THREE-PHASE POSITIVE AND NEGATIVE SEQUENCES ESTIMATOR TO GENERATE CURRENT REFERENCE FOR SELECTIVE ACTIVE FILTERS F. Ronchi, A. Tilli Dept. of Electronics, Computer Science and Systems (DEIS) University of Bologna Viale Risorgimento n., 46 Bologna, ITALY fax: +9 5 97 e-mail: fronchi, atilli @deis.unibo.it Keywords: active power filters, harmonic distortion, positive and negative sequences. Abstract This article deals the on-line estimation of three-phase three-wire signal harmonics. It is performed in the d-q synchronous reference frame and exploits the Luenberger observer concept to isolate positive and negative sequences of each harmonic. This estimator is suitable to be implemented as reference generator for selective active power filters. Simulation and experimental results are presented. Introduction The use of nonlinear devices, e.g. power electronics, generatearmonics, subharmonics, and interharmonics in voltage and current mains spectra. It is necessary to measure and decrease thiarmonic distortion, thus reducing the power losses and the risk of equipment damage or malfunctioning. Current harmonicave been traditionally compensated passive filters, which have several drawbacks: their operation depends on the network impedance, they have to be tuned on fixed frequencies, etc. Active power filters [],[5] based on digital controllers can be more expensive respect to the passive ones, but they are less network-dependant and can be tuned on different frequencies simply changing software parameters. Besides, shunt active filters have the goal to generate current equal but opposite to the harmonic currents in the load waveform. This leads to a cheaper design of the filter components respect to series active filters [6]. The performance of active power filters (APF) is based on the inverter parameters, control algorithm and on the method of obtaining current reference. In conventional load current detection methods [],[4], the generation of the active filter current reference is based on the harmonic detection of load currents, using the well known instantaneous power theory [], time-domain correlation techniques [7], FFT, etc. The objective of this paper is to provide an on-line method to estimate load current harmonics, separating positive and negative sequences. An approach based on Luenberger observer is proposed. The estimated harmonics can be used both to simply monitor the load and to generate current references for the APF controller, to perform selective compensation. Moreover, in order to cope the delay of the voltage-source inverter current loop, each estimated harmonic can be processed inverting the transfer function of the closed-loop control system APF. The current control is usually [8] performed in a d-q synchronous reference frame. Hence the estimation and isolation of positive and negative sequences of each harmonic is carried out in this reference frame. This paper is organized as follows. In section () some
concepts about positive, negative sequences and harmonics description are presented; in section () the estimation-isolation algorithm is reported. The last two paragraphs show simulation and experimental results. Conclusions summarize the contents ofpsfrag the replacements paper. b β a α Preliminaries According to Fortescue s theorem [], an unbalanced set of N phasors can be resolved into N systems of phasors called the symmetrical components of the original phasors. For a three-phase systems (i.e. N ), the three sets are:. Positive sequence. Three phasors I p a I p b I p c, equal in magnitude, o apart, the same sequence (a-b-c) as the original phasors.. Negative sequence. Three phasors I n a I n b I n c, equal in magnitude, o apart, the opposite sequence (a-c-b) of the original phasors.. Zero sequence. Three identical phasors Ia Ib Ic : equal in magnitude, no relative phase displacement. Defining r e j π the operator that rotates a phasor of o, the relationships among the sequence components for a-b-c are: I p a I n a I a r r r r The system under study is three-phase three-wire: the sum of the three a-b-c currents is identically zero due to Kirchoff s current law and therefore there is no zero sequence current. Positive and negative sequences correspond to two vectors rotating in opposite directions in the complex plane. In order to better understand the direction of rotation of the vector, the α β fixed reference frame is considered. The matrix that changes coordinates from a-b-c to α β is, according to Fig. αβ T abc k c I a I b Ic c Figure : Fixed reference frames k c arbitrarily chosen constant, typically k c. Let consider the positive sequence i p a t Re I p a e jωt I p cos ωt i p b t Re I p b e jωt I p cos ωt π i p c t Re I p c e jωt I p cos ωt π It can be calculated that i p α t i p β t k ci p cos ωt k ci p sin ωt that correspond to a phasor rotating frequency ω in the complex plain. Consider the negative sequence i n a t Re I n a e jωt I n cos ωt i n b t Re I n b e jωt I n cos ωt π i n c t Re I n c e jωt I n cos ωt π It can be calculated that i n α t i n β t k ci n cos ωt k ci n sin ωt that correspond to a phasor rotating frequency ω in the complex plain. The effect of changing from a fixed reference frame to the rotating d-q frame is the following. Consider the matrix that describes the change of reference frame from the fixed α β one to the d-q one,
