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1242 IEEE TRANSACTIONS ON SMART GRID, VOL. 8, NO. 3, MAY 2017
Feasible Range and Optimal Value of the
Virtual Impedance for Droop-Based
Control of Microgrids
Xiangyu Wu, Student Member, IEEE, Chen Shen, Senior Member, IEEE, and Reza Iravani, Fellow, IEEE
Abstract—This paper presents a systematic method to deter-
mine the feasible range and optimal value of the virtual
impedance of the droop-based control to enhance a microgrid
system performance with respect to power decoupling, reac-
tive power sharing, system damping, and node voltage profile.
A modified power flow analysis and an augmented small-signal
dynamic model of the droop-based controlled microgrid, con-
sidering the impact of the virtual impedance, are developed.
Subsequently, based on the developed methods, the feasible range
of the virtual impedance, which can satisfy all the system per-
formances requirements, is determined and presented. Based on
a particle swarm optimization technique, an optimization pro-
cess is introduced to select a virtual impedance value within
the feasible range to achieve the overall optimal microgrid per-
formance. Finally, simulation results in the PSCAD/EMTDC
platform are provided to validate the feasibility and effectiveness
of the proposed methods.
Index Terms—Droop-control, microgrid, optimal virtual
impedance, power decoupling, power flow, power sharing, small
signal stability.
I. INTRODUCTION
WITH THE increasing degree of utilization of distributed
generation (DG) units, the concept of microgrid to
facilitate grid integration of various DG units has emerged
as a viable technical and economical option. The droop-based
control method has been identified as a viable approach for
control and operation of DG units within a microgrid [1]–[3].
However, the basic droop-based control exhibits the following
limitations:
•coupling between active power and reactive power com-
ponents under low X/R ratios [4]–[8],
•improper reactive power sharing among DG units due to
the line voltage drop [9], [10],
Manuscript received June 15, 2015; revised October 5, 2015; accepted
January 12, 2016. Date of publication February 3, 2016; date of current ver-
sion April 19, 2017. This work was supported in part by the National High
Technology Research and Development Program of China (863 Program)
under Grant 2012AA050217, and in part by the Foundation for Innovative
Research Groups of the National Natural Science Foundation of China under
Grant 51321005. Paper no. TSG-00682-2015.
X. Wu and C. Shen are with the Department of Electrical
Engineering, Tsinghua University, Beijing 100084, China (e-mail:
wuxiangyu639@163.com;shenchen@mail.tsinghua.edu.cn).
R. Iravani is with the Department of Electrical and Computer
Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail:
iravani@ecf.utoronto.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/TSG.2016.2519454
•dynamic interactions among DG units and poor damping
of dynamic modes when DG units operate as voltage-
controlled units [11].
One approach to address the above limitations is to embed
“virtual impedance” in the control system of each DG unit.
The virtual impedance can (i) provide power decoupling by
artificially changing the X/R ratio [4]–[8], (ii) impose accurate
reactive power sharing through dynamic regulation of voltage
drop [5], [9], [10], and (iii) enhance dampings of the oscilla-
tory modes [11]. However, if the virtual impedance is deter-
mined based on any of the above performance requirement,
e.g., active/reactive power decoupling, the other requirements
may not be satisfied or even degraded. Furthermore, due to the
voltage drop associated with the virtual impedance, the node
voltages may violate their limits. Thus the virtual impedance
must be selected from a feasible region that meets all the
requirements.
Reference [11] introduces a feasible range for the virtual
impedance to satisfy all performance criteria. The limitations
of [11] are as follows.
•The line impedance and the virtual impedance are com-
bined and treated as one entity while they have different
impacts on the node voltages, power losses, power decou-
pling, and system damping. This is specially the case
when the real part of the virtual impedance is negative.
•Node voltage limits are not taken into account.
•Proper/accurate reactive power sharing is not considered.
•Only identical DG units is assumed.
Although a virtual impedance inside the feasible range can
guarantee that the system performance criteria are satisfied,
still there is a need to determine the virtual impedance (within
the feasible range) that can provide the overall system optimal
performance. The notion of optimal droop-based control has
been investigated [12]–[14], however, to the best of our knowl-
edge no systematic methodology to design the optimal virtual
impedance to achieve the overall optimal system performance
has been reported.
This paper overcomes the limitation of [11] by proposing
a systematic approach to identify the feasible range for the
virtual impedances of a droop-based DG controllers to simul-
taneously satisfy system performance criteria with respect to
power decoupling, reactive power sharing, system damping,
and node voltage profile. This paper also presents a particle
swarm optimization (PSO) approach to identify the opti-
mal virtual impedance from the feasible range to optimally
1949-3053 c
2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL 1243
Fig. 1. Virtual impedance implementation structure.
achieve all the performance criteria. The proposed method
is independent of the DG parameters and the microgrid
topology, which is a salient improvement as compared with
those of [11] and [15]. This paper also provides a modified
power flow analysis that includes the impact of the virtual
impedance and overcomes the shortcomings of the existing
power flow approaches [12], [16], [17]. This paper presents
analytical and time-domain simulation results to verify the
proposed concepts.
The rest of the paper is organized as follows. Section II
presents a general overview of the subject. Section III
describes the proposed power flow analysis and highlights its
features. Section IV presents a small-signal dynamic model
of a droop-based controlled microgrid which utilizes the
virtual impedance. The feasible range of the virtual impedance
is illustrated in Section V. Optimal design of the virtual
impedance value is presented in Section VI. Case studies and
simulation results are provided in Section VII. Conclusions
are stated in Section VIII.
