Wind power ramp events bring great harm to system operation as more and more wind power integrated into power grid. Copying with these new events, a series of studies has been carried out, such as ramp definition, ramp prediction and classification. Currently, there are four mainstream definitions widely used in ramp analysis (Zha et al., 2016). These definitions are defined as follows.

Definition 1

When the change of wind power in time duration ∆t exceeds a given threshold Wp _val , there is a ramp event occurring, as expressed as below. | W p ( t + Δ t ) − W p ( t ) | ≥ W p v a l (1) where: Wp(t+∆t) and Wp(t) are wind power values at time t+∆t and t, respectively; Wp _val represents the threshold of ramp amplitude. When the criterion in Eq. 1 meets under a given time period, a ramp event is identified.

Definition 2

When the largest difference of wind power in time duration ∆t exceeds the given threshold Wp _val , then a ramp event is regarded as occurring. | max i ∈ [ t , t + Δ t ] W p i − min i ∈ [ t , t + Δ t ] W p i | ≥ W p v a l (2) where, max i ∈ [ t , t + Δ t ] W p i and min i ∈ [ t , t + Δ t ] W p i represent the maximum and minimum value of wind power in a time duration ∆t.

Definition 3

When the average change of wind power in a given time duration ∆t exceeds the given threshold Wp _val , a ramp event is identified. ( ∑ i = 1 h | W p ( t + Δ t + i ) − W p ( t + i ) | ) / h ≥ W p v a l (3) where, h is the time horizon in ramp identification.

Definition 4

When the ratio between the wind power change |Wp(t+∆t)-Wp(t)| and time duration ∆t exceeds a given threshold R _val , then a ramp event is regarded as occurring. | W p ( t + Δ t ) − W p ( t ) | / Δ t > R v a l (4) where, this definition in Eq. 4 pays more attention on ramp rate while that in Eq. 1 emphasizes only ramp amplitude. Summarizing the above four definitions, we can see they mainly focus on three major characteristics (Zha et al., 2016), such as ramp amplitude Var, ramp duration T, and ramp rate R. Based on these three characteristics, the variance of wind power in a given period could be determined as ramps or non-ramps. Therefore, they consist of the initial characteristic vector in ramp study, as C _v0 = [Var, T, R]. As we know, sometimes ramp direction is also used in characteristic analysis. However, through above numerical definitions, ramp direction is not necessary, and could be expressed by sign of R when needed. Considering that ramp analysis in some cases needs the detailed information about power range rather than only fluctuation amplitude, so we replace ramp amplitude Var with maximum and minimum values. Assuming there are N studied ramp events in dataset, the initial characteristic matrix X ₀ is expressed as below. X 0 = [ W p 1 , max W p 1 , min T 1 R 1 W p 2 , max W p 2 , min T 2 R 1 ⋮ ⋮ ⋮ ⋮ W p N , max W p N , min T N R N ] (5) where, Wp _i,min and Wp _i,max represent the minimum and maximum value of wind power in the ith studied ramp event.

As the research on ramp events get more in depth, profound characteristics are needed to describe ramps’ formation and categories. For example, the initial characteristics in Eq. 4 is used to identify the basic formation of ramps and non-ramps. The detailed fluctuation inside ramps needs structural characteristics, and the essence of ramps’ occurrence involves energy characteristics. In order to extract more profound information in ramp study, wavelet transformation is applied in this paper. Wavelet transformation is an advanced mathematical technique in signal analysis (Mohanty et al., 2015). It has advantages at decomposing a signal into various time and frequency domains, so it is useful to study the structural characteristics of ramps in different domains. Wavelet transformation also has advantages at detecting abrupt changed values (e.g., in edge detection) and analyzing signal in a specific time window. While ramp events certainly have large power charge in finite time durations, so it is relatively suitable to utilize wavelet transformation in ramp analysis (Escalante Soberanis and Mérida, 2015). On the other hand, due to the multiresolution feature of wavelet decomposition, energy of a given signal could be allocated into different frequency spaces. In this way, we are inspired to extract characteristics for expressing ramps’ energy characteristics. This is also a key reason for considering wavelet transformation in extracting ramp characteristics.

