Complex Frequency

Abstract

The paper introduces the concept of complex frequency. The imaginary part of the complex frequency is the variation with respect of a synchronous reference of the local bus frequency as commonly defined in power system studies. The real part is defined based on the variation of the voltage magnitude. The latter term is crucial for the correct interpretation and analysis of the variation of the frequency at each bus of the network. The paper also develops a set of differential equations that describe the link between complex powers and complex frequencies at network buses in transient conditions. No simplifications are assumed except for constant elements of the network admittance matrix. A variety of analytical and numerical examples show the applications and potentials of the proposed concept.

Full text

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Complex Frequency
Federico Milano, IEEE Fellow
Abstract— The paper introduces the concept of complex fre-
quency. The imaginary part of the complex frequency is the
variation with respect of a synchronous reference of the local
bus frequency as commonly defined in power system studies.
The real part is defined based on the variation of the voltage
magnitude. The latter term is crucial for the correct interpre-
tation and analysis of the variation of the frequency at each
bus of the network. The paper also develops a set of differential
equations that describe the link between complex powers and
complex frequencies at network buses in transient conditions. No
simplifications are assumed except for constant elements of the
network admittance matrix. A variety of analytical and numerical
examples show the applications and potentials of the proposed
concept.
Index Terms— Power system dynamics, converter-interfaced
generation, frequency control, low-inertia systems.
I. INTRODUCTION
A well-known and accepted definition of the frequency
of a signal x(t) = Xm(t) cos(ϑ(t)) is given in the IEEE
Std. IEC/IEEE 60255-118-1 [1], as follows:
f(t) = 1
2π˙
ϑ(t) = 1
2π˙
θ(t) + fo,(1)
where θis the phase difference, in radians, between the angular
position ϑ, also in radians, of the signal x(t)and the phase
due to the reference nominal frequency fo, expressed in Hz.
If the magnitude Xmof the signal is constant, this definition
is adequate. However, if Xmchanges with time, the definition
of the frequency in (1) does not provide a meaningful way
to separate the effects of the variations of ϑand Xm. Thus,
the definition of frequency in the most general conditions is a
highly controversial concept that has been discussed at length
in the literature (see the interesting discussion in [2] and the
references therein).
This paper provides a novel interpretation of “frequency” as
complex quantity, i.e., composed of a real and an imaginary
part. This complex frequency takes into account the time
dependency of both ϑand Xm. The focus is on the theory,
modeling, simulation and some application aspects of the
proposed definition. The proposed complex frequency allows
a neat and compact representation as well as a consistent
interpretation of frequency variations in ac power systems.
The complex frequency is also capable of explaining the
interactions among active and reactive power injections at
buses and flows in network branches. It is important to note
that the proposed approach does not attempt to substitute
F. Milano is with the School of Electrical & Electronic Engineering,
University College Dublin, Belfield, Ireland. E-mail: federico.milano@ucd.ie
This work was supported by Science Foundation Ireland, by funding
F. Milano under project AMPSAS, Grant No. SFI/15/IA/3074; and by the
European Commission by funding F. Milano under project EdgeFLEX, Grant
No. 883710.
the modelling approaches that go beyond the classical phasor
representation or that focus on analysis of non-sinusoidal
signals (see, for example, [3] for a state-of-the-art survey on
this topic and the several references therein). On the contrary,
the proposed concept of “complex frequency” is compatible
with the approaches that have been proposed in the literature
as it allows interpreting angle and magnitude variations as
complementary compoenents of the same phenomenon, pro-
vided that one accepts to extend the domain of frequency to
the complex numbers.
This paper focuses on electro-mechanical transients in high-
voltage transmission systems. Thus, the starting point is sim-
ilar to that of [4]–[7], that is, the transient conditions during
which the magnitude and the phase angle of bus voltage
phasors change according to the inertial response of syn-
chronous machines and the frequency control of synchronous
and non-synchronous devices. On the other hand, harmonics,
unbalanced conditions and electro-magnetic transients are not
taken into consideration.
The resulting formulation is exact, in the measure that power
system models based on the dqo transform for voltage and
angle stability analysis are exact; general, as it provides a
framework to study the dynamic effect of any device on the
local frequency variations at network buses; and systematic,
because it provides with the tools to determine analytically
the impact of each device on bus frequencies.
The remainder of the paper is organized as follows. Section
II provides the background for the proposed theoretical frame-
work. Section III provides the formal definition of complex
frequency and its link with complex power injections, voltage
and current dynamic phasors and network topology. The
special cases of constant power and constant current injections
as well as constant impedances are discussed in Section III-
C. Section III-D discusses a variety of relevant approximated
expressions that link the complex frequency to bus power
injections. Section IV illustrates some applications of the
analytical expressions derived in Section III to simulation, state
estimation and control. Finally, Section V draws conclusions
and outlines future work.
II. BACKGROU ND
The starting point is the set of equations that describe the
complex power injections, in per unit, at the nnetwork buses
of the system, say ¯
s∈Cn, as follows:
¯
s(t) = p(t) + q(t) = ¯
v(t)◦¯
ı∗(t),(2)
where p∈Rn×1and q∈Rn×1are the active and reactive
power injections at network buses, respectively; ¯
v∈Cn×1
and ¯
ı∈Cn×1are the dynamic voltage and injection current
phasors at network buses; ∗indicates the conjugate of a
arXiv:2105.07769v1 [eess.SY] 17 May 2021

