High Performance Control of a Single-Phase Shunt Active Filter

Transcription

1 High Performance Control of a Single-Phase Shunt Active Filter R. Costa-Castelló, R. Griñó, R. Caroner, E. Fossas Abstract Shunt active power filters are evices connecte in parallel with nonlinear an reactive loas which are in charge of compensating these characteristics in orer to assure the quality of the istribution network. This work analyzes the ynamics of boost-converter use as an active filter an proposes a control system which guarantees close-loop performance (power factor close to 1 an current harmonics compensation). Propose controller is hierarchically ecompose in two control loops, one in charge of shaping the current an the other in charge of assuring the power balance. Differently from other works both control loops are analytically tune. The work escribes both the analytical evelopment an the experimental results showing the goo performance of the closeloop system. Inex Terms Active power filters, igital repetitive control I. INTRODUCTION Active filters are evices which allow to coexist nonlinear loas an goo energy quality in istribution networks. A principal effort in the esign an control of these evices has been evelope in the past years. One research line eals with topologies an architectures [1], [2], several types of topologies have been propose incluing parallel (shunt active filters), serial connections an hybri serial-parallel connections. Besies the architecture, the behaviour principle has also been a research topic: passive, active (using switching converters) an mixe passive-active evices have been propose. Aitionally, converter base active filters may be base on a voltage or a current bus [3]. Most use active filters are connecte in parallel an correspon to active components base on a voltage bus, although a passive serial connecte filter is usually ae to compensate switching noise. Another important research line relate with active filters is their control, many approaches have been propose [4], [5], [6], [7], [8], [9], [10]. Most of the propose control schemes are base on two hierarchical control loops, an inner one in charge of assuring the esire current an an outer one in charge of etermining the require sinusoial shape as well as the appropriate power balance an converter operation point. The current control loop nees to be fast an precise in orer to assure the esire energy flow quality. Several approaches have been propose: hysteresis base control [11], [12], [13], eabeat controllers [14], Park transformation combine with linear controllers [15] an repetitive control [16], [17] are the most relevant options. This work was supporte in part by the Ministerio e Eucación y Ciencia (MEC) uner project DPI C The authors are with the Instituto e Organización y Control e Sistemas Inustriales, Universitat Politècnica e Catalunya, Barcelona, Spain. E- mail:{ramon.costa,roberto.grino,rafel.caroner,enric.fossas}@upc.eu Several approaches have been propose for the outer control loop: selecting the esire switching pattern by optimization, the optimization can be base on genetic algorithms [18], [19], neural networks [14] or Fourier series analysis [13]; using Lyapunov Functions [20]; or using a PI controller to etermine the amplitue of the network sinusoial current. Using a PI controller is the most common approach by far but, since the plant is nonlinear, this PI controller is usually experimentally tune. This work presents a new controller for a single-phase shunt active filter that uses the traitional two control loops ecomposition. In our work the current controller is compose by a feeforwar action in charge of assuring very fast transient response an a feeback control law in charge of assuring close-loop stability an very goo harmonic performance. The feeback control law is base on the use of a repetitive o-harmonic controller [21]. The outer control law is base on the exact computation of the sinusoial current network amplitue; in orer to improve robustness this computation is combine with a feeback control law with a PI controller. One of the contributions of this paper is the analytical tuning of the PI, which is unusual. The complete system results in a simple control law which offers very goo results, both in transient an steay-state behavior. II. PROBLEM FORMULATION A. Physical moel of the boost converter Fig. 1. i f u r L L u General loa C v 1 1 C v 2 2 r C,1 r C,2 Single-phase shunt filter connecte to the network-loa system /07/$20.00 '2007 IEEE 3350

