Wind power time series simulation model based on typical daily output processes and Markov algorithm

0 Introduction

The construction of a future power grid with new energy sources is one of the key strategic methods to promote the realization of international and domestic carbon peak and neutralization goals [1].This means that wind energy and photovoltaic power generation become the main power sources instead of coal power and thermal power.However,due to the weak controllability and high randomness of wind power output with respect to time and space, the access to high proportions of wind power introduces new problems to power grid planning and operation [2-4].Wind power time series modeling is a key technique for new energy allocation planning, operation mode calculation, and security assessment [5], [6].It is anticipated to be extremely important for the construction of a future power grid with new energy sources.

Studies on the simulation of wind power time series have been implemented by both domestic and overseas academics.Numerous studies have shown that the Markov status transition model compared with other wind power time series simulation models can better represent the time-varying characteristics and fluctuation of wind power output.Accordingly, it can maintain the original probability distribution and autocorrelation characteristics of the series [7].

Simulation models are mainly classified into two classes according to their research objectives: transformation models and direct generation models [8].Moreover, [9] shows that the effect of direct generation models is better than that of transformation models; consequently, errors in the conversion of wind speed into wind power are avoided.Wu et al.[10]proposed a wind power time series simulation model based on Markov chain and Monte Carlo simulation.They verified the availability of the presented model considering the average, standard deviation, probability density function, and autocorrelation coefficient (ACC).Zhao et al.[11] leveraged the seasonal and dry-wet features of wind power and presented a wind power time series simulation model based on the modified Markov algorithm.Huang et al.[12] proposed a new wind power time series simulation model using particle swarm optimization-based K-means and Markov chain-Monte Carlo method.Zhu et al.[13] proposed a three-dimensional status transition probability matrix and a solution dimension correction model to simulate the wind power status series by sampling.Xu et al.[14] proposed a method for optimizing power status based on the original Markov chain-Monte Carlo algorithm and used it to model the wind power time series.

In summary, domestic and overseas academics have performed considerable research on the simulation of wind power time series and achieved significant results.However,the accuracy of existing wind power time series modeling methods based on Markov algorithm that all generate power time series using a moment in time as a unit can still be improved.This is particularly true in terms of wind power output probability distribution and autocorrelation characteristics.These methods ignore the diurnal fluctuation of wind power [15-17] and are unable to consider both calculation accuracy and efficiency.

To solve the aforementioned problems, this paper presents a wind power time series simulation model based on typical daily output processes and Markov algorithm.The main highlight of the study is that the power time series is generated in daily time units, which can retain the characteristics of the daily output of wind power.In addition, for generating a one-year time series, this method only implements 365 simulations.In contrast, traditional methods require 365 × 24 × 4 simulation times (assuming that the time series is generated every 15 min).This means that the proposed model not only ensures simulation accuracy but also improves simulation efficiency.The main contributions of this work are as follows:

1) A typical daily output process classification method based on time series similarity and modified K-means clustering algorithm is presented.The proposed model can effectively identify the time series similarity of wind power and realize the effective differentiation of typical daily output processes of wind power.

2) A wind power time series simulation model based on typical daily output processes and Markov algorithm is constructed.The statistical, probability distribution, and autocorrelation characteristics of the series generated by this model are better than those produced by traditional modeling methods.Moreover, the computational efficiency is improved.

The remaining sections of this paper are structured as follows.Section 2 introduces the typical daily output process classification method based on time series similarity and modified K-means clustering algorithm.Section 3 describes the wind power time series simulation model based on typical daily output processes and Markov algorithm.Section 4 presents the case study, and Section 5 discusses the research conclusions.

1 Typical daily output process classification method based on time series similarity and modified K-means clustering algorithm

Wind power output is influenced by wind speed,air temperature, air pressure, humidity, and many other meteorological factors.The fluctuation rules of wind power output have distinct variations under different weather conditions; however, they are similar under the same weather condition.By reducing the large number of multitime-scale wind power output processes and extracting typical output processes, the workload of characterizing the dynamic characteristics of wind power output can be greatly reduced [18-20].Accordingly, this paper proposes a typical daily output process classification method based on time series similarity and modified K-means clustering algorithm.The typical daily output processes are used as the status variables of the wind power time series simulation based on Markov algorithm presented in Section 3.

