Modeling and Analysis of Harmonic Stability in an AC Power-Electronics-Based Power System

Transcription

1 Aalborg Univeritet Modeling and Analyi of Haronic Stability in an AC Power-Electronic-Baed Power Syte Wang, Xiongfei; Blaabjerg, Frede; Wu, Weiin Publihed in: I E E E Tranaction on Power Electronic DOI (link to publication fro Publiher): 1.119/TPEL Publication date: 214 Docuent Verion Early verion, alo known a pre-print Link to publication fro Aalborg Univerity Citation for publihed verion (APA): Wang, X., Blaabjerg, F., & Wu, W. (214). Modeling and Analyi of Haronic Stability in an AC Power- Electronic-Baed Power Syte. I E E E Tranaction on Power Electronic, 29(12), General right Copyright and oral right for the publication ade acceible in the public portal are retained by the author and/or other copyright owner and it i a condition of acceing publication that uer recognie and abide by the legal requireent aociated with thee right.? Uer ay download and print one copy of any publication fro the public portal for the purpoe of private tudy or reearch.? You ay not further ditribute the aterial or ue it for any profit-aking activity or coercial gain? You ay freely ditribute the URL identifying the publication in the public portal? Take down policy If you believe that thi docuent breache copyright pleae contact u at vbn@aub.aau.dk providing detail, and we will reove acce to the work iediately and invetigate your clai.

2 Modeling and Analyi of Haronic Stability in an AC Power-Electronic-Baed Power Syte Xiongfei Wang, Meber, IEEE, Frede Blaabjerg, Fellow, IEEE, and Weiin Wu Abtract Thi paper addree the haronic tability caued by the interaction aong the wideband control of power converter and paive coponent in an AC power-electronicbaed power yte. The ipedance-baed analytical approach i eployed and expanded to a ehed and balanced threephae network which i doinated by ultiple current- and voltage- controlled inverter with LCL- and LC-filter. A ethod of deriving the ipedance ratio for different inverter i propoed by ean of the nodal adittance atrix, and thu the contribution of each inverter to the haronic tability of the overall power yte can be eaily predicted through Nyquit diagra. Tie-doain iulation and experiental tet on a three-inverter-baed power yte are preented. The reult validate the effectivene of the theoretical approach. Index Ter Current-controlled inverter, voltage-controlled inverter, ipedance-baed analyi, haronic tability, powerelectronic-baed power yte I. INTRODUCTION The proportion of power electronic apparatu in electric power yte keep growing in recent year, driven by the rapid developent of renewable power ource and variablepeed drive [1]. A a conequence, power-electronic-baed power yte are becoing iportant coponent of electrical grid, uch a renewable power plant [2], [3], icrogrid [4], and electric railway yte [5]. Thee yte poe uperior feature to build the odern power grid, including the full controllability, the utainability, and the iproved efficiency, but bring alo new challenge. Highorder haronic tend to be aggravated by the high-frequency witching operation of power converter, which ay trigger the parallel and erie reonance in the power yte [6]. The interaction of the wideband control yte for power converter with each other and with paive coponent ay anifet intability phenoena of a power-electronic-baed power yte in the different frequency range [2]-[5]. Continuou reearch effort have been ade to invetigate the intability in AC power-electronic-baed power yte. Manucript received Septeber 8, 213; revied Deceber 17, 213; accepted February 8, 214. Thi work wa upported by European Reearch Council (ERC) under the European Union Seventh Fraework Progra (FP/27-213)/ERC Grant Agreeent n. [ Harony]. X. Wang and F. Blaabjerg are with the Departent of Energy Technology, Aalborg Univerity, 922 Aalborg Eat, Denark (e-ail: xwa@et.aau.dk; fbl@et.aau.dk; li@et.aau.dk; zch@et.aau.dk). W. Wu i with the Departent of Electrical Engineering, Shanghai Maritie Univerity, Shanghai, China (e-ail: wwu@cle.htu.edu.cn) However, any of the tudie focu on the low-frequency ocillation caued by the contant power control for converter a contant power load [7]-[9] or contant power generator [1], [11], the Phae-Locked Loop (PLL) for gridconnected converter [12]-[14], and the droop-baed power control for converter in ilanded icrogrid [15]-[17]. Apart fro uch ocillation aociated with the outer power control and grid ynchronization loop, the interaction of the fat inner current or voltage control loop ay alo reult in haronic intability phenoena, naely haronic-frequency ocillation (typically fro hundred of Hz to everal khz), due to the inductive or capacitive behavior of converter in thi frequency range [18]-[22]. Further, thi haronic intability ay alo be generated or agnified by the control of converter in interaction with haronic reonance condition introduced by the high-order power filter for converter and paraitic capacitor of power cable [23]-[26]. Such phenoena have been frequently reported in renewable energy yte and high-peed railway [2], [3], [5], [26]-[3], and are challenging the yte tability and power quality. It i therefore iportant to develop the effective odeling and analyi approach for the haronic tability proble in AC power-electronic-baed power yte. A general analytical approach for the haronic tability proble i to build the tate-pace odel of the power yte, and identify the ocillatory ode baed on the eigenvalue and eigenvector of the tate atrix [31]. However, unlike conventional power yte where the dynaic are ainly deterined by the rotating achine, the all tie contant of power converter require the detailed odel of load and network dynaic in power-electronic-baed power yte [7]. Thu, the forulation of the yte atrice ay becoe coplicated, and the virtual reitor are uually needed to avoid the ill-conditioned proble [32]. To overcoe thee liit, the Coponent Connection Method (CCM) i introduced for the tability analyi of AC power yte including the High-Voltage Direct Current (HVDC) traniion line [33]. The CCM i baically a particular for of tate-pace odel, where the power yte coponent and network dynaic are eparately odeled by a et of two vector-atrix equation. Thi reult in the parity of the tate equation and reduce the coputation burden of forulating the yte tranfer atrice. Further, the coponent interaction and the critical yte paraeter for the different ocillatory ode can be ore eaily deterined [34], [35].

