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IEEE TRANSACTIONS ON SMART GRID 1
A Distributed Framework for Stability Evaluation
and Enhancement of Inverter-Based Microgrids
Yue Song, Student Member, IEEE, David J. Hill, Life Fellow, IEEE, Tao Liu, Member, IEEE,
and Yu Zheng, Member, IEEE
Abstract—In this paper, a distributed framework is established
to evaluate and improve the small-disturbance stability of a
microgrid with inverter-based distributed generators (DGs). First,
we propose a distributed protocol such that a selected pilot DG
bus collects necessary data from finite steps of communication
between physically neighbouring buses. Then, we design an
algorithm that enables the pilot DG bus to reconstruct the system
dynamic Jacobian matrix from the data and carry out stability
evaluation locally. Further, if the evaluation result does not meet
a preset requirement, we develop a distributed successive linear
programming algorithm to enhance the system stability via sparse
communication among a minority of DG buses. The proposed
algorithms have high efficiency and versatility—they apply to
lossy microgrids with either radial or meshed topology under
various operation scenarios. Moreover, the control actions for
stability enhancement preserve the predefined economic dispatch
and power sharing schemes. The results are validated by the case
study of a 14-bus microgrid.
Index Terms—Microgrid, small-disturbance stability, dis-
tributed method, inverter-based distributed generator
I. INTRODUCTION
MICROGRIDS are low-voltage or medium-voltage
power networks connecting distributed generators
(DGs), loads and storage devices, which can operate in either
islanded mode or grid-connected mode. The microgrid concept
is recognized widely as a key component in future grids. It
not only provides a promising solution to better integrate and
utilize the emerging distributed renewable generation, but also
benefits the users by providing reliable and quality power
supply [1]. Unlike the synchronous generators in traditional
power grids, DGs in microgrids are commonly interfaced with
the network via droop-controlled inverters. These inverters
bring more flexibility to system control but on the other hand,
pose challenges to system stability as it leads to low inertia
and less capacity to resist disturbances. Also, a proper power
sharing among DGs usually requires high droop gains that
make frequency and voltage sensitive to power injections [2].
Manuscript received October 27, 2016; revised February 19, 2017; accepted
March 13, 2017. This work was supported by the Hong Kong PhD Fellowship
Scheme (HKPFS), the Research Grants Council of the Hong Kong Special
Administrative Region under the General Research Fund (GRF) through
Project No. 17202414 and the Theme-based Research Scheme (TRS) through
Project No. T23-701/14-N.
Y. Song, T. Liu and Y. Zheng are with the Department of Electrical and
Electronic Engineering, The University of Hong Kong, Hong Kong (e-mail:
yuesong@eee.hku.hk; taoliu@eee.hku.hk; zhy9639@hotmail.com).
D. J. Hill is with the Department of Electrical and Electronic Engineering,
The University of Hong Kong, Hong Kong, and also with the School of
Electrical and Information Engineering, The University of Sydney, Sydney,
NSW 2006, Australia (e-mail: dhill@eee.hku.hk; david.hill@sydney.edu.au).
However, high gains may cause large frequency and voltage
deviations when the system is subjected to disturbances, which
may introduce negative impacts on system stability. Therefore,
stability issues of microgrids are appealing for careful study
for the sake of secure operation.
This paper considers the small-disturbance stability problem
that refers to the ability of a microgrid to maintain frequency
synchronism and steady-state voltages under small distur-
bances [3, 4]. As a basis of stability analysis, the microgrid
modeling can be generally classified into two types. The first
one attempts to establish a comprehensive model that includes
control loop details in each inverter [5, 6]. This method leads to
a very accurate but inevitably high-dimensional model, so that
the computation is not scalable to larger-sized microgrids and
order reduction may be needed [7]. The second one, called the
phasor-based model, makes a trade-off between accuracy and
efficiency in characterizing system-wide dynamics [8, 9]. This
method takes nodal phase angles and voltage magnitudes as
state variables and neglects the elements whose dynamics are
insignificant, so that good scalability and satisfactory precision
are achieved. Given a microgrid model, small-disturbance
stability is studied by the system dynamic Jacobian, i.e., the
Jacobian matrix of system dynamical equations around an
equilibrium point. Much attention has been paid on the impact
of parameter settings on the eigenvalue profile of the system
dynamic Jacobian, such as droop gain [5], DG capacity [10],
network topology [11], and communication delay [12].
So far the stability analysis of microgrids has been mainly
carried out in a centralized manner, which requires a con-
trol center to collect global information of the system via
complicated measurement and communication architectures.
However, the centralized methods may not work properly due
to the nature of microgrids. For instance, microgrid operation
presents changing features. DGs have intermittency and plug-
and-play capabilities [13], and network topology changes via
reconfiguration [14]. Frequent redesign is needed to adapt
centralized methods to these changes, which is not efficient.
Also, the computational burden of centralized methods may
become too heavy to afford with the increasing number of
devices. On the other hand, the distributed scheme seems a
more promising alternative due to its high flexibility and scala-
bility [15]. Distributed methods have already gained popularity
in microgrid energy management, such as economic dispatch
[16, 17] and voltage control [18, 19]. This success shows the
great potential of distributed methods in microgrid application.
Nevertheless, to the authors’ knowledge, the results on the
distributed analysis of microgrid stability are few. Zhang and
2 IEEE TRANSACTIONS ON SMART GRID
Xie have reported a distributed criterion for small-disturbance
stability of interconnected microgrids [20]. Simpson-Porco and
Bullo have proposed a distributed algorithm to monitor voltage
stability [21]. Therefore, to evaluate and enhance microgrid
stability in distributed ways deserves further investigation.
With the phasor-based model, a distributed computation
framework for small-disturbance stability of inverter-based
microgrids is developed in this paper. We first show that
the system dynamic Jacobian can be constructed by several
sub-matrices with special structural properties. Then, these
properties lead to a stability evaluation algorithm based on
a distributed communication protocol. By this algorithm, a
selected pilot bus can recover the system dynamic Jacobian
from the data obtained in finite steps of communication
between neighbouring buses. Further, we propose a distributed
successive linear programming (SLP) algorithm that outputs
a satisfactory solution to enhance system stability via a few
steps of sparse communication among a minority of DG buses.
The algorithms are computationally efficient and applicable to
microgrids with arbitrary network topology and heterogeneous
line X/R ratios. Moreover, they are flexible to accommodate
the nature of microgrid operation including DG plug-and-
play and topological change. The control actions given by
the stability enhancement algorithm have no influence on the
power flow and power sharing schemes, and thus can be
seamlessly embedded into an energy management system.
The rest of the paper is organized as follows. The dynamic
model of an inverter-based microgrid is formulated in Section
II. In Section III, the small-disturbance model and distributed
stability evaluation approach are presented. Then, the dis-
tributed approach to enhance system stability is developed in
Section IV. Section V discusses the possible extension of the
proposed approaches. Section VI gives a numerical study on
a 14-bus microgrid and Section VII concludes the paper.
Notations: We introduce the following notations that will
be used throughout the paper. Denote Rand Cas the set
of real numbers and set of complex numbers, respectively.
For simplicity, a vector x= [x1, x2, ..., xp]T∈Cpis
denoted as x= [xi]∈Cp, and a diagonal matrix A=
diag{a1, a2, ..., ap} ∈ Cp×pis denoted as A=diag{ai} ∈
Cp×p. The notation Ip∈Rp×pdenotes the identity matrix,
1p∈Rpdenotes a vector with all entries being one, and
ei∈Rpdenotes a vector with the i-th entry being one and
the other entries being zero. The notation |x|takes the modulus
of x∈C.
II. MODELING OF INVERTER-BASED MICROGRID
We consider an islanded microgrid in this paper. Suppose
the islanded microgrid has totally nbuses including gDG
buses and d=n−gload buses, where a DG bus refers to a
bus that connects a DG and may connect a load, and a load
bus refers to a bus connecting a load only. The phasor-based
model is adopted as we concern with the stability issue of
the entire system. The DG inverters are modeled as voltage
sources with the conventional P-ωand Q-V droop controllers
whose diagrams are depicted in Fig. 1. Thus the dynamics of
DG bus ican be formulated as
˙
θi=ωbωi(1a)
ωi=ω∗−Kpi(Pm
Gi −P∗
Gi)(1b)
Vi=V∗
i−Kqi(Qm
Gi −Q∗
Gi)(1c)
F−1
pi ˙
Pm
Gi =PGi −Pm
Gi (1d)
F−1
qi ˙
Qm
Gi =QGi −Qm
Gi (1e)
where θi, ωi, Videnote the phase angle, angular frequency
and voltage magnitude of bus i, respectively; ωbdenotes the
base frequency that is equal to 100πor 120πrad/s; PGi
and QGi denote the active and reactive power generation;
Kpi and Kqi denote the frequency and voltage droop gains;
Fpi and Fqi denote the cut-off frequencies of the low-pass
filters used in power measurement; Pm
Gi and Qm
Gi denote the
measured active and reactive power; ω∗,P∗
Gi and Q∗
Gi denote
the rated frequency, active generation and reactive generation,
respectively. We note that all these variables except θiare in
per-unit values. Then, substituting the Pm
Gi, Qm
Gi in (1b) and
(1c) into (1d) and (1e) gives [9]
˙
θi=ωbωi
F−1
pi ˙ωi=−ωi+Kpi(P∗
i−PGi)
F−1
qi ˙
Vi=−Vi+Kqi(Q∗
i−QGi)
(2)
where P∗
i=P∗
Gi +K−1
pi ω∗,Q∗
i=Q∗
Gi +K−1
qi V∗
i.
