Shunt Active Power Filter Based on Proportional Integral and Multi Vector Resonant Controllers for Compensating Nonlinear Loads

,e current tracking control strategy determines the compensation performance of shunt active power filter (SAPF). Due to inadequate compensation of the main harmonic by traditional proportional integral (PI) control, a control algorithm based on PI and multi vector resonant (VR) controllers is proposed to control SAPF. ,e mathematical model of SAPF is built, and basic principle of VR controller is introduced. Under the synchronous reference frame, the proposed control method based on pole zero cancellation is designed, which narrows the order of the control system and improves the system dynamic response and the control accuracy. ,en the feasibility of the method is demonstrated by analyzing the closed loop frequency characteristics of the system. Finally, the simulation and experimental results are carried out to verify the performance of the proposed method.


Introduction
In recent years, with the intensification of the global energy crisis, the renewable energy and power electric technology have been integrated, which has promoted the development of distributed generators (DGs) [1,2].e speed driver of small motor, voltage source converter, and a large number of nonlinear loads including electric arc furnace, high-power rectifier, transducer, and fluorescent are connected to the grid, which will cause harmonic pollution unavoidably [3][4][5].e power grid operates in this state for a long run, which will bring great harm to power system and power consumers, mainly manifested in the following respects: (1) e line loss of the system is increasing greatly, and the operating efficiency of the grid is reduced.A lot of 3rd harmonics will also increase the neutral current, leading to the damage of the custom power devices.(2) It makes the motor vibrate and produce noise, which affects the safety operation in production.(3) It makes the relay protection device act by mistake, which interrupts the power supply and makes the extent loss to production.(4) It interferes with measuring instruments and communication systems [6][7][8].erefore, it is urgent to control the power harmonic pollution to improve the power quality.
As a new type of power electronic device that can suppress harmonics dynamically, a three-phase three-wire SAPF with excellent performance has been widely used in the field of power quality.e real-time and accurate compensation for the fast changing harmonic currents is an important guarantee for the operation of SAPF safely and reliably.In the past years, a lot of research and analysis have been done on the detection and tracking control strategy of the harmonic currents.In [9,10], it transforms the harmonic component to the DC component through multisynchronous rotating coordinate transformation; thus, traditional PI controller can be utilized to track the harmonics with zero steady-state error; however, it is too complicated.e hysteresis current control method presented in [11,12] with quickly dynamic respond performance is simple and easy to realize without carrier modulation, but the control accuracy is inversely proportional to switching frequency.In addition, the switching frequency is not fixed, which sets a higher request to switching device.In [13,14], the repetitive control theory was developed to solve harmonic problems, which has the advantages of good robustness, simple structure, and easy implementation.However, dynamic performance is limited if the load changes suddenly.In order to overcome the e ect of computational delay and control delay on compensation performance, the dead-beat control in [15,16] is used to track the reference current, which plays a certain e ect, but it highly depends on the system parameters.When the system parameters change, the compensation performance will be greatly a ected.In [17,18], vector resonant(VR) control method is proposed, where proportional controller is to adjust the control bandwidth to improve the dynamic response speed, while the resonant controller is to select frequency for speci c frequency signal to improve the control accuracy of current.However, it can not compensate the phase delay of controlled object.
Viewing of the shortcomings of above methods, a current control method based on PI and VR controller is proposed to compensate the selected harmonics accurately.
e proportional control can improve the dynamic performance of the current loop and the fundamental current.e VR controller can control a group of positive and negative sequence harmonic currents with zero steady-state error by the idea of zero pole cancellation.Finally, the feasibility and validity of the proposed strategy are con rmed by simulation based on Matlab/Simulink and a prototype of the SAPF using TMS320F28335 as control core.

