INTEGRATION of geographically dispersed photovoltaic

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1 1740 IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 7, NO. 6, NOVEMBER 2017 Probabilistic Load Flow for Power Grids With High PV Penetrations Using Copula-Based Modeling of Spatially Correlated Solar Irradiance Joakim Widén, Member, IEEE, Mahmoud Shepero, Student Member, IEEE, and Joakim Munkhammar Abstract This paper presents and applies an improved model for the instantaneous power generation from distributed photovoltaic (PV) systems intended for probabilistic load flow (PLF) simulations. The model combines a probability distribution model for the instantaneous solar irradiance at individual sites with an improved spatial correlation model and uses a Gaussian copula to allow correlated sampling from the distributions for an arbitrary set of distributed PV systems. We show that the model realistically reproduces the spatially distributed clear-sky index over a set of sites, based on comparisons with irradiance sensor network data. We also demonstrate that the probability distributions for system parameters such as customer voltage, substation loading, and power losses, obtained from PLF simulations with the model, differ substantially from those of a nonspatial approach. The results show that spatial irradiance modeling needs to be incorporated in PLF simulations in order not to overestimate the grid impacts of PV systems. Index Terms Copula, photovoltaic (PV), probabilistic load flow (PLF). I. INTRODUCTION INTEGRATION of geographically dispersed photovoltaic (PV) systems into distribution grids can have several adverse effects on power delivery, such as voltage rise, line overloading, and network protector trips [1], [2], and may require the use of one or several mitigation strategies [3]. When designing and operating electricity distribution grids with high PV penetrations, it is important to take into account that solar irradiance fluctuates over both space and time, and is partly unpredictable. Solar irradiance has a characteristic, arguably trimodal, probability distribution [4], and dispersed sites are correlated more or less due to cloud pattern variability [5]. Therefore, rather than designing grids and planning operation strategies for handling high PV penetrations based on worst-case conditions, a probabilistic approach is often more appropriate. Probabilistic load flow (PLF), in which system parameters such as power generation and demand are sampled from probability distributions [6], Manuscript received August 21, 2017; accepted August 31, Date of publication September 26, 2017; date of current version October 19, This work was supported by the Swedish Energy Agency within the two research programmes SolEl-programmet and Electricity and fuel from the sun. (Corresponding author: Joakim Widén.) The authors are with the Department of Engineering Sciences, Uppsala University, Uppsala SE , Sweden ( joakim.widen@angstrom.uu.se; mahmoud.shepero@angstrom.uu.se; joakim.munkhammar@angstrom.uu.se). Digital Object Identifier /JPHOTOV has been used extensively over the last decades for this reason, also for distribution systems with PV [7], [8]. However, in existing studies, whereas temporal variability has been given much consideration, there is a lack of approaches for dispersed PV generation that also take spatial variability and dependence into account. This is despite the fact that the so-called dispersion-smoothing effect, i.e., that spatial dispersion of generators evens out fluctuations in the aggregate output, is now well documented, as well as the factors influencing it [9]. Some existing studies have used methods for correlating simulation variables, e.g., wind turbines and PV systems [10], but have seldom included spatial correlations among PV systems. In fact, many studies use PV system data for one site as representative of a whole network [11], [12], although it has been shown that the difference in simulated grid impact between using singlesite versus spatially resolved data can be substantial [13], [14]. Time-series models such as the wavelet variability model [15] have been proposed to represent the smoothing effect from geographic dispersion of PV systems, but no method has yet been proposed for a purely probabilistic approach that would be more suitable for PLF simulations. For this reason, the authors have developed a copula-based approach in simulating solar irradiance over dispersed sets of PV systems with varying degrees of spatial correlation, in which marginal distributions for solar irradiance at individual sites are joined by a Gaussian copula and a spatial correlation model. In [16], we introduced this approach and showed that it could accurately reproduce the smoothing effect in spatially averaged irradiance over a sensor network in Hawai