wind_turbine_model.sagews

Modeling the energy produced by a wind turbine

(Content prepared with help from former student Amara Yeb, EC '15)

Introduction The overall objective here is to explore the feasibility of meeting the energy needs of our campus (Earlham College) using wind turbines. For simplicity, we will model the operation of one turbine. As a first step, we must estimate the average or typical energy usage of our campus. Shown below is the annual electricity usage for the years 2008-2013. Year Electric Consumption (kWh) 2008-2009 13,571,150 2009-2010 13,754,406 2010-2011 12,657,243 2011-2012 13,674,575 2012-2013 11,703,801

Note that the campus also consumes energy in other forms, such as natural gas, particularly for heating. But for the purpose of this study we only consider electric consumption.

Wind energy modeling: Some background

In a nutshell, here is how a wind turbine works:

• It converts the wind’s kinetic energy into rotational KE in the turbine.
• The rotational KE is converted into electrical energy.
• Conversion depends on wind speed and swept area of turbine.
• Efficiency losses occur at each stage.
  
  
  $P_w=$ wind power; $P_e=$ electric power output;
  $C_p$, $C_t$, $C_g$, are constant efficiency coefficients

It is theoretically possible, from a simplified 1-D form of energy conservation laws, to estimate

$~~~~~~~~~~~~\displaystyle P_w = \frac{1}{2}\rho A u^3 ~~~~~~~~~~~~ \displaystyle P_e = C_p C_t C_g P_w$

where $\rho=$ air density, $A=$ swept area, $u=$ wind speed.

However, in practice, it is very difficult to estimate reliable values for all the needed parameters. Therefore, a more common way to model the electric power output of a wind turbine is to use a "power curve," such as the one shown in the sketch. From the curve, the electric power is a function of $u$, and has the form $~~~~~~~~P_e(u)=\left\{\begin{array}{ll} a + b u^\alpha, & \mbox{ if } u_c \le u \le u_r \\ P_{er}, & \mbox{ if } u_r [removed]u_f \end{array}\right.$

Wind turbine power curve

where
$~~~~u_c=$ cut-in wind speed, where electric power output rises above 0
$~~~~u_r=$ rated wind speed
$~~~~u_f=$ furling wind speed, at which turbine is shut off to prevent damage
$~~~~\alpha$ is a fitting parameter whose value is typically close to 2
$~~~~P_{er}$ is the rated power output under optimal wind conditions.
Numerical values of all these parameters depend on design, manufacture and installation properties of the wind turbine.

How to estimate total power output?

By integration, of course! But we need one more piece of information, i.e., the distribution of actual wind speeds at the turbine location. This typically comes from historical meteorological data, in the form of a probability density function. For example, a Weibull distribution is frequently used to model wind speed data.

Weibull pdf (probability density function): $\displaystyle f(u)=\frac{\frac{k}{c}(\frac{u}{c})^{k-1}}{e^{(\frac{u}{c})^k}}$ $k$ and $c$ are model parameters that depend on the median and variability in wind speeds. Estimated power output: $\displaystyle P_{tot}=\int_0^\infty P_e(u) f(u) du$ where $P_e(u)$ and $f(u)$ are both given by expressions above.

Wind speed distribution

Simulation studies

We want to compute the integral: $\displaystyle P_{tot}=\int_0^\infty P_e(u) f(u) du$

with $~~~~~P_e(u)=\left\{\begin{array}{ll} a + b u^\alpha, & \mbox{ if } u_c \le u \le u_r \\ P_{er}, & \mbox{ if } u_r [removed]u_f \end{array}\right.$       and $~~~~~~\displaystyle f(u)=\frac{\frac{k}{c}(\frac{u}{c})^{k-1}}{e^{(\frac{u}{c})^k}}$

The wind turbine we are investigating for installation is manufactured by General Electric, with model number GE 1.7-103. Based on the model and installation, we have the following estimates for the parameters
$~~~u_c ~ \sim ~ 3\mbox{ to }4 ~m/s$; $~~u_r ~ \sim ~ 12\mbox{ to }14 ~m/s$ $~~u_f = 20 ~m/s$;
$~~P_{er} ~ \sim ~ 3000\mbox{ to }4000 ~kW$ per hour, depending on value or $u_r$;
$~~\alpha ~ \sim ~ 2\mbox{ to }2.3$; $~~k = 2.323$; $~~c=6.52 ~m/s$.
$~~a$ and $b$ can be found if we know $u_c$, $u_r$, by curve-fitting.

