1 Introduction

Volatility, intermittency, randomness, and uncertainty of wind power can threaten the security and stability of power grid, and even cause serious losses [1]. Large fluctuations of wind power, such as significant increases or decreases in a short period, will form the so-called wind power ‘ramp event’ [2–4]. Wind power ramp events are particularly important in the management and the dispatch of wind power. It is necessary to regulate the output of traditional generators in power grid to make up for the substantial changes of wind power, such as using grid ancillary services or restricting and cutting down the ramp rate of wind turbine output. However, all the measures will cause major economic losses to the operation of power grid. The research of ramp event forecasting is an effective method to solve these issues which can help the power system operators, especially dispatching operators, developing wind power integration schemes and ancillary decisions in advance [5].

According to the difference of ramp directions, ramp events can be generally divided into two basic types: up-ramp and down-ramp [6]. The characteristic of the up-ramps is sharply increasing. Strong low pressure air systems (or cyclones), low-level jets, thunderstorms, wind gusts or similar atmospheric phenomena will induce up-ramps [7]. Likewise when the wind power suddenly drops or high speed gusts of wind make wind turbines reach cut-out limits (usually 22~25 m/s), wind turbines would be shut down for protecting themselves from damages. All this phenomena could induce down-ramps [8]. Down-ramps are the reverse physical process of up-ramps [9]. Based on the property of each ramp event, ramp events are parameterized by the following variables: ramp start time, ramp duration, ramp rate, ramp swing, and ramp end time.

There are two major problems for forecasting wind power ramp events. As for the statistical or time-series models, the first problem is that the deterministic point prediction methods can obtain different results after simulations, such as the autoregressive moving average (ARMA) [10], neural networks (NNs) [11] and support vector machine (SVM) [12]. The second problem is the difficulty to forecast the ramp characteristics with respect to physical models [13]. These models use the surface roughness, orography, obstacles, atmosphere pressure, and temperature of a wind plant site, which is too complex to be simulated.

In this paper, a new wind power ramp event forecasting strategy is proposed and its efficiency is demonstrated by using several numerical experiments of an actual wind plant. The main contributions of this paper can be summarized as follows.

  1. 1)

    A novel and efficient forecast method for wind power ramp events is proposed. The forecast method is composed of a NN probability generating technique to generate forecast scenarios and a modified genetic algorithm to optimize model parameters. All the cumulative distribution features and higher order moment features of actual wind power can be considered to generate the forecast of wind power ramp events.

  2. 2)

    Less previous researches in the area of wind power ramp events focused on forecast methods. In this paper, an effective feature detecting technique, based on the forecasted statistical scenarios captured bands and the dynamic programming recursion algorithm, is used to detect all the forecasted ramp characteristics and compared with the actual ramp characteristics to verify the validity.

This paper is organized as follows. Section 2 reviews previous work on wind power ramp event forecast. Section 3 formulates the NN probability generating model and introduces a method to solve the optimal model parameters. Section 4 presents a ramp event detection algorithm. Section 5 presents the results of experiments in forecasting ramp events of an actual wind plant. Section 6 comes to the conclusion.

2 Previous work

The field of wind power ramp event detection and forecasting is a recent research hotspot in the sustainable energy sector. Especially as representatives, Raffi Sevilian from Stanford University and Nicholas Cutler from New South Wales University bringing this issue to a new research level. Sevilian defines a family of scoring functions with ramp event definitions and uses a dynamic programming recursion to detect all ramp events. It is also mentioned that the identified ramp information could be used in forecasting and simulation [14]. Cutler forecasts ramp events with the engineering application for the first time and evaluates the efficiency of the Wind Power Prediction Tool (WPPT) and the Mesoscale Limited Area Prediction System (MesoLAPS) for ramp event forecasting [15]. But sudden drops in an interval, determining a premature termination or start for ramp events cannot be excluded. Ramp events in [12] are grouped in classes and the SVM method is used to forecast and classify ramp events. This method maybe forecast ramp events successfully, but its lack of forecasting structure will make it undesirable for establishing models in future control applications, such as dispatching or unit commitment issues. Reference [16] used the ramp durations and ramp intensity of the predicted ramp events as additional variables to improve the reliabilities for the forecasted quantiles. It maps the number of ensemble members for forecasting a specific ramp event to a probability of that ramp actually occurring to produce confidence intervals of ramps occurring. But these confidence intervals of ramp events cannot be used conveniently without specific scenarios when referring to the dispatching or unit commitment issues. Reference [17] showed a user-friendly way of forecasting up-ramps and down-ramps with the uncertain start time and combines the numerical weather prediction (NWP) to bring the forecasting errors down. But for forecasting ramp events, better effect relies too much on better NWP forecasts. Better NWP forecast has a ramp capture nearly 10% higher than the combination and the other NWP forecasts.

