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Comparative Study of Collocation Method-Spline, Finite Difference and Finite Element Method for Solving Flow of Electricity Ina Cable of Transmission Lines

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Comparative Study of Collocation Method-Spline, Finite Difference and Finite Element Method for Solving Flow of Electricity Ina Cable of Transmission Lines Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 COMPARATIVE STUDY OF COLLOCATION METHOD-SPLINE, FINITE DIFFERENCE AND FINITE ELEMENT METHOD FOR SOLVING FLOW OF ELECTRICITY INA CABLE OF TRANSMISSION LINES 2 Dr A.K.Pathak1 Dr. H.D.Doctor Lecturer, Prof & Head Of H&S.S. Department Mathematics Department S.T.B.S.College Of Diploma Engg. Veer Narmad South Gujarat Uni. Surat, India Surat, India [email protected] [email protected] Abstract The present work describes spline collocation ,finite difference and finite element approximation for solving flow of electricity in a cable of transmission lines problem. The mathematical problem gives rise to solve a linear partial differential equation which is of parabolic type. The method involves the solution of algebraic linear equation which can be written in the matrix form is main advantage. The solution are obtained by these two methods and compared with analytic solution to demonstrate the justification and simplicity of the spline approximation converge to finite difference approximation as well as analytic solution. Keywords: Spline collocation, Finite difference, finite element method, Partial differential equation. 1. INTRODUCTION: The present work describes spline collocation, finite difference and finite element method for solving flow of electricity in a cable of transmission lines problem. The mathematical problem gives rise to solve a linear partial differential equation which is of parabolic type. The method involves the solution of algebraic linear equation which can be written in the matrix form is main advantage. The solution are obtained by these three methods and compared with analytic solution to demonstrate the justification and simplicity of the spline approximation converge to finite difference, finite element approximation as well as analytic solution. 2. FINITE DIFFERENCE METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATION: 2.1 Explicit Method: Consider parabolic partial differential equation having one space variable. 2 ut = c u xx ; 0<x<1, t>0 (2.1.1) It can be replaced by finite difference approximation which can written as ui, j+1 = ui, j + r[ui-1, j - 2ui, j + ui+1, j ] . (2.1.2) where r= c 2 Dt/ h2 It is called the EXPLICIT formula it can be shown that this formula is valid only for 0< r £ 1/2. 2.2 Implicit Method: Equation (1.2.1.1) can be replaced by finite difference approximation which can written as 2(1+r)ui, j+1 - r(ui-1, j+1 + ui+1, j+1 ) = 2(1- r)ui, j + r(ui-1, j + ui+1, j ) (2.2.1) where r= c 2 Dt / h2 It is called the IMPLICIT formula it can be shown that this formula is valid for all finite value of r. 36 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 3 FINITE ELEMENT METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATION: Basic concept of finite element method is that to develop a finite element formulation of equation (2.1.1) First we have to know how to formulate the problem as a variational problem and then derive an algebraic system of equation associated with the variational problem and we obtain the equation . [ M 1 ]{u} +[K] {u}= {F} ( 3.1) and usingq family of approximation we can write equation (1.3.1) as set of linear algebraic equation in matrix form L1 L1 [K]{U}n+1 = {F} ( 3.2) L1 where [K]=[ M 1 ] +qDt[K] L1 1 {F}= ([M]- (1-q)Dt[K]){U}n + Dt(q{F}n+1 + (1-q ){F}n ) (3.3) L1 L Here D t is time step and [K]and{F} requires the knowledge of initial and boundary condition and by solving equation (1.3.2) we get {U}n+1 . 