Energy-Based State-Space Representation of Modular Multilevel Converters with a Constant Equilibrium Point in Steady-State Operation

Transcription

1 Energy-ased State-Spae Representaton of Modular Multlevel Converters wth a Constant Equlbrum Pont n Steady-State Operaton Glbert ergna-daz, Jon re Suul, Member IEEE, and Salvatore D'ro bstrat he nternal urrents and voltages of Modular Multlevel Converters (MMCs) ontan multple frequeny omponents n steady state operaton and reman tme-perod even when transformed nto a synhronously rotatng referene frame. hs prevents a straghtforward state-spae representaton where a onstant equlbrum pont s reahed and all state varables onverge to onstant values under steady-state ondtons. Suh steady-state tme-nvarant (SSI) representatons are needed for lnearzaton and egenvaluebased analyss of small-sgnal stablty. hs paper presents an energy-based model of an MMC wth a modulaton strategy where the nserton ndes are ompensated for the osllatons n the sum arm voltage. he formulaton of the model allows for dervng, by the applaton of Park transformatons at three dfferent frequenes, a SSI representaton that aurately aptures the nternal dynams of the MMC. hs model an be smplfed to a redued order model that mantans aurate reproduton of the external behavor at the a- and d-sdes whle negletng some of the nternal dynams. he valdty and auray of these two SSI MMC models are verfed by tmedoman smulatons and ther utlzaton for egenvalue-based analyss of MMC dynams s demonstrated by examples. Index erms HVDC ransmsson, Modular Multlevel Converter, Park ransformatons, State-spae Modellng NOMENCLURE 1) MMC and system varables u, l Current n upper (u) and lower (l) arm, v Crulatng urrent, and a grd-sde urrent w u, w l ggregated apator energy n upper (u) and lower (l) arm w Σ, w Δ Capator energy sum and dfferene between upper and lower arms Manusrpt reeved Marh, 016; revsed Deember 19, 016, Marh 8, 017 and May 0, 017; aepted June 30, 017. Date of publaton: xxxxxx xx, 017; date of urrent verson: ugust 30, 014. hs work was supported by the projet by the projet Proteton and Fault Handlng n Offshore HVDC Grds, (ProOfGrds), fnaned by the Researh Counl of Norway s RENERGI program and the ndustry partners; EDF, Natonal Grd, Semens, Statkraft, Statnett, Statol and NVE. Glbert ergna-daz was wth SINEF Energy Researh, 7465 rondhem, Norway and s now wth the Department of Eletr Power Engneerng, Norwegan Unversty of Sene and ehnology, 7495 rondhem, Norway, (E-mal: glbert.bergna@ntnu.no) Jon re Suul and Salvatore D'ro are wth SINEF Energy Researh, 7465 rondhem, Norway, (E-mal: Jon..Suul@sntef.no, salvatore.daro@sntef.no) Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Objet Identfer n u, n l Upper and lower arm nserton ndexes v u, v l Voltage of the th sub-module apator n the upper or lower arm σ v uσ, v l Upper and lower arm apator voltage sum v u, v l Upper and lower arm output voltages u, e v Voltages drvng rulatng and a-sde urrents v o, v g Voltage at the pont of ommon ouplng and voltage of a-grd hévenn equvalent v d Voltage at the d termnals of the MMC Indates referene values n the ontrol system ) Man system parameters R a, L a MMC arm resstane and ndutane R f, L f Equvalent MMC output resstane and ndutane, representng transformer seres mpedane and any addtonal flters C o Equvalent apatane at onneton to a grd R eq, L eq Equvalent a resstane and ndutane defned as R eq = R a / + R f, L eq = L a / + L f R g, L g Equvalent grd-sde resstane and ndutane C Capatane of a MMC sub-module N Number of sub-modules n an arm C eq Equvalent MMC arm apatane defned as C eq = C /N C d Equvalent apatane at the d termnals 3) Referene frame orentatons ab Natural three-phase oordnates z -ω Synhronous referene frame rotatng at ω z +ω Synhronous referene frame rotatng at +ω +3ω Synhronous referene frame rotatng at +3ω I. INRODUCION he Modular Multlevel Converter (MMC) s emergng as a preferred topology for Voltage Soure Converter (VSC) -based HVDC transmsson shemes [1]-[5]. However, the modellng and the ontrol of the MMC s n general more hallengng than for two- or three-level VSC onfguratons, sne the MMC s haraterzed by a hgh number of ndependent swthng elements and by addtonal nternal dynams related to the rulatng urrents flowng through the submodules of eah phase [6]. Moreover, eah phase of an MMC behaves as a sngle-phase mult-level onverter, where the double frequeny osllatons n the power flow ause orrespondng flutuatons n the sub-module apator voltages. hus, even n steady state operaton, the nternal urrents and voltages of an MMC wll ontan multple frequeny omponents [7]. Sgnfant efforts have reently been dedated towards modellng and analyss of the MMC topology and ts ontrol.

2 Steady-State me-perod (SSP) Models MMC Swthng/ Dsontnous Models IG+ Dode ON/OFF Resstors Ideal Swthes sw(t) Swthng funton (Smplfaton) veragng MMC veraged Contnuous Model n ab Coordnates + n u(t) v σ Contnuous u (t) Inserton Indexes d ab f ( ab, ab, t) x = x u Contnuous non-lnear State-Spae representaton (Mnor smplfaton) ransformaton Steady-State me-invarant (SSI) Models Non-Lnear SSI State-Spae Models Lnearzed Small-Sgnal State-Spae Models d z z z f (, ) x = x u d 0 0 d z Steady State: 0 x =0 z z z x x x 0 Lnearzaton x = x x + x u z z z Explt representaton of sub-modules and swthng events - Computatonally ntensve - hévenn reduton for faster smulaton Sutable for Eletro- Magnet ransent (EM) smulatons - ssessment of omponent stress - Desgn valdaton - Evaluaton of submodule ontrol strateges - Fault studes ssumes deal apator voltage balanng Equvalent arm apatane and ontnuous-tme averaged arm voltages - Faster and less omputatonally ntensve tme-doman smulatons - Explt analytal expressons for arm voltage and energy balane Sutable for analyss and desgn of ontrollers - Current ontrol - Sum voltage or energy balane ontrol - Outer loop power flow ontrollers nalyss of state-spae models n tme-perod framework n overvew of dfferent types of models, how they orgnate from the MMC topology and ther typal range of applaton s shown n Fg. 1. Indeed, detaled swthng models wth explt representaton of all sub-module apators of an MMC, nludng models wth hévenn equvalent representaton of eah arm aordng to [8], are ntended for tme-doman smulatons. If the ndvdual representaton of eah sub-module apator voltage s not neessary, smplfed swthng funton models an be ntrodued to redue the requred smulaton tme [9], [10]. Contnuous tme average models an be obtaned by approxmatng the swthng effets wth a ontnuous sgnal and assumng perfet balanng between the sub-module apator voltages [6], [7], [11],[1]. Suh average models allow for effent tme-doman smulaton and lead to smple analytal expressons for representng eah arm of an MMC. hus, they are ommonly utlzed n mathematal analyss for ontrol system desgn and for understandng the nternal dynams of eah phase of the MMC. Sne suh models represent the phase and arm quanttes of the MMC, steadystate operaton s haraterzed by an orbt of the state-spae varables and not by a onstant equlbrum pont. hus, the models wll nherently have Steady-State me-perod (SSP) haratersts, as ndated n the left part of Fg. 1. Stablty analyss based on SSP average models requres advaned methods spefally developed for tme-perod systems, as reently appled to an MMC n [13]. lthough the varous SSP models ndated to the left of Fg. 1 are sutable for most purposes related to tme-doman smulaton and ontroller desgn, or for dynam analyss of Derved from average model n statonary ab oordnates - ransformed nto synhronously rotatng z referene frames - Smlar to statonary frame average models n auray and omputatonal requrements Non-lnear system analyss and ontrol desgn requrng onstant varables n steady-state - Calulatng steady-state operaton - Estmatng regon of attraton - Passvty based ontrol Prerequste for lnearzaton at equlbrum Fg. 1 Overvew of MMC modellng approahes and ther areas of applaton Can be obtaned from a non-lnear SSI modes at any spef equlbrum pont Vald for a small regon around the lnearzaton pont nalyss of system dynams and stablty by lnear methods - Small-sgnal stablty assessment by egenvalue analyss - Partpaton fator analyss - Egenvalue parametr senstvty Compatble wth models used for analyss of small-sgnal stablty n large-sale power systems eah arm or phase of an MMC, they are not easly applable n establshed methods for system-orented analyss. Indeed, SSP average models of MMCs annot be lnearzed and utlzed for tradtonal egenvalue-based analyss ommonly appled n studes of small-sgnal stablty of power systems [14]. Instead, methods for system analyss that depend on lnearzaton, as well as many establshed tehnques for nonlnear stablty assessment or ontrol system desgn, assume as a prerequste the avalablty of a state-spae model where all stable operatng ponts are haraterzed by an equlbrum pont and all state varables onverge to onstant values n steady state operaton [14], [15]. hus, models to be utlzed for suh purposes should have Steady-State me-invarant (SSI) haratersts. Whle SSI representatons of two-level VSCs an be easly derved by applyng the Park transformaton, the multple frequeny omponents appearng n the arm urrents and apator voltages of the MMC prevent SSI representaton by transformaton nto a sngle Synhronous Referene Frame (SRF). hus, dervaton of MMC models wth SSI haratersts s stll objet of researh. Fg. 1 ndates how suh SSI-models should be derved from a orrespondng SSP average model by applyng approprate referene frame transformaton and smplfatons. he fgure also shows how a non-lnear SSI state-spae model s needed for obtanng a lnearzed small-sgnal model, as well as for alulatng the equlbrum pont where the model an be lnearzed. In the ontext of Fg. 1, several dfferent approahes for SSI state-spae representaton of three-phase MMCs have

