Small Signal Stability Analysis of Microgrids
Considering Comprehensive Load Models –
A Sensitivity Based Approach
Sayed Mohammad Amelian
System and Energy Study Center
Monenco Iran Consulting Engineers
Tehran, Iran
amelian.mohammad@monenco.com
Abstract— Microgrid concept provides an appropriate context
for installing distributed generation resources and providing
reliability and power quality for sensitive loads. Most of the
literature have analyzed the effects of different power control
strategies of inverter-based distributed energy resources on the
stability of microgrids and only a few of them have addressed
various load models in their formulation and analysis. This
paper proposes a well-structured formulation for state-space
representation of comprehensive static and dynamic load models
in an islanded microgrids environment. Impact of considering
induction motor models on root loci of eigenvalues is analyzed
and sensitivity calculation is performed to account for the effect
of different load model parameters on system oscillatory modes.
Keywords- Microgrid, state-space model, small-signal stability,
sensitivity analysis, induction motor.
I.
INTRODUCTION
Microgrids are absorbing more attention as they enable
integration of renewable and distributed energy resources in
power systems [1]. An important issue in islanded microgrids,
however, is still to maintain stability confronting various
disturbances in islanded mode, because distributed generations
(DG) are usually operating near their maximum capacity and
besides, in microgrids with inverter-based DGs, there is lack
of inertia and system oscillations becomes a problem. Hence,
small-signal stability of the system is of great concern [2].
Many publications have proposed different formulation for
state-space representations of islanded microgrids to analyze
its small-signal stability [2],[3]. Their focus, however, has
been mainly on different power control methods and how
these affect system's stability. Therefore, in almost all of them,
the simplest static models are considered for load modeling
[2]. However, evidences exist from the analysis of large power
systems that dynamic loads are a source of added oscillation
and destabilization as it was demonstrated in [4] that a
reduced, simple system is more prone to instability when
induction motors are considered. The IEEE Task Force
reported that the lack of induction motors in the load models
could be the reason for a major inconsistency between
simulations and field measured data [5].
Almost all of the early works on microgrids have
considered different static load models, only, and studied load
model effects on transient response of system during transfer
from grid-connected to islanded mode [6], during fault
occurrence in microgrid [7], on non-detection zones of
islanding detection techniques [8]. Small-signal stability
analysis of microgrid based on perturbation theory considering
only dynamic load models with uncertainty of microgrid
parameters is studied in [9]. Composite load modeling in an
islanded microgrid has been addressed in [10], but static load
type is only regarded as constant impedance, and besides, the
formulation is somehow complex as it wants to model
induction motor dynamics in a similar way with those of the
network, and there is not any clear conclusion about the effect
of different load model parameters on system stability.
This paper advances one of the mainly used state-space
representations for microgrids with inverter-base DGs and
incorporates detailed static and dynamic load models, i.e.,
composite load models into the formulation by a wellstructured method. Next, small-signal-based comparison is
made between two cases of considering simple RL load types
and full-order induction motor models. In addition, sensitivity
analysis is performed to conclude about the effect of different
load model parameters on system's eigenvalues as a measure
of system stability in terms of damping and stability margin.
II.
GENERAL MICROGRID MODEL
A systematic approach for constructing microgrid model,
adopted from [2], is used in this paper. It divides the whole
system into three major sub-modules; inverter, network and
load. It is considered that state-less impedance models of the
network are inadequate for use with inverter models which
include high frequency modes. Instead a dynamic model of the
network is formed on the common reference frame.
A. Inverter-Based Source Model
In this approach, all of the DG resources are considered to
be interfaced to the grid through inverters. Each inverter is
modeled on its individual reference frame whose rotation
frequency is set by its local power sharing controller. The
inverter model includes dynamic equations of power
controller, voltage and current controllers, output filter, and
coupling inductor. It has an outer power loop based on droop
control to share the fundamental real and reactive powers with
other DGs. Internal voltage and current controllers, designed
to decline high frequency disturbances and damp the output
LC filter to avoid resonance with external network. The statespace model of an individual inverter is constructed by
including the controllers, output filter and coupling inductor
on a synchronous reference frame. One of the inverters' frames
can be arbitrarily chosen as the common reference frame and
all other inverters are translated to this frame. To allow
simpler system representation, d-q axis components of
voltages and currents are used to form system equations.
Reference frame of one of the inverters is taken as the
common frame. To translate the variables from an individual
inverter reference frame into the common frame, an angle δ is
defined for each inverter as the angle between an individual
inverter reference frame and the common reference frame.
     com 


