Problem 93 - Applied Statistics and Probability for Engineers

Question

An article in International Journal of Electrical Power \& Energy Systems ["Stochastic Optimal Load Flow Using a Combined Quasi-Newton and Conjugate Gradient Technique" (1989, Vol.11(2), pp. 85-93)] considered the problem of optimal power flow in electric power systems and included the effects of uncertain variables in the problem formulation. The method treats the system power demand as a normal random variable with 0 mean and unit variance. (a) What is the power demand value exceeded with \(95 \%\) probability? (b) What is the probability that the power demand is positive? (c) What is the probability that the power demand is more than -1 and less than \(1 ?\)

Step-by-Step Solution

Answer

(a) -1.645; (b) 0.5; (c) 0.6826.

Step 1: Understanding the Normal Distribution

We have a normal random variable representing power demand with mean \(\mu = 0\) and variance \(\sigma^2 = 1\). To solve this problem, we need to understand the properties of a standard normal distribution, \(Z\), which is \(N(0,1)\).

Step 2: Calculate the Z-value for 95% Probability

To find the power demand value that is exceeded with 95% probability, we find the Z-score corresponding to the 5th percentile (since 100% - 95% = 5%). From Z-tables or using statistical software, \(Z_{0.05} = -1.645\), meaning there is a 95% probability that the demand is below \(-1.645\). Therefore, the power demand exceeded with 95% probability is \(-1.645\).

Step 3: Probability of Positive Power Demand

The probability that the power demand is positive translates to finding \(P(Z > 0)\). Since the distribution is symmetric around zero, \(P(Z > 0) = 0.5\).

Step 4: Probability of Demand Between -1 and 1

To find the probability that the power demand is between -1 and 1, we calculate \(P(-1 < Z < 1)\). This can be found using Z-tables: \(P(Z < 1) - P(Z < -1)\). \(P(Z < 1)\) is approximately 0.8413 and \(P(Z < -1)\) is approximately 0.1587. Subtracting these values gives \(0.8413 - 0.1587 = 0.6826\). Therefore, the probability is approximately 0.6826.

Key Concepts

Optimal Power Flow

Optimal Power Flow (OPF) refers to the process of optimizing the operation of electrical power systems by determining the most efficient operating conditions. This involves managing the power generation, distribution, and consumption in order to minimize costs or losses while meeting the required demand. In essence, OPF aims to find the best way to run the electricity network to make sure everyone has power, and it's done in the cheapest way possible.

There are multiple objectives when dealing with Optimal Power Flow, including:

• Minimizing the cost of power generation
• Reducing transmission losses
• Maintaining system reliability and security
• Ensuring that all power demands are met without violating the system's operational limits

Incorporating stochastic elements, like random variable power demands, into the OPF makes it more robust. It allows system operators to plan for uncertainties, such as fluctuating demand levels.

Probability Theory

Probability Theory provides a mathematical framework to quantify uncertainty. It allows us to make informed deductions and predictions about various scenarios via probabilities. In the context of the normal distribution problem provided, probability theory helps determine the likelihood of certain power demand values occurring.

Key principles include:

• Probability Space: The set of all possible outcomes and associated probabilities.
• Random Variables: Variables whose outcomes are determined by chance.
• Probability Distribution: Describes how probabilities are distributed over the values of a random variable.

Using probability theory, we can calculate specific probabilities related to the power demand and understand the behavior of demand under uncertainties.

Random Variables

A random variable is a fundamental concept in probability and statistics, representing a variable that can take on different values, each with a certain probability. In this exercise, the power demand is modeled as a normal random variable with a mean of 0 and a standard deviation of 1.

Random variables can be either discrete or continuous:

• Discrete random variables have specific outcomes, like flipping a coin.
• Continuous random variables can take any value within a range, such as the power demand in this problem.

By modeling the power demand as a random variable, we account for its inherent uncertainty and can use statistical methods to predict its behavior under various conditions.

Z-score

The Z-score, also known as the standard score, is a measure that describes a value's position relative to the mean of a group of values, expressed in terms of standard deviations. It is a crucial concept in statistics, especially when dealing with normal distributions.

The Z-score formula is:\[Z = \frac{(X - \mu)}{\sigma}\]Where:

• \(X\) is the raw score
• \(\mu\) is the mean of the distribution
• \(\sigma\) is the standard deviation of the distribution

In the given problem, Z-scores help determine the probability of the power demand falling below certain levels or within specified ranges. For instance, converting a demand value to a Z-score allows us to use Z-tables to find probabilities related to that score. This is essential in scenarios like finding the 5th percentile or determining the probability of demand being between specific values.