3 been reently proposed, wth the am of obtanng lnearzed models for small-sgnal power system stablty analyss. frst approah has been to apply dynam phasor modellng to all the nternal eletral states of the MMC, as dsussed n [16] and [17], resultng n omplated hgh order models. nother approah has been to neglet parts of the nternal dynams of the MMC, and model manly the a-sde dynams n a SRF together wth a smplfed d-sde representaton, as n the models proposed n [18]-[0]. mong these studes, only the model from [19] nludes a representaton of the nternal energy storage apaty of the sub-module apators and ther dynam mpat on the power transfer between the a and d termnals. However, [19] dd not derve any SSI state-spae representaton that ould be sutable for lnearzaton. n approah based on further smplfatons was appled n [18] and [0], assumng an deal power balane between the a- and d-sdes of the MMC n a smlar way as for two-level VSCs. hs mples sgnfant nauraes n the model, sne the transent responses of the nternal varables and ther ontrollers are not represented. hus, suh models are only sutable for studyng slow dynams. o address these lmtatons, more detaled dynam state spae models have been proposed n [1]-[6]. wo dfferent sets of assumptons and approxmatons are appled n the dervaton of these publatons:. he models presented n [1]-[4] assume that the MMC s operated wth a Crulatng Current Suppresson Controller (CCSC) mplemented n a negatve sequene double frequeny SRF, for elmnatng the seond harmon omponents of the rulatng urrent [7]. he dfferent frequeny omponents of the arm urrents and the equvalent arm apator voltages are modelled by separate state-varables n ther assoated SRFs by applyng phasor-based harmon superposton. hus, the ouplngs between the varous frequeny omponents are trunated as a frst step of the model dervaton. hese models have revealed nstablty problems assoated wth nteraton between the rulatng urrents, the nternal apator voltages and the d-sde voltage as dsussed n [], [4]. However, the modellng approahes from [1]-[4] are not drely sutable for representng MMCs wth energy-based ontrol strateges as wll be explaned n seton II of ths paper. he approah presented n [5], [6] s based on a smplfed representaton of the MMC, where only the aggregated dynams of the zero sequene rulatng urrent and the total energy stored n the apators of the MMC are modelled. hs approah s vald when the modulaton ndes for the MMC arms are alulated to ompensate for the voltage osllatons n the nternal equvalent arm apator voltages, as assumed n [6], [7], [8]. hs modulaton strategy wll be referred to as Compensated Modulaton (CM) and ts mplatons for the modellng wll be further elaborated n seton II. hese resultng models an aurately represent the external behavour of the MMC at the a- and d-sdes, but do not nlude the nternal dynams. hs paper demonstrates how an energy-based modellng approah nspred by [5]-[6] an apture also the nternal urrent and energy dynams of an MMC. he resultng model s derved from an average model wth the sum and the dfferene of the arm energes n eah phase as state varables and results n a omplete and aurate SSI representaton of the MMC under the assumpton of ompensated modulaton. hus, the model overs a ase that has not been prevously studed n the avalable lterature. Furthermore, the man ontrbuton of the presented approah s that t nherently takes nto aount the ouplng between the varous frequeny omponents of the MMC dynams by a SSI state-spae representaton. It s also shown how the detaled SSI model an be smplfed to the redued order model from [5]-[6] by gnorng the states representng the osllatng omponents of the nternal MMC varables. he valdty of these two models are demonstrated by tme-doman smulatons n omparson to the SSP nonlnear tme-doman model of the MMC that was used as startng pont for the model dervaton. Fnally, t s demonstrated how these state-spae models an be lnearzed and utlzed for analyzng the small-sgnal dynams and ontrol system tunng of the MMC by applyng egenvaluebased tehnques. II. MMC OPOLOGY ND INSERION INDEX CLCULION he model and the defntons that wll be used as a startng pont for dervng an MMC model wth SSI haratersts are brefly outlned n the followng. hs seton also dentfes how the dervatons presented n ths manusrpt ontrbutes to the SSI representaton of MMCs beyond what s avalable n prevous lterature.. verage Model of the hree-phase Modular Multlevel Converter he general topology of a three-phase MMC s shown n Fg.. In ths ase, operaton n a able-based HVDC transmsson system s assumed, resultng n an equvalent apatane C d at the d termnals. he followng nomenlature and onventons are appled for modellng of the MMC: tal lower ase letters 'x' represent sngle varables, tal-bold letters 'x' represent vetors and matres, whereas non-tal bold letters 'x' represent the omplex spae vetor x = x d +j x q. Wth the above onventons, the man expressons assoated wth a gener phase k a,b, of an MMC are gven by (1)-(5) [6]. uk lk vk uk lk, k (1) N vulk, vulk,, vulk, nulk, vulk, () 1 vlk vuk nlk vlk nuk vuk evk (3) vlk vuk nlk vlk nuk vuk uk (4) C w u, lk vu, lk, wk wuk wlk, w k wuk wlk (5) N ssumng a fast apator voltage balanng algorthm, eah arm output voltage v u,lk an be expressed by the produt of the nserton ndex n resultng from a modulaton algorthm and the sum arm apator voltage v σ u,lk, as expressed by the

4 Cd v d DC rm e v ua la L a L a va ub lb La L seond part of () [7]. hus, the voltage e v, whh drves the a sde urrents from the MMC, an be expressed by (3). Smlarly, the nternal voltage of eah leg u, whh drves the rulatng urrent, s defned as u and an be expressed by (4). he energy w stored n the apators of eah arm s gven by (5), whh also defnes the sum energy w Σ and the energy dfferene w Δ between the upper and lower arms [6], [7].. Calulaton of nserton ndexes: defnton of ompensated vs. un-ompensated modulaton he spefaton of how the upper and lower arm nserton ndexes are alulated s rtal for the development of MMC models. ommon approah for alulatng the nserton ndexes s gven by [7], [30]: ev u ev u nu, nl Vd, nom Vd, nom (6) lternatvely, the measured d voltage v d an be used as the denomnator n (6) [9], [13]. However, as long as the value n the denomnator s onstant durng steady-state operaton, the nserton ndex alulaton aordng to (6) wll not nlude any ompensaton for the ontnuous osllatons n the arm apator voltages. hus, the nfluene of these osllatons wll have to be ompensated by the ontrol loops. Suh approahes for nserton ndex alulaton an be referred to as "Un- Compensated Modulaton" (UCM) [9]. hs paper wll onsder the ase when the nserton ndexes are alulated by dvdng the referene ontrol voltages e v and u by the measured or estmated tme-varyng aggregated voltage n the orrespondng arm, v σ u,l [6], [7]. s defned n [9], ths approah wll be referred as Compensated Modulaton (CM) and an be expressed by: ev u ev u nu, n l vu vl (7) Wth the CM approah, the dvson of the output of the ontrollers (.e. ± e v + u ) by v σ u,l wll ompensate for the nonlnearty aused by the produt of the nserton ndes and the tme-perod sum arm voltages n (3) and (4). hus, t an be onfrmed by substtutng (7) nto (3) and (4) that the voltages e v and u that are drvng the grd-sde urrents and the rulatng urrents respetvely, wll be equal to the voltage a u vb l Leg La L a Fg. opology of a three-phase MMC v L f Submodule a b C vo n referene outputs, e v and u, of the orrespondng ontrollers, as expressed by: ev ev; u u (8) s wll be shown n the followng setons, ths haraterst s useful for dervng an energy-based SSI representaton of MMCs wth CM-based ontrol system mplementatons. C. Seleton of SSI modellng approah aordng to nserton ndex alulaton It s demonstrated n [9] that energy-based models are not sutable for dervng SSI representaton of MMCs wth UCMbased ontrol, whle voltage-based formulatons are unsutable for MMCs wth CM-based ontrol [9]. Indeed, voltage-based modellng approahes dependng on harmon superposton were appled for obtanng the SSI representatons and the orrespondng lnearzed models of MMCs wth UCM-based ontrol n [17], [1]-[4]. he resultng models represent the nternal dynams of an MMC n z-varables assoated wth the SRFs orrespondng to eah osllaton frequeny of the state varables n steady-state. n alternatve voltage-based modellng approah for avodng the approxmatons assoated wth harmon superposton was proposed n [9]. In ontrast to the voltage-based MMC models n [17], [1]- [4], smplfed energy-based MMC models for the ase of CM-based ontrols have been proposed n [5], [6]. However, no energy-based models wth SSI representaton of the nternal dynams of the MMC n approprate z referene frames are avalable n the lterature. n overvew of how voltage-based or energy-based modellng approahes are sutable for dervng SSI representatons aordng to the seleted strategy for nserton ndex alulaton s shown n Fg. 3. s ndated n the fgure, the man ontrbuton of ths paper s to fll a gap n the avalable lterature by presentng the detaled dervaton of an energy-based state-spae model wth SSI representaton of the nternal dynams of an MMC wth CM-based ontrol. Furthermore, t wll be shown how smplfaton of the derved model results n the zero-sequene models from [5], [6]. III. MMC SE-SPCE MODELLING FOR OINING IME- INVRINCE IN SEDY-SE In the followng subsetons, a proedure for dervng a detaled energy-based SSI representaton of an MMC wth CM-based ontrol s presented. It s frst shown how the average model n the statonary referene frame should be expressed to obtan separaton of the domnant frequenes appearng n the MMC steady-state operaton. On ths bass, step-by-step dervatons are presented for transformng the three-phase varables of the average model nto a set of SRFs. he resultng model wll nherently nlude the ouplng between the dfferent frequeny omponents, even f all statevarables wll settle to onstant values n steady-state operaton. Fnally, t wll be shown how the derved model an be smplfed to the redued order model of an MMC wth CMbased ontrol frst presented n [5].. Mathematal dervaton of a steady-state tme-nvarant MMC model based on energy formulaton o aheve SSI haratersts wthout dependng on harmon superposton, the MMC varables should be

