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A Copula Function Based Monte Carlo Simulation Method of Multivariate Wind Speed and PV Power Spatio-Temporal Series
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www.elsevier.com/locate/procedia
Energy (2019) 000–000 213–218 EnergyProcedia Procedia159 00 (2017) www.elsevier.com/locate/procedia
Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, REM 2018, 29–30 September 2018, Rhodes, Greece TheFunction 15th International Symposium District Heating and Cooling A Copula Based Monte on Carlo Simulation Method of Multivariate Wind Speed and PV Power Spatio-Temporal Series Assessing the feasibility of using the heat demand-outdoor a b a, Chaoqiong Pan , Can Wang Zhaoa, Jinhao Wang , Zhaohong * temperature function for aa, Ziyan long-term district heat demandBieforecast State Key Laboratory of Electrical Insulation and Power Equipment, Shaanxi Key Laboratory of Smart Grid, Xi’an Jiaotong University, Xi’an a,b,c a a b c c 710049, China b State Grid Shanxi Electric Power Company, Taiyuan 030001, China a IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal b Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, France c Département Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France
a
I. Andrić
*, A. Pina , P. Ferrão , J. Fournier ., B. Lacarrière , O. Le Corre
Abstract
As high penetration of uncertain wind and solar energy into power system, spatio-temporal correlation of wind and solar energy is ofAbstract great necessity for planning and operation of wind farms and PV plants. A Copula function and first order Markov process based Monte Carlo simulation method of multivariate wind speed and PV power spatio-temporal series is proposed in this paper. In the District heating networks are commonly in the literature onespeed of theand most solutions for decreasing the proposed approach, either spatial correlationaddressed or temporal correlation of as wind PV effective power series is considered including greenhouse gas emissions theand building sector. These systems require investments which are returned through the heat cross-correlation between allfrom series self-correlation of every series. Thehigh results of case study based on National Renewable sales. Due to the (NREL) changed ofclimate conditions and of building policies, demand in the future could decrease, Energy Laboratory the U.S. Department Energyrenovation are analyzed, whichheat verify the scientific and feasibility of the prolonging the investment return period. proposed approach. The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand ©forecast. 2019 The Authors. Published Ltd. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665 Copyright © 2018 Elsevier Ltd. by AllElsevier rights reserved. This is an open accessinarticle under the CC-BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) buildings that vary both construction period andthetypology. weatherofscenarios (low,Energy medium, high) and and threeForum, district Selection and peer-review under responsibility of scientificThree committee the Applied Symposium Selection and peer-review under responsibility of the scientific committee of thetheApplied Energy Symposium and Forum, renovation scenarios were developed (shallow, intermediate, deep). To estimate error, obtained heat demand values were Renewable Energy Integration with REM 2018. Renewable Energy Integration withMini/Microgrids, Mini/Microgrids, REM 2018. compared with results from a dynamic heat demand model, previously developed and validated by the authors. The results showed thatcorrelation; when onlyCopula weather change considered, the margin ofCarlo errorsimulation; could be acceptable for some applications Keywords: spatio-temporal function; firstisorder Markov process; Monte renewable energy (the error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered). value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the 1.The Introduction decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the With the seriousness of energy crisis and environmental pollution, wind and solar energy as alternatives to fossil coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and fuels is popularly promoted by many countries. In addition to the intermittency and fluctuation of wind and solar improve the accuracy of heat demand estimations. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. * Corresponding author. Tel.: +86-029-82668655; fax: +86-029-82665489. E-mail address: [email protected] Keywords: Heat demand; Forecast; Climate change
1876-6102 Copyright © 2018 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, REM 2018. 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. 1876-6102 © 2019 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. This is an open access article under the CC-BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, REM 2018. 10.1016/j.egypro.2018.12.053
214 2
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
