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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.
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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).
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:,, and
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.
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:
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.
Frequency nadir
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 ˙pGDG˙ωG.(3)
Within the inertial response time scale, we can assume that:
˙pUC 0,˙pSFC 0,(4)
|˙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:
where indicates an estimated quantities and it is further
assumed that ˙pPFC 0and DG0. 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.
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
The generalized inertia estimation formula is:
Mbb M
bb =˙pbb
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 =ωBxeq ˙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:
bb =
,|¨ωbb| ≥ o,
bb(tt),|¨ωbb |< o,
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.
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:
bb =γ¨ωbb ˙pbb +M
bb ¨ωbb,(11)
γ(x) =
1, x x,
0,x< x < x,
1, x ≤ −x,
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 ˙
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
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
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 DG0.
With this in mind, we rewrite (11) as:
G=γ(¨ωG) [ ˙pG+M
where D
Gis the estimated value of damping, which is not
known. The following equation allows estimating the damping:
G=γ(∆ωG) [∆pG+M
where ωG=ωGωG,o, with ωG,o =ωG(0), or equivalently:
ωG=Z˙ωGdt , (15)
pG=Z˙pGdt . (16)
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 =RωGωref ,(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 ωGωG,ref .(19)
Substituting (17) and (19) into (1), we have:
Z˙pGdt +MG˙ωG+DG+RZ˙ωGdt = 0 .(20)
Comparing (18)-(20) with (13)-(14), we can deduce that D
in (13) and (14) actually tracks DG+R. Since ˙ωGvaries much
slower than ¨ωGwithin the first seconds after a contingency,
Gwill take more time to converge than M
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
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:
M=γ(¨ω)[ ˙pM¨ωD˙ω],(21)
D=γZ˙ωdtZ˙pdt M˙ωDZ˙ωdt,(22)
where ωis the internal frequency of the non-synchronous
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 pD(ωωgrid).(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 ωωgrid
and accordingly:
D(ωωgrid)pFFR .(25)
Comparing (25) with (20), we can deduce, for CIG sources,
the Dof 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 Mcan 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 ˙ωpassing through the PI filter
improves the robustness of the estimator against measurement
PI Filter
Fig. 4: Real-time loop for inertia estimation (21)-(22).
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.
7 9 32 8
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= 106are 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
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.
Fig. 6: Trajectories of estimated inertia of G3 without TG as obtained
with E0, E1 and E2.
1.56 1.565 1.57 1.575 1.58 1.585 1.59 1.595 1.6
Fig. 7: Trajectories of the dynamic variations of G3 as obtained with
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
˙pand denominator ¨wof (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.
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.
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.
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 104for 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
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= 104. 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.
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
Fig. 12: Estimated inertia of the VSG with adaptive inertia as obtained
with E2: Mis the actual inertia of the adaptive VSG; Mis the
estimated inertia as obtained with E2; and ˜
Mis the filtered estimated
Fig. 13: Estimated droop gain of different VSGs through estimator
E2 with TD= 104.
adaptive VSG, the estimated inertia Mobtained by E2 with
TM= 5 ·105s, TD= 104s and the filtered estimated
inertia ˜
M. The filter utilized to obtain ˜
Min 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 ·104and 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
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
Fig. 14: Output active power of the WPP without ESS.
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
Fig. 16: Output active power of the WPP with ESS.
Time [s]
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 ·104and 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
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
Time [s]
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
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.
... The discrete methods estimate inertia upon the occurrence of an event, such as loss of generation or fault [1], [7]- [16]. The methods presented in [1], [7]- [10] use the swing equation to estimate the inertia of areas [1], whole system [7]- [9], or individual sources [10] in the event of a power imbalance. Similarly, the papers [11] and [12] use energy-based methods to estimate inertia. ...
... The discrete methods estimate inertia upon the occurrence of an event, such as loss of generation or fault [1], [7]- [16]. The methods presented in [1], [7]- [10] use the swing equation to estimate the inertia of areas [1], whole system [7]- [9], or individual sources [10] in the event of a power imbalance. Similarly, the papers [11] and [12] use energy-based methods to estimate inertia. ...
