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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 =−RωG−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+RZ˙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∗=γZ˙ωdtZ˙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