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Robust Control of DC Converter in DC Microgrid Applications Using Linear
Matrix Inequality (LMI)
Conference Paper · October 2019
DOI: 10.1109/CPERE45374.2019.8980220
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2019 IEEE Conference on Power Electronics and Renewable Energy (CPERE)
Robust Control of DC Converter in DC Microgrid
Applications Using Linear Matrix Inequality (LMI)
Magdi A. Mosa
Electrical Power and Machines Engineering
Helwan University
Cairo, Egypt
magdimosa@yahoo.com
Ahmed A. Ali
Electrical Power and Machines Engineering
Helwan University
Cairo, Egypt
ahmedtawoos33@gmail.com
A. M. Abdel Ghany
Electrical Power and Machines Engineering
Helwan University
Cairo, Egypt
ghanyghany@hotmail.com
Adel A. El Samahy
Electrical Power and Machines Engineering
Helwan University
Cairo, Egypt
el_samahya@yahoo.com
Abstract— The aim of this paper is development of a
distributed robust controller for distributed generators (DGs)
interfaced to a DC microgrid through a boost converter. The
suggested controller is robust static state feedback local
controller based on H2/H∞/pole-placement/polytope technique.
The proposed technique considers the system robustness
against uncertainty, the system speed, and the system damping
simultaneously, through construction a set of linear matrix
inequalities (LMI) and polytopic representation of the system.
The control problem is solved using LMI optimization
approach, Using MATLAB LMI toolbox, because of LMI
ability to handle multi-objective design problems. To test the
effectiveness of the proposed technique, different simulation
cases are carried out under diverse disturbances such as
generation outage and variation of constant power loads
(CPL). Eventually, the results of the suggested technique are
compared with three other tuning techniques known as
H2/H∞, H2/H∞/pole-placement, and H2 techniques. The
comparisons prove the superiority of the proposed technique.
Keywords: LMI, Polytope, H2/H∞, pole-placement, Boost
converter, DC microgrid.
I.
INTRODUCTION
Today, increasing the price of electricity, the cost of
fossil fuels, development of remote areas, and increasing the
environmental concern, enhance the applications of
distributed generators (DGs) which are based on renewable
energies such as solar and wind or small scale conventional
generation units such as microturbines or non-conventional
generation units such as fuel cells [1]. Integration of these
DGs with loads in a local manner to improve the system
reliability and attain a common objective is defined as
modern microgrid [2]. Connection of DGs to common DC
bus is called DC microgrid which provides several
advantages compared to AC microgrid such as simple
control, no reactive power, higher cables capacity, etc. [3].
Each DG is interfaced to the DC bus using power electronics
converter according to the DG's DC voltage as shown in Fig.
1 [4]. In order to enhance the merits of DGs applications in
microgrid, a suitable control system is required. The control
system should maintain the proper operation of the microgrid
at presence of different types of disturbances such as
generation outage, variation of available generation capacity,
variation of loads, etc. [5].
The control system can be implemented in centralized
manner, which gathers all real-time data from different DGs
978-1-7281-0910-7/19/$31.00 ©2019 IEEE
Helmy M. El Zoghpy
Electrical Power and Machines Engineering
Helwan University
Cairo, Egypt
helmy_028123288@yahoo.com
and provides a suitable control signal to each DG. This
scheme limits the scalability of the microgrid, suitable only
for small number of DGs, needs communication network,
and suffers from single point failure [6]. On the other hand,
the control system can be distributed along the microgrid.
The later scheme is based on local controllers built inside the
control unit of each DG. The local controllers perform its
task fast and the overall system does not suffer from single
point failure. However, decentralized controllers must be
robust against interactions between DGs connected to the DC
microgrid [7].
