# Explicit Numerical Integrators

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## Modified Euler Method

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

Predictor stage:

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

Corrector stage:

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

## 4th-Order Runge Kutta Method

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

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