2
complex quantity; and ◦is the Hadamard product, i.e. the
element-by-element product of two vectors.1It is important to
note that (2) is valid in transient conditions [8]. On the other
hand, in steady-state, balanced conditions, (2) expresses the
well-known power flow equations.
In this context, dynamic phasor means the dq-axis compo-
nents of the well-known dqo transform of the voltages and
currents. For example, for the voltage, one has:
¯
v(t) = vd(t) + vq(t).(3)
where the components vd,k and vq,k of the k-th element of
the vector ¯
vare calculated as follows:


vd,k(t)
vq,k(t)
vo,k(t)

=P(t)

va,k(t)
vb,k(t)
vc,k(t)

,(4)
where
P(t) = q2
3

cos(α(t)) cos(α0(t)) cos(α00(t))
sin(α(t)) sin(α0(t)) sin(α00(t))
1
√2
1
√2
1
√2

,(5)
and αis the angle between the phase aand the q-axis, with
˙α=ωo, and α0=α−2π
3and α00 =α+2π
3. The same
transformation (4) is applied to the abc currents. Since no
assumption is made on the abc quantities, the d- and q-axis
components of the dynamic phasors ¯
vand ¯
ıand, hence, (2)
are valid in transient conditions, i.e., for non-sinusoidal abc
quantities.
The vo,k is the o-axis or zero component and is null for
balanced systems. If the system is not balanced and the o-
axis components are not null, then the vector pin (2) does
not represent the total active power injections at network
buses as it does not include the term vo◦ıo. The hypothesis
of balanced system is not necessary for the developments
presented below. However, since the focus is on high-voltage
transmission systems, in the remainder of this paper, balanced,
positive sequence operating conditions are assumed.
For the purposes of the developments given below, it is
convenient to rewrite (3) in polar form:
¯
v(t) = v(t)◦∠θ(t),(6)
where v=|¯
v|,∠θ= cos(θ) + sin(θ)and
θ(t) = ϑ(t)−θo(t),(7)
namely, θis the vector of bus voltage phase angles referred
to the rotating dq-axis reference frame, ϑare the bus voltage
phase angles referred to a constant reference and θo=Rtωodt
is the angle of the rotating dq-axis reference frame and ωois
the angular frequency in rad/s of the dq-axis reference frame.
From (1), the time derivative of θgives:
ω(t) = ˙
θ(t) = ˙
ϑ(t)−ωo(t),(8)
where ωis the vector of frequency deviations with respect
to the reference frequency at the network buses. In [1], it is
assumed that ωo= 2πfois constant and equal to the nominal
1The Hadarmard product of two column vectors xand zcan be also written
as x◦z= diag(x)z, where diag(x)is a diagonal matrix whose element
(i, i)is the i-th element of the vector x.
angular frequency of the grid, e.g., ωo= 2π60 rad/s in North
American transmission grids. Note, however, that ωobeing
constant is not a requirement of the derivations given in the
remainder of this paper. As a matter of fact, in the examples
presented in Section IV, ωois set to be equal to the frequency
of the Center of Inertia (CoI).
We now introduce the only approximation of the whole
derivations given in this section, i.e., we assume that the link
between current injections and voltages is given by
¯
ı(t)≈¯
Y¯
v(t),(9)
where ¯
Y=G+B∈Cn×nis the conventional admittance
matrix of the network. It is important not to confuse (9)
with the conventional relationship between current and voltage
phasors (in which case (9) is an exact equality). ¯
ıand ¯
v
are “dynamic” complex quantities and, hence, (9) represents
an approximation of the dynamics of the grid. In turn, to
obtain (9), it is assumed that, for network inductances and
capacitances the relationships between voltages and currents
can be approximated with:
¯v=L˙
¯ı≈ωoL¯ı=X¯ı ,
¯ı=C˙
¯v≈ωoC¯v=B¯v , (10)
where L,C,X,Bare the inductance, capacitance, reac-
tance and susceptance, respectively. The approximation above
assumes that electro-magnetic transients in the elements of
the transmission lines and transformers are fast and can
be assumed to be in Quasi-Steady-State (QSS). This is the
conventional approximation utilized in RMS models for angle
and voltage stability analysis [9]. The focus of this paper is,
in fact, on the time scales of electro-mechanical and primary
frequency and voltage control transients, which are a few
orders of magnitude slower than electro-magnetic dynamics.
Merging (2) and (9) becomes:
¯
s(t) = ¯
v(t)◦[¯
Y¯
v(t)]∗.(11)
These equations resemble the well-known power flow equa-
tions except for the fact that the voltages and, hence, the power
injections at buses are time-varying quantities.
A. Time Derivative of Algebraic Equations
An important aspect of the developments discussed in the
next section is whether (11) can be differentiated with respect
to an independent variable and, in particular, with respect to
time. With this aim, observe that (9) leads to the well-known
QSS model for power system angle and voltage transient
stability analysis, as follows [9], [10]:
˙
x=f(x,y,z),
0=g(x,y,z),(12)
where f∈Rnx+ny+nz7→ Rnxare the differential equations;
g∈Rnx+ny+nz7→ Rnyare the algebraic equations, x∈ X ⊂
Rnxare the state variables; y∈ Y ⊂ Rnyare the algebraic
variables; and z∈ Z ⊂ Rnzare discrete variables that defines
events such as line outages and faults. In practice, discrete
variables can be modelled as if-then rules that modify the
structure of fand g, and, hence, they do not require additional
equations.

