eeh
power systems
laboratory
Ekaterina Telegina
Impact of Rotational Inertia Changes on
Power System Stability
Master Thesis
PSL1510
EEH – Power Systems Laboratory
ETH Zurich
Examiner: Prof. Dr. Göran Andersson
Supervisor: Theodor S. Borsche
Zurich, November 11, 2015
ii
Abstract
High shares of converter-connected renewable generation and consumer devices lead to reduction of rotational inertia in modern power systems. Low
level of inertia in a power system affects the system operation and its stability margin. Inertial response, inherent to rotating machines, degradates
with the rise of inverter-connected RES. Since inertia level defines the rate
of frequency deviation in the first seconds after a disturbance, reduced inertia results in faster frequency dynamics. Operation of primary frequency
control and protection systems becomes more challenging due to the larger
and faster transient frequency deviations. One of the measures to mitigate
the effects of reduced inertia is implementation of faster primary frequency
control. Another possible solution is provision of artificial rotational inertia
in the system. The latter option also allows to provide additional damping
for inter-area oscillations.
This work investigates the impact of inertia changes on damping of system modes and frequency response of a power system. It expands an optimization algorithm proposed in [1]. The algorithm serves for optimization
of rotational inertia and damping levels in a system to enable the assessment of optimal artificial inertia and damping procurement volumes. The
algorithm is focused on improvement of damping of the system modes under a transient frequency overshoot constraint. For the analysis of system
modes, the system state matrix is computed based on a detailed model of
synchronous machine, including voltage dynamics and operation of primary
frequency control. Sensitivities of damping ratio and frequency overshoot
to inertia and damping are derived and incorporated in the algorithm. The
algorithm is implemented for two test systems, optimal solutions are found
for cases with various optimization parameters. Transient simulations are
accomplished to illustrate the results of small-signal stability analysis.
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Acknowledgements
First and foremost, I would like to thank my supervisor Theodor Borsche for
his continuous support and guidance during my work on this master thesis.
Thank you for offering such an interesting research topic. It has been a
pleasure working with you.
I would also like to thank Professor Dr. Göran Andersson for giving me
the opportunity to write a master thesis at the Power System Laboratory.
The “Power System Analysis” and “Power System Dynamics and Control”
courses that he taught further improved my knowledge on the subject of
power system operation and stability which was pivotal for the successful
completion of the present work.
My sincere appreciation goes to my friends for their patience and love.
Special thanks to Elena for her invaluable support during the hard times
and to Anton for his encouragement and understanding.
Finally, I am deeply grateful to my family for their constant love and
support. You always motivated me to work hard and do my best.
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Contents
List of Figures
viii
List of Tables
x
List of Acronyms
xiii
List of Symbols
xv
1 Introduction
1.1 Background and Literature Overview . . . . . . . . . . . . . .
1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . .
1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Power System Stability Fundamentals
2.1 Definitions and Classification . . . . .
2.2 State-Space Representation . . . . . .
2.3 Small-Signal Stability . . . . . . . . .
2.4 Transient Stability . . . . . . . . . . .
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1
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3 Modelling of Power System
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3.1 Synchronous Machine Modelling . . . . . . . . . . . . . . . . 13
3.1.1 Swing Equation . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2 Representation of Synchronous Machine Rotor Circuits Dynamics . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3 Effects of Excitation System and Automatic Voltage
Regulation . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.4 Power System Stabilizer . . . . . . . . . . . . . . . . . 17
3.1.5 Primary Frequency Conrol . . . . . . . . . . . . . . . 19
3.1.6 Full Set of Differential and Algebraic Equations . . . . 20
3.2 Transmission Network Modelling . . . . . . . . . . . . . . . . 23
3.3 Load Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.1 Static Load Models . . . . . . . . . . . . . . . . . . . . 24
3.3.2 Load Damping . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Overall System Equations . . . . . . . . . . . . . . . . . . . . 26
vii
3.4.1
3.4.2
Small-Signal Stability . . . . . . . . . . . . . . . . . .
Transient Stability . . . . . . . . . . . . . . . . . . . .
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4 Impact of Inertia and Damping
4.1 Sensitivity of Damping Ratio . . . . . . . . . . . . .
4.1.1 State Matrix Sensitivity to Rotational Inertia
4.1.2 State Matrix Sensitivity to Damping . . . . .
4.2 Sensitivity of Transient Overshoot . . . . . . . . . .
4.3 Optimization Algorithm . . . . . . . . . . . . . . . .
4.4 Implementation in MATLAB . . . . . . . . . . . . .
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5 Simulation Results
5.1 IEEE Two-Area Test System . . . . . . .
5.1.1 System Description . . . . . . . . .
5.1.2 Small-Signal Stability Analysis . .
5.1.3 Optimization . . . . . . . . . . . .
5.1.4 Transient Stability Analysis . . . .
5.2 IEEE South East Australian Test System
5.2.1 System Description . . . . . . . . .
5.2.2 Small-Signal Stability Analysis . .
5.2.3 Optimization . . . . . . . . . . . .
5.2.4 Transient Stability Analysis . . . .
5.3 Discussion of Simulation Results . . . . .
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6 Conclusions and outlook
89
A Runge-Kutta Methods
91
B Calculation of Initial Steady State
93
C Transmission Network Modelling
95
D Structure of MATLAB input arrays
99
E IEEE South East Australian System
101
Bibliography
107
viii
List of Figures
3.1
3.2
3.3
Thyristor excitation system with AVR [2] . . . . . . . . . . .
Thyristor excitation system with AVR and PSS [2] . . . . . .
Reference frame transformation . . . . . . . . . . . . . . . . .
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17
27
4.1
Structure of the developed optimization program . . . . . . .
51
Two-area test system [2] . . . . . . . . . . . . . . . . . . . . .
Frequency response to disturbances at buses 1 (blue) and 3
(green) of the two-area system in Base Case . . . . . . . . . .
5.3 Frequency response to disturbances at buses 1 (blue) and 3
(green) of the two-area system with the inertia of all machines
reduced by 50% . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Frequency response to disturbances at buses 1 (blue) and 3
(green) of the two-area system with damping of all the machines reduced by 50% . . . . . . . . . . . . . . . . . . . . . .
5.5 Results of transient overshoot computation in the two-area
system for three different cases. Left: disturbance at bus 1.
Right: disturbance at bus 3. . . . . . . . . . . . . . . . . . . .
5.6 Transient frequency of G1 after a short cirtcuit at bus 9 and
disconnection of a circuit of the line 8-9 of the two-area test
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Transient frequency of G1 after a short cirtcuit at bus 9 and
disconnection of a circuit of the line 8-9 of the two-area test
system (first 5 seconds) . . . . . . . . . . . . . . . . . . . . .
5.8 Rotor angles of the generators G1-G4 of the two-area test
system after a short circuit at bus 9 in Base Case (left) and
Low-Inertia Case (right) . . . . . . . . . . . . . . . . . . . . .
5.9 Rotor angular velocity of the generators of the five-area test
system after a short circuit at bus 217 and disconnection of
a circuit of the line 217-215 in Base Case (left) and Case 1
(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Transient frequency response to a disturbance in the two-area
test system with different values of the time constant Tt . . .
56
5.1
5.2
ix
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86
C.1 A shunt connected to bus k [3] . . . . . . . . . . . . . . . . .
C.2 Lumped-circuit model of a transmission line [3] . . . . . . . .
C.3 Unified branch model [3] . . . . . . . . . . . . . . . . . . . . .
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96
97
E.1 IEEE South East Australian five-area test system [4] . . . . . 102
x
List of Tables
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
System modes with manual excitation control . . . . . . . .
Rotational inertia constant M and damping coefficients of
the two-area system generators in Base Case and Low-Inertia
Case, calculated on the rated MVA base (900 MVA) . . . . .
Eigenvalues of the two-area system in Base Case (left) and
Low-Inertia Case (right). . . . . . . . . . . . . . . . . . . . . .
Results of transient overshoot computation in the two-area
system in Base Case . . . . . . . . . . . . . . . . . . . . . . .
Results of transient overshoot computation in the two-area
system in Low-Inertia Case . . . . . . . . . . . . . . . . . . .
Results of transient overshoot computation in the two-area
system with the damping of all the machines reduced by 50%
Results of transient overshoot computation in the two-area
system with the inertia and damping of all the machines reduced by 50% . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameters of the optimization program for two-area test system (Case 1 - Case 4) . . . . . . . . . . . . . . . . . . . . . .
Parameters of the optimization program for two-area test system (Case 5 - Case 8) . . . . . . . . . . . . . . . . . . . . . .
Optimization results of the two-area test system (Case 1) . .
Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 1) .
Optimization results of the two-area tests system (Case 2) . .
Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 2) .
Optimization results of the two-area test system (Case 3) . .
Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 3) .
Optimization results of the two-area test system (Case 4) . .
Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 4) .
Optimization results of the two-area test system (Case 5) . .
xi
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72
5.19 Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 5) .
5.20 Optimization results of the two-area test system (Case 6) . .
5.21 Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 6) .
5.22 Optimization results of the two-area test system (Case 7) . .
5.23 Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 7) .
5.24 Optimization results of the two-area test system (Case 8) . .
5.25 Values of the inertia constants M and damping coefficients
KD on 900 MVA base in the two-area test system (Case 8) .
5.26 Steady-state operating condition of the five-area test system .
5.27 Rotational inertia constants M and damping coefficients of
the five-area test system generators in Base Case and LowInertia Case, calculated on 100 MVA base . . . . . . . . . . .
5.28 Results of transient overshoot computation in the five-area
test system in Base Case . . . . . . . . . . . . . . . . . . . . .
5.29 Parameters of the optimization program for the five-area test
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.30 Optimization results of the five-area test system (Case 1) . .
5.31 Values of the inertia constants M and damping coefficients
KD on 100 MVA base in the five-area test system (Case 1) . .
5.32 Optimization results for the five-area test system (Case 2) . .
5.33 Values of the inertia constants M and damping coefficients
KD on 100 MVA base in the five-area test system (Case 2) . .
5.34 Optimization results for the five-area test system (Case 3) . .
5.35 Values of the inertia constants M and damping coefficients
KD on 100 MVA base in the five-area test system (Case 3) . .
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83
D.1 Bus data structure (BUSES) . . . . . . . . . . . . . . . . . . 99
D.2 Branch data structure (LINES) . . . . . . . . . . . . . . . . . 100
D.3 Generator data structure (GENS) . . . . . . . . . . . . . . . . 100
E.1 Power flow input data for IEEE South Australian test system
[4] calculated on 100 MVA base . . . . . . . . . . . . . . . . .
E.2 Parameters of the branches of IEEE South Australian test
system [4] calculated on 100 MVA base . . . . . . . . . . . . .
E.3 Parameters of the aggregated synchornous machines of IEEE
South East Australian . . . . . . . . . . . . . . . . . . . . . .
E.4 Eigenvalues of the South East Australian test system in Base
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.5 Eigenvalues of the South East Australian system in LowInertia case . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
103
104
104
105
106
List of Acronyms
AC
AVR
BESS
HVDC
IEEE
PFC
PSS
RES
R-K
SMIB
SVC
Alternating Current
Automatic Voltage Regulator
Battery Energy Storage System
High Voltage Direct Current
Institute of Electrical and Electronics Engineers
Primary Frequency Control
Power System Stabilizer
Renewable Energy Sources
Runge-Kutta
Single Machine Infinite Bus
Static VAR Compensator
xiii
xiv
List of Symbols
A
AS
B
C
D
The
The
The
The
The
state matrix
system state matrix
control matrix
output matrix
feedforward matrix
u
x
y
Ig
V
The
The
The
The
The
vector of input variables
state variables vector
vector of output variables
vector of generator currents
vector of nodal voltages
λi
σi
ωi
ζi
φi
ψi
A state matrix eigenvalue
A real part of an eigenvalue
An imaginary part of an eigenvalue
A system mode damping ratio
A state matrix right eigenvector
A state matrix left eigenvector
J
KD
KA
KSTAB
M
Rfd
R1d
R1q
R2q
The total moment of inertia of a synchronous machine
A damping coefficient
The AVR gain
The PSS gain
The mechanical starting time of a synchronous machine (rotational inertia constant)
The resistance of the field circuit of a synchronous machine
The resistance of the d-axis damping circuit of a synchronous machine
The resistance of the first q-axis damping circuit of a synchronous machine
The resistance of the second q-axis damping circuit of a synchronous machine
xv
S
TR
Tt
TW
T1
T2
Xfd
X1d
X1q
X2q
Xadu
Xads
Xaqs
δ
Efd
ed
eq
id
iq
ifd
i1d
i1q
i2q
Ψad
Ψaq
Ψfd
Ψ1d
Ψ1q
Ψ2q
v1
Vref
v2
vs
∆Pm
∆ωr
ω0
The droop of PFC
The AVR time constant
The turbine time constant
The PSS washout block time constant
A PSS phase compensation block time constant
A PSS phase compensation block time constant
The inductance of the field circuit of a synchronous machine
The inductance of the d-axis damping circuit of a synchronous machine
The inductance of the first q-axis damping circuit of a synchronous machine
The inductance of the second q-axis damping circuit of a synchronous machine
The unsaturated mutual inductance between the stator and d-axis rotor circuits
of a synchronous machine
The saturated mutual inductance between the stator and d-axis rotor circuits
of a synchronous machine
The saturated mutual inductance between the stator and q-axis rotor circuits
of a synchronous machine
The rotor angle of a synchronous machine
The field circuit voltage of a synchronous machine
The d-axis component of the terminal voltage of a synchronous machine
The q-axis component of the terminal voltage of a synchronous machine
The d-axis component of the terminal current of a synchronous machine
The q-axis component of the terminal current of a synchronous machine
The current of the field circuit of a synchronous machine
The current of the d-axis damping circuit of a synchronous machine
The current of the first q-axis damping circuit of a synchronous machine
The current of the second q-axis damping circuit of a synchronous machine
The mutual flux linkage between the stator and d-axis rotor circuits
of a synchronous machine
The mutual flux linkage between the stator and q-axis rotor circuits
of a synchronous machine
The flux linkage of the field circuit of a synchronous machine
The flux linkage of the d-axis damping circuit of a synchronous machine
The flux linkage of the first q-axis damping circuit of a synchronous machine
The flux linkage of the second q-axis damping circuit of a synchronous machine
The AVR output voltage
The AVR reference voltage
The PSS washout block output voltage
The PSS phase compensation block output voltage
The adjustment of the mechanical power of a machine by means of PFC
The relative angular velocity of the rotor of a synchronous machine
The synchronous electrical angular velocity
xvi
Vk
θk
Y
Ykm
αkm
The
The
The
The
The
G(s)
k
Mpl
A transfer function
The approximated magnitude of the transient overshoot at bus l
after a disturbance at bus k
A residue of the transfer function at pole s = λi
The first peak time of the dominating oscillatory mode after a disturbance
at bus k observed at bus l
The time domain response to a disturbance at bus k observed at bus l
Rlik
tkpl
ylk (t)
magnitude of the nodal voltage at bus k
angle of the nodal voltage at bus k
admittance matrix of a transmission network
magnitude of Y element in the k-th row and m-th column
angle of Y element in the k-th row and m-th column
xvii
xviii
Chapter 1
Introduction
1.1
Background and Literature Overview
High penetration of renewable energy sources (RES), such as wind and photovoltaic power plants, creates a number of challenges for the operation of
power systems. First of all, intermittent generators introduce uncertainty
into dispatch schedule of a power system, which makes balancing between
generation and load more complicated. Furthermore, they affect the dynamic behaviour of the system since they normally do not provide any rotational inertia.
Inertia is an inherent property of synchronous generators, and frequency
dynamics of the system within the first seconds after a disturbance is governed by inertial response of the rotating machines. For reliable operation of
a power system, the operating frequency should be kept close to its nominal
value. To ensure this, generated power should match power demanded by
the load devices. Any disturbance in the grid leads to an imbalance between
produced and consumed electrical power. Before the activation of primary
frequency control, this imbalance is compensated by the kinetic energy released to the grid (or drawn from it) by rotating masses. In case of a severe
disturbance, if the power mismatch is not eliminated sufficiently fast by the
protection systems, generators of the system might lose synchronism with
the rest of the system. The loss of stability may lead to major consequences,
such as damage of equipment and widespread outages.
Inertia of the machines defines the rate of their acceleration or deceleration and, thus, the rate of the frequency deviation. High level of rotational
inertia in the system prevents the system frequency from changing too fast
after a disturbance.
Power output of converter-connected RES is usually decoupled from the
system frequency, and they do not contribute to the inertial response. The
same is true for the operation of converter-connected motor loads. This leads
to reduction of inertia levels and thus results in faster frequency dynamics.
1
2
CHAPTER 1. INTRODUCTION
The speed of primary frequency control might become insufficient to prevent
large frequency deviations. Furthermore, rotational inertia starts to vary in
time and space which complicates the dynamics of the system [5].
Reduced levels of inertia lead to low frequency in the Nordic power system [6] in the last years. Lower level of the system frequency after the loss
of a large production unit is believed to be caused by a reduction of number
of on-line synchronous generators which affects the amount of inertia and,
thus, power regulation.
To mitigate arising difficulties, [5] proposes faster primary frequency control and the procurement of synthetic rotational inertia. Utilization of battery energy storage systems (BESS) for provision of fast primary frequency
control is investigated in [7] and [8]. [7] shows the advantages of faster
frequency control for a system with reduced inertia levels.
Synthetical inertial response, as a new ancillary service, is recommended
by Irish TSOs in [9] and by an Independent System Operator of Texas, U.S.,
ERCOT [10]. Provision of inertial response by wind turbines is proposed in
[11]. In case of a large generation unit loss, the power output of the wind
turbine can be increased by about 5-10% of rated power for several seconds.
Inertial response as a service provided by RES is also suggested by [12].
In [1], effects of inertia changes on damping of power system modes
and frequency transients are investigated. Lower inertia improves damping
of power system modes but may lead to higher frequency deviations. [1]
proposes an optimization algorithm that allows to find a trade-off between
improved damping of oscillatory modes and sufficiently limited transient
frequency deviations by adjusting inertia and damping levels at the system
nodes. The algorithm is based on the “Classical Model” [2] of a synchronous
machine.
1.2
Research Objectives
The aim of the present thesis is to investigate the impact of inertia changes on
damping of oscillatory modes and frequency stability using a detailed model
of synchronous machine, including operation of automatic voltage regulator (AVR), power system stabilizer (PSS), and primary frequency control
(PFC).
Within this work, a detailed model of synchronous machine is incorporated in the multi-machine stability analysis, along with the interconnecting
transmission network model and aggregated load model. System equations
are derived and linearized for the small-signal stability analysis; and the
system state matrix is computed. Sensitivities of damping ratios and transient frequency overshoot are derived based on [1]. Optimization algorithm
is formulated as in [1] and tested on two test systems using a number of
different cases. The objective of the optimization is maximization of the
1.3. STRUCTURE
3
minimal damping ratio of system modes under a transient frequency deviation constraint. Procurement of both artificial inertia and damping incurs
costs. The optimization program defines optimal levels of inertia and damping which can be used as a planning tool for synthetical inertial response
and fast frequency response provision. It can also serve to define stability
margin of a power system under different RES-share conditions. Results of
transient simulations are provided to compare the time-domain response of
the test systems in different inertia cases.
1.3
Structure
This thesis is organized as follows: Chapter 2 briefly reviews the power system stability fundamentals. Chapter 3 presents modelling of synchronous
machine, transmission network, and aggregated load for the rotor angle stability studies. System equations are formulated and system state matrix is
derived. Chapter 4 develops an optimization algorithm focused on improvement of the damping of system modes under a transient frequency overshoot
constraint. Sensitivities of damping ratio and frequency overshoot to inertia and damping changes are derived. Furthermore, implemenation of the
algorithm in MATLAB is described. Chapter 5 investigates the small-signal
stability of two test systems for various RES penetration cases and implements the developed optimization algorithm. The impact of rotational
inertia changes on stability of the test systems is illustrated by providing
the results of transient simulations. Finally, a conclusion and an outlook of
the present work are given in Chapter 6.
4
CHAPTER 1. INTRODUCTION
Chapter 2
Power System Stability
Fundamentals
2.1
Definitions and Classification
Power systems are designed to provide a reliable access to electrical energy.
Power system stability, as an ability of a power system to withstand diverse
disturbances, is crucial to the reliability of power supply. The following
definition of power system stability was elaborated by IEEE/CIGRE Task
Force [13]:
Power system stability is the ability of an electric power system, for
a given initial operating condition, to regain a state of operating equilibrium
after being subjected to a physical disturbance, with most system variables
bounded so that practically the entire system remains intact”.
The three main categories of power system stability are rotor angle stability, voltage stability and frequency stability. The main focus of this work
is on rotor angle stability and frequency stability.
Frequency stability refers to [13] “the ability of a power system to
maintain steady frequency following a severe system upset resulting in a
significant imbalance between generation and load”. An example of shortterm frequency instability is the formation of an island with insufficient
generation followed by the blackout of this island within a few seconds due
to a rapid decrease of frequency [13]. For the reliable operation of the system,
the probability of large frequency excursions should be minimized.
Rotor angle stability is defined by [13] as ”the ability of synchronous
machine of an interconnected power system to remain in synchronism after being subjected to a disturbance. It depends on the ability to maintain/restore equilibrium between electromagnetic torque and mechanical
torque of each synchronous machine in the system. Instability that may
result occurs in the form of increasing angular swings of some generators
leading to their loss of synchronism with other generators”.
5
6
CHAPTER 2. POWER SYSTEM STABILITY FUNDAMENTALS
Rotor angle stability analysis involves the analysis of the effect of small
disturbances on the system of interest (small-signal stability) and the dynamic behaviour of the system subjected to a large disturbance (transient
stability).
Small-signal stability is the ability of the power system to maintain
synchronism under small disturbances. A great number of small disturbances occur in a system during its normal operation. They are primarily
caused by the constant variation of demanded and generated power. The
disturbances are considered to be sufficiently small to enable linearization
of the system equations for the purposes of analysis.
Small-signal stability problems could be divided in two groups, local and
global. Local problems are associated with rotor angle oscillations of a small
part of the system. As an example, generators of a certain power plant may
be oscillating against the rest of the power system. This type of oscillations
is called local plant mode oscillations. Other local problems that might
occur in a power system include interplant mode oscillations, control modes
and torsional mode oscillations [2].
Global small-signal stability problems are caused by oscillations involving a large group of generators. The oscillations of generators in one area
swinging against generators in another area are reffered to as interarea
mode oscillations. In large power systems, usually there are two forms of
interarea oscillations [2]:
• An oscillation mode with a very low frequency (0.1-0.3 Hz) that involves all the generators in the system. Generators of the interconnected system are split in two groups, with one of the groups swinging
against another.
• Higher frequency oscillation modes (0.4-0.7 Hz) representing the swings
of subgroups of machines against each other.
Transient stability is the ability of the power system to maintain synchronism when subjected to a severe transient disturbance, e.g. a short
circuit on a transmission line. Whether a system remains stable or not after
a large disturbance, depends on the initial state of this system and the severity of the disturbance. If a disturbance leads to the rotor angle separation
of a part of the machines, the system loses its stability.
Both small-signal stability of the system under possible operating conditions and transient stability in various contigency scenarios should be thoroughly analyzed to ensure the secure operation of a power system. Such
analysis is based on the state-space representation of the power system and
its dynamic behaviour.
2.2. STATE-SPACE REPRESENTATION
2.2
7
State-Space Representation
The state of a system represents the minimal amount of information about
the system at any instant in time t0 that is necessary so that its future
behaviour can be determined without reference to the input before t0 [2].
The variables chosen to describe the state of a system are referred to as the
state variables. The choice of the state variables is not unique, any chosen
set will give the same information about the system.
The system state may be represented in an n-dimensional Euclidean
space referred to as the state space.
For the purpose of stability analysis, a power system in a dynamic state
can be described by a set of first order differential and algebraic equations
ẋ = f (x, u, t)
(2.1)
y = g(x, u, t)
where x is the state vector with the state variables as elements, u is the
vector of inputs to the system, y is the vector of output variables, f and g
are vectors of nonlinear functions relating ẋ and y to x and u. With n as
the order of the system of differential equations, r as the number of inputs,
and m as the number of output variables, the vectors have the following
form
 