II. VIRTUAL IMPEDANCE IMPLEMENTATION
The investigated virtual impedance concept for a droop-
based control [6] is as shown in Fig. 1for DGi.DGiis
represented by a dc voltage source, a voltage source inverter,
filter LfCf, and inductance Lc, and connected to the point
of common coupling (PCC) through a feeder line. DGi
includes an internal dq-current controller through which pro-
vides control over its terminal voltage, voi.Fig.2shows
a block-representation of the power controller of Fig. 1. Based
on Fig. 2, the P/f and Q/V droop equations for DGiare
ωi=ω0−miPi−P∗
i,(1)
|Eoi|=V0−niQi,(2)
where miand niare active and reactive power droop coef-
ficients, respectively, ωiis the frequency of DGi,ω0is the
rated system frequency, V0is the no-load voltage for reac-
tive power droop control, E0iis the voltage reference from the
power controller, Piand Qiare the output active and reactive
power components of DGiat ω0, respectively, and P∗
iis the
rated active power. miand niare determined by the following
Fig. 2. Details of power controller block.
Fig. 3. Equivalent circuit for virtual impedance implementation.
Fig. 4. Details of voltage controller block.
equations based on the concept of [6]
mi=ω0−ωmin
Pimax −P∗
i
,(3-1)
ni=Vn−Vmin
Qimax ,(3-2)
where ωmin and Vmin are the minimum allowable operating
frequency and voltage magnitude of the microgrid, Vnis the
rated voltage magnitude, and Pimax and Qimax are the maximal
output active and reactive power components of DGi.
Fig. 3illustrates an equivalent circuit for realization of the
virtual impedance Z0i=R0i+jX0i.FromFig.3
v∗
oi =Eoi −ioiZ0i,(4)
where v∗
oi is the voltage reference for voi.Fig.4shows the
details of the voltage control block diagram of Fig. 1. Details
of the current controller are given in [18].
III. POWER FLOW ANALYSIS INCLUDING
VIRTUAL IMPEDANCE
References [12] and [16] provide a power flow method for
an islanded microgrid based on the droop control approach
without including the effect of the virtual impedance on the
power flow. This shortcoming is addressed in [17], however,
it considers Eoi,Fig.3, as the voltage reference for voi and
not v∗
oi in the power flow model. Thus, it does not accurately
1244 IEEE TRANSACTIONS ON SMART GRID, VOL. 8, NO. 3, MAY 2017
represent the virtual impedance. This paper also overcomes
this limitation through taking v∗
oi as the voltage reference for
voi of DGiand includes the effect of the virtual impedance on
the power flow (hereinafter referred to as “virtual impedance
based power flow”).
Consider the two types of nodes in the virtual impedance
power flow, i.e., PQ nodes and DG nodes. The DG nodes
utilize the control method of Fig. 1. The loads and the net-
work nodes are represented by PQ nodes. For the PQ node
number j,
P∗
j−Pj=0,(5)
Q∗
j−Qj=0,(6)
where P∗
jand Q∗
jare the pre-specified power components,
and Pjand Qjare the injected power components and
the unknowns are the real and imaginary voltage compo-
nents (VlD j and VlQ j ) of the node. For DGi, assuming that
voi accurately tracks its reference v∗
oi in the steady state,
from (1), (2), (4), we deduce
ω−ω0+miPi−P∗
i=0,(7)
|Eoi|−V0+niQi=0,(8)
Eoi −voi −ioiZ0i=0.(9)
Let δidenote the angle of Eoi and node 1 is taken as the
reference node by setting δ1=0. voi and ioi can be expressed
as voDi +jvoQi and ioDi +jioQi. Then, (9) can be rewritten as
|Eoi|cos δi−voDi −ioDiR0i+ioQiX0i=0,(10)
|Eoi|sin δi−voQi −ioQiR0i−ioDiX0i=0.(11)
There are 4 equations for each DG node, and the correspond-
ing 4 unknowns are |Eoi|,δi,voDi and voQi. There are only
3 unknowns for node 1 since δ1is set to 0.
The power flow can be calculated from (5)-(8),
(10) and (11). If the number of PQ nodes and DG nodes
are NPQ and NDG, then the number of power flow equations
is 2NPQ +4NDG. Considering that system frequency ωis
also an unknown, then the total number of unknowns is
also 2NPQ +4NDG. The unknowns are given by xas in the
Appendix. Assume Z0denotes the virtual resistances R0iand
virtual reactances X0iof all the DG units, i.e.,
Z0=R01, ... R0NDG ,X01,...X0NDG .(12)
Then the power flow equations, including virtual impedances,
can be expressed as
g(x,Z0)=0.(13)
IV. SMALL SIGNAL DYNAMIC MODEL OF A
DROOP-BASED CONTROLLED MICROGRID
INCLUDING VIRTUAL IMPEDANCE
The small-signal dynamic model presented in this sec-
tion is based on the model of [18]. However, the effect of
the virtual impedance is not included in [18]. In this paper,
the small-signal model of a droop-based controlled microgrid
is augmented to take into account the effect of the virtual
impedance. Due to space limitations, developments of [18]are
not repeated here.
