# An Appropriate Wind Model for Wind Integrated Power Systems Reliability Evaluation Considering Wind Speed Correlations

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## Abstract

**:**

## 1. Introduction

## 2. Wind Data Modeling

_{i}(i = 1, 2, 3…s) and θ

_{j}(j = 1, 2, 3…m) are autoregressive and moving average coefficients of the wind model respectively. These coefficients can be calculated using the historical data of a site [12]. α

_{t}is a Normally and Independently Distributed(NID) white noise process with zero mean and σ

^{2}variance generally expressed in the form α

_{t}∈NID (0, σ

^{2}). Equation (1) represents a time series for y, which can be generated using value of α

_{t}randomly generated each time interval and previous values of y and α. An hourly time interval is used in this study. The time series of y obtained using Equation (1) can then be used to calculate the simulated hourly wind speeds using Equation(2).

_{t}= Simulated wind speed for hour t. µ

_{t}= Hourly mean wind speed for hour t. σ

_{t}= Hourly standard deviation of wind speed for hour t.

_{t}can be generated using a suitable normally distributed random number generator. For the initial four calculations, i.e., y

_{1}to y

_{4}, the preceding values of y and α that at the right hand side of Equation (3) were assumed to be zero if they were not yet obtained.

^{T}= Transpose of matrix G.

_{ij}is the desired correlation between i

^{th}and j

^{th}column of matrix X. Upper triangular matrix G can then be calculated using Cholesky Decomposition. Matrix X

_{c}which contains p correlated random number series as defined in matrix A can be deduced using Equation (7).

_{1}and X

_{2}are series of uncorrelated random numbers. ζ is the desired correlation coefficient. Series X

_{c}calculated using Equation (8) has a correlation of ζ with series X

_{1}. A particular case was considered taking ζ = 0.5. The scatter plots of 1000 pairs of uncorrelated and correlated random numbers thus obtained are compared in Figure 1.

**Figure 1.**Generation of correlated random numbers using Cholesky decomposition, (

**A**) Uncorrelated random numbers; (

**B**) Correlated random numbers with ζ= 0.5.

_{1}and X

_{c}in Equations (2) and (3). The hourly mean and hourly standard deviation values of Swift Current site were obtained from Environment Canada. Two pairs of wind speed data series were simulated using ζ = 0 and ζ = 0.5. The scatter plots for 1000 data pairs for two cases are shown in Figure 2.

**Figure 2.**Simulated wind speeds using uncorrelated and correlated random numbers. (

**A**) Using uncorrelated random numbers; (

**B**) Correlated random numbers with ζ= 0.5.

## 3. Wind Power Modeling

_{ci}) is the minimum wind speed that is required for a WTG to generate any power. Rated speed (V

_{r}) is the wind speed required for a WTG to generate maximum or rated power. Cut-out speed (V

_{co}) is the maximum wind speed that the WTG can safely handle, i.e., the WTG is shut down for safety reason at the cut-out speed. The relationship between wind speed (v) and the corresponding output power of a WTG is presented in [3] and can be expressed as Equation (9).

_{r}is the rated capacity of the WTG, and constants A, B and C depend on V

_{ci}, V

_{r}and V

_{co}[3]. The V

_{ci}, V

_{r}and V

_{co}of 14.4 km/h, 46.8 km/h, and 90 km/h respectively were used in this study. The simulated hourly wind speeds were converted to hourly power output values using (9). The effect of the FOR of the WTGs in a wind farm can be considered in the evaluation of the power output model of the wind farm. It has however been shown in [13] that the FOR of a WTG can be neglected during reliability evaluation of WECS without losing reasonable accuracy. The effect of FOR has therefore been neglected in this study to simplify the wind model. Another assumption that has been made in this study is that all the WTGs in a wind farm experience same amount of wind speed in any particular hour. Five pairs of hourly power output series were developed using the above mentioned procedure for correlations of 0, 0.25, 0.5, 0.75 and 1. Both the wind farms were assumed to have equal installed capacities for all the cases in this study. The total hourly power generated by two wind farms denoted by P

_{ρi}can be calculated using Equation (10).

_{ρji}= power output of wind farm j at hour I; i = 1 to 8760 × N; N = number of simulated years. NWF = number of wind farms considered. ρ = correlation between wind farms which can be in the form a correlation matrix if more than two wind farms are considered.

_{rjk}= installed capacity of WTG k in wind farm j; NWTG

_{j}= number of WTG in wind farm j.

_{i }for a state i is given by Equation (13). The probability of CO

_{i}outage state, P (CO

_{i}) is given by Equation (14):

_{i}is the number of wind power data points in the interval i.

## 4. Impact of Wind Penetration and Correlation

Correlation | Penetration | |||
---|---|---|---|---|

5% | 10% | 15% | 20% | |

8.1065 | 6.9114 | 5.9108 | 5.0427 | |

0.25 | 8.1121 | 6.9442 | 5.9772 | 5.1453 |

0.5 | 8.1186 | 6.9803 | 6.049 | 5.2543 |

0.75 | 8.1231 | 7.0133 | 6.1161 | 5.3572 |

1.0 | 8.1263 | 7.044 | 6.1798 | 5.4549 |

## 5. Appropriate Wind Capacity Model Considering Wind Correlation and Penetration

Penetration Level | Minimum NoS |
---|---|

up to 10% | 7 states |

up to 15% | 9 states |

up to 20% | 11 states |

## 6. Conclusions

## Conflict of Interest

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**MDPI and ACS Style**

Karki, R.; Dhungana, D.; Billinton, R. An Appropriate Wind Model for Wind Integrated Power Systems Reliability Evaluation Considering Wind Speed Correlations. *Appl. Sci.* **2013**, *3*, 107-121.
https://doi.org/10.3390/app3010107

**AMA Style**

Karki R, Dhungana D, Billinton R. An Appropriate Wind Model for Wind Integrated Power Systems Reliability Evaluation Considering Wind Speed Correlations. *Applied Sciences*. 2013; 3(1):107-121.
https://doi.org/10.3390/app3010107

**Chicago/Turabian Style**

Karki, Rajesh, Dinesh Dhungana, and Roy Billinton. 2013. "An Appropriate Wind Model for Wind Integrated Power Systems Reliability Evaluation Considering Wind Speed Correlations" *Applied Sciences* 3, no. 1: 107-121.
https://doi.org/10.3390/app3010107