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1 A. Bayo-Sala, J. Beerten, J. Rimez an D. Van Hertem, Impeance-bae tability aement of parallel VSC HVDC gri connection, Proc. IET International Conference on AC an DC Power Tranmiion ACDC 2015, 11th e., Birmingham, UK, Feb , 2015, 9 page. Digital Object Ientifier: /cp URL (IET Digital Library): URL (IEEE Xplore Digital Library): IET. Thi paper i a potprint of a paper ubmitte to an accepte for publication in Proc. IET International Conference on AC an DC Power Tranmiion 2015 an i ubject to Intitution of Engineering an Technology Copyright. The copy of recor i available at IET Digital Library.

2 Impeance-bae tability aement of parallel VSC HVDC gri connection A. Bayo-Sala, J. Beerten, J. Rimez, D. Van Hertem Elia Sytem Operator, Belgium, ELECTA, KU Leuven, Belgium, NTNU, Norway Keywor: Stability tuy, control interaction, VSC HVDC, Impeance-bae analyi Abtract The number of converter connecte to the ytem i likely to be increae. Thee evice may interact with each other through the network. TSO are concerne about the impact of thee interaction an it influence on the tability. Thi paper preent the tability analyi of a ytem with two converter. The problem i tuie in the frequency omain by uing an impeance-bae approach. A moel for aeing the cloe-loop tability of the complete ytem i evelope. Thi approach how to have potential for tuying the interaction in larger network. The relative tability uner ifferent gri parameter i aree. Simulation reult how that the tability i compromie when a parallel converter i connecte. 1 Introuction Current invetment in power generation are encouraging the intallation of offhore win farm in the North Sea. In parallel, gri planning an market integration are riving the nee for more interconnection between countrie. High Voltage Direct Current (HVDC) line are een a the preferre olution for accommoating the increae in renewable energie an aitional interconnection capacity ue to their controllability an aequacy for long unergroun an unerea connection. The technology ue for the connection of the converter-bae win generator an tranferring power through HVDC line i the voltage ource converter (VSC). A future planning for the extenion to an offhore gri, increaing number of VSC will be cloely connecte to each other at the AC ie. Thee VSC, which might be built by ifferent manufacturer, coul impact on the operation of each other. The latter give rie to crucial concern uch a the influence of one converter in other evice an it range of impact. One of the outtaning reearch quetion i the tuy of poibly avere control interaction between uch converter. When a new converter i connecte to the gri, it ha an influence on the AC network characteritic. The gri will change towar a ytem increaingly ominate an hence influence by uch evice. Thee converter are commonly eigne for operating inepenently, i.e., without coniering the impact of future connecte converter. Currently the VSC control eign i mainly coniering the implification of the gri repone at the Point of Common Coupling (PCC). However the VSC might interact with other converter leaing to a poor repone an reult in tability concern an more uncertaintie uring the eign phae. A a reult, connection tuie for uch converter might require coniering the characteritic an parameter of VSC cloely connecte. Such a future cenario will thu require a continuou tuy an the flexibility of the converter to be aapte to the new conition an requirement a well a urge for a better knowlege in orer to anticipate future poible iue. Lately the ubject of interaction between converter i gaining a lot of interet an trigger ome outtaning reearch quetion. Firt the influence of the integration of a new VSC in the operation of the ret of connecte converter i unknown. Secon the concept of electrical proximity or the range of thee avere interaction i unknown an relevant. Bae on thi proximity of influence, tability tuie will conier or neglect the impact of a etermine converter an hence aing complexity or reucing the moel to be ue in the evaluation. The integration of thee evice in the AC gri involve everal tuie with high complexity an require a eep intericiplinary knowlege. Harnefor et al. [1] ue the promiing impeance-bae approach in tability tuie for VSC-bae ytem. In the reference the VSC ha been foun to a an equivalent negative reitance at the PCC an thereby thu egrae the amping of the overall ytem. Interaction within the control banwith caue power quality an tability iue [2-4]. Subynchronou interaction between a VSC an a nearby ynchronou generator are aree with the impeance-bae approach in [5]. The ame methoology i ue for evaluating the tability in a VSC-bae infee connection [6]. Kocewiak et al.