Abstract: A study is conducted to obtain a deeper insight into the primary and subharmonic resonances

of the swing equation. The primary resonance, which can result in increased oscillatory responses,

voltage instability, and potential system collapse, happens when the external disturbance frequency

coincides with the natural frequency of the system. Subharmonic resonance occurs when the disturbance

frequency is an integer fraction of the natural frequency, leading to low-frequency oscillations and possible

equipment damage. The purpose of this study is to provide an extension of the existing literature of the

eﬀects of primary resonance and further provide a thorough understanding of subharmonic resonance on

the stability of a certain power system paradigm. Motivated by the rich nonlinear dynamical behaviour

exhibited by this evergreen model, analytical and numerical techniques are employed to examine the

underlying principles, creating an eﬃcient control solution for this resonant-related problem. The main

objective of this research is to provide a comprehensive understanding of the primary and subharmonic

resonances considering the dynamical and bifurcational behaviour of the underlying swing equation,

whereby both analytical and numerical techniques are employed, allowing for an identiﬁcation of certain

precursors to chaos that may lead and cater for the safe operation of practical problems.

Key-Words: nonlinear dynamics, swing equation, resonance

1 Introduction

The swing equation can be considered as a foun-

dational model for analysing the dynamic be-

haviour of power systems, particularly the oscil-

latory motion of synchronous generators. To pre-

serve the stability and dependability of power in-

frastructures, it is necessary to comprehend the

resonance phenomena that can occur in this non-

linear system. Primary resonance and subhar-

monic resonance are two signiﬁcant resonance

types undergone in the swing equation. This pa-

per is an expansion of prior work, [1] and builds

on its conclusions in order to explain the subhar-

monic resonance.

Primary and subharmonic resonances play a

crucial role in determining the stability of dynam-

ical system. The idea of disturbances, which are

abrupt changes to the system’s operating quanti-

ties, is strongly related to the concept of stability

in a power system. A minor perturbation can

nonetheless have a fascinating and varied impact

on a system’s dynamics, [1]. The dynamical be-

haviour of this system is observed through alter-

ing the variables in the equation whilst keeping

other factors constant. The primary resonance

is considered to be important when studying the

swing equation. Under primary resonance condi-

tions, a small-amplitude excitation may result in

a relatively large-amplitude response if the forc-

ing frequency is close to the linearised natural fre-

quency, [2]. Additionally, nonlinear dynamic be-

haviours such as saddle-node bifurcations could

be present in the steady-state forced response of

the nonlinear system, [3].

Primary resonance is when the excitation fre-

quency is approximately close to the natural fre-

quency of the system. The subharmonic reso-

nance on the other hand, is when the excita-

tion frequency is a multiple of the natural fre-

quency, [4]. Numerous studies have been con-

ducted in order to examine these resonances in

nonlinear power systems, to comprehend the un-

derlying principles, and to create eﬃcient control

schemes. For instance, to investigate the eﬀect

of primary and subharmonic resonance on the

stability of power systems, researchers have used

mathematical modelling, simulation studies, and

experimental validations. To reduce the negative

impacts of resonance and improve system stabil-

ity, these studies have helped to develop cutting-

edge control techniques such as adaptive control,

robust control, and damping controllers.

An Investigation into the Primary and Subharmonic

Resonances of the Swing Equation

ANASTASIA SOFRONIOU, BHAIRAVI PREMNATH

School of Computing and Engineering, University of West London,

St. Mary’s Road, W5 5RF, UNITED KINGDOM

Received: October 21, 2022. Revised: May 16, 2023. Accepted: July 15, 2023. Published: August 11, 2023.

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1.1 Brief Literature Review

A vital component of guaranteeing the depend-

able and eﬀective operation of electric circuits is

the stability of power systems, [5]. When a power

system is stable, it can continue to operate within

reasonable bounds and retain its balance in the

face of perturbations. The swing equation is cru-

cial to understanding the dynamic behaviour of

power systems among other stability issues, [6].

Resonance at the primary and subharmonic levels

is another important element that might impact

system stability. Transient stability and steady-

state stability are the two main subtypes of power

system stability. The capacity of the system to

return to a stable operating point following a sig-

niﬁcant disruption, such as a fault or a sudden

loss load, is referred to as transient stability, [7].

