Shunt APF Based on Current Loop Repetitive Control

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1 Volume 5, Number 4, June 200 Shunt APF Based on Current oop Repetitive Control College of Geophysics and Information China University of Petroleum, Beijing, 02249,China, doi: 0.456/jcit.vol5.issue4.4 Abstract Three-phase shunt active power filter (APF) is often used to compensate the harmonic and reactive current derived from non-linear load. But many non-ideal factors, such as the limited bandwidth of output current loop and the time lag of signal sensing circuit and reference current generation etc, will deteriorate the compensation effect. Proportion-Integral (PI) controller cannot be used simply to control the output current without any steady error due to its limited current tracking capability. A novel output current control architecture in the synchronous rotating frame with ordinary PI and advanced repetitive controller (RC) in parallel is put forward in this paper to enhance the compensation characteristic of three-phase shunt APF. The advanced repetitive controller in parallel is used to eliminate the steady current tracking error caused by cyclic disturbance signal to improve the compensation precision of APF, while the ordinary PI controller is used to ensure the dynamic response performance of APF. The feasibility of this output current controller architecture is verified completely by theoretic analysis and experimental results in detail. eywords: PI control, repetitive control, shunt active power filter, current compensating. Introduction With widely application of power electronic devices and other non-linear loads in the utility, nowadays more and more harmonic and reactive current are injected into the electrical network, resulting in the worry about the secure operation of the utility and the normal utilization of other related apparatus. Current harmonic is the main reason for voltage harmonics and damage to electrical equipment. Therefore, using active power filter to reduce the current harmonics is becoming a research hotspot[-3]. The effect of APF depends largely on the performance of its controller. PI control is the most commonly used method in the industrial, but conventional integrator can only guarantee that the system non-steady-state error only when the reference is constant. APF s reference and feedback currents are harmonic stacking, conventional integrator can not completely eliminate them. For this reason, synchronous coordinate system of the PI control was put out [8-9]. However, the APF s reference signal is multi-frequency and periodic signal, if you are using PI controller, multi-frequency must be carried out under the coordinate transformation. As the multi-frequency control characteristics of APF, the repetitive control technology has been introduced into the APF control to eliminate the cyclical load harmonic. Repeat control theory derived from the internal model principle of control theory. It is a kind of control scheme which can eliminate the periodic error through the use of the laws of the cyclical nature of the load disturbance. But the main drawback is the repetitive control can not be achieved in a shorter cycle of dynamic response, so each control scheme has its advantages and disadvantages. Combined into a composite control scheme will be an inevitable trend in control system design. In order to enhance the dynamic response ability and improve the steady compensation accuracy of APF as far as possible, this paper bring forward a novel output current controller architecture with traditional PI and advanced RC controller in parallel for three-phase shunt APF. The fast PI controller is used to ensure the dynamic response ability of APF, while the advanced RC based on internal-mode principle is used to improve the steady compensation accuracy of APF. The feasibility of the combined current controller is verified by the theoretic analysis and experimental results. 2. Synchronous coordinates APF description and PI controller 29

2 Shunt APF Based on Current oop Repetitive Control 2. APF structure The structure of a three-phase shunt APF is shown in Figure,which is mainly made up of a voltage source converter with a large capacitor to maintain the voltage constant at the DC side and three inductors to output the compensation current at the AC side in connection with the utility. u sa i sa i a u sb i sb i b u sc i sc i cc icb ica i c i dc u ra u rb u rc Figure. Three-phase shunt APF system In fact, this kind of APF in nature is Boost circuit as the same as the high frequency reversible PWM rectifier, which means the DC voltage level must be higher than the peak-topeak value of the utility line-to-line voltage so as to control the output current completely. 2.2 APF Mathematical model Following the direction of voltage and current shown in Figure 2, ignoring the line resistance and the utility internal equivalent inductance, we can list the differential equation of output current based on V theorem as follows. dica sa ca ra u i Ru dt dica usb icbrurb () dt dica usc iccr urc dt Where u sa, u sb and u sc is the utility voltage. u ra, u rb and u rc is the PWM voltage. i ca, i cb and i cc is the output current. R is the equivalent resistance of connection inductor. Through synchronous rotating transformation, Equation in abc stationary frame can be expressed in d-q frame as follows. It s obvious that there exists cross coupling term between d-axis and q-axis current. d icd /dt usd icd Ricq urd (2) d icq /dt usq icqricd urq 2.3 PI controller It s necessary to give some remarks on the physical meaning of Equation (3) before choosing the architecture of current controller. Firstly, a utility voltage feed-forward path should be added in the controller to offset the effect of the utility voltage and provide a stable operation point for controller. Secondly, a state feedback cross decoupling technique can be applied in the controller to eliminate the coupling between d-axis and q-axis current so that they can be controlled individually even in the case of step response. Thirdly, a traditional PI adjuster may be used to decrease current tracking error. An ordinary PI controller architecture 30

