University of Maribor Faculty of Energy Technology Volume 8 / Issue 2 OCTOBER 2015 www.fe.um.si/en/jet.html Journal of ENERGY TECHNOLOGY ✓—____л Ш VOLUME 8 / Issue 2 Revija Journal of Energy Technology (JET) je indeksirana v bazi INSPEC©. The Journal of Energy Technology (JET) is indexed and abstracted in database INSPEC©. /_____ ш JOURNAL OF ENERGY TECHNOLOGY Ustanovitelj / FOUNDER Fakulteta za energetiko, UNIVERZA V MARIBORU / FACULTY OF ENERGY TECHNOLOGY, UNIVERSITY OF MARIBOR Izdajatelj / PUBLISHER Fakulteta za energetiko, UNIVERZA V MARIBORU / FACULTY OF ENERGY TECHNOLOGY, UNIVERSITY OF MARIBOR Glavni in odgovorni urednik / EDITOR-IN-CHIEF Jurij AVSEC Souredniki / CO-EDITORS Bruno CVIKL Miralem HADŽISELIMOVIĆ Gorazd HREN Zdravko PRAUNSEIS Sebastijan SEME Bojan ŠTUMBERGER Janez USENIK Peter VIRTIČ Ivan ŽAGAR Uredniški odbor / EDITORIAL BOARD Zasl. prof. dr. Dali ĐONLAGIĆ, Univerza v Mariboru, Slovenija, predsednik / University of Maribor, Slovenia, President Prof. ddr. Denis ĐONLAGIĆ, Univerza v Mariboru, Slovenija / University of Maribor, Slovenia Doc. dr. Željko HEDERIĆ, Sveučilište Josipa Jurja Strossmayera u Osijeku, Hrvatska / Josip Juraj Strossmayer University Osijek, Croatia Prof. dr. Ivan Aleksander KODELI, Institut Jožef Stefan, Slovenija / Jožef Stefan Institute, Slovenia Prof. dr. Milan MARČIČ, Univerza v Mariboru, Slovenija / University of Maribor, Slovenia Prof. dr. Greg NATERER, University of Ontario, Kanada / University of Ontario, Canada Prof. dr. Enrico NOBILE, Università degli Studi di Trieste, Italia / University of Trieste, Italy Prof. dr. Brane ŠIROK, Univerza v Ljubljani, Slovenija / University of Ljubljana, Slovenia Znan. sod. dr. Luka SNOJ, Institut Jožef Stefan, Slovenija / Jožef Stefan Institute, Slovenia Prof. dr. Mykhailo ZAGIRNYAK, Kremenchuk Mykhailo Ostrohradskyi National University, Ukrajina / Kremenchuk Mykhailo Ostrohradskyi National University, Ukraine, Tehnični urednik / TECHNICAL EDITOR Sonja Novak Tehnična podpora / TECHNICAL SUPPORT Tamara BREČKO BOGOVČIČ Izhajanje revije / PUBLISHING Revija izhaja štirikrat letno v nakladi 150 izvodov. Članki so dostopni na spletni strani revije -www.fe.um.si/si/jet.html / The journal is published four times a year. Articles are available at the journal's home page - www.fe.um.si/en/jet.htiril. Cena posameznega izvoda revije (brez DDV) / Price per issue (VAT not included in price): 50,00 EUR Informacije o naročninah / Subscription information: http://www.fe.um.si/en/jet/ subscriptions.html Lektoriranje / LANGUAGE EDITING Terry T. JACKSON Oblikovanje in tisk / DESIGN AND PRINT Fotografika, Boštjan Colarič s.p. Naslovna fotografija / COVER PHOTOGRAPH Jurij AVSEC Oblikovanje znaka revije / JOURNAL AND LOGO DESIGN Andrej PREDIN Ustanovni urednik / FOUNDING EDITOR Andrej PREDIN Izdajanje revije JET finančno podpira Javna agencija za raziskovalno dejavnost Republike Slovenije iz sredstev državnega proračuna iz naslova razpisa za sofinanciranje domačih znanstvenih periodičnih publikacij / The Journal of Energy Technology is co-financed by the Slovenian Research Agency. Spoštovani bralci revije Journal of energy technology (JET) Učinkovita raba energije in razvoj energetskih tehnologij so gonilo človeškega razvoja. Danes predstavlja energetika, poleg proizvodnje hrane in zdravja ljudi, enega najpomembnejših poslovnih stebrov. Žal se ideja znanstvenika Nikole Tesle o energiji, ki bi bila zastonj, do sedaj še ni uresničila. Gospodarski razvoj v svetu tudi v tem desetletju sloni na pretežni izrabi fosilnih virov, kar pa žal posledično povzroča ekološke probleme. Upajmo, da se bodo v Parizu predstavniki držav dogovorili o omejitvah izpusta toplogrednih plinov. Prihodnost energetike je prav gotovo v izdatnejši izrabi obnovljivih virov v povezavi z alternativno energetiko. O tem pišejo tudi nekateri avtorji člankov v tem in prejšnjih izvodih revije JET. Jurij AVSEC odgovorni urednik revije JET Dear Readers of the Journal of Energy Technology (JET) Efficient use of energy and development of new energy technologies is the driving force of human development. Today, energy technology is one of the most important pillars of business, in addition to food production and human health. Unfortunately, the idea of the scientist Nikola Tesla that energy would be free has not materialized thus far. The economic development in this decade in the world is still based on the dominant exploitation of fossil resources, which unfortunately has led to some ecological problems. Currently, Paris will be occupied by representatives of almost all nations; hopefully restrictions on the greenhouse gas emissions will be agreed on. The future is undoubtedly in the increase of exploitation of renewable energy sources. Several articles in this and previous issues of JET discuss this. Jurij AVSEC Editor-in-chief of JET Table of Contents / Kazalo Determination of the conditions for the existence of higher-order differential electromagnetic invariants / Določitev pogojev za obstoj diferencialnih elektromagnetnih invariant višjega reda Boris Nevzlin, Valentina Zagirnyak, Veronika Zahorulko......................11 The inverted distorted parabola-like shape of the bias-dependent electric field at an electron-injecting metal/organic interface deduced using the current-voltage method / Obrnjena, deformirani paraboli podobna odvisnost električnega polja od pritisnjene napetosti na vmesni ploskvi kovina/organski polprevodnik izpeljana z uporabo metode tokovne karakteristike Matjaž Koželj, Bruno Cvikl......................................17 Numerical and experimental investigations of transient cavitating pipe flow / Numerične in eksperimentalne raziskave prehodnega kavitacijskega toka v cevi Anton Bergant, Uroš Karadžić.....................................31 Integer programming and Gröbner bases / Celoštevilsko programiranje in Gröbnerjeve baze Brigita Ferčec, Matej Mencinger...................................43 Electric cars in Slovenia / Električni avtomobili v Sloveniji Gregor Srpčič..............................................59 Instructions for authors........................................71 ш JOUrnal Of JE1 Volume 8 (2015) p.p. 11-16 Issue 2, October 2015 Typology of article 1.01 Technology www.fe.um.si/en/jet.html DETERMINATION OF THE CONDITIONS FOR THE EXISTENCE OF HIGHER-ORDER DIFFERENTIAL ELECTROMAGNETIC INVARIANTS DOLOČITEV POGOJEV ZA OBSTOJ DIFERENCIALNIH ELEKTROMAGNETNIH INVARIANT VIŠJEGA REDA Boris Nevzlin1, Valentina ZagirnyakR, Veronika Zahorulko2 Keywords: four-element dipole, electromagnetic invariants, differential transformations, conditions for the existence of higher-order invariants Abstract A four-element dipole representation by first-order electromagnetic invariants according to differential transformation and increments is well known. The paper deals with a most general description of the conditions of existence of an electromagnetic invariant for a four-element dipole with active-reactive components in a differential form and as increments of any order. It is shown analytically that invariants exist at mutual transformations of increments into differentials and differentials into increments. R Corresponding author: Valentina Zagirnyak, (Eng.), Tel.: +38 05366 36218, Fax: +38 05366 36000, Mailing address: Kremenchuk Mykhailo Ostrohradskyi National University, Manufacturing Engineering Department, Vul. Per-shotravneva, 20, 39600, Kremenchuk, Ukraine, E-mail address: mzagirn@kdu.edu.ua 1 Kremenchuk Mykhailo Ostrohradskyi National University, Electric machines and Apparatus Department, Vul. Per-shotravneva, 20, 39600, Kremenchuk, Ukraine 2 Kremenchuk Mykhailo Ostrohradskyi National University Electric machines and Apparatus Department, Vul. Per-shotravneva, 20, 39600, Kremenchuk, Ukraine Povzetek Predstavitev dipola, sestavljenega iz štirih elementov, z elektromagnetno invarianto prvega reda je z vidika diferencialne transformacije in inkrementov že dobro poznana. Članek obravnava splošen opis pogojev za obstoj elektromagnetnih invariant dipola, sestavljenega iz štirih elementov, z delovno in jalovo komponento v diferencialni obliki in kot inkrementi poljubnega reda. Analitično je dokazano, da pri medsebojni transformaciji inkrementov v diference in diferenc v inkremente invariante obstajajo. 1 INTRODUCTION A four-element dipole with active-reactive components (Fig. 1, a, b, c) is known [1, 2] to represent electromagnetic invariants according to differential transformation and increments that are of the form: дС_ dm dg_ dm dC dg С - С 2 M Cj + C2 m2 - m j gj + g/ g2 - g1 w2 -m j R2 - R Aa AR a -a2 Rj + R2 j dl dm dR dm AC dL_ ~dR Cj +C2 gj + g 2 dR da dm dR dm da ~dR aj +a2 Rj + R2 L+ l2 Rj + R2 L2 L R2 - Rj AL AR L +l2 Rj + R2 (1) where m -arbitrary circular frequency; m2, m j -circular frequencies and m2 >m j ; C2, a2, L2, g2, R2 -values C, a, L, g, R at frequency m2 ; C, aj, L, gj, Rj -values C, a, L, g, R at frequency m j . Conditions for invariants existence consist of, respectively: C g2 - C2gj Ф 0, aR2 -a2Rj ф 0, LR2 -L2Rj Ф 0. (2) The same papers [1, 2] state that (1) there exist invariants not only according to frequency m, but also according to order a of derivatives and increments: d "N d "g" N dg N +N 2 gj + g2 d" a d "R da dR aj +a Rj + R 2 2 d "L d"R SL_ dR L +L 2 R + R (3) 2 but conditions for their existence are not given. The purpose of this paper: determination of conditions for the existence of higher order invariants. 