Content uploaded by Johanna Vorwerk
Author content
All content in this area was uploaded by Johanna Vorwerk on Apr 11, 2023
Content may be subject to copyright.
Analytical Discussion on Dynamics of Inverter-
Based Resources under Small-Signal Conditions
Johanna Vorwerk∗§, Mehdi Ghazavi Dozein‡, Pierluigi Mancarella‡, and Gabriela Hug∗
∗Power Systems Laboratory, ETH Z¨
urich, Z¨
urich, Switzerland
‡School of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, Australia
§vorwerkj@ethz.ch
Abstract—Grid integration of inverter-based resources (IBRs)
and the consequent phase-out of synchronous machines cause a
reduction in system strength and inertia, which may pose system
instability challenges, including small-signal and inverter-driven
instability. There is an ongoing discussion on the contributing
factors to IBR stability, synchronism, and response specifications
(e.g., response time, damping) under various grid conditions
and characteristics. This paper presents an analytical discussion
highlighting how grid characteristics such as short-circuit ratio,
R/X ratio, and system loading conditions may affect IBR stability
and synchronism under small-signal conditions. Furthermore,
the contributing factors to IBR response characteristics imposed
by the external grid are identified. The analytical discussion
highlights the need to consider such factors when determining
grid-code requirements for IBR response specifications and
stability conditions.
Index Terms—- Power system dynamics, IBR, Small-signal
analysis, Weak grid, IBR stability.
I. INTRODUCTION
Power systems worldwide undergo a substantial transition
as the penetration of inverter-based resources (IBRs), includ-
ing generation and loads, is expected to rise tremendously
in the coming years. The reduction of inertia associated
with replacing synchronous machines increases the need for
frequency stability support and fast frequency response pro-
vision. In addition, the phase-out of synchronous machines
reduces the short-circuit current, which weakens the system
and can result in system strength challenges, including IBR
instability and disconnection [1, 2]. Hence, it is essential to
understand how to ensure IBR stability and synchronism under
various grid characteristics and operating conditions.
There is an ongoing discussion aiming to define response
specifications for IBRs, also called IBR grid-code require-
ments, to guarantee their inverter-driven stability while en-
hancing overall power system stability during abnormal con-
ditions such as system contingencies and faults [3, 4]. In the
literature, IBR stability and its response specifications are
mainly discussed on a device level, while potential issues
emerging from the external system are yet to be understood.
For example, [5, 6] identify how converter control schemes
enhance IBR stability but neglect system-level impacts. On
the other hand, grid codes define IBR response functionalities
such as power factor control, ride-through response, momen-
tary cessation, or angle jump requirements while ignoring
how the IBR performance may be influenced by external grid
operating conditions [7–9].
Literature suggests that IBR control schemes and their
stable operation may be affected by the grid characteristics at
the connection point, which substantially vary with the voltage
level. In particular, the short-circuit ratio (SCR) significantly
affects the stable space for droop and virtual inertia emulation
control parameters, as highlighted through eigenvalue analysis
in [10]. Furthermore, [11] presents fundamental transfer func-
tions to analyze the limitations of dynamic support from IBRs
in weak systems. Although the authors acknowledged the
dependency on grid impedance and pre-disturbance operating
conditions, they do not explore the effect of external grid
characteristics and operating points on IBR stability.
Recent events in the Australian power system highlight that
IBR stability and response compliance are severe concerns
in the real-life renewable-rich system. In 2020, voltage os-
cillations caused challenges in maintaining stable IBR oper-
ation in the West Murray TN [12]. Significant oscillations
were observed at the active/reactive power outputs of IBRs,
which led to significant renewable curtailment enforced by the
Australian system operator to ensure system-level stability. In
addition, the events reported in [13] suggest that the external
system operating conditions may influence and threaten stable
IBR operation and response. Even so, an analytical and
systematic discussion of such dynamic challenges and their
root cause is lacking.
Extending the IBR mathematical framework derived in [11],
this work explores how diverse grid conditions typical for
different voltage levels might lead to undesirable or unstable
operating conditions of IBRs. As such, the contributions of
the presented work are threefold:
•First, we present an analytical discussion highlighting
the contributing factors to IBR stability and synchronism
under small-signal conditions, particularly those imposed
by the external power system characteristics, including
R/X ratio, grid strength, and voltage angle.