3 Problem formulation

3.1 NN probability generating model

Assuming that the current moment is t, the input variables X contains (n + 1) inputs, where n inputs contain the current wind power (at time t) and historical wind power data (at time t − 1, t − 2, …, t – n + 1) and 1 input contains the possible wind power value of the next moment (at time t + 1). The output variable O for the next moment contains the probability \(p_{{\tilde{X}_{t + 1} ,m}}\) of the mth possible wind power value. The NN probability generating model is shown in Fig. 1.

Fig. 1
figure 1

NN probability generating model diagram

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Due to the strongly nonlinear mapping relationship between the input and output variables, this model is established instead of the specific analytic expressions. Probability generating model has (n + 1) input layer nodes, 1 is the output layer node, and l are hidden layer nodes, a total of [(n + 1)l + l × 1] weight parameters and (l + 1) threshold parameters. Thus, there are [(n + 1)l + l × 1 + l + 1] model parameters to solve. The calculation process of the model is as follows:

Step 1: Neural network initialization

The network input and output \(\left( {X|O} \right) = ( {X_{t - n + 1} ,X_{t - n + 2} , \ldots ,X_{t - 1} ,X_{t} ,\tilde{X}_{t + 1,m} |p_{{\tilde{X}_{t + 1} ,m}} } )\) needs to be determined. After initializing connection weight parameters (\(\omega_{ij}\) and \(\omega_{jk}\)) among the input, hidden and output layer and threshold parameters (\(\alpha\) and \(\beta\)) of the hidden layer and output layer, the learning rate and neuron excitation function are given.

Step 2: Hidden layer calculation

Hidden layer outputs H are calculated based on the input variables X, connection weight parameters \(\omega_{ij}\) and threshold parameters \(\alpha\) as follows:

$$h_{j} = f\left( {\sum\limits_{i = 1}^{n + 1} {\omega_{ij} x_{i} } - \alpha_{j} } \right)\quad \, j = 1,2, \ldots ,l$$
(1)

where l is the hidden layer nodes; f is the hidden layer excitation function.

There are many types of the excitation function formulations and this paper uses the function as:

$$f(x) = \frac{1}{{1 + e^{ - x} }}$$
(2)

Step 3: Output layer calculation

According to the hidden layer outputs H, connection weights \(\omega_{jk}\) and threshold parameter \(\beta\), the probability \(p_{{\tilde{X}_{t + 1,m} }}\)of the mth possible wind power value at the next moment is calculated as the neural network output:

$$p_{{\tilde{X}_{t + 1,m} }} = \sum\limits_{j = 1}^{l} {h_{j} \omega_{jk} } - \beta \quad k = 1$$
(3)
$$P_{{\tilde{X}_{t + 1} }} = \left\{ {p_{{\tilde{X}_{t + 1,1} }} ,p_{{\tilde{X}_{t + 1,2} }} , \ldots ,p_{{\tilde{X}_{t + 1,m} }} , \ldots ,p_{{\tilde{X}_{t + 1,M} }} } \right\}$$
(4)
$$\sum\limits_{m = 1}^{M} {p_{{\tilde{X}_{t + 1,m} }} } = 1$$
(5)

where \(P_{{\tilde{X}_{t + 1} }}\)is the totality of all the probability values; M is the total number of possible wind power values at the next moment.

3.2 Solving optimal model parameters

The improved genetic algorithm is adopted to solve all the parameters of the NN probability generating model. One individual in the population represents all the weight and threshold parameters of one determinate neural network. Individual fitness value is calculated by the improved fitness function. And then the individual corresponding to the best fitness value is found by the selection, crossover and mutation operations. Eventually the optimal individual is created and assigned to the optimal weight and threshold parameters of the NN probability generating model.