[4] applie frequency metho to tuy the impact of win farm component an number of turbine on the tability an interference at the collector point. Control interaction between a VSC an a STATCOM connecte at the ame point in teay-tate are tuie in [7]. An harmonic tability tuy with the impeance-bae approach in an ilane ytem compoe by three converter i aree in [8]. However interaction between VSC connecte at ifferent point in the gri have not been tuie o far. The role of the 1

3 i12 Z l Z 1 i1 Z 2 i2 I1 vc Y1 vc I2 vc Y2 vc u pcc 1 E g1 + u pcc 2 E g2 + Fig. 1: Sytem with two parallel-connecte VSC. network interconnecting them an the impact of the itance on the interaction i not clear. Thi paper invetigate the interaction between two converter connecte at ifferent point of the gri. For oing o, the problem i tuie in the frequency omain a a Multi-Input Multi-Output (MIMO) ytem. The cloe-loop repone of thi couple ytem i evaluate. The main contribution of thi paper i to preent an analytical methoology for tuying the cloe-loop interaction in the frequency omain. Thi will allow to tuy the egree of coupling an/or potential iue in future connection by mean of impeance moel of the ifferent component, in an eaily aaptable way by only changing the frequency-epenant impeance an without the limitation of only relying on time-conuming EMT imulation tuie for a large number of cenario or highly complex an large tate-pace matrice. 2 Moel of the AC network with parallel-connecte converter The ytem in tuy i hown in Figure 1. It conit of two converter connecte to the AC ytem at ifferent PCC an interconnecte through a tranmiion line. The problem i tuie in the frequency omain which allow to give a quick an imple check to conition which give rie to tability iue. The whole network i tranforme into an equivalent circuit. The element which compoe thi ytem are converte into the reciprocal circuit equivalent moel a explaine in the ection. The VSC i tranforme into a current ource behin an amittance, which characterize the VSC frequency repone, AC network are converte into voltage ource behin an equivalent impeance an the tranmiion line i calculate a the impeance een from the repective PCC. The equivalent circuit of the tuie ytem i epicte in Figure 2. Next the moelling of the ifferent component i etaile. 2.1 Input-amittance moelling of the VSC A way to characterize the converter ynamic i to conier the converter a a frequency-epenant amittance een from the PCC. The repreentation of the VSC a an input-amittance Fig. 2: Equivalent circuit of the ytem. ha become generalize an wiely ue in tability tuie [1,5-6,8-10]. In thi methoology, the converter i repreente by it Norton equivalent circuit a a current-controlle ource through an amittance a een in Figure 2. The amittance Y vc characterize the frequency repone of the reactor, commutation of witching evice an it control trategy an parameter. The input-amittance moel allow to tuy the repone in the frequency-omain. The injecte controlle current to the PCC i with thi approach hence obtaine a function of the external et-point an the PCC voltage u regare a the iturbance, i vc = G() i ref + Y vc ()u (1) where G() i the outer loop an i ref repreent the external et-point. The equivalent amittance can be erive in two way. One conit on canning the harmonic repone from the converter at the connection point. By thi, non-linearitie of witching evice an the harmonic generation can be repreente. The other conit on analytically moelling the repone in the frequency omain. Thi olution i bae on the erivation of the orinary ifferential equation which efine the moel. Since the ytem i non-linear, the lineariation aroun one operating point i performe an thu it i only vali in the cloe proximity to thi operation point. In the paper, the equivalent amittance i analytically obtaine from a general an baic moel. Y vc conit of a tranfer function from the input voltage u to the output current i. Since equation an control are implemente into the ynchronou frame, the amittance Y vc i a matrix of TF compoe by the iagonal element relating an q component repectively an the coupling in the non-iagonal term a, ( ) ( Δi Y vc = Δi q Y q vc Yvc Yvc qq )( ) Δu Δu q The control i implemente in the reference frame. The VSC moel i epicte in Figure 3. The baic control principle for the VSC i bae on a two-level cacae control. Active an reactive power are inepenently controlle. Since the tuy i only focue on the ynamic in the AC ytem, the DC link voltage i coniere contant an thu the ynamic from the DC ie are not coniere in thi paper. The AC gri ynamic in the rotatory frame are given by, (2) 2

