Fluorescent Lamp Modelling for Voltage Fluctuations

Transcription

1 Fluoresen amp Modelling for Volage Fluuaions ETE C. Carrillo, J. Cidrás Absra The wide use of fluoresen lamps means heir influene on he power sysem needs o be sudied. A se of models ha enables he evaluaion of he elerial behaviour during seady sae is presened in his paper. These models have differen levels of omplexiy; saring by aking ino aoun all he non-lineariies of he lamp, passing hrough negleing he non-lineariies of he reaane, supposing he volage wave shape of he ube o be square and, finally, reaing he variaions in he RMS soure volage as an seady-sae phenomenon. The mos omplex model needs an ieraive algorihm o ge resuls, however analyial expressions an be ahieved wih he oher models. Several siuaions an be analysed wih he differen models bu speial aenion is paid o he sinusoidal volage soure and he modulaed volage soure ases. Also, he sudy presened in his paper is foused on obaining he waveshape of urren, volage and power in he ube. These waveforms allow he sudy of he normal seady-sae funioning and he fliker phenomena. In his paper he simulaion resuls for he above models in he wo kinds of siuaions are shown. As a resul, a performane es beween he models is made and i is used o validae hem. Inroduion The fliker problem is well doumened wih referene o inandesen lamps. The inernaional sandards (e.g. IEC6868 are based on his kind of lamp. However, fluoresen lamps are widely used and he fliker problem relaed o hem is very imporan. Moreover, he physial mehanism of elerial energy onversion ino ligh differs drasially beween he wo ypes of lamps. The fluoresen lamp is a disharge lamp ha has a nonliner behaviour, and he inandesen lamp has a linear behaviour. The firs one needs an ar inside he ube, he seond one uses he Joule-heaing proess. This paper is inended o over he seady-sae sudy of fluoresen lamps. Therefore a se of numerial and analyial simulaion models is presened. The models have differen degrees of omplexiy and auray. Firs, he Complee Model [] is presened. In his model, he mos imporan non-lineariies of he fluoresen lamp are represened. These are, he sauraion and hyseresis effe on he reaane, and he ar phenomenon inside he ube. An ieraive mehod is needed o analyse his approximaion, hen simpler models are proposed in order o ge an easy mehod o analyse he fluoresen lamp hrough analyial expressions. Approximae Model I is inended o be a simpler opion. This simplifiaion is ahieved supposing he reaane o be linear. So, he sauraion effe and he hyseresis effe are no aken ino aoun in his model. Then analyial expressions have been ahieved and he sudy is made quikly and easily. A deeper simplifiaion is made abou he las model. The wave shape of he ube is supposed o be square and he resisane of he reaane is negleed. In his way he Approximae Model II is shown. Finally, a las model is presened, he Approximae Model III [6]. This model omes from a simplifiaion on he dynami behaviour of he Approximae Model II, he RMS variaions on he soure volage are reaed as a sai phenomenon. In his way he simples expressions are reahed. All he above models are sudied under wo siuaions: he volage soure being a sinusoidal soure and a modulaed soure. These siuaions help he sudy of he normal seady-sae behaviour, i. e. harmoni analysis, and he fliker phenomenon. Speial aenion is paid o he urren and power in he ube beause hey are losely relaed o he fliker and harmoni sudy. The resuls for eah Approximae Model are ompared wih he Complee Model in