Dynamic system simulation

Charging Capacitor The capacitor is initially uncharged There is no current while switch S is open (Fig.b) If the switch is closed at t= 0 (Fig.c) the charge begins to flow, setting up a current in the circ uit, and the capacitor begins to charge Note that during charging, charges do not jump across the capacitor plates because the gap between the plates represents an open circuit The charge is transferred between each plate and its connecting wire due to E by the battery As the plates become charged, the potential difference across the capacitor increases Once the maximum charged is reached, the current in the circuit is zero

Charging Capacitor (2) Apply Kirchhoff’s loop rule to the circuit after the switch is closed Note that q and I are instantaneous values that depend on time At the instant the switch is closed (t = 0) the charge on the capacitor is zero. The initial current At this time, the potential difference from the battery terminals appears entirely across the resistor When the charge of capacitor is maximum Q, The charge stop flowing and the current stop flowing as well. The V battery appears entirely across the capacitor

Charging Capacitor (3) The current is, substitute to voltage equation The equation is called Ordinary Differential Equation (ODE) How to solve this equation? Solve mean we can express the equation into q(t)=….

Solution of ODE Using Deterministic Approach Using Numerical approach: 1.Euler’s method 2.Heun’s method 3.Predictor-corrector method 4.Runge-kutta method 5.Etc.

Deterministic Approach The current is, substitute to voltage equation Integrating this expression we can write this expression as

Deterministic Approach If you integrate to obtain the solution, then you use exact/deterministic method. However in practical use, we often cannot integrate the function directly. The numerical approach is often preferable.

Numerical approach

Numerical approach (2)

Numerical approach (3)

Solution in Matlab Using ODE solver (m-file) Using Simulink

State space of charging capacitor

State space in practical use In practical use, the A matrix consists of many states space Simulating the power system is just solving the differential equation of system states and (sometimes) algebraic equation related to load flow. Normally we use states space in power system simulation such as rotor speed, rotor angle, Flux-linkage change, etc.

State space in practical use (2) Example: state space of synchronous generator with PSS

Order greater than 1 (n>1) Suppose second order (n=2) equation We need to write second order equation into n order first order differential equation These equations can be solved simultaneously Homework 1: how to solve this equation for a=b=c=1 using Matlab (use function: ode45)? With all initial states are zero

Transformer Simulation Equivalent circuit

Transformer Simulation (2) Voltage Equation The flux linkage per second Mutual flux linkage

Transformer Simulation (3) The current can be expressed as Eqn is now Collecting mutual flux linkage

Transformer Simulation (4) Define Eqn 4.33 can be expressed as

Transformer Simulation (5) The flux linkage in integral form

Transformer Simulation (6)

Implementation in Simulink Homework 2: Build this block in Simulink with all initial values of flux linkage are zero

Rules for student Maksimal terlambat 20 min Tidak boleh titip absen Tidak boleh menggunakan barang elektronik kec berhubungan dengan kegiatan perkuliahan

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