i, USA. However, only empirical correlations and marginal distributions were used, limited to the exact network layout of the studied network. Therefore, in [17], a general model was proposed with analytical marginal distribution functions and an analytical correlation model. However, the logarithmic correlation model used is only statistically and physically valid within a certain spatial limit and was only validated for the 1 km range of the Hawai i network. In this paper, we complement the Hawai i network data with sensor data from a wider network constructed for this study in Uppsala, Sweden, and fit a two-component exponential model to the network correlations, which is both physically motivated and statistically valid for any dispersion. We also apply the copula approach to PLF simulations of the entire medium-voltage distribution grid of a Swedish distribution system operator (DSO) in order to demonstrate the differences between using the spatial IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 WIDÉN et al.: PROBABILISTIC LOAD FLOW FOR POWER GRIDS WITH HIGH PV PENETRATIONS USING COPULA-BASED MODELING 1741 TABLE I NUMBER OF DAYS IN DAILY CLEAR-SKYINDEX (DCSI) CATEGORIES IN THE HAWAI I AND UPPSALA SENSOR NETWORKS DCSI category Hawai i Uppsala (0.7%) 19 (5.3%) (0.7%) 48 (13%) (1.7%) 34 (9.5%) (2.5%) 46 (13%) (7.7%) 38 (11%) (15%) 58 (16%) (24%) 55 (15%) (28%) 34 (9.5%) (18%) 25 (7.0%) Total 401 (100%) 357 (100%) data provided by the model and a nonspatial approach with one site representing the whole network. In Section II, the sensor network and DSO data are described, Section III presents the models and methods used, results are presented in Section IV, and conclusions are drawn in Section V. II. DATA A. Irradiance Sensor Networks Two sensor networks for solar irradiance were utilized for correlation model fitting and model evaluation, one on the island of Oahu, Hawai i, and one in the city of Uppsala, Sweden. The data are almost entirely analyzed in terms of the clear-sky index (CSI), which is the ratio κ = G/G c between global horizontal irradiance G and the theoretical clear-sky irradiance G c.data from both networks were categorized into intervals of daily clear-sky index (DCSI), denoted κ, which is the ratio between daily G and G c values, as shown in Table I. 1) Hawai i Network: The network consists of 17 pyranometers, dispersed up to around 1 km, measuring global horizontal irradiance with a 1 s resolution [18]. 401 days (March 20, 2010 October 15, 2010 and October 19, 2010 April 27, 2011) were used after exclusion of dates with missing data. Corresponding clear-sky irradiance was determined via CAMS McClear [19]. Data for times with sun elevation <20 were excluded from the analyses to avoid too low-standing sun. 2) Uppsala Network: The network consists of 24 HOBO Pendant Temperature/Light Data Loggers [20] recalibrated to measure irradiance, installed at 13 different sites dispersed up to around 10 km, and mounted horizontally and/or in planes of PV systems. Data were sampled every fifth minute over one year from March 24, 2016 to March 23, 2017, but only 357 of these were included in order to use the same DCSI categories as for the Hawai i data. Since the loggers do not allow perfect synchronization, all data were resampled to the same nearest 5 min interval midpoints in order to get the same time stamp for all data. This results in a lower time accuracy than for the Hawai i network, but it should still be accurate enough for the evaluations in this paper. Clear-sky irradiance was calculated in the plane of the sensors with the Perez Ineichen model [21] and a lower threshold for in-plane clear-sky irradiance of 300 W/m 2 was used to exclude times with high incidence angles. B. Distribution Grid Data For fitting load models and performing PLF simulations, grid data for the whole medium-voltage distribution network (609 lines and 589 buses) of the Swedish DSO Herrljunga Elektriska were provided. The data used were grid topology and cable impedances for the utility s two medium-voltage grids, along with metered hourly consumption from 2014 for almost all of the 3921 individual customers. For a map of substations in the grid, see Fig. 4. III. METHODS A. Instantaneous Clear-Sky Index Model The probabilistic model for solar irradiance over a network of sites, developed and fitted to the Hawai i network data in [17], is based on describing the network in either of two states: sunny, in which the sky is clear over all sites, and cloudy, in which at least one of the stations is obscured by a cloud. A random variable for the instantaneous CSI at a site i is then κ i = Sκ s,i +(1 S)κ c,i (1) where S is a random variable indicating if the state is sunny (S = 1) or cloudy (S = 0), i.e., S is Bernoulli distributed with some probability s for the sunny state. The random variables κ s,i