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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 26, NO. 3, JULY 2011
An Adaptive Feedforward Compensation for
Stability Enhancement in Droop-Controlled
Inverter-Based Microgrids
Mohammad B. Delghavi, Student Member, IEEE, and Amirnaser Yazdani, Senior Member, IEEE
Abstract—This paper proposes an adaptive feedforward compensation that alters the dynamic coupling between a distributedresource unit and the host microgrid, so that the robustness of the
system stability to droop coefficients and network dynamic uncertainties is enhanced. The proposed feedforward strategy preserves the steady-state effect that the conventional droop mechanism exhibits and, therefore, does not compromise the steady-state
power sharing regime of the microgrid or the voltage/frequency
regulation. The feedforward compensation is adaptive as it is modified periodically according to the system steady-state operating
point which, in turn, is estimated through an online recursive leastsquare estimation technique. This paper presents a discrete-time
mathematical model and analytical framework for the proposed
feedforward compensation. The effectiveness of the proposed control is demonstrated through time-domain simulation studies, in
the PSCAD/EMTDC software environment, conducted on a detailed switched model of a sample two-unit microgrid.
Index Terms—Adaptive control, current control, distributed
generation (DG), distributed resource (DR), droop, dynamics,
feedforward, microgrid, model, power sharing.
I. INTRODUCTION
HE microgrid concept [1] involves the coexistence of multiple distributed energy resource (DR) units within a prespecified part of an electrical network. One critical control task
within the microgrid framework is the regulation of the network frequency and voltage magnitude, in the islanded (offgrid) mode of operation. Fundamentally, this objective is fulfilled through a shared contribution of real and reactive power
by a multitude of dispatchable embedded DR units. The most
widely adopted technique to ensure power sharing and coordinated voltage/frequency regulation is to droop the frequency
and magnitude of the point-of-coupling voltage of each DR unit,
versus the real and reactive powers that the DR unit delivers to
the network [2]–[4]. The popularity of the method can be attributed to its ease of implementation, based merely on local
voltage and current information, in addition to the facts that it
T
Manuscript received August 16, 2010; revised January 16, 2011; accepted
February 20, 2011. Date of publication April 07, 2011; date of current version
June 24, 2011. This work was supported in part by the Natural Sciences and
Engineering Research Council (NSERC) of Canada. Paper no. TPWRD-006172010.
The authors are with the University of Western Ontario, London, ON N6A
5B9 Canada (e-mail: mdelghav@uwo.ca; ayazdan2@uwo.ca).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRD.2011.2119497
enables decentralized control of multiple DR units, readily accommodates the grid-connected mode of operation, and enables
plug-and-play operation of the DR units.
The prime issue with respect to the conventional droop-based
control is that, it is, in essence, a steady-state measure that is
taken to prevent the DR units from competing against each
other, for individually imposing the network frequency and
voltage; any such competition would inevitably result in a
network collapse. Consequently, the transient performance and
stability of the droop-based decentralized control highly depend
on the droop coefficients, and on dynamic properties of the
network, DR units, and embedded loads [5]. Even in terms of
steady-state performance, the droop technique is effective the
most for highly inductive networks, such as high-voltage transmission networks, but performs rather poorly when adopted for
distribution networks [6]. These dependencies, combined with
the fact that the droop coefficients are commonly determined
based merely on steady-state criteria [2]–[4], give rise to the
likelihood of inaccurate power sharings, unsatisfactory transient performances, or even instabilities, in the islanded mode
of operation, as recently noticed by researchers [5], [7]–[9].
References [5] and [10] report two studies on dynamic characteristics of islanded microgrids that embed droop-controlled
electronically coupled DR units. In [5], the sensitivity of the
overall system eigenvalues to droop coefficients is shown,
whereas [10] places the emphasis on the controller parameters.
