J. Beerten, S. D Arco nd J.A. Suul, Frequency-dependent cle modelling for smll-signl stility nlysis of VSC-HVDC systems, IET Genertion, Trnsmission & Distriution, vol, no. 6, Apr. 26, pp. 37-38. Digitl Oject Identifier:.49/iet-gtd.25.868 URL (IET Digitl Lirry): http://digitl-lirry.theiet.org/content/journls/.49/iet-gtd.25.868 URL (IEEE Xplore Digitl Lirry): http://ieeexplore.ieee.org/xpl/rticledetils.jsp?rnumer=74675 26 IET. This pper is postprint of pper sumitted to nd ccepted for puliction in IET Genertion, Trnsmission & Distriution nd is suject to Institution of Engineering nd Technology Copyright. The copy of record is ville t IET Digitl Lirry.
Frequency-Dependent Cle Modelling for Smll-Signl Stility Anlysis of VSC-HVDC Systems Jef Beerten, Slvtore D Arco 2, Jon Are Suul 2,3 Deprtment of Electricl Engineering (ESAT), Division ELECTA & Energyville, University of Leuven (KU Leuven), Belgium 2 SINTEF Energy Reserch, Trondheim, Norwy 3 Dept. Electric Power Engineering, Norwegin University of Science nd Technology (NTNU), Trondheim, Norwy *jef.eerten@est.kuleuven.e Astrct: Stte-of-the-rt time-domin models of power cles ccount for the frequency dependency of physicl prmeters to enle ccurte trnsient simultions of high voltge trnsmission schemes. Due to their formultion, these models cnnot e directly converted into stte-spce form s required for smll-signl eigenvlue nlysis. Thus, dc cles re commonly represented in HVDC power system stility studies y cscded pi-section equivlents tht neglect the frequency-dependent effects. This pper demonstrtes how the conventionl cscded pi-section model is unle to ccurtely represent the dmping chrcteristic of the cle nd how this cn led to incorrect stility ssessments. Furthermore, n lterntive model consisting of cscded pi-sections with multiple prllel rnches is explored, which llows for stte-spce representtion while ccounting for the frequency dependency of the cle prmeters. The performnce of the proposed model is enchmrked ginst stte-of-the-rt cle models oth in the frequency domin nd in the time domin. Finlly, the pper provides comprtive exmple of the impct of the cle modelling on the smll-signl dynmics of point-to-point VSC HVDC trnsmission scheme.. Introduction The present trends for improving cross order power mrket integrtion nd the ccelerting development of lrge-scle power production from renewle energy sources re generting growing demnd for High Voltge Direct Current (HVDC) trnsmission systems []. Moreover, Voltge Source Converter (VSC) HVDC technology is incresingly preferred due to the recent dvnces in efficiency nd power rting [2, 3] comined with the inherent cpility for rective power or voltge control. Thus, even point-to-point HVDC trnsmission schemes for ulk power trnsfer re currently eing developed nd uilt using VSC technology [4, 5]. Furthermore, VSC HVDC cn e especilly relevnt for the design of multi-terminl nd even meshed grid configurtions envisioned s future offshore trnsmission grid in the North Se region nd s n overly trnsmission grid in minlnd Europe [6]. With the incresing penetrtion of HVDC trnsmission schemes in the existing c grids, their ccurte representtion ecomes criticl when ssessing power system dynmics nd stility [7, 8]. For lrgescle power systems, the smll-signl stility is commonly studied using lineristion nd corresponding
eigenvlue nlysis [9]. The underlying theory is well developed for stility phenomen relted to the synchronous mchines nd their controllers, nd tools re ville in commercilly-grded power system softwre to study smll-signl stility phenomen in c power systems. Since the min dynmics of trditionl lrge-scle power systems hve een dominted y the inertil dynmics of the synchronous genertors, these softwre tools re representing the c grid y lgeric phsor models without ccounting for ny electromgnetic trnsients. However, VSC HVDC systems re chrcterised y fster control loops nd system dynmics. This hs triggered reserch efforts in smll-signl modelling nd nlysis of VSC HVDC systems during the lst decde [ 5]. The focus hs lrgely een on model development for interction studies with the surrounding c grid, whilst incorporting the dynmics of the c- nd dc- sides of the HVDC converter sttion. The ehviour of the dc cle in this respect, however, hs een given less ttention. Stte-of-the-rt frequency-dependent cle models for electromgnetic trnsient (EMT) simultions cnnot e directly trnslted into stte-spce representtion needed for smll-signl stility nlysis. Therefore, it hs een common prctice to undertke smll-signl stility studies with the dc cles represented s either single pi-equivlent circuit [ 2, 5] or y multiple pi-equivlent circuits [3, 4]. In some cses the internl dynmics of dc cles hve een deliertely ignored y representing the dc trnsmission system s resistive network [6]. In [7] the effect of this omission on the time-domin response ws nlysed, ut gin single pi-equivlent cle model ws used s sis for comprison. Recently, it ws demonstrted in [8, 9] tht conventionl pi-equivlent representtions for dc cle cn led to wrong stility ssessment. The results presented in [9] lso indicte tht leving out the cle current dynmics y omitting the inductnce cn t lest void flse conclusions regrding the stility of intermedite or high frequency oscilltions while retining resonly ccurte representtion of the slower dynmics relted to the overll power flow control. The need for representing the frequency-dependent chrcteristics of the dc cle for n ccurte ssessment of smll-signl dynmics of HVDC trnsmission schemes ws lso confirmed y the findings in [9]. To ccount for the effect of frequency-dependent cle prmeters on the oscilltion modes nd dmping in smll-signl studies, n HVDC cle model sed on pi-equivlent sections with multiple prllel RL-rnches ws proposed in [9] s n lterntive to the conventionl cscded pi-section stte-spce model [2]. The model is sed on the pproch presented in [2], which ws pplied for stte-spce representtion of trnsmission lines in [22]. The model prmeter vlues were determined y vector fitting [23] nd the order of the model could esily e dpted ccording to the ccurcy requirements. Strting from the pproch presented in [9], this pper extends the nlysis of the proposed model nd 2
