Abstract: A research investigation is undertaken to gain a more comprehensive understanding of the

primary and subharmonic resonances exhibited by the swing equation. The occurrence of the primary

resonance is characterised by ampliﬁed oscillatory reactions, voltage instability, and the possibility for

system failure. The phenomenon of subharmonic resonance arises when the frequency of disturbance is

a whole-number fraction of the natural frequency. This results in the occurrence of low-frequency oscil-

lations and the potential for detrimental eﬀects on equipment. The objective of this study is to expand

upon the current literature regarding the impacts of primary resonance and enhance comprehension of

subharmonic resonance in relation to the stability of a speciﬁc power system model. The analytical and

numerical tools are utilised to investigate the fundamental principles of this resonant-related problem,

aiming to provide an eﬀective control solution. This choice is driven by the model’s complex nonlin-

ear dynamical behaviour, which oﬀers valuable insights for further analysis. This analysis includes the

Floquet Method, the Method of strained parameters, and the concept of tangent instability in order

to provide an extension to existing literature relating to primary and subharmonic resonances, taking

into account the dynamic and bifurcation characteristics of the swing equation. This objective will be

achieved through the utilisation of both analytical and numerical methods, enabling the identiﬁcation

of speciﬁc indicators of chaos that can contribute to the safe operation of real-world scenarios.

Key-Words: nonlinear dynamics, swing equation, resonance, bifurcation, power system

Received: March 15, 2023. Revised: August 14, 2023. Accepted: September 16, 2023. Published: October 13, 2023.

1 Introduction

The swing equation is widely regarded as a funda-

mental model used to analyse the dynamic char-

acteristics of power systems, with a focus on the

oscillatory movement exhibited by synchronous

generators. In order to ensure the stability and

reliability of electricity infrastructures, it is im-

perative to have a comprehensive understanding

of the resonance events that may arise within this

nonlinear system. This equation is subject to two

important types of resonance: primary resonance

and subharmonic resonance. This manuscript

serves as an extension of the preceding research

conducted by the authors cited in, [1], [2] whereby

their ﬁndings are further developed to provide a

thorough elucidation of the phenomenon known

as subharmonic resonance.

The stability of a dynamical system is signif-

icantly inﬂuenced by primary and subharmonic

resonances. The notion of disturbances, charac-

terised by sudden alterations in the operational

variables of a system, is closely interconnected

with the notion of stability within a power sys-

tem. A slight disturbance can yet have a diverse

inﬂuence on the dynamics of a system, [1]. The

dynamic behaviour of the system is examined by

manipulating the variables in the equation while

holding all other elements constant. The signiﬁ-

cance of the primary resonance is deemed crucial

in the analysis of the swing equation. When fun-

damental resonance conditions are met, a minor

perturbation can lead to a signiﬁcant response

if the frequency of the external force is in close

proximity to the linearised natural frequency, [3].

Furthermore, it is worth noting that the steady-

state forced response of the nonlinear system may

exhibit nonlinear dynamic phenomena, such as

saddle-node bifurcations and period doubling bi-

furcations, [4].

The Floquet approach is a signiﬁcant tool in

the analysis of power system stability, particu-

larly in the context of tiny disruptions, [5], [6].

The mathematical methodology employed in as-

sessing the stability of periodic solutions, such as

those seen in the swing equation, involves the

examination of the eigenvalues of the linearized

equations governing the system. Tangent insta-

bility, conversely, refers to a phenomena in which

minor disturbances in the operational parameters

of a power system can result in prolonged oscilla-

Addressing 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

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tions or instability, [7]. The strained parameters

method is a strategy in control theory that is em-

ployed to mitigate tangent instability by modi-

fying system characteristics in order to uphold

stability, [8]. Collectively, these notions oﬀer a

comprehensive theoretical structure for examin-

ing and managing the stability of a mathemati-

cal problem, thereby guaranteeing their depend-

able functioning in the presence of dynamic dis-

turbances.

The phenomenon of primary resonance occurs

when the frequency of excitation is in close prox-

imity to the natural frequency of the system. On

the contrary, subharmonic resonance occurs when

the frequency of stimulation is a multiple of the

natural frequency, [9], [10]. A wide range of re-

search have been undertaken to investigate the

resonances present in nonlinear power systems,

with the aim of comprehending the fundamen-

tal principles governing them and developing ef-

fective control strategies. Researchers have em-

ployed mathematical modelling, computer stud-

ies, and experimental validations to examine the

impact of primary and subharmonic resonance on

the stability of power systems, [1], [2] . In or-

der to mitigate the adverse eﬀects of resonance

and enhance the stability of systems, researchers

have made signiﬁcant contributions through the

advancement of sophisticated control approaches,

including adaptive control, robust control, and

damping controllers.

