Analysis of derivative control based virtual
inertia in multi-area high-voltage direct
current interconnected power systems
Elyas Rakhshani
1,2
✉
, Daniel Remon
1
, Antoni Mir Cantarellas
1
, Pedro Rodriguez
1,2
1
Abengoa Research, Abengoa, Campus Palmas Altas, Seville, Spain
2
Electrical Engineering Department, Technical University of Catalonia (UPC), Barcelona, Spain
✉ E-mail: elyas.rakhshani@gmail.com
Abstract: Due to increasing level o f power converter-based component and consequently the lack of inertia, automatic
generation control (AGC) of interconnected systems is experiencing different challenges. To cope with this challenging
issue, a derivative control-based virtual inertia for simulating the dynamic effects of inertia emulations by HVDC (high-
voltage direct current) interconnected systems is introduced and reflected in the multi-area AGC system. Derivative
control technique is used for higher level applications of inertia emulation. The virtual inertia will add an additional
degree of freedom to the system dynamics which makes a considerable improvement on first overshoot responses in
addition to damping characteristics of HVDC links. Complete trajectory sensitivities are used to analyse the effects of
virtual inertia and derivative control gains on the system stability. The effectiveness of the proposed concept on
dynamic improvements is tested through Matlab simulation of two-area test system for different contingencies.
1 Introduction
Automatic generation control (AGC) of a multi-area power system
during load and resource variation is known as a very important
mechanism that could facilitates frequency restoration and tie-line
power flow control between authority areas of AC/DC
interconnected systems [1, 2]. The need for transmitting power
over long distances with lower losses and higher stability has been
always the main challenge for transmission technologies. Due to
several limitations associated with AC lines especially for long
distance connections and in parallel recent developments of
renewable energy integration and super-grid interconnections in
modern power systems, attracts a lot of attention to HVDC
(high-voltage direct current) transmission, which is known as a
proven tool for dealing with the new challenges of future power
system [3, 4]. The HVDC interconnection is one of the main
applications of power converters in multi-area interconnected
power systems which could bring beneficial advantages like: Fast
and bidirectional controllability, power oscillation damping (POD)
and frequency stability support [5, 6]. For these reasons, in some
parts of the world, HVDC or hybrid interconnections, consisting
of parallel AC and DC interconnections became already the
preferred solution [7, 8] and it is necessary to consider DC links
on frequency stability analyses of interconnected power systems.
Usually, conventional generators can provide inertia and governor
responses against frequency deviations. Until now several classical
and advanced control techniques have been implemented to solve
load frequency control (LFC) problem [9–12]. However, in the case
of renewable generations, the lack of sufficient inertia will be the
main limitation of grid connected rene w able electrical energy sys tems
which gives rise to negative impa cts on the power sys tem operation
[13]. In such complex AC/DC system with the lack of inertia,
controlling the power exchange through tie-lines of multi-area
interconnected systems makes frequency regulation more complex
with several stability problems [14]. It is obvious tha t the matter of
modelling and control considering the methods of providing virtual
inertia to the sys tem is critical and the role of advanced technologies
such as the use of modern power processing systems, energy storage,
and advanced converters in HVDC links will be essential.
Grid-scale energy storage is widely believed to have the potential
to provide more flexibility which led to research investigation on
both the technical and economic issues surrounding energy storage
system (ESS) applications [15–21]. In most of these applications,
the matter of inertia and different control methodologies for
providing inertia is missing. Recently, several efforts have been
carried out to perform virtual inertia with different control
methodologies [ 22–25]. This virtual inertia is emulated by
advanced control of power converters considering a short-term
energy storage [25] which gave rise to the possibility of having a
huge amount of converter-based components without comprising
the system stability.
In the generation level, applications of energy storage devices are
very important in order to recover the lack of inertia in power
electronic parts of generation [26–29] and facilitating renewable
sources, especially photovoltaic power plants, to act as a
conventional generator for frequency support issues. New concepts
like synthetic inertia [30, 31] and virtual synchronous generator
are some of the main researches in this part [
32]. One application
of synthetic inertia emulation for wind power is reported in [31].