energy, spatio-temporal correlation of wind and solar energy is of great necessity for power system planning and operation [1]. Thus, it has become a key problem that considering spatio-temporal correlation of wind farms and PV plants and guiding the suitable planning and expansion of wind farms and PV plants in the aspect of accelerating renewable energy development and utilization. In past years, the study of spatio-temporal correlation of wind and solar energy which was applied in reliability evaluation [2], stochastic optimal power flow [3], and power market [4] and so on has provided important insights for power system planning and operation. Reference [5] researched reliability assessment of generating systems based on Copula-ARMA model considering multivariate wind speed correlation. Reference [6] contributed a Markov chain Monte Carlo (MCMC) method for the direct generation of synthetic time series of wind power output. Long-term correlations and cross-correlations of wind speed and solar radiation temporal series from Brazil were analyzed in reference [7]. An integration study of photovoltaics and wind turbines distributed in a distribution network is investigated based on the stochastic modeling using Archimedean Copulas in reference [8]. Reference [9] proposed an algorithm to generate synthetic hourly cloudiness data and solar radiation data for any time of the year at any location in the south west region of Western Australia (WA). Reference [10] analyzed three popular simulation methods of spatially correlated wind power in small geographic areas. Though some achievements have been made with general knowledge of spatio-temporal correlation of wind and solar energy, most researchers only focused on the analysis of spatio-temporal correlation, for instance, judgement of series linear or nonlinear correlation, calculation of Pearson correlation coefficient or rank correlation coefficient and so on. Some researchers only considered correlation of wind energy or solar energy to generate corresponding temporal series but failed to consider synthetic correlation of wind and solar energy. Without considering spatio-temporal correlation of wind and solar energy when generating temporal series of wind and solar energy, it may lead to inaccurate modeling, power imbalance and increase the risk of power system during operation. This paper proposes a Copula function and first order Markov process based Monte Carlo simulation method of multivariate wind speed and PV power spatio-temporal series. In this paper, spatio-temporal correlation model of wind speed and PV power is established and wind speed and PV power series are generated using Copula function and first order Markov process based on data from the wind speed and PV power dataset of National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy. It is implemented to evaluate the wind farms and PV plants sitting, planning and operation, which contributes to power system planning and operation. And the proposed approach can be easily extended to other cases in different regions. 2. Simulation algorithm of multivariate wind speed and PV power spatio-temporal series The proposed simulation method is divided into two parts including generation of spatial correlation series and generation of temporal correlation series in this paper. Each part consists of parameter estimations of all kinds of Copula functions, determination of the best fit Copula function for history wind speed and PV power series and simulations of the needed data. The detailed flow chart of proposed algorithm is shown in Fig. 1. Start Input history wind speed and PV power data Estimate the marginal cumulative distribution functions of wind speed and PV power series Estimate parameters of Copula functions for spatial correlation Determine the best fit Copula function for spatial correlation Simulate spatial correlation series Estimate parameters of Copula functions for temporal correlation Determine the best fit Copula function for temporal correlation
Simulate temporal correlation series Output simulation wind speed and PV power data
End
Fig. 1. The simulation process of wind speed and PV power spatio-temporal series.
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
215 3
In all steps, the first and utmost step is to estimate the marginal probability density functions of wind speed and PV power series, which can be get by kernel density estimation method shown in equation (1). Then transform the marginal probability density functions into the marginal cumulative distribution functions by indefinite integral. x x 1 N ) fˆN ( x) k( i Nh i 1 h
(1)
In equation (1), fˆN is the kernel density estimation at any point x, k is the kernel function, h is the window width, N is the sample size and xi is one of the samples. The two key parts of kernel density estimation are selections of kernel function and window width. According to empirical validation, the window width is calculated by equation (2) and Gaussian kernel function drawn in equation (3) is selected as the kernel function.
h 0.9ˆ N 1 5
(2)
In equation (2), ˆ is standard deviation of the sample.