1.1 Brief Literature Review

Ensuring the reliability and eﬃciency of the func-

tioning of electric circuits depends upon the sta-

bility of power systems, hence a deeper under-

standing is required to prevent chaos happening

in the system, [11], [12]. When a power sys-

tem exhibits stability, it is capable of maintaining

its operational state within acceptable limits and

preserving its equilibrium despite encountering

disturbances. The comprehension of the dynamic

behaviour of power systems and other stability

concerns is signiﬁcantly enhanced by the study of

the swing equation, [13], [14]. The presence of

resonance at both fundamental and subharmonic

frequencies is a signiﬁcant factor that might po-

tentially inﬂuence the stability of a system. Tran-

sient stability and steady-state stability are the

primary classiﬁcations of this power system’s sta-

bility. Transient stability is the term used to de-

scribe the ability of a system to regain a stable

operational state after experiencing a notable dis-

turbance, such as a fault or an abrupt reduction

in load, [15]. The topic of steady-state stabil-

ity, commonly referred to as small-signal stabil-

ity, pertains to the system’s capacity to maintain

stability even when confronted with little distur-

bances, such as mild ﬂuctuations in power con-

sumption or generation, [16].

The swing equation is a fundamental dynamic

equation employed for the purpose of simulating

the behaviour of synchronous generators within

a power system. This paper elucidates the tran-

sient behaviour of synchronous machines, specif-

ically focusing on the speed dynamics and rotor

angle stability. The swing equation is based on

the premise that the electrical output of a gener-

ator is inversely proportional to the angle formed

between its rotor and the voltage of the system at

its terminal, [17], [18]. Primary resonance occurs

when the natural frequency of a power system

aligns with the frequency of an externally im-

posed disturbance. The phenomenon described

has the potential to induce oscillations that lack

stability leading to instability within the system,

[19], [20]. The phenomenon of primary resonance

often arises in the context of electromechanical

oscillation modes characterised by low frequen-

cies. This phenomenon is commonly observed in

the interaction between generators and their re-

spective control systems, [21]. According to the

cited source, [22], the occurrence of substantial

oscillations in generator rotor angles has the po-

tential to result in cascading failures and subse-

quent blackouts if not promptly addressed. Sub-

harmonic resonance refers to a phenomena ob-

served in power systems, when the system’s re-

sponse exhibits oscillations at frequencies that are

lower than the frequency of the external distur-

bance delivered to it, [23], [24]. This arises when

the natural frequency of a power system decreases

to a level below the frequency of a disturbance.

Power electronic components, such as voltage

source converters or thyristor-controlled reactors,

have the potential to induce subharmonic reso-

nance when interacting with the power system,

[25]. If left unmitigated, this phenomenon has the

potential to lead to persistent oscillations and in-

stability. According to the literature, [26], [27], it

is imperative to consider subharmonic resonance

while designing and operating power electronic

equipment that is connected to the grid.

Understanding the unique attributes and im-

plications for power system stability necessitates

a thorough examination and comparison of pri-

mary and subharmonic resonance. In the study

conducted by the authors cited in, [28], a com-

bination of analytical and experimental method-

ologies was employed to undertake a comprehen-

sive analysis and comparison of the two resonance

events. The study conducted by the researchers

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shed insight on the similarities and diﬀerences be-

tween primary and subharmonic resonance, em-

phasising the importance of conducting a compre-

hensive investigation, [29]. The advancement of

categorization methodologies has facilitated en-

hanced discernment and distinction between fun-

damental and subharmonic resonance phenom-

ena. Previous studies have provided evidence to

support the notion that machine learning meth-

ods, including neural networks and support vec-

tor machines, has the capability to eﬀectively

classify diﬀerent resonance variations. The re-

searchers in the cited studies, [30], [31] introduced

a methodology based on neural networks to clas-

sify resonance phenomena in real-time, enabling

prompt detection and response to signiﬁcant sta-

bility events. The authors in the cited publication

not only conducted an analysis on the impact of

control strategies on subharmonic resonance, but

also underscored the need of accounting for vari-

ations in system parameters when assessing the

dynamic characteristics of main and subharmonic

resonances.

The investigation of power system stability has

been the focus of substantial scholarly inquiry,

resulting in the development of many method-

ologies aimed at comprehending the complex dy-

namics of synchronous machines. Signiﬁcant at-

tention has been given to the Floquet method,

method of strained parameters, and examination

of tangent instability within the framework of the

swing equation. The Floquet approach, which is

based on the principles of linear periodic systems,

has demonstrated its eﬃcacy as a valuable tech-

nique for evaluating the stability of systems under

periodic disturbances, [32]. This method provides

valuable information regarding stability bound-

aries and bifurcation occurrences. In the context

of system analysis, the utilisation of strained pa-

rameters enables a detailed investigation into the

dynamics of the system across diﬀerent opera-

tional scenarios. This approach facilitates a more

profound understanding of the swing equation’s

responsiveness to alterations in its parameters,

[33]. Furthermore, the investigation of tangent

instability has provided a deeper understanding

of the crucial signiﬁcance of bifurcations in the

dynamics of power systems, hence providing sig-

niﬁcant knowledge regarding the occurrence of

chaotic phenomena, [34]. The utilisation of these

methodologies collectively contributes to the ad-

vancement of comprehensive stability assessment

procedures for power systems, hence augmenting

their dependability and ability to withstand ad-

verse conditions.

