On-Line Inertia Estimation for Synchronous and Non-Synchronous Devices

Abstract

This paper proposes an on-line estimation method able to track the inertia of synchronous machines as well as the equivalent, possibly time-varying inertia from the converter- interfaced generators. For power electronics devices, the droop gain of the Fast Frequency Response (FFR) is also determined as a byproduct of the inertia estimation. The proposed method is shown to be robust against noise and to track accurately the inertia of synchronous generators, virtual synchronous genera- tors with constant and adaptive inertia, and wind power plants with inclusion of energy storage-based frequency control.

Full text

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On-line Inertia Estimation for Synchronous and
Non-Synchronous Devices
Muyang Liu, Member, IEEE, Junru Chen, Member, IEEE, Federico Milano, Fellow, IEEE
Abstract— This paper proposes an on-line estimation method
able to track the inertia of synchronous machines as well as
the equivalent, possibly time-varying inertia from the converter-
interfaced generators. For power electronics devices, the droop
gain of the Fast Frequency Response (FFR) is also determined
as a byproduct of the inertia estimation. The proposed method
is shown to be robust against noise and to track accurately the
inertia of synchronous generators, virtual synchronous genera-
tors with constant and adaptive inertia, and wind power plants
with inclusion of energy storage-based frequency control.
Index Terms— Inertia estimation, power system dynamics,
Fast Frequency Response (FFR), equivalent inertia, Converter-
Interfaced Generation (CIG).
I. INTRODUCTION
A. Motivation
The replacement of Synchronous Generators (SGs) with
non-synchronous devices, namely Converter-Interfaced Gen-
eration (CIG) sources, such as wind and solar, decreases the
inertia of the power system [1]. This creates operation and
security issues as a minimum inertia is required in the system
[2]. Advanced control schemes that make non-synchronous
devices provide inertia support have been developed in recent
year. Examples are the virtual synchronous generator control
[3] and inertial response control [4]. The objective of these
controls is to emulate the inertia response in the SGs and thus
enforce the non-synchronous devices boosting the power at
the instant of the contingency, and therefore, leading to the
concept of equivalent inertia. The equivalent inertia of non-
synchronous devices, unlike the inertia constant of SG, may
be variable [5], and even be specially designed as time-varying
[6]. A general method to fast and accurately estimate both
the constant and non-constant (equivalent) inertia, however, is
still missing. This paper aims at developing an on-line inertia
estimation method that can accurately track the (equivalent)
inertia of both synchronous and non-synchronous devices.
B. Literature Review
Efforts have been made to improve the accuracy to estimate
the inertia constant for SGs via off-line tests [7]–[9]. Similar
techniques also developed for the off-line identification for
the inertia of non-synchronous renewable turbines [10], [11].
The authors are with AMPSAS, School of Electrical and Electronic
Engineering, University College Dublin, Ireland. E-mails: muyang.liu@ucd.ie,
junru.chen.1@ucdconnect.ie, and federico.milano@ucd.ie).
This work is supported by the Science Foundation Ireland, by funding
Muyang Liu and Federico Milano under the Investigator Program Grant
No. SFI/15/IA/3074; and by the European commission, by funding Junru Chen
and Federico Milano under the project EdgeFLEX, Grant No. 883710.
The inertia of these devices can be uncertain, or even time-
varying, due to the ever-changing renewables and converter
controls [12]. Off-line tests, therefore, are not enough to track
the presented inertia of the non-synchronous devices.
The accurate and precise on-line monitoring for the dynamic
behavior of the power system becomes feasible with the
development of the smart grid techniques [13], especially,
the wide application of Phasor Measurement Units (PMUs)
[8], [14]. For example, reference [15] presents a Bayesian
framework based on the data collected with PMUs to estimate
the inertia of the generators with high accuracy. The high com-
putational burden of the Bayesian method, however, makes its
utilization impractical for on-line monitoring. Several PMU-
based estimation methods for the equivalent inertia constant
of a power system have been developed [16]–[19]. Most of
them, however, are not adequate tools for the on-line inertia
estimation of single devices, especially non-synchronous de-
vices with non-constant inertia control.
Reference [16] proposes an on-line identification algorithm
for the equivalent inertia of an entire power system by ana-
lyzing its dynamic response to a designed microperturbation.
