1
Grid-Forming Inverters: A Critical Asset for the
Power Grid
Robert Lasseter, Life Fellow, IEEE, Zhe Chen and Dinesh. Pattabiraman, Student Members IEEE
Abstract— Increasing inverter-based sources the system’s inertia is reduced and frequency stability becomes a concern. Understanding low-inertia systems and their stability properties is of
crucial importance. This work introduces fundamental ways to
integrate high levels of RE and DER in the power system while
creating a more flexible power system. Using RE and DER in the
distribution system has many advantages such as; reducing the
physical and electrical distance between generation and loads,
bringing sources closer to loads contributes to enhancement of
the voltage profile, reduction to distribution and transmission
bottlenecks, improved reliability, lower losses and enhances the
potential use of waste heat. A basic issue for high penetration
of DER is the technical complexity of controlling hundreds of
thousands to millions of inverters. This is addressed through
autonomous techniques using local measurements eliminating the
need for fast control systems. The key issues addressed in this
paper includes using inverter damping to stabilize frequency in
systems with low or no inertia, autonomous operation, methods for relieving inverter overload, energy reserves and their
implementation in PV systems. This work provides important
insight to the interactions between inverter bases sources and
the high-power system. The distinction between GFM inverter
and GFL inverter is profound. GFM inverters provides damping
to frequency swings in a mixed system while GFL inverter can
aggravate frequency problems with increased penetration. Rather
than acting as source of inertia, the GFM inverter acts as a
source of damping to the system. On the other hand, application
of inverters in the power system have two major issues. One is
the complexity of controlling hundreds of thousands to millions
of inverters. This is addressed through autonomous techniques
using local measurements. The other is the potential of high overcurrent in GFM inverters and techniques for explicitly protecting
against overloading. To exploit the innate damping of GFM
inverters energy reserves are critical.
Index Terms— Grid-following, Grid-forming, Inverter damping, Low-inertia power systems, Reserves, Renewables
I. I NTRODUCTION
Massive penetration of variable renewable energy (RE) in
addition to other distributed energy resources (DER), poses
major operational challenges to utility system operators. Power
system operation traditionally assumes synchronous generators
provide frequency stability via their stored kinetic energy.
With increasing inverter-based sources the system’s inertia is
reduced and frequency stability becomes a concern. For example, operators in Ireland [1], Texas [2] and South Australia
This paper was submitted for review 21 September 2019. This work is
supported in part by the U.S. Dept. of Energy Office of Energy Efficiency and
Renewable Energy and the Office of Electricity through contracts administered
by the Lawrence Berkeley National Laboratory. Additional support through
Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC).
The authors are with the Electrical and Computer Engineering Department,
University of Wisconsin, Madison, WI 53706 USA. Robert Lasseter (email: rlasseter@gmail.com). Zhe Chen (e-mail: zchen275@wisc.edu). Dinesh
Pattabiraman (e-mail: dinesh.pattabiraman@wisc.edu)
[3] are already facing obstacles regarding high penetration
of inverter-based sources during certain periods of the day.
Understanding low-inertia systems and their stability properties is of crucial importance. We need to find fundamental
ways to integrate high levels of RE and DER in the power
system while creating a more flexible power system.
Traditionally, inverter-based-sources such as photovoltaics
(PV) and Wind have been deemed to possess zero inertia; they
are typically operated at their rated power output and are not
expected to respond dynamically to frequency changes [4].
With increasing inverter penetration levels due to growth in
installations of variable renewable energy sources, the total
stored mechanical energy is reduced. This can result in larger
frequency swings as a larger fraction of kinetic energy storage
is decommissioned. Larger deviations can cause reliability
issues such as frequency-based tripping of loads and legacy
equipment in the system. Smaller islanded power systems
such as Australia, Hawaii, face imminent low inertia related
issues. This does not need to be the case. Inverters can
be controlled to increase frequency damping with increased
penetration. Fundamentally grid-forming inverter frequency
control could be very advantageous particularly for islanded
power systems with frequency issues. Compared to large
synchronous machines, inverter-based resources are able to
change their output much faster arresting system’s frequency
changes before any load shedding is triggered.
DER and RE bring a level of variability and nuance never
before seen by network operators. This tangible convergence
of DER interconnections to networks ill-suited to integrate
variable demand side behaviors represents ground zero for the
disruption of the global energy landscape caused by DER.
We need to rethink our distribution system including the
integration of high levels of DER, to provide a smarter and
more flexible distribution system. Our overarching objective is
to transform the installation of a very large number of inverter
based sources from a major potential liability to a critical asset
for both the power grid and utility customers.
