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© Science and Research Pioneers Institute
7th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE-2017),
7-8 September 2017, Tbilisi, Georgia
A model for cascading failures in power system
based on complex network
Ronak Mirzadaei
Mohammad khansari
Faculty of New Sciences and Technologies
University of Tehran
Tehran, Iran
E-mail: r.mirzadaei@ut.ac.ir
Faculty of New Sciences and Technologies
University of Tehran
Tehran, Iran
E-mail: m.khansari@ut.ac.ir
Abstract— Cascading failure is a well know phenomenon in
real systems that could cause to blackout and collapse in
many infrastructure networks like power system. Although
most failures may occur locally but sometimes they can
propagate and can largely affect the entire network and lead
to global collapse and blackout. So according to the
importance of network stability, in such condition quick
analysis in order to forecast and prevent cascading failure is
so important. In conventional models, by applying
differential equations, power system dynamic has been
studied. In this paper a novel model based on “Centrality
measures” and “Complex network” for analyzing cascading
failure in power system has been presented. The main
advantage of this model is removing complicated and timeconsuming differential equation calculation and therefore
decreasing time of calculations compared to conventional
models. For validation, the proposed model for IEEE 30 bus
has been implemented and result comparison with power
system simulator software shows that the proposed model
corresponds to conventional models based on components’
removal order and largest connected component.
component, it leads to component outage and this process
would continue.
Keywords- power grid; cascading failures; complex network;
centrality measures
I.
INTRODUCTION
For many realistic infrastructure networks, e.g. the
power grid and Internet, the occurrence of some disastrous
failures may be due to the faults at only one or a few
network components. Because of the coupling property of
components in the network, these faults could be extended
to a large scale. In this case, many components lose their
functionalities consequently and may lead to entire
collapse of the network. Such phenomenon is known as
the network cascading failure or ‘‘network avalanche’’
[1]. On the other hand, the occurrence of faults leads to
removal or exit of a component from the system and thus
the load of that component should be handled by other
components. If extra load exceeds capacity of a
Because of important role of cascading failures in
infrastructure networks, a lot of studies have been done for
prediction of this process [1]. Thus, much research efforts
on the investigation of cascading failures in complex
networks have been carried out from different aspects.
Most of previous researches have followed two
general approaches: in the first approach, cascading
failures are merely studied based on the relation between
system components, but due the improvement and
development of networks, a general and integrated view
towards system is needed. The second approach, using
graph concepts and definitions, is merely focused on
network structural characteristics and doesn’t consider
dynamics between components. Recently some models
have been developed based on these two approaches by
combining dynamics of infrastructure network, like load
flow, trip of components, and graph theory. In fact,
combination of these two approaches contributes to
coming up with a more realistic model of cascading
failure [1-8].
In general, four parameters are defined by most of
models for cascading failure: loading, capacities of
components, probability of trip and load redistribution.
DC or AC load flow methods have been used for loading
by models which act based on circuit rules and focus on
interactions of components. These models use real
capacities of components and load flow for load
redistribution. In other cascading failure models, instead
of real loading and capacities, degree and betweenness
centrality measures are used for loading. In all models
there are two definite or probable removal of components
for probability of trip parameter [1-8].
In this paper, in addition to physical quantities for
parameters like loading and capacity, network centrality
measures are used for load redistribution in order to model
cascading failure process. As a result, the proposed
method eliminates complex and time-consuming
SRPioneers
© Science and Research Pioneers Institute
7th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE-2017),
7-8 September 2017, Tbilisi, Georgia
calculations of electrical DC or AC load flow. In addition,
weight, capacity and load for both nodes and edges have
been considered for investigating cascading failure
process at both node and edge levels of network.
The paper is organized as follows. In part II necessary
concepts are defined and in the next part, the proposed
model is provided. Results and comparison are done based
on components’ removal order and largest connected
component in part IV. Finally conclusion and future works
are presented in part V.
II.
