http://openelectrical.org/index.php?action=history&feed=atom&title=Explicit_Numerical_Integrators 2026-04-28T02:51:59Z Revision history for this page on the wiki MediaWiki 1.35.0 http://openelectrical.org/index.php?title=Explicit_Numerical_Integrators&diff=138&oldid=prev 2020-11-22T08:14:03Z

<p>Created page with &quot;== Modified Euler Method == The modified Euler (or Heun&#039;s) method is a two-stage [http://en.wikipedia.org/wiki/Predictor%E2%80%93corrector_method predictor-corrector method]:...&quot;</p> <p><b>New page</b></p><div>== Modified Euler Method ==<br /> <br /> The modified Euler (or Heun's) method is a two-stage [http://en.wikipedia.org/wiki/Predictor%E2%80%93corrector_method predictor-corrector method]:<br /> <br /> Predictor stage:<br /> : &lt;math&gt; \boldsymbol{\tilde{x}}(t + \Delta t) = \boldsymbol{x}(t) + \Delta t \boldsymbol{f}(\boldsymbol{x}(t), t) \, &lt;/math&gt;<br /> <br /> Corrector stage:<br /> : &lt;math&gt; \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] \, &lt;/math&gt;<br /> <br /> == 4th-Order Runge Kutta Method ==<br /> <br /> The 4th-order Runge-Kutta algorithm is one of the most popular numerical integration methods for power systems. <br /> <br /> : &lt;math&gt;\boldsymbol{k}_{1} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t), t) \, &lt;/math&gt;<br /> <br /> : &lt;math&gt;\boldsymbol{k}_{2} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \frac{\boldsymbol{k}_{1}}{2}, t + \frac{\Delta t}{2}) \, &lt;/math&gt;<br /> <br /> : &lt;math&gt;\boldsymbol{k}_{3} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \frac{\boldsymbol{k}_{2}}{2}, t + \frac{\Delta t}{2}) \, &lt;/math&gt;<br /> <br /> : &lt;math&gt;\boldsymbol{k}_{4} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \boldsymbol{k}_{3}, t + \Delta t) \, &lt;/math&gt;<br /> <br /> : &lt;math&gt; \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) \, &lt;/math&gt;<br /> <br /> [[Category:Modelling / Analysis]]</div>

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