4.1. Description of Dataset
The grid-connected PV power station built by the National Institute of Standards and Technology (NIST) in Gaithersburg, MD campus can provide the high-resolution, low uncertainty, comprehensive PV output power data for extended, continuous time periods. There is a single inverter at the station that is connected to the local grid via the NIST campus grid [
35]. In this paper, the data of 70 days in the third quarter of 2015 were selected for simulation. Sampling was done daily from 6:00 am to 7:00 pm every 5 min, and 157 sampling points were included in one day set. In order to obtain an appropriate prediction accuracy with an affordable computation burden, historical data of 62 days were used as the training dataset, and 8 days of data under different weather conditions were chosen as the forecasting dataset. The training dataset includes different weather conditions, and all the dataset only includes historical PV power data. To reflect the prediction performance of the proposed model, the selected forecasting dataset include 2 sunny days, 2 cloudy days, 2 overcast days, and 2 abrupt weather days like sunny to cloudy and cloudy to sunny weather [
36]. For this dataset, the ultra-short-term PV power prediction was carried out with the step length of 5 min.
To quantify errors, the mean absolute percentage error (MAPE) and the root-mean-squared error (RMSE) were used as the main two metrics. In particular, MAPE and RMSE are defined as follows:
where
and
are the
s-th value in the predicted time series and the actual series of measured PV power, respectively, and
N denotes the number of samples in test set.
However, since in some extreme weather conditions or at certain points in time, the actual PV power may fall to zero, the sum of squares due to error (SSE) defined by Equation (32) was used to represent the error in PV power prediction.
The above three evaluation metrics give the prediction information of point-wise error, however, they are not sufficient to distinguish the prediction behavior between different prediction methods. In the variability of PV power, repercussions from large ramping events are of primary concern. Hence it is useful to use ramp metric to quantify the ability of prediction methods to capture the ramp events. In this paper, we use the Ramp score proposed by Vallance et al [
37] as another metric. The Ramp metric is defined as follows:
where
SD(
T(t)) and
SD(
R(t)) are the slopes of the test series and real series ramps, respectively, and the t
max and t
min are the bounds of the period to be predicted.
4.2. Benchmark Models for Numerical Comparison
For comparison, the proposed model was compared with a persistence model (PM) [
38] commonly used as a benchmark model for ultra-short-term PV power prediction. In addition, the performance of the LSTM-based model, LiAENN-based model and the BPNN-based model were also compared to the proposed model.
It is noteworthy to mention that for a fair comparison, the setting of key parameters was tested in the search of optimum values. For the proposed model, the statistic curve obtained with the C-C method is shown in
Figure 6.
As can be seen from
Figure 6, since
has no zero crossing, the first local minimum value of
can be chosen to determine optimal delay amount
in the time series phase space reconstruction. Determine the global minimum value of
, which corresponds to the average trajectory cycle optimal estimate
. From it, we have
and
. We then calculate the optimal dimension
via Equation (8).
For decay rate
, due to the sensitivity of the PV power chaotic system to the initial value, the
should be set as a relatively small value. Take values at intervals of 0.05 within 0 to 1, and each training is repeated 10 times.
= 0 is unreliable, meaning that the model barely learns new pattern-target samples during each training iteration. With the continuous increase of
, the error jump range increases, and finally, the performance of the model tends to be unstable. When
= 0.55, the training fails. Then, constrain the range of
from 0 to 0.05 with step size 0.01. Finally, the value 0.01 is achieved as the optimum decay rate. For the BPNN-based model, we choose BPNN with three-layer network structure, the logsig function is used for the neural-transfer function of hidden layer, and the purelin function is used for the neuron transfer function of output layer. The weights and thresholds of the network are initialized by rand function. The number of neurons in the hidden layer is determined by trail according to the empirical formulas [
39]. Namely, that,
where
G,
l,
H are the number of neurons in the input layer, the hidden layer, and the output layer, respectively; and a is a constant between 0–10.
Here, eight different structures (5-9-1; 5-10-1; 5-11-1; 5-12-1; 5-13-1; 5-14-1; 5-15-1; 5-16-1) of the BPNN-based model were considered. For each structure, the experiment was repeated 20 times, and the results are presented in
Table 2.
As can be seen from
Table 2, the best architecture of the BPNN-based model for PV power prediction is 5-11-1 (5 inputs, 11 hidden neurons, and 1 output).
Table 3 lists the final parameters of the successfully trained models, including the BPNN-based model, the LiAENN-based model, and the proposed model.
4.3. Numerical Results and Analysis
The simulations were carried out, aimed at testing the performance of the proposed model and comparing its performance with the benchmark models. Training and testing of the prediction models were implemented in MATLAB. For a fairer comparison, each model was run 30 times independently.
Figure 7 shows the prediction results of PV power under five typical weather conditions. It is clear that the five prediction models coincide well with the actual value in sunny weather from
Figure 7b,c. It can be seen from the
Figure 7b that the actual power curve is not completely smooth, so the prediction curves of each prediction model have different degrees of deviation throughout the prediction interval. Between 6:00 am to 7:00 am and 15:00 pm to 19:00 pm, the prediction results of the LiAENN-based model and the BPNN-based model both show significant deviations, and the BPNN-based model is the most significant. The prediction results of the PM and the proposed model are relatively close. Overall, the prediction curve of the proposed model is closer to the actual curve. However, the prediction error of the LSTM-based model, PM and the proposed model is mainly reflected in the stage of steep rise and fall of power. In order to further compare the prediction performance of the three prediction models, the prediction curve of the stage with large power fluctuation between 11:00 and 12:00 was selected to be enlarged. From the partial enlarged drawing, it can be seen that each prediction model has a certain lag when tracking PV output. During the power climbing phase, the predicted value is generally lower than the actual value, and during the power decline phase, the predicted value is generally higher than the actual value. The strong inertia effect of the PM model in a short period of time makes the dislocation between the predicted curve and the actual curve most obvious. Compared with the proposed model, the prediction error increases significantly. Compared with the LSTM-based model, the prediction ability of the proposed model at the power inflection point is better than that of LSTM model, which can detect ramp events better. The PV output power curve of
Figure 7c is smoother than that on the first sunny day. The large prediction deviation of the benchmark models appears near the peak value. Combining two sunny test days, the proposed model outperforms all of the benchmark models in sunny weather.
