A Comprehensive Analysis into the Eects of Quasiperiodicity
on the Swing Equation
Abstract: This research studies the case of quasiperiodicity occurring within the swing equation, a fun-
damental model that characterises the behaviour of rotor of the machine in synchronous generators in
electrical systems. Quasiperiodicity is explained by intricate patterns and understanding the stability
of power systems. Bifurcation analysis, frequency domain techniques and numerical simulations are em-
ployed to study the swing equation in detail. The objective of this study is to provide a comprehensive
understanding of the dynamical behaviour of the equation for the case of quasiperiodicity, using both
analytical and numerical methods, when changes are made to the variables of the system. The results
show the comparison of primary resonance and quasiperiodicity in the swing equation and analyses the
rate at which stability is lost. This will help with the system losing its stability and identies precursors
to chaos which will prevent unavoidable circumstances in the real world.
Key-Words: nonlinear dynamics, swing equation, quasi-periodicity, power system, chaos.
Received: April 12, 2023. Revised: October 24, 2023. Accepted: November 23, 2023. Published: December 31, 2023.
1 Introduction
The concept of quasiperiodicity describes a type
of motion that is characterised by the presence of
two or more frequencies that are not rational mul-
tiples of one another, which results in the frequen-
cies being inconsistent with one another. That
is when the ratios of frequencies are an irrational
value, quasiperiodicity occurs within the nonlin-
ear system. In light of this, it may be deduced
that the system does not completely return to its
initial state, although it does come close to pe-
riodic motion. The phenomenon of quasiperiodic
motion is widely observed in dynamical systems
that display perturbations of integrable systems,
[1]. An example of this can be found in the case
of a double pendulum, where the motion displays
quasiperiodicity when the amplitudes are at their
lowest, [2].
The concept of quasiperiodicity is a multi-
faceted phenomenon, lacking a universally agreed-
upon description. Nevertheless, a frequently em-
ployed methodology involves the establishment of
a denition for quasiperiodic motion in the follow-
ing manner: A dynamical system is classied as
quasiperiodic when it possesses a solution that can
be expressed as the combination of two or more
frequencies that are not in a rational ratio with
each other, [3].
An alternative perspective on quasiperiodicity
is to conceptualise it as a form of motion that has
characteristics closely resembling periodicity. A
periodic signal is dened as a signal that exhibits
repetitive behaviour, recurring exactly after a spe-
cic duration of time, [4]. In contrast, a quasiperi-
odic signal does not exhibit perfect repetition, al-
though it does exhibit approximate repetition at
consistent intervals, [5]. The concept of quasiperi-
odicity holds signicant importance within the
realm of dynamical systems theory. It can be seen
in many biological and physical systems. This in-
cludes the studies of the motion of planets, human
heartbeat and respiratory cycles, [6].
The swing equation in hand, studies the dy-
namical behaviour of the rotor of the machine and
the eects of external force, [7], [8]. Changing
and altering some parameters in the equation ex-
hibit quasiperiodicity within the system. Hence
the system struggles to return to the initial state
showing minute changes and cascading to chaos,
[9]. Analysing the fundamental principles of chaos
theory will provide vital knowledge towards con-
trolling the nonlinear system, [10].
This study focuses on the gap in understanding
quasiperiodicity and the swing equation, demon-
strating analytical approaches to enhance knowl-
edge for researchers and scholars. Hence empha-
sising developments in the analysis of quasiperiod-
icity in the swing equation using Hamilton’s prin-
ciple and focusing on comprehending this method
to provides fresh insights to persistent issues re-
lating to stability of dynamical systems.
ANASTASIA SOFRONIOU, BHAIRAVI PREMNATH
School of Computing and Engineering
University of West London
St. Mary’s Road, W5 5RF
UNITED KINGDOM
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1.1 Brief Literature Review
The quasiperiodic solutions of discrete dynamical
systems dened by mixed-type functional equa-
tions are studied by [11]. The xed-point theorem
is employed by the authors to prove the existence
of quasiperiodic solutions. Additionally, a study
was conducted by the authors, [12], which focuses
on quasiperiodic solutions for fractional dieren-
tial equations. Following that, they use a vari-
ational method to show that quasiperiodic solu-
tions exist, and they provide numerical examples
to support their ndings.