demonstrtes its dvntges nd limittions y mens of time-domin comprisons with stte-of-the-rt EMT model. Such comprisons re presented for the model of dc cle s well s for the nlysis of point-to-point HVDC link, verifying the ility of the proposed cle model to ccurtely represent the smll-signl dynmics of the cle nd its impct t system level. Furthermore, it is shown tht incresing the order of the conventionl cscded pi-section model y dding more sections does not improve the results, while model with prllel RL-rnches effectively cn retin the dmping chrcteristics of the ctul cle, therey preventing the prediction of non-existent instilities. 2. Conventionl cle modelling for stte-spce representtion In generl, cle models cn e clssified in two min sugroups, sed either on lumped prmeters or on distriuted prmeters. In prticulr, the models with distriuted nd frequency-dependent prmeters hve proved to provide n ccurte representtion of the complex ehviour of power lines nd cles [24]. Arguly the most estlished exmple of such model is the universl line model (ULM) lso known s widend model [25]. The model enefits from efficient numericl implementtions tilored to EMT nlysis. Thus, the ULM is t present the preferred numericl implementtion for time domin simultion, s reflected y its vilility in the stndrd lirries of commercil EMT softwre. Furthermore, it my e ssumed s reference to enchmrk the performnces of lterntive models, s mde cler y the time domin comprison presented in [26]. Although suitle for time domin simultions, the ULM, nd distriuted prmeter models in generl, cnnot e trnslted into stte-spce representtion for integrtion into lrger models for performing smll-signl eigenvlue nlysis of power systems. For this clss of pplictions, lumped prmeters models re more common since they cn e effectively nd conveniently expressed in stte-spce comptile formt. However, lumped prmeter representtion sed on stndrd pi-equivlents fils to ccount for the frequency dependency of the prmeters. This section summrises the conceptul steps tht led to the formultion of the conventionl cscded pi-section model for power cles. Moreover, the simplifying hypotheses tht re emedded in the modelling pproch re explicitly highlighted together with the resulting limittions tht rise. 2.. Stedy-stte cle model For converter control interctions studies, the model of HVDC cles cn e reduced y mens of Kron reduction. This ssumes n idel grounding for oth the rmour nd sheth conductors, nmely tht they re t ground potentil long the entire length of the cle. In relity, the sheths of suse cles re usully grounded t oth ends, while onshore cles cn lso e grounded t dditionl points long the 3
length of the cle. Therefore, the reduction only pplies when the voltges in rmour nd sheth remin smll compred to the conductor voltge [27], which is relistic ssumption for control studies. As consequence, the nlyticl representtion of suse cle with three conducting lyers (conductor, sheth nd rmour) reduces to tht of n equivlent conductor. The cle cn then e ccurtely descried y the stedy-stte pi-model, shown in Fig.. v in i in Z π i out v out Y π 2 Y π 2 Fig.. Stedy-stte pi-model of the cle. The series nd prllel elements of this circuit re given y sinh γl Z π = z l γl Y π = y l tnh γl 2 γl 2 () (2) where z nd y re the cle impednce nd dmittnce per unit length, respectively z(ω) = r(ω) + j ω l(ω) (3) y(ω) = g(ω) + j ω c(ω) (4) nd γ the propgtion constnt defined s γ = zy (5) The frequency dependency of the conductnce g nd the cpcitnce c is normlly omitted for EMT models of power cles [27], simplifying (4) to y = g + j ω c (6) On the contrry, omitting the frequency dependency of the resistnce r nd inductnce l is much more severe, s discussed in the reminder of the pper. Figs. 2 2 show the frequency dependency of r nd l for 32 kv XLPE cle for VSC HVDC trnsmission with dt from [28]. Figs. 2c 2d show the impednce for 4
Resistnce [Ω/km] 2 4 2 2 4 Frequency [Hz] Inductnce [mh/km] 4 2 2 4 Frequency [Hz] Mgnitude [Ω] 2 4 2 2 4 Frequency [Hz] c Fig. 2. Frequency dependence of cle prmeters nd cle impednce. Cle prmeters resistnce r Cle prmeters inductnce l c Stedy-stte cle model impednce mgnitude d Stedy-stte cle model impednce ngle Phse ngle [deg.] 5 5 4 2 2 4 Frequency [Hz] d 3 km long cle with these prmeters, using the nlyticl formultion from () (6), with the cle short-circuited t one end. 2.2. Constnt prmeter pproximtion An underlying ssumption in the conventionl cscded pi-section model is tht the frequency dependency of ll prmeters cn e omitted, hence lso simplifying (3) to z = r + j ω l (7) It is importnt to stress tht the resulting model still relies on the nlyticl formultion from () (2), ut with the pproximtion from (6) (7). Figs. 3 3 show the effect of removing frequency dependency of r nd l. In this exmple, the single vlues of these prmeters re otined y weighted fitting over the frequency rnge of µhz up to MHz [23]. The process is descried in more detil in Section 3.. The overll result is n ccurte representtion of predominntly the low-frequency rnge. 5
