This document describes research into using a fractional order PID (FOPID) controller to control voltage fluctuations in an islanded microgrid with a single power source. The proposed FOPID controller has more tuning parameters than a standard PID controller, allowing it to better regulate the microgrid's output voltage under different load conditions and uncertainties. The controller is designed using an optimization technique to maximize system performance. Simulation results show the FOPID controller is effective at reducing voltage fluctuations and provides a fast, robust response for the microgrid system.

1. Fractional Order Robust PID Controller Design for Voltage
Control of Islanded Microgrid
Abstract—This paper emblems the application of fractional
order PID (FOPID) controller into a single phase islanded
microgrid having a single power source to control the fluc-
tuations in its output voltage. The proffered controller has
adaptability in devise as the controller provides more parame-
ters than integer Order PID controller (IOPID)to tune it. The
proffered controller is devised using Nelder-Mead optimization
technique. Use of the optimization technique gives increased
performance of the system. The controller is applied into the
system under uncertainties and different load settings. After
assessment of performance, it is observed that the application
of the proffered controller can cut down the voltage fluctuations
of the system and afford fast response with robust performance.
Index Terms—Control, Microgrid, Fractional Order PID.
1. Introduction
The appetence of an uninterrupted flow of low-cost elec-
tricity is rising with the advancement of modern technology
and distributed energy production systems are being set up
in low exhaustion areas for the realization of the appe-
tence [1]. These types of energy supplies are worthwhile
as they are occupied adjacent to the consumer loads [2].
Green energy appliances like geothermal power, PV solar
generators, wind generators, biogas, hydroelectric plants are
some of the distributed generators [3]. These distributed
energy generators get associated to energy storages as well
as loads and construct a system which is known as microgrid
system [1]. Microgrid gets connected to the main grid to
meet the appetence of the electric energy. The construction
of microgrid includes slightly one distributed energy gen-
erator, energy reservoir and consumer loads. The system
gets more responsive, robust and rapid than conventional
synchronous generators because of getting interfaced with
power electronic converters [4]. Grid connected mode and
islanded mode are two operational conditions followed by
the microgrid [5], [6]. Point of common coupling (PCC) is
a system that is used by microgrid to get connected to the
main grid in grid connected mode. The main gets support
by using the grid connected mode and it also reduces the
burden on nonrenewable energy resources. In this mode,
Voltage and frequency ratings of the supply line comes from
the main grid and the microgrid system can maintain both
discharging and receiving network with the main grid. In
the act of breakdown of the regular grid, the islanded mode
supports the significant loads and supply continuous power
to the associated loads. These breakdown occurs due to the
poor power quality, voltage collapse, faults in the regular
grid and also for natural calamities. It supports both main
grid and regional loads by increasing the accuracy of the
generated electric power.
Sustainable energy resources make the islanded microgrid
system disparate from the regular grid control [7]. These
resources are the instigators for DG units. The conduction
of the sustainable energy resources rely on different factors
like weather, speed of wind, power of sunlight, speed of
water flow etc and these factors make the output voltage of
the microgrid system to fluctuate and cause risky action in
use [8], [9].
Number of control schemes have been proffered to control
the microgrid voltage. A leading scheme is droop control
method [10].It restricts the precision in sharing power due
to divergence in voltage. Decentralized control [11] and
Distributed control [12] schemes have been brought in to
improve the precision in sharing power by removing voltage
divergence. It suffers the low bandwidth in communication
network.
Hierarchical control scheme provides steady-state perfor-
mance and advanced bandwidth in the control [13] of mi-
crogrid voltage. The performance of this controller depends
on 3 consecutive levels. The levels can be termed as pri-
mary control, secondary control and tertiary control. It may
deviates the performance due to fall down of one level that
ensure the poor performance of MG.
Linear quadratic regulator (LQR) [14], [15] can be
used in getting better voltage regulation and simultaneous
load sharing in microgrids. The system gets shortage in
robustness while working with LQR controller, if the plant
dynamics is shifted [16].
Model Predictive Controller (MPC) have been proffered
for robust performance in microgrid [17] . The design of
MPC controller depends on order of the system. Controller
of higher order is needed for controlling higher order system
and this phenomenon can feel necessity for having progres-
sive system of digital signal processing.