•Furthermore, we perform a small-signal analysis of the
derived transfer functions to investigate how the system-
level characteristics and operating conditions, such as
R/X ratio, grid strength, and voltage angle, impact IBR
response characteristics.
•Finally, we present implications of the results for real-
world systems that highlight the need to further analyze
the dependencies to ensure IBR stability and desirable
dynamic response under small-signal conditions.
II. GE NE RAL SMAL L-SI GNAL MO DE L FOR IBRS
This section derives transfer functions that describe the
current dynamics of an arbitrary grid-connected IBR. Then,
the validity of the derived transfer functions and required as-
sumptions are discussed. Finally, potential operating regimes
for the given system are assessed to define parameterizations
for the subsequent small-signal analysis.
The following notation is adopted for the presented work:
Phasor approximation is assumed to hold for the provided
system and dq-decomposition of currents and voltages applies
and is indicated through corresponding subscripts. All system
quantities and variables are formulated in per unit, represented
by small letters (e.g. v, r, ℓ), except for the voltage angle.
A. Derivation of Transfer Functions
Various IBRs provide system services at different voltage
levels in modern power systems. In general, both inverter-
interfaced load and generation may adapt their active and/or
reactive power setpoints following system disturbances. Fig. 1
shows a simplified but general grid connection of an IBR
and defines the convention applied in this work: An IBR is
connected to the power grid in parallel to a load. The load
accounts for local consumption while il= 0 corresponds to a
no-local-loading condition.
The transmission network (TN) is considered with an
equivalent impedance (rTN,ℓTN) in series with a voltage
source. The entire corridor, including lines and transformers
in between the IBR terminal and the interface to the TN, is
represented by the equivalent impedance (rDN,ℓDN). The TN
internal voltage source is modeled with its magnitude edand
provides the reference angle. The voltage at the IBR terminal
is described by its magnitude Vand phase angle θ.
Applying the sign convention in Fig. 1 and phasor approx-
imation, i.e. i(t) = idqejωst, the dynamic equations for the
current are formulated as:
ℓ
ωb
did
dt =ed−vcos(−θ)−rid+ωsℓiq,
ℓ
ωb
diq
dt =−vsin(−θ)−riq−ωsℓid,
(1)
where r=rDN +rTN and ℓ=ℓDN +ℓTN are the resistance
and inductance seen from the IBR point of common coupling
(PCC) in per unit, the base frequency is ωb= 2πf , and the
synchronous frequency ωsis expressed in per unit.
Small-signal disturbances cause deviations in the voltage
magnitude and angle at the PCC that are followed by an
initially uncontrolled and, depending on the control time con-
stants, delayed control reaction of the IBR. While traditionally
voltage and angle were decoupled and studied individually,
weak systems exhibit disruptions in both quantities simulta-
neously [11]. Mathematically, such a small-signal disturbance
is described by the pre-contingency quantity identified through
subscript 0 and the deviation from the pre-fault condition in-
dicated through ∆-notation. Replacing the per-unit inductance
with its per-unit reactance x=ℓ, using the grid impedance
z=√x2+r2and expressing the resistance through the
R/X ratio k=r/x, the small-signal model is expressed in
terms of the grid impedance and R/X ratio at the IBR’s PCC:
∆id=G1∆θ+G2∆v, ∆iq=G3∆θ+G4∆v,
G1=v0ωb√1 + k2
z
(s+kωb) sin θ0+ωsωbcos θ0
(s+kωb)2+ (ωsωb)2,
G2=−ωb√1 + k2
z
(s+kωb) cos θ0−ωsωbsin θ0
(s+kωb)2+ (ωsωb)2,
G3=v0ωb√1 + k2
z
(s+kωb) cos θ0−ωsωbsin θ0
(s+kωb)2+ (ωsωb)2,
G4=ωb√1 + k2
z
(s+kωb) sin θ0+ωsωbcos θ0
(s+kωb)2+ (ωsωb)2.