Mgnitude [Ω] 3 2 dt r, l const 4 2 2 4 Frequency [Hz] Phse ngle [deg.] 5 5 dt r, l const 4 2 2 4 Frequency [Hz] Mgnitude [Ω] 3 2 dt π 5 π 5 π 4 2 2 4 Frequency [Hz] c Phse ngle [deg.] 5 5 dt π 5 π 5 π 4 2 2 4 Frequency [Hz] Fig. 3. Cle impednce constnt prmeter pproximtion nd conventionl pi-section model pproximtion. Constnt prmeter pproximtion impednce mgnitude Constnt prmeter pproximtion impednce ngle c Conventionl pi-section model impednce mgnitude d Conventionl pi-section model impednce ngle d 2.3. Conventionl cscded pi-section model As the model from the previous section still relies on the nlyticl formultion from () (2), n pproximtion is needed to express it in stte-spce formt. A common pproch is to use cscded pi-sections. The pproximtion stems in the fct tht for short lines, the hyperolic correction fctors in () (2) pproch, resulting in series impednce z l. For longer lines, these correction fctors cn e pproximted y cscding multiple pi-sections in series. This pproch is tken in [2] to present stte-spce model for trnsmission lines nd is commonly used pproch to model HVDC cles for smll-signl stility studies. Figs. 3c 3d show the pproximtion of the cle with respectively, 5 nd 5 cscded pi-equivlents, using the constnt r nd l vlues from Section 2.2. These results clerly show tht y cscding severl rnches the cle response etter resemles the ehviour of the nlyticl equivlent pi-model with constnt prmeters (Figs. 3 3), insted of the ctul cle impednce (Figs. 2c 2d). 3. Frequency-dependent cscded pi-model The previous discussion clerly highlights tht incresing the numer of pi-sections results in etter pproximtion of the hyperolic correction fctors from () (2), ut this does not result in good pproximtion of the ctul ehviour of the cle since it does not llow to tke into ccount the frequency 6
dependency of the distriuted prmeters. To overcome these drwcks, the ide of modelling the cle y mens of prllel series RL-rnches from [2] hs een used. In [22], this ide hs een explored to include trnsmission lines in stte-spce model. To some extent, the method presented in this pper lso shows similrities to the pproch tken in [8], which involved modelling the cle screen y including coupled inductor. However, the proposed method is different since it ssumes the screen to e t ground potentil nd is more generl in the sense tht the pproch cn e extended to n ritrry numer of rnches, therey improving the fitting in the frequency domin. The modelling pproch consists of two susequent steps. First, the frequency dependency of the series elements re fitted with prllel rnches y using vector fitting [23]. Second, the hyperolic correction fctors from () (2) re pproximted with multiple pi-sections, resulting in the model from Fig. 4. R L R L R L v in i, R2 L 2. i,2. v i p, R2 L 2. i p,2. v p i p, R2 L 2. i p,2. v out C 2 G 2 R N i,n L N C G R N L N i p,n C G R N i p,n L N C 2 G 2 u i l l 2 l N r r 2 r N Fig. 4. Cscded pi-section model with prllel series rnches. Full model Approximte representtion of the frequency-dependent series prmeters 7
The elements of the pi-equivlent scheme re given y R i = r i l π (8) L i = l i l π (9) G = c l π () C = g l π () where l π = l/p is the length of pi-section nd p represents the numer of pi-sections. 3.. Vector fitting of series impednce elements The gol of the first step, the vector fitting, is to provide n dequte description of the series elements in the frequency domin y mens of rtionl pproximtion of order N. The prolem tkes the generl form of sum of prtil frctions [23] f(s) = N i= c i s i + d + s e (2) Considering N prllel rnches with resistnce r i nd inductnce l i s in Fig. 4, the series dmittnce of the cle, y s (s) = [r(s) + s l(s)], is in fct pproximted s y s (s) N i= l i s + r i (3) A comprison of (2) nd (3) shows tht this model nturlly leds to simplified version of the rtionl formultion from (2) with d =, e = nd l i = c i (4) r i = i c i (5) Hence, the dc resistnce per unit length of the equivlent cle model is given y r dc = N i= i c i (6) Figs. 5 5 show the pproximtion of the series impednce using 5 prllel rnches. The fitting of the series impednce dt to prllel rnches clerly provides vlid mens to tke into ccount the 8
Mgnitude [Ω/km] 2 2 4 6 dt model devition 4 2 2 4 Frequency [Hz] Phse ngle [deg.] 5 dt model devition 4 2 2 4 Frequency [Hz] Mgnitude [Ω] 2 dt 5// π 5// 5 π 5// 5 π 4 2 2 4 Frequency [Hz] c Phse ngle [deg.] 5 5 dt 5// π 5// 5 π 5// 5 π 4 2 2 4 Frequency [Hz] Fig. 5. Cscded pi-section model pproximtion Series impednce z nd cle impednce using 5 prllel rnches. Series impednce impednce mgnitude Series impednce impednce ngle c Cle impednce impednce mgnitude d Cle impednce impednce ngle d frequency-dependence of the cle prmeters. 3.2. Cscded pi-section model with prllel rnches The hyperolic correction fctors in () (2) re now tken into ccount y using cscded piequivlents with prllel series rnches. The numer of pi-sections depends on the modelling needs nd consequentilly on the ndwidth of the cle model demnded in the stte-spce representtion. This is illustrted in Figs. 5c 5d, which show the results of using, 5 nd 5 pi-sections for cle model with 5 prllel rnches. The picture indictes tht the method s such does not pose ny theoreticl restrictions on the level of detil tht cn e represented, ut the model order increses with the numer of pi-equivlents nd the numer of prllel rnches. 3.3. Stte-spce representtion The liner model from Fig. 4 cn e directly written in stte-spce form x c = A c x c + B c u c (7) 9
