This paper presents a model predictive control for shunt active power filters in synchronous reference frame using space vector pulsewidth modulation (SVPWM). The three phase load currents are transformed into synchronous rotating reference frame in order to reduce the order of the control system. The proposed current controller calculates reference current command for harmonic current components in synchronous frame. The fundamental load current components are transformed into dc components revealing only the harmonics. The predictive current controller will add robustness and fast compensation to generate commands to the SVPWM which minimizes switching frequency while maintaining fast harmonic compensation. By using the model predictive control, the optimal switching state to be applied to the next sampling time is selected. The filter current contains only the harmonic components, which are the reference compensating currents. In this method the supply current will be equal to the fundamental component of load current and a part of the current at fundamental frequency for losses of the inverter. Mathematical analysis and the feasibility of the suggested approach are verified through simulation results under steady state and transient conditions for nonlinear load. The effectiveness of the proposed controller is confirmed through experimental validation.
1. Introduction
Power electronic devices impose nonlinear loads on the ac mains, such as singlephase and threephase diode rectifiers, thyristor converters and electronic appliances
[1]
. Aluminum smelter plants and large electrolysis chemical plants, in which large DC rectifiers with high power ratings, are usually used
[2
,
3]
. They generate considerable amounts of characteristic and noncharacteristic harmonics, which will be harmful for other loads connected to the same bus. In such plants, auxiliary control, protection measuring systems and kWh counters severely suffer from these harmonics
[4]
. Also, large DC drives used in cement factories or in electric traction application generate enormous harmonics. DC arc furnaces generate several harmonics in the feeding system
[5
,
6]
. Electrolysis of water and wind mill generators are other examples of harmonic pollution. Many international agencies have implemented firm harmonic restrictions to electronic equipment
[7
,
8]
. To overcome these problems, classically, passive LC filters are used to eliminate the current harmonics and to improve the power factor. However, passive LC filters are bulky, load dependent and inflexible
[9
,
10]
. As a result, a vast number of power factor correction techniques have been developed in compliance with these regulations
[11
,
12]
. As an alternative, parallel harmonic correction techniques, shunt active power filters (APFs), have been explored by many researchers and considered as a possible solution for reducing current harmonics and improving the power quality
[13

14]
. The APF is required to generate a matched reactive and harmonic current to compensate for the negative effect of nonlinear loads on the line, thus it handles only the fraction of the total power supplied to the load.
Many articles have been published that focus on obtaining the current reference for threephase or singlephase APF. Standard APFs configurations require the measurement of both load and filter currents with reference current regulators implemented by hysteresis and PWM modulators. Ozdemir et al. proposed a simplified control algorithm for shunt APF without load and filter current measurement in
[15]
. In
[16]
, Chen developed a statespace model of the fourleg APF based on H controller for current tracking from the passivity point of view based on 4leg VSI. In
[17]
, APF is combined with thyristor switched capacitor for the purpose of reducing cost. A model predictive control (MPC) is presented where Fuzzy model predictive control
[18]
and neural network predictive control
[19]
are used in shunt APFs, but the control designs are still complicated. Mohanty evaluates the performance of shunt APF for two different control strategies namely hysteresis current control and space vector pulse width modulation
[20]
. The same concept is used in
[21]
by Mendalek, where the voltage level of the dc side is regulated using a linearizing feedback control. In
[22]
, Wang developed a simplified MPC method for shunt APFs. Da Silva in
[23]
proposed a compensation algorithm used to extract the reference currents. However, still three predictive current control models for each phase are used. Moreover, the crosscoupling terms between
dq
axis and synchronous frame system have not been taken into account. Many other attempts are proposed and discussed several techniques in order to improve the efficiency and the performance of APF
[24

35]
. In
[36]
, Vatani et al. proposed finite control set MPC based on pq theory to control a three phase Neutral Point Clamped multilevel converter to act as a shunt APF. The predictions are used to estimate the reference current two steps ahead. Nevertheless, multilevel converters have drawbacks such as complex topologies and bulky circuitry structures, higher costs, complexity of the control system and higher active power losses.
In this paper a model based MPC for shunt APF in synchronous reference frame
dq0
is presented. The proposed MPC has the advantage of exhibiting faster performance due to the removal of the trajectory reference stage; which is normally adopted in conventional MPC to make the system arrive to its reference value softly. In addition, handling of the control variables in
dq0
space, leads to reduction of the required number of current controllers to two instead of three. Moreover,
dq0
transformation allows for the control procedure to proceed with constant quantities instead of time variant quantities.