For load modeling, a generic form is adopted. The active
and reactive load at bus iare expressed as
PLi =PLi(Vi, ωi)
QLi =QLi(Vi, ωi).(3)
The expression in (3) includes a wide range of load models
extensively used in the study of low-voltage and medium-
voltage networks [22].
(a) The active power-frequency droop.
(b) The reactive power-voltage droop.
Figure 1. The droop control scheme of inverter.
Next we formulate the dynamics of the whole microgrid
with interconnected DGs and loads. We number the DG buses
as VG={1,2, ..., g}, and load buses as VL={g+ 1, ..., n}.
Without loss of generality,we take bus 1 as the angle reference
bus, and introduce new variables αi=θi−θ1,i∈ VG∪ VL
that represent the relative angles with respect to bus 1 and α1
is fixed to zero. The active and reactive power injection at bus
SONG et al.: A DISTRIBUTED FRAMEWORK FOR STABILITY EVALUATION AND ENHANCEMENT OF INVERTER-BASED MICROGRIDS 3
i, denoted Pinj
iand Qinj
i, can be expressed in terms of αias
Pinj
i=PGi −PLi
=V2
iGii +
n
X
j=1
ViVj|Yij |sin(αij −ϕij)
Qinj
i=QGi −QLi
=−V2
iBii −
n
X
j=1
ViVj|Yij|cos(αij −ϕij )
(4)
where Yij denotes the (i, j )entry of the admittance matrix
Y∈Cn×nof the power network; Gij , Bij denote the real
part and imaginary part of Yij ;ϕij =−tan−1Gij
Bij is the
phase shift caused by the resistance of line (i, j ), if Yij =
0, i.e., bus iand bus jare not connected by a line, we set
ϕij = 0; the notation αij is the short for αi−αj; and we
set PGi =QGi = 0 (PLi =QLi = 0) if bus ihas no DG
(load). Then, substituting (4) into (2) and (3) gives the system
dynamics
˙αi=ωb(ωi−ω1), i ∈(VG∪ VL)\{1}(5a)
˙ωi=−fi, i ∈ VG(5b)
˙
Vi=−hi, i ∈ VG(5c)
0 = −fi, i ∈ VL(5d)
0 = −hi, i ∈ VL(5e)
where the functions fiand hiare
fi=KpiFpi (Pinj
i+PLi +K−1
pi ωi−P∗
i), i ∈ VG
Pinj
i+PLi, i ∈ VL
hi=KqiFqi(Qinj
i+QLi +K−1
qi Vi−Q∗
i), i ∈ VG
Qinj
i+QLi, i ∈ VL.
(6)
Remark 1: To focus on the algorithms for stability eval-
uation and enhancement, we adopt the simplified model (5)
which implicitly assumes that all DGs are droop controlled and
all loads are frequency dependent. Also, it should be pointed
out that the proposed algorithms apply to more general cases.
It will be shown in Section V that they can be extended to grid-
connected microgrids, grid-feeding DGs (i.e., constant-power
sources [9, 23]) and frequency-independent loads with some
slight modifications.
Remark 2: It is reasonable to model the power network by
steady-state power flow due to strong time scale separation in
microgrids. By (2), the frequency and voltage time constants
of DGs are mainly determined by F−1
pi , F −1
qi , which are com-
monly in the order of tens or hundreds of milliseconds [5, 24].
On the other hand, the time constant of the power network
is mainly determined by the inductance-to-resistance ratio of
transmission line, which is no more than 3 milliseconds as
X/R is around or less than unity at distribution level [25]. The
network dynamics decay much faster than DG dynamics, and
thus can be neglected.
III. DISTRIBUTED STABILITY EVALUATION OF MICROGRID
A. Small-disturbance analysis in centralized way
Let (ω0,α0,V0)be an equilibrium point of system (5),
where ω0=ω01n,α0= [α0
i]∈Rnand V0= [V0
i]∈Rn
denote the synchronized frequencies, steady-state bus angles
and voltage magnitudes, respectively. The synchronized fre-
quency ω0is not necessarily equal to the rated frequency ω∗
as there is no slack bus in the islanded microgrid to balance
power deficit. Also note that the equilibrium point satisfies the
following power flow equations
0 = fi(ω0,α0,V0), i ∈ VG∪ VL(7a)
0 = hi(ω0,α0,V0), i ∈ VG∪ VL.(7b)
We observe that the values of Fpi , Fqi do not influence the
solution to (7). This property will be further utilized in the
distributed stability enhancement.
Linearizing (5) around the equilibrium point gives the small-
disturbance model
∆˙
x
0=A B
C D∆x
∆y(8)
where x=αT
rωT
GVT
GT∈Rn+2g−1and y=
VT
LωT
LT∈R2ddenote state variables and algebraic
variables, respectively. In xand y, the vector αris de-
fined as αr= [α2, .., αn]∈Rn−1and the variables with
subscript “G” and “L” refer to the components indexed by
DG buses and load buses, respectively. The sub-matrices
A∈R(n+2g−1)×(n+2g−1) and B∈R(n+2g−1)×2drepresent
the Jacobian matrices of right-hand side (RHS) of differen-
tial equations (5a)-(5c) with respect to state variables and
algebraic variables, respectively; and C∈R2d×(n+2g−1) and
D∈R2d×2dare the Jacobian matrices of RHS of algebraic
equations (5d)-(5e) with respect to state variables and algebraic
variables, respectively. By (5), the sub-matrices A,B,C,D
in (8) have the following form
A B
C D =−
0−ωbTG0 0 −ωbTL
∂fG
∂αrSf
P G
∂fG
∂VG
∂fG
∂VL
0
∂hG
∂αrSf
QG
∂hG
∂VG
∂hG
∂VL
0
∂fL
∂αr
0∂fL
∂VG
∂fL
∂VLSf
P L
∂hL
∂αr
0∂hL
∂VG
∂hL
∂VLSf
QL
(9)
where TG∈R(n−1)×gand TL∈R(n−1)×dare sub-
matrices of T=−1n−1In−1∈R(n−1)×nsuch that
T=TGTL; the matrices with superscript “f” describe
the bus frequency sensitivities
Sf
P G =diag{Fpi +KpiFpi Sf
pi} ∈ Rg×g, i ∈ VG
Sf
QG =diag{KqiFqi Sf
qi} ∈ Rg×g, i ∈ VG
Sf
P L =diag{Sf
pi} ∈ Rd×d, i ∈ VL
Sf
QL =diag{Sf
qi} ∈ Rd×d, i ∈ VL
(10)
with Sf
pi =∂PLi
∂ωi
(V0
i,ω0
i)and Sf
qi =∂QLi
∂ωi
(V0
i,ω0
i)being load
frequency sensitivities; the matrix ∂fG
∂αr∈Rg×(n−1) is the
Jacobian matrix of fi,i∈ VGwith respect to the vector αr,
and similar interpretations apply to the other matrices with
partial derivative operators. We assume that the expressions
of A,B,C,Din the following context take values at the
equilibrium point (ω0,α0,V0), and we drop the superscript
“0” on the corresponding variables for simplicity.
4 IEEE TRANSACTIONS ON SMART GRID
We assume the matrix Dis nonsingular. Then eliminating
yin (8) gives
˙
x= (A−BD−1C)x=Jdx(11)
where Jdis the system dynamic Jacobian. The small-
disturbance stability is characterized by the spectrum of Jd.
We refer to the eigenvalue with the maximum real part as the
critical eigenvalue, denoted as λc(Jd)∈C. To achieve an
adequate stability level, a margin should be reserved for the
critical eigenvalue, i.e.,
Re{λc(Jd)} ≤ −λ∗
c(12)
where λ∗
c>0is a preset threshold.
Remark 3: The system has a singularity-induced bifurcation
(SIB) and losses causality if the matrix Dis singular [26].
However, it is commonly adopted in the literature [7, 24]
and supported by numerical experiments on IEEE test systems
that the Jacobian matrix of algebraic equations with respect to
algebraic variables is nonsingular (in our case it refers to D
in (8)). So we will not develop the idea of SIB in this paper.
Furthermore, it will be seen later that the proposed distributed
stability evaluation algorithm is also able to recover the matrix
Dand detect SIB if it does occur.
B. Reformulation of system dynamic Jacobian
Stability evaluation is commonly done in a centralized way
as the sub-matrices A,B,C,Dneed global information.
In this subsection, we reformulate A,B,C,Dinto a new
form with clear structural properties, which make it possible
to obtain them in a distributed way.
By a proper elementary transform, the block expression (9)
of A,B,C,Dcan be reformulated as follows
A B
C D=−E1
−ωbT0
Sf
P G 0
0Sf
P L
Sf
QG 0Jr
pf
0Sf
QL
E2(13)
where E1,E2are the elementary matrices
E1=
In−10000
0Ig000
0 0 0 Ig0
0 0 Id0 0
0 0 0 0 Id
∈R(3n−1)×(3n−1)
E2=
0Ig000
0 0 0 0 Id
In−10000
0 0 Ig0 0
0 0 0 Id0
∈R(3n−1)×(3n−1).