3 Volume 5, Number 4, June 200 with utility voltage feed-forward and output current feedback cross decoupling is shown in Figure 2. urd usd icq urd (3) urq usq icd urq According to the Equation (2) and Equation (3) we can derive as follows, dicd icdrurd dt (4) dicq icqrurq dt From the Equation (4) can be seen that by controlling the u rd * u rq * can be independently controlled i cd i cq. * i cd Pi s Ii u u rd rd s R i cd u sd * i cq Pi s Ii u rq u rq usq s Figure 2. Current control block diagram of APF in d-q frame R i cq As illustrated in Figure 2, the right side of the dotted line is the plant, while the left side of it is the controller, where i cd * and i cq * is the harmonic compensation current reference in d-q frame calculated by the instantaneous reactive power theory or any other methods. With the help of both utility voltage feed-forward and output current feedback cross decoupling control, Figure 2 can be simplified as Figure 3. i * cd Pi s Ii u rd * s R i * cq s R Ii Pi s i cq Figure 3. Simplified current control block diagram of APF in d-q frame The transfer function of closed-loop current control system can be deduced from Figure 3 as follows, in which Pi and Ii is the proportional and integral parameter of PI controller respectively. I () cd () s Icq s ( Pi ) s Ii (5) * * 2 Icd () s Icq () s s [( Pi R)] s Ii Obviously, because the control plant is first-order, the current control closed-loop is a typical second-order system which can be corrected by PI controller into a first-order element by means of Siemens methods which is used widely in the actual tuning of PI parameters. The core thought of this design is to let the zero of the controller offset the pole of the plant in the s- plane. That is to say, R (6) Ii Pi u rq * i cd 3

4 Shunt APF Based on Current oop Repetitive Control Substituting equation (6) into (5) yields the following results, I () cd () s Icq s Pi (7) * * Icd () s Icq () s Pi s ( Pi ) s T f s T f is the time constant of the closed-loop current control, which is inversely proportional to the parameter Pi of PI controller and is in proportion to the connection inductance. In order to make the output current track the compensation reference as best as possible, the current loop response delay time is requested to be as short as possible, i.e. the current loop bandwidth is wide enough. We can increase gain of PI controller, or reduce the inductance, or increase sampling and switching frequency to make T f shorter. However, all these means cannot improve the performance of harmonic compensation effectively and may cause other problems such as oscillation and instability of the APF and much more switching loss. In addition, the value of T f cannot be too small, due to the limit by the operating frequency of the switching devices. That is why the signal cannot be non-static error tracking by digital controller, and the higher the signal frequency the more difficult to tracking the signal by digital PI controller. From the above analysis we can found that digital PI controller in current loop has significant limitations. So we need to find a suitable controller to improve steady-state performance of APF and enhance its harmonics tracking accuracy. 3. Based on PI control and repetitive control parameters of the composite control system design and stability analysis Based on the above analysis, this paper proposed a composite control method which is the composition of digital PI control and digital repetitive control the control. The structure is shown in Figure 4. There are two main components: ) Digital PI controller. Real-time adjust instruction error to improve the Active power filter s dynamic performance. 2)Digital Repetitive Controller. Used to eliminate the cyclical nature of the system tracking error and improve the APF s steady-state compensation precision. Due to the existence of Repetitive Controller can guarantee the output waveform amplitude and phase was no difference on the track for a given signal. Where, z - on behalf of one-beat delay, because the sampling, calculation delay, and digital control delay the impact of one-beat lag control. N is a fundamental cycle of the sampling frequency, kr for the gain coefficient, Q(z) for the compensation factor, G p (z) as the controlled object, S(z) to compensate for part of the object, zk is ahead of the compensation factor, k for the step ahead of G p (z) and S (z) for phase compensation. z I P z Digital PI Controller z r(k) e(k) z N z r k S(z) G ( z ) P y(k) Q(z) Digital Repetitive controller RP( z) Figure 4. PI based on digital control and digital repetitive control of the composite control In the composite control program, PI controller and repeat controller in parallel, co-ordination, complementarity and common output of the system have an impact. On the one hand, when the system is in stable running state, the system's tracking error is small, PI controller s effect is very small which has little impact on the system. At this time, the system runs most of the required control action provided by repeat controller and control action is the result of the cumulative error of the past. On the other hand, when the tracking error suddenly become larger because of the disturbance of the load, due to a reference cycle delay, 32

