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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1,
JANUARY 2013 121
Real-Time Implementation of ANFIS Controlfor Renewable
Interfacing Inverter in
3P4W Distribution NetworkMukhtiar Singh, Member, IEEE, and
Ambrish Chandra, Senior Member, IEEE
AbstractPower electronics plays an important role in
con-trolling the grid-connected renewable energy sources. This
pa-per presents a novel adaptive neuro-fuzzy control approach
forthe renewable interfacing inverter. The main objective is
toachieve smooth bidirectional power flow and nonlinear unbal-anced
load compensation simultaneously, where the
conventionalproportional-integral controller may fail due to the
rapid changein the dynamics of the highly nonlinear system. The
combinedcapability of neuro-fuzzy controller in handling the
uncertaintiesand learning from the processes is proved to be
advantageouswhile controlling the inverter under fluctuating
operating condi-tions. The inverter is actively controlled to
compensate the har-monics, reactive power, and the current
imbalance of a three-phasefour-wire (3P4W) nonlinear load with
generated renewable powerinjection into the grid simultaneously.
This enables the grid toalways supply/absorb a balanced set of
fundamental currents atunity power factor even in the presence of
the 3P4W nonlinearunbalanced load at the point of common coupling.
The proposedsystem is developed and simulated in
MATLAB/SimPowerSystemenvironment under different operating
conditions. The digitalsignal processing and control
engineering-based laboratory exper-imental results are also
provided to validate the proposed controlapproach.
Index TermsDistributed generation, grid
interconnection,neuro-fuzzy control, nonlinear load, power quality,
renewableenergy, unbalanced load.
I. INTRODUCTION
THE increase in global energy demand, air pollution,
globalwarming, and the rapid evaporation of fossil fuel hasmade it
necessary to look toward renewable sources as a futureenergy
solution. However, the higher penetration level of
theseintermittent renewable energy sources (RESs) poses a
greatthreat to network security. Therefore, the RESs are requiredto
comply with strict technical and regulatory frameworks toensure the
safe, reliable, and efficient operation of the overallnetwork. With
the advancement in power electronics and digital
Manuscript received September 18, 2009; revised April 27, 2011
andSeptember 14, 2011; accepted November 20, 2011. Date of
publicationJanuary 26, 2012; date of current version September 6,
2012.
M. Singh is with the Department of Electrical Engineering,
DeenbandhuChhotu Ram University of Science and Technology, Murthal,
Haryana 131039,India (e-mail: [email protected]).
A. Chandra is with the Department of Electrical Engineering,
Ecole detechnologie superieure, Universite du Quebec, Montreal, QC
H3C 1K3, Canada(e-mail: [email protected]).
Color versions of one or more of the figures in this paper are
available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2012.2186103
control technology, the RES can now be actively controlled
toenhance the system stability with an improved power quality atthe
point of common coupling (PCC).
Recently, a lot of control strategies for renewable
interfacinginverter have been introduced [1][7]. Some control
strategiesfor grid-connected inverters incorporating power quality
so-lution have also been investigated by researchers. In [8],
aninverter operates as an active inductor at a certain frequencyto
absorb the harmonic current. However, the exact calculationof
network inductance in real time is very difficult and
maydeteriorate the control performance. A similar approach inwhich
a shunt active filter acts as an active conductance to dampout the
harmonics in distribution network is proposed in [9]. In[10], a
control strategy for renewable interfacing inverter basedon pq
theory is proposed. A similar decoupled current controltechnique
using PI regulator in dq reference frame is presentedin [11]. In
both of these strategies, the load and inverter currentsensing is
required to compensate the load current harmonics.