The Sage code below computes the integral, which gives the average power generated per hour. Multiplying this by $24\times 365$ gives the average power generated per year in $kWh$.

# Define my own integration function, because for some reason 
# the builtin "integral" function hangs on my computation.  
# This is a very simple, mid-point rule based integration 
# function. The inputs are: the integrand, the end-points, 
# and "n," the number of (uniform) integration intervals.
def my_integral(fcn, a, b, n):
    deltax = (b-a)*1.0/n
    xs = [ a + deltax*i for i in range(n+1) ]
    ysmid = [ fcn( (xs[i]+xs[i+1])/2.0 ) for i in range(n) ]
    return deltax*sum(ysmid)

# Now we're ready to compute the integral:
u = var('u')
uc = 4            # cut-in speed
ur = 12           # rated speed
uf = 20           # furling speed
per = 3157        # rated power output
alpha = 2.2       # exponent in power curve

c = 6.52          # speed parameter in Weibull distribution
k = 2.323         # exponent in Weibull distribution
f(u) = ( (k/c)*(u/c)^(k-1) ) / ( exp((u/c)^k) ) # the Weibull distribution

a = (per*uc^alpha) / (uc^alpha - ur^alpha)
b = - per / (uc^alpha - ur^alpha)

# Next, define piecewise function to be integrated:
piece1(u) = 0*f(u)
piece2(u) = (a + b*u^alpha)*f(u)
piece3(u) = per*f(u)
pe = piecewise( [[(0,uc), piece1], [(uc,ur), piece2], [(ur,uf), piece3] ] )

# Integrate and find power generated per hour:
pe_per_hour = my_integral(pe, 0, 20, 200)
pe_per_year = pe_per_hour*24*365

print "Estimated power generated per hour = %.1f kW" % pe_per_hour
print "Estimated power generated per year = %i kWh" % pe_per_year

Estimated power generated per hour = 608.2 kW Estimated power generated per year = 5327728 kWh

Exercise 1

Mathematically modeling real-world processes often involves making a variety of approximations at different stages. In order for the model to be useful, it is important to estimate the impact of our approximations on the model's results.

One key area of approximation in most math models is numerical estimates of parameter values. For example, in the wind turbine model, the values of $u_c$, $u_f$, etc., are empirical estimates that will necessarily have some errors. A question of practical interest is: How much will a small error in the value of $u_c$ affect the predicted power output?

In this exercise we will vary $u_c$ up and down by 10% and study the effect on the power output. Use the Sage code given above to do this.

Exercise 2

Suppose we were to install the same wind turbine somewhere on the campus of Kasetsart University to provide a portion of the electric power used on campus. Estimate how many kWh of energy would be generated per year.

Steps and hints:

Our model has 2 key components: the wind turbine power curve, and the probability density function (pdf) of wind speeds.

Assume the wind turbine power curve remains the same, since we are using the same turbine and installation type.

The pdf model of wind speeds will depend on local data from Bangkok and Kasetsart. To keep things simple, let's use a different pdf model (called the Rayleigh distribution) that only requires knowledge of mean wind speeds. It is given by $$f(u)=\frac{\pi}{2} \left(\frac{u}{u_m}\right) \exp\left(\frac{-\pi}{4} \left(\frac{u}{u_m}\right)^2 \right)$$ So, the only additional work to be done is, find a reasonable value for $u_m$, the mean wind speed at Kasetsart.

After that, compute the integral: $\displaystyle P_{tot}=\int_0^\infty P_e(u) f(u) du$

f(u) = ( (pi/2)*(u/um) ) * ( exp((-pi/4)*(u/um)^2) )