When using environmental power sources like wind and solar power, it is important to remember that they are often unpredictable and intermittent. For instance, should an island be powered by sunlight and wind, on a a calm, dark evening these sources will not be available. Unless intermittent power on the island is deemed acceptable, an energy reserve must be maintained for periods when the primary generators are not working, either by storing excess energy produced during active periods, or importing energy to the island in the form of fuel or charged batteries. Needless to say, unless the island can produce sufficient revenue to cover the cost of importing energy (say, by selling excess fish caught or vegetables grown on site), this method will not allow self sufficiency.
Electrical energy would be stored in rechargeable batteries, whose DC output would need to be converted to mains AC to run standard equipment on the island (unless the equipment is built to use DC,) or electrical power could be generated as needed by driving a generator from imported (or grown) fuel, or possibly recovered biomass. An engine burning imported fuel could equally well provide mechanical energy directly (say, for driving propellers should the island be navigable). It is very important to have a method of starting the generators and engines in the absence of other energy sources, say a starter motor, pull cord or starting handle, if the latter approach is taken.
OTEC is an acronym standing for Ocean Thermal Energy Conversion. It uses the difference in temperature between the cold water at depth and the much warmer water at the surface. 
Off-shore the wind has few obstacles that cause turbulence; because of this the wind tends to blow slightly stronger. (Mention anchoring to seabed or floating platforms (and anchor problems) and economies of scale)
Advantage: 24 hours, except when there is no wind.
Wave Power GeneratorsEdit
These types of generators that harness the energy created by sea-waves is, so far, too expensive and impractical for the average island-maker. Many of the various designs for the generator are still frustrated by the practical challenges of the sea, chiefly storm damage and saltwater corrosion. The salter's duck is one example of a wave based generation system.
Another method, which does not require tethering to the sea floor like salter's duck and hence allows the island to be moved without dismantling, is to build an island of several modular, independently floating units linked by pivoted arms, which could also serve as bridges. In the presence of waves of the correct wavelength, twice the length of each arm for example, the relative vertical movement of the floating units would apply an oscillating turning moment to the arms, from which power could be extracted by attaching pistons to them to pump seawater through a turbine or compress air into a tank. Of course, this method of power generation would require the island to be located in an area with an abundance of waves, and the occupants to endure the wave motion of the floating units. One solution might be to make one large floating island to be used as a central habitation area, of sufficiently great buoyant volume that wave effects are cancelled out, and surround it by small auxiliary islands on arms which would not be occupied for significant periods, storage for food, fresh water and machinery, for example. These smaller islands would then move relative to the stationary central hub to generate power.
There are many currents in the sea, the most obvious one to most people is the gulf stream. This flows year round at around 4 knots. A single turbine placed in this stream could produce significant amounts of power. It is unknown how much power we could remove from a current such as this before it may have an appreciable effect on the world. (Mention Betz theorem as related to wind and adjust to water as just another fluid dynamics problem)
Solar panels may not be a good choice when considering that they will take up space and sunlight on the island which could otherwise be used for growing food, and are also only usable during fairly bright daylight.
Solar panels have been around since the early 70s for commercial purposes. Only now with economies and power crises around the world are they becoming very popular. Photovoltaic panels are made up of many small cells each having a voltage of around 0.4-0.6V these are then connected in series or parallel to produce a panel of a certain size and specification. Current photovoltaic panels use a single layer of very pure silicon to convert solar radiation into electricity. Next generation panels incorporating up to five layers of slightly less pure, and hence much cheaper, silicon are just around the corner. Photovoltaic panels are fairly inefficient, converting around 10% of the light falling on them into usable electricity, and hence a very large area of panels will be required to provide a sizeable power output. It might be worth calculating the relative efficiency of plants converting light into chemical energy, and the subsequent efficiency of burning this (for example, as plant oils, ethanol or methanol) in an engine to produce electricity, and comparing this to solar panels. Photovoltaic cells also have a problem with temperature, becoming less efficient the hotter they get.
Solar furnace or Steam powered turbinesEdit
Sun power is free, however most techniques for converting it into usable power are very expensive to implement. One technique that can be fairly cheap to build if designed carefully is the concept of the solar furnace. In the solar furnace many mirrors are used to focus sunlight at a single point. This point will get very hot very quickly and this temperature rise can be used to turn water into steam. This steam can then be used to turn a turbine and produce electricity. Design of a solar furnace is critical, since you are potentially dealing with superheated steam and explosion risk is a distinct possibility. However it is a valid concept and a floating island could use its very nature to assist. To avoid the problems of dealing with superheated steam, peltier devices or hot air engines could be substituted, at the cost of scale.
Instead of turning all the mirrors to face the sun it may be easier to turn the island. This depends on a number of things and the mirrors will still need to track the sun's elevation in the sky.
Geothermal (ocean vents)Edit
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There has been much publicity in the last few years about crops of fast growing wood such as willow being grown strictly as an alternative fuel for burning in power stations. The same idea could possibly be used with kelp or seaweed in the ocean. However it may not be a particularly efficient process.
Fossil Fuels should be used as a last resort. Keeping an adequate fuel supply for full time habitation is difficult, and requires a regular resupply. Generators are loud, and would be a significant annoyance in a small space. Furthermore, their use is detrimental to the environment. Thus their use should be limited to backup generators for emergencies.
If the island is close enough to the mainland and permanently anchored in one place, it may be possible to simply arrange for a mains power connection. This will likely be expensive upfront.