A METHOD TO DETERMINE A SINGLE NUMERICAL VALUE FOR REACTIVE POWER IN NON-LINEAR SYSTEMS

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1 Joural of Sciece ad Arts Year 13, No. 4(5), , 13 ORIGINAL PAPER A MEHOD O DEERMINE A SINGLE NUMERICAL VALUE FOR REACIVE POWER IN NON-LINEAR SYSEMS ALI KHAI DIZABADI 1, MOHAMMAD J. JAFARIAN, AREF DOROUDI Mauscrit received: ; Acceted aer: ; Published olie: Abstract. Classical aaret ower decomositios do ot fulfill the coditios caused by the resece of harmoics. I harmoic olluted system, the roosed ower defiitios should be able to determie a accurate value for caacitace ower to imrove ower factor of system. I this aer we modify, by meas of istataeous ower equatios, the method suggested by Khalsa-Zhag ad rooses a ew defiitio for reactive ower i o-siusoidal system. he ew defiitio correctly gives a sigle umerical equivalet value of reactive ower ad ca be easily alied to calculate the caacitace ower. he alicatio of this defiitio is illustrated with a simle examle Keywords: Active ower; No-active ower; Reactive ower No-siusoidal coditio. 1. INRODUCION I 197, whe Budeau reseted his aer as "Reactive ad fictitious owers" [1], he has ever suosed beig the iitiator of oe of the most cotroversial issues i ower system studies. By widesread use of ower electroic loads ad harmoic ollutio resulted from these devices, his ure theoretical idea has become a ractical issue for electrical egieers. Although may theories have bee roosed to determie ower arameters i o-siusoidal coditios [-1], but oe of them are widely used i ractice ad they have ot bee able to obtai ublic accetatio. Cosequetly, the researchers cotiuously attemt to attai a commo defiitio about this subject. However, a roer ad comrehesive theory should have the followig characteristics: 1. o reset a commo defiitio ad formula for ower i siusoidal ad osiusoidal coditios.. Caability of measurig the roosed ower decomositios i ractical coditios. 3. Ca be alied to reactive ower comesatio ad ower factor imrovemet. Amog some of the roosed theories, e.g. Fryze's method [] does ot have the first ad third feature, Sheherd ad Zakikhai's reactive ower defiitio [4] caot be alied for ower factor calculatio ad ower comoets omiated by Czarecki [1] caot be easily measured ad so it does t have the secod feature. Because of distributed reactive loads i moder ower etworks ad related comesatio roblems the third feature is so imortat ad if a method does ot cover it, the method will ot be ractical ad desirable. 1 Deartmet of Power,Qaemshahr braces, Islamic Azad Uiversity, Qaemshahr, Ira. alikhatidizabadi@yahoo.com Deartmet of Power Egieerig, Faculty of Egieerig, Shahed Uiversity, ehra, Ira ISSN:

2 376 A method to determie Geerally egieers i the idustry are used to a sigle umerical equivalet value for the reactive ower. But may of the researchers defie istataeous reactive ower i the time domai. Oe of the ew methods i this field which was reseted by Khalsa Zhag, based o eergy trasfer, rovides a ew formula for reactive ower i o-siusoidal coditios [11]. he method gives a sigle umerical value for o-active ower, but this value is ot suitable for reactive ower comesator desig by simle caacitor. Furthermore the method has some limitatios. I this aer, the Khalsa Zhag's method is first modified to remove its disadvatages ad the based o istataeous ower equatios, the method is develoed so that oe ca easily aly it to reactive ower comesatio. his aer will be cotiued i the secod ad third sectios by exlaiig the ower formula i o-siusoidal coditios ad reactive ower comesatio cocet. I sectio four the Khalsa Zhag's method is itroduced ad i sectio five it is develoed. Sectio six resets simulatios ad fially coclusios are give i sectio seve.. POWER DEFINIION IN UNSINUSOIDAL CONDIIONS he o-siusoidal voltage ad curret geerated by oliear loads could be aalyzed usig Fourier series. he o-siusoidal voltage ad curret are exressed; V () t. V Si t 1 ( ω ) = (1) I ( t ) =. I Si t ( ω α) () 1 As i siusoidal coditio the istataeous aaret ower could simly be calculated with roduct of istataeous voltages ad currets.. ( ω ) ( ω α) (3) S ( t ) = V Si t. I Si t 1 1 Decomositio of istataeous aaret ower i o-siusoidal coditios is ot a clear-cut rocess, as doe for siusoidal coditio [1]. 3. REACIVE POWER COMPENSAION I ure siusoidal system, classical aaret ower decomositio has bee emloyed to facilitate a tool for system iciecy imrovemet by meas of reactive ower comesatio. However, the classical decomositio does ot fulfill the coditios caused by the resece of o-siusoidal voltage ad/or curret. Cosequetly, ew ower defiitios should be itroduced to rovide a tool for ower factor imrovemet. I fact, i harmoic olluted system the o-active ower is divided ito two comoets; reactive ower ad distortio ower. I this coditio, eve i the case of ure resistive load, the ower factor does ot attai uit value. herefore, i o-siusoidal coditios comesatio reactive ower beig so highlighted ad the roosed ower defiitios geerally bare the aims of

