A microgrid (MG) with integrated renewable energy resources can benefit both utility companies and customers. As a result, they are attracting a great deal of attention. The control of a MG is very important for the stable operation of a MG. The droopcontrol method is popular since it avoids circulating currents among the converters without using any critical communication between them. Traditional droop control methods have the drawback of an inherent tradeoff between power sharing and voltage and frequency regulation. An adaptive droop control method is proposed, which can operate in both the island mode and the gridconnected mode. It can also ensure smooth switching between these two modes. Furthermore, the voltage and frequency of a MG can be restored by using the proposed droop controller. Meanwhile, the active power can be dispatched appropriately in both operating modes based on the capacity or running cost of the Distributed Generators (DGs). The global information (such as the average voltage and output active power of the MG and so on) required by the proposed droop control method to restore the voltage and frequency deviations can be acquired distributedly based on the Multi Agent System (MAS). Simulation studies in PSCAD demonstrate the effectiveness of the proposed control method.
I. INTRODUCTION
In order to meet the increasing demand for electricity, Distributed Generation (DG) based on Renewable Energy Sources (RES) (such as photovoltaic, fuel cells and windpower) and storage systems is emerging as a promising new technology
[1]
. By using Distributed Generators (DGs), the distance between the generation and the loads is reduced, which can reduce losses, and postpones investments in new transmission and large scale generation systems
[2]
. Microgrids (MGs) are attracting a great deal of attention since they integrate these DGs into the main grid reliably and cleanly
[1]
,
[3]

[5]
. For utility companies, MGs can operate as a single dispatchable unit to provide power or ancillary services. For customers, they can improve power quality by supporting local voltage and frequencies
[6]
,
[7]
.
The droop control methods are presented as wireless control techniques that avoid circulating currents among the converters without using any critical communications between them. A lot of studies suggest that droop control methods are the best option for controlling the DGs in MGs
[3]
,
[8]

[10]
. Furthermore, droopcontrol methods are often applied in many experimental MGs because of the advantages they offer, such as flexibility and absence of critical communications
[1]
,
[11]
.
For droop control methods, an inherent tradeoff between power sharing and voltage and frequency regulation is one of its drawbacks. This is due to the fact that if the droop coefficients are increased, then good power sharing is achieved at the expense of degrading the voltage/frequency regulation. On the other hand, if the droop coefficients are decreased, then good voltage/frequency regulation is achieved at the expense of degrading the power sharing
[12]
. Nowadays, many researchers are focusing on improving droop control and many improved methods have been proposed. A modified droop controller was discussed in
[13]

[18]
, but the frequency and voltage amplitude experience deviations in the steady state. A restoration control was proposed in
[19]
,
[20]
, but it must be carried out by means of a central controller, which requires some form of communication between the restoration control and the DGs, which is complex and not reliable for a single point of failure.
In order to restore the MG voltage magnitude and frequency while achieving accurate power sharing, centralized active load sharing techniques and average load sharing techniques have been proposed in
[20]
and
[21]
, respectively. However, these techniques reduce system reliability and expandability due to their dependency on critical intercommunication lines among modules
[22]
.
The Multi Agent System (MAS) is one of the most popular distributed control strategies. The convergence, robustness and stability analysis of the multi agent based algorithm was discussed in
[23]

[25]
, which is helpful for application to MGs. In recent years, several global information discovery algorithms based on MAS have been proposed and applied to MGs
[25]

[27]
.
In this paper, an adaptive droop control method is proposed, which can operate in both the island mode and the gridconnected mode. In addition, it can shift smoothly between these two modes. Furthermore, the voltage and frequency of a MG can be restored. Meanwhile, the active power can be dispatched appropriately in both of the operating modes based on the capacity or the running cost of the DGs. Based on the MAS, the global information (which is required by the proposed droop control method to restore the voltage and frequency deviation ) can be acquire distributedly.
The global information discovered in this paper includes the status information (such as the state of each of the DGs and the weight factor to the dispatch active load) and the performance metric (such as the average voltage and the output active power of the MG). During the information discovery process, the agents only need to communicate with their corresponding neighbors. Furthermore, the global information discovery algorithm is independent of the system configuration. Thus, it can be applied to MG systems with any structure, such as radial, mesh and mixed.
The main novel contributions of this paper are:

1) A MAS based adaptive droop control method for microgrids was proposed and the stability of this control architecture was analyzed through a smallsignal model.

2) A global information discovery algorithm based on the MAS is applied for the DGs to discovery global information distributedly. In addition, the frequency and voltage deviation can be restored with the adaptive droop controller using the acquired global information, which overcomes the drawback of traditional droop control methods.

3) The transient and steady state performances of the adaptive droopcontrolled MG are verified by simulations. It is shown to have better performance than traditional droop control methods.
This paper is organized as follows. In section II, both the MG system model and the control method are presented. Section III proposes the MASbased global information discovery algorithm. Section IV presents simulations and results. Finally, Section V concludes this paper.
II. MICROGRID SYSTEM MODEL AND CONTROL METHOD
 A. Voltage Source Inverter Model
In MGs, Voltage Source Inverters (VSIs) are very popular because they can provide ridethrough capability and powerquality enhancement to DGs
[20]
. These inverters are required to operate in both the gridconnected mode and the island mode, and they often change their behavior from voltage to current sources in the gridconnected mode. In order to operate in both the gridconnected and islanded modes, VSIs are required to control the exported or imported power to the main grid to stabilize the MG
[28]
. The circuit structure of a threephase VSI is shown in
Fig. 1
.
Circuit structure of threephase VSI.
In
Fig. 1
,
L_{f}
is the inductance of the LC filter,
R_{f}
is the resistance of the inductor,
C_{f}
is the capacitance of the LC filter,
u_{a}
,
u_{b}
,
u_{c}
are the three phase output voltages of the inverter,
u_{oa}
,
u_{ob}
,
u_{oc}
are the voltages of the capacitor,
Z_{line}
is the impedance of the transmissionline,
i_{a}
is the current of the filter inductor, and
i_{oa}
is the current in the transmissionline.
By using a
d

q
transformation, the threephase voltage and current can be transformed to the
d

q
rotating reference frame. Then the active and reactive powers can be decoupled by using a
d

q
transformation
[17]
. Details of the
abc
/
dq
transformation and its sign convention are shown in Eq. (1), and the
dq
/
abc
transformation is shown in Eq. (2).
By using the former
d

