5. Forecast error modelling for wind integration studies
5.3. Choice of parameter sets
In this section, we propose possible parameter sets for a wind forecast error process that is compatible with wind process models based on Equations (5.2)–(5.3). The model gives us a rather rich set of parameters with which to impose the desired properties of the forecast error process. These are:
1. The parameters of the time series process for Y(k, i), i = 0 . . . Nf, represented here by the MA weightings ψyj, j = 0 . . . Nf. These can be tuned to match desired auto-
correlation properties for the forecast error over different time horizons.
2. The horizon-dependent scale factors syi. These can be calibrated to desired RMS errors in the normalised wind level (X) domain using Equation (5.12). In practice, however, we will usually wish to prescribe RMS errors in the forecast wind power, rather than errors in the forecast normalised wind level. We assume here that the wind power point forecast for a future time, whose RMS error has been measured, is the median of the forecast wind power distribution for that time. In other words, the probability that the realised wind power exceeds the forecast is 0.5. The RMS error in the forecast wind power i timesteps ahead, ∆Pirmse, is then related to the
CHAPTER 5. FORECAST ERROR MODELLING FOR WIND INTEGRATION STUDIES
parameters of the wind realisation and wind forecast error models by
(∆Pirmse)2= 1 Nd Nd−1
∑
j=0 Z +∞ −∞ Z +∞ −∞ W(x+µ(j))−W(x+z+µ(j)) 2 fixz(x, z)dz dx (5.29) where fixz(x, z)is the joint distribution of X(k+i)and Z(k, i)over all k, a bivariate normal that can be written (using the fact that X(k)has unit variance):fixz(x, z) = 1 si φ z−x ˆE[X(k+i)Z(k, i)] si φ(x) (5.30)
where si is the standard deviation of Z(k, i)conditional on X(k+i), given by
si =
q ˆ
E[Z(k, i)2]−Eˆ[X(k+i)Z(k, i)]2 (5.31)
and ˆEZ(k, i)2, ˆE[X(k+i)Z(k, i)]are given by Equations (5.18), (5.19) respectively.
Equation (5.29) can be evaluated using numerical quadrature methods and embed- ded in a non-linear root-finding algorithm, enabling the scaling volatilities syi to be tuned to reproduce the desired RMSE profile in the power output domain.
3. ρxy, the correlation between ǫx(k+i)and ǫy(k, i). This can be used to ensure that the forecast errors are unbiased, as shown in section 5.2.2.
4. ρyy, the correlation between ǫy(k−1, i+1)and ǫy(k, i). This can be used to create a correlation between the errors in the forecasts made on successive timesteps, as shown in section 5.2.2. If realistic data are not available to infer this correlation, we can test its effect on the results using a sensitivity study.
The first task is to choose the autoregressive parameters (or MA weightings) for the normalised forecast error process Y(k, i). A reasonable choice is to use the same param- eters as are used by the underlying realised wind process: ϕy1 = ϕx1, ϕy2 = ϕx2 . (If the realised wind data is obtained directly from a historic time series, then an autoregres- sive model will have to be fitted to it in order to obtain these parameters.) By choosing the same parameters for the forecast error process as for the realised wind process, we ensure that, at least for the case where all siy are the same, the forecast normalised wind
level F(k, i)is also an autoregressive process and has the same autoregressive parameters as the realised wind, but with a lower volatility due to the negative value of ρxy. Further- more, we can then manipulate the volatility term structure parameters (while adjusting
ρxy to maintain approximately unbiased forecasts) to obtain any desired forecast quality ranging from perfect forecasts to statistical-only ones. For perfect forecasts, we simply choose syi = 0 for all i. In this case the correlation ρxy is immaterial. An upper bound on
the unbiased RMSE is provided by a forecast that is based only on the underlying time- series process for the realised wind, which assumes that the future innovations will be zero. This is the forecast assumption used in chapter 4. In the case of an AR(2) process, it
CHAPTER 5. FORECAST ERROR MODELLING FOR WIND INTEGRATION STUDIES
can be written as Equation (4.24), repeated here:
F(k, i) = X(k), i≤0 ϕ1xF(k, i−1) +ϕ2xF(k, i−2), i>0. (5.32) This worst-case, “statistical-only” forecast can be obtained by setting syi = σxfor all i and
ρxy =−1; it provides an upper bound on the values of σiz. A corollary is that the method cannot be used to provide persistence forecasts, defined by F(k, i) = X(k)for all i ≥ 0. In any case, persistence forecasts cannot have trending or mean reversion properties and are therefore always biased if used with wind regimes where these properties are present. When using version 2 of the model, one must also choose a value of ρyywhich deter- mines how the forecast errors evolve over successive forecasts. In section 5.7 we investi- gate the effect of switching from version 1, where ρyyis always equal to(ρxy)2, to version 2, with ρyyset to 0.99.