Problem 161 - Applied Statistics and Probability for Engineers

Question

An article in the Journal of Geophysical Research ["Spatial and Temporal Distributions of U.S. of Winds and Wind Power at 80 m Derived from Measurements" (2003, vol. 108)] considered wind speed at stations throughout the United States. A Weibull distribution can be used to model the distribution of wind speeds at a given location. Every location is characterized by a particular shape and scale parameter. For a station at Amarillo, Texas, the mean wind speed at \(80 \mathrm{~m}\) (the hub height of large wind turbines) in 2000 was \(10.3 \mathrm{~m} / \mathrm{s}\) with a standard deviation of \(4.9 \mathrm{~m} / \mathrm{s} .\) Determine the shape and scale parameters of a Weibull distribution with these properties.

Step-by-Step Solution

Answer

The shape parameter \(k \approx 2.5\) and the scale parameter \(\lambda \approx 11.48\).

Step 1: Understand the Weibull Distribution

The Weibull distribution is a continuous probability distribution defined by two parameters: the shape parameter \(k\) and the scale parameter \(\lambda\). The probability density function (PDF) is given by \[ f(x|k,\lambda) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-\left(\frac{x}{\lambda}\right)^k} \] where \(x \geq 0\). The mean \(\mu\) and standard deviation \(\sigma\) of the Weibull distribution are used to determine these parameters.

Step 2: Express the Mean and Standard Deviation in Terms of Parameters

The mean \(\mu\) and standard deviation \(\sigma\) of the Weibull distribution can be expressed as follows: - The mean \(\mu = \lambda \Gamma(1 + 1/k)\)- The variance \(\sigma^2 = \lambda^2 (\Gamma(1 + 2/k) - (\Gamma(1 + 1/k))^2)\) We need to solve these equations to find both \(\lambda\) and \(k\).

Step 3: Use the Given Mean and Standard Deviation

The mean wind speed at the station is given as \(\mu = 10.3\) m/s and the standard deviation as \(\sigma = 4.9\) m/s. Plug these values into the expressions from Step 2:- \(10.3 = \lambda \Gamma(1 + 1/k)\)- \(4.9^2 = \lambda^2 (\Gamma(1 + 2/k) - (\Gamma(1 + 1/k))^2)\)

Step 4: Solve for Shape Parameter \(k\)

Solving these equations involves finding an approximate solution for \(k\) using numerical methods. Typically, this involves testing several values for \(k\) until both the mean and variance equations are satisfied. A trial and error or an iterative method is used to narrow down the correct \(k\). Through this process, \(k\) is found to be approximately 2.5 for this problem.

Step 5: Solve for Scale Parameter \(\lambda\)

Once \(k\) is found, substitute it back into one of the equations to solve for \(\lambda\). Using \(\mu = \lambda \Gamma(1 + 1/k)\):\(10.3 = \lambda \Gamma(1 + 1/2.5)\)Solving gives \(\lambda \approx 11.48\).

Key Concepts

Probability Distribution

Probability distributions are mathematical functions that provide the probabilities of occurrence of different possible outcomes in an experiment. In simple terms, they tell us how the probabilities are distributed across various outcomes. One such distribution is the Weibull distribution.

The Weibull distribution is particularly useful because it can effectively model various types of data, such as life data, and in this case, wind speeds. It is defined by two parameters: the shape parameter and the scale parameter. These parameters influence the distribution's form, affecting its probability density and cumulative distribution functions. Understanding how these parameters work helps in accurately predicting and analyzing data behaviors under different conditions.

For instance, the Weibull distribution can adjust its form to fit numerous situations. Whether the probability of failure over time or variations in wind speeds, it helps in situations where data may not strictly follow standard distributions like normal or exponential.

Shape Parameter

The shape parameter, usually denoted as \(k\), is a crucial part of the Weibull distribution. It dictates the shape of the distribution's curve, influencing its skewness and the rate at which probabilities change. In essence, the shape parameter describes how sharp or flat the distribution appears.
The value of \(k\) affects the distribution significantly:

• If \(k < 1\), the distribution has a decreasing failure rate, meaning most events occur early.
• If \(k = 1\), the distribution simplifies to an exponential form, indicating a constant failure rate.
• If \(k > 1\), it suggests an increasing failure rate, with events occurring later and less frequently at the start.

This flexibility in form allows the Weibull distribution to tailor its properties to match real-world data closely, such as wind speed variations.

Scale Parameter

The scale parameter, denoted as \(\lambda\), is another essential aspect of the Weibull distribution. This parameter sets the scale of the distribution, determining the "spreading" of the data. It effectively shifts the distribution left or right on the horizontal axis, affecting overall variance.
This parameter impacts the distribution's range and expectation level. Larger values of \(\lambda\) spread the data wider apart, leading to increased variance. Conversely, smaller values imply a tighter clustering of data points around the mean. These adjustments make the Weibull distribution useful in applications where data must be stretched or compressed to fit specific patterns, as is often necessary in wind speed analysis.

Wind Speed Modeling

Wind speed modeling is critical for numerous practical applications, such as wind energy assessment and forecasting weather conditions. Accurately modeling wind speeds allows for efficient design and operation of wind turbines, thereby optimizing energy capture.
Using the Weibull distribution for wind speed modeling is advantageous due to its flexibility and ability to fit a wide range of wind data profiles. By adjusting its shape and scale parameters, the distribution can mimic real wind speed observations at different locations efficiently. This modeling helps in predicting how often certain wind speeds occur, aiding in the planning and management of wind resources.

By understanding and applying these models, we can design more efficient wind power systems and make better investment decisions in renewable energy sources.