Using RE and DER in the distribution system reduces
the physical and electrical distance between generation and
loads. The benefits include enhanced voltage profiles, reduced
distribution and transmission bottlenecks, improved reliability,
enhanced potential use of waste heat and lower losses. The
basic issue for high penetration of DER is the technical
complexity of controlling hundreds of thousands to millions of inverters. This complexity is greatly reduced using
autonomous techniques to eliminate the need for fast control
systems. The key issues addressed in this paper includes
inverter damping to stabilize frequency in systems with low
or no inertia, autonomous techniques for relieving inverter
Digital Object Identifier: 10.1109/JESTPE.2020.2959271
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Fig. 2.
Fig. 1.
Block Diagram of a typical Grid Forming Inverter
Block Diagram of a typical Grid Following Inverter
overload, use of energy reserves and implementation of PV
systems.
II. GFL AND GFM I NVERTERS
There are two basic control technologies for utility-based
inverters. They are grid-following (GFL) and grid-forming
(GFM). Grid-following inverters control the output of real and
reactive power by injecting a current at a given phase angle.
A phase locked loop (PLL) is used to track the grid phase
angle in real time. The grid-following inverter cannot directly
provide regulate system voltage and frequency. Voltage and
frequency reference is provided externally either by a gridforming inverter or the power system. Fundamentally if the
GFL inverter loses a voltage/frequency source it must shut
down.
Grid-forming inverters are intrinsically different from gridfollowing inverters. A grid-forming inverter is a controllable
voltage source behind a coupling reactance much like grid
tied synchronous generators. Voltage source inverters with
droop characteristic allows for direct control of voltage and
frequency. During contingencies the droop-controlled gridforming sources will increase or decrease their output power
instantaneously to balance loads and maintain local voltage
and frequency. There is no significant delay between the
change of output power and the change of output frequency
in droop-controlled grid-forming inverters. Therefore, GFM
sources respond much faster to any contingencies than the
response of the GFL sources. Providing primary frequency
control from inverter-based resources could be very advantageous particularly for “low-inertia” power systems. Compared
to large synchronous machines, inverter-based resources are
able to change their output much faster thus arresting system’s
frequency changes before any load shedding is triggered.
An excellent example is the O’ahu, power system in Hawaii
[5]. Peak load summer case with a total load of about 1080
MW was used in this example. The system has 16 synchronous
generators with a total output of 660 MW and transmissionconnected renewable sources of 80 MW. The rest of the
generation fleet is comprised of distributed PV with a total
of 360 MW. The initiating contingency is the loss of a 200
MW synchronous generator unit in the system. The response
Fig. 3. System frequency regulation with grid-following and grid-forming
inverters
with PV grid-following inverters incorporating FrequencyWatt function is indicated in the red “FW” trace in Fig. 3.
The response of grid-forming inverters with droop control is
indicated in the blue “CERTS” trace. The frequency damping
provided by the GFM inverters is stunning.
Magnitude of the imbalance between generation and load,
implies a sufficient level of untapped capacity or headroom for
frequency control. This can take the form of storage, spinning
reserves, and/or variable renewable energy sources that are
operated sufficiently below maximum available power level.
Inverter-interfaced batteries can also be deployed. A drawback
is that unused capacity may impose additional operating costs.
III. G ENERATORS /I NVERTER DYNAMICS .
1) Generator Model: As demonstrated in the earlier work
[6] using small signal analysis, the inertia dependent slow
electromechanical mode is responsible for the swing type
response observed in power systems. A simple mathematical
formulation is presented to satisfactorily capture this behavior
in a system with all generators. This approach is later expanded
to accommodate mixed source behavior in the following
sections.
The slow frequency dynamic response is predominantly
affected by the mechanical inertia and turbine-governor
dynamics of generators. In this time scale, the inter-generator
oscillatory modes can be ignored as they are typically faster. A
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TABLE I
T HEORETICAL S TUDY PARAMETERS
Fig. 4.
Aggregate generator model for slow frequency dynamics studies
simple aggregate model can capture the frequency dynamics
by assuming that the transient frequency change is same at
all buses [7], which is justifiable at quasi-steady state in
this relatively slower timescale. This power system inertial
aggregate model can be developed by summing up generator rotor speed differential equations. The aggregated inertia
constant
and damping values are then written as Mg,a =
Dg,i (Note- M 2H /ωs ). The
Mg,i and Dg,a =
aggregate mechanical power PM,a =
PM,i is written as
a function of frequency change PM,a (ω). The governorturbine system can be reduced to an aggregate equivalent
structure. This approach is proposed in the model reduction
package “DYNRED” to aggregate turbine-governor transfer
functions of coherent machines by summing up individual
mechanical power output responses by perturbing a speed
input [8]. Such an approach is also used in [9] to aggregate
governor turbine dynamics.
