Dynamic Model, Control and Stability Analysis of MMC in HVDC Transmission Systems

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1 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery Dynamc Model, Control and Stablty Analyss of MMC n HVDC Transmsson Systems Majd Mehrasa, Edrs Pouresmael, Member, IEEE, Sasan Zabh, and João P. S. Catalão, Senor Member, IEEE Abstract A control technque s proposed n ths paper for control of modular multlevel converters (MMC) n hgh-voltage drect current (HVDC) transmsson systems. Sx ndependent dynamcal state varables are consdered n the proposed control technque, ncludng two ac currents, three crculatng currents, and the -lnk voltage, for effectvely attanng the swtchng state functons of MMCs, as well as for an accurate control of the crculatng currents. Several analytcal expressons are derved based on the reference values of the state varables for obtanng the MMC swtchng functons under steady state operatng conons. In adon, dynamc parts of the swtchng functons are accomplshed by drect Lyapunov method (DLM) to guarantee a stable operaton of the proposed technque for control of MMCs n HVDC systems. Moreover, the capablty curve (CC) of MMC s developed to valdate maxmum power njecton from MMCs nto the power grd and/or loads. The mpacts of the varatons of MMC output and -lnk currents on the stablty of -lnk voltage are also evaluated n detal by small-sgnal analyss. Index Terms Hgh-Voltage Drect Current (HVDC) systems, modular multlevel converter (MMC), drect Lyapunov method (DLM), stablty analyss. Ines k a,b,c, Abbrevatons HVDC MMC DLM VSC MPC CC KVL KCL SMs Varables k ulk crk I. NOMENCLATURE Hgh-Voltage Drect-Current Modular Multlevel Converter Drect Lyapunov Method Voltage-Source Converter Model Pretve Control Capablty Curve Krchhoff's Voltage Law Krchhoff's Current Law Sub-Modules MMC currents Upper and Lower arm currents Crculatng currents of MMCs lnk currents of MMCs dq MMC currents n dq frame crdq Crculatng currents n dq Reference currents of MMC dq Reference crculatng current crdq Reference lnk current MMC currents varatons lnk currents varatons Iavdq Average values of MMC currents vk Output voltages of MMC vulk Upper and Lower arm voltages of MMC v, lnk voltage vdq MMC output voltages n dq frame v Reference MMC output voltages Δ dq Δ dq v vtdq Δv Δv P ΔP Q ΔQ uk(,) u dq() Reference lnk voltage Termnal voltages of MMC lnk voltage varaton MMC effects on lnk voltage MMC actve power MMC actve power varaton MMC reactve power MMC reactve power varaton MMC swtchng functons Reference MMC swtchng functons udq(,) MMC swtchng functon n dq frame Δudq(,) Dynamc of MMC swtchng functon (-ψ, -χ) The center of d-q curve r The radus of d-q curve (-ψ, -χ ) The center of P-Q curve r The radus of P-Q curve Parameters L MMC nductance R MMC resstance Lul Upper and lower arm nductance Rul Upper and lower arm resstance Lt Equvalent nductance of MMC arm Rt Equvalent resstance of MMC arm C The equvalent capactance of MMCs R The total swtchng loss of MMCs ω Angular frequency of MMC voltage α(-5) Coeffcents of DLM controller Ths work was supported by FEDER funds (European Unon) through COMPETE, and by Portuguese funds through FCT, under Projects FCOMP4-FEDER-8 (Ref. PTDC/EEA-EEL/859/), UID/CEC/5/ and SFRH/BPD/744/4. Also, the research leadng to these results receved fundng from the EU Seventh Framework Programme FP7/7 under grant agreement no Majd Mehrasa s wth Young Researchers and Elte Club, Sar Branch, Islamc Azad Unversty, Sar, Iran. E-mal: m.majdmehrasa@gmal.com. Edrs Pouresmael and João P. S. Catalão are wth INESC TEC and the Faculty of Engneerng of the Unversty of Porto, Porto 4-465, Portugal, also wth C-MAST, Unversty of Bera Interor, Covlhã 6, Portugal, and also wth INESC-ID, Insttuto Superor Técnco, Unversty of Lsbon, Lsbon 49, Portugal. E-mals: edrs.pouresmael@gmal.com; catalao@ub.pt. Sasan Zabh s wth ABB Australa, McroGrd, Renewable Integraton and Dstrbuted Generaton CoC, 46 Export Drve, Berrmah, Northern Terrtory 88, Australa. E-mal: sasanzabh@gmal.com (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery D II. INTRODUCTION stngushed features of MMCs, ncludng decentralzed energy storages, modular structure, easy redundant SMs, smple fault dentfcaton and clearance promoted the utlzaton of MMCs n hgh and medum voltage/power applcatons [-]. Attentons have been attracted to desgnng proper controllers [-5], dervng comprehensve general and nner dynamc models [6-9] and presentng effectve modulaton methods for the new approach []. The most sgnfcant technology concerned to connectng remotely located off-shore wnd farms nterest the major ndustral centers and up-to-dated researchers n usng the dfferent knds of MMC n VSC-HVDC transmsson systems [6]. Analyzng detaled mathematcal models of MMC utlzed n HVDC applcatons offers smultaneous control of actve and reactve power and desred lnk voltage n varous operatng conons. In [7] an open-loop strategy s desgned for controllng the total amount of energy stored nsde the MMC. The control technque employs the steady-state solutons of the dynamc equatons to make the system globally asymptotcally stable [7]. Generc voltage-based and energy-based control structures for MMC nverters are presented n [8] that nclude voltage balancng between the upper and lower arms. Then, an mproved pulse wh modulaton based control technque s also proposed n the same reference n order to balance the voltage among arm capactors. The new technque overcomes some major dsadvantages correspondng to the appled voltage balancng methods, such as voltage sortng algorthm, extra swtchng actons, and nterference wth output voltage. In [9], a dgtal plug-n repettve controller s desgned to control a carrerphase-shft pulse-wh-modulaton (CPS-PWM)-based MMC. The mproved crculatng current control method wth ts stablty analyss has the merts of smplcty, versatlty, and better performance of crculatng harmonc current elmnaton n comparson wth the traonal proportonal ntegral controller [9]. Three cost functons based on an MPC are presented n [] that result n a reduced number of states consdered for the ac-sde current, crculatng current, and capactor voltage-balancng controls of an MMC. The duty of the frst cost functon s controllng the ac-sde current wthout consderng redundancy. The second one s for the control of the -lnk current rpple, the transent characterstcs of the unbalanced voltage conon, and the crculatng current. Fnally, the last one s desgned for reachng the capactor voltage balancng and reducng the swtchng frequency of the SM []. In adon to the modelng and control schemes analyzed n [], a swtchng-cycle state-space model based on the unused swtchng states of an MMC and the correspondng control method s proposed n []. By calculatng the average voltage of all SMs n one arm durng each control cycle and comparng t wth the capactor voltage of each SM, the swtchng state of each SM n MMCs s obtaned n []. In ths method, a lttle sortng of the capactor voltages s employed and consequently the calculaton burden on the controller s sgnfcantly decreased. In order to nvestgate the mpact of the voltage-balancng control on the swtchng frequency n an MMC, the dynamc relatons between the SM s capactor unbalanced voltage and converter swtchng frequency are acheved n [4]. Furthermore, by consderng negatve effects of the unbalanced voltage on the SM capactor voltage rpple and voltage/current harmoncs, the desgn nteracton between swtchng frequency and SM capactance, as well as the selecton of unbalanced voltage, are also accomplshed n [4]. A control technque targetng ndependent management of capactor s average voltage n each MMC arm s performed n [5]. In ths method, a decomposton of arms energy n dfferent components s consdered based on the symmetres of MMC arms. By consderng the effects of ac and systems, a dynamc MMC model wth four ndependent components of upper and lower arm currents are ntroduced n [6]. By usng ths model, dynamcal analyss of currents and also desgn and mplementaton of current controllers are become smplfed. A dynamc model, control and stablty analyss of MMC- HVDC transmsson systems s presented n ths paper. The man contrbutons are fourfold: () obtanng a comprehensve dynamc model n d-q frame for MMC-based HVDC system wth sx ndependent dynamcal state varables, ncludng two ac currents, three crculatng currents, and the -lnk voltage. () developng the dynamc parts of swtchng functons by the use of DLM to reach globally asymptotcal stablty. () dervng a detaled capablty curve (CC) based on actve and reactve power of the MMC for the proposed system, nvestgatng the mpacts of varous values of the -lnk currents on CC; t can be used to verfy the maxmum capacty of nterfaced MMC for njecton of actve and reactve power nto the power grd. (4) performng a comprehensve nvestgaton of MMC output and -lnk current varatons effects on -lnk voltage stablty by usng small-sgnal analyss. The rest of ths paper s organzed nto seven sectons. Followng the ntroducton, the dynamc model of MMCbased HVDC s presented n Secton III. Steady state analyss of the proposed model s provded n Secton IV, whle dynamc stablty analyss s assessed n Secton V. In Secton VI, capablty curve analyss of MMC s executed, and -lnk voltage stablty analyss s performed n Secton VII. Smulaton results and concluson are presented n Sectons VIII and IX. III. THE PROPOSED MMC-BASED HVDC MODEL The proposed MMC-based HVDC transmsson system wth two three-phase transformers utlzed for the ams of nsulaton and voltage converson are llustrated n Fg.. Each MMC s composed of sx SMs n ts ether upper or lower arms along wth relevant resstance and nductance to mmc arm losses and lmt arm-current harmoncs and fault currents, respectvely. R s the total swtchng loss of MMCs. Consderng each SM to be an IGBT half-brdge converter, rudmentary operatonal manner of SMs can be explctly seen throughout dynamc analyss of MMC. Furthermore, the two ac systems are lnked to the transformers through resstances and nductances of ac sde as shown n Fg (c) 6 IEEE. 