(2)
The transfer functions are similar. Namely, it holds G3=
−v0G2and G1=v0G4. This proportionality suggests that
amplitudes differ for each pair while the dynamic properties
are equivalent. In other words, the d-axis current response to a
change in angle is equivalent to a change of the q-axis current
following a disruption in the voltage magnitude. As such, the
TN equivalent
V∠−θ
PCC
rDN ℓDN
idq rTN ℓTN
ed
iIBR
IBR
il
Fig. 1. General connection of an IBR to the grid. Note the selected convention
of currents and voltage arrows.
dynamic coupling between the following quantities is similar:
(i) ∆id-∆θand ∆iq-∆V, (ii) ∆id-∆Vand ∆iq-∆θ.
B. Validity and Assumptions
Phasor approximation is required for the formulation pro-
vided in (1), assuming that currents and voltages are narrow-
band signals around the nominal frequency. This assumption
might be invalid for power systems with high penetration
of IBRs in weak grid conditions. Especially during system
transients, physical states in inverter-dominated power systems
are more likely to be broad-band signals, as shown for real
events in Australia [14,15].
The formulation in (1) captures the reaction of the IBR
currents to a change of the voltage magnitude and angle
without making any assumptions on the inverter control. Thus,
they apply for any IBR before the control affects the currents.
Ancillary services are subsequent as they exhibit slower
reactions. As such, the derived transfer functions capture the
effect of grid characteristics on the initial response of any
IBR, independent of the implemented control scheme.
Besides employing (2) to assess the dynamic response of
IBRs, we can study them in reversed direction and assess
how small changes in the IBR current affect the local grid.
In this case, the control option of the IBR matters. If the
IBR performs grid-feeding control, it synchronizes with the
PCC’s voltage angle, and the loading condition determines
the angle. However, if a grid-forming concept is adopted, the
IBR and the loading condition affect the PCC angle. This grid-
forming case is not captured in the transfer functions. Thus
when assessing (2) in reversed fashion, the results are valid
for grid-feeding control only.
C. Operating Regimes
Due to different load and generation patterns, distinct
operating conditions for IBR operation exist. If the IBR in
Fig. 1 consumes power (iIBR >0), e.g., as a load or a
battery, the current idq is strictly positive. Power flows from
the upstream network towards the IBR terminal, the voltage
phasor at the IBR terminal lags compared to the Thevenin
equivalent, and the angle θis positive. Supposing the IBR is
a generator, e.g., a PV system or a generating battery, two
possibilities exist: If the local load exceeds generation at the
IBR terminal, i.e., |iIBR|<|il|, power still flows from the
upstream network into the IBR terminal. Thus current idq
and angle θare still positive. However, if the load is smaller
than the IBR generation, i.e., |iIBR|>|il|, the power flow is
inverted, resulting in an adverse current and a negative angle.
As a result, positive and negative currents idq and angles θ
are realistic and depend on the operating condition.
While grid codes define acceptable ranges for voltage
magnitudes and typically require V0∈[0.9,1.1]p.u., the initial
angle depends on the loading condition of the grid. To ensure
angular stability, voltage angles are required to stay within
a secure angle region, within ±90◦∓ε, where εrepresents
an additional stability margin [16]. While small angle values,
around 0◦reference value, indicate low loading conditions,
larger angle values indicate high loading patterns.
As suggested by (2), the dynamics of the current being
exchanged between IBR and the external grid follow second-
order transfer functions right after the contingency and solely
depend on some distinct grid characteristics. Besides the pre-
fault voltage angle and magnitude, these include the R/X ratio
and the grid impedance at the PCC. The latter two generally
differ between voltage levels but can also substantially vary
within one voltage level. TNs exhibit low R/X ratios, and
system losses are usually insignificant. On the other hand,
distribution networks (DNs) typically exhibit non-negligible
losses that manifest in higher R/X ratios, above 0.5 or closer
to 1 [16, 17].
The grid impedance correlates with the short-circuit power
at the PCC and traditionally changes with the size of the
system and grid strength [17]. Weak systems exhibit low fault
levels, which manifests itself in low short-circuit ratios (SCR),
e.g., SCR values less than 3 p.u., and high grid equivalent
impedance, while the opposite holds for strong grids [8].