Proportional-Integral-Derivative (PID) controllers have
been proffered to regulate voltage in microgrid [5]. The PID
controller has the major drawback of having lower band-
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2. Voltage Source
Inverter
dc
v
L
C
line
R
Load
G
v
L
i G
i
C
i
dc
s v
v
)LJSinglePhaseIslandedMicrogrid
width and very low in robustness when the plant dynamics
is shifted [18].
The intention of this article is solely to compensate the
fluctuation of microgrid voltage in various load dynamics.
Variation in load dynamics may endure a lot of fluctuation
in microgrid voltage that may cause risky activity of the MG
system . Motivated by this fluctuation problem in microgrid,
a high performance robust apprehend controller is proffered
in this article. benefaction of this article is to devise the
fractional order PID (FOPID) controller for maintaining the
performance of microgrid system. The proffered controller
may provide more tuning parameters compared to integer
order control. The number of tuning parameter provides the
minimum level of steady state error and higher bandwidth.
The benefits of proposed controller is its design flexibility,
obvious implemention and independence of the system or-
der. Parameter values of proffered controller has been found
using trial and error method with respect to the appropriate
performance indication.
The remaining of the article has been standardized as
follows. Modeling of the islanded microgrid is presented in
section II. Section III describes the devision of the FOPID
controller to control the above system. Section IV describes
the evaluation of microgrid perfomance using the proposed
controller. Conclusion of the article has been carried out in
Section V.
2. Modelling of Microgrid
Schematic of a single phase islanded microgrid sourced
by a single power generator is expounded in Fig. 1. It is
observed from the Fig. 1 is that the inductor voltage vL is,
vL = L
diL
dt
(1)
Hence,
diL
dt
=
vL
L
(2)
where iL is the inductor current.
Here,
vs = vL + vG (3)
Hence,
vL = vs − vG (4)
From Eq. (2),
diL
dt
=
vs − vG
L
(5)
where vG is the grid voltage and vs is the voltage after
inverted by VSI. The voltage is the multiplication output of
the duty cycle (α) with dc source voltage (vdc).
The laplace transformation of the Eq. (2)
VL(s) = sLiL(s) (6)
Hence,
IL(s) =
VL(s)
sL
=
Vs(s) − VG(s)
sL
(7)
The capacitor voltage which can be termed as the grid
voltage can be accessed as,
dvG
dt
=
1
C
iC (8)
where iC can be termed as the current through capacitor.
The state space representation of a system can be shown as,
dx
dt
= Ax + Bu (9)
and
y = Cx + Du (10)
Where, x = [iL; vG] represents the state vector, A =
[0, −1/L; 1/c, 0] represents the state coefficient matrix, B =
[1/L; 0] represents the control vector, u = vs represents
the input control variable, y = [0 : vG] represents the
output vector, C = [0, 1] represents the output coefficient
vector, D = 0 represents the transient vector. Here, d = iG
represents disturbance that is created as a result of unknown
composition in microgrid system. It is originated during the
turn on or off of the load during the activation period of
microgrid. From equation (9) and (10)it is accessed,
d
dt
iL
vG
=
0 − 1
L
1
C 0
iL
vG
+
iL
0
vs
+
0
− 1
C
iG
and the system output,
y =
vG
=
0 1
iL
vG
The ratings of the components of the microgrid plant are
presented by Table 1.
TABLE 1: Component ratings of microgrid plant
Components Ratings
Source DC voltage (Vdc) 300V
Capacitance (Ct) 15μF
Inductance (Lt) 2mH
Wire Resistance (Rline) 0.45 Ω
Load Resistance (R) 40 Ω
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3. 3. Design of Controller
The devise of PID controller of fractional order using
trial and error method is discussed abruptly in this section.
3.0.1. Fractional Order Controller. Fractional Order Con-
trol (FOC) is an abstraction of the Integer Order Con-
trol(IOC) into the order of fraction. [19]. The Order ab-
straction goes here:
Dα
=
⎧
⎨
⎩
dα
dtα , ifα 0
1, ifα = 0
t
0
dτ−α
, ifα 0
⎫
⎬
⎭
(11)
where α Re is the order of the fractional order system.