The ability of the system to remain stable in the

face of little disruptions, such as slight changes

in power demand or generation, is the subject of

steady-state stability, also known as small-signal

stability, [8].

A key dynamic equation used to simulate the

behaviour of synchronous generators in a power

system is the swing equation. It describes the

speed dynamics and rotor angle stability of syn-

chronous machines under transient situations.

The swing equation is predicated on the idea that

a generator’s electrical output is inversely pro-

portional to the angle between its rotor and the

system’s voltage at its terminal, [9].

Primary resonance happens when the natural

frequency of a power system coincides with the

frequency of an applied external disturbance. It

is an occurrence that could result in unstable os-

cillations and bring about system instability, [10].

Primary resonance is frequently linked to low-

frequency electromechanical modes of oscillation,

which are frequently exempliﬁed by the interac-

tion between generators and their corresponding

control systems, [11]. It can cause signiﬁcant

oscillations in generator rotor angles, which, if

left unchecked, might eventually cause cascad-

ing failures and blackouts, [12]. Subharmonic

resonance is a phenomenon where a power sys-

tem’s response shows oscillations at frequencies

lower than the applied external disturbance’s fre-

quency, [13]. It happens when a power system’s

inherent frequency falls below the disturbance fre-

quency. Power electronic components, such as

voltage source converters or thyristor-controlled

reactors, can interact with the power system to

cause subharmonic resonance, [14]. If not re-

duced, it may result in long-lasting oscillations

and instability. Power electronic equipment that

is connected to the grid must be designed and op-

erated in a way that takes subharmonic resonance

into account, [15].

A comparative analysis of primary and sub-

harmonic resonance is essential for comprehend-

ing their distinctive characteristics and implica-

tions for the stability of a power system. The au-

thors in [16], used both analytical and experimen-

tal techniques, to carry out a comparison between

the two resonance phenomena. Their research

illuminated the similarities and distinctions be-

tween primary and subharmonic resonance, high-

lighting the signiﬁcance of a thorough analysis,

[16]. The development of classiﬁcation techniques

has enabled improved identiﬁcation and diﬀeren-

tiation of primary and subharmonic resonance.

It has been demonstrated that machine learn-

ing algorithms, such as neural networks and sup-

port vector machines, can accurately classify res-

onance varieties. The authors in [17], presented

a neural network-based method for the classiﬁca-

tion of resonance phenomena in real time, allow-

ing for rapid response to critical stability events.

The authors in [18], not only examined the eﬀect

of control strategies on subharmonic resonance,

but also emphasised the signiﬁcance of consider-

ing system parameter variations when evaluating

the dynamic behaviour of primary and subhar-

monic resonances.

For electrical circuits and grids to operate reli-

ably and eﬃciently, power system stability plays

a key role. When examining the dynamic be-

haviour of power systems, particularly when ex-

amining rotor angle stability and speed dynamics,

the swing equation is crucial. Resonance on the

primary and subharmonic scales is a major phe-

nomenon that can impact the stability of a power

system, [19]. Subharmonic resonance involves os-

cillations at frequencies lower than the distur-

bance frequency, whereas primary resonance hap-

pens when the system’s intrinsic frequency co-

incides with the frequency of an applied exter-

nal disturbance hence it is vital when studying

about a system’s stability, [20]. Eﬀective stabil-

ity analysis, regulation, and mitigation measures

in power systems depend on a thorough under-

standing of these phenomena. To ensure the sta-

bility and resilience of electricity systems in the

face of changing grid circumstances and diﬃcul-

ties, more study and breakthroughs in these ﬁelds

are essential.

Basins of attraction are regions in the state

space where the trajectories of the system con-

verge to particular attractors. Studies have ex-

amined the basins of attraction relating to pri-

mary and subharmonic resonance in power sys-

tems. Various techniques, including bifurcation

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analysis, numerical simulations, and Lyapunov

exponent calculations, have been utilised in these

studies to determine the boundaries and charac-

teristics of the basins of attraction, [21] which will

also be employed here within. To obtain insight

into the stability boundaries and robustness of

power systems, researchers have investigated the

eﬀects of system parameters, initial conditions,

and control strategies on systems.