1
k ( x)
2
e
x2 2
(3)
This paper adopts the maximum likelihood estimation method to estimation unknown parameters of all kinds of Copula functions according to history wind speed and PV power data. The joint probability density function of n variables is shown as equation (4) based on Copula function. The logarithm likelihood function is shown in equation (5), in which is the correlation coefficient of Copula function. The estimation value of correlation coefficient ˆ needs to meet equation (6). f x1 ,
, xn
n C F1 x1 , F1 x1
N
In L( ) In( f ( x1,t ,
, Fn xn Fn
n
Fi xi
i 1
xi
x n
(4)
, xn ,t ))
t 1
(5)
ˆ =arg max In L( )
(6)
The best fit Copula function for history wind speed and PV power data is determined by comparing Euclidean distances between the empirical Copula function and each theoretical Copula function. The Copula function with minimum Euclidean distance among Copula families is the most close to the actual correlation of history data. The empirical Copula function is expressed as follows: N
i i Ce ( 1 , , n ) N N
I (x t 1
1, t
x1(i1 ) , N
, xn,t xn (in ) )
,1 i1 , i2 ,
, in N ,
(7)
where I is the indicator function, whose value is 1 when the condition in parentheses is met, otherwise that is 0 and x1(i1 ) , , xn (in ) is order statistics of the sample. Then the Euclidean distance is formulated as equation (8).
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
216 4
d (C , Ce )
N
N
i1
(C ( N ,
i1 1 in 1
,
in i ) Ce ( 1 , N N
,
in 2 )) N
(8)
The steps involved in generating multivariate wind speed and PV power spatial correlation series based on condition distribution of Copula function are as follows: Generate n random numbers z1 , z2 , , zn from U 0,1 Generate n values u1 , u2 , , un as equation (9) Repeat the above T times
zi C , ui 1 ) i (ui | u1 ,
i 1C (u1 , u1
, ui ,1, ui 1
,1)
i 1C (u1 , , ui 1 ,1, u1 ui 1
,1)
(9)
The steps involved in generating multivariate wind speed and PV power temporal correlation series based on condition distribution of Copula function and first order Markov process are as follows: Generate ( wk ,1 , wk ,2 , , wk ,T ), k 1, 2, , n by equation (10) Generate wind speed and PV power series ( xk ,1 , xk ,2 , , xk ,T ), k 1, 2, equation (11) uk ,t C k ( wk , t | wk , t 1 )
, n using inverse transformation as
Ck ( wk ,t , wk ,t 1 )
(10)
wk ,t 1
xk ,t Fk 1 ( wk ,t )
(11)
3. Case study This paper simulates 4 wind speed and PV power spatio-temporal series based on wind speed and PV power data of 2017 from National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy. The history data is from NREL Solar Radiation Research Laboratory Baseline Measurement System (BMS) and NREL National Wind Technology Center M2 Tower (M2), which are near to each other to ensure strong correlation of the data. In this paper, wind speed and PV power series measured every 1 hour were provided. Gumbel Copula function is selected to describe the spatial and temporal correlations of obtained wind speed and PV power data. And the correlation coefficients of spatial and temporal correlations are listed in Table 1. Table 1. The correlation coefficients of spatial and temporal correlations Correlation
Correlation coefficient
Spatial correlation of all series
1.282
Temporal correlation of Wind speed series from BMS
2.604
Temporal correlation of Wind speed series from M2
3.034
Temporal correlation of PV power series from BMS
3.877
Temporal correlation of PV power series from M2
3.777
The statistical properties of history and simulation wind speed and PV power series are shown in Table 2, in which the statistical properties of history data and simulation data generated through proposed method are almost same. The traditional method is to generate data through Weibull and Beta distribution. Fig. 2 depicts the 4-dimensional empirical distribution function value scatterplot of history and simulation data, which indicates history data and simulation wind speed and PV power data generated through proposed method have more similar probability distribution
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
217 5
characteristics and spatial correlation. Fig. 3 describes the self-correlation coefficients of history and simulation wind speed and PV power data for a certain delay time, which imply history data and simulation data generated through proposed method have more similar temporal correlation.