Basins of attraction refer to spatial regions in-

side the state space wherein the system’s trajecto-

ries converge towards speciﬁc attractors. Numer-

ous investigations have been conducted to anal-

yse the basins of attraction associated with fun-

damental and subharmonic resonance phenom-

ena in power systems. In these investigations,

researchers have applied a range of approaches

such as bifurcation analysis, numerical simula-

tions, and Lyapunov exponent calculations to as-

certain the borders and properties of the basins

of attraction, [35], [36]. These same techniques

will also be implemented in the present study.

2 Methodology

2.1 Analytical Work

The swing equation was developed from the Law

of Rotation, a key principle used to characterise

the motion of revolving bodies. This law is rooted

in Newtonian mechanics, a foundational frame-

work in physics. Synchronous generators demon-

strate rotating characteristics when they are in-

terconnected with the electrical grid within the

framework of power systems. The derivation of

the equation governing the dynamic motion of the

generator rotor can be achieved by applying New-

ton’s second law of motion to the synchronous

generator. This analysis takes into account the

mechanical and electrical torques acting on the

rotor, as well as the inertia of the rotating mass

and the damping eﬀects. This approach has been

discussed in previous studies, [1], [2], [13]. The

swing equation is a second-order nonlinear diﬀer-

ential equation that describes the temporal vari-

ation of the angle deviation of a generator’s rotor

from its synchronous position.

The equation governing the motion of the ro-

tor of the machine under the study is the swing

equation, which includes a damping term is as

follows, [13]:

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 Power,

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VG= Voltage of machine,

XG= Transient Reactance,

VB= Voltage of bus,

θB= phase of bus,

VB1and θB1magnitudes assumed to be small.

In order to enhance one’s understanding of

the phenomenon of subharmonic resonance, it is

imperative to undertake a comprehensive mathe-

matical examination of the swing equation. Sev-

eral mathematical techniques, such as algebraic

procedures, Taylor expansion, and substitution,

are utilised in order to accomplish this task. The

aim of this study is to derive a comprehensive

equation suitable for perturbation analysis, fo-

cusing speciﬁcally on the investigation of sub-

harmonic resonance inside the swing equation.

The utilisation of Taylor expansion facilitates the

reduction of complexity associated with speciﬁc

nonlinear variables present in the swing equation,

hence enabling easier manipulation and analysis.

Considering the following transformations,

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,

[2]:

d2η

dt2+ωRD

dη

dt +Kη =α2η2+α3η3+G1ηcos (Ωt+ϕv)

+ G2η2cos (Ωt+ϕv) + G3η3cos (Ωt+ϕv) +

Qcos (Ωt+ϕe).

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,[2].

The proposed solution has the potential to be

utilised for the anticipation of the commencement

of intricate dynamics and the assessment of sta-

bility. Therefore, the accuracy of the solution

diminishes with increasing excitation amplitude

due to its failure to incorporate the frequency

shift caused by the external stimulation. Intro-

ducing a small dimensionless parameter ε, which

is used as a bookkeeping device, [2].

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 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

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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

2+ε9η3

3+. . .

+εg1εη1+ε2η2+ε3η3cos (Ωt+ϕv) +

εg2ε2η2

1+ε4η2

1+ε6η2

3cos (Ωt+ϕv)

+εg3ε3η3

1+ε6η3

2+ε9η3

3+. . .+

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+µD0η1+

4Ω

η3+ση2= 2α2η1η2+α3η3

1+g1η2cos(ΩT0+ϕv)

+ g2η2

1cos(ΩT0+ϕv) (14)

As also seen in, [2] the 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)

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) and rear-

ranging the terms gives the following,

0η2+1

4Ω2η2=e1

2iΩT0[−σA + 2α2N¯

A−

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, as in, [2].

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−A

2Ω2Γei(3

2ΩT0+ϕee)+

Ω2[α2(A¯

A+N¯

N) + 12g1Neiϕv]−4

15Ω2[α2N2

+ 12g1Neiϕv]ei2ΩT0+ ¯c(21)

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

gives,

0η3+1

4Ω2η3=−iΩD2A−D2

1A−2µD1A−

−8α2

3Ω2[−(2iµΩ + σ)N¯

A+α2A2¯

A]−α2A¯

Ω2Γeiϕee+

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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,

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,

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∆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, [2], [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.

To facilitate the comparison between analyt-

ical ﬁndings and numerical simulations in the

context of subharmonic resonance, Figure 1 is

provided. This ﬁgure showcases phase portraits

and time histories at a frequency of Ω = 26.01

rad/sec.

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.

The perturbation analysis was simulated and

compared to its numerical counterpart using

the Runge-Kutta and Newton Raphson meth-

ods. The computed numerical errors of the New-

ton Raphson technique and the Runge-Kutta

method, when compared to the actual simulation

error, were found to be 0.0995 and 0.0419, re-

spectively. These results indicate that the New-

ton Raphson method exhibits a better ﬁt, as ev-

idenced by its lower error value.