 
 
x1
u1
f1
 x2 
u2 
 f2 

 
 
x=
(2.2)
 ...  u =  ...  f =  ... 
xn
ur
fn


 
y1
g1
 y2 
 g2 

 
y=
 ...  g =  ... 
ym
gm
(2.3)
In the rotor angle stability analysis, Equations (2.1) should represent
the dynamics of the power system in the time-scale relevant to rotor swings
(0.01 s - 10 s). The dynamic behaviour of the power system components,
namely generators, transmission network, static and dynamic loads, static
VAR compensators (SVC), etc., should be reflected adequately to the analysis scope. Among the mentioned components, modelling of the synchronous
generators plays certainly the most important role for the investigation of
the rotor angle stability. Quite often the dynamic behaviour of a system
is described only by the differential equations associated with synchronous
generators, whereas all the other components are represented by algebraic
equations. For instance, the transient processes occuring in transmission
8
CHAPTER 2. POWER SYSTEM STABILITY FUNDAMENTALS
lines after a contigency decay too fast to be included in the analysis of
electro-mechanical swings.
Since power systems are highly nonlinear, their stability after disturbances depends not only on their parameters but also on the characteristics
of the disturbance and on the initial operating state of the system. Therefore, to find a unique solution of the system equations within the transient
stability analysis, one should specify the initial conditions and accurately
model the disturbance. Thus, to get a general view on the dynamic features
of the system by means of transient stability analysis, a great number of
disturbances in different locations should be investigated.
However, Henri Poincare showed that if the linearized form of the nonlinear system is stable, so is the non-linear system stable at the steady-state
operating condition at which the system is linearized [14]. Furthermore,
the dynamic features of the system at the given operating condition can be
assessed from linear control system theory, and the response of the system
to small disurbances can be approximated. Therefore, small-signal stability
analysis is used to investigate the dynamic characteristics of the system,
with the main focus on the system modes.
2.3
Small-Signal Stability
To investigate the effect of small disturbances on a power system, the system
equations (2.1) could be linearized around the initial operating point of the
system.
Linearization of (2.1) around an equilibrium point with x = x0 and
u = u0 and implementation of Taylor’s series expansion yield
∆ẋ = A∆x + B∆u
(2.4)
∆y = C∆x + D∆u
(2.5)
where

∂f1
∂x1

A =  ...
∂fn
∂x1

∂g1
∂x1

C =  ...
∂gm
∂x1
...
...
...
...
...
...
A is the state matrix, n × n
B is the control matrix, n × r
∂f1
∂xn


... 
∂fn
∂xn
∂g1
∂xn


∂f1
∂u1

B =  ...
∂fn
∂u1

∂g1
∂u1


...  D =  ...
∂gm
∂xn
∂gm
∂u1
...
...
...
...
...
...
∂f1
∂ur


... 
∂fn
∂ur
∂g1
∂ur


... 
∂gm
∂ur
(2.6)
2.3. SMALL-SIGNAL STABILITY
9
C is the output matrix, m × n
D is the feedforward matrix, m × r
The system (2.4) is a system of linear differential equations in terms of
perturbed variables. The perturbations of the variables from their initial
values must be sufficiently small to enable the approximation of the nonlinear
functions with the first term of Taylor’s series expansion.
Analysis of the state matrix A allows to draw the conclusions about
the stability of an underlying nonlinear system, as stated in the theorem
formulated by Alexander Lyapunov.
Lyapunov’s first method [2]
The stability in the small of a nonlinear system is given by the roots of the
characteristic equation of the system of first approximations, i.e., by the
eigenvalues of A:
• If all the eigenvalues have negative real parts, the original system is
asymptotically stable, i.e. it returns to the original state after being
subjected to a small perturbation.
• If at least one of the eigenvalues has a positive real part, the original
system is unstable.
• If the eigenvalues have real parts equal to zero, it is not possible on
the basis of the first approximation to say anything in general.
The characteristic equation of the state matrix A is given by
det(A − λI) = 0
(2.7)
where I is the identity matrix, and λ = λ1 , λ2 , ..., λn are eigenvalues of the
state matrix
If the column vector φi satisfies
Aφi = λi φi
(2.8)
it is referred to as the right eigenvector of the state matrix A associated
with λi . The n-row vector ψ i that satisfies
ψ i A = λi ψ i
(2.9)
is called the left eigenvector of A associated with λi .
The product of the right and left eigenvectors associated with the same
eigenvalue is a non-zero constant. Often the eigenvectors are normalized as
follows
ψ i φi = 1
(2.10)
10
CHAPTER 2. POWER SYSTEM STABILITY FUNDAMENTALS
The natural response of the system (when ∆u = 0) is given by the solution
of
∆ẋ = A∆x
(2.11)
∆xi (t) = φi1 c1 eλ1 t + φi2 c2 eλ2 t + ... + φin cn eλn t
(2.12)
as
Thus, the natural response of the system can be represented as a linear
combination of n dynamic modes. In Equation (2.12), ci = ψ i ∆x(0) is the
magnitude of excitation of the i-th mode defined by the initial conditions.
Each eigenvalue is associated with a dynamic mode, and the characteristics
of the eigenvalues are related to the nature of the modes:
• A real eigenvalue is associated with a non-oscillatory mode. Negative
real eigenvalues correspond to exponential decay modes. The smaller
the magnitude of a negative eigenvalue, the longer it takes for the
mode to decay. Positive eigenvalues represent aperiodic instability.
• A conjugate pair of complex eigenvalues is associated with an oscillatory mode. The imaginary part of a complex eigenvalue represents
the frequency of the oscillations, and the real part is associated with
the damping of the oscillations. A negative real part gives an exponentially decaying magnitude of the mode. Complex eigenvalues with
a positive real part represent oscillations with a growing magnitude,
i.e. an unstable oscillatory mode .
A conjugate pair of complex eigenvalues can be presented as
λ = σ ± jω
(2.13)
The damping of the oscillations is evaluated by means of the damping ratio
ζ = −√
σ
σ2 + ω2
(2.14)
The damping ratio of a decaying oscillatory mode should stay within the
limits
0<ζ<1
(2.15)
Ensuring that the damping of oscillatory modes in the system is sufficient
for a stable operation of the system within a range of possible operating
conditions is one of the concerns of the system operators. Another primary
concern with regards to stability is the stability of the system after major
disturbances.
2.4. TRANSIENT STABILITY
2.4
11
Transient Stability
In transient stability analysis, nonlinear ordinary differential equations of
the form
dx
= f (x, t)
(2.16)
dt
should be solved to investigate the effect of the large disturbances of interest
on stability of the system. The solution of (2.16) is the change of the state
variables x in time t from their initial values x0 at t0 .
It would be a challenging task to find an analytical solution of (2.16) even
for a very simple system [3]. Therefore, a number of qualitative methods
was developed that serve to define whether a system can remain stable after
a given disturbance (e.g. Equal Area Criterion, see [3]). However, when
the main purpose of the research is to trace the behaviour of the state
variables after a contigency, these methods will not give sufficient results.
In this case, (2.16) should be solved by the methods of numerical integration.
The numerical integration methods used in this work are the second order
Runge-Kutta (R-K) method and the fourth order R-K method, presented in
Appendix A.
Dynamic phenomena in power systems have a complex electromagnetic
and mechanical nature. The simplest model of electro-mechanical swings in
a power system represents solely the motion mechanics of the synchronous
machine rotors and is based on the swing equation:
J
d2 δm
= Tm − Te
dt2
(2.17)
where
J is the total moment of inertia of the synchronous machine
δm is the mechanical angle of the rotor
Tm is the mechanical torque on the rotor
Te is the electrical torque on the rotor
If a power system is in a normal operational state, the balance between
generated and consumed power is maintained, and all the synchronous generators are rotating with the same electrical angular velocity. However, a disturbance, such as a transmission line failure, can lead to imbalance between
electromagentic and mechanical torques at the rotor of a machine. This imbalance causes acceleration (if more power is generated than demanded) or
deceleration (when generated power is not enough to cover the demand) of
the rotor of a synchronous generator. In case of a severe disturbance, one or
more generators can lose synchronism with the rest of the system. This may
have major consequences for operation of the system, including damage of
the equipment, economical losses, and substantial outages.
12
CHAPTER 2. POWER SYSTEM STABILITY FUNDAMENTALS
A major contigency, as a rule, triggers the relay protection of the power
system. This is necessary, above all, for the following purposes:
• to isolate the fault and thus ensure normal operational conditions for
as much equipment as possible,
• to avoid the damage of the equipment by the high currents,
• to prevent the loss of synchronism of the generators.
An example of a severe contigency is a short circuit on a transmission line
close to a generator. When it occurs, it should be cleared by opening the
circuit-breakers at both ends of the line. But this can not happen immediately because of the time necessary for the operation of a circuit-breaker.
Meanwhile, the rotor of the generator would accelerate due to the imbalance
between mechanical and electrical power (Pm > Pe ). Depending on the level
of damping, the magnitude of the disturbance, and the fault clearing time,
the rotor will settle at a new equilibrium point or the generator will fall out
of step. The faster the fault is cleared, the less kinetic energy the rotor gets
for acceleration. The critical fault clearing time is the maximal duration of a disturbance during which the system does not lose its synchronism.
It is an important characteristic for design and operation of a power system, which depends on many factors, including the rotational inertia of the
generators in the system.
Chapter 3
Modelling of Power System
3.1
Synchronous Machine Modelling
In the present thesis, modelling of synchronous machines and their excitation
systems is based on [2]. The adopted synchronous machine model involves
the effects of AVR and PSS on the field voltage. Furthermore, the model is
augmented by implementation of PFC.
The structure of this section repeats the development of a model from
the “Classical Model” to the tenth order model, which incroporates voltage
and speed control. In the end of the section, a complete set of differential
and algebraic equations for synchronous machine representation in stability
studies is presented. For the sake of brevity, derivation of these equations is
not included in the present work and could be reviewed in [2].
3.1.1
Swing Equation
Changes in electrical state of a system affect the rotation of electrical machines and thus cause electro-mechanical oscillations.
The mechanical power Pm = Tm ωm , with ωm denoting the mechanical
angular velocity of the rotor, is provided to a synchronous machine by a
turbine and can be adjusted by changing the gate opening of the turbine.
To maintain a constant angular velocity of the rotor, the applied mechanical power should be balanced with the electrical power extracted from the
machine.
The electrical power Pe = Te ωm is a function of both rotor angle δ and
its time derivative δ̇. The latter contribution is associated with the damping
of electromechanical oscillations due to the currents induced in the rotor
circuits during transients.
Equation (2.17) could be rewritten in terms of power as
ωm J
d2 δm
= Pm − Pe
dt2
13
(3.1)
14
CHAPTER 3. MODELLING OF POWER SYSTEM
To express the moment of inertia in electrical p.u. quantities, the inertia
constant H should be introduced as
H=
2J
1 ωm
stored energy at rated speed in MW · s
=
2 S
MVA rating
(3.2)
where S is the MVA rating of the machine. The inertia constant shows how
much time it would take for a machine to decelerate from synchronous speed
to standstill if rated power is extracted from it and no mechanical power is
fed into it [3].
Another quantity that is broadly used in the literature is called the
mechanical starting time M , defined as
M = 2H
(3.3)
Rewriting Equation (3.1) in p.u. of the synchronous machine rating and
taking account of damping by introducing the term −KD δ̇ yield
2H d2 δ
= Pm − Pe − KD δ̇
ω0 dt2
(3.4)
where
KD - damping coefficient in p.u. torque/p.u. speed deviation
ω0 - synchronous electrical angular velocity of the rotor
Equation (3.4) is commonly reffered to as swing equation, as it represents swings in rotor angle during disturbances.
Using the following notation for the relative angular velocity in p.u.
∆ωr =
1 dδ
ω0 dt
(3.5)
the swing equation can be rewritten in the form of a system of first order
differential equations:
1
(Pm − Pe − KD ∆ωr )
M
pδ = ω0 ∆ωr
p∆ωr =
(3.6)
(3.7)
where p stands for the differential operator d/dt.
The quantities δ and ∆ωr are in this case state variables and
x = [δ ∆ωr ]T
(3.8)
is the state vector.
Differential equations (3.6) are fundamental for power system dynamics analysis and, by supplementing them with a set of algebraic equations,
3.1. SYNCHRONOUS MACHINE MODELLING
15
one can analyze the stability of a system. This modelling approach was
widely used in the early stability studies. Therefore, it is often referred to
as ”Classical Model”. However, such a model does not take into account
the electromagnetic dynamics of the machine, such as dynamics of the rotor
circuits and effects of the voltage control devices on the field voltage. To incorporate the specified dynamic effects in the model, additional differential
equations are formulated further in this section.
3.1.2
Representation of Synchronous Machine Rotor Circuits Dynamics
A disturbance in a power system leads to the rise of transient processes
associated with a change in electrical quantities. Transients in the stator
windings decay rapidly and thus can be neglected in most of the cases,
whereas transients in the rotor circuits could not be neglected when the
system is subjected to a disturbance [2]. Dynamics of the rotor circuits
could be presented in form of the flux variation differential equations (3.93.12). The flux variations in the rotor circuits originate in the armature
reaction, i.e. in the effect of the stator field on the rotor currents.
ω0 Rfd
Efd − ω0 Rfd ifd
Xadu
= −ω0 R1d i1d
pΨfd =
pΨ1d
(3.9)
(3.10)
pΨ1q = −ω0 R1q i1q
(3.11)
pΨ2q = −ω0 R2q i2q
(3.12)
where the subscripts ”fd”, ”1d”, ”1q”, ”2q” stand for the quantities of the field
circuit, d-axis damping circuit, and q-axis damping circuits respectively. Ψ
denotes the flux linkage of a circuit, i designates the circuit current, R is the
resistance of a circuit, Efd is the exciter output voltage, ω0 is the synchronous
angular velocity, and Xadu stands for the unsaturated mutual impedance.
Thus, the state vector should be augmented by the flux linkages of the
rotor circuits
x = [δ ∆ωr Ψfd Ψ1d Ψ1q Ψ2q ]T
(3.13)
In a simplified stability analysis, the field voltage Efd might be assumed
constant (manually adjusted), but in modern power systems this assumption
does not conform with the reality due to the operation of AVR. If the field
voltage is controlled by AVR, the field flux variations are also caused by the
field voltage variations, in addition to the armature reaction. Modelling of
the excitation system and AVR for the system stability analysis is covered
by the next section.
16
3.1.3
CHAPTER 3. MODELLING OF POWER SYSTEM
Effects of Excitation System and Automatic Voltage
Regulation
The excitation system of a synchronous machine provides its field winding with direct current and performs control and protective functions by
changing the field voltage. AVR controls the generator stator terminal
voltage by adjusting the exciter output voltage and thus the field current.
Modern producers offer various types of excitation systems and AVRs. In
the present thesis, the excitation system called potential-source controlledrectifier (thyristor) excitation system is considered. This system is supplied
with power through a transformer from the generator terminals or the station auxiliary bus, and is regulated by a controlled rectifier.
A block diagram providing a simplified illustration of the operational
principle of this system is shown in Figure 3.1.
Et
Terminal voltage
transducer
1
1  sTR
Vref
v1

∑

EFMAX
Exciter
KA
EFMIN
E fd
Figure 3.1: Thyristor excitation system with AVR [2]
The first block of the diagram represents the terminal voltage transducer.
It measures terminal voltage of the machine (Et ), rectifies and filters it with
an output
1
v1 =
Et
(3.14)
1 + pTR
Equation (3.14) could be rearranged to get the time derivative of v1 at the
left side:
1
(Et − v1 )
(3.15)
pv1 =
TR
This differential equation supplements the swing equation and Equations
(3.9-3.12) in modelling of the dynamic behaviour of a synchronous machine.
The voltage v1 should be therefore added to the state vector (3.13)
x = [δ ∆ωr Ψfd Ψ1d Ψ1q Ψ2q v1 ]T
(3.16)
The output quantity of the terminal voltage transducer v1 is compared
to the reference voltage Vref , that could be adjusted manually or by means
of Secondary Voltage Regulation of the grid.
3.1. SYNCHRONOUS MACHINE MODELLING
17
The residual signal (Vref − v1 ) is amplified by an exciter with a high gain
KA (block 2) yielding the output voltage
Efd = KA (Vref − v1 )
(3.17)
The value of the exciter output voltage is subject to a limitation
EFMIN ≤ Efd ≤ EFMAX
(3.18)
Since Efd is not assumed manually adjusted anymore, Equation (3.9)
should be changed to take account of (3.17):
pΨfd =
ω0 Rfd
KA (Vref − v1 ) − ω0 Rfd ifd
Xadu
(3.19)
The operation of AVR may significantly affect stability of the system.
In many cases, a high gain exciter introduces negative damping, thus endangering system stability. At the same time, a high response AVR has a
positive effect on the synchronizing torque. An effective way to benefit from
this advantage, while keeping damping torque at acceptable level, is to use
a PSS.
3.1.4
Power System Stabilizer
In Figure 3.2, the block diagram of the thyristor excitation system is extended to include the three blocks (a gain block, a washout block, and a
phase compensation block) that represent PSS.
Terminal voltage
transducer
1
1  sTR
Et
r
Gain
KSTAB
Washout
sTW
1 sTW
v2
Phase
compensation
1  sT1
1  sT2
Vref
v1

