This paper presents a probabilistic power flow (PPF) analysis method for distribution network incorporating the randomness and correlation of photovoltaic (PV) generation. Based on the multivariate kernel density estimation theory, the probabilistic model of PV generation is proposed without any assumption of theoretical parametric distribution, which can accurately capture not only the randomness but also the correlation of PV resources at adjacent locations. The PPF method is developed by combining the proposed PV model and Monte Carlo technique to evaluate the influence of the randomness and correlation of PV generation on the performance of distribution networks. The historical power output data of three neighboring PV generators in Oregon, USA, and 34bus/69bus radial distribution networks are used to demonstrate the correctness, effectiveness, and application of the proposed PV model and PPF method.
1. Introduction
In recent years, the development of solar photovoltaic (PV) generation has been steadily increasing around the world; the total global PV capacity reached 100 GW in 2012
[1]
. PV generation is becoming an important renewable energy resource. The increasing amount of PV generation significantly affects the performance of distribution networks.
Compared with conventional generation, PV generation is a variable resource with two characteristics 1) Randomness: The power output of PV generator at each time point is random because of uncertain solar irradiation and other related weather conditions. 2) Correlation: PV generation at different sites can be assumed to be independent if they are far away from each other. However, the power outputs of PV generators at adjacent locations may be strongly correlated owing to common effects such as solar irradiation, temperature and other environmental factors. In addition, residential rooftop PV generation is not allowed to provide voltage control. PV generation can potentially cause various power quality issues in distribution networks, such as fluctuation of bus voltage magnitudes, line flows, and voltage violations
[2]
. Hence, the randomness and correlation of PV generation must be considered and statistically characterized.
The probabilistic power flow (PPF) analysis, which was first proposed by Borkowska in 1974
[3]
, is an important and popular approach to evaluate the performance of power systems while considering the uncertainties of load demand and renewable generation
[4]
. The techniques for solving the PPF can be classified into two categories: analytical solution and Monte Carlo simulation
[5]
. There are different analytical probabilistic power flow methods such as convolution method
[6

9]
, the point estimation method
[10
,
11]
and the cumulant method
[2
,
12]
. The necessitated unavoidable approximations always exit in the analytical methods
[5]
. By contrast, Monte Carlo methods can handle various complex conditions without simplification
[5]
; such methods have been widely used in solving the PPF
[4
,
13

15]
.
In recent years, extensive research has been devoted to PPF analysis for power systems with PV generation. The probabilistic model of PV generation is derived from the probabilistic density function of the clearness index, which is used to denote the impact of clouds on solar irradiation
[14]
. The probabilistic models of global and diffuse irradiation are established in
[15]
. Based on the functional relationship of PV power output and irradiation, the probabilistic PV model is derived. The Monte Carlo technique is used to analyze the PPF of distribution networks
[14
,
15]
. Note that the correlation of PV generation is not considered in the above PPF methods. A cumulantbased PPF algorithm is presented in
[2]
to consider the correlation of PV generation. The corresponding probabilistic model of PV generation is acquired by combining the Beta distribution of solar irradiation and the normal distribution of the forecast error of the PV cell temperature. Based on Nataf transformation and Latin Hypercube Sampling, hypercube sampling, a PPF method is proposed in
[4]
to address correlated power sources while assuming that PV power follows Beta distribution. The assumption of theoretical parametric distribution for PV generation is necessary in all aforementioned PPF methods. However, it is well known from theory and experience that there may be a substantial gap between an assumed theoretical parametric distribution and the physical behavior of actual PV power. In other words, an assumed parametric distribution model may not always achieve satisfactory results
[16]
. This has been confirmed by recent investigations into considerable PV power data
[17]
. Consequently, the assumption of unsuitable distribution may lead to biased analysis results of the probability distributions of bus voltage magnitudes, line flows, and so on in the aforementioned PPF methods.
This paper presents a Monte Carlo based PPF method for distribution networks, which can accurately capture the randomness and correlations of PV generators at adjacent locations without any assumption of theoretical parametric distribution and evaluate the integration of PV generation on the performance of distribution network. The main contributions of the paper include the following :

The probabilistic model of PV generation is established using the nonparametric kernel density estimation, which can accurately capture the randomness and correlation of PV generation without any assumption.

The actual power data of three PV generators at adjacent locations are used to validate the proposed PV model.

Combined with the proposed PV model, the PPF method is developed on the basis of Monte Carlo technique to evaluate the influence of the randomness and correlation of PV generation on the performance of distribution networks.

