Probabilistic Distribution Power Flow Based on Finite Smoothing of Data Samples Considering Plug-in Hybrid Electric Vehicles

Probabilistic Distribution Power Flow Based on Finite Smoothing of Data Samples Consideri ng Plug- in Hybrid Electric V ehicles Mohamm adhadi Rouhani 1 Mohammad Mohamm adi 1 1 School of Electrical and Computer Engin eering, Shiraz University, Sh iraz, Iran E- mai l: mh_rouhani@s hirazu .ac.ir , m. mohammadi@shirazu.ac.ir Corresp onding Aut hor Info rmation : Name: M. Moh ammadi Address : School of Electrical and Computer Engin eering, Shiraz University, Shiraz, Iran Tel: (+98) 7116133278 Fax: (+98 ) 7116133278 E- mail: m. mohammadi@shirazu.ac.ir Abstract: The ever increasing penetration of plug - in hybrid electric ve hicles in dis tribution sy stems has triggered the need for a more accurate and at the sa me time fast solution to probabilistic distr ibution power flow problem. In this paper a novel algorithm is introduced based on f inite sample points to determine probabilistic density function of probabilistic distribution power flow results. A modified probabilistic charging behavior of plug - in hybrid electric vehicles at ch ar ging stations and their overlap with residential peak load is evaluated in probabilistic distribution power flow problem. The proposed algorithm is faster than Monte Carlo Sim ulation and at the same tim e keeps adequate accuracy. It is applied to solve prob abilistic distr ibution power flow for two dimensionally different test systems and is compared with recent probabilistic solutions. Simulation re sults show the accuracy and efficiency of the proposed algorithm to calculate probability density function of uncertain outputs. 1 Index Terms - Data samples, distribution power flow, electric vehicles charging station, load uncertainty, plug - in hybrid electric vehic le, probab ility density f unction, prob abilistic distribution power flow . 1. Introduction A. Motivation & proposed algorithm In the near future, t he emergence of smart grid high - lights the impact of various uncertainties on operation and planning. D istribution sy stems as the infras tructures of micro grids nowadays encounters many integrated uncertainties. Penetration of Electric Vehicl es (EV)s, for instance is the ever increasing concern among smart grid planners. Since their charging behavior is intrinsical ly stochastic, it would in fluence the operating point of power generators either at distribution or t ransmission level. Concept of Probabilistic Power Flow (PPF) has been a grea t challenge since 1970s [1] . It was initially introduced to provide a sol ution to power systems probabilistic analysis where loads an d branch flows vary over time and deterministic power flow is not able to handle all these uncertainties in its fra mework. Then numerical and several analytical m ethods have been introduced to solve PPF problems. Analytical methods, such as, convolution technique, fast Fourier transform, and Cumulants provide Probab ility Density F unction (P DF) and/or stat istica l moments with the expen se of simplifications and mathematical assumptions in their PPF equations [3], [4]. These methods are applicable to cas es where inpu t random variables fol low normal distrib ution s, otherwise it is very time - consuming to calculate PPF outputs using traditional analytical techniques. Monte Carl o Simulation (MCS), a numerical method, is the most accurate method to provide PDF for probabilistic analysis results. This method, howe ver deals with a grea t 2 number of random samples that lengthens the sim ulation process and occupies large quantity of memory [5]. Point Estimate Methods (PEM)s have been i ntroduced as another probabilistic analysis method to PPF problems. They are based on probability of some points to determine statistical character istics of PPF resul ts. These methods are rapid and functional to problem s where few number of uncertainties exis t increases. Moreover, PEM do not provide PDF of PPF results which restricts its ap plication in many system planning and operation [6], [7]. A distribution system is generally fed at one node and is radial ly expanded in a tree shape st ructure to feed residential loads. Distribution P ower Flow (DPF) calculates steady state value of node v olta ges and branch flows. Distribution systems du e to their ill - conditioned