In this reference, derivation of frequency is used to modify the
references of wind farm generation for providing synthetic inertia
emulation. Small-signal analysis and frequency response
estimation, both with considering the effects of energy storage and
synthetic inertia are also reported by the authors [26, 29],
respectively. As reported in [29], estimation of frequency response
is considered as complementary task of synthetic inertia to
increase the reliability of the system with high wind penetration of
power. Most of the works are mainly related to the generation
parts of the system and there are no works which focus on
transmission and HVDC parts of multi area systems.
In fact, with high penetration of renewables, applications of
HVDC interconnections are increased. For two-area AGC power
system, AC–DC parallel tie-lines and superconducting magnetic
energy storage (SMES) units are proposed in [20, 21]. The HVDC
link is used as system interconnection in parallel with AC tie-line
to effectively damp the frequency oscillations of AC system while
the SMES unit provides bulk energy storage for achieving
combined benefits. In these references, there is not any detail
about inertia emulation methods and detailed analysis to know the
effects and sufficient amount of inertia during contingencies,
especially for high level control applications like the AGC of
interconnected systems. In [28], a generic modelling is proposed,
but it is not providing enough information about modelling virtual
inertia by transmission lines for large area power system. This type
of modelling is useful to model each component individually to
see the effects on the frequency for isolated system. An Inertia
Emulation Control (INEC) system is also proposed in reference
[22], which allows voltage source converter (VSC)-HVDC system
to perform an inertial response in a similar fashion to synchronous
machines.
In this paper, a new application of virtual inertia emulation by
converter-based components in multi-area AGC interconnected
AC/DC systems is presented. The main objective of this paper is
to propose a new approach of frequency stability analysis in
multi-area AGC system adding the matter of virtual inertia in AC/
DC interconnected systems. In this way, a supplementary power
modulation controller (SPMC) is also presented in a coordinated
manner for controlling the HVDC set-points during AGC
operation. As explained before, traditional LFC models have been
modified and revised to add different functionalities in
reformulation of conventional power systems. Most of those
modifications are related to AGC in a deregulated market scenario
[33], different types of power plants like renewable generation [34]
and recently the demand side dynamic models [35]. Therefore, the
general model of multi-area AGC system will be modified by
introducing derivative control technique in AC/DC interconnected
AGC model for high level frequency control studies. It should be
mentioned that, the importance of the virtual inertia in providing
ancillary services like emulating the damping and inertia for
frequency control improvements is another motivation of this paper.
Another goal of this paper is to propose a model which is very
useful for pre-evaluation of dynamic effects of converter stations
in higher level control design for power systems applications. In
modern power system, it would be valuable to have a clear idea
about the required energy through the transmission line, with
proper dynamic analysis considering worst case studies.
2 Dynamic model of multi-area AGC system
2.1 Conventional frequency regulation
In this section, a general overview on modelling and mathematics
behind the mechanism of frequency regulation for multi-area
interconnected power systems is presented.
2.1.1 Generator-load dynamic model: This model is based on
swing equation of synchronous generator where the relation between
inertia and load variation will be indicated. The stored energy in the
mechanical part of generator is related to rated power (MW) and its
inertia constant (W
0
ke
= H × S
b
). Considering that kinetic energy is
proportional to square of speed (fundamental frequency), the
energy for the ( f
0
+ Δf ) could be calculated as follows
W
ke
≃ HS
b
1 +
2Df
f
0
d
dt
(W
ke
) =
2HS
b
f
0
d
dt
(Df )
(1)
Then the power balance equation could be written as
DP
g
− DP
L
=
2HS
b
f
0
d
dt
(Df ) (2)
and in per unit, Δf is the same as Δω
DP
g
(pu) − DP
L
(pu) = 2H
d
dt
D
v
(3)
In the dynamic analysis of frequency stability, the most important
part of damping is the one related to the sensitivity of load to the
frequency variation which is defined as follows
∂P
L
∂f
Df = D · Df
where the D (pu/Hz) is the sensitivity of load change for 1% of
frequency change and in per unit. Therefore, the complete
equation considering the damping in pu will be as follows
DP
g
(pu) − DP
L
(pu) = 2H
d
dt
D
v
(pu) + D(pu) · D
v
(pu) (4)
Assuming that T
p
=2H/D and K
p
=1/D, the equation could be
written in the Laplace domain as follows
D
v
(s) = DP
g
(s) − DP
L
(s)
K
P
1 + sT
P
(5)
where T
p
is power system time constant and K
p
is the gain of power
system [2].
