Powered by TCPDF (www.tcpdf.org) Thi i an electronic reprint of the original article. Thi reprint may differ from the original in pagination and typographic detail. Author(): Title: Kukkola, Jarno & Hinkkanen, Marko State oberver for grid-voltage enorle control of a grid-connected converter equipped with an LCL filter Year: 24 Verion: Pot print Pleae cite the original verion: Kukkola, Jarno & Hinkkanen, Marko. 24. State oberver for grid-voltage enorle control of a grid-connected converter equipped with an LCL filter. 6th European Conference on Power Electronic and Application (EPE'4-ECCE Europe). ISBN 978--4799-34-2 (electronic). DOI:.9/epe.24.6974. Right: 24 Intitute of Electrical & Electronic Engineer (IEEE). Peronal ue of thi material i permitted. Permiion from IEEE mut be obtained for all other ue, in any current or future media, including reprinting/republihing thi material for advertiing or promotional purpoe, creating new collective work, for reale or reditribution to erver or lit, or reue of any copyrighted component of thi work in other work. All material upplied via Aaltodoc i protected by copyright and other intellectual property right, and duplication or ale of all or part of any of the repoitory collection i not permitted, except that material may be duplicated by you for your reearch ue or educational purpoe in electronic or print form. You mut obtain permiion for any other ue. Electronic or print copie may not be offered, whether for ale or otherwie to anyone who i not an authoried uer.
State Oberver for Grid-Voltage Senorle Control of a Grid-Connected Converter Equipped With an LCL Filter Jarno Kukkola and Marko Hinkkanen Aalto Univerity School of Electrical Engineering Department of Electrical Engineering and Automation P.O. Box 3, FI-76 Aalto Epoo, Finland E-mail: jarno.kukkola@aalto.fi, marko.hinkkanen@aalto.fi Acknowledgment The author would like to thank ABB Oy, Walter Ahltröm Foundation, and Finnih Foundation for Technology Promotion for the financial upport. Keyword Adaptive oberver, grid-connected converter, grid-voltage enorle, enorle control, mall-ignal linearization Abtract Thi paper propoe a imple grid-voltage enorle alternative to the conventional PLL ynchronization for a grid-connected converter equipped with an LCL filter. The grid-voltage magnitude and angle are etimated baed only on the converter current and DC-voltage meaurement. Reduced number of enor decreae cot and amount of enor wiring. The analytically derived deign i validated experimentally.. Introduction Three-phae grid converter equipped with an AC-ide LCL filter are increaingly ued to connect variou renewable and ditributed energy ource a well a motor drive to the electric grid. Interet for uing LCL filter ha increaed, becaue they afford better grid-current quality, lower cot, and maller phyical ize in comparion with the L filter. The control cheme of thee converter i uually implemented in the grid-voltage or grid-flux ynchronou reference frame (dq), in the natural reference frame (abc), or in the tationary reference frame (αβ). Regardle of the reference frame, ynchronization with the grid i needed for intantaneou active and reactive power control. Conventionally, ynchronization i achieved uing ome variant of the phae-locked loop (PLL) to detect the angle of the meaured grid voltage. If the grid voltage i not meaured, the ynchronization can be baed on: ) control error (i c,ref i c ) of current control []; 2) DC-link power balance [2]; 3) intantaneou power theory [3]; 4) duality between the grid-connected converter and the PWM inverter-fed electric motor [4, 5]; 5) direct etimation [6, 7]; or 6) adaptive model-baed etimation [8]. The above-mentioned method have been developed for a grid-connected converter uing the conventional