The output variables of an inverter are the output currents
represented as a vector. Using the transformation technique,
the small-signal output current on the common reference
frame can be obtained. Similarly, the input signal to the
inverter model is the bus voltage which is expressed on the
common reference frame. It is to be noted that the inverter
whose reference frame is taken as the common reference
frame has to provide its reference frequency Δωcom to all of the
sub-modules of the model.
The compact form of linearized state-space model of each
inverter can be written as follows:


 x invi   A INV i  x invi   B INV i  v bDQi 
 B i com  com  

 i  C INV i 
 i

  x invi 
 oDQi   C INV ci 
where there are 13 states, three inputs (d-q components of bus
voltage and common frame frequency), and two outputs (d-q
components of output current) in each inverter model:
x invi  [  i , Pi , Q i , dqi ,  dqi ,
i ldqi , v odqi , i odqi ]T
A mg


A INV  B INV R N M INV C INV c


  B 1NET R N M INV C INV c  B 2 NET C INV 
 B 1LOA D R N M INV C INV c  B 2OA D C INV 
Details of state space matrices (AINVi, BINVi, Biωcom, CINVωi,
and CINVci) can be found in [2].
The modeling approach continues to form a sub-model of
all the individual DG inverters and combine them with the
network and individual load models:


 x INV   A INV  x INV   B INV
 i oDQ   C INV c  x INV 
 v bDQ 


Again, state space matrices (AINV, BINV, CINV) are formed
from the above-mentioned matrices and detailed in [2].
B. Network and RL-Type Load Models
State equations of network and loads are represented on
the common reference frame. State equations of line current
connected between two nodes are written based on basic
circuit theories, then linearized, and finally combined to form
the small-signal state-space model of the network:


 i lineDQ   A NET  i lineDQ   B 1NET  v bDQ 
 
 B 2 NET 
where precise array combination of state and input matrices
(A and Bs) can be found in [2], and they are mainly formed by
branch impedances and operation point currents.
Similarly, the small-signal state-space model of generally
considered RL loads is formed:


 i loadDQ   A load  i loadDQ   B1LOAD  v bDQ 
 
 B 2 LOAD 
where, again, the precise array combination of state and input
matrices (A and Bs) can be found in [2], and they are mainly
formed by load impedances and operation point currents.
C. Complete Microgrid Model
As can be seen from (3)-(5), node voltages are treated as
inputs to each subsystem. To ensure the node voltage is well
defined, a virtual resistor is assumed between each node and
ground with sufficiently large resistance, such that its
introduction would have minimum influence on system's
dynamic stability. Finally, the complete microgrid smallsignal state-space model and hence the system state matrix can
be obtained as follows:

 x INV 


 i lineDQ   A mg
 i loadDQ 



 x INV 


 i lineDQ  
 i loadDQ 



The complete system state matrix Amg is given in (7).
B INV R N M NET
A NET  B 1NET R N M NET
B 1LOA D R N M NET