Reference [7] proposes a method for improving the transient
power sharing among the DR units, in which the magnitude and
frequency of each DR unit voltage are also drooped versus the
derivatives/integrals of the real and reactive output powers. The
method proposed in [7] is further advanced in [8] by making
the droop coefficients dependant on the operating point, to
mitigate the eigenvalue migration phenomenon and to improve
the damping of the critical eigenmodes. Based on a slightly
different approach proposed in [9], the droop coefficients of a
DR unit are made functions of the DR unit real and reactive
output powers. In all three references [7]–[9], it is assumed
that a corresponding tie reactor interfaces each DR unit with
a common ac bus, which, in turn, is connected to a lumped
load. Thus, the common bus voltage is taken as the reference
voltage, and the real and reactive powers that a DR unit exchanges with the network depend only on the magnitude and
phase angle of the DR unit terminal voltage, relative to those
of the common bus voltage; however, the load is considered
an independent power sink and, therefore, exhibits no dynamic
interactions with the DR units. Although simple, this model,
0885-8977/$26.00 © 2011 IEEE
DELGHAVI AND YAZDANI: ADAPTIVE FEEDFORWARD COMPENSATION FOR STABILITY ENHANCEMENT
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Fig. 1. Schematic diagram of the electronically interfaced DR unit.
which implicitly assumes a quasi steady-state condition, offers
an approximate account of dynamic interactions among the DR
units, but obscures the existence of such interactions with the
network/loads. The reason is that the DR unit terminal voltages
are functions of dynamic, as well as steady-state, power flows
within the overall network and, thus, depend also on the loads.
Reciprocally, the loads real and reactive powers are functions
of the network voltage and frequency. Consequently, the aforementioned simplified model may not guarantee system stability
and/or a satisfactory performance, in a real-life scenario.
On the other hand, due to the diversities in network topologies and equipment, dynamics of a real-life network are typically governed by complex dynamic models [5], [10], [11],
even for relatively small networks with loads of prespecified
(e.g., RL) configurations. Consequently, the incorporation of
network/load dynamics into the control design process is understandably involved [12], does not promise sufficient generality, and expectedly renders the design prone to case-by-case
refinements and compromises the capability for plug-and-play
operation; the authors of this paper consider these issues to be
the main reasons behind the widespread adoption of the simplified model described earlier. The objective of the study reported
in this paper has thus been to circumvent the aforementioned
difficulties by making the control design, to a large extent, independent of the droop coefficients and network/load dynamics.
This paper proposes an adaptive feedforward compensation
that alters the dynamic coupling between a DR unit and the
host network, such that the system stability is desensitized to
the droop coefficients and network dynamics. The proposed
feedforward strategy preserves the steady-state effect that the
conventional droop mechanism exhibits and therefore does not
compromise the steady-state power sharing amongst the DR
units or the voltage/frequency regulation. Rather, it reshapes the
contribution of the load dynamics in the control process, such
that the system stability is enhanced. The proposed feedforward
compensation is adaptive since it is periodically modified
based on the system steady-state operating point, which, in
turn, is estimated through the recursive least square (RLS)
technique. The effectiveness of the proposed droop strategy is
demonstrated through time-domain simulation studies, in the
PSCAD/EMTDC software environment [13], conducted on a
detailed switched model of a sample two-unit microgrid.
II. STRUCTURE OF THE DR UNIT
In this paper, it is assumed that the DR units are all dispatchable and of the electronically coupled type, and adopt the proposed adaptive feedforward strategy (not necessarily with identical parameters). Thus, hereafter, a DR unit under study is referred to as the “DR unit,” and the microgrid without the DR
unit is referred to as the “rest of the microgrid,” irrespective of
whether it embeds any other DR unit.
Fig. 1 shows a schematic diagram of the DR unit. The power
circuit of the DR unit consists of a conditioned prime energy
source, a current-controlled voltage-sourced converter (VSC),
filter. The per-phase resistance, inducand a three-phase
tance, and capacitance of the filter are denoted by , , and
, respectively. The resistance represents the ohmic loss of
the filter inductor and also includes the effect of the on-state
,
resistance of the VSC valves. The three-phase variables
, and
are referred to as the “DR unit terminal voltage,”
“VSC ac-side current,” and “DR unit terminal current,” respectively. The VSC dc side is parallelled with a dc-link capacitor
and a voltage source. The latter represents the effect of either a
dispatchable mover-generator-rectifier set or an energy storage
device (e.g., a battery bank).