3
The set of Differential-Algebraic Equations (DAEs) in (12)
is continuous except for a finite set of points where the discrete
variables zchange their value. The implicit function theorem
indicates that, if the Jacobian matrix ∂g/∂yis not singular,
there exists a function φsuch that:
y=φ(x,z).(13)
Equation (13) is often utilized to reduce the set of DAEs in
(12) into a set of Ordinary Differential Equations (ODEs)
that depends only on xand z. In this work, however, (13)
is utilized the other way round, i.e., to guarantee that it is
possible to define the time derivative of yexcept for the finite
number of points where an element of vector zchanges value.
This condition leads to:
˙
y=∂φ
∂x˙
x=∂g
∂y−1∂g
∂x˙
x
=∂g
∂y−1∂g
∂xf(x,φ(x,z),z).
(14)
The condition (14) implies that the set of DAEs in (12) is
assumed to be index 1 [11], which is the form of DAEs
that describes most physical systems, including power systems
[10].
The voltage magnitudes vand phase angles θthat appears
in (11), and hence also the real and imaginary parts of the
complex power ¯
s, are algebraic variables in the conventional
formulation of QSS models. Thus, the assumption of index-
1 DAEs allows rewriting the current injections at bus and,
hence, the complex power ¯
sas functions of state and discrete
variables, as well as of the bus voltages ¯
v, namely ¯
s(¯
v,x,z),
which are smooth, except at the points where the elements of z
transition from one value to another. Then, the time derivatives
of ¯
scan be computed with the chain rule as:
˙
¯s=∂¯
s
∂¯
v˙
¯v+∂¯
s
∂x˙
x.(15)
The next section of this paper elaborates on (15) and deduces
an expression that involves the concept of complex frequency.
III. DERIVATION
For the sake of the derivation, it is convenient to drop
the dependency on time and rewrite (11) in an element-wise
notation. For a network with nbuses, one has:
ph=vhPn
k=1vk[Ghk cos θhk +Bhk sin θhk ],
qh=vhPn
k=1vk[Ghk sin θhk −Bhk cos θhk ],(16)
where Ghk and Bhk are the real and imaginary parts of the
element (h, k)of the network admittance matrix, i.e. ¯
Yhk =
Ghk +Bhk;vhand vkdenote the voltage magnitudes at
buses hand k, respectively; and θhk =θh−θk, where
θhand θkare the voltage phase angles at buses hand k,
respectively. Equations (16) and all equations with subindex h
in the remainder of this section are valid for h= 1,2, . . . , n.
Equations (16) can be equivalently written as:
ph=Pn
k=1phk ,and qh=Pn
k=1qhk ,(17)
where
phk =vhvk[Ghk cos θhk +Bhk sin θhk],
qhk =vhvk[Ghk sin θhk −Bhk cos θhk].(18)
Differentiating (16) and writing the active power injections
as the sum of two components:
dph=Pn
k=1
∂ph
∂θhk
dθhk +Pn
k=1
∂ph
∂vk
dvk≡dp0
h+dp00
h,(19)
In (19), dphis the total variation of power at bus h;dp0
his the
quota of the active power that depends on bus voltage phase
angle variations; and dp00
his the quota of active power that
depends on bus voltage magnitude variations. From (11) and
using the same procedure that leads to (19), one can define
two components also for the reactive power, as follows:
dqh=Pn
k=1
∂qh
∂θhk
dθhk +Pn
k=1
∂qh
∂vk
dvk≡dq0
h+dq00
h.(20)
From (17), it is relevant to observe that:
∂ph
∂θhk
=−qhk ,and ∂qh
∂θhk
=phk ,(21)
which leads to rewrite dp0
hand dq0
has:
dp0
h=−Pn
k=1qhk dθhk ,
dq0
h=Pn
k=1phk dθhk ,(22)
Recalling that θhk =θh−θk, one has:
dp0
h=−qhdθh+Pn
k=1qhk dθk,
dq0
h=phdθh−Pn
k=1phk dθk,(23)
where the identities (17) have been used.
In the same vein, from (17), (19) and (20), dp00
hand dq00
h
can be rewritten as:
dp00
h=ph
vh
dvh+Pn
k=1
phk
vk
dvk,
dq00
h=qh
vh
dvh+Pn
k=1
qhk
vk
dvk.
(24)
Let us define the quantity:
uh≡ln (vh),(25)
where vhis expressed in per unit and uhis dimensionless.
The differential of (25) gives:
duh=dvh
vh
.(26)
Then, (24) can be rewritten as:
dp00
h=phduh+Pn
k=1phk duk,
dq00
h=qhduh+Pn
k=1qhk duk.(27)
Equations (23) and (27) can be expressed in terms of complex
powers variations:
d¯s0
h=dp0
h+dq0
h=¯shdθh−Pn
k=1 ¯shk dθk,
d¯s00
h=dp00
h+dq00
h= ¯shduh+Pn
k=1 ¯shk duk,(28)
and, finally, defining the complex quantity:
¯
ζh≡uh+ θh,(29)

