Download A Noniterative Optimized Algorithm for Shunt Active Power Filter

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013
A Noniterative Optimized Algorithm for Shunt
Active Power Filter Under Distorted and
Unbalanced Supply Voltages
Parag Kanjiya, Vinod Khadkikar, Member, IEEE, and Hatem H. Zeineldin, Member, IEEE
Abstract—In this paper, a single-step noniterative optimized
algorithm for a three-phase four-wire shunt active power filter
under distorted and unbalanced supply conditions is proposed.
The main objective of the proposed algorithm is to optimally
determine the conductance factors to maximize the supply-side
power factor subject to predefined source current total harmonic
distortion (THD) limits and average power balance constraint.
Unlike previous methods, the proposed algorithm is simple and
fast as it does not incorporate complex iterative optimization
techniques (such as Newton–Raphson and sequential quadratic
programming), hence making it more effective under dynamic
load conditions. Moreover, separate limits on odd and even THDs
have been considered. A mathematical expression for determining
the optimal conductance factors is derived using the Lagrangian
formulation. The effectiveness of the proposed single-step noniterative optimized algorithm is evaluated through comparison
with an iterative optimization-based control algorithm and then
validated using a real-time hardware-in-the-loop experimental
system. The real-time experimental results demonstrate that the
proposed method is capable of providing load compensation under
steady-state and dynamic load conditions, thus making it more
effective for practical applications.
Index Terms—Active power filter (APF), harmonic compensation, optimized control, power quality, unity power factor (UPF).
I. I NTRODUCTION
T
HE INCREASING demand of power-electronics-based
nonlinear loads has raised several power quality problems.
The uneven distribution of dynamically changing single-phase
loads gives rise to the additional problems of excessive neutral current and current unbalance (UB) [1]. The combined
effects of the above on today’s power distribution systems
result in increased voltage and current distortions, unbalanced
supply voltages, excessive neutral currents, poor power factor,
increased losses, and reduced overall efficiency.
Manuscript received May 31, 2012; revised August 2, 2012 and October 19,
2012; accepted November 29, 2012. Date of publication December 19, 2012;
date of current version June 21, 2013. This work was supported by the Masdar
Institute of Science and Technology under MISRG Internal Grant 10PAAA2.
P. Kanjiya and V. Khadkikar are with the Masdar Institute of Science
and Technology, Abu Dhabi, United Arab Emirates (e-mail: pkanjiya@
masdar.ac.ae; [email protected]).
H. H. Zeineldin is with the Masdar Institute of Science and Technology,
Abu Dhabi, United Arab Emirates, and also with Cairo University, Giza, Egypt
(e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2012.2235394
Active power filters (APFs) are widely used to overcome
such power quality problems [2]–[5]. There are two main
APF control strategies for load compensation when the supply
voltages are unbalanced and distorted [6]–[13]: 1) harmonicfree (HF) source currents and 2) unity-power-factor (UPF)
source currents. The HF strategy results in sinusoidal source
currents [6], while the UPF strategy can achieve minimum rootmean-square (rms) source current magnitudes. Additionally,
the UPF strategy can provide effective damping to avoid any
resonance [7], [8]. Both HF and UPF control strategies under
sinusoidal and balanced supply condition will lead to identical
performance. However, when the supply voltages are distorted
and unbalanced, HF and UPF operations cannot be achieved
simultaneously. For this reason, previous literature in the area
of shunt APFs under distorted and unbalanced supply condition use either the HF or the UPF control strategy [6]–[13].
Recently, several approaches have been proposed to combine
the advantages of both control strategies using nonlinear optimization techniques [15]–[20].
Papers on optimization-based shunt APF control under distorted and unbalanced voltages can be classified into two main
categories. In the first category, the distorted and unbalanced
supply voltages of each phase are processed through a set
of filters (such as bandpass). The filter gains are optimized
considering both voltage total harmonic distortion (THD) and
voltage UB limit constraints to achieve the desired compensated voltages. These compensated voltages are then multiplied
with constant conductance factors to obtain the desired source
currents. In [15], the compensated voltages are obtained in the
α−β−0 reference frame, whereas in [16]–[18], the compensated voltages are generated in the a−b−c stationary reference
frame to avoid the complex transformation from one frame to
the other. As stated in [16]–[18], due to computational delay,
the studied approach is not suitable for loads that operate
dynamically.
In the second category, to reduce the complexity and dimensionality of the optimization problem, the authors in [19]
and [20] formulated the optimization problem considering the
conductance factors as variables. The conductance factors for
each harmonic order are optimally determined. These conductance factors are then multiplied with a balanced set of supply
voltages to obtain the desired source currents. In [19], the
supply voltages are considered as distorted and balanced. In
[20], to enhance the performance of the system under unbalanced and distorted voltages, the balanced set of voltages is
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KANJIYA et al.: NONITERATIVE OPTIMIZED ALGORITHM FOR SHUNT ACTIVE POWER FILTER
extracted using instantaneous symmetrical components along
with complex Fourier transform.
One of the main disadvantages of all the aforementioned
optimization-based approaches is the use of iterative techniques
for solving the optimization problem. The use of an iterative technique can result in a computational delay, which can
constrain the applicability of these control approaches under
dynamic load conditions. As a result, the methods proposed in
[15]–[20] focused on steady-state load compensation only. In
[21], Pogaku and Green have considered only one conductance
factor to compute the reference current for a distributed generator inverter controller in a microgrid application to provide
adjustable damping at harmonic frequencies to mitigate voltage
distortion.
To overcome the aforementioned challenges, a single-step
noniterative optimized control algorithm for shunt APFs under
distorted and unbalanced supply conditions is proposed in this
paper. The algorithm is based on direct calculation of conductance factors without incorporating any iterative optimization
technique. It is shown that there is no need to compute the
conductance factors for each harmonic order separately. Three
conductance factors (for the fundamental, odd, and even harmonics) are sufficient to achieve the desired performance. Since
the algorithm is based on direct calculation of the conductance
factors (only three) without incorporating any iterative technique, it can effectively work under steady-state as well as
dynamic load conditions. The Lagrangian formulation is utilized to develop the proposed single-step noniterative approach.
Moreover, to extract a balanced set of voltages from unbalanced and distorted supply voltages, a novel and simple balanced voltage extractor based on synchronous reference frame
(SRF) theory is proposed. The performance of the proposed
single-step algorithm is evaluated through comparison with
the Newton–Raphson (NR)-based optimization-based control
algorithm (OCA). A real-time hardware-in-the-loop (HIL) test
bed system is developed using an OPAL-RT simulator and
a digital signal processor (DSP) DS1103 from dSPACE to
validate the performance of the proposed algorithm for practical
applications. The real-time experimental results show the compensation effectiveness of the proposed algorithm, particularly
under a dynamic load condition.