which rotates synchronously the voltage mains. dq T αβ cos ω m t sin ω m t sin ω m t cos ω m t Harmonics rotating frequency n ω m (indicated as PR, Positive Rotating) are shifted into frequency n ω m in the d-q reference frame. i p d t i p q t k ci p cos n ω m t k ci p sin n ω m t In the same way, harmonics rotating frequency n ω m (indicated as NR, Negative Rotating) are shifted into frequency n ω m in the d-q reference frame. A linear time-invariant (LTI) state-space representation of the PR and NR sequences at frequency ω is given by the following oscillators A A r A A c ẋ t ω ω ω ω Ax t if ω (PR) if ω (NR) In a digital control system, signals are sampled period T s. Hence the following discrete-time model of previous oscillators is considered where A A r A A c i α k i β k A coω m T s sin hω m T s i α k i β k if ω (PR) if ω (NR) Estimation Let consider the three-phase signal u, its coordinates in the d-q reference frame are: u I d M n! I dn cos nω m t ϕ dn I q M n! I qn cos nω m t ϕ qn " At first, let assume that there is only one d-q harmonic in the signal u whose PR and NR components have to be estimated. Let hω m be its frequency. The proposed estimator based on the Luenberger observer scheme is the following: ˆx t A hˆx t# K $ u t# C hˆx t&% () where A h A rh A ch C h A rh A ch hω m hω m hω m hω m and ˆx '$ ˆx dr ˆx qr ˆx dc ˆx qc % T is the state of the estimator, where: ˆx dr is the d-component of the estimated positive sequence; ˆx qr is the q-component of the estimated positive sequence; ˆx dc is the d-component of the estimated negative sequence; ˆx qc is the q-component of the estimated negative sequence. The system A h C h is completely observable, therefore all the eigenvalues of the estimator dynamic matrix can be arbitrarily chosen by means of K. Let u be decomposed in the sum of of u, the part to be estimated, and u, the part to be rejected, u C h x x $ x dr x qr x dc T x qc % ẋ A h x
/ / 4 Defining x ˆx x estimation error M h A h KC h () the estimation error dynamic can be described as follows x M h x Ku and then, Laplace-transforming X s( '$ si M h %*) K U s The matrix M h must be shaped in order to ensure that. M h is Hurwitz. This guarantees the convergence of the harmoni estimation.. +$ jωi M h % ) K,+.- ω nω m n h All the harmonics different from the h one have to be greatly decreased. A way to satisfy these requirements is to impose complex conjugate eigenvalues for M h, the peak frequency in hω m and a very low damp. Choosing K k k k k k k k k () it can be found that the characteristic polynomial of M h is 4 s k s hω m hω m k 5 Therefore, k and k can be chosen to impose polynomial damp δ and natural frequency ω n δ ) h ω m k δω n (4) ωn k ω (5) ω Starting from the previous result, let consider the general case of many harmonics to be estimated. The matrixes A h C h M h of () and () have to be replaced by the following A C M : A h 6676 A j 666 A C... A z (6) C h C j 666 C z 5 (7) M A KC (8) where h j 6676 z are the numbers of the harmonics considered for estimation. The system A C is still completely observable and therefore all the estimator dynamic matrix eigenvalues can be arbitrarily chosen by means of K. The conditions to take in account are:. M must be Hurwitz. This guarantees the convergence of harmonic j 666 z estimation.. +$ jωi M% ) K,+.- ω nω m n h j 6766 z This guarantees that all the harmonics different from the h j 666 z ones are greatly decreased. These conditions can be satisfied forcing complex conjugate eigenvalues for M, the peak frequencies at the harmonics that have to be estimated, and a very low damp. In particular, if all the harmonics from to h M have to be estimated, then the rejection must be ensured only for the harmonicigher than h M ; hence $ j ω I M% ) K can be shaped as a low pass filter that greatly attenuatearmonics higher than h M. In order to implement the proposed estimator on a DSP board, its time-discrete version must be investigated. All the considerations made about the continuous-time estimators can be repeated for the discrete-time ones. In particular, in the case of one d-q harmonic PR and NR components to be isolated, the discrete-time solution is ˆx k 8 A h ˆx k# K $ u k9 C h ˆx k:% (9)
; A h A rh A ch C h A rh A ch The considerations about the placement of the eigenvalues are the same seen for the continuous-time estimators. In particular, if there is only one d-q harmonic to divide into its PR and NR components, and the following characteristic polynomial is desired z cos ω n δ T s e) δω n T s z e) δω n T s< the same gain matrix K reported in () can be considered, k cos ω n δ δω T s e) n T s δω e) n T s k c k h If the harmonics to estimate are several, then the matrixes to consider have the same structure of (6), (7) and (8). 4 Simulation results In order to verify the performances of the estimator, the following simulation has been executed. Let consider the three-phase signal u '$ u α u β % T, u α t cos 4ω m t cos 5ω m t cos 7ω m t π 5 cos 8ω m t π 5 cos ω m t cos ω m t cos ω m t cos ω m t π 5 cos 4ω m t π 5 u β t( sin 4ω m t sin 5ω m t sin 7ω m t π 5 sin 8ω m t π 5 sin ω m t sin ω m t sin ω m t sin ω m t π 5 sin 4ω m t π 5 ω m π5 rad s. Assume that the goal is to estimate harmonics 5 7. The 5 th harmonic has NR sequency only, while the 7 th one has the PR one only. Hence both are mapped as a 6 th harmonic in the d-q reference frame. The th harmonic is completely NR, the th one is PR. Hence both are mapped as a th harmonic into the d-q reference frame. A continuous-time estimator having dynamic matrix (8) is considered. A C A r6 A c6 A r A c In order to ensure high rejection to the other harmonics, the matrix K is chosen to obtain for M the following characteristic polynomial: s δω n6 s ω n6 s δω n s ω n δ 6 ω n6 ω n 6ω m δ ω m δ All the estimator state variableave initial values equal to zero. Each of the resulting estimated harmonics is converted from d-q to α β coordinates and compared the original ones. The α component of the resulting estimation errors is presented in Fig.. The β component is similar and therefore is omitted.