... Let s i be the current state and a j be the action taken by the algorithm. The new value for Q(s i , a j ) is computed using the update rule given in (10). where lr is the learning rate (0 < lr ≤ 1) that determines how quickly the Q-values change, and r is the reward for the selected action decided by the reward mechanism. ...
Full-text available
With the growing emphasis on mitigating climate change, the power industry is moving towards renewable energy sources as an alternative to fossil fuel-based power plants. The transition to renewable energy has created numerous challenges, one of which is the low levels of inertia that impact the stability of power systems. Therefore, inertia monitoring has become an integral part of power system operation to dispatch renewable energy sources while maintaining frequency stability. This article presents an online method to continuously estimate the inertia of a power system. The inertia is computed from PMU (Phasor Measurement Unit) data using small variations in frequency and power under ambient conditions. The method uses electrical and kinetic energy variations to compute inertia. In addition, a Q-learning-based method is presented to identify mechanical power changes to discard invalid inertia estimates. The method is demonstrated using the IEEE-39 bus system to monitor the regional inertia of the test system.
... To mitigate the frequency instability induced by IBRs, some inverters are equipped with inertia emulation control, i.e., virtual synchronous generator (VSG) control, which may introduce time-varying inertia. [5] develops a dynamic estimator to track the time-varying inertia from VSG, but it suffers from numerical oscillations. [6] utilizes ambient measurements to estimate non-synchronous generator inertia constant, however, it assumes the knowledge of generator rotor speed and angle, difficult to obtain in practice. ...
... In all scenarios, L is 10, β is 0.99 while γ is 0.95. Besides, the state-of-art methods in [8] (Method 1) and [5] (Method 2) are compared. The initial value of ω, δ, and V s can be obtained from the power flow solutions-based initialization that is widely used in power system transient simulations. ...
... It is worth pointing out that Method 2 is subject to numerical oscillations especially when the virtual resistance is small. This is because the dynamic estimator in Method 2 also cannot estimate the damping factor well, which has been shown in [5]. On the other hand, by comparing Method 1 with EAUKF, it can be shown that the adaptive adjustment of Q k and R k is critical to enhancing its capability of handling both constant and time-varying inertia. ...
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This letter proposes a data-driven inertia estimator for inverter-based resources (IBRs) with grid-forming control. It is able to track both constant and time-varying inertia. By utilizing the Thevenin equivalent, the virtual frequency inside IBRs is first estimated with only its terminal voltage and current phasor measurements. The virtual frequency and the measurements are then used together to derive the state-space swing equation model. Then, an enhanced adaptive Unscented Kalman filter (EAUKF) is developed to estimate IBR inertia. Numerical results on the modified IEEE 39-bus power system demonstrate that the proposed inertia estimator remarkably outperforms the existing state-of-art methods both in tracking speed and accuracy.
... The Inertia Response of the Synchronous Generators Most of the traditional thermal power and hydropower plants are synchronous generators, which have a fast inertia response and high reliability. The inertia time constant of synchronous generators usually falls within the range of 2-9 s, which depends on the size, rated power, and type of machines [25]. ...
... Most of the traditional thermal power and hydropower plants are synchronous generators, which have a fast inertia response and high reliability. The inertia time constant of synchronous generators usually falls within the range of 2-9 s, which depends on the size, rated power, and type of machines [25]. ...
Full-text available
Frequency stability is an important factor for the safety and stability of the power system operation. In a traditional power system, the operation stability is ensured by the inertia response, primary frequency modulation, and secondary frequency modulation. In recent years, in order to achieve the goal of carbon neutralization and carbon peaking, China has made great efforts in new energy development. With large-scale new energy connected to the power grid, the proportion of traditional conventional synchronous units has gradually declined. At the same time, a large number of power electronic devices have been used in the power grid, which led to the capability decline of the inertia response and primary frequency modulation. For example, the East China Power Grid has experienced a sharp frequency drop in such an environment. In order to solve the above problems, the operation principle and control mode of various new energy resources are analyzed in this paper. Moreover, the process and principle of power grid frequency response are studied and the evaluation index of frequency response capability is proposed. The research results can quantitatively evaluate the system inertia response and primary frequency modulation level and provides a judgment tool for dispatching operators and system planners.