structure are m ore prone to loa d variations and consequently more frequent steady state changes over time [8]. U ncertain presence of EVs, and unpredicted behavior of residential lo ads bring many complexities to distribu tion systems analysis. Probabilis tic Distribution Power Flow (PDPF) is alloca ted to characterize th ese uncertainties exist in distribution systems. Most of the probabilistic techniques used for PPF problem are applica ble for PDPF yet for some techniqu es, such as, PE Ms and analytical m ethods parametric approaches ar e applied to model uncertain ties in the system. In other words, all input random variables are modeled by standard distributions. Nevertheless, some of the s tochastic variables in distribution systems, such as, charging behavio r of EVs are dependent on several nonparametric factors and it is cost - inefficient to characterize all th e uncertain features with standard distributions. EVs have been nominated as efficient alternatives to fossil fuel vehicl es to reduce carbon emission. Among different types of EVs, Plug - in Hybrid Elec tric Vehicl es (PHEV)s are more economic since technology has constrained batter ies to a limited charge capacity an d lifespan. PHEVs c omprises of a drive tr ain that contains an internal combustion engine, an electrical motor, a battery storage system [9]. Charging behavior of PHEV s is probabilist ic from 3 many perspectives; the fl eet of PHEVs at the station, charging v oltage and curren t level, sta te of charge, battery c apacity, charging time duration etc. Their impact on dist ribution systems is remarkable when hundreds of EVs in the future enter chargi ng stations throughout urban places. Therefore, there is a demand to analyze this kind of behavior, especially when this probabilistic charging process coincides with the peak - demand period [10]. This could cause volta ge stabil ity problems and reduce distribution systems reliabil ity. In this pa per a new algorithm based on finite data samplings of PDPF input random variables is proposed to determine PDF of the results. F inite Smoothing of Data Samples (FSDS) algorithm does not require mathematical assumptions and simplificatio ns. It is a pplicable to probabilis tic problem s where no standard information is available on the distribution of input random variables. In other words, the proposed algorithm is a distributio n - free technique. Hence one can readily use the proposed algorithm to eval uate the probabilistic impact of correlated PHEVs charging be havior on distribution system during peak - demand periods. The proposed algorithm im plements Forward/Backward Sweep (FBS) power flow for each input data sample and provides PDF of PDPF node voltages. The FBS dist ribution power flow is an efficient method to determine voltages and branch flows of distribution systems [8]. B. Literature review Many literatures have investig ated PPF in power system. PPF methodologies classified into three methods along with their characteristics a re tabulated in Table I [12] - [1 6]. Table I A n umerical approach based on MCS is investigated in [17]. Ahmed et al. [18] investigated different wind turbine models in a PDPF study. A PPF using PEM is investigated in [12] to 4 investigate uncertain behavior of distributed generations in the future smart grid. PPF modeling and interactions of renewable energy and Plug - in EVs to power grid is described in [13]. PPF calculation considering probabilistic charging demand of PHEVs is presented in [19] . Many literatures have investig ated PHE V modeling and their impact on power grid. In [20] the impact of PHEVs on distribution systems was assesse d. Tan et al. [ 21] described differen t aspects of PHEVs’ imp act on distr ibution syst ems. In [ 22] a new stochastic m odel for PHEV utili zation was prese nted. Sharma et al. [23] proposed a model of P EV charging in an unbalanced, distribution system and a smart distr ibution power flow was also presented for smart charging of PEVs. A 24 - hour travel of a plug - in electric vehicle between several points during a trip in a city is modeled in [24]. Load uncertainties is another problem that should be included in PPF studies. Since the generating schedule of generator are based on the demand, their modeling and behavior is important in operation and planning [25]. C. Paper content The remainder of this paper is as fo llows: Section 2 briefly introduces concept of PDPF using FBS power flow and describes the problem exists in dis tribution networks with the im pact of uncertain PHEVs and residential lo ads. The proposed a lgorithm is explained in detail in this section. Section 3 provi des two different case studies consid ering several uncertain parameters with the presence of PHEVs. Simulation results of various probabilistic solutions are displayed and discussed in this se ction. Section 4 summarizes the m ain concluding points r emarked in this paper. 