3 Apart fro the tate-pace analyi, the ipedance-baed approach, which i originally introduced for the deign of input filter in DC-DC converter [36], provide another attractive way to analyze the haronic tability proble. Siilarly to the CCM, the ipedance-baed analyi i alo built upon the odel of coponent or ubyte. However, intead of yteatically analyzing the eigenpropertie of the tate atrix a in the CCM and other tate-pace odel, the ipedance-baed approach locally predict the yte tability at each Point of Connection (PoC) of coponent, baed on the ratio of the output ipedance of coponent and the equivalent yte ipedance [37]. Thu, the forulation of the yte atrice can be avoided, and the contribution of each coponent to the yte tability can be readily aeed in the frequency-doain. Alo, the output ipedance of coponent or ubyte can be accordingly rehaped to tabilize the overall power yte [38]. Therefore, the ipedance-baed ethod provide a ore traightforward and deign-oriented tability analyi copared to the CCM and other tate-pace odel. Several application of the ipedance-baed approach for the haronic tability analyi can be found in AC power-electronic-baed power yte, e.g. the cacaded our-load inverter yte [39], the parallel grid-connected converter with LCL-filter [22], and the parallel uninterruptible power upply inverter with LC-filter [25]. However, in all thee cae, the network dynaic are often overlooked which affect the derivation of the equivalent yte ipedance, and few of the have conidered the yte with ultiple voltage- and current-controlled converter. Thi paper attept to fill in thi gap by expanding the ipedance-baed analyi to a three-phae ehed and balanced power yte, where a voltage-controlled and two current-controlled inverter with LC- and LCL-filter are interconnected. The haronic intability that reult fro the interaction aong the inner voltage and current control loop of thee inverter and paive coponent i tudied, while the dynaic ipact of the outer power control and grid ynchronization loop are neglected by deigning the control bandwidth lower than the yte fundaental frequency. By the help of the nodal adittance atrix, a iple ethod of deriving the ipedance ratio at the PoC of inverter i propoed to ae how each inverter contribute to the haronic intability of the power yte. The theoretical analyi in the frequency-doain i perfored baed on the linearized odel of inverter with inner control loop. The reult are further validated by the nonlinear tie-doain iulation and experiental tet on a three-inverter-baed power yte. II. SYSTEM MODELING AND ANALYSIS TECHNIQUES Thi ection firt decribe the tructure of the built powerelectronic-baed power yte in thi work, and then review the CCM and the ipedance-baed approach for odeling and LCL-filter Current-controlled inverter 3 Cable 3 LC-filter Cable 1 Fig. 1. Siplified one-line diagra of a three-phae power-electronic-baed AC power yte. analyi of haronic tability in the power yte. A. Syte Decription Fig. 1 how a iplified one-line diagra for the balanced three-phae power-electronic-baed power yte which i conidered in thi work, where a voltage-controlled and two current-controlled inverter are interconnected a a ehed power network through power cable. The voltage-controlled inverter regulate the yte frequency and voltage aplitude. The current-controlled inverter operate with unity power factor. In uch a yte, the preence of hunt capacitor in the LC- and LCL-filter of inverter and power cable bring in reonant frequencie, which ay interact with the inner voltage and current control loop of voltage- and currentcontrolled inverter reulting in the haronic-frequency ocillation and unexpected haronic ditortion. On the other hand, the dynaic interaction between the inner control loop of inverter ay alo trigger the exiting reonant frequencie in the power yte. Thi conequently neceitate the ue of CCM or ipedance-baed analyi to reveal how inverter interact with each other and with the haronic reonance condition in the yte. Since thi work i concerned with the haronic intability owing to the dynaic of inner control loop, the DC-link voltage of inverter are aued to be contant. Alo, the grid ynchronization loop for the current-controlled inverter i deigned with the bandwidth lower than yte fundaental frequency, thu only the ubynchronou ocillation ay be induced by grid ynchronization [12]-[14]. Under thee auption, only the inner voltage and current control loop are odeled in thi work. The previou tudie have hown that the inner control loop theelve can be iply odeled by the ingle-input and ingle-output tranfer function in the tationary αβ-frae [2], [25]-[27], [39]-[43]. B. CCM Fig. 2 how the block diagra of the CCM applied for the built power yte, where the CCM decopoe the overall yte into three ubyte by inverter and the connection Voltage-controlled inverter 1 LCL-filter Current-controlled inverter 2

