SMALL WORLD NETWORK

-Naveen Teja Vempati -Ravi Teja Mitta

INTRODUCTION

Every person in this world is some way or other connected by some means is the idea behind evolution of small world phenomenon.

Later Harvard approach and MIT approach to study small world had been developed.

Through MIT approach widening of the chain and breakage of links is studied.

Out of many chains found between source and target, the shortest path found to be with in six intermediaries.

Six degrees of separation came into existence as a result of Harvard approach.

DEFINITION

A mathematical graph based on concept that every target node can be reached from every other node by less number of steps.

The Topology of the network is studied based on the

Clustering Coefficient C(p): measures the degree to which nodes cluster together.

Characteristic path length L(p): measures the separation between two nodes in the graph where L(p) is directly proportional to logarithm of nodes between them.

C(p) >> C random and L(p)>> L random.

Rewired edges must connect to vertices that would otherwise be farther than L random.

FEATURES OF SMALL WORLD NETWORK

RESEMBLANCE

Many technological networks were found to lie between regular and random networks.

By analogy people found following systems in accordance with small world phenomenon.

Neural network of worm Caernorhabditis elegan.

Power grid of Western United States.

Collaboration graph of film actors.

SIMULATION RESULTS OF DISEASE SPREADING

1.Charactersitic path length L(P) and clustering coefficient C(P) of small world network. 2. Critical infectiousness r half at which disease spreads resembles clustering coefficient curve.

Fig 1

3. Time to spread resembles L(p) curve

Fig2. Fig3.

APPLICATION TO SMART GRID

In the recent past, Black-Outs occurred repeatedly in United States, UK, Australia, Russia and many other countries.

All these black-outs experienced a cascading proceeding.

Redistribution of load in case of failure and there by corresponding neighborhood lines are overloaded leading to cascading failures.

Many kinds of complex systems in nature and society are tested to study the cause of black-out.

Fortunately collective dynamics of small world network gave good explanation about black-out.

PRIMARY ASSUMPTIONS

All edges in reality high voltage transmission lines are assumed to be undirected(bidirectional).

Nodes of the network treated to be identical and featureless

a) Generators

b) Transformers

c) Substations

o All transmission lines are assumed to be identical ignoring the fact that voltage varies considerably.

o The node of ground is ignored, the parallel lines are combined and graph is made to look simple.

STATISTIC PARAMETERS OF ELECTRICAL NETWORKS

o From the above results we observe that the Chinese power gird exhibits similar properties as that of the WSPG there by confirming that it is also a small world network.

TYPES OF CASCADING FAILURES

1. Failure of node:

o High centrality is characteristic of high degree node which is basically defined as the load capability.

o Removal of vertices with high centrality will worsen the networks functionality.

o Changes the balance of flows leads to a global redistribution of loads over the entire network

o It can be either random breakdown or intentional attacks.

o Chinese and US power grids exhibit heterogeneous distribution of loads per node.

2.Failure of Tie line:

o Inner vertices within local clusters are connected together tightly while there exists few external tie lines connecting them(low coupling)

o Tie lines are respected as main contributors to characteristic path length of small-world network.

o For example in case of Chinese power gird there exists only single line between Sichan-Chongquing province power grid and Center Chinese power grid.

o So failure of these lines can change the structure and property of networks to a great extent.

CASCADING FAILURE MODES ANALYSIS

Practically both failure of nodes and tie lines are often mixed together in cascading.

So Monte-Carlo simulation was used to test the different effects of different failure events.

In this approach even equal area criterion was used to determine the stability of the system.

If the system is unstable, author assumed that a cascading failure occurs.

In all the failure events occurred there exists at least one or more critical node or critical line.

SAMPLE AND STATISTICAL DATA ANALYSIS

o The graph explains that there exists few critical lines .

o When simulation was applied to Northern Chinese power grid it was observed that there are 48 events lead to system unstable, and they are all multiple failure events.

IMPROVEMENTS PROPOSED FOR ABOVE MODEL

The magnitude of line impedance is taken as the weight of the edges which was assumed un-weighted in the above model.

Line impedance which reflects the electrical connection degree between nodes cant be ignored as it will differ amongst each other.

In this model real power flow in edges is introduced as network flow in small world model since it is an essential factor in vulnerability analysis.

FORMULAE

Considering weight of edges:

o Considering Power flow :

COMPARISON

o Vg=F(S,T), Where Vg indicates online tool for vulnerability analysis given as a function of state quantities and topology factors.

FLOW CHART FOR VULNERABILITY ANALYSIS

DIFFERENT TYPES OF ATTACKS

Fig4. The distribution of node degree of power grid Fig5. The curve of global efficiency under different attacks

o In this example it was assumed that the maximum degree of the substation is 14.The degree of seven substations in eight at 500KV is higher than 6, while another one is 5.

o Fig5 shows that as the high degree nodes are taken off overall efficiency declines rapidly and when it reaches to 8.5% loss.

o When attacks on the nodes are random, the change in the overall efficiency is relatively slight. Until about 80% nodes lose, in both the cases the power grid collapses.

CONCLUSION

The nodes at high voltage level as well as the connective nodes have higher node degree, and they connect more closely. Once they break down, a serious blackout probably follows.

The vulnerability of the system under attack is related to the types of the nodes. Under general attack, it is strong; if the nodes of high degree are under attack, it is vulnerable and a chain failure is tend to happen.

Multiple failure happening on the nodes of low degree can easily lead to regional blackout. So we should strengthen monitoring and consolidating the key nodes and lines, make sound prevention and control measures against the potential failure.

REFERENCES

Stanley Milgram” The Small-World problem” Psychology study, vol. 1,no.1,May 1967

Duncan J.Watts & Steven H. Strogatz “Collective dynamics of small-world networks” Vol393, 4th June 1998.

Lu Zongxiang, Meng Zhongwei and Zhou Shuangxi. “Cascading failure analysis of bulk power system using small-world network model.” Proceedings of the 8th International Conference on Probabilistic Methods Applied to Power Systems, Iowa State University , Ames Iowa, pp. 635-640,2004.

Jian Ding, Xiaomin Bai, Wei Zhao, Zhu Fang, Zaihua Li and Min Liu “The improvement of the Small-world network model and its application research in Bulk power system.”2006 International conference on power system technology.

Li Fu, Wenjie Huang, Sheng Xiao, Yuan Li and Shifan Guo “ Vulnerability Assessment for power grid based on Small-world topological model”2010 IEEE.