Robust management of a power grid

Power grids are networks that host flow of high-voltage energy. The nodes of these networks can be of three types:

  • Generators: connected to power plants;
  • Transmission nodes: only used as intermediary between power plants and distribution;
  • Distribution nodes: connected to a relatively small distribution network, each serving a city or a suburban area.

The network links are associated with a given capacity that is decided at the design step and are sufficient to a stable functioning of the network.

A cause for the major blackouts occurred in a recent past (north-east USA, Brasil, Paris) is the failure of one single link: an event such as a lightning or a cut that causes the capacity of that link to be reduced to zero.

Although the fault in one link would not disrupt the distribution, that fault triggers an avalanche effect in the rest of the network: the flow of high-voltage current finds other ways around the first failed link, possibly causing other links to fail due to capacity overload. This phenomenon repeats on other links of the network until the whole network collapses.

The purpose of this project is to understand how to modify the capacity of one or more network links in such a way that a set of likely fault conditions do not have an avalanche effect on the whole network.

Definition:

  • G=(V,E): power network;
  • O\subset V: power plants
  • D\subset V\setminus O: distribution nodes
  • u_{ij}: current capacity at link \{i,j\}
  • P_i^{\max}: maximum power of plant i\in O
  • c_{ij}: unitary extra capacity to be possibly installed on link \{i,j\}; either we install that capacity of we leave it as it is, u_{ij}
  • w_{ij}: (fixed) cost of installing extra capacity at link \{i,j\}
  • A set of fault scenarios will be provided

Formulate and solve an Optimization model for the problem of finding how much extra capacity to be installed at each link in such a way that an avalanche effect does not take place in any of the fault scenarios. Simulate the behavior of the initial system of nodes and links that shuts off a link when its capacity is overloaded. Simulate what happens to the improved, robust system.

Data: See the 12-node network serving the west states – ignore the Canadian and Mexican nodes. The two “Eastern” nodes can be thought of as the sources of energy in O\subset V, while all other nodes are in D=V\setminus O. The available amount is 100 and 120 for the two supplying nodes (100 for the one in the North, 120 in the South), while the demand can be split as follows:

  • Oregon: 22
  • California: 30
  • Nevada: 24
  • Arizona: 23
  • Montana: 21
  • Idaho: 21
  • Utah: 19
  • New Mexico: 24
  • Colorado: 20
  • Wyoming: 16

Compute the capacity necessary to transfer all energy – this is a simple min-cost-flow problem. The capacity obtained will be the initial u_{ij}. Set c_{ij} = \frac{1}{3}u_{ij} and w_{ij} = \frac{1}{10}(u_{ij})^2.
The case scenarios that have to be taken into account consist in all the possible failures of one (at a time) of the links having u_{ij}>0.