Computing reliability indices of a wind power system via Markov chain modelling of wind speed

Abstract

Statistical modelling of wind speed is of great importance in the evaluation of wind farm performance and power production. Various models have been proposed in the literature depending on the corresponding time scale. For hourly observed wind speed data, the dependence among successive wind speed values is inevitable. Such a dependence has been well modelled by Markov chains. In this paper, the use of Markov chains for modelling wind speed data is discussed in the context of the previously proposed likelihood ratio test. The main steps for Markov chain based modelling methodology of wind speed are presented and the limiting distribution of the Markov chain is utilized to compute wind speed probabilities. The computational formulas for reliability indices of a wind farm consisting of a specified number of wind turbines are presented through the limiting distribution of a Markov chain. A case study that is based on real data set is also presented.

Keywords

Expected energy not supplied loss of load probability hybrid system reliability renewable energy wind speed

Introduction

Modelling statistical behaviour of wind speed is of special importance in wind energy studies. The choice of the suitable statistical model for wind speed regime at a certain location is important for the assessment of wind resources, performance of wind farms and their power production. Some well-known statistical distributions such as Weibull, Gamma and lognormal have been used for modelling wind speed when the data set includes daily averages. Manifestly, a statistical dependence among successive wind speed values may occur especially when hourly observations are considered. That is, the wind speed value for the next hour may depend on the wind speed values at previous hour(s). Thus, a proper dependence model is necessary for better forecasting of wind turbine/farm power production.

Markov chain is a useful tool for modelling dependence for sequential data. Indeed, Markov chains have been used to model and simulate the wind speed in a discrete way. Sahin and Sen¹ used first-order Markov chain approach for wind speed modelling. Ettoumi et al.² have fitted first-order three-state Markov chain for the wind speed data collected at Oran (Algeria). Shamshad et al.³ considered first and second order Markov chain models for synthetic generation of wind speed time series. Song et al.⁴ presented a way to estimate the first order (one step) Markov chain transition matrix from wind speed time series by two steps which are based on estimators of transition matrices and an evolutionary algorithm that searches for the first order Markov chain transition matrix which can best match these basic estimators. Tang et al.⁵ proposed a way for the categorization of steps in Markov chain based wind speed modelling. Nguyen et al.⁶ used first-order Markov chain to model wind speed and investigated the impact of correlation between wind speed and wind turbine availability on wind farm availability. Zheng et al.⁷ studied optimal preventive maintenance for wind turbines considering the effects of wind speed and used Markov chains for modelling the wind speed.

In the literature, mostly first order Markov chain has been used for modelling wind speed. Under first order Markov dependence, when the current state of the wind speed is exactly known, the probability of the future behaviour of the wind speed is not altered by the additional knowledge concerned with its past behaviour. That is, given the values $v_{n}, v_{n - 1}, . . ., v_{1}$ of the wind speed, it is assumed that $v_{n}$ contains all information about the future value $v_{n + 1} .$ However, in some cases, the value of $v_{n + 1}$ may depend not only on $v_{n}$ but also on the last $r$ wind speed values $v_{n}, . . ., v_{n - r + 1}$ . Such a dependence can be modelled by higher order Markov chains, in particular Markov chain of order $r$ . The determination of $r$ is a statistical problem and certain statistical tests can be used for finding the most suitable value of $r$ . Thus, for the wind speed data collected at a certain site, the first step should include the test if a Markov chain modelling is suitable for the data under concern. In the literature, this step has been mostly omitted in Markov chain modelling of the wind speed.

Various reliability indices have been proposed to evaluate the performance of power systems. Among the others, the Loss-of-Load Probability, Loss-of-Load Expectation and Expected Energy not Supplied have been widely used in reliability assessment of wind power systems (see, e.g. Tina et al.,⁸ Moharil and Kulkarni,⁹ Billinton and Huang,¹⁰ Chen et al.,¹¹Čepin,¹² Nguyen et al.,⁶ Eryilmaz et al.,¹³ Ghaedi and Gorginpour,¹⁴ Zhang et al.¹⁵). These reliability indices have also been used to choose the optimal wind turbine model for a wind farm (see, e.g. Fotuhi-Firuzabad and Dobakhshari¹⁶, Nemes and Munteanu,¹⁷ Mohiley and Moharil¹⁸).

The reliability indices depend on the aggregate power produced by a wind farm which is in fact a function of wind speed at the wind farm’s location. Thus, to compute and evaluate any reliability index, a proper statistical model should be developed for the wind speed random variable.

