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1

Applications of the Frenet Frame to Electric Circuits

Federico Milano, Fellow, IEEE, Georgios Tzounas, Member, IEEE,

Ioannis Dassios, and Taulant K¨

erc¸i, Student Member, IEEE

Abstract—The paper discusses the relationships between elec-

trical quantities, such as voltages, currents, and frequency, and

geometrical ones, namely curvature and torsion. The proposed

approach is based on the Frenet frame utilized in differential

geometry and provides a general framework for the deﬁnition of

the time derivative of electrical quantities in stationary as well

as transient conditions. As a byproduct, the proposed approach

uniﬁes and generalizes the time- and phasor-domain frameworks.

Other noteworthy results are a new interpretation of the link be-

tween frequency and the time derivatives of voltage and current;

and a deﬁnition of the rate of change of frequency that includes

the novel concept of “torsional frequency.” Several numerical

examples based on balanced, unbalanced, harmonically-distorted

and transient voltages illustrate the ﬁndings of the paper.

Index Terms—Differential geometry, Frenet frame, curvature,

torsion, time derivative, frequency, Rate of Change of Frequency

(RoCoF), Park transform.

I. NOTATIO N

In this paper, scalars are indicated with normal font,

e.g. x, whereas vectors are indicated in bold face, e.g. x=

(x1, x2, x3). All vectors have order 3, unless otherwise indi-

cated.

Scalars:

slength of a curve

ttime

Vvoltage magnitude

wangular frequency

ηsymmetric part of the geometric RoCoF

θvoltage phase angle

κcurvature

ξtorsional frequency

ρsymmetric part of the geometric frequency

τtorsion

ωmagnitude of vector ω

Vectors:

0null vector

Bbinormal vector of the Frenet frame

eii-th vector of an orthonormal basis

ıcurrent vector

nnormal vector before normalization

Nnormal vector of the Frenet frame

qelectric charge vector

F. Milano, G. Tzounas, I. Dassios and T. K¨

erc¸i are with School of Electrical

and Electronic Engineering, University College Dublin, Dublin, D04V1W8,

Ireland. E-mails: {federico.milano, ioannis.dassios, georgios.tzounas}@ucd.ie

This work is supported by the European Commission by funding F. Milano,

G. Tzounas and T. K¨

erc¸i under project edgeFLEX, Grant No. 883710; and by

Science Foundation Ireland by funding F. Milano and I. Dassios under project

AMPSAS, Grant No. SFI/15/IA/3074.

Ttangent vector of the Frenet frame

vvoltage vector

φmagnetic ﬂux vector

ωantisymmetric part of the geometric frequency

Derivatives:

x0,x0derivative of a scalar/vector with respect to t

˙x, ˙

xderivative of a scalar/vector with respect to s

Dx

ttime derivative operator applied to vector x

II. INTRODUCTION

A. Motivation

The study and simulation of circuit dynamics has tradition-

ally been approached using different frameworks. Stationary

AC circuits are conveniently studied using quantities such as

phasors and impedances; circuits with harmonic contents are

studied using Fourier analysis or similar frequency-domain

approaches; rotating machines and power electronic devices

are often studied using Park and/or Clarke transforms; generic

transients are studied using a time-domain analysis [1]. In this

paper, we propose an approach based on differential geometry,

more speciﬁcally on the Frenet frame [2]. This approach leads

to the deﬁnition of a framework that admits, as special cases,

the circuit analysis transformations mentioned above.

B. Literature Review

Differential geometry ﬁnds applications in several ﬁelds

of science and engineering. Some examples are the use of

differential geometric properties, such as that of curvature, in

image segmentation and three-dimensional object description

[3], as well as in robotic control along geodesic paths [4].

Another relevant example are the utilities of the Frenet frame

in the area of autonomous vehicle driving [5], [6]. Moreover,

there is a number of applications that are based on the theory

of geometric algebra, for example the use of quaternions in

computer graphics and visualization [7], [8] and in the control

of multi-agent networked systems [9].

The utilization of concepts of geometric algebra in circuit

and power system analysis is limited. There is a group of

works that elaborate on the concept of instantaneous power

[10]–[15] that provide an interpretation of the active and reac-

tive power as the inner and cross (or wedge in the polyphase

case) products, respectively, of voltage and currents. More

recently, some studies, including [16]–[20], have attempted to

extend the instantaneous power theory to a systematic study of

electrical quantities or circuits in the framework of geometric

algebra. In the same vein, but using a novel perspective, [21]

makes an additional step by proposing to interpret voltages and

currents as the time derivative of a multi-dimensional curve.

arXiv:2112.03633v1 [math.DG] 7 Dec 2021

2

This interpretation allows the deﬁnition of the “geometric

frequency” as the result of an inner and an outer product.

In this work, we exploit differential geometry rather than

geometric algebra. We are interested in the geometrical “mean-

ing” of the time derivative of electrical quantities such as

voltage, current and frequency. With this aim, the formulas

obtained in the paper are deduced through the Frenet frame

[2]. The importance in circuit analysis of the time derivatives

of voltages and currents is apparent as they are required in

the constitutive equations of capacitors and inductors. The

relevance of the Rate of Change of Frequency (RoCoF), on the

other hand, is due to the increasing penetration, in the electric

grid all around the world, of renewable energy sources and

the consequent shift from synchronous to non-synchronous

generation. The RoCoF is, in turn, strictly related to the

amount of available inertia in the system [22]. The ability to

estimate accurately the RoCoF is thus becoming an important

aspect of the measurements utilized by system operators. As

a matter of fact, several works discuss the estimation of the

RoCoF from an instrumentation point of view [23]–[27].

C. Contributions

We apply differential geometry to deﬁne a general frame-

work for the deﬁnition of electrical quantities and their time

derivatives. The speciﬁc contributions of the paper are the

following.

•The derivation of the expressions of the tangent, normal

and binormal vectors of the Frenet frame in terms of the

voltage (or current) of an electrical circuit.

•A novel interpretation of the time derivative of any order

of voltage and current in electrical circuits.

•An expression of the RoCoF which involves the deﬁnition

of the novel concept of “torsional frequency,” which is

also proposed and deﬁned in the paper.

•An example that shows that analytic signals commonly

utilized in signal processing are a special case of the

proposed framework in two dimensions.

The meaning and derivation of the vectors of the Frenet frame

when applied to electric quantities such as voltage, current and

frequency are duly discussed in the paper.