4 where X cc i the VSC equivalent amittance only coniering the current controller an reactor an H cc i the tranfer function from the current reference to i. The iturbance from the couple current are compenate for by coniering a correct ecoupling in the control. The reference current in Equation (4) are compute by the outer loop which track the active an reactive power et-point, P ref an Q ref, repectively, Fig. 3: VSC moel an control. ( ) i R + L ( ) ( ) ( ) ( ) i u = u c i q t i q u ωliq q u c + (3) q ωli where R + L i the VSC reactor, from thi point on X r, u are the voltage at the PCC an u c the moulate voltage at the VSC terminal. The inner current control provie the voltage reference u c,ref ent to the witching valve, ( ) ( )( ) Δu c,ref G Δu c,ref =- cc () Δi ref q G q Δi cc() Δi ref q Δi q ( ) ( ) Δiq Δû +wl + Δi Δû (4) q where û are the meaure gri voltage an G cc repreent a PI controller, G cc () =K cc p + Kcc i Noie an harmonic in the meaure gri voltage, û, are attenuate through a low-pa filter H lp for a cut-off frequency α ff in orer to improve the harmonic rejection capability of the current controller. Δû = (5) α ff + α ff Δu (6) The PWM witching elay D pwm, hown a the converter in Figure 3, from u c,ref to u c i moelle a a firt-orer Paé approximation. 1 D pwm () = (7) t pwm +1 Equation (4) i ubtitute in (3) an it term are arrange by coniering u an iref a the input an i a the output. The tranfer function of the inner controller an AC ynamic are given by, 1 ( ) 1 Dpwm H lp i = X r + D pwm G cc }{{} X cc ( u Dpwm G cc + X r + D pwm G cc ) } {{ } H cc i ref In the matrix form, ( ) ( ) Xcc 0 i = u Hcc 0 0 X + i ref cc 0 H (9) cc 1 From now on the term in TF a function of Laplace operator i neglecte ue to reunancy. (8) i ref = G p (P ref P ) (10) i ref q = G q (Q ref Q) (11) where G p an G q are the PI control of the active an reactive power controller repectively. Active an reactive power are meaure uing a econ-orer low-pa filter F 2n, F 2n = 2 +2ω 0 ξ + ω0 2 (12) The implemente active power control for the component i hown in Figure 4. Equation (P = u i + u qi q an Q = u i q + u qi ) introuce non-linearitie an thu the ( ynamic moel i linearie ) aroun the operating point i = P0 u,i q0 = Q0 u a, P 0 ΔP = F 2n u Δu + F 2n u Q 0 Δi + F 2n u Δu q (13) P 0 ΔQ = F 2n u Δu q F 2n u Q 0 Δi q + F 2n u Δu (14) Then active an reactive reference current in Equation (10) an (11) are erive (coniering P ref an Q ref a contant) an linearie in the form, ω 2 0 Δi ref = G p (ΔP ) (15) Δi ref q = G q (ΔQ) (16) By ubtituting Equation (15), (16), (13) an (14) in the equivalent amittance in Equation (8), the outer loop are inclue a, ( ) Δi = Δi q P X cch ccg 0 p u 1+H ccg pf 2n u H ccg q Q 0 u 1H ccg qf 2n u H ccg p Q 0 u 1+H ccg pf 2n u P X cch ccg 0 q u 1H ccg qf 2n u ( ) Δu Δu q (17) Note that after the aition of outer loop an the lineariation, non-iagonal element appear in the matrix. The ij term in Equation (17) are enominate a, ( ) ( )( ) Δi G G = Δu Δi q G q G qq Δu q (18) The PLL ha been inclue in the moel uing the approach preente in [1]. 3

5 P ref u Δθ H lp u iref + u c,ref i L+R 1 G p G cc D pwm i ωli q ωli q P m F 2n X u Fig. 4: -component loop for the active power. The matrix of cloe-loop input-amittance tranfer function Y VSC i finally given by, Y vc = G Y vc = G ( Q0 u )G pll Y q vc = G q Y qq vc = G qq (1 u G pll)+( P0 u )G pll (19) Fig. 5: Magnitue of the equivalent VSC amittance Y vc. where G pll i, C pll G pll = + u C (20) pll an C pll i the PLL PI controller. The VSC input-amittance i moelle with the parameter hown in Table 1. Amittance magnitue an phae angle are plotte in Figure VSC parameter S n V n Reactor Inner control P control Q control H lp F 2n t h Table 1: VSC control parameter. Value 