Structural Analysis of Electrical Networks

1. Structural Analysis of Electrical Networks Jiangzhuo Chen Joint work with Karla Atkins, V. S. Anil Kumar, and Achla Marathe Network Dynamics and Simulation Science Lab Virginia Bioinformatics Institute Virginia Tech Network Dynamics and Simulation Science Laboratory

2. 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

3. 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

4. Topology of A Real Power Grid Network Dynamics and Simulation Science Laboratory

5. Topology of Different Networks Real grid Complete graph Binary tree Ring Network Dynamics and Simulation Science Laboratory

6. 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

7. Summary of Grids in Our Experiments Network Dynamics and Simulation Science Laboratory

8. 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

9. Max Component Size: Greedy Node Attack Binary tree Network Dynamics and Simulation Science Laboratory

10. Max Component Size: Greedy Node Attack Binary tree Network Dynamics and Simulation Science Laboratory

11. Max Component Size: Greedy Node Attack Binary tree Network Dynamics and Simulation Science Laboratory

12. Max Component Size: Greedy Node Attack Network Dynamics and Simulation Science Laboratory

13. 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

14. Number of Components: Greedy Node Attack Network Dynamics and Simulation Science Laboratory

15. 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

16. 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

17. 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

18. 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

19. 1 1 generator load serving node 1 1 Flow capacity = 2 Flow Vulnerability Network Dynamics and Simulation Science Laboratory

20. 1 1 generator load serving node 1 1 Flow capacity = 2 generator load serving node 1 1 Flow capacity = 1 Flow Vulnerability Network Dynamics and Simulation Science Laboratory

21. 1 1 generator load serving node 1 1 Flow capacity = 2 generator load serving node 1 1 Flow capacity = 1 Flow Vulnerability Flow vulnerability with one node deletion = 50% Network Dynamics and Simulation Science Laboratory

22. 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

23. 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

24. 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

25. 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. Network Dynamics and Simulation Science Laboratory