Studies also look at the possibility of quasiperi-
odicity, which is dened by irrational frequency,
in the context of a non-autonomous dierential
equation, [13]. A methodology based on the con-
cept of averaging is employed by the researchers
to ascertain the presence of quasiperiodic solu-
tions. The existence of quasiperiodic solutions for
a non-autonomous fractional dierential equation
with a nonlinear component is examined by [14].
The presence of quasiperiodic solutions is veri-
ed by the researchers using an approach based
on the concept of xed points. Research focusing
on quasiperiodic solutions in a particular class of
abrupt eects non-autonomous dierential equa-
tions have been carried out, [15]. To prove that
quasiperiodic solutions exist, the authors employ
a methodology grounded in the concepts of upper
and lower solutions theory.
The method of averaging to examine the oc-
currence of quasiperiodicity in the swing equation
when it is inuenced by a sinusoidal driving force
is utilised for study, [16]. The author showcased
that the swing equation has the capacity to ex-
hibit quasiperiodic dynamics across a wide range
of driving force amplitudes and frequencies. In
another publication, the aforementioned author
also extended his research to include the impact
of damping coecient, [17]. The author illus-
trated that the implementation of damping can
reduce quasiperiodic behaviour, while also gener-
ating new types of quasiperiodic behaviour. In a
study, author provided a comprehensive explana-
tion of his research on the occurrence of quasiperi-
odicity in the swing equation, [18]. In addition,
the author examined various other uses of the av-
eraging technique in the study of nonlinear dy-
namical systems.
Intermittency, in the context of quasiperiod-
icity in the swing equation, pertains to the un-
predictable and sudden shifts between regular and
chaotic patterns observed in the system, [19]. It is
characterized by intermittent bursts of chaos al-
ternating with periods of regular, quasiperiodic
motion, in contrast to the continuous irregulari-
ties observed in traditional chaotic dynamics, [20].
Within the framework of the swing equation, these
sudden patterns can be witnessed as abrupt tran-
sitions between stable quasiperiodic paths and
chaotic behaviour, emphasizing the system’s sus-
ceptibility to specic changes in parameters or ini-
tial conditions, [21]. Intermittency in the swing
equation has important consequences for power
systems, since it can cause abrupt and unforeseen
uctuations in the pendulum-like motion. A thor-
ough study is vital to understand the alterations to
the system and hence analyse the stability of the
system within power networks, [22], [23]. Hence
this provides the researchers and scholars to de-
velop and control unavoidable eects within the
systems present in power plants
The torus phenomena is explained and exhib-
ited through the case of quasiperiodicity on the
swing equation. This phenomena is when the
nonlinear system shows both periodic and non-
periodic characteristics, [24], [25]. The stability of
the system can be studied in detail by analysing
the torus structure of a dynamical system provid-
ing key comprehension of the behaviour and the
challenges faced within electrical and electronic
elds, [26]. Hence it is important to understand
the principles related to torus structure to elabo-
rate on the chaos theory.
Chaos in the eld of quasiperiodicity in the
swing equation denotes the occurrence of irreg-
ular and apparently unexpected behaviour in a
system that, given specic circumstances, is an-
ticipated to display more organized and peri-
odic motion, [27]. Chaos in the swing equation
is characterised by the disruption of the antic-
ipated quasiperiodic paths, resulting in unpre-
dictable and non-repetitive movement, [28]. Un-
derstanding the chaos for the case of quasiperiod-
icity within the swing equation provides evidence
of sudden changes to the variables of the system,
[29]. Analysing the concept of chaos in power grid
systems and studying the parameters that cause
this behaviour will allow to reduce the adverse ef-
fects that can take place within short duration of
time within the systems, [30], [31], [32].
2 Methodology
2.1 Analytical Work
The swing equation is formulated from the Law
of Rotation which explains the motion of rotating
systems. It derived with the help of Newton’s sec-
ond law of motion in synchronous generators and
applied on the rotor of the swing equation. The
analytical work shown below studies both mechan-
ical and electrical torques on the rotor. Previous
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study done related to this concept are referenced
by [7], [8], [23], [33]. The swing equation is a
second-order dierential equation, depicting the
change in angle of the rotor of the machine from
its synchronous position relative to time.
