10 Wind Energy—Principles
“Wind is solar energy once removed. The sun heats the atmosphere unevenly; air moves to equalize the imbalance; and we harvest the kinetic energy of that motion.”
While solar PV converts photons directly to electrons, wind turbines extract kinetic energy from moving air and convert it to rotational motion, then to electricity. The physics is fundamentally different (fluid dynamics rather than semiconductor physics) but the thermodynamic principles from Chapter 2 still apply.
Wind is solar energy once removed, or “second-order solar.” The atmospheric heat engine operates between the warm tropics (~290 K) and cold poles (~250 K), giving it a Carnot efficiency of approximately \(1 - 250/290 \approx 14\%\). In practice, only about 2% of incoming solar energy is converted to wind kinetic energy. But 2% of a very large number (the sun delivers \(1.7 \times 10^{17}\) W to Earth) is still a very large number: roughly \(3.4 \times 10^{15}\) W, or about 200 times current global power consumption. The same winds that powered Columbus’s caravels and the clipper ships now power turbines. Different technology, same principle.
10.0.1 What Does Wind Feel Like?
Before working with wind speeds as abstract numbers, it helps to build physical intuition:
| Wind Speed | Beaufort | Felt Experience |
|---|---|---|
| 2 m/s | 2 | Smoke drifts gently, leaves rustle |
| 6 m/s | 4 | Brisk cycling headwind, dust lifts |
| 10 m/s | 5 | Small trees sway, flags extend |
| 15 m/s | 7 | Hard to walk against, whole trees move |
| 25 m/s | 10 | Structural damage, trees uprooted |
Most wind turbines operate in the 3-25 m/s range (cut-in to cut-out), with rated power typically at 11-14 m/s. Below 3 m/s, there isn’t enough energy to bother; above 25 m/s, the loads threaten structural damage and turbines shut down for self-protection.
10.1 The Origin of Wind
10.1.1 Solar-Driven Circulation
Wind exists because the sun heats Earth’s surface unevenly. The equator receives more solar energy per square meter than the poles. Land heats and cools faster than water. Mountains and valleys create local temperature gradients. These temperature differences create pressure differences, and air flows from high to low pressure.
At the largest scale, Earth’s atmospheric circulation consists of:
Hadley cells: Hot air rises at the equator, flows poleward at high altitude, descends at ~30° latitude, and returns to the equator at the surface. The surface return flow, deflected by Earth’s rotation (Coriolis effect), creates the trade winds.
Ferrel cells: Between ~30° and ~60° latitude, surface winds generally flow toward the poles (deflected to become westerlies in both hemispheres).
Polar cells: Cold polar air descends and flows toward lower latitudes.
These large-scale patterns create the prevailing winds—the trade winds that powered transoceanic sailing, the westerlies that dominate mid-latitude weather.
10.1.2 Local Wind Patterns
Superimposed on global circulation are local effects:
Sea breezes: During the day, land heats faster than water, creating low pressure over land. Air flows from sea to land. At night, the pattern reverses.
Mountain-valley winds: By day, slopes heat and air rises up valleys. By night, air cools and drains down valleys.
Thermal lows: Hot deserts create persistent low pressure, drawing in air from cooler surroundings.
Terrain acceleration can dramatically amplify wind in specific locations. Mountain passes, ridgelines, and coastal funnels act as natural nozzles. California’s Tehachapi Pass and Altamont Pass became early wind farm sites because mountains funnel the prevailing westerlies, accelerating wind speeds. A ridge facing the prevailing wind can have 2× the speed of a sheltered valley a few kilometers away. Since power scales as \(v^3\), that is 8× the available power.
For wind energy, local terrain and climate determine whether a site has good or poor wind resources. Global patterns set the baseline; local geography amplifies or diminishes it. The westerlies (30-60° latitude) are why Northern Europe, the US Great Plains, and Patagonia have excellent wind resources. New York City sits at 40°N, squarely in the westerlies belt.
