Introducing a new system performance function to formulate reliability analysis problems of wind turbines

Abstract

Consideration of safety is one of the current requirements to design a new system that is often implemented by defining the system failure probability. Renewable energy systems (RESs) do have the same requirements when it comes to safety and reliability. When designing a wind turbine, as a RES, its reliability is of the highest importance. So, efficient reliability models are required to ensure the turbine is working safely to generate electricity. A new model is introduced in this paper by taking into account the wind speed and the wind angle as two effective factors. These two random variables are reported to follow the Gaussian and Weibull probability distributions, and so are employed to define a limit-state function for the turbines. This limit-state function, which is also called system performance function, will then be used to find out the system failure probability.

Keywords

Wind turbines renewable energy reliability analysis performance function

Introduction

Increasing global warming in the last couple of years has made it clear that it is important to move away from fossil fuels as soon as possible. In this case, renewable energy systems (RESs), such as wind turbines and solar panels, play significant roles to reduce carbon emission globally (Morawski, 2022). However, each RES is associated with several challenges regarding their economic efficiency, engineering design, reliability features, etc, which need to be addressed properly as these challenges may bring a whole renewable energy station to shut down (Hill, 2022).

Given the random nature of renewable energy resources, such as availability of irradiance and wind, machine learning and data mining algorithms are used to predict this availability in order to study system behavior in the future (Shakya, 2021; Vennila et al., 2022). Although several machine learning techniques are used to predict solar panel energy production, a new finding shows that a hybrid method that incorporates machine learning techniques and statistical methods is a preferred method for PV energy predictions.

The Internet of Things (IoT) is also applied to monitor solar power stations (Shakya, 2021). Since huge areas are required to install large-scale solar power stations, it would be an important task to analyze and monitor functionality of the panels. For this purpose, algorithms are proposed that employ the IoT methods to enhance these systems’ reliability. Furthermore, a widely applied technique to set up optimization models of solar PV cells is to estimate their parameters. In this case, a newly introduced method uses stochastic fractal search algorithms for estimating current-voltage parameters for modeling purposes (Rezk et al., 2021).

When it comes to the wind power generation by wind turbines, several challenges are associated with the random nature of the wind. Although the wind power has great potentials to help environment due to its low emission compared with traditional fossil fuels, its randomness and intermittent characteristics have brought new barriers to research and industry in order to design these systems. The challenges increase uncertainty level and so need to be addressed by relevant techniques, such as reliability analysis models (Deng et al., 2021).

Offshore wind power has also attracted researchers’ attention to concentrate on this renewable energy resource. It is reported that there are rich resources of wind available in (mostly untouched) offshore locations. The existing offshore wind turbines are often constructed with fixed structures, such as monopile, jacket, and tripod support. It has been found that semi-probabilistic nature of the existing frameworks takes uncertainty of these systems into account (Okpokparoro and Sriramula, 2021).

On the top of the regular challenges of wind turbines, there are research findings showing that offshore turbines would introduce more reliability related issues to the process. To manage and overcome these problems, the Gaussian process regression is reported to be one of the most efficient methods for this purpose (Wilkie and Galasso, 2021). Despite of the mentioned great potential of the wind resources, these power generating stations need huge maintenance costs that makes it challenging to deal with these systems and so the related projects have been abandoned in some cases (Clark and DuPont, 2018).

Wind turbine electricity power formulation

In the last couple of years, there have been reports about an increasing trend on installation of solar panels and wind turbines in the residential areas, which has however brought the need to regulate how to use these systems. It is anticipated that small size RESs will be used more and more in the future around the world as part of global plans to move away from fossil fuels toward low carbon emission resources (Costello, 2022; Zhang et al., 2021).

There have been numerous researches about wind turbine power generation in the last decade among which the main focus in many cases were on reliability aspect of these systems. Reliability of a system, in general, can be investigated by a reliability function that is defined using the system failure rate $λ$ as $R (t) = e^{- λ t}$ (Ezzati and Rasouli, 2014). However, reliability analysis problems are also being employed widely to take system reliability into account by finding a reliability index.

