering efforts on the components that participate most. From this, large cost

savings can be achieved as decreased power consumption yields lower utility costs, equipment

can be operated more efficiently, stability control efforts are minimized, and equipment life

cycles are prolonged. In addition, the evolution of Smart Grid technology allows for the

application of these techniques in a more ‘real time’ fashion to achieve more continual system

87

improvement that matches the dynamic nature of the power system. Again, this will result in

increased reliability and efficiency in power systems that yield cost savings.

Moving forward, future work includes expanding the investigative process by using more

detailed generator models that may include governors and exciters, dynamic load modeling, and

a more general linearization technique that includes more states which will drive studies to use

more detailed power system stabilizers.

88

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School of Electrical Engineering, 2012.

[13] H. N. Vijaiyan, A. K. Ramasamy, A. M. Teng and S. K. Ahmed, "Effects of Variations in

Generator Inputs for Small Signal Stability Studies of a Three Machine Nine Bus Network,"

World Academy of Science, Engineering and Technology, vol. 5, no. 2, pp. 249-252, 2011.

[14] J. Rommes, N. Martins and F. D. Freitas, "Computing Rightmost Eigenvalues for Small-Signal

Stability Assessment of Large -Scale Power Systems," IEEE Transactions on Power Systems,

vol. 25, no. 2, pp. 929-938, 2010.

[15] A. A. Lerm, C. A. Canizares and N. Mithulananthan, "Effects of Limits in Small Signal Stability

Analysis of Power Systems," in IEEE-PES Summer Meeting, Vancouver, BC, 2001.

[16] K. Devender and S. S. Balwinder, "Power, Transient Stability of a Multi Machine System,"

International Journal of Engineering and Advanced Technology (IJEAT, vol. 2, no. 4, pp. 759-

762, 2013.