∑


EFMAX
Exciter
KA
EFMIN
E fd
vs
Figure 3.2: Thyristor excitation system with AVR and PSS [2]
A gain block senses the value of the angular velocity deviation from
the synchronous speed (∆ωr ,) and with the gain KSTAB , it sets the level
of damping introduced by the PSS. The output signal of the gain block is
processed by the washout block with a time constant TW that serves as a
high-pass filter.
18
CHAPTER 3. MODELLING OF POWER SYSTEM
The main purpose of a washout block is to eliminate the influence of
steady-state or slow changes in the system frequency on the operation of
PSS. According to Figure 3.2, the output voltage of the washout block v2 is
defined as
pTW
(KSTAB ∆ωr )
(3.20)
v2 =
1 + pTW
Hence
pv2 = KSTAB p∆ωr −
1
TW v2
(3.21)
Substition for p∆ωr , given by (3.6), yields
pv2 =
KSTAB
1
v2
(Pm − Pe − KD ∆ωr ) −
M
TW
(3.22)
A phase compensation block serves to compensate for the phase lag between the exciter input and the air-gap torque of the generator. The phase
characteristic of the system depends on its state, and the settings of PSS
should be acceptable for a wide range of possible system conditions.
From Figure 3.2,
1 + pT1
vs =
v2
(3.23)
1 + pT2
Hence
pvs =
1
1
T1
pv2 + v2 − vs
T2
T2
T2
(3.24)
With pv2 given by (3.22), (3.24) can be rewritten as
pvs =
T1 KST AB
1
T1 1
1
(Pm − Pe − KD ∆ωr ) + ( −
)v2 − vs
T2 M
T2 T2 TW
T2
(3.25)
The value of vs is subject to a constraint
vs
min
≤ vs ≤ vs
max
(3.26)
A new expression for the exciter output voltage according to Figure 3.2 is
Efd = KA (Vref + vs − v1 )
(3.27)
Thus, the differential equation for the flux linkage of the field winding should
be adjusted once more:
pΨfd =
ω0 Rfd
KA (Vref + vs − v1 ) − ω0 Rfd ifd
Xadu
(3.28)
The system of the synchronous machine differential equations is now
expanded with (3.22) and (3.25), and v2 and vs should be added to the state
vector
x = [δ ∆ωr Ψfd Ψ1d Ψ1q Ψ2q v1 v2 vs ]T
(3.29)
3.1. SYNCHRONOUS MACHINE MODELLING
3.1.5
19
Primary Frequency Conrol
A disturbance such as the loss of a generator leads to negative values of the
residual ∆Pm −∆Pe and, consequently, to a decrease of the system frequency.
According to the swing equation (3.6), the angular velocity deviation will
rise untill the disbalance between the mechanical and electrical torque is
eliminated.
Positive values of ∆Pm − ∆Pe can be provoked by the loss of a bulk
load, e.g. in case of the islanding of an area with a lot of generation units.
Furthermore, unpredictable variations of load within the normal operation
of the system may affect the system frequency too. Nevertheless, the system
frequency should be kept at an acceptable level. First of all, low values of
the system frequency may threaten a normal operation of the system. If
system frequency is below 47 - 48 Hz (with 50 Hz as the nominal system
frequency), steam turbines can be damaged, and, therefore, they should be
disconnected by the protection system. This would lead to a further decrease
of the frequency and may result in a collapse of the system. In addition,
the maintanence of the nominal frequency is required to ensure satisfactory
operation of many consumer devices.
To compensate the power disbalance and control the frequency, it is
necessary to provide a power system with frequency control. The control reserves are divided among primary, secondary and tertiatry frequency control.
The first two operate automatically, while the tertiary control is activated
manually to release control reserves used by the primary and secondary control in response to a disturbance. Primary frequency control serves to adjust
the turbine power of the machine in order to achieve a balance between the
mechanical and electrical power. The resulting frequency may significantly
differ from 50 Hz. To bring the frequency back to its nominal value, secondary frequency control adjusts the power setpoints of the generators. Since
the main research objective of the present thesis is to investigate short-term
stability, only the primary frequency conrol, as the fastest control structure,
is included into the power system modelling.
The dynamic characteristic of the primary control loop describes the
adjustment of the turbine power ∆Pm in response to the speed deviation
from its nominal value ∆ωr :
1
1
p∆Pm = − ∆Pm −
∆ωr
(3.30)
Tt
STt
where S denotes the droop, a decrease in frequency associated with the
power demand increase, and Tt is the turbine time constant. The latter
value might significantly affect the short-term stability of the system. The
faster the reaction of the frequency control, the less threatening is a power
mismatch for the system.
In the interconnected European power system, primary control reserves
should be deployed within the first 30 s after the activation signal. Thus, the
20
CHAPTER 3. MODELLING OF POWER SYSTEM
turbine time constants should not exceed 10-15 s. Typical values of Tt of the
high-pressure steam turbine are 0.1-0.4s, a re-heater has a larger time delay
(4-11s). The time constant of the delay between the intermediate and low
pressure turbines is in the order of 0.3-0.6s [3]. It should be noted, that (3.30)
describes only one turbine stage and therefore represents a simplified model
of a turbine control. A faster primary frequency control can be provided by
Battery Energy Storage Systems, as shown in [7].
The mechanical power output change ∆Pm completes the state vector
that now consists of 10 state variables:
x = [δ ∆ωr Ψfd Ψ1d Ψ1q Ψ2q v1 v2 vs ∆Pm ]T
(3.31)
Differential equations (3.6,3.22,3.25) should be adjusted to account for the
change in Pm .
3.1.6
Full Set of Differential and Algebraic Equations
The full set of the first order differential equations modelling the dynamic
behaviour of a synchronous machine for the purpose of the stability analysis
is presented by (3.32).
Differential Equations
1
(Pm + ∆Pm − Pe − KD ∆ωr )
M
pδ = ω0 ∆ωr
ω0 Rfd
pΨfd =
KA (Vref + vs − v1 ) − ω0 Rfd ifd
Xadu
pΨ1d = −ω0 R1d i1d
p∆ωr =
pΨ1q = −ω0 R1q i1q
pΨ2q = −ω0 R2q i2q
1
(Et − v1 )
pv1 =
TR
KSTAB
1
pv2 =
(Pm + ∆Pm − Pe − KD ∆ωr ) −
v2
M
TW
T1 KSTAB
1
T1 1
1
pvs =
(Pm + ∆Pm − Pe − KD ∆ωr ) + ( −
)v2 − vs
T2 M
T2 T2 TW
T2
1
1
p∆Pm = − ∆Pm −
∆ωr
Tt
STt
(3.32)
To find a unique solution of this system of differential equations, boundary conditions of the problem should be specified. The mode of operation of
a synchronous machine depends on the power demanded from it and, therefore, on the operational state and parameters of other system elements. The
3.1. SYNCHRONOUS MACHINE MODELLING
21
boundary conditions should relate the internal variables of the machine with
the demanded power output and, thus, with the rest of the power system.
This is achieved if boundary conditions are represented by the stator voltage
equations (3.33).
Stator Voltage Components
ed = −Ra id + Xl iq − Ψaq
(3.33)
eq = −Ra iq − Xl id + Ψad
The demanded power output and the terminal voltage magnitude setpoint determine the generator currents id and iq , and internal variables of
the machine.
Equations (3.32) and (3.33) should be expressed in terms of the state
variables, currents, and terminal voltage magnitudes. Thus, the internal
variables of the machine (rotor currents, flux linkages Ψad and Ψaq , and
electrical power demand Pe ) should be eliminated from (3.32) and (3.33) by
means of Equations (3.34),(3.35) and (3.37).
Rotor Currents
1
(Ψfd − Ψad )
Xfd
1
=
(Ψ1d − Ψad )
X1d
1
(Ψ1q − Ψaq )
=
X1q
1
=
(Ψ2q − Ψaq )
X2q
(3.34)
Ψfd
Ψ1d
+
)
Xfd X1d
Ψ1q
Ψ2q
00
= Xaqs
(−iq +
+
)
X1q X2q
(3.35)
ifd =
i1d
i1q
i2q
Flux Linkages
00
Ψad = Xads
(−id +
Ψaq
where
00
Xads
=
00
Xaqs
=
1
1
Xads
+
1
Xfd
+
1
X1d
1
1
Xaqs
+
1
X1q
+
1
X2q
(3.36)
22
CHAPTER 3. MODELLING OF POWER SYSTEM
Electrical Torque
Since, as already mentioned, in p.u Pe = Te ,
Te = Pe = Ψad iq − Ψaq id
(3.37)
System of Differential and Algebraic Equations for Representation of a Synchronous Machine in Power System Stability Studies
p∆ωr =
Ψ2q
Ψ1q
Ψfd
1
Ψ1d
00
00
(Pm + ∆Pm − Xads
(−id +
+
)id −
+
)iq + Xaqs
(−iq +
M
Xfd X1d
X1q X2q
− KD ∆ωr )
pδ = ω0 ∆ωr
ω0 Rfd
Ψfd
Ψ1d
1
00
pΨfd =
KA (Vref + vs − v1 ) − ω0 Rfd
(Ψfd − Xads
(−id +
+
))
Xadu
Xfd
Xfd X1d
Ψfd
Ψ1d
1
00
pΨ1d = −ω0 R1d
(Ψ1d − Xads
(−id +
+
))
X1d
Xfd X1d
Ψ1q
Ψ2q
1
00
pΨ1q = −ω0 R1q
(Ψ1q − Xaqs
(−iq +
+
))
X1q
X1q X2q
Ψ1q
Ψ2q
1
00
(Ψ2q − Xaqs
(−iq +
+
))
pΨ2q = −ω0 R2q
X2q
X1q X2q
1
(Et − v1 )
pv1 =
TR
KSTAB
Ψfd
Ψ1d
00
pv2 =
(−id +
+
)iq +
(Pm + ∆Pm − Xads
M
Xfd X1d
Ψ1q
Ψ2q
1
00
+ Xaqs
(−iq +
v2
+
)id − KD ∆ωr ) −
X1q X2q
TW
T1 KSTAB
Ψfd
Ψ1d
00
pvs =
(Pm + ∆Pm − Xads
(−id +
+
)iq +
T2 M
Xfd X1d
Ψ1q
Ψ2q
1
T1 1
1
00
+ Xaqs
(−iq +
+
)id − KD ∆ωr ) + ( −
)v2 − vs
X1q X2q
T2 T2 TW
T2
1
1
p∆Pm = − ∆Pm −
∆ωr
Tt
STt
Ψ1q
Ψ2q
00
ed = −Ra id + Xl iq − Xaqs
(−iq +
+
)
X1q X2q
Ψfd
Ψ1d
00
eq = −Ra iq − Xl id + Xads
(−id +
+
)
Xfd X1d
(3.38)
The system (3.38) models the dynamic behaviour of a synchronous generator but it should be supplemented by the initial values of the machine
state variables, since stability of a system significantly depends on its initial
operational state. The expressions for the calculation of the synchronous
machine initial setpoint are presented in Appendix B.
3.2. TRANSMISSION NETWORK MODELLING
23
As the synchonous machine state variables depend on the state of the
interconneting transmission network, the next step in developing a dynamic
power system model is to formulate equations representing the operation of
a transmission grid.
3.2
Transmission Network Modelling
A transmission network connects power plants to the substations supplying
demand centers with electrical energy. If a power system is assumed to
operate in a balanced steady state, each AC power system component can
be represented by its single-phase equivalent. The corresponding models
of AC transmission lines, transformers and shunt devices are presented in
Appendix C. Modelling of the transmission network in the present work is
based on [3] , [2], and [15].
To couple the network model with the generator and load models, the
equations representing power or current injections in the grid nodes should
be formulated. It is a common practice to use the current injection equations, as, for instance, it is done in [2]. However, in this work the power
injection equations were adapted from [15], as they seem to be more intuivite. These equations will be further referred to as Network Equations. A
transmission network can be represented by its admittance matrix (for its
elements see Appendix C)
Y = G + jB
(3.39)
From Kirchhoff’s Current Law, the expression for nodal current injections can be derived as
I =YE
(3.40)
where
I is the current injection vector with elements Ik , k = 1, 2, ..., N
E is the nodal voltage vector with elements Uk ejθk
The complex value of the current injection at bus k is given by
Ik =
X
Vm Ykm ej(θm +αkm )
(3.41)
m∈K
where
Ykm and αkm are the magnitude and angle of the complex element of admittance matrix in k-th row and m-th column.
The admittance matrix Y is usally very sparse but its size can be reduced by means of network reduction. There are several network reduction
techniques, one of the most common is application of Kron’s reduction formula.
24
CHAPTER 3. MODELLING OF POWER SYSTEM
If the current injection at node k, Ik = 0, node k can be eliminated
from the matrix by replacing the elements of the remaining n − 1 rows and
columns with
yik ykj
0
yij
= yij −
(3.42)
ykk
for i = 1, 2, ..., k − 1, k + 1, ...n and j = 1, 2, ..., k − 1, k + 1, ..., n [2].
The complex power injection at bus k is given by
Sk = Pk + jQk = Ek Ik∗
(3.43)
applying (3.41), it yields
Sk = Vk
X
Vm Ykm ej(θk −θm −αkm )
(3.44)
m∈K
Decomposing it into real and imaginary part results in separate equations
for active and reactive power injections, as follows
X
Pk = Vk
Vm Ykm cos(θk − θm − αkm )
(3.45)
m∈K
Qk = Vk
X
Vm Ykm sin(θk − θm − αkm )
(3.46)
m∈K
These equations will be used to represent the coupling of the generator
and load buses with the transmission network.
3.3
Load Modelling
Since any changes in the load demand in a power system should be followed
up by adjusting the power output of the generators, adequate load representation becomes an important step in the power system modelling for stability
studies. Thus, unrealistic models of the load dynamic behaviour could lead
to incorrect evaluation of the power system stability. However, the exact
modelling of loads seems to be impossible since each load bus represents a
changing in time composition of thousands of consumer devices. Therefore,
the load models used in system studies should be a compromise between
simplicity and accuracy. A common practice is to use static load models
such as the polynomial model.
3.3.1
Static Load Models
A static load model expresses the characteristics of the load at any instant
of time as algebraic functions of the bus voltage magnitude and frequency
at that instant [2]. One of the static models which is widely used is the
polynomial model:
P
= P0 [p1 V̄ 2 + p2 V̄ + p3 ]
2
Q = Q0 [q1 V̄ + q2 V̄ + q3 ]
(3.47)
(3.48)
3.3. LOAD MODELLING
25
where
V̄ = VV0 is the relative voltage magnitude at the load bus, P and Q are active
and reactive components of the load when the bus voltage magnitude is V ,
and the subscript 0 stands for their values at the initial operating point.
This model is composed of the following components:
• constant impedance (proportional to the square of the voltage magnitude)
• constant current (proportional to the voltage magnitude)
• constant power (does not vary with changes in the voltage magnitude)
The coefficients p1 to p3 and q1 to q3 define the proportion of each component.
This model relates the demanded power to the bus voltage magnitude
but not to its frequency. The frequency dependence of the load can be
represented by multiplying the right parts of Equations (3.47) and (3.48) by
special factors as follows:
P
= P0 [p1 V̄ 2 + p2 V̄ + p3 ](1 + Kpf ∆f )
2
Q = Q0 [q1 V̄ + q2 V̄ + q3 ](1 + Kqf ∆f )
(3.49)
(3.50)
Utilization of these equations is quite complicated because load bus frequency is not a state variable in stability analysis. Its approximation as an
average frequency of generator buses yields incorrect results and therefore
should be avoided [16]. However, it can be computed by taking the numerical derivative of the bus voltage angle. This approach is not applicable to the
small-signal stability analysis, since this type of analysis does not implicate
calculation of the state variables at more than one time instant.
Another way to model the frequency dependence of the load, based on
[1], is presented further.
3.3.2
Load Damping
The load damping could be represented by a damping coefficient
KD =
∆P
∆P
=
∆f
∆ωr
(3.51)
where
∆P is the change of active power demand due to the change of the bus
frequency ∆f or relative angular velocity ∆ωr , which are equal in p.u. Since
the voltage frequency is a derivative of the voltage angle,
pθ = ω0 ∆ωr =
ω0
∆P
KD
(3.52)
26
CHAPTER 3. MODELLING OF POWER SYSTEM
and actual power injection can be represented by Equation (3.46)
∆P = PL − PL0 = Vk
X
Vm Ykm cos(θk − θm − αkm ) − PL0 ,
(3.53)
m∈K
the differential equation for the load bus voltage angle can be rewritten as
pθk =
X
ω0
(Vk
Vm Ykm cos(θk − θm − αkm ) − PL0 )
KDk
(3.54)
m∈K
Hence, the load bus voltage angle becomes a state variable and its changes
are described by the differential equation (3.54).
3.4
Overall System Equations
In power system stability analysis, the equations (3.38,3.45-3.48) should be
solved simultaneously. In this work, the modelling of the power electronic
equipment, such as HVDC converters, static var compensators, is not covered. If these components are in focus of the analysis, the corresponding
equations should be added to the system model. The transient occuring
in both transmission network and stators of synchronous machines were
neglected, which is a common practice [2], resulting in algebraic, and not
differential, network and stator voltage equations. The synchronous machine motion mechanics, dynamics of rotor circuits, excitation system and
control devices are represented by a set of differential equations. The synchronous machines connected to the same bus are modelled by an equivalent
aggregated synchronous machine, since the dynamic behaviour of individual
machines is out of the focus of the current thesis. This simplification still
provides a sufficient level of accuracy [2].
Each set of synchronous machine equations has its own d − q reference
frame that rotates with the rotor of machine. To enable the simultaneous
solution of these equations for an interconnected multimachine system, voltages and currents should be expressed in a common reference frame. Such
a common reference frame R − I can be chosen to be rotating with the
synchronous speed.
3.4. OVERALL SYSTEM EQUATIONS
I
q
eq
27
Et
EI


ER
ed
d
R
r
0
Figure 3.3: Reference frame transformation
A new reference frame requires a transformation of the algebraic equations (3.33,3.45,3.46), which serve as an interface for the interconnected
generators. From Figure 3.3,
ed = Et sin(δ − θ)
(3.55)
eq = Et cos(δ − θ)
(3.56)
Hence, (3.33) can be expressed in the common reference frame as
Ψ1q
Ψ2q
+
) (3.57)
X1q X2q
Ψfd
Ψ1d
00
(−id +
+
) (3.58)
Et cos(δ − θ) = −Ra iq − Xl id + Xads
Xfd X1d
00
Et sin(δ − θ) = −Ra id + Xl iq − Xaqs
(−iq +
Network equations (3.45 and 3.46) for the generator buses should be
rewritten considering
Sg = Eg Ig∗ = Vg e(jθ) (id − jiq )e−j(δ−π/2)
(3.59)
Pg = Vg [id sin(δ − θ) + iq cos(δ − θ)]
(3.60)
Qg = Vg [id cos(δ − θ) − iq sin(δ − θ)]
(3.61)
and thus
that results in
Vk [idk sin(δk − θk ) + iqk cos(δk − θk )] =
X
Vk
Vm Ykm cos(θk − θm − αkm )
(3.62)
m∈K
Vk [idk cos(δk − θk ) − iqk sin(δk − θk )] =
X
Vk
Vm Ykm sin(θk − θm − αkm )
m∈K
(3.63)
28
CHAPTER 3. MODELLING OF POWER SYSTEM
The network equations for the load buses should be adjusted to include the
static load characteristrics (3.47, 3.48). If the active power component of
the load demand is modelled by a constant current characteristic, and the
reactive power component is represented by a constant impedance, (3.47,
3.48) become
Pk0 ( VVk0 ) = Vk
k
Q0k ( VVk0 )2
k
X
Vm Ykm cos(θk − θm − αkm )
m∈K
= Vk
X
(3.64)
Vm Ykm sin(θk − θm − αkm )
m∈K
Thus, for the purpose of stability analysis, a power system can be modelled by 10 · ng differential equations and 4 · ng + 2 · nL algebraic equations,
where ng is the number of the generator buses, and nL is the number of the
load buses. If the load damping modelling approach described in Section
3.3.2 is adopted, the number of differential equations becomes 10 · ng + nL ,
whereas the number of the algebraic equations is reduced to 4 · ng + nL .
The system equations are expressed in terms of the state variables, the
generator currents id and iq , the complex bus voltages with the magnitude
Vk and angle θk , and the parameters of the system components.
For the small-signal stability analysis, the system equations should be
linearized to take the form of Equations (2.4). The results of the linearization
are presented in Section 3.4.1. The transient stability analysis by means of
the formulated system equations is shortly discussed in 3.4.2.
3.4.1
Small-Signal Stability
Since application of the load damping model changes the structure of the
system equations by adding new differential equations, this section will cover
two cases: with and without load damping, starting with the latter.
No Load Damping
Linearization of the system equations results in the following set of equations
expressed in terms of the perturbed variables:
∆ẋ = A∆x + F1 ∆Ig + F2 ∆Vg + B∆u
(3.65)
0 = C1 ∆x + G1 ∆Ig + G2 ∆Vg
(3.66)
0 = C2 ∆x + G3 ∆Ig + G4 ∆Vg + G5 ∆VL
(3.67)
0 = G6 ∆Vg + G7 ∆VL + D∆u
(3.68)
Apart from the differential equations (3.65), this system includes stator voltage equations (3.66), generator bus network equations (3.67), and load bus
3.4. OVERALL SYSTEM EQUATIONS
29
network equations (3.68). The state vector is composed by the state vectors
of ng synchronous machines:

∆x1
 ∆x2 


∆x =  . 
 .. 

(3.69)
∆xng
where an individual state vector xi is defined by (3.31), and i = 1, 2, ..., ng .
The matrix A is a block diagonal matrix composed of the submatrices
Agi associated with individual generators


Ag1
0
···
0

.. 
 0
Ag2 0
. 


A= .

.
.
.
 .
.
0
0 
0
···
0 Agng
(3.70)
The non-zero entries of each Agi matrix are expressed in terms of the
30
CHAPTER 3. MODELLING OF POWER SYSTEM
machine parameters and initial values of the currents id and iq as
A(1,2)
= ω0
g
00
1 Xads
iq
M Xfd
00
1 Xaqs
A(2,5)
=
id
g
M X1q
1
A(2,10)
=
g
M
00
Rfd Xads
A(3,4)
=
ω
0
g
Xfd X1d
Rfd
A(3,9)
= KA ω0
g
Xadu
X 00
R1d
A(4,4)
= −ω0
(1 − ads )
g
X1d
X1d
00
R1q Xaqs
A(5,6)
= ω0
g
X1q X2q
00
Xaqs
R2q
A(6,6)
=
ω
(1
−
)
0
g
X2q
X2q
A(2,3)
=−
g
A(8,2)
= KSTAB A(2,2)
g
g
A(8,4)
= KSTAB A(2,4)
g
g
A(8,6)
= KSTAB A(2,6)
g
g
A(8,10)
= KSTAB A(2,10)
g
g
T
1 (8,3)
A(9,3)
=
A
g
T2 g
T1 (8,5)
A
A(9,5)
=
g
T2 g
T1
1
A(9,8)
= − TW +
g
T2
T2
T
1 (8,10)
A(9,10)
=
A
g
T2 g
1
A(10,10)
=−
g
Tt
1
KD
M
00
1 Xads
Ag(2,4) = −
iq
M X1d
00
1 Xaqs
Ag(2,6) =
id
M X2q
X 00
Rfd
A[ g (3,3) = −ω0
(1 − ads )
Xfd
Xfd
Rfd
Ag(3,7) = −KA ω0
Xadu
00
R1d Xads
Ag(4,3) = ω0
X1d Xfd
00
Xaqs
R1q
(1 −
)
Ag(5,5) = −ω0
X1q
X1q
00
R2q Xaqs
Ag(6,5) = −ω0
X2q X1q
1
Ag(7,7) = −
TR
Ag(8,3) = KSTAB A(2,3)
g
Ag(2,2) = −
(3.71)
A(8,5)
= KSTAB A(2,5)
g
g
1
A(8,8)
=−
g
TW
T
1
A(8,2)
Ag(9,2) =
T2 g
T1 (8,4)
Ag(9,4) =
A
T2 g
T1 (8,6)
Ag(9,6) =
A
T2 g
1
Ag(9,9) = −
T2
1
A(10,2)
=−
g
STt
C1 and C2 are block diagonal matrices with the block elements C1−g and
C2−g respectively, where
"
#
X 00
X 00
−V cos(δ − θ) 0
0
0
− Xaqs
− Xaqs
0 0 0 0
1q
2q
C1−g =
00
X 00
Xads
V sin(δ − θ) 0 Xads
0
0
0 0 0 0
X1d
fd
(3.72)
3.4. OVERALL SYSTEM EQUATIONS
31
id V cos(δ − θ) − iq V sin(δ − θ) 0 0 0 0 0 0 0 0 0
C2−g =
−id V sin(δ − θ) − iq V cos(δ − θ) 0 0 0 0 0 0 0 0 0
(3.73)
In (3.72) and (3.73), V and θ denote the terminal voltage magnitude and
angle of the generator in question.
The generator current vector ∆Ig is given by 1


∆id1
 ∆iq1 


 ∆id2 




∆Ig =  ∆iq2 
 .. 
 . 


∆idn 
g
∆iqng
(3.74)
The matrices F1 , G1 , and G3 , that all also have a block diagonal structure,
are comprised by individual generator matrices of the form

0
00 i )
− 1 (−X 00 iq − Ψaq ) − 1 (Ψad + Xaqs
d 
ads
M
 M
Rfd 00


−ω
X
0
0


Xfd ads


R1d 00
0
−ω0 X
X


ads
1d


R1q 00

0
−ω0 X1q Xaqs 


=

R
00


0
−ω0 X2q
X
aqs
2q




0
0


(2,1)


T1 (2,2)
KSTAB F1−g1
F


T2 1−g1


(8,1)
(8,2)
T
T
1
1


T2 F1−g1
T2 F1−g1
0
0

F1−g
0
00
−Ra
Xl + Xaqs
=
00
−Xl − Xads
Ra
(3.75)
V sin(δ − θ) V cos(δ − θ)
G1−g
G3−g = g
Vg cos(δ − θ) −V sin(δ − θ)
(3.76)
The elements of the generator voltage vector Vg are the voltage angles
1
The notation for the generator current vector and the voltage vectors was adapted
from [15], whereas in other sources (e.g.[2]) the preference is given to the real and imaginary
components of current and voltage.
32
CHAPTER 3. MODELLING OF POWER SYSTEM
and magnitudes of the generator buses:


∆θ1
 ∆V1 


 ∆θ2 




∆Vg =  ∆V2 
 .. 
 . 


 ∆θng 
∆Vng
(3.77)
Each block of the block diagonal matrix F2 has only one non-zero element:
1
(7,2)
(3.78)
F2−g =
TR
The coefficients of the voltage variables in the stator voltage equations are
defined by another block diagonal matrix G2 , comprised of
V cos(δ − θ) − sin(δ − θ)
(3.79)
G2−g =
−V sin(δ − θ) − cos(δ − θ)
The vector of the voltage magnitudes and angles of the load buses is defined
as


∆θng +1
 ∆Vng +1 


 ∆θng +2 




∆VL =  ∆Vng +2 
(3.80)


.
.


.


