At present, several machine-learning approaches have been studied to develop models for PV power forecasting. These models use on-site power-measured data in combination with numerical weather prediction or satellite-derived data. These methods commonly minimize data used while increasing the overall accuracy of the solution. However, these approaches rarely provide a physical description of the PV system’s components and configuration. In this study, we focus on methods without exogenous data, in other words, methods that include exclusively on-site PV power-measured data and on-site measured weather data.

Mandal et al (2012) offered a solution with a mix of wavelet transform (WT) and different machine learning techniques. A PV system model is created by dividing the PV power time-series data between the ill-mode and non-linear fluctuations (spikes in power). After that, four components are extracted by downsampling and then filtered using low-pass filters. Finally, the PV system model is obtained after a wavelet reconstruction, including some upsampling after feeding each one of the individual components from the TW into different neural networks. A mean absolute percentage deviation (MAPD) of 2.38% can be achieved for clear-sky periods of PV power forecasting. This method has a horizon of 12 h, and its performance decreases considerably when cloudy or rainy days are forecasted (Mandal et al, 2012).

Almeida et al (2015) proposed an alternative model for PV power output forecasting that includes on-site weather data. They applied a random forest (RF) model to create a PV model and then used numerical weather prediction data to forecast the PV power output. The results suggested an MAPD of 9.5% between the forecasted and the on-site measured data (Almeida et al, 2015).

This contribution proposed a data-driven approach based on the use of artificial neural networks (ANNs) trained with measured PV power to create day-ahead forecasts of PV power. Unlike other methods, this solution does not require any weather data and offers an MAPD of 6.64% for the best-case scenario.

The suggestions from González Ordiano et al (2017) on using weather-free approaches based on machine learning techniques are particularly relevant. This contribution proposed a data-driven approach based on the use of ANNs trained with measured PV power to create day-ahead forecasts of PV power. Unlike other methods, this solution does not require any weather data and offers an MAPD of 6.64%, for the best-case scenario (González Ordiano et al, 2017).

In contrast to the various machine learning solutions, there are only a few physics-based or hybrid methods in the literature that provide a description of a specific PV system or subsystem’s configuration.

The article by Ogliari et al (2017) presents a model for next-day PV power forecasting based on the well-known single-diode model, which can consider either three or five parameters. In addition, the authors propose two other approaches, an ANN model and a hybrid model combining both physical and ANN models, for next-day forecasting. The ANN and hybrid models use historical weather data for training, with two different training approaches. However, it is important to note that the physical parameters required for the single-diode model need to be determined from the PV module’s datasheet or from previous experiments (Ogliari et al, 2017).

It is important to mention that the physical model used in Ogliari et al can only describe the behavior of a PV module under outdoor conditions without considering the losses from the rest of the components of a PV system or subsystem, i.e., cabling losses and inverter efficiency (DC-to-AC ratio). Ogliari et al reported achieving mean absolute deviations (MADs) of 19.1 W and 20.2 W for the three- and five-parameter physical models, respectively. In the case of the hybrid method, the MAD is 12.46 W for the best case of the first training approach and 12.5 W for the best case of the second training approach.

As shown in the orange rectangle in Figure 1, the adapted PV system simulation tool can simulate AC PV power with only on-site measured T mod and G as inputs. Comparable to Guzman1, the simulation tool proposed here neglects some PV power performance losses, such as soiling, degradation, snow, and inter-row shading. However, surrounding shading loss effects (horizon line) can be included in the G on-site measurements, in addition to specific objects projecting shadow to a particular part of the PV system or subsystem. The PV system simulation model proposed here comprises two main steps, DC PV power and AC PV power:

Using equations suggested by Heydenreich et al (2008)(Eqs. 5, 6), we simulated DC PV power (Heydenreich et al, 2008). First, the DC PV power at a temperature of 25°C, or standard test conditions (STCs) Eq. 5, was simulated, and later, we translated that simulated DC PV power to the on-site measured temperature and irradiance conditions Eq. 6. η G , T S T C = a G + b ln G + 1 + c ln 2 G + e G + 1 − 1 , (5) where T S T C is 25°C, and a , b , and c are constants defining the PV module efficiency curve at a specified G . η G , T mod = η G , T S T C 1 + γ T mod − T S T C , (6) where γ represents the PV module temperature coefficient, T mod represents the PV module temperature measured on-site, and in both equations, and η represents the PV module efficiency at specific conditions.