1000 MVA 380 kv [pu] G cc = G p = G q = α ff =20Hz ω 0 = 140 Hz ξ = m 5 an 6 repectively. The paivity of a ytem, or the poitive reitance of an electrical component, i efine by the region with a phae angle between -90 an 90 eg. Such repone i een in Figure 6 where the angle i within thi range above 2 Hz. The angle i lagging -90 eg when ω ue to the inuctive behaviour of the converter. The negative-conuctance region i in thi cae ituate in the frequency range uner 2 Hz. Negative-conuctance region inicate frequencie at which the converter i een a a negative amping from the ytem. Thu the converter may caue ocillation at thoe frequencie to grow. Fig. 6: Phae angle of the equivalent VSC amittance Y vc. Different moification in the control uch a parameter, aitional filtering or ue of other trategie will affect the frequency repone of the converter an thu it interaction with the ytem. However the influence of thee parameter in the interaction i not tuie in the paper. 2.2 AC network One of the mot baic an imple way of moelling the AC network i by mean of repreenting it by a voltage ource behin an impeance Z ac. Z ac i a erie RL repreenting the hort-circuit impeance at the noe, which i an approximation of the ytem trength at the PCC. The hort-circuit impeance, Z c i efine by the quotient V 2 S c, where S c i the hort-circuit power. The equivalent reitance R g an inuctance L g of the Thevenin moel of the network are obtaine from the magnitue of the impeance an 4

6 the angle efine by the X/R ratio which efine the quotient between the reactance an reitance of the network. Since the analyi i performe in the frame, all the network ie ynamic are converte into the rotatory frame being the reulting matrix equal to, Z AC () = ( Z AC Z q AC Z AC Zqq AC where the i, j element of the matrix are, ) (21) Z AC = Zqq AC = R g + L g (22) Z AC = Zq AC = ωl g (23) 2.3 AC tranmiion line In the paper witching an moulation tranient are not coniere an thu neither the reulting harmonic at thoe frequencie. Due to the focu on the frequencie aroun the funamental an pecially at lower value covering the banwith of the control ytem, the equivalent π moel of the tranmiion line with lumpe parameter i coniere to be accurate. The erie impeance an hunt amittance are equal to, Z erie = R + jx [Ω/km] (24) Y hunt = jωc [S/km] (25) π parameter R X ωc 2 Value 29[mΩ/km] 0.32[Ω/km] 1.9 [μs/km] Table 2: Parameter of the tranmiion line π moel. 3 Interaction between parallel-connecte converter A a common practice in the VSC control eign, the AC ynamic are implifie by an equivalent network at the PCC an hence parallel influence with other electronic evice are often neglecte. When two converter are being integrate at ifferent point of the gri an following the tanar approach, the control eign only conier the characteritic at the PCC. Thu both converter woul operate correctly uner the aumption of an inepenent operation. However unexpecte or unconiere mutual coupling between both converter might lea to a poor repone or poe a threat to the tability of the ytem becaue of interaction between them. In thi ection, the methoology for tuying the interaction i evelope an explaine. To that en, a MIMO ytem compoe by two Single-Input Single-Output (SISO) ytem i formulate. Each of the SISO ytem correpon to one VSC. Their coupling are function of the network impeance Z E g u pcc i vc Y vc Fig. 7: Cloe-loop repone of the input-amittance with the gri impeance. of the circuit. The complete ytem i ketche in Figure 8. But before tackling the cloe-loop repone of thi ytem, the tability tuy of a VSC connecte to the AC gri with an impeance-bae approach i introuce. The open-loop repone of the converter i efine by, i vc = G i ref + Y vc u pcc (26) When the converter i integrate into the ytem, the controlle current i injecte to the AC gri through an impeance which itort the PCC voltage. The iturbance i alo propagate through the VSC control by mean of the fee-forwar voltage. Thi voltage iturbance i efine by, u pcc = E g Z i vc (27) Then the cloe-loop repone i inicate by the feeback epicte in Figure 7 an calculate from Equation (26) an (27) a, i vc = G 1+ZY vc i ref + Y 1+ZY vc E g (28) Thi moel i extene to the ytem epicte in Figure 2 with two converter. Hence the current injecte at the PCC from the VSC moelle a a current ource with the amittance in parallel i, i vc,k = I vc,k + Y vc,k u pcc,k k =1, 2 (29) The PCC voltage i efine a, u pcc,k = E gk Z k i,k (30) where i,k i the current injecte to the AC gri equal to i,k = i vc,k + i 12 an the current through the line i 12 i, 2 i 12 = Z l (u pcc,1 u pcc,2 ) (31) Subtituting (30) into (31), the PCC voltage are obtaine a function of the Thevenin voltage an the output current a, ( u pcc1 = 1 Z ) ( ) 1 Z 2 E g1 + 1 Z 1 i vc1 + Z 1 E g2 Z 1Z 2 i vc2 (32) 2 The tranmiion line i coniere ymmetrical in the paper. Thu impeance Z l een at both en are equal. Otherwie equation mut conier the repective impeance een at the point of tuy. 5