 ∆θng +n 
L
∆Vng +nL
The matrices G4 -G7 represent the coefficients of the voltage variables
in the network equations. The odd- and even-numbered rows of the matrices correspond to the active power equations (3.62) and the reactive power
equations (3.63) respectively, whereas the odd- and even-numbered columns
refer to the voltage angle and the voltage magnitude coefficients.
The matrix G4 contains the elements that show the sensitivity of the
generator nodal equations to the voltage components of all the generators.
The off-diagonal elements of G4 , with k = 1, 2, ..., ng and m = 1, 2, ..., ng
are given as
(2k−1,2m−1)
G4
(2k−1,2m)
= −Vk Vm Ykm sin(θk − θm − αkm )
G4
= −Vk Ykm cos(θk − θm − αkm )
(2k,2m−1)
G4
= Vk Vm Ykm cos(θk − θm − αkm )
(2k−1,2m)
G4
= −Vk Ykm sin(θk − θm − αkm )
(3.81)
3.4. OVERALL SYSTEM EQUATIONS
33
The diagonal entries of the matrix are defined by the following expressions:
(2k−1,2k−1)
G4
= −idk Vk cos(δk − θk ) + iqk Vk sin(δk − θk )−
−Vk
X
Vm Ykm sin(θk − θm − αkm )
m∈K
(2k−1,2k)
G4
= idk sin(δk − θk ) + iqk cos(δk − θk )−
−
X
Vm Ykm cos(θk − θm − αkm )
m∈K
(2k,2k−1)
G4
(3.82)
= idk Vk sin(δk − θk ) + iqk Vk cos(δk − θk )−
−Vk
X
Vm Ykm cos(θk − θm − αkm )
m∈K
(2k,2k)
G4
= idk cos(δk − θk ) − iqk sin(δk − θk )−
−
X
Vm Ykm sin(θk − θm − αkm )
m∈K
The entries of G5 and G6 and the off-diagonal elements of G7 are similar
to the off-diagonal elements of G4 (3.81) with the only difference in the
indexation. For G5 , that represents the sensitivities of the generator network
equations to the load voltages, k = 1, 2, ..., ng whereas m = ng + 1, ng +
2, ...ng + nL . The indices in G6 and G7 , incorporating the sensitivities of
the load network equations, are k = ng + 1, ng + 2, ...ng + nL (G6 and G7 ),
m = 1, 2, ..., ng (G6 ) and m = ng + 1, ng + 2, ...ng + nL (G7 ).
The diagonal elements of G7 , i.e. sensitivities of the network equations
of the load buses to the voltages at the load buses are given by
(2k−1,2k−1)
G7
= Vk
X
Vm Ykm sin(θk − θm − αkm )
m∈K
(2k−1,2k)
G7
=
dPLk
dVk
−
X
Vm Ykm cos(θk − θm − αkm )
m∈K
(2k,2k−1)
G7
= −Vk
X
(3.83)
Vm Ykm cos θk − θm − αkm )
m∈K
(2k,2k)
G7
=
dQLk
dVk
−
X
Vm Ykm sin(θk − θm − αkm )
m∈K
dQLk
Lk
where dP
dVk and dVk are the sensitivities of the static load characteristics to
the voltage at the corresponding load bus. With the constant current and
the constant impedance characteristics for the active and reactive power
34
CHAPTER 3. MODELLING OF POWER SYSTEM
components respectively, as in (3.64),they become
dPLk
dVk
dQLk
dVk
= Pk0 /Vk0
(3.84)
= 2Q0k Vk /(Vk0 )2
(3.85)
The input vector ∆u may contain different quantities, e.g. the turbine
setpoint changes ∆Pm set . Another example would be ∆u as a vector of
the load changes ∆PL set which explains the appearance of the D matrix in
(3.68). Since the load power demand is not explicitly given in (3.65), the last
term of (3.65) should disappear in this case. However, ∆u can be defined
as
∆Pm set
∆u =
(3.86)
∆PL set
which yields non-zero elements in the both matrices B and D.
To enable the small-signal analysis, the system state matrix AS should
be derived. The successive elimination of ∆Vl and ∆Ig while assuming
∆u = 0 will give
∆ẋ = A0 ∆x + F 0 ∆Vg
0
(3.87)
0
0 = C ∆x + G ∆Vg
(3.88)
where
A0 = A − F1 G1 −1 C1
F
0
C
0
G
0
= F2 − F1 G1
−1
= C2 − G3 G1
G2
−1
= G4 − G3 G1
(3.89)
(3.90)
C1
−1
G2 − G5 G7
(3.91)
−1
G6
(3.92)
Equation (3.88) can be rewritten as
∆Vg = −G0
−1
C 0 ∆x
(3.93)
Substitution of ∆Vg in (3.87) gives
∆ẋ = (A0 − F 0 G0−1 C 0 )∆x
(3.94)
AS = A0 − F 0 G0−1 C 0
(3.95)
Thus
The analysis of the eigenvalues of the system state matrix AS could
show, whether the system is stable or unstable at the given operating point.
3.4. OVERALL SYSTEM EQUATIONS
35
Load Damping Case
If the load damping modelling is to be adopted, the system equations and,
therefore, the previously defined matrices should be adjusted. The load bus
voltage angles become state variables, and they should be added to the state
vector:
∆x = [∆xTold ∆θng +1 ∆θng +2 · · · ∆θng +nL ]T
(3.96)
The vector of the load bus voltage components, on the contrary, gets reduced:


∆Vng +1
 ∆Vn +2 
g


∆VL = 
(3.97)

..