An equivalent linearized and per-unitized aggregate transfer
function is used here for studying the frequency dynamics. In
this work, a simple aggregated first order transfer function with
one pole and zero and a 5% governor speed droop (M p,gen )
is used for the turbine-governor response of the aggregated
system. This results in a convenient second order transfer
function model, enabling derivation of simple closed form
expressions. The transfer function model for the aggregate
generator is shown in fig. 4.
Assuming negligible mechanical damping (Dg,a = 0), the
characteristic equation can then be derived from the system
transfer function as,
1
1
TB
2
s+
(1)
+
s +
TA
Mg,a T A M p,gen
Mg,a T A M p,gen
The roots of the second order characteristic equation give the
dominant electromechanical mode that affects the frequency
dynamics. The oscillation frequency and damping values can
be calculated as,
1
ωn = Mg,a T A M p,gen
Mg,a M p,gen
1
TB
ζ =
+
√
2
Mg,a T A M p,gen
TA
(2)
For a nominal choice of parameters given in Table I at 1pu
power output, one can calculate ωn ≈ 0.6r ad/s and ζ ≈ 0.37,
which indicate a slow and poorly damped response. It is clear
from the expressions a lower value of inertia would yield a
Fig. 5.
GFM inverter model and its inertia equivalent structure
higher oscillation frequency and smaller damping factor value
(hence, a faster and poorly damped response).
2) GFM Inverter Model: Only the frequency/power dynamics of the GFM inverter are modeled, with a first-order power
measurement filter as in Fig. 5 [10]. Voltage control and other
dynamics are relatively fast and can be ignored. Limit controls
are ignored in the theoretical model. GFM inverter equations
can be simply rewritten to obtain structural resemblance to
a generator unit, with inertia and damping coefficients [11],
where
Mi TM /M p,inv Di 1/M p,inv
(3)
Typical values of these virtual terms at unit power output
(=1pu) are significantly
different from their generator counter
parts (Mi = 0.16 ωs Di = 20/ωs pu), with significantly lower
inertia and higher damping.
3) GFL Inverter Model: The GFL inverter model is simplified for the relevant timescales. Voltage related dynamics are
not modeled for the theoretical study. Delays due to frequency
measurement (phase-locked-loop), current control loop etc.
are not included as they are relatively fast and do not to
play a role in this timescale. PLL is not as fast as the other
faster control blocks such as current control, they are still
slower than the power controller modeled in this work. In
general, such PLL models are often ignored in bulk-system
transient stability simulation tools for this reason. However,
they are generally used for studying voltage stability issues
which can occur under weak grid conditions. The terminal
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Fig. 6.
GFL inverter model for theoretical study
voltage angle is used as the feedback input to determine
the frequency and determine a power command using the
frequency-watt function (droop). Consequently, a simple firstorder response to approximate the power control response
(Fig. 5). TL = 0.5s is used as the time constant for the
power response in this work. TL is sometimes set based on
requirements by applicable standards or can be limited by
inverter power response bandwidth.
The GFL case is structurally different from a generator or
a GFM inverter; the inputs and outputs are switched in the
GFL case. Hence, the inertia equivalent structure cannot be
formulated for the GFL case. However, the same definition
for damping as in the GFM case is adopted (Di 1/M p,inv ).
4) Mixed Generator/Inverter Model: In the mixed source
case, generators and inverters are each aggregated into respective equivalent units and modeled as a two-source structure.
Each aggregated unit is connected through a line impedance to
the aggregate load modeled as an impedance. The linearized
mixed source system model for the GFM case is constructed as
shown in Error! Reference source not found.7 using the aggregate generator and inverter models. The model is derived as
function of inverter penetration level(x); the generator power
rating reduces to (1-x) pu and the inverter rating to ‘x’ pu
to meet a constant 1pu system load. This scales the generator
inertia and damping values by (1-x) and the inverter inertia and
damping terms by a factor of x. The Kx coefficient terms are
obtained from the small signal power-angle relationship based
on external connections between the generator and inverter.
Pg
Pi
=
K1
K3
K2
K4
δg
δi
(4)
A similar approach can be used to develop the mixed source
system model for the GFL case. As mentioned earlier, the
inputs and outputs are reversed with inverter power being the
output and the terminal inverter angle being a feedback input
[11].
5) Reduced Order Mixed Source System Models: The aggregate model for the GFM case in Fig. 7, can be simplified due
to the fast time constants
involved in the inverter dynamics.