10.2 Wind Power Physics
10.2.1 Kinetic Energy of Moving Air
The kinetic energy of air moving at velocity \(v\) is: \[E_k = \frac{1}{2}mv^2\]
For a wind turbine, we care about the power—energy per unit time—available in the wind passing through the rotor’s swept area:
\[P_{wind} = \frac{1}{2}\rho A v^3\]
where:
- \(\rho\) = air density (~1.225 kg/m3 at sea level, 15°C)
- \(A\) = rotor swept area (πR2)
- \(v\) = wind speed
The cubic dependence on velocity is crucial: doubling wind speed increases available power by a factor of eight. This explains why site selection is paramount—a site with 7 m/s average wind has twice the energy of a 5.5 m/s site.
For a modern large turbine with 80-meter blade radius: \[A = \pi \times 80^2 = 20,106 \text{ m}^2\]
At 10 m/s wind speed: \[P_{wind} = 0.5 \times 1.225 \times 20,106 \times 10^3 = 12.3 \text{ MW}\]
At 8 m/s: \[P_{wind} = 0.5 \times 1.225 \times 20,106 \times 8^3 = 6.3 \text{ MW}\]
The 20% reduction in wind speed (10 to 8 m/s) causes a 49% reduction in power. This sensitivity explains why wind developers obsess over site meteorology.
10.2.2 The Betz Limit
A wind turbine cannot extract all the kinetic energy from the wind. If it did, the air would stop completely behind the rotor, blocking further flow. The fundamental limit was derived by Albert Betz in 1919.
The analysis considers air flowing through an imaginary “stream tube” that passes through the rotor:
- Upstream: wind at velocity \(v_1\), pressure \(p_1\)
- At rotor: velocity \(v_2\), pressure drops as energy is extracted
- Downstream: velocity \(v_3\), expanded stream tube
Conservation of mass requires: \[\rho A_1 v_1 = \rho A v_2 = \rho A_3 v_3\]
The power extracted equals the change in kinetic energy: \[P = \frac{1}{2}\rho A v_2 (v_1^2 - v_3^2)\]
The fraction of available power extracted (power coefficient \(C_p\)) depends on the ratio of downstream to upstream velocity. Maximizing \(C_p\):
\[C_{p,max} = \frac{16}{27} \approx 0.593\]
This is the Betz limit: no wind turbine can convert more than 59.3% of the wind’s kinetic energy to mechanical power, regardless of design.
The optimal condition occurs when downstream velocity is one-third of upstream velocity—the rotor slows the wind by two-thirds.
10.2.3 Real-World Power Coefficients
Modern turbines approach but don’t reach the Betz limit:
| Turbine type | Typical \(C_p\) |
|---|---|
| Traditional windmill | 0.10-0.20 |
| Early wind turbines (1980s) | 0.25-0.35 |
| Modern turbines | 0.45-0.50 |
| Theoretical maximum | 0.593 |
The best modern turbines achieve about 85% of the Betz limit—a remarkable engineering achievement. Further improvement is possible but limited; the physics ceiling is close.
Additional losses reduce the conversion from mechanical to electrical power:
- Gearbox: 2-3% loss (for geared turbines)
- Generator: 3-5% loss
- Power electronics: 2-3% loss
- Transformer: 1-2% loss
Overall wind-to-wire efficiency for a modern turbine: 40-45% of the power in the wind becomes electricity.
10.3 The Wind Resource
10.3.1 Wind Speed Distributions
Wind doesn’t blow at constant speed. At any site, wind speed varies continuously, and we characterize the resource by its statistical distribution.
The Weibull distribution fits wind speed data well: \[f(v) = \frac{k}{c}\left(\frac{v}{c}\right)^{k-1} \exp\left[-\left(\frac{v}{c}\right)^k\right]\]
where:
- \(c\) = scale parameter (related to mean wind speed)
- \(k\) = shape parameter (typically 1.5-3 for wind)
When \(k = 2\), this becomes the Rayleigh distribution, a common simplification: \[f(v) = \frac{\pi v}{2\bar{v}^2} \exp\left[-\frac{\pi v^2}{4\bar{v}^2}\right]\]
The shape parameter \(k\) matters:
- Low \(k\) (~1.5): Wide distribution, highly variable winds
- High \(k\) (~3): Narrow distribution, steady winds
For energy production, steady winds (high \(k\)) are preferable—the turbine operates near its optimal range more often.
