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Proximity-to-Separation Based Energy Function Control Strategy for
Power System Stability
Teck-Wai Chan B.Eng (Hons.), MSc
A thesis submitted in partial fulfilment of the requirements for the degree of
Doctor of Philosophy
Research Concentration in Electrical Energy
School of Electrical and Electronic Systems Engineering
Queensland University of Technology
2003
Statement of Original Authorship
The work contained in this thesis has not been previously submitted for a
degree or diploma at any other higher education institution. To the best of
my knowledge and belief, the thesis contains no material previously
published or written by another except where due reference is made.
Signed:_____________________________
Date: ______________________________
Acknowledgement List
I like to express my heartfelt gratitude to my principal supervisor, Professor
Gerard Ledwich for his valuable guidance and selfless supports during these
three years of my PhD research. His patience and unfailing encouragement
have been the major contributing factors in the completion of my PhD
research. Professor Ledwich’s constructive comments on our conference
papers and prospective journal paper have benefited me significantly. It has
been my greatest pleasure to expand my learning horizon during these three
years of PhD research under the guidance of Professor Gerard Ledwich.
Finally, I must say that I am fortunate to have Professor Gerard Ledwich as
my principal supervisor who has always been there for me.
I also like to thank my associate supervisor, Dr. Edward Palmer, for his
patience and encouragement during the course of the research. His
constructive comments on our conference paper and prospective journal
paper have benefited me significantly. Finally, I must say that I am fortunate
to have Dr. Palmer as my associate supervisor who has always been there
for me.
Finally, I like to thank my family for their support during my entire research
period that has enabled me to concentrate my work at QUT.
Dedication
This work is dedicated to my two PhD supervisors, my late father, my
mother, my late elder sister, my brother, my younger sister, my wife and
son.
List of Publications
1. T. W. Chan and G. Ledwich, "Multi-mode damping using single HVDC link," Australasian Universities Power Engineering Conference 2001, Perth, Australia, pp. 483-488, September 2001.
2. T. W. Chan, G. Ledwich, and E. W. Palmer, "Is velocity feedback
always best for machine stability control?," Australasian Universities Power Engineering Conference 2002, Melbourne, Australia, October 2002.
3. T. W. Chan, G. Ledwich, and E. W. Palmer, "Is velocity feedback
always best for machine stability control ?," Journal of Electrical & Electronics Engineering, Australia, Vo. 22, No. 3, pp. 195-202, 2002.
4. T. W. Chan, G. Ledwich, and E. W. Palmer, “Strengthening
Multimachine Synchronism," under review by the IEEE Transactions on Power System, 2003.
i
Table of Contents
Abstract v
Keywords vii
List of Illustrations and Diagrams viii
List of Tables xiii
List of Abbreviations xiv
Table of Symbols xvi
Chapter 1. Introduction 1 1.1. Power System Oscillation 5 1.2. Small Signal Analysis 6
1.2.1. Eigenvalues Analysis 9 1.2.2. Participation Matrix P 12
1.3. Direct Method and Total Energy 13 1.4. Aim of the Research 17 1.5. References 18
Chapter 2. Existing Control Methods 21 2.1. Introduction 21 2.2. Energy Function and Unstable Equilibrium Point (UEP) 25
2.2.1. Closest UEP Method 27 2.2.2. UEP in the Direction of Fault Trajectory 28 2.2.3. Controlling UEP Method 29 2.2.4. PEBS based Controlling UEP Method 30 2.2.5. BCU Method 31 2.2.6. Mode of Disturbance Method 31 2.2.7. Critical Cluster Method 33 2.2.8. Cutset Energy Function 34
2.3. Methods of Stability Assessments 36 2.4. Evaluation of Transient Energy 39 2.5. Energy function based Switching Control 39 2.6. Control of DC link and TCSC 41
ii
2.6.1. History of HVDC Link 42 2.6.2. Considerations in the Use of DC Link to Damp
Oscillations 43 2.6.3. Using Different Control Schemes to Supplement the
Control of Inverter 44 2.6.4. Discontinuous Control of Thyristor Control Series
Compensation (TCSC) 47 2.7. Wide Area Control 49
2.7.1. Centralized and Wide Area Control 50 2.8. Discussion 51 2.9. References 54
Chapter 3. Kinetic Energy Reduction for Power System Stability Design 59
3.1. Introduction 60 3.2. Energy Function Based Switching Control 62 3.3. Signum Function (Bang-bang Control) 68
3.3.1. A High Gain Feedback Control Problem 72 3.3.2. Reducing the Number of Mode by Switching
Control 75 3.3.3. Location of Zeros and Bang-bang Control 78
3.4. Exponential Convergence Introduced by a Saturation Function 79
3.5. Conclusion 83 3.6. References 87
Chapter 4. Weighted Energy Control 89 4.1. Introduction 90 4.2. Looking at a Two-area Energy Problems 91 4.3. A Proximity to Separation weighting Based on UEP 95 4.4. Conclusion 98 4.5. References 100 Chapter 5. Optimal Switching Near Separation 101 5.1. Introduction 102 5.2. A Velocity Proportional Control Based on Energy 104 5.3. Characterization of a First Swing Switching Stability
Problem 108
iii
5.3.1. Recognizing a Partly Stable region (PS region) 109 5.4. Undesirable Effect of Saturation Function 112 5.5. An Angle Look-ahead Control 114
5.5.1. An Optimal Switching Line 118
5.6. An Optimal Look-ahead ∆T that Satisfy Minimum Time Criteria 121
5.7. Conclusion 125 5.8. References 127 Chapter 6. Towards improving the Transfer Capacity 129 6.1. Introduction 130 6.2. Total Energy Decomposition – Cutset Energy 132
6.2.1. General Algebraic Expression for Cutsets Kinetic Energy 138
6.2.2. Decomposition of Total Potential Energy 140 6.3. Weighting of Cutset Energy 144 6.4. Detection of Proximity to Angle Separation Based on the
Boundary of Partly Stable Regions 146 6.5. A Cutset Based Energy Control that Enhances the
Survival of Power System 147 6.5.1. Derivative of Cutset Kinetic Energy 147 6.5.2. Cutset Angle Look-ahead Dependant Terms 148 6.5.3. A Cutset Energy Based Control 150
6.6. Case study 1 (Classical Three-machine 9-bus System) 152 6.6.1. Total Kinetic Energy Reduction Control 153 6.6.2. Determining Feasible Cutsets and Its Corresponding
UEPs 156 6.6.3. Determining the Cutset Energy Equation 159 6.6.4. Determining the Proximity to Separation Prediction 160 6.6.5. Determining the Cutset Energy Reduction Control 165 6.6.6. Results 166
6.6.6.1. Energy Evaluated at an Unstable Local Minimum (ULM) is not Critical Energy 167
6.6.6.2. The Uncertainty in the Type of Power System Separation in a Multiple UEPs Operating Condition 172
6.6.6.3. The Ease of Using Cutset Energy to Predict a Potential Separation 177
6.6.6.4. The Value of using Cutset energy Control 179 6.7. Case Study 2 (Detailed Six-machine 21-bus System) 186 6.8. Implementation Issues 198
iv
6.8.1. Response Time of SVC and Time Delay in Data Transmission 198
6.8.2. GPS Jitter 200 6.8.3. Measurement Noise 200 6.8.4. Multidimensional Issue of Cutsets 201 6.8.5. Summary of Implementation Issues 204
6.9. Conclusion 206 6.10. References 209 Chapter 7. Conclusion and Recommendation 211 7.1. Conclusion 211 7.2. Recommendations 217 7.3. References 224
v
Abstract
The issue of angle instability has been widely discussed in the power
engineering literature. Many control techniques have been proposed to
provide the complementary synchronizing and damping torques through
generators and/or network connected power apparatus such as FACTs,
braking resistors and DC links. The synchronizing torque component keeps
all generators in synchronism while damping torque reduces oscillations and
returns the power system to its pre-fault operating condition. One of the
main factors limiting the transfer capacity of the electrical transmission
network is the separation of the power system at weak links which can be
understood by analogy with a large spring-mass system. However, this
weak-links related problem is not dealt with in existing control designs
because it is non-trivial during transient period to determine credible weak
links in a large power system which may consist of hundreds of strong and
weak links. The difficulty of identifying weak links has limited the
performance of existing controls when it comes to the synchronization of
generators and damping of oscillations. Such circumstances also restrict the
operation of power systems close to its transient stability limits. These
considerations have led to the primary research question in this thesis, “To
what extent can the synchronization of generators and damping of
oscillations be maximized to fully extend the transient stability limits of
power systems and to improve the transfer capacity of the network?” With
the recent advances in power electronics technology, the extension of
vi
transfer capacity is becoming more readily achievable. Complementary to
the use of power electronics technology to improve transfer capacity, this
research develops an improved control strategy by examining the dynamics
of the modes of separation associated with the strong and weak links of the
reduced transmission network. The theoretical framework of the control
strategy is based on Energy Decomposition and Unstable Equilibrium
Points. This thesis recognizes that under extreme loadings of the
transmission network containing strong and weak links, weak-links are most
likely to dictate the transient stability limits of the power system. We
conclude that in order to fully extend the transient stability limits of power
system while maximizing the value of control resources, it is crucial for the
control strategy to aim its control effort at the energy component that is
most likely to cause a separation. The improvement in the synchronization
amongst generators remains the most important step in the improvement of
the transfer capacity of the power system network.
vii
Keywords
Lyapunov, power system, stability, switching, energy function-based
control, bang-bang control, saturation function, energy in phase portrait,
partly stable region, energy weighting, cutset, energy decomposition, cutset
energy, proximity-to-critical cutset energy, proximity-to-partly stable
region, cutset energy-based control, quantified transient stability limits and
transfer capacity.
viii
List of Illustrations and Diagram
Figure 2.1: A back-to-back DC link interconnecting two AC power
systems. -------------------------------------------------------- 45
Figure 3.1: Single line diagram of a four-machine two-area power
system. --------------------------------------------------------- 62
Figure 3.2: The response of machine speeds. Due to the bang-bang
control, continuous oscillations are formed. -------------- 69
Figure 3.3: The system response is kept on a switching hyperplane
S beyond 3.5 seconds effectively resulting in zero
control. --------------------------------------------------------- 69
Figure 3.4: The remaining total kinetic energy in the power system
results in continuous oscillations. -------------------------- 71
Figure 3.5: A general negative feedback transfer function block. -- 73
Figure 3.6: A Root Locus diagram showing a non-divergent control
effect. ----------------------------------------------------------- 74
Figure 3.7: A dual LC circuit with a control voltage V. --------------- 76
Figure 3.8: Under a bang-bang control, the capacitor voltages of
dual LC circuit have one remaining mode undamped. -- 78
Figure 3.9: The machine speeds converge owing to the soft
switching in linear region. ----------------------------------- 81
Figure 3.10: The system responses decay exponentially near an
origin as the continuous control in the linear region
dominates. ----------------------------------------------------- 82
Figure 3.11: The total kinetic energy converges exponentially
towards an origin reaching the system solution. --------- 82
Figure 3.12: The energy based control that uses the saturation
function control law. ----------------------------------------- 83
Figure 4.1: Single line diagram of a four-machine two-area system. 91
ix
Figure 4.2: Control effort directed to save area 2 from separation
around 0.25s. -------------------------------------------------- 97
Figure 4.3: Weightings of the two areas that indicated the high risk
separation in the area. ---------------------------------------- 98
Figure 5.1: Two-machine system with breaking resistor at bus 3. --- 105
Figure 5.2: Control chatters and angle hovers at maximum as the
forced damping is effectively zero. Faults at bus 4 are
cleared at 1.0459 seconds. ----------------------------------- 109
Figure 5.3: Effects of δ& bang-bang control switching in the PS
region causing control chattering. Fault cleared at
1.0459 seconds. ----------------------------------------------- 110
Figure 5.4: A close-up view of the partly stable region showing two
sets of trajectories directing towards the 0=δ&
switching line. ------------------------------------------------ 111
Figure 5.5: Undesirable effect of using a δ& saturation function at
first swing when the severe faults at bus 4 are cleared at
1.067 seconds. ------------------------------------------------ 113
Figure 5.6: The use of switching line ST in the δ& bang-bang control
avoids the chattering of control. ---------------------------- 117
Figure 5.7: Understanding a divider line at a reduced UEP. ---------- 119
Figure 5.8: Understanding an optimal slope. --------------------------- 119
Figure 5.9: Illustration of the approximated switching line and the
total switching line ST. --------------------------------------- 121
Figure 5.10: Delayed switching performance using ∆T=1.2
(dashed) and ∆T=1.7 (solid). -------------------------------- 123
Figure 5.11: Insignificant settling time between the two examples
of switching at different instances. ------------------------- 125
Figure 6.1: A three-machine 6-bus system is used to illustrate the
possible separations. ----------------------------------------- 132
Figure 6.2: A three-machine 9-bus system with a controllable SVC
installed at bus 5. --------------------------------------------- 153
x
Figure 6.3: Unstable system trajectory on a total potential energy
surface. The fault at bus 7 is cleared at 233ms. Under
no SVC control, power system separates at the UEP 2
associated with the cutset (23/1). --------------------------- 167
Figure 6.4: Relative angles show the power system separation
between generator 1 and the rest of the system. The
fault at bus 7 is cleared at 233ms. -------------------------- 168
Figure 6.5: Generator angles are separating when the fault at bus 7
is cleared at 300ms. Cutset (1/23) or (23/1) is
confirmed as the only separation possible for the power
system. --------------------------------------------------------- 170
Figure 6.6: Unstable system trajectory on a total potential energy
surface. The fault at bus 9 is cleared at 300ms. Under
no SVC control, power system separates at the UEP 2
associated with the cutset (23/1). --------------------------- 171
Figure 6.7: Unstable system trajectory on a total potential energy
surface. The fault at bus 7 is cleared at 400ms. Under
no SVC control, power system separates at the UEP 2
associated with the cutset (23/1). --------------------------- 173
Figure 6.8: Separation of generator 3 from the rest of the system
associates with cutset (3/12). The fault at bus 9 is
cleared at 463ms. --------------------------------------------- 175
Figure 6.9: System trajectory on potential energy surface. The fault
at bus 9 is cleared 381ms and an unexpected separation
associated with cutset (2/13) occurs. ----------------------- 176
Figure 6.10: The total energy diagram shows the difficulty of
determining when the power system has separated. The
fault at bus 9 is cleared at 381ms. -------------------------- 177
Figure 6.11: Cutset energy responsible for the various types of
separations. The fault at bus 9 is cleared at 381ms. The
power system separates at the UEP 2. --------------------- 178
xi
Figure 6.12: Generator 2 has separated from the rest of the system
at around 1.2s which is predicted by the “cutset energy
2”associated with cutset (2/13). ---------------------------- 179
Figure 6.13: The system trajectory that is controlled using Uen
(dashed) and U# (solid line with dots) are shown on the
potential energy surface. The fault at bus 9 is cleared at
390ms. --------------------------------------------------------- 181
Figure 6.14: The response of machine angles under the influence of
the Uen (dotted) and U# (solid) controls. The fault at bus
9 is cleared at 390.4ms. -------------------------------------- 182
Figure 6.15: The switching of control values due to the Uen (dotted)
and U# (solid) controls. -------------------------------------- 183
Figure 6.16: Cutset energy of the power system under the influence
of the energy control Uen. The fault at bus 9 is cleared
at 390.4ms. ---------------------------------------------------- 184
Figure 6.17: Cutset energy of the power system under the influence
of the cutset energy control U#. The fault at bus 9 is
cleared 390.4ms. ---------------------------------------------- 185
Figure 6.18: A six-machine 21-bus test system with the Left side of
the generations far from the Central and Right areas. --- 187
Figure 6.19: SVC control using local measurements of power and
voltage to approximate a velocity and positional
feedback. --------- 189
Figure 6.20: COA angle difference for the Central-Right area
(dotted lines) and the Central-Left area (solid lines)
under the influence of different SVC controls. Line
trips at 150ms for the fault at bus 12. ---------------------- 190
Figure 6.21: COA angle difference for the Central-Right area
(dotted lines) and the Central-Left area (solid lines)
under the influence of different SVC controls. Line
trips at 200ms for the fault at bus 12. ---------------------- 192
xii
Figure 6.22: Responses of SVC controls at bus 18 and 13. Line
trips at 204ms. SVC installed at bus 18 and 13 respond
to the inter-area mode associated with the Central-Left
area (solid line) and the Central-Right area (dotted line)
respectively. --------------------------------------------------- 193
Figure 6.23: Quantifying the transient stability limits of the power
system under different SVC controls. Fault is removed
by tripping one of the faulty parallel lines between bus
12 and 13. P18-19 refers to the steady state power flow. 195
Figure 6.24: Wrong selection of K2 in the voltage error control loop
leads to the separation at central-Left areas (dotted).
Line tripped at 199ms. --------------------------------------- 197
Figure 6.25: A comparison between the damping performance of
(mode) and (cutset) controls. Line trips at 199ms. ------- 198
Figure 6.24: Ten-machine 39-bus New England system. ------------- 204
xiii
List of Tables
Table 6.1: The relationship between the unstable operating points
and cutsets. ---------------------------------------------------- 158
Table 6.2: The reduced unstable operating points of the power
system in Figure 6.2. ----------------------------------------- 163
Table 6.3: System data of Figure 6.2 modeified to reduced the
loading of lines between bus 5 & 7 and bus 6 & 9. This
modified loading yielded six UEPs. ----------------------- 172
Table 6.4: The relationship between the Unstable Equilibrium
Points (UEPs) and cutsets. ---------------------------------- 172
xiv
List of Abbreviations
AC: Alternating Current
ACC: Acceleration
AC/DC: Alternating Current and Direct Current
BCU: Boundary of stability based Controlling Unstable equilibrium
point
COA: Centra-Of-Area
DC: Direct Current
DFP: Davidon-Fletcher-Powell
DSA: Dynamic Security Assessment
EHV: Extra High Voltage
EOP: Equilibrium Operating Point
FACT: Flexible AC Transmission system
GPS: Global Positioning System
HVDC: High Voltage Direct Current
KE: Kinetic Energy
LC: Inductor and Capacitor
LQR: Linear Quadratic Regulator
MOD: Mode of Disturbance
NR: Newton Ralphson
PEBS: Potential Energy Boundary Surface
PS: Partly Stable
PSS: Power System Stabilizer
SCS: Series Capacitor Compensation
SCR: Short-circuit ratio
SEP: Stable Equilibrium operating Point
SMIB: Single Machine Infinite Bus
SVC: Static VAR Compensator
TCSC: Thyristor Controlled Series Compensation
xv
TNSP: Transmission Network Service Provider
UEP: Unstable Equilibrium Point
ULM: Unstable Local Minima
MPC: Model Predictive Control
xvi
Table of Symbols
Symbol Meaning
x State variables
xo The initial conditions of the state variables
y Output variables
yo The initial conditions of the output variables
u The input variables. In the switching perspective, it
represents a switching input
uo The initial conditions of the input variables
∆x The small perturbations of state variables
∆y The small changes in the output variables
∆u The small perturbations of input variables
x& The derivatives of state variables
ox& The initial conditions of the derivatives of state variables
f(.) An objective function f
g(.) An objective function g
i
i
xf
∂∂
Derivative of the ith function fi with respect to the ith variable
xi
i
i
xg
∂∂
Derivative of the ith function gi with respect to the ith variable
xi
[.] A matrix of variables or a column vector or a row vector
A An nxn system matrix
B An nxn input matrix
C An mxn output matrix
D An mxn feed forward matrix
v Left eigenvector
xvii
V Matrix containing rows of left eigenvectors
w Right eigenvector
W Matrix containing columns of right eigenvector
λ Eigenvalue
x∆ Vector of state variables transformed by V from its original
base to eigenvector base
Λ A matrix containing eigenvalues in its diagonal entries and
zeros in its off-diagonal entries
P Participation matrix
V A Lyapunov function. In energy perspective, it is also used to
represent total energy
V& A Lyapunov function
∑=
n
iix
1 Summation of the variable x from its ith to nth elements
mi Inertial constant of ith machine
δi The ith machine angle
δij The angle difference between the ith and jth generator angle
siδ The ith machine angle at an equilibrium point
uiδ The ith machine angle at an equilibrium point
ωi, iδ& Angular velocity of the ith machine angle δi
ωo Angular velocity of the network reference frame
iδ&& Angular acceleration of the ith machine angle δi
Pmi Mechanical power of the ith machine
vi Voltage at the ith bus
Bij The admittance of the transmission line between the ith and jth
buses
∆VP.E. Small change in potential energy
UEPV .. Potential energy evaluated at an unstable equilibrium point
clEPV .. Potential energy evaluated at fault clearing
VK.E.|corr Corrected transient kinetic energy
xviii
Meq Equivalent inertial constant of a group of machine
Mcr Sum of all machine inertial constant in a critical machine
group
Msys Sum of all machine inertial constant in a system machine
group
syscr ωω ~,~ Centre of area angular velocity of a critical machine group
and a system machine group respectively
η(δc) Extended equal area criterion’s stability margin at critical
angle
Adec(δc) Extended equal area criterion’s deceleration area between
critical angle δc and angle at a stable equilibrium point δs
Aacc(δc) Extended equal area criterion’s acceleration area between
critical angle δc and angle at a stable equilibrium point δs okσ Inter-nodal angle at a stable equilibrium point of the kth line
lkµ Vulnerability coefficient of the kth line evaluated between o
kσ
and (π- okσ )
ukµ Vulnerability coefficient of the kth line evaluated between o
kσ
and (-π- okσ )
−+ii υυ , Vulnerability indices of the ith line
U Vector of switching control input
∆ijα Changes in line reactance between the ith and jth buses due to
control Uα
maxc
minc X,X The minimum and maximum series compensating
capacitance
scX The steady state series compensating capacitance
Y Reduced admittance matrix of a power system network
Yout Reduced admittance matrix Y without controller
Yin Reduced admittance matrix Y with controller
xix
Yactual Reduced admittance matrix Y taking into consideration the
switching operation
∆Y Changes in the reduced admittance matrix Y
gij Shunt conductance at the ith row and jth column of Y
bij Transfer admittance at the ith row and jth column of Y
∆gij Changes in the shunt conductance at the ith row and jth
column of Y
∆bij Changes in the Transfer admittance at the ith row and jth
column of Y
Pdci Power transfer across a DC link at the ith area
Pcoai Centre of area power or perfect governor term in the ith area
Pei Electrical power output of the ith generator
G2 Shunt resistor
ii θδ ,~
The ith machine angle measured in the centre of area frame
θij The angle difference between the ith and jth buses measured in
the centre of area frame
δcoa Angle for the centre of area
icoaδ Angle for the centre of area of the ith area
sgn(.) Signum function or bang-bang control function
sat(.) Saturation function
t time
S Switching surface
Vkei Kinetic energy of the ith generator
Vpe Total potential energy
K Gain
z(s) Feedback signal of a negative feedback control loop.
ϕi Angle of departure in Root Locus analysis
αz Angle from the closed loop pole of interest to all of the finite
zero
xx
αp Angle from the closed loop pole of interest to the rest of the
poles
wt Weighting
SV Switching surface derived from the derivative of kinetic
energy
γ A scalar multiplier for angle look-ahead control
∆T Angle look-ahead duration
SP Switching surface derived from angle look-ahead control
ST Total switching surface
(i/jk) #gA The ith group of generators that separates from the rest of the
power system that contain the jth and kth groups of generators.
The subscript ‘#’ distinguishes the commonly used notation
(i/jk) as the cutset, g refers to the gth cutset in the set A of the
list of possible cutset and A is the Ath set of cutsets where the
ith generator group is the critical group of generators that
tends to separate from the rest of the system.
ηi The product of the ith generator’s angular velocity and square
root of ith generator’s inertial constant
ijp The square of the sum of ηi and ηj
ijn The square of the difference of ηi and ηj
Vke#i The ith cutset kinetic energy
µ The total number of separations
sυ The vector of indices of the generators separating from the
rest of the system
rυ The vector of indices of the remaining generators that do not
appear in vector sυ .
ns The total number of generators in vector sυ
nr The total number of generators in vector rυ
Ω The kinetic energy decomposition scaling coefficient
β The total number of the types of separation including the
cutset (ijk)
xxi
njC Combinatory notation of choosing j elements from n
elements
Fυ A vector evaluated to eliminate the repeating types of
separation
trunc(.) truncating function that rounds a positive non-integer to the
nearest integer towards zero
ijl The energy stored in the lines interconnecting the ith and jth
generators
τ The decomposition scaling coefficient for the energy stored
in transmission lines
σi Square root of the shaft energy of ith generator
ijs+ The square of the sum of σi and σj
ijs- The square of the difference of σi and σj
λ The decomposition scaling coefficient of the shaft energy
sisn )(δ Stable equilibrium angle of the ith generator selected from the
vector ns
wt#i Weighting of the ith cutset
εi Proximity to critical cutset energy coefficient
Vpe#i The ith cutset potential energy cri
ipeV # The critical ith cutset potential energy
)(#
lineipeV The energy stored in lines associated with the ith cutset
)(#
shaftipeV The shaft energy associated with the ith cutset
Vke#i The ith cutset kinetic energy
Vt#i The ith cutset energy evaluated from the sum of Vke#i and Vpe#i
Proxuep#i The closeness to angle separation index evaluated at ith cutset
unstable equilibrium point
[θsys] Angle vector of the system trajectory
xxii
ruep
i#θ Angle vector of the ith cutset’s unstable equilibrium point
reduced due to switching operation
SV# Switching surface derived from the derivative of cutset
kinetic energy
SP# Switching surface derived from the cutset based angle look-
ahead control
S# Weighted switching surface of the sum of SV# and SP#
wuepijθ The angle difference between the ith and jth generators
operating at unstable equilibrium point associated with the
wth cutset
U# Vector of switching control input derived from the sum of SV
and S# under the condition of the saturation function
Uen vector of switching control input derived from the sum of SV
under the condition of the saturation function
δij/kh Angle difference between δij and δkh
∠ Bus angle
E Generator terminal voltage
Texc Exciter time constant
J Performance index
Q Weighting matrix for the minimization of performance index
R Weighting matrix for the control values used in the
minimization of performance index
xT The superscript T refers to the transpose of a matrix or vector
1
Chapter 1
Introduction
A power system is one of the most complex systems ever built, consisting of
hundreds of generators, protection switches and thousands of kilometers of
transmission lines. It is constructed to generate and deliver electricity to
serve the needs of mankind. When operating this power system, one would
encounter certain power system dynamics such as electromagnetic changes
in electric machines, electromechanical dynamics between rotating masses
and the thermodynamic changes of boilers [1]. These dynamics could occur
in various overlapping time frames from microseconds to hours.
One example of a large modern power system formed by interconnecting
the power systems of various states is the South-Eastern Australian power
network made up of the power systems of Queensland, New South Wales,
Victoria and South Australia. Although this interconnection to create a large
longitudinal power system may experience complicated electromechanical
dynamics amongst its generators giving rise to potential power system
separations at its weak interconnecting links, it offers attractive advantages.
2
For instance, one of the states can maintain a reliable operation of its power
system while saving a large amount of spinning reserves when it is
interconnected to this ‘large power system’. Another advantage is the
economic benefit for all these states in terms of lower electricity prices as a
result of the interstate generation competition and reduced capital
investment in controllers. However, before these states can enjoy the full
benefits of an interconnected power system, the entire power system must
have the abilities to withstand (survive) severe disturbances and allow high
power transfers between regions. To achieve these abilities, an appropriate
control that synchronizes electric machines during transient at post-fault is
necessary. It is understood from the excitation control and power system
stabilizer (PSS) control loops [2, Page 254-277] that in small signal
analysis, voltage feedback to the excitation system is a crucial feedback
control that tends to improve transient stability by increasing the
synchronizing torques in the power system. The PSS control is introduced to
increase the damping torques in the power system to damp the subsequent
oscillations. This means there can be a conflict between synchronizing and
damping torques.
In this chapter, the fundamentals of small signal analysis, in particular
eigenvalues analysis and participation factors are introduced. Its benefits
and limitations will also be discussed. We also introduce the Lyapunov’s
direct method as a promising non-linear analysis tool that can be used to
overcome the restrictions found in small signal analysis, particularly, when
3
the consequences of severe disturbances are being considered. The benefit
of using the direct method to predict transient stability at post-fault without
the need to perform a lengthy time domain simulation during the post-fault
period is discussed.
In Chapter 2, literature review associated with energy function based control
is described. In particular, the literature review focuses on stability
prediction methods associated with Unstable Equilibrium point (UEP). The
literature review is also extended to understand the problems associated
with the control of DC link and the control of Thyristor Controlled Series
Compensator (TCSC) associated with the damping of multiple (n-1) modes.
The literature review in wide area control in association centralized and
decentralized control aids in the reorganization of the benefits of remote
measurement for control.
Chapter 3 describes the energy function based switching control. The
problem of using this high gain velocity feedback control (or a velocity
based bang-bang control) near an equilibrium operating point is discussed.
Chapter 4 describes the problem of energy function based switching control
associated with the operation of power system in which some areas are
strongly (or closely) linked while some areas are weakly (or distantly)
linked. This is referred to as the strong and weak links in this thesis. We will
discuss on how Unstable Equilibrium Points (UEPs) can be associated with
4
these strong and weak links and how UEP associated weightings can be
used in energy function based switching control to direct control efforts to
the weak area. The benefits of giving the weak area a higher control priority
in terms of saving a power system from angle instability are also discussed.
Chapter 5 describes the problem associated with switching near an Unstable
Equilibrium Point (UEP). The undesirable effects of machine angle
hovering near a UEP and the effect of proposing phase shift in association
with the robustness constraint in control implementation is discussed.
Chapter 6 describes the decomposition of total energy that characterizes the
different types of power system separations. These different types of power
system separations are defined as the different modes of separation in this
thesis. This chapter extends the controls proposed in the earlier chapters
using the Energy Decomposition technique to maximize the transient
stability limits of a multi-machine power system. The process of how a
switching control based on Energy Decomposed can be used to extend the
transient stability limits and improve the transfer capacity of the electrical
transmission network is discussed.
5
1.1. Power System Oscillation
In a power system, electricity is generated from rotating machines, known
as synchronous machines. The key feature of the power system operation is
that the generators at every power station must run in synchronism with
each other. This permits efficient transfer of power between generators and
loads. As energy can not be stored in the network for the late subsequent use
by loads, the total power generated must balance the consumption of power
by loads. Power flow from a generator depends on the angle of its rotor
compared with angle of other generators hence the power flow along
transmission lines is largely controlled by the angle between the voltages on
the two ends of the line. For instance, the transfer of power from bus A to B
along a transmission line is only possible when the angle of the voltage at
bus A is higher than the angle of the voltage at bus B. When there are small
disturbances in the power system network such as the changes in loads,
these cause an imbalance between the total power generated by synchronous
machines and the total power to be consumed by loads. In such case, each
synchronous machine is automatically controlled to adjust its generated
power to achieve the required power balance in the network. The generated
power at each synchronous machine relies on the control of its governor. As
the control of governor does not give rise to an instantaneous change in the
output power of the synchronous machine, this give rise to oscillating power
output amongst these synchronous machines in the power system. These
oscillations amongst synchronous machines are termed electromechanical
6
oscillations. Under the influence of the damper winding in each
synchronous machine, small oscillations caused by small changes in loads
can be sufficiently damped. When severe disturbances such as the outage of
lines occur, high magnitude oscillations may be noticeable, and the use of
controlled devices such as an exciter, power system stabilizer or FACTs can
be used to damp these oscillations. Generally, electromechanical oscillations
are classified into local mode and inter-area mode oscillations. Local mode
oscillation is commonly associated with an oscillating frequency range of
0.8 to 2.5Hz whereas inter-area oscillations are associated with 0.1 to 0.7Hz.
In general, inter-area oscillations are more poorly damped than local
oscillations. The focus of this study is to develop a control methodology that
is applicable to the control of these oscillations.
1.2. Small Signal Analysis
Small signal analysis [3, Page 18-23] can be applied to the study of the
nonlinear power system dynamics around an equilibrium operating point. At
an equilibrium operating point, a power system is subject to relatively small
disturbances such that the nonlinear power system dynamics, represented in
its fundamental form as a set of swing equations [1, Page 141-144], can be
linearized to approximate the non-linearity in the power system.
Equilibrium operating points are usually obtained from load flow analysis
[3, 4]. For the purpose of this study, boiler dynamics and governor dynamics
7
are not considered as they are largely outside the range of frequency being
examined.
The set of swing equations that describes the dynamic of a power system is
basically a set of first order differential equations, which are not an explicit
function of time [1, 3, 5]. Its state derivative dtdx
and output y represented in
a compact form is
),(),(
uxgyuxfx
==&
(1.1)
where x and u are the vectors of state variables and system inputs
respectively.
Under small perturbations, (1.1) becomes
yyuuxxgyxxuuxxfx
ooo
ooo
∆+=∆+∆+=∆+=∆+∆+=
),(),( &&&
(1.2)
where xo, yo and uo are the vectors of the initial conditions of states, outputs
and inputs at an equilibrium operating point respectively. The notations ∆x,
∆y and ∆u refers to the vectors of the small perturbations of states, outputs
and inputs around the equilibrium operating point respectively.
Expanding (1.2) by Taylor series and neglecting the second order
derivatives, an ith state derivative is
8
nn
iin
n
iiooii
nn
iin
n
iiooii
uug
uug
xxg
xxg
uxgy
uuf
uuf
xxf
xxf
uxfx
∆∂∂
++∆∂∂
+∆∂∂
++∆∂∂
+=
∆∂∂
++∆∂∂
+∆∂∂
++∆∂∂
+=
....),(
....),(
11
11
11
11
&
(1.3)
At an equilibrium operating point where the sum of all electrical power
generated meets the sum of all demands and losses in the transmission
network, machine accelerations are zero. This leads to 0=ox& . The compact
form of (1.3) is
uDxCyuBxAx
∆+∆=∆∆+∆=∆&
(1.4)
where
∂∂
∂∂
∂∂
∂∂
=
∂∂
∂∂
∂∂
∂∂
=
n
nn
n
n
nn
n
uf
uf
uf
uf
B
xf
xf
xf
xf
A
...
.........
...
,
...
.........
...