The current-regulated voltage source inverters have a verywide
range of applications such as the grid synchronization ofRES,
static reactive power compensation, uninterruptible powersupply,
active power filters (APF), and adjustable speed drives.However, in
the case of the very first application, the installedinverter
rating has a very low utilization factor due to the inter-mittent
nature of RES. According to [12] and [13], the expectedRES output
during peak is nearly 60% of the rated output,yet the annual
capacity factor may be in the 20%30% range.Therefore, the authors
have incorporated the APF features inthe RES interfacing inverter
to maximize its utilization withoutany additional hardware cost.
Moreover, the proposed controlstrategy requires only the grid
current sensing, which furtherreduces the cost and complexity. The
grid-interfacing inverterinjects the generated active power from
renewable as well ascompensates the load reactive power, current
harmonics, andload imbalance in a three-phase four-wire (3P4W)
system. Thisenables the grid to always supply a balanced set of
sinusoidalcurrents at unity power factor (UPF).
Since the inverter works under highly fluctuating
operatingconditions, it is not possible to set the optimal value of
gainsfor the conventional PI regulator [14][16]. This may lead toa
false operation of the inverter. To alleviate this problem,an
adaptive neuro-fuzzy controller is developed, which haswell-known
advantages in modeling and control of a highlynonlinear system
[17], [18]. An adaptive error backpropagationmethod is used to
update the weights of the system for the fastconvergence of
control.
0278-0046/$31.00 2012 IEEE
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122 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1,
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Fig. 1. Schematic and control description of proposed
renewable-based dis-tributed generation system.
This paper is organized as follows: Section II presents
thesystem description and control algorithm for the inverter.
InSection III, the simulation results are discussed, while
theexperimental results under different operating conditions
arepresented and discussed thoroughly in Section IV. Section
Vfinally concludes this paper.
II. SYSTEM CONFIGURATION AND CONTROL
The system under consideration with control description isshown
in Fig. 1, where a RES is connected on the dc linkof a
grid-interfacing four-leg inverter. The fourth leg of theinverter
is utilized to compensate the neutral current of 3P4Wnetwork. Here,
the inverter is a key element since it deliversthe power from
renewable to grid and also solves the power-quality problem arising
due to unbalanced nonlinear load atPCC. The duty ratio of inverter
switches are varied in a powercycle such that the combination of
load and inverter-injectedpower appears as balanced resistive load
to the grid, resultinginto the UPF grid operation.
The renewable source may be a dc source or an ac sourcewith
rectifier coupled to a dc link [19]. The regulation of
dc-linkvoltage carries the information regarding the exchange of
activepower in between renewable source and grid. The error
betweenreference dc-link voltage (V dc) and actual dc-link voltage
(Vdc)is given to the neuro-fuzzy controller, and the same error is
usedto update the weights. The output of neuro-fuzzy controller
isfurther modified by subtracting the renewable injected
current(iRen). This results into the reference d-axis current (id),
whilethe reference q-axis current (iq) is set to zero for UPF
gridoperation. The grid-synchronizing angle () obtained fromphase
lock loop is used to generate the reference grid currents(ia, ib,
and ic). The reference grid neutral current in is set to
Fig. 2. Optimized ANFIS architecture suggested by
MATLAB/anfiseditor.
Fig. 3. Schematic of the proposed ANFIS-based control
architecture.
zero to achieve balanced grid-current operation. The
hysteresiscurrent controller is utilized to force the actual grid
currents totrack the reference grid currents accurately. This
enables thegrid to supply/absorb only the fundamental active power,
whilethe RES-interfacing inverter fulfills the unbalance, reactive,
andnonlinear current requirements of 3P4W load at PCC.
Design of Adaptive Neuro-Fuzzy ControllerAn optimized
adaptive-network-based fuzzy inference sys-
tem (ANFIS) having a 1:3:3:3:1 architecture is generated fromthe
initial data using MATLAB/anfiseditor as shown in Fig. 2.