2 Fig. 1 presents the system architecture. A loa is connecte to the power source, in parallel an active filter is applie in orer to fulfill the require performance, i.e. to guarantee unity power factor in the network sie. A boost converter with the ac neutral wire connecte irectly to the mipoint of the c bus is use as active filter. The average (at the switching frequency) moel of the boost converter is given by L i f v 1 C 1 v 2 C 2 = r L i f v 1 v 2 ( 1) (1) = v 1 r C,1 i f (2) = v 2 r C,2 i f ( 1) (3) where is the uty ratio, i f is the inuctor current an v 1, v 2 are the DC capacitor voltages, respectively; = V n 2sin(ωn t) is the voltage source that represent the ac-line source; L is the converter inuctor, r L is the inuctor parasitic resistances, C 1,C 2 are the converter capacitors an r C,1, r C,2 are the parasitic resistances of the capacitors. The control variable,, takes its values in the set [0, 1] an represents the average value of the PWM (pulse-wih moulate) control signal injecte to the actual plant. B. Loa Description Due to the nature of the voltage source, the loa current, in steay-state, is usually a perioic signal with only oharmonics in its Fourier series expansion, so the current can be written as: = a n sin(ω n (2 n 1)t)b n cos(ω n (2 n 1)t) n=0 (4) where a n,b n R are the real Fourier series coefficients of the loa current. Hence a zero c component of is presume. C. Control objectives The active filter goal is to assure that the loa is seen as a resistive one. This goal can be state 1 as i n = I sin(ω nt), i.e. the source current must have a sinusoial shape in phase with the network voltage. Another collateral goal, necessary for a correct operation of the converter, is to assure constant average value of the c bus voltage 2,i.e.<v 1 v 2 > 0= 2v,wherev must fulfill the boost conition (v > 2v s ). It woul also be esirable that this voltage woul be almost equally istribute among both capacitors (v 1 v 2 ). These two objectives efine a non-stanar control problem: the secon one is a regulation objective for the mean value of v 1 v 2, while the first one is not a tracking specification because only a shape an not a signal is efine, that is I is not known a priori an it must take the appropriate value to maintain the power balance of the whole system. This special form of the problem specifications implies the particular structure of the controller loops escribe in the next section. 1 x represents the steay-state value of signal x(t). 2 <x> 0 means the c value, or mean value, of signal x(t) D. Rewriting the Equations sin (ω n t) I current control voltage control α Variable change Boost converter v 2 v 1 E T ET i f Fig. 2. Block iagram of the controller showing the current control loop (inner) an the voltage (or energy) control loop (outer). It is stanar for this kin of systems to linearize the current ynamics by the partial state feeback α = v 1 v 2 ( ( 1). Moreover, the change of variables i f = i f, E C = 1 2 C1 v1 2 C ) 2v2 2, D = C1 v 1 C 2 v 2 makes appear two more meaningful variables. Namely, E C, the energy store in the converter capacitors an D, the charge unbalance between them. Assuming that the two c bus capacitors are equal (C = C 1 = C 2, r C = r C,1 = r C,2 ) the system ynamics in the new variables results in: L i f E c D = r L i f α (5) = 2E c r C C i f α (6) = 1 r C C D i f (7) It is important to note that the state feeback together with the change of variables results in a state an input iffeomorphism so the linearization is complete an formally correct [22]. In aition, eq. (7) is linear too. This new system (5)-(7) nees a controller to fulfill the esire performance. This controller will be efine using a two step approach, first of all a current controller which forces the sine wave shape, afterwars the sine wave amplitue will be efine by an outer control loop to fulfill the appropriate active power balance for the whole system. This balance is achieve if the energy 3 store in the active filter capacitors, E c, is equal to a reference value, E c. The full control scheme for the system is epicte in Fig. 2. The specific controller esigns will be presente in sections III-A an III-B. III. CONTROL DESIGN A. Current Loop Taking benefit from the fact that current equation (5) is linear, a linear controller is esigne to force a sinusoial 3 A similar reasoning can be one with the c bus capacitor voltages as it is implie by the change of variables use. 3351