90

Appendix A

Figure 35: IEEE 39 Bus System

91

Table 17: Generator Data

92

93

94

Table 18: Branch Data

95

96

97

Table 19: Power and Voltage Data

98

Appendix B

Test 2 – Load Bus 4

System Manipulation and Evaluation

Case a – Increment of 1000MW


1 1.0454 0.0087 0 0

2 1.0395 -0.0477 0 0

3 1.0113 -0.1445 -3.22 -0.024

4 0.9781 -0.2422 -14.9999 -1.84

5 0.9864 -0.1745 0 0

6 0.9905 -0.1584 -0.0001 0.0001

7 0.979 -0.1838 -2.338 -0.84

8 0.9776 -0.1857 -5.22 -1.76

9 1.0155 -0.0475 0 0

10 1.0021 -0.1189 0 0

11 0.9969 -0.1324 0 0

12 0.9834 -0.1344 -0.075 -0.88

13 0.9974 -0.1345 0 0

14 0.9908 -0.172 0 0

15 0.9997 -0.1366 -3.2 -1.53

16 1.0197 -0.0933 -3.29 -0.323

17 1.0199 -0.1108 0 0

18 1.0151 -0.1313 -1.58 -0.3

19 1.0448 0.0105 0 0

20 0.9882 -0.0024 -6.28 -1.03

21 1.0224 -0.0446 -2.74 -1.15

22 1.0443 0.04 0 0

23 1.039 0.0367 -2.475 -0.846

24 1.0259 -0.0894 -3.086 0.92

25 1.05 -0.0249 -2.24 -0.472

26 1.042 -0.0559 -1.39 -0.17

27 1.0255 -0.1016 -2.81 -0.755

28 1.0439 0.0156 -2.06 -0.276

29 1.0454 0.0671 -2.835 -0.269

30 1.0475 0.004 3.0366 2.0064

31 0.982 0 5.7356 2.6656

32 0.9831 0.0328 6.9574 2.8152

33 0.9972 0.1071 6.6854 1.4478

34 1.0123 0.0944 5.4122 1.8378

35 1.0493 0.1331 6.9446 2.5574

36 1.0635 0.1828 5.9373 1.2805

37 1.0278 0.1011 5.7105 0.3636

38 1.0265 0.1973 8.7408 0.5556

39 1.03 0.0441 5.3481 -1.08

99


1 -0.2798 0.5738 speed 1 1

2 0 0.1275 angle 5 1

3 -0.2000 + 4.3190i 0.1982 speed 1 0.0463

4 -0.2000 - 4.3190i 0.1982 speed 1 0.0463

5 -0.2955 + 6.0485i 0.3512 speed 9 0.0488

6 -0.2955 - 6.0485i 0.3512 speed 9 0.0488

7 -0.2984 + 6.4915i 0.1455 angle 5 0.0459

8 -0.2984 - 6.4915i 0.1455 angle 5 0.0459

9 -0.3072 + 7.2823i 0.1925 speed 6 0.0421

10 -0.3072 - 7.2823i 0.1925 speed 6 0.0421

11 -0.2680 + 8.4651i 0.255 speed 3 0.0316

12 -0.2680 - 8.4651i 0.255 speed 3 0.0316

13 -0.2809 + 8.7417i 0.2197 angle 2 0.0321

14 -0.2809 - 8.7417i 0.2197 angle 2 0.0321

15 -0.3498 + 9.2709i 0.3646 angle 8 0.0377

16 -0.3498 - 9.2709i 0.3646 angle 8 0.0377

17 -0.3354 + 9.3949i 0.3862 speed 4 0.0357

18 -0.3354 - 9.3949i 0.3862 speed 4 0.0357

19 -0.3232 + 9.8353i 0.3029 angle 7 0.0328

20 -0.3232 - 9.8353i 0.3029 angle 7 0.0328

100

101

Case b – Increment of 2000MW


1 1.0305 0.1687 -0.0001 0.0001

2 1.0164 -0.0014 -0.0004 0.0004

3 0.9686 -0.1534 -3.2216 -0.0233

4 0.9205 -0.3492 -24.9906 -1.8385

5 0.9385 -0.2174 -0.0012 0.0007

6 0.9466 -0.1948 -0.0032 0.0014

7 0.9305 -0.2085 -2.3373 -0.8399

8 0.9271 -0.2032 -5.2188 -1.7593

9 0.9759 0.0897 -0.0006 0.0004

10 0.9663 -0.1597 -0.0006 0.0003

11 0.9582 -0.1718 -0.0001 0

12 0.9432 -0.1772 -0.075 -0.8799

13 0.9579 -0.1806 -0.0004 0.0001

14 0.9437 -0.232 -0.0013 0.0005

15 0.9658 -0.154 -3.2 -1.5298

16 0.9944 -0.0919 -3.2901 -0.3228

17 0.9906 -0.1075 0 0

18 0.9805 -0.133 -1.5799 -0.3

19 1.0348 0.0274 0.0001 0.0001

20 0.9825 0.0188 -6.28 -1.03

21 1.0036 -0.0353 -2.7401 -1.1499

22 1.0337 0.0576 0.0001 0.0001

23 1.028 0.0544 -2.475 -0.8459

24 1.0025 -0.0861 -3.086 0.9201

25 1.032 0.0177 -2.2398 -0.4721

26 1.0213 -0.0256 -1.39 -0.1699

27 1 -0.0844 -2.81 -0.7549

28 1.0319 0.0572 -2.06 -0.276

29 1.0369 0.1125 -2.8349 -0.2689

30 1.0475 0.0609 3.5732 3.3472

31 0.982 0 6.7255 4.4793

32 0.9831 0.0082 7.4144 4.5583

33 0.9972 0.1298 7.0508 2.1248

34 1.0123 0.1219 5.7444 2.1727

35 1.0493 0.1576 7.3892 3.3621

36 1.0635 0.2103 6.2746 1.762