In the present paper, we compute well-known reliability indices for a wind farm consisting of a specified number of wind turbines when the wind speed is modelled by a Markov chain. To this end, we first summarize the existing hypothesis testing procedure to determine the order of the Markov dependence in wind speed modelling. Then, the long term distribution of the Markov chain is obtained via estimation of the transition probabilities and is used to compute discrete probability distribution corresponding to the wind speed. Finally, based on this distribution the reliability indices are computed. The main contribution of the paper lies in presenting and applying Markov chain methodology for wind speed modelling, and computing well-known reliability indices of wind and hybrid power systems based on the discrete wind speed distribution obtained via Markov chains. The new expressions for the Loss-of-Load Probability and Expected Energy not Supplied of a hybrid system that consists of specified number of wind turbines and a photovoltaic system that has a certain area are also obtained. These expressions contain reliability values of a wind turbine and photovoltaic system. Therefore, they enable us to evaluate the performance of a hybrid system with respect to reliability values of hybrid system components. The works appeared in the literature mostly focus on Markov chain based modelling of wind speed. In the present paper, we not only obtain a discrete distribution of the wind speed via Markov chains, but also compute and evaluate reliability indices for wind and wind-solar based power systems.

Based on the above perspective, the results and application-oriented findings of our study are reported in the following order. In Section 3, we present the theoretical framework for modelling wind speed via Markov chains. Section 4 contains equations for reliability indices which are based on the limiting distribution of the wind speed. Finally, in Section 5, we obtain numerical results which are based on real wind speed data.

Notation and abbreviations

In the following, we list the notation and abbreviations that will be used throughout the paper.

$V$ : Wind speed random variable

$v_{ci}$ : The cut-in wind speed

$v_{r}$ : The rated wind speed

$v_{co}$ : The cut-out wind speed

$P_{r}$ : The rated power of a single wind turbine

$g (v)$ : The power generated by the wind turbine at wind speed $v$

$n$ : The number of wind turbines within a wind farm

$p$ : The reliability value of the wind turbine

$P_{WF}$ : The power output of the wind farm

$S$ : Solar irradiation random variable

$g^{*} (s)$ : The power output of the photovoltaic system at solar irradiation $s$

$θ$ : The reliability value of the photovoltaic system

$P_{PV}$ : The power output of the photovoltaic system

$A_{PV}$ : The area of the photovoltaic system

$η_{PV}$ : The electrical efficiency of the photovoltaic system

$LR$ : Likelihood ratio

$LOLP$ : Loss-of-Load Probability

$LOLE$ : Loss-of-Load Expectation

$EENS$ : Expected Energy not Supplied

PVS: Photovoltaic system

Markov chain modelling of wind speed

A series of hourly wind speed measurements is converted to a sequence which is defined on the state space ${1, 2, . . ., s} .$ That is, a number from the set ${1, 2, . . ., s}$ is assigned for each observed average hourly wind speed value. For appropriately chosen limits $j_{l}$ and $j_{u}$ , the wind speed $v$ is in state $j$ if

v \in [j_{l}, j_{u}), j = 1, 2, . . ., s - 1,

and in state $s$ if $v \in [s_{l}, \infty)$ . The choice of the number $s$ of states and the intervals depend on target applications and some factors such as wind speed variation and wind turbine specifications. Various methods have been proposed for state categorization. See, for example, Ettoumi et al.,² Shamshad et al.,³ Tang et al.⁵

Let the sequence ${X_{n}}_{n \geq 1}$ consist of successive values of the wind speed after discretization. If the sequence ${X_{n}}_{n \geq 1}$ follows first order homogeneous Markov chain, then we have

\begin{matrix} P {X_{n + 1} = j | X_{n} = i, . . ., X_{1} = i_{1}} \\ = P {X_{n + 1} = j | X_{n} = i} \end{matrix}

for all time points $n$ and all states $i_{1}, . . ., i, j$ . The probability $p_{ij} = P {X_{n + 1} = j | X_{n} = i}$ is called transition probability and it denotes the conditional probability of being in state $j$ at time $n + 1$ given that the process is in state $i$ at time $n$ . The matrix $P$ which contains conditional probabilities $p_{ij}$ for all states $i$ and $j$ is called transition probability matrix of the Markov chain. The elements $p_{ij}$ of any transition probability matrix $P$ satisfy the conditions

p_{ij} \geq 0 and \sum_{j} p_{ij} = 1 .