1.1 Time series similarity

Clustering algorithm is typically employed to classify the typical daily output processes of wind power.It is mainly classified into two types depending on the clustering method: clustering based on eigenvalues and clustering based on time series similarity.Because the eigenvaluebased clustering ignores the dynamic information characteristics of wind power output, it cannot represent the time series similarity among different wind power series.Therefore, this study uses the typical daily output process classification method based on time series similarity measurement.Further, a time series similarity index is utilized to measure the time series similarity among different wind power series.

Time series similarity measures the temporal resemblance among wind power series from three perspectives: absolute level, fluctuation level, and discrete degree [21], [22].It is specifically divided into three parts:absolute distance, fluctuation distance, and discrete distance.

1) Absolute distance.The absolute distance between wind power time series i and j is given by (1):

where Pi(t) and Pj(t) are the power values of wind power time series i and j at time point t ( t∈[1,T]), respectively.The absolute distance mainly represents the degree of absolute difference between two wind power time series throughout period T.

2) Fluctuation distance.The fluctuation distance between wind power time series i and j is given by (2):

where ΔPi(t) and ΔPj(t) are the power fluctuations of wind power time series i and j at time point t, respectively.The fluctuation distance index mainly represents the similarity degree of variation trend between two wind power time series throughout period T.

3) Discrete distance.The discrete distance between wind power time series i and j is given by (3):

where α is 0.1 (based on experience); hi is the specific standard deviation value of the average of wind power time series i (also known as coefficient of variation); STDi andare the standard deviation and average of wind power time series i throughout period T, respectively.The discrete distance mainly represents the degree of discrete difference between two wind power time series throughout period T.

With the absolute distance, fluctuation distance, and discrete distance derived, the weight of each distance index is obtained by the entropy method.The time series similarity index is determined as follows:

where z(dij,A), z(dij,V), and z(dij,D) are the normalized values of the three distance indicators, and w1, w2, and w3 are the distance weights, respectively.

1.2 Modified K-means clustering algorithm

In the commonly used K-means clustering algorithm, the similarity measure among samples is generally Euclidean distance.However, the Euclidean distance can only reflect the absolute difference among wind power time series and cannot represent the time series fluctuation characteristics.Therefore, this study employs the proposed time series similarity index to replace the Euclidean distance.The typical daily output process classification is derived based on the modified K-means clustering algorithm with time series similarity.In addition, the optimal clustering number is selected by the elbow method [23].The specific implementation procedure is follows:

1) Data preprocessing: the measured time series of wind power is converted into a time series matrix of wind power in daily unit and then normalized:

where M is the time series matrix of wind power; n is the number of days; Pn(1) is the power unit value at the first moment of the n th day; and Pn(t) is time t unit value of the n th day.

2) Data input: the input of the modified K-means clustering algorithm is the wind power time series matrix.

3) Modified K-means clustering analysis: the clustering number range is set as [1, m]; the clustering results under different clustering numbers are obtained using the modified K-means clustering algorithm with time series similarity.

4) Typical daily output process classification: calculate the sum of squared errors (SSE) under different clustering numbers; use the elbow method to select the best number of clusters; and obtain the optimal typical daily output processes of wind power.

2 Wind power time series simulation model based on typical daily output processes and Markov algorithm

2.1 Markov algorithm

The core of Markov algorithm is discrete-time Markov chain, which relates to the stochastic process of transferring from the previous status to the subsequent status in the status space.The probability of occurrence of the random process is called transition probability.A process belongs to a Markov chain if it satisfies the two following conditions: a status corresponding to the process at any time exists in the status space of Markov algorithm, and the process has no aftereffect.The latter means that the probability of a certain status at a later moment is only related to the current status;it is unaffected by all statuses before the current one (i.e.,the process has no memory) [24], [25].

In this study, the typical daily output processes are defined as status variables to form the status space of Markov algorithm:

where SN is the N th status (i.e., the N th typical daily output process), and N is the number of typical daily output processes.