Modeling of SAPF
e structure of two-level three-phase three-wire SAPF used in this paper is shown in Figure 1 [19].e mathematical model can be described in three-phase ABC static coordinate system as follows: where e a , e b , and e c are the 3-phase source voltage; u a , u b , and u c are the 3-phase output voltage of SAPF; i sa , i sb , and i sc are source current; i Fa , i Fb , and i Fc are the compensation currents of SAPF; S a − S * c are fully controlled switch devices; L is AC-link inductance; C is DC-link capacitance; u dc is DClink voltage; and R is the equivalent loss of the AC-link inductance and switch device.
By means of Clarke transform and Park transform, the mathematical model under the synchronous reference frame is given as follows: (2) e d , e q u d , u q and i Fd , i Fq are d-axis components and q-axis components of the three-phase source voltage, AC-link output voltage, and compensation current of the SAPF under synchronous reference frame, respectively.ω s represents the voltage angle frequency detected by a software phase-locked loop.

Basic Principle of VR Controller
e current controller based on VR control has a higher gain at its resonant frequency, which can control the AC current at the resonant frequency with high precision.
e transfer function of the VR controller can be derived as [20] G VRh (s) where K ph and K ih are proportional coe cient and the integral coe cient, respectively, and hω s is the resonant angle frequency.e absolute value of h represents the order of the harmonic currents, the sign of which represents the rotation direction in sequence component diagram.When h is positive, it represents that harmonic currents are in positive sequence.Otherwise, it represents that harmonic currents are in negative sequence.e bode diagram of VR controller is shown in Figure 2 for K ph 1, K ih 50, h 6, ω s 100π.On one hand, the open loop gain of VR controller is in nity at its two resonant points ± hω s , but for other frequencies, the gain is rapidly attenuated.On the contrary, the VR controller is phaseleading about 180 °at the frequency of 0 ∼ hω s .Once the angular frequency exceeds the resonant frequency, the phase jumps from 180 °to 0 °.But at negative frequency, it appears on the opposite performance.On the basis of the analysis above, it shows that VR controller has a good control precision for the signal at the resonant frequency.

VR Controller for Compensating Harmonic Currents.
For the three-phase uncontrollable recti er, the main harmonic currents consist of 5th of negative sequence, 7th of positive sequence, 11th of negative sequence, 13th of positive Nonlinear load Shunt active power filter 2 Journal of Electrical and Computer Engineering harmonic, and so on, such that h ± 6n + 1, n is 1, 2, . . . .Since the synchronous reference frame provides a frequency shift of −50 Hz, the main harmonic current in fundamental frame transforms to 6th of positive and negative sequence harmonic, 12th of positive and negative sequence harmonic, and so on.e harmonic orders become h ± 6n.us the multi-VR controller can compensate the selected harmonic based on formula (3).e equivalent gain of the SAPF K PWM is set to 1. e current control block diagram is shown in Figure 3, where P(s) 1/(Ls + R) is the transfer function of the controlled object.
Taking the d-axis current as an example, current open loop transfer function of the multi-VR controller is e basic idea of VR control is to adopt the zero point of the control to o set the poles of the controlled object, so as to compensate for the phase lag of the controlled object, thus improve the control accuracy of the harmonic currents.e zero of the controller can e ectively cancel the poles of the controlled object on condition that K ph K ih * L/R, and the simpli ed open loop transfer function is given as e closed loop transfer function is e values of K ph , K ih , L, andR in formulae ( 4)-( 6) are both greater than 0. e bode diagram of harmonic current close loop control is depicted in Figure 4, at the resonant frequency, the amplitude gain of the transfer function is 1 without phase delay, which indicates that the VR controller has frequency selection function on a pair of harmonic currents and can be controlled with zero steady-state error.erefore, the system is stable even when the circuit parameters and VR parameters change.
Whether SAPF can operate stably is a ected not only by its own system parameters, but also by external disturbances, and the grid voltage is the most important factor for SAPF.Next, the tracking performance of compensation current is analyzed, when the grid voltage is disturbed.e current control block is shown in Figure 5.
When controlling any order of harmonic, the transfer function of compensation current caused by grid voltage disturbance is At the resonant frequency, the amplitude gain caused by the grid voltage disturbance tends to 0, that is, the steadystate error is 0, which indicates that the compensation current is a ected by the voltage disturbance very little by using the VR controller.