Fig. 2. The 4-dimensional empirical distribution function value scatterplot of history and simulation data.
Fig. 3. The self-correlation coefficients of history and simulation data with different shifted time intervals.
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
218 6
Table 2. The statistical properties of history and simulation data. Statistical property
History data
Proposed method simulation data
Traditional method simulation data
Mean value of wind speed from BMS
2.411
2.379
2.556
Mean value of wind speed from M2
3.360
3.613
3.460
Mean value of PV power from BMS
99.261
95.648
109.892
Mean value of PV power from M2
86.673
82.753
94.254
Standard deviation of wind speed from BMS
1.938
1.727
1.942
Standard deviation of wind speed from M2
2.485
2.369
2.655
Standard deviation of PV power from BMS
155.642
161.422
162.516
Standard deviation of PV power from M2
143.598
151.022
134.988
4. Conclusion Multivariate wind speed and PV power spatio-temporal series are generated using a Monte Carlo simulation method based on Copula function and first order Markov process in this paper. Both spatial and temporal correlations are considered to make the modeling, simulation and analysis more accurate. The proposed study can provide detailed spatial and temporal correlation information for the modeling, simulation and analysis of wind and solar energy. The results of case study on the data from National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy validate the proposed approach and indicate that it is of significance to the planning and operation of wind and PV power bases as well as power system. Acknowledgements This work was supported by National Natural Science Foundation of China (U1610122, 51637008). References [1] Qin, Zhilong, W. Li, and X. Xiong. "Generation System Reliability Evaluation Incorporating Correlations of Wind Speeds with Different Distributions." IEEE Transactions on Power Systems 28.1 (2013): 551-558. [2] Billinton, R., H. Chen, and R. Ghajar. "Time-series models for reliability evaluation of power systems including wind energy." Microelectronics Reliability 36.9 (1996): 1253-1261. [3] Xie, Z. Q., et al. "Quasi-Monte Carlo Based Probabilistic Optimal Power Flow Considering the Correlation of Wind Speeds Using Copula Function." IEEE Transactions on Power Systems 33.2 (2017): 2239-2247. [4] Elberg, Christina, and S. Hagspiel. "Spatial dependencies of wind power and interrelations with spot price dynamics." European Journal of Operational Research 241.1 (2015): 260-272. [5] Li, Yudun, K. Xie, and B. Hu. "Copula-ARMA Model for Multivariate Wind Speed and Its Applications in Reliability Assessment of Generating Systems." Journal of Electrical Engineering & Technology 8.3 (2013): 421-427. [6] Papaefthymiou, George, and B. Klockl. "MCMC for Wind Power Simulation." IEEE Transactions on Energy Conversion 23.1 (2008): 234-240. [7] Anjos, Priscilla Sales Dos, et al. "Long-term correlations and cross-correlations in wind speed and solar radiation temporal series from Fernando de Noronha Island, Brazil." Physica A Statistical Mechanics & Its Applications 424 (2015): 90-96. [8] Haghi, H. Valizadeh, et al. "Using Copulas for analysis of large datasets in renewable distributed generation: PV and wind power integration in Iran." Renewable Energy 35.9 (2010): 1991-2000. [9] Laslett, Dean, C. Creagh, and P. Jennings. "A method for generating synthetic hourly solar radiation data for any location in the south west of Western Australia, in a world wide web page." Renewable Energy 68.7 (2014): 87-102. [10] Díaz, Guzmán, P. G. Casielles, and J. Coto. "Simulation of spatially correlated wind power in small geographic areas—Sampling methods and evaluation." International Journal of Electrical Power & Energy Systems 63 (2014): 513-522.