II. E XTRACTION OF BALANCED S ET OF VOLTAGES
F ROM D ISTORTED AND U NBALANCED S UPPLY
VOLTAGES U SING SRF T HEORY
One of the main objectives of the four-leg shunt APF is to
achieve balanced source currents by compensating unbalanced
load currents. In order to generate the balanced reference source
currents to control the shunt APF under distorted and unbalanced supply conditions, a balanced set of voltages needs to be
extracted.
In [27], a voltage detector approach is given, where the sum
of all harmonic components is extracted (by subtracting fundamental positive-sequence voltages from measured voltages)
to control the shunt APF for damping the voltage harmonic
propagation in distribution systems. The extracted voltages
by this method contain unbalanced harmonics as it does not
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extract the individual balanced harmonics and cannot be used
here directly. To optimally compensate the load currents under
unbalanced and distorted supply conditions using a shunt APF,
the individual balanced harmonic components were extracted
in [20]. An instantaneous symmetrical component theory combined with complex Fourier transform is utilized to extract the
balanced set of voltages in [20]. This approach is complex,
and the use of a fixed-frequency moving-average technique to
carry out integration into the Fourier transform may affect the
extraction under the following: 1) supply frequency variations
[22] and 2) the presence of interharmonics into the supply
voltages. A new approach to extract the balanced set of voltages
utilizing SRF theory is proposed in this paper and discussed in
the following.
Let the three distorted and unbalanced supply voltages at the
point of common coupling be represented as follows:
vsx (t) =
h
√ 2
Vxn sin(nωt + θxn ),
x = a, b, c
(1)
n=1
where subscript s denotes the supply, subscript x denotes the
phase of the system, n denotes the harmonic order, h denotes
the maximum harmonic order (the choice of h will depend
on the maximum harmonic order to be expected in the supply
voltages, and it is selected by the user), V denotes the rms value
of the voltage, and θ denotes the phase angle. For the nth-order
harmonic, the voltages given in (1) can be converted into the
SRF using Park’s transformation as
⎡
⎤T ⎡
⎤
sin n(ωt)
cos n(ωt)
vsa (t)
2⎣
vdn
sin n(ωt − 120) cos n(ωt − 120) ⎦ ⎣ vsb (t) ⎦
=
vqn
3
sin n(ωt + 120) cos n(ωt + 120)
vsc (t)
(2)
where ω is the fundamental angular frequency of the supply
voltages which can be obtained using the phase-locked loop.
vdn and vqn are the direct- and quadrature-axis voltage components of the nth-order harmonic voltage. With this approach,
there is no need to extract the positive-, negative-, and zerosequence components separately.
The components vdn and vqn can be represented as
v̄dn + ṽdn
vdn
=
(3)
vqn
v̄qn + ṽqn
where v̄dn and v̄qn are the dc components corresponding to the
balanced part of the nth-order harmonic voltage present in the
supply voltages while ṽdn and ṽqn are the ac components corresponding to the unbalanced part of the nth harmonics. The zerosequence component does not contain any information about
the balanced part of the voltages; hence, it is not considered
in (2).
The dc direct- and quadrature-axis components v̄dn and v̄qn
can be obtained after processing vdn and vqn through low-pass
filters (LPFs). The use of LPFs over fixed-frequency movingaverage filters into the proposed SRF-theory-based extractor
gives robustness to the extraction against supply frequency
variations and the presence of interharmonics. The balanced
set of supply voltages in the stationary reference frame for the
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013
nth harmonic is then obtained using inverse Park transform as
follows:
⎤ ⎡
⎤
⎡ sin n(ωt)
cos n(ωt)
vsan (t)
v̄
⎣ vsbn
(t) ⎦ = ⎣ sin n(ωt − 120) cos n(ωt − 120) ⎦ dn .
v̄qn
(t)
sin n(ωt + 120) cos n(ωt + 120)
vscn
(4)
The above procedure is carried out for n = 1, 2, . . . , h.
Finally, the three-phase balanced set of voltages can be expressed as follows:
(t) =
vsx
h
vsxn
(t),
x = a, b, c.
(5)
n=1
Since all the quantities on the right-hand side of (5) are
balanced, vsx
represents the balanced set of supply voltages
derived from the distorted and unbalanced supply voltages.
III. P ROPOSED S INGLE -S TEP N ONITERATIVE
O PTIMIZED A LGORITHM
To achieve UPF operation, under distorted and unbalanced
supply conditions, the following conditions should be satisfied:
1) The source currents should have the same harmonic content
as the supply voltages, and 2) all the phase currents should be in
phase with their respective phase voltages. This suggests that,
in the case of UPF operation, the source current THD should
equate to the source voltage THD value while maintaining inphase relationship with respect to the individual phase voltage.
Therefore, the source current THD and UB factor may not
be within the acceptable limit. By controlling the harmonic
ratios and balancing the average power equally among the
three phases, the source current THD and UB factor can be
maintained within the specified limits. For the source current
to meet all the aforementioned constraints while maximizing
the power factor, the APF control algorithm is formulated as an
optimization problem where the main variable is the conductance factor Gn for each individual harmonic order. Using (5),
the desired source currents can be expressed as follows:
+
i∗sx (t) = G1 vsx1
h
Gn vsxn
,
x = a, b, c
(6)
n=2
where ∗ denotes the reference or desired quantity. G1 and Gn
are the conductance factors for the fundamental and nth-order
harmonic components. The values of these conductance factors
can be controlled to maximize the power factor while satisfying
the average power balance and THD constraints. Since the
same conductance factor value will be applied to each phase
to achieve balanced source currents, the optimization problem
can be solved and formulated considering one phase. The next
sections will highlight the proposed problem formulation for
determining the optimal conductance factors.
A. Objective Function
There are various objectives that can be applied to the shunt
APF problem which include minimization of the source current
THD, minimization of the APF kilovoltampere rating, or maximization of the source power factor [16], [18]. The most practical objective is to maximize the source power factor, which
consequently reduces the cost of electricity consumption under
power-factor-based tariff (the most commonly used electricity
tariff plan for industrial users). Therefore, the objective chosen
here is to maximize the power factor which can be achieved
by minimizing the square of the apparent power calculated
using the extracted balanced set of voltages and the desired
source current of any phase [19], [20]. Since both the extracted
voltages and the desired source currents are balanced, the threephase rms quantities are equal and thus can be represented as
follows:
= Vsbn
= Vscn
= Vsn
Vsan
,
for n = 1, 2, 3, . . . , h
∗
∗
∗
∗
Isan
= Isbn
= Iscn
= Isn
(7)
where the rms value of the balanced nth-order harmonic voltage
of any phase can be computed as follows:
2 + v̄ 2
v̄dn
qn
.