.
∆Vng +nL
Since the active power injections at the load buses are already defined by
the new differential equations, the corresponding algebraic equations should
be excluded from the load bus network equations.
Thus, (3.65-3.68) become
∆ẋ = A∆x + F1 ∆Ig + F2 ∆Vg + F3 ∆VL + B∆u
0 = C1 ∆x + G1 ∆Ig + G2 ∆Vg
(3.98)
(3.99)
0 = C2 ∆x + G3 ∆Ig + G4 ∆Vg + G5 ∆VL
(3.100)
0 = C3Q ∆x + G6Q ∆Vg + G7Q ∆VL
(3.101)
The matrices A and F2 should be extended to account for the new differential equations. The block AL contains nL × nL elements of the following
form
ω0
(k,m)
AL
=
Vk Vm Ykm sin(θk − θm − αkm )
(3.102)
KDk
where k = ng + 1, ng + 2, ..., ng + nL , m = ng + 1, ng + 2, ..., ng + nL and
m 6= k
and
X
ω0
(k,k)
AL
=−
Vm Ykm sin(θk − θm − αkm )
(3.103)
Vk
KDk
m∈K
where m = 1, 2, ..., ng + nL
AL should be added at the diagonal of A (3.70),
The matrices F2 and F3 should be augmented with nL additional rows
each. The elements of these rows are as follows:
ω0
(k,2m−1)
F2−L
=
Vk Vm Ykm sin(θk − θm − αkm )
(3.104)
KDk
ω0
(k,2m)
F2−L
=
Vk Ykm cos(θk − θm − αkm )
(3.105)
KDk
where k = ng + 1, ng + 2, ..., ng + nL ,m = 1, 2, ..., ng The first 10ng × nL
entries of the new matrix F3 are zeros, since the derivatives of the generator
36
CHAPTER 3. MODELLING OF POWER SYSTEM
state variables are not explicitely influenced by the voltage magnitude at
the load buses. The last nL rows of F3 have the elements defined as
(k,m)
F3−L =
ω0
Vk Ykm cos(θk − θm − αkm )
KDk
(3.106)
with k = ng + 1, ng + 2, ..., ng + nL , m = ng + 1, ng + 2, ..., ng + nL
(k,k)
F3−L =
X
ω0
Vk
Ykm cos(θk − θm − αkm )
KDk
(3.107)
m∈K
with m = 1, 2, ..., ng + nL
Besides, nL rows with zero entries should be added to the matrix F1 .
Now the system state matrix can be computed by using a similar approach as in the no damping case:
C4 = C2 − G5 G7Q −1 G3Q
G8 = G4 − G5 G7Q
A
0
F
0
C
0
G
0
= A − F1 G3
−1
= F2 − F1 G3
G8 − F3 G7Q G6Q
−1
= G2 − G1 G3
G6Q
C4 − F3 G7Q C3Q
−1
= C1 − G1 G3
−1
C4
−1
G8
(3.108)
(3.109)
(3.110)
(3.111)
(3.112)
(3.113)
∆ẋ = (A0 − F 0 G0−1 C 0 )∆x
(3.114)
AS = A0 − F 0 G0−1 C 0
(3.115)
Hence
3.4.2
Transient Stability
For the transient stability analysis, the system equations expressed in the
form
ẋ = f (x, Ig , V )
(3.116)
0 = g(x, Ig , V )
(3.117)
should be solved to model the response of the system to a given disturbance. In (3.116,3.117), f and g are nonlinear functions. The solution of a
nonlinear sysem of differential and algebraic equations can be obtained by
implementing the methods of numerical integration. In this work, a partitioned approach with explicit integration is used. The term partitioned
solution refers to the separate solution of differential and algebraic equations. In explicit integration methods, such as R-K methods, the value of x
at any t can be estimated from its value at the previous time step.
This approach includes the following steps [2]:
3.4. OVERALL SYSTEM EQUATIONS
37
1. Calculate the initial state of the system (before a disturbance).
2. Change the admittance matrix to model the given disturbance, e.g.
add a shunt element with a large conductance at the faulty bus to
model a three-phase short circuit.
3. The state variables x cannot change instantaneously after the disturbance. The algebraic equations (3.117) should be solved with the
known x to find the voltages and currents at the instant after the disturbance. The Newton-Raphson algorithm, used in this thesis for the
solution of the algebraic equations, is broadly used in power system
analysis and extensively covered in the literature (e.g.[17]).
4. The time derivatives f (x, Ig , V ) can be estimated now by using the
known values of x, Ig , and V .
5. To find the values of the state variables x at the next time instant, RK numerical integration method can be applied (see Appendix A) by
using the time derivative values from the previous step in the formulae
(A.2).
6. The steps 3-5 should be repeated to obtain the time response of the
system to the given disturbance. If the fault is assumed to be cleared
at some time point, the admittance matrix will be changed again, and
the network variables will change stepwise, but not the state variables.
According to [2], the advantages of this approach are its flexibility, simplicity, reliability, and robustness. However, it might become numerically
unstable if the time step is bigger than the time constant of the smallest
eigenvalue.
38
CHAPTER 3. MODELLING OF POWER SYSTEM
Chapter 4
Impact of Rotational Inertia
and Damping on Power
System Stability
In this chapter, sensitivities of the system state matrix to inertia and damping are derived. They are further implemented in computation of damping
ratio sensitivities and sensitivities of transient frequency overshoot based on
[1]. In the following sections, an algorithm for inertia and damping optimization and its implementation in MATLAB are presented.
4.1
Sensitivity of Damping Ratio
The damping ratio of an oscillatory mode shows how fast the associated
oscillations will decay after a small disturbance. To ensure the secure operation of a power system, the oscillatory modes should be sufficiently damped
under any possible normal operating conditions of the system.
With the time dependent inertia level, it becomes very important to
understand how the changes in inertia affect the oscillatory modes. Furthermore, when damping level in the system can be adjusted, it would be
valuable to know how the system reacts to the changes in damping.
The sensitivity of the damping ratio of the i-th dynamic mode to a
parameter η (M or KD ) could be derived from the definition of damping
ratio (2.14) as
(i)
(i)
(σ (i) ∂ω∂η − ω (i) ∂σ∂η )
∂ζ (i)
∂
−σ (i)
=
(p
)=ω
∂η
∂η
(σ (i)2 + ω (i)2 )3/2
σ (i)2 + ω (i)2
(4.1)
where
∂σ (i)
∂η
= Re(
39
∂λ(i)
)
∂η
(4.2)
40
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
∂ω (i)
∂η
= Im(
∂λ(i)
)
∂η
(4.3)
The sensitivity of the eigenvalue λi to η is determined by the values of the
right and left eigenvectors, φ(i) and ψ (i) , calculated using the normalization
(2.10), and by the sensitivity of the state matrix to this parameter as [1]
∂λ(i)
∂AS (i)
= ψ (i)T
φ
∂η
∂η
(4.4)
Thus, to find the sensitivities of the damping ratios to the inertia and
damping coefficients, the expressions for the sensitivity of the system state
matrix should be derived.
4.1.1
State Matrix Sensitivity to Rotational Inertia
No Load Damping
The sensitivity of the system state matrix to the inertia of one of the
syncronous machines, M , is determined by
∂AS
∂A0 ∂F 0 0 0−1 0 ∂A0 ∂F 0 0 0−1 0
=
−
F G C =
−
F G C
∂M
∂M
∂M
∂M
∂M
(4.5)
with
∂A0
∂M
∂F 0
∂M
∂C 0
∂M
∂G0
∂M
=
=
∂
∂F1
∂A
(A − F1 G1 −1 C1 ) =
−
G1 −1 C1
∂M
∂M
∂M
∂
∂F1
(F2 − F1 G1 −1 G2 ) = −
G1 −1 G2
∂M
∂M
(4.6)
(4.7)
= 0
(4.8)
= 0
(4.9)
∂A
1
In (4.6), ∂M
and ∂F
∂M are block diagonal matrices. However, their only
blocks that contain non-zero elements are the ones that correspond to the
generator in question, since the inertia level of a generator does not explicitly
influence the state variables of the other generators. The non-zero elements
4.1. SENSITIVITY OF DAMPING RATIO
of the submatrix
∂Ag
∂M
can be derived from (3.71) that yields
(2,2)
∂Ag
1
= 2 KD
∂M
M
(2,3)
∂Ag
1 X 00
= 2 ads iq
∂M
M Xfd
(2,4)
∂Ag
∂M
=
=−
00
1 Xaqs
id
M 2 X1q
=−
00
1 Xaqs
id
M 2 X2q
(2,6)
∂Ag
∂M
(8,2)
00
1 Xads
iq
M 2 X1d
(2,5)
∂Ag
∂M
(2,10)
∂Ag
∂M
1
=− 2
M
(2,1)
∂M
(8,1)
∂F1−g
∂M
(9,1)
∂F1−g
∂M
(2,2)
∂Ag
∂Ag
= KSTAB
∂M
∂M
(8,3)
(2,3)
∂Ag
∂Ag
= KSTAB
∂M
∂M
(8,4)
(2,4)
∂Ag
∂Ag
= KSTAB
∂M
∂M
(8,5)
(2,5)
∂Ag
∂Ag
= KSTAB
∂M
∂M
(8,6)
(2,6)
∂Ag
∂Ag
= KSTAB
∂M
∂M
(8,10)
(2,10)
∂Ag
∂Ag
= KSTAB
∂M
∂M
The non-zero elements of the block
machine in the system as
∂F1−g
41
1
00
= 2 (−Xads
iq − Ψaq )
M
(2,1)
∂F1−g
= KSTAB
∂M
(8,1)
T1 ∂F1−g
=
T2 ∂M
∂F1−g
∂M
(9,2)
∂Ag
∂M
(8,2)
=
T1 ∂Ag
T2 ∂M
=
T1 ∂Ag
T2 ∂M
=
T1 ∂Ag
T2 ∂M
=
T1 ∂Ag
T2 ∂M
=
T1 ∂Ag
T2 ∂M
(8,3)
(9,3)
∂Ag
∂M
(8,4)
(9,4)
∂Ag
∂M
(9,5)
∂Ag
∂M
(8,5)
(8,6)
(9,6)
∂Ag
∂M
(9,10)
∂Ag
∂M
8,10)
=
T1 ∂Ag
T2 ∂M
(4.10)
are computed for each synchronous
(2,2)
∂F1−g
∂M
(8,2)
∂F1−g
∂M
(9,2)
∂F1−g
∂M
1
00
(Ψad + Xaqs
id )
M2
(2,2)
∂F1−g
= KSTAB
∂M
(8,2)
T1 ∂F1−g
=
T2 ∂M
=
(4.11)
Load Damping Case
In case damping is provided at the load buses, and it is to be modelled
as in Section 3.3, Equations (4.6) and (4.7) should be modified according to
the adjusted structure of the set of the system equations (3.98-3.101).
∂A0
∂M
∂F 0
∂M
∂A
∂F1
−
G3 −1 C4
∂M
∂M
∂F1
= −
G3 −1 G8
∂M
=
(4.12)
(4.13)
∂A
1
The non-zero elements of ∂M
and ∂F
∂M can be determined by means of
(4.10) and (4.11).
∂A
1
The size of ∂M
and ∂F
∂M should be similar to that of A and F1 respectively. Therefore, if the load damping differential equations are introduced,
∂F1
∂A
∂M should be augmented by nL zero rows and nL zero columns, while ∂M
gets nL zero rows.
42
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
4.1.2
State Matrix Sensitivity to Damping
No Load Damping
It could be assumed that the damping at a generator bus could be
changed, e.g. by means of BESS.
∂AS
∂A0
∂A
=
=
∂KD
∂KD
∂KD
(4.14)
∂A
The only non-zero elements of a block ∂KDg , representing the sensitivity of
the state variables of the generator in question to the associated damping
coefficient, are
(2,2)
1
∂Ag
=−
∂KD
M
(8,2)
∂Ag
1
= −KSTAB
∂KD
M
(4.15)
(9,2)
T1
1
∂Ag
= − KSTAB
∂KD
T2
M
Load Damping Case
∂AS
∂A0
∂F 0 0 −1 0
=
−
G C
∂KD
∂KD ∂KD
(4.16)
where
∂A0
∂A
∂F3
=
−
G7Q −1 C3Q
∂KD
∂KD ∂KD
∂F 0
∂F2
∂F3
=
−
G7Q −1 G6Q
∂KD
∂KD ∂KD
(4.17)
(4.18)
The sensitivities of the matrix A to the load damping at the bus in question
∂A
are given by the matrix ∂K
of the same size as A. Generator state variables
D
are not explicitly affected by the change of load damping, thus the first 10·ng
∂A
rows and the first 10·ng columns of ∂K
will always have only zero elements.
D
Non-zero entries will appear in the row (10ng + k) for the k-th load bus:
∂A (9ng +k,9ng +m)
ω0
=−
Vk Vm Ykm sin(θk − θm − αkm )
∂KDk
KDk2
(4.19)
where k = ng + 1, ng + 2, ..., ng + nL , m = ng + 1, ng + 2, ..., ng + nL and
m 6= k
and
X
ω0
∂A (9ng +k,9ng +k)
= 2 Vk
Vm Ykm sin(θk − θm − αkm )
∂KDk
KDk
m∈K
(4.20)
4.2. SENSITIVITY OF TRANSIENT OVERSHOOT
43
where m = 1, 2, ..., ng + nL
The derivative of F2 with respect to the damping at a given load bus
has the following non-zero elements:
∂F2 (9ng +k,2m−1)
ω0
= − 2 Vk Vm Ykm sin(θk − θm − αkm ) (4.21)
∂KDk
KDk
∂F2 (9ng +k,2m)
ω0
= − 2 Vk Ykm cos(θk − θm − αkm )
∂KDk
KDk
(4.22)
where k = ng + 1, ng + 2, ..., ng + nL , m = 1, 2, ..., ng
Finally, non-zero entries of
∂F3
∂KDk
are defined by
∂F3 (9ng +k,m)
ω0
= − 2 Vk Ykm cos(θk − θm − αkm )
∂KDk
KDk
(4.23)
where k = ng + 1, ng + 2, ..., ng + nL , m = ng + 1, ng + 2, ..., ng + nL
X
∂F3 (9ng +k,k)
ω0
Ykm cos(θk − θm − αkm )
= − 2 Vk
∂KDk
KDk
(4.24)
m∈K
where m = 1, 2, ..., ng + nL
4.2
Sensitivity of Transient Overshoot
The frequency response of a system to a disturbance depends on the level
of inertia and damping in this system. Inertial response of synchoronous
machines is an inherent reaction to an imbalance of mechanical and electrical
torques at their rotors. After a major disturbance, such as the loss of bulk
generation units, the rotational inertia of the remaining machines reduces
the rate of frequency drop. This provides more time for the control actions,
aimed at settling the frequency at an acceptable level and ensuring the
stability of the system. In case of high penetration of RES, the grid inertia
significantly decreases. This affects the frequency response, causing larger
frequency deviations, and requires faster operation of the frequency control.
In this section, the sensitivity of a transient overshoot after a disturbance
to the inertia of the generators will be derived based on [1] and on the system
equations formulated in Chapter 3.
The sensitivities of the right and the left eigenvectors to a parameter η
are given by
X η
∂φ(k)
=
ckj φ(j)
∂η
j∈N
,
X η
∂ψ (k)
=
dkj ψ (j)
∂η
j∈N
(4.25)
44
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
where N is the set of the system modes, and the off-diagonal elements cηkj
and dηkj can be expressed as
cηkj
=
dηkj
=
(k)
S
ψ (j)T ∂A
∂η φ
(λ(k) − λ(j) )ψ (j)T φ(j)
(j)
S
ψ (k)T ∂A
∂η φ
(λ(k) − λ(j) )ψ (j)T φ(j)
k 6= j
(4.26)
k 6= j
(4.27)
The derivative of the product φ(i) ψ (i)T , which is given by
∂φ(i) ψ (i)T
∂φ(i) (i)T
∂ψ (i)T
=
ψ
+ φ(i)
∂η
∂η
∂η
(4.28)
can be rewritten noting that ckj = −dkj as [1]
∂φ(i) ψ (i)T X (j) η (i)T
=
[φ cij ψ
− φ(i) cηji ψ (j)T ]
∂η
(4.29)
j\i
The derivative (4.29) will be employed in the latter derivations.
The frequency response can be estimated by means of an open-loop transfer function between the input (disturbance ∆u = ∆Pk ) and the output
(angular velocity deviation ∆y = ∆ωr ) variables. The open-loop transfer
function could be obtained from
∆ẋ = AS ∆x + bk ∆u
(4.30)
∆y = cl ∆x
(4.31)
where cl is a matrix mapping the frequency of node l on the output ylk and
bk shows the contribution of a disturbance at the k-th node to the deviation
of the state variables. Hence, the transfer function G(s) can be expressed
as
G(s) =
∆Y (s)
= cl (sI − A)(−1) bk
∆U (s)
= cl φ(sI − Λ)(−1) ψ T bk
X Rk
li
=
(i)
s
−
λ
i∈N
(4.32)
where Rlik is a residue of G(s) at pole s = λ(i)
Rlik = cl φ(i) ψ (i)T bk
(4.33)
The matrix bk is not as easily derived as in [1] because in this thesis, a more
complex model of a power system is employed. The derivation of bk will be
covered later in this section.
4.2. SENSITIVITY OF TRANSIENT OVERSHOOT
With the short-hand Klik =
k
Rli
,
λ(i)
45
the step response Y (s) is given by
X
λ(i)
1
Klik
Ylk (s) = G(s) = −
s
s(s − λ(i) )
(4.34)
i∈N
which in time domain yields
ylk (t) = L−1 [Ylk (s)] = −
X
Klik (1 − eλ
(i) t
)
(4.35)
i∈N
After dividing the eigenvalues into real and complex conjugate, represented
by the sets Λ0 and Lambda+ , respectively, the time-domain response can be
rewritten as
ylk (t) = −
X
Klik (1 − eλ
(i) t
)−2
X
(KlikRe − eσ
(i)t
||Klik || sin(ω (i) t − βlik ))
i∈Λ+
i∈Λ0
(4.36)
with
βlik = arctan(KlikRe , KlikIm )
(4.37)
The dominating mode i could be defined by finding the largest Kljk
i = argmaxKljk
(4.38)
j
The first peak time and magnitude of the dominating mode i could be approximated by
1
(0.5π − βlik )
ω (i)
= ylk (tkpl )
tkpl =
k
Mpl
(4.39)
(4.40)
l to inertia and
The next step would be to estimate the sensitivity of Mpk
damping. The derivative of the transfer function residue Rlik to inertia or
damping of the j-th generator is given by
∂Rlik
∂
∂φ(i) ψ (i)T k
∂bk
=
(cl φ(i) ψ (i)T bk ) = cl
b + cl φ(i) ψ (i)T
∂ηj
∂ηj
∂ηj
∂ηj
(4.41)
The gain of the step-response Klik is affected by inertia or damping changes
as follows:
∂Klik
∂ Rlik
=
=
∂ηj
∂ηj λ(i)
k
∂Rli
(i)
∂ηj λ
(i)
− Rlik ∂λ
∂ηj
(λ(i) )2
(4.42)
46
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
The change of the first peak time can be estimated by
∂ arctan( xy )
=
∂η
∂ 1
∂η ω (i)
∂x
∂η y
y2
= −
−
+
1
∂y
∂η x
x2
(4.43)
∂ω (i)
(ω (i) )2 ∂η
(4.44)
KiRe
∂ω (i)
1 ∂ arctan( KiIm )
= − (i) 2
(0.5π − β) − (i)
(4.45)
∂ηj
(ω ) ∂ηj
ω
∂tkpl
1
∂ηj
Finally, the derivative of the overshoot is given by
k
∂Mpl
∂ηj
=
∂ylk (tkpl )
∂ηj
− Klik
∂λ(i)
∂ηj
X ∂K k
(i) k
li
[
(1 − eλ tpl )
∂ηj
=−
i∈N
λ(i) tkpl
tkpl e
− Klik λ
∂tkpl
(i)
∂ηj
λ(i) tkpl
e
(4.46)
]
As a next step, the matrix bk is derived.
No Load Damping
To enable the calculation of the open-loop transfer function as in (4.32),
the system frequency response should be described by
∆ẋ = A∆x + bk ∆Pk
∆ωr = cl ∆x
(4.47)
(4.48)
where ∆Pk is the vector of disturbances. If we assume the disturbance to
happen at one of the load buses, the system equations should be adjusted
as follows:
∆ẋ = A∆x + F1 ∆Ig + F2 ∆Vg
(4.49)
0 = C1 ∆x + G1 ∆Ig + G2 ∆Vg
(4.50)
0 = C2 ∆x + G3 ∆Ig + G4 ∆Vg + G5 ∆VL
(4.51)
k
0 = G6 ∆Vg + G7 ∆VL + d ∆Pk
(4.52)
Rearrangement of (4.50) and (4.52) yields
∆VL = −G7 −1 (G6 ∆Vg + dk ∆Pk )
∆Ig = −G1
−1
(C1 ∆x + G2 ∆Vg )
(4.53)
(4.54)
Substitution of VL and Ig in (4.49) and (4.51) gives
∆ẋ = A0 ∆x + F 0 ∆Vg
0
0
0 = C ∆x + G ∆Vg −
(4.55)
k
G5 G−1
7 d ∆Pk
(4.56)
4.2. SENSITIVITY OF TRANSIENT OVERSHOOT
47
where A0 , C 0 , F 0 and G0 are the shortcuts adopted in (3.89-3.92). Now
(4.49) can be rewritten as
k
∆ẋ = AS ∆x + F 0 G0−1 G5 G−1
7 d ∆Pk
(4.57)
Thus, the desired matrix bk could be computed as
k
bk = F 0 G0−1 G5 G−1
7 d
(4.58)
where dk is a 2nL · 2nL matrix with only one non-zero element
dk(2k−1,2k−1) = 1
(4.59)
If the distrubances at generator buses are to be considered, the term dk ∆Pk
appears in (4.51), and (4.60) becomes
∆ẋ = A0 ∆x + F 0 ∆Vg
0
0
(4.60)
k
0 = C ∆x + G ∆Vg − d ∆Pk
(4.61)
∆ẋ = AS ∆x + F 0 G0−1 dk ∆Pk
(4.62)
bk = F 0 G0−1 dk
(4.63)
that yields
and
Now
∂bk
∂Mj
and
∂bk
∂KD
should be calculated to be used in (4.41)
∂bk
∂Mj
=
∂F 0 0−1
k
G G5 G−1
7 d
∂Mj
(4.64)
∂bk
∂Mj
=
∂F 0 0−1 k
G d
∂Mj
(4.65)
∂bk
∂KD
= 0
(4.66)
where equation (4.64) corresponds to the disturbances at the load buses,
and equation (4.65) corresponds to the disturbances at the generator buses.
∂F 0
The derivative ∂M
is given by (4.7).
j
Load Damping Case
Since the active power injection at the load buses is in this case represented by the differential equations (3.54), disturbances at the load buses
could be modelled by directly adding the term bk ∆Pk to Equation (3.98).
The equation becomes
∆ẋ = AS ∆x + bk ∆Pk
(4.67)
48
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
where bk has size (10ng + nL ) · nL and the non-zero element of bk is given
by
ω0
bk(10ng +k,k) =
(4.68)
KDk
A disturbance at a generator bus could be modelled in the same way as in
the no load damping case, with (4.62), while the matrices A0 , F 0 , C 0 and
G0 should be adopted from (3.110-3.113).
The derivatives of bk with respect to M and KD for the case of a disturbance at a load bus are given by
∂bk
∂Mj
∂bk
∂KD
= 0
(10ng +k,k)
=
ω0
2
KDk
(4.69)
(4.70)
If a disturbance occurs at a generator bus, taking the derivatives of bk yields
0
∂bk
∂Mj
=
∂F 0 0−1 k
G d
∂Mj
(4.71)
∂bk
∂KD
=
∂F 0 0−1 k
G d
∂KD
(4.72)
0
∂F
∂F
with ∂M
and ∂K
as defined by (4.18).
j
D
By comparing Equations (4.64) and (4.71), it could be concluded that
the load damping model, suggested in Section 3.3.2, does not adequately
reflect the sensitivity of the generator frequency response to the rotational
inertia for the load bus case.
The expressions for bk and its derivatives can be used to calculate the
k and its sensitivities.
transient overshoot Mpl
Knowing how the changes of inertia and damping affect the eigenvalues
of the system state matrix and the magnitude of the transient overshoot,
it is now possible to formulate an optimization program focused on the
improvement of the system stability.
4.3
Optimization Algorithm
The optimization algorithm described in this section was proposed in [1].
Its objective is the maximization of the worst-case (minimal) damping ratio
of any mode in the system. The larger the damping ratios are, the faster
the oscillatory modes decay which is advantageous for the system stability.
At the same time, to avoid large deviations of the frequency due to reduced
inertia, the frequency overshoot after a disturbance should be constrained.
Damping and inertia are assumed to be adjustable within some bounds and
4.3. OPTIMIZATION ALGORITHM
49
associated with a cost. The sets K and M include the nodes with damping
and inertia, respectively. The total amount of inertia and damping that can
be added is assumed to be limited.
The sensitivities of the damping ratios and the overshoot are non-linear,
therefore, each step of the optimization will be associated with a solution
of a linearized problem. Superscript ν denotes the number of the iteration
with 0 being the first iteration. After each iteration, the system state matrix,
along with the sensitivities of damping ratios and transient overshoot should
be computed all over again.
The change of the damping from the current iteration to the next is given
by
ν+1
ν+1
ν
∆KDj
= KDj
− KDj
(4.73)
A similar expression is applied to the value of inertia. To enable the calculaν+1
tion of the absolute change in inertia and damping, KDj
and Mjν+1 should
+
−
and Mj+ , and negative parts,KDj
and Mj− ,
be split in positive parts, KDj
as
(
ν+1
ν+1
0 >0
0
− KDj
− KDj
if KDj
KDj
+
=
KDj
ν+1
0 ≤ 0 (4.74)
0
if KDj − KDj
(
ν+1
0 >0
− KDj
0
if KDj
−
=
KDj
ν+1
0 | if K ν+1 − K 0 ≤ 0 (4.75)
− KDj
|KDj
Dj
Dj
Mj+
Mj−
M −ν+1
−Mj0
j
0
if
if
Mjν+1 − Mj0 > 0
(4.76)
Mjν+1 − Mj0 ≤ 0
0
ν+1
|Mj − Mj0 |
if
if
Mjν+1 − Mj0 > 0
(4.77)
Mjν+1 − Mj0 ≤ 0
=
=
The absolute change could be now estimated as
+
−
|KDj | = KDj
+ KDj
(4.78)
Mj−
(4.79)
|Mj | =
Mj+
+
The objective function penalizes the minimal damping ratio with cost
cζ ≥ 0. The purpose of the slack variable kpl is to ensure the feasibility of
the problem, and it is penalized by the cost c . Procurement of additional
inertia and damping implicates economic costs cMi and cKi . However, the
accurate calculation of cMj and cKj would be a complex task without a
well-developed inertia and damping procurement market.
min [−cζ ζ min +
KDj ,Mj
XX
k
l
(c kpl ) +
X
i∈K
cKj |KDj | +
X
cMj |Mj |] ∀i ∈ Λ+
i∈M
(4.80)
50
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
s.t.
ζiν+1 = ζiν +
X ∂ζ ν
X ∂ζ ν
ν+1
i
i
∆KDj
∆Mjν+1 (4.81)
+
∂KDj
∂Mj
j∈K
(4.82)
|Mj | ≤ M tot
(4.84)
ζ
X
j∈M
ζiν+1
tot
KD
min
≤
|KDj | ≤
(4.83)
j∈K
X
j∈M
ν+1
min
max
KDj
≤ KDj
≤ KDj
(4.85)
Mjmin ≤ Mjν+1 ≤ Mjmax
(4.86)
min
∆KDj
∆Mjmin
≤
≤
0 ≤
+
KDj
−
−
KDj
−
Mj−
=
0 ≤
Mj+
k
∆Pk
fp − Mpl
≥
=
ν+1
max
∆KDj
≤ ∆KDj
∆Mjν+1 ≤ ∆Mjmax
−
+
,
0 ≤ KDj
KDj
ν+1
0
− KDj
KDj
Mj+ ,
0 ≤ Mj−
Mjν+1 − Mj0
k
X ∂Mpl
j∈M
∂Mj
∆Pk ∆Mjν+1 +
(4.87)
(4.88)
(4.89)
(4.90)
(4.91)
(4.92)
k
X ∂Mpl
j∈K
∂KDj
ν+1
− kpl (4.93)
∆Pk ∆KDj
With (4.81), the value of the damping ratios are computed at the step ν + 1
using the previous value ζiν and the changes related to the adjustment of Mj
and KDj . The constraint (4.82) serves to set ζ min to the lowest damping
ratio value. The inequality constraints (4.83) and (4.84) limit the total
change of inertia and damping to the values available for procurement. The
individual values of Mj and KDj at each bus are limited by (4.85) and
k computed at each iteration are valid only
(4.86). Sensitivities of ζ and Mpl
ν . Therefore, the
for a small range of values around the initial Mjν and KDj
ν+1
ν+1
steps ∆KDj and ∆Mj
should be limited by (4.87) and (4.88). Equalities
(4.89-4.92) split KDj and Mj as in (4.78 and 4.79).
The magnitude of the transient overshoot after a disturbance ∆Pk is
k ∆P . The change of inertia and damping by ∆K ν+1 and
given by Mpl
k
Dj
ν+1
∆Mj
results in additional terms of the overshoot as shown in (4.93).
The total magnitude of the overshoot should not exceed the limit fp .
At each iteration ν, the program finds the optimal levels of damping and
inertia. These values are used to calculate the system state matrix and its
eigenvalues, the new values of sensitivities, and the approximate magnitude
4.4. IMPLEMENTATION IN MATLAB
51
of transient overshoot. All these quantities are implemented in optimization
at iteration ν + 1. This optimization program can be used to facilitate the
provision of synthetic inertia and fast frequency response. It could also serve
as a reference tool for planning of power systems.
4.4
Implementation in MATLAB
To analyze the impact of inertia changes on the stability of test power systems, the proposed optimization algorithm was implemented in the engineering environment MATLAB, along with a supplementary transient simulations tool .
Optimization Program
START
DATA.mat
BUSES
LINES
GENS
 0
NO
min (dm;dk) <0,1
YES
Reindexation
Power Flow
Computation
Calculation of the
initial state of the
generators
  1
Optimization of
the inertia and
damping levels
Estimation of
the transient
overshoot and
sensitivities
Kron Reduction
Calculation of
the eigenvalue
and eigenvector
sensitivities
Computation of
the system state
matrix and
eigenvalues
END
Figure 4.1: Structure of the developed optimization program
The flow chart of the developed MATLAB program for inertia and damping optimization is presented in Figure 4.1. The input data for the optimization program should be saved in the file DATA.mat and should include the
following arrays:
1. BUSES with information on the nodes of the investigated power system, such as active and reactive power injections, voltage magnitudes
of PV-buses, susceptance of the shunt devices, etc.
2. LINES where topology of the grid and parameters of the transmission
lines are stored.
52
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
3. GENS where parameters of the on-line generators are listed.
The required structure of the arrays is described in Appendix D.
The MATLAB file run optimization.m is the master file of the optimization program. At the start of the master file, the user could choose the model
order for the synchronous machines: 6,7,9 or 10. Each option represents the
number of the differential equations describing a machine. The option 6
corresponds to an unregulated machine, by entering 7 the model with AVR
would be chosen, with 9 PSS would be implemented, and 10 stands or the
full model employed in this work. The ability to choose the model could be
useful, if the effects of a particular regulation system on the system stability
are to be analyzed. Besides that, the user could choose, whether he or she
would like to include the load damping modelling by typing ’on’ or ’off’ in
response to a corresponding inquiry.
After the user has made his choice, the program starts to process the
data. First of all, the nodes and the lines of the system are re-indexed by
reindexation.m to ensure the sequential numbering of the elements starting with ”1”. The next step is calculation of the initial steady state of the
system by means of power flow computation accomplished by the package
MATPOWER 5.1. This tool is called upon by pf.m where the necessary
power system data is first processed to get the MATPOWER format. Resulting from MATPOWER computations are the voltage magnitudes and
angles, along with active and reactive power injections at the nodes of the
system.
Next, the nodes are sorted into three categories: generator nodes, load
buses and other buses (with no power injections). To exclude the other
buses from the analysis, Kron reduction is implemented via kron.m. The
MATPOWER output data is used as an input to evaluate the generator
variables id , iq , and δ which is done by gencurrents.m. These variables represent the interface between synchronous machines and transmission network, and after getting their values, the algorithm proceeds with computation of coefficient matrices for the linearized network equations (G4 − G7 ,