L filter, but in the cae of the reonant LCL filter, only a few [9 3] grid-voltage enorle ynchronization method exit. For the grid-voltage enorle ynchronization in the cae of the LCL filter, the Kalman filter ha been ued to directly detect the grid voltage together with LCL filter tate [9]. Intead of direct etimation, in [], the grid voltage ha been adapted uing the grid-current etimation error of the Kalman filter. However, the proce noie parameter for the Kalman filter, which determine the dynamic of the oberver, have to be determined by trial and error [9]. In [], power-theory-baed grid-voltage etimation
ha been propoed for the converter with an LCL filter, but intead of the grid voltage, the capacitor voltage of the LCL filter ha been etimated. In the cae of the grid converter equipped with an LC filter, a neural-network grid-voltage etimator ha been preented in [4], and a teepet-decent adaptation algorithm ha been ued for the grid-voltage etimation. Moreover, the erie reitance and inductance have been identified in parallel with the grid-voltage etimator in [4]. The parallel etimator provide the elf-tuning feature, but high expertie of modern control theory i needed when deigning and analyzing the neural-network etimator. Similar algorithm have been ued without neural network [2], whereupon the additional value of neural network with thee relatively imple model can be challenged. Furthermore, in the cae of LCL filter, the teepet-decent algorithm ha been ued with the adaptive oberver [2, 3] to directly etimate the grid voltage from the converter current etimation error of the Kalman filter [2], or to give an etimate of the aggregated uncertaintie, which include information about the grid voltage [3]. In thi paper, an etimation method for the grid-voltage magnitude and angle i propoed. The propoed method i baed on the full-order oberver [5], which i augmented with adaptation loop for the grid voltage magnitude and angle. Only the AC-ide converter current and the DC-ide voltage are meaured. In the adaptation loop, grid-voltage etimation i baed on regulating the error between the meaured and etimated converter current to zero uing proportional-integral (PI) controller. Grid-voltage and tate etimation reduce the number of enor, which provide cot advantage and decreae the amount of enor wiring. Analytical deign rule, enabling automatic tuning of the etimation method, are alo propoed. Firt, the ytem model and the oberver are defined. Then, adaptation loop are formulated and their dynamic are analyzed by mean of mall-ignal linearization. The expreion for the oberver and adaptation gain are given a function of ytem parameter and deired dynamic. Finally, imulation and experimental reult are preented. 2. Model The equivalent circuit model of the LCL filter model, between converter and grid, i preented in Fig.. The grid voltage in the tationary reference frame i u g = e jϑg u g, () where ϑ g = ω g dt i the angle, ω g i the angular frequency, and u g i the amplitude of the grid voltage. When the converter-ide current i c, the capacitor voltage u f, and the grid-ide current i g of the LCL filter are elected a tate variable, the LCL filter dynamic in a reference frame rotating at the angular frequency of ˆω g are d dt i c u f i g }{{} x = jˆω g L fc C f jˆω g C f jˆω g L fg } {{ } A i c u f i g L fc + u c + }{{} B c L fg } {{ } B g u g, i c = [ ] x, }{{} C c where u c and u g are the converter and grid voltage, repectively, and L fc, C f, and L fg are the LCL filter parameter. The loe of the filter are neglected, which repreent the wort-cae ituation for the (2) i L c fc L i fg g u c C f u f u g Fig. : The grid-connected LCL filter. The upercript refer to the tationary reference frame, which i ued for the ake of implicity.