1
1
1
1
1
1
1
1
∂∂
∂∂
∂∂
∂∂
=
∂∂
∂∂
∂∂
∂∂
=
n
mm
n
n
mm
n
ug
ug
ug
ug
D
xg
xg
xg
xg
C
...
.........
...
,
...
.........
...
1
1
1
1
1
1
1
1
Matrix A is a nxn state matrix, matrix B is a nxn input matrix, matrix C is a
mxn output matrix, matrix D is a mxn feed forward matrix, ∆x is the vector
of state variables perturbed at an equilibrium operating point and ∆u is the
input vector of input states perturbed at an equilibrium operating point.
The above process of linearizing the swing equation is described in [5, Page
209-219] and is applicable to detailed machine models that describe the
9
dynamics of stator transient and the flux linkages between stator and rotor
circuits. Treatment of the linearized detailed machine model and the
electrical torques expressed in flux linkages terms are found in [3, 5-7]. The
machine-network interface equations that interfaces between the set of first
order differential equations describing machine dynamics and the set of
algebraic equations describing the power balance in the transmission
network are elaborated in [5].
One advantage of linearizing a set of non-linear power system equations
represented by a fundamental set of swing equations or detailed machine
model, is the availability of linear analysis tools such as the eigenvalues
analysis [3], the eigenvalues sensitivity analysis based on participation
factor [8], the residue method [9] and pole-zero analysis [10]. These
analysis tools determine system stability around an equilibrium operating
point. In control design, for example, the residue method and the poles-
zeros movement analysis are used to tune Power System Stabilizers (PSS) in
[9] and [10] respectively.
1.2.1. Eigenvalue Analysis
One of the most effective linear analysis tools is eigenvalue analysis. In
eigenvalue analysis, each of the system eigenvalues obtained from the
determinant 0Idet =− λA describes a mode of oscillation such as the local
10
mode, inter-area mode, control mode or torsional mode [3]. Amongst these
modes, the control mode is usually well damped [11] and the torsional mode
is usually excited by a poorly designed series compensation control [12].
The electromechanical oscillations of the power system are mainly
associated with the local and inter-area modes and it is common that local
modes are oscillating in a frequency range of 0.8 to 2.5 Hz while the inter-
area modes are in the frequency range of 0.1 to 0.7 Hz [13].
Although eigenvalues are easily obtained from a system matrix A (1.4), the
relation between states and modes are not easily observable as these state
equations in (1.4) are interdependent. In order to overcome this problem,
state equations can be decoupled using eigenvectors [3].
Considering the state space form of a linearized dynamical system
xAx ∆=∆& , matrix V is chosen as an operator that transforms a vector of
state variables ∆x from its original base to an eigenvector base represented
as x∆
xWx
orxx
1
V
−=∆
∆= (1.5)
Matrix V is orthogonal to matrix W that contain columns of system
eigenvectors. Each row vector in matrix V is referred to as a left eigenvector
and each column vector in matrix W is referred to as a right eigenvector.
Eigenvectors v and w are normalised to 1
11
1i
i
v
1v
0v
−=
=
=
w
w
w
iT
iT
(1.6)
Transforming the dynamical system xAx ∆=∆& , the new set of system
equations with eigenvector as a base is
( )( ) xAW
xAx∆=
∆=∆ −
VVV 1&
(1.7)
Referring to (1.7), matrix A is transformed by matrix V and W to a matrix
that contains system eigenvalues [3] in its diagonal entries
==
n
AW?000...000?
?1
V (1.8)
Looking at (1.4) and (1.8), it is clear that every state equation x&∆ has been
decoupled and each state variable ix∆ is directly related to a mode λ .
Eigenvectors v and w are defined in [3, 14]. Considering an ith column of the
right eigenvector wi, its kth element measures the activity of the state variable
kx∆ in the ith mode whereas for an ith row of the left eigenvector vi, its kth
element weighs the contribution of this activity to the ith mode.
12
1.2.2. Participation Matrix P
While it is easy to study the relation between states and modes by observing
the right and left eigenvectors independently, there are problems associated
with the units and scale of state variables [3]. The participation matrix P is
proposed in [3, 8] to overcome this dimension problem based on the product
of the left and right eigenvectors. For each element of Pki, the kth row is
associated with the kth state variable and the ith column is associated with the
ith mode
variablesstateofrowskth
pppp
pppppppppppp
modesofcolumnsith
P
nnnnn
n
n
n
..................
...
...
...
321
3333231
2232221
1131211
44444 844444 76
= (1.9)
In the P matrix (1.9), the influence of state variables on the ith mode is
considered by observing the weighted values of an ith column whereas the
contribution of a kth state to various modes is observed from the kth row;
[Pki].
We have thus far described some of the most useful linear analysis tools to
predict the dynamical stability of a power system around an equilibrium
operating point. When severe disturbances occur in a power system and
result in a large change in machine states, it is erroneous to use a set of
13
linearized power system equations to approximate the set of non-linear
power system equations. However, it is common for TNSP companies to
perform numerous small signal analyses based on several equilibrium
operating points in the hope of providing an extensive coverage of small
signal stability analysis [15].
1.3. Direct Method and Total Energy
In this section, the Lyapunov’s direct method and energy function are
explained. The benefits of using these non-linear analyses in the prediction
of transient stability are described.
In a large interconnected power system consisting of thousands of
generators, TNSP companies are facing problems [15] such as the
substantially long time involved in observing and predicting the result of
angle instability in a numerical simulation while examining a set of
contingencies, and the number of credible contingencies that needs to be
selected from a long list of possible contingencies for transient stability
study. This arises because during the actual operating conditions, the
behaviour of power system may change and result in a different set of
unexpected contingencies compared to the set of contingencies that is
examined during an off-line transient stability study. Although small signal
analysis based on linearized power system equations is capable of
determining system small signal stability, it is time consuming as several
14
equilibrium operating points associated with a contingency must be
considered, and the selection of equilibrium operating point is solely based
on the experience in operating the power system.
The above inconveniences found in transient stability study using the time
domain simulation and small signal analysis make Lyapunov’s direct
method more suitable for the transient stability analysis. This is mainly due
to the fact that the stability criterion of the direct method is capable of
predicting the stability of power system at post-fault without performing
time consuming simulations.
The Lyapunov’s direct method applied on a nonlinear system of )(xfx =&
requires a Lyapunov function V [16, Page 23-24] that satisfies the properties
in (1.10) to determine if a dynamical system will remain stable along its
post-fault trajectory without going through a simulation of the post-fault
period.
)( ,0)(
,0)(
0)(
xfxoftrajectorythealongxV
xxforxV
xVs
s
=<
≠>
=
&& (1.10)
where xs is the system states at an equilibrium operating point. The stability
criterion requires that if (1.10) is satisfied then x will be asymptotically
stable as time t progresses.
15
For a simple second order system, a Lyapunov function V is based on the
total energy of the nonlinear system and if V is found to be positive definite
for all x and 0<V& , the system will be globally stable.
For a power system, a candidate Lyapunov function V based on the total
energy neglecting the energy dissipated in lines is
( ) ( )∑ ∑ −−−−∑== +==
n
i
n
ij
sijijijji
siimi
n
iii BvvPmV
1 11
2 )cos()cos(21
δδδδω (1.11)
where ( )jiij δδδ −= , ( )sj
si
sij δδδ −= , Pmi is the mechanical power output
of electric machines, mi is the machine inertia constant, ωi is the machine
angular velocity and siδ is the machine angle at an equilibrium operating
point.
Examining (1.11), it is clear that the candidate Lyapunov function V is not
positive definite owing to the second and third terms (i.e. the potential
energy associated with the shaft energy and the energy stored in lines).
However, if the potential energy (i.e. the second and third terms of V) is
bounded within some angle limits forming a local region in angle space, this
results in a positive definite V. Relative to this local region, V becomes
positive semi-definite (i.e., 0≥V ). This local region is referred to as the
region of attraction [16, Page 49].
16
As V is positive semi-definite and bounded within a local region in angle
space, and derivative V& is negative semi-definite (i.e. 0≤V& ) relative to this
local region
( )∑ ∑∑= +==
−−+−=n
i
n
ijjijiijjiimi
n
iiii BvvPmV
1 11))(sin( δδδδδωω &&&&& (1.12)
where all parameters have been defined earlier in association with equation
(1.11).
Based on the above mentioned stability criterions, the task of determining
the transient stability of a large power system is seemingly achievable as
long as the system trajectory is in the local region (or region of attraction) at
post-fault.
Considering the Lyapunov stability criteria of 0<V& , for asymptotic
stability. It is often difficult to satisfy this strict requirement in control
design such as in the energy function-based switching control design that
aims at achieving a most negative V& to force the convergence of system
trajectory. This gives rise to a relaxed Lyapunov criteria of 0≤V& . Consider
the system trajectory of an unforced system with no damping inside the
region of attraction at post-fault, this system trajectory will not be unstable
but oscillate continuously with a constant energy. This response satisfies to
the relaxed Lyapunov criteria of 0≤V& but the continuous oscillations can
still exist. A controller that continuously yields a negative V& while keeping
17
the system trajectory inside the region of attraction will result in a system
trajectory that converges asymptotically.
A control that is designed based on this relaxed Lyapunov stability criteria,
relative to a local region of attraction, is referred to as an energy function
based control [17, 18].
1.4. Contribution of Thesis
The main objective of this research is to develop an effective way of
improving the transfer capacity of the power transmission system. The
control proposed emphasizes the weak links in a power system and is
generally independent of the structure of the power system, for instance, the
control is applicable in a longitudinal or meshed power system.
The purpose of this proposed control is to provide synchronization between
generators during first swing while damping the remaining oscillations at
subsequent swings. This has the benefits of maximizing the control
resources through the use of a control strategy that recognizes the existence
of strong and weak links in a large interconnected power system. In general,
the control strategy determines the appropriate control efforts when one
potential mode of separation becomes severely strained to reduce angle
instability in a large power system.
18
1.5. References
[1] J. Machowski, J. W. Bialek, and J. R. Bumby, Power System Dynamics and Stability: John Wiley & Sons Ltd., 1997.
[2] P. W. Sauer and M. A. Pai, Power System Dynamics and Stability: Prentice Hall, Inc, 1998.
[3] P. Kundur, Power System Stability and Control: McGraw-Hill, Inc, 1994.
[4] J. Arrillaga and C. P. Arnold, Computer Analysis of Power Systems: John Wiley & Sons, 1990.
[5] P. M. Anderson and A. A. Fouad, Power System Control and Stability: IEEE Press, 1994.
[6] P. C. Krause, O. Wasynczuk, and S. D. Schdhoff, Analysis of Electric Machinery: IEEE Press, 1995.
[7] R. P. Schulz, "Synchrounous Machine Modelling," The Symposium - Adequacy and Philosophy of Modelling System Dynamic Perfromnace, San Francisco, July 9-14 1972.
[8] I. J. Perez-Arriaga, G. C. Verghese, and F. C. Schweppe, "Selective Modal Analysis with Applications to Electric Power Systems, Part I and II," IEEE Transactions on Power Apparatus and Systems, vol. PAS-101, No. 9 September 1982.
[9] D. R. Ostojic, "Stabilization of Multimodal Electromecahnical Oscillations by Coordinated Application of Power System Stabilizers," IEEE Transactions on Power Systems, vol. 6, No. 4 November 1991.
[10] J. H. Chow, J. J. Sanchez-Gasca, H. Ren, and S. Wang, "Power System Damping Controller Design- Using Multiple Input Signals," in IEEE Control Systems Magazine, August 2000, pp. 82-90.
[11] J. V. Milanovic, "The Influence of Loads on Power System Electromechanical Oscillation," PhD thesis The University of Newcastle, NSW, Australia, 1996.
[12] N. Ozay and A. N. Guven, "Investigation of Subsynchronous Resonance Risk in the 380KV Turkish Electric Network," IEEE International Symposium on Circuits and Systems, Espoo, Finland, July 1988.
[13] M. Pavella and P. G. Murthy, Transient Stability of Power System Theory and Practice: John Wiley & Sons, 1994.
[14] B. porter and R. Crossley, Modal control: theory and application: Taylor & Francis, London, 1972.
[15] C. Taylor, "Advanced Angle Stability Controls," Cigre Technical Brochure Cigre TF38.02.17, No. 155, pp. 2-2 to 2-8, April 2000.
[16] M. A. Pai, Power System Stability - Analysis by the Direct Method of Lyapunov, vol. 3: North-Holland Publishing Company, 1981.
[17] M. A. Pai, Energy Function Analysis For Power System Stability: Kluwer Academic Publishers, 1989.
19
[18] G. Ledwich, J. Fernandez-Vargas, and X. Yu, "Switching Control of Multi-machine Power Systems," IEEE / KTH Stockholm Power Tech Conference, Stockholm, Sweden, pp. 138-142, June 1995.
20
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21
Chapter 2
Existing Control Methods
2.1. Introduction
The dynamics of a power system are non-linear when there are large
changes in the machine states during severe disturbances. However, it is
acceptable to linearize [1 (page 209-219), 2 (page 265-266)] these dynamics
of power systems around an equilibrium operating point (EOP) when
disturbances and states changes are small; this is known as small signal
analysis or small signal stability [3 (page 222-235)]. In the study of power
system stability, several linear and non-linear methods mentioned in [4] can
be used to assess the stability of a linearized or non-linear power system
and, to some extent, these methods can also be used to design its controllers.
Eigenvalue sensitivity, participation factors, Prony analysis [5, 6] (modified
for transfer function identification) and Linear Quadratic Regulator (LQR)
22
[7] are some of the examples of linear control design technique [4] whereas
adaptive control, cost function, discontinuous control, energy (Lyapunov)
function, normal forms, fuzzy logic and neural network are known as non-
linear control design techniques [4].
One of the limitations of linear control design techniques such as eigenvalue
analysis is the simplification of the set of non-linear power system equations
by linearizing it around an equilibrium operating point, for instance, the
linearization of the swing equations [8, 9 (page141-144)]. This process of
linearizing the swing equations simplifies the non-linear dynamics of the
power system. A concern over the robustness of linear control design
techniques raised by C. Taylor in [4] is the limited set of possible operating
conditions used in stability analysis. These can become crucial as power
system blackouts can be caused by unexpected cascading disturbances. A
power system model should be robust enough to handle these unforeseen
disturbances. Consequently, non-linear control techniques such as the
energy (Lyapunov) function method [10, 11] becomes attractive as its
accuracy is independent of the network structure and has a large region of
validity based on a non-linear system [4].
Another limitation of linear control design techniques is that they require
model reduction as detailed models involve extensive computation
mentioned in [4]. These inherent limitations of linear control techniques and
the need to consider all possible operating conditions, limit the TNSP
23
companies from using them exclusively in the assessment of the transient
stability of power system. According to [12], TNSP companies are still
relying on the time domain simulation method to assess transient stability as
part of dynamic security assessment (DSA). The main problem they
encounter is the extremely long simulation time required for a typical
transient stability study of a large scale power system. For example, a power
system can consist of 500 buses and 100 machines. In order to reduce the
number of these transient stability studies, power system planners limit
these studies to a few likely scenarios of fault occurrences. From the power
system operators’ perspective, it remains difficult to judge if a power system
is stable even if machine models are simplified, without first looking at a
time domain simulation of machine rotor angles and velocities. These
difficulties can, however, be eliminated if transient stability indicators are
computed by the Lyapunov’s direct method using energy function analysis.
Generally, linear analysis as a control design tool provides a very effective
control of large complex systems and it has desirable properties for small
variations around a single operating point. The robustness of these control
designs becomes uncertain when the operating state of a power system
changes significantly, for example, when severe disturbances cause large
changes in machine states. Hence, the performance of a controller designed
from linear analysis is not guaranteed at post-fault. In contrast, controllers
designed from non-linear analysis such as energy function are capable of
handling a power system's transient instability since energy function does
24
not use a linearized power system model. Using energy functions as an
analysis tool has the advantage of treating the source of the problem that
causes severe oscillations: the kinetic energy that is injected into
synchronous machines during the fault period.
From the above considerations, it is desirable to carry out a literature review
in the area of non-linear control techniques derived from energy (Lyapunov)
functions. As the control of power system is concerned with the prediction
of transient stability limits, the literature reviews emphasize on certain areas
such as Controlling Unstable Equilibrium Point (UEP), evaluation of
transient energy and methods of stability assessment. The control of power
apparatus such as DC links and Thyristor Controlled Series Compensators
(TCSCs) [13, 14] will also be covered in the literature review as this may
provide insights to the problems encountered in the control of DC links and
TCSC with respect to the damping of unstable modes of separation. Other
areas such as the construction of energy function with the transfer
conductance losses [12, 15, 16], Lyapunov direct method and Lyapunov
stability theorem [17] will not be covered in the literature review.
25
2.2. Energy Function and Unstable Equilibrium Point
(UEP)
A key concept in power system dynamics is the equilibrium point. At an
equilibrium point, the electrical power flow along the line from each
generators match the mechanical power driving the generators resulting in
zero acceleration in generators. A Stable Equilibrium Point (SEP) is an
operating condition where in response to small disturbances the operating
point will return to the SEP. An Unstable Equilibrium Point (UEP) [18] is
an operating condition where it is not guaranteed if the operating point
would return to the UEP when subject to small disturbances. At an UEP,
when the operating point is subject to small disturbances and moves away
from the UEP instead of returning to it, the power system is said to have
separated into two groups of machines. When a power system is subject to a
severe disturbance, it causes the operating point originally at the SEP to be
driven towards a UEP. The trajectory commonly moves beyond the UEP
and the system separates into two groups. The particular separation is
characteristic of the specific UEP. The UEP characterizing the separation is
described as the Controlling UEP associated with this disturbance. In a
multi-machine power system, there are multiple UEPs where a power
system separation can occur, and it is important to find the correct
Controlling UEP in a transient stability assessment because faults occurring
at different locations in a power system give different Controlling UEPs. In
other words, any UEP can become a Controlling UEP depending on the
26
location of severe faults and this makes the UEP a unique descriptor of
power system separation.
In a particular operating condition where some UEPs are lying close to each
other, it becomes difficult to determine if a particular UEP is a Controlling
UEP associated with a fault [19]. Several methods were developed to
determine a Controlling UEP amongst many UEPs. These methods are
categorized into two major groups: empirical approaches and theoretical
approaches. The empirical approaches are associated with the Closest UEP
[20, 21], Controlling UEP in the direction of fault-on trajectory [22], Mode
of Disturbance (MOD), Cutset Energy function and critical cluster
identification. The theoretical approaches are Closest UEP [19, 23],
Controlling UEP [19, 24], Potential Energy Boundary Surface (PEBS) [25]
and Boundary of stability based Controlling Unstable equilibrium point
(BCU) [24, 26]. In principal, these different approaches differentiate a
Controlling UEP from a UEP, and they will be reviewed in later sections.
The iterative search algorithms for determining Controlling UEP such as the
Newton-Raphson method [20, 21], gradient system based method [27],
Shadowing method [28], gradient method and ray method [29] and Dynamic
gradient method [29] that detects an exit point even if it is far from a PEBS
will not be covered. The reason for not covering these search algorithms
arises from the need to understand the different interpretations of the
27
Controlling UEP and not to analyse how these search algorithms reach a
Controlling UEP.
2.2.1. Closest UEP Method
In this approach, the method of finding a Closest UEP and declaring it a
Controlling UEP was first proposed by two groups of researchers that used
two different methodologies.
In [20, 21], an empirical approach was mentioned. The discussion of this
approach is based on a particular disturbance. The UEPs associated with the
post-fault network arising from this disturbance are determined based on a
single-machine-infinite-bus analogy giving UEP angles of sδπ − and
sδπ −− , where sδ is the machine operating angle at the SEP. These UEPs
are associated with the various types of possible machine separations in the
form of a group of generators separated from the rest of the power system.
A Newton-Raphson algorithm is used to obtain a better approximation of
the UEPs in order to evaluate the various potential energy at these UEPs.
The UEP with the lowest potential energy which is in the direction of the
fault trajectory is defined as a Closest UEP. This UEP is designated as the
Controlling UEP associated with the disturbance that yields this fault
trajectory. As this Closest UEP method is based on a UEP that is closest to a
fault-on trajectory, this concept is useful since it differentiates the UEP with
28
the lowest potential energy from a group of UEPs that are in the direction of
a fault trajectory while determining a Controlling UEP.
The theoretical approach in [19, 23] conditionally defines a Closest UEP as
a Controlling UEP. The Closest UEP found is responsible for system
instability if it is a type-one equilibrium point that belonged to a sub-set of a
stable manifold (i.e. a set of states that converged to the type-one UEP as
time approached positive infinity) around a post-fault equilibrium point. The
energy at the Closest UEP is used as the critical energy. It is noted in [19]
that if a Closest UEP is selected without checking to ensure that it is on a
stable manifold of a post-fault equilibrium point, then an incorrect Closest
UEP with a lower energy would have been found. This approach generally
applies to low order system such as a Single-Machine-Infinite-Bus power
system where it is not too demanding in the computational aspect to find
stable and unstable manifolds of a post-fault equilibrium point and check
that the Closest UEP is on the stability boundary.
2.2.2. UEP in the Direction of a Fault Trajectory
The empirical approach in [22] is based on a fault trajectory and the
Controlling UEP is determined in a different way. This method
approximates a fault trajectory with a non-linear expression in similar form
with a conventional potential energy equation in [17]. At the fault clearance
time, a linearly projected directional vector is constructed based on a post-
29
fault equilibrium point and a particular singular surface point. This singular
surface point is derived from a set of approximated potential energy
equations for the fault trajectory when a maximum power mismatch along a
fault trajectory has occurred. This is similar to a fault clearance time when
the system trajectory is at a maximum. As the directional vector is
minimized as a one dimensional minimization problem, an intersection
between the directional vector and the boundary of separation is found. The
intersection is used as an initial guessing point in an iterative algorithm that
searched an exact UEP. This approach is classified as a Controlling UEP in
the direction of a fault trajectory [18] and is useful in identifying the
Controlling UEP amongst a group of UEPs where some of them are outside
the region of attraction. It appears that in a multi-machine power system
where one or more UEP lies close to each other than the linear projection of
a directional vector may result in an incorrect Controlling UEP.
2.2.3. Controlling UEP Method
The Controlling UEP method in [19, 24] relies on the exit point of fault-on
trajectory that crosses a stability boundary. If an exit point belonges to a
stable manifold, defined as an invariant sets of a system where every
trajectory started from this set of states will remains in it for all time t [18,
page 1500], of an UEP then a Controlling UEP is found. In [17, page 166],
invariant set is defined for a power system problem as “the set of all
trajectories of the post-fault system … whose initial conditions … lies on
30
the faulted trajectory … for the fault i.” However, this method is considered
to be a nontrivial approach in [25] as it requires all type-one UEP’s on a
stability boundary to be identified prior to the determination of their
respective stable manifolds. This approach is limited in its application
because it requires the stable manifold of every post-fault equilibrium point
to be found.
2.2.4. PEBS based Controlling UEP Method
This Potential Energy Boundary Surface (PEBS) method in [30] is
theoretically treated in [25] to approximate a PEBS to a stability boundary
in a ) ,( ?δ domain such that a constant energy surface obtained from PEBS
can be used as a close approximation to the stability boundary (a stability
boundary is a union of stable manifolds where type-one UEPs can be found)
of a post-fault equilibrium point )0 ,( sδ . The key to using the PEBS method
is to detect a PEBS crossing in a gradient system (or simply the Potential
Energy surface). Since PEBS is also a union of the stable manifolds of the
type-one UEPs [25] in a gradient system, any stable manifold that containes
this PEBS crossing point will lead to a Controlling UEP. The constant
energy surface of a Controlling UEP in a gradient system is used as a local
approximation to a stability boundary in ) ,( ?δ domain. However, PEBS
will fail if a fault-on trajectory passes the stable manifold before crossing
31
the constant energy surfaces. The potential energy surface is useful as it can
confirm visually if UEP is on the stability boundary.
2.2.5. BCU Method
This BCU method is known as the Boundary of stability based Controlling
Unstable equilibrium point method (BCU) and is proposed in [24, 26]. It is
related to the PEBS method mentioned in [25] but the BCU method does not
involve tedious steps in finding a stable manifold that contained an exit
point (or PEBS crossing point) instead it uses its controlling UEP's constant
energy surface as an approximation to a stability boundary. While solving a
set of non-linear algebraic equations based on a gradient system, an initial
guess point close to an exit point at fault clearance is used to reach a
Controlling UEP. This method is cumbersome as it requires one to solve a
set of non-linear algebraic equations at fault clearance which could be time
consuming.
2.2.6. Mode of Disturbance Method
This Mode of Disturbance (MOD) method in [31-33] is an empirical
approach. It emphasizes that the Controlling UEP associated with a
candidate mode has the lowest normalized potential energy margin. In the
MOD test, a candidate mode is defined as the separation between two
32
groups of machines formed during transient stage; one consisted of the
critical machines and the other contained the remaining machines. A
normalized potential energy margin is evaluated by the ratio of the potential
energy margin ..EPV∆ and corrEKV |.. . ..EPV∆ is
...... EPcl
EPU
EP VVV −=∆
where U indicates the energy evaluated at the UEP and cl indicates the
energy evaluated a fault clearing.
The corrected transient kinetic energy responsible for separation corrEKV |.. is
2~
|.. 21
= eqeqcorrEK MV ω
where syscr
syscreq MM
MMM
+=
* and )(
~~~
syscreq ωωω −= .
The crM is the sum of all machines inertia in a critical machine group and
sysM is the sum of all machine inertia in a system machine group. The
notation ~
crω is the centre of area motion of a critical machine group and
~
sysω is a centre of area motion of a system machine group.
The screening of all candidate modes in [33] is based on the disturbances
effect on various machines during a MOD test. The method is simplified in
[34] by ranking kinetic energy and acceleration of all machines at fault
clearance. A short-listing of all machine indices with kinetic energy (KE)
and acceleration (ACC) higher than 50 percent of the respective maximum
value of KE and ACC resultes in a small set of candidate machines. Three
33
distinct machine indices groupings are then formed such that group A
containes the common machine indices found from a KE and ACC list,
group B containes all machine indices from KE list and group C containes
all machine indices from ACC list. Three further machine groups are formed
with each group selected from groups A, B and C. The process of evaluating
the normalized potential energy margins ( ..EPV∆ ) of these three machine
groups and associating a Controlling UEP to the critical machine group with
the lowest ..EPV∆ is being called the MOD testing method. This method can
exclude machines that may lead to different potential separations when the
short-listing of machines is based on 50 percents of KE and ACC.
2.2.7. Critical Cluster Method
The critical cluster method in [35, 36] is extended from the Extended Equal
Area Criterion transient stability studies. This method uses a machine initial
acceleration at fault clearance to select the likely candidates of critical
cluster. A group of machines that is accelerated above a pre-defined level at
fault clearing is selected and the combination of these machines forms
candidate critical clusters. Each candidate’s critical clearing angle is then
evaluated by solving a set of non-linear algebraic equations until the
stability margin η vanishes.
)()()( cacccdecc AA δδδη −=
34
where )(⋅η is the stability margin, )(⋅decA is a deceleration area and )(⋅accA
is an acceleration area.
A critical clearing angle in a fourth-order Taylor series expansion computes
a critical clearing time and corrective factors are used in the series to avoid a
deteriorating accuracy due to a truncation of higher order terms. An actual
critical cluster is then identified from a candidate list as the one that has the
lowest critical clearing time.
An improved critical cluster identification technique is proposed in [37, 38]
which uses both machine acceleration and pre-fault transfer admittance
between a machine and a fault location to evaluate a product term. This
acceleration and pre-fault transfer conductance product term selectes likely
candidates of critical cluster. The selection of critical cluster based on
acceleration can exclude potential machines that cause separation. The use
of pre-fault admittance as one of the selection criteria may have indicated
the vulnerability in the interconnection between two machines but its
influence can be small compares to machine accelerations.
2.2.8. Cutset Energy Function
The cutset energy function in [39-41] examines the vulnerability of
transmission lines between two groups of machines. The main step of this
35
method is to obtain all possible cutsets of two internal nodes corresponding
to machine terminal. Every cutset is identified by considering every possible
flow between two internal nodes with both positive and negative flows
between nodes considered. In this aspect, a cutset consists of transmission
lines indices (i.e. ith line or jth line), and each kth transmission line has two
vulnerability coefficients lkµ and u
kµ
∫ −=lk
ok
duu ok
lk
σ
σσµ )sin(sin
∫ −=uk
ok
duu ok
uk
σ
σσµ )sin(sin
where okσ (inter-nodal angle) is a stable equilibrium point of kth line,
ok
lk σπσ −−= and o
kuk σπσ −= .
From the vulnerability coefficient, vulnerability index of each ith line is
∑∑−+
+=+
ii C
lkk
C
ukki bb µµυ
∑∑−+
− +=iC
ukk
iC
lkki bb µµυ
where kb is a line admittance, +iC is a cutset with positive line flows
referenced to an ith node and −iC is a cutset with negative line flows
referenced to an ith node.
Both +iυ and −
iυ are interpreted as two modes of angle separation and the
lowest value of iυ (or +iυ ) becomes a vulnerability index of a cutset. As
36
shown in [41], a large cutset list can be reduced to a smaller list of candidate
cutsets that contains only the lines with faults. A Controlling UEP is related
to a critical cutset that is being selected from a group of fault related cutsets.
The critical cutset selected has the lowest vulnerability index among those
in the group. This cutset energy function evaluates the vulnerability of a
cutset based on the vulnerability coefficients and these approaches may
helps in the understanding of how a power system separates. The evaluation
of vulnerability indices iυ and +iυ of every possible cutset based on the
flow direction determined during steady state may not agree with the power
swing direction during transient. It appears that the lists of possible cutsets
between two internal nodes are large and it may become computational
demanding when In a multi-machine power system, it may be too
computationally demanding to determine a list of critical cutsets since the
possible list of cutsets between two internal nodes (i.e. generator) are long
for the relatively practical example in [12, page 68].
2.3. Methods of Stability Assessments
In our earlier sections, we have reviewed the methods of determining a
Controlling UEP. This section looks at the methods of transient stability
assessments using critical energy [19-22, 24, 25, 32, 33, 36-41] and
convergence analysis [42, 43].
37
For the approach that requires one to find a Controlling UEP such as in [19-
22, 24, 25], an energy function is used to evaluate the critical energy at the
Controlling UEP. As the total energy evaluated at post-fault exceeds this
critical energy, it implies that the power system will be unstable. In these
references [19-22, 24, 25], different types of energy function are considered.
For the cutset approach in [40, 41], cutset energy function is used to
evaluate the critical energy at the vulnerable cutset. At post-fault, if the total
energy exceeds this critical energy evaluated at the vulnerable cutset, the
power system will become unstable.
For the Mode of Disturbance (MOD) approach in [32, 33, 39], a critical
MOD is selected amongst all candidate MODs that is in the direction of the
fault-on trajectory and a corrected energy function is used to evaluate the
critical energy or the normalised potential energy margin associated with
this critical MOD. As a corrected kinetic energy evaluated at post-fault
exceeds the energy of the critical MOD, transient instability will occur.
In the extended equal area criterion approach in [36-38], the approach
determines the critical cluster as the one with the lowest critical clearing
time among all candidate clusters. The critical clearing times of all clusters
are being evaluated based on the equal area criterion of a two-Machine-
equivalent model. If the fault clearing time exceeds this critical clearing
time of the critical cluster, power system will become unstable.
38
Lyapunov stability concept is used in the convergence analysis such as in
[42, 43],. At post-fault, if the total kinetic energy in the power system
reduces monotonically then the power system will be stable. The Lyapunov
function is based on the energy function
∫
−∑−∑++∑
=dt?T-
haftplied by sEnergy supuctorsred in indEnergy storotationEnergy of
V
oCOA
where first term is referred to as the kinetic energy, the second and third
terms are referred to as the potential energy with the energy dissipated from
the transfer conductance neglected and the last term is the centre of area
motion. Centre of area motion refers to the perfect governors [15, 17] added
to all generators to ensure a balance between the shaft and delivered power.
This allows swing equations to focus on the differences between generators
angles. In [15], since the presence of TCOA ensured that a centre of area does
not accelerate, the centre of area motion term in the aforesaid energy
function equation is effectively zero.
If the rate of change of Lyapunov function V& is negative along the direction
of the fault trajectory, the power system will be asymptotically stable. V& is
in the form of
( )∑=
+=n
iiii fmV
1)(δδδ &&&&
39
where )(δf is a function of δ , m is machine inertia, n is the total number
of machines and both δ& and δ&& are the respective machine angle and
angular velocity derivatives.
2.4. Evaluation of Transient Energy
The transient energy that is responsible for a power system separation can
be evaluated at a Controlling UEP [19-22, 24, 25], a critical MOD [32, 33,
39], a critical cutset [40, 41] or a critical cluster [36-38] using different
energy functions. These different energy functions are generally capable of
predicting instability at a fault clearance and informing an operator
accordingly. However, in a large power system, experiences in the selection
of a critical set of separations amongst the many types of separations are
often required. This is because a critical system separation depends on the
location of faults.
2.5. Energy Function Based Switching Control
Energy function based switching controls in [42, 43] use Lyapunov stability
criteria as the basis to derive the control law. Considering a delta-connected
three-machine system such as in [42], the rate of change of energy function
V& is
40
[ ])sin(dd?)sin(dd?)sin(dd?U? )f(dV 232323a131313a121212aa&&&& ++−= ,
where ija? is the change in line reactance between an ith and a jth buses due
to a capacitor switching (by Uα) on a line.
The switching control Uα can be determined by keeping the rate of change
of energy V& most negative to yield an asymptotic behaviour in the system
trajectory. Hence the control law is designed to respond to the g terms
++=
•••
)sin(dd?)sin(dd?)sin(dd? 232323a131313a121212ag
A non-linear control law for the control of series capacitor becomes
<>
= 0 gfor 0
0 gfor 1αU
The control switching between 1 and 0 represents the respective switching
in and out of the series capacitor. The case where the control switches
between +1 and -1 was not considered in [42] as the series capacitor
compensator consisted of only capacitor rather than both capacitor and
inductor in the case of SVC control. This energy function based control is
used in the research as it has attractive properties with regards to the
reduction of disturbance energy.
