Voltage unbalance assessment in secondary radial distribution networks with single-phase photovoltaic systems

Electrical Power and Energy Systems 64 (2015) 646–654

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Voltage unbalance assessment in secondary radial distribution networks with single-phase photovoltaic systems F.J. Ruiz-Rodriguez a, J.C. Hernández b,⇑, F. Jurado a a b

Dept. of Electrical Engineering, University of Jaén, 23700 EPS Linares, Jaén, Spain Dept. of Electrical Engineering, University of Jaén, 23071 EPS Jaén, Spain

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 10 September 2013 Received in revised form 21 July 2014 Accepted 23 July 2014

This research describes a way to analyze voltage unbalance sensitivity for different maximum sizes of a single-phase photovoltaic system (SPPVS) with multiple PV penetration levels in a typical secondary radial distribution network (SRDN) in Spain. This analysis effectively assesses current requirements as specified in regulations concerning maximum size to be connected. It thus helps distribution network operators to define optimal limits, depending on their context. A stochastic assessment method is proposed to account for any random combination of SPPVSs in an SRDN. In addition, this method evaluates weekly voltage unbalance during a one-year time period, on the basis of 10-min intervals. More specifically, the voltage unbalance in SRDNs with SPPVSs is assessed for each 10-min interval by means of a probabilistic radial three-phase load flow (RTPLF). The results obtained show the maximum sizes of the SPPVS to be connected as a function of the PV penetration level in the SRDN, where high PV penetrations can produce voltage unbalance problems. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Secondary area networks Voltage unbalance Single-phase photovoltaic systems Stochastic assessment Probabilistic load flow

Introduction Nowadays, the scientific community is involved into an intense investigation to find new photovoltaic devices (third generation PV technology) which provide low-cost solar electricity. These devices are composed of materials characterized by low preparation cost, minimum environmental impact, light weight and wide availability [1–6]. The performances of many of these devices are lower than those of silicon cells, however the trade-off between energy generated and invested capital can be profitable. Several countries have reached the PV grid parity as the levelised cost of electricity for the PV technology in these countries can be compared with their local retail electricity prices in a competitive way. In most of the cases, this parity has been reached without any current subsidy or feed-in tariff incentive, but it has also been necessary a favourable regulatory framework, through net-metering or self-consumption laws [7,8]. The grid parity paradigm is possible in the rest of the world, but future research works are required in the smart grid research to mature this outcome [9,10]. The integration of high PV penetration levels in low voltage radial distribution networks can cause inadmissible voltage unbalances [11–14]. In order to minimize this problem as well as other ⇑ Corresponding author. Tel.: +34 953 212463; fax: +34 953 212478. E-mail addresses: [email protected] (F.J. (J.C. Hernández), [email protected] (F. Jurado).

Ruiz-Rodriguez),

http://dx.doi.org/10.1016/j.ijepes.2014.07.071 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

[email protected]

adverse impacts, there are various technical regulations for the interconnection of PV systems [15]. These connection requirements are based on a deterministic analysis. However, this type of analysis has the disadvantage of not being able to objectively specify the location where the voltage unbalance in a secondary radial distribution network (SRDN) could surpass the standard limit during a given time interval. Furthermore, it is also incapable of determining the frequency of such an event. The reason for this lies in the fact that the variables in an SRDN with PV systems are subject to uncertainties stemming from the inherent randomness of PV power outputs and loads. Therefore, probabilistic techniques are the tools that can best assess the impact of uncertainty on SRDN variables. Among all these techniques, the most frequently used are probabilistic load flow based on Monte Carlo simulation, analytical methods, and approximation methods. Although analytical methods are computationally more effective than Monte Carlo simulations, they must be based on certain mathematical assumptions. Firstly, it is necessary to define linear models to handle the nonlinearities of the balanced power system, either for meshed [16] or radial [17,18] configurations. Then, convolution techniques must be implemented to obtain the uncertainty of random outputs [16–19]. Finally, Gram–Charlier [17,19] or Cornish–Fisher [16,18] expansions estimate the probability functions of random outputs. An example of an approximation method is the point-estimate method (PEM), which directly provides the first statistical moments

F.J. Ruiz-Rodriguez et al. / Electrical Power and Energy Systems 64 (2015) 646–654