5 Steady-State me-invarant (SSI) Non-Lnear Models Compensated Modulaton MMC z Model hs Researh Contrbuton Smplfed Zero Sequene Model Un-Compensated Modulaton MMC z Model Lnearzed State-Spae Models d 0 0 x= x x+ x u Lnearzed State- Spae representaton Lnearzed MMC z Model under CM Lnearzed MMC zero-sequene Model under CM Lnearzed MMC z Model under UCM Fg. 3 Summary of relatons between dfferent models of MMCs wth SSI haratersts expressed suh that state varables assoated wth the dfferent frequeny omponents an be separated and transformed nto ther orrespondng SRFs whle retanng the ouplng wth varables assoated wth other frequeny omponents. y hoosng a -Δ energy-based formulaton aordng to (5) and onsderng the steady-state haratersts of the MMC aordng to [6], [7], the varables of the MMC an be separated nto two groups, where eah group s assoated wth a sngle frequeny as: ab 1 z ab 1 z ab 1 z : ; u u ; w w (9) ab 1 ab 1 ab 1 z : ; e e ; w w v v v v hus, the varables an be lassfed as those ontanng osllatons at ω (, w and u ), and those osllatng at +ω ( v, w Δ and e v ). Furthermore, (9)shows how the statonary frame varables an be expressed from ther equvalent SSI z varables. he transformaton matrxes ω and ω are representng the Park transformatons, wth phase angles synhronzed wth the grd voltage and ts orrespondng negatve sequene double frequeny, respetvely. he formulaton of the MMC varables suh that ths ntal separaton of frequeny omponents an be aheved onsttutes the bass for the proposed modellng approah, as llustrated n Fg. 4. hs fgure ndates that Park transformatons at dfferent frequenes wll be used to derve dynam equatons for equvalent z varables that are SSI n ther respetve referene frames. In addton, a Park transformaton 3ω at three tmes the grd frequeny wll be used to ensure a SSI representaton of the zero sequene of the energy dfferene, as wll be dsussed n subseton III..3). In the remander of ths seton, the mathematal dervaton of SSI state equatons representng the dynams of a CM-ontrolled MMC wll be desrbed and expressed by usng the defntons n (9) aordng to the approah llustrated by Fg. 4. lthough the mathematal dervatons nvolve several steps, the resultng model s relatvely smple as summarzed n seton III.. Smlar proedures an also be applable for obtanng SSI haratersts of voltage-based MMC models for the ase of UCM-based ontrol, and for SSI representaton of statonary frame ontrol systems. 1) Energy Sum z Dynams he dynams of the energy sum w k for a gener phase k an be expressed aordng to the defnton ntrodued n [6]. When represented on vetor form, the sum energy dynams for the three phases are gven by MMC State varables for Current and Energy Voltage Varables SSI - Frequeny ransformaton Overvew 3 SSI ab w ab v ab e v ab z w w d ab ab ab w pv p (10) ab ab where p v and p are the vetors defned n (11). ab pv evava evbvb evv ab p uaa ubb u (11) Sne eah omponent of the vetor rows of w, p v and p osllates at twe the fundamental grd frequeny, (10) an be rewrtten n a SRF at ω, as: d z z z z w pv p J w (1) z-ω z-ω where p v and p are expressed n (13) and (14) respetvely. hese equatons show how the SRF varables are obtaned from multplaton of the orgnal vetor n phase oordnates by the ampltude-nvarant Park transformaton matrx. Furthermore, J ω s the ross-ouplng matrx obtaned by replang h= n (15). z ab p p e e e (13) v v va va vb vb v v z ab uaa ubb u w p p (14) 0 h 0 J h h 0 0 ; hn 0 (15) he grd-sde and rulatng urrents vk and k, whh appear n equatons (13) and (14) along wth the orrespondng voltages e vk and u k, an be expressed n ther respetve z rotatng frames at +ω and ω by usng the defntons gven n ab ab (9). Hene, substtutng the expressons for e v and v resultng from (9) nto equaton (13), and solvng the produt between -ω and the resultng vetor, yelds n: evd vd evq vq z 1 p v evd vq evq vd (16) evd vd evq vq Indeed, all varables n ths expresson wll settle to a onstant value n ther assoated SRF. hus, (16) s a steady-state tme- w z 90 w z 3 w z ab z ab u u v z e v Fg. 4 he proposed modellng approah based on three Park transformatons to aheve SSI representaton of MMC varables 3

6 nvarant expresson for the z omponents of the power flow from the grd-sde of the MMC. z-ω smlar proedure s repeated for p gven n (14). ab ab Replang eah row of u v and as defned n (9) nto (14), and expandng the multplaton wth -ω results n (17). It s mportant to note that unlke (16), equaton (17) ontans a set of 6 th order harmon terms. However, the ampltudes of the 6 th harmon terms are all defned by produts between d- and q-axs omponents of the rulatng urrents and the orrespondng voltages. Sne the ampltudes of u and are small, the produts between u and wll be very small ompared to any of the terms ontanng a multplaton wth the zero sequene voltage u z or the zero sequene urrent z. hus, these 6 th order harmon terms wll have neglgble nfluene on the power omponents defned by (17), and an be dsarded to aheve tme-nvarane n steady state. It should be noted that ths s the only approxmaton ntrodued n the dervatons of the SSI equatons for representng the MMC, and that tme-doman smulatons onfrm that ths smplfaton s not ompromsng the auray of the model. he energy sum dynams n z oordnates an now be expressed by (18), where (16) and the frst term of (17) have been substtuted nto (1). w d d z d w w q w z (18) 1 e vd vd evq vq uzd u d z w q 1 e vd vq evq vd uzq u q z w d 1 evd vd evq vq ud d u q q u z z mong these equatons, the omponents, w Σ,d, ω and w Σ,q, ω represent the seond harmon osllaton supermposed to the average sum energy n the three phases. Indeed, when expressed as a spae vetor or on omplex vetor, ω form (.e. w Σ = w Σ,d, ω + j w Σ,q, ω), these -omponents represent the three-phase seond harmon energy osllatons wthn the MMC. hus, the ampltude of the sum energy osllatons and the phase angle wth respet the referene frame orentaton of the model (.e. the phase angle deteted by a Phase Loked Loop) s gven by: w 1 q w w d w q, w tan (19) w d he varable w Σz represent the zero sequene omponent of the sum energy n the three phases, whh s assoated wth the average value or d-omponent of the total energy stored nsde the MMC. Consderng the relatonshps n (5), the dfferent omponents of the sum energy an also be dretly assoated wth the arm energes and the orrespondng sum arm voltages. ) Energy Dfferene z- Dynams he dervaton of the steady-state tme-nvarant equatons for the energy dfferene dynams of the MMC s relatvely smlar to the ase for the energy sum regardng ts omponents, yet very dfferent regardng the zero-sequene. fter presentng the frst steps of the dervaton, ths seton onsders only the -dynams, whereas the zero-sequene dynams are addressed n the subsequent seton. s for the energy sum, the dynam equaton for the energy dfferene w Δk of a gener phase k s defned aordng to [6]. When expressed on vetor form, the dynams of the three phases are gven by d ab ab ab w p 1 p (0) ab ab where p Δ1 and p Δ are defned by ab p1 evaa evbb ev (1) ab p uava ubvb uv Sne the man frequeny omponent of the energy dfferene dynams n steady state s the fundamental frequeny of the grd voltage, (0) an be re-wrtten n the SRF rotatng at +ω, yeldng n d z z z z w p p J w () 1 z+ω z+ω where p Δ1 and p Δ are expressed n (3) and (4) respetvely. hese equatons are obtaned by multplyng the orgnal vetor n the statonary ab referene frame by the Park transformaton matrx at +ω;.e., +ω. z ab 1 1 va a vb b v p p e e e (3) p p u u u (4) z ab a va b vb v ab Substtutng nto (3) the expressons for the voltage e v ab and the rulatng urrent that an be obtaned from the z+ω defntons gven n (9), the ndvdual elements of p Δ1 an be expressed as a funton of the z urrent and voltage omponents: evd d e vq q e vd z evd q e vq d e vq z z p 1 (5) evd d e os 3... vq q t... evd q e sn 3 vq d t Contrary to the power expressons gven n (16), (17) only the d- and q-axs omponents of (5) are tme-nvarant n steady state. Indeed, the zero-sequene omponent p Δ1z gven n (5) s tme-perod, wth thrd harmon osllatons n steady state. he orgn of ths thrd harmon omponent s the multplaton of varables ontanng fundamental frequeny and double frequeny omponents. Indeed, the zero sequene omponent of (5) shows that the ampltude of the thrd harmon osllatons depends on produts between the rulatng urrents and the a-sde voltage. hus, they annot be negleted n a detaled model of the MMC. z+ω z+ω Smlarly as for p Δ1, t s possble to express p Δ as a 1 1 u os 6 sn6 d d q q q d d q z d u d u u t u u t z z p 1 sn 6 1 uzq u q z ud d u q q t u os 6 q d u d q t 1 u 1 d d u q q u z z 0 0 (17)

