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lecture22 - Solving Non-linear ODE Objective Time domain...

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Power Systems I Solving Non-linear ODE Objective Time domain solution of a system of differential equations Given a function or a system of functions: f ( x ) or F(x) Seek a time domain solution x ( t ) or x( t ) which satisfy f ( x ) or F(x) Integration of the differential equations Linear equations - Closed form solutions: Laplace transforms Non-linear equations - Frequently no closed form solutions: Numerical integration Taylor Series Euler Runga-Kutta
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Power Systems I Solving Non-linear ODE Taylor Series Consider Then by expansion ʹஒ ʹஒ = ʹஒ ʹஒ ʹஒ ʹஒ = ʹஒ ʹஒ = ʹஒ + + ʹஒ ʹஒ ʹஒ + ʹஒ ʹஒ + ʹஒ + = + t c c x k x t c t c c x k x t c t c t c k x k x x h x h x h x h x h t x iv 3 2 1 2 3 2 1 1 3 3 2 2 1 2 1 6 2 3 2 ! 4 ! 3 ! 2 ) ( ) ( x f dt dx =
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Power Systems I Solving Non-linear ODE Euler’s Method First term of the Taylor’s series only is used ( ) ( ) k k k k k k k k x kh x t x h k c h k c kh c c x c h x t c t c t c c x c h x x h t t t c t c t c c x c x t c t c t c c x c x x y h t e x h x h t x = + = + = + = = ʹஒ = + + + + + + ʹஒ = + ʹஒ + = + + + ) ( ) ˆ ( 0 ) ( ) , ( ) ( 3 3 30 2 2 20 10 00 01 3 30 2 20 10 00 01 1 1 3 30 2 20 10 00 01 3 30 2 20 10 00 01
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Power Systems I Euler’s Method x 0 x 1 x 2 t 0 t 1 t 2 Δ t Δ t t x
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