q
transformation, the mathematical equation of the inverter circuit can be obtained from
Fig. 1
as Eqns. (3)(6). In this paper, the subscripts
d
and
q
represent the
d
 axis component and the
q
 axis component of some variable after the
d

q
transformation.
Eqns. (3)(4) demonstrate that ,
i_{d}
,
i_{q}
are not only affected by,
u_{od}
,
u_{oq}
, but that they are affected by the coupling currents
w
·
L_{f}
·
i_{q}
and
w
·
L_{f}
·
i_{d}
. Reference
[29]
shows that the feedforward quantities,
u_{od}
,
u_{oq}
and cross decoupled quantities
w
·
L_{f}
·
i_{q}
and
w
·
L_{f}
·
i_{d}
are used to achieve independent current control in the
d

q
axis. Thus, the inner current control loop is designed as Eqns. (7)(8)
[29]
to eliminate the current coupling.
Applying Eqns. (7)(8) into Eqns. (3)(4) the following are obtained:
As a result,
i_{d}
and
i_{d}
can be controlled separately without coupling.
Eqns. (5)(6) indicate that,
u_{od}
,
u_{oq}
are affected by,
i_{od}
,
i_{oq}
and the coupling voltages
w
·
C_{f}
·
u_{oq}
and
w
·
C_{f}
·
u_{od}
. Reference
[29]
indicates that the feedforward quantities,
i_{od}
,
i_{oq}
and cross decoupled quantities
w
·
C_{f}
·
u_{oq}
and
w
·
C_{f}
·
u_{oq}
are used to achieve independent voltage control in the
d

q
axis. To eliminate the voltage coupling, the outer voltage control loop is designed as Eqns. (11)(12)
[29]
.
Applying Eqns. (11)(12) into Eqns. (5)(6), the following are obtained:
u_{od}
and
u_{oq}
can be controlled separately without coupling.
 B. Adaptive Droop Control
The values of the active and reactive powers flowing between two AC voltage sources, which are connected in parallel through the line impedance as shown in
Fig. 2
, can be calculated as Eqns. (15)(16)
[30]
,
[31]
. In
Fig. 2
,
R
and
X
are the line impedance and inductance, respectively.
U
_{1}
and
U
_{2}
are the RootMeanSquare (RMS) value of the AC voltage sources, and their phases are
θ
_{1}
and
θ
_{2}
, respectively. If
X
>>
R
and
δ
=
θ
_{1}

θ
_{2}
is small, then Eqns. (15)(16) can be simplified as Eqns. (17)(18).
Power flow and vector diagram between the main grid and the inverter interfaced DG.
As shown by Eqns. (17)(18), the active power, flowing from voltage source 1 to 2 through a highly inductive line impedance, can be controlled by varying the phase
δ
. The reactive power supplied by source 1 can be controlled by controlling the magnitude of source 1 (
U
_{1}
). This is the basis of the conventional
P
vs.
f
and
Q
vs.
U
droop controls. As shown in Eqns. (19)(20), in most cases,
f_{oi}
and
U_{oi}
are the nominal values
[17]
.
Where,
i
is the DG index,
mp_{i}
and
mq_{i}
are the droop parameters, and
U_{i}
,
f_{i}
,
P_{i}
, and
Q_{i}
are the bus RMS voltage, the system frequency, and the locally measured active and reactive powers, respectively. The subscript
o
represents the preset values of the normal operating points.
The angle
δ
is generated by dynamically controlling the frequency of source 1. The relationship between the frequencies of the two interconnected AC voltage sources and
δ
is given by Eq. (21)
For directly coupled DG units, the impedance between two DGs is just the low voltage distribution feeder and has the relationship
R
>>
X
[32]
,
[33]
. Eqns. (15)(16) can then be simplified as Eqns. (22)(23). The droop can be based on
f
(
P
) as in Eq. (19) or reversely,
P
(
f
), where the frequency is measured and drooped to determine
P
. Unlike measurements of power, the accurate measurement of frequency is difficult. Therefore, the
f
(
P
) strategy is often chosen
[34]
. In this paper, the
f
(
P
) strategy is applied with frequency as a function of the delivered active power.
In order to prevent the inverter from filling up beyond its capacity, the problem of reaching the maximum output power needs to be addressed. It is possible to take advantage of two assumptions: 1) the inverter can transiently sustain overshoots of the value
P
_{max}
for short periods of time; 2) the inverter can transiently sustain the condition of
P
< 0 for brief periods of time.
Fig. 3
show the proposed
P
vs.
f
and
Q
vs.
U
droop control diagrams. In
Fig. 3
,
P
_{max}
is the allowed maximum output power,
P_{i}
is the output active power of the
i
th inverter, and
errP
_{max}
is the value of
P
_{max}
minus
P
_{min}
. If the power is larger than its maximum,
errP
_{max}
is negative and the integral starts generating a negative value for the offset. This offset in turns translates the droop curve down. It keeps on translating down until the offset exactly matches the value where
errP
_{max}
= 0 . Until it does, the integral keeps on increasing the offset. When power output matches the maximum power, the error become zero and the integral stops increasing the offset. As a result, a new steady state is reached with the offset Δ
f
_{max}
. When preventing the injections of power from exceeding the minimum value, the approach used to calculate the offset Δ
f
_{min}
is very similar to the approach used to calculated Δ
f
_{max}
. To prevent the reactive power from reaching the maximum or minimum values, the same strategy is used, as shown in
Fig. 3
.