7 funton of z urrents and voltages. hs s obtaned by ab ab replang the expressons for v and u aordng to (9), z+ω nto (4). y solvng for the ndvdual elements of p Δ, (4) an be expressed as a funton of the z urrent and voltage omponents, as gven by 1 1 ud u z vd uq vq 1 1 uq vd ud u z vq z (6) p 1 ud os3... vd uq vq t 1... uq sn 3 vd ud vq t s for p Δ1z, the zero-sequene omponent p Δz, expressed n (6), s not tme-nvarant n steady state. hus, the zero sequene omponents n (5) and (6) wll be further analyzed n the followng sub-seton. Consderng only the d- and q-axs omponents of the power vetors from (5) and (6), and substtutng the obtaned expressons nto () results n the dynam equatons for the d- and q-axs energy dfferene as expressed by (7). hese two state equatons do not requre any further smplfatons sne all ther elements are already SSI. Indeed, w Δ,d,+ω and w Δ,q,+ω represent the fundamental frequeny osllatons of the energy dfferene between the upper and the lower arms of the MMC. hus, the ampltude and phase angle of these osllatons s aurately represented by the energy,+ω dfferene omponents (.e. w Δ = w Δ,d,+ω + j w Δ,q,+ω). ased on (5), t an also be understood how these sgnals are dretly assoated to the fundamental frequeny osllaton n the sum arm energes and the orrespondng varatons n the sum arm voltages. 3) he energy dfferene zero-sequene dynams Sne the zero sequene omponents n (5) and (6) are tme-perod n steady state, further reformulaton s neessary to obtan a SSI representaton of the zero sequene energy dfferene dynams of the MMC. hs an be obtaned by defnng a vrtual sgnal w β Δz whh s 90 shfted wth respet to the orgnal "sngle-phase" tme-perod zero sequene energy dfferene sgnal w Δz gven n (6). hs approah s oneptually smlar to the ommonly appled strategy of generatng a vrtual two-phase system for representng snglephase systems n a SRF [31]. However, sne the ampltudes of the dfferent sne and osne omponents are defned by SSI varables, the sgnal w β Δz an be dentfed wthn the model, and wthout ausng any addtonal delay. he atual and vrtual energy dfferene zero-sequene varables an be labelled as w α Δz and w β Δz and together they defne an orthogonal αβ-system. hs αβ system an be expressed by (8), wth p α Δ1z and p α Δz defned by (5) and (6), whereas p β Δ1z and p β Δz are reated by replang the "os(3ωt)" and "sn(3ωt)" terms that appear n the α-sgnal by " sn(3ωt)" and "os(3ωt)," respetvely. hus, the ampltude of the β-sgnals wll be dental to the α-sgnal ampltude. d w p p (8) z 1z z hs orthogonal system an be represented by varables defned n a SRF at 3ω. Hene, the αβ-vetors on the rght hand sde of (8) an be expressed by (9), where p d3ω Δ1z, p q3ω Δ1z, p d3ω Δz and p q3ω Δz, are defned by (30). 1 d 3 q z p 1z p 1z 3p1z p 1z 3 1z p p p p 1 d 3 q3 1 3 z p z p z 3 pz p z 3 z d3 d3 1 1z vd d ; vq q z d vd q vq q3 q3 1 1z vd q ; vq d z q vd d vq p e e p u u, (9) (30) p e e p u u he dynams of the energy dfferene zero-sequene αβ vetor w αβ Δz from (8) an be transformed nto the rotatng referene frame at +3ω by means of +3ω and the defntons gven n (9)-(30), yeldng n: d w z p 1z pz J 3 w z (31) Introdung the power defntons n (30), (31) an be expressed by (3). It s possble to onfrm by smple nspeton that the zerosequene dynams of the energy dfferene expressed n the form of (3) are SSI as long as the d- and q-axs omponents of e v, u, v and n ther assoated SRFs are SSI. herefore, ths equaton preserves tme-nvarane when the rulatng urrent s ontrolled to njet a nd harmon omponent (for energy shapng) as well as for suppresson of the nd harmon rulatng urrent aordng to [7]. When onsderng the zero sequene energy dfferene dynams n (3), t should be kept n mnd that ths s a orthogonal vetor representaton of a sngle phase snusodal sgnal. Indeed, sne the thrd harmon osllaton s a zero sequene omponent, the same sgnal appears n all the three phases of the MMC. he ampltude of ths sgnal and the phase angle wth respet to the thrd harmon SRF an be found dretly from the vetor ampltude and phase angle of the,+3ω defned zero sequene energy varables (.e. w Δz = d,+3ω q,+3ω w Δz + j w Δz ). It an also be understood from (5) how these thrd harmon osllatons wll appear n the sum arm energes and n the orrespondng sum arm voltages. 4) Crulatng urrent dynams he dynams of the rulatng urrents are realled n (33) n vetor representaton for a three-phase MMC [7]. d v ab ab ab d La Ra u (33) Equaton (33) an be easly expressed n the SRF rotatng at ω, yeldng n (34) d d z vd q vq q d w evd d e d vq q evd u u u w z w w q evd q e vq d e vq z uq vd ud u z vq w d (7) 1 q d 3 3 d vd d vq q d vd q vq 3 z d w e e u u w 3 z w z q 3 w 1 d3 z evd q e 3 vq d u q vd ud vq w z (3)

8 d z R 1 a z z v d IJ u (34) La La La he equatons for the -omponents of the rulatng urrents have the same form as for any SRF representaton of urrents n a three phase system. However, the zero sequene omponent s a d-sgnal, representng the d-omponent of the rulatng urrents of the three phases and s dretly assoated to the power transfer between the a- and d-sdes of the MMC.. Summary of derved model wth SSI representaton of MMC nternal dynams he ndvdual equatons desrbng the nternal dynams of the MMC as represented by SSI state varables, as derved n the prevous subsetons, are summarzed here. he resultng SSI state equatons are olleted n (35) and result dretly from (18), (7), (3) and (34) by expressng the -omponents wth omplex vetor notaton. he algebra equatons lnkng the ontroller outputs, u and e v, wth the rest of the system are gven by (36). d 1 w ev v uz u z jw d 1 w Re Re z ev v u u z z d 1 w ev ev z u v uzv jw d w z ev u v j 3w z d R 1 a j u La La d Ra 1 vd 1 u z z z La La La (35) u u ; u u ; e e (36),, z z v v hese equatons defne a non-lnear SSI state-spae representaton of the average model of an MMC wth energybased formulaton aordng to [6], [7]. he only smplfaton ntrodued durng the dervaton s that the 6 th harmon terms n (17) have been negleted. hus, the developed SSI equatons preserve the dynams and the non-lnear relatonshps of the model t s derved from, and nherts the same lmtatons as the analytal average models n the statonary frame. s for any other analytal average model, ths mples that the developed SSI representaton of the MMC s not representng any physal saturaton lmts wthn the model, lke for nstane the over-modulaton lmt that an be reahed f the voltage referene for the onverter s hgher than the avalable voltage n the nternal apators. However, as long as the onverter s operated wthn ts lmtatons, the derved model s ontanng detaled nformaton about the dynam haratersts as well as the steady-state operatng ondtons of the MMC. hus, the model nherently nludes the dynam ouplng between the varous frequeny omponents, whh an be learly noted by onsderng that several of the state equatons n (35) are defned by varables from SRFs at dfferent frequenes. It should also be noted that the model n (35) effetvely represents the MMC by 10 SSI state-equatons. he grd-sde urrents are not nluded n these equatons, as they ontans only a fundamental frequeny omponent and an be dretly modelled n the SRF at the fundamental frequeny. Consderng the MMC topology from Fg., representaton of the 6 equvalent arm apator voltages and the 3 rulatng urrents as state-varables wll mply a model wth 9 states. hus, the derved SSI representaton of the MMC nludes only one addtonal state equaton, sne two state varables are requred to obtan a SSI representaton of the zero sequene energy dfferene, w Δz. C. Smplfed zero-sequene model of MMC Observng the struture of the model n (35), t an be noted that the dynams of the zero sequene urrent z do not depend on any of the -varables. Furthermore, the dynam equaton for the zero sequene sum energy w,z ontans terms dependng on the produt of the -omponents of u and. Sne the -omponents of u are sgnfantly smaller than the zero sequene omponent, u z, and the ampltude of the a-sde voltages, e v, the nfluene of these terms on the sum energy dynams wll be very small. Under the assumpton of ompensated modulaton, ths mples that a smplfed model for representng only the zero-sequene omponent of the MMC nternal varables an be obtaned, as gven by (37). d w 1 z e vd vd e vq vq u z z (37) d R 1 1 z z uz vd L L L hs smplfaton and reduton of the equatons from (35) s dretly resultng n the model proposed by [5], [6]. It an also be understood from the struture of the detaled model n (35) that the smplfed model n (37) wll be sutable as a "marosop" model of the a- and d-sde dynams of the MMC by onsderng only the zero sequene omponents of the energy-sum and the rulatng urrent. hus, the dervaton of the detaled model provdes a theoretal bass for verfyng the auray and for understandng the level of approxmaton mpled by the smplfed models n [5], [6]. he zero-sequene-based redued order MMC model n (37) has a lower number of equatons and s muh smpler to mplement than the detaled model n (35). However, t wll be verfed that under CM-based ontrol, the zero-sequene model s aurately representng the dynams of the states that nfluene the external behavour at the a- and d-sdes (.e. v d, u z, z, w z, e v and v ). Indeed, these varables reman pratally unaffeted by the dynams of the negleted z nternal varables (w, w Δ, and u ) as long as ther dynams are stable and the nserton ndexes are alulated aordng to (6). hus, the zero-sequene model only preserves nformaton about the power balane between the a-sde, the nternally stored energy and the d-termnals. Hene, t s expeted that ths zero-sequene model wll be of most nterest for large-sale power system stablty studes, when the nternal dynams of the MMC are of lmted nterest.