2 MATERIAL AND RESULTS OF RESEARCH Second derivatives with respect to С and g: Q2N _ -2 (Cj + C2 ) (Cjg2 - C2gj )2 [(gj + g2 )2 - 3m2 (Cj + C2 )2 ] dm2 [(gj +g2)2 +m2 (Cj +C2)2] m a CO о - CO m m - m д2g 2 (gj + g2 )(Cjg2 - C2g! ) \(gj + g2 ) - 3»2 (Cj + C2 ) ] — =-[-. (4) ^ [(gj +g2 )2 +»2 (Cj +C2 )2 J The second condition for the existence of a second-order differential invariant follows from (4): gj + g2 Cj + C2), (5) which is supplementary to (2). Analogously, for a third-order differential invariant, the second condition for existence (the first one, as before, is (2)) is of the form: gj + g 2 *ro(Cj + C 2 ) , (6) and for the fourth one: (gj + g2 )4 - 5Ш 2 (Cj + C2 )2 [2 (gj + g2 )2 - ®2 (Cj + C2 )2 ] * 0. (7) Condition (7), obviously, always exists at n/2 (gj + g 2 ) = ®(Cj + C2 ). (8) Cj C2 gj g2 a- -Ф C, a) R aj a 2 R 2 a, R б) Lj Rj ^2 J^YV R 2 -Ф L, R *- в) Figure 1: Circuits of fout-slsmsst dipolss: e) Ci, С2 esd С - cosdssssts cepecitiss esd tstmisel cepecitescs of ths circuit et etbittety ftsqusscy, gj, g2 esd x - ths sems coscstsisg cosductioity; b) aj, a 2 esd a - potsstiel cosfficissts of cosdssssts esd oostell potsstiel coefficient nf the circuit, Rb R2 and R - the same concerning resistances; c) Lb L2, and L -branches inductances and overall inductance of the circuit, Ri, R2, and R - the same concerning resistances Let us determine a derivative of the (n+i)-th order by a mathematical induction method, [3], in accordance with which a formula is considered true for any transformation, if it is proved that it, being true for n-th transformation, is also true for (n+1)-th transformation. As invariant of the n-th order is of the form: \2/,~r v> \2 (( + N2 )(( + N2 )2 (( g 2 - ^ )2 fn ( C-2, gl, g 2, ®) (gl +g2)2 +®2 (N1 +N2) 9nN d"N d(0n _ dng 9V (gl + g2)(Ni + N2)2 (Ng -N2gl)2 fn (Cl, C, gl, g2, () d(n (9) [(gl +g2 )2 +®2 (Nl +N2 )2 where fn (Cl, C2, gl, g2, ю) -the determined function for the derivative of the n-th order, e.g. for n = 3 fn = -24 ю [(gl + g2 )2 -ю2 (Cl + C2 )2 ] . Invariant of the (n+1)-th order is obtained: (( +N2 )(( +N2 )2 ((g -Ng )2 { (gl + g 2 )2 +ю2 (( +N2 )2 a"+lN ön+lN = ö(n+l rjn+l ДИ + l 9 g 9_g (gl + g2 )2 +ю2 (Nl +N2 )2 n+2 (gl + g2 )(( +N2 )2 (%2 -N2gl )2{ (gl + g2)2 +®2 (Nl +N2)2 X --» [(gl + g2)2 +ю2 (Ni +N2 )2 <{[(gl + g2)2 +®2 (Nl +N2)2]A[fn(Cl, C2, gl, g2, m)]- (gl + g2)2 +ю2 (Ni + N2)2]—[fn(Cl, C2, gl, g2, ю)]- -2ю (n + l)(Nl + N2 )2 fn (Cl, C2, gl, g2, ю )} Nl + N2 -2ю (n + l)(Nl + N2 )2 fn (Cl, C2, gl, g2, ю )} gl + g2 [(gl +g 2 )2 +ю2 (Nl +N2 )2 ]j~[fn (Cl, C2, gl, g 2, ю) (10) In this case, the condition for existence of (n+i)-th invariant, except (2), is of the form: \2 1_9_ " /J (ii) * 2(n + l)rn(Nl + N2)2 fn (Cl, C2, gl, g2, ю). Thus, it can be stated that as a relation of any order derivatives of С and g with respect to frequency represents an invariant with respect to frequency, it is also an invariant with respect to the order of differential transformation with the condition for existence (2) for all invariants and (5) or (6), or (7)-(8) - for invariants of the determined orders (in a general form (11)). > > Analogously to the circuits shown in Fig. 1, b, c, higher-order invariants are of the form (3) and, accordingly, conditions for their existence are of the form (except (2): ю(( +R2 )ф^3 (aj +a2 ), (12) Rj + R2 Фл/3ю(L + L2) - second-order invariant, i(Rj +R2 )^aj +a2, (13) Rj + R2 Ф -\/3ю(L + L2) - third-order invariant, V2ro(Rj + R2)*aj +a2, 42(Rj + R2)*i(L + L2) or (14) ю2 (Rj +R2 )4-5 (aj +a2 )2 \2 (Rj +R2 )2 ю2-(aj +a2 )2 ]ф 0, -order invariant. [и2 (Rj + R2 )2 + (a j + a2 )2 ] \_fn (( , a2, Rj, R2, ю )] : (Rj + R2 )4 - 5ю2 (L + L2 )2 [2 (Rj +R2 )2-ю2 (L +L2 )2 ]ф 0 - fourth-c In the general form, the condition for existence: d_ J ÖTO 1 Ф 2ю (n + j )(a j +a2)2 fn (a j, a2, Rj, R2, ю); \(j +R2 )2 +ю2 (L +L2 )2 Шл (L, L2, Rj, R2, ю)]Ф [ (15) Ф 2ю (n + j )(L + L2)2 fn (L, L2, Rj, R2, ю). Let us determine conditions for the existence of invariants according to increments. Second-order increment (otherwise -finite differences of the second order [4]) can be determined as: Ai2 - A A3 -A2 - ((2 -Aj ) (Aj +^2 )(Aj g2 - A2gj )2 (Ю3 -Ю j ) l2 A2A _ Ai2 -Ai j _ ю3 -ю2 - (ю2 -ю j) _ (ю3 + ю j -2ю2) A2g Agi2 -Agi j g3 -g2-(g2 -gj) (g + g2)(A g2 - A2g)2 (ю3 -юj ) > Аю2 -Аю j ю3 -ю2 -(ю2 -ю j) (ю3 +ю j -2ю2 )х \2 /,~т \2 Г —■ --> X{(gj + g 2 ) -(Aj + A2 ) \ю2 (ю3 +Ю !) + Ю3Ю j]} 3 хЩ« + g 2 )2 -ю2 (A +À2 )2 Aj + A2 (16) X{(gj + g 2 )2 -(Aj +À2 )2 [l2 (Ю3 +Ij ) + l3lj ]} gj + g2 — 3 xn[(gj + g 2 )2 -ю2 (Aj +A2 ) i_j where Q, С2, С3, gj, g2, g3 -values С and g, respectively, at frequencies Ю2'13 . The condition for existence in this case: (gj +g2)2 Ф(Al +A2)2 \ю2 (Ю3 ) +®3®j ]. (17) Analogously, for other invariants: A2a _ aj + a2 A2L _ L + L2 A2R Rj + R2 A2R Rj + R2 (18) > conditions for existence: («j +a2 ) +R2 f[w2 (3 ) + ЮзЮ1 ] ; (R1 +R2 )2 +L2 )2 [®2 (3 +®1 ) + ®3®1 ] ■ Obviously, value Ю2 (®3 +®1 )+®3®1 at approximation Дю^-0, where ®3 = ®2 +Аю =ю1 + 2Дю tends to value 3ю2 then conditions (17) and (19) turn into (5) and (12), which confirms the correctness of the performed transformations. For increments of any n-th order, the condition for the existence of an invariant is of the form, e.g. with respect to С and g, fn C C2 ' a, g2> ®1 0 . (20) It should be noted that invariants also exist in mutual transformations of increments into differentials and differentials into increments, i.e. (omitting lengthy intermediate transformations): Д(дС ) = д(ДС )= C1 + C2; Д(да) = д(Да) = a1 + a2; Д(* ) = 0(Д?) "" g1 + g 2 ; Д(8К ) "д(дЯ ) "" R + R2; (21) Д(дЬ) = ö(L) =_ L+ L2 Д(дЯ ) = д(ДЯ )= R + R' In this case, the condition for existence of an invariant e.g. with respect to С and g in transformations of increments into derivatives: (1 +g2)4 * 2<»1<»2 ((1 +N2)2 [( +g2)2 +(ю12 +Ю1Ю2 + <»2)(( +N)2], (22) and transformation of derivatives into increments provide an invariant existing at any positive real values C1, C2, g1; g2 (meeting (2). Analogous results also take place for circuits 6,R, and L, R. 3 CONCLUSIONS Conditions for the existence of invariants of a four-element dipole with active-reactive components at differential transformations and in increments of any order and also for mutual differential-difference transformations have been determined. References [1] B. I. Nevzlin: Idrntificntinn and application nf an electromagnetic inonrinnt nf a mathematical mndrl nf tOr controlled rnoirnnmrnt with actior-rractior cnmpnnrnts, Herald of East-Ukrainian State. Univ., Iss. 2, p.p. 155-161, 1997 [2] B. I. Nevzlin: Abokt extension of tOr scope of rxistrncr of an electromagnetic invariant, Herald of East-Ukrainian State. Univ., Iss. 4., p.p. 12-14, 1997 [3] Mathematical encyclopedia. - M. : Sov. Encyclopedia, Vol. 3, p.p. 563-564, 1982 [4] Mathematical encyclopedia. - M. : Sov. Encyclopedia, Vol. 2, p.p. 1026, 1979 ш Journal Of JET Volume 8 (2015) p.p. 17-30 Issue 2, October 2015 Typology of article 1.01 Technology www.fe.um.si/en/jet.html THE INVERTED DISTORTED PARABOLA-LIKE SHAPE OF THE BIAS-DEPENDENT ELECTRIC FIELD AT AN ELECTRON-INJECTING METAL/ORGANIC INTERFACE DEDUCED USING THE CURRENT-VOLTAGE METHOD OBRNJENA, DEFORMIRANI PARABOLI PODOBNA ODVISNOST ELEKTRIČNEGA POLJA OD PRITISNJENE NAPETOSTI NA VMESNI PLOSKVI KOVINA/ORGANSKI POLPREVODNIK IZPELJANA Z UPORABO METODE TOKOVNE KARAKTERISTIKE Matjaž KoželjR Bruno Cvikl1 Keywords: Metal-organic interface, Electric field, Bias-dependent interfacial field, Current density modelling, Effective electron mobility, Organic semiconductors Abstract Using the recently derived expression for the traditional Mott-Gurney charge-drift model extended by the non-zero electric field at the charge-injecting interface E.nt, the published dependence of the R Corresponding author: Matjaž Koželj, MSc, Tel.: +386 1 588 5277, Fax: +386 1 588 5376, Mailing address: Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia E-mail address: matjaz.kozelj@ijs.si 1 Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia, and University of Maribor, Faculty of Energy Technology, Hočevarjev trg 1, 8270 Krško, Slovenia current density on the applied electric field j-Ea for two good-ohmic-contact, electron-only, metal/ organic structures is analysed. It is argued that the Mott-Gurney law with the well-known empirical exponential bias-dependent mobility included, in spite of a very good fit to the j-Ea measurements, represents an unsatisfactory method for data analyses. It is shown that the internal electric field at the electron-injecting interface is strongly bias dependent, and in such a way is coupled to the electron current within the organic bulk. The bias dependence of the interfacial field resembles an inverted, distorted, parabola-like-shaped curve, the maximum of which is organic-material dependent. Beyond the maximum, which occurs at high values of the externally applied electric field Ea, the interfacial electric field E exhibits a rapid decrease towards zero, and only at this limit can the traditional Mott-Gurney law be applied. In contrast to the present notion, it is found that the (large) electron effective mobility for the two samples investigated does not change with the bias, but it is the total effective mobility (its product with the specific non-linear algebraic function of Ea) that is bias dependent. The effective mobility may be uniquely determined, providing the applied electric field spans a sufficiently wide Ea interval. It is argued that an appropriate width of this interval may be tested by the judicious application of the derived expression in the limit E ® 0. The Alq3 bias-dependent interfacial electric field at the electron injecting cathode/organic junction results in a non-linear response of the corresponding free electron density, nfrJi=200 nm), at this