PI Controller for Compensating Fundamental Current.
e previous section demonstrates that the VR controller with double resonant frequency can compensate the harmonic with zero steady-state error, but from bode diagram of harmonic close loop control, the amplitude gain of the current near 0 Hz is very small, which means that the VR Journal of Electrical and Computer Engineering controller is not ideal for the low frequency signal including DC current signal and has the problem of slow dynamic response.However, in addition to controlling the harmonic currents, the SAPF needs to control the fundamental current from voltage controller to compensate for its loss.e DC current obtained fundamental current by the synchronous reference frame.So, it is necessary to ameliorate the VR controller to improve the gain of the low frequency signals.
For h 0 in formula (3), the VR controller can be transformed to PI controller, as shown in the following formula: e fundamental current control block diagram based on PI controller is given in Figure 6.
e fundamental current closed loop transfer function based on the PI controller is For K p0 /K i0 L/R, the zero point of the PI controller is used to o set the pole of the controlled object; thus, the phase delay of the controlled object is compensated.at is, e closed loop transfer function shown in formula ( 10) is a typical rst-order inertia link, and its control bandwidth is ω K p0 /L; that is, the cut-o frequency of the transfer function is f c K p0 /2πL.When L is constant, the bigger K p0 is, the bigger bandwidth ω is, which represents that the response speed of current loop is faster; when K p0 is constant, the bigger L is, the bigger bandwidth ω is, which represents that the response speed of current loop is slower.
e bode diagram of fundamental close loop control is given in Figure 7 for L 3 mH and K p0 1, 10, 20.At the frequency of 0 Hz, the amplitude gain of the fundamental current is 1 without phase delay.However, the gain is rapidly attenuated outside 0 Hz.It indicates that the PI controller has the function of frequency selection for the DC current under the synchronous reference frame, which can achieve high precision control.

e In uence of Digital Control
Delay.Digital control delay, including computation delay and PWM update link delay, will a ect tracking performance of the current loop and even cause instability of the system.us, it is necessary to study the e ect of control delay on the tracking performance of the current loop.e delay of the digital control is generally 1.5 times of the sample period Ts [21], and the transfer function can be expressed as Bode diagram of closed loop current considering digital control delay is shown in Figure 8 for L 3mH, T s 10 −4 s, K ph 0.4.Even if a ected by the digital control delay, the current loop has 35 °of phase margin, which is still stable.In addition, the phase shift is still 0 at every positive and negative sequence harmonic frequency, which indicates that the digital control delay has less in uence on the system and the current loop still has high static control accuracy.

Design of Control System.
e control block diagram of the three-phase three-wire SAPF for the proposed method is shown in Figure 9. e PI control is used to maintain DClink voltage stability in the external voltage loop.e difference between the reference voltage u * dc and the actual voltage u dc is input to the PI controller and the output of PI controller as a part of active power loss is added to the active source current component.PI    extracted from the load current i Labc by instantaneous reactive power theory is added to the output of the PI controller i * sd and i * sq .Compensation current i Cabc is turned to i cd and i cq based on synchronous coordinate transformation; then the di erence between i cd , i cq and i * cd , i * cq is input to the current controller; besides, the control pulse signal required by SAPF is obtained by the space vector pulse width modulation (SVPWM) to compensate the harmonic currents [22].