To simulate AC PV power output, we used the inverter model proposed by Schmidt and Sauer (1994), accounting for 1% of cabling losses (Schmidt and Sauer, 1994).

To define the best climatic conditions for the parameter extraction, in the GA optimization training phase, we created two different training sets:

• The first set includes all possible conditions measured on-site i.e., overcast and clear-sky-like moments. This training dataset will be referred to in this work as all-sky.

We detected and filtered clear-sky-like moments based on the two-step process described in detail in Guzman1. First, based on the on-site measured PV power data, a statistical clear-sky curve was created following the method implemented and proposed by Stein et al (2012) and Reno and Hansen (2016). Next, with the “detect_clearsky” function from the PVLib library (Holmgren et al, 2020), we detected clear-sky-like moments. Training datasets of both clear-sky and all-sky moments comprised only on-site measured PV power data and their correspondent on-site measured T mod and G being used in this study.

In general terms, we based our GA algorithm on the technique proposed by Holland (1975). Although GA optimization is considered a traditional optimization algorithm, it has been rarely implemented in the solar industry. Moreover, inherently, the GA optimization offers an accurate and time-efficient deterministic approximation of the real parameters of unknown PV systems and subsystems. Additionally, the GA performs effectively for problems relating to dynamic environments where an optimal answer can evolve over time. In these situations, the solution space might be too big to thoroughly explore, and the ideal answer can change as the situation changes (Mori and Kita, 2000).

The novelty of this work relies on the fact that we can extract the main characteristics of an unknown PV system or subsystem by implementing a similar GA optimization to the one proposed by Holland (1975) within a stepwise process. In this work, we opted for a stepwise optimization process to reduce the compensation between parameters.

To extract the most accurate PV system parameters, we minimized the MAD (see Eq. 4) between simulated AC PV power and on-site measured AC PV power. The MAD of the best member of the population and population mean MAD are two key performance indexes to monitor throughout the GA optimization. The optimization process is interrupted if neither of the two performance indexes improves any more. A detailed description of this process can be found in Guzman1.

In the GA optimization, to create the initial population of PV system parameters to be evaluated and optimized, we began by defining the initial parameters for this work as 1-kilowatt peak (kW_p) for nominal power, −0.43 %/°C for the power temperature coefficient, and a ratio of 1 for the DC to AC power.

Next, we normalized the on-site measured AC PV power by the maximum measured value. By normalizing on-site measured AC PV power, we compared it with the simulated AC PV power of a 1-kWp installed capacity system. It is important to mention that the measured G must be in the same plane of array of the PV system or subsystem. After the AC PV power is normalized, the subsequent steps are followed to complete the optimization process for the assumed 1-kWp PV system:

• With cross-validation optimization, we selected the best set of a, b, and c parameters for (Eq. 5). The cross-validation optimization process was based on a database comprising 107 sets of three parameters, described in Guzman1, including the results from Fraunhofer ISE CalLab efficiency measurements of 107 PV modules at different irradiance levels. We simulated DC PV power, using each one of the 107 sets of parameters, and compared it with the normalized AC power. We considered the lower MAD as the optimum set of parameters for the simulated PV system.

The results obtained from applying the PV models with the optimized parameters in this study accurately reflect the real behavior of the tested PV system. As a result, by using these optimized parameters, we can create a digital twin of the PV system or subsystem and simulate its performance by changing the irradiance and temperature conditions to current (now-cast) or future (forecast) values.

Figure 2 shows two digital twins created from on-site measurements of a system located in southwest Germany with a training dataset of 30 days length (before the day to be tested), in which all-sky conditions are considered. On the left-hand side, subplot (a) shows a PV power simulation for a clear-sky-like day: measured PV power, the measured G . Subplot (c) shows the deviation between the simulated and measured PV power in W/kW_p. On the right-hand side, subplot (b) shows overcast-like day results: simulated PV power, measured PV power, and measured G . Subplot (d) shows the deviation between the simulated and measured PV power in W/kW_p. On-site data for the optimization have been measured during the year 2020, and the results presented in this example considered only daytime measurements.