The 34bus/69bus radial distribution network is adopted to demonstrate the effectiveness and validity of the presented PPF method.
The rest of the paper is organized as follows. The probabilistic model of PV generation is proposed in Section II, and the probabilistic load model is presented in Section III. The deterministic power flow is briefly summarized in Section IV. The procedure of the PPF method that considers the correlation of PV generation is proposed in Section V. Case studies are provided in Section VI, followed by conclusions in Section VII.
2. Probabilistic Model of PV Generation Considering Correlation
Assume that
n
PV generators are geographically close, i.e., the power outputs of the
n
PV generators are correlated with each other. Let
p_{i}
(
i
= 1, 2 ,…,
n
) represents the power output of the
i
th PV generator and
P
= [
p
_{1}
,
p
_{2}
,…,
p_{n}
]. Additionally,
m
day samples of PV generators are available and the jth day sample is
P_{j}
= [
P
_{1j}
,
P
_{2j}
,…,
P_{nj}
],
j
= 1, 2 ,..,
m
. The joint probability density function of
P
is assumed as
f(P)
, which should consider the randomness and correlation of PV power outputs. Nonparametric kernel density estimation theory with multiple variables is introduced to estimate
f(P)
[18]
. The multivariate kernel density estimation of
f(P)
can be denoted as follows
[19]
:
where
H
is the
n×n
symmetric positive definite matrix, and
K
(•) denotes a multivariate kernel function operating on n arguments. To simplify the calculation,
H
can be assumed to be a diagonal matrix
H
= diag [
h
1,
h
2 ,…,
h
_{n}
]. Thus,
K
(•) can be expressed in the following form of multiplicative kernel functions:
where
k
(•) is a univariate kernel function. The kernel estimation theory indicates that the type of kernel function has very little effect on the accuracy of kernel density estimation. In this paper, the following Gaussian function, which is widely recommended in mathematics books, is selected as the kernel function.
The following analytical expression is obtained by substituting (2) and (3) into (1).
Bandwidth vector
H
is a crucial factor in accurately estimating
f(P)
. Different methods can be used to estimate multiple optimum bandwidth vectors in (4). For example, a method of utilizing a multivariate normal distribution as the reference distribution is a popular one. However, this method is only appropriate for the reference probability density function and may fail to consider multimodal densities
[18]
, which may be the case for PV power output. Hence, the CrossValidation (CV) method
[19]
, which requires no assumption or reference distribution, is applied to select the optimum bandwidth vector whose components are multiple bandwidths.
In the CV method, the estimation error of
f_{H}
(
P
) can be represented by the integrated squared error
ISE
. The bandwidth vector
H
that minimizes the
ISE
is the optimum bandwidth.
In (5), the first term can be easily calculated from the data, and the last term does not depend on
H
and can be ignored as far as the minimization over
H
is concerned. Hence, only the second term of (5) is unknown and must be estimated.
Here,
E
(
f_{H}
(
P
)), the expected value of
f_{H}
(
P
), can be estimated by a leaveoneout estimator, which is also the unbiased estimation of
E
(
f_{H}
(
P
)), i.e.,
[19]
where
By substituting the unbiased estimation of
E
(
f_{H}
(
P
)) in (7) into (5) and ignoring the last term in (5), the following analytical expression can be obtained:
By choosing the Gaussian function in (3) as the kernel function and substituting (4) into (9), (9) can be transformed into the following:
This equation is a nonconstraint optimization problem. In this study, the interior point method is adopted to solve the optimum bandwidth vector
H
.
3. Probabilistic Model of Load
The normal distribution is the most popular load distribution and is widely used
[14].
The probability density function
f
(
P_{L}
) for the normal distribution of load power
P_{L}
is given by the following expression:
where
μ
and
σ
are the mean and standard deviation of load power
P_{L}
, respectively. Thus, an expected value and a standard deviation specify the load at each load point.
4. Probabilistic Power Flow method considering the correlation of PV generation
Combining the probabilistic models of PV generation and load presented in Sections 2 and 3, the PPF analysis method for distribution networks, which considers the correlation of PV generation, is developed based on the Monte Carlo technique. The flow chart is shown in
Fig. 1
. The procedure is as follows:

1) Input the historical data of PV power output and the electrical and geometrical parameters of the distribution network. Initialize the maximum iteration timekmax.

2) Based on the proposed probabilistic PV model in Section 2, generate stochastic samples of PV power outputsPGusing the rejection sampling method[21].

3) Generate stochastic samples of bus loadsPLbased on probabilistic load model in Section 3.

4) Use the NewtonRaphson method to solve the power flow with stochastic samples of PV and load powers.

5) Save the result of each power flow calculation, such as bus voltage magnitudes, line flows, and network loss.