7 Z 2 1 Z 1 u pcc1 E g1 1 Z 1 Y vc1 i vc1 Z 1 Z 1 Z 2 Z 2 Z 1 Z 2 u pcc2 E g2 1 Z 2 Y vc2 i vc2 Z 2 2 Z 2 Fig. 8: MIMO ytem of interacting parallel-connecte converter. u pcc2 = ( 1 Z ) ( ) 2 Z 2 E g2 + 2 Z 2 i vc2 + Z 2 E g1 Z 1Z 2 i vc1 (33) where = Z 1 + Z 2 + Z l. From Equation (32) an (33), one can oberve that PCC voltage are influence by both AC ytem, the feeback from the current injecte to the ame PCC an alo the controlle current from the other VSC, which i alo influence by the current from the firt converter. The complete MIMO ytem i epicte in Figure 8 where the tranfer function relating the ifferent variable are obtaine from Equation (32) to (33). The cloe-loop tability from the tiff gri voltage to the controlle current i tuie from the ytem efine by, i vc1 = G 11 E g1 + G 12 E g2 i vc2 = G 21 E g1 + G 22 E g2 (34) Thee tranfer function relating input an output in Equation (34) are obtaine by ubtituting Equation (32) an (33) into (29) an iolating the current cro-coupling. Interaction are thu reflecte into the converter by mean of the cloe-loop amittance of the ytem G ij. Each of thee tranfer function have been erive by coniering all the poible path an hence are influence by all the element which compoe the network. 4 Cae tuy an reult The ytem compoe by two converter, two AC network an a tranmiion line from Figure 1 i tuie. The ifferent component are ubtitute by their repective moel evelope in ection 2. The MIMO ytem preente in ection 3 i evelope by ubtituting the equivalent moel of the VSC amittance, AC network an tranmiion line impeance in Equation (32) an (33). Tranfer function relating the ifferent input an output in Equation (34) are erive. Then the interaction an egree of coupling between both ubytem are evaluate by mean of tuying the repone of the cloe-loop amittance G ii. The parameter which have an influence in the coupling between both ubytem, i.e., line an gri impeance are change an the cloe-loop repone of the couple ytem i compare with the cloeloop repone of the converter operating inepenently, i.e., by neglecting the parallel coupling a in Equation (28). The complete ytem i expree in frame. Thu all the variable which appear in the MIMO ytem are relate to the ame component of the matrice. Each of the matrix term mut be hanle eparately. Only the reult of the -component in amittance an impeance are hown. Special focu mut be pai to the range of low frequencie an within the control banwith ince ocillation in high frequencie are ampe enough by the inherent high impeance in thi range an the efficient rejection capability of the VSC to iturbance above the control banwith. 4.1 Control interaction between two converter Firt the effect of interaction in the repone of one VSC i illutrate. It uffice to check the iagonal cloe-loop tranfer function of one converter, i.e., G ii. For limiting the tuy an avoiing repetition, only the firt ubytem i coniere. However the ame reult can be extrapolate to the other converter. The tranfer function G 11 in Equation (34), which relate the controlle current injection of the firt converter to the 6

8 Fig. 9: Nyquit iagram of the interconnecte ytem in oli an the ytem without the influence from the parallel VSC in ahe for a line length of 25 km an a SCR in both ytem equal to 3. AC gri voltage of the firt ytem, i influence by the element of the econ ubytem uch a the gri impeance an the VSC amittance of the econ converter. The enominator of thi tranfer function, enote a 1+L() for following common notation in literature, i, 1+L() =1+Y vc1 [ Z 1 Z2 1 Y vc2 Z 2 1Z2 ( 2 )) (35) Z 2 (1 Y 2 vc2 Z 2 The loop tranfer function, i.e., L() in Equation (35), allow to tuy the cloe-loop tability with the Nyquit Criterion. For a converter inepenently connecte the loop tranfer function, an coming back