Wind Farm Wake Power Loss Calculator

What this calculator does

This calculator estimates how much power a downstream wind turbine may lose when it sits in the wake of a single upstream turbine. It uses the classic Jensen (PARK) top-hat wake model to predict the wake wind speed at a downstream distance (given in rotor diameters), then converts that wind-speed deficit into a power-loss estimate using the standard turbine power equation.

Use it for early-stage layout trade-offs (e.g., “What happens if spacing changes from 6D to 8D?”) and for building intuition about how C_T and spacing affect wake losses.

Inputs (what they mean)

• Free stream wind speed (m/s) (U): The undisturbed inflow wind speed upstream of the first turbine.
• Turbine rotor diameter (m) (D): Rotor diameter; swept area is A = π(D/2)².
• Downstream spacing (rotor diameters) (s): Center-to-center downstream distance expressed as multiples of rotor diameter. The physical distance is x = s·D.
• Thrust coefficient (C_T): Describes how strongly the rotor extracts momentum (and therefore creates a wake). Typical operating values are often ~0.6–0.9, depending on turbine control and wind speed.
• Power coefficient (C_P): Aerodynamic conversion efficiency used in the power equation. Typical values might be ~0.35–0.50 (and vary with wind speed and turbine control).

Model equations (Jensen/PARK)

The Jensen model assumes a linearly expanding wake with a uniform (“top-hat”) velocity deficit across the wake cross-section. Wake radius grows with downstream distance:

Wake radius: R_w = R + kx, where R = D/2 and k is the wake expansion constant.

Using axial induction a (actuator disk concept), thrust coefficient is related by:

Thrust relation: C_T = 4a(1 − a)

The Jensen centerline (and top-hat) wake wind speed at distance x is often written as:

Finally, turbine power (ignoring cut-in/cut-out, rated power limits, and control region changes) is modeled as:

Baseline power: P = ½ ρ A U³ C_P

Waked power: P_w = ½ ρ A U_w³ C_P

So the power ratio is simply P_w/P = (U_w/U)³.

Outputs (how to interpret results)

• Wake wind speed (U_w): Predicted wind speed at the downstream turbine in the upstream turbine’s wake.
• Velocity deficit: Often reported as 1 − U_w/U. Small changes here can cause larger power changes because power scales with U³.
• Baseline power (P): What the downstream turbine would produce in free stream wind.
• Waked power (P_w): What it would produce under the wake wind speed.
• Percent power loss: (1 − P_w/P)×100%, the key planning metric for spacing trade-offs.

Worked example

Suppose:

• Free stream wind speed U = 10 m/s
• Rotor diameter D = 100 m
• Spacing s = 7D → x = 700 m
• C_T = 0.80
• C_P = 0.45

First compute induction factor a from C_T = 4a(1−a). For C_T=0.8, a common solution is a ≈ 0.276 (the physically relevant root in normal operating conditions). With a wake expansion constant k (commonly around 0.075 onshore or 0.04 offshore; many simple calculators pick a fixed default), you can compute U_w using the Jensen equation. Then compute baseline power and waked power using the same C_P. The reported loss is typically substantial because the cube-law amplifies even moderate speed deficits.

Typical parameter guidance

Limitations & assumptions (important)

• Single-wake, aligned flow: This is a one-upstream-to-one-downstream estimate. Real wind farms require wake superposition across many turbines and wind directions.
• Top-hat profile: The Jensen model assumes a uniform deficit across the wake cross-section; real wakes have non-uniform (often Gaussian-like) profiles.
• Fixed wake expansion constant (k): If k is fixed in the implementation, results won’t reflect site turbulence intensity, stability, or atmospheric conditions.
• No partial wake overlap: If the downstream turbine is not fully immersed in the wake (lateral offset, yaw misalignment, wind veer), the effective deficit can be smaller than predicted.
• Steady-state physics: Ignores turbulence-driven unsteady loading and time-varying inflow.
• Simplified power model: Uses P ∝ U³ with constant C_P and no rated power cap, cut-in/cut-out behavior, or control region transitions.
• Flat terrain / no blockage: Does not include complex terrain, blockage effects, or wind shear across the rotor.

When to use something more advanced

If you need bankable energy estimates, directional/sector analysis, turbulence impacts, yaw/veer, or multi-row farm performance, consider more advanced engineering wake models (e.g., Gaussian/Bastankhah-type) or validated farm tools that support wake superposition and calibration to site measurements.