41
2.6. Control of DC Link and TCSC
We have reviewed the energy related controls and use of critical energy at
UEP to predict system instability. In this section, the control of DC link and
Thyristor controlled series capacitor (TCSC) are covered.
In general, power apparatus such as braking resistors, DC links and
quadrature booster system are used to modulate real power flows in power
system. These power apparatus are capable of supplying and absorbing real
power flows. Braking resistor cannot supply real power and can only absorb
real power for short duration. A reactive power-modulating device is
generally referred to as a Flexible AC Transmission System (FACT) device
which provides the shunt compensation via a Static VAR Compensator
(SVC) or series compensation via a TCSC. As SVC are installed near loads
to provide the voltage modulation of load which is more efficient in the
support of voltage at the receiving end of a transmission line, this
modulation of reactive power flows in SVC also changes the real power
flows to the loads. Increasing the value of SVC will increase the voltage at
the SVC bus. This in turn will increase the voltage at the load buses in
proximity. This increase of voltage will result in the increase of power flow
to the load. Thus SVC is seen as an indirect modulator of real power in the
power system. Insertion of series capacitor can help reduce the effective line
impedance. One difficulty with the use of fixed capacitors is the potential
result of causing sub-synchronous resonance (SSR). When line resonance
42
matches the generator shaft resonance, there is potential for shaft damage
[44]. The control of series reactive element in TCSC avoids this issue by
active control. In many cases, it has been found that TCSC 13, 14, 44, 45] is
much more effective in modulating power flow than a SVC [46-49] of
equivalent rating. In this part of the literature review, the control of DC link
and TCSC are reviewed.
2.6.1. History of HVDC Link
The HVDC system has long been recognized as a preferred tool to transfer
bulk power over long distances as compared to a HVAC system. The
records in [45] shows the different justifications used in various countries
when a HVDC system is being implemented.
From a system stability point of view, both the shunt/ series compensations
and HVDC link systems offer good contributions in damping oscillations
[46, 47]. Several reasons for the use of HVDC link are the ease of future
expansion, the ease of transferring bulk power over long distances without
the need of intermediate reactive compensations and the ease of controlling
two systems with different frequency [45, 48, 49]. Apart from these reasons,
recent developments in semiconductor technology [50, 51] have further
encouraged the use of HVDC systems, for instance, the use of Insulated
Gate Bipolar Transistor (IGBT) based converters allows the connection of
HVDC device to weak AC system. This is due to the possibility of
43
modulating reactive power independently from real power. The records in
[45-49] have shown that HVDC system is being extensively used to control
large power transfer and provide fast control in the damping of oscillations.
2.6.2. Considerations in the Use of DC Link to Damp
Oscillations
When a HVDC link is used to damp oscillations in a large power system, its
damping action will reduce the disturbances at one end but will also produce
some disturbances at the other end [48] (page 138-140). This control action
could result in a frequency variation above a normal frequency.
When two AC power systems interconnected with a DC link are
significantly different in size, a system inertia based weighting [48] can be
used in a DC link control to share disturbances in a pre-determined manner
and damp system oscillations simultaneously. In the control of power
system during the transient period, some level of DC link capacity can be
provided [48] to overcome the overloading of a DC link during the first
swing whereas in the dynamic stability context, damping of oscillations can
be achieved without overloading the DC link.
44
2.6.3. Using Different Control Schemes to Supplement the
Control of Inverter
In response to disturbances in the AC system at the inverter side of a HVDC
system, large variations in the AC current and AC voltage are expected.
Different control schemes are used to overcome these issues as well as those
mentioned in [47, 52]. In [46], a DC link design is reported to emphasize the
damping of both the inter-area and local area modes.
In [47], a back-to-back HVDC link interconnecting two different power
systems was considered as shown in Figure 2.1. The control strategy is
based on several schemes such as the use of low voltage DC current limiter,
power stabilizer and an inverter current control. A DC current limiter at an
inverter station limits a large variation in DC current that follows after the
fault clearance. As the disturbances in weak AC system on the inverter side
can cause a large DC current variation and a severe voltage fluctuation, a
power stabilizer is installed at the inverter station to damp power
oscillations separately. The power stabilizer is designed using a linear
analysis to enhance the low damping factor at an existing bus that
experiences an oscillation around 1Hz. An integrated control action
consisted of a low voltage DC current limiter and a power stabilizer are used
to damp the inverter AC voltage and this can result in a limiting effect on
the DC current. This has a rapid damping effect on the bus 5’s (Figure 2.1)
power fluctuations. In order to reduce the voltage fluctuation at the inverter
45
side during the fault recovery period to boost control performance during
the transient period, the inverter is designed to operate in a current control
mode instead of the gamma control mode, used during a steady state
operation. The use of the above control schemes in [47] prevents a weak AC
system at the inverter side from separating when additional power is being
injected into it by the new DC link. This is a typical operating situation [47]
whereby a DC link can deteriorate an existing weak AC system at the
inverter station. Figure 2.1 shows a simplified back-to-back DC link
configuration that interconnects two AC power systems. This coordination
between power stabilizer and control of inverter in current control mode has
shown that it is possible to design each controller separately for different
purposes.
Figure 2.1: A back-to-back DC link interconnecting two AC power systems
In [52], it was shown that Root Locus analysis could be used to determine
the best feedback signal for inverter control which used a locally available
feedback signal. Locally available DC and AC variables such as direct
current, direct voltage, AC voltage (magnitude and angle) and AC current
Back to back DC system Receiving end Sending end
4 5 6
Power flow Power flow
1 2 3
46
(magnitude and angle) are selected. The analysis based on the movement of
eigenvalues using these variables as feedback signals and their effect on
SCR (short circuit ratio) reduction suggests that an AC current angle is the
preferred feedback signal. It is also important that this AC current angle
signal results in a pair of stable zeros whereas other signals gives rise to
unstable zeros. The stable zeros allow a wider range of control gain to be
used. This choice of feedback signal is in contrast to the conventional
HVDC system design that uses direct current as the feedback signal at
rectifier and the gamma angle as the feedback signal at inverter. The process
identifies the relations between SCR and eigenvalues pair such that low
system strength with low SCR such as 1.1 can unstablize a pair of critically
stable eigenvalues. The use of Eigenvalue analysis to select control input is
important in control design to ensure the feedback control yields both stable
zeros and stable close-loops as the control gain increases.
In [46], it was shown that the damping of the inter-area and local area
modes could be effective when an active power modulation scheme was
used in the rectifier and a voltage modulation scheme was used in an
inverter. The test system consists of a DC link installed in parallel with an
AC tie line. The intention is to investigate the effect of dual modulation on
machine interactions responses at the sending end of the DC link. The first
problem encountered in this paper is the increase in DC link power transfer
that resulted in the reduction of the power transfer across the parallel AC tie
line [46]. This is due to the reactive loading at the HVDC terminal stations
47
that causes a reduction in AC voltage and affects the loadings of AC lines
and magnitude of local loads. This reduction in the AC line loadings defeats
the purposes of the HVDC system. The second problem in this paper is
concerned with the control mode interactions that occurred when the
controllers are not being coordinated properly. It is found that the control
mode interactions limits the gain of the controller such that any further
increases in the gain will lead to the a mode of oscillation becoming
destabilized. The effect of the dual modulation using power modulation at
the rectifier station and voltage modulation at an inverter station reduces the
control mode interactions and improves the AC power transfer limit by
reducing the reactive power constraints at the inverter side. This dual
modulation control is designed from the linear-quadratic-Gaussian control
theory. The different modulation controls used in rectifier and inverter is
effective in reducing the effect of control interaction.
2.6.4. Discontinuous Control of Thyristor Controlled Series
Compensation (TCSC)
Time optimal control of TCSC is used to stabilize a two-area system
interconnected with three parallel AC tie lines such as in [53]. The time
optimal control requires Xcmax, Xc
min and Xcs to be selected where Xc
max,
Xcmin and Xc
s are the maximum, the minimum and the steady state series
compensating capacitance respectively. A bang-bang switching control that
48
switches between the Xcmax and Xc
min is based on a control strategy of
single-switch approach. The Xcmin is selected to include a post-fault
equilibrium point under its control influence while the Xcmax is selected to
enlarge a stability domain. The single-switch approach at post-fault forced a
system trajectory to approach a time-optimal switching line in an enlarged
stability domain which is created from the influence of Xcmax. As the
trajectory intersects the switching line, Xcmin is switched in and drives the
system trajectory to a post-fault stable equilibrium point (SEP). When the
system reaches the post-fault SEP, a steady state Xcs replaces the Xc
min. This
single-switch approach requires a fairly precise design of switching line that
brings the trajectory towards the post-fault SEP and is dependent on the
nature of a particular fault it was designed to response to.
In [42], a discontinuous switching control based on the function is derived
for a hypothetical case of a delta-connected three-machine system where a
TCSC is applied between its line 1 and 2. The control law is designed to
switch between the limits of 1 and 0
<>
=0dsind0 for 0dsind1 for
U1212
1212a )(
)(&&
where 12d& is the angular velocity difference of machine 1 and 2. It is shown
that the oscillation associated with machine 1 and 2 is well damped but the
system remains oscillatory owing to the presence of a remaining oscillatory
mode associated with the oscillation between machine 2 and 3. It is found
that this remaining mode of oscillation is not significantly influenced by the
49
modulation of line 1 and 2. When two TCSCs are used, each installed
between the line 1 and 2 and the line 1 and 3, both the modes are well
damped and the energy decayes to zero. This has shown that n modes
requires n controllers in order to achieve overall system stability.
2.7. Wide Area Control
Wide area control is a control strategy where controllers are designed to act
on remote data obtained from wide area measurement such as those used in
the regional protection and control scheme in [54] and the transmission
highway strategy in [55]. Other examples of controls that are designed
based on wide area information are [56-59]. In general, wide area control
consists of two main categories: event-based and response-based [54, 60].
An event-based wide area control initiates control action upon detection of
power apparatus outage while a response-based wide area control only
initiates remedy action when specific system responses reaches some pre-
defined threshold condition.
This made the response-based control relatively slower than the event-based
control, however, the advantage of the response-based control is its ability
to response to some of the events that are not easily detected or not
sufficiently well defined.
50
A wide area response-based control is different from a conventional
centralized control. In wide area control, Global Positioning System (GPS)
gives a reliable measurement of data based on common time stamps
whereas the conventional supervisory centre performs data analysis,
ignoring the time skew factor, on all collected data measurement prior to
sending them out to relevant control centres.
2.7.1. Centralized and Wide Area Control
In [61], the results of applying power modulation to a HVDC system using
linear decentralized and centralized approaches were compared. It is shown
that a centralized modulation based on remote data allows better
performance of multi-terminal HVDC system in damping oscillations. The
decentralized control system pre-assigns a terminal for voltage-control while
a centralized control uses its supervisory program to determine and assign a
new voltage-controlling terminal for an effective modulation during the
transient period. When a disturbance such as load rejection is applied, a
decentralized modulation scheme that pre-assigns a terminal for control will
reach its limit at a first swing [61]. This is because the HVDC terminal pre-
assigned for voltage control has to be overloaded prior to any further
improvement in system damping. On the other hand, a centralized
modulation scheme shows the possibility of increasing system damping
when its supervisory program performs a weighted distribution of
modulating signals to all terminals [61] and allows all terminals to
51
participate in modulation to share disturbances. This avoids overloading of a
terminal. This form of wide area control using centralized concept appears
to be useful when many controllers are involved and coordination of these
controllers becomes critical.
In [62], a decentralized controller that used remote data was presented. This
paper uses a Global positioning system (GPS) to obtain synchronized
measurements of remote generators' data to design its controllers for the
control of a three-machine nine-bus system. When the test system is
subjected to a 100ms earth fault, significant damping is observed. It is
indicated that a decentralized control can be made to perform well using
remote data measurement via GPS. This form of wide area control based on
decentralized control concept uses remote measurements can be
comparatively fast in response time as compared to the wide area control
designed based on the concept of centralized control where all decision are
weighted centrally.
2.8. Discussion
It has been shown that energy (Lyapunov) function method is capable of
determining whether a system converges or diverges by inspecting the rate
of change of energy V& based on the Lyapunov stability theorem. The
controlled power system will show monotonic convergence provided the
52
conditions such as a positive definite energy function V , a semi-negative
definite rate of change of energy function V& , a fast governor control and the
post-fault trajectory that lies inside the region of attraction are being
satisfied. The possibility of deriving a switching control law from a V&
equation as in [42] is promising in the control of power system especially
when such a control law is insensitive to system structure and system
operating conditions.
The evaluation of critical energy and critical separation group using various
methods such as in [19-22, 24, 25, 32, 33, 36-41] are useful as this literature
review make us understand the different ways of determining a Controlling
UEP. It is also clear that finding a correct Controlling UEP that is in the
direction of a fault trajectory (or closest to a fault trajectory) is crucial in
determining the correct critical energy. In control design, if the lowest
energy is not the critical energy, this may give rise to conservative results in
assessing stability.
The ability of a controller to recognize a Controlling UEP and the critical
energy that is responsible for a power system separation may give an
efficient damping performance in a control design since control efforts can
be directed to the correct mode of separation.
In the control of a DC link, stabilization of AC voltage at an inverter station
is critical. The use of a gamma control during steady state and the use of a
53
current control during the transient period reduce inverter voltage
fluctuation. Control mode interaction that limits the gain of controllers is
one of the concerns in control design, for instance, the control of DC link in
the literature review. The coordination between the power modulation at the
rectifier and the voltage modulation at the inverter appears to reduce this
control mode interaction and allows a larger range of gain to use in control
design. It is also noted in the literature review concerning the DC linked
power system that a large power system consists of strong and weak areas
and a control that can recognize this problem and direct appropriate efforts
to these areas correctly may extend the limit of power transfer in the
transmission network.
The improvements in communication technology have made remote
measurements feasible. It has also made the coordination of controllers
much easier in terms of using remote data that is on a common time
reference. The wide area control in [61] shows that conventional
decentralized control performs less satisfactorily than the centralized
control. It is also shown in [62] that a decentralized control of Power
System Stabilizers (PSSs) that uses remote measurements (or wide area
information) can result in satisfactory damping performance. From a wide
area control perspective, it is possible that a control strategy that uses
remote measurements may achieve multi-mode damping results.
54
2.9. References
[1] P. M. Anderson and A. A. Fouad, Power System Control and Stability:
IEEE Press, 1994. [2] A. R. Bergen, Power Systems Analysis: Prentice Hall, 1986. [3] P. W. Sauer and M. A. Pai, Power System Dynamics and Stability:
Prentice Hall, Inc, 1998. [4] C. Taylor, "Advanced Angle Stability Controls," Cigre Technical
Brochure Cigre TF38.02.17, No. 155, pp. 2-2 to 2-8, April 2000. [5] P. S. Dolan, J. R. Smith, and W. A. Mittelstadt, "Prony Analysis and
Modelling of a TCSC Under Modulation Control," 4th IEEE conference on Control Applications, Albany, NY, USA, pp. 239-245, September 1995.
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[7] B. D. O. Anderson and J. B. Moore, Optimal Control, Linera Quadratic Methods 1991.
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Dynamics and Stability: John Wiley & Sons Ltd., 1997. [10] P. D. Aylett, "The Energy-Integral Criterion of Transient Stability
Limits of Power Systems," Proceedings IEE, vol. 105(C), pp. 527-536, July 1958.
[11] M. C. Magnusson, "The Transient -Energy Method of Calculating Stability," American Institute of Electrical Engineers Transactions, vol. 66, pp. 747-755, 1947.
[12] M. A. Pai, Energy Function Analysis For Power System Stability: Kluwer Academic Publishers, 1989.
[13] C. Gama, "Brazilian North-South Interconnection control-application and operating experience with a TCSC," IEEE Power Engineering Society Summer Meeting, Edmonton, Alta. , Canada,vol. 2, pp. 1103 -1108, July 1999.
[14] G. N. Taranto, J.-K. Shiau, J. H. Chow, and H. A. Othman, "A robust decentralized control design for damping controllers in FACTS applications," Control Applications, Albany, NY , USA, pp. 233 -238, Sepetember 1995.
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[16] A. A. Fouad and S. E. Stanton, "Transient Stability of A Multi-Machine Power System. Part I: Investigation of System Trajectories," IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, No. 7, pp. 3417-3424, July 1981.
55
[17] M. A. Pai, Power System Stability - Analysis by the Direct Method of Lyapunov, vol. 3: North-Holland Publishing Company, 1981.
[18] H. D. Chiang, C.-C. Chu, and G. Cauley, "Direct Stability Analysis of Electric Power Systems Using Energy Functions: Theory, Applications, and Perspective," Proceedings of the IEEE, vol. 83, No. 11, pp. 1497-1528, November 1995.
[19] H. D. Chiang, F. F. Wu, and P. P. Varaiya, "Foundations Of Direct Methods For Power System Transient Stability Analysis," IEEE Transactions on Circuits and Systems, vol. CAS-34, No.2, pp. 160-173, February 1987.
[20] F. S. Prabhakara and A. H. El-Abiad, "A simplified determination of transient stability regions for Lyapunov methods," IEEE Transactions on Power Apparatus and Systems, vol. PAS-94, No. 2, pp. 672-680, March/April 1975.
[21] C. L. Gupta and A. H. El-Abiad, "Determination of the closest unstable equilibrium state for Liapunov methods in transient stability studies," IEEE Transactions on Power Apparatus and Systems, vol. PAS-95, No. 5, pp. pp. 1699-1712, September/ October 1976.
[22] T. Athay, R. Podmore, and S. Virmani, "A practical method for the direct analysis of transient stability," IEEE Transactions on Power Apparatus and Systems, vol. PAS-98, No. 2, pp. 573-584, March/April 1979.
[23] H. D. Chiang and J. S. Thorp, "The Closest Unstable Equilibrium Point Method for Power System Dynamic Security Assessment," IEEE Transactions on Circuits and Systems, vol. 36, No. 9, pp. 1187-1199, September 1989.
[24] H. D. Chiang, F. F. Wu, and P. P. Varaiya, "A BCU Method for Direct Analysis of Power System Transient Stability," IEEE Transactions on Power Systems, vol. 9, No. 3, pp. 1194-1208, August 1994.
[25] H. D. Chiang, F. F. Wu, and P. P.Varaiya, "Foundations Of The Potential Eneegy Boundary Surface Method For Power System Transient Stability Analysis," IEEE Transactions on Circuits and Systems, vol. 35, No. 6, pp. 712-728, June 1988.
[26] H. D. Chiang and C.-c. Chu, "Theoritical Foundation Of The BCU Method For Direct Stability ANalysis Of Network-Reduction Powerr System Models With Small Transfer Conductances.," IEEE Transactions on Circuits and Systems - I: Fundamental Theory And Application, vol. 42, No.5, pp. 252-265, May 1995.
[27] C.-W. Liu and J. S. Thorp, "A novel method to compute the closest unstable equilibrium point for the transient stability region estimate in power systems," IEEE Transactions on circuits and Systems-I: Fundamental Theory And Applications, vol. 44, No. 7, pp. 630-635, July 1997.
[28] R. T. Treinen, V. Vittal, and W. Kliemann, "An Improved Technique to Determine the Controlling Unstable Equilibrium Point in a Power System," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 43, No. 4, pp. 313-323, 1996.
56
[29] J. T. Scruggs and L. Mili, "Dynamic Gradient Method for PEBS Detection in Power System Transient Stability Assessment," Internation Journal of Electrical Power and Energy Systems, vol. 23, pp. 155-165, 2001.
[30] N. Kakimoto, Y. Ohsawa, and M. Hayashi, "Transient Stability Analysis of Electric Power System via Lure-Type Lyapunov Function, Part I and Part II," Transactions of IEE of Japan, vol. 98, pp. 516, 1978.
[31] A. A. Fouad and S. E. Stanton, "Transient Stability of A Multi-Machine Power System. Part II: Critical Transient Energy," IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, No. 7, pp. 3417-3424, July 1981.
[32] A. A. Fouad and V. Vittal, "The Transient Energy Function Method," International Journal of Electrical Power and Energy Systems, vol. 10, No. 4, pp. 233-246, October 1988.
[33] A. A. Fouad, V. Vittal, and T. K. Oh, "Critical Energy For Direct Transient Stability Assessment of A Multimachine Power System," IEEE Transactions on Power Apparatus and Systems, vol. PAS-103, No. 8, pp. 2199-2206, August 1984.
[34] A. A. Fouad and V. Vittal, Power System Transient Stability Analysis - Using the Transient Energy Function Method: Prentice-Hall, Inc., 1992.
[35] Y. Xue, T. V. Vutsem, and M. Ribbens-Pavella, "Extended Equal Area Criterion Justifications, Generalizations, Applications," IEEE Transactions on Power Systems, vol. 4, No. 1, pp. 44-52, February 1989.
[36] Y. Xue, T. V. Cutsem, and M. Ribbens-Pavella, "A Simplified Direct method For Fast Transient Stability Assessment of Large Power System," IEEE Transactions on Power Systems, vol. 3, No. 2, pp. 400-412, May 1988.
[37] Y. Xue, et al., "Extended Equal Area Criterion Revisted," IEEE Transactions on Power Systems, vol. 7, No. 3, pp. 1012-1022, Augist 1992.
[38] Y. Xue and M. Pavella, "Critical-cluster identification in Transient Stability Studies," IEE Proceedings-C, vol. 140, No. 6, pp. 481-489, November 1993.
[39] G. Cai, G. Mu, Z. Liu, and Z. Lin, "The Trajectory Based Network Transient Energy Evaluation and Its Application to Enhancement of Transient Stability," POWERCON '98,vol. 2, pp. 1369-1373 1998.
[40] A. R. Bergen and D. J. Hill, "A structure Preserving Model For Power System Stability Analysis," IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, No. 1, pp. 25-35, January 1981.
[41] K. S. Chandrashekhar and D. J. Hill, "Cutset Stability Criterion For Power Systems Using A Structure-Preserving Model," Internation Journal of Electrical Power and Energy Systems, vol. 8, No. 3, pp. 146-157, July 1986.
57
[42] G. Ledwich, J. Fernandez-Vargas, and X. Yu, "Switching Control of Multi-machine Power Systems," IEEE / KTH Stockholm Power Tech Conference, Stockholm, Sweden, pp. 138-142, June 1995.
[43] E. Palmer and G. Ledwich, "Switching control for power systems with line lossess," IEE Proceedings- Generation, Transmission, Distributions, vol. 146, No.5, pp. 435-440, September 1999.
[44] P. M. Anderson, B. L. Agrawal, and J. E. Vanness, Subsynchronous resonance in Power System: IEEE Press, 1989.
[45] C. Adamson, et al., High Voltage Direct Current Converters and Systems: Macdonald & Co. Ltd., 1965.
[46] C. E. Grund, E. M. Pollard, H. Patel, S. L. Nilsson, and J. Reeve, "Power Modulation Controls for HVDC Systems," Cigre, vol. 14-03, pp. 1-8, 1982.
[47] A. E. Hammad, J. Gagnon, and D. McCallum, "Improving The Dynamic Performance Of A Complex AC/DC System By HVDC Modifications," IEEE Transactions on Power Delivery, vol. 5, No. 4, pp. 1934-1943, 1990.
[48] J. Arrillaga, High Voltage Direct Current Transmission-2nd Edition: IEE, 1998.
[49] E. W. Kimbark, Direct Current Transmission, Vol. I: John Wiley & Sons, Inc, 1971.
[50] G. Asplund, et al., "DC transmission based on voltage source converter," Cigre Conference, Paris, France, pp. 10 1998.
[51] M. Chamia and B. Normark, "An Alternative Means Of Transporting Energy In A Deregulated World," in ABB review, vol. ABB review supplement, 4 1999, pp. 1-13.
[52] D. Jovcic, N. Pahalawaththa, and M. Zavahir, "Inverter Controller For HVDC System Connected To Weak AC System," IEE Proceedings - Generation, Transmission, Distribution, vol. 146, No. 3, pp. 235-240, 1999.
[53] D. N. Kosterev and W. J. Kolodziej, "Bang-Bang Series Capacitor Transient Stability Control," IEEE Transactions on Power Systems, vol. 10, No. 2, pp. 915-924, May 1995.
[54] D. G. Hart, M. Subramanian, D. Novosel, and M. Ingram, "Real-time wide area measurement for adaptive protection and control," NSF/DOE/EPRI Sponsored Workshop on Future Research Directions for Complex Interactive Electric Networks, Washington D.C., pp. 1-7, 16-17 November 2000.
[55] J. F. Christensen, "New control strategies for utilizing power system network more effectively," IFAC/Cigre Symposium, Beijing, China,vol. ELECTRA No. 173, pp. 5-16, August 1997.
[56] H. Ni, G. T. Heydt, and L. Mili, "Power system stability agents using robust wide area control," IEEE Transactions on Power Systems, vol. 17, No. 4, pp. 1123 -1131, November 2002.
[57] J. F. Hauer and C. W. Taylor, "Information, reliability, and control in the new power system," American Control Conference, Philadelphia, PA , USA,vol. 5, pp. 2986-2991, June 1998.
58
[58] I. Kamwa and R. Grondin, "PMU configuration for system dynamic performance measurement in large, multiarea power systems," IEEE Transactions on Power Systems, vol. 17, No. 2, pp. 385 -394, May 2002.
[59] A. A. Grobovoy, "Russian Far East interconnected power system emergency stability control," IEEE Power Engineering Society Summer Meeting, Vancouver, BC, Canada,vol. 2, pp. 824-829 2001.
[60] C. W. Taylor, "Response-based, Feedforward Wide area control," NSF/DOE/EPRI Sponsored Workshop on Future Research Directions for Complex Interactive Electric Networks, Washington D.C., pp. 1-6, 16-17 November 2000.
[61] D. P. Carroll and C. M. Ong, "Coordinated Power Modulation in Multiterminal HVDC Systems," IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, No. 3, pp. 1351-1361, March 1981.
[62] E. Haryadi, A. S. Sabzevary, and S. Iwamoto, "Robust Stabilizer with GPS for Multimachine oscillation ,," 14th Triennial World Congress,1999, IFAC, Beijing, P.R. China, pp. 237-242 1999.
59
Chapter 3
Kinetic Energy Reduction for Power
System Stability Design
The key idea in this chapter is to extend the energy function control in [1] to
a complex power system such as a DC link interconnecting two power
systems. This chapter also examines the problem of control chattering while
using a bang-bang control near a stable equilibrium operating condition. The
undesirable result due to this control chattering is described and an effective
switching solution is proposed. A four-machine two-area DC link power
system is used to demonstrate the control design procedure.
A simple saturation function applied to the four-machine two-area power
system shows that a single HVDC controller can be effective in damping
multi-mode oscillations without compromising large signal performance. A
systematic procedure that derives this control law using the nonlinear
structure of energy function based on Lyapunov’s stability criteria is
60
outlined. It is shown, using this simplified example of the four-machine
two-area interconnected power system, that all oscillations (or modes of
separations) are well damped. In a power system, a good damping
contribution by a particular device to a mode of separation is only possible
if the device controller has strong influence over it.
3.1. Introduction
A power system has non-linear characteristics and will response in
unexpected ways when it is subject to different disturbances. In particular,
different fault locations in a power system may result in different types of
machine angle separations in the power system. In the normal operation of
power system, linearly tuned controllers are effective in damping small
disturbances such as small load changes. However, in contingency planning,
where severe disturbances such as line faults are considered, linear analysis
are generally limited to only few contingencies as discussed in [2]. In a
typical contingency planning exercise, several operating points need to be
considered for each contingency. This cumbersome approach is exacerbated
when most of the contingencies are examined using the time domain
analysis, which is time consuming. A generally robust and non-linear
analysis tool such as the Lyapunov energy function [3, 4] has been used to
supplement contingency analysis by computing stability indices. These
stability indices relieve TNSP companies from the need to determine the
61
transient stability of a power system from a post-fault time domain
simulation.
This chapter investigates the possibility of using energy functions to derive
a non-linear control law that yields good damping performance. The energy
function control in [1, 5] damps disturbances in a pure AC power system
using series capacitor compensation (SCS). In this chapter, this energy
function based control is extended to the control of a HVDC link in a mixed
AC/DC network. Through the formulation of a HVDC link problem, we
seek a control approach that gives a good reduction of total system energy.
This chapter is organized into three parts. Firstly, a switching control law
based on energy function is derived for a power system consisting of two
AC segments jointed by a DC link. Secondly, an initial condition on a g(x)
surface or switching hyperplane S [5] is used to demonstrate that under the
influence of a signum function (or a bang-bang control), a stable limit cycle
can be formed. The occurrence of the stable limit cycle in this power system
control application can be characterized as a high gain speed feedback
control problem. An example of a dual LC (i.e. inductor-capacitor) circuit is
used to illustrate how switching control can result in the reduction of the
number oscillatory modes. Thirdly, the use of a simple saturation function to
avoid a stable limit cycle and achieve a multi-mode damping via a single
HVDC controller is explained.
62
3.2. Energy Function Based Switching Control
A simplified structure of a four-machine two-area HVDC linked power
system is shown in Figure 3.1 where the DC link between the two areas is
capable of real power modulation. The sign of the shunt resistor G2 indicates
real power withdrawal or injection between buses 2 and 3. The converter
dynamics and voltage variation at buses 2 and 3 are to be ignored in this
case.
DC Link
2115MW
Small system
Generator 4 Generator 3
3 4 X34=j0.025 1485MW
-G2 m1=1.56 m2=0.922
Generator 2 4454MW
Generator 1
Large system
1 2 X12=j0.008 5049MW
G2 m3=0.315 m4=0.126
Figure 3.1: Single line diagram of a four-machine two-area power system.
The sets of swing equations for the power system in Figure 3.1 is
−−−−−=
−−−−=
121212112
212112222
2222
11121221
121221112
1111
sin
cos
sin
cos
dccoa
m
coa
m
PmPbvv
gvvgvPm
mPbvv
gvvgvPm
δδδ
δδδ
&&
&&
63
−−−−=
+−−−−=
42434334
434334442
4444
232343443
343443332
3333
sincos
sincos
mPbvvgvvgvPm
PmPbvvgvvgvPm
coa
m
dccoa
m
δδδ
δδδ
&&
&&
(3.1)
o
o
o
o
ωωδ
ωωδ
ωωδ
ωωδ
−=
−=
−=
−=
44
33
22
11
&
&
&
&
where
• Pcoa1 and Pcoa2 are the perfect governor terms of the large and small
power systems interconnected via a DC link,
• all δδ &&& , and ω are the respective machine angular acceleration, machine
angle and machine angular velocity in an inertial frame,
• Pmi is the ith machine mechanical power,
• vi is the ith bus voltage,
• mi is the ith machine inertial constant,
• δij is the angle difference between the ith and jth buses,
• g is the shunt conductance element of reduced Y admittance matrix [6],
• b is the transfer admittance element of reduced Y admittance matrix
and
• Pdc is the simple representation of DC power transfer across a DC link.
The perfect governor term (Pcoa) is expressed as
43
243432
21
121211
mmPPPPP
P
mmPPPPP
P
dceemmcoa
dceemmcoa
++−−+
=
+−−−+
=
(3.2)
where Pe is the electrical power of a machine and the remaining parameters
have been defined earlier in association with (3.1).
64
The purposes of including perfect governor terms [5, 7, 8] for each
generators is to ensure that the sets of swing equations (3.1) focus on angle
differences and neglect the governor dynamic in a power system during the
analysis. Under the assumption of perfect governors’ action, all angles are
identical to measurements as referenced to a centre-of-area (COA) frame.
The Pdc1 and Pdc2 terms in (3.2) are the steady state real powers drawn or
injected at bus 2 and 3 respectively through the shunt resistor G2. Pdc1 and
Pdc2 are calculated from
22GvP idc = (3.3)
where vi is the corresponding bus voltage.
The energy function V [3] of the power system in Figure 3.1 can be written
in the form
0
)()(
cos
cos
)cos(cos
)cos(cos
))((21
332221
4
1
4
3
2
1
2
1
4
1
4
3
2
1
2
1
4
1
24
1
2
=
−−−+
∑ ∫∑+
∑ ∫∑+
∑ −∑−
∑ −∑−
∑ −−−∑
=
=
+=
−
=
=
+=
−
=
=
+=
=
=
=
+=
=
=
=
=
=
=
sdc
sdc
n
ijijijji
n
i
n
ijijijji
n
i
n
ij
sijijijji
n
i
n
ij
sijijijji
n
i
n
i
siiiiimi
n
iii
PP
gvv
gvv
bvv
bvv
gvPm
V
δδδδ
δ
δ
δδ
δδ
δδω
(3.4)
65
The derivative of (3.4) is expressed as
( )∑ ++−==
n
iiiimiiii fgvPmV
1
2 )(δδδ &&&& (3.5)
where f(δ) is a function of δ.
Assuming that all bus voltages vi and vj are 1.0 p.u. and the switching in and
out of G2 has resulted in a real power modulation between the two areas, an
admittance matrix Yactual [1] for the switching of G2 in the network is
outin
outactual
YYYYUYY
−=∆∆+= *
(3.6)
where the terms Yin and Yout represent the network admittance matrix when
G2 is switched in and out respectively. U is a switching input and Y∆ is the
change in the network admittance matrix
−=
−−
−−
−
−−+−
+−−
=∆
00000000000000
0000
0000
0000
0000
2
2
4443
3433
2221
1211
4443
34332
22221
1211
GG
YYYY
YYYY
YYYYG
YGYYY
Y
Substituting (3.6) into (3.1), the first four equations in (3.1) can be rewritten
as (3.7) based on the assumption that all bus voltages vi and vj are 1.0 p.u.
