Abstract: Motivated by the nonlinear dynamics of mathematical models encountered in power systems,
an investigation into the dynamical behaviour of the swing equation is carried out. This paper examines
analytically and numerically the development of oscillatory periodic solutions, whereby increases of the
control parameter, lead to a cascade of period doubling bifurcations, before eventually loss in stability
is exhibited and effective forerunners to chaos revealed. Gaining an understanding on the dynamical
behaviour of the system can help to produce a deeper insight of the bifurcations entailed, with the
appearance of the triggered sequence of the first period doubling’s acting as precursors of imminent
danger and difficult operations of a practical system.
Key-Words: nonlinear dynamics, swing equation, chaos
Received: September 27, 2022. Revised: November 16, 2022. Accepted: December 15, 2022. Published: January 25, 2023.
1 Introduction
Stability in a power system is closely tied to the
concept of disturbances, which are sudden or se-
quential changes to the system’s parameters or
operating quantities. Even a small disturbance
can have an interesting and rich effect in terms of
the dynamics of a system. Linearising the equa-
tions that represent a system [1], undertaking
eigenvalue and frequency response methods can
be employed to study the stability of the system
[2, 3].
In this study, a single degree of freedom system
is considered that will allow for the study of non-
linear dynamics to be examined and to acknowl-
edge even chaotic attractors. This study focuses
on the nonlinear aspect of solving a system us-
ing methods such as perturbation techniques and
nonlinear methods. Initially in this study, an infi-
nite busbar is represented with the assumption of
constant voltage and frequency. A busbar system
is a metallic strip/bar used for high-current power
distribution. It is usually used in panel boards,
switchgear, and home circuits. In general, the
busbars are uninsulated and receive support from
the air by insulated pillars, allowing for enough
cooling for the conductors [4]. If a classical repre-
sentation is considered, that is with a fixed volt-
age behind a transient reactance, then the busbar
system is reduced to a second-order differential
equation but with constant coefficients. As this
resulting equation does not offer much useful or
novel information about the response of the sys-
tem, the analogous swing equation is considered
here within, which includes parametric and ex-
ternal excitations allowing for the techniques of
perturbation theory to be employed under this
new formulation of the extended busbar system
[5, 6].
This newly formulated swing equation will be
analysed analytically and numerically to obtain a
better understanding of the stability of the model.
1.1 Brief Literature Review
A power system is stable at a particular op-
erating condition when it is able to maintain
a steady state. When the system experiences
a small disturbance, it is able to return to its
pre-disturbance operating conditions or achieve a
steady state once again. However, in the event of
a large disturbance, the equations that describe
the system’s behavior can no longer be linearised,
and it becomes necessary to use numerical sim-
ulation techniques based on geometric methods
to analyze the system’s behavior, which is now
considered to be a part of nonlinear dynamics
[6, 7]. The focus of this paper is the nonlinear as-
pect of systems which can be addressed through
various dynamical and perturbation techniques
[8, 9]. Researchers have studied the swing equa-
tion which showed the rotor of the machine’s mo-
tion [6, 7, 10]. Although power systems have
been studied for quite some time now, the growth
An Insight into the Dynamical Behaviour of the Swing Equation
ANASTASIA SOFRONIOU, BHAIRAVI PREMNATH, KEVIN JAGADISSEN MUNISAMI
School of Computing and Engineering
University of West London
St. Mary’s Road, W5 5RF
UNITED KINGDOM
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
70
Volume 22, 2023
of the topic is tremendous. The power system
in electric applications has seen ongoing develop-
ment in many areas [11]. With this growth, the
conservation of energy and renewing the existing
energy have been under the radar by many insti-
tutions. To help with the environmental concerns
the power systems must be studied further, and
new techniques should be introduced [12].
The swing equation which is studied initially
in this research work will play a vital part in
the analysis of the dynamics of a power system
[13]. It does exhibit similar characteristics as
other power systems, but it is imperative to anal-
yse it first in detail for a better understanding of
the concepts. Recent research has found that the
generalised form of the swing equation also helps
with understanding transient stability in power-
electronic power systems [14]. During any slight
disturbance, the rotor of the machine will show
some motion with respect to the synchronously
rotating air gap. This in turn starts a relative mo-
tion allowing for the swing equation to describe
and model this relative motion [15, 16]. Although
Tamura et al. [10] initiated the quasi-infinite bus-
bar which is formulated in phase and magnitude,
Hamdan and Nayfeh [3, 8] improved the idea to
have quadratic and cubic nonlinearities. This
helps in applying techniques such as perturba-
tion analysis to the single-machine-quasi-infinite
busbar system.
