Available via license: CC BY 4.0

Content may be subject to copyright.

1

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

REFERENCES

[1] “IEEE/IEC International Standard - Measuring relays and protection

equipment - Part 118-1: Synchrophasor for power systems - Measure-

ments,” pp. 1–78, 2018.

[2] H. Kirkham, W. Dickerson, and A. Phadke, “Deﬁning power system

frequency,” in IEEE PES General Meeting, 2018, pp. 1–5.

[3] M. Paolone, T. Gaunt, X. Guillaud, M. Liserre, S. Meliopoulos,

A. Monti, T. Van Cutsem, V. Vittal, and C. Vournas, “Fundamentals

of power systems modelling in the presence of converter-interfaced

generation,” Electric Power Systems Research, vol. 189, p. 106811, 2020.

[4] J. Nutaro and V. Protopopescu, “Calculating frequency at loads in

simulations of electro-mechanical transients,” IEEE Trans. on Smart

Grid, vol. 3, no. 1, pp. 233–240, 2012.

[5] F. Milano and ´

A. Ortega, “Frequency divider,” IEEE Trans. on Power

Systems, vol. 32, no. 2, pp. 1493–1501, 2017.

[6] H. Golpˆ

ıra and A. R. Messina, “A center-of-gravity-based approach to

estimate slow power and frequency variations,” IEEE Trans. on Power

Systems, vol. 33, no. 1, pp. 1026–1035, 2018.

[7] F. Milano and ´

A. Ortega, “A method for evaluating frequency regulation

in an electrical grid Part I: Theory,” IEEE Trans. on Power Systems,

2020, in press.

[8] ——, Frequency Variations in Power Systems: Modeling, State Estima-

tion, and Control. Hoboken, NJ: Wiley, 2020.

[9] P. W. Sauer and M. A. Pai, Power System Dynamics and Stability. Upper

Saddle River, NJ: Prentice Hall, 1998.

[10] F. Milano, Power System Modelling and Scripting. London, UK:

Springer, 2010.

[11] U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary

Differential Equations and Differential-Algebraic Equations, 1st ed.

USA: Society for Industrial and Applied Mathematics, 1998.

[12] H. Stephani, Differential Equations: Their Solution Using Symmetries,

M. MacCallum, Ed. Cambridge University Press, 1990.

[13] ´

A. Ortega and F. Milano, “Estimation of voltage dependent load models

through power and frequency measurements,” IEEE Trans. on Power

Systems, vol. 35, no. 4, pp. 3308–3311, 2020.

[14] B. Stott and O. Alsac¸, “Fast decoupled load ﬂow,” IEEE Trans. on Power

Systems, vol. PAS-93, no. 3, pp. 859–869, 1974.

[15] F. Milano, “A Python-based software tool for power system analysis,”

in IEEE PES General Meeting, 2013, pp. 1–5.

[16] ´

A. Ortega and F. Milano, “Comparison of different PLL implementations

for frequency estimation and control,” in ICHQP, 2018, pp. 1–6.

[17] A. Derviˇ

skadi´

c, G. Frigo and M. Paolone, “Beyond phasors: Modeling

of power system signals using the Hilbert transform,” in IEEE Trans. on

Power Systems, vol. 35, no. 4, pp. 2971-2980, 2020.

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.