2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeC05.5 O-harmonic Repetitive Control of an Active Filter uner Varying Network Frequency: Control Design an Stability Analysis Josep M. Olm, Germán A. Ramos, Ramon Costa-Castelló, Rafael Caroner Abstract This work eals with the esign an analysis of a controller for a shunt active power filter. The esign is base on combine feeforwar an feeback actions, the last using repetitive control, an aims at the obtention of a goo closeloop performance in spite of the possible frequency variations that may occur in the electrical network. As these changes affect the performance of the controller, the proposal inclues a compensation technique consisting of an aaptive change of the igital controller s sampling time accoring to the network frequency variation. However, this implies structural changes in the close-loop system that may estabilize the overall system. Hence, this article is also concerne with close-loop stability of the resulting system, which is analyze using a robust control approach through the small gain theorem. Experimental results that inicate goo performance of the close-loop system are provie. I. INTRODUCTION The control of shunt active filters can be carrie out using ifferent approaches [1], [2]. Most of them 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 shape as well as the appropriate power balance. In this work, the current controller is compose of a feeforwar action that provies very fast transient response an an o-harmonic repetitive control law yieling close-loop stability an a very goo harmonic correction performance [3]. The outer control law is base on the exact computation of the sinusoial current network amplitue an, in orer to improve robustness, this computation is combine with a feeback control law incluing an analytically tune PI controller [3]. This control algorithm epens on network voltage frequency an shows a ramatic performance ecay when this value is not properly know or changes in time. This article proposes an aaptation of the controller sampling rate accoring to the isturbance/reference perio [4], [5], [6], [7]. In this paper, both the inner an the outer loop aapt their sampling frequency to the perio of the signal being tracke. This allows to preserve the steay-state performance while maintaining a low computational cost. Nevertheless, R. Costa-Castelló an R. Caroner are with the Institute of Inustrial an Control Engineering, Universitat Politècnica e Catalunya, Av. Diagonal, 647 08028 Barcelona, Spain ramon.costa@upc.eu, rafael.caroner@upc.eu J. M. Olm, corresponing author, is with the Department of Applie Mathematics IV, Universitat Politècnica e Catalunya, Ava. Víctor Balaguer, s/n, 08800 Vilanova a Geltrú, Spain josep.olm@upc.eu G.A. Ramos is with the Department of Electrical an Electronic Engineering, Universia Nacional e Colombia, Bogotá DC, Colombia e-mail: garamosf@unal.eu.co This work is partially supporte by the spanish Ministerio e Eucación y Ciencia (MEC) uner project DPI2007-62582 the structural changes inuce by this operation, which transforms an original Linear Time Invariant system (LTI) into a Linear Time-Varying (LTV) one, may estabilize the close-loop system. Hence, we use the small-gain theorembase technique introuce in [8] to fin out stability margins where reliable performance is efinitely ensure. This aaptive proceure, along with the introuction of feeforwar paths, yiel very goo performance both in transient an in steay-state behavior, as well as robustness in front of network frequency variations for which stability margins are erive. A. The boost converter II. STATEMENT OF THE PROBLEM The system architecture is epicte in Fig. 1. A loa is connecte to the power source, while an active filter is applie in parallel in orer to fulfill the esire behavior, i.e. to guarantee unity power factor at 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 t C 1 v 1 t C 2 v 2 t = r L i f v 1 1 2 = v 1 1 i f r C,1 2 = v 2 1 i f r C,2 2 1 v 2 (1) 2 where is the uty ratio, i f is the inuctor current an v 1, v 2 are the c capacitor voltages; = V