order o ge a validaion mehod. The loss of auray is he prie for he simpliiy, hus he Approximae Model III is expeed o be he simples and mos inaurae model. Fluoresen amp Cirui The main lamp elemens are shown in he Fig.. These are: reaane, ube and sarer. This irui has a srongly non-linear behaviour during he seady sae, as shown see in Fig.. The ar inside he ube is he main ause of he disored volage shape. Thus he ube is he mos imporan non-lineariy in he irui. e( Reaane u( Tube Fig.. Fluoresen lamp equivalen irui C Sarer ETE Vol., No., Marh/April 9

2 u e V U a U a u( e(. Fig.. Volage and urren waveforms The hyseresis and he sauraion in he iron ore reaane are he oher non-lineariies aken ino aoun for he fluoresen lamp sudy. The sarer is open during normal ube operaion, so i has no been modelled. In he following models all he parameers are obained from he measuremens on a ypial 36- ube (see Fig.. 3 Complee Model γ..6 A.3 In his par, he seps o ge a simulaion mehod for he fluoresen lamp are depied []. Firsly, a desripion and modelling of he differen elemens are made. The sudy of reaane and ube have been emphasised, beause hese are he main non-linear elemens. s i Reaane = Fig.. Equivalen reaane model The presen non-lineariies preven a dire soluion for a given volage supply. Insead, an ieraive approah is required for aurae derivaion of urrens and volages in he lamp irui. The power in he ube is he main resul from he ieraive analysis. The sudy of he fliker is based on his power, and his is he reason for is imporane. 3. Model Desripion R r (Coil resisane R h (Hyseresis s (Sauraion In his seion he ube and reaane model are depied, beause hey are he main non-lineariies of he lamp irui. As is shown in Fig. 3, he ube model is derived from a sraigh lines approximaion of is volage-urren haraerisi. In his way, an easy model for ompuer implemenaion is obained. The reaane is he following non-linear elemen for he modelling. The model mus over he oil resisane and he sauraion and hyseresis effes in he iron ore. A series resisane is used o model he oil resisane (Fig.. The sauraion effe is modelled by a non-linear induane, and i has he following equaion []: a.6 A b.6 A.3 U = V U = V U = V i ( = kφ( kφ (, ( where φ ( is he flux in he iron ore, is he urren aross he induane, and k, k are onsans. Finally, a resisane in parallel wih he non-linear induane represens he hyseresis. Is value an be alulaed from he equaion [3]: ω Rh = φmax, ( di where φ max is he maximum flux, ω is he fundamenal frequeny of he soure and d i is one half of he hyseresis widh a φ = (see Fig. 5. As a resul, he model of he reaane is obained. Measured V u( Fig. 3. Volage-urren haraerisi of he fluoresen ube and is approximaion a Measured (U = V b Approximae d i Approximaion φ max Fig. 5. Comparison beween he hyseresis haraerisi and is approximaion φ ( ETE Vol., No., Marh/April

3 Sar Calulaion of iniial urrens Frequeny domain paper is o ahieve models wih abiliy o represen power fluuaions. So, he appliaion of he menioned urve is ou of he sope of he paper. Approximae Model I ETE Calulaion of he urren aross he sarer apaior No Calulaion of he volage of he ube Calulaion of he urren aross he ballas Error < 5 End Fig. 6. Simulaion flow diagram 3. Ieraive Harmoni Analysis One he lamp has been modelled an ieraive algorihm is used o derive he seady-sae volage and urren wave shapes in he irui. The algorihm alled Ieraive Harmoni Analysis [] is illusraed in Fig. 6. The Ieraive Harmoni Analysis involves a hybrid ehnique whih formulaes he non-linear elemens in he ime domain (sauraion and ube response, and he linear sysem in he frequeny domain. The ommon urren is used as he inerfae beween he wo sysems in eah ieraion. Operaions in he ime domain and in he frequeny domain are mahed in order o obain he seady-sae response of he lamp irui. The urren, insead of he volage, has been hosen as an inerfae o improve he Ieraive Harmoni Analysis rae of onvergene. Sine he ube volage has a shape whih is oo disored o be ahieved wih a ieraive algorihm. 