This Takagi-Sugeno-Kang fuzzy model-based ANFIS archi-tecture
has one input and one output, which is further tunedonline using
the error backpropagation method as shown inFig. 3. The error
between reference dc-link voltage and actualdc-link voltage ( = V
dc Vdc) is given to the neuro-fuzzycontroller, and the same error
is used to tune the preconditionand consequent parameters. The
control of dc-link voltage givesthe active power current component
(id ), which is furthermodified to take into account the active
current componentinjected from RES (iRen). The node functions of
each layer inthe ANFIS architecture are described as follows:
Layer 1: This layer is also known as the fuzzification
layerwhere each node is represented by a square. Here, three
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SINGH AND CHANDRA: IMPLEMENTATION OF ANFIS CONTROL FOR RENEWABLE
INTERFACING INVERTER 123
Fig. 4. Fuzzy membership functions.
membership functions are assigned to each input. The
trape-zoidal and triangular membership functions are used to
reducethe computation burden as shown in Fig. 4, and their
corre-sponding node equations are given as follows:
A1() =
1 b1a1b1a1 b1a1
0 a1(1)
A2() ={
1 a20.5b2 | a2| 0.5b20 | a2| 0.5b2 (2)
A3() =
0 a3a3b3a3 a3b3
1 b3(3)
where the value of the parameters {ai, bi} changes with
thechange in error and accordingly generates the linguistic value
ofeach membership function. Parameters in this layer are referredas
premise parameters or precondition parameters.
Layer 2: Every node in this layer is a circle labeled as which
multiplies the incoming signals and forwards it to thenext
layer
i = Ai(1) Bi(2) , i = 1, 2, 3. (4)However, in our case, there is
only one input, so this layer
can be ignored and the output of the first layer will directly
passto the third layer.
Here, the output of each node represents the firing strengthof a
rule.
Layer 3: Every node in this layer is represented by a
circle.This layer calculates the normalized firing strength of each
ruleas given in the following:
i =i
1 + 2 + 3, i = 1, 2, 3. (5)
Layer 4: Every node in this layer is a square node with a
nodefunction
Oi = i fi = i(ai0 + a
i1
), i = 1, 2, 3 (6)
where the parameters {ai0, ai1} are tuned as the function ofthe
input (). The parameters in this layer are also referred
asconsequent parameters.
Layer 5: This layer is also called the output layer
whichcomputes the output as given in the following:
Y = 1 f1 + 2 f2 + 3 f3. (7)
The output from this layer is multiplied with the
normalizingfactor to obtain the active power current component (id
). Thedetailed algorithm for the training of the ANFIS architecture
isgiven in Appendix II.
III. SIMULATION RESULTS AND DISCUSSION
An extensive simulation study has been carried out for
therenewable interfacing inverter in order to verify the
proposedcontrol strategy. The system under consideration is
simulatedusing the SimPowerSystem tool box of MATLAB/Simulink.
AnIGBT-based four-leg current-controlled voltage source inverteris
actively controlled to achieve the balanced sinusoidal gridcurrents
at UPF despite the highly unbalanced nonlinear loadat the PCC under
varying renewable generating conditions. ARES with variable output
power is connected on the dc linkof the grid-interfacing inverter.
An unbalanced 3P4W nonlin-ear variable load, whose harmonics,
unbalance, and reactivepower are to be compensated, is connected on
the PCC. Thewaveforms of grid voltage (Vg), grid current (ig),
unbalancedload current (il), injected inverter currents (iinv), and
dc-linkvoltage (Vdc) are shown in Fig. 5. In Fig. 6, the traces of
phasea grid current (iga), phase a load current (ila), and phase
ainverter current (iinva) are shown w.r.t. phase a grid
voltage(Vga). In addition, the waveforms of grid neutral current
(ign),load neutral current (iln), and inverter neutral current
(iinvn)are also shown in the same diagram. Fig. 7 shows the traces
ofphase a grid voltage (Vga) and phase a grid current (iga) onthe
same plot, phase a load current (ila), and phase a invertercurrent
(iinva).