n 2sin(ωn t) is the voltage source, 1 L is the converter inuctor, r L is the inuctor parasitic resistance, 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 value in the close real interval [1,1] an represents the average value of the Pulse-With Moulation (PWM) control signal injecte to the actual system. Due to the nature of the voltage source, the steay-state loa current is usually a perioic signal with only oharmonics in its Fourier series expansion, so it can be written as = k=0 a k sin(ω n (2k1)t)b k cos(ω n (2k1)t). B. Control objectives The active filter goal is to assure that the loa is seen as a resistive one. This can be state as i n = I sin(ω nt), i.e. the 1 ω n = 2π/T p ra/s is the network frequency. (2) (3) 978-1-4244-7425-7/10/$26.00 2010 AACC 1749
i f General loa Distribution Network frequency estimator E c Tp sampling time i f Generic Loa Fig. 1. system. u v 1 r L L u C 1 v C 2 2 r C,1 r C,2 Single-phase shunt active filter connecte to the network-loa source current must have a sinusoial shape in phase with the network voltage 2. Another collateral goal, necessary for a correct operation of the converter, is to assure constant average value of the c bus voltage 3, i.e. < v 1 v 2 > 0 = v, where v must fulfill the boost conition (v > 2 2 ). It is also esirable for this voltage to be almost equally istribute among both capacitors (v 1 v 2 ). C. Transforming the plant equations It is stanar for this type of systems to linearize the current ynamics by the partial state feeback α = 1 2 v 1 1 2 v 2. Moreover, the change of variables i f = i f, E C = 1 ( C1 v 2 2 1C 2 v 2 ) 2, D= C1 v 1 C 2 v 2 introuces 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 using the new variables is L i f t E C t D t = r L i f α (4) = 2E C r C C i f α (5) = 1 r C C Di f. (6) It is important to note that (4) an (6) are linear an ecouple with respect to state variable E C. The partial state feeback an the change of variables will be applie as the lowest level control action on the close-loop system. III. CONTROL DESIGN The controller is esigne using a two level approach, as portraye in Fig. 2: first, an inner current controller forces the sine wave shape for the network current an, secon, an outer control loop yiels the appropriate active power balance for E c energy control Fig. 2. I current control α variable change v 1 v 2 PWM moulator S1,S2 Global architecture of the control system. Boost Converter the whole system. The output of this loop is the amplitue of the sinusoial reference for the current control loop. The active power balance is achieve if the energy store in the active filter capacitors, E C, is equal to a reference value, E C. A. The current loop controller Taking avantage of the linearity of (4), a linear controller is esigne to force a sinusoial shape in. This controller consists of two parts, as picture in Fig. 3: A feeforwar controller which fixes the esire steay state: i n = I sin(ω nt) (7) A feeback controller which compensates uncertainties an assures close-loop stability. As previously state, the current control goal is to assure that i n = I sin(ω nt)=i f (t) (t), where I is constant in steay state. From the circuit topology an (4), = r L t L α L L t r L L. (8) In orer to force to achieve the esire value (7), it is necessary that α takes the value ( α f f = L ) t r L (r L sin(ω n t)lω n cos(ω n t))i = F( )M(I,t,ω n ), (9) thus efining the nominal control action that may keep the system tracking the esire trajectory. Hence, it is use as a I sin carrier extraction i re f Feeforwar Controller Feeback Controller - sin,cos F(z) internal moel M G x (z) G c(z) α f f α f b α i G n p (z) i f 2 x represents the steay-state value of signal x(t). 3 < x> 0 means the c value, or mean value, of the signal x(t). Fig. 3. Current control block iagram. 1750