Two main approaches to obtain a reliability index of a power system are based on analytical and simulation methods. Analytical methods employ a mathematical model in order to evaluate system reliability, while simulation methods apply simulation techniques to estimate the index. One of the simulation methods is based on probabilistic approaches where variables involved in the process are considered as random variables and so their performance is evaluated by using probability distribution functions (Tina et al., 2006).

For RESs, other effective factors, such as intermetallic compound (IMC) thickness, are also studied in the available solar panels’ reliability analysis problems (Zarmai and Oduoza, 2021). Also, for the wind turbines, one of the first things that is considered when studying their reliability is the random variables involved in the related models. It can be seen in the literature that the wind speed and the wind angle are the main random variables when formulating reliability analysis problems of wind turbines (Al Sanad et al., 2021; Eryilmaz et al., 2021). Therefore, in this paper, these two random variables are employed to propose a new reliability analysis problem for the wind turbines.

There are two thresholds for each wind turbine that need to be considered when investigating its generated electricity power. These boundaries are the cut-in and cut-off speeds that are respectively represented by $V_{ci}$ and $V_{co}$ . So, the wind speed, which is defined by a random variable $v$ and measured by $\frac{m}{s}$ , must be between these two thresholds.

In other words, low speed winds, which are below the cut-in speed $V_{ci}$ do not have enough power to overcome friction and so, the turbine does not rotate and then no power generated. Also, strong winds, which are above the cut-off speed $V_{co}$ make real danger to the turbine and therefore the turbine stops rotating and then no power generated. It means $P_{w} = 0$ if $v \leq V_{ci}$ or $V_{co} \leq v$ where $P_{w}$ is the power generated by the wind turbine (Askarzadeh and Dos Santos Coelho, 2015; Eryilmaz et al., 2021).

It is also reported that to setting up a reliability analysis problem for wind turbines, a wind energy conversion system (WECS) can be used, which is accompanied by a PV system with a parallel connection (Deng et al., 2021). It is shown that one of the factors affecting the WECS power output is the wind speed probability distribution of a selected site. In this case, the wind speed, as a random variable, follows the Weibull distribution. To describe this random variable, meteorogical data is required that is collected over a long term.

In this model, to formulate the wind speed during the $j^{th}$ hour of the $m^{th}$ month, a scale parameter $α_{w}$ and a shape parameter $β_{w}$ need to be used in the Weibull density and distribution functions. So,

f_{v} (v) = \frac{β_{w}}{{α_{w}}^{β_{w}}} v^{β_{w} - 1} e^{- \frac{v}{{α_{w}}^{β_{w}}}}

F_{v} (v) = 1 - e^{- \frac{v}{{α_{w}}^{β_{w}}}}

A typical WECS starts generating power when the wind speed passed the cut-in wind speed $V_{ci}$ . Then, as the wind speed increases, the WECS keeps generating power with an increasing linear trend until the rated wind speed $V_{R}$ is reached. Between $V_{R}$ and the next threshold, which is the cut-off wind speed $V_{co}$ , the rated power capacity $P_{R}$ is generated, while no power is generated if the wind speed passes $V_{co}$ , because the WECS stops working due to safety reasons. It can then be formulated as below (Askarzadeh and Dos Santos Coelho, 2015; Eryilmaz et al., 2021; Tina et al., 2006):

P_{w} (v) = {\begin{matrix} 0 if v < V_{ci} \\ \frac{v - V_{ci}}{V_{R} - V_{ci}} P_{R} if V_{ci} \leq v \leq V_{R} \\ \begin{matrix} P_{R} if V_{R} \leq v \leq V_{co} \\ 0 if V_{co} < v \end{matrix} \end{matrix}

The fraction $\frac{v - V_{ci}}{V_{R} - V_{ci}}$ in the above formula can also be replaced by a quadratic polynomial to estimate the coefficient that needs to be multiplied into the rated capacity (Zhang et al., 2021).