The definitions representing fractional order derivative goes
here:
The order of the system is defined by Riemann and Liouville
[20] that generalizes the following definition:
Dα
t f(t) =
dm
dtm
[
1
Γ(m − α)
t
0
f(τ)
(t − τ)(α + 1 − m)
dτ]
(12)
where mZ+
, (m − 1) α ≤ m, α − m − 1 1, Dα
t f(t)
representing the derivative of the function of time f(t) of the
order of α (positive non-integer number) in the limit 0 to t,
1
Γ(m−a) represents the inverse fractional factorial of (m−α)
and τ is a variable in complex plane.
Another definition is proposed by Grunwald and Letnikov
[21] that generalizes the following definition: ,
Dα
f(t) = lim
h→0
1
hα
t−α
h
m=0
(−1)m Γ(α + 1)
m!Γ(α − m + 1)
f(t − mh)
(13)
where, Dα
t f(t) representing the α (positive non-integer
number) order derivative of the function of time f(t) in the
limit 0 to t.
3.0.2. Fractional Order PID Controller. The parallel form
of the fractional PID controller, called the PIλ
Dμ
controller
[22] has the following transfer function:
C(s) = Kp +
Ki
sλ
+ Kdsμ (14)
where Kp, Ki, Kd, λ and μ are the proportional gain, in-
tegral gain, derivative gain, order of the integral component
and order of the derivative component respectively.
The fundamental purpose of the controller design is to
take care of the unwanted fluctuations in the output voltage
of the islanded microgrid. The encouragement is to gain
stable output voltage without fluctuation and robust perfor-
mance while the system undergoes uncertainty. It can be
seen that the fractional order PID controller gives more flexi-
bility in tuning than conventional PID controller because the
conventional PID controller gives three parameters to tune
the controller where the fractional order PID controller gives
five parameters for getting it tuned. The order of the integral
)
(
)
( s
v
s
y G
Plant
G(s)
Controller
C(s)
)
(
)
( s
v
s
u s
-
)LJClosed-loopsystem
and differential components of the PID controller can be set
at a fractional order to have an optimum response from the
system.
Fig. 2 serves as the block diagram of the system with the
proferred controller in closed loop. Where, G(s) represents
transfer function of the microgrid system which is the ratio
of input supply voltage vs(s) to grid output voltage vG(s)
according to the single phase microgrid system in Fig.
1 . C(s) represents the transfer function of the proffered
controller. u(s) represents the reference voltage and y(s)
represents the output from the system.
The closed-loop transfer function of the feedforward
interconnection of the plant G(s) with the controller C(s)
can be obtained as:
Gc(s) =
G(s) ∗ C(s)
1 − G(s) ∗ C(s)
(15)
Parameters of the controller has been selected using trial
and error method. The values are presented in Table 2.
TABLE 2: Parameter values for Fractional Order PID Con-
troller
Parameter Value
Kp 240
Ki 4.8 × 106
Kd
3 × 10−3
λ 1 × 10−2
μ 1.99
4. Evaluation of Performance
Performance of the microgrid under the application of
FOPID is evaluated in this section.
4.1. Step and frequency analysis
Fig. 3(a) represents the analogy of the open-loop step
response with closed-loop step response.It is observed that
oscillation is created in the open-loop response whereas the
closed loop system gives an instant response with low steady
state error. The obtained result ensure the high performance
of the fractional order PID controller. Fig. 3(b) represents
the analogy of the open-loop bode plot with closed-loop
bode plot. Form the figure, it is observed that resonance
frequency 5600 rad/s occurs at 40 dB without controller. In
the closed loop response, the controller is able to 40 dB
damping of voltage fluctuations that establishes a speedy
and snug activity of microgrid.
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4. Plant
G(s)
Controller
C(s)
-
Plant
G(s)
Controller
C(s)
-
Magnitude
(dB)
(a) (b)
(a) (b)
Magnitude
(dB)
(c)
)LJ (a) The addition of Uncertainty into the system, (b) Step response of the analogy of open-loop and closed-loop
system under uncertainty,(c) Bodediagram of the analogy of open-loopand closed-loop system underuncertainty.The
blueline(-)isforrepresentingtheopen-loopresponseandtheredline(- - )isforrepresentingtheclosed-loopresponse.
(a) (b) (c) (d)
)LJSchematicofdifferentloads(a)Unknownload,(b)Dynamicload,(c)Non-linearload,(d)Harmonicload.