feeforwar action. As the system is igitally implemente, the operator F is approximate by F(z)= (Lr L )zl. z This action is combine with a feeback controller to overcome moel uncertainties, isturbances an measurement noise. The iscrete-time moel of (4), once filtere by an antialiasing evice with time constant τ, answers to: G p (z)= Z [ 1 Lsr L 1 τs1 1eTs s ] (10) As the signal to be tracke an rejecte in this system is an o-harmonic perioic one, it is convenient to esign a controller which allows to track/reject this type of signal. A technique that has been prove to be specially suitable for this case is o-harmonic repetitive control [9]. Repetitive controllers are compose of an internal moel, which assures steay-state performance, an a stabilizing controller, G x (z), which assures close-loop stability. Traitionally, repetitive controllers are implemente in a plug-in fashion, i.e. the repetitive compensator is use to augment an existing nominal controller, G c (z) (see Fig. 3). This nominal compensator is esigne to stabilize the plant, G p (z), an provies isturbance attenuation across a broa frequency spectrum. The internal moel use in o-harmonic repetitive control [9] has the form G im (z)= H(z) z N 2 H(z), where N = T p an H(z) is a low pass filter use to improve system robustness. It is important to note that N correspons to the iscrete-time perio of the signal to be tracke/rejecte an its value is structurally introuce in the control system. In this work the values T p = 1 50 s an N = 400 have been selecte to obtain a goo reconstruction of the continuoustime signals. The close-loop system of Fig. 3 is stable if the following conitions are fulfille [9]: 1) The close loop system without the repetitive controller is stable, i.e. G o (z)= G c(z)g p (z) 1G c (z)g p (z) is stable. It is avisable to esign the controller G c (z) with a high enough robustness margin. In this work, the lag controller G c (z)= 0.6305z0.629 z0.9985 provies a phase margin of 140 o. 2) H(z) < 1. H(z) is esigne to have gain close to 1 in the esire banwith an attenuate the gain out of it. The first orer linear-phase FIR filter has prove to be goo enough in this application. 3) 1G o (z)g x (z) < 1, where G x (z) is a esign filter to be chosen. A trivial structure 4 which is often use is [10]: G x (z)= k r G o (z). In this application k r = 0.3 has been selecte [11]. The repetitive controller efines the feeback law α f b = G c (z)[1g x (z)g im (z)] ( i re f ) that will be use with the feeforwar action given in (9), this yieling α = α f b α f f. Fig. 3 shows the complete current control loop that will be use in the system. Uner the combine action of the feeforwar an the feeback control action, one can assume that the network current is (t) I (t)sin(ω n t), which will be taken as a fact. B. The energy shaping controller Following [12], the outer controller that assures a mean value 5 of the energy store in the capacitors, i.e. E c (t) Tp close to the esire reference value Ec, is mae up of two parts (see Fig. 4): A feeforwar term which makes I f f = a 0. This assures the energy balance in the ieal case (r L = 0 an r C = 0) an takes into account characteristics an changes instantaneously. I f f is calculate using an amplitue moulator with a scale signal of the source voltage as a carrier an a mean value extraction. This last operation has been implemente through the filter P(z)= 1 N 1zN 1z 1, which correspons to a goo approximation of the corresponing continuous-time mean value extraction operation. A feeback term which compensates issipative effects an system uncertainties. The ynamics of the plant can be moele by the iscrete-time integrator (z1) 2(z1) 4 There is no problem with the improperness of G x (z) because the internal moel provies the repetitive controller with a high positive relative egree. 5 f(t) Tp = T 1 t p ttp f(τ)τ. E c Active component extraction PI I f b I f f I Vn 2 (z1) 2(z1) E c Tp H(z)= 1 4 z 1 2 1 4 z1 Fig. 4. Simplifie 50Hz energy (voltage) control loop. 1751