DIFINITIONS
A. Power grid
Power grid is one of the most important infrastructure
networks and it has three types of substations:
1) Generator substations or plants: which are
manufacturer and source of power.
2) Transmission substations: which receive power
and transmit it to other substations.
3) Distribution substations: which distribute power in
local distribution networks [9].
Power grid is shown as a single line diagram which
consists of buses and transmission lines. Buses are
substations which have acted as nodes in graph, and
transmission lines that connect different substations at
different voltage levels (high voltage, middle voltage and
low voltage) corresponding to edges [9]. In general,
distance is considered as the cost of physical quantities’
transmission between two nodes. Electric distance
between nodes i and j is equal to total impedance of
transmission lines between these two nodes and is defined
as follows.
Zi =
j
Uij
Iij
network is being laid or when a power network is
undergoing expansion.
C. Complex Network
Most of systems and surrounding phenomena have
complex structures and relations, and for understanding,
analysis and forecasting future behavior, recognition of
their corresponding network is needed. Graph theory is
used for network mathematical display in such a way that
nodes, components of network and edges show relations
between components. Thus each network corresponds
with a graph and graph features show network
characteristics. If there is a dynamic in network, it would
become a weighted network [11].
D. Shortest path in weighted graph
Several paths may exist between each of two desired
nodes in graph. These paths consist of consecutive edges.
The shortest path is the one that passes through edges with
the least weight [11].
E. Centrality
Centrality is a measure which determines the
importance of edge or node in graph. Centrality has
different definition based on network type and function.
One type of centrality is betweenness centrality which is
defined in next part [11].
F. Betweenness Centrality in weighted graph
Betweenness centrality of a node (edge) equals the
number of shortest paths through which node (edge)
passes. Node (edge) betweenness centrality is calculated
by the following equation.
n
=
Bi
st
(1)
j
In (1), U i is the potential difference between two
j
nodes, and I i is the flowed current between nodes i and j.
Inversion of impedance is called admittance or
conductivity between two nodes [10]. Power consists of
active and reactive power. Active power is generated in
plant and consumed in loads. This kind of power is
generated and consumed completely. Reactive power is
generated by plant as well but is saved as electric and
magnetic fields in inductor and capacitor of network [9].
B. Load flow
The starting point of any analysis of power system will
be the computation of complex voltages at all the busses.
Once the complex voltages have been computed, the
power coming out of a bus and the power flowing in all
the transmission lines can be calculated. Load flow
analysis is a computational tool for this purpose. Load
flow is normally used in planning studies when a power
1
=
nsti , nsti 
∑
0
In (2), if node (edge) i is in the shortest path from node
i
s to node t, nst would have the value of one, otherwise it
would be zero. This measure is provided for weighted and
un-weighted networks. In such a way that the number of
possible shortest paths would be obtained from that node
or edge in weighted network [11].
G. Largest connected component(LCC)
This measure illustrates the number of nodes in the
largest connected component of the total number of nodes
in the network. In fact the less this amount, the less the
efficiency and robustness of network [11].
III.
THE PROPOSED MODEL FOR CASCADING
FAILUERS
On one hand, in the proposed model physical
quantities of network like load of components, their
capacities and impedance have been considered. On the
other hand, centrality measures for load redistribution
have been applied for facilitating calculations. In this
model, in contrast to previous ones, cascading failure
(2)
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© Science and Research Pioneers Institute
7th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE-2017),
7-8 September 2017, Tbilisi, Georgia
process has been considered at both node and edge levels.
In this way, the proposed model is made of two parts
(cascading failure on nodes and cascading failure on
edges). Due to the nature of power grid, network
dynamics can contribute to the two-way transmission
between nodes. Therefore, the network is considered as
undirected.
nodes i and j. According to (3), the weight of edges
forming the second shortest path becomes more than the
weight of edges forming the first shortest path and
therefore the former has less important role in load
redistribution. This process applies to the next shortest
paths, and as the same way edges in the next paths would
have more weight.
