Aalborg Universitet
Vulnerability Evaluation of Power System Integrated with Large-scale Distributed
Generation Based on Complex Network Theory
Liu, Leo; Xu, Quan; Chen, Zhe; Bak, Claus Leth
Published in:
Proceedings of the 47th International Universities Power Engineering Conference (UPEC 2012)
DOI (link to publication from Publisher):
10.1109/UPEC.2012.6398605
Publication date:
2012
Document Version
Early version, also known as pre-print
Link to publication from Aalborg University
Citation for published version (APA):
Liu, L., Xu, Q., Chen, Z., & Bak, C. L. (2012). Vulnerability Evaluation of Power System Integrated with Large-
scale Distributed Generation Based on Complex Network Theory. In Proceedings of the 47th International
Universities Power Engineering Conference (UPEC 2012) IEEE Press. UPEC 2012
https://doi.org/10.1109/UPEC.2012.6398605
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Vulnerability Evaluation of Power System
Integrated with Large-scale Distributed Generation
Based on Complex Network Theory
Chengxi Liu
Aalborg University,
Denmark
cli@et.aau.dk
Quan Xu
Southwest Jiaotong
University, China
qxu@et.aau.dk
Zhe Chen
Aalborg University,
Denmark
zch@et.aau.dk
Claus Leth Bak
Aalborg University,
Denmark
clb@et.aau.dk
Abstract- As the most wide-area industrial network, the
power system can be modeled as a graph with edges and
vertices, which represent the lines and buses of the power grid
respectively. Further methodologies such as complex network
theory may help in identifying the vulnerability of power grid,
analyzing the contingency, preventing cascading blackouts and
so on. When power system is integrated with distributed
generation (DG), decentralized generation at distribution level
replaces some of the centralized generation at transmission level.
DG units are able to improve the reliability of the power system,
shorten the electrical distance between the sources and loads,
alleviate the long-distance large-capacity transmission, and
increase the efficiency. This paper proposes several vulnerability
indices, such as structural vulnerability index (SVI), contingency
vulnerability index (CVI) and operational vulnerability index
(OVI) to evaluate the impact of DG to power system
vulnerability. The simulation in DIgSILENT/PowerFactory is
conducted to assess the vulnerability of a 93-bus test power
system, identify the vulnerable lines and buses, evaluate the
improvement of the vulnerability index when the network is
integrated with DG units, and may further to optimize the
planning DG units in the future.
Index Terms—Complex network theory, contingency
vulnerability index, distribution generation, operational
vulnerability index, structural vulnerability index.
I. INTRODUCTION
The networks of power system, often called power grids,
have been regarded as one of the most important
infrastructures whose security should be paid more and more
concern. However, in recent years, several large blackouts
occurred in USA and Europe, which have resulted in direct
loss up to billions of dollars [1-3]. These blackouts expose the
potential problems of current analysis methods for power
systems. So far, most work of power system analysis has
focused on only one aspect of such blackouts. Although,
these approaches have made impressive advances in the
understanding of each aspect, such as voltage stability
analysis, transient stability analysis and frequency stability
analysis, it does not provide a framework for understanding
the overall phenomena. It is reasonable to go beyond these
traditional deterministic bottom-up descriptions, instead to be
in favour of statistical top-down approaches. The
vulnerability is the ability of a network continuing to provide
key services during random failures or intentional attacks.
Further technology, i.e. complex network theory provides a
feasible way to study the vulnerability of power grids, which
has drawn the link between the topological structure and the
vulnerability of networks [4].
The first systematic study about complex network theory
appeared in late 1990s, having the goal of studying the
properties of large networks that behave as complex systems
[5-8]. Complex network theory has received considerable
attention recently since the investigation of the small-world
networks [5] and the scale-free networks [7], as their
characteristics have been discovered in many real networks
including power grids. With its recent considerable progress,
complex network theory can be of interest in assessing the
vulnerability of power grids [9-14]. The concept of global
efficiency was widely used to assess the vulnerability or
locate critical components for networked infrastructures [16,
17]. Furthermore, the cascading failure model was also
directly applied to power grids analysis [18-20]. These above
studies provided a new direction for analyzing the power
grids.