III. DYNAMIC PROPE RTI ES O F IBR RESPONSES UNDER
SMA LL -S IGNA L CONDITIONS
This section contains a detailed small-signal analysis of the
second-order IBR response developed in (2) considering the
operating regimes discussed in Section II-C. First, it compares
the derived model against a standardized 2nd order system
formulation that permits general conclusions regarding the
speed and damping of IBR responses under diverse grid
conditions. Then, it provides exemplary bode plots and step
responses for distinct grid characteristics.
A. Comparison to 2nd Order Standard Form
General conclusions arise when comparing systems to a
standardized 2nd order formulation commonly studied in con-
trol theory. Analytically formulating its coefficients permits
quantifying the system damping, response time, and overshoot
depending on grid quantities.
1) Standard 2nd Order System with one Zero: Any 2nd
order system with a zero can be expressed in terms of the
following standard form [18]:
H(s) = Ks/(αζωn)+1
(s/ωn)2+ (2ζ/ωn)s+ 1 ,(3)
where Kis the steady-state gain, ζthe damping factor, ωn
the natural frequency, and αpositions the zero.
These coefficients provide an estimate for time-domain
specification like the rise time tr, settling time ts, over-
shoot Mpand peak time tpof a system. A general introduction
to such systems and the effect of the coefficients on the
dynamic performance is provided in [18].
2) Analysis of the Coefficients: By rearranging (2), the
coefficients of the standard 2nd order system and the time-
domain specifications can be extracted. Table I displays the
analytic results. While resonance frequency, damping factor,
and time-domain characteristics solely depend on the R/X ra-
tio, the position of the zero, hence α, additionally depends on
the voltage angle. Note that the peak time is a constant.
Fig. 2 depicts the time-domain characteristics for different
grid characteristic parameters, while Fig. 3 displays the coeffi-
cients. The expected rise times are below 6 ms, and the settling
times do not exceed 100 ms for most R/X ratios, ensuring that
the analysis is suitable for any controlled or uncontrolled IBR
in grids with R/X¡0.15 as discussed in Section II-B.
4.0
4.5
5.0
5.5
trin ms
0
100
200
tsin ms
0.0 0.2 0.4 0.6 0.8 1.0
k=r/x
0.0
0.5
1.0
Mp
Fig. 2. Time domain characteristics of the transfer functions. Note that these
are only valid when |α| ≤ 3and may substantially vary otherwise.
Some meaningful differences are observed for different
R/X ratios. The natural frequency and damping factor rise
as kincreases, causing lower overshoots, and shorter rise,
and settling times. Consequently, oscillatory time responses
with high overshoots of up to two times the new steady-state
(i.e. Mp= 1) are expected for TNs with small R/X. Note that
increased damping for higher R/X, hence DNs, correlates with
diminishing overshoots and shorter settling times. As such,
IBRs placed in DNs exhibit higher damping, and the time
domain response settles quicker than for IBRs in TNs.
The location of the zero, hence the value of α, may affect
the dynamic properties. While its placement does not affect
the system’s stability, an RHP zero causes non-minimum-
phase behavior, an initially inverted response. As such, the
zero significantly affects the dynamic properties of the system
if it is close to the poles, i.e., α≈1. In that case, the rise
time of the system is significantly reduced, indicating a faster
responding system. In addition, the overshoot is enhanced. On
the other hand, a large αindicates that the zero is far from
the poles of the system, and the effect of the zero diminishes.
When the zero moves to the origin of the complex plane, it
causes a direct, almost derivative behavior between the input
and the output with immediate responses [18].
Fig. 3 suggests under which conditions the zero affects the
time-domain characteristics. In all cases, operating conditions
exist that cause non-minimum-phase behavior, as suggested
TABLE I
COE FFICI EN TS A ND TI ME -DOMAIN PERFORMANCE CHARACTERISTICS.