network equations.m).
In the following stage, the function sensitivities.m computes the system
state matrix AS , its eigenvalues and eigenvectors, and sensitivities of damping ratios to rotational inertia of generators and to damping. To derive
AS , the program has to build the matrices Ag ,FX−g ,CX−g and GX−g for
each generator bus, along with the derivatives of Ag and FX−g used in the
further analysis. This step is carried out by gen equations.m. Then, the
transient overshoot magnitudes and their sensitivities to inertia and damping are estimated by transient overshoot.m.
After the eigenvalues, overshoot magnitudes, and their sensitivities have
been computed, the parameters of the optimization should be defined. Among
these parameters are the costs introduced in the objective function, the total
4.4. IMPLEMENTATION IN MATLAB
53
avaliable inertia and damping, the parameters of the equality constraints,
and the number of iterations.
The optimization is carried out by means of the free of charge optimization package YALMIP. YALMIP is a modelling language for advanced
modelling and solution of convex and nonconvex optimization problems [18].
In optimization.m, with optimization parameters as an input, the specified
objective function is maximized subject to the given constraints by IBM
ILOG CPLEX solver. Results of the optimization problem solution, the updated values of inertia and damping at the nodes, are used to calculate the
new eigenvalues, overshoot, and sensitivities by means of sensitivities.m and
transient overshoot.m. Next, another optimization round is carried out, and
further, by repetitive optimization solution and calculation of eigenvalues,
transient overshoot, and their sensitivities, the local optimum of the objective function is found. The number of the iterations should be manually
adapted to ensure that the solution has converged. If from one iteration
to another, the minimal damping ratio does not get any improvement, the
maximal size of the steps ∆K and ∆M is decreased by 10% , as an attempt
to push the solution into another direction. The reduction of the steps is
accumulated in factors dm and dk . When one of them becomes smaller
than 0.1, the user receives the message that declares the termination of the
whole optimization process and displays the achieved improvement of minimal damping ratio in percent.
Transient Simulations
The master file of the transient simulations is called run transient. It requires the same input data as the optimization program and starts the computations by calling reindexation.m, pf.m, and kron.m. The next step is
estimation of the initial values of the generator variables in initial x gen.m.
Following this, the bus and the branch where a disturbance occurs is specified, and the admittance matrix of the system is altered to involve a shunt
element at the faulty bus. The new values of the network variables (voltages
and currents) are calculated at the next step by means of Newton-Raphson
iteration algorithm implemented in newtraph.m. The latter has two subfunctions, algebraic.m, where the right-side parts of the network equations
(3.117) are evaluated, and jacobian.m computing the Jacobian associated
with these equations.
After that, the time step and the number of the time intervals before
the fault is cleared are specified. For each time interval, the values of the
state variables are computed by means of numerical integration, while the
values of the network variables are estimated in newtraph.m. Numerical
integration is carried out by the second order R-K method (rungekutta2.m)
54
CHAPTER 4. IMPACT OF INERTIA AND DAMPING
or the fourth order R-K method in Gill’s modification (rungekuttagill.m).
More than once at each time step, R-K methods require the calculation of
the state variable derivatives with respect to time which is implemented in
deriv gen.
After the fault is assumed to be cleared, the admittance matrix should
be adjusted according to the new network conditions. Following that, newtraph.m is executed again to find the updated value of the network variables,
that abruptly altered after the admittance matrix changed. Finally, the new
time step and the number of the time intervals are specified, and R-K numerical integration along with Newton-Raphson algorithm are implemented
to folllow the behaviour of the state and network variables after the fault
was cleared.
Chapter 5
Simulation Results
The proposed optimization algorithm and the developed transient simulation
program were implemented for two test systems, IEEE two-area test system
[2] and IEEE South East Australian test system [4]. Both systems are often
used for testing small-signal stability analysis programs. In the present
chapter, the results of optimization are presented and illustrated by the
transient simulation results.
5.1
5.1.1
IEEE Two-Area Test System
System Description
The two-area system shown in Figure 5.1 is often used for benchmarking of
small-signal stability analysis tools. This simplified power system consists
of two areas interconnected by a weak tie link. Each area includes 2 generators supplying an aggregated load bus. Eventhough this system is far less
complex than the real-life power systems, it is already a step ahead from the
single machine infinite bus (SMIB) representation, as it allows to investigate
interarea oscillations.
In the present work, the system operating state and parameters listed
in Example 12.6 of [2] were adopted. This allowed to assess the modelling
accuracy by comparing the obtained results with those in [2]. The investigated operating state is described by the following data:
G1: P = 700 MW Q = 185 MVAr
Et = 1.03∠20.2
G2: P = 700 MW Q = 235 MVAr
Et = 1.01∠10.5
G3: P = 719 MW Q = 176 MVAr
Et = 1.03∠ − 6.8
G4: P = 700 MW Q = 202 MVAr
Et = 1.01∠ − 17.0
Bus 7: PL = 967 MW QL = 100 MVAr QC = 200 MVAr
Bus 9: PL = 1767 MW QL = 100 MVAr QC = 350 MVAr
Nominal frequency in the system is 60 Hz. Active load components are
modelled by constant current characteristics, and reactive load components
have constant impedance characteristics.
55
56
G1
CHAPTER 5. SIMULATION RESULTS
1
5
6
25 km
10 km
L7
7
400 MW
8
9
110 km
110 km
C7
C9
10 km
10
11
25 km
3
L9
2
G2
4
G4
Area 1
Area 2
Figure 5.1: Two-area test system [2]
5.1.2
Small-Signal Stability Analysis
To validate the developed eigenvalue calculation routine, the system state
matrix eigenvalues obtained in the simulations were compared to the ones
listed in Example 12.6 of [2]. In accordance with the outline of Example 12.6,
it was assumed that all four generators are operated on manual excitation
control, and there are no PFC devices. Hence, each synchronous machine
was represented by 6 state variables. All the damping coefficients were set
to zero, thus, No Load Damping case was modelled.
The calculated eigenvalues are presented at the left side of Table 5.1,
and the corresponding values from [2] are given at the right side of the
table. As it could be seen in Table 5.1, the calculation results exhibit three
decimal place accuracy which could serve as a verification for the developed
eigenvalue computation program.
The first two eigenvalues in Table 5.1 represent the zero eigenvalues due
to the redundant state variables. The appearence of the zero eigenvalues
is explained in [2]. One of these zero eigenvalues is caused by the lack
of uniqueness of absolute rotor angle. The other zero eigenvalue results
from the assumption that the generator torques are independent of speed
deviation (speed governors are not modelled and KD = 0).
All the non-zero eigenvalues of the system have negative real parts that
means that the system is stable in the given operational condition.
Each mode of the system can be characterized by the state variables
that contribute the most to this mode. The level of contribution of a state
variable to a mode can be assessed by means of the participation matrix
analysis, described in [2]. The dominant states of the system modes in the
investegated case are given in Table 5.1. The rotor angle oscillatory modes
of the two-area system are represented by three conjugate pairs of complex
G3
5.1. IEEE TWO-AREA TEST SYSTEM
57
eigenvalues. Conjugate pairs λ = −0.492 ± 6.83 and λ = −0.506 ± 7.02
are associated with the local intermachine oscillations between generators
G1 and G2, and generators G3 and G4 respectively. The third rotor angle
mode, described by the conjugate pair λ = −0.111 ± 3.43, is the interarea
mode, with generators G1 and G2 swinging against G3 and G4. As it could
be seen in Table 5.1, this oscillatory mode has the lowest damping ratio.
Table 5.1: System modes with manual excitation control
Eigenvalues
Real
Imaginary
Damping
Ratio
Eigenvalues [2]
Real
Imaginary
Damping
Ratio
Dominant
States
1.33E-07
-1.33E-07
-0.099
-0.111
-0.111
-0.116
0
0
0
-3.43
3.43
0
1
1
1
0.032
0.032
1
-7.60E-04
-7.60E-04
-0.096
-0.111
-0.111
-0.117
2.20E-03
-2.20E-03
0
-3.43
3.43
0
0.327
0.327
1
0.032
0.032
1
∆ω and ∆δ
of G1, G2, G3, G4
-0.265
0
1
-0.265
0
1
∆Ψfd of G3 and G4
-0.276
0
1
-0.276
0
1
∆Ψfd of G1 and G2
-0.492
-0.492
-6.83
6.83
0.072
0.072
-0.492
-0.492
-6.82
6.82
0.072
0.072
∆ω and ∆δ
of G1 and G2
-0.506
-0.506
-7.02
7.02
0.072
0.072
-0.506
-0.506
-7.02
7.02
0.072
0.072
∆ω and ∆δ
of G3 and G4
-3.428
-4.139
-5.288
-5.303
-31.03
-32.45
-34.07
-35.53
-37.89
-37.89
-38.01
-38.01
0
0
0
0
0
0
0
0
-0.142
0.142
-0.037
0.037
1
1
1
1
1
1
1
1
1
1
1
1
-3.428
-4.139
-5.287
-5.303
-31.03
-32.45
-34.07
-35.53
-37.89
-37.89
-38.01
-38.01
0
0
0
0
0
0
0
0
-0.142
0.142
-0.038
0.038
1
1
1
1
1
1
1
1
1
1
1
1
flux linkages of
d- and q-axis
damping circuits
The assumption of the manual excitation control simplifies the analysis,
however, it is extremely important to know how the control devices influence
on the small-signal stability. The eigenvalue computation in MATLAB,
as well as the results given in [2] show that if the excitation is controlled
by means of AVR with a high gain without PSS, the investigated system
becomes unstable, with an unstable interarea oscillation mode represented
by a conjugate pair λ = 0.0301 ± 3.84.
The operation of PSS with given parameters eliminates the negative effect of AVR on the damping torque. The interarea oscillatory mode of the
two-area system with PSSs modelled as shown in Figure 3.2 is represented
by complex eigenvalues λ = −0.663 ± 3.286. It should be noted, that these
58
CHAPTER 5. SIMULATION RESULTS
values differ from the ones presented in [2]. This discrepancy could be explained by the difference between the implemented PSS model and a more
detailed model chosen by the author of [2].
Implementation of PFC introduces additional damping of the oscillations. Speed governors were not included into the model used for the smallsignal stability analysis of the two-area system in [2], therefore, the parameters of PFC were chosen at our own discretion as follows:
• Droop S = 2%
• Turbine time constant Tt = 10s
The value of the turbine time constant is set in accordance with the maximal
time of full primary control reserve deployment allowed in interconnected
European power system which is 30 s. Normally, the values of Tt of nonreheat steam turbines are significantly lower than 10 s, but to investigate
the “worst-case” scenario, the chosen value of Tt represents steam turbines,
equipped with a re-heater.
The eigenvalues of the state matrix of the given two-area system with
AVR, PSS, and PFC are presented in Table 5.3. In the case that will be
further referred to as Base Case, the rotational inertia and damping coefficients of the machines are set to the values given in Table 5.2 in accordance
with Example 12.6 of [2]. This case represents a system with conventional
generation, and thus “conventional” level of rotational inertia.
It should be noted, that the small-signal stability analysis of the test
system in Load Damping case, i.e. with incorporation of the frequency
dependency of the load modelled as in Section 3.3.2, yielded positive eigenvalues. This result contradicts the expectations from the effect of the load
damping on the stability of the system. Modelling of an aggregated load is
a complex task since it requires an adequate reflection of both voltage and
frequency dependency of the consumed power and due to the diversity of
the consumer devices. The model proposed in Section 3.3.2 does not seem
to offer an appropriate description of the voltage dependence of the active
component of demanded power and, thus, it should be further elaborated. It
is not included in the optimization analysis conducted in the present work.
In case of the high penetration of RES, the level of inertia is significantly
lower. For example, according to [5], in 2012, the share of inverter-connected
RES infeed in the German power system has reached maximal value of 50%.
Consequently, the aggregated inertia of the system lost half of its value
during the times with such a high RES share, changing from H = 6s to
H = 3 − 4s (or from M = 12s to M = 6 − 8s). This highly reduced inertia
scenario is reflected in the present thesis by Low-Inertia Case with all the
inertia constants reduced by 50% compared to Base Case.
5.1. IEEE TWO-AREA TEST SYSTEM
59
Table 5.2: Rotational inertia constant M and damping coefficients of the
two-area system generators in Base Case and Low-Inertia Case, calculated
on the rated MVA base (900 MVA)
Generator
G1
G2
G3
G4
KD
M base case [s]
M low inertia [s]
1
13
6.5
1
13
6.5
1
12.35
6.175
1
12.35
6.175
60
CHAPTER 5. SIMULATION RESULTS
Table 5.3: Eigenvalues of the two-area system in Base Case (left) and LowInertia Case (right).
Eigenvalues
Real
Imaginary
-3.56E-14
-0.100
-0.100
-0.100
-0.071
-0.071
-0.778
-0.795
-0.804
-1.681
-0.696
-0.696
-3.714
-3.839
-4.383
-4.383
-3.545
-3.545
-3.795
-3.795
-16.372
-16.372
-16.205
-16.205
-18.148
-18.148
-17.515
-17.515
-32.724
-33.119
-37.928
-38.080
-51.491
-51.491
-52.928
-53.080
-94.601
-95.725
-97.488
-97.545
0
0
0
0
-0.116
0.116
0
0
0
0
-3.283
3.283
0
0
-0.043
0.043
-5.140
5.140
-5.117
5.117
-14.082
14.082
-14.683
14.683
-19.579
19.579
-24.341
24.341
0
0
0
0
-0.058
0.058
0
0
0
0
0
0
Damping
Ratio
1
1
1
1
0.522
0.522
1
1
1
1
0.207
0.207
1
1
1
1
0.568
0.568
0.596
0.596
0.758
0.758
0.741
0.741
0.680
0.680
0.584
0.584
1
1
1
1
1
1
1
1
1
1
1
1
Eigenvalues
Real
Imaginary
-1.24E-13
-0.100
-0.100
-0.100
-0.077
-0.077
-0.780
-0.795
-0.805
-2.371
-1.273
-1.273
-3.402
-3.509
-4.370
-4.370
-5.123
-5.123
-5.287
-5.287
-13.819
-13.819
-13.618
-13.618
-17.349
-17.349
-17.235
-17.235
-32.714
-32.944
-37.831
-37.992
-52.624
-52.624
-54.558
-54.754
-94.663
-95.778
-97.541
-97.599
0
0
0
0
-0.128
0.128
0
0
0
0
-4.163
-4.163
0.000
0.000
-0.017
0.017
-5.270
5.270
-5.133
5.133
-22.603
22.603
-23.873
23.873
-23.609
23.609
-27.021
27.021
0
0
0
0
-0.135
0.135
0
0
0
0
0
0
Damping
Ratio
1
1
1
1
0.515
0.515
1
1
1
1
0.292
0.292
1
1
1
1
0.697
0.697
0.717
0.717
0.522
0.522
0.495
0.495
0.592
0.592
0.538
0.538
1
1
1
1
1
1
1
1
1
1
1
1
5.1. IEEE TWO-AREA TEST SYSTEM
61
The interarea oscillatory mode in Base Case is given by a conjugate
pair of eigenvalues λ = −0.696 ± 3.28. As could be seen in Table 5.3, this
mode has the worst damping ratio among all the system modes (0.207).
The decay of interarea oscillations can be accelerated by the reduction of
inertia in the system, as lower inertia is associated with a faster damping of
oscillations. In accordance with the expectations, the damping ratio of the
mode of interest increases by roughly 41% in Low-Inertia Case compared to
Base Case. However, when inertia level in the system is too low, the inertial
response of the machines is reduced, and the system becomes less resilient
to large disturbances. To ensure the stable operation of the system, the
frequency nadir after possible large-scale disturbances should be limited to
acceptable values.
The reaction of the system to sudden load changes at the generator buses
was investigated to assess the level of the frequency deviations in the system.
The frequency response of the two-area system was estimated by applying
an open-loop transfer function G(s) defined by (4.32). As proposed in [1],
to facilitate the assessment of the system frequency response, the transient
overshoot after a disturbance was approximated by the first peak magnitude
k , given by (4.40). The results of the
of the dominating oscillatory mode Mpl
transient overshoot calculations in Hz are presented in Table 5.4.
Table 5.4: Results of transient overshoot computation in the two-area system
in Base Case
Node
Mpmin [Hz]
min y(t) [Hz]
1
2
3
4
-0.348
-0.347
-0.281
-0.275
-0.395
-0.395
-0.310
-0.310
Table 5.5: Results of transient overshoot computation in the two-area system
in Low-Inertia Case
Node
Mpmin [Hz]
min y(t) [Hz]
1
2
3
4
-0.380
-0.380
-0.299
-0.292
-0.414
-0.414
-0.330
-0.324
As it could be seen in Table 5.4, the approximated values of the transient
frequency overshoot Mpmin deviate by roughly 12% from the actual values of
overshoot (min y(t)) after the disturbances at the generator nodes.
62
CHAPTER 5. SIMULATION RESULTS
Figure 5.2 illustrates the transient frequency deviations at bus 1 after
the load change at bus 1 (blue curve) and bus 3 (green curve).
0
Transient Frequency [Hz]
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
−0.4
−0.45
−0.5
0
20
40
60
Time [s]
80
100
Figure 5.2: Frequency response to disturbances at buses 1 (blue) and 3
(green) of the two-area system in Base Case
In this graph, it is easy to recognize the operation of PFC. After a fast
drop, the frequency starts to increase due to the PFC, and after some time it
will settle, however, it will not get back to 60 Hz. The first part of the curve
(before the frequency reaches its minimum) shows the reaction of the system
to the disturbance before the deployment of the primary control reserve. The
steepness of the curve, i.e. the rate of the frequency deviation, is determined
by the rotational inertia and damping levels in the system. If the frequency
in a real-life system decreases too fast, the system PFC may not have enough
time to restore the frequency at an acceptable level. In this case, when the
frequency becomes critically low, the generator protection will disconnect
the machines which will lead to further complications and possibly to the
loss of the system stability.
It is, therefore, valuable to know how the changes in inertia and damping
affect the frequency nadir. If the rotational inertia of the generators of the
investigated system is reduced by 50 %, the absolute values of the frequency
overshoot increase by 6-10 %, as it could be seen in Table 5.5.
5.1. IEEE TWO-AREA TEST SYSTEM
63
0
Transient Frequency [Hz]
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
−0.4
−0.45
−0.5
0
20
40
60
Time [s]
80
100
Figure 5.3: Frequency response to disturbances at buses 1 (blue) and 3
(green) of the two-area system with the inertia of all machines reduced by
50%
Figure 5.3 shows the frequency response to the disturbances at buses 1
and 3 in Low-Inertia case. The reduction of the inertia is associated with a
high penetration level of RES. However, it seems necessary to note, that the
modelling of RES was not covered in the present work, and all the generation
units are represented by synchronous machines. Nevertheless, in this stage
of the research, a simple reduction of inertia constants of the machines is
assumed to be sufficiently accurate in representing the changes in inertia
due to the intermittant generation.
Another parameter that affects the value of the frequency nadir is the
damping coefficient KD . As already discussed in Chapter 3, KD represents
the relation of the electrical torque at the rotor of a generator to the frequency deviation. From Equation (3.6), it is clear that the higher is KD ,
the smaller is the rate of the frequency deviation.The effect of the reduced
damping at the generator buses could be seen in Table 5.6 and Figure 5.4.
64
CHAPTER 5. SIMULATION RESULTS
Table 5.6: Results of transient overshoot computation in the two-area system
with the damping of all the machines reduced by 50%
Node
Mpmin [Hz]
min y(t) [Hz]
1
2
3
4
-0.388
-0.388
-0.310
-0.308
-0.438
-0.438
-0.348
-0.342
0
Transient Frequency [Hz]
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
−0.4
−0.45
−0.5
0
20
40
60
Time [s]
80
100
Figure 5.4: Frequency response to disturbances at buses 1 (blue) and 3
(green) of the two-area system with damping of all the machines reduced by
50%
If both the rotational inertia and damping coefficients of the machines are
reduced by 50% compared to Base Case, the absolute value of the transient
frequency overshoot increases by 15-17%, as shown in Table 5.7.
5.1. IEEE TWO-AREA TEST SYSTEM
65
Node
Mpmin
min y(t)
1
2
3
4
-0.404
-0.403
-0.322
-0.314
-0.460
-0.460
-0.368
-0.348
0
0
−0.05
−0.05
−0.1
Transient Frequency [Hz]
Transient Frequency [Hz]
Table 5.7: Results of transient overshoot computation in the two-area system
with the inertia and damping of all the machines reduced by 50%
−0.15
−0.2
−0.25
−0.3
−0.35
−0.4
−0.45
−0.5
0
20
base case
low−inertia case
low inertia and damping
40
60
80
100
Time [s]
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
−0.4
−0.45
−0.5
0
20
base case
low−inertia case
low inertia and damping
40
60
80
100
Time [s]
Figure 5.5: Results of transient overshoot computation in the two-area system for three different cases. Left: disturbance at bus 1. Right: disturbance
at bus 3.
The results of the frequency response computation for three discussed
cases are compared in Figure 5.5. The red lines, representing the case of
low inertia and low damping, as well as the blue lines of Low-Inertia Case
are noticeably steeper than the green curves of Base Case. This difference
in the frequency rate reflects that the inertial response, which is a natural
limitation of the frequency change rate, decreases due to the inertia level
reduction in the system. Reduced damping causes amplification of the transient frequency oscillations which is illustrated by the higher magnitude of
the red line oscillations peaks in Figure 5.5.
Thus, in the given operational state, the reduction of rotational inertia
of the generators of the IEEE two-area system by 50% leads to
• improvement of the minimal damping ratio by 40%
• increase in the transient frequency overshoot maginutede by 10%
Low damping at the generator buses aggravates the situation in Low-Inertia
Case, further increasing the amplitude of the transient frequency deviations,
whereas increased damping levels help to eliminate the effects of reduced
66
CHAPTER 5. SIMULATION RESULTS
inertia on the transient frequency. For instance, a 40% increase in damping
coefficients in Low-Inertia case allows to fully mitigate the effect of the 50%
inertia reduction on the frequency nadir.
The proposed optimization program has been used to further investigate
the impact of inertia and damping changes on the parameters of interest
and to find the optimal levels of inertia and damping.
5.1. IEEE TWO-AREA TEST SYSTEM
5.1.3
67
Optimization
The optimization of inertia and damping in the two-area system has been
accomplished in several stages on a “simple-to-complex” basis. The parameters of the optimization in the investigated cases are presented in Tables 5.8
and 5.9.
Table 5.8: Parameters of the optimization program for two-area test system
(Case 1 - Case 4)
Parameter
Case 1
Case 2
Case 3
Case 4
Kjmin
Mjmin
Kjmax
Mjmax
K tot
0.25Kjbase
0.25Mjbase
4Kjbase
4M base
X j
Kjmax
0.25Mjbase
4Mjbase
-
0.25Mjbase
4Mjbase
-
0.25Mjbase
2Mjbase
-
j∈K
X
X
X
X
M tot
Mjmax
j∈M
cζ
cKj
cMj
c
fp [Hz]
100
0
0
0
-
Mjmax
j∈M
100
0
0
0
-
Mjmax
j∈M
100
0
0
0
-0.312
Mjmax
j∈M
100
0
0
0
-0.312
Table 5.9: Parameters of the optimization program for two-area test system
(Case 5 - Case 8)
Parameter
Case 5
Case 6
Case 7
Case 8
Kjmin
Mjmin
Kjmax
Mjmax
K tot
0.25Mjbase
4Mjbase
X
0.25Kjbase
0.25Mjbase
4Kjbase
4M base
X j
Kjmax
j∈K
X
Mjmax
j∈M
0.25Kjlow inert.
0.25Mjlow inert.
4Kjlow inert.
4Mjlow inert.
X
Kjmax
M tot
0.25Kjbase
0.25Mjbase
4Kjbase
4M base
X j
Kjmax
j∈K
X
Mjmax
j∈M
100
0
0.015
0
-0.312
100
0.01
0.015
0
-0.312
100
0.01
0.015
15
-0.312
Mjmax
j∈M
cζ
cKj
cMj
c
fp [Hz]
100
0
0.015
0
-0.312
j∈K
X
Mjmax
j∈M
68
CHAPTER 5. SIMULATION RESULTS
Case 1 In Case 1, the minimal damping ratio was optimized without
putting a constraint on the frequency overshoot. Furthermore, the costs
of inertia and damping procurement were set to zero. The results of this
simplified optimization case could serve as a validation of the general performance of the algorithm.
By means of the developed optimization program, the minimal damping
ratio in Case 1 was increased by 94% compared to Base Case. The optimization results are summarized in Table 5.10 and Table 5.11. The inertia constants were significantly reduced compared to Base Case, whereas the damping coefficients were increased to the maximal possible values. This outcome
meets all the expectations and conforms with the findings of Low-Inertia case
analysis (Table 5.3). It should be noted, that the inertia constants M did
not reach the minimal values Mjmin = 0.25Mjbase . A simple computation of
the eigenvalues with Mjmin = 0.25Mjbase yields ζ min = 0.3378, showing that
a further reduction of M from the values listed in Table 5.11 does not lead to
the enhancement of the minimal damping ratio. The changes in inertia and
damping that improve the damping ratio of one oscillatory mode may cause
a significant reduction of another damping ratio. Therefore, at some point,
a further enhancement of the minimal damping ratio becomes impossible,
since there are several damping ratios that compete with each other. Thus,
it can be concluded that the optimization program has successfully found
the optimal solution of Case 1.
Table 5.10: Optimization results of the two-area test system (Case 1)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2074
0.4024
90
94%
-0.258
Table 5.11: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 1)
Generator
M [s]
M/M base
KD
G1
G2
G3
G4
3.2500
4.6222
3.1435
4.7548
0.25
0.34
0.25
0.38
4
4
4
4
The results of Case 1 implicate that a high share of RES should be seen
as a positive condition for the stability of the two-area test system with
5.1. IEEE TWO-AREA TEST SYSTEM
69
regard to the damping of the oscillatory modes. However, as previously discussed, low values of inertia complicate the operation of the system, since