Although the complex network theory has made so much
progress, few researchers have used it to explore the impact
of large-scale DG to transmission system [21]. In case of
Danish power system, a significant proportion of today’s
installed capacity is decentralized generation (about 40% of
total capacity), such as wind turbines and combined heat and
power (CHP) units, which are mostly connected to the
distribution system, as shown in Fig. 1 [22]. Further, more
onshore wind farms are expected to be connected to the
distribution system below 100kV. Compared with
conventional power grid which is only supplied by
centralized power plant (CPP), DG units mainly supply part
of the local load, while contributing much less to remote
loads.
In Section II, some principles of complex network theory
are introduced. In Section III, three vulnerability indices i.e.
SVI, CVI and OVI are proposed to explore the DG impact on
power grid. The simulation and result of a 93-bus test system
are shown in Section IV. Section V gives concluding
comments.
Fig. 1. Production capacities at each voltage level in western Danish power
system [22].
II. PRINCIPLE OF COMPLEX NETWORK THEORY
The complex network theory has gained wide acceptance
and has been successfully applied in analysis of power
systems. In the complex network theory, each bus of the
power system which may be a power source or a power sink,
can be modelled as a vertex (or node), and each transmission
line and transformer can be modelled as an edge (or line), in
which power flow may be transmitted between its terminal
buses in the forward or reverse direction.
So the power grid can be abstracted as a directed and
weighted graph Y = {B, L, W} where B (dim{B}=N
B
) is the
set of vertices (or nodes) and L (dim{L}=N
L
) is the set of
edges (or lines) with an associated set of weight W. Each
vertex B
i
can be identified by i, and each edge L
ij
represents a
connection going from vertex i to vertex j with associated
weight w
ij
[23].
The following efficiency index (EI) has been widely
applied to evaluate the transmission efficiency of a power
grid [24].
,
11
gl
iV jV
gl ij
EI
nn d
∈∈
=
∑
(1)
where n
g
and n
l
represent the number of the generator and
the number of load respectively, V
g
and V
l
are the sets of
generators and the sets of loads respectively, the geodesic
distance
(, )
ij g l
diV jV∈∈represents the least number of
transmission lines in the shortest transmission path between a
specific generator G
i
and a specific load L
j
. The distribution
of geodesic distance is usually used to measure the network
connectivity. The lower number of d
ij
means the lower
distance or closer connectivity between sources and loads,
which implies the higher efficiency of the power grid.
III.
VULNERABILITY ASSESSMENT INDICES
A.
Equivalent Impedance Between Generation and Load
In the basic circuit theory, the node current equation is used
to compute the voltages of the nodes and the currents of
branches in a network.
=ZI V
(2)
where
Z is the node impedance matrix. It can be written in
the extended form as given in (3).
11 12 1 1 1
21 22 2 2 2
12
z
n
n
nn nnn n
z
zI V
z
zzIV
z
zzIV
⎡
⎤⎡ ⎤ ⎡ ⎤
⎢
⎥⎢ ⎥ ⎢ ⎥
⎢
⎥⎢ ⎥ ⎢ ⎥
=
⎢
⎥⎢ ⎥ ⎢ ⎥
⎢
⎥⎢ ⎥ ⎢ ⎥
⎣
⎦⎣ ⎦ ⎣ ⎦
"
"
## ## #
"
(3)
The equivalent impedance Z
eqij
between generation bus i
and load bus j can represent the difficulty in transmitting a
unit current from bus i and bus j, as shown in Fig. 2.
1
i
I =
1
j
I =−
1
i
I =
1
j
I =−
Fig. 2. Equivalent impedance in the equivalent electrical circuit.
Assuming a unit current is injected at bus i and withdrawn
at bus j, while no other current is injected or withdrawn at
other buses.
