MODELING of POWER SYSTEM ELEMENTS by WAVELETS and TLM A Case Study-Transformer Internal Fault Study

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1 MODEING of POWER SYSEM EEMENS by WAVEES and M A ase Study-ransformer Internal Fault Study Guven Onblgn e-mal: gonblg@omu.edu.tr O. M. U, Faculty of Engneerng, Department of Electrcal & Electroncs Engneerng, 5539, Samsun, urey Oan Ozgonenel e-mal: oanoz@omu.edu.tr O. M. U, Faculty of Engneerng, Department of Electrcal & Electroncs Engneerng, 5539, Samsun, urey Vel urmenoglu e-mal: turmen67@hotmal.com Ordu Unversty, echncal and Vocatonal Hgher School Ordu, urey Key Words: Wavelets, M, transformer, nternal fault, modelng ABSRA In ths paper wavelet based transent smulaton and transmsson lne method (M) are proposed for modelng and solvng power networs. he wavelet doman equvalents of electrcal crcut components are defned for solvng the networ to fnd currents and voltages of the nodes. Durng the computer smulatons, a number of smulatons are performed to get the best level of wavelet decomposton. he modelng algorthm s then compared to another method so called M to see the ablty of solvng ntegro-dfferental equatons. MAAB s used for modelng and requred computer calculatons. I. INRODUION omputer aded smulaton for solvng power electronc crcuts taes a very mportant role and helps the user to get the best performance from the crcut. here are a lot of methods for analyzng those crcuts. Some of them can be lsted such as aplace ransform Approxmaton [], Fourer Seres Approxmaton [2], State Varable Analyss [3], Analyss by Neglectng Harmoncs [4], Analyss usng Boolean Algebra [5], Averagng Approxmaton [6], omplex Varable Analyss [7], Parc vector analyss [8], Swtchng Functon Analyss [9]. Wavelet based smulaton, usng steady state equatons, was developed by Gandell [0] and u []. On the other hand, u [2] and Zheng [3] had remarable contrbutons at power system transent analyss. All these methods have been approved n the soluton of power electroncs crcuts snce for a long tme. However, the nown methods summarzed above use steady state equatons of the crcut and hence, numercal soluton of ntegral and dervatve equatons taes tme for the convergence. It s possble that a coeffcent matrx can be used n wavelet doman nstead of numercal ntegral and dervatve processes. As t s proposed n ths wor, crcut equatons are turned to ther basc algebrac forms by replacng ntegral and dervatve matrces. he transform coeffcent matrces and crcut component models have got sparse matrx characterstc. herefore, the proposed algorthm s as fast as the conventonal methods. Haar wavelets are used for wavelet based modelng and computer smulaton snce they have orthogonal property and ther nverse transform matrces can easly be obtaned. he M method was frst developed n early 970s for modelng two-dmensonal feld problems. Snce then t has been extended to cover three dmensonal problems and crcut smulatons. For crcut smulaton, the M method can be used to develop a dscrete crcut model drectly from the system wthout settng up any ntegrodfferental equatons. he M model algorthm s dscrete n nature and deally suted for mplementaton on computers [4], [5]. In ths paper, the two proposed algorthms are compared accordng to ther computatonal effcency. It s seen that both technques are smlar to each other by defnng crcut components accordng to based on modelng aspect. II. MODEING EHNIQUES 2. Wavelet Based near Resstor Model he V-I characterstc of a resstor, R, s descrbed as v ( t) R * ( t) n contnuous tme doman. et WV and WI be the DW coeffcent vectors of the voltage V and the current I, set the coarsest resoluton level as zero, t has WV DW * V [ cv.. WV WI DW * I [ c.. WI s s and WV s calculated by usng Eq. (2). ] ] () WV R* U * WI (2) Where U s an dentty matrx c and cv are the D components of voltage and current waveforms, WI s and

