29-09-2016, 12:34 PM
A Noniterative Optimized Algorithm for Shunt Active Power Filter Under Distorted and Unbalanced Supply Voltages
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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.
INTRODUCTION
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
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 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. EXTRACTION OF BALANCED SET OF VOLTAGES
FROM DISTORTED AND UNBALANCED SUPPLY
VOLTAGES USING SRF THEORY
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 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) = √
2
h
n=1
Vxn sin(nωt + θxn), x = a, b, c (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
vdn
vqn
= 2
3
⎡
⎣
sin n(ωt) cos n(ωt)
sin n(ωt − 120) cos n(ωt − 120)
sin n(ωt + 120) cos n(ωt + 120)
⎤
⎦
T⎡
⎣
vsa(t)
vsb(t)
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
vdn
vqn
=
v¯dn + ˜vdn
v¯qn + ˜vqn
(3)
where v¯dn and v¯qn are the dc components corresponding to the
balanced part of the nth-order harmonic voltage present in the
supply voltages while v˜dn and v˜qn 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 v¯dn and v¯qn
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.