ScienceDirect Availableonline onlineatatwww.sciencedirect.com www.sciencedirect.com Available Energy Procedia 00 (2018) 000–000
ScienceDirect ScienceDirect
www.elsevier.com/locate/procedia
Energy (2019) 000–000 213–218 EnergyProcedia Procedia159 00 (2017) www.elsevier.com/locate/procedia
Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, REM 2018, 29–30 September 2018, Rhodes, Greece TheFunction 15th International Symposium District Heating and Cooling A Copula Based Monte on Carlo Simulation Method of Multivariate Wind Speed and PV Power Spatio-Temporal Series Assessing the feasibility of using the heat demand-outdoor a b a, Chaoqiong Pan , Can Wang Zhaoa, Jinhao Wang , Zhaohong * temperature function for aa, Ziyan long-term district heat demandBieforecast State Key Laboratory of Electrical Insulation and Power Equipment, Shaanxi Key Laboratory of Smart Grid, Xi’an Jiaotong University, Xi’an a,b,c a a b c c 710049, China b State Grid Shanxi Electric Power Company, Taiyuan 030001, China a IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal b Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, France c Département Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France
a
I. Andrić
*, A. Pina , P. Ferrão , J. Fournier ., B. Lacarrière , O. Le Corre
Abstract
As high penetration of uncertain wind and solar energy into power system, spatio-temporal correlation of wind and solar energy is ofAbstract great necessity for planning and operation of wind farms and PV plants. A Copula function and first order Markov process based Monte Carlo simulation method of multivariate wind speed and PV power spatio-temporal series is proposed in this paper. In the District heating networks are commonly in the literature onespeed of theand most solutions for decreasing the proposed approach, either spatial correlationaddressed or temporal correlation of as wind PV effective power series is considered including greenhouse gas emissions theand building sector. These systems require investments which are returned through the heat cross-correlation between allfrom series self-correlation of every series. Thehigh results of case study based on National Renewable sales. Due to the (NREL) changed ofclimate conditions and of building policies, demand in the future could decrease, Energy Laboratory the U.S. Department Energyrenovation are analyzed, whichheat verify the scientific and feasibility of the prolonging the investment return period. proposed approach. The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand ©forecast. 2019 The Authors. Published Ltd. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665 Copyright © 2018 Elsevier Ltd. by AllElsevier rights reserved. This is an open accessinarticle under the CC-BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) buildings that vary both construction period andthetypology. weatherofscenarios (low,Energy medium, high) and and threeForum, district Selection and peer-review under responsibility of scientificThree committee the Applied Symposium Selection and peer-review under responsibility of the scientific committee of thetheApplied Energy Symposium and Forum, renovation scenarios were developed (shallow, intermediate, deep). To estimate error, obtained heat demand values were Renewable Energy Integration with REM 2018. Renewable Energy Integration withMini/Microgrids, Mini/Microgrids, REM 2018. compared with results from a dynamic heat demand model, previously developed and validated by the authors. The results showed thatcorrelation; when onlyCopula weather change considered, the margin ofCarlo errorsimulation; could be acceptable for some applications Keywords: spatio-temporal function; firstisorder Markov process; Monte renewable energy (the error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered). value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the 1.The Introduction decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the With the seriousness of energy crisis and environmental pollution, wind and solar energy as alternatives to fossil coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and fuels is popularly promoted by many countries. In addition to the intermittency and fluctuation of wind and solar improve the accuracy of heat demand estimations. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. * Corresponding author. Tel.: +86-029-82668655; fax: +86-029-82665489. E-mail address: [email protected] Keywords: Heat demand; Forecast; Climate change
1876-6102 Copyright © 2018 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, REM 2018. 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. 1876-6102 © 2019 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. This is an open access article under the CC-BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the Applied Energy Symposium and Forum, Renewable Energy Integration with Mini/Microgrids, REM 2018. 10.1016/j.egypro.2018.12.053
214 2
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
