Adjust rotor diameter, Cp coefficient and wind parameters to compute output in real time. Understand the 59.3% Betz limit and Weibull-based annual energy production interactively.
Weibull AEP:
$${\rm AEP} = 8760 \int_0^\infty P(v)\,f(v)\,dv$$The core equation calculates the mechanical power extracted from the wind by the rotor. It depends on the air density, the area swept by the blades, the cube of the wind speed, and the turbine's efficiency (Cp).
$$P(v) = \frac{1}{2}\rho C_p A v^3$$Where:
• $P(v)$ = Mechanical power (Watts)
• $\rho$ = Air density (approx. 1.225 kg/m³ at sea level)
• $C_p$ = Power coefficient (efficiency, 0 ≤ $C_p$ ≤ 16/27)
• $A$ = Swept area = $\pi (D/2)^2$ (m²)
• $v$ = Wind speed (m/s)
Note the $v^3$ relationship: doubling the wind speed gives 8x the power!
To estimate yearly energy output, we integrate the power curve $P(v)$ over the probability of all wind speeds, described by the Weibull distribution $f(v)$, and multiply by the hours in a year.
$${\rm AEP}= 8760 \int_0^\infty P(v)\,f(v)\,dv$$Where:
• AEP = Annual Energy Production (kWh or MWh)
• $8760$ = Total hours in a year (365 × 24)
• $f(v)$ = Weibull probability density function: $f(v) = \frac{k}{\lambda}\left(\frac{v}{\lambda}\right)^{k-1}e^{-(v/\lambda)^k}$
• $k$ = Weibull shape parameter (describes wind variability)
• $\lambda$ = Weibull scale parameter, related to the mean wind speed $v_m$
This integral is what the simulator calculates when it gives you the final AEP value.
Wind Farm Site Selection & Feasibility: Developers use this exact type of simulation, with local wind data fitted to a Weibull distribution, to predict the energy yield and financial return of a potential wind farm. They vary turbine models (different D and Cp) to find the most profitable configuration for a specific site.
Turbine Design & Performance Optimization: Engineers constantly strive to improve the Cp of rotor blades through advanced aerodynamic design and materials. Every 0.01 increase in Cp translates to significant additional energy over a turbine's 20+ year lifespan, directly impacting the cost of wind energy.
Energy Production Forecasting for Grid Operators: Grid operators need to know how much power wind farms will produce in the coming hours and days. Models based on the power curve and forecasted wind speeds (with their statistical distribution) are essential for maintaining grid stability and integrating renewable energy.
Financial Modeling & Power Purchase Agreements (PPAs): The calculated AEP is the fundamental number used to secure project financing and negotiate long-term contracts to sell electricity. Accurate AEP estimation is critical—overestimation leads to investor losses, while underestimation leaves money on the table.
When you start using this simulator, there are several pitfalls that beginners often fall into. The first is the misconception that annual energy production is determined solely by the average wind speed. While the average wind speed is certainly important, even with the same average speed of 7 m/s, the energy production can differ significantly depending on whether the Weibull distribution's shape parameter 'k' is high (e.g., k=3.0) or low (e.g., k=1.8). A lower 'k' indicates greater wind speed variability and more hours of strong winds. Since power output is proportional to the cube of the wind speed, the actual energy yield tends to be higher than what you might predict from the average speed alone. Conversely, a site with a high average wind speed but a very high 'k' (meaning the wind speed is almost constant) can sometimes have lower energy potential.
The second point is ignoring the air density, ρ. The simulator uses a standard value of 1.225 kg/m³, but in reality, it varies with altitude and temperature. For example, on a plateau at 1000m altitude, the air density drops to about 1.112 kg/m³, a decrease of nearly 10%. This means that for the same wind speed, the obtainable power decreases proportionally by about 10%. This factor is often omitted in rough estimates but is an essential correction for precise evaluation.
The third point is trusting the simulation results too literally as the actual energy yield. This tool calculates the "theoretically available energy," but in practice, various losses occur, such as turbine failures, downtime for maintenance, transmission losses, deviations from the ideal power curve, and even interference between turbines (wake effects). In practical work, you derive a more realistic "net AEP" by multiplying the AEP from the simulation by an "availability factor" and "various loss factors" (typically totaling around 85-92%).
The calculation principles behind this wind power simulator are deeply connected to various CAE fields. The first to mention is Fluid Dynamics (CFD: Computational Fluid Dynamics). The value of the power coefficient Cp is determined by the blade shape, number, and angle. To explore the optimal shape, 3D analysis of airflow using CFD simulation is essential. CFD allows for the design of blades achieving efficiency close to the Betz limit and detailed evaluation of turbulence effects.
Next is Probability & Statistics Engineering and Reliability Engineering. The Weibull distribution is a probability model widely used not only for wind speeds but also for predicting the lifespan of mechanical components and failure analysis. The same mathematical framework serves as the basis for "fatigue life analysis," which predicts the service life of key wind turbine components (bearings, gearboxes, etc.) under fluctuating loads (wind speeds).
Furthermore, it connects with Power System Engineering, which deals with delivering the generated electricity to the grid. The AEP calculated by the simulator and the time-varying output patterns are crucial input data for power system stability analysis and planning the optimal combination with other power sources (thermal, solar, etc.)—known as the generation mix. In particular, how to smooth the output fluctuations ("flicker") associated with wind speed variations is a major challenge in grid operation.
If you're interested in the calculations of this simulator and want to learn more, consider taking the next steps. First, solidify the mathematical background. The keys are the "Weibull distribution" and "integration." Trying a simple case by hand—understanding the derivation of the Weibull probability density function $$f(v) = \frac{k}{\lambda} \left( \frac{v}{\lambda} \right)^{k-1} \exp\left[-\left( \frac{v}{\lambda} \right)^k \right]$$ and why integrating it with the power curve $P(v)$ yields the annual energy production—will significantly deepen your understanding.
Next, learn about modeling more realistic "power curves". This simulator likely uses an ideal curve, but an actual turbine's curve is a piecewise function defined by "cut-in wind speed," "rated wind speed," and "cut-out wind speed." For instance, explore how to incorporate into a model the effects of "pitch control" or "stall control," which keep the output constant above the rated wind speed.
Finally, I recommend engaging with real-world data outside the tool. Obtain "wind resource maps" published by agencies like the Japan Meteorological Agency or NEDO, or long-term wind speed time-series data for specific sites, and try fitting a Weibull distribution yourself to estimate the parameters (k, λ). This can be done with simple statistical processing in Excel or Python. Experiencing the gap between theory and real data is the best way to develop practical engineering sense.