Dynamic behavior of the harmonic detection part of an active power filter (APF) has an essential role in filter compensation performances during transient conditions. Instantaneous power (
p–q
) theory is extensively used to design harmonic detectors for active filters. Large overshoot of
p–q
theory method deteriorates filter response at a large and rapid load change. In this study the harmonic estimation of an APF during transient conditions for balanced threephase nonlinear loads is conducted. A novel fuzzy instantaneous power (FIP) theory is proposed to improve conventional
p–q
theory dynamic performances during transient conditions to adapt automatically to any random and rapid nonlinear load change. Adding fuzzy rules in
p–q
theory improves the decomposition of the alternating current components of active and reactive power signals and develops correct reference during rapid and random current variation. Modifying
p–q
theory internal highpass filter performance using fuzzy rules without any drawback is a prospect. In the simulated system using MATLAB/SIMULINK, the shunt active filter is connected to a rapidly timevarying nonlinear load. The harmonic detection parts of the shunt active filter are developed for FIP theorybased and
p–q
theorybased algorithms. The harmonic detector hardware is also developed using the TMS320F28335 digital signal processor and connected to a laboratory nonlinear load. The software is developed for FIP theorybased and
p–q
theorybased algorithms. The simulation and experimental tests results verify the ability of the new technique in harmonic detection of rapid changing nonlinear loads.
I. INTRODUCTION
In industrial drive systems that control high acceleration processes, the use of a line reactor to improve rapid and sharp current variation is impossible. Large values of current harmonics are injected to the alternating current (AC) line and must be filtered. An ideal active power filter (APF) must have adaptability and flexibility to any kind of load current variation. The dynamic performance of an APF highly depends on the harmonic detection part of the filter closedloop control system. One of the most important properties of harmonic detection is tracking ability in steadystate and transient conditions. Although many different methods are proposed for harmonic detection, instantaneous power (
p–q
) theory is extensively used for this purpose
[1]

[13]
. Recently, some methods have been applied to improve the dynamic behavior of harmonic estimation.
The frequencybased and neuralbased methods have several drawbacks. For example, the frequencybased methods are time consuming, require large computational effort, and experience initial adaptation delay, and the neuralbased methods requires initial training and have poor response in untrained conditions during harmonic detection application
[14]

[23]
. The ability to control the APFs in real time, the simplicity of the calculations, and the flexibility to apply to threephase systems with or without neutral wire are the main reasons for the extensive use of the
p–q
method.
p–q
theory needs an additional phaselocked loop (PLL) circuit for synchronization. Therefore, the
p–q
method is frequency variant
[24]

[26]
. However, some problems occur because of misinterpretations of this theory. One of the main drawbacks of
p–q
theory method is its large overshoot.
p–q
theory method does not respond well in large and rapid load change systems, such as highpower speed drives
[3]
,
[9]
,
[10]
,
[13]
. A combination of AC and direct current (DC) signals and invisible signals with intermediate frequency less than the main harmonic tend to deteriorate the method performance during transient conditions. Compromising between cutting frequencies of the highpass filter (HPF) and filter order do not lead to successful solutions. Increasing phase error and deterioration of output quality are the result of this compromise. Fuzzy logic is an alternative approach to handle this type of problem
[17]

[23]
. However, many issues on fuzzy logic application in different parts of the APF exist. Harmonic detector, current controller, and DC bus voltage controller are improved with fuzzybased configuration
[19]
. Fuzzy logicbased shunt APF shows better dynamic response and higher control precision compared with other filters
[21]
. Modifying filter performances using fuzzy rules without any drawback is a prospect. In this study, fuzzy instantaneous power (FIP) theory is introduced. First,
p–q
theory will be reviewed, and then FIP theory will be explained in Section 2. In Sections 3 and 4, the application of
p–q
theory and FIP theory in harmonic detection with rapid load change is simulated and implemented using digital signal processor (DSP), and the results are discussed.
II. HARMONIC DETECTION METHODS AND PROBLEM
Different algorithms are used for harmonic detection, and most debates focus on detection accuracy, speed, filter stability, and easy and inexpensive implementation. In a rapidly timevarying condition, the most important property of a harmonic detector is the speed of the algorithm
[3]

[6]
,
[9]
. Based on classification relative to the domain where the mathematical model is developed, the two major directions are the timedomain and frequencydomain methods. The timedomain methods are mainly used to gain more speed or fewer calculations compared with the frequencydomain methods. Furthermore, in timedomain methods, the three important dynamic specifications of detection algorithms, that is, phase error, settling time, and overshoot of
p–q
theory method, are better than other methods
[9]
,
[13]
.
