K-means++ clustering algorithm is an improved version of K-means algorithm. This algorithm separates the K initial cluster centers more from each other. In this work, which is selected as the classifier due to its higher efficiency and improved robustness compared with others (e.g., standard K-means, K-medoids, Gaussian mixture models, etc.) [30]. The running process of K-means++ is as follows:

Step 1: Randomly select a sample as the first cluster center C₁;

Step 2: Calculate the probability of each sample being selected as the next cluster center:

D ( x ) 2 ∑ x ∈ X D ( x ) 2 (1)

where, D(x) represents the distance between the sample and the nearest cluster center.

Then use the roulette method to select the next cluster center;

Step 3: Repeat step 2 until K cluster centers are selected;

Step 4: For each sample x_i in the datasets, calculate its distance to K cluster centers, and then put it into the class corresponding to the smallest distance cluster center;

Step 5: For each cluster, recalculate its cluster center C_i:

c i = 1 c i ∑ x ∈ c i x (2)

Step 6: Repeat steps 4 and 5 until the position of the cluster center does not change.

In this part, the historical power data is directly clustered by season to obtain different power generation modes due to the diversity of weather. Moreover, the aging of the equipment itself and its own parameters will be different under different weather, it is difficult for us to accurately measure these changes. The characteristics of historical power data will integrate these changes into it. After clustering the historical power data, the centroid value of each cluster is calculated by the minimum, average and maximum of global horizontal irradiance (GHI), diffuse horizontal irradiance (DHI), relative humidity (RH) and temperature (T)(12 meteorological factor eigenvalues).

The basic idea of the grey relational analysis algorithm is to judge the correlation degree by comparing the geometric similarity between the reference sequence and several data columns. Generally, the more consistent the change tendency of the reference sequence and the comparison sequence, the higher the degree of correlation between the two variables. The flow of the GRA algorithm is as follows:

Step 1: Determine the reference sequence y and the comparison sequence x_i:

y = { y ( k ) | k = 1 , 2 , ⋯ , n } (3)

x i = { x i ( k ) | k = 1 , 2 , ⋯ , n } , i = 1 , 2 , ⋯ , m (4)

where, n and m represent the dimension of the eigenvalues and the number of comparison sequence, respectively.

Step 2: Non-dimensionalization of variables:

d j * ( k ) = D j ( k ) − D a v ( k ) D max ( k ) − D min ( k ) , k = 1 , 2 , ⋯ , n ;   i = 0 , 1 , 2 , ⋯ , m ;   j = 1 , 2 , ⋯ , m + 1 (5)

where, D_j(k) contains reference sequence and comparison sequence, D_av(k), D_min(k) and D_max(k) are the average, minimum and maximum values of each column, j represents sum of the number of reference sequence and comparison sequence.

Non-dimensionalization is used to solve the problem that the columns cannot be compared due to the different dimensions.

Step 3: Calculate correlation coefficient ξ_i(k):

ξ i ( k ) = min i min k | y ( k ) − x i ( k ) | + ρ max i max k | y ( k ) − x i ( k ) | | y ( k ) − x i ( k ) | + ρ max i max k | y ( k ) − x i ( k ) | (6)

where, ρ is called the resolution coefficient, here, ρ is 0.5.

Step 4: Calculate correlation degree.

Calculate the average value of the correlation coefficient at each moment (that is, each point in the curve) r_i:

r i = 1 n ∑ k = 1 n ξ i ( k ) , k = 1 , 2 , ⋯ , n (7)

Step 5: Sort correlation degree.

After determining the cluster to which the prediction day belongs, the correlation between the prediction day and each sample in the cluster is calculated by GRA based on 12 meteorological factor eigenvalues, and the date with the correlation degree greater than the threshold (an appropriate correlation value that takes into account the similarity and the number of samples) is regarded as the similar days. Based on GRA global matching results: for the ideal weather, the sample with the highest correlation in the 7 days before the forecast date is set as the nearest neighbor similar day; for non-ideal weather, the sample with the highest correlation in the 30 days before the prediction is set as the nearest neighbor similar day.

Based on the structural risk minimization theory, the support vector machine constructs a hyperplane in the feature space, thereby overcoming the local optimal problem and requiring fewer training samples [30]. When the data type is complex, support vector regression is used. For a set of data { ( X i , Y i ) , i = 1 , 2 , ⋯ , n } , X_i is the input variable of the sample, and Y_i is the target value. The support vector machine equation based on Vapnik theory is as follows [31]:

f ( x ) = ω T ϕ ( x ) + b (8)

where, ω is a vector of weight coefficients, Φ(x) is the nonlinear mapping function and b denotes a bias constant.