When the status of wind power is Si at time t, the probability of the transfer of wind power status to status Sj at the next time is given by (7):

The Markov chain satisfies the Markov property of (8)for arbitrary i,j,a,b ∈ N, as follows:

The conditional probability, is called the status transition probability.It is obtained according to the historical wind power time series:

where mij is the frequency of transition from status Si to Sj of the historical wind power time series.

The matrix is composed of all status transition probabilities.It is known as the status transition probability matrix, P, in which the sum of each row element is 1.The matrix is

The cumulative status transition probability matrix, Q,is obtained from matrix P.The cumulative status transition probability of row i and column j in matrix Q is the sum of all status transition probabilities before column j of row i in matrix P.The cumulative status transition probability matrix is

Matrices P and Q are the core of the Markov chain model.Given any initial status, the status number of the next can be obtained through matrix Q; details are presented in Section 3.2.

In the traditional single-point simulation model of wind power time series by Markov algorithm, a value is randomly selected within the wind power range corresponding to the next status number.This value is considered as the actual simulated power of the status.The model ignores the typical daily output characteristics and tends to reduce modeling efficiency (i.e., more simulation times are required).Therefore, to retain the typical daily output characteristics and consider both calculation accuracy and efficiency, the typical daily output processes are used as status variables in this study.One process is randomly selected from the set of typical daily output processes corresponding to the status number of the next day as the wind power simulation result.

2.2 Modeling process

The specific modeling processes of wind power time series simulation model based on typical daily output processes and Markov algorithm are as follows.The modeling process is shown in Fig.1.

Fig.1 Modeling process

1) Typical daily output process classification: using the typical daily output process classification method based on the modified K-means clustering algorithm with time series similarity, N types of typical daily wind power processes are clustered; the cluster numbers of daily wind power time series data are simultaneously obtained.

2) Calculation of status transition probability matrix, P,and cumulative status transition probability matrix, Q: the typical daily output processes of wind power are considered as new status variables in the Markov chain.This means that the number obtained by clustering is the status number,and two adjacent cluster numbers form a group to count the frequency of different status transitions.Matrix P is obtained using (9) and (10), and matrix Q is obtained by(11).

3) Initial status category determination: based on the time series similarity index, the typical daily output process cluster and initial status number of the wind power series on the last day of the existing historical data are determined.If the simulation series is not required to connect with the original time series, a random integer, [1, N] (named Sinitial),can be generated as the initial status number.

4) Generation of next-day status number: generate a stochastic digit between 0 and 1; compare this digit with the cumulative status transition probability in the row of the initial status, Sinitial, in the status transition probability matrix,Q; and find the first cumulative status transition probability greater than the stochastic digit.The corresponding number is the simulation status number of the next day.

5) Wind power time series simulation: after the status number is obtained, a process is randomly selected from the typical daily output process cluster corresponding to the status number as the simulation result.Repeat step 4 until the simulation generates the wind power time series of the required length of time (H number of days).

3 Case study

3.1 Database

In this study, the effectiveness and applicability of the proposed model are validated based on the measured wind power data of a wind farm in China.The data cover 365 d,and the time resolution is 15 min.

3.2 Classification results of typical daily output processes

The measured wind power time series is converted into a wind power time series matrix with day as the unit of time.Then, the normalization processing is implemented on the 365 × 96 power matrix.The original K-means clustering algorithm based on eigenvalues (mean and variance)[12] and the method of clustering into the 365 × 96 time series matrix directly based on the Euclidean distance are compared with the modified K-means clustering algorithm based on the time series similarity presented in this paper.The value range of the clustering number, k, is set to [1], [40],and the best clustering number is determined by the elbow method, as shown in Fig.2.

Fig.2 Optimal number of clusters

The clustering results of the original K-means clustering algorithm based on eigenvalues are shown in Fig.3.The results of directly clustering the 365 × 96 time series matrix are shown in Fig.4 and Fig.5.In Fig.4, the original algorithm based on Euclidean distance is used, whereas in Fig.5, the modified algorithm based on time similarity is employed.The comparison among Fig.3-5 shows that the effect of the direct clustering of the daily wind power time series to obtain typical output processes is better than that of the method based on eigenvalues.The comparison between Fig.4 and Fig.5 shows that the modified algorithm based on time series similarity can better divide the typical daily output processes and effectively represent the time series similarity of wind power time series.This means that the time series fluctuation of the same type of daily output processes is similar, and significant variations among different types of typical daily output processes exist.