Time constant Tm = Mi Di ≈ 8ms is insignificant compared
to the generator inertia and other time constants in Table I and
can be ignored. The K matrix values is a rank 1 matrix and
is related by K 1 K 4 = K 2 K 3 . Further, in the two-source case
it turns out that K 1 ≈ K 4 and K 2 ≈ K 3 for a wide range
of interconnecting impedance values. These simplifications
lead to the reconstructed block diagram in Fig. 6, where the
inverter is represented by a linear function x Di and a first order
response with
time constant T . The value can be derived as
T = x Di K 4 , which can be considered a delay in feedback
path due to network connections. The value of K4 is dependent
Fig. 7.
Mixed Source System with GFM Inverter
Fig. 8.
Reduced order model of the mixed source (GFM or GFL)
on the load and network admittances and is in the range
of 0.5 to 1.5 at varying penetration levels. In general, this
network delay value |x Di /K 4 | < 0.05s is insignificant at all
penetration levels and can be ignored (T = 0).
The model for the mixed source case with GFL inverter can
be developed using a similar procedure. Surprisingly, simple
algebraic manipulations lead to the same model structure in
Fig. 6, albeit the value of time constant T . The time constant
for the first order inverter response can be derived as,
T = TL + x
Di
≈ TL
K4
(5)
In this case, the time constant is affected by the added
delay due to the inverter power response which can be much
larger than the network delay. Again, this network delay value
|x Di /K 4 | is considered insignificant at all penetration levels
and can be ignored.
Hence, both GFM and GFL cases lead to a similar structure
except for the values of the T. In the mixed source GFM case,
T → 0 and the first order delay is effectively neglected.
Hence, the GFM inverter can be represented by a simple
linear damping feedback for the generator frequency and is
an indicator for virtual damping provided by GFM inverters.
The damping and oscillation frequency of electromechanical modes are shown in Fig.9, which indicates overdamped
response (ζ >1) beyond 25% penetration. See Appendix for
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the closed form expressions. For the GFL case, time response
is dominated by TL and cannot be ignored as it is comparable
to generator time constants. This results in attenuating the
damping feedback on the generator frequency that can be
provided by the inverter. Qualitatively, the attenuation of
damping power feedback due to GFL inverter is clear, when
evaluated at the value of oscillation frequency (ωn ) of the
electromechanical mode.
De f f ( j ωn ) ≈ x Di
1
2
TL ωn2 + 1
(6)
For a finite value TL, the damping term on generator frequency
is reduced along with the generator inertia.
6) System Dynamics: Variation of oscillation frequency and
damping of electromechanical mode with variation in inverter
penetration level is shown Fig. 9. As the penetration level
of GFL inverters increases, increase in damping is initially
observed up to ∼20%, beyond which the damping declines.
Theoretically, the oscillation frequency, (ωn ), is small at lower
penetration levels, and damping feedback from (6) is effective.
With increasing penetration levels, ωn increases significantly
in the GFL case, and attenuates the damping feedback. From
this study, it is quite clear that such delays result in reduced
damping from inverters and contribute negatively to the system
frequency response.
The GFL inverter plot of damping vs penetration level in
Fig.9(b), is sensitive to system inertia (H ) and the power
response time constant (TL ). For larger inertia values, the peak
point at which the decline in damping begins shifts right,
occurring at higher penetration levels and a higher value of
damping. For values of TL (slower power response), the peak
damping value decreases, but the penetration level at which
the decline begins remains roughly the same.
Comparisons of GFM and GFL models derived earlier are
simulated to verify the frequency dynamics. The aggregated
two-source system case is simulated, for a 5% load decrease
event and the generator frequency plots are shown in Fig.10.
Simulations are repeated at inverter penetration levels of
20%, 50% and 80% for GFM and GFL cases. Time domain
responses with the aggregate GFM model (Fig. 7), visibly
overlaps with that of the simplified reduced model from Fig.
8, (with T = 0). Minor differences in the fast transient
are observed at 80% GFM penetration, which is relatively
insignificant and is inconsequential for the frequency dynamic
response.
It is clear that the aggregate GFM inverter provides damping
feedback on the generator frequency to the system. Rather than
acting as source of inertia, the GFM inverter acts as a source of
damping to the system. This helps clarify concerns regarding
low system inertia or emulating virtual inertia through inverters. Improvement in frequency dynamic performance can be
directly achieved by using inverters for damping the inertial
response, rather than adding inertia to the system. Artificially
introducing delays in the form of inertia is unnecessary and
may not be as helpful.