The theory of discrete wavelet transformation is described as below. Assuming a discrete signal (e.g., wind power time series of a ramp event) is expressed as {x(t); t = 1,2, ⋯, T}, it could be reconstructed by elements of wavelet transformation, as expressed below. x ( t ) = ∑ j ∈ Z ∑ k ∈ Z δ j , k ⋅ ψ j , k ( t ) (6) where: x(t) is actually the signal of wind power; δ _j,k is the wavelet coefficient; ψ _j,k is the child wavelet transformed from mother wavelet, denoted as follows. { ψ j , k ( t ) = a − j / 2 ψ ( a − j ⋅ t − k b ) δ j , k = ∑ − ∞ ∞ ψ j , k ( t ) ⋅ x ( t ) a > 1 , b > 0 (7) where: j and k represent the scale and shift parameters of child wavelet ψ _j,k ; ψ(t) is the mother wavelet function; a and b are real parameters. From Eqs 6, 7 it implies that the reproduction of original signal could be realized by the weighted sum of wavelet components at different scales.

To in-depth explain the meaning of each wavelet component in Eq. 6, we could operate wavelet decompose step by step. Assuming the original signal with finite energy is projected on a space L, x(t) with one-level-decomposition could be expressed as below. x ( t ) = x 1 ( t ) + ∑ k d k ⋅ ψ k ( x ) (8) where, x ₁(t) is the estimation of the original signal reflecting variation in time domain; ∑ k d ( k ) ⋅ ψ k ( x ) are detailed signals expressed by wavelet functions which contain frequency-domain information. By estimating the approximate signal iteratively, x _j (t) at the jth decomposition level is expressed as below. x j ( t ) = x j + 1 ( t ) + ∑ k d j , k ⋅ ψ j , k ( t ) (9) Similarly, we could utilize a series of wavelet functions to describe signal x _j (t), as ∑ k c j , k ϕ j , k ( t ) . Combining these formulas, transformation of the original signal in Eq. 6 could be rewritten in details as below x ( t ) = ∑ k c j n , k ϕ j n , k ( t ) + ∑ j = j 1 j n ∑ k d j , k ψ j , k ( t ) (10) where, the auxiliary function ϕ is called father wavelet; c _jn,k is the coefficients of wavelets ϕ j n , k ( t ) ; d _j,k are the coefficients of wavelets ψ j , k ( t ) . The formula in Eq. 10 is generally called as the multi-resolution analysis of wavelet transformation (Doucoure et al., 2016).

On the other hand, by corresponding each wavelet component to a frequency space, we could also divide the space L to series of energy subspaces. As shown in Figure 2, V_j0 represents the original signal space L with a frequency band (0∼f). According to Eq. 10, the frequency band is also divided step by step. For example, V _j0 could be divided as orthogonal sum of a low-frequency space V_j1 (0∼f/2) and a high-frequency space W_j1 (f/2∼f). The relationship reflecting the division of frequency spaces in wavelet transformation is presented as below. { V j ⊕ W j = V j − 1 L = W j 1 ⊕ W j 2 ⊕ ⋯ ⊕ W j n ⊕ V j n (11) where: ⊕ is a denoted operator calculating the orthogonal sum; n is the number of wavelet decomposition.

Considering signal’s energy is generally expressed at frequency subspaces, therefore we could utilize the multi-resolution of wavelet decomposition to analyze the energy distribution of a signal (Ashrafian et al., 2017), as described in the following formula. | P { x } | 2 = ∑ k | c j n , k | 2 + ∑ j ∑ k | d j , k | 2 (12) where: |P_{x}|² represent the energy of the given signal {x(t)}; ∑ k | c j n , k | 2 and ∑ k | d j , k | 2 represent the energy of different subspaces. Since wind process generally involved atmosphere movements (Mohanty et al., 2015), the occurrence of ramp events could be comprehensibly regarded as the speedy energy release or accumulation in atmosphere systems. Therefore, the energy expressions based on wavelet transform could be utilized to extract ramp energy characteristics which is meaningful in studying ramp’s harms on power systems.