energy, spatio-temporal correlation of wind and solar energy is of great necessity for power system planning and operation [1]. Thus, it has become a key problem that considering spatio-temporal correlation of wind farms and PV plants and guiding the suitable planning and expansion of wind farms and PV plants in the aspect of accelerating renewable energy development and utilization. In past years, the study of spatio-temporal correlation of wind and solar energy which was applied in reliability evaluation [2], stochastic optimal power flow [3], and power market [4] and so on has provided important insights for power system planning and operation. Reference [5] researched reliability assessment of generating systems based on Copula-ARMA model considering multivariate wind speed correlation. Reference [6] contributed a Markov chain Monte Carlo (MCMC) method for the direct generation of synthetic time series of wind power output. Long-term correlations and cross-correlations of wind speed and solar radiation temporal series from Brazil were analyzed in reference [7]. An integration study of photovoltaics and wind turbines distributed in a distribution network is investigated based on the stochastic modeling using Archimedean Copulas in reference [8]. Reference [9] proposed an algorithm to generate synthetic hourly cloudiness data and solar radiation data for any time of the year at any location in the south west region of Western Australia (WA). Reference [10] analyzed three popular simulation methods of spatially correlated wind power in small geographic areas. Though some achievements have been made with general knowledge of spatio-temporal correlation of wind and solar energy, most researchers only focused on the analysis of spatio-temporal correlation, for instance, judgement of series linear or nonlinear correlation, calculation of Pearson correlation coefficient or rank correlation coefficient and so on. Some researchers only considered correlation of wind energy or solar energy to generate corresponding temporal series but failed to consider synthetic correlation of wind and solar energy. Without considering spatio-temporal correlation of wind and solar energy when generating temporal series of wind and solar energy, it may lead to inaccurate modeling, power imbalance and increase the risk of power system during operation. This paper proposes a Copula function and first order Markov process based Monte Carlo simulation method of multivariate wind speed and PV power spatio-temporal series. In this paper, spatio-temporal correlation model of wind speed and PV power is established and wind speed and PV power series are generated using Copula function and first order Markov process based on data from the wind speed and PV power dataset of National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy. It is implemented to evaluate the wind farms and PV plants sitting, planning and operation, which contributes to power system planning and operation. And the proposed approach can be easily extended to other cases in different regions. 2. Simulation algorithm of multivariate wind speed and PV power spatio-temporal series The proposed simulation method is divided into two parts including generation of spatial correlation series and generation of temporal correlation series in this paper. Each part consists of parameter estimations of all kinds of Copula functions, determination of the best fit Copula function for history wind speed and PV power series and simulations of the needed data. The detailed flow chart of proposed algorithm is shown in Fig. 1. Start Input history wind speed and PV power data Estimate the marginal cumulative distribution functions of wind speed and PV power series Estimate parameters of Copula functions for spatial correlation Determine the best fit Copula function for spatial correlation Simulate spatial correlation series Estimate parameters of Copula functions for temporal correlation Determine the best fit Copula function for temporal correlation
Simulate temporal correlation series Output simulation wind speed and PV power data
End
Fig. 1. The simulation process of wind speed and PV power spatio-temporal series.
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
215 3
In all steps, the first and utmost step is to estimate the marginal probability density functions of wind speed and PV power series, which can be get by kernel density estimation method shown in equation (1). Then transform the marginal probability density functions into the marginal cumulative distribution functions by indefinite integral. x x 1 N ) fˆN ( x) k( i Nh i 1 h
(1)
In equation (1), fˆN is the kernel density estimation at any point x, k is the kernel function, h is the window width, N is the sample size and xi is one of the samples. The two key parts of kernel density estimation are selections of kernel function and window width. According to empirical validation, the window width is calculated by equation (2) and Gaussian kernel function drawn in equation (3) is selected as the kernel function.
h 0.9ˆ N 1 5
(2)
In equation (2), ˆ is standard deviation of the sample.
