Real-time simulation of induction motor IGBT drive on a PC-cluster

Transcription

1 Inenaional Confeence on Powe Sysems Tansiens IPST 23 in Hon Kon Real-ime simulaion of inducion moo IGBT dive on a PC-cluse Chisian Dufou, Simon Aouida, Jean Bélane Opal-RT Technoloies Inc.,1751 Richadson, ueau 2525, Monéal QC, Canada H3K 1G6 Phone ( [email protected]) hp:// Asac This pape pesens a eal-ime moo dive simulao capale o accuaely simulae a doule IGBT ide invee conneced o an inducion moo houh an inephase ansfome. The simulao also includes a DClink model wih eeneaion capailiy and wo choppes. The pape will deail he implemenaion of he model and simulaion esuls and imin. The whole sysem, wih exenal I/O IGBT fie opic ae sinals, execues in ealime a 8 µs ime sep unde he RT-La eal-ime disiued simulaion sofwae on a PC-cluse composed of 3 Penium pocessos unnin a 1 GHz. The same model, u his ime wih only one ide and no inephase ansfome, uns in eal-ime a 65 µs ime sep. Keywods eal-ime simulaion, hadwae in he loop, ime sampin, I/O, dives. I. INTRODUCTION Tesin, ineaion and validaion of complex conolled sysems have een adiionally made in a sysemaic way consisin of analyzin he ehavio of individual componens, mosly y simulaion, efoe complee ineaion on analo simulaos o eal appaaus. In seveal cases, he ineaion is diecly made wih pooypes of he eal equipmens. A his sae, eal cauions wee o e aken ecause of he powe levels: a simple conolle malfuncion could damae he pooype o he eal sysem and ceae pojec delays and cos inceases. A moe aduae esin/ineaion appoach was needed o diminish he poailiy of damae and pesonal injuies and also o decease oal developmen coss and ime o make. Wih he inceasin complexiy and coss of pojecs, as well as he advances in compue science, i has ecome advanaeous o make a moe complee and adual appoach duin he couse of sysem ineaion. An excellen way o achieve ha oal is o use Hadwae-inhe-Loop (HIL) diial simulaion. In HIL simulaion, a device unde es is un fully conneced o a eal-ime simulaed dynamic equivalen of an appaaus. Thus, if he appaaus equivalen is coecly modeled, he device unde es will ehave as if i was conneced o he eal appaaus. The devices unde es, ofen a conolle, can heefoe e esed fo a wide ane of paamees wihou any isk o he main sysem. Sofwae packae fo he simulaion of elecical cicuis usually ely on inemediay ouines ha compue discee ime ieaion equaions of he cicui. This is done ased on a nodal volae appoach like EMTP [5] o on he sae-space fomalism of he SimPoweSysem lockse (PSB) [4] fo Simulink and ARTEMIS [2][3]. Those appoaches have he advanae of implicily solvin any causaliy polems ha may lie in he newok equaions. Ohewise, if he newok is simple enouh and suffes fom no causaliy polems, modelin can e done usin individual ineaos fo each elemen of he newoks. In he case of a volae souce invee divin an moo, he pesence of a hue capaciy in fon of he ide iself pemis he decouplin of he invee devices fom any cicui en amon of he capaciy, povided he capacio volae will no chane much duin a sinle ime sep. The DC-link-invee-moo se-up hen ecomes simple enouh o e compued diecly fom is individual diffeenial equaions, wihou he use of EMTP o ARTEMIS ype aloihms. An impoan consideaion fo he choice of HIL simulaion is he minimum samplin ime ha can e achieved unde ealisic condiions. Even wih GHz-speed pocesso echnoloy, pacical ime sep achievale y hih-end HIL simulaos like RT-LAB [1], RTDS [9] and HYPERSIM [1] aely oes unde 25 µs fo a vaiey of easons, like he I/O access ime and ine-pocesso communicaion laency. This pefomance limiaion found in mos hih-end commecial eal-ime simulao availale oday may e accepale fo powe newok simulaion ha can e simulaed wih ime sep of aou 