Model Bias in Wake Modeling

This page explains what model bias is, why it matters for wind farm performance prediction, and how WIFA-UQ addresses it.

What is Model Bias?

Model bias is the systematic difference between a model's predictions and reality. Unlike random noise, bias is predictable and often depends on operating conditions.

For wake models:

bias = P_model - P_reference

Where: - P_model is the power predicted by a wake model (PyWake, FOXES, etc.) - P_reference is the "true" power (from LES, SCADA, or measurements)

In WIFA-UQ, we normalize by rated power for comparability across turbines and farms:

normalized_bias = (P_model - P_reference) / P_rated

Why Wake Models Have Bias

Engineering wake models make simplifying assumptions that introduce systematic errors:

1. Wake Deficit Representation

Most models use analytical functions (Gaussian, Jensen, etc.) to describe the velocity deficit:

ΔU/U∞ = f(x, r; k, σ, ...)

These parameterizations capture average behavior but miss: - Meandering and unsteady effects - Complex turbine interactions - Near-wake transitions

2. Atmospheric Simplifications

Models typically assume: - Neutral stratification (or simple stability corrections) - Logarithmic inflow profiles - Homogeneous turbulence

Reality includes: - Strong stratification effects on wake recovery - Low-level jets and complex shear profiles - Heterogeneous turbulence from terrain and thermal effects

3. Uncertain Parameters

Key parameters like wake expansion rate (k_b) and turbulence intensity (TI) are: - Derived from limited measurements - Site-specific and condition-dependent - Often set to literature defaults

4. Blockage Effects

Large wind farms create: - Upstream induction (global blockage) - Local speedup/slowdown between turbines - Farm-scale momentum extraction

These effects are challenging to model accurately.

How Bias Manifests

Bias varies systematically with physical conditions:

Atmospheric Dependence

Condition	Typical Bias Pattern
Stable atmosphere	Models often underpredict wake losses (wakes persist longer)
Convective atmosphere	Models may overpredict wake losses (faster mixing)
High wind veer	Direction changes cause wake-turbine misalignment
Low ABL height	Wake interactions with capping inversion

Layout Dependence

Layout Feature	Bias Effect
Deep arrays	Cumulative wake interactions amplify errors
Tight spacing	Near-wake effects harder to model
Irregular layouts	Superposition models struggle with complex interactions

Parameter Sensitivity

Small changes in model parameters can shift bias:

Δbias/Δk_b ≈ 0.5-2.0  (per 0.01 change in k_b)

This sensitivity is why calibration is valuable.

The Cost of Ignoring Bias

Uncorrected bias leads to:

Financial Impact

Energy yield estimates: 5-10% bias → millions in financing errors
O&M planning: Wrong load predictions → suboptimal maintenance
Curtailment strategies: Incorrect wake predictions → lost revenue

Technical Impact

Array optimization: Biased models → suboptimal layouts
Control strategies: Wake steering based on wrong models
Lifetime estimation: Incorrect load calculations

WIFA-UQ's Approach to Bias

WIFA-UQ addresses bias through a multi-step strategy:

Step 1: Characterize the Bias Landscape

Generate a database spanning: - Multiple parameter samples (k_b, α, TI corrections, etc.) - Multiple flow cases (different atmospheric conditions) - Multiple metrics (farm-average power, per-turbine, etc.)

This reveals how bias depends on parameters and conditions.

Step 2: Calibrate Parameters

Find parameter settings that reduce systematic bias:

Global calibration: Single best parameter set

k_b* = argmin_k Σ |bias(k, case)|

Local calibration: Condition-dependent parameters

k_b*(case) = f(ABL_height, wind_veer, ...)

Step 3: Learn Residual Bias

Even after calibration, residual bias remains. Learn it as a function of features:

residual_bias = ML_model(ABL_height, wind_veer, lapse_rate, blockage_ratio, ...)

Step 4: Apply Correction

The corrected prediction is:

P_corrected = P_model(calibrated_params) - predicted_residual_bias

Bias vs. Uncertainty

It's important to distinguish:

Concept	Definition	Treatment
Bias	Systematic, predictable error	Calibration + ML correction
Uncertainty	Random variability, epistemic gaps	Probabilistic methods (PCE, Bayesian)

WIFA-UQ addresses both: - Bias correction reduces the mean error - Uncertainty quantification characterizes the remaining spread

Measuring Success

WIFA-UQ evaluates bias correction via cross-validation:

Metrics

Metric	Formula	Interpretation
RMSE	√(Σ(pred-true)²/n)	Overall prediction error
R²	1 - SS_res/SS_tot	Variance explained
MAE	Σ\|pred-true\|/n	Average absolute error

Visualization

The standard diagnostic plot shows three panels:

ML Model Performance: Predicted vs true bias (should align with 1:1)
Uncorrected Model: Raw model vs reference (shows original bias)
Corrected Model: After bias correction (should be tighter around 1:1)

Example: Bias Reduction

A typical WIFA-UQ workflow might achieve:

Stage	Farm-Average RMSE
Uncalibrated model	8-12% of rated
After calibration	5-8% of rated
After ML correction	2-4% of rated

The exact improvement depends on: - Quality and quantity of reference data - Richness of physical features - Consistency of bias patterns

Key Takeaways

Bias is systematic — It follows patterns that can be learned
Bias depends on conditions — Atmospheric state, layout, and parameters all matter
Calibration helps but isn't enough — Residual bias remains
ML can capture complex patterns — Features like ABL height and wind veer are predictive
Cross-validation is essential — Ensure corrections generalize to new conditions