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Steady state evaluation of distributed secondary frequency control
strategies for microgrids in the presence of clock drifts*
Ajay Krishna
1
, Christian A. Hans
1
, Johannes Schiffer
2
, J
¨
org Raisch
1,3
and Thomas Kral
4
Abstract— Secondary frequency control, i.e., the task of
restoring the network frequency to its nominal value following a
disturbance, is an important control objective in microgrids. In
the present paper, we compare distributed secondary control
strategies with regard to their behaviour under the explicit
consideration of clock drifts. In particular we show that, if
not considered in the tuning procedure, the presence of clock
drifts may impair an accurate frequency restoration and power
sharing. As a consequence, we derive tuning criteria such that
zero steady state frequency deviation and power sharing is
achieved even in the presence of clock drifts. Furthermore, the
effects of clock drifts of the individual inverters on the different
control strategies are discussed analytically and in a numerical
case study.
I. INTRODUCTION
Electric power systems are currently facing various chal-
lenges that mostly arise from an increase in spatially dis-
tributed renewable energy sources (RES). As a consequence,
power generation is moving from a relatively small number
of large scale power stations to a very large number of
small scale distributed units. A promising way to tackle
the challenges that arise from this structural change is the
decomposition of the overall grid into regional entities called
microgrids (MGs). MGs typically consist of renewable and
storage units, as well as conventional generators, and loads.
In a general setting, MGs may interact with each other, but
- by matching generation and consumption within the MG
as far as possible - transmitted power is reduced and trans-
mission losses are decreased. MGs can usually be operated
in two modes, either connected to the grid or electrically
isolated (islanded) [1].
Motivated by existing control strategies in conventional
power systems, a hierarchical control approach has also been
advocated for MGs [2]. Thereby, one typically distinguishes
primary and secondary control layers (as in conventional
power systems), while the top control level, which is mostly
referred to as operational management or tertiary control, is
mainly concerned with generation scheduling.
*The project leading to this manuscript has received funding from the
German Academic Exchange Service (DAAD), the German Federal Ministry
for Economic Affairs and Energy (BMWi), Project No. 0325713A and the
European Union’s Horizon 2020 research and innovation programme under
the Marie Skłodowska-Curie grant agreement No. 734832.
1
Fachgebiet Regelungssysteme, Technische Universit
¨
at Berlin, Germany,
{krishna, hans, raisch}@control.tu-berlin.de
2
School of Electronic and Electrical Engineering, University of Leeds,
UK, j.schiffer@leeds.ac.uk
3
Max-Planck-Institut f
¨
ur Dynamik komplexer technischer Systeme,
Magdeburg, Germany
4
Younicos AG, Berlin, Germany, thomas.kral@younicos.com
Primary control is typically implemented in the form of
decentralised proportional (droop) control. Its major objec-
tives are active power sharing as well as frequency and
voltage stability [1]. In MGs, this task is mostly assigned
to conventional generators and grid forming inverters [3].
Despite many advantages, a major drawback of this control
law is that voltage amplitudes and frequencies usually deviate
from their nominal values at steady state [4].
Secondary frequency control aims at removing stationary
frequency deviations. There are two prominent implementa-
tion approaches: centralised and distributed controllers. Cen-
tral approaches are widely used in existing power systems
[5] and have been implemented and studied for MGs in
[1], [2]. However, a major disadvantage of such approaches
is that the central control unit represents a single point of
failure. This issue motivated distributed strategies which use
locally available as well as neighbouring information that is
exchanged over a communication network [6].
Recently, various distributed secondary frequency control
strategies have been proposed. A distributed averaging pro-
portional integral secondary controller was presented in [7].
For this, an optimal tuning strategy using the input-output
H
2
-norm has been provided by [8]. A related consensus
based distributed frequency controller was proposed by [9].
Therein, a so called pinning control is used to ensure zero
steady state frequency error. Another distributed frequency
control approach is presented in [10]. Here also, pinning con-
trol is used to achieve frequency convergence. This way, the
reference frequency value only needs to be provided for one
inverter. Consensus based distributed frequency control along
with a weight calculation procedure for optimal convergence
speed is presented in [11]. All the above mentioned control
laws can achieve frequency synchronisation. Furthermore,
the communication layer can be designed such that the
controllers are resilient to communication path failures. In
[12], conditions for robust non-linear stability of MGs op-
erated with a distributed averaging integral controller under
fast-varying time-delays and switching communication topol-
ogy are derived. Furthermore, various secondary frequency
control policies are compared in [13]. In particular, the
effects of communication properties on different strategies,
such as, centralized, decentralized, averaging and consensus
strategies, are analysed in a quantitative way.
