Análisis comparativo

We use a univariate, autoregressive model, representing the forecast error in the aggre- gated wind output as a single value. Hence, the model is not suitable in its current form for systems where transmission constraints are a concern between buses with correlated wind feed-in. However, many studies use univariate wind data and for these it is con- venient to model the aggregate error with a single process, rather than summing up the errors from several correlated regions.

We draw on the wind model developed in chapter 2, and operate in the domain of the

normalised wind level (“X-domain”) instead of the aggregate wind power. The normalised

wind level has the advantage that it is Gaussian, allowing Gaussian time series models and linear correlations to be used to generate the forecast data. The normalised wind level at timestep k, X(k), is related to the power output Pw(k)by

Pw(k) =W(X(k) +µ(k mod Nd)) (5.2)

where Nd is the number of timesteps per day, µ(_·)is a diurnal shift, and W(_·)is a non- linear transformation function. As described in chapter 2, µ and W are chosen to ensure that, if X(k)_∼N(0, 1), the diurnal variation and asymptotic distribution of Pw(k)match historic data. In order to apply the model from chapter 2, therefore, we need to have available the transformation and diurnal shift functions for the realised wind power. If the wind power has been generated using the model, then these functions will already be available. If, however, the realised wind power comes directly from historic data, then some analysis will have to be performed in order to calibrate them.

In chapter 2 we showed that this model can provide a good fit to the aggregate power output for a large-scale wind fleet if we model X(k)as a stable, second-order autoregres- sive [AR(2)] process with hourly timesteps:

X(k) = ϕ₁xX(k−1) +ϕ₂xX(k−2) +σxǫx(k) (5.3) where ǫx(k)are independent N(0,1) innovations and the parameters are chosen such that

X(k) ∼ N(0, 1). In order to maintain generality and simplify the algebra, we represent the time series here as the equivalent Moving Average (MA) process, i.e. the weighted sum of the Gaussian innovations that have been realised at timestep k:

X(k) =σx ∞

∑

CHAPTER 5. FORECAST ERROR MODELLING FOR WIND INTEGRATION STUDIES

For an AR(p) process whose autoregressive (ϕ) parameters are known, the equivalent MA parameters can be calculated recursively using

ψi =          0 : i<_0, 1 : i=0, ∑p_j=1ϕjψi₋j : i>0. (5.5)

Let F(k, i)be the the median forecast for X(k+i), predicted at timestep k for horizon

i timesteps ahead, so that the forecast median wind power output is

Pwf(k, i) =W F(k, i) +µ((k+i)mod Nd)
, i=1 . . . Nf (5.6) where Nf is the longest forecast horizon in timesteps. In chapter 4, F(k, i)was generated statistically, by assuming that X is perfectly characterised by the fitted AR(2) process, and setting future innovation terms to zero. We now assume that a meteorological forecast is also available, which could be more accurate than the statistical-only one used in chapter 4. The aim of the forecast model is as follows. Given the realisation of the normalised wind level, X(k), generate an autocorrelated forecast time series F(k, i)whose error at i timesteps ahead is normally distributed with an exogenous standard deviation σ_iz. As a consequence of the nonlinearity, the distribution of forecast errors in the wind power domain, after transformation with Equation (5.6), will not be Gaussian. Some properties of the distribution of wind power forecast errors will be compared with publicly available forecast data in section 5.4. For now, we merely note that given F(k, i), the mode, median and mean of the distribution of X(k+i)is F(k, i); whereas we can use the monotonicity of W(·)to infer that the median (50th percentile) of Pw(k+i)is Pwf(k, i), but not the mode or mean.

Like S ¨oder [109], we synthesise the wind forecasts using a Gaussian time series pro- cess to synthesise meteorological forecast errors with the requisite statistical properties. Let Z(k, i)be the forecast error in the normalised wind level, defined according to

Z(k, i) =F(k, i)−X(k+i). (5.7) Note that the generation of the forecast from the forecast error requires a priori knowl- edge of the subsequent realisation. However, this knowledge will be hidden from the scheduling algorithm, which will only be given information about the forecast in order to arrive at its scheduling decisions.

We decompose Z(k, i) into a horizon-dependent scaling factor sy_i and a time series process Y(k, i):

Z(k, i) =sy_i Y(k, i) (5.8) where the underlying time-series process Y(k, i)can be written as an autoregressive pro-

CHAPTER 5. FORECAST ERROR MODELLING FOR WIND INTEGRATION STUDIES

cess of order p and unit volatility, driven by N(0,1) innovations ǫy(k, i):

Y(k, i) =    0 : i≤0 ∑p_j=1ϕy_j Y(k, i−j) + ǫy(k, i): 1≤ i≤ Nf (5.9)

or, equivalently (provided that its parameters are chosen to make the process stable), as an MA process whose parameters can be calculated from the autoregressive parameters using (5.5): Y(k, i) =    0 : i≤0 ∑i_j−₌1₀ψy_j ǫy(k, i−j): 1≤i≤ Nf. (5.10) For fixed k, the innovations ǫy(k, i)are always uncorrelated between all pairs of horizons. Using the operator ˆE[·]to mean the sample average value across a large number of values of k but fixed i, j, this can be written

ˆ

E[ǫy(k, i)ǫy(k, j)] =0 : 1≤i< _Nf_{, i}< _j< _Nf_{, (5.11)} although some other correlations will be set to non-zero values in section 5.2.2. The fore- cast normalised wind level for an i-timestep horizon is normally distributed with zero mean, and an RMS value of [cf Equation (4.33) on page 79]

σ_iz = q ˆ E[Z(k, i)2_{] =s}y i v u u t i−1

∑

from which the scale factors sy_i can be derived to satisfy any desired profile of RMS fore- cast errors.