Americn Journl of Applied Sciences 6 (8): 1539-1547, 2009 ISSN 1546-9239 2009 Science Publictions Exponentil-Hyperbolic Model for Actul Operting Conditions of Three Phse Arc Furnces 1 Mhdi Bnejd, 2 Rhmt-Allh Hooshmnd nd 2 Mhdi Torbin Esfhni 1 Fculty of Electricl nd Robotic Engineering, Shhrood University of Technology, Shhrood, Irn 2 Deprtment of Electricl Engineering, University of Isfhn, Isfhn, Irn Abstrct: Problem sttement: The Electric Arc Furnce (EAF) is non-liner lod nd cretes power qulity relted problem. Therefore, ccurte modeling of the EAF is essentil. Approch: In this study, n optiml model for EAF in time domin clled exponentil-hyperbolic, ws proposed to describe the behvior of the EAF for ll of the operting conditions nd it does not need the initil conditions s they needed for the existing methods of modeling of the EAF. Then, the behvior of the proposed model of EAF on the power system ws studied using the PSCAD softwre. In order to nlyze the proposed method, severl chrcteristics for different operting conditions were investigted. Results: In the simultion, the prmeters were tken from the EAF of the Mobrkeh Steel Mking Compny (Isfhn-Irn). The results of the simultion ccurtely showed the behvior of the EAF of the compny. Conclusion: The finding of this study showed tht the proposed exponentil-hyperbolic model ws cpble in modeling of EAF for different operting conditions. Key words: Electric rc furnce, furnce modeling, unblnced voltge, unblnced current, flicker INTRODUCTION The EAF is inherently nonliner nd time-vrint lods nd it cn cuse power qulity problems such s voltge flicker, odd nd even hrmonics s well s unblnced currents nd voltges. These problems should be solved in the EAF. In this regrd, n optiml model is necessry to consider the mentioned problems. Also, it should be mentioned tht the time response of the EAF hs gret effect on power qulity. The dynmic specifictions of the EAF t ny instnt of time re ffected by conditions of the furnce t tht time nd previous instnts of the time. The reson for tht is when the rc is creted, the sudden chnge in the electrons, ions nd gs temperture (tht my occur due to sudden chnge of current) is impossible. Therefore, the sudden chnge of the current will not led to sudden chnge of the rc chrcteristic. Thus, this phenomenon hppens grdully. In fct, there is hysteresis phenomenon in the dynmic of the rc chrcteristic due to the effects of the current in the previous instnts of time on the present time. In brief, the time response of n electric furnce depends on the length of rc, positions of electrodes nd topology the externl circuit. The importnt issue in the modeling of the rc is the simultion of rc [1-14]. There hve been severl methods to describe the electric rc. The blnced stedy stte equtions re used in [1,2]. The time domin methods bsed on the differentil equtions re presented by in [3-5]. The methods used in [6,7,12-14] re bsed on the linerlized methods nd pproximtion. Other methods such s methods bsed on frequency response [7], -I chrcteristic [9,13], nonliner differentil equtions [9,10], re used to nlyze electric rc of the furnces. In the following, the dvntges nd disdvntges of the mentioned methods re explined briefly. The equtions of stedy stte re very helpful in computtionl work. However, it dels only with the blnced three phse currents. It lso uses n pproximted step model for the wveform of oltge- Current Chrcteristic (IC) of the EAF. In the time domin nlysis, the prmeters re determined using the hrmonic source voltges nd the unblnced three phse currents. This method lso uses n pproximted step model for IC [4]. Other method in nlyzing the rc model in time domin is formed bsed on the Cssi- Myer eqution [5,9]. In this method, Cssi nd Myer equtions re used for the low nd high current of the Corresponding Author: Mhdi Bnejd, Fculty of Electricl nd Robotic Engineering, Shhrood University of Technology, Shhrood, Irn 1539
rc, respectively. In the method of lineriztion nd pproximtion of the rc voltge is determined using the current of in the -I chrcteristics of the rc [2,13]. When the EAF model in the time domin is compred with the model in the frequency domin [8], it cn be concluded tht modeling in the time domin is more efficient in studying the rc furnce elements nd the EAF performnce. However, the frequency domin models re more suitble for hrmonic nlysis of the externl network. The externl network is the model of liner system t ech hrmonic. In the bove explined methods, there re some limittions such s need to initil conditions for solving the differentil equtions, blnced operting condition of the three phses of the EAF nd use of sophisticted mthemticl equtions for the estimtion of the rc model. Considering the mentioned limittions, this study presents new model for the EAF in the time domin. The min spect of the new model is modeling of