The equation analysing the rotor’s motion of
the machine including a damping term is given by
[23].
2H
ωR
d2θ
dt2+Ddθ
dt =Pm−
VGVB
XG
sin (θ−θB) (1)
VB=VB0+VB1 cos (Ωt+ϕv) (2)
θB=θB0+θB1cos(Ωt+ϕ0) (3)
with
ωR=Constant angular velocity,
H= Inertia,
D= Damping,
Pm= Mechanical Power,
VG= Voltage of machine,
XG= Transient Reactance,
VB= Voltage of busbar system,
θB= phase of busbar system,
VB1and θB1magnitudes assumed to be small.
A deeper understanding of this equation is es-
sential to understand the concept of quasiperiod-
icity and its dynamical behaviour. This analytical
work uses Taylor expansion and algebraic methods
to formulate the equation to obtain more results
using the digital computers, [7], [8], [33]. Hence
changes can be made to the variables of the swing
equation to observe and analyse the intricate be-
haviour of this system.
The Swing Equation Model
The swing equation, equation (1), explains
both electrical and mechanical torque of the ro-
tor of the machine and studies the behaviour of
the angle of the rotor and speed when a small
change is introduced. Analysing the acceleration
of the machine and the torques provides a strong
foundation for the engineers to overcome dicul-
ties within the systems, [23]. Hence modelling this
concept to obtain real time values will be ideal to
study the equation in detail.
The rotor of the machine used by the swing
equation, explains the intricate behaviour of both
electrical and mechanical elements of the system.
Hence studying the stability of this machine is vi-
tal to comprehend the abrupt alterations to the
parameters of the equation. Stability can be ob-
served through changing the load and inputs of the
systems over time and hence reducing the cascade
of chaos within power systems, [32].
Hamilton’s Principle
Hamilton’s principle studies the dynamics of
the swing equation system and considering this
for the case of quasiperiodicity, this principle uses
Langrangian multiples to formulate the equation,
[34], [35]. It also provides deeper insight into
the behaviour of the variable change within the
nonlinear systems, [36]. Hence this principle pro-
vides a better understanding of the parameters
and chaos theory of the swing equation, [37].
Cascading of chaos within the context of swing
equation can be analysed using Hamilton’s Prin-
ciple. Transition to chaos in the case of quasiperi-
odicity disrupts stable periodic orbits, leading to
bifurcations. The system’s vulnerability to distur-
bances in quasiperiodic cases accelerate the shift
towards chaos. Hamilton’s principle, focusing on
action minimisation and helps in understanding
how quasiperiodicity can lead to loss of synchro-
nisation within the nonlinear dynamical system.
Using this principle the swing equation can
be studied through variational calculus, aiding in
understanding the relationships between dierent
frequencies on the system’s quasiperiodic motion.
Hamilton’s method also identies key factors inu-
encing quasiperiodicity by systematically deriving
equations of motion to investigate stability and en-
ables researchers to understand the analysis of the
sudden dynamical behaviour of the swing equation
and nd methods to reduce chaos within the sys-
tem.
Consider equation (1) and rearranging the
terms to obtain the following,
d2θ
dt2=−
ωRD
2H
dθ
dt +ωR
2HPm−
ωRVGVB
2HXG
sin (θ−θB) (4)
Substituting equation (2) and equation (3) into
equation (4) and expanding the brackets to obtain,
d2θ
dt2=−
ωRD
2H
dθ
dt +ωR
2HPm−
ωRVGVB0
2HXG
sin (θ−(θB0+θB1cos(Ωt+ϕθ)) −
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ωRVGVB1
2HXG
cos (Ωt+ϕv) sin (θ−(θB0+
θB1cos(Ωt+ϕθ)).(5)
Simplifying equation (5) further, assuming
θB0, VB0, ϕθand θB1to be very small,
d2θ
dt2=−pdθ
dt +q−r sinθ +f sin(Ωt) (6)
where
p=ωRD
2H, q =ωR
2HPm, r =ωRVG
2HXG
,
f=ωRVGVB1
2HXG
cos (Ωt+ϕv).