Adaptive P vs. f and Q vs. U droop control diagram.
As shown previously, for directly coupled DG units, the impedance between two DGs is just the low voltage distribution feeder and it has the relationship
R
>>
X
. If the
f
(
P
) strategy is to be used, a physical coupling inductor or a virtual coupling inductor should be added between the inverter terminals. The physical coupling inductor increases the size and weight of the system. The virtual impedance creates a “voltage drop” without generating real active and reactive powers losses and it is insensitive to the nature of the line impedance
[31]
. These physical inductances can be replaced by virtual impedances
[1]
and consecutive virtual impedance variations
[30]
,
[35]
,
[36]
. Therefore, a virtual impedance loop is added to the real line impedance as shown in
Fig. 4
.
Virtual impedance in series with the real line impedance.
In
Fig. 4
,
Z_{v}
is the virtual impedance,
R_{v}
is the resistive part of the virtual impedance,
L_{v}
is the inductive part of the virtual impedance,
Z
_{line}
is the line impedance,
R_{line}
is the resistive part of the line impedance, and
Z_{line}
is the inductive part of the line impedance.
 C. Synchronization Conditions and Frequency Restoration of Droop Controlled MGs
The grid voltage
U_{grid}
rotates at 50 Hz in the counter clockwise direction, while the MG Voltage
U_{MG}
rotates at a frequency lower than 50 Hz in the same direction. If the vectors are strobed 50 times a second, the vector
U_{grid}
stays still, while vector
U_{MG}
draws back in the clockwise direction
[2]
.
Fig. 5
shows the voltage vector synchronization process of
U_{grid}
and
U_{MG}
. As
Fig. 2
shows, if the grid injects the active power of
P_{steady}
to the MG in the steady state, then from Eq. (17), the time to connect the MG to the grid is when
δ
satisfies Eq. (24).
Voltage vector synchronization process.
For droop control methods, the inherent tradeoff between power sharing and voltage and frequency regulation is one of its drawbacks. In this paper, for the application of the voltage and frequency restoration and DG status identification, a MAS based global information discovery algorithm is used to acquire the global information. This algorithm is described in section III. There are two kinds of global information to be discovered: status information (such as the state of each DG and the weight factor to dispatch an active load) and performance metric (such as the average voltage and the output active power of a MG).
The frequency and voltage need to be restored only if the MG is operating in the island mode. In the gridconnected mode, the frequency and voltage are maintained by the main grid. In the island mode, it is assumed that, the DGs in the MG are all running properly and within the upper and lower limits.
From
Fig. 3
, the
P
vs.
f
and
Q
vs.
U
droop curves can be derived as Eqns. (25)(26).
P_{oi}
and
Q_{oi}
are set to be zero, in this paper. In the island mode,
f_{syn}
is zero too, and the function of
f_{syn}
is described later. Then Eq. (25) can be rewritten as Eq. (27).
Assume that:
Where,
n
is the number of DGs running in the MG, and
λ_{i}
is the weight factor to share all of the active power consumed by the load, which is decided based on the capacity or the running cost of the DGs.
If:
Then:
After restoration of the frequency, Δ
f_{i}
is zero. Thus,
P_{loadrefi}
is rewritten as Eq. (32).
Through the global information discovery process implemented by the MAS (as mentioned before),
P_{total}
and
λ_{total}
l can be acquired by all of the DGs in the MG.
λ_{i}
is a local parameter, which can be adjusted on line.
The frequency of the MG is restored by
P_{loadrefi}
in the island mode. In order to change to the gridconnected mode from the island mode, the synchronization conditions need to be met. Therefore,
f_{syn}
is used to find a suitable angle d to connect the MG to the main grid.
f_{syn}
is set to be a small value (0.2 Hz for example) when the MG going to connect to the main grid. At other times, it is set to be zero. When
f_{syn}
is bigger, it is able to find a suitable time to connect to the main grid more quickly, as described in
Fig. 5
. However, it must be lower than 0.5 Hz to satisfy the standard of frequency variation.
Unlike the frequency, the voltage of a MG is not a global parameter
[37]
. The frequencies of all of the DGs are equal in the steady state, but the amplitudes and phase angles of the voltages for the DGs are different from each other. In order to restore the voltage, a global information discovery algorithm is implemented based on a MAS to derive the average voltage
U_{aver}
of the DGs. A proportionalintegral (PI) controller is used to calculate
U_{loadrefi}
and to restore the average voltage of the MG to its preset value. The relationship between
U_{aver}
and
U_{loadrefi}
is given by Eqns. (33)(34).
Where,
K_{pres}
and
K_{ires}
are the PI controller parameters used to restore the average voltage of the MG.
 D. SmallSignal Model Based Stability Analysis of a Microgrid System
Fig. 6
shows a control block diagram of inverter interfaced DGs in the
d

