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Braess Paradox in AC Power Networks

Adding an extra road in a road network can increase traffic congestion. This counter-intuitive outcome is called a Braess Paradox.  Braess Paradoxes, where increased connectivity degrades the performance of a network, has been shown to occur in electric power transmission networks.[1]

An AC power network consists of “buses” (vertices or nodes) and “transmission lines” (edges or links). Buses are of two types – “generators” (power sources) and “loads” (power sinks). To a good approximation, power flowing along a link from node i to node j in a high voltage AC transmission network is given by

(1)   \begin{equation*}P_{ij} = K_{ij} \sin(\theta_j-\theta_i)\end{equation*}

where \theta_i is the voltage phase at node i and K_{ij} is the capacity of the transmission line linking i and j. Maximum power flows when the phase difference is 90^\circ.

The capacity of the transmission network as a whole is the maximum total power consumption that it allows. Assume for simplicity that all transmission lines have the same capacity K and that all generators and all loads are identical. This simplification means that, if there are N_g generators and N_c cities, each generator produces power g and each load consumes power g N_g/N_c. Given a network topology and g/K, the set of \theta‘s which satisfy Equation 1 can easily be found, at least for smaller networks. Eventually, when g/K is increased to a critical value, no solution is possible. The capacity of the network has been reached.

Figure 1 shows a 5-node 3-generator 5-link network operating at it’s critical capacity. Each generator produces power K and each load consumes 1.5K. Power flows are indicated in red, with arrows  on links giving the directions of flow. Node arrows indicate phase angles. Note the 30^\circ phase difference between generators A and B and between the cities.

Figure 1

Suppose that a new link is added between A & B, coupling the phases of these two generators.  The large phase angle between A & B of Figure 1 produces a current which cannot be sustained without disrupting the phases on all of the other nodes.

The new critical state with phases reorganised to maximise capacity in the new network is shown in Figure 2. Generators A & B are now nearly in phase, with only a small current flowing between them. The capacity of the network is reduced to 2.91K, 3\% lower than the less connected network of Figure 1.

Figure 2

A more extreme case is shown below, where adding an additional three links to a simple power network degrades the capacity of the network by 1\over 3.

Figure 3

Calculations using R‘s excellent igraph and nleqslv packages.

[1]  Braess’s paradox in oscillator networks, desynchronization and power outageD. Witthaut and M. Timme, 2012.

 

On the speed of ships

speedofships

  • The graph shows annual median speed of ships from 1750 to 2014. It is based on historical location data from 617,754 vessels in the ICOADS dataset.
  • Ship speeds increased by 0.2kn/decade up to 1880. This reflects incremental improvements in sailing ship technology, which reached a peak with Sovereign of the Seas (1852, capable of 22kn).
  • The impact of new technologies such as steam paddle ships (SS Sirius, 1837), steel hulls and screw propellors (RMS Oceanic, 1871) is not evident until the late 1800s. Then, between 1881 and 1905, median speed of ships increased at an astonishing rate ~ 3kn/decade, a near step change compared to the past. The ocean liner SS City of Paris (1889) had a cruising speed of 20kn.
  • From 1906 to 2014, speeds increased at a modest 0.3kn/decade on average, comparable to the rate during the sail ship era.
  • A striking feature of the graph is that the median speed of ships is much more variable in the modern era compared to 18th and 19th centuries. Political and economic factors, rather than technology, became dominant driving forces e.g. oil supply and financial crises. A dramatic example is the sharp decline in speed of ships at the end of 2008 following the collapse of the Lehman Brothers.

R code

Ship trajectories from ICOADS  were stored in a dataframe ships with ~100 million rows.

Rendered by QuickLaTeX.com

Median speeds were calculated using dplyr

The speed of a vessel was calculated whenever it’s location coordinates were available within consecutive days in ICOADS dataset.Periods of zero speed (ship docked or at anchor) were omitted. Buoy and drifter data were excluded.   Insufficient data was available for the years 1864-87.