37 1.0278 0.1524 6.0209 1.1787

38 1.0265 0.25 9.1816 1.1393

39 1.03 0.2716 11.7363 1.5928

102


1 -0.2585 0.6117 speed 1 1

2 0 0.1308 angle 9 1

3 -0.2135 + 4.4167i 0.1774 speed 1 0.0483

4 -0.2135 - 4.4167i 0.1774 speed 1 0.0483

5 -0.2925 + 6.0946i 0.3625 speed 9 0.0479

6 -0.2925 - 6.0946i 0.3625 speed 9 0.0479

7 -0.2973 + 6.5768i 0.145 angle 5 0.0452

8 -0.2973 - 6.5768i 0.145 angle 5 0.0452

9 -0.3066 + 7.3359i 0.1867 speed 6 0.0418

10 -0.3066 - 7.3359i 0.1867 speed 6 0.0418

11 -0.2592 + 8.7572i 0.2161 angle 3 0.0296

12 -0.2592 - 8.7572i 0.2161 angle 3 0.0296

13 -0.3006 + 9.1556i 0.3018 angle 2 0.0328

14 -0.3006 - 9.1556i 0.3018 angle 2 0.0328

15 -0.3297 + 9.4788i 0.2378 speed 4 0.0348

16 -0.3297 - 9.4788i 0.2378 speed 4 0.0348

17 -0.3476 + 9.5459i 0.2449 angle 8 0.0364

18 -0.3476 - 9.5459i 0.2449 angle 8 0.0364

19 -0.3221 + 9.9537i 0.2972 angle 7 0.0323

20 -0.3221 - 9.9537i 0.2972 angle 7 0.0323

103

104

Case c – Increment of 3000MW


1 0.984 0.3771 -0.0022 0.0008

2 0.9496 0.0366 -0.0097 0.002

3 0.8436 -0.2118 -3.2503 -0.0212

4 0.745 -0.6159 -34.8482 -1.8289

5 0.7888 -0.3388 -0.0208 0.0036

6 0.809 -0.2991 -0.026 0.0075

7 0.779 -0.3031 -2.3304 -0.8395

8 0.7699 -0.2874 -5.2102 -1.7565

9 0.8637 0.2823 -0.0159 -0.0012

10 0.8553 -0.2678 -0.0076 0.0013

11 0.8379 -0.2788 -0.0006 0

12 0.8184 -0.2909 -0.0749 -0.8794

13 0.8364 -0.3004 -0.0044 0.0007

14 0.8004 -0.3848 -0.0239 0.0035

15 0.8676 -0.2299 -3.2038 -1.5291

16 0.9233 -0.1361 -3.2922 -0.3226

17 0.9074 -0.148 -0.0006 0.0001

18 0.8808 -0.1814 -1.5793 -0.2999

19 1.0076 0.0063 -0.0008 0

20 0.9674 0.0017 -6.2799 -1.03

21 0.952 -0.0662 -2.7404 -1.1499

22 1.0054 0.0399 -0.0006 0

23 0.9985 0.0367 -2.4755 -0.846

24 0.937 -0.1276 -3.0857 0.9199

25 0.9806 0.0426 -2.239 -0.4727

26 0.9648 -0.0227 -1.3906 -0.1699

27 0.9293 -0.1034 -2.8098 -0.7549

28 1.0011 0.0759 -2.06 -0.276

29 1.0154 0.1367 -2.8352 -0.269

30 1.0475 0.1133 4.1095 7.1613

31 0.982 0 8.7494 10.1937

32 0.9831 -0.0662 7.8686 9.8265

33 0.9972 0.1157 7.4161 3.9307

34 1.0123 0.1118 6.0765 3.0341

35 1.0493 0.149 7.8337 5.4503

36 1.0635 0.2055 6.6118 2.9755

37 1.0278 0.1904 6.3311 3.4503

38 1.0265 0.2828 9.6223 2.5627

39 1.03 0.5679 18.1234 9.165

105


1 -0.219 0.6816 speed 1 1

2 0 0.1415 angle 9 -1

3 -0.2364 + 4.3865i 0.1408 speed 1 0.0538

4 -0.2364 - 4.3865i 0.1408 speed 1 0.0538

5 -0.2873 + 6.1001i 0.3831 speed 9 0.047

6 -0.2873 - 6.1001i 0.3831 speed 9 0.047

7 -0.2939 + 6.6532i 0.1475 speed 3 0.0441

8 -0.2939 - 6.6532i 0.1475 speed 3 0.0441

9 -0.3094 + 7.3384i 0.1751 speed 6 0.0421

10 -0.3094 - 7.3384i 0.1751 speed 6 0.0421

11 -0.2761 + 9.2352i 0.2 speed 10 0.0299

12 -0.2761 - 9.2352i 0.2 speed 10 0.0299

13 -0.2890 +10.4041i 0.333 angle 2 0.0278

14 -0.2890 -10.4041i 0.333 angle 2 0.0278

15 -0.3292 + 9.7131i 0.3959 angle 4 0.0339

16 -0.3292 - 9.7131i 0.3959 angle 4 0.0339

17 -0.3480 + 9.8576i 0.3558 angle 8 0.0353

18 -0.3480 - 9.8576i 0.3558 angle 8 0.0353

19 -0.3197 +10.1566i 0.284 angle 7 0.0315

20 -0.3197 -10.1566i 0.284 angle 7 0.0315

106

107

System Control Analysis

Point of Instability – 5800MW

108

109

110

Eigenvalue/Mode

1 1.1009

2 -0.0304 + 2.9593i

3 -0.0323 + 3.6084i

4 -0.0396 + 6.5347i

5 -0.0352 + 5.0067i

6 -0.0333 + 6.2174i

7 -0.0378 + 6.7679i

8 -0.0394 + 6.2225i

9 -0.0430 + 4.8011i

10 -0.0105 + 5.0523i

11 -1.1009

12 -0.0304 - 2.9593i

13 -0.0323 - 3.6084i

14 -0.0396 - 6.5347i