q
rotating reference frame. As shown in
Fig. 6
, the DG controllers involve three types of controllers: the droop controllers, the voltage controllers and the current controllers. The droop controller calculates the output voltage reference, whereas the voltage and current controllers control the inverter voltage and current, respectively. As described in section II, voltage controllers and current controllers are all PI controllers.
Control block for inverterinterfaced DGs.
From
Fig. 6
, the mathematical equations are derived as Eqns. (35)(42).
Fig. 7
shows a MG equivalent circuit model. The mathematical equations of the MG equivalent circuit can be obtained from
Fig. 7
, as Eqns. (43)(50).
Microgrid equivalent circuit model.
Because the PWM circuits and output filters have much faster dynamics than the inverter output controller, it is reasonable to assume that the actual inverter output voltage can follow the inverter voltage reference fast enough
[17]
. Thus, Eqns. (51)(52) can be derived.
From Eqns. (3)(6) and Eqns. (35)(52), a smallsignal closedloop model can be obtained as shown in Eqns. (53)(54).
Where:
Fig. 8
shows the closeloop dominant eigenvalue locus when the PI parameters change. As shown in
Fig. 8
(a), the dominant complex poles move to the right when increasing
K_{pi}
from 1 to 100 while the other parameters are fixed. When
K_{pi}
is smaller than 15, the system is under damped. On the other hand, when
K_{pi}
is bigger than 15, the poles are near the imaginary axis. As a result, the stability margin is lower.
Fig. 8
(b) shows that the dominant complex poles move to the left when increasing
K_{li}
from 1 to 100 while the other parameters are fixed. This makes the system quicker but more oscillatory.
Fig. 8
(c) indicates that the dominant complex poles move to the right when increasing
K_{pu}
from 1 to 100 while the other parameters are fixed. When
K_{pu}
is smaller than 3, the system is under damped. On the other hand, when
K_{pu}
is bigger than 3, the poles are near the imaginary axis. Thus, the stability margin is lower. Fig.8 (d) shows that the dominant complex poles move to the left when increasing
K_{lu}
from 1 to 100 while the other parameters are fixed. This makes the system quicker but more oscillatory. This helps us to find suitable parameters for the inverter controllers.
Closedloop eigenvalue analysis. (a) Dominant pole locus when K_{pi} varies ( K_{Ii} = 10, K_{pu} = 3, K_{Iu} = 10 ); (b) Dominant pole locus when K_{Ii} varies ( K_{pi} = 15, K_{pu} = 3, K_{Iu} = 10 ); (c) Dominant pole locus when K_{pu} varies ( K_{pi} = 15, K_{Iu} = 10, K_{Ii} = 10 ); (d) Dominant pole locus when K_{Iu} varies ( K_{pi} = 15, K_{pu} = 3, K_{Ii} = 10 ).
III. MASBASED GLOBAL INFORMATION DISCOVERY ALGORITHM
As described in section II, the traditional droop control has the drawback of a tradeoff between power sharing and voltage and frequency regulation. Several proposed restoration controls must be carried out by means of a central controller, which requires some form of communication between the restoration control and the DGs, which is complex and not reliable for a single point of failure.
In section II, an adaptive droop control method is proposed. This method can restore the voltage and frequency. It can also properly dispatch the active power by using the status information (such as the state of each DG and the weight factor to dispatch the active load) and the performance metric (such as the average voltage and the output active power of the MG). This information is discovered by a MAS based global information discovery algorithm.
In recent years, global information discovery algorithms based on the MAS have attracted a great deal of interest in various areas
[26]
,
[38]
,
[39]
. These algorithms are based on multi agent coordination and do not need a powerful central controller. Thus, they are robust to the single point failures.
A MG, as shown in
Fig. 9
, is taken for research. The MG is coupled with the main grid through the PCC, and can operate either in the island mode or the gridconnected mode. There are three feeders in the MG. Feeder 1 has one load and two DGs; Feeder 2 also has one load and two DGs; while feeder 3 has one load and one DG. The length and parameters of the transmission line, the capacities of the DGs and the load demand are shown in
Fig. 9
. In this paper, it is assumed that each DG unit in the MG is assigned with an agent. Every agent knows the local information (the agent can measure the information locally and acquire it without any communication), such as the local voltage, the output active power of a certain DG, and so on. However, it does not have direct access to the global information (this information cannot be acquired directly, by communicating with the corresponding neighbor agents, each agent must acquire the global information through the MASbased global information discovery algorithm), such as the average voltage of the MG, the total output active powers of all of the DGs, and so on. As
Fig. 6
shows, with the local information of
P_{i}
,
U_{i}
, and
λ_{i}
, agent
i
can acquire the global information of
P_{total}
,
U_{aver}
,
λ_{total}
by communicating with its neighbors
j
and
k
. With this global information,
P_{loadref}
and
U_{aver}
can be calculated by Eq. (32) and Eq. (34), respectively. The adaptive droop controller can restore the frequency and voltage deviation of the MG with
P_{loadref}
and
U_{aver}
. This overcomes the main drawback of traditional droop control methods (the tradeoff between power sharing and voltage and frequency regulation).
Topology of a microgrid.
In order to improve the robustness against communication failures, the topology of the communication network is designed to satisfy the
N
1 rule, which means that the rest of the communication network can work normally, with the breakdown of any one agent. Each agent can communicate with its neighbor agents. For example, in
Fig. 9
, DG 1 can communicate with DG 2 and DG 3. All of the agents constitute a multi agent system. The original data of each DG for the case study are summarized in
Table I
.
DATA OF EACH AGENT TO COMMUNICATE
DATA OF EACH AGENT TO COMMUNICATE
In the global information discovery algorithm, the information discovery process for the ith agent is represented as Eq. (55).
Where,
are the pieces of information (average voltage, average active power, and so on) discovered by agents
i
and
j
at iteration
k
, respectively;
i
= 1,2,...,
n
;
n
is the number of agents in the MG;
is updated to
at iteration
k
+ 1 ;
a_{ij}
is the coefficient for the information exchanged between the neighboring agents
i
and
j
; and
N_{i}
is the set of neighbors for agent
i
.
The coefficient
a_{ij}
plays an important role for the convergence speed. Y. Xu and W. Liu summarize and discuss several methods for designing the coupling coefficients
[25]
.
a_{ij}
is updated based on Eq. (56), as proposed in
[25]
.
Where,
n_{i}
is the number of neighbors for agent
i
,
n_{j}
is the number of neighbors for agent
j
, and
N_{i}
is the set of neighbors for agent
i
.
ε
is a very small number. The value of
ε
is set to 0.3 in this paper. Based on the consensus theory, the overall information discovery process can be modeled as a discrete time linear system as shows in Eq. (57).
Where,
is the vector of the discovered information at the kth iteration; and
D
is the Laplacian matrix of the multiagent network graph
[39]
. The convergence speed is determined by the second largest eigenvalue of
D
.
The global information discovery algorithm can guarantee the convergence for systems of any size and topology, as shown in
[25]
,
[27]
. In this paper, the information discovery process terminates once the following Equation is valid:
Where,
γ
is a small real number that defines the precision requirement. The value of
γ
is set to 0.01 in this paper.
For the application of the voltage and frequency restoration and the DG status identification, there are two kinds of information to be discovered: the status information and the performance metric
[40]
. In order to acquire information on the number of DGs and which DG are running in this MG, each agent in the MG is assigned a unique state vector beforehand. In the beginning, agent
i
's
N
 dimensional (
N
is the total number of agents in the MG and it is a constant for a given MG) state vector
s_{i}
has only one nonzero element at (
i
,1) with the value of
i
, that is
s_{i}
= [0,0,...,i,...,0]
^{T}
[27]
. The
i
th elements of the state vector will converge to
i
/
M
(
M
is the total number of working agents in the MG,
M
≤
N
), by applying the global information discovery algorithm to the same elements of the state vectors, as shown in
Fig. 10
(e). When DG 4 is disconnected (but agent 4 is still working normally), agent 4’s state vector
s
_{4}
is reset to [0,0,0,0,0,0]
^{T}