Fig.3 Results of original K-means clustering algorithm based on eigenvalues

Fig.4 Results of K-means clustering algorithm based on Euclidean distance

Fig.5 Results of modified K-means clustering algorithm with time series similarity

The results show that the time series similarity index can guarantee similar fluctuation characteristics.Moreover, the degree of dispersion for each wind power output process in the same cluster from three perspectives (i.e., absolute distance, fluctuation distance, and discrete distance) is also ensured.The modified K-means clustering algorithm with time series similarity can better classify the typical daily output processes, proving the superiority and rationality of the indexes proposed in this paper.

To quantitatively evaluate the effectiveness of the presented typical daily output process classification method, this study employs the commonly used clustering effect evaluation index, Davies-Bouldin Index (DBI), for comparative analysis, as listed in Table 1.The DBI of the modified K-means clustering algorithm using time series similarity may be distinctly smaller than those of other methods.In summary, the proposed method can effectively characterize the time series similarity of wind power time series and effectively classify the typical daily output processes of wind power.

Table 1 DBIs of two clustering algorithms

Clustering algorithm DBI Original K-means clustering algorithm based on eigenvalues 1.53 K-means clustering algorithm based on Euclidean distance 1.31 Modified K-means clustering algorithm with time series similarity 1.28

3.3 Simulation results of wind power time series

A wind power time series simulation model based onMarkov algorithm is constructed by considering the typical daily output processes as status variables.The comparison models and new model are described as follows.

1) “History” pertains to the historical wind power time series.

2) “Tra” refers to the traditional Markov algorithm model.

3) “Char” refers to the model obtaining the typical daily output processes according to the eigenvalues (mean and variance) of the daily wind power output processes [12];simulation is based on the modified Markov algorithm.

4) “Time” pertains to the model obtaining the typical daily output processes based on the modified K-means clustering algorithm with time series similarity; simulation is based on the modified Markov algorithm.

The simulation results of different wind power time series models are shown in Fig.6.

Fig.6 Partial simulation results of wind power time series

The simulation results of a three-day time series demonstrating the simulation output are shown in Fig.6.Note that the obtained results can be compared on the same day.The wind power output processes simulated by the new model are intuitively observed to be close to the historical output processes.This observation is the highlight of the study; this means that the proposed model can retain the daily output characteristics of the wind farm.

The comparisons of the monthly average value,standard deviation, and absolute deviation (ADF) between the simulation results of different methods and historical wind power time series are summarized in Tables 2-4,respectively.The ADF calculation formulae are as follows:

where PH(t) is the measured wind power (actual operation data of the wind farm); PS(t) is the simulated wind power per unit at time t; ΔPH(t) is the measured wind power fluctuation; and ΔPS(t) is the simulated wind power fluctuation per unit at time t.

Based on Tables 2-4, the two wind power series obtained by the two methods that utilize the typical daily output processes as Markov status variables are observed to have similar statistical characteristics as the historical data.This proves the feasibility of using typical daily output processes as status variables.The tables further indicate that the method based on the time series similarity and Markov algorithm proposed in this paper exhibits the best performance.

Table 2 Average value of simulation results of different models and historical wind power time series

Month History Tra Char Time 1 0.325 0.208 0.232 0.255 2 0.366 0.236 0.266 0.291 3 0.235 0.220 0.191 0.314 4 0.191 0.251 0.206 0.201 5 0.144 0.236 0.261 0.166 6 0.125 0.219 0.172 0.135 7 0.133 0.211 0.244 0.106 8 0.186 0.147 0.215 0.172 9 0.322 0.204 0.261 0.259 10 0.298 0.281 0.238 0.281 11 0.269 0.188 0.249 0.233 12 0.241 0.201 0.268 0.311 Average 0.236 0.216 0.233 0.227

Table 3 Standard deviation of simulation results of different models and historical wind power time series

?

