Optimal sizing of energy storage considering the spatial‐temporal correlation of wind power forecast errors
2018; Institution of Engineering and Technology; Volume: 13; Issue: 4 Linguagem: Inglês
10.1049/iet-rpg.2018.5438
ISSN1752-1424
AutoresChengfu Wang, Teng Qijun, Xiaoyi Liu, Feng Zhang, Suoying He, Zhengtang Liang, Xiaoming Dong,
Tópico(s)Integrated Energy Systems Optimization
ResumoIET Renewable Power GenerationVolume 13, Issue 4 p. 530-538 Research ArticleFree Access Optimal sizing of energy storage considering the spatial-temporal correlation of wind power forecast errors Chengfu Wang, Corresponding Author Chengfu Wang wangcf@sdu.edu.cn Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorQijun Teng, Qijun Teng State Grid Qingdao Power Supply Company, Qingdao, 266000 Shandong Province, People's Republic of ChinaSearch for more papers by this authorXiaoyi Liu, Xiaoyi Liu Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorFeng Zhang, Feng Zhang Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorSuoying He, Suoying He School of Energy and Power Engineering, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorZhengtang Liang, Zhengtang Liang Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorXiaoming Dong, Xiaoming Dong Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this author Chengfu Wang, Corresponding Author Chengfu Wang wangcf@sdu.edu.cn Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorQijun Teng, Qijun Teng State Grid Qingdao Power Supply Company, Qingdao, 266000 Shandong Province, People's Republic of ChinaSearch for more papers by this authorXiaoyi Liu, Xiaoyi Liu Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorFeng Zhang, Feng Zhang Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorSuoying He, Suoying He School of Energy and Power Engineering, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorZhengtang Liang, Zhengtang Liang Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this authorXiaoming Dong, Xiaoming Dong Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan, 250061 Shandong Province, People's Republic of ChinaSearch for more papers by this author First published: 11 January 2019 https://doi.org/10.1049/iet-rpg.2018.5438Citations: 13AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Energy storage is considered as an effective approach to deal with the power deviation that caused by the stochastic wind power forecast error. Because of the spatial-temporal correlation of forecast errors for wind farms, which locate close to each other and integrate into the same regional power grid, energy storage could be deployed effectively and economically. Therefore, this study proposes an optimisation method to size the capacity of energy storage system (ESS) considering the spatial-temporal correlation of forecast errors for multiple nearby wind farms. The copula theory based correlation model of high-dimensional forecast errors is built to capture the spatial-temporal characteristics of forecast error. Then, according to multiple scenarios technique, an optimal ESS sizing model is established to minimise the investment and operation costs of the ESS. Meanwhile, an operation strategy considering the trend of prediction and correlation of errors is used to implement rolling operation strategy of ESS. The case studies demonstrate the effectiveness of the new method and illustrate the significant impact of the correlation of forecast error on the capacity of ESS. Nomenclature Constants k/ρ degree of freedom/correlation matrix for t location-scale distribution per unit prices of the surplus/shortage energies per unit price of the rated energy/rated power of storage installation cost of energy storage Ksu/Ksh/Kin coefficients of the surplus energy/shortage energy/investment costs T time period (T = 24 h in this paper) charge/discharge efficiency H time horizon of rolling correction Variables SOC at time t in the ith scenario rated energy capacity in the ith scenario output power at time t in the ith scenario actual wind power at time t forecast wind power at time t probability of the ith scenario Pi(t) wind power of ith scenario at time t γh error correlation coefficient between time t and t + h charge/discharge power at time t in the ith scenario rated charge power in the ith scenario rated discharge power in the ith scenario binary variables which show the states of charge and discharge at time t in the ith scenario upper/lower boundary of forecast error range of the expected power output reference range of the updated output power surplus energy incurred by the curtailment of wind power shortage energy caused by load shedding 1 Introduction Owing to increasingly severe energy resource and environmental problems, renewable energy, especially wind energy, has attracted considerable attention at the global scale and exhibited a rapidly increasing trend [1, 2]. However, due to the uncertainty and fluctuation of wind power generation, the large-scale integration of wind power introduces significant challenges to the operation of power system. With the development of energy storage technology, the installation of energy storage system (ESS) can be treated as an effective method to mitigate the randomness and uncertainty of wind power generation. Nevertheless, how to size energy storage remains a significant challenge for the application of ESS since the high cost of energy storage device [3]. To determine the optimal size of ESS for wind farms, the balance between the economy of ESS sizing and the resulting performance of wind