# Symbolic Math Toolbox

## Modeling the Power Generated by a Wind Turbine

This example uses Symbolic Math Toolbox to derive an analytical expression for the average power generated by a wind turbine.

### Background

The total power delivered to a wind turbine can be estimated by taking the derivative of the wind's kinetic energy. This results in the following expression:

(1)

where A = swept area of turbine blades, in m2
ρ = air density, in kg/m3
u = wind speed, in m/s

The process of converting wind power to electrical power results in efficiency losses, as described in the diagram below.

The electrical power output of a practical wind turbine can be described using the following equation:

(2)

where Ctot = overall efficiency = CpCtCg

Overall efficiency is typically between 0.3 and 0.5, and varies with both wind speed and rotational speed of the turbine. For a fixed rotational speed, the turbine operates most efficiently at what's known as the rated wind speed. At this wind speed, the electrical power generated by a wind turbine is near its maximum (Per), and overall efficiency is denoted CtotR.

(3)

For a fixed rotational speed, the electrical power output of a wind turbine can be estimated using the profile below.

Where
uc = cut-in speed, the speed at which the electrical power output rises above zero and power production starts
ur = rated wind speed
uf = furling wind speed, the speed at which the turbine is shut down to prevent structural damage

### Derive Equation for Average Power

#### I. Define Piecewise Expression for Power

We define a piecewise function that describes wind turbine power.

Pe := piecewise([u < u_c, 0],[u_c <= u <= u_r, a+b*u^k],
[u_r <= u <= u_f, Per], [u > u_f, 0])

Where variables a and b are defined as follows:

a := Per*u_c^k/(u_c^k-u_r^k)

b := Per/(u_r^k-u_c^k)

#### II. Define External Wind Conditions

The rated power output offers a good indication of a how much power a wind turbine is capable of producing, however we'd like to estimate how much power (on average) the wind turbine will actually deliver. To calculate average power, we need to account for external wind conditions. A Weibull distribution does a good job of modeling the variance in wind; therefore the wind profile can be estimated using the following probability density function:

(4)

In general, larger 'c' values indicate a higher median wind speed, and larger 'k' values indicate reduced variability. We call a custom function called `windProfile.mu` to illustrate the variability in wind for a wind farm site with c=12.5 and k=2.2.

`read`(NOTEBOOKPATH."windProfile.mu");
`windProfile`(2.2,12.5)

#### III. Calculate Average Power

The average power output from a wind turbine can be obtained using the following integral:

(5)

Power is zero when wind speed is less than the cut in speed and greater than furling speed. Therefore, the integral can be expressed as follows:

(6)

There are two distinct integrals in equation (6). We plug equation (4) into these integrals and simplify them using the substitutions:

, and

. This simplifies our original integrals to the following:

(7)

(8)

Solving these integrals and then replacing x with

yields:

int1 := int(exp(-x),x):
int1 := subs(int1,x=(u/c)^k)

int2 := int(x*exp(-x)*c^k,x):
int2 := subs(int2,x=(u/c)^k)

Subsituting the results into equation (6) yields an equation for average power output of the wind turbine.

Peavg := (subs(a*int1,u=u_r)-subs(a*int1,u=u_c)) +
(subs(b*int2,u=u_r)-subs(b*int2,u=u_c)) +
(subs(Per*int1,u=u_f)-subs(Per*int1,u=u_r))

Click on image to see enlarged view.

We could read this equation into MATLAB using the `getVar` function in Symbolic Math Toolbox, and perform simulation studies to determine the average power generated for various wind turbine configurations and wind farm sites.