with x c R nc the cle stte vrile vector, u c R 2 the cle input vector nd A c R nc nc, B c R nc 2 the cle coefficient mtrices. Considering the cle seprtely, Fig. 4 cn e represented y stte-spce model y considering externl current sources t oth cle ends s inputs nd the internl currents nd voltges s sttes. In this pper, however, the cle is connected to converter model with cpcitor t the dc side. Since in this cse the dc voltges t oth cle ends re lredy stte vriles of the converter, the shunt elements (G/2 nd C/2 in Fig. 4) of the pi-equivlents t the cle ends need to e treted s prt of the converter insted, y dding C/2 to the converter cpcitnce. Hence, the cle cn e written s s stte-spce model with vectors x c nd u c mtrices A c nd B c given y [ x c = A c = i, i,n v i 2, i 2,N v 2 v p i p, i p,n ] T R L L.... R N L N L N C... C G C C... C L R L L..... L N R N L N L N... G... C C C C C............ G... C C C C C L R L. L N... R N L N (8) (9)
B c = L L N L L N T (2) u c = v in (2) for the generl cse of cle with p pi-equivlents nd N prllel rnches. The first nd lst sulock of A c in (9) differ s the voltges t the eginning nd t the end of the cle re treted s inputs to the model (see B c in (2)). Similrly, in cse only one pi-equivlent is used, mtrices A c nd B c respectively simplify to n N N nd n N 2 mtrix, with vector x c only retining the N currents s stte vriles. 3.4. Time domin verifiction v out In this section different modelling pproches re compred y simulting cle section in PSCAD/ EMTDC. Cle geometries re tken from the open 4-terminl test network presented in [28]. The cle hs een implemented s ULM cle model, s well s using model relying on conventionl cscded pi-section pproximtion nd the proposed cscded pi-section model with prllel rnches. Voltge [V].5.5.2.4.6.8.2.4.6.8 2 Time [s] 2 Voltge [V] 2.2.4.6.8..2.4 Time [s] ULM // 5 π 5// π 5// 5 π 5// 5 π 9// 5 π 9// 5 π Current [A].2..232.23.228.67.68.69.7.2.4.6.8.2.4.6.8 2 Time [s] c Fig. 6. Time-domin cle model verifiction open circuit nd short circuit response. Open circuit step response (detil) Open circuit step response c Short circuit step response (full response & detil) d Short circuit step response 5// 9// Current [A].2..2.4.6.8..2.4 Time [s] d ULM // 5 π 5// π 5// 5 π 5// 5 π 9// 5 π 9// 5 π
Figs. 6 6 show the open-circuit response of the different cle models. The ULM clerly shows the expected reflection ptterns of of the voltge t the cle ends. The conventionl cscded pi-section model with 5 pi-equivlents poorly represents the time-response of the cle, resulting in dominnt oscilltion with higher mplitude, much longer settling time nd different oscilltion frequency compred to the ULM s reflection pttern. The conventionl pi-section model lso results in high frequency oscilltions superimposed to the poorly dmped low frequency oscilltion. The equivlent high-frequency dynmics re negligile in the ULM response, nd re well dmped for the models with prllel rnches. It cn e seen from the figures tht ll models with prllel rnches provide resonly ccurte representtion of the open-end cle dynmics, with the most ccurte results oserved for the models with 5 pi-sections. The difference in response etween the models with 5 nd 9 prllel rnches is negligile. It is lso noted tht mong the models with prllel rnches, the cse with single pi-section nd 5 prllel rnches (in red) hs the lowest ccurcy in representing the reflections t the cle ends: the model does not ccurtely represent the delyed increse in the voltge due to the trvelling wve effect nd hs n initil pek vlue in the time response tht is slightly lower thn the other models. However, the oscilltion frequency nd settling time re much closer to the ULM model thn those of the conventionl cscded pi-section model. Thus, single section model with multiple prllel rnches cn more ccurtely represent the cle dynmics thn conventionl model with multiple cscded pi-sections. Figs. 6c 6d show the current through the cle for the different models fter step in the voltge of V t one cle end with the other cle end short-circuited. No noticele difference is oserved with respect to the numer of pi-equivlents dded. Indeed, the three curves representing the different models using 5 prllel rnches (in mgent nd red), s well s the two models using 9 prllel rnches (in lue) re lrgely overlpping. In generl, ll models correctly represent the stedy-stte ehviour (due to good fitting t low frequencies), ut the rise time is significntly different for the conventionl cscded pi-section model with 5 pi-equivlents. Furthermore, the resonnces re lso triggered to much higher extent thn in the other, more ccurte models (Fig. 6d). The sme oscilltion frequencies s oserved in Fig. 6 re excited for the conventionl cscded pi-section model, while the ULM nd the pi-section models with multiple prllel rnches show smooth nd well-dmped response. Figs. 6c 6d indicte tht ll models using prllel rnches give rther good pproximtion of the short circuit response of the system. However, the numer of prllel rnches does lter the step response slightly nd correspondence is est for the model with 9 prllel rnches (Fig. 6c detil). 2
4. Test system modelling 4.. Reference configurtion In order to vlidte the proposed modelling pproch, simultions re crried out for ±32 kv, 9 MW two-terminl VSC HVDC link with length of 3 km. Fig 7 shows the system configurtion, with converters nd respectively set to constnt voltge nd constnt power control. v dc control = 3 km p c control = v dc + v dc,f ) i K dc ( + ld τ dc s p c + p c,f ) i K P ( + ld τ P s Fig. 7. System modelling nd control implementtion. Two-terminl test system configurtion Converter model c Constnt dc voltge control loop d Constnt power control loop c d The converter model used in this study is sed on the model descried in [5], ssuming n verged 3