4 Fig. 2. Block diagra of the CCM applied for the built power yte. network. Two current-controlled inverter are odeled by the Norton equivalent circuit [22], while the voltage-controlled inverter i repreented by the Thevenin equivalent circuit [23]. Conequently, the copoite inverter odel can be derived in the following y() G ( ) u() G ( ) d() (1) cl where y() and d() are the output vector and diturbance vector of inverter, repectively. cd y () [ V(), i (), i ()] T 1 g2 g3 (2) u () [ V(), i (), i ()] (3) T 1 g2 g3 G cl () and G cd () denote the cloed-loop reference-to-output and the cloed-loop diturbance-to-output tranfer atrice, repectively, which depict the unterinated dynaic behavior of inverter een fro the PoC and can be given by G ( ) diag[ G ( ), G ( ), G ( )] (4) cl clv,1 cli,2 cli,3 G ( ) diag[ Z ( ), Y ( ), Y ( )] (5) cd ov,1 oi,2 oi,3 where G clv,1 (), G cli,2 (), and G cli,3 () are the voltage and current reference-to-output tranfer function of voltage- and currentcontrolled inverter, repectively. Z ov,1 (), Y oi,2 (), and Y oi,3 () are the cloed-loop output ipedance and adittance of the voltage- and current-controlled inverter, repectively. The dynaic of the connection work can be repreented by the tranfer atrix G nw () a d () G ( ) y () (6) nw Thu, the cloed-loop repone of the overall power yte can be derived a 1 y () I+G ( )G ( ) G ( ) u () (7) nw cd cl where the tranfer atrix [I + G nw ()G cd ()] -1 predict the haronic intability of the power yte, provided that the unterinated behavior of inverter G cl () are table. In addition to the eigenvalue technique, the tranfer atrix can alo be evaluated in the ultivariable frequency-doain by ean of the generalized Nyquit tability criterion [44], [45]. It i obviou that the ain uperior feature of the CCM copared to the other tate-pace odel i to decopoe the power yte into ultiple decentralized feedback loop by inverter, and thu the effect of inverter controller and the aociated phyical coponent on the yte ocillatory ode i ore intuitively revealed. Further, the decentralized tabilizing control loop ay be developed by ean of the CCM [46]. C. Ipedance-Baed Approach Fig. 3 depict the equivalent circuit of the voltage- and current-controlled inverter applied for the ipedance-baed analyi. It i intereting to note that thi approach alo eparately odel the internal dynaic of inverter by ean of the output ipedance and adittance. However, differently fro the CCM, there i no need of tacking the reference-tooutput and diturbance-to-output tranfer function of inverter a tranfer atrice in the ipedance-baed approach [38]. Intead, the ipact of a given inverter on the overall yte tability i deterined by a inor feedback loop copoed by the ratio of the inverter output ipedance or adittance, Z ov,1 or Y oi, and the equivalent ipedance or adittance for the ret of the yte, Z lv,1 or Y li, [4]. Further, due to the calar type odel for inner control loop of inverter, the ipedance ratio can alo be depicted by ingleinput and ingle-output tranfer function [39], which ignificantly iplify the haronic tability analyi copared to the CCM and other tate-pace odel. Hence, the ipedance-baed approach i preferred in thi work. Fro Fig. 3, the cloed-loop tranfer function of inverter can be derived a follow V1 () 1 G clv,1() V1 () Zov,1() 1 Z () lv,1 ig() 1 G cli, () i () Yoi, () g 1 Y () li, It i clear that if the unterinated dynaic behavior of inverter, G clv,1 and G cli, are table, the tability of voltage and current at the PoC of inverter will be dependent on the inor feedback loop copoed by the following ipedance ratio ov,1 oi, c lv,1 li, (8) (9) Z () Y () Tv(), T () (1) Z () Y () which are alo tered a the inor feedback loop gain [4]. Baed on thee inor feedback loop gain, the pecification

5 (a) Fig. 4. Voltage-controlled inverter with the ultiloop control chee. (b) Fig. 3. Ipedance-baed equivalent odel for (a) the voltage- and (b) current-controlled inverter. of inverter output ipedance to preerve the yte tability can be derived. III. MODELING OF INVERTERS A. Voltage-Controlled Inverter Fig. 4 depict the iplified one-line diagra of voltagecontrolled inverter and the ultiloop voltage control chee. The control yte i ipleented in the tationary frae, including the inner Proportional (P) current controller and outer Proportional Reonant (PR) voltage controller. It i worth entioning that the three-phae inverter without neutral wire can be tranfored into two independent inglephae yte in the tationary αβ-frae [42]. Further, due to the auption of the contant DC-link voltage and balanced three-phae operation, the voltage-controlled inverter can be linearized baed on the LC-filter and odeled a a real calar yte by ingle-input and ingle-output tranfer function [23]-[25], [42], [43]. Fig. 5 how the block diagra of the ultiloop voltage control yte, where the following two tranfer function are ued to decribe the effect of the inverter output voltage V PWM,1 and grid current i g1 on the filter inductor current i L1, repectively. 