5. Forecast error modelling for wind integration studies
5.6. Implementation in FREDA
5.6.1. Construction of wind forecast time series
The forecast error model has been implemented in FREDA, a power system simulation tool that is described in chapter 4. At each timestep k, a forecast for the normalised wind level F(k, i), i =1 . . . Nf is generated using the method described in section 5.2 and
converted to a forecast wind output using Equation (5.6).
The wind power time series in FREDA can be read from historic data or generated from the time series model described in chapters 2 and 3. If historic data are used, and a non-trivial forecast error process is chosen, we will generally have to fit the model to the data anyway in order to calculate the innovations (residuals) and the autoregressive parameters, which are used in the forecast error generation. The innovations ǫx(k) of the normalised wind level X(k)are required to generate the innovations ǫy(k, i)of the
forecast error process Y(k, i), from equations (5.14) or (5.28), if a non-zero value of ρxy is used. The autoregressive parameters are required because we will often choose the autoregressive parameters of the forecast error process to be the same as those for the realised wind process, for reasons discussed in section 5.3.
The procedure for setting up the realised and forecast wind power time series within FREDA is as follows:
CHAPTER 5. FORECAST ERROR MODELLING FOR WIND INTEGRATION STUDIES
1. Read in the historic wind power time series (if used), realised wind model param- eters ϕxj, σx, W(·), µ(·), and forecast error parameters ϕyj, ρxy, ρyy, syi. If the au- toregressive parameters ϕyj are not specified, their default values are the same as the autoregressive parameters ϕxj of the realised wind model. If the ρyy parame- ter is not specified, the forecast error process is generated using version 1 of the model (5.14). If it is specified then version 2 is used (5.28). The correlation ρxy can be specified directly, or a flag can be used to indicate that it is to be generated us- ing Equation (5.21), to create approximately unbiased forecasts. The scale factors,
syi, can either be specified directly or fitted to a user-specified RMSE profile using Equation (5.29).
2. Generate the time series of realised wind innovations ǫx(k)and normalised wind levels, X(k). If historic wind data are being used then the historic power output levels Phw(k)must be converted to normalised wind levels using
X(k) =W−1(Phw(k))−µ(k mod Nd). (5.33)
The innovations (residuals) then follow from
ǫx(k) =X(k)−
p
∑
j=1ϕxj X(k−j). (5.34)
If the realised wind is to be synthesised by the model, then the ǫx(k)are first gen- erated as independent N(0,1) random variables. The normalised wind levels X(k) follow from Equation (5.3) and the corresponding realised wind power outputs from Equation (5.2).
3. For each timestep k, generate the innovations for the forecast normalised wind level, ǫy(k, i), i = 1 . . . Nf, using Equations (5.14) or (5.28) as appropriate, and use them to create normalised wind forecasts F(k, i)from Equations (5.9), (5.8) and (5.7). The median wind power forecast made at timestep k for i timesteps ahead, Pwf(k, i), is then calculated from Equation (5.6).
5.6.2. Calculation of nodal forecast errors in scenario tree
FREDA uses a Stochastic Unit Commitment algorithm to arrive at the scheduling deci- sions. The set of possible wind power paths over the next few hours is discretised into a scenario tree of linked nodes, with each node having an associated wind power level (and hence a forecast error) and a probability weighting. The topology of the scenario tree is user-specified, along with the nodal probability quantiles, and the nodal probabil- ity weightings are precalculated as in section 4.4.2. However, the nodal forecast errors must now be dictated by the forecast error model, rather than from the uncertainties in the realised wind model which produce the statistical-only forecasts from Equations (4.34)–(4.36).
CHAPTER 5. FORECAST ERROR MODELLING FOR WIND INTEGRATION STUDIES
The following equations define the nodal forecast errors in the more generalised model considered here. The statistical-only forecasts of chapter 4 are in fact a special case of this model with ρxy =−1 and syi = σx for all i. Any undefined symbols are in the List of Symbols on page 12.
Provided that node n’s ancestors are one timestep apart, we can derive my(n, i), the expected Y value at i timesteps after node n, as follows:
my(n, i) = p
∑
j=1 ϕyj × 0 : ℓ(n) +i−j≤0, y(a(j−i)(n)): i−j≤0, my(n, i−j): i−j>0. (5.35)The standard deviation of the Y value at i timesteps ahead, σiy, is independent of the fore- cast wind level and is most conveniently calculated from the equivalent MA parameters (ψyj), which can be derived from the autoregressive parameters using Equation (5.5):
σiy= v u u t i−1
∑
j=0 (ψyj)2. (5.36)We can now calculate the nodal forecast errors analogously to chapter 4, in which the forecast error corresponds to the user-specified nodal probability quantile, conditional on the wind forecast error state variables at the node’s progenitor. We use the Gaussian properties of the process to write
y(n) =my b(n), h(n)
+σhy(n)Φ−1 q(n), (5.37) from which we can calculate the nodal forecast errors in normalised wind level, by mul- tiplying by the appropriate scale factor:
z(n) =syℓ(n)y(n). (5.38)
It can be seen that the calculation of my(n, i)in Equation (5.35) for all nodes other than the root node is conditional on the nodal Y values on the scenario leading up to node n, i.e. y(n′): n′ ∈ A(n). These values will have been obtained from Equation (5.37), so that the nodal Y values can be built up for the whole tree using a recursive procedure starting at the root node.
In the special case of a scenario tree with a fan topology, i.e. branching at the root node only, Equation (5.38) simplifies to
z(n) =syℓ(n)σ y ℓ(n)Φ−
1 q(n)
(5.39) which can be calculated without recursion. This simplified version will be used for all the studies in the current and subsequent chapters.
CHAPTER 5. FORECAST ERROR MODELLING FOR WIND INTEGRATION STUDIES
The nodal forecast errors z(n)are precalculated for all n ∈ N before the simulation begins. During the simulation, the nodal wind powers are calculated at each timestep k from Equation (4.36), repeated here:
Pwn(n) =WF k,ℓ(n)
−z(n) +µ (k+ ℓ(n))mod Nd
. (5.40)