D. Organization

The remainder of the paper is organized as follows. Section

III outlines the concepts of differential geometry that are

needed for the derivations of the theoretical results of this

work, which are given in Section IV. Section V illustrates the

formulas of the time derivatives through a series of examples.

The examples are aimed at showing that the formulas derived

in Section IV admit as special cases widely utilized frame-

works such as DC circuits, phasors and Park transform, as

well as illustrate the formulas in unbalanced cases that lead

to the birth of time-variant curvature and torsion. Section VI

draws conclusions and outlines future work.

III. FRENET FRAME OF SPACE CURVES

Let us consider a space curve x: [0,+∞)→R3with

x= (x1, x2, x3). Where x1=x1(t),x2=x2(t),x3=x3(t),

is the set of parametric equations for the curve. Equivalently:

x=x1e1+x2e2+x3e3,(1)

where (e1,e2,e3)is an orthonormal basis. The length sof the

curve is deﬁned as:

s=Zt

0px0(r)·x0(r)dr +s0,(2)

from which one obtains the expression:

s0=ds

dt =√x0·x0=|x0|,(3)

where

x0=d

dt(x1e1) + d

dt(x2e2) + d

dt(x3e3),(4)

and ·represents the inner product of two vectors, which

in three dimensions, for a= (a1, a2, a3),b= (b1, b2, b3),

becomes:

a·b=a1b1+a2b2+a3b3.(5)

The length sis an invariant of the curve. It is relevant to

observe that, according to the chain rule, the derivative of x

with respect to scan be written as:

˙

x=dx

ds =dx

dt

dt

ds =x0

s0=x0

|x0|.(6)

The vector ˙

xhas magnitude 1 and is tangent to the curve x.

The Frenet frame is deﬁned by the tangent vector T, the

normal vector Nand the binormal vector B, as follows:

T=˙

x,

N=¨

x

|¨

x|,

B=T×N,

(7)

where ×represents the cross product, which in three dimen-

sions can be written as the determinant of a matrix, as follows:

a×b=

e1e2e3

a1a2a3

b1b2b3

.(8)

The vectors in (7) are orthonormal, i.e. T=N×Band N=

B×T, and have relevant properties, which can be expressed

as follows [2]: ˙

T=κN,

˙

N=−κT+τB,

˙

B=−τN,

(9)

where κand τare the curvature and the torsion, respectively,

which are given by:

κ=|¨

x|=|x0×x00|

|x0|3,(10)

and

τ=˙

x·¨

x×...

x

κ2=x0·x00 ×x000

|x0×x00|2.(11)

The quantities deﬁned above, namely κand τ, as well as (9),

are utilized in the following section.

3

IV. ELECTRICAL QUANTITIES IN THE FRENET FRAME

This section presents the main theoretical results of the

paper. In particular, the Frenet frame as well as of the curvature

and torsion of a space curve are expressed in terms of electrical

quantities. Then the expressions of the time derivatives of the

vectors of voltage, current as well as the frequency of these

quantities are derived based on the Frenet frame. A general

expression for higher-order derivatives is also presented at the

end of the section.

A. Voltage and its Time Derivative in the Frenet Frame

The starting assumption of the discussion given in this

section is that the vector of the voltage, v, is the time derivative

of a space curve. From a physical point of view, this means

assuming that the vector that describes the magnetic ﬂux, say

ϕ, is formally deﬁned as:

ϕ=−x.(12)

Then, Faraday’s law gives:

v=−ϕ0=x0.(13)

Then one can rewrite the expressions of the vectors T,Nand

Bof the Frenet frame in terms of the vector for the voltage

and its derivatives.

Let us observe ﬁrst that the derivative of the length s,

according to (3) and (13), becomes [21]:

s0=|v|=v , (14)

and, then

˙

x=−˙

ϕ=−ϕ0

s0=v

v,(15)

and:

¨

x=−¨

ϕ=v0

v2−v0v

v3,(16)

and:

...

x=−...

ϕ=v00

v3−3v0v0

v4+ 3(v0)2v

v5−v00 v

v4,(17)

where v0=d

dt (v)and v00 =d2

dt2(v). It is relevant to observe

that, from the property ˙

x·¨

x= 0, as these vectors are

orthogonal by construction, and from (15) and (16), one

obtains [21]:

ρ=v0

v.(18)

As it is well known, in time-frequency analysis and signal

processing, the quantity ρis deﬁned as the instantaneous

bandwidth [28]. In this work, however, we rather use the

interpretation of ρgiven in [21], namely, the symmetric part

of the geometric frequency.1It is also relevant to note that,

from a geometrical point of view, ρ v =v0can be viewed as

the “radial” component of the velocity v. In this vein, ρcan

be deﬁned as radial frequency.

On the other hand, from (10), (13) and (15)-(18), one has:

κ=|v×v0|

v3=|ω|

v=ω

v,(19)

1In this work, the terms symmetric and antisymmetric do not refer to the

properties of a matrix but rather to the effect of operators.

where the vector ωis deﬁned as the antisymmetric component

of the geometric frequency, as follows [21]:2

ω=v×v0

v2.(20)

From a geometrical point of view, in 3 dimensions, ω v =|ω|v

can be interpreted as the azimuthal component of the velocity

v. Then, ωcan be deﬁned as azimuthal frequency.

Then, using the deﬁnition of ωabove, the torsion given in

(11) can be rewritten as:

τ=v·v0×v00

ω2v4.(21)

The vectors of the Frenet frame can be written as:

T=v

v,N=n

n,B=ω

ω,(22)

where nis the normal vector before normalization, as follows:

n=v0−ρv,

n=|n|=p|v0|2−(ρv)2.(23)

Note that, from the following property of the scalar triple

product:

a·b×c=c·a×b,(24)

the expression of the torsion can be rewritten as follows:

τ=v00 ·v×v0

ω2v4=v00 ·ω

κ2,(25)

which indicates that the torsion is null, apart from the obvious

cases v00 =0and ω=0, if v00 is perpendicular to ω. This

happens if the voltage vector is unbalanced, as illustrated in

Section V.