Floquet Method

Let u(t) be a small disturbance (arbitrary),

then,

ˆη(t) = η(t) + u(t) (40)

The stability of η(t) depends on the

growth/decay of u(t).

Substituting equation (26) into equation (I)

and eliminating any nonlinear terms with ζ(t) will

give

d2u

dt2+ωRD

dt +u(K−2α2η−3α3η2) = 0 (41)

Behaviour of u(t) is obtained from the Floquet

theory. If u1(t) , u2(t) are solutions to equation

(41) then u1(t+T), u2(t+T) are also solutions

to the equation.

Therefore, they are represented as linear com-

binations as shown below,

u1(t+T) = a11u1(t) + a12u2(t)

u2(t+T) = a21u1(t) + a22u2(t)

Then two linearly independent solutions are

calculated for the initial conditions stated below,

u1(0) = 1

u2(0) = 0

˙u1(0) = 0

˙u2(0) = 1

Then the monodromy matrix is obtained,

A= u1(T) ˙u1(T)

u2(T) ˙u2(T)

The eigenvalues are also called the Floquet

multipliers. Behaviour of u(t) and the stability

of η(t) depends on the eigen values. If both mul-

tipliers lie inside the unit circle, then it is stable.

Analytical solution predicts saddle-node bifurca-

tion accurately.

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Method of Strained Parameters

Considering equation (8) and substituting in

equation (41) leads to,

¨u+ωRD

2H˙u+u(K−2α2[εa cos (Ωt+β+φe) +

ε2a2α2

6Ω2[3 −cos (2Ωt+ 2β+ 2φe)]]−

3α3[[εacos (Ωt+β+φe) + εa2α2

6Ω2[3 −cos(2Ωt+

2β+ 2φe)]]2) = 0

Expanding the brackets,

¨u+ωRD

2H˙u+uK −2uα2εa cos (Ωt+β+φe) +

ε2a2uα2

Ω2−ε2a2uα2

3Ω2cos (2Ωt+ 2β+ 2φe)−

3α3uε2a2cos2(Ωt+β+φe)−

3α3ua3ε2α2

Ω2cos (Ωt+β+φe) +

α3uε2a3α2

Ω2cos (Ωt+β+φe)cos(2Ωt+ 2β+

2φe)−α3uε2a4α2

12Ω4( (9 + cos(2Ωt+ 2β+ 2φe)−

6cos(2Ωt+ 2β+ 2φe) ) = 0

From the formulated equation (7),

ωRD

2H= 2ε2µ

Also considering,

Φ = Ωt+β+φe

Then the equation will become,

¨u+ 2ε2µ˙u+uK −2uα2εa cosΦ−

α2

2uε2a2

Ω2

α2uε2α2a2

3Ω2cos2Φ −

3α3uε2a2cos2Φ−3α3ua3ε2α2

Ω2cosΦ +

α3uε2α2a3

Ω2cosΦcos2Φ −3α3uε2a4α2

4Ω4−

cos22Φ α3uε2a4α2

12Ω4+α3uε2a4α2

2Ω4= 0

(Cancelling out ε2terms)

¨u+ 2µ˙u+uK −2uα2a cosΦ + α2

2ua2

Ω2+α2

2ua2

3Ω2cos2Φ

- 3α3ua2(cos 2Φ

2+1

2)−3α3ua3α2

Ω2cos Φ +

α3ua3α2

Ω2cos Φcos 2Φ −3α3ua4α2

4Ω4−

cos22Φ α3ua4α2

12Ω4+α3ua4α2

2Ω2cos 2Φ = 0

Simplifying further, the equation below is ob-

tained,

...

u+ 2µ˙u+uK∗=χu cos Φ+Λu cos 2Φ (42)

Where: K∗=K+3α3

2−α2

3Ω2a2−19α2

2αea4

24Ω4

χ= 2α2a+5α2α3a

2Ω2

Λ=(3α3

2−α2

3Ω2)a2−α2

2α3a4

2Ω4

Φ = Ωt+β+φe(43)

Introducing εwhich is a small dimensionless

parameter as bookkeeping device. Then order

damping and parametric term at 0(ε), hence the

equation (42) becomes,

...

u+ 2µε ˙u+uK∗=εχu cos Φ + εΛu cos 2Φ (44)

in accordance to, [2], [13].

Uniform expansion of solutions shown below

are considered,

u(t;ε) = εu1(t) + ε2u2(t) +

.... (45)

K∗=1

4Ω2+ε δ1+ε2δ2+

.... (46)

This determines δ1,δ2and results in periodic

expansion, while K∗deﬁnes transition curves sep-

arating stability from instability giving the curve

for period-doubling bifurcation.

