Accurate and reduced SISO Sequence Impedance Models of Grid-tied Voltage Source Converter for Small Signal Stability Analysis

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1 Aurate and redued IO equene Imedane Model o Grid-tied Voltage oure Converter or mall ignal tability Analyi Chen hang, Xu Cai, Atle Rygg and Marta Molina Deartment o Wind Power Reearh Center, hanghai Jiao Tong Univerity, hanghai, China. Deartment o Engineering Cyberneti, Norwegian Univerity o iene and Tehnology, Trondheim, Norway. Abtrat Imedane model are widely ued in aeing mall ignal tability o grid-tied voltage oure onverter (VC) ytem. Reent reearh ha roven that imedane model o grid-tied VC in both dq and equene domain are generally Multi-Inut Multi-Outut (MIMO) ytem, and the generalized Nyquit riterion ha to be alied or tability analyi to thee MIMO ytem. However, inding ingle-inut and ingle-outut (IO) equivalent or thi ytem i alway aealing beaue o the imliity and the onveniene or hyial interretation when aeing the tability, omared to MIMO ytem. Thi aer reent two tye o IO imedane model o grid-tied VC ytem, one i derived rom the trong grid aumtion, and the other i rom the loed-loo equivalene. The auray o thee model i aeed with reet to the meaured imedane in PCAD/EMTDC, and their eet on the tability aement were analyzed a well. It i roven that the aurate IO model give idential reult a the MIMO (matrix-baed) imedane model with reet to tability analyi. However, the redued IO model may lead to wrong reult i the bandwidth o hae loked loo i large. Index term IO, P, equene imedane, ouling, VC, tability analyi. I. INTRODUCTION Nowaday, voltage oure onverter (VC) have been widely ued in the grid-integration o renewable energie [] a well a the lexible ower tranmiion ytem []. Oillation at both low requenie [3] and high requenie [4] have been oberved in VC-baed ytem, in artiular in weak grid ondition [5]. uh kind o mall ignal tability iue an be eetively aeed by imedane-baed analyi aroah. Imedane model o e.g., three-hae VC [6]-[9], ingle-hae VC [], and multi-modular-onverter [] et., have been develoed rigorouly in reent literature. Although imedane are generally a derition o the inut-outut harateriti in requeny domain, the exat ormulation an be dierent due to the modeling tehnique ued. For a tyial two-level and three-hae grid-tied VC, the imedane an be extrated either in dq ynhronouly rotating rame [8] or in three hae tationary rame [7]. In dq rame, the grid-tied VC ytem i time-invariant i the three-hae grid i balaned. It allow linearization diretly, thereby erorming alae tranormation on the reultant linear time invariant (TI) model yield the dq imedane [9]. However, i een rom three hae tationary rame, the grid-tied VC i time-varying inherently. Thereore, the harmoni linearization method in [] ha to be alied to obtain the equene imedane [7]. Or more generally, by linearizing the time-varying ytem along a teady eriodi trajetory yield a linear time eriodi (TP) ytem. In order to rereent the TP

