Impact of Wind Correlation and Load Correlation on Probabilistic Load Flow of Radial Distribution

Systems Sooraj Narayan K, Ashwani Kumar Department of Electrical Engineering

NIT Kurukshetra Haryana, India

soorajn14@gmail.com, ashwa_ks@yahoo.co.in

AbstractThis paper presents a probabilistic analysis of radial distribution systems considering correlated nodal load demands and correlated wind power sources. The loads and wind power sources are modeled as probability distribution functions. Cholesky factorization method is used to generate correlated nodal power samples and correlated wind power injection samples. Monte Carlo Simulation (MCS) is used for the Probabilistic Load Flow (PLF) procedure. A comparison is carried out for various scenarios of correlations. The power loss variation and voltage profile variation for the various scenarios are observed for the various scenarios. The results are analyzed for an IEEE 33 bus radial distribution system.

KeywordsProbabilistic Analysis, Radial Distribution Network, Correlated Nodal Load, Correlated Wind Power, Cholesky Factorization.

I. INTRODUCTION Utility deregulation and rise of competitive electricity

market has significantly scaled up the interest in Distributed Generation (DG) in the recent past [1]. The performance benefits that come with the integration of DG into the distribution system especially solar and wind energy sources has been well studied and analyzed [2]. The renewable distributed sources of power are both cost-effective and environmentally viable [3].

Uncertainty and intermittency of wind power source means that the prediction of wind speed and the power produced by the wind turbine can be implemented in a probabilistic manner [4]. Since the loads connected to a distribution system may also vary with time, weather and other factors, it can also be modeled as a probabilistic variable [5]. Correlation or dependence between wind power sources integrated at various nodes in a distribution system may be included in the analysis of the system for obtaining more realistic results [6]. Moments and Cornish-Fisher expansion were used in [7] for probabilistic load flow with correlated wind power. In [8], correlation between generation, wind power and loads were considered for the probabilistic load flow procedure. Uncertainty of wind power and load correlation was considered for probabilistic optimal load flow in [9]. In order to solve probabilistic load flow with load correlation using DC load flow, an analytical method was proposed in [10]. Hybrid Latin Hypercube Sampling along with Cholesky Decomposition was used in

[11] for probabilistic analysis and corresponding load flow evaluation.

This paper performs a comparison study to analyze the impacts of both load power correlation and wind power correlation on probabilistic load flow of radial distribution systems with integrated wind power sources. A comparison between four different cases of correlation is considered for performance studies.

The rest of the paper is as follows: Section II discusses the load power modeling, substation voltages modeling and wind power modeling. Section III discusses correlation and dependence between random variables and also the generation of correlated random numbers using Cholesky Factorization. Section IV deals with the computational procedure of the study. Section V deals with various case studies of load flow with correlated wind power and load power. An IEEE 33 bus radial distribution test system is used for study in this paper. A program was developed in MATLAB 7.1 the study. The program was run on an Intel Core(TM) i7-3770 3.70 GHz processor. The results are analyzed and discussed in detail.

II. PROBABILISTIC MODELLING

A. Modeling of Load The probabilistic nature of load at each node in a radial

distribution system can be implemented in the load flow studies by modeling the loads as random variables distributed with a variance about a mean value. In this paper, the load demands at each bus are assumed to be random variables with Normal distribution [12].

, , , (1) where , is the active load demand at bus number and , ,

are the mean and standard deviation values of each load power respectively.

B. Modeling of Substation Voltage The substation voltage can also be modeled, similar to the

load modeling, as a normally distributed random variable. [13].

(2)

where is the substation bus voltage and , are the mean and standard deviation values of substation voltage respectively.

C. Wind Power Modeling The real power output from a wind turbine is given by the

following equation [14]. 0, ,, 0, (3) where is the power output of wind turbine in MW, is the wind velocity in m/s, is the cut-in speed of the wind turbine in m/s, is the cut-out speed of the wind turbine in m/s, is the rated speed of the wind turbine in m/s, is the rated

power output of the wind turbine in MW, and .