Fig. 9. Variation of oscillation frequency and damping function of Inverter
penetration level
IV. GFM I NVERTER : C OMPLEXITY AND OVERLOAD
C ONTROL
7) Generalized Overload Control: Application of inverters
in the power system has two major issues. One is the complexity of controlling hundreds of thousands to millions of
inverters. The other is the potential of high over current since
GFM inverters have no direct control of current. The complexity issue requires designs that allow each GFM inverter to
operate as an autonomous energy source. This concept is not
new to the bulk-power system. Most large generators track
fast load changes autonomously using frequency droop, not
signals from a centralized controller. Relative to microgrids
these autonomous concepts have been demonstrated in the
CERTS microgrid at the American Electrical Power’s test site
[12].
Autonomous GFM sources can also use frequency to explicitly protect themselves from self-overloading [13]. A load
increase or fault can cause a GFM source to exceed its power
rating (Pmax ). A sustained overload can stall prime movers,
collapse PV dc bus voltage or cause the inverter to shut
down on overcurrent. In systems with massive penetration
of GFM inverters we must expect that some sources will
reach their power limits before other sources. Step loads result
in all GFM sources instantaneously increase their output to
compensate for the extra load. However, the sources which
have reached their power limits must restrict their overload.
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Fig. 10.
Simulation of aggregated system response to 5% load decrease
This is achieved by the overloaded inverter rapidly decreasing
its output frequency changing the loading between sources due
to the change in system phase angles. If there are no adequate
reserves, relief comes from frequency load tripping. The basic
overload function is to force the overloaded source’s frequency
down faster than the droop for power levels greater than its
Pmax .
An overload mitigation controller is shown in Fig 11. The
top half are the droop controls, where Mp and Mq are the
slopes of P vs. f droop, and Q vs. V droop, respectively. The
dotted box in Fig 9 contains the overload mitigation controller.
Pmax is the maximum power of the source, the error between
Pmax and P is the input of the PI controller, where Kppmax and
Kipmax are the proportional and integral gains, respectively.
Fig. 11.
Overload mitigation controller
Fig. 12.
state.
Demonstration of the overload issue of microsources in transient
The output of the PI controller will regulate the frequency.
When the droop-controlled source becomes overloaded, the
PI controller will reduce the frequency rapidly to mitigate its
overload.
The basic overload function is demonstrated in a two-source
microgrid, Fig. 12. The two sources initially operate at 60 Hz
with unit-1 near its maximum power P1max , and unit-2 well
below its rating. Loss of the grid at 0.2 seconds results in
overloading of unit-1. To prevent this overload, unit-1 rapidly
decreases the frequency command in order to transfer the
extra power to the unit-2. After the transient, the system’s
new frequency is fb . Unit-1 is at its maximum value while
unit-2 has increased its output power to accommodate the
load increase. The frequency of unit -1 drops faster than unit2, resulting in a needed power-angle change for the loading
transfer, Fig 12(b). Overload power is transferred to unit 2 in
less than 0.2 seconds.
An alternative to frequency-based overload control is for the
overloaded GFM sources to switch to GFL mode. This will
force unit-2 to provide the extra power. The problem with this
approach occurs when all sources are overloaded resulting in
no voltage or frequency control forcing the system to shut
down. In contrast GFM’s frequency-based overload control
will continue to force frequency down until frequency-based
shedding of non-critical loads provides relief. At this point the
microgrid can recover with reduced loads while maintaining
stable voltage and frequency [14]. Inverter fault protection
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is ultimately a manufacture’s design issue, but there are
techniques that can be used to reduce the likelihood of inverter
tripping on overcurrent. They include wye delta transformers,
local protection design and the droop controllers. Overcurrent
due to power is mitigated by the overload controller discussed
in this section. Over current due to reactive power is reduced
by the reactive power droop vs. voltage by reducing the output
voltage with increase reactive power [22].
8) PV Overload Control: Using GFM inverter with a PV
panel implies that the Pmax in Fig. 11 is also the Maximum
Power Point, [15]. Setting the maximum power limit at the
MPP protects the PV source from trying to delivering more
power than is available. When demand exceeds the MPP the
frequency decreases rapidly. This frequency drop causes a
rapid change in the PV source’s phase angle that causes other
sources that have not reached their limits to automatically
assume load tracking responsibilities. The maximum power
limit is designed to respond to dynamic changes in the PV
power capacity caused by varying insolation and temperature.
The interaction between synchronous generators and PV
operating at the MPP can be illustrated using the mixed
two-source power system introduced in Table I. For 50%
penetration the expected response to a 5% load increase
would be for both sources to equally share the increased load.