Fig. 1 also shows the control components of the DR unit. It
,
, and
are sampled and digitized
is noted that
by corresponding sample-and-hold (S/H) and analog-to-digital
converters (ADCs). The sampled variables are then provided to
–to– frame transformation blocks. Fig. 1 furrespective
ther indicates that the DR unit is controlled in a rotating
frame whose axis makes an angle with respect to the stationary axis (i.e., the axis). is obtained from a phase-locked
loop (PLL) which also determines , that is, the frequency of
the DR unit terminal voltage. In the grid-connected mode of operation, the DR unit terminal voltage is dictated by the rest of the
microgrid, and represents the power system frequency. In the
islanded mode, however, the DR unit, in conjunction with the
other DR units, must contribute to the regulation of the network
voltage and frequency, based on the proper commands given to
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Fig. 2. Block diagram of the current-control scheme of the DR unit.
Fig. 3. Block diagram of the amplitude regulation scheme.
the amplitude and frequency setpoints of the DR unit terminal
voltage; the setpoints are, in turn, determined by a droop-based
power sharing mechanism, which enables decentralized control
of the network voltage and frequency, and ensures proper power
sharing among the DR units.
III. BASIC CONTROL
The control schemes of the DR unit of Fig. 1 are extensively
discussed in [14], but are also briefly reviewed here for ease
of reference. In the following formulations, the - and -axis
are denoted by
components of a three-phase signal
and
, respectively.
A. Current-Control Scheme
The function of the current-control scheme is to regulate the
- and -axis components of the VSC ac-side current
, by
means of the pulsewidth modulation (PWM) switching strategy.
This is primarily to ensure that the DR unit is protected against
network faults, but also enables the regulation of the amplitude
.
and frequency of the DR unit terminal voltage
Fig. 2 shows a block diagram of the current-control scheme
and illustrates that two respective compensators
process the error signals
and
and, based on the method described
and
. These
in [14], deliver the signals
two last signals are then delayed by one sampling period and
and
for the PWM scheme
generate the signals
of the VSC; the delays are essential in a microprocessor-based
implementation due to the fact that the control signals, which
are calculated based on the feedbacks at the th sampling instant, are practically computed during the time interval between
th sampling instants and, consequently,
the th and
th sampling instant.
cannot be released earlier than the
Finally, the modulating signals
,
, and
are
,
, and
, and determine the
calculated from
switching instants of the VSC valves. The goal is achieved
based on the symmetrical regular sampled PWM technique
[15], which renders the VSC switching frequency equal to
the control system sampling frequency. As explained in [14],
ensures that
and
track their
proper design of
corresponding reference commands in two sampling periods,
that is
Fig. 4. Block diagrams of the d- and q -axis closed loops equivalent to the amplitude regulation scheme of Fig. 3.
B. Amplitude Regulation Scheme
Fig. 3 shows a block diagram of the amplitude regulation
scheme whose function is to regulate
and
(i.e.,
the – and –axis components of the DR unit terminal voltage)
and
. This obat their respective setpoints
and
. In
jective is accomplished by the control of
the scheme of Fig. 3, the – and –axis compensators
process the error signals
and
, respectively, and generate the setand
for the current-control scheme. As
points
will be discussed in Section III-C,
is indirectly employed in the frequency regulation scheme and has zero steady, that is,
,
state value. Hence, the amplitude of
is predominantly determined by
and, as such,
effectively serves as the setpoint for the amplitude of the DR unit
terminal voltage. As explained in [14], the control scheme of
Fig. 3 transforms the closed-loop system to the two decoupled
control loops of Fig. 4 in which the effective control plant is
. ( is the sampling period.)