3.3 ower in he Tube Yes Time domain Frequeny domain Time domain: Sauraion alulaion Frequeny domain: Flux Hyseresis osses Any volage or urren an be alulaed via Ieraive Harmoni Analysis. However, he urren and volage in he ube are pariularly ineresing in order o obain he eleri power. hen here is a fliker phenomenon, his is losely onneed wih he eleri power in he ube. Moreover, he human eye an dee osillaions beween Hz and 35 Hz, as an approximaion of a Buerworh filer [5] wih a u frequeny of 35 Hz. So he power variaions beween Hz and 35 Hz are relaed o luminous flux variaions ha he eye an dee. Therefore, he power p( relaed o he visible luminous flux variaions is alulaed by filering he insananeous power p i ( in he ube. The filer is an ideal lowpass one wih a u-off frequeny of 35 Hz. The power ( ould be weighed aording o he flikermeer urve. However, he main purpose of his ih he Complee Model, he numeri alulaion of he power in he ube an now be obained. However, some simplifiaions on he model are needed o ahieve an analyial power alulaion. In he Approximae Model I wo simplifiaions are made: The reaane is supposed o be linear. Thus, is model is an R- ombinaion, negleing he hyseresis and sauraion effes. The volage wave shape of he ube is supposed o be onsan. This simplifiaion is more realisi when he soure has small fluuaions. I mus no be forgoen o ake ino aoun ha in he ube, he phase shif beween he urren and volage is equal o zero (see Fig.. The equivalen lamp irui wih he above simplifiaions is shown in he Fig. 7. The irui behaviour follows he differenial equaion: d e u Ri i ( ( ( = (, (3 d where e( is he soure volage. One ondiion for he equaion soluion is he zero value for he phase shif beween and u(. If he phase referene is he urren or volage in he ube, he harmoni sperum of u( is onsan. However, he usual phase referene is he volage soure, so he harmoni faoring is: u = Un sin( nω θn nγ, ( where ω is he fundamenal frequeny of he soure, U n and φ n are onsan values beause of he above simplifiaions and γ is he phase shif beween he soure volage e( and volage in he ube u( (or he urren aross he ube. As is shown in eq. (, he volage u( only depends on he phase shif γ.. Sinusoidal Volage Soure The usual and simples siuaion in whih o sudy he fluoresen lamp behaviour is when he volage soure is pure sinusoidal, hus: e = Usin ( ω, (5 where U is he RMS soure volage. e( n= Reaane X R u( Volage of he ube Fig. 7. Approximae Model I (U n onsan and X = R ETE Vol., No., Marh/April

4 The frequeny domain is more appropriae in order o sudy he seady-sae behaviour. So, eq. (5 gives: E( ω U( ω I( ω =, R jω (6 where E(ω and U (ω are he Fourier ransforms of e( (see eq. (5 and u( (see eq. (, and R- are he values for he reaane simplifiaion. The soluion of eq. (6 gives an expression for he urren: U i ( = Z os ω θ ( ω ω Un n n n n n Z ( n os ( ω θ ( ω θ γ, = ω (7 where Z (ω and θ (ω represen he module and phase shif of he -R impedane. Their expressions are: Z ( ω = R ( ω, π ω θ( ω = an. R (8 A his momen, he parameer γ an be