Parameter Valid for Expression
Resonance ωnG1to G4ωbp(k2+ω2
s)
Damping ζ G1to G4k/p(k2+ω2
s)
αG1, G41 + ωs/(ktan θ0)
G2, G31−ωs/k tan θ0
Rise time tr
G1to G4
1.8/(ωbpk2+ω2
s)
Peak time tpπ/ωb
Settling time ts4.6/(kωb)
Overshoot Mpexp(−πk)
1.0
1.2
1.4
ωnin p.u.
0.0 0.2 0.4 0.6 0.8 1.0
k=r/x
0.0
0.2
0.4
0.6
ζ
−20
0
20
αfor G1and G4
k: 0.05 0.1 0.5 1
−40 −20 0 20 40 60 80
θ0in ◦
−10
0
10
αfor G2and G3
k: 0.05 0.1 0.5 1
Fig. 3. Coefficients of the 2nd order standard transfer function for different
R/X ratios and initial angles. All are independent of the initial voltage and
grid impedance as indicated by Tab. I. The shaded areas mark α∈[−3,3].
by α < 0. For G1and G4, the zero is only expected to affect
the time-domain characteristics for high-loading scenarios,
i.e., significant positive and negative initial angles and high
R/X ratios. For small initial angles, a singularity occurs, i.e.,
αapproaches infinity for 0◦, indicating that the conclusions
for low loading conditions and the time-domain response
characteristics are extremely sensitive to the parameters of
the system.
For G2and G3,αheavily depends on the R/X ratio. For
small R/X, the zero is only close to the origin for low loading
conditions and quickly moves away from the origin as the
angle increases or decreases. Thus, for TNs, an instant reaction
between idand V, or iqand θ, is only expected for low
loading conditions, and the analytic description of the time-
domain characteristics is valid. For IBRs in DNs, the situation
differs. αstays close to unity for almost all loading conditions.
Thus, the overshoot under these conditions is expected to
increase compared to the values provided in Fig. 2.
B. Bode Plot Analysis
Fig. 4 provides the bode diagrams for all transfer functions
and different R/X values. G1and G4are identical, thus shown
in the same plot. Despite the same gain for G2and G3, their
phases are shifted by 180◦due to the inversion in (2).
All amplitude responses exhibit a resonance close to the
nominal frequency. On the one hand, this ensures controllabil-
ity. If the system operated above its resonance, the gain would
be reduced, preventing currents from following a voltage angle
or magnitude change. Large control gains would be required
to ensure suitable reactions in such a scenario. On the other
hand, the resonance is enhanced for low R/X ratios, making
overshoots and oscillatory behaviors more likely for TNs. At
the same time, due to the high inherent gains in TNs, smaller
−40
−20
0
20
Gain in dB
G1=∆id
∆θ,G4=∆iq
∆VG2=∆id
∆V
k=r/x: 0.05 0.1 0.5 0.8 1
G3=∆iq
∆θ
10−1100101
ωin p.u.
−200
0
200
Phase in ◦
10−1100101
ωin p.u.
10−1100101
ωin p.u.
Fig. 4. Bode plots of the transfer functions for different R/X ratios for
v0= 1 p.u. G1and G2on the left, G3in the middle and G4on the right.
−0.1
0.0
0.1
G1=∆id
∆θ
k=r/x: 0.05 0.1 0.5 0.8 1
−0.1
0.0
0.1
G2=∆id
∆V
−0.1
0.0
0.1
G3=∆iq
∆θ
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
t in s
−0.1
0.0
0.1
G4=∆iq
∆V
Fig. 5. Step response of the transfer functions G1to G4for a 0.1 p.u.
input step for different R/X ratios and the constant parameters v0= 1 p.u.,
θ0= 10◦and z= 1 p.u.
control reactions are required to control the output variable,
while higher gains might be required for DNs.
C. Exemplary Step Response
Fig. 5 displays dynamic responses to a 0.1 p.u.step in
the input and varying R/X ratios while keeping all other
parameters constant at v0= 1 p.u., θ0= 10◦and z= 1 p.u..
Some of the previous findings are evident: G1and G4exhibit
the same response, because v0= 1 p.u.. Furthermore, G2is
the inversion of G3, and the peak times are approximately
equal as suggested by Tab. I.
Fig. 5 clearly illustrates the dependence of the damping on
the R/X ratio. TNs with low R/X ratio are prone to oscillations
and higher overshoots of up to twice the post-fault steady-
state magnitude. Similar findings hold for the settling time.