4
the total complex power variation is given by:
d¯sh=d¯s0
h+d¯s00
h
=dp0
h+dp00
h+(dq0
h+dq00
h)
= ¯shd¯
ζh+Pn
k=1 ¯shk d¯
ζ∗
k.
(30)
Equation (30) can be written in a compact matrix form, as
follows:
d¯
s=¯
s◦d¯
ζ+¯
Sd¯
ζ∗,(31)
where ¯
S∈Cn×nis a matrix whose (h, k)-th element is ¯shk .
The expression (31) has been obtained in general, i.e., as-
suming a differentiation with respect to a generic independent
parameter. If this parameter is the time t, (31) can be rewritten
as a set of differential equations:
˙
¯s−¯
s◦¯
η=¯
S¯
η∗(32)
where
¯
η=˙
¯
ζ=˙
u+˙
θ,(33)
and recalling the derivative of θgiven in (8) and defining
%≡˙
u, one obtains:
¯
η≡%+ω(34)
We define ¯
ηas the vector of complex frequencies of the buses
of an AC grid. Note that both real and imaginary part of (34)
have, in fact, the dimension of s−1, as uis dimensionless
and ωis expressed in rad/s. In (34), the imaginary part is
the usual angular frequency (relative to the reference ωo). On
the other hand, it is more involved to determine the physical
meaning of the real part of (34), %. From the definition of uh,
one has vh= exp(uh), that is, the magnitude of the voltage
is expressed as a function whose derivative is equal to the
function itself. This concept is key in the theory of Lie groups
and algebra, which defines the space of linear transformations
of generalized “rates of change” [12].
Equation (32) is the sought expression of the relationship
between frequency variations and power flows in an ac grid.
It contains the information on how power injections of the
devices connected to the grid impact on the frequency at their
point of connection as well as on the rest of the grid. In (32),
the elements of ¯
sare the inputs or boundary conditions at
network buses and depend on the devices connected to grid,
whereas ¯
Sdepends only on network quantities.
Another way to write (32) is by splitting ˙
¯sinto its compo-
nents ˙
¯s0and ˙
¯s00. According the definitions of ¯
s0and ¯
s00,˙
¯s0
does not depend on %, whereas ˙
¯s00 does not depend on ω, as
follows: ˙
¯s0=¯
s◦ω−¯
Sω,
˙
¯s00 =¯
s◦%+¯
S%.(35)
It is important to note that, in general, the expressions of ˙
¯s0
and ˙
¯s00 are not known a priori. These components, however,
can be determined using (32), (35) and:
˙
¯s=˙
¯s0+˙
¯s00 .(36)
With this regard, Section IV-B explains through an example
how to calculate ωand %based on (32). In the following, (32),
(35) and (36) are utilized to discuss relevant special cases.
A. Alternative Expressions
We now derive (32) in an alternative and more compact
formulation as a function of the currents. First, observe that:
˙
¯v=¯
v◦¯
η.(37)
The proof of (37) is given in the Appendix. Then, recalling (2),
the time derivative of ¯
swith respect to the dq-axis reference
frame can be written as:2
˙
¯s=d
dt(¯
v◦¯
ı∗)
=˙
¯v◦¯
ı∗+¯
v◦˙
¯ı∗
=¯
v◦¯
η◦¯
ı∗+¯
v◦˙
¯ı∗
=¯
s◦¯
η+¯
v◦˙
¯ı∗.
(38)
Substituting (38) into (32), one obtains:
¯
v◦˙
¯ı∗=¯
S¯
η∗(39)
Yet another way to write (32) can be obtained by deriving
with respect of time (9), or, which is the same, dividing each
row hof ¯
Sby the corresponding voltage ¯vhin (39). This leads
to (see also footnote 1):
˙
¯ı=¯
Y[¯
v◦¯
η] = ¯
Ydiag(¯
v)¯
η,(40)
and, defining ¯
I=¯
Ydiag(¯
v), one obtains:
˙
¯ı=¯
I¯
η(41)
As per (32), the right-hand sides of (39) and (41) depend
exclusively on network quantities, whereas the left-hand side
is device dependent. While equivalent, the relevant feature of
(39) and (41) with respect to (32) is that the complex frequency
vector only appears once.
B. Remarks on the Complex Frequency of Current Injections
Equation (41) also indicates that, in transient conditions, the
frequency of the current injections at network buses is not the
same as the frequency of the voltages at the same buses. Let
assume that ¯ıh=ıh∠βh, then, one has:
˙
¯ıh= ¯ıh˙ıh
ıh
+˙
βh= ¯ıh¯
ξh,(42)
where ¯
ξhis the complex frequency of the current injection at
bus hwhich is determined using the same procedure that leads
to (37) and that is described in the Appendix. Note that, in
general, ¯
ξh6= ¯ηh. A relevant special case is that of constant
power factor devices (either generators or loads), for which
2Note that the Hadamard product is commutative. Note also that, in a
rotating frame such as the one defined by the dqo transform, the total time
derivative of (2) is given by:
˙
¯s+ωo¯
s=d
dt (¯
v◦¯
ı∗) + ωo(¯
v◦¯
ı∗),
where ωois due to the angular rotation of dq-axis. However, since (2) has
to be always satisfied, the terms that are multipled by ωocan be removed
for the expression above, thus leading to (32). In this paper, only the time
derivative with respect to the rotating dq-axis is considered. This means that
the imaginary part of the complex frequency represents the variations with
respect to ωo.

