# Time Domain Simulations of Power Systems

In time domain simulations of power systems, a dynamic model is defined as any model that can be described by a set of differential-algebraic equations (DAE), commonly presented in the following form:

${\displaystyle {\dot {\boldsymbol {x}}}={\boldsymbol {f}}({\boldsymbol {x}},{\boldsymbol {y}},{\boldsymbol {p}},t)\,}$
${\displaystyle {\boldsymbol {0}}={\boldsymbol {g}}({\boldsymbol {x}},{\boldsymbol {y}},{\boldsymbol {p}},t)\,}$

where ${\displaystyle {\boldsymbol {x}}\,}$ is a vector of state variables

${\displaystyle {\boldsymbol {y}}\,}$ is a vector of algebraic variables
${\displaystyle {\boldsymbol {p}}\,}$ is a vector of constant or controllable parameters
${\displaystyle t\,}$ is time (an independent variable)
${\displaystyle {\boldsymbol {f}}(.)\,}$ are the differential equations
${\displaystyle {\boldsymbol {g}}(.)\,}$ are the algebraic equations