B 1NET R N M load

A load  B 1LOA D R N M load 

B INV R N M NET

III.
LOAD MODELING
In an islanded microgrid with inverter DGs, voltage and
frequency changes occur very often, due to lack of inertia and
reactive power production, whereas DG resources are usually
of small capacity and aimed to produce as much as active
power they can. Therefore, detailed modeling of voltage and
frequency dependency of the loads seems to be necessary for
precise analysis of system behavior, confronting disturbances.
Therefore, in this section, first, static load type modeling
based on standard models is provided. Next, a complete
induction motor model is presented for dynamic parts of loads.
Finally, the composite load model is presented and embedded
in the systematic formulation of the previous section.
A. Static Load Model
For static load models, general model presented in [5] is
used (reactive part of the load is also modeled by
correspondingly same formulae and parameters):
P
V
V
 K Pz ( ) 2  K Pi ( )  K Pp
P0
V0
V0

V n
 K P 1 ( ) pv 1 (1  n pf 1 (f  f 0 )) 
V0
Figure 1. Equivalent circuit of a symmetrical induction machine with
balanced source in an arbitrary reference frame.
As a result, complete 5-order model of induction motors is
considered here. Fig. 1 shows equivalent circuit of a
symmetrical induction motor in an arbitrary reference frame.
Principal equations of the model are:

V n
 K P 2 ( ) pv 2 (1  n pf 2 (f  f 0 ))
V0

where K Pz  1  (K Pi  K Pp  K P1  K P 2 ).
Parameters KPi, KPp, KP1, KP2, npv1, npv2, npf1, and npf2
represent the static characteristics of loads. Voltage magnitude
of connecting bus is represented by V, consisting of DQ
components in synchronous reference frame. Powers, current
and voltage components are related together as following:

VQ
VD
Q 2
V 2
V 
V
V
i Q  P Q2  Q D2
V
V
iD  P
where

i D 
Y v 11 Y v 12  v D  Y f 1 
i  Y v .v Y f .f  
     f  

Y v 21 Y v 22  v Q  Y f 2 
 Q
where Yv11-Yv22, Yf1, and Yf2 can be found in the appendix.
B. Dynamic Load Model- Induction Motor Model
Most of the dynamic load models are based on induction
motor model [11]. In these models, usually used for
conventional power systems, stator circuit transients are
neglected due to the fact that time constants of rotating
machines are much larger than those of the electrical branches.
In the case of microgrid, however, as DG resources are mostly
connected through inverters, whose response times are very
small and network dynamics would influence the system
stability, this assumption doesn’t seem to be correct, anymore.
d  ds

  R s i ds   qs  v ds
b
dt
d  dr
  r
 qr  v dr
  R r i dr 
b
dt
d  qs
dt
d  qr
dt
  R s i qs 

 v
b ds qs
  R r i qr 
  r
 dr  v qr
b


d r
1

T e T m 
2H
dt
 ds   X s  X m  i ds  X m i dr

Re-arranging the equations and linearizing them for smallsignal state-space representation, we could obtain:
1

 b
1

 b
 1

 b
1

 b
1

 b
 dr  X m i ds   X r  X m  i dr 
 qs   X s  X m  i qs  X m i qr
 qr  X m i qs   X r  X m  i qr

ψ, ω, X, and i are flux, speed, reactance, and current
symbols, respectively, where subscripts s, r, and m denote
stator, rotor, and magnetizing branch variables, respectively,
and ωb is the base angular speed. Also, Te and Tm are electrical
and mechanical torques, defined by the following equations:

T e   qr i dr  dr i qr 



T m  T 0 ( r ) 
b

Substituting fluxes in (11) by the current components,
linearizing and re-arranging them into the state-space form, we
obtain:

x IM  A IM x IM  B 1IM v  B 2 IM 
y  C IM x IM


where xIM is the model’s space vector consisting of stator and
rotor d-q current components as well as mechanical speed, Δv
is the stator voltage vector as model’s input, and y is the stator
currents vector as model’s output:
T
x IM   i ds , i dr , i qs , i qr , r 
T
v   v ds , v qs 