obained and, in his way, he volage e( of eq. ( and urren of eq. (7 are ompleely known. For his purpose he zero shif ondiion is used: γ i =. ω Subsiuion of eq. (9 in eq. (7 resuls in: γ = θ ( ω (9 U ω n Z( os ( θ θ ω = ( ω os n ( n. n U Z n ( The proedure shown in Seion 3.3 is used in order o ge he power in he ube: U = Z( ω Usin γ θ θ ( ω U sin θ ( ω. ( [ ( ( ]. Modulaed Volage Soure In order o sudy he fliker phenomenon in he fluoresen lamp, we assume ha he supply volage is modulaed by sinusoidal volage of a lower frequeny: a a e = U os( Ω sin ( ω, ( where Ω is he modulaion frequeny (Ω ω / and a is he modulaion faor ( < a <.. roeeding as in he previous seion he following expression for he urren is ahieved: i ( = U a Z( ω os ( ω θ( ω au os (( ω Ω θ( ω Ω Z( ω Ω au os (( ω Ω θ( ω Ω Z( ω Ω Un n n n Z ( n os ( ω θ γ θ( ω ω. n = (3 A his poin, he use of he ondiion i( γ /ω = resuls in: f( γ = i( γ ω = U a os ( γ θ( ω Z( ω ω Ω os γ θ( ω Ω au ω Z( ω Ω ω Ω os γ θ( ω Ω au ω Z( ω Ω Un θ θ ω Z( nω ( n ( n =. n = ( The above non-linear equaion mus be solved by an ieraive mehod. The well-known Newon-Raphson mehod is hosen, whih means ha, before alulaion, he following ondiions are needed: The derivaive, for whih he equaion is: d f ( γ f ( γ = dγ = U( a sin ( γ θ( ω Z( ω au ω Ω Z( ω Ω ω ω Ω os γ θ( ω Ω ω au ω Ω Z( ω Ω ω ω Ω os γ θ( ω Ω. ω (5 An iniial guess for γ. This value is alulaed from eq. (, supposing a = and he a Newon-Raphson algorihm is applied. The power in he ube is: p ( = (, (6a Ω [ U = U( a sin γ θ θ( ω Z( ω U sin θ ( ω, ( ] (6b ETE Vol., No., Marh/April

5 (6 where is he mean power, and Ω ( is he alernaing omponen. The fluuaions ha he eye an see are represened by Ω (, whose fundamenal frequeny is simply Ω and whose ampliude is Ω. 5 Approximae Model II In order o obain simpler analyi power and urren expressions, more simplifiaions have been be made on he previous model. In his model he volage of he ube is supposed o be a square wave. This wave has a zero phase shif wih regard o he urren, as is shown in Fig. 8. The oil resisane is negleed in order o ahieve a higher degree of simplifiaion. The square volage in he ube is: (7 where U a is he mainenane volage of he ar inside he ube (see Fig.. The above expression omes from Fourier faoring of a square wave whose exreme values are Ua and -Ua. The resuling wave forms afer he new simplifiaions are shown in Fig. 9. Reaane u( Tube in Approximae Model II Fig. 8. Approximae Model II and Approximae Model III u e Ω auu sin Ω ( Ω ( = θ γ θ ω 8 Z ω Ω V U a U a auu sin Ω ( Ω θ γ θ ω, 8 Z ( ω Ω U sin n ω a u ( = π n n = u( e( γ Tube in Approximae Model III.6.. s.3 Fig. 9. Volage and urren waveforms in he simplified irui [( γ ],.6 A.3.3 i 5. Sinusoidal Volage Soure If he soure volage expression is eq. (5, an expression for he urren is reahed wih similar seps o hose in previous seions: U i = os( ω ω U os n a [( ( ω γ ]. (8 πω n= ( n The phase shif beween (or u( and e( is: os γ = U a π. (9 U So, he power is: UU πω a = sin γ. 5. Modulaed Volage Soure ( hen he soure volage has he expression shown in eq. (, hen he urren is: U a i = os( ω ω au os( ω Ω os( ω Ω ω Ω ω Ω U os[ ( n ( ω γ a ]. πω n n = ( The Newon-Raphson algorihm is used again in order o obain he phase shif γ. The following expressions are used: U a f ( γ = osγ ω γ γ os ( ω Ω os ( ω Ω au ω ω ω ω Ω Ω Ua π = ; ( ω d f( γ U a a Ω f ( γ = = os sin. dγ ω ω γ γ So he power in he ube is: p ( = (, Ω a UU a = sin γ, πω (3 (a (b ETE Vol., No., Marh/April 3