When the PV inverter is operating at its MPP this sharing
is not possible and all the load increase must be provided
by the synchronous generators. Details shown in Fig. 13
are the power outputs from each source, the load voltage
and the PV inverter’s dc voltage. The power plots show not
only how the transient overload of the PV source has been
transferred to the generator, but also the frequency swings
caused by the transient PV overloading. With application of
the 5% load increase, both sources increase their output power
instantaneously to meet this positive load step, resulting in
transient overloading of PV source. The Pmax controller shifts
the excess load of PV source to the generator. The PV dc
bus voltage recovers to its initial value to restore delivery of
its maximum power. In the new steady-state operating points,
the generator compensates for the load change by increasing
its output power by 5%. Since the PV source operates at the
MPP, the steady-state frequency of the system is determined
by the generator’s droop characteristic, which changes from
60 Hz to 59.7 Hz in this case. However, it is found that the
system including the PV source needs more than 20 seconds
to reach steady state. The dynamic damping seen in Fig. 10
is not possible for PV operating at MPP since there is no
reserves.
V. R ESERVES FOR L OW I NERTIA S YSTEMS
9) Damping Reserves: It is clear from Figs 9, and 10 that
aggregated GFM inverters can provide damping feedback to
the system. Rather than acting as a source of inertia, the GFM
inverter acts as a source of damping to the system. Tradition
power system relies on spinning reserve to compensate for
power shortages or frequency drops within a given period of
time. Traditionally, spinning reserve is a concept for systems
with large synchronous generators. If the largest generator in
Fig. 13. Simulated dynamic responses of the mixed-source system operating
at the MPP for a positive 5% load step.
the power system is tripped, the remaining generators need to
increase their output to recover the power shortage. However,
leveraging traditional generation assets for creating reserved
capacity creates a number of inefficiencies. For example,
because these generators are operated below their rated values,
the utility is not maximizing their power output that could
be used for base load supply. Also, it requires the use of
additional fuel to ramp these generators up in the event that
their reserved generation potential is needed, which increases
emissions while reducing the net efficiency of the power
system [16].
In principle, GFM distributed energy resources can be
implemented with “spinning reserve” assets. In autonomous
operation the reserve allocation between DERs and synchronous generators is dependent on the penetration ratio and
the percent droop. Let us return to the mixed two-source
power system. In a system with 50% penetration and 5%
droop the autonomous response to a 5% step load is for
each source to increase its output by 2.5% or one half of
the applied load. In a more complex bulk power system this
distribution is also dependent on load and source locations
and the impedance distribution. In this mixed two-source
system the synchronous generator and grid-forming inverter
both need to have reserves greater than 2.5%. Communication
would allow different allocations, but the complexity could be
enormous.
The bulk system could easily have hundreds of thousands
to millions of inverters throughout a large geographic area
making centralized, secondary and tertiary control impractical without appropriate aggregate-control schemes and system partitioning. An alternative is to operates each source
autonomously. In this case each autonomous PV source must
have the necessary reserve capacity to ensure that it does not
exceed its maximum available power. Generally, for a mixedsource system with n source, each inverter or generator may
have a unique penetration level, x j , with a droop of M p, j
so that the power change in each source Pi after the load
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change is determined as follows:
Pi = PL
xi
n
m=1
xm
n
j =1, j =i
M p, j
n
j =1, j =m
M p, j
where n is the source number, the subscripts m, i , and j
represent the source m, i , and j , respectively. The power
change constraint is:
ω = P1
M p,1
M p,n
= . . . = Pn
x1
xn
where x m is the instantaneous source penetration, nm=1 x m =
1.
For our mixed two-source power system the distribution
of the load change between the generator and inverter are
a function pentation and droop. For equal droops and 50%
penetration the load allocation is equal. For a 1% inverter
droop and 5% generator droop the allocation of the load
is 5/6 for the inverter and 1/6 for the generator radically
changing the required reserves. For the first example the
reserves requirements are equal. For the second example the
inverter reserve need is 5 time that of the generator. For the
example shown in fig.13 the generator’s reserves requirement
is the full load increase.
10) Grid-forming PV Sources with Reserves: The frequency
dynamics seen in Fig. 13. can be improved by introducing
power reserve margins to the PV source, [17]. This reserve is
achieved by dispatching the PV power below its MaximumPower-Point. Using the same initial conditions and load step
used in the above simulation Fig. 14(a) presents the time
response of the system having adequate power reserve capabilities for the PV source. In this case droop and penetration are
equal. These two sources evenly share the 5% load increase.
Compared to Fig. 13, the dynamic performance is obviously
improved. The frequency reduction is less than half and the
system achieves steady-state in half the time. Clearly better
system dynamic performance is achieved using PV power
reserves. In fig.14(b) the penetration levels remain equal but
the inverter droop is changed from 5% to 1%. The allocation
of the load seen in fig. 14(b) is far from equal. The inverter
provides 5/6 of the need power for the load increase while the
generator provides 1/6. The change in frequency is defined
by the 1% droop achieving steady state is less than a second.