. By applying the global information discovery algorithm to the same elements of the state vectors, all of the elements except for the 4th element of the state vectors will converge to the same value (
i
/
M
) as before, as can be seen in
Fig. 10
(f). By checking the positions of the zeros, all of the agents will know that DG 4 is now disconnected. As a result, the number of DGs connected to the MG
n
is updated. Furthermore, by comparing the ith element with
x_{i}
/
M
, where
x_{i}
can be either
i
or 0 (when the corresponding DG is disconnected) and
M
can be any integer between 1 and
N
, it can find the
M
match of the converged state vector [
x
_{1}
/
M
,...,
x_{i}
/
M
,...,
x_{N}
/
M
]
^{T}
. Thus, the number of working agents
M
can also be found. Multiply the discovered average performance metrics by
M
, and the total value of the according metrics are derived.
Information discovery processes of status information and performance metrics. (a) Average active power discovery process. (b) Average weight factor discovery process. (c) Average voltage discovery process. (d) Synchronization signal discovery process. (e) DGs’ state vector discovery process, when all DGs are connected to the MG. (f) DGs’ state vector discovery process, when DG4 is disconnected from the MG.
In order to deal with the fault of an agent loss, all of the agents need send heartbeat packets to
i
th's neighbors. Then each agent knows who its neighbors are and the number of neighbors at the present time. By using Eq. (56),
a_{ij}
can be updated. When the MG is expanded, it only needs to update its’ neighbors’ information. Thus, the MAS control offers a plug and play performance which involves low costs related to modifications in the MG’s architecture.
Agent 1 is responsible for synchronize with the main grid. Agent 1’s synchronization signal
Syn
will be 1, when the micriogrid is going to change from the island mode to the gridconnected mode, otherwise
Syn
will be 0. The other agents’ synchronization signals
Syn
are always initialized to be zero. Thus, when an agent finds that
Syn
converges to a nonzero value,
f_{syn}
in (25) is set to 0.2Hz for all the droop controllers to synchronize the MG with the main grid, as described previously. The information discovery processes of the status information and the performance metrics are shown in
Fig. 10
.
Fig. 10
(a) shows the average active power discovery process. All of the agents converge to the same value of 21.22=(47.8+6.4+26.5+15.9+9.5)/5 within 30 iterations.
Fig. 10
(b) describes the average weight factor discovery process. It shows that all of the agents converge to the same value within 30 iterations.
Fig. 10
(c) and
Fig. 10
(d) indicate the average voltage and synchronization signal discovery processes, respectively. They also show that all of the agents converge to the same value within 30 iterations.
Fig. 10
(e) shows the updating process of the
i
th element of the
i
th agents’ state vectors when all of the DGs are connected to the MG. From
Fig. 10
(e), it can be seen that all of the agents converge to the same value of
i
/
M
.
Fig. 10
(f) shows the updating process of the element of the agents’ state vectors when DG4 is disconnected from the MG. From
Fig. 10
(f), it can be seen that all of the agents except for agent 4 converge to the same value of
i
/
M
as before.
The data to communicate is represented by 16 bits for one piece of data. There are 9 pieces of data to communicate for each agent, and it is supposes that the maximum iteration to convergence is 50 for the global information discovery algorithm (which is rational considering the results in
Fig. 10
). Suppose that one agent has 10 neighbors as a maximum in the MG. In order to get the global information 5 times per second, the communicating rate should be higher than 16 × 9 × 50 × 10 × 5 × 2 = 720kbps. Nowadays, a communication rate of 720kbps is easy to fulfill. Thus, it is reasonable to get global information 5 times per second.
IV. SIMULATION AND RESULTS
The MG system model in
Fig. 9
is simulated by using PSCAD/EMTDC. This model contains five inverterinterfaced DGs with PWM voltage source inverters. The DGs in the MG are coordinated via
P
vs.
f
and
Q
vs.
U
adaptive droop controllers and the global information is discovered by a MAS.
All of the loads are connected in a star configuration and they can be connected and disconnected with a switch. The parameters of the loads in
Fig. 9
are listed in
Table II
.
LOADS DATA SUMMARY
In this paper, the inverter is fed by a DC bus voltage source of 750V, which emulates the combined effects of a prime mover and energy storage. The carrier frequency of the VSI is 5000Hz. The rated power factors of the DGs are all set as 0.8, and the minimum operating frequency is 49.5Hz, which is 0.5 Hz below the nominal frequency.
The fixed slops
mp_{i}
of the
P
vs.
f
droop are chosen to be the value of the slop obtained for
P_{oi}
= 0 in Eq. (25). They are list in
Table III
DGS DATA SUMMARY
When the value of the maximum offset expected for the voltage, Δ
E
, is known, with the help of
Q
_{max}
, the slopes
mq_{i}
of the
Q
vs.
U
droop can be determined by Δ
E
/
Q
_{max}
. The maximum offset for the voltage is taken as the difference in voltage between the best and worst case scenarios
[4]
. The best case scenario is when the MG is fully energized from the grid, without micro sources or loads. The worst case scenario is when the MG is energized from the grid, without micro sources and with all of the loads. The voltages in the best case scenario are represented as
V
_{max}
, in the worst case scenario they are represented as
V
_{min}
,
V
_{max}
,
V
_{min}
, and the slopes
mq_{i}
are list in
Table III
.
The virtual impedance can generate good system transient and steadystate responses if designed properly. The effects of virtual impedance have been verified in some published papers. The virtual impedance increases the system damping and enhances the system’s transient response but it also affects the inverters’ voltage regulation (which will cause voltage drop)
[41]
,
[42]
. As Eq. (17) shows, the phase angle difference
δ
is affected by the coupling inductor
X
. The angle difference
δ
cannot be allowed to get too small to avoid errors caused by an imprecise measurement of
δ
. However, in order to have a good linear approximation between
P
and
δ
,
δ
cannot be allowed to get too big. In this paper, the method to calculate the virtual impedance is from reference
[2]
.
δ
is set to be 7 degrees at full ratings. This guarantees operation in the linear region of the sinusoidal characteristic. This means that the virtual impedance
X
can be calculated as Eq. (59) as shown in
[2]
. The Virtual impedances
X
are calculated and listed in
Table III
.
The values of
K_{pres}
and
K_{ires}
in Eq. (34) are designed by experiential adjustments according to the simulation results. Within certain limits, when the value of
K_{pres}
is bigger, the speed is higher for restoring the voltage. However, it cannot be too big to maintain system stability. Within certain limits, when the value of
K_{ires}
is bigger, the steady state deviation is lower. However, it cannot be too big either to maintain system stability. The appropriate values of
K_{pres}
,
K_{ires}
and the other parameters used in this paper are given in
Table IV
.
OTHER PARAMETERS IN THIS PAPER
OTHER PARAMETERS IN THIS PAPER
In order to show the decoupling effect of the decoupling method introduced in section IIA, a comparison study was carried out. In the comparison study, the performance of the decoupling method is compared with the method without a feedforward decoupling network. The results are shown in
Fig. 11
. As shown in the figure, when the reference voltages are given a step change at 0.1 s, the decoupling method has a faster response speed and a lower steady state deviation than the method without a feedforward decoupling network.
Compare the feedforward decoupling control methods with the coupling control methods.
In order to verify the proposed adaptive droop control method, the simulation sequence was set as follows: 1) 0 − 2 s, the MG is operating in the island mode and all of the DGs and loads are connected; 2) at 3 s, DG 4 is disconnected; 3) 2 − 6.8 s,
f_{syn}
is set to 0.2Hz for all of the DGs to synchronize the MG with the main grid, in order to be connected to the main grid; 4) 6.8 − 13s, the MG is connected to the main grid, all of DGs are set to output 10kW of active power and DG 1, DG 2, DG 3 and DG 5 are set to output 10kVar, 5kVar, 8kVar and 0kVar of reactive power, respectively; 5) 13 − 14s, it is assumed that the DGs have not detected the island to analyze the effect of the time delay to the system; 6) at 14s, all of the DGs get the information of the island, recover the frequency and dispatch active power; 7) at 17s, load 2 is disconnected. In this simulation, the global information is acquired once every 0.2 seconds. The simulation results of the MG are shown in
Figs. 12