ω and b can be obtained by the following formula:

minimize : 1 2 ‖ ω ‖ 2 + C ∑ i = 1 n ξ i + ξ i * (9)

subject   to { y i − 〈 ω , ϕ ( x i ) 〉 − b ≤ ε + ξ i 〈 ω , ϕ ( x i ) 〉 + b − y i ≤ ε + ξ i * ξ i ≥ 0 , ξ i * ≥ 0 (10)

where, ξ i and ξ i * are slack variables, and C denotes the penalty variable, ε is the insensitive loss function.

By introducing Lagrangian multipliers and optimal constraints, (8) can be transformed into:

f ( x , a i a i * ) = ∑ i = 1 n ( a i − a i * ) K ( x , x i ) + b (11)

where, K(x, x_i) = Φ(x_i)Φ(x_j) is the kernel function.

In this paper, the radial basis function (RBF) kernel is applied to construct the SVR model. The RBF kernel is presented as:

K ( x i , x i ) = exp ( − γ ‖ x i − x i ‖ 2 ) (12)

where, γ is the kernel parameter.

The flow chart of the hybrid K means-GRA-LSTM model is shown in Figure 1, and the workflow is as follows:

Step 1: Obtain historical photovoltaic output power and meteorological factor data, and deal with the missing and abnormal data in the data set.

Step 2: Use the multi-index K-means++ algorithm to cluster historical photovoltaic power data by season, and calculate the 12 meteorological factor eigenvalues as the central value of each cluster. According to the average power of each cluster, it is divided into ideal weather cluster and non-ideal weather cluster.

Step 3: The Euclidean distance, Pearson correlation coefficient and GRA correlation between the 12 meteorological eigenvalues of the forecast day and the centroid value of each cluster are calculated to determine the cluster to which the forecast day belongs.

Step 4: Calculate the correlation between the predicted day and each sample in the matched cluster through GRA to obtain the similar days (as the training set) and the nearest neighbor similar day (as the validation set) and normalize them.

Step 5: Select appropriate input features and similar days are used to train SVR. Determine the C and γ of SVR through grid search and cross-validation, and use the nearest neighbor similar day to test.

Step 6: Use the trained model to predict the prediction day.

If the ground truth labels are not known, evaluation must be performed using the model itself. The Silhouette Coefficient is an example of such an evaluation, the score is higher when clusters are dense and well separated. Silhouette Coefficient S(i) is defined as follows [32]:

S ( i ) = b ( i ) − a ( i ) max { a ( i ) , b ( i ) } (13)

where, a(i) is the mean distance between a sample and all other points in the same cluster, b(i) is the mean distance between a sample and all other points in the next nearest cluster. Average the Silhouette Coefficient of all points, which is the total Silhouette Coefficient of the clustering result.

Davies-Bouldin index is defined as follows [33]:

DBI = 1 n ∑ i = 1 n max j ≠ i ( S i ¯ + S j ¯ ‖ ω i − ω j ‖ 2 ) (14)

where, S i ¯ is the average distance from the points in the cluster to the cluster centroid, ‖ ω i − ω j ‖ 2 is the distance between the centroid of cluster i and j.

The Davies-Bouldin index is lower if the model clusters have better separation.

SSE is also an effective metric, that is, the sum of squared errors of the distance between the centroid of each cluster and the points in the cluster. SSE is defined as follows:

SSE = ∑ i = 1 K ∑ d i s t ( x , c i ) 2 (15)

In order to evaluate the performance of the proposed method HKGSVR for photovoltaic power generation forecasting, the root mean square error (RMSE), average absolute error (MAE) and coefficient of determination (R²) indicators were calculated. The mean absolute error can better reflect the difference between the predicted value and the true value. RMSE is used to measure the deviation between the predicted value and the actual value, so it is more sensitive to outliers (that is, if the predicted value of a point is very different from the true value, the RMSE of the curve will be very large). R² is used to test the fit of the predicted value to the true value, and is generally used to evaluate the prediction performance of the model. They are defined as follows [14].

1) The RMSE is defined as:

RMSE = 1 N ∑ i = 1 N P f i − P a i (16)

where, P_ai and P_fi are the actual and predicted value at i hour. N refers to the number of hours a sample contains.