2 Methodology

2.1 Analytical Work

The swing equation was derived from the Law

of Rotation, which is fundamental in character-

ising the motion of rotating bodies and is based

on Newtonian mechanics. Synchronous genera-

tors exhibit rotational behaviour when connected

to the electrical grid in the context of power sys-

tems. By applying Newton’s second law of mo-

tion to the synchronous generator while taking

into account the mechanical and electrical torques

acting on the rotor and by also considering the

inertia of the rotating mass and the damping ef-

fects, the equation governing the dynamic motion

of the generator rotor can be derived, [1], [6]. The

swing equation is a nonlinear diﬀerential equation

of the second order that represents the angle devi-

ation of a generator’s rotor from its synchronous

position as a function of time.

The damping term of the swing equation that

characterises the motion of the rotor of the ma-

chine used in this investigation is as follows, [6]:

ωR

d2θ

dt2+Ddθ

dt =Pm−VGVB

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,

θB=phase of bus.

VB1and θB1magnitudes assumed to be small.

To achieve a deeper comprehension of the phe-

nomenon of subharmonic resonance, it is neces-

sary to conduct a thorough mathematical anal-

ysis of the swing equation. Various mathemati-

cal techniques, including algebraic methods, Tay-

lor expansion, and substitution, are employed to

achieve this objective. The objective is to develop

a ﬁnal equation for use in perturbation analysis,

with a particular emphasis on understanding sub-

harmonic resonance in the swing equation. Alge-

braic techniques such as simplifying complex ex-

pressions, factoring, combining like elements, and

rearranging the equation are adopted to make fur-

ther analysis easier. Using Taylor expansion, re-

duces the complexity of certain nonlinear terms

in the swing equation, making it simpler to ma-

nipulate and analyse.

Allowing consideration for the transforma-

tions,

θ−θB=δ0+η(4)

δ0=θ0−θB0(5)

η= ∆θ−θB1cos(ωt +ϕ0) (6)

After manipulating equation (1), the following

is obtained which is used for further analysis with

regard to primary and subharmonic resonances:

d2η

dt2+ωRD

dη

dt +Kη =α2η2+α3η3+

G1ηcos (Ωt + ϕv) + G2η2cos (Ωt + ϕv) +

G3η3cos (Ωt + ϕv) + Qcos (Ωt + ϕe).

(7)

Primary Resonance

The primary resonance is observed when the

natural frequency is approximately equal to the

excitation frequency of the system. The pertur-

bation analysis for this case has been carried out

previously by the authors, [1], thus this paper

initially provides an extension to that work by

considering the basins of attraction of the swing

equation under a variation of the parameters VB1

and θB1. The notation employed here within is

consistent to, [1].

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Perturbation Analysis for Subharmonic

resonance

This method uses multiple scales to determine

second order approximate expression for period-

two solutions for the case Ω ≃2ω0.

This solution can be used to predict the on-

set of the complex dynamics and stability. Hence

the solution becomes less accurate as excitation

amplitude increases because of its inability to ac-

count for frequency shift due to the external exci-

tation. Introducing a small dimensionless param-

eter ε, which is used as a bookkeeping device.

Let

η=O(ε)then ωRD

2H=O(ε)

G1=O(ε)Q=O(ε)

and

VB1=O(ε)and θB1= 0(ε)

Then the ﬁnal equation from swing equation

derivation above has the following coeﬃcients,

G1=εg1

G2=εg2

G3=εg3

Q=εq

After some mathematical operations equation

(7) is formulated as follows,

¨η+ 2εµ ˙η+ω2

0η=α2η2+α3η3+εg1ηcos(Ωt + ϕv)

+εg2η2cos (Ωt + ϕv) + εg3η3cos (Ωt + ϕv)+

εq cos (Ωt + ϕe)

where µ=ωRD

4H.