site. The possibility for an investigation of the electric field at the charge-injecting metal/organic interface using the j-V method is therefore outlined. Povzetek V prispevku je analizirana odvisnost tokovne gostote od pritisnjenega električnega polja j-Ea za primer dveh struktur kovina/organski polprevodnik, katerih značilnost so dobri ohmski kontakti in elektronsko prevajanje toka, pri čemer smo uporabili nedavno izpeljano enačbo za konvencionalni Mott-Gurney model dopolnjen z od nič različnim električnim poljem na vmesni ploskvi. Dokazano je, da Mott-Gurneyev zakon z vključeno empirično eksponentno odvisnostjo, kljub dobrem ujemanju z j-Ea meritvami, ne predstavlja zadovoljivo metodo za analizo podatkov. Pokazali smo, da je električno polje na vmesni ploskvi, na kateri poteka vbrizgavanje elektronov, močno odvisno od pri-tisnjene napetosti in tako povezano z elektronskim tokom skozi organski polprevodnik. Odvisnost električnega polja od pritisnjene napetosti je podobna obrnjeni, deformirani paraboli podobni krivulji, katere maksimum je odvisen od vrste organskega materiala. Konvencionalni Mott-Gurneyev zakon je možno uporabiti v limiti, ko električno polje na vmesni ploskvi, E , preseže maksimum in se potem hitro zmanjša do nule, kar se zgodi pri visokih vrednostih pritisnjene napetosti. Za razliko od trenutno veljavne razlage smo ugotovili, da se (velika) efektivna elektronska mobilnost v dveh raziskanih vzorcih ne spreminja s pritisnjeno napetostjo, pač pa je celotna efektivna mobilnost (produkt efektivne mobilnosti s specifično nelinearno algebraično funkcijo Ea) odvisna od pritisnjene napetosti. Efektivno mobilnost je možno določiti izključno pod pogojem, da se pritisnjeno električno polje spreminja v dovolj širokem intervalu. Dokazano je, da je primernost tega intervala možno preveriti s primerno uporabo izpeljanega izraza v limiti E ® 0. Električno polje na vmesni ploskvi katoda/organski polprevodnik Alq3, na kateri poteka vbrizgavanje elektronov, ki je odvisno od pritisnjene napetosti, ima za posledico nelinearni odziv ustrezne gostote prostih elektronov, nfnJi=200 nm), na tem mestu. Podana je možnost za raziskave električnega polja na vmesni ploskvi kovina/ organski polprevodnik, kjer poteka vbrizgavanje naboja, s pomočjo j-V metode. 1 INTRODUCTION An understanding of the intrinsic charge-transport properties is vital to the optimum operation of any electronic device that is based on organic semiconductors. Such devices currently used in practical applications include flat-panel displays, organic solar cells, flexible electronics, solidstate lighting, etc. An important step towards the widespread additional application of organic electronic devices is the continuous striving for a fuller understanding of the numerous factors that are currently limiting their performance. Considerable research has been devoted to uncovering the exact operation mechanisms of such devices and to a precise determination of their electrical and optical properties. In this respect, two important issues that have received a great deal of attention recently are the effects of the organic electron and hole layers on the efficiencies of devices, [1], and the exact role of chemical impurities, [2], which may either hinder or enhance the performance of the device. Both issues are intimately related to an investigation of the intrinsic charge-carrier mobility, [3-5], which is often measured using the current-voltage method, admittance spectroscopy, the time-of-flight method and transient electroluminescence. This work is focused on a determination of the electron mobility within a single organic layer using the current-voltage method. It represents the extension of a related investigation on hole charge carrying, [6], with the aim being to dispense with the charge-density singularity at the charge-injecting metal/organic interface that characterizes the well-known Mott-Gurney law. The Mott-Gurney expression, incorporated with the empirical exponential bias-dependent mobility, [7], is of paramount importance for the charge-carrier mobility determination using the current-voltage method. It has been shown recently, [6], that the existence of a non-zero interfacial electric field at the charge-injecting metal/organic junction causes the extinction of the singularity of the free-charge density that the Mott-Gurney law predicts. The investigation of the published j-V data obtained on two distinct organic structures, characterized by a series of single organic layers that differ in thickness, has shown that the effective hole mobility (to be distinguished from the total hole mobility) is bias independent. It was shown in the literature that the well-known empirical exponential bias-dependent mobility, [7], is an artefact that should be replaced by a derived, nonlinear algebraic expression that depends only on the ratio of the interfacial field to the externally applied electric field [6]. However, assuming a non-zero bias-independent interfacial electric field at the eleslcer-irreslird cathode/organic interface, Eird, it is not possible to describe the electron-only current-voltage, j-V, data of the good-ohmic-contact, single-layer organic structures of Yasuda et al., [8]. However, these data have been analysed by the authors in terms of the Mott-Gurney law that includes the empirical exponential bias-dependent mobility, and a very good fit was obtained; see Figs. 4 and 5 of Ref. [8]. In this work, it is shown that the electron-only current density within the two organic structures investigated in Ref. [8] is strongly coupled to the bias-dependent internal electric field existing at the electron-injecting interface. The bias dependence of the interfacial electric field, Eiol, is explicitly revealed for the published j-V data [8] for two metal/organic structures. For small applied fields Ee, it is shown that the interfacial electric field of both structures coincides and exhibits a Иг-ос irsceese with an increasing external electric field Ee, a behaviour that is apparently independent of the organic composition. At a certain value of Ee, which is organic-material dependent, the straight line transforms into an inverted distorted parabola-like curve, rapidly decreasing to a value close to zero. This small value of Eint apparently occurs when the last, the maximum, value of the current density in the j-Vdiagram is reached. If Eint turns out to be negligibly small at the maximum value Eamax of the Ea interval, then the traditional Mott-Gurney limit is attained. The effective mobility within the electron-only, single-layer, metal/organic structure investigated in this work turns out to be bias independent. As shown earlier in [6], apart from the Ea2 term, the additional external bias dependence of the current density is provided by the previously derived, non-linear algebraic function of the argument X(Ea) = E|ntE^, which is a implicitly and explicitly dependent on the external electric field. This fact offers an indication that the processes that determine the electron mobility differ from the ones that determine the hole mobility. In this work, it is once again confirmed that the empirical exponential bias-dependent function for the effective mobility is redundant, within the range of the j-V measurements. 2 THEORETICAL OUTLINE It can be easily verified that the drift-current density in a single-layer organic structure is, irrespective of the sign of the charge carriers, described by the expression, [6], . = ssoMeff E 2 j = 2L a 1 8 2 I 64 4 (2.1) where Ea is the externally applied electric field, defined as Ea = Va/L, and Va is the applied bias on the anode placed at the origin of the frame of reference (the cathode at x = L is at zero potential), j is the current density through the metal/organic structure, jueff is the effective mobility, e is the dielectric constant and e0 is the permittivity of free space, [6]. The current density, j, Eq. (2.1) replaces the well-known generalized Mott-Gurney model (in the sense that the empirical, [7], exponential bias-dependent charge mobility is included in the expression) of charged traps in the organic layer for completely empty (or equivalently, completely full) or the stated model, if the non-zert interfacial electric field, Eint, occurring at the charge-injecting metal/organic interface is taken into account. Here, the parameter X denotes the ratio of the non-zert electric field at the charge-injecting interface, Eint, to the externally applied electric field, Ea, X = (2.2) Ea where Eint might be bias dependent The introduction of the interfacial field results in the disappearance of the free-charge-density singularity at the stated interface, a serious shortcoming of the above-mentioned, generalized Mott-Gurney model. Of equal importance is the fact that in Eq. (2.1) the effective mobility |eff is bias independent and the bias dependence is described in terms of the non-linear algebraic expression, which is a function of the parameter A. Evidently, for Eint = 0, Eq. (2.1) reduces to the (original) Mott-Gurney model with a bias-independent effective mobility. In Ref. [6], Eq. (2.1) was tested on the published j-V data obtained on two different sets of single-layer, hole-only, metal/organic structures and good agreements with the measurements were obtained. For holes, it was shown that the interfacial electric field is bias independent and just slightly smaller than the initial value of the externally applied electric field. It was also explicitly shown that the effective mobility of the holes is thickness dependent, 8 but bias independent. Furthermore, it