to Equation (28), i Y vc1 Z 1. Therefore one ee by comparing both loop TF the influence of the other converter amittance Y vc2. The cloe-loop tability i now tuie by plotting the Nyquit curve of both loop TF. Figure 9 epict the Nyquit iagram of the TF in Equation (35) an the ytem without coniering the econ converter amittance Y vc2. In the figure the curve from the loop tranfer function of Equation (35), i.e., the ytem coniering Y vc2, get cloer to the critical point (1+j0). The ytem i not untable ue to the curve i not encircling the critical point. However thi itance, epicte with an arrow in the figure, give an iea of the relative tability. Thi itance inicate the peak of the enitivity function an hence how prone the ytem i to intabilitie uner uncertaintie. In other wor, thi meaure of relative tability inicate that the ytem i cloer to be untable given a variation in the parameter of the plant. A een by the ifference between the critical point an both curve, the fact of neglecting the influence of the other converter will lea to a very ignificant ifference in the Fig. 10: Phae angle of cloe-loop amittance G 11 for a line length of 0.5, 10, 30 an 60 km, for P=1GW an SCR=3. tability aement. 4.2 Impact of tranmiion line length Now the effect of the tranmiion line i aree. Thi i the element which interconnect both VSC an hence coupling the ynamic of both parallel ytem. Firt the iagonal cloe-loop amittance G 11 for the ifferent tranmiion line length are calculate. Figure 10 how the reult for the tranmiion line. The paivity in the ytem i tuie from the phae plot of thee TF. In the figure a new negative-conuctance region appear in the range of frequencie between 55 an 250 Hz for the ytem interconnecte by a 0.5-km line. Thi negative region oe not appear in the original amittance Y vc1. An ince the line an gri impeance are paive ytem, the negative conuctance can only be caue by the other VSC. Thee region i ignificantly ampe for the 10-km line an oe not appear in the ret of cae neither in the cloe-loop repone neglecting the coupling. Then parallel connection of converter lea to a new point of potential intabilitie aroun the control banwith. The lat reult an evaluation bae on the paivity of the ytem are compare with another frequency-omain analyi. A one before, the Nyquit curve for the ifferent line length an for the ytem neglecting the influence of the econ converter are plotte. Figure 11 how the reult. Smaller line length have the Nyquit curve L(jω) cloer to the critical point. Therefore thi inicate that the horter the line, the more eteriorate relative tability i. However variation in the line length o not have a ignificant impact on the relative tability a een by the proximity between the ifferent Nyquit curve. In cae of neglecting the influence of the parallel-connecte converter, the Nyquit curve i 7

9 Fig. 11: Nyquit curve for the ifferent loop TF with variable line length. further from the ret of curve. Thi fact give an iea of the conequence of implifying by neglecting the impact of a parallel-connecte VSC in the relative tability of one converter. To um up, both theorie, the Nyquit Criterion an the paivity of the ytem, inicate the relative tability eterioration an hence a higher enitivity uner uncertainty. However the ifference between the tuie tranmiion line are not ignificant. Thi i ue to the implifie moel coniting of two AC ytem an one line interconnecting both. Since gri impeance are the main limiting factor to the propagation of ynamic, the line will conuct all the poible iturbance. Alo it magnitue i ignificantly maller than the magnitue of the gri impeance. Fig. 12: Phae angle of G 11 cloe-loop amittance for SCR equal to 2.5, 7.5 an 25 for a line of 25km an P=1GW. 4.3 Short circuit capacity Finally the lat element which compoe the moel, the gri impeance, i tuie. The AC ytem trength i a characteritic which play a ignificant role in the propagation of ynamic. Stiffer network eaily aborb itortion at the noe an thu o nor lea to tability iue in the VSC connecte to the gri neither propagate the ynamic to other point of the ytem. It impact on the influence between converter i aree by varying the gri impeance magnitue. In thi cae, only the SCR from the gri to which the econ converter i connecte i moifie. Figure 12 how the phae angle uner ifferent network impeance. Sytem connecte to low SCR approach cloer to the