Accurate estimation of non-synchronous inertia in renewable energy (RE)-integrated power systems is challenging through conventional approaches, as emulated inertia from inverter-based resources (IBRs) is fundamentally different from the inherent inertial response of synchronous generators. In this context, this paper presents a novel realistic approach for estimating the non-synchronous inertial response in a large-scale RE-integrated power system. The method incorporates an optimal number of frequency and voltage monitoring nodes while ensuring an accurate estimation of non-synchronous inertia. In the proposed method, the synchronous inertial response for a frequency disturbance is estimated first using the conventional approach, followed by an estimate of the loads' inertial contribution. The latter is calculated using center-of-inertia frequency and voltage measurement across the power system with a voltage-controlled zone approach. Finally, the non-synchronous inertial response is estimated by segregating the aggregate inertial response of synchronous generators and the loads from the overall estimated inertia of the system. The proposed approach can help the system operator determine inertial contributions from the loads and IBRs depending on loading conditions and the availability of virtual inertia from RE sources, estimate the minimum required synchronous inertia, and select the appropriate proportional gains and time constants of virtual inertial controllers. The proposed method is validated by implementing it on a modified IEEE 39-bus system and a real-life Gujarat State grid model (in India).
This paper proposes a fast topology identification method to avoid estimation errors caused by network topology changes. The algorithm applies a deep neural network to determine the switching state of the branches that are relevant for the execution of a dynamic state estimator. The proposed technique only requires data from the phasor measurement units (PMUs) that are used by the dynamic state estimator. The proposed methodology is demonstrated working in conjunction with a frequency divider-based synchronous machine rotor speed estimator. A centralized and a decentralized approach are proposed using a modified version of the New England test system and the Institute of Electrical and Electronics Engineers (IEEE) 118-bus test system, respectively. The numerical results in both test systems show that the method demonstrate the reliability and the low computational burden of the proposed algorithm. The method achieves a satisfactory speed, the decentralized approach simplifies the training process and the algorithm proves to be robust in the face of wrong input data.
Abstract With the high‐proportion integration of renewable energy and power electronic equipment, the inertia supporting ability of new power system continues to decline, which seriously threatens the frequency stability of power grids. In order to clarify the operation boundary, and realise the rapid analysis and prediction of the minimum inertia demand of new power systems, this study proposes a minimum inertia demand estimation method based on deep neural network (DNN). Firstly, this study establishes the system frequency response model of new power systems containing diverse inertia resources including renewable energy, induction machine and so on. Considering the constraints of rate of change of frequency and maximum frequency deviation, the minimum inertia demand estimation model is established to ensure the system frequency stability. DNN is introduced to effectively map non‐linear relations in complex situations, which can quickly estimate and predict the minimum inertia of new power systems. Adam algorithm is utilised to optimise the input weight matrix and hidden layer feature vector of the network to improve accuracy. Finally, the simulations and analysis are conducted in IEEE‐39 system to verify the accuracy and generalisation ability of the proposed method in this paper.
Full-text available
The second part of this two-part paper discusses how to determine whether a device connected to the grid is providing inertial response and/or frequency control. The proposed technique is based on the index proposed in Part I of this paper. This part first discusses the dynamic behavior in terms of the rate of change of controlled power of a variety of non-synchronous devices that do and do not regulate the frequency. These include passive loads, energy storage systems and thermostatically controlled loads. Then a case study based on a real-world dynamic model of the all-island Irish transmission system discusses an application, based on a statistical analysis, of the proposed technique to wind power plants with and without frequency control. The properties and the robustness with respect to noise and other measurement issues of the proposed technique are also thoroughly discussed.
Full-text available
The first part of this two-part paper proposes a technique that consists in the measurement, through phasor measurement units, of bus frequency variations to estimate the rate of change of regulated power, and in the definition of a local index that is able to discriminate between devices that modify the frequency at the connection bus and devices that do not. A taxonomy of devices based on their ability to modify locally the frequency is proposed. A byproduct of such an index is to estimate the inertia or equivalent inertia of the monitored device. The proposed index is shown to be a relevant consequence of the concept of frequency divider formula recently published by the authors on the IEEE Transactions on Power Systems. The properties of the proposed index is illustrated through examples based on the synchronous machine and its controllers.