4 SPLINE COLLOCATION METHOD: For solving linear and nonlinear differential equation with the help of the numerical methods required much computational work and time. Bickley [1968] Suggested the method of spline function containing truncated power polynomials to solve a linear boundary value problem Ahlberg et al [1967] used cardinal splines for solving differential equations. Doctor et al [1983,1984] have shown that the method of spline collocation is quite useful for the solution of physical phenomena which give rise to linear parabolic one dimensional partial differential equation. The method demonstrates the use of spline function. Spline functions are piecewise polynomial and their successive derivatives are continuous. They were used for data interpolation initially. In 1967, Blue [1969] suggested the use of spline function for the solution of B.V.P. y” = f(x,y,y’) (4.1) with boundary conditions G1 [Y(0),Y’(0)] G2 [Y(1),y’(1)] (4.2) The following recurrence relations were used. 2 S”( xi-1 )+4s”( xi )+s”( xi+1 ) = 6/ h (f( xi-1 )-2f( xi )+f( xi+1 ) (4.3) 4.1 Spline Formula To Solve Parabolic Liner Partial Differential Equation Having One Space Variable: Consider parabolic partial differential equation having one space variable. 2 ut = c u xx 0<x<1, t>0 (4.1.1) with dirichlilet boundary conditions u (0,1) = 0 u(1,t)=0 (4.1.1a) Along with initial condition u(x,0)= f(x) :0<x<1 (4.1.1b) We discretizing left side of equation (4.1.1) by forward difference formula and replacing right side by the second derivative s”( xi ) at j th level explicit scheme in finite difference . We get 2 (ui, j+1 - ui, j )/ c Dt = S"i, j (4.1.2) 37 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 whereS"i, j denotes s”( xi )at j th level Substitute values of S"i, j from (4.1.2) in to equation (4.3) and get ui-1, j+1 + 4 ui, j+1 + ui+1, j+1 = (1+6r) ui-1, j +(4-12r) ui, j +(1+6r) (4.1.3) Where r = c 2 Dt / h 2 .This equation known as cubic Spline Explicit formula to solve equation (4.1.1) The finite difference replacement of (4..1.1) corresponding to implicit scheme is ui, j+1 - ui, j =½(S" + S" ) (4.1.4) c 2 Dt i, j i, j+1 whereS"i, j , S"i, j+1 denote second derivatives at x= xi at the time level j and j+1 respectively. We use the relationship (4.3) and rewrite it in following forms. 2 s"i-1, j +s"i, j +s"i+1, j = (6 / h )(ui-1, j - 2ui, j - ui+1, j ) (4.1.5) 2 s"i-1, j+1 +s"i, j+1 +s"i+1, j+1 = (6 / h )(ui-1, j+1 - 2ui, j+1 - ui+1, j+1 ) (4.1.6) i=1, 2, 3... n-1 putting the values of S"i, j+1 etc from (4.1.4) in to (4.1.5) We get which gives (1- 3r)ui-1, j+1 + (4 + 6r)ui, j+1 + (1- 3r)ui+1, j+1 = (1+ 3r)ui-1, j + (4 - 6r)ui, j + (1+ 3r)ui+1, j (4.1.7) Where r= c 2 Dt / h2 , i=1, 2, 3... (n-1) Above equation (4.1.7) is known as cubic spline implicit formula to solve equation (4.1.1) The convergence and stability of these methods totally depend upon the value of r. For explicit scheme we have to choose the value of r such that 0< r £1/2. Implicit scheme has advantage that there no limitation on the value of r. 5 FLOW OF ELECTRICCITY IN A CABLE OF TRANSMISSION LINES: This is the study of the derivation of partial differential equations from physical principals, we assume the flow of electricity in a cable of transmission line. We assume the cable to be imperfectly insulated so that there is both capacitance and current leakage to ground. Figure (3.1 a) shows such a cable with the electromotive source x=0 and load x=1. Figure (5.1a) Figure (5.1b) Here the current returns through the earth. Consider an element PQ of the cable with P at a distance x from the source .e(x, t) the potential, i (x, t) the current at P at any time t. Let R denote the resistance. L the inductance, G the conductance (leak ness) and C the capacitance per unit length of the cable. Let Dx be the length of the element PQ and i+ D i , e+ D e be the current and the electromotive force, respectively, at Q. Figure (3.1b) gives the electrical circuit for the 38 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 element PQ. Since the drop in voltage from P to Q is due to the resistance RDx and inductance L Dx , We have - De = (R Dx )i + (L Dx ) ¶i / ¶t (5.1) Again, since the drop in the current - Di is due to the conductance G Dx and capacitance C Dx ,we have - Di = (G Dx ) e + (C Dx ) ¶e/ ¶t (5.2) Dividing equation (5.1) and (5.2) by Dx and taking the limit as Dx ® 0, We get ¶e / ¶x + L ( ¶i / ¶t ) + R i = 0 (5.3) and ¶i / ¶x + c ¶e/ ¶t + Ge = 0 (5.4) Differentiating equation (5.3) with respect to x and equation (5.4) with respect to t, we get ¶ 2 e / ¶ x 2 + L ( ¶ 2 i / ¶ x ¶ t ) + R ¶ i / ¶ x = 0 and C ( ¶ 2 i / ¶ x ¶ t ) + C ( ¶ 2 e / ¶ x 2 ) + ¶ e / ¶ x = 0 If we eliminate the term ¶ 2i / ¶x¶t between these two equation and then substitute for ¶i / ¶t from equation (5.3) we obtained ¶ 2e / ¶x 2 = Lc(¶ 2e / ¶t 2 + (Rc + LG)¶e / ¶t + RGe (5.5) ¶ 2i / ¶x 2 = Lc¶ 2i / ¶t 2 + (Rc + LG)¶i / ¶t + RGi (5.6) Equation (5.5) and (5.6) are known as the telephone equations.
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