of random outputs based on only a few deterministic load flows. Gram–Charlier or Cornish–Fisher expansions are then used to characterize random outputs. The first PEM in [20] was subsequently followed by new and improved versions. In fact, this research study used one of the schemes of Hong’s PEM [21]. The number of deterministic simulations performed (e.g. K*m or K*m + 1) grows linearly with the number of input random variables (m) and the scheme selected (K = 2, 3, 4). According to [22,23], the 2m + 1 scheme provides the best performance for a high number of random inputs. Although reference [22] focuses on deterministic balanced load flows and reference [23] focuses on deterministic three-phase load flows, both studies only consider meshed configurations [24]. In the context of four-wire SRDNs, phase domain analysis is necessary, not only for incorporating the load unbalance and line impedance asymmetries, but also to assess the impact of singlephase photovoltaic systems (SPPVSs). The contribution of this paper is the use of Hong’s PEM (2m + 1 scheme) [22,23] to solve the probabilistic three-phase power flow problem in radial configurations. The use of a specific radial load flow (e.g. [25]) is mandatory in radial configurations since the conventional load flow [24] gives convergence problems. This PEM for radial configurations is then combined with the Cornish–Fisher expansion. The advantages of this tool for the assessment of the impact of SPPVSs on SRDNs are the following:  This PEM uses deterministic routines, but has a lower simulation cost than that of the Monte Carlo simulation.  The simulation cost of this PEM is only slightly higher than the cost of the analytical method based on convolution techniques. However, when the analytical method is applied to unbalanced power systems and radial configurations, its computational cost is much higher than that of the PEM.  The Cornish–Fisher expansion used in our study performs better for non-Gaussian PV random variables [18]. Aim of the study Voltage unbalance [26] is a growing power quality concern in SRDNs with SPPVSs because of their variable size and location [11,12,27–31]. Such PV systems are currently being connected on the basis of the ‘‘fit and inform’’ principle, which mainly depends on the perspectives and financial conditions of homeowners. Within this context, even if the voltage unbalance in an SRDN without PV is within standard limits, there is no guarantee that it will remain so. Therefore, the number of admissible SPPVSs or their maximum size in SRDNs must be analyzed in such a way as to keep the voltage unbalance within standard limits. Currently, the connection of SPPVSs in SRDNs is subject to requirements regarding maximum size (power). The objective of such regulations is to limit voltage unbalance. In most national regulations, this limit is approximately 5 kVA: 3 kW in the Endesa Utility Company [32]; 3.4 kVA in the UK [33]; 4.6 kVA in Austria and Germany [33]; 5 kW in Spain [34]; and 6.6 kW in Italy [35]. However, in Norway and France [33], the limit is much higher (15 kVA and 18 kVA, respectively). This difference in standards indicates that the impact of variables such as the type of SRDN and PV penetration level is in urgent need of clarification. This knowledge will help distribution network operators to define optimal limits for specific scenarios. In our study, this information was obtained by means of a voltage unbalance sensitivity analysis in an SRDN of the maximum size of SPPVS and of the PV penetration level. PV penetration is defined as follows:

PV penetration ¼

Annual PV capacity factor  Installed PV power Maximum SRDN power ð1Þ

647

where the annual PV capacity factor is the ratio of annually produced energy to the energy that could have been produced if the PV had operated continuously at full power. A stochastic assessment method was used to account for any random combination of SPPVSs in an SRDN. In addition, this method can be used to evaluate the voltage unbalance in a oneyear time period, for 10-min intervals, according to regulations [36,37]. More specifically, the voltage unbalance in SRDNs with SPPVSs is assessed by a probabilistic radial three-phase load flow (RTPLF). Voltage unbalance Voltage unbalance can be characterized by different variables [38]. Generally addressed is the ratio of the fundamental nega! tive-sequence component ð U 2 Þ to the fundamental positive!1 sequence component ð U Þ. Therefore, the voltage unbalance factor ! ! (VUF) is defined as n ¼ j U 2 = U 1 j: Voltage unbalance is time-variant in power systems. In this scenario, the use of indices is the most useful way of reducing the voltage unbalance to a single number [38]. Although the PV systems can influence the unbalance levels of all the nodes of a radial distribution network [11,12], the use of a site index is preferred [39,40]. This site index is obtained from a certain percentile (e.g. 95th, 99th) of the statistical characterization (cumulative distribution function or CDF) of the measurements of voltage unbalance over a long observation period (e.g. a one-day or one-week period) with a given time average (e.g. 3-s or 10-min interval). The most commonly used voltage unbalance index is the 95th weekly percentile of the variable n,10-min (VUF based on a 10-min mean), i.e., n,10-min,95w [38]. As voltage unbalance can have various adverse effects [41], the allowable compatibility level in low voltage supply systems is usually limited to 2% [36]. In the same context, different regulations, utility guidelines, and international standards suggest allowable planning/compatibility levels in the 1–5% interval: 1.3% in the UK [33]; 2% in France, Germany [33], and the EU [37]; and 2–2.5% in IEEE standard [42]. In this paper, a 2% limit is assumed for the site index 95th weekly percentile of the variable 10-min VUF at any kth SRDN node (nk,10-min), i.e., nk;10- min;95w . Probabilistic PV and load model Probabilistic PV system model The probabilistic PV system model in [18] is specified for a 10-min interval. It provides information regarding the marginal distribution (probability density function [PDF] and cumulative distribution function [CDF]) of the PV random power for each ith 10-min interval, mth month, and jth SRDN node uj;10-mini ðmÞ. Thus, the random variables hourly diffuse fractions and daily clearness index are used to build uj;10-mini ðmÞ. Furthermore, this model accounts for the stochastic interdependence of the PV distributions corresponding to close locations (nodes) due to similar meteorological conditions [43]. This dependence is modelled separately from marginal distributions with a specific rank correlation matrix [18]. The model generates dependent PV power outputs based on multivariate dependent random numbers [18]. Probabilistic load model Currently, certain distribution network operators are involved in the massive deployment of smart meters in the SRDNs to measure electrical load. This makes it possible to statistically characterize the load at each node by using measurements obtained and