they require a faster reaction of control devices to large disturbances. In
Case 1, the optimization resulted into the maximal allowed values of KD
which illustrates that procurement of additional damping improves the minimal damping ratio. Furthermore, higher damping levels also improve the
transient frequency performance of the system and compensate for a poor
inertial response of the system. This could be seen by comparing the overshoot value in Case 1 Mpmin = −0.258 Hz with the values obtained for
Low-Inertia case (see Table 5.5). Thus, higher damping levels in the system
have a significant positive impact on the stability and operation of the test
system.
Case 2 To illustrate the impact of rotational inertia changes on the frequency response of the test system, Case 2, a modified version of Case 1,
was investigated. In this version, the damping coefficients were assumed to
be constant, so the minimal damping ratio was maximized only by changing
the values of M . The optimization results for Case 2 are given in Tables 5.12
and 5.13.
Table 5.12: Optimization results of the two-area tests system (Case 2)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2074
0.3719
62
79%
-0.398
Table 5.13: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 2)
Generator
M [s]
M/M base
KD
G1
G2
G3
G4
3.2500
3.8726
3.0875
4.2606
0.25
0.30
0.25
0.35
1
1
1
1
The minimal damping ratio grew by 79% with respect to Base Case. At
the same time, the maximal magnitude of the transient frequency overshoot
increased by roughly 14%.
70
CHAPTER 5. SIMULATION RESULTS
Case 3 The limitation of the overshoot magnitude is first considered in
Case 3, where the frequency overshoot constraint (4.93), ignored in Case 1
and Case 2, was activated. The damping level is kept constant in Case 3
to enable the investigation of the inertia impact on the frequency response.
The absolute value of |fp | = 0.312 Hz was chosen, which is smaller than
0.348 Hz, obtained in Base Case, but larger than 0.258 Hz of Case 1.
Table 5.14: Optimization results of the two-area test system (Case 3)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2074
0.4030
61
94%
-0.294
Table 5.15: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 3)
Generator
M [s]
M/M base
KD
G1
G2
G3
G4
49.1459
7.3904
3.1622
4.8596
3.78
0.57
0.26
0.39
1
1
1
1
It should be noted, that the 94% improvement, obtained in Case 3,
repeats the result of Case 1. However, in Case 3, the damping coefficients
remained at their initial level KD = 1, whereas in Case 1 their maximization
significantly enhances the minimal damping ratio, as it could be seen in
comparison to Case 2. It could be concluded, that the introduction of the
frequency overshoot constraint pushes the solution of the highly nonlinear
problem into another direction of the solving.
The results, presented in Tables 5.14 and 5.15, show that in order to
keep the transient frequency in the acceptable range while maximizing the
damping ratio, the program suggests to significantly increase the rotational
inertia at bus 1.
First of all, the results of Case 3 indicate a high participation of the G1
states in the dominating oscillatory mode. Furthermore, as it could be seen
in Tables 5.5-5.7, the frequency overshoot at buses 1 and 2 has a greater
magnitude than at nodes 3 and 4, hence, it is more likely for the frequency
of G1 or G2 to violate the overshoot constraint.
5.1. IEEE TWO-AREA TEST SYSTEM
71
Case 4 To check how the availability of inertia at bus 1 affects the optimal solution, Case 4 was designed, with the total available inertia at bus 1
reduced by 50% to 2M1base . The results of the Case 4 simulations can be
found in Tables 5.16 and 5.17.
Table 5.16: Optimization results of the two-area test system (Case 4)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2074
0.2829
47
36%
-0.312
Table 5.17: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 4)
Generator
M [s]
M/M base
KD
G1
G2
G3
G4
26.0000
12.6602
5.4356
7.8022
2.00
0.97
0.42
0.60
1
1
1
1
The restriction imposed on the bus 1 inertia reserves reduces the improvement of the minimal damping ratio from 94% to 36%. The solution
is obtained by deploying the whole available inertia reserve at bus 1 and
reducing the rotational inertia at buses 3 and 4.
Case 5 Another possibility to limit the inertia changes is to impose the
costs on each additional inertia unit. In the developed program, it is assumed
that both reduction and increase of the inertia could be regarded as a service
and could be rated with the same costs.
In Case 5, the damping at the generator buses remains unchanged,
whereas the rotational inertia changes are penalized with cMj = 0.015.
The transient frequency overshoot is restricted at the same level as before,
fp = −0.312 Hz.
72
CHAPTER 5. SIMULATION RESULTS
Table 5.18: Optimization results of the two-area test system (Case 5)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2074
0.3832
54
85%
-0.312
Table 5.19: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 5)
Generator
M [s]
M/M base
KD
G1
G2
G3
G4
34.0543
12.9336
3.0875
4.4721
2.62
1.00
0.25
0.35
1
1
1
1
The obtained results (see Tables 5.18 and 5.19) are similar to those of
Case 4. The inertia of G1 is increased by 162%, while it is set to the minimal
value at bus 3 and close to the minimal value at bus 4. However, the inertia
level at bus 2 remains intact which can be seen as a reaction to the costs
imposed on the changes.
Case 6 Case 6 illustrates the optimization of both inertia and damping
under the same conditions as in Case 5. The damping changes, in contrast
to the inertia changes, are not penalized by costs in this case. The results
in Tables 5.20 and 5.21 show that by setting the damping at the maximal
values, the improvement level of 86% can be achieved, which is close to
the performance of the program in Case 1. At the same time, the required
adjustments of inertia in Case 6 are much smaller than in Case 1. This
discrepancy draws attention to the important role of the damping for the
stability of the investigated two-area test system.
Table 5.20: Optimization results of the two-area test system (Case 6)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2075
0.3796
88
83%
-0.252
5.1. IEEE TWO-AREA TEST SYSTEM
73
Table 5.21: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 6)
Generator
M [s]
M/M base
KD
G1
G2
G3
G4
13.0000
13.0000
4.1744
3.3003
1.00
1.00
0.34
0.27
4
4
4
4
However, procurement of damping at the generator buses implies costs
cKj 6= 0. Non-zero costs of damping were introduced in Case 7.
Case 7 The optimization results in Case 7 are presented in Tables 5.22 and
5.23. The following parameters were used in this case: cMj = 0.015, cKj =
0.01, fp = −0.312 Hz. Since the absolute value of fp is smaller than the
magnitude of the frequency overshoot in Base Case, the slack variable was
introduced to ensure the feasibility of the frequency overshoot constraint.
The costs c associated with the slack variable were set at the value c = 15.
Table 5.22: Optimization results of the two-area test system (Case 7)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2074
0.3768
88
82%
-0.294
Table 5.23: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 7)
Generator
M [s]
M/M base
KD
G1
G2
G3
G4
13.0000
13.0000
3.0691
3.3077
1.00
1.00
0.25
0.35
2.06
1.00
4.00
4.00
Case 8 The last optimization case for the considered test system, Case 8,
is based on Low-Inertia Case. The initial values of the rotational inertia are
already reduced by 50%, compared to the previous optimization cases. As
can be seen in Tables 5.24 and 5.25, improvement of the minimal damping
74
CHAPTER 5. SIMULATION RESULTS
ratio is achieved by a further reduction of the rotational inertia, and the
frequency overshoot constraint is satisfied by increased damping levels.
Table 5.24: Optimization results of the two-area test system (Case 8)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.2926
0.3840
190
38%
-0.312
Table 5.25: Values of the inertia constants M and damping coefficients KD
on 900 MVA base in the two-area test system (Case 8)
5.1.4
Generator
M [s]
M/M low
G1
G2
G3
G4
3.4298
4.0404
2.9882
4.4418
0.53
0.62
0.46
1.68
inert
KD
2.14
2.14
2.14
2.14
Transient Stability Analysis
The results obtained for the two-area test system by means of the optimization program could be illustrated by the results of the transient simulations
implemented in MATLAB. The developed transient simulation program allows to observe the behaviour of the system state variables after large symmetrical disturbances. One of the disturbances of interest is a three-phase
short circuit at one of the circuits of the line 8-9 close to bus 9. Such a
disturbance can be classified as an overfrequency event, since it leads to the
acceleration of the synchronous machines due to reduction of the electrical
torque at their rotors. The fault is cleared after 0.01 s by a disconnection
of the faulty circuit. The disconnection of the circuit changes the topology
of the system and decreases the transmission capacity between two areas.
The frequency of the generators rises, while the PFCs with Tt = 10s are
gradually adjusting the mechanical torque. After a sufficient deployment of
the primary control reserves is achieved, the frequency decreases and settles
at a new steady value.
As shown previously, inertia and damping in the test system influence on
the transient response of the system. The results of the simulations of the
discussed disturbance conform with this statement. Figure 5.6 illustrates the
transient frequency of G1 after the disturbance of interest in 5 different cases.
The presented cases can be divided in two groups, namely the ones with
5.1. IEEE TWO-AREA TEST SYSTEM
75
KDj = 1 (Base Case, Low-Inertia Case, Case 2) and those with improved
damping KDj = 4 (Low Inertia and Low Damping, Case 1). The peaks of
the first group lines lay considerably higher than those of the second group
lines. This stands as a clear illustration of the influence of damping on the
transient frequency in the investigated system.
0.18
Transient Frequency [Hz]
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
base case
low−inertia case
low inertia and high damping
case 1
case 2
15
20
25
30
35
40
Time [s]
Figure 5.6: Transient frequency of G1 after a short cirtcuit at bus 9 and
disconnection of a circuit of the line 8-9 of the two-area test system
Among the cases of the first group, Base Case implicates the highest values of the rotational inertia and, thus, demonstrates the strongest inertial
response. Therefore, as illustrated by the blue line in the graph, Base Case is
associated with the slowest rise and the smallest magnitude of the transient
frequency compared to the other cases in the group. Case 2, characterized by very low levels of inertia and an improved minimal damping ratio,
demonstrates the highest rate and magnitude of the frequency deviation.
However, the difference between the peak values of the transient frequency
in Base Case and Case 2 is merely 0.01 Hz which is 6.25% of the magnitude
in Base Case.
Figure 5.6 does not give a clear impression on the transients occuring
right after the short circuit. The transient frequency within the first 5
seconds after the fault is shown in Figure 5.7. This graph demonstrates that
the lower is the inertia, the steeper is the growth of the frequency after the
76
CHAPTER 5. SIMULATION RESULTS
disturbance. In this case, the rate of the frequency change is independent
from the damping in the system. Thus, the inertial response of the system,
along with the initial state of the system and severeness of the disturbance,
define how much time the protection systems have to clear the fault before
the system stability is lost.
0.12
Transient Frequency [Hz]
0.1
0.08
0.06
0.04
0.02
0
0
1
base case
low−inertia case
low inertia and high damping
case 1
case 2
2
3
4
5
Time [s]
Figure 5.7: Transient frequency of G1 after a short cirtcuit at bus 9 and
disconnection of a circuit of the line 8-9 of the two-area test system (first 5
seconds)
The effect of the inertial response on the critical fault clearing time can
be observed by comparing two plots in Figure 5.8. Both plots represent the
rotor angles of the four generators after the considered disturbance. The
fault was cleared in both cases after 0.012 s, but the rotational inertia of the
generators in the left graph was set to the values of Base Case, whereas at
the right, the same event within Low-Inertia Case is modelled. With inertia
values of Base Case, the system remains stable. In contrast, in Low-Inertia
Case, the generators G3 and G4 fall out of step, which implicates islanding
of the two areas.
5.2. IEEE SOUTH EAST AUSTRALIAN TEST SYSTEM
0.8
8000
7000
Relative Rotor Angle [rad]
Relative Rotor Angle [rad]
0.6
0.4
0.2
0
−0.2
G1
G2
G3
G4
−0.4
−0.6
0
77
10
20
30
40
50
Time [s]
60
70
6000
5000
4000
3000
2000
1000
G1
G2
G3
G4
0
80
−1000
0
10
20
30
40
50
Time [s]
60
70
80
Figure 5.8: Rotor angles of the generators G1-G4 of the two-area test system
after a short circuit at bus 9 in Base Case (left) and Low-Inertia Case (right)
5.2
5.2.1
IEEE South East Australian Test System
System Description
A simplified 14-generator test system, based on the southern and eastern
Australian power networks, is shown in Figure E.1. It consists of 5 areas,
with areas 1 and 2 more closely electrically coupled. This test system will
be further reffered to as the five-area test system. The system is characterized by its long tie lines and, according to [4], demonstrates 3 inter-area
oscillatory modes and 10 local-area modes, with some of these modes being
unstable without PSSs. The parameters of the grid elements and the power
flow data are adopted from [4]. The initial steady-state operating condition
of the system corresponds to the case 2 (medium-heavy loading) of [4].
The parameters of the grid and the detailed power flow data are presented in Appendix E. The parameters of the PFCs repeat those of the
two-area system. It should be noted, that the types and models of AVR and
PSS used in the present thesis (see Chapter 2) differ from those implemented
by the authors of [4]. Modelling of SVC is not included in the present work,
therefore, all the SVCs were represented by uncontrolled reactive shunts.
Furthermore, some of the generator parameters, listed in Table E.3, such as
saturation constants, were approximated by using the typical data from [2]
and [19], since they are not given in [4]. Due to the mentioned discrepancies
in the modelling, benchmarking of the results presented in this thesis with
those of [4] is not possible.
The initial operating condition of the investigated system is described in
Table 5.26. Power flow occurs from south (Area 5) to north (Area 4).
78
CHAPTER 5. SIMULATION RESULTS
Table 5.26: Steady-state operating condition of the five-area test system
Load Condition
Total generation [MW]
Total load [MW]
Medium-Heavy
21590
21000
Inter-area flows
Area
Area
Area
Area
5.2.2
4
2
1
3
to
to
to
to
Area
Area
Area
Area
2
1
3
5
[MW]
[MW]
[MW]
[MW]
-500
-1120
-1000
-500
Small-Signal Stability Analysis
Similarly as in case of the two-area test system, the analysis of the smallsignal stability of the five-area test system starts with calculation of the
eigenvalues of the system matrix in Base Case and Low-Inertia Case. Complete lists of the eigenvalues in the two considered cases can be found in
Appendix E.
The values of M and KD used in the two cases are presented in Table 5.27. The parameters are given on 100 MVA base to conform with the
p.u. base of the data in Appendix E. The values of KD in Table 5.27 correspond to KD = 1 on rated power of the generators. In Base Case, the
Table 5.27: Rotational inertia constants M and damping coefficients of the
five-area test system generators in Base Case and Low-Inertia Case, calculated on 100 MVA base
Generator
G1
G2
G3
G4
G5
G6
G7
Node
KD
M Base [s]
M Low Inertia [s]
101
9.999
71.993
35.996
201
33.335
213.344
106.672
202
22.224
124.454
62.227
203
16.668
106.675
53.338
204
26.668
138.674
69.337
301
53.336
298.682
149.341
302
17.776
124.432
62.216
Generator
G8
G9
G10
G11
G12
G13
G14
Node
KD
M Base [s]
M Low Inertia [s]
401
17.776
106.656
53.328
402
9.999
79.992
39.996
403
17.776
92.435
46.218
404
19.998
103.990
51.995
501
6.666
46.662
23.331
502
10.000
80.000
40.000
503
8.335
125.025
62.513
min = 0.096, as
five-area system has a very low minimal damping ratio, ζbase
could be seen in Table E.4. Furthermore, the system exhibits three more
oscillatory modes with damping ratios lower than 0.200. Thus, oscillations
occuring after small disturbances in this system decay at a very slow rate,
and changes of the operating state of the system might lead to its instability.
In Low-Inertia Case, the system is unstable with an expanding oscillatory
5.2. IEEE SOUTH EAST AUSTRALIAN TEST SYSTEM
79
mode λ = 1.812 ± 39.368 (see Table E.5). This might cause considerable
concerns, if the system has a high installed capacity of RES.
Frequency response of the five-area test system was estimated in the
same way as it was done for the two-area system, i.e. by applying steplike
load changes to the generator buses. The results of the transient overshoot
approximation are shown in Table 5.28. The magnitude of the overshoot
depends on the node where disturbance occured. As could be seen in Table 5.28, disturbances that take place in the same area cause frequency
deviations of the same amplitude. The magnitude of the overshoot has the
highest value in Area 5 and decreases in accordance with the power flow
direction.
Table 5.28: Results of transient overshoot computation in the five-area test
system in Base Case
Node
Generator
Mpmin [Hz]
Node
Generator
Mpmin [Hz]
101
201
202
203
204
301
302
G1
G2
G3
G4
G5
G6
G7
-0.040
-0.040
-0.040
-0.040
-0.040
-0.049
-0.047
401
402
403
404
501
502
503
G8
G9
G10
G11
G12
G13
G14
-0.032
-0.035
-0.035
-0.035
-0.050
-0.052
-0.052
80
5.2.3
CHAPTER 5. SIMULATION RESULTS
Optimization
Poor damping of oscillatory modes in the five-area system can be improved
by providing synthetical inertia and additional damping. By means of the
developed optimization program, the required amount of inertia and damping at the generator nodes can be estimated.
The optimization was carried out for three cases with different parameters presented in Table 5.29.
Table 5.29: Parameters of the optimization program for the five-area test
system
Parameter
Case 1
Case 2
Case 3
Kjmin
Mjmin
Kjmax
Mjmax
cζ
cKj
cMj
c
fp [Hz]
0.25Kjbase
0.25Mjbase
4Kjbase
4Mjbase
100
0
0
0
-
0.25Kjbase
0.25Mjbase
4Kjbase
4Mjbase
100
0.001
0.015
0
-
0.25Kjbase
0.25Mjbase
4Kjbase
4Mjbase
100
0.001
0.015
15
-0.045
Case 1 In Case 1, costs of inertia and damping provision were set to
zero, and the frequency overshoot was not constrained. The results of the
optimization in Case 1 are shown in Tables 5.30 and 5.31. The developed
program allows to improve the minimal damping ratio by 82%. Any further
improvement of the initially lowest damping ratio leads to a decrease in
another critical damping ratio which limits the possible advances of the
algorithm.
To achieve an 82% improvement of the minimal damping ratio, the rotational inertia at five buses was reduced, whereas at the rest of the nodes, it
was increased, with the maximal value of 3.62M base at G12. Furthermore,
adjusted values of damping at majority of the buses are lower than those
in Base Case. Only the damping values of the generators in Area 5 are
significantly increased.
5.2. IEEE SOUTH EAST AUSTRALIAN TEST SYSTEM
81
Table 5.30: Optimization results of the five-area test system (Case 1)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.0960
0.1743
185
82%
-0.047
Table 5.31: Values of the inertia constants M and damping coefficients KD
on 100 MVA base in the five-area test system (Case 1)
Generator
M [s]
M/M base
KD
base
KD /KD
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
30.06
500.74
257.49
183.60
311.87
193.20
172.32
155.43
52.79
39.10
114.45
169.12
72.55
149.22
0.42
2.35
2.07
1.72
2.25
0.65
1.38
1.46
0.66
0.42
1.10
3.62
0.91
1.19
8.82
27.07
11.14
20.77
13.34
26.70
13.58
13.65
5.03
16.98
10.06
23.75
34.20
28.65
0.88
0.81
0.50
1.25
0.50
0.50
0.76
0.77
0.50
0.96
0.50
3.56
3.42
3.44
Case 2 In Case 2, changes of inertia and damping are penalized by costs
cMj = 0.015 and cKj = 0.01, respectively. As shown in Tables 5.32 and 5.33,
introduction of the costs significantly affects the optimization results. In
contrast to Case 1, inertia is changed only at two generators in Area 2, G2
and G4, and two generators in Area 5, G12 and G14. The values of inertia of
the corresponding aggregated machines are increased, furthermore, damping
of G2 and G12 is considerably higher than in Base Case. Inertia and damping at the rest of nodes stay intact. From the results of Case 1 and Case 2,
it could be concluded that generators G2, G4, G12, and G14 participate the
most in the critical oscillatory modes. However, the obtained results do not
conform with the expectation that damping ratios are improved by inertia
reduction.
82
CHAPTER 5. SIMULATION RESULTS
Table 5.32: Optimization results for the five-area test system (Case 2)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.0960
0.1511
150
58%
-0.049
Table 5.33: Values of the inertia constants M and damping coefficients KD
on 100 MVA base in the five-area test system (Case 2)
Generator
M [s]
M/M base
KD
base
KD /KD
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
71.99
319.95
124.45
115.53
138.67
298.68
124.43
106.66
79.99
92.44
103.99
123.20
80.00
141.52
1.00
1.50
1.00
1.08
1.00
1.00
1.00
1.00
1.00
1.00
1.00
2.64
1.00
1.13
10.00
71.73
22.22
16.67
26.67
53.34
17.78
17.78
10.00
17.78
20.00
17.50
10.00
8.34
1.00
2.15
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
2.63
1.00
1.00
Case 3 Optimization under the frequency overshoot constraint was considered in Case 3. The threshold fp was set to -0.045 Hz. The optimization
results for Case 3 are presented in Tables 5.34 and 5.35. Inertia and damping of generator G12 are significantly increased compared with Case 2. The
optimization of inertia and damping levels in the five-area system with constrained transient frequency allowed to improve the minimal damping ratio
of the system by 66%.
5.2. IEEE SOUTH EAST AUSTRALIAN TEST SYSTEM
83
Table 5.34: Optimization results for the five-area test system (Case 3)
ζ0min
min
ζopt
Number of Iterations
Improvement
Mpmin , [Hz]
0.0960
0.1590
150
66%
-0.045
Table 5.35: Values of the inertia constants M and damping coefficients KD
on 100 MVA base in the five-area test system (Case 3)
Generator
M [s]
M/M base
KD
base
KD /KD
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
71.99
363.11
145.29
121.11
138.67
298.68
124.43
106.66
79.99
92.43
103.99
154.57
71.12
146.87
1.00
1.70
1.17
1.14
1.00
1.00
1.00
1.00
1.00
1.00
1.00
3.31
0.89
1.17
10.00
33.30
22.23
16.67
26.66
53.34
17.77
17.75
10.00
17.78
19.99
20.97
12.98
8.33
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
3.146
1.298
1.000
84
CHAPTER 5. SIMULATION RESULTS
5.2.4
Transient Stability Analysis
In Case 1, rotational inertia of several generators is significantly reduced.
This might affect the frequency stability of the system after large disturbances. For further investigation, a short circuit at bus 217 was modelled
by means of the developed transient simulations tool. The short circuit is
eliminated after 0.005 s by disconnection of one of the circuits of the line
217-215. Thus, the transmisission capacity between Area 1 and Area 2 is reduced. Figure 5.9 illustrates the resulting rotor angular velocity excursions
in Base Case (left) and Case 1 (right).
4
x 10−4
0.03
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
3.5
3
2.5
2
1.5
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
0.02
0.01
0
1
−0.01
0.5
−0.02
0
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
−0.03
0
0.5
1
1.5
2
2.5
3
3.5
Figure 5.9: Rotor angular velocity of the generators of the five-area test
system after a short circuit at bus 217 and disconnection of a circuit of the
line 217-215 in Base Case (left) and Case 1 (right)
Generator G1 has the electrically closest location to the faulty bus. The
fault leads to a large power imbalance at the rotor of this generator, with
mechanical power exceeding the power that can be transmitted from the
generator to other areas or consumed by the load at bus 102. In Case 1,
with inertia of G1 reduced by 60%, the disturnance leads to the acceleration
of G1 and, consequently, this generator loses synchronism with the grid.
This illustrates the importance of the transient frequency restriction for the
inertia optimization.
4
5.3. DISCUSSION OF SIMULATION RESULTS
5.3
85
Discussion of Simulation Results
Case analysis has demonstrated that damping of the critical oscillatory mode
in the IEEE two-area test system is significantly improved at reduced levels
of inertia. Thus, the minimal damping ratio could be increased by 79%
compared to the “convenitional” Base Case by reducing the inertia constants
by 65%-75%. However, a further reduction of the inertia would not lead
to any advancements which indicates that there is a specific level of the
RES penetration in the test system, optimal for damping of the interarea
oscillatory mode.
However, high shares of RES generation and low inertia levels raise a
common concern over the transient frequency response of the system. The
transient simulations have shown that the magnitude of the frequency overshoot after a short circuit greatly depends on the inertia levels in the system,
as it is defined by the inertial response of the synchronous machines. Rotational inertia levels may become crucial for the system stability because low
inertia in the system leads to reduction of the critical fault clearing time.
Nevertheless, the magnitude of the frequency overshoot after 10s of seconds following the fault clearance shows a weaker relation to the inertia levels. In this case, the frequency response is mainly affected by the damping
levels which also agrees with the results of optimization under the frequency
overshoot constraint.
The results of Case 3 and Case 5 show that the frequency overshoot
in the test system can also be limited by a considerable increase of the
rotational inertia at bus 1. Procurement of inertia at this bus could be
seen as an effective measure of securing an acceptable level of the frequency
overshoot. However, if the limitation imposed on the frequency magnitude