(8)
=
Vsn
2
Using (7), the objective function f , which is the square of the
apparent power of any phase, is given by
f=
h
n=1
2
Vsn
h
2
G2n Vsn
.
(9)
n=1
B. Equality Constraint
For the desired source currents to be balanced, the three
source currents should supply the demanded average total
power equally. Therefore, the equality constraint for any phase
can be written as
PLavg + PLoss 2
−
Gn Vsn
=0
3
n=1
h
(10)
where PLavg is the demanded average load power, which is
computed using instantaneous three-phase source voltages and
load currents. This instantaneous power PL (t) is then processed
by an LPF to obtain PLavg . PLoss is the average power required
to overcome the losses in the shunt APF and thus to maintain
the dc-link voltage.
C. Inequality Constraints
The approaches discussed in [19] and [20] consider one THD
limit on the source current THD (THDi ). As per IEEE Standard
519, the limit on the current distortion for individual even
harmonics is 25% of the limit for odd harmonics. To address
this, different THD limits for the current distortion due to odd
and even harmonics are considered in this paper.
The upper bounds on the source current harmonic distortion
due to odd and even harmonics are denoted as THDi,max _o and
KANJIYA et al.: NONITERATIVE OPTIMIZED ALGORITHM FOR SHUNT ACTIVE POWER FILTER
THDi,max _e , respectively. The inequality constraints on current
distortion due to odd and even harmonics are given by
THD2i_o ≤ THD2i,max _o
(11)
≤ THD2i,max _e
(12)
THD2i_e
where
THDi_o =
THDi_e =
h
2 2
o=3,5,... Go Vso
G1 Vs1
h
2 2
e=2,4,... Ge Vse
G1 Vs1
(13)
(14)
The source-side power factor is maximum (UPF) when both
the source voltage and current THDs are equal (UPF operation).
Let THDspec_o and THDspec_e be the user-defined (prespecified) THD limits on the source current harmonic distortion
due to odd and even harmonics, respectively. For the cases
where the source voltage THDs due to odd and even harmonics (THDv_o and THDv_e ) are greater than THDspec_o and
THDspec_e , respectively, the source current THDs should be
equal to the prespecified THD values to achieve maximum
power factor. On the other hand, for the cases where THDv_o
and THDv_e are less than THDspec_o and THDspec_e , respectively, the source current THDs should be equal to the supply
voltage THDs to maximize the source-side power factor. This
can be mathematically represented as follows:
THDi,max _o = THDspec_o , if THDspec_o < THDv_o
= THDv_o ,
otherwise
THDi,max _e = THDspec_e ,
= THDv_e ,
(15)
if THDspec_e < THDv_e
.
otherwise
(16)
For optimality, the source current THDs will always reach
their upper limits as per (15) and (16), and thus, the inequality
constraints given in (11) and (12) can be reformulated as
equality constraints as follows:
h
G2o Vso
− THD2i,max _o G21 Vs1
=0
2
2
(17)
o=3,5,...
h
G2e Vse
− THD2i,max _e G21 Vs1
= 0.
2
of the problem, and thus, a simple single-step noniterative
solution is achievable.
D. Proposed Single-Step Noniterative Solution
The optimization problem presented in the previous sections
is a constrained nonlinear optimization problem where the main
variables are the conductance factors. Using the Lagrangian
function, the problem can be transformed into an unconstrained
optimization problem as follows [25]:
L=
.
2
h
2
h
G2n Vsn
2
n=1
+ λ1
PLavg + PLoss 2
−
Gn Vsn
3
n=1
h
h
+ λ2
2
G2o Vso
−
2
THD2i,max _o G21 Vs1
−
2
THD2i,max _e G21 Vs1
o=3,5,...
h
+ λ3
2
G2e Vse
.
(19)
e=2,4,...
By applying the Karush–Kuhn–Tucker optimality conditions
to (19) [25], the following set of equations can be derived:
h
∂L
2
2
2
2
= 2G1 Vs1
Vsn
− λ1 Vs1
− 2G1 λ2 Vs1
THD2i,max _o
∂G1
n=1
− 2G1 λ3 Vs1
THD2i,max _e = 0
2
(20)
h
∂L
2
2
2
2
= 2Go Vso
Vsn
− λ1 Vso
− 2Go λ2 Vso
= 0,
∂Go
n=1
o = 3, 5, . . . , h
(21)
h
∂L
2
2
2
2
= 2Ge Vse
Vsn
− λ1 Vse
− 2Ge λ3 Vse
= 0,
∂Ge
n=1
e = 2, 4, . . . , h
(22)
∂L
PLavg + PLoss 2
−
=
Gn Vsn
=0
∂λ1
3
n=1
h
(18)
In the following section, it will be proven that, by converting
the inequality constraints given in (11) and (12) into equality
constraints as per (17) and (18) [by selecting THDi,max _o/e
according to the conditions given in (15) and (16)], the closedform solution of the optimum conductance factors is possible
without incorporating any iterative optimization technique. The
identification of the optimum source current THDs to achieve
the maximum power factor significantly reduces the complexity
Vsn
n=1
∂L
=
∂λ2
e=2,4,...
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∂L
=
∂λ3
h
(23)
G2o Vso
− THD2i,max _o G21 Vs1
=0
(24)
G2e Vse
− THD2i,max _e G21 Vs1
= 0.
(25)
2
2
o=3,5,...
h
2
2
e=2,4,...
From (21) and (22), it can be deduced that the conductance
factors for all odd harmonics as well as for all even harmonics
will be equal and can be represented as follows:
G3 = G5 = · · · = GH _o
(26)
G2 = G4 = · · · = GH _e .
(27)
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013
By substituting (26) and (27) into (23)–(25), the following
equations can be derived:
PLavg+PLoss
2
−Vs1
G1+GH _o THD2v_o+GH _e THD2v_e =0
3
(28)
THD2v_o G2H _o − THD2i,max _o G21=0
(29)
THD2v_e G2H _e − THD2i,max _e G21=0
(30)
where
THDv_o =
THDv_e =
h
o=3,5,...
Vs1
2
Vso
h
e=2,4,...
Vs1
2
Vse
(31)
.
(32)
From (29) and (30), GH _o and GH _e can be represented in
terms of G1 as follows:
GH _o =
THDi,max _o
G1
THDv_o
(33)
GH _e =
THDi,max _e
G1 .
THDv_e
(34)
Furthermore, by substituting (33) and (34) into (28), an
expression for G1 can be derived as follows:
G1 =
PLavg +PLoss
.