In this work, we used a database comprising 16 real on-site measured PV systems installed in Germany, which are part of the Fraunhofer ISE monitoring portfolio. The geographical location of those systems can be observed in Figure 3.

The database used here, collected between 2018 and 2020, consists of a time series of approximately 567,500 points of 5-min resolution including three main features: measured PV power, measured T mod , and measured G . Table 1 shows the nominal power and the G sensor installed at each location of the 16 PV systems. PV systems from 36 kWp up to 1,202 kWp were considered in this publication.

Additionally in Section 4.3.1, we used daily observations of snow depth data from the Deutscher Wetterdienst (German Weather Service) climate data center (CDC) (Kaspar et al, 2013) for the year 2020, for all 16 locations.

To evaluate the effect of seasonality and the length of training data on the parametrization of the digital twin and the accuracy of now-casting, we randomly selected 1 day per week of the year 2020 (a total of 52 days). For each selected day, we used six different training dataset lengths, including 30, 60, 90, 120, 150, and 182 days prior to the selected day. We also identified clear-sky conditions within all the training datasets and trained the GA using both all-sky and clear-sky moments. Each combination of training dataset length, selected day, and all/clear-sky moments resulted in a different set of parameters (digital twin) for the PV system under consideration.

Figure 4 shows plots of optimization results for nominal power and the DC-to-AC ratio, considering 52 randomly chosen days and different training set lengths: 30, 60, 90, 120, 150, and 182°days. Optimization results for the temperature coefficient have been reported constantly throughout the 52°days (−0.4%/°C) and, therefore, are not shown in the figure. In general, while considering all-sky moments, the DC-to-AC ratio (see subplot c)) remains more constant than while considering only clear-sky moments. In contrast to that, optimization for nominal power remains more stable while considering only clear-sky moments (see subplot b)).

The results of a quantitative analysis are included in Table 2. The analysis includes results from the GA optimization considering clear-sky and all-sky conditions for all the different training length datasets.

One of the advantages of parametrizing a PV system or subsystem based on only on-site measured data (including G ) is the low variability of optimized parameters over time. As shown in Table 2, a training length of a minimum of 120 days is a common agreement between the best results, considering both clear-sky and all-sky moments.

The parametrization results for nominal power shown in Figure 4 are on the side of an underestimation. As mentioned previously, using only clear-sky moments seems to be more stable. As shown in Table 2, the best conditions for an accurate nominal power parametrization are as follows: all-sky conditions in combination with a 120-day training length. This combination shows a mean value of 510.55 kW_p with an MAPD of only 7.93% and an MBD of only −42.45 kW_p. In contrast to our previous publication (Guzman Razo et al, 2020), these values have reduced considerably from 10.69% and −84.24 kW_p, respectively. This is most likely due to the increase in accuracy of G which was measured on-site instead of using satellite-based irradiance information.

Given that the reported nominal power value is 553 kW_p, there is still a deviation between it and the value extracted using the GA model. One possible explanation for this deviation could be attributed to a degradation rate, which according to Jordan et al (2016), is expected to be between 0.8% and 0.9% per year since the installation year (Jordan et al, 2016). For system ID5, which has been installed for 10 years to the date of the experimental data, a degradation rate of between 8% and 9% can be expected, resulting in a nominal power between 508.76 kW_p and 503.23 kW_p. The value calculated using the proposed GA model shows good agreement with the expected value. However, this should not be taken lightly as additional power losses can exist and require further investigation.

It seems that there is a trade-off between using all-sky conditions and clear-sky moments for parametrization of the AC-to-DC ratio; see Figure 4. Although all-sky conditions provide more stable results in a particular period, clear-sky moments tend to result in spreader results on the side of overestimation. However, regardless of the condition used, a training length of 182 days seems to be the most suitable for accurate parametrization; see Table 2. The best-case scenario shows a mean value of 1.06 with a 5.7% MAPD and an MBD of 0.02, which is an improvement from the MBD of −0.14 achieved in Guzman1.