u d u d,ref DC-voltage control q ref p ref Current reference calculation i c,ref Current control u c,ref e j( ˆϑ g+ϑ d ) SVPWM û g ˆx Adaptive oberver ˆϑg i c e j ˆϑ g i c Gate ignal Etimated grid-voltage reference frame Stationary reference frame p g, q g LCL Fig. 2: Block diagram of the control ytem. reonance of the LCL filter. Switching-cycle-averaged complex-valued pace vector are ued (e.g., the converter current i c = i cd + ji cq ). Complex, matrix, and vector quantitie are marked with boldface ymbol. 3. Adaptive Oberver The block diagram of the control ytem i hown in Fig. 2. The power balance i regulated with a DC-voltage controller together with an inner tate-pace current controller [5]. The full-order oberver etimate the tate variable x = [i c u f i g ] T for the tate-pace controller. The oberver gain vector L = [l l 2 l 3 ] T can be expreed by deired dynamic behavior and ytem parameter, a in [5], contrary to the Kalman filter whereby the knowledge of the noie covariance i needed [9, 2]. The ynchronou reference frame of the control ytem i tied to the etimated grid voltage, i.e., û g = û g + j, and it i rotating at etimated grid-voltage angular frequency of ˆω g. The magnitude and angle of the grid voltage are etimated uing an adaptation loop, explained later. In the elected reference frame, the full-order oberver i ˆx = ˆx + ˆB c u c + ˆB g û g + L(i c îc), î c = C cˆx, (3) where the etimated quantitie are marked with ˆ, and Â, ˆBc and ˆB g conit of nominal (etimated) ytem parameter ˆL fc, Ĉf, and ˆL fg. If accurate circuit parameter are aumed, i.e., ˆB c = B c, ˆB g = B g, and  = A, the dynamic of the etimation error e = x ˆx are obtained from (2) and (3) a ė = (A LC c )e + B g (e j(ϑg ˆϑ g) u g û g ), (4) where ˆϑ g i etimated angle of the grid voltage. A can be een from the equation, the etimation error dynamic are nonlinear and are affected by the angle difference ϑ g ˆϑ g between the reference frame of the grid voltage and the oberver and the difference u g û g between the grid-voltage magnitude and it etimate. In the following, the error dynamic are linearized in order to examine thee relationhip more cloely. 3.. Linearization The nonlinear dynamic of (4) can be linearized uing mall-ignal approach. The operation point quantitie are marked by ubcript and mall-ignal deviation by, e.g. ˆω g = ˆω g + ˆω g. Accurate circuit parameter etimate and grid-voltage etimation are aumed. Hence, e =, ω g = ˆω g, ϑ g = ˆϑ g,
and û g = u g hold in the operation point. Then, the linearized etimation error dynamic are dẽ dt = (A L C c )ẽ + B g [ e j( ϑ g ˆϑ g) }{{} (u g + ũ g ) û g û g ] +j( ϑ g ˆϑg) (5) = (A L C c )ẽ + B g [ũ g û g + j( ϑ g ˆϑg )u g ]. Further, if the grid voltage, the angular peed, and the angle error are denoted by ũ = ũ g û g, ω = ω g ˆω g, and ϑ = ϑ g ˆϑg, repectively, the etimation error dynamic are [ẽ ] [ ] [ẽ ] [ ] [ ] d A L = C c jb g u g Bg + ũ + ω dt ϑ ϑ }{{}}{{}}{{} A L B u B ω The converter current i the only meaured tate variable. Therefore, let u examine the effect of the etimation error of the grid voltage and angular peed on the etimation error of the converter current ĩ = ĩc î c. From (6), the tranfer function from the grid-voltage error to the current error i ĩ() ũ() = C c (I A L ) B u = P () where C c = [C c ], and ( P () = C f L fc L fg [ 3 + (3jω g + l ) 2 + 3ωg 2 + L fc + L fg + 2jω g l l ) 2 C f L fc L fg L fc jωg 3 ωgl 2 jω ] g l 2 + jω g (L fc + L fg ) + L fc l + L fg l 3. L fc Similarly, the tranfer function from the angular-peed error to the current error i ju g ĩ() ω() = C c (I A L ) B ω = P () and the tranfer function from the angle error to the current error i obtained from (9) uing the relationhip ϑ g = ω g dt, leading to ϑ = ω/. It i to be noted that the oberver gain (l, l 2, and l 3 ) trongly affect the characteritic polynomial of the tranfer function. Further, thi i een in the orientation of the current error ĩ in the complex plane. 3.2. Selection of the Full-Order Oberver Gain The direct pole placement i a traightforward method for the election of the full-order oberver gain. Thi can be done auming firt the perfect grid-voltage orientation and known or meaured grid voltage. Then, the characteritic polynomial of the etimator error dynamic (4) i et to correpond the thirdorder dynamic a follow det(i A + LC c ) = ( + α o )( 2 + 2ζ o2 ω o2 + ω 2 o2), () where α o determine the firt-order pole and ζ o2 and ω o2 the econd-order pole pair of the etimator dynamic. Thi give the gain l = α o + 2ζ o2 ω o2 3jω g l 2 = L fc ( 2α o ζ o2 ω o2 + ω 2 o2 + 3ω 2 g L fc + L fg L fc L fg C f 2jω g l ) l 3 = α o ω 2 o2c f L fc + jω g ( ω 2 gc f L fc L fc L fg ) ( + ωgc 2 f L fc L ) fc l + jω g C f l 2. L fg (6) (7) (8) (9) (a) (b) (c) The oberver gain are function of the ytem parameter, the grid angular frequency at the operating point and the deired pole of the etimation error dynamic. The pole can be een a tuning parameter and their election can be baed on the open-loop ytem, a decribed in [5].