We are now ready to present one of the main results of this

paper. Recalling that the Frenet vectors are orthonormal and,

in particular, N=B×T, one has:

n

n=ω

ω×v

v.(26)

Noting that nis equal to the azimuthal speed, i.e. n=ωv

(see the proof in the Appendix), the expression above can be

simpliﬁed as:

n=ω×v,(27)

and, from (23):

v0=ρv+ω×v.(28)

The previous expression is relevant as it allows the deﬁnition

of a time derivative operator. Equation (28) can be, in fact,

written as: d

dt −[ρ+ω×]v=0,(29)

which leads to deﬁne the operator:

Dv

t:= d

dt −[ρ+ω×].(30)

In (30), the upper symbol xin the notation Dx

tindicates

the vector upon which the operator is applied, whereas the

2Note that in [21] the wedge product is used rather than the cross product,

as the deﬁnition of the geometric frequency is given for a voltage vector of

arbitrary dimension n. However, for simplicity but without lack of generality,

this paper focuses on the cases for which n≤3.

4

subindex tindicates the independent variable with respect to

which the operator is calculated, i.e. time.

Equation (28) and the operator (30) can be interpreted as

follows: the time derivative of a vector can be split into two

components, one symmetric (ρ) and one antisymmetric (ω×).

This operator has been obtained without any assumption on

the time dependence of vnor on its dimension. For dimensions

greater than 3, in fact, it is sufﬁcient to substitute the cross

product with the wedge product [21].

While it has been obtained considering the vector of the

voltage, the time derivative operator deﬁned in (30) can be

applied to any vector that represents the ﬁrst time derivative

of a space curve. In particular, if one considers the vector of

the current ı=q0, being qthe vector of the electric charge

in a given point of a circuit [21], then:

Dı

tı=0.(31)

For simplicity, in the remainder of this work, we focus

exclusively on the voltage. However, all examples provided

in Section V can be equivalently applied to currents.

B. Derivative of the Vector Frequency in the Frenet Frame

To obtain an expression of ω0, we start with (9) and rewrite

the vectors in terms of the derivative with respect to time,

recalling that, for the chain rule:

T0=v˙

T,N0=v˙

N,B0=v˙

B.(32)

Then, one obtains:

T0=ωN,

N0=−ωT+ξB,

B0=−ξN,

(33)

where ω=vκ from (19) and:

ξ=vτ , (34)

where ξcan be deﬁned as the torsional frequency and has the

unit of s−1, as ωand ρ. From (22) and letting ω0=d

dt ω, the

third equation of (33) can be rewritten as follows:

d

dt

ω

ω=ω0

ω−ωω0

ω2=−ξn

n,(35)

or, equivalently,

ω0=ω0ω

ω−ωξ n

n,(36)

or, equivalently,

ω0=ηω+τv×ω,(37)

where η=ω0/ω and [τv×]are the symmetric and antisym-

metric parts, respectively, of the time derivative of ω.

Equation (37) expresses the generalization of the RoCoF,

commonly utilized in power system studies as a metric of the

severity of a transient following a contingency or a power

imbalance. For a balanced system, τis null and, thus, (36)

leads to:

|ω0|=ω0,(38)

which represents the conventional deﬁnition of RoCoF. How-

ever, (36) shows that, when the torsion is not null, e.g., in

unbalanced cases, ω0is a vector with richer information than

the usual understanding of the RoCoF. Section V illustrates

(36) through numerical examples.

Finally, the time derivative of the symmetric part of the

geometric frequency deﬁned in [21] is:

ρ0=v·v00

v2+ω2−ρ2=v00

v−ρ2,(39)

which is obtained from the ﬁrst equation of (20) and (18).

C. Higher-Order Time Derivatives

The vectors of the Frenet frame constitute a basis for 3-

dimensional systems, hence any vector, including any time

derivatives of the voltage, current, and frequency can be

written in terms of N,Tand B. This can be deduced from

(33). For example, the second time derivative of Tbecomes:

T00 =ω0N+ωN0=ω0N−ω2T+ωξ B,(40)

or, equivalently, in terms of (v,n,ω):

v00 = (ρ0+ρ2−ω2)v−(2ρ−η)n−vξ ω.(41)

While the complexity of the expressions of the coefﬁcients

of higher-order derivatives increases, the following general

expression for the r-th derivative holds:

v(r)=arv+brn+crω.(42)

For example, for the ﬁrst two derivatives, one has:

a1=ρ , b1= 1 , c1= 0 ,

a2=ρ0+ρ2−ω2, b2= 2ρ−η , c2=vξ .

The following remarks are relevant.

•Expressions with the same structure as (42) can be written

also for n(r)and ω(r).

•Since n=ω×v, equation (42) is, in turn, a function

exclusively of vand ω.

V. EX AM PL ES

This section illustrates the theoretical results above through

special cases that are relevant in circuit and power system

analysis. The ﬁrst two examples are aimed at illustrating the

properties of (30) in the special cases of DC circuits and

stationary AC circuits. The subsequent examples show the

effect of imbalances and harmonics on the various components

of the frequency of three-phase voltages, as well as on the time

derivative of the frequency itself. In all examples below, we

assume that voltages are curves in three dimensions, where

the basis, unless otherwise stated, is given by the following

orthonormal vectors:

e1= (1,0,0) ,

e2= (0,1,0) ,

e3= (0,0,1) .

(43)

As already said in the theoretical sections, systems with

dimensions higher than 3 can be also considered by using

the wedge product rather than the cross product and the

generalization of the Frenet-Serret formulas. However, the

5

study of voltages with dimensions higher than three is beyond

the scope of this paper.

In Sections V-C to V-E, we show state-space three-

dimensional plots of the voltage vas this quantity has a clear

physical meaning and it is widely used in practice. However, it

is important to keep in mind that, in the proposed framework,

the voltage is, in effect, the time derivative of a position (ﬂux)

vector x=−ϕ. The curvature and torsion, thus, are those of

such a ﬂux vector, not of the voltage.

In the following, we do not discuss explicitly the behavior

of the vectors T,Nand B. However, it is relevant to observe

that the voltage vector vand the antisymmetric part of the

geometric frequency ω, are the tangent and binormal vectors,

namely Tand B, of the Frenet frame before normalization.

Moreover, n, which is the normal vector Nof the Frenet frame

before normalization, has an important role in the deﬁnition of

the time derivative of the voltage, v0, as it is the antisymmetric

part of such a derivative. Thus, the discussions on the behavior

of v,v0and ωin the examples below are, indirectly, also

discussions of the behavior of the vectors of the Frenet frame.

