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This is where the output variables of the wind power model, which include prediction of 6- hour, three-hour or 72-hour ahead as the case may be, are presented for performance measure comparison especially with other algorithms.

Pre-Processing Neural Network Post-Processing

Sets of inputs

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4.6.1 Descriptive Statistics

To identify or eliminate a candidate distribution, descriptive statistics are applied. For instance, the sample mean and median times will be close for a symmetrical or nearly symmetrical distribution, such as the normal or Weibull with a shape parameter between 3 and 4. If the mean is considerably larger than the median, then the exponential or lognormal distribution will provide a better fit.

The mean, 𝜇 or wind variability, Median, 𝑡_𝑚𝑒𝑑, Mode,𝑡_{𝑚𝑜𝑑𝑒}, and variance, 𝜎2,of the

lognormal distribution are discussed in [62], hence, realisation of Eq. (4.23) μ = t_medexp1

2𝜎

2 _(4.23)

The mean of the wind speed data over the wind farm 𝜇′ is described in terms of 𝑡𝑚𝑒𝑑, and 𝜎

is given by Eq. (4.24) μ′= ln(tmed) − 1 2ln ( σ2 tmed+ 1) (4.24)

Furthermore, the median of this distribution is given by Eq. (4.25):

t

= e

The mode is estimated using Eq. (4.26) while the standard deviation is obtained using Eq. (4.27):

tmode= tmed

exp(S2₎ (4.26)

σ2 = t_med2 exp(S2)[exp(S2_{) − 1] (4.27)}

4.6.2 Probability Plots

A probability plot is utilized to provide a better visual test of a distribution than comparison of a histogram with a probability density function. Initial estimates of the parameters of the

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distribution fitted are made possible with probability plots, such as exponential, normal distributions, and lognormal plots.

4.6.2.1 Exponential Plots

The cumulative distribution function plot for the exponential distribution is expressed by the computation of Eq. (4.28)

F(t) = 1 − e−λt _(4.28)

Taking the natural logarithm of both sides ln[1 − F(t)] = −λt

− ln[1 − F(t)] = ln [ 1

1−F(t)] = λt

An accurate fit to the observed times data may be obtained by performing a least-squares fit of Eq. (4.29) λ̂ = b = ∑ni=1xiyi ∑ni=1xi2 (4.29) where yi = ln 1 [1−F(t)] and xi = ti

The estimate for wind power from adjacent turbines is expressed by =1

𝑏

4.6.2.2 Normal Distribution Plots

For the normal distribution, which represents random wind speed variable selection from different turbines is estimated with Eq. (4.30) to form the tradition bell-shaped graph shown in Figure 5.7.

F(t) = ϕ (t−μ

σ ) = ϕ(z) (4.30)

The inverse function is rewritten as Eq. (4.31), which is linear in time 𝑡.

z_i = ϕ−1_{[F(t)] =}ti−μ

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The points (ti, ℱ̂(ti)) are plotted with appropriate transformation of the vertical scale. A least

squares fit is obtained by setting

𝑥_𝑖 = 𝑡_𝑖 and 𝑦_𝑖 = 𝑧_𝑖 (4.32)

The values of 𝑧𝑖 are obtained from a Table of standardized normal probabilities, based on the

corresponding values of ℱ̂ (𝑡_𝑖). From the least-squares fit and Eq. (4.33),

𝜎̂ = 1

𝑏 and 𝜇̂ = −𝑎𝜎̂ = −

𝑎

𝑏 (4.33)

4.6.2.3 Lognormal Plots.

The lognormal plot is made with the relationship of the distribution with the normal distribution. Since F(t) = ϕ (1 Sln t tmed) = ϕ(z) and Z = ϕ −1_{[F(t)] =}1 Slnt − 1

Slntmed, The points (𝑙𝑛𝑡𝑖, 𝑍𝑖)

are plotted. For the least squares fit, Eq. (4.35) is presented, where;

xi = lnti and yi = Zi (4.35)

The shape parameter 𝑆 is the reciprocal of the slope of the plotted line and 𝑡_𝑚𝑒𝑑, the median,is

obtained from the intercept of the fitted line. That is

Ŝ = 1

b and tmed = e

−Ŝa _(4.36)

4.6.2.4 Parameter Estimation

Probability plots and least squares data fitting only provides estimates of the distribution parameters and does not provide best results. For this reason, a maximum likelihood

estimator (MLE) 𝑃 is defined in Eq. (4.37).

P ̂ = n n+∑ni=1(xi−1) =_∑ n xi n i=1 (4.37)

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If the probability of a failure remains a constant 𝑃 and each trial is independent, then

P_r{X = x} = f(x) = (1 − P)x−1_{P x = 1,2, … (4.38)}

Where 𝑋 is the variable representing the number of trials necessary to obtain the first failure

and 𝑥 represents the sample size, f(x) is the likelihood function and represents the probability

of obtaining the observed sample.

4.6.3 Stationarity Test.

Prior to building predictive models for training time series algorithms, a stationarity test is one of the required steps. The research employed the Dickey-Fuller Test (DFT) to study stationarity considering daily wind variations within five days of historical data. From the test, the series is not stationary as in Figure3. The non-stationarity is attributed to trend and issues of normal distribution.

To obtain stationarity, the first level DFT is computed (d = 1) where d = differencing which is

the difference between current series (𝛾_𝑡) and previous series (γ_t−1) as in Δγ_t = γ_t− γ_t−1.

From our differenced data, Figure 3.6 is generated. This figure implies that maximum wind speed is experienced from 4AM to noontime leaving the afternoon time with low wind speeds.

The insight gained in the figure led to data split; 06-14-2003 to 06-17-2003, equivalent to 80% of the data for training while the rest in 6-17-2003 to 06-18-2003 equivalent to 20%, within the high wind time is set for testing the regularised models.

The non-linear nature of RNN and improvements in LSTM as demonstrated by [70, 91] shows the dynamic nature of wind in relation to geography, climate, landforms, seasonality can be addressed although in [34], statistical wind power transform appears strenuous.

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