.lOllRNAI, OF ,S( lENC E & TEC HNOLtKiY * No 79B - 2010 ASSESSMENT Ol EM:CrR()rVlA(;Ni;H( TRANSIENTS IN POWER SYSTEMS li\ \VAVELi;r-!E(IINIQUE-BASEI) ALGORITHM (ilAl TllUAl" Dfl'A TRLN KY TIlUA'f WAVl l l ! I S DL DANH (ilA QUA DO DIEN lU' IKEN Hi: IIIONC Dll.'N \guy en Nhan Bon, Quyen 11 uy Anh i nivcrsitv ol Technical T.ducation of llo Chi Minh Citv Nguyen Huu Phuc llo Chi Minli University of Technology ABSTRACT In this paper we use a method to simulate the transient in power system based on wavelets technique This approach deals with nonlinear power circuit in a time domain with high numerical accuracy, unlike the method of working in the frequency domain Furthermore, the wavelet property of localization in both the time and frequency domains makes a uniform approximation possible, which is generally not found in time-marching methods Numerical results show the good features of the approach in power system transient simulation Keywords: wavelet approximation, multiresolution, fast discrete wavelet transform, electromagnetic transient, power system transient TOM TAT Bai bao tnnh bay mdt phwang phdp di danh gia qua dien tir dwa tren ky thuat wavelets giai he phwang tnnh phi tuyin ciia he thing dien Phwang phap mai giai quyit tinh phi tuyin cua mang dien miin thdi gian vdi chinh xac cao, khdng giong phuang phdp tinh toan miin tin so Tinh chit wavelets dinh vi miin thai gian va tin so tao nen sw xip xl khdng ddng nhit khdng giong cac phwang phap tinh toan miin thai gian trwdc day Cac kit qua tinh todn md phdng hira hen cdc dp dung tinh todn he thing dien support and higher fVcqucncy components are onlv used near the singularities, thus prov iding a uniform error distribution for the wavelet approximations, a fast wavelet collocation mcthod(l\\'CM) is developed which works in the time domain and provides a uniform error distribution The method is extremelv powerflil in treating singularitv because of the wavelet propertv localization, in both frequency and time domains and has a fourth-order convergence rate [2] Although the complete mathematical discussion of the wavelet collocation method for partial differential equations (PDE's) was given in [2] and the method's general properties for ODE's were presented in [1 I INTRODUCTION The existing numerical methods for the circuit simulation can be classified into two classes: time-marching and ("fequency-domain methods The time-marching method is the most popular one because in the time domain circuit nonlinearity can be easily handled and design engineers often need to see the signal waveform directly However, the method suffers from the difficulty in effectivciv handling the singularity which often develops in high-speed circuits and the problem of nonuniform error distribution The motivation for using wavelets for the simulation of fast transient circuits relies on the following observation Wavelet basis functions will provide a nice adaptive approach, which maintains a uniform approximation to resolve the singularifies of a function [2] The key feature in the wavelet approximation is the fact that wavelet basis functions with compact This paper will focus on the FWCM for linear ODE's The extension of the FWCM to nonlinear systems is discussed in a sequel paper [3], due to limitations in space here Therefore, we shall emphasize potenfial application of 76 JOURNAL OF SCIENCE & TECHNOLOGY * No 79B - 2010 electromagnetic in power system In Section 11 we first introduce the basic concept of wavelets and function expansion by using wavelets We then develop an interpolation scheme for solving ordinary differential equations In Section III we compare the FWCM with the other simulation methods, based on both theoretic analysis and numerical experiments Then we develop a further application on power system In Section I"V conclusion is given I'„={//,(/).;/:(/).//,(L-/).;7,{/,-0 i/''o.-i(').i//,H,(0.1//o,|(0. -.V/„„„ iC).!//,,,,,,, : ( ' ) (1) V^.-i(').V/,„(0.i//,,(/). '.V/o,„, ,(/).i/V», '-,iL-t) The (PoA') = where interior scaling 9i'-l^lO J '-\ h.I Jl.V.l|.|\.l] And (18) at the boUom of this page where / M=l-A h ^h + fi, I/O (19) h - ( ^ - ' ' =p „ 3//, -''• d = -c h.*\' lis -\ III ASSESSMENT AND APPLICATION IN POWER SYSTEMS = ^'(.v(/ + A/),y(/ + A/)) -1- i*\ (21) 'AV'" •V' F[=S/^f G\=V,gv^ = V t; (matrices depending on the algebraic Table I Comparing time consuming For a generic time i, and assumed time step Ar one has to solve the follow ing O = /;,(.V(/ + A/),.V(/ + A / ) / ( ) X • ] ' fhis session demonstrates a simple RLC circuit, fhen this approach is applied in electric jiower svstem lie shows the solution b; adding the second-level wavelets in the time range from 2.0 to 6.0 (not the entire time interval) to increase the accuraev of the solution obtained, by using onlv up to level-one wavelets It is seen that accuracy in the timrange concerned has been increased (^•v.,)" 3/ /;: and state Jacobian matrices of the system) The loop stops if the variable increment is below a certain fixed tolerance GQ or if the maximum number o^ iteration is reached In the latter cas the time step \/ is reduced and the NewtonRaphson technique repeated again Figure •! depicts the block diagram of the time domain based on FWXAl d b I _y where F.'