energies
Article
Power System Stabilizer Tuning Algorithm in a Multimachine
System Based on S-Domain and Time Domain System
Performance Measures
Predrag Marić 1, *, Ružica Kljajić 1 , Harold R. Chamorro 2 and Hrvoje Glavaš 1
1
2
*
Citation: Marić, P.; Kljajić, R.;
Chamorro, H.R.; Glavaš, H. Power
System Stabilizer Tuning Algorithm
in a Multimachine System Based on
S-Domain and Time Domain System
Faculty of Electrical Engineering, Computer Science and Information Technology Osijek, University of Osijek,
31000 Osijek, Croatia; ruzica.kljajic@ferit.hr (R.K.); hrvoje.glavas@ferit.hr (H.G.)
KTH Royal Institute of Technology, 100 44 Stockholm, Sweden; hr.chamo@ieee.org
Correspondence: predrag.maric@ferit.hr
Abstract: One of the main characteristics of power systems is keeping voltages within given limits,
done by implementing fast automatic voltage regulators (AVR), which can raise generator voltage
(i.e., excitation voltage) in a short time to ceiling voltage limits while simultaneously affecting
the damping component of the synchronous generator electromagnetic torque. The efficient way
to increase damping in the power system is to implement a power system stabilizer (PSS) in the
excitation circuit of the synchronous generator. This paper proposes an enhanced algorithm for PSS
tuning in the multimachine system. The algorithm is based on the analysis of system participation
factors and the pole placement method while respecting the time domain behavior of the system
after being subdued with a small disturbance. The observed time-domain outputs, namely active
power, speed, and rotor angle of the synchronous generator, have been classified and validated with
proposed weight functions based on the minimal square deviation between the initial values in a
steady-state and all sampled values during the transitional process. The system weight function
proposed in this algorithm comprises s-domain and time-domain indices and represents a novel
approach for PSS tuning. The proposed algorithm performance is validated on IEEE 14-bus system
with a detailed presentation of the results in a graphical and table form.
Performance Measures. Energies 2021,
14, 5644. https://doi.org/10.3390/
en14185644
Keywords: multi-machine stability; s-domain analysis; participation analysis; PSS tuning; timedomain analysis
Academic Editor: Abu-Siada Ahmed
Received: 8 August 2021
1. Introduction
Accepted: 3 September 2021
Low-frequency oscillations, also known as electromechanical oscillations, were first
noticed in the American West system. They can appear in large interconnected systems
within the frequency range between 0.01 to 3 Hz [1], mainly caused by an imbalance
between energy demand and generation. Electromechanical oscillations represent characteristic system responses (low-frequency modes) to a disturbance in the system and, if
not properly damped, can lead to system instability [2]. PSSs are a practical solution for
electromechanical oscillations damping and their implementation as an additional block in
generator excitation systems is simple. PSS is the most efficient for local oscillatory modes
with frequencies 6–12 rads−1 and can also be used to improve transient stability mostly
combined with AVR [3,4] or flexible AC transmission system devices (FACTS) [5–7].
However, PSS tuning is a complex task. One of the most common methods for the PSS
tuning is a pole placement method where the dominant pair of the poles of the observed
power system is damped and shifted to the left side of the S-plane by selecting the proper
PSS parameters. One of the challenges for PSS implementation in a multi-machine system
is to find an optimal PSS location. Participation factor analysis shows the impact of each
generator on the system state variables and thus indicates the location where PSS should
be implemented [8]. Participation factors can be found for both monotone and oscillatory
Published: 8 September 2021
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Energies 2021, 14, 5644. https://doi.org/10.3390/en14185644
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Energies 2021, 14, 5644
2 of 29
modes. In comparison with oscillatory modes, participation factors are very low for
monotone modes [9]. In papers [10,11], the participation analysis is applied on a large
multi-machine system to identify critical generators and modes in the system. Several
papers combine both the participation factor analysis in the multi-machine system and
advance methods for PSS tuning [12–14]. The author [15] presents global pole placement
problems as the composition of separate global and local pole-placement problems used to
solve local oscillatory modes, usually undamped. This solution was applicable for multiple
operating conditions, which led to a robust design technique [16]. In paper [17], extensive
research on system stability for various combinations of system parameters is performed. In
a three-part paper [18–20], PSS application and a complete tuning process are overviewed.
Over the years, application and tuning processes evolved into more sophisticated methods,
mostly heuristic methods [21–23], like genetic algorithms [24], fuzzy logic [25,26], or other
optimization techniques [27,28]. Paper [29] compares three different types of stabilizers,
conventional PSS (CPSS), fuzzy, and adaptive neuro-fuzzy inference system (ANFIS) based
PSS. All three types show appropriate responses to a system disturbance, giving a slight
advantage in performance to ANFIS. Artificial intelligence methods are mainly applied on
single machine infinite bus (SMIB) system models, as in [30–34]. Additionally, many novel
approaches for PSS and AVR tuning are applied on SMIB, for example, nonlinear robust
PSS-AVR controller [35]. However, PSS tuning in multi-machine systems is much more
complex in comparison to SMIB since one generator can deteriorate the performance of
other generators in the system.
In most papers, tuning of the PSS parameters is based on results in the S-domain [36–38],
and, afterwards, results were verified in the time domain by comparing the system response
to disturbances before and after PSS implementation. Due to the possible inertia decrease
with high penetration of converter-interfaced renewable energy sources, the modern power
system may become more vulnerable to disturbances that can initiate low-frequency oscillations. From the consumer side, the amplitude of active power low-frequency oscillations is
particularly unfavorable since they can lead to load shedding. The PSS parameters obtained
with the traditional pole-placement method show satisfactory results regarding s-domain
system performance measures; however, this paper points to the improvement of damping
of the low-frequency oscillations when the time-domain system performance measures are
included in the PSS tuning too. Hence, the paper proposes the algorithm for the PSS1A
tuning considering both the s-domain and time-domain system performance measures. The
synthesis of these measures is achieved through the proposed weight functions that output
comparable absolute values; the s-domain weight function is proposed as the damping ratio of
the damping oscillatory mode since the damping ratio takes values between 0 and 1; the time
domain weight functions are proposed as the distance between value 1 and the sum of the
square differences from the initial steady-state value and the sampled values of the observed
variable during the time-domain simulation of the disturbance in the system. The proposed
time-domain weight functions are related to active power, speed, and the rotor angle of the
generator where the PSS has been implemented according to the participation factor analysis
and represents the generator with the most significant contribution to the dominant system
mode. The output of the weight functions closer to value 1 indicates the higher function
weight. The sum of the s-domain and time-domain weight functions constitutes a system
performance weight function. The maximum of the system performance weight function
is obtained with the PSS parameters variations in an ascending and a descending order in
separate clusters. The PSS parameters that correspond to the maximum of the system weight
function are classified as optimal.
This paper consists of six sections. The Introduction provides a literature overview of
the PSS tuning techniques and briefly highlights the novelties proposed in the paper. The
second section gives a brief description of the PSS1A structure. The proposed algorithm
for PSS1A tuning, considering both the s-domain and time domain system performance
measures, is the scope of Section 3. In the fourth section, the application of the proposed algorithm is examined on the performance of the IEEE 14-bus system model after initiation of
rithm for
s 2021, 14, x FOR PEER REVIEW
PSS1A tuning, considering both the s-domain and time domain system
3 ofperfor30
mance measures, is the scope of Section 3. In the fourth section, the application of the
proposed algorithm is examined on the performance of the IEEE 14-bus system model
afterrithm
initiation
of thetuning,
disturbances
withboth
a different
duration
within
threesystem
case studies.
for PSS1A
considering
the s-domain
and time
domain
perfor- The
Energies 2021,Discussion
14, 5644
3 of 29
follows
the
Results,
presented
in
figures
and
tables
for
all
case
studies.
mance measures, is the scope of Section 3. In the fourth section, the application of theFinal
remarks
and algorithm
possible benefits
fromonthe
algorithm
are given
proposed
is examined
theapplication
performanceofofthe
theproposed
IEEE 14-bus
system model
after
initiation of the disturbances with a different duration within three case studies. The
in the
Conclusion.
the disturbances with a different duration within three case studies. The Discussion follows
Discussion follows
the Results,
presented
in figures
and for
tables
for all
case studies.
Final and possible
the Results,
presented
in figures
and tables
all case
studies.
Final remarks
remarks and
the application
of the proposed
given
2. Materials
andpossible
Methods
benefitsbenefits
from thefrom
application
of the proposed
algorithmalgorithm
are givenare
in the
Conclusion.
in the Conclusion.
The authors conducted an extensive literature review on the topic of PSS tuning, cov2. Materials and Methods
ering
the fundamentals
of tuning and modern optimization methods. Based on this
2. both
Materials
and Methods
The authors conducted an extensive literature review on the topic of PSS tuning,
research, it was found that commonly used PSS tuning procedures based solely on the sThe authors
conducted
extensive
literature
themodern
topic of optimization
PSS tuning, covcovering
bothanthe
fundamentals
of review
tuningon
and
methods. Based
domain
measures
canmodern
be improved
by involving
the
time-domain
eringsystem
both theperformance
fundamentals
of tuning
and
optimization
Based
on this based solely
on
this research,
it was found
that commonly
usedmethods.
PSS tuning
procedures
system
performance
measures.
Tosystem
address
this
the can
authors
proposed
method
research,
it wasonfound
that commonly
used
PSSdeficiency,
tuning
procedures
based
solely on
sthe
s-domain
performance
measures
be
improved
byathe
involving
the timethatdomain
includes
both
domain
system
performance
measures
to
find
the
optimal
PSS
paramsystemdomain
performance
canmeasures.
be improved
by involving
the time-domain
system measures
performance
To address
this deficiency,
the authors proposed a
eters.
The results
were measures.
verified
on
IEEE
14–bus
test performance
system
consisting
ofafive
generasystem
performance
To the
address
this
deficiency,
the authors
proposed
method
method
that includes
both
domain
system
measures
to
find
the optimal PSS
both
domain
system
performance
measures
to find
the
optimal
paramparameters.
The
results
were
verified
on are
the
IEEE
14–bus
testPSS
system
consisting
tors that
withincludes
parameters
publicly
available.
All generators
equipped
with
standard
IEEE of five
eters.
The
results
were
verified
on
the
IEEE
14–bus
test
system
consisting
of
five
generagenerators
with
parameters
publicly
available.
All
generators
are
equipped
with standard
AVRs models with parameters available in the Appendix A.
IEEE AVRs
models
with parameters
available
in the Appendix
A.
tors with parameters
publicly
available.
All generators
are equipped
with standard
IEEE
AVRs
models with parameters available in the Appendix A.
3. PSS
Structure
3. PSS Structure
system stabilizer
provides
an additional
torque
in a phase
withingen3.Power
PSS Structure
Power system
stabilizer
providesdamping
an additional
damping
torque
a phase with
erator speed
variation
and
it
is
simply
implemented
as
a
part
of
the
synchronous
generagenerator
speed
variation
and
it
is
simply
implemented
as
a
part
of
the
Power system stabilizer provides an additional damping torque in a phase with gen- synchronous
tor excitation
system
[18]—Figure
1.system
generator
excitation
[18]—Figure
erator speed
variation
and
it is simply
implemented
as a1.part of the synchronous generator excitation system [18]—Figure 1.
Figure
1. Synchronous
generator
Figure 1.excitation
Synchronous
generator excitation system.
Figure
1. Synchronous
generator
excitation system.
system.
Theof
main
of the PSS
presented
in known
Figure
2in(also
known
in [39] as PSS1A),
main
structure
ofthe
thestructure
PSSpresented
presented
in
2 2(also
[39]
as PSS1A),
TheThe
main
structure
PSS
in Figure
Figure
(also
known
in [39]
as PSS1A),
consists
of the gain,
wash-out
filter, and
phase compensation
blocks (lead-lag
filters). Some
consists
of
the
gain,
wash-out
filter,
and
phase
compensation
blocks
(lead-lag
filters).
consists of the gain, wash-out filter, and phase compensation blocks (lead-lag filters).
additional
blocks,
such
as
high-frequency
torsional
filters
or
transducer
time
Some
additional
blocks,such
suchasashigh-frequency
high-frequency torsional
oror
transducer
timetime
con-con- constant
Some
additional
blocks,
torsionalfilters
filters
transducer
compensation,
may
be
implemented
[39].
Torsional
filter
block
can
be
implemented by
compensation, may be implemented [39]. Torsional filter block can be implemented
stantstant
compensation,
may constants
be implemented
[39].
Torsional
filter
block
can
be implemented
defining
A
and
A
.
If
the
torsional
filter
is
to
be
neglected,
these
2
by defining constants A1 and A2. If 1the torsional
filter is to be neglected, these constants constants are
by defining constants
A
1 1
and
Athis
2. Ifblock
the torsional
filter is to betorsional
neglected,
these
constants
set
to
be
and
is
used
to
modes,
which
are the result of
are set to be 1 and this block is used to compensatecompensate
torsional modes, which
are the
result
are set
to
be
1
and
this
block
is
used
to
compensate
torsional
modes,
which
are
the
result
rotational
movement.