As mentioned previously, a constant parameter has been estimated for the 52 test days for the power temperature coefficient. Therefore, the parametrization results of the Heydenreich et al model for PV system ID5, considering both scenario clear-sky and all-sky moments, are the following:

• Heydenreich a: 0.004326

Although the optimal conditions for parametrizing the presented model are defined based on clear-sky or all-sky conditions and the length of the training dataset, the accuracy of the now-casting must also be evaluated. For instance, if the nominal power is close to the reported value, it would receive a higher score. However, these values do not account for additional losses that are commonly present in PV systems under outdoor conditions, such as soiling and degradation.

Therefore, in the following subsection, we evaluated the accuracy of now-casting using the parameters calculated with different training dataset lengths.

To select the optimal training length for the GA algorithm presented here, in this section, we evaluate simulated PV power with parameters from the digital twin considering only daytime values. We randomly selected 52°days from the year 2020 of System ID 5 and generated a set of parameters for each of the 52°days in combination with different training lengths (30°days, 60°days, 90°days, 120°days, 150°days, and 182°days). Finally, we compared the simulated PV power with the on-site measured PV power.

To generalize and correlate the results offered here with any other PV system or subsystem, we measured the deviation in W/kW_p installed. Table 3 shows the results of now-casting for all the training datasets (30, 60, 90, 120, 150, and 182°days) and all the conditions (all-sky and clear-sky).

We evaluated the now-casting results based on the MAPD parameter. As shown in Table 3, the best combination of conditions and training length is achieved with all-sky conditions and a 182-day training dataset, resulting in an MAPD of 0.91%.

Figure 5 shows that the high variation in the parameterization, particularly in the DC-to-AC ratio, is not necessarily reflected in the now-casting results. The now-casting deviation using all-sky conditions (subfigure (a)) is consistently underestimated compared to the deviation using only clear-sky conditions (subfigure (b)), particularly for short training datasets (30–90°days). However, the absolute deviation of the now-casting is similar for both all-sky and clear-sky conditions, particularly for longer training data sets (120–182°days).

Based on the results presented here, we considered that 182 days (or 6 months) based on all-sky conditions is the minimum required length to train GA optimization. With these condition-training lengths, it is possible to achieve an MBD of 3.27 kW_p and an MAPE of only 0.91%. Previous publications, including Guzman1, reported an MAPE from 6% to 10% for now-casting tests (Ding et al, 2011; Mandal et al, 2012; Kaspar et al, 2013; Monteiro et al, 2013; Ibrahim et al, 2015; Landelius et al, 2019; Guzman Razo et al, 2020).

According to the results presented in the previous subsections, a training dataset length of 182°days is suggested to achieve high accuracy for both the digital twin parametrization and now-casting. To validate this suggestion, a 182-day training dataset was selected for each of the 16 monitored PV systems for the year 2020. Similar to the experiment described in the previous subsections, a random day from each week of the year 2020 was selected (52°days in total) to create a digital twin for each PV system on each selected day.

Figure 6 shows the PV power now-casting results for all 16 real PV systems. As observed, the mean deviation of all 16 real PV systems, in W/kW_p installed, indicates an under-casting, −1.14, which is influenced by high-deviation peaks in winter, particularly for System ID15. Additionally, the figure shows that the MAD for all 16 real PV systems is merely 1.81%, indicating a good benchmark for short-term PV power forecasting.

In general terms, the now-casting accuracy of all the digital twins created throughout the year 2020 for all 16 real PV suggests that their performance is time-independent, indicating that they can be implemented at any time of the year. However, there are some exceptions in winter which will be clarified in the following subsection.

We acknowledge that the GA optimization method presented here has some limitations. In addition to the loss factors mentioned in the aforementioned subsections (degradation, power clipping, and soiling), it is also important to mention that some shading effects, such as inter-row, can directly impact the optimization results. Furthermore, measuring G in a different POA will lead to biased PV power simulation, which ultimately will lead to incorrect parametrization. Additional filters and flags have to be implemented within the database to ensure that the on-site measured data are valid. Additionally, special PV system configurations, such as single- and double-axis tracked systems and bi-facial systems, require modifications to the PV power simulation model and optimization logic.