The input stage of the MPC is the
dq
synchronous reference frame harmonic current and the output stage is synchronous frame harmonic current commands which are the input commands to the space vector pulse width modulation (SVPWM) generator. In this way the supply current will be equal to the fundamental component of load current and a part of the current at fundamental frequency for losses of the inverter system. A decoupling stage outputs the synchronous frame voltages which are the input commands to the SVPWM generator is utilized to let the currents injected by the filter track rapidly their references. Additionally, the dc voltage level is not need to be regulated in the proposed system.
2. Design and Modelling of the APF with MPC
From the shunt APF circuit connected in parallel with the electric system shown in
Fig. 1
, the stepwise model equations can be derived. The nonlinear load draws currents
i
_{L(abc)}
and the supply currents denotes
i
_{S(abc)}
. The APF will be supplying only the harmonic currents
. The dynamic analytical model of the APF filter is developed in its original threephase
abc
frame and the model is then transformed to the synchronous reference frame.
Shunt active power filter connected to electric system.
The APF system of equations can be described as:
where
v _{abc}
and
u_{abc}
are the source and APF voltages, respectively.
R
and
L
are the smoothing inductor inductance and resistance.
Transforming the above currents in synchronous reference frame using the transformation matrix:
where
u
_{dq0}
could be obtained from
Hence,
u
_{dq0}
is found for the threephase system to be
 2.1 Modeling synchronous predictive controller
In order to obtain a closed loop mathematical form for the controller, discretizing Eq. (4) becomes necessary as in
Assuming a threephase balanced power source, the
v_{dq0}
term can be removed from Eq. (5)
Hence, solving Eq. (6) for
v_{dq0}
and replacing
k
=
k
+ 1 results in the predictive system of equations
where
and
and the middle term associated with
is the crosscoupling term between the
d
and
q
axis, and it acts as a feedback loop between them. The first order differential equation for the APF model is used as the predictive current controller without the source disturbance. The single step predictive model of the active power filter output current in synchronous reference frame is reduced to
d
and
d
components for a balanced three phase system.
The crosscoupling term after manipulation turns into an angular coefficient labeled crosscoupling angular correction and expressed as
Where
T_{f}
is the fundamental frequency and
τ
is the RL time constant. It is noted here that in the predictive control and error correction,
, where
τ_{er}
is the error coefficient in the time constant.
 2.2 Controller feedback correction
In MPC modelling, feedback correction is necessary to account for the drift in the predictive model equations due to the nonlinear nature of the APF. Therefore, an error
e
(
k
) is added to the output current of the APF to account for such discrepancy
where the error is calculated as the difference between the actual current at
k
and the predicted current
i
_{dq0}
at the same discrete instance. Where
δ
is the correction coefficient. The reference trajectory block is necessary for applications that requires the output to arrive at reference values softly. While in APF implementation, this is omitted as the predictive controller is required to quickly respond to reference commands and is required to fast track any changes in the reference values.
 2.3 Controller dynamic optimization
MPC dynamic optimization requires an objective function that will track changes in the commands online. The weighted quadratic performance index is commonly used in dynamic optimization as its objective function and expressed as
where the parameters
q
is the weighting coefficient of the predictive error and
λ
is the predictive control variable. It is noted that the control variables of the APF are its voltage
u
_{dq0}
and are set as the input commands to the APF. The optimal performance of the controller can be achieved by differentiating the objective function
J
with respect to the voltage comaned and equating that to zero (
dJ
/
du
= 0) .
where
After substitution,
dJ
/
du_{dq0}
= 0 , it can be found to be (16)
Fig. 2
debicts the block diagram of the proposed MPC. Implementaion of the controller is fairlly easy. The voltage at the point of the filter connection is considered as a disturbance and omitted by the closed loop. The colsed loop consists of feedback correction and dynamic optimization. Therefore, no voltage sensors are required by the MPC controller. Hence, cost reduction is is likely attained. Furthermore, calculating the control variables of the next sampling, at instant
k
+1, is carried out at instent
k
, that enables rapid tracking and fast dynamic reponce.
Predictive controller for the APF
 2.4 Controller design parameters
The model predictive controller is realized as in
Fig. 3
. The values for the resistance and inductance used are 0.225 Ω and 2
m
H respectively, which results in:
Block diagram of MPC for simulation and practical implementation.
The dynamic optimization parameters (
λ
,
q
) were chosen after an in depth evaluation as in
[22]
and after a number of trial and evaluation attempts, a value of
λ
=1 and
q
=5000 is selected which allows fast tracking of the command signal with relatively minimal ripple.
The final block diagram of the simulation and practical implementation of proposed MPC for shunt APF is illustrated in
Fig. 3
.
3. Simulation Results
Implementation of the proposed MPC for APF in synchronous reference frame has been carried out in PSIM software to study the performance of the proposed control strategy. The feasibility of the suggested predictive current control is verified through simulation results under steady state and transient conditions for nonlinear load. The simulation results will be discussed in the following:
The command current was obtained by sampling the load current including harmonics and then subtracting the fundamental desired current, resulting in the harmonic contents required to be removed.