15 -0.0352 - 5.0067i

16 -0.0333 - 6.2174i

17 -0.0378 - 6.7679i

18 -0.0394 - 6.2225i

19 -0.0430 - 4.8011i

20 -0.0105 - 5.0523i

111

112

Power system stabilizers Locations: Generators 4, 9, and 10

113

114

115

Eigenvalue/Mode

1 -0.0051 + 2.4615i

2 -0.0443 + 4.9124i

3 -0.0468 + 4.7588i

4 -0.0998

5 -0.0503 + 5.6053i

6 -0.0481 + 7.1137i

7 -0.0547 + 7.6789i

8 -0.0572 + 8.1127i

9 -0.9967

10 -0.9993

11 -0.0051 - 2.4615i

12 -0.0443 - 4.9124i

13 -0.0468 - 4.7588i

14 -0.378

15 -0.0503 - 5.6053i

16 -0.0481 - 7.1137i

17 -0.0547 - 7.6789i

18 -0.0572 - 8.1127i

19 -0.2358 - 6.4564i

20 -0.1409 - 8.2181i

116

117

Test 3 – Load Bus 7

System Manipulation and Evaluation

Case a – Increment of 1000MW


1 1.0464 0.0428 0 0

2 1.0416 0.0166 0 0

3 1.015 -0.0696 -3.22 -0.024

4 0.9821 -0.1466 -5 -1.84

5 0.9792 -0.1684 0 0

6 0.982 -0.1606 -0.0003 0.0001

7 0.9638 -0.2417 -12.3377 -0.84

8 0.9643 -0.2221 -5.2199 -1.7599

9 1.0071 -0.0515 0 0

10 0.9991 -0.0899 0 0

11 0.9917 -0.1133 0 0

12 0.98 -0.1055 -0.075 -0.88

13 0.9961 -0.0958 0 0

14 0.993 -0.1078 0 0

15 1.0013 -0.0691 -3.2 -1.53

16 1.0211 -0.0245 -3.29 -0.323

17 1.022 -0.0407 0 0

18 1.0179 -0.0593 -1.58 -0.3

19 1.0453 0.0792 0 0

20 0.9885 0.0663 -6.28 -1.03

21 1.0234 0.0241 -2.74 -1.15

22 1.0449 0.1087 0 0

23 1.0396 0.1053 -2.475 -0.846

24 1.0272 -0.0206 -3.086 0.92

25 1.0518 0.0405 -2.24 -0.472

26 1.0437 0.0118 -1.39 -0.17

27 1.0274 -0.0327 -2.81 -0.755

28 1.0447 0.0831 -2.06 -0.276

29 1.046 0.1346 -2.835 -0.269

30 1.0475 0.0682 3.0366 1.8914

31 0.982 0 5.7635 2.9865

32 0.9831 0.0623 6.9574 2.9561

33 0.9972 0.1758 6.6854 1.4136

34 1.0123 0.1631 5.4122 1.8219

35 1.0493 0.2016 6.9446 2.5189

36 1.0635 0.2514 5.9373 1.2586

37 1.0278 0.1663 5.7105 0.2859

38 1.0265 0.2648 8.7408 0.5183

39 1.03 0.0597 5.3481 -0.7187

118


1 -0.2832 0.5683 speed 1 1

2 0 0.1298 angle 5 -1

3 -0.1989 + 4.3023i 0.2006 speed 1 0.0462

4 -0.1989 - 4.3023i 0.2006 speed 1 0.0462

5 -0.2952 + 6.0233i 0.3545 speed 9 0.049

6 -0.2952 - 6.0233i 0.3545 speed 9 0.049

7 -0.2987 + 6.5054i 0.1508 angle 5 0.0459

8 -0.2987 - 6.5054i 0.1508 angle 5 0.0459

9 -0.3059 + 7.2823i 0.1895 speed 6 0.042

10 -0.3059 - 7.2823i 0.1895 speed 6 0.042

11 -0.2626 + 8.4791i 0.2407 angle 3 0.031

12 -0.2626 - 8.4791i 0.2407 angle 3 0.031

13 -0.2882 + 8.7757i 0.2623 angle 2 0.0328

14 -0.2882 - 8.7757i 0.2623 angle 2 0.0328

15 -0.3495 + 9.2429i 0.3664 angle 8 0.0378

16 -0.3495 - 9.2429i 0.3664 angle 8 0.0378

17 -0.3346 + 9.3831i 0.3897 speed 4 0.0356

18 -0.3346 - 9.3831i 0.3897 speed 4 0.0356

19 -0.3232 + 9.8250i 0.303 angle 7 0.0329

20 -0.3232 - 9.8250i 0.303 angle 7 0.0329

119

120

Case b – Increment of 2000MW


1 1.0369 0.2515 -0.0002 0.0001

2 1.0223 0.1412 -0.0006 0.0004

3 0.9765 0.0113 -3.222 -0.0234

4 0.9226 -0.14 -4.9995 -1.8388

5 0.9075 -0.2108 -0.0043 0.0013

6 0.9126 -0.2063 -0.0201 0.0071

7 0.8754 -0.3534 -22.2932 -0.8377

8 0.8759 -0.2961 -5.2162 -1.7536

9 0.9392 0.0926 -0.0083 0.0026

10 0.9503 -0.099 -0.0023 0.0009

11 0.9353 -0.1338 -0.0024 0.0005

12 0.9253 -0.1171 -0.075 -0.8798

13 0.9463 -0.0984 -0.0001 0

14 0.9421 -0.0931 -0.0009 0.0002

15 0.966 -0.0067 -3.2002 -1.5297

16 0.9959 0.0587 -3.2902 -0.3227

17 0.9941 0.0463 0 0

18 0.9858 0.0249 -1.5799 -0.3

19 1.0353 0.1779 0.0001 0.0001

20 0.9828 0.1694 -6.28 -1.03

21 1.0046 0.1152 -2.7401 -1.1499

22 1.0343 0.208 0.0001 0.0001