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3 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery v ua ua ub uc uc ub ua S mk C sm S mk L t L t R L v v abct abc s s L abc R N Rt v R R t v v abct abc abc R L N Ls R s v abctr s R t R t s Pl jql v Pl jq v l P jq s s L t L t Load P jq Load v la lb la lc lc lb la Fg.. General model of the proposed MMC-based HVDC system. AC Flter A. The mathematcal model As can be seen n Fg., groundng ponts are consdered at each neutral pont of ac systems and transformers wth Y connecton. Ascertanng another groundng pont n the -lnk voltage of MMCs, () and () are obtaned by applyng KVL law to the loop ncludng -lnk and MMC ac-sde voltages as, dk duk v vk L Rk Lu Ruuk vuk () dk dlk v vk L Rk Ll Rllk vlk () Followng varables are defned as, k uk lk, u k v uk v lk crk uk lk vuk vlk, uk, Subtractng and summng up () from and to (), besdes usng the defned terms n (), the dynamc equatons of MMCs can be acheved as, L Lt dk R Rt k u k v k rk v Lt Rt crk Rt uk The equvalent crcuts of (4) and (5) are drawn n Fg.. The output currents of MMCs can be controlled by accurate analyss of the crcut shown n Fg. (a) and thus the swtchng functon of u k s a key factor to regulate MMCs actve and reactve power acqured by output currents and voltages. As shown n Fg. (b), mtgaton of crculatng currents s dependng on the approprate adjustment of lnk voltage of MMCs. In adon, the swtchng functon of u k plays an mportant role n effectve mnmzaton of undesrable dstortons caused by MMC s crculatng currents. The dynamc relatons between -lnk voltage and the upper or lower arms currents can be derved by applyng a KCL to the -lnk of Fg., dv v (6) C ua ub uc R () (4) (5) vlk v R t L t uk k tk vk R L O N v v lk uk crk / v Rt Lt Rt Fg.. Equvalent crcuts of: (a) Dynamc model based on MMC output currents, (b) dynamc model based on crculatng currents. dv v C la lb lc R By addng (6) and (7) and also usng the relatonshp of crculatng current n (), the dynamc relaton of -lnk voltage and crculatng currents s deduced as, dv v C cra crb crc R By applyng Park s transformaton to the (4), (5), and (8), general dynamc equatons of the proposed model n dq reference frame and based on a selected set of state varables ncludng MMC s output currents, crculatng currents and also -lnk voltage can be expressed as, L Lt d d R Rt L Lt d q d d u v L Lt dq R Rt L Lt q d uq vq rd Lt Rt crd Lt crq ud rq Lt Rt crq Ltcrd uq r v Lt Rtcr u Rt dv v C cr R The needs for reachng well-desgned current control loops and guaranteeng desrably balanced operaton of -lnk and SM voltages verfy that the dfferent parts of (9) should be accurately dentfed for a fne desgn of the proposed controller to attan respectve ams. The followng sectons wll cover all mentoned ponts. (7) (8) (9) (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

4 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery 4 IV. STEADY STATE ANALYSIS The state varables of the proposed model should be kept n ther desred values n steady state operatng conon, regardless of experencng new crcumstances such as a step load change. Consequently, the reference values and are calculated as demonstrated n Fg.. As a matter of fact, the q component of ac voltages should be equal to zero for balanced and snusodal ac systems. Ths means that the reference values of MMCs ac voltages are approached to v v and v. Based on two frst terms of (9) and wth respect to q the above ponts, the frst swtchng state functons of MMC n steady state operatng conon are derved as shown n Fg.4: L Lt d d R Rt L Lt ud d q d v () L L d t q R Rt L Lt uq q d In the same conon, the crculatng currents of MMCs should be governed to become zero, d d d q. crd crq cr As a result, the second swtchng functons of MMCs are obtaned n accordance to (9) and gven n Fg. 5.,, v () ud uq u Rt Combnng () and () leads to the man upper and lower swtchng functons of MMCs n steady state operaton. Usng the last term of (9), the dynamc of lnk voltage n steady state can be expressed as, dv v () CR C C Equaton () shows the dynamc relaton between lnk voltage and currents of MMCs. Under the steady state operaton, lnk voltage wll be equal to, v R () Equaton () shows that lnk voltage s dependent on currents of MMCs n steady state operatng conon. Snce currents of MMCs are related to the lower and upper MMCs currents, t s understood from () that a proper control of output and crculatng currents of MMCs yelds a balanced value for lnk voltage of MMCs. V. DYNAMIC STABILITY ANALYSIS An accurate operaton of the system can be provded by takng all possble dynamc changes nto account. Dynamc presentaton of all state varables nvolved n the proposed HVDC system can be stated