Since the microperturbation signal affects the frequency re-
sponse of the system, it may lead to the unexceptional action
of the protective relays and thus increases the potential risk
of the power system stability. The same limit also exists for
the perturbation-needed inertia estimation method proposed in
[17]. Reference [18] obtains the system inertia by analyzing
the frequency signal via rotational invariance techniques. The
analysis requires a precise model that may not be available in
real-world applications. Reference [19] avoids the limitations
of [16]–[18] by proposing an on-line inertia estimator based
on the extension and mixture of a dynamic regressor. While
this regressor is designed under the assumption that the inertia
is constant. Time-varying equivalent inertia, therefore, can
prevent above estimation techniques to converge.
C. Contributions
This paper takes inspiration from the inertia estimation for-
mula proposed in [20] and [21], which is able to on-line track
the physical or equivalent inertia of a device. Such a formula,
however, is prone to numerical issues. With this regard, the
specific contributions of this work are the following:
•A discussion of the numerical issues of the inertia esti-
mation formula proposed in [20], [21] and the proposal
of two new formulas with improved numerical stability.
•As a byproduct of the above, a formula able to estimate,
under certain conditions, the damping of SGs and the
droop gain of FFR controls.

•The design of on-line inertia estimators that are based on
the proposed formulas.
The accuracy of the inertia estimators on tracking the
constant or non-constant inertia of synchronous and non-
synchronous devices is duly tested via the revised WSCC 9-
bus system under several scenarios.
D. Organization
The remainder of the paper is organized as follows. Section
II reviews the basic concepts developed in [20] and leads to the
on-line inertia estimation discussed in this paper. Section III
proposes the improved inertia estimation methods with higher
accuracy. The WSCC 9-bus system, adequately modified to
include non-synchronous generation, serves to investigate the
performance of the proposed inertia estimators on different
devices, including SG, Virtual Synchronous Generator (VSG)
and Wind Power Plant (WPP). Conclusions are drawn in
Section V.
II. TECHNICAL BACK GROU ND
Subsection II-A recalls the definition of the inertia constant
of SG and outlines the frequency evaluation of the power sys-
tem dominated by SGs. Subsection II-B presents the developed
inertia estimation formula of [20] and [21] and discusses its
numerical issues.
A. Inertia constant and system frequency evaluation
The inertia constant conditions the dynamic of SGs through
the well-known swing equation:
MG˙ωG=pm−pG−DG(ωG−ωo),(1)
where ωGis the rotor speed of the SG; ωois the reference
angular speed; DGis the damping; MGis the mechanical
starting time; pGis the electrical power of the SG injected into
the grid; and pmis the mechanical power of the SG. The inertia
constant is defined as HG=MG/2[22]. To avoid carrying
around the factor “2”, the estimation technique described in
the remainder of this paper are aimed at determining MG.
For the derivation of the inertia estimation formula dis-
cussed in the next section, it is convenient to split the me-
chanical power into three components:
pm=pUC +pPFC +pSFC ,(2)
where pUC is the power set point obtained by solving of the
unit commitment problem; pPFC is the active power regulated
by the Primary Frequency Control (PFC) and pSFC is the
active power regulated by the Secondary Frequency Control
(SFC). For a typical SG, the PFC is achieved through Turbine
Governor (TG), and the SFC is achieved through Automatic
Generation Control (AGC).
Figure 1 shows a typical frequency evolution of a power
system following a contingency [23]. As we can see in Fig. 1,
the evolution of the frequency can be divided into three time
scales, namely the inertial response, the PFC and the SFC.
These time scales differ by an order of magnitude from each
other: Tinertia ≈1s, TPFC ≈10 s and TSFC ≈100 s.
Tinertia
RoCoF
Frequency
Time
Frequency nadir
TPFC
TSFC
Reference frequency
Steady-state frequency after PFC
Fig. 1: Time scales of the frequency response and regulation of
synchronous machines.
During the period of inertial response, the dynamic behavior
of the frequency mainly depends on the inertia of the system
and is characterized by a relatively high ˙ω, often called Rate
of Change of Frequency (RoCoF) [24]. Following the inertial
response, the frequency gradually recovers to the nominal via
the PFC and SFC. The inertia estimation approach proposed in
this paper takes advantage of the fact that the inertial response
is the fastest among the frequency response of the synchronous
machine and the one with highest ˙ω.
B. Existing inertia estimation formulation
Differentiating (1) with respect to time and taking into
account (2), we can deduce:
MG¨ωG= ˙pUC + ˙pPFC + ˙pSFC −˙pG−DG˙ωG.(3)
Within the inertial response time scale, we can assume that:
˙pUC ≈0,˙pSFC ≈0,(4)
and:
|˙pPFC|  | ˙pG|.(5)
Since pGis the SG grid power injection, it is always
measurable by the Transmission System Operators (TSOs).