2. P robabilisti c D istributi on P ower Flow A. Overview 5 Power flow study is done to calcu late steady state operatin g point of generators to equalize generations and consumptions. In ra dial distribu tion sys tems, node vol tages vary remarkably with respect to different values of residential loads. Among the various DPF methods, FBS [8] is rapid and provides re latively acceptab le results. PPF has been introduced to deal with uncertainties exist in power grid due to numerous factors, such as branch outages, extreme weather conditions, consumer behaviors [3]. These uncertainties convert determ inistic parameters of p ower flow equations in to probabilis tic vari ables. Henc e statistical methods should be considered for this type of problem. In a distribution system impact of PHEVs’ charging demand is a big challenge and needs to be considered in future operation and planning. Therefore, PDPF can be applied to deal with the se uncertai nties exist a nd will soon emerge in distrib ution systems. S imilar to PPF analysis, in PDPF stud y the statistical formulation is based on DPF eq uations, except that all th e input and/or system data parameters are co nsidered to be probabilistic. It is con venient to rea r range DPF equations such t hat the outputs be function of the inputs. For a distribution sy stem, the input s are (re)active load s, voltage vecto r at root node, and distribution line data. The outputs are node voltages and power line flows. A rearranged form of DPF equations is as follows:  =  (  ) ( 1 ) Y follows pr obabilistic behavior if only U is considered to be a random vector. The purpose of PDPF study is to characterize unc ertain behavior of output vectors with respect to available statistical in formation of input vecto rs. B. Problem defini tion The main problem with the recent proposed P(D)PF solutions is their inability to provide accurate output results w hen parameters become severely uncerta in that cannot be modeled appr opriately by standard distributions. On the other hand, Point Estim ation Method (PEM) is not 6 applicable when the number of input random variables are extremely high [6], [7] o r Cumulants and approximation expansions are efficient only for inputs with norma l di stributions [ 4]. Another problem is the uncertain PHEVs’ charging impact on radial systems, especially du ring the peak - demand interval. No accurate EV charging modeling is available due to customer behavior, grid, weather condition uncertainties [19]. Residential loads are also another sources of unc ertainty in distribution grid. Loads behavior is not precise and needs to be statistically analyzed. Unpredictab le residentia l loads have floating peak value that may change over time. Charging periods of PH EVs intersect with the peak load many tim es. Therefore, it is important to consider probabilistic behavior of uncertain loads in DPF problem. C. Probabilistic modeling 1) Probabilistic PHEV’s charging demand modeling: As aforementioned seve ral uncertain pa rameters exist to analyze charging behavior of PHEVs at charging stati ons. The energy consumption probabilistic model for PHEVs is derived from [19]. The operating st atus of a PHEV is described by the fraction of th e total powe r input to the drive tra in su pplied by th e batt ery. Battery capacity is an other key param eter of a PHEV. A correlation exists among the battery capacity and operating status of a PHEV. State of charge is th e percentage of energy remained in a battery when a PHEV arrives to a chargi ng station. The energy consumption per mile has a relation with the opera ting status o f the PHE V and is formulize d with respe ct to the P HEV type. The daily driven m iles of a PHEV which can be deri ved using a lognormal distribution. Hence the daily recharge of a