C. Frequency Regulation Scheme
The objective of the frequency regulation scheme is to regulate , that is, the frequency of the DR unit terminal voltage, at
its setpoint
. As Fig. 1 shows, a PLL processes
by
and determines
in such a way that
the filter
is forced to zero [16]. In the grid-connected mode,
is imposed by the rest of the microgrid and becomes equal to the
power system angular frequency. However, in the islanded mode
DELGHAVI AND YAZDANI: ADAPTIVE FEEDFORWARD COMPENSATION FOR STABILITY ENHANCEMENT
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The calculated values of the real- and reactive-power outputs
of the DR unit are formulated as
(2)
Fig. 5. Block diagram of the frequency regulation loop.
and
(3)
Filtering each calculated value
or
by a corresponding discrete-time first-order low-pass filter yields
(4)
and
Fig. 6. Block diagram of a microgrid embedding a droop-controlled DR unit.
of operation,
is regulated by
. As explained in [14],
, through the control of
is of the form
(1)
is the gain.
for which
Fig. 5 shows a block diagram of the frequency regulation
closed loop including the frequency regulation scheme. As Fig.
processes the error
5 shows, a compensator
and determines one component of
;
the other component of
, that is,
is an auxiliary
signal of zero steady-state value which plays an important role in
the proposed adaptive feedforward strategy, as will be discussed
is then tracked by
in Section IV. The setpoint
through the action of the –axis amplitude regulation loop of
is regulated at
. It should be noted
Fig. 4(b), and
at
1,
(and, therefore,
that due to the pole of
) settles at zero in a steady state. Therefore, in order
for
to track
with no steady-state error,
is
sufficient to be a pure gain .
IV. PROPOSED ADAPTIVE FEEDFORWARD COMPENSATION
Fig. 6 illustrates a generic block diagram of a microgrid,
whether in the islanded mode or in the grid-connected mode, in
which a DR unit is interfaced with the rest of the microgrid. As
and
are the responses of the rest
Fig. 6 indicates,
,
,
of the microgrid to the DR unit outputs (i.e.,
). In turn,
and
are the responses of the
and
DR unit to the setpoints
and
, respectively,
is an internal variable of the DR unit control (see
while
Section III-C). It is noted that (filtered measures of) the DR unit
and
real and reactive powers determine the setpoints
, based on two respective droop characteristics.
Fig. 6 also illustrates the proposed feedforward compensation.
As the figure indicates, the proposed feedforward compensation
with a signal
and determines the
augments
signal
for the frequency regulation scheme of the DR
and
.
unit (see Section III-C), based on measures of
As such, the feedforward compensation alters the dynamic coupling between the DR unit and the rest of the microgrid, by maand
, based on
and
.
nipulating
(5)
where
and
signify the filtered versions of
and
, respectively, and
is a constant parameter. This
, where
is the
parameter can be determined as
corner frequency of an equivalent continues-time, first-order,
should be
low-pass filter. From the viewpoint of dynamics,
adequately large to not significantly alter the controller dynamic
properties, but reasonably small to ensure adequate filtering.
Then, as Fig. 6 indicates, the amplitude and frequency setpoints
and
of the DR unit terminal voltage are drooped versus
as
(6)
(7)
and are the real- and reactive-power droop coeffiwhere
is an auxiliary signal with zero
cients, respectively, and
steady-state value, similar to the signal
which was introand
are the no-load
duced in Section III-C. The setpoints
frequency and amplitude of the DR unit terminal voltage, respectively.
and
in (4) and (5), from (2) and
Substituting for
(3), one deduces
(8)
(9)
To enhance system stability, the dynamic coupling between
the DR unit and rest of the microgrid [see Fig. 6] should be
weakened as much as possible, but must exist in steady and
quasi-steady states for real- and reactive-power sharing (i.e.,
conventional droop strategy). Thus, a small-signal model of the
system is developed so that the steady-state and dynamic characteristics can be treated separately. To that end, (8) and (9) are
linearized about a steady-state operating point to yield
(10)
(11)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 26, NO. 3, JULY 2011
where “ ” and the subscript “ ” denote the small-signal perturbation and the steady-state value, of a variable, respectively. It
is zero, as exshould be noted that the steady-state value of
plained in Section III-C and, therefore, does not appear in (10)
and
have zero
or (11). It is also recalled that
steady-state values.
Defining
method of stability [17]. To that end, consider the following positive semidefinite function, as a candidate Lyapunov function:
(12)
(13)
(14)
(15)
(27)
one can rewrite (10) and (11) as
(16)
(26)
Then
Since
(note that
, based on (12), and
), then
and the system is stable [17].