Finding the derivative to apply the Hamilton’s
Principle,
L(θ, ˙
θ, t) = 1
2˙
θ2+q−p˙
θ−cos θ+fsin (Ωt) sin θ
Calculating partial derivatives and applying
to the Hamilton’s Principle and substituting to
the Euler-Lagrange equating derives the following
equation,
¨
θ−sinθ +fsin(Ωt)cosθ = 0 (7)
Plotting equation (7) with angle against time
for the case of Hamilton’s Principle and then com-
paring this with Method of Strained Parameters
and Floquet Theory for further analysis within the
context of quasiperiodicity.
The graph, Figure 1, depicts the progressive de-
cline in stability over time, resulting in a state of
instability characterized by quasiperiodicity. Over
time, the system’s movement becomes more ir-
regular and unpredictable, demonstrating the sys-
tem’s sensitivity to the quasiperiodicity for all the
methods considered. The observed behaviour indi-
cates that the intricate interaction between exter-
nal forces and the inherent dynamics of the system
might result in chaotic motion.
Basins of Attractions for the case of
Quasiperiodicity
The phenomenon of quasiperiodicity is of ut-
most importance in understanding the stability
characteristics of a nonlinear system. Therefore,
Fig. 1: Simulation of the Swing Equation with
the Hamilton’s Principle comparing with Method
of Strained Parameters and Floquet Theory under
quasiperiodicity, [8], [33].
it is essential to conduct a thorough examination
of the basins of attraction associated with the pri-
mary resonance in order to acquire a thorough un-
derstanding of the system. The concept of basins
of attraction is utilised in order to delineate the
stable and unstable regions within a system, facil-
itating the analysis of modications made to the
said system, [38]. The plots illustrate the alter-
ations in the basins of attraction as variables are
modied. When drawing inferences from these
graphs, it is important to take into account the
boundary conditions as well, [39].
Studies of the basins of attraction of quasiperi-
odicity have revealed signicant ndings regarding
the stability characteristics of power systems. The
impact of parameter uctuations, including sys-
tem damping, excitation levels, and control gains,
on the conguration and amplitude of the basins
of attraction has been investigated, [39], [40]. Fur-
thermore, scholarly investigations have mostly fo-
cused on the identication of crucial borders that
demarcate stable and unstable regions within the
state space, [41], [42].
2.2 Numerical Analysis
Graphical Representation
The equations (1), (2), and (3) were solved us-
ing the fourth-order Runge-Kutta method in Mat-
lab. The primary aim was to examine the inu-
ence of modifying the excitation frequency Ωon
the occurrence of quasiperiodicity with irrational
values, [7], [8].
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Fig. 2: Phase portrait, frequency-domain plot and
Poincaré map when Ω= 2πrads−1.
Fig. 3: Phase portrait, frequency-domain plot and
Poincaré map when Ω=πrads−1.
Fig. 4: Phase portrait, frequency-domain plot and
Poincaré map when Ω= 2π/3 rads−1.
Fig. 5: Phase portrait, frequency-domain plot and
Poincaré map when Ω=π/2 rads−1.
Fig. 6: Phase portrait, frequency-domain plot and
Poincaré map when Ω= 2π/8 rads−1.
Figure 2, Figure 3, Figure 4, Figure 5, and
Figure 6 were produced to illustrate the changes
in excitation frequency in the swing equation
(1). These gures include the phase portraits,
frequency-domain plots, and Poincaré maps. The
variations in excitation frequency are also docu-
mented in [2]. As the system undergoes a decrease,
its stability decreases and it transitions towards a
state of chaos through a cascading process. Every
plot illustrates the progressive decline of coordina-
tion within the system.
As the parameter Ωdecreases systematically, it
becomes apparent that the graphs experience dy-
namic alterations. At approximately 2π/8 rads−1,
a chaotic attractor is observed, as shown in Figure
6.
The torus phenomena is depicted by Figure 2,
Figure 3, Figure 4 and Figure 5, signifying that
the movement of the synchronous generator is not
strictly periodic, but rather exhibits a more intri-
cate, torus-shaped conguration in phase space.
Golden Ratio Number
The golden ratio (1+√5
2), has captivated sci-
entists and mathematicians due to its visually ap-
pealing qualities and distinctive mathematical im-
portance, [43]. Using the golden ratio as the angu-
lar frequency (Ω) in the swing equation allows for
an intriguing investigation into the dynamics of
quasiperiodicity. The golden ratio is an irrational
number. When used as the driving frequency,
it creates a non-commensurate relationship with
other system characteristics. This might poten-
tially result in complex quasiperiodic motion. By
introducing the golden ratio, the system’s reaction
is anticipated to display captivating patterns and
frequencies, demonstrating the intrinsic intricacy
of quasiperiodic behaviour.
