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 Cellular Automata And Electric Power

The motivation for developing Capow as an EPRI project is that cellular automata are parallel systems and the electric power grid is a parallel system. By investigating complex cellular automata we get a better qualitative idea about possible behaviors of a physical parallel system like the power grid. So as to try and make the analogy closer, we specifically looked at cellular automata which embody wave motions and oscillations similar to those of electrical systems.

Regarding the parallelism of the electric power grid, note that generators, loads, wires, and switching stations interact with each other locally. The laws of physics cause each element of the system to update itself locally and independently in real time. The behavior of a circuit breaker, for instance, depends on the preset parameters of the circuit breaker and on the current in the wires coming into the breaker. The current in a wire depends on the wire’s physical parameters and on the generators, switches or loads connected to the wire.

The electric power grid is like a large parallel computer which has been assembled and programmed with an uncertain knowledge of what the full system will do. Although it was built and designed to distribute power, the grid also embodies an unknown and unpredictable computation system. This is why there is sometimes no one really compelling explanation for a given power surge or outage. An anomalous event can emerge as the unpredictable result of the system’s massively parallel "computation".

Unpredictability is an essential aspect of parallel systems. It is very commonly the case that it is impossible to predict the behavior of a parallel system by any means other than actually simulating the system in action. This is known, for instance, to be true of cellular automata. Most Cellular automata computations are "algorithmically incompressible", meaning that there is no faster way to predict the eventual output of a CA than actually running the CA.

Exploring cellular automata with CAPOW gives us a better intuition about parallel systems such as the power grid.

The Wave Equation mode of CAPOW can be used as a model for an electrical waveform travelling down a transmission line. Switching to a nonlinear wave equation such as Quadratic Wave or Cubic Wave allows us to model nonlinear effects of physical transmission lines.

The Oscillators and Diverse Oscillators rules allow us to examine behavior of electrical oscillators. In addition we have a family of Oscillator Chaotic.DLL rule which simulates chaotic oscillations.

If we think of these oscillators as generators or loads, we can imagine coupling them together in a network connected by the Wave Equation. The rules Wave Oscillators, Diverse Wave Oscillators demonstrate this effect. An inspiration for the Diverse Wave Oscillators rule was to represent the dynamics of a power grid to which a large number of differently responding oscillatory loads are coupled.

In order for these representations to be meaningful we must think of the CAPOW display as representing a power grid roughly the size of a state. This is because there is an issue of dimensional scale in our representation of oscillating loads by these cellular automata. A CA of necessity represents the speed of light (or the transmission of electrical signals) at a scale in which "light" travels at one space cell per update. Focus on one the spacetime representation of a one-dimensional CA rules. If c is the speed of light, then whenever one horizontal screen inch represents WorldXperCM units of physical distance, then one vertical screen inch represents WorldXperCM/c time units WorldTperCM. Conversely WorldXperCM is c*WorldTperCM. Since c is roughly 3 x 10 ^ 10 cm/sec, if one horizontal screen centimeter represents 1 centimeter of physical distance, then a vertical screen centimeter represents 0.33 x 10 ^ -10 seconds, which is much smaller that the oscillation cycle times considered in power engineering. In electrical power systems we are very often interested in oscillation frequencies of an order of sixty cycles per second, with an oscillation cycle of 0.016. If we want to spread such a cycle over the height of a typical computer screen, we might want a vertical centimeter to represent something like 0.001 seconds of time, that is, we would want a WorldTperCM of 0.001, which gives a WorldXperCM of 3 x 10^7 centimeters, which is 300 kilometers, a reasonable length scale for state-wide or nation-wide power grid.

The two-dimensional CAs we investigate are also of interest in modeling the electrical power grid. Here we should regard the CA rules as giving a "satellite-view" of a power grid. Imagine looking down at a dense power grid from several hundred kilometers above the Earth’s surface. At this distance we might think of each cell as a generator or load node that is connected to an adjacent node; if the grid is sufficiently dense, as in a city, we can abstract away from representing the connecting wires.

In the two-dimensional CAs we can represent, as before, linear and nonlinear waves as well as chaotic and non-chaotic oscillations which can be linked together by waves. The 2D Oscillator Wave.DLL and the 2D Oscillator Wave Chaotic.DLL are examples of such rules.

An additional kind of CA phenomenon arises in two-dimensional system, this is the reaction-diffusion kind of rule. 2D Hodge.DLL is a good example of such a rule. This type of rule produces a pattern of spirals of excitation and inhibition similar to that seen in the spread of rolling power black-outs in which the circuit-breakers repeatedly attempt to reconnect the circuit. The spiral patterns of the 2D Hodge.DLL rule are similar to a satellite view of the lights going on and off in a city experiencing a rolling power black-out.