Based on (2), in the local dq frame of DGi, we deduce
Eodi =V0−niQi,Eoqi =0.(14)
Similarly, (4) in the local dq frame can be expressed as
v∗
odi =Eodi −R0iiodi +X0iioqi
v∗
oqi =Eoqi −R0iioqi −X0iiodi ,(15)
and linearized as
v∗
odi
v∗
oqi =CPvi⎡
⎣
δi
Pi
Qi
⎤
⎦+EPvi⎡
⎣
ildqi
vodqi
iodqi
⎤
⎦,(16)
where CPvi and EPvi are
CPvi =00−ni
00 0
,
EPvi =0000−R0iX0i
0000−X0i−R0i.(17)
Considering the dynamics of the power controller, voltage
controller, current controller and LCL filter, the small signal
dynamic model of DGiunder the control method of Fig. 1is
˙
Xinvi=Ainvi[Xinvi]+BinvivbDQi +Biωcom[ωcom],
(18)
where Xinvi includes the state variables of DGi, and details
of Ainvi,Binvi and Biωcom are given in the Appendix. The
last column of Ainvi is different from that of [18] due to the
introduction of EPvi, which reflects the impact of the virtual
impedance on the system small-signal dynamic model. The
overall model of the islanded microgrid can be expressed as:
⎡
⎣
˙
Xinv
˙
ilineDQ
˙
iloadDQ
⎤
⎦=Amg⎡
⎣
Xinv
ilineDQ
iloadDQ
⎤
⎦,(19)
where Xinv,ilineDQ and iloadDQ are the state variables of DGs,
lines and loads, respectively, and Amg is the system state
matrix. The system steady-state information, including the
virtual impedance, is embedded in Amg. Therefore, the vir-
tual impedance based power flow should be calculated first to
construct Amg and perform eigenvalue analysis.
V. FEASIBLE RANGE OF VIRTUAL IMPEDANCE
Based on the virtual impedance based power flow model
and the small-signal dynamic model of the previous sections,
this section proposes a methodology to determine the feasible
range of the virtual impedance which can satisfy all the system
performance requirements. The feasible range is determined by
a set of constraints which consider node voltage limits, power
decoupling, system damping and reactive power sharing as
follows.
A. Node Voltage Limits
All node voltages in a microgrid should be maintained
within a permissible range. Therefore, the lowest (highest)
WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL 1245
node voltage μLV (μHV) should be larger (smaller) than the
minimum (maximum) value μ0_LV (μ0_HV), i.e.,
μLV ≥μ0_LV ,(20)
μHV ≤μ0_HV ,(21)
where (20) and (21) specify lower and upper voltage limit.
B. Power Decoupling Constraint
Under low X/R ratios, the P/f and Q/V droop-based con-
trols can exhibit strong coupling between P and Q [5], [6].
This paper provides a solution to this issue by introducing the
virtual impedance to the droop control system. By emulating
the effect of a physical impedance based on considering the
voltage drop on the virtual impedance in the droop control
(Fig. 1and Fig. 3), the virtual impedance introduces a pre-
dominantly inductive impedance to increase the X/R ratio and
effectively decouple P and Q [11]. To maintain a satisfactory
power decoupling performance, the virtual impedance must
be confined by a power decoupling constraint. The power
decoupling constraint presented in this paper is based on the
concept of [11]. Assume φoi is the angle difference between
Eoi and Uin Fig. 3. The measure of decoupling of voltage
and angle for DGiare [11]
dec_Eoi =
∂Qi
∂Eoi
∂Pi
∂Eoi
,(22)
dec_φoi =
∂Pi
∂φoi
∂Qi
∂φoi
.(23)
Large dec_Eoi and dec_φoi provide a higher decoupling value
of voltage and angle corresponding to Piand Qi, respectively.
The minimum value of voltage and angle decoupling of all
the DGs in the system, μdec, is defined as
μdec =mindec_Eo1,dec_φo1,...dec_EoNDG ,dec_φoNDG .
(24)
If μdec is selected above the minimum allowable value of
μ0_dec, i.e.,
μdec ≥μ0_dec,(25)
it can guarantee that all the voltage and angle decoupling val-
ues are above μ0_dec.If(25) is satisfied with the implemented
virtual impedances, then a satisfactory power decoupling per-
formance is achieved. μ0_dec is usually selected as 1 to ensure
that for all the DGs: 1) the coupling between Piand φoi is
stronger than that of Qiand φoi; 2) the coupling between Qi
and Eoi is stronger than that of Piand Eoi.
C. System Damping Constraint
The virtual impedance of the droop-based control approach
of Fig. 1effectively increases the electrical distance between
DGiand the other DG units and thus enhances the damping
of the dynamic interaction modes of the DG units.
Let eigkand ξkbe the kth mode and its damping ratio
respectively. Assume that the number of modes is Neand only
the modes which have real parts larger than σ(least damped
modes) are considered. σis selected based on multiple case
studies and parameters of the study system. For the study
system in Section VII, σis selected as -100. μdamp1 is the
minimum damping ratio of the modes which have real parts
larger than σ, i.e.,
μdamp1=min(ξ1,...ξ
i,...ξ
NA), i∈A
A=j|Reeigj>σ,j=1,...Ne,(26)
where Ais the set of the modes with real parts larger than σ,
and NA is the number of elements in A. Assume ξ(eigkA)=
μdamp1, and kA ∈A. Then μdamp2 is defined as the absolute
value of the real part of the eigenvalue with the minimum
damping ratio, i.e.,
μdamp2=|Re(eigkA)|.(27)
μdamp1 and μdamp2 should be larger than their respective min-
imum allowable values μ0_damp1 and μ0_damp2 to ensure that
the system has adequate damping, i.e.,
μdamp1≥μ0_damp1,(28-a)
μdamp2≥μ0_damp2.(28-b)
μ0_damp1 and μ0_damp2 are determined based on the engineer-
ing judgement and knowledge of the system. Equation (28)
constitutes the system damping constraints.