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recorded over various years. Accordingly, the stochastic load model used in this research is directly based on smart meter measurements. The load model, based on a 10-min mean, obtains a typical daily profile for each dth day of the week and kth SRDN node for each month. Indeed, measurements for a given month over a period of several years were used to define typical daily profiles. This approach takes into account the real (reactive) power consumed by the kth SRDN node pdk;10-mini ðmÞðqdk;10-mini ðmÞÞ as a Gaussian random variable, which changes each mth month, dth day of the week, and ith 10-min interval.

Deterministic radial three-phase load flow The conventional method [24] applied to the resolution of three-phase load flows is not valid for SRDNs because of its poor convergence. Therefore, a specific method is needed for radial configurations. The method in this research [25] consists of an iterative process, which arrives at the solution by covering all the SRDN nodes by first moving forward and backward, and then backward and forward. The iterative process ends when convergence is reached. For example, in a 3-node SRDN (Fig. 1) the steps taken to solve the deterministic RTPLF are the following:  Step 1: Forward sweep The unbalance current at the extreme load node of the SRDN, for example, node 3, ½Iabc load is determined by: 3

½Iabc 3 ¼ ½Iabc load ¼ ð½Sabc 3 =½V abc 3 Þ 3

ð2Þ

in which:  load  load  Iload a3 ¼ ðSa3 =V a3 Þ ; Ib3 ¼ ðSb3 =V b3 Þ ; Ic3 ¼ ðSc3 =V c3 Þ

ð3Þ

where S3 (V3) is the complex power (voltage) of load node 3, and a, b, and c are the phases of the system. For the first iteration, the extreme load node voltage equals that of the source node (i.e., node 1). From the current at node 3, the voltage and current at node 2 in matrix form can be determined as follows:

½V abc 2 ¼ ½a  ½V abc 3 þ ½b  ½Iabc 3

ð4Þ

½Iabc line ¼ ½c  ½V abc 3 þ ½d  ½Iabc 3 2

where the coefficient values of matrices [a], [b], [c] and [d] are stated in [25]. From the voltage at node 2, the current produced by the load conditions at this node ½Iabc load is obtained by Eq. (3). Thus, the total 2 current per phase at node 2 is the following:

½Iabc 2 ¼ ½Iabc load þ ½Iabc line 2 2

ð5Þ

Voltages and currents per phase at source node [Vabc]1, ½Iabc 1 ¼ ½Iabc line are obtained by Eq. (4). 1

[Vabc]2

[Vabc]3

[Iabc]line 1

[Iabc]line 2

1

2

Source

[Iabc]load 2 ,

3 [Sabc]2

When the convergence criterion is verified, the following result is obtained:

  ½Mismatches ¼ ½V abc Source  ½V abc 1 =V base

ð6Þ

where Vbase is the voltage base in the SRDN. When the highest value in the matrix [Mismatches] is less than 0.001 p.u., the iterative process is finished.  Step 3: Backward sweep The reference voltage [Vabc]Source is considered at node 1. The voltages at the successive nodes are calculated with the currents in step 1. Voltages of nodes 2 and 3 are given by:

Probabilistic radial three-phase load flow

[Vabc]1

 Step 2: Verification of the convergence criterion

[Iabc]load 3 , [Sabc]3

Fig. 1. Example of SRDN with 3 nodes.

½V abc 2 ¼ ½A1   ½V abc Source  ½B1   ½Iabc 2 ½V abc 3 ¼ ½A2   ½V abc 2  ½B2   ½Iabc 3

ð7Þ

where matrices [A1], [A2], [B1], [B2] are calculated according to [25]. With the preceding voltages, the iterative process begins again at the first step. Probabilistic radial three-phase load flow The formulation of a deterministic RTPLF is based on parameters with fixed values. However, certain input parameters of a SRDN with PV systems are subject to uncertainties (e.g. loads, PV power outputs). One good way to characterize the sources of uncertainty of this type of SRDN is to represent the input data as random variables. In this respect, the Monte Carlo simulation and PEM are simulation techniques that make it possible to continue using the deterministic load flow routine. Monte Carlo simulation The Monte Carlo simulation generates random values of random inputs from their distribution functions. With these values, the steady state variables are evaluated by solving a deterministic RTPLF. Random outputs are then reconstructed from these deterministic data. Hong’s PEM: 2m + 1 scheme The PEM uses only a few deterministic RTPLFs to solve the probabilistic RTPLF. Thus, the PEM concentrates the statistical information provided by the first few moments of a problem input random variable xr at K deterministic locations ^ xr;K (a specified value of the random input). By using these locations and the function h, which relates input and output variables, the random outputs can be obtained. Obviously, a weighting factor xr,K, which accounts for the relative importance of each evaluation in the random outputs, must be included. Pair ð^ xr;K ; xr;K Þ is known as the Kth concentration. There are different schemes in Hong’s PEM, which are characterized by different K values in the deterministic evaluation of function h (K = 2m, 2m + 1, 3m + 1, etc., where m is the random input number). This study uses the 2m + 1 scheme since it provides the best performance for a high number of random inputs [22,23]. The procedure used to compute the raw statistical moments of random outputs for the RTPLF can be summarized as follows:  The rotational transformation method is used to transform a set of correlated random inputs (u1, . . ., um) into a set of uncorrelated random inputs (x1, . . ., xm) [44]. This is a necessary first step in the 2m + 1 scheme in the presence of the correlation between random inputs. The method is based on the eigenvalues and corresponding eigenvectors of the covariance matrix.  The Kth concentration (^ xr;K ; xr;K ) of each random input xr is obtained from the statistical input data (e.g. PDF of xr, i.e., f xr ). The two deterministic locations are determined by:

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^xr;K ¼ lxr þ dr;K rxr

ðK ¼ 1; 2Þ

ð8Þ

where dr,K is the standard location; and lxr and rxr are the mean and standard deviation of the random input xr. The two standard locations and weights are the following [21]:

dr;K ¼ kr;3 =2 þ ð1Þ3K 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kr;4  3=4  k2r;3 ðK ¼ 1; 2Þ

xr;K ¼ ð1Þ3K =ðdr;K  ðdr;1  dr;2 ÞÞ

ðK ¼ 1; 2Þ

ð12Þ

where z(r, K) is a vector of output variables associated with the Kth concentration of random input xr.  The vector z(r, K) is used to estimate the statistical moments around zero of the output random variables z (z1, . . ., zov, where ov is the random output number):

Eðzj Þ ffi

ov X 2 X



xr;K ðzðr; KÞÞj þ xo hð. . . ; lxr ; . . .Þ

j

ðmÞ

),

or

the

corresponding

CDFs

;

(F nweek;1 ðmÞ ; . . . ; k;10- min P -i = 1,

), and weights -1, . . ., -I, such that -i P 0 and ðmÞ

random variable nweek k;10- min ðmÞ can be represented by writing either the PDF, f nweek ðmÞ , or the CDF, F nweek ðmÞ , as a sum:

ð11Þ

lx1 ; lx2 ; . . . ; ^xr;K ; . . . ; lxm

k;10- min

ðmÞ

ð10Þ

 The solution for each of the preceding deterministic RTPLFs is:

zðr; KÞ ¼ h

. . . ; f nweek;I F nweek;I

k;10- min

r¼1



k;10- min

the finite mixture distribution for all the PV locations of every kth

 The function h must be evaluated only two times for each random input xr at the points specified for the Kth location ^ xr;K of the random input xr and the mean of the m  1 remaining inputs, namely (lx1, lx2, . . ., ^ xr;K , . . ., lxm).  One additional evaluation of function h at the point specified for the m random input means (lx1, lx2, . . ., lxr, . . ., lxm) is required with the following specific weighting factor xo:



set I of PDFs for each kth random variable nweek;I k;10- min ðmÞ, (f nweek;1

ð9Þ

where kr;3 and kr;4 are the skewness and kurtosis of the random input xr.

m  .  X xo ¼ 1  1 kr;4  k2r;3

location has the same probability of occurrence. Thus, given a finite

ð13Þ

r¼1 K¼1

Stochastic assessment method to evaluate the weekly voltage unbalance in an SRDN Voltage unbalance assessment due to the uncertainty regarding the PV location This research proposes a 3000-trial stochastic assessment of the voltage unbalance in an SRDN for any random combination of locations for SPPVSs of a given size and PV penetration level (left side of Fig. 2, dashed-line shapes). The number of trials, I = 3000, has the advantage of providing an acceptable coverage of different stochastic PV locations in the SRDN (see section ‘Voltage unbalance assessment due to the uncertainty in the PV location’). The two random inputs of this stochastic assessment determine the random location of SPPVSs in the SRDN, in other words, the node/s and phase where each SPPVS is connected. The uncertainty of PV locations is modelled by drawing random numbers distributed uniformly in the interval (1, nB), which represent the nodes with PV. Another random number, distributed uniformly in the interval (1), (3), represents the phase in which the SPPVS is connected at each node. Previously, a PV penetration level (0%, 5%, 10%, and 15%) and an SPPVS size (5, 10, 15 kW) are specified in order to perform a sensitivity analysis of these variables. Obviously, the number of SPPVSs to be connected in the SRDN is determined by both variables. For each Ith PV location (I = 3000 trials), the random variable weekly VUF is calculated based on the 10-min mean in the mth month for each kth SRDN node nweek;I k;10- min ðmÞ. Then, a finite mixture distribution [45] is applied to all of the PV locations because each