power fluctuation smoothing should be respected. Accordingly, concerning recent research, large amount of models have been built to economise the capacity or cost of ESS [4, 5]. Nevertheless, the scheme of power system operation and control are pre-deployed referring to the wind power prediction data, this prediction data is thus expected as closer to actual wind power output as possible [6]. Therefore, the wind power forecasts or forecast errors have been considered during ESS sizing [7, 8]. A sizing method for ESS composed of different types of storage devices has been presented in [9] to reduce the time-varying components of the forecast errors. An energy storage sizing method has been proposed in [10] to effectively reduce the negative impact of short-term wind power forecast error. Baker et al. [11] have proposed an ESS optimal sizing tool allowing for the forecast errors based on two-stage stochastic model predictive control. Overall, the studies above have verified the effectiveness of considering forecast error during ESS sizing and analysed the effect of error distribution on ESS sizing. However, the spatial-temporal correlation of forecast error has a remarkable impact on the temporality and accuracy of error distribution, and is ignored in previous research. Thus, this factor needs to be further considered to improve the accuracy of ESS sizing. In [12-15], it has been verified that the correlation of wind power and its forecast error have a significant effect on the distribution of forecast error. Accordingly, a spatial-temporal correlation model for wind speed and wind power based on copula function has been proposed in [16]. The proposed nonlinear modelling method can address the dependence accurately and can identify the zero dependence of wind data. Moreover, a simplified ESS model is applied to demonstrate the advantages and disadvantages of different correlation modelling approaches. However, it may be more accurate and effective to directly use spatial-temporal dependencies of forecast errors than using dependencies of wind power or speed during ESS sizing. In order to capture the autocorrelation of day-ahead forecast error, a data-fitted autoregressive process has been used to model forecast error [17]. However, the proposed linear model has been not fit for the non-linear relationship of dataset [18], and the influences of spatial-temporal correlation of forecast errors on ESS sizing have not been mentioned either. In [19], the influence of forecast error autocorrelation to the sizing of ESS has been analysed in detail. In [20], the temporal correlation of forecast error based on Copula theory has been used to size the energy storage power and capacity. The works mentioned above have proved that the storage capacity would be misestimated if the autocorrelation of forecast errors is not taken into consideration. That is, it is necessary to consider the temporal correlation into rational energy storage sizing. However, there is also a strong spatial correlation between wind farms in the same regional power grid. These spatial correlations have not been considered in previous research of ESS. Therefore, it is necessary to further study the ESS sizing considering the spatial-temporal correlation of the forecast errors for multiple nearby wind farms. On the other hand, the effect of different control strategies on ESS sizing is significant [21, 22]. In [21], an optimal control strategy for ESS has been built based on traditional feedback control method. In [23], an optimal sizing model for battery storage system has been proposed and the optimisation results of four controllers (simple, fuzzy, ANN and Adv. ANN) have been compared. In [22], a variable-interval reference signal optimisation approach and a sizing method for BESS planning have been presented to smooth wind power and the fuzzy control-based scheme is used to decrease BESS capacity. The control strategies mentioned above are all based on known and deterministic information during the operation and control of ESS. However, the uncertain information such as forecast data should be considered during the operation of ESS to better deal with the wind power fluctuation. Consequently, to reduce the effect of forecast error fluctuation, a novel optimal ESS sizing method is proposed considering the spatial-temporal correlation of forecast errors for multiple nearby wind farms. In order to improve the economic and operating benefits, the proposed method minimises the investment and operating costs of the ESS. In this method, a correlation model of multi-dimensional forecast errors based on Copula theory is built to guarantee the accuracy of correlation analysis. Meanwhile, an operation strategy including the trend of prediction and correlation of error is used to address optimal operation of ESS during energy storage sizing. The remaining of this paper is organised as follows. The data source and a simple demonstration are described in Section 2. The spatial-temporal correlation of forecast error is modelled in Section 3. A new optimal energy storage sizing method is proposed in Section 4. The simulation results using different cases are demonstrated in Section 5. The conclusion is given in Section 6. 2 Data source and simple demonstration In this section, the research data and a simple demonstration will be shown to briefly illustrate the motivation of this paper. 