1 Z () YLi (), G () Z () Z () Z () Z () Cf,1 ii Lf,1 Cf,1 Lf,1 Cf,1 (11) where Z Lf,1 () and Z Cf,1 () are the ipedance of the filter inductor and capacitor, repectively. Thu, the dynaic behavior of the inner current control loop can be given by T () G () i () i () i () c,1 ii L1 L1 g1 1 Tc,1 ( ) 1 Tc,1 ( ) (12) Fig. 5. Block diagra of the ultiloop voltage control yte. where T c,1 () i the open-loop gain of the inner current control loop, which i expreed a T () G () G () Y () (13) c,1 ci PWM Li G K G e (14) j1.5t ci () pi, PWM () where G ci () i the P current controller and G PWM () depict the effect of the digital coputation delay (T ) and the Pule Width Modulation (PWM) delay (.5T ) [42]. Then, by including the outer voltage control loop, the voltage referenceto-output tranfer function and cloed-loop output ipedance are derived in the following V () G () V () Z () i () (15) 1 clv,1 1 ov,1 g1 T () Zoi() G (), Z () 1 ( ) 1 T ( ) v clv,1 ov,1 Tv Gvi () Gci () GPWM () ZCf,1() Tv () Z () Z () G () G () Lf,1 Cf,1 ci PWM ZCf,1() ZLf,1() Gci () GPWM () Zoi () Z () Z () G () G () Lf,1 Cf,1 ci PWM K rv vi() pv G K v (16) (17) (18) (19) where G vi () i the PR voltage controller and ω 1 denote the yte fundaental frequency. T v () i the open-loop gain of

6 the control yte. Z oi () i the open-loop output ipedance obtained with the outer voltage control loop open and the inner current loop cloed. B. Current-Controlled Inverter Fig. 6 how the iplified one-line diagra of the currentcontrolled inverter and the aociated control yte. The grid-ide inductor current of the LCL-filter i controlled for the inherent reonance daping effect of the coputation and odulation delay [47]. The PR controller in the tationary αβ-frae i adopted for grid current control. The ynchronou reference frae PLL i ued for grid ynchronization [48]. It i iportant to ention that the PLL ha an iportant effect on the output adittance of inverter in addition to the current control loop. The incluion of PLL effect will introduce an unbalanced three-phae inverter odel which need to be decribed by tranfer atrice [12]. However, it ha been found in recent tudie that the PLL only affect the output adittance within it control bandwidth where the negative reitance behavior ay be introduced [14], and thi proble can be avoided by reducing the bandwidth of PLL [13]. Hence, in thi work, the bandwidth of the PLL i deigned to be lower than the yte fundaental frequency in order not to bring in any haronic-frequency ocillation. Thu, the current control loop itelf i linearized baed on the LCL-filter and i odeled a a real calar yte. Fig. 7 depict the block diagra of the grid current control loop. The LCL-filter in itelf i a two-input and ingle-output yte, where the PoC voltage V and inverter output voltage V PWM, are the two input and grid current i g i the output. Thu, the following two tranfer function are ued to odel the LCL-filter plant. Y gi, i () V g () () PWM, V ( ) ZCf, () Z () Z () Z () Z () Z () Z () Y Cf, Lf, Lg, Lf, Cf, Lg, o, ig() () V () VPWM, ( ) Cf, Lf, Lg, Lf, Cf, Lg, ZLf, () ZCf,() Z () Z () Z () Z () Z () Z () (2) (21) where Z Lf, (), Z Lg, () and Z Cf, () denote the ipedance of the LCL-filter inductor and capacitor, repectively. Y o, () i the open-loop output adittance. Fro Fig. 7, the cloed-loop repone of the current control loop can be derived a follow i () G () i () Y () V () (22) g cli, g oi, T () Y () G (), Y () 1 T ( ) 1 T ( ) c, o, cli, oi, c, c, (23) Fig. 6. Current-controlled inverter (=2, 3) with grid current control chee. Fig. 7. Block diagra of the grid current control loop. where G cli, () and Y oi, () are the current reference-to-output tranfer function and cloed-loop output adittance, repectively. T c, () i the open-loop gain of current control loop, which i given by T () G () G () Y () (24) c, cgi PWM gi, G () K cgi pgi K rgi where G cgi () i the PR current controller. (25) IV. ANALYSIS OF HARMONIC STABILITY To perfor the ipedance-baed analyi of haronic tability in the built power yte, a ethod of deriving the ipedance ratio at each PoC of inverter i propoed in thi ection. It i noted fro Fig. 3 that the equivalent yte ipedance ee fro the PoC of inverter i indipenable in the ipedance ratio. Hence, an equivalent yte ipedance derivation procedure i developed by the help of the nodal adittance atrix and ued in the following analyi. A. Syte Equivalent Circuit Fig. 8 depict the ipedance-baed equivalent circuit for the power yte hown in Fig. 1. The power cable are repreented a the Π-ection odel to include the effect of paraitic capacitor. Alo, to facilitate the forulation of nodal adittance atrix, the Thevenin odel of voltage-controlled inverter i converted to the Norton circuit where Y ov,1 =1/Z ov,1. To obtain the equivalent yte ipedance at each PoC of inverter, the yte nodal adittance atrix (Y nc ) including the cloed-loop output adittance of inverter, a highlighted