3.3. Adaptation Law Since the grid voltage i not meaured, the oberver in (3) i upplemented with the adaptation of the magnitude and angle of the grid voltage. Here, quai-teady-tate analyi i ued to deign the adaptation loop. The etimation error dynamic of the full-order oberver in (6) are conidered to be much fater than the adaptation mechanim of the grid voltage and it angle. The different time cale enable conidering the current etimation error ĩ = ĩc î c being in teady tate from the viewpoint of the adaptation. If the gain () are inerted in (7), the tranfer function reduce to G iu () = ĩ() ũ() = C f L fc L fg ( + α o )( 2 + 2ζ o2 ω o2 + ω 2 o2 ) (2) Similarly, if the gain () are inerted in (9), the tranfer function become ju g G iω () = ĩ() ω() = C f L fc L fg ( + α o )( 2 + 2ζ o2 ω o2 + ωo2 2 ) (3) Auming thee tranfer function to be in teady tate (i.e., = ) lead to u g ĩ = C f L fc L fg α o ωo2 2 ũ j ϑ. C f L fc L fg α o ωo2 2 (4) Thi equation how that the d component ĩ d of the etimation error i affected by the grid-voltage etimation error ũ and the q component ĩ q i affected by the angle error ( ϑ = ω/). Thee obervation point out that the etimation of the grid voltage and it angular peed can be baed on regulating the error between the converter current and etimated current to be zero. Hence, the adaptation law for the grid voltage and it angular peed can be contructed a follow û g = k pu ĩ d + k iu ĩ d dt, ˆω g = k pω ĩ q + k iω ĩ q dt (5) where k pu, k iu, k pω, and k iω are the gain of the adaptation-loop controller. It i worth noting that the election of the oberver gain ha ignificant effect on the formulation of the adaptation law. If zero oberver gain are inerted in (7) and (9), the current component for the adaptation loop hould be elected oppoite. Thi can be eaily hown with a imilar quai-teady-tate analyi, which give u g ĩ = ω g (L fc + L fg ωg 2 C fl fc L fg ) ϑ + j ω g (L fc + L fg ωg 2 C fl fc L fg )ũ (6) in the cae of zero oberver gain. The d component ĩ d of the converter current error i affected by the angle error ϑ and the q component ĩ q by the voltage error, which i oppoite to (4). 3.4. Adaptation-Loop Gain Fig. 3(a) and 3(b) how the linearized adaptation loop of the grid voltage and it angular peed. The gain of the adaptation-loop controller can be elected to give the approximate bandwidth of α u for the etimation dynamic of the amplitude of the grid voltage and the natural frequency of ω ω for the etimation dynamic of the angular frequency of the grid voltage. Conidering the etimation error of the converter current in teady tate a in (4) and electing k pu = and k iu = α u C f L fc L fg α o ω 2 o2, (7) in (5), the cloed-loop tranfer function of the linearized grid-voltage adaptation loop i G u () = û g () ũ g () = α u + α u (8)
with the bandwidth of α u. Similarly, uing the teady-tate approximation (4), the gain for the adaptation law of the angular peed in (5) can be elected a k pω = 2ζ ω ω ω C f L fc L fg α o ω 2 o2/u g and k iω = ω 2 ω C f L fc L fg α o ω 2 o2/u g. (9) Then, the cloed-loop tranfer function of the approximated and linearized adaptation loop of the angular peed i G ω () = ˆω g () ω g () = 2ζ ωω ω + ωω 2 2 + 2ζ ω ω ω + ωω 2 with the natural frequency of ω ω and damping of ζ ω. (2) 3.5. Small-Signal Stability The quai-teady-tate approximation can be ued to give imple tuning for adaptation loop. However, if the bandwidth of the adaptation loop are increaed cloe to the bandwidth of the full-order oberver, the quai-teady-tate aumption i not valid anymore and the tability