Magnitue (B) 90 80 70 60 50 40 30 20 10 0 10 Fig. 5. Open Loop Internal Moel Gain Open Loop Controller Nominal Frequency (50Hz) Frequency increase(51hz) Frequency ecrease(49hz) Frequency (ra/sec) 10 3 Open-loop transfer function gain iagram. an the losses in the inuctor an capacitors parasitic resistances can be consiere as an aitive isturbance. So, the PI controller I f b = k i (z1) 2(z1) E k p E, (11) where E = Ec E c (t) Tp, will regulate E c (t) Tp to the esire value Ec with null steay-state error. C. Network frequency variations Most control algorithms in the previous section contain the ratio N = T p, i.e. the perio T p of the signals to be tracke or attenuate over the sampling perio. In systems where the perio of the signal T p is kept constant, N an are esigne a priori accoring to the esire number of samples per perio an the technological constrains over. However, in this case, the electrical istribution network frequency can unergo fluctuations an, then, T p can not be assume constant. If T p varies, the value of N or shoul be change in orer to preserve the ratio N = T p. If this is not the case the control algorithm performance may ramatically ecay. As an example, Fig. 5 shows the feeback control open-loop gain esigne for a nominal frequency of 50Hz with the gain for 49Hz, 50Hz an 51Hz (an some of their harmonics) highlighte. Note that while for the 50Hz signal the gain is important, it ecays for the other frequencies. Something similar occurs with the phase lag of the close-loop control system. While for the nominal frequency the phase is almost zero, this is not the case for the other frequencies. It is worth emphasizing that, for the active filter, this woul imply a reuction of the harmonic rejection capabilities an the introuction of reactive current in the system. Clearly, both effects woul contribute to the reuction of the system performance. To overcome this problem, the sample time will be aaptively varie in orer to maintain a constant value for N. However, the change of implies changes in the system ynamics an, particularly in the plant moel, G p (z). It is important to check that these changes oes not imply a loss of close-loop stability. Next section is evote to this subject. IV. STABILITY ANALYSIS Let the iscrete-time state-space representations of the blocks G i be enote by (A i,b i,c i,d i ), i {im,x,c, p}. The close-loop system state equations are erive uner the following assumptions: In repetitive controllers with the structure of Figure 3 it is D im = 0. The continuous-time plant G p (s) has, at least, relative egree 1, so D p = 0. The representations corresponing to blocks G im (z), G x (z) an G c (z) are obtaine from the nominal sampling time = T an remain constant t. Only the plant iscrete-time moel matrices A p, B p, vary accoring to sampling rate upating: A p = A p ( ), B p = B p ( ), while C p is maintaine constant. Hence, assuming that (A,B,C,0) stans for the continuous-time plant state-space representation, i.e. G p (s) = C(sIA) 1 B, then T A p (T) e AT, B p (T) e Ar Br. (12) 0 Let the system be sample at time instants{t 0,t 1,...}, with t 0 = 0, t k1 > t k, the sampling perios being T k = t k1 t k. Let also x k x(t k ), r k r(t k ), y k y(t k ). The state equations are given by the iscrete-time LTV system: where x k1 = Φ(T k )x k Π(T k )r k, y k = ϒx k, (13) ( Φ(T) K B p (T)M L A p (T)B p (T)N Π(T) ( B im 0 B c B p (T)C c ), ), (14) K, L, M, N being constant matrices available from [13]. Assume that G im (z), G x (z) an G c (z) are esigne to provie stability for a nominal sampling time = T. Hence, when T k = T, k, the overall system is stable by construction. A methoology for stuying the close-loop behavior uner non-uniform sampling perio is evelope below. Next result allows to reuce the stability analysis of (13) to that of its zero-input response. Proposition 1 ([14]): Let the sampling perio, T k, take values in a compact subset T R. Then, the uniform exponential stability of x k1 = Φ(T k )x k (15) implies the uniform Boune Input-Boune Output (BIBO) stability of system (13). Proposition 2 ([14]): Let the sampling perio, T k, take values in a compact subset T R. If there exists a matrix P such that L Tk (P)=Φ(T k ) PΦ(T k )P<0, s.t. P=P > 0, (16) T k T, then (15) is uniformly exponentially stable. 1752
