Lorenz Wind Disturbance Model Based on Grey Generated Components

Transcription

1 Energies 214, 7, ; doi:1.339/en Article OPEN ACCESS energies ISSN Lorenz Wind Disturbance Model Based on Grey Generated Comonents Yagang Zhang 1,2, *, Jingyun Yang 1, Kangcheng Wang 1 and Yinding Wang State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 1226, China; s: yangjingyun614@163.com (J.Y.); kcwang@foxmail.com (K.W.) Interdiscilinary Mathematics Institute, University of South Carolina, Columbia, SC 2928, USA; wang227@ .sc.edu * Author to whom corresondence should be addressed; yagangzhang@gmail.com; Tel.: External Editor: Frede Blaabjerg Received: 21 August 214; in revised form: 1 October 214 / Acceted: 27 October 214 / Published: 7 November 214 Abstract: In order to meet the needs of wind seed rediction in wind farms, we consider the influence of random atmosheric disturbances on wind variations. Considering a simlified fluid convection mode, a Lorenz system can be emloyed as an atmosheric disturbance model. Here Lorenz disturbance is defined as the Euroean norm of the solutions of the Lorenz equation. Grey generating and accumulated generating models are emloyed to exlore the relationshi between wind seed and its related disturbance series. We conclude that a linear or quadric olynomial generating model are otimal through the verification of short-term wind seed rediction in the Sotavento wind farm. The new roosed model not only greatly imroves the recision of short-term wind seed rediction, but also has great significance for the maintenance and stability of wind ower system oeration. Keywords: wind disturbance model; Lorenz equation; olynomial generating function; accumulated generating model; Rayleigh number; short-term wind seed rediction

2 Energies 214, Introduction Wind energy is one of the most oular and otential renewable energies worldwide [1,2]. Raid develoment of wind energy has contributed greatly to energy suly systems. The Global Wind Energy Council (GWEC) has rovided the latest and comrehensive statistics about the develoment of wind ower industries. By the end of 213, global cumulative and new installed wind ower caacities have exceeded 318 GW and 35 GW, resectively. China occuied the largest share of both arts [3]. As a result how to effectively utilize wind resources has received increasing attention [4,5]. Wind ower generation in wind farms could make full use of wind energy on a large-scale [6,7], but considering the stochastic volatility nature of wind energy, integration of wind ower into ower systems becomes a challenge [8 11]. Necessary measures should be taken to maintain the normal oeration of electric systems and decrease losses that could affect eole s daily rodivity and life. Therefore, high recision wind seed and ower rediction is urgently needed. Scholars at home and abroad have done lots of studies on wind forecasting and achieved excellent results [5,12 14]. In this aer a novel wind disturbance model to imrove the erformance of conventional back roagation (BP) neural networks is roosed. Wind formation and variation in the atmoshere are tyical nonlinear rocesses [15]. As Lorenz said in his 28 lecture at the University of Rome named the Butterfly Effect, the real atmosheric state of motion is actually the observed state lus a small erturbation. This random erturbation is the key consideration that affects wind seed forecasting recision. This roblem could not be resolved by linear theory. Lorenz extracted a three-variable system called the Lorenz system from a fluid convection model in 1963 [16 19]. The Lorenz system is frequently used to study nonlinear science since it was first roosed due to its good erformance in accomodating chaos [2,21]. Here it is used as a tyical atmosheric disturbance model. Different initial conditions and values of arameters could result in different evolutions of a Lorenz system. Various tyes of Lorenz disturbances are used to establish wind disturbance models in this article. This aer is organized as follows: Section 2 briefly introduces the related content on the toic of nonlinear Lorenz disturbances. Section 3 resents the modeling rocess and the verification on disturbance model through short-term wind seed rediction. Section 4 concludes the aer. 2. Nonlinear Lorenz Disturbance and Data Prerocessing Forms 2.1. Different Tyes of Lorenz Disturbance and Normalization Constants Different Tyes of Disturbance in a Lorenz System The Lorenz system was extracted from the seven-variable fluid convection model of Saltzman [17 19,22]. A Lorenz system could exhibit chaotic state in a simlest way in nonlinear systems. It has enriched eole s acknowledgement of the dynamics of nonlinear systems and layed a vital role on the develoment of chaos theory. The fluid convection model simlified by Saltzman can be exressed as follows [17]:

3 Energies 214, , 2 4 g t x, z x, T t x, z H x (1) where ψ is a two-dimensional stream function, θ is a temerature difference from that occurring in equilibrium state, g,ς,υ,κ denote the acceleration of gravity, the coefficient of thermal exansion, the kinematic viscosity, and the thermal conductivity, resectively. The convection Equation (1) describes the following system. The fluid moves between two fixed arallel surfaces, whose temerature difference is ket constant by external heating. The whole rocedure was assumed to develo in the vertical lane. If the solution of Equation (1) was unstable, convective motion would develo. This system was similar with what Lorenz studied at that time. Then Equation (1) was further simlified into the following three-variable system called Lorenz system given by: x xy y xzrx y z xybz where x is roortional to fluid intensity, y is roortional to the temerature difference between the ascending current and the descending current, z is roortional to the temerature difference in the vertical direction comared to the equilibrium state, σ, r and b are all ositive arameters. We can obtain various Lorenz attractors by taking different values of r. Those attractors in hase sace corresond to various forms of disturbance in a real atmosheric system. The values of r can be divided into four intervals as shown in Table 1 [16]. Table 1. The actual fluid motions in Lorenz system when arameters σ and b are resectively equal to 1 and 8/3. Items to be Comared The Values of r and its Corresonding Fluid Motions Rayleigh Number (r) < r < 1 1 < r < < r < r > Actual fluid motion Heat conduction Regular convection Transient chaos Chaos Normalization Constant of Lorenz Disturbance It is convenient to discuss the solutions of Lorenz system in R 3. Here we define the Euroean norm of vectors in R 3 as the atmosheric disturbance. Let P(x,y,z) be an arbitrary vector in R 3. The Euroean norm of P(x,y,z) can be exressed as: L x y +z (3) There will be a roblem if taking Formula (3) as the form of the Lorenz disturbance. Let the initial condition and arameters σ, b, r be (,1,),1,8/3,45, resectively. The Lorenz disturbance defined by Formula (3) is demonstrated in Figure 1a. We can clearly see that the disturbance fluctuates within interval (,9), which is larger than the range of the actual wind seed changes. Therefore, we need to (2)

4 Energies 214, define a normalization constant to reduce this range to a reasonable small interval like (,3), which is decided by the deviations occurred in revious wind seed redictions. Let the normalization constant be 5 for the disturbance shown in Figure 1a. The normalized disturbance is shown in Figure 1b. Generally the normalization constant varies according to different size of Lorenz disturbance and wind fluctuations. Figure 1. (a) The initial Lorenz disturbance defined by Formula (3). (b) The normalized Lorenz disturbance. Lorenz disturbance Normalized Lorenz disturbance t (a) t (b) 2.2. Data Prerocessing It is necessary to rerocess the scattered samle data before building the disturbance model. Figure 2 shows the distribution of wind seed sequence and its corresonding disturbance series in a certain eriod. For the calculation of the two arrays we can refer to the modeling rocess in Section 3.3. We can see that the oints are distributed extremely irregularly. A further calculation of the correlation coefficient can quantitatively describe the relationshi of the two arrays. The calculated result is.1713, which is less than.3 and roves the irrelevance between the two arrays. The correlation coefficient is exressed by: r x x y y x x y y 2 2 where x and y denote the two arrays to be analyzed, x and y denote the average of x and y, resectively. In this section grey generating is alied to discover the inherent relationshi between variables x and y. Grey generating commonly includes Accumulated Generating Oeration (AGO), Inverse AGO (IAGO), Mean (MEAN), and Effect Measure (EM), etc. AGO not only does well in establishing a grey model, but also can be used to reduce the randomness of discrete sequences [23,24]. AGO means adding original data in sequence to obtain the generating series [25,26]. (4)

5 Energies 214, Figure 2. Wind seed versus the corresonding disturbance series for a eriod of time Lorenz disturbance Wind seed (m/s) Definition 1 Let x be raw series: 1 x is AGO series of x, denoted as: 1, 2,, x x x x n (5) x 1 AGOx (6) Provided that: where: , 2,, x x x x n (7) k 1 m1 x k x m, k 1,2,, n (8) Definition 2 y is IAGO series of x, denoted as: y IAGOx (9) Provided that: where: 1, 2,, y y y y n (1) y(1) x (1), y k x k x k1, k 2,3,, n (11) Based on the ways of data rocessing like Formulas (5) (8), the same treatment could be alied to the arrays in Figure 2. Figure 3 shows the generated data of Figure 2. It is clear that the irregular data