non-paive region. Specifically the cae with the highet gri impeance, SCR equal to 2.5, lag over -90 an hence inicate a region of negative amping. A in the previou ection, the problem i alo analye with the Nyquit criterion. The Nyquit curve of the loop tranfer function Fig. 13: Nyquit curve for SCR equal to 2.5, 7.5 an 25. in the enominator i plotte for ifferent SCR. Figure 13 how the reult. A large an ignificant ifference in the itance from the critical point to the Nyquit curve i een. Sytem with higher gri impeance get cloer to the critical point an are thu more enitive to uncertaintie. Therefore interaction between converter are highly epenant on the trength of the ytem. 5 Concluion Interaction between voltage ource converter connecte at ifferent point in the gri are tuie in thi paper. The focu i on the methoology to evaluate the tability in uch a couple ytem. An increaing number of converter will be integrate to the ytem in the future an unexpecte 8

10 interaction between them might lea to tability iue. The importance of thi tuy reult from the nee of evaluating the influence of thee interaction in the tability aement an the impact of the electrical itance between evice. The aement by mean of EMT tuie of uch a large, complex an with a wie range of cenario make it an aruou tak. However frequency-omain metho allow to etect potential point of intability an it expanion i not a complex a in time-omain metho. The paper challenge the tuy of the cloe-loop repone of a ytem with two converter an evaluate the role of, in thi cae, the tranmiion line an the impact of the AC ytem trength. The impeance-bae approach i expane for incluing both converter an tability i tuie from the cloe-loop tranfer function of the complete ytem. Simulation reult reveal the pronene to intability of one converter ue to the influence of the other VSC. The relative tability of one converter i eteriorate ue to interaction from the other converter. Reult how that horter line an higher gri impeance impoe limitation in the tability margin an the ytem i more enitive uner uncertaintie. In the tuie moel, gri impeance have larger impact on the influence between converter whilt the tranmiion line ha a minor effect. Therefore interaction between converter are mainly epenant of the trength of the ytem in uch a topology of the network an the nee of coniering all the ynamic i more important in weaker ytem. Acknowlegement The reearch leaing to thee reult ha receive funing from the People Programme (Marie Curie Action) of the European Union Seventh Framework Programme FP7/ / uner REA grant agreement no , project title MEDOW. Jef Beerten i fune by a pot-octoral fellowhip from the Reearch Founation Flaner (FWO-Vlaaneren). Reference [1] L. Harnefor, M. Bongiorno, S. Lunberg. Input- Amittance Calculation an Shaping for Controlle Voltage-Source Converter, IEEE Tran. on In. Elec., vol.54 n.6, pp , (2007). [2] J. Enlin, P. Heke. Harmonic interaction between a large number of itribute power inverter an the itribution network, IEEE Tran. on Power Electronic, vol.19 no.6, pp , (2004). [3] P. Brogan. The tability of multiple, high power, active front en voltage ource converter when connecte to win farm collector ytem, EPEC Proceeing, (2010). [4] Ł. Kocewiak, J. Hjerril, C. Leth Bak. Win turbine converter control interaction with complex win farm ytem, IET Renewable Power Generation, vol.7 no.4, pp , (2013). [5] K. Alawaa, Y. Abel-Ray, W. Xu. Moeling, analyi an uppreion of the impact of full-cale win-power converter on ubynchronou amping, IEEE Sytem Journal, vol.7 no.4, pp , (2013). [6] C. Wan, M. Huang, C.K. Te, X. Ruan. Stability of interacting gri-connecte power converter, Journal of Moern Power Sytem an Clean Energy, Springer, vol.1 no.3, pp , (Dec. 2013). [7] L. Shen, M. Barne, J.V. Milanović, K.R.W. Bell, M. Belivani. Potential Interaction between VSC HVDC an STATCOM, 18 th PSCC in Wroclaw, Polan, (2014). [8] X. Wang, F. Blaabjerg, W. Wu. Moeling an Analyi of Harmonic Stability in an AC Power-Electronic- Bae Power Sytem, IEEE Tran. on Power Electronic, vol.29 no.12, pp , (Dec. 2014). [9] J. Sun. Impeance-Bae Stability Criterion for Gri- Connecte Inverter, IEEE Tran. on Power Electronic, vol.26 no.11, pp , (2011). [10] M. Cepee, J. Sun. Impeance moeling an analyi of gri-connecte voltage-ource converter, IEEE Tran. Power Electronic, vol.29 no.3, pp , (Mar. 2014). 9