23 1.0285 0.2048 -2.475 -0.8459

24 1.0038 0.0645 -3.086 0.9201

25 1.0367 0.1627 -2.2398 -0.4721

26 1.0248 0.1237 -1.39 -0.1699

27 1.0037 0.0671 -2.81 -0.7549

28 1.0337 0.2062 -2.06 -0.276

29 1.0381 0.2615 -2.8349 -0.2689

30 1.0475 0.2032 3.5732 3.0093

31 0.982 0 6.8631 5.7831

32 0.9831 0.0717 7.4137 5.3013

33 0.9972 0.2803 7.0508 2.0904

34 1.0123 0.2724 5.7444 2.1568

35 1.0493 0.308 7.3892 3.3232

36 1.0635 0.3607 6.2746 1.7399

37 1.0278 0.2969 6.0209 0.9739

38 1.0265 0.3988 9.1816 1.0591

39 1.03 0.3191 11.7356 3.0385

121


1 -0.2656 0.6005 speed 1 1

2 0 0.1345 angle 9 -1

3 -0.2125 + 4.4201i 0.1813 speed 1 0.048

4 -0.2125 - 4.4201i 0.1813 speed 1 0.048

5 -0.2916 + 6.0466i 0.3698 speed 9 0.0482

6 -0.2916 - 6.0466i 0.3698 speed 9 0.0482

7 -0.2981 + 6.6280i 0.1556 angle 5 0.0449

8 -0.2981 - 6.6280i 0.1556 angle 5 0.0449

9 -0.3024 + 7.3409i 0.1758 speed 6 0.0412

10 -0.3024 - 7.3409i 0.1758 speed 6 0.0412

11 -0.2538 + 8.7326i 0.2548 speed 10 0.0291

12 -0.2538 - 8.7326i 0.2548 speed 10 0.0291

13 -0.3224 + 9.3257i 0.3061 speed 2 0.0346

14 -0.3224 - 9.3257i 0.3061 speed 2 0.0346

15 -0.3408 + 9.5249i 0.224 speed 4 0.0358

16 -0.3408 - 9.5249i 0.224 speed 4 0.0358

17 -0.3219 + 9.4428i 0.1898 angle 4 0.0341

18 -0.3219 - 9.4428i 0.1898 angle 4 0.0341

19 -0.3221 + 9.9336i 0.2972 angle 7 0.0324

20 -0.3221 - 9.9336i 0.2972 angle 7 0.0324

122

123

Case c – Increment of 3000MW


1 1.0175 0.527 -0.0525 0.0515

2 0.9924 0.3096 -0.1461 0.1736

3 0.9189 0.123 -3.5573 0.1785

4 0.8346 -0.1248 -4.9936 -1.5854

5 0.8033 -0.2607 -0.3186 0.1202

6 0.8123 -0.2613 -1.393 0.5191

7 0.7473 -0.4903 -27.7525 -0.6754

8 0.7463 -0.3847 -4.7437 -0.9683

9 0.834 0.2977 -1.0809 0.741

10 0.8799 -0.1064 -0.2077 0.1405

11 0.8539 -0.1558 -0.2299 0.0836

12 0.8464 -0.1274 -0.0751 -0.8564

13 0.874 -0.097 0.0021 0.0034

14 0.8674 -0.0683 -0.1065 0.0646

15 0.915 0.0804 -3.2757 -1.4387

16 0.9602 0.172 -3.3721 -0.2225

17 0.954 0.1648 -0.0072 0.0099

18 0.9389 0.1403 -1.564 -0.2912

19 1.0217 0.3101 0.0316 0.0646

20 0.9752 0.306 -6.2716 -1.021

21 0.9789 0.2381 -2.7685 -1.1056

22 1.0203 0.341 0.0282 0.0548

23 1.0139 0.3379 -2.4638 -0.7943

24 0.971 0.1801 -3.0937 0.9506

25 1.0137 0.3267 -2.1795 -0.475

26 0.9981 0.2739 -1.4129 -0.1198

27 0.9698 0.2018 -2.8247 -0.7336

28 1.0192 0.37 -2.0695 -0.2581

29 1.028 0.4299 -2.7879 -0.2137

30 1.0475 0.383 4.1075 4.7425

31 0.982 0 7.7042 9.7665

32 0.9831 0.0861 7.7311 8.6483

33 0.9972 0.4189 7.4292 3.0043

34 1.0123 0.4156 6.0747 2.5997

35 1.0493 0.4486 7.8403 4.3788

36 1.0635 0.5045 6.6169 2.3657

37 1.0278 0.4708 6.3394 2.0112

38 1.0265 0.5751 9.638 1.7515

39 1.03 0.6565 17.899 9.3788

124


1 -0.2395 0.6475 speed 1 1

2 0 0.1457 angle 9 1

3 -0.2305 + 4.5092i 0.1549 speed 1 0.051

4 -0.2305 - 4.5092i 0.1549 speed 1 0.051

5 -0.2876 + 6.0832i 0.3861 speed 9 0.0472

6 -0.2876 - 6.0832i 0.3861 speed 9 0.0472

7 -0.2993 + 6.7879i 0.1623 angle 5 0.0441

8 -0.2993 - 6.7879i 0.1623 angle 5 0.0441

9 -0.2966 + 7.4170i 0.152 speed 6 0.04

10 -0.2966 - 7.4170i 0.152 speed 6 0.04

11 -0.2542 + 8.9753i 0.2591 speed 10 0.0283

12 -0.2542 - 8.9753i 0.2591 speed 10 0.0283

13 -0.3328 + 9.6112i 0.3483 speed 4 0.0346

14 -0.3328 - 9.6112i 0.3483 speed 4 0.0346

15 -0.3621 + 9.7194i 0.3728 angle 8 0.0372

16 -0.3621 - 9.7194i 0.3728 angle 8 0.0372

17 -0.2947 +10.1887i 0.3678 angle 2 0.0289

18 -0.2947 -10.1887i 0.3678 angle 2 0.0289