Then, reference [20] discusses how to estimate ˙pG, abbreviated
as Rate of Change of Power (RoCoP), based on PMUs mea-
surements. Finally, based on the estimation technique proposed
in [25], we can assume to be able to estimate ωGand, thus,
be able to calculate ¨ωG. In the following, we can thus assume
that ˙pGand ¨ωGare measurable and known.
With these assumptions, the inertia estimation formula is
proposed as a byproduct of the RoCoP:
MG≈M∗
G=−˙pG
¨ωG
,(6)
where ∗indicates an estimated quantities and it is further
assumed that ˙pPFC ≈0and DG≈0. The former assumption
holds in the time scale of the inertial response of SG. Note
that neglecting the damping and PFC is acceptable for SGs
but might not be adequate for non-synchronous devices. With
this in mind, Section III-B proposes a method to eliminate the
impact of damping and PFC on the inertia estimation of CIG.
2

Reference [21] extends the estimation formula (6) to evalu-
ate the (equivalent) inertia of any device that is able to modify
the frequency at its point of connection with the grid, namely
those devices whose power injection satisfies the condition:
|˙pbb|> p,(7)
where the subindex bb indicates a black box device; and
pis an empirical threshold to exclude the small frequency
fluctuations due to, for example, the stochastic variations
of ever-changing renewable sources such as wind and solar
generation.
The generalized inertia estimation formula is:
Mbb ≈M∗
bb =−˙pbb
¨ωbb
,(8)
where, ˙pbb can be obtained through the RoCoP estimation
method proposed in [21]; and, according to Frequency Divider
Formula (FDF) [26], the internal frequency of the device ωbb
can always be obtained through:
ωbb =ωB−xeq ˙pbb ,(9)
where ωBis the bus frequency the device connected to, and
xeq is the equivalent impedance of the device.
Although (8) proves to be fast and accurate in some scenar-
ios, it may fail due to numerical issues. Equation (8), in fact,
utilizes the second derivatives of the frequency signal as the
denominator, which might change sign and, thus, cross zero
in the first seconds after a contingency and therefore lead to
a singularity of (8).
A simple heuristic to remove the singularity consists in
holding the current value of the estimated inertia if the
denominator is close to zero:
M∗
bb =


−˙pbb
¨ωbb
,|¨ωbb| ≥ o,
M∗
bb(t−∆t),|¨ωbb |< o,
(10)
where ∆tis the sampling time and ois a positive threshold
to avoid the numerical issue. In the reminder of this paper,
we use (10), rather than (8), to compare the inertia estimation
technique proposed in this paper with the one discussed in
[25]. A large oleads to estimation error, while a small o
cannot avoid numerical issues. According to a comprehensive
set of numerical tests, we have concluded that a proper o
is hard to find, if it exists at all, and is device dependent.
Therefore, in the following section, we propose a new formula
with enhanced numerical stability.
III. PROP OSE D ON-LIN E INERT IA EST IMATORS
This section elaborates on (8) and proposes two novel
inertia estimation formulas. The first formula is presented
in Subsection III-A and avoids the numerical issue of (8).
The second formula is presented in Subsection III-B and
accounts for damping and PFC through an additional formula,
which can also be utilized to estimate the droop gain of
the FFR control of non-synchronous devices, as discussed in
Subsection III-C. Finally, Subsection III-D provides the design
of the inertia estimators based on the proposed formulas.
A. Improved formula with better numerical stability
As discussed in Section II-B, the fragile numerical stability
of (8) is due to the division by ¨ω. Therefore, we propose the
following differential equation that avoids such a division:
TM˙
M∗
bb =γ¨ωbb ˙pbb +M∗
bb ¨ωbb,(11)
where
γ(x) = 




−1, x ≥x,
0,−x< x < x,
1, x ≤ −x,
(12)
and xis a small positive threshold closing to zero.
The rationale behind (11) is as follows. At the equilibrium
point, M∗
bb ¨ωbb =−˙pbb. According to (8), this conditions is
obtained for M∗
bb =Mbb, which is the sought inertia value.