16
.
Fig. 12
shows the active powers and frequencies of the DGs. In the island mode, the frequency can be restored and the active powers can be simultaneously dispatched as desired. In the gridconnected mode, the system frequency is provided by the main grid, the output active power of each of the DGs can follow up the preset value, while the grid provides the residual load demand. For the time delay of the island mode detection during 13 − 14s, the frequency is not restored but the system can still work properly. At 17s, when load 2 is disconnected, all of the DGs will decrease the output active power while the frequency is higher than the normal value for a short time, then it is restored to the normal value.
Active powers and the frequencies of DGs.
Fig. 13
shows the reactive powers and voltages of the DGs. In the island mode, when DG 4 is disconnected at 3 s, the voltages are decreased and restored for a while. In the gridconnected mode, the voltages are provided by the main grid and the output reactive power of each DG can follow up the preset value.
Reactive Powers and Voltages of DGs.
Fig. 14
shows the active power ratios for each of the DGs. The dashed line represents the preset ratios and the solid line represents the actual active power ratios. Only the moment when the MG is in the island mode needs to be seen, because in the gridconnected mode, each of the DGs is set to output 10kW of active power. In this figure, it can be seen that the DGs’ active power ratios can follow the preset values most of the time. During the time delay of the island mode detection of 13 − 14s the actual active power ratios cannot follow the preset value. The difference between the actual ratios and actual ratios at other times is caused by the time delay of the global information discovery.
Active power ratios of DGs.
Fig. 15
and
Fig. 16
show the output voltage and current of DG 1 when the microgrid connects to and disconnects from the main grid, respectively. As the reference values of the output active and reactive powers are changed between the island mode and the gridconnected mode in the study case, the magnitudes of the currents of,
i_{oa}
,
i_{ob}
and
i_{oc}
changed accordingly. In the steady state, the total harmonic distortions (THD) of DG1’s currents are 0.81% and 2.30% when the microgrid is in the island mode and the gridconnected mode, respectively. The current THD value is smaller than the international standard
[43]
during the island and gridconnected operation modes.
Output voltage and current of DG1 when the microgrid shits to gridconnected mode from island mode.
Output voltage and current of DG1 when the microgrid shifts to island mode from gridconnected mode.
In order to verify the validity of proposed adaptive droop control method, comparative results with conventional droop control methods are shown in
Fig. 17
. From
Fig. 17
, it can be seen that with the proposed adaptive droop control method, the microgrid can restore the frequency and voltage deviation. Meanwhile the microgrid controlled by the conventional droop control method had frequency and voltage deviations in the steady state.
Compare the adaptive droop control method with the conventional droop control method.
V. CONCLUSIONS
In this paper, an adaptive droop control method is proposed for MGs. This adaptive droop controller can restore the voltage and frequency with global information. Meanwhile, the active power can be dispatched properly, according to the capacity or running cost of each DG. Furthermore, the MG can shift between the island mode and the gridconnected mode smoothly by using the proposed control method. The proposed adaptive droop control method can overcome the drawback of the tradeoff between the power sharing and the voltage and frequency regulation experienced in the traditional droop control. By using a global information discovery algorithm based on the MAS, each agent only communicates with its direct neighbors, and the global information that is needed for each of the DGs in the MG can be acquired. The smallsignal mode of the DGs in the MG is also set up. The proper current and voltage controller parameters for the inverterinterfaced DG can be found by using the smallsignal mode. The simulation results demonstrate that the proposed method is effective.
Acknowledgements
This work was supported by the Natural Science Foundation of China under contact (61100159, 61233007), the National High Technology Research and Development Program of China (863 Program: 2011AA040103), Foundation of Chinese Academy of Sciences under contact (KGCX2EW104)，financial support of the Strategic Priority Research Program of the Chinese Academy of Sciences under contact XDA06021100.
BIO
Zhongwen Li was born in Henan, China, in 1989. He received his B.S. degree in Control Science and Engineering from Zhengzhou University, Zhengzhou, China, in 2011. He is currently working toward his Ph.D. degree at the Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current research interests include the control of inverter interfaced distributed generators and the optimization of microgrids.
Chuanzhi Zang received his B.S. degree in Application Mathematics and his M.S. degree in Control Theory and Control Engineering from Northeastern University, Shenyang, China, in 1999 and 2002, respectively, and his Ph.D. degree in Mechatronic Engineering from the Graduate School of the Chinese Academy of Sciences, Shenyang, China, in 2006. He is an Associate Professor in the Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current research interest include wireless sensor networks, control theory and smart grids.
