Technical Article

Using Probability Models to Understand Electrical System Reliability Engineering

November 02, 2023 by Bob Odhiambo

Reliability engineering in electrical systems is essential to ensuring system efficiency. This article explores how probability models are used in electrical system reliability.

Reliability engineering is the foundation of reliable electricity generation, utilization, and distribution in electrical systems. Probability engineering is the discipline that ensures all electrical device components and systems function as intended and consistently. This makes reliability engineering significant in all electrical systems, from ensuring safety while operating electrical equipment to reducing power supply interruption.

As this article explores reliability engineering, it will dive deep into the theoretical foundations and use of probability models to optimize electrical systems for reliability.


Power substation uses reliability engineering to ensure uninterrupted operation. Image used courtesy of Unsplash


Probability Theory in Electrical System Reliability

In complex electrical systems such as high-voltage transmission, converters, and even voltage regulation devices like transformers, failures can occur that cause losses through downtime. To predict and solve this failure factor, probability theory can determine electrical system reliability.

Probability can be used to model events such as circuit breaker activation and semiconductor failure and anticipate their outcome using a mathematical scale represented by 0 and 1, where 0 implies impossibility of occurrence, while 1 implies high certainty.


Mathematical Models for Electrical System Reliability

To understand and predict how electrical systems behave, reliability engineering uses mathematical models with probability distributions, including:

  • Weibull distribution
  • Exponential distribution
  • Markov model
  • Log-normal distribution


Exponential Distribution Model

By assuming a constant failure rate, the time probability between the failure of the electrical systems or components can be determined using the exponential distribution model. The building blocks of this distribution model include the probability density function (PDF), the mean time between failures (MTBF), and the reliability function.

In high-voltage transmission systems, components like high-voltage transformers form a major part of uninterrupted power transmission due to the use of PDF of the exponential distribution. Mathematically represented as \(f(x)=\lambda*E^{(-\lambda x)}\), PDF can be used by engineers to get probability metrics essential for predicting transformer failure within a particular timeframe using functions like MTBF. The PDF formula evaluates how likely an electrical system is to fail per unit of time, in which () represents the number of failures per hour. Due to the memorylessness characteristic of the exponential distribution model, the probability of an electrical system failing in the next time interval is deemed wholly independent of the past.

Example calculation of PDF for exponential distribution: Consider a high-voltage transformer with a failure rate of 0.0001 failures per hour. Calculate the mean time between failures.

MTBF for the transformer can be calculated using the formula:



This implies the transformer has a working time of 10,000 hours before expected failure.


Weibull Distribution Model in High-Voltage Transmission Reliability

The Weibull distribution model can be used in high-voltage transmission to predict failure patterns resulting from aging effects and environmental stresses. Offering flexibility, the model can work out various failure patterns in components within high-voltage transmission systems like transformers, insulators, and even circuit breakers. It provides a proactive maintenance strategy for engineers as components are more prone to wear and can handle varying failure rates over time. To mathematically express reliability engineering, the model uses the Weibull probability density function to analyze high-voltage transmission systems and is determined using the following formula:


The probability density function is stated by f(x), indicating a system component will likely fail at a specified time (x). The scale parameter is represented by (α) and (β) represents the shape parameter that shows a failure pattern in the distribution in which a value greater than one represents wear-out failure and equal to one shows early life failure.

From the graph in Figure 1, the reliability evaluation is assessed using a scale parameter of 1. 

When λ= 1, K =0.5, the shape parameter is less than 1, meaning the electrical component is likely in its early life failure and likely to fail as soon as it enters service. 

When λ= 1, K =5, the shape parameter is greater than 1, which means the electrical component will likely fail due to being worn out or aging over time. 

When λ= 1, K = 1, the shape parameter equals 1, meaning that the electrical component has a constant failure pattern over time.


Figure 1. Graph of Weibull Distribution Function where the scale and shape parameters can be used to determine the time to failure. Image used courtesy of Bob Odhiambo


Example calculation in estimating transformer reliability: Consider a transformer whose scale parameter (α) is 10,000 hours, shape parameter (β) is 1.5, and specific time (t) is 5000 hours. Find an estimate of the transformer's reliability using the Weibull reliability function.


Using the formula below, substitute the parameters in the above question to approximate the function.



Following the above calculations, the probability of the transformer still being able to work at 5,000 hours is 70.71%. 


Log-Normal Distribution in the Reliability of High-Voltage Transmission

Systems in high-voltage transmission often experience fluctuating stress levels resulting from operation demands and changing environmental conditions. To predict the lifetime of the components in the transmission system, Log-Normal distribution is used. The reliability model can predict the lifecycle of components like insulators and transformers where voltage, load, and temperature variations may cause varying stress rates over time.

To evaluate the Log-Normal probability density function, use the formula:


where (x) is a positive continuous value representing the age of the component in the electrical system or the time-to-failure, while () represents the central tendency of distribution and is the mean of the natural log of (x). () controls the spread of the distribution and is the Natural Log standard deviation of (x).


Reliability Assessment: Component- and System-Level Reliability

Electrical system reliability assessment examines component-level and system-level reliability. In this case, a whole electrical system can be assessed using metrics such as mean time between failure and system availability. Compared to component-specific reliability using specific measures of rate of failure and MTBF, the reliability can be improved by implementing redundancy strategies like N+1 and N+M as a backup in the event of a component failure. Another method is using parallel systems to provide a failover system that can share loads in the event of failure.


Maintenance and Reliability-Centered Maintenance

Maintenance becomes part of the system's routine to ensure that electrical systems stay operational for long periods. Preventive and predictive maintenance help reduce unplanned failures in electrical systems, reducing downtime. Predictive maintenance uses real-time data from electrical components, allowing for efficient and cost-effective prediction. One additional maintenance method, reliability-centered maintenance, focuses on individual electrical system component needs—specifically, how critical a component is and the type of failure modes in the electrical system. This allows for good resource allocation for electrical system maintenance, making it even more cost-effective using probability models like Weibull distribution and others.


Takeaways of Reliability Engineering in Electrical Systems

Regarding providing electrical services, reliability engineering is important in ensuring consumers get what they pay for without interruptions. Researchers and engineers can draw valuable insights from this article, enabling them to handle maintenance based on probability model evaluations and ensuring that electrical systems last far into the future.