5 µs. Howeve, a minimum ime sep as low as 25 µs is no accepale fo some sinen simulaion needs, like moo- IGBT-dive simulaion. In his case, IGBT un-on and un-off ime mus e sampled pecisely o ensue pope ineaion of he moo fluxes. Fo example, he ain acion simulaion applicaion descied in [6] equied a 1 µs swich imin esoluion. The soluion o his polem is found y samplin he I/O ae sinals wih hih fequency coune cads and y insein his imin infomaion in he simulaion pocess. The pape will descie an implemenaion of his soluion o simulae a complex IGBT moo dive sysem used in lae minin off-hihway ucks. II. PARALLEL IGBT BRIDGE MOTOR DRIVE MODEL The simulao is aimed a he eal-ime simulaion of one wheel moo dive of an off-hihway vehicle used in minin faciliies. The simulaed model compises wo IGBT ides conneced o an inephase ansfome ha feeds he inducion moo. The IGBT ides ae fed y a DC-link composed of a capaciance and wo choppes. A key paamee in he model is he value of he DC-link capaciy ha is lae enouh o assume ha is volae chanes neliily 1

2 Inenaional Confeence on Powe Sysems Tansiens IPST 23 in Hon Kon duin one sep of simulaion. The DC-link is modeled o e ale o simulae ove-volae ha can occu in eeneaion mode, i.e. akin. When in he eeneaion mode, he capacio volae of he DC-link is compued fom he ineaion of ides and choppes cuens. Ohewise, he diode in he model ses he DC-link volae o a use defined value. The complee model is deailed in Fiue 1. Bide conol is made fom exenal fie opic IGBT ae sinals which ae ime samped wih NI662 ime cads unnin a 2MHz. Measuemens have shown ha he cad povides a ± 15 ns effecive ae pulse imin esoluion. This esoluion allows fo pecise IGBT le dead-ime compuaion as well as accuae modelin of swichin delays and volae unalances eween ides. The model povides analo oupus of use seleced model sinals like machine cuens and DC-link volae as well as incemenal speed senso sinals. The model is desin in he Simulink sofwae fom The Mahwoks inc. The ineaion aloihms ae seleced o e compaile wih he Real-Time Wokshop code eneao and RT- LAB sofwae since ha he ojecive is o ineconnec he eal-ime simulaed plan model wih acual IGBT conol sysems. Real-ime simulao + Vdc (se poin) DC-Link model DC-Link volae Fiue 1 Choppe ae sinals IGBT invees 6 6 IGBT ide ae pulses Exenal Conolle inephase ansfome Moo cuens Real-ime simulao model Inducion Moo Moo speed encode sinals A. Inephase ansfome model The inephase ansfome (IPT) limis cuen spikes in he IGBTs in cases whee hee would e a mismach eween coesponden IGBTs ise-fall imes o a conolle-eneaed fiin delay eween hem. The cuen oin houh he IPT, and heefoe houh he coespondin IGBTs, is compued y ineain he volae diffeence a he IPT eminals. Nomally, if cousin IGBTs of oh ides un on and off a he same ime, he volae a he IPT is null and only he moo cuen oes houh he IPT, equally divided eween oh ides. If a volae is pesen a he IPT eminals hen a cuen ows fom one ide eminal o he ohe. This cuen is supeimposed on he moo cuen o compue each ide cuen. B. Moo model wih IGBT ae ime sampin The inducion moo is modeled in he Pak domain wih saionay efeence fame: ψ & = [ R / L, ω ] ψ + V s (1) The moo equaions ae disceized wih he Eule- Cauchy mehod, a explici second ode mehod. As he moo fluxes do no chane auply (saes vaiales of a physical sysem ae linked o he eney of he sysem and hus canno e disconinuous), pecise ineaion of he moo flux equaion is hihly dependen on he ineaion of he sao volae V s which oles quasi-insanly eween zeo and he DC-link volae dependin on he IGBT conducion saes. The moo phase volae ineaion is compued wih he imesamp of he ime oad. Duin each ime sep, each phase of he inducion moo is conneced o he posiive volae of he DC link (V + ) o else o ound (V ). Fo each ime sep, le he up : ime he uppe IGBT of a anch is on. The coespondin phase is conneced o V +. low :ime he lowe IGBT of a anch is on. The coespondin phase is conneced o V. d :ime oh IGBTs ae off. One ani-paallel diode connecs he phase o V o V +, dependin on he phase cuen. hen he ineaed sao volae, a each phase, duin each ime sep is: + h + V sd = V * up + V *( low + d ) (2) (fo cuen enein he moo phase) + h + V sd = V * ( up + d ) + V * low (3) (fo cuen oin ou of he moo phase, i.e. causin he uppe diode o un ON) C. Effec of IGBT fiin pulse ime samp esoluion Pecise deeminaion of he IGBT un on-off imes is impoan fo accuae simulaion of he complee dive appaaus. Fiue 2 shows he effecs of he pecision of he ime samp esoluion on he simulaed machine elecical oque when he simulaion ineaion ime sep is se a 8 µs. The fiue shows ha a 2 µs ime samp esoluion may e adequae fo his off-hihway vehicle applicaions wih a PWM ase fequency of 68 Hz. A ime samp esoluion of 2 µs poduces a.2% nonchaaceisic oque ipple. A ime samp esoluion of 5 µs (cuve ) poduced a less accepale ipple of aou 1% a 6 Hz on he elecical oque. This ipple can iniiae conol insailiy o simply lead o false conol pefomance evaluaion. The quesion hee is: "How o disinuish eween a eal polem caused a conolle insailiy o a polem caused y he simulao?" Of couse, he esul wihou ime samp compensaion 2

3 Inenaional Confeence on Powe Sysems Tansiens IPST 23 in Hon Kon (oom cuve) is oally unaccepale. Bu aain, nonexpeienced eninees may elieve ha an unsale conolle causes his oscillaion. Elecical oque (l.f) (wih 1 Hz osevaion file) m & & PWM caie: 68 Hz, PWM ase fequency: 6 Hz Vdc=12V Moo speed: 1125 pm ** case wih 4us ise ime delay on 2e ide IGBTs ().2 us ime samp esoluion () 2 us ime samp esoluion () 5 us ime samp esoluion (m) 5 us ime samp esoluion ** No ime samp Fiue 2 Effec of ime sep esoluion on he oque Tale 1. Machine paamees Ineia 12 l.s 2 Sao leakae inducance.217 mh Roo leakae inducance.274 mh Muual inducance mh Sao esisance 9.3 mω Roo esisance 8.7 mω Pai of poles 3 D. Real-ime disiued simulaion unde RT-LAB RT-LAB sofwae[1] fom OPAL-RT Technoloies inc. enales eal-ime simulaion of Simulink models on lowcos ae PC cluses wih QNX opeain sysems fo maximum eliailiy. Hih-end sinle-, dual- o quad- Penium pocessos oads ae suppoed on each node of he PC cluses. The IGBT inducion moo dive model has een implemened in a PC-cluse confiuaion composed of 3 Penium III pocessos unnin a 1 GHz. Two of hose pocessos ae insalled on a dual-cpu compue wih shaed memoy while he ohe one is conneced houh a fas FieWie 4 (Mis/s) eal-ime link. The fis CPU of he dual-cpu uni compues he machine model, inephase ansfome model and he ime-samp ased volaes of he wo IGBT ides. The second CPU eads he IGBT ae sinals fom he I/O and oupus model vaiale values o D/A cads. The hid CPU compues low-ae hemal models used o compue IGBT swichin delays and machine windin esisance evoluion. This pocesso can also e used o implemen he conol aloihm, if fully diial simulaion is desied. Fiue 3 shows he hadwae confiuaion of he RT- LAB Elecical Enineein Simulao used in his pape. I can e seen ha he acual hadwae conolle is conneced o he simulao houh fie opic ineface and Naional Insumens NI662 ime cads. RT-La disiued simulaion kene l CPU3 CPU1 CPU2 Fiue 3 Daa acquisiion node (Window XP) machine hemal model (QNX Neuino) IM moo dive compuaion (QNX Neuino) shaed memoy Timed sampin eadin & analo oupu NI662 (Times) NI6713 (D/A) Ehene 1/1 Mps FieWie 4Mps PCI us Console compue Sinle CPU compue Dual CPU compue Sinal condiionin -Diial Fie opic I/O -Quadaue encodes -Powe Analo Oupus To conolle Hadwae confiuaion of he RT-La simulao III. VALIDATION OF