5 Volume 5, Number 4, June 200 repeat controller output does not change, but the PI controller can track the error and can ensure the response speed of current tracking. In this cycle, the output compensation current is decided by the PI controller. Although the compensation performance poor, but after a cycle, Repetitive Controller plays a regulatory role which makes the tracking error gradually decreases according with the attenuation factor. With the tracking error decreases, PI regulator control effect of gradually weakened until the system reached a new steady state of operation. 3. Design of digital PI controller parameters For the PI controller of parameters we must ensure that the control system s open-loop bandwidth is greater than maximum harmonic frequency which is necessary to compensate. But in the digital control system, due to the impact of sampling period and calculate delay, The output signal is usually a switch cycle delay which is equivalent to string in the system into a timedelay link, causing the system phase lag. So if we want to make the system stable, the system openloop gain kp should be small enough to ensure a large phase margin. Meanwhile, the signals which are used in digital control exist the quantization error, resulting in a phenomenon that the bigger the kp the lower of the control precision, and even cause system oscillation. Proportional coefficient kp should be selected compromise based on the above two aspects. Integral coefficient makes the output of the system has phase deviation with a given offset, so you can design integral coefficient under the phase margin. 3.2 Repetitive Controller Parameter Design Repetitive controller is equivalent to a pure integral which is means that Q(z)=. In order to overcome imprecise object model to enhance the system robustness Q(z) is usually taken as the lowpass filter or slightly smaller than, it makes internal model into a quasi-periodic integral part. The primary role of the compensator S(z) is to transform the control object properties to meet the requirements of repetitive control. For the control object G p (z) characteristics of the compensator set S(z). After learning of the previous cycle error information, the compensator S(z) the task is to go the next cycle, how to give the appropriate amount of control. In order to rapidly and effectively to offset errors, the given signal must have a correct phase and amplitude.s(z) can be designed as two-order low-pass filter in the form of S(z)=ω 2 /(s 2 +2ξωs+ω 2 ). The phase lag produced by S(z) have to be compensate appropriately in order to eliminate the impact of high-frequency interference and improve the system stability. 3.3 Composite control system design In Figure 4, PI controller is expressed as: D(z)= p + Iz /(z-) (8) According to the control block diagram we can deduce the relationship between the tracking error and the given signal. N z Q() z (9) ez () g rz () ( k z D() z GP ()) z () () [ N zszg ( ( ) r P z z Q z )] z D( z) G ( z) P The characteristic equation of the composite control system is, k N zszg r ( ) P( z) ( z D( z) GP ( z)) g[ z ( Q( z) )] (0) z D( z) GP ( z) It is obvious that the characteristic equation in the composite control system (+z - D(z)G P (z)) is a characteristic equation only containing PI controller. In other words, based on PI controller and 33

6 Shunt APF Based on Current oop Repetitive Control repetitive controller composite controller system is one prerequisite for a stable system only in the digital PI controller is stable. Further can be obtained as, GP ( z) CP ( z) () z D( z) GP ( z) We can see that C P (z) is shown in Figure 5, the feedback branch of the PI-controlled closed-loop transfer function. r(k) G ( ) P z y(k) z D( z) Figure 5. Diagram of PI controller feedback channel Another factor in the characteristic equation [z N -(Q(z)-z k r S(z)C p (z))] which is shown in Figure 5 after the repetitive control system uses the characteristic equation. In other words, based on PI controller and repetitive controller of the composite controller system stability condition is through the feedback channel of PI controller system is stable under the repetitive controller. For repetitive controller, C p (z) can be seen as the control object. With regard to steady-state error, assuming that Q(z)= and r(z) completely repetitive, substituting into equation (9) yields, e(z)=0 (2) As can be seen, because the system contains a repetitive controller, ideally, the system can achieve zero steady-state error. Of course, as noted above, in order to take into account the system stability, steady-state error to zero cannot be achieved only in ensuring the stability of the system under the premise of the smallest tracking error as far as possible. 4. Experiment results 4. Digital PI controller Figure 6 shows the waveforms of both load and utility current with only a PI current controller when shunt APF switches on, where waveform is the utility current and waveform 2 is the load current. The frequency spectrum of relative waveform is shown in Figure 7 and 8. It can be learned from these figures that the value of total harmonic distortion (THD) can be reduced from 28.5% to 5% with the help of harmonic compensation of shunt APF with PI controller. Especially, the components of 5th, 7th, th and 3th harmonic have declined greatly, but the components of 7th and 9th harmonic is suppressed a little. Figure 6. Steady waveforms of load and utility current with PI current controller 34

Volume 5, Number 4, June 200 THD/% 8.0 3.5 9.0 4.5 0 2 6 0 4 8 f/hz Figure 7. Magnitude spectrum of load current waveform THD/% 7.2 5.34 3.56.78 0 2 6 0 4 8 f/hz Figure 8.