During a transient, M∗
bb ¨ωbb 6=−˙pbb. Let us consider the
case M∗
bb ¨ωbb >−˙pbb. Then the sign of ˙
M∗
bb is adjusted
through the function γ(¨ωbb)in roder to make M∗
bb converge
to Mbb. The sign of γis decided based on the sign of ¨ωbb. If
¨ωbb >0,M∗
bb has to decrease to decrease M∗
bb ¨ωbb and thus
γ(¨ωbb) = −1. Otherwise, if ¨ωbb <0,γ( ¨ωbb)=1to increase
M∗
bb. The time constant TMdecides the rate of change speed
of M∗
bb. To avoid chattering around the the equilibrium point,
a small deadband is included in (12), namely (−x, x). A
proper choice of xcan effectively reduce the impact of
frequency fluctuations and noise, and therefore, the deadband
for RoCoP, namely (7) is no longer needed.
Compared to (8), the inertia estimation formula (11) not
only avoids numerical issues, but also allows filtering spikes
and noises by adjusting TM. Using a proper initial guess on
M∗
bb can improve the speed of the estimation (11), but it is
not essential for convergence. Finally, note that all results
presented in this paper are obtained assuming the initial
condition M∗
bb(0) = 0, where t= 0 corresponds to the time
at which the contingency occurs. This value serves to show
that the proposed method is fast, effective and is suitable for
on-line applications as it does not require storing historical
data. In practice, however, any value of M∗
bb as obtained from
previous estimations can be used.
B. Improved formula with damping estimation
This subsection focuses exclusively on SGs. The accuracy
of (11) can be increased by removing the assumption DG≈0.
With this in mind, we rewrite (11) as:
TM˙
M∗
G=γ(¨ωG) [ ˙pG+M∗
G¨ωG+D∗
G˙ωG],(13)
where D∗
Gis the estimated value of damping, which is not
known. The following equation allows estimating the damping:
TD˙
D∗
G=γ(∆ωG) [∆pG+M∗
G˙ωG+D∗
G∆ωG],(14)
where ∆ωG=ωG−ωG,o, with ωG,o =ωG(0), or equivalently:
∆ωG=Z˙ωGdt , (15)
and
∆pG=Z˙pGdt . (16)
3

According to (12), the proposed inertia estimation formulas
(13)-(14) introduce two thresholds related to the frequency
variations of the device, namely ¨ωGand ∆ωG. If properly set,
these two thresholds can remove small frequency fluctuations
resulting from the stochastic renewable energy sources in a
more effective way than (7).
Note that even though the integrals in (15) and (16) are
presented as indefinite integrals, in practice, they are calculated
with a fixed initial time. In particular, t= 0 s is used as the
initial time when the disturbance that triggers the variations of
the frequency occurs, namely at the moment the step change
from 0to ±1of the function γoccurs.
The on-line estimator based on (13)-(14) allows to eliminate
the impact of damping on the accuracy of inertia estimation.
However, the estimated damping D∗
Gmay never converge to
the actual DGdue to the effect of PFC. In the first seconds
following a contingency, we have:
pPFC =−RωG−ωref ,(17)
where ωref is the reference of the frequency, Ris the droop
gain of TG [22].
Substituting (17) into (3), we have:
˙pG+MG¨ωG+DG+R˙ωG= 0 .(18)
Let us consider another reasonable assumption that ωG,o ≈
ωG,ref . Therefore, one can always assume:
Z˙ωGdt ≈ωG−ωG,ref .(19)
Substituting (17) and (19) into (1), we have:
Z˙pGdt +MG˙ωG+DG+RZ˙ωGdt = 0 .(20)
Comparing (18)-(20) with (13)-(14), we can deduce that D∗
G
in (13) and (14) actually tracks DG+R. Since ˙ωGvaries much
slower than ¨ωGwithin the first seconds after a contingency,
D∗
Gwill take more time to converge than M∗
G.
The discussion above proves that D∗
Gcannot accurately esti-
mate the damping of SGs but effectively improve the accuracy
of inertia estimation by eliminating the impact of damping
and PFC through taking their resulted power variations into
account.
C. Applications to non-synchronous devices with FFR
Equations (13)-(14) can be generalized for any device that
regulates the frequency. Dropping for simplicity the subindex
G, we have:
TM˙
M∗=γ(¨ω)[ ˙p−M∗¨ω−D∗˙ω],(21)
TD˙
D∗=γZ˙ωdtZ˙pdt −M∗˙ω−D∗Z˙ωdt,(22)
where ωis the internal frequency of the non-synchronous
device.
Note that the time constants TMand TDshould be small
enough to accurately track the time-varying inertia. Small time
constants, however, make (21)-(22) more sensitive to noise and
may introduce spurious oscillations. This issue can be solved
through an additional filter. An example that illustrates this
point is given in Section IV-B.