In this paper, a distributed controller is designed at each
DG. The proposed local controller comprises of two loops,
the outer loop controls the output voltage of the DG’s
converter and the inner loop enhances the system response
by controlling the converter’s current. The current controller
is a conventional PI controller tuned using frequency
response according to the required crossover frequency and
phase margin. However, the proposed voltage controller is
robust static state feedback (SSF) controller. The
conventional SSF controller is usually tuned based on pole
placement using a well-known Ackerman’s formula [8,9] or
optimal state feedback based on linear quadratic regulator
(LQR) rule by solving Algebraic Riccati Equation (ARE)
[9]. Pole placement approach gives direct information about
the system damping and speed but does not give direct
information about the control input and state signals. On the
contrary, The LQR method gives an idea about the input and
state signal integration, however does not provides clear
information about the system damping and speed.
Furthermore, the two methods do not give a clear idea about
the system robustness against uncertainty.
Herein, static state feedback controller is tuned using H2,
H2/H∞, H2/H∞/Pole-placement, and the proposed H2/H∞/poleplacement/polytope techniques. H2 is the most similar to
LQR, so that it considers the state and input signal in the
design stage. H∞ enhances the stability of the control system
against presence of uncertainty and disturbances. Poleplacement takes into account the system speed and damping.
Polytope ensures the system stability against structure
uncertainty. Tuning the state feedback controller using the
aforementioned techniques needs special tools capable of
dealing with the requirements and constraints of each
controller. The applied tool is linear matrix inequality (LMI)
approach.
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2019 IEEE Conference on Power Electronics and Renewable Energy (CPERE)
The paper is organized as follow: Section 2 describes the
proposed local controller at each DG. Section 3 demonstrates
the design procedure for the current loop. Section 4
illustrates the design of robust voltage control using H2, H∞,
and mixed H2/H∞ objectives. Section 5 develops the polytope
representation model of the controlled DC boost converter
connected to the microgrid. Section 6 elaborates the
constraints to allocate the closed loop poles in a certain
region in the complex plane. Section 7 applies different
control technique on a small microgrid consists of only two
distributed generators and different characteristics load.
Lastly, section 8 provides the conclusion and contribution of
this work.
III.
CURRENT CONTROLLER
The current control loop of DGi is shown in Fig. 3.
Where: G (s) is the transfer function between the converter
( )
inductor current and duty cycle of DG number i, and
is the PI current controller transfer function which is given
by (1). Where K , K are the controller’s proportional and
integral gains and the last i in symbol refer to the converter
number. The controller gains are deigned to achieve a
crossover frequency of 1 kHz and a phase margin of 60˚. The
crossover frequency is selected to be smaller than the
converter switching frequency (20 kHz) [12]. The controller
gains are obtained using (2) and (3) [9,12,13].where: w and
⌀ are the crossover frequency and gain margin of current
loop of DGi respectively.
i L_refi +
Gci(s)
-
Gpi(s)
i Li
Fig. 3 Current control loop of DGi
K
S
cos(π + ⌀ − ∠G (jw))
G (s) = K +
Fig. 1 Example of modern DC microgrid
II.
K
PROPOSED CONTROL SYSTEM
The proposed local control system of each boost
converter consists of two loops. The inner loop controls the
converter inductor current and the outer loop controls the
converter output voltage as shown in Fig. 2. The current
controller is proportional integral (PI) controller, while the
voltage control loop is static state feedback (SSF) controller
with additional integrator to eliminate the steady state error.
The voltage and current controllers operate in cascaded
manner since the voltage controller provides the reference
current to the inner current loop in order to eliminate the
error in the converter output voltage, after that the current
controller gives the final control signal (converter’s duty
cycle) in order to achieve the required reference current. The
droop gain, block of Fig. 2, is used to share the power
between DGs inside the microgrid. This droop scheme is
similar to governor control of conventional power station,
while here the converter output voltage at steady state decay
with increasing the converter output power. The variation
should be kept with ±5%. The details of droop scheme can
be found in [10,11].
Voltage controller
vo +
-
vref +
-

=
Current controller
K
iLref +
d
PI
-
v
Converter
Inner Current loop
Outer voltage loop
kd
p
Droop gain
Fig. 2 Proposed control scheme
iL
K =
G (jw)
−w sin(π + ⌀
− ∠G (jw))
G (jw)
(1)
(2)
(3)
ROBUST VOLTAGE CONTROLLER
IV.