66
( ) ( )( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
−∆+−∆+−∆+−
=
+−∆+−∆+−∆+−
=
−−∆+−∆+−∆+−
=
−∆+−
∆+−∆+−=
2434343
4343434444444
22343434
3434343333333
11212121
2121212222222
1121212
1212121111111
sin*
cos**
sin*
cos**
sin*
cos**
sin*
cos**
coa
m
dccoa
m
dccoa
m
coa
m
Pbub
guggugPm
PPbub
guggugPm
PPbub
guggugPm
Pbub
guggugPm
δδ
δ
δδ
δ
δδ
δ
δ
δδ
&&
&&
&&
&&
(3.7)
Substituting (3.7) into the rate of change of energy function V& (3.5) and
considering only the controllable terms in the equation, the simplified V&
reduces to
( ) ( )( ) ( )
∆+∆+∆+∆+
∆+∆+∆+∆+
∆+∆+∆+∆
−=
4343434343
2121212121
444333222111
bgbg
bgbg
gggg
uV
δδ
δδ
δδδδ
&&
&&
&&&&
& (3.8)
Since all the elements in Y∆ in (3.6) are zero except for the 22g∆ and 33g∆
elements, the rate of change of energy function V& is simplified further into
−−= 322
~~δδ &&& uGV (3.9)
where the speed iδ~& is measured in the COA frame. The conversion from
inertial frame to the COA frame is evaluated from the expression
COAii δδδ &&& −=~
where iδ& is in inertia frame and ∑
∑=
=
=x
jj
x
iii
COAm
m
1
1δ
δ
&& .
67
Similarly, the conversion of machine angle iδ in inertia frame to the COA
frame is via the expression
COAii δδδ −=~
where ∑
∑=
=
=x
jj
x
iii
COAm
m
1
1δ
δ .
The control law in (3.10) is selected based on (3.9) to keep the derivative of
energy V& most negative. This aims to satisfy the necessary condition of
Lyapunov stability [5] for asymptotic stability such that when a Lyapunov
function V is positive definite and a Lyapunov function derivative V& is
negative semi-definite, a system trajectory is forced towards a stable
equilibrium operating condition asymptotically. A control law that makes
maximum contribution to a non-divergent system trajectory is
=<−
>==
0 0 for S 0 1 for S
0 1 for S Usgn(S) andU (3.10)
where S is the switching hyperplane in (3.11) or a g(x) surface [5]. It should
be noted that the speed measurement 2~δ& and 3
~δ& in (3.9) refer to the large
and small power systems centre of area (COA) respectively.
0~~
32 =−= δδ &&S (3.11)
68
The COAs for the small (COAS) and large (COAL) area are evaluated using
the expression
∑
∑
=
== 2
1
2
1
jj
iii
COA
m
m
S
δδ and
∑
∑
=
== 4
3
4
3
jj
iii
COA
m
m
L
δδ
For simplicity, the “~” notation that represents the machine angle and speed
in COA frame (i.e. iδ~
and iδ~& ) will be dropped in the following sections.
Instead both iδ and iδ& would simply refer to the machine angle and speed
measured in COA frame.
3.3. Signum Function (Bang-bang Control)
The control law derived in (3.10) is a bang-bang control [1, 5] and is
referred to as a signum function. As large disturbances occur in the four-
machine two-area power system in Figure 3.1, a bang-bang controller is
desirable as the control switches between its upper and lower limits to give
maximum available influence. This result is shown in Figure 3.2.
It is expected from observing Figure 3.2 that the continuous oscillations
after t=4s will be damped only by machine damper windings.
69
Figure 3.2: The response of machine speeds. Due to the bang-bang control,
continuous oscillations are formed.
Figure 3.3: The system response is kept on a switching hyperplane S beyond
3.5 seconds effectively resulting in zero control.
0 2 4 6 8 - 6
- 4
- 2
0
2
4
6
Time (sec.)
S=pd2
- pd3
0
2
4
6
8
10- 6
- 4
- 2
0
2
4
6
S= 32 δδ && −
Switc
hing
hyp
erpl
ane
S
0 2 4 6 8 1-8
-6
-4
-2
2
4
6
8
Time (sec.)
pd
pd
pd
pd
0 2 4 6 8 10-8
-6
-4
-2
0
2
4
6
8
Rat
e of
cha
nge
of m
achi
ne a
ngle
1δ&
2δ&
3δ&
4δ&
70
An examination of the switching hyperplane diagram in Figure 3.3 suggests
that during the period between t=0 to t=3.5 seconds, the system trajectory
does not remain on the switching hyperplane S, it reaches and leaves the
hyperplane S. During this duration, the control effort is insufficiently strong
to keep the system trajectory on the hyperplane S. Beyond 3.5 seconds, a
bang-bang switching control effort has sufficient strength to hold a system
trajectory on a switching hyperplane S as machine speeds approach zero. As
machine speeds approaches zero, the signal ( 32 δδ && − ) would be close to
zero and the control U would chatter between its +1 and –1 limits when the
signal 32 δδ && − changes sign. This chattering of U effectively results in zero
damping of some modes of the system. This agrees with the discussion in
[5] that when a system trajectory reaches and stays on a g(x) surface at a
point where it is not a stable operating condition, it oscillates continuously
on the surface.
From the total kinetic energy diagram in Figure 3.4, residue kinetic energy
can be observed in the power system. Correlating this observation with that
of Figure 3.3, it is understandable that continuous oscillations are possible
as there is still energy in the power system. From an energy perspective,
when an unforced and undamped power system has a solution that is not at
velocity origin, it will have a constant energy leading to a system trajectory
which follows a closed orbit inside a separatrix [7, page 110] defined as the
boundary of instability in phase plane. This non-system solution [7, page27]
resembles a continuous oscillation. Since the power system in Figure 3.1 did
71
not incorporate the effect of damper winding and it has effectively zero
control damping owing to the chattering of U beyond 3.5 seconds, it is non-
divergent inside a separatrix. However, the system operates in a limit cycle.
This is clear as seen in Figure 3.4 that the constant oscillation of kinetic
energy corresponds to a constant value of total energy.
Figure 3.4: The Remaining total kinetic energy in the power system results
in continuous oscillations.
In the next section, feedback control theory is used to characterize the cause
of a limit cycle in a power system under the influence of a bang-bang
control.
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9
1
t
Vke1 + Vke2 + Vke3 + Vke4
0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
Time (sec.)
Tot
al K
inet
ic e
nerg
y
(Vke
1 +
Vke
2 +
Vke
3 +
Vke
4)
Vke1 + Vke2 + Vke3 + Vke4
72
3.3.1. A High Gain Feedback Control Problem
An alternative explanation for the occurrence of a limit cycle or a
continuous oscillation on a g(x) surface can be given by considering a high
gain feedback control problem using Root Locus analysis.
The power system in Figure 3.1 is first linearised to the form of
CxyBuAxx
=+=&
(3.12)
where A and x are the respective system matrix and states vector
respectively.
Vectors B and C are constructed by considering (3.1), (3.6) and (3.7), where
all bus voltages are assumed to be 1.0 p.u.
[ ]00000110
00000
0
0000
3
33
2
22
4
434344
3
343433
2
212122
1
121211
−=
∆−
∆−
=
∆−∆−∆−
∆−∆−∆−
∆−∆−∆−
∆−∆−∆−
=
C
mg
mg
mbgg
mbgg
mbgg
mbgg
B (3.13)
where iig∆ and ijg∆ are elements of the change in admittance matrix Y∆
which have been defined earlier in association with (3.6).
73
A Root Locus analysis that examines the characteristic equation in (3.14)
considers various gains represented by the notation K from 0 to infinity.
0)(1 =+ SKG (3.14)
where G(s) is an open loop transfer function.
A general unity negative feedback diagram is shown in Figure 3.5. This
corresponds to a control equation U(s) in the form of
)()( sKzsU = (3.15)
where function z(s) resemble the feedback signal of )( 32 δδ && − obtained from
(3.9). For simplicity, the notations U(s) and z(s) are referred in the following
sections as U and z.
Figure 3.5: A general negative feedback transfer function blocks.
If U is to remain bounded as K approaches infinity, all closed loop poles
would move from its open loop pole position towards the open loop zeros.
Using velocity feedback, all open loop zeros are on the j? axis and as K
approaches infinity the closed loop responses become undamped since all
closed loop poles have moved to the open loop zeros on the j? axis. This
will result in continuous oscillations. The movement of closed loop poles,
K Y(s)
U(s)z (s)R(s)=0C
G(s)
BuAx x& = BuAx = BuAx = K Y(s)
U(s)z (s)R(s)=0_ C
G(s)
++
74
when gain K increases, is shown in Figure 3.6. In Root Locus analysis, the
angle of departure from an eigenvalue refers to the direction of movement
of the eigenvalue as gain K increases from 0. This information is useful as it
determines if a feedback control is divergent or non-divergent.
The angle of departure iϕ (3.18) of every closed loop poles, it is noted that
such control yields non-divergent system responses
180aa pzi −∑−∑=ϕ (3.16)
where αz is the angle from the ith closed loop pole of interest to all of the
finite zeros and αp is the angle from the ith closed loop pole of interest to the
rest of the closed loop poles. A Root Locus diagram in Figure 3.6 shows the
non-divergent system modes for the linearized system described under the
influence of the control U.
Figure 3.6: A Root Locus diagram showing a non-divergent control effect.
- 2 -1.5 -1 -0.5 0 0.5 1-20
-15
-10
-5
0
5
10
15
20
Imag
Axi
s
real axis- 2 -1.5 -1 -0.5 0 0.5 1
-20
-15
-10
-5
0
5
10
15
20
Imag
Axi
s
real axis
Four common mode poles and three finite zeros are located near origin.
One common mode pole approaches the infinity zero as gain K increases to infinity.
75
For this system, similarities are found between the high gain feedback
control problem and the bang-bang control problem, in particular, these
control problems lead to a non-divergent system trajectory. As the state
feedback signal z in (3.15) is based on ( )32 δδ && − which is similar to the
switching surface 32 δδ && −=S in (3.11), when the system response of the
linearized system described by (3.12) and (3.15) is undamped as gain K
approaches to infinity, this implies that the signum (or bang-bang function)
resembles an infinite gain feedback control. This also implies that the
system trajectory that is kept on the g(x) surface by the bang-bang control
represents an infinite gain feedback control. These relation characterize a
bang-bang control (or signum function) using a high gain feedback control
problem. It is reiterated that using a signum function corresponds to using a
feedback control with infinite gain as the system trajectory converges.
3.3.2. Reducing the Number of Mode by Switching Control
In this section, we propose a simple dual LC example [5] to substantiate that
(i) a bang-bang switching control has a mode reduction ability and, (ii) the
oscillation frequency of a limit cycle is the frequency of a remaining mode
which is also the frequency of an open loop zeros on j? axis. By virtue of
the above (ii), we can further support our earlier statement to characterise
bang-bang control using a high gain feedback control problem.
76
Figure 3.7 shows a simple dual LC circuit where U is a reversible control
voltage v1 and v2 are the capacitor 1 and 2's pre-charged voltages
respectively.
Figure 3.7: A dual LC circuit with a control voltage V.
The state equations for Figure 3.7 is
+
−
−
=
00
1
1
001
0
0001
1000
01
00
2
1
2
1
2
1
2
1
2
1
2
1
2
1
L
L
U
VVII
C
C
L
L
VVII
&&&&
(3.17)
From the sum of the capacitor energy )21
( 2CV and inductor energy
)21
( 2LI , the rate of change of total energy W& in the circuit after
substituting relevant terms from (3.17) yields
I1
L1 L2
C2I2 C1
v2U v1
77
)( )()(
21
222111222111
2221112211
222111222111
IIUIILIILILUIILUI
IILIILVIVI
IILIILVVCVVCW
+=++−+−=
+++=
+++=
&&&&
&&
&&&&&
(3.18)
The derived bang-bang control law is
)sgn( 21 IIU +−= (3.19)
and the switching surface S is
21 IIS += (3.20)
A Root Locus analysis on the dual LC system in Figure 3.7 showed that (i)
two oscillatory modes exist, and (ii) under a high gain feedback control, one
mode is damped while the other ends up at the open loop zeros on j? axis
as the power system converges. The zeros of the G(s) function are located
approximately at (0, ± j16.0357) as shown in Figure 3.6.
The remaining mode is compared with a time domain solution under a bang-
bang control and a limit cycle is observed in Figure 3.8. It is noted that the
continuous oscillation has a resonance frequency of 16.1 rad/sec which
approximates the frequency of the remaining mode at ± j16.0357 rad/sec.
This reaffirm our earlier assertation that a bang-bang control can be
described by a high gain feedback control problem.
78
Figure 3.8: Under a bang-bang control, the capacitor voltages of dual LC
circuit has one remaining mode undamped.
3.3.3. Location of Zeros and Bang-bang Control
A Root Locus analysis of the characteristic equation (3.14) shows that root
loci are traced by considering the responses of transfer function G(s) in
Figure 3.5. When K= 0, the roots of the characteristic equation (3.14) give
the poles of G(s) and when K= ∞ , the roots are the zeros of G(s). This
implies that the location of open loop zeros depends on the G(s) function
and a bang-bang control simply forces the open loop poles to the open loop
zeros, as discussed in the earlier sections.
We compare the switching equations in (3.11) of the two-area DC link
power system with that of the dual LC circuit in (3.21) and understand that
0
2
4
6
8
10
- 20
- 15
- 10
- 5
0
5
10
15
20
V1 V2
0
2
4
6
8
10
- 20
- 15
- 10
- 5
0
5
10
15
20
Time (sec.)
V1 V2
Cap
acito
r vol
tage
(v 1
and
v2)
79
both switching equations contain only velocity terms. It is noted that any
velocity based control law with poles on j? axis will give rise to zeros on
j? axis. This is evident from the Root Locus diagram (Figure 3.6) of the
linearized power system represented by equation (3.12 - 3.13).
21 qqS && += (3.21)
Equation (3.21) is written in a velocity form by considering the initial
capacitor charges (q1, q2) and current ( 21 qq && + ) as the respective position
and velocity states.
3.4. Exponential Convergence Introduced by a
Saturation Function
A 'full region' switching function or simply a saturation function is proposed
to yield a system converging behaviour and the derivation of the saturation
function will be described in the following paragraphs.
An inspection of the Root Locus diagram in Figure 3.6 shows that if a finite
gain K is used in a control design instead of an infinite gain K, system poles
will be damped at some damping factor. This leads to the use of a 'full
region' control law that allows switching inside both non-linear and linear
regions (i.e. 11 +≥≥− U and 11 +>>− U ) which would result in the
80
respective infinite gain switching and finite gain switching at appropriate
instances. Such control law is known as a saturation function and it has the
capability to remove the remaining energy in the power system as noted in
Figure 3.4. The control law based on a saturation function is
<≤≤
>==
-1 S -1 for 1S 1 S for -
1 1 for S Usat(S) andU (3.22)
where S is the same switching hyperplane or g(x) surface as derived in
(3.11).
The responses of the power system in Figure 3.1 under the influence of the
saturation function control law in equation (3.22) are shown in Figure 3.9,
3.10 and 3.11. The behaviour of the finite time convergence owing to the
switching in a non-linear region and the exponential convergence due to the
switching in a linear region can be observed from the machine speed (Figure
3.9) and the kinetic energy (Figure 3.11) diagrams.
Comparing Figure 3.2 and 3.9, their differences are explained in the
following paragraphs. The reasons for the state convergence in Figure 3.9
can be summarised as: (i) from a Root Locus analysis in Figure 3.6 a gain K
of 1 introduces approximately a system-damping factor of 0.07 in the large
power system and a 0.05 system damping factor in the small power system.
This guarantees an exponentially converging system states when switching
in a linear region (i.e. 11 +>>− U ), (ii) in energy context, equation (3.9)
and (3.10) describe the rate of dissipation of energy which guarantee a finite
81
time convergence of system states in a non-linear region, and (iii) when the
control effort U no longer switches when operating in the linear region, the
system trajectory will continue to intersect with closed orbits inside a
separatrix of an unforced power system towards an origin.
Figure 3.9: The machine speeds converge owing to the soft switching in
linear region.
The control based on the saturation function in equation (3.22) is shown in
Figure 3.12. It is observed that the saturation function approximates a bang-
bang control before t=3 seconds giving rise to a finite time convergence of
system states towards the S=0 surface. Then the control leaves saturation
and becomes linear resulting in the exponential convergence of system
states.
0 2 4 6 8 10-8
-6
-4
-2
0
2
4
6
8
t
pd pd pd pd
4
0 2 4 6 8 10- 8
- 6
- 4
- 2
0
2
4
6
8
Time (sec.)
Rat
e of
cha
nge
of m
achi
ne a
ngle
1δ&
2δ&
3δ&
4δ&
82
Figure 3.10: The system responses decay exponentially near an origin as the
continuos control in the linear region dominates.
Figure 3.11: The total kinetic energy converges exponentially towards an
origin reaching the system solution.
0 2 4 6 8 10
-6
-4
-2
0
2
4
6
S=pd2
-pd3
Exponentially decaying signal.
0 2 4 6 8 10-6
-4
-2
0
2
4
6
S= 32 δδ && − -
Exponentially decaying signal. Switc
hing
hyp
erpl
ane
S
Time (sec)
0 2 4 6 8 1
0
1
2
3
4
5
6
7
8
9
1
Vke1 + Vke2 + Vke3 + Vke4
Exponentially decaying energy
0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
1
Tot
al k
inet
ic e
nerg
y V
ke Vke1 + Vke2 + Vke3 + Vke4
Exponentially decaying energy
Time (sec)
83
Figure 3.12: The energy based control that uses the saturation function
control law.
3.5. Conclusion
A full region switching control function or a saturation function that is
derived based on energy function and the Lyapunov’s stability criteria is
proposed and verified on a four-machine two-area HVDC link power
system in Figure 3.1. A saturation function is also tested on a dual LC
circuit and system convergence is being achieved. The possibility of
multimode damping using a single HVDC controller to achieve
multimachine stability has thus been addressed.
0 1 2 3 4 5 6 7 8 9 10 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (sec.)
Con
trol
U (b
ased
on
satu
ratio
n fu
nctio
n)
84
Traditionally, HVDC link is only used for control when it is directly
installed in parallel with AC lines. This chapter shows that one HVDC link
can rapidly reduce the oscillation energy injected into the power system
during a fault damping several modes. In general, it is expected in a power
system that oscillation modes receive good damping if the controller has
strong influence over them.
The bang-bang control characterised by a high gain velocity feedback
control problem is different from a sliding mode control. In this aspect, we
see similarity such that both types of switching action tend to attract a
system trajectory to a switching line. In the case of sliding mode control [9],
large control rating is required to attract a system trajectory to a switching
surface and hold it onto the switching surface (or a sliding surface). The
sliding mode control ensures that the system trajectory on the sliding surface
slides towards an origin exponentially and the rate of sliding motion
depends on a sliding equation. In general, a sliding mode control is a high
gain control system that can be designed to introduce system stability in a
system of any order. The proposed bang-bang control derived from the total
energy function approach does not use large control rating as in the sliding
mode control case to attract the system trajectory towards the switching line
keeping the system trajectory on the switching line. Instead, this proposed
bang-bang control drives the system trajectory towards the switching line
letting it passes the switching line in response to switching between control
limits. This is observed from the system response in Figure 3.3. The effect
85
of this bang-bang control based on total energy function approach has the
effect of finite time reduction in kinetic energy. In general, high gain control
such as the bang-bang control derived from total energy approach is capable
of introducing system stability to a system of any order provided at fault
clearing the system trajectory lies inside the separatrix of the system [7,
page 110]. However, the ability of a sliding mode control to introduce
system stability to a system of any order depends on the ability of the
control to attract the system trajectory to the switching line. With regards to
the implementation of high gain control in high order system, this is not the
interest of study in this thesis. However, the proposed total energy function
approach in switching control application does not have difficulty in its
implementation in high order system. The issue being discussed is in
relation to the use of high gain feedback control to characterize a bang-bang
control in order to resolve the issue of control chattering near a stable
equilibrium point. The consequence of control chattering with respect to the
control damping introduced in the system is an effectively zero control
damping being introduced. Through the understanding of a simple
linearized high gain feedback control problem, the bang-bang switching
function is replaced by a saturation switching function to allow finite gain
switching near a stable equilibrium point. This saturation switching function
resolves the issue of control chattering near a stable equilibrium point which
is approximated as the region of the non-linear system. In general, the
proposed energy function control based on a saturation function
approximates a bang-bang control and provides continuous control value in
86
a linear region preventing a system trajectory from being kept on a
switching surface near a stable equilibrium point (or a linear region).
The control design based on energy function is outlined and extended to the
control of a HVDC link power system which is also applicable to the control
of Flexible AC Transmission (FACT) devices such as a Static Var
Compensator (SVC). The proposed saturation function overcomes the
deficiencies in a linearly design controller. It is evident that for a linearly
tuned controller, it will experience unanticipated performance during severe
disturbances and underperforms when it saturates. When an energy function
control design uses a saturation function instead of a signum function, the
benefit of the maximum energy reduction obtained from a velocity based
bang-bang control is being retained and the problem of control chattering
near a stable equilibrium operating condition has been mitigated.
87
3.6. References
[1] G. Ledwich, J. Fernandez-Vargas, and X. Yu, "Switching Control of Multi-machine Power Systems," IEEE / KTH Stockholm Power Tech Conference, Stockholm, Sweden, pp. 138-142, June 1995.
[2] C. Taylor, "Advanced Angle Stability Controls," Cigre Technical Brochure Cigre TF38.02.17, No. 155, pp. 2-2 to 2-8, April 2000.
[3] M. A. Pai, Energy Function Analysis For Power System Stability: Kluwer Academic Publishers, 1989.
[4] H. D. Chiang, C.-C. Chu, and G. Cauley, "Direct Stability Analysis of Electric Power Systems Using Energy Functions: Theory, Applications, and Perspective," Proceedings of the IEEE, vol. 83, No. 11, pp. 1497-1528, November 1995.
[5] E. W. Palmer, "Multi-Mode Damping of Power System Oscillations," PhD thesis in Electrical and Computer Engineering The University of Newcastle, 1998, pp. 196.
[6] K. Prabhashankar and W. Janischewsyj, "Digital Simulation of multimachine Power Systems for Stability Studies," IEEE Transactions on Power Apparatus and Systems, vol. PAS-87, No. 1, pp. 73-80, January 1968.
[7] M. A. Pai, Power System Stability - Analysis by the Direct Method of Lyapunov, vol. 3: North-Holland Publishing Company, 1981.
[8] G. Ledwich and E. Palmer, "Energy Function For Power Systems with Transmission Losses," IEEE Transactions on Power Systems, vol. 12, No. 2, pp. 785-790, May 1997.
[9] M. V. D. Wal, B. D. Jager, and F. Veldpaus, "The slippery road to sliding control: conventional versus dynamical sliding mode control," International Journal of Robust and Nonlinear Control, vol. 8, pp. 535-549, 1998.
88
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89
Chapter 4
Weighted Energy Control
The energy function based control aims at maximizing the rate of reduction
of disturbance energy. The saturation function introduced in the earlier
chapter approximates a bang-bang control and yields at lease exponential
damping modes near a stable equilibrium operating condition. In a power
system, there are hundreds of strong and weak links and the power system
could separate anyway in the transmission network. In over the total set of
disturbances, separations at weak links are more likely to occur than at the
strong links but the total energy function does not differentiate that. In
control application, this total energy approach underperforms when it comes
to preventing a power system separation at a weak link at first swing. In this
chapter, the strong and weak area operating conditions in a large power
system is introduced to emphasize the relationship between transient
stability limits and the weak area of a power system.
90
One of the major limitations of energy function based control is the control
direction that targets on the portion of the system with greatest disturbance
energy instead of the area with the highest probability of separation. In this
thesis, the term “power system separation” refers to two coherent groups of
generators that separate from each other resulting in system instability.
4.1. Introduction
The energy function based control introduced in the earlier chapter yielded
appropriate control directions for the control of a two area DC link power
system. One of the major deficiencies of the energy function based control
arises from the emphasis of its control directions on the area with the
highest disturbance energy, which is undesirable from the strong and weak
areas perspective. It is understandable from a practical aspect that the
strong and weak areas in a two-area DC linked power system have different
transient stability limits; the weaker area has a lower stability limit owing to
its higher network reactance. When we consider the two-area DC link power
system as one large power system, it is apparent that severe disturbances on
a g(x) surface may introduce different disturbance energy in these two areas,
with the larger area having higher disturbance energy than the smaller area
since the larger area has a higher aggregated inertia constant. Although
disturbance energy in the smaller area is low, it may be dangerously close to
its low transient stability limit. This becomes an obvious control problem
91
when a power system is on the verge of separating into two coherent groups
of generators. Hence it is desirable to have a control that is capable of
recognizing both the strong and weak areas circumstances and directs the
correct control efforts to these areas that are in need of it.
4.2. Looking at a Two-area Energy Problems
The two-area DC link power system as shown in Figure 4.1 is used as an
intuitive example to demonstrate the strong and weak areas case in a large
power system.
Figure 4.1: Single line diagram of a four-machine two-area system
A simplified process of deriving energy function based control law using
only total kinetic energy instead of an energy function [1] is proposed.
m2=0.922
2 3 5 4
m1=1.5
1 j0.01 j0.0144 j0.0016
j0.016
j0.01
m4=0.922
2 3 5 4
m3=1.56
1 j0.01 j0.095 j0.045
j0.05
j0.01
DC Link G2
-G2
Generator 1Pm1=20p.u.
Generator 2Pm2=5p.u.
Generator 3Pm3=8p.u.
Generator 4Pm4=2p.u.
Large system (area 1)
Small system (area 2)
92
A total kinetic energy for the power system is in the form
∑==
=
4
1
2
21 n
iiike mV δ& (4.1)
where n is the number of machines, m is machine inertia constant and δ& is
machine angular velocity.
Taking the derivative of the total kinetic energy keV , we obtained keV& as in
(4.2). Equation (4.2) has a similar switching surface (or controllable terms)
compared to the derivative of energy function V& introduced in the earlier
chapter.
( )
∆−∆−∆−∆−
−−−−
+++−+
=
∑==
=
)()()()(
)()()()(
)()()()(
44332211
44332211
44332211
4
1
eeee
eeee
mdcmdcmm
n
iiiike
PPPPU
PPPP
PPPPPP
mV
δδδδ
δδδδ
δδδδ
δδ
&&&&
&&&&
&&&&
&&&&
(4.2)
∆+∆+
∆+∆+
∆+∆+
∆+∆+
∆+∆+∆+∆
−=
4343434343
2121212121
4343434343
2121212121
444333222111
sinsin
sinsin
coscos
coscos
),(
δδδδ
δδδδ
δδδδ
δδδδ
δδδδ
δδ
bb
bb
gg
gg
gggg
U
f
&&
&&
&&
&&
&&&&
&
SUf
Ugf
*),(
),(),(
−=
−=
δδ
δδδδ&
&&
where
93
• Pe is the machine electrical power output
∑ ∑=
=
=
≠=
++=4
1
4
1
))cossin(n
i
n
ij
ijijijijiiei gbgP δδ ,
• ∆Pe is the change in machine electrical power output due to the power
modulation in the network caused by the DC link
∑ ∑=
=
=
≠=
∆+∆+∆=∆4
1
4
1
))cossin(n
i
n
ij
ijijijijiiei gbgP δδ ,
• ),( δδ&gS = is the switching surface and
• U is the control effort that switches between the 1± limits.
Examining (4.2), when the power system in Figure 4.1 has no control (i.e.
U=0) and is operating at a SEP as defined in Chapter 2, the sum of machine
speeds is zero since the power supplies (Pmi) equals the power demands
(Pei±Pdc) in both areas. This results in 0=keV& in the large power system. At
post-fault when the operating point is deviated from the SEP causing the
changes in generator’ electrical power from Pei to Pei*, the aim of U is to
introduce changes in the network electrical power (∆Pei) to minimize Pei*
and reduce keV& . This leads to the derivation of the control law that keeps the
keV& most negative.
The derived bang-bang control law becomes
=<−
>==
for S for S for S
(S) and UU0001
01sgn (4.3)
94
The total kinetic energy equation (4.1) describes the two areas’ energy
problem and its derivative (4.2) indicates that the direction of the control
effort is determined by the dependant terms associated with area 1 and 2’s
machine angular velocity ( 21 and δδ && ) and ( 43 and δδ && ) respectively. From
these dependant terms, it is possible to identify the energy associated with
each of the area and it becomes feasible to weight between the two areas in
terms of their energy. Hence the weighted energy control is
++−+
++
++−+
+
=
∆∆
∆∆
∆∆
∆∆
)(cos)(sin*
)(cos)(sin*
sgn
431212431212
444333
211212211212
222111
δδδδδδ
δδ
δδδδδδ
δδ
&&&&
&&
&&&&
&&
GB
GGwtB
GB
GGwtA
U (4.4)
where wtA and wtB are the respective weighting factor for area 1 and area
2.
The evaluation of an area’s weighting factor is based on the concept of UEP
and this will be discussed in the following section.
95
4.3. A Proximity to Separation Weighting Based on
UEP
An Unstable Equilibrium Point (UEP) [2] describes the transient instability
of an area and in the above two-area DC link problem (Figure 4.1), there is
only one UEP in each area as a two-machine configuration leads to only one
type of power system separation.
A two-machine system can be made equivalent to a Single-Machine-
Infinite-Bus (SMIB) system and an approximated UEP for each area is
evaluated using the expression of
)( si pUEP δ−= (4.5)
where sδ is a stable operating angle difference between two machines
obtained from a loadflow solution and UEPi refers to the ith area’s UEP.
A centre of area (COA) [3] angle frame is used in this two-machine power
system representing an area and the COA angle coaδ for each area is
evaluated using the expression of
)( 21
2211mmmm
coa ++
=δδ
δ (4.6)
where mi and iδ refers to the ith machine inertia constant and angle
respectively.
96
Substituting (4.5) into (4.6), the approximated UEP ),( 21uu δδ referenced to
the COA is
2112
12211
/
/)(
mm
mm
uu
su
δδ
δπδ
−=
−= (4.7)
The energy evaluated at a UEP is the critical energy [4] of a power system
separation. As each area has one UEP that describes the power system
separation at the area, the proximity to this unique critical energy will yield
a warning indicator for each area.
(t)-VVindex
(t)-VVindex
arae23uepB
arae12uepA
2
1
=
= (4.8)
where both uepV12 and uepV34 are energy evaluated at the UEP of area 1
and area 2 respectively.
A dynamic weighting approach that is suitable for a two-area energy
problem in (4.4) is
1)index20*exp(wtB1)index20*exp(wtA
B
A
+−=+−=
(4.9)
These exponential weighting factors flexibly direct the control in (4.4) when
the power system in Figure 4.1 is close to separation in an area.
From Figure 4.2, it is apparent that placing a higher weighting on a
vulnerable area has significant advantages as the control is directed to keep
97
both areas synchronized and prevented the power system separation in area
2 around 0.25s where disturbances on a g(x) surface are examined. The
disturbances on a g(x) resemble the occurrence of disturbances in both the
large and small areas. The control in equation (4.4) is based heavily on area
2’s disturbance energy in the form of 43 and δδ && oscillations.
Figure 4.2: Control effort directed to save area 2 from separation around
0.25s.
The dynamic weightings for the two areas are shown in Figure 4.3. The
weighting of area 2 outweighs that of area 1 at the first swing near 0.25s
when area 2 is close to separation. As area 2 becomes no longer in risk of
separating, its weighting reduces significantly.
0 0.5 1 1.5 2 2.5 3
-6
-4
-2
0
2
4
Time (sec.)
Effect of weighted bang-bang control1δ&2δ&
3δ&
4δ&
Mac
hine
ang
ular
vel
ocity
98
Figure 4.3: Weightings of the two areas that indicate the high risk separation
in the area.
4.4. Conclusion
An energy function based control law has a switching surface S that is
similar to that of a total kinetic energy function control. This significantly
simplifies the process of formulating a V function and deriving a V& control
law that yields stable results.
The large and small area scenario mentioned in this chapter is common in a
large power system. The proposed idea of weighted energy control is
attractive as it emphasizes the probability of separation instead of the size of
disturbance energy. It is clear that when a control uses the size of
0 0.5 1 1.5 2 2.5 30
20
40
60
80
100
120
140
160
180
200
Time (sec.)
Response of weighting factors in the two
Area 1 weightingArea 2 weighting
Wei
ghtin
gs
99
disturbance energy as the main determining factor for its control direction,
the weak area is often neglected during the first swing. Using a two-area
power system with each area represented by a two-machine system, the
weighting of energy based control is obvious since each area has only one
UEP. In each area, the comparison between the post-fault energy and the
critical energy evaluated at respective UEPs give rise to a proximity-to-
separation indicator that can be used to redirect control to a weak area at the
first swing. This increases the chance of survival in a weak area.
When it is required to consider the effect of machine interactions (or area
interactions) in a large power system during the transient period, a multiple
machines model is used instead of aggregated machines in the power
system. In a multi-machine power system, there are various ways the power
system can be separated hence several UEPs must be considered. This
complicates the use of weighted energy control in a multiple UEPs
environment and a control approach that identifies this multiple separation
situation is required. The extension of weighted energy control will be
discussed in the later chapter.
100
4.5. References
[1] M. A. Pai, Energy Function Analysis For Power System Stability: Kluwer Academic Publishers, 1989.
[2] H. D. Chiang, C.-C. Chu, and G. Cauley, "Direct Stability Analysis of Electric Power Systems Using Energy Functions: Theory, Applications, and Perspective," Proceedings of the IEEE, vol. 83, No. 11, pp. 1497-1528, November 1995.
[3] G. Ledwich and E. Palmer, "Energy Function For Power Systems with Transmission Losses," IEEE Transactions on Power Systems, vol. 12, No. 2, pp. 785-790, May 1997.
[4] C. L. Gupta and A. H. El-Abiad, "Determination of the closest unstable equilibrium state for Liapunov methods in transient stability studies," IEEE Transactions on Power Apparatus and Systems, vol. PAS-95, No. 5, pp. pp. 1699-1712, September/ October 1976.
101
Chapter 5
Optimal Switching Near Separation
The advantages of finite-time damping performance introduced by the total
kinetic energy reduction control and the advantages of using saturation
function based switching near a stable equilibrium operating condition have
been introduced in earlier chapters. In the earlier chapters, while it is
appreciated that the use of weighted energy based control prevents a weak
area from separating, it is also note that the rate of changes of machine
angles associated with the weak area tends to hover close to zero near a
separation. This low rate of change of machine angles can result in possible
hovering of machine angles near separation which is not desirable since it is
necessary to yield from the control the overall greatest reduction in
disturbance energy.