The formulas (21)-(22) can be utilized to obtain the droop
gain of the FFR that is modeled as:
pFFR =−R(ωgrid −ωref ),(23)
where ωgrid is the grid frequency.
Here we should highlight that in contrast to SG, the primary
response in CIG is instant along with the inertia response after
the contingency. The damping is the friction of the rotational
change of the device to the grid frequency, while the droop is
the frequency deviation of the grid frequency to the nominal
one, as follows:
M˙ω=pUC +pFFR −p−D(ω−ωgrid).(24)
In CIG, the device tracks the grid frequency change simul-
taneously, e.g. via the Phase-Locked Loop (PLL) with time
constant below 0.1s. Therefore, we can assume that ω≈ωgrid
and accordingly:
D(ω−ωgrid)pFFR .(25)
Comparing (25) with (20), we can deduce, for CIG sources,
the D∗of the estimator (21)-(22) actually tracks R.
D. Design of real-time loop
The proposed inertia estimation formulas can be used to
fulfill the real-time measuring of the inertia through the
estimators fed by the RoCoP and RoCoF signals.
Figure 2 shows the structure of a real-time inertia estimator
based on (11). If |¨ω|< ¨ωin γ( ¨ω)(see (12)), dM∗= 0
holds. This condition indicates that the estimated M∗can be
held after the inertial response with a proper ¨ω.
PI Filter
Fig. 2: Real-time loop for inertia estimation (11).
The control scheme of the PI filter included in Fig. 2 is
shown by Fig. 3. The parameters of the PI filter are selected
as Kp= 50,Ki= 1 and Tf= 0.0001 for all the simulation
results shown in the remainder of the paper.
Fig. 3: Control scheme of PI filter.
The real-time loop of the inertia estimator based on (21)-
(22) is shown in Fig. 4. Instead of directly taking the input
˙ωfor computing dD∗, the ˙ω∗passing through the PI filter
improves the robustness of the estimator against measurement
noise.
4

PI Filter
Fig. 4: Real-time loop for inertia estimation (21)-(22).
IV. CAS E STUDY
The WSCC 9-bus system shown in Fig. 5 is utilized in
this section to investigate the performance and accuracy of the
proposed on-line inertia and damping estimators. To test the
performance of the estimators with SGs, the standard WSCC
9-bus system described in [27] is used. Then the machine
connected at bus 2 is substituted for a VSG and a Doubly-Fed
Induction Generator (DFIG) to test the estimation of equivalent
inertia constant and droop gains of non-synchronous devices.
G
65
4
7 9 32 8
1
G
G
Fig. 5: WSCC 9-bus system.
This section considers and compares three on-line inertia
estimators. The estimators are denoted as E0 based on (8),
E1 based on (11) (see Fig. 2) and E2 based on (21)-(22)
(see Fig. 4). Three different devices are considered with the
following objectives:
1) Verify the accuracy of the proposed estimators to eval-
uate the inertia constant of SGs;
2) Test the accuracy of the estimators on tracking the
constant and time-varying inertia of the grid-forming
CIG via a VSGs with known inertia;
3) Illustrate the capability of the estimators to evaluate the
inertia support from the stochastic renewable source,
i.e. the WPP, without and with co-located Energy Stor-
age System (ESS) in grid-following control.
All scenarios are triggered by a sudden load change, i.e. an
increase of 20% load connecting to Bus 5, occurring at t= 1
s. The thresholds o=p=¨ω=∆ω= 10−6are used in
Section IV-A and IV-B. The time step for all time domain
simulations is 1ms. This is also assumed to be the sampling
time of the measurements utilized in the proposed estimators.
All simulations are obtained using the Python-based software
tool Dome [28].
A. Inertia estimation for SGs
This subsection discusses the performances of the on-line
inertia estimators for evaluating the inertia constant of the SG
connected to Bus 3 (denoted as G3). The actual mechanical
starting time MGand damping DGof G3 are 6.02 s and 1.0
respectively. The results discussed in this section are obtained
with TM= 0.01 for E1, and TM= 0.001 and TD= 0.001
for E2.
1) No primary frequency control: In this first scenario, we
assume that G3 has no TG. This is, of course, not realistic,
but allows us better illustrating the transient behavior for the
estimators. TGs are included in all subsequent scenarios.