2.1 Data source The data in this paper are obtained from four real wind farms located in the northern part of Hebei Province, China. The geographical positions of the wind farms (including wind farms A, B, C, and D) are shown in Fig. 1. In this paper, wind farm A is selected as energy storage sizing target, which named TWF. Other wind farms named RWFs, are used to analyse the effects of spatial-temporal correlation on energy storage sizing for TWF A. The data of each wind farm contain wind power data from 1 January 2014, to 31 December 2014 [24]. The wind power data have been normalised according to the installed capacities of each wind farm. The time interval between adjacent sampling points is 1 h. Fig. 1Open in figure viewerPowerPoint Geographical position of the selected wind farms In order to obtain the wind power forecast and error data, the forecast method in [25] is used to predict data of TWF and RWFs. The combination forecast method proposed in [25] is effective to correct the short-term wind power forecast by building an error prediction model using the regression learning algorithm (including Support Vector Machine and Extreme Learning Machine), to obtain better prediction results. Based on the method mentioned above, the wind power forecast data of 8760 sampling points, i.e. one year, are generated. The forecast is generated once per day by blocks of 24 h, and the forecasted series start at midnight (0 o'clock). Correspondingly, the forecast error can be obtained as the difference between the forecast value and actual value. In order to further illustrate the forecast error characteristics, the MAE and RMSE results of forecast error are shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint MAE and RMSE of 24-h wind power forecast errors of TWF Fig. 2 shows the MAE and RMSE of wind power forecast errors of TWF A. As observed, with the prediction horizon increasing, MAE and RMSE values gradually increase, implying the prediction accuracy decreases with the increase of the prediction horizon. Generally, the RMSE and MAE are different for the different forecast methods. However, the tendency above is usually similar to other forecast methods. 2.2 Simple demonstration In order to clearly explain the relationship between the correlation of forecast error and the ESS size, a basic and simple example is shown briefly to demonstrate the impact of various correlation levels of forecast error on ESS size. In this demonstration, an ESS is installed in a 1 MW wind farm to mitigate the power deviation between 24 h-ahead forecast errors and the actual values. For simplicity, the wind power forecast values are chosen as the expected output power of wind farm, which means all of the error fluctuation is complemented by energy storage. ESS size is determined by minimising the power and energy capacity of ESS. Actually, the storage power size equals to the max absolute value of the error sequence, and the storage capacity size equals to the absolute value of the cumulated energy in this section. According to the correlation levels of forecast error, four tests are designed as follows: Test 1: For the most extreme case, the linear correlation coefficients of the 24 h-ahead forecast errors are set to 1, implying the sequence of forecast error can be assumed as [1, 1, …, 1]24 MW. Therefore, at least 24 MWh capacity of ESS is needed to compensate the forecast error completely. Test 2: The linear correlation coefficients of adjacent time intervals are set as opposite number but with the same absolute value 1, namely, the sequence of forecast error can be assumed as [1, −1, …1, −1]24 MW. For this extreme case, only 1 MWh ESS capacity is required to completely compensate the forecast error. Test 3: To ignore the temporal correlation, the forecast errors of 24 h-ahead are set as a stochastic sequence within [−1, 1]. For generality, 1000 stochastic sequences are generated to determine the capacity of ESS and the results are shown in Table 1. Test 4: According to the error data of TWF A, a joint probability distribution of forecast errors for 24 time periods are built, and the forecast error sequences containing the temporal correlation of TWF A can be generated to calculate the capacity of ESS. Similar to test 3, 1000 sequences are generated to determine ESS capacity and the results are shown in Table 1. Table 1. Sizing result of simple demonstration Type Test 1 Test 2 Test 3 Test 4 Ave. Range Ave. Range capacity, MWh 24 1 2.25 [1.12, 4.04] 4.22 [2.38, 6.86] power, MW 1 1 0.49 [0.12, 0.86] 0.27 [0.11, 0.54] In Table 1, the Ave. is average of the results over 1000 error sequences. The range is the interval of the results over 1000 error sequences. From Table 1, it can be seen that the effects of correlation of wind power forecast error on optimal ESS size are significant. If the correlation of forecast error is not fully considered, the ESS capacity may be seriously misestimated. With this simple demonstration, it can be concluded that the optimal size of ESS has a strong relationship with the correlation of forecast error, and it needs to be considered carefully during the procedure of ESS sizing. Besides, the other factors that may impact the ESS size, such as investment cost of ESS, penalty cost for wind curtailment, will be considered in the following. 