The main purpose of the proposed control strategy is to
injectthe generated renewable active power, load harmonics, and
re-active power in such a way that only the injection/absorption
ofthe active power takes place in the grid. Initially, the
generatedactive power is more than the load active power demand,
sothe extra generated power is being injected into the grid.
Thisfact can be verified from the traces of different currents,
wherethe current supplied from the renewable is more than the
loadcurrent, so the difference of these is being injected into the
gridas evident from the out-of-phase relation of the grid
voltage(Vga) and grid current (iga). In addition, the inverter is
alsosupplying the harmonics, neutral current, and reactive
currentcomponent of the load current demand. This results into
theperfectly balanced sinusoidal grid current even in the
presenceof a 3P4W unbalanced nonlinear load at PCC as shown inFig.
5. This fact can also be visualized from Figs. 6 and 7,where the
phase a grid current (iga) is purely sinusoidal andin phase
opposition with the phase a grid voltage (Vga). Here,it can also be
noticed that the load neutral current (iln) is fullysupplied by the
inverter neutral current (iinvn). This results intothe zero value
of the grid neutral current (in).
At t = 0.375 s, there is a sudden increase in the load
powerdemand, and the generated renewable active power is not
suf-ficient enough to meet this enhanced demand. At this
instant,the renewable interfacing inverter supplies the generated
activepower and total load reactive power demand, while the
gridsupplies only the deficient amount of load active power.
Thisfact can be verified from Figs. 6 and 7, where the phase a
grid
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124 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1,
JANUARY 2013
Fig. 5. Simulation results. (a) Grid voltages. (b) Grid
currents. (c) Unbalancedload currents. (d) Inverter currents.
current, which was in the opposite phase to the grid
voltagebefore t = 0.375 s, is now in phase with the grid voltageand
the load neutral current is still being supplied from theinverter.
Thus, from the simulation results, it is clear that thegrid always
works at UPF under fluctuating renewable powergeneration and
dynamic load conditions with an unbalancednonlinear load at PCC. It
can also be noticed that the dc-linkvoltage is almost constant at
300 V under both steady state anddynamic conditions, except
negligible deviation due to a changein injected active power. Here,
the dc-link voltage is shown ona very small scale, just to
demonstrate the performance of theproposed ANFIS controller in
controlling the dc-link voltage.
IV. EXPERIMENTAL RESULTS AND DISCUSSION
The proposed adaptive neuro-fuzzy controller is imple-mented in
real time on a four-leg IGBT-based inverter us-
Fig. 6. Simulation results: Phase a grid voltage, grid current,
load current,inverter current, load neutral current, and inverter
neutral current.
Fig. 7. Simulation results. (a) Phase a grid voltage and
current. (b) Loadcurrent. (c) Inverter current.
ing digital signal processing and control engineering
DS1104,whereas the RES is emulated with an auxiliary inverter
con-nected on a dc link. It takes a sampling time of 75 s torealize
the proposed ANFIS controller in real time. The 3P4Wnonlinear load
is composed of three-phase nonlinear RL load,one-phase RL nonlinear
load connected in between phase a andneutral, and a single-phase RL
load in between phase b andneutral.
An extensive experimental study is carried out to highlightthe
performance of the inverter as a multiobjective device.The inverter
operation is mainly divided into two parts: activefilter operation
and renewable interfacing operation. All theexperimental results
are captured with an oscilloscope in realtime as shown in Figs.
810.
A. Active Filter Operation
In this mode of operation, only the active filtering
capabilitiesof the inverter are demonstrated. In Fig. 8(a), the
traces of3P4W grid currents are shown before and after
compensation.Initially, the grid supplies an unbalanced nonlinear
load currentwith a high neutral current, which is highly
undesirable. Inorder to compensate this unbalanced nonlinear
current, theinverter currents are injected in such a way that the
combinationof load and inverter current appears as a balanced set
of fun-damental currents. The traces of the injected inverter
currentsare shown in Fig. 8(b), just before and after
compensation.Here, it can be easily noticed that the grid currents
are perfectlybalanced with a sinusoidal profile. Moreover, the
inverter is
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SINGH AND CHANDRA: IMPLEMENTATION OF ANFIS CONTROL FOR RENEWABLE
INTERFACING INVERTER 125
Fig. 8. Experimental results: (a) Grid currents and (b) inverter
currents, justbefore and after compensation.
successfully able to supply the load neutral current
demandlocally, as evident from the zero value of the grid neutral
current(ign).