Substituting equations (31) and (32) into (30),

¨u+ 2εµ ˙u+ ( 1

4Ω2+εδ1+ε2δ2+....)u=εχ (εu1(t)

+ε2u2(t) + ....) + εΛ (εu1(t) +

ε2u2(t) + ......)cos 2Φ

Comparing the coeﬃcients of equal powers of

εin the above equation,

Consider ε0/: ¨u0+1

4Ω2u0= 0 (47)

ε1/: ¨u1+1

4Ω2u1=−2µ˙u0−δ1u0+

χu0cosΦ + Λ u0cos 2Φ (48)

ε2/: ¨u2+1

4Ω2u2=−2µ˙u1−δ1u1−

δ2u2+χu1cosΦ + Λ u1cos 2Φ (49)

Given u0=acos 1

2Φ+ bsin 1

2Φ (50)

Substituting equation (50) into equation (48)

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¨u1+1

4Ω2u1=−2µ˙u0−δ1(a cos 1

2Φ + b cos 1

2Φ )

+χcos Φ (a cos 1

2Φ + b sin 1

2Φ ) +

Λcos 2Φ (acos 1

2Φ + bsin 1

2Φ)

Replacing with,

˙u0=−a

2sin 1

2Φ + b

2cos 1

2Φ

gives,

¨u1+1

4Ω2u1=µaΩsin 1

2Φ−µb Ωcos 1

2Φ

-δ1a cos1

2Φ−δ1a sin1

2Φ + aχ cosΦcos 1

2Φ

+ bχ cosΦ sin 1

2Φ + aΛcos 2Φ cos 1

2Φ +

bΛcos 2Φ sin1

2Φ

Employing trigonometric identities,

aχcosΦcos1

2Φ = aχ

2cos3

2Φ + cos1

2Φ

bχ cosΦsin 1

2Φ = bχ

2sin3

2Φ−sin1

2Φ

aΛ cos 2Φ cos 1

2Φ = aΛ

2cos5

2Φ + cos3

2Φ

bΛ cos 2Φ sin 1

2Φ = bΛ

2sin5

2Φ−sin3

2Φ

Substituting the above into the equation and

rearranging,

¨u1+1

4Ω2u1=cos 1

2Φ [(1

2χ−δ1)a−µbΩ]+

sin1

2Φ[µaΩ−(12χ+δ1)b]+

2(χ+ Λ)cos3

2Φ + b

2(Φ −Λ)sin3

2Φ + aΛ

2cos5

2Φ−

bΛ

2sin5

2Φ (51)

For eliminating secular terms in equation (51),

consider

2χ−δ1)a−µbΩ = 0 (52)

µaΩ−(1

2χ+δ1)b= 0 (53)

It is also given that for non-trivial solution to

exist, the following should be satisﬁed,

δ2

1=1

4χ2−µ2Ω2(54)

Using equations (52) and (53), equation (51)

becomes

u1=D cos 1

2Φ + E sin 1

2Φ−

(X+ Λ) a

4Ω2cos 3

2Φ−(x−Λ) b

4Ω2sin 3

2Φ + .... (55)

D and E constants.

Substituting equations (51) and (55) into

equation (50) the following equations are ob-

tained,

2χ−δ1)D−µΩE= [δ2+(X+ Λ) Λ

8Ω2]a(56)

µΩD−1

2(χ+δ1)E= [ δ2+(X−Λ) Λ

8Ω2]b(57)

Given that equations (56) and (57) have non-

trivial solution, the inhomogeneous equations

have solution if and only if consistency (solvabil-

ity) condition is satisﬁed,

δ2=−χ2+ 4Λδ1+ Λ2

8Ω2(58)

Then equation (32) becomes transition curves

determining period doubling as shown below,

K∗=1

4Ω2±ε(1

4χ2−µ2Ω2)1/2−

ε2(χ2+ 4Λδ1+ Λ2

8Ω2)1/2+.... (59)

Tangent Instability

Initially the points corresponding to vertical

tangents are in the frequency-response curves

given by the equation below, [1],

µ2+ ( σ

2Ω +αea2

Ω)2=g2

4Ω2a2

Rearranging the above equation,

4µ2Ω2+ (σ+ 2αea2)2= (g

a)2(60)

Also assume: s=a2

x= Ω2

σ=ω2

0−Ω2

Then equation (46) is written as: 4µ2xs +

s(ω2

0−x−2αes)2=g2(61)

Taking the ﬁrst derivative of equation (61):

4µ2xds

dx+ (ω2

0−x−2αes)2ds

dx−4αesω2

0−x−2αesds

+4µ2s−2sω2

0−x−2αes= 0 (62)

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Equating the coeﬃcient term of ds

dx = 0;

4µ2x+(ω2

0−x−2αes)2−4sαeω2

0−x−2αes= 0

(63)

Substituting equation (61) into (63)

g2= 4sαeω2

0−x−2αes(64)

Let z= ω2

0−x−2αes(65)

This equation then can be formulated:

2z+x−ω2

0= 4αes

Substituting equation (65) into (63)

4µ2x+z2+ 4sαez= 0

4µ2x+z2+z2z+x−ω2

0 = 0

Expanding the brackets and rearranging:

3z2+ 2zx−ω2

0+ 4µ2x= 0 (66)

In order to calculate the tangent instability

using MATLAB, the variable zis determined by

solving equation (66) with speciﬁc values assigned

to the parameter Ω. Subsequently, by solving

equation (65) for the variable sand substituting

the obtained value into equation (64), the vari-

able gcan be determined.