model of 2-level converter, including filter us dynmics. The connection to the grid is tken into ccount using complex impednce, which represents the comintion of the trnsformer nd the grid Thevenin impednce. Fig. 7 shows the verged model tht is used nd indictes the different converter control loops. A phse-locked loop (PLL) is used to synchronise the dq reference frme to the voltge t the filter us, while other control loops include decoupled inner current controllers nd ctive power or dc voltge control. The structure of the outer control loops re depicted in Figs. 7c 7d. The ctive power PI controller hs een tuned to otin n equivlent time constnt of 25 ms (hence times slower thn inner current controller), while the dc voltge controller hs een tuned ccording to symmetric optimum tuning. Further fetures of the model include n ctive dmping lgorithm to prevent filter us voltge oscilltions from entering the control loops, s well s dynmics relted to the low-pss filtering of dc voltge nd ctive power mesurements. The first-order mesurement filters on the dc voltge nd c power signls hve een tuned in order to otin 4 db ttenution t the switching frequency, ssumed to e 2. khz. Outer control loops for the rective power hve een left out of the study. The current t the dc side is considered s n input to the converter model, s re the c voltge source nd the references for the controllers. 4.2. System stte-spce modelling In order to write the non-liner converter model in liner stte-spce form, the equtions re linerised round stedy-stte opertion point x R n ẋ = A x + B u ; x() = x (22) with x R n the converter stte vector, u R m its input vector nd A R n n, B R n m the coefficient mtrices. The different components in the HVDC system re first modelled s independent liner time-invrint susystems. With ll different susystems descried in the form of (22), stte-spce model of the entire system is ssemled. In order to find the stedy-stte opertion point for the entire system, which is needed to linerise the converter equtions, dc power flow solution is clculted, ccounting for the dc system losses. The overll system mtrix A t cn therefter e ssemled y ccounting for the stte vriles of the different models tht re input vriles to the model of other components. More specificlly, these re the dc currents t the cle ends nd the converter dc voltges. The totl stte-spce model is given y ẋ t = A t x t + B t u t (23) 4
with x t = u t = [ x x x c ] T (24) [ u r u r ] T (25) For this two-terminl HVDC system, suscripts, nd c respectively refer to the first nd second converter nd the cle connecting the two converters. Without overlpping sttes in the modelling of the components, x t R nt with n t =n +n +n c. The totl input vector u t R mt only consists of reduced versions u r Rmr, u r Rmr input vectors u R m of the converter nd u R m, since the dc currents nd voltges re no longer inputs to susystem models, ut stte vriles of the cles nd converters respectively. 5. Impct of cle model on system interctions 5.. Identifiction of interction modes Following the procedure presented in [9], this pper uses the concept of interction modes to identify the system interctions in the test system nd to ddress the effect of the cle modelling on these modes. These interction modes re defined s those system modes in which the two converters prticipte. Using prticiption fctors s defined in [9], let p ki denote the prticiption fctor of stte vrile x k in mode i, p i R nt the vector with the prticiption fctors for ll system sttes ssocited with mode i, nd p α,i R nα the vector with the prticiption fctors for ll sttes of susystem α. A prmeter η αi is now defined s mesure for the overll prticiption for ech susystem α in mode i such tht η αi = p α,i p i (26) with denoting the L -norm. η i, η i nd η ci re mesure for the degree to which the 2 converters nd the cle prticipte in ech mode. Using threshold χ, we define n interction mode i s mode for which oth η i > χ nd η i > χ, resulting in suset of interction modes S. 5.2. Interction nlysis with cscded pi-section model with prllel rnches Fig. 8 shows the eigenvlues of A (converter ), the dc voltge controlling converter. The ndwidth of oth converter models f B is equl to 58 Hz. The sttes tht re minly ssocited with these modes re resulting from the LC circuit t the c side nd hence, little impct on their position cn e expected when 5
4 2 A 2 5 C Img Img Img E F G 2-5 D 4 5 4 3 2 Rel - 6 4 2 Rel 2 B 3 2 Rel c 2 A C 9// 5 π 9// 5 π 5// 5 π 3// 5 π 3// π Img E F G D 22.55 22.6 G (detil) 2 B 22.65.85.8 35 3 25 2 5 5 Rel Fig. 8. System, converter nd interction modes nd cle model reduction. Eigenvlues A (converter ) Eigenvlues A t (system) c Interction modes Eigenvlues A t with η >5% nd η >5% d Cle model reduction preserving interction modes d connecting the cle. The conservtive ssumption tht the model for the cle needs to e ccurte until this frequency, results in cle model with 9 prllel rnches nd 5 pi-sections. Thus, the impednce ngle devition is limited to less thn.5 in the low frequency region nd is limited to 2 t f B. Similrly, the impednce mgnitude devition is limited to 7% t f B. Fig. 8 shows the resulting eigenvlues for the comined system. The eigenvlues with rel prts lower thn - 3 nd imginry prts over 4 hve not een depicted in this figure s these modes re well-dmped nd minly relted to internl cle sttes. Fig. 8c shows eigenvlues tht re the result of the interction study fter defining η αi for the different components α for mode i with threshold χ = 5%. In totl, 7 interction modes re identified etween the two converters. 6