as, d d, q q, crd crd x x x 4 crq crq, 5 cr cr, 6 x x x v v (4) Total dynamc saved energy s a basc requrement for DLM. Followng the ponts dscussed above, the dynamc energy functon of the proposed model can be calculated as, L Lt L Lt Lt Lt Lt C H( x ) (5) x x x x4 x5 x6 4 4 d v d P v q q k pp kp / s d d v q Fg.. Calculaton of MMC output currents. d d d v d v d q v d u d u d u d q q Q v d d v q q v q q v q kpq kq / s u q u q q u q Fg. 4. Swtchng functons based on MMC output currents (a) d-component, (b) q-component. crd crd crq cr 4 v v u d u d R t / crq crq v u 4 / u q R t u q Fg. 5. Swtchng functons based on crculatng currents (a) d-component, (b) q-component, (c) -component. The tme-based dervaton of (5) can be expressed as, L Lt L Lt H ( x ) x x x x Lt x x L x x L x x C x x t 4 4 t Each part of (6) can be obtaned from (9) and (4) as, L Lt R Rt L Lt x x x x x u u x v v x d d d d L Lt R Rt L Lt x x x x x u u x v v x q q q q L x x R x L x x u u x t t t 4 d d L x x R x L x x u u x t 4 4 t 4 t 4 q q 4 L x x R x u u x t t v v x R x 5 t 5 x R c x x x x x 6 x u (6) (7) (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

5 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery 5 In adon, the MMCs swtchng functons are extended to (8) wth dynamc components whch are used by the proposed controller durng dynamc changes, u u u (8) dq() dq() dq() The frst part of (8), u dq (), s the dynamc part of the MMC swtchng functons n d-q reference frame that can be acheved by DLM. Ths part s responsble to mantan the stablty of the proposed model aganst load varatons. Second part of (8) s related to the steady state part of MMC swtchng functons shown as u. Ths part s employed dq() so that the state varables of the proposed model follow a specal reference values wthout any dynamc change. By substtuton of (7) and (8) n (6), the summarzed dervaton of MMCs total saved energy s attaned as (9), R R R R H ( x ) x x R x R x R x t t t t 4 t 5 u v v x u v v x u x d d d q q q d u x u x R x x q 4 6 t 6 5 (9) x 6 x6 R Accordng to DLM, a tme-varyng system wth certan state varables wll become asymptotcally globally stable, f the total saved energy functon of system s postve and ts dervatve s defntely negatve. Therefore, takng nto account DLM prncple and all terms present n (9), the dynamc components of the MMCs swtchng functons are,, u x v v u x v v d d d q q q u x, u x d q 4 4 () u 5 x5 x6 Rt x6 The coeffcents of are the effectve factors for regulatng the dynamc parts of the proposed controller that should be chosen approprately [7]. Terms of () guarantee the ultmate desgned controller operaton aganst any sudden dynamc changes. As can be seen n (), due to presence of steady state values n (), the accurate performance of dynamc parts of swtchng functon are hghly relant on the correct functonng of the proposed model n steady state conons. Consderng (), all terms avalable n (9) can evdently dentfed to be negatve values or zero except for the last term that s, x 6 x6 () R By assumng balanced MMCs crculatng currents, equaton (8) can be rewrtten as, dv v C () R Eq. () can also be restated wth respect to (4) and () as, C x 6 dx 6 () In order to nvestgate the mpact of () on (9), the varous possble amounts that exst for () are dscussed n ths secton. Fg. 6 shows the varous states of (). Notcng the reference value demonstrated n red, two possble constant and fluctuated states are consdered for -lnk voltage as shown n Fg. 6. The constant states specfed wth state and state can be more or less than the reference value (for equal value, x6 ). For fluctuated cases, three states are consdered. As can be seen n Fg. 6, for the states of and, () s equal to zero (dx 6/=). Moreover, for fluctuated states, snce the sgn of dx 6/ s varyng due to the varaton of voltage slopes, the ultmate value of () becomes peroally postve or negatve as depcted n Fg. 6. Ths s natng that () s always close to zero n other states and consequently not able to notceably mpact the negatve value of (9). Therefore, the whole term of (9) s defntely negatve or zero. VI. CAPABILITY CURVE ANALYSIS OF THE MMCS Identfyng maxmum capablty of each MMC n actve and reactve power njecton durng operatng conon of HVDC system leads to a more accurate desgn for the controller. The relaton between the -lnk voltage and the ac sde voltage of each MMC shown n Fg. can be acheved as, v v v (4) td d tq q In adon, the relaton between the ac sde and the output voltage of each MMC n d-q frame can be drven by applyng KVL s law to Fg., d d vtd v d L R d L q d t (5) dq vtq v q L R q Ld d t By assumng d dq/=i avdq and substtutng (5) n (4), the followng crcle s obtaned as, d q r L I v LI d a vq v, R R a vd LI avd v d LI a vq vq 4 Rv d c r 4 R Equaton (6) s a crcle wth the center of (-ψ, -χ) and radus of r. Ths crcle descrbes a gven area of MMC output current based on a dq frame n whch the maxmum and mnmum values of the current can be accurately calculated. By substtutng d=p /v d and q.=-q /v d n (6), the followng relaton s obtaned as, v v v Fg. 6. Dfferent states of v and dv/. q v (6) (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