(14)
The matrix
Jr
pf =
∂fG
∂αr
∂fG
∂VG
∂fG
∂VL
∂fL
∂αr
∂fL
∂VG
∂fL
∂VL
∂hG
∂αr
∂hG
∂VG
∂hG
∂VL
∂hL
∂αr
∂hL
∂VG
∂hL
∂VL
∈R2n×(2n−1) (15)
is called the reduced power flow Jacobian as it is the power
flow Jacobian Jpf ∈R2n×2n
Jpf =
∂fG
∂αG
∂fG
∂αL
∂fG
∂VG
∂fG
∂VL
∂fL
∂αG
∂fL
∂αL
∂fL
∂VG
∂fL
∂VL
∂hG
∂αG
∂hG
∂αL
∂hG
∂VG
∂hG
∂VL
∂hL
∂αG
∂hL
∂αL
∂hL
∂VG
∂hL
∂VL
=fαfV
hαhV(16)
with the first column deleted, where fα∈Rn×nis the
Jacobian matrix of fi,i= 1, ...n with respect to αi,i= 1, ...n,
and similar interpretations apply to fV,hα,hV∈Rn×n.
The deletion of the first column is caused by taking bus
1 as the angle reference. Note that (13) is not a similarity
transformation as E16=E−1
2. It cannot be achieved by only
reordering the variables of x,yin (8).
Next we derive the detailed expression of power flow
Jacobian entries. According to (4) and (6), the entries (fα)ij
(fV)ij ,(hα)ij,(hV)ij ,i∈ VGcan be expressed as
(fα)ij =KpiFpi Pn
k=1 ViVk|Yik|cos(αik −ϕik ), i =j
−KpiFpi ViVj|Yij |cos(αij −ϕij ), i 6=j
(fV)ij =
KpiFpi [SV
pi + 2ViGii+
Pn
k=1 Vk|Yik|sin(αik −ϕik )], i =j
KpiFpi Vi|Yij |sin(αij −ϕij ), i 6=j
(hα)ij =KqiFqi Pn
k=1 ViVk|Yik|sin(αik −ϕik ), i =j
−KqiFqiViVj|Yij |sin(αij −ϕij ), i 6=j
(hV)ij =
KqiFqi[SV
qi +K−1
qi −2ViBii−
Pn
k=1 Vk|Yik|cos(αik −ϕik)], i =j
−KqiFqiVi|Yij |cos(αij −ϕij ), i 6=j(17)
where SV
pi =∂PLi
∂Vi,SV
qi =∂QLi
∂Vi,i∈ VGare load voltage
sensitivities. And the entries (fα)ij (fV)ij ,(hα)ij ,(hV)ij,
i∈ VLcan be expressed as
(fα)ij =Pn
k=1 ViVk|Yik|cos(αik −ϕik ), i =j
−ViVj|Yij |cos(αij −ϕij), i 6=j
(fV)ij =
SV
pi + 2ViGii+
Pn
k=1 Vk|Yik|sin(αik −ϕik ), i =j
Vi|Yij |sin(αij −ϕij), i 6=j
(hα)ij =Pn
k=1 ViVk|Yik|sin(αik −ϕik ), i =j
−ViVj|Yij|sin(αij −ϕij ), i 6=j
(hV)ij =
SV
qi −2ViBii−
Pn
k=1 Vk|Yik|cos(αik −ϕik), i =j
−Vi|Yij |cos(αij −ϕij), i 6=j
(18)
where SV
pi =∂PLi
∂Vi,SV
qi =∂QLi
∂Vi,i∈ VL. Further, we introduce
the active and reactive power flow across line (i, j)at the bus
iterminal1
Pij =−V2
i(Gij −Gsi
ij ) + ViVj|Yij |sin(αij −ϕij)
Qij =V2
i(Bij −Bsi
ij )−ViVj|Yij |cos(αij −ϕij)(19)
where Gsi
ij , Bsi
ij are the equivalent shunt conductance and
capacitance at bus iterminal in the π-equivalent circuit of line
1The power flows across a line at its two terminals may not be exact oppo-
site numbers due to the line loss and shunt components in the corresponding
π-equivalent circuit.
SONG et al.: A DISTRIBUTED FRAMEWORK FOR STABILITY EVALUATION AND ENHANCEMENT OF INVERTER-BASED MICROGRIDS 5
(i, j). Combining (17), (18) and (19), (fα)ij (fV)ij ,(hα)ij ,
(hV)ij ,i∈ VGcan be rewritten as
(fα)ij =KpiFpi [−Qinj
i−V2
iBii], i =j
KpiFpi [Qij −V2
i(Bij −Bsi
ij )], i 6=j
(fV)ij =KpiFpi [V−1
iPinj
i+ViGii +SV
pi], i =j
KpiFpi [V−1
jPij +Vi(Gij −Gsi
ij )], i 6=j
(hα)ij =KqiFqi [Pinj
i−V2
iGii], i =j
KqiFq i[−Pij −V2
i(Gij −Gsi
ij )], i 6=j
(hV)ij =KqiFqi [V−1
iQinj
i−ViBii +SV
qi +K−1
qi ], i =j,
KqiFqi[V−1
jQij −Vi(Bij −Bsi
ij )], i 6=j(20)
and (fα)ij (fV)ij ,(hα)ij,(hV)ij ,i∈ VLcan be rewritten
as
(fα)ij =−Qinj
i−V2
iBii, i =j
Qij −V2
i(Bij −Bsi
ij ), i 6=j
(fV)ij =V−1
iPinj
i+ViGii +SV
pi, i =j
V−1
jPij +Vi(Gij −Gsi
ij ), i 6=j
(hα)ij =Pinj
i−V2
iGii, i =j
−Pij −V2
i(Gij −Gsi
ij ), i 6=j
(hV)ij =V−1
iQinj
i−ViBii +SV
qi , i =j,
V−1
jQij −Vi(Bij −Bsi
ij ), i 6=j.
(21)
By the above derivation, the system dynamic Jacobian can
be reconstructed by Jpf ,Sf
P G,Sf
QG,Sf
P L,Sf
QL,T,E1and
E2. The entries of Jpf are closely related to power network
structure, Sf
P G,Sf
QG,Sf
P L,Sf
QL are diagonal matrices, and T,
E1and E2are constant matrices. This inspires a distributed
scheme that only requires finite steps of communication be-
tween neighbouring buses, which will be detailed in the next
two subsections.
C. Distributed framework requirements & discussion
The assumptions adopted in the distributed scheme to be
established are listed below.
(A1) Each bus has data processing capabilities.
(A2) Each bus iknows the following quantities:
(A2-1) its own bus number iand total number of buses n;
(A2-2) DG parameters Kpi, Fpi , Kqi , Fqi ;
(A2-3) load sensitivities Sf
pi, S f
qi,SV
pi, S V
qi ;
(A2-4) line parameters Gii, Bii ,Gij , Bij , Gsi
ij , Bsi
ij ,j∈ Ni
where Niis the set of neighbouring buses of bus i;
(A2-5) the steady-state voltage magnitude Viand power
flow quantities Pinj
i, Qinj
i,Pij , Qij ,j∈ Ni.
(A3) The buses can communicate over the distributed commu-
nication network depicted in Fig. 2.
Assumption (A1) can be realized by installing a micropro-
cessor at each bus, which is practical. For Assumption (A2),
we discuss the required quantities group by group as follows.
(A2-1). This assumption can be relaxed as the number of
buses can be obtained via a distributed consensus algorithm.
We refer to [27] for the details.
(A2-2). It is reasonable to assume that each bus can get the
parameters of its local DG.
Figure 2. Illustration of communication structure.
(A2-3). The load sensitivities need to be carefully identified
as they have significant impacts on system stability [28]. With
the increasing deployment of advanced metering infrastructure,
these parameters can be estimated with satisfactory precision
by measurement-based load model identification. We refer to
[29–32] where various novel identification techniques have
been developed.
(A2-4). The line parameters can be known a priori, and
we assume each bus has stored the parameters of its adjacent
lines. Moreover, line impedance can be online updated with
high precision by e.g., the method in [33].
(A2-5). The nodal quantities Vi,Pinj
i, Qinj
ican be directly
obtained by the local voltage and power injection measurement.
To obtain the branch power flow, a convenient way is to install
a PMU at each bus so that Pij , Qij can be calculated by (19).
Another possible way is to use current sensors that are installed
in each line. This method is more practical for low-voltage
networks [34, 35] as current sensors are much cheaper than
PMUs. In this case Pij , Qij can be calculated by
Pij =ViIij cos δi−ij
Qij =ViIij sin δi−ij.(22)
where Iij denotes the magnitude of line (i, j )current that is
measured by the current sensor at bus iterminal, and δi−ij
denotes the phase difference between Viand Iij , which can
be obtained locally by applying the phase difference detector
[36] to the signals of Viand Iij . Note that the centralized
stability evaluation also requires the parameters of DGs, loads
and lines as inputs to compute the equilibrium point, so the
only extra assumption for the distributed approaches is (A2-5).
As it is not realistic for distributed approaches that each bus
knows the equilibrium point, we adopt (A2-5) to “replace” the
equilibrium point computation so that the stability evaluation
can still be achieved by the proposed algorithm.
For Assumption (A3), we further explain the communica-
tion network in Fig. 2 and give a general idea of the distributed
scheme. The communication network consists of two parts.