Fig. 4
shows the grid voltage and currents with and without APF. The simulation was started at t = 0.0s, where the APF was turned off between 0 < t < 0.4s. During this period of time, the current supplied by the grid is the nonlinear load current, which is non sinusoidal and includes harmonic contents. At t=0.4s the active power filter is turned on. During this period, the APF samples the grid current and supplies the appropriate harmonic currents required by the nonlinear load. After t=0.4s, the grid currents are shown to be sinusoidal at the operating point of 50A rms. The harmonic contents in the grid currents are nearly eliminated and the grid currents become almost sinusoidal as shown in
Fig. 4(b)
. The grid currents for a step change in the nonlinear load current at t = 0.8 sec. is illustrated in
Fig. 4(c)
, which shows a fast dynamic response of the proposed MPC.
Grid Voltages and currents during APF connection
The function of the APF predictive controller is investigated under a step change in the nonlinear load profile. A step change in the load current is introduced at t = 0.8s is shown in
Fig. 5
. As it can be seen from the
Fig. 5(a)
, the threephase grid currents move toward the new operating point of 105A rms. The grid currents remain sinusoidal as the predictive controller of the APF dynamically adjusts to the new operating point. It can be seen here that the predictive controller is able to handle a step change and adjusts for the new operation conditions to reach steady state values within three cycles. It also instantaneously supplies the required harmonics on demand and partial fundamental current during the sudden load change. This means, the APF acts as a smoother for the grid current during the sudden change in operating point. As it can be seen, the grid current remains sinusoidal and within 0.048 sec meets the required current. During this process, the APF supplies the harmonic current component required by the load in addition to a transitional component of the fundamental current. In this case, no harmonic component exhibited by the grid and the function of the active power filter fulfills the requirement of the step change in load. It is worthy to note that this particular APF is designed to meet the power requirement of this specific application.
Grid currents at step change in load.
To illustrate the fast performance characteristics of the predictive controller of the APF,
Fig. 6(a)
shows the load harmonic component and active power filter currents. Between 0 < t < 0.4s, it can be seen that the APF is disconnected and the load currents including its harmonic contents are totally supplied by the grid. After t = 0.4s, the APF is connected and instantly tracks the harmonic component of the load. At t = 0.8s a step change in the load occurs and
Fig. 6(b)
shows the APF controller compensation for the sudden change in load requirement and in addition, it transitions the grid current into its new operating point without any major disturbance or noise, see
Fig. 5
.
APF and load currents.
In parallel to the grid current above,
Fig. 7
shows onephase of the APF injected currents for the same particular periods. It is worth mentioning here that the APF tracking of the harmonic components in grid currents is instantaneous. As seen from the lower part of
Fig. 7
, the APF supplies both the fundamental and harmonic current during next three cycles after the step change in load current.
Active power filter injected currents tracking capability.
Fig. 8
shows the harmonic contents of the grid current (Phase A) before and after the APF connection. Before the filter connection, the THD content was %20 while the active power filter reduced the content for the same period to less than %5.
Total harmonic distortion content in grid current.
4. Experimental results
The performance of the proposed active power filter is verified experimentally with the configuration shown in
Fig. 9
. The experimental platform consists of grid emulator to supply the nonlinear load, three phase dc/ac SVPWM inverter acting as APF with a 10KHz switching frequency to generate and inject the required harmonics. A dSPACE DS1103based digital signal processor is used to generate the gating signals and implement the proposed control scheme. The APF is practically implemented using sing Powerex CM150TX24 intelligent power module (IGBT) produced by Mitsubishi Co. Ltd. The nonlinear load is represented by a threephase full wave diodebridge rectifier using 6RI75G120 by Fuji Electric with resistive load. The grid voltages are measured using CYVS411D07380V6_01 high accuracy AC voltage sensors. Two sets of three CYCS411D478A5 high accuracy AC current sensors are used for measuring the load and the APF injected currents. A DSO7104A Agilent digital oscilloscope is used to display and capture the output waveforms and a Fluke 43B power quality analyzer is used for harmonic calculations. The dSPACE DS1103 controller board provides a realtime interface (RTI) between the hardware and the computer model created in Matlab Simulink.
Test rig photograph
A prototype of power rating of 2 kW is practically examined and the experimental results will be discussed in the following.
Fig. 9
demonstrates the overall appearance of the experimental setup with the design specifications and circuit parameters used in the simulation and experimental tests are listed in
Table 1
.
Design specifications and circuit constants
Design specifications and circuit constants
The distorted load current of nonlinear load of threephase diodebridge rectifier is compared with its fundamental signal to generate the required feedback signalof the proposed controller.