19 -0.3208 +10.0746i 0.2903 angle 7 0.0318

20 -0.3208 -10.0746i 0.2903 angle 7 0.0318

125

126

System Control Analysis

Point of Instability – 5000MW

127

128

129

Eigenvalue/Mode

1 1.8855

2 -0.0318 + 3.0376i

3 -0.0336 + 3.6748i

4 -0.0402 + 7.4569i

5 -0.0354 + 6.0392i

6 -0.0339 + 6.7365i

7 -0.0385 + 7.3215i

8 -0.0408 + 5.6757i

9 -0.0435 + 4.9482i

10 -0.0109 + 4.9434i

11 -1.8855

12 -0.0318 - 3.0376i

13 -0.0336 - 3.6748i

14 -0.0402 - 7.4569i

15 -0.0354 - 6.0392i

16 -0.0339 - 6.7365i

17 -0.0385 - 7.3215i

18 -0.0408 - 5.6757i

19 -0.0435 - 4.9482i

20 -0.0109 - 4.9434i

130

131

Power system stabilizers Locations: Generators 1,2,4,6,7,10

132

133

134

Eigenvalue/Mode

1 -0.6247 - 0.0073i

2 -0.5958 - 0.0134i

3 -0.0489 + 5.5381i

4 -0.6099 - 0.0167i

5 -0.0538 + 5.7197i

6 -0.6072 - 0.0162i

7 -0.6130 - 0.0138i

8 -0.0599 + 8.2948i

9 -0.0648 + 4.7306i

10 -0.6097 - 0.0088i

11 -0.6247 + 0.0073i

12 -0.5958 + 0.0134i

13 -0.0489 - 5.5381i

14 -0.6099 + 0.0167i

15 -0.0538 - 5.7197i

16 -0.6072 + 0.0162i

17 -0.6130 + 0.0138i

18 -0.0599 - 8.2948i

19 -0.0648 - 4.7306i

20 -0.6097 + 0.0088i

135

136

MATLAB Script:

function ybus = build(linedata); % Returns ybus

fb = linedata(:,1); % From bus number...

tb = linedata(:,2); % To bus number...

r = linedata(:,3); % Resistance, R...

x = linedata(:,4); % Reactance, X...

b = linedata(:,5); % Ground Admittance, B/2...

a = linedata(:,6); % Xfmr tap setting (ratio of Vfb/Vtb)

z = r + 1i*x; % Z matrix...

y = 1./z; % To get inverse of each element...

b = 1i*b; % Make B imaginary...

nbus = max(max(fb),max(tb)); % no. of buses...

nbranch = length(fb); % no. of branches...

ybus = zeros(nbus,nbus); % Initialise YBus...

% Formation of the Off Diagonal Elements...

for k=1:nbranch

if (a(k)==0)

ybus(fb(k),tb(k)) = -y(k);

ybus(tb(k),fb(k)) = ybus(fb(k),tb(k));

else

ybus(fb(k),tb(k)) = -y(k)/a(k);

ybus(tb(k),fb(k)) = ybus(fb(k),tb(k));

end

end

% Formation of Diagonal Elements....

137

for m=1:nbus

for n=1:nbranch

if (fb(n) == m || tb(n) == m)&&a(n)==0

ybus(m,m) = ybus(m,m) + y(n) + b(n);

end

if (fb(n) == m)&&a(n)~=0

ybus(m,m) = ybus(m,m) + y(n)/a(n) + y(n)*((1-

a(n))/a(n)^2);

end

if (tb(n) == m)&&a(n)~=0

ybus(m,m) = ybus(m,m) + y(n)/a(n) + y(n)*((a(n)-1)/a(n));

end

end

end

138

function [V theta P Q] = loadflow(lfdata , ybus0)

%naming constants and unknowns

V_SW=[];

theta_SW=[];

V_PV=[];

theta_PV=[];

V_PQ=[];

theta_PQ=[];

Pi_PV=[];

Pi_PQ=[];

Qi_PQ=[];

P_SW=[];

P_PV=[];

P_PQ=[];

Q_SW=[];

Q_PV=[];

Q_PQ=[];

%rearrange ybus based on bustype

bustype = lfdata(:,2)';

busindex=lfdata(:,1)';

busindex=[bustype' busindex'];

busindex=sortrows(busindex,1);

busindex(:,1)=[];

busindex=busindex';

ybus=[bustype' ybus0];

ybus=sortrows(ybus,1);

ybus(:,1)=[];

ybus=[bustype;ybus];

ybus=ybus';


139

ybus=ybus';

ybus(1,:)=[];

%initialize unknowns (V=1.0, theta=0.0, P=Pi, Q=Qi for PQ bus=3)

%determining constants (V and theta for swing bus=1 and V for PV bus=2)

for k=1:length(bustype)

if (bustype(k)==1)

V_SW=[V_SW lfdata(k,3)];

theta_SW=[theta_SW lfdata(k,4)];

P_SW=[P_SW 0];

Q_SW=[Q_SW 0];

end

if (bustype(k)==2)

V_PV=[V_PV lfdata(k,3)];

theta_PV=[theta_PV 0];

P_PV=[P_PV 0];

Q_PV=[Q_PV 0];

Pi_PV=[Pi_PV lfdata(k,5)];

end

if (bustype(k)==3)

V_PQ=[V_PQ 1];

theta_PQ=[theta_PQ 0];

P_PQ=[P_PQ 0];

Q_PQ=[Q_PQ 0];

Pi_PQ=[Pi_PQ -lfdata(k,5)];