Part 1, which is marked by blue dash lines, has the same
structure as the power grid. It enables each bus to communicate
with its physically neighbouring buses. Part 2 of the commu-
nication network, which is marked by red dash lines, is a star
network connecting all DG buses. The hub of this star network
is selected as the pilot DG bus that can communicate with
the other DGs. A distributed communication protocol using
Part 1 of the communication network and an algorithm for
stability evaluation are proposed. The pilot DG bus can recover
the system dynamic Jacobian from the data obtained during
the communication protocol, and carry out stability analysis
6 IEEE TRANSACTIONS ON SMART GRID
locally. The details will be given in the next subsection. In
addition, if the evaluated system stability level is below the
preset requirement defined in (12), a distributed algorithm via
Part 2 of the communication network is triggered to solve an
optimization problem to enhance stability. The details will be
given in Section IV.
D. Distributed scheme to compute system dynamic Jacobian
We now introduce the communication protocol for sta-
bility evaluation in detail. At step kof the protocol,
bus ihas four data vectors for communication, say
pa
i(k),qa
i(k),pb
i(k),qb
i(k)∈R2n, which are updated by the
power flow Jacobian Jpf as follows
Xa(k+ 1) Xb(k+ 1)=fαfV
hαhVXa(k)Xb(k)
(23)
where
Xa(k) = pa
1(k), ..., pa
n(k),qa
1(k), ..., qa
n(k)T∈R2n×2n
Xb(k) = pb
1(k), ..., pb
n(k),qb
1(k), ..., qb
n(k)T∈R2n×2n
(24)
denote the data matrices at step k. The initial values are defined
as Xa(0) = I2nand Xb(0) ∈R2n×2n
Xb(0) =
Sf
P G 0 0 [0g×(d−1) −gind]
0Sf
P L 0[0d×(d−1) lind ]
0 0 Sf
QG 0
000 Sf
QL
(25)
where gind = [i]∈Rg,i∈ VGand lind = [i]∈Rd,i∈ VL.
The entries of gind and lind denote the indices of DG buses
and load buses, respectively. It will be seen later that the
settings of Xa(0) and Xb(0) are used to recover the power
flow Jacobian Jpf and sub-matrices Sf
P G,Sf
P L,Sf
QG,Sf
QL
in the distributed stability evaluation algorithm. Moreover, at
step k, the pilot DG bus receives the data vectors of its
neighbouring buses via Part 1 of the communication network,
and stores them together with its own data vectors, which
will be used for further process. Denote the pilot bus and its
neighbouring buses as the set {j1, ..., jnt}where ntis the
cardinality. Then, we can express the entire protocol as the
discrete system below
Xa(k+ 1) Xb(k+ 1)=Jpf Xa(k)Xb(k)
za
t(k)zb
t(k)=CtXa(k)Xb(k)(26)
where Ct=ej1, ..., ejnt,en+j1, ..., en+jntT∈R2nt×2n.
The matrix Ctselects the data vectors of the pilot DG bus
and its neighbouring buses, so that za
t(k),zb
t(k)∈R2nt×2n
represent the data collected by the pilot DG bus.
Next we show that equation (23) can be realized via
the aforementioned communication network. First, the initial
values Xa(0),Xb(0) can be assigned locally as each bus
knows its own bus number and parameters of its local DG
and load. Second, according to (20) and (21), for each of the
sub-matrices fα,fV,hαand hV, the nonzero entries of its
i-th row are functions of Vi, Vj, P inj
i, Qinj
i, Pij , Qij , i.e., the
information with respect to bus iand its neighbours. Due to
this structural property, equation (23) can be implemented over
Part 1 of the communication network in Fig. 2. Moreover, as
discussed in Assumption (A2-5), Pij , Qij can be measured
without synchrophasors so that PMUs are not necessarily
required.
We can obtain za
t(k),zb
t(k),k= 0, ..., 2nby implementing
protocol (26) for 2nsteps. Further, we define the following
matrices in terms of za
t(k),zb
t(k)
Za
t1=za
t(0)Tza
t(1)T··· za
t(2n−1)TT∈R4ntn×2n
Za
t2=za
t(1)Tza
t(2)T··· za
t(2n)TT∈R4ntn×2n
Za
t=(Za
t1)T(Za
t2)TT∈R8ntn×2n
Zb
t1=zb
t(0)Tzb
t(1)T··· zb
t(2n−1)TT∈R4ntn×2n.
(27)
Let
Za
t=˜
Uz˜
U0
zΣz
0˜
VT
z=˜
UzΣz˜
VT
z(28)
be the singular value decomposition (SVD) of Za
t, where
˜
Uz∈R8ntn×2n,˜
U0
z∈R8ntn×(8ntn−2n)and Σz,˜
Vz∈
R2n×2n. The result below shows that Jpf and Xb(0) can be
recovered from these matrices.
Theorem 1: Suppose the discrete system (26) is observable,
i.e., rank(Ot) = 2n, where
Ot=CT
t(CtJpf )T··· (CtJ2n−1
pf )TT∈R4ntn×2n.
(29)
Then it follows that
1) The power flow Jacobian Jpf can be re-expressed as
Jpf = ( ˜
UT
z1Za
t1)−1R˜
UT
z1Za
t1(30)
where ˜
Uz1,˜
Uz2∈R4ntn×2nare sub-matrices of ˜
Uzsuch
that ˜
Uz=˜
UT
z1˜
UT
z2Tand
R= ( ˜
UT
z1˜
Uz2)( ˜
UT
z1˜
Uz1)−1∈R2n×2n.(31)
2) The initial value Xb(0) can be re-expressed as
Xb(0) = (Za
t1)†Zb
t1(32)
where the superscript †denotes the Moore-Penrose in-
verse.
Proof: 1) It follows from (26) that Xa(k) =
Jk
pf Xa(0) = Jk
pf so that za
t(k) = CtJk
pf . Then by (27) we
have Za
t1=Ot,Za
t2=OtJpf and thus Za
t2=Za
t1Jpf .
Let Jpf =UJΛJU−1
Jbe the Jordan decomposition of Jpf
where ΛJis the Jordan normal form of Jpf and UJis
the transformation matrix. Substituting Za
t2=Za
t1Jpf and
Jpf =UJΛJU−1
Jinto (28) gives
˜
Uz1
˜
Uz2Σz˜
VT
z=Za
t=Za
t1
Za
t2=Za
t1
Za
t1UJΛJU−1
J.(33)
If system (26) is observable, it follows from (33) that
rank(Za
t) = rank(Za
t1) = rank(Ot) = 2n, and thus Σzis
nonsingular. So equation (33) can be reformulated as
˜
Uz1=Za
t1(˜
VT
z)−1Σ−1
z(34a)
˜
Uz2=Za
t1UJΛJU−1
J(˜
VT
z)−1Σ−1
z.(34b)
SONG et al.: A DISTRIBUTED FRAMEWORK FOR STABILITY EVALUATION AND ENHANCEMENT OF INVERTER-BASED MICROGRIDS 7
Substituting (34) into (31) gives
ΛJ=U−1
RRUR(35)
where
UR=Σ−1
z˜
VT
z(Za
t1)TZa
t1UJ.(36)
By (34a) and (36), UJcan be expressed as
UJ= ( ˜
UT
z1Za
t1)−1UR.(37)
Then, substituting (35) and (37) into Jpf =UJΛJU−1
Jgives
(30).
2) It follows from (26) that zb
t(k) = CtJk
pf Xb(0). Thus
we have Zb
t1=OtXb(0) = Za
t1Xb(0) so that (Za
t1)†Zb
t1=
(Za
t1)†Za
t1Xb(0). As mentioned before, Za
t1has full column
rank, so we have (Za
t1)†Za
t1=I2n, which leads to (32).
A similar theorem has been reported in [37, Theorem
3], which requires the matrix for data update to be real
symmetric. Theorem 1 makes a generalization that applies
even if the matrix for data update is asymmetric or even not
diagonalizable, such as the power flow Jacobian Jpf in (26).
The structural property of Jpf has been utilized for other dis-
tributed schemes as well. We refer to [21] where a continuous-
time communication protocol between neighbouring buses is
designed to compute voltage stability indices. In addition, the
assumption on the observability of discrete system (26), which
is the precondition for Theorem 1, holds in practice. This is
supported by [37, Theorem 1], saying that the discrete system
(26) is observable if CtUJhas no column with all zero entries
where UJ∈C2n×2nis the eigenvector matrix of Jpf . The
experiments on numerous IEEE test systems in MATPOWER
package [38] show that UJis a full matrix, which indicates
that the claim is true.
By Theorem 1, the pilot DG bus can recover the sub-
matrices needed in the construction of system dynamic Jaco-
bian Jdby the following steps. First, the power flow Jacobian
Jpf and the initial value Xb(0) can be obtained by (30)
and (32), respectively. Second, it follows from (25) that the
negative entries of the last column of Xb(0) give the indices
of DG buses. Thus, the matrices Sf
P G,Sf
QG,Sf
P L,Sf
QL can
be extracted from Xb(0) according to the indices of DG buses,
and E1and E2can be formed by (14) as gis known. Finally,
Jdcan be reconstructed by (11) and (13), and the system
stability can be evaluated by applying eigen-algorithm to Jd.
The above manipulations are summarized as Algorithm 1.
We make some important remarks on this algorithm.