A. DC Voltage

As a ﬁrst example, we show the effect of (30) on DC

voltages. From the geometrical point of view, a DC voltage

is equivalent to the time derivative of a curve with one

dimension, i.e. a straight line. Using the vector notation, in

DC, the voltage is a curve that has only one component along

one direction, say e1of the basis. Then:

v=vdc e1+ 0 e2+ 0 e3.(44)

It is immediate to show that, for a curve in one dimension,

κ=τ= 0,ω=0and, the operator (30) is reformulated as:

Dv

t=d

dt −ρ . (45)

Thinking in terms of curves, this result comes with no surprise,

as a straight line cannot rotate or twist. Only the radial

component of the velocity, thus, can be nonnull. It is also

relevant to note that, according to (30), expression (45) states

that the antisymmetric component of the time derivative of DC

quantities is always null.

B. Stationary Single-Phase AC Voltage

Let us consider a stationary single-phase voltage with con-

stant angular frequency woand magnitude V. Then the voltage

vector can be written as:

v=Vcos(wot+α)e1+Vsin(wot+α)e2+ 0 e3,(46)

where αis a constant phase shift.

Note that the representation of the voltage in (46) is that

of an analytic signal, that is, the ﬁrst component is the

signal itself, namely u(t) = Vcos(wot+α), whereas the

second component is the Hilbert transform of the signal, i.e.,

H[u(t)] = Vsin(wot+α)[29], [30]. It is important to note

that analytic signals are deﬁned as complex quantities rather

than vectors but, nonetheless, they are intrinsically two dimen-

sional quantities. The utilization of the Frenet framework and

the interpretation of the signal as a curve is more general and,

as shown in this example, admits analytic signals as a special

case.

It is also relevant to note that, a single phase AC voltage

represents, from the geometric point of view, a plane curve.

This is also consistent with the fact that an AC signal requires

two independent quantities to be deﬁned, e.g., magnitude and

phase angle and, hence, the coordinate basis requires two

dimensions to be complete. Applying the deﬁnitions given in

the previous section, one can easily ﬁnd that:

ρ= 0 ,ω=woe3, ξ = 0 .(47)

These results were expected for a plane curve, which can rotate

(κ=wo/V 6= 0) but cannot twist (τ=ξ/V = 0). On the

other hand, ρ= 0 is a consequence of the fact that V= const.

Moreover, from (30), one obtains:

Dv

t=d

dt −wo[e3×].(48)

It is worth to further elaborate on (48). We note ﬁrst that,

since ρ,κand τare geometric invariants, same results can be

obtained using any other orthonormal basis. In particular, if

one chooses a basis that rotates at constant angular speed wo:

ˆ

e1= (cos(wot),0,0) ,

ˆ

e2= (0,sin(wot),0) ,

ˆ

e3= (0,0,1) ,

(49)

then the voltage vector becomes:

v=Vcos(α)ˆ

e1+Vsin(α)ˆ

e2=v1ˆ

e1+v2ˆ

e2,

with v=|v|=Vand:

v0=woˆ

e3×v=−wov2ˆ

e1+wov1ˆ

e2,(50)

where we have used the identities:

−ˆ

e1=ˆ

e3×ˆ

e2,ˆ

e2=ˆ

e3×ˆ

e1.

Equation (50) has a striking formal similarity with the well-

known expression in phasor-domain where the derivative is

given by wo, where is the imaginary unit. As a matter of

fact, in R2, the cross (wedge) vector is isomorphic to complex

numbers. Thus, one can deﬁne the following correspondences:

ˆ

e1⇒1,

ˆ

e2⇒ ,

wo[ˆ

e3×]⇒ wo,

(51)

which leads to rewrite the vector vas the well-known phasors,

namely ¯v=v1+ v2. We note also that, in two dimensions,

|ω|=wohence the azimuthal frequency coincides with the

well-known angular frequency of circuit analysis and with

the instantaneous frequency as commonly deﬁned in time-

frequency analysis and signal processing based on analytic

signals [28], [29].

Noteworthy, this observation can be generalized for any

signal u(t)for which the Hilbert transforms H[u(t)] exists.

With this assumption, the analytic signal associated with u(t)

is the following complex quantity:

¯u(t) = u(t) + jH[u(t)] = u(t) + jˆu(t),(52)

6

the instantaneous frequency of which is deﬁned as [28]:

φ0(t) = ˆu0(t)u(t)−u0(t)ˆu(t)

u2(t) + ˆu2(t),(53)

where

φ(t) = arctan ˆu(t)

u(t)(54)

is the phase angle of ¯u(t). Using the approach proposed in

this work, on the other hand, we deﬁne the vector:

u(t) = u(t)e1+ ˆu(t)e2+ 0 e3,(55)

which leads to:

ρ=u0u+ ˆu0ˆu

u2+ ˆu2,ω=ˆu0u−u0ˆu

u2+ ˆu2e3, ξ = 0 ,(56)

where the time dependency has been omitted for simplicity.

Both formulations, thus, leads to the same expression for the

instantaneous frequency, i.e., φ0=|ω|. However, the proposed

vector-based approach is more general as it allows deﬁning the

quantities ρand ξand is not limited to two dimensions.

C. Three-Phase AC Voltages

In this section, we consider a three-phase AC system.

A possible approach is to utilize the same coordinates that

we have considered for the single-phase AC voltage of the

previous example. This approach is the one commonly utilized

in circuit analysis. In this section, however, we show that

choosing as coordinates the voltages of each phase leads to

interesting results. Hence, we deﬁne the voltage vector as:

v=vae1+vbe2+vce3.(57)

Let us ﬁrst consider the case of voltages represented by a

single harmonic:

va=Vasin(θa),

vb=Vbsin(θb),

vc=Vcsin(θc).

(58)

The expressions of ρ,ωand ξfor (58) are:

ρ=PiV2

iθ0

isin(2θi) + ViV0

i(1 −cos(2θi))

v2,(59)

ω=Pijk (rjk +uj k)ei

v2,(60)

ξ=vPi(pisin(θi) + qicos(θi)) ωi

Pjk (rjk +uj k)2,(61)

where i∈ {a, b, c},ijk ∈ {abc, bca, cab},jk ∈ {bc, ca, ab}

and:

v=qPiV2

i(1 −cos(2θi)) ,

rjk = (VjV0

k−VkV0

j) sin(θj) sin(θk),

ujk =VjVk(θ0

ksin(θj) cos(θk)−θ0

jsin(θk) cos(θj)) ,

pi=ViV00

i+ (V0

i)2−Vi(θ0

i)2,

qi=V0

iθ0

i−Viθ00

i.

The following remarks are relevant.

Remark 1: In general, ρ,ωand ξdepend on the time

derivatives of both the magnitudes and the phase angles of

the three-phase voltages. This result is counter-intuitive. In

the common understanding, in fact, the angular frequency is

deﬁned as the time derivative of the sole phase angle of the

voltage (see [31]). The conventional expression of the angular

frequency is obtained if V0

i= 0. This result reiterates that the

common deﬁnition of “frequency” is, in effect, a special case

of the framework proposed in this paper.