k;10- min

f nweek

k;10- min

ðmÞ

F nweek

ðmÞ k;10- min

¼

k;10- min

I X

-p f nweek;p

k;10- min

p¼1

¼

I X

-p F nweek;p

ðmÞ

k;10- min

p¼1

ðmÞ

ðk ¼ 1 . . . nB Þ

ð14Þ

ðk ¼ 1 . . . nB Þ

ð15Þ

As all -i are equal, -i is 1/nB. Once the mixture distribution for all of the PV locations is built (f nweek ðmÞ ; . . . ; f nweek ðmÞ ), (F nweek ðmÞ ; . . . ; F nweek ðmÞ ), the mixture 1;10- min

nB ;10- min

1;10- min

nB ;10- min

distribution is selected at month m* with the worst weekly VUF, i.e., (f nweek ðm Þ ; . . . ; f nweek ðm Þ ), (F nweek ðm Þ ; . . . ; F nweek ðm Þ ). 1;10- min

nB ;10- min

1;10- min

nB ;10- min

Voltage unbalance assessment due to time varying load and PV profiles The node loads in the SRDN have a daily, weekly, and monthly variation. PV power output also has a daily and monthly variation. Hence, the voltage unbalance assessment of both series at each Ith PV location should extend throughout a one-year time period (right side of Fig. 2, continuous-line shapes). It is thus possible to ascertain the worst weekly VUF for the twelve months of the year, according to regulations [37]. Given an Ith PV location, the probabilistic PV system model in section ‘Probabilistic PV system model’ determines correlated PV random power for the ith 10-min interval, mth month, and jth SRDN node uIj;10-mini ðmÞ. The set of correlated PV random power outputs is transformed into a set of uncorrelated PV random outputs pv Ij;10-mini ðmÞ. Taking PV power and load concentrations as inputs, the probabilistic RTPLF (see section ‘Probabilistic radial three-phase load flow’) provides the raw statistical moments of the VUF for each SRDN node at an Ith PV location, ith 10-min interval, dth day of the week, and mth month (e.g. at a kth SRDN node nd;I k;10-mini ðmÞÞ. Then, the Cornish–Fisher expansion reconstructs solutions for random variable VUFs in terms of PDFs and CDFs, (e.g. PDFs -f nd;I ; . . . ; f nd;I -). This process is repeated ðmÞ ðmÞ 1;10-mini

nB ;10-mini

12,096 times, which corresponds to an annual evaluation, based on 10-min intervals in which seven days of each month are considered. When all simulations are run, a finite mixture distribution is applied to the seven days of each mth month, i.e., only one week per month (e.g. PDFs -f nweek;I ðmÞ ; . . . ; f nweek;I ðmÞ -). 1;10- min

nB ;10- min

Case studies: rural SRDN A representative SRDN for a rural and urban environment should reflect all potential load profiles and design characteristics. For this reason, the characteristics of the SRDNs selected in this study are very close to the average of the wide range of SRDNs in Andalusia (Spain). Initial simulations showed that voltage profiles and voltage unbalance were not a source of problems in the urban SRDN. For example, its worst voltage drop (voltage unbalance) based on the10-min mean, (i.e., the worst weekly 95th percentile throughout the year) was 0.69% (0.11%) as compared to 6.58% (1.64%), which was the value in the rural SRDN. Therefore, the voltage

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F.J. Ruiz-Rodriguez et al. / Electrical Power and Energy Systems 64 (2015) 646–654

Fig. 2. Flowchart of the stochastic assessment method for weekly voltage unbalance in a SRDN.

unbalance sensitivity analysis, implemented in MATLAB, was only performed on the rural SRDN. Test SRDN data and PV input data The rural SRDN is a 30-branch, 31-node, 0.4-kV SRDN located in a rural area of Andalusia. It is a four-wire SRDN which has a multigrounded neutral system. Its total area is 0.0478 km2, and it supplies a 114.36-kVA annual mean load distributed in an unbalanced pattern. The most distant nodes (1–31) are 478 m away from each other. Maximum SRDN power is 211 kVA. The single-phase diagram is shown in Fig. 3 and the SRDN data are given in Appendix A. It is important to highlight that this rural SRDN has unbalanced lines and line sections carrying a mixture of single-phase, doublephase, and three-phase loads. Data for real and reactive load profiles were collected at each node in 10-min intervals by using smart meters over a two-years time period. Global irradiation data for the hourly diffuse fraction and the daily clearness index was obtained from [43]. The spatial PV dependence structure was modelled with a rank correlation matrix based on the distance between nodes [43]. Annual PV capacity factor is approximately 0.15 at the SRDN site. Base case To study the effect of different loading conditions on voltage unbalance, two scenarios were considered. Load scenario 1 represented the lightly loaded SRDN. This was a 101.15-kVA load in