3 shape in the network current. This controller will be esigne in two parts: A feeforwar controller which fixes the esire steay state : = I sin (ω n t) (8) The control action relate with this action is compute by inverting the plant ynamics in steay state an forcing the output to be the esire one. A feeback controller which compensates uncertainties an assures close-loop stability. This feeback controller is esigne applying the o-harmonic repetitive control technique [23], [21]. This control technique allows to obtain perfect steaystate tracking/rejection of a certain perioic signal an all its o-harmonics. The complete control action is obtaine by aing both control actions. Uner this control action the output is the esire one also in the case of uncertainties an isturbances. B. Energy Shaping (Voltage Loop) E T Fig. 3. I fb Active component extraction PI I ff I E Tp 2C Simplifie 50Hz moel with PI Controller 1 z1 As the source voltage is assume to be = V n 2sin(ωn t) the esire network current is I sin (ω n t), with I locally constant. As a consequence the esire power flow seen from the net is: E T p n (t) (t) (t) I 2Vn sin 2 (ω n t) (9) Aitionally, the active filter goal is not consuming power so ieally the following relationship is esire: p n (t) =I 2Vn sin 2 (ω n t) p l (t)p f (t) (10) where p l (t) an p f (t) are the instantaneous power of the loa an the filter, respectively. However, as I is esigne locally constant the power requirements can not be fulfille instantaneously. What can be achieve is an energy compensation within one perio t t p n p l p f, (11) tt p tt p that yiels the I ieal value. From the power flow point of view the active filter reistributes the power flow within one perio in orer to assure the state power balance: t p f 0 (12) tt p Hence, the total energy store in the converter (E T ) shoul not suffer variations within a perio, i.e. t Ė T =0 (13) tt p The store energy in the converter can be ecompose in the energy store in the inuctors (E L = 1 2 L(i f ) 2 ), an the energy store in the capacitors (E c = 1 2 C 1V C 2V2 2). Aitionally, it is important to note that some energy is lost in the parasitic resistors of the inuctors, the capacitors an the switches. Noting that i f I sin (ω n t) is an o-harmonic signal an without taking into account the parasitic resistance of the inuctors an capacitors, it can be easily proven that inepenently of the value of I an the loa currents the variation of energy in the inuctors on one perio is zero. Thus, t t Ė T = Ė c (14) tt p tt p In case r L 0 the next relationship can be state t 2Vs π (I a 0 ) Ė T = E T (t) E T (t T p ). tt p ω This energy balance can be seen as a linear iscrete-time system (with sampling time T p ) with an input I an a constant isturbance a 0. So, applying the z-transform E T (z) = 1 z 1 2Vs π ω n (I (z) a 0 U s (z)) (15) where U s (z) is the z-transform of the step signal. The value of a 0 correspons to the active component of (the loa current). So, in orer to assure the esire energy balance (E T 0), a close-loop system is propose. The control action will be compose of two main parts: A feeforwar term : I ff = a 0. A feeback term which is in charge of compensating the issipative terms effects an the uncertainties in the system. Thus, a classical PI controller will regulate E T to the esire value ET ( I fb (z) = z 1 k p k i z 1 without steay-state error, i.e. ) (ET (z) E T (z)) (16) Fig. 3 shows the complete close-loop scheme. IV. EXPERIMENTAL SETUP AND IMPLEMENTATION ISSUES The experimental setup use to test the esigne controller has the following parts: Active filter: half-brige boost converter (split-capacitor c bus) with IGBT switches (nominal current 100 A) an the following parameters: r =0.3 Ω, L =0.8 mh, C 1 = C 2 = 9900 μf an r C = 8200 Ω. The switching frequency of the converter is 20 khz an a synchronous (regular) centere-pulse single-upate moe pulse-wih moulation strategy is use to map the controller s output to the IGBT gate signals (see Figure 4 ). Rectifier (non-linear loa): Full-wave ioe rectifier with a filter capacitor C = 4500 μf. The active power with 3352

4 Fig. 4. Fig. 5. Power Converter Picture Nonlinear loa: voltage an loa current (92 V/iv, 19.2 A/iv). nominal c resistor is P = 4.56 kw an its reactive power is approximately zero. Fig. 5 shows the shape of the ac mains voltage an current an Fig. 6 the harmonic content of the voltage an the current for the rectifier with the nominal c resistor. It is worth to remark that the total harmonic istortion 4 (THD) of this current is about 63.9% an its maximum erivative is about 70kA/s. Analog circuitry of feeback channels: the ac mains voltage, the ac mains current an the c bus voltages are sense with a voltage transformer, a hall-effect sensor an two isolation amplifiers, respectively. All the signals from the sensors pass through the corresponing gain conitioning stages to aapt their values to A/D converter input taking avantage of their full ynamic range. In aition, all the feeback channels inclue a first orer low-pass filter with unity c gain an 4.3 khz cutoff frequency 5. Control harware an DSP implementation: the control boar has been internally evelope an is base on an ADSP floating-point DSP processor with an ADSP fixe-point mixe-signal DSP processor that acts as coprocessor, both from Analog Devices. The ADSP an the ADSP communicate each other using a high-spee synchronous serial channel in DMA moe. The ADSP eals with the PWM generation an the A/D conversions with its integrate 14 bits eight high-spee A/D channels (Figure 7). The controller has been implemente running at the IGBT switching frequency. Technologically this swithching frequency is limite to 20 khz so this frequency has been selecte as the sampling one. The nominal voltage of the ac mains is V n = 230 V RMS an its nominal frequency is 50 Hz. Fig. 6. Fig. 7. Nonlinear loa: voltage an loa current (RMS an THD). Control Setup Picture V. EXPERIMENTAL RESULTS This section shows some of the experimental results obtaine for the active filter operation with the esigne control system. The results are presente by means of oscilloscope an power analyzer screen umps of the ac mains electrical variables an the active filter semi-bus c voltages when necessary. Apart from the selecte experiments collecte in this section, a lot of numerical simulations, incluing mainly capacitive or inuctive loas, have been carrie out showing the same goo performance as it will be shown below. Also, it is worth noting that several numerical simulations incluing loas that work as generators at some time perios 6 (thus imposing a negative active power flow to the source) have been carrie out without problems. The voltage loop of the overall controller assures the active power balance an, after a transient, in steay state the input to the AM moulator 4 In this work the THD figures an the harmonic content are always taken with respect to the funamental harmonic (50 Hz) an they have been obtaine using a Power Quality Analyzer Fluke 43 instrument. 5 The oscilloscope screens in the figures of this section an the following show the voltages an currents after the corresponing analog low-pass filters. 6 This problem was establishe as a har one by Depenbrock an Stau [24]. 3353