The power produced by a wind turbine is dependent on the wind speed, which is intermittent. Hence, it can be modeled as a random variable. Weibull distribution function has been used to model wind speed in this paper. The Weibull distribution function is a two parameter function which is used to describe wind speed mathematically as:

,0 (4) where is the wind speed, is the shape parameter and is the scale parameter [12].

III. CORRELATION AND DEPENDENCE In probability theory, correlation factor () is a term that is

used to show the degree of dependence between two random variables. The value of may vary from -1 to +1. When two random variables have high correlation or high dependence, the absolute value of the correlation factor tends to be close to 1 [10]. Similarly, if there is no dependence between two variables, the absolute value of correlation factor tends to be close to 0. In this paper, the correlation is implemented by considering correlated load demands and correlated wind power injections.

A. Correlated Load Power Demands Various factors that may affect the nodal load demands in a

distribution system, like social or environmental, may cause the loads in a system to vary in a similar manner [8]. Generally, the variation in loads may be due to interchangeable reasons. Hence, a certain degree of dependence may be assumed between various load demands.

B. Correlated Wind Power Injections Since the distribution systems cover a comparatively lesser

area geographically, the various wind turbines connected to the system may be assumed to be part of the same wind farm. Hence, the inclusion of correlation between the power outputs produced by these wind turbines becomes an essentiality for load flow computation [7].

C. Generation of Correlated Random Numbers Cholesky Factorization method [9] is used to generate

correlated random numbers in this paper. It is used to decompose a symmetric positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. Let be a matrix containing non-correlated random numbers as its columns. Cholesky Factorization is done to obtain a lower triangular matrix such that:

(5)

where is the symmetric covariance matrix. The matrix of correlated random numbers as its columns ( can be obtained by the following transformation: (6)

IV. COMPUTATIONAL PROCEDURE Monte-Carlo Simulation is used to generate wind speed

samples from the Weibull parameters of the site. A large sample size of 10000 or 20000 is normally required for convergence of PLF using MCS. From the wind speed samples obtained, wind power samples are also obtained from Equation 3. The MCS is also used to generate normally distributed samples of nodal power demands with the respective mean and variance. A wind farm with suitable Weibull parameters is selected along with a wind turbine. The candidate nodes where the wind turbine is to be placed are the farthest nodes in the system. Then, the correlated wind speed and power samples, along with the correlated load power samples, are generated using Cholesky Factorization. Various scenarios are introduced and PLF is carried out for the required number of MCS samples in order to investigate the impacts of correlation on voltage profile and power losses.

V. SIMULATION CASE STUDIES AND RESULTS The studies were conducted on an IEEE 33 bus radial

distribution test system [14]. The base power of the system is 100 MVA and the base voltage is 12.66 KV. The total connected active power load is 3.72 MW and reactive power load is 2.30 MVAR. The wind turbines are placed at the candidate nodes 18, 25 and 33, since these nodes are the farthest nodes in the system and hence is bound to experience severe voltage deviations [15]. A wind farm having the Weibull parameters =1.75 and =8.78 [12] is selected for the study. The wind turbine selected has the parameters 3 m/s, =11.5 m/s, =20 m/s and =2.0 MW [16]. The same wind turbine is placed on all the three candidate nodes. For the PLF, the values of power demand at each bus are assumed to be their respective mean values. A standard deviation of 10% is set for every node. The mean and standard deviation of substation voltage is set to 1.0 pu and 1.5% respectively [13]. The sample size of MCS is taken as 15000. A correlation factor of 0.98 is taken for both wind power correlation and nodal load power correlation. Four cases are included in the study, apart from the base scenario. The base case is the scenario where no wind power is injected and basic PLF is carried out. The total active power losses in the system for the base case are 210.9824 MW.