B. Renewable Interfacing OperationIn this mode of operation, the
bidirectional power flow
capabilities of the renewable interfacing inverter are
discussed.Here, the main objective is not only the grid interfacing
ofthe renewable but also to compensate the 3P4W nonlinearunbalanced
load at PCC simultaneously. The inverter suppliesthe renewable
injected current and the nonlinear unbalancedcomponent of load
current. This enables the grid to alwayssupply/absorb only the
balanced set of currents at UPF. InFig. 9, the traces of grid
voltage (Vg), grid current (ig), loadcurrent (il), and inverter
injected current (iinv) are shown. InFig. 9(a), initially, the
inverter current is supporting the loadcurrent partially and it
goes on increasing. At middle stage,the inverter current is almost
equal to the load current, and thisforces the grid current to be
almost zero. In the last stage, theinverter current is more than
the load demand, and at this stage,the grid absorbs this excessive
amount of current as evident bythe out-of-phase relationship of the
grid voltage and current.Similarly, in Fig. 9(b), the grid current
is again shown from
Fig. 9. Experimental results: Inverter performance under the
renewable inter-facing mode of operation.
Fig. 10. Experimental results: Inverter performance under the
renewableinterfacing mode of operation.
absorption mode to supplying mode with the correspondingchange
in the inverter current. Fig. 10 shows the traces ofgrid voltage
(Vg) and grid current (ig) on the same axis, dc-link voltage (Vdc),
and inverter injected current (iinv), where
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126 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1,
JANUARY 2013
TABLE IINVERTER PERFORMANCE AS A COMPENSATING DEVICE
it can be noticed that the dc-link voltage is almost
constantirrespective of any kind of variation in injected inverter
current.
A comparative table showing the total harmonic distortions(THDs)
and unbalance factor (UF) before and after compensa-tion is given
in Table I, where the percentage UF is calculatedseparately for
each phase using
%UFabc =|iabc iavg.|
iavg. 100. (8)
Here, it can be noticed that the grid current is highly
unbalancedwith the UFs of 20.48%, 0.41%, and 21.07% in phase a,
phaseb, and phase c, respectively, resulting into the flow of a
1.1-Acurrent in neutral wire. The percent THDs present in phase
a,phase b, and phase c currents are 14.7%, 18.2%, and
23.2%,respectively. However, once after the interconnection of
therenewable interfacing inverter, the grid currents become
almostbalanced and harmonic free with a very low UF of 0.7%, a
verylow level of THDs of 2.9%, and an almost zero current in
grid-side neutral wire.
V. CONCLUSION
This paper has presented a novel adaptive neuro-fuzzycontrol
algorithm for the renewable interfacing inverter. Thecontroller
works satisfactorily under the dynamic operatingconditions. It has
also been shown that the inverter is ableto perform all the duties
of the shunt APF while maintain-ing the smooth bidirectional power
flow simultaneously. Thesimulation results supported by the
experimental results areprovided to validate the fact that the
renewable interfacinginverter can act as a multioperation device in
order to utilizeits maximum rating. The current unbalance, current
harmonics,and load reactive power demand of an unbalanced
nonlinearload at PCC are compensated effectively such that the
grid-side currents are always maintained as a balanced set (0%
UF)of sinusoidal current (2.7% THD) at UPF. Moreover, the
loadneutral current is restricted to flow toward the grid side
(almostzero) by supporting it locally from the fourth leg of the
inverter.When the power generated from the renewable is more
thanthe total load power demand, the grid-interfacing inverter
withthe proposed control approach successfully fulfills the
totalload demand (active, reactive, and harmonics) and delivers
theremaining active power to the main grid at UPF operation.