Moreover, when the number of decomposition levels is high (e.g., j _n →∞), the value of c _jn,k is small, which implies the energy of V _jn becoming small to be ignored. In that case, the energy characteristics of W _j are mainly selected into the characteristic vector for ramp study in this paper. { P j = ∑ k | d j , k | 2 ; C v = [ W p max , W p min , T , R , P j ] ; j = 1,2 , ⋯ , j n ; (13) where: P _j represents the jth energy characteristic and j is the decomposition level. C _v is the characteristic vector combining initial vector C _v0 and P _j . By utilizing this vector including basic ramp characteristics and wavelet energy characteristics as inputs, some advanced ramp study could be implemented besides the identification of ramps and non-ramps.

First, we propose to utilize dispersion matrix (Gu et al., 2017) to create selection indicator. Dispersion matrix is a mathematic tool based on feature distances of different classes, its elements are denoted as below. { B = ∑ g = 1 G n g ( μ g − μ 0 ) T ( μ g − μ 0 ) W = ∑ g = 1 G ∑ k = 1 n g ( x g , k − μ g ) T ( x g , k − μ g ) ( k = 1,2 , ⋯ , n g ; g = 1,2 , ⋯ , G ;                     n 1 + n 2 + ⋯ + n g = N ) (14) where: W and B are the dispersion matrix representing intra-class and inter-classes, respectively; x _g,k is characteristic vector of the kth ramp in the gth class; μ _g and μ _o represent the average vector of the gth class and all classes, respectively; n _g and N are the number of ramps the gth class and all classes, respectively; G is the number of classes. A Wilks criterion function λ _p could be selected as the reference indicator in feature selection, it is defined as below. λ p = | W | / | T | (15) where: T is total dispersion matrix, calculated as T = B + W ; p is the dimension of feature space. When the value of λ _p is small, implying a small value of | W | and a large value of | T |, it illustrates that the characteristic is effective to distinguish different classes of samples. Generally, the statistical indicator λ _p is assumed to obey the Wilks distribution. By deciding a testing level α and its corresponding threshold λ _(α), the hypothesis testing of λ _p could be implemented. For the convenience of calculation, the value of Wilks distribution function could be estimated by the following two common distribution functions.

1) Bartlett approximation.

Combining the selected Wilks indicator λ _p and ramps’ categories information, a supervised heuristic method could realize high-performance feature selection. In this paper, we propose to utilize SFFS (sequential floating forward search) and SFBS (sequential floating backward search) algorithms (Gan et al., 2014) which retain the strengths and improve the weakness of SFS (sequential forward selection) and SBS (sequential backward selection).

Considering the specificity of ramp studies, the basic characteristic vector C _v0 contains basic ramp characteristics identifying ramps and non-ramps, so it is necessary included in the characteristic subset. Actually, the task of feature selection is to select optimal newly-extracted characteristics in ramp study.

Therefore, in this paper the initial subset is X ₀, the characteristics that need to be processed are wavelet energy characteristics P _i (i = 1,2, ⋯, j _n ). Assuming after the mth selection step, the feature subset is denoted as X _m . If we consider to add a new characteristic x _r at the (m+1) ^th step by SFFS algorithm, the updated dispersion matrix could be calculated as below. { W ' = ( W 11 W 12 W 21 W 22 ) T ' = ( T 11 T 12 T 21 T 22 ) (18) where, W ₁₁ and T ₁₁ are the intra-class and the total dispersion matrix of X _m , respectively; The rest of sub-matrix are newly introduced matrix related with x _r , calculated as the following formulas. { W 12 = ( w 1 r , w 2 r , ⋯ , w p r ) T ; W 21 = W 12 T ; W 22 = w r r ; T 12 = ( t 1 r , t 2 r , ⋯ , t p r ) T ; T 21 = T 12 T ; T 22 = t r r ; (19) Based on calculation of these sub-matrixes, the updated indicator λ_m+1 could be calculated as below. { λ m + 1 = | W ' | | T ' | = | W 11 | | W 22 − W 21 W 11 − 1 W 12 | | T 11 | | T 22 − T 21 T 11 − 1 T 12 | = λ m ⋅ A r A r = | W 22 − W 21 W 11 − 1 W 12 | | T 22 − T 21 T 11 − 1 T 12 | (20) By substituting Eqs 15–18, the expression λ m / λ m + 1 − 1 could be replaced by ( 1 − A r ) / A r . Then, the estimated value of testing x _r is denoted as F _1r , expressed as below F 1 r = 1 − A r A r ⋅ N − p m − g g − 1 ∼ F ( g − 1 , N − p m − g ) (21) where: p _m is the number of characteristics in set X _m . If F _1r >F _α (g-1, N-p _m -g) at given testing level α, then the hypothesis is correct and x _r is added into characteristic vector.