2 ytem in a requeny domain ramework a the TI ytem, the generalized harmoni balane aroah [3] hould be adoted. Deite the dierent model in dq and equene domain, both o them are ouled due to the reene o nonzero o diagonal term in the imedane matrie. Thereore, reent reearhe have hown a great interet in the interretation o thee ouling and their onequene when aeing the tability. In [4] and [5], requeny ouling were identiied in equene domain, i.e. oitive and negative equene are ouled and earated by twie undamental requeny and their imat on low requeny tability were emhaized. Thi intereting roerty o VC wa alo identiied rom dq imedane and introdued a dq aymmetry in [6]. Due to thi imliit binding between dq and equene imedane, the relationhi between equene imedane and dq imedane wa urther invetigated in [7], and they identiied that dq imedane an be tranormed into modiied equene imedane by mean o ymmetrial deomoition method [8]. imilarly, the omlex ae vetor method [6] wa ued to derive the VC imedane in tationary rame diretly rom dq rame in [9] lately. However, the requeny ouling in the oregoing review are in ingle requeny ouling orm, i.e. a ingle requeny erturbation indue a ingle requeny ouling earated by twie undamental requeny. Thi ondition i true i either the onverter or grid imedane i dq aymmetri [6] or the mirror requeny ouling eet in [7]. But i the aymmetry wa reented in hae domain, e.g., aymmetri three hae grid, there tend to be multile requeny ouling between VC and grid. In order to inlude thee ouling harmoni tate ae method [] an be adoted, more oten the trunated harmoni traner untion or numeri and tability tudy in [3] wa ued. At reent, both the ingle and multile requeny ouling ae an only be atured by matrix-baed imedane whih are multi-inut and multi-outut (MIMO) ytem by nature, thereore the generalized Nyquit riterion (GNC) [] hould be adoted or tability analyi, whih obure the advantage o alying equene imedane. On the other hand, inding the ingle-inut and ingle-outut (IO) equivalent o grid-tied VC are alway aealing, due to the imliity and onveniene or hyial interretation. Thi aer aim to ahieve thi objetive by exloring urther the roertie o ingle requeny ouling rom a hyial oint o view. The ret o the aer i organized a ollow: etion II, the equene imedane model o grid-tied VC wa develoed rom the knowledge o dq imedane. In etion III, the ytem blok diagram in orm o omlex traner untion wa etablihed. Aording to the blok diagram, the roertie o equene ouling an be revealed intuitively. ubequently two IO equene imedane model o grid-tied VC with dierent auraie were etablihed. Thee model were omared with the meaured imedane in PCAD/EMTDC. etion IV erorm the tability analyi uing the rooed IO model, the auray in rediting tability were analyzed both by numerial and time domain imulation. etion V draw the inal onluion. II. Modeling o grid-tied VC in (modiied) equene domain A. Toology and ontrol heme o grid-tied VC Fig. reent the grid-tied VC ytem to be analyzed in thi aer. It ontitute a tyial two-level VC, a -tye ilter and a Thevenin equivalent grid. For the ontrol ytem, only urrent ontroller and the hae-loked-loo are onidered or the imliity in the orthoming roerty analyi. Note that, the voltage eed-orward term are not hown in the urrent ontrol loo. I the bandwidth o thee eed-orward term are not areully hoen, they an aet both the tranient [6]

3 and mall ignal reone [5] o VC. In thi regard, the eed-orward term are viewed a imedane haing eet and will not be diued in thi aer ine our ou i on modeling. V d i a i b u a u b u R i R ll ll u d u q H H i q i d re i q re i d u d uq H ll ki Hk ll ki Hll k ll ll Fig. hemati o the grid tied VC ytem B. ymmetrial deomoition o dq imedane Taking a dq imedane model in [9] a an examle: U I (), d dd dq d Uq qd qq Iq t ine () i a TI ytem, a omlex exonential inut (e.g. e ) will give an outut in the ame orm [3], thu variable uh a dq urrent and voltage in () an be written exliitly with variable a below. Ud t dd dq Id t e e, j () Uq qd qq Iq Where j i a tranlation rom -domain to requeny-domain. I d, I q and Ud, Uq are the Fourier oeiient o urrent and voltage at requeny. They an be urther deomoed into oitive and negative omonent a: Alying the matrix A and it invere A U j Ud Ud n j A (3) U Uq Uq to () yield: U dd dq I I A A, j n qd qq (4) U In In Where eah element in n n i generally a omlex traner untion. Thi