A total energy control approach that is derived from Lyapunov based energy
function design and applied in a complex power system is shown to provide
102
satisfactory energy reduction performance when severe disturbances are
encountered. Most energy-based controllers use a control proportional to
velocity and this provides satisfactory damping but not a good first swing
stability. This chapter addresses this problem of undesirable switching at
first swing by using a Single-Machine-Infinite-Bus (SMIB) example to
examine the cause of this problem. A constant energy surface diagram is
used in the analysis to demonstrate the key issue of switching control. An
angle look-ahead control is derived for the SMIB system as an effective
solution to this first swing switching stability problem. It is shown that
through the understanding of this problem, controller design can be made
effective in limiting the first swing without the loss of damping
performance.
5.1. Introduction
Lyapunov based energy reduction control is effective in damping
oscillations [1, 2]. This total energy approach [3] is applied in a two-area
DC link power system and had shown that velocity proportional control can
be used effectively in the control of complex power system. The δ& bang-
bang control in [3] gives satisfactory damping performances when severe
disturbances are encountered. However, this type of velocity based (δ& )
control also gives undesirable switching decisions at first swing and is
referred to as the first swing switching stability problem in this chapter.
103
Conventional analyses of a first swing stability problem investigate whether
a system is close to separating. This information is not sufficient in
providing a clear understanding of the first swing stability problem in
switching control.
This chapter uses constant energy surface in phase portrait as an explicit
analysis tool to understand this first swing switching stability problem.
Through the formulation of a SMIB example, we seek to characterize this
first swing switching stability problem and disclose the causes of the
undesirable switching at first swing when δ& bang-bang control [3] is being
used. An improved understanding of the problem leads us to the
development of an angle look-ahead control which is found to be effective.
This chapter is organized into three parts. Firstly, a SMIB example will be
formulated to derive a δ& bang-bang control. Secondly, disturbances
introduced near the control limits will be examined to demonstrate that
undesirable switching instances can occur and compromise the performance
of the controller. The characterization of a first swing switching stability
problem is then proposed and consequently, the existence of a partly stable
region is recognized. The constant energy surface diagram is used as an
analysis tool as it gives clear illustration of the problem. Thirdly, angle
look–ahead dependant terms are derived and its influence is then examined
and used as an approximate, but effective, remedy to the first swing
switching stability problem. Minimum time control and robustness
104
requirements are discussed in relation to the proposed solution. Fourthly, an
energy reduction switching line which implements an approximate
minimum time criteria will be proposed. An optimal look-ahead duration is
found at the reduced control limit.
5.2. A Velocity Proportional Control Based on Energy
A two-machine power system is shown in Figure 5.1 where the conductance
G2 is capable of real power modulation. The sign of G2 indicates a real
power withdrawal or injection at the connected bus. The device dynamics
describing the particular implementation of the controllable admittance G2
and the voltage variation at bus 3 will not be considered in this example.
The set of swing equations based on the reduced equation of the shown in
Figure 5.1 is
( ) ( )( )
( ) ( )( )
−∆+−∆+−∆+−
=
−∆+−∆+−∆+−
=
12212121
2121212222222
11121212
1212121111111
sin*
cos**
sin*
cos**
coa
m
coa
m
PmbUb
gUggUgPm
PmbUb
gUggUgPm
δδ
δ
δδ
δ
&&
&&
o
o
ωωδ
ωωδ
−=
−=
22
11&
& (5.1)
where pcoa1 is the perfect governor term [3-5], )*( gUg ∆+ and )*( bUb ∆+
are the respective changes in the shunt conductance and transfer admittance
in the reduced admittance matrix actualY [1] owing to control switching and
all δδ &&& , and ω are expressed in inertia frame.
105
Figure 5.1: Two-machine system with breaking resistor at bus 3
The perfect governor terms (pcoa1) in (5.1) is
21
21211 mm
PPPPP eemm
coa +−−+
= (5.2)
The assumption of the perfect governor assumes there is no motion of the
COA [1]. The angles are equal to those measured in COA frame with no
governors. The changes in the reduced admittance matrix actualY through the
switching in and out of the conductance G2 is
outin
outactual
YYYYUYY
−=∆∆+= *
(5.3)
where Yin and Yout represent the network reduced admittance matrix for the
case as G2 switches in and out respectively, U is the switching input and
Y∆ is the change in reduced admittance matrix.
j 0.074 j 0.001 j 0.001
G2 (Braking resistor)
Pm1=5.0 Pm2=- 5.0 1 243
Gen 1 Gen 2
106
Without the loss of generality, the inertia of machine 2 is set to approach
infinity and gives rise to a classical SMIB power system. The use of SMIB
power system simplifies the analysis of undesirable switching during the
first swing.
The kinetic energy function Vke for the power system using a total energy
approach [3] is in the form of
222
211 2
121
ωω mmVke += (5.4)
The derivative of Vke is
( )∑==
=
2
1
n
iiiike mV δδ &&&& (5.5)
Considering that all bus voltages are 1.0 p.u. and substituting the machine
angle acceleration from (5.1) into (5.5), the simplified keV& with the transfer
conductance terms neglected is
( )( )
∆+
∆+∆+∆−=
21212
12121222111
sin
sin ),(
δδ
δδδδδδ
b
bggUfVke &
&&&&& (5.6)
where ∆gii and ∆bij are the elements of ∆Y given in (5.3).
We maximize the reduction of total kinetic energy by making keV& (5.6) as
negative as possible and a δ& bang-bang control law is chosen as
) sgn (SU V = (5.7)
107
where the sgn function is defined as
=<−
>=
0 0 for S 0 1 for S
0 1 for SU
V
V
V
The switching surface SV based on (5.6) is
∑ ∑ ∆+∆==
=
=
≠=
2
1
2
1)sin(
n
ii
n
ij
ijijiiV bgS δδ & (5.8)
Considering the classical SMIB version of Figure 5.1 mentioned earlier (by
assuming an infinite machine 2 inertia), we simplify (5.8) into
111δ&gSV ∆= (5.9)
where both ∆bij and 2δ& are neglected since ∆bij is significantly small due to
the nature of real power modulation and that 2δ& is zero when machine 2 has
infinite inertial constant in the SMIB case. As the δ& bang-bang control U
derived in (5.7) switches between its upper and lower limits as SV (5.9)
changes sign, we call 0=VS as the switching line. As this corresponds to
01=δ& provided 011 ≠∆g , the switching line for the δ& bang-bang control in
phase plane for all bounded δ is at
0=δ& (5.10)
Having defined the switching lines for the δ& bang-bang control law, the
characterisation of the first swing switching stability problem using the
classical SMIB case of Figure 5.1 will be discussed in the following section.
108
5.3. Characterization of a First Swing Switching
Stability Problem
In switching control, stability limits can be visualized as the extension or
reduction of an existing stability limit of an unforced system. The δ&
switching control law (5.7) derived from total energy approach [3] aims at
maximising the reduction of total kinetic energy. As far as the switching
performance is concerned, the first swing switching response that occurs
between the upper and lower control limits has resulted in an undesirable
switching behaviour, which deteriorates control performance. Thus,
characterization of this first swing switching problem becomes important.
We examine this problem of first swing switching control by applying
severe faults at the bus 4 of the power system in Figure 5.1. The control
performances of a δ& bang-bang control law (5.7) result in a chattering
control U at first swing switching when high control effort is directed to
yield zero velocity. The result is shown in Figure 5.2.
Looking at Figure 5.2, it is found that machine angle hovers as the result of
an effectively zero damping in the power system when the control chatters
and these circumstances occur near an unstable equilibrium operating
condition.
109
Figure 5.2: Control chatters and angle hovers at maximum as the forced
damping is effectively zero. Faults at bus 4 are cleared at 1.0459 seconds.
5.3.1. Recognizing a Partly Stable Region (PS Region)
The characterization of machine angle hovering is illustrated using a
constant energy surface diagram as an analysis tool. An understanding of
the switching operation helps in the development of control strategy.
It is evident from the constant energy surface diagram in Figure 5.3 that a
partly stable region is responsible for the undesirable switching effect when
the δ& bang-bang control (5.7) is used. In particular, in Figure 5.3, the
switching inside the PS region gives rise to a limit cycle instead of the
energy reduction effect. This becomes obvious when we look at the various
trajectory directions in the PS region in Figure 5.4.
0 0.5 1 1.5 2 2.5
0.5
1
1.5
2
2.5
Time (sec.)
Mac
hine
ang
le,
δ 12
(rad
.)
0 0.5 1 1.5 2 2.5
-1
-0.5
0
0.5
1
Con
trol
U
Time (sec.)
Machine angle
hovers
Controller output
chatters
110
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+
U-
Uo
U+
curves U-
curves
Fault -on trajectory
Post fault trajectory U
ocurve
Limit cycle
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+
U-
Uo
U+
curves U-
curves
Fault -on trajectory
Post fault trajectory U
ocurve
Limit cycle
PS region
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+
U-
Uo
U+
curves U-
curves
Fault -on trajectory
Post fault trajectory U
ocurve
Limit cycle
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6U+U-Uo
U+ curves U- curves
Fault-on trajectory-
Post-fault trajectory Uo curve
Limit cycle
PS region
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+
U-
Uo
U+
curves U-
curves
Fault -on trajectory
Post fault trajectory U
ocurve
Limit cycle
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+
U-
Uo
U+
curves U-
curves
Fault -on trajectory
Post fault trajectory U
ocurve
Limit cycle
PS region
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+
U-
Uo
U+
curves U-
curves
Fault -on trajectory
Post fault trajectory U
ocurve
Limit cycle
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6U+U-Uo
U+ curves U- curves
Fault-on trajectory-
Post-fault trajectory Uo curve
Limit cycle
PS region
Figure 5.3: Effect of δ& bang-bang control switching in the PS region
causing control chattering. Fault cleared at 1.0459 seconds.
In Figure 5.4, it is seen that the system trajectory follows one of the U+
curves towards the 0=δ& switching line derived earlier in (5.10). As δ&
became negative, the system trajectory is switched to the U- trajectory in the
direction towards the 0=δ& switching line. The repeated switching results in
a constant magnitude oscillation or a limit cycle.
111
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+
curves
U-
curves
Uo
curve
Limit cycle
Post faultTrajectory
PS region U+U-Uo
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+
curves
U-
curves
Uo
curve
Limit cycle
Post faultTrajectory
PS region U+U-Uo
U+U-Uo
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+
curves
U-
curves
Uo
curve
Limit cycle
Post faultTrajectory
PS region U+U-Uo
U+U-Uo
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+ curves
U- curves
Uo curves
Limit cycle
Post-faulttrajectory
PS region U+UoU-
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+
curves
U-
curves
Uo
curve
Limit cycle
Post faultTrajectory
PS region U+U-Uo
U+U-Uo
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+
curves
U-
curves
Uo
curve
Limit cycle
Post faultTrajectory
PS region U+U-Uo
U+U-Uo
U+U-Uo
U+U-Uo
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+
curves
U-
curves
Uo
curve
Limit cycle
Post faultTrajectory
PS region U+U-Uo
U+U-Uo
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2
-1
-0.5
0
0.5
1
U+ curves
U- curves
Uo curves
Limit cycle
Post-faulttrajectory
PS region U+UoU-
Figure 5.4: A close-up view of the partly stable region showing two sets of
trajectories directing towards the 0=δ& switching line.
It is apparent that switching in a PS region is highly undesirable and the
indication of control chattering leading to the hovering of machine angle has
provided a distinctive characterisation of a partly stable region in this
control switching application.
Observing both Figure 5.3 and 5.4, it is noted that one possible solution to
the problem of control chattering is to introduce a delay in switching (or a
phase shift) to avoid the switching in the PS region. For instance, the
switching decision at the 0=δ& switching line could be adequately delayed
until the system trajectory moves out of the PS region. In control context,
such a strategy is equivalent to introducing a phase shift to the 0=δ&
switching line at. Instead of switching at 0=δ& regardless of the angular
112
position, both velocity and position proportional control are used when a
system trajectory is close to a PS region. This forces switching outside a PS
region while the system trajectory remains non-divergent since it is inside
the region of attraction extended by the U+ curve in Figure 5.4. As delayed
switching can be in a form of look-ahead control, angle look-ahead terms
are discussed in the next section.
5.4. The Undesirable Effect of Saturation Function
From the earlier sections, it is noted that switching in the partly stable
region (PS region) using a δ& bang-bang control results in control chattering
followed by the hovering of machine angle. Before an angle look-ahead
term is examined, the deficiency of using a saturation function at first swing
is studied.
It is understood from the earlier chapter that a saturation function is capable
of preventing control chattering near a stable equilibrium operating
condition and it has the form of
≤≤−−<−
>==
1 S 1S for 1 1 for S
1 1 for SU) sat(S U
V
V
V
V
and
where SV has been defined earlier in association with equation (5.8).
113
The result of using a saturation function at first swing is shown in Figure 5.5
when a severe fault which occurred at bus 4 is cleared at a critical clearing
time of 1.067 seconds.
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6Constant energy surface in phase portrait
Machine angle
Mac
hien
ang
ular
vel
ocity
Uo U- U+ System trajectory
Figure 5.5: Undesirable effect of using a δ& saturation function at first swing
when the severe faults at bus 4 are cleared at 1.067 seconds.
(a): Control output U reduces its value at the wrong instances.
(b): System trajectory in phase portrait becomes unstable.
0 0.5 1 1.5 2 2.5 3 3.5 4-0.5
0
0.5
1
Time
Con
trol
out
put U
U
(a)
(b)
114
The result in Figure 5.5 has indicated that switching between the upper and
lower control limits at first swing is crucial to the survival of the power
system when control effort is most needed. The use of a δ& saturation
function control at first swing while switching in a linear region becomes
undesirable as the control value reduces. The result of reduced control value
at critical moments is shown in the subplot (1) of Figure 5.5 and the
consequences of an unstable system trajectory due to a reduced control
value is shown in Figure 5.5 (b).
5.5. An Angle Look-ahead Control
In this section, angle look-ahead dependant terms are being derived for the
SMIB system in Figure 5.1 and its integration with the existing velocity
proportional bang-bang control (5.7) is discussed.
First, we consider an angle error cost function of
2)()( sg δδηδ −= (5.11)
where η is the scalar multiplier and sδ is the machine angle at the stable
equilibrium operating condition.
Applying the Taylor series expansion to (5.11) with the third and higher
order terms neglected, we have
115
2)()(
2TgTgtgTtg
∆+∆+=+ &&& (5.12)
The first term of (5.12) is similar to (5.11) while the second and third terms
are expanded into
( )2
)(2)(22
)(22
22 TT
g
TTg
s
s
∆+−=
∆
∆−=∆
ηδδδδ
δδδη
&&&&&
&& (5.13)
Considering that all bus voltages are 1.0 p.u., (5.1) is substituted into (5.13)
and simplified to
( )),())((222 11
1
22δδδδ
η &&& fgUmTT
g s +∆−−∆
=∆
(5.14)
Substituting equations (5.13 - 5.14) into (5.12) and considering only the
controllable terms that are function of U, equation (5.12) reduces to
))(()( 111
2
gmT
UTtg s ∆−∆
−=+ δδη
(5.15)
From (5.15), we obtain the position proportional dependant term SP for the
SMIB power system
∑∑=
=
=
≠=
∆+∆−∆=
2
1
2
1
2 )sin)((n
i
n
ij i
ijijiisii
P m
bgTS
δδδη (5.16)
116
We understand that the total energy approach in (5.4) yields the velocity
dependant term SV in (5.7) and the angle error cost function look-ahead in
(5.12) yields the position dependant term SP in (5.16). Incorporating (5.12)
with (5.4), we obtain a total switching surface ST that contains both a
velocity (5.7) and a position (5.16) dependant term
∑ ∑
∆+∆−∆+
∆+∆
==
=
=
≠=
2
1
2
1 2 )sin)((
)sin(n
i
n
ij
i
ijijiisii
ijijiii
T
m
bgT
bg
S δδδη
δδ&
(5.17)
The new control law is
)sgn(SU T= (5.18)
where the sgn function has been defined earlier in association with (5.7).
Based on the switching surface ST (5.18), an improved controller
performance is achieved for the first swing switching stability problem due
to the incorrect switching near or inside the partly stable region. This
improvement is shown in Figure 5.6.
117
Optimal switching line
Vertical limit
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+ curves
U- curvesUo curve
Fault trajectory-
Post-fault trajectory
U+UoU-
OptimalSwitching line
Vertical limit
PS region
Optimal switching line
Vertical limit
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+ curves
U- curvesUo curve
Fault trajectory-
Post-fault trajectory
U+UoU-
OptimalSwitching line
Vertical limit
PS region
Optimal switching line
Vertical limit
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6 U+ curves
U- curvesUo curve
Fault trajectory-
Post-fault trajectory
U+UoU-
OptimalSwitching line
Vertical limit
PS region
Figure 5.6: The use of total switching line ST in the δ& bang-bang control
avoids the chattering of control.
(a): The effect of using ST avoided the hovering of machine angle and
chattering of control U.
(b): The system trajectory becomes stable as switching is being delayed
appropriately.
0 1 2 3 4 5
-0.8
-0.4
0
0.4
0.8
U
Time (sec.)
Con
trol
ler o
utpu
t U
-
0 1 2 3 4 5- -0.50
0.5
1
1.5
2
2.5
3
Time (sec.)
Mac
hine
ang
le 1
Delayedswitching
0 1 2 3 4 5
-0.8
-0.4
0
0.4
0.8
-
0 1 2 3 4 5
-0.8
-0.4
0
0.4
0.8
-
0 1 2 3 4 5
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
0
0.5
1
1.5
2
2.5
3
Delayedswitching Delayedswitching
(a)
(b)
118
Comparing Figure 5.6 (a) with Figure 5.2, it is apparent that a slight phase
shift introduced in the δ& bang-bang control (subplot (1) of Figure 5.6)
significantly reduces the hovering of machine angle near the Unstable
Equilibrium Point [4] (UEP). With reference to the PS region as shown in
the subplot (2) of Figure 5.6, it is obvious that in order to avoid switching
inside a PS region, we need an optimal switching line consisting of both
vertical limit and optimal slope. In the next section, we will discuss the
vertical limit and optimal slope are being established.
5.5.1. An Optimal Switching Line
In Figure 5.6, an optimal switching line is found for the first swing
switching stability problem. From the above characterization of the first
swing switching stability problem, it is recognised that switching on the
0=δ& switching line inside the PS region is detrimental as illustrated in
Figure 5.7. However, it is acceptable to switch at 0=δ& before the reduced
UEP, which is simply a reduced transient stability limit due to the lower
control limit of 1−=U . This provides us with a key line at the reduced
UEP. Similarly, it is also unsafe to switch inside the PS region when 0≠δ&
which provides us with the optimal slope as shown in Figure 5.8.
119
The Figure 5.7 and 5.8 explained the construction of the optimal switching
line portrayed in Figure 5.6 as the boundary of the PS region for desirable
switching results.
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U- curve
Uo curve
U+ curve
Post-fault trajectory
Fault trajectory
2
4
6
Switching onBefore the vertical limit. The vertical limit
Switching oninside the PS region
U+UoU-
0=•δ
= 0•δ
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U- curve
Uo curve
U+ curve
Post-fault trajectory
Fault trajectory
2
4
6
Switching onBefore the vertical limit. The vertical limit
Switching oninside the PS region
U+UoU-
0=•δ
= 0•δ
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U- curve
Uo curve
U+ curve
Post-fault trajectory
Fault trajectory
2
4
6
2
4
6
Switching onBefore the vertical limit. The vertical limit
Switching oninside the PS region
U+UoU-
0=•δ
= 0•δ
Figure 5.7: Understanding a divider line at a reduced UEP.
Figure 5.8: Understanding an optimal slope.
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
Switching on
optimal slope
Switching inside
PS region, offthe optimal slope
The optimal slope
U+curve
U- curve
Faul
-on trajectory
Post fault trajectory
U ocurve
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
Switching on the
optimal slope
Switching inside the
PS region, off the
Optimal slope
The optimal slope
U+
U- curve
Fault
Post-fault trajectory
Uo curve
120
The implementation of this optimal switching line (Figure 5.6) requires
accurate switching on the line or else we may be riskily switching in the PS
region. The use of optimal switching line becomes impractical when we
consider the robustness of a control design to accommodate modelling
errors and control tolerances. One possible solution is to replace the optimal
switching line in Figure 5.6 by the approximated switching line as shown in
Figure 5.9 which uses a divider line at a reduced UEP instead of an optimal
slope. This approximated switching line forces switching on the divider line
at the reduced UEP (Figure 5.7) instead of the optimal slope (Figure 5.8) for
relatively large disturbances.
However, this type of approximated switching line will again become
impractical when the robustness of a control design is considered, in
particular, when disturbances occur in the vicinity of the reduced UEP. By
comparing this approximated switching line with a total switching line
based on ST as derived in (5.16), it is obvious that ST is capable of solving
this problem associated with the robustness of the control design. The total
switching line ST is illustrated in Figure 5.9.
In view of the minimum time criteria, we must also consider the issue of
optimal look-ahead T∆ affecting the switching results of a total switching
line ST. In the next section, a snapshot of the switching in the vicinity of the
approximated switching line (Figure 5.9) will be discussed in order to
determine the optimal look-ahead T∆ .
121
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U+ curve
Uo curve U- curve
The approximated switching line
Total switching line ST
U+U oU-
Stable equilibrium
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U+ curveUo curve
U- curve
The approximated switching line
Total switching line S
T
U+U oU-
Stable equilibrium
Machine angle
Mac
hine
ang
ular
vel
ocity
Constant energy surfaces in phase portrait plane
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U+ curve
Uo curve U- curve
The approximated switching line
Total switching line ST
U+U oU-
Stable equilibrium
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U+ curveUo curve
U- curve
The approximated switching line
Total switching line S
T
U+U oU-
Stable equilibrium
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U+ curve
Uo curve U- curve
The approximated switching line
Total switching line ST
U+U oU-
Stable equilibrium
-2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
U+ curveUo curve
U- curve
The approximated switching line
Total switching line S
T
U+U oU-
Stable equilibrium
Figure 5.9: Illustration of the approximated switching line and the total
switching line ST.
5.6. An Optimal Look-ahead T∆ that Satisfies the
Minimum Time Criteria
Using the total switching line ST, we seek to investigate the differences in
switching results when the control switches in the vicinity of the
approximated switching line. It is seen from Figure 5.10 that while using an
optimal look-ahead T∆ of 1.2, the switching that is initiated in the region
between the approximated switching line (in Figure 5.9) and the optimal
slope (in Figure 5.8), defined as the sensitive region in this chapter, causes
control chattering. This is easily understood by observing the trajectories
122
direction in Figure 5.10 (a) that in this sensitive region, the system trajectory
is supposed to switch to the U- curve that brings it away from the ST
switching line instead of directing towards the ST switching line. This results
in the repeated switching operation around the ST switching line which
appears as control chattering (Figure 5.10 (b)). The ST switching line
provides sufficient condition for sliding and the system trajectory chatters
towards the stable equilibrium operating point. This is different from the
explicitly designed sliding mode control.
However, this chattering behaviour does not occur when an optimal look-
ahead ∆T of 1.7 is used, the system trajectory switches beyond the vertical
limit (i.e., the reduced UEP) at the total switching line ST. The system
trajectory follows the U- curve directing away from the ST switching line. A
desirable result can be obtained using ∆T=1.7 where the near minimum-time
energy reduction is achieved without any control chattering.
123
Machine angle
Mac
hine
ang
ular
Vel
ocity
Constant energy surfaces in phase portrait plane
2.2 2.4 2.6 2.8 3 3.2
-2
-1.5
-1
-0.5
0
0.5
1
Uocurve
U+curve
U- curves
Verticallimit
Post fault trajectory
Post fault trajectory
U+UoU-
)7.1( ?? TSTST )2.1( ??T
Uocurve
U+curve
U- curves
Post fault trajectory B
Post fault trajectory Aswitching before the divider line
U+UoU -
+o-
)7.1( )7.1∆T=
)2.1( )2.1
ST
STThe approximated switching line
Controller chatters
switching after the divider line
∆T=
2.2 2.4 2.6 2.8 3 3.2
-2
-1.5
-1
-0.5
0
0.5
1
Uocurve
U+curve
U- curves
Verticallimit
Post fault trajectory
Post fault trajectory
U+UoU-
)7.1( ?? TSTST )2.1( ??T
Uocurve
U+curve
U- curves
Post fault trajectory B
Post fault trajectory Aswitching before the divider line
U+UoU -
+o-
)7.1( )7.1∆T=
)2.1( )2.1
ST
STThe approximated switching line
Controller chatters
switching after the divider line
∆T=
Machine angle
Mac
hine
ang
ular
Vel
ocity
Constant energy surfaces in phase portrait plane
2.2 2.4 2.6 2.8 3 3.2
-2
-1.5
-1
-0.5
0
0.5
1
Uocurve
U+curve
U- curves
Verticallimit
Post fault trajectory
Post fault trajectory
U+UoU-
)7.1( ?? TSTST )2.1( ??T
Uocurve
U+curve
U- curves
Post fault trajectory B
Post fault trajectory Aswitching before the divider line
U+UoU -
+o-
)7.1( )7.1∆T=
)2.1( )2.1
ST
STThe approximated switching line
Controller chatters
switching after the divider line
∆T=
2.2 2.4 2.6 2.8 3 3.2
-2
-1.5
-1
-0.5
0
0.5
1
Uocurve
U+curve
U- curves
Verticallimit
Post fault trajectory
Post fault trajectory
U+UoU-
)7.1( ?? TSTST )2.1( ??T
Uocurve
U+curve
U- curves
Post fault trajectory B
Post fault trajectory Aswitching before the divider line
U+UoU -
+o-
)7.1( )7.1∆T=
)2.1( )2.1
ST
STThe approximated switching line
Controller chatters
switching after the divider line
∆T=
2.2 2.4 2.6 2.8 3 3.2
-2
-1.5
-1
-0.5
0
0.5
1
Uocurve
U+curve
U- curves
Verticallimit
Post fault trajectory
Post fault trajectory
U+UoU-
)7.1( ?? TSTST )2.1( ??T
Uocurve
U+curve
U- curves
Post fault trajectory B
Post fault trajectory Aswitching before the divider line
U+UoU -
+o-
)7.1( )7.1∆T=
)2.1( )2.1
ST
STThe approximated switching line
Controller chatters
switching after the divider line
∆T=
2.2 2.4 2.6 2.8 3 3.2
-2
-1.5
-1
-0.5
0
0.5
1
Uocurve
U+curve
U- curves
Verticallimit
Post fault trajectory
Post fault trajectory
U+UoU-
)7.1( ?? TSTST )2.1( ??T
Uocurve
U+curve
U- curves
Post fault trajectory B
Post fault trajectory Aswitching before the divider line
U+UoU -
+o-
)7.1( )7.1∆T=
)2.1( )2.1
ST
STThe approximated switching line
Controller chatters
switching after the divider line
∆T=
Figure 5.10: Delayed switching performance using ∆T=1.2 (dashed) and
∆T=1.7 (solid).
(a) A close-up view of the two different cases of delayed switching near
PS region.
(b) The delayed switching using ∆T=1.2 (dashed) experiences minor
control chattering.
-0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
Machine angle (radian)
Mac
hine
spe
ed (
rad.
/sec
.)
Minor chattering of control effort U when ∆T=1.2 is used.
(a)
(b)
124
We have shown that with the appropriate use of look-ahead T∆ , an optimal
switching line ST that satisfies the minimum time criteria is found near the
reduced UEP. It is seen from Figure 5.10 that switching based on a ∆T of
1.2 resulted in a higher energy reduction whereas the use of a larger look-
ahead of ∆T of 1.7 has a lower energy reduction effect. In spite of these
differences in energy reduction performance, their differences in settling
time are small. A time domain response is shown in Figure 5.11.
An optimal switching line ST that depends on the appropriate use of ∆T as
clarified in Figure 5.10 and satisfies the minimum time requirement without
the significantly prolonged settling time is proposed to counter the first
swing switching stability problem. The robustness of a control design has
been addressed. Verification of the energy control based on the total
switching line ST using a simple SMIB power system yields satisfactory
results which overcomes the first swing switching stability problem and
makes the energy control robust.
125
Figure 5.11: Insignificant settling time between the two examples of
switching at different instances.
5.7. Conclusion
In switching control applications, this partly stable region has not previously
been identified and this chapter addresses it hoping to maximize a switching
control effect during the most crucial phase of transient stability.
A total energy based proportional control has been outlined. The effect of a
δ& saturation control applied near a stable operating condition was found
satisfactory in earlier chapter. However, it is noted in this chapter that a δ&
saturation control applied at a critical first swing is undesirable and causes a
0
1
2
3
4
5
6 -
0.5
0
0.5
1
1.5
2
2.5
3
Time (sec.)
Mac
hine
ang
le
Time plot of machine angle
Machine angle response
due to switching with
Machine angle response
due to switching with
2 . 1 = ∆ T
7 . 1 = ∆ T
0
1
2
3
4
5
6 -
0.5
0
0.5
1
1.5
2
2.5
3
Machine angle response due to switching with
Machine angle response due to switching with
1.2= ∆ T
=∆ T 1.7
126
power system separation where machine 1 separates from machine 2. An
optimal switching line ST overcomes the detrimental first swing switching
stability problem with an optimal look-ahead duration. The optimal look-
ahead duration has prevented a δ& bang-bang control from switching inside
a partly stable region. This control design is shown to be robust with respect
to measurement errors. This optimal look-ahead approach can also be
extended to a δ& saturation control to prevent the control from switching
into linear regions thereby reducing the required control strength at first
swing.
The optimal switching using the total switching line ST with the appropriate
angle look-ahead duration ∆T is, however, not easily applicable to a multi-
machine power system. The reason is because multi-machine power system
gives rise to these multiple UEPs. Each of these multiple UEPs gives rise to
a possibility for a mode of separation. In the next chapter, an Energy
Decomposition is introduced to identify each UEP for control
implementation, with the major control strategy extended from SMIB
concepts of optimal switching and PS region.
127
5.8. References
[1] G. Ledwich, J. Fernandez-Vargas, and X. Yu, "Switching Control of Multi-machine Power Systems," IEEE / KTH Stockholm Power Tech Conference, Stockholm, Sweden, pp. 138-142, June 1995,
[2] E. Palmer and G. Ledwich, "Switching control for power systems with line lossess," IEE Proceedings- Generation, Transmission, Distributions, vol. 146, No.5, pp. 435-440, September 1999.
[3] T. W. Chan and G. Ledwich, "Multi-mode damping using single HVDC link," Aupec 2001, Perth, Australia, pp. 483-488, September 2001,
[4] M. A. Pai, Power System Stability - Analysis by the Direct Method of Lyapunov, vol. 3: North-Holland Publishing Company, 1981.
[5] G. Ledwich and E. Palmer, "Energy Function For Power Systems with Transmission Losses," IEEE Transactions on Power Systems, vol. 12, No. 2, pp. 785-790, May 1997.
128
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129
Chapter 6.
Towards Improving the Transfer Capacity
The transfer capacity of an electrical transmission system is limited by two
concerns, thermal rating and transient stability. One of the problems
associated with the control of a large power system is the need to consider
several modes of separation (referred to as cutsets in this thesis). In the
presence of weak links, several modes of separation may be at risk during
major faults. This may cause a power system to separate eventually at a
UEP instead of the Controlling UEP. In this chapter, we develop the
concepts of energy decomposition. In particular the energy term associated
with each of the mode of separation. The control strategy based on these
decomposed energy aims at any cutsets that are most likely to cause a power
system separation. In essence, the proposed control maximizes the
synchronism amongst generators and damps the subsequent oscillations.
Finally, the control proposed is demonstrated on a detailed six-machine 21-
bus system.
130
6.1. Introduction
There are increasing pressures to operate power systems closer to stability
limits. The main reasons for this arises from the increases in load demand
within urban areas that are far from large generators, restrictions on
transmission network expansion and geographical constraints that place
large generators far from load centers. This desire to maximize the use of
existing assets increases the desirability of control which can maximize the
retention of synchronism.
In a large power system, there are multiple links between generations. Faults
on different parts of the network stress these links differently such that some
links can withstand severe faults while others break more easily, and we
termed these as strong and weak links respectively. Generally, system
instability is characterized by a system separating into two groups of
generators [1] as some weak links break (or are disconnected by circuit
breakers) and result in one generator or a group of coherent generators
separating from the rest of the system. The enumeration of the generators in
a coherent group which can separate from the rest of the system
characterizes cutsets in this thesis. This is a different concept to the group of
transmission lines necessary to be broken for power system separation in the
structured preserving cutsets introduced in [2]. In a structured preserving
131
cutset [2], line indices are used and one or more structured preserving
cutsets can be described as the same separation between two groups of
generators. On the other hand, the definition of cutsets used in this chapter is
concerned with the separation of a power system between two coherent
groups of generators and it adopts generator indices. For brevity, the phrase
of “power system separation” will be used interchangeably with
“separation” and “cutset” in this chapter.
In a transient stability and energy context, the concept of energy function [1]
is widely used to predict angle instability [2-4] as well as evaluating the
critical energy of a separation. A recent extension of energy function using
remote measurements and derivative of energy function in the design of
energy based switching control [5, 6] has shown good damping
performance. In terms of avoiding a power system separation, the retention
of synchronism and damping of subsequent oscillations are considered to be
important. With the issue of the damping of subsequent oscillations being
well understood and managed, the ability of a control to provide the
successful retention of synchronism becomes the focus of this chapter. This
control must be capable of directing the correct effort to avoid a high risk
separation amongst all types of feasible separations. Hence, the
characterization of feasible cutsets (or separations) became critical in
achieving this objective. Ultimately, this control aims to provide a high
chance of survival for the power system in its normal configuration.
132
With regards to this survival issue, this chapter introduces two concepts; (i)
a total energy decomposition technique that characterizes the transient
energy of every cutset (or separation), and (ii) a control strategy based on
the weighting of separation energy combined with the proximity-to-
separation detection.
6.2. Total Energy Decomposition – Cutset Energy
This section describes a total energy decomposition technique that is used to
characterize cutsets. We illustrate the total energy decomposition using a
three-machine 6 bus power system as shown in Figure 6.1
j0.06251
3
4 5 2
0.0085+j0.072
0.1219+j0.1008
j0.0586
6 j0.0576
Figure 6.1: A three-machine 6-bus system is used to illustrate the possible
separations.