T
y   i ds , i qs 
State matrix AIM, input matrices B1IM and B2IM, and output
matrix CIM can be found in the appendix.
C. Composite Load Model
Combining static and induction motor models from (10)
and (14), final composite load model can be obtained:

i load  i static  i IM  Y v .v Y f .f  C IM x IM 
Replacing this equation in (6), we will have:
v bDQ  R N [M INV i oDQ  M NET i lineDQ
 M load (Y v .v bDQ Y f .f  C IM x IM )]
 v bDQ  R NS [M INV i oDQ  M NET i lineDQ
 M load Y f .f  C IM x IM ]

where R NS  (I 2 m 2 m  R N M loadY v )1 R N .
IV.



MODAL AND SENSITIVITY ANALYSIS
Once the complete small-signal model has been formed,
eigenvalues (or modes) could be identified that indicate the
frequency and damping of the system oscillations. Sensitivity
analysis is then conducted which provides the sensitivity of
different modes to the load model parameters and points out
the role of each in forming of these modes. States associated
with modes that are not of interest in a particular problem can
then be considered for removal from the model in order to
simplify the analysis. This represents a systematic approach to
finding appropriate model parameters and avoids the danger of
neglecting a system feature that later turns out to be important.
Sensitivity factor is the measure of the association between
system parameters and modes and is equal to sensitivity of the
eigenvalue λi to the parameter Kj of the system state matrix.
Sensitivity factors can be calculated using A as system state
matrix, ωi and i, as right and left eigenvectors, respectively:
A
iT
i
K j
i




K j
iT i
Right eigenvector gives the mode shape, i.e., the relative
activity of the state variables when a particular mode is
excited. The left eigenvector identifies which combination of
the original state variables displays only the ith mode.
V.
Figure 2. Sample medium voltage microgrid

ANALYSIS RESULTS
A medium voltage distribution network with DG resources
is used for analysis of the derived model. Modeling is
performed in Matlab program. Capacities of DG resources and
system information (lines and loads) are detailed in [12].
Using the state-space modeling of sections II and III, small
signal stability analysis of the microgrid with detailed load
models will be performed and sensitivity analysis will also be
accomplished as in section IV. As there are lots of publication
relating to critical modes of islanded microgrids associated
with inverter power controls, here we will focus on system
modes which are directly attributed to load models [2],[3].
Typical values for load model parameters are used as of
corresponding benchmark values in [5]. Fig. 3 shows the
microgrid eigenvalues participated mostly by the loads that are
completely modeled as induction motors in this case.
For each load, there are five eigenvalues (two complex
conjugate pairs and one real). The pairs with frequency close
to the rated value (2π×50 Hz=314 rad/s) are related to stator
windings. Their frequency is closer to nominal value specially
in larger motors case. Real part of this pair is larger in small
motors, which shows better damping of lower rated motors’
disturbances. The other pair is attributed to the rotor circuit
and its real part as well as its frequency decrease with the
increase of motor capacity. This represents lower damping and
more influence on system dynamics. The only real eigenvalue
is related to mechanical part of the motors.
300
200
Imaginary