6 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery 6 d, d, d P Q r v v r v r (7) The relaton descrbed n (7) s the capablty curve of MMC as a crcle wth the center of (-ψ, -χ ) and radus of r. Capablty curve of MMCs are plotted n Fg. 7. The smallest crcles shown n Fg. 7(a) and Fg. 7(b) are typcal MMC CC wth > and < respectvely. By ncreasng the postve values of and decreasng the negatve values of, CC can vary as depcted n Fg. 7 for dfferent -lnk current values. As shown n these fgures, the postve and negatve areas of CC are sgnfcantly altered for both actve and reactve power by changng -lnk currents. Ths has to be noted whle desgnng any control process. VII. DC-LINK VOLTAGE STABILITY ANALYSIS How the changes of dfferent varables avalable n the proposed HVDC system affect the -lnk voltage stablty s dscussed n ths secton. Wth respect to (), next equaton can be nferred as, dv v P P (8) C R v Applyng small sgnal lnearzaton to (8), the relaton between lnk voltage and MMC actve power varaton s obtaned as, P P (9) P P C s v R v v By substtutng (5) n (4), () s acheved as, L d d L d () q v P Rd Rq Another relaton between Δv and ΔP can be derved by the use of small sgnal lnearzaton for () as, P v v Ls R Ls R () d d q q By substtutng () n (9), the effects of actve power of MMC and MMC on the lnk voltage can be stated as () and () respectvely, Δv ΔP = = Δv = Ls+R Δ Δ d d d Ls+R q v - Δ q = f f f Δ q Δ Δ Δ Δ Δ () Q(VAR) (a) (b) Fg. 7. Capablty curve of MMCs (a) ncreasng >, (b) decreasng <. Thus, the effects of the MMC d-q components and -lnk currents varaton on -lnk voltage stablty can be evaluated by (5). The Nyqust dagrams of each fj for varous postve ncreasng values of -lnk current are separately depcted n Fg. 8. As realzed from Fg. 8 (a) and Fg. 8 (b), f and f cannot lead to a notceable nstablty n -lnk voltage. But, accordng to Fg. 8 (c), f sgnfcantly ncreases the nstablty margns n both generaton and control processes of -lnk voltage n HVDC system. For ths case, regulatng -lnk current at desred value s a vtal operaton n order to reach a stable -lnk voltage. The same dscusson s governed for varous negatve decreasng values of -lnk current as shown n Fg. 9. However, by decreasng the negatve values of -lnk current, f dagram s gone to the rght-hand part and ts magntude s drastcally decreased as llustrated n Fg. 9 (c) and consequently mproves the stablzng propertes of f. VIII. SIMULATION RESULTS The purpose of ths secton s to assess the capablty of the proposed control scheme at reachng the desred values of MMC currents, voltages, and actve and reactve power under both dynamc and steady state operatng conons. System parameters and the MMC rated values are lsted n Table I. Imagnary Axs.5 x P(W) x Nyqust Dagram ncreasng Q(VAR) P(W) x 4 Imagnary Axs.5 x ncreasng Nyqust Dagram v P v Ls d d Ls R d R q v q f f f q () Consderng steady state operatonal conon of (8), s equal to, v (4) Cv s R Usng () and (), each part of -lnk voltage varatons can be rewrtten as follows, v f f f (5) d q Real Axs (a) Imagnary Axs ncreasng Nyqust Dagram Real Axs Real Axs (c) Fg. 8. Nyqust dagram of -lnk voltage varatons for > due to (a) d- component varatons of MMC current (f) (b) q-component varatons of MMC current (f) (c) -lnk current varatons (f). (b) (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

7 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery 7 Imagnary Axs decreasng Nyqust Dagram Real Axs (a) Imagnary Axs Nyqust Dagram Nyqust Dagram (b) decreasng Real Axs abccr v abc abc v uabc v labc decreasng Imagnary Axs Real Axs (c) Fg.9. Nyqust dagram of -lnk voltage varatons for < due to: (a) d-component varatons of MMCs currents (f), (b) q-component varatons of MMCs currents (f), and (c) -lnk current varatons (f). Fg.. Overall structure of the proposed controller. u dq u dq vudq v ldq SmPower package of Matlab software s utlzed to execute ths assessment process as structured n Fg. For the modulaton method, SLPWM technque s selected to synthesze gate swtchng sgnals for MMCs. To show effectvely the mpact of DLM on the stablty of the proposed controller, two smulaton processes wll be consdered. As a common operaton n both processes, frstly the proposed HVDC system works n steady state and each MMC s responsble to supply actve and reactve power requred by the respectve loads. Then, n the second tme of each smulaton, load changes take place at t=.4s and t=.6 s for MMC and MMC, respectvely, n whch, n the frst process DLM s not used, whle the completed proposed controller wth DLM s employed n the second process. The results are presented and dscussed n the followng secton. A. -lnk and ac voltages evaluaton Fgs. and shows SM voltages and also and ac sde voltages of MMCs n two smulaton processes: wthout and wth DLM. As can be observed, approprate steady-state operaton for -lnk voltage, upper and lower SMs voltages, and ac sde voltages of MMCs are acheved wth DLM. TABLE I UNITS FOR MAGNETIC PROPERTIES fac 6 HZ C 4mF fs khz Cf 65µF v 8kV P MW vc kv Q MVAR L 45mH Transformer 8kV/kV ( /Y) power ratng R.