From Figure 6.1, there are six possible separations (or cutsets) associated
with the separation between two groups of generators, and it is possible to
have one or more coherent generators in each group of generators. These six
possible separations are also referred to as feasible separations as these
133
separations describe two groups of generators that are separated from each
other. These separations are categorized into set A and B,
)3/12( , )12/3( )2/13( , )13/2(
)1/23( , )32/1(
)123( ,(123)
cutsets Possible
3#3#
2#2#
1#1#
0#0#
BA
BA
BA
BA
(6.1)
where the cutsets of set A and B are identified by the subscripts (#0A, #1A,
#2A, #3A) and (#0B, #1B, #2B, #3B) respectively, and the cutset (i/jk) refers
to the separation between two groups of coherent generators; one contained
the ith generator while the other contains the jth and kth coherent generators.
From the potential energy [1] aspect, both cutsets of set A and B needs to be
considered as the nth cutset (i/jk)#nA of set A and nth cutset (jk/i)#nB of set B
having different potential energy evaluated at their corresponding UEP. The
purpose of including the cutsets (123)#0A and (123)#0B that are not a mode of
separation is for completeness so that energy decomposition can be
validated.
Considering a hypothetical situation where the generator 2 in Figure 6.1 is
connected directly to bus 5, the cutsets (2/13)#2A and (13/2)#2B are no longer
considered as feasible cutsets as they do not result in a separation forming
two groups of generators. This hypothetical example makes it easy to
understand that in a large power system, it is possible to have fewer feasible
cutsets than possible cutsets. In addition, it is also understandable from
examining the longitudinal network configuration of the power system in
134
Figure 6.1 that it is possible to have fewer feasible cutsets in a longitudinal
power system than a meshed power system.
Some mathematical assumptions are used in the Energy Decomposition of
kinetic energy. In a group of generators, for any two coherent generators
that swing in the same direction, the square of the sums of generators’
angular velocity in the form of 2)( jjii mm ωω + is used whereas for two
generators, each from a different group of generators, that oscillate against
each other, the square of the differences of generators’ angular velocity in
the form of 2)( jjii mm ωω − is used. For instance, as a separation occurs
associated with the cutset (1/23), the oscillation between generator 1 and the
coherent generators (i.e., generators 2 and 3) is described by the terms
22211 )( ωω mm − and
23311 )( ωω mm − while the oscillating behaviour
of the two coherent generators 2 and 3 is described by 2
3322 )( ωω mm + .
The reason for using the product term iim ω in the above mathematical
assumption is to enable the inertial and velocity product term of )( 2iim ω to
be formed after the decomposition process. This provides a convenient
mathematical mapping between the cutset kinetic energy and the total
kinetic energy. On the other hand, the squares of the sums or differences of
angular velocity terms 2)( jjii mm ωω ± are found to be able to represent
135
separation amongst generators fairly well, in that, after a decomposition
process the chosen signal responds strongly to separation. For instance, as
generator 1 is being separated from generator 2, the term
22211 )( ωω mm − becomes large whereas the term
22211 )( ωω mm +
remains comparatively small.
For the ease of handling the above terms, we define a new notation,
iii m ωη = (6.2)
where m and ω are the respective generator inertial constant and generator
angular velocity.
By substituting equation (6.2) into the square of the sums and the square of
the differences terms, these terms become compact. New cutset notations
are used and hereafter referred to as the p and n terms,
( ) ( )22 , jinjip ijij ηηηη −=+= (6.3)
where i and j refers to the respective ith and jth generators. The ijp refers to
the ith and jth generators that swing in the same direction whereas ijn refers to
the ith and jth generators that swing against each other. Examining the
equation (6.3), it has a symmetrical properties of ijn=jin and ijp=jip.
Expanding equation (6.1) using the square of the sums and the square of the
differences terms based on the new cutset notation η , we have
136
B
B
B
B
A
A
A
A
3# 2
212
322
31
2# 2
312
232
21
1# 2
322
132
12
0# 2
322
312
21
3# 2
212
232
13
2# 2
312
322
12
1# 2
322
312
21
0# 2
322
312
21
)(,)(,)(
)(,)(,)(
)(,)(,)(
)(,)(,)(
)(,)(,)(
)(,)(,)(
)(,)(,)(
)(,)(,)(
ηηηηηη
ηηηηηη
ηηηηηη
ηηηηηη
ηηηηηη
ηηηηηη
ηηηηηη
ηηηηηη
+−−
+−−
+−−
+++
+−−
+−−
+−−
+++
(6.4)
In order to map all cutsets in equation (6.4) to the total kinetic energy Vke
[1], we use the symmetrical properties of ijn=jin and ijp=jip, and sum all the
cutsets components in equation (6.4). The sum of all cutset kinetic energy
becomes
−+−+−+
++++++=
232
231
221
232
231
221
#)()()(4
)()()(4
ηηηηηη
ηηηηηηkeV (6.5)
where Vke# is the sum of all cutset kinetic energy and η is a cutset notation
defined earlier in association with equation (6.2).
A direct expansion and manipulation of the (ηi+ηj)2 and (ηi-ηj)2 terms in
equation (6.5) has resulted in
)21
21
21
(32
)(16
23
22
21
23
22
21#
ηηη
ηηη
++=
++=keV (6.6)
137
Replacing the η notation in equation (6.6) with the iim ω terms in
equation (6.2), the sum of all cutset kinetic energy in masses and machine
angular velocity is
)21
21
21
(32 233
222
211# ωωω mmmVke ++= (6.7)
Comparing equation (6.7) with the total kinetic energy Vke, we establish a
mapping relation between Vke# and Vke,
keke VV 32# = (6.8)
To express the total kinetic energy in terms of the p and n terms from
equation (6.3), we rewrite equation (6.4) by replacing all η terms with the p
and n terms
BpnnBpnn
BpnnBpppApnn
ApnnApnnAppp
3# 2#
1# 0# 3#
2# 1# 0#
12,23,13,13,32,12
23,31,21,23,13,12,12,32,31
13,23,21,23,13,12,23,13,12
(6.9)
Knowing that the sum of all cutset kinetic energy Vke# in equation (6.5) is
derived from the summation of all cutsets in equation (6.4), the total kinetic
energy expressed using the p and n terms is obtained by summing all cutsets
in equation (6.9).
138
++++++
+++++++
+++++++
++++++
=
BpnnBpnn
BpnnBppp
ApnnApnn
ApnnAppp
keV
3#2#
1#0#
3#2#
1#0#
)122313()133212(
)233121()231312(
)123231()132321(
)231312()231312(
321
(6.10)
where 21n=12n (i.e. ijn=jin) and 12p=21p (i.e. ijp=jip).
6.2.1. General Algebraic Expression for Cutset Kinetic Energy
By referring to equation (6.10) and the derivation that yields equation (6.8),
an algebraic expression for an n-machine system is
∑
∑
−∑+
+∑
+∑+
+∑
+∑
Ω=
=
==
−
+=
−
=
+==
µ
υυ
υυ
υυ
υυ
υυ
υυ
ω
ω
ω
ω
ω
ω
0
1
2
)()(
)()(
1
1
2
)()(
)()(
1
1
2
)()(
)()(
1
1
w
w
rn
j jrjr
isissn
i
snn
ij jrjr
irirsnn
i
sn
ij jsjs
isissn
i
ke
m
m
m
m
m
m
V (6.11)
where µ is the total number of separations, w is the counting index for the
wth cutset, sυ is the vector of indices of the generators separated from the
rest of the system, rυ is the vector of indices of the remaining generators
that do not appear in vector sυ (or machines in rest of the system), sn is the
total number of generators in vector sυ , rn is the total number of
generators in vector rυ , n is the total number of generators in the system,
139
Ω is the kinetic energy decomposition scaling coefficient, m is generator
inertial constant and ω is generator angular velocity. Equation (6.11) shows
that the total kinetic energy is expressed as the sum of the µ cutset kinetic
energy.
The total number of separation µ is derived from
∑==
−
βυµ
11*)(2
j
njF Cj (6.12)
where β refers to the total number of the types of separation including the
cutsets (ijk) that is not a mode of separation mentioned in association with
equation (6.1).
For the three-machine six-bus system in Figure 6.1, for instance, there are
two types of cutsets; the (ijk) cutset and the cutset (i/jk) or (jk/i). The total
number of the types of separations β including the (ijk) types of cutsets is
2123
12
=
+=
+= trunc
ntruncβ (6.13)
where the truncating function (.)trunc rounds a non-integer to the nearest
integer towards zero.
The notation Cnj 1− in equation (6.12) is a combinatory notation that
evaluates the total number of p or n terms,
( )))!1(()!1(!1
−−−=−
jnjnCn
j (6.14)
140
and Fυ eliminates any repetition in the evaluation of the total number of p
and n terms,
==<=≤=
= n for evenn for odd
ß for i.k
ß i fork
ß i fork
k
k?
i
i
i
F
50
1
11
β
M (6.15)
Referring to the cutsets in equation (6.1), for instance, vectors sυ and rυ
for each cutset are
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] B sA s
B sA s
B sA s
3#r3#r
2#r2#r
1#r1#r
3 ?,21?,21 ?,3?
2 ?,31?,31 ?,2?
1 ?,32?,32 ?,1?
====
====
====
(6.16)
Using the notations in equations (6.13 - 6.15), the decomposition scaling
coefficient Ω for total kinetic energy is
[ ]C
nCCCj
nj
jnjnjF
2
1
)1(1
111 )1(2***)(22
−∑
=Ω=
−−−−
βυ
(6.17)
In the next section, we will discuss the decomposition of total potential
energy.
6.2.2. Decomposition of Total Potential Energy
It is well known that the total potential energy [1] consists of energy stored
141
in transmission lines and shaft energy. The energy stored in the
interconnecting lines between two groups of generators is responsible for
their separation (or cutset), and angle difference terms such as
)cos(cos sij
uijijB θθ − is used to describe this energy. From a cutset
perspective, we define this energy as the cutset potential (line) energy. We
can observe from equation (6.9) that these angle difference terms associate
with the n terms in every cutset. It is recalled that the n terms in every cutset
describe two generators that swing against each other.
Taking the power system in Figure 6.1 as an example, we define a new
cutset notation lij for the cutset potential (line) energy,
)cos(cos sijijijl Bij θθ −= (6.18)
where ij refers to the ith and jth generators.
We sum all cutsets in equation (6.9) in the same way as we did to yield
equation (6.10), except that this time we consider only the n terms.
Replacing the n terms with the new notation in equation (6.18), we arrive at
an algebraic expression for the decomposition of the energy stored in
transmission lines,
( )w
rn
j
sjris
ujrisjris
sn
iwlinepe BV
∑ −∑∑==== 1
)()()()()()(10)(
coscos1
υυυυυυµ
θθτ (6.19)
where B is the admittance from the reduced Y admittance matrix, uθ is the
generator angle, sθ is the generator angle at a stable equilibrium point and τ
142
is the decomposition scaling coefficient for the energy stored in
transmission lines. The rest of the parameters had been defined earlier in
association with equation (6.11).
Using the notations in equations (6.13 - 6.15), the decomposition scaling
coefficient for the energy stored in lines is
( ))1(4 −Ω= nτ (6.20)
To decompose the total shaft energy, we first define a new cutset notation
iσ for the cutset potential (shaft) energy associated with two groups of
generators,
))(( si
uiiimii gP θθσ −−= (6.21)
where Pmi is the ith generator mechanical power, iig is the shunt
conductance from the reduced Y admittance matrix, uiθ is the ith generator
angle and siθ is the ith generator angle at a stable equilibrium point.
The process of decomposing the total shaft energy is similar to the process
used to decompose the total kinetic energy. We use the mathematical
assumption of the square of the sums and the square of the differences of iσ
to represent the shaft energy associated with the generators that swing in the
same direction and oscillate against each other respectively. A new set of
cutset notation is
143
( ) ( )22 , jisjis ijij σσσσ −=+= −+ (6.22)
We sum all cutsets in equation (6.9) in the same way as we did to yield
equation (6.10) considering both the p and n terms and replacing them with
the new notation in equation (6.22). As a result, we obtain an algebraic
expression for the decomposition of the total shaft energy
w
rn
jjris
sn
i
snn
ijjrir
snn
i
sn
ijjsis
sn
i
wshaftpeV
∑ −∑+
+∑ +∑+
+∑ +∑
∑=
==
−
+=
−
=
+==
=
2
1)()(
1
2
1)()(
1
2
1)()(
1
0)(
)(
)(
)(
1
υυ
υυ
υυ
µ
σσ
σσ
σσ
λ (6.23)
where iσ is described in equation (6.21) and λ is the decomposition
scaling coefficient of the cutset potential (shaft) energy. The rest of the
parameters had been defined earlier in association with equation (6.11).
Using the notations in equations (6.13-6.15), the decomposition scaling
coefficient for the total shaft energy is
2Ω=λ (6.24)
One is to note that the sum of (6.19) and (6.23) results in total potential
energy neglecting the energy loss in transmission lines.
The usefulness of cutset energy helping to identify at which cutset a power
144
system separates is demonstrated in the following case study 1.
6.3. Weighting of Cutset Energy
After elaborating the decomposition of total energy, this section describes
the concept behind the control strategy of weighting the cutset energy (or
separation energy).
In our earlier section, we have decomposed total energy into key separation
(or cutset) energy using equations (6.11), (6.19) and (6.23). In an n-machine
power system, there are various types of separations and it is common to
describe these separations using the concept of Unstable Equilibrium Points
(UEPs) [3]. Close approximation of UEPs are searched using gradient
methods such as Newton-Raphson and Davidon-Fletcher-Powell methods
[7]. These gradient methods, however, requires initial guesses which can be
derived from the list of possible cutsets in equation (6.1). The initial guesses
for the cutsets in set A are derived from )( siδπ − whereas for the cutsets in
set B, )( siδπ −− [8] is used. Such an approach of obtaining initial guesses is
found satisfactory in yielding the UEPs of a classical model three-machine 9
bus system by verifying the results in a potential energy surface [3]. An
alternative approach to obtain initial guess such as in [9] is also feasible.
The corresponding UEPs for the cutsets in set A and B are
145
( )( ) for set B
for set A
snsn
sisnnrn
sirn
snrn
sirnnsn
sisn
- ,...,,,...,
- ,...,,,...,
)()()()(
)()()()(
δδδπδπ
δδδπδπ
−−−−
−− (6.25)
Substituting the approximated UEPs into the cutset potential energy
equations in (6.19) and (6.23), the critical cutset potential energy of each
cutset is obtained. This is similar to the concept of using critical potential
energy [4] to predict system instability but in our control application, we
weigh these critical cutset potential energy terms. For the critical cutset
potential energy obtained, we weigh the ith cutset energy
( )ipeikeit VVV ### += based on the proximity of the ith cutset energy to its ith
critical cutset potential energy criipeV # using an exponential weighting,
ii expwt ε−=# (6.26)
where iε is the proximity to critical cutset energy coefficient,
( ) criipeit
criipei VVV ### −=ε
for an ith cutset and iwt # is the weighting of an ith cutset energy itV # . ikeV #
and ipeV # are the respective ith cutset kinetic energy and potential energy.
146
6.4. Detection of Proximity to Angle Separation Based
on the Boundary of Partly Stable Regions
In this section, we will use UEPs to forewarn and avoid control switching in
partly stable (PS) regions [10]. Its benefit is an extended stability limit as the
available control resources are maximized. In switching control, it is clear
that a PS region [10] exists for every cutset and bang-bang switching inside
such region causes generator angles to hover near a UEP owing to the result
of incorrect switching. These control responses are undesirable and may
result in instability. The use of switching control based on saturation
function also gives rise to undesirable results. In particular, when the control
values [10] reduces during the first swing this can increase the risk of
instability. Although the result of switching between U=-1 and U=1 has
created a PS region, it has also defined an upper and a lower boundary for
the region [10]. These boundaries were conveniently found at the extended
UEP (uepex) and reduced UEP (uepr) respectively. Comparing this reduced
UEP with the system trajectory, it will indicate to us whether the system
trajectory is close to the PS region.
A reduced UEP (uepr) boundary of a PS region in angle space is evaluated
using the norm of the vector. The closeness to angle separation index
iuepprox # becomes
147
||||||||
][][#
#
##
ruepiruep
i
ruepi
sys
iuepprox θθ
θθ−
⋅= (6.27)
where ][ sysθ is the vector of system trajectory in angle space, ][ #ruep
iθ is the
vector of ith cutset reduced uepr in angle space and |||| #ruep
iθ is the norm of
the vector of ith cutset reduced UEP (uepr) in angle space. As this proximity
indicator iuepprox # approaches zero towards positive values, it means that
the system trajectory in angle space is close to an ith cutset’s PS region.
6.5. A Cutset Based Energy Control that Enhances the
Survival of a Power System
This section elaborates the procedure of designing a cutset energy based
control that aims at high risk separations and yield correctly directed control
efforts during critical instances.
6.5.1. Derivative of Cutset Kinetic Energy
The benefit of energy based control design [5, 6, 10] using remote
measurements in switching control is obvious. It reduces kinetic energy
effectively and gives good performance in damping the subsequent
148
oscillations. Its control law ensures the most negative derivative of kinetic
energy. A cutset energy based control uses remote measurements and it
extends the benefit of the energy based control.
A wth cutset kinetic energy derivative obtained from equation (6.11) is in
the form of
( )wijijiiwwke gbggufV ∆∆∆−= ,,,,*),(# θθθθ &&& (6.28)
where ijii bg ∆∆ , and ijg∆ are the changes in shunt conductance, transfer
admittance and transfer conductance of the reduced Y admittance matrix, w
is the counting index for the wth cutset and uw is the control value associated
with the wth cutset.
Based on the controllable terms in equation (6.28), the switching surface
#VS of the cutset kinetic energy derivative is
( )∑ ∆∆∆==
µθθ
1# ,,,,
w wijijiiV gbggS & (6.29)
where µ has been defined earlier in association with equation (6.11) and all
parameters has been defined in association with equation (6.28).
6.5.2. Cutset Angle Look-ahead Dependant Terms
Angle look-ahead dependant terms [10] are found to be effective in
switching control where an energy-based velocity control is corrected to
149
avoid switching in a PS region. The result of finding a partly stable region
(PS) in [10] is based on a Single-Machine-Infinite-Bus (SMIB) system. This
simple SMIB system implied that every cutset had a PS region and
switching inside it must be avoided.
This section proposed a cutset angle look-ahead term to forewarn the
closeness of a system trajectory to the PS region of every cutset using the
result of total energy decomposition. The angle look-ahead dependant terms
based on a SMIB system in [10] are repeated here
∑ ∑
∆+∆−∆=
=
=
=
≠=
2
1
2
1
2 )sin)((n
i
n
ij i
ijijiisii
P m
bgTS
θθθη (6.30)
where SP is a position based switching surface, η is a scaling factor, 2T∆ is
a look-ahead duration, θi is the ith generator angle, θiS is the ith generator
angle at a stable equilibrium point and mi is the ith generator inertial
constant. In equation (6.30), ∆gii and ∆bij are the respective changes in shunt
conductance and admittance of the reduced Y admittance matrix. It should
be noted that the term )sin( ijijii bg θ∆+∆ in equation (6.30) is the change
in the ith generator electrical power eiP∆ .
A general expression of cutset based angle look-ahead term extended from
equation (6.30) for a power system containing multiple cutsets is
150
( )
w
w
sn
i
rn
jjr
jr
is
is
sjrisjrisjris
wP
m
Pe
m
Pe
B
TS ∑ ∑ ∑
∆−
∆
−
∆== = =
µ
υ
υ
υ
υ
υυυυυυ θθ
0 1 1)(
)(
)(
)(
)()()()()()(2
# (6.31)
where w counts the wth cutset.
In equation (6.31), wT∆ is the wth cutset look-ahead duration and this is set
according to the wth closeness-to-angle-separation index wuepprox # as
discussed in equation (6.27). This minimizes control chattering inside a PS
region when 0≠θ& [10]
<
>=∆
0 1
0
#
#
wuep
wuepw proxfor
proxforvaluehighT (6.32)
It is important to note that the difference between equations (6.30) and
(6.31) is that equation (6.30) is to be used in a SMIB system since it has
only one mode of separation. However, equation (6.31) can be used in both
SMIB and power system with multiple cutsets.
6.5.3. A Cutset Energy Based Control
Applying the cutset energy weighting in equation (6.26) to both the
derivative of cutset kinetic energy in equation (6.29) and the cutset-based
angle look-ahead in equation (6.31), we have a cutset energy-based
switching surface,
151
)( #### PV SSwtS += (6.33)
In the next sections, two case studies will be used to elaborate the design of
cutset energy-based control.
152
6.6. Case Study 1 (Classical Three-machine 9-bus
Power System)
The difficulty in determining the Controlling UEP that is responsible for a
power system separation when power system is subject to severe faults that
could occur anywhere in the transmission network will be illustrated in this
section. In this case study, a three-machine 9-bus power system is used to
illustrate the benefit of using Energy Decomposed to predict system
separations and design the control of SVC devices. As total energy is
decomposed to emphasize the dynamical energy between any two
separating groups of generators (or simply referred to in this thesis as
cutsets), transient energy of all feasible cutsets were evaluated to provide an
overall prediction of system instability due to angle separation.
The three-machine 9-bus system as shown in Figure 6.2 uses classical
machine models and the system data is obtained from [11, Page 38-39]. The
controllable SVC installed at bus 9 is represented by the change in shunt
admittance at bus 8 and is capable of modulating the network power flows.
Generally, the response time of a SVC is less than 30ms and for a SVC light
apparatus it is less than 3ms. In view of this relatively fast response time of
SVC compared to the oscillations frequency in the range of 1.25 to 2.2 Hz,
the SVC dynamics are excluded from the control algorithm. All loads are
represented as constant impedances to emphasize the analysis of generators
153
interactions.
Pm1 =0.71 H 1 =0.125
3 j0.0625
1.25+j0.5
2
1.04 ∠ 0 °
4
5 6
7 8 9
0.03
2+j0
.161
j0.0576
0.01+j0.085 0.017+j0.092
0.03
9+j0
.17
0.0085+j0.072 0.0119+j0.1008 j0.0586
0.9+j0.3
1.0+j0 .35
1.025 ∠ 9.4 ° 1.025 ∠ 4.7 °
∠ - 2.2 °
∠ - 4.0 ° ∠ - 3.7 °
∠ 3.8 ° ∠ 0.7 ° ∠ 2.0 °
Pm2 =1.63 H 2 =0.03
Pm3 =0.85 H 3 =0.01
1
SVC
Figure 6.2: A three-machine 9-bus system with a controllable SVC installed
at bus 5.
6.6.1. Total Kinetic Energy Reduction Control
The set of swing equation for the three-machine 9-bus system in Figure 6.2
is
coaeem
coaeem
coaeem
PPUPPm
PPUPPm
PPUPPm
−∆−−=
−∆−−=
−∆−−=
33333
22222
11111
δ
δ
δ
&&
&&
&&
(6.34)
where
• mi is the ith generator’s inertial constant,
154
• iδ&& is the ith generator’s angular acceleration,
• Pmi is the ith generator’s mechanical power output,
• Pei is the ith generator’s electrical power,
• U is the control of the shunt admittance at bus 9,
• ∆Pei is the change in ith generator’s electrical power due to the change in
shunt admittance in the network and
• Pcoa [12] is the centre of area power added to each generator creating the
perfect governor terms to emphasize the study of electromechanical
oscillation.
The ith generator’s electrical power Pe1 and change in power ∆ Pe1 in (6.34)
are
( )
( )∑ ∑ −∆+−∆+∆=∆
∑ ∑ −+−+=
=
=
=
+=
=
=
=
+=
3
1
3
1
2
3
1
3
1
2
)cos()sin(
)cos()sin(
n
i
n
ijjiijjijiijjiiiiei
n
i
n
ijjiijjijiijjiiiiei
gvvbvvgvP
gvvbvvgvP
δδδδ
δδδδ
where
• vi is the ith generator bus voltage,
• bij is the line admittance between the ith and jth generator bus,
• gij is the line conductance between the ith and jth generator bus,
• gii is the shunt conductance at ith generator bus and
• iδ is the ith generator’s angle.
The admittances and conductance are based a reduced Y admittance of the
network eliminating all buses except the generator buses. The ∆ symbol
indicates the corresponding change in the admittances and conductance as
155
the shunt admittance at bus 9 changes under the control of U from U=0 to
U=±1.
The total kinetic energy of the power system is
233
222
211 2
121
21
ωωω mmmV ++= (6.35)
where ωi is the ith generator’s angular velocity and all parameters has been
defined earlier in association with (6.34).
Taking the derivative of the total kinetic energy V in (6.35) and substituting
the swing equation in (6.34), the derivative of total kinetic energy V& is
( )332211
333222111
),( eee PPPUf
mmmV
∆+∆+∆+=
++=
δδδδδ
δδδδδδ&&&&
&&&&&&&&&& (6.36)
A δ& bang-bang control law based on the total kinetic energy reduction [13]
has the form of
)( Ven SsgnU = and
−<−=>
=1 1
0 0
1 1
V
V
V
en
Sfor
Sfor
Sfor
U (6.37)
where ( ) ( )∑
∑ ∑ ∆+∆+∆=
=≠
=≠
=
n
ii
n
ij
n
ij
ijijijijiiV gbgS1 1 1
cossin δδδ & .
The switching surface VS is derived from the controllable terms (or
function of U) found in the derivative of total kinetic energy.
156
Before the cutset energy based control law is derived, it is necessary to
illustrate on how the feasible cutsets of the power system are determined.
This can reduce the need to consider the large amount of cutsets when a
large power system such as a 100 machines system was being studied.
6.6.2. Determining Feasible Cutsets and Its Corresponding UEPs
Examining the network structure of Figure 6.2, it is apparent that there are
six possible separations (or cutsets) and each cutset is associated with the
separation between two groups of generators,
Possible cutsets
BBBB
AAAA
3#2#1#0#
3#2#1#0#
)3/12(,)2/13(,)1/23(,)123(
)12/3(,)13/2(,)23/1(,)123( (6.38)
where (i/jk) refers to the separation between the ith machine group and the jth
and kth machine group, and (jjk) refers to no separation between the three
generators.
In (6.38), the difference between a cutset (i/jk) and cutset (jk/i) is their
different potential energy evaluated at their corresponding UEP. The UEP
for the cutset (i/jk) and (jk/i) are found using a gradient search algorithm
such as the Davidon-Fletcher-Powell (DFP) method [7], which requires
initial guesses. These initial guesses for cutsets (i/jk) and (jk/i) are derived
from ( )iδπ − and ( )iδπ −− respectively. The purpose of including the
157
cutsets (123)#0A and (123)#0B, which are not a mode of separation in the
decomposition, is for the completeness of the decomposition so that energy
decomposition can be validated. The number of feasible cutsets in a power
system is determined by examining the physical structure of the power
system, for example, the power system in Figure 6.2 has six feasible cutsets
Feasible cutsets
BBB
AAA
3#2#1#
3#2#1#
)3/12(,)2/13(,)1/23(
)12/3(,)13/2(,)23/1( (6.39)
One important aspect in determining the number of feasible cutsets amongst
the set of possible cutsets in the power system is to analyze the number of
ways the power system breaks into two groups of coherent generators. For
example, if the lines between bus 8 and 5 of the power system in Figure 6.2
are removed, this longitudinal power system will give four feasible cutsets.
This is because both cutset (3/12) and (12/3) does not result in two separate
groups of coherent generators. From this example, it is apparent that a
longitudinal power system often results in fewer feasible cutsets than a
meshed power system.
Referring to (6.39), given that there are six feasible cutsets, six unstable
operating points are found using the DFP method and are shown in Table
6.1.
158
δ 21 δ 31 δ 23
cutset(1/23) -3.5719 -3.9126 0.3407 UEP 1
cutset(2/13) 3.2165 1.3582 1.8583 ULM 1
cutset(3/12) 1.2787 3.5598 -2.2811 ULM 2
cutset(23/1) 2.7113 2.3706 0.3407 UEP 2
cutset(13/2) -3.0667 1.3582 -4.4249 ULM 3
cutset(12/3) 1.2787 -2.7234 4.0021 ULM 4
Table 6.1: The relationship between the unstable operating points and
cutsets.
In Table 6.1, UEP refers to the Unstable Equilibrium Point [14] which is an
unstable operating point in the operation of a power system. The remaining
unstable operating points are referred to as unstable local minima (i.e., ULM
refers to Unstable Local minimum) in this thesis as these points are not
equilibrium points since they can not be searched by the Newton-Ralphson
(NR) search algorithm [7]. Generally, a NR algorithm is able to converge to
an unstable operating point starting from an initial nearby point (or guess
point) only if the unstable operating point is an equilibrium point. This
convergence to the equilibrium point occurs only when a solution exists,
which means the power mismatch at the equilibrium point is approximately
zero or close to zero depending on the desire solution tolerance. When a NR
method can only converge to these ULM, it implies that these unstable local
minima are not equilibrium point. This diverging problem in Newton-
Ralpson method is encountered in [15].
159
The characteristics of UEP and ULM will be elaborated further in the later
sections.
6.6.3. Determining the Cutset Energy Equation
In this section, the construction of cutset energy equation for the three-
machine 9-bus system in Figure 6.2 is explained. Examining (6.38), we
determines the vectors sυ and rυ of each cutset as follows:
[ ] [ ][ ] [ ][ ] [ ][ ] [ ][ ] [ ][ ] [ ]
( )( )( )( )( )( )
3B
2B
1
3
2A
1
r
r
r
r
r
r
23,13,1232,12,1331,21,23
12,23,1313,23,1223,13,12
3 ? 21?32 ? 31?2
1 ? 32?121? 3?3
31? 2?232 ? 1?1
cutsetnnpcutsetnnp
BcutsetnnpAcutsetpnn
cutsetpnnAcutsetpnn
s
s
s
s
s
s
B: cutset B: cutset
B: cutset A: cutset
A: cutset A: cutset
⇔
====
====
====
(6.40)
where sυ , rυ has been defined earlier in association with equation (6.11).
Both the n and p terms has been defined earlier in association with equation
(6.3) and are repeated here in (6.41) for easy reference.
( )( )2
2
jjiin
jjiip
mmij
mmij
ωω
ωω
−=
+= (6.41)
The cutset kinetic energy #keV associated with all of the feasible cutsets in
(6.39) can be evaluated from the general algebraic expression in equation
(6.11). The total kinetic energy decomposition scaling coefficient Ω for this
power system is evaluated from (6.17) to be 32=Ω .
160
The cutset potential (line) energy )(
#line
ipeV associated with the energy stored in
the interconnecting lines between two groups of machines in all of the
feasible cutsets in (6.39) can be evaluated from the general algebraic
expression in equation (6.19). The total potential energy decomposition
scaling coefficient τ for the total energy stored in the transmission lines of
this power system is evaluated from (6.20) to be 4=τ .
The cutset potential (shaft) energy )(
#shaft
ipeV associated with the generators
mechanical power in all of the feasible cutsets in (6.39) can be evaluated
from equation (6.23). The total potential decomposition coefficient λ for
the total shaft energy of this power system is evaluated from (6.24) to be
16=λ .
In the next section, we will explain on how the proximity-to-separation
prediction is being derived.
6.6.4. Determining the Proximity-to-Separation Prediction
The proximity-to-separation prediction involves two processes. One is to
predict the closeness of an ith cutset energy to its ith cutset critical energy
evaluated at its corresponding UEP, and this process is referred to as the
161
proximity-to-critical cutset energy prediction. This process determines
creditable separations which is similar to the familiar critical energy
approach [16, 17]. The other process predicted how close is a system
trajectory to the boundary of partly stable (PS) regions [10] in angle space
and is referred to as the proximity-to-cutset PS region prediction.
With regards to the proximity-to-critical cutset energy prediction, an ith
cutset energy is evaluated from
)(#
)(###
shaftipe
lineipeikei VVVV −−= (6.42)
where ikeV # , )(
#line
ipeV and )(
#shaft
ipeV has been defined in the earlier section
(3.1.3) as the ith cutset’s kinetic and potential energy. The summation of
)(#
lineipeV and
)(#
shaftipeV forms the ith cutset potential energy.
An ith cutset’s critical energy criipeV # is evaluated from substituting its
associated UEP in the form of ( )233121 ,, δδδ in Table 6.1 into its cutset
potential energy )(
#)(
#shaft
ipeline
ipe VV + , defined earlier in associated with (6.42).
The general algebraic expression of an wth critical cutset potential energy is
162
∑ ∑
−+
+
−
+
+∑ ∑
−+
+
−
−+
+
∑
−
−∑
=
−
=
−
+=
= +=
==
w
snn
i
snn
ij sjr
wuepjrjrm
sir
wuepirirm
sn
i
sn
ij sjs
wuepjsjsm
sis
wuepisism
w
rn
j sjris
wuepjris
jrissn
i
criwpe
P
P
P
P
n
B
V
1 1)()()(
)()()(
1 1)()()(
)()()(
1)()(
)()()()(
1
#
)1(1
cos
cos1
υυυ
υυυ
υυυ
υυυ
υυ
υυυυ
θθ
θθ
θθ
θθ
λ
θ
θ
τ
(6.43)
where the wuepjris )()( υυθ , wuep
is )(υθ and wuepir )(υθ refers to the generator’s
operating angle at the UEP that corresponds to the wth cutset. For instance,
the UEP for cutset (1/23) is described by the unstable operating angles of
( )204.0,5738.3,3335.3 233121 =−=−= δδδ . The symbol w is the counting
index for the wth cutset and the remaining notations has been defined earlier
in association with equations (6.19, 6.21 and 6.23).
The relative angle in COA frame ijθ equals to the relative angle in inertial
frame ijδ . This is shown in
ijji
ji
jiij
ii
coacoa
coacoacoa
coacoa
δδδ
δδδδ
θθθ
δδθ
=−=
+−−=
−=
−=
)()(
)()()(
),()(
(6.44)
Substituting (6.43) into the proximity-to-critical cutset energy coefficient
defined in association with (6.26), the exponential weightings as defined in
163
(6.26) for each of the feasible cutsets in (6.39) are evaluated continuously
during the transient period. It must be noted that criipeV # is evaluated prior to
the simulation whereas iV# is evaluated continuously during the transient
period.