4 ormula ha been rigorouly develoed in [7] by the ame author a thi aer. Additionally, a imilar imedane model wa derived in [9] uing omlex ae vetor method. C. Frequeny notation eiiation The requeny notation o imedane an be dierent due to dierent modeling tehnique ued. Thi etion will lariy the requeny notation ued in [4], [7] and [9]. Eentially, the requeny notation in thi aer i the ame a [7], i.e. the requeny (e.g. in (4)) i reerred to the dq rame and i alway oitive. Thi i due to at that ymmetrial deomoition method only utilize the haor o oitive requeny. Thereore, the term modiied reer to the requeny notation in thi work. In [4], a dierent requeny notation i ued, whih i reerred to the tationary rame or hae domain. However, the variable in [4] (denoted by [4] ) an be related with the variable in thi aer (denoted by mod ) by requeny hit, e.g. mod [4] j or the oitive equene and mod [4] j or negative equene. ubtitute mod into (4) (admittane i ued or omarion) yield: I 4 Y 4 j Y 4 U4 U4, Po In4 j Yn4 j J 4 (5) I 4 j Yn 4 j Jn 4 Un4 Un4, Neg In4 Y4 j Yn 4 hiting the requeny o negative equene omonent by 4 4 j yield: I 4 Y 4 J 4 j U 4 (6) In4 j J4 Yn 4 jun4 j A a reult, the admittane in (6) ha the ame truture a in 错误! 未找到引用源, whih how that the (modiied) equene imedane an be related to the hae domain equene imedane via requeny notation tranlation. On the other hand, a omlex-ae-vetor aroah in [9] wa adoted, whih lead to another requeny notation. It allow the variable to be negative or the rereentation o negative equene or reverely rotated ae vetor. However, the model in [9] an alo be obtained uing the model in thi aer, e.g., uoe a omlex ae vetor in tationary rame a reulting equene omonent uing (4) i U I and j t I I e, the Un n j I, hene the vetor in dq rame an be derived a j t * jt v U e U e, while in tationary rame, the dq n vetor i jt t * j t v v e U e U e. ubtitute the equene haor into the vetor yield: dq n

5 Conjugating the vetor in (7) yield: t j t * * v j Ie j I e (7) n j t * * * j t t n e v j I e j I e (8) Rewriting (7) and (8) in a omat matrix orm yield: * j n j v I jt * * jt * e n j j e (9) v I Note that the omlex onjugate oerator * denote onjugation o the untion, wherea the uer line denote onjugation o the variable. For the omlex traner untion in (4), * *, n n. Thereore, (9) ha the ame truture a in [9], whih in thi aer i derived rom the (modiied) equene imedane with areully examined requeny notation. In general, the above model are related via linear tranormation. Thereore, tability analyi hould be idential in all o thee model. D. ytem blok o grid-tied VC in equene domain Alying the tranormation method in etion II.B, the ytem blok o grid tied-vc in (modiied) equene domain i hown bellow: Table dq [3] and equene imedane blok Blok Variable dq domain Variable equene domain Ciruit inut outut Inut outut Eqv. Control i i d q u u d q H H i u n n i u H H Feedorward i i d q u u d q i n i u n u j j P u u d q i i d q q ll I T V Id Tll V u n u i n i T I I ll * * V I I u u d q u u d q VqTll V Vd Tll V u n u u n u T V V ll * * V V V Filter i i d q u u d q i i u n u

6 Grid i i d q u u d q i i u n u In table., I Id jiq and V Vd jvq are urrent and voltage haor o VC at teady tate reetively. V i the nominal voltage at PCC. T ll, R R, VH ll, j, j, VH ll j. H j, H H.The uerrit or dq variable denote ytem domain, wherea the uerrit denote onverter domain [9]. ubrit and denote variable in the grid ide and onverter ide reetively. Traner untion in bold tye denote omlex traner untion. E. equene imedane model o grid-tied VC ytem Combining the equene traner untion blok in Table. in aordane with ytem hemati in Fig., the VC (oad) admittane and grid (oure) imedane an be written a (generation notation i ued or the urrent diretion): Where Y n i Y Y u u n n n Y n () i Y Y u u u i i n n n () u i i i i, i n i n () u u u, u n u n (3) inj Tll G ll H I V, Y G V n ll H n n, j j G ll H *, j j * Y Y, Y Y. Note that, the uerrit in aital orm denote matrix, wherea the n in lower-ae denote element in the matrix. The equene equivalent iruit an be draw in aordane with (4)-(8):