6 auua sin Ω γ sin Ω γ Ω ( =. π ω Ω ω Ω ( where is he mean power in he ube and Ω ( is he alernaing power in he ube. 6 Approximae Model III Anoher simplifiaion level is when he modulaion frequeny is very low (Ω <<. Then he volage fluuaions an be onsidered as a sai phenomenon. This simplifiaion is referred o as Approximae Model III [6]. 6. Sinusoidal Volage Soure ih he soure as sinusoidal he resuls are he same as hose shown in eq. ( for Approximae Model II. 6. Modulaed Volage Soure hen a modulaed volage soure is applied o his model, he power an be obained by simply subsiuing eq. ( in eq. ( whih resuls in: 6.3 Simulaion The simpliiy of he Approximae Model III allows an easy way o ge a simulaion equivalen. This equivalen is depied in Fig Resuls All he models have been simulaed under several modulaion frequenies Ω = 3.5 Hz, 6.5 Hz, Hz 5. Hz, and modulaion faors a =. p.u.,.5 p.u.,. p.u.,.5 p.u.,. p. u. In he following figures he resuls from hese simulaions are represened. 8 Conlusions a a p = 8UU os( U. a Ω π a πω (5 The resuls from measuremens (Measured ower on a ypial 36- ube under he same ondiions as he simulaion models are inluded (Fig. and Fig.. The analysis of he simulaion resuls presened in he above-menioned figures leads o he following onlusions: As an be shown in Fig., he mean power ( is he mos imporan power omponen. The seond one is he omponen Ω ( whose osillaion frequeny is Ω. Differen erms of frequeny are presen only in he Complee Model and in he Measured ower, bu hey an be negleed. a p b Aproximae I Aproximae II Aproximae III 3 Complee Measured...6 s Fig.. ower wih he differen models (Ω = Hz; a =. a Time-domain power b Frequeny-domain power ( Hz 35 Hz a b Ω, p Complee Approximae I Approximae II Approximae III Measured Hz 35 f f Mod = Hz f Mod = 5. Hz f Mod = Hz f Mod = 5. Hz p.u p.u.. Fig.. Measured power a Time-domain power b Frequeny-domain power ( Hz 35 Hz ETE Vol., No., Marh/April

7 a b 5. Ω, p f Mod = Hz f Mod = 5. Hz f Mod = Hz f Mod = 5. Hz p.u p.u.. Fig.. Mean and alernaing power wih he Complee Model a Time-domain power b Frequeny-domain power ( Hz 35 Hz a b 5. Ω, p f Mod = Hz f Mod = 5. Hz p.u.. Fig. 3. Mean and alernaing power wih Approximaion I a Time-domain power b Frequeny-domain power ( Hz 35 Hz f Mod = Hz f Mod = 5. Hz p.u a b 5. Ω, p f Mod = Hz f Mod = 5. Hz , p.u.. ETE p.u.. Fig.. Mean and alernaing power wih Approximaion II a Time-domain power b Frequeny-domain power ( Hz 35 Hz a b 5. Ω, p f Mod = Hz f Mod = 5. Hz p.u.. Fig. 5. Mean and alernaing power wih Approximaion III a Time-domain power b Frequeny-domain power ( Hz 35 Hz ,.5..5 p.u ETE Vol., No., Marh/April 5

8 Mean power (Ω Approximaion error Complee % Approximaion I % Approximaion II % Approximaion III % Alernaing power β (Ω Approximaion error Complee 79 5 % Approximaion I % Approximaion II % Approximaion III % Ω = Hz and a =. p.u. Tab.. Approximaion by he eas-square Mehod of he Mean and Alernaing ower Coeffiiens (Ω and β (Ω In he simulaion models, he mean power and he alernaing power ampliude Ω have linear dependene on he modulaion faor a (see Fig., Fig. 3, Fig. and Fig. 5. If he modulaion frequeny Ω is onsan hen he mean power and he alernaing power ampliude are: = ( Ω a, = a (6 a= Ω, p β( Ω, where a = is he mean power (wih a =, is he mean power and Ω,p is he alernaing power ampliude. The oeffiiens (Ω, β (Ω are alulaed via he eas-square Mehod, and he resuls are shown in Tab.. This linear behaviour in no so lear in he Measured ower (Fig.. The mean power dereases wih he modulaion faor and inreases wih he modulaion frequeny. hereas