The solution to this above equation should be

in the form of,

η(t;ε) = εη1(T0, T1, T2) + ε2η2(T0, T1, T2) +

ε3η3(T0, T1, T2) + ....... (8)

First derivative of this equation will be,

dt =D0+εD1+ε2D2+.... (9)

Second derivative of the equation is,

dt2=D2

O+ 2εD0D1+ε2(2D0D2+D2

1) +....(10)

where

Dn=∂

∂Tn

Also considering the equation where σis in-

troduced as a detuning parameter:

ω2

0=1

4Ω

+εσ (11)

and substituting equations (8), (9), (10) and

(11) into (7) gives

¨η+ 2εµ ˙η+ (1

4Ω2+εσ)[εη1(T0, T1,T2)+

ε2η2(T0, T1,T2) + ε3η3(T0, T1,T2) + ....] =

α2(ε2η2

1+ε4η2

2+ε6η2

3+....) + α3(ε3η3

1+ε6η3

ε9η3

3+....) + εg1(εη1+ε2η2+

ε3η3) cos (Ωt + ϕv) + εg2(ε2η2

1+ε4η2

ε6η2

3) cos (Ωt + ϕv) + εg3(ε3η3

1+ε6η3

2+ε9η3

....) + εq cos (Ωt + ϕe)

Equating coeﬃcients of like powers of ε,

ε/ :η1D2

0+1

4η1Ω2=qcos(Ωt+ϕe) (12)

ε2/:η2D2

0+1

4η2Ω2+ 2D0D1η1+ση1=α2η2

+ g1η1cos(ΩT0+ϕv) (13)

ε3/:D2

0η3+ 2D0D1η2+ (D2

1+ 2D0D2)η1+

2µD0η1+1

4Ω2η3+ση2= 2α2η1η2+α3η3

g1η2cos(ΩT0+ϕv) + g2η2

1cos(ΩT0+ϕv) (14)

Solution to equation (12) can be in two forms,

(i)η1=a(T0, T1,T2)cos [1

2ΩT0+β(T0, T1,T2)]

+ 2Λcos(ΩT0+ϕe).(15)

(ii)η1=A(T1, T2)e1

2iΩT0+¯

A(T1, T2)e

−1

2iΩT0

+ ΛeiΩT0+¯

Λe−iΩT0(16)

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It is given that

N=−2q

3Ω2eiϕe(17a)

Comparing coeﬃcients in equations (15) and

(16) gives:

A=1

2aeiβ (17b)

Substituting equation (16) in (13) gives the

following,

0η2+1

4Ω2η2=−2µD0(Ae1

2iΩT0+¯

−1

2iΩT0

+ NeiΩT0+¯

Ne−iΩT0)−2D0D1(Ae1

2iΩT0+

−1

2iΩT0+NeiΩT0+¯

Ne−iΩT0)−

σ(Ae1

2iΩT0+¯

−1

2iΩT0+NeiΩT0+¯

Ne−iΩT0) +

α2(Ae1

2iΩT0+¯

−1

2iΩT0+NeiΩT0+¯

Ne−iΩT0)2+

g1cos(ΩT0+ϕv)(Ae1

2iΩT0+¯

−1

2iΩT0+NeiΩT0+

Ne−iΩT0)

Rearranging the terms,

0η2+1

4Ω2η2=e1

2iΩT0[−σA + 2α2N¯

- Ωi(D1A+µA) + 1

2g1¯

Aeiϕv] + eiΩT0[−σN +

α2A2−2iµΩN]+ e3

2iΩT0[1

2Af1eiϕv]+e2iΩT0[α2N2+

12g1Neiϕv] + [α2(A¯

A+N¯

N) + 12Ng1eiϕv] +

¯c. (18)

where ¯cis the complex conjugate.

Eliminating the secular terms,

−iΩD1A−iΩµA −σA +¯

AΓeiϕee = 0 (19)

where

Γeϕee = 2α2N+1

2g1eiϕv(20)

The solution of equation (18) is of the form,

η2=−4

3Ω2[α2A2−(2iµΩ + σ)N]eiT0−

2Ω2Γei(3

2ΩT0+ϕee)+4

2[α2(A¯

A+N¯

N)+

12g1Neiϕv]−4

15Ω2[α2N2+ 12g1Neiϕv]

ei2ΩT0+ ¯c(21)

Substituting equations (16) and (21) into (14);

0η3+1

4Ω2η3=−iΩD2A−D2

1A−2µD1A−

−8α2

3Ω2[−(2iµΩ + σ)N¯

A+α2A2¯

A]−α2A¯

Ω2Γeiϕee+

8α2

Ω2[2α2A2¯

A+2α2AN ¯

N+1

2g1A(¯

Neiϕv+Ne−iϕv)]

+ 6α3AN ¯

N+ 3α3A2¯

A−A1g1Γ

4Ω2ei(ϕee−ϕv)+

g2A(¯

Neiϕv+Ne−iϕv) + NST + ¯c(22)

where NST is the not signiﬁcant terms and ¯c

is the complex conjugate.