Figure 7 visually represent the inuence of the
golden ratio on the swing equation. The phase
portrait oers a perceptive representation of the
system’s trajectory as it evolves over time, show-
casing the changes in the system’s tilt and an-
gular velocity. It also exemplies the torus phe-
nomena for the case considered. Poincaré maps
provide a concise depiction of the system’s be-
haviour by showing where the trajectory intersects
with a particular plane. Moreover, frequency do-
main charts facilitate the examination of the spec-
tral characteristics of the system, revealing promi-
nent frequencies and possible resonances. Utilis-
ing the golden ratio in these analyses oers a dis-
tinct perspective to see and comprehend the com-
plex quasiperiodic patterns that arise in the swing
equation, providing a fascinating examination of
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the system’s dynamic behaviour.
Fig. 7: Phase portrait, frequency-domain plot and
Poincaré map when Ω=1+√5
2rads−1.
Bifurcation and Lyapunov Exponents
Figure 8 depicts the bifurcation diagram as-
sociated with the phenomenon of quasiperiodic-
ity. The construction technique entailed evalu-
ating the swing equation for a specic angular
frequency value of Ω=π/2 rads−1, and sub-
sequently performing numerical time integration
using the widely recognised fourth order Runge-
Kutta method. The forcing parameter, repre-
sented by the symbol r, is systematically incre-
mented, and the time integration process is sub-
sequently prolonged. The obtained data is sub-
sequently utilised to generate a graph illustrating
the highest magnitude of the oscillatory solution
in relation to r, as referenced in [7].
r=VGVB
XG
sin(θ−θB)
Fig. 8: Bifurcation diagram for the case of
Quasiperiodicity where Ω=π/2 rads−1.
Figure 8 also depicts the occurrence of the ini-
tial period doubling right before attaining a value
of r that is equal to 1.085 in the case of quasiperi-
odicity. Furthermore, it is evident that at approx-
imately r = 1.94, the initial occurrence of period
doubling in a sequence of successive period dou-
blings is seen, nally leading to the formation of
chaotic behaviour. The results of this numerical
research demonstrate that an augmentation in the
value of parameter r results in a gradual deterio-
ration of synchronisation in the swing equation,
specically in relation to quasiperiodicity.
The Lyapunov exponent as shown in Figure
9, generally demonstrates positive values in the
region around the values of r = 1.9. The de-
picted behaviour is being examined. In this sce-
nario, two points that are initially very near to-
gether, separated by an extremely small distance,
gradually move apart from each other over time.
The quantication of this divergence is accom-
plished through the utilisation of Lyapunov expo-
nents. The behaviour observed in the bifurcation
diagram provides additional conrmation of the
previously reported phenomenon. More precisely,
when the value of r reaches a particular threshold,
a sequence of period doubling occurs, ultimately
resulting in chaotic behaviour. Hence, it may be
inferred that the presence of a positive Lyapunov
exponent signies the existence of a chaotic attrac-
tor.
Fig. 9: Lyapunov exponents as r is varied.
Basins of attractions for the case of
Quasiperiodicity
Figure 10, Figure 11, Figure 12 and Figure 13
illustrate the basins of attraction for the case of
quasiperiodicity. These images depict the uctu-
ations in the variables VB1and θB1while keep-
ing the value of Ωconstant at π/2rads−1. The
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system’s stability is susceptible to alteration as
the variable is increased. The stable portions of
the system are shown by the presence of red and
green colours, while the other colours represent
the unstable regions. As the independent variable
increases, the system experiences a stage of degra-
dation marked by the existence of unstable zones.
Therefore, it is essential to thoroughly investigate
the inuence of additional factors in the system to
ensure the validity and strength of the conclusions
in this specic study.
Fig. 10: Basins of attractions when VB1is 0.051
rad and 0.062 rad respectively for Ω=π/2rads−1.
Fig. 11: Basins of attractions when VB1is 0.071
rad and 0.151 rad respectively for Ω=π/2rads−1.