It has also been found that climate change-related issues encourage researches and industries to shift their activities from single renewable source systems toward hybrid systems. However, a hybrid renewable energy system (HRES) would bring more complexity that is originated from huge number of parameters and variables. Among these parameters, there are reliability indices that represent random nature of the system and are used for reliability analysis purposes (Tina et al., 2006). Reliability of HRESs is also considered when studying how to assess long term performance of these systems. In this case, the expected energy not supplied (EENS) is expressed for this purpose that enables us to calculate the energy index of reliability (EIR) by $EIR = 1 - \frac{EENS}{L}$ where $L$ is the local load that exceeds the generated power (Eryilmaz et al., 2021).

Moreover, linear correlation methods, like the Pearson correlation coefficient, have widely been employed to model wind power structures, but it is found that these methods are not capable of reflecting nonlinear characteristics of wind power (Deng et al., 2021). Another consideration is that the power generated by a wind turbine needs to be properly converted by energy converters in order to get injected into a grid successfully. This means these converters’ reliability is one of the most important aspects of RESs reliability evaluation (Zhang et al., 2021).

In the next section, a new system performance function or limit-state function is proposed for the wind turbines that employs the two random variables mentioned earlier in this section as the wind speed and the wind angle. It will then be used in this paper to propose the new reliability analysis problem.

Limit-state function of wind turbines

Wind power generating stations, onshore and offshore, heavily depend on environmental factors, and these factors have random characteristics by nature. This randomness brings uncertainty to the systems that needs to be investigated to ensure system safety. Therefore, a system performance function or limit-state function for the wind turbines, which can be used to define system failure surface, is required that can then be used to formulate a reliability analysis problem for the system.

Given there are two random variables in the reliability analysis problem of a wind turbine as the wind speed and the wind angle, and based on the fact that these two factors (i.e. the random variables) are not always in their optimal situation, it can be expected to see these variables affect system reliability and also, they can even lead to system failure.

The above-mentioned limit-state functions can be used to set up a first-order (direct) reliability analysis problem to evaluate system reliability. This problem is generally modeled as below (Ezzati et al., 2014):

min ‖ (u_{1}, u_{2}, \dots, u_{n}) ‖

s . t . G_{U} (u_{1}, u_{2}, \dots, u_{n}) = 0

where ‖.‖ represents the Euclidean norm, $(u_{1}, u_{2}, \dots, u_{n})$ is used to show the design variable after standard normalization, and $G_{U}$ is the system performance function. Methods such as the conjugate gradient direction-based (CGDB) method can be applied to solve this problem to find $‖ (u_{1}, u_{2}, \dots, u_{n}) ‖$ that is indeed the system reliability index $β$ (Artin and Salimzadeh, 2022).

This reliability analysis problem can be employed to design a probabilistic constraint based on the system failure probability as $P {G (t) \leq 0] \leq Φ (- β)$ to use in a reliability-based design optimization (RBDO) problem. This strategy has been employed by researchers to set up an RBDO model for electricity power networks (Ezzati, 2015).

The following probability density function (PDF) is available for the electricity power generated by the WECS (Tina et al., 2006).

f_{P_{w}} (P_{w}) = {\begin{matrix} F_{1} = 1 - F_{3} if P_{w} = 0 \\ F_{2} if V_{ci} \leq v \leq V_{R} \\ F_{3} if P_{w} = P_{R} \end{matrix}

where

F_{2} = \frac{V_{R} - V_{ci}}{P_{R}} \frac{β_{w}}{α_{w}^{β_{w}}} (V_{ci} + (V_{R} - V_{ci}) \frac{P_{w}}{P_{R}})^{β_{w} - 1} e^{- (\frac{V_{ci} + (V_{R} - V_{ci}) \frac{P_{w}}{P_{R}}}{α_{w}}) β_{w}}

and

F_{3} = F_{v} (V_{F}) - F_{v} (V_{ci})

In this model, $F_{v}$ refers to the probability distribution function of the WECS.