Qi_PQ=[Qi_PQ -lfdata(k,6)];

end

end

error=1; % dummy value for error

while(abs(error(1))>0.01)

140

% all voltage and angles grouped in order of bustype

theta=[theta_SW theta_PV theta_PQ];

V=[V_SW V_PV V_PQ];

%re-calculating P and Q (reset P and Q to zero)

P=zeros(1,length(bustype));

Q=zeros(1,length(bustype));

for i=1:length(bustype)

for j=1:length(bustype)

P(i)=P(i)+abs(V(i))*abs(V(j))*(real(ybus(i,j))*cos(theta(i)-

theta(j))+imag(ybus(i,j))*sin(theta(i)-theta(j)));

end

end



Q(i)=Q(i)+abs(V(i))*abs(V(j))*(real(ybus(i,j))*sin(theta(i)-theta(j))-

imag(ybus(i,j))*cos(theta(i)-theta(j)));

end

end

P_SW=P(1:length(P_SW));

P_PV=P(length(P_SW)+1:length(P_SW)+length(P_PV));

P_PQ=P(length(P_SW)+length(P_PV)+1:length(P_SW)+length(P_PV)+length(P_P

Q));

Q_SW=Q(1:length(Q_SW));

Q_PV=Q(length(Q_SW)+1:length(Q_SW)+length(Q_PV));

141

Q_PQ=Q(length(Q_SW)+length(Q_PV)+1:length(Q_SW)+length(Q_PV)+length(Q_P

Q));

%develop Jacobian

for p=length(P_SW)+1:length(P_SW)+length(P_PV)+length(P_PQ)

for

q=length(theta_SW)+1:length(theta_SW)+length(theta_PV)+length(theta_PQ)

if(p~=q)

J11(p-length(P_SW),q-

length(theta_SW))=abs(V(p))*abs(V(q))*(real(ybus(p,q))*sin(theta(p)-

theta(q))-imag(ybus(p,q))*cos(theta(p)-theta(q)));

end

if(p==q)

J11(p-length(P_SW),q-length(theta_SW))=-Q(p)-

imag(ybus(p,p))*V(p)^2;

end

end

end

for

p=length(Q_SW)+length(Q_PV)+1:length(Q_SW)+length(Q_PV)+length(Q_PQ)

for

q=length(theta_SW)+1:length(theta_SW)+length(theta_PV)+length(theta_PQ)

if(p~=q)

J21(p-length(Q_SW)-length(Q_PV),q-length(theta_SW))=-

abs(V(p))*abs(V(q))*(real(ybus(p,q))*cos(theta(p)-

theta(q))+imag(ybus(p,q))*sin(theta(p)-theta(q)));

end

if(p==q)

J21(p-length(Q_SW)-length(Q_PV),q-

length(theta_SW))=P(p)-real(ybus(p,p))*V(p)^2;

142

end

end

end

for p=length(P_SW)+1:length(P_SW)+length(P_PV)+length(P_PQ)

for

q=length(V_SW)+length(V_PV)+1:length(V_SW)+length(V_PV)+length(V_PQ)

if(p~=q)

J12(p-length(P_SW),q-length(V_SW)-

length(V_PV))=abs(V(p))*(real(ybus(p,q))*cos(theta(p)-

theta(q))+imag(ybus(p,q))*sin(theta(p)-theta(q)));

end

if(p==q)

J12(p-length(P_SW),q-length(V_SW)-

length(V_PV))=P(p)/abs(V(p))+real(ybus(p,p))*abs(V(p));

end

end

end

for

p=length(Q_SW)+length(Q_PV)+1:length(Q_SW)+length(Q_PV)+length(Q_PQ)

for

q=length(V_SW)+length(V_PV)+1:length(V_SW)+length(V_PV)+length(V_PQ)

if(p~=q)

J22(p-length(Q_SW)-length(Q_PV),q-length(V_SW)-

length(V_PV))=abs(V(p))*(real(ybus(p,q))*sin(theta(p)-theta(q))-

imag(ybus(p,q))*cos(theta(p)-theta(q)));

end

if(p==q)

J22(p-length(Q_SW)-length(Q_PV),q-length(V_SW)-

length(V_PV))=Q(p)/abs(V(p))-imag(ybus(p,p))*abs(V(p));

143

end

end

end

%Jacobian

J=[J11 J12;J21 J22];

%x=angles and voltages

x=[theta_PV theta_PQ V_PQ];

%f(x)=P(x)-Pi and Q(x)-Qi

f_x=[P_PV-Pi_PV P_PQ-Pi_PQ Q_PQ-Qi_PQ];

%formula xk+1=xk-[J^-1*f(x)]

%precision of solution

error=inv(J)*f_x';

x=x-error';

theta_PV=x(1:length(theta_PV));

theta_PQ=x(length(theta_PV)+1:length(theta_PV)+length(theta_PQ));

V_PQ=x(length(theta_PV)+length(theta_PQ)+1:end);

end

%rearrange voltages and angles based on the bus indexing

theta=[theta_SW theta_PV theta_PQ];

V=[V_SW V_PV V_PQ];

V=[busindex' V'];

V=sortrows(V,1);

144

V(:,1)=[];