5 is negative giving a current reference shifte π ra from the network voltage that the current loop tracks without ifficulty. v 1,v 2 A. Active filter operation with no loa v 1,v 2 Fig. 10. Active filter with the nonlinear loa: voltage, current an semi-bus c voltages (92 V/iv, 19.2 A/iv an 74.5 V/iv, respectively). Fig. 8. Active filter with the no loa: voltage, current an semi-bus c voltages (92 V/iv, 19.2 A/iv an 74.5 V/iv, respectively). Fig. 11. Active filter with the nonlinear Loa : RMS an THD %f values of mains current an cos φ, PF. Fig. 9. Active filter with no loa: RMS an THD values of current (t) an P, Q, cos φ an PF. This subsection presents some results of the no loa operation of the active filter. Fig. 8 shows the network voltage an current an the semi-bus c voltages. The RMS value of the current is about 0.68 A an its THD value is 6.8%. Then, the resulting active power consume by the filter to cover its losses without compensating any loa is about 0.15 kw. Itis worth to note that the funamental component of the current is in phase with the voltage (cos φ =1),seeFig.9.So,almost no reactive power is consume by the filter. The low power factor (PF) is ue to the high value of the switching ripple with respect to the funamental component of the current. B. Active filter operation with the nonlinear loa In this experiment the ioe rectifier previously escribe is connecte to the network. This nonlinear loa has not reactive power at the funamental frequency, however the active filter must work to compensate all the generate higher orer harmonics. Fig. 10 shows the current that appears with a goo sinusoial shape an in phase with the gri voltage. This figure also shows the values of each semi-bus of the active filter c bus. As it can be seen in Fig. 11 the THD of the current is very low (0.6 %) an the power factor is 1. C. Active filter transient response This section presents the results for the following experiments: 1) the full nonlinear loa is applie to the network with the active filter in operation (Fig. 12); 2) the full nonlinear loa is isconnecte from the ac mains with the active filter in operation (Fig. 13). In each case, the overshoot in the c bus voltage is almost imperceptible. Therefore, there is no problem with the maximum loa variations expecte in the system. VI. CONCLUSIONS The paper shows the esign an implementation of a controller for a current active filter. The controller consists of a current control loop an an outer c bus voltage control loop. The current references for the inner control loop is 3354

6 v 1,v 2 Fig. 12. Source current () an semi-bus c voltages (v 1,v 2 ): from no-loa to full nonlinear loa (19.2 A/iv an 74.5 V/iv, respectively). v 1,v 2 Fig. 13. Source current () an semi-bus c voltages (v 1,v 2 ): from full nonlinear loa to no-loa (19.2 A/iv an 74.5 V/iv, respectively). create passing the output of the voltage controller through an AM moulator that uses as a carrier a filtere version of the network voltage. Both, the current an the outer controllers are base on the combination effect of a feeforwar an a feeback control law. This combination allows to obtain almost perfect response both in transient an steay-state operation. As a conclusion the propose control scheme constitutes a step forwar in the active filter control. Both the transient an steay-state behaviour are very goo in the network current shape an the active filter semi-bus c voltages. REFERENCES [1] H. 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