1. A Model of Power Transmission Disturbances in Simple Systems
B. A. Carreras, V. E. Lynch, and M. L. Sachtjen
Oak Ridge National Laboratory. Oak Ridge, TN 37830
Ian Dobson,
University of Wisconsin, Madison, WI
D. E. Newman,
University of Alaska, Fairbanks, AK 99775
January 2001

2. Motivation
To model electric power system blackouts in a way that incorporates long and short time scale dynamics
Focus on global behavior
Incorporation of “management” of the grid
Investigate the role of different levels of inhomogeneity
Two tasks ahead:
Building and understanding the model
Application of the model to understand power transmission grid dynamics.

3. Model
1 day loop
A detailed description of the model has been given by Ian Dobson
Secular increase on demand
Random fluctuation of loads
Upgrade of lines after blackout
Possible random outage
LP calculation
If power shed,
it is a blackout
Are any overload lines?
1 minute
loop no
Yes, test for outage
Yes
No outage
Line outage

4. Networks
We consider networks: square, hexagon, tree-like, and IEEE standard.
Each network is characterized by a number of nodes Nd and a number of connecting lines Nl
Nodes are generators and loads. Generators are given a maximum operation power, Pimax.
Lines are characterized by their impedance zl and by a power flow, Fl, the maximum power flow that they can carry, Flmax.
Tree network with 46 nodes 69 lines

5. Time evolution The system evolves to steady state.
A measure of the state of the system is the average fraction of overloads. It is defined as
200 days

6. Regimes of Operation
This model is very rich. It has several dynamical regimes depending on the parameters. We do not yet have a systematic classification of those regimes.
In a general sense, we can define two broad regimes of operation:
A SOC-like regime for high reliability of components, low daily fluctuations of loads, and high generator capability margin.
A Gaussian regime for low reliability of components, high daily fluctuations of loads, and low generator capability margin.
The boundaries also depend of details of the implementation of the dynamic rules (random fluctuations in space and time, or only in time, or by regions in space,…)

7. Regimes of Operation
Decreasing the probability of outages, p2, increases the averaged fraction of overload and brings the system closer to the critical point.
If the level of the daily fluctuations of the loads is high, the time averaged value of <M> stays below 0.7.

8. Steady State Solution
In steady state and in the Gaussian regime, the time-average fraction of overloads is close to 0.5 and a weak function of m.
The dependence of <M> on the lines is characteristic function of each network
The plot shows this function for the IEEE 118 bus network.

9. Steady State Solution
In steady state and in the SOC regime, the time-average fraction of overloads is close to 0.9.
The plot shows <M> as a function of the line for the Tree 46 network.

10. Dynamics on the Fast Time Scale
At the beginning of each day all powers loads, generators power limits, line impedance, and line power flow limits are updated following the slow time scales rules.
After, random events are allowed (random increase or decrease on demand and random failure of a line)
With this input a solutions is found. If the solution has some line overload. We assign a probability of failure to overloaded lines and iterate the solution till no new outages take place. This iteration represents the cascading events.
Measures of the size of a cascading event: total power shed, number of overloads, number of outages, duration of the event.

11. Frequency of Blackouts
The frequency of blackouts scales approximately as a –0.8 power of m – 1 for all networks considered.
Frequency
of blackouts
in the US power
grid

12. Frequency of Blackouts
The frequency of blackouts also depends on the power generator capability margin ∆P/P.
When ∆P/P is comparable to the averaged daily load fluctuation, there is a change of regime.

13. PDF of Load Shed
A measure of the blackout size is the load shed. Because the power demand is continuously increasing, we normalize the load shed to the power delivered.
The functional form of the PDF depends on the regime:
SOC-like Gaussian

14. Conclusions
This approach offers a new way of looking at the blackout dynamics by combining long time scales (governed by grid management and marked issues) and short time scales (controlled by random failures of the system).
This model shows possible operation in different regimes:
A SOC-like regime where the PDF of the normalized load shed has algebraic tail
A Gaussian regime where the PDF is essentially exponential
We are still analyzing the capabilities of the model, to move in a near future to application to realistic situations.