The
first
block
is
used
to
compensate
transducer
time
of rotational movement. The first block is used to compensate transducer time constant constant and
can be by
neglected
setting
T6 to 1. transducer time constant
of rotational
The
firstby
block
istime
used
to compensate
and can bemovement.
neglected
setting
time
constant
Tconstant
6 to 1.
and can be neglected by setting time constant T6 to 1.
Figure
2. Type
PSS1A
single-input
power
system
stabilizer.
Reproduced
from
IEEE: 2016.
Figure
2. Type
PSS1A
single-input
power
system
stabilizer.
Reproduced
from
[39],[39],
Copyright
2016 IEEE.
Figure 2. Type PSS1A single-input power system stabilizer. Reproduced from [39], IEEE: 2016.
Gain KS definesGain
a circuit
gain and
damping
Gain increase
to damping
KS defines
a circuit
gain torque.
and damping
torque. leads
Gain increase
leads to damping
increase until it
reachesuntil
a maximum
after which
a further
gain aincrease
in
increase
it reachespoint
a maximum
point
after which
further results
gain increase
results in
Gain KS defines a circuit gain and damping torque. Gain increase leads to damping
less damping torque
[40]. Gain
should
tuned
following
the maximum
damping
but damping but
less damping
torque
[40].beGain
should
be tuned
following the
maximum
increase until it reaches
a maximum point after which a further gain increase results in
also all the other limitations need to be taken into consideration. The second block is the
less damping torque
[40]. filter
Gain(defined
should by
beits
tuned
following
maximum
damping
but defines
wash-out
time constant
T5 ).the
This
is a high-pass
filter which
the operating margin for PSS. PSS should not respond to small and slow changes which
Energies 2021, 14, 5644
4 of 29
are mostly caused by a change of the system operating point. Wash-out filter eliminates
these disturbances and only reacts to transient disturbances [40]. The value of T5 must
be in a range in which it does not change observed frequencies but also does not lead to
unwanted voltage trips. The range for T5 is 1 to 20 s [41]; for local modes, it should be 1–2 s
and 10–20 s for interarea modes [42]. Input in an excitation system and electrical torque as
an output signal has a phase shift, which needs to be compensated. Phase compensation
is achieved with cascade lead-lag filters, which are presented with constants T1 –T4 . To
simplify the process, it is assumed to be T2 = T4 and T1 = T3 . T2 and T4 usually have fixed
values and the other two-time constants need to be tuned. PSS output voltage needs to be
limited to restrict terminal voltage fluctuations [43]. The output voltage is limited between
values V STmin and V STmax .
4. PSS Tuning Algorithm
The first step in PSS tuning in the proposed algorithm is the system modal analysis
in order to identify a dominant oscillatory mode since it determines the dynamic system
properties. Each oscillatory mode is the system eigenvalue represented as a complex
pair with the real part—the damping constant, and the imaginary part—the oscillation
(damping) frequency. The dominant system oscillatory mode is identified as the complex
pair closest to the imaginary axis on the system S-plane. The pole placement method used
in this algorithm aims to nullify the imaginary part of the dominant system mode and to
significantly shift it left from the initial location on the S-plane integrating the PSS into
the system [1]. The most efficient PSS application would be for the damping frequencies
range of the dominant system mode within 1.5–12 rads−1 , i.e., for the inter-area and local
oscillatory modes [44]. Hence, the system oscillatory modes with damping frequencies
outside this range will not be the subject for this paper’s analysis.
Once the appropriate dominant system mode is identified, the proposed algorithm
tends to find an optimal location for the PSS implementation in the multi-machine system.
Participation factors determine the contributions of each active component in the system
oscillatory properties [41]. Since the PSS affects the system damping, the participation of
the state variable speed for the dominant oscillatory mode is chosen as a criterion for the
PSS allocation, which implies that the excitation system of the synchronous machine with
the highest participation of the state variable speed in the dominant oscillatory mode is the
most appropriate for the PSS implementation.
The PSS tuning optimization is based on the optimal selection of the PSS parameters:
the washout filter time constant-T5 , gain-K, and lead-lag time constants-T2 = T4 and
T1 = T3 . According to the suggestions in literature [18–20,39,45], the lead-lag-T2 = T4
constants should be set to fixed value T2 = T4 = 0.02 s. The washout filter time constant
is set to T5 = 10 s according to the suggested values in [18–20,40]. The initial gain-Kinit is
determined according to the damping of the dominant oscillatory mode-αdom represented
as [45]:
K
− αdom = d
(1)
4H
and should be at least equal to coefficient-Kd :
Kinit = Kd
(2)
While coefficient-H represents an inertia time constant of the synchronous machine
with the highest participation in the dominant oscillatory mode.
As recommended in [18–20,39], the initial value for time constants T1 and T3 should
be T1 = T3 = 0.2 s.
The PSS tuning procedure proposed in this paper is the optimization problem to find
the PSS gain-K and the PSS time constant-T = T1 = T3 for the maximal left-shift and the
maximal damping ratio ζ dom of the dominant oscillatory mode in the S-plane considering
the time-domain behavior of the rotor angle, speed, and active power of the synchronous
Energies 2021, 14, 5644
5 of 29
machine with the highest participation in the dominant oscillatory mode after initiating a
(small) disturbance in the system.
The time-domain indices presented in this paper are related to the time-domain
measures of the system performance after initiating a small disturbance. The proposed
measures are based on the sum of the square differences between the initial values of the
rotor angle-δ (related to synchronizing torque); speed-ω (related to damping torque); active
power-P in steady-state-δinit , ωinit , Pinit ; and all of the simulated values δt , ωt , Pt during
the time domain simulation period tsim − tinit after a (small) disturbance has been subdued
at the time instant t = tinit :
t = tsim δt − δinit 2
(3)
∑
δinit
t=t
init
t = tsim ∑
t = tinit
ωt − ωinit
ωinit
t = tsim ∑
t = tinit
Pt − Pinit
Pinit
2
(4)
2
(5)
The smaller outputs of the sum of the square differences (3)–(5) correspond to better
time-domain measures of the system performance. Based on (3)–(5), the weight functions
Γδ ,Γω , and Γ P are introduced:
Γδ = 1 −
t = tsim ∑
t = tinit
Γω = 1 −
t = tsim ∑
t = tinit
ΓP = 1 −
t = tsim ∑
t = tinit
δt − δinit
δinit
2
(6)
ωt − ωinit
ωinit
Pt − Pinit
Pinit
2
(7)
2
(8)
Hence, the output of each weight function closer to 1 represents better time-domain
system performance. Since the damping ratio ζdom of the dominant system mode falls into
the interval (0, 1) and indicates better system dynamic properties when it tends to a value
of 1, as the s-domain system performance measure, the damping ratio is comparable with
the proposed weight functions (6)–(8). It follows that a new weight function based on the
damping ratio-Γ(ζ ) can be presented as:
Γζi = ζ dom
(9)
Comprising both the time-domain and s-domain measures, the system performance
function can be represented as the weight function-Γ:
Γ=
∑
Γζ,δ,ω,P
(10)
ζ,δ,ω,P
The optimal PSS time constant T is selected as Ti,j value obtained for the maximal
output of weight function Γ j (i ) which represents the system performance weight functionΓ calculated in the iterative procedure for the j-th cluster composed of five PSS time
constants-Ti,j increasing the initial value-Tinit for the increment of 0.1 s in each subsequent
i-th iteration:
T = Ti,j f or maxΓ j (i )
(11)
i =4
Ti,j = Tinit + ∑ i ·0, 1
i =0
(12)
Energies 2021, 14, 5644
6 of 29
Until Γ j (4) > Γ j (3), a new cluster, composed of five Ti,j will be defined according
to (12), where the initial PSS time constant value will be the last value from the previous
cluster j−1:
Tinit = T4,j−1
(13)
After the PSS time constant T has been determined, the PSS gain K will be obtained in
iterative procedures with invariant T.
The PSS gain K is chosen as one of the values obtained in the following procedure:
If
maxΓi,um (i ) > maxΓi,ln (i )
(14)
K = Ku
else
K = Kl
where -Ku is the PSS gain obtained for the maximal output of weight function Γi,um (i),
which represents the system performance weight function-Γ calculated in the iterative
procedure for the u-th cluster composed of five PSS gain constants-Ki,um increasing the
initial PSS gain-Kinit for increment 1 in each subsequent i-th iteration:
Ku = Ki,um f or maxΓi,um (i )
(15)
i =4
Ki,um = Kinit + ∑ i ·1
(16)
i =0
Until Γ4,um (4) > Γ3,um (3), a new cluster composed of five PSS gain constants-Ki,um, ,
will be defined according to (16) where the initial PSS time constant value will be the last
value from the previous cluster u−1:
Kinit = K4,um−1
(17)
Kl is the PSS gain obtained for the maximal output of weight function Γi,ln (i) which
represents the system performance weight function-Γ calculated in the iterative procedure
for the l-th cluster composed of five PSS gain constants-Ki,ln decreasing the initial PSS
gain-Kinit for decrement 1 in each subsequent i-th iteration:
Kl = Ki,ln f or maxΓi,ln (i )
(18)
i =4
Ki,ln = Kinit − ∑ i ·1
(19)
i =0
Until Γ4,ln (4) > Γ3,ln (3), a new cluster, composed of five PSS gain constants -Ki,ln , will
be defined according to (19) where the initial PSS time constant value will be the last value
from the previous cluster l−1:
Kinit = K4,l −1
(20)
More detailed insight into the proposed algorithm is presented in Figure 3. The
proposed principle makes both the s-domain and time-domain measures of the system
performance comparable and can be considered beneficial for the PSS fine-tuning.
Energies 2021, 14, x FOR PEER REVIEW
Energies 2021, 14, 5644
7 of 30
7 of 29
MODAL ANALYSIS
OF THE SYSTEM
j=1
Tinit = Tij
N – EIGENVALUES
Tinit = T4j
j=j+1
m=1
m=m+1
n=1
n=n+1
λ = α ±ϳω
1.5 ≤ ϳω ≤
≤ 12 rad s-1
NO
YES
FOR i = 0 TO i = 4
Tij = Tinit + 0.1 i
FOR i = 0 TO i = 4
Ki,um = Kinit + i·
1
FOR i = 0 TO i = 4
Ki,ln = Kinit - i·
1
MODAL ANALYSIS
(PSS WITH Tij)
MODAL ANALYSIS
(PSS WITH Ki,um)
MODAL ANALYSIS
(PSS WITH Ki,ln)
FIND DOMINANT
OSCILLATORY MODE
FIND DOMINANT
OSCILLATORY MODE
FIND DOMINANT
OSCILLATORY MODE
Γζi = ζ dom
Γζi = ζ dom
Γζi = ζ dom
TIME-DOMAIN
SIMULATION OF
SMALL DISTURBANCE
TIME-DOMAIN
SIMULATION OF
SMALL DISTURBANCE
TIME-DOMAIN
SIMULATION OF
SMALL DISTURBANCE
CALCULATE
Γδ
Γω
Γp
Γj(i)= Γδ + Γω+ Γp+Γζ
CALCULATE
Γδ,um
Γω,um
Γp,um
Γj,um= Γδ,um + Γω,um+ Γp,um+Γζi
CALCULATE
Γδ,ln
Γω,ln
Γp,ln
Γj,ln= Γδ,ln + Γω,ln+ Γp,ln+Γζi
FIND DOMINANT
OSCILLATORY MODE
λdom = αdom ±ϳωdom
EXECUTE PARTICIPATION
ANALYSIS
COMPARE PARTICIPATONS OF
STATE VARIABLES
GENERATOR SPEED
MAX PARTICIPATION
PSS LOCATION
αdom =
Kd
4H
Kinit = Kd
T5 = 10s
T1 = T3 = 0.2 s
T2 = T4 = 0.02 s
Γj(4) > Γj(3)
YES
YES
Γ4,um(4) > Γ2,um(3)
Γ4,ln(4) > Γ2,ln(3)
YES
Tinit = T1 = T3
COMPARE Γj(i) FOR
max Γj(4) =
T= Tij
K = Ku
NO
NO
NO
YES
Ku = Ki,um
Ku = Ki,ln
FOR max Γi,um(i)
FOR max Γi,ln(i)
IF
max Γj,um(i) >
max Γj,ln(i)
K = Kl
NO
END
Figure 3.
3. PSS
Figure
PSS tuning
tuning algorithm.
algorithm.
5. PSS Tuning Algorithm Impact on the System Performance
5. PSS Tuning Algorithm Impact on the System Performance
The proposed algorithm’s impact on the system performance has been analyzed on
The 14–bus
proposed
algorithm’s impact
system performance
has been
analyzed
on the
the IEEE
multi-machine
systemoninthe
DIgSILENT
PowerFactory
simulation
interface.
IEEE
14–bus
multi-machine
system
in
DIgSILENT
PowerFactory
simulation
interface.
The single line diagram system model is presented in Figure 4. All the generators in The
the
single line
diagram system
model(parameters
is presentedare
in Figure
4. the
All Appendix
the generators
in the model
model
are equipped
with AVRs
given in
A). Busbars
2, 4,
are equipped
with AVRs
(parameters
are given
in the (parameters
Appendix A).are
Busbars
2, the
4, and
5 are
and
5 are equipped
with static
compensation
devices
given in
Appenequipped
with static compensation devices (parameters are given in the Appendix A).
dix
A).