Similarly, if we utilize the SFBS algorithm to delete a characteristic x _r form the set X _m , the final testing value is defined as F _2r , expressed as below F 2 r = 1 − A r A r ⋅ N − ( p m − 1 ) − g g − 1 ∼ F ( g − 1 , N − ( p m − 1 ) − g ) (22) If the formula F _2r ≤ F _α (g-1, N-(p _m -1)-g) is satisfied at a given level α, then the variable x _r is regarded as invalid and removed from X _m .

In ramp classification study, class labels of historical ramp events are taken as the output. The classification model can be constructed by data mining algorithms. To accurately classify ramp events, four data mining algorithms are applied to train ramp classification model, including support vector machine (SVM), neural networks (NN), random forests algorithm (RF) and boosted trees (BT) (Chen et al., 2017; Ouyang et al., 2017c; He et al., 2017; Ouyang, 2021). The optimal classification model could be determined by the comparison of their performance.

To evaluate the classification performance, the confusion matrix introduced from information retrieval (IR) field is widely applied (He et al., 2013b). The detailed expression is presented in the following table.

In Table 1, four types of events are defined, such as true positive event (TP), false negative event (FN), false positive event (FP) and true negative event (TN). Based on these events, several indicators could be defined to evaluate classification performance. Four representative indicators are defined as below. { Re c = n u m TP / ( n u m TP + n u m FN ) Pr e = n u m TP / ( n u m TP + n u m FP ) A c c = ( n u m TP + n u m TN ) / n u m All E r r = 1 − A c c (23) where, num _X represents the number of the specific event X; Pre represents precision indicator implying the percentage of TP in classified true events; Rec represents recall indicator implying the percentage of TP in observed true events; Acc is the classification accuracy, and Err is the classification error. By utilizing these four indicators, we could complete selection of the optimal classification model and the evaluation of classification performance.

Ramp prediction is usually divided into two types: event prediction and regression prediction. Event prediction includes ramp detection and ramp classification study. Regression prediction mainly focus on utilizing traditional regression models to predict ramp characteristics, e.g., the ramp rate prediction in (Zheng and Kusiak, 2009). In this paper, we consider predicting two characteristics: ramp amplitude and ramp rate, which implies values of these two characteristics are taken as the output in modeling. Since the above study on ramp characteristics’ extraction and selection are based on historical ramp events which randomly occur in wind power time series. Therefore, we propose to utilize these characteristics extracted from wind power in a given time window as inputs, then predict one appointed characteristic in the predicted time window, e.g., to predict ramp amplitude or ramp rate in the future 1-h horizon. Since this type of prediction is still based on regression models, the performance indicator could be decided by the commonly used root-mean-square error (RMSE), which is defined as below. R M S E = ∑ k = 1 K ( y ^ k − y k ) 2 / K (24) where, y _k and y ^ k are the kth values predicted and observed of an appointed ramp characteristic; K is the number of tested samples.

Through analysis on ramp definitions, basic ramp characteristics (amplitude, duration, and ramp rate) could be extracted, as (5). These basic characteristics could be used to identify ramps and non-ramps, as two typical signals in Figure 4. To identify more detailed division of ramp categories, wavelet transformation is proposed to extract profound characteristics.

In order to illustrate the feasibility of wavelet characteristics, we firstly utilize wavelet coefficients as an index to qualitatively analyze ramps and non-ramps.