3 Modelling the spatial-temporal correlation of forecast error In this paper, energy storage is used to complement the wind power forecast error and reduce the effect of lower accuracy forecast result on power system dispatch. Moreover, according to the forecast value and distribution of error, the system operator can make pre-decision for the energy storage. Therefore, compared to the actual wind power data, forecast error is more suitable and practical to be used in the ESS sizing. In this part, the distribution fitting and correlation of forecast error will be illustrated to support the following ESS sizing model. The KDE which is more precise than other fitting methods (such as Normal and t location-scale), is chosen as the fitting method. And the nonlinear copula method is used to analyse the correlation between multiple wind farms and establish the correlation model of multi-dimensional forecast errors. 3.1 Fitting method of wind power forecast error In order to model the correlation of forecast error, the probability distribution of forecast error needs to be fitted for 24 time periods of one day. Since the fitting accuracy can affect the result of joint probability distribution and the size of ESS, an improved evaluation index in [26] is used to choose the fitting method. Because this index uses the difference between the cumulative probability at the upper and lower boundaries of each histogram column, similar fitted curves can be clearly distinguished. The formula of this index is shown as follows: (1) where NH is the number of bins in the histogram, F is the CDF of fitting data, is the probability interval between the upper and lower boundaries corresponding to histogram i, and PNi is the probability density of histogram i. The fitting precision is more accurate when the index I is closer to zero. The calculations of indexes for the different fitting methods are shown in Table 2. Table 2. Fitted indexes for different time period and method Distribution type Fitting for 24 time periods Fitting for one day Max Min Average normal 0.0077 0.0016 0.00329 0.0060 t location-scale 0.0013 0.00092 0.001 0.0019 KDE 0.00029 0.00012 0.000194 0.0006 In Table 2, the fitting methods of normal, t location-scale and KDE are compared. The t location-scale distribution is a generalisation of Student's t-distribution with two parameters for location and scale. From Table 2, it can be appreciated that the KDE is superior to other distribution-based methods in any case. Therefore, in this paper, the KDE-based method is selected to model the marginal cumulative distribution function (CDF) of forecast error in each time period. 3.2 Analysis of the spatial-temporal correlation of forecast error for multiple wind farms To clearly illustrate the spatial-temporal correlation of forecast errorfor multiple wind farms, this subsection gives the dependence structures of theforecast error for the TWF and RWFs. Fig. 3 shows the linear dependence structures of theforecast errors. It can be seen that the forecast error of each wind farm has astronger temporal correlation between adjacent forecast time periods. Meanwhile,considering the geographic positions of the four wind farms in Fig. 1, the distance between TWF A and RWF B isthe closest, and their maximal linear correlation coefficient is 0.6435, whichis corresponding to the 24th time periods of both TWF A and RWF B. Thus, theirspatial-temporal correlation of forecast errors is the largest. The distancebetween TWF A and RWF C is longer than that between A and B, and the maximallinear correlation coefficient is 0.4338, which is corresponding to the 24thperiod of A and 20th period of C. TWF A and RWF D are separated with the longestdistance, and their maximal linear correlation coefficient is only 0.06536,which implies that there is a little spatial correlation between them.Therefore, the wind farms with closer location have a higher spatial-temporalcorrelation, and this observation needs to be considered when optimising ESSsize. Fig. 3Open in figure viewerPowerPoint Linear dependence structures of the forecast errors for theTWF and RWFs 3.3 Copula theory based modelling for the forecast error of multiple wind farms Copula theory provides a method to build a joint distribution model for different individual variables. In this joint distribution model, the nonlinear dependence of different random variables can be identified by copula function [27]. Mathematically, Sklar's theorem stresses that any multivariate joint distribution can be expressed to marginal distribution functions of each individual variable and a copula function that describes the dependence structure of the variables [28]. Assuming that are L-dimensional random variables with marginal CDFs , respectively, and the transformed variables follow a uniform distribution: (2) According to Sklar's theorem, the joint distribution of can be described as (3) where C is a copula function. The t-copula can be thought of as representing the dependence structure implicit in a multivariate t distribution [29]. Therefore, t-copula is chosen to establish the multivariate JCDF of hourly day-ahead forecast error. It allows any marginal distribution and positive definite dependence matrix. Specifically, the t-copula function is expressed as follows: (4) where is the inverse CDF of the one-dimensional t location-scale distribution; is the CDF of the multivariate t distribution. Subsequently, the detailed procedure for generating a copula model of the multiple-dimensional forecast errors for N wind farms can be summarised as follows: Step 1: Generate a matrix of forecast errors for N wind farms. Assuming that there are R observations of wind power forecast error for each period, and the number of time periods is T for each single wind farm, the formulation of the matrix E is shown as follows: (5) where is the Rth observation of wind power forecast error of the Tth time period for wind farm N. In this paper, the length of T is 24 h. Step 2: Calculate the marginal CDF for each set of forecast error using the KDE-based fitting method. The expression of fitting results, marginal function (MF) is shown in (6). (6) Step 3: Using the maximum likelihood estimation method, the parameters of the t-copula function are estimated, which include the degree of freedom k and the correlation matrixρ, of which the rank is R. Step 4: Formulate the JCDF of the forecast error for N wind farms using the obtained t-copula functions. 