APPENDIX ISYSTEM PARAMETERS
Three-phase supply (rms) : Vg = 30 V, 60 HzThree-phase nonlinear
load : R = 26.66 , L = 10 mHOne-phase linear load(AN) : R = 26.66 ,
L = 10 mHOne-phase nonlinear load(BN) : R = 56 , L = 10 mHDC-link
capacitance and voltage : Cdc = 3000 F, Vdc = 75 VCoupling
inductance : Lsh = 2.0 mHSource impedance ratio : X/R = 7
APPENDIX II
A. Online Training of the ANFIS ArchitectureThe ANFIS structure
is tuned with a gradient descent tech-
nique to reduce the error (usually a cost function given by
thesquared error), where the weights are iterated by propagatingthe
error from output layer to input layer. This backward tripof such a
calculation is termed as backpropagation [20].The training
algorithm is completed in two stages, known asthe precondition
parameter tuning and the consequent para-meter tuning, where the
objective function to be minimized isdefined as
2 = (V dc Vdc)2 . (A1)Precondition Parameter Tuning: The
precondition parame-
ters are required to update the fuzzy membership functions
asdiscussed in the previous section for Layer1. To minimize
theerror function 2 by the gradient descent method, the change
ineach precondition parameter must be proportional to the rate
ofchange of the error function w.r.t. that particular
preconditionparameter, i.e.,
aAi = 2
aAi= 1, 2, 3 (A2)
where is the constant of proportionality defined as the
learn-ing rate. Therefore, the new value of the consequent
parameteris given as
aAi(n + 1) = aAi(n) + aAi , i = 1, 2, 3 (A3)or
aAi(n + 1) = aAi(n) 2
aAi, i = 1, 2, 3. (A4)
Now the partial derivative term in (A4) can be found by thechain
rule of differentiation as follows:
2
aA1=
2
Vdc Vdc
id i
d
1 1A1
A1aA1
(A5)
where
2
Vdc=2 (V dc Vdc)=2 (A5a)
Vdcid
=J (A5b)
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SINGH AND CHANDRA: IMPLEMENTATION OF ANFIS CONTROL FOR RENEWABLE
INTERFACING INVERTER 127
id1
=
1(1 f1 + 2 f2 + 3 f)=f1 (A5c)
1A1
=
A1
(A1
A1 + A2 + A3
)=
A2 + A3(A1 + A2 + A3)
2
=1
(A1 + A2 + A3)
(A2
(A1 + A2 + A3)
+A3
(A1 + A2 + A3)
)
=2 + 3
A1 + A2 + A3(A5d)
A1aA1
=
aA1
( aA1
bA1 aA1
)=
bA1(bA1 aA1)2
= bA1 + aA1 aA1
(bA1 aA1)2
=1
(bA1 aA1)(
aA1(bA1 aA1)
bA1 aA1(bA1 aA1)
)
=A1 1
bA1 aA1(A5e)
where J is Jacobian matrix, which can be taken as constant,being
a single-inputsingle-output ANFIS architecture, and canbe included
in the learning rate. In computing all the terms of(A5) and putting
in (A4), we can find the updated value of theparameter aA1 as
follows:
aA1(n + 1) = aA1(n) + 2 (n) f1(n) 2(n) + 3(n)A1(n) + A2(n) +
A3(n)
A1(n) 1bA1(n) aA1(n)
. (A6)
Similarly
bA1(n + 1) = bA1(n) 2 (n) f1(n)
2(n) + 3(n)A1(n) + A2(n) + A3(n)
A1(n)bA1(n) aA1(n)
. (A7)
In the same manner, the precondition parameters for the
re-maining fuzzy membership functions can be derived as
follows:
bA2(n + 1) = bA2(n) + 2 (n) f2(n)
1(n) + 3(n)A1(n) + A2(n) + A3(n)
1 A2(n)bA2(n)
(A8)
aA3(n + 1) = aA3(n) + 2 (n) f3(n)
1(n) + 2(n)A1(n) + A2(n) + A3(n)
A3(n) 1bA3(n) aA3(n)
(A9)
bA3(n + 1) = bA3(n) 2 (n) f3(n)
1(n) + 1(n)A1(n) + A2(n) + A3(n)
A3(n)bA3(n) aA3(n)
. (A10)
Consequent Parameter Tuning: To tune the consequent pa-rameters
as discussed in Layer 4, the following updated lawsare
developed:
a0i(n + 1) = a0i(n) c 2
a0i, i = 1, 2, 3 (A11)
a1i(n + 1) = a1i(n) c 2
a1i, i = 1, 2, 3 (A12)
where c is the learning rate for the consequent parameters.