7 Y u u inj i Y Y u n n u i n n u inj Y u n Fig. equene equivalent iruit o grid-tied VC ytem At thi oint, the derived equene imedane model an be ued to ae tability with the hel o GNC [4]. It ha been roven in [7] that thi analyi give idential reult a the GNC baed on dq domain imedane. III. IO equivalent o grid-tied VC ytem A. Analyi o ouled equene loo Combining the equene domain ontrol blok in Table in aordane with the ytem oniguration in Fig., yield equene domain ontrol blok a hown in Fig.3: re i re i n H H I * I u n u V * V i n i Tll V u n u u inj n u inj Fig. 3 ytem blok diagram o grid tied VC in equene domain From Fig.3 we an learly identiy that the oitive and negative equene loo are ouled through ix ath, all aued by the P loo gaint ll. Conidering dierent ath will give equene model with dierent auraie, e.g.: Cae : Neglet all the ath, give a imlet model with oitive and negative equene deouled. Although thi may not be eetive or tability analyi diretly, it i ueul or identiying the intrini roertie in the grid-vc ytem, uh a intrini oillation oint [], and the ouling eet o P an be introdued a additional daming oure to the intrini oint []. Cae : Conider ath and earately. Thi lead to another deouled equene model a hown in [7]. The loo imedane rom erturbation voltage u to the urrent reone inj i an be alulated diretly rom Fig.3, i/ Y. imilarly, the negative ounterart i

8 / Y. Thi ondition i atiied i the grid i relatively trong. (ee the roo in () and ()). The oregoing analyi give two deouled equene imedane model o grid-tied VC, both o whih are IO ytem. However, both o them neglet the ouling to ome extent. The ollowing etion will derive an aurate IO-model o the ytem where the oitive and negative loo are viewed a ubytem o a loed-loo ytem. B. Aurate and redued IO model o grid-tied VC In thi etion, we regard the VC and grid a a loed-loo ytem intead o VC and grid ubytem. Due to the linearity, oitive and negative equene erturbation to the loed-loo ytem an be analyzed earately. Taking the oitive equene erturbation a an examle, the loed-loo ytem wa drawn in Fig.4. u inj B n u n i Y C i Fig. 4 The loed-loo rereentation o grid-tied VC ytem olving the linear ytem in Fig.4 yield the oitive equene loo imedane: loo Where, Y. ubtitute () and () into (4) yield: uinj (4) i C B n n loo Y eq Alying thi method to ind the negative equene loo imedane by relaing the matrix B T, n C, and u u, yield: inj inj (5) n n n loo Y eq (6) (5) and (6) are two IO ytem. More imortantly, or eah IO ytem, the equene ouling are aurately inluded. A hyial interretation i that the negative equene iruit in Fig. i augmented into the oitive equene network (and vie vera) by olving the mutually deendent voltage-ontrolled-urrent-oure, whih introdued the ouling (Fig.3). Further, onidering a relatively trong grid,, and, (6) an be redued to:, (5) and

9 det loo (7) Y det n loo (8) Y (7) and (8) i equivalent to the loo imedane (etion III.A Cae ) obtained diretly rom ontrol blok in Fig.3. C. Comarion o analytial model with imedane meaurement There are two analytial model to be omared, i.e. the aurate model (model A) rom (5) and (6), and the redued model (model B) rom (7) and (8). Imedane meaurement are arried out in PCAD/EMTDC with VC and grid model hown in Fig.hown (iruit arameter are in the Aendix A). The multi-run module in PCAD i ued, during eah run, a ingle tone harmoni voltage i injeted into the grid. The requeny injeted i rom Hz to Hz with an inrement o Hz. The amling requeny and amling window ued or Fourier analyi are khz and.5 e reetively. All the igure and data are ot-roeed in Matlab. (a) Cloed loo imedane lot (ondition ) (CR=4, CC=Hz, P=5Hz, nominal urrent (low out), horizontal axi i requeny in dq rame, vertial axi or uer lot are magnitude and lower lot are hae angle)

10 (b) Cloed loo imedane lot (ondition ) (CR=8, CC=Hz, P=Hz, nominal urrent (low out), horizontal axi i requeny in dq rame, vertial axi or uer lot are magnitude and lower lot are hae angle) () Cloed loo imedane lot (ondition 3) (CR=8, CC=Hz, P=Hz, nominal urrent (low in), horizontal axi i requeny in dq rame, vertial axi or uer lot are magnitude and lower lot are hae angle)