of the loop can be lot. In the following, the mall-ignal tability of the adaptation loop i analyzed. With the elected tuning, the cloed-loop tranfer function of the adaptation loop of the grid voltage magnitude i ( ) Re{G iu ()} k pu + k iu k G u () = ( ) iu = + Re{G iu ()} k pu + k iu / Re{G iu ()} + k iu (2) α u α o ωo2 2 = 4 + (α o + 2ζ o2 ω o2 ) 3 + (2α o ζ o2 ω o2 + ωo2 2 )2 + α o ωo2 2 + α u α o ωo2 2. Since the degree of characteritic polynomial i four, analytical calculation of the pole become difficult. However, the tability can be teted uing the Routh-Hurwitz tability criterion [6]. Thi give neceary and ufficient condition for the tability of the ytem. A an example, the polynomial P () = a 4 4 + a 3 3 + a 2 2 + a + a, ha the root with negative real part and i thu table (Hurwitz polynomial), if the following condition are fulfilled: a i > for all i =... 4, a 2 a 3 > a a 4, and a a 2 a 3 > a a 2 3 +a2 a 4. According to polynomial of the example, the tability criterion for the magnitude adaptation i: all the tuning parameter α o, ω o2, ζ o2, and α u (i.e., deired pole of the oberver and the adaptation loop) mut be poitive and it i neceary that α u < (a 2 a 3 a )/a 2 3, (22) ũ g ω g ũ ω G iu () G iω () ĩ ĩ Re{ } (a) Im{ } k pu + k iu k pω + k iω ûg ˆωg ˆϑ g Damping ratio ζω.8.6.4.2 Stable area 2 3 4 5 Natural frequency ω ω /(2π) (Hz) (b) (c) Fig. 3: (a) Block diagram of the linearized adaptation loop of grid voltage etimation; (b) Block diagram of the linearized adaptation loop of the grid angle etimation. The tranfer function G iu () and G iω () are obtained from (6).; (c) Region of the table operation of the angle adaptation loop when the tuning of the full-order oberver i: α o = 2π rad/, ω o2 = 2π 42 rad/, and ζ o2 =.5. With the elected tuning, the table area i limited by the boundary condition of (25).
where a 3 = α o + 2ζ o2 ω o2, a 2 = 2α o ζ o2 ω o2 + ωo2 2 and a = α o ωo2 2. The equation determine the upper limit for the tuning parameter α u, which i the approximate bandwidth for the etimation of the grid-voltage magnitude. A an example, if α o = 2π rad/, ω o2 = 2π 42 rad/ and ζ o2 =.5 are elected, the condition α u < 2π 75 rad/ mut hold. When the elected tuning i ued, the cloed-loop tranfer function of the angle adaptation loop i ( ) Im{G iω ()} k pω + k iω G ω () = ( ) + Im{G iω ()} k pω + k iω (23) = 2ζ ω ω ω α o ω 2 o2 + ω2 ω α o ω 2 o2 2 ( + α o )( 2 + 2ζ o2 ω o2 + ω 2 o2 ) + 2ζ ωω ω α o ω 2 o2 + ω2 ω α o ω 2 o2 The Routh-Hurwitz tability criterion for thi ytem i: all the tuning parameter α o, ω o2, ζ o2, ω ω, and ζ ω mut be poitive and it i neceary that ω 2 ω 2e ζ ω ω ω + f e > (24) and ω 3 ω + 4e ζ ω ω 2 ω ω ω ( f 2 α o ω 2 o2 e + 4ζ2 ω e 2 + f e ) + 2ζ ω f >, (25) where e = α o + 2ζ o2 ω o2 and f = 2ζ o2 ω o2 (ω 2 o2 + α2 o + 2α oζ o2 ω o2 ). The boundary condition of the equation (25) i illutrated in Fig. 3(c). It i to be noted that the derived tability condition determine the local tability of the nonlinear etimation error dynamic in ideal condition. In the true ytem, error due to the dicretization and parameter uncertaintie naturally hrink the table operation area. However, the preented condition can be een a an uppermot limit for the deign and they can be ued to perceive the limitation of the tuning. 