the proposed method with good pproximtion without need to the initil conditions of the EAF performnce. Also, the proposed method cn be used to describe the different operting conditions of the EAF nd power system. Finlly, the proposed method presents suitble model with very good pproximtion for the IC. In order to increse the ccurcy of the lod model, rndom noise is employed to estblish new model of the furnce lod. Then, unblnced sitution of the currents nd voltges, the effects of the furnce lod nd voltge flicker re studied in the new model. It should be noted tht in this study directly considers the effect of voltge flickering on the voltge rc furnce, in the frequency rnge of the humn vision (between 4-14 Hz). In the simultion of this study, the dt belongs to rel EAF tken from Mobrkeh Steel Mking Compny (Isfhn-Irn). MATERIALS AND METHODS Fig. 1: Digrm of n EAF connected to rest of power system Fig. 2: Circuit digrm of n EAF connected to rest of power system Fig. 3: The Actul -I model of electric rc furnces In Fig. 2, X P is the series rectnce for compenstion of voltge flicker. The elements of X C nd R C re the rectnce nd resistnce of the connected line between furnce electrodes nd T S. This impednce is the dominnt prt of the totl impednce seen by the rc furnce. The performnce of electric digrm for the EAF: Figure 1 shows the electric digrm of source which supplies n EAF. In order to chnge the ctive input power of the rc furnce, furnce trnsformer, T F, is used. This trnsformer is connected to point of connection, PC. This trnsformer is equipped with tp chnger locted t the secondry winding. This tp chnger vries the voltge of the furnce. The rc furnce is lso connected to the point of common coupling, PCC, through the substtion trnsformer, T S. The simplified digrm of Fig. 1 is shown in Fig. 2. 1540 Modeling of the EAF's lod: In this prt, the modeling of the EAF is performed using the estimtion of the voltge nd current of the electric rc. This modeling is lso bsed on Fig. 3. In the other word the modeling is bsed on the system identifiction. As cn be seen from Fig. 3, the electric rc consists of four mjor prts s: di dt rc Are1 > 0, v rc,irc > 0 (1)
di dt rc Are2 < 0, v rc,irc > 0 di dt rc Are3 < 0, v rc,irc < 0 di dt rc Are4 > 0, v rc,irc < 0 (2) (3) (4) According to bove equtions, Eq. 1 is similr to Eq. 4 nd 2 is lso similr to Eq. 3. However, the sign of voltge nd the current in the similr ones re opposite. Thus, the rc voltge cn be expressed s function of the rc current in their region. In this cse we cn describe the rc voltge with two different functions. In these descriptions, the rc voltge is dependent on the rc current. For this purpose in the following, the equtions of EAF re nlyzed using three different models bsed on the IC of the electric rc. Then, the best model mong the three models which hs description the performnce of the furnce is selected. Model 1: Hyperbolic model: In this model the IC of the EAF is considered to be in the form of = (I ) nd it cn be described s: C (I ) = t + D + I (5) where, I nd re rc current nd voltge of phse. Also, t is the threshold mgnitude to which voltge pproches s current increses. This voltge depends on the rc length. The constnts C nd D re corresponding to the rc power nd rc current, respectively. These constnts cn tke different vlues which depend on the sign of the derivtive of the rc current. Since Eq. 5 hs behvior similr to hyperbolic function, it is clled hyperbolic model. There re two pths to increse or decrese of current. The first pth is relted to the incresing stte of the current nd the second pth is ssocited with decresing stte of the current. In this regrd, the constnts C nd D re clssified into two groups. The constnts for the first group (or first pth) re C nd D. Also constnts of the second group (or second pth) re C b nd D b. (i) = < i/ τ t (1 e ), i 0 -i/ τ t (1 e ), i 0 (6) In the Eq. 6, the current time constnt (τ) is used to describe the positive nd negtive currents. The exponentil function is lso used to model the IC of the rc. This model cn be used in optimiztion nd stbility issues of the EAF. Model 3: Proposed model: Exponentil-hyperbolic model: In this model the IC of the EAF described using the following equtions: C di t + 0, i > 0 D + i dt (i) = i/ τ di t (1 e ) < 0, i > 0 dt (7) In Eq. 7, τ is the current time constnt in ka. As cn be seen s in from this eqution, for the positive current nd regrding the hysteris property of the rc, there re two cses. To increse nd decrese the current of the EAF, the hyperbolic eqution nd exponentil-hyperbolic form of the eqution re used, respectively. The proposed method hs the cpbility of describing the EAF behvior in time domin using differentil eqution [5]. In ddition, it is ble to nlyze the behviors in the frequency domin without solving the sophisticted differentil equtions. Moreover, the