Basins of Attractions

(i) Primary Resonance

The phenomenon of resonance is of utmost im-

portance in understanding the stability charac-

teristics of a nonlinear system. Therefore, it is

imperative 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, fa-

cilitating the analysis of modiﬁcations made to

said system, [37]. The plots illustrate the alter-

ations in the basins of attraction as variables are

modiﬁed. When drawing inferences from these

graphs, it is imperative to take into account the

boundary conditions as well, [38].

Studies of the basins of attraction of primary

resonance have revealed signiﬁcant ﬁndings re-

garding the stability characteristics of power sys-

tems. The impact of parameter ﬂuctuations, in-

cluding system damping, excitation levels, and

control gains, on the conﬁguration and ampli-

tude of the basins of attraction linked to primary

resonance has been investigated, [39], [40]. Fur-

thermore, scholarly investigations have mostly fo-

cused on the identiﬁcation of crucial borders that

demarcate stable and unstable regions within the

state space, [41], [42].

(ii) Subharmonic Resonance

This study examines the subharmonic reso-

nance phenomenon and its implications for identi-

fying stable zones within the system. The basins

of attraction associated with subharmonic reso-

nance illustrate the areas of stability and insta-

bility in a dynamical system when the excitation

frequency is nearly twice the natural frequency,

[43]. This analysis aims to identify the points of

chaos and instability within the system, provid-

ing a foundation for future studies, [2], [44].

Extensive research has been conducted on the

origins of attraction associated with subharmonic

resonance. The authors in, [45] and [46] con-

ducted a study to examine the impact of dif-

ferent parameters, including the amplitude and

frequency of the subharmonic component, on

the basins of attraction. The investigation of

transitions between diverse subharmonic reso-

nant states and the impact of control tactics on

the stability boundaries has been examined in

previous studies, [47], [48]. Therefore, it is im-

perative to do additional research on the basins

of attraction in order to examine the stability in

the event of parameter changes, [49], [50].

2.2 Numerical Analysis

Graphical Representation

The equations (1), (2), and (3) were solved

using the fourth-order Runge-Kutta technique in

Matlab. The main objective was to investigate

the impact of altering the excitation frequency

Ω on the phenomenon of subharmonic resonance,

[2].

Fig. 2: Phase portrait, frequency-domain plot

and Poincar´e map when Ω = 26.01 rad/sec, [2].

Fig. 3: Phase portrait, frequency-domain plot

and Poincar´e map when Ω = 21.04 rad/sec, [2].

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Fig. 4: Phase portrait, frequency-domain plot

and Poincar´e map when Ω = 19.37 rad/sec, [2].

Fig. 5: Phase portrait (loss of synchronism) when

Ω = 19.37251 rad/sec, [2].

The phase portraits, frequency-domain plots,

and Poincar´e maps were generated for the swing

equation (1) to yield Figure 2, Figure 3, Figure

4, and Figure 5, which depict the variations in

excitation frequency reﬂected also in, [2]. As the

system experiences a reduction, its stability di-

minishes and it undergoes a cascading process

towards a state of chaos. Each plot depicts the

many occurrences of period doubling and the sub-

sequent loss of synchrony within the system.

Moreover, as depicted in Figure 3, the period-

one orbit undergoes deformation until the angular

frequency Ω attains a value of 21.04 rad/sec. At

this critical threshold, the period-one attractor

ceases to be stable and is subsequently replaced

by a period-two attractor. The frequency-domain

plot and Poincar´e map illustrate the appearance

of the period doubling bifurcation, [2].

As the parameter Ω is systematically reduced,

it becomes evident that the graphs undergo dy-

namic changes, such as the emergence of period-

doubling solutions. Eventually, as Ω approaches

a value of around 19.37 rad/sec, a chaotic attrac-

tor is detected, as depicted in Figure 4. The sys-

tem then experiences a loss of synchronism, as

depicted in Figure 5, when the angular velocity

(Ω) is reduced to 19.37251 rad/sec, [2].

Figure 6 illustrates the bifurcation diagram

and the associated Lyapunov exponents for the

instances of primary and subharmonic resonances

as echoed in, [2], respectively. The construction

process involved calculating the swing equation

for a particular angular frequency value of Ω =

8.27 rad/sec for the primary resonance and Ω

= 19.416 rad/sec for the subharmonic resonance,

followed by numerical time integration using the

well-known fourth order Runge-Kutta method.

The value of the forcing parameter, denoted as

r, is incrementally increased, and the time inte-

gration process is continued. The resulting data

is then used to create a plot that shows the max-

imum amplitude of the oscillatory solution as a

function of r, [1].

r=VGVB

sin(θ−θB)

Fig. 6: Bifurcation diagrams and Lyapunov

Exponents for Primary and Subharmonic Reso-

nances, [1], [2].