The proximity-to-cutset PS region prediction differentiates a high-risk
separation from its neighboring low risk separations in angle space. It is
described in the earlier section (6.4) that the reduced UEP is used as the
boundary of a PS region. This three-machine 9-bus power system in Figure
6.2 has the following reduced unstable operating points as shown in Table
6.2.
δ 21 δ 31 δ 23
Reduced unstable
operating points
cutset(1/23) -3.8062 -3.9795 0.1733 UEP1
cutset(2/13) 2.4773 2.3039 0.1734 ULM1
cutset(3/12) 1.3428 3.4655 -2.1227 ULM2
cutset(23/1) 2.477 2.3037 0.1733 UEP2
cutset(13/2) -3.8059 2.3039 -6.1098 ULM3
cutset(12/3) 1.3428 -2.8177 4.1605 ULM4
Table 6.2: The reduced unstable operating points of the power system in
Figure 6.2.
The closeness to angle separation index defined in (6.27) is repeated in
(6.45) for easy reference.
164
||||||||
][][#
#
##
ruepiruep
i
ruepi
sys
iuepprox θθ
θθ−
⋅= (6.45)
The notations in (6.45) are defined for this power system as
i) 2
233121#
++=
riuepr
iuepriuepruep
i δδδθ is the vector magnitude of an ith
cutset’s reduced UEP in angle space. The symbol riuep
jkδ is the angle
difference between a jth and kth machine associated with an ith cutset’s
reduced UEP obtainable from Table 6.2. For instance, the vector magnitude
of cutset (1/23)’s reduced UEP in angle space is
( )2# 1733.09795.38062.3 +−−=
ruepiθ .
ii) ][ sysθ is the vector of system trajectory in angle space
[ ]Tsystemsystemsystem233121 δδδ and ][ #
ruepiθ is the vector of the ith cutset’s
reduced UEP obtained from Table 6.2
riuepr
iuepriuep
233121 δδδ .
Although ][ #
ruepiθ is evaluated prior to the simulation, equation (6.45) is
continuously evaluated when ][ sysθ changes continuously during the
transient period.
165
6.6.5. Determining the Cutset Energy Reduction Control
In this section, the use of cutset energy based control consisting of the
derivative of cutset kinetic energy and the cutset based angle look-ahead
control as described in earlier sections is elaborated.
Expanding (6.11), wth cutset kinetic energy for the power system is
w
n
i
n
ijjiji
n
iiiwke mmmV
∑ ∑ ±+∑==
=
=
+=
=
=
3
1
3
1
3
1
2# 2*2
321
ωωω (6.46)
where the ± sign of the second term will follow the sign in the p and n
terms in (6.3) such that a ( )p12 will yield a positive second term (i.e.
+ 21212 ωωmm ) whereas a ( )n12 will result in a negative second term (i.e. -
21212 ωωmm ).
Taking the derivative of the wth cutset kinetic energy and considering only
the controllable terms (or function of U) found in the set of swing equations
in (6.34), the general algebraic equation of the derivative of cutset kinetic
energy is
w
n
iei
i
jej
j
in
ijji
n
ieiiwke P
mP
mmmPuV
∑
∆+∆∑±
∑ ∆=
=
=
=
+=
=
=
3
1
3
1
3
1# 24
321
)(δδ
δ&&
&&
(6.47)
where eiP∆ has been defined earlier in association with (6.34).
In order to maximize the reduction of cutset kinetic energy, a suitable
166
switching surface #VS that targets control at cutset kinetic energy is
∑==
=
6
1## )(
µ
wwkeV uVS & (6.48)
Considering the cutset-based dependent terms #VS defined in (6.48) and
#PS defined in (6.31) together with the energy-based dependent term VS
defined in (6.37), the complete cutset energy-based control law is
)( ## SSsatU V += and
−<−=>
=1 1
0 0
1 1
#
#
#
#
Sfor
Sfor
Sfor
U (6.49)
where )( #### PV SSwtS += is the cutset-based switching surface defined
in (6.33) and sat(.) is the saturation function proposed in [13].
The next section illustrates the advantage of using cutset energy to predict a
particular separation by comparing the information obtained from total
energy and cutset energy.
6.6.6. Results
Before elaborating on the benefit of using the decomposed energy (i.e.,
cutset energy), the issue of critical energy at an unstable local minimum
(ULM) and the uncertainty in the type of power separations in a multiple
UEPs operating condition are discussed.
167
6.6.6.1. Energy Evaluated at an Unstable Local Minimum
(ULM)
When severe faults occur at the bus 7 of the three-machine nine-bus system
in Figure 6.2, the critical clearing time (tcr) under no SVC control is 232ms.
As the faults are cleared beyond tcr, the system separated into two groups of
generators; generator 2 and 3 forms a coherent group and is separated from
generator 1. This is shown in Figure 6.3.
Figure 6.3: Unstable system trajectory on a total potential energy surface.
The fault at bus 7 is cleared at 233ms. Under no SVC control, power system
separates at the UEP 2 associated with the cutset (23/1).
In Figure 6.3, UEP refers to Unstable Equilibrium Point and ULM refers to
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
-1.7363-0.770191.162
0.19592
1.162
1.162
1.162
-2.7024
3.0943
2.1282
2.1282
2.1282
0.19592
-1.7363
1.162
-0.77019
3.0943
3.0943
0.19592
2.1282
3.0943
1.162
4.0604
0.19592
4.0604
4.0604
2.1282
5.0265
1.162
4.0604
5.0265
Total potential energy surface
δ 21
5.9926
5.9926
7.9249
3.0943
2.1282
5.02655.9926
8.891
4.0604
6.9588
6.9588
5.0265
9.8571
7.92498.891
6.9588
9.8571
9.8571
6.9588
7.9249
δ31
UEP 2
ULM 1
ULM 2
UEP 1 ULM 3
ULM 4
SEP
ULM of cutset (2/13) UEP of cutset (1/23) ULM of cutset (3/12) Initial guess points System trajectory w/o SVC control
168
Unstable Local Minima defined earlier in association with Table 6.1. The
difference between a UEP and ULM is that a system trajectory in angle
space has approximately zero acceleration at UEP and non-zero acceleration
at a ULM. These are clarified in earlier section that the failure of Newton-
Raphson method to converge on a ULM has indicated non-zero acceleration
at a ULM.
The power system separation at the UEP 2 associated with the cutset (23/1)
is observed in Figure 6.4 in the form of relative angles. In Figure 6.4, the
relative angle between generator 1 and 2 ( 21δ ) increases while the relative
angle between generator 2 and 3 ( 23δ ) remains relatively small as generator
1 separates from the rest of the power system.
Figure 6.4: Relative angles show the power system separation between
generator 1 and the rest of the system. The fault at bus 7 was cleared at
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-2
-1
0
1
2
3
4
5δ 21δ 32
Mac
hine
ang
le d
iffer
ence
s (r
ad.)
Time (sec.)
169
233ms.
Observing the power system in Figure 6.2, the fault at bus 7 is most likely to
cause generator 2 to separate from the rest of the system. It is seen in the
total potential energy surface that the system trajectory is driven towards the
ULM 1 during the fault duration intending to cause a power system at the
cutset (2/13). However, a counter intuitive separation occurs at cutset (23/1)
instead.
Using the concept of the Controlling UEP in the direction of a fault
trajectory [17] and the critical energy at a UEP [18], we seek to understand
the characteristic of Unstable Local Minima (ULMs).
When the fault at bus 7 is cleared at a long fault clearing time of 300ms, it is
intended to cause the post-fault energy to exceed the energy evaluated at a
ULM 1 and is supposed to cause a separation at cutset (2/13), since a
Controlling UEP in the direction of the fault trajectory should result in a
separation pattern associated with the Controlling UEP. However, it turns
out that the post-fault total energy exceeds both the energy evaluated at UEP
2 and ULM 1, and generator 1 has separated from the rest of the power
system (i.e. cutset (23/1)). This implied that the ULM 1 (i.e. cutset (2/13)) is
not the Controlling UEP and the energy evaluated at ULM 1 is not a critical
energy. It appears that UEP 2 is the Controlling UEP. The result of
generator angles is shown in Figure 6.5.
170
Figure 6.5: Generator angles are separating when the fault at bus 7 is cleared
at 300ms. Cutset (1/23) or (23/1) is confirmed as the only separation
possible for the power system.
Similarly, when the fault at bus 9 is cleared at 300ms which is intended to
cause the energy evaluated at ULM 2 to be exceeded and a separation at the
cutset (3/12), it turns out that the post-fault energy exceeds both the energy
evaluated at UEP 2 and ULM 2, and generator 1 has separated from the rest
of the power system (i.e. cutset (23/1)). The unstable system trajectory in
the potential energy surface is shown in Figure 6.6.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4
-2
0
2
4
6
8
10 δ δ δ
123
Mac
hine
ang
les
(rad
.)
Time (sec.)
171
Figure 6.6: Unstable system trajectory on a total potential energy surface.
The Fault at bus 9 is cleared at 300ms. Under no SVC control, power
system separates at the UEP 2 associated with the cutset (23/1).
The above exercises suggest that the power system in Figure 6.2 appears to
have only one type of power system separation which is associated with
UEP 2 or UEP 1. It is also suggested that ULM is not a feasible cutset
because a power system would have already been separated at a UEP before
the system trajectory reaches close to a ULM.
In the next section, the uncertainty of separation in a multiple UEP
environment is discussed.
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
-1.7363-0.770191.162
0.19592
1.1621.162
1.162
-2.7024
3.0943
2.1282
2.1282
2.1282
0.19592
-1.7363
1.162
-0.77019
3.0943
3.0943
0.19592
2.1282
3.0943
1.162
4.0604
0.19592
4.0604
4.0604
2.1282
5.0265
1.162
4.0604
5.0265
Total potential energy surface
δ21
5.9926
5.9926
7.9249
3.0943
2.1282
5.0265
5.9926
8.891
4.0604
6.9588
6.9588
5.0265
9.8571
7.92498.891
6.9588
9.8571
9.8571
6.9588
7.9249
δ31 UEP 2
ULM 1
ULM 2
UEP 1 ULM 3
ULM 4
SEP
172
6.6.6.2. The Uncertainty of Power System Separations in a
Multiple UEPs Operating Condition
In this section, the system data of the three-machine system in Figure 6.2 is
modified to yield six UEPs by reducing the loading on the transmission
lines between bus 5 & 7 and bus 6 & 9. The generators’ mechanical power
(Pmi), generators’ bus voltages (vi) and generators’ bus angle are shown in
Table 6.3.
Pm1= 1.34 Pm2= 1.20 Pm3= 0.62
v1= 1.0∠0° v2= 1.025∠0.9° v3= 1.025∠2.2°
Table 6.3: System data of Figure 6.2 are modified to reduce the loading of
lines between bus 5 & 7 and bus 6 & 9. This modified loading yields six
UEPs.
These six UEPs are shown in Table 6.4.
δ 21 δ 31 δ 23
cutset(1/23) -3.3253 -3.5602 0.2349 UEP 1
cutset(2/13) 3.3681 1.145 2.2231 UEP 2
cutset(3/12) 0.8111 3.4106 -2.5995 UEP 3
cutset(23/1) 2.9579 2.723 0.2349 UEP 4
cutset(13/2) -2.9151 1.145 -4.0601 UEP 5
cutset(12/3) 0.8111 -2.8726 3.6837 UEP 6
Table 6.4: The relationship between the Unstable Equilibrium Points
(UEPs) and cutsets.
173
Before we illustrates the purpose of this section (i.e. the uncertainty in the
types of separations in a multiple UEP environment), we examine the
concept of Controlling UEP in the direction of fault trajectory using this
configuration of power dispatch.
When the fault at bus 7 is cleared at 400ms with the intention to cause the
separation at cutset (2/13), the concept of Controlling UEP in the direction
of fault trajectory concept is verified. The result as shown in Figure 6.7
agrees with the above mentioned concept and the power system has
separated at cutset (2/13).
Figure 6.7: Unstable system trajectory on a total potential energy surface.
The Fault at bus 7 is cleared at 400ms. Under no SVC control, power
system separates at the UEP 2 associated with the cutset (23/1).
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
1.41790.815150.21239
2.0207
2.6234
2.0207
1.41792.0207
0.81515
0.21239
2.0207
1.4179
0.81515
2.6234
0.81515
2.6234
1.4179
2.6234
1.41792.0207
2.6234 2.0207
3.2262
3.2262
3.2262
0.21239
2.6234
1.4179
0.81515
2.62343.829
3.829
3.829
3.829
2.0207
3.2262
4.4317
2.6234
3.2262
3.8293.2262
2.6234
4.4317
3.2262
4.4317
3.2262
4.4317
4.4317
3.2262
5.0345
5.0345
5.6372
5.6372
4.4317
6.24
3.829
5.0345
5.6372
7.4455
4.4317
6.8428
5.0345
6.24
6.24
4.4317
5.0345
5.6372
6.8428
7.4455
7.4455
6.8428
6.24
6.24
6.24
6.24
UEP 3
Total potential energy surface
δ31
δ 21
UEP 2 UEP 4 SEP
UEP 6 UEP 1
UEP 5
174
When the fault at bus 9 is cleared at 400ms, the power system separates at
the UEP3 which agrees with the concept of Controlling UEP in the direction
of fault trajectory. This example of simulating the fault at bus 9 and has
suggested that all UEP are feasible separations.
The following part of this section illustrates that in a multiple UEP
environment, the concept of Controlling UEP in the direction of a fault
trajectory appears flawed. Prior to performing a short-circuit test on bus 9, it
is obvious that the fault at bus 9 is most likely to cause generator 3 to
separate from the rest of the system. When the fault at bus 9 is cleared at
463ms, the system trajectory is being driven towards UEP 2 but results in a
separation associated with cutset (3/12). The result of an angle separation at
cutset (3/12) due to a long fault clearing time of 463ms is shown in Figure
6.8 which agrees with the concept of the Controlling UEP in the direction of
fault trajectory [17].
175
Figure 6.8: Separation of generator 3 from the rest of the system associates
with cutset (3/12). The Fault at bus 9 is cleared at 463ms.
However, when the fault at bus 9 is cleared at a critical fault clearing time of
381ms, the power system does not separate at cutset (3/12) instead the
power system separates at cutset (2/13). The result is shown in Figure 6.9.
This uncertainty in the mode of separation is referred to as the shifting of
modes in this thesis.
In Figure 6.9, it is apparent that a control that aims at a Controlling UEP in
the direction of a simulated fault trajectory by considering its critical energy
is undesirable. One desirable way of overcoming this shifting of modes is to
consider all UEPs in the development of the control strategy. However, this
suggestion has its limitation when total energy is used as the design
framework. It is understandable when we look at the total energy diagram in
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
-2
0
2
4
6
8
Time (sec.)
Mac
hine
ang
le (
rad.
)
δ1 δ2 δ3
176
Figure 6.10. (Note: Figure 6.10 does not include the energy dissipated in the
network transfer conductance.)
Figure 6.9: System trajectory on potential energy surface. The fault at bus 9
is cleared at 381ms and an unexpected separation associated with cutset
(2/13) occurs.
The total energy diagram in Figure 6.10 offers no specific information
pertaining to how the power system separates. It becomes difficult when one
needs to design a control that aims at a separation. This control design
objective becomes exacerbated given a multi-machine system in which
several possible separations can be encountered. The next section illustrates
how cutset energy could alleviate these difficulties.
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
1.4179 0.81515
0.21239
2.0207
2.6234
1.4179
2.0207 1.4179
2.0207
0.81515
0.81515
0.21239
2.0207
2.6234
2.6234
1.4179
0.81515 1.4179 2.0207
2.6234 2.6234
3.829
2.0207
3.2262
3.2262
3.2262 2.6234
0.81515
2.6234
3.829
3.829
4.4317
1.4179 2.0207
3.2262
2.6234
3.829
3.2262
2.6234 3.2262
3.8294.4317
3.2262
4.4317
3.2262
4.4317
4.4317
4.4317
3.2262 3.829
5.0345
5.0345 6.24
5.0345
5.6372
6.8428
4.4317
5.6372
7.4455
6.24
5.0345
6.24
5.6372
6.8428
4.4317 5.0345
5.6372
6.8428
7.4455
6.24
7.4455
6.24
6.24
6.24
UEP 3
UEP 2
UEP 6
UEP 1
UEP 5
SEP UEP 4
δ21
δ31
177
Figure 6.10: The total energy diagram shows the difficulty of determining
when the power system has separated. The fault at bus 9 is cleared at
381ms.
6.6.6.3. The Ease of Using Cutset Energy to Predict a Potential
Separations
The ability of cutset energy to predict which type of separation that is most
likely to occur is an advantage that should be exploited. Referring to the
case associated with Figure 6.7, the cutset energy of the six types of
separations are shown in Figure 6.11.
0.2
0.4
0.6
0.8
1 1.2
1.4
0
1
2
3
4
5
Time (sec.)
Ene
rgy
Total energy Total potential energy Total kinetic energy Critical energy at UEP4 Critical energy at UEP2 Critical energy at UEP3
178
Figure 6.11: Cutset energy responsible for the various types of separations.
The fault at bus 9 is cleared at 381ms. The power system separated at the
UEP 2.
In Figure 6.11, “cutset energy 1” describes the transient energy associated
with the separation between generator 1 and the rest of the system (i.e.
cutset (1/23) & (23/1)). The “Cutset energy 2” describes the transient energy
associated with the separation between generator 2 and the rest of the
system (i.e. cutset (2/13) & (13/2)) whereas “cutset energy 3” describes the
transient energy associated with the separation between generator 3 and the
rest of the system (i.e. cutset (3/12) & (12/3)).
It is observed that “cutset energy 1” is dangerously close to its critical cutset
energy evaluated at UEP 4 at around 1s when “cutset energy 2” and “cutset
energy 3” are less critical since they are far from their critical cutset energy
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Time (sec.)
Cutset energy and critical cutset energy
cutset energy 1 cutset energy 2 cutset energy 3 critical cutset energy at UEP 1critical cutset energy at UEP 4critical cutset energy at UEP 2critical cutset energy at UEP 5critical cutset energy at UEP 3critical cutset energy at UEP 6
Cut
set e
nerg
y
179
evaluated at their corresponding UEPs. As time progresses, generator 2 had
separates from the rest of the system at around 1.2s which is when the
“cutset energy 2” exceeds its critical cutset energy evaluated at UEP 2. This
is observed in Figure 6.12.
Figure 6.12: Generator 2 has separated from the rest of the system at around
1.2s which is predicted by the “cutset energy 2” associated with cutset
(2/13).
6.6.6.4. The value of cutset energy control
From the earlier sections, it is realized that it is difficult to predict which
types of power system separations will occur when severe faults are
0 0.5 1 1.5 2-4
-3
-2
-1
0
1
2
3
4
5
6
δ (
rad.
)
Time (sec.)
δ1δ2δ3
Mac
hine
ang
le,
Generator separates from the rest of the system
180
encountered in the power system. The undesirable consequences include the
hovering behavior of machine angles near UEPs, as shown in Figure 6.9. It
is also understood from the earlier sections that it is difficult to predict the
types of power system separations that are likely to occur from total energy.
Hence, using total energy to derive the switching control law leading to the
energy control Uen as defined in (6.37) will be less than perfect. In this
section, we will illustrate the problem of the energy control Uen at post-fault
and the result of cutset energy control U# as defined in (6.49) in solving this
problem.
When the fault at bus 9 is cleared at a critical clearing time of 390.4ms, the
system trajectory survives the separation at cutset (2/13) and hovers near the
UEPs (i.e. the system trajectory hovers near the boundary of the region of
attraction). This undesirable effect as shown in Figure 6.13 may lead to a
separation at either the cutset (23/1) or (2/13) which is easily understood by
observing the potential energy surface diagram in Figure 6.9. On the
contrary, when the cutset energy control U# is used instead of the energy
control Uen, the problem of machine angles hovering near UEPs is avoided.
The result of U# control is shown in Figure 6.13 to enable a comparison
with that of the Uen control.
181
Figure 6.13: The system trajectory that is controlled using Uen (dashed) and
U# (solid line with dots) are shown on the potential energy surface. The
Fault at bus 9 is cleared at 390.4ms.
The hovering of machine angles near UEPs when the Uen control is used can
be observed from the machine angles in Figure 6.14. In Figure 6.14,
machine angle 1 and 2 hovers for a longer time than machine angle 3 when
the system trajectory is much closer to the UEP 2 and 4 respectively. This is
because the energy control Uen does not yield the correct switching control
when the system trajectory is near these UEPs instead it operates in linear
region at the crucial instances of first swing. The switching of control values
for Uen is shown in Figure 6.15.
The result of U# control is also shown in Figure 6.14 where it is obvious that
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
1.41790.81515
0.21239
2.0207
2.6234
1.4179
2.02071.4179
2.0207
0.81515
0.81515
0.212392.0207
2.6234
2.6234
1.4179
0.815151.41792.020
7 2.6234
2.6234
3.829
2.0207
3.2262
3.2262
3.2262
2.6234
0.81515
2.6234
3.829
3.829
4.4317
1.41792.0207
3.2262
2.6234
3.829
3.2262
2.62343.2262
3.8294.4317
3.2262
4.4317
3.2262
4.4317
4.4317
4.4317
3.2262
3.829
5.0345
5.03456.24
5.0345
5.6372
6.8428
4.4317
5.6372
7.4455
6.24
5.0345
6.24
5.6372
6.8428
4.43175.0345
5.6372
6.8428
7.4455
6.24
7.4455
6.24
6.24
6.24
Total potential energy surface
δ21
δ31
UEP 3
UEP 2
UEP 6 UEP 1
UEP 5
SEP
UEP 4
Trajectory due to cutset energy control
Trajectory due toEnergy control
182
this control avoids the hovering of the machine angles. In general, both the
Uen and U# controls has similar performance when it comes to the damping
of subsequent oscillations.
Figure 6.14: The response of machine angles under the influence of Uen
(dotted) and U# (solid) controls. The Fault at bus 9 is cleared at 390.4ms.
(a): The effect of Uen (dotted) and U# (solid) controls on machine angle 1.
(b): The effect of Uen (dotted) and U# (solid) controls on machine angle 2.
(c): The effect of Uen (dotted) and U# (solid) controls on machine angle 3.
It is obvious from the comparison of the switching of control values
between Uen and U# controls as shown in Figure 6.15 that the U# control has
applied different phase shifts near the three main UEPs (i.e. UEP 2, 3 and 4)
at appropriate timings to avoid the PS regions of these UEPs. The U# control
has this ability to recognize the different types of cutsets because its control
law is derived from the cutset energy instead of total energy. It is evident
from our earlier discussion that total energy lacks the information on power
0 2 4 6 8 10 -1
0
1
δ1 (
rad.
)
0 2 4 6 8 10 -5
0
5
δ2
(rad
.)
0 2 4 6 8 10 -5
0
5
Time (sec.)
δ3
(rad
.)
183
system separations.
Figure 6.15: The switching of control values for the Uen (dotted) and U#
(solid) controls.
The cutset energy representing the different cutsets (or power system
separations) when the SVC in the power system is being controlled using
Uen is shown in Figure 6.16. In Figure 6.16, we observes that both the
“cutset energy 1” associated with cutset (23/1) and “cutset energy 2”
associated with cutset (2/13) are close to their respective critical cutset
energy. This implies that the power system may separate at either UEP 4 or
UEP 2.
0 1 2 3 4 5 6 7 8 9 10-1
-0.5
0
0.5
1
Con
trol U
en
0 1 2 3 4 5 6 7 8 9 10-1
-0.5
0
0.5
1
Time (sec.)
Con
trol U
#
184
Figure 6.16: Cutset energy of the power system when the energy control Uen
is used. The fault at bus 9 is cleared at 390.4ms.
The influence of U# control on the power system is shown in Figure 6.17. It
is obvious that the “cutset energy 1” associated with cutset (23/1) has been
reduced owing to the appropriate phase being introduced at the crucial time
of first swing. The consequences of the reduced “cutset energy 1” has
resulted in a substantial reduction in the “cutset energy 2”.
0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (sec.)
Cut
set e
nerg
y
Cutset energy and critical cutset energy
cutset energy1cutset energy2cutset energy3critical cutset energy at UEP 1critical cutset energy at UEP 4 critical cutset energy at UEP 2critical cutset energy at UEP 5critical cutset energy at UEP 3critical cutset energy at UEP 6
185
Figure 6.17: Cutset energy of the power system when the energy control U#
is used. The fault at bus 9 is cleared at 390.4ms.
0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (sec.)
Cutset energy and critical cutset energy
cutset energy1cutset energy2cutset energy3critical cutset energy at UEP 1critical cutset energy at UEP 4 critical cutset energy at UEP 2critical cutset energy at UEP 5critical cutset energy at UEP 3critical cutset energy at UEP 6
Cut
set e
nerg
y
186
6.7. Case Study 2 (Detailed Six-machine 21-bus Power
System)
In this case study, a six-machine 21-one bus longitudinal power system
using detailed machine model is used as shown in Figure 6.18. The system
data for the detailed machine models and line parameters are obtained from
[19] with the line reactance between bus 18 and 19 being modified to 0.488
per unit to demonstrate the control of a stressed power system using cutset
energy based control. This case study is used to demonstrate the
performance of the cutset energy based control in retaining synchronism
when compared against two different SVC controls; the energy based
approach [5, 6, 13] using remote measurements and the mode frequency
tuning approach using local measurements. The study of relaying
coordination and short-time rating of 275KV transmission lines, for
example, due to power swings are not within the scope of this chapter. The
main objective of the case study in this chapter is to emphasize the
importance of improving the synchronization amongst generators in order to
yield a significant improvement in the transfer capacity of transmission
network. In this example, the control of SVC devices is based on the
proximity-to-separation strategies and Energy Decomposition. This
improved transfer capacity demonstrates the ability of this control
methodology to maximize the use of existing assets in extending the
transient stability limits of the power systems.
187
As the cutset energy-based control and energy-based control uses remote
measurements, time delay in data transmission is unavoidable. The time
delay in these remote measurements based on Global Positioning System
(GPS) synchronization using ground based communication is expected to be
less than 30ms for a typical data transmission distance of 2000km. These
delays are to be neglected considering its insignificant influences in the
control of 1.5 to 1.8 Hz oscillations. The SVC dynamics describing the
implementation of the controllable reduced Y admittance is neglected in this
chapter as the response time of a typical SVC is approximately 2 cycles for
small changes and 1 cycle for major changes.
Figure 6.18: A six-machine 21-bus test system with Left side of the
generations far from the Central and Right areas.
In this case study, the performances of cutset energy-based control such as
that derived in equation (6.49) for the three-machine 9-bus system are
1
5
6
7
8
12
21 Swing bus H 1 =10.47
Pm 2 =5.3 H 2 =10.47
Pm 3 =6.0 H 3 =10.47
Pm 4 =6.0 H 4 =10.47
Pm 5 =0.8 H 5 =2.47
Pm 6 =0.8 H 6 =2.47
Right Central Left
∠ 0.0 °
∠ - 37 °
∠ - 37 ° ∠ - 38 °
∠ - 38 ° 2 9 ∠ -3.5 °
∠ - 9.2 °
∠ - 25 °
∠ - 18 ° ∠ - 25 ° 3 10
13 14 15 16 17 18 19 20
15+j4.93 - j2.3 4.4+j1.6 5+j1.6 - j1.3 - j2.0
svc svc
∠ - 22 ° ∠ - 18 ° ∠ - 15 ° ∠ - 12 °
∠ - 17 °
∠ - 33 ° ∠ - 44 ° ∠ - 39 °
∠ - 6.4 ° 4 11
∠ - 18 ° ∠ - 25 ° 1
5
6
7
8
12
21 Swing bus H 1 =10.47
Pm 2 =5.3 H 2 =10.47
Pm 3 =6.0 H 3 =10.47
Pm 4 =6.0 H 4 =10.47
Pm 5 =0.8 H 5 =2.47
Pm 6 =0.8 H 6 =2.47
Right Central Left
∠ 0.0 °
∠ - °
∠ - ° ∠ - °
∠ - ° 2 9 ∠
∠ - °
∠ - ° ∠ - 25 ° 3 10
13 14 15 16 17 18 19 20
- - j1.3 - j2.0
∠ - ° ∠ - ° ∠ - ° ∠ - °
∠ - °
∠ - ° ∠ - ° ∠ - °
∠ - ° 4 11
∠ - ° ∠ - 25 °
188
compared against two other forms of control algorithm.
For the energy based control, referred to as the Uen control, or simply
referred to as (energy) in this thesis, it is derived based on [13] having a
control law of
) sat (SU Ven = and
−<−≤≤−>
=1 1
11 1 1
SforSforSSfor
Uen (6.50)
where the switching surface SV is
∑ ∑ ∆+∑ ∆+∆==
=
•=
≠=
=
≠=
6
1
6
1
6
1)cossin(
n
iiij
n
ij
ijn
ij
ijijiiV gbgS δθθ (6.51)
For the mode frequency tuning approach referred to as the Umode control, or
simply referred to as (mode) in this thesis, it aims at two mode frequencies
of approximately 9.8 rad/sec. and 11 rad/sec., which coarsely associated
with the separation at the Right-Central link and the Left-Central link. The
9.8 rad/sec mode frequency is obtained from the eigenvalues of the
perturbed system in Figure 6.18 when a SVC (–j1.3) is placed at bus 13. The
11 rad/sec. mode frequency is obtained when a SVC (-j2.0) is placed at bus
18. It is found to be effective when each SVC is assigned to handle a mode.
The local measurements of the real power flows on line 18-19 and line 13-
14 feeding through a 90° phase shift block as shown in Figure 6.19
approximate a velocity and positional feedback control. As shown in Figure
6.19, the real power (Powerij) and voltage (vi) measurements at bus 18 and
189
13 are used for the control of SVC installed at bus 18 and 13 respectively.
This block diagram is derived from the excitation controller with stabilizer
signal as used in [20].
Figure 6.19: SVC control using local measurements of power and voltage to
approximate a velocity and positional feedback.
For the cutset energy-based control referred to as the U#, or simply referred
to as (cutset) in this thesis, it uses both switching surface #S (6.33) and VS
(6.51) forming the control law of
( ))( ##### PVVV SSwtSsat) Ssat (SU ++=+= (6.52)
and
( )( ) ( )
( )
−<+−≤+≤−+
>+
=1 1
1 1
1 1
#
##
#
#
SSforSSforSS
SSfor
U
V
VV
V
(6.53)
It is found that the clearing of the severe fault at bus 12 by tripping one of
the parallel lines between bus 12 and 13 excites two inter-area modes; one
mode of separation is associated with the oscillation between the Right and
2) 100( + s
K1sPoweri j
2) 100( + s
Poweri j+ -
Vi
Vref + - K2
modeU
190
Central-Left areas while the other is associated with the oscillation between
the Left and Central-Right areas. The results of using no control (0) and
using different control approaches of Umode (mode), Uen (energy) or U#
(cutset) are shown in Figure 6.20 with one of the lines between bus 12 and
13 cleared at 150 ms. As it is found that local area modes in the Left area
(i.e., involving generator 5 and 6), Central area (i.e., involving generators 1
and 2) and Right area (i.e., involving generators 3 and 4) are not excited,
they are not shown. Instead, the inter-area modes (in the range of 1.5 to 1.8
Hz); one associated with the Central and Left areas oscillation and the other
associated with the Central and Right areas oscillation, are shown in Figure
6.20.
Figure 6.20: COA angle difference for the Central-Right area (dotted lines)
and the Central-Left area (solid lines) under the influence of different SVC
0 2 4 6 8 10 12 14 16 18 20
0
1
2
0 2 4 6 8 10 12 14 16 18 20
0
1
2
0 2 4 6 8 10 12 14 16 18 20
0
1
2
CO
A a
ngle
diff
eren
ce,
δ ij/
kh (
radi
ans)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
Time (sec.)
(a)
(b)
(c)
(d)
191
controls. Line trips at 150ms for the fault at bus 12.
(a): Under no control (0),
(b): SVCs control using the (mode) control.
(c): SVCs control using the (energy) control.
(d): SVCs control using the (cutset) control.
The angle difference between the Central and Left area is represented by
34/12δ in Central-of-area [12] (COA) frame whereas 56/12δ represents the
Central and Right areas interaction in COA frame. They are evaluated from
hk
hhkk
ji
jjiikhij mm
mmmm
mm
++
−+
+=
δδδδδ
****/ (6.54)
It is obvious that all control approaches are capable of damping the
subsequent oscillations satisfactorily. In the “no control” case in Figure
6.20, beating phenomena are observed as a lower frequency component that
is formed when certain frequency components in the power system vanishes
intermittently [21]. These beating phenomena does not influence the
damping performance of the SVCs, particularly in the case of the (mode)
control, since the control of each SVC is designed separately to target the
specific oscillations. In terms of the general damping performance, energy-
based (energy) and cutset energy-based (cutset) controls gives a better result
than the (mode) control. A comparison between the (energy) and (cutset)
controls shows that the energy-based and cutset energy-based controls have
insignificant differences when the system is not in danger of separating.
Considering the retention of synchronism (or the survival of a power
192
system) in relation to the maximization of control resources, the cutset
energy based (cutset) control outperforms the rest of the controls in
preventing a power system separation. The results of two inter-area modes
under different controls are shown in Figure 6.21 for the fault at bus 12 and
line cleared at 200ms.
Figure 6.21: COA angle difference for the Central-Right area (dotted lines)
and the Central-Left area (solid lines) under the influence of different SVC
controls. Line trips at 200ms for the fault at bus 12.
(a): Under no control (0),
(b): SVCs control using the (mode) control.
(c): SVCs control using the (energy) control.
(d): SVCs control using the (cutset) control.
0 1 2 3 4 5 6
0 2 4
0 1 2 3 4 5 6
0 2 4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 2 4
CO
A a
ngle
diff
eren
ce,
δ ij/kh
(ra
dian
s)
0 1 2 3 4 5 6-10 1 2
Time (sec.)
(a)
(b)
(c)
(d)
193
A comparison between Figure 6.20d and 6.21d has shown that a prolonged
tripping of one of the lines between bus 12 and 13 causes greater excursion
of generator angles. These results of large angle swings are, however,
correctly synchronized under the control of SVCs using (cutset) control. The
control responses associated with the fourth plot of Figure 6.21 is shown in
Figure 6.22.