1
k ( x)
2
e
x2 2
(3)
This paper adopts the maximum likelihood estimation method to estimation unknown parameters of all kinds of Copula functions according to history wind speed and PV power data. The joint probability density function of n variables is shown as equation (4) based on Copula function. The logarithm likelihood function is shown in equation (5), in which is the correlation coefficient of Copula function. The estimation value of correlation coefficient ˆ needs to meet equation (6). f x1 ,
, xn
n C F1 x1 , F1 x1
N
In L( ) In( f ( x1,t ,
, Fn xn Fn
n
Fi xi
i 1
xi
x n
(4)
, xn ,t ))
t 1
(5)
ˆ =arg max In L( )
(6)
The best fit Copula function for history wind speed and PV power data is determined by comparing Euclidean distances between the empirical Copula function and each theoretical Copula function. The Copula function with minimum Euclidean distance among Copula families is the most close to the actual correlation of history data. The empirical Copula function is expressed as follows: N
i i Ce ( 1 , , n ) N N
I (x t 1
1, t
x1(i1 ) , N
, xn,t xn (in ) )
,1 i1 , i2 ,
, in N ,
(7)
where I is the indicator function, whose value is 1 when the condition in parentheses is met, otherwise that is 0 and x1(i1 ) , , xn (in ) is order statistics of the sample. Then the Euclidean distance is formulated as equation (8).
Chaoqiong Pan et al. / Energy Procedia 159 (2019) 213–218 Chaoqiong Pan et al. / Energy Procedia 00 (2018) 000–000
216 4
d (C , Ce )
N
N
i1
(C ( N ,
i1 1 in 1
,
in i ) Ce ( 1 , N N
,
in 2 )) N
(8)
The steps involved in generating multivariate wind speed and PV power spatial correlation series based on condition distribution of Copula function are as follows: Generate n random numbers z1 , z2 , , zn from U 0,1 Generate n values u1 , u2 , , un as equation (9) Repeat the above T times
zi C , ui 1 ) i (ui | u1 ,
i 1C (u1 , u1
, ui ,1, ui 1
,1)
i 1C (u1 , , ui 1 ,1, u1 ui 1
,1)
(9)
The steps involved in generating multivariate wind speed and PV power temporal correlation series based on condition distribution of Copula function and first order Markov process are as follows: Generate ( wk ,1 , wk ,2 , , wk ,T ), k 1, 2, , n by equation (10) Generate wind speed and PV power series ( xk ,1 , xk ,2 , , xk ,T ), k 1, 2, equation (11) uk ,t C k ( wk , t | wk , t 1 )
, n using inverse transformation as
Ck ( wk ,t , wk ,t 1 )
(10)
wk ,t 1
xk ,t Fk 1 ( wk ,t )
(11)
3. Case study This paper simulates 4 wind speed and PV power spatio-temporal series based on wind speed and PV power data of 2017 from National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy. The history data is from NREL Solar Radiation Research Laboratory Baseline Measurement System (BMS) and NREL National Wind Technology Center M2 Tower (M2), which are near to each other to ensure strong correlation of the data. In this paper, wind speed and PV power series measured every 1 hour were provided. Gumbel Copula function is selected to describe the spatial and temporal correlations of obtained wind speed and PV power data. And the correlation coefficients of spatial and temporal correlations are listed in Table 1. Table 1. The correlation coefficients of spatial and temporal correlations Correlation
Correlation coefficient
Spatial correlation of all series
1.282
Temporal correlation of Wind speed series from BMS
2.604
Temporal correlation of Wind speed series from M2
3.034
Temporal correlation of PV power series from BMS
3.877
Temporal correlation of PV power series from M2
3.777
The statistical properties of history and simulation wind speed and PV power series are shown in Table 2, in which the statistical properties of history data and simulation data generated through proposed method are almost same. The traditional method is to generate data through Weibull and Beta distribution. Fig. 2 depicts the 4-dimensional empirical distribution function value scatterplot of history and simulation data, which indicates history data and simulation wind speed and PV power data generated through proposed method have more similar probability distribution
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characteristics and spatial correlation. Fig. 3 describes the self-correlation coefficients of history and simulation wind speed and PV power data for a certain delay time, which imply history data and simulation data generated through proposed method have more similar temporal correlation.