is too strict, it might become too expensive to comply with it by increasing
inertia solely. Thus, procurement of additional damping would become an
appropriate measure to limit the frequency deviations that develop in the
first minute after a large disturbance.
The magnitude of these frequency oscillations is also strongly related
to the speed of the PFC operation. Figure 5.10 illustrates the transient
frequency after a sudden load change at bus 1 for three different values of
Tt . It is clear from the plot, that the reduction of the PFC time constant
from Tt = 10 s to Tt = 3 s allows to significantly reduce the frequency
deviations. Instantaneous reaction of PFC, modelled by Tt = 1 s allows to
prevent any major frequency oscillations and causes only a slight deviation
of the settling frequency from 60 Hz.
86
CHAPTER 5. SIMULATION RESULTS
0
Transient Frequency [Hz]
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
−0.4
Tt = 1s
−0.45
Tt = 3s
Tt = 10s
−0.5
0
20
40
60
Time [s]
80
100
Figure 5.10: Transient frequency response to a disturbance in the two-area
test system with different values of the time constant Tt
Such a fast PFC could be provided by BESS [7]. The damping procurement might be also accomplished by means of BESS [20]. However, if the
storage systems are used for damping, it requires from them an immediate reaction to any oscillations of the frequency, whereas PFC reserves are
activated outside of a dead-band around the nominal frequency. In Continental Europe, this dead-band is ±10mHz [7]. The potential of the battery
systems in provision of these two services simultaneously should be further
investigated.
IEEE South East Australian test system has demonstrated very poor
damping of the oscillatory modes. Improvement of the minimal damping
ratio by 82% can be achieved by inertia and damping adjustment. The
solution implicates increased level of inertia at particular buses which does
not meet the expectations that high inertia would worsen the damping.
Furthermore, in Low-Inertia Case, with inertia constant reduced by 50%,
the test system becomes unstable.
The discrepancy between the results and the expectations can be originated in the simplicity of the employed AVR and PSS models and lack of
tuning of thier parameters. Alternative models of the control devices were
5.3. DISCUSSION OF SIMULATION RESULTS
87
not implemented due to a different focus of the present work. To further
improve the optimization program, various types of AVR and PSS should
be added to its modelling capabilities. Nevertheless, the developed program
proved to enable the inertia and damping optimization for a more complex
power system than the two-area test system.
88
CHAPTER 5. SIMULATION RESULTS
Chapter 6
Conclusions and outlook
The present work investigated the impact of rotational inertia changes on
damping of oscillatory modes and frequency stability of a power system.
The analysis of system stability was based on a multimachine power system
representation with incorporation of a detailed synchronous machine model.
This model takes into account the voltage dynamics and includes the effects
of AVR, PSS and PFC operation. Based on the implemented synchronous
machine model, along with the models of the interconnecting transmission
network and aggregated load, a set of system equations was formulated.
Linearization of these equations enabled the calculation of the system state
matrix and the small-signal stability analysis of a power system.
Further, the sensitivities of the damping ratios of the system oscillatory
modes to inertia and damping were calculated. This allows to evaluate
the changes in the system modes due to incremental changes of inertia and
damping at the system nodes. To assess the frequency dynamics of a power
system, the magnitude of the transient overshoot in response to a steplike
load change was approximated by using a transfer function. The sensitivities
of this magnitude to inertia and damping were estimated by computing the
sensitivities of the eigenvector product.
The calculated sensitivities were employed in an algorithm for optimization of inertia and damping. The objective of this algorithm, based on [1], is
the maximization of the minimal damping ratio associated with the system
oscillatory modes. The improvement is achieved by adjusting rotational inertia and damping at the system nodes. At the same time, the algorithm
allows to limit the magnitude of the transient frequency overshoot which
ensures acceptable levels of the frequency deviations in the system. The
changes of inertia and damping are penalized with costs which serves as a
simplified representation of remuneration for provision of synthetic inertia
and additional damping.
The proposed algorithm was implemented in the engineering environment MATLAB. To assess the performance of the developed program, a
89
90
CHAPTER 6. CONCLUSIONS AND OUTLOOK
case study was conducted for two test systems, IEEE two-area test system
and IEEE South East Australian test system. The case analysis of the former system included eight cases with different optimization parameters. It
was shown, that a reduction of rotational inertia significantly improves the
damping of the oscillatory modes. However, a simultaneous limitation of
the transient frequency deviations required the provision of additional inertia and damping at particular nodes. Introduction of costs for inertia and
damping procurement affected the optimal solution by leaving only the most
effective changes of inertia and damping. The influence of PFC speed on
the frequency nadir was shown. The impact of inertia on frequency deviations after a severe disturbance was illustrated by the results of transient
simulations implemented in MATLAB.
The minimal damping ratio of IEEE South East Australian test system
was optimized in three different cases. As an optimal solution in case of
non-zero costs, higher levels of rotational inertia and damping at particular
system nodes are suggested. However, the modelling accuracy in case of
this system should be improved by incorporation of realistic models and
parameters of AVR and PSS.
Detailed modelling of AVR, PSS, and turbine governors can be proposed
as as an objective for the future work. Implementation of the RES models
is another possible enhancement of the modelling framework. Furthermore,
the algorithm could be adjusted to consider the provision of inertia and
damping at buses with no generation.
Another possible field of research is the economical aspects of inertia and
damping procurement. Economical factors clearly have a great influence on
the optimal inertia and damping levels. Furthermore, pricing at potential
inertia-as-a-service markets requires an adequate estimation of the economical losses due to complications in system operation and outages caused by
a certain inertia level.
If the focus of analysis is to be shifted from the maximization of the
damping of oscillations to minimization of the transient frequency overshoot
in a system with low inertia, the optimization program could be adjusted
respectively, by setting the frequency overshoot magnitude as a main objective while restraining the minimal damping ratio. This would allow to
optimize the operation of the system during a high RES penetration with
regard to the frequency response. As shown in the case analysis of the IEEE
two-area test system, with the optimization program focused on the minimal damping ratio, the already low inertia levels are proposed to be further
reduced. To find a proper balance between two objectives, the consequences
of both poor oscillatory mode damping and high frequency overshoot should
be evaluated for each particular power system.
Appendix A
Runge-Kutta Methods of
Numerical Integration
R-K methods used for numerical integration in the present work were proposed for power system transient simulations in [2]. Depending on the
number of evaluations of the first derivative in Taylor series solution, RK methods of different orders could be used for this purpose. In this thesis,
the second order R-K method and Gill’s version of the fourth order R-K
method were implemented.
Second order R-K method
Consider the first-order differential equation
dx
= f (x, t)
dt
with initial condition xn at the moment tn .
The second-order R-K formula for the value of x at the moment t =
tn + ∆t is [2]
k1 + k2
xn+1 = xn + ∆x = xn +
(A.1)
2
where
k1 = f (xn , tn )∆t
(A.2)
k2 = f (xn + k1 , tn + ∆t)∆t
Gill’s version of fourth order R-K method
In Gill’s version of the fourth order R-K method, solution of the differential equation is obtained by a four-step approximation of x. Each stage is
91
92
APPENDIX A. RUNGE-KUTTA METHODS
denoted by j = 1, 2, 3, 4 and described by
kj = aj [f (xj−1 , t) − bj qj−1 ]
xj = xj−1 + kj ∆t
(A.3)
qj = qj−1 + 3kj − cj f (xj−1 , t)
For the first time interval q0 = 0, in further calculations the value of q0 is
given by q4 of the previous step. The values of a, b, and c are given by
√
a1 = 1/2,
b1 = 2, c1 = a1 , a2 = 1 − 0.5, b2 = 1, c2 = a2
√
b4 = 2, c4 = 1/2
a3 = 1 + 0.5, b3 = 1, c3 = a3 , a4 = 1/6,
Solution at each time step is represented by x4 . The advantage of Gill’s
version of fourth order R-K method is minimization of the roundoff errors
achieved by implementation of the q variable. Furthermore, it requires less
storage capacity than the original R-K methods.
Appendix B
Calculation of Initial Steady
State
Complex voltage and current at the terminals of a synchronous machine in
the initial operational state can be denoted as
Vt = V ejθ
It = Iejγ
(B.1)
The initial value of the rotor angle δ can be estimated by [2]
δ = arg(Vt + (Ra + jXqs )It )
(B.2)
Further, d-q components of voltage and current can be calculated as
id = Re(It ej(γ−δ+0.5π) )
j(γ−δ+0.5π)
iq = Im(It e
ed = Re(Vt e
)
(B.3)
(B.4)
j(γ−δ+0.5π)
)
(B.5)
j(γ−δ+0.5π)
)
(B.6)
eq = Im(Vt e
Field circuit current and voltage are given by
eq + Ra iq + Xds id
Xads
= Rfd ifd
Xadu
=
Rfd efd
ifd =
(B.7)
efd
(B.8)
Efd
(B.9)
The mutual flux linkages Ψad and Ψaq are calculated as follows
Ψad = Xads (−id + ifd )
(B.10)
Ψaq = −Xaqs iq
(B.11)
93
94
APPENDIX B. CALCULATION OF INITIAL STEADY STATE
Initial values of the flux linkages of the rotor circuits are defined by
Ψfd = Ψad + Xfd ifd
(B.12)
Ψ1d = Xads (ifd − id )
(B.13)
Ψ1q = −Xaqs iq
(B.14)
Ψ2q = −Xaqs iq
(B.15)
The relative rotor angular velocity is equal to zero in a steady state
∆ωr = 0
(B.16)
The excitation quantities are determined as follows
v1 = V
(B.17)
v2 = 0
(B.18)
vs = 0
(B.19)
The AVR reference is given by
Vref =
Efd
+ v1
KA
(B.20)
The mechanical power is not regulated in the initial steady state
∆Pm = 0
(B.21)
Appendix C
Modelling of Transmission
Network Elements
Reactive Shunt Devices
Reactive shunt devices (shunt capacitors and reactors) can be represented
by corresponding shunt admittance yksh .
Figure C.1: A shunt connected to bus k [3]
With the sign convention from [3], the current injection from the shunt
is defined by
Iksh = −yksh Ek
(C.1)
where Ek is the complex voltage at bus k.
Transmission Lines and Transformers
Depending on the goals of analysis, transmission lines could be modelled
by either differential or algebraic equations. Since the network transients
are out of the focus of the present work, an algebraic model, namely the
lumped-circuit model of a transmission line, is used (see Figure C.2).
95
96
APPENDIX C. TRANSMISSION NETWORK MODELLING
Figure C.2: Lumped-circuit model of a transmission line [3]
This model is characterized by its series impedance
zkm = rkm + jxkm
(C.2)
sh
sh
+ jbsh
= gkm
ykm
km
(C.3)
and shunt admittance
The series admittance of the line model is given by
−1
= gkm + jbkm
ykm = zkm
(C.4)
where
rkm
+ x2km
xkm
=− 2
rkm + x2km
gkm =
bkm
2
rkm
(C.5)
(C.6)
Another π-model, derived in [3], is used to represent transformers. In case
of transformers, the π-model incorporates complex tap ratios
tkm = akm ejφkm
(C.7)
where akm is the turns ratio. For in-phase transformers, considered in this
thesis, tkm = akm which means that tkm ∈ R.
In [3], a unified branch model for lines, in-phase transformers, and phaseshifting transformers was developed to facilitate the modelling routine. This
model is shown in Figure C.3.
97
Figure C.3: Unified branch model [3]
The general expression for the branch current in this model is given by
Ikm = a2km (yk msh + ykm )Ek − t∗km tmk ykm Em
(C.8)
where Ek and Em are the complex node voltages.
Admittance Matrix Elements
On the base of the unified branch model, the elements of the admittance
matrix Y can be derived as
Ykm = −t∗km tmk ykm
X
sh
Ykk = yksh +
a2km (ykm
+ ykm )
(C.9)
(C.10)
m∈Ωk
where Ωk is the set of nodes adjacent to k, k = 1, 2, ..., N , m = 1, 2, ..., N ,
m 6= k, with N representing the number of nodes in the network.
98
APPENDIX C. TRANSMISSION NETWORK MODELLING
Appendix D
Structure of MATLAB input
arrays
Table D.1: Bus data structure (BUSES)
Column
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Parameter
Bus number
Active power generation [p.u.]
Reactive power generation [p.u.]
Active power demand [p.u.]
Reactive power demand [p.u.]
Voltage magnitude [p.u.]
Damping coefficient
Shunt susceptance [p.u. injected at V=1.0 p.u.]
Slack bus = 1, Other buses =0
Area
Maximal reactive power [p.u.]
Minimal reactive power [p.u.]
Total MVA base of generator [p.u,]
Maximal active power [p.u.]
Base Voltage [kV]
99
100
APPENDIX D. STRUCTURE OF MATLAB INPUT ARRAYS
Table D.2: Branch data structure (LINES)
Column
1
2
3
4
5
6
7
Parameter
Branch number
”From” bus number
”To” bus number
Transformer tap ratio
Branch resistance [p.u.]
Branch reactance [p.u.]
Total line charging susceptance [p.u.]
Table D.3: Generator data structure (GENS)
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Parameter
Aggregated Generator No.
Node
Power rating [MVA]
Inertia constant H [s]
Number of generators
Number of generators on-line
Synchronous reactance Xd [p.u.]
Synchronous reactance Xq [p.u.]
Stator leakage inductance Xl [p.u.]
Stator resistance Ra [p.u.]
Transient reactance Xd0 [p.u.]
Transient reactance Xq0 [p.u.]
Subtransient reactance Xd00 [p.u.]
Subtransient reactance Xq00 [p.u.]
0 [s]
Transient OC time constanct Td0
0 [s]
Transient OC time constanct Tq0
00 [s]
Subtransient OC time constant Td0
00 [s]
Subtransient OC time constant Tq0
Saturation constant Asat
Saturation constant Bsat
Saturation constant ψt
AVR constant KA
AVR time constant TR [s]
PSS constant KSTAB
PSS time constant TW [s]
PSS time constant T1 [s]
PSS time constant T2 [s]
Damping coefficient KD
Voltage base [kV]
101
102
APPENDIX E. IEEE SOUTH EAST AUSTRALIAN SYSTEM
Appendix E
IEEE South East Australian
System
Figure E.1: IEEE South East Australian five-area test system [4]
103
Table E.1: Power flow input data for IEEE South Australian test system [4]
calculated on 100 MVA base
Bus No.
Pg [p.u.]
Qg [p.u.]
Pl [p.u.]
Ql [p.u.]
Vi [p.u.]
Qshunt [p.u.]
Area
Base Voltage [kV]
101
102
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
501
502
503
504
505
506
507
508
509
4.788
0
28
19.2
13.5
15.84
0
0
0
0
0
0
0
0
0
0
0
0
0
46.8
15.32
0
0
0
0
0
0
0
0
0
0
0
0
0
14
8.7
14
14.64
0
0
0
0
0
0
0
0
0
0
0
0
6
8
6.9
0
0
0
0
0
0
1.632
0
1.945
2.42
2.472
0.712
0.418
0
0
0
0
0
0
0
0
0
0
0
0
11.28
2.532
0
0
0
0
0
0
0
0
0
0
1.294
0
0
4.66
0.942
1.888
2.388
0
0
0
0
0
0
0
0.639
0
0
0
0
-0.176
2.12
1.845
0
0
0
0.368
0
0.502
0
-3.8
0
0
0
0
-3.3
-1.1
-16
-1.8
0
0
-14.45
-14.1
0
0
-4.1
-15.65
-10.7
0
0
0
0
0
-12.3
-6.5
-6.55
-1.95
0
0
-1.15
-24.05
-2.5
0
0
0
0
0
-12.15
-9.05
0
-1.85
-3.1
-6.5
-7
-15.35
0
0
0
0
0
0
0
-2
0
0
-7.1
-5.2
-0.7
0
-0.38
0
0
0
0
-0.33
-0.11
-1.6
-0.18
0
0
-1.45
-1.4
0
0
-0.4
-1.55
-1.1
0
0
0
0
0
-1.23
-0.65
-0.66
-0.2
0
0
-0.12
-2.4
-0.25
0
0
0
0
0
-1.2
-0.9
0
-0.2
-0.3
-0.65
-0.7
-1.55
0
0
0
0
0
0
0
-0.4
0
0
-1.4
-1.05
-0.15
1.000
0
0
0
0
0
0
0
0
0
0
0
0
0
1.5
0
0
0
1.5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.6
0
0.3
0
0
-0.3
-0.6
-0.6
0
0
0
-0.9
0
0
0
0
0
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
15
330
20
20
20
20
330
330
330
330
330
500
330
330
500
330
330
330
330
20
20
500
500
500
500
500
500
330
330
330
220
220
220
275
20
20
20
20
275
275
275
275
275
275
275
275
275
330
330
330
20
15
15
275
275
275
275
275
275
1.000
1.000
1.000
1.000
1.055
1.000
1.000
1.015
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.040
1.027
104
APPENDIX E. IEEE SOUTH EAST AUSTRALIAN SYSTEM
Table E.2: Parameters of the branches of IEEE South Australian test system
[4] calculated on 100 MVA base
Line No.
From Bus
To Bus
Tap Ratio
R [p.u.]
X [p.u.]
Bsh [p.u.]
Line No.
From Bus
To Bus
Tap Ratio
R [p.u.]
X [p.u.]
Bsh [p.u.]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
102
102
205
205
206
206
206
207
207
208
209
210
211
211
212
214
214
215
215
216
303
303
304
305
305
306
307
309
310
312
313
315
405
405
405
406
407
217
309
206
416
207
212
215
208
209
211
212
213
212
214
217
216
217
216
217
217
304
305
305
306
307
307
308
310
311
313
314
509
406
408
409
407
408
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.002022
0.001865
0.004800
0.001850
0.002250
0.003300
0.003300
0.000900
0.000800
0.001033
0.004500
0.000500
0.000700
0.001900
0.007000
0.001000
0.004900
0.002550
0.003600
0.005100
0.001000
0.000550
0.000300
0.000200
0.000150
0.000100
0.001150
0.004500
0.000000
0.002000
0.000500
0.003500
0.001950
0.005400
0.006000
0.000300
0.004200
0.016066
0.014771
0.038000
0.023000
0.017800
0.026350
0.026350
0.007000
0.006200
0.008267
0.035600
0.007250
0.005400
0.015500
0.055800
0.007700
0.038800
0.020150
0.028700
0.040300
0.014000
0.008000
0.004000
0.003000
0.002250
0.001200
0.016250
0.035667
-0.016850
0.015000
0.005000
0.025000
0.023750
0.050000
0.040667
0.003800
0.051300
3.268
1.634
1.862
1.460
0.874
1.292
1.292
0.342
0.076
0.912
0.437
3.080
0.266
0.190
0.684
0.095
0.475
0.988
1.406
0.494
1.480
3.400
0.424
0.320
0.894
0.127
6.890
1.748
0.000
0.900
0.520
0.380
0.762
0.189
2.370
0.124
0.412
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
69
70
71
72
73
74
408
409
410
410
410
411
414
415
504
504
505
505
506
506
507
507
101
201
202
203
204
209
213
301
302
304
305
305
308
401
403
404
413
501
502
503
410
411
411
412
413
412
415
416
507
508
507
508
507
508
508
509
102
206
209
208
215
210
214
303
312
313
311
314
315
410
407
405
414
504
505
506
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.948
0.948
0.948
0.948
0.948
0.99
1
0.935
0.952
0.961
1
1
0.96
0.939
0.952
0.952
1
0.952
0.93
0.93
0.005500
0.005150
0.004300
0.001075
0.002000
0.000600
0.001000
0.001850
0.011500
0.013000
0.000800
0.002500
0.000800
0.015000
0.002000
0.003000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.064000
0.035450
0.053200
0.013300
0.024700
0.012500
0.012500
0.023000
0.075000
0.009500
0.008500
0.028000
0.008500
0.110000
0.019000
0.022000
0.012000
0.004800
0.007200
0.010200
0.006000
0.006800
0.006800
0.003000
0.008450
0.016000
0.012000
0.012150
0.013500
0.008450
0.008450
0.008500
0.002667
0.025500
0.016000
0.020000
2.019
0.920
0.427
1.708
0.800
0.780
0.780
1.460
1.120
1.740
0.060
0.170
0.060
1.800
0.090
0.900
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Table E.3: Parameters of the aggregated synchornous machines of IEEE
South East Australian
Aggregated Generator No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Node
Power Rating [MVA]
H [s]
Number of generators
Number of generators on-line
Synchronous reactance Xd [p.u.]
Synchronous reactance Xq [p.u.]
Stator leakage inductance Xl [p.u.]
Stator resistance Ra [p.u.]
Transient reactance Xd0 [p.u.]
Transient reactance Xq0 [p.u.]
Subtransient reactance Xd00 [p.u.]
Subtransient reactance Xq00 [p.u.]
0 [s]
Transient OC time constanct Td0
0 [s]
Transient OC time constanct Tq0
00 [s]
Subtransient OC time constant Td0
00 [s]
Subtransient OC time constant Tq0
Saturation constant Asat
Saturation constant Bsat
Saturation constant ψt
AVR constant KA
AVR time constant TR [s]
PSS constant KSTAB
PSS time constant TW [s]
PSS time constant T1 [s]
PSS time constant T2 [s]
101
333.3
3.6
12
3
1.1
0.65
0.15
0.0025
0.25
0.55
0.25
0.25
8.5
0.4
0.05
0.2
0.015
9.6
0.9
200
0.1
20
1.4
0.15
0.02
201
666.7
3.2
6
5
1.8
1.75
0.2
0.0025
0.3
0.7
0.21
0.21
8.5
0.3
0.04
0.08
0.015
9.6
0.9
400
0.02
20
1.4
0.15
0.02
202
555.6
2.8
5
4
2.2
2.1
0.15
0.0025
0.3
0.5
0.2
0.21
4.5
1.5
0.04
0.06
0.015
9.6
0.9
400
0.02
20
1.4
0.15
0.02
203
555.6
3.2
4
3
1.8
1.75
0.2
0.0025
0.3
0.7
0.21
0.21
8.5
0.3
0.04
0.08
0.015
9.6
0.9
300
0.01
20
1.4
0.15
0.02
204
666.7
2.6
6
4
2.3
1.7
0.2
0.0025
0.3
0.4
0.25
0.25
5
2
0.03
0.25
0.015
9.6
0.9
400
0.02
20
1.4
0.15
0.02
301
666.7
2.8
8
8
2.7
1.5
0.2
0.0025
0.3
0.85
0.25
0.25
7.5
0.85
0.04
0.12
0.015
9.6
0.9
400
0.05
20
1.4
0.15
0.02
302
444.4
3.5
4
4
2
1.8
0.15
0.0025
0.25
0.55
0.2
0.2
7.5
0.4
0.04
0.25
0.015
9.6
0.9
200
0.05
20
1.4
0.15
0.02
401
444.4
3
4
4
1.9
1.8
0.2
0.0025
0.3
0.55
0.26
0.26
6.5
1.4
0.035
0.04
0.015
9.6
0.9
300
0.1
20
1.4
0.15
0.02
402
333.3
4
3
3
2.2
1.4
0.2
0.0025
0.32
0.75
0.24
0.24
9
1.4
0.04
0.13
0.015
9.6
0.9
300
0.05
20
1.4
0.15
0.02
403
444.4
2.6
4
4
2.3
1.7
0.2
0.0025
0.3
0.4
0.25
0.25
5
2
0.03
0.25
0.015
9.6
0.9
300
0.01
20
1.4
0.15
0.02
404
333.3
2.6
6
6
2.3
1.7
0.2
0.0025
0.3
0.4
0.25
0.25
5
2
0.03
0.25
0.015
9.6
0.9
250
0.2
20
1.4
0.15
0.02
501
333.3
3.5
2
2
2.2
1.7
0.2
0.0025
0.3
0.8
0.24
0.24
7.5
1.5
0.025
0.1
0.015
9.6
0.9
1000
0.04
20
1.4
0.15
0.02
502
250
4
4
4
2
1.5
0.2
0.0025
0.3
0.8
0.22
0.22
7.5
3
0.04
0.2
0.015
9.6
0.9
400
0.5
20
1.4
0.15
0.02
503
166.7
7.5
6
5
2.3
2
0.2
0.0025
0.25
0.35
0.17
0.17
5
1
0.022
0.035
0.015
9.6
0.9
300
0.01
20
1.4
0.15
0.02
105
Table E.4: Eigenvalues of the South East Australian test system in Base
Case
Eigenvalue
Rel
Imaginary
4.57E-13
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.035
-0.035
-0.493
-0.677
-0.677
-0.725
-0.725
-0.754
-0.761
-0.771
-0.771
-0.800
-0.825
-0.837
-0.846
-0.846
-0.905
-0.925
-0.948
-0.948
-1.189
-1.281
-0.475
-0.475
-1.387
-1.663
-1.729
-2.149
-2.344
-2.539
-0.625
-0.625
0
0
0
0
0
0
0
-1.56E-07
1.56E-07
0
0
0
0
0
-0.053
0.053
0
-0.100
0.100
-0.143
0.143
0
0
-0.055
0.055
0
0
0
-0.0266
0.0266
0
0
-0.125
0.125
0
0
-1.213
1.213
0
0
0
0
0
0
-2.473
2.473
Damping
Ratio
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.549
0.549
1
0.989
0.989
0.981
0.981
1
1
0.997
0.997
1
1
1
0.999
0.999
1
1
0.991
0.991
1
1
0.364
0.364
1
1
1
1
1
1
0.245
0.245
Eigenvalue
Rel
Imaginary
-0.638
-0.638
-1.290
-1.290
-4.067
-1.032
-1.032
-4.683
-2.521
-2.521
-1.981
-1.981
-3.024
-3.024
-2.365
-2.365
-2.160
-2.160
-4.127
-4.127
-2.435
-2.435
-2.010
-2.010
-6.883
-6.903
-6.953
-7.864
-9.688
-11.791
-12.884
-13.910
-15.837
-16.561
-7.886
-7.886
-20.000
-7.651
-7.651
-23.233
-23.233
-6.100
-6.100
-5.255
-5.255
-25.880
-28.226
-2.751
2.751
-3.390
3.390
0
-4.345
4.345
0
-3.946
3.946
-4.728
4.728
-4.289
4.289
-4.720
4.720
-4.946
4.946
-3.618
3.618
-5.300
5.300
-5.603
5.603
0
0
0
0
0
0
0
0
0
0
-15.535
15.535
0
-21.558
21.558
-0.004
0.004
-23.599
23.599
-24.518
24.518
0
0
Damping
Ratio
0.226
0.226
0.356
0.356
1
0.231
0.231
1
0.538
0.538
0.387
0.387
0.576
0.576
0.448
0.448
0.400
0.400
0.752
0.752
0.417
0.417
0.338
0.338
1
1
1
1
1
1
1
1
1
1
0.453
0.453
1
0.334
0.334
1.000
1.000
0.250
0.250
0.210
0.210
1
1
Eigenvalue
Rel
Imaginary
-28.886
-29.313
-6.192
-6.192
-8.563
-8.563
-30.781
-3.402
-3.402
-31.379
-31.438
-10.882
-10.882
-7.069
-7.069
-4.224
-4.224
-37.717
-42.761
-16.877
-16.877
-48.896
-7.404
-7.404
-50.000
-50.000
-50.000
-50.000
-50.000
-52.194
-11.885
-11.885
-53.951
-56.069
-57.587
-57.795
-5.647
-5.647
-59.302
-60.553
-62.474
-65.097
-68.293
-75.534
-92.866
-103.288
-141.098
0
0
-28.747
28.747
-28.819
28.819
0
-31.147
31.147
0
0
-29.944
29.944
-32.543
32.543
-34.133
34.133
0
0
-44.072
44.072
0
-48.448
48.448
0
0
0
0
0
0
-50.909
50.909
0
0
0
0
-58.540
58.540
0
0
0
0
0
0
0
0
0
Damping
Ratio
1
1
0.211
0.211
0.285
0.285
1
0.109
0.109
1
1
0.342
0.342
0.212
0.212
0.123
0.123
1
1
0.358
0.358
1
0.151
0.151
1
1
1
1
1
1
0.227
0.227
1
1
1
1
0.096
0.096
1
1
1
1
1
1
1
1
1
106
APPENDIX E. IEEE SOUTH EAST AUSTRALIAN SYSTEM
Table E.5: Eigenvalues of the South East Australian system in Low-Inertia
case
Eigenvalue
Rel
Imaginary
5.12E-13
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.033
-0.037
-0.037
-0.493
-0.676
-0.676
-0.724
-0.724
-0.754
-0.762
-0.771
-0.771
-0.801
-0.827
-0.838
-0.850
-0.850
-0.904
-0.927
-0.947
-0.947
-1.271
-1.360
-1.360
-0.724
-0.724
-1.655
-1.691
-2.229
-2.522
-0.505
-0.505
-3.117
0
0
0
0
0
0
0
-1.56E-07
1.56E-07
0
0
0
0
0
-0.056
0.056
0
-0.100
0.100
-0.143
0.143
0
0
-0.056
0.056
0
0
0
-0.030
0.030
0
0
-0.129
0.129
0
-0.038
0.038
-1.298
1.298
0
0
0
0
-2.758
2.758
0
Damping
Ratio
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.554
0.554
1
0.989
0.989
0.981
0.981
1
1
0.997
0.997
1
1
1
0.999
0.999
1
1
0.991
0.991
1
1.000
1.000
0.487
0.487
1
1
1
1
0.180
0.180
1
Eigenvalue
Rel
Imaginary
-1.293
-1.293
-3.700
-1.687
-1.687
-4.576
-1.165
-1.165
-3.028
-3.028
-2.393
-2.393
-3.477
-3.477
-2.882
-2.882
-2.636
-2.636
-4.697
-4.697
-2.950
-2.950
-2.510
-2.510
-6.855
-6.896
-6.921
-7.563
-8.576
-11.476
-12.034
-12.866
-15.389
-15.722
-18.789
-20.000
-22.639
-12.170
-12.170
-25.351
-25.795
-26.339
-28.519
-29.111
-29.111
-7.778
-7.778
-2.966
2.966
0
-3.884
3.884
0
-4.646
4.646
-4.250
4.250
-5.054
5.054
-4.467
4.467
-4.900
4.900
-5.330
5.330
-3.709
3.709
-5.642
5.642
-6.107
6.107
0
0
0
0
0
0
0
0
0
0
0
0
0
-21.993
21.993
0
0
0
0
-0.081
0.081
-28.753
28.753
Damping
Ratio
0.400
0.400
1
0.398
0.398
1
0.243
0.243
0.580
0.580
0.428
0.428
0.614
0.614
0.507
0.507
0.443
0.443
0.785
0.785
0.463
0.463
0.380
0.380
1
1
1
1
1
1
1
1
1
1
1
1
1
0.484
0.484
1
1
1
1
1.000
1.000
0.261
0.261
Eigenvalue
Rel
Imaginary
-31.345
-5.959
-5.959
-4.286
-4.286
-34.515
-2.721
-2.721
-37.796
-1.265
-1.265
1.812
1.812
-8.347
-8.347
-11.864
-11.864
-10.765
-10.765
-48.484
-50.000
-50.000
-50.000
-50.000
-50.000
-52.313
-56.124
-58.495
-14.378
-14.378
-60.061
-61.470
-61.601
-6.345
-6.345
-63.369
-10.377
-10.377
-66.093
-71.711
-73.283
-4.147
-4.147
-80.751
-93.441
-103.459
-144.905
0
-30.809
30.809
-33.750
33.750
0
-36.133
36.133
0
-39.190
39.190
-39.368
39.368
-41.850
41.850
-41.653
41.653
-42.572
42.572
0
0
0
0
0
0
0
0
0
-57.587
57.587
0
0
0
-62.552
62.552
0
-64.836
64.836
0
0
0
-77.155
77.155
0
0
0
0
Damping
Ratio
1
0.190
0.190
0.126
0.126
1
0.075
0.075
1
0.032
0.032
-0.046
-0.046
0.196
0.196
0.274
0.274
0.245
0.245
1
1
1
1
1
1
1
1
1
0.242
0.242
1
1
1
0.101
0.101
1
0.158
0.158
1
1
1
0.054
0.054
1
1
1
1
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