Figure 6.22: Responses of SVC controls at bus 18 and bus 13. Line trips at
200ms for a fault at bus 12. SVC installed at bus 18 and 13 respond to the
inter-area mode associated with the Central-Left area (solid line) and the
Central-Right area (dotted line) respectively.
In Figure 6.22, both the SVC installed at bus 18 and 13 respond strongly
during the first one and a half seconds as the respective control values (U18
and U13) saturates to yield the maximum control. As time progressed, the
0 1 2 3 4 5 6-1
-0.5
0
0.5
1 SVC control at bus 18 (U18)
0 1 2 3 4 5 6-1
-0.5
0
0.5
1
SVC control at bus 13 (U13)
Time (sec.)Res
pons
es o
f SV
C c
ontr
ols
usin
g th
e (c
utse
t) c
ontr
ol
194
separation risk at the Central-Right area becomes less serious than that at
the Central-Left area as the SVC at bus 18 saturates for another second. This
strong response of U18 to the inter-area mode associated with the Central-
Left area agrees with the SVC placement concept in [22].
A comparison between Figure 6.22 and Figure 6.21d shows that when the
faults at bus 12 are most likely to break the Central-Right link around 0.4s,
the SVC at bus 13 saturates (i.e., U=1) to synchronize the Central and Right
areas. At around 0.6s as the Central-Left link is close to separation, SVC at
bus 18 remains at U=1 to synchronize the Central and Left areas while the
SVC at bus 12 reduces its strength to coordinate in the saving of the two
areas. This case study has highlighted that apart from emphasizing the
probable separation at the Central-Right link due to the faults at bus 12, it is
also essential to direct the control to the separation that is associated with
another weak link (i.e. Central-Left area). A cutset-based energy control
using the benefits of decomposed energy is capable of achieving these
demanding control requirements. Next, we consider the benefits of the
extended transient stability limits from the investment perspective.
In practice, it is difficult to visualize the benefit of the extended stability
limits expressed in critical fault clearing time. A familiar way of quantifying
the transient stability limits maximized from the use of different controls is
shown in Figure 6.23. A familiar way of quantifying the transient stability
limits maximized from the use of different controls is shown in Figure 6.23.
195
This figure compares the improvement provided by different forms of
controllers showing the improvement from the different control approaches.
This means if we double the size of the controller, we can double the actual
improvement but we expect to retain the same relative benefit through the
use of advanced control.
Relative to a fault clearing time, the comparisons between the no control
case (0) and the three controlled cases has shown that using a cutset energy-
based (cutset), energy (energy) and mode (mode) controls, the increased
transfer between bus 18 and 19 is 3.5%, 2.1% and 2.7% respectively. The
fault clearing time considered has neglected the possibility of high short-
circuit current at these transient stability limits which may or may not
exceed the short-time short-circuit rating and thermal rating of Extra High
Voltage lines.
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.20.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22 no control(energy) control(cutset) control(mode) control
Crit
ical
faul
t cle
arin
g tim
e, tc
l (se
c.)
Power flow between bus 18 and 19, (P18-19) per unit
A fault clearing time of 80ms
196
Figure 6.23: Quantifying the transient stability limits of the power system
under different SVC controls. One of the parallel lines between bus 12 and
13 is removed for the fault at bus 12. P18-19 refers to the steady state power
flow.
This improvement in transfer capacity depends on the size of the SVCs. The
use of (mode) control increases the benefit of the SVCs by a factor of 1.2 as
compared with the (energy) control whereas the advanced cutset energy-
based control improves the benefit of the SVC by a further 120% relative to
the (mode) control. Thus for this example, a cutset energy-based SVC of 2/3
of its size can achieve the stability improvement provided by the
conventionally controlled SVC.
In this case study, although the use of (mode) control results in a
comparatively higher transient stability limit than (energy) control, as
shown in Figure 6.23, by synchronizing generators at first swing, this
synchronization feature of the (mode) control is sometime unreliable
because the selection of K2 (Figure 6.19) involves the trying out of a range
of K2. Generally, K2 and voltage variation has no distinct relationship with
the separation at UEP. On the contrary, the (cutset) control synchronizes
generators at first swing based on the recognition of partly stable regions
that are unique to every separation at UEP. The result of selecting a wrong
K2 leading to the wrong synchronization of generators at first swing is
shown in Figure 6.24. It is shown that the use of K2=10 results in the control
197
switching (i.e. U=1) around 1.05s (Figure 6.24b) which leads to the
separation at Central-Left area (Figure 6.24a).
The second weakness of (mode) control is its generally poor damping
performance as a result of the use of voltage feedback signal )(2 iref VVK −
in the Umode control loop. The poor damping performance of (mode) control
is shown in Figure 6.25a whereas the result of (cutset) controls is shown in
Figure 6.25b. From Figure 6.25, it is understandable that (mode) control
achieves good first swing by compromising the damping performance.
Figure 6.24: Wrong selection of K2 in the voltage error control loop leads to
the separation at central-Left areas (dotted). Line trips at 199ms.
(a) Separation at the Central-Left area (dotted).
(b) Wrong selection of K2=10 for the control of SVC at bus 18 leads to the
separation at Central-Left area. The (mode) control is used.
(c) No change in K2 for the control of SVC at bus 13.
0 1 2 3 4 5 6
0
2
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-1
0
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-1
0
1
Time (sec.)
SVC control at bus 13, K2=15
SVC control at bus 18, K2=10
SVC control at bus 18, K2=5
δ 12coa- δ 56coa δ 12coa- δ 34coa
198
Figure 6.25: A comparison between the damping performance of (mode)
and (cutset) controls. Line tripped at 199ms.
(a) Damping performance of (mode) control.
(b) Damping performance of (cutset) control.
6.8. Implementation issues
There are several issues which can influence the control implementation and
this section is mainly to elaborate these practical considerations such as
measurement noises, device response time, delayed data transmission, GPS
jitter and quantity of cutsets. This set of errors considered is specific to the
implementation of this type of cutest control.
6.8.1. Response Time of SVC and Time delay in Data
Transmission
0 1 2 3 4 5 6 7 8 9 10-2
0
2
4
0 1 2 3 4 5 6 7 8 9 10-2
-10123
Time (sec.)
SVCs are controlledusing (mode) control
SVCs are controlledusing (cutset) control
199
The cutset energy-based control and energy-based control proposed in this
chapter uses remote measurements of machine angle δ and rate of change
of machine angle δ& in COA frame. The effect or control from time delay
experienced in data transmission and response time of control devices are
discussed in this subsection using the six-machine 21-bus power system as
an example.
For the six-machine 21-bus power system as shown in Figure 6.18, the
control of the SVCs installed at bus 18 has high influence on the inter-area
oscillations associated with the interaction between Central and Left areas
(i.e. Central-Left ). The control of SVC at bus 13 influences the interaction
between Central and Right areas (i.e. Central-Right). The frequencies of
Central-Left and Central-Right oscillations are approximately in the range
of 1.5 to 1.8 Hz as obtained from eigenvalue analysis. As the remote
measurements of δ and δ& in COA frame are based on Global positioning
System (GPS) uses ground based communication, the expected time delay
in the fibre optic network is less than 10ms considering an effective data
transmission rate of 2x108 m/sec. and maximum distance of 2000km. This
time delay is considerably small relatively to the control of low frequency
oscillations (1.5 Hz to 1.8 Hz) where an approximately 10ms delayed in
critical switching will not significantly deteriorate the controlled responses
in the power system.
The response time of power electronics based control devices such as SVC
200
is approximately 20ms for large changes and 40ms for small changes. For
the control of power system at critical first swing, large changes in δ and δ&
are expected and the SVC response is 20ms. Hence, simulating a delay of
30ms gives rise to a 2.5 percent changes in the critical clearing time.
6.8.2. GPS Jitter
The commercial use of GPS receiver often experiences time pulse jitter [23]
of less than 1 nanosecond on data measurements. In this control application,
the effect of 1ns GPS jitter results in a negligible phase shift of 0.018x10-3
degrees in the measured data. This inaccuracy due to phase shift does not
influence the control performance significantly.
6.8.3. Measurement Noise
In the measurement of field data, measurement noise is a common issue. For
the GPS based measurement unit, a high sampling rate of typically 10 KHz
minimizes the measurement noises presence in the analogue signal resulting
in a low inaccuracy of ± 0.01 degrees in phase measurement. During
transient, relatively low measurement noise is expected and with the phase
measurement in the order of 80 degrees, a ± 0.012 percents inaccuracy due
to noise does not affect the control performance significantly.
201
As the cutset energy-based control uses saturation function instead of the
bang-bang switching, as a result it is less sensitive to measurement noise. It
is understandable by considering the bang-bang function that even when
low noise is presents in the data measurements near switching, it can cause
control chattering whereas a saturation function is relatively insensitive to
low noise measurement.
6.8.4. Multidimensional Issue of Cutsets
The Energy Decomposition is to characterize total energy into energy
components that are likely to cause particular separations in a power system.
We have shown by considering all the possible cutsets that the sum of cutset
energy components equals to total energy and the decomposition scaling
coefficient for total kinetic energy, total energy stored in transmission lines
and total shaft energy (i.e. Ω, τ and λ) can be evaluated respectively from
(6.17), (6.20) and (6.24). Applying this Energy Decomposition in the cutset
energy control, it is only required to consider the cutset energy components
that are likely to cause separations in a power system, which associate with
the feasible cutsets. The following section elaborates on the construction of
feasible cutsets.
Considering a 10 machine 39-bus New England system as shown in Figure
6.24, the number of possible cutsets obtained is µ=1024 using the equation
(6.12). The decomposition scaling coefficients (i.e. Ω, τ and λ ) that are
202
evaluated to be Ω =18432, τ =512 and λ =9216 are used in the computation
of cutset energy associated with the list of feasible cutsets. Examining
Figure 6.24, 33 cutsets are defined feasible as shown below using the
notation developed in (6.1):
)36,35,33,34,32,31,39,37,30/ 38()38,35,33,34,32,31,39,37,30/ 36()38,36,33,34,32,31,39,37,30/ 35()38,36,35,34,32,31,39,37,30/ 33()38,36,35,33,32,31,39,37,30/ 34()38,36,35,33,34,31,39,37,30/ 32()38,36,35,33,34,32,39,37,30/ 31()38,36,35,33,34,32,31,37,30/ 39()38,36,35,33,34,32,31,39,30/ 37()38,36,35,33,34,32,31,39,37/ 30(
)36,35,33,34,32,31,39,30 38,37()35,33,34,32,31,39,37,30 38,36()38,33,34,32,31,39,37,30 36,35()38,36,35,32,31,39,37,30 33,34()38,36,35,33,34,39,37,30 32,31()38,36,35,33,34,32,37,30 31,39()38,36,35,33,34,32,31,37 39,30()38,36,35,33,34,32,31,39 37,30(
),35,33,34,32,31,39,30 37,38,36(),33,34,32,31,39,37,30 38,36,35(
)38,36,35,33,34,37,30 32,31,39()38,36,35,33,34,32,37 31,39,30()38,36,35,33,34,32,37 31,39,30()38,36,35,33,34,32,31 39,37,30(
203
),33,34,32,31,39,30 37,38,36,35()38,32,31,39,37,30 36,35,33,34()38,36,31,39,37,30 35,33,34,32()38,36,35,39,37,30 33,34,32,31()38,36,35,33,34,37 32,31,39,30()38,36,35,33,34,32 31,39,37,30(
)38,31,39,37,30 36,35,33,34,32()38,36,39,37,30 35,33,34,32,31()38,36,35,37,30 33,34,32,31,39()38,36,35,33,34 32,31,39,37,30(
where 30 to 38 are generators’ indices. It is obvious from observing Figure
6.24 that remaining cutsets such as )3,35,36,3830,37,39,3 / 31,32,34( are
not feasible cutsets that can give rise to a power system separation in the
form of two coherent groups of generators. Based on these 33 feasible
cutsets, 33 cutset energy componenets are evaluated in the control algorithm
to compare against 66 (i.e. 2x33) critical cutset energies evaluated at the 66
numbers of UEPs. Using the Davidon-Fletcher-Powell method, the 66
numbers of UEPs are searched from initial guesses obtained from equation
(6.25).
204
Figure 6.24: Ten-machine 39-bus New England system.
6.8.5. Summary of Implementation Issues
The inaccuracy in remote measurement is caused by the GPS jitter giving
rides to a phase shift of 0.018x103 degrees and measurement noise that gives
rise to a ±0.01 degrees. These errors in measurement do not have severe
influence on the implemented control. In particular, when the unexpected
phase shifts introduced are negligibly small and the use of saturation
function avoids instance switching between the ±1 limits.
G30
39
1
2
25
37
29
17
26
9
3
38
16
5
4
18
27
28
3624
35
22
21
20
34
23
19
33
10
11
13
14
15
8 31
126
32
7
G
GG
G
G G G GG
205
The response time of devices and delayed data transmission have some
impacts on the control performance. Comparing between the ideal (cutset)
control and practical (cutset) control subject to a 30ms delay, the critical
clearing time obtained from the practical (cutset) control is reduced by 2.5
%. In general, these undesirable results can be reduced by increasing the
feedback gain of the control loop.
In Energy Decomposition, the concern over the number of possible cutsets
obtained from a relatively large power system such as a Ten-machine 39-
bus meshed system that results in 1024 possible cutsets can be reduced
significantly. The approach is to consider only the feasible cutsets that are
likely to cause a power system separation in the form of two coherent
groups of generators.
Although these issues of feasibility and performance of control may give
rise to some difficulties in implementation, their effects are generally minor
and are easily overcome. Hence, the cutset energy control can be
implemented for the benefit of a realistic power system.
206
6.9. Conclusion
In the case study 1, the classical three-machine 9-bus power system is used.
It is shown that the energy based control SV is capable of damping the
system oscillations fairly well however it causes the system trajectory to
hover near the UEPs at the boundary of the region of attraction. This
hovering behavior can lead to a system separation in the PS region of the
UEPs as explained in the earlier chapter pertaining to PS region.
As the decomposed energy (i.e. cutset energy) is able to predict at which
cutset the power system will separate, the cutset energy-based control is
thus designed with the control strategy that aimed at separations, which are
likely to occur. This has inevitably equipped the control with the ability to
introduce phase shifts near a UEP’s PS region when the system trajectory is
critical of separating. This feature has maintained high control values at
critical instances, which is particularly important for the control of a multi-
machine power system.
Although the energy evaluated at unstable local minima is not critical
energy, it is important to note that under the different power dispatches and
loading at weak links, a UEP can become a ULM and vice versa. Hence, all
unstable operating points shall be considered in the design of a cutset based
energy control.
207
The case study 2 focuses on the synchronization of generators and damping
of subsequent oscillations. The quantification of the extended transient
stability limits using different controls as seen in Figure 6.23 does not
include the short-circuit rating study on the EHV lines in observing whether
factors such as the variations of short-circuit current and weather conditions
would result in the short-circuit rating or thermal rating of lines being
exceeded. The aim of this quantification is to demonstrate the possibility of
extending the transient stability limits and improving the network transfer
capacity limits using a (cutset) control. The result of this chapter is to be
seen as a contribution to future discussions amongst engineers in the areas
of relay protection, thermal rating of overhead lines and transient stability
analysis.
It is found that the control performance of both energy-based control design
in [6] and the proposed cutset-based energy control are insensitive to an
energy function approximation using classical model while transient
stability study is based on detailed machine models. Assuming a linear
relation between the investments in SVC and transfer capacity, investments
that are spent in the extra SVC capacity to cater for a higher transfer
capacity can be saved using the proposed cutset energy-based (cutset)
control. By including cutset information in energy-based control design,
these controls will yield high system survival, maximize control resources
and improve network transfer capacity.
208
This chapter develops a method of decomposing energy into key separation
energy (or cutset energy) and its application in control design has been
demonstrated. The cutset energy based control (cutset) is capable of
retaining system synchronism to sustain a high system survival while
damping the subsequent oscillations satisfactorily. In spite of the
longitudinal power system chosen in the case study, the proposed total
energy decomposition and cutset energy based control are generally suitable
for any configurations in power system layout. As such it is also applicable
in the control of a meshed power system.
The main outcome of this work is the control of FACT devices that
maximizes the probability of retaining synchronism after disturbances as
well as providing a well coordinated multi-machine damping. This is
demonstrated using a simple system containing full order generator models.
This example showed at least a 120 % improvement in control effectiveness
when compared to a conventional design.
209
6.10. References
[1] M. A. Pai, Energy Function Analysis For Power System Stability: Kluwer Academic Publishers, 1989.
[2] A. R. Bergen and D. J. Hill, "A structure Preserving Model For Power System Stability Analysis," IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, No. 1, pp. 25-35, January 1981.
[3] A. A. Fouad and V. Vittal, "The Transient Energy Function Method," International Journal of Electrical Power and Energy Systems, vol. 10, No. 4, pp. 233-246, October 1988.
[4] A. A. Fouad, V. Vittal, and T. K. Oh, "Critical Energy For Direct Transient Stability Assessment of A Multimachine Power System," IEEE Transactions on Power Apparatus and Systems, vol. PAS-103, No. 8, pp. 2199-2206, August 1984.
[5] G. Ledwich, J. Fernandez-Vargas, and X. Yu, "Switching Control of Multi-machine Power Systems," IEEE / KTH Stockholm Power Tech Conference, Stockholm, Sweden, pp. 138-142, June 1995.
[6] E. Palmer and G. Ledwich, "Switching control for power systems with line lossess," IEE Proceedings- Generation, Transmission, Distributions, vol. 146, No.5, pp. 435-440, September 1999.
[7] B. D. Bunday, Basic optimisation methods: Edward Arnold (Publisher) Ltd, 1984.
[8] M. A. Pai, Power System Stability - Analysis by the Direct Method of Lyapunov, vol. 3: North-Holland Publishing Company, 1981.
[9] C.-W. Liu and J. S. Thorp, "A novel method to compute the closest unstable equilibrium point for the transient stability region estimate in power systems," IEEE Transactions on circuits and Systems-I: Fundamental Theory And Applications, vol. 44, No. 7, pp. 630-635, July 1997.
[10] T. W. Chan, G. Ledwich, and E. W. Palmer, "Is velocity feedback always best for machine stability control ?," Aupec 2002, Melbourne, Australia, October 2002. (Available at http://www.itee.uq.edu.au/~aupec/aupec02/Final-Papers/T-W-Chan1-808.pdf).
[11] P. M. Anderson and A. A. Fouad, Power System Control and Stability: IEEE Press, 1994.
[12] G. Ledwich and E. Palmer, "Energy Function For Power Systems with Transmission Losses," IEEE Transactions on Power Systems, vol. 12, No. 2, pp. 785-790, May 1997.
[13] T. W. Chan and G. Ledwich, "Multi-mode damping using single HVDC link," Aupec 2001, Perth, Australia, pp. 483-488, September 2001.
210
[14] H. D. Chiang, C.-C. Chu, and G. Cauley, "Direct Stability Analysis of Electric Power Systems Using Energy Functions: Theory, Applications, and Perspective," Proceedings of the IEEE, vol. 83, No. 11, pp. 1497-1528, November 1995.
[15] J. S. Thorp and S. A. Naqavi, "Load Flow Fractals," 28th Conference on Decision and Control, Tampa, Florida, U.S.A 1989.
[16] A. A. Fouad and S. E. Stanton, "Transient Stability of A Multi-Machine Power System. Part II: Critical Transient Energy," IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, No. 7, pp. 3417-3424, July 1981.
[17] T. Athay, R. Podmore, and S. Virmani, "A practical method for the direct analysis of transient stability," IEEE Transactions on Power Apparatus and Systems, vol. PAS-98, No. 2, pp. 573-584, March/April 1979.
[18] C. L. Gupta and A. H. El-Abiad, "Determination of the closest unstable equilibrium state for Liapunov methods in transient stability studies," IEEE Transactions on Power Apparatus and Systems, vol. PAS-95, No. 5, pp. pp. 1699-1712, September/ October 1976.
[19] E. W. Palmer, "Multi-Mode Damping of Power System Oscillations," PhD thesis in Electrical and Computer Engineering The University of Newcastle, 1998, pp. 196.
[20] P. Kundur, Power System Stability and Control: McGraw-Hill, Inc, 1994.
[21] S. Kim and Y. Park, "On-line Fundamental Frequency Tracking Method For Harmonic Signal and Application to ANC," Journal of Sound and Vibration, vol. 241, N0. 4, pp. 681-691, 2001.
[22] E. W. Palmer and G. Ledwich, "Optimal Placement of Angle Transducers In Power Systems," IEEE Transactions on Power Systems, vol. 11, No. 12, pp. 788-793, May 1996.
[23] T. N. Osterdock, "Using A New GPS Frequency Reference In Frequency Calibration Operations," 47th IEEE International Frequency Control Symposium, Salt Lake City, UT, USA, pp. 33-39, February-April 1993.
211
Chapter 7.
Conclusion and Recommendations
7.1. Conclusion
It is mentioned in the thesis that modern power systems are frequently
formed by interconnecting the power systems of various regions in spite of
the fact that these interconnecting practices may give rise to complicated
electromechanical dynamics amongst the generators of a large
interconnected power system. From the small signal analysis perspective,
the provision of synchronizing and damping torques through the excitation
and power system stabilizer (PSS) control loops are crucial in the
improvement of transient stability in the power system. However, it is clear
that using small signal analysis to predict the stability of a power system is
limited to operating conditions subject to small disturbances. The objective
of this thesis is to investigate whether the synchronization of generators and
damping of oscillations can be maximized via a non-linear approach to
212
extend fully the transient stability of power system and improve the network
transfer capacity. The approach taken in the analyses of generators
interaction and development of control methodology was based on
Lyapunov energy function.
The Lyapunov stability criteria and Lyapunov energy function have been
introduced in this thesis. It is explained that energy function predicts the
transient stability of power systems regardless of the size of disturbance as
long as the Lyapunov function is positive definite when it is bounded within
some angle limits, its derivative is negative definite and system trajectory at
fault clearance lies inside the region of attraction.
From the literature review on the control of thyristor controlled series
compensation (TCSC), it has been found that energy control based on
Lyapunov’s stability criteria is capable of reducing the disturbance energy
that is injected into accelerating the rotor of generators during a fault period.
It is learnt from this literature review that the design of a control law based
on the Lyapunov’s stability criteria guarantees a non-divergent trajectory
inside a region of attraction and the damping of n modes requires n numbers
of controllers.
The literature review on the control of DC link has suggested that the
common problems encountered in the control of DC link in power systems
are mainly associated with the control mode interactions and voltage
213
variations at inverter stations. In particular, the voltage variations at the
inverter are usually associated with the limitations of the gamma control.
During transient period, current control of inverter is being used to reduce
these voltage fluctuations. The control mode interactions are largely excited
by the limitations of the controls at the rectifier and inverter. This problem
can be overcome using the respective power modulation control and voltage
modulation control at the rectifier and inverter stations. The literature
review on the control of DC link has highlighted that the choice of feedback
signal is largely limited by the strength of the system (or size) at the inverter
side. From this literature review, it is found that adequate understanding of
the dynamics of control interactions and impact of reactive power
‘consumption’ of converters are essential in the development of an effective
control methodology. In addition, it is important to consider the strengths of
weak power system at its inverter side in a control design.
The literature review on wide area control, comparing the effectiveness of
control strategy in centralized and decentralized controls, has emphasized
the importance of using wide area information (or remote measurements) as
the control feedback signals. The performance of the decentralized control
of PSS is found to be satisfactory when remote measurements are being
used as feedback signals. This literature review has indicated that it is likely
for a control to influence distant modes or multiple modes if wide area
information is being utilized in control design.
214
As transient stability limits are associated with the separation of power
system into two coherent groups of generators, it is important to understand
the literature in the area of Controlling Unstable Equilibrium Point (UEP) to
gain an understanding of the different interpretations of these unique
descriptors of power system separations. It is evident that Controlling UEP
depends on the characteristic of respective faults. The literature review on
the methods of stability assessment has shown that the comparison between
the total energy evaluated at post-fault and critical energy evaluated at a
Controlling UEP has helped to determine if a power system is stable at post-
fault. In the prediction of power system stability, if the energy evaluated at a
UEP is the lowest amongst a set of credible UEPs and it is not a critical
energy, then it is not a Controlling UEP. It has appeared from these
literature reviews that a desirable control design should not be based on a
Controlling UEP which is fault-dependent instead it should be based on any
credible UEPs.
The above-mentioned research objective and literature reviews have
motivated the following developments of the control methodology that is
effective in the synchronization of generators and damping of oscillations in
power systems.
In a hypothetical case, an energy function control based on Lyapunov’s
stability criteria has been applied to the control of two independently
operated AC power systems which is connected by a DC link. This two-area
215
DC link power system has shown that energy from each of these areas can
be summed to represent the energy of one large power system. When this
total energy is converging under the influence of energy function control,
both these independent AC power systems will converge to their stable
equilibrium operating points. Using bang-bang control, the common
problem of control chattering near a stable equilibrium operating point
(SEP) has been encountered and characterized as a high gain feedback
control problem. The Root Locus analysis has shown that the bang-bang
control can be described as a velocity feedback control with an infinite gain.
The use of finite gain switching near the SEP will provide the necessary
system damping that is required for the convergence of system trajectory. A
saturation function has been found to be suitable in satisfying these
requirements especially when both the benefits of finite-time damping
introduced by the bang-bang control and exponential damping introduced by
the use of finite gain in switching must be reaped.
Using the same hypothetical case of a four-machine two-area DC linked
power system where one area is weaker than the other, it is highlighted that
the survival of a large power system from severe faults is commonly
determined by the transient stability limits of its weaker area. Hence, it is
important that the determining factors of strong and weak areas are to be
considered in control design using the form of weighted energy control. The
proximity-to-critical energy approach is found to be capable of directing
216
most of the control efforts to a weak area when it is most needed during the
crucial instances.
While it is possible to subject a power system to larger disturbances by
weighing the energy of its weak area to a larger extent, it has given rise to
the problem of control chattering near a UEP. Using a Single-Machine-
Infinite-Bus system (SMIB) as an example, the energy of the power system
in phase portrait has been used as an analysis tool to provide an insight to
the cause of this control problem near a UEP. It has been found that the
repetitive switching inside a partly stable region has resulted in the control
chattering near a UEP with machine angles hovering at maximum. In order
to overcome this undesirable switching, phase shift is being introduced via
an angle look-ahead control.
In a multi-machine power system, energy function control has limits in fully
synchronizing the generators and keeping the different types of angle
separations as small as possible in a power system since this control design
is based on total energy. From this total energy, it has become difficult to
determine at which points in time a power system separation has occurred.
This problem has been overcome by decomposing total energy into cutset
energy that is associated with the different types of separations. Using the
proximity to critical cutset energy and partly stable region as the major
control strategy, the weighting between the cutset dependent terms and
energy function dependent terms has given rise to a cutset energy control.
217
This cutset energy control is capable of targeting control efforts when a
power system is close to a separation. It has shown that this control has the
ability to fully extend the transient stability limits of the power system thus
improving the transfer capacity of electrical transmission network
significantly.
The main outcome of this research is the control of FACTs that maximizes
the probability of retaining synchronism after disturbances as well as
providing a well coordinated multi-machine damping. This has been
demonstrated using a classical three-machine 9-bus power system and a
detailed six-machine 21-bus power system. One of the major strengths of
cutset energy control lies in its ability to recognize the threat of weak links
in power system and synchronize generators by targeting the control efforts
at these weak links. In general, improvement in the synchronization
amongst generators remains the most important aspect in improving the
survival of power system which in turn improves the network transfer
capacity through improvement on the transient stability limits of power
system.
7.2. Recommendations For Future Works
The major recommendations of this research mentioned in the thesis are as
follows:
218
1) One of the major limitations in the control algorithm is the recognition
of feasible cutsets for large power systems. As it is understood from the
Energy Decomposition in Chapter 6 that although possible cutsets are
used in the Energy Decomposition to derive cutset energy, the control
strategy only needs to evaluate the proximity-to-separation
circumstances for all feasible cutsets. The feasible cutsets refers to the
separation of a power system into two coherent groups of generators.
In a large longitudinal power systems which may consist of hundreds of
generators and thousands of transmission lines, the advantages of using
a list of feasible cutsets as the basis for the evaluation of the proximity-
to-separation circumstances in the power system are the reduced burden
in the proximity-to-separation evaluation. This is understandable when
we consider a large longitudinal power system and found that the lists
of feasible cutsets is often smaller than that of the possible cutsets.
It is anticipated that the construction of an incident matrix [1, page 64]
based on Graph Theory [2] may help in the finding of feasible cutsets
of a large power system. The concept of identifying feasible cutsets is
based on the numbers of ways a power system could break into two
coherent groups of generators. One possible way of constructing an
incident matrix disregarding the power flow for the 5-bus network [1,
page 63-64] as shown in Figure 7.1 is
219
000010000100010000100010110100000011000100001111
654321
87654321
where all columns are lines indices and all rows are nodes reference.
This figure is not available online. Please consult the hardcopy thesis
available at the QUT Library.
Figure 7.1: 5 Bus network extracted from [1].
2) As we have learnt from the control methodology in this thesis that the
control is capable of yielding and managing the pure damping and
synchronizing torque components in power systems via the network
connected devices such as SVC, it is most desirable to extend this
concept to the excitation control as an additional mean of synchronizing
generators and controlling the voltage profile of power system through
generators.
Thus far we have considered the case where a change of control
variable (SVC value) results in instantaneous change of real power
220
flow. One form of control, which is used in power system, is adjusting
the field of generator through an exciter. Because the change of voltage
applied to the exciter does not result in an immediate change to field
flux, we have a case where the power system doe not response
instantaneously to control variables. The issue that needs to be
researched is the development of a control process other than
maximizing V& at each instance of time. To give some directions to the
research that could be useful, let us consider the following steps in the
derivation of a control law using a SMIB case. The Lyapunov function
in the form of kinetic energy Vke in a power system is
221 δ&mVke =
where m is the machine inertial constant and δ& is the rate of change of
generator angle.
The time constant between a control input u state influences changes in
the generator terminal voltage E can be expressed crudely as
uTs
KEexc+
=
where Texc is the exciter time constant and K is the gain.
The derivative of kinetic energy keV& is
−=
=
xEVP
mV
m
keδδ
δδsin&
&&&&
221
It is noted from the keV& equation that the controllable terms describing
the instantaneous changes in the network’s power flow, such as when
SVC switches between its limits, no longer exist. This is because the
changes in excitation control do not result in instantaneous changes in
the network’s power flow. Taking the second derivative of the kinetic
energy, we have
)(sincossin
)(sincossin
22uET
xV
xEV
xEV
uETx
Vx
EVx
EVV
exc
excke
+−++
=
+−++=
δδδδδ
δδδδδδ
&&
&&&&&&
Examining the keV&& equation, it appears that the control law could be
used to maximize the reduction in the second derivative of kinetic
energy instead of first derivative of Lyapunov function. This type of
control law based on the maximum reduction of keV&& does not guarantee
the asymptotic convergence of system trajectory towards a stable
equilibrium operating condition.
The above problem in the control of excitation based on cutset energy is
likely to be resolved when Model Predictive Control is considered.
Model Predictive Control is discussed in the following sections.
The receding horizon control strategy is also known as Model
Predictive Control (MPC). MPC generally solves an optimization
222
problem based on a set of state measurement to find optimal control
values. Optimal control values are usually obtained by solving a
discrete-time optimal problem under a given horizon (or sampling
window) [3], with the first control value being applied to the control of
a process. The next control value is obtained by running an
optimization problem again and a new set of state measurements is
used. One key aspect of this recursive optimization approach is to
consider the minimization of a performance index such as
( )∫ += uRuxQxJ
where x is the state vector referenced to a stable operating condition, u
is a vector of control values, Q is a weighting matrix that emphasizes
the relative importance of states over the control effort u and R is a
weighting matrix for the control values (or control effort).
In control context, velocity feedback generally yields satisfying control
performance and the appropriate use of the state penalty term (xTQx) to
penalize the diversion of machine angular velocity in the control of
power system during a transient is advantageous and this type of
performance index [4] is
dtJ ft
o 2∫= ω
Another types of performance index based on mass scaled machine
angles error [5] was used to penalize the divergence of machine angles
and found satisfactory
223
∫ ∑ −= To
icoaii dtMJ 2)( δδ
One advantage of the Model Predictive Control is the ability to handle
the constraints of a stable system under saturated control. MPC with
control constraints incorporates the practical aspect of an actuator in its
search routine to find optimal control values. In [6], the control
constraints were defined in an optimization problem using a different
type of performance index
∑ −==
m
iidiitu
yyminJ1
2**
)()(ˆ)( ττω
hiili utuu ≤≤ )(
where *diy is the ith output reference trajectory, *ˆ iy is the ith delay-free
controlled output, ω is the weightings set according to the relative
importance of controlled output, J is the minimized performance index
and u is the control effort bounded within an upper )( hiu and lower
)( liu value.
Judging from these brief discussions on MPC, it appears that as long as
the MPC’s receding horizon (or sampling window) is sufficiently
designed to include the major dynamics of voltage variations, the
control law derived from keV&& can be used in conjunction with the MPC
control to yield asymptotic convergence in the power system.
224
7.3. References
[1] M. A. Pai, Energy Function Analysis For Power System Stability: Kluwer Academic Publishers, 1989.
[2] E. Kreyszig, Advanced Engineering Mathematics, 8th ed: Wiley, New York, 1999.
[3] M. Larsson, D. J. Hill, and G. Olsson, "Emergency voltage control using search and predictive control," Electrical Power and Energy Systems, vol. 24, pp. 121-130, 2002.
[4] H.-C. Chang and M.-H. Wang, "Neural Network-Based Self-Organizing Fuzzy Controller for Transient Stability of Multimachine Power Systems," IEEE Transactions on Energy Conversion, vol. 10, No. 2, pp. 339-347, June 1995.
[5] S. M. Rovnyak, C. W. Taylor, and J. S. Thorp, "Performance Index and Classifier Approaches to Real-Time, Discrete-Event Control," Control Engineering Practice, vol. 5, No. 1, pp. 91-99, 1997.
[6] H. M. Kanter, W. D. Seider, and M. Soroush, "Nonlinear Feedback Control of Stable Processes," American Control Conference, Arlington, VA, pp. 3624-3629, June 25-27 2001.