D1A=−(µ+iσ

Ω)¯

A+i

Ω¯

AΓeiϕee (23)

1A= [µ2−2iµσ

Ω+Γ2

−σ2

Ω2]A+

2iµ

Ω¯

AΓeiϕee (24)

Eliminating the secular terms in equation (22)

and then substituting equations (19) and (24);

-iΩD2A+ [µ2−Γ2

−σ2

Ω2−α2¯

NΓ

Ω2eiϕee +

(6α3+16α2

Ω2)N¯

N+ ( ¯

Neiϕv+Ne−iϕv) (4α2f1

Ω2+

f2)−Γf1

4Ω2ei(ϕee −ϕv)]A+ (3α3+40(α2)2

32 )A2¯

A+

3Ω2(2iµΩ + σ)NA = 0 (25)

Using method of reconstitution, the derivative

of A with respect to tis found and substitut-

ing equation (19) and (25) into equation (9) and

equating ε= 1, gives the following,

iΩ( ˙

A+µeA) + σeA−4αeA2¯

A−ˆ

Γeiˆ

ϕe=

0 (26)

where µe=µ−2α2qΓ

3Ω5sin (ϕee −ϕe) +

Γg1

4Ω3sin (ϕee −ϕv).(27)

Also σe=σ−µ2+Γ2

−σ2

Ω2−(2q

3Ω2)2(6α3+

16α2

Ω2) + 4q

3Ω2(g2+4α2g1

Ω2)cos(ϕv−ϕe)−

2qΓα2

3Ω4cos(ϕee −ϕe) + Γg1

4Ω2cos (ϕee −ϕv) (28)

where αe=10α2

3Ω2+3

4α3(29)

and

Γeiˆ

ϕe= Γeiϕee −16α2q

9Ω4(2iµΩ + σ)eiϕe.(30)

Separating the real and imaginary parts gives

the equations below,

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Ω( ˙a+µea)−aˆ

Γsinγ = 0 (31)

-Ωa˙

β+σea−αea3−aˆ

Γcosγ = 0 (32)

where γ=ˆ

ϕe−2β. (33)

Therefore

η=acos[1

2cos(Ωt+ˆ

ϕe−β)]−4q

3Ω2cos(Ωt+ϕe)+

32µq2

9Ω3sin(Ωt+ϕe)−16σq

9Ω4cos(Ωt+ϕe)−

2a2α2

3Ω2cos(Ωt+ˆ

ϕe−γ)−32α2q

135Ω6cos[2(Ωt+ϕe)]−

ag1

4Ω2cos[3

2Ωt+ϕv+1

2(ϕe−γ)]+ 2α2

Ω2(a2+16q2

9Ω4)−

8g1q

3Ω4cos(ϕv−ϕe)+2α2aq

3Ω4cos[3

2Ωt+ϕe+1

2(ˆ

ϕe−γ)]+

8g1q

45Ω4cos(2Ωt+ϕe+ϕv) + .... (34)

∆θ=θB1cos(Ωt+ϕθ) + acos[1

2(Ωt+ˆ

ϕe−β)]−

3Ω2cos(Ωt+ϕe) + 32µq

9Ω3sin(Ωt+ϕe)−

16σq

9Ω4cos(Ωt+ϕe)−2a2α2

3Ω2cos(Ωt+ˆ

ϕe−γ)+

2α2aq

3Ω4cos[3

2Ωt+ˆ

ϕe+1

2(ϕe−γ)]−

ag1

4Ω2cos[3

2Ωt+ϕv+1

2(ϕe−γ)]+ 2α2

Ω2(a2+16q2

9Ω4)+

8g1q

3Ω4cos(ϕv−ϕe) + 32α2q2

135Ω6cos[2(Ωt+ϕe)]+

8g1q

45Ω4cos(2Ωt+ϕe+ϕv)+.... (35)