the 10-min interval at 4:20 a.m. on a Sunday morning in April. Load scenario 2 represented the heavily loaded SRDN. This was a 149.47-kVA load at 9:10 p.m. on a Monday night in July. Fig. 4 shows the VUF variation (mean, 95th percentile, and standard deviation) for every kth SRDN node in the load scenario 1 and load scenario 2. In general, the VUF increased with the distance from the source node. It was assumed that the source node was a balanced three-phase source. As we moved along the SRDN, unbalanced load currents caused unequal voltage drops, which produced unbalanced phase voltages. The voltage unbalance magnitude for the test SRDN with a low X/R ratio was mostly due to unequal voltage magnitude. As can be observed, during the light load condition, the VUF at node 31 was intermediate and reached a 1.04% (1.09%) value for the mean (95th percentile). There was no appreciable voltage drop along the SRDN, and as a result the phase voltages remained nearly balanced. However, during the heavy load condition, the VUF reached a higher mean (95th percentile), i.e., 1.59% (1.69%). Accuracy of the PEM to assess the voltage unbalance in the SRDN for a specific 10-min interval and PV location To verify the accuracy of Hong’s PEM (2m + 1 scheme), this method and the Monte Carlo simulation were used to obtain the results of VUFs in the test SRDN at a specific I*th PV location and for a specific i*th 10-min interval. In particular, the 10-min interval at 12:00 a.m. on a Monday in July was analyzed. A 10% PV penetration level with 10-kWp SPPVSs was considered. This amounted to

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Fig. 3. Single-line diagram of the rural SRDN and relevant phase impedance matrixes.

0.07 VUF th Mean 95 percentil Standard desviation

0.03

0.01

1.0

95th baja

95th alta

0.5

VUF -standard desviation- (%)

0.05

Load scenario 1 Load scenario 2

th

VUF -mean, 95 percentile- (%)

1.5

0.0 1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

SRDN node number Fig. 4. Variation of VUF (mean, 95th percentile, and standard deviation) along the SRDN for two load scenarios without PV.

15 SPPVSs, which were located at the following nodes and phases: 2a, 3c, 4b, 7a, 7c, 10c, 12a, 16b, 18c, 20b, 20a, 25b, 28c, 30a, and 30c. A Monte Carlo simulation with 10,000 trials was used as a reference. Table 1 shows the individual maximum relative errors of the first seven moments about zero for the VUFs in the SRDN. The individual maximum relative error e for the r-order moment about zero (lr) of the VUF (n) in the PEM for the i*th 10-min interval, d*th day, m*th month, and I*th PV location, i.e., elr   is d ;I n ðm Þ max;10mini

given as follows [18]:

elr d ;I n

k;10mini

¼

i

max;10mini

i

lr;MCS   nd ;I

ðm Þ

elr d ;I n

      r;PEM r;MCS 100  l d ;I  l   ;I d n ðm Þ   nk;10min  ðm Þ k;10min  k;10mini

ðm Þ

8 < ¼ max elr   k : nd ;I

k;10mini

ð16Þ

ðm Þ

ðm Þ

9 = ;

ðk ¼ 1 . . . nB Þ

ð17Þ

The values in Table 1 shows the high accuracy level of the PEM for all moments. The individual relative error always achieved its maximum value for each moment in an SRDN node with an SPPVS.

Table 1 Individual maximum relative errors of the first seven moments about zero for the VUFs in the SRDN. r-Order moment about zero

1st

2nd

3rd

4th

5th

6th

7th

Individual maximum relative error (%)

0.265

0.510

0.725

0.864

0.854

0.624

0.667

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F.J. Ruiz-Rodriguez et al. / Electrical Power and Energy Systems 64 (2015) 646–654

Voltage unbalance assessment due to time varying load and PV profiles for a specific PV location

1.0 Weekly VUF Skewness Kurtosis

0.8

Without PV

"cdf sin pv"

PV scenario 1

"CDF PV scenario 1"

PV scenario 2

0.6

CDF

PV scenario 3

"CDF PV scenario 2"

"CDF PV scenario 3"

0.16

2.10

-0.34

1.96

-0.13

2.49

0.51

3.01

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Without PV PV scenario 1

"mean PV sin pv"

percentil sin PV

"mean PV scenario 1"

95th pv scenario 1

PV scenario 2

"mean PV scenario 2"

95th pv scenario 2

PV scenario 3

"mean PV scenario 3"

95th pv scenario 3

1.5

1.0

0.5 0

3

6

9

2.0

th

Weekly VUF -mean, 95 percentile- (%)

2.0

1.8 Percentile 95th 95th percentil

1.6

Mean Mean

1.4

1.2 1

10

100

10,000

1,000

Number of PV locations Fig. 7. Weekly VUF (mean, 95th percentile) for node 31 in July for a varying number of PV locations (fixed SPPVS size and PV penetration).

1.0

0.8 SRDN node #31 Month of July, weekly VUF is the worst

0.6

Without PV 5%-PV penetration, 5-kW SPPVS

CDF sin PV

CDF 5%-PV penetration, 5kW

5%-PV penetration, 10-kW SPPVS

CDF 5%-PV penetration, 10k

5%-PV penetration, 15-kW SPPVS

CDF 5%-PV penetration, 15k

10%-PV penetration, 5-kW SPPVS

CDF 10%-PV penetration, 5kW

10%-PV penetration, 10-kW SPPVS

CDF 10%-PV penetration, 10k

10%-PV penetration, 15-kW SPPVS

CDF 10%-PV penetration, 15k

15%-PV penetration, 5-kW SPPVS

CDF 15%-PV penetration, 5kW

15%-PV penetration, 10-kW SPPVS

CDF 15%-PV penetration, 10k

15%-PV penetration, 15-kW SPPVS

CDF 15%-PV penetration, 15k

0.0 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

Weekly VUF Fig. 8. Stochastic assessment of weekly VUF based on a 10-min mean for node 31 in the worst month (July) with different PV penetration levels and SPPVS sizes.

