Explicit Numerical Integrators

Modified Euler Method

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

Predictor stage:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tilde{x}}(t + \Delta t) = \boldsymbol{x}(t) + \Delta t \boldsymbol{f}(\boldsymbol{x}(t), t) \, }

Corrector stage:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{k}_{1} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t), t) \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{k}_{2} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \frac{\boldsymbol{k}_{1}}{2}, t + \frac{\Delta t}{2}) \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{k}_{3} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \frac{\boldsymbol{k}_{2}}{2}, t + \frac{\Delta t}{2}) \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{k}_{4} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \boldsymbol{k}_{3}, t + \Delta t) \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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) \, }