proposed method cn describe different operting conditions of the EAF such s initil melting (scrp stge), mild melting (pltting stge) nd refinement of the EAF. The results gree with ctul conditions of the EAF in the steel industries. The Unblnced sitution in the proposed method: In order to investigte the effect of unblnced situtions on the proposed exponentil-hyperbolic model of lod in the EAF, different vlues for the voltge t in Eq. 7 re considered for different phses. The effect of the different vlues is more observble t the primry side of PCC. To consider the effect of even hrmonics which re produced in the erly stge of the chrging the furnce, the different vlues of t re considered for positive nd negtive prt of ech phse current. Model 2: Complete exponentil model: In this model the IC of the EAF is pproximted by n exponentil function s follows: 1541 Investigtion the effect of rndom voltge flicker in the proposed method: In order to study the effect of voltge flicker on the systems with the EAF, it is
sufficient to consider t in the time-vrint form of. This section considers the rndom vritions for the voltge t. In this regrd, the voltge t is modulted with rndom signl t different phses. This signl hs the men of zero with the frequency bnd in the rnge of 4-14 Hz. Thus, in this cse the voltge t for different phses cn be written s Am. J. Applied Sci., 6 (8): 1539-1547, 2009 = + k N (t) t1 t 01 1 1 = + k N (t) t 2 t 02 2 2 = + k N (t) t 3 t 03 3 3 (8) where, N j (t), (j = 1,2,3) is bnd limited white noise with zero men nd vrince of 1. Also, the modultion index nd vrince of the rndom signl re shown by k j (j = 1,2,3) nd k j N j (t), respectively. It should be noted tht the flicker intensity cn be chnged by k j. Fig. 4: The IC of the rc the in the hyperbolic model RESULTS Comprison the different models of the furnce lod: In order to compre the proposed exponentilhyperbolic model with the other models, firstly the results of simultions hyperbolic lod model re presented. In Model 1, the prmeters of EAF chrcteristics re chosen s: C C t b = 250 = 190000 W = 39000 W D = D = 5000 A b (9) The vlues in Eq. 9 re chosen with respect to rel EAF tken from Mobrkeh Steel Mking Compny (Isfhn-Irn). Using these prmeters, the IC of the EAF is derived nd shown of Fig. 4. The wveforms of the current nd voltge of the rc re illustrted in Fig. 5. The second simultion is ssocited with exponentil model. This model is formed bsed on Eq. 6. The constnt prmeters in this model re chosen s: Fig. 5: Wveforms of the rc voltge nd current in the hyperbolic model Fig. 6: The IC of the rc the in the exponentil model t = 250 τ = 10kA (10) With the bove vlues for the prmeters, the IC of the rc is obtined s shown in Fig. 6. Also, the wveforms of the current nd voltge of the rc re given in Fig. 7. 1542 Results for the proposed exponentil-hyperbolic method: In this prt, the results for the proposed exponentil-hyperbolic method re studied. In the proposed exponentil-hyperbolic model, the prmeters re combintion of Eq. 9 nd 10. The derived IC of the rc is depicted in Fig. 8. At the points of this chrcteristic where the length of the rc is not vried with time, this chrcteristic is ssumed to be constnt (t
Fig. 7: Wveforms of the rc voltge nd current in the exponentil model Fig. 9: Wveforms of rc voltge nd current in the proposed exponentil-hyperbolic model Fig. 8: The IC of the rc the proposed exponentilhyperbolic model refining process). In this sitution, the EAF does not crete ny flicker t PCC. At this stge, the EAF produces only odd hrmonics in voltge nd current, becuse the IC hs symmetric behvior. This sitution describes the ctul performnce of the EAF t the plting period. In this period, the level of melting mteril is nerly constnt nd the melting is distributed uniformly in the furnce. Figure 9 shows the wveform of voltge nd current for the proposed exponentil-hyperbolic model of the EAF. Also, the rc conductnce nd three phse current of the primry side of PCC re shown in Fig. 10 nd 11, respectively. The results indicte tht if the furnce lod does not produce ny flicker, then the rc voltge, current nd voltge nd current of the primry side of PCC oscillte similrly. These wveforms present the ctul performnce of the electric system supplying the rc system. When the EAF is in the melting process (or scrp stge), the IC of the rc is in the form of Fig. 12. Finlly, for the refining stge of the rc mteril (t the melt down stge), the IC of the furnce is given in Fig. 13. 1543 Fig. 10: Wveform of the rc conductnce in the sitution of the proposed exponentilhyperbolic model Fig. 11: Wveform of the three phse current t the primry side of the PCC bus in proposed exponentil-hyperbolic model Results for unblnced three phse furnce lod: performnce in the unblnce lod sitution which mostly hppens in the initil working condition of the EAF (t the scrp melting stge).
