Explicit Numerical Integrators

Modified Euler Method

The modified Euler (or Heun's) method is a two-stage predictor-corrector method:

Predictor stage:

${\displaystyle {\boldsymbol {\tilde {x}}}(t+\Delta t)={\boldsymbol {x}}(t)+\Delta t{\boldsymbol {f}}({\boldsymbol {x}}(t),t)\,}$

Corrector stage:

${\displaystyle {\boldsymbol {x}}(t+\Delta t)={\boldsymbol {x}}(t)+{\frac {\Delta t}{2}}\left[{\boldsymbol {f}}({\boldsymbol {x}}(t),t)+{\boldsymbol {f}}({\boldsymbol {\tilde {x}}}(t+\Delta t),t)\right]\,}$

4th-Order Runge Kutta Method

The 4th-order Runge-Kutta algorithm is one of the most popular numerical integration methods for power systems.

${\displaystyle {\boldsymbol {k}}_{1}=\Delta t{\boldsymbol {f}}({\boldsymbol {x}}(t),t)\,}$
${\displaystyle {\boldsymbol {k}}_{2}=\Delta t{\boldsymbol {f}}({\boldsymbol {x}}(t)+{\frac {{\boldsymbol {k}}_{1}}{2}},t+{\frac {\Delta t}{2}})\,}$
${\displaystyle {\boldsymbol {k}}_{3}=\Delta t{\boldsymbol {f}}({\boldsymbol {x}}(t)+{\frac {{\boldsymbol {k}}_{2}}{2}},t+{\frac {\Delta t}{2}})\,}$
${\displaystyle {\boldsymbol {k}}_{4}=\Delta t{\boldsymbol {f}}({\boldsymbol {x}}(t)+{\boldsymbol {k}}_{3},t+\Delta t)\,}$
${\displaystyle {\boldsymbol {x}}(t+\Delta t)={\boldsymbol {x}}(t)+{\frac {1}{6}}\left({\boldsymbol {k}}_{1}+2{\boldsymbol {k}}_{2}+2{\boldsymbol {k}}_{3}+{\boldsymbol {k}}_{4}\right)\,}$