PR, 6 th harmonic.5 magnitude rag replacements rag replacements -.5.5.5.5 -.5.5.5.5 -.5.5.5.5 NR, 6 th harmonic PR, th harmonic NR, th harmonic -.5.5.5.5 Figure : Estimation errors, α axis 4 4.97.975.98.985.99.995 4 5-5 - 5-5 - v ma i la.97.975.98.985.99.995 4 i lb.97.975.98.985.99.995 4 PSfrag replacements PSfrag replacements.5 4 6 8 4 6 8 4 5 5-5 - phase -5 4 6 8 4 6 8 4 Figure 4: FFT of current signal to be processed, phase a 4 - -4.966.968.97.97.974.976.978.98.98.984.986.5 -.5 -.966.968.97.97.974.976.978.98.98.984.986 v ma PR NR - -.966.968.97.97.974.976.978.98.98.984.986 Figure 5: harmonics estimated in simulation, d axis Figure : Phase voltage v ma and current signal to be processed In order to make comparisons among simulation and experimental results, the three-phase signal of Fig. is considered. In a fixed reference frame, this signal is characterized by the following harmonics: 5 7 7 9 666 the fundamental at f m 5 Hz. The goal is to estimate the 5 th and the 7 th harmonics, that are shown in Fig. 4. They both are shifted into 6 th harmonic on the d-q reference frame. The three-phase signal is sampled frequency f s 7 khz and then converted from the fixed a-b-c reference frame to the rotating d-q one. A time-discrete estimator having the structure of (9) is considered, h 6. The gain matrix K is chosen to obtain for M 6 a characteristical polynomial damp δ 6 and natural frequency ω n 6ω m δ. The d-components of the estimator outputs are shown in Fig. 5. They match the amplitudes in Fig. 4. In order to verify if also the phases are well estimated, they have been converted into a-b-c coordinates and subtracted from the original signals. The resultave 5 th and 7 th near to zero, therefore estimations are good. 5 Experimental results The current signal to be processed is the one described in the previous section and is shown in Fig.. The three-phase signals are acquired and sampled frequency f s 7kHz. The DSP on-line executes all the calculus to change reference frame from
rag replacements 4 - -4..4.6.8...4.6.8..5 -.5 -..4.6.8...4.6.8. v ma PR NR - -..4.6.8...4.6.8. Figure 6: harmonics estimated in laboratory a-b-c to d-q and implements a time-discrete estimator having the structure of (9), h 6. The d- components of estimated harmonics are shown in Fig. 6. They match the ones resulting from simulation. 6 Conclusions A positive and negative sequence on-line estimator has been presented. It allows to separate the positive from the negative sequence, that hence can be separately processed by the active power filter controller. Simulation and experimental resultave been presented. [4] L. Malesani, L. Rossetto, P. Tenti. Active filter power filter hybrid energy storage, IEEE Trans.Power Electron., volume 6, pp. 9 97, (July 997). [5] P. Mattavelli. A closed-loop selective harmonic compensation for active filters, IEEE Trans.Ind.Applicat., volume 7, pp. 8 89, (January/February ). [6] F. Ronchi, A.Tilli Design methodology for shunt active filters, Proc. th EPE-PEMC, (Cavtat & Dubrovnik, Croatia, September ), to be published,. [7] G. L. Van Harmelen, J.H.R. Enslin. Realtime dynamic control of dynamic power filters in supplies high contamination, IEEE Trans. Power Electron., volume 8, pp. 8, (July 99). [8] P. Verdelho, G. D. Marques. An active power filter and unbalanced current compensator, IEEE Trans.Ind.Applicat., volume 44, pp. 8, (June 997). References [] H. Akagi. New trends in active filters for power conditioning, IEEE Trans.Ind.Applicat., volume, pp., (Nov./Dec.996). [] H. Akagi, A. Nabae. Control strategy of active power filters using multiple voltage source PWM converters, IEEE Trans.Ind.Applicat., volume IA-, pp. 46 465, (May/June 986). [] C.L.Fortescue newblock Method of symmetrical coordinates applied to the solution of polyphase networks, Trans.AIEE volume 7, pp. 7 4, (98).