The provided ﬁgure, Figure 6, illustrates the

occurrence of the initial period doubling just prior

to reaching a value of r equal to 0.9 in the case

of primary resonance. Additionally, it can be ob-

served that at about r = 2.36, the ﬁrst instance of

period doubling in a series of subsequent period

doublings is displayed, ultimately resulting in the

emergence of chaotic behaviour. The ﬁndings of

this numerical analysis indicate that an increase

in the value of parameter r leads to a progressive

loss of synchronisation in the swing equation, [1].

The phenomenon of subharmonic resonance is

characterised by the emergence of a chaotic zone

when the value of rexceeds around 2.1, [2]. In

this region, the Lyapunov exponent tends to ex-

hibit positive values. The behaviour under con-

sideration is illustrated. In this scenario, two

points in close proximity, initially separated by

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an inﬁnitesimally small distance, tend to move

apart from each other over a period of time. This

divergence is quantitatively assessed using the

Lyapunov exponents. The behaviour shown in

bifurcation diagram further conﬁrms the afore-

mentioned phenomenon. Speciﬁcally, once the

value of r approaches a certain threshold, a series

of period doubling occurs, eventually leading to

chaotic behaviour. Consequently, it can be con-

cluded that the presence of a positive Lyapunov

exponent is indicative of the existence of a chaotic

attractor.

In order to assess the soundness of the analyt-

ical solution, a comparison is made between the

analytical solution and the numerical simulation.

Additionally, a frequency domain plot for equa-

tion (39) is depicted in Figure 7, [2]. The ﬁndings

demonstrate a signiﬁcant correlation between the

two analyses conducted on the swing equation

pertaining to subharmonic resonance. Therefore,

this paper’s analysis is being validated.

Fig. 7: Frequency domain plot for Subharmonic

Resonance, [2].

Figure 8 below compares the numerical re-

sults with the analytical methods such as Floquet

method, Method of strained parameters and Tan-

gent instability. The application of the Floquet

method in MATLAB allows for a comprehensive

examination of the stability and dynamic charac-

teristics of the swing equation, hence providing

signiﬁcant insights into the transient stability of

power systems. This technique predicts the sad-

dle node bifurcation as shown in the ﬁgure but

with 9.21% error compared to the numerical re-

sult. Method of strained parameters and tangent

instability were also solved with computing the

solutions for the equation (59) and equation (64)

respectively. Method of strained parameters pre-

dicts the period-doubling bifurcation with an er-

ror of 10.32% when compared to the numerical

analysis. Finally the tangent instability method

also predicts the saddle node bifurcation with an

error of 12.5% compared to its numerical coun-

terpart.

Fig. 8: Bifurcation diagram showing a compar-

ison of diﬀerent analytical methods for Primary

Resonance.

Fig. 9: Bifurcation diagram showing a compar-

ison of diﬀerent analytical methods for Subhar-

monic Resonance.

Similarly, Figure 9 presents the contrast be-

tween the numerical simulation and the analyt-

ical methodologies that have been examined for

the case of subharmonic resonance. The time con-

sidered is twice that of the primary resonance in

this analysis. The Method of Strained Param-

eters and Tangent Instability have been utilised

to forecast saddle node bifurcations, yielding er-

rors of 0.091% and 5.43% respectively, whilst the

Floquet method predicts the period doubling bi-

furcation with 0.102% error compared to the nu-

merical analysis. Some of these predicted meth-

ods provide coherency with speciﬁc results from,

[13].

(i) Basins of attractions for Primary

Resonance

The following ﬁgures, namely Figure 10, Fig-

ure 11, and Figure 12, depict the basins of attrac-

tion pertaining to the primary resonance. These

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ﬁgures illustrate the variations in the variables

VB1and θB1while maintaining a constant value

of Ω at 8.27 rad/sec. The stability of the system

is subject to change as the variable is raised. The

stable portion of the system is shown by the pres-

ence of red and green colours, while the remain-

ing colours reﬂect the unstable regions. As the

independent variable is incremented, the system

undergoes a state of corruption characterised by

the presence of unstable regions. Consequently,

it is imperative to conduct a more comprehen-

sive examination of the impact of other variables

within the system in order to obtain reliable and

robust ﬁndings in this particular study.

Fig. 10: Basins of attractions when VB1is 0.071

rad and 0.151 rad respectively, [2].

Fig. 11: Basins of attractions when θB1is 0.191

rad and 0.181 rad respectively.

Fig. 12: Basins of attractions when θB1is 0.141

rad and 0.181 rad respectively.

(ii) Basins of attractions for Subhar-

monic Resonance

Figure 13 and Figure 14 depict the basins of

attraction pertaining to the subharmonic reso-

nance phenomenon in the swing equation of the

dynamical system. The variations in VB1and

θB1are considered, while keeping Ω constant at

a value of 19.375 rad/sec. According the authors,

[2], [13], the corruption of the system occurs as

the variable is manipulated. It is vital to study

these changes in order to gain a deep understand-

ing of the stability of the system.