 A. Instantaneous Power (p–q) Theory
p–q
theory determines the harmonic distortion from the instantaneous power calculation in a threephase system, which is the multiplication of the instantaneous values of the currents and voltages
[9]
. The calculations may be conducted in the
α–β
coordinates, as shown in Eq. (1), for three balanced sinusoidal voltage supplies and a balanced load:
The values of the instantaneous powers
p
and
q
, which are the real and imaginary powers, respectively, contain DC and AC components depending on the existing active, reactive, and distorted powers in the system. The values of
p
and
q
can be expressed based on Eq. (1) in terms of the DC components plus the AC components, as follows:
where
is the DC component of the instantaneous power
p
and is related to the conventional fundamental active current;
is the AC component of the instantaneous power
p
and is related to the harmonic currents;
is the DC component of the imaginary instantaneous power
q
and is related to the reactive power generated by the fundamental components of voltages and currents; and
is the AC component of the instantaneous imaginary power
q
and is related to the harmonic currents.
The zero sequence power exists only in threephase systems with neutral wire. Furthermore, the systems must have unbalanced voltages and currents and/or third harmonics in the voltage and current of at least one phase.
where
is the DC component of the instantaneous zero sequence power and corresponds to the energy per time unity, which is transferred from the power supply to the load through the zero sequence components of voltage and current and
is the AC component of the instantaneous zero sequence power and corresponds to the energy per time unity that is exchanged between the power supply and the load through the zero sequence components. A shunt APF, as shown in
Fig. 1
, can theoretically compensate for the AC component of the active power and also the DC and AC components of the reactive and zero sequence power.
Compensation of power components p, q, and p_{0} in α–β–0 coordinates.
To compensate for current harmonics generated by nonlinear loads, the reference signal of the shunt APF must include the values of
and
. In this case, the reference currents required by the APF are calculated from the following expression for a balanced system:
Fig. 2
shows the principle diagram of
p–q
theory for determining the compensated reference currents in the balanced condition. The DC components of
p
and
q
represent the active and reactive powers and must be removed by HPF (with a cutting frequency between 5 and 35 Hz) to retain only the AC signals for harmonic current compensation. The AC components are transferred to the abc frame and represent the harmonic distortion, which are considered references for the current controller. The HPF is usually a digital filter. The numerical implementation of the HPF influences the dynamics and accuracy of the entire APF
[12]
.
Principle flow diagram of p–q theory for the calculation of reference currents in the balanced condition.
Flow diagram of p–q theory for the calculation of reference currents in the unbalanced threephase fourwire system.
When the threephase system is unbalanced, the reference currents of the APF, as shown in
Fig. 1
, are determined as follows:
Given that the zero sequence current must be compensated, the reference compensation current in the 0 coordinate is
i
_{0}
itself:
 B. Some Problems of p–q Theory
The implementation of
p–q
theory encounters some problems in real application. The two main problems are large overshoot and dependence on voltage shape. In harmonic detection based on
p–q
theory with rapid load change, the AC components of
p
and
q
contain erroneous terms because of the rapid change of harmonic current amplitude and phase and the nondeterministic relationship between them. In some industrial systems such as DC drives during transient conditions, the phase angle of the drive current changes rapidly. By this variation, not only the harmonic current varies but also the reactive power changes considerably. Large variation of reactive power in rapid transient conditions cannot be distinguished from harmonic variation based on
p–q
theory and results in high overshoot in detection response
[13]
. The HPF filtering capability has an essential role in the quality and accuracy of the detected harmonics. Increasing the filter order or decreasing the cutting frequency increases the phase error and degrades the overshoot. In the rapidly timevarying condition, the DC component of
p
and
q
may vary with a frequency less than the main harmonic frequency (intermediate frequency). To consider these phenomena, Eqs. (2) and (3) can be rewritten as follows:
and
in rapid load change can have step reference changes (as shown in
Fig. 4
for the desired active power). However, the actual active power has variations because of the system dynamics, as shown in
Fig. 4
. The difference between the actual value of
and its estimated value (which is equal to its desired value) is called Δ
p
. The terms Δ
p
and Δ
q
correspond to the “slow” variation of the DC component of
p
and
q
and are not related to harmonics. However, these components become related to harmonics when they pass through the HPF, thus increasing the values of the AC components of
p
and
q
and producing undesired responses.