D. Reactive Power Sharing Constraint
The reason for undesirable reactive power sharing is the
different voltage drops on the line impedances of the DG
units [6]. Proper selection of the virtual impedance can adjust
the effective voltage drop on each line impedance and its cor-
responding virtual impedance to achieve the desired reactive
power sharing [19]. For the study system in Fig. 6, without
considering the local load of each DG unit, i.e., Load1 to
Load4, the DG unit equivalent impedance is designed in
inverse proportion to the DG maximal power to eliminate the
reactive power sharing error [19], i.e.,
Re1P1 max =Re2P2 max =Re3P3 max =Re4P4max ,(29)
Xe1Q1 max =Xe2Q2 max =Xe3Q3 max =Xe4Q4max ,(30)
where Rei (i=1,2,3,4) is the equivalent resistance of DGi, and
Xei (i=1,2,3,4) is the equivalent reactance of DGi.TheDG
equivalent impedance in (29) and (30) consists of three series
parts, i.e.,
Rei +jXei =(RLinei +jXLinei)+jωLc+(R0i+jX0i),(31)
where RLinei and XLinei are the feeder resistance and reac-
tance of the ith line in Fig. 6.From(31), the proper vir-
tual impedance can adjust the DG equivalent impedance to
satisfy (29) and (30). Therefore, the reactive power sharing
error can be eliminated.
Assume that the desired reactive power sharing ratios of the
DGs are:
Q1:Q2:...:QNDG =kq1:kq2:...:kqNDG .(32)
Then we define the reactive power sharing error μQS as
μQS =
NDG
i=1
Q1−kq1
kqiQi
.(33)
1246 IEEE TRANSACTIONS ON SMART GRID, VOL. 8, NO. 3, MAY 2017
μQS is the sum of the reactive power sharing error between
DG1and the other DG units. A smaller μQS indicates that
reactive power sharing is more accurate. Therefore, the reac-
tive power sharing constraint should impose μQS to be smaller
than a maximum permissible value μ0_QS, i.e.,
μQS ≤μ0_QS.(34)
μ0_QS can be approximated based on the empirical expression:
μ0_QS =0.25Q1 max(NDG −1), (35)
where Q1max is the maximum output reactive power of DG1.
Equation (35) indicates that the average reactive power shar-
ing error of DG1and DGiis limited to 25% of Q1max.If(34)
is satisfied with the implemented virtual impedances, then
a satisfactory reactive power sharing performance is achieved.
Equations (20), (21), (25), (28) and (34) describe the virtual
impedance constraints with respect to the defined performance
criteria. The identification of the feasible range is based on
a point-by-point approach [20].
VI. OPTIMAL DESIGN OF VIRTUAL IMPEDANCE
The feasible range of the virtual impedances that meet
the imposed performance constraints was specified in
Section V. This section provides an approach to select fea-
sible virtual impedances that also ensure the overall optimal
performance of the system.
A. System Performance Evaluation
To assess the overall system performance, the comprehen-
sive assessment index Jis defined as
J=c1fvol +c2fdec +c3fdamp +c4fQS,(36)
where fvol,f
dec,f
damp and fQS are the assessment indices with
respect to the node voltage, power decoupling, system damp-
ing, and reactive power sharing performances, respectively,
and ci(i=1...4) is the corresponding weighting factor subject
to 4
i=1
ci=1. In addition, fvol and fdamp are each divided into
two sub-indices, i.e.,
fvol =c11 fLV +c12 fHV ,(37)
fdamp =c31 fdamp1+c32 fdamp2,(38)
where c11,c12,c31 and c32 are the sub-weighting factors, and
c11 +c12 =1 and c31 +c32 =1.
Variables μLV ,μHV ,μdec,μdamp1,μdamp2 and μQS,pre-
sented in Section V, can be used to evaluate the corresponding
system performances. However, their dimensions are not the
same, and thus index Jof (36) cannot be readily used.
Therefore, fLV ,fHV ,fdec,fdamp1,fdamp2and fQS are used to
normalize μLV ,μHV,μdec,μdamp1,μdamp2 and μQS within
the ranges of 0 to 1, respectively. After normalization, these
assessment indices can be added, including appropriate weigh-
ing factors, to calculate J. Note that Jalso varies within 0 to 1.
Fig. 5. The two kinds of normalization function.
Two normalization functions fx1and fx2are introduced as
fx1=1−e−1
τ(μ−μ0),μ ≥μ0,(39)
fx2=1−e1
τ(μ−μ0),μ ≤μ0,(40)
where μ0and τdetermine each function. Fig. 5graphically
shows fx1and fx2. A pre-specified point (μs, 0.95) on the plots
of Fig. 5is used to determine τfor (39) and (40), i.e.,
fx1:τ=μ0−μs
ln 0.05 ,fx2:τ=μs−μ0
ln 0.05 .(41)
Higher μLV ,μdec,μdamp1and μdamp2provide higher system
performance. Fig. 5(a) also shows that a higher μprovides
a higher fx1. Therefore, fx1is appropriate for normalizing μLV ,
μdec,μdamp1and μdamp2. Similarly, lower μHV and μQS pro-
vide higher system performance. Fig. 5(b) shows that a lower
μprovides a higher fx2. Therefore, fx2is appropriate for nor-
malizing μHV and μQS. By substituting μLV ,μ0_LV and τLV
into fx1,fLV can be obtained from (42-a) and similarly other
indices are deduced as
fLV =1−e−1
τLV (μLV −μ0_LV ),μ
LV ≥μ0_LV ,(42-a)
fHV =1−e1
τHV (μHV −μ0_HV ),μ
HV ≤μ0_HV ,(42-b)
fdec =1−e−1
τdec (μdec−μ0_dec ),μ
dec ≥μ0_dec,(42-c)
fdamp1=1−e−1
τdamp1(μdamp1−μ0_damp1),
μdamp1≥μ0_damp1,(42-d)
fdamp2=1−e−1
τdamp2(μdamp2−μ0_damp2),
μdamp2≥μ0_damp2,(42-e)
fQS =1−e1
τQS (μQS−μ0_QS ),μ
QS ≤μ0_QS.(42-f)
From (42) one observes that the definitions of assessment
indices are consistent with the constraints of the feasible range
of the virtual impedance. This ensures that the comprehensive
assessment is defined inside the feasible range.