2.1.2 Governor-turbine model: As shown in Fig. 1, the output
of generator (ΔP
m
) in frequency regulation is adjusted by droop
governor action and for modelling the governor action a simple
first-order function can be used [14]
DP
g
= DP
ref
−
1
R
Df (6)
DP
m
=
1
1 + T
tg
s
DP
g
(7)
where ΔP
ref
is coming from area error determining set-points, T
tg
is
the time constant and ΔP
g
is the input signal for turbine-governor
system.
2.1.3 Frequency regulation of interconnected systems: A
proper control strategy for active power/frequency issue in
large-scale interconnected power systems is shown in Fig. 2.A
low-order linearised model is used for modelling the
load-generation dynamic behaviours. The angular frequency
deviation (Δω
i
) for the ith area in pu could be as follows
D
v
i
=
K
pi
1 + sT
pi
[DP
mi
− DP
Li
− DP
tie,i
] (8)
where ΔP
mi
is generated power deviation for each generation unit,
ΔP
tie,i
is the AC tie-line power change and ΔP
L
is the load change
in each area
DP
mi
=
1
1 + sT
tgi
−D
v
i
R
i
− K
I
apf
i
DP
refi
(9)
DP
ref i
=
ACE
i
s
(10)
and T
tgi
is the time constant of turbine-governor of the ith unit, R
i
isFig. 1 Turbine-governor model
known as droop characteristic, ΔP
refi
is the reference set-point of
units through AGC operation and ACE is the area control error
expressed as a linear combination of tie-line power flow
and weighted frequency deviations adding bias factors (β
i
) for
each area
ACE
i
=
b
i
D
v
i
+ DP
tie,i
(11)
In fact, control centres will collect the relevant frequency and power
flow information for ACE to identify the area participation factors
(apf) and the appropriate set-point adjustments for each generating
unit in AGC.
For modelling the interconnections between N areas in multi-area
AGC system, the tie-line power change between area i and the rest of
area could be presented as follows [10]
DP
tie,i
=
N
j=1
j=i
DP
tieAC,ij
=
1
s
N
j=1
j=i
T
ij
D
v
i
−
N
j=1
j=i
T
ij
D
v
j
⎡
⎢
⎣
⎤
⎥
⎦
(12)
DP
tieAC,ji
=
a
ij
DP
tieAC,ij
(13)
a
ij
=−
P
ri
P
rj
(14)
where T
ij
is the synchronising coefficient between areas and P
ri
is the
rated power of each area.
2.2 AC/DC transmission model
2.2.1 Brief description of HVDC systems: In two-terminal
HVDC transmission, two VSCs are the core of the system.
Usually one of the VSCs controls the DC voltage and the other
will controls the active power flow. The main control blocks of an
averaged VSC, as shown in Fig. 3, are inner and outer control
loops. The inner current control of HVDC is a fast dynamic
control system for controlling the AC current in this system. The
references are provided by outer control.
As shown in Fig. 3, the average model includes of controlled
voltage sources on the AC side and controlled current sources on
the DC side of converters. A controlled current source is also used
for modelling the relationship between AC part and dynamic
variations of the DC voltage caused in DC link.
2.2.2 SPMC-based HVDC control in AGC operation: For
frequency control analyses, this VSC could be modelled in terms
of power transferred from one area to another area through the DC
link. Each converter station could be modelled as a first-order
transfer function consisting a proper time constant which are
related to control blocks presented in Fig. 4. For HVDC system
with two VSC stations, the second-order transfer function will be
approximated by equivalent first-order transfer function imitating
the overall time response of HVDC system as follows
1
1 + sT
1
×
1
1 + sT
2
=
1
1 + (T
1
+ T
2
)s + (T
1
T
2
)s
2
1
1 + sT
DC
(15)
where T
1
and T
2
are the time constants of converters and T
DC
is the
equivalent time constant of overall HVDC control system
T
DC
= T
1
+ T
2
(16)
Therefore, the incremental power flow through the HVDC
transmission system could be modelled by a linear first-order
model with a proper time constant
T
DC
dDP
DC
dt
= Dx
r
− DP
DC
(17)
where Δx
r
is the reference control signal of DC power and ΔP
DC
will
be the real DC power flow through the system.
Fig. 2 Basic frame of ith area in AGC implementation of multi-area AC/DC interconnected power system
Fig. 3 Structure of average modelling for each VSC
As mentioned in [12–14], the authors focusing on this type of
higher level control design, the proper time response could be
between 100 and 500 ms. In this study, the time constant is
assumed 300 ms for T
DC
.