Microgrids are suffering from many sources of
disturbances such as plug in or plug out of generation units,
variation of the loading conditions, or variation in the
environmental conditions such as PV and wind. In addition,
the model of DG is suffering from many sources of
uncertainties due to non-linearity of the DG's models, nonlinearity in constant power loads, and interaction between
generation units. Moreover, presence of constant power load
deteriorates the system damping [14]. Subsequently, the
controller design should take into considerations these
factors.
In order to overcome the system nonlinearity and
uncertainty, the voltage controller may be adaptive controller
or robust controller. In this work, a robust static state
feedback based on H2/H∞/Pole-placement/Polytope controller
is suggested. The design problem is modeled as a multiobjective optimizing problem subjects to a set of linear
matrix inequality constrains (LMI). Primarily, to generalize
the design procedure, the system model should be obtained
in standard form as depicted in Fig. 4 [15]. where:
x, w, and u are the system state variables, the exogenous
input, and the control input respectively. z and z are the
system performance indicators. K is the state feedback gain
matrix and ∆(s) is the uncertainty model. The generalized
sate space model is described by (4) to (6) and the
corresponding closed loop state space model is given by (7)
to (10).
=
386
+
+
(4)
2019 IEEE Conference on Power Electronics and Renewable Energy (CPERE)
=
=
+
+
+
+
z∞
Controlled system
(Plant)
u
x
z2
Nominal plant
including the
controller G(s)
k
Fig. 4 Generalized model of the controlled system
=
=
(9)
+
(10)
Where:
=
+
=
=
,
+
,
+
=
,
,
=
The performance indicators are selected to minimize the
error of voltage loop e , minimize the effect of disturbances
on the output voltage ∆v , and limit the controller output
to avoid the saturation as follow:
signal i
z = e + Ri
(11)
z = ∆v
(12)
Where: R represents weighting factor. Now based on the
standard sate space model and the proposed system variables,
the static state feedback matrix K is determined using
different techniques. The voltage control loop should be
slower than inner current loop, robust stable against grid
interaction, and robust stable against constant power loads
applications.
A. H2 controller
H2 controller enhances the system speed through
minimization of the second norm of the transfer function
from the exogenous input to the system performance output
[16]. This optimization problem can be represented in time
domain or frequency domain. In order to combine the H2
controller with other objectives such as system robustness
requirements, the LMI optimization approaches emerge as
powerful tool. In LMI approaches, each objective is
represented by one or more matrix inequality.
The
equivalent LMI model of H2 controller problem is
summarized as follow: find the symmetrical positive definite
matrices S and Y, and matrix L that minimize the trace of
matrix Y without violation the constraints of (14) and (15)
[17].
( )
+
.
.
.
(13)
0
+
+
0
+
+
0
+
(14)
<0
(15)
(16)
1
‖G(jw)‖
(18)
The origin of H∞ control problem is constructed in
frequency domain as given by (18). In order to combine the
H∞ and H2 objectives, the LMI approach is used. There are
many equivalent LMI mathematical representations of H∞
control problem [7,16,20], one of them is given by (19)
[16,17]. Inequality (19) means, the system is stable against
uncertainty of magnitude ‖∆(jw)‖
γ
if there is a
matrix L and symmetrical positive definite matrix S 0
attain (19).
(8)
+
=
‖∆(jw)‖ <
(7)
+
(17)
B. H2 / H∞ controller
H∞ controller minimizes the effect of disturbances on the
system output. These disturbances appear due to the
exogenous input to system or presence of uncertainties in the
system model. According to small gain theorem [18], the
range of uncertainty while the system holds its stability is
given by (18), where G(s) is the nominal plant model
including the controller models and ∆(s) is the uncertainty
transfer function [19].
Δ (s )
w
=
Then the controller gains
(5)
(6)
AS + B L + SA + L B
B
C S
B
−I
D
SC
D
−γ
(19)
<0
H∞ controller boosts the system disturbance rejection and
reinforces the system damping, while the system response
becomes slower. On the other hand, H2 controller enhances
the system speed [16]. Therefore, the mixing between H2 and
H∞ controllers considers both of the system speed and
disturbance rejection. This problem is known as mixed H2/
H∞ control problem [16]. The LMI representation of mixed
H2 and H∞ problem as follow:
( )+
+
.