2
3Vs1 (1+THDi,max _o THDv_o +THDi,max _e THDv_e )
(35)
Equations (33)–(35) present closed-form mathematical formulas for determining the optimal values for the conductance
factors. By substituting the values of G1 , GH _o , and GH _e
from (33)–(35) into (6), the reference source currents can be
rewritten as
i∗sx (t) = G1 vsx1
+ GH _o
h
vsxo
o=3,5,...
+ GH _e
h
vsxe
,
x = a, b, c.
(36)
e=2,4,...
Equation (36) suggests that the proposed algorithm may have
superior performance compared to the algorithms given in [19]
and [20] due to the direct calculation of the conductance factors
G1 , GH _o , and GH _e using (33)–(35). The distinguishing
features of the proposed single-step optimized approach can be
summarized as follows.
1) The proposed approach involves only three conductance
factors (one for the fundamental harmonic and two other
for odd and even harmonics) as opposed to the h conductance factors in [19] and [20]. This will reduce the
complexity of the problem.
2) The different conductance factors for the odd and even
harmonics facilitate the selection of separate THD limits
on odd and even harmonics.
Fig. 1. Flowchart to determine the conductance factors G1 , GH _o , and
G H _e .
3) Mathematical formulas that can calculate the optimal
conductance factors directly and thus avoid the use of
iterative techniques are derived. The direct calculation of
G1 , GH _o , and GH _e can greatly reduce the computation
time and thus achieve faster response time when compared to other methods [19], [20].
4) As opposed to the previous approaches in [15]–[20],
since the proposed method does not involve an iterative
approach, it can be more effective for dynamic load
variations.
The flowchart of the proposed single-step noniterative solution procedure is shown in Fig. 1. The algorithm will
start by measuring the source voltage and extracting the
rms value of each balanced harmonic component. This will
be used to determine THDv_o and THDv_e . As mentioned
previously in (15) and (16), THDv_o and THDv_e will be
compared with THDspec_o and THDspec_e , respectively, to
determine THDi,max _o and THDi,max _e . For the cases where
THDspec_o is less than THDv_o , THDi,max _o will be set equal
to THDspec_o . Otherwise, the value of THDi,max _o will be
set equal to THDv_o . Similarly, THDi,max _e will be selected
by comparing THDspec_e with THDv_e . Using the derived
formulas in (33)–(35), the optimal conductance factors are
determined. The flowchart is an integral part of the overall
control block of the APF which is discussed in the next section.
The noniterative solution of the optimization problem with
constraints on different power quality factors, such as distortion
factor (DF) and K-factor (KF), is also discussed briefly in
Appendix III.
KANJIYA et al.: NONITERATIVE OPTIMIZED ALGORITHM FOR SHUNT ACTIVE POWER FILTER
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(36). The reference source neutral current is set equal to zero.
These reference source currents are compared with the actual
measured currents using the hysteresis current controller which
determine the switching signals for the VSI.
V. S IMULATION R ESULTS
To verify the performance of the proposed single-step noniterative optimized control algorithm, a detailed simulation study
is carried out under the MATLAB/SIMULINK environment.
The simulated test system data are given in Appendix I. The
performance of the proposed single-step noniterative optimized
control algorithm is validated and compared with that of an iterative OCA given in [20]. The simulation studies for both steadystate and dynamic conditions are performed and discussed in
the following.
Fig. 2.
Four-leg VSI-based shunt APF system configuration.
A. Case 1: Unbalanced Supply Voltages Distorted With Odd
Harmonics Only—Steady-State Load Condition
IV. OVERALL C ONTROL B LOCK OF S HUNT APF
Fig. 2 represents the system under study which consists of
an equivalent grid behind an impedance, combined single- and
three-phase loads, and the shunt APF. The four-leg voltagesource inverter (VSI) topology is utilized to realize a threephase four-wire (3P4W) APF system. The overall control block
diagram of the proposed single-step noniterative optimized
control algorithm to control the shunt APF under distorted and
unbalanced supply conditions is shown in Fig. 3.
The inputs to the block of single-step calculation of G1 ,
GH _o , and GH _e are the specified source current THD limits (THDspec_o and THDspec_e ), the demanded average total
power (PLavg + PLoss ), and the rms values of the balanced
, n = 1, 2, . . . , h). The demanded
set of harmonic voltages (Vsn
instantaneous load power is calculated as
PL (t) = vsa iLa + vsb iLb + vsc iLc .
(37)
The instantaneous power PL (t) is then processed by an
LPF to obtain the average power PLavg . To maintain the dclink capacitor voltage constant at a prespecified value, a small
amount of active power should be drawn from the grid. To
accomplish this, a discrete proportional–integral (PI) controller
is used, which can be given as follows:
PLoss (k) = Ploss (k − 1) + Kp {e(k) − e(k − 1)} + Ki e(k)
(38)
where
f
∗
e(k) = vdc
− vdc
(k)
(39)
f
(k) is the measured instantaneous voltage across
where vdc
∗
is the
the dc-link capacitor processed through the LPF, vdc
reference dc-link voltage, and k is the sample number. Kp and
Ki are the proportional and integral gains, respectively, of the
PI controller. The rms values of the balanced set of harmonic
voltages are calculated as in (8) using the extracted balanced
set of harmonic voltages in the SRF. After calculating the
conductance factors G1 , GH _o , and GH _e as shown in Fig. 1,
the three-phase reference source currents are calculated as in
In order to test the proposed approach under distorted and
unbalanced supply voltage conditions, the unbalanced threephase supply voltages are expressed in Table I.
Fig. 4 shows the simulated results with the proposed singlestep optimized algorithm under steady-state load condition.
The profiles of the distorted–unbalanced supply voltages and
the nonlinear distorted–unbalanced load currents are given in
Fig. 4(a) and (b). Table II gives the rms and THD values of
each phase voltage and load current with the source voltage and
load current UB factors. The values of the power factor and the
source current UB, with APF compensation, using the proposed
control algorithm and OCA are also presented in Table II.
The UB factor is calculated as follows:
UB =
avg(rms a, b, c) − min(rms a, b, c)
× 100.
avg(rms a, b, c)
(40)
The extracted balanced set of voltages using the d−q transformation method is depicted in Fig. 4(e). The performance of
the proposed optimized control algorithm is evaluated for the
following three conditions: 1) HF (0% THD) operation; 2) UPF
operation; and 3) a specified-THD-limit operation.
Fig. 4(f) gives the compensated source currents when the
APF is controlled under HF condition using the proposed
optimized control. For this case, the THD limits THDspec_o
and THDspec_e are set equal to 0%. The source currents after
the compensation are balanced and sinusoidal. As noticed from
Table II, the THD of the source currents with HF mode of
operation is around 1.7%. The power factors in the HF mode
measured with respect to the unbalanced supply voltages are
0.955 lagging (phase a), 0.905 lagging (phase b), and 0.884
lagging (phase c), while the power factor of all phases measured
with respect to the extracted balanced set of voltages is 0.969
lagging.