B. Virtual Impedance Optimization
To optimize virtual resistances and virtual reactances of
all droop-controlled DGs of an islanded microgrid, index J
must be maximized. Since Jrepresents the overall system per-
formance, the maximum Jindicates that the overall system
WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL 1247
Fig. 6. Schematic diagram of the studied microgrid.
performance is optimal within the feasible range. Therefore,
the optimization problem is formulated as
Max J =c1fvol +c2fdec +c3fdamp +c4fQS,
s.t.
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
g(x,Z0)=0,
μLV ≥μ0_LV ,
μHV ≤μ0_HV ,
μdec ≥μ0_dec,
μdamp1≥μ0_damp1,
μdamp2≥μ0_damp2,
μQS ≤μ0_QS,
(43)
where g(x,Z0)=0 is the power flow constraint. The vir-
tual impedance optimization problem is a non-linear and
non-convex optimization problem which also includes eigen-
values constraints. In contrast to the classical gradient-based
optimization methods, the heuristic optimization technique
provides an effective solution method for this class of prob-
lems. In this study, a particle swarm optimization (PSO) [21]
is adopted to solve the virtual impedance optimization prob-
lem. PSO has been widely used as one of the promising
optimization technique due to its implementation simplicity
and computational efficiency. Compared with other heuris-
tic optimization techniques, such as genetic algorithm and
ant colony optimization, PSO provides a flexible and well-
balanced mechanism to enhance the global and local explo-
ration abilities [21].
VII. STUDY RESULTS AND DISCUSSION
A. Study System
A single-line diagram of a 9-bus studied microgrid is pre-
sented in Fig. 6[22]. It includes four DG units and five loads
which are connected to a 0.38kV and 50Hz distribution sys-
tem. Each DG unit is connected to its load bus through the
coupling inductance Lc. Load1∼Load4 are the local loads of
DG units. Line1∼Line4 are the feeder lines between each
DG unit and the PCC bus. Load5 is located at the PCC bus.
The microgrid is connected to the utility grid through a cir-
cuit breaker (CB) and a 10kV/0.38kV /Ygtransformer. The
TAB L E I
ELECTRICAL PARAMETERS OF THE MICROGRID
TAB L E I I
CONTROL PARAMETERS OF DG UNITS
microgrid is operated in the islanded mode, i.e., the CB is
open. The structure of each DG unit is as that of Fig. 1. Each
three-phase load is represented by a series RL branch at each
phase. Each feeder line is modelled by a lumped, series RL
branch at each phase. It should be noted that when there are
other types of DG units in the microgrid, e.g., solar_PV, wind
power units or diesels, their impacts on the system perfor-
mances can be considered by incorporating their steady-state
and dynamic models into the power flow analysis (Section III)
and the small-signal dynamic stability analysis (Section IV).
However, the basic concept of the feasible range and optimal
value of the virtual impedance remains the same.
Electrical and control parameters of the system are shown
in Table Iand Table II, respectively. The modulation strategy
of the inverter is a SPWM with the modulation frequency
of 3500Hz.
Table Iindicates that X/R ratios of the lines are small [23]
and thus the output active and reactive power components of
the DG units are tightly coupled in the absence of the vir-
tual impedance. From Table II one observes that the ratios of
the four DG capacities are 4:6:5:3. Thus the ratios of droop
coefficients of the DGs are also selected as 4:6:5:3, for both
the active and reactive power components. Kpv and Kiv are the
proportional and integral parameters of the voltage controller
presented in Fig. 4, and Kpc and Kic are the proportional and
integral parameters of the current controller in Fig. 1[18]. The
difference of voltage and current control parameters of the DG
units reflect their different response times.
Parameters of normalization functions in (42) for the study
system of Fig. 6are shown in Table III. In Table III,μ0and μs
in fLV and fHV are per unit values and μ0is also the constraint
value of the feasible range of the virtual impedance.
B. Feasible Range of Virtual Impedance
The microgrid operating point, for the study results of this
subsection and the following Sections VII-C and VII-D, cor-
responds to the maximum active power load scenario, Table I,
since it constitutes the most demanding operational condition
1248 IEEE TRANSACTIONS ON SMART GRID, VOL. 8, NO. 3, MAY 2017
TABLE III
PARAMETERS OF NORMALIZATION FUNCTIONS IN (36)
under different active power loads. To facilitate visualization
of the feasible range and the optimal virtual impedance, the
results of Fig. 7are deduced based on the assumption that
the virtual impedances of all DG units are identical and all
feeder lines have the same length. It should be noted that the
developments of Section III to Section VI are not subject to
these assumptions.
Fig. 7(a) shows the feasible range of the virtual impedance
when all lines are 300m long. The real (imaginary) part of
the virtual impedance are specified on the R0-axis (X0-axis)
of Fig. 7(a) and the boundaries corresponding to voltage lim-
its (20), (21), power decoupling (25), system damping (28),
and reactive power sharing (34) are also plotted on the R0X0
plane. The virtual impedance inside the feasible range can
also satisfy the system performance criteria with respect to
the node voltage profile, system damping, power decoupling,
and reactive power sharing. Fig. 7(a) concludes:
•Further away from one boundary line indicates a “better”
performance with respect to the corresponding criterion.