and κ c,i are the CSI in the sunny and cloudy states, respectively. The CSI in the sunny state has a very nearly Gaussian distribution and is modeled as κ s,i N(μ s,σ 2 s) (2) with mean value μ s and standard deviation σ s. In the cloudy state, the CSI is modeled by a bimodal Gaussian mixture distribution 2 f κ c (κ) = w k f N (κ μ k,σk 2 ) (3) k=1 where w 1, w 2 are the weights of the Gaussian density function f N with mean values μ 1, μ 2 and standard deviations σ 1, σ 2.The indices 1 and 2 denote cloudy substates overcast and broken clouds, following the three-state interpretation of Hollands and Suehrcke [4]. As shown in [17], all model parameters (except for the sunny state distribution) have a characteristic dependence on the DCSI. The fraction of time in the sunny state, i.e., the probability s, has a nonlinear appearance modeled by the quadratic function in Fig. 1(a). The component proportions, mean values, and standard deviations of the cloudy state mixture distribution are modeled as functions of the DCSI as indicated in Fig. 1(b) (d). The sunny-state distribution is very closely Gaussian with mean value μ s = 1.00 and standard deviation σ s = B. Spatial Correlation Model To allow correlated sampling of solar irradiance at a set of sites, the spatial correlation ρ ij = Cov(κ i,κ j ) Var(κi )Var(κ j ) (4)

3 1742 IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 7, NO. 6, NOVEMBER 2017 distribution function (cdf) can, according to Sklar s theorem [22], be expressed by the marginal distributions, as modeled above, and a copula function C joining them F κ 1,...,κ N (κ 1,...,κ N )=C(F κ 1 (κ 1 ),...,F κ N (κ N )). (6) Since a random variable is related to a uniformly distributed random variable through its inverse cdf, i.e., κ i = Fκ 1 i (U i ),the equation above can be expressed as F κ 1,...,κ N (F 1 κ 1 (u 1 ),...,F 1 κ N (u N )) = C(u 1,...,u N ). (7) Thus, the copula C is a multivariate probability distribution with uniform marginals, and realizations of κ 1,...,κ N, distributed according to F κ 1,...,κ N, can be obtained by sampling values from the copula and feeding them through the inverse marginal distributions F κ 1,...,F κ N. In this paper, a Gaussian copula is used, which means that the copula function is defined as C(u 1,...,u N )=Φ Σ (Φ 1 (u 1 ),...,Φ 1 (u N )) (8) Fig. 1. Parameters in the distribution model for the instantaneous clear-sky index, as fitted to the Hawai i data and modeled as functions of the daily clearsky index κ in [17]. The subfigures show (a) fraction of time spent in the sunny state and (b) component proportions, (c) mean values and (d) standard deviations for the mixture distribution for the cloudy state. between any pair of sites i and j is modeled. While the model proposed in [17] was based on a logarithmic correlation model that would yield unrealistic correlations for large site separations, here a more general two-component exponential model is proposed, based on the hypothesis that the correlation decreases very fast for distances below 1 km (as previously seen in the Hawai i network [17]) and slower for larger distances, supposedly due to fast fluctuations in solar irradiance decorrelating over short distances and more slow fluctuations over larger distances. In this model, the correlation between two sites separated by a distance d is ρ(d) =(1 β)e k1d + βe k 2d (5) where β weights the respective components and k 1, k 2 describe how fast the correlation components decrease with distance. As for the CSI distributions, a dependence between the parameters and the DCSI is presumed. To find this dependence, the two components were fitted separately to the Hawai i and Uppsala network data in each DCSI category. When fitting the model to the Hawai i data (<1 km), it was assumed that e k 2d 1, so that a function ρ 1 (d) =(1 β)e k 1d + β could be fitted. For the Uppsala data (only for distances >1 km),itwasassumed that e k 1d 0, so that a function ρ 2 (d) =βe k 2d could be fitted. Model fitting results and parameter models are presented in Section IV-A. C. Copula-Based Correlated Sampling When sampling irradiance over a spatial network of N sites, we want to be able to get correlated samples from the joint multivariate distribution of κ 1,...,κ N. The cumulative where Φ Σ is the joint cdf of a multivariate Gaussian distribution with zero means and correlation matrix Σ, and Φ 1 is the inverse cdf for a standard Gaussian distribution. For obtaining a set of N correlated CSI values (one for each site), the complete procedure is to sample from the multivariate Gaussian distribution with the elements of Σ set to the correlations given by the spatial correlation model, feed the sampled values through the Gaussian cdf to obtain correlated, uniformly distributed numbers, and then feed these through the inverse cdf of the CSI. In-depth descriptions of