Wind Turbine Power Calculations

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RWE npower is a prominent integrated UK energy company and is a part of the RWE Group, one of Europe's leading utilities. We possess and manage a diverse portfolio of power plants, including gas-fired combined cycle gas turbines, oil, and coal-fired power stations, as well as Combined Heat and Power plants on industrial sites that provide both electrical power and heat. RWE npower also boasts a robust in-house operations and engineering capability that supports our existing assets and fosters the development of new power plants.

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Our retail business, npower, stands as one of the largest suppliers of electricity and gas in the UK.

In the UK, RWE is at the forefront of renewable energy production. npower renewables leads the UK wind power market and excels in hydroelectric generation. We developed the UK's inaugural major offshore wind farm, North Hoyle, located off the North Wales coast, which commenced operations in 2003. Under the RWE Power International brand, RWE npower offers specialized services encompassing all aspects of power plant ownership and operation, spanning construction, commissioning, operations and maintenance, and eventual decommissioning.

Introduction

Scenario:

Wind turbines operate by converting the kinetic energy present in the wind into rotational kinetic energy within the turbine, subsequently transforming it into electrical energy. This electrical energy can then be supplied, via the national grid, to serve various purposes throughout the UK. The energy available for conversion primarily hinges on wind speed and the swept area of the turbine. When planning a wind farm, it is imperative to have a precise understanding of the expected power and energy output of each wind turbine to determine its economic viability.

Problem Statement:

Recognizing the critical economic significance of knowing the power and energy generated by different types of wind turbines under varying conditions, this exemplar aims to calculate the rotational kinetic power produced by a wind turbine at its rated wind speed. The rated wind speed represents the minimum wind speed at which a wind turbine achieves its maximum rated power output.

Mathematical Model

The following table outlines the definitions of the various variables employed in this model:

Variable	Definition	Units
E	Kinetic Energy	Joules (J)
ρ	Density	Kilograms per cubic meter (kg/m³)
m	Mass	Kilograms (kg)
A	Swept Area	Square meters (m²)
v	Wind Speed	Meters per second (m/s)
pC	Power Coefficient	Unitless
P	Power	Watts (W)
r	Radius	Meters (m)
dt	Change in time	Seconds (s)
dm	Mass flow rate	Kilograms per second (kg/s)
x	Distance	Meters (m)
dE	Energy Flow Rate	Joules per second (J/s)
t	Time	Seconds (s)

Under constant acceleration, the kinetic energy of an object with mass m and velocity v is equal to the work done W when displacing that object from rest to a distance s under a force F, i.e.:

^Fs_WE = (1)

According to Newton’s Law, we have:

^ma_F = (2)

Hence,

^mas_E = … (1)

Using the third equation of motion:

^as_u^v₂₂₂⁺⁼

we get:

( )

^suva₂²²⁻⁼

Since the initial velocity of the object is zero, i.e., 0=u, we get:

^suva₂²⁼

Substituting it into equation (1), we find that the kinetic energy of an object in motion is:

²₂₁^mvE = … (2)

The power in the wind is determined by the rate of change of energy:

^dt_dmv^dt_dEP²²¹⁼ … (3)

As the mass flow rate is given by:

^dt_dxA^dt_dm^ρ⁼

and the rate of change of distance is given by:

^v_dt^dx⁼

we have:

^Av_dt^dm^ρ⁼

Hence, from equation (3), the power can be defined as:

³₂₁^AvP^ρ⁼ … (4)

Calculations with Given Data

In 1919, the German physicist Albert Betz determined that no wind turbine can convert more than 16/27 (approximately 59.3%) of the kinetic energy of the wind into mechanical energy to turn a rotor. This fundamental principle is known as the Betz Limit or Betz' Law. Consequently, the theoretical maximum power efficiency for any wind turbine design is 0.59, meaning that no more than 59% of the energy carried by the wind can be extracted by a wind turbine. This efficiency is quantified by the "power coefficient," defined as:

⁵⁹⁰_maxp =

However, it's essential to note that wind turbines cannot operate at this maximum limit. The power coefficient (pC) is unique to each turbine type and is a function of the wind speed at which the turbine operates. When considering various engineering requirements, such as strength and durability, the real-world efficiency falls well below the Betz Limit. In practice, values of 0.35 to 0.45 are common even in well-designed wind turbines. Furthermore, when factoring in other components of a complete wind turbine system, such as the gearbox, bearings, and generator, only 10-30% of the power from the wind is actually converted into usable electricity. Therefore, we need to incorporate the power coefficient into equation (4), and the extractable power from the wind is given by:

^pavail_CAvP³²¹^ρ = …(5)

The swept area of the turbine can be determined from the length of the turbine blades using the equation for the area of a circle:

^2rA_π = … (6)

Here, the radius is equivalent to the blade length.

Now, let's perform calculations using the provided data:

Blade length, l = 52 m
Wind speed, v = 12 m/s
Air density, ρ = 1.23 kg/m³
Power Coefficient, pC = 0.4

Inserting the value for the blade length as the radius of the swept area into equation (8), we have:

²₂²₈₄₉₅⁵²₅₂_m_rA_mr^l⁼⁼^π^π

We can then calculate the power converted from the wind into rotational energy in the turbine using equation (7):

^MW.^..^CρAvP_pavail⁶³^40128495231²¹²¹³³⁼^××××⁼

Conclusion

The power coefficient, pC, is typically determined by turbine designers, and it's essential to comprehend the relationship between all the involved factors. This equation is especially valuable for calculating power at wind speeds other than the rated wind speed.

Understanding how a turbine behaves under different wind speeds is critical for assessing potential income losses during turbine downtime. It also aids in identifying issues when the turbine's power output falls below estimated values.

Predicting the energy production of a turbine holds significance in the energy market since energy is sold before it's generated. Accurate energy calculations are vital for market energy balance and a company's income forecasts.