Fig. 2. The 4-dimensional empirical distribution function value scatterplot of history and simulation data.
Fig. 3. The self-correlation coefficients of history and simulation data with different shifted time intervals.
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Table 2. The statistical properties of history and simulation data. Statistical property
History data
Proposed method simulation data
Traditional method simulation data
Mean value of wind speed from BMS
2.411
2.379
2.556
Mean value of wind speed from M2
3.360
3.613
3.460
Mean value of PV power from BMS
99.261
95.648
109.892
Mean value of PV power from M2
86.673
82.753
94.254
Standard deviation of wind speed from BMS
1.938
1.727
1.942
Standard deviation of wind speed from M2
2.485
2.369
2.655
Standard deviation of PV power from BMS
155.642
161.422
162.516
Standard deviation of PV power from M2
143.598
151.022
134.988
4. Conclusion Multivariate wind speed and PV power spatio-temporal series are generated using a Monte Carlo simulation method based on Copula function and first order Markov process in this paper. Both spatial and temporal correlations are considered to make the modeling, simulation and analysis more accurate. The proposed study can provide detailed spatial and temporal correlation information for the modeling, simulation and analysis of wind and solar energy. The results of case study on the data from National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy validate the proposed approach and indicate that it is of significance to the planning and operation of wind and PV power bases as well as power system. Acknowledgements This work was supported by National Natural Science Foundation of China (U1610122, 51637008). References [1] Qin, Zhilong, W. Li, and X. Xiong. "Generation System Reliability Evaluation Incorporating Correlations of Wind Speeds with Different Distributions." IEEE Transactions on Power Systems 28.1 (2013): 551-558. [2] Billinton, R., H. Chen, and R. Ghajar. "Time-series models for reliability evaluation of power systems including wind energy." Microelectronics Reliability 36.9 (1996): 1253-1261. [3] Xie, Z. Q., et al. "Quasi-Monte Carlo Based Probabilistic Optimal Power Flow Considering the Correlation of Wind Speeds Using Copula Function." IEEE Transactions on Power Systems 33.2 (2017): 2239-2247. [4] Elberg, Christina, and S. Hagspiel. "Spatial dependencies of wind power and interrelations with spot price dynamics." European Journal of Operational Research 241.1 (2015): 260-272. [5] Li, Yudun, K. Xie, and B. Hu. "Copula-ARMA Model for Multivariate Wind Speed and Its Applications in Reliability Assessment of Generating Systems." Journal of Electrical Engineering & Technology 8.3 (2013): 421-427. [6] Papaefthymiou, George, and B. Klockl. "MCMC for Wind Power Simulation." IEEE Transactions on Energy Conversion 23.1 (2008): 234-240. [7] Anjos, Priscilla Sales Dos, et al. "Long-term correlations and cross-correlations in wind speed and solar radiation temporal series from Fernando de Noronha Island, Brazil." Physica A Statistical Mechanics & Its Applications 424 (2015): 90-96. [8] Haghi, H. Valizadeh, et al. "Using Copulas for analysis of large datasets in renewable distributed generation: PV and wind power integration in Iran." Renewable Energy 35.9 (2010): 1991-2000. [9] Laslett, Dean, C. Creagh, and P. Jennings. "A method for generating synthetic hourly solar radiation data for any location in the south west of Western Australia, in a world wide web page." Renewable Energy 68.7 (2014): 87-102. [10] Díaz, Guzmán, P. G. Casielles, and J. Coto. "Simulation of spatially correlated wind power in small geographic areas—Sampling methods and evaluation." International Journal of Electrical Power & Energy Systems 63 (2014): 513-522.