One of the simplest methods for predicting the future behavior of a time series is the so-called persistence model. Persistence implies that future values of the time series are calculated on the assumption that conditions remain unchanged between “current” time t and future time t + Th. For a stationary time series—one whose mean and variance do not change over time—a straightforward implementation of the persistence model is simply
y(t + Th ) = y(t)
which may be referred as “dull persistence.”
However, solar irradiance at the ground level and other related atmospheric phenomena are clearly nonstationary because of diurnal, seasonal, and interannual cycles. For solar applications, dull-persistence models perform poorly for time horizons involving appreciable variations in the diurnal cycle, which limits their use to intrahour applications. A simple and effective way to circumvent this limitation is to detrend the data: to decompose them into (1) a trend component (made up of the clear-sky expected value for the variable at hand) and (2) a random component (made up of random fluctuations about the clear-sky component); that is,
y(t) =ycs(t) +yst(t)
where ycs(t) denotes the clear-sky component of the variable y(t), and yst(t) is the stochastic component of the time series. Depending on the variable under consideration, ycs(t) may be known exactly or may be modeled, or it may be approximated from experimental results.
An alternative way of describing the variable with respect to clear-sky conditions that is often used in the literature is the clear-sky index:
which returns the variable’s ratio with respect to the clear-sky expected value. Figure 15.2 exemplifies the output of these operations for GHI measured every 30 s for Merced, California.
The new detrended variables are more suitable for forecasting. Once they are determined, there are several options to define the persistence model. Two useful definitions are
• Stochastic component persistence
yp1 (t + Th) = (t + TH) + yst (/)
• Clear-sky index persistence
ky t ycs t + Th, if ycs^)S0
ycs (t + TH), otherwise (at night)
The first model (pl) assumes that the absolute value of the stochastic component remains unchanged between times t and t + TH, whereas the second model (p2) assumes that the fraction relative to clear-sky conditions remains the same during the interval between t and t + TH. Figure 15.3 shows the schematic for the three persistence models applied to GHI forecasting.