This event needs to be followed with local re-dispatch of the
generator to return the PV to a state with necessary reserves.
VI. I MPLEMENTATION OF PV P OWER R ESERVE C ONTROL
The concept of PV power reserve is to operate the PV at
a power level lower than the available maximum power from
the PV array. The reserve power value is
P = Pmpp − Ppv,o
(7)
where P is the amount of reserved power and Ppv,o is the
value of output power from the PV source. It is implemented
by adjusting the power setpoint of the PV source to insure
needed reserved power. It also needs to point out that due
to the non-monotonic characteristics of the PV array’s P-V
Fig. 14.
reserves
Fig. 15.
Simulated dynamic responses of the mixed-source system with
Single-diode electrical equivalent circuit of the PV panel
curve, there are two possible operating points associated with
the desired PV power Ppv,o . The operating point with voltage
less than MPP can introduce over-modulation issues and
instability for grid-forming PV sources. The control system
must insure operation over the entire range of available needed
reserved power at a voltage higher than the MPP. In order
to implement the power reserve concept on grid-forming PV
sources, tracking the time-variant maximum available power is
a key requirement. A model-based MPP estimation is proposed
that avoids the need for irradiance and temperature sensors
[18] and requires much less complexity than control-based
methods [19] and curve-fitting algorithms [20].
The equivalent circuit of the PV panel in Fig. 15 is described
9
by the well-known nonlinear current-voltage equation [21] as
V panel +Rs I panel V panel + Rs I panel
N BVT
−1 −
I panel = I ph − Is e
(8)
Rsh
where
Is = K panel AT 3 e
−E g
VT
(9)
is the diode’s saturation current [A], I ph is the photocurrent
source [A], N represents the number of series-connected cells,
B is the diode ideality factor, and Rs and Rsh are the series
and shunt resistances [], respectively. In (8), K panel is the
thermal coefficient [A/(m2 ·deg-K3)], A is the surface area of
a single cell [m2 ], E g is the energy gap for silicon [eV], and
VT = kT /q is the temperature equivalent voltage [V], where
k is the Boltzmann constant [eV/deg-K], q is the electron
charge [coulomb], and T is the temperature [deg-K]. This
equation can also be expressed as voltage in terms of current
using the Lambert-W function. Without additional hardware
requirements, model-based MPP estimates can be derived by
manipulating this current-voltage equation.
Using PV voltage and current measurements, the PV panel
equivalent circuit can be used to play the role of light irradiation and temperature sensors. First, with voltage and current
measurements at the PV panel terminals, operation region on
the PV I-V curve can be decided. If measurements are made
to the right of the MPP, the PV panel’s open-circuit voltage
Voc can be estimated to be the intercept of the curve-fitted
voltage-source region of the I-V curve with the voltage axis.
The short-circuit current Isc can also be estimated using a
similar process if measurements are made to the left of the
MPP. (8) can be expressed in terms of the PV panel’s opencircuit voltage Voc as:
V
oc
Voc
N
BV
T
I ph = Is e
−1 +
(10)
Rsh
And the short-circuit current Isc can be estimated by solving
the following rearrangement of (8):
RI
s sc
Rs
N BVT
Isc = I ph − Is e
−1 −
Isc
(11)
Rsh
(9)-(11) and equations in [21] used for translation of parameters that are dependent on atmospheric conditions can be
used to roughly estimate likely ranges of the temperature
and irradiance values to reduce the search time. Furtherly,
by substituting (9) into (8) and replacing V panel and I panel
with a few sets of sampled voltage and current readings,
the temperature and irradiance values can be estimated. This
process can be accelerated using the Newton-Raphson method.
Next, circuit parameters that are relevant to the temperature
and irradiance, such as I ph , Is , Rsh , and VT , are updated and
full P-V /I -V curve can be generated. Since the MPP current
(Impp ) has a nearly proportional relationship with Isc , Impp
can be approximately expressed as:
Impp = K Isc Isc wher e 0.78 < K Isc < 0.92
(12)
where the value of K I sc is essentially constant, and its value
can be estimated from the specified PV panel’s datasheet. The
Fig. 16.
June 7th , 2019
MPP voltage (Vmpp ) can be approximately expressed by using
the nonlinear Lambert-W function based on [21]:
⎫
⎧
√
Rs + Rsh Rs +Rs2 ⎪
⎪
⎪
⎪
I
1
+
⎨ mpp
⎬
Rsh
∼
Vmpp = N BVT W
− Rs Impp
⎪
⎪
Is
⎪
⎪
⎩
⎭
(13)
After Impp and Vmpp are estimated using (12) and (13),
respectively, the MPP power (Pmpp ) can be calculated from
their product [23].