single PHEV can be evaluat ed considering the daily driven mileage, state of charge, a nd energy consum ption per mile. For fleet of PHEVs plugged at diffe rent charging stations a queuing algorithm is defined. In this algorithm e ach custom er arrives to a c harging station at a specific time and ch arges its vehicle d uring a random interval. Depending on the total 7 capacity of each chargin g station, the total number of PHE Vs being charged at the sam e time can be evaluated employing queuing theory. Since only th ree c harging power levels exists f or charging PHEVs, considering voltage and charging current level of the station, the power consumed by substantial PHEVs charging at the same time can be evaluated. 2) Probabilistic residential load modeling: Loads natu rally vary randomly. Load prediction in short- term analysis has been a great challenge [26]. Load forecast uncert ainty has a considerable impact on the reliab ility assessm ent in a generating plan. Loads uncertainty can also influence capacity reserve o f generations [27]. Several works have been done investigating uncertain load modeling in power grids. Load uncertainty can be described by probability distribution whose parameter can be obtained from historical data. It is frequent to model loads by a no rmal distribution [25]. D. Solution meth od Nonparametric methods can be used to determine probability density of random variables i n a probabilistic system where no i nformation is available on inputs’ distribution. Application of nonparametric algorithms h as been investigated in [28] to quantify uncertainties associated wit h wind power. The proposed a lgorithm in this paper is a nonparametric based method that are used to estimate PDF of probabil istic output random variables [29]. 1) Finite Smoothing of Data Samples density estimation, mathemati cal definition & formulation: The objective is to provide an approximate PDF for a set of random samples whose distribution is unknown. For this purpose  is defined and centered on a test point. The probability that k points out of total random samples k n fall within  is determined assuming that f(x) be flat inside  . This is determined as follows:  [    |   ] =   󰆹 (  )  dτ   󰆹 (  ) .   ( 2 ) 8 Since  󰆹 (  ) is unknown, (2) is rearranged to estimate  󰆹 (  ) :  󰆹 (  )   [    |   ] . 1   ( 3 ) Random samples may either fall into  or out of it. Then the probability that a vector of random samples X (kn× 1) calculated from P(x) fall within  is obtained as f ollows:  =  [    ] =   (  )   ( 4 ) It should be noted that P(x) follows either a standard distribution or a dataset with unknown distribution. The probability t hat k points out of total points k n fall in < can be given by binomial distribution:  [  =  ] =        (1   )    ( 5 ) In order to estimate the density function of a random vari able, two assum ptions need to be taken into account: • Maximum likelihood estimation for probability α is    . • P(x) is approximately f lat inside reg ion  . Considering the maximum probability for α , P(x) at point x can be estim ated as follows:  (  )       ( 6 ) It is clear fr om (6) that the accuracy of P(x) relies on two computationa l parameters: • Total number of random samples k n . • V  volume of  . Steps of the proposed algorithm to estimate PDF of random variable x are described as a flowchart in Fig. 1. Fig. 1 As shown in this flowchart, a region  is considered around each test point and a random vector 9 X with size D is considered (S tep 1).  is centered on test point x and finite samp le data fall in  is evaluated (Step 2). To formulate attribution of each random sample a window function is defined to  (Step 3). k is calcul ated in (Step 4). PDF for test point x is determined by averaging the finite sample data f all within  in (Step 5). It has been shown that the value of λ influences the accuracy of the proposed algorith m; a n excessive value of λ can smooth out the structure of estimated PDF and in contrast, a sm all a mount produces an irregular spiky PDF [30]. In the latter, the procedure to evaluate optimized value for computational param eters is described. 