11 Fig. 5 Comarion o analytial IO imedane model with imulation meaurement Fig.5 (a) illutrate that both aurate and redued model ahieve a good math with the meaured imedane under a mall P bandwidth ondition. However, i P bandwidth wa inreaed to a relatively large value, the hae o redued model dier rom the meaurement, artiularly or the negative equene imedane a hown in Fig.5 (b). On the ontrary, the aurate IO model an trak the meaured imedane even at the tranition etion in Fig.5 (b). Fig.5 () lot the imedane with invered ower low diretion. It alo rove that the aurate model i uerior to the redued model in aturing the tranition o imedane harateriti. Note that the bandwidth or P and urrent ontroller under a eii grid ondition are areully hoen in aordane to the analyi in [] to aure a table oerational oint. From the above omarion, we an identiy that the P ha a great imat on the imedane within it ontrol bandwidth (normally at low requenie). The aurate model i aable o aturing thi harateriti hange but the redued model aot. However, the redued model an till be eetive i thee model direanie have negligible eet on mall ignal tability. Thereore, the eet o model direanie on the mall ignal tability hould be urther jutiied. Thi i erormed in the next etion. IV. mall ignal tability analyi Thi etion intend to analyze the validity o rooed IO model in rediting mall ignal tability. Beneit rom the IO roertie, the rooed model an be ued in ombination with lai Nyquit riterion (NC) [3] or tability analyi. A. Numerial tability analyi Three ombination o numerial lot are onidered in thi art: ) Model A IO with (NC). (To be omared). ) Model B IO with (NC). (To be omared). 3) Model C MIMO (() and ()) with (GNC). (Reerene model). In ) the lou o minor loo gain are l Y and eq l Y in n eq and aordane with (5) and (6). In ) the minor loo gain are l Y n l Y in aordane with (7) and (8). In 3) the eigenvalue loi an be alulated rom det Y, where diag l l (, ). Note that the model above are generally omlex traner untion, thu the lou or the negative requenie are not the onjugated o the lou o oitive requenie [6]. However, the omlex traner untion or IO ytem have the roerty * j n j l l, thu the negative requeny lot an be obtained by onjugating the negative equene lou.

12 Im Im P = 5 Hz P = 5 Hz P = Hz 4 table Model A 4 Untable Model A 5 Untable Model A Poitive Poitive Poitive Negative Negative Negative (a) Re (d) Re (g) Re table Model B Untable Model B table Model B Poitive Poitive Poitive Negative Negative Negative (b) Re (e) Re (h) Re Im Im Im Im table Model C Untable Model C Untable Model C Im Im Im () Re () Re (i) Re Fig. 6 Numerial tability omarion (Grid CR i 4. Current ontroller bandwidth i Hz. The VC i loaded with nominal urrent (low out). The P bandwidth i hoen a 5 Hz, 5 Hz and Hz reetively) A hown in Fig.6, the Model A mathe the tability reult o Model C in all P bandwidth ondition, i.e. rom the irt olumn to the third olumn o ig.6 the tability reult are table, untable and untable, whih rove the validity o Model A. On the other hand, the Model B redit the wrong tability reult under large P bandwidth ondition, a hown in Fig.6 (h) omared with (g) and (i). It i intuitively illutrated in Fig.7 (g) and (h) that the negative equene lou o Model B dier largely rom Model A Thi wa alo identiied in the oregoing analyi in Fig.5. Moreover, although the hange in the oitive equene lou o Model B were mall omared with Model A, it onequene on tability are large. In Fig.7 (h), the oitive equene lou doe not enirle oint at (-,j), wherea the oitive equene lou in Fig.7 (g) doe, thereore Model B ail to redit the tability aurately. Thi inding an alo be jutiied by the aivity o oitive and negative loo imedane o Model A and B.

13 Fig. 7 aivity analyi A hown in Fig.7 (a), the oitive equene omonent o Model A ha negative reitane eet a the real axi i being roed, wherea the negative omonent o Model A ha oitive reitane eet. On the other hand, in Fig.7 (b), both the oitive and negative omonent o Model B exhibit oitive reitane eet. A oitive reitane eet imlie a table reonane, thereore the aivity analyi [4] give the ame tability reult a the ae in Fig.6 (g) and (h). B. imulation tudy By varying P bandwidth, the marginally table ondition o grid-tied VC wa ound baed on Model A, a hown in Fig.8, whih i 8 Hz. And the oitive and negative equene oillation requenie an be determined diretly rom the lot, whih i aroximately Hz and 6 Hz reetively (reerred to dq rame). However, due to oitive reitane eet at negative equene reonane (Fig.7 a), it i uoed to be table. Thereore, only the oitive equene oillation at Hz i exeted. The arameter value orreonding to marginally table ondition are then alied in the imulation model (Fig.) built in PCAD/EMTDC to erorm time domain veriiation.