Most of cascading failures happen in power grid
because of the following reasons: Short circuit of
transmission lines, fault at power stations and damage to
the transmission lines or substations, overloading of
electricity mains, bad function of protective device and
bad climate that could be modeled as the form of edge
removal [12].
After updating weights of all edges in the diffusion
network, edge betweenness centrality (weighted) is
calculated for each edge of this network. This centrality
shows the number of shortest paths between each pair of
nodes (based on weight). As a result, there will be
multiple paths in the diffusion network in the way that in
every path, each edge has its own betweenness centrality.
Edge betweenness centrality is an estimation of the load
that each edge can pass through. Because each path is
composed of some edges in series, and in serial path,
current flows for all edges are equal and also the
betweenness centrality would define the portion of current
through which the edge could pass so the minimum
betweenness centrality in each path is considered for all of
the edges in that path.
In power grid nodes and edges intrinsically are
weighted components, but in previous models just one of
them has considered as weighted. In the proposed model,
both nodes and edges have been considered weighted and
cascading failure process at both node and edge level of
network has been studied. An important advantage of this
model is usage of betweenness centrality instead of load
flow.
A. Edge Level
In this part, network edges would be weighted with
impedance. Thus basic network would be an undirected
weighted network. Initial load of edges has been obtained
by calculating load flow in ETAP software (for simulating
power grid). Edge capacity illustrates tolerance of edge
load that is inversely related to impedance (and directly
related to admittance). Above data are used in edge level
of proposed model.
At first, network works normally. At the end, a
weighted network including edges with definite load and
capacity is obtained. Failure process starts with removal of
an edge. In order to study the failure process, a new
network is being created from original network based on
shortest paths between two points of removed edge and it
would be considered as diffusion network. These shortest
paths is being chosen based on the weight of their edges
so that the first shortest path has the lowest weight.
Electrical current in power grid desires to flow through
shortest paths, for this reason diffusion network consists of
shortest paths between two points of deleted edge. In fact,
calculations are done only on a part of this network which
is effective in supplying electrical load of removed edge.
(Calculation is done in a smaller graph.) First shortest path
is more important in diffusion network due to its less
weight. That is why edges’ weight in this path is set to
have a greater role in load redistribution. According to the
above, the new weight of each edge can be obtained from
the following equation.
wij = wij e ( k −1)W
k
(3)
In (3), wij is the weight of edge between nodes i and j.
The index k represents the k-shortest path between two
Next equations is used for calculating new load on
edges in the network.
I Di =
Bi
∑B
k
I deleted − edge
(4)
k
In (4), Bi , Bk and I deleted − edge are betweenness
centrality of edge i, the lowest betweenness centrality in
path k, and load of removed edge respectively. I Di is the
portion of load of edge i from extra load which is the
result of removal.
I new=
I Di + I old −i
−i
In (5), I new−i is the value of new load of edge i. With
the above operations, loads of edges in all shortest paths
between two nodes of deleted edge are updated, in other
words, load flow is done in the network. In this stage, load
of network edges should not exceed their capacity. If load
exceeds the capacity in one or more edges, that edge or the
edge with highest load will be selected and removed. After
removal of new edge, previous process would be
continued and new load for network edges will be
calculated. Process continues until load of all edges won’t
exceed their capacities.
It is essential to note that if a path is not found between
two nodes, load of removed edge will have no effect on
other edges and process will stop.
B. Node Level
At this level, type, load and capacity are dedicated to
nodes. Nodes are generally divided into two categories:
generators and loads. Load of each node is obtained from
(5)
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© Science and Research Pioneers Institute
7th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE-2017),
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total active and reactive power substation (generator or
load). Generators generate power, thus they have certain
capacity and following equation is used to calculate node
capacity.
C=
Ii × α
i
IEEE system bus 30 and its corresponding graph are
illustrated in (a) and (b) of fig.1 respectively.
The basic network (graph) is illustrated in fig.1.
(6)
In (6), Ci , I i and α are capacity of node i, load of
node i and tolerance parameter respectively (in this model
α is set to 1.1) but capacity is not considered for loads.