4 Method for optimal ESS sizing Based on the proposed spatial-temporal correlation model of forecast error, the spatial-temporal characteristics of forecast error can be accurately captured by establishing the JCDF of forecast error. Apparently, the forecast error is a random variable with known distribution. It is difficult to solve the stochastic programming model with the random variable of the forecast error. Therefore, this paper adopts multi-scenario technology to establish and solve the model conveniently. Meanwhile, an operation strategy of ESS considering the trend of prediction and correlation of error is also explained in this section. 4.1 Methods for generating multi-dimensional scenarios The uncertainty of the output power of a wind farm can be described by the multiple scenarios technique, which includes the procedures of scenarios generating and reduction. Firstly, multi-dimensional scenarios can be generated based on the JCDF, which is formulated by the aforementioned t-copula function, and the detailed procedure for generating the scenarios is shown in Fig. 4. Fig. 4Open in figure viewerPowerPoint Scenario generation procedure In Fig. 4, M is the number of scenarios of forecast error. The time period T of each scenario, is 24 h as well. When the M × T dimensional scenarios of forecast error are generated, the wind power data Pi(t) can be calculated according to the error data of each scenario and original forecast data. Compared with the correlation analysis of wind power forecast value, the spatial-temporal correlation of the error is considered, and the same time, the temporal characteristic of original wind power forecast is not destroyed by using this wind power data. Secondly, to reduce the calculation burden and improve the solving efficiency during the sizing procedure of Section 4.2, the number of generated scenarios by the proposed method should be tuned. A simultaneous backward reduction technique is used to reduce the number of scenarios in this paper. The purpose of this technique is to identify a smaller set of scenarios that could reasonably approximate the scenarios of the actual system. The details of this technique can be referred to [30]. 4.2 Optimal ESS sizing model The aim of sizing ESS is to mitigate the wind power forecast error and decrease the negative impact of stochastic nature of wind power. Considering the higher cost of ESS devices nowadays, optimisation of ESS size is a critical problem for the improvement of economic performance and the further application in future. To determine the optimal ESS size, the objective function is to minimise the investment cost and operation cost of the ESS. This function can achieve optimal comprehensive benefits via the trade-off between the investment cost and the operation cost. Specifically, the operation cost includes surplus power cost and shortage power cost, both of which vary along with ESS energy capacity and charge/discharge power. To efficiently analyse the effect of spatial-temporal correlation of forecast errors on ESS sizing, the monthly data which show the most significant wind characteristic during the whole year is selected for calculation. The objective function for the ith scenario is formulated as follows: (7) where and are incurred by the curtailment of wind power and load shedding due to insufficient capacity of ESS. Here, three coefficients Ksu, Ksh and Kin need to be set differently, because the costs of abandoning wind, loss load and installation of energy storage are not proportion during the ESS sizing. So that the different kinds of costs mentioned above can be adjusted in a reasonable range. Moreover, decision-makers can adjust those relationship according to the actual situation and the preferences of decision during ESS sizing. The surplus energy and shortage energy for each scenario are expressed as below: (8) (9) In (8) and (9) (10) (11) (12) (13) (14) (15) (16) where and are the Boolean variables that describe the surplus energy incurred by the charge power limitation and capacity limitation in the ith scenario, respectively; and are the Boolean variables that describe the shortage energy incurred by the discharge power limitation and minimum capacity limitation in the ith scenario, respectively; is taken as the reference interval for energy storage output power, the detailed introduction can be found in Section 5.1. The objective function is subject to the following constraints: (17) (18) (19) (20) (21) (22) where (17) and (18) are charge power constraints, (19) and (20) are discharge power constraints, (21) and (22) are output power constraints. After optimal results of energy capacity and power of each scenario are obtained, the rated energy capacity and the charge/discharge power of ESS can be calculated as follows: (23) (24) where CBESSN and PBESSN are the rated capacity and power of energy storage. 4.3 Rolling operation strategy The operation
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