Fig. 12: The IC of the rc in the scrp stge (or melting process in the proposed exponentilhyperbolic model Fig. 14: Wveform of the three phse current t the primry side of the PCC bus in the unblnced sitution for different voltges for three phse in the proposed exponentil-hyperbolic model Fig. 13: The IC of the rc in the refining stge (or melt down process) stge in the proposed exponentil-hyperbolic model In order to study the unblnced sitution in the proposed exponentil-hyperbolic model, the voltge t in different phses re considered s: t tb tc = 250 = 350 = 450 (11) The wveforms of the three phse currents t the primry side re obtined using the vlues given in Eq. 11. The obtined currents re shown in Fig. 14. Also, in order to show the even hrmonics nd symmetry of the rc, the voltges t for positive nd negtive current re chosen s: t1 t2 = 250 = 120 (12) where, t1 nd t2 re ssocited with positive current nd negtive current, respectively. With these vlues, the wveform of the rc voltge nd current re determined nd they re shown in Fig. 15. 1544 Fig. 15: Wveforms of the rc voltge nd current in the cse of even hrmonics genertion with different voltges t in proposed exponentilhyperbolic model Results for nlysis of voltge flicker: In this prt, the simultion of the voltge flicker using the rndom voltge is crried out bsed on Eq. 8. The vlues for the prmeters used in Eq. 8 re: = = = 250 t 01 t 02 t 03 k = k = k = 1 1 2 3 (13) Also in Eq. 8, N 1 (t), N 2 (t) nd N 3 (t) re three white noise voltges with zero men nd vrince equls 1 in the limited bnd (between 4-14Hz). The results of the simultion re plotted in Fig. 16-20. Figure 16 shows the time vritions of the rndom flicker. The IC of the rc in the sitution of rndom flicker is given in Fig. 17. The rc voltge nd current depicted in Fig. 18. Figure 19 shows the wveform of the rc conductnce in this sitution. Also, Fig. 20 illustrtes the wveforms of the three phse current, t the primry side of the PCC bus for the sitution.
Fig. 16: Wveform of vritions of the rndom flicker Fig. 18: Wveform of the rc voltge nd current in the sitution of rndom flicker Fig. 17: The IC of the rc in sitution of rndom flicker DISCUSSION Fig. 19: Wveform of the rc conductnce in the sitution of rndom flicker Discussion for the unblnced three phse furnce lod: The simultion results presented in Fig. 14 nd 15 shows tht when three phses for the furnce lod become unblnced, the three phse currents t the primry side of PCC lso becomes unblnced, remrkbly. However, s shown in Fig. 15, the three phse voltges t the primry side of PCC re not nerly chnged s shown in Fig. 15. Also, different vlues for thee phse voltges of the furnce lod re chosen in the situtions tht current increses or decreses, the symmetry sitution occurs of incresing nd decresing current leds to symmetry in the voltge nd current of the furnce. Consequently, this results in producing even hrmonics s cn be seen in Fig. 15. Discussion for voltge flicker: Similr to the previous prt nd bsed on Fig. 16-20, when the rndom flicker is pplied, the lod specifictions of EAF re vried rndomly. Thus, voltge, current, 1545 Fig. 20: Wveform of the three phse current t the primry side of the PCC bus in the sitution of rndom flicker conductnce nd the three phse current of the primry side of the PCC bus, re vried rndomly. However, the vritions of the voltge t the primry side of the PCC