As it is well known, bifurcation occurs when
a small change to a parameter value of a system
causes a change in the behaviour whether this is
a topological or qualitative change occurring in
both discrete and continuous systems. A bifurca-
tion has significant effects on power systems, in-
cluding oscillation and voltage collapse [17, 18].
Eigenvalue analysis may be further utilised to
consider stability and to determine the nature
of the system [19]. Bifurcations can be studied
using both mathematical models and computer
simulations involving oscillators [20]. Some au-
thors have pointed out the limitations of using
physical oscillators for this purpose and have sug-
gested computer algorithms as an alternative for
more accurate and efficient analysis of bifurca-
tions. Matlab software, specifically the packages
MATCONT and CLMATCONT, can be used to
analyse dynamical systems with bifurcations. In
a study [21, 22], the unique nature of a para-
metrically pressurized system was characterized
using a pinched cylinder, and the mechanism of
symmetry-breaking pitchfork bifurcation was ex-
amined. It has been shown that the stability and
behavior of the swing equation can be affected by
various factors, and that increasing the time de-
lay can cause limit cycle branches to move and
combine through bifurcations [23].
In [24, 25] bifurcation analysis is employed
to estimate the boundary of the chaotic precur-
sors of a parametrically excited pendulum sys-
tem, considering the effect of a bias term inclusion
in the model that breaks the symmetry of the sys-
tem, gaining deeper insights into bifurcations en-
tailed with the purpose of growing a higher reali-
sation for any unique problem. The authors also
explain that the easy uneven equation of move-
ment proposed in the study ends in diverse non-
linear phenomena, inclusive of cascades of period
doubling bifurcations, which had been tested and
compared with different models.
2 Methodology
2.1 Analytical Work
The swing equation studied here depicts the mo-
tion of rotor of machine as reproduced below as
Figure 1 [26].
Fig. 1: Swing equation describing the motion of
the rotor of the machine. Figure reproduced from
[26].
Considering the damping term, the swing
equation that describes the motion of the rotor
of the machine employed in this study is as fol-
lows [6, 7, 10],
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 P ower,
VG=V oltage of machine,
XG=T ransient Reactance,
VB=V oltage of bus,
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
71
Volume 22, 2023
θB=phase of bus.
VB1and θB1magnitudes assumed to be small.
Mathematical analysis is performed on this
equation for further investigation whereby alge-
braic techniques, Taylor expansion and substitu-
tion are undertaken so as to obtain a final equa-
tion that will be used for the perturbation anal-
ysis.
Allowing consideration for the transforma-
tions,
θ−θB=δ0+η(4)
δ0=θ0−θB0(5)
η= ∆θ−θB1cos(ωt +ϕ0) (6)
equation (4) becomes,
sin (θ−θB) = sin (δ0+η) (7)
with first differential and second differential
equations of (4) being substituted into equations
(1), (2) and (3) to derive the modified swing equa-
tion with excitation to:
d2η
dt2+ωRD
2H
dη
dt +Kη =α2η2+α3η3+
G1ηcos (Ωt + ϕv) + G2η2cos (Ωt + ϕv) +
G3η3cos (Ωt + ϕv) + Q1cos (Ωt + ϕθ) +
Q2sin (Ωt + ϕθ) + Q3cos (Ωt + ϕv) (8)
and with,
α2=1
2Ktan δ0, α3=1
6K,
G1=−VB1
VB0
K, G2=−VB1
2VB0
Ktan δ0,
G3=−VB1
6VB0
K,
Q1= Ω2θB1, Q2=ΩDωRθB1
2H,
Q3=−VB1
VB0
Ktan δ0,
K=VGVB0ωRcos δ0
2HXG
,
here
Q cos (Ωt + ϕe) = Q1cos (Ωt + ϕθ)+
Q2sin (Ωt + ϕθ) + Q3cos (Ωt + ϕv),
equation (8) reduces to,
d2η
dt2+ωRD
2H
dη
dt +Kη =α2η2+α3η3+
G1ηcos (Ωt + ϕv) + G2η2cos (Ωt + ϕv) +
G3η3cos (Ωt + ϕv) + Qcos (Ωt + ϕe).