Remark 2: It is possible to have ρ= 0 and ω6=0(for

example, the obvious case of balanced and stationary three-

phase voltages, which is illustrated below) but also ρ6= 0 and

ω=0(for example a balanced voltage with V0

i6= 0).

Remark 3: The torsional frequency ξis non-null if and

only if ω6=0. This is consequence of the expression of the

torsion given in (25). The condition ξ6= 0 holds for V00

i6= 0

and/or θ00

i6= 0.

Remark 4: A question might arise on why the base

considered in Section V-B for the single-phase AC system

is not also utilized for the three-phase system. One can,

of course, consider each phase of the three-phase system

separately, effectively considering each phase as a single-

phase system as in the previous example. This is the common

approach in three-phase circuit analysis based on phasors.

However, in this and in the following examples, the three-

phase voltages are assumed to form a three-dimensional vector.

It is relevant to note that, since ρ,κand τare geometric

invariants, it does not matter which coordinates one chooses

as long as these coordinates form a complete basis for the

system.

Stationary Voltages: Assuming that the voltage is station-

ary with constant angular frequency wo, the three components

of the vector are:

va=Vasin(wot+θao),

vb=Vbsin(wot+θbo),

vc=Vcsin(wot+θco),

(62)

where Viis the voltage magnitude of phase i; and θio denotes

the voltage angle of phase iat t= 0 s. Let wo= 100πrad/s,

θao = 0 rad. Next, we consider relevant special cases of (58).

Positive and negative sequence voltages: From (59)-(61),

one has for the stationary positive sequence with Va=Vb=

Vc= const.and θbo =−θco =−2π/3rad:

ρ=ξ= 0 ,ω=wo

√3(e1+e2+e3).

Analogously, the stationary negative sequence, namely Va=

Vb=Vc= const.and θbo =−θco = 2π/3rad, leads to:

ρ=ξ= 0 ,ω=−wo

√3(e1+e2+e3).

Hence, for the stationary positive and negative sequences, the

time derivative operator (30) becomes:

Dv

t=d

dt ∓wo

√3[(e1+e2+e3)×],(63)

and, as in the example of Section V.B, ω=|ω|=wo, namely

the azimuthal and angular frequencies coincide.

7

0.000 0.005 0.010 0.015 0.020

Time [s]

−20

−10

0

10

20

Voltage [V]

va

vb

vc

(a) E0: voltage components

0.000 0.005 0.010 0.015 0.020

Time [s]

−20

−10

0

10

20

Voltage [V]

(b) E1: voltage components

0.000 0.005 0.010 0.015 0.020

Time [s]

−20

−10

0

10

20

Voltage [V]

(c) E2: voltage components

0.000 0.005 0.010 0.015 0.020

Time [s]

−2

−1

0

1

2

ρ[pu(rad/s)]

E0

E1

E2

(d) E0-E2: ρ

0.000 0.005 0.010 0.015 0.020

Time [s]

0.8

1.0

1.2

1.4

ω[pu(rad/s)]

(e) E0-E2: ω

0.000 0.005 0.010 0.015 0.020

Time [s]

−1.0

−0.5

0.0

0.5

1.0

ξ[pu(rad/s)]

(f) E0-E2: ξ

Fig. 1: E0-E2: Three-phase AC voltage components, and geometric invariants ρ,ωand ξ.

va[V]

−10 −50510

vb[V]

−10−5

0

5

10

vc[V]

−10

−5

0

5

10

E0

E1

E2

Fig. 2: Three-phase voltage in the space (va, vb, vc), E0-E2.

Unbalanced voltage: Unbalanced voltages are character-

ized by Va6=Vb6=Vcand/or θbo 6=−θco 6=−2π/3rad.

Then, assuming that the voltage magnitudes and phase shifts

are constants, from (59)-(61), one has:

ρ=woPiV2

isin(2θi)

v2, ξ = 0 ,

ω=woPijk VjVksin(θj−θk)ei

v2.

In this case, thus, the components of the time derivative in

(28) depend on the parameters of the voltage.

Zero-sequence voltage: This example allows us dis-

cussing the issue of the choice of the basis for the voltage

vector. A zero-sequence voltage is composed of three equal

AC voltages. In this case, thus, we cannot utilize the same

coordinates we have used in the previous two examples,

namely (va, vb, vc), as these are linearly dependent and do

not form a basis. One has to proceed as discussed in Section

V-B for a single-phase AC voltage, for which ω=woe3.

Thus, when considering three-phase voltages, one should thus

ﬁrst remove the zero-sequence from the voltage signals. This

is common practice, anyway, for the zero-sequence to be

generally treated separately or simply just removed from the

positive and negative ones in power system measurements and

protections.

Next, we illustrate (62) through the following examples.

E0 : Va=Vb=Vc= 12 V ,

θbo =−θco =−2π/3 rad .

E1 : Va= 12 V , Vb= 8 V , Vc= 12 V ,

θbo =−θco =−2π/3 rad .

E2 : Va=Vb=Vc= 12 V ,

θbo =−2π/3 rad , θco = 1.5π/3 rad .

Figure 1 shows the phase voltages as well as the symmetric

component of the geometric frequency (ρ), the Euclidean norm

of the vector component (ω), and the torsional frequency (ξ)

for examples E0 to E2. Figure 2 illustrates the curve formed

by the three-phase voltage in the space (va, vb, vc). Notice that

ξ= 0 holds for the three cases. This is consistent with the

fact that each curve in Fig. 2 lies in a plane.

D. Three-Phase AC Voltages with Harmonics

In this section, we consider the case of three-phase voltages

with harmonics. The conventional Fourier analysis deﬁne a

basis that consists of as many dimensions as harmonics. While

the proposed approach can be also utilized in an arbitrary

n-dimensional space, we illustrate the consequences of the

proposed approach in the same three-dimensional space we

have utilized in the previous section.