Let $π_{i} = lim_{n \to \infty} P {X_{n} = i}$ denote the probability that the Markov chain ${X_{n}}_{n \geq 1}$ is in state $i$ in the long run, $i = 1, 2, . . ., s$ . If the Markov chain ${X_{n}}_{n \geq 1}$ is regular, then the limiting distribution $(π_{1}, π_{2}, . . ., π_{s})$ is the unique nonnegative solutions of the equations (Taylor and Karlin¹⁹):

π_{j} = \sum_{k = 1}^{s} π_{k} p_{kj}, j = 1, 2, . . ., s

\sum_{k = 1}^{s} π_{k} = 1 .

(1)

For a Markov chain defined on the states ${1, 2, . . ., s}$ , the transition probability from state $i$ to state $j$ can be estimated from

{\hat{p}}_{ij} = \frac{n_{ij}}{\sum_{j = 1}^{s} n_{ij}},

(2)

where $n_{ij}$ is the number of transitions from state $i$ to state $j$ in $X_{1}, . . ., X_{n} .$

In the case when the future behaviour of the process ${X_{n}}_{n \geq 1}$ depends not only on the current state but also on the last $r$ values of the process, the process ${X_{n}}_{n \geq 1}$ is called Markov chain of order $r$ . In this case, we have

\begin{matrix} P {X_{n + 1} = j | X_{n} = i_{n}, . . ., X_{1} = i_{1}} \\ = P {X_{n + 1} = j | X_{n} = i_{n}, . . ., X_{n - r + 1} = i_{n - r + 1}}, \end{matrix}

for all time points $n$ and all states $i_{1}, . . ., i_{n}, j$ . Such a Markov chain is studied via the following transition probabilities

P {X_{n + 1} = j | X_{n} = i_{n}, . . ., X_{n - r + 1} = i_{n - r + 1}} .

Manifestly, when $r = 1$ we have a first order Markov chain. Assume that the sequence ${X_{n}}_{n \geq 1}$ which is obtained by converting hourly wind speed series into a discrete valued series with $s$ possible states is modelled by a Markov chain of order $r$ . The determination of the value of $r$ is of special importance for further developments. For an observed sequence of wind speed series, the method of likelihood ratio can be used to test if the order of the Markov chain ${X_{n}}_{n \geq 1}$ is $r - 1$ or $r$ . In this case, the null and alternative hypotheses are given by

H_{0} : Order is r - 1 versus H_{1} : Order is r .

(3)

For testing $H_{0}$ against $H_{1}$ , the likelihood ratio test is given by (see, e.g. Menendez et al.²⁰)

\begin{matrix} LR = - 2 (\sum_{a_{2}, . . ., a_{r + 1}} n_{a_{2}, . . ., a_{r}, a_{r + 1}} \log \frac{n_{a_{2}, . . ., a_{r}, a_{r + 1}}}{n_{a_{2}, . . ., a_{r}}} \\ - \sum_{a_{1}, . . ., a_{r + 1}} n_{a_{1}, . . ., a_{r}, a_{r + 1}} \log \frac{n_{a_{1}, . . ., a_{r}, a_{r + 1}}}{n_{a_{1}, . . ., a_{r}}}), \end{matrix}

(4)

for $r > 1$ , where

\begin{matrix} n_{a_{1}, . . ., a_{r}, a_{r + 1}} = \sum_{k = 1}^{n - r} I (X_{k} = a_{1}, . . ., X_{k + r - 1} \\ = a_{r}, X_{k + r} = a_{r + 1}), \end{matrix}

and $I (A) = 1$ if $A$ occurs and $I (A) = 0$ if $A$ does not occur. If $r = 1$ , then

LR = - 2 (\sum_{a_{1}} n_{a_{1}} \log \frac{n_{a_{1}}}{n} - \sum_{a_{1}, a_{2}} n_{a_{1}, a_{2}} \log \frac{n_{a_{1}, a_{2}}}{n_{a_{1}}}),

(5)

where

\begin{matrix} n_{a_{1}} = \sum_{k = 1}^{n} I {X_{k} = a_{1}}, \end{matrix}

\begin{matrix} n_{a_{1}, a_{2}} = \sum_{k = 1}^{n - 1} I {X_{k} = a_{1}, X_{k + 1} = a_{2}} . \end{matrix}

It should be noted that in the special case when $r = 1$ , the test is of the null hypothesis that $X_{1}, X_{2}, . . .$ consists of independent and identical trials against the alternative hypothesis that trials are from a first-order Markov chain. The asymptotic distribution of (4) and (5) is chi-squared with $(s^{r} - s^{r - 1}) (s - 1)$ degrees of freedom under $H_{0}$ (Hoel²¹), where $s$ is the number of states for the Markov chain under concern.