In this work, we explicitly consider the effect of clock
drifts in secondary frequency control. The term clock drifts
describes the fact that all units, operated with different
processors have a slightly different “understanding” of time,
i.e., their clock rates are not synchronized [14]. Most of
the distributed control approaches, as they make use of the
internal frequencies that are calculated by the controls of
the inverters, are influenced by clock drifts. In practice,
even if the units are synchronized to a global frequency,
the internal frequencies of the inverters are slightly different
[15]. As external synchronization units that could hamper
this problem are expensive, they are not used in most of the
applications. Whereas, a widely chosen alternative approach
to tackle this problem is the use of a central secondary
controller with a very accurate measurement. To enable a
design of distributed controllers that fulfil the requirement
of zero steady state frequency error and power sharing,
conditions on the tuning in the presence of clock drifts must
be derived. However, to the best knowledge of the authors,
none of the publications on secondary frequency control
investigates the effect of clock drifts.
Motivated by this fact, we compare a set of different
distributed control strategies proposed in the literature [7],
[10] with regard to their steady state performance in terms
of frequency restoration and power sharing under explicit
consideration of clock drifts. Furthermore, we identify a
suitable parametrisation for a distributed control strategy that
achieves zero steady state frequency error and steady state
power sharing in the presence of clock drifts.
The remainder of this paper is structured as follows. In
Section II, we provide the model for the electrical network
and the distributed units. Then, in Section III a central and a
distributed secondary frequency controller are introduced. In
Section IV, the distributed controller is parametrized to re-
semble different control laws reported in the literature. These
control laws are then analysed regarding their steady state
behaviour. Finally, in Section V, we compare performance
of the different controllers in a case study.
II. MODEL OF A MICROGRID
In this section, the employed MG model is introduced.
We start by introducing some notation and basics on graph
theory.
A. Preliminaries and notations
Throughout the paper, the identity matrix of size N × N
is denoted by I
N
. Furthermore, 1
N
∈ R
N
is the vector of
all ones and 0
N
∈ R
N
is the vector of all zeros. The matrix
of all ones is denoted by 1
N×N
∈ R
N×N
and the matrix of
all zeros by 0
N×N
∈ R
N×N
. The N × N diagonal matrix
with entries a
j
, j = 1, . . . , N is denoted by diag (a
j
).
1) Graph theory: A finite undirected graph G is a tuple
G = (J , E), where J is a finite set of vertices with
J = {1, . . . , J} and J ∈ N is the total number of vertices.
Furthermore, E ⊆ [J ]
2
is the set of edges where [J ]
2
represents the set of all two-element subsets of J . The entries
of the adjacency matrix A ∈ R
J×J
of G are a
ij
= a
ji
= 1
if {i, j} ∈ E and a
ij
= a
ji
= 0 otherwise. The set of
neighbouring nodes of node i is given by J
i
= {j ∈
J | a
ij
6= 0}.
An ordered sequence of nodes such that any pair of
consecutive nodes in the sequence is connected by an edge
is called a path. If there exists a path between every pair
of distinct nodes, then the graph G is called connected.
The diagonal degree matrix D ∈ R
J×J
is given by
D = diag
P
j∈J
a
ij
. The Laplacian matrix L ∈ R
J×J
of
an undirected graph is given by L = D − A. If and only if a
graph G is connected, then L is positive semi-definite, with a
simple zero eigenvalue and a corresponding right eigenvector
1
J
[16]. Thus, L1
J
= 0
J
and 1
T
J
L = 0
T
J
[17].
B. Network modelling
The electrical network of the considered microgrid is
assumed to be connected. In this network, vertices at which
only loads and no other units are connected, are called
passive nodes. Using Kron-reduction [18], the original net-
work containing passive nodes is reduced to a lower dimen-
sional network that contains only nodes where grid forming
units, i.e., grid forming inverters or rotating generators, are
connected. We assume this reduction has been carried out.
Then, each grid forming unit i ∈ J is connected to a node
i ∈ J . This work focusses on secondary frequency control.
Therefore, we assume that the voltage amplitudes V
i
∈ R
≥0
at all buses are constant [19].