Figure 6 shows the estimated mechanical starting time M∗
G
of G3 through the three estimators. According to Fig. 6, both
E1 and E2 can accurately estimate the inertia constant after
roughly 80 ms. This period can be decreased with smaller time
constants TM, which, however, can lead to small oscillations.
012345
Time [s]
−2
0
2
4
6
8
10
M∗
G[MW s /MVA]
E0
E1
E2
Fig. 6: Trajectories of estimated inertia of G3 without TG as obtained
with E0, E1 and E2.
012345
Time [s]
−2.5
−2
−1.5
−1
−0.5
0
0.5
pu (MW /s)
−˙p∗
¨ωM
1.56 1.565 1.57 1.575 1.58 1.585 1.59 1.595 1.6
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
−˙p∗
¨ωM
Fig. 7: Trajectories of the dynamic variations of G3 as obtained with
E0.
E0 shows a faster response comparing with E1 and E2, but
the worst accuracy for introducing spurious spikes. Section
III-A briefly explains the cause of the spurious spikes, which
can be further clarified by Fig. 7. As we can see in Fig. 7,
there is a small phase differences between the nominator
−˙p∗and denominator ¨w∗of (8). It means that they do not
cross zero at the same time, and thus when the denominator
goes to zero, the numerator is small but no null, hence the
large estimation errors and, eventually, the spikes. Given the
intrinsic numerical issues of E1, we consider exclusively E1
and E2 in the remainder of the paper.
5

2) Effect of primary frequency control: Figure 8 shows the
estimated inertia of G3 with TG. In this scenario, E1 and E2
obtain the inertia constant with good accuracy. E2 shows a
slightly smaller estimation error than E1.
012345
Time [s]
−1
0
1
2
3
4
5
6
7
M∗
G[MW s /MVA]
E1
E2
Fig. 8: Estimated inertia of G3 with TG as obtained with E1 and E2.
For the sake of example, Fig. 9 shows the estimated
damping coefficient of G3 with and without TG through the
estimator E2. As expected, E2 can accurately estimate the
damping Dof G3 only if the PFC is not included. This result
is consistent with the discussion in Section III-B. Clearly, PFC
is always presented in conventional power plants. But this is
not a drawback of the proposed estimation approach as, in
practice, the damping of synchronous machines is very small
and its estimation is not necessary. Much more relevant is the
estimation of the FFR droop gain of non-synchronous devices.
This is discussed in Section IV-B.
012345
Time [s]
−1
0
1
2
3
4
5
6
D∗
G[MW /MVA]
No TG
With TG
Fig. 9: Trajectories of the estimated damping of G3 with and without
TG and without measurement noise as obtained with E2.
3) Impact of measurement noise: This section investigates
the robustness of the proposed estimators E1 and E2 against
measurement noise. Noise is added to both RoCoP and RoCoF
measurements fed into the estimators. The noise is modeled as
an Ornstein-Uhlenbeck stochastic process [29]. The standard
deviation of the measurement noise are selected according
to the expected maximum PMU error at the fundamental
frequency [30] and relevant tests for RoCoP measurement [31],
namely 10−4for RoCoF signal and 0.01 for RoCoP signal.
Figure 10 shows the inertia estimated with E1 and E2. Both
estimators prove to be robust against measurement noise.
B. Inertia estimation for VSGs
The power-electronics-based VSG control is regarded as
one of the most effective methods to improve the frequency
012345
Time [s]
−1
0
1
2
3
4
5
6
7
M∗
G[MW s /MVA]
E1
E2
Fig. 10: Estimated inertia of G3 with TG and measurement noise as
obtained with E1 and E2.
stability of the low-inertia system in recent years [1]. Since
the equivalent inertia of VSGs is imposed by the control of
the converter and is thus known a priori, the VSG represents
a good test to evaluate the accuracy of the inertia estimators
proposed in this work.
1) VSG with constant inertia: We first consider the VSG
described in [32]. In this scenario, the inertia and FFR droop
gain are constant, i.e. MVSG = 20 s and RVSG = 20.
Figure 11 shows the trajectories of the equivalent inertia as
obtained with E1 with TM= 0.001 and E2 with TM= 0.001
and TD= 10−4. E2 obtains the accurate MVSG roughly
60 ms after the contingency, while the estimated inertia of
E1 oscillates around the actual value of the inertia. The
amplitude of such an oscillation decreases as ˙ωdecreases. This
is because, in the power electronics device, the droop/damping
and the inertia response “pollutes” the inertia estimation as in
E1. The additional loop included in E2 for the droop/damping
estimation can avoid this issue. Therefore, for the CIG with
FFR, E2 performs better than E1. Since the remainder of this
section focuses on CIGs, only E2 is considered.