(8a)
Perturbation Analysis
Initially, the focus of the analysis is on primary
resonance. To study this, a technique called mul-
tiple scales is used to find a uniform solution for
equation (8a). A small, dimensionless parameter
εis introduced to account for the effects of damp-
ing, nonlinearities, and the excitation frequency,
which occur in a specific order.
Letting
η=O(ε),ωRD
2H=Oε2
and
VB1=O(ε3)and θB1=O(ε3),
then the final equation from the swing equa-
tion derivation above has the following coeffi-
cients,
G1=ε3g1, G2=ε3g2, G3=ε3g3
Q=ε3q.
Also, considering the equation with detuning
parameter σ.
ω2
0= Ω2+E2σ
to allow for the derived final swing equation
(8a) to be re-written as,
¨η+ 2ε2µ˙η+ (Ω2+E2σ)η=α2η2+α3η3+
ε3g1ηcos (Ωt + ϕv) + ε3g2η2cos (Ωt + ϕv) +
ε3g3η3cos (Ωt + ϕv)+ ε3qcos (Ωt + ϕe) (9)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
72
Volume 22, 2023
The solution to this above equation is of the
form:
η(t;ε) = εη1(T0, T1, T2) + ε2η2(T0, T1, T2) +
ε3η3(T0, T1, T2) + ....... (10)
where T0is a fast scale describing motions of
frequencies and T1,T2are slow scales describing
amplitude variation [26].
The first derivative of this equation will be,
d
dt =D0+εD1+ε2D2+.... (11)
The second derivative of the equation is,
d2
dt2=D2
O+ 20D1+ε2(2D0D2+D2
1) +...... (12)
where
Dn=∂
∂Tn
.
Equation (10) can be rewritten as;
η=εη1+ε2η2+ε3η3+....... (13)
Finding the first derivative with respect to t
for equation (13) and substituting equation (11)
gives,
η(D0+εD1+ε2D2+......) = εη1(D0+
εD1+ε2D2+......) + ε2η2(D0+εD1+
ε2D2+......) + ε3η3+ (D0+εD1+ε2D2+
......) (11a)
Differentiating for the second derivative with
respect to tfor equation (13) and substituting
equation (12) to obtain,
η(D2
0+ 2εD0D1+ε2(2D0D2+D2
1) +
....) = εη1(D2
0+ 2εD0D1+ε2(2D0D2+D2
1) +
....) + ε2η2(D2
0+ 2εD0D1+ε2(2D0D2+D2
1) +
....) + ε3η3(D2
0+ 2εD0D1+ε2(2D0D2+D2
1) +
....) (11b)
Substituting equations (11a), (11b) and (13)
into equation (9) and comparing coefficients of ε
gives,
ε1/:η1D2
0+η1Ω2= 0 (14)
ε2/:η1D2
0+η2Ω2+ 2D0D1η1=α2η2
1(15)
ε3/:D2
0η3+ 2D0D1η2+ (D2
1+ 2D0D2)η1+
20η1+ Ω2η3+ση1= 2α2η1η2+α3η3
1+
qcos(Ωt+ ϕe) (16)
From equations (14), (15) and (16) it can be
seen that the parametric terms do not have key
effects on the system. Hence only the external
forcing term remains [26].
The solution to equation (14) is of the form:
η1=A(T1, T2)eiΩT0+¯
A(T1, T2)e−iΩT0(17)
where A is an undetermined function. Given
that
Dn=∂
∂Tn
, D0=∂
∂T0
by integration,
T0=1
D0
.
Substituting equation (17) into (15),
η2D2
0+η2Ω2=−2D0D1(A(T1, T2)eiΩT0+
¯
A(T1, T2)e−iΩT0+α2(A(T1, T2)eiΩT0+
¯
A(T1, T2)e−iΩ T0)2
and expanding the brackets,
η2D2
0+η2Ω2=−2D0D1A(T1, T2)eiΩT0
- 2D0D1¯
A(T1, T2)e−iΩT0+α2((A2e2iΩT0+
¯
A2e−2iΩ T0+ 2A¯
A).