Table 4 ADF of simulation results of different models and historical wind power time series

Month Tra Char Time 1 0.047 0.046 0.046 2 0.052 0.056 0.047 3 0.047 0.042 0.039 4 0.042 0.041 0.044 5 0.039 0.039 0.032 6 0.041 0.030 0.037 7 0.039 0.038 0.035 8 0.039 0.039 0.043 9 0.043 0.041 0.045 10 0.039 0.043 0.036 11 0.040 0.039 0.036 12 0.037 0.041 0.040 Average 0.042 0.041 0.040

In addition, the daily mean value and daily standard deviation of the simulation results of different models are compared with those of the historical wind power series.Because showing the comparison of 365 days of data is inconvenient, this paper presents the root mean square error (RMSE) of the daily mean value and daily standard deviation between the simulation results of different methods and historical wind power series [26], as listed in Table 5.The RMSE is computed using

Table 5 RMSE of daily mean value and daily standard deviation between simulation results of different models and historical wind power series

Model Daily mean value Daily standard deviation Tra 0.3181 0.1211 Char 0.3535 0.1208 Time 0.3017 0.1181

where T= 365; t is the t th day; X(t) represents the daily mean value or daily standard deviation of simulation results of different models; and Y(t) represents the daily mean value or daily standard deviation of the historical wind power series.

The difference of the daily mean and daily standard deviation between the results of the proposed model and historical series is small.This means that the daily output process characteristics of wind power are retained.

The simulation accuracy of the proposed model does not considerably exceed those of the traditional methods.Nevertheless, it can significantly reduce the required number of simulations, that is, it can consider both modeling accuracy and efficiency.

To further verify the applicability and effectiveness of the proposed model, the wind power output probability distribution characteristics and autocorrelation characteristic are evaluated [27], [28].The probability distribution and ACC are shown in Fig.7 and Fig.8, respectively.

Fig.7 Probability distribution of simulation results of different models and historical wind power series

Fig.8 ACC of simulation results of different models and historical wind power series

As shown in Fig.7, the simulation results of the traditional Markov algorithm inadequately reproduce the probability distribution of small power values (0-0.2).In contrast, the proposed model is consistent with the probability distribution of the historical wind power time series in the entire range.As shown in Fig.8, the ACC of the proposed method approaches that of the historical wind power time series.This confirms that the presented model can effectively reflect the time-varying characteristics of wind power.

In summary, the statistical characteristics, probability distribution, and autocorrelation characteristic of the wind power time series generated by the proposed model are better than those yielded by traditional modeling methods.Moreover, the presented model can effectively improve the modeling efficiency.

4 Conclusions

A wind power time series simulation model based on typical daily output processes and Markov algorithm is proposed in this paper.First, a typical daily output process classification method based on time series similarity and modified K-means clustering algorithm is presented.Second, by considering the typical daily output processes as status variables, a wind power time series simulation model based on Markov algorithm is constructed.Finally,a case based on the actual power time series of a wind farm in China is analyzed.The effectiveness and applicability of the presented model are verified through comparison with traditional modeling methods.The conclusions drawn are as follows:

1) A typical daily output process classification method based on time series similarity and modified K-means clustering algorithm is proposed; compared with the original K-means clustering algorithm, this method can efficaciously represent the temporal similarity of wind power time series and realize the effective classification of the typical daily output processes of wind power.

2) A wind power time series simulation model based on typical daily output processes and Markov algorithm is constructed.The proposed model uses the typical daily output processes as the status variables of Markov algorithm.Consequently, the number of simulations is effectively reduced.

3) Compared with the statistical characteristics,probability distribution characteristics, and autocorrelation characteristic obtained by the traditional modeling method,those of the wind power time series generated by the proposed model can be modified.Moreover, both modeling accuracy and efficiency can be considered simultaneously.

Acknowledgements

This work was supported by the China Datang Corporation project “Study on the performance improvement scheme of in-service wind farms”, the Fundamental Research Funds for the Central Universities (2020MS021), and the Foundation of State Key Laboratory “Real-time prediction of offshore wind power and load reduction control method ” (LAPS2020-07).

Declaration of Competing Interest

We declare that we have no conflict of interest.