The required cle ndwidth is now reduced to 297 Hz, corresponding to the frequency of the highest interction mode of interest. Fig. 8d shows the effect of lowering the numer of prllel rnches nd pi-sections. The figure shows tht model with 5 prllel rnches nd 5 pi-equivlents provides good compromise to still ccurtely represent the eigenvlues of interest. The picture indictes tht keeping the numer of prllel rnches equl to 9 whilst lowering the numer of pi-sections, leves the poles C to G lrgely unltered nd only impcts A nd B, the poles with the highest imginry prts. Lowering the numer of prllel rnches to 5 impcts the poles C nd D s well. The numer of prllel rnches cn e lowered to 3, which forms lower limit to still represent the complex conjugte poles A nd B. It is cler from this picture tht including numer of prllel rnches whilst only using one pi-equivlent still llows resonly ccurte representtion of the poles C nd D. It is lso cler from Fig. 8d (see detil) tht ll models ccurtely represent the interction modes F nd G, with the lowest ccurcy for the simplest models with only three prllel rnches. These models re slightly conservtive with respect to the dmping of modes C, D, F nd G nd cn in this cse still e used for stility ssessment. 5.3. Conventionl stte-spce cle modelling effects on interction modes The result of the simplifictions tht re commonly encountered in stte-spce representtion nd their effect on the plcements of the poles involved in the interction modes re shown in Fig. 9. It is cler from 2 2 2 Img 22.25 22.4 E G (detil) A C F G D B Img 22.25 22.4 E G (detil) C F G D Img 22.25 22.4 E G (detil) F G 2 22.55 3 2 Rel ref.: 9// 5 π.4..8 // 5 π 2 22.55 3 2 Rel ref.: 9// 5 π.4..8 // π 2 22.55 3 2 Rel ref.: 9// 5 π c.4..8 // π, no L Fig. 9. Conventionl cle stte-spce representtions nd their effect on the interction modes (reference cse: 9 prllel rnches, 5 cscded pi-sections). Cscded pi-section pproximtion Single pi-section pproximtion c Single pi-section pproximtion without inductnce this figure tht the simplifictions sed on using one prllel rnch (nd either one or multiple pi-sections) give misleding impressions out the reltive stility of the cle modes (denoted A,B, C nd D in Fig. 8c): 7
the corresponding eigenvlues not only pper t different frequencies, ut re lso poorly dmped. The representtion of the lower order cle modes is lso less ccurte thn the representtions from Fig. 8d, nd even thn the model using one pi-section nd 3 prllel rnches. Similrly, the representtion of the rel pole (E in Fig. 8c) is less ccurte. It cn e noted tht the simplest representtion, only using the resistive cle vlue nd leving out the current s stte vrile (Fig. 9c), results in similr vlues for modes E, F nd G, ut does not include the wrongly predicted oscilltory modes from the conventionl (cscded) pi-section models. 6. Cle modelling effects on dynmic response nd system stility The stte-spce models re verified ginst time-domin model using MATLAB/Simulink. The overll im is to illustrte how the typicl time response of the two-terminl system chnges when the frequency dependency of the cle prmeters is ccounted for in the stte-spce model or when, lterntively, conventionl cscded pi-model is used. The enchmrk study cse is non-liner three-phse verged model in MATLAB/Simulink SimPowerSystems, with the cle implemented using the WideBnd Line model from the OPAL-RT ARTEMiS-SSN lirry. The results of the enchmrk model hve een compred ginst the response of liner stte-spce model using respectively cle model with 5 prllel rnches nd 5 pi-sections, s well s the conventionl cscded pi-model with 5 pi-sections. Fig. shows the response of the system when sujected to % step chnge of the equivlent c grid voltge t the power controlling converter sttion. From Fig. it is cler tht the step chnge in c voltge cuses n oscilltion of the dc voltge in the system. Compring the AC Voltge [p.u.]..8.6.4.2 ULM 5// 5 π // 5 π.65.6.55.5.6.8.2..2.3.4.5.6 Time [s] DC Voltge [p.u.].99.98.982.98 ULM 5// 5 π // 5 π.98.6.8.2..2.3.4.5.6 Time [s] Fig.. Time-domin comprison of linerised model with non-liner 2-terminl model with ULM cle model c voltge step chnge (. p.u.) t power controlling converter. Filter cpcitor voltge (d-component) DC voltge results of the conventionl cscded pi-section model (in red) with the others, it is cler tht the perturtion t the c side triggers poorly dmped oscilltion t the dc side which is not present in the ULM model 8
nd in the model with prllel rnches. The smll differences etween the ULM model (lck) nd the cscded pi-section model with prllel rnches re minly result of the linerised stte-spce modelling when operting wy from the lineristion point. On the c side, the two linerised models show good mtch with the non-liner verged model. DC Voltge [p.u.].6.4.2.6 ULM 5// 5 π // 5 π.5.4.6.8.2.22.98..2.3.4.5.6 Time [s] DC Voltge [p.u.].6.4.2.6.5 ULM 5// 5 π // 5 π.4.3.6.8.2.22.98..2.3.4.5.6 Time [s] DC Voltge [p.u.].8.6.4.2 ULM 5// 5 π // 5 π.6.8.2.22.98..2.3.4.5.6 Time [s] Fig.. Time-domin comprison of linerised model with non-liner 2-terminl model with ULM cle model dc voltge reference step chnge (.5 p.u.), voltge t thepower controlling converter. Symmetric optimum controller tuning Incresed dc voltge controller gins (K dc scling fctor 3) c Incresed dc voltge controller gins (K dc scling fctor 4.85) c.4.2 In next step, the effect of the cle modelling on the stility nlysis of the system is investigted y scling the controller gin K dc of the dc voltge PI controller (Fig. 7c). Fig. shows the dc