Due to D0=∂
∂T0and
∂(2D0D1A eiΩT0)
∂T0
= 2iΩD1A eiΩT0
∂(2D0D1¯
A e−iΩT0)
∂T0
=−2iΩD1¯
Ae−iΩT0
substituting into the equation and rearranging
leads to,
η2D2
0+η2Ω2=−2iΩD1A eiΩT0+
α2(A2e2iΩT0+¯
A2e−2iΩT0) + ¯c(18)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
73
Volume 22, 2023
where ¯cis the complex conjugate.In this equa-
tion D1A= 0, to avoid secular terms η2and
hence A=A(T2) so that replacing equation (14)
into (18) and simplifying,
η2=−α2(A2e2iΩT0)
3Ω2−α2(¯
A2e−2iΩT0)
3Ω2+2α2A¯
A
Ω2(19)
which is also echoed in [26].
Replacing equations (17) and (19) into equa-
tion (16),
D2
0η3+ 2D0D1(−α2(A2e2iΩT0)
3Ω2−
α2(¯
A2e−2iΩT0)
3Ω2+2α2A¯
A
Ω2) + (D2
1+
2D0D2) (A eiΩT0+¯
Ae−iΩT0) + 2µD0(AeiΩT0+
¯
A e−iΩT0) + Ω2η3
+σ(AeiΩT0+¯
Ae−iΩT0)=2α2(A eiΩT0+
¯
Ae−iΩT0)(−α2(A2e2iΩT0)
3Ω2−α2(¯
A2e−2iΩT0)
3Ω2+
2α2A¯
A
Ω2)
+α3(AeiΩT0+¯
A e−iΩT0)3+qcos(Ωt + ϕe)
and importing the cubic bracket separately of,
α3(AeiΩT0+¯
A e−iΩT0)3=α3A3e3iΩT0+
3α3A2¯
AeiΩT0+ 3α3A¯
A2¯
AeiΩT0+α3¯
A3e3iΩT0,
eliminating terms that lead to secular terms
and D1A= 0 then,
2iµΩ (A′+µA) + −1
2geiϕ + 8αeA2¯
A= 0 (20)
where
αe=−3
8α3−5α2
2
12Ω2.
Expressing A in polar form,
A=1
2ae−i(β+ϕe)(21)
and substituting equation (21) into equation
(20) gives,
2iµΩ (1
2a′e−i(β+ϕe)+µ(1
2ae−i(β+ϕe))) +
σ(1
2ae−i(β+ϕe))−1
2geiϕ +
8αe(1
2ae−i(β+ϕe))2(1
2a′e−i(β+ϕe)) = 0.
Separating real and imaginary parts,
Ω(a′+µa) + 1
2qsin β= 0 (22)
−Ωaβ′+αea3−1
2qcos β+1
2= 0.(23)
Equation (21) can also be written in the form:
A=1
2a cos(β+ϕe)
Substituting A and its conjugate into equation
(17) leads to:
η1=a cos(2Ωt+β+ϕe)
Similarly replacing into equation (19) gives:
η2=α2a2
2Ω2−α2a2
6Ω2cos (2Ωt+ 2β+ 2ϕe)
Substituting the above to derivations for η1
and η2in equation (10) to obtain the second ap-
proximation,
η=εa cos(Ωt+β+ϕe) + ε2a2α2
6Ω2[3
- cos(2Ωt+ 2β+ 2ϕe)] +..... (24)
Setting ε= 1 and letting ” a” be the pertur-
bation parameter, using equation (24), equation
(6) maybe rewritten as,
∆θ=θB1cos(Ωt+ϕθ) + acos(Ωt+β+ϕe) +
a2α2
6Ω2(3 −cos(2Ωt+ 2β+ 2ϕe)) + ....
with a2α2
2Ω2defined as the drift term, which be-
cause of its quadratic nonlinearity the oscillatory
motion is not centered as seen also in [26].
To understand the character of equations (22)
and (23), fixed points are found in align with a′=
β′= 0 to reduce to:
µa =−qsin β
2Ω (24a)
aσ
2Ω +αea3
Ω=qcos β
2Ω (24b)
Squaring and adding equations (24a) and
(24b) will give,
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
74
Volume 22, 2023
µ2+ ( σ
2Ω +αea2
Ω)2=q2
4Ω2a2(25)
which is an implicit equation for α(amplitude)
as a function of σ(the tuning parameter).