Fig. 13: Basins of attractions when VB1is 0 rad

and 0.051 rad respectively, [2].

Fig. 14: Basins of attractions when θB1is 0.191

rad and 0.181 rad respectively, [2].

3 Discussion

The objective of this study is to analyse the dy-

namic characteristics of the swing equation under

diﬀerent variations of control parameters. This

study compares analytical methods, particularly

perturbation techniques, with numerical simula-

tion in order to verify the accuracy of the per-

turbed solution for subharmonic resonance and

the basins of attraction associated with these phe-

nomena.

The examination of the primary and sub-

harmonic resonances of the swing equation in-

volves the utilisation of many analytical methods,

namely the Floquet method, method of strained

parameters, and analytical techniques. These ap-

proaches give unique perspectives and contribute

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signiﬁcant knowledge to the understanding of this

paper’s power system stability. The analytical

approach generally depends on the utilisation of

mathematical modelling and manual computa-

tions, resulting in accurate outcomes within the

context of simpliﬁed assumptions. However, it

may encounter diﬃculties in accurately represent-

ing the many interconnections and non-linear dy-

namics that manifest in power systems found in

real-world scenarios. On the other hand, the Flo-

quet method and the method of strained param-

eters utilise numerical and computational tech-

niques to eﬀectively address complex dynamics.

These methodologies provide a methodical inves-

tigation of the system’s reaction to diverse condi-

tions and external disturbances, which can be ef-

fectively depicted through graphical illustrations.

Researchers can enhance their comprehension of

the system’s behaviour in the vicinity of the pri-

mary resonance by graphing the response of the

swing equation across various parameter values or

forcing frequencies. Graphical analyses serve as

a vital supplement to analytical techniques, pro-

viding a more holistic perspective on the stabil-

ity attributes. This aids power system engineers

in making well-informed judgements to guarantee

the dependable functioning of the grid.

The anticipated response of the system is de-

termined by employing the swing equation in di-

verse scenarios, including instances involving load

alterations. The data is utilised by power sys-

tem management in order to guarantee the sta-

bility and reliability of the system. The use of

this approach extends to the design and analysis

of control systems for power systems, namely in

the areas of autonomous generation control and

load frequency management. For instance in the

case to mitigate the occurrence of blackouts and

the consequential catastrophic consequences they

may entail.

4 Conclusion

To summarise, the present study employed var-

ious analytical techniques, including bifurcation

diagrams, Lyapunov exponents, phase portraits,

frequency domain plots, and Poincar´e maps, to

investigate the dynamics of the swing equation

in the context of subharmonic resonance. The

occurrence of period doubling in a sequence sug-

gests an impending state of turbulence, which

presents potential threats to power systems and

operational diﬃculties. According to research,

chaos can be induced by the collapse of quasi-

periodic torus structures and the presence of in-

termittency. Period doubling is a widely recog-

nised illustration.

This study focused on investigating the impact

of parameter modiﬁcations on the dynamics of

the system, speciﬁcally highlighting the observed

changes before and after the onset of chaotic be-

haviour. It also highlights and incorporates dif-

ferent methods used to study the stability of the

system, such as the Floquet method, Method

of strained parameters and Tangent instability.

The identiﬁcation of pre-chaotic motion patterns

serves to elucidate the transitory dynamics of a

system prior to its entry into a state of chaos.

Furthermore, an examination of the basins of at-

traction pertaining to primary and subharmonic

resonances has substantiated the inherent insta-

bility of the system, resulting in the manifestation

of chaotic phenomena when subjected to subhar-

monic resonance circumstances.

This study makes a valuable contribution to

the current scholarly understanding of the swing

equation, namely by oﬀering an extension to the

most recent literature authored by the same in-

dividuals as this paper, [1], [2]. This research

enhances the understanding of the fundamental

principles and system stability of the swing equa-

tion through a speciﬁc emphasis on primary and

subharmonic resonances. The discoveries assist

power system engineers and researchers in devel-

oping improved control strategies and preventive

measures to address the chaotic eﬀects caused by

subharmonic resonance.

The present study illuminates the dynamic be-

haviour of the swing equation and its reaction to

subharmonic resonance, thereby shedding light

on several elements of system stability. This

study has the potential to contribute to the de-

velopment of more resilient and secure power in-

frastructures, particularly as power systems con-

tinue to expand and encounter increasingly com-

plex challenges.

In further research, the incorporation of quasi

periodic conditions within the swing equation

framework hold the potential to advance the

knowledge of this intricate power system. The

aforementioned approach has the potential to of-

fer signiﬁcant insights on the enduring stability

and adaptability of the system.

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DOI: 10.37394/232011.2023.18.19

Anastasia Sofroniou, Bhairavi Premnath

E-ISSN: 2224-3429

212

Volume 18, 2023

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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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E-ISSN: 2224-3429

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