4. Control Sytem The block diagram of the control ytem compriing cacaded DC-voltage and current control loop i hown in Fig. 2. State-pace current control [5] i implemented in the etimated grid-voltage reference frame. The current controller and the oberver are dicretized uing Tutin method. The PI regulator of the adaptation loop and DC-voltage control are dicretized uing the forward Euler method. The component of the current reference i c,ref = i cd,ref + ji cq,ref are calculated in the teady tate. From (2), the grid current i i g = (i c jˆω g C f u g )/( ˆω 2 gc f L fg ). The complex power at the point of common coupling i g = 3/2 u g i g = p g + jq g. Auming i c = i c,ref, the current reference become i cd,ref = 2 3 ˆω gc 2 f L fg p ref, i cq,ref = 2 ˆω gc 2 f L fg q ref + ˆω g C f û g, (26) û g 3 û g where the peak-value caling of the pace vector i ued, and p ref and q ref are the reference for the active and reactive power, repectively. The active power reference i the output of DC-voltage control, which i implemented indirectly by mean of DC-capacitor energy control [7,8]. A pure PI-controller i ued for the capacitor energy control, a in [7], but the tuning i baed on the parameter of the ytem, a in [8]. The pace-vector pule-width modulation (SVPWM) i ued and a imple current feedforward dead-time compenation i added [9]. The effect of the delay on the ynchronou-to-tationary reference frame tranformation i compenated for by modifying the tranformation e j ˆϑ g e j( ˆϑ g+ϑ d ), where ϑ d = (3/2) ˆω g T a decribed in [2]. It i worth noticing that the tuning of the control ytem can be automated, if the equivalent circuit parameter of the ytem are known or etimated. All the oberver gain (), (7), and (9) are function
Table I: Sytem parameter. Parameter Value Parameter Value u g 2/3 4 V ( p.u.) ωg 2π 5 rad/ i N 2 8 A ( p.u.) Cf µf (.4 p.u.) L fc 2.94 mh (.72 p.u.) L fg.96 mh (.48 p.u.) f w 6 khz T /(2f w ) of the equivalent circuit parameter and deired dynamic performance pecification. Either nominal or etimated value could be ued for the operation-point grid voltage in the gain expreion (nominal value are ued in thi paper). A the input value for the tuning, the dynamic performance pecification mut be determined: ) deired bandwidth and the reonance damping of the LCL filter for the tate-pace controller, 2) convergence dynamic of the etimation error for the full-order oberver, 3) approximate bandwidth for the controller in the adaptation loop. The robutne of the control ytem depend on the elected performance pecification. The more robut ytem i deired, the lower dynamic performance hould be pecified. In thi paper, the following performance pecification are ued. The current controller i tuned to give an approximate bandwidth of 2π 5 rad/ a in [5]. The dominant dynamic of the oberver i et twice a fat a the current control, i.e, α o = 2π rad/ and the econd-order pole of the oberver ω o2 i et to correpond to the reonance frequency ω p = 2π 47 rad/ of the LCL filter in the ynchronou reference frame, i.e., ω o2 = ω p ω g. The damping of thi pole pair i et to ζ o2 =.5. The tuning of the adaptation loop i et, with a good damping, approximately one decade below the theoretical limit obtained from (22), (24) and (25). Thu, the tuning parameter for the adaptation loop are: α u = 2π rad/, ω ω = 2π 5 rad/ and ζ ω =.9. The DC-voltage regulator bandwidth [8] i et approximately one decade below the current controller bandwidth. The ytem parameter are given in Table I. 5. Experimental Reult The propoed grid-voltage enorle cheme wa experimentally teted uing a 2.5-kVA, 4-V, gridconnected converter equipped with an LCL filter. The control of the converter wa implemented on dspace DS6, DS22, and DS522 board. The witching frequency of the converter wa f w = 6 khz, and ynchronou ampling, twice per carrier, wa