The smaller the R/X ratio, the longer oscillations exist. On the
contrary, DNs, with their higher k, exhibit smaller overshoots,
and shorter settling times and thus provide more damping.
In addition, Fig. 5 confirms the standard P θ, and QV -
decoupling applied in traditional TN control concepts. Consid-
ering a typical R/X ratio of k= 0.1, the steady-state deviation
−0.1
0.0
0.1
G1=∆id
∆θ
θ0: -20◦-10◦10◦20◦50◦75◦
−0.1
0.0
0.1
G2=∆id
∆V
−0.1
0.0
0.1
G3=∆iq
∆θ
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
t in s
−0.1
0.0
0.1
G4=∆iq
∆V
Fig. 6. Step response of the transfer functions G1to G4for a 0.1 p.u.
input step for different initial angles θand constant parameters v0= 1 p.u.,
k= 0.5and z= 1 p.u.
−40 −20 0 20 40
θ0in ◦
−4
−2
0
2
4
∆Vss
∆id
TN, k=0.013
SCR : 1 3 5 10 infinite bus
−40 −20 0 20 40
θ0in ◦
DN, k=0.5
−40 −20 0 20 40
θ0in ◦
−1
0
1
2
3
4
∆Vss
∆iq
TN, k=0.013
SCR : 1 3 5 10 infinite bus
−40 −20 0 20 40
θ0in ◦
DN, k=0.5
Fig. 7. Steady-state voltage deviation depending on the initial angle for
different short circuit ratios (SCR= 1/x). Dependency for ∆idon the top
and ∆iqon the bottom.
following a change in voltage magnitude is almost zero for
the d-axis current, while it is maximal for the q-axis current.
This indicates adequate controllability of voltage magnitudes
through q-axis current, which is typically aligned with reactive
power control. Similar findings hold for a change in the angle,
ergo frequency, the d-axis current, and active power control.
The decoupling is no longer valid for R/X ratios in the DN
range, i.e. closer to unity. A disruption in voltage magnitude
is followed by a significant steady-state deviation in both
currents. The same holds for an angle disruption.
Fig. 6 shows step responses for different pre-fault loading
conditions expressed in terms of varying θ0, for k= 0.5.
As predicted from the previous analysis, θmainly affects
the zero placement and thus does not significantly affect
the damping and time constants - however, the steady-state
reaction changes. A general trend appears for G2and G3: the
more the angle deviates from zero, i.e., the higher loaded the
system, the larger the steady-state deviation. In other words,
the higher the exchange between IBR and the external system,
the more inaccurate the assumption of decoupling between P
and Vas well as Qand θ.
IV. EFFE CTS OF IBR CO NT RO L
System operators typically require IBRs to contribute to
system frequency and voltage control. Hence, they adjust
their power consumption or production following a change
in local voltage magnitude or frequency. Especially in weaker
systems, the voltage deviations caused by IBRs that provide
frequency response are of concern [11]. Considering a grid-
following IBR, the derived transfer functions are reformulated
to compute the steady-state voltage magnitude change follow-
ing an adjustment in current. Extending the formulation in
[11], this section first formally derives the expected steady-
state voltage magnitude deviation caused by IBR current
control before analyzing it systematically under different grid
operating conditions.
A. Derivation
The voltage change following an IBR current adjustment is
obtained by inverting G2and G4. Computing s→ ∞ provides
the expected deviation in steady-state voltage magnitude ∆vss
for a change in the two currents:
∆vss = lim
s→0
1
G2
∆id=−1
η
k2+ω2
s
kcos θ0−ωssin θ0
∆id,
∆vss = lim
s→0
1
G4
∆iq=1
η
k2+ω2
s
ksin θ0+ωscos θ0
∆iq,
(4)
where η=x−1is the short-circuit ratio (SCR) of the grid.
B. Critical Initial Operating Conditions
For both currents, a condition exists so that the denominator
in (4) vanishes. Consequently, the change in voltage at this
point is theoretically infinite. In real systems, it indicates that
the local voltage is extremely sensitive to changes in current.