voltge response t the power controlling converter fter 5% step chnge in the dc voltge reference with different controller tuning. Under symmetric optimum controller tuning (Fig. ) the erroneous dc voltge oscilltions re minly present during the initil phse of the step response nd do not significntly influence the time response. When incresing oth proportionl nd integrl gins y fctors of respectively 3 (Fig. ) nd 4.85 (Fig. c), the oscilltions ecome more nd more persistent nd will eventully led to instility in the conventionl cscded pi-section model when incresing the controller gins even further. Fig. 2 shows clerly tht dding more pi-sections, which results in n incresed ndwidth of the cle model, does not result in more relistic time-domin response, hence confirming the results from Fig. 9. There is lmost no noticele difference etween the models with 5, 5 nd 25 pi-sections (Fig. 2) nd lso the model with single pi-section indictes similr oscilltory pttern (Figs. 2-2c). The time domin response of the single pi-section model with the inductnce omitted does not suffer from the wrongly predicted oscilltion, ut fces slightly lower overshoot nd somewht fster response thn the ULM model (Fig. 2). None of the conventionl pi-section models succeed in representing the system response s ccurtely s the cscded pi-model with prllel rnches (Fig. ). Finlly, Fig. 3 shows the effect of this prmeter vrition on the loction of the overll system modes 9
DC Voltge [p.u.].8.6.4.2 ULM // 5 π // 5 π // 25 π.98..2.3.4.5.6 Time [s] DC Voltge [p.u.].8.6.4.2 ULM // π // π, no L.98..2.3.4.5.6 Time [s] DC Voltge [p.u.].7.6.5.4.3.2..2.4.6 Time [s] Fig. 2. Time-domin comprison of different conventionl cscded pi-section models with ULM cle model for K dc scling fctor equl to 3 dc voltge reference step chnge (.5 p.u.), voltge t the power controlling converter. Cscded pi-section model Single pi-section model, with nd without inductnce c Detil, including ll models from () nd () c (Figs. 3 nd 3c) nd on the interction modes (Figs. 3 nd 3d) in prticulr when chnging gin K dc of the dc voltge controller (Fig. 7c) t converter from respectively. to times the originl settings. The intensity of the colour indictes the prticiption of the converters in these modes: the lighter the colour, the more the mode is relted to the cle, the drker the colour, the more it is relted to the converters. Chnging the gins slightly lters the interction pttern (Figs. 3 nd 3d) in terms of new poles ppering compred to the results from the previous sections. These rel poles re relted to the integrtor of the dc voltge controller t converter nd the dc filtered voltge t this converter, s well s with the dc voltges t oth cle ends. Compring the interction pttern from Figs. 3 nd 3d confirms the time-domin nlysis. Nmely, n increse of the controller gins cuses the poles which re strongly linked to the first cle resonnce (modes C nd D from Fig. 9, encircled in red in Fig. 3d) to move into the right-hnd plne, hence mking the system unstle for the cse of the conventionl cscded pi-section model. On the other hnd, in the model with prllel rnches (Fig. 3), the poles linked to the first cle resonnce (modes C nd D from Fig. 8d) do not trigger ny instility nd re still well dmped, even for the higher controller gin settings. From the overll system poles in Figs. 3 nd 3c, it is cler tht no other system instilities re triggered when chnging the gins, other thn the wrongly represented dc resonnce in cse of using conventionl cscded pi-section models. Investigtion of the prticiption fctors of these unstle modes shows tht they re out eqully linked with the dc voltges t oth converters, nd with internl cle stte vriles. This confirms tht the pole ecomes unstle ecuse of wrongly represented interction etween the converter controls nd cle resonnce tht is well dmped in relity, ut poorly dmped in conventionl cscded pi-section model. 2
6 2 5 Img 4 2 2 4 Kdc scling fctor Img 5 5 25 Kdc scling fctor 6 5 4 3 2 Rel 2 3 2 Rel 4 Img 2 2 4 5 4 3 2 Rel c Kdc scling fctor Img 5 5 8 6 4 2 2 Rel Fig. 3. Chnge of system modes nd interction eigenvlues when chnging the voltge controller gin K dc. All modes 5 prllel rnches, 5 pi-sections Interction modes 5 prllel rnches, 5 pi-sections c All modes conventionl cscded pi-section model, 5 pi-sections d Interction modes conventionl cscded pi-section model, 5 pi-sections d Kdc scling fctor 7. Conclusion In this pper, the effect of the frequency dependency of HVDC cles on the smll-signl stility hs een ssessed. The trditionl stte-spce models generlly encountered in literture using the conventionl cscded pi-section modelling result in poor representtion of the cle modes in the frequency domin. This cn led to flse conclusions on the dynmic response nd stility mrgin of HVDC systems. As n lterntive, model with prllel rnches sed on vector fitting of the series elements of the cle hs een proposed to ccount for the frequency dependency of the cle prmeters. The model llows for n n ccurte representtion of the cle in the frequency domin nd provides time domin response similr to tht of widend cle models. Furthermore, the pproch cn lso e pplied for other line or cle configurtions. It is shown tht the model ccurtely represent the system interction modes with lumped prmeter model designed to cover the frequency rnge t which such interctions cn occur. The study leds to the generl conclusion tht the cle should preferly e modelled y comintion of prllel 2