2) The MAE is expressed as:

MAE = 1 N ∑ i = 1 N | P f i − P a i | (17)

3) The R² is given as:

R 2 = ( N ∑ i = 1 N P f i P a i − ∑ i = 1 N P f i ∑ i = 1 N P a i ) 2 ( N ∑ i = 1 N P f i 2 − ( ∑ i = 1 N P f i ) 2 ) ( N ∑ i = 1 N P a i 2 − ( ∑ i = 1 N P a i ) 2 ) (18)

In this paper, the general datasets on the DKASC (Desert Knowledge Australia Solar Center) website are used for related experiments. The photovoltaic array is composed of 22 polycrystalline silicon photovoltaic panels with a rated power of 265 W, whose total rated power is 5.83 kW. The photovoltaic array is located at the Desert Knowledge Precinct in Alice Springs, a town in the Northern Territory that enjoys one of the country’s highest solar resources in an arid desert environment. The configuration information of the photovoltaic array is shown in Table 1. Meteorology (global horizontal irradiance, diffuse horizontal irradiance, relative humidity and temperature) and historical power data of PV arrays from March 1, 2018 to February 29, 2020 were used in the experiment. The experiment uses data with an interval of 1 hour from 7:00 to 18:00 every day.

In order to obtain the appropriate number of clusters for each season, SSE, DBI and Silhouette Coefficient (S) are used for evaluation. Taking autumn as an example, the experimental results are shown in Figure 2.

It can be seen from Figure 2 that SSE decreases as the number of clusters increases. When the number of clusters is 3, the downward trend begins to slow down. DBI has the best performance when the value of K is 3. When the value of K is 2 and 3, the value of S is 0.71 and 0.64, respectively. Then, as the value of K increases, the value of S drops sharply. So the value of K is chosen between 2 and 3. When K = 2, the blue cluster and the red cluster merge into one cluster. However, the blue clusters are mostly smooth arcs, while the red clusters are mostly polylines. Therefore, the value of K is chosen to be 3. The blue cluster is selected as the ideal weather cluster (most of the curves are smooth and the average power is larger in the cluster), and the green and red clusters are non-ideal weather clusters (most of them are broken lines in the clusters, and the average power is small, the average power of the green cluster is 123.09 kW, and the average power of the red cluster is 301.90 kW).

The evaluation of clustering results in each season is shown in Table 2. In order to prevent local optima or other abnormal situations, 100 rounds of experiments were carried out. Considering all indicators and clustering results comprehensively, the number of clusters in spring is 3, the number of clusters in summer is 2, the number of clusters in autumn is 3, and the number of clusters in winter is 3.

The clustering results of each season are divided into ideal weather clusters and non-ideal weather clusters by comparing the average power and geometric shape (arc and polyline) of each cluster. The average power of each cluster in the four seasons is shown in Table 3. The ideal weather clusters are mostly smooth

arcs, and the average power is relatively large. The non-ideal weather clusters are mostly broken lines, and the average power is small. Therefore, spring cluster 1, summer cluster 1, autumn cluster 1 and winter cluster 2 and 3 are divided into ideal weather clusters, and the rest are non-ideal weather clusters.

The similar days are obtained by calculating the GRA correlation between the predicted days and the samples in the matching clusters. In order to improve the prediction accuracy while reducing the computational cost and speeding up the training speed of the model, it is necessary to select an appropriate correlation threshold. A higher correlation threshold can improve the prediction accuracy, but too few training samples may cause overfitting. After comprehensive consideration, the similar day correlation threshold of each forecast day, the nearest neighbor similar day and its correlation degree are shown in Table 4 and Table 5. It can be seen that the nearest neighbor similar days of ideal weather are mostly adjacent days, while the time intervals of nearest neighbor similar days of non-ideal weather are relatively long.

This part is mainly to explore the optimal C and γ of SVR, which are usually related to the characteristics of power generation in different seasons. Grid search and cross-validation are used to find the optimal number of C and γ for SVR. This experiment uses PyCharm (python3.6) to train and optimize the SVR model on a Win 7 System personal computer with Intel core i5-3230CPU, 2.60 GHz processor and 4 GB RAM.

It can be seen from Table 6 and Table 7 that the optimal training time of the ideal weather model for each season is 2.0342, 1.9506, 2.3272, 0.6826 s, and the average time is 1.74865 s. And the optimal training time of the model for each season of non-ideal weather is 0.2490, 0.2400, 0.2240, 0.2060 s, and the average time is 0.22975 s. The number of similar days matched has a greater impact on the training optimization time. Comparing Table 6 and Table 7, it can be found that because the data complexity of non-ideal weather is higher than that of ideal weather, the C of non-ideal weather is generally larger than that of ideal weather, and the γ of non-ideal weather is generally smaller than that of ideal weather.

In order to select the appropriate input feature, GRA and Pearson correlation analysis is performed between the power generation and various meteorological factors. The historical power and meteorological data for the year from March 1, 2018 to February 28, 2019 are used for analysis. The result is shown in Figure 3. The definition of Pearson correlation coefficient is as follows:

ρ X , Y = N ∑ X Y − ∑ X ∑ Y N ∑ X 2 − ( ∑ X ) 2 N ∑ Y 2 − ( ∑ Y ) 2 (19)

where, X and Y are meteorological factors and photovoltaic output power respectively, and N is the number of sampling points per day.

