The research direction of this paper is prediction accuracy under the combination of solar and wind energy systems. In essence, the optimal solution of a hybrid wind and solar energy system does not affect the accuracy of the prediction. Based on Lazard’s latest annual levelized cost of energy analysis [
27], the mean value cost of wind and solar energy is 0.045 and 0.05 American dollars per kilowatt hour (kWh), respectively. Those two costs of generating electricity are growing progressively smaller, and eventually the unit generating cost of solar energy may be same or even less than that of wind energy. Therefore, we can assume that the unit cost of wind and solar energy can be regarded as the same. The purpose of setting this assumption is to simplify the problem to avoid discussing the proportions of wind and solar, because the problem of proportions does not affect the performance of predictability of the hybrid solar and wind system. Based on this assumption, we use double the capacity of Starfish Hill wind farm, or installing the same capacity of solar panels from a prediction point of view, to determine the best choice. Under this assumption, the following sections will deal with the set-up for testing the predictability of wind, solar and a hybrid of these.
2.1. Data
The Starfish Hill wind farm energy output and Adelaide Airport measured global solar radiation from August 2009 to January 2010; 30 min of data were used to demonstrate how the CARDS model [
27] was employed for prediction performance of a hybrid solar and wind system. Starfish Hill wind farm is near Cape Jervis, South Australia, approximately 88 km south of Adelaide. Adelaide is the capital of South Australia, latitude 34°51′ S, longitude 138°33′ E and the time zone is AEST (UTC+9.5). The wind farm data set consists of 8784 (48 × 183) half-hourly wind power output values in total. The Starfish Hill wind farm installed capacity is 34.5 MW. The wind farm is located south of Adelaide at latitude 35°34′ S and longitude 138°9′ E. The average half-hourly wind-power output was 5.4 MW and the maximum value was 16.9 MW. The solar data set consists of 8784 (48 × 183) half-hourly global radiation values in total. The average half-hourly global radiation was 258 W/m
2 including the night-time period, and the maximum value was 1139 W/m
2. In order to correspond to the same installed capacity of the Starfish Hill wind farm of 34.5 MW, and according to the Adelaide Airport-measured global solar radiation 30-min data, the energy flow of assumed solar power plant is illustrated in
Figure 1. The installed capacity of the solar farm is 34.5 MW, the measured solar panels area about 202,941 m
2, and the efficiency rate of solar panels about 17%. Note that the two locations are sufficiently close to be able consider them as a combined solar and wind system, as though the total output feeds into the grid through a single connection.
Table 1 displays the Starfish Hill wind farm and assumed solar farm energy output general information.
2.2. The Coupled Autoregressive and Dynamical System (CARDS) for Solar
The CARDS model is built on the Fourier series, autoregressive (AR) methods, a dynamical system model known as the Lucheroni model [
28], and a fixed component [
26]. The Lucheroni model is a resonating model for the power market which exploits the simultaneous presence of a Hopf critical point, periodic forcing and noise in a two-dimensional first order non-autonomous stochastic differential equation system for the log price and the derivative of the log price. The fixed component is a variable construct in our previous paper for considering measured data at time
t and
t − 1 to further correct the problem of the model change delay.
Here, a Fourier series is mainly used to capture the periodic change characteristics of data, which is the non-linear part. The autoregressive component is used to model the linear part of the data. The Lucheroni model is also used to capture the high-frequency part of the data, and the fixed component is used to further improve the accuracy of the linear part. The CARDS model is built by using the Visual Basic® program (VB is a universal object-based programming language developed by Microsoft®).
The first step is using the Fourier series to deseason the data. Following [
29], and using the results of Power Spectrum analysis, the Fourier series can be written as
St, and the result is shown in
Table 2.
Here,
is the mean of the data,
and
are coefficients of the yearly cycle, and
and
are coefficients of the daily cycle and its harmonics, n and associated beat frequencies,
m.
Table 1 shows the frequency of yearly, daily and twice-daily cycles, with the percentage of variance of the original time series explained by the contribution of the Fourier series component at each frequency. All significant values are indicated in bold. From
Table 1, it can easily be seen that, overall, the
St can represent over 80% of the variance of the data.
Figure 2 displays the three-day Fourier series compared with assumed solar farm energy-flow data.
According to the CARDS model steps, the next step is to model the deseasoned data Rt, where is assumed solar farm energy flow based on Adelaide airport-measured global solar radiation 30-min data.
The Box–Jenkins methodology comprises a joint investigation of the sample autocorrelation coupled with the partial autocorrelation function. See Boland [
30] for details on how a slowly decaying autoregressive function and a partial autocorrelation function having the bars going to zero abruptly indicate an autoregressive process.
Figure 3 shows that the deseasoned data follow a typical AR(2) model, so
can be written as,
Figure 4 shows AR(2) as compared with deseasoned data.
Figure 4 shows that the AR(2) cannot capture the maximum value of high-frequency data very well. Therefore, the Lucheroni model is used to make up for the AR model [
26,
28].
Based on that, the model can be written as,
Here,
is the noise after a model is developed for
and for forecasting we do not include it in the Lucheroni model, and
is Lucheroni model forecast.
is time span, which for our data is 1/48, because it corresponds to half-hourly data. The new component
is introduced to help us find the future vale of
. The ordinary least squares (OLS) is used to estimate all the other components
к,
λ,
ε,
γ and
b. The coefficients are
к = −8.41,
λ = −0.94 × 10
−15,
ε = 0.15,
γ = −0.0096 and
b = 2.68.
Figure 5 shows that the Lucheroni model captures the peak value better than the AR model (compared with
Figure 4). Therefore, the Lucheroni model is used for capturing the rising part of residual solar radiation values and AR(2) for falling values. After combining the AR and Lucheroni models, the fixed component is used to further improve the accuracy of the combined AR and Lucheroni model (for more detail see [
26]).
Based on three different models and one component, the general function of the CARDS and Fourier series models can be written as,
Here, et is assumed to be white noise with mean zero and variance . Equation (6) shows that the Fourier series is used to capture the seasonal component of the data; the combined AR and Lucheroni model is used for the linear part which is the deseasoned component of the data; and the fixed component is used to further improve the forecasting accuracy by modifying the forecasting of the combined AR and Lucheroni model.
Figure 6 shows the three-day Fourier series model plus CARDS model compared with measured Adelaide airport global solar radiation 30-min data generating an assumed energy flow of a solar power plant.
2.3. The Modified CARDS for Wind
Wind energy was found to have fewer seasonal features than solar radiation.
Table 3 shows that only the yearly cycle gives a slightly significant contribution. Therefore, there is no need to use the Fourier series method to capture the seasonal component of the Starfish Hill wind farm energy output data. The autoregressive method can model wind energy very well, so there is no need to use the combination model [
31]. In order to ensure that the predictability is compared under the same benchmark, we still use the CARDS model as the foundation method to test the wind-energy data. Note that the autoregressive model performs quite well compared with other models to forecast short-term wind-energy data, such as the self-exciting threshold autoregressive (SETAR) and smooth transition autoregressive (STAR) models [
32].
Through partial autocorrelation function analysis, the Starfish Hill wind farm energy output is following a typical AR(4) model (see
Figure 7). The AR(4) function can be written as:
After an AR(4) model is estimated, the fixed component is used to further improve the autoregressive model, and the general function of the modified CARDS models can be written as,
Here, Equation (8) indicates that for modified CARDS, we only need the AR model to capture the linear component and then use the fixed component to further improve the forecasting accuracy by modifying the forecasting of the AR model.
Figure 8 shows how for seven days the CARDS model fits with Starfish Hill wind farm energy data.