How Does Spare Capacity and Topology Affect the Security of Resource Networks?

1. How Does Spare Capacity and Topology Affect the Security of Resource Networks? Qian-Chuan Zhao zhaoqc@tsinghua.edu.cn Qing-Shan Jia jiaqs@tsinghua.edu.cn Yang Cao caoy02@mails.tsinghua.edu.cn

2. Outline • How much spare capacity is necessary? • Models: Balanced partition & node-weighted graph • The necessary condition on spare capacity • Critical value of the linearly increasing speed • How topology affects security: An upper bound of resource network security • Motivation • Upper bounds of security • Identification of critical transmission lines

3. Motivation • Real life examples: • Aug. 14, 2003 blackout of the North American electric power network in USA and Canada [CanadaUSPSOTF2004] • Summer of 2005, Hurricanes Katrina & Rita, record prices in the US oil market • Jan. 2006, gas pricing dispute between Russia & Ukraine, shortage in Ukraine and throughout western Europe • …

8. Hurricane Rita

11. Gas Pricing Dispute

12. Motivation • Necessary condition: balance between generation & consumption • Why do local contingencies lead to global performance degradation? • Practical solution: Spare capacity (e.g., backup generators, load shedding), costly, growing network capacity • Interesting question: How much spare capacity is necessary for the security of resource network?

13. Related work in Physics • Network structure + node/edge dynamics  network complexity • Network resilience w.r.t. remove of nodes or edges • Results • Many real life networks & artificial network, robust to random failure, fragile to intentional attacks • Critical components  cascading failure

14. Difference in the model

15. Balanced partition Tolerance = 0 (exact balance), network (a) has 1 BP, (b) has none. Tolerance = 1, (a) & (b) has BPs.

16. The question • Fix the topology • Given the separation contingency, network capacity (i.e., total power consumption) & tolerance (i.e., spare capacity) • Consider all generation patterns and load patterns • Measure the BP probability • When the network capacity, how fast to increase the tolerance s.t. the BP prob.>0?

17. Experiment I: Complete Graph

18. Experiment I: Result T(P)=0.5Pb

19. Theoretical Result I • Theorem 1: When the tolerance increases slower than a linear function of the growing network capacity, then the BP probability will converge to zero. • Proof (see [ZhaoJiaCao2007])

20. Experiment II: Critical value T(P)=aP Critical value: a=0.43

21. Theoretical Result II • There is a critical value of the linearly increasing speed of the tolerance ac. If a<ac, BP prob. Still decreases to zero when the network capacity . • Proof (See [ZhaoJiaCao2007]).

22. Partial Conclusion • How much spare capacity is necessary for the security of resource networks? • At least linearly increase the spare capacity when the network capacity increases. • The linear increasing speed should be no less than a critical value.

23. How Topology Affects the Security

24. Motivation • Fixing the topology, considering single and multiple line outages, what’s the probability for a power network to collapse (i.e., large load shedding)? • Formulate a decision problem: will a line outage contingency cause a collapse (i.e., a large load shedding)? • Use Ordered Binary Decision Diagram (OBDD) to count the line outages leading to collapse. • The sum of the probabilities of these line outages gives a lower bound of collapse probability, i.e., an upper bound of security. • By counting the appearance of each line in the fatal line outages, we identify the critical lines.

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27. Conclusions • Necessary condition of security: resource balance condition. • How much spare capacity is necessary for the security of resource networks? • Linear increasing, with rate greater than a critical value. • How topology affects security: an upper bound of security. • OBDD-based fast enumeration. • Identifies critical transmission lines.

28. Thank you! Q&A