Thederivative terms in (A11) and (A12), can be found by the
chainrule as already discussed in the case of precondition
parametersas follows:
2
a0i=
2
Vdc Vdc
id i
d
fi fia0i
, i = 1, 2, 3 (A13)2
a1i=
2
Vdc Vdc
id i
d
fi fia1i
, i = 1, 2, 3. (A14)
In the aforementioned (A13) and (A14), the first two termson the
right-hand side are already known and the last two termscan be
derived as
idfi
=i
A1 + A2 + A3, i = 1, 2, 3 (A15)
fia0i
=1, i = 1, 2, 3 (A16)fia1i
= , i = 1, 2, 3. (A17)
In substituting the terms derived in (A15)(A17) into (A13)and
(A14), the updated value of the consequent parameters canbe derived
as follows:
a0i(n + 1) = a0i(n) + 2 c iA1 + A2 + A3
, i = 1, 2, 3 (A18)a1i(n + 1) = a1i(n) + 2 c
ia1 + a2 + a3
, i = 1, 2, 3. (A19)
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Mukhtiar Singh (M11) received the B.Tech. andM.Tech. degrees in
electrical engineering from theNational Institute of Technology,
Kurukshetra (for-merly known as Regional Engineering
College,Kurukshetra), Haryana, India, in 1999 and 2001,
re-spectively, and the Ph.D. degree from Ecole de tech-nologie
superieure, Universite du Quebec, Montreal,QC, Canada, under the
National Overseas Scholar-ship funded by the Government of India in
2010.
He was a Faculty Member with BhagwanMahaveer Institute
Engineering and Technology,
Sonepat, and the Krishna Institute of Engineering and
Technology, Ghaziabad,India, in 20002002. Since 2002, he has been
an Assistant Professor with theDepartment of Electrical
Engineering, Deenbandhu Chhotu Ram University ofScience and
Technology, Murthal, Haryana. His research interests include
therenewable energy sources, smart grid, power quality, energy
storage systems,electric vehicles, and power electronics and
drives.
Ambrish Chandra (SM99) was born in India in1955. He received the
B.E. degree from the Uni-versity of Roorkee [currently, Indian
Institute ofTechnology (IIT)], Roorkee, India, in 1977, theM.Tech.
degree from IIT, New Delhi, India, in 1980,and the Ph.D. degree
from the University of Calgary,Calgary, AB, Canada, in 1987.
He was a Lecturer and later a Reader with theUniversity of
Roorkee. Since 1994, he has been aProfessor with the Department of
Electrical Engi-neering, cole de technologie suprieure,
Universit
du Qubec, Montreal, QC, Canada. His main research interests are
power qual-ity, active filters, static reactive power compensation,
flexible ac transmissionsystems, and renewable energy
resources.
Dr. Chandra is a member of the Ordre des Ingnieurs du Qubec,
Montreal.