•The negative virtual resistance facilitates power decou-
pling, reactive power sharing, and compliance with the
lower voltage limit. However, it increases the DG voltage
based on (4). The upper voltage boundary may influence
the feasible range, specially subject to a large negative
virtual resistance.
Fig. 7(b) and (c) show the feasible ranges of the virtual
impedance when the lengths of feeder lines all are either 150m
or 50m, respectively. Comparison of Fig. 7(a), (b) and (c) indi-
cates that when all the line lengths become longer, the feasible
range of the virtual reactance does not significantly vary, while
the feasible value of the virtual resistance becomes smaller.
This is because a longer feeder line length needs a smaller
and even negative virtual resistance to mitigate the effect of
the line resistance.
Fig. 8corresponds to the scenario when the lengths of
line1 to line4 are selected at 150m, 300m, 225m and 50m,
and the virtual impedances of all DG units remain identical.
Fig. 8indicates that there is no feasible region for the virtual
impedance.
C. Optimal Virtual Impedance
1) Equal Feeder Line Lengths and Equal Virtual
Impedances: Corresponding to equal feeder line lengths at
either 50m, 150m or 300m, i.e., cases of Fig. 7(c), (b) and (a),
Table IV shows the real- and imaginary- part of the optimal
virtual impedance and index J(43), and Table Vshows
the associated weighting factors (43). The weighting factors
Fig. 7. Feasible range of virtual impedance with equal feeder line length.
TAB L E I V
OPTIMAL VIRTUAL IMPEDANCE VALUE IN EQUAL FEEDER LINE LENGTH
are determined based on the subjective preference of the
microgrid operator. The locations of the optimal virtual
impedances are identified by “O” Fig. 7.
2) Unequal Virtual Impedances: This section in contrast to
the previous section determines the optimal virtual impedances
of the DG units assuming that they are not necessarily equal.
Table VI shows four different feeder line length scenarios
WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL 1249
Fig. 8. Feasible range of virtual impedance with unequal feeder line length.
TAB L E V
WEIGHTING FACTORS OF OBJECTIVE FUNCTION
TAB L E V I
FOUR TYPICAL LINE LENGTH SCENARIOS
TAB L E V I I
WEIGHTING FACTORS OF OBJECTIVE FUNCTION
that are considered and Table VII shows the weighting fac-
tors of the optimization cost function (43). The optimal virtual
impedances, corresponding to the four scenarios of Table VI
are given in Table VIII. The values of index J, Table VIII,
indicate that in all cases Jis larger than 0.91, which reveals
the overall system performance is highly desirable. Table IX
(Table X) provide values of μand fxfrom (42-a) to (42-f)
for scenario 2 of Table VIII, with (without) considering the
optimal virtual impedance. Table IX indicates that (i) the sys-
tem performance meets all the required indices, (ii) the error
of reactive power sharing is 0.999kVar and corresponds to
a high degree of reactive power sharing, and (iii) the power
decoupling degree, μdec, is 1.976 and indicates a high degree
of power decoupling. Table Xindicates that without the use
of the virtual impedance, the system does not meet the per-
formance requirements in term of power decoupling, system
damping and reactive power sharing.
D. Simulation Results
Time-domain simulation studies in the PSCAD/EMTDC
platform are carried out to verify the results of the proposed
virtual impedance design approach.
Table XI compares the power flow results obtained from
PSCAD and the proposed power flow analysis for the case
TABLE VIII
OPTIMIZATION RESULTS IN DIFFERENT LINE LENGTH SCENARIOS
TAB L E I X
OPTIMIZATION RESULTS OF SCENARIO 2WITH
OPTIMAL VIRTUAL IMPEDANCE
TAB L E X
RESULTS OF SCENARIO 2WITHOUT VIRTUAL IMPEDANCE
TAB L E X I
POWER FLOW RESULTS FOR SCENARIO 2OF TABLE VIII
of Scenario 2 of Table VIII. Table XI shows that the maxi-
mum deviation of the two methods, associated with the real
and imaginary components of all the nodes voltages, are
0.14% and 1.28%, respectively. The good agreement between
the results demonstrates the accuracy and the validity of the
proposed approach.
Fig. 9compares the output real power components of the
DG units obtained from PSCAD (solid line) and the pro-
posed small-signal dynamic model (dashed line, index “lin”)
for the case of Scenario 2 of Table VIII when the load power
of load5 at time 0.5s is changed from 145kW+10kVar to
165kW+10kVar. The good agreement between the results also
demonstrates the accuracy and the validity of the proposed
small-signal dynamic model.
1250 IEEE TRANSACTIONS ON SMART GRID, VOL. 8, NO. 3, MAY 2017
Fig. 9. Output active power of the four DG units for Scenario 2 of Table VIII.
Fig. 10. Output active power of the DG units.
Fig. 11. Lowest node voltage of the system.
Fig. 10 (a) and (b) show the PSCAD-based simulation
results when the optimal virtual impedance (point “O” of
Fig. 7(b)) and a non-optimal virtual impedance (point A of
Fig. 7(b)) which does not satisfy the system damping crite-
rion are used in the control of DG units. Comparison of the
corresponding results reveals the damping effect of the opti-
mal virtual impedance on an oscillatory mode of the system
that results in fluctuation of real-power components.