5
βh=θh+ϕho, where ϕho is the constant power factor angle.
Then: ˙
βh=˙
θh+ ˙ϕho =ωh.(43)
On the other hand, Re{¯
ξh}=%honly if ıh=kvh. Since
constant impedances have constant power factor and their
current is proportional to the voltage, they satisfy the condition
¯
ξh= ¯ηh∀t. Finally, (41) is also particularly suitable for the
calculation of ¯
ηin a software tool as it only requires to express
the current injections at each bus has a function of vhand/or
the state variables of the device connected at the bus. Some
examples of this procedure are given in Section IV.
C. Special Cases
Three cases are considered in this section, namely, con-
stant power injection, constant current injection, and constant
admittance load. These cases illustrate the utilization of the
formulas deduced above, namely (32), (39) and (41).
1) Constant Power Injection: We illustrate first an appli-
cation of (32) for a constant power injection, say ¯sh= ¯sho.
From (36), the boundary condition at the h-th bus is:
˙
¯sh= 0 ⇒˙
¯s0
h=−˙
¯s00
h,(44)
and, from (35):
¯sho ωh−Pn
k=1 ¯shk ωk=−¯sho %h−Pn
k=1 ¯shk %k,(45)
and, from (32):
−¯sho ¯ηh=Pn
k=1 ¯shk ¯η∗
k.(46)
2) Constant Admittance Load: This case illustrates an
application of (39). For a constant admittance load, the current
consumption at the h-th bus is:
¯ıh=−¯
Yho ¯vh,(47)
where the negative sign indicates that the current is drawn
from bus h. From (39) and (47) and recalling that ˙
¯vh= ¯vh¯ηh
(see the Appendix), one obtains:
−¯
Y∗
ho v2
h¯η∗
h=Pn
k=1 ¯shk ¯η∗
k.(48)
The same result can be obtained also from (32). The power
consumption at bus hcan be written as:
¯sh= ¯vh¯ı∗
h=−¯
Y∗
ho v2
h,(49)
which indicates that the power consumption ¯shin (49) does
not depend on θh. Hence, from (36), ¯sh= ¯s00
hand ¯s0
h= 0.
From the first equation of (35), one has:
−¯
Y∗
ho v2
hωh= ¯shωh=Pn
k=1 ¯shk ωk.(50)
Then, from the time derivative of (49) and the second equation
of (35), one has:
−2¯
Y∗
ho vh˙vh=−¯
Y∗
ho v2
h%h+Pn
k=1 ¯shk %k.(51)
Observing that, from (26), %h= ˙vh/vh, then:
vh˙vh=v2
h%h,(52)
and, hence, (51) can be rewritten as:
−¯
Y∗
ho v2
h%h= ¯sh%h=Pn
k=1 ¯shk %k,(53)
The expressions (50) and (53) have the same structure and
can be merged into (48). Moreover, in the summations on the
right-hand-sides of (48), the term ¯shh is given by:
¯shh =¯
Y∗
hh v2
h,(54)
and, defining ¯
Yh,tot =¯
Yho +¯
Yhh, one obtains:
−¯
Y∗
h,tot v2
h¯η∗
h=Pn
h6=k¯shk ¯η∗
k,(55)
which indicates that the two components of the complex
frequency, %hand ωhat the bus hof a constant admittance load
are linear combinations of the %kand ωkat the neighboring
buses. This, in turn and as expected, means that constant
admittance loads are passive devices and cannot modify the
frequency at their point of connection but rather “take” the
frequency that is imposed by the rest of the grid. This
conclusion generalizes the results of the appendix of [7] that
considers the simplified case of an admittance load connected
to the rest of the system through a lossless line.
3) Constant Current Injection: This last example shows an
application of (41). For a constant current injection, we have
two cases. If the magnitude and phase angle of the current are
constant, then ˙
¯ıh= 0 and, from (41), one obtains immediately:
0 = Pn
k=1 ¯ıhk ¯ηk,(56)
where, ¯ıhk is the (h, k)element of ¯
I. Equivalently, from (39),
the condition ˙
¯ıh= 0 leads to:
0 = Pn
k=1 ¯shk ¯η∗
k.(57)
On the other hand, it is unlikely that a device is able to impose
the phase angle of its current injection independently from the
phase angle of its bus voltage. More likely, a device imposes
a constant magnitude and power factor. In this case, the phase
angle of the current βhdepends on the phase angle θhof the
voltage at bus has discussed in Section III-B. Hence, ¯s00
h= 0
and ¯sh= ¯s0
h. This result generalizes the one obtained in [13].
From (41), a current injection with constant magnitude and
power factor leads to:
˙
¯ıh=Pn
k=1 ¯ıhk ωk,
0 = Pn
k=1 ¯ıhk %k,(58)
and, since ˙
βh=ωh(see Section III-B and (43)), the first
equation of (58) can be rewritten as:
Re{¯
ξh}=˙ıh
ıh
=−ωh−Pn
k=1
¯ıhk
¯ıh
ωk.(59)
D. Approximated Expressions
The mathematical developments carried out so far have
assumed no simplifications except for neglecting the electro-
magnetic dynamics of network branches. All formulas that
have been deduced are thus accurate in the measure that
the effect of electro-magnetic transients are negligible. It is,
however, relevant to explore whether the expressions (32), (39)
and (41) can be approximated while retaining the information
on the relationship between power injections and frequency
variations at network buses.