To test the controller performance, the odd and even THD
limits were specified as 100%. As discussed in the previous
sections and shown in the flowchart (refer to Fig. 1), the proposed algorithm compares the specified THD limits THDspec_o
and THDspec_e of 100% with the balanced set of source voltage
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Fig. 3. Overall control block diagram for the proposed noniterative optimized control of the four-leg VSI-based shunt APF.
TABLE I
D ISTORTED AND U NBALANCED S UPPLY VOLTAGES (C ASE 1)
THDs THDv_o and THDv_e which, in this case, are 24% and
0%, respectively. The specified THD limits of 100% are greater
than the balanced set of supply voltage THDs, and therefore,
the control algorithm operates in the UPF mode to achieve the
maximum possible power factor operation. Fig. 4(g) illustrates
the compensated source currents when the APF system operates
in the UPF operation. The UPF with respect to the balanced set
of voltages is achieved in this mode of operation with the source
current THD close to 24% (equal to the value of the balanced
set of supply voltage THDs). The power factors of each phase
measured with respect to the unbalanced supply voltages are
given in Table II.
Finally, the performance of the proposed single-step optimized control algorithm is evaluated considering constraints on
source current THD levels while maximizing the power factor.
The THD limits of the source currents are specified as 5%.
Fig. 4(h) depicts the simulation results under the optimized
mode of operation. From Table II, the source current THD is
achieved around 5.6% with the source-side power factors, measured with respect to the unbalanced supply voltages, as 0.967
lagging (phase a), 0.912 lagging (phase b), and 0.900 lagging
(phase c). The power factor of all the phases is measured to
be 0.983 lagging with respect to the extracted balanced set
of voltages. Note that the slight increase in the actual source
current THD values (greater than specified 5%) is due to the
switching operation of the VSI.
It can be viewed from the previously discussed three modes
that the compensated source current profile in the optimized
mode is in between HF and UPF operations. In all the afore-
Fig. 4. Simulation results under unbalanced supply voltages distorted with
odd harmonics using the proposed single-step optimized control algorithm.
(a) Supply voltages. (b) Load currents. (c) Load neutral current. (d) DC-link
voltage. (e) Extracted balanced set of voltages. (f) Source currents (HF mode).
(g) Source currents (UPF mode). (h) Source currents (optimized mode).
mentioned operating modes, the fourth leg of the shunt APF
effectively compensates the load neutral current [Fig. 4(c)] and
thus reduces the source-side neutral current to zero [shown with
cyan color in Fig. 4(f)–(h)].
KANJIYA et al.: NONITERATIVE OPTIMIZED ALGORITHM FOR SHUNT ACTIVE POWER FILTER
5383
TABLE II
P ERFORMANCE I NDICES W ITH THE P ROPOSED O PTIMIZED A PPROACH AND THE OCA A PPROACH (S TEADY-S TATE C ONDITION ; C ASE 1)
Fig. 5 gives the simulated results for the system under consideration, under three operating modes (HF, UPF, and optimized),
with an iterative OCA method. The performance indices of the
OCA method are also provided in Table II. As viewed from
Figs. 4 and 5 and Table II, the performance of the proposed
single-step optimized algorithm is identical to that of the OCA
method since the supply voltages are distorted with only odd
harmonics and the THD limit in the case of OCA was set the
same as THDspec_o . The proposed optimized method, however,
requires minimum computational burden compared to the OCA
method since the model relies on only three control variables
(G1 , GH _o , and GH _e ) and the direct computation of the
variables without any iterative technique. Table III gives the
comparison between NR-based OCA and the proposed singlestep optimized control method. The steady-state per-unit values
used to determine the conductance factors using both methods
are given in Appendix II.
The CPU used for this study has a Core i5 processor with
4-GB RAM. For the NR method, the optimization problem
with seven conductance factors is formulated and solved using
MATLAB optimization toolbox [19], [20]. The optimum value
is found to be 0.088 which was reached in 0.11 s or 5.5 cycles
(average) utilizing eight iterations (average) compared to 10 μs
or almost instantaneous (Inst.) for the proposed single-step
method. For both cases, 0.11 s and 10 μs represent the times
required to compute the conductance factors which do not
include the time required for the extraction of the balanced set
of voltages.
This demonstrates that the proposed optimized algorithm
computes the conductance factors almost instantaneously. The
extremely low computational time makes the proposed method
capable of compensating dynamically changing loads. The
results by the NR method, with maximum numbers of iterations
set equal to four and two iterations, are shown in Table III. It
can be noticed that, despite the reduction in the computational
time, the solution is not optimum. Moreover, it is worthy
to note that the execution time and the number of iterations
taken by the NR method to find the optimum value are highly
dependent on the initial guess of the control variables and, in
some cases, might not guarantee the global optimal solution.
The NR method was run 50 times with random initial guesses
Fig. 5. Simulation results under unbalanced supply voltages distorted with
odd harmonics using OCA. (a) Supply voltages. (b) Load currents. (c) Load
neutral current. (d) DC-link voltage. (e) Extracted balanced set of voltages.
(f) Source currents (HF mode). (g) Source currents (UPF mode). (h) Source
currents (optimized mode).
for all control variables, and it was found that the maximum
number of iterations taken was 11 with an execution time of
7.5 cycles, while the minimum number of iterations recorded
was three with an execution time of 2.5 cycles. The average
number of iterations and the average execution time found
for 50 runs of the NR method were eight and 5.5 cycles,
respectively. It is important to note that the sampling time of
the controller for the shunt APF with the NR method should be
higher than the maximum optimization execution time.
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TABLE III
OVERALL C OMPARISON B ETWEEN THE I TERATIVE NR A PPROACH AND
THE P ROPOSED N ONITERATIVE S INGLE -S TEP O PTIMIZED A PPROACH
Fig. 7. Simulation results for dynamic load condition under unbalanced
supply voltages distorted with odd harmonics using an iterative optimization
method. (a) Source currents (optimized mode). (b) DC-link voltage.
TABLE IV
P ERFORMANCE I NDICES W ITH THE P ROPOSED O PTIMIZED A PPROACH
AND THE OCA A PPROACH (A FTER L OAD C HANGE )
Fig. 6. Simulation results for dynamic load condition under unbalanced
supply voltages distorted with odd harmonics using the proposed algorithm.
(a) Supply voltages. (b) Load currents. (c) Source currents (optimized mode).
(d) DC-link voltage.
B. Case 2: Unbalanced Supply Voltages Distorted With Odd
Harmonics Only—Dynamic Load Condition
The performance of the proposed control algorithm, considering the distorted and unbalanced supply voltages given in
Table I, during a sudden load change condition is illustrated
in Fig. 6.