Modal analysis of the system indicates, on 55 modes (eigenvalues), 22 modes are
oscillatory while 33 modes are monotone. Damping frequencies of five oscillatory modes
fall in the range of 8.6–13 rads−1 (1.3–2.1 Hz), which corresponds to local modes and will be
further analyzed in Table 1. Other oscillatory modes correspond to low-frequency modes
mainly caused by AVRs and are supposed to be well-damped after the PSS implementation.
rgies 2021, 14, x FOR PEER REVIEW
8 of
Energies 2021, 14, 5644
8 of 29
Figure
IEEE
14 system
bus system
modelininDIgSILENT
DIgSILENT PowerFactory
simulation
interface.
Figure 4.
IEEE4. 14
bus
model
PowerFactory
simulation
interface.
Table 1. Local
oscillatory
modes
for theindicates,
analyzed system
without
the PSS
implementation.22 modes are
Modal
analysis
of the
system
on 55
modes
(eigenvalues),
cillatory Mode
whileNumber
33 modes are monotone.
Damping
frequencies
of fiveΞ oscillatory mod
α [s−1 ]
±jω [rad
s−1 ]
−1
fall in the range
(1.3–2.1 Hz), which
corresponds0.069933275444
to local modes and w
Mode 1of 8.6–13 rads
−0.6029112884
8.6001286177
Mode
2
−
3.0888275889
9.1371914398
0.32024636109
be further analyzed in Table 1. Other oscillatory modes correspond to low-frequen
Mode 3
−3.3668021306
10.953437301
0.30026554697
modes mainly
caused by AVRs
and are supposed
to be well-damped
after the PSS imp
Mode 4
−3.7886935644
11.819709815
0.30524242934
mentation. Mode 5
−4.5222909467
12.584279377
0.33818650996
Table 1. Local
oscillatorydiagram
modes of
forthe
theanalyzed
analyzed
system
without the
PSS implementation.
The root-locus
system
is presented
in Figure
5; monotone
modes are marked green, low-frequency oscillatory modes are marked red, while local
Mode
Number
α [s−1]
±jω [rad s−1]
Ξ
oscillatory
modes are marked blue.
The damping
closest−0.6029112884
to the horizontal axis on the
S-plane indicates Mode
1 as the
Mode
1
8.6001286177
0.069933275444
dominant
oscillatory
mode.
According
to
the
algorithm
proposed
in
the
previous
chapter,
Mode 2
−3.0888275889
9.1371914398
0.32024636109
the PSS location will be determined considering the highest participation of the generator
Mode
0.30026554697
speed
state3variable in the −3.3668021306
dominant oscillatory mode. 10.953437301
As can be seen in Table 2, the
highest
Mode 4 the value of−3.7886935644
11.819709815
0.30524242934
participation,
0.954, belongs to synchronous
generator 1 (Gen1, highlighted
in
Figure
4),
which
implies
that
the
excitation
system
of
this
generator
is
the
most
suitable
for
Mode 5
−4.5222909467
12.584279377
0.33818650996
the PSS implementation. The participation of other synchronous generators (Gen 2–Gen 5
and External Grid) in the dominant system mode can almost be neglected.
The root-locus diagram of the analyzed system is presented in Figure 5; monoto
modes are marked green, low-frequency oscillatory modes are marked red, while lo
oscillatory modes are marked blue.
14, x FOR PEER REVIEW
9 of 30
Energies 2021, 14, 5644
9 of 29
Figure 5.Figure
Root locus
diagram
of the of
analyzed
system
without
the the
PSSPSS
implementation.
5. Root
locus diagram
the analyzed
system
without
implementation.
The damping
closest
to the horizontal
axisspeed
on the
indicates
Mode
1 as the
Table
2. Participation
of the generator
stateS-plane
variable for
the dominant
oscillatory
mode.
dominant oscillatory mode. According to the algorithm proposed in the previous chapter,
Generator Unit
Participation in Mode 1
the PSS location will be determined considering the highest participation of the generator
Gen1
0.954
speed state variable in the dominant oscillatory mode. As can be seen in Table
2, the highGen2
0.120
est participation, the value of 0.954,
(Gen1, highGen3belongs to synchronous generator 10.005
lighted in Figure 4), which impliesGen4
that the excitation system of this generator
0.003is the most
Gen5The participation of other synchronous
0.002generators
suitable for the PSS implementation.
External Grid
(Gen 2–Gen 5 and External Grid) in the dominant system mode can almost 0.046
be neglected.
After
the PSS location
has been
determined,
the tuning
process begins
Table 2. Participation of
the generator
speed state
variable
for the dominant
oscillatory
mode. with washout
filter time constant setting to T5 = 10 s according to the proposed algorithm, i.e., the
Generator
Mode
1
suggestedUnit
values in [18–20,39]. This valueParticipation
will not affectinthe
regulation
loop regarding
the Gen
phase
gain is set to K = Kinit = 10. The
1 shift. According to (1) and (2), the initial PSS
0.954
value
for the time constants T1 and T3 is initially set
to T1 = T3 = 0.2 s (recommended
Gen2
0.120
in [18–20,39]).
Gen3
0.005
0.003
5.1. Gen4
PSS Lead-Lag Tuning
Gen5
0.002
According to (11) and (12) in the proposed algorithm,
modal analyses and time-domain
External
Grid
simulations
are made in clusters containing five samples,0.046
changing T = T1 = T3 in each iteration
while maintaining the PSS gain as a constant value. Table 3 shows the modal analyses results
forlocation
time constants
of cluster
1 − Ti,1the
=T
= 0.2–0.6
s for begins
the dominant
oscillatory mode.
After the PSS
has been
determined,
tuning
process
with washout
Increasing
constants T shifts this pole further to the left side of the S-plane and at
filter time constant
settingPSS
to Ttime
5 = 10 s according to the proposed algorithm, i.e., the sugthe same time increases the damping ratio. The maximum value of the damping ratio and
gested values in [18–20,39]. This value will not affect the regulation loop regarding the
the minimum value of the damping are fulfilled with the time constant(s) T = T3,1 = 0.5 s. A
phase shift. According
(1) and
initial PSS
gainthe
is set
to 𝐾 =oscillatory
𝐾𝑖𝑛𝑖𝑡 = 10.pole
Theback
value
further to
increase
of (2),
timethe
constant(s)
pushes
dominant
in the right
for the time constants
T
1 and T3 is initially set to T1 = T3 = 0.2 s (recommended in [18–20,39].
part of the S-plane while at the same time the damping ratio decreases (Figure 6). Mode T00
represents the dominant pole without implemented PSS.
5.1. PSS Lead-Lag Tuning
According to (11) and (12) in the proposed algorithm, modal analyses and time-domain simulations are made in clusters containing five samples, changing T = T1 = T3 in each
iteration while maintaining the PSS gain as a constant value. Table 3 shows the modal
analyses results for time constants of cluster 1−𝑇𝑖,1 = T = 0.2 s–0.6 s for the dominant oscillatory mode. Increasing PSS time constants T shifts this pole further to the left side of the
Energies 2021, 14, 5644
damping ratio and the minimum value of the damping are fulfilled with the time con
stant(s) T = T3,1 = 0.5 s. A further increase of time constant(s) pushes the dominant oscilla
tory pole back in the right part of the S-plane while at the same time the damping rati
10 of 29
decreases (Figure 6). Mode T00 represents the dominant pole without implemented
PSS.
Table 3. Dominant oscillatory pole for PSS parameters K = 10, T = Ti,1 = 0.2 s–0.6 s (cluster 1).
Table 3. Dominant oscillatory pole for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s (cluster 1).
I
0
1
2
3
4
i
0
1
2
3
4
Ti,1 [s]T [s]
i,1
0.2 0.2
0.3 0.3
0.4 0.4
0.5
0.5 0.6
0.6
α [sα−1[s]−1 ]
−0.846729311
−0.846729311
−1.306793920
−1.306793920
−2.059476510
−2.059476510
−2.442664922
−2.442664922
−2.024097138
−2.024097138
−1
±jω
[rad
±jω
[rad
s−1s] ]
8.6001286177
8.6001286177
9.1371914398
9.1371914398
10.953437301
10.953437301
11.819709815
11.819709815
12.584279377
12.584279377
ξ
0.094299372
0.094299372
0.141835333
0.141835333
0.214871539
0.214871539
0.507928180
0.507928180
0.425562142
0.425562142
ξ
Figure6.6.Dominant
Dominant
oscillatory
movement
the S-plane
after
the
PSS implementation
with
Figure
oscillatory
polepole
movement
in theinS-plane
after the
PSS
implementation
with
=
10
and
the
increasing
time
constants
from
T
=
0.2
s
to
T
=
0.6
s.
K = 10 and the increasing time constants from T = 0.2 s to T = 0.6 s.
The
variables
in the
simulations
are rotor
Theobserved
observed
variables
in time-domain
the time-domain
simulations
areangle,
rotorspeed,
angle,and
speed, an
active
power
of
Gen
1
after
initiating
a
(small)
disturbance
(3-phase
short-circuit)
on
BUS
active power of Gen 1 after initiating a (small) disturbance (3-phase short-circuit) on BU
4 with the duration of 30 ms at the time instance tinit = 0.1 s from the simulation begin4 with the duration of 30 ms at the time instance tinit = 0.1 s from the simulation beginning
ning. Figure 7 shows the rotor angle response with different PSS time constant(s)-T = Ti,1
Figure 7 shows the rotor angle response with different PSS time constant(s)-T = Ti,1 setting
setting. As can be seen from the figure, increasing the PSS time constant(s) results in a
As cansettling
be seen
from
figure,overshoot
increasing
the
PSS timeofconstant(s)
results
in a shorte
shorter
time
andthe
decreased
and
undershoot
the rotor angle
during
settling
time and
decreased
undershoot
of the speed
rotor angle
the
transitional
process.
Similarovershoot
behavior isand
shown
by the generator
(Figureduring
8) and the tran
sitional
process.
Similar
behavior
is
shown
by
the
generator
speed
(Figure
8) and the ac
the active power (Figure 9).
tive power (Figure 9).
FOR PEER REVIEW -1.175
Energies 2021, 14, 5644
-1.180
11 of 30
11 of 29
-1.185
[rad] angle [rad]
Rotor angleRotor
-1.190
-1.175
-1.195
-1.180
-1.200
-1.185
-1.205
-1.190
-1.210
-1.195
-1.215
-1.200
-1.220
-1.205
-1.225
-1.210
-1.230
-1.215 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time [s]
-1.220
T = 0.2 s
-1.225
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
-1.230
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure 7. Gen1 rotor angle response on a small disturbance with PSS parameters K = 10, and T = Ti,1
Time [s]
= 0.2 s–0.6 s.
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
Visual observation of the mentioned figures implies the most favorable time-domain
Figure 7.
Gen1
angle
response
onwith
a small
with
parameters
K
= 10, and T =KT=
Figure
Gen1rotor
rotor
angle
response
on PSS
adisturbance
small
disturbance
with
PSSconstant(s)-T
parameters
=i ,110,
and
Ts.=s.Ti,1
system
performance
measures
gain-K
= 10PSS
and
time
T=4,10.2–0.6
= 0.6
= 0.2 s–0.6 s.
1.0004
[rads-1]
Generator
-1]
[radsspeed
Generator speed
Visual observation of the mentioned figures implies the most favorable time-domain
system1.0003
performance measures with PSS gain-K = 10 and time constant(s)-T = T4,1 = 0.6 s.
1.0002
1.0001
1.0004
1.0000
1.0003
0.9999
1.0002
0.9998
1.0001
0.9997
1.0000
0.9996
0.9999
0.9995
0.9998
0.9994
0.9997
0.9996
0.9995
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time [s]
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
0.9994
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure
8. Gen1
speed
ona asmall
small
disturbance
with
PSS parameters
K T= =10,
T = Ti,
Figure
8. Gen1
speedresponse
response on
disturbance
with PSS
parameters
K = 10, and
Ti ,and
s. 1 = 0.2
1 = 0.2–0.6
Time [s]
s–0.6 s.
Visual observation of the mentioned figures implies the most favorable time-domain
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
system performance measures with PSS gain-K = 10 and time constant(s)-T = T4,1 = 0.6 s.
Table 4 shows results of the weight functions calculated following the proposed
first cluster.
Figure 8. Gen1 speedalgorithm
responsefor
onthe
a small
disturbance with PSS parameters K = 10, and T = Ti,1 = 0.2
s–0.6 s.
Energies
14, x FOR PEER REVIEW
Energies
2021,2021,
14, 5644
12 of 30
12 of 29
465
Active power [MW]
460
455
450
445
440
435
430
425
420
415
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time [s]
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
Figure 9. Gen1 active power response on a small disturbance with PSS parameters K = 10, and T =
Figure
9. Gen1 active power response on a small disturbance with PSS parameters K = 10, and
Ti,1 = 0.2 s–0.6 s.
T = Ti ,1 = 0.2–0.6 s.