7 ipedance atrix are the equivalent yte ipedance een fro the equivalent current ource of inverter, which include the cloed-loop output ipedance of inverter and the equivalent yte ipedance at the PoC of inverter. Conequently, the equivalent yte ipedance at the PoC of inverter can be derived by the following relationhip Z Z 1 1 Z, Z, Z ov,1 lv, Zov,1 Zlv,1 Yoi,2 Yli,2 Yoi,3 Yli,3 (3) Fig. 8. Ipedance-baed equivalent circuit for the built power yte. by the dot-dahed line in Fig. 8, i derived a Yov,1 Gclv,1V1 V1 Gcli,2i g 2 V Y nc 2 (26) Gcli,3i g3 V3 Yov,1 2Yp Yp Yp Y nc Yp Yoi,2 2Yp YL,1 Yp (27) Yp Yp Yoi,3 2Yp Y L,2 where Y p i the cable adittance, Y L,1 and Y L,2 are the adittance for load 1 and 2 connected to Bue 2 and 3, repectively. Then, by inverting Y nc, the nodal ipedance atrix (Z nc ) i given by Z Z Z Z Y (28) Z21 Z22 Z nc nc 23 Z31 Z32 Z33 where the eleent are expanded in (29). It i worth noting that the diagonal eleent of the nodal Further, coparing the ter Z 11 in (3) with (8), it i een that the cloed-loop repone of the voltage-controlled inverter can alo be expreed by V ( ) G ( ) Z ( ) Y ( ) 1 clv,1 11 ov,1 V1 ( ) (31) where Z 11 Y ov,1 i the cloed-loop gain of the inor feedback loop. Siilarly, the cloed-loop repone of the currentcontrolled inverter can alo be found by ean of the nodal adittance atrix (Y no ) derived at the PoC of inverter, which i highlighted by the dahed line in Fig. 8. i 2Y Y Y V g1 p p p 1 ig2 Yp 2Yp YL,1 Y p V 2 i g3 Yp Yp 2Yp YL,2V3 Y no (32) Together with (26), (28) and (29), the cloed-loop repone of the equivalent current ource can be derived a ig,1 Yov,1 Gclv,1V1 ig2 YnoZ nc Gcli,2ig2 (33) i g3 Gcli,3i g3 Z 11 Y Y Y Y Y Y 2Y Y 2Y Y Y Y 2Y Y 2Y Y 3Y Y 2 oi,2 oi,3 oi,2 L,2 oi,3 L,1 oi,2 p oi,3 p L,1 L,2 L,1 p L,2 p p Z 22 Y Y Y Y 2Y Y 2Y Y 2Y Y 3Y Y no 2 ov,1 oi,3 ov,1 L,2 ov,1 p oi,3 p L,2 p p no Z 33 Y Y Y Y 2Y Y 2Y Y 2Y Y 3Y Y 2 ov,1 oi,2 ov,1 L,1 ov,1 p oi,2 p L,1 p p no (29) Y Y Y Y 3Y Y Y Y Y 3Y Y Y 3Y Z Z Z Z Z Z oi,3 p L,2 p p oi,2 p L,1 p p ov,1 p p 12 21, 13 31, Yno Yno Yno Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y 2 no 3 p ( ov,1 oi,2 oi,3 L,1 L,2 ) 2 p( ov,1 oi,2 ov,1 oi,3 ov,1 L,1 ov,1 L,2 ) (2 Y Y )( Y Y Y Y Y Y Y Y ) p ov,1 oi,2 oi,3 oi,2 L,2 oi,3 L,1 L,1 L,2

8 Coparing (33) with (9), it can be found that the econd and third diagonal eleent of the cloed-loop tranfer atrix Y no Z nc repreent the cloed-loop gain of the inor feedback loop for the current-controlled inverter. B. Ipedance-Baed Analyi Table I lit the ain electrical paraeter of the tudied power yte. Table II give the paraeter of the voltage and current controller for inverter. The cable ue the ae Π- ection odel, and two current-controlled inverter are alo deigned with the ae paraeter for the ake of iplicity. Fig. 9 how the frequency repone of inner control loop gain, T v () and T c, (), for the voltage- and current-controlled inverter, repectively. The table unterinated dynaic behavior of inverter at the PoC are oberved with the controller paraeter lited in Table II. Thi provide a theoretical bai for uing the inor feedback loop to ae the haronic tability of the power yte. Power cable (Π-ection) RL load 1, 2 TABLE I MAIN ELECTRICAL PARAMETERS OF POWER SYSTEM Voltage-controlled inverter 1 Current-controlled inverter 2, 3 Electrical Contant Value (p.u. a ) Serie inductance (L p).5 Sere reitance (R p ).13 Shunt capacitance (C p).1 Reitance (R 1 =R 2 ).12 Inductance (L 1 =L 2) 5.6 Filter inductor (L f,1 ).3 Filter capacitor (C f,1).13 DC-link voltage (V dc,1 ) 1.88 Filter inductor (L f,2 = L f,3).3 Filter capacitor (C f,2 = C f,3 ).24 Filter inductor (L g,2 = L g,3).35 DC-link voltage (V dc,2 =V dc,3 ) 1.88 Active power (P 2 = P 3).1 Reactive power (Q 2 = Q 3 ) a. Syte PU bae voltage: 4 V, bae frequency: 5 Hz, and bae power: 1 kva. Voltage-controlled inverter 1 Current-controlled inverter 2, 3 TABLE II CONTROLLER PARAMETERS OF INVERTERS Controller paraeter Value Current controller K pi 8 Voltage controller K pv.1 K rv 1 Sapling period T d,1 1 μ Current controller K pgi,2 = K pgi,3 15 K rgi,2= K rgi,3 6 Sapling period T d,2 = T d,3 1 μ Magnitude (db) Phae (deg) Frequency (Hz) Fig. 9. Frequency repone for the open-loop gain of the voltage-controlled inverter, T v(), and the current-controlled inverter, T c(). Fig. 1 depict the Nyquit diagra of the inor feedback loop gain of the voltage- and current-controlled inverter. Since the ipedance ratio are decribed by ingle-input and ingle-output tranfer function, the Nyquit tability criterion can be directly ued to evaluate