As shown in Fig. 5, in order to implement AGC action, a
supplementary modulation controller is designed as higher level
damping controller to improve the performance of power system
during load changes. The inputs of the proposed SPMC are
coming from AGC control centre and the outputs of AGC will
generate the new set-points for VSC stations and generation units
in all areas.
The Δx
r
is the input signal of a HVDC system which will be
generated by different control signals. These signals are frequency
deviations of each interconnected areas and AC power flow
deviations between area i and k (ΔP
tie,ik
). This power modulation
controller is modelled as a proportional controller
Dx
r
= K
fi
D
v
i
+ K
fk
D
v
k
+ K
AC
DP
tie,ik
(18)
Considering this new state in DC link, the ACE signal of each area in
AGC operation which contains one additional HVDC link will be
changed as follows
ACE
i
= B
i
D
v
i
+ DP
tie
(19)
DP
tie
= DP
tie,DC
+ DP
tie,i
(20)
where ΔP
tie
is the total tie-line power deviations, ΔP
tie,i
is the tie-line
power exchange between area i and other areas and ΔP
tie,DC
is the
DC power deviation in the HVDC link between area i and area k.
3 Derivative control-based virtual inertia
3.1 Virtual inertia concept
If the derivation signal of the grid frequency is used proportionally
for modifying the active power reference of a converter, a virtual
inertia in the power system could be emulated which will
contribute in enhancing the inertial response of the system against
the change in the power demand. The general control law for the
active power of the power electronics converter will be as follows
P
emulate
= k
a
v
0
d(D ˙
v
)
dt
(21)
where P
emulate
is the emulated power, k
a
is the proportional
conversion gain and ω
0
is the nominal grid frequency. This
concept could be used for different application like AGC analysis
considering a system for providing this virtual inertia to the system
[7, 8]. In order to emulate such inertia, an energy source is
required. Such energy can be supported from neighbouring area of
interconnected system or from an installed ESS. In this study, it is
assumed that inertia is emulated through an installed ESS.
3.2 Brief description of energy storage for LFC
As it was explained in the introduction, a variety of battery
technologies are being scaled up for grid application [15]. As
reported in [17, 18], battery and super-capacitor units can be used
for high power applications. Application of SMES as bulk energy
storage with HVDC links is also presented in [19, 20]. The
application of redox flow type batteries is successfully presented in
[36] for two-area interconnected LFC systems. The converter of
ESS element will be controlled to keep the storage element
charged during normal operation and then help the system during
contingencies. In fact, it is assumed that during normal conditions
the ESS is fully charged. In the event of a contingency in any
area, the ESS automatically will receive an error signal coming
from derivation of frequency error. Then it will inject the required
power within a very short time. Since the response of ESS is
much faster than mechanical part of governors, it is expected that
it can affect in a very fast response. It will improve the first
overshoot until the rest of conventional reserve in the
interconnected areas starts to recover the AGC action. Therefore,
the output of ESS will be zero for the steady-state condition.
3.3 Derivative-based virtual inertia for AGC model
To emulate sufficient virtual inertia with power electronic-based
component, the control scheme shown in Fig. 6 is proposed. The
derivative of the system frequency is obtained and the power
reference for converters will be modified. This energy can be
provided by storage devices or from reserved capacity of
neighbour area. In this study, we assumed that ESS is providing
sufficient energy for emulating inertia. This control concept is the
derivative control which calculates the rate of change of frequency
(ROCOF) during contingencies. The derivative control is sensitive
to the noise in the frequency measurements. To solve this
problem, a low-pass filter could be added to the control. This filter
could also simulate the dynamics of storage device which should
be fast. The control law for active power emulation in Laplace
with per unit value will be shown in Fig. 6. Where J
i
is the control
gains of inertia emulation controller and T
ESS
is also the time
constant of added filter for imitating the dynamic control for
electronic storage devices. Therefore, the emulated power (ΔP
ESS
)
Fig. 5 Control actions of interconnected area with HVDC model Fig. 6 Block diagram of derivative inertia emulation
Fig. 4 Block diagram of the HVDC system
in term of frequency deviations can be defined like this
DP
ESS1
(s) =
J
1
1 + sT
ESS,1
[s D
v
1
(s)] (22)
DP
ESS2
(s) =
J
2
1 + sT
ESS,2
[s D
v
2
(s)] (23)
The control structure for hybrid AC/DC link for interconnected areas
with the ability of storing energy and inertia emulation by derivative
control is presented in Fig. 7.