Table 4 shows results of the weight functions calculated following the proposed alTable
4. Weight
for K = 10, T = Ti,1 = 0.2–0.6 s (cluster 1).
gorithm
for the functions
first cluster.
i 4. Weight
Ti,1functions
[s]
Γωis–0.6 s (cluster
Γδi1).
Table
for KΓζi
= 10, T = Ti,1 = 0.2
i0
1
0
2
1
3
24
3
4
Ti,1 [s]0.2
0.3
0.2
0.4
0.3
0.5
0.4 0.6
0.5
0.6
ΓPi
Γ1 (i)
Γζi 0.094299 Γωi0.999995596 Γ0.989720196
δi
Γ0.936069374
Pi
Γ3.02008453797
1(i)
0.141835
0.999996902 0.991708475 0.955296445
3.08883715433
0.094299
0.999995596 0.989720196 0.936069374 3.02008453797
0.214872
0.999997748 0.993060487 0.968084688
3.17601446330
0.141835
0.999996902 0.991708475 0.955296445 3.08883715433
0.507928
0.999998262 0.993814845 0.976163024
3.47790431049
0.214872
0.999997748
0.993060487
0.968084688
3.17601446330
0.425562
0.999998447
0.993415907
0.980405629
3.39938212468
0.507928
0.999998262 0.993814845 0.976163024 3.47790431049
0.425562
0.999998447 0.993415907 0.980405629 3.39938212468
Except for Γδi , the time-domain weight functions-Γωi and ΓPi increase with increasing
the PSS
time
the system
performance
weight
function-Γ1 (i)
Except
forconstant(s)-T
Γδi, the time-domain
weight functions-Γ
ωi and
ΓPi increase with
increasing
i,1 . However,
the PSS time
constant(s). However,
system performance
weight function–Γ
1(i) obobtained
maximum
for TΓi,11 (3),
which the
corresponds
to PSS parameters
K = 10 and
T = T3,1 =
tained
maximum
1(3),awhich
corresponds
to 2)
PSS
K =of
10PSS
andtime
T = Tconstant(s)-T
3,1 = 0.5
0.5
s. Since
Γ1 (4) <for
Γ1Γ(3),
new cluster
(cluster
ofparameters
five samples
i,2
s. Since
1(4)
< Γ1(3), T
a 3,1
new
cluster
(cluster for
2) of
five
samples
of PSS time constant(s)-Ti,2
will
not Γbe
created,
will
be selected
the
PSS
time constant(s)-T:
will not be created, T3,1 will be selected for the PSS time constant(s)-T:
max
(i)==Γ1Γ(3)
T =s 0.5 s
1 (3)
max Γ
Γ11(i)
→→
T3,1T=3,1
T ==0.5
Energies 2021, 14, x FOR PEER REVIEW
The weight
from
Table
1 are
illustrated
in Figures
10–14.10–14.
The
weightfunctions
functions
from
Table
1 are
illustrated
in Figures
Γζ vs PSS time constant(s)- T
0.6
0.5
0.4
0.3
0.2
0.1
0
NO PSS
T = 0.2 s
T = 0.3 s
T = 0.4 s
Figure 10. Γζi for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
T = 0.5 s
Figure 10. Γζi for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
Γω vs PSS time constant(s)- T
0.9999990
0.9999985
0.9999980
0.9999975
0.9999970
T = 0.6 s
13 of 30
0.3
0.1
0.2
0
Energies 2021, 14, 5644
0.1
NO PSS
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
Figure 10. Γζi for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
0
NO PSS
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
Γω vs
PSS time constant(s)- T
Figure 10. Γζi for PSS parameters
K = 10, T = Ti,1 = 0.2–0.6 s.
0.9999990
Γω vs PSS time constant(s)- T
0.9999985
0.9999990
0.9999980
0.9999985
0.9999975
0.9999980
0.9999970
0.9999975
0.9999965
0.9999970
0.9999960
0.9999965
0.9999955
0.9999960
0.9999950
0.9999955
0.9999950
0.9999945
0.9999945
0.9999940
0.9999940
T = 0.2 s
T = 0.2 s
T = 0.3 s
T = 0.3 s
T = 0.4 s
T = 0.4 s
T = 0.5 s
T = 0.5 s
T = 0.6 s
T = 0.6 s
Figure11.
11.ΓΓωifor
forPSS
PSSparameters
parametersKK==10,
10,TT== TTi,, 1 =
= 0.2–0.6 s.
Figure
ωi
i 1 0.2–0.6 s.
Figure 11. Γωi for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
Γ vs PSS time consant(s)- T
δ PSS time consant(s)- T
Γδ vs
0.995
0.995
0.994
0.994
0.993
0.993
0.992
0.992
0.991
0.991
0.99
0.99
0.989
0.989
0.988
0.988
0.987
0.987
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
Energies 2021, 14, x FOR PEER REVIEW
Figure 12. Γδi for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
Figure 12. Γδi for PSS parameters K = 10, T = Ti ,1 = 0.2–0.6 s.
Figure 12. Γδi for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
T = 0.614s
ΓP vs PSS time constant(s) -T
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
Figure 13.
13. ΓΓPi for
PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
Figure
Pi for PSS parameters K = 10, T = Ti ,1 = 0.2–0.6 s.
Γ1 vs PSS time constant(s) -T
3.600
3.500
3.400
3.300
T = 0.6 s
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13 of 29
0.93
0.92
0.91
T = 0.2 s
Energies 2021, 14, 5644
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
14 of 29
Figure 13. ΓPi for PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
Γ1 vs PSS time constant(s) -T
3.600
3.500
3.400
3.300
3.200
3.100
3.000
2.900
2.800
2.700
T = 0.2 s
T = 0.3 s
T = 0.4 s
T = 0.5 s
T = 0.6 s
Figure 14.
14. ΓΓ1 for
PSS parameters K = 10, T = Ti,1 = 0.2–0.6 s.
Figure
1 for PSS parameters K = 10, T = Ti ,1 = 0.2–0.6 s.
5.2. PSS
PSS Gain
5.2.
GainTuning
Tuning
Performed time-domain
simulations
imply
the insignificant
influence
of the (small)
Performed
time-domain
simulations
imply
the insignificant
influence
of the (small)
disturbance
duration
on
the
PSS
time
constant
tuning.
However,
the
PSS
gain
tuning
disturbance duration on the PSS time constant tuning. However, the PSS
gain will
tuning will
be studied for different durations of the same (small) disturbance (3-phase short-circuit)
be
studied for different durations of the same (small) disturbance (3-phase short-circuit)
on BUS 4.
on BUS 4.
5.3. Case Study 1–Disturbance Duration 30 ms
5.3. Case Study 1—Disturbance Duration 30 ms
Following the proposed algorithm, the PSS gain is varied from the initial value K
Following
proposed
algorithm,
the PSS
gain
is varied
from for
thethe
initial
= 𝐾𝑖𝑛𝑖𝑡
= 10. Tablethe
5 presents
the dominant
oscillatory
pole
s-domain
measures
clus- value K
=ter
Kinit
=
10.
Table
5
presents
the
dominant
oscillatory
pole
s-domain
measures
for
the cluster
where the PSS initial value is decreased by 1 in each subsequent iteration (lower cluswhere
ter). the PSS initial value is decreased by 1 in each subsequent iteration (lower cluster).
Table5.5. Dominant
Dominant oscillatory
forfor
PSS
parameters
T=T
T13 ==0.5
i,ln =s,6–10,
1 (lower
Table
oscillatorypole
pole
PSS
parameters
T1 == T
T3 s,= K0.5
Ki,ln n= =6–10,
n = 1 (lower
cluster 1).
1).
cluster
i
i
0
1
2
3
4
Ki,l1
NO PSS
α [s−1]
±jω [rad s−1]
ξ
Ki,l1
ξ
α [s−1 ]
±jω [rad s−1 ]
−0.602911288
±8.600128618
0.069933275
NO PSS
−0.602911288
±8.600128618
0.069933275
6
7
8
9
10
−2.044897210
−2.323979821
−2.615513529
−2.536071719
−2.442664922
±9.123657619
±9.251852650
±9.361915009
±4.118411476
±4.142534779
0.218705293
0.243622413
0.269074414
0.524347042
0.50792818
Table 6 presents the dominant oscillatory pole s-domain measures for the cluster where
the PSS initial value is increased by 1 in each subsequent iteration (upper cluster).
Table 6. Dominant oscillatory pole for PSS parameters T = 0.5 s, Ki,um = 10–14, m = 1 (upper cluster 1).
i
Ki,u1
α [s−1 ]
±jω [rad s−1 ]
ξ
0
1
2
3
4
10
11
12
13
14
−2.442664922
−2.350630618
−2.262346344
−2.179202113
−2.101785372
±4.142534779
±4.154693103
±4.156618791
±4.150363212
±4.137873414
0.50792818
0.492426473
0.478053895
0.464877617
0.452866896
Increasing the PSS gain in the lower cluster 1 resulted in the initial dominant system
oscillatory pole (system without the PSS) better damping ratio and movement to the left
side of the S-plane until a complex pair of the eigenvalues obtained for PSS gain K3,l1 = 9
Energies 2021, 14, 5644
15 of 29
became a new dominant oscillatory pole. However, the further increase of the PSS gain in
lower cluster 1 resulted in the dominant system oscillatory pole decreased damping ratio
Energies 2021, 14, x FOR PEER REVIEW
16 of 30
and movement to right-side on the S-plane, which was continued in the first upper cluster 1
too (Figure 15).
Energies 2021, 14, x FOR PEER REVIEW
16 of 30
Figure
Dominantoscillatory
oscillatory
pole
before
implementation-K
and
the PSS impleFigure 15.
15. Dominant
pole
before
the the
PSS PSS
implementation-K
0,0 and 0,0
after
theafter
PSS implementation
with
T
=
0.5
s,
PSS
gain
increase
in
the
lower
cluster
1-K
=
6–10
and
the
upper
mentation with T = 0.5 s, PSS gain increase in the lower cluster 1- Ki,ln = 6–10
1 cluster
i,lnand the upper cluster
=
10–14.
1KKi,um
=
10–14.
i,um
Figure
Dominant
oscillatory
pole before
the PSSof
implementation-K
0,0 and
thecluster
PSS impleOn15.the
other hand,
a continual
increase
the PSS gain in
bothafter
lower
1 and upper
mentation with T = 0.5 s, PSS gain increase in the lower cluster 1- Ki,ln = 6–10 and the upper cluster 1
cluster-1.185
1 resulted in better time-domain measures of the system performance (Figures 16–18).
Ki,um = 10–14.
-1.190
[rad] angle [rad]
Rotor angleRotor
-1.195
-1.200
-1.185
-1.205
-1.190
-1.210
-1.195
-1.215
-1.200
-1.220
-1.205
-1.225
-1.210
-1.230
-1.215
-1.220
-1.225
-1.230
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time [s]
K=6
K0 = 11 1
K=7
2
3 K = 412
K=8
5
6
K7 = 13 8
K=9
9
14
10 K =11
K = 10
12
13
14
15
Time [s] with PSS parameters T = 0.5 s, the PSS
Figure 16. Gen1 rotor angle response on a small disturbance
= 6the lower cluster
K =1-K
7 i,ln = 6–10 and Kthe
= 8upper cluster 1 KK=i,um9 = 10–14.
K = 10
gain increaseK in
K = 11
K = 12
K = 13
K = 14
Figure 16. Gen1 rotor angle response on a small disturbance with PSS parameters T = 0.5 s, the PSS
Figure 16. Gen1 rotor angle response on a small disturbance with PSS parameters T = 0.5 s, the PSS
gain increase in the lower cluster 1-Ki,ln = 6–10 and the upper cluster 1 Ki,um = 10–14.
gain increase in the lower cluster 1-Ki,ln = 6–10 and the upper cluster 1 Ki,um = 10–14.
Energies 2021, 14, x FOR PEER REVIEW
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2021,
x FOR PEER REVIEW
Energies
2021,
14,14,
5644
17 of 30
17 of1630of 29
1.0003
1.0003
1.0001
1.0001
Generator speed [rads-1]
Generator speed [rads-1]
1.0002
1.0002
1.0000
1.0000
0.9999
0.9999
0.9998
0.9998
0.9997
0.9997
0.9996
0.9996
0.9995
0.9995
0 0 1
1 2
2 3 3 4 4
K=6
K=6
K = 11
K = 11
5 5
6 6
77
88
99
Time[s][s]
Time
K=7
K=7
K = 12
K = 12
K=8
K=8
K = 13
K = 13
10
10
11
11
12
12
13
13
K=9
K=9
K = 14
K = 14
14
14
15
15
K = 10
K = 10
Figure17.
17. Gen1
Gen1 speed
speed response
on aasmall
with
PSS parameters
T = 0.5
s, the
PSSPSS
gaingain
Figure
response
smalldisturbance
disturbance
with
parameters
= 0.5
s, the
Figure
17. Gen1
speed
oni,lnon
a=small
with
PSSPSS
T =T0.5
s, the
PSS gain
increase
in the
lowerresponse
cluster 1-K
6–10, disturbance
and the upper
cluster
1parameters
Ki,um = 10–14.
increase
in the
lower
cluster
= 6–10,
the upper
cluster
1 K=i,um
= 10–14.
increase
in the
lower
cluster
1-K1-K
i,ln =i,ln
6–10,
andand
the upper
cluster
1 Ki,um
10–14.