0 1 2 3 4 5
Time [s]
0
5
10
15
20
25
MVSG [MW s /MVA]
E1
E2
Fig. 11: Estimated inertia of the VSG with constant inertia as obtained
with E1 and E2.
2) VSG with adaptive inertia: In this scenario, we consider
an adaptive VSG, which can tune its inertia with respect to
the grid state. The detailed model of the adaptive VSG can be
found in [6]. The adaptive VSG has the same droop gain as
the VSG with constant inertia discussed above.
In order to track the time-varying inertia of the adaptive
VSG, we need to decrease the time constant of the estimator.
A smaller time constant, however, may lead to spurious oscil-
lations in the estimated result and thus an extra filter is needed.
Figure 12 shows the trajectories of the actual inertia Mof the
6

0 2 4 6 8 10
Time [s]
0
5
10
15
20
25
MVSG [MW s /MVA]
M
M∗
˜
M∗
Fig. 12: Estimated inertia of the VSG with adaptive inertia as obtained
with E2: Mis the actual inertia of the adaptive VSG; M∗is the
estimated inertia as obtained with E2; and ˜
M∗is the filtered estimated
inertia.
0123456
Time [s]
0
5
10
15
20
D∗
VSG [MW /MVA]
Constant inertia
Adaptive inertia
Fig. 13: Estimated droop gain of different VSGs through estimator
E2 with TD= 10−4.
adaptive VSG, the estimated inertia M∗obtained by E2 with
TM= 5 ·10−5s, TD= 10−4s and the filtered estimated
inertia ˜
M∗. The filter utilized to obtain ˜
M∗in Fig. 12 is a
basic average filter [33] with time constant T= 0.25 s. Figure
12 shows that the estimator E2 can accurately track the time-
varying inertia with proper parameters and filter.
Figure 13 shows the estimated droop gain of the VSGs with
constant and adaptive inertia through E2. E2 can accurately
estimate the droop gain for these two kinds of VSG. This
result is consistent with the discussion in Section III-C. The
oscillations shown in the estimated inertia for adaptive VSG
have no impact on the droop gain estimation.
C. Inertia estimation for WPPs
This subsection focuses on WPPs modeled as DFIGs. The
detailed model of the DFIG can be found in [34]. The wind
speed is modeled as an Ornstein-Uhlenbeck stochastic process
that fitted with real-world wind speed measurement data [35].
The trajectories of the wind speed obtained from 500 Monte
Carlo simulations. In all the figures shown in this section,
µand σrepresent the mean and standard deviation of the
simulated time series.
All the trajectories of the estimated inertia presented in this
subsection are obtained through the estimator E2 with TM=
0.001 and TD= 0.001. In oder to depress the impact of the
stochastic wind, we set ¨ω= 2 ·10−4and ∆ω= 0.1.
1) WPP without ESS: We first consider the case of a
DFIG without ESS. Figure 14 shows the trajectories of the
output active power of the WPP following the sudden load
increase. The WPP has limited response to the contingency.
The active power of the WPP varies following the dynamics
of the wind speed, while the mean remains the same before
and after the occurrence of the contingency. Accordingly,
the estimated inertia of the WPP are within the small range
M∗
WPP ∈[−0.28,0.1] and the mean is almost zero, according
to Fig. 15. Figure 15 also shows that the inertia estimation is
not biased by the stochastic wind dynamics resulted before
the contingency. The values of ¨ωand ∆ω, therefore, are
adequate.
0123456
Time [s]
1.54
1.56
1.58
1.6
1.62
1.64
1.66
1.68
1.7
pWPP [pu(MW)]
µ
µ±3σ
Fig. 14: Output active power of the WPP without ESS.
0123456
Time [s]
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
M∗
WPP [MW s /MVA]
µ
µ±3σ
Fig. 15: Estimated inertia of the WPP without ESS.
As expected, the results shown in Figs. 14 and 15 lead to
conclude that the WPP without frequency control nor ESS
does not provide any significant inertia support to the system.
2) WPP with ESS: In this scenario, we consider the DFIG
coupled with an ESS. The ESS is modeled as a Grid-Following
Converter (GFC) with RoCoF control. The detailed model of
the GFC can be found in [36]. Due to the short-term analysis,
the storage limits of the ESS is not considered. The gain of
the RoCoF control in the ESS is 40.