2) Computational parameter tuning for the proposed algorithm : To catch an optimized value for λ , two indices are defined: • Mean Integrated Squared Error (MISE) which is com monly used in density estimation to evaluate the accuracy of the estimated PDF . MISE is defined as follo ws:    (  ;   ) =   [    ;  󰆹    (  )]   ( 7 ) • Maximum Probable Point Tracking (MPPT) is introduced i n this paper to evaluate PDF estimation by comparing points with maximum probable point. It is implem ented in the tuning process to track the most probable point of a random variable. MPPT is defined as follows:     = |         | ( 8 ) These two indices are combined in M C to evaluate the effect on λ tuning. MC value is defined as foll ows:   =     +     ( 9 ) a and b are effective co efficients represent ing the impact of each index on λ . For a specific random variable, optimized λ corresponds to minimum value of M C . In order to enhance runtime and accuracy of PPF calculation at the same time, number of random 10 samples should be optimally chosen. Therefore, a convergence coefficient is def ined to determine optimized number of random samples k n that meets the convergen ce coefficient. E. Probabilistic distribution power flow using the proposed algorithm Input parameters for PDPF study (slack bus voltage vector, (re)active power at load buses, branch data ... ) are set and an optimized number of sample points are initialized. Considering (1), PDPF outputs are calculated using FBS power flow for each set of random samp les. The detailed procedure for PDF e stimation in PDPF problem using the proposed algorithm is represented in Fig. 2. Fig. 2 In this flowchart, the number of random samples are set (k n ). For each set of random sam ples a FBS distribution power flow is calculated. Considering a specific te st point x , the value of λ is optimized. Then PDF at the test point is calculated considering attribution of each sample point to the  through a suitable window function. Output PDF and probability moments are then calculated. 3. Case studies The proposed algorithm is executed for modified 34 and 123- node IEEE test cases. Th e data of branches and nodes and their configurations are derived from [31]. The model used for PHE Vs charging at an EV charging station is derived from [19]. A datase t of random samples are derived fro m PHE Vs charging behavior PDFs as inputs to the PDPF problem . It is taken in to account that uncertain residenti al loads are established in all distribution nodes of case stud ies. These lo ads along with PHEVs probabili stic charging behavior are considered for a peak - demand period (18:00 11 PM). For simulati on evaluation purposes, MCS with 5000 iterations is considered as th e most accurate base reference . In order to evaluate the efficiency of the proposed algorithm, it is compared with TPEM and UT [32]. TPEM implements 2n points where n is the number of uncertain variables and provides statistical moments. UT uses 2 n + 1 sigma points and ca lculates means and covariances o f random variables. Several indices fo r PDPF output re sults of v arious solutions are applied. Using MCS, histogram, and the proposed al gorithm, the PDF fo r selected results are determined . The effectiven ess of the pr oposed algorithm is characterized by comparing relative erro r of the statistica l characteris tic value o f the output results. The relative erro r is defined as follows:   = |       | ( 10 ) For further com parison of techniques, average, minimum, and maximum relative indices of statistical moments are c alculated. They can be form ulated as follows:    = 1         ( 11 )    = min {    ,    , … ,    } ( 12 )    = max {    ,    , … ,    } ( 13 ) A. 34- node IEEE test system 1) System description: Three charging stations are included in the test system. T he locations of charging stations are ch osen arbitrarily; a level -1 charging station at node 5 and 28 each consisting of one station at each sid e of the road and a level -3 charging station at node 15 that includes two stations at each side of the road and is available only 45% of times due to maintenance. Expected values o f (re)active power of balanced three phase residential loads are derived from [33] and value of Standard Deviations (STD) are considered to be 5% of the expected values. 