465
460
Active power [MW]
465
Active power [MW]
460
455
455 450
450 445
445 440
440 435
435 430
430 425
425 420
0
420
0
1
1
K=6
2
2
3
3
4
4
5
5
K=7
K = 6K = 11
K =K7= 12
K = 11
K = 12
6
6
7
8
Time8 [s] 9
7
K=8
Time [s]
9
10
10
11
12
13
14
15
11
12
13
14
15
K=9
K K= =8 13
KK==914
K = 13
K = 14
K = 10
K = 10
Figure 18. Gen1 active power response on a small disturbance with PSS parameters T = 0,5 s, the
PSS gain increase in the lower cluster 1-Ki,ln = 6–10 and the upper cluster 1 Ki,um = 10–14.
Figure
Gen1
active
power
responseonona asmall
smalldisturbance
disturbance with
with PSS parameters
Figure
18. 18.
Gen1
active
power
response
parameters TT==0,5
0,5s,s,the
thePSS
in
the
lower
cluster
1-K
and
the
upper
cluster
1
K
=
10–14.
PSSgain
gainincrease
increase
in
the
lower
cluster
1-K
i,ln== 6–10
6–10
and
the
upper
cluster
1
K
i,um =
10–14.
i,ln
i,um
Tables 7 and 8 give an overview of the proposed weight functions. The damping ratio
of the dominant oscillatory mode reached the maximum in the lower cluster 1 for the PSS
Tables
7 and
8 give
overview
of the
proposed
weight
functions.
The
damping
ratio
Tables
8 give
an an
overview
of the
proposed
weight
functions.
The
damping
ratio
gain
K3,l17=and
9. Weight
functions
representing
the generator
speed-Γ
ωi and
active
power-Γ
Pi
ofincrease
the
dominant
mode
reached
the
maximum
in the
lower
cluster
1 for
PSS
of the
dominant
mode
reached
maximum
inrepresenting
the
lower
cluster
1 for
thethe
PSS
with oscillatory
theoscillatory
PSS gain
increase.
The the
weight
function
the
rotor-angle-Γ
δi
gain
K
=
9.
Weight
functions
representing
the
generator
speed-Γ
and
active
power-Γ
ωi
3,l1
gain
K
3,l1
=
9.
Weight
functions
representing
the
generator
speed-Γ
ωi
and
active
power-Γ
Pi
increases with the PSS gain increase before it reaches the maximum in upper cluster 1, for Pi
increase
gain
increase.
The
weight
function
representing
rotor-angle-Γ
increase
thethe
PSSPSS
gain
increase.
The
weight
function
representing
thethe
rotor-angle-Γ
δi δi
K1,u1 =with
11.with
increases
with
the
PSS
gain
increase
before
it
reaches
the
maximum
in
upper
cluster
1,
increasesSince
withΓthe
gain the
increase
before
theΓ4,l1
maximum
in according
upper cluster
1, for
i,l1(i)PSS
reached
maximum
forit Γreaches
1,l1(1) and
(4) < Γ3,l1(3),
to (18),
it for
K
=
11.
1,u1
K1,u1follows:
= 11.
Since
(i) reached
maximum
for Γ(1)
(1) and
Γ4,l1<(4)
Γ3,l1
(3), according
to it
(18),
1,l1and
Since
Γi,l1Γ(i)i,l1reached
the the
maximum
for
Γ4,l1(4)
Γ3,l1<(3),
according
to (18),
𝐾𝑙 Γ=1,l1𝐾
1,𝑙1 = 9
it
follows:
follows:
Kl = for
K1,l1
=(0)
9 and Γ4,u1(4) < Γ3,u1(3), according to
Further, Γi,u1(i) reached the maximum
Γ0,u1
𝐾𝑙 = 𝐾1,𝑙1 = 9
(15), it follows:
Further, Γi,u1 (i) reached the maximum for Γ0,u1 (0) and Γ4,u1 (4) < Γ3,u1 (3), according to
Further,
Γi,u1(i) reached the maximum for Γ0,u1(0) and Γ4,u1(4) < Γ3,u1(3), according to
(15),
it follows:
(15), it follows:
Ku = K0,u1 = 10
𝑚𝑎𝑥𝛤𝑖,𝑢𝑚 (𝑖 ) < 𝑚𝑎𝑥𝛤𝑖,𝑙𝑛 (𝑖 )
Which implies that the PSS gain K is chosen as:
Energies 2021, 14, 5644
17 of 29
𝐾 = 𝐾𝑙 = 𝐾1,𝑙1 = 9
Further, according to (14):
Table 7. Weight functions for PSS parameters T = 0.5 s, Ki,ln = 6–10, n = 1 (lower cluster 1), t = 30 ms.
i
0
1
2
3
4
Ki,l1
10
9
8
7
6
(i ) < maxΓ
maxΓ
(i )
i,ln
Γζi,l1
Γωi,l1
Γi,um
δi,l1
ΓPi,l1
Γi,l1(i)
Which implies
that the PSS0.9938148445
gain K is chosen 0.9764722440
as:
0.507928
0.9999982622
3.8037042787
0.524347
0.9999981598 0.9937118105 0.9749639780 3.8180089833
K = Kl = K1,l1 = 9
0.269074
0.9999980102 0.9934398148 0.9729530840 3.5597830173
Table 7. Weight
functions for PSS parameters
T = 0.5 s,0.9705117330
Ki,ln = 6–10, n = 1 (lower
cluster 1), t = 30 ms.
0.243622
0.9999978261
0.9930707387
3.5307066213
0.218705
0.9926580155
i
K 0.9999976055
Γ
Γ
Γ 0.9675305460
Γ 3.5014016407
Γ (i)
i,l1
ζi,l1
ωi,l1
i,l1
Pi,l1
δi,l1
0
10
0.507928
0.9999982622 0.9938148445
0.9764722440
3.8037042787
Table 8. Weight functions1 for PSS
0.5 s, Ki,um = 10–14,
m = 1 (upper
cluster 1); t =3.8180089833
30 ms.
0.9937118105
0.9749639780
9 parameters
0.524347 T = 0.9999981598
2
8
0.269074
0.9999980102 0.9934398148
0.9729530840
3.5597830173
3
7
0.9999978261
3.5307066213
i
Ki,u1
Γζi,u1
Γ0.243622
ωi,u1
Γδi,u1 0.9930707387ΓPi,u1 0.9705117330Γi,u1(i)
4
6
0.218705
0.9999976055 0.9926580155
0.9675305460
3.5014016407
0
1
2
3
4
10
11
12
13
14
0.507928
0.9999982622 0.9938148445 0.976472244 3.80370427871
0.492426
0.9999983415 0.9938523457 0.977678002 3.78984782950
Table 8. Weight functions for PSS parameters T = 0.5 s, Ki,um = 10–14, m = 1 (upper cluster 1); t = 30 ms.
0.478054
0.9999984003 0.9938187092 0.978677412 3.77677422066
i
Ki,u10.9999984437
Γζi,u1
Γωi,u1
Γδi,u1
Γi,u1 (i)
0.464878
0.9937383989
0.979498216ΓPi,u1
3.76461208143
0
10 0.9999984808
0.507928
0.9999982622
0.9938148445
0.976472244
3.80370427871
0.452867
0.9936327709
0.980223698
3.75346307772
1
11
0.492426
0.9999983415 0.9938523457
0.977678002
3.78984782950
2
12
0.478054
0.9999984003 0.9938187092
0.978677412
3.77677422066
It can be noticed 3that the
weight
functions0.9937383989
𝛤𝑖,𝑢𝑚 and 𝛤𝑖,𝑙𝑛0.979498216
are mainly3.76461208143
influ13 overall
0.464878
0.9999984437
0.9999984808
0.9936327709
0.980223698
3.75346307772
4
14
0.452867
enced by the s-domain system performance measures. However, the weight functions re-
lated to time-domain system performance measures increase with the PSS gain increase,
It can be and
noticed
that the overallofweight
functions Γi,um
and Γi,lnafter
are mainly
influenced
resulting in better over-shoots
under-shoots
the simulated
variable
initiating
by the s-domain system performance measures. However, the weight functions related to
a (small) disturbance.
High amplitudes of the active power oscillations can cause the loss
time-domain system performance measures increase with the PSS gain increase, resulting
of load and trigger the
protection
devices
in the system.
Decreased
ofinitiating
the active
in better
over-shoots
and under-shoots
of the
simulatedamplitude
variable after
a (small)
power oscillations obtained
with
the
PSS gain
supports
the idea
tuning
PSSof load
disturbance.
High
amplitudes
of increase
the active power
oscillations
canof
cause
the loss
and trigger
the protection
devices in
the system.
Decreasedsystem
amplitude
of the active power
taking into consideration
both
the s-domain
and
time-domain
performance
oscillations obtained with the PSS gain increase supports the idea of tuning PSS taking into
measures.
consideration both the s-domain and time-domain system performance measures.
The weight functions
Tables
7 and
8 Tables
are presented
Figuresin19–23.
theFor
sake
Thefrom
weight
functions
from
7 and 8 arein
presented
FiguresFor
19–23.
the sake
of better visibility, the
results
for
both
lower
cluster
1
and
upper
cluster
1
are
shown
on a on a
of better visibility, the results for both lower cluster 1 and upper cluster 1 are shown
single diagram.
single diagram.
Γζ vs PSS gain
0.600
0.500
0.400
0.300
0.200
0.100
0.000
NO PSS
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
Figure 19. ΓζFigure
(Γζi,l1 and
from
7 and
Table
8) 8)
for
= 0.5
= 6–14.
19. ΓΓ
(Γζi,l1
and Table
Γζi,u1 from
Tables
7 and
forPSS
PSSparameters
parameters T T
= 0.5
s, Ks,=K
6–14.
ζ ζi,u1
4,
FOR
PEER
REVIEW
1, xx14,
x FOR
PEER
REVIEW
4,
FOR
PEER
REVIEW
Energies
2021, 14, 5644
1919
of of
30 30
19
of
30
ΓΓωω vs
PSS gain
gain
ω vs PSS gain
0.999999
0.999999
0.999999
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999998
0.999997
0.999997
0.999997
0.999997
0.999997
0.999997
0.999997
0.999997
0.999997
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
FigureFigure
20. Γω 20.
(Γωi,l1
and
Γ
ωi,u1
from
Table
7
and
Table
8)
for
PSS
parameters
T
=
0.5
s,
K
=
6–14.
Γω (Γωi,l1 and Γωi,u1 from Tables 7 and 8) for PSS parameters T = 0.5 s, K = 6–14.
Figure 20. Γω (Γωi,l1 and Γωi,u1 from Table 7 and Table 8) for PSS parameters T = 0.5 s, K = 6–14.
Figure 20. Γω (Γωi,l1 and Γωi,u1 from Table 7 and Table 8) for PSS parameters T = 0.5 s, K = 6–14.
Γ vs PSS gain
ΓΓδδ vs
PSS gain
δ vs PSS gain
0.9940
0.9940
0.9938
0.9940
0.9938
0.9936
0.9938
0.9936
0.9934
0.9936
0.9934
0.9932
0.9934
0.9932
0.9930
0.9932
0.9930
0.9928
0.9930
0.9928
0.9926
0.9928
0.9926
0.9924
0.9926
0.9922
0.9924
0.9924
0.9920
0.9922
0.9922
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
0.9920
0.9920
K=8 7 andK=9
K=11
K=12
Figure 21. Γδ K=6
(Γδi,l1 and ΓK=7
δi,u1 from Table
Table 8) K=10
for PSS parameters
T
= 0.5 s, KK=13
= 6–14. K=14
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
Figure 21.Figure
Γδ (Γδi,l1
δi,u1 from Table 7 and Table 8) for PSS parameters T = 0.5 s, K = 6–14.
21.and
Γδ (ΓΓ
Γδi,u1
from
Tables 7 and
forPSS
PSSparameters
parameters TT == 0.5
Figure 21. Γδ (Γδi,l1
and
Γδi,l1
δi,u1 and
from
Table
7 and
8)8)for
0.5s,s,KK==6–14.
6–14.
ΓP Table
vs PSS
gain
0.985
ΓΓP vs
PSS gain
P vs PSS gain
0.985
0.985
0.980
0.980
0.975
0.980
0.975
0.970
0.975
0.970
0.965
0.970
0.965
0.960
0.965
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
0.960
0.960
Figure 22. ΓP (ΓPi,l1 and ΓPi,u1 from Table 7 and Table 8) for PSS parameters T = 0.5 s, K = 6–14.
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
Due
protection
and
circuit-breaker
opening T
time,
Figure 22.
ΓPto
(ΓPi,l1
and ΓPi,u1activation
from Tabletime
7 and
Table
8) for PSS parameters
= 0.5ins,the
K = real
6–14.power
Figure
22.
Γ
P (ΓPi,l1
and
Γ
Pi,u1 and
from
Table
7
and
Table
8)
for
PSS
parameters
T
0.5
s,
K
=
6–14.hence
Figure
22.