Letting ˙a=˙

β= 0 in equations (31), (32) and

(33),

Ωµea−ˆ

Γa sinγ = 0 (36)

σea−αea3+ˆ

Γa cosγ = 0 (37)

When a=0,

∆θ=θB1cos(Ωt+ϕθ)−4q

3Ω2cos(Ωt+ϕe) +

32µq

9Ω3sin(Ωt+ϕe)−16σq

9Ω4cos(Ωt+ϕe) + 32α2q2

9Ω6−

8g1q

3Ω4cos(ϕv−ϕe) + 32α2q2

135Ω6cos[2(Ωt+ϕe)]+

8g1q

45Ω4cos(2Ωt+ϕe+ϕv) (38)

which is similarly echoed in [6].

When a = 0 , eliminating γto obtain the fre-

quency response equation,

a2=1

αe[σe±p(ˆ

Γ2−Ω2µ2

e)] (39)

The frequency response plot is obtained with

regard to equation (39) which shows the numeri-

cal simulation and perturbation solution.

In order to compare the analytical results with

the numerical simulations for the case of subhar-

monic resonance, the following ﬁgure, Figure 1,

presents phase portraits and time histories when

Ω = 26.01 rad/sec.

Using the Runge-Kutta and Newton Raphson

methods, the perturbation analysis was simulated

and compared to its numerical counterpart. It

was determined that the Newton Raphson tech-

nique approximates the numerical answer more

accurately. The computed numerical error of the

Newton Raphson approach and the Runge-Kutta

method in comparison to the actual simulation er-

ror were 0.0995 and 0.0419, respectively, demon-

strating that the Newton Raphson method is a

better ﬁt due to the lower error number.

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Fig. 1: Perturbed solution employing Runge-

Kutta and Newton Raphson algorithms in com-

parison to numerical simulations for the case of

subahrmonic resonance in the phase plane and

time history for Ω = 26.01 rad/sec.

Basins of Attractions

(i) Primary Resonance

This resonance plays a vital role in under-

standing the stability of a nonlinear system.

Hence it is important to study the basins of at-

traction of the primary resonance to obtain more

in-depth information about the system. Basins

of attraction shows the stable and unstable re-

gions and helps to analyse the changes made to

the system, [22]. Plots show the changes in the

basins of attraction when variables are altered.

It is also necessary to consider boundary condi-

tions when analysing these graphs when arriving

at conclusions, [23].

Important insights into the stability behaviour

of power systems have been uncovered by studies

of the basins of attraction of primary resonance.

The eﬀect of parameter variations, including sys-

tem damping, excitation levels, and control gains,

on the shape and magnitude of the basins of at-

traction associated with primary resonance has

been studied, [24], [25]. In addition, research ef-

forts have concentrated on identifying the critical

boundaries separating stable and unstable regions

in the state space, [26].

(ii) Subharmonic Resonance

The subharmonic resonance analysed in this

study further provides evidence on the stable re-

gions of the system. The basins of attractions for

the subharmonic resonance depicts the stable and

unstable regions when the excitation frequency

is approximately double the natural frequency of

the dynamical system, [27]. This analysis will

show the chaos and instability points of the sys-

tem for further studies, [28].

Subharmonic resonance’s sources of attraction

have also been studied extensively. In [29], [30],

the authors investigated the eﬀects of various pa-

rameters, such as the amplitude and frequency

of the subharmonic component, on the basins of

attraction. Transitions between distinct subhar-

monic resonant states and the eﬀect of control

strategies on the stability boundaries have been

studied, [31], [32]. Hence further investigation on

the basins of attraction in necessary to analyse

the stability when there is a change in parame-

ters, [33].

2.2 Numerical Analysis

Graphical Representation

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 for the subhar-

monic resonance.

Fig. 2: Phase portrait, frequency-domain plot

and Poincar´e map when Ω = 26.01 rad/sec.

Fig. 3: Phase portrait, frequency-domain plot

and Poincar´e map when Ω = 21.042 rad/sec.

Fig. 4: Phase portrait, frequency-domain plot

and Poincar´e map when Ω = 19.4162 rad/sec.