1
i
I =
and 1
j
I =− (4)
j
0
1
0
1
0
ii ij i
ji jj
z
zV
z
zV
⎡
⎤⎡ ⎤ ⎡ ⎤
⎢
⎥⎢ ⎥ ⎢ ⎥
⎢
⎥⎢ ⎥ ⎢ ⎥
⎢
⎥⎢ ⎥ ⎢ ⎥
=
⎢
⎥⎢ ⎥ ⎢ ⎥
−
⎢
⎥⎢ ⎥ ⎢ ⎥
⎢
⎥⎢ ⎥ ⎢ ⎥
⎣
⎦⎣ ⎦ ⎣ ⎦
## #
"""
## #
"""
## #
(5)
So the equivalent impedance Z
eqij
can be calculated as
/1 ( ) ( ) 2
ijeq ij i j ii ij ij jj ii ij jj
Z
VVVzzzzzzz==−=−−−=−+
(6)
where z
ij
is the ith, jth element of the impedance matrix.
B.
Structural Vulnerability Index (SVI)
The efficiency index EI in (1) assumes that the electric
power is only transmitted through the shortest path, in which
ij
d does not represent the electrical characteristic. However,
this assumption may be far away from the reality in power
systems. The power flow from a specific generation at bus i
to a specific load at bus j is distributed all over various
transmission lines as determined by topology and electrical
performance in the network.
Besides, the capacity of generator and the load in (1) are
not considered, which should act as the weight of the
transmission relationship from the generation i and a load j.
From the structural point of view, the main factors of the
power system vulnerability should be based on the inherent
characteristics, such as the topological relationship, the
impedance of the transmission lines and transformers as well
as the capacity of the generators. The varying power system
operation conditions has little affect on the structural
characteristics. Therefore, this paper proposes a novel SVI for
power system with large amount of DG.
1
exp( )
gl
gi
iVjV
g
lljijeq
P
SVI
nn P Z
∈∈
=
∑
(7)
where P
gi
is the capacity of generation at node i, P
lj
is the
maximum load at node j, and Z
ijeq
is the electric distance
(equivalent impedance) between node i and node j.
Furthermore, compared with the conventional power
system, the power system integrated with large amount of DG
has relative more generation capacity in the low voltage level.
The DG mainly supports the relatively local load demand,
contributing much less for remote loads. This characteristic is
taken in to account for SVI in (7), where P
gi
/exp(Z
eqij
)
approximately represents the contribution from generation
bus i to load bus j. The contribution from the generation to
the load should exponentially decrease with the increase of
the impedance between them. When the DG and load are
connected to the same node, Z
eqij
=0, P
gi
/exp(Z
eqij
) =1, which
means that DG has the highest priority to satisfy the local
load demand.
So SVI is more effective and more accurate than EI as
defined by (1) for evaluating the changes of power system
transmission efficiency before and after integrated with DG
units. The higher SVI means the higher the transmission
efficiency, which was widely used to assess the vulnerability
or locate critical components for networked infrastructures.
C.
Contingency Vulnerability Index (CVI)
The contingency in the power system is most possibly a
fault followed by a trip of transmission line or transformer by
protection devices. When contingencies take place in the
power grid, the tripping of transmission lines or transformers
possibly result in the severe deterioration of power
transmission performance. It is because that the deletion of
the components increase the electrical distance between
sources and loads. So the SVI will likely decrease after the
removal of transmission lines or transformers.
CVI is used to evaluate the criticality of a contingency as
defined by (8), which is the reduction percentage of SVI
related to the network structure variation. Furthermore, the
decrease in the percentage of SVI in N-1 contingency can
also be used to identify the vulnerable point in the power grid.
/
0
0
100%
SVI SVI
CVI
SVI
−
=×
(8)
The higher CVI represents the contingency is more
critical or the power is more vulnerable after removal of this
component. So the operator from TSO should pay more
attention on the contingencies in this transmission line or load
and CVI also provides an index for preventing cascading trips.
D.