Γ
(Γ
Γ
from
Tables
7
and
8)
for
PSS
parameters
=
0.5
s,
K
=
6–14.
P
Pi,l1
Pi,u1
system, disturbance duration lasts longer than the one simulated in this case study,
theDue
proposed
algorithm
performance
examined for opening
longer disturbance
to protection
activation
time will
and be
circuit-breaker
time, in thedurations.
real power
Due to protection activation time and circuit-breaker opening time, in the real power
system, disturbance duration lasts longer than the one simulated in this case study, hence
system, disturbance duration lasts longer than the one simulated in this case study, hence
the proposed algorithm performance will be examined for longer disturbance durations.
the proposed algorithm performance will be examined for longer disturbance durations.
18 of 29
5.4. Case Study 2-Disturbance Duration 150 ms
The
Energies 2021, 14, 5644
observations from the aforementioned case study will be supplemented with the 19 of 29
OR PEER REVIEW new time-domain system performance measures for the disturbance duration of 15020ms.
of 30
The simulation results for the lower cluster 1 and the upper cluster 1 are presented in Figures
23–25.
5.4. Case-1.14
Study 2-Disturbance Duration 150 ms
Rotor angle [rad]
-1.16
The
observations from the aforementioned case study will be supplemented with the
-1.18
new time-domain system performance measures for the disturbance duration of 150 ms.
-1.20
The simulation
results for the lower cluster 1 and the upper cluster 1 are presented in Figures
23–25. -1.22
-1.14-1.24
-1.26
-1.16
-1.28
Rotor angle [rad]
-1.18
-1.30
-1.20
0
-1.22
1
2
3
4
K=6
K = 11
-1.24
5
6
7
8
9
Time [s]
K=7
K = 12
10
K=8
K = 13
11
12
13
14
K=9
K 0 14
15
K = 10
-1.26
Figure 23.
Gen1 rotor angle response on a small disturbance with PSS parameters T = 0.5 s, the PSS gain increase in the
Figure 23. Gen1 rotor angle response on a small disturbance with PSS parameters T = 0.5 s, the PSS
lower cluster
-1.281-Ki,ln = 6–10, and the upper cluster 1 Ki,um = 10–14.
gain increase in the lower cluster 1-Ki,ln = 6–10, and the upper cluster 1 Ki,um = 10–14.
-1.30
Due to protection activation time and circuit-breaker opening time, in the real power
1.0015
0
1
2system,
3 disturbance
4
5 duration
6
7lasts longer
8
9than 10
11 simulated
12
13
14case15
the one
in this
study, hence
Time [s] will be examined for longer disturbance durations.
the proposed algorithm performance
Generator speed [rads-1]
K=6
1.0010
K=7
K = 12
K = 11
K=8
K = 13
K=9
K 0 14
5.4. Case Study 2—Disturbance Duration 150 ms
1.0005
K = 10
The observations from the aforementioned case study will be supplemented with the
1.0000
time-domain
system
performance
measures
for the
disturbanceT duration
of 150
Figure 23. Gen1 rotor new
angle
response on
a small
disturbance
with PSS
parameters
= 0.5 s, the
PSSms. The
simulation
results
for
the
lower
cluster
1
and
the
upper
cluster
1
are
presented
in
Figures
23–25.
gain increase
0.9995in the lower cluster 1-Ki,ln = 6–10, and the upper cluster 1 Ki,um = 10–14.
0.9990
1.0015
0.9985
Generator speed [rads-1]
1.0010
0.9980
1.0005
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time [s]
1.0000
K=6
K=7
K=8
K=9
0.9995
K = 11
K = 12
K = 13
K = 14
K = 10
Figure
0.9990 24. Gen1 speed response on a small disturbance with PSS parameters T = 0.5 s, the PSS gain
increase in the lower cluster 1-Ki,ln = 6–10, and the upper cluster 1 Ki,um = 10–14.
0.9985
0.9980
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time [s]
K=6
K=7
K=8
K=9
K = 11
K = 12
K = 13
K = 14
K = 10
Figure
Gen1
response
on disturbance
a small disturbance
with PSSTparameters
T =gain
0.5increase
s, the PSS
gain
Figure
24. 24.
Gen1
speedspeed
response
on a small
with PSS parameters
= 0.5 s, the PSS
in the
lower
increase
the lower
cluster
i,ln =
the upper cluster 1 Ki,um = 10–14.
cluster
1-Ki,lnin
= 6–10,
and the
upper 1-K
cluster
1 6–10,
Ki,um =and
10–14.
x FOR PEER REVIEW
21 of 30
Energies 2021, 14, 5644
20 of 29
Active power [MW]
490
470
450
430
410
390
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time [s]
K=6
K = 11
K=7
K = 12
K=8
K = 13
K=9
K = 14
K = 10
Figure
25. active
Gen1power
activeresponse
power on
response
on a smallwith
disturbance
with T
PSS
parameters
T =increase
0.5 s, the
Figure
25. Gen1
a small disturbance
PSS parameters
= 0.5
s, the PSS gain
in the
gain
in
the
lower
cluster
1-K
i,ln ==6–10,
and
the
upper
cluster
1
K
i,um = 10–14.
lowerPSS
cluster
1-Kincrease
=
6–10,
and
the
upper
cluster
1
K
10–14.
i,ln
i,um
Gen1 rotor
angle
settling
time andundershoot
the maximal undershoot
increase
with the increase
Gen1 rotor angle settling
time
and
the maximal
increase with
the increase
of the PSS gain in both upper cluster 1 and lower cluster 1 in this study case. The active
of the PSS gain in both upper cluster 1 and lower cluster 1 in this study case. The active
power and the generator speed undershoot decrease with the PSS gain increase while
power and the generator
speed
undershoot
decrease
with the
PSS9 gain
increase
the of the
the settling
times
remain almost
the same.
Tables
and 10
give anwhile
overview
settling times remain
almost
the same.
Tables 9 and 10 give an overview of the proposed
proposed
weight
functions.
weight functions.
Table 9. Weight functions for PSS parameters T = 0.5 s, Ki,ln = 6–10, n = 1 (lower cluster 1), t = 150 ms.
Table 9. Weight functions for PSS parameters T = 0.5 s, Ki,ln = 6–10, n = 1 (lower cluster 1), t = 150 ms.
i
i
0
1
2
3
4
Ki,l1
10
9
8
7
6
Ki,l1
Γζi,l1
Γωi,l1
Γδi,l1
ΓPi,l1
Γi,l1 (i)
10
0.507928
0.9999717530
Γ0ζi,l1
Γωi,l1
Γδi,l1 0.8671126304
ΓPi,l1 0.6144343930Γi,l1(i)2.9894469569
1
9
0.524347
0.9999706671 0.8723687207
0.5977923551
2.9944787854
0.507928
0.8671126304
0.6144343930
2.9894469569
0.9999693749
0.8766792552
0.5791968020
2.7249198464
2
8 0.9999717530
0.269074
3
7 0.9999706671
0.243622
0.9999679219
0.8801760142
0.5594264612
2.6831928107
0.524347
0.8723687207
0.5977923551
2.9944787854
4
6
0.218705
0.9999662783 0.8826227186
0.5379485031
2.6392427924
0.269074
0.9999693749 0.8766792552 0.5791968020 2.7249198464
0.243622
0.9999679219 0.8801760142 0.5594264612 2.6831928107
Table
10.
Weight
functions for PSS
parameters T = 0.5379485031
0.5 s, Ki,um = 10–14,
m = 1 (upper cluster 1);
0.218705
0.9999662783
0.8826227186
2.6392427924
t = 150 ms.
Table 10. Weight functions
for
PSS parameters
T = Γ0.5 s, Ki,um = 10–14,
cluster 1); t = 150
i
Ki,u1
Γζi,u1
Γδi,u1 m = 1 (upper
ΓPi,u1
Γi,u1 (i)
ωi,u1
ms.
i
0
1
2
3
4
Ki,u1
10
11
12
13
14
0
1
Γζi,u1
2
3
0.507928
4
10
11
12
13
14
0.507928
0.492426
Γωi,u1
0.478054
0.464878
0.9999717530
0.452867
0.9999717530 0.8671126304
0.6144343930
2.9894469569
0.9999726603 0.8609451785
0.6293255630
2.9826698748
Γδi,u1
ΓPi,u1 0.6432479360Γi,u1(i)2.9752607011
0.9999734435 0.8539854267
0.9999741856
0.8468810809
0.6563139020
2.9680467857
0.8671126304
0.6144343930
2.9894469569
0.9999748504 0.8395454183
0.6686698430
2.9610570069
0.492426
0.9999726603 0.8609451785 0.6293255630 2.9826698748
0.478054
0.9999734435 0.8539854267 0.6432479360 2.9752607011
Speed weight function Γω as well as active power weight function ΓP increase with
0.464878
0.9999741856 0.8468810809 0.6563139020 2.9680467857
the PSS gain increase. As in the previous study case:
0.452867
0.9999748504
0.8395454183
0.6686698430
2.9610570069
Γ (i) reached
the maximum
for Γ (1) and
Γ (4) < Γ (3),
and according to (18),
i,l1
1,l1
4,l1
3,l1
it follows:
Kl = K
Speed weight function Γω as well as active power
weight
1,l1 = 9 function ΓP increase with
the PSS gain increase. As in the previous study case:
Γi,l1(i) reached the maximum for Γ1,l1(1) and Γ4,l1(4) < Γ3,l1(3), and according to (18), it
follows:
𝐾𝑙 = 𝐾1,𝑙1 = 9
Γi,u1(i) reached the maximum for Γ0,u1(0) and Γ4,u1(4) < Γ3,u1(3), and according to (15), it
follows:
Energies 2021, 14, 5644
21 of 29
Γi,u1 (i) reached the maximum for Γ0,u1 (0) and Γ4,u1 (4) < Γ3,u1 (3), and according to (15),
it follows:
Ku = K0,u1 = 10
Energies 2021, 14, x FOR PEER REVIEW
Energies 2021, 14, x FOR PEER REVIEW
22 of 30
22 of 30
Further, according to (14):
maxΓi,um (i ) < maxΓi,ln (i )
𝑚𝑎𝑥𝛤
Which implies that the PSS gain
K𝑖,𝑢𝑚
is: (i) < 𝑚𝑎𝑥𝛤𝑖,𝑙𝑛 (i)
𝑚𝑎𝑥𝛤𝑖,𝑢𝑚 (i) < 𝑚𝑎𝑥𝛤𝑖,𝑙𝑛 (i)
Which implies that the PSS gain K is:
Which implies that the PSS gainK K=is:Kl = K1,l1 = 9
K = 𝐾𝑙 = 𝐾1,𝑙1 = 9
K = 𝐾𝑙 outputs
= 𝐾1,𝑙1 =of9the Γi,um (i) and Γi,ln (i) functions is
The difference between the maximal
The difference between the maximal outputs of the Γi,um
(i) and Γi,ln(i) functions is
smaller
than
in the previous
case
study,
implying
that the
disturbance
duration
affects is
the
The
difference
betweencase
thestudy,
maximal
outputs
thedisturbance
Γi,um(i) andduration
Γ
i,ln(i) functions
smaller
than
in the previous
implying
thatofthe
affects the
weight
functions
related
to the
time-domain
system
performance
measures
andaffects
emphasize
smaller
than
in
the
previous
case
study,
implying
that
the
disturbance
duration
the
weight functions related to the time-domain system performance measures and emphatheir
significance
in the PSS
tuning process.system
As in the previous measures
case study, the
weight
weight
functions
related
to the
size
their
significance
in the
PSStime-domain
tuning process. As inperformance
the previous case study,and
theemphaweight
functions
from
Tables
9
and
10
are
illustrated
in
Figures
26–29.
size their significance
the PSS
tuning
process.
in the26–29.
previous case study, the weight
functions
from Tablesin
9 and
10 are
illustrated
in As
Figures
functions from Tables 9 and 10 are illustrated in Figures 26–29.
0.600
0.600
0.500
0.500
0.400
0.400
0.300
0.300
0.200
0.200
0.100
0.100
0.000
0.000
NO PSS
NO PSS
K=6
K=6
K=7
K=7
K=8
K=8
K=9
K=9
K=10
K=10
K=11
K=11
K=12
K=12
K=13
K=13
K=14
K=14
Figure26.
26. ΓΓζ (Γ
ζi,l1 and Γζi,u1 from Table 9 and Table 10) for PSS parameters T = 0.5 s, K = 6–14.
Figure
ζ (Γζi,l1 and Γζi,u1 from Tables 9 and 10) for PSS parameters T = 0.5 s, K = 6–14.
Figure 26. Γζ (Γζi,l1 and Γζi,u1 from Table 9 and Table 10) for PSS parameters T = 0.5 s, K = 6–14.
0.999976
0.999976
0.999974
0.999974
0.999972
0.999972
0.999970
0.999970
0.999968
0.999968
0.999966
0.999966
0.999964
0.999964
0.999962
0.999962
0.999960
0.999960
Γω vs PSS gain
Γω vs PSS gain
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
Figure 27. Γω (Γωi,l1 and Γωi,u1 from Table 9 and Table 10) for PSS parameters T = 0.5 s, K = 6–14.
