Wind energy for dummies
Part 2: Wind farm layout and the wake effect
Introduction
In part 1 of this series on wind energy, we went through some basic concepts of potential wind power. We know that it depends on the air density, the rotor diameter and most importantly, the wind speed. With these foundations, we can now examine how these factors are taken into account when designing a wind farm with optimal electricity production.
In this article, we will focus on one (seemingly small) aspect: the placement of each turbine in a wind farm.
The wake effect
Till now we have considered the properties of a single turbine. As you can guess, the installation of many of them in one area presents additional physical challenges.
One of the main ones is the wake effect: a wind turbine placed too close behind another one will receive the turbulent air exiting from the first one and thus have a degraded efficiency. For example, in an offshore wind farm in Denmark, a drop of 12% in efficiency has been observed when comparing the efficiency of a single turbine and that of the whole farm [5].
Ideally, turbines should be placed as far apart as possible. However, land area for wind farm is a limited resource that also needs to be optimised. Finding the best layout under all the physical, economical and social constraints is a hard combinatorial problem. For this reason, until recently, the placement of turbines is based on some battle-tested heuristics.
The heuristic approach
Wind farm usually has a rectangular layout with some offsetting between rows. A traditional consensus for the reasonable distance between two turbines in a row is 3 to 5 times the rotor diameter. The distance between rows is then 5 to 9 times the rotor diameter [2].
More recent studies suggest that this kind of heuristics leads to sub-optimal efficiency, especially in large wind farms with hundreds or thousands of turbines [10].
With the advance of optimisation algorithms and computing power, researchers are looking into a more analytical approach to the problem.
The modelling approach
Finding the best turbine layout can be formulated as an optimisation problem:
We want to find the optimal distances between wind turbines in a wind farm to minimise the total wake loss.
Cost, land-usage and several others physical or economical constraints could also be included for a more realistic model. In the scope of this article, we will keep it simple and focus on the model of wake deficit, which is a central piece of the framework.
The Jensen wake model
Researchers have proposed several wake effect models in the literature. The Jensen’s approach [9] is the oldest one and the most widely used because of its simplicity and thus fast computation time.
The Jensen model assumes the wind speed to be constant within the wake, which expands radially at the rate k_w*x; k_w being the wake decay coefficient and x the distance between the two turbines.
It estimates the downstream wind speed deficit δ between two turbines, which is the ratio between the downstream and upstream wind speeds u2, u1:
This is then expressed as:
where C_t is a predetermined thrust coefficient, r the rotor radius.
The first parenthesis can be considered as a constant, the downstream wind speed deficit δ thus mainly depends on the ratio between the distance between turbines and the rotor diameter.
If we apply the numbers in the heuristic approach here, where:
- the distance between turbines is 5 to 9 times the rotor diameter,
- with awake decay coefficient of 0.075 (typical for onshore wind farms),
- and a thrust coefficient of 1 (for simplicity)
The estimated wind speed deficit will be between 0.528 and 0.356.
More generally, if we draw the graph of wind speed deficit vs the distance between turbines, we observe that when the said distance is significantly bigger than the rotor radius, δ will tend toward zero (ie. no wake deficit), which aligns with our intuition.
Generalising to a wind farm
We have the wake model for a turbine. Within a wind farm, the speed deficit at the nth wind farm is the quadratic sum of the square of the local deficit [12].
Supposing we have 50 wind turbines in a row with a distance of 9 times the rotor diameter between them, using the same coefficients as above, we can compute the estimated wind speed deficit at each one of them:
We see that the downstream turbines have higher deficits, which stabilise quite quickly though.
The speed at the nth turbine can be then computed as:
We can plug that estimated wind speed in the power formula that we already derived in the previous article to have an estimation of the wind power at the nth turbine:
The objective function
With these elements, we can now define the objective function for our optimisation problem. The simplest approach is to maximise the total estimated power output:
In practice, as some turbines are exposed to more wake effects than others, they tend to have more significant problems, which is not great for the general maintenance of a wind farm. Therefore, we present a modified version of the above objective function to control for the variation in wake deficit among turbines:
As stated in the previous section, this objective function are optimised with respect to some contextual constraints (cost, land-usage, noise, etc.), which in turn will need a model for their own. However, that is out of scope for this article.
Conclusion
In this article we have familiarised ourself with the notion of wake effect. It is one of the main challenges when designing a wind farm with optimal energy output.
Historically, people have employed a heuristic approach based on the rotor diameter to determine the ideal distance between wind turbines. More recent studies have proposed analytical solutions where each factor (power, wake deficit, cost etc.) is modelled using mathematical functions. However, these constrained optimisation problems are usually computationally intractable without significant simplifications in the design of the mathematical models, which is why the simple ones (like Jensen’s) is still preferred. This remains an active area of research with a lot of room for improvements.
References
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[2] Masters, Gilbert M. Renewable and efficient electric power systems. John Wiley & Sons, 2013.
[3] Ghaisas, Niranjan S., and Cristina L. Archer. “Geometry-based models for studying the effects of wind farm layout.” Journal of Atmospheric and Oceanic Technology 33.3 (2016): 481–501.
[4] Nitsch, Felix, Olga Turkovska, and Johannes Schmidt. “Observation-based estimates of land availability for wind power: a case study for Czechia.” Energy, sustainability and society 9.1 (2019): 1–13.
[5] Sorensen, P., and T. Nielsen. “Recalibrating wind turbine wake model parameters — validating the wake model performance for large offshore wind farms.” European Wind Energy Conference and Exhibition, EWEA. 2006.
[6] B. Saavedra-Moreno, S. Salcedo-Sanz, A. Paniagua-Tineo, L. Prieto, A. Portilla- Figueras, Seeding evolutionary algorithms with heuristics for optimal wind turbines positioning in wind farms, Renew. Energy 36 (2011) 2838e2844, https://doi.org/10.1016/j.renene.2011.04.018.
[7] S. Chowdhury, J. Zhang, A. Messac, L. Castillo, Unrestricted wind farm layout optimization (UWFLO): investigating key factors influencing the maximum power generation, Renew. Energy 38 (2012) 16e30, https://doi.org/10.1016/ j.renene.2011.06.033.
[8] Kuo, Jim YJ, et al. “Wind farm layout optimization on complex terrains–Integrating a CFD wake model with mixed-integer programming.” Applied Energy 178 (2016): 404–414.
[9] Jensen, Niels Otto. A note on wind generator interaction. Vol. 2411. Roskilde, Denmark: Risø National Laboratory, 1983.
[10] Meyers, Johan, and Charles Meneveau. “Optimal turbine spacing in fully developed wind farm boundary layers.” Wind energy 15.2 (2012): 305–317.
[11] Peña, Alfredo, Pierre‐Elouan Réthoré, and M. Paul van der Laan. “On the application of the Jensen wake model using a turbulence‐dependent wake decay coefficient: the Sexbierum case.” Wind Energy 19.4 (2016): 763–776.
[12] Katic I, Højstrup J, Jensen NO. A simple model for cluster efficiency. Proceedings of the European Wind Energy Association Conference & Exhibition, Rome, 1986; 407–410.