Remark 4: The novelties of Algorithm 1 are reflected in its
two-step structure such that the whole task is split into the
distributed communication and computation at the pilot DG
bus. The communication only needs finite steps that aims to
generate necessary data for the recovery of system dynamic
Jacobian. The computation is of the “centralized” type, i.e.,
carried out at the pilot DG bus locally, thus many sophisticated
techniques for eigen-algorithm acceleration can be adopted
to achieve very high efficiency [39]. Moreover, since the
complete spectrum of the system dynamic Jacobian can be
obtained by Algorithm 1, the eigenvalues other than the critical
Algorithm 1 (Distributed stability evaluation)
1: Initialization:
•Each bus igets the local parameters listed in the assumptions
(A1)-(A3).
2: Distributed communication:
•Implement protocol (26) for 2nsteps, and the pilot DG bus
collects the data za
t(k),zb
t(k),k= 0, ..., 2n.
3: Matrix recovery at the pilot DG bus:
•Recover Jpf and Xb(0) by (30) and (32).
•Get the indices of DG buses from the number of negative
entries in Xb(0). Extract Sf
P G,Sf
QG,Sf
P L and Sf
QL from
Xb(0). Form E1and E2by (14).
•Form A,B,C,Dby (13). Construct Jdby (11).
4: Stability evaluation at the pilot DG bus:
•Apply eigen-algorithm to Jdand check if inequality (12) holds.
eigenvalue can also be monitored and controlled if they are of
interest.
Remark 5: Algorithm 1 requires no assumptions on the
network topology and line losses. So it is applicable to a
wide range of microgrids, i.e., the network can be either radial
or meshed, the transmission lines can be either lossless or
lossy, and the X/R ratio of transmission lines can be either
identical or non-identical. Furthermore, Algorithm 1 is flexible
to accommodate the operational characteristics of microgrids.
For instance, it can apply to plug-and-play DGs with slight
modification. When a DG is newly connected to a load bus
i, then bus ican participate in the protocol as a DG bus by
including the DG parameters in the relevant entries in (26) and
assign the i-th entry of the last column of Xb(0) as −irather
than i. After recovering Jpf and Xb(0) by Algorithm 1, the
pilot DG bus can learn from the last column of Xb(0) that bus
iis changed to a DG bus. Then the pilot DG bus augments
the sensitivities of bus iinto Sf
P G,Sf
QG as the (g+1)-th main
diagonals, and moves the i-th row and column of each of
fα,fV,hα,hVin Jpf to be the (g+1)-th row and column.
Thus the system dynamic Jacobian can still be constructed.
Similar manipulations apply when a DG at some bus is
disconnected from the grid. In addition, Algorithm 1 applies
in case of network reconfiguration as long as the transmission
lines are switched on/off together with the corresponding
communication links. This is because the changes of topology
and system status are directly reflected into the voltage and
power flow measurements Vi,Vj,Pinj
i,Qinj
i,Pij and Qij in
(20) and (21) that are used in the communication.
Remark 6: In practice, it is inevitable that measurements
have error due to the consistent perturbations in the system
and measuring asynchronism. Also the execution of Algorithm
1 has latency that comes from communication delay. We show
that Algorithm 1 can still achieve satisfactory performance
with these factors included. For a certain load and generation
pattern that typically lasts for 10-15 minutes [40, 41], it is
reasonable to regard that the system variables slightly fluctuate
around their steady-state values. Thus the measurement error
can be described by small-magnitude noise, and the statistics
over a large number of numerical tests in Section VI will
show that Algorithm 1 still works under such circumstances.
For the execution issues, the latency of one communication
step is typically in the order of tens of milliseconds [42].
8 IEEE TRANSACTIONS ON SMART GRID
For a microgrid with tens of buses, Algorithm 1 takes a few
seconds to finish the finite-step communication and eigen-
decomposition at the pilot DG bus. So Algorithm 1 is fast
enough to give stability evaluation for a certain load and
generation pattern.
Remark 7: Communication failure between neighbouring
buses may influence the performance of Algorithm 1 as the
power flow Jacobian entries exactly depend on the network
structure. So a wired architecture such as power line communi-
cation is more suitable to Part 1 of the communication network
due to its high reliability [43]. Also, we can use some more
redundant and reliable communication schemes. For instance,
each bus sends information to its neighbours periodically until
it receives their acknowledgement signal. This method can
reduce the possibility of communication failures at the cost of
extra communication delay. However, this cost is acceptable
since the time scale of Algorithm 1 is much smaller than
that of a load and generation pattern as mentioned before. In
addition, the local eigenvalue computation at the pilot DG bus
brings improved efficiency, but on the other hand, leads to the
vulnerability to single point of failure. To address this issue,
we can adopt a multi-pilot scheme, i.e., to select several pilot
DG buses to collect the data during protocol (26) and carry
out stability evaluation. Thus, the failure of one pilot DG can
be covered by other backup DGs. These pilot DGs can also
work simultaneously and cross check the results via mutual
communication. From this viewpoint, a wireless architecture is
more suitable to Part 2 of the communication network among
DGs. In a wireless network, it is easy to set up and change
(if needed) multiple pilot DGs that are able to communicate
with other DGs [43]. We will further discuss the multi-pilot
scheme in the next section.
Remark 8: It is also worth pointing out that protocol (26)
may generate data with large numbers as the iteration is
unstable. However, it will not cause problems as the required
communication has only finite steps. In addition, the large data
can be avoided by using the matrix M=I−βJpf to replace
Jpf in (26) for data update, where βis a small positive number.
The matrix Mpreserves the same structural property as Jpf ,
so that the protocol can still be implemented distributively.
Then, the pilot DG bus recovers the matrix Mby Algorithm
1, and obtains Jpf by Jpf =β−1(I−M).
IV. DISTRIBUTED STABILITY ENHANCEMENT
A. Formulation of the optimization problem
If the evaluation algorithm gives that the system stability
does not satisfy the preset requirement (12), an optimization
problem to enhance system stability needs to be set up. We
take the cut-off frequencies Fpi, Fqi of DG low-pass filters as
decision variables by the following reasons. First, the low-
pass filters are commonly digital filters, so that their cut-
off frequencies are adjustable. Second, equation (2) can be
rewritten as
˙
θi=ωbωi(38a)
K−1
pi F−1
pi ˙ωi=P∗
i−K−1
pi ωi−PGi (38b)
K−1
qi F−1
qi ˙
Vi=Q∗
i−K−1
qi Vi−QGi.(38c)
The form (38a), (38b) are similar to the swing equation of
a synchronous generator and the item K−1
pi F−1
pi effectively
gives virtual inertia [44]. Overall, (38) implies that the cut-
off frequencies, which are traditionally used to eliminate high-
frequency noise, are also effective for stability improvement.
There is an emerging literature that considers cut-off frequency
tuning as a new tool to regulate system performance [45–
47]. Furthermore, the equilibrium point remains the same after
changing cut-off frequencies as aforementioned. This brings a
merit that stability control actions are decoupled from other
problems such as economic dispatch and power sharing that
are tuned by droop gains [16, 48]. It is a desirable property for
microgrid operation as the control actions can be seamlessly
combined with other predefined schemes.
We analyze how cut-off frequencies affect the system dy-
namic Jacobian before formulating the optimization problem.
Suppose the cut-off frequencies are changed from Fpi, Fqi to
F′
pi, F ′
qi. By (5), (9) and (11), this is equivalent to multiplying
F′
pi/Fpi and F′
qi /Fqi to the corresponding rows of A,B.
So the system dynamic Jacobian is changed from Jdto
J′
d=RJd+Jd, where R=diag{0n−1,Rp,Rq}with
Rp=diag{Rpi},Rq=diag{Rqi } ∈ Rg×gsuch that
Rpi =F′
pi/Fpi −1,Rqi =F′
qi /Fqi −1,i∈ VG. Thus,
by taking Rpi, Rq i as the decision variables, we formulate
an optimization problem to improve the stability margin and
damping ratio of the critical eigenvalue as follows
min
Rpi,Rqi
Re{λc(RJd+Jd)} − wcςc(39a)
s.t. Fmin
pi ≤Fpi(1 + Rpi )≤Fmax
pi , i ∈ VG(39b)
Fmin
qi ≤Fpi(1 + Rqi)≤Fmax
qi , i ∈ VG(39c)
where the variables with superscripts “max” and “min” denote
the upper and lower bounds of the corresponding variables,
respectively; ςc=−Re{λc(RJd+Jd)}
|λc(RJd+Jd)|is the damping ratio of
the critical eigenvalue; and wc≥0is the weight factor of
the damping ratio. The value of wcdepends on the priority
between the stability margin and damping ratio. For instance,
if the real part of the critical eigenvalue is very close to zero,
then it is more important to improve the stability margin so that
wccould be set to a small number. If the real part of critical
eigenvalue is just a bit greater than −λ∗
c, then wccould be set
larger to balance the two objectives.
As it is unreasonable to assume that the pilot DG bus knows
the private parameters such as Fpi, F min
pi , F max
pi of all DGs,
problem (39) cannot be solved locally. A distributed algorithm
that makes use of Part 2 of the communication network in Fig.
2 will be developed.