SIMULATOR The secion pesens validaion ess made on he IGBT inducion moo dive model. The ess ae made on a inducion moo wih he chaaceisic lised in Tale 1. Hadwae-in-he-loop ess wee conduced on he simulao wih IGBT ae sinals capued fom he simulao's I/O. The ae sinals ae eneaed y a hihpecision pulse eneao implemened in hadwae. Ohe ess wee conduced off-line wih RT-Evens lockse [7] eneaed IGBT ae sinals fo accuae su-µsecond esoluion of he swichin evens. A. Tes 1: Effec of Diode volae dop and IGBT swichin delays The poposed IGBT ide model can finely simulae he IGBTs and diodes volaes dops and swichin delays. Fiue 4 shows he effecs on he simulaed machine elecical oque of he ides device volae dops and swichin delays. In he poposed model, he followin paamees can e individually se fo each device in each ides: V f,v ce :volaes offse fo IGBT and diode. R on :ON esisances of IGBT and diodes T don :IGBT un on delay (fixed) T :IGBT un on delay due o cuen ise in he device T doff :IGBT un off delay (fixed) T F :IGBT un off delay due cuen fall in he device T :Ani-paallel diode un off delay due o evese ecovey 3

4 Inenaional Confeence on Powe Sysems Tansiens IPST 23 in Hon Kon Elecical oque (f.l) 1.52 x Null volae dop & swichin delays 4us IGBT un on delay only 4us IGBT un on delay & 3V IGBT/diode volae offse Simulink Ts=5 us Vdc=9 V Fixed moo speed: 3 pm PWM caie: 68 Hz PWM oupu fequency: Hz Toque osevaion file cu off fequency: 1 Hz Fiue 4 Influence of IGBT paamees on moo oque I is impoan o poin ou ha all simulaion esuls of Fiue 4 wee made wih a fixed ime sep of 5 µs. These esuls demonsae ha he ime samped ide aloihm is ale o ake in accoun swichin delays ha ae a facion of he ime sep. B. Tes 2: Effec on small ide chaaceisic unalances An impoan aspec of he paallel IGBT ide confiuaion is he cuen mismach eween ides caused y diffeence in he IGBT volaes dop and swichin delays. The IGBT ide models ae desined o ake ino accoun all individual IGBT and diode volae dop, swichin delays. The model also akes in accoun hose paamees dependence on device cuens. Cuen (A) () cuen ide 1 () cuen ide 2 () cuen diffeence 1 Simulink Ts=5us Vdc(se poin)=12v Fixed moo speed: 384 pm PWM caie:68hz PWM oupu fequency: Hz Fiue 5 Bide cuens wih unalanced IGBT chaaceisics This es was made off-line wih PWM ime samped ae sinals eneaed fom wihin Simulink and is made o demonsae he ailiy of he ide model o ake ino accoun fine vaiaion in individual IGBT chaaceisics. Those small unalances cause aup volae spikes a he inephase ansfome eminal and cause cuens imalances ino i. Tale 2 liss he swichin devices paamees aken ino accoun y he ide model. The second ide, fo he cuen es, has is uppe and lowe IGBT/diode chaaceisics invesed (i.e. he cause of he imalance). Tale 2 Fis ide paamees Bide No. 1 volae dops and swichin delays Uppe Lowe Diode paams V f (V) R on (mω) T (µs) I (A) (a T ) 1 1 IGBT paams V ce (V) R on (mω) T don (µs) T doff (µs) T (µs) I (A) (a T ) 1 1 T f (µs).5 2. I f (A) (a T f ) 1 1 Fiue 5 shows he simulaion a 5 µs, wih ime samps, of he ides cuens fo he ides paamees lised in Tale 1. The same model simulaed he VisSim [8] simulaion sofwae a a.1µs ime sep (no shown) closely mached he 164A peak cuen diffeence eween phases A of oh ides. C. Tes3: Hadwae-in-he-loop: Squae wave IGBT paen eneaed fom exenal conolle This es makes a one-ide simulaion wih a squae wave modulaion of he inducion moo volae. The esulin elecical oque is shown a Fiue 6. Toque (f.l) () inenal IGBT pulses () exenal IGBT pulses fom I/O () exenal IGBT pulses wihou ime samps 55 squae wave fequency: 18 Hz Vdc=12V Moo speed: 1125 pm Fiue 6 Elecical oque fo squae wave volae modulaion a he inducion machine The opical IGBT ae sinals wee eneaed fom an exenal hadwae cad conneced o I/O and compaed 4