100
Rotor Modes
Stator
Modes
0
Mechanical
Modes
-100
-200
-300
-16
-14
-12
-10
-8
Real
-6
-4
-2
Figure 3. System modes participated by induction motor states
0
Sensitivity analysis of the microgrid critical modes with
regard to load model parameters is performed next. Using
(18), the sensitivity of each critical mode is calculated related
to each load model parameter and the calculated values are
summed for each single parameter. Result of this analysis is
important as it implies which of the load model parameters are
more effective on the islanded microgrid stability and hence
should be accurately modeled and valued, when planning and
operating microgrids with certain types of consumers.
Table II presents the sensitivity of critical models to
induction motor parameters, which are obtained considering
100% dynamic models for differently rated industrial loads.
It can be seen that in all the cases, sensitivity to the rotor
resistance (Rr) is the largest and to the magnetizing reactance
(Xm) is the lowest. In addition, total sensitivity is larger for
load no. 5, which was anticipated as it is bigger than the other
two, hence affects the system dynamics more.
TABLE I.
CHANGE OF MODES AFTER CONSIDERING INDUCTION
MOTOR LOAD MODEL
Eigenvalue
No.
Load
(kW)
RL Load Model
3
350
-129.917 ± j313.2126
Induction Motor Load Model
5
507.5
-23.6918 ± j311.899
7
53.9
-195.7602 ± j313.2167
Load
Number
3
5
7
TABLE III.
Changes of root loci of the critical modes are presented at
last to show the parameter effects, visually. Loads no. 3, 5, 7
are modeled as differently rated induction motors and the
system eigenvalues are obtained for three different cases of
changing stator reactance (Xs), stator resistance (Rs), and rotor
resistance (Rr) from 0.9 to 1.1 of rated value. The results are
provided in Fig. 4. As can be observed, increasing Xs shifts the
decreases modes' damping and stability margin. In load no. 5
(as largest motor), this displacement is lesser, specially for
rotor circuit and mechanical modes. Changes in Rs displace
the modes more than previous case, hence confirming the
obtained results, but it doesn’t cause any special effect on
rotor and mechanical modes as the stator resistance represents
only losses and has nothing to do with mutual interactions of
magnetic fields of stator and rotor currents. Changes in Rr has
a significant impact on mechanical mode due to related effect
on motor slip and hence the mechanical speed. This mode has
moved into the right-half plane for 0.9 rated value in smaller
motor (no. 7), and has become unstable. Rotor reactance (Xr)
has the same impact as Xs, but mutual reactance (Xm) and
inertia (H) have almost no impact on mode locations.
1.0 Xs
200
Xr
Xm
Rs
Rr
H
Total
0.264
0.342
0.037
0.364
0.484
0.049
0.028
0.034
0.004
3.148
4.549
0.477
3.930
5.040
0.496
2.135
2.915
0.314
9.868
13.364
1.376
SENSITIVITY OF CRITICAL MODES TO STATIC LOAD MODEL
PARAMETERS
Parameter
Sensitivity
Parameter
Sensitivity
KPz
KQz
KPi
KQi
KPp
KQp
KP1
KQ1
KP2
0.265
0.024
0.130
0.011
0.260
0.021
4.557
0.677
8.659
KQ2
npv1
nqv1
npv2
nqv2
npf1
nqf1
npf2
nqf2
0.297
0.026
0.005
0.066
0.008
0.863
0.122
2.159
0.169
1.1 Xs
100
0
-100
-200
SENSITIVITY OF CRITICAL MODES TO INDUCTION MOTOR
(DYNAMIC LOAD MODEL) PARAMETERS
Xs
0.9 Xs
300
-300
-14
-12
-10
-8
-6
-4
-2
0
Real
a) Change of stator reactance from 0.9 to 1.1 rated value
Imaginary
TABLE II.
-8.099 ± j313.12i
-9.896 ± j22.614i
-2.1666
-7.105 ± j313.14i
-6.046 ± j21.015i
-0.2846
-11.676 ± j313i
-12.046 ± j24.171i
-1.741
Similarly, sensitivity analysis of the critical modes related
to static load model parameters is also performed and the
results are presented in Table III. The largest effects are due to
frequency dependent terms (KP1, KP2, KQ1, KQ2), which
signifies the importance of modeling of frequency dependency
of loads in islanded microgrids, where the frequency is also
changing more often because of lack of inertia. This frequency
dependency has been completely ignored in conventional
power systems [11]. High sensitivity to other frequency
dependency coefficients (npf1, npf2) is another evidence for this.
Imaginary
Next, changes in root loci of induction motor load modes
are analyzed, comparing with simplified RL load model case.
Table I represents these changes, only for industrial loads. As
can be seen, modeling the loads with induction motors has
caused the eigenvalues to become much closer to imaginary
axis and decreased their damping. As a result, modeling the
large industrial loads of the microgrid merely by simple RL
branches could never represent the accurate system model.
300
0.9 Rr
200
1.0 Rr
1.1 Rr
100
0
-100
-200
-300
-12
-10
-8
-6
Real
-4
-2
0
b) Change of rotor resistance from 0.9 to 1.1 rated value
Figure 4. Displacement of system modes due to changes in induction motor
parameters
VI.
CONCLUSION
This paper addressed small-signal stability in converterbased microgrids with comprehensive static and dynamic load
models. An integrated load model formulation is adapted into
state-space representation approach of the microgrid. Small
signal stability along with sensitivity analysis of different load
model parameters on mode damping and system stability
margin has been performed. The analysis has revealed that
selection of only the simple static load models cannot be
satisfying in an advanced analysis aiming to provide an
accurate view on the microgrid stability. It has been shown
that considering induction motor loads notably decreases
stability margin for load modes. Also, considering frequency
dependency in static load models is very crucial. Finally, the
importance of rotor resistance value of the induction motor
model on system stability has been highlighted.
A IM
 r X m2