Denoting the vector of phase angles of all nodes in the
grid by δ = (δ
1
, . . . , δ
J
) ∈ R
J
, the active power injection of
unit i is given by
P
i
(δ) = G
ii
V
2
+ V
2
P
j∈J
i
|Y
ij
| sin (δ
i
− δ
j
+ φ
ij
), (1)
where G
ii
=
ˆ
G
ii
+
P
J
j=1,j6=i
G
ij
. Here, G
ii
∈ R is the self-
conductance,
ˆ
G
ii
∈ R denotes the shunt conductance at node
i and G
ij
∈ R
≥0
the conductance of the line connecting
nodes i and j [18]. With the susceptance B
ij
∈ R, the
absolute value of the admittance is given by |Y
ij
| = (G
2
ij
+
B
2
ij
)
1
2
. Moreover, φ
ij
= arctan (G
ij
/B
ij
) is the admittance
angle. Note that if there is no direct electrical connection
between nodes i and j then Y
ij
= 0.
Usually, grid forming units such as synchronous genera-
tors and grid forming inverters are employed for frequency
restoration. Also, all the loads and grid feeding units can
be described by a constant impedance G
ii
V
2
∀i ∈ J .
Therefore this work will focus on grid forming units which
will be simply referred to as units in the following.
C. Droop controlled units
A widely used control approach in MGs is droop control,
implemented on grid-forming inverters and synchronous gen-
erators. To realize this low level control, each unit is typically
equipped with its own digital controller with individual pro-
cessor clock. The time signal of all controllers slightly vary
from each other because of the so called clock drifts [15].
As has been shown in [15], clock drifts can be incorporated
in the model of a grid-forming inverter by introducing a
(constant) unknown scaling factor in the model. Then, the
dynamics of the ith unit equipped with frequency droop
control is given by
(1 + µ
i
)
˙
δ
i
= (1 + µ
i
)ω
i
= ¯ω
i
, (2a)
= ω
d
− k
i
(P
m
i
− P
d
i
) + ξ
i
, (2b)
(1 + µ
i
)τ
i
˙
P
m
i
= −P
m
i
+ P
i
, (2c)
where µ
i
∈ R, is the clock drift factor, ω
i
∈ R is the actual
electrical frequency and ¯ω
i
∈ R is the internal frequency
of the ith unit. Note that only the internal frequency ¯ω
i
is
available to every unit. Furthermore, ω
d
∈ R is the frequency
set point, k
i
∈ R
>0
the droop coefficient, P
d
i
∈ R the active
power set point from a higher control level, e.g, energy
management [20], and ξ
i
∈ R is the control input. The
measured active power P
m
i
∈ R is obtained by filtering the
power output P
i
in (1) by a first order low pass filter with
time constant τ
i
∈ R
>0
.
The model (2) can be used to model both, droop controlled
inverters and synchronous generators (see, e.g., [21]). How-
ever, using (2) without any secondary control, i.e., ξ
i
= 0, the
steady state frequency error is typically non-zero. To achieve
the desired ¯ω
i
= ω
d
, secondary control as described in the
next section can be used.
III. SECONDARY FREQUENCY CONTROL
Secondary frequency control aims at driving the frequency
value at steady state to a desired value. Strategies, that change
the input ξ
i
to achieve this goal are introduced in this section.
The study starts with a widely used central control scheme.
Then, a distributed secondary control law is presented.
A. Central control
A standard approach for frequency secondary control is to
measure the frequency at a single bus bar where an accurate
frequency measurement can be realised. This frequency value
is then used in a standard central frequency controller (see,
e.g., [5]). Such control law can be described by
˙
ξ
c
= ω
d
− ω
c
, ξ
i
= b
i
ξ
c
, ∀i ∈ J , (3)
where ξ
c
∈ R is the integrated frequency error and ω
c
is
the frequency measured at one bus, ξ
i
∈ R is the secondary
control input and b
i
∈ R
≥0
is the controller gain of unit i.
Usually in this control scheme, clock drifts are addressed
using an accurate central frequency measurement with µ
c
=
0 and hence, ¯ω
c
= ω
c
. Despite their popularity, central
controllers are vulnerable to single point failures which need
to be addressed by redundant communication or computation
infrastructure [10].
B. Distributed consensus based control
A generalised representation of a consensus based dis-
tributed secondary frequency control scheme explicitly con-
sidering clock drifts is
(1 + µ
i
)
˙
ξ
i
= −
b
i
(¯ω
i
− ω
d
)+
c
i
P
j∈J
G
i
a
ij
(¯ω
i
− ¯ω
j
) + d
i
P
j∈J
G
i
a
ij
(ξ
i
− ξ
j
)
, (4)
where b
i
∈ R is called pinning gain and c
i
∈ R as well
as d
i
∈ R are controller gains. Furthermore, a
ij
∈ R
≥0
are entries from the adjacency matrix (see Section II-A)
that describes the communication structure of the secondary
controller and J
G
i
is the set of neighbouring units of unit
i for the communication network. By parametrising (4),
different control strategies can be implemented, e.g., [10],
[7] or a controller similar to the one described by [9]. For all
considered strategies, we assume the communication graph
is connected and undirected.
