Modeling and Simulation of Vehicular Power Systems - webfiles its ...

⎡ I ( t ) ⎤⎢ ⎥ =⎣0⎦⎡v( t ) ⎤× ⎢ ⎥⎣ y ( t ) ⎦G[ v ( t ), v ( t − h ), I ( t ), I ( t − h ), y ( t ), y ( t − h ), t ]⎡b1− ⎢⎣b2[ v ( t ), v ( t − h ), I ( t ), I ( t − h ), y ( t ), y ( t − h ), t ][ ] ⎥ ⎤v ( t ), v ( t − h ), i ( t ), i ( t − h ), y ( t ), y ( t − h ), t ⎦( 3-1)where I is a vector of through variables, V is a vector of across variables, h is time step ofthe simulation is a vector of state variables, G is Jacobin matrix and b 1 and b 2 areconstants depending generally on past history values of through, across variables andinternal states and values of these quantities at time instant. The full description of RCFand the modeling of some components like DC machines, DC/DC converters and vehicledynamics were given in [8].Once all the components are modeled and connected to each other according to theexisting constraints between the modules, the equations could be solved by numericalintegration methods, (rectangular, trapezoidal, second order Gears method) and the statevariables are found in each time step.3.2 Numerical Integration MethodsAs the state space dynamic equations are solved by numerical integrations there arenumerical integration methods available to solve the dynamic equations. Some numericalintegration methods like, the rectangular, the trapezoidal, Simpson’s, Rounge Kutta’s,Gear’s, backward Euler’s can be found in [6], [8].Among the mentioned methods, the trapezoidal and rectangular methods are commonlyused. In some transient simulation programs like EMTS( Electromagnetic TransientSimulation), the trapezoidal methods are used because of their merits of low distortionand absolute-stability [8]. Most of the transient simulation programs follow up the chartshown in figure 3-2 to converge to the final result.13