To create the dynamic condition, at time t = 0.2 s, the load
is changed from L1 to L1+L2 (Appendix I) and again brought
back to L1 at time t = 0.3 s. The change in the load current
profile can be viewed from Fig. 6(b). The compensated source
current profile is shown in Fig. 6(c). As noticed, the APF system
with the proposed single-step optimized approach achieves
the new steady-state condition within one cycle and without
affecting the APF compensation capability during both load
increase and decrease. Furthermore, the dc-link controller, as
shown in Fig. 6(d), effectively regulates the dc-bus voltage at
the set reference value.
Under the same dynamic condition, the performance of the
iterative OCA method is given in Fig. 7. As observed from
Fig. 7(a), the compensated source currents are not balanced.
In addition, the dc-link voltage, shown in Fig. 7(b), settles
at a new operating point, lower than the set reference value.
This is mainly because of the ten-cycle computational delay
(maximum execution time of 7.5 cycles plus 2.5-cycle safety
margin) in calculating the new conductance factors. Therefore,
for a duration of ten cycles, the OCA-based controller tries
to compensate the source currents based on the previously
computed conductance factors, and thus, the source currents
become more distorted and unbalanced. Table IV presents the
compensated current THDs, power factors, rms values, and UB
factors under the new operating condition using the proposed
optimized algorithm and OCA. Thus, this dynamic condition
demonstrates the true capability and enhanced performance
of the proposed single-step optimized algorithm over other
optimization-based approaches.
C. Case 3: Unbalanced Supply Voltages Distorted With Both
Odd and Even Harmonics—Steady-State Load Condition
The superiority of the proposed algorithm over OCA in terms
of computation time was highlighted in the previous two cases.
In order to show the advantage of having different THD limits
on odd and even harmonics, the performance of the proposed
algorithm is evaluated and compared with that of OCA under
unbalanced supply voltages distorted with both even and odd
harmonics (Table V).
KANJIYA et al.: NONITERATIVE OPTIMIZED ALGORITHM FOR SHUNT ACTIVE POWER FILTER
TABLE V
D ISTORTED AND U NBALANCED S UPPLY VOLTAGES (C ASE 3)
5385
its are imposed on odd and even harmonics with the proposed method, the measured second and fourth harmonics in
the source currents are 0.95% and 0.88%, respectively (below
the limit). The measured odd harmonics (fifth and seventh)
in the source currents are within the individual odd harmonic
limit as per IEEE Standard 519 with both algorithms. The KF of
the source current is slightly higher with the proposed algorithm
due to higher amount of higher order harmonics (fifth and
seventh) compared to OCA. It is worthy to note that considering
different THD limits on odd and even harmonics is one step
toward the optimized control of the shunt APF taking into
account individual harmonic constraints. As seen from the results, the proposed approach improves upon existing methods.
Future work would be to implement a noniterative approach
that considers directly constraints on individual harmonics.
VI. R EAL -T IME HIL I MPLEMENTATION
Fig. 8. Simulation results under unbalanced supply voltages distorted with
both odd and even harmonics using the proposed optimized control algorithm
and OCA. (a) Supply voltages. (b) Load currents. (c) Source currents (proposed
algorithm). (d) Source currents (OCA).
The source current THD limit in the case of OCA is set
equal to 5%, while the THD limits (THDspec_o and THDspec_e )
in the case of the proposed algorithm are set equal to 4.85%
and 1.21%, respectively. THDspec_o is set equal to four times
THDspec_e to comply with IEEE Standard 519. The aforementioned limits for the proposed algorithm are calculated using
the following to maintain an overall THD limit (THDspec_all )
of 5%:
THD2spec_all = THD2spec_o + THD2spec_e .
(41)
The profiles of the supply voltages and load currents are
given in Fig. 8(a) and (b). The compensated source currents
using the proposed algorithm are shown in Fig. 8(c), while
Fig. 8(d) depicts the source currents using OCA. Note that
there is a slight difference in the shape of the source current
waveforms with the proposed algorithm and OCA due to different odd and even harmonic levels in the source currents. The
different performance indices with the proposed algorithm and
OCA are provided in Table VI. The performances of the shunt
APF with both algorithms are almost identical in terms of rms
value, power factor, and THD of the source currents; the main
differences lie in the individual harmonic distortion (IHD), the
filter’s kilovoltampere rating, and the KF as shown in Table VI.
The average IHDs of the three-phase source currents for both
algorithms are shown shaded in Table VI. IEEE Standard 519
recommends that the individual even and odd harmonics in the
source current should be less than 1% and 4%, respectively,
for a system with short-circuit ratio less than 20. The second
and fourth harmonics in the source currents are measured to
be 2.88% and 2.62%, respectively, with OCA (shown bold)
which are above the individual even harmonic limit as per
IEEE Standard 519. On the other hand, since individual lim-
A real-time HIL system is built to validate the feasibility
of the proposed single-step optimized approach for practical
applications. Fig. 9 illustrates the developed laboratory experimental setup. The real-time HIL system is composed of an
OPAL-RT digital simulator and a rapid prototyping DSP board
from dSPACE, DS1103. The OPAL-RT is a real-time simulation platform based on two Intel Xeon QuadCore 2.40-GHz
processors (total of eight cores or CPUs) working under the RTLAB software environment. OPAL-RT is equipped with analog
inputs/outputs (16 each) and digital inputs/outputs (32 each).
The DS1103 has 20 analog-to-digital converter (ADC) ports,
eight digital-to-analog converter (DAC) ports, and 32 digital
inputs/outputs. As shown in Fig. 9, the OPAL-RT represents the
power system where all the power circuit components, such as
the 3P4W unbalanced–distorted source, the unbalanced load,
and the shunt APF, are implemented. The DS1103, on the
other hand, represents the digital controller for the shunt APF.
In an actual practical system, the OPAL-RT will be replaced
by the actual power source, load, and inverter, whereas the
digital controller will remain the same. The necessary seven
signals (three supply voltages, three load currents, and the dcbus voltage) are measured and taken out of OPAL-RT through
its DAC ports. These real-time signals are available in the
MATLAB/SIMULINK platform on the host computer through
the ADC ports of DS1103 and are utilized to generate the
reference source currents based on the proposed control algorithm (indirect control). For the neutral current compensation,
the source neutral current is directly considered as zero. These
generated reference source currents (total of four, namely, three
phase currents and one neutral current) are taken out of the
DSP through DAC ports. The actual source current signals
(total of four, namely, three phase currents and one neutral
current) are also taken out from the OPAL-RT. An external
analog hysteresis current control board is developed to perform
pulsewidth modulation. The actual and reference source current
signals are then compared, and the eight necessary switching
pulses for the shunt inverter are generated. Finally, these eight
switching/gate pulses are transferred to OPAL-RT using digital
input–output (I/O) ports and utilized to control the shunt APF
inverter in real time. It should be noted that all the signals are
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TABLE VI
P ERFORMANCE I NDICES FOR THE P ROPOSED O PTIMIZED A PPROACH AND THE OCA A PPROACH (S TEADY-S TATE C ONDITION ; C ASE 3)
Fig. 9. Laboratory real-time HIL system representation.
normalized on a 5-V scale (5 V = 1 p.u.; the base values in
Appendix I). The maximum limits on the I/O signals for OPALRT are ±16 and ±10 V for dSPACE. The sampling times for
both OPAL-RT and DS1103 systems were 20 μs each.