Fig. 12: Basins of attractions when θB1is 0.101
rad and 0.05 rad respectively for Ω=π/2rads−1.
Stability Reduction
The provided graph, Figure 14, examines two
instances of stability degradation in the swing
equation using the Lyapunov exponents. In the
rst scenario, the estimated value of Ωis to be
Fig. 13: Basins of attractions when θB1is 0.07
rad and 0.181 rad respectively for Ω=π/2rads−1.
Fig. 14: Comparing the reduction in stability re-
gion for Primary Resonance (Ω= 8.27 rads−1.)
and the case of Quasiperiodicity (Ω= 8.27 + 2π/8
rads−1).
8.27 rads−1, which represents the primary reso-
nance frequency. Under these circumstances, the
system experiences a driving force that matches its
resonant frequency, leading to a steady decrease in
its amplitude. The second scenario examines the
value of Ωas 8.27 + 2π
8rads−1, representing a
quasiperiodicity frequency which is an irrational
number closer to the value of primary resonance.
This frequency is chosen in particular to illustrate
an example of the behaviour of the dynamical sys-
tem and its stability behaviour for quasi periodic-
ity. In this scenario, the frequency at which the
driving force is delivered causes the machine to
exhibit a quasiperiodic response.
Lyapunov exponents can be used to analyse pri-
mary and quasiperiodic processes. Changing the
parameter VB1in the swing equation may cause
the Lyapunov exponents to vary, indicating transi-
tions between stability and instability. The graph
illustrates how an increase in a parameter leads
to a loss in stability by showing the change in the
number of Lyapunov exponents for each case.
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The Lyapunov exponents provide useful in-
sights into the stability of a dynamical system,
particularly the rate at which neighbouring tra-
jectories either converge or diverge over time,
[44]. Within the framework of the swing equa-
tion, which describes the movement of a system
aected by an external force represented by VB1,
Lyapunov exponents can be employed to evalu-
ate the eect of increasing VB1on the stability of
the system. Increasing VB1allows for the calcu-
lation of Lyapunov exponents at each iteration.
Decreased stability is indicated by a transition
from negative exponents to less negative or pos-
itive exponents. Negative exponents suggest sta-
ble trajectories, showing that disturbances in the
system decrease gradually over time, [10], [45]. If
the Lyapunov exponents decline or change sign as
VB1grows, it indicates a deterioration in stabil-
ity. This suggests a greater vulnerability to initial
conditions and a more unpredictable behaviour of
the dynamical system.
The swing equation’s stability as a function of
changing the parameter VB1is accurately mea-
sured using the Lyapunov exponents as shown in
this study. Changes in the exponents depict a
deterioration in stability as VB1increases, hence
providing a thorough understanding of the case of
stability of the swing equation.
For power systems to be durable and reliable, it
is essential to study the swing equation’s stability
reduction. Through an awareness of the elements
that can cause the swing equation to become less
stable, such as abrupt changes in load or network
disruptions, control techniques can be developed
by engineers and operators to stop cascade fail-
ures. Furthermore, the signicance of preserving
stability in light of the growing integration of re-
newable energy sources and the expanding com-
plexity of contemporary power systems cannot be
emphasised. Stability reduction research advances
the understanding of system dynamics and aids in
the creation of sophisticated control strategies and
grid management procedures, all of which help to
maintain the dependability power infrastructure.
3 Discussion
This work primarily aims to thoroughly analyse
the dynamic characteristics displayed by the swing
equation when control parameters are varied, with
a specic emphasis on the complex phenomena
of quasiperiodicity. This investigation involves a
comparison of analytical methods, specically per-
turbation techniques, with numerical simulations
in order to verify the precision of the perturbed so-
lutions and the corresponding basins of attraction.
This study attempts to gain a thorough under-
standing of the quasiperiodic dynamics and their
impact on power system stability by utilising the
analytical tool, the Hamilton’s Principle.
Analytical tools are crucial in analysing the res-
onances that are naturally present in the swing
equation. By employing mathematical modeling
and computations, these methods provide accu-
rate insights based on reduced assumptions. Nev-
ertheless, their eectiveness may decrease when
faced with the intricacy of actual power networks.
The integration of numerical and computational
tool in the Hamilton’s Principle overcomes this
constraint, enabling a more detailed investiga-
tion of the system’s reaction to various situations.