Therefore, the transient response of the system is bound to decay
to zero and, in other words, the large-signal system response is
bound to settle, in a steady state. It should be emphasized that
the stability does not depend on the droop coefficients or on any
network properties.
Expressing (22) and (23) in the matrix form
(28)
(17)
The DR unit amplitude regulation scheme (Section III-B) enand
at their respective
sures rapid regulation of
by
in (7), and linsetpoints. Thus, replacing
earizing the resultant equation, one deduces
(18)
.
In addition, as Fig. 5 shows,
On the other hand, the frequency regulation loop ensures that
(Section III-C), the amplitude regulation scheme
ensures that
(Section III-B), and the steady-state
is zero (Section III-C). Hence, one finds
value of
(19)
and
Substituting for
(18) and (19), one finds
in (16) and (17), based on
(20)
(21)
Let
and
be determined in such a way that
(22)
(23)
Then, (20) and (21) assume the following forms:
(24)
(25)
Equations (24) and (25) govern the small-signal dynamics of
the real and reactive powers that the DR unit delivers to the
rest of the microgrid. The stability of the system described by
(24) and (25) can be established by means of Lyapunov’s direct
one can calculate
and
as
(29)
where
(30)
(31)
One difficulty that arises in implementing the proposed feedand
, one
forward compensation is that to determine
needs knowledge about the steady-state values of the DR unit
terminal voltage and current. On the other hand, in practice,
the system operating point changes constantly as the loads shed
back and forth, the power flow within the microgrid changes,
etc. Consequently, the operating point needs to be estimated,
for example, by means of an identification algorithm. This is
explained next.
V. RECURSIVE LEAST SQUARE IDENTIFICATION SCHEME
Online estimation of system parameters, that is, online
system identification, plays a crucial role in most adaptive
control schemes [18]. For the adaptive feedforward compensation of Section IV, the parameters to be estimated are the
steady-state values of the DR unit terminal voltage and DR
unit terminal current (see (13)–(15)). It should be pointed out
that these parameters do not bear a rigorous physical meaning;
rather, they are the byproducts of the mathematical formulation
(i.e., the linearization process).
In this paper, the RLS identification method with an exponential forgetting algorithm [18] has been employed for estimating
the required parameters; the estimates of parameters are updated
at every sampling instant.
Equations (29)–(31) indicate that the proposed adaptive feedforward compensation requires the parameters , , and .
DELGHAVI AND YAZDANI: ADAPTIVE FEEDFORWARD COMPENSATION FOR STABILITY ENHANCEMENT
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(39)
Fig. 7. Block diagram of the scheme for the calculation of feedforward coefficients based on estimated parameters.
is the vector of estimated
where
is the error vector,
is known as Kalman
parameters,
gain,
is referred to as the error covariance, and is known
as the forgetting factor and determines the speed of adaptation.
The estimation algorithm is executed at every sampling period,
through the following steps:
• a new set of data is acquired and the prediction error is
computed, based on (36), using the old estimated parameters;
• Kalman gain is updated based on (37);
is calculated based on
• estimated parameters vector
(38);
is updated for the next sampling period, based on
•
(39).
Once the parameters , , and , are estimated, and
are calculated, based on (30) and (31), and
and
are generated based on (29). The small-signal perturbations of
the variables involved (see Fig. 7) are extracted by high-pass
filtering of the variables to eliminate their steady-state components. Each high-pass filter has the following transfer function:
(40)
is the filter pole.
where
The process is illustrated by the block diagram of Fig. 7.
VI. TEST CASES AND RESULTS
Fig. 8. Schematic diagram of the test microgrid.
These parameters can be estimated if the RLS algorithm is applied to (16). Let us rewrite (16) as
(32)
Let us define the vector of parameters
(33)
and the regression vector
(34)
Then, (32) can be rewritten as
(35)
The estimated parameters of the plant, denoted by
calculated in the th sampling period, as [18]
, are then
(36)
(37)
(38)
To demonstrate the effectiveness of the proposed control
strategy, a detailed switched model of a two-unit test microgrid
has been simulated in the PSCAD/EMTDC software environment. The test microgrid is based on a typical 12.47-kV North
American distribution network [19], of which a simplified
schematic diagram is illustrated in Fig. 8.