The performance of the shunt APF real-time HIL system is
evaluated for both steady-state and dynamic conditions. Fig. 10
gives the real-time test system results. The source voltage
and load current profiles in steady state are given in Fig. 10(a)
and (b), respectively. It is worthy to note that identical system
parameters and load conditions are considered (the same as
discussed in Section V-A and B). The real-time experimental results with the proposed optimized algorithm for three
different modes of operation are given in Fig. 10(c)–(e). As
noticed from Fig. 10(c) for the case of HF operation, the
source currents are achieved as balanced and sinusoidal. The
actual THD values of the source currents are noticed as 3.9%.
The harmonics are mostly due to the sampling time of the
OPAL-RT and the switching operation of the inverter. Fig. 10(d)
gives the source currents when the optimized algorithm is
operated to achieve maximum power factor at 5% THD limit.
In this case, the THD of the source currents is measured to be
5.94%. The harmonic spectrum of the phase-a supply voltage,
load current, and source current up to the seventh harmonic
during 5%-THD-limit operation is given in Table VII. The
source current profiles when the THD limits are defined as
100% are illustrated in Fig. 10(e). In this case, the proposed
algorithm determines the maximum THD limits by comparing
them with a balanced set of source voltages (THDv_o = 24%
and THDv_e = 0%). The compensated source currents have a
THD of 24.47%. Note that the source currents are identical to
the extracted distorted–balanced set of supply voltage profiles
shown in Fig. 4(e).
The dynamic performance of the proposed noniterative optimized control algorithm is shown in Fig. 10(f)–(i). Initially,
the load on the system is a three-phase diode bridge rectifier
with a resistor. Suddenly, an unbalanced load is connected to
the system. The load current profile due to this dynamic load
change is shown in Fig. 10(f). Prior to the load change, the
controller is working under HF mode. As seen from Fig. 10(g),
the shunt APF system, together with the proposed algorithm,
maintains the desired performance. Additionally, as noticed
from the three-phase source currents in Fig. 10(h) and (i),
the dynamic performance can be achieved in different modes
of operation. The load neutral current compensation during
dynamic load change is depicted in Fig. 11. It can be seen that
the source neutral current is achieved equal to zero by injecting
a compensating neutral current opposite to the load neutral
current through the fourth leg of the shunt APF. This study
thus validates that the proposed optimized control algorithm
can perform well under dynamic conditions.
VII. C ONCLUSION
A single-step noniterative optimized control algorithm has
been proposed for a 3P4W shunt APF to achieve an optimum
performance between power factor and THD. The proposed
optimized approach is simple to implement and does not require complex iterative optimization techniques to determine
the conductance factors. It is shown mathematically that only
three conductance factors (one for the fundamental harmonic
and two other for odd and even harmonics) are sufficient to
determine the desired reference source currents. The proposed
algorithm determines the conductance factors in 10 μs. Because
of the smaller computational time, the proposed algorithm
KANJIYA et al.: NONITERATIVE OPTIMIZED ALGORITHM FOR SHUNT ACTIVE POWER FILTER
5387
Fig. 10. Real-time experimental results (scale: X-axis = 10 ms/div and Y -axis = 1 p.u./div for all the quantities except for Vdc where Y -axis = 0.2 p.u./div).
(a) Source voltages (vsa , vsb , and vsc ). (b) Load currents (iLa , iLb , iLc , and iLn ). (c) Source currents (HF mode). (d) Source currents (optimized mode).
(e) Source currents (UPF mode). (f) Load currents (load change). (g) Phase-a performance (HF-mode load change). (h) Source currents (HF-mode load change).
(i) Source currents (optimized-mode load change).
TABLE VII
H ARMONIC S PECTRUM OF D IFFERENT Q UANTITIES
D URING 5%-THD-L IMIT O PERATION
performs satisfactorily under dynamically changing load conditions (other optimization-based approaches are limited to
steady-state conditions). The performance of the proposed algorithm is validated by a real-time HIL experimental prototype.
The satisfactory real-time experimental results for steady-state
as well as dynamic conditions demonstrate the feasibility of the
proposed algorithm for practical implementation.
Fig. 11. Real-time experimental results: Neutral current compensation (scale:
X-axis = 10 ms/div and Y -axis = 1 p.u./div).
A PPENDIX II
A PPENDIX I
The system data for simulation as well as experimental study
are shown in Table VIII.
The steady-state per-unit values used to determine the conductance factors are as follows: THDspecified = 0.05, PLavg =
= 0.6566, Vs5
= 0.1389, Vs7
= 0.1, and Vs2
=
0.8645, Vs1
Vs3 = Vs4 = Vs6 = 0.
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TABLE VIII
S YSTEM DATA FOR S IMULATION AS W ELL AS E XPERIMENTAL S TUDY
TABLE IX
P ERFORMANCE I NDICES ACHIEVED FOR THE O PTIMIZATION
P ROBLEM W ITH DF C ONSTRAINT
when DFv is less than DFspec , the source current DF should
equate to DFspec to achieve maximum possible power factor
satisfying the DF constraint. The appropriate value of DFmin
can be selected by comparing DFv with DFspec . With this
selected value of DFmin , the inequality constraint in (44) can
be reformulated as equality constraint and expressed as
G21 V1
2
A PPENDIX III
O PTIMIZATION P ROBLEM F ORMULATION AND I TS
S OLUTION W ITH A LTERNATE P OWER
Q UALITY C ONSTRAINTS
From (45), GH can be represented in terms of G1 as follows:
A. Constraint on Source Current DF
The DF describes how the harmonic distortion of the source
current affects the effective source power factor. The current DF
(DFi ) is defined as the ratio of the fundamental rms current to
the total rms current
I1rms
.
Irms
(42)
The lower the DF, the higher the current distortion; therefore,
the lower bound on the source current DF is considered and
given as
DF ≥ DFmin .
(43)
As discussed before, to achieve maximum power factor, the
conductance factors for all the harmonics should be equal, and
here, it is denoted as GH . Using the fundamental conductance
factor G1 and the harmonic conductance factor GH , the constraint on the source current DF can be rewritten as
G21 V12
G21 V12
≥ DF2min .