Graphical representations, obtained from numeri-
cal calculations, provide a visual depiction of how
the swing equation behaves under dierent param-
eter values and forcing frequencies for the case
of quasiperiodicity. These visual representations
enhance the understanding gained from analyt-
ical approaches. This comprehensive methodol-
ogy enables power system engineers to make well-
informed judgments, assuring the dependable op-
eration of the grid in the face of quasiperiodic dy-
namics.
Understanding the expected reactions of
the system, especially when quasiperiodicity is
present, is of utmost relevance in real-life situa-
tions. Load uctuations, which frequently hap-
pen in power systems, provide a relevant illustra-
tion. The information obtained from these situa-
tions is crucial for the management of the power
system, assisting in the maintenance of system
stability and dependability. Moreover, the nd-
ings obtained from this research have implications
for the development and evaluation of control sys-
tems, namely in the areas of autonomous genera-
tion control and load frequency management. Un-
derstanding quasiperiodic dynamics in the swing
equation is crucial for eectively reducing the like-
lihood of blackouts and the severe repercussions
they can have. This information has practical ap-
plications and highlights the importance of under-
standing this topic.
4 Conclusion
In conclusion, this extensive study utilised a wide
range of analytical methods, including bifurcation
diagrams, Lyapunov exponents, phase portraits,
frequency domain plots, and Poincaré maps, to
thoroughly examine the complex dynamics of the
swing equation in the domain of quasiperiodicity.
The occurrence of intricate behaviours, such as
the repetition of periods in sequences of bifurca-
tions, suggests an upcoming shift towards turbu-
lence, which could pose risks to power systems and
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.28
Anastasia Sofroniou, Bhairavi Premnath
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Volume 18, 2023
create operational diculties.
The research ndings highlight the importance
of chaos induction caused by the collapse of
quasiperiodic torus structures and the existence of
intermittency in the swing equation. Period dou-
bling, a widely recognised occurrence, exemplies
the system’s vulnerability to quasiperiodic transi-
tions. The study focuses on analysing the eects of
parameter variations on the behaviour of the sys-
tem, providing insights into the changes observed
before and after chaotic behaviour occurs.
This study expands upon the recent academic
research conducted by the same group of re-
searchers, further developing their previous nd-
ings. It aims to enhance existing approaches by
oering a more profound understanding of the
underlying mathematics, rather than replacing
them. This research contributes to the improve-
ment of control strategies and preventive measures
for power systems by enhancing the understand-
ing of fundamental principles and system stability,
with a specic focus on quasiperiodicity. It aims
to mitigate the chaotic eects caused by the phe-
nomena of quasiperiodicity beneting power sys-
tem engineers and researchers.
The ndings obtained from this work provide a
clear grasp of how the swing equation behaves in
the presence of quasiperiodic conditions, thereby
making signicant contributions to the compre-
hension of system stability. These discoveries
could lead to improvements in the creation of
power infrastructures that are more durable and
safe, especially as power systems face more intri-
cate issues throughout expansion.
In the future, scholars might look at how to
include quasiperiodic circumstances in the frame-
work of swing equations. This may provide impor-
tant new information about the long-term stability
and exibility of electricity systems. These initia-
tives have the potential to deepen the knowledge
of these intricate nonlinear systems and produce
improvements that increase their robustness.
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Contribution of individual
authors to the creation of a
scientic article (ghostwriting
policy)
All authors contributed to the development of this
paper. Conceptualisation, Anastasia Sofroniou;
Methodology, Anastasia Sofroniou and Bhairavi
Premnath; Analytical and Numerical Analysis
Bhairavi Premnath; Validation, Anastasia Sofro-
niou and Bhairavi Premnath; Writing-original
draft preparation, Bhairavi Premnath and Anas-
tasia Sofroniou; Writing-review and editing, Anas-
tasia Sofroniou and Bhairavi Premnath; Supervi-
sor, Anastasia Sofroniou.
Sources of Funding for Research
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or Scientic Article Itself
No funding was received for conducting this study.
Conict of Interest
The authors have no conicts of interest to declare
that are relevant to the content of this article.
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.28
Anastasia Sofroniou, Bhairavi Premnath
E-ISSN: 2224-3429
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