Figure 2, Figure 3, Figure 4, Figure 5 and

Figure 6 were obtained by plotting the phase

portraits, frequency-domain plots, and Poincar´e

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Fig. 5: Phase portrait, frequency-domain plot

and Poincar´e map when Ω = 19.375 rad/sec.

Fig. 6: Phase portrait (loss of synchronism) when

Ω = 19.37251 rad/sec.

maps when this excitation frequency is varied in

the swing equation (1). As it is decreased the sys-

tem begins to lose stability and cascades towards

chaos. Each plot represents the diﬀerent period

doubling and how the system loses its synchro-

nism. Figure 5 shows that there exists only one

steady-state attractor when there is a large Ω, Ω

= 26.01 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.

Furthermore, in Figure 3, the period-one or-

bit deforms until Ω reaches 21.042 rad/sec, at

which point the period-one attractor loses stabil-

ity and is replaced by a period-two attractor. The

frequency-domain plot and Poincar´e map show

the occurence of the period doubling bifurcation.

As Ω is decreased further to 19.4162 rad/sec, the

phase portrait illustrates an attractor with two

loops.

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 = 19.375 rad/sec a chaotic attractor is ex-

hibited as exempliﬁed in Figure 5. The system

then loses the synchronism as shown in Figure 6

when the Ω is decreased to the value of 19.37251

rad/sec.

The bifurcation diagram presented as Figure

7, was constructed by solving the swing equation

for a speciﬁc value of Ω = 19.416 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, [1]:

r=VGVB

sin (θ−θB).

Fig. 7: Bifurcation diagram when r value is var-

ied and constant Ω = 19.4162 rad/sec.

Figure 7 indicates the initial period doubling

occurrence just before r = 0.975, also justiﬁed by

the Poincar´e maps of Figure 8 and at around r

approximately 2.1, 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. The corresponding

Poincar´e maps are plotted as shown, Figure 8.

They clearly depict the points where period dou-

bling occurs and how as r is increased the phe-

nomenon of chaos is veriﬁed.

Considering the subharmonic resonance, it is

observed that at approximately r > 2.1, the

chaotic region has commenced where the Lya-

punov exponent generally takes positive values.

This behaviour is depicted and presented as Fig-

ure 9, where it is the case when two nearby points,

initially separated by an inﬁnitesimal distance,

typically diverge from each other over time and

this is quantitatively measured by the Lyapunov

exponents. The bifurcation diagram of Figure

7, also veriﬁes this behaviour, where at approx-

imately the same value of r, the cascade of pe-

riod doubling sequence leads to chaos such that

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Fig. 8: Poincar´e maps for the diﬀerent r values.

is suﬃces to say that a chaotic attractor can be

identiﬁed by a positive Lyapunov exponent. This

is further validated through the Poincar´e maps

shown in Figure 8.

Fig. 9: Lyapunov exponents as r is varied.

To analyse the validity of the analytical so-

lution, it is compared to the numerical simula-

tion and the frequency domain plot for equation

(39) is plotted as shown below in Figure 10. This

shows a strong concurrence between the two anal-

ysis performed on the swing equation for the sub-

harmonic resonance. Hence validating the analy-

sis studied in this paper.

Fig. 10: Frequency domain plot for subharmonic

resonance.

(i) Basins of attractions for Primary

Resonance

The ﬁgures below, Figure 11 and Figure 12,

show the basins of attractions for the primary

resonance when the variable VB1is varied whilst

Ω= 19.375 rad/sec. As the variable is increased

the stability of the system changes. The red and

green colour show the stable region and the other

colours represent the unstable regions of the sys-

tem. As the variable is increased the system en-

ters a corrupt state with unstable regions, hence a

further analysis on the aﬀect of other variables in

the system should be considered for sound results

in this study.

Fig. 11: Basins of attractions when VB1is 0.051

rad and 0.062 rad respectively.

Fig. 12: Basins of attractions when VB1is 0.071

rad and 0.151 rad respectively.

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(ii) Basins of attractions for Subhar-

monic Resonance

Figure 13 and Figure 14 represent the basins

of attraction for the subharmonic resonance when

VB1and θB1are varied in the swing equation of

the dynamical system. As the variable is changed

the system becomes fractal and it becomes cor-

rupt.

Initially only the variable VB1is varied when

others are ﬁxed to observe the eﬀect of this par-

ticular variable. Even when VB1= 0 the system

is still corrupted and this is due to the eﬀect of

θB1.