b b 

v Q 0 
1 v D 0 
[ 2  P0 v D 0 aPv   2 Q 0 v D 0 aQv
2 
V 0 V 0 
V 0 
2v
 P0   P0 v D 0  Q 0 v Q 0  ( D2 0 )]
V0
Y v 12 
Y v 12 
Y v 21 
v Q 0 
1 v D 0 
[
P v a   2 Q 0 v Q 0 aQv
2 
2  0 Q 0 Pv
V 0 V 0 
V 0 
2v Q 0
 Q 0   P0 v D 0  Q 0 v Q 0  ( 2 )]
V0
v D 0 
1 v Q 0 
[
 P0 v D 0 aPv   2 Q 0 v D 0 aQv
V 02  V 02 
V 0 
2v
 Q 0   P0 v Q 0  Q 0 v D 0  ( D2 0 )]
V0
1 v Q 0
[
V 02  V 02

v D 0
 P0 v Q 0 aPv   2

V 0

Q 0 v Q 0 aQv

2v Q 0
 P0   P0 v Q 0  Q 0 v D 0  ( 2 )]
V0
Yf1 
1
 P0 v D 0 aPf  Q 0 v Q 0 aQf 
V 02 
Yf 2 
1
 P0 v Q 0 aPf  Q 0 v D 0 aQf 
V 02 
aPv  2K Pz  K Pi  K P 1n pv 1  K P 2 n pv 2 , aPf  K P 1n pf 1  K P 2 n pf 2
aQv  2K Qz  K Qi  K Q 1nqv 1  K Q 2 n qv 2 , aQf  K Q 1nqf 1  K Q 2 nqf 2
B 1IM
X r  X m



 Xm



 b 
0



0


0

0
0
X r X m

Xm

0






 , B 2 IM  b






  qs 0 


 b 
  qr 0 


 b 
  
  ds 0 
 b 


   dr 0 
 b 


 0 
1 0 0 0 0 
C IM  
 , where   X s X m  X r X m  X s X r .
0 0 1 0 0 
0
X r X m
Rr

r X m ( X r  X m )
b


r X
  r

b 
b

Xm
i qs 0
2H
 X m (X r  X m )
b


r X m2   r

b 
b
X r X m
Rs

0
0
X s X m
Rr


Xm
i dr 0
2H

2
m

r X m ( X r  X m )
b

APPENDIX
Y v 11 
X r X m

Rs
 



0


 r X m2


 b  
b b 

 r X m ( X r  X m )


 b

Xm
i qr 0


2H
Xm
i ds 0
2H



 qr 0 


b 


0

 dr 0 

b


T 0 r 1 


2H b 
0
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