As Fig. 8 shows, the microgrid is divided into two subnetworks: Subnetwork 1 and Subnetwork 2, each embedding threephase (residential and industrial) loads as well as smaller singlephase networks (not shown in Fig. 8). The two subnetworks are
interfaced, respectively, with Bus 1 and Bus 12 of the upstream
network, through the corresponding switches S3 and S4. The
subnetworks can also be interconnected by the switch S5. In
turn, Bus 1 and Bus 12 are energized through the corresponding
transformers Tr3 and Tr4, from a 115-kV transmission system.
When S5 is open, the two subnetworks demand the real and reactive powers of (3.1 MW, 2.4 MVAr) and (1.2 MW, 0.8 MVAr),
respectively. As shown in Fig. 8, the part of the system that is
outside the microgrid boundaries is referred to as the “grid.”
Fig. 8 also shows that the microgrid embeds a 4-MVA DR
unit: DR1, and a 2-MVA DR unit: DR2. The two DR units are
connected to Bus 2 and Bus 13, respectively. Each DR unit is
interfaced with the host bus through a corresponding isolation
transformer and a disconnect switch (S1 and S2, respectively,
for DR1 and DR2); the transformers have a solidly grounded
wye winding configuration at their low-voltage sides. The control algorithm of each DR unit (which includes the PWM, signal
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transformation and conditioning, phase-angle extraction, current control, amplitude and frequency regulation, power calculation and droop, adaptive feedforward compensation, and online parameter estimation schemes) is implemented by a Fortran
code that is linked to the PSCAD/EMTDC model of the DR unit.
Parameters of the loads and transmission lines can be found in
[19]. Other system parameters are given in Appendix A.
The case studies demonstrate the DR unit performances in
response to the connection of their corresponding host subnetworks to the grid, despite (intentional) inaccurate synchronization, and in response to the interconnection of the two subnetworks, when both of them are isolated from the grid. More
important, two study cases are dedicated to highlighting the
fact that the proposed feedforward compensation makes the dynamic performance and stability of the DR units insensitive
to the droop coefficients and load/network dynamic properties.
For each case study, the response under the proposed control
is compared with its counterpart under the conventional droopbased control; the latter control strategy is invoked by setting
and
to zero. In the graphs
the auxiliary signals
to follow, the real powers, reactive powers, and currents are expressed in megawatts, megavoltamperes-reactive, and kiloamperes, respectively.
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 26, NO. 3, JULY 2011
Fig. 9. Sample responses of the DR units to the connection of their host subnetworks to the grid, under the conventional control (Case 1).
A. Case 1: Transition to the Grid-Connected Mode Following
Inaccurate Synchronization
In this case, the switches S3, S4, and S5 are initially open (but,
S1 and S2 are closed), the DR units are disabled, and, therefore,
0, the
Subnetwork 1 and Subnetwork 2 are de-energized. At
(see
DR units are turned on, and their respective values of
(7), Section IV) are ramped up from zero to 0.52 kV, in about
35 ms; hence, the (islanded) subnetworks get energized by their
respective DR units. Subsequently, S3 and S4 are closed, at
0.75 s, and the subnetworks are connected to the grid, while,
right before the closure of S3 and S4, the voltages of Bus 2
and Bus 13 have phase displacements of about 24 relative to
the voltages of Bus 1 and Bus 12, respectively; the phase shifts
are introduced intentionally to subject the DR units to a rather
2.5 s, S3 and S4 are opened again
severe disturbance. At
2.5 s
and isolate the subnetworks from the grid. Thus, from
onwards, the two subnetworks and DR units operate under the
same conditions that prevailed from
0 to
0.75 s. Figs.
9 and 10 illustrate the system response to the aforementioned
sequence of events.