Full-text available
Nowadays, power system inertia is changing as a consequence of replacing conventional units by renewable energy sources, mainly wind and photovoltaic power plants. This fact affects significantly the grid frequency response under power imbalances. As a result, new frequency control strategies for renewable plants are being developed to emulate the behaviour of conventional power plants under such contingencies. These approaches are usually called 'virtual inertia emulation techniques'. In this study, an analysis of power system inertia estimation from frequency excursions is carried out by considering different inertia estimation methodologies, discussing the applicability and coherence of these methodologies under the new supply-side circumstances. The modelled power system involves conventional units and wind power plants including wind frequency control strategies in line with current mix generation scenarios. Results show that all methodologies considered provide an accurate result to estimate the equivalent inertia based on rotational generation units directly connected to the grid. However, significant discrepancies are found when frequency control strategies are included in wind power plants decoupled from the grid. In this way, authors consider that it is necessary to define alternative inertia estimation methodologies by including virtual inertia emulation. Extensive discussion and results are also provided in this study.
A virtual synchronous generator (VSG) control has been proposed as a means to control a voltage source converter interfaced generation and storage to retain the dynamics of a conventional synchronous generator. The storage is used to provide the inertia power and droop power in the VSG control to improve the frequency stability. Since the parameters in the VSG control can be varied, it is necessary for it to be tuned to be adaptive, in order to achieve an optimal response to grid frequency changes. However, the storage cannot provide infinite power and the converter has a strict power limitation which must be observed. The adaptive VSG control should consider these limitations, which have not been considered previously. This paper proposes an adaptive VSG control aimed at obtaining the optimal grid supporting services during frequency transients, accounting for converter and storage capacity limitations. The proposed control has been validated via hardware-in-The-loop testing. It is then implemented in storage co-located with wind farms in a modified IEEE 39-bus system. The results show that the proposed control stabilizes the system faster and has better cooperation with other VSGs, considering storage and converter limits.
System inertia plays a vital role in controlling the angular stability of the system during a disturbance. Due to increased penetration of power electronic interfaced sources, such as Solar Photovoltaic (SPV) source, the overall system inertia reduces and varies depending on their operating conditions. In this paper, an approach for online inertia estimation in the power system network with SPV sources is proposed, using the synchronized measurements from Phasor Measurement Units (PMUs). An equivalent swing equation is used to emulate the network dynamics. A relationship between the inertia constant and the roots of this equation is determined. In order to numerically obtain the roots, the Estimation of Signal Parameter via Rotational Invariance Techniques (ESPRIT) method is first used to find the modes present in the frequency signal. A new formulation is proposed to extract an equivalent mode from all the obtained modes. Also, to avoid phase step error, Rate Of Change Of Frequency (ROCOF) is estimated from the equivalent mode of the frequency signal. Results obtained for the 39 bus New England system for various test cases, using Real-Time Digital Simulator (RTDS), prove the efficacy and superiority of the proposed approach over the existing approaches in the literature.
The paper presents a systematic method to build dynamic stochastic models from wind speed measurement data. The resulting models fit any probability distribution and any autocorrelation that can be approximated through a weighted sum of decaying exponential and/or damped sinusoidal functions. The proposed method is tested by means of real-world wind speed measurement data with sampling rates ranging from seconds to hours. The statistical properties of the wind speed time series and the synthetic stochastic processes generated with the Stochastic Differential Equation (SDE)-based models are compared. Results indicate that the proposed method is simple to implement, robust and can accurately capture simultaneously the autocorrelation and probability distribution of wind speed measurement data.
The increasing penetration of power-electronic-interfaced devices is expected to have a significant effect on the overall system inertia and a crucial impact on the system dynamics. In the future, the reduction of inertia will have drastic consequences on protection and real-time control and will play a crucial role in the system operation. Therefore, in a highly deregulated and uncertain environment, it is necessary for Transmission System Operators to be able to monitor the system inertia in real time. We address this problem by developing and validating an online inertia estimation algorithm. The estimator is derived using the recently proposed dynamic regressor extension and mixing procedure. The performance of the estimator is demonstrated via several test cases using the 1013-machine ENTSO-E dynamic model.