3 A method to determie 377 accurate ad easy determiatio of maximum ower factor realizable with basic caacitive comesatio ad criterio to measure the degree of load s o-liearity. 4. KHALSA- ZHANG'S MEHOD Khalsa- Zhag's roosed a ew defiitio for active ad o-active ower that is based o eergy trasfer ad rovides a uique solutio for them uder both siusoidal ad o-siusoidal coditios. his defiitio gives a sigle umerical value of both active ad o-active ower termed ective active ower ad ective o-active ower. he ective ower of istataeous active or o-active ower was defied as the amlitude of a equivalet siusoidal ower waveform that has the same eergy trasfer as the corresodig active or o-active istataeous forms i either siusoidal or o-siusoidal case. he authors roosed that istataeous aaret ower ( S( t )) is made u two comoets: istataeous active ower ( Pt ( )) ad istataeous o-active ower ( Qt ( )). St () = Pt () + Qt () (4) where Pt () is fully absorbed ad cosumed by the load, but Qt () vibrates betwee source ad load. I their theory, the active eergy trasfer ( E ) due to flow of istataeous active ower ( P ) is defied by: E = P () t dt (5) where is the fudametal eriod. he eergy trasfer due to flow of istataeous oactive ower ( Qt ( )) is bidirectioal ad caot be defied i a similar maer to that Pt () usig Eq. (5), because its value will be zero. he authors ostulated that the ositive ad egative areas i Qt () waveform are equal irresective of the waveform. With this cosideratio, the o-active eergy trasfer ( E Q ) is defied as twice the area of the ositive area: E Q = Q os () t dt (6) where: Q(t) : Q(t) > Q os = (7) : Q(t) < hey itroduced P ad Q are ective active ower ad o-active ower, resectively as follows: E = P. (8) EQ = Q. (9) π ISSN:

4 378 A method to determie he they cocluded: E 1 P = = P(t)dt (1) π π = Q = os (11) Q E Q dt 4.1. KHALSA- ZHANG'S MEHOD LIMIAIONS Khalsa- Zhag's defiitio has a umber of limitatios as follows: 1. he method requires kowledge of the load imedace ad it does ot idicate how we ca determie the ower decomositios whe the load is ukow.. he defiitio rovides useless iformatio to determie the caacitor ower for ower factor imrovemet. 3. he limit of itegratio i Eq. (5) ad Eq. (6) may ot be obtaied, e.g. whe o liear load or system s voltage have a high third harmoic. I this coditio ad i oe eriod, the istataeous o-active ower caot be searated ito four idividual subarea i the time itervals[ /4],[/4 /],[/ 3/4], ad[3/4 ]. Hece, the eergy trasfer caot be obtaied by determiig the area of Qt () i the time itervals [ /4] ad[/ 3/4]. As a illustratio of the last limitatio cosider the sigle hase circuit show i Fig. 1. he circuit cosists of a 6 Hz source voltage ( v( t)) of 11 volts rms ad egligible imedace, ad a o-liear load which has bee show with a curret source (i) ad a imedace with ( L =.15 H) ad ( R = Ω ) (Norto equivalet circuit). I aalysis, the curret source has oly third harmoic with amlitude of 3 A with zero hase agle. Fig. shows the istataeous o-active ower that ca be searated i eight uequal regios i oe eriod. As seem, the waveform has four ositive ortios above the x-axis. he to determie the eergy trasfer (by Eq. (6)), the times that Qt () crosses zero oit should be secified. I other words, sice the waveform of Qt () varies with harmoics amlitude ad is ukow, the limits of itegral of Eq. (6) is ot redictable. 5. HE PROPOSED MEHOD I this sectio, we modified the Khalsa- Zhag's method by resetig a ew defiitio for o-active ower. his defiitio is based o the istataeous ower equatios istead of eergy trasfer. Furthermore, the ew defiitio correctly gives a sigle umerical equivalet value of reactive ower ad ca be easily alied to comesatio roblem ACIVE POWER We first searate active ad o-active owers from each other whe tye ad value of load are ukow. I this case, we roose a curret based decomositio i which curret is divided ito two comoets amely active curret ( I ( t)) ad o-active curret ( IQ ( t )). Hece:

5 A method to determie 379 S() t = V () t I() t = V ()[ t I () t + I ()] t (1) he istataeous active ower ( Pt ( )) is give by: Pt () V() t. I () t Q = (13) I () t has the same waveform ad hase as source voltage ad ca be calculated as follows [8]: rms V t rms V rms V () t V () I () t = I () t =. P (14) R V V rms 1 P = V () t I () t dt R = (15) where R is load resistace ad V rms deotes rms value of voltage. he by searatig I () t from I() t ad substitutig it i Eq. (13) oe ca calculate the active ower without kowig about the value of load resistace. O the other had, ay o-siusoidal istataeous active ower ca be equivalet by a siusoidal active ower with amlitude of P where P is obtaied by the followig equatio: 5.. NON ACIVE POWER 1 P P t dt = () (16) Khalsa- Zhag's decomositio iherits a roblem which is the lack of direct determiatio of comesatio caacitor ower. o achieve this goal, i additio to calculate istataeous o-active ower we should searate the reactive ower from istataeous o-active ower. Hece, we first itroduce ective aaret ower as follows: 1 S S t dt = () (17) Istataeous o-active ower ca be derived by subtractig istataeous active ower from istataeous aaret ower: ad ective o-active ower is give by: Qt () = St ()- Pt () (18) 1 Q Q t dt = () (19) With this defiitio, we do't have to recogize the time iterval whe Q () t > or the times that Qt () crosses zero oit. We ca omiate S ad Q as umerical measures of ISSN:

6 38 A method to determie aaret ad o-active ower. Istataeous o-active ower is made u two comoets: istataeous reactive ower that comesate by addig caacitor ad istataeous distortio ower which should be costat durig the variatio of caacitace [8]. he oe ca divide the o-active curret ito: 1. A reactive curret comoet ( IQcom ( t)) which has the same waveform ad hase as itegral of source voltage.. A distortio curret ( Id ( t)) which remais of the total curret after the active ad reactive curret comoets have bee extracted. Cosequetly, if V deotes the rms value of itegral of V () t, oe ca obtai comesated curret from itegral of active curret Eq. (14 ) ad Eq.(15) as follows: V () t dt 1 I Qcom () t = ( V () t dt ).() i t dt V () ad the reactive ower the is give by: Q () t = V (). t I () t (1) com Qcom ad ective of this ower is: 1 Qcom = Q comdt () For ower factor imrovemet we ca use Q com to calculate comesator caacitors ower. Istataeous distortio ower ad the ective of this ower will also be obtaied as follows: Q () t = Qt () Q () t (3) d com Q d 1 = Q d dt (4) his arameter should be costat whe we use caacitor for reactive ower comesatio. 6. COMPUAION EXAMPLE Behavior of the roosed ower decomositios o reactive ower comesatio rocess is aalyzed by usig the circuit show i Fig. 1. At first we assumed a siusoidal voltage source with f = 6 HZ ad Vrms = 11 V, a liear load with R = Ω, L =.15H ad i =. he values for S, P, Q, Q com, Qd of the system calculated with the roosed method are give i able 1.

