# Predictive Modeling of Photovoltaic Energy Yield Using an ARIMA Approach
## 2 Methods and Design
### Methods  
In this subsection, we introduce the ARIMA method, a fundamental approach for time series forecasting. ARIMA models are widely used for predicting various time series, including energy yields, by analyzing historical data patterns. It calibrates quickly and achieves a good fit even with a relatively small amount of data. This method falls under the umbrella of regression analysis, allowing for the exploration of relationships between a target variable and influencing (in some cases, exogenous) factors that exhibit changes over time, making it applicable across diverse fields (see [29; 30]). The core goal of the ARIMA technique is to forecast future trends in PV yields by focusing on the changes between successive values within the series rather than solely on the observed values themselves. ARIMA models are constructed using three key parameters: \(p\), \(d\), and \(q\)[31]. The autoregressive parameter, \(p\), accounts for the influence of observations from the preceding \(p\) time points. The integrated parameter, \(d\), reflects the underlying trend in the data by incorporating the necessary differencing to achieve stationarity. The moving average parameter, \(q\), smooths short-term fluctuations by incorporating \(q\) lagged forecast errors into the model. Mathematically, an ARIMA model with parameters \(p\), \(d\), and \(q\)[29] can be expressed as follows:  
\[Y_{t}=C+\varphi_{1}\Delta^{d}Y_{t-1}+\ldots+\varphi_{p}\Delta^{d}Y_{t-p}- \theta_{1}\epsilon_{t-1}\;-\ldots\theta_{q}\epsilon_{t-q}\;+\epsilon_{t},\]  
where \(C,\varphi_{i},i=\overline{1,p},\) and \(\theta_{j},j=\overline{1,q},\) are the parameters to be estimated. Here, \(\epsilon_{j}\) is a randomly distributed error term with a mean of zero and a variance \(\sigma^{2}\). In the absence of specific distribution information, a normal distribution is typically assumed by default. The difference operator \(\Delta\) is defined as follows:  
\[\Delta^{0}Y_{t}=Y_{t},\;\Delta^{1}Y_{t}=Y_{t}-Y_{t-1},\ldots,\Delta^{k}Y_{t}= \Delta^{k-1}Y_{t}-\Delta^{k-1}Y_{t-1}.\]  
To identify the appropriate parameters \((p,\;d,\;q)\), the autocorrelation function (ACF) and partial autocorrelation function (PACF) are examined (see [32]). These functions, which depend on a predefined number of lags, are computed for each time point \(t\), excluding instances where computation is not feasible. During the calibration, all the available data are employed, while the residuals' ACF and PACF are used to validate the models. During the fitting procedure, few of the very first data points are used to initiate the moving average (MA) component, and the rest of the data are used for calibration. The calculations are performed by means of IBM SPSS Statistics software ([[https://www.ibm.com/products/spss-statistics](https://www.ibm.com/products/spss-statistics)]([https://www.ibm.com/products/spss-statistics](https://www.ibm.com/products/spss-statistics))). If the ACF and PACF of \(Y_{t}\) show more than one minor deviation beyond the  
Figure 3: The generated power (kW) from the PV system between 2013 and 2023 for ten years. confidence intervals, the model parameters are adjusted. Significant spikes and repetitive patterns in these functions signal the estimated values of \(p\) and \(q\), serving as the basis for parameter selection in this study. The input variables are the time series of total energy and specific energy yield, given in Tables 1 and 2. The output (target) variables are the forecasts, which are thoroughly described in the next section. Essentially, the ARIMA method finds a relationship between the input and the output variable and uses this knowledge to make predictions of the target variable. The organization of the algorithm is displayed via the flowchart in Figure 4.  
In the process of predicting, when a suitable model configuration is chosen and calibrated, the forecast is obtained at once.