In Figure 4, the two typical ramp and non-ramp events are identified by characteristics in Eq. 5. The function applied here is Haar wavelet which was validated useful in ramp analysis (Gallego et al., 2013). It is seen from Figure 4 that wavelet coefficients which reflect signal’s energy could obviously distinguish ramps and non-ramps. For example, ramps have larger coefficients than non-ramps. It verifies the validity of wavelet transformation in ramp characteristics analysis, therefore we could further extract more profound wavelet characteristics for ramp recognition.

According to the heuristic methods in selecting characteristics, the class labels are required. Therefore, we propose to construct ramp classification according to a specific context. Considering wind process are formed by different meteorological phenomena, so the categories of ramp events are related to division wind processes to some extent. In (Wang et al., 2013), five types of wind process are discussed, namely small wind, small fluctuation wind, large fluctuation wind, double peak wind and multi-peaks wind. However, there are a few ramp events attributed to the small wind. Double peaks wind can be regarded as a special type of multi-peaks wind. Based on these assumptions, we can group historical ramp events into the following category library L, denoted as below. L ={A, B, C} { A={small wind,small fluctuation wind}; B=large fluctuation wind; C={double peaks wind, multi-peaks wind}; (26) Combing with the description of wind process in (Wang et al., 2013) and the constructed library L, historical ramp events of training set are classified into three classes, as presented below.

Table 2 shows the statistical results of ramp events belonging to three categories in L, where number 1, 2 and 3 are defined as the class labels of A, B, C. It is seen that most ramp events are associated with the large fluctuation wind, a few of ramp events associated with the small wind and multi-peak wind. These results agree with the concept of ramp events involving a large change of wind power. Therefore, the constructed application context is reasonable for studying ramp characteristics. Based on the constructed library of ramp categories, then the extraction and selection of ramp characteristics could be implemented.

Assuming each signal is decomposed into five wavelet layers, the energy of each wavelet layer is extracted as ramp characteristics by Eq. 13, expressed as P = [P₁,P₂,P₃,P₄,P₅]. Here, the number of decomposition layers is set as 5 since ramp events have only three categories in this paper. In other application context which requires more refined ramp classification, the level of decomposition could be higher. As the description in Feature Selection, the purpose of feature selection is to delete redundant characteristics, reduce dimension and computation cost. Combining ramp basic characteristics and extracted wavelet energy characteristics, the characteristic vector is expressed as C _v = [Wp _max , Wp _min , T, R, P ₁, P ₂, P ₃, P ₄, P ₅]. For all historical ramp events, the feature set constructed by C _v is denoted as X _p0. Then, according to the selection algorithm in Feature Selection, the process is shown in the following table.

At each step of Table 3, the value of λ _p is calculated for the rest characteristics P _i first. Then the minimum one is applied for F _r testing based on Eqs 15, 16. If the result is satisfied, adding the corresponding characteristic into characteristic vector C _v . It is seen from Table 3 that P ₃ and P ₄ is selected, so the final characteristic is re-written as C _vp = [Wp _max , Wp _min , T, R, P ₃, P ₄], and the final feature set for all ramp events as X _p .

By taking the feature set X _p as inputs and the class labels from Table 2 as output, five data mining algorithms (SVM, NN, RF, BT, ELM) are applied in ramp classification modeling. The classification results of three categories of ramps are shown in following figure.

For the convenience of presentation, Figure 5 utilizes two characteristics (Wp _max ×P ₃ ) to show ramp classification results, as X-axis representing values of Wp _max and Y-axis representing values of P ₃ . Red, blue and green points represent ramps of Class1, Class2 and Class3, respectively. Black points represent ramp events classified into incorrect class. Based on these classification results, the performance indicators defined in Eq. 23 could be calculated, as presented below.