6 Energies 214, in Figure 2 are converted into a monotonically increasing sequence in Figure 3. This conversion will realize a high recision curve fitting called accumulated generating model, namely the new roosed wind disturbance model in this research. Figure 3. The oints in this figure denote the AGO series of the data shown in Figure AGO of wind seed (m/s) 3. Wind Disturbance Model Based on Grey Generation 3.1. Data Descrition The modeling data set is derived from the Sotavento wind farm from 1 January to 28 February in 214. The observations contain 8196 grous of wind seed and wind direction. Figure 4 shows the distribution of wind seed in the above two months. The average, maximum, and minimum wind seed are 1.83 m/s, m/s, and 2.6 m/s, resectively. We can see that the range of wind fluctuation is quite large. Figure 4. The distribution of wind seed recorded from January to February in 214 in Sotavento wind farm Wind seed (m/s) Number of wind seed from January to Feburary in 214

7 Energies 214, Error Criteria Selecting a reasonable set of error indicators can objectively evaluate the level of wind seed redictions and the effectiveness of forecasting models. Some common error criteria include absolute error (AE), mean absolute error (MAE), mean absolute ercentage error (MAPE), root mean-square error (RMSE), standard deviation of error (SDE) [8,15,27 29], etc. Comared to MAE, RMSE is more sensitive to large data samles and is robust when dealing with large errors [3]. Here we use MAE and RMSE, given by: 1 M (12) MAE y k f k M k 1 1 M k 1 2 RMSE y k f k (13) M where y(k) and f(k) resectively denote the observed data and redicted value, M is samle size Wind Disturbance model When Rayleigh Number Equals to BP Neural Network BP neural network is widely used in wind forecasting and does well in dealing with nonlinear roblems [31 33]. Thus, it is adoted as the basic forecasting model in this study. Based on a gradient descent algorithm, the BP neural network obtains the minimum mean square error between outut vectors and the samle values through constantly adjust the weights and biases in networks. This articular network could erfectly learn and store the maing relations between inut and outut variables. The structure of the BP neural network used in this aer is shown in Figure 5. A BP network generally consists of three layers, i.e., inut layer, hidden layer, and outut layer. The three inut vectors of BP network are wind seed (V(t)) and sine-cosine of wind direction ((sin D, cos D)) at time t. The hidden layer has three sigmoid neurons followed by an outut layer of one linear neuron. The oututs of hidden layer and outut layer are ruled by the following two formulas: 3 j j ij i j i1 y f w x b (14) 3 y f w y b j j (15) j1 where x i is inut vector with i = 1,2,3, y j and y are the outut vectors of hidden layer and outut layer with j = 1,2,3, f j and f are transfer functions of nodes, w ij and w j are connection weights, b j and b are biases.

8 Energies 214, Figure 5. Three-layer BP neural network Modeling Process and Discussion We attemt to establish a disturbance model that it is able to outut a valid disturbance sequence according to a given wind seed series. Then we could use this relevant disturbance sequence to reduce the nonlinear comonents contained in the wind seed series. Polynomial functions are some of the most widely used and simlest fitting functions. Hence, we use a olynomial generating function to construct the disturbance model, which is based on the rincile of least squares. The modeling rocedure can be divided into the following four stes: Ste one: Determine the wind seed and sine-cosine of wind direction as the inut variables of BP network. Train the BP network with the training data and redict the subsequent wind seed series. Denote the initial forecasting result and the real wind seed as W I and W R, resectively. Ste two: In order to seek the minimum deviation between W I and W R, we need to seek a certain disturbance sequence called D L from the normalized disturbance series shown in Figure 1b. D L is said to be the best comensation to W I. Ste three: According to the data rerocessing method in Section 2.2, we aly AGO to W I and D L, resectively. Then we can obtain two arrays that ossess a clear relationshi as shown in Figure 3. No more than fifth degree olynomials are alied to analyze the generated data. Table 2 shows the detailed statistics of the disturbance models, including the olynomial exression, accumulated generating model, and the fitting error (RMSE). The olynomial exressions are used to describe the features of related disturbance models. f (x) and x denote and 1-AGO of wind seed, resectively.