Ω MMC load I 75MW,-5MVAR Lt mh MMC load II 85MW,8MVAR Rt Ω MMC load I MW,7MVAR n 6 MMC load II 5MW,5MVAR Fg. shows that not usng DLM n the proposed controller leads to the dvergence of MMCs voltages from ts desred values when the load changes happen for MMC and MMC at t=.4s and t=.6s, respectvely. Fg. demonstrates the accurate operaton of the proposed controller ncluded DLM n both dynamc and steady states. Accordng to ths fgure, n response to a transent varaton, the lnk voltage s kept n desred value wth small devatons. v (kv) v sm (kv) v sm (kv) v a (kv) v a (kv) Fg.. SM voltages and and ac sde voltages of MMCs wthout DLM (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

8 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery 8 v (kv) v sm (kv) v sm (kv) v a (kv) v a (kv) Tme [s] Fg.. SM voltages and and ac sde voltages of MMCs wth DLM. also nfluenced by the functon of keepng output voltages of MMC snusodal and balanced. The d and q components of MMC currents are shown n Fgs. 5 and 6. As can be seen, these components become unstable when DLM s not consdered n the proposed control technque. When DLM s used n dynamc state operaton of the proposed HVDC system, these currents move on ther desred values wth a small transent response tme. C. Actve and reactve power sharng assessment The proposed control technque of MMC s also responsble to provde the actve and reactve power demanded from the proposed HVDC system. MMCs actve and reactve power waveforms are llustrated n Fgs. 7 and 8. Frstly, MMC and MMC are amed to supply loads of MW+j()MVAR and MW+j.7MVAR respectvely. Then, another set of loads as 8MW+j9MVAR and 7MW+j4MVAR are connected to MMC and MMC respectvely at t=.4s and t=.6s, respectvely. Fg. 7 verfes that the proposed controller wthout DLM cannot lead to a stable actve and reactve power sharng for MMCs n dynamc operatng conon.. In adon, durng both dynamc and steady operaton of the proposed controller, the upper and lower SMs voltages follow the reference value of v / 6 wth acceptable fluctuatons. The approprate ac sde voltages of MMCs are also obtaned as shown n Fg.. The MMC should act as an nverter and consequently generaton of snusodal and balanced ac voltages s a man duty of MMC, whch s completely performed as depcted n Fg.. B. Analyss of MMC currents To verfy sutable performance of the proposed control technque for mnmzng MMC crculatng currents and regulatng -lnk and ac currents of MMCs, Fgs. and 4 can be referred to. Fg. shows the smulaton results of the proposed MMCbased HVDC system under operaton of the proposed controller wthout DLM. As t can be understood from ths fgure, when load changes take place for MMCs, the controller wthout DLM s not able to keep the proposed system n stable operaton and consequently the currents of MMC and MMC become unstable at t=.4 s and t=.6 s, respectvely. From crculatng current waveforms of Fg. 4, t can be derved that mnmzng these currents n both MMCs are properly done by the steady state secton of the proposed controller and subsequently, n case of sudden loads changes, DLM fully provdes dynamc control requrements n order to keep the currents at the mnmzed values. Moreover, Fg. 4 shows the lnk currents of MMC and MMC respectvely n both dynamc and steady states. Consderng the dynamc step change tme of t=.4s and t=.6s for MMC and MMC respectvely, the duraton of transent tme and transent error values are nsgnfcant n the second smulaton process wth DLM. Also the ac-sde currents of MMCs are shown n Fg. 4. As can be seen, current waveforms and ther changes are proportonal to the nstantaneous needs of the load. They are cra (ka) cra (ka) (A) (A) a (ka) x 4 x a (ka) Fg.. Crculatng, -lnk and ac-sde currents of the nterfaced MMCs wthout DLM (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

9 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery 9. cra (ka) d (A) cra (ka) (A) (A) a (ka) a (ka) Fg. 4. Crculatng, -lnk and ac-sde currents of the nterfaced MMCs wth DLM. d (A) q (A) d (A) q (A) x x x x Fg. 5. d and q components of MMCs currents wthout DLM. q (A) d (A) q (A) Fg. 6. d and q components of MMCs currents wth DLM. On the other hand, as can be seen n Fg. 8, MMCs actve power orentaton s n the drecton of ts respectve load actve power. Also the dynamc change of loads s hghly compensated wth a fast transent response, due to the proper controlled reacton operated by the desgned DLM as depcted n Fg. 8. Moreover, accordng to Fg. 8, the reactve power njecton by MMC s due to the presence of respectve reactve loads n both steady state and dynamc operatng conons