bus re not nerly vried. In the other word, in this cse, furnce lod flicker leds to little vrition in the voltge of the bus supplying the EAF. Discussion for hrmonic nlysis for different models: In this prt, the results of hrmonic nlysis for the different models of the EAF re discussed. Moreover, the effects of furnce lod on voltge hrmonics re investigted using the proposed exponentil-hyperbolic model. So, Tble 1 illustrtes the generted voltge hrmonics t the mild melting stge for the different models; hyperbolic, exponentil nd proposed exponentil-hyperbolic model. Becuse of the presence of the odd symmetry in the models, no even hrmonics is present in the rc voltge. Also, Tble 2 shows the voltge hrmonics of the EAF for different operting condition. As cn be shown in Tble 2, becuse of lck of odd symmetry in the electric rc, the even hrmonics re present in the rc voltge. Thus, these even hrmonics re injected to the power system. In ddition, when there is voltge flicker, the hrmonics of the electric rc voltge re present. As cn be shown in Tble 2, the mgnitude of hrmonics is influenced the type nd intensity of the flicker. Tble 1: Comprison of the hrmonic mgnitude between different models of furnce lod in the plte stge (in volt) Model -------------------------------------------------------------------- Bulk d proposed model: Hrmonic Hyperbolic Exponentil Exponentil-hyperbolic H 1 251.62 258.32 255.33 H 3 85.18 91.34 92.22 H 5 50.71 58.68 61.57 H 7 40.55 46.91 52.33 H 9 29.42 35.12 48.19 H 11 26.33 31.32 36.52 H 13 19.12 29.34 30.13 Tble 2: Comprison of hrmonic mgnitude between different operting conditions with proposed model (in volt) Stte ---------------------------------------------------------------- Hrmonic Blnced Unblnced Rndom flicker H 1 251.62 255.32 261.32 H 2-16.68 14.36 H 3 92.22 85.34 88.74 H 4-15.91 8.65 H 5 61.57 58.36 65.26 H 6-15.42 8.22 H 7 52.33 40.12 49.36 H 8-14.96 8.46 H 9 48.19 35.16 51.33 H 10-14.55 7.98 H 11 36.52 29.11 40.34 H 13 30.13 18.51 34.61 Am. J. Applied Sci., 6 (8): 1539-1547, 2009 1546 CONCLUSION This study, firstly investigtes the existing hyperbolic nd exponentil models for EAF. After tht, the exponentil-hyperbolic model is proposed. The proposed model hs no limittion of the existing models nd does not require ny initil conditions or specil needs for modeling of the rc. Most of the specifictions of the EAF cn be described by the proposed method. This study lso proposes three phse structure of the electric EAF which covers power qulity spects such s: voltge flicker nd voltge nd current unblnced sitution. Since this model considers the rectnce nd resistnce of the flexible cbles, the proposed modeling hs high ccurcy. The proposed method is simulted on prmeter corresponding to rel EAF nd the results of the simultion show the outstnding spects of the proposed method in well describing the behvior of the EAF when compred to the existing methods. REFERENCES 1. Myordomo, J.G., L.F. Beites, R. Asensi, M. Izzeddine, L. Zbl nd J. Amntegui, 1997. A new frequency domin rc furnce model for itertive hrmonic nlysis. IEEE. Trns. Power Delivery, 12: 1771-1778. http://ieeexplore.ieee.org/xpl/freebs_ll.jsp?rnum ber=634204 2. Beites, L.F., J.G. Myordomo, A. Hernndes nd R. Asensi, 2001. Hrmonics, interhrmonic, unblnces of rc furnces: A new frequency domin pproch. IEEE. Trns. Power Delivery, 16: 661-668. http://ieeexplore.ieee.org/xpl/freebs_ll.jsp?rnum ber=956754 3. Medin, A., M.A. Gomez-Mrtinez nd C.R. Fuerte-Esquivel, 2005. Appliction of bifurctions theory to ssess nonliner oscilltions produced by AC electric rc furnces. IEEE. Trns. Power Delivery, 20: 801-806. http://ieeexplore.ieee.org/xpl/freebs_ll.jsp?rnum ber=1413319 4. Collntes-Bellido, R. nd T. Gomez, 1997. Identifiction nd modelling of three phse rc furnce forvoltge disturbnce simultion. IEEE. Trns. Power Delivery, 12: 1812-1817. http://ieeexplore.ieee.org/xpl/freebs_ll.jsp?rnum ber=634210 5. Mokhtri, H. nd M. Hejri, 2002. A new three phse time-domin model for electric rc furnces using MATLAB. Proceeding of the Asi Pcific Trnsmission nd Distribution Conference nd Exhibition, Oct. 6-10, IEEE Xplore Press, USA., pp: 2078-2083. DOI: 10.1109/TDC.2002.1177781
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