was shown that the empirical exponential bias-dependent effective hole mobility is redundant in j-V experiments since it represents merely an approximation of the derived algebraic expression represented by Eq. (2.1). It can be easily shown that the expression for the current density j, i.e., Eq. (2.1), is invariant with respect to the sign of a drifting charge carrier within a single organic layer and is consequently also valid for electrons. Consequently, the spatial dependence of the internal electric field E(x) reads, E(x) = Eint2 + ^(L-x) V SSoMeff (2.3) and the spatial dependence of the electric potential V(x), taking into account that V(L) = 0, is then: V(x) ee0Meff bXEo ) 2 + 2EE0Meoo Eint + j{Ea ) (L" x ) - Ei3 and the free-electron (number) density no (x) reads: AEo )0 _ qMeff so(x) Е0в e e (x ) q Cx E2nt (L-x ) (2.4) (2.5) EE, 0 M-OO In Eq. (2.5) д denotes the ratio of the free to the total (free and the bound) charge densities [9], д = o and the effective mobility [9] is defined as noo = n д. (o0 + nb ) Evidently, in the limit Еое = 0, Eq. (2.1) reduces to: j(Ee) = 9 EE0MLEf2 8 L (2.6) the well-known Mott-Gurney expression, which in combination with the empirical exponential bias-dependent mobility of Gil, [7] n-oo = По exp (^VE7 ) (2.7) has been in current-voltage, j-V, experiments exclusively used for the determination of the charge mobility. For Еое Ф 0, the curve evaluated by the combination of Eqs. (2.6) and (2.7) may, within the relevant interval of Ee, coincide with Eq. (2.1), see Ref. [6]. Here, the two parameters n0 and у are determined from the fit to the experimental data, but have no clear physical meaning. However, as pointed out in Ref. [6], the combination of Eqs. (2.6) and (2.7) is incomplete, exhibiting a singularity of the free-charge density at the charge-injecting interface, see Eq. (2.5), and it should be substituted by the corresponding Eq. (2.1). This assertion was already empirically proved for hole charge carriers, [6], and so its extension to the electron current density is the subject of the work presented here. 1 3 BIAS DEPENDENCE OF THE INTERFACIAL ELECTRIC FIELD Yasuda et al. [8] used the current-voltage method to investigate the electron mobility of six different electron-only, single-layer organic structures characterized by a quasi-ohmic contact. It should be emphasized that all the data were analysed in terms of Eqs. (2.6) and (2.7), since an excellent agreement between the calculated curves and the measurements was obtained, see Figs. 4 and 5 of Ref. [8]. For the purposes of this work, the interval of the externally applied electric field constitutes indispensable information for the application of Eq. (2.1). For two of the six organic structures, i.e., tris-(8-hydroxyquinoline), denoted as Alq3, and 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline, known as the BCP organic material, this information is clearly revealed, see Figure 2 of Ref. [8]. As a result of the excellent agreement between the data and the calculated fit shown in Ref. [8], in this work the data will be represented in terms of Eqs. (2.6), and (2.7) for the following values of the parameters, [8]: ß0 = 4.7 x 10-13 m2/Vs and y = 6.9 x 104 (m/V)1/2 valid within the interval of the bias 1 V < Va < 20 V for the Alq3 organic, and ßo = 2.3 x 1012 m2/Vs and y = 11.0 x 10-4 (m/V)1/2 valid within the interval of the bias 3 V < Va < 15 V for the BCP structure. In both cases, an average value [8] L = 200 nm was taken as the layer thickness in the calculations. It is clear that because of the considerable disagreement between the two curves shown in Figure 1, the interfacial electric field Eint may not be constant in the experiment of Yasuda et al. [8]. In order to test the hypothesis, the current density of Eq. (2.1) at a given Ea should equal the corresponding data (i.e., using Eqs. (2.6), and (2.7) for convenience) and the parameter 1 is then deduced as a real root of the expression, Ueff 9 - 312 + -314 + 313 _ 271 8 2 I 64 4 8 ßoexp W ) (3.1) The obtained value of 1 and the associated value of the interfacial electric field at the electron-injecting interface Eint are presented in Table 1. Using these values in Eq. (2.1), the two curves presented in Figure 1 then coincide to an excellent degree. The deduced bias dependence of Eint for the Alq3 organic (circles) is presented in Figure 2. 9 4 2 4 6 8 10 £a[x 107 V/m] Figure 1: Tle ceem-lempecelpce j-Ee Cole ef lle eleslcer-erly, aedC-elmie-serlesI, AiaAlq3(200 rm) slcaslace ece willir lle semplele irlecvel 5 MV/m = V<51,52>-To understand the concept of an affine variety better we need the notion of an ideal. An ideal in the polynomial ring is a subset I of satisfying i. if f,gEl then / + g G I and ii. if f GI and h G k[x1,^,xn], then hf G I. Let A,..,fs be polynomials of k[x1,^,xn]. We denote xlx2x3 >xlx2x|0. This monomial order is called a lexicographic (monomial) onden. More precisely, xa x2 >x3, the leading term of / is LT(/) = 2x|x2x|, whilst the leading coefficient is LC(/) = 2 and the leading monomial LM(/) = x1 x2 x|. Finally, note that any vector с £ E" defines an monomial order < in E[x1,_,xn] in the following way: vs ( с • а < с - ß or fP > x" < xp o lc • а = (?• ß and a y. Hi - ж + У Я! : i __r_ fi xy-1 \fx2y + xy2 + y2 x2y — X xy2 + I + y2 xy2 - у x + y2 + y ■уГ+ у -> X у2- 1 y + i 1 —> X + у О X + у + 1 Figure 1: The scheme of multivariable division procedure On the first step, the leading term LT(f1) = xy divides the leading term LT(f) =x2y. Thus, we divide x2y by xy, leaving x and then subtract x ■ f1 from f. Next, we repeat the same process on x y2 + x + y2. We divide f by LT(f1) = xy again. Note that neither LT(f1) = xy nor LT(f2) = y2 divides LT(x +y2 +y) = x. However, x + y2 +y is not the remainder since LT(f2) divides y2. Thus, if we move x to the remainder column, we can continue with the process. If we can divide by LT(f1) or LT(f2), we proceed as usual, otherwise, we move the leading term of the intermediate dividend (the polynomial under the radical sign) to the remainder column. We continue dividing in such a way. Now the polynomial under the radical is y2 +y. It is not divisible by LT(f1) but we can divide it by LT(f2) yielding 1 and the subtract 1 ■ y2 from y2 +y. The obtained polynomial under the radical is y +1 and neither LT(f1) nor LT(f2) is divisible by LT(y + 1) = y. Therefore, we move y to the remainder column and obtain 1, which is also moved to the remainder column. Thus, the remainder is x + y + 1 and this concludes the example. Thus, we can write f in the form f = x2y + xy2 + y2 = (x + y) • (xy — 1) + 1 • (y2 — 1) + X + у +1. In contrast, if we change the order of polynomials f and f2 in F, i.e. if we divide f by the ordered set F = {f2,f}, we obtain f = x2y + xy2 +y2 = х - (xy — 1) + (x + 1) • (y2 — 1) + 2x + 1. Obviously, this multivariable division is very sensitive on the order of fi,f2. The order affects the multi-quotients q,1,ql2, as well as the remainder r. Dividing the polynomial f with the (ordered) set F = {f1,f2}, one can simply write: f = {{q,1,ql2}, r} instead of f = ql1f1 + ql2f2 + r. Using this notation, in the first case, we have f = {{x + y, 1}, x + y +1} and in the second case we have f = {{x, x + 1},2x+ 1}. Figure 2 shows the last results obtained by the MATHEMATICA computer algebra system. The multi-quotients and the remainder are also changed if we use another monomial order. lm[l]:= PolynomialReduoe [x л 2 у + x у л2 + у л2, {х у - 1, у л2 - 1} , {х, у] ] 1ш[2]:= PolynomialReduoe[xA2 у + х у л2 + у л2, {у л2 - 1, х у - 1} , {х, yj] Figure 2: Results obtained by MATHEMATICA fon the case above Now, we present the basic definitions and properties of Gröbner bases. For any ideal, I we define <£T(I)> = (LT(/): /£I \ {0}> = (LM(/): /£I \ {0}>. A Gröbner basis of an ideal Icfc[;e1,,_,xn] is a finite subset G = {flfi,...,flft} of I such that = <ЩЛ).....LT(,t)>. It is a special generating set for ideal for which the multivariable division algorithm for a given f returns the remainder r = 0 if and only if f £ ,[4]. Using a Gröbner basis, we obtain the uniqueness of the remainder, which was not assured when we divided by an arbitrary set of polynomials, [2]. We now describe an algorithm for computing a Gröbner basis of a polynomial ring. Let f,, be from fc[x1,.,x„] with LT(f) = axa and LT(,) = fox^. The least common multiple of xa and x^, denoted LCM(xa,x^), is the monomial xr = x^ •••x,n such that ^ = max(a_/-,^J), 1< 7 < n. The so-called 5—polynomial of f and , is = x^ x^ = ZTčf)-f — Щ£)- ^ Buchcbengen's basic observation was the following criterion, [2]. Let / be a nonzero ideal in fc[x1,.,x„] and let < be a fixed monomial order on fc[x1,.,x„]. Then, G = {,1,,2,.,,Е} is a Gröbner basis for / with respect to < if and only if for all i V*,'0. This criterion is the essence of the famous Buchöeneen's blggnithm, which produces a Gröbner basis for the nonzero ideal I = (f1,.,fs). Buchberger's algorithm is shown below, [11]. Buchberger's Algorithm Input: A set of polynomials {f1,.,fs} £fc[x1,_,xn]\{0). Output: A Gröbner basis G of the ideal (f1,.,fs). Procedure: G: = {f1,.,fs}. Step 1. For each pair gi,gj EG,i ^j, compute the S — polynomial Sg.gj and compute the remainder r;j- of division Sg.gj by the set G. Step 2. Check if all rij are equal to zero . If "yes", then G is a Gröbner basis, otherwise add all nonzero rij to G and return to step 1. It is proved in [2] that the algorithm terminates and returns a Gröbner basis of the ideal I = y is computed in systems MATHEMATICA and SINGULAR, [9], respectively. We see that in both cases the result is {y3 —y2,xy2 —y2,x2 y — y2,x3 — y}. In[6]:= GroebnerBasis [ |-к3 + у, хгу-у2}, {x, у}] Figure 3: Output gf gnäöaen basis ia system MATHEMATICA SINGULAR A Computer Algebra System for Polynomial Computations by: U. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann FB Mathematik der Uniuersitaet, D-67653 Kaiserslautern > ring r1 = 0, (X,y) ,1p; > poly f1=-x3+y; > poly f2=x2*y-y2; > ideal I=f1 ,f2; > ideal G=groebner(I) ; > G; G[1]=y3-y2 G[2]=xy2-y2 G[3]=x2y-y2 G[i|]=x3-y > I Figure 4: Output of Gnöbnen basis in system SINGULAR A reason to use more special systems than MATHEMATICA offers is to compute the Gröbner basis with respect to some special monomial order. In Figure 5 we compute the Gröbner basis of (—x3 + y,x2y — y2) using SINGULAR with respect to the weighted monomial order with weight vector с = (1,3). Note, that the result G<(13) = (x3 — y,y2 —x2y) is not the same as the Gröbner basis with respect to lexicohraphic monomial order G<)CT(v>r) = (—xs + X6, —X3 +y). SINGULAR Й Computer Algebra System for Polynomial Computations by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann FB Mathematik der Uniuersitaet, D-67653 Kaiserslautern > ring r1=B, (x,y),Wp(1,3); // ** redefining r1 ** > poly f1=-x3+y; > poly f2=x2*y-y2; > ideal I=f1,f2; > ideal gI=groebner(I); > gì; gl[1]=x3-y gl[2]=y2-x2y Figure 5: Gröbner basis G<( ) of (x3 — y, —x2y + y2) computed in SINGULAR Recall that to solve a system of linear equations, an effective method is to reduce it to the form in which an initial string of variables is missing from some of the equations, that is, the so-called "row-echelon" form. The next definition and theorem provide a way to eliminate a group of variables from a system of nonlinear polynomials. Moreover, it provides a way to find all solutions of a polynomial system in the case that the solution set is finite, or in other words, to find the variety of a polynomial ideal in the case that the variety is zero-dimensional. For any ideal / = (/i,..,/š) c fc[x1,x2,^,x„] the i—th elimination ideal /г is the ideal of fc[xi+1,xi+2,.., xn] defined by / / uersion 3-1-' \ Jan 2012 \ ICA ) / / uersion 3-1-4 0< \ Jan 2012 \ Il — I nk[Xi+1,Xi+2, — ,Xn]. In the case of solving a system of nonlinear equations (1.1), this means that I — (f1,.,fs), but the elements of Il are the equations (polynomials) that follow from f1 — 0,...,fs — 0 and eliminate the variables x1,.,xl. Concerning the Gröbner bases and elimination ideals, we have the following Theorem, ([4]). If G — {g1,.,gt} is a Gröbner basis for I — (f1,.,fs) c k[x1,.,xn] with respect to lexicographic order with x1 > — >xn, then for each 0 <1 < n the set Gl — G П k[xi+1,xi+2,..., x„] is a Gröbner basis for the I— th elimination ideal. Gröbner basis theory allows one to find all solutions of a system (1.1) if the system has only a finite number of solutions. In such case a Gröbner basis with respect to the lexicographic order is always in a "row-echelon" form, as can be seen using the following example, [5]. Consider the polynomials Д —x2 + yz + x f2 —z2 + xy + z f3 —y2 + xz + y. With respect to the lexicographic order with x >y >z, the Gröbner basis of ideal (f1,f2,f3) is G — {д1,д2,дз-94.95-9б}, where g1 — x + x2 + yz g2 — xy + z + z2 g3 — z + xz + yz + z2 + 2 yz2 gA — y + y2 —z — yz — z2 — 2 yz2 g5 —z2 + 2yz2 +z3 + 2yz3 g6 — z2 + 3z3 + 2z4. Thus, the system A —f2 —f3 — 0 is equivalent to the system g1 — g2 — g3 — g4 — gs —g& — For a generic system (1.1), a Gröbner basis may have significantly more complex structure than obtained in this example. However, if the system has only a finite number of solutions (i.e. the ideal is zero-dimensional), then any reduced Gröbner basis {g-i^,... ,gn] must contain a polynomial in one variable, let say, g1(x1). Then, there is a subset of the Gröbner basis depending on this variable and one more variable, say, g2(x1,x2),.,gt(x1,x2), etc. Thus, we first solve (perhaps only numerically) the equation g1(x1) — 0. Then, for each solution xl of g1(x1) — 0, we find the solutions of g2(xl,x2) — ••• — gt(xl,x2) — 0, which is a system of polynomials in a single variable x2. Continuing the process, we obtain in this way all solutions to (1.1). Thus, in the case of the finite number of solutions, Gröbner basis computations theoretically provide the complete solution to the problem (see e.g. Section 2.2. of [1] for more details). 3 GRÖBNER BASIS AND INTEGER LINEAR PROGRAMMING In previous sections, we considered the process of multi-division and the Gröbner bases theory. Before the formal definition of the integer linear programming problem, note that the essence of the problem concerns the integer solutions of a linear system Ax = b (constraints), which optimizes the so-called cost function. Therefore, it would be quite convenient if the rows of the matrix equation Ax = b (i.e. the equations) could be presented (in the first approximation) as exponents of some new variables. In order to make things more straightforward, let us consider Ax = b (where: [A];j- = a;j- and b = (b1(...,bm)) with the following restrictions: atj EZ, b; EZ and Cj £ E with i = 1,2,... ,n and j = 1,2,., т. We want to find a solution x = (x1,x2,...,xn)EZn to a11x1 +a12x2 + "■ + a1nxn =b1 ! (3.1) am1x1 + am2x2 + "■ + amnxn = bm, which minimizes the cost function c(x1,x2,...,xn) = Hn=:L Cjxj. We call (3.1) an integer (linear) program (IP). In the matrix form, we have minimize с • x subject to Ax = b, whene A EZmxn and b = (b1,. ,bm) EZm. Note that all coefficients (including the solution vector) are generally allowed to be from the set Z, but for now let the coefficients be limited to be natural numbers: N. The main mathematical idea which makes use of Gröbner bases when solving IP (3.1) is to associate new variables Xk for к = 1,2,..., т (one variable to each equation) to (3.1) to represent the к—th equation in (3.1): X->Xm>Y1 > — > Yn. Then a Gröbner basis for KHk[Y1,., Yn] will be precisely the polynomials of the Gröbner basis of К that do not have any X variables. Obviously, for a given homomorphism Ф, any monomial f £ k[Y1,..., Yn] is not in the image of Ф. Let us denote the Gröbner basis of K = {Y1—f1,..., Yn—fn) obtained by the elimination theorem by G. According to the above results, we have f £ Кег(Ф) o 3h £ k[Y1,.,Yn] s.t. f ^ h, which is then the key to the solution of IP (3.1) constrained by the cost function c(x1,x2,...,xn)= Z,n=1 Cjxj. Finally note that, if take care that the monomial order which is used to compute the Gröbner basis, GX2 >Y± >Y2 >Y3. We obtain Г — f V 6 -L- V 3v v V 4 v 2v v V V 2 v 2v 2 у V 3 v Y 2v G 1 Y2 ' Y3'A2X2 ~ X1 y3'X2 2 ,X2 Y2 ~ Х1Г3'А2 ~ X3'~X2 Y2 + X1Y3,—Y1+X1Y2,X1X2 — Y2}. Then, we divide monomial XlX2 by G and obtain Y11Y21Y31. Therefore, the non-negative integer solution is (x1,x2,x3) — (1,1,1). Example 2 ([8]): We show how to optimize the cost function with respect to some constraints Ax — b, and the coefficients are now allowed to be also negative integers. Following (3.1), we have to minimize the cost function с ■ x — 1000x1 +x2 +x3 + 100x4 subject to 3x1 2x2 hx^ X4 — 1 4x1 +x2—x3 — 5. The solution to the above example obtained with system SINGULAR is shown in Figure 6. Note that the weighted term order is used with С = (1000002,1000001,1000000,1000,1,1,100) to ensure that X1 > X2 > W > Y1 > Y2 > Y3 > Y4 and to ensure the weight order (1000,1,1,100), corresponding to с = (1000,1,1,100). Note that, for example, the monomials XSWg and XA2W2 are: X%g =X1"1Xf =X1"1X2"1 -XfXf = W1X26, XA w2 =x1"2x21 = xj-2x2-2 -x22x2 = w2x|. The optimal solution x = (1,3,2,0) is obtained from the result of the multivariable division: W1 X26 Д Y11Y23Y32Y4°. SINGULAR / fi Computer Algebra System for Polynomial Computations / 0< by: W. Decker, G.-M. Greuel, G. PFister, H. Schoenemann \ FB Mathematik der Uniuersitaet, D-67653 Kaiserslautern \ > ring r1 = 0,(N1 ,X2,U,V1,V2,V3,V4) ,Up(1000002,1 000001 ,1000000,1000,1 ,1 ,100) ; > poly F1=V1-N13*N2ii; > poly f1=V1-Nl"3*N2"ii; > poly f2=V2-X2"3*lT2; > poly F3=Y3-X1~2*U; > poly fii=V4-N2*W; > poly F5=X1*X2*W-1; > ideal I=F1,f2,f3,F4,f5; > ideal gI=groebner( I ) ; > reduce(W*X2~ó,gI); Y1*Y2~3*Y3"2 > I Figure 6: Computing the optimal solution in system SINGULAR uersion 3-Jan 2012 References [1] W.W. Adams, P. Loustaunau: An introduction to Gröbner bases: Graduate Studies in Mathematics. Vol. 3, Providence, RI: American Mathematical Society, 1994 [2] B. Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restlasseringes nach einem nulldimensionalen Polynomideal. PhD Thesis, Mathematical Institute, University of Innsbruck, Austria, 1965 [3] P. Conti, C. Traverso: Buchberger algorithm and integer programming, Applied algebra, falgebraic algorithms and error-correcting codes (New Orleans, LA, 1991), Lecture Notes in Comput. Sci. vol. 539, p. 130-139, 1991 [4] D. Cox, J. Little, D. O'Shea: Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geomety and Commutative Algebra. New York: Springer, 2007 [5] S.R. Czapor: Gröbner basis methods for solving algebraic equations. Ph.D Thesis. University of Waterloo, Canada, 1988 [6] V.F. Edneral, A. Mahdi, V.G. Romanovski, D.S. Shafer: The center problem on a center manifold in R3, Nonlinear Anal., vol. 75, p. 2614-2622, 2012 [7] B. Ferčec, M. Mencinger: Isochronicity of centers at a center manifold, AIP conference proceedings, 1468. Melville, N.Y.: American Institute of Physics, p. 148-157, 2012 [8] S. Flory, E. Michel: Integer Programming with Gröbner basis. (http://www.iwr.uni-heidel-berg.de/groups/amj/People/Eberhard.Michel/Documents/Else/DiscreteOptimization.pdf) [9] G.M. Greuel, G. Pfister, H. Schönemann: Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2005; http://www.singular.uni-kl.de. [10] V.G. Romanovski, M. Mencinger, B. Ferčec: Investigation of center manifolds of 3-dim systems using computer algebra. Program. comput. softw., vol. 39, no. 2, p. 67-73, 2013 [11] V.G. Romanovski, D.S. Shafer: The center and cyclicity problems: A computational algebra approach. Boston: Birkhauser Verlag, 2009 [12] C. Wendler: Groebner Bases with an Application to Integer Programming; (http://docu-ments.kenyon.edu/math/CWendler.pdf), 2004 ш Journal Of JET Volume 55 (20 15) p.p. 59-70 Issue 2, 0ctpber2015 Energy Pypology pf га^Р)с^б) 1.02 Technology www.fe.um.si/en/jet.html ELECTRIC CARS IN SLOVENIA ELEKTRIČNI AVTOMOBILI V SLOVENIJI Gregor Srpčič« Keywords: Electric vehicles, electric cars, charging stations, financial incentives Abstract This article deals with electric vehicles in Slovenia. The introduction of electric cars is necessary in order to achieve the objectives of the European Union's strategy for reducing CO2 emissions from road vehicles. In the first part of the article, the development of electric cars and their functioning is described. A key component of electric cars are batteries; therefore, various types of batteries, battery charging, and types of charging stations, which occur in Slovenia, are described in this article. An extensive network of charging stations is essential for the use of electric vehicles. A map of charging stations and the number of charging stations in Slovenia are presented. Currently, the most charging stations are located in central Slovenia; this is in the area around Ljubljana. In addition, to an extensive network of charging stations, the introduction of electric vehicles also requires financial incentives to buyers of electric vehicles. Slovenian environmental public funds or Eco Fund grants provide financial incentives to buyers of electric cars. In Slovenia, there are two types of financial incentives. Povzetek Članek govori o električnih avtomobilih v Sloveniji. Uvajanje električnih avtomobilov je nujno za dosego ciljev strategije Evropske unije za zmanjšanje emisij CO2 iz cestnih vozil. V prvem delu članka je opisan razvojelektričnih avtomobilov in delovanje le-teh. Ključni sestavni del električnega avtomobila so baterije, zato so v članku opisani tudi različni tipi baterij, polnjenje baterij ter tipi polnilnih postaj, ki se pojavljajo v Sloveniji. Dobro razvejana mreža polnilnih postaj je nujna za uporabo električnih vozil. V članku je prikazan zemljevid in število polnilnih postaj v Sloveniji. Trenutno je največ polnilnih postaj lociranih v osrednji Sloveniji, torej v okolici Ljubljane. Poleg dobro razvejane mreže polnilnih postajso pri uvajanju električnih vozil я Corresponding author: Gregor Srpčič, Tel.: +386 40 520 511, Mailing address: Faculty of energy technology, Hočevarjev trg 1, 8270 Krško, Slovenija E-mail address: grega.srpcic@um.si potrebne tudi finančne spodbude kupcem električnih vozil. V Sloveniji za finančne spodbude za nakup električnih vozil skrbi Slovenski okolijski javni sklad oziroma Eko sklad. V Sloveniji sta na voljo dva tipa finančnih vzpodbud. 1 INTRODUCTION The European automotive industry is a world leader in developing clean and energy efficient technologies based on combustion engines, because it has invested heavily in research and development since the year 2000. It is also one of the key European industrial sectors, since it is competitive, innovative and supports a wide range of related industries, [1]. The European Union has set a long-term strategy to reduce CO2 emissions from road vehicles and much has been realized already. Regulation (EC) No. 443/2009 on the setting emission performance standards for new cars requires that the target of reducing average CO2 emissions of the new cars to 130 g/km be fully meet by 2015. The automotive industry will need to invest even more in emission abatement technologies, including intelligent traffic management technologies and further improve the efficiency of internal combustion engines, [1]. Providing long-term sustainable mobility calls for more energy-efficient vehicles that are powered by alternative energy sources. Electric vehicles offer a solution from the dependency on fossil fuels and for reducing CO2 emissions. Electric vehicles are the first choice in transition to a more efficient traffic, [1]. 2 DEVELOPMENT OF ELECTRIC VEHICLES Electric vehicles embody our recent green-oriented mentality, but they are by no means a new innovation. They have been on the market for more than a century and have an interesting history of development, which continues. The first countries to develop an electric propulsion system were France and England. In 1835, Professor Sibrandus Stratingh drew a design of an electric car, which was later realized by his assistant Christopher Becker. In America, Thomas Davenport and Robert Davison, who made the electric vehicle more useful, contributed to the development of electric vehicles. In the following years, the development of electric vehicles went mainly towards the larger capacity of batteries for storing electricity, which was a prerequisite for a greater practicality of electric vehicles. Over the years, the highest possible speed of electric vehicles increased; in 1899, the limit of 100 km/h was broken. The speed achievement was credited to the Belgian Camille Jenatzy, whose vehicle was named 'Never Satisfied'. Electric vehicles had, in comparison to other technologies in the industry, a number of advantages. Compared to vehicles powered by gasoline, they were quieter, did not spread any unpleasant odours in the surroundings and were not causing vibrations when functioning, [2], [3], [4]. A great advantage over petrol vehicles was also that there was no need to shift gears when driving, which was causing a lot of clumsiness in the competitive technology, [4]. The dominance of electric vehicles lasted somewhere until the beginning of the 1920s, with peak production in 1912, [3]. The reasons for the turnaround in the favour of the industry of vehicles with internal combustion engines are different. The construction of long roads between cities required a greater range of vehicles, which electric vehicles were not able to handle. In Texas, new oil reserves were discovered, which reduced the cost of use of vehicles with internal combustion engines. The refinement of the ignition system of a petrol engine in 1912, which was presented by Charles Kettering, also had a significant impact on the turnaround. Ultimately the industry of petrol engines obtained its dominance with mass production of internal combustion cars from the Henry Ford factory, which offered vehicles to its customers that were more than half the price of electric cars, [2], [4]. Electric vehicles have been used only for specific purposes, such as vehicles for transporting milk, golf cars, and trucks. In the 1970s, the oil crisis has led to renewed interest in electric vehicles, which would alleviate the dependence of the transport sector on the situation on the oil market. The California Agency for Clean Air demanded that car manufacturers invest in the development of vehicles with low emission levels; the main objective were electric vehicles with zero emissions, [2], [5]. The biggest sales success was experienced by the EV1 model from General Motors, which represented the only car that met all the objectives of the Office for Energy of United States of America upon its arrival on the market. It was produced from 1996 to 2002 and was offered to customers through a lease agreement. As a reason for halting the production of the EV1, General Motors stated a lack of profitability. The public blamed pressure of oil lobbies and the fear of automobile companies about any strict rules regarding automotive emissions in other countries. Some of the EV1 autos are kept in various technical museums, but most were scrapped and recycled, although users of EV1 created strong publicity against these measures, [2], [5]. In 2004, the company Tesla Motors began developing an electric sports car, the Tesla Roadster, which came on the market in 2008. The Roadster was the first car with a built-in Li-ion battery. It boasts record-breaking driving performance, since it can travel 320 km on a single charge and accelerate from 0 to 100 km/h in just 4 s. [2], [5]. 