3.3 Multi-objective fitness function

The core part of the improved genetic algorithm for solving model parameters is to establish a set of multi-objective function, which consists of the cumulative density function (CDF) and the higher order moment autocorrelation function. The absolute error E 1 is calculated between the forecasted and expected CDFs, and the absolute error E 2 is calculated between the forecasted and expected higher order moment autocorrelation functions. The final individual fitness F is the sum of absolute errors E 1 and E 2 (with the equal weight) as:

$$\hbox{min}\,F = E_{1} \left( {Y_{\text{cdf}} ,O_{\text{cdf}} } \right) + E_{2} \left( {Y_{\text{corr}} ,O_{\text{corr}} } \right)$$
(6)
$$E_{1} = \sum {\big|Y_{\text{cdf}} - O_{\text{cdf}} \big|} \,$$
(7)
$$E_{2} = \sum {\big|Y_{\text{corr}} - O_{\text{corr}} \big|}$$
(8)

where \(Y_{\text{cdf}}\) is the expected CDF of mounts of actual historical wind power; \(O_{\text{cdf}}\) is the forecasted CDF of the scenario sequence sampled by the NN probability generating model; \(Y_{\text{corr}}\) is the expected higher order moment autocorrelation function value of mounts of actual historical wind power; \(O_{\text{corr}}\) is the forecasted higher order moment autocorrelation function value of the scenario sequence sampled by the NN probability generating model.

3.4 Cumulative density fitness function

Based on the probability \(p_{{\tilde{X}_{t + 1,m} }}\) of the mth possible wind power value and the definition of the cumulative density function, CDF can be calculated as:

$$O_{\text{cdf}} \left( {\tilde{X}_{t + 1,m} } \right) = \sum\limits_{i = 1}^{m} {p_{{\tilde{X}_{t + 1,i} }} }$$
(9)

After all the M possible wind power forecasted value and their possibilities have been prepared, the method for generating the forecasted CDF will be performed. The horizontal axis is divided into (M − 1) equal and non-overlapping intervals (with the width 1/(M − 1) of each interval). CDF generation and iterative schemes are shown in Fig. 2. The probability of each sample point has been generated by the NN probability generating model, as shown in Fig. 2a. The iterative process of solving CDF is shown in Fig. 2b. Equation (9) is modified as

Fig. 2
figure 2

CDF generation and iterative schemes

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$$O_{\text{cdf}} \left( {\tilde{X}_{t + 1,m} } \right) = O_{\text{cdf}} \left( {\tilde{X}_{t + 1,m - 1} } \right) + p_{{\tilde{X}_{t + 1,m} }}$$
(10)

3.5 Higher order moment autocorrelation fitness function

In fact, even if the forecasted and expected distribution of CDF is entirely fitting, the probability paper test result shows tnat the distribution is just quasi [18], as shown in Fig. 3.

Fig. 3
figure 3

CDF curve and its probability paper test

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Figure 3a shows that the two CDFs fit completely, while the splashes in Fig. 3b do not gather in a straight line. This phenomenon illustrates that the distribution of the forecasted CDF cannot obey that of the expected CDF. Even though the forecasted random sequence could satisfy the distribution properties (such as the first order moment statistical property of a random variable) of a certain actual wind power, the higher order moment [19] statistical properties are not consistent. Therefore, the higher order moment autocorrelation fitness function needs to be considered to establish the time sequence [18] between the forecasted and actual wind powers, and maintains the higher order moment features of the actual wind power sequence. The iterative process of the secondary order moment autocorrelation function is shown in Fig. 4.

Fig. 4
figure 4

Iterative schemes of secondary order moment autocorrelation function

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3.6 Population initialization

Each individual is a real number string, that is, using real number coding method. Each individual consists of connection weights between the input layer and hidden layer, thresholds in the hidden layer, connection weights between the hidden layer and output layer, and the threshold in the output layer. Each individual contains all the NN weight and threshold parameters, and corresponds to a determinate NN probability generating model.

4 Detecting algorithm for ramp events

After solving optimal parameters of the forecasting model, forecasting scenarios and scenarios captured bands are sampled in quantity [20, 21]. Predicted characteristic values in all the scenarios are recognized by the detecting algorithm and the extensive statistical analysis is performed to calculate the probabilistic estimated values. The performance of the statistical scenarios forecasting model is evaluated by comparing the forecasted values with actual ones of ramp event characteristics.

Based on the large number of forecasting statistical scenarios, a latest detecting algorithm is adopted for the statistical analysis of prediction effects. The algorithm is proposed by Dr. Raffi Sevlian from Stanford University in 2012 [22]. It is also the first time that a set of the complete theoretical system for detecting ramp events is directly put forward (detecting effects of a certain period of the actual wind power, as shown in Fig. 5).