Fig. 9 illustrates the waveforms of the real- and reactivepower outputs of the DR units, in addition to those of the - and
-axis components of the terminal current of DR1, under the
conventional droop-based control (hereafter, the conventional
control). Similarly, Fig. 10 illustrates the waveforms of the same
variables, but under the proposed feedforward compensation
(hereafter, the proposed control). A comparison between Figs. 9
and 10 reveals that while the responses of the DR units to the
switching incidents exhibit remarkable transient excursions and
ringings under the conventional control, they are remarkably
smooth and damped under the proposed control, despite the
severity of the disturbances.
Fig. 10. Sample responses of the DR units to the connection of their host subnetworks to the grid, under the proposed control (Case 1).
Fig. 11. Estimated parameters of DR1 under the proposed control (Case 1).
Fig. 11 illustrates the waveforms of the estimated parameters
of DR1 (for instance), and indicates that subsequent to each disturbance incident, they smoothly converge to their steady-state
DELGHAVI AND YAZDANI: ADAPTIVE FEEDFORWARD COMPENSATION FOR STABILITY ENHANCEMENT
Fig. 12. Sample responses of the DR units to the interconnection of the two
subnetworks, under the conventional control (Case 2).
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Fig. 13. Sample responses of the DR units to the interconnection of the two
subnetworks, under the proposed control (Case 2).
values. As the figure shows, the settling values of , , and
are, respectively, 0.0022, 0.0207, and 0.0153, at the end
of the time intervals over which Subnetwork 1 is isolated from
the grid. On the other hand, under the off-grid condition, the
and
are 4.45 kA and 3.29 kA, resettling values of
is
spectively (see Fig. 10), and the steady-state value of
0.475 kV (not shown in the graphs). Substituting for these three
steady-state values in (13)–(15), one calculates , , and as
0.00221, 0.02069, and 0.01529, respectively. It is noted that
the calculated values are very close to the values estimated by
the RLS identification scheme of DR1. It should be emphasized
do not bear clear physical meanings, but are
that , , and
the byproducts of the mathematical formulation set forth in Section IV.
B. Case 2: Network Topological Change in the Islanded Mode
This case study demonstrates the responses of the DR units to
a network topological change, resulting in a power-flow change,
in the islanded mode of operation. In this case, the system continues from the same steady state as that under the operating
conditions of Case 1, that is, Subnetwork 1 and Subnetwork 2
are isolated from the grid (switches S3 and S4 are open) and
independently energized by their respective DR units. At
4.0 s, the switch S5 is closed (see Fig. 8), while, right before
the switching incident, the voltages of Bus 8 and Bus 14 are
phase-displaced by about 43 ; subsequent to the closure of S5,
the aggregate of the loads of Subnetwork 1 and Subnetwork 2 is
shared by the two DR units. Figs. 12 and 13 illustrate the system
response to the disturbance under the conventional and proposed
controls, respectively.
As Fig. 12 indicates, subsequent to the switching incident,
the variables plotted in the figure experience remarkable fluctuations under the conventional control. By contrast, despite the
disturbance severity, the system response is well damped under
the proposed control, as Fig. 13 shows. A comparison between
Figs. 12 and 13 confirms that the proposed control does not alter
the steady-state power sharing regime that would exist under the
conventional control.
Fig. 14. Sample responses of the DR units to stepwise increase in their droop
coefficients, under the conventional control (Case 3).
C. Case 3: Stepwise Increase in Droop Coefficients
As has been analytically shown in [5], a sufficient increase in
the droop coefficients has a destabilizing effect on the conventional control. This case study confirms the conclusion of [5],
and also demonstrates the effectiveness of the proposed control
in maintaining the system stability despite an increase in the
droop coefficients.
In this case, the system continues from the same steady state
as that under the operating conditions of Case 2, that is, Subnetwork 1 and Subnetwork 2 are isolated from the grid, but the
switch S5 is closed; therefore, the two DR units share the aggre6.5 s, the frequency/
gate of the two subnetwork loads. At
real-power droop coefficients of DR1 and DR2 are stepped up,
2.0 to 12.0 (rad/s)/MW, and from
4.0 to 24.0
from
(rad/s)/MW, respectively.
