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.) [37]. The running process of K-means++ is as follows [38]:

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, K-means++ clustering method is used to directly cluster the historical power data of each season. The reason for selecting historical power data for clustering is that the aging of the equipment itself and its own indicators are different under different weather conditions. It is difficult for us to finely 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). The Euclidean distance between the 12 meteorological factor characteristic values of the forecast day and each cluster centroid is calculated to determine the cluster to which the forecast day belongs.

Grey relational analysis refers to the quantitative description and comparison method of the development and change of a system. The basic idea is to judge the correlation degree by comparing the geometrical similarity between the reference data column and several data columns. It reflects the degree of correlation between the curves. 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 [39]:

Step 1: Determine the reference sequence y that reflects the characteristics of the system behavior and the comparison sequence x_i that affects the system behavior:

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. 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.

SVM obtains the ability to linearly analyze the nonlinear characteristics of the sample by mapping low-dimensional data to high-dimensional space. Based on the structural risk minimization theory, SVM constructs the optimal classification surface in the feature space, thereby overcoming the local optimal problem and requiring fewer training samples. When the data type is complex, SVR can be used, which was first developed by Vapnik et al. [40]. The definition of SVR is as follows:

f ( x ) = ω T ϕ ( x ) + b (8)

where ω is a vector of weight coefficients, ϕ ( x ) is the nonlinear mapping function (mapping x to a high-dimensional feature space), and b denotes a bias constant. In addition, b and ω 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.

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 ) (11)

where γ is the kernel parameter.

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:

S ( i ) = b ( i ) − a ( i ) max { a ( i ) , b ( i ) } (12)

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:

DBI = 1 n ∑ i = 1 n max j ≠ i ( S i ¯ + S j ¯ ‖ ω i − ω j ‖ 2 ) (13)

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.

Sum of squared errors (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 (14)

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. They are defined as follows.

1) The RMSE is defined as:

RMSE = 1 N ∑ i = 1 N P f i − P a i (15)

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 | (16)

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 ) (17)

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 geographic location, physical object and configuration information of the photovoltaic array are shown in Figure 3, Figure 4, and 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 summer as an example, the experimental results are shown in Figure 5, Figure 6, Figure 7, and Figure 8. It can be seen from Figure 5 that SSE decreases as the number of clusters increases, and when it reaches 2, the downward trend begins to slow down. Observe Figure 6, DBI achieves the best value in 3 clusters. Figure 7 shows that the maximum value of S appears in 2 clusters.

After comprehensively considering each evaluation index and cluster observation results, the number of clusters in summer clustering is set to 2, and the result is shown in Figure 8. In Figure 8, the blue cluster is defined as ideal weather cluster, and the red cluster is non-ideal weather cluster. 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. Finally, 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 selection of the threshold for similar days is not only related to the season, but also to the weather conditions on the forecast day. For ideal weather, it is observed that when the correlation degree is greater than 0.85, the power curve of similar days in each season is in a relatively ideal state (as shown in Figure 9, the power curve of August 19, 2018 (winter) with a predicted day correlation degree of 0.85). In order to increase the speed of training the model and reduce the computational cost, a higher correlation threshold is used in the experiment.

Under ideal weather conditions, the forecast days of each season, the correlation threshold of similar days, and the selection of nearest neighbor similar days are shown in Table 3. The thresholds of similar days in each season are 0.90, 0.88, 0.89 and 0.86.

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. The optimal SVR structure for each season under ideal weather is shown in Table 4. Observing the following table shows that the models training time is 2.0342 s, 1.9506 s, 2.3272 s and 0.6826 s respectively.

Figure 10 shows the prediction results of the HKGSVR model in each season under ideal weather conditions. The feature combination is the power of the nearest neighbor similar day and the 12 meteorological factor eigenvalues of the forecast day (NP_W). Because the prediction accuracy is high under ideal weather condition, feature selection is considered from the difficulty of obtaining and the accuracy of the data. The daily weather eigenvalues of the forecast day are easier to obtain and more accurate than the hourly forecast value.

It can be seen from Figure 10 that the forecast performance of each season is superior under ideal weather conditions. From the evaluation metrics in Table 5, it can be seen that 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 model has high prediction accuracy for ideal weather.

As shown in Figure 11, HKGSVR’s forecast results in each season are significantly better than SVR. Through the comparison of Table 5 and Table 6, it can be found that the training and optimization time of HKGSVR is much shorter than that of SVR within the same search range. Compared with SVR, the proposed model has an optimization of 74.99%, 73.86%, 69.18%, and 91.05% in training and optimization time for each season. 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.