10.3.2 Wind Shear
Wind speed increases with height above ground due to surface friction. This wind shear is typically modeled as: \[v(z) = v_{ref} \left(\frac{z}{z_{ref}}\right)^{\alpha}\]
where \(\alpha\) is the wind shear exponent, typically 0.1-0.25 depending on terrain:
- Water surfaces: α ≈ 0.10
- Flat grassland: α ≈ 0.14
- Crops, hedges: α ≈ 0.20
- Forest: α ≈ 0.25-0.30
Wind speed at 10 m height: 6 m/s Wind shear exponent: 0.14 (flat terrain)
At 100 m (modern turbine hub height): \[v_{100} = 6 \times \left(\frac{100}{10}\right)^{0.14} = 6 \times 1.38 = 8.3 \text{ m/s}\]
Power increase from 6 to 8.3 m/s: \[\left(\frac{8.3}{6}\right)^3 = 2.6\]
Going from a 10 m measurement tower to a 100 m turbine increases available power by 160%. This is why turbines have grown taller over decades—reaching higher, more energetic winds.
10.3.3 Measuring Wind Resources
Wind resource assessment requires careful measurement:
Meteorological towers: Anemometers at multiple heights measure wind speed for 1-2 years before project commitment. Standard heights: 40, 60, 80 m.
Remote sensing: Lidar (light detection and ranging) and sodar (sound detection and ranging) measure wind profiles without physical towers. Increasingly used for offshore sites where towers are expensive.
Long-term correlation: Short-term measurements are correlated with nearby long-term weather stations to estimate the site’s true long-term resource.
Uncertainty quantification: Project financing requires not just expected production but confidence intervals. Banks typically require P90 estimates (90% probability of achieving at least this level).
10.3.4 Global Wind Resources
The best onshore wind resources are found in:
- Great Plains of North America: Flat terrain, strong pressure gradients
- Northern Europe: Persistent westerlies, offshore potential
- Patagonia: Persistent strong winds, low population
- Inner Mongolia and Xinjiang: Remote but enormous resources
- Coastal areas globally: Land-sea temperature gradients
Offshore wind resources are generally superior:
- Higher average speeds (7-10 m/s vs. 5-8 m/s onshore)
- Lower turbulence (smoother, more consistent flow)
- Less wind shear (friction-free surface)
- Near load centers (coastal cities)
But offshore installation and maintenance are far more expensive than onshore—a tradeoff we’ll examine in Chapter 11.
10.4 Power Curves and Capacity Factors
10.4.1 The Turbine Power Curve
Every wind turbine has a characteristic power curve relating wind speed to electrical output:
- Cut-in speed (3-4 m/s): Below this, turbine doesn’t operate (insufficient torque to overcome friction)
- Rated speed (11-14 m/s): Turbine reaches maximum (rated) power
- Rated power region: Above rated speed, output is limited by generator capacity
- Cut-out speed (25 m/s): Turbine shuts down to prevent damage
Between cut-in and rated speed, output follows approximately \(v^3\) (Betz physics). Above rated speed, pitch control adjusts blade angle to limit power capture.
10.4.2 Capacity Factor
The capacity factor is the ratio of actual energy production to theoretical maximum: \[CF = \frac{\text{Actual production (kWh)}}{\text{Rated power (kW)} \times \text{Hours}}\]
Capacity factor integrates the wind speed distribution with the power curve. For onshore wind:
- Poor site: CF = 20-25%
- Average site: CF = 30-35%
- Excellent site: CF = 40-45%
For offshore wind:
- Average site: CF = 40-45%
- Excellent site: CF = 50-55%
Higher capacity factors mean more energy per MW of installed capacity, reducing the cost per kWh.