Furthermore, the variable θB1is changed to

observe the transitions in the basins of attractions

for subharmonic resonance.

Fig. 13: Basins of attractions when VB1is 0 rad

and 0.051 rad respectively.

Fig. 14: Basins of attractions when VB1is 0.151

rad and 0.21 rad respectively.

Fig. 15: Basins of attractions when θB1is 0.191

rad and 0.181 rad respectively.

Fig. 16: Basins of attractions when θB1is 0.151

rad and 0.141 rad respectively.

Figure 15 and Figure 16 depict the system

when the variable is varied whilst others are kept

constant. In this instance as θB1is decreased the

basins of attractions change and the stable and

unstable regions can be observed.

3 Discussion

This paper examines the dynamical behaviour of

the swing equation as control parameters are var-

ied. Analytical methods, speciﬁcally perturba-

tion techniques, are contrasted with numerical

simulation to validate the perturbed solution for

subharmonic resonance and the basins of attrac-

tion of these phenomena.

The system’s behaviour is predicted using the

swing equation under a variety of circumstances,

such as shifting loads. Power system managers

use this data to ensure the stability and depend-

ability of the system. It can be applied to the

design and analysis of power system control sys-

tems, such as automatic generation control and

load frequency management, to minimise black-

outs and, more importantly, their potentially

catastrophic impacts.

4 Conclusion

In conclusion, the thorough numerical analysis

performed in this study, utilising a variety of

mathematical tools such as bifurcation diagrams,

Lyapunov exponents, phase portraits, frequency

domain plots, and Poincar´e maps, provided cru-

cial insights into the behaviour of the swing equa-

tion under subharmonic resonance. The presence

of the ﬁrst period doubling in a sequence has been

identiﬁed as a key indicator of impending chaos,

signalling potential dangers and operational diﬃ-

culties for power systems. While period doubling

is a well-known scenario for chaotic behaviour,

the research has shown that other phenomena,

such as intermittency or the collapse of quasi-

periodic torus structures, can also result in sys-

temic chaos.

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Notably, the study has considered the eﬀects

of various parameter variations on the dynami-

cal behaviour of the system, eﬀectively depict-

ing pre-chaotic and post-chaotic changes. The

identiﬁcation of pre-chaos motion patterns pro-

vides a clearer comprehension of the transitional

behaviour of a system prior to its entry into a

chaotic regime. In addition, investigating the

basins of attraction for primary and subharmonic

resonances has validated the system’s loss of sta-

bility, which results in chaotic behaviour under

subharmonic resonance conditions.

This research contributes considerably to the

existing literature on the swing equation, partic-

ularly in the power systems domain. By con-

centrating on primary and subharmonic reso-

nances, this study expands the understanding

of the swing equation’s fundamental aspects and

their implications for system stability. The ﬁnd-

ings provide valuable direction for power system

engineers and researchers, allowing them to de-

velop more eﬀective control strategies and protec-

tive measures to mitigate the dangers associated

with subharmonic resonance-induced chaos.

This study improves comprehension of the dy-

namic behaviour of the swing equation and its re-

sponse to subharmonic resonance, shedding light

on the critical factors that determine system

stability. As power systems continue to evolve

and confront increasingly complex challenges, the

ﬁndings of this study can help to advance the de-

velopment of more resilient and secure power in-

frastructures.

Future research in this area could investigate

innovative control methods and technologies to

assure the stability and dependability of power

systems under subharmonic resonance conditions.

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Contribution of individual

authors to the creation of a

scientiﬁc article (ghostwriting

policy)

All authors contributed to the development of

this paper. Conceptualisation, Anastasia Sofro-

niou; Methodology, Anastasia Sofroniou and

Bhairavi Premnath; Analytical and Numerical

Analysis Bhairavi Premnath; Validation, Anas-

tasia Sofroniou and Bhairavi Premnath; Writing-

original draft preparation, Bhairavi Premnath

and Anastasia Sofroniou; Writing-review and

editing, Anastasia Sofroniou and Bhairavi Prem-

nath; Supervisor, Anastasia Sofroniou.

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 de-

clare that are relevant to the content of this arti-

cle.

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