Applications of the Frenet Frame to Electric Circuits

Abstract

The paper discusses the relationships between electrical 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 between 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.

Full text

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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

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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.

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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׈
e2,ˆ
e2=ˆ
e3׈
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.