Combining the unit model (2) with the power flow equa-
tions (1) and the distributed control (4), the dynamics of the
closed-loop MG system can be written as
(I
J
+ µ)
˙
δ = (I
J
+ µ)ω = ¯ω, (5a)
= 1
J
ω
d
− k(P
m
− P
d
) + ξ, (5b)
(I
J
+ µ)τ
˙
P
m
= −P
m
+ P (δ), (5c)
(I
J
+ µ)
˙
ξ = −
(B + CL)(¯ω − 1
J
ω
d
) + DLξ
, (5d)
where
µ = diag(µ
1
, . . . , µ
J
),
ω = [ω
1
, . . . , ω
J
]
T
,
¯ω = [¯ω
1
, . . . , ¯ω
J
]
T
,
k = diag(k
1
, . . . , k
J
),
P
d
= [P
d
1
, . . . , P
d
J
]
T
,
P
m
= [P
m
1
, . . . , P
m
J
]
T
,
ξ = [ξ
1
, . . . , ξ
J
]
T
,
τ = diag(τ
1
, . . . , τ
J
),
P (δ ) = [P
1
(δ), . . . , P
J
(δ)]
T
,
B = diag(b
1
, . . . , b
J
),
C = diag(c
1
, . . . , c
J
),
D = diag(d
1
, . . . , d
J
).
Note that (5) is non-linear due to P (δ) from (1). For the
subsequent analysis, it is convenient to introduce the notion
below.
Definition 1: The system (5) admits a synchronised mo-
tion if it has a solution for all t ≥ 0 of the form
δ
s
(t) =δ
s
0
+ ω
s
t, ω
s
= 1
J
ω
∗
, (6a)
with ω
∗
∈ R and δ
s
0
∈ R
J
such that
|δ
s
0,i
− δ
s
0,j
| <
π
2
∀ i ∈ J , j ∈ J
i
. (6b)
With Definition 1, we can now analyse the steady state be-
haviour of the closed-loop system (5) for different parametri-
sations of the controller (4) in the next section.
IV. STEADY-STATE BEHAVIOUR OF DISTRIBUTED
SECONDARY CONTROL STRATEGIES
The control strategies described in Section IV aim at
driving the steady state error of the frequency to zero. In
this section we will analyse under which parametrization
of the control (4), this can be achieved. Furthermore, we
will investigate how power sharing, i.e., that the units share
variations in load power in a desired manner, can be ensured.
Before analysing these properties, we derive an analytic
expression for the steady state frequency and a condition
for power sharing.
Lemma 1: Suppose that (5) admits a synchronised motion
(see Definition 1) where D is non-singular and that at
least one of the matrices B and C is non-zero. Then, the
corresponding synchronised electrical frequency is given by
ω
∗
=
1
T
J
D
−1
B1
J
1
T
J
D
−1
(B + CL)(I
J
+ µ)1
J
ω
d
. (7)
Furthermore, ω
∗
= ω
d
if and only if
1
T
J
D
−1
(B + CL)µ1
J
= 0. (8)
Proof: Along any synchronised motion, the electrical
frequencies at all nodes of (5) have to be identical, i.e.,
˙
δ
s
= ω
s
= 1
J
ω
∗
, (9)
which directly implies from (5a) that
¯ω
s
= (I
J
+ µ)1
J
ω
∗
. (10)
Furthermore,
˙
ξ
s
= 0
J
. Hence,
(I
J
+ µ)
˙
ξ
s
= 0
J
= (B + CL)(¯ω
s
− 1
J
ω
d
) + DLξ
s
. (11)
Multiplying (11) from the left with 1
T
J
D
−1
and recalling the
fact from Section II-A that 1
T
J
L = 0
T
J
as the graph induced
by the communication network is undirected and connected
yields
0 = 1
T
J
D
−1
(B + CL)(¯ω
s
− 1
J
ω
d
).
Using (10) and L1
J
= 0
J
leads to
0 = 1
T
J
D
−1
(B + CL)(I
J
+ µ)1
J
ω
∗
− B1
J
ω
d
.
Unless B = 0
J×J
or ω
d
= 0, the above equation is solvable
if 1
T
J
D
−1
(B+CL)( I
J
+µ)1
J
is non-zero. Then, (7) follows
immediately.