Figure27.
27. ΓΓωω (Γ
and Γωi,u1 from Table 9 and Table 10) for PSS parameters T = 0.5 s, K = 6–14.
Figure
(Γωi,l1
ωi,l1 and Γωi,u1 from Tables 9 and 10) for PSS parameters T = 0.5 s, K = 6–14.
s 2021, 14, x FOR
PEER
REVIEW
Energies
2021,
x FOR PEER REVIEW
Energies
2021,
14,14,
5644
Energies 2021, 14, x FOR PEER REVIEW
0.890
0.880
0.870
0.860
0.850
0.840
0.830
0.820
0.810
23 of 30
23 22
of of
3029
23 of 30
Γδ vs PSS gain
Γδ vs PSS gain
Γδ vs PSS gain
0.890
0.890
0.880
0.880
0.870
0.870
0.860
0.860
0.850
0.850
0.840
0.840
0.830
0.830
0.820
0.820
0.810
0.810
K=7
K=6
K=7
K=8
K=14
K=8
K=9
K=10 K=9
K=11 K=10
K=12 K=11
K=13 K=12
K=14 K=13
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
Figure
28.ΓΓδ (Γ
(Γδi,l1
and
from
Table
9 and
Table
PSS
0.5s,s,KK==6–14.
6–14.
Figure 28. Γδ (Γδi,l1
and Γ28.
δi,u1 from
Table
9Γδi,u1
and
Table
10)
for
PSS
parameters
T parameters
= parameters
0.5 s, K = 6–14.
from
Tables
9 and
10)10)
forfor
PSS
TT==0.5
and
Γδi,u1
Figure
δi,l1
Figure 28. Γδδ (Γδi,l1
and Γδi,u1
from Table 9 and Table 10) for PSS parameters T = 0.5 s, K = 6–14.
K=6
ΓP vs PSS gain
ΓP vs PSS gain
Γ vs PSS gain
0.680
0.680
0.660
0.680
0.660
P
0.660
0.640
0.640
0.620
0.640
0.620
0.620
0.600
0.600
0.580
0.600
0.580
0.580
0.560
0.560
0.540
0.560
0.540
0.540
0.520
0.520
0.500
0.520
0.500
0.500
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=7
K=8
K=9
K=10 K=9
K=11 K=10
K=12 K=11
K=13 K=12
K=14 K=13
K=6
K=7
K=8
K=14
Figure 29. ΓP (ΓPi,l1 and ΓPi,u1 from Table 9 and Table 10) for PSS parameters T = 0.5 s, K = 6–14.
Figure 29. ΓP (ΓPi,l1
and Γ29.
Pi,u1
Table
9ΓPi,u1
and
Table
10)
for
PSS
parameters
T =parameters
0.5 s, K = 6–14.
Figure
29. Γfrom
ΓP (Γ
and
Table
9 and
10)for
forPSS
PSS
parameters
0.5 s,
s, K
K=
= 6–14.
Figure
(ΓPi,l1
and
Γ from
from
Tables
9 Table
and
10)
TT== 0.5
6–14.
K=6
P
Pi,l1
Pi,u1
-1.10
-1.10
-1.10
-1.15
-1.15
-1.15
-1.20
-1.25
-1.30
[rad]
angle
Rotor
[rad]
angle
Rotor
Rotor angle [rad]
5.5. Case Study 3–Disturbance Duration-220 ms
3—Disturbance
Duration-220msms
5.5. Case
Case Study
StudyDuration-220
3–Disturbancems
Duration-220
5.5. Case Study 5.5.
3–Disturbance
New time-domain system performance measures for the (small) disturbance duraNew
time-domain
system
performance
measures
for
the
(small)disturbance
disturbanceduration
duratime-domain
system
performance
for
the
New time-domain
system
measures
formeasures
the presented
(small)
disturbance
duration New
of
220
ms areperformance
obtained
from
the simulations
in (small)
Figures
30–32.
tion
ofms
220are
ms
are obtained
the
simulations
presented
in Figures
30–32.
220
obtained
fromfrom
the simulations
presented
in30–32.
Figures
30–32.
tion of 220 ms of
are
obtained
from
the simulations
presented
in Figures
-1.20
-1.20
-1.25
-1.25
-1.30
-1.30
-1.35
-1.35 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
-1.35
0
1
2
3
4
5
6 Time
7 [s]8
9
10 11 12 13 14 15
0
1
2
3K = 6 4
5
6 K =7
8
9 Time
12 13
15
K10
= [s]
8 11
K = 9 14
K = 10
KK == 611
KKTime
== 712 [s]
KK == 813
KK == 914
K = 10
K=6
K=7
K=8
K=9
K = 10
K = 11
K = 12
K = 13
K = 14
K = 11 Figure 30. Gen1
K = rotor
12 angle response
K = 13
K = 14 with PSS parameters T = 0.5 s, the PSS
on a small disturbance
Figure
30. Gen1
rotor
angle
response
a small
with PSS
gain increase
in the
lower
cluster
1-Ki,lnon
= 6–10,
anddisturbance
the upper cluster
1 Ki,parameters
um = 10–14. T = 0.5 s, the PSS
Figure
30. Gen1
angleonresponse
on
acluster
small 1-K
disturbance
with
PSSupper
parameters
TK=i,um
0.5
s,gain
the increase
PSS
Figure 30. Gen1
rotor
angle rotor
response
ainsmall
disturbance
with
T =cluster
0.5 s, 1the
PSS
in the
gain
increase
the lower
i,ln = PSS
6–10,parameters
and the
= 10–14.
increase
thethe
lower
cluster
1-K1i,lnK=,6–10,
and the upper cluster 1 Ki,um = 10–14.
lower clustergain
1-Ki,ln
= 6–10,inand
upper
cluster
=
10–14.
i um
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14,
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4, xxFOR
24 of
of 30
30
24
23 of 29
1.0015
1.0015
-1
Generator
[rads-1] ]
speed[rads
Generatorspeed
1.0010
1.0010
1.0005
1.0005
1.0000
1.0000
0.9995
0.9995
0.9990
0.9990
0.9985
0.9985
0.9980
0.9980
0.9975
0.9975
00
11
22
33
KK==66
11
KK==11
44
55
66
KK==77
12
KK==12
77
88
Time[s]
[s]
Time
99
10
10
KK==88
13
KK==13
11
11
12
12
13
13
KK==99
14
KK==14
14
14
15
15
10
KK==10
Figure Figure
31. Gen1
on a smallon
disturbance
with PSS parameters
Tparameters
= 0.5 s, the PSS
gain
the lower
31.speed
Gen1response
speedresponse
response
smalldisturbance
disturbance
withPSS
PSSparameters
0.5s,s,increase
thePSS
PSSin
gain
Figure 31.
Gen1
speed
on aasmall
with
TT==0.5
the
gain
cluster 1-K
=
6–10,
and
the
upper
cluster
1
K
,
=
10–14.
um
i,ln
increase
in the lower cluster 1-Ki,ln = i6–10, and the upper cluster 1 Ki,um = 10–14.
increase in the lower cluster 1-Ki,ln = 6–10, and the upper cluster 1 Ki,um = 10–14.
530
530
510
510
Active
[MW]
power[MW]
Activepower
490
490
470
470
450
450
430
430
410
410
390
390
370
370
00
11
22
33
KK==66
11
KK==11
44
55
KK==77
12
KK==12
66
77
88
Time[s]
[s]
Time
99
10
10
KK==88
13
KK==13
11
11
12
12
13
13
KK==99
14
KK==14
14
14
15
15
10
KK==10
Figure
32.
Gen1
active
poweron
response
on aa small
smallwith
disturbance
with PSS
PSS
parameters
0.5
thein the
FigureFigure
32. Gen1
active
power
response
a small disturbance
PSS parameters
T =parameters
0.5
s, the PSSTT
gain
increase
32.
Gen1
active
power
response
on
disturbance
with
== 0.5
s,s, the
PSS
gain
increase
in
the
lower
cluster
1-K
i,ln
=
6–10,
and
the
upper
cluster
1
K
i,um = 10–14.
lower cluster
1-K
=
6–10,
and
the
upper
cluster
1
K
,
=
10–14.
um
i,ln
i
PSS gain increase in the lower cluster 1-Ki,ln = 6–10, and the upper cluster 1 Ki,um = 10–14.
11 overview
and 12 give
an overview
of the weight
As incase,
the previous
Tables 11
11 and
and 12
12Tables
give an
an
of the
the
weight functions.
functions.
As in
infunctions.
the previous
previous
Tables
overview
of
weight
the
case, with the
case, give
the weight
functions
related
to generator speedAs
and active
power increase
the weight
weight functions
functions related
related to
to generator
generator speed
speed and
and active
active power
power increase
increase with
with the
the PSS
the
PSS gain increase
in both lower cluster
1 and upper
cluster 1 while
the rotorPSS
angle weight
gainincrease
increasein
in both
both
lower
cluster11and
andupper
upper cluster
cluster11 while
while the
therotor
rotorangle
angle weight
weightfuncfuncgain
lower
cluster
function
decreases.
tiondecreases.
decreases.
The damping ratio of the dominant oscillatory mode reached the maximum in the
tion
lower cluster 1 for the PSS gain K1,l1 = 9.
Table
11.
Weight
functions
forTT==0.5
0.5s,s,KKi,ln
i,ln = 6–10, n = 1 (lower cluster 1); t = 220 ms.
Table 11. Weight functions for
= 6–10, n = 1 (lower cluster 1); t = 220 ms.
ii
00
11
22
33
i,l1
KKi,l1
10
10
99
88
77
ζi,l1
ΓΓζi,l1
0.507928
0.507928
0.524347
0.524347
0.269074
0.269074
0.243622
0.243622
ωi,l1
ΓΓωi,l1
0.9999430777
0.9999430777
0.9999411008
0.9999411008
0.9999389630
0.9999389630
0.9999366821
0.9999366821
δi,l1
ΓΓδi,l1
0.7031884278
0.7031884278
0.7163057813
0.7163057813
0.7280097024
0.7280097024
0.7383721799
0.7383721799
Pi,l1
ΓΓPi,l1
0.211588861
0.211588861
0.183731701
0.183731701
0.154920355
0.154920355
0.124432751
0.124432751
i,l1(i)
ΓΓi,l1
(i)
2.42264854628
2.42264854628
2.42432562562
2.42432562562
2.15194343419
2.15194343419
2.10636402651
2.10636402651
Energies 2021, 14, 5644
24 of 29
Table 11. Weight functions for T = 0.5 s, Ki,ln = 6–10, n = 1 (lower cluster 1); t = 220 ms.
25 of 30
i
Ki,l1
Γζi,l1
Γωi,l1
Γδi,l1
ΓPi,l1
Γi,l1 (i)
14, x FOR PEER REVIEW
0
10
0.507928
0.9999430777 0.7031884278
0.211588861
2.42264854628
1
9
0.524347
0.9999411008 0.7163057813
0.183731701
2.42432562562
Table 12. Weight functions
for8 T = 0.5,
Ki,um = 10–14,
m = 1 (upper
cluster 1); t = 220
ms.
2
0.269074
0.9999389630
0.7280097024
0.154920355
2.15194343419
3
7
0.243622
0.9999366821 0.7383721799
0.124432751
2.10636402651
i
Ki,u1
Γωi,u1
Γδi,u1 0.7469502268
ΓPi,u1 0.091936998
Γi,u1(i)2.05752669831
4Γζi,u1 6
0.218705
0.9999341808
0
1
2
3
4
10
11
12
13
14
0.507928
0.9999430777 0.7031884278 0.211588861 2.42264854628
0.492426
0.9999449201 0.6889357381 0.238297313 2.41960444481
Table 12. Weight functions for T = 0.5, Ki,um = 10–14, m = 1 (upper cluster 1); t = 220 ms.
0.478054
0.9999466013 0.6733378546 0.263934882 2.41527323297
i
K
Γζi,u1
Γωi,u1
Γδi,u1
Γi,u1 (i)
0.464878 i,u1 0.9999480314
0.6559384555
0.286974980 ΓPi,u1
2.40773908371
0
10 0.9999493362
0.507928
0.9999430777
0.7031884278
0.211588861
2.42264854628
0.452867
0.6376144533
0.308911693
2.39934237831
1
11
0.492426
2
12
0.478054
The damping ratio
of13the dominant
3
0.464878
14 K1,l1
0.452867
lower cluster 1 for the4 PSS gain
= 9.