2 WV s represent the wavelets coeffcents. he equvalent crcut of resstor n wavelet doman s shown n Fg.. v ( t) R * ( t) v ( t) ( d / dt) WD * WI * * (0) * WV 0 WV R* U * WI Fgure. Resstor model n wavelet doman [2], [3] Resstor, R, s modeled as constant varable n M and ts value s added nto the system mpedance such as Z ( R + Z + Z Zc). system sw Wavelet Based near Inductor Model he V-I characterstc of an nductor,, s descrbed n d contnuous tme doman s v ( t). et us suppose dt that the mnmum tme nterval n the fnest resoluton level s the samplng cycle,. For the transent study n the dscrete tme nterval (0 -?), the followng equaton (3) s wrtten. v (3) * D * I * (0) * v0 Where (0) s the ntal value of the crcut current, and v [ ] (4) (5) D.... WV * WD * WI * (0) * WV 0 Fgure 2. ransent model of nductor n wavelet doman [2], [3] Equvalent crcut of nductor n wavelet doman s shown n Fg. 2. As t s shown n Fg. 2, equvalent crcut s formed by a wavelet doman mpedance and a voltage source whch s determned by the ntal value of current through the nductor. M based modelng s smlar to the wavelet based one summarzed above. he basc M technque models lnear reactve components as transmsson lne segments (called stubs). he stub model representng the nductor s termnated by a short crcut because, to emphasze nductve behavor, current and, hence, storage n the magnetc feld must be maxmzed. he M model for a capactor s a stub wth ts far end and open crcut. It emphaszes voltage dfferences, storage n the electrc feld and, hence, manly capactve behavor. o model an nductor usng a transmsson lne stub s gven n Fg. 3. he stub model representng the nductor s termnated by a short crcut to emphasze nductve behavor and current. he varables and t are treated as nductor current and tme. d / dt s then equal to the voltage V across the nductor. et WV 0 be the DW coeffcents vector of v 0 wavelet doman followng equaton s obtaned., then n WV (6) * WD * WI * (0)* WV 0 where WD s called the transent dfferental operator. *WD represents the wavelet doman mpedance of an nductor. Fgure 3. a) nductance, b) stub model of nductance, c) M equvalent crcut Any dfferental terms n the form of d / dt can be V replaced by the dscrete transform Z. + 2., where Z 2., t s the tme step and pulse. It s assumed that t taes one tme step V s the ncdent t for the

3 pulse to mae a round-trp to travel to one end and be reflected bac as the ncdent pulse n the next tme step. In summary, d V ( t). (7) dt Accordng to the M theory, the voltage across the port of the transmsson s also equal to the sum of the ncdent pulse V and reflected pulsev r. hus, at any tme step, n r nv n V Z V (8a ) n n V + (8b) where the subscrpt n donates the n th tme step. For a short-crcuted transmsson lne, the pulse s reflected and nverted when t encounters the shortcrcuted end. herefore, the reflecton coeffcent s - as the reflected pulse becomes the ncdent pulse n the next tme step, from Eq. (8b) r nv n+v (9) r nv nv It s clearly seen from Eq. (6) and Eq. (8a and 8b) that the component of to * WD * WI n wavelet doman s smlar 2. * Z * n M doman whle the component of * (0) * WV 0 s smlar to successvely. nv n r nv V, 2.3 Wavelet Based apactor Model he V-I characterstc of a capactor,, n contnuous tme doman s represented by the followng equaton. t ( t) v( t) v(0) + dt (0) 0 For a fnte length sgnal, t yelds to Eq. () V v( 0) * V0 + IN * I () where V0[..]. (2)... IN IN s called the transent dfferental operator. hus, n wavelet doman Eq. () s expressed as, WV v( 0) * WV 0 + WIN * WI (3) where WIN s wavelet doman representaton of IN, and WV 0 [ j ] (4) j 2 -j/2 s a constant that s determned by the fnest resoluton level J. Moreover, WIN / forms the transent mpedance of capactor n wavelet doman. Fgure 4. ransent model of a capactance n wavelet doman [2], [3]. Fg. 4 shows the equvalent crcut of a capactor n wavelet doman. In transent analyss, the capactor s modeled as transent mpedance and a voltage source that s calculated from the ntal value of the capactor voltage. In a smlar manner capactance can be smulated n M. Fg. 5 shows a capactance n tme doman, ts M representaton, and also ts hevenn equvalent. Fgure 5. A capactance, a) n tme doman, b) ts M representaton, and c) ts hevenn equvalent. he characterstc mpedance, Z Z c where t s the samplng nterval. 2 apactor voltage s then calculated as /*WIN *WI, s calculated as V 2 V + I. Z (5) hs also equals to the sum of ncdent and reflected r voltages, V V + V, by defnton. At the next v(0)*wv 0 smulaton tme step, ncdent voltage s updated by usng Eq. (6). r + V V V V (6)