Fig. 14 depicts the responses of the DR units under the
conventional control and indicates that the system becomes
6.5 s. By contrast, as Fig. 15 shows, the
oscillatory after
system remains stable under the proposed control, despite the
disturbance.
1772
Fig. 15. Sample responses of the DR units to stepwise increase in their droop
coefficients, under the proposed control (Case 3).
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 26, NO. 3, JULY 2011
Fig. 17. Sample responses of the DR units to the introduction of an asynchronous machine to Subnetwork 1, under the proposed control (Case 4).
Fig. 16 shows that under the conventional control, the connection of the asynchronous machine to the subnetwork results
in oscillations and instability. However, under the proposed control, the microgrid remains stable, as Fig. 17 indicates. Fig. 17
also shows that, from
6.5 s onwards, the two DR units deliver comparatively smaller (real) powers, due to the power contribution of the asynchronous machine.
VII. CONCLUSION
Fig. 16. Sample responses of the DR units to the introduction of an asynchronous machine to Subnetwork 1, under the conventional control (Case 4).
D. Case 4: Change in Load/Network Properties
As pointed out in the introduction of this paper, the performance and stability of the conventional control depend on the
load/network dynamic properties. This case study supports the
claim and further demonstrates that the proposed control is robust to load/network characteristic variations.
In this case study, the system continues from the same steady
state as that under the operating conditions of Case 2, that is,
Subnetwork 1 and Subnetwork 2 are isolated from the grid,
but connected to each other through S5, and the two DR units
6.5 s,
share the aggregate load of the two subnetworks. At
an asynchronous machine, spun by a mechanical torque of
1.0 per unit at angular speed close to 377 rad/s, gets connected to Bus 6 of Subnetwork 1; the connection is enabled by
a circuit breaker and a voltage-matching transformer (none of
these components are shown in Fig. 8). Parameters of the asynchronous machine are given in Appendix B. Figs. 16 and 17
illustrate the responses of the DR units, under the conventional
and proposed controls, respectively.
This paper proposed an adaptive feedforward compensation
strategy that alters the dynamic coupling between a DR unit
and the host microgrid so that the system stability is made insensitive to droop coefficients and load/network dynamic characteristics. The proposed feedforward compensation preserves
the steady-state effect that the conventional droop mechanism
exhibits and, therefore, does not compromise the steady-state
power sharing regime and voltage/frequency regulation offered
by the conventional droop-based control. The feedforward compensation is adaptive since it is periodically modified according
to the system steady-state operating point which, in turn, is estimated through an online RLS estimation technique. This paper
presented a discrete-time mathematical model and analytical
framework for the proposed feedforward compensation. The effectiveness of the proposed control was demonstrated through
time-domain simulation studies, in the PSCAD/EMTDC software environment, conducted on a detailed switched model of a
sample two-unit microgrid.
APPENDIX A
SYSTEM PARAMETERS
The system parameters are given in Table I. The transfer funcand
are given by
tions
for
for
(41)
(42)
DELGHAVI AND YAZDANI: ADAPTIVE FEEDFORWARD COMPENSATION FOR STABILITY ENHANCEMENT
TABLE I
DR UNITS CIRCUIT AND CONTROL PARAMETERS
TABLE II
ASYNCHRONOUS MACHINE PARAMETERS (CASE 4)
for
for
(43)
(44)
APPENDIX B
ASYNCHRONOUS MACHINE PARAMETERS
Asynchronous machine parameters are given in Table II.
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Mohammad B. Delghavi (S’09) received the M.Sc. degree from the Iran University of Science and Technology (IUST), Tehran, Iran, in 1996 and is currently
pursuing the Ph.D. degree at the University of Western Ontario (UWO), London,
ON, Canada.
Currently, he is a Research Assistant with UWO. His research interests include switching power converters, distributed generation, and microgrids.
Amirnaser Yazdani (M’05–SM’09) received the Ph.D. degree in electrical engineering from University of Toronto, Toronto, ON, Canada, in 2005.
Currently, he is an Assistant Professor with the University of Western Ontario, London, ON, Canada. His research interests include modeling and control
of electronic power converters, renewable electric power systems, distributed
generation and storage, and microgrids.