The Maximum Power Principle

Lotka and Odum

In 1922, Lotka proposed a ‘law of maximum energy’ for biological systems. He reasoned that what was most important to the survival of an organism was a large energetic output in the form of growth, reproduction, and maintenance. Organisms with a high output relative to their size should outcompete other species.

In 1955, Odum and Pinkerton built on Lotka’s work with the ‘maximum power principle’, stating that ‘systems perform at an optimum efficiency for maximum power output, which is always less than the maximum efficiency.

The electrical Ohm’s Law provides a way of thinking about the relationship between maximum power and maximum efficiency. In electronic devices, the output resistance of the power source should match the input resistance of the load to maximise the power throughput for maximum power. However, the point of optimised power does not match the point of optimised efficiency. In the case of loudspeakers, for example, the maximum sound output is achieved when the speakers match the impedance (AC resistance) of the amplifier. At this point, half the energy is dissipated in the speakers and half in the amplifier. Speakers with a much higher impedance will improve the system efficiency but with less volume, requiring a larger amplifier to reproduce an equivalent volume (negative feedback in amplifiers actually makes this a bit more complicated). The same applies to antennas, that require the antenna impedance to match the transmitter at the designated frequencies.

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In the biological realm, Hall recounts the relationship between a tree’s leaf area index (LAI) and the energy capture. The highest efficiency is achieved with a relatively low LAI since the topmost leaves capture the most sunlight, and each leaf is energetically expensive to maintain. But the usefulness of the high efficiency is offset by the limited leaf area and therefore total energy capture; an efficient plant would be short and outcompeted in a forest.  But there is a limit to which a plant can grow before the marginal gains of additional leaf area contribute to energy capture.

Perhaps the most sobering outcome of the Maximum Power Principle is thinking about the role of humanity, and what this means for human appropriation of net primary production of biomass for liquid and other fuels. William Rees cites Lotka’s maximum power principle in posing whether humans are unsustainable by nature, noting that –

by virtue of cumulative knowledge and technology, homo sapiens has become, directly or indirectly, the dominant macro-consumer in all major terrestrial and accessible marine ecosystems on the planet.

Role of energy efficiency and maximising energy throughput

The operation of electrical generators provides an example of the economic trade-off between maximising power and maximising efficiency; the revenue of an electrical generator depends on energy throughput, but the efficiency defines the fuel cost per unit of electricity. An efficient plant will increase electricity output for a given quantity fuel, but beyond the optimal power/efficiency, the additional gains in output do not justify the additional cost of improved efficiency. These trade-offs are routine in engineering practice.

The limits of energy efficiency as a goal in itself

The concept of ‘energy efficiency’ is really a human construct that is often useful to conceptualize the performance of energy systems. But energy efficiency targets, in themselves, can be counterproductive. The Australian building regulations typify this problem. The Building Code specifies deemed-to-satisfy building requirements for thermal performance, such as insulation R-value, and double glazing. But the Code merely institutionalises energy efficiency as a goal in itself, rather than per-capita energy consumption. This is because larger homes generate a better thermal efficiency score than equivalent smaller homes, since geometrically, larger homes gain proportionally more interior space relative to exterior fabric area (better know as Galileo’s square-cube Law). But rating tools do not penalise larger homes even though it is obvious that they consume more energy, nor account for functional use or the number of occupants. This leads to the perverse outcome that the Code favours homes that consume more energy, but do so ‘more efficiently’.

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