Table 4 presents values of four performance indicators in classification of three ramp categories. According to the definitions of four matrixes, a classification system performs well with large value of Pre, Rec, Acc, and small value of Err. In Table 4, it is seen that NN, RF and ELM algorithms outperform on three classes, respectively. However, it is difficult to choose the best to classify all three ramp categories. To determine the final optimal model in ramp classification, the receiver operating characteristic curve (ROC) is introduced to compare performance further. ROC space is constructed by Recall (Rec) in the X-axis and false alarm (F) in the Y-axis. The definition of F is also based on Table 1, as expressed below. F = n u m F P / ( n u m F P + n u m T N ) (27) where, F calculates the percentage of FP in observed false events. According to these two indicators’ definitions, it is easily comprehended that a classifier having a large Rec and a small F performs better, which implies the upper-left corner of the ROC space means the better performance. For a discrete classification system, a classifier is usually represented by a point in ROC space. Therefore, points representing all classifiers in Table 4 are shown in the following figure.

In Figure 6, three points having the same type present classifiers of three classes by a same algorithm. The points representing SVM and BT perform worse than the other three algorithms again. By comparing the points of NN, RF and ELM, it is seen that points of RF obviously are concentrated and closer to the upper-left corner, implying their classifiers have a better performance. Therefore, RF algorithm is finally chosen for modeling ramp classification in this paper.

Then, taking these trained RF models as ramp classifiers, 100 ramp events are selected from July to December as testing samples. To discuss the performance with and without wavelet characteristics in ramp classification, three different input sets are considered in the case study, such as initial feature set X ₀ consisting of vector C _v0 , feature set X _p0 consisting of vector C _v = [C _v0 , P ] which contains basic ramp characteristics and all wavelet characteristics, refined feature set X _p consisting of C _vp = [C _v0 , P ₃, P ₄] which reduces dimension by feature selection. Classification performance of testing data is presented in the following table.

Table 5 shows values of four indicators at classification of three ramp categories. Since three ramp categories have imbalance distribution as Table 2II, so the classification performance of Class 2 is the best. By comparing classification performance of using three feature sets, it is seen that using X _p0 has better performance than using X ₀, having an average improvement of 10.17% (8.14% on Pre, 14.77% on Rec, 1.80% on Acc, 15.98% on Err). These results imply wavelet characteristics are useful in ramp classification. Also, using X _p improves a little again (with an average improvement of 6.05%) than using X _p0, which implies that the proposed feature selection approach is feasible and effectual in ramp classification. While, for comparing with other feature selection methods, the commonly used PCA is applied. After the analysis of PCA on P = [P ₁, P ₂, P ₃, P ₄, P ₅], two principal components are selected to keep a same dimension with X _p , these two components contribute 84.00% explanation in classification, and consist of the feature set X _pca. Table V also presents the performance of X _pca. By comparing X _p and X _pca, it is seen that using X _p has an average outperformance of 3.69% than using X _pca. Summarizing all these results, the proposed approach on selecting wavelet characteristics is validated to be feasible.

On the other hand, by utilizing these selected ramp characteristics, we could do some try on the study of ramp prediction. Since ramp classification has validated the effectiveness on distinguishing different categories, so two basic ramp characteristics are taken as target output in prediction, such as ramp amplitude (Var) and ramp rate (R). As the design of ramp prediction in Ramp Anomaly Analysis and Evaluation, the historical feature extraction window and the predicted time window are set as the same for convenience. In this paper, ramp prediction is designed to predict two variables (Var and R) within future horizon of 1, 2, ⋯, 5 h, the prediction performance is presented in the following table.

Table 6 shows the performance of ramp prediction by RMSE of two ramp characteristics. In these two variables’ prediction, a typical NN with three layers is used in modeling. For comparison study, four feature sets discussed in Table 5 are also utilized as inputs of prediction models. It is seen from results of Table 6 that models using feature sets containing wavelet characteristics (e.g., X _p0, X _p , X _pca) outperforms than that only containing basic ramp characteristics (e.g., X ₀). Through the proposed feature selection in this paper, the model using X _p has an improvement of 5.97% than using X ₀, 1.29% than using X _p0, and 2.70% than using X _pca on prediction of Var. Similarly, using X _p has an improvement of 16.30% than using X ₀, 8.24% than using X _p0, and 13.07% than using X _pca on prediction of R. Through the discussion on results of Table VI, it is concluded that the proposed approach on extracting and selecting ramp characteristics is also useful for constructing inputs of ramp prediction, and acquires good prediction performance.