5 Inenaional Confeence on Powe Sysems Tansiens IPST 23 in Hon Kon wih an exac eplica of he sinals eneaed wih he RT- Evens oolox. Fiue 6 also shows he effecs on he simulaed machine elecical oque when nelecin he pecise in-sep imin infomaion, i.e. ime samp, of he IGBT ae sinals (ace wih excessive jie). Tes esuls show he impoance of he ime samped aveain echnique o accuaely simulae he machine elecical oque. D. Tes 4: Hadwae-in-he-loop wih PWM pulse paen eneaed fom exenal space veco conolle. The simulao has een esed wih an exenal PWM conolle eneain fie opic IGBT ae pulse sinals. The conolle PWM aloihm is space veco ype and has is dead ime se o 4 µs. Fiue 7 shows he simulao simulaion esuls (ace ) oained wih he exenal hadwae space veco conolle. One can oseve a 2.5% ipple a 6 Hz on he elecical oque esuls ha may e caused y he conolle fiin pulse unalance o aloihm. The ide-moo model uns a 8 µs ime sep fo all cases pesened in Fiue 7. The ace and pesens he esul oained wih inenally eneaed PWM fiin sequence wih sinusoidalianula echnique. The inenal PWM sequence eneao is uild wih RT-Evens lockse and is heefoe fully compensaed fo in-sep evens. One can oseve ha he 6-Hz ipple has disappeaed wih he inenally eneaed PWM pulses, demonsain ha he plan simulao povides ood esul and ha he ipple measued in ace is caused y he conolle. Aain, he wos case is when he ime samp infomaion is no used (ace m) Toque (f.l) m () exenal IGBT pulses fom I/O () inenal IGBT pulses (m) inenal IGBT pulses wo/ ime samps () inenal IGBT pulses (4 us ise ime) PWM caie: 68 Hz, PWM ase fequency: 6 Hz Vdc=12V Moo speed: 1125 pm Toque osevaion file cu off fequency: 2 Hz Fiue 7 Elecical oque fo PWM volae paen a he inducion machine eneaed fom he I/O The diffeence eween aces and can e explained in a nume of ways. Fo example, inenal and exenal IGBT paens ae eneaed wih diffeen aloihms, namely space-veco vs. sinusoidal-ianula mehod. Fo example, he space-veco aloihm someimes eneaes isin and fallin IGBT ae sinals wihin a sinle ime sep. Alhouh he ide aloihm is ale o handle such evens, he discee-ime RT-Evens PWM inenal aloihm was no ale o eneae hem in he fis place. E. Tes 5: Toque conol chaaceisic accuacy and small sinal lineaiy An impoan aspec of he HIL simulaion is he fideliy of ineacion of he simulaed model wih he conolle. In he case of an IGBT dive invee, one ciical aspec fo he accuacy of simulaion is he pecise capue of IGBT ae pulse imins and hei handlin in model ineaion. Elecical oque (l.f) () 8 us SimPoweSysems modeled ide () 1 us SimPoweSysems modeled ide () 8 us Opal RT ime samped ide 2 V dc = 12 V 1 PWM caie feq.: 68 Hz PWM oupu feq.: 6Hz Moo speed : 1125 pm Toque osevaion file cu off feq.: 1 Hz PWM modulaion index Fiue 8 Elecical oque as a funcion of PWM modulaion faco (68 Hz caie) To show he poin, he ime-samped ide is compaed o he ide model of SimPoweSysems lockse (PSB). Elecical oque (l.f) Toque osevaion file cu off feq.: 1 Hz PWM modulaion index Fiue 9 () 25 us SimPoweSysems modeled ide () 1 us SimPoweSysems modeled ide () 8 us Opal RT ime samped ide V dc = 12 V PWM caie feq.: 5 khz PWM oupu feq.: 6Hz Moo speed : 1125 pm Elecical oque as a funcion of PWM modulaion faco wih 5 khz PWM caie The PSB is no desined o compensae he sysem soluion fo in-sep swichin evens. Wih he moo chaaceisics of Tale 1, he PWM modulaion index has 5