7 A method to determie 381 R V L Fig. 1. he circuit used for the aalysis. Istataeous No Active Power VA S.1 ime.15 Fig.. Istataeous o-active ower of system of Fig. 1. able 1. Power decomositios with ad without o-liear load. S (VA) P (Watt) Q (VA) Q com (Var) Q d (VA) Without o-lier load With o-liear load As exected, ective distortio ower is zero ad ective reactive ower is equal to ective o-active ower. I secod case, we assumed a o-lier load i that curret source ( i ) has 3 rd ad 5 th harmoics with amlitudes of 1A ad.5a ad zero hase agles. Other arameters are as the same as the first case. he results have bee also show i able 1. As it ca be see, i site of fixed P, the o-active ower has chaged ad is ot equal to reactive ower. Fig. 3. Istataeous o-active. Fig. 4. Istataeous reactive ower. Fig. 5. Istataeous distortio ower. ISSN:

8 38 A method to determie Figs. 3 5 show istataeous o-active ower, istataeous reactive ower ad istataeous distortio ower of the system, resectively. he behavior of roosed the ower the comositios o reactive ower comesatio is aalyzed with addig a caacitor to the system Fig 6. We assumed two differet values for caacitor ower. hese values are Qca = Qcom = kvar (equals to ective reactive ower) ad Qca = Q = 45 kvar (equals to ective o-active ower). he results have bee show i able. V c R L i Fig. 6. he circuit used for the aalysis. able. Power decomositios i two deferet values for caacitor ower. I c S P Q Q com Q d (A) (VA) (Watt) (VA) (Var) (VA) Q com =134.5 (kvar) Q com =45 (kvar) Note that for the case Q ca = kvar, the caacitor curret ca be obtaied from Eq. (). O the other had, i the case of Q ca = 34.5 kvar, the caacitive curret should be obtaied from the follwig equatio : ( ( )- ( )) 1 Ic = I t I t (5) he results comariso of the two cases show that with our roosed method the reactive ower comletely comesated whereas we caot achieve to this goal by ective o-active ower ad sice i secod case the caacitace ower has bee chose higher tha that we eed, source curret leads source voltage ad therefore the system goes to caacitive mode. Fig. 7 shows the istataeous fudametal curret ad voltage of the source without comesator caacitor. As it ca be see, the curret lags source voltage. Figs. 8 ad 9 show the istataeous fudametal currets ad voltages with comesator caacitor icluded i two cases Q c =134.5 kvar, Q c = 45 kvar. Fig. 7. Istataeous fudametal curret without comesator caacitor.

9 A method to determie 383 Fig. 8. Istataeous fudametal curret with comesator caacitor i first case. Fig. 9. Istataeous fudametal curret with comesator caacitor i secod case. Fig. 8 reveals that by correct comesatig the istataeous fudametal curret ad voltage cross the zero oit simultaeously, but as show i Fig. 9 icorrect comesatio causes the curret leads the voltage. O the other had, table I shows that the distortio ower has ot chaged i both cases after addig the caacitor to the system. 6. CONCLUSION I this aer we develo the KHalsa-Zhag method to roose a ew defiitio for reactive ower i o-siusoidal coditios. he ew defiitio ca be directly alied to calculate the caacitace ower for reactive ower comesatio ad ower factor imrovemet. A simle hase circuit was used ad the results showed the ew defiitio correctly rovides a sigle value for caacitor ower. ISSN:

10 384 A method to determie REFERENCES [1] Budeau, C.I., Reactive ad Fictitious Powers, Publicatio No. of the Rumaia Natioal Ist. Bucuresti, 197. [] Fryze, S., Wirk-, Blid-, ud Scheileistug i Elektrische Stromkreise Mit Nichts iusoidalformigem Verfauf vo Strom ud Saug, Elektrotechische Zeitschriji 53(5) 596, 193. [3] Kimbark, E.W., Direct Curret rasmissio, J. Wiley ad Sos, [4] Sheherd, W., Zakikhai, P., J. Phys. D: Al. Phys., 6, 185, [5] Sheherd, W., Zad, P., Eergy Flow ad Power Factor i No-siusoidal Circuits, Cambridge Uiversity Press, [6] Sharo, D., Reactive ower defiitio ad ower-factor imrovemet i oliear systems, Proc. Ist. Electr. Eg., 1, 74, [7] Deebrock, M., Wirk-udBlidleistug, EG-Fachtagug Blidleistug, Aache, [8] Kusters, N.L., Moore, W.J.M., IEEE ras. Power Aaratus Syst., 99(5), 1845, 198. [9] Czarecki, L.S., IEEE ras. Istrum. Meas. 39(), 34, 199. [1] Czarecki, L.S., IEEE ras.powerdeliv., 8(1), 437, [11] Khalsa, H., Zhag, J., It. J.of Emergig Electric Power Systems, 7(4), 1, 6. [1] Erha Balci, M., Haka H.M., Electric Power Systems Research, 78, 318, 8.