The performance of the MPP estimation has been evaluated
for a PV array consisting of series- and parallel-connected
SunPower X21-345 PV panels that are installed on the rooftop
of the Wisconsin Energy Institute at UW-Madison. Circuit
parameter characterization that accounts for deviations from
datasheets due to a variety of reasons such as aging, degradation, cable resistances, and so on is determined using curve
scan operation.
Fig 16 plots the estimation performance for this PV array
operating in MPPT mode during two different days, which
is compared with measured MPPs. These figures demonstrate
that the MPPs can be estimated with acceptable errors by using
this proposed method in hardware PV source equipment in
the presence of real-world disturbances such as measurement
noises.
VII. C ONCLUSION
This work provides important insight to the interactions
between inverter bases sources and the bulk-power system.
The distinction between GFM inverter and GFL inverter
is profound. GFM inverters provides damping to frequency
swings in a mixed system while GFL inverter can aggravate
frequency problems with increased penetration. Rather than
acting as source of inertia, the GFM inverter acts as a source
of damping to the system. On the other hand, application of
inverters in the power system have two major issues. One is the
10
TB . Hence, the damping factor (ζ ) improves significantly
with increasing inverter penetration. As we have seen GFM
inverters can damp frequency swings for changes in load or
generation.
R EFERENCES
Fig. 16.
June17th, 2019
complexity of controlling hundreds of thousands to millions of
inverters. This was addressed through autonomous techniques
using local measurements. The other is the potential of high
over-current in GFM inverters and techniques for explicitly
protecting against overloading. To exploit the innate damping
of GFM inverters energy reserves was shown to be critical.
In autonomous operation the reserve allocation between GFM
sources and synchronous generators is shown to be approximately equal to the penetration ratio with equal droops. If
was seen that unequal droops can have profound effects on
the needed reserves and can greatly reduce frequency swings.
The last section derives and test a reserve allocation algorithm
for PV sources. The test demonstrates that the MPPs can
be estimated with acceptable errors by using this proposed
method.
A PPENDIX
Closed form expressions for the mixed source GFM case
can be easily obtained due to the second order nature of the
system model. The characteristic equation for the mixed source
GFM case can be derived from the transfer function as,
Di
1
TB
x
2
+
+
s +
TA
Mg,a T A Rg
1 − x Mg,a
x
1
1
Di
+
+
(1)
Mg,a T A Rg
1−x
Closed form expressions are obtained for oscillation frequency
and damping, as a function of inverter penetration level (x).
1
Mg,a T A Rg (1 − x)
√
TB
x
TA
ζ ≈ 1−x +
(2)
√
TB 1 − x
2 Mg,a T A Rg
√
With
√ increasing values of x from 0 to 1, 1 − x decreases and
1/ 1 − x increases and leads to an increase
in ωn , indicating
√
a faster response. The increase in x/ √1 − x term more than
compensates for the decrease due to 1 − x term as T A ≫
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Robert H. Lasseter (F’s 92) received the Ph.D.
in Physics from the University of Pennsylvania,
Philadelphia in 1971. He was a Consulting Engineer
at General Electric until he joined the Department of
Electrical and Computer Engineer at University of
Wisconsin-Madison in 1980. Dr. Lasseter is internationally recognized as one of the earliest and
most influential pioneers in the microgrid field. His
professional career during the past 40 years has been
dedicated to applying power electronics to utility
systems. He is the technical lead for the Consortium
for Electrical Reliability Technology Solutions’ (CERTS) Microgrid Project.
CERTS’s microgrid architecture is widely implemented and recognized for its
plug-and-play flexibility.
Zhe Chen (S’14) received the B.S. degree in
Electrical Engineering and Automation from the
Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2012, and the M.S. degree
in Electrical Engineering from the University of
Wisconsin-Madison, WI, USA, in 2016, where she is
currently working toward the Ph.D. degree in Electrical Engineering. Her research interests include
modeling, analysis, and control of renewable energy
sources, microgrids, power electronics, and electric
machines.
Dinesh Pattabiraman received his B.Tech. degree
in electrical engineering from the National Institute
of Technology, Tiruchirappalli, India and his M.S
degree from the University of Wisconsin, Madison,
USA. Currently, he is pursuing his Ph.D. in Electrical Engineering at the University of Wisconsin,
Madison. His research interests include utility applications of power electronics, inverter control, power
system dynamic response, microgrid control etc.