B. Distributed successive linear programming
The objective (39a) is a nonconvex and implicit function
with respect to the decision variables. This prevents the
application of a distributed primal-dual subgradient algorithm
that works well for convex problems [49]. In addition, it is
not necessary for all DGs to participate in the optimization
as the critical eigenvalue is usually attributed to a minority
of decision variables [50, 51]. Based on these properties,
we design a distributed SLP algorithm to quickly obtain a
SONG et al.: A DISTRIBUTED FRAMEWORK FOR STABILITY EVALUATION AND ENHANCEMENT OF INVERTER-BASED MICROGRIDS 9
satisfactory solution. Given a starting point, the SLP algorithm
solves a nonlinear optimization problem via a sequence of
linear programs. It is particularly attractive for large, sparse
nonlinear programs such as optimal power flow [52] due to its
efficiency, flexibility and easy implementation. We recall that
a conventional SLP commonly follows the procedure below
[53]
Step 1: For the current decision variables xk, solve the lin-
earized model of original problem with a given trust-
region size. Denote the solution as ∆xk.
Step 2: Check if the solution is successful, i.e., whether the
objective function is decreased at xk+ ∆xk.
Step 3: If the solution is successful, update the decision
variables xk+1 =xk+ ∆xk, go to Step 1. Otherwise
go to Step 4.
Step 4: Set xk+1 =xk. Reduce the trust-region size. Go to
Step 1.
The proposed distributed SLP generally follows the procedure.
In particular, in each iteration, we set that only those decision
variables with high control sensitivities update their values via
Part 2 of the communication network, while the other variables
are fixed.
We now establish the algorithm in detail. First, we derive the
linearized model of problem (39). The derivatives of objective
(39a) to Rpi and Rqi can be expressed as
dpi =Re{spi}+wc
(1 −ς2
c)Re{spi} − ςcp1−ς2
cIm{spi}
|λc|
dqi =Re{sqi }+wc
(1 −ς2
c)Re{sqi} − ςcp1−ς2
cIm{sqi}
|λc|(40)
where spi and sqi denote the derivative of λc(RJd+Jd)to
Rpi and Rqi, respectively; spi and sqi can be calculated by
spi =lT∂R
∂Rpi Jdu
lTu, sqi =lT∂R
∂Rqi Jdu
lTu(41)
with u,lbeing the right and left eigenvector of the critical
eigenvalue. Thus the linearized problem at certain Fpi , Fqi can
be formulated as
min
Rpi,Rqi X
i∈Vp
G
dpiRpi +X
i∈Vq
G
dqiRq i (42a)
s.t. F min
pi ≤Fpi(1 + Rpi )≤Fmax
pi , i ∈ V p
G(42b)
Fmin
qi ≤Fqi (1 + Rqi )≤Fmax
qi , i ∈ Vq
G(42c)
−∆R≤Rpi ≤∆R, i ∈ V p
G(42d)
−∆R≤Rqi ≤∆R, i ∈ V q
G(42e)
where ∆Rdenotes the trust-region size of the linearization;
and the sets Vp
G,Vq
Gconsist of DG buses with high control
sensitivities
Vp
G={i|i∈ VG,|dpi|> d∗}
Vq
G={i|i∈ VG,|dqi |> d∗}(43)
where d∗is a threshold to select top ranking variables. The
other variables are not included into (42) as their control
effects are less significant. We can obtain that the solution
to (42) is
Rpi =
min{∆R,Fmax
pi
Fpi −1}, i ∈ V p
G, dpi <0
−min{∆R,1−Fmin
pi
Fpi }, i ∈ V p
G, dpi >0
0, i /∈ Vp
G
Rqi =
min{∆R,Rmax
qi
Fqi −1}, i ∈ V q
G, dqi <0
−min{∆R,1−Rmin
qi
Fqi }, i ∈ V q
G, dqi >0
0, i /∈ Vq
G.
(44)
The explicit expression of (44) leads to the distributed SLP
as follows. We take the current cut-off frequencies Fpi , Fqi
as the starting point. This is because the pilot DG bus has
already obtained Jdat the current Fpi, Fqi by Algorithm 1
and it can directly calculate DG control sensitivities dpi, dq i
and find the top-ranking DGs Vp
G,Vq
Gwithout communication.
The pilot DG bus sets trust-region size ∆Rand forms the
solution (44) after obtaining Fmax
pi
Fpi ,Fmin
pi
Fpi ,i∈ Vp
Gand Fmax
qi
Fqi ,
Fmin
qi
Fqi ,i∈ Vq
Gvia Part 2 of the communication network. Then it
checks whether the solution should be accepted. If the solution
is successful, i.e., the objective of the original problem (39a)
is decreased, the pilot DG bus updates the system dynamic
Jacobian by
Jd←RJd+Jd(45)
and the corresponding DG buses update their cut-off frequen-
cies by
Fpi ←Fpi(1 + Rpi ), i ∈ V p
G
Fqi ←Fqi (1 + Rqi ), i ∈ V q
G
(46)
where Rpi, Rq i are received from the pilot DG bus. Then
these manipulations are repeated by using the updated Jdand
Fpi, Fqi . If the solution is unsuccessful, i.e., the linearization
with the current trust-region size is not “precise” enough, the
pilot DG bus reduces the trust-region size in (44) to regenerate
another solution and checks it again. We summarize the above
iterative process as Algorithm 2.
We make some important remarks below.
Remark 9: The adopted SLP idea leads to an easy and
efficient implementation of the distributed approach. The lin-
earized problem (42) can be interpreted as the linear search
in an “effective” subset of the feasible region of the original
problem (39). The dimension reduction of (42) can reduce
communication cost, and it has a low impact on optimality
as the excluded variables have less significant contributions.
Moreover, solution (44) is an explicit function of the decision
variables. It can be obtained without iterations, which further
speeds up the algorithm.
Remark 10: We further explain the stopping criteria in
Line 16, 18 and 20 in Algorithm 2. The solution satisfying
Line 16 is called a “satisfactory” solution as it achieves
an adequate system stability margin. Although this solution
may not be an optimum, it is desirable if the algorithm can
quickly output such a solution that reaches a balance between
optimality and efficiency. The condition in Line 18 implies
that there exists no feasible decreasing direction around the
10 IEEE TRANSACTIONS ON SMART GRID
Algorithm 2 (Distributed SLP-based stability enhancement)
1: Invoke Algorithm 1 to obtain Jdat the current Fpi, Fq i.
2: if Re{λc(Jd)}>−λ∗
cthen
3: Initialization: The pilot DG bus sets algorithm parameters—
the weight wc, initial trust-region size ∆0
R, maximum iterations
kmax, iteration counter k= 0 and two small positive numbers
dǫ,∆ǫ.
4: loop
5: At the current Jd, the pilot DG bus finds dpi, dqi ,i∈ V
by (40), and determines Vp
G,Vq
G, e.g., DGs with top two dpi and
dqi. The pilot DG bus obtains Fmax
pi
Fpi ,Fmin
pi
Fpi ,i∈ Vp
Gand Fmax
qi
Fqi ,
Fmin
qi
Fqi ,i∈ Vq
Gvia Part 2 of the communication network. Let
∆R= ∆0
R.
6: The pilot DG bus forms the solution by (44).
7: if objective function (39a) is decreased then
8: succ = 1.⊲the solution is successful
9: The pilot DG bus updates Jdby (45).
10: DG buses update Fpi, Fqi by (46).
11: k←k+ 1.
12: else
13: succ = 0.⊲the solution is not successful
14: ∆R←γ∆R(0< γ < 1);
15: end if
16: if Re{λc(Jd)} ≤ −λ∗
cthen
17: Stop and output a satisfactory solution.
18: else if |∆R| ≤ ∆ǫor |dpi|,|dqi | ≤ dǫ,∀i∈ V then
19: Stop and output a local optimal solution.
20: else if k≥kmax then
21: Stop and output an improved solution.
22: else if succ = 0 then
23: Go to Line 6.
24: end if
25: end loop
26: end if
solution, i.e., it reaches an optimum. By [53], the algorithm
will converge to a local optimum if we set kmax → ∞ and
neglect the criterion in Line 16. Considering that the SLP has
global optimum searching ability [53], the dissatisfaction of
Line 16 and satisfaction of Line 18 imply that the stability
requirement (12) may not be achieved by the current control
resources. However, the algorithm can still output a solution
that improves stability. The stopping criterion in Line 20 is
used to avoid excessive iterations in practice. It will be seen
in a case study that the algorithm usually satisfies Line 16
so that a satisfactory solution is obtained with only a few
communication steps, which is the best situation as discussed.
Remark 11: In Algorithm 2, the information sent from
one DG bus to the pilot bus, such as Fmax
pi
Fpi ,Fmax
qi
Fqi , does not
disclose DG private parameters (e.g., Fpi , Fqi ) so that data
privacy is preserved. Note that Part 2 of the communication
network may become dense as the number of DGs increases.
However, communication sparsity is still preserved since only
a minority of links (between the pilot DG bus and those
indexed by Vp
G∪ Vq
G) are activated at each communication
step. From another perspective, Part 2 of the communication
network has g-1 links in total. It requires no more resources
than other communication structures in the literature [12, 19],
which are usually connected graphs among DG buses and
have at least g-1 links. In addition, the single point of failure
problem in Algorithm 2 can also be addressed by the multi-
pilot scheme mentioned in Remark 7. The multi-pilot scheme
can further lead to a more “distributed” structure of Part 2 of
the communication network as shown in Fig. 3. The microgrid
is divided into several subregions. There exists a pilot DG bus
in each subregion that can communicate with other DGs within
this region. There also exists a communication graph that
connect those pilot DG buses. Under this structure, the stability
enhancement can be done by each pilot DG bus rotationally
performing Algorithm 2 with its available DGs. The structure
in the multi-pilot scheme in Fig. 3 avoids long-distance and
dense communication at one pilot DG bus that may occur in
Fig. 2. Newly installed DGs can just establish links to their
closest pilot DG, so that high scalability can be achieved.