Fig. 5
figure 5

Detected ramp events during a certain period

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The detecting algorithm relies on the definition of a family of appropriate scoring functions. A dynamic programming recursive algorithm is used to identify and characterize all the ramp events needed for the power system operation, especially for the ancillary services and ramp event forecasting.

5 Experimental results

The actual wind power data is a time series from the Bonneville Power Administration (BPA). 184000 wind power samples are measured from the wind turbine output and sampled every 30 s with sampling time from January 1st, 2005 to December 31st, 2006. The selected wind plant capacity is 24 MW. It is noted that all the data are normalized on the basis of the rated capacity of this wind plant [23]. The experiments are executed on a desktop with i7-2640M CPU at 2.80 GHz with 3 GB RAM. To simplify the calculating process, NN probability generating model consists of 6 input layer nodes (five wind power values of historical moments and one possible wind power value of the next moment), 4 hidden layer nodes and 1 output layer node. The initial weight and threshold parameters are random numbers generated by MATLAB 2010a, and the optimal weight and threshold parameters are shown in Table 1.

Table 1 Optimal weight and threshold parameters
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5.1 Forecasting results of statistical scenarios

D% statistical scenarios captured band (SSCB) is defined to evaluate the validity of the forecasting model. The index definition refers to the ratio of n capt which is the number of actual wind power value falling into the SSCB and the total number of actual wind power data n ttl. When the actual wind power value falls into the SSCB, it means that the value is successfully captured and marked as 1. Whereas when the actual wind power value falls outside the SSCB, it fails to be captured and marked as 0.

$$D = \frac{{\sum\limits_{i = 1}^{{n_{\text{capt}} }} { \, 1} }}{{n_{\text{ttl}} }} \times 100\%$$
(11)

Six SSCBs are taken an example for the quantitative description shown from Figs. 6, 7, 8 and 9. Different SSCBs forecasting actual wind power are respectively described with different confidence intervals 30%, 50%, 70% and 90%.

Fig. 6
figure 6

Prediction results of different statistical scenarios in 90% confidence interval

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Fig. 7
figure 7

Prediction results of different statistical scenarios in 70% confidence interval

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Fig. 8
figure 8

Prediction results of different statistical scenarios in 50% confidence interval

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Fig. 9
figure 9

Prediction results of different statistical scenarios in 30% confidence interval

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Meanwhile, in order to further verify the robustness of the statistical scenarios forecasting model, time window experiments are repeated ten thousand times. The statistical proportion of each SSCB occurring with different confidence intervals is shown in Fig. 10.

Fig. 10
figure 10

Statistical proportion of the SSCBs

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As shown from Figs. 6, 7, 8 and 9, different SSCBs in the same confidence interval can capture different actual wind power values. When the confidence interval is more than 50%, 100% SSCB with the highest percentage verifies the forecasting model with better effects. When the confidence interval equals to 30%, 90% SSCB occurs with the highest percentage. It illustrates that the percentage of 100% SSCB fully captures the actual wind power value occupies less and less with the shrinking of the confidence interval, which means that the confidence interval should not be too narrow in ramp event forecasting model. Especially, the proportion of 100% SSCB partially equals to the corresponding confidence interval value. After the SSCB is formulated, the ramp event detecting algorithm can be fully used to extract the forecasted characteristics of ramp events.

5.2 Independent distribution experiments of the single characteristic

Detected ramp events in Fig. 5 and SSCBs in Fig. 7a are taken for example. Sevlian detecting algorithm is used to identify all the ramp events (up-ramps and down-ramps) and characteristics. The statistical analysis is made through a large amount of forecasted characteristic data. Methods for solving model parameters with the single objective function as the fitness function are compared. Statistical results of three ramp event characteristics are shown in Fig. 11 and Table 2. In Fig. 11, solid, dashed and dotted lines represent the multi-objective, CDF and secondary order moment autocorrelation fitness functions, respectively.

Fig. 11
figure 11

Probability density estimation charts of the characteristics

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Table 2 Statistical results of three ramp characteristics
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The corresponding maximum probability density value can be calculated from ramp characteristic statistics and is most likely to occur. Therefore, these statistical results are recorded in Table 2.