0.9999449201 0.6889357381
0.9999466013 0.6733378546
oscillatory
mode
reached
0.9999480314
0.6559384555
0.9999493362 0.6376144533
0.238297313
0.263934882
the0.286974980
maximum
0.308911693
2.41960444481
2.41527323297
in2.40773908371
the
2.39934237831
Γi,l1(i) reached the maximum for Γ1,l1(1) and Γ4,l1(4)< Γ3,l1(3) and, according to (18), it
follows:
Γi,l1 (i ) reached the maximum for Γ1,l1 (1) and Γ4,l1 (4) < Γ3,l1 (3) and, according to
(18), it follows:
𝐾𝑙 = 𝐾1,𝑙1 = 9
Kl = K1,l1 = 9
Further, Γi,u1(i) reached
formaximum
Γ0,u1(0) and
4,u1(4) < Γ3,u1(3) and, according
Further, the
Γi,u1maximum
(i) reached the
for ΓΓ0,u1
(0) and Γ4,u1 (4) < Γ3,u1 (3) and, according
to (15), it follows:to (15), it follows:
K = K0,u1 = 10
𝐾𝑢 = 𝐾0,𝑢1 = u10
Further, according to (14):
Further, according to (14):
maxΓ
(i ) < maxΓi,ln (i ).
i,um
𝑚𝑎𝑥𝛤𝑖,𝑢𝑚 (i) < 𝑚𝑎𝑥𝛤
𝑖,𝑙𝑛 (i)
Which
thatKthe
PSS gainas:
K is chosen as:
Which implies that
theimplies
PSS gain
is chosen
K ==Kl9= K1,l1 = 9
K = 𝐾𝑙 = 𝐾1,𝑙1
Here, it can alsoHere,
be noticed
thatbedifference
functions
Γ1,l1(1) →
1,l1 →
= K1,l1 = 9
it can also
noticed thatbetween
differenceweight
between
weight functions
Γ1,l1K(1)
→ K0,l1 smaller
= 10 becomes
smaller
the fault increase.
duration increase.
The weight
9 and Γ0,u1(0) → Kand
0,l1 =Γ10
becomes
as the
fault as
duration
The weight
func- functions
0,u1 (0)
from
Tables
11
and
12
are
presented
in
Figures
33–37.
tions from Tables 11 and 12 are presented in Figures 33–37.
0.600
0.500
0.400
0.300
0.200
0.100
0.000
NO PSS
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
Figure Figure
33. Γζ (Γ
Γζi,u1
from
Table
and 11
Table
forPSS
PSSparameters
parameters
= 0.5
= 6–14.
and
Γζi,u1
from11
Tables
and 12) for
T =T 0.5
s, Ks,=K6–14.
33.ζi,l1Γζand
(Γζi,l1
Energies
2021,
x FOR PEER REVIEW
Energies
2021,
14,14,
5644
Energies 2021, 14, x FOR PEER REVIEW
Energies 2021, 14, x FOR PEER REVIEW
26 of 25
30of 29
26 of 30
26 of 30
Γω vs PSS gain
Γω vs PSS gain
Γω vs PSS gain
0.99996
0.99996
0.99996
0.99995
0.99995
0.99995
0.99995
0.99995
0.99995
0.99994
0.99994
0.99994
0.99994
0.99994
0.99994
0.99993
0.99993
0.99993
0.99993
0.99993
0.99993
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=6 and ΓK=7
K=9 TableK=10
K=11
K=12 T = 0.5
K=13
K=14
Figure 34.
34. Γ
ωi,u1 from K=8
TableTables
11 and
12)
for
PSS
parameters
s, s,
K =K6–14.
Figure
Γωω(Γ(Γωi,l1
and
Γ
from
11
and
12)
for
PSS
parameters
T
=
0.5
= 6–14.
ωi,l1
Figure 34. Γω (Γωi,l1
and Γωi,u1ωi,u1
from Table 11 and Table 12) for PSS parameters T = 0.5 s, K = 6–14.
Figure 34. Γω (Γωi,l1 and Γωi,u1 from Table 11 and Table 12) for PSS parameters T = 0.5 s, K = 6–14.
Γδ vs PSS gain
0.760
Γδ vs PSS gain
0.760
Γδ vs PSS gain
0.740
0.760
0.740
0.720
0.740
0.720
0.700
0.720
0.700
0.680
0.700
0.680
0.660
0.680
0.660
0.640
0.660
0.640
0.620
0.640
0.620
0.600
0.620
0.600
0.580
0.600
0.580
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
0.580
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=10
K=11
K=12 T = 0.5
K=13
K=14
Figure 35. ΓK=6
δ (Γδi,l1 andK=7
Γδi,u1 fromK=8
Table 11 K=9
and Table
12) for PSS
parameters
s, K = 6–14.
Figure 35.
35. Γ
Γδ (Γ
δi,l1 and Γδi,u1 from Table 11 and Table 12) for PSS parameters T = 0.5 s, K = 6–14.
Figure
(Γ
and
Γ
from
Tables
11
and
12)
for
PSS
parameters
T
=
0.5
s,
K
=
6–14.
δi,l1
Figure 35. Γδδ (Γδi,l1
and Γδi,u1δi,u1
from Table 11 and Table 12) for PSS parameters T = 0.5 s, K = 6–14.
Γ
vs
PSS
gain
P
0.350
ΓP vs PSS gain
0.350
ΓP vs PSS gain
0.350
0.300
0.300
0.300
0.250
0.250
0.250
0.200
0.200
0.200
0.150
0.150
0.150
0.100
0.100
0.100
0.050
0.050
0.050
0.000
0.000
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
0.000
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
K=11
K=12 T = 0.5
K=13
K=14
Figure 36. ΓK=6
P (ΓPi,l1 andK=7
ΓPi,u1 fromK=8
Table 11 K=9
and TableK=10
12) for PSS
parameters
s, K = 6–14.
Figure 36. ΓP (ΓPi,l1 and ΓPi,u1 from Table 11 and Table 12) for PSS parameters T = 0.5 s, K = 6–14.
Figure36.
36. ΓΓP (Γ
ΓPi,u1
Table
11 and
12) for
for PSS
PSS parameters
6–14.
Figure
(ΓPi,l1 and
and
Γ fromfrom
Tables
11Table
and 12)
parametersTT==0.5
0.5s,s,KK= =
6–14.
P
Pi,l1
Pi,u1
Energies 2021, 14, x FOR PEER REVIEW
Energies 2021, 14, 5644
27 of 30
The difference between the outputs of the weight functions Γi,l1(i) and Γi,u1(i) becomes
insignificant with further PSS gain increase after the maximum has been reached for K26=of 29
9 (Figure 37).
Weight functions Γi.l1(i) and Γi,u1(i)
2.500
2.400
2.300
2.200
2.100
2.000
1.900
1.800
K=6
K=7
K=8
K=9
K=10
K=11
K=12
K=13
K=14
Figure 37. Weight functions: Γi,l1(i) –(i = 0–4 for K = 6–10), Γi,u1(i) (i = 0–4 for K = 10–14), T = 0.5 s.
Figure 37. Weight functions: Γi ,l1 (i)—(i = 0–4 for K = 6–10), Γi,u1 (i) (i = 0–4 for K = 10–14), T = 0.5 s.
Hence,
the weight
functions
representing
the time-domain
The
difference
between
the outputs
of the weight
functions Γi,l1system
(i) and performance
Γi,u1 (i) becomes
measures
have
a
more
significant
influence
on
the
PSS
parameters
selection.
The weight
insignificant with further PSS gain increase after the maximum has been reached
for K = 9
function
associated
with
the
generator
active
power
oscillations
is
of
particular
im(Figure 37).
portance since the active power oscillations can lead to the load loss. All the simulation
Hence, the weight functions representing the time-domain system performance mearesults indicate the lower amplitude of active power oscillations if the PSS parameters are
sures have a more significant influence on the PSS parameters selection. The weight
selected according to a higher value of this function. Therefore, the impact of the proposed
function associated with the generator active power oscillations is of particular importance
weight function associated with the generator active power oscillations can be emphasince the active power oscillations can lead to the load loss. All the simulation results
sized by multiplying it with the corrective factor. Even multiplied with value 1.1, the
indicate the lower amplitude of active power oscillations if the PSS parameters are selected
weight functions ΓPi,l1 and ΓPi,u1 will increase the PSS gain from K = 9 to K = 10. With the
according to a higher value of this function. Therefore, the impact of the proposed weight
further increase of this corrective factor, the reduction of the active power oscillations amfunction associated with the generator active power oscillations can be emphasized by mulplitudes will be more emphasized. The decision of the most appropriate corrective factor
tiplying
it with the corrective factor. Even multiplied with value 1.1, the weight functions
value will be subjected to future works applying machine learning methods.
ΓPi,l1 and ΓPi ,u1 will increase the PSS gain from K = 9 to K = 10. With the further increase of
this
corrective factor, the reduction of the active power oscillations amplitudes will be more
6. Conclusions
emphasized. The decision of the most appropriate corrective factor value will be subjected
In this paper, an algorithm for the PSS1A tuning comprising both the s-domain and
to future works applying machine learning methods.
the time domain system performance measures was proposed and tested on multi-maIEEE 14-bus system model. The location of the PSS implementation is determined
6.chine
Conclusions
by analyzing the participation of the generator speeds state variables in the dominant osIn this
paper,
algorithm
for the
tuning
comprising
both method
the s-domain
cillatory
mode
of theananalyzed
system.
ThePSS1A
well-known
phase
compensation
for
and
the
time
domain
system
performance
measures
was
proposed
and
tested
onupmultithe PSS tuning based on the s-domain system performance measures has only been
machine
IEEE
14-bus
system
model.
The
location
of
the
PSS
implementation
is
determined
graded with the time-domain system performance measures incorporating the weight
by
analyzing
the participation
of the generator
speeds
state system
variables
in the
dominant
functions
comparable
with the damping
ratio of the
dominant
mode
as the
prooscillatory
mode
of
the
analyzed
system.
The
well-known
phase
compensation
posed weight function related with the s-domain system performance measures. Formethod
diffor
the system
PSS tuning
based on
the s-domain
system
performance
measures
hasduration
only been
ferent
disturbance
durations,
three case
studies
were examined.
As the
upgraded
the time-domain
system
performance
measuressystem
incorporating
of system with
disturbances
increases, the
impact
of the time-domain
measuresthe
on weight
PSS
functions
comparable
with
the damping
ratio of
system mode
as the proposed
tuning increases
as well.
Besides
the damping
ofthe
thedominant
electromechanical
oscillations
in the
weight
related
with the s-domain
system
performance
measures.
For different
power function
system, the
implementation
of the PSS,
according
to the proposed
algorithm,
can
system
durations,
three case
studies
were examined.
As modern
the duration
of syssystem
reducedisturbance
the amplitudes
of the active
power
oscillations
to which the
power
disturbances
increases,
thereduction
impact ofare
the
time-domain
system Hence,
measures
tuning
tems with possible
inertia
particularly
vulnerable.
the on
PSSPSS
implementationasand
tuning,
according
to the of
proposed
algorithm, can oscillations
be beneficialinfor
increases
well.
Besides
the damping
the electromechanical
theboth
power
small-signal
and transient stability
the according
power system.
system,
the implementation
of thein
PSS,
to the proposed algorithm, can reduce
the amplitudes of the active power oscillations to which the modern power systems with
possible inertia reduction are particularly vulnerable. Hence, the PSS implementation and
tuning, according to the proposed algorithm, can be beneficial for both small-signal and
transient stability in the power system.
Author Contributions: Conceptualization, P.M. and R.K.; methodology, R.K.; software, P.M.; validation, P.M. and H.R.C.; formal analysis, H.R.C. and H.G.; investigation, R.K.; resources, R.K.;
Energies 2021, 14, 5644
27 of 29
data curation, H.G.; writing—original draft preparation, R.K.; writing—review and editing, P.M.;
visualization, H.G.; supervision, H.R.C.; project administration, P.M.; funding acquisition, P.M., R.K.
and H.G. All authors have read and agreed to the published version of the manuscript.
Funding: The APC was funded by Faculty of Electrical Engineering, Computer Science, and Information Technology Osijek.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: IEEE 14-bus test system data are available online. AVR and PSS
standard models are available online; AVRs parameters are given in Appendix A. All other data
presented in this study are available on request.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
AVR parameters (AVR, IEEE URST5B):
AVR1, AVR2, AVR3: Tr = 0.01 s, Tb1 = 10 s, Tc1 = 2 s, Tb2 = 1 s, Tc2 = 1 s, Kr = 200 p.u.,
T1 = 0.01 s, Kc = 0.1 p.u., Vmin = −3 p.u., Vmax = 3.4 p.u.
AVR4, AVR5: Tr = 0.01 s, Tb1 = 10 s, Tc1 = 2 s, Tb2 = 1 s, Tc2 = 1 s, Kr = 150 p.u., T1 = 0.01 s,
Kc = 0.1 p.u., Vmin = −3 p.u., Vmax = 3.4 p.u.
Static compensation devices parameters:
SC: SB = 100 MVA, UB = 13.8 kV, UN = 1.07 p.u., Qmax = 0.24 p.u., Qmin = −0.06 p.u.,
Vmax = 1.2 p.u., Vmin = 0.8 p.u.
SC1: SB = 100 MVA, UB = 69 kV, UN = 1.01 p.u., Qmax = 0.4 p.u., Qmin = 0.00 p.u.,
Vmax = 1.2 p.u., Vmin = 0.8 p.u.
SC2: SB = 100 MVA, UB = 18 kV, UN = 1.09 p.u., Qmax = 0.24 p.u., Qmin = −0.06 p.u.,
Vmax = 1.2 p.u., Vmin = 0.8 p.u.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
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