# Towards Better Dynamic Simulation

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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