9 p C p C v d v d,z w w w z, w w, w z w z C Power Controller wth DC Droop PLL SRF@ & Zero Seq. PI Controllers PLL SRF@+ Proportonal Controllers PLL SRF@3 Proportonal Controllers,z,, z vd vq,z v C-sde Current Control Crulatng Current Control PLL PLL z Phase PLL Loked Loop vd e v Equaton 35, u u z Z g v o L eq v g, e v v DQ Seq. Energy Sum Zero Seq. Energy Sum Dynams DQ Seq. Energy Dff. Zero Seq. Energy Dff. u z L 3 a 3 z d, s o C d v d C f DQ Seq. Current SRF PI ontrollers n the double frequeny negatve sequene referene frame, aordng to [7], ontrols the -omponents of the rulatng urrents. n dental PI-ontroller struture wth energy deouplng terms at ω PLL regulates the omponents of the sum energy, by provdng urrent referenes for the rulatng urrent ontrollers. Furthermore, a smple proportonal ontroller wth deouplng terms at ω PLL regulates the energy dfferene -omponent dynams. Smlarly, the -omponents of the zero sequene of the energy dfferene are ontrolled by an addtonal proportonal ontroller wth deouplng terms at 3ω PLL. he ontrbuton of eah energy ontroller s added to form the referene for the rulatng urrent as llustrated n Fg. 5. It should be noted that the derved MMC model ould be ombned wth dfferent ontrol system mplementatons. However, aurate SSI representaton of ommonly appled ontrol loops mplemented n the statonary referene frame would requre smlar dervatons as presented for the MMC topology. Suh dervatons and subsequent analyss are beyond the sope of ths manusrpt, but an example of how an SSI representaton of statonary frame per-phase energy-based ontrol strateges an be obtaned s presented n [34]. Fg. 5 Overvew of the derved tme-nvarant MMC model wth representaton of the nternal z dynams, nludng a-sde and d-sde dynams as well as all elements of the appled ontrol system IV. SIMPLIFIED CONROL SYSEM FOR SSI REPRESENION he MMC model under CM annot be tested or valdated wthout ntrodung a losed loop ontrol sheme. herefore, ths seton brefly ntrodues a smplfed losed loop ontrol system adapted to the SSI representaton. Note that the purpose of the added ontrollers s to enable omparson of the derved SSI representaton wth an establshed MMC average model, wthout requrng sgnfant efforts n modellng of the ontrol system. n overvew of the entre model of an MMC HVDC termnal, nludng the ontrol loops as well as the a- and dsde eletral dynams s shown n Fg. 5. Conventonal SRF PI urrent ontrollers wth deouplng terms are appled for ontrollng the a-sde urrents of the MMC [5], [3]. n asde PI power ontroller, based on feedbak of a low-passfltered measurement of the power flowng from the MMC nto the grd, s provdng the d-axs atve urrent referene to the urrent ontrollers. However, a d voltage droop funton based on a low-pass-fltered measurement of the voltage at the d termnals s atng on the atve power referene. For smplty, the q-axs urrent referene s kept equal to zero. Furthermore, a SRF Phase Loked Loop (PLL) s utlzed to synhronze the ontrol system of the MMC to the measured grd voltage v o. he state equatons for the a-sde ontrollers of a three-phase VSC, the d voltage droop funton and the PLL from [5] and [3] an be utlzed for state-spae representaton of the system wthout modfatons. he zero sequene omponents of the rulatng urrent and the sum energy are ontrolled by PI ontrollers, utlzng the same equatons as n [5]. Furthermore, a set of deoupled V. MODELS OF MMCS INCLUDING C-SIDE ND DC-SIDE GRID DYNMICS y ombnng the SSI state-spae representaton of the MMC dynams derved n seton III. wth the smplfed ontrol struture ntrodued n seton IV, t s possble to establsh state spae models of an MMC ntegrated nto any aor d grd onfguraton. For smplty, only the onfguraton from Fg. 5 wll be studed here, although the derved models an be dretly utlzed for studes of larger system onfguratons, for nstane n pont-to-pont or mult-termnal HVDC transmsson shemes by smlar approahes as dsussed n [35], [36].. MMC models wth a-sde and d-sde grd dynams he equatons of the a-sde dynams nluded n the model result dretly from the rut dagram ndated n Fg. 5, the average modellng of eah arm of the MMC topology and the assumpton of CM-based ontrol [6], [7], [1], [33]. hus, the a-sde model represented n the SRF s the same as for a -L VSC, and the same approah as n [5], [3] an be appled for obtanng a SSI state-spae representaton nludng the PLL dynams. he d-sde s modelled wth a apator representng the equvalent apatane of an HVDC able, and a urrent soure d,s representng the able urrent, as shown n Fg.. hus, the eletral dynams at the d termnals an be modelled by the same equatons as n [5].. Non-lnear state-spae models wth SSI soluton general SSI state-spae model of the studed system an be expressed on standard form aordng to [15], [14]: x f x, u, y g x, u (38) Models nludng the detaled MMC dynams aordng to (35) and Fg. 5, as well as models based on the smplfed zerosequene representaton of the MMC from (37) an be easly developed on the same form.

10 p C p C v d v d w z C Power Controller wth DC Droop Zero Seq. Energy Sum Controller w z 1) Model wth detaled representaton of MMC ased on all presented dervatons and referenes, a SSI representaton of the entre system from Fg. 5 an be represented by the state vetor x and the nput vetor u as defned by (39) and (40), respetvely. hus, the SSI state spae representaton nludng a- and d-sde nterfaes, as well as the grd synhronzaton dynams on the a-sde results n a model wth 34 states and 13 nput varables. In addton to the state varables already explaned, the states γ and φ are assoated to the a-sde urrent ontrol, whle all varables wth subsrpt 'PLL' are assoated to the Phase Loked Loop used for grd synhronzaton, as desrbed n detal n [3]. he low-pass fltered d voltage s gven by v d,f, the low-pass fltered measurement of the a-sde power flow s defned by the state p a,m, and ρ defnes the ntegrator state of the PI-ontroller for the a-sde power flow. he states κ are ntegrator states of the PI ontrollers regulatng the sum energy omponents, whle ξ defnes the PI ontroller states for the d- q- and z-omponents of the rulatng urrents. ) Smplfed zero-sequene MMC Model non-lnear SSI state-spae model nludng a representaton of the MMC by the smplfed zero sequene representaton from (37) [5], [6], an be establshed n the same way as for the detaled model. he only dfferene wll be that the states w Σd, w Σq, w Δd, w Δq, w Δzd, w Δzq, κ Σd, κ Σq ξ d and ξ q as well as the nput sgnals w Σ,d, w Σ,q, w Δ,d, w Δ,q, w Δz,d and w Δz,q wll be elmnated. hus, wth the smplfed MMC representaton, the struture of the model wll be redued to the smplfed onfguraton shown n Fg. 6. C. Lnearzed small-sgnal models s mentoned, the need for obtanng a lnearzed state- x spae model for ondutng egenvalue-based studes of smallsgnal stablty s among the man motvatons for dervng a SSI representaton of the MMC. However, a non-lnear SSI representaton n the form of (38) s also neessary for alulatng the steady-state operatng pont. hus, any feasble steady-state operatng ondton of the system an be found by solvng for the values of the state varables when mposng ẋ = 0. Subsequently, the model an be lnearzed at the seleted steady-state operatng pont. For a gener lnearzaton pont x 0, the lnearzed small-sgnal state-spae model an be obtaned by onsderng the frst order dervatves wth respet to all state varables and nput sgnals [14], [15], and an be expressed as: x x0xx0u (41). ycx0xdx0u where the prefx Δ denotes small-sgnal devatons around the steady-state operatng pont. VI. MODEL VLIDION Y IME-DOMIN SIMULION o valdate the derved SSI equatons, the detaled as well as the smplfed representaton of the MMC, and the orrespondng small-sgnal models, results from tme-doman smulaton of fve dfferent models wll be shown and dsussed n ths seton. hese models orrespond to the followng ases: 1) he referene ase s a rut-based average model of a three-phase MMC, where eah arm s represented by a ontrolled voltage soure and where the nternal arm voltage dynams are represented by an equvalent arm apatane as shown n Fg. 7 [7], [1] [33]. hs model nludes nonlnear effets, exept for the swthng operatons and the dynams of the submodule apator voltage balanng algorthms. Sne ths model s well-establshed for analyss and smulaton of MMCs, and has been prevously verfed by laboratory-sale experments n [7], [1], t wll be used as a benhmark referene. he model s smulated n Matlab/Smulnk wth the SmPowerSystem toolbox, and operated wth the ontrol strategy presented n seton IV. Smulaton results obtaned wth ths model wll be denoted as "M," sne t an be onsdered as an veraged rm Model. ) non-lnear state-spae model nludng the derved SSI representaton of the MMC nternal z dynams, as depted n Fg. 5. he parts of ths model that represent the MMC dynams are summarzed n (35), whle the assumed ontrol system mplementaton and the nluded a- and d-sde dynams are brefly desrbed n seton IV and seton V., respetvely. Results from ths model wll be denoted as "DQZ". 3) he smplfed tme-nvarant MMC model desrbed n seton III.C. hs model s based on the zero-sequene omponents of the energy sum and the rulatng v v v v v v od, oq, vd, vq, d q od, oq, d q PLLd, PLLq, PLL PLL d d, f p... vd vq z PLL C-sde Current Control Crulatng u z Current Control z PLL PLL v Phase Loked Loop vd ev Z g v o L eq v g Zero Seq. Energy Sum Dynams d w 1 e e 4 u z vd vd vq vq z z L 3 a d, s u z o, e v Fg. 6 Overvew of the smplfed tme-nvarant MMC model 3 z v C d v d C f a, m, z, d, q, z, d, q, d, q z, d z, q, z, d, q d q z p w w w w w w w u vq vg g d, s pa vd w, d w, q w, d w, d w z, d w z, d w, z (40)... (39)