6 Inenaional Confeence on Powe Sysems Tansiens IPST 23 in Hon Kon een scanned and he esulin elecical oque ploed in Fiue 8. Close exam of he esuls shows ha he ime sampin echnique allows fo accuae epoducion of he oque chaaceisic a ime sep compaile wih HIL simulaion. Someimes even moe ciical fo conolle esin, he small-sinal lineaiy is peseved fo he ime samped ide while he uncompensaed ide causes he chaaceisic o have ude slope disconinuiies. Even wih a PWM caie fequency of 5 khz (Fiue 9), he Opal- RT Time-Samped ide sill shows lineaiy in is oque conol chaaceisic while uncompensaed simulaion is no linea even a smalle ime seps (25 µs). F. Tes 6: Measuemen of ime samp esoluion wih he NI662 The impoance of he esoluion of he ime sampin of he volae applicaions o he moo has een esalished in Secion IIC. This es measues he acual pecision of he NI662 imin oad used. The maximum esoluion of he NI662 unnin a 2MHz is 5 ns. Bu even occuin nea he einnin o he end of he ime sep may hu his specificaion ecause of he synchonizaion commands ha ae sen o he cad a each ime seps. The es has een made wih he IGBT moo dive wih fie opic IGBT ae inpu o measue he accuacy of he ime samp eadin fom he NI662. A a 8 µs ime sep, a squae wave IGBT ae sinal is sen houh he fie opic inpu and ead y he Simulink dive model ime samp eadin mechanism. If he squae wave isin fons occu a a ae ha is exacly an inee muliple of he ime sep, hen he ime samp ead fom he I/O will e consan in ime. Deviaion fom pefec incemen (ns) Time samp non lineaiy Ts = 8 us Fequency inpu = 26 Hz isin ede only Non null ime samp nume Fiue 1 Time samp deviaion fom consan incemens linea incease. This diffeence is shown in Fiue 1. Resuls show ha he NI662 cad has an effecive ime samp esoluion of ± 15 ns. IV. COMPUTATIONAL SPEED UNDER RT-LAB Tale 3 shows he oainale had eal-ime sep fo wo dive confiuaions unnin unde RT-La on Penium III, 1-GHz pocessos. In oh cases, he IGBT ae sinals ae comin fom he I/O. Tale 3: Real-ime sep size oainale on 1 GHz PC Bide confiuaion Had eal-ime sep Dual ide + IPT 8 µs Sinle ide 65 µs In all cases, he compuaional ime of he elecical pa of he sysem (CPU1 in Fiue 3: ides, IPT, DClink and d-q modeled inducion moo) is unde 2µs. In fac, he main compuaional uden comes fom he I/O laency of he ime samp eadin, houh he PCI us on CPU2, fom he 5 NI662 cads (3 fo sinle ide case) and analo oupus. Theefoe, samplin imes elow 25µs ae expeced wih fase FPGA-ased I/O echnoloy cuenly unde developmen a Opal-RT Technoloies. V. REFERENCES [1] RT-LAB 6.1, Opal-RT Technoloies inc Richadson, ueau 2525, Moneal Qc H3K 1G6 [2] C. Dufou, J. Bélane, S. Aouida, "Accuae Simulaion of a 6-Pulse Invee Wih Real Time Even Compensaion in ARTEMIS", Poceedins of he ELECTRIMACS 22 confeence, Moneal, Canada, Auus18-21, 22 [3] ARTEMIS Add-On fo he Powe Sysem Blockse fo Simulink, v2.3, Opal-RT Technoloies Inc., Moneal, Qc, Canada [4] Powe Sysem Blockse vesion 2. fo Simulink, The MahWoks Inc. Nawick, MA, USA [5] EMTP Theoy Book, H.W. Dommel edio, 2nd ediion, Micoan Powe Analysis Copoaion, May [6] Pee Tewiesch, Thomas Kelle, Eich Scheien, "Rail Vehicle Conol Sysem Ineaion Tesin Usin Diial Hadwae-inhe-Loop Simulaion", IEEE Tansacions on Conol Sysem Technoloy, Vol. 7, No. 3, May 1999, pp [7] C.A. Raah, M. Adoune, J. Belane and K. Bus, "Simulain Hyid Dynamic Sysems", IEEE Rooics and Auomaion Maazine, Vol. 9, No. 2, June 22, pp [8] VisSim v.4.5, Visual Soluions Inc., Wesfod, Massachuses, USA [9] P.G. McLaen and al.,"a Real-Time Diial Simulao fo Tesin Relays", IEEE Tansacions on Powe Delivey, Vol.7 No.1, Jan.1992, pp [1] V.Q. Do, J.-C. Soumane, e al., "HYPERSIM, an Ineaed Real-Time Simulao fo Powe Newoks and Conol Sysems", Tansacions of he ICDS '99 Confeence, Vaseas, Sweden, 1999 The es hen consiss of sendin a squae wave wih a vey small diffeence o he exac inee peiod aio. This esuls in a scannin of he ime samp fom o 1 (in nomalized value). In he ideal case, he incease of ime samp values is consan fom an even o anohe. The ead ime samp value can hen e compaed wih he ideal 6