14 (a) oo imedane lot (b) Time domain imulation (Beore, the P bandwidth i et to 5 Hz to ahieve a table oerational oint, aterward the P bandwidth i et to Hz. The oillation i obervable ater everal eond) () Fourier analyi o hae urrent (amling rate i khz, amling window i e) Fig. 8 marginal table imulation (Current ontroller bandwidth i Hz, CR i 4, VC i loaded with nominal urrent (low out)) Fig.8 (b) reent that the ytem i mall ignal untable a redited by Model A in Fig.8 (a). In Fig.8 (), the mirror requenie 4 Hz and 6 Hz are originated rom the oitive equene oillation in dq rame, whih i Hz eiied by Fig.8 (a). The oillatory behavior hown in Fig.8 (b) i imilar to the ield meaurement o grid-tied hotovoltai inverter ytem in [5]. V. CONCUION Thi aer invetigated the IO equene imedane model o grid-tied VC ytem. In the eretive o hyial undertanding, the equene ouling ath were revealed intuitively rom the etablihed equene blok diagram. ubequently, two IO model have been rooed, one wa

15 derived rom the trong grid aumtion (reer to redued IO model), i.e. the oitive and negative ytem loo were onidered earately. The other one wa derived rom the loed-loo analyi o grid-tied VC ytem (reer to aurate IO model), i.e. the negative equene loo wa augmented into the oitive equene loo i oitive equene erturbation wa analyzed. By mean o numerial and time domain analyi, the aurate model give idential tability analyi reult a the matrix imedane model, wherea the redued model an lead to wrong tability reult i P bandwidth wa relatively large. A onequential beneit rom the imliity o the rooed IO model i that, the tability an be aeed by the lai Nyquit riterion or aivity analyi. In addition, the imedane meaurement an be imle omared with dq imedane or matrix equene imedane. In theory, the rooed IO model i eetive or any ingle requeny ouling ytem aued by dq aymmetry, e.g. VC inlude ower ontroller or d voltage ontroller et., however, thi hould be jutiied in the uture work. ACKNOWEDGEMENT Thi aer and it related reearh are uorted by grant rom the Power Eletroni iene and Eduation Develoment Program o Delta Environmental & Eduational Foundation (DREM65). REFERENCE: [] Teodoreu R, ierre M, Rodriguez P. Grid onverter or hotovoltai and wind ower ytem [M]. John Wiley & on,. [] Flourentzou N, Agelidi V G, Demetriade G D. VC-Baed HVDC Power Tranmiion ytem: An Overview [J]. Power Eletroni IEEE Tranation on, 9, 4(3):59-6. [3] Dong D, Wen B, Boroyevih D, et al. Analyi o hae-loked loo low-requeny tability in three-hae grid-oeted ower onverter onidering imedane interation[j]. IEEE Tranation on Indutrial Eletroni, 5, 6(): 3-3. [4] Kunjumuhammed,. P., Pal, B. C., Oate, C., & Dyke, K. J. (6). Eletrial oillation in wind arm ytem: Analyi and inight baed on detailed modeling. IEEE Tranation on utainable Energy, 7(), 5-6. [5] Yang D, Ruan X, Wu H. Imedane haing o the Grid-Coeted Inverter with C Filter to Imrove It Adatability to the Weak Grid Condition[J]. Power Eletroni IEEE Tranation on, 4, 9(): [6] Harneor, Bongiorno M, undberg. Inut-Admittane Calulation and haing or Controlled Voltage-oure Converter [J]. IEEE Tranation on Indutrial Eletroni, 7, 54(6): [7] Ceede M, un J. Imedane Modeling and Analyi o Grid-Coeted Voltage-oure Converter [J]. Power Eletroni IEEE Tranation on, 4, 9(3):54-6. [8] M. Belkhayat, tability riteria or AC ower ytem with regulated load. Purdue Univerity, 997. [9] Wen, B., Boroyevih, D., Burgo, R., Mattavelli, P., & hen,. (5). mall-ignal tability analyi o three-hae a ytem in the reene o ontant ower load baed on meaured dq rame imedane. IEEE Tranation on Power Eletroni, 3(), [] hah, Para. On Imedane Modeling o ingle-phae Voltage oure Converter[C]// IEEE Energy Converion Congre and Exoition. IEEE, 6.