= \ ,./ • • • = y '' > U'\ Ai-r\ n-l " I " , ": h - / • • _Av'_ //: = e, -I r Av' Methodology (20) Wavelets technique Euler method (Runge - Kutta) Wavelets technique (calculating on power system) Euler method (Runge - Kutta) (calculating on power system) where yfve/?""),77 being the number of buses in the network, are algrebaic variables, g(ge/?"")are the algebraic equations for the active and the reactive power balance at each bus (the algebraic equation g internally stored 80 Time of Calculating (s)_ 0.311^ 0.5600 0.0900 2.5440 JOURNAL OF SCIENCE & TECHNOLOGY * No 79B - 2010 h ' Ktul uluilAii \ \ H t r l ( ( uppf-oKlmHllnii ' itilh icuhnE riinillKii, rli* tn\r - 'ihiaitrc'ix-iitia rut*>-».inViru ^1.0.1) ; I ; i \ v\_y I H Fig Voltages at Nodes of Svstem 9-bus IEEE Fig Solution of Wavelet Technicpie The time consuming of the based-wavelet technique is fast Result from table shows that time consuming of wavelet technique is short than that of other method Moreover level of j changes easily to adapt specified cases ' r ; aaa =( >.: _ — \i • Fig Voltages at Nodes of Svstem ~r t5 ' - r0 -i:£} J : 3] ^: - r ^ " - - 14-bus IEEE ; : ~JZ':.i ^H^s^sMN^—™:! •0 ^ — " : •s —t T m h l ; i , -r I \ i - , I i i c ; e a ; ; i "Hi,» Fig Voltages at Nodes of 30-bus IEEE System Fig 2: 14-bus IEEE System Based on this algorithm on fig 4,5,6 9bus IEEE system, 14-bus IEEE system, 30-bus IEEE systems are tested These results show voltages at buses with fauh at a bus on each system A three phase fault occure at / = s The fauh is then cleared by opening a line at / = 1.07 s Finally the line is recluse at / = s The simulation results are considered as the same as those of MATLAB program or ATP-EMTP program It proved that the it is accurate and practical wavelet-based methodology Wavelet technique improves accuracy and time consuming in power system transient and creates calculation grid (node calculation matrix) on the Sobolev space associated with accuracy and stability We presented in this paper a methodology for transient power system simulation The basic properties of the wavelet method for solving differential equations are demonstrated The wavelet can be extended to the high nonlinear case Furthermore, the wavelet-based method can be applied to solve the nonlinear partial differential equations by expanding the space variables CONCLUSION 81 .lOUKNAL OF S( IKNC E & Tl,( HNOLOGY * No 79B - 2010 SUirl Si.1 up (k'liviilivc ciUMlinii 111 pinvci s>-.lcni Yes Solve by \V iiMjiclIcchniquc-HiisLiI Method Solution for Transient ol" Time Domain in l \ m c r Ss^tLMll Yes Output inal I'xam lind Figure Proposed Inised-li dvelets- Technique Algorithm method becomes mature The wavelet-based methodology has demonstrated a promising direction in the development of high-speed transient simulation of power system We are investigating the use of this method to develop some practical transient simulations in power systems ni The paper is an idea to use the wavelet technique for power system transient The experimental data are presented simply for demonstrating the feasibilitv and properties of the wavelet-based method, not for showing its efficiency in practical power system transient simulations Many implementation details and more benchmark testing are needed before the 82 JOURNAL OF SCIENCE & TECHNOLOGY * No 79B - 2010 REFERENCES J Mahseredjian; Computation of Power System Transients : Overview and Challenges, IEEE PES General Meeting 2007 June 24-28 Tampa, USA W Cai and J Z Wang; "Adaptive multi-resolution collocation methods for initial boundary value problems of nonlinear PDE's," SIAM J.Numer AnaL Vol 33, no 3, pp 937-970, 1996 D Zhou, W Cai, and W Zhang; "An adaptive wavelet method for nonlinear circuit simulation," IEEE Trans Computer-AidedDesign, vol 46 Aug 1999 Phuc H.N Bon N.N Anh Q.H.; " Wavclet-fourier- based artificial intelligent technology in assessment and classification power quality disturbances" pp495-500, 1SEE2007 in October 2007 Hochiminh Cit> University of Technology Phuc H.N., Bon N.N Khanh Q.T.; " Discrete wavelets transform technique application in identification of power quality disturbances"" pp 159-164, ISEE2005 in October 2005, Hochiminh Citv Universitv of Technolouv Author s address: Nguven Nhan Bon -Tel.: (+84)903.871.443, Email: nnbon@hcmute.edu.vn; nnbon2009(rt}gmail.corn Universitv of Technical Education of Ho Chi Minh City 01, Vo Van Nsan Str., Thu Due District, Ho Chi Minh City 83 ... any function wavelets where indexes j and k respectively, represent the operations of dilation and translation The dilation generates highfrequency wavelets and the translation moves wavelets to... collocation points, to determine the expansion coefficients Unlike the wavelets used in the image processing, we need to construct the wavelets which are not only able to represent a signal in the interior... electromagnetic in power system In Section 11 we first introduce the basic concept of wavelets and function expansion by using wavelets We then develop an interpolation scheme for solving ordinary differential