It can be seen from Figure 5 that the Pearson correlation coefficients between photovoltaic power generation and T, RH, GHI, and DHI are 0.35, −0.41, 0.97, and 0.35, respectively. GHI has the greatest impact on photovoltaic output, and there is a negative correlation between relative humidity and photovoltaic power. The GRA correlations between photovoltaic power generation and T, RH, GHI, and DHI are 0.68, 0.60, 0.88 and 0.67, respectively. GHI still has the largest impact on photovoltaic output.

Based on the above analysis, the paper proposes 10 feature combinations. The prefix N represents the nearest neighbor similar day, P, G, and M respectively represent Power, GHI and meteorological factor eigenvalues. For example, NG_MG represents the nearest neighbor day GHI and predicted day meteorological factor eigenvalues and GHI.

For ideal weather, due to its high prediction accuracy, the main consideration for the selection of its input features is to select a feature combination that is easier to obtain and requires less data accuracy while ensuring sufficient prediction accuracy. Therefore, the input features of the ideal weather are power of the nearest neighbor similar day and 12 meteorological factor eigenvalues of the forecast day (NP_M). For non-ideal weather, the main goal of feature selection is to improve the prediction accuracy. Tables 8-10 show the evaluation of 10 feature combinations of non-ideal weather in each season. The best performance of the evaluation indicators in the table is bolded.

It can be seen from Tables 8-10 that the MAE of NG_MG feature combination in each season is 1.3733, 2.0817, 1.6475, and 2.2323 kW, respectively. And

the average MAE is 1.8337 kW, which is the smallest among all feature combinations. The RMSE of the NG_MG feature combination in each season was 1.4699, 2.6625, 1.8700, 2.5492 kW. And the average RMSE was 2.1379 kW, the best performance among all the feature combinations. The R² of each season of NG_MG feature combination is 99.18%, 97.31%, 99.26% and 98.14%. And the average R² is 98.47%, which is the highest degree of fit among all feature combinations. From the perspective of comprehensive performance, NG_MG feature combination has higher prediction accuracy and robustness, so NG_MG is selected as the input feature of non-ideal weather.

Figure 4 shows the prediction results of the models in each season under ideal

weather. The feature combination is NP_M. The average value of R² is 0.9966. The MAE are 1.4521, 1.4661, 0.7120, and 0.2132 kW respectively. The average value of RMSE is 0.9608 kW. The proposed model’s MAE enhancement with respect to the SVR model is 45.51%, 25.37%, 81.21%, 91.72%, respectively. The presented model’s RMSE improvement relative to the SVR model is 41.63%, 38.80%, 77.97%, 90.37%, respectively. The average R² of the proposed model is also better than the SVR model. And this method reduced the average training time by 77.27% compared with the standard SVR model.

Under non-ideal weather, the forecast results of the HKGSVR model and the other five forecast models for the four seasons are shown in Figures 5-8. It can be seen from the figure that the HKGSVR model has the highest degree of fit in each season.

From the spring forecast results in Figure 5, it can be seen that HKGSVR has the highest degree of fit, HKGLSTM is the second, and the SVR trend is more consistent with the predicted day. From the summer forecast results in Figure 6, it can be seen that each point of HKGSVR has a high degree of fit, and HKGLSTM has a good performance except for one point that has a lower degree of fit. Through the observation of the autumn forecast results in Figure 7, HKGSVR still performs best, and the trends of SVR and HKGARIMA are more consistent with the forecasted day. From the winter forecast results in Figure 8, it is found that both HKGSVR and SVR perform better, and HKGLSTM and HKGBP have poor performance due to less training data.

According to the MAE value of each model given in Figure 9, the HKGSVR's average value of MAE is 1.8337 kW, which is the minimum value of all models. The presented model’s average MAE improvement relative to the compared five models (SVR, HKGLSTM, HKGBP, HKGLR, HKGARIMA) are 26.61%, 41.80%, 58.18%, 57.36%, 52.98%, respectively.

Observation in Figure 10 finds that the average RMSE of the HKGSVR model is 2.1379 kW, which is the best value among all models. The proposed model’s average RMSE enhancement with respect to the compared five models is 24.13%, 52.12%, 59.87%, 61.37%, 52.66%, respectively.

Comparing the R² values of the models shown in Table 11 shows that the average R² of the proposed model is 0.9847, which is better than other models.

Under non-ideal weather, compared to the standard SVR (the average training optimization time is 11.5389 s), the average training optimization time of the HKGSVR model is 0.2225 s, which is 98.07% less than the standard SVR.