In order to compare the analytical results with
the numerical simulations for the case of pri-
mary resonance, the following figure, Figure 2,
presents phase portraits and time histories when
Ω = 8.61 rad/sec. Runge-Kutta and Newton
Raphson method were both employed for the sim-
ulation of the perturbation analysis and both
compared with its numerical counterpart, to con-
clude that the Newton Raphson algorithm gives
a better approximation to the numerical solution.
The calculated numerical error of the Runge-
Kutta method versus the Newton Raphson tech-
nique compared to the actual simulation error
was 0.0884 and 0.0747 respectively, exemplifying
that the Newton Raphson method is a more ap-
propriate fit due to the smaller error value.
Fig. 2: Perturbed solution employing Runge-
Kutta and Neton Raphson algorithms in com-
parison to numerical simulations for the case of
primary resonance in the phase plane and time
history for Ω = 8.61 rad/sec.
2.2 Numerical Analysis
Graphical Representation
The free undamped oscillation of the machine
is given by,
d2θ
dt2−ωRPm
2H+ωRVGVB0
2HXG
sin (θ−θB0) = 0.
When the damping term is introduced into
the system, the phase plane alters diagrammat-
ically. Matlab was employed to numerically de-
termine the phase space plots for the undamped
and damped oscillator.
Figures 3 and 4 pictorially show the phase por-
traits for a variation of c values, c being the value
of the solution of the differential equation. The
Fig. 3: Phase portrait: undamped oscillation for
the different stated c values.
damped oscillation is given by the following equa-
tion, with D representing the damping term,
d2θ
dt2+ωRD
2H
dθ
dt −ωRPm
2H+ωRVGVB0
2HXG
sin (θ−θB0)
= 0
Fig. 4: Phase portrait: damped oscillation for
the different stated c values.
The phase portraits of free oscillations show
saddle point separating its closed orbits from the
trajectories which go off to infinity [6, 26, 27].
The equations (1), (2) and (3) were configured
and solved using the fourth-order Runge-Kutta
method in Matlab, focusing on the effect of vary-
ing the excitation frequency Ω.
Fig. 5: Phase portrait, frequency-domain plot
and Poincar´e map when Ω = 8.61 rad/sec.
Figures 5, 6, 7, 8 and 9 were obtained by
plotting the phase portraits, frequency-domain
plots, and Poincar´e maps when this excitation
frequency is varied in the swing equation (1). As
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
75
Volume 22, 2023
Fig. 6: Phase portrait, frequency-domain plot
and Poincar´e map when Ω = 8.43 rad/sec.
Fig. 7: Phase portrait, frequency-domain plot
and Poincar´e map when Ω = 8.282 rad/sec.
Fig. 8: Phase portrait, frequency-domain plot
and Poincar´e map when Ω = 8.275 rad/sec.
Fig. 9: Phase portrait (loss of synchronism) when
Ω = 8.2601 rad/sec.
it is decreased the system begins to lose stability
and cascades towards chaos. Each plot represents
the different period doubling and how the system
loses its synchronism. Figure 5 shows that there
exists only one steady-state attractor when there
is a large Ω, Ω = 8.61 rad/sec. The phase orbit
has a closed form and is a period-one attractor.
This can be verified using the frequency-domain
plot and the Poincar´e map.
As the value of Ω is decreased it can be ob-
served that the graphs undergo dynamical trans-
formations including period-doubling solutions
and eventually as Ω is decreased to further around
Ω = 8.2601 rad/sec a chaotic attractor is exhib-
ited as exemplified in Figure 9.
The bifurcation diagram presented as Figure
10, was constructed by solving the swing equa-
tion for a specific value of Ω = 8.27 rad/sec and
by numerical time integration using the classical
fourth order Runge-Kutta algorithm. The forcing
r value is incremented slightly and time integra-
tion continues plotting the maximum amplitude
of the oscillatory solution versus r,
r=VGVB
XG
sin (θ−θB).