Figure 16 shows the trajectories of the output active power
of the DFIG with the ESS obtained from 500 Monte Carlo
simulations. The active power of the WPP with ESS increases
after the occurrence of the contingency, while its magnitude
vary slightly depending on the stochastic wind speed.
Figure 17 shows the estimated inertia of the WPP with
ESS through the on-line inertia estimator E2 in 500 tests.
Consistently with the uncertain active power injection shown
in Fig. 16, the equivalent inertia provided by the WPP varies
within the range M∗
WPP ∈[33.1,45.8] according to Fig. 17.
The average value of the WPP inertia is 40 s, which is
consistent with the RoCoF control gain. These results indicate
7

0123456
Time [s]
1.55
1.6
1.65
1.7
1.75
1.8
1.85
pWPP [pu(MW)]
µ
µ±3σ
Fig. 16: Output active power of the WPP with ESS.
0123456
Time [s]
−10
0
10
20
30
40
50
M∗
WPP [MW s /MVA]
µ
µ±3σ
Fig. 17: Estimated inertia of the WPP with ESS.
that the WPP can provide an inertial response through the
RoCoF control of its ESS.
3) Inertia estimation of SG in the high-wind-penetration
system: In this scenario, we consider again the system dis-
cussed in Section IV-C.2 but, in this case, we focus on the
estimation of the inertia of the synchronous generator G3 via
estimator E2. Since the system includes a stochastic energy
source, the thresholds are ¨ω= 2 ·10−4and ∆ω= 0.1, and
the time constants are TM=TD= 0.001 s.
Figure 18 shows the estimated inertia of G3 in the revised
WSCC 9-bus system with high wind penetration and FFR en-
ergy storage through the on-line inertia estimator E2 obtained
with 500 simulations. E2 shows a satisfactory accuracy, even
though small fluctuations are introduced compared to the ideal
scenario discussed in Section IV-A. In the vast majority of
Monte Carlo realizations, the thresholds avoid to trigger the
inertia estimation before the occurrence of the contingency.
In general, thus, and as shown in Fig. 18, the accuracy of
inertia estimation following the contingency is not affected by
noise. These results also demonstrate that E2 is able to obtain
an accurate estimation of the inertia of a specific device even
if the system include other devices with faster dynamics and
controllers.
V. CONCLUSION
This paper elaborates on the inertia estimation method (E0)
discussed in [20] and proposes two on-line inertia estimation
formulas for both synchronous and non-synchronous devices.
The first proposed method (E1) avoids the potential numerical
issues of E0 by changing the structure of the formula. The
second method (E2) further improves the accuracy of E1 by
including an additional equation to eliminate the effect of
0123456
Time [s]
−1
0
1
2
3
4
5
6
7
M∗
WPP [MW s /MVA]
µ
Fig. 18: Estimated inertia of G3 in the modified WSCC 9-bus system
with inclusion of a WPP and an ESS.
damping and/or droop. E1 is simpler and shows satisfactory
accuracy for the inertia estimation of the SG. On the other
hand, E2 works better for non-synchronous devices, including
time-varying inertia response and stochastic sources.
The work presented in this paper can be extended in various
directions. We aim at further validating the proposed inertia
estimation using measurements of real-world grids. We also
aim at improving its robustness against large measurement
errors, due to, e.g., cyber attacks. We will explore other
applications of the proposed estimators, e.g., tracking the
inertia of sub-networks rather than single devices. This can
done by modeling the sub-network as a multi-port device.
Finally, we are considering the development of advanced
controllers that track the inertia by means of the estimators
proposed in this work.
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Muyang Liu (S’17-M’20) received the ME and
Ph.D. in Electrical Energy Engineering from Uni-
versity College Dublin, Ireland in 2016 and 2019.
Since December 2019, she is a senior researcher with
University College Dublin. Her scholarship is funded
through the SFI Investigator Award with title “Ad-
vanced Modeling for Power System Analysis and
Simulation.” Her current research interests include
power system modeling and stability analysis.
Junru Chen (S’17-M’20) received the ME and
Ph.D. degree in Electrical Energy Engineering from
University College Dublin in 2016 and 2019. He was
exchanging student at Kiel University (Germany) in
2018 and at Tallinn University of Technology (Esto-
nia). He is currently a senior researcher at University
College Dublin and a visiting scholar at Aalborg
University, Denmark. His current research interests
in Power electronics control, modeling, stability and
application.
Federico Milano (S’02, M’04, SM’09, F’16) re-
ceived from the University 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 Waterloo, 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.
9