A. Non-Correlated Load and Non-Correlated Wind Power (Case 1) In this case, the wind speed generated and the power

injections at nodes 18, 25 and 33 are completely independent, as shown in Fig. 1 and Fig. 2 respectively. The load power demands are also non-correlated, as shown in Fig. 3. The load flow is performed by subtracting wind power injections from their corresponding nodes for each MCS sample. The total real power loss is observed to be 183.6966 KW.

B. Correlated Load and Non-Correlated Wind Power(Case 2) In this case, the wind speed generated and the power

injections at nodes 18, 25 and 33 are completely independent, but the load power demands are correlated with =0.98, as shown in Fig. 4. The total real power loss is observed to be 183.106 KW.

Fig 1. Scatter diagram of non-correlated wind speed generated at the candidate nodes

Fig 2. Scatter diagram of non-correlated wind power output at the candidate nodes

Fig 3. Scatter diagram of non-correlated real power demand at the candidate nodes

Fig 4. Scatter diagram of correlated real power demand at the candidate nodes

C. Non-Correlated Load and Correlated Wind Power(Case 3) In this case, the wind speed generated and the power

injections at nodes 18, 25 and 33 are correlated with =0.98, as shown in Fig. 5 and Fig. 6 respectively. The load power demands are independent in this case. The total real power loss is observed to be 179.9314 KW.

D. Correlated Load and Correlated Wind Power (Case 4) In this case, the wind speed generated and the power

injections at nodes 18, 25 and 33 are correlated with =0.98. The load power demands are also correlated with =0.98. The total real power loss is observed to be 179.3219 KW.

Fig. 7 shows the comparison of voltage profiles for the four cases. Fig. 8 shows the comparison of total real power losses for the four cases. The real losses are seen to be minimum in Case 4 and maximum in Case 1. Fig. 9 shows the comparison of total reactive power losses for the four cases. The reactive losses are seen to be minimum in Case 3 and maximum in Case 2.

Fig 5. Scatter diagram of correlated wind speed generated at the candidate nodes

Fig 6. Scatter diagram of correlated wind power output at the candidate node

Fig 7. Comparison of voltage profile for various cases

Fig 8. Comparison of total real power loss for various ca

Fig 9. Comparison of total reactive power loss for variou

Fig. 10 shows the comparison of line reathe four cases. Similarly, Fig. 11 shows the creactive power losses for the four cases. FCumulative Distribution Function (CDF) of number 25 for the various cases. It is observCDF is inclined more towards unity in Caother cases. Hence the inclusion of correlatthe voltage CDF and reduced total active pow

Fig 10. Comparison of branch real power losses for vari

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VI. CONC This paper analyzes the iminjections and correlated load value of correlation factor on ascenarios are considered and thon a probabilistic perspective. Aand real power losses is carriedis observed to minimum whenload power demands are correof a distribution system is obimpacts of correlation are cointerpretation of the results canthe real power loss is observedwind samples and load powercorrelated. It shows that there ipower loss and the various possible within the system. conducted to investigate the whstudies on real power loss and system. A plausible applicatiothe implications of wind farmwhere correlation has to be takcollective change in load demanaccommodated in load flow stuThe load flow procedure is bresults if all the possible corconsideration. The study mconsidering correlation betwdemands. Consideration of oth

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CLUSIONS mpacts of correlated wind power

power demands for a specific a radial distribution system. Four he load flow study is conducted A comparison of voltage profile

d out. The total active power loss n both the wind speeds and the elated. The general performance bserved to be altered when the onsidered. The most important n be observed from the fact that d to be minimum when both the r injections are assumed to be is a relationship between the real

combinations of correlations Further studies have to be

holesome impacts of correlation voltage profile variations of the

on of this research is to analyze s in large distribution networks

ken into consideration. Also, the nds due to various factors can be udies using correlation analysis.

bound to provide more realistic rrelation factors are taken into

may be further extended by ween wind power and load her renewable sources like solar

17 19 21 23 25 27 29 31ch number

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and biomass in the load flow study is another possible adjunction to the research.