the interaction aong the inner control loop of inverter and the network dynaic. It i een that the inor feedback loop for the voltage-controlled inverter i untable wherea the inor feedback loop for the current-controlled inverter are table. Thi iplie that the ipedance interaction at the PoC of the voltage-controlled inverter (Bu 1) caue the haronic intability in the power yte, and the adittance interaction at the PoC of currentcontrolled inverter have no contribution to the haronic intability. Fig. 11 how the Nyquit diagra of the inor feedback loop gain after adjuting the controller paraeter for the voltage-controlled inverter. In thi cae, the proportional gain of the P current controller and the PR voltage controller are reduced (K pi =5, K pv =.5). It i clear that all inor feedback loop becoe table. Thi fact again indicate that the voltagecontrolled inverter with the controller paraeter given in Table II caue the haronic intability in the power yte even if it ha a table unterinated dynaic behavior. V. SIMULATION AND EXPERIMENTAL RESULTS To validate the ipedance-baed tability analyi in the frequency-doain, the power yte in Fig. 1 i built in the nonlinear tie-doain iulation by uing MATLAB and PLECS Blocket, and the experiental tet. A. Siulation Reult Fig. 12 how the iulated grid current of inverter with the electrical contant and controller paraeter given in Table I and II. The iulated bu voltage are hown in Fig. 13. It i een that the haronic-frequency ocillation occur in the power yte, which i confir the frequency-doain

9 Iaginary Axi Real Axi (a) Iaginary Axi Real Axi (a) T v () T c () Iaginary Axi 2-2 Iaginary Axi (-1, j) Real Axi (b) Fig. 1. Nyquit plot of the inor feedback loop gain of inverter in the untable cae (a) Full view. (b) Zoo on (-1, j). analyi in Fig. 1. In contrat, Fig. 14 how the iulated grid current of inverter after reducing the proportional gain of controller for the voltage-controlled inverter (K pi =5, K pv =.5). The iulated voltage at each bu of the yte i hown in Fig. 15. It i obviou that the haronic intability phenoenon hown in Fig. 12 and 13 becoe tabilized in thi cae. Thi agree well with the theoretical analyi in Fig. 11 and alo verifie that the haronic intability in the power yte i caued by the voltage-controlled inverter rather than the current-controlled inverter. B. Experiental Reult Fig. 16 how a hardware picture of the built powerelectronic-baed power yte in laboratory. Three Danfo frequency converter are eployed to operate a a voltagecontrolled inverter and two current-controlled inverter. The Real Axi (b) Fig. 11. Nyquit plot of the inor feedback loop gain of inverter in the table cae (a) Full view. (b) Zoo on (-1, j). control algorith of inverter are ipleented in DS16 dspace yte, where the DS511 digital wavefor output board i adopted to generate witching pule in ynchronou with the apling circuit [42]. The current tranducer LA 55- P and voltage tranducer LV 25-P are ued to acquire current and voltage ignal, repectively, for the digital control circuit. In the experiental tet, the bu voltage of the power yte and grid current of inverter are recorded by uing the digital ocillocope with the 25-kS/ apling rate and the 1-kHz effective bandwidth. The current probe with the 1-kHz bandwidth i adopted for the current eaureent. The differential ode voltage probe with the 25-MHz bandwidth and axiu 1-V Root Mean Square (RMS) i ued for voltage eaureent. Firt, the untable cae that i baed on the controller paraeter given in Table II i teted. Fig. 17 and 18 depict

10 2 ig1 2 ig1 Current (A) Current (A) -2 1 ig2-2 1 ig2 Current (A) Current (A) -1 1 ig3-1 1 ig3 Current (A) Current (A) Tie () Tie () Fig. 12. Siulated grid current of inverter in the untable cae. Fig. 14. Siulated grid current of inverter in the table cae. 4 V1 Voltage (V) V2 Voltage (V) V3 Voltage (V) Tie () Fig. 13. Siulated bu voltage in the untable cae. the eaured grid current of inverter and the eaured bu voltage, repectively. It i oberved that the ae haronicfrequency ocillation arie in the experiental tet a in the iulation reult hown in Fig. 12 and 13. Further, in both the iulation and experiental tet, thi haronic-frequency ocillation propagate into the whole power yte and lead to the unexpected haronic diturbance to the current control loop of the current-controlled inverter. Conequently, even though no untable condition i brought by the currentcontrolled inverter, a hown in Fig. 1, the haronicfrequency ocillation are till preent in the grid current of current-controlled inverter. Hence, it i difficult to identify the ource of haronic intability in the power yte by erely oberving the iulation and experiental reult. The ipedance-baed analyi in the frequency-doain thu Fig. 15. Siulated bu voltage in the table cae. Fig. 16. Hardware picture of the built laboratory tet etup.