6) Take the coefficient of varianceηas the convergence criterion. Ifηis less than the given precisionε, or ifkis up tokmax, then exit. Otherwise, letk=k+ 1 and proceed to Step 2.
Flow chart of PPF analysis method
5. Case Studies
The case studies include two parts: verification of the proposed probabilistic PV model that incorporates the correlation among power outputs of multiple PV generators and the results and analysis of PPF for two distribution networks that incorporates multiple PV generators.
 5.1 Data, network, and study conditions
The power output data (20092011) of three PV generators at adjacent locations in the Oregon, USA, are used, which is available online
[22]
. The three PV generators are strongly correlated with one another. This strong correlation enables the testing of the correctness, effectiveness, and adaptability of the proposed PV model against the actual PV power output data. In this paper, P1, P2, and P3 denote the three PV generators.
The proposed PPF algorithm for distribution networks has been implemented using MATLAB and tested on the 34bus radial distribution network, which consists of 33 lines and 20 loads, and 69bus radial distribution network, which consists of 68 lines and 48 loads. A full description of the test distribution networks can be found in
[23]
and
[24]
.
 5.2 Verification of the proposed probabilistic model of PV generation
The verification of the proposed PV model includes three aspects. First, the correlation coefficients of PV generation obtained using the PV model should be consistent with those obtained from the original PV power output data. Second, the mean and variance of PV power output obtained using the PV model should be sufficiently close to that obtained from the original PV power output data. Third, the randomness of the power output of a single PV generator should be evaluated correctly.
The stochastic samples are generated by applying the proposed PV model using the historical data of the three PV generators P1, P2, and P3. To analyze the test result conveniently, the stochastic and historical data are both normalized, i.e., the data are divided by their maximum. Hence, all data fall in the interval [0, 1].
The correlation coefficients for each pair of variables in the PV power outputs of P1, P2, and P3 are directly estimated from the historical data and given in
Table 1
. The correlation coefficients calculated using the stochastic samples generated by the proposed PV model are given in
Table 2
. The percentages in the brackets in
Table 2
represent the relative errors of the correlation coefficients obtained using the proposed PV model against those obtained using the historical data of PV power output. It can be observed that the proposed PV model can accurately represent the correlations between the power outputs of multiple PV generators.
The mean and variance values of the power outputs of P1, P2, and P3 are estimated using the historical data and stochastic samples generated by the proposed PV model. The mean values are shown in
Table 3
, and the variance values are presented in
Table 4
. The percentages in the brackets in
Tables 3
and
4
represent the relative errors of the indices obtained using the proposed PV model against those obtained using the historical data. It can be seen that the results obtained using the proposed PV model match the reality very well. Take the mean values as an example. The maximum error of the proposed PV model is only 4.11%. Hence, the mean and variance of the historical data can be reflected by the proposed PV model with high accuracy.
The discrete probability distributions of the power outputs of the three PV generators P1, P2, and P3 are acquired using the historical data and stochastic samples generated by the proposed PV model respectively. The histograms of the three discrete distributions of P1, P2, and P3 are shown in
Figs 2

4
. As revealed in the figures, the theoretical parametric distribution (such as the beta and normal distribution) cannot be used to establish the probabilistic model of PV power outputs because the histograms of P1, P2, and P3 have approximately two peaks. Furthermore, the probability distributions of the PV power outputs obtained using the proposed PV model basically match those estimated using the original historical data. This condition is attributable to the fact that the proposed PV model based on nonparametric kernel density estimation theory does not need any assumption of theoretical parametric distribution. Consequently, the probability distributions of the PV power outputs, regardless of whether they can be described by certain parametric distribution, can be accurately reflected by the proposed PV model.
Original correlation matrix of PV power outputs
Original correlation matrix of PV power outputs
Calculated correlation matrix of PV power outputs
Calculated correlation matrix of PV power outputs
Mean values of PV power outputs
Mean values of PV power outputs
Variance values of PV power outputs
Variance values of PV power outputs
The test results reveal that the randomness (represented by mean, variance, and probability distribution) of PV generation and the correlation between power outputs of multiple PV generators can be accurately captured by the proposed PV model.
Probability distribution of power output of P1
Probability distribution of power output of P2
Probability distribution of power output of P3
 5.3 Probabilistic power flow analysis for 34bus distribution network containing PV generation
The PPF studies for distribution networks that consider the correlations of PV generation are performed using the 34bus test system incorporating the three PV generators. The 34bus radial distribution network is shown in
Fig. 5.
The numbers with underline denote the load bus numbers and the numbers behind ‘L’ denote the line numbers.
To verify the proposed PPF method and analyze the impacts of the randomness and correlation of PV generation on the PPF, the following three cases are studied. The randomness of PV generation is considered in both cases 1 and 2 but the correlation of PV generation is only considered in case 1. No PV generator is included in case 3.
Case 1: Three PV generators P1 (45 kW), P2 (18 kW), and P3 (60 kW) that are correlated with one another are added to buses 26, 27, and 34, respectively. The total capacity of the added PV power is 123 kW. The PV generation penetration is 29.7%, which is obtained by dividing the capacity of integrated PV generators with the mean value of total network active loads. The proposed probabilistic PV model is used to model the randomness and correlation of PV generation.
Case 2: The connection location and capacity of the PV generators are the same as those in Case 1. The probabilistic model of the PV generation in 0 is used to model the power outputs of P1, P2, and P3 respectively, that is, only the randomness of PV generation is considered, and the power outputs of the different PV generators are regarded as independent of one another.
34bus distribution network
Mean values of bus voltage magnitudes
Case 3: No PV generator is connected to the distribution network.
The mean and variance values of the voltage magnitudes/angles at each bus are obtained based on the proposed PPF analysis method under three different cases and the results are given in
Figs. 6

