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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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) \, }