As shown in Fig. 7, if the active power through the converter is
controlled using the derivative of the frequency, a virtual inertia
could be emulated, thus enhancing the inertial response of
conventional generator to changes in the power demand. Equations
for frequency variations in two-area interconnected AGC system
could be modified as follows
D
v
1
(s) =
K
p1
1 + sT
p1
(DP
m1
+ DP
m2
− DP
tieAC,12
− DP
tie,DC
+ DP
ESS1
− DP
L1
) (24)
D
v
2
(s) =
K
p2
1 + sT
p2
(DP
m3
+ DP
m4
+ DP
tieAC,12
+ DP
tie,DC
+ DP
ESS2
− DP
L2
) (25)
It is obvious that with neglecting the damping (D
i
) and dynamics of
filter in the emulated power signal, these equations for the ith area
could be simplified as follows
M
i
+ J
i
dD
v
i
dt
= DP
a
(26)
where ΔP
a
is the total accelerating power. Considering (26) for
better understanding of added inertia, it is clear that using
derivative control technique, the total inertia (M
i
+ J
i
) of the system
can be increased.
The rest of the equation for the two-area system will be as follows
DP
m1
(s) =
1
1 + sT
tg1
D
v
1
R
2
× 2
p
− K
I1
DP
ref1
(27)
DP
m2
(s) =
1
1 + sT
tg2
D
v
2
R
2
× 2
p
− K
I1
DP
ref1
(28)
DP
m3
(s) =
1
1 + sT
tg3
D
v
3
R
3
× 2
p
− K
I2
DP
ref2
(29)
DP
m4
(s) =
1
1 + sT
tg4
D
v
4
R
4
× 2
p
− K
I2
DP
ref2
(30)
DP
ref1
=
ACE
1
s
=
1
s
b
1
2
p
D
v
1
+ DP
tieAC,12
+ DP
tie,DC
(31)
DP
ref2
=
ACE
2
s
=
1
s
b
2
2
p
D
v
2
− DP
tieAC,12
− DP
tie,DC
(32)
DP
tie,AC
=
T
12
s
D
v
1
− D
v
2
(33)
DP
tie,DC
=
1
1 + sT
DC
K
f 1
D
v
1
+ K
f 2
D
v
2
+ K
AC
DP
tieAC,12
(34)
To perform detailed analysis for the two-area power system, the
complete state-space presentation of the studied system should
be performed. The state space form can be obtained considering
(22)–(34)
˙
x = Ax + Bu (35)
where the state matrix A is partitioned as follows
A =
A
11
A
12
A
21
A
22
A
31
A
32
⎡
⎣
⎤
⎦
(12×12)
(36)
Therefore, each sub-matrix will be as follows: (see (equation 37) at
the bottom of the next page)
As identified in state-space presentation of global system, the
parameters of derivative control (J
1
and J
2
) are appeared in
sub-matrices A
31
and A
32
which are related to derivative control
state variables. In fact, these parameters are presented in 11th and
12th rows of the global system matrix A and could be used in
analysing the system performance. These elements are as follows
a
11,1
=
−J
1
T
ess1
T
p1
, a
11,3
=
J
1
K
p1
T
ess1
T
p1
, a
11,4
=
J
1
K
p1
T
ess1
T
p1
,
a
11,9
=
J
1
K
p1
T
ess1
T
p1
, a
11,10
=
J
1
K
p1
T
ess1
T
p1
, a
11,11
=
−J
1
K
p1
T
ess1
T
p1
+
−1
T
ess1
,
a
12,2
=
−J
2
T
ess2
T
p2
, a
12,5
=
J
2
K
p2
T
ess2
T
p2
, a
12,6
=
J
2
K
p2
T
ess2
T
p2
,
a
12,9
=
J
2
K
p2
T
ess2
T
p2
, a
12,10
=
J
2
K
p2
T
ess2
T
p2
, a
12,12
=
−J
2
K
p2
T
ess2
T
p2
+
−1
T
ess2
(38)
Finally, the B matrix can be presented as follows where the control
Fig. 7 Proposed model for two-area AC/DC interconnected system