.
(20)
0
+
+
+
+
+
+
+
(21)
(22)
<0
+
.
<0
−
(23)
−
.
0
0
Then
(24)
=
(25)
Where: μ and ρ are weighting factors, if = 1 and =
0 it is the H2 control problem while if = 0 and = 1 it
becomes H∞ control problem.
V.
POLYTOPE REPRESENTATION
In microgrid applications, DGs subject to several sources
of disturbances such as variation of loads, nonlinearity of
constant power loads, variation of grid configuration, etc.
The control system is robust stable if it preserves the system
stability at all operating condition. However, design of the
control system at specific operating point does not guarantee
the system stability at another point. Therefore, the DG is
represented by as set of models belong to ( , ℬ), each pair
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2019 IEEE Conference on Power Electronics and Renewable Energy (CPERE)
represents the DG's model at particular operating condition
[21]. Theoretically, the DG has infinite number of models,
whilst to manage the problem a few numbers of models are
developed and called vertices [22].
= {A , A A , … , A }
(26)
ℬ = {B , B , B , … , B }
(27)
The set of distributed generator models are called
polytope representation, if for any operating condition the
system matrix is related to the vertices using (28) [22,23]. If
the control system is stable at all vertices of the polytope, it
will be stable at any model satisfies (28). Consequently, the
system will be robust stable [23].
( )=
,
≥ 0 ∀ ∈ {1,2, … ,
}
=1
(28)
In this paper, the boost converter of Fig. 5 is modeled
using two vertices(A , B ), (A , B ). Since, A1 and A2 are the
system matrices at minimum and maximum system
resistance respectively. Equations (29) and (30) represent
the vertices of the polytope model of boost converter
including the inductor current control loop.
Jω
ωn
Jω d
−Jωd
Fig. 6 different pole placement regions
A. minimum decay time (α)
The polytope system poles lie to the left of vertical line at
α in complex plan if there is symmetrical positive definite
matrix S satisfies (32) [17,25]. Special case, if α=0 the LMI
region is the open left side of the complex plane, which is
equivalent to Lyapunov stability inequality.
SA
SA
A =
0
0
−1
−∑
A =
0
0
−1
VI.
−
−1
0
0
−
−1
0
0
0
0
0 B = 0
1
0
0
0
(29)
−ω S
< 0 ∀q ∈ {1,2, … , m}
(33)
sin(β) A S + SA
cos(β) A S − SA
cos(β) SA
sin(β) A S + SA
−A S
<0
(34)
∀q ∈ {1,2, … , m}
(30)
D. Maximum damped natural frequency ( )
The polytope system closed loop roots are placed in a
strip between (−Jω and Jω ) in the complex plane, if exist
a symmetrical positive definite matrix S fulfills (35). Table1
gives the parameters of suggested LMI regions.
REGIONAL POLE PLACEMENT
Design of H2 or H∞ controllers does not provide
information about the pole location or the system dynamics
directly [24]. In this section, the controller is designed to
optimize the H2/ H∞ controller while the polytope system
poles lie in certain region in the complex plan. The system
poles lies in region if there is symmetrical positive definite
matrix S satisfies (31) [24,17]. Where: T and M define the
LMI region and ⨂ is the Kronecker product of matrices. The
complex region satisfies (31), denoted as , is called LMI
region. The poles regions can be defined using different
segments such as maximum natural frequency ω ,
minimum decay time α, minimum damping ratio ξ, and
maximum damped natural frequency ω as shown in Fig. 6.
ℳ (A, S) ≔ T⨂S + M⨂(SA) + M ⨂(A S) < 0
A S
C. Minimum damping ratio ( )
The polytope system eigen values are located in the cone
of angle β in complex plane, if there is a symmetrical
positive definite matrix S achieves (34) [17]
Fig. 5 Fuel cell connected to the microgrid using boost converter
0
0 B = 0
1
0 0
0
0 0
0 0
(32)
B. Maximum natural frequency ( )
The polytope system closed loop poles belong to a
circular disk of radius ω and center at the origin, if there is
a symmetrical positive definite matrix S attains (33) [17].