rnches nd pi-sections. In cse very simple model is sought for, it is etter to model the cle using only prllel rnches insted of merely cscding pi-sections. Acknowledgements The uthors would like to thnk B. Gustvsen (SINTEF Energy Reserch) for his ssistnce with the vector fitting tool nd W. Leterme (KU Leuven) for providing the cle dt. Jef Beerten is funded y postdoctorl reserch grnt from the Reserch Foundtion Flnders (FWO). The work of SINTEF Energy Reserch in this pper ws supported y the project Protection nd Fult Hndling in Offshore HVDC Grids ProOfGrids," finnced y the Norwegin Reserch Council together with industry prtners EDF, Ntionl Grid, Siemens, Sttkrft, Sttnett, Sttoil nd NVE. References [] Europen Network of Trnsmission System Opertors for Electricity (ENTSO-E), Ten-yer network development pln 24, Finl Report, 24. [2] P. Judge, M. Merlin, P. Mitcheson, nd T. Green, Power loss nd therml chrcteriztion of IGBT modules in the lternte rm converter, in Proc. IEEE ECCE 23, Denver, USA, Sep. 5 9, 23, pp. 725 73. [3] M. Brnes nd A. Beddrd, Voltge source converter HVDC links the stte of the rt nd issues going forwrd, Energy Procedi, vol. 24, pp. 8 22, Aug. 22, selected ppers from Deep Se Offshore Wind R&D Conference, Trondheim, Norwy, Jn. 9 2, 22. [4] P. Frncos, S. Verdugo, H. Alvrez, S. Guyomrch, nd J. Loncle, INELFE Europe s first integrted onshore HVDC interconnection, in Proc. IEEE PES GM 22, Sn Diego, USA, Jul. 22-26, 22, 8 pges. [5] M. Cllvik, P. Lunderg, nd O. Hnsson, NORDLINK pioneering VSC-HVDC interconnection etween Norwy nd Germny, White Pper, ville online, Mr. 25. [6] D. Vn Hertem nd M. Ghndhri, Multi-terminl VSC HVDC for the Europen supergrid: Ostcles, Renew. Sust. Energy Rev., vol. 4, no. 9, pp. 356 363, Dec. 2. [7] U. N. Gnnrthn, A. M. Gole, nd R. P. Jysinghe, Efficient modeling of modulr multilevel HVDC converters (MMC) on electromgnetic trnsient simultion progrms, IEEE Trns. Power Del., vol. 26, no., pp. 36 324, Jn. 2. [8] H. Sd, J. Perlt, S. Dennetiere, J. Mhseredjin, J. Jtskevich, J. Mrtinez, A. Dvoudi, M. Seedifrd, V. Sood, X. Wng, J. Cno, nd A. Mehrizi-Sni, Dynmic verged nd simplified models for MMC-sed HVDC trnsmission systems, IEEE Trns. Power Del., vol. 28, no. 3, pp. 723 73, Jul. 23. [9] P. Kundur, Power System Stility nd Control. McGrw-Hill Inc, New York, 993. [] E. Prieto-Arujo, F. D. Binchi, A. Junyent-Ferre, nd O. Gomis-Bellmunt, Methodology for droop control 22
dynmic nlysis of multiterminl VSC-HVDC grids for offshore wind frms, IEEE Trns. Power Del., vol. 26, no. 4, pp. 2476 2485, 2. [] R. Pinto, S. Rodrigues, P. Buer, nd J. Pierik, Opertion nd control of multi-terminl DC network, in Proc. IEEE ECCE Asi 23, Melourne, Austrli, Jun. 3-6, 23, pp. 474 48. [2] G. Pinres, L. Tjernerg, L. A. Tun, C. Breitholtz, nd A.-A. Edris, On the nlysis of the dc dynmics of multi-terminl VSC-HVDC systems using smll signl modeling, in Proc. IEEE PowerTech 23, Grenole, Frnce, Jun. 6-2, 23, 6 pges. [3] W. Wng, M. Brnes, nd O. Mrjnovic, Droop control modelling nd nlysis of multi-terminl VSC-HVDC for offshore wind frms, in Proc. IET ACDC 22, Birminghm, UK, Dec. 4 6, 22, 6 pges. [4] N. Chudhuri, R. Mjumder, B. Chudhuri, nd J. Pn, Stility nlysis of VSC MTDC grids connected to multimchine AC systems, IEEE Trns. Power Del., vol. 26, no. 4, pp. 2774 2784, Oct. 2. [5] S. D Arco, J. A. Suul, nd M. Molins, Implementtion nd nlysis of control scheme for dmping of oscilltions in VSC-sed HVDC grids, in Proc. PEMC 24, Antly, Turkey, Sep., 2-24 24, 8 pges. [6] G. O. Klcon, G. P. Adm, O. Any-Lr, S. Lo, nd K. Uhlen, Smll-signl stility nlysis of multi-terminl VSC-sed DC trnsmission systems, IEEE Trns. Power Syst., vol. 27, no. 4, pp. 88 83, Nov. 22. [7] J. Beerten, S. Cole, nd R. Belmns, Modeling of multi-terminl VSC HVDC systems with distriuted DC voltge control, IEEE Trns. Power Syst., vol. 29, no., pp. 34 42, Jn 24. [8] P. Rult, Dynmic modeling nd control of multi-terminl HVDC grids, Ph.D. disserttion, Ecole Centrle de Lille, Mr. 24. [9] J. Beerten, S. D Arco, nd J. A. Suul, Cle model order reduction for HVDC systems interoperility nlysis, in Proc. IET ACDC 25, Birminghm, UK, Fe., -2 25, pges. [2] J. Rosendo Mcís, A. Gómez Expósito, nd A. Bchiller Soler, A comprison of techniques for stte-spce trnsient nlysis of trnsmission lines, IEEE Trns. Power Del., vol. 2, no. 2, pp. 894 93, Apr. 25. [2] A. Semlyen nd A. Deri, Time domin modelling of frequency dependent three-phse trnsmission line impednce, IEEE Trns. Power App. Syst., vol. PAS-4, no. 6, pp. 549 555, Jun. 985. [22] N. Grci nd E. Ach, Trnsmission line model with frequency dependency nd propgtion effects: A model order reduction nd stte-spce pproch, in Proc. IEEE PES GM 28, Pittsurgh, USA, Jul., 2-24 28, 7 pges. [23] B. Gustvsen nd A. Semlyen, Rtionl pproximtion of frequency domin responses y vector fitting, IEEE Trns. Power Del., vol. 4, no. 3, pp. 52 6, Jul. 999. [24] J. Mrti, L. Mrti, nd H. Dommel, Trnsmission line models for stedy-stte nd trnsients nlysis, in Proc. Athens Power Tech (APT) 993, vol. 2, Sep. 993, pp. 744 75. [25] A. Morched, B. Gustvsen, nd M. Trtii, A universl model for ccurte clcultion of electromgnetic trnsients on overhed lines nd underground cles, IEEE Trns. Power Del., vol. 4, no. 3, pp. 32 38, Jul. 23
999. [26] A. Beddrd nd M. Brnes, HVDC cle modelling for VSC-HVDC pplictions, in Proc. IEEE PES GM 24, Wshington DC, USA, Jul. 27-3 24, 5 pges. [27] B. Gustvsen, T. Nod, J. Nredo, F. Urie, nd J. Mrtinez-Velsco, Power System Trnsients: Prmeter Determintion. Tylor nd Frncis Group, 2, ch. 3, pp. 37 75. [28] W. Leterme, N. Ahmed, J. Beerten, L. Ängquist, D. Vn Hertem, nd S. Norrg, A new HVDC grid test system for HVDC grid dynmics nd protection studies in EMT-type softwre, in Proc. IET ACDC 25, Birminghm, UK, Fe., -2 25, 7 pges. 24