Today’s popular commercial power system dynamics simulation (PSDS) software use explicit integration to handle the time-increment response of power system controls. The basis of this goes back to as early as 1960, when an integration technique known as the Fortran Analog Computer Equivalent, or FACE, was proposed. Explicit integration allows for simpler programming by assuming the response to stimulus can be represented in the next time step, thus avoiding iteration. However, explicit integration leads to a slight numerical error that is cumulative. This error leads to phenomena such as the “lie” and “drift” (illustrated in a case study below) encountered during dynamic simulation.
While the error was tolerable when studying transient stability problems that could be conducted within simulation times of 2-5 seconds, today’s brand of simulation problems present a concern. As transmission systems are stressed, there is an increasing need to study voltage stability. The transients in voltage stability, which may include self-adjusting load, may take anywhere from seconds to minutes to resolve, extending the simulation time and number of time steps required to conduct the analysis. In addition, the increase in induction motor and generator equipment, combined with other existing voltage-varying or fast-acting controls such as FACTS and HVDC, lead to an increased requirement for even smaller explicit time steps to assure network convergence in the simulation.
To improve PSDS, the best alternative is to use an implicit integration algorithm. the implicit method offers a “stable” numerical solution. “Stable” in this context means the simulation progresses smoothly with each time step converging within an acceptable tolerance. An implicit solution further allows adaptive stepping, wherein the size of the time step is varied in accordance with the simulated dynamic response. As the simulation approaches steady-state, the time step approaches infinity. A trigger can be set such as to automatically stop the simulation when the response has sufficiently damped. This latter is a distinct advantage that would improve dynamic simulation performance in online and batch-mode applications. Furthermore, implicit integration allows multirate simulation wherein multiples of the time step can be applied to the set of state variables in the simulation as a function of the rate of change.
In the implementation of the implicit algorithm just described, it is important to introduce a frequency filter as the time step size increases to avoid inadvertent sampling of higher frequency control response. The filter has a net effect of introducing damping to the simulation. Effective implementation of the filter would minimize damping. In future implementations, we believe it is possible to conduct a dynamic simulation focusing only on control response of tap-changers, switched shunts, AGC, time-varying load models and special protection schemes such as to replace the steady-state contingency analysis now presently performed with power flows.
In addition to the improved simulation accuracy, and being able to avoid the lie and drift phenomena, the implicit algorithm with adaptive and multirate stepping (IIAMS) requires less computation time. This is perhaps a lesser consideration with today’s breed of fast and powerful processors, but could be an important characteristic that can bring to reality real-time dynamic security assessment.
Case Study
The following case study illustrates the application of the IIAMS to power system dynamic simulation. The case study comes out
of a larger work to examine the dynamic behavior of power systems operating at the voltage ledge (see “The In-Between Voltage State,” Techblog of December 2005).
The study system is representative of today’s modern systems, that is, one with large amounts of induction generators in the form of wind farms and other renewables. The combination of initial conditions and contingency define an operating state that would be on the ledge of the P-V curve; i.e., a condition that is close to the voltage stability limit for transfers on the study system.
Simulating the contingency with PSDS using explicit integration leads us to a “lie” (see 
Figure 1 (click on the figure at right to enlarge) for a plot of terminal voltage of an induction generator). The “lie” is a term, perhaps not too commonly used, to describe a time plot of stability simulations which has become numerically unstable. It’s called the “lie” because the sudden and sharp swings in the time plot resemble the plot of nervous response of someone who has become untruthful during a lie detector test. The “lie” is typically a consequence of the network solution failing to converge at some point during a dynamic simulation. After the first instance of non-convergence, the next time steps are suspect and likely to fail convergence as well. There is no recovery other than to solve the cause of non-convergence. The “lie” is numerical rather than representative of any actual physical event on the power system. Options to resolve it include:
- decelerate the iterative network solution
- reduce the convergence tolerance
- reduce the size of the time step from the standard half cycle (about 8.3 msec for a 60 Hz system)
In the study case, the source of the lie is the fairly large components of induction 
generators whose time constants are the same
order of magnitude as the explicit time step. In addition, the combination of induction generators, taking into account the impedance of their network connections, could lead to even lower equivalent time constants. Figure 2 shows the induction generator’s terminal voltage using a quarter cycle time step. The lie has been moved back a few seconds but is still present. With an eighth of a cycle time 
step, the simulation runs far enough to allow an assessment of voltage stability (see Figure 3) before the lie shows up again.
Simulating the same contingency using IIAMS gives a similar response for terminal voltage as the explicit integration with a very
small time step. This is shown in Figure 4. The plots are nearly coincident,
and for purposes of the voltage evaluation, lead to the same assessment of operation on the voltage ledge.
However, differences can be seen when viewing plots of rotor angle at a neighboring generator, as shown in Figure 5. The swing
curve for the IIAMS starts to deviate from explicit solution with a small time step at around 5 seconds into the simulation.
This is a 
combination of “drift” in the explicit solution and damping from the high frequency filter of the IIAMS. Drift is the accumulated error as the number of time steps increase in the explicit integration method.
Timing comparisons for the various solution methods applied to the same contingency are as follows:
| Solution Method | Relative Simulation Time |
|---|---|
| Explicit Integration with Half-Cycle Time Step | 1.8 |
| Explicit Integration with Quarter-Cycle Time Step | 4.1 |
| Explicit Integration with Eighth-Cycle Time Step | 9.0 |
| Implicit Integration with Adaptive and Multirate Time Step (IIAMS) | 1.0 |
Conclusions
Improvements in dynamic simulation performance and timing are possible using an implicit integration approach with adaptive and multirate stepping (IIAMS). The characteristics of IIAMS could help avoid the instances of lie and drift that is common in power system dynamic simulation based on explicit integration. In particular, voltage stability studies and studies of systems with large induction load and generator components are more feasible with IIAMS. In addition, IIAMS may improve online and batch-mode applications by providing indicators when near steady-state conditions are reached. The IIAMS approach may be extended to simulate what is currently termed steady-state contingency analysis.
The relative timing benefits to using IIAMS may make it feasible for use in online dynamic security assessment.
References
- J. J. Sanchez-Gasca, et al, “Extended-term Dynamic Simulation Using Variable Time-Step Integration,” IEEE Computer Applications in Power, pp. 23-28, Oct. 1993.
- A. Kurita, et al, “Multiple Time Scale Power System Dynamic Simulation,” IEEE Transactions on Power System, vol. 8, Feb, 1993.
- M. L. Crow and J. G. Chen, “The Multirate Method for Simulation of Power System Dynamics,” IEEE Transactions on power Systems, vol 9, Aug. 1994.
- Jingjia Chen, Mariesa L. Crow, Badrul H. Chowdhury, Levent Acar, “An Error Analysis of the Multirate Method for Power System Transient Stability Simulation,” 2004 Power Systems Conference & Exposition (PSCE), Oct, 2004.
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