2) Discussion & result s: Computational parameters of FSDS density estimation λ , k n are 12 evaluated by using the procedure described in section II for the voltage at node 15. Figs. 3 and 4 illustrate M C with respect to λ and STD sequence of index M C for different values of k n respectively. Coefficients in (9) are a = 1, b = 0.05. Table II shows selected values of computational indices with respect to λ . Table II For MPPT optimum = 0.012 the output PDF calculated is distorted fro m the actual PDF, while for MISE optimum = 0.754 the output PDF becomes over - smoothed. Hence a combination of both indices (M C ) can provide a m ore accurate PDF ( λ optimum = 0.00285, M Cmin = 0.8434). A large number of random samples can also be considered to increase the accuracy of the p roposed algorithm yet this is time - consuming and inefficient. Thus for this test cas e it is assumed that k n = 45, as shown in Fig. 4 provides efficient accuracy and runtime to estim ate output PDFs using t he pr oposed algorithm. Fig. 4 PDF of voltage at node 15 is illustrated in Fig. 5 for different techniques. Fig. 5 The statistical charact eristics for the voltage s at charging station s are tabulated in Table III. Table III Note that th e UT method provides only expected values and covariance matrix of output resul ts. Average, minimum, and maximum error indices of voltages in the 34 - node IEEE test system are depicted in Table IV. Figs. 6 and 7 show expected and STD values of nodes res pectively using various methods. Fig. 6 13 Fig. 7 Fig 5 indicates that the proposed algorithm accurately tracks the ac tual PDF, especially as the voltage at node 15, due to m ulti - modal behavior of EV charging stations at this node, has different values. The e rror indices in T able III show that statistic al characteri stics evaluat ed by FSDS algorithm is closer to the actual values with respect to other probabilistic soluti ons. Table IV shows that moments calculated by the proposed m ethod for all voltages are more precise compared to TPEM solution. Table IV Note that the com putation elapsed times for MCS, FSDS, UT, and TP EM are 8.45, 0.144, 0.1452, 0.11 seconds respectively. B. 123- node test system A detailed explanation on the derivation of this test case is described in Appendix A. This test case is divided in to six regions. Level - 3 charging stations are located at nodes 33, and 104 at each node two EV charging stations are located at both sides of the road. Level - 1 charging stations are placed at nodes 4, 55, 77, and 116 at each node an EV charging station is placed at both sides of the road. The models used for t hese charging stations are s imilar to the previous test case . Al l residential loads are assum ed to be balanced and follow normal distribution with expected values as base data and ST D is considered to be 10% of the expected values. An i ndustrial loa d described in [31] at node 34 is m odified and implemented in the tes t case. It is o perative only 45% of tim es. 1) Discussion & results: Optimized value for computational parame ters for this test case is also calculated. For the voltage at node 34 (due to its high uncertainty) λ op ti mum = 0. 0003 and k n = 400. Hence PDF of the voltage at node 34 is shown in Fig. 8. Fig. 8 14 Statistical character istics of the PDPF sol utions are calculated for voltages a t charging stations and are depicted in Tabl e V. Table V Statistical moments o f PDPF outputs are calculated using diff erent probabi listic so lutions. Average of error indices for all voltages in the 123 - node IEEE test system are presented in Tabl e VI. Table VI Expected and STD values of 123 nodes in this test system are illustrated in Figs. 9 and 1 0 respectively . Fig. 9 Fig. 10 PDF of the voltage at node 34 shown in Fig. 8 indicates that for any uncertainty in the system, the proposed algorithm can approximately provide accurate results. Statistical charac teristics and moments in Tabl e V and VI respectively prove that FSDS is m ore accurate and effici ent compared to other probabilistic solutions. The elapsed computation times for the four probabilistic solutions as reported for the previous case study are 17.6248, 1.8281, 0.731, 0.6863 seconds re spectively. 