6
Except during faults, which in any case have to last few tens
of milliseconds to prevent generators to loose synchronism,
one can assume that vh≈1pu and that bus voltage phase
angle differences are small, hence sin(θh−θk)≈θh−θk
and cos(θh−θk)≈1. These assumptions, which, in turn, are
the well-known approximation utilized in the fast decoupled
power flow method [14], lead to:
¯shk ≈¯
Y∗
hk .(60)
Equation (60) allows simplifying (32) as:
˙
¯s−¯
s◦¯
η≈¯
Y∗¯
η∗,(61)
and (39) and (41) as:
˙
¯ı≈¯
Y¯
η.(62)
One can further simplify the expressions above for high-
voltage transmission systems, for which ¯
Y≈B:
˙
¯s−¯
s◦¯
η≈ −B¯
η∗,(63)
and (39) and (41) as:
˙
¯ı≈B¯
η.(64)
Then, approximating the term ¯
s◦¯
η≈¯
Y∗
diag ¯
η, where ¯
Ydiag
is a matrix obtained using the diagonal elements of ¯
Y, and
splitting the real and imaginary part of ¯
η, (63) leads to:
˙
p0≈B0ω,(65)
where B0
hk =−Bhk and B0
hh =Pn
h6=kBhk are the elements
of B0, and
˙
q00 ≈B00%,(66)
where B00
hk =−Bhk and B00
hh =−2Bhh are the elements
of B00. Equation (65) is the expression deduced in [7] and
that, with due simplifications, leads to the frequency divider
formulas presented in [5].
Considering the resistive parts of the network branches, the
following dual expressions hold:
˙
p00 ≈G00%,˙
q0≈G0ω,(67)
where the elements of G0and G00 are defined as G0
hk =
G00
hk =−Ghk,G0
hh =Pn
h6=kGhk, and G00
hh =−2Ghh.
Interestingly, from (67), it descends that, in lossy networks,
the reactive power can be utilized to regulate the frequency.
Combining together (65), (66) and (67) leads to the follow-
ing approximated expressions:
˙
¯s0≈¯
Y0∗ ω,˙
¯s00 ≈¯
Y00 %.(68)
IV. EXA MPL ES
The examples presented below apply the theory developed
in the previous section and discuss applications to power sys-
tem modeling, state estimation and control. In particular, three
devices are discussed, namely, the synchronous machine; the
voltage dependent load; and a converter-interfaced generator
with frequency and voltage control capability. The objective
is to illustrate the methodological approach discussed above
and showcase some problems that the proposed definition of
complex frequency and the expression (32) make possible to
solve. In the following, all simulation results are obtained
using the software tool Dome [15].
A. Ratio between ωand %
This first example illustrates the transient behavior of the
two components of the complex frequency. Figure 1 shows the
transient behavior of the components of the complex frequency
at a generator and a load bus, bus 2 and bus 8, respectively, for
the well-known WSCC 9-bus system [9]. The estimation of ωh
and %hat the buses of the grid is obtained through a numerical
differentiation based on a synchronous-reference frame Phase-
Locked Loop (PLL) model [16]. Simulation results show that
|ωh||%h|. This inequality holds for all networks and
scenarios that we have tested for the preparation of this work.
It is important to note that the inequality |ωh||%h|does
not imply ˙
¯s0
h˙
¯s00
h. As a matter of fact, taking as an example
constant impedance loads, ˙
¯s0
h= 0 and ˙
¯s00
h=˙
¯sh. The rationale
behind this observation can be explained by rewriting (35)
using (17) and an element-by-element notation:
˙
¯s0
h=Pn
k=1 ¯shk (ωh−ωk),
˙
¯s00
h=Pn
k=1 ¯shk (%h+%k),(69)
which indicates that ˙
¯s0
his proportional to the difference of the
elements of ω, whereas ˙
¯s00
his proportional to the sum of the
elements of %. Hence, even if the elements of %are small
relatively to those of ω, their effect on the system are not
necessarily negligible.
B. Implementation of Equation (32)
The implementation of (32) in a software tool for the
simulation of power systems can be useful to determine the
“exact” frequency variations at network buses in a QSS model
for transient stability analysis. This topic has been discussed
and solved under various hypotheses in [4]–[6]. In particular,
the Frequency Divider Formula (FDF) proposed in [5] is based
on (65), which is an approximation of (32).
The determination of ¯
ηrequires the solution of the set of
complex differnetial equations (32). These can be rewritten as
a set of 2nreal equations, with unknowns (%,ω). The right-
hand side of (32) is linear with respect to ¯
η, so it remains to
determine the dependency of the elements of ˙
¯s−¯
s◦¯
η∗or,
alternatively, of ¯
v◦˙
¯ı∗from (39), on ¯
ηand, eventually, on the
time derivative of the state variables ˙
x.
We illustrate the procedure using a conventional 4th order
model of the synchronous machine. The stator voltage of the
machine with respect to its dq-axis reference frame is linked
to the grid voltage vhwith the following equations [9]:
¯vs=vs,d+ vs,q= ¯vh∠(π
2−δr),(70)
where δris th rotor angle of the machine. The time derivative
of (70) gives (see the Appendix for the derivative of ¯vh):
˙
¯vs= ¯vs(¯ηh− ωr),(71)
where ωris the deviation in rad/s of the angular speed of the
machine with respect to ωo. Then, the stator electrical and
magnetic equations are:
X0
dıs,d+Raıs,q=−vs,q+e0
r,q
Raıs,d−X0
qıs,q=−vs,d+e0
r,d
(72)

