Wind farm optimization algorithms

I have always been amazed by the number of published papers, master thesis and documents focusing on the use of algorithms to optimize the layout of a wind farm. Some of them were proposed more than 25 years ago, showing a continuous, sustained interest in the topic.

I guess that the reason for such abundance is the stimulating difficulty of the problem and the fact that there are huge investments behind a wind farm.

From a mathematical perspective the problem is complex due to the type of variables involved, both discrete (you can have 30 or 31 turbine but not 30.5) and continuous (for instance, the length of cables). Additionally there are strong links between variables (for instance higher turbines = higher tower and foundation cost) so finding the “sweet spot” that maximize earnings is not a simple task.

Generally speaking, these algorithm try to maximize the profitability of the investment, usually expressed in terms of Net Present Value (NPV). Basically they compare the value of all expenditures during the life of the project “in today money” with all the earning “in today money” using a certain discount rate for cash flows in the future.

Expenses belong to two categories, capital expenses (CAPEX) and operational expenses (OPEX), while net earnings are function of the amount of power produced, the price of electricity and the electrical losses.

Therefore even a simplified model should try to minimize these expenses:

• Wind turbine
  • Model (power curve)
  • Tower
  • Installation
• Civil works
  • Foundations
  • Roads
• Electrical works
  • MV cables
  • Substation
• Operation & Maintenance

While maximizing the production, a mainly a function of:

• Wind
• Wind shear (of the speed of the wind increase with height)
• Wake effect (how turbine interact with each other creating turbulences)

The interaction between all these variables is what makes the problem interesting.

To give a few examples,

1. Packing the turbines densely in a small area will lower the cost of roads and cables but will create huge production losses due to the turbulences inducted by the turbines upwind.
2. Using a higher tower should increase the production – unless the wind shear is low, in which case the additional tower and installation costs would off weight the benefits
3. A certain position could be extremely productive – but it could be very far away from the substation (increasing the electrical losses ) or on the top of a steep hill (increasing the earthworks cost)

Additionally you have to decide the level of complexity of the model. For instance the foundation cost can be considered as:

• A lump sum, equal for all turbine models. Under such assumption, you would see a benefit decreasing the number of turbines but not switching to a different WTG model.
• A function if the wind turbine model (greater loads = greater foundation).
• A function of wind turbine model, geotechnical parameters of the soil and unit cost of concrete of still. This latter option, although more precise, would probably make the model very difficult to handle.

I believe that a reasonable compromise between complexity of the model and quality of the result can be achieved using nested algorithms as proposed by these researchers.

In the first steps, only the variables related to the turbines (power curve, wind resource, availability and cost) are considered. Once the turbine model and the layout are fixed the civil and electrical works can be considered, defining the optimum position of the substation (to minimize cable length) and the shortest roads connecting the wind turbines.