In this part, new parameter called free capacity of
generator substations (nodes) are defined which are
obtained from the difference between total capacity of
generators and their total load.
If an edge is removed from a network and the node on
both ends of this edge becomes isolated, more detailed
examination and calculation is needed because if this node
is a generator, the load of generator node would be
deducted from free capacity and the load of isolated node
would be removed from network. If the node is of load
type, the load would be removed from network and this
value would be added to free capacity.
In each stage of repetition of failure process (edge
removal), value of free capacity should be larger than zero
because otherwise generators cannot handle loads and
network would stop working.
IV.
(a)
Failure process is started with removal of one edge.
After the process is stopped, edges’ removal order and
largest connected component is reported according to
table I. Network edges are divided into two groups based
on vulnerability. If any of the edges in group 1 is removed,
failure cascades all over the graph and network become
unstable. If any of the edges in group 2 is removed, failure
process will be stopped and network would work
normally.
TABLE I.
SIMULATION RESULTS
The network under simulation is a standard test
network IEEE (Institute of Electrical and Electronics
Engineers) 30 bus. This network includes 30 nodes and 41
edges [13]. Nodes are generator and distribution
substations. All the impedance and transformer lines of
network are considered as edges. Each node has the
characteristics of type, load and capacity. Each edge has
the characteristics of weight, capacity and load. Cascading
failure starts with the removal of one edge. To study the
vulnerability of network edge, process is modeled by
eliminating each edge. Simultaneously with the removal
of an edge in the proposed model, the corresponding line
of ETAP software is removed and then load flow is
carried. ETAP software has been used in order to examine
and assess results. ETAP offers a suite of fully integrated
electrical engineering software solutions including arc
flash, load flow, short circuit, transient stability, relay
coordination, cable ampacity, optimal power flow, and
more. In this paper, an open source programming
language R is used for modeling cascading failure, which
includes many packages for calculating centrality.
The results of the two models are evaluated with two
criteria, edges’ removal order and largest connected
component (LCC). In this network, at first the proposed
model is being explained and then result validation with of
ETAP software will be done.
The results show that the proposed model corresponds
to conventional models based on two mentioned criteria.
(b)
Fig. 1. (a) IEEE system bus 30 .(b) its corresponding graph in R (b)
CLASIFICATION EDGES IN IEEE 30 BUS
Measure
G
r Deleted
edge
o
u
p
1
1
2
1
3
2
4
2
6
3
4
4
6
2
5
5
7
6
7
9
11
12
13
25
26
2
Other
edges
Edges’ removal order
Edge 1-3
Edge 1-2
Isolation
node 1
Isolation
node 1
Unstable
Unstable
Edge 1-3
Edge 1-2
Isolation
node 1
Unstable
Edge 2-5
Edge 2-4
Edge 1-3
Isolation
edge 1-2
Edge 1-2
Isolation
edge 1-3
Unstable
Edge 1-3
Edge 1-2
Edge 2-6
Edge 5-7
Edge 2-5
Edge 2-5
No path
No path
No path
Isolation
node 5
Isolation
edge 5-7
Isolation
node 11
Isolation
node 13
Isolation
node 26
LCC
Isolation
node 1
Isolation
node 5
Unstable
Unstable
End of process
End of process
End of process
End of process
End of process
End of process
End of process
To understand the above table, work process will be
explained with removing some edges in two models (the
proposed model and ETAP software).
29/
30
29/
30
29/
30
28/
30
28/
30
29/
30
29/
30
29/
30
28/
30
29/
30
29/
30
29/
30
30/
30
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© Science and Research Pioneers Institute
7th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE-2017),
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A. The proposed model
If edge 1-3 from group 1 is chosen and removed for
the start of cascading failure, diffusion network would
consist of shortest paths between nodes 1 and 3. New
loads of edges are calculated through edge betweenness
centrality in diffusion network. Then, loads of all edges
are compared with their capacities. In this stage, edge load
1-2 exceeds its capacity and will be removed. In result,
node 1 which is a generator type will be isolated and the
value of free capacity would become negative. Therefore,
other generators cannot supply the loads and normal
operation of network will be stopped. In this case, LCC
will be 29/30 and network will be changed according to
fig.2.
show that substation 1 (main generator) is overloaded and
network collapse is occurred according to fig.4.