But it cannot be enlarged without any limit considering the stability of APF.

7 Volume 5, Number 4, June 200 THD/% f/hz Figure 7. Magnitude spectrum of load current waveform THD/% f/hz Figure 8. Magnitude spectrum of utility current waveform As analyzed before, the bandwidth of current control loop can be extended by increasing the P parameter. But it cannot be enlarged without any limit considering the stability of APF. If P is enlarged to its one and a half times, the waveform quality of the utility current will be improved a little, as shown in Figure 9. From frequency spectrum of utility current shown in Figure 0, we can see the THD value of utility current has decreased to 2%. If we continue to increase gain to 2 times of the old value, oscillation occurs between the utility and APF. Through experimental research, a simple PI controller, harmonic compensation is not enough bandwidth, even if the increase in P parameters, or increase switching frequency, improving the effectiveness of the waveform is limited. Therefore, it is verified that the bandwidth is not wide enough when only PI current controller is used to compensate harmonic current. Even though we can improve the waveform quality by increasing the gain of controller, the effect is limited. In addition, from the compensated waveform of utility current, we can see some current spikes appearing at some fixed point. The phenomena are relative to the current commutation between three phases of the harmonic source. At the commutation time of three-phase diode rectifier, one phase current sudden alter its direction immediately, making compensating reference of APF change abruptly and exceed the bandwidth range of PI current controller. APF cannot response very fast to counteract such spikes. Figure 9. Waveforms of load & utility current with.5 times gain THD/% f/hz Figure 0. Magnitude spectrum of utility current waveform with.5 times gain 35

is is the utility current and waveform i is the load current.

8 Shunt APF Based on Current oop Repetitive Control 4.2 Composite controller under steady-state Figure shows the waveforms of both load and utility current with PI and repetitive current controller in parallel when shunt APF switches on, where waveform is is the utility current and waveform i is the load current. Compared with APF controlled by only PI current controller, the spikes in utility current are generally eliminated and the waveform is near perfect sine wave. The magnitude spectrum of utility current is shown in Figure 2. The value of THD of utility current can be reduced to 2.79%, fully meeting the regulations of international harmonic standards. Above all, almost all harmonic currents in the spectrum are suppressed greatly by the shunt APF with PI and repetitive controller in parallel. Figure. Steady waveforms of load and utility current with composite controller THD/% f/hz Figure 2. Magnitude spectrum of utility current waveform 4.3 Dynamic waveforms with composite controller In order to test dynamic response characteristic of this PI and repetitive current controller in parallel, we record some dynamic waveforms when load switch on and off, shown in Figure 3~4 respectively. As shown in Figure 3, from waveform 2, we can see that the composite controller responses the current reference very quickly. When load is switch on suddenly, reference distortion lasts only half a period. This is not because the response speed of controller is not fast enough, but a low pass filter is used in the harmonic detection algorithm, resulting in the distortion of reference. However, from the second fundamental cycle, the effect of repetitive control becomes more and more obvious, and APF begins to approach steady state till the fifth cycle. In Figure 4, waveform 2 shows the output current of APF when load is switched off. In addition to the effect of half period instruction distortion caused by a low pass filter in the harmonic detection, the combined current controller also responses very quickly. The restore time of repetitive controller is relatively long, so the compensation amount will decrease slowly in a fundamental period and cannot response as rapidly as PI controller. This means that repetitive controller is in favor of improving the steady state performance of APF but has no advantage in dynamic response speed. 36

9 Volume 5, Number 4, June 200 Figure 3. Dynamic waveforms of load and utility current with composite controller l during load on Figure 4. Dynamic waveforms of load and output current with composite controller during load off 5. Conclusion Based on the detailed performance analysis of current control in synchronous rotating frame for three-phase shunt APF, this paper propose a novel composite controller architecture with PI and repetitive controller in parallel to improve the steady state compensation precision and ensure the dynamic response performance. Experimental results in detail shows that this kind of current controller can decrease the value of THD of utility current from almost 30% to nearly 3% in steady state and the dynamic response time of APF is only half fundamental period (0ms). The feasibility of this control method is verified completely by both theoretical analysis and laboratory test. The composite controller is so useful that it can be also extended to the control of other power electronics devices such as high frequency PWM reversible rectifier and static var compensator etc. 6. References [] Akagi H,anazawa Y,Nanae A.Generalized theory of the instant- taneous reactive power in threephase circuits[c].ieee&jieeproceedingsipec. Japan,983. [2] Santolo Meo,Aldo Perfetto.Comparison of different control techniques for active filter applications[c].fourth IEEE International Caracas Conference on Devices,Circuits and Systems,Aruba,

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