Fig. 11 shows the PCC bus voltage (lowest node voltage) for
the study system when the virtual impedance at time t=2.5s
is changed from its optimal value (point “O” on Fig. 7(b)) to
a non-optimal value (point B on Fig. 7(b)) which does not
satisfy the lower voltage limit. Fig. 11 shows that the voltage
decreases to 0.914, which is lower than the allowable value
of 0.93. This is consistent with the position of point B in
Fig. 7(b).
Fig. 12 shows the simulation results for the case study of
Scenario 2 of Table VIII, when a 2.5kW+20kVar shunt reactor
is connected to the PCC bus at time 1s. The total reactive
power load of the system increases by 40%, from 50kVar to
70kVar, after the shunt reactor connection.
Fig. 12. Simulation results when a 2.5kW+20kVar shunt reactor is connected
to the PCC bus at time 1s.
Fig. 12 (a) shows that the output active power components
of the DG units almost remain unchanged. Fig. 12 (b) shows
that the output reactive power components of the DG units
increase to 15.1kVar, 23.0kVar, 18.8kVar and 10.3kVar respec-
tively due to the extra reactive power consumption by the shunt
reactor. The error of reactive power sharing increases from
0.999kVar (Table IX) to 1.193kVar. Fig. 12(c) shows that the
PCC bus voltage decreases to 0.96 per unit due to the connec-
tion of the shunt reactor. The results of Fig. 12 indicate that the
overall system performance is degraded after the shunt reac-
tor connection. However, the system performance satisfies the
node voltage limits (20) and the reactive power sharing con-
straint (34). Since the operating condition is changed, a new
optimal system performance in the new operating condition
can be obtained by incorporating the shunt reactor model
into the system model and then updating the optimal virtual
impedance.
VIII. CONCLUSION
This paper presents a methodology to (i) specify the fea-
sible range of the virtual impedance and (ii) identify the
virtual impedance that provides the overall optimal perfor-
mance for a microgrid. This paper also provides power flow
and linearized dynamics models of the microgrid that incor-
porates the impacts of the virtual impedance. The analytical
and time-domain simulation study results conclude that:
•The virtual impedance has an impact on the power
flow results. It can increase or decrease the node voltages,
depending on the value of the virtual impedance.
•The virtual impedance can enhance the damping of
oscillatory modes through virtually increasing the electrical
distance between one DG unit and the other DG units.
•The feasible range of the virtual reactance is not highly
sensitive to the variation of the feeder lengths, whereas the
feasible value of the virtual resistance becomes smaller as the
feeder lengths increase.
WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL 1251
•A feasible range for the virtual impedance may not exist
if the lengths of feeders within the microgrid are significantly
different.
APPENDIX
Matrices Ainvi,Binvi and Biwcom of (18)are
Ainvi =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
APi 0
BV1iCPvi 0
BC1iDV1iCPvi BC1iCVi
BLCL1iDC1iDV1iCPvi
+BLCL2iT−1
Vi 00
+BLCL3iCPwi
BLCL1iDC1iCVi
0BPi
0BV1iEPVi +BV2i
0BC1iDV1iEPVi +BC1iDV2i+BC2i
BLCL1iCCi ALCLi +BLCL1iDC1iDV1iEPvi
+BLCL1i(DC1iDV2i+DC2i)
⎤
⎥
⎥
⎥
⎥
⎦
,
(A1)
Binvi =⎡
⎢
⎢
⎣
0
0
0
BLCL2iT−1
S
⎤
⎥
⎥
⎦
,Biωcom=⎡
⎢
⎢
⎣
BPωcom
0
0
0
⎤
⎥
⎥
⎦
.(A2)
Vector xin (13)is
x=vlD1, ... vlDNPQ ,vlQ1, ... vlQNPQ ,
|Eo1|, ...
EoNDG
,δ
2,...δ
NDG ,
voD1, ... voDNDG ,voQ1,...voQNDG ,ω
.
(A3)
REFERENCES
[1] A. H. Etemadi, E. J. Davison, and R. Iravani, “A generalized
decentralized robust control of islanded microgrids,” IEEE Trans.
Power Syst., vol. 29, no. 6, pp. 3102–3113, Nov. 2014.
[2] D. E. Olivares et al., “Trends in microgrid control,” IEEE Trans. Smart
Grid, vol. 5, no. 4, pp. 1905–1919, Jul. 2014.
[3] R. Majumder, “Some aspects of stability in microgrids,” IEEE Trans.
Power Syst., vol. 28, no. 3, pp. 3243–3252, Aug. 2013.
[4] J. M. Guerrero, J. Matas, L. G. de Vicuña, M. Castilla, and J. Miret,
“Wireless-control strategy for parallel operation of distributed-generation
inverters,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1461–1470,
Oct. 2006.
[5] J. M. Guerrero, L. G. de Vicuna, J. Matas, M. Castilla, and J. Miret,
“Output impedance design of parallel-connected UPS inverters with
wireless load-sharing control,” IEEE Trans. Ind. Electron., vol. 52, no. 4,
pp. 1126–1135, Aug. 2005.
[6] Y. W. Li and C.-N. Kao, “An accurate power control strategy for
power-electronics-interfaced distributed generation units operating in
a low-voltage multibus microgrid,” IEEE Trans. Power Electron., vol. 24,
no. 12, pp. 2977–2988, Dec. 2009.
[7] W. Yao, M. Chen, J. Matas, J. M. Guerrero, and Z.-M. Qian, “Design
and analysis of the droop control method for parallel inverters consider-
ing the impact of the complex impedance on the power sharing,” IEEE
Trans. Ind. Electron., vol. 58, no. 2, pp. 576–588, Feb. 2011.