11 Fig. 17. Meaured grid current of inverter in the untable cae. Fig. 19. Meaured grid current of inverter in the table cae. Fig. 18. Meaured bu voltage in the untable cae. Fig. 2. Meaured bu voltage in the table cae. becoe iportant to reveal how each inverter contribute to the haronic tability of the power yte. Then, the table cae with the reduced proportional gain of the controller for the voltage-controlled inverter (Kpi=5, Kpv=.5) i teted. Fig. 19 and 2 how the eaured grid current of inverter and bu voltage, repectively, where the table operation of the power yte i clearly oberved. Thi atche well with the iulation reult hown in Fig. 14

12 and 15 and further validate the ipedance-baed tability analyi in Fig. 11. Together with Fig. 17 and 18, the experiental tet point out that the interaction between the voltage control loop of the voltage-controlled inverter and the ret of network reult in haronic-frequency ocillation propagating into the power yte. VI. CONCLUSIONS Thi paper ha dicued a odeling and analyi procedure for the haronic tability proble in AC power-electronicbaed power yte. Two attractive tability analyi ethod, i.e., the CCM and ipedance-baed approach have been briefly reviewed, and have found that the ipedancebaed approach provide a ore coputationally efficient and deign-oriented analyi tool than the CCM. The ipedancebaed approach wa expanded to a three-phae ehed and balanced network, where the haronic intability reulted fro the interaction of the inner control loop for the voltageand current-controlled inverter wa tudied. A ethod for deriving ipedance ratio wa developed baed on the yte nodal adittance atrix. Tie-doain iulation and experiental reult have hown that the propoed approach could be a proiing way to addre the haronic intability in AC power-electronic-baed power yte. REFERENCES [1] F. Blaabjerg, Z. Chen, and S. B. Kjaer, Power electronic a efficient interface in dipered power generation yte, IEEE Tran. Power Electron., vol. 19, no. 5, pp , Sep. 24. [2] P. Brogan, The tability of ultiple, high power, active front end voltage ourced converter when connected to wind far collector yte, in Proc. EPE 21, pp [3] J. H. Enlin and P. J. Heke, Haronic interaction between a large nuber of ditributed power inverter and the ditribution network, IEEE Tran. Power Electron., vol. 19, no. 6, pp , Nov. 24. [4] J. Rocabert, A. Luna, F. Blaabjerg, and P. Rodriguez, Control of power converter in AC icrogrid, IEEE Tran. Power Electron., vol. 27, no. 11, pp , Nov [5] E. Mollertedt and B. Bernhardon, Out of control becaue of haronic An analyi of the haronic repone of an inverter locootive, IEEE Control Syt. Mag., vol. 2, no. 4, pp. 7-81, Aug. 2. [6] Z. Shuai, D. Liu, J. Shen, C. Tu, Y. Cheng, and A. Luo, Serie and parallel reonance proble of wideband frequency haronic and it eliination trategy, IEEE Tran. Power Electron., vol. 29, no. 4, pp , Apr [7] N. Bottrell, M. Prodanovic, and T. C. Green, Dynaic tability of a icrogrid with an active load, IEEE Tran. Power Electron., vol. 28, no. 11, pp , Nov [8] N. Jelani, M. Molina, and S. Bologani, Reactive power ancillary ervice by contant power load in ditributed AC yte, IEEE Tran. Power Del., vol. 28, no. 2, pp , Apr [9] B. Wen, D. Boroyevich, P. Mattavelli, Z. Shen, and R. Burgo, Experiental verification of the generalized Nyquit tability criterion for balanced three-phae AC yte in the preence of contant power load, in Proc. IEEE ECCE 212, pp [1] T. Meo, J. Jokipii, J. Puukko, and T. Suntio, Deterining the value of DC-link capacitance to enure table operation of a three-phae photovoltaic inverter, IEEE Tran. Power Electron., vol. 29, no. 2, pp , Feb [11] J. M. Epi and J. Catello, Wind turbine generation yte with optiized dc-link deign and control, IEEE Tran. Ind. Electron., vol. 6, no. 3, pp , Mar [12] L. Harnefor, M. Bongiorno, and S. Lundberg, Input-adittance calculation and haping for controlled voltage-ource converter, IEEE Tran. Ind. Electron., vol. 54, no. 6, pp , Dec. 27. [13] B. Wen, D. Boroyevich, P. Mattavelli, Z. Shen, and R. Burgo, Influence of phae-locked loop on input adittance of three-phae voltage-ource converter, in Proc. IEEE APEC 213, pp [14] T. Meo, J. Jokipii, A. Makinen, T. Suntio, Modeling the grid ynchronization induced negative-reitor-like behavior in the output ipedance of a three-phae photovoltaic inverter, in Proc. IEEE PEDG 213, pp.1-8. [15] E. A. Coelho, P. Cortizo, and P. F. Gracia, Sall ignal tability for parallel-connected inverter in tand-alone ac upply yte, IEEE Tran. Ind. Appl., vol. 38, no. 2, pp , Mar./Apr. 22. [16] M. Marwali, J. W. Jung, and A. Keyhani, Stability analyi of load haring control for ditributed generation yte, IEEE Tran. Energ. Conver., vol. 22, no. 3, pp , Sept. 27. [17] E. Barklund, N. Pogaku, M. Prodanvoic, C. H. Araburo, and T. C. Green, Energy anageent in autonoou icrogrid uing tabilitycontrained droop control of inverter, IEEE Tran. Power Electron., vol. 23, no. 5, pp , Sept. 28. [18] J. D. Ainworth, Haronic