Since the point only depends on the initial angle and R/X, we
define the critical angle θcrit
0as the angle at which the gain
becomes infinite for a specific k:
θcrit
0,id= arctan (k/ωs), θcrit
0,iq= arctan (−ωs/k).(5)
C. Effect of Grid Characteristics
Generally, (4) indicates that the voltage deviation solely
depends on the SCR, R/X ratio, and the initial loading
condition. Fig. 7 displays the expected change in voltage
magnitude depending on grid characteristics.
In all cases, weaker grids with low SCR exhibit higher
voltage deviations following a current control action. At the
same time, this conclusion suggests that smaller control gains
are required for local voltage control. However, when minor
current adaptions significantly impact local voltages, they limit
the amount of current that can be provided for ancillary
services while still keeping the voltage magnitude within
bounds. In some conditions, the gain exceeds 1, indicating
0.0 0.2 0.4 0.6 0.8 1.0
k=r/x
−50
0
50
θcrit
0in ◦
∆id∆iq
Fig. 8. Critical angle as a function of R/X ratio for control with ∆idand
∆iq.
that the change in voltage magnitude is higher than the initial
change in current.
For IBR control with id, Fig. 7 illustrates the consequences
of operating at the critical angle. For k= 0.013, the critical
angle is close to 0, as indicated by Fig. 8. During low
loading conditions, small changes in d-axis current cause
large voltage magnitude deviations and endanger IBR stability.
Grid-following devices need a voltage reference signal to syn-
chronize, but difficulties arise when large voltage oscillations
occur. Consequently, devices face issues synchronizing and
may disconnect during such conditions.
The position of the singularity and critical angle shifts with
the R/X ratio as suggested by Fig. 7 and Fig. 8. For higher
k, the critical operating condition for IBR control through id
shifts to higher initial loading.
V. IMPLICATION FOR REAL -WORL D SYS TE MS
The analytical discussion suggests that the R/X, external
grid strength, and voltage angle at the PCC significantly affect
the IBR performance and that some loading conditions might
cause small-signal instability at the IBR terminal.
Distinct differences in IBR operation in TNs and DNs
occur. Higher damping is expected for IBRs in DN systems,
while those in TN systems are more prone to oscillations due
to lower R/X, higher resonance gain, and higher damping fac-
tor. Our analytical discussions highlight that system damping
is a key contributing factor to the secure operation of IBR-
rich TNs. This also means that system damping should be
considered in future system planning for security purposes.
In addition, our analysis implies that small-signal oscillations
may not be a severe issue for IBRs connected to DNs.
Furthermore, in DNs, a coupling between idand V, and iqand
θis expected, which is different from operation in strong TN
systems. Under these conditions, decoupling frequency and
active power studies from voltage and reactive power studies
is no longer valid. Since decoupling no longer holds, there
may be a need for detailed dynamic modeling of IBR-rich
DNs.
Independent of the voltage level, low loading conditions
appear critical. The traditionally studied coupling between id
and θand iqand Vexhibits high sensitivity during these
conditions, no matter the R/X ratio. As a result, IBRs might
face problems in staying synchronized. From an industry
perspective, this issue is now being discussed and referred
to as minimum operational demand security challenges [19].
In addition, the results indicate that for a specific grid
impedance and R/X ratio, critical operating conditions in
terms of a critical voltage angle exist. These conditions should
be tested when IBRs are integrated into ancillary service
markets and are supposed to participate in frequency/voltage
control. The analytic results indicate that voltages are ex-
tremely sensitive to changes in IBR current for some loading
conditions. Under these conditions, the IBRs might destabilize
themselves and harm other units in the area.
VI. CONCLUSION
This paper presents analytical discussions on IBR small-
signal stability in varying grid conditions. The derived transfer
functions and subsequent small-signal analysis hold for any
inverter-based technology.
When IBRs are used for grid support in a grid-following
control fashion, the grid characteristics at the PCC severely in-
fluence the IBR’s small-signal stability and synchronism. The
analytic derivation indicates that critical operating conditions
exist given a particular R/X and grid strength at the PCC. The
mentioned system-level aspects should be further investigated
and considered when defining IBR grid-code requirements in
future system operations. We have also highlighted that system
damping is a key contributing factor to the secure operation
of IBR-rich TNs. At the same time, it may be a manageable
issue in IBR-rich DNs. It is then advisable to include system
damping when planning the security of future TNs.