9
respectively.
Fig. 10
shows the probability density curves of the voltage magnitude at bus 26 under three different cases. The values in the horizontal and vertical axis denote the voltage magnitude and probability density respectively in
Fig. 10
. The following observations can be made:
Variance values of bus voltage magnitudes
Mean values of bus voltage angles
Variance values of bus voltage angles
The voltage level of the distribution network is improved by the connection of the PV generators, especially the PV connection bus. Take bus 26 as an example. The probability of voltage magnitude in the interval [0.98, 1] becomes extremely bigger than that in the cases without a connected PV generator.
The fluctuation of bus voltage increases because the variance of bus voltage increases with the connection of PV generators. This increase is caused by PV generation being a variable resource whose production is influenced by everchanging weather conditions.
Neglecting the correlation of PV generation leads to a biased estimation of bus voltage magnitudes and angles. The correlation has little impact on the mean values of the bus voltage magnitudes and angles, but significantly influences the variance values or probability distributions of bus voltage magnitudes and angles.
Probability density curves of voltage magnitude at bus 26
Mean values of line flows
Variance values of line flows
The mean and variance values of line flows are shown in
Figs. 11
and
12
, respectively. The probability density and cumulative probability curves of line flow through line 23 under three different cases are given in
Figs. 13
and
14
respectively. The values in the horizontal axis denote the line flow and the values in the vertical axis denote the probability density and cumulative probability respectively in
Figs. 13
and
14
. As shown in
Figs. 11
and
12
, the mean values of line flows are decreased, whereas the variance values are increased by the connection of PV generation. The correlation of PV generation has little impact on the mean values of line flow. However, neglecting the correlation of PV generation has great impact on the underestimation of line flow variance.
Figs. 13
and
14
show that the probability distribution of line flow through line 23 that are connected to three correlated PV generators is significantly influenced by the correlation of PV generation. That is, ignoring the correlation leads to a biased analysis of line flows.
Probability density curves of line flow through line 23
Cumulative probability curves of line flow through line 23
Probability density curves of network loss rate
Cumulative probability curves of network loss rate
The network loss rate is also calculated by the proposed PPF method and the probability density and cumulative probability curves of loss rate under three PV connection cases are given in
Figs. 15
and
16
respectively. The values in the horizontal axis denote the network loss rate and the values in the vertical axis denote the probability density and cumulative probability respectively in
Figs. 15
and
16
. It can be observed that the loss rate is decreased by the PV generation. The probability that the loss rate falls in interval [4%, 4.5%] under case 3 is distinct from that under cases 1 and 2. Neglecting the correlation of PV generation has significant impact on the probability distribution of loss rate. The correlation is related to the synchronism of the PV power output values and affects the total power output of the multiple PV generators and network loss in each simulation. Hence, the probabilistic distribution of network loss is associated with the correlation of PV generation.
 5.4 Probabilistic power flow analysis for 69bus distribution network containing PV generation
The proposed PPF method is also tested by the 69bus distribution network incorporating three PV generators.
Fig. 17
shows the 69bus radial distribution network. The numbers with underline denote the load bus numbers.
69bus distribution network
The following three cases are developed to analyze the impacts of the randomness and correlation of PV generation on the performance of distribution network. The randomness of PV generation is considered in both cases 4 and 5 but the correlation of PV generation is only considered in case 4. No PV generator is included in case 6.
Mean values of bus voltage magnitudes
Variance values of bus voltage magnitudes
Mean values of bus voltage angles
Variance values of bus voltage angles
Case 4: Three PV generators P4 (300 kW), P5 (300 kW), and P6 (200 kW) that are correlated with one another are added to buses 38, 39, and 53, respectively. The total capacity of the added PV power is 800 kW. The PV generation penetration is 21.1%. The proposed probabilistic PV model is used to model the randomness and correlation of PV generation.
Case 5: The connection location and capacity of the PV generators are the same as those in Case 4. The probabilistic model of the PV generation in
[17]
is used to model the power outputs of P4, P5, and P6 respectively, that is, only the randomness of PV generation is considered, and the power outputs of the different PV generators are regarded as independent of one another.
Case 6: No PV generator is connected to the distribution network.
Mean values of line flows
Variance values of line flows
Probability density curves of network loss rate
The mean and variance values of bus voltage magnitudes/angle are shown in
Figs. 18