Operational Vulnerability Index (OVI)
OVI is the index based on the operational conditions. The
most significant impact of the DG on power grid is that to
reduce the long-distance large-capacity power transmission
thus to increase the power transmission efficiency. OVI is
proposed to evaluate the operational vulnerability, as defined
in (9).
ll
l
l
l
z
p
OVI
z
×
=
∑
∑
(9)
where p
l
represents the active power in the transmission
line l, and the impedance z
l
is the weight of line l. The smaller
the OVI implies the less amount of the long-distance large-
capacity transmission for active power and high transmission
efficiency is in the network.
IV.
SIMULATION AND RESULT
A.
Test Power System
A 93-bus test model that representing a power system
integrated with large amount of DG units is used to evaluate
the effect of the DG to power system vulnerability, as shown
in Fig. 3. The model has 124 components (edges) composed
of 112 lines and 12 transformers. Almost every node in the
distribution system underneath the transmission level is
integrated with DG units, such as PV, onshore wind farms
and CHP plants, however, 12 CPPs are connected to the
transmission system. The test system is simulated by
DIgSILENT/PowerFactory, in which the balanced positive
sequence AC load flow is calculated by adjusting the power
output of DG and CPP.
Fig. 3. The test power system.
B. Structural Vulnerability Assessment
In order to evaluate the impact of DG penetration level for
power transmission efficiency in the structural vulnerability
assessment, the share of DG output is gradually increased to
replace the generation of CPP. The load is the maximum load
of each load bus, remaining invariant before and after DG
integration.
As shown in Fig. 4, with the increase in the penetration
level of DG from 0% to 100%, the SVI linearly increases
from 0.006 to 0.021, which means that the transmission
efficiency improves with the increase of DG penetration level.
Fig. 4. The SVI with respect to the DG penetration level.
C. Contingency Vulnerability Assessment
In order to evaluate the impact of DG on the ability of
power system to resists the N-1 contingency, SVI is
calculated before and after the removal of one component
(transmission line or transformer). As shown in Fig. 5, the x-
axis shows the number of components corresponding to every
transformers and lines in Fig. 3. It is obvious that the SVI
integrated with 100% DG is much higher than that without
DG before and after N-1 contingency. Besides, the decrease
of SVI after N-1 contingency with DG is generally lower than
that of the case without DG integration. This means the
power grid is stronger with DG integration regarding N-1
contingency than that without DG. This result also testifies
that DG helps to improve the vulnerability of the power grid
after contingencies.
Fig. 5. The SVI before and after N-1 contingency.
As mentioned in the Section III, CVI can also be used to
evaluate the criticality of a contingency. From another point
of view, CVI helps to identify the vulnerable points of a
power grid. As shown in Fig. 6, without DG integration, Line
71-72, Line 42-50 and Line 16-11 are the 3 most vulnerable
components in the power grid. They have been marked in the
test power system in Fig. 3. The blue line in Fig.6 shows that
the criticality of the contingencies in these 3 lines is evidently
reduced because of DG integration, which means that the
power system is more resistant to contingencies with DG
integration.
Fig. 6. The CVI with and without DG integration.
Fig. 7 shows the descending ordering of CVI. It is
important to note here that, with DG integration, most of
components have much lower CVI, but some components
have higher CVI, which means the power grid is more
vulnerable after contingency take places at only a few points
after DG integration. The maximum values and mean values
of CVI with and without DG integration are shown in Table I.
Fig. 7. The descending sort of CVI.
TABLE I STATISTIC DATA OF CVI
CVI
Maximum CVI Average of CVI
0% DG integration
19.71%
2.49%
100 % DG integration
1.72%
0.38%
Fig. 8 shows the descending ordering of CVI for the N-1
contingency in 10 most critical components. It can be seen
that, with DG integration, the vulnerable points in the power
grid have been changed.
Fig. 8. The 10 most vulnerable components.
![Fig. 1. Production capacities at each voltage level in western Danish power system [22].](/figures/fig-1-production-capacities-at-each-voltage-level-in-western-1mraukb6.webp)