We are interested in obtaining exactly the order of the converted wind speed series ${X_{n}}_{n \geq 1} .$ To this end, we will consider the test given by (3) for successive values of $r$ . For $r = 1,$ we first test

$H_{0} : Order is 0 versus H_{1} : Order is 1$ ,

that is, independence against first order Markov chain. If $H_{0}$ is accepted, then the sequence ${X_{n}}_{n \geq 1}$ consists of independent trials. Otherwise, we assume that the order of the underlying Markov chain is $1$ . Then, we test $H_{0} :$ Order is $1$ versus $H_{1} :$ Order is $2 .$ If we accept the null hypothesis, then we conclude that our sequence has order $1$ . In the case of rejection of $H_{0}$ , we test $H_{0} :$ Order is $2$ versus $H_{1} :$ Order is $3$ and we continue until we accept the null hypothesis. At that moment, it is stopped and concluded that the order of the Markov chain is the order that has been tested in the null hypothesis. If the underlying Markov chain has order $r$ , then the $r$ th order transition probabilities are estimated via

{\hat{p}}_{a_{1},..., a_{r}; a_{r + 1}} = \frac{n_{a_{1},..., a_{r}, a_{r + 1}}}{n_{a_{1},..., a_{r}}},

(6)

where

n_{a_{1}, . . ., a_{r}} = \sum_{a_{r + 1} \in {1, 2, . . ., s}} n_{a_{1}, . . ., a_{r}, a_{r + 1}} .

The process for modelling wind speed series via Markov chain is summarized below.

Step 1. A series of hourly wind speed measurements is converted to a sequence which is defined on the state space ${1, 2, . . ., s} .$ This is a state categorization step.

Step 2. The test given by (3) is performed for successive values of $r$ and the value of $r$ is obtained.

Step 3. The estimates of the transition probabilities $p_{a_{1}, . . ., a_{r}; a_{r + 1}}$ are obtained via (6) for $a_{1}, a_{2}, . . ., a_{r + 1} \in {1, 2, . . ., s} .$

Step 4. The probability of being in state $i$ in the long run, that is, $π_{i}$ for $i = 1, 2, . . ., s$ is obtained using (1) when $r = 1$ and the equations obtained in Adke and Deshmukh²² if $r > 1 .$

Reliability indices

After obtaining the limiting distribution $(π_{1}, π_{2}, . . ., π_{s})$ described in the previous section, the reliability indices of a wind power system can be computed by well-known equations. Note that $π_{j}$ is in fact the probability that the wind speed is in state $j$ in the long term. Let ${\bar{v}}_{j}$ denote representative wind speed value (which can be computed by averaging lower and upper limits of the corresponding wind speed interval) corresponding to the state $j$ , $j = 1, 2, . . ., s,$ that is.

{\bar{v}}_{j} = {\begin{matrix} \frac{j_{l} + j_{u}}{2}, & if v_{ci} \leq j_{l} < j_{u} < v_{co} \\ 0 & otherwise, \end{matrix}

where $v_{ci}$ and $v_{co}$ denote respectively the cut-in and cut-out wind speed values of the wind turbine. Then

π_{j} = P {V = {\bar{v}}_{j}} .

If the relationship between the wind speed $v$ and power output of a single wind turbine is denoted by $g (v),$ then $g ({\bar{v}}_{j})$ denote the wind turbine power output at wind speed ${\bar{v}}_{j}$ . Table 1 summarizes the definition of wind speed intervals, the representative wind speed, its probability and power output for each state. For a given wind speed data set, the entries of Table 1 can be obtained by following the steps 1–4 described in Section 2.

Table 1.

Wind speed probability distribution and wind turbine power output.