Figure 10 indicates the initial period doubling
occurrence just before r = 0.9, also justified by
the Poincar´e maps of Figure 11 and at around r
approximately 2.36, the first period doubling in a
sequence of period doubles is exhibited leading to
chaotic behaviour. This numerical analysis shows
that the swing equation moves towards loss of
synchronisation as the value of r is increased.
Fig. 10: Bifurcation diagram when r value is var-
ied and constant Ω = 8.27 rad/sec.
The corresponding Poincar´e maps are plotted
as shown below, Figure 11. They clearly depict
the points where period doubling occurs and how
as r is increased the phenomenon of chaos is ver-
ified.
Fig. 11: Poincar´e maps for the different r values.
It is observed that at approximately r > 2.4,
the chaotic region has commenced where the Lya-
punov exponent generally takes positive values.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
76
Volume 22, 2023
This behaviour is depicted and presented as Fig-
ure 12, where it is the case when two nearby
points, initially separated by an infinitesimal dis-
tance, typically diverge from each other over time
and this is quantitatively measured by the Lya-
punov exponents. The bifurcation diagram of
Figure 10, also verifies this behaviour, where at
approximately the same value of r, the cascade
of period doubling sequence leads to chaos such
that is suffices to say that a chaotic attractor can
be identified by a positive Lyapunov exponent.
Fig. 12: Lyapunov exponents as r is varied.
3 Discussion and Conclusion
This paper highlights the dynamical behaviour of
the swing equation as control parameters are var-
ied. The entailed work provides analytical meth-
ods, specifically perturbation techniques which
are compared with the numerical simulation to
verify the validity of the perturbed solution for
the case of primary resonance.
The swing equation is used to predict the be-
havior of the system under various conditions,
such as changes in load. This information is used
by power system operators to maintain the stabil-
ity and reliability of the system. It may be used
in the design and analysis of control systems for
power systems, such as automatic generation con-
trol and load frequency control, preventing black-
outs and even more so, catastrophic effects.
The numerical analysis incorporating a nu-
merically constructed bifurcation diagram, Lya-
punov exponents, phase portrait, frequency do-
main plots and Poincar´e maps, all confirm that
an appearance of the first period doubling in its
sequence triggers chaos, and should be regarded
as a precursor of the imminent danger and diffi-
cult operations of a practical system. It is im-
portant to note that the bifurcation diagrams
do examine pre-chaotic or post-chaotic changes
in a dynamical system under different parame-
ter variations and whilst period doubling is the
most recognised scenario for chaotic behaviour,
there are also other scenarios to reach this phe-
nomenon, such as intermittency or the break-up
of the quasi periodic torus structure. It is bene-
ficial however, to identify a pre-chaos pattern of
motion in order to offer a better understanding
of the under study chaotic physical phenomenon.
The aim of this paper is to compliment current
literature of this model utilised in power systems,
yet gaining a better understanding of the under-
lying swing equation.
References:
[1] IEEE Task Force, “Proposed terms and def-
initions for power system Stability,” /EEE
Trans. Power Appar. Syst., Vol. PAS-101,
1982, pp. 1894-1898.
[2] F. M. Hughes and A. M. A. Hamdan, “De-
sign of turboalternator excitation controllers
using multivariable frequency response meth-
ods.” Proceedings of the IEE, Vol. 123,1976,
pp. 901-905.
[3] H. M. A. Hamdan, A. M. A. Hamdan, and B.
Kahhaleh, “Damping of power system oscil-
lations using a current feedback signal,” Pro-
ceedings of the IEE, Vol. 136, Part C,1989,
pp. 137-144.
[4] Bao, Y.J., Cheng, K.W.E., Ding, K. and
Wang, D.H., 2013, December. The study on
the busbar system and its fault analysis. In
2013 5th International Conference on Power
Electronics Systems and Applications (PESA)
(pp. 1-7). IEEE.
[5] P. Berge, Y. Pomeau, and C. Vidal, Order
Within Chaos, 1984, Wiley-Interscience, New
York.
[6] P.M. Anderson and A. A. Fouad, Power Sys-
tem Control and Stability , 1977, The lowa
State University Press.
[7] Pai, Power System Stability-Analysis by the
Direct Method of Lyapunov,1981, North Hol-
land, New York.
[8] A. H. Nayfeh, Perturbation Methods , 1973,
Wiley-Interscience, New York.