4. Conclusion This paper proposes a new algorithm based on nonparametric methods to solve probabilistic distribution power flow problem based on finite smoothing of data samples. The proposed algorithm can be used where no information is available on the probabilis tic chara cteristics of input random variables. Plug - in hybrid electric vehicles charging behavior duri ng peak -demand interval alon g with unc ertain resid ential loads are investigated in case studies. Modif ied 34 and 123- node I EEE test cases are exam ined and the efficiency of the proposed algorithm is validated 15 by comparing it with two point estimation method, unscented transform, and Monte Carlo simulation. Backw ard/forward sweep algori thm is used to com pute distribution power flow problem. The following concluding rem arks are drawn from the simu lation results: 1) The proposed algorithm is faster than Monte Carlo Simulation and some other algorithms. 2) It provides accurate output probability density function with respect to Monte Carlo simulation and histogram. 3) The probability moments calculated using the proposed algorithm are more accurate than those computed using other techniques. 4) It can be used for any type of systems with any uncertainties. This paper exclusively focuses on p robabilistic distribution power flow . With the be neficial ch aracteristics of the proposed algorithm, it can be applied to other probabilistic power system analysis concepts. 5. Appendix A - Modi fied IEEE 123 - node test case derivation Branch numbering is considered from the root node to the rest of the radial network. The conductors of this test case are A CSR type and the resistanc e is 1.120 (ohm/mi), the diameter is 0.398 (in), GMR is 0.00446 (ft.). The overhead line spacing ID is assumed to be 500 [31]. L in e data of this distribution system is derived from Table 4 of [31]. In this model all residential loads are assumed to be balanc ed on three phases. 16 R EFERENCES [1] P. C hen, Z. Chen, a nd B. B ak - Jensen “Probabilistic load flow: A review,” in Proc. 3rd Int. Conf. Electr. Utility Deregulation Restructuring Power Techno l., pp. 1586 - 1591, Nanjing, China , Apr. 2008. [2] Y. K. T ung, and B . C. Ye n, “Hydros ystem s Engineerin g Uncertai nty Analys is,” McGra w - Hill, New Yo rk, 2005. [3] G. J. Anders, “P robabili ty concepts i n elec tric power s ystem s,” 1989. [4] M. Fan, V. Vittal, G. T. Heydt, and R. 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Raoof at “Optimal stochastic reactive pow er scheduling in a microg rid considering vo ltage droop scheme of DG s and uncertainty of w ind farms,” Energy, vo l. 45, no. 1, pp. 994 - 1006, 2 012. 19 Table Captions: Table I: Properties of V arious Probabilistic D istribution Power Flow Methodologies Table II: Val ue of MISE, MPPT and M C with respect to λ for volt age at node 15 for IEEE 34 - node test system Table III: Statistical chara cteristics of voltages at IEEE 34 - node t est sy stem Table IV: Average of error indices for statistical moments of all voltages in the 34 - node IEEE test system Table V: Statistic al characte ristics o f voltages at IEE E 123 - node test system 20 Table I Properties of V arious Probabilistic D istribution Power Flow Methodologies PDPF Methodology 1 2 3 4 5 6 7 *MCS high (-) high (-) yes (+) yes (+) yes (+) no (+) no (+) **Convolution high (-) high (-) no (-) yes (+) yes (+) yes ( -) yes ( -) **FFT high (-) high (-) no (-) yes (+) yes (+) yes ( -) yes ( -) **Cumulants and Gram-Charlier low (+) low (+) × yes (+) yes (+) yes ( -) no (+) ***PEMs low (+) low (+ ) × no (-) yes (+) yes ( -) no (+) ***UT low (+) low (+) yes (+) no (-) yes (+) yes ( -) no (+) *FSDS low (+) low (+ ) yes (+) yes (+) yes (+) no (+) no (+) * Numerical methods × Varies by method (+/-) 1. Computatio n time 2. Memory occupation 3. A pplicable for multivariate paramete rs ** Analytical methods 4. Provides PDF 5. Provides statistical moments *** Approximation methods 6. Dependent on input distribution functions 7. Mathematical assumptions and si mplifications Table II Value of MI SE, MPPT and M C with respect to λ for volt age at node 15 for IEEE 34 - no de test sy stem Values: 1 2 3 4 5 6 7 8 9 λ 1.805 2.637 2.749 2.813 2.853 3.205 3.357 3.533 4.005 ×   MISE 1.534 0.882 0.846 0.830 0.819 0.767 0.758 0.754 0.773 MPPT 7.017 0.583 0.012 0.331 0.478 1.886 2.398 2.932 4.129 M C 1.885 0.911 0.846 0.843 0.841 0.861 0.878 0.901 0.979 Table III Statistic al cha racteri stics o f voltage at nod e 15 of IEEE 34 - node test s ystem Method Mean ε μ ( %) STD ε STD ( %) Skewness ε Skew. (%) Chargin g type 1 Node 5 MCS 0.9882 - 0.004 - - 1.132 - TPE 0.9725 1.58 0.004 10.94 - 2.33 105.63 FSDS 0.9879 0.03 0.004 20.52 - 1.46 28.91 UT 0.9725 1.585 0.0007 84.96 × × Node 28 MCS 0.9481 - 0.018