+ G2H hn=2 Vn2
(45)
n=2
In the previous discussion, the optimization problem aiming
at maximizing the power factor subject to the power balance
and THD constraints was introduced. However, there are various power quality constraints other than THD such as DF and
KF that are of technical interest in certain conditions. This
section provides a brief discussion on how to formulate and
solve an optimization problem to compute optimal conductance
factors considering DF and KF.
DFi =
h
2
1 − DF2min − DF2min G2H
Vn = 0.
(44)
The maximum source-side power factor can be achieved
when both the source voltage and current DFs are equal. Let
DFspec be the lower bound on the DF specified by the user. To
achieve the maximum possible power factor under the condition
where the source voltage DF (DFv ) is higher than DFspec , the
source current DF should equate to DFv . On the other hand,
GH =
XDF
G1
THDv
(46)
where
XDF =
1 − DF2min
.
DFmin
(47)
Using G1 and GH , the power balance constraint given in (10)
can be rewritten as
PLavg + PLoss
2
− Vs1
G1 + GH THD2v = 0.
3
(48)
By substituting (46) into (48), an expression to compute G1
can be derived as follows:
G1 =
PLavg + PLoss
.
2
3Vs1 (1 + XDF THDv )
(49)
Equations (46) and (49) present closed-form mathematical
expressions to compute the optimal conductance factors which
maximize the source-side power factor satisfying the power
balance and DF constraints.
Using the steady-state per-unit values provided in
Appendix II, the optimization problem with DF constraint is
solved using (46) and (49). The results achieved with different
values of DFspec are tabulated in Table IX. First, DFspec is
specified equal to 0%. As discussed earlier, the proposed
algorithm compares DFspec with DFv to determine DFmin . As
seen from Table IX, the specified DF limit of 0% is less than
DFv , and therefore, the control algorithm chose DFmin equal
to DFv . With this value of DFmin , the optimum conductance
factors which ensure the source current DF (DFi ) within the
limit (equal to DFv ) are computed and provided in Table IX.
The conductance factors and DFi corresponding to DFspec
equal to 98% are also provided in Table IX. Note that DFspec
is higher than DFv in this case; therefore, the control algorithm
chose DFmin equal to DFspec , and the source current DF (DFi )
is achieved equal to DFspec .
KANJIYA et al.: NONITERATIVE OPTIMIZED ALGORITHM FOR SHUNT ACTIVE POWER FILTER
TABLE X
P ERFORMANCE I NDICES ACHIEVED FOR THE O PTIMIZATION
P ROBLEM W ITH KF C ONSTRAINT
[4]
[5]
B. Constraint on Source Current KF
Another important power quality factor is the KF, particularly when a nonlinear load is supplied through a transformer.
The KF is a weighting of the harmonic currents according to their effects on transformer heating, as derived from
ANSI/IEEE C57.110. The KF for current is defined as follows:
2
h
2 In
n
n=1
I1
(50)
KFi = 2 .
h
n=1
In
I1
[6]
[7]
[8]
[9]
[10]
To limit the transformer heating, the upper bound on KF is
given by
KFi ≤ KFmax .
(51)
Let KFspec be the user-defined maximum limit on KF. By
comparing KFspec with the source voltage KF (KFv ) and
following a similar procedure discussed in the previous section,
the inequality constraint in (51) can be converted into equality
constraint as follows:
h
h
2 2
2
2 2
2 2
2
2
n Vn − KFmax G1 V1 +GH
Vn = 0.
G1 V1 + GH
n=2
n=2
(52)
From (52), GH can be represented in terms of G1 as follows:
GH = XKF G1
where
XKF = V1
[11]
[12]
[13]
[14]
[15]
(53)
[16]
h
KFmax − 1
2 2
n=2 n Vn
− KFmax
h
n=2
Vn2
.
(54)
The expression for G1 can be derived by substituting GH
from (53) into the power balance constraint in (48)
PLavg + PLoss
.
G1 =
(55)
2
3Vs1 1 + XKF THD2v
Equations (53) and (55) present closed-form mathematical
expressions to compute the optimal conductance factors which
maximize the source-side power factor satisfying the power
balance and KF constraints.
Using the same steady-state per-unit values in Appendix II,
the optimization problem with KF constraint is solved, and the
results for KFspec equal to 10% and 2% are provided in Table X.
[17]
[18]
[19]
[20]
[21]
[22]
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Parag Kanjiya received the B.E. degree from Birla
Vishvakarma Mahavidyalaya Engineering College,
Gujarat Technological University (formerly known
as Sardar Patel University), Vallabh Vidhyanagar,
India, in 2009 and the M.Tech. degree from the
Indian Institute of Technology Delhi (IITD), New
Delhi, India, in 2011.
Since October 2011, he has been a Research
Engineer with the Masdar Institute of Science and
Technology, Abu Dhabi, United Arab Emirates. His
research interests include applications of power electronics in distribution systems, power quality enhancement, HVdc, flexible ac
transmission systems, and power system optimization.
Mr. Kanjiya was the recipient of the K.S. Prakasa Rao Memorial Award for
achieving the highest cumulative grade point average at IITD in August 2011.
Vinod Khadkikar (S’06–M’09) received the B.E.
degree in electrical engineering from the Government College of Engineering, Dr. Babasaheb
Ambedkar Marathwada University, Aurangabad, India, in 2000, the M.Tech. degree in electrical engineering from the Indian Institute of Technology
Delhi, New Delhi, India, in 2002, and the Ph.D.
degree in electrical engineering from the École
de Technologie Supérieure, Montréal, QC, Canada,
in 2008.
From December 2008 to March 2010, he was a
Postdoctoral Fellow with the University of Western Ontario, London, ON,
Canada. Since April 2010, he has been an Assistant Professor with the Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates.
From April 2010 to December 2010, he was a Visiting Faculty member with
the Massachusetts Institute of Technology, Cambridge. His research interests
include applications of power electronics in distribution systems and renewable
energy resources, grid interconnection issues, power quality enhancement,
active power filters, and electric vehicles.
Hatem H. Zeineldin (M’06) received the B.Sc. and
M.Sc. degrees in electrical engineering from Cairo
University, Giza, Egypt, in 1999 and 2002, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Waterloo,
Waterloo, ON, Canada, in 2006.
He worked for Smith and Andersen Electrical Engineering Inc., where he was involved with projects
involving distribution system design, protection, and
distributed generation. He then was a Visiting Professor at the Massachusetts Institute of Technology,
Cambridge. He is currently an Associate Professor with the Masdar Institute
of Science and Technology, Abu Dhabi, United Arab Emirates, and a Faculty
Member with Cairo University. His research interests include power system
protection, distributed generation, and deregulation.