21
respectively. The mean and variance values of line flows are given in
Figs. 22
and
23
respectively.
Fig. 24
is probability density curves of network loss rate under three different PV integration cases. It can be observed that the fluctuations of bus voltage magnitudes/angles and line flows are increased, but the line flows and network loss are decreased with the connection of PV generation. Moreover, neglecting the correlations of PV generation has little impact on the mean values of bus voltage magnitudes/angles and line flows but significantly influences the variance values or probability distributions of bus voltage magnitudes/angles, line flows, and network loss.
6. Conclusions
This paper presents a PPF analysis method for distribution networks that incorporates the correlation of PV generation. Based on nonparametric kernel density estimation theory, the probabilistic model of PV generation is proposed without any assumption of theoretical parametric distribution. Combined with the proposed PV model, the PPF method is developed based on the Monte Carlo technique to evaluate the influence of the randomness and correlation of PV generation on the performance of distribution networks.
The historical power output data of three neighboring PV generators located in Oregon, USA, and 34bus/69bus radial distribution network are used to demonstrate the correctness, effectiveness, and application of the proposed PV model and PPF method. Based on the simulation results and the discussion in previous sections, the following conclusions can be drawn:
The randomness and correlation of PV generation can be captured by the proposed PV model with high accuracy.
The PPF method can efficiently calculate the impact of the randomness and correlation of PV generation on distribution networks in terms of bus voltage, line flows, and network loss.
The line flows and network loss are decreased since the PV generation is close to loads. Meanwhile, the fluctuations of bus voltage magnitudes/angles, line flows and network loss are increased with the connection of PV generation
The correlations of PV generation must be considered in the analysis of PPF for distribution networks. Neglecting or underestimating the correlations of PV generation has little impact on the mean values of bus voltage magnitudes/ angles and line flows but significantly influences the variance values or probability distributions of bus voltage magnitudes/angles, line flows, and network loss.
The correlations of PV generation must be considered in the analysis of PPF for distribution networks. Neglecting or underestimating the correlations of PV generation has little impact on the mean values of bus voltage magnitudes/ angles and line flows but significantly influences the variance values or probability distributions of bus voltage magnitudes/angles, line flows, and network loss.
The correlation of PV generation has an important impact on the PPF analysis for distribution networks with PV generation. Accurately modeling of PV generation correlations will become highly important as the PV generation penetration level increases in the future.
Acknowledgements
The authors are grateful for the support from the National Natural Science Foundation of China (51177178, 51307186) and The National High Technology Research and Development of China 863 Program (2011AA05A107).
BIO
Zhouyang Ren He is currently working towards the Ph.D. degree at the College of Electrical Engineering, Chongqing University, Chongqing, China. His research interests include probabilistic applications to power systems and optimal operation of Microgrids.
Wei Yan He received the Ph.D. degree in electrical engineering from Chongqing University, Chongqing, China, in 1999. Currently, he is a Professor in the Electrical Engineering College at Chongqing University. His research interests include optimal operation and control in power systems.
Xia Zhao She received the Ph.D. degree in electrical engineering from Chongqing University, Chongqing, China, in 2009. Currently, she is an associate professor in the Electrical Engineering College at Chongqing University. Her research interests include power system risk analysis, distributed generation modeling and analysis.
Xueqian Zhao He received Master's degree in electrical engineering from Chongqing University, Chongqing, China, in 2012. His research interests include the planning and operation of distribution systems.
Juan Yu She received the Ph.D. degree in electrical engineering from Chongqing University, Chongqing, China, in 2007. Currently, she is an associate professor in the Electrical Engineering College at Chongqing University. Her research interests include reactive optimal problem, static equivalent, voltage stability and risk assessment in power systems.
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