this sampling process can be found in [16] and [17]. D. Photovoltaic System Model PV system output power at dispersed sites is calculated with simple yet accurate standard models for splitting G into beam and diffuse components, transposing these to the tilted planes of the PV arrays, and calculating PV array and inverter output. A detailed account of this modeling procedure can be found in [17]. E. Load Model Customer loads are modeled as random variables with lognormal distributions, meaning that the logarithm of such a variable L has a Gaussian distribution: ln(l) N(μ, σ 2 ). (9) For each of the 39 customer types defined by the DSO, distributions were fitted to actual hourly active power consumption, one for each month of the year and each hour of the day. To avoid unrealistically high sampled values from the distributions, a maximum allowed power was defined as a factor λ times the mean value of the distribution: P max = λe(l) =λe μ+ 1 2 σ 2. (10) Reactive power is modeled based on the active power assuming a constant power factor cos θ.

4 WIDÉN et al.: PROBABILISTIC LOAD FLOW FOR POWER GRIDS WITH HIGH PV PENETRATIONS USING COPULA-BASED MODELING 1743 Fig. 3. Probability density functions for the spatially averaged instantaneous clear-sky index in the Uppsala sensor network, measured and sampled from the spatial clear-sky index model in different daily clear-sky index (DCSI) categories. Fig. 2. Correlation model fitting results. Values of k 1 as fitted to the Hawai i network data are shown in (a), and values of k 2 and β as fitted to the Uppsala network data in (b), with β values obtained for Hawai i included as reference. Also shown are functions modeling the dependence on the daily clear-sky index (DCSI). Examples of correlation data and the resulting correlation model for different DCSI bins are shown in (c). Note the logarithmic scale in (c). F. Probabilistic Load Flow Three-phase balanced PLF is simulated by sampling values from the spatial PV system and load models, as defined above, and using Newton s method [23] to calculate the power flow in the grid for each set of samples. The whole simulation procedure was implemented in MATLAB. IV. RESULTS A. Correlation Model Fitting and Performance The results of fitting the two correlation model components to the Hawai i and Uppsala sensor network data can be seen in Fig. 2. The fast-decreasing component as well as β depend strongly on the DCSI, while for the slow-decreasing component a systematic dependence on the DCSI is less evident. Based on visual inspection of the resulting model parameters in each category, the following simple functions were fitted to describe the dependence on the DCSI: { for κ <0.301 β( κ) = κ for κ (11) { 0 for κ <0.169 k 1 ( κ) = 3.31 κ for κ (12) k 2 ( κ) = (13) As can be seen, a linear fit was used for k 1 and a constant value for k 2. β was fitted to the Uppsala data, since β will strongly affect the decorrelation for large distances. The β values obtained for Hawai i are included as reference and are very similar. A linear function was fitted to the seven highest categories and a constant value to the two lowest. Fig. 2(c) shows the correlations in three of the DCSI categories for the Hawai i and Uppsala sensor data along with the modeled correlation. It can be seen that the data for Uppsala seem to continue the decreasing correlation trend in the Hawai i data, albeit with a wider spread (due to the much lower amount of data in the Uppsala dataset) and with the hypothesized slower decline. The exception is the lowest category, in which the results are however uncertain due to the low amount of data (cf., Table I). Hence, when categorized by the DCSI, the irradiance at very different geographical locations seem to exhibit the same correlation patterns. In order to test whether the marginal CSI distributions derived from the Hawai i data can be used in combination with the correlation model and the Gaussian copula to reproduce the spatially correlated CSI in the Uppsala network, the instantaneous 1 CSI averaged over all sites, N ΣN i=1 κ i, was compared between the network data and the model. The model was applied to the network sites with κ set to the midpoint of each DCSI category, and values per category were sampled. The resulting empirical and modeled probability density functions are shown in Fig. 3, showing that the distributions within each category are highly similar. Note that over this network the bimodality of the spatially averaged CSI has almost disappeared, both in the model and data. These results confirm that the model is representative for this location in Sweden and should be possible to use for realistic PLF simulations in Swedish grids, but they also suggest that the model as outlined here should be generally valid, even if more comparisons to other locations are needed to confirm this.