Fig. 4. Network resulted from removing link 1-3 in ETAP
In ETAP software, transmission line between
substation 9 and 11 has been removed, then load flow is
performed. In result, it can be found that none of the
substations are overloaded and load of removed line is
handled by other lines. So network will operate in stable
condition according to fig.5.
Fig. 2. Graph resulted from removing edge 1-3 in R
If edge 9-11 from group 2 is removed, because there is
no path between these two nodes, diffusion network
would not be formed and load of removed edge will not be
distributed and failure process will be stopped. On the
other hand, node 11 will be isolated. This node is a load
type and removing it will keep value of free capacity
positive. In this case, LCC will be 29/30 and network will
be changed according to fig.3.
Fig. 5. Network resulted from removing link 9-11 in ETAP
The main advantage of the proposed model is reducing
time-consuming calculation of load flow since the average
time for making diffusion network and its calculation is 70
millisecond but the time of each load flow is more than
500 millisecond.
V.
Fig. 3. Graph resulted from removing edge 9-11 in R
B. Validation with ETAP
To evaluate the results in ETAP software, transmission
line between substation 1 and 3 has been removed. After
removing, load flow is performed. Results of simulation
CONCLUSION AND FUTURE WORK
In most of previous cascading failure models, just
initial loading that causes to cascading failure, based on
network centrality measures, has been studied. In these
models, cascading failure process starts only with the
removal of one node which is less probable [2, 4-6, 8].
Cascading failure usually happens in cases such as
unexpected accidents and short circuit that leads to edge
removal. In the proposed model, at first power grid is in
normal operation mode and network stability will be
examined by line outages. Therefore, in proposed model
SRPioneers
© Science and Research Pioneers Institute
7th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE-2017),
7-8 September 2017, Tbilisi, Georgia
cascading failure starts with one edge removal and
diffusion process is studied at both edge and node levels
[12]. Combining study of cascading failure at node and
edge levels simultaneously would lead us to gain better
results in comparison with previous cascading failure
models.
In some cascading failure models degree and
betweenness centrality measures had been used for
initializing capacities and component loading. But these
initialized values are differed from real values. In the
proposed model physical and real quantities have been
used for initial load and capacities and in this way,
accuracy of calculations has been increased [1-8].
Since electrical current in power grid flows through
shortest paths and considering edge betweenness centrality
which calculates shortest paths, in the proposed model
edge betweenness centrality measure in diffusion network
has been used instead of complicated and time-consuming
differential equitation calculations of electrical power
flow.
For validation results in IEEE 30 bus dataset based on
components’ removal order and largest connected
component, ETAP software has been used. It can be found
that simulation results and the proposed model results are
similar. Edges in this network are divided into two groups
based on vulnerability. If any of the edges in group 1 is
removed, failure cascades all over the graph and network
become unstable. If any of the edges in group 2 is
removed, failure process will be stopped and network
would work normally. Stability of edges in group 1 is less
than group 2, so damaging these edges in group 1 will
disturb the network and it should be protected against
various damages.
This model can be applied on other standard test
systems and real networks with defined parameters.
This model can be used for all the infrastructure
networks in which there are dynamic quantities, load and
capacity of system components which play important roles
in cascading failure. With adjusting these values and
capacities, this can contribute to design and development
of infrastructure systems and model network dynamics
(such as consecutive trip of lines).
In power systems, fluctuations from cascading failure
cause network components to have different interaction on
each other and make changes in load diffusion, frequency,
domain, and voltage phase and protection device function,
control, operator procedures and alarm and monitoring
systems [12]. As a result, other dynamics can be added to
edges and nodes characteristics in order to obtain a more
exact model of cascading failure process.
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