Let the voltage vector in (57) also include a harmonic

8

0.000 0.005 0.010 0.015 0.020

Time [s]

−20

−10

0

10

20

Voltage [V]

va

vb

vc

(a) E3: voltage components

0.000 0.005 0.010 0.015 0.020

Time [s]

−20

−10

0

10

20

Voltage [V]

(b) E4: voltage components

0.000 0.005 0.010 0.015 0.020

Time [s]

−20

−10

0

10

20

Voltage [V]

(c) E5: voltage components

0.000 0.005 0.010 0.015 0.020

Time [s]

−1

0

1

ρ[pu(rad/s)]

E3

E4

E5

(d) E3-E5: ρ

0.000 0.005 0.010 0.015 0.020

Time [s]

1

2

ω[pu(rad/s)]

(e) E3-E5: ω

0.000 0.005 0.010 0.015 0.020

Time [s]

−10

−5

0

5

10

ξ[pu(rad/s)]

(f) E3-E5: ξ

Fig. 3: Three-phase voltage components, and geometric invariants ρ,ωand ξ, E3-E5.

voltage component. Then, the three-phase voltage becomes:

va=Vasin(wot+θao) + Va,h sin(hwot+θao,h),

vb=Vbsin(wot+θbo) + Vb,h sin(hwot+θbo,h),

vc=Vcsin(wot+θco) + Vc,h sin(hwot+θco,h),

(64)

where Vi,h is the magnitude of the h-th harmonic of phase i;

θio,h is the angle of the h-th harmonic of phase iat t= 0 s.

Let Va=Vb=Vc= 12 V, θbo =−θco =−2π/3rad. For the

sake of example, we consider the following cases of (64): E3,

balanced voltage; E4, phase imbalance in the harmonic; E5,

magnitude imbalance in the harmonic. The following values

are used.

E3 : Va,11 =Vb,11 =Vc,11 = 0.5 V ,

θbo,11 =−θco,11 =−2π/3 rad .

E4 : Va,11 =Vb,11 =Vc,11 = 0.5 V ,

θbo,11 =−2.7π/3 rad θco,11 = 2.7π/3 rad .

E5 : Va,11 = 0.5 V , Vb,11 = 0.9 V , Vc,11 = 1.3 V ,

θbo,11 =−θco,11 =−2π/3 rad .

The phase voltages for cases E3-E5 as well as the trajec-

tories of ρ,ωand ξare shown in Fig. 3, while the curve

that the vector vforms in the (va, vb, vb)space is illustrated

for each case in Fig. 4. For the balanced case, i.e. E3, a

plane curve is obtained which implies a null torsion, whereas

the curves in E4 and E5 are three-dimensional and thus the

torsion in these cases is non-zero. It is relevant to note that

the proposed approach, differently from Fourier analysis or

any other approach based on the projections of the signal

on a kernel function (e.g., the periodic small-signal stability

analysis [32], wavelets and the Hilbert-Huang transform [33]),

does not need to increase the size of the basis to take into

account harmonics.

va[V]

−10 −50510

vb[V]

−8

−6

−4

−2

02468

vc[V]

−10

−5

0

5

10

E3

E4

E5

Fig. 4: Three-phase voltage in the (va, vb, vc)space, E3-E5.

E. Three-Phase AC Voltages with Time-Variant Angular Fre-

quency

In the previous section, we have discussed the difference

between the Fourier approach and the proposed geometric

approach. The latter utilizes a space that has always same

dimensions regardless the number of harmonics present in the

voltage. In this section, we further illustrate the beneﬁt of this

frugality of dimensions. We consider in fact a three-phase

voltage with time-varying angular frequency. In this case,

the Fourier transform would require a basis with inﬁnitely

many dimensions as the angular frequency varies continuously.

The geometric approach, on the other hand, retains the three

dimensions.

Let us consider that each component of the voltage vector

in (57) has a time-varying frequency wi,i={a, b, c}. Then,

9

012345

Time [s]

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

ρ[pu(rad/s)]

E6

E7

(a) E6 and E7: ρ.

012345

Time [s]

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

ω[pu(rad/s)]

(b) E6 and E7: ω.

012345

Time [s]

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

ξ[pu(rad/s)]

(c) E6 and E7: ξ.

012345

Time [s]

−0.2

−0.1

0.0

0.1

0.2

ρ[pu(rad/s)]

E6

E8

(d) E6 and E8: ρ.

012345

Time [s]

0.8

0.9

1.0

1.1

1.2

ω[pu(rad/s)]

(e) E6 and E8: ω.

012345

Time [s]

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

ξ[pu(rad/s)]

(f) E6 and E8: ξ.

Fig. 5: Geometric frequency components and torsional frequency, E6-E8.

the three AC voltage components are:

va=Vasin(wot+θa(t) + θao),

vb=Vbsin(wot+θb(t) + θbo),

vc=Vcsin(wot+θc(t) + θco).

(65)

Let Va=Vb=Vc= 12 V, wo= 100πrad/s, θbo =−θco =

−2π/3rad. As an example, we consider the following cases of

(65): E6, balanced θi; E7, frequency imbalance in θi; and E8,

magnitude imbalance in θi. The following values are assumed.

E6 : θa=θb=θc=πsin(0.4πt).

E7 : θa=θb=πsin(0.4πt),

θc=πsin(0.44πt).

E8 : θa=θb=πsin(0.4πt),

θc= 1.1πsin(0.4πt).

Example E6 is representative of the transient following a

contingency in a power system, where the oscillations of the

phase angles of the voltages are due to the electro-mechanical

swings of the synchronous machines. On the other hand,

examples E7 and E8 do not represent a situation that can occur

in a power system but are nevertheless relevant to show the

effects of the torsional frequency.

Figure 5 shows ρ,ωand ξfor cases E6-E8. In the bal-

anced case, i.e. E6, the three-phase voltages show the same

frequency variation and null ρand ξ. The three-dimensional

representation of vin the space va-vb-vcis shown in Figs. 6

and 7. For E7 and E8, the period of the voltage is varying

with time yet not in the same way in all three phases, thus

leading to the curves shown in Figs. 6 and 7.

Figure 8 shows how the magnitude of the rate of change

of the vector frequency (|ω0|) compares to the magnitude of

its symmetric component, i.e. |ηω|(see equation (36)) for the

three examples E6-E8. These two quantities are equal only

at the time instants when the torsion is null or, equivalently,

when ξ= 0 (see Fig. 5f). It is also interesting to observe how

unbalanced harmonics lead to large variations of |ω0|.

va[V]

−10 −50510

vb[V]

−10.0

−7.5

−5.0

−2.5

0.0

2.5

5.0

7.5

10.0

vc[V]

−15

−10

−5

0

5

10

15

E6

E7

Fig. 6: Three-phase voltage in (va, vb, vc)space, E6 and E7.

va[V]

−10 −50510

vb[V]

−10.0

−7.5

−5.0

−2.5

0.0

2.5

5.0

7.5

10.0

vc[V]

−15

−10

−5

0

5

10

15

E6

E8

Fig. 7: Three-phase voltage in (va, vb, vc)space, E6 and E8.