State	Wind speed interval	Representative wind speed	Probability	Wind turbine power output
$1$	$[1_{l} {, 1}_{u})$	${\bar{v}}_{1}$	$π_{1}$	$g ({\bar{v}}_{1})$
$2$	$[2_{l} {, 2}_{u})$	${\bar{v}}_{2}$	$π_{2}$	$g ({\bar{v}}_{2})$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$s$	$[s_{l}, s_{u})$	${\bar{v}}_{s}$	$π_{s}$	$g ({\bar{v}}_{s})$

The aggregate power output of a wind farm that consists of $n$ identical wind turbines can be represented as

P_{WF} = S_{n} \cdot g (V),

where $S_{n}$ is the total number of available wind turbines whose probability mass function is

P {S_{n} = l} = (\begin{matrix} n \\ l \end{matrix}) p^{l} (1 - p)^{n - l},

for $l = 0, 1, . . ., n,$ where $p$ is the availability of a wind turbine. Then, the cumulative distribution function of the aggregate power produced by a wind farm that consists of $n$ identical wind turbines is (Giorsetto and Utsurogi²³):

\begin{matrix} P {P_{WF} \leq x} = \sum_{l = 0}^{n} \sum_{j = 1}^{s} u (x - l \cdot g ({\bar{v}}_{j})) \end{matrix}

\begin{matrix} P) P {V = {\bar{v}}_{j}} (\begin{matrix} n \\ l \end{matrix}) p^{l} {(1 - p)}^{n - l}, \end{matrix}

where $u (x) = 1$ if $x \geq 0$ and $u (x) = 0$ if $x < 0$ . Using the limiting distribution of the underlying Markov chain, we obtain

P {P_{WF} \leq x} = \sum_{l = 0}^{n} \sum_{j = 1}^{s} u (x - l \cdot g ({\bar{v}}_{j})) π_{j} (\begin{matrix} n \\ l \end{matrix}) p^{l} {(1 - p)}^{n - l} .

(7)

The distribution given by (7) can be easily computed using the entries presented in Table 1.

The relationship between wind speed and the power output of a wind turbine depends on the wind turbine characteristics. According to the widely used power equation, (see, e.g. Louie and Sloughter²⁴)

g (v) = {\begin{matrix} 0, & if v < v_{ci} or v \geq v_{co} \\ P_{r} \frac{(v^{3} - v_{ci}^{3})}{(v_{r}^{3} - v_{ci}^{3})}, & if v_{ci} \leq v < v_{r} \\ P_{r}, & if v_{r} \leq v < v_{co}, \end{matrix}

(8)

where $v_{ci}, v_{r}$ and $v_{co}$ denote respectively the cut-in, rated and cut-out wind speed values, and $P_{r}$ is the rated power of the wind turbine. As it is clear from equation (7), the computation of the cumulative distribution function of the wind farm output needs the following inputs:

Wind turbine characteristics ( $v_{ci}, v_{r}, v_{co}$ and $P_{r}$ )

Wind turbine availability ( $p$ )

The long term distribution of the wind speed ( $π = (π_{1}, . . ., π_{s})$ )

For a given load $L$ , the LOLP is defined by

LOLP = P {P_{WF} < L} .

Using (7), by conditioning on the number of available wind turbines and the number of wind speed states, the LOLP can be computed from

LOLP = \sum_{l = 0}^{n} \sum_{j = 1}^{s} h (L - l \cdot g ({\bar{v}}_{j})) π_{j} (\begin{matrix} n \\ l \end{matrix}) p^{l} (1 - p)^{n - l},

(9)

where $h (x) = 1$ if $x > 0$ and $h (x) = 0$ if $x \leq 0 .$

LOLE is defined to be the expected period during which the load demand is greater than available power generation. It can be defined as

LOLE = \sum_{i = 1}^{T} P {P_{WF}^{i} < L_{i}},

where $i$ is the equally sized time step (e.g. hour, day), $T$ is the total number of time steps, and $L_{i}$ is the load demand at time period $i$ . If $L_{i} = L$ for all $i$ , then

LOLE = T \cdot LOLP .

(10)

The EENS is defined to be the expected energy that will not be supplied when the local load $L$ exceeds the available generation. It can be expressed as

EENS = E (max (L - P_{WF}, 0)) .

By conditioning on the number of available wind turbines and the number of wind speed states,

E E N S = \sum_{l = 0}^{n} \sum_{j = 1}^{s} u (L - l \cdot g ({\bar{v}}_{j})) (L - l \cdot g ({\bar{v}}_{j})) π_{j} (\begin{matrix} n \\ l \end{matrix}) p^{l} {(1 - p)}^{n - l} .