[9] A. H. Nayfeh, Introduction to Perturbation
Techniques , 1981, Wiley-Interscience, New
York.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
77
Volume 22, 2023
[10] T. Tamura and Y. Yorino, “Possibility of
Auto- and Hetero- Parametric Resonances in
Power Systems and their Relationship with
Long-Term Dynamics,” /EEE Transactions
on Power Systems, Vol. PWRS-2, 1987 pp.
890-897.
[11] Kothari, D.P., March. Power system opti-
mization. In 2012 2nd National conference on
computational intelligence and signal process-
ing (CISP) ,2012,(pp. 18-21). IEEE.
[12] M. Zhao, X. Yuan, J. Hu, and Y. Yan, “Volt-
age dynamics of current control time-scale in
a VSC-connected weak grid,” IEEE Trans.
Power Syst. 31, 2015, 2925–2937.
[13] Qiu, Q., Ma, R., Kurths, J. and Zhan, M.
Swing equation in power systems: Approxi-
mate analytical solution and bifurcation curve
estimate. Chaos: An Interdisciplinary Journal
of Nonlinear Science, 30(1), 2020, p.013110.
[14] Ma, R., Li, J., Kurths, J., Cheng, S. and
Zhan, M. Generalized Swing Equation and
Transient Synchronous Stability With PLL-
Based VSC. IEEE Transactions on Energy
Conversion, 37(2),2021, pp.1428-1441.
[15] Padhi, S., and B. P. Mishra. ”Solution of
swing equation for transient stability analysis
in dual-machine system.”,2015, IOSR Journal
of Engineering 5.
[16] David Crawford, J. 1989, Introduction to bi-
furcation theory.
[17] Chiang, H. D. et al. ‘Chaos in a sim-
ple power system’, IEEE Transactions on
Power Systems, 8(4), 1993,pp. 1407–1417. doi:
10.1109/59.260940.
[18] Chitnis, N., Cushing, J. M. and Hyman, J.
M. ‘Bifurcation analysis of a mathematical
model for malaria transmission’, SIAM Jour-
nal on Applied Mathematics, 67(1), 2006, pp.
24–45. doi: 10.1137/050638941.
[19] Crandall, M. G. and Rabinowitz+, P. H.
1971, Bifurcation from Simple Eigenvalues,
Journal of Functional Analysis.
[20] Sieber, J. and Krauskopf, B. ‘Control based
bifurcation analysis for experiments’, Nonlin-
ear Dynamics, 51(3),2008, pp. 365–377. doi:
10.1007/s11071-007-9217-2.
[21] Miles, John W. ”Nonlinear faraday res-
onance.” Journal of Fluid Mechanics 146
(1984): 285-302.
[22] Bishop, S. R., Sofroniou, A. and Shi,
P. ‘Symmetry-breaking in the response of
the parametrically excited pendulum model’,
Chaos, Solitons and Fractals, 25(2), 2005,pp.
257–264. doi: 10.1016/j.chaos.2004.11.005.
[23] Scholl, Tessina H., Lutz Gr¨oll, and Veit Ha-
genmeyer. ”Time delay in the swing equation:
A variety of bifurcations.” Chaos: An Inter-
disciplinary Journal of Nonlinear Science 29,
no. 12 (2019): 123118.
[24] Sofroniou, A. and Bishop, S. ‘Dynamics of
a Parametrically Excited System with Two
Forcing Terms’, Mathematics, 2(3), 2014, pp.
172–195. doi: 10.3390/math2030172.
[25] Sofroniou, Anastasia. The parametrically
excited pendulum system and applications
to ship dynamics, 2006 University of Lon-
don, University College London (United King-
dom).
[26] Nayfeh, Mahir Ali. ”Nonlinear dynamics in
power systems.” , 1990 PhD diss., Virginia
Tech.
[27] A.H.Nayfeh and D.T. Mook, Nonlinear Os-
cillations,1979, New York.
Contribution of individual
authors to the creation of a
scientific 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, All
authors; Supervisors, Anastasia Sofroniou and
Kevin J. Munisami.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.9
Anastasia Sofroniou, Bhairavi Premnath,
Kevin Jagadissen Munisami
E-ISSN: 2224-2880
78
Volume 22, 2023
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