10

012345

Time [s]

0.0000

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

0.0175

0.0200

RoCoF [pu(rad/s)/s]

|ηω|

|ω0|

(a) E6

012345

Time [s]

0

200

400

600

800

RoCoF [pu(rad/s)/s]

|ηω|

|ω0|

(b) E7

012345

Time [s]

0

20

40

60

80

100

120

140

RoCoF [pu(rad/s)/s]

|ηω|

|ω0|

(c) E8

Fig. 8: Magnitudes of ω0and its symmetrical component ω0=|ηω|, E6-E8.

F. Park Transform

In this ﬁnal example, we consider the time derivative of the

voltage in the Park reference frame. Let us consider a voltage

vector in the dqo coordinates:

v=vded+vqeq+voeo.(66)

The time derivative of this vector in the “inertial” reference is

given by:

v0= (v0

d−wdqvq)ed+ (v0

q+wdqvd)eq+v0

oeo,(67)

where wdq is the angular speed of the Park reference frame

and we have assumed a choice of the dqo-axis such that e0

d=

wdqeq,e0

q=−wdqed, and e0

o=0[1].

The time derivative in the inertial reference in (67) is

composed of two parts, namely the derivative in the rotating

dqo-axis frame plus the effect of rotation:

v0=ˆ

v0+r×v,(68)

where:

ˆ

v0=v0

ded+v0

qeq+v0

oeo,(69)

and

r=wdq eo.(70)

Next, we apply (28) and show that the proposed approach

leads to the same results as (67) but with a different structure

of the components of v0with respect to (68). The quantities

ρand ωare:

ρ=vdv0

d+vqv0

q+vov0

o

v2,(71)

and

ω=vqv0

o−vov0

q−wdqvovd

v2ed+

vov0

d−vdv0

o−wdqvovq

v2eq+

vdv0

q−vqv0

d+wdq(v2

d+v2

q)

v2eo,

(72)

where v2=v2

d+v2

q+v2

o. Although requiring some tedious

algebraic manipulations, it is not difﬁcult to show that, in fact,

ρv+ω×vis equal to the right-hand side of (67). On the

other hand, it is straightforward to observe that:

ˆ

v06=ρv,r×v6=ω×v,(73)

i.e., ˆ

v0is not equal to the symmetric component of the time

derivative and the term r×vis not equal to the antisymmetric

component of the time derivative.

A relevant case is for vo= 0, namely balanced conditions,

which leads to:

ρ=vdv0

d+vqv0

q

v2,ω=vdv0

q−vqv0

d+wdqv2

v2eo,(74)

where v2=v2

d+v2

q. We note that ρ=v0/v, as expected, and:

vdv0

q−v0

dvq

v2=d

dt arctan vq

vd= ∆ω , (75)

which is the deviation of the angular frequency of vwith

respect to wdq. Hence the inertial- and rotating-frame time

derivatives of the voltage can be written as:

v0=ρv+ (wdq + ∆ω)eo×v,(76)

and:

ˆ

v0=ρv+ ∆ωeo×v.(77)

The following remarks are relevant.

Remark 5: The only case for which ρv=ˆ

v0and r×v=

ω×vis if the angular speed of the Park transform is the actual

frequency of v. In this case, in fact, ∆ω= 0. This result is

consistent with the commonly-used Park-Concordia model of

the synchronous machine [34].

Remark 6: The Clarke transform can be viewed as a

special case of the Park transform, with wdq = 0. In this

case (68) leads to v0=ˆ

v0. This result is consistent with (76).

Remark 7: In stationary conditions, i.e., for v0

d=v0

q= 0,

(76) leads to the same expression as (50), thus conﬁrming that

balanced stationary three-phase voltages are equivalent to a

single-phase phasor.

G. IEEE 39-Bus System

This example shows the application of the proposed for-

mulas to estimate ω,ρ, and ξat a bus of a power system

following a contingency. To this aim, we use the model

of the IEEE 39-bus system for Electro-Magnetic Transient

(EMT) simulations provided by DIgSILENT PowerFactory.

The system model is based on the original IEEE 39-bus

benchmark network, which has been modiﬁed to capture the

behavior during EMTs of the power network, namely, the

frequency dependency of transmission lines and the non-linear

saturation of transformers.

11

The system is numerically integrated assuming a phase-to-

phase fault between phases aand bat terminal bus 3 of the

system at t= 0.2s. The fault is cleared at t= 0.3s. The

integration time step considered is 10−5s. The phase voltages

at bus 26 following the contingency are shown in Fig. 9,

while the curve formed by the three-phase voltage in the space

(va, vb, vc)is illustrated in Fig. 10. Figure 11 shows ω,ρ, and ξ

following the contingency, where ﬁrst-order ﬁltering has been

applied to smooth the numerical noise in the calculation of the

voltage vector time derivatives. Before the occurrence of the

fault, the three phases are balanced and thus, the corresponding

part of the curve in Fig. 10 is circular and lies in a plane.

The same holds after the fault clearance. Results also indicate

that before the occurrence and after the clearance of the fault,

both ρand ξare null, which is consistent to the discussion

of Section V-E (e.g., example E6). On the other hand, the

voltage phases are unbalanced during the fault, which gives

rise to the non-circular and non-planar sections observed in

Fig. 10. For this part, both ρand ξare non-zero as shown

in Fig. 11. This is again consistent with Section V-E and in

particular with the discussion of example E8. Furthermore, ω

accurately captures the primary frequency response at bus 26

following the contingency (see Fig. 11b).

Finally, for the sake of comparison, we mention that the

differences of the IEEE 39-bus system with respect to the

results of E8 are that (i) the frequency oscillation is damped

and reaches a new steady state condition, whereas this does

not hold for E8, and (ii) the imbalance occurs only for few

voltage cycles, whereas in E8 voltages are unbalanced during

the whole simulation, which is the reason why the curve in

Fig. 10 does not appear like a compact three-dimensional

object as is the case in Fig. 7.

0.15 0.20 0.25 0.30 0.35 0.40

Time [s]

−300

−200

−100

0

100

200

300

Voltage [kV]

va

vb

vc

Fig. 9: Three-phase voltage at bus 26, IEEE 39-bus system.

