See Also
The simulation of a transient stability event involves solving a set of differential and algebraic equations. PowerWorld Simulator uses a second order Runge-Kutta integration method to simulate the differential equations. A few special topics related to the integration method are as follows.
Handling Ignored States
In a commercial software package, each model must be implemented in a generic manner which accounts for all possible configurations that model may take. Because of this, Simulator must implement the maximum number of system state variables possible for each model. The software must then determine from the input data whether or not to ignore a particular state. As an example consider the ESAC1A exciter shown in the block diagram below.
If the parameters Tc and Tb are both entered with a value of zero, this indicates that the state associated with the lead lag block will not be a dynamic state and instead will always be equal to the input value. Simulator's integration engine automatically handles these situations and the derivative of particular states will be reported as Ignored in the interface in these situations as shown in Transient Stability Data: Object Dialogs. Also the propagation of values will automatically be handled. For example, any change to the input to the Tc block will immediately propagate to the output of that block.
Sub-Interval Integration
Like most commercial transient stability software, PowerWorld Simulator uses an explicit integration approach that alternates between solving the differential equations and solving the algebraic network power balance equations. This approach can be quite useful provided there are no model dynamics with time constants faster than the transient stability time step (using ½ cycle). However for some transient stability models, such as the EXST1 exciter model, and during induction motor startup, these conditions can be violated. To avoid numerical instabilities with these models, we have implemented a technique known as sub-interval integration. When using this approach, within each transient stability time step the differential equations for some specific models are integrated using a much smaller time step with the assumption that during these shorter time steps the algebraic (network) variables remain fixed. This approach has proven to be quite successful in integrating very fast dynamics.
As an example, the two figures below show the internal sub-transient voltages for an induction motor startup for one second of simulation time. In order to show the fast voltage transients that can occur during motor startup, the first figure is integrated with a very small time step (1/20 cycle) without the subinterval integration. Note the oscillations that occur during the first 0.1 seconds, with a frequency on the order of 60 Hz. The second figure repeats the first except using the new sub-interval functionality and a time step of one cycle. Because of the longer time step, the faster oscillations are no longer visible, but the response is numerically stable and is quite similar to the case with the much smaller time step. Without subinterval integration this case is numerically unstable with a time step of 1/5 cycle. Two key advantages of the subinterval integration approach with the longer time step are 1) that network algebraic equations are evaluated much less frequently, and 2) the subinterval integration is only applied to the handful of models for which it is required.
For Sub-Interval models in PowerWorld information Go Here.