of the proposed HVDC system, even though the scenaro s dfferent for MMC. Fg. 8 also shows that due to presence of flter capactance at the output of MMC for the am of achevng the desred snusodal voltages, MMC consumes reactve power. IX. CONCLUSION Ths paper presented a d-q frame based model of MMC- HVDC wth sx ndependent dynamcal state varables, ncludng ac and crculatng currents and also -lnk voltage, to effectvely obtan the swtchng functons of MMC as well as for accurate crculatng current control. Based on the reference values of selected state varables, the MMC swtchng functons under steady state were obtaned to regulate the MMC operaton n ths state. Moreover, DLM was employed to develop dynamc parts of swtchng functons to reach globally asymptotcally stablty. In fact, usng DLM leads to a proper operaton of the desgned controller and a better stablzaton of MMC-HVDC aganst dynamc changes. Then, capablty curve analyss of the nterfaced MMCs n the proposed system was carred out by consderng the effects of -lnk current changes. Furthermore, effects of MMC output and -lnk current varatons on the -lnk voltage stablty were evaluated n detal. Fnally, the valy of the proposed controller for the proposed MMC-HVDC system was thoroughly verfed and demonstrated by analyzng the smulaton results acheved n Matlab/Smulnk envronment modelng (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

10 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery P (MW) Q (MVAR) P (MW) P l (MW) P f (MW) Q (MVAR) Q l (MVAR) Q f (MVAR) Fg.7. MMC and MMC actve and reactve power waveforms wthout DLM. REFERENCES [] H. Fehr, A. Gensor, and M. Muller, Analyss and trajectory trackng control of a modular multlevel converter, IEEE Trans. Power Electron., vol., no., pp , Jan. 4. [] U. N. Gnanarathna, A.M. Gole, and R. P. Jayasnghe, Effcent modelng of modular multlevel HVDC converters (MMC) on electromagnetc transent smulaton programs, IEEE Trans. Power Del.,vol. 6, no., pp. 6 4, Jan.. [] M. Vatan, M. Hovd, M. Saeedfard, Control of the Modular Multlevel Converter Based on a Dscrete-Tme Blnear Model Usng the Sum of Squares Decomposton Method, IEEE Trans. Power Del., vol., no. 5, pp , Oct. 5. [4] E. Pouresmael, M. Mehrasa, M. A. Shokrdehak, E. Rodrgues, and J. P. S. Catalao, Control of Mult Modular Converters for Integraton of Dstrbuted Generaton Sources nto the Power Grd, n Proc. IEEE Internatonal Conference on Smart Energy Grd Engneerng (SEGE), pp. -6, Aug. 5. P (MW) Q (MVAR) P (MW) P l (MW) P f (MW) Q (MVAR) Q l (MVAR) Q f (MVAR) Fg.8. MMC and MMC actve and reactve power waveforms wth DLM. [5] S. Du, J. Lu, and T. Lu, Modulaton and Close-loop based DC Capactor Voltage Control for MMC wth Fundamental Swtchng Frequency, IEEE Trans. Power Electron., vol., no., pp. 7-8, Jan. 5. [6] H. Saad, J. Peralta, S. Dennetere, J. Mahseredjan, J. Jatskevch, J.A, Martnez, A. Davoud, M. Saeedfard, V. Sood, X. Wang, J. Cano, and A. Mehrz-San, Dynamc Averaged and Smplfed Models for MMC- Based HVDC Transmsson Systems, IEEE Trans. Power Del., vol. 8, no., pp. 7-7, July.. [7] M. Saeedfard, and R. Iravan, Dynamc Performance of a Modular Multlevel Back-to-Back HVDC System, IEEE Trans. Power Del., vol. 5, no. 4, pp. 9-9, Oct.. [8] A. Beddard, M. Barnes, and R. Preece, Comparson of Detaled Modelng Technques for MMC Employed on VSC-HVDC Schemes, IEEE Trans. Power Del., vol., no., pp , Aprl. 5. [9] L. Harnefors, A. Antonopoulos, S. Norrga, L. Angqust and H-P Nee, Dynamc Analyss of Modular Multlevel Converters, IEEE Trans. Industral Elec., vol. 6, no. 7, pp , July.. [] P. Hu, and D. Jang, A Level-Increased Nearest Level Modulaton Method for Modular Multlevel Converters, IEEE Trans. Power Electron., vol., no. 4, pp , Aprl. 5. [] B. L, R. Yang, D. Xu, G. Wang, W. Wang, and D. Xu, Analyss of the Phase-Shfted Carrer Modulaton for Modular Multlevel Converters, IEEE Trans. Power Electron., vol., no., pp. 97 -, Jan (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

11 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully eed. Content may change pror to fnal publcaton. Ctaton nformaton: DOI.9/TPWRD , IEEE Transactons on Power Delvery [] A. Dekka, B. Wu, and N.R Zargar, A Novel Modulaton Scheme and Voltage Balancng Algorthm for Modular Multlevel Converter, IEEE Appled Power Electroncs Conference and Exposton (APEC), pp. 7, March. 5. [] W. Wang, A. Beddard, M. Barnes, and O. Marjanovc, Analyss of Actve Power Control for VSC HVDC, IEEE Trans. Power Del., vol. 9, no. 4, pp , Aug. 4. [4] S. Lu, Z. Xu, W. Hua, G. Tang and Y. Xue, Electromechancal Transent Modelng of Modular Multlevel Converter Based Mult- Termnal HVDC Systems, IEEE Trans. Power System., vol. 9, no., pp. 7-8, Jan. 4. [5] G. Bergna, E. Berne, P. Egrot, P. LefrancA. Arzandé, J.C Vanner, and M. 