VI. CONCLUSIONS

The paper elaborates on the geometrical interpretation of

electric quantities and deduces several expressions that link

the time derivatives of the voltage, current and frequency

in electrical circuits with the Frenet frame. Among these

expressions, we mention in particular (28) and (37). Equation

(28) indicates that the time derivative of the voltage (and

the current) is composed of two parts, one symmetric, that

va[kV]

−300

−200

−100 0100 200 300

vb[kV]

−200

−100

0

100

200

vc[kV]

−300

−200

−100

0

100

200

300

Fig. 10: Three-phase voltage at bus 26 in the space (va, vb, vc), IEEE

39-bus system.

depends only on the magnitude, and one antisymmetric that

depends on the “rotation” of the quantity itself. Equation (37)

shows that the time derivative of the vector frequency is more

complex than the common notion of RoCoF and includes a

“rotational” and a “torsional” component. The latter is deﬁned

in this paper for the ﬁrst time. It is interesting to note that

the antisymmetric component of the RoCoF may affect the

implementation and/or performance of existing controllers.

Since the proposed approach allows separating the symmetric

and antisymmetric terms, it appears as a useful tool for the

study of power system transients and the design of controllers.

More in general, we believe that the proposed approach may

ﬁnd relevant applications in estimation, control and stability

analysis of power systems.

The proposed theory is certainly more complex than the

current conventional approach based on phasors. However, it

shows added values from the theoretical point of view, as

follows.

•It is a generalization of the conventional approach. The

conventional approach, in fact, appears to be a special

case of the proposed theory.

•It is an example of interdisciplinary approach. Differential

geometry and the Frenet frame, in fact, were originally

developed for mechanical systems. Their applications,

under certain hypotheses, to electrical circuits appears as

an interesting advance which paves the way to several

further developments.

An interesting byproduct of the latter point is that the

proposed theory allows “visualizing” electrical quantities. This

is important, as, in the experience of the ﬁrst author, students

always struggle with the lack of visual aid when studying

circuit theory. Such a support is a given in mechanical en-

gineering. Thus, the ability to re-utilize well-known concepts

such as curvature and torsion also adds a didactic value to the

proposed approach.

We anticipate several future work directions. Among these,

we mention the development of a geometric framework for

circuit analysis; the applications of the formulas to estimate

12

0246810

Time [s]

−0.125

−0.100

−0.075

−0.050

−0.025

0.000

0.025

0.050

0.075

ρ[pu(rad/s)]

(a) ρ

0246810

Time [s]

0.9900

0.9925

0.9950

0.9975

1.0000

1.0025

1.0050

1.0075

1.0100

ω[pu(rad/s)]

(b) ω

0246810

Time [s]

−0.100

−0.075

−0.050

−0.025

0.000

0.025

0.050

0.075

0.100

ξ[pu(rad/s)]

(c) ξ

Fig. 11: Geometric invariants ρ,ωand ξat bus 26, IEEE 39-bus system.

unbalanced conditions in three-phase circuits as well as to

circuits with more than three phases using Cartan’s extensions

of the Frenet framework (see, e.g., [35]); and the development

of active controllers to reduce the effect of harmonics and

imbalances.

APPENDIX

In this appendix, we prove the identity n=ωv. From (20)

on has:

ω=|ω|

v2=|v×v0|

v2.(78)

Let us focus on the term |v×v0|. This can be written as:

|v×v0|=p(v×v0)·(v×v0).(79)

From the following identity of the triple scalar product:

a·b×c=b·c×a,(80)

equation (79) can be rewritten as:

|v×v0|=pv·v0×(v×v0).(81)

Then, from the following identity of the triple vector product:

(a×b)×c= (a·c)b−(b·c)a,(82)

equation (81) can be rewritten as:

|v×v0|=pv·[(v0·v0)v−(v0·v)v0].(83)

From (20) and (18), the previous expression is equivalent to:

|v×v0|=p(v0·v0)(v·v)−(v·v0)2

=p|v0|2v2−ρ2v4,(84)

and, hence, (78) becomes:

ω=p|v0|2−ρ2v2

v,(85)

which, recalling the deﬁnition of ngiven in (23), demonstrates

that n=ωv and, hence, n=ω×v. From this relationship and

the properties of the vectors of the Frenet frame, the following

relationships follow:

v=n×ω

ω2,ω=v×n

v2.(86)

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Federico Milano (F’16) received from the Univ. of

Genoa, Italy, the ME and Ph.D. in Electrical Engi-

neering in 1999 and 2003, respectively. From 2001

to 2002 he was with the University of Waterloo,

Canada, as a Visiting Scholar. From 2003 to 2013,

he was with the Universtiy of Castilla-La Mancha,

Spain. In 2013, he joined the University College

Dublin, Ireland, where he is currently Professor

of Power Systems Protection and Control. He is

an IEEE PES Distinguished Lecturer, an editor of

the IEEE Transactions on Power Systems and an

IET Fellow. He is the chair of the IEEE Power System Stability Controls

Subcommittee. His research interests include power system modelling, control

and stability analysis.

Georgios Tzounas (M’21) received from National

Technical University of Athens, Greece, the Diploma

(ME) in Electrical and Computer Engineering in

2017, and the Ph.D. in Electrical Engineering from

University College Dublin, Ireland, in 2021. He is

currently a post doctoral researcher with Univer-

sity College Dublin, working on the Horizon 2020

project edgeFLEX. His research interests include

modelling, stability analysis, and automatic control

of power systems.

Ioannis Dassios received his Ph.D. in Applied

Mathematics from the Dpt of Mathematics, Univ. of

Athens, Greece, in 2013. He worked as a Postdoc-

toral Research and Teaching Fellow in Optimization

at the School of Mathematics, Univ. of Edinburgh,

UK. He also worked as a Research Associate at

the Modelling and Simulation Centre, University

of Manchester, UK, and as a Research Fellow at

MACSI, Univ. of Limerick, Ireland. He is currently

a UCD Research Fellow at UCD, Ireland.

Taulant K¨

erc¸i (S’18) received from the Polytech-

nic University of Tirana, Albania, the BSc. and

MSc. degree in Electrical Engineering in 2011 and

2013, respectively. From June 2013 to October 2013,

he was with the Albanian DSO at the metering

and new connection department. From November

2013 to January 2018, he was with the TSO at the

SCADA/EMS ofﬁce. Since February 2018, he is a

Ph.D. candidate with UCD, Ireland. In September

2021, he joined the Irish TSO, EirGrid. His research

interests include power system dynamics and co-

simulation of power systems and electricity markets.