(11)

Case study

Our case study is based on a series of wind speed data collected at every hour at School of Foreign Languages of Atilim University (39.813°N, 32.726°E). The hourly wind speed data collected between 1 March 2020 and 28 February 2021 have been converted to a sequence which is defined on the state space ${1, 2, . . ., 11}$ . In our case study, we use a wind turbine with $v_{ci} = 2, v_{r} = 11$ , $v_{co} = 21$ and $P_{r} = 5.5$ kW. Based on cut-in, rated and cut-out wind speeds, the wind speed intervals corresponding to the states are chosen to be $[0, 2), [2, 3), [3, 4), [4, 5), [5, 6), [6, 7), [7, 8), [8, 9), [9, 10), [10, 11)$ and $[11, 21)$ with step size of 1m/s except for the first and last intervals. Following the methodology described in Section 3, we first test .

$H_{0} : Order is 0 versus H_{1} : Order is 1$ ,

that is, independence against first order Markov chain. The value of (5) has been found to be $LR = 9140.351$ . The corresponding critical value is $c = χ_{100, 0.05}^{2} \approx 124.342 .$ Since $LR > c$ , $H_{0}$ cannot be accepted. Next, we test (3) for $r = 2$ , that is,

$H_{0} : Order is 1 vers us H_{1} : Order is 2$ .

For $r = 2$ , the value of (4) is found to be $LR = 757.4643$ . The corresponding critical value is $c = χ_{1100, 0.05}^{2} \approx 1178.803$ . Therefore, $H_{0}$ cannot be rejected. Thus, a first-order Markov chain with 11 states fits the discretized wind speed data. The estimate of the transition probability matrix is given in Table 2.

Table 2.

Estimated transition probability matrix.

	1	2	3	4	5	6	7	8	9	10	11
1	0.8080	0.1213	0.0531	0.0145	0.0013	0.0006	0.0009	0.0003	0	0	0
2	0.2776	0.3824	0.2276	0.0937	0.0118	0.0062	0	0	0.0007	0	0
3	0.0988	0.2377	0.3593	0.2613	0.0358	0.0064	0	0	0.0007	0	0
4	0.0382	0.0863	0.2201	0.4784	0.1251	0.0413	0.0081	0.0025	0	0	0
5	0.0159	0.0335	0.0494	0.3686	0.3298	0.1287	0.0511	0.0212	0.0018	0	0
6	0.0069	0.0278	0.0347	0.1945	0.2847	0.2813	0.1111	0.0382	0.0139	0.0069	0
7	0.0066	0.0329	0.0395	0.1250	0.1579	0.2039	0.2829	0.0855	0.0461	0.0131	0.0066
8	0	0.0274	0.0137	0.0274	0.0411	0.1233	0.3013	0.2877	0.0822	0.0685	0.0274
9	0	0	0	0.0357	0	0.2143	0.2500	0.2143	0.1429	0.1071	0.0357
10	0	0	0.0500	0	0.0500	0.0500	0.1500	0.1500	0.1000	0.2500	0.2000
11	0	0	0	0.0910	0	0	0	0.1818	0.1818	0.2727	0.2727

Based on the transition probability matrix given in Table 2, we obtain the limiting distribution of the Markov chain using (1). Table 3 contains representative wind speed values ( ${\bar{v}}_{j}$ ) and the corresponding probabilities ( $π_{j}$ ) as a solution of the equations in (1). The wind turbine output values ( $g ({\bar{v}}_{j})$ ) corresponding to the representative wind speed values are also presented. According to Table 3, if the wind turbine is full reliable, that is, $p = 1$ , then its expected power output is

\sum_{j = 1}^{11} g ({\bar{v}}_{j}) π_{j} = 0.25 kW

Table 3.

Wind turbine power output and its probability for a wind turbine with 5.5 kW rated power.

State	Wind speed interval	Representative wind speed	Probability	Wind turbine power output
$1$	$[0, 2)$	1	$0.3640$	$0.0000$
$2$	$[2, 3)$	2.5	$0.1645$	$0.0317$
$3$	$[3, 4)$	3.5	$0.1593$	$0.1450$
$4$	$[4, 5)$	4.5	$0.1824$	$0.3456$
$5$	$[5, 6)$	5.5	$0.0648$	$0.6584$
$6$	$[6, 7)$	6.5	$0.0327$	$1.1084$
$7$	$[7, 8)$	7.5	$0.0173$	$1.7206$
$8$	$[8, 9)$	8.5	$0.0083$	$2.5198$
$9$	$[9, 10)$	9.5	$0.0032$	$3.5310$
$10$	$[10, 11)$	10.5	$0.0023$	$4.7792$
$11$	$[11, 21)$	16	$0.0012$	$5.5000$