REFERENCES
[1] M. G. Dozein, System dynamics of low-carbon grids: Fundamentals,
challenges, and mitigation solutions. PhD thesis, The University of
Melbourne, Australia, 2021.
[2] N. Hatziargyriou, J. Milanovic, C. Rahmann, V. Ajjarapu, C. Canizares,
I. Erlich, D. Hill, I. Hiskens, I. Kamwa, B. Pal, P. Pourbeik, J. Sanchez-
Gasca, A. Stankovic, T. Van Cutsem, V. Vittal, and C. Vournas,
“Definition and classification of power system stability – revisited &
extended,” IEEE Trans. Power Sys., vol. 36, no. 4, pp. 3271–3281, 2021.
[3] “Ieee standard for interconnection and interoperability of inverter-based
resources (ibrs) interconnecting with associated transmission electric
power systems,” IEEE Std 2800-2022, 2022.
[4] National Grid ESO, “NOA Stability Pathfinder Phase 3, technical
performance requirements,” Nov 2020.
[5] M. Chen, D. Zhou, A. Tayyebi, E. Prieto-Araujo, F. D¨
orfler, and
F. Blaabjerg, “Generalized multivariable grid-forming control design for
power converters,” IEEE Trans. Smart Grid, vol. 13, no. 4, pp. 2873–
2885, 2022.
[6] D. B. Rathnayake and B. Bahrani, “Multivariable control design for
grid-forming inverters with decoupled active and reactive power loops,”
IEEE Trans. Power Electron., vol. 38, no. 2, pp. 1635–1649, 2023.
[7] “Ieee standard for interconnection and interoperability of distributed
energy resources with associated electric power systems interfaces,”
IEEE Std 1547-2018 (Revision of IEEE Std 1547-2003), 2018.
[8] Australian Energy Market Commission (AEMC), “Investigation into
system strength frameworks in the NEM,” 2020.
[9] NERC, “PSB-connected inverter-based resource performance,” 2018.
[10] U. Markovic, J. Vorwerk, P. Aristidou, and G. Hug, “Stability analysis
of converter control modes in low-inertia power systems,” in 2018 IEEE
PES ISGT-Europe, 2018.
[11] M. Ghazavi Dozein, B. C. Pal, and P. Mancarella, “Dynamics of
inverter-based resources in weak distribution grids,” IEEE Trans. Power
Sys., vol. 37, no. 5, pp. 3682–3692, 2022.
[12] AEMO, “West Murray,” 2018. www.aemo.com.au/en/energy-
systems/electricity/national-electricity-market-nem/participate-in-the-
market/network-connections/west-murray.
[13] L. Fan, Z. Miao, S. Shah, Y. Cheng, J. Rose, S.-H. Huang, B. Pal,
X. Xie, N. Modi, S. Wang, and S. Zhu, “Real-world 20-hz ibr subsyn-
chronous oscillations: Signatures and mechanism analysis,” IEEE Trans.
Energy Convers., pp. 1–11, 2022.
[14] M. Paolone, T. Gaunt, X. Guillaud, M. Liserre, S. Meliopoulos,
A. Monti, T. Van Cutsem, V. Vittal, and C. Vournas, “Fundamentals
of power systems modelling in the presence of converter-interfaced
generation,” Electric Power Systems Research, vol. 189, 2020.
[15] AEMO, “Final report-Queensland and South Australia system separa-
tion on 25 august 2018,” 2019.
[16] P. Kundur, Power System Stability and Control. McGraw-Hill, 1994.
[17] A. J. Schwab, Elektroenergiesysteme. Berlin: Springer Vieweg, 2020.
[18] G. F. Franklin, Feedback control of dynamic systems. Always learning,
Boston: Pearson, seventh edition, global edition. ed., 2015.
[19] AEMO, “Smart Meter Backstop Mechanism Capability Trial, Phase 2
Evaluation Report,” 2022.