Peng Zeng received his B.S. degree in Computer Science from Shandong University, Shandong, China, in 1998, and his Ph.D. degree in Mechatronic Engineering from the Graduate School of the Chinese Academy of Sciences, Shenyang, China, in 2005. From 2005 to 2007, he was an Associate Professor with the Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China, where he was engaged in research on wireless sensor networks. He is presently a Professor with the Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current research interests include wireless sensor networks for industrial automation, smart grids, and demand response.
Haibin Yu received his B.S., and M.S. degrees in Automation Engineering and his Ph.D. degree in Control Theory and Control Engineering from Northeastern University, Shenyang, China, in 1984, 1987, and 1997, respectively. He has more than 25 years of research experience in fieldbus, industrial wireless networks, wireless sensor networks, network controlled systems, and network manufacturing. He is presently the President and a Professor of the Shenyang Institute of Automation (SIA), Chinese Academy of Sciences (CAS), Shenyang, China. His current research interests include basic and applied research in the areas of automation control systems, advanced manufacturing techniques and smart grids. Dr. Yu is a Senior Member of the Instrumentation, Systems, and Automation Society.
Hepeng Li received his B.S. degree in Applications Mathematics and his M.S. degree in Control Theory and Control Engineering from Northeastern University, Shenyang, China, in 2010 and 2012, respectively. He is presently a Research Assistant at the Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current research interests include the control and optimization of microgrids.
Shuhui Li received his B.S. and M.S. degrees in Electrical Engineering from Southwest Jiaotong University, Chengdu, China, in 1983 and 1988, respectively, and his Ph.D. degree in Electrical Engineering from Texas Tech University, Lubbock, TX, USA, in 1999. From 1988 to 1995, he was with the School of Electrical Engineering at Southwest Jiaotong University, where his research interests included the modeling and simulation of large dynamic systems, dynamic process simulations of electrified railways, power electronics, power systems, and power system harmonics. From 1995 to 1999, he was engaged in research on wind power, artificial neural networks, and the application of massive parallel processing. He joined Texas A&M University, Kingsville, TX, USA, as an Assistant Professor, in 1999, and then as an Associate Professor, in 2003. He worked with the Oak Ridge National Laboratory, Oak Ridge, TN, on simulation system development on supercomputers, from 2004 and 2006. He joined the University of Alabama, Tuscaloosa, AL, USA, as an Associate Professor, in 2006. His current research interests include renewable energy systems, smart grids, smart microgrids, power electronics, power systems, electric machines and drives, FACTS, distributed generation, and the application of artificial neural networks in energy systems.
Planas E.
,
GildeMurob A.
,
Andreua J.
,
Kortabarriaa I.
,
de Alegríaa I. M.
2013
“General aspects, hierarchical controls and droop methods in microgrids: A review,”
Renewable & Sustainable Energy Reviews
17
147 
159
DOI : 10.1016/j.rser.2012.09.032
EETD
http://eetd.lbl.gov/sites/all/files/publications/ctrldeesignmicrogridcomponents.pdf
Zamora R.
,
Srivastava A. K.
2010
“Controls for microgrids with storage: Review, challenges, and research needs,”
Renewable and Sustainable Energy Reviews
14
(7)
2009 
2018
DOI : 10.1016/j.rser.2010.03.019
Alanne K.
,
Saari A.
2006
“Distributed energy generation and sustainable development,”
Renewable & Sustainable Energy Reviews
10
(6)
539 
558
DOI : 10.1016/j.rser.2004.11.004
MICROGRIDS
http://microgrids.eu/micro2000/presentations/16.pdf
Huang J. Y.
,
Chuanwen J.
,
Rong X.
2008
“A review on distributed energy resources and MicroGrid,”
Renewable & Sustainable Energy Reviews
12
(9)
2472 
2483
DOI : 10.1016/j.rser.2007.06.004
Justo J. J.
,
Mwasilu F.
,
Lee J.
,
Jung J.W.
2013
“ACmicrogrids versus DCmicrogrids with distributed energy resources: a review,”
Renewable & Sustainable Energy Reviews
24
387 
405
DOI : 10.1016/j.rser.2013.03.067
Peas Lopes J. A.
,
Moreira C. L.
,
Madureira A. G.
2006
“Defining control strategies for MicroGrids islanded operation,”
IEEE Trans. Power Syst.
21
(2)
916 
924
DOI : 10.1109/TPWRS.2006.873018
Katiraei F.
,
Iravani R.
,
Hatziargyriou N.
,
Dimeas A.
2008
“Microgrids management,”
IEEE Power Energy Mag.
6
(3)
54 
65
DOI : 10.1109/MPE.2008.918702
Ustun T. S.
,
Ozansoy C.
,
Zayegh A.
2011
“Recent developments in microgrids and example cases around the worlda review,”
Renewable and Sustainable Energy Reviews
15
(8)
4030 
4041
DOI : 10.1016/j.rser.2011.07.033
Lidula N. W. A.
,
Rajapakse A.D.
2011
“Microgrids research: A review of experimental microgrids and test systems,”
Renewable & Sustainable Energy Reviews
15
(1)
186 
202
DOI : 10.1016/j.rser.2010.09.041
Guerrero J. M.
,
de Vicuna L.G.
,
Matas J.
,
Miret J.
,
Castilla M.
“A wireless controller for parallel inverters in distributed online UPS systems,”
in Proc. IECON
2003
1637 
1642
Barklund E.
,
Pogaku N.
,
Prodanovic M.
,
HernandezAramburo C.
,
Green T. C.
2008
“Energy management in autonomous microgrid using stabilityconstrained droop control of inverters,”
IEEE Trans. Power Electron.
23
(5)
2346 
2352
DOI : 10.1109/TPEL.2008.2001910
Diaz G.
,
GonzalezMoran C.
,
GomezAleixandre J.
,
Diez A.
2010
“Scheduling of droop coefficients for frequency and voltage regulation in isolated microgrids,”
IEEE Trans. Power Syst.
25
(1)
489 
496
DOI : 10.1109/TPWRS.2009.2030425
Majumder R.
,
Chaudhuri B.