7
0 2.557.5 10 12.5 15 17.5 20
Time [s]
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
ωh[Hz]
Bus 2
Bus 8
0 2.557.5 10 12.5 15 17.5 20
Time [s]
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
%h[Hz]
Bus 2
Bus 8
(a) Outage of the load at bus 5.
0 2.557.5 10 12.5 15 17.5 20
Time [s]
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
ωh[Hz]
Bus 2
Bus 8
0 2.557.5 10 12.5 15 17.5 20
Time [s]
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
%h[Hz]
Bus 2
Bus 8
(b) Fault at bus 7 cleared by tripping line 5-7.
Fig. 1: Transient behavior of the components of the complex fre-
quency for the WSCC 9-bus system.
where ¯ıs=ıs,d+ ıs,qis the stator current. From (72), one
can thus obtain the expressions of ıs,dand ıs,qas a function
of vs,d,vs,qand the state variables e0
r,dand e0
r,q. In this case,
these relationship are linear, so, the time derivatives of the
components of the current give:
˙ıs,d=∂ ıs,d
∂ωhωh+∂ ıs,d
∂%h%h+∂ ıs,d
∂ωrωr+∂ ıs,d
∂e0
r,d˙e0
r,d+∂ıs,d
∂e0
r,q˙e0
r,q,
˙ıs,q=∂ ıs,q
∂ωhωh+∂ ıs,q
∂%h%h+∂ ıs,q
∂ωrωr+∂ ıs,q
∂e0
r,d˙e0
r,d+∂ıs,q
∂e0
r,q˙e0
r,q,
(73)
where the time derivative of vs,dand vs,qhave been substituted
with the real and imaginary parts of the left-hand side of (71).
Finally, from (39), one obtains:
Re{¯vh˙
¯ı∗
h}= Re{¯vs˙
¯ı∗
s}=vs,d˙ıs,d+vs,q˙ıs,q,
Im{¯vh˙
¯ı∗
h}= Im{¯vs˙
¯ı∗
s}=vs,q˙ıs,d−vs,d˙ıs,q.(74)
Finally, substituting (73) into (74), one obtains the expressions
of the left-hand side of (39) at the buses of the synchronous
machines.
The very same procedure can be applied to any device of
the grid and, in the vast majority of the cases, the resulting
expressions are linear with respect to (%,ω). Hence, at each
time step of a time domain simulation one need to solve a
problem of the type Aχ=b, where χ= [%T,ωT]T, and:
A=[H+ diag(p)−P%] [K−diag(q)−Pω]
[K+ diag(q)−Q%]−[H−diag(p) + Qω](75)
and
b=Px˙
x
Qx˙
x,(76)
where H= Re{¯
S}and K= Im{¯
S};P%,Pω,Q%and Qω
are the Jacobian n×nmatrices of pand qwith respect to %
and ω, respectively; and Pxand Qxare the Jacobian n×nx
matrices of pand qwith respect to the state variables x,
respectively.
Figure 2 shows the imaginary part of the complex frequency
as obtained for the WSCC 9-bus system using a 4th and a
2nd order model of the synchronous machines, as well as
for a scenario where the machine at bus 3 is substituted
for a Converter-Interfaced Generator (CIG). The model of
the CIG is shown in Fig. 3. The results obtained using
PLLs matches well the “exact” results obtained with proposed
method (indicated with CF in the legends) except for the
numerical spikes that following the load disconnection and
a small delay that is due to the control loop of the PLL. The
proposed formula is also robust with respect to noise as shown
in Fig. 2b. Figure 2 also shows the results obtained using the
FDF proposed in [5]. The FDF matches closely the results
of the proposed method when considering the simplified 2nd
order model of the machines but introduces some errors if
the machine model includes rotor flux dynamics or when
considering the CIG.
C. Voltage Dependent Load (VDL)
The power consumption of a VDL is:
¯sh=ph+jqh=−povγp
h− qovγq
h,(77)
then:
˙
¯sh=−poγp˙vhvγp−1
h− qoγq˙vhvγq−1
h
= (γpph+ γqqh)%h,(78)
where it has been assumed that the exponents γpand γqare
constant and that poand qovary “slowly” with respect to vh.
If γp=γq=γ, then ˙
¯sh=γ¯sh%h, which generalizes the
results obtained in Section III-C.2.

8
0 0.511.522.5 3
Time [s]
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
ωbus 1 [Hz]
CF
PLL
FDF
(a) 4th order machine models
0 0.511.522.5 3
Time [s]
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
ωbus 3 [Hz]
CF
PLL
(b) 4th order machine models & noise
0 0.511.522.5 3
Time [s]
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
ωbus 1 [Hz]
CF
PLL
FDF
(c) 2nd order machine models
0 0.511.522.5 3
Time [s]
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
ωbus 3 [Hz]
CF
PLL
FDF
(d) CIG connected at bus 3
Fig. 2: Comparison of bus frequencies using PLL, FDF and the
proposed approach based on the complex frequency. The plots refers
to the WSCC 9-bus system following the disconnection at t= 1 s
of the load at bus 5.
As an application, we utilize (78) to estimate the parameters
γpand γqof a VDL using a similar technique as the one
proposed in [13]. From (32) and (78), one obtains:
(γpph+ γqqh)%h=Pn
k=1[¯shk(¯ηh+ ¯η∗
k)] ,(79)
¯vh
¯sref
h¯ıref
h¯ıh
1
Converter
Current
set point
controller 1 + sT
ωh
ωref
vh
vref
h
1 + sTp
pref
h
1 + sTq
qref
h
+ +
−−
KpKq
Fig. 3: Simplified scheme of a CIG and its controllers.
and, splitting real and imaginary parts:
γp= (ph%h)−1Re {Pn
k=1 ¯shk (¯ηh+ ¯η∗
k)},
γq= (qh%h)−1Im {Pn
k=1 ¯shk (¯ηh+ ¯η∗
k)},(80)
where the right-hand sides can be determined based on mea-
surements. The fact that %h→0in steady-state can create
numerical issues, which can be solved, as discussed in [13],
using finite differences over a period of time ∆t, namely
¯ηh≈∆¯
ζh/∆t,¯η∗
k≈∆¯
ζ∗
k/∆t, and %h≈∆uh/∆t, as follows:
ˆγp≈(ph∆uh)−1Re Pn
k=1 ¯shk (∆¯
ζh+ ∆¯
ζ∗
k),
ˆγq≈(qh∆uh)−1Im Pn
k=1 ¯shk (∆¯
ζh+ ∆¯
ζ∗
k).(81)
Equations (80) and (81) generalize the empirical formulas to
estimate γpand γqproposed in [13]. The latter, in fact, can
be obtained from (81) by approximating ¯shk ≈ −jBhk .
Figure 4 shows the results obtained for the WSCC 9-bus
system where the bus connected at bus 8 is a VDL with γp= 2
and γq= 1.5. The results show that (81) is, as expected,
more precise than the approximated estimations based on the
expression proposed in [13]. Equation (81) is, in fact, an
exact expression and its accuracy depends exclusively on the
accuracy of the measurements of ¯
ζhand ¯
ζk, which can be
obtained, for example, with PMUs. If the R/X ratio of the
transmission lines of the system is changed to resemble that
of a distribution system, (81) appears also numerically more
robust than its approximated counterparts (see Fig. 4b).
D. Converter-Interfaced Generation
This last example illustrates an application of the approx-
imated expression (67). We focus in particular on the link
between ωand ˙
qthrough the resistances of network branches.
Using again the WSCC 9-bus system and substituting two
synchronous machines with CIGs, we compare the dynamic
response of the system following a load variation using the
control scheme of Fig. 3 (Control 1), and the control scheme
shown in Fig. 5 (Control 2). The latter regulates the frequency
by both the active and reactive powers of the CIGs.
Figure 6 shows that Control 2 is more effective than Control
1 to reduce the variations of the frequency. This result indicates
that the relationship between qand ωis not weak and can
be exploited to improve the frequency response of low-inertia
systems. This result is predicted by (67) and hence by (32).