To show that ω
∗
= ω
d
if and only if (8) is satisfied, we
note that according to (7), ω
∗
= ω
d
if and only if
1
T
J
D
−1
(B + CL)(I
J
+ µ)1
J
= 1
T
J
D
−1
B1
J
.
Recalling the fact that CL1
J
= 0
J
, the above equation is
equivalent to (8).
Next, we investigate under which conditions power sharing
can be achieved with the control (4) in the presence of
clock drifts. In this work, we are interested in power sharing
relative to the set-points P
d
i
. Therefore, we employ the
definition below, which is in a similar spirit to that introduced
in [21].
Definition 2: Let χ
i
∈ R
>0
and χ
j
∈ R
>0
. The units
at nodes i ∈ J and j ∈ J share their active powers
proportionally if
P
s
i
− P
d
i
χ
i
=
P
s
j
− P
d
j
χ
j
. (12)
In vector notation with X = d iag (χ
1
, . . . , χ
J
) and any
arbitrary constant γ ∈ R, (12) can be expressed as X
−1
(P
s
−
P
d
) = γ1
J
. Note that χ
i
and χ
j
are parameters that can
be chosen by the designer and don’t necessarily have to be
equal. In practice, a typical choice for χ
i
is the nominal
power rating of the unit at node i.
Lemma 2: Assume that the system (5) possesses a syn-
chronized motion (see Definition 1). Then, active power
sharing along this motion can be achieved if and only if
k, B, C and D are chosen such that
B + (C + D) L
F1
J
ω
d
+ γDLkX 1
J
= 0
J
, (13)
where
F =
1
T
J
D
−1
B1
J
1
T
J
D
−1
(B+CL)(I
J
+µ)1
J
(I
J
+ µ) − I
J
. (14)
Proof: Along a synchronised motion, (5) becomes
¯ω
s
= 1
J
ω
d
− k(P
m
− P
d
) + ξ
s
, (15a)
0
J
= −P
m
+ P
s
, (15b)
0
J
= (B + CL)(¯ω
s
− 1
J
ω
d
) + DLξ
s
. (15c)
Using (15b), we can rewrite (15a) as
ξ
s
= (¯ω
s
− 1
J
ω
d
) + k(P
s
− P
d
).
Inserting this equation in (15c) results in
0
J
=
B + (C + D) L
(¯ω
s
− 1
J
ω
d
) + DLk(P
s
− P
d
).
Following Definition 2, with P
s
− P
d
= γX 1
J
, we have
0
J
=
B + (C + D) L
(¯ω
s
− 1
J
ω
d
) + γDLkX 1
J
.
Furthermore, using (10) yields
0
J
=
B + (C + D) L
(I
J
+ µ)1
J
ω
∗
− 1
J
ω
d
+
γDLkX 1
J
. (16)
Substituting (7), power sharing is achieved if and only if
B + (C + D) L
F1
J
ω
d
+ γDLkX 1
J
= 0
J
,
with F given in (14), completing the proof.
Since the coefficients µ
i
are unknown, Lemma 2 reveals
that unlike in the case of ideal clocks [9], [7], [10], when
taking clock drifts explicitly into account, it is hard to derive
necessary and sufficient conditions for the controller gains
B, C, D and k to guarantee power sharing. However, based
on Lemmata 1 and 2 we can provide the following tuning
criterion that ensures power sharing.
Lemma 3: Assume that the system (5) possesses a syn-
chronized motion (see Definition 1). Then, active power
sharing along this motion can be achieved if k, B, C and
D are chosen such that
Bµ = 0
J×J
, and (C + D) = 0
J×J
, (17a)
as well as
kX = αI
J
(17b)
with α ∈ R.
Proof: For Bµ = 0
J×J
, (8) becomes
1
T
J
D
−1
CLµ1
J
= 0.
Furthermore, with (C + D) = 0
J×J
⇔ D
−1
C = −I
J
, and
recalling the fact that 1
T
J
L = 0
J
, we can show that (8) holds.
Thus, ω
∗
= ω
d
if (17a) holds. Furthermore, with Bµ = 0
J
and (C + D) = 0
J×J
,(13) becomes
0
J
= γDLkX 1
J
.
Inserting (17b) yields
0
J
= γαDL1
J
.
This completes the proof.
Using the derived conditions from Lemmata 1–3, in the
following we will investigate whether and how a zero steady
state frequency error and power sharing can be reached.
Therefore, we will compare different parametrizations of the
control law (4).