5 1744 IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 7, NO. 6, NOVEMBER 2017 Fig. 4. Geographical dispersion of secondary substations in the two mediumvoltage grids (Herrljunga and Ljung) of the DSO Herrljunga Elektriska, used in the PLF simulation. Also shown are the locations of primary substations and the substation buses that are most sensitive in terms of voltage variations, far out on radial feeders. B. Comparison of Spatial and Nonspatial PLF In order to quantify the benefit of using spatially correlated solar irradiance provided by the model instead of a nonspatial model (i.e., using the irradiance at one site as representative of all sites), the two medium-voltage grids of the DSO Herrljunga Elektriska were simulated in a high-penetration PV scenario with two approaches: 1) Modeling the CSI at each PV system as N individual random variables, joined by the Gaussian copula with the correlation from the spatial correlation model. 2) Using one random variable for the CSI as a representative for the whole network. In the first case, correlated values were sampled from the joint N-dimensional distribution, using the Gaussian copula. In the second case, one value was sampled directly from the CSI distribution for one site and applied to all sites. The geographical dispersion of all 334 secondary substations in the two medium-voltage grids for which coordinates could be obtained (a few substations were missing exact coordinates) are shown in Fig. 4. In the power flow simulations, these are modeled as individual load and generator buses. In each such bus, a total PV generation capacity of 75 kw p, facing south and tilted 30, is assumed to be connected, corresponding to one fairly typical rural roof- or ground-mounted PV system per substation. These systems, 500 m 2 at a module efficiency of 15%, are small enough to be considered points with no internal smoothing. Should larger systems have been simulated they could have been divided into a discrete number of points to represent internal dispersion. For each substation, all customers in the underlying low-voltage grid were modeled with the probabilistic load model (with λ = 10 and cos θ = 0.9) and aggregated (after sampling) to one total load per substation. The substations that could not be assigned coordinates were excluded from load and generation modeling and were assigned zero load and generation. Fig. 5. Probability density functions for secondary substation voltages, power flow through the primary substation, and total power losses in the two studied medium-voltage grids (Ljung and Herrljunga) obtained through PLF simulations with spatial and nonspatial irradiance modeling. For both grids, individual load flow solutions were determined based on sampled generation and load data. For the voltages, the main plots show probabilities calculated over all substation buses and the smaller inset graphs show the probabilities for voltage at the most sensitive buses (cf., Fig. 4). To test the model in a scenario with high PV yield, the whole medium-voltage network was simulated for the specific date and time of July 15th, 12.00, and a DCSI of κ = 0.7 was assumed, corresponding to a day with a fair amount of broken clouds, and also a κ value in the range in which the dispersion-smoothing effect should be high [17]. About sets of values were sampled from the CSI and load models; in each set, 334 correlated samples (1 sample in the nonspatial model) were obtained from the clear-sky irradiance model and more than 3000 values from the load model. One power flow solution was iterated for each sample set yielding currents in all lines and voltages in all buses. The results are summarized in Fig. 5, showing probability density functions for secondary substation voltage, power flow through the primary substation (simulation slack bus), and total power losses. For the voltages, the smaller inset graphs show the probability densities for the voltage at the most sensitive buses, far out on radial feeders (see the map in Fig. 4). In particular, for the primary substation load and the losses, the nonspatial model consistently overestimates the probability for high values and underestimates the probability for low values. For the voltage, the differences between spatial and nonspatial modeling are small for the voltage distributions calculated over all substation buses in the grid. However, the probability distributions for the individual, most sensitive, buses are more dissimilar. This is because the former shows the spread of voltages both between buses due to grid structure and at individual points due