Figure 11 and Table 2 illustrate that the proposed method for the multi-objective fitness function is more accurate than the other two methods only using the CDF or secondary order moment autocorrelation (SecMo) fitness function separately. This is because the multi-objective fitness function considers not only the wind power distribution characteristics of historical wind power data but also the correlation degree between the predicted and actual wind power data. The reason that secondary order moment features are more important is that each characteristic of ramp events is closely related to the temporality of actual wind power data. During the iterative process of forecasting output data, secondary order moment features can make sorting features of output data fully conform to that of actual wind power data and guarantee the strong correlations between them. It can be seen that probability density estimation values calculated by the three methods are different, which means that statistical results of multi-objective fitness functions are more concentrated and have statistical significance.

5.3 Analysis of forecasting results on different SSCBs

The evaluation of statistical scenarios forecasting models is based on the deviation between the forecasted and actual values of wind power ramp event characteristics rather than the pure wind power. In order to evaluate the robustness of the statistical scenarios forecasting model, two evaluation criteria are defined to characterize the error relationship between forecasted and actual values of ramp event characteristics. The normalized mean absolute error (E NMAE) and the root mean square error (E RMSE) criteria are defined as;

$$E_{\text{NMAE}} = \frac{1}{{N_{\text{S}} }}\sum\limits_{t = 1}^{{N_{\text{S}} }} {\frac{{\left| {R_{\text{P}} (t) - R_{\text{PF}} (t)} \right|}}{{P_{\text{N}} }}} \times 100\%$$
(12)
$$E_{\text{RMSE}} = \sqrt {\frac{1}{{N_{\text{S}} }}\sum\limits_{t = 1}^{{N_{\text{S}} }} {\left( {R_{\text{P}} (t) - R_{\text{PF}} (t)} \right)^{2} } }$$
(13)

where \(R_{\text{P}} (t)\) and \(R_{\text{PF}} (t)\) are actual and forecasted values of ramp event characteristics at time t, respectively; \(P_{\text{N}}\) is the nameplate capacity of a wind turbine or wind plant; \(N_{\text{S}}\) indicates the number of samples in the forecasting horizon. 70% confidence interval is taken and statistical results of forecasting error criteria are shown in Table 3.

Table 3 Forecasting error criteria statistics of different SSCBs
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Table 3 indicates the prediction effect of the 100% SSCB prefers on other SSCBs with respect to ramp event characteristics. It can capture the variation trend of actual wind power with smaller errors in a set of confidence interval, which verifies the veracity of the 100% SSCB. Furthermore, the 100% SSCB occupies 70.12% of all the SSCBs, as shown in Fig. 10, which further verifies the robustness of the statistical scenarios forecasting model. It is worth nothing that although the 0% SSCB cannot forecast wind power totally (due to the 0% probability), there is also a small probability to identify ramp event characteristics, which is the difference between wind power forecasting and ramp event forecasting.

6 Conclusions

In this paper, a new forecasting strategy for statistical wind power ramp events is proposed. The improved genetic algorithm based on multi-objective fitness functions is preformed to optimize parameters of the NN probability generating model in order to produce a number of statistical scenarios. After detecting ramp event characteristics in the statistical scenarios captured bands, the efficiency of the forecasting model can be evaluated by ramp features. Through the simulation, the calculation and the statistical analysis, conclusions are presented as follows:

  1. 1)

    When the confidence interval is greater than 50%, the 100% SSCB cannot completely capture the wind power and occupy the highest proportion among all the SSCBs. This phenomenon verifies the robustness of the statistical scenarios forecasting model.

  2. 2)

    Multi-objective fitness functions can consider distribution and timing characteristics of the actual wind power. Results by using the method are more accurate than that by using single objective function.

  3. 3)

    Correlations of ramp-up characteristics are lower than that of ramp-down characteristics. Ramp duration canot exceed 150 min in the period of 400 min normally. Moreover along with the last ramp event occurring, the occurring possibility of the next ramp event is higher.

  4. 4)

    SSCBs are very robust for detecting ramp event characteristics and can be used as a method for forecasting wind power ramp events.

However, there are still some large errors in the SSCBs. In order to simplify the calculation process, only the secondary order moment as a representative of higher order moments and 6 inputs 4 hidden nodes as an assumption of more variables are used in the multi-objective fitness functions because of our computational conditions. Nevertheless, as a line of thinking of using higher order moment for wind power ramp even forecasting, the secondary order moment is still effective. In the next step work, some higher order moment, such as the third order moment, will be performed to improve the forecasting effects. Besides, more inputs and hidden nodes will be tested with larger computable servers, especially parallel computing.