9 Energies 214, Table 2. Detailed statistics of grey generating models based on various olynomials. Polynomial Exression f x x.648, Accumulated Generating Model AGO of wind seed (m/s) Fitting Error (RMSE) f x x x , AGO of wind seed (m/s) f x x x x ,.2 7 =.143, f x x x x x , , AGO of wind seed (m/s) AGO of wind seed (m/s) f x x x x x x , Ste four: , , AGO of wind seed (m/s).4496 In this ste we will use the disturbance models to redict wind seed for a eriod of time in the future. We retrain the BP network and make an initial rediction of the future wind seed, which is similar to Ste one. We denote the initial rediction in redicted eriod as W IP, and then inut the AGOW series into the disturbance models established in ste three, resectively. The generated IP

10 Energies 214, oututs are in turn written as D, i1,2,,5, which needs to be reduced by IAGO. The comensating i formula is given by: W W IAGOD i IP i (16) where W i denotes the wind rediction result corresonding to each disturbance model with i 1, 2,, 5. The rediction results are resented in Table 3. The indicator (I), which means initial result, is used to distinguish the result of initial redictions and disturbed redictions. We can draw the following conclusions through a comrehensive analysis of Tables 2 and 3. On the one hand, seen from the figures of generating models and the fitting errors in Table 2, the fitting does become much better with a higher degree of olynomial. On the other hand, coefficients of high-order terms of generating functions aroximate to zero. Hence, it is not wise to use high order olynomials. Besides, the MAE and RMSE results in Table 3 suggest that the forecasting errors become aarently worse starting from the third degree olynomial model. Therefore, a good fitting cannot romise a good forecasting erformance. The universality of disturbance models is seriously reduced due to over-fitting roblem. It turns out that linear or quadratic olynomials could work out much better wind redictions comared to other forms of olynomials. Table 3. MAE and RMSE results of wind seed rediction in February 214. Degree of Polynomial Wind Forecasting Error MAE (I) (m/s) MAE (m/s) RMSE (I) (m/s) RMSE (m/s) Wind Disturbance Models When Rayleigh Numbers are Equal to.7, 12, and 16, Resectively A Lorenz system would resent various evolutions when taking different Rayleigh numbers. Each evolution corresonds to a concrete fluid motion referred in Table 1. In order to verify the universality of Lorenz fluid motions in founding a disturbance model, we can resectively choose four values of Rayleigh number from each interval in Table 1, such as.7, 12, 16, and 45. We have made a detailed analysis when the Rayleigh number is equal to 45. Now we continue to verify the other three cases. According to the conclusions in Section 3.3, a linear or quadratic olynomial generating model is adoted in this section. Similar stes are alied to the modeling rocesses in this section. The otimal disturbance models are shown in Table 4. Persistence model (PM) is introduced as a reference with indicator (P). Table 5 resents the related wind rediction results. Figures 6 9 are the wind seed forecast curves corresonding to each disturbance model in Table 5. The red curves, blue curves, and black curves resectively denote the initial wind redictions of BP neural network, the imroved redictions of disturbance model, and the actual wind seed series.

11 Energies 214, Rayleigh Number.7 Table 4. The otimal disturbance models when taking different Rayleigh number values. Polynomial Exression f x x x , Accumulated Generating Model AGO of wind seed (m/s) Error (RMSE) f x x.888, AGO of wind seed (m/s) f x x x , AGO of wind seed (m/s) f x x x , AGO of wind seed (m/s) Although the imrovements are not exactly the same, these results have fully roved that all forms of Lorenz disturbance flow could contribute to the redicted level of a BP neural network. Table 5 resents another imortant discovery. The models could be easily groued into two categories by the errors in Table 5. The forecasting accuracy tends to be high when the Rayleigh numbers equal 16 and 45. MAE and RMSE are resectively about.56 m/s and.78 m/s. The same quantities are about.14 m/s and.18 m/s when the Rayleigh numbers are equal to.7 and 12. Theoretically seaking, when the Rayleigh number is larger than 13.97, a Lorenz system has transient chaotic or chaotic solutions, whose disturbance forms are quite comlex. This kind of comlexity greatly hels to describe the real wind variations.