A 3 MW turbine with 35% capacity factor: \[E = 3,000 \text{ kW} \times 0.35 \times 8,760 \text{ h/yr} = 9,198,000 \text{ kWh/yr}\]
At \(40/MWh average price:\)\(\text{Revenue} = 9,198 \text{ MWh} \times 40 = \$368,000/\text{yr}\)$
If the turbine costs $3 million installed, simple payback is about 8 years (before O&M, financing, taxes).
10.4.3 Why Capacity Factor Matters
Unlike solar (where capacity factor is primarily geographic and seasonal), wind capacity factors are sensitive to:
Turbine design choices: Larger rotors on a given generator rating increase capacity factor by capturing more energy in low winds. This is why “low specific power” turbines (more rotor area per MW) are popular in moderate wind sites.
Site selection: Within a region, micro-siting can vary capacity factors by 10+ percentage points.
Technology vintage: Modern turbines have higher capacity factors than older ones at the same site due to taller towers, larger rotors, and better control systems.
10.5 Variability and Forecasting
10.5.1 Temporal Variability
Wind varies on multiple timescales:
Seconds to minutes: Turbulence and gusts cause rapid fluctuations. Modern turbines smooth some of this variation; aggregating many turbines smooths more.
Hours: Weather fronts passing, diurnal patterns. This is the timescale most challenging for grid operators.
Days: Weather system cycles, typically 3-7 days in mid-latitudes.
Seasonal: Winter winds are often stronger than summer (at least in mid-latitudes), unlike solar which peaks in summer.
10.5.2 Forecasting
Wind forecasting has improved dramatically:
| Forecast horizon | Error (% of installed capacity) |
|---|---|
| 1 hour | 5-8% |
| 4 hours | 8-12% |
| Day ahead | 15-20% |
| Week ahead | 25-30% |
Short-term forecasts use:
- Numerical weather prediction (NWP) models
- Statistical methods correlating current conditions to future output
- Machine learning combining multiple inputs
Better forecasting enables better grid integration: operators can plan backup generation or storage dispatch with hours or days of lead time.
10.5.3 Smoothing Through Aggregation
Individual turbines are highly variable. But aggregating turbines across geographic areas reduces variability:
- Wind farm smoothing: A 100-turbine farm has less relative variability than one turbine
- Regional smoothing: Winds in Texas don’t correlate perfectly with winds in Oklahoma
- Continental smoothing: At continental scale, weather systems provide natural diversification
Studies show that interconnected grids with geographically dispersed wind farms can significantly reduce overall variability—one reason why transmission is so important for wind integration.
Wind turbines use land differently than solar:
Turbine footprint: Each turbine’s foundation is only ~500 m2—tiny.
Spacing: But turbines must be spaced ~5-10 rotor diameters apart to avoid wake interference. A 100 MW wind farm might span 10-20 km2.
Compatible uses: The land between turbines can often continue in agriculture or grazing. Farmers receive lease payments while farming continues.
Visual impact: Tall turbines are visible for kilometers. Some communities object; others embrace them.
The effective power density (2-3 W/m2 for wind vs. 5-8 W/m2 for solar) is lower, but the dual-use potential is higher. The Equity dimension includes aesthetic preferences and local economic benefits—these are genuinely contested.
10.6 Comparing Solar and Wind Physics
| Attribute | Solar PV | Wind |
|---|---|---|
| Energy source | Electromagnetic radiation | Kinetic energy of air |
| Theoretical limit | Shockley-Queisser (33% single junction) | Betz (59.3%) |
| Practical efficiency | 20-24% (commercial modules) | 40-50% (rotor to grid) |
| Scale dependence | Nearly scale-invariant | Strong economies of scale |
| Resource predictability | Daily cycle highly predictable | Weather-driven, less predictable |
| Night operation | No | Yes |
| Winter performance | Poor (temperate latitudes) | Often better |
| Land intensity | Higher (panels cover ground) | Lower (compatible with farming) |
The physics differences explain why:
- Solar panels can go on every rooftop; wind turbines cannot
- Wind turbines have grown to enormous size; solar panels haven’t
- Wind and solar complement each other in a portfolio
MacKay estimated wind farm power density at ~2 W/m2. Let’s check.
