IEEE Electrification Magazine - December 2019 - 59

Many of
the classical
synchronous
generators on ac
power systems are
being replaced by
power electronic
converters, which
do not inherently
respond to power
perturbations.

part of the kinetic energy stored in
its inertia to the grid. The power
exchange with the grid is, therefore,
dependent on the difference between
the angles of the machine (d 1) and
the grid side (d 2), which is also shown
in Figure 2(a).
The situation for dc systems is
quite different because there is no
electrical frequency and the power
exchange is determined by the voltage difference-and not the angle difference-between the device output
and the grid side. The dc bus voltage
can be established by using a power
converter controlled as a voltage
source. However, when the scale of
the dc grid increases, it is more convenient to connect several devices in
parallel to regulate the dc bus so that
they share all of the power perturbations in the system.
This way, the regulation does not rely on a single device
and the stability of the system can be preserved by the
systems connected in parallel. If we look back at the early
stages of power systems, when dc systems were still popular, we can find research papers in which dc machines
were proposed as load equalizers to cope with voltage
variations on dc grids. By looking at Figure 2(b), we can
deduce that the concept with dc machines is very similar
to that of the previously described synchronous machine.
The electromotive force (EMF) of the machine is proportional to its shaft speed, and its mechanical system resembles that of the synchronous machine. When a power
variation occurs at the grid side, the electrical power (and
therefore the torque) of the dc machine increases, causing
its shaft to slow down. The inertia of the machine opposes
that variation, and the EMF of the dc machine decreases
as a reflection of the machine's speed.
This system is equivalent to a capacitor directly connected to a dc grid [Figure 2(c)]. In this case, the energy is
not stored mechanically in rotational inertia but rather as
the electrostatic charge of the capacitor. Initially the capacitor and the dc bus will be charged at the same potential.
When a power variation occurs at the dc bus, the voltage of
the capacitor will vary depending on the current magnitude and on the capacitor's capacity. Capacitors hence
oppose voltage variations on the grid by releasing their
stored energy. This way, we can observe that a capacitor
connected to a dc bus is analogous to a synchronous
machine or dc machine connected to an ac or dc grid,
respectively. Hence, the bigger the capacitor value or the
rotational inertia, the more these systems will oppose variations in the frequency or voltage and the lower is the RoC
of these variables.
Figure 2 clearly shows that there is an analogy in terms of
how power is controlled and exchanged in ac and dc

systems. No matter whether the energy
is stored mechanically in a rotating
mass or an electrostatic field, there are
devices for ac as well as dc systems
that can be directly connected to
oppose variations of the bus frequency
or voltage. Table 1 shows the most relevant parameters of ac and dc systems that are analogous and that can
be used to improve the dynamic performance of these grids.

Virtual Inertia Concept

With the massive integration of RESs
and ESSs into the power system, many
of the classical synchronous generators
on ac power systems are being replaced by power electronic converters,
which do not inherently respond to
power perturbations. As a result, the
inertial responses of power systems are deteriorating. Moreover, different parts of power systems are being interconnected via ICs, and there will be parts of the grid that are
decoupled from the parts where synchronous machines are
connected. The main consequence of this transition is a
poorer inertial response for keeping the system stable and
operating normally. As shown in Figure 3, grids dominated
by converters suffer from a more severe RoCoF under the
same power perturbation. This might cause the disconnection of other parts of the grid to avoid equipment damage,
which, in turn, might lead to a chain reaction and even a
blackout. Therefore, various system operators have begun to
require that newly installed RES-based power plants and
ESSs provide some kind of inertial response under frequency and voltage deviations in the grid.

TABLE 1. The variables of ac and dc systems

that are analogous.

ac

	

dc

#

Parameter

SM Machine

dc Machine

Capacitor

1

Power
transfer
variable

~m

E

VC

2

Energy
storage
variable

J

J

C

3

Dynamic
equation

/x = J~o m

/x = J~o m

/ I C = CVo C

4

Relation
between
#1 and #3

-

E = K~ m

-

5

Stored
energy

1
E J = 2 J~ 2m

1
E J = 2 J~ 2m

1
E C = 2 CV 2C

IEEE Elec trific ation Magazine / D EC EM BE R 2 0 1 9

59



IEEE Electrification Magazine - December 2019

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https://www.nxtbook.com/nxtbooks/pes/electrification_december2022
https://www.nxtbook.com/nxtbooks/pes/electrification_september2022
https://www.nxtbook.com/nxtbooks/pes/electrification_june2022
https://www.nxtbook.com/nxtbooks/pes/electrification_march2022
https://www.nxtbook.com/nxtbooks/pes/electrification_december2021
https://www.nxtbook.com/nxtbooks/pes/electrification_september2021
https://www.nxtbook.com/nxtbooks/pes/electrification_june2021
https://www.nxtbook.com/nxtbooks/pes/electrification_march2021
https://www.nxtbook.com/nxtbooks/pes/electrification_december2020
https://www.nxtbook.com/nxtbooks/pes/electrification_september2020
https://www.nxtbook.com/nxtbooks/pes/electrification_june2020
https://www.nxtbook.com/nxtbooks/pes/electrification_march2020
https://www.nxtbook.com/nxtbooks/pes/electrification_december2019
https://www.nxtbook.com/nxtbooks/pes/electrification_september2019
https://www.nxtbook.com/nxtbooks/pes/electrification_june2019
https://www.nxtbook.com/nxtbooks/pes/electrification_march2019
https://www.nxtbook.com/nxtbooks/pes/electrification_december2018
https://www.nxtbook.com/nxtbooks/pes/electrification_september2018
https://www.nxtbook.com/nxtbooks/pes/electrification_june2018
https://www.nxtbook.com/nxtbooks/pes/electrification_december2017
https://www.nxtbook.com/nxtbooks/pes/electrification_september2017
https://www.nxtbook.com/nxtbooks/pes/electrification_march2018
https://www.nxtbook.com/nxtbooks/pes/electrification_june2017
https://www.nxtbook.com/nxtbooks/pes/electrification_march2017
https://www.nxtbook.com/nxtbooks/pes/electrification_june2016
https://www.nxtbook.com/nxtbooks/pes/electrification_december2016
https://www.nxtbook.com/nxtbooks/pes/electrification_september2016
https://www.nxtbook.com/nxtbooks/pes/electrification_december2015
https://www.nxtbook.com/nxtbooks/pes/electrification_march2016
https://www.nxtbook.com/nxtbooks/pes/electrification_march2015
https://www.nxtbook.com/nxtbooks/pes/electrification_june2015
https://www.nxtbook.com/nxtbooks/pes/electrification_september2015
https://www.nxtbook.com/nxtbooks/pes/electrification_march2014
https://www.nxtbook.com/nxtbooks/pes/electrification_june2014
https://www.nxtbook.com/nxtbooks/pes/electrification_september2014
https://www.nxtbook.com/nxtbooks/pes/electrification_december2014
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