Simulation and Experimental Results
In order to verify the correctness and e ectiveness of the proposed current control method based on PI and multi-VR control in this paper, both classical PI controller and PI + multi-VR controller are designed according to the above-mentioned method in Matlab/Simulink.e simulation model parameters are shown in Table 1.
Since both of the power supply and the nonlinear load are three-phase symmetry, the A phase is only analyzed in Current controller (Equations ( 3) and ( 8)) Current controller (Equations ( 3) and ( 8  the simulation waveform.e waveform of the load current and harmonic currents are depicted in Figures 10 and 11, and the FFT analysis of the nonlinear load current are shown in Figure 12.From Figure 11, the total harmonic distortion (THD) before compensation is up to 28.92%, and the load current's harmonic spectrum contains harmonics of the order h 6n ± 1, n 1, 2, ... .e compensation performance using PI controller is shown in Figures 13 and 14, respectively.e compensation results of the proposed control strategy based on PI and 6th VR controller are shown in Figures 15 and 16, respectively.Comparing the four gures, PI controller can compensate a part of harmonic, which is not too fruitful.After adopting the 6th of the VR controller, the 5th harmonic and 7th harmonic can be compensated e ectively.5th harmonic is reduced to 0.28% from 2.08%.7th harmonic is reduced to 0.11% from 2.16%, but distortion rate of other harmonics remains the same.e simulation results show that the 6th resonant controller has good inhibitory e ects on 5th harmonic and 7th harmonic, which can further improve the compensation performance for some speci c harmonics.e compensation results using PI controller and multi-VR controller are shown in Figures 17 and 18.When the PI and multi-VR controllers are introduced in the control system, distortion rate of each harmonic decreases remarkably.
e experimental waveforms based on PI + multi-VR controllers proposed in the case of abrupt loading is shown in Figure 19.It is known from Figure 19 that at 0.3 s, a three-phase diode recti er of 20 kW is added to the system, and the SAPF can still track the abrupt load current quickly.
After about one fundamental period, the system reaches stability.It shows that the proposed control strategy has good dynamic performance.
In order to further verify the e ectiveness of the proposed strategy, a SAPF prototype of 15 kVA is built.e main control chip is TMS320F28335 of TI company.PM50RL1B120 produced by Mitsubshi is used as power devices of the main circuit.In the process of prototype testing, the experimental waveform is recorded by the DPO3014 digital uorescence oscilloscope produced by the Tektronix Inc in the United States, and the harmonic data are recorded by PM3000A Power Quality Analyzer produced by Voltech Inc. Experimental setups of SAPF are shown as Figure 20.
e experimental parameters are as follows:  2, the THD of the source current reduces from 7.34% to 2.86% by using the proposed control strategy.e above results show that the proposed control strategy has good steady performance and control accuracy.
Experimental dynamic results are shown in Figure 23.Before the time of 0.04 s, the SAPF has been invested in the system to nish the harmonic current detection.Besides, voltage controller has been working steadily and DC-link voltage has been stabilized at 750 V.Moreover, the basic control loop of current controller or PI control has also been run.At the time of 0.04 s, the multi-VR controllers start running.After about 3 fundamental periods, the system enters the new steady state.Experimental results show that the proposed strategy also has good dynamic performance.Journal of Electrical and Computer Engineering

Conclusion
With development of power electronic technology, a large number of power electronic devices are integrated into the power grid, which causes the harmonic pollution.rough the study on SAPF mathematical model and the principle of VR controller, a current control strategy based on PI and multi-VR controller is proposed in this paper.rough detailed analysis on frequency response characteristic of current closed loop, in that the PI and VR controller can compensate the harmonic currents with zero steady-state error, little phase delay and good dynamic performance are proved.In addition, under the synchronous reference frame, the proposed method is simple enough to compensate the harmonic, which reduces the computation and is better adapted to frequency fluctuation.e simulation and experimental results show that the proposed control strategy is correct and effective, which improves the power quality.

Figure 6 :Frequency
Figure 6: Fundamental current control block diagram based on PI controller.

Figure 23 :
Figure 23: Experimental dynamic waveform after adding VR controller.

Table 2 :
Compensation results with two control algorithms.