3 WHAT ACTUALLY IS AN ELECTRIC VEHICLE? An electric vehicle is a vehicle that is powered solely by electricity stored in batteries located inside the vehicle. In a simple propulsion system, an accumulator powers an electric motor that enables the rotation of the wheels through mechanical transmission. This kind of a drive system reflects the simplicity of construction and good efficiency, [6]. An electric vehicle does not generate greenhouse gases. It can be powered by electricity generated from renewable energy sources; not only is the level of pollution zero, but the use of electricity generated from renewable energy sources also reduces the level of emissions, [2], [7]. Advantages of electric vehicles over conventional vehicles are lower maintenance costs, since the electric vehicle has fewer moving parts, thus allowing better utilization of energy. Electric vehicles do not need engine oil, a clutch or a transmission, and enable linear accelerations [7]. Due to the silent operation of the electric motor, it does not cause noise pollution and offers a more comfortable ride for passengers, [2]. Several types of electric vehicles are defined: battery electric vehicles, plug-in hybrids, and extended-range electric vehicles. A battery electric vehicle is a vehicle that uses only the energy stored in the batteries for propulsion. Plug-in hybrid vehicles use battery power as the main source for short distances, but at the same time, the internal combustion engine is running when the batteries are depleted. Extended-range electric vehicles use battery power as the main source for short distances, but when the battery is depleted, an internal combustion engine, which provides power to the electric motor, begins to operate, [2], [8]. Ball et al. [8] also distinguish different classes or categories of electric vehicles. In the L6e category are four-wheelers with a maximum authorized carrying capacity of 550 kg together with batteries and a maximum output of 4 kW of the drive motor. Such vehicles are especially suitable for urban driving, [9]. They are allowed to be driven by persons over the age of fifteen years, pensioners and drivers without B and B1 driving license. In the L7e category of vehicles are four-wheelers with a maximum weight of 400 kg and a maximum specified drive motor power of 15 kW. In this category, the maximum prescribed vehicle weight of 400 kg is without batteries. Electric vehicles in this category meet the definition of an urban electric car and are well suited for driving in urban areas. In the M1 category of vehicles are four-wheeled passenger vehicles that are intended for the transport of up to eight passengers, together with the operator of the vehicle, [2], [9]. 4 BATTERIES AND CHARGING OF BATTERIES There are several types of batteries, whose most important characteristics are specific power, specific energy, charging efficiency, the number of charging cycles and, of course, price, [2], [10]. Lead acid batteries have the longest history of use; consequently, they have a well-developed recycling process, [10]. This type of battery is very affordable; they have high voltage galvanic cells and are still in use today. Their drawback is their weight, fast draining and the impossibility of refilling if they are empty for too long, [2]. The advantage of NiMH batteries is fast charging. Their weaknesses are high price and rapid self-discharge. The use of Ni-MH batteries in electric vehicles was most widespread at the turn of the century when they have replaced lead acid batteries in the well-known EV1 model of General Motors [2], [11]. A significant leap forward regarding capacity was made with the introduction of Li-Ion batteries. Their advantages are a highly adaptable design, small weight in comparison with the other types of batteries, a small percentage of self-discharge and no memory effect, [11]. One disadvantage of Li-Ion batteries is that they lose their original capacity over time, [2]. At present, Li-FePO4 batteries have the best characteristics. They are based on the technology of Li-Ion batteries, but they differ in the selected cathode material. Their advantages are stability and security, because when in use there is no possibility of overcharging or short-circuit shock. Physical damage to the battery cannot cause an explosion. They provide a high number of refilling cycles, they are resistant to high temperatures and allow discharges with high currents. Li-FePO4 batteries also allow rapid charging and are half the weight of lead acid batteries, [12]. Table 1: Cherectsristics af bettsriss, [13] Type Specific energy [Wh/kg] Energy density [Wh/l] Specific power [W/kg] Charging efficiency [%] Lifetime [cycles] Pb 35-40 70 100-150 68 300-500 Ni-MH 50-60 175 200 76 600-1000 Li-Ion 80-90 200 <1000 80 1200 110 220 <3000 90 2000 FePO4 Table 1 clearly shows that Li-FePO4 batteries have the best characteristics, they exceed other types of batteries in all characteristics that are given in Table 1. Several different ways of charging electric cars exist. Typically, electric vehicles are recharged at home through a single-phase 220 V socket. Charging takes place via a power adapter that is already installed in the vehicle. This way of charging has a power supply of 3 kW and is usually very slow. For faster charging of electric vehicles, manufacturers of charging stations have developed a power adapter, of 22 or 43 kW, which is installed in the vehicle. Charging also takes place through three-phase socket outlets with a voltage of 400 V and the use of an external charger with an AC-DC converter. The charging power is 50 kW, [2], [14]. Table 2: Technical cherectsristics af charging stations, [14] Charging time [h] Power supply Voltage Maximum current 6-8 Single phase - 3.3 kW 220 VAC 16 A 2-3 Three-phase - 10 kW 400 VAC 16 A 3-4 Single phase - 7 kW 220 VAC 32 A 1-2 Three-phase - 24 kW 400 VAC 32 A 0.3-0.5 Three-phase - 43 kW 400 VAC 63 A 0.3-0.5 Direct current - 50 kW 400-500 VDC 100—125 A Table 2 shows that the charging process takes the longest time at a voltage of 220 V, a maximum current of 16 A and a single-phase power supply with the power of 3.3 kW, which represents a home socket outlet. The battery of an electric vehicle is recharged in the shortest time with a DC power supply with power of 50 kW, voltage of 400 to 500 V and current of 100 to 125 A. [2], [14] 5 CHARGING STATIONS Electric vehicles need charging stations, where batteries are filled with electric energy. In most cases, the manufacturer of electric vehicles or batteries provides a suitable charger, through which energy is transferred from the energy grid to the battery. The charging of electric vehicles is similar to charging the battery in a mobile phone; the only difference is that greater electrical power is needed for charging electric vehicles. Several alternative methods for charging electric vehicles are developed, including those that obtain energy from renewable energy sources, [2], [15]. The existing infrastructure of gas stations in developed countries makes it easy to introduce an additional offer of electric charging stations. On the existing electrical installation charging stations that enable the transmission of electricity into electric vehicles could be connected. In addition to petrol stations, it is reasonable to install fast charging stations in all areas with many parking spaces. These are, for example, public parking spaces and parking spaces of hotels, airports, restaurants, shopping centres, and similar facilities, as well as parking spaces used by employees of businesses during working hours, [2], [14]. Charging stations are roughly divided into two categories, which differ according to the use of the charger. A smart charging station is connected directly to the battery, which is recognized by the charging station itself. Other charging stations represent just a source of electrical power. There are two methods of charging: traditional or conductive connection and inductive connection, [2], [15]. In the case of conductive charging, the connector provides a secure connection between the charging station and the charging connector on the vehicle. The electric vehicle is connected to the primary outlet that usually has a voltage of 220 V at home; electric current flows from the charging station through the charging adapter into the battery and thus charges the battery of the electric vehicle. The charger is usually located outside the vehicle, or it may be installed in the vehicle. In the case of inductive charging, the charger has no direct electrical connection to the vehicle, as the energy is transmitted through a magnetic field, [15]. The identification of the user is of major importance for the application of charging stations. User identification allows charging for use, prohibition of use, activation or deactivation of the device. Several different methods of identification have been developed. The charging station can identify the user via a personal identification code (PIN), through radio frequency identification (RFID), over messages sent from a mobile phone or via a key belonging to a specific individual. [2], [16] Borzen (Slovenian power market organizer) and SODO (electricity distribution system operator) are responsible for the infrastructure of charging stations for electric vehicles in Slovenia according to the proposal of the National Energy program (NEP), [2]. The most active companies in this area are electricity distribution companies Elektro Maribor and Elektro Ljubljana. The first public charging station in Slovenia was activated on 9 April 2009 in Ljubljana, at Castle Kodeljevo, [2]. In summer 2015, the map of public charging stations in Slovenia (on the website of the company Elektro Ljubljana) lists 127 public charging stations; 58 of them are located in the vicinity of Ljubljana, and the fewest charging stations are in the northwest and southeast of Slovenia, [17]. Figure 1: Detailed mep af charging stations in Slarsnie (Saurcs: http://www.elektro-crpalke.si/, 31.8.2015) Slovenia's largest provider of public charging stations is Elektro Ljubljana, which enables the charging of electric cars, electric motorcycles, and bicycles with an electric motor. Various charging stations have different ways of connection. Connecting via the infrared port (IR ID), activation by using users' mobile phone (GSM) or the charging station is activated, and the vehicle merely has to be connected (Plug & Charge), [18]. In public charging stations in Slovenia six types of connectors can be found (Figure 2): normal socket, 3-pin socket, 5-pin socket, 7-pin socket, CHAdeMO DC Quick charger and Tesla Supercharger. [17] normal socket CHAdeMO DC Quick charger Tesla Supercharger Figure 2: Types af cannsctars, [17] Figure 3: Etrel's charging station (Source: http://polnjenjeev.etrel.si/grid/evcharging_we_our_solutions, 31.8.2015) The charging station consists of a robust housing that resists unfavourable weather conditions, and the ergonomic design is easy to use. Charging the user for the service is ensured by using user identification and embedded counters. The charging station allows users to book a charging station, see the consumed energy and send messages to the users' mobile phone about the completion or an unexpected interruption of charging. For the users of electric vehicles, a web portal is also available. This web portal provides users with information about the occupancy of charging stations, with the ability to book a charging station and with the supervision of charging of users' electric vehicle. The portal also functions as a mobile application, through which the user searches for and books free charging stations and also supervises the recharging of his electric vehicle, [2], [19]. 