The above formula of power generated by a wind turbine calculates output power of a wind turbine at hourly intervals based on hourly wind speed data collected at a specific location. Based on the existing literature, the random variable $v$ may follow two probability distribution functions as the Weibull distribution (Tina et al., 2006) or the Normal distribution (Ochoa et al., 2019).

Setting up reliability analysis problem

A reliability analysis problem can be designed for a system once its performance function (i.e. the limit-state function) is made available. The system performance function however depends on random variables that affect system functionality and output. Therefore, all this process starts with an investigation on random variables and their probability distribution. For this purpose, three scenarios are discussed in this paper.

In the first scenario, random variables of wind turbines need to be identified that can be used to set up the system performance function and then to formulate the reliability analysis problem. Two random variables are available for this purpose.

The first random variable in this process is the wind speed for which the collected data in various locations show that at least two probability distribution function are identified to study the wind speed. The probability distributions found for this purpose are the Gaussian (Normal) distribution and the Weibull distribution functions. The other random variable of a wind turbine performance function can be the wind angle. So, if studies could be done on analyzing the wind angle characteristics and particularly to determining its probability distribution function(s), then this random variable along with the first one, that is, the wind speed, would have been used to formulate the performance function and then set up the reliability analysis problem.

In the second scenario, the cut-in and cut-off speeds are assumed as two thresholds to set up the performance function. In other words, the fact that the wind speed $v$ needs to be within the range of $V_{ci}$ and $V_{co}$ can be interpreted in terms of system safety and system failure. So, the system is considered to be safe, which means it can generate power, if $V_{ci} \leq v \leq V_{co}$ , and also the system fails otherwise, which no power is generated. It should be noted that the system failure in this concept means the wind turbine stops generating electricity power, not a total failure that means the turbine needs to go under maintenance or replacement. However, the latter definition of system failure can also be used in one of the future works to design a new reliability analysis problem for the wind turbines.

Therefore, the system performance function can be defined in two different ways as $G_{1} (v) = v - V_{ci}$ and $G_{2} (v) = V_{co} - v$ . In both cases, $G_{i} (v) > 0, i = 1, 2$ shows the system is safe (i.e. power is being generated) and $G_{i} (v) < 0$ indicates system failure (i.e. no power is generated due to either low speed wind or strong wind). Therefore, the failure surface or limit-state function is defined as $G_{i} (v) = 0$ .

Another approach (i.e. the third scenario) to define the system performance function is to employ the power function or the probability density function. For this purpose, any of these two functions (or even maybe both of them) could be re-designed to form a system performance function, which can then be used to model the reliability analysis problem.

Conclusion and future works

As wind turbines are being installed more and more around the world to help environment, their safety is also becoming more important every day. Recent failures of wind turbines due to various reasons and in different countries signifies how crucial is to invest on safety and reliability of these systems.

Reliability of a system, such as wind turbines, can be discussed and studied via a wide range of approaches and strategies. The main idea of this paper however is to focus on reliability analysis of wind turbines and the fact that how a reliability analysis problem could be set up. Three scenarios are discussed in this paper in order to formulate a system performance function for the wind turbines and then use this function to set up a reliability analysis problem.

Either of the scenarios introduced in this paper can be used in the future works to design a model of the wind turbines’ reliability analysis problem. The outstanding work of each scenario is summarized here.

The main future work of the first scenario is to collect data for the wind angle and find out its distribution function. Then, both random variables required to set up a performance function would be available.

The second scenario is almost ready to use as system safety and failure and therefore the limit-state function is already defined. So, designing the reliability analysis problem is the major part of this job in the future.

The last approach relies on the existing power functions of a wind turbine. So, it is required to reformulate these functions to ensure they can represent system performance function. This can also be done by defining limit-state function based on these functions.

Footnotes

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Sorena Artin

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