he alernaing power ampliude inreases wih he modulaion faor and wih he modulaion frequeny. hen low values of modulaion faor and modulaion frequeny are used, he lowes differenes beween he Approximae models and Complee Model are ahieved. However, in all simulaions he Complee Model has had he lowes error wih relaion o he Measured ower. The differene beween he models and he measuremens are no negligible beause of he dependene on he frequeny and ampliude of haraerisi of he ube (see Fig. 3. In summary The lamp behaviour under volage fluuaions anno be derived from he seady sae wih sinusoidal volage. High errors are obained using he differen models, however, he omplee model is he neares o he ube behaviour. 9 is of Symbols 9. Symbols ω F(ω ime frequeny Fourier ransform of f( K modulaion faor Ω modulaion frequeny u, U volage in he ube U a mainenane volage of he ar inside he ube e, E soure volage i, I urren aross he ube p, power omposed by frequeny omponens beween Hz and 35 Hz of he insananeous power p i ( p i power before filering φ flux in he iron ore of he reaane γ phase shif beween he soure volage e and he volage in he ube u f(γ urren aross he ube when = γ /ω Z, θ module and phase of he reaane impedane d i one half of he hyseresis when φ( = k, k sauraion urve onsans X reaane R oil resisane of he reaane R h resisane ha represens he hyseresis in he reaane induane for he reaane n harmoni order 9. Subsrips mean power fundamenal omponen n n-h harmoni s wih sauraion Ω alernaing power k k-h ieraion in he Newon-Raphson algorihm max maximum p peak value or ampliude 9.3 Abbreviaions IEC RMS Inernaional Eleroehnial Commission roo mean square Referenes [] Cidrás, J.; Carrillo, C.; Arrillaga, J.: An Ieraive Algorihm for he Analysis of he Harmoni Currens rodued by Fluoresen amps. 7h In. Conf. on Harmon. and Qualiy of ower (ICHQ, as Vegas/USA 996, ro. pp [] ason, N.; Robbies, A.; Arrillaga, J.: Represening Transformer Sauraion in Ieraive Harmoni Analysis. In. Conf. on Harmon. ower Sys. (ICHS-VI, Bologna /Ialy 99, ro. pp [3] Eleromagnei Transien rogram (EMT/AT Referene manual. euven Cener, Belgium,987 [] Arrillaga, J.; ason, N. R.; Eggleson, J. F.; Callaghan: C. D.: Comparisons of Seady and Dynami Models for he Calulaion of AC/DC Sysem Harmonis. ro. of IEE 3-C (987 no., pp [5] IEC/TR 6868 (986-9: Flikermeer Funional and design speifiaions. Offenbah Berlin/Germany: VDE VERAG, 986 [6] Emanuel, E.; ereo,.: The Response of Fluoresen amp wih Magnei Ballas o Volage Disorion. IEEE Trans. on ower Delivery RD- (997 no., pp Manusrip reeived on Augus 5, ETE Vol., No., Marh/April

9 The Auhors Camilo Carrillo (967 reeived his degree in elerial engineering from he Universiy of Vigo (Spain in 99. Sine hese year he has been a leurer in he Deparmen of Elerial Engineering. His main fields of researh are: harmonis, fliker and wind energy sysems. (Dpo. Enxeñería Eléria, ETSEIM, Universidade de Vigo, agoas Marosende S/n, 36 Vigo /Spain, hone: , Fax: , arrillo@uvigo.es José Cidrás (957 has reeived his degree in Elerial Engineering from he Universiy of as almas de G.C. (Spain. He obained a hd in elerial engineering from he Universiy of Saniago (Spain in 987. He is professor and head of he Deparmen of Elerial Engineering of he Universiy of Vigo (Spain, and leads some invesigaion projes on wind energy, phoovolais and planning of power sysems. He is a Member of IEEE. (Dpo. Enxeñería Eléria, ETSEIM, Universidade de Vigo, agoas Marosende s/nº, 36 Vigo/Spain, hone: , Fax: , jidras@uvigo.es ETE Vol., No., Marh/April 7