In abrupt weather, the clouds change suddenly, and the PV power suddenly rises or falls with large fluctuation. The prediction results of each model fluctuated to a large extent. In the power smoothing phase, each model coincides well with the actual value. In the stage of large power fluctuations, as shown during 11:00 am to 16:00 am in
Figure 7a and 9:00 am to 11:00 am in
Figure 7f, both the LiAENN-based model and the BPNN-based model have large prediction errors. From the partial enlarged drawings, it can be seen that when the power rises and falls sharply, the prediction curve of the LSTM-based model is smoother than that of the proposed model, and the ability to detect slope events is poor. Although the prediction results based on the PM can reflect the overall trend of PV power, due to the inertia effect of the PM, when the PV power sharply rises and falls, especially at the inflection point, the tracking effect is obviously inferior to the proposed model. The proposed model can still track the original power curve well, although its prediction curve has some fluctuations. This shows that reconstructing the chaotic phase space to extract the original PV power information, and reconstructing the extended signal and emotional parameters, makes the model more sensitive to abrupt changes and fluctuations of PV power.
On cloudy days, effected by the randomness behavior of the clouds, power fluctuates greatly as the PV output is large and the prediction performance of each model is the worst in cloudy weather. From
Figure 7d,e and the partial enlarged drawings, it can be seen that the proposed model still outperforms all of the benchmark models, and the BPNN-based model performs the worst. It is shown that the proposed model successfully eliminates large prediction errors, especially when the PV power fluctuates sharply.
On overcast days, the PV output is low, and the PV power fluctuation is relatively small as the cloud cover is relatively uniform. From
Figure 7g,h, it can be clearly seen that the predicted values of the four models are generally smaller than the actual values in the power climbing stage. The predicted values of the four models are generally larger than the actual values in the power downhill stage. The prediction deviation is mainly reflected near the peak points and valley points; the BPNN-based model is the worst, followed by the LiAENN-based model. From the partial enlarged drawings, overall, the prediction curves of the proposed model, LSTM-based model and PM are close to each other. The prediction curve of the proposed model is closer to the actual curve means that the proposed model can improve the prediction accuracy of PV output on overcast days, but the accuracy is limited. There is still room for improvement.
To closely compare the effectiveness of the proposed model and the benchmark models, the comparison of prediction errors among different models under different weather conditions is summarized in
Table 4. As can be seen from
Table 4, the prediction performance of each model has the least difference in sunny weather. The proposed model outperforms all of the benchmark models under different weather conditions in general, except for individual metrics that are slightly higher than those of the LSTM-based model and PM, which are shown in bold font in the table. Focusing on the average of four metrics under various weather conditions in
Table 5, the improvement in the average MAPE of the proposed model with respect to the other four models is 27.84%, 23.04%, 31.65%, and 44.90%; the improvement in the average RMSE of the proposed model with respect to the other four models is 1.82%, 7.25%, 13.77%, and 22.81%; the improvement in the average SSE of the proposed model with respect to the other four models is 2.58%, 12.71%, 23.93%, and 37.38%; the improvement in the average Ramp score of the proposed model with respect to the other four models is 30.85%, 19.55%, 31.48%, and 42.84%.
As a comparison, the distributions of relative error for the proposed model and benchmark models over an 8-day period are depicted in
Figure 8. The percentage of the relative error is divided into 10 bins and the reduction in prediction error is highlighted in the figure. The largest proportion of reduction in prediction errors associated with the proposed model lies in the first bin; compared with the LSTM-based model, PM, LiAENN-based model and BPNN-based model, it has 9.24%, 5.34%, 14.89% and 20.39% improvement, respectively. This result validates the effectiveness of the LERENN-based model in reducing large prediction errors.
At present, the minimum time resolution of power dispatching is 15 min. To verify prediction performance comprehensively, the three-step prediction of the first, second, fourth, sixth and seventh test days was implemented.
In order to analyze the performance of each model for the three-step prediction, the prediction errors of the four models under different typical weather conditions are given in
Table 6.
As can be seen from
Table 6, except on overcast days, the proposed prediction model has individual metrics slightly higher than the LSTM model. Overall, the proposed prediction model has the highest prediction accuracy, and it can still detect ramp events well. Comparing
Table 4 and
Table 6, the prediction accuracy of all five of the models deteriorates, along with the increase of the prediction steps. The deterioration of each model is different. Compared with single-step prediction, in three-step prediction the RMSE mean value of the proposed model, the LSTM-based model, PM, the LiAENN-based model, and the BPNN-based model are increased by 17.30%, 19.67%, 21.32%, 20.29% and 23.98%, respectively. Overall, the prediction performance of the BPNN-based model is the worst. The proposed model is less affected by the increase of the prediction steps. Namely, the proposed model can improve the prediction accuracy, and it is still robust to power fluctuations and weather changes.