The expected output of the wind turbine is quite low when it is compared with its rated power. This is due to the dominance of low wind speeds in the area. Indeed, as it is clear from Table 3, the probability of the state 1 which corresponds to the lowest representative wind speed is the highest. Thus, a renewable energy system that consists of only wind turbines leads to high LOLP, LOLE and EENS values and hence a less reliability system. In other words, a reliable system cannot be established only using wind turbines. To obtain a required reliability level, many wind turbines should be used. This may not feasible due to constraints such as cost and the area of the wind farm location. In Table 4, LOLP, LOLE and EENS values are computed using the data presented in Table 3 and the equations (9) to (11) when the availability of each wind turbine is $p = 0.97 .$ As observed from Table 4, the LOLP, LOLE and EENS values are quite high even if a wind farm consists of large number of wind turbines. To obtain a system that has a higher reliability value, Eryilmaz et al.¹³ considered a hybrid system that consists of wind turbines and solar panels. Hybrid energy systems that consist of both solar and wind sources together are more advantageous than only solar or wind energy based systems since they have high system efficiency and power reliability.

Table 4.

LOLP, LOLE and EENS values for a wind farm consisting of $n$ identical wind turbines having 5.5 kW rated power when the load is $L = 5$ kW.

$n$ (The number of wind turbines)	$LOLP$	$LOLE$ (h/year)	$EENS$ (kW)
$30$	$0.6878$	$6025.10$	$2.6152$
$50$	$0.5285$	$4629.66$	$2.3896$

To observe the efficacy the Markov chain based modelling, we have performed a Monte Carlo simulation study. The first step in our simulation is based on the generation of hourly wind speed states from the Markov chain having transition probabilities presented in Table 2. That is, a series of wind speed states and corresponding representative wind speed values have been generated for one year period. Then, for $L = 5$ kW, we have computed LOLP and LOLE when $n = 30$ , $v_{ci} = 2, v_{r} = 11$ , $v_{co} = 21$ , $P_{r} = 5.5$ kW and $p = 0.97 .$ The Monte Carlo estimates of LOLP and LOLE were found to be $0.6900$ and $6044.4$ , respectively. These values are quite close to the entries presented in Table 4.

Hybrid system

Assume that a PVS having a certain area is integrated into a wind farm. For a fixed solar irradiation $S =$ s(kW/ $m^{2}$ ) $, the$ basic equation for the power output of the PVS is

P_{PV} = S \cdot η_{PV} \cdot A_{PV},

where $A_{PV}$ is the module area and $η_{PV}$ is the electrical efficiency of the module. Similar to the wind speed modelling, assume that there are $m$ states for the solar irradiation. Let ${\bar{s}}_{i}$ denote the representative solar irradiation corresponding to the state $i$ , $i = 1, 2, . . ., m$ . Define

ρ_{i} = P {S = {\bar{s}}_{i}},

for $i = 1, 2, . . ., m$ with $\sum_{i = 1}^{m} ρ_{i} = 1$ . The power output of the PVS at solar irradiation ${\bar{s}}_{i}$ is

g^{*} ({\bar{s}}_{i}) = {\bar{s}}_{i} \cdot η_{PV} \cdot A_{PV} .

Thus, for a hybrid system that consists of $n$ identical wind turbines and PVS, the LOLP can be computed from

\begin{matrix} L O L P_{h y b r i d} = \sum_{l = 0}^{n} \sum_{i = 1}^{m} \sum_{j = 1}^{s} h (L - l \cdot g ({\bar{v}}_{j}) - g^{*} ({\bar{s}}_{i})) ρ_{i} π_{j} (\begin{matrix} n \\ l \end{matrix}) p^{l} {(1 - p)}^{n - l} θ \\ + \sum_{l = 0}^{n} \sum_{j = 1}^{s} h (L - l \cdot g ({\bar{v}}_{j})) π_{j} (\begin{matrix} n \\ l \end{matrix}) p^{l} {(1 - p)}^{n - l} (1 - θ), \end{matrix}