11 +30 kv d C eq C eq C eq d, s C d v d a u e v a l L a La a v b u b l La La b v u l La La v L f ransformer 310/380 kv ab v o Grd equvalent C o L g ab o R g C soure a v g b n C eq C eq C eq d 30 kv urrent, as defned by (37), and orresponds to the model proposed n [5]. he a- and d-sde dynams nluded are the same as for the other models, and the smulated ontrol system s a smplfed verson of what was dsussed n seton IV, resultng n the same ontrol struture as dsussed n [5]. n overvew of the model s shown n Fg. 6, and results from the model wll be denoted as "ZERO". 4) he small-sgnal state-spae model obtaned from lnearzaton of the model n ase. he model wll be lnearzed at the ntal steady-sate operatng pont of the detaled nonlnear model, and the values of the state varables wll be alulated as x = x 0 + Δx. Results obtaned from ths model wll be denoted as "ssdqz". 5) he small-sgnal model obtaned from lnearzaton of the model n ase 3. he results wll be presented n the same way as for ase 4, and the results wll be denoted as "sszero". ll smulatons are based on the MMC HVDC termnal onfguraton shown n the prevous fgures, wth parameters gven n able I. It should be noted that the a-sde ndutane L f for the MMC n ths ase s the equvalent leakage ndutane of a transformer onnetng the MMC to a smplfed model of a 380 kv transmsson system, as ndated n Fg. 7. Smlarly, R f s the equvalent seres resstane of the transformer. In able I, all parameters of the a-system are referred to the onverter-sde of the transformer, sne the transformer s expltly represented only n the benhmark model. Furthermore, the equvalent arm apatane C eq Fg. 7 Smulated referene model LE I PRMEERS OF SIMULED SYSEM orresponds to an MMC wth 400 sub-modules per arm, where eah sub-module has a apatane of about 8500 µf. ddtonally, a droop gan of 10 pu determnes the ouplng between the d voltage and the a-sde power referene. It should be onsdered that the referene model s a onventonal tme-doman smulaton model of a three-phase MMC representng arm or phase quanttes, whle the other 4 models wth SSI haratersts represent the MMC dynams by varables transformed nto a set of SRFs. Sne the omparson of transent and steady-state responses s smpler wth a SSI representaton, the results obtaned from the referene model are transformed nto the approprate SRFs by usng the phase angle from the smulated PLL. ll results are plotted n per unt quanttes, wth base values derved from the nomnal kv ratng of the MMC and the peak value of the nomnal phase voltage, as spefed n able I. o exte the MMC dynams n the dfferent models, a 10 % step reduton s ntrodued n the d sde urrent soure d,s, whh s ntally at 0.85 pu, orrespondng to a d power of 1.08 [pu]. he step s mposed at the smulaton tme t = 0 s and the urrent soure s returned to ts ntal value at t = s. he frst set of results s presented n Fg. 8, for a ase when the nd harmon omponents of the energy sum are regulated to zero. In ths fgure, some of the varables whh are ommon to all the smulated models are shown;.e., the sgnals that are represented n both the "DQZ" and the "ZERO" models. hese varables are, n Fg. 8 a); the zero-sequene energy sum w z, b) the zero-sequene of the rulatng urrent z, ) the voltage at the MMC d termnals v d, d) the atve omponent of the a- Referenes [pu] MMC Parameters Per Unt System Controller Parameters w Δ, z+ω 0 Ra w d-ω 0 La w q-ω 0 Lf w z 1.5 Rf d,s 0.85 Co Ω (0.5%) Lg H (8%) H (16.48%) 0.80 Ω (0.85%).871uF (8.87%) Rg Ceq Cd H (11.1%) Ω (0.01%) uf (80%) 1.67 uf (5.1637%) v d 1.5 Vn,l-l kvrms Sn 1000 MV Sb VnIn 3 = Sn kp,v.6010 kp,w Vb, Ib /3Vn,l-l, In k,v k,w ωb πfn kp,= kp,z kp,w Zb, Lb, Cb Vb/Ib, Zb/ωb, 1/(ωbZb) k, = k,z.1875 kp,w z 0. fn 50 Hz Vbd, Ibd Vb, Sb/Vbd kp,w z 10 kp,pa 1 Zbd, Lbd, Cbd Vbd/Ibd, Zb/ωb, 1/(ωbZb) k,w z 10 k,pa 50

12 a) Zero-Sequene Energy Sum, w z b) Zero-sequene rulatng urrent, z ) Voltage at d-termnals, vd d) tve a-sde urrent omponent, v,d e) Phase dsplaement between vg and PLL, δθpll Fg. 8 me-doman valdaton of tme-nvarant MMC models: Complete z and smplfed zero-sequene model valdaton by omparson to MMC benhmark model wth sum energy osllatons ontrolled to zero. urrent (onverter sde) v,d, and e) the phase shft between the PLL orentaton and the equvalent grd voltage, δθpll. From Fg. 8 t an be ntally onluded that the detaled SSI representaton as well as the smplfed zero sequene model (.e. "DQZ" and "ZERO") obtan a hgh degree of auray, as they apture the dynam response of the referene model wthout any noteable devaton. he results presented n the fgure also onfrm that the model s aurate for both fast and slow dynams. Smlarly, ther lnearzed small-sgnal versons ("ssdqz" and "sszero") aurately apture the system dynams, partularly for the event ourrng at t = s, as the system s then returnng to the operatng pont around whh t was lnearzed. From the urves n Fg. 8, t an be noted that energy sum reahes the desred value of n steady-state, whh orresponds to the square of the desred d termnal voltage;.e., 1.5. hs results from havng a zero-sequene energy sum referene n real values defned as W z=(½ C/N(1.5Vd,b)), and a base value for the energy defned as W b=(½c/n(vd,b)). hus, the energy base value orresponds to the energy n one phase when the upper and lower arm voltages are equal to the base value for the d voltage. In addton to the zero-sequene energy, ths energy base value s further used to sale the omponents of the energy sum, as well as all omponents of the energy dfferene. Moreover, t an be noted that the rulatng urrent settles to 0.15 pu after t = s, whh orresponds to one fourth of the fnal value of ds. hs salng s a result of applyng the a-sde base value to the zero sequene rulatng urrent, and the salng of the Park transformaton [14], [5]. Note that the smplfed zero-sequene SSI model does not requre any nformaton on how the nd harmon rulatng urrents are ontrolled to provde an aurate representaton of the marosop varables presented n Fg. 8, as long as the nternal varables are stable and the losses assoated wth the nternal MMC dynams are neglgble. hus, the auray of the smplfed model s not sgnfantly nfluened by whether a onstant rulatng urrent ontrol strategy or a onstant energy sum ontrol s used. y ontrast, Fg. 9 shows the MMC energy state varables that have been negleted n the smplfed zero-sequene model: the -omponents of a) the energy sum w, b) energy dfferene wδ, and ) zero-sequene energy dfferene wδz. In addton, the -omponents of the rulatng urrent are gven n Fg. 10. he results n Fg. 9 and Fg. 10 demonstrate that the detaled SSI model aurately aptures the nternal dynams of the average model t was derved from. In addton, ts lnearzed small-sgnal model s able to represent the dynam behavor wth hgh auray. s a pont of referene, the behavor of the rulatng urrent, the energy sum and the energy dfferene n eah phase have been plotted n Fg. 11-a), -b) and -). hese are exatly the same smulaton results that have been transformed nto ther assoated SRFs for the omparson of the models n Fg. 9 and Fg. 10. Sne t was demonstrated that all the models provde the same results, only the referene model s plotted for the sake of larty. he waveforms are as expeted, wth the energy settlng to a onstant value n steady state. Furthermore, t s worth notng that all osllatng varables settle to balaned three-phase sgnals wth a ommon average value n steady state, sne ther respetve -omponents are ontrolled to onstant values by the MMC ontrollers. Fnally, Fg. 11 d) and e) show respetvely the arm urrents and aggregated voltages of the phase a, to llustrate the atual waveforms of the referene model. Indeed, the d-omponent, the fundamental frequeny omponent and the seond harmon omponent n the arm urrents an be learly seen from Fg. 11 d). When studyng the results n Fg. 11 e) n omparson to the results n From Fg. 9 and Fg. 10, t should be remembered that the atual waveform of the sum arm voltage s related to the sum arm energy, and the per phase sum energy and energy dfferene varables aordng to (5). Sne the average value of the sum arm voltage s muh hgher than the osllatng omponents, the nfluene of the square root relatonshp

13 a) d- and q-axs omponents of the energy sum w b) d- and q-axs omponents of the energy dfferene, wδ ) d- and q-axs omponents of the zero sequene energy dfferene, wδz Fg. 9 me-doman valdaton of detaled SSI representaton of the nternal MMC energy varables between the sum arm energy and the sum arm voltage annot be easly noted from Fg. 11 e). hus, t an be seen from the urves n Fg. 11 e) that the transent response n the equvalent sum arm voltage ontans: 1) d-bas wth ts orrespondng transents ) fundamental frequeny omponent