Desired and actual variation of .
The current reference calculations of Eqs. (1) to (5) are also affected by zero sequence components because of an existing unbalanced system. Therefore, an Δ
p
_{0}
component must be added to provide a complete analysis of the unbalanced systems
[10]
:
 C. FIP Theory
When a system is too complex or too poorly understood to be described in precise mathematical terms, fuzzy modeling provides the ability to linguistically specify approximate relationships between the inputs and desired outputs. The relationships are represented by a set of fuzzy if–then rules in which the antecedent is an approximate representation of the state of the system and the consequent provides a range of potential responses
[18]

[23]
. With the use of the fuzzy algorithm, decision making can be easy when the subject is the outcome of two or more parameter relationships and interactions. Improving the filtering capability without changing the cutoff frequency or filter order is a new approach to achieve a reasonable response with rapid load change. The use of Eq. (7) to calculate the reference current with
,
, and
during transition leads to high overshoot in the output. Attenuating these signals is a simple approach to correct the overshoot problem during rapid load change. The attenuator coefficients are determined by using the fuzzy modifier. The fuzzy modifier contains three fuzzy controllers with two inputs and one output.
Fig. 5
shows the principle diagram of the simplified fuzzy modifier for the
q
variable, where
G_{q}
is the fuzzy modifier coefficient and 0.1 <
G_{q}
≤ 1.
Principle diagram of the simplified fuzzy modifier for q.
The fuzzy logic controller (FLC) based on the signal value and its rate of change determines the modifier coefficient, and its inference rules with nine rules are shown in
Table I
.
FUZZY INTERFACE RULE
The performance of any FLC can be improved by increasing the number of rules present in the rule base of the controller, but with increased computation cost. The required computational time slows down the process considerably
[22]
. Some studies regarding the reduction of the rule base number have been reported. Some issues on the design and rule base size reduction for the fuzzy control exist, and some concepts regarding resizing of the rule base by removing inconsistent and redundant rules have been proposed
[19]
,
[20]
. Many studies have proposed the 49rule FLC for the control of the shunt APF. The 49rule FLC has the drawback of a large number of fuzzy sets and control rules. This drawback is overcome by approximation using the simpler ninerule FLC
[20]
,
[22]
and the simplest fourrule FLC
[19]
. In this application with a compromise between transient and steadystate behaviors of the APF, a ninerule FLC is selected. The behavior of the control surface, which relates the input and output variables of the system, is governed by a set of rules. When a set of input variables are read, each rule that has any degree of truth in its premise is fired and contributes to the formation of the control surface by approximately modifying it. When all rules are fired, the resulting control surface is expressed as a fuzzy set to represent the output.
Fig. 6
shows the fuzzy controller surface viewer, which graphically defines the fuzzy inference rules. Using FIP theorybased harmonic detector, the principle flow diagram shown in
Fig. 3
can be modified, as shown in
Fig. 7
.
Fuzzy controller surface viewer.
Principle flow diagram of FIP theory (unbalanced fourwire system).
In this case, Eq. (7) can be rewritten as follows:
For a balanced threephase system,
p
_{0}
is not considered. Expanding this method to balanced threephase systems modifies Eqs. (14) and (15) and
Figs. 7
and
8
.
Principle flow diagram of FIP theory (for balanced systems).
 D. Effect of the New Harmonic Detection Method on the Stability of the Current Control System
In the new method, the cutoff frequencies and order of HPFs and lowpass filters are unchanged. Only the amplitude of the filter output signal is adaptively varied. Therefore, no differences in the phase of the filtered signals in the two methods were observed.
Fig. 9
shows the block diagram of the shunt APF control system. The dynamic performance of the harmonic detector has no effect on the characteristic equation and stability of the closedloop control system because it is out of the control loop, which sets the control system reference signal value. The performance of this block fixes the overall harmonic compensation quality. As such, hardware generation using the two harmonic detection methods can be conducted individually, and the active filter can be omitted
[13]
.