−ω S
0
∀q ∈ {1,2, … , m}
+ A S < 2αS
(31)
−2ω S
SA
A S − SA
−A S
−2ω S
< 0 ∀q ∈ {1,2, … , m}
(35)
TABLE 1 LMI REGION PARAMETERS
6283.2 rad/sec
-100 sec-1.
0.8
3141.6 rad/sec
α
ξ
VII. SIMULATION RESULTS
In this section, the different control techniques are
applied on a microgrid comprises of two Fuel cell generators,
CPL, and resistive load as shown in Fig. 7. The controllers’
gains, using different techniques, are collected in table 2. Fig.
8 exhibits the eigen values of the polytope representation,
where o and * marks indicate the closed loop poles at vertex
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2019 IEEE Conference on Power Electronics and Renewable Energy (CPERE)
1 and vertex 2 respectively. Four different simulation cases
are performed to compare between the applied techniques.
450
H2
PCC Voltage (Volt)
440
H2/H
∞
430
H2/H /PP
∞
420
H2/H /PP/poly
∞
410
400
420
402
390
410
400
380
370
0.5
400
0.6
0.5
1 1.05 1.1
1
Time (Sec)
1.5
Fig. 9 System response at connection and disconnection of Two DGs
B. Case 2: Changing of the resistive load
Fig. 10 displays the system response for step change in
the resistive load. The load increases by 44 kW at 0.6 sec
and 0.8 sec, after that decreases again at 1 sec. During steady
state, at all loading conditions, under different tuning
techniques, the microgrid grid voltages is kept within the
permissible limits, while the proposed technique, H2/
H∞/pole placement/ polytope, is capable of keeping the
system more damped and fast dynamic response.
Fig. 7 DC microgrid consists of two fuel cells and DC load
TABLE 2 CONTROLLER GAINS UNDER DIFFERENT TUNIING TECHNIQUES
Controller type
Controller gains
H2
-0.1929, -0.00, -10.7880, 100
H2/H∞
-2.6410 , -0.0005, -144.9271, 100.0005
H2/ H∞/pole placement
-4.2, -0.000, -309.64 , 2106.3
H2/ H∞/pole placement/
-22.9, -0.3, -2379.4, 4369.0
polytope
420
FC Eigen Values
8000
410
PCC Voltage (Volt)
6000
4000
Imaginary
2000
0
Desired
Poles
Region
390
380
H2/H
∞
-2000
200
H2/H /PP
370
∞
0
-4000
H2/H /PP/poly
-200
-6000
360
-300-200-100 0 100
0
Real
∞
0.6
0.7
0.8
0.9
Time (Sec)
1
1.1
1.2
Fig. 10 Response of the controlled system when step changing in the
resistive load with a step of 44 kW
2,000 4,000 6,000 8,000
Fig. 8 Closed loop eigen values of polytope representation for boost
converter
A. Case 1: Plug in and plug out the generators
Fig. 9 shows the DC bus voltage for DGs connection at
0.5 sec and disconnection at 1 sec. The system response is
more damped and less oscillated at DGs connection, while
the damping and oscillation become worse at disconnection
due to changing in the operation condition. The system
response with H2 controller has higher overshoot, oscillation
and long settling time. Adding of H∞ controller improves the
system damping, while the system still has high overshoot.
Inserting regional pole placement constraints in the design
problem of H2/H∞ enhances the system speed. Employing the
concept of polytope improves the system behavior at
connection and disconnection instants as shown in figure 9.
At instant of DGs disconnection, the system controlled by H2
controller goes outside the limits for a period of time. From
now H2 controller is omitted from the results because H2
controller only is not robust. Recall, according to droop
control concept, the steady state DG's voltage depends on the
amount of delivered power from the DG; therefore the
voltage during DG's integration differs than the voltage
before integration. During steady state periods, the microgrid
voltage is maintained within the permissible range between
380 volt and 420 volt.