The power in the wind through a rotor: \(P_{wind} = \frac{1}{2}\rho A v^3\). Apply the Betz limit (×0.59) and real turbine efficiency (×0.80) to get \(P_{rotor} \approx 0.24 \times \rho A v^3\).
But turbines must be spaced ~7D × 5D apart to avoid wake interference. For a 160 m rotor, that is 1,120 m × 800 m = 0.9 km2 per turbine. Dividing rotor power by land area, and accounting for ~30% capacity factor, yields roughly 2-3 W/m2.
Real-world check: Whitelee Wind Farm in Scotland occupies 55 km2 with 322 MW installed capacity at 33% capacity factor: \(322 \times 0.33 / 55 = 1.9\) W/m2. MacKay’s estimate checks out.
US average electricity demand: ~450 GW. If wind provides 30%, that is ~135 GW average output.
At 2 W/m2: \(135 \times 10^9 / 2 = 67{,}500\) km2.
That is about the size of West Virginia, or 0.7% of US land area. The wind turbines themselves occupy less than 1% of that footprint; the rest can remain farmland. Transmission and permitting, not land availability, are the real constraints.
A key insight about turbine scaling: spacing scales with rotor diameter (\(D\)), while swept area scales with \(D^2\). Power per unit land area is therefore proportional to \(v^3\) but independent of turbine size. Bigger turbines do not help with power density; they access higher, steadier winds.
10.7 Key Concepts
- Power proportional to velocity cubed: Small wind speed differences have large power impacts
- Betz limit: Maximum 59.3% extraction efficiency, achieved at 2/3 velocity reduction
- Wind shear: Wind speed increases with height; taller turbines capture more energy
- Weibull distribution: Statistical description of wind speed variability
- Capacity factor: Actual production divided by theoretical maximum
- Aggregation smoothing: Geographic diversification reduces variability
10.8 Exercises
Betz limit derivation: A wind turbine slows air from upstream velocity \(v_1\) to downstream velocity \(v_3 = v_1/3\) (optimal condition). What is the air velocity at the rotor plane? (Hint: use the average of upstream and downstream velocities as an approximation.)
Power scaling: If a turbine’s rotor diameter doubles while wind speed stays constant, by what factor does the power output increase? If hub height also doubles (in terrain with α = 0.15), what is the total power increase?
Capacity factor calculation: A 2.5 MW turbine operates at a site with Rayleigh-distributed winds (mean 7 m/s). The turbine’s power curve shows: cut-in 3 m/s, rated at 12 m/s, cut-out 25 m/s. Estimate the capacity factor by assuming the turbine produces rated power whenever wind exceeds 12 m/s and zero otherwise. (This underestimates; why?)
Wind shear value: Wind at 50 m is measured at 6.5 m/s. Local terrain gives α = 0.20. What is the wind speed at 150 m? What is the ratio of power available at 150 m vs. 50 m?
Offshore advantage: An onshore site has 7 m/s average wind at hub height (80 m). A nearby offshore site has 9 m/s at the same height. If turbines and installation are identical, what is the ratio of capacity factors (approximately)?
Variability reduction: Two wind farms, each with individual output varying ±30% around the mean, have output correlation of 0.5. By how much is the combined variability reduced compared to a single farm? (Hint: use portfolio theory from finance.)
This chapter establishes the Principles of wind energy—the physics that constrains all wind technology:
- The Betz limit sets an absolute efficiency ceiling
- The cubic power-velocity relationship makes site selection critical
- Wind shear rewards height, driving turbine scale increases
- Inherent variability creates integration challenges
Chapter 11 will show how these Principles become Technologies (large rotors, tall towers, pitch control) that have been commercialized as Products (specific turbine models from specific manufacturers). The historical and policy journey follows in Chapter 12.