16 [] Jing, Xu C, Molina M. Imedane modeling o modular multilevel onverter[c]// IECON 5 -, Conerene o the IEEE Indutrial Eletroni oiety. IEEE, 5:8-85. [] un J. mall-ignal Method or AC Ditributed Power ytem A Review[C]// Eletri hi Tehnologie ymoium, 9. Et. IEEE, 9:44-5. [3] Möllertedt E. Dynami analyi o harmoni in eletrial ytem [J]. PhD Thee,. [4] Bakhhizadeh, M. K., Wang, X., Blaabjerg, F., Hjerrild, J., Koewiak, Ł., Bak, C.., & Heelbæk, B. (6). Couling in hae domain imedane modeling o grid-oeted onverter. IEEE Tranation on Power Eletroni, 3(), [5] hah, Para. equene Domain Traner Matrix Model o Three-Phae Voltage oure Converter[C]//IEEE PE General Meeting, Boton U. IEEE. 6. [6] Harneor. Modeling o Three-Phae Dynami ytem Uing Comlex Traner Funtion and Traner Matrie [J]. IEEE Tranation on Indutrial Eletroni, 7, 54(4): [7] A. Rygg, M. Molina,. Chen, and X. Cai, A modiied equene domain imedane deinition and it equivalene to the dq-domain imedane deinition or the tability analyi o a ower eletroni ytem, IEEE Journal o Emerging and eleted Toi in Power Eletroni, vol. PP, no. 99,.,6 [8] G. C. Paa, "ymmetrial omonent in the time domain and their aliation to ower network alulation," in IEEE Tranation on Power ytem, vol. 5, no.,. 5-58, May. [9] Wang X, Harneor, Blaabjerg F, PC oh. A Uniied Imedane Model o Voltage-oure Converter with Phae-oked oo Eet[C]// IEEE Energy Converion Congre and Exoition. IEEE, 6. [] Kwon, J., Wang, X., Blaabjerg, F., Bak, C.., ularea, V.., & Bua, C. (6). Harmoni Interation Analyi in Grid-oeted Converter uing Harmoni tate ae (H) Modeling. IEEE Tranation on Power Eletroni. [] C. Deoer and Y.-T. Wang, On the generalized nyquit tability riterion, Automati Control, IEEE Tranation on, vol. 5,no., , Ar 98. [] Chen hang, Xu Cai, heng i, Atle Rygg, Marta Molina. Proertie and Phyial Interretation o the Dynami Interation between Voltage oure Converter and Grid: Eletrial Oillation and It tability Control [J], IET ower eletroni, DOI htt://digital-library.theiet.org/ontent/journal/.49/iet-el [3] un J. Imedane-Baed tability Criterion or Grid-Coeted Inverter [J]. IEEE Tranation on Power Eletroni,, 6(): [4] Harneor,., Wang, X., Yee, A. G., & Blaabjerg, F. (6). Paivity-baed tability aement o grid-oeted VC An overview. IEEE Journal o emerging and eleted toi in Power Eletroni, 4(), 6-5. [5] C. i, "Untable Oeration o Photovoltai Inverter rom Field Exeriene," in IEEE Tranation on Power Delivery, vol.pp, no.99,.- doi:.9/tpwrd APPENDIX: A. Ciruit arameter ued in tability analyi and imulation TABE.I Ciruit arameter o the grid-tied VC ytem Name Value Name Value Nominal ower MVA ilter imedane (.5+.j).u.

17 Nominal voltage.69kv Current reerene (.5 +j).u. DC link voltage withing requeny.kv.4 khz Grid imedane Grid X/R ratio reiroal o CR 5 Fundamental requeny 5Hz