Block diagram of the shunt APF.
III. SIMULATION OF THE NONLINEAR LOAD WITH ACTIVE POWER FILTER AND THE RESULTS
In this paper, harmonic cancelation is conducted using a pure shunt type APF as
Fig. 10
. A voltage source pulsewidth modulation inverter, with insulated gate bipolar transistor switches, series inductor (
L_{F}
), and energy storage capacitor (
C_{dc}
) on the DC bus, is used as a shunt active filter.
Shunt APF.
Fig. 11
shows the simulation of the programmable nonlinear load. The load is a threephase bridgecontrolled rectifier with programmable current source, which is the same as the threephase AC voltage controller of the preheat section in a galvanization process that works in the phase angle control mode with rapid current changing capability. In this model, a PLLbased voltage reference generator and a hysteresisbased controller for current control are used
[5]
.
Fig. 12
shows the SIMULINK model of
p–q
and FIP theorybased reference current generator, with the fuzzy modifier as their only difference.
SIMULINK model of the programmable nonlinear load with shunt APF.
(a) FIP theorybased reference current generator. (b) p–q theorybased reference current generator.
 A. Comparison of p–q TheoryBased and FIP TheoryBased Compensation Methods in Rapid Load Change
The load current is changed from 20 A to 200 A as a step function at time 0.1 s, as shown in
Fig. 13
. The simulated source voltage and load current are shown in
Figs. 13
(a) and
13
(b), respectively.
Figs. 13
(c) and
13
(d) show the source currents that are compensated by the
p–q
and FIP methods, respectively.
Simulated system signals for p–q and FIP theorybased APF compensation. (a) Source voltage. (b) Load current. (c) p–q theorybased compensated source current. (d) FIP theorybased compensated source current.
As shown in
Figs. 14
(a) and
14
(b), the DC part of the estimated active power in
p–q
theorybased and FIP theorybased methods has 23% and 5% overshoot, respectively.
Figs. 14
(b) and
14
(c) show that the DC part of q in the two methods has 20% and 3% overshoot, respectively. The higher overshoot in the
p–q
reference current estimation method results in a higher overshoot in the compensated current. The unwanted part of variation can be omitted by the fuzzy modifier.
Simulated system signals for p–q and FIP theorybased current reference generator. (a) p–q theorybased estimated . (b) FIP theorybased estimated . (c) p–q theorybased estimated . (d) FIP theorybased estimated .
For more details,
Fig. 15
shows one phase current and compares
p–q
and FIP theorybased dynamic performances.
Step change of the load and source current of phase “b” in (a) p–q theorybased compensation method and (b) FIP theorybased compensation method.
P–QTHEORYBASED AND FIP THEORYBASED DYNAMIC PERFORMANCE
P–Q THEORYBASED AND FIP THEORYBASED DYNAMIC PERFORMANCE
As shown in
Table III
, the FIP theorybased compensation method with nine rules has similarity to the
p–q
theorybased method in steady state. The total harmonic distortion (THD) of the supply current in both methods is below the 5% limit imposed by the IEEE 519 standard, where the THD of the load current is 23.2% at 200 A and 28.67% at 20 A.With an increase in the number of rules of the FLC, the THD of the source current in the FIP theorybased method decreases to a value similar to that of the THD of the
p–q
theorybased method. However, the computational time increases, which slows down the process considerably. The ninerule FLC satisfies the acceptable transient and steadystate condition performances.
THD OF THE LOAD AND SOURCE CURRENT IN STEADY STATE
THD OF THE LOAD AND SOURCE CURRENT IN STEADY STATE
 B. Comparison of p–q TheoryBased and FIP TheoryBased Nonlinear Balance Load with Rapid Change
In some applications, load current has random variation. For instance, highpower traction DC drives in four quadrant operation have rapid and random load current variation during directionchanging rotation. The results of rapid current reference are [20 200 400 200 100], as shown in
Fig. 15
. The FIP theorybased method is more stable and more adaptable to rapid current change.
 C. Comparison of p–q TheoryBased and FIP TheoryBased Compensation Behavior during Transient Load Change
In the simulated system, nonlinear load (threephase rectifier with constant current source) is replaced with nonlinear load (threephase rectifier with resistive capacitive load) with transient behavior, as shown in
Fig. 17
.