4.2. Robustness of FOPID controller
The robustness analysis of FOPID controller is verified
in this section. Fig. 4(a) represents the integration of uncer-
tainty to test the the robustness of FOPID controller. Here,
G(s) represents the microgrid system, Wi(s) and Δi(s) rep-
resent the plant variatons. The value of uncertainty is chosen
as twenty five percent of the amplitude of the reference
signal. The obtained result for uncertainty is presented in
4(b) and 4(c). All of the results for uncertainty confirms the
closedloop system remain stable with high bandwidth and
less time require to reach steady-state value. This perfor-
mance ensures that the proffered controller is robust against
uncertainties.
4.3. Performance against Unknown Load
Fig. 5(a) represents the schematic of an unknown load
that is having a parallel connection with microgrid. Fig. 6(a)
represents the performance of the system under unknown
load conditions with and without getting connected to the
controller. The analogy of the open-loop and closed-loop
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5. (a) (b)
(c) (d)
)LJPerformanceevaluationwithdifferentloads(a)Unknownload,(b)Dynamicload,(c)Non-linearload,(d)Harmonic
load.Thesolidline(-)isforrepresentingtheReferenceSinewave,solidline(-)isforrepresentingtheopen-loopoutput
waveformanddashedline(- - )isforrepresentingtheclosed-loopoutputwaveform.
output voltage show that the closed loop voltage has high
in performance than open-loop voltage.
4.4. Performance against Dynamic Load
A single phase induction motor is taken as a dynamic
load that is shown in Fig. 5(b). Analogy of performance
of open-loop and closed-loop system under dynamic load
is presented in Fig. 6(b) . It is seen that the open-loop
signal oscillates and even if the oscillation is reduced, it
can not track the reference signal. Initially, the closed loop
response also oscillates but it tracks the reference voltage at
0.005 seconds. The comparison ensures that the closed-loop
is better as compared to open-loop system.
4.5. Performance against Nonlinear Load Dynam-
ics
Fig. 5(c) represents the model of a non-linear load which
is a two-phase four-pulse diode-bridge rectifier. Fig. 6(c)
shows the analogy of the open-loop and closed-loop voltage
under the admittance of non-linear load which ensures the
high performance of proffered controller.
4.6. Performance against Harmonic Load
Fig.5(d) represents the schamatic of a load that creates
harmonics. The schematic consists of a resistance connected
in series with a current source having amplitude of 7A
and frequency of 150 Hz. Fig.6(d) shows the result of
investigation. The investigation shows that the open-loop
voltage oscillates and even if the oscillation is reduced,
the response can not track the reference signal. Initially,
the closed voltage also oscillates but it tracks the reference
voltage at 0.0057 seconds.
So,it is observed that the proffered controller provides
balanced, efficient and high accuracy voltages.
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6. 5. Conclusion
In the article, the design of Fractional-Order PID Con-
troller is proffered for a single phase islanded microgrid sys-
tem which runs with a single energy source. The application
of FOPID to the microgrid system makes it spectacle that
the proffered controller certifies a fast, robust and precise
operation of the microgrid system.
References
[1] D. Pullaguram, S. Mishra, N. Senroy, and M. Mukherjee, “Design and
tuning of robust fractional order controller for autonomous microgrid
vsc system,” IEEE Transactions on Industry Applications, vol. 54,
no. 1, pp. 91–101, 2018.
[2] C. K. Sao and P. W. Lehn, “Control and power management of
converter fed microgrids,” IEEE Transactions on Power Systems,
vol. 23, no. 3, pp. 1088–1098, 2008.
[3] M. Armin, P. N. Roy, S. K. Sarkar, and S. K. Das, “Lmi-based robust
pid controller design for voltage control of islanded microgrid,” Asian
Journal of Control, 2018.
[4] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis
and testing of autonomous operation of an inverter-based microgrid,”
IEEE Transactions on power electronics, vol. 22, no. 2, pp. 613–625,
2007.
[5] M. A. Hossain and H. R. Pota, “Voltage tracking of a single-phase
inverter in an islanded microgrid,” International Journal of Renewable
Energy Research (IJRER), vol. 5, no. 3, pp. 806–814, 2015.