,
Ghosh A.
,
Majumder R.
,
Ledwich G.
,
Zare F.
2010
“Improvement of stability and load sharing in an autonomous microgrid using supplementary droop control loop,”
IEEE Trans. Power Syst.
25
(2)
796 
808
DOI : 10.1109/TPWRS.2009.2032049
Hasanzadeh A.
,
Onar O. C.
,
Mokhtari H.
,
Khaligh A.
2010
“A proportionalresonant controllerbased wireless control strategy with a reduced number of sensors for paralleloperated UPSs,”
IEEE Trans. Power Del.
25
(1)
468 
478
DOI : 10.1109/TPWRD.2009.2034911
Chung I. Y.
,
Liu W.
,
Cartes D. A.
,
Collins E. G.
,
Moon S.I.
2010
“Control methods of inverter interfaced distributed generators in a microgrid system,”
IEEE Trans. Ind. Appl.
46
(3)
1078 
1088
DOI : 10.1109/TIA.2010.2044970
Golestan S.
,
Joorabian M.
,
Rastegar H.
,
Roshan A.
,
Guerrero J. M.
“Droop based control of parallelconnected singlephase inverters in DQ rotating frame,”
in Proc. ICIT
2009
1 
6
Chandrokar M. C.
,
Divan D. M.
,
Banerjee B.
“Control of distributed UPS systems,”
in Proc. PESC
1994
197 
204
Guerrero J. M.
,
Vasquez J. C.
,
Matas J.
,
de Vicuña L. G.
,
Castilla M.
2011
“Hierarchical control of droopcontrolled AC and DC microgridsa general approach toward standardization,”
IEEE Trans. Ind. Electron.
58
(1)
158 
172
DOI : 10.1109/TIE.2010.2066534
Sun X.
,
Lee Y.S.
,
Xu D.
2003
“Modeling, analysis, and implementation of parallel multiinverter systems with instantaneous averagecurrentsharing scheme,”
IEEE Trans. Power Electron.
18
(3)
844 
856
DOI : 10.1109/TPEL.2003.810867
Kim J.
,
Guerrero J. M.
,
Rodriguez P.
,
Teodorescu R.
,
Nam K.
2011
“Mode adaptive droop control with virtual output impedances for an inverterbased flexible AC microgrid,”
IEEE Trans. Power Electron.
26
(3)
689 
701
DOI : 10.1109/TPEL.2010.2091685
Jain P.
,
Ranade S. J.
“Capacity discovery in customerdriven microgrids,”
in Proc. NAPS
2009
1 
6
Jain P.
,
Ranade S. J.
,
Srivastava S. K.
“Island identification in customerdriven microgrids,”
in Proc. IEEE PES T&D
2010
1 
7
Xu Y. L.
,
Liu X.
2011
“Novel multiagent based Load restoration algorithm for microgrids,”
IEEE Trans. Smart Grid
2
(1)
152 
161
DOI : 10.1109/TSG.2010.2099675
Xu Y. L.
,
Zhang W.
,
Liu W.
,
Ferrese F.
2012
“Multiagentbased reinforcement learning for optimal reactive power dispatch,”
IEEE Trans. Syst., Man, Cybern. C, Appl. Rev.
42
(6)
1742 
1751
DOI : 10.1109/TSMCC.2012.2218596
Xu Y. L.
,
Liu W.
,
Gong J.
2011
“Stable multiagentbased load shedding algorithm for power systems,”
IEEE Trans. Power Syst.
26
(4)
2006 
2014
DOI : 10.1109/TPWRS.2011.2107754
Guerrero J. M.
,
Vasquez J. C.
,
Matas J.
,
Castilla M.
,
de Vicuna L. G.
2009
“Control strategy for flexible microgrid based on parallel lineinteractive UPS systems,”
IEEE Trans. Ind. Electron.
56
(3)
726 
736
DOI : 10.1109/TIE.2008.2009274
Hu X.
,
Cheng Q.
“Application of UPFC based on improved doubleloop decoupling PI control in photovoltaic systems,”
in Proc. CDCIEM
2012
153 
157
de Brabandere K.
,
Bolsens B.
,
Van den Keybus J.
,
Woyte A.
,
Driesen J.
,
Belmans R.
2007
“A voltage and frequency droop control method for parallel inverters,”
IEEE Trans. Power Electron.
22
(4)
1107 
1115
DOI : 10.1109/TPEL.2007.900456
Guerrero J. M.
,
Matas J.
,
de Vicuna L. G.
,
Castilla M.
,
Miret J.
2007
“Decentralized control for parallel operation of distributed generation inverters using resistive output impedance,”
IEEE Trans. Ind. Electron.
54
(2)
994 
1004
DOI : 10.1109/TIE.2007.892621
Li Y.
,
Li Y. W.
2011
“Power management of inverter interfaced autonomous microgrid based on virtual frequencyvoltage frame,”
IEEE Trans. Smart Grid
2
(1)
30 
40
DOI : 10.1109/TSG.2010.2095046
Strunz K.
,
Fletcher R. H.
,
Campbell R.
,
Gao F.
“Developing benchmark models for lowvoltage distribution feeders,”
in Proc. IEEE Power Energy Soc. Gen. Meet., PES
2009
1 
3
Vandoorn T. L.
,
Meersman B.
,
de Kooning J. D. M.
,
Vandevelde L.
2012
“Analogy between conventional grid control and islanded microgrid control based on a global DCLink voltage droop,”
IEEE Trans. Power Del.
27
(3)
1405 
1414
DOI : 10.1109/TPWRD.2012.2193904
Guerrero J. M.
,
de Vicuna L. G.
,
Matas J.
,
Castilla M.
,
Miret J.
2005
“Output impedance design of parallelconnected UPS inverters with wireless loadsharing control,”
IEEE Trans. Ind. Electron.
52
(4)
11269 
1135
DOI : 10.1109/TIE.2005.851634
Zhang X. T.
,
Zhang H.
,
Guerrero J. M.
,
Ma X.
“Reactive power compensation for parallel inverters without control interconnections in microgrid,”
in Proc. IECON
2008
922 
925
Vandoorn T. L.
,
de Kooning J. D. M.
,
Meersman B.
,
Guerrero J. M.
,
Vandevelde L.
2012
“Automatic powersharing modification of P/V droop controllers in lowvoltage resistive microgrids,”
IEEE Trans. Power Del.
27
(4)
2318 
2325
DOI : 10.1109/TPWRD.2012.2212919
Xiao F.
,
Wang L.
,
Jia Y.
“Fast information sharing in networks of autonomous agents,”
in Proc. American Control Conference
2008
4388 
4393
OlfatiSaber R.
,
Fax J. A.
,
Murray R. M.
“Consensus and cooperation in networked multiagent systems,”
in Proc. IEEE
Jan. 2007
Vol. 95, No. 1
215 
233
Liang H.
,
Choi B. J.
,
Zhuang W.
,
Shen X.
,
Awad A. S. A.
,
Abdr A.
2012
“Multiagent coordination in microgrids via wireless networks,”
IEEE Wireless Communications
19
(3)
14 
22
DOI : 10.1109/MWC.2012.6231155
Boubakri A. E.
,
Lopes L. A. C.
“Phase shift based equivalent virtual impedance loop for droopcontrolled inverters,”
in Proc. EPE’14ECCE
2014
1 
8
Li C. D.
,
Chaudhary S. K.
,
Vasquez J. C.
,
Guerrero J. M.
“Power flow analysis algorithm for islanded LV microgrids including distributed generator units with droop control and virtual impedance loop,”
in Proc. APEC
2014
3181 
3185
1993
IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems, IEEE Std. 5191992
1 
112