Rapid change of load and source current of phase “b” in (a) p–q theorybased compensation and (b) FIP theorybased compensation.
SIMULINK model of nonlinear load with transient behavior.
After closing the breaker at
t
= 0.1 s, the load current varies from 50 A to 100 A as a step function with transient behavior. The load current has a settling time of 100 ms and increases up to 268 A in transient conditions. The compensated source current using the
p–q
theorybased method[
Fig. 18
(a)] has a peak value of 305 A. For the FIP theorybased compensated source current [
Fig. 18
(b)], the peak value is 271 A. At the steadystate condition, no differences were observed.
Transient behavior of the load and source current of phase “b” in (a) p–q theorybased and (b) FIP theorybased compensation methods.
IV. EXPERIMENTAL SETUP AND RESULTS
A laboratorycontrolled rectifier DC drive with a 3 hp DC motor coupled with a 2.2 kW AC generator used as nonlinear load. The harmonic detection methods were implemented using TMS320F28335 DSP. Hardwareintheloop simulation with a serial communications interface is a utility of this family of processors for many tasks such as online data logging in SIMULINK environments using Texas Instrument Code Composer Studio
[27]
.
Fig. 19
shows the experimental setup.
Experimental system setup.
To evaluate the accuracy and dynamic behavior of the two methods, the harmonic component of the load current is removed, which results in the “expected” source current. This signal resembles the reference current for source current compensation. The dynamic performance of this current is the same as the closedloop source current. In investigating the effectiveness of the two methods during rapid load change, the breaker is closed and the AC generator is loaded with a resistive load. The rectifier effective current varies from 0.4 A to 1.5 A as a step function. The expected compensated source current using
p–q
theory has a peak value of 2.65 A [
Fig. 20
(a)]. For the FIP theorybased expected compensated source current, the peak value is 2.2 A [
Fig. 20
(b)]. During transient conditions, no differences were observed. and the peak value is 2.15 A. The overshoot of the first method is 23.2% and that of the second method is 2.3%. This result shows that the second method performs better during transient conditions.
(a) p–q theorybased and (b) FIP theorybased compensation behavior in step change of the load current.
The simulation and experimental results show that the FIP theorybased method is more stable than the conventional
p–q
theorybased method. In addition, the FIP theorybased method has a fast, reliable, and good tracking ability.
V. CONCLUSIONS
FIP theory combines
p–q
theory and fuzzy logic techniques to efficiently compensate for the source current in transient conditions. This configuration improves the dynamic behavior of the harmonic detector without any drawback. Through correct selection of the fuzzy modifier (number of rules of the FLC), the response has low overshoot for any condition and the tracking ability of the harmonic detector improves. Moreover, other behaviors and performances, such as settling time and stability of the generated compensated reference currents, are enhanced. Compared with other methods, the FIP theorybased method has a unique behavior in random current and harmonic variation conditions. The FIP theorybased method has a precise compensation capability and is computationally efficient for realtime implementation. The results of the simulation and experimental tests show the ability of the new technique in dynamic harmonic detection.
BIO
Nasser Eskandarian obtained his M.S. in Control Engineering from the University of Tehran, Iran. In 1989, he joined the Department of Electrical and Computer Engineering at Semnan University, Semnan, Iran. His research interests include computer control, robotics, power electronics, and electric vehicles. He has managed several industrial projects, some of which have won international awards. He is currently working toward his Ph.D. degree in power electronics.
Yousef Alinejad Beromi obtained his B.S. in Electrical Engineering from the K.N.T. University of Tehran, Tehran, Iran and his M.S. and Ph.D. in Electrical Engineering from the University of Wales College of Cardiff (UWCC), Cardiff, UK, in 1992. He is an associate professor at the Faculty of Electrical and Computer Engineering, Semnan University, Semnan, Iran.
Shahrokh Farhangi obtained his B.S., M.S., and Ph.D. in Electrical Engineering from the University of Tehran, Tehran, Iran, with honors. He is a professor at the School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran. His research interests include design and modeling of power electronic converters, drives, and photovoltaic and renewable energy systems. He has published more than 90 papers in conference proceedings and journals. Dr. Farhangi has managed several research and industrial projects, some of which have won national and international awards. He was selected as a Distinguished Engineer in electrical engineering by the Iran Academy of Sciences in 2008. He is currently the Dean of the School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran.
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