Figure 3. Illustration of communication structure in the multi-pilot scheme.
V. EXTENSION OF THE PROPOSED METHODS
The above results have been established in the context of an
islanded microgrid with droop-controlled DGs and frequency-
dependent loads. We further show that the proposed methods
can be extended to other scenarios.
1) Grid-connected microgrids. In this case, the microgrid is
integrated to the main grid via the point of common coupling
that works as a slack bus, which brings two major differences.
First, zero frequency deviation is achieved at the equilibrium
point, however, it causes no difficulties in the analysis. Second,
by taking the slack bus as the angle reference, the system
dynamics can be expressed as
˙αi=ωb(ωi−1), i ∈ VG∪ VL
˙ωi=−fi, i ∈ VG
˙
Vi=−hi, i ∈ VG
0 = −fi, i ∈ VL
0 = −hi, i ∈ VL.
(47)
where the angle and voltage dynamics of the slack bus are not
included as they are given as constants. For such a system,
we can still reformulate the four sub-matrices of the small-
disturbance model as
A′B′
C′D′=−E′
1
−ωbIn0
Sf
P G 0
0Sf
P L
Sf
QG 0Jpf
0Sf
QL
E′
2
(48)
where E′
1,E′
2∈R3n×3nare elementary matrices (different
from the E1,E2in (13)), and Jpf ∈R2n×2nis the power
flow Jacobian with respect to non-slack buses. Comparing (13)
and (48), the methods still apply with slight modifications.
SONG et al.: A DISTRIBUTED FRAMEWORK FOR STABILITY EVALUATION AND ENHANCEMENT OF INVERTER-BASED MICROGRIDS 11
2) Frequency-independent loads. We first consider the case
where there is only one frequency-independent load. Assume
the load at bus i∈ VLis frequency independent, then
ωishould be deleted from system (5) and αibecomes an
algebraic variable. Without loss of generality, we list αias
the last algebraic variable, then the four sub-matrices of the
small-disturbance model become
A′B′
C′D′=
−
0−ωbT′
G0 0 −ωbT′
L0
∂fG
∂α′
rSf
P G
∂fG
∂VG
∂fG
∂VL
0∂fG
∂αi
∂hG
∂α′
rSf
QG
∂hG
∂VG
∂hG
∂VL
0∂hG
∂αi
∂fL
∂α′
r
0∂fL
∂VG
∂fL
∂VL(Sf
P L)′∂fL
∂αi
∂hL
∂α′
r
0∂hL
∂VG
∂hL
∂VL(Sf
QL)′∂hL
∂αi
(49)
where α′
r= [α2, ..., αi−1, αi+1, ..., αn]∈Rn−2;T′
G∈
R(n−2)×gis obtained by deleting the i-th row of the TG
defined in (9); T′
L∈R(n−2)×(d−1) is obtained by deleting
the i-th row and (i-g)-th column of the TLdefined in (9); and
(Sf
P L)′,(Sf
QL)′∈Rd×(d−1) are obtained by deleting the (i-
g)-th column of the Sf
P L,Sf
QL defined in (9), respectively.
Comparing (9) and (49), the pilot DG bus can operate as
follows to include the frequency-independent load. First, bus
iparticipates in protocol (26) with Sf
pi, S f
qi being zero, and
the pilot DG bus obtains the A,B,C,Ddefined in (9) by
Algorithm 1. Second, the pilot DG bus can identify from the
zero main diagonals of Sf
P L,Sf
QL that the load at bus iis
frequency independent. Finally, the pilot DG bus obtains (49)
by adjusting the corresponding rows and columns of A,B,
C,Dso that the stability evaluation and enhancement are
still achieved. For the case of multiple frequency independent
loads, the above manipulations apply in a similar way.
3) Other DG inverter control schemes. The proposed meth-
ods are directly applicable when some DGs work as virtual
synchronous machines (VSMs). This is because the dynamical
equations of a VSM-controlled DG are equivalent to those of
a conventional droop-controlled DG [44]. The methods also
apply to microgrids with the presence of grid-feeding DGs
that can be modeled as constant-power sources [9]. Those
DGs can be regarded as negative frequency-independent loads,
and thus can be treated the same way as discussed in the
last paragraph. Moreover, P-V and Q-ωdroop control may be
adopted for a dominantly resistive microgrid, which leads to
the DG dynamics below
˙
θi=ωbωi
F−1
qi ˙ωi=−ωi+Kqi (Q∗
Gi +K−1
qi ω∗−QGi)
F−1
pi ˙
Vi=−Vi+Kpi(P∗
Gi +K−1
pi V∗
i−PGi).
(50)
However, the system dynamics can still be reformulated into
the form of (13) by some other elementary transform, and thus
the methods still work.
As discussed above, the proposed distributed stability evalu-
ation and enhancement algorithms constitute a general frame-
work. It provides a new perspective for microgrid stability
monitoring and control.
VI. CASE STUDY
We take the 14-bus system in Fig. 4 to demonstrate Al-
gorithm 1 and Algorithm 2. The system is derived from the
IEEE 14-bus system. The readers can refer to MATPOWER
package [38] for the original parameters. The generators at
bus 1, 2, 3, 4, 5 are DGs. To mimic a more realistic microgrid,
we triple the line resistance and reduce the generations, loads
and charging capacitances of lines to 10% of their original
values, and a 1.0+j0.2 MVA load is also added to bus 7.
Assume Kpi = 0.6,Kqi = 7.0and Fpi =Fqi = 15 rad/s
for each DG, and Sf
pi = 3.0, Sf
qi = 1.0,SV
pi =SV
qi = 0.02 for
each load. In practice, those parameters can be obtained by
online identification as aforementioned. We do not discuss the
detailed load modeling here as we need load sensitivities only.
Centralized computation gives that the critical eigenvalue is
-0.371 ±j384.654 under these settings.
We select bus 1 as the pilot DG bus and use the matrix
M=I−10−3Jpf for data update in protocol (26). To
simulate real situations where measurements are with error
due to perturbation and asynchronism, each of the voltage and
power flow quantities used in protocol (26) is the actual value
multiplied by a coefficient η. We assume each ηindependently
follows a normal distribution with the mean ¯η= 1.0and
certain standard deviation σηrepresenting measurement error
level. We compare the critical eigenvalue given by Algorithm
1 with the centralized computation result (-0.371 ±j384.654)
under four scenarios ση= 0.0005,0.0010,0.0015,0.0020.
Specially, the measurement error level in case of ση= 0.0010
is in compliance with IEEE Standard 1459-2000 and recent
power measuring techniques at a three-sigma level [54]. For
each standard deviation ση, we execute Algorithm 1 for 10000
times and depict the probability distribution of the error of
critical eigenvalue in Fig. 5. For the case of ση= 0.0010 (see
the red curves in Fig. 5), the error of the real part is within 15%
and the error of the imaginary part is within 0.2% for almost
all samples, which is satisfactory. And the precision is still
acceptable for higher measurement error levels. In addition,
we test the case where bus 4 is also a pilot DG bus. The error
statistics of the stability evaluation at bus 4 are shown in Fig. 6,
which is almost the same as Fig. 5. It verifies the feasibility of
the aforementioned multi-pilot scheme to addresses the single
point of failure problem.
Figure 4. Diagram of the 14-bus microgrid.
We turn to the distributed stability enhancement. The close-
ness of the critical eigenvalue to the origin indicates that
the system stability is weak. Assume Fmin
qi =Fmin
qi = 5,
Fmax
qi =Fmax
qi = 30 for each DG. By invoking Algorithm
12 IEEE TRANSACTIONS ON SMART GRID
−50 −40 −30 −20 −10 0 10 20 30 40 50
0
0.02
0.04
0.06
0.08
0.1
0.12
Error in percentage %
Probability density
ση=0.0005
ση=0.0010
ση=0.0015
ση=0.0020
(a) Error of the real part.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0
1
2
3
4
5
6
7
8
9
10
Error in percentage %
Probability density
ση=0.0005
ση=0.0010
ση=0.0015
ση=0.0020
(b) Error of the imaginary part.
Figure 5. Accuracy of the critical eigenvalue valuation at bus 1.
−50 −40 −30 −20 −10 0 10 20 30 40 50
0
0.02
0.04
0.06
0.08
0.1
0.12
Error in percentage %
Probability density
ση=0.0005
ση=0.0010
ση=0.0015
ση=0.0020
(a) Error of the real part.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0
1
2
3
4
5
6
7
8
9
10
Error in percentage %
Probability density
ση=0.0005
ση=0.0010
ση=0.0015
ση=0.0020
(b) Error of the imaginary part.
Figure 6. Accuracy of the critical eigenvalue valuation at bus 4.
2 with bus 1 being the pilot DG bus, the system stability
returns to the preset level after two communication steps.