6 WIDÉN et al.: PROBABILISTIC LOAD FLOW FOR POWER GRIDS WITH HIGH PV PENETRATIONS USING COPULA-BASED MODELING 1745 to irradiance variability, while the latter shows only the spread of voltages at individual grid points. Thus, spatial modeling is important, in particular when analyzing the voltage at specific buses rather than the overall probability for voltage violations anywhere in the grid. V. CONCLUSION The copula-based spatial model of solar irradiance applied in this paper is shown to realistically represent the spatially distributed instantaneous CSI observed in a solar irradiance sensor network on a scale of 1 10 km. In particular, the improved two-component exponential correlation model proposed in this paper allows correlated spatial sampling over larger areas than the previous one. From the PLF simulations, the conclusion is that for grids on a spatial scale, such as the one studied here ( km), in which there is substantial decorrelation of irradiance with distance, a spatial model is required in order to not overestimate the negative impact of PV on power grids, in terms of voltage violations (in particular at ends of radial feeders), overloading, and power losses. Future research should investigate the universality of the applied CSI distribution model and the proposed correlation model, further test the validity of the three-state interpretation of the CSI, and investigate the benefits of spatial PV modeling for different grid types. ACKNOWLEDGMENT The authors would like to thank A. Mannikoff and T. Erikson of Herrljunga Elektriska AB for kindly providing power grid data. REFERENCES [1] R. A. Walling, R. Saint, R. C. Dugan, J. Burke, and L. A. Kojovic, Summary of distributed resources impact on power delivery systems, IEEE Trans. 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Available: Accessed on: May 2, [19] SoDa Service, Cams McClear service for estimating irradiation under Clear-Sky, [Online]. Available: Accessed on: May 2, [20] HOBO Data Loggers, HOBO pendant temperature/light 64K data logger, [Online]. Available: data-loggers/ua Accessed on: May 2, [21] R.Perez et al., A new operational model for satellite-derived irradiances: Description and validation, Sol. Energy, vol. 73,no.5,pp ,2002. [22] R. B. Nelsen, An Introduction to Copulas (Springer Series in Statistics). New York, NY, USA: Springer, [23] J. J. Grainger and W. D. Stevenson, Power System Analysis. New York, NY, USA: McGraw-Hill, Joakim Widén (S 10 M 10) was born in Sweden in He received the M.Sc. and Ph.D. degrees from Uppsala University, Uppsala, Sweden, in 2005 and 2010, respectively. He is currently an Associate Professor in the Built Environment Energy Systems Group (BEESG) in the Department of Engineering Sciences, Uppsala University, leading research on different aspects of integration of solar power into power systems. Mahmoud Shepero (S 17) received the M.Sc. degree in energy technologies from Uppsala University, Uppsala, Sweden, in He is currently working toward the Ph.D. degree in modeling of electric vehicles charging on city scale, in the Built Environment Energy Systems group (BEESG) in the Department of Engineering Sciences, Uppsala University. His current research focuses on the spatial planning of electric vehicles chargers and on the impacts of charging the electric vehicles on the electricity grid. Joakim Munkhammar received the Ph.D. degree in engineering physics from Uppsala University, Uppsala, Sweden, in He leads the Transport Energy Systems Group as a part of the Built Environment Energy Systems Group in the Department of Engineering Sciences, Uppsala University. His research is currently focused on mathematical modeling of photovoltaic power generation and electrification of transport.