intability between controlled tatic converter and a.c. network, Proc. Int. Elect. Eng., vol. 114, pp , Jul [19] F. Wang, J. Duarte, M. Hendrix, and P. Ribeiro, Modelling and analyi of grid haronic ditortion ipact of aggregated DG inverter, IEEE Tran. Power Electron., vol. 26, no. 3, pp , Mar [2] R. Turner, S. Walton, and R. Duke, Stability and bandwidth iplication of digitally controlled grid-connected parallel inverter, IEEE Tran. Ind. Electron., vol. 57, no. 11, pp , Nov. 21. [21] L. Harnefor, L. Zhang, and M. Bongiorno, Frequency-doain paivity-baed current controller deign, IET Power Electron., vol. 1, no. 4, pp , Dec. 28. [22] X. Wang, F. Blaabjerg, M. Lierre, Z. Chen, J. He, and Y. Li, An active daper for tabilizing power-electronic-baed AC yte, IEEE Tran. Power Electron., Early Acce Article, 213. [23] X. Wang, F. Blaabjerg, Z. Chen, and W. Wu, Reonance analyi in parallel voltage-controlled ditributed generation inverter, in Proc. IEEE APEC 213, pp [24] X. Wang, F. Blaabjerg, and Z. Chen, Autonoou control of inverterinterfaced ditributed generation unit for haronic current filtering and reonance daping in an ilanded icrogrid, IEEE Tran. Ind. Appl., Early Acce Article, 213. [25] M. Corradini, P. Mattavelli, M. Corradin, and F. Polo, Analyi of parallel operation of uninterruptible power upplie loaded through long wiring cable, IEEE Tran. Power Electron., vol. 25, no. 4, pp , Apr. 21. [26] S. Zhang, S. Jiang, X. Lu, B. Ge, and F. Z. Peng, Reonance iue and daping technique for grid-connected inverter with long traniion cable, IEEE Tran. Power Electron., vol. 29, no. 1, pp , Jan [27] J. Agorreta, M. Borrega, J. Lopez, and L. Marroyo, Modeling and control of N-paralleled grid-connected inverter with LCL filter coupled due to grid ipedance in PV plant, IEEE Tran. Power Electron., vol. 26, no. 3, pp , Mar [28] L. H. Kocewiak, J. Hjerrild, C. L. Bak, Wind turbine converter control interaction with coplex wind far yte, IET Renew. Power. Gen., vol. 7, no. 4, pp , Jul [29] Z. He, H. Hu, Y. Zhang, and S. Gao, Haronic reonance aeent to traction power-upply yte conidering train odel in China highpeed railway, IEEE Tran. Power Del., vol. PP, no. 99, pp. 1-9, Early Acce Article, 213. [3] H. Lee, C. Lee, and G. Jang, Haronic analyi of the Korean highpeed railway uing the eight-port repreentation odel, IEEE Tran. Power Del., vol. 26, no. 2, pp , Apr. 26.

13 [31] P. Kundur, Power Syte Stability and Control, McGraw-Hill Profeional, [32] N. Pogaku, M. Prodanovic, and T. C. Green, Modeling, analyi and teting of an inverter-baed icrogrid, IEEE Tran. Power Electron., vol. 22, no. 2, pp , Mar. 27. [33] G. Gaba, S. Lefebver, and D. Mukhedkar, Coparative analyi and tudy of the dynaic tability of AC/DC yte, IEEE Tran. Power Syt., vol. 3, no. 3, pp , Aug [34] S. Lefebver and R. Dube, Control yte analyi and deign for an aerogenerator with eigenvalue ethod, IEEE Tran. Power Syt., vol. 3, no. 4, pp , Nov [35] S. Lefebver, Tuning of tabilizer in ulti-achine power yte, IEEE Tran. Power Appar. Syt., vol. PAS-12, no. 2, pp , Feb [36] R. Middlebrook, Input filter conideration in deign and application of witching regulator, in Proc. IEEE IAS 76, pp , [37] M. Belkhayat. Stability Criteria for AC Power Syte with Regulated Load. Ph.D. thei, Purdue Univerity, Dec [38] J. Sun, Sall-ignal ethod for AC ditributed power yte a review, IEEE Tran. Power Electron., vol. 24, no. 11, pp , Nov., 29. [39] R. Turner, S. Walton, and R. Duke, A cae tudy on the application of the Nyquit tability criterion a applied to interconnected load and ource on grid, IEEE Tran. Ind. Electron., vol. 6, no. 7, pp , Jul [4] J. Sun, Ipedance-baed tability criterion for grid-connected inverter, IEEE Tran. Power Electron., vol. 26, no. 11, pp , Nov., 211. [41] L. Harnefor, Modeling of three-phae dynaic yte uing coplex tranfer function and tranfer atrice, IEEE Tran. Ind. Electron., vol. 54, no. 4, pp , Aug. 27. [42] S. Buo and P. Mattavelli, Digital Control in Power Electronic, Morgen & Claypool Publiher, 26. [43] S. Hiti, D. Boroyevich, and C. Cuadro, Sall-ignal odeling and control of three-phae PWM converter, in Proc. IEEE IAS 1994, pp [44] Z. Yao, P. G. Therond, and B. Davat, Stability analyi of power yte by the generalized Nyquit criterion, in Proc. Int. Conf. CONTROL 94, vol.1, pp , Mar [45] A. MacFarlane and I. Potlethwaite, The generalized Nyquit tability criterion and ultivariable root loci, in Int. Jour. of Control, vol. 25, pp , Jan [46] S. Lefebvre, D. P. Carroll, and R. A. DeCarlo, Decentralized power odulation of ultiterinal HVDC yte, IEEE Tran. Power Appr. Syt., vol. PAS-1, no. 7, pp , Jul [47] J. Yin, S. Duan, and B. Liu, Stability analyi of grid-connected inverter with LCL filter adopting a digital ingle-loop controller with inherent daping characteritic IEEE Tran. Ind. Infor., vol. 9, no. 2, pp , May 213. [48] S. K. Chung, A phae tracking yte for three phae utility interface inverter, IEEE Tran. Power Electron., vol. 15, no. 3, pp , May 2.