VUF th Mean 95 percentil

th

VUF -mean, 95 percentile- (%)

2.0

1.8

2.2

0.2

The voltage unbalance assessment in the test SRDN, given a certain SPPVS size and PV penetration level, requires a high number of different stochastic PV locations, which represent the widely

1.6

Fig. 6. CDF of the weekly VUF based on a 10-min mean for node 31 in July for four PV scenarios.

0.4

Voltage unbalance assessment due to the uncertainty in the PV location

1.4

Weekly VUF

CDF

Firstly, this section analyzes the voltage unbalance variation in the test SRDN during a full day (Monday) in July for a specific PV location. The weekly evaluation for the same month is then presented. Let us assume three scenarios for a specific PV location, given a 10% PV penetration level and three potential SPPVS sizes: (i) 5-kW SPPVS (PV scenario 1 composed of 30 SPPVSs); (ii) 10-kW SPPVS (PV scenario 2 composed of 15 SPPVSs); (iii) 15-kW SPPVS (PV scenario 3 composed of 10 SPPVSs). Another PV scenario without PV is also included. Fig. 5 shows the daily VUF variation (mean, 95th percentile) based on a 10-min mean for node 31, which has the highest VUF in the four PV scenarios. The unbalanced location of the SPPVSs in the phases further increased the already existing unbalance in the SRDN. For example at midday, the mean (95th percentile) of the random variable VUF increased from 1.18% (1.25%) without PV to 1.58% (1.85%) in the worst PV scenario. This was PV scenario 3 with the largest SPPVS size (15 kW). Thus, many small SPPVSs distributed throughout the SRDN helped to maintain its balanced condition. However, as the SPPVSs became larger in size, their number also decreased, which heightened the probability of unbalance conditions. Fig. 6 shows the CDF of weekly VUF based on a 10-min mean for node 31 in July for the four PV scenarios. PV scenario 1 had a worse weekly 95th percentile than PV scenario 2 in spite of its better performance at midday on Monday (Fig. 5). This occurred because of the influence of the other days of the week. However, the higher probability of a lower weekly VUF is evident in PV scenario 1. Additionally, the distribution functions moved away from the normal distribution (see skewness and kurtosis) as a result of non-Gaussian distributions for each 10-min PV power output throughout the week. The SPPVSs led to a shift in the distribution towards higher voltage unbalances only in PV scenery 3. The dispersion of resulting distributions became greater as the number of SPPVSs increased. In any case, the voltage unbalance at node 31, evaluated by the weekly 95th percentile of the VUF, in PV scenario 1 (1.59%), PV scenario 2 (1.50%), and PV scenario 3 (1.72%), was lower than the 2% voltage unbalance limit.

12

15

18

21

24

Time (hours) Fig. 5. Daily variation of VUF (mean, 95th percentile) based on a 10-min mean for node 31 of the SRDN on a Monday in July for four PV scenarios.

diverging values that voltage unbalance can reach at each node. It is a known fact that connecting an SPPVS at the beginning or end of an SRDN often leads to a significant difference in voltage unbalance. Since this number of PV locations may be very high, it is important to determine the lowest number of different PV locations that originate different values of voltage unbalance. Based on the assumption of a 10% PV penetration level and a 10-kW SPPVS size, Fig. 7 shows the weekly VUF (mean, 95th percentile) for node 31, i.e., nweek 31;10- min;l ð7Þ, as a function of the number of different random PV locations. Even though the number of potential combinations is very high, only 1000 random PV locations generated widely diverging values for voltage unbalances at this node. In fact, mean or higher-order moments varied very little after 1000 trials.

653

F.J. Ruiz-Rodriguez et al. / Electrical Power and Energy Systems 64 (2015) 646–654 Table 2 SRDN data. Nodes Sending node

Branch parameters Receiving node

Length

Configuration (see Fig. 3)

m 1 2 3 4 5 3 7 8 8 10 10 8 13 13 15 16 16 18 19 19 15 22 22 24 25 24 27 28 28 30

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

38 90 50 6 13 17 29 11 65 28 11 114 113 36 6 17 16 50 5 5 26 18 104 54 6 40 16 10 17 4