(Parameter settings: λ∗
c= 10.0,wc= 100,∆0
R= 0.4,
∆ǫ=dǫ= 10−4,kmax = 20,γ= 0.5, and select top two
|dpi|,|dqi |as Vp
G,Vq
G.) The eigenvalue profiles and responses
to a small disturbance before and after the optimization are
depicted in Fig. 7(a) and Fig. 7(b), respectively, which show
the significant enhancement on system stability. The decision
variables and critical eigenvalues after each communication
step are listed in Table I. We observe that the solution at each
step is successful by using the initial trust-region size. The
required communication is sparse as only one communication
link is activated at each step—bus 1 to bus 2 at step 1, and
bus 1 to bus 3 at step 2. We note that the system dynamic
Jacobian used in Algorithm 2 is arbitrarily selected from the
10000 tests when ση= 0.0010, which includes measurement
error. The control error caused by this “contaminated” system
dynamic Jacobian are shown in the last row in Table I. It can
be seen that satisfactory control effects are achieved despite
of measurement error.
We set up another experiment as a comparison. Suppose
the pilot DG bus receives the private parameters from all
the other DGs so that it can solve the optimization problem
locally. We also assume that the pilot DG bus uses the same
“contaminated” system dynamic Jacobian as in the last case.
The problem is solved by the centralized SLP where wc,∆0
R
are the same as those in the last case and the stopping criterion
Re{λc(Jd)} ≤ λ∗
cis ignored. Consequently, the centralized
SLP converges to a local optimum after getting seven success-
ful solutions, as shown in Table II. The final solution is a
little better than the one in Table I. However, this approach
impacts data privacy and takes more communication cost than
Algorithm 2 as all links are activated to send parameters to
the pilot DG bus. We also observe that the trust-region size is
reduced at some steps, i.e., some solutions generated by using
the initial trust-region size are not successful. It implies that
the “valid” trust-region size of those low-sensitivity variables
are very small. It brings a dilemma such that a large trust-
region size may impact the precision of linearization, while
a carefully tuned small trust-region size may adds much
complexity and impacts efficiency. Overall, it can be seen that
Algorithm 2 reaches a balance between the optimality and
efficiency.
−105−103−101−10−1
−400
−300
−200
−100
0
100
200
300
400
Real part
Imaginary part
Original settings
After optimization
(a) Eigenvalue profile.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
Time/s
Voltage deviation at bus 1/p.u.
Original settings
After optimization
(b) Voltage deviation under small dis-
turbance.
Figure 7. System stability before and after optimization.
Table I
DECIS ION VARIABL ES AFTE R EACH COM MUNI CATION ST EP IN THE
DIST RIBUTED SLP (PART ICIPATION O F HIGH -SENS ITIV ITY DGS ON LY)
Step 0 1 2
Fp115.00 9.0015.40
Fp215.00 21.00 21.00
Fp315.00 15.00 21.00
Fp415.00 15.00 15.00
Fp515.00 15.00 15.00
Fq115.00 21.00 29.40
Fq215.00 21.00 21.00
Fq315.00 15.00 21.00
Fq415.00 15.00 15.00
Fq515.00 15.00 15.00
λc-0.354 ±j384.524
(-0.371 ±j384.654)2-4.630 ±j184.447
(-4.686 ±j184.191)
-11.389 ±j106.587
(-11.512 ±j107.091)
1The data in bold means it is updated in the corresponding communication step.
2The data in brackets are the actual values with measurement error excluded.
Table II
DECISION VARIABLES AF TER EACH STEP IN THE CENTRALIZED SLP
(PARTI CIPATIO N OF A LL DGS)
Step 0 1 2 3 4 5 6 7
Fp115.00 9.00 12.60 13.23 13.40 13.44 13.46 13.47
Fp215.00 21.00 29.40 30.00 30.00 30.00 30.00 30.00
Fp315.00 21.00 29.40 30.00 30.00 30.00 30.00 30.00
Fp415.00 9.00 12.60 13.23 13.40 13.44 13.46 13.47
Fp515.00 9.00 12.60 13.23 13.40 13.44 13.46 13.47
Fq115.00 21.00 29.40 30.00 30.00 30.00 30.00 30.00
Fq215.00 21.00 12.60 13.23 13.40 13.44 13.46 13.47
Fq315.00 21.00 12.60 11.97 11.82 11.78 11.77 11.76
Fq415.00 9.00 5.40 5.13 5.07 5.05 5.04 5.04
Fq515.00 9.00 12.60 13.23 13.40 13.44 13.46 13.47
Final
λc
-11.800 ±j110.577 (-11.811 ±j110.416)1
1The data in brackets are the actual values with measurement error excluded.
SONG et al.: A DISTRIBUTED FRAMEWORK FOR STABILITY EVALUATION AND ENHANCEMENT OF INVERTER-BASED MICROGRIDS 13
VII. CONCLUSION
We have developed a distributed framework to evaluate
and enhance the small-disturbance stability of a microgrid.
For stability evaluation, an algorithm is designed such that
a selected pilot DG bus collects the data from a finite-step
communication between physically neighbouring buses and
recovers the system dynamic Jacobian by the observability
of the communication protocol. For stability enhancement,
a stability optimization problem is set up and solved by a
distributed SLP algorithm via sparse communication among
a minority of DG buses. These two algorithms apply to
various microgrid operation scenarios. The power dispatch and
power sharing schemes are preserved under the control actions
given by the stability enhancement algorithm. In addition,
the algorithms can be extended to multi-pilot versions that
overcome the failure of single pilot DG and improve com-
munication network scalability. The simulation results show
that the proposed methods have high efficiency and achieve
satisfactory performance in case of voltage and power flow
measurement error in the communication.
ACKNOWLEDGEMENT
The authors wish to thank the reviewers and editor for their
valuable comments that improve this manuscript.
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Yue Song (S’14) received B.S. and M.S. degrees in
electrical engineering from Shanghai Jiao Tong Uni-
versity, China, in 2011 and 2014, respectively. He
is currently working toward the Ph.D. degree in the
Department of Electrical and Electronic Engineering,
the University of Hong Kong.
He received the Hong Kong Ph.D. Fellowship
from the Research Grants Council of Hong Kong,
and CLP Fellowship in Electrical Engineering from
the University of Hong Kong during his Ph.D. stud-
ies. His research interests are in power systems,
stability analysis, and dynamical networks.
David J. Hill (S’72-M’76-SM’91-F’93-LF’14) re-
ceived the B.E. degree in electrical engineering and
the B.Sc. degree in mathematics from the University
of Queensland, Australia, in 1972 and 1974, respec-
tively, and the Ph.D. degree in electrical engineering
from the University of Newcastle, Australia, in 1976.
He is the Chair of Electrical Engineering with the
Department of Electrical and Electronic Engineering,
University of Hong Kong, where he directs the Cen-
tre for Electrical Energy Systems and is the Program
Coordinator for the Multiuniversity RGC Theme-
Based Research Scheme Project on Sustainable Power Delivery Structures
for High Renewables. He is also a part-time Professor and the Director of the
Centre for Future Energy Networks, University of Sydney, Australia. From
2005 to 2010, he was an Australian Research Council Federation Fellow with
the Australian National University, and since 2006, he has been a Theme
Leader of Complex Networks and the Deputy Director with the ARC Centre
of Excellence for Mathematics and Statistics of Complex Systems.
He has held various positions at the University of Sydney since 1994,
including the Chair of Electrical Engineering until 2002 and again from
2010 to 2013, along with an ARC Professorial Fellowship. He has also held
academic and substantial visiting positions at the University of Melbourne; the
University of California at Berkeley; the University of Newcastle, Australia;
the University of Lund, Sweden; the University of Munich; the City University
of Hong Kong; and Hong Kong Polytechnic University. From 1996 to 1999
and 2001 to 2004, he served as the Head of the respective departments in
Sydney and Hong Kong.
His general research interests are in control systems, complex networks,
power systems, and stability analysis. His work is now mainly on control and
planning of future energy networks, and basic stability and control questions
for dynamic networks. Prof. Hill is a fellow of the Society for Industrial
and Applied Mathematics, USA; the Australian Academy of Science; and the
Australian Academy of Technological Sciences and Engineering. He is also
a Foreign Member of the Royal Swedish Academy of Engineering Sciences.
Tao Liu (M’13) received the B.E. degree from
Northeastern University, China, in 2003 and the
Ph.D. degree from the Australian National Univer-
sity, Australia, in 2011.
From January 2012 to May 2012, he was a Re-
search Fellow in the Research School of Engineering
at the Australian National University. During this
period, he also held a visiting scholar position in the
Centre for Future Energy Networks at the University
of Sydney, Australia. From June 2012 to August
2013, he worked as a postdoctoral fellow at the
University of Groningen, the Netherlands. He moved to the University of
Hong Kong in September 2013, and worked also as a postdoctoral fellow
until June 2015. Now he is a Research Assistant Professor in the Department
of Electrical and Electronic Engineering at the same University.
His research interests are in power systems, dynamical networks, distributed
control, event-triggered control and switched systems.
Yu Zheng (M’15) obtained his B.E and Ph. D
degrees from Shanghai Jiao Tong University, China
and The University of Newcastle, Australia, in 2009
and 2015, respectively. He is now a senior re-
search assistant at the University of Hong Kong,
Hong Kong. His research interests include power
electronic applied in power system, power system
planning, and smart grid.