(12)

where $θ$ denotes the reliability of the PVS. The equation (12) is obtained considering the two possible cases which respectively correspond to operation and failure of the PVS. Similarly, the EENS for the hybrid system is

\begin{matrix} EEN S_{hybrid} = E (max (L - P_{WF} - P_{PV}, 0)) \\ = \sum_{l = 0}^{n} \sum_{j = 1}^{s} u (L - l \cdot g ({\bar{v}}_{j}) - g^{*} ({\bar{s}}_{i})) \\ (L - l \cdot g ({\bar{v}}_{j}) - g^{*} ({\bar{s}}_{i})) π_{j} ρ_{i} (\begin{matrix} n \\ l \end{matrix}) p^{l} {(1 - p)}^{n - l} θ \\ + \sum_{l = 0}^{n} \sum_{j = 1}^{s} u (L - l \cdot g ({\bar{v}}_{j})) (L - l \cdot g ({\bar{v}}_{j})) \\ π_{j} (\begin{matrix} n \\ l \end{matrix}) p^{l} {(1 - p)}^{n - l} (1 - θ) . \end{matrix}

(13)

For the same area, the solar irradiation distribution has been empirically estimated based on the data collected between 1 March 2020 and 28 February 2021. Table 5 contains representative solar irradiation values ( ${\bar{s}}_{i}$ ) and the corresponding probabilities ( $ρ_{i}$ ). The power output values ( $g^{*} ({\bar{s}}_{i})$ ) corresponding to the representative solar irradiation values are also presented.

Table 5.

PVS power output and its probability when $A_{PV} =$ 235 $m^{2}$ and $η_{PV} = 0.227$ .

State	Representative solar irradiation (kW/ $m^{2}$ )	Probability	PVS power output (kW)
$1$	$0.1$	$0.4027$	$5.3345$
$2$	$0.3$	$0.2100$	$16.0035$
$3$	$0.5$	$0.1650$	$26.6725$
$4$	$0.7$	$0.1381$	$37.3415$
$5$	$0.9$	$0.0842$	$48.0105$

In Table 6, we compute LOLP, LOLE and EENS values for the hybrid system that consists of $n$ identical wind turbines with $v_{ci} = 2, v_{r} = 11$ , $v_{co} = 21$ and $P_{r} = 5.5$ kW, and PVS with total area $A_{PV} =$ 235 $m^{2}$ and $η_{PV} = 0.227$ . The reliability of the PVS is chosen to be $θ = 0.95$ . As it is observed from Table 6, a significant increase in reliability occurs with the inclusion of PVS.

Table 6.

LOLP, LOLE and EENS values for the hybrid system.

$n$ (The number of WTs)	$LOL P_{hybrid}$	$LOL E_{hybrid}$ (h/year)	$EEN S_{hybrid}$ (kW)	$EI R_{hybrid}$
1	0.0499	437.1240	0.2379	0.9524
5	0.0470	411.7200	0.2014	0.9597
10	0.0435	381.0600	0.1733	0.9653
30	0.0344	301.344	0.1308	0.9738
50	0.0264	231.264	0.1195	0.9761

Based on EENS value, Energy Index of Reliability (EIR) is also calculated and included in Table 6. The EIR can be computed from

EI R_{hybrid} = 1 - \frac{EEN S_{hybrid}}{L} .

Summary and conclusions

In the literature, Markov chains have been directly used as a modelling tool without checking its suitability for a given wind speed data. For hourly collected wind speed data, a Markov chain of order $r$ may be suitable after discretization process. As explained clearly in the present paper, the value of $r$ can be determined using a well-known likelihood ratio test. After the choice of the most suitable value of $r$ , the limiting distribution of the Markov chain can be used to find the long term wind speed probability distribution. In this paper, the limiting distribution of the Markov chain has been used to compute the well-known reliability indices such as LOLP, LOLE and EENS. The equations for these reliability indices have been modified for a hybrid system consisting of a specified number of wind turbines and PVS. Illustrative computational results have been presented for real data set.

In our developments, it has been assumed that the wind turbine reliability is not influenced by the wind speed. In the literature, based on particular data sets, a significant correlation between wind speeds and the failure rate of the components of a wind turbine has been observed (Tavner et al.²⁵). As pointed out by Nguyen et al.,⁶ although the higher speed of wind within cut-in and cut-out speed values produces higher power output, it also degrades the reliability of wind turbines. The evaluation of reliability indices LOLP, LOLE and EENS considering the dependence between wind speed and wind turbine reliability under the present Markov chain based setup will be among our future research problems.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under project TUBITAK 1001-119F182.

ORCID iD

Serkan Eryilmaz

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