C. Case 3: Changing of constant power load
In this case, the power value of CPL increases by 44 kW
at 0.6 sec and 0.8 sec, and decreases again at 1sec.
430
420
PCC Voltage (Volt)
-8000
-8,000 -6,000 -4,000 -2,000
400
410
400
390
380
H2/H
∞
370
H2/H /PP
∞
360
H2/H /PP/poly
∞
350
0.6
0.7
0.8
0.9
Time (Sec)
1
1.1
1.2
Fig. 11 Response of controlled system with step changing in constant
power load with a step of 44 kW
Fig. 11 presents the system response at step changes in
constant power loads using different techniques. The
technique
produces
higher
H2/H∞/pole-placement
oscillation, compared to resistive load of Fig. 10 with
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2019 IEEE Conference on Power Electronics and Renewable Energy (CPERE)
increasing the load due to the negative incremental
resistance of CPLs. For the same reason, H2/H∞ has higher
overshoot and undershoot compared to the same technique
with the resistive loading condition. It is clear, the proposed
H2/H∞/pole-placement/polytope technique provides better
response compared to other techniques, and keeps the
microgrid voltage within acceptable limits during transient
and steady state intervals.
D. Case 4: Change of fuel cell voltage
Herein, the microgrid total load is kept constant around
50 kW and fuel cell stack voltage is reduced at 0.4 sec by
20% and returned back to its nominal value at 0.6 sec. Fig.
12 shows that, the proposed technique is capable of
achieving a well damped and fast response compared to other
techniques. The DC output voltage is the same during steady
state periods because the load is maintained constant.
[4]
[5]
[6]
[7]
[8]
[9]
[10]
410
PCC Voltage (Volt)
[11]
405
400
[12]
H2/H
∞
395
H2/H /PP
∞
[13]
H2/H /PP/poly
∞
390
0.3
0.4
0.5
0.6
Time (Sec)
0.7
0.8
[14]
Fig. 12 Response of controller at 20 % change in th fuel celle output
voltage
[15]
VIII. CONCLUSION
[16]
Based on the presented work, merging of polytope
system's representation with regional pole placement and
H2/H∞ technique not only enhances the system robustness,
but also keeps the system dynamics with the desired
damping and speed. The suggested H2/H∞./pole
placement/polytope is capable of manipulating the structured
uncertainty and unstructured uncertainty throughout H∞
Controller and polytope representations respectively.
Moreover, Using of H2/H∞./pole placement /polytope, as
local controller for DGs in DC microgrid application, proves
its capability of maintaining the microgrid stability during
generation outage, generation in, variation of constant power
load, and variation of the DG's characteristics. Eventually,
the proposed controller depends on the local parameters of
the corresponding DG, therefore strengthens the scalability
feature of the microgrid.
[17]
[18]
[19]
[20]
[21]
[22]
REFERENCE
[1]
[2]
[3]
Faisal A. Mohamed and Heikki N. Koivo, "System modeling and
online optimal management of microgrid with battery storage," in
IEEE, International Conference on Power Engineering, Energy and
Electrical Drives, Setubal, Portugal, 2007.
Tomislav Dragičević, Xiaonan Lu, Juan C. Vasquez, and Josep M.
Guerrero, "DC microgrids-part I: Review of control strategies and
stabilization techniques," IEEE Trans. on Power Electronics, pp. 4876
– 4891, 2016.
Ahmed T. Elsayed, Ahmed A. Mohamed, and Osama A. Mohammed,
"DC microgrids and distribution systems: An overview," Electric
Power Systems Research, pp. 407–417, 2015.
View publication stats
[23]
[24]
[25]
390
Moataz Elsied, Amrane Oukaour, Hamid Gualous, and Radwan
Hassan, "Energy management and optimization in microgrid system
based on green energy," Energy, pp. 139-151, 2015.
Renke Han, Michele Tucci, Raffaele Soloperto, and Josep M.