Fig. 10
shows a phase A distorted load current along with its harmonic contents and its fundamental component. The shunt active power filter is connected with the series reactor to eliminate the grid current harmonics.
Fig. 11
shows the threephase load currents, APF injected currents and grid currents.
The load current (upper trace), its harmonics (middle) and fundamental component (lower)
Three phase system currents
Fig. 12
shows the wafeforms of threephase grid currents at a step change in load, which reflects the rapid tracking performance and fast dynamic reponse of proposed MPC controller under sudden load change. The transition to new operating point appears smooth due to the interaction of the APF which is responsible for eliminating the harmonic currents in addition to ensuring soft transition of the grid current fundamental value to the new operating point without any undesired harmonic components.
Threephase grid currents at step change in load.
Fig. 13
shows the grid voltage and current, phase A. It is clear to obvious that the grid current become sinusoidal and in phase with the grid voltage.
Grid voltage and current, phase A
Harmonic spectrums of the grid current before and after connecting the proposed MPC for APF are shown in
Figs. 14(a)
and
(b)
, respectively. The total harmonic distortion (THD) in the grid current is computed and it is found to be 28.2% before harmonic compensation and 3% after harmonic current compensation that is that means a highquality sinusoidal waveforms within the limit of the of 519 IEEE harmonic standard of grid current is obtained using the proposed APF.
Harmonic spectrum of the grid current
5. Conclusion
A predictive current control for a threephase three wire current source shunt active power filter has been presented in this paper. The control approach uses the predicted value of the direct and quadrature error in synchronous reference frame and the optimal switching state to be applied to the next sampling time is selected to control the inverter with minimum switching frequency while maintaining fast harmonic compensation. Moreover, this control approach does not require supply voltage sensing, which generally reduces the system complexity. Mathematical analyses have been presented and simulation results are performed under steady state and transient conditions for nonlinear loads to validate the theoretical development and confirm the performance of proposed approach. The presented simulation results indicate that the presented control approach provides fast dynamic response and good tracking of the harmonic compensation to its reference value. The proposed predictive current control is simple and very easy to implement compared to classical PWM techniques. As assessment comparing both predictive and classical PI control techniques is considered for a future study, as well the operation at unity input power factor, allowing harmonic current compensation and reactive power compensation.
Acknowledgements
This project was supported in full by the Public Authority of Applied Education & Training, Kuwait. Project # TS1305.
BIO
A. K. AlOthman is an associate professor in the department of electrical engineering at the college of technological studies in Kuwait. His interests include power systems dynamics, state estimation, robust regression, evolutionary programming, application of heuristic optimization techniques to power systems and integration of renewable energy sources into power grid.
Mishel AlSharidah was born in Kuwait in 1971. He received the B.S. degree from the University of Arizona (1995), M.S. degree from Portland State University (1999), and Ph.D. degree from the University of British Columbia(2012). He joined the College of Technological Studies, Shuwaikh, Kuwait, in 1999. Since 2012, he has been with the electrical engineering department where he is currently an assistant professor. His main areas of research interest are power electronics, electric vehicle motor drives, grid interconnection of distributed generation & renewable power conditioning. Dr. AlSharidah is a member of the Institute of Electrical and Electronics Engineers of Canadian, the IEEE Power and Energy Society, and the Power Electronics Society.
Nabil A. Ahmed received the B.Sc. and M.Sc degrees in Electrical Eng. from Assiut Univ., Egypt and the Ph.D. from the Univ. of Toyama, Japan in 1989, 1994 and 2000, respectively. Since 1989, he has been with Assiut University, where he is currently a Professor. He was a Post Doctoral fellow at the Elec. Eng. Saving Research Center, Kyungnam University, Korea from October 2004 to April 2005 and he was a JSPS post doctorate fellow at Sophia University, Japan from July 2005 to September 2006. He is now an Associate Professor at the College of Technological Studies, Public Authority of applied Education and Training, Kuwait. His research interests are in the area of power electronics, integration of renewable energy systems and soft switching converters. Prof. Ahmed is the recipient of Egypt State Encouraging of Research Prize 2005, Japan Monbusho Scholarship 19962000, JSPS Fellowship 20052007, best Paper Awards from ICEMS’05 and IATC’06 conferences and best Presentation award from ICEMS’04 conference. Prof. Ahmed is a member of the IEEE Industrial Electronics Society, and the Institute of Electrical of Engineering of Japan IEEJ.
Bader N. Alajmi received the B.Sc. degree and M.Sc. from California State University, Fresno in 2001 and 2006, respectively. He received the Ph.D. degree in electrical engineering from the Electrical Department, Strathclyde University, Glasgow, UK, 2013. His research interests are digital control of power electronic systems, power quality, microgrids, distributed generation and renewable energy. Filtering and signal processing.
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