9
2 6 10 14
Time [s]
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
ˆ
γp
Paper [8]
Equation (79)
2 6 10 14
Time [s]
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
ˆ
γq
Paper [8]
Equation (79)
(a) Ratio of transmission lines: R/X 1.
2 6 10 14
Time [s]
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
ˆ
γp
Paper [8]
Equation (79)
2 6 10 14
Time [s]
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
ˆ
γq
Paper [8]
Equation (79)
(b) Ratio of transmission lines: R/X ≈1.
Fig. 4: Estimation of the exponents of the VDL connected at bus 8
following the disconnection of 15% of the load at bus 5 at t= 1 s
for the WSCC 9-bus system.
The simplified converter-interfaced generator model shown
in Fig. 5, with differential equations:
Tp˙ph=Kp(ωref −ωh)−ph,
Tq˙qh=Kq(vref −vh)−qh.(82)
ωh
ωref
ωh
ωref
1 + sTp
pref
h
1 + sTq
qref
h
+ +
−−
KpKq
Fig. 5: Alternative frequency control through active and reactive
powers for the CIG of Fig. 3.
V. CONCLUSIONS
The paper introduces a new physical quantity, namely, the
complex frequency. This quantity allows writing the differen-
tial of power flow equations in terms of complex powers and
voltages at network buses. The most significant property of
the newly defined complex frequency presented is its ability
to give a much more robust and clean indication of frequency
than what generally accepted in the literature, especially to
describe the behavior of the frequency at buses close to a
disturbance. For example, it is well known that many other
methods result in meaningless spikes in frequency at the
inception and clearing of faults and other sudden disturbances.
The proposed definition provides a solution to this issue.
Noteworthy byproducts of the definition of the complex
frequency are the expressions (32), (39) and (41). These equa-
tions express the formal link between the complex power and
complex frequency variations and are a relevant contribution
0246810
Time [s]
59.5
59.6
59.7
59.8
59.9
60
60.1
ωCoI [Hz]
Control 1
Control 2
0246810
Time [s]
0.995
1
1.005
1.01
1.015
1.02
1.025
vbus 5 [pu(kV)]
Control 1
Control 2
Fig. 6: Frequency of the CoI and voltage at bus 5 following the
connection of p= 0.25 pu at bus 5 at t= 1 s for the WSCC 9-bus
system with high penetration of CIG.
of the paper. The paper shows that the expressions above
are a generalization of the FDF proposed in [5]. The several
analytical and numerical examples discussed Sections III and
IV show the several prospective applications of the proposed
theoretical approach.
Future work will focus on further elaborating the expression
(32) and exploiting its features for the control and state
estimation of power systems. It also appears relevant to
combine the proposed complex freqeuncy approach to some
of the techniques described in [3]. Particularly interesting, for
example, is the “beyond-phasor” approach based on the Hilbert
transform proposed in [17].
APPENDIX
This appendix provides the proof of (37). With this aim, let
us consider the h-th element of (37), namely ¯vh=vh∠θh=
vh(cos θh+sin θh). The time derivative of ¯vhwith respect
to the dq-axis reference frame gives:
˙
¯vh= ˙vh∠θh+ vhωh∠θh
=vh(%h∠θh+ ωh∠θh)
=vh∠θh(%h+ ωh)
= ¯vh¯ηh,
where the following identities hold:
d
dt∠θh=ωh(−sin θh+cos θh)
= ωh(cos θh+sin θh)
= ωh∠θh.

10
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Federico Milano (F’16) received from the Univer-
sity of Genoa, Italy, the ME and Ph.D. in Electrical
Engineering in 1999 and 2003, respectively. From
2001 to 2002, he was with the Univ. of Water-
loo, Canada. From 2003 to 2013, he was with the
Univ. of Castilla-La Mancha, Spain. In 2013, he
joined the Univ. College Dublin, Ireland, where he
is currently Professor of Power Systems Control and
Protections and Head of Electrical Engineering. His
research interests include power systems modeling,
control and stability analysis.