6 FINANCIAL INCENTIVES FOR THE PURCHASE OF ELECTRIC VEHICLES IN SLOVENIA In Slovenia in 2015, there are currently two financial incentives for the purchase of electric vehicles. 6.1 Public tender 31SUB-EVOB15 Public tender 31SUB-EVOB15 concerns non-refundable financial incentives to individuals for purchase or investment into electric vehicles. An individual is eligible for a grant: • in the case of purchasing a new electric vehicle, • in the case of purchasing a new hybrid vehicle, • in the case of purchasing a new electric vehicle with a range extender, or • in the case of converting a vehicle with internal combustion engine into an electric vehicle, [20]. Grants may be awarded for the purchase of vehicles in categories L7e, L6e, N1 and M1 with electric propulsion without CO2 emissions. Incentives can also be granted for the purchase of hybrid vehicles and vehicles with a range extender; however, CO2 emissions must not exceed 50 g of CO2 emissions per km, [20]. The total budget for the public tender is €500,000. The amount of the financial incentive is: • €5,000 for a new or a processed electric vehicle without CO2 emissions in the category M1; • €3,000 for a new or a processed electric vehicle without CO2 emissions in the category L7e; • €3,000 for a new hybrid vehicle or an electric vehicle with a range extender with CO2 emissions less than 50 g of CO2 emissions per km. • €2,000 for a new or a processed electric vehicle without CO2 emissions in the category L6e, [20]. Each natural person that has a permanent residence in Slovenia is entitled to the financial incentive, [20]. 6.2 Public call for loans for environmental investments 53PO15 The subject of the public call is loans, for environmental investments, of the Eco Fund. Among the environmental investments are, [21]: • Purchase of an electric vehicle with zero CO2 emissions. • Purchase of a hybrid vehicle or a vehicle with a range extender. CO2 emissions of the mentioned vehicle types must not exceed 110 g/km, [21]. The total budget for the mentioned public tender is ten million euros. Loans are available for companies, entrepreneurs and natural persons with permanent residence in Slovenia. Loans are not available to legal entities that, [21]: • do not have settled overdue financial obligations to the Slovenian Eco found; • do not have settled tax or other financial obligations to the Republic of Slovenia; • have a blocked bank account, [21]. The minimum amount of loan is €25,000, and the maximum amount is two million euros. The total debt of the borrower at the Eco Fund may not exceed ten million euros. The interest rate on the loan is EURIBOR plus a minimum of 1.5%. The repayment period is five years for the purpose of purchasing electric vehicles, [21]. 7 CONCLUSION In 2014, Slovenia had 1,412,315 registered road vehicles. Unfortunately, the Statistical Office of the Republic of Slovenia did not collect data about what powered them before 2014, and there is not yet any data for 2015. However, in 2014, around 53% of registered road vehicles were using gasoline, around 47% of registered road vehicles used diesel, and only 0.001% of all registered road vehicles were electric vehicles. In 2014, there were 153 registered electric vehicles and 86% of those were less powerful passenger cars, [22]. As already mentioned in Chapter 5, there are already 127 public charging stations in Slovenia. The network of charging stations in Slovenia is constantly expanding. Electric charging stations are and will be part of the Slovenian power grid; therefore, they represent a consumer of electrical energy. However, even in the case of a widespread use of electric vehicles and thus an extensive network of charging stations, the latter would not represent a significant load on the Slovenian network. If there were 500,000 electric cars in Slovenia and each of them would travel 20,000 km per year, with a consumption of 100 Wh/km, we would require an average power of 114 MW to charge all the electric cars. This represents only 5% of the total energy produced, [2], [15]. As said in Chapter 6, in Slovenia there are also financial incentives for the purchase of electric cars and thus long-term sustainable mobility. Electric vehicles will still cover the current target markets. A rapid increase in the use of electric vehicles is expected when battery technologies will improve. Studies show that the market for electric vehicles will comprise 1 to 2.7 % of the total market in 2020 and 11 to 30% of the total market in 2030, [1]. For hybrid cars, the expected sale is 4% in 2020 and 5 to 20% in 2030. For buyers of electric vehicles, the economic efficiency of vehicles is most important. The price of electric vehicles will have to decrease due to the progress in the electric vehicle technology and market demand. Electric vehicles have enormous potential for the solution to the challenges of the European Commission such as climate changes, independence from fossil fuels, local air quality, and the storage of renewable energy over the smart grid, [1]. References [1] Informacija o trendih uveljavljanja baterijskih električnih vozil s predlogom ukrepa zagotavljanja infrastrukture, sistemskega okolja in vzpostavitve demonstracijskega projekta, Služba vlade Republike Slovenije za podnebne spremembe, http://www.vlada.si/fileadmin/ dokumenti/Slikce/fotoarhiv/2010/SPS/_Informacija_o_trendih_uveljavljanja_baterijskih_ elektri_350nih_vozil_.pdf (7th September 2015) [2] L. Juteršek: Smiselnost nakupa električnega avtomobila, Fakulteta za logistiko, Univerza v Mariboru, Maribor, 2012. [3] M. 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Aleksič: Kratka zgodovina električnega vozila, http://www.mladina.si/45593/kratka_zgo-dovina_elektricnega_avtomobila/ (7th September 2015) [5] History of electric vehicle, Wikipedia, https://en.wikipedia.org/wiki/History_of_the_elec-tric_vehicle (7th September 2015) [6] Zakon o dopolnitvah zakona o davku od dohodkov pravnih oseb - pojasnilo oziroma definicija hibridnega in električnega pogona vozila, Davčna uprava Republike Slovenije, http:// www.durs.gov.si/si/davki_predpisi_in_pojasnila/davek_od_dohodkov_pravnih_oseb_po-jasnila/davcne_olajsave/zakon_o_dopolnitvah_zakona_o_davku_od_dohodkov_prav-nih_oseb_pojasnilo_oziroma_definicija_hibridnega_in_elektricnega_pogona_vozila/ (7th September 2015) [7] E-mobilnost, Elektro Maribor, http://www.elektro-maribor.si/index.php/obnovljivi-viri/66--e-mobil (7th September 2015) [8] R.Ball,Na.Keers,M.Alexander,E.Bower:Mobileenergyresourcesingridsofelectricity,http:// www.transport-research.info/Upload/Documents/201402/20140203_154622_76425_De-liverable_2.1_Modelling_Electric_Storage_devices_for_Electric_Vehicles.pdf (7th September 2015) [9] Program ukrepov za zagotavljanje infrastrukture in sistemskega okolja za vstop baterijskih električnih vozil na slovenski trg, Služba vlade Republike Slovenije za podnebne spremembe, http://www.arhiv.svps.gov.si/fileadmin/svps.gov.si/pageuploads/EBV_Program_ukre-pov.pdf (7th September 2015) [10] G. 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No material can be published without written permission of the publisher. Chapter examples: 1 MAIN CHAPTER (Arial bold, 12pt, after paragraph 6pt space) 1.1 Section (Arial bold, 11pt, after paragraph 6pt space) 1.1.1 Sub-section (Arial bold, 10pt, after paragraph 6pt space) Example of Equation (lined 2 cm from left margin, equation number in normal brackets (section. equation number), lined right margin, paragraph space 6pt before in after line): Equation (1.1) Tables should have a legend that includes the title of the table at the top of the table. Each table should be cited in the text. Table legend example: Table 1: Name of the table (centred, on top of the table) Figures and images should be labelled sequentially numbered (Arabic numbers) and cited in the text - Fig.1 or Figure 1. The legend should be below the image, picture, photo or drawing. Figure legend example: Figure 1: Name of the figure (centred, on bottom of figure, photo, or drawing) References [1] N. Surname: Title, Publisher or Journal Title, Vol., Iss., p.p., Year of Publication Examples: [2] J. Usenik: Mathematical model of the power supply system control, Journal of Energy Technology, Vol. 2, Iss. 3, p.p. 29 - 46, 2009 [3] J.J. DiStefano, A.R. Stubberud, I.J.Williams: Theory and Problems of Feedback and Control Systems, McGraw-Hill Book Company, 1987 Example of reference-1 citation: In text [1], text continue. Nomenclature (Symbols) (Symbol meaning) t time ISSN 1855-5748 771855 574008 D ГО —t (П у t го о 3" D о o ю у < о о о r-f о сг го 2 о у o сг о Ö) о о t у o D ГО ю у t го о 3" D о o ю у 9 9771855574008