The stability analysis follows the approach propose in [8], where the non-uniform sampling is viewe as a nominal sampling perio affecte by an aitive isturbance. Then, the actual problem is to quantify the amount of isturbance ue to aperioic sampling that the system can accommoate while preserving stability. Proposition 3: Let T = T be a fixe sampling perio an efine θ k = T k T. Then, the matrix Φ(T k ) may be written as Φ(T k )=Φ( T) (θ k )Ψ( T), (17) where ( ) 0 0 θ (θ), (θ) e 0 (θ) Ar r, (18) 0 ( )( ) 0 0 0 I Ψ(T). (19) A p (T)A A p (T)B M N Notice that, using Proposition 3, the original system (15) can take the form x k1 = [ Φ( T) (θ k )Ψ( T) ] x k, (20) which allows the following interpretation [8]: (20) can be regare as the LTI system { xk1 = Φ( T)x Σ := k u k (21) v k = Ψ( T)x k, G T(z) = Ψ( T)[zIΦ( T)] 1 being its associate iscretetime transfer function, receiving the time-varying output feeback control action u k = (θ k )v k. From now on, let R = [ ρ(r R) ] 1/2 enote the 2-norm of a real matrix R, with ρ( ) staning for the spectral raius. Theorem 4 ([8]): Assume that T = T is a nominal sampling perio. Let γ T =(1ε) G T(z), ε > 0, (22) be an upper boun of the H -norm of system Σ (21), an let also T R be compact. If γ T (T T) 1, T T, (23) then system (13) is uniformly BIBO stable in T. The application of the above results to the active filter is carrie out straightforwar. The continuous-time plant is 1 G p (s)= 2.8544 10 8 s 2 0.000081784s0.5, (24) where L=0.8mH, r L = 0.5Ω an τ = 3.568 10 5 s. The controller is constructe for a nominal frequency of ν = 50 Hz, an N = 400 is selecte to obtain a goo reconstruction of the continuous-time signals; this yiels a nominal sampling perio of T = T p N 1 =(N ν) 1 = 0.05 ms. Notice that the close-loop system orer is high, because the imensions of Φ are now 209 209. Uner these assumptions, (10) becomes G p (z)= 0.02855z0.01783 z 2 1.215z0.2387. (25) These settings yiel G T(z) = 2.6530 10 4. In orer to efine γ T (see (22)), ε = 0.0001 has been selecte. Recall Fig. 6. Nonlinear loa connecte to the gri (50Hz):, an vs time (The active filter is off). now that the continuous-time plant matrix can be obtaine from (24). Then, accoring to Theorem 4, the stability interval for the network frequency obtaine from a numeric computation of norm bouns for the matrix exponential (see (23)) is [42.4676,59.3855] Hz. It is worth mentioning that line frequency variations use to be, at most, a 10% of its nominal value, i.e. they can be expecte to lie insie the interval [45,55] Hz. V. EXPERIMENTAL SETUP AND RESULTS A. Experimental setup The experimental setup is compose of a full-brige ioe rectifier (nonlinear loa), the previously escribe singlephase active filter, the regular istribution network an ac power source (PACIFIC Smartsource, 140-AMX-UPC12) that acts as a variable frequency ac source 6. The active filter is connecte in a shunt manner with the rectifier to compensate its istorte current. The active filter controller has been implemente on a DSP base harware, i.e. within a igital framework, with a nominal sampling frequency equal to the switching frequency of 20 khz. The network frequency is obtaine from the network voltage zero crossings through some aitional harware an a igital lowpass filter that runs in the DSP. With this information, the sampling frequency is upate to maintain the ratio N = 400. B. Experimental results Fig. 6 shows the waveforms of, an when the nonlinear loa is connecte to the network. The rectifier current has a total harmonic istortion(thd) of 62.6% an RMS value of 19.56A. As Fig. 7 shows, when the active filter is connecte in parallel with the rectifier the shape of the current at the source port is nearly sinusoial with a THD of 1.2% while the power factor (PF) an cosφ at the port are unitary. The figure shows that the mean value of v 1 is maintaine almost constant 7. In the next experiment the network frequency of the system changes from 48Hz to 53Hz in a 20 cycles ramp 6 Use only in the varying frequency experiments or when not working at the nominal frequency. 7 v 2 is not show ue the limite number of channels in the instrumentation. 1753