REFERENCES [1] R.C. Dugan and T.E. Mcdermont, Distributed generation, IEEE

Ind.Applicat.Mag.,pp.19-25, Mar./Apr. 2002. [2] L. F. Ochoa, A. Padilha-Feltrin, and G. P. Harrison, Evaluating

distributed generation impacts with a multiobjective index, IEEE Transaction Power Delivery., vol. 21, no. 3, pp. 14521458, July 2006.

[3] Kejun Qian, Chengke Zhou, Yue Yuan, Xiaodan Shi and Allan. M., "Analysis of the environmental benefits of Distributed Generation," Power and Energy Society General Meeting - Conversion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE , vol., no., pp. 1-5, 20-24 July 2008.

[4] Morteza Aien, Reza Ramezani and S. Mohsen Ghavami, Probabilistic load flow considering wind generation uncertainty, Engineering,Technology & Applied Science Research., vol 1,pp. 126-132, 2011.

[5] Chen P, Chen Z and Bak-Jensen B, Probabilistic load flow: a review Third International Conference on Electric utility deregulation and restructuring and power technologies, pp. 158691, April 2008.

[6] Xue Li, Jia Cao and Pan Lu, "Probabilistic load flow computation in power system including wind farms with correlated parameters," Renewable Power Generation Conference (RPG 2013), 2nd IET , vol., no., pp. 1-4, 9-11 Sept. 2013.

[7] Julio Usaola, Probabilistic load flow with correlated wind power injections, Int. J. Electr. Power Energy Syst., vol. 80, pp. 528-536, May 2010.

[8] D. Villanueva, A. Feijo and J. Pazos, "Probabilistic Load Flow Considering Correlation between Generation, Loads and Wind Power," Smart Grid and Renewable Energy, vol. 2, no. 1,pp. 12-20, February 2011.

[9] Xue Li, Jia Cao and Dajun Du, Probabilistic optimal power flow for power systems considering wind uncertainty and load correlation, Int. J. Neurocomputing., vol. 148, pp. 240-247, Januray 2015.

[10] Daniel Villanueva, Andres. E. Feijoo and Jose. L. Pazos, An analytical method to solve the probabilistic load flow considering load demand correlation using the DC load flow, Int. J. Electr. Power Energy Syst., vol. 110, pp. 1-8, May 2014.

[11] Yu.H, Chung. C.Y., Wong. K.P., Lee. H.W. and Zhang, J.H., "Probabilistic Load Flow Evaluation With Hybrid Latin Hypercube Sampling and Cholesky Decomposition," IEEE Transactions on Power Systems, vol.24, no.2, pp. 661-667, May 2009

[12] Alireza Soroudi, Morteza Alen and Mehdi Ehsan, A probabilistic modeling of photo voltaic modules and wind power generation impact on distribtuion networks, IEEE Systems Journal, vol. 6, no. 2, pp. 254-259, June 2012

[13] Vichakorn Hengsritawat, Thavatchai Tayjasanant and Natthaphob Nimpitiwan, Optimal sizing of photovoltaic distributed generators in a distribution system with consideration of solar radiation and harmonic distortion, Int. J. Electr. Power Energy Syst., vol. 39, pp. 36-47, July 2012.

[14] Rakesh Ranjan and Das, Simple and Efficient Computer Algorithm to Solve Radial Distribution Networks, Electric Power Components and Systems, 31:1, pp. 95-107, June 2010

[15] Emingolu U. and Hocaoglu M.H., A voltage stability index for radial distribution networks, Universities Power Engineering Conference, vol., no., pp.408-413, Sept. 2007.

[16] VestasProductsandServices,http://www.vestas.com/en/products_and_services/turbines/v110-2_0_mw#!at-a-glance.html

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