Guerrero, "Plug-and-play design of current controllers for gridfeeding converters in DC microgrids," in 11th Asian Control
Conference (ASCC), Gold Coast Convention Centre, Australia, 2017.
Antonis Tsikalakis, "Large scale integration of micro-generation to
low voltage grids, deliverable_DC1," 2004.
H. Bevrani, Y. Mitani, and K. Tsuji, "Robust decentralised loadfrequency control using an iterative linear matrix inequalities
algorithm," IEE Proc.-Gener. Transm. Distrib, 2004.
Ahmed Bensenouci, Abdel Ghany M. Abdel Ghany, and Ahmed
M.A. Alzahrani, "Proportional-integral state-feedback control for
wind power plant," in 2nd International Conference on Electrical and
Electronics Engineering, 2008
Katsuhiko Ogata, "Modern Control Engineering" New Jersey:
Pearson Education, Inc, 2010.
Inam Ullah Nutkani, Poh Chiang Loh, and Frede Blaabjerg, "Droop
scheme with consideration of operating costs," IEEE Trans. On Power
Electronics, pp. 1047 – 1052, 2014.
Josep M. Guerrero, Juan C. Vasquez, José Matas, Luis García de
Vicuña, and Miguel Castilla, "Hierarchical control of droop
controlled AC and DC microgrids — A general approach toward
standardization," IEEE trans. on industrial electronics, pp. 158–172,
2011.
Omar Hegazy, Joeri Van Mierlo, and Philippe Lataire, "Analysis,
Modeling, and Implementation of a Multi device Interleaved DC/DC
Converter for Fuel Cell Hybrid Electric Vehicles," IEEE Trans on
power electronics, pp. 4445- 4458, 2012.
Uğur Demiroğlua and Bilal Şenolb, "Analytical design of PI
controllers for first order plus time delay systems," International
scientific and vocational journal (ISVOS), pp. 40-47, 2018.
Dong Chen and Lie Xu, Technologies and applications for smart
charging of electric and plug-in hybrid vehicle.: springer, 2017.
Raoul Herzog and Jurg Keller, "Advanced control an overview on
robust control," 2011.
A.M. Abdel Ghany, "Design of Robust PID Controllers Using a
Mixed H2/H∞ with Pole-Placement and H∞ Techniques based on
Simulated Annealing Optimization for a Power System Stabilizer,"
Engineering Research Journal 121, 2009.
M. Chilali, P. Gahinet, and P. Apkarian, "Robust pole placement in
LMI regions," IEEE Trans. on Automatic Control, pp. 2257–2270,
1999.
Dennis S. Bernstein and David C. Hyland, "Small Gain Versus
Positive Real Modeling of Real Parameter Uncertainty," in the 30 th
IEEE Conference on Decision and Control, Brighton, UK, 1991.
Leonidas D. Dritsas, "Notes on robust and H8 control," 2011.
Yong Yan Cao, James Lama, and You-Xiam Sun, "Static Output
Feedback Stabilization: An ILMI Approach," Automatica, pp. 16411645, 1998.
Mahdieh S. Sadabadi, Qobad Shafiee, and Alireza Karimi, "Plug-andPlay Robust Voltage Control of DC Microgrids," IEEE, Trans.on
Smart Grid, pp. 6886–6896, 2018.
Ricardo C. L. F. Oliveira, Maur´icio C. de Oliveira, and Pedro L. D.
Peres, "Robust State Feedback LMI Methods for Continuous-Time
Linear Systems: Discussions, Extensions and Numerical
Comparisons," in IEEE International Symposium on Computer-Aided
Control System Design (CACSD), Denver, CO, USA, 2011.
D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou, "A new
robust D-stability condition for real convex polytopic uncertainty,"
2000.
Mahmoud Chilali and Pascal Gahinet, "H8 Design with Pole Place
Constraints: An LMI Approach," IEEE Trans on automatic control,
pp. 358-367, 1996.
Feng Zheng, Qing-Guo Wang, and Tong Heng Lee, "On the design of
multivariable PID controllers via LMI approach," Automatica, pp.
517–526, 2002.