14 Fg. 10 me-doman valdaton of detaled SSI representaton of the d- and q-axs omponents of the nternal rulatng urrents a) Crulatng Current n the three phases b) Energy Sum for the three phases d) rm urrents of phase a ) Energy Dfferene for the three phases e) Equvalent arm voltages of phase a Fg. 11 MMC nternal varables n the statonary ΣΔ representaton, and an example of arm quanttes obtaned from the referene model 3) seond harmon omponent (whh n ths ase s slowly regulated to zero to redue the apator voltage osllatons as seen from Fg. 11 b)) 4) thrd harmon omponent. From the results n Fg. 9 and Fg. 10, t should be lear that these omponents are all aurately represented n ther approprate SRFs by the derved SSI state-spae equatons. VII. NLYSIS OF MMC LL-SIGNL DYNMICS For demonstratng the potental applablty of the derved SSI representaton of the MMC, an example of small-sgnal egenvalue analyss s presented n ths seton. hs example wll demonstrate how the nonlnear state-spae model s neessary for alulatng the steady-state operatng pont needed for lnearzaton, and how the lnearzed small-sgnal model an be utlzed for revealng the dynam propertes, senstvtes and stablty lmtatons of the modelled system. It s mportant to that note the obtaned results rely on the SSI modellng approah, and that smlar results annot be dretly obtaned from the onventonal average model n the statonary referene frame.. Egenvalue analyss for dentfyng soures of osllatons s a frst example of small-sgnal analyss, the egenvalues are alulated for the small sgnal state-spae model representng the detaled nternal dynams of the MMC as well as for the smplfed model, when the system s lnearzed at the same operatng pont as used for the smulatons n the prevous seton. he resultng egenvalues are plotted n the omplex plane for omparson, as shown n Fg. 1. From the varous sales shown n Fg. 1 a)-d), t an be learly seen that all egenvalues that exst n the smplfed "sszero" model are also present n the detaled "ssdqz model." hs learly onfrms that the smplfatons assoated wth the zero

15 a) ll egenvalues b) Egenvalues wth Re(λ)> 500 ) Egenvalues wth Re(λ)> 70 d) Egenvalues wth Re(λ)> 5 Fg. 1 Comparson of egenvalue for detaled "ssdqz" model and smplfed "sszero" small-sgnal models sequene model only mples that some of the system dynams are not represented, whle the dynams nluded n the model aurately orresponds to the detaled model. For further assessng the nformaton that an be obtaned from the small-sgnal models, the egenvalues of the "ssdqz" model are lsted n able II. hs table also lsts the tmeonstant, the osllaton frequeny f, and the dampng fator ζ of eah mode, whh are defned from the real and magnary part of the egenvalue, aordng to [14]: at a jb z t z0, e os bt, 1 b, a, (4) f a a ab hs equaton also defnes the general form of the tmeresponse z(t) assoated wth an ndvdual mode λ. y onsderng the transent responses resultng from the tme-doman smulatons, t an be onfrmed how the osllatory omponents n the SSI state varables are eah dretly assoated to one of the dentfed modes. he hgh frequeny osllaton at about 1400 Hz whh an be seen n the Fg. 8 ) and d) s, for nstane dretly orrespondng to the osllaton mode gven by the egenvalues λ,3. Smlarly, the relatvely damped osllaton wth a frequeny slghtly above 50 Hz whh an be noted n the zoomed plots of Fg. 9 orresponds to the mode defned by the egenvalues λ 19,0. lthough t s possble to dentfy some dstnt egenvalues n the tme-doman response of the system, ths does not expltly reveal whh varables are nvolved n eah osllaton model. hus, partpaton fator analyss an be utlzed to dentfy whh states are ontrbutng to the dfferent modes [14]. Suh analyss an reveal whh state varables are nvolved n ausng poorly damped osllatons or nstablty problems and ndate potental nteratons between the varous state varables. he results from suh partpaton fator analyss are summarzed n the rghtmost olumn of able II, where all state varables wth a partpaton hgher than 10 % are lsted for eah mode. For nstane, t an be noted that the egenvalues wth the hghest tme onstant (.e. longest settlng tme of the transent) n ths ase are assoated wth the voltages and urrents on the a-sde (λ 4,5 ) and the ntegrator states of the energy ontrollers (λ 5,6, λ 7 ).. ssessment of small-sgnal dynams n the full expeted operatng range Sne the non-lnear SSI state-spae equatons an be solved for any feasble ombnaton of nput varables, t an be utlzed as startng pont for assessng the small-sgnal stablty haratersts of the system over ts entre range of expeted operatng ondtons. s an llustraton, a ase where the power LE II EIGENVLUES OF HE DEILED MMC MODEL ND HEIR MIN PRICIPING SES Mode me Osllaton Dampng onstant frequeny fator Man partpatng states λ s - - vpll,d λ, ± j s 1395 Hz vd,,z λ4, ± j s Hz vo,d, vo,q, o,d, o,q λ x10-4 s - - v,q λ7, ± j s Hz vo,d, vo,q, o,d, o,q λ9, ± j x10-4 s Hz 0.96 v,d, vd,f, w,z λ11, ± j x10-4 s Hz 0.951,d,,q, wd, wq, wδz,d, wδz,q λ13, ± j s Hz wd, wq, wδz,d, wδz,q λ s - - v,d, o,d, vd,f, pa,m,,d,,q, w,z,wd, wδz,q, λ s - - vpll,q λ17, ± j s 14.1 Hz pa,m,,d,,q, w,z,wd, wq, wδ,d, wδ,q, wδz,d λ19, ± j s Hz wd, wq, wδ,d, wδ,q λ1, ± j s Hz ρp, pa,m, w,z, ξz λ3, ± j s 5.91 Hz εpll, δθpll λ5, ± j s Hz κ,d, κ,q, λ s - - κ,z, λ8, ± j s Hz d, q, ξz, λ s - - d, q, ξz, λ s - - ξd, ξq, λ3, ± j s - 1 d, q, λ s - - ξd, ξq

16 Mode - nstablty Mode - nstablty Im Fg. 13 Egenvalue trajetory for operatng ondtons between pa = 1.0 pu (blue olor) and pa = 1.0 pu (red olor) referene s hanged from 1.0 pu to 1.0 pu, whle the d-sde nput urrent d,s s hanged to provde a power equal to the referene value (.e. d,s = p a /v d ) s studed and the results are presented n Fg. 13. hs fgure shows the trajetory of the egenvalues wth real part hgher than 500 as the power flow s hanged from 1.0 pu (blue olor) to 1.0 pu (red olor). he hange of the egenvalue loatons an be onsdered as a measure of how the non-lneartes of the system nfluene the small sgnal dynams. Indeed, the results demonstrate that the system s approahng the stablty lmt when the power transfer s nreasng. If the stablty margn beomes very small, t wll also ndate that any hange of ontroller tunng or system parameters an easly ause stablty problems. C. Influene of nternal varables on stablty of the MMC s demonstrated n seton VI and VII., the smplfed MMC state-spae model s aurately representng the termnal dynams of the MMC as long as all the nternal dynams are stable. However, the nternal dynams of the MMC an possbly ompromse the overall system stablty f the ontrol loops are not tuned properly. lthough the ontrol systems used n ths paper s a smplfed mplementaton, the onsequenes of mproper ontroller tunng an easly be demonstrated. s an example, Fg. 14 a) shows the egenvalue trajetory when hangng the gan of the ontrollers for the d- and q- axs energy sum from half of ts ntal value to 4 tmes ts ntal value from able I. It an be seen from the fgure that the system has one unstable mode for low values of the gan k p,w (Mode ), and that another mode beomes unstable at very hgh values of k p,w (Mode ). Partpaton fator analyss s utlzed to reveal the results of the nstablty dentfed n Fg. 14 a), and the results are plotted as bar dagrams for the two dentfed unstable modes n Fg. 14 b). hs fgure ndates that the Mode nstablty s assoated wth a lak of ontrol of the nternal dynams of the MMC due to the low gans, sne the partpatng states are w d, w q, w Δzd and w Δzq. However for hgh values of k p,w the unstable mode (Mode ) s assoated to the output voltage, the output urrent and the zero sequene sum energy w z. hs ndates that a wrong tunng of the nternal ontrollers an also ause stablty problems to appear on the termnals of the MMC. hus, the smplfed zero-sequene model of the MMC should only be used when t an be assumed that the nternal Re a) Egenvalue trajetory for a hange of kp,w and k,w from 0.5 to 4 tmes the ntal value (olor gradent: blue to red) blak trangles denote nstablty Mode - nstablty Partpaton Fators wsd wsq wdzd wdzq Mode - nstablty vod voq od oq wsz b) Partpaton fator analyss of the unstable egenvalues at low values of kp,w (Mode ) and at hgh values of kp,w (Mode ) Fg. 14 Example of egenvalue analyss revealng potental nstablty of the MMC resultng from wrong tunng of the nternal ontrollers Im Partpaton Fators x Re Im Re a) ll egenvalues wth Re(λ) > 500 b) ll egenvalues wth Re(λ) > 100 Fg. 15 Egenvalue trajetory when hangng the grd ndutane lg from 0.01 pu (blue olor) to 0.6 pu. (orange/red olor) blak trangles ndate nstablty dynams of the MMC are not ausng any stablty problems that an nfluene the overall operaton of the system. D. Senstvty to operaton under weak a grd ondtons he developed SSI models an also be utlzed for evaluatng the senstvty wth respet to parameter varatons n the physal system or n the ontroller tunng. s an example of how external network parameters an nfluene the operaton of the MMC, the mpat of varatons n the grd-sde ndutane of the assumed a-system have been nvestgated. he egenvalue trajetory resultng from hangng the grd ndutane between 0.01 pu and 0.6 pu are shown n Fg. 15. In ths ase, d,s s set to 0.5 pu and p a s set to 0.4 pu, whle all