Node loads

3  150/95 mm2 4  150 mm2 Al 4  50 mm2 Al 4  50 mm2 Al 4  25 mm2 Al 4  150 mm2 Al 4  150 mm2 Al 4  25 mm2 Al 2  25 mm2 Al 2  25 mm2 Al 2  25 mm2 Al 4  150 mm2 Al 4  25 mm2 Al 4  150 mm2 Al 4  150 mm2 Al 4  25 mm2 Al 2  25 mm2 Al 2  25 mm2 Al 4  25 mm2 Al 4  25 mm2 Al 2  25 mm2 Al 4  25 mm2 Al 2  25 mm2 Al 2  25 mm2 Al 4  25 mm2 Al 2  25 mm2 Al 2  25 mm2 Al 4  25 mm2 Al 2  25 mm2 Al 4  25 mm2 Al

Given the different PV penetration levels and SPPVS sizes in this paper, 3000 trials were considered. Stochastic assessment of voltage unbalance for varying SPPVS sizes and PV penetration levels This section discusses the impact of SPPVS size on voltage unbalance in the test SRDN for different PV penetrations. Fig. 8 shows the results of stochastic assessment of the weekly VUF based on a 10-min mean for node 31 in the test SRDN with 5%, 10%, and 15% PV penetration levels as well as with SPPVS sizes of 5-kW, 10-kW, and 15-kW. As can be observed, whenever the PV penetration level was below 5%, the connection of large SPPVSs (up to 15 kW) did not deteriorate the voltage unbalance in the SRDN. Thus, for the worst node (node 31) and month (July, 7*), the weekly 95th percentile of the VUF for a 5-kW SPPVS (1.67%), a 10-kW SPPVS (1.78%) and a 15-kW SPPVS (1.90%) was lower than the 2% voltage unbalance limit. When the PV penetration level increased up to 10%, the connection of large SPPVSs (15-kW) led to an inadmissible voltage  unbalance in the SRDN. Thus, index nweek 31;10- min;95w ð7 Þ was 2.08% higher than 2%. However, the connection of smaller SPPVSs (e.g. 5-kW or 10-kW) was permitted since the indices of voltage unbalance (1.73% and 1.98%) met the 2% requirement. For the 15% PV penetration level, only the connection of small SPPVSs (5-kW) was allowed since voltage unbalance reached an admissible level,  i.e., index nweek 31;10- min;95w ð7 Þ was 1.69%. Conclusion A voltage unbalance sensitivity analysis for different maximum sizes of SPPVSs and PV penetration levels was performed on a typical SRDN in Spain. The objective was to verify current requirements as specified in the regulations regarding the maximum

Phase a

Phase b

Phase c

kW

kvar

kW

kvar

kW

kvar

0 5.42 0 0 0 3.42 0 0 0 0 0 0.77 0 0 5.09 0 0 7.39 0 0 3.76 0 2.63 0 1.55 0 0 0 0 0

0 1.99 0 0 0 1.42 0 0 0 0 0 0.29 0 0 1.59 0 0 1.53 0 0 1.28 0 1.03 0 0.43 0 0 0 0 0

0 0 0 0 3.58 0 0 4.45 0 0 2.59 2.63 0 0 0 7.12 0 0 0 1.98 0 0 0 0 0 3.41 0 2.55 0 7.39

0 0 0 0 1.40 0 0 1.25 0 0 0.77 1.03 0 0 0 2.62 0 0 0 0.67 0 0 0 0 0 1.49 0 0.87 0 1.80

0 4.45 0 2.67 0 0 1.74 0 2.59 0 2.52 0 0 1.77 0 0 2.79 3.41 0 0 0 0 0 0 5.8 0 4.30 0 2.59 6.62

0 2.03 0 0.98 0 0 0.60 0 1.01 0 1.32 0 0 0.43 0 0 0.95 1.80 0 0 0 0 0 0 3.06 0 2.26 0 1.01 3.48

size to be connected. For this purpose, a stochastic assessment method was used, which was able to account for any random combination of SPPVSs in the SRDN. In addition, this method evaluated any weekly voltage unbalance during a one-year time period, assuming 10-min intervals. More specifically, the voltage unbalance in SRDNs with SPPVSs for each 10-min interval was assessed by using a probabilistic RTPLF. The results obtained in this research study show that the connection of large SPPVSs (of a size up to 15 kW) is permissible for a 5% PV penetration level. However, current maximum size levels in regulations (approximately 5 kW) only apply to high PV penetration levels, such as a 15% PV penetration. This tool should also be applied in sensitivity analyzes for other SRDN variables (e.g. voltage regulation, line losses, line flow, etc.). This would be particularly valuable for the analysis of line flow in urban SRDNs, which is the limiting factor. Acknowledgment The authors would like to thank J. de la Cruz of ENDESA Distribución Eléctrica S.L.U., who provided the practical information and data regarding SRDNs. Appendix A The loads in Table 2 refer to annual means for 10-min intervals. References [1] Lee JW, Lee TY, Yoo PJ, Graatzel M, Mhaisalkard S, Park NG. Rutile TiO2-based perovskite solar cells. J Mater Chem A 2014;2:9251–9. [2] Bella F, Imperiyka M, Ahmadda A. Photochemically produced quasi-linear copolymers for stable and efficient electrolytes in dye-sensitized solar cells. J Photoch Photobio A: Chem 2014;289:73–80.

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