ued. The DC-link voltage wa regulated to 65 V and the converter under tet wa loaded with another back-to-back connected converter (2.5 kva, 4 V) with an LCL filter. An iolation tranformer wa ued in the grid connection of the loading converter. Fig. 4(a) illutrate the operation of the converter when an impule diturbance of 6 wa applied into the etimated grid-voltage angle ˆϑ g inide the control ytem. The converter wa upplying a power of Angle (deg) 8 9 9 8 ˆϑ g ϑ g Angle error (deg) 6 4 2 2 Meaured ϑ g ˆϑ g Small-ignal imulation..2.3.4 Time ()..2.3.4 Time () (a) (b) Fig. 4: Experimental reult when a tep diturbance in the angle etimate of the grid voltage i applied: (a) etimated and meaured angle, ˆϑ g and ϑ g, repectively; (b) imulated and meaured angle error.
Grid voltage (V) 5 4 3 2 û g u g Voltage error (V) 4 3 2 Meaured u g û g Small-ignal imulation..2.3.4 Time () (a)..2.3.4 Time () (b) Fig. 5: Experimental reult when a tep diturbance in the amplitude etimate of the grid voltage i applied: (a) etimated and meaured amplitude of the grid voltage; (b) imulated and meaured amplitude error. Power, current (p.u.).2.8.4.4 p g i ga.8 i gb i gc.2..2.3.4.5.6 Time () (a) Power (p.u.).8.6.4.2 q g,ref q g p g.5..5.2 Time () (b) Fig. 6: Experimental reult: (a) Operation with both power direction. Phae current and the power p g are meaured from the grid-ide of the LCL filter; (b) Waveform of the active power p g and reactive power q g injected to the grid at the grid-ide of the LCL filter, when a tep of 5 kvar (.4 p.u.) i applied into the reactive power reference q g,ref. A 2-m period i hown. 5 kw (.4 p.u.) to the grid. Fig. 4(b) compare the meaured and imulated waveform of the etimated grid-voltage angle error during the diturbance. In the imulation, an angular peed impule correponding a diturbance of 6 in the angle etimate wa applied in ω g in the mall-ignal model hown in Fig. 3(b). A can be een from the Fig. 4(b), the agreement with the mall-ignal imulation and the meaurement i good. Meaured dynamic behavior of the etimation of the grid voltage amplitude i hown in Fig. 5(a), when a diturbance of 3 V wa applied in the amplitude etimate. The converter wa upplying the power of 5 kw. Experimental reult are compared with the mall-ignal imulation in Fig. 5(b). In the imulation, tep of 3 V wa applied in the ignal ũ g in the mall-ignal model hown in Fig. 3(a). The reult of Fig. 5(b) indicate good agreement with the mall-ignal imulation and the meaurement. Fig. 6(a) demontrate the ability to operate with both power direction. The loading condition of the converter under tet wa changed from kw to kw by changing the direction of the active power of the load connected to the DC bu. Further, dynamic performance of the current control i illutrated in Fig. 6(b), when a tep of 5 kvar wa applied into the reference of the reactive power q g. At the ame time, the converter wa upplying the active power of p g 5 kw. A Fig. 6(b) how, the reonance of the LCL filter i effectively damped and the reactive power rapidly follow it reference according to the deigned dynamic of the current control. 6. Concluion Thi paper preent an adaptive full-order oberver for the grid-voltage enorle control of a gridconnected converter equipped with an LCL filter. Converter-ide current of the LCL filter and DC voltage are only meaured variable, and the full-order oberver i ued for tate etimation. Further, the magnitude and angle of the grid voltage are etimated uing the converter current etimation error a an
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