V=V';

theta=[busindex' theta'];

theta=sortrows(theta,1);

theta(:,1)=[];

theta=theta';

%Calculating final solutions for P, Q

P=zeros(1,length(bustype));

Q=zeros(1,length(bustype));



P(i)=P(i)+abs(V(i))*abs(V(j))*(real(ybus0(i,j))*cos(theta(i)-

theta(j))+imag(ybus0(i,j))*sin(theta(i)-theta(j)));

end

end



Q(i)=Q(i)+abs(V(i))*abs(V(j))*(real(ybus0 (i,j))*sin(theta(i)-

theta(j))-imag(ybus0(i,j))*cos(theta(i)-theta(j)));

end

end

145

function [M lamda Pf] = statematrix1(WR ,gen_data, load_data, P ,Q ,V

,theta ,ybus)

%determine the gen and load bus connections

load_conn=load_data(:,2);

gen_conn = gen_data(:,2);

%generator inertia constant

H=gen_data(:,4);

%generator admittances

xd=gen_data(:,3);

yd=-1i./xd;

%number of loads and gens

n_loads=max(load_data(:,1));

n_gens=max(gen_data(:,1));

%initializations

yload=[];

E_complex=[];

%determine Load admittances

for k=1:n_loads

yload=[yload -

P(load_conn(k))/V(load_conn(k))^2+1i*(Q(load_conn(k))/V(load_conn(k))^2

)];

end

%modifying the ybus with the load and generator admittances

genindex=0;

loadindex=0;

for k=1:length(ybus)

146

if any(gen_conn==k)

genindex=genindex+1;

ybus(k,k)=ybus(k,k)+yd(genindex);

end

if any(load_conn==k)

loadindex=loadindex+1;

ybus(k,k)=ybus(k,k)+yload(loadindex);

end

end

%rearranging ybus with the generator connected buses on the top

s=[];

genindex=0;

for k=1:length(ybus)

if any(gen_conn==k)

genindex=genindex+1;

s=[s gen_data(genindex,1)];

else

s=[s (n_gens+1)];

end

end

ybus=[s' ybus];


ybus(:,1)=[];

ybus=[s;ybus];

ybus=ybus';


ybus=ybus';

ybus(1,:)=[];

%adding generator impedance matrix to ybus

147

gen_imp_matrix=zeros(length(yd));

for i=1:length(yd)

for j=1:length(yd)

if(i==j)

gen_imp_matrix(i,i)=yd(i);

end

end

end

extra=zeros(length(yd),(length(ybus)-length(yd)));

temp1=[gen_imp_matrix -gen_imp_matrix extra];

temp2=[-gen_imp_matrix;extra'];

temp3=[temp2 ybus];

temp4=[temp1; temp3];

ybus_temp=temp4;

%KRON reduction of ybus

n=n_gens;

r=length(ybus_temp);

ybus_red=ybus_temp(1:n,1:n)-

ybus_temp(1:n,n+1:r)*inv(ybus_temp(n+1:r,n+1:r))*ybus_temp(n+1:r,1:n);

%finding E and delta for generators

for k=1:n

E_complex=[E_complex

V(gen_conn(k))+Q(gen_conn(k))*xd(k)/V(gen_conn(k))+1i*P(gen_conn(k))*xd

(k)/V(gen_conn(k))]; %#ok<AGROW>

end

E=abs(E_complex);

delta=angle(E_complex);

delta0=zeros(1,length(delta));

148

for k=1:length(theta)

if any(gen_conn==k)

index=1;

while(gen_conn(index)~=k)

index=index+1;

end

delta0(index)=delta(index)+theta(k);

end

end

%develop Ps matrix nxn (diagonal entries are meaningless and never

used)

for i=1:n

for j=1:n

Ps(i,j)=E(i)*E(j)*(imag(ybus_red(i,j))*cos(delta0(i)-

delta0(j))-real(ybus_red(i,j))*sin(delta0(i)-delta0(j)));

end

end

A=zeros(n);

%develop A matrix n-1xn-1 with the nth generator as a reference

for i=1:n

for j=1:n

if(i~=j)

A(i,j)=((WR/2)/H(i))*Ps(i,j);

end

if(i==j)

for k=1:n

if (k~=i)

A(i,i)=A(i,i)-((WR/2)/H(i))*Ps(i,k);

end

end

end

end

end

149

d=0.1; %Damping factor

D=zeros(n);

for i=1:n

for j=1:n

if (i==j)

D(i,j)=-(WR*d)/(2*H(i));

else

D(i,j)=0;

end

end

end

%develop M matrix

M=[zeros(n) eye(n);A D];

%right eigenvector matrix R

%left eigenvector matrix L

%diagonal eigenvalue matrix lamda

[R lamda]=eig(M);

L=inv(R);

%participation factor matrix Pf state:k and mode:i

for i=1:length(R)

for k=1:length(R)

Pf(k,i)=abs(L(i,k)*R(k,i));

end

end

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POWER SYSTEM STABILITY RESPONSE AND CONTROL USING …rx915z833/... · POWER SYSTEM STABILITY RESPONSE AND CONTROL USING SMALL SIGNAL ANALYSIS A Thesis Presented By Mark Germanos to

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