[6] S. K. Sarkar, M. H. K. Roni, D. Datta, S. K. Das, and H. R. Pota,
“Improved design of high-performance controller for voltage control
of islanded microgrid,” IEEE Systems Journal, no. 99, pp. 1–10, 2018.
[7] M. Babazadeh and H. Karimi, “A robust two-degree-of-freedom
control strategy for an islanded microgrid,” IEEE transactions on
power delivery, vol. 28, no. 3, pp. 1339–1347, 2013.
[8] M. Hamzeh, M. Ghafouri, H. Karimi, K. Sheshyekani, and J. M.
Guerrero, “Power oscillations damping in dc microgrids,” IEEE
Transactions on Energy Conversion, vol. 31, no. 3, pp. 970–980,
2016.
[9] T. L. Vandoorn, B. R. L. Degroote, B. Meersman, and L. Vandevelde,
“Voltage control in islanded microgrids by means of a linear-quadratic
regulator,” in Proc. IEEE Benelux Young Researchers Symposium in
Electrical Power Engineering (YRS10), Leuven, Belgium, 2010.
[10] Q. Salem and J. Xie, “Transition from grid-connected to islanded
drooped microg id based on islanding detection scheme,” Interna-
tional Journal of Power and Energy Systems, vol. 36, no. 3, 2016.
[11] Q. Shafiee, J. M. Guerrero, and J. C. Vasquez, “Distributed secondary
control for islanded microgridsa novel approach,” IEEE Transactions
on power electronics, vol. 29, no. 2, pp. 1018–1031, 2014.
[12] J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. D. Vicuña, and
M. Castilla, “Hierarchical control of droop-controlled ac and dc
microgridsa general approach toward standardization,” IEEE Trans-
actions on industrial electronics, vol. 58, no. 1, pp. 158–172, 2011.
[13] Y. A.-R. I. Mohamed and A. A. Radwan, “Hierarchical control system
for robust microgrid operation and seamless mode transfer in active
distribution systems,” IEEE Transactions on Smart Grid, vol. 2, no. 2,
pp. 352–362, 2011.
[14] S. Alepuz, J. Salaet, A. Gilabert, J. Bordonau, and J. Peracaula, “Con-
trol of three-level vsis with a lqr-based gain-scheduling technique
applied to dc-link neutral voltage and power regulation,” in IECON 02
[Industrial Electronics Society, IEEE 2002 28th Annual Conference
of the], vol. 2. IEEE, 2002, pp. 914–919.
[15] M. Rahman, S. K. Sarkar, S. K. Das, and Y. Miao, “A comparative
study of lqr, lqg, and integral lqg controller for frequency control
of interconnected smart grid,” in Electrical Information and Commu-
nication Technology (EICT), 2017 3rd International Conference on.
IEEE, 2017, pp. 1–6.
[16] S. K. Sarkar and S. K. Das, “Nonlinear modeling and control of dc
machines: a partial feedback linearization approach,” in Electrical,
Computer Telecommunication Engineering (ICECTE), Interna-
tional Conference on. IEEE, 2016, pp. 1–4.
[17] S. K. Sarkar, F. R. Badal, S. K. Das, and Y. Miao, “Discrete time
model predictive controller design for voltage control of an islanded
microgrid,” in Electrical Information and Communication Technology
(EICT), 2017 3rd International Conference on. IEEE, 2017, pp. 1–6.
[18] S. K. Sarkar and S. K. Das, “High performance nonlinear controller
design for ac and dc machines: partial feedback linearization ap-
proach,” International Journal of Dynamics and Control, pp. 1–15,
2017.
[19] J. L. Adams, T. T. Hartley, and C. F. Lorenzo, “Fractional-order
system identification using complex order-distributions,” IFAC Pro-
ceedings Volumes, vol. 39, no. 11, pp. 200–205, 2006.
[20] K. Oldham and J. Spanier, The fractional calculus theory and appli-
cations of differentiation and integration to arbitrary order. Elsevier,
1974, vol. 111.
[21] C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu-Batlle,
Fractional-order systems and controls: fundamentals and applica-
tions. Springer Science Business Media, 2010.
[22] I. Podlubny, “Fractional-order systems and pi/sup/spl
lambda//d/sup/spl mu//-controllers,” IEEE Transactions on automatic
control, vol. 44, no. 1, pp. 208–214, 1999.
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