Network Dynamics and Simulation Science Laboratory

Structural Analysis of Electrical Networks

Jiangzhuo Chen

Joint work with Karla Atkins, V. S. Anil Kumar, and Achla Marathe

Network Dynamics and Simulation Science LabVirginia Bioinformatics Institute

Virginia Tech

Network Dynamics and Simulation Science Laboratory

Vulnerable Electrical Networks

• 2003 northeast blackout and 2003 Italy blackout (and many others): cascading failure of large power grids from local failure.

• Damage to other critical infrastructures: transportation, communication, financial networks, etc.

• It is important to study robustness of power grids, identify potential points of vulnerabilities, and build redundancies to make the infrastructure more robust.

Network Dynamics and Simulation Science Laboratory

Methodology: Structural Analysis

• Instead of simulations, we study vulnerability of an electrical network by structural analysis.

• Structural properties of a power grid: degree distribution, shortest path distribution, flow capacity, etc.

• Generic viewpoint to investigate the fundamental causes of vulnerabilities.

• Structural analysis of real power grids also helps building more realistic synthetic grids for simulations.

• This method is not limited to power grids. It can be applied to other infrastructure networks.

Network Dynamics and Simulation Science Laboratory

Topology of A Real Power Grid

Network Dynamics and Simulation Science Laboratory

Topology of Different Networks

Real grid

Ring

Complete graph

Binary tree

Network Dynamics and Simulation Science Laboratory

Experiment Set-up

• Data– A real grid of a large US city– A random grid: G(n,m) graph– Synthetic grids: standard IEEE test cases

• Compute basic structural measures– degree distribution– shortest path distribution– size of minimum dominating set (subset of nodes that have

all other nodes as neighbors)• Robustness under attacks: random or targeted

– Removal of random nodes or random transmission lines– Removal of high degree nodes or high capacity links

Network Dynamics and Simulation Science Laboratory

Summary of Grids in Our Experiments

Grid Lines Nodes Avg. Deg. Load Serving Nodes Generators Dom. Set

Real 776 662 2.34 328 41 45

Random 776 662 2.34 328 41 80.55

14-bus 20 14 2.86 11 5 3

30-bus 41 30 2.73 21 6 5

57-bus 80 57 2.81 42 7 10

118-bus 186 118 3.15 91 54 10

145-bus 453 145 6.25 51 50 8

162-bus 284 162 3.51 89 17 18

300-bus 411 300 2.74 188 69 25

Network Dynamics and Simulation Science Laboratory

Degree and Shortest Path Distributions

• Higher avg. degree (145-bus) means larger link redundancy, i.e., more robust to link failure.

• Shortest path distribution reflects how far apart generators and load serving nodes are. One expects it to become flatter with larger mean as network size increases. Real, 162-bus, and 145-bus grids seem to have shorter shortest paths, thus more robust.

Network Dynamics and Simulation Science Laboratory

Max Component Size: Greedy Node Attack

Binary tree

Network Dynamics and Simulation Science Laboratory

Max Component Size: Greedy Node Attack

Binary tree

Network Dynamics and Simulation Science Laboratory

Max Component Size: Greedy Node Attack

Binary tree

Network Dynamics and Simulation Science Laboratory

Max Component Size: Greedy Node Attack

Network Dynamics and Simulation Science Laboratory

Max Component Size: Greedy Node Attack

• Real grid seems most vulnerable: removal of 10% high degree nodes max component size decreases by 90%.

• 162-bus network has a component with half of its nodes, even if 20% of nodes are removed.

Network Dynamics and Simulation Science Laboratory

Number of Components: Greedy Node Attack

Network Dynamics and Simulation Science Laboratory

Number of Components: Greedy Node Attack

• Again, real grid breaks into (normalized) more pieces than other grids (except random grid, which starts with about 72 components), when high degree nodes are removed.

• 162-bus network still seems to be most robust.

Network Dynamics and Simulation Science Laboratory

Max Component Size: Random Node Attack

• Real grid is most vulnerable even with random node deletion.• Curve of 162-bus grid looks linear: decrease in max component

size mainly comes directly from deletion of nodes.

Network Dynamics and Simulation Science Laboratory

Number of Components: Random Node Attack

• Real grid continues to be most vulnerable to random node removal; while 162-bus grid still seems to be robust.

Network Dynamics and Simulation Science Laboratory

Flow Vulnerability

• Flow capacity of a power grid is the maximum (normal) flow that can be sent from generators to load serving nodes, subject to constraints on transmission line capacity, generator capacity, and substation capacity.

• We have capacity information only for real grid (thus also random grid) and 162-bus grid.

• Flow vulnerability of a power grid is the percentage decrease in its flow capacity, when nodes or links are removed.

Network Dynamics and Simulation Science Laboratory

Flow Vulnerability

generator1

1

1

1load serving node

Flow capacity = 2

Network Dynamics and Simulation Science Laboratory

Flow Vulnerability

generator1

1

1

1load serving node

Flow capacity = 2

generator1 1

Flow capacity = 1

load serving node

Network Dynamics and Simulation Science Laboratory

Flow Vulnerability

generator1

1

1

1load serving node

Flow capacity = 2

generator1 1

Flow capacity = 1

load serving node

Flow vulnerability with one node deletion = 50%

Network Dynamics and Simulation Science Laboratory

Flow Vulnerability: Node Attacks

• All seem to be robust to random node attack.• Random grid is robust to targeted node attack, because it has no particular structure (small flow

capacity in base case).• Surprisingly, 162-bus grid is most vulnerable to high degree node deletion. Probably although it

breaks into small number of pieces with large sizes, generators and load serving nodes are separated and flow cannot be transmitted.

Network Dynamics and Simulation Science Laboratory

Flow Vulnerability: Link Attacks

• Real grid is very robust to link attacks. Its flow capacity is intact even if 40 high capacity links are removed.

• 162-bus grid is vulnerable to greedy link deletion, too. It is possibly out of the same reason: many generator nodes or load serving nodes are isolated.

Network Dynamics and Simulation Science Laboratory

Comparison with Other Infrastructure Networks

• Wireless network seems to be very robust to targeted node attacks. An almost linear curve means that decrease of its max component size is almost completely due to removal of nodes.

• Power grid and transportation network seem to have similar vulnerability under greedy node attacks.

Network Dynamics and Simulation Science Laboratory

Conclusions and Future Work

• Grids are robust or vulnerable if different structural measures are used. We need a systematic quantification of vulnerability, which probably integrates different measures.

• Extend the study to other real electrical networks.

• Combine structural analysis with simulations.