12 Energies 214, Rayleigh Number Table 5. MAE and RMSE results of wind seed rediction in February 214. Wind Forecasting Error MAE (I) (m/s) MAE (m/s) MAE (P) (m/s) RMSE (I) (m/s) RMSE (m/s) RMSE (P) (m/s) Besides, the forecasting errors of PM are used to evaluate the existing results. PM is esecially suitable for short-term wind rediction [34]. In this aer, PM behaves better than the BP network on short-term wind rediction. Seen from Table 5, excet for RMSE (P) when Rayleigh numbers are equal to 12 and 45, all errors of PM are smaller than the corresonding errors of a conventional BP network. Both MAE and RMSE of disturbance models greatly outerform PM and the BP neural network. Figure 6. Wind seed forecasting results of the quadratic olynomial generating model when Rayleigh number equals Wind seed (m/s) Initial rediction 6 2-Disturbed rediction Actual wind seed Predicted eriod Figure 7. Wind seed forecasting results of the linear olynomial generating model when Rayleigh number equals Initial rediction 1-Disturbed rediction Actual wind seed Wind seed (m/s) Predicted eriod

13 Energies 214, Figure 8. Wind seed forecasting results of the quadratic olynomial generating model when Rayleigh number equals Wind seed (m/s) Initial rediction 6 2-Disturbed rediction Actual wind seed Predicted eriod Figure 9. Wind seed forecasting results of the quadratic olynomial generating model when Rayleigh number equals Initial rediction 2-Disturbed rediction Actual wind seed Wind seed (m/s) Predicted eriod 4. Conclusions With the raid develoment of the wind ower industry, wind seed and ower generation forecasting methods have become a focus of research in the fields of ower systems and renewable energy. After a deeer study on the atmosheric dynamics system and wind rediction methods, in this aer we introduce the Lorenz system to reduce the negative imact of atmosheric disturbances on wind forecasting. Based on grey generating and an accumulated generating model, we roose a new wind disturbance model to imrove the wind forecasting recision of a BP network. PM is also introduced for comarison. The best imrovements (r = 45) in this aer are measured by the following two criteria: MAE with.535 m/s and RMSE with.789 m/s. In [32], the new roosed model IS-PSO-BP obtained good wind rediction erformance based on different training numbers and data sources, whose MAE is.16 m/s and RMSE with.4123 m/s. In [34] its authors adoted

14 Energies 214, low-quality measurements as exogenous information to refine the one-hour-ahead wind redictions. The smallest MAE of BP neural network in [34] is.18 m/s and RMSE with.2236 m/s. All the above statistics and analysis suggest that we have roosed a valid and feasible wind disturbance model based on a BP neural network. This model fully considers the basic features of nonlinear atmosheric systems. Given an aroriate Lorenz disturbance sequence, it can be used to redict any tye of wind seed series without seasonal limitations. Nevertheless there is no unified algorithm or rincile at the moment for the selection of Rayleigh numbers. In addition, in view of the advantages and features of grey system theory [26,35,36], the alication of the Lorenz system to the grey model is worthy of further exloration. Acknowledgments The authors thank the anonymous referees for their helful comments and suggestions. This research was suorted artly by the National Key Basic Research Project (973 Program) of China (212CB2152), the NSFC ( ), the Secialized Research Fund for the Doctoral Program of Higher Education ( ) and the Fundamental Research Funds for the Central Universities (214ZD43). Author Contributions This aer is a result of the full collaboration of all the authors. However, Jingyun Yang wrote Case Study and Methodology, Kangcheng Wang erformed the exeriments. All authors discussed the results and commented on the manuscrit. Conflicts of Interest The authors declare no conflict of interest. References 1. Cheng, M.; Zhu, Y. The state of the art of wind energy conversion systems and technologies: A review. Energy Convers. Manag. 214, 88, Irwanto, M.; Gomesh, N.; Mamat, M.R.; Yusoff, Y.M. Assessment of wind ower generation otential in Perlis, Malaysia. Renew. Sustain. Energy Rev. 214, 38, Global Wind Energy Council (GWEC). Available online: htt:// (accessed on 12 June 214). 4. Jung, J.; Broadwater, R.P. Current status and future advances for wind seed and ower forecasting. Renew. Sustain. Energy Rev. 214, 31, Fan, C.; Liu, S. Wind seed forecasting method: Grey related weighted combination with revised arameter. Energy Proced. 211, 5, Chandel, S.S.; Ramasamy, P.; Murthy, K.S.R. Wind ower otential assessment of 12 locations in western Himalayan region of India. Renew. Sustain. Energy Rev. 214, 39, Rasheed, A.; Kristoffer Suld, J.; Kvamsdal, T. A multiscale wind and ower forecast system for wind farms. Energy Proced. 214, 53,

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