4 Fgure 6. he proposed smulaton algorthm It s clearly seen from Eq. (3) and Eq. (5 and 6) that the component of n wavelet doman s * WIN * WI smlar to * Z * n M doman whle the 2 component of v ( 0) * WV 0 s smlar to r + V V V V, successvely. he proposed computer smulaton algorthm s descrbed as follows.. he netlst s composed accordng to node numbers of the model crcut. 2. Wavelet transform matrx s bult (no requred for M). 3. rcut component model n wavelet doman s formed (system matrx calculated for M). 4. he crcut s solved. Snce the crcut components are modeled n the form of mpedance or admttance, the crcut can be solved by mpedance or admttance matrx method (M currents and voltages calculated). 5. he desred outcomes are obtaned. he proposed procedure s shown n Fg. 6.

5 III. APPIAION At ths case a turn to turn fault s modeled by M. et us suppose that /0 porton of the prmary wndng (of a sngle phase transformer) has a turn to turn fault. he hevenn equvalent of the M system s shown n Fg. 7. Resstance of prmary wndng s the summaton of R + R + and nductance of the prmary wndng s 22 R33 the summaton of As n secton, before the fault, the system equaton s as the same wth (8a, n matrx form) but the only dfference s that load s modeled as a seres R and crcut. However durng the fault, the followng equatons are derved and modeled. Fgure 9. Prmary current of the transformer durng nternal short crcut n M based modelng Fgure 7. A urn to turn fault at prmary sde (a M example). Fg. 8 shows prmary current of the transformer n wavelet based soluton whereas Fg. 9 shows the current n M based soluton. Fgure 8. Prmary current of the transformer durng nternal short crcut n wavelet based modelng IV ONUSION In ths study, the use of wavelets for the transent analyss of power electronc crcuts and smlartes wth the M method are presented. Frst, crcut models are extracted by usng Haar wavelet functon at the desred order. hese models have two common types, transent and steady state. One of the man advantages of the proposed technques, wavelets and M, s dscrete n nature and can be adopted n nonlnear components easly. Results are compared wth the MAAB solutons and observed that the outcomes are consstent wth each other at the level of 5 or above. Hgher level analyss taes long tme to converge. In ths study, optmum decomposton level s set 9. It has been observed that M based soluton of modelng s faster than wavelet based modelng due to requrng less comp utatonal manpulaton. For a future wor, a hysteress model of transformer can be used to get more accurate smulatons both n M and wavelet based modelng. Besdes dfferent mother wavelet famles can be tred for computer smulatons. REFERENES. aeuch,. J. (968), heory of SR rcut and Applcaton to Motor ontrol, oyo Electrcal Engneerng ollege Press, oyo 2. nos Jacovdes, Analyss of Inducton Motor Drves wth a Non-snusodal Supply Voltage Usng Fourer Analyss, IEEE rans. On Ind. App., A-9, No. 6,74-7(973) 3. P. H. Naya and R.G. Hoft, omputer-aded Steadystate Analyss of hyrstor D Drves on Wea Power Systems, IEEE on. Rec. 976 th Annu. Meet. Ind. App. Soc., P.. Krause and.a. po, Analyss and Smplfed Representatons of a Rectfer-Inverter Inducton Motor Drve, IEEE rans. on Power App. And Sys., PAS-88, No.5, (969) 5. M. Rammorty and b. Hango, Applcaton of State- Space echnques to Steady-State Analyss of yrstor

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