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 53706
D. E. Newman,
University of Alaska, Fairbanks, AK 99775
January 2001
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.
Model
•
1 day loop
Secular increase on demand
Random fluctuation of loads
Upgrade of lines after blackout
Possible random outage
A detailed description
of the model has been
given by Ian Dobson
LP calculation
If power shed,
it is a blackout
no
No outage
Are any overload lines?
1 minute
loop
Yes, test for outage
Line outage
Yes
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
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
Fij  Power flow between
70
60
50
40
30
20
10
0
2 104
4 104 6 104
Time (days)
8 104
20
15
10
5
6.06 10
4
4
6.07 10
Time (days)
200 days
Total overloads
1
0.9
0.8
0.7
0.6
0.5
0.4
0 100
nodes i and j
Total overloads
<M>
Fij
1
M 

Number lines Lines Fijmax
0
6.08 104
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,…)
Regimes of Operation
•
Decreasing the probability of outages, p2, increases the
averaged fraction of overload and brings the system closer to
the critical point.
1
If the level of the
daily fluctuations of
the loads is high, the
time averaged value
of <M> stays below
0.7.
0.8
<M>
•
0.6
0.4
0.2
0
10-3
No flutuation of loads
Maximum fluctuation 30%
10-2
p2
10-1
100
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.
Fraction of Overload
1
m.
m.
m.
0.8
0.6
0.4
0.2
0
0
40
80
120
Lines
160
Steady State Solution
•
In steady state and in the SOC regime, the time-average fraction of
overloads is close to 0.9.
1
The plot shows
<M> as a function
of the line for the
Tree 46 network.
0.6
M
•
0.8
0.4
5%
10%
15%
20%
0.2
0
0
10 20 30 40 50 60 70
Lines
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.
Frequency of Blackouts
The frequency of blackouts scales approximately as a –0.8 power
of m – 1 for all networks considered.
10 0
Blackout frequency
•
f = 0.0019 ( m.
Frequency
of blackouts
in the US power
grid
10 -1
10 -2
10 -3
10 -4
Tree 190
Tree 94
Tree 46
Hexagon 61
Square 49
IEEE 188
10 -3
10 -2
m
10 -1
10 0
Frequency of Blackouts
The frequency of blackouts also depends on the power generator
capability margin ∆P/P.
P

P
P
max
j
jG
  Pi
 Pi
iL
iL
•
When ∆P/P is
comparable to the
averaged
daily
load fluctuation,
there is a change
of regime.
100
Frequency of Blackout
•
Maximum daily
fluctuation of the loads
20%
30%
40%
10-1
f = 0.03039 * (² P/P)-0.578
10-2
10-2
10-1
² P/P
100
PDF of Load Shed
103
Probability distribution
•
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
Probability distribution
•
x-0.5
102
10
x-1.5
1
100
10-1
10-4
-3
10
-2
10
-1
10
0
10
Load shed/Power delivered
102
PDF = 26.124 e-25.92(Ls/Pd)
101
100
Tree 190
10-1
10-4
10-3
10-2
10-1
Load shed/Power delivered
100
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.