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Power Del., vol., no., pp. - 8, Feb. 5. [4] Y. L, E. A. Jones, and F. Wang, The Impact of Voltage-Balancng Control on Swtchng Frequency of the Modular Multlevel Converter, IEEE Trans. Power Electron., vol., no. 4, pp , Aprl 6. [5] R. Lzana, M.A Perez, S. Bernet, J.R Espnoza, and J. Rodrguez, Control of Arm Capactor Voltages n Modular Multlevel Converters, IEEE Trans. Power Electron., vol., no., pp , Feb 6. [6] R. Lzana, M.A Perez, D. Arancba, J.R Espnoza, and J. Rodrguez, Decoupled Currents Model and Control of Modular Multlevel Converters, IEEE Trans. Industral Electron., vol. 6, no. 9, pp , Sep 5. [7] M. Mehrasa, E. Pouresmael, M.F. Akorede, B.N Jørgensen, and J.P.S Catalao, Multlevel Converter Control Approach of Actve Power Flter for Harmoncs Elmnaton n Electrc Grds, Energy., vol. 84, pp. 7-7, May 5. Majd Mehrasa receved the B.Sc. and M.Sc. degrees n electrcal engneerng from the Unversty of Mazandaran, Babol, Iran, n 6 and 9, respectvely. He s currently wth the Young Researchers and Elte Club, Sar Branch, Islamc Azad Unversty, Sar, Iran. Hs current research nterests nclude power electronc applcatons to power system and the use of nonlnear control theores for varous power converters. Edrs Pouresmael (M 4) receved the B.Sc. and M.Sc. degrees n Electrcal Engneerng from the Unversty of Mazandaran, Babol, Iran, n and 5, respectvely. He receved the Ph.D. degree n Electrcal Engneerng wth honor from the Techncal Unversty of Catalona. Barcelona Tech. (UPC), Barcelona, Span, n. After hs Ph.D. he joned the Department of Electrcal & Computer Engneerng, at the Unversty of Waterloo, Canada, as a Postdoctoral Research Fellow, and later he joned the Department of Electromechancal Engneerng, at the Unversty of Bera Interor, Portugal. He was an Assocate Professor n Energy Technology Engneerng at the Unversty of Southern Denmark (SDU), Denmark, and Senor Researcher at Abengoa Research, Span. He s currently a Senor Researcher wth the Insttute Super Tecn Lsbon (INESC-ID), Portugal. Hs research nterests nclude applcaton and control of power converters n ntellgent power systems, stablty analyss of power converters n power system, and ntegraton of large-scale renewable energy sources nto the lownerta power grd. Sasan Zabh receved the B.Sc. (Eng.) and M.Sc. (Eng.) degrees n Electrcal Engneerng Telecommuncaton and Power respectvely n and 6 from Techncal Unverstes n Iran. After years n ndustry and academa, he started a Ph.D. program n Power-Electroncs at Queensland Unversty of Technology (QUT), Brsbane, Australa. He completed hs PhD n, and from to, he was a lecturer wth QUT. Snce he s workng as R&D desgn specalst for ABB/Australa n CoC for McroGrd, renewable ntegraton, and dstrbuted generaton. Hs research nterests nclude power-electroncs topologes, control and applcatons, pulsed power, McroGrd, renewable energy ntegraton, and dstrbuted generaton. João P. S. Catalão (M 4-SM ) receved the M.Sc. degree from the Insttuto Superor Técnco (IST), Lsbon, Portugal, n, and the Ph.D. degree and Habltaton for Full Professor ("Agregação") from the Unversty of Bera Interor (UBI), Covlha, Portugal, n 7 and, respectvely. Currently, he s a Professor at the Faculty of Engneerng of the Unversty of Porto (FEUP), Porto, Portugal, and Researcher at INESC TEC, INESC-ID/IST-UL, and C-MAST/UBI. He was the Prmary Coordnator of the EU-funded FP7 project SNGULAR ("Smart and Sustanable Insular Electrcty Grds Under Large- Scale Renewable Integraton"), a 5.-mllon-euro project nvolvng ndustry partners. He has authored or coauthored more than 475 publcatons, ncludng 57 journal papers, 8 conference proceedngs papers, book chapters, and 4 techncal reports, wth an h-ndex of 7 and over ctatons (accordng to Google Scholar), havng supervsed more than 45 postdocs, Ph.D. and M.Sc. students. He s the Eor of the books enttled Electrc Power Systems: Advanced Forecastng Technques and Optmal Generaton Schedulng and Smart and Sustanable Power Systems: Operatons, Plannng and Economcs of Insular Electrcty Grds (Boca Raton, FL, USA: CRC Press, and 5, respectvely). Hs research nterests nclude power system operatons and plannng, hydro and thermal schedulng, wnd and prce forecastng, dstrbuted renewable generaton, demand response and smart grds. Prof. Catalão s an Eor of the IEEE TRANSACTIONS ON SMART GRID, an Eor of the IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, and an Assocate Eor of the IET Renewable Power Generaton. He was the Guest Eor-n-Chef for the Specal Secton on "Real-Tme Demand Response" of the IEEE TRANSACTIONS ON SMART GRID, publshed n December, and the Guest Eor-n-Chef for the Specal Secton on "Reserve and Flexblty for Handlng Varablty and Uncertanty of Renewable Generaton" of the IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, publshed n Aprl 6. He was the recpent of the Scentfc Mert Award UBI-FE/Santander Unverstes and the Scentfc Award UTL/Santander Totta. Also, he has won 4 Best Paper Awards at IEEE Conferences (c) 6 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.