17 Fg. 16 Parametr senstvty for rtal egenvalue ausng nstablty wth nreasng grd ndutane other parameters are as gven n able I. From ths fgure, t an be seen that one of the egenvalues prevously dentfed to be assoated wth the a-sde eletral system s rossng nto the rght half-plane ausng nstablty for hgh values of the grd ndutane. ordng to the results n Fg. 15, the ontrol system should be re-tuned to ensure robustness wth respet to the grd mpedane for operatng the MMC n weak grd ondtons. For dentfyng the ontroller parameters that an be utlzed to aheve a wder stablty range, t s useful to alulate the parametr senstvty of the egenvalue ausng the stablty problems. he parametr senstvty α n,k of the egenvalue λ n to varatons n parameter ρ k s defned s defned as: Φn Ψn n k nk, (43) k ΦΨ n n where Ψ n and Φ n are the left and rght egenvetors assoated to the egenvalue λ n [14]. he real parts of the parametr senstvty for the egenvalues dentfed from Fg. 15 to ause nstablty have been alulated and are plotted n Fg. 16 for the ase of a grd ndutane of 0.6 pu (.e. n the unstable regon). From ths fgure, t an be seen that the egenvalue loaton s espeally senstve to the value k p,z of the proportonal gan for the zero sequene urrent ontroller, and to the value k p,pa for the proportonal gan of the a-sde atve power ontroller. Sne the plots ndate the dervatve of the egenvalue real part wth respet to the parameter, ether of these parameters ould be redued to mprove the stablty of the system. hs nformaton allows for smple re-tunng of the ontrollers, sne the loaton of the egenvalues, and the parametr senstvty an be easly realulated after hangng any parameter value. y redung the gans of k p,,z and k p,pa to 80 % of ther ntal values, t s possble to aheve a reasonable stablty margn for the entre operatng range for grd ndutanes up to 0.5 pu (.e. SCR ). In ase very hgh grd mpedane values, the parameters of the PLL wll also start to nfluene the stablty of the system, as dsussed n [37], but further nvestgatons towards the a-sde grd nteratons s beyond the sope of ths paper. n example of a tme-doman smulaton from the referene model desrbed n seton VI s presented n Fg. 17 to verfy the results from the presented egenvalue analyss. hs fgure shows a ase wth grd ndutane of 0.5 pu, when the d-sde urrent d,s s nreased from 0.4 pu to 0.5 pu, orrespondng to a hange of atve power flow from about 0.5 pu to 0.6 pu. Wth the ntal tunng of the system, labelled as Case, the operaton wth d,s equal to 0.5 pu would be slghtly beyond the stablty lmt aordng to Fg. 15, whle Fg. 17 me doman verfaton of how system wth ntal tunng experenes stablty problems as predted by egenvalue analyss durng weak grd ondtons and how re-tuned system mantans stablty wthout sgnfant osllatons the operaton wth d,s of 0.4 ould be found to be stable. hs s learly verfed n the urve for Case n Fg. 17, sne the system s stable before the step n d,s whle t beomes unstable wth an nreasng osllaton at about 310 Hz after the step. hs osllaton frequeny orresponds aurately to the magnary part of the unstable egenvalues from Fg. 15. he ase wth k p,,z and k p,pa redued to 80 % of ther ntal values s labelled as Case, and the result from smulatng the same step n d,s for ths ase s also shown n Fg. 17, learly verfyng that the system has been stablzed. VIII. CONCLUSIONS hs paper presents a modellng approah for obtanng a Steady-State me-invarant (SSI) state-spae representaton of MMCs. he presented approah s sutable for MMCs wth ontrol strateges utlzng on-lne ompensaton for the arm voltage osllatons n the alulaton of the arm nserton ndes, referred to n ths paper as ompensated modulaton. he derved model aptures the MMC nternal dynams whle mposng steady-state tme-nvarane on eah varable. hs was aheved by an energy-based -Δ formulaton whh enabled separaton of the MMC varables aordng to ther osllaton frequenes. proedure for dervng equvalent SSI z representaton of all state varables by applyng three dfferent Park transformatons was presented, referrng eah varable to ts assoated SRF, rotatng at one, twe or three tmes the grd fundamental frequeny. he resultng model an be suted for detal-orented studes, as t aptures the dynams of the seond harmon rulatng urrents and the nternal energy dynams of the MMC. he paper also demonstrates how the developed detaled model an be smplfed due to the haratersts of the ompensated modulaton. hs yelds n a MMC representaton based only on the zero-sequene of the energy-sum and the zero-sequene of the rulatng urrent. hs model orresponds to prevously proposed MMC models for CM-

18 based ontrol, derved by physal onsderatons and approxmatons, but the presented dervatons provde explt dentfaton of the requred smplfatons. he smplfed model s aurately representng the nterfae varables on the a- and d- sde dynams of the MMC, whh are the man varables of onern from a marosop pont of vew and wll be vald under the assumpton that the negleted nternal varables are properly tuned and therefore stablzed. hus, ths model s suted for power system-orented studes. he fous of ths paper has been to derve SSI models that an aurately represent the dynams of a MMC, and a smplfed ontrol system was ntrodued only for verfyng the derved models. Utlzaton of the presented models an enable a wde range of studes related to analyss and ontrol system desgn for the MMC. s an example of applablty, the presented SSI models have been lnearzed and assessed by means of small-sgnal egenvalues-based tehnques. 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19 [30] G. ergna,. Gares, E. erne, P. Egrot,. rzande, J.-C. Vanner, M. Molnas, " Generalzed Power Control pproah n C Frame for Modular Multlevel Converter HVDC Lnks ased on Mathematal Optmzaton," n IEEE ransatons on Power Delvery, vol.9, no.1, Feb. 014, pp [31] R. eodoresu, M. Lserre, P. Rodríguez, "Grd Converters for Photovolta and Wnd Power Systems," IEEE/John Wley & Sons, Chhester, UK, 011 [3] S. D'ro, J.. Suul, M. Molnas, "Implementaton and analyss of a ontrol sheme for dampng of osllatons n VSC-based HVDC grds," n Proeedngs of the 16 th Internatonal Power Eletrons and Moton Control Conferene and Exposton, PEMC 014, ntalya, urkey, 1-4 September 014, pp [33] H. Saad, X. Gullaud, J. Mahseredjan, S. Dennetère, S. Nguefeu, "MMC Capator Voltage Deouplng and alanng Controls," n IEEE rans. on Pow. Delvery, Vol. 30, No., prl 015, pp [34] J. Freytes, G. ergna, J.. Suul, S. D'ro, H. Saad, X. Gullaud, "State-spae modellng wth Steady-State me Invarant Representaton of Energy ased Controllers for Modular Multlevel Converters," n Proeedngs of the 1 th IEEE PES Powereh Conferene, Manhester, UK, 18- June 017, 7 pp. [35] G. O. Kalon, G. P. dam, O. naya-lara, S. Lo, K. Uhlen, "Smallsgnal stablty analyss of mult-termnal VSC-based DC transmsson systems," n IEEE ransatons on Power Systems, vol. 7, no. 4, November 01, pp [36] J. eerten, S. D'ro, J.. Suul, "Identfaton and Small-Sgnal nalyss of Interaton Modes n VSC MDC Systems," n IEEE rans. on Power Delvery, Vol. 31, No., prl 016, pp [37] J. Z. Zhou, H. Dng, S. Fan, Y. Zhang,. M. Gole, "Impat of Short- Crut Rato and Phase-Loked-Loop Parameters on the Small-Sgnal ehavor of a VSC-HVDC Converter," n IEEE ransatons on Power Delvery, Vol. 9, No. 5, Otober 014, pp Glbert ergna-daz reeved hs eletral power engneerng degree from the Smón olívar Unversty, n Caraas, Venezuela, n 008, a Researh Master from SUPÉLEC (Éole Supéreure d Életrté), n Pars, Frane, n 010; and a jont PhD degree between SUPÉLEC and the Norwegan Unversty of Sene and ehnology (NNU) n 015. In Marh 014 he joned SINEF Energy Researh as a researh sentst, workng on tops related to modelng and ontrol of HVDC transmsson systems. From ugust 016 he started a post-dotoral fellowshp at NNU, workng on ontrol and modellng of power eletron systems. Jon re Suul (M 11) reeved the M.S. degree n energy and envronmental engneerng and the Ph.D. degree n eletr power engneerng from the Norwegan Unversty of Sene and ehnology (NNU), rondhem, Norway, n 006 and 01, respetvely. From 006 to 007, he was wth SINEF Energy Researh, rondhem, where he was workng wth smulaton of power eletron onverters and marne propulson systems untl startng hs PhD studes. From 01, he resumed a poston as a Researh Sentst at SINEF Energy Researh, frst n part-tme poston whle also workng as a part-tme postdotoral researher at the Department of Eletr Power Engneerng of NNU untl 016. Hs researh nterests are manly related to analyss and ontrol of power eletron onverters n power systems and for renewable energy applatons. Salvatore D ro reeved the M.S. and Ph.D. degrees n eletral engneerng from the Unversty of Naples Federo II, Naples, Italy, n 00 and 005, respetvely. From 006 to 007, he was a postdotoral researher at the Unversty of South Carolna, Columba, SC, US. In 008, he joned L, Veldhoven, the Netherlands, as a Power Eletrons Desgner, where he worked untl 010. From 010 to 01, he was a postdotoral researher n the Department of Eletr Power Engneerng at the Norwegan Unversty of Sene and ehnology (NNU), rondhem, Norway. In 01, he joned SINEF Energy Researh where he urrently works as a Researh Sentst. He s the author of more than 50 sentf papers and s the holder of one patent. Hs man researh atvtes are related to ontrol and analyss of power-eletron onverson systems for power system applatons, nludng real-tme smulaton and rapd prototypng of onverter ontrol systems.