v 1 v 1, v 2 Fig. 7. Nonlinear loa an the active filter connecte to gretwork (50Hz). (top),, an v 1 vs time; (bottom) PF, cosφ an THD for. manner (in this experiment the ac power source is use to fee the nonlinear loa). The response of,, an the semibus voltage v 1 is plotte in Fig. 8 (top). As it can be seen, all variables are boune an the mean value of v 1 is almost constant. Aitionally, Fig. 8 epicts the system behavior in the steay state at 52Hz. As it can be shown the PF an the cosφ return to unitary values an the THD for is 0.4% (in this experiment the ac power source is use to fee the nonlinear loa). VI. CONCLUSIONS This work shows the architecture, some esign issues an a stability analysis for an active filter igital controller base on repetitive control. The controller inclues a mechanism to follow possible network frequency variations without losing the avantages of the repetitive control an maintaining its low computational cost. In turn, the stability analysis is base on the small-gain theorem. Theoretical an experimental results prove that the controlle system has a goo performance an that, using the frequency aaptation mechanism, it is able to cope with more aggressive frequency changes than the usual ones in electrical istribution networks. REFERENCES [1] S. Buso, L. Malesani, an P. Mattavelli, Comparison of current control techniques for active filters applications, IEEE Trans. on Inustrial Electronics, vol. 45, pp. 722 729, october 1998. [2] P. Mattavelli, A close-loop selective harmonic compensation for active filters, IEEE Trans. on Inustry Applications, vol. 37, no. 1, pp. 81 89, january 2001. [3] R. Costa-Castelló, R. Griñó, R. Caroner Parpal, an E. Fossas, High-performance control of a single-phase shunt active filter, IEEE Transactions on Control Systems Technology, vol. 17, no. 6, pp. 1318 1329, Nov. 2009. Fig. 8. Nonlinear loa an the active filter: (top),, an v 1 vs time when the network frequency changes from 48Hz to 53Hz in a 20 cycles ramp manner; (bottom),, PF, cosφ an THD for when the system reaches the steay state at 52 Hz. [4] T. J. Manayathara, T. Tsao, an J. Bentsman, Rejection of unknown perioic loa isturbances in continous steel casting process using learning repetitive control approach, IEEE Transactions on Control Systems Technology, vol. 4, no. 3, pp. 259 265, May 1996. [5] J. Liu an Y. Yang, Frequency aaptive control technique for rejecting perioic runout, Control Engineering Practice, vol. 12, pp. 31 40, 2004. [6] R. Costa-Castelló, S. Malo, an R. Griñó, High performance repetitive control of an active filter uner varying network frequency, in IFAC WORLD CONGRESS 2008, Seoul. Korea, Jul.6-11, 2008, pp. 1 6. [7] G. A. Ramos, J. M. Olm, an R. Costa-Castelló, Digital repetitive control uner time-varying sampling perio: An lmi stability analysis, in Proc. IEEE Control Applications, (CCA). Intelligent Control, (ISIC), Saint Petersburg, Russia, Jul. 8 10, 2009, pp. 782 787. [8] H. Fujioka, Stability analysis for a class of networke-embee control systems: A iscrete-time approach, in Proceeings of the 2008 American Control Conference, pp. 4997 5002. [9] R. Griñó an R. Costa-Castelló, Digital repetitive plug-in controller for o-harmonic perioic references an isturbances, Automatica, vol. 19, no. 4, pp. 1060 1068, July 2004. [10] M. Tomizuka, T. Tsao, an K. Chew, Analysis an synthesis of iscrete-time repetitive controllers, Journal of Dynamic Systems, Measurement, an Control, vol. 111, pp. 353 358, September 1989. [11] G. Hillerström an R. C. Lee, Trae-offs in repetitive control, University of Cambrige, Tech. Rep. CUED/F-INFENG/TR 294, June 1997. [12] R. Costa-Castelló, R. Griñó, R. Caroner, an E. Fossas, High performance control of a single-phase shunt active filter, in Proc. of the 2007 IEEE International Symposium on Inustrial Electronics (ISIE 2007), june 2007, pp. 3350 3355. [13] G. Ramos, J. Olm, an R. Costa-Castelló, Non-uniform sampling in igital repetitive control systems: an LMI stability analysis, Technical Report n. IOC-DT-P-2009-01, Universitat Politècnica e Catalunya, 2009, available online at: http://hl.hanle.net/2117/2651. [14] W. Rugh, Linear system theory, 2n E. Prentice-Hall, Inc., Upper Sale River, NJ, 1996. 1754