[8] J. M. Guerrero, J. Matas, L. G. de Vicuna, M. Castilla, and J. Miret,
“Decentralized control for parallel operation of distributed generation
inverters using resistive output impedance,” IEEE Trans. Ind. Electron.,
vol. 54, no. 2, pp. 994–1004, Apr. 2007.
[9] J. W. He, Y. W. Li, and F. Blaabjerg, “An enhanced islanding microgrid
reactive power, imbalance power, and harmonic power sharing scheme,”
IEEE Trans. Power Electron., vol. 30, no. 6, pp. 3389–3401, Jun. 2015.
[10] H. Mahmood, D. Michaelson, and J. Jiang, “Accurate reactive power
sharing in an islanded microgrid using adaptive virtual impedances,”
IEEE Trans. Power Electron., vol. 30, no. 3, pp. 1605–1617, Mar. 2015.
[11] J. W. He and Y. W. Li, “Analysis, design, and implementation of virtual
impedance for power electronics interfaced distributed generation,” IEEE
Trans. Ind. Appl., vol. 47, no. 6, pp. 2525–2538, Nov./Dec. 2011.
[12] A. Elrayyah, Y. Sozer, and M. E. Elbuluk, “A novel load-flow analysis
for stable and optimized microgrid operation,” IEEE Trans. Power Del.,
vol. 29, no. 4, pp. 1709–1717, Aug. 2014.
[13] M. A. Hassan and M. A. Abido, “Optimal design of microgrids
in autonomous and grid-connected modes using particle swarm opti-
mization,” IEEE Trans. Power Electron., vol. 26, no. 3, pp. 755–769,
Mar. 2011.
[14] I.-Y. Chung, W. X. Liu, D. A. Cartes, E. G. Collins, Jr., and
S.-I. Moon, “Control methods of inverter-interfaced distributed gener-
ators in a microgrid system,” IEEE Trans. Ind. Appl., vol. 46, no. 3,
pp. 1078–1088, May/Jun. 2010.
[15] E. Planas, A. Gil-de-Muro, J. Andreu, I. Kortabarria, and
I. Martiìnez de Alegriìa, “Design and implementation of a droop
control in d-q frame for islanded microgrids,” IET Renew. Power
Gener., vol. 7, no. 5, pp. 458–474, Sep. 2013.
[16] M. M. A. Abdelaziz, H. E. Farag, E. F. El-Saadany, and
Y. A.-R. I. Mohamed, “A novel and generalized three-phase power flow
algorithm for islanded microgrids using a newton trust region method,”
IEEE Trans. Power Syst., vol. 28, no. 1, pp. 190–201, Feb. 2013.
[17] C. Li, S. K. Chaudhary, J. C. Vasquez, and J. M. Guerrero, “Power
flow analysis algorithm for islanded LV microgrids including distributed
generator units with droop control and virtual impedance loop,” in Proc.
Appl. Power Electron. Conf. Expo. (APEC), Fort Worth, TX, USA, 2014,
pp. 3181–3185.
[18] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis and
testing of autonomous operation of an inverter-based microgrid,” IEEE
Trans. Power Electron., vol. 22, no. 2, pp. 613–625, Mar. 2007.
[19] J. W. He, Y. W. Li, J. M. Guerrero, F. Blaabjerg, and J. C. Vasquez,
“An islanding microgrid power sharing approach using enhanced vir-
tual impedance control scheme,” IEEE Trans. Power Electron., vol. 28,
no. 11, pp. 5272–5282, Nov. 2013.
[20] Q. Sun, “Power system small signal stability region and low-frequency
oscillations,” Ph.D. dissertation, Dept. Elect. Eng. Autom., Tianjin Univ.,
Tianjin, China, 2007.
[21] M. A. Abido, “Optimal design of power-system stabilizers using par-
ticle swarm optimization,” IEEE Trans. Energy Conv., vol. 17, no. 3,
pp. 406–413, Sep. 2002.
[22] Q. Shafiee et al., “Robust networked control scheme for distributed
secondary control of islanded microgrids,” IEEE Trans. Ind. Electron.,
vol. 61, no. 10, pp. 5363–5374, Oct. 2014.
[23] A. Engler, “Applicability of droops in low voltage grids,” Int. J. Distrib.
Energy Resour., vol. 1, no. 1, pp. 1–6, 2005.
Xiangyu Wu (S’13) received the B.S. degree
from the Department of Electrical Engineering,
Zhejiang University, Hangzhou, China, in 2012. He
is currently pursuing the Ph.D. degree in electri-
cal engineering from Tsinghua University, Beijing,
China. In 2015, he worked as a Visiting Scholar at
the University of Toronto, Toronto, ON, Canada. His
research interests are microgrid control and stability
analysis.
Chen Shen (M’98–SM’07) received the B.E.
degree in electrical power engineering and the
Ph.D. degree in electrical power engineering from
Tsinghua University in 1993 and 1998, respectively.
Since 2009, he has been a Professor in the Electrical
Engineering Department with Tsinghua University,
where he is currently the Director of Power System
Research Institute with the Department of Electrical
Engineering. His research focuses on power sys-
tems analysis and control, including fast modeling
and simulation of smart grids, stability analysis of
power systems with wind generation, emergency control and risk assessment
of power systems, planning, simulation operation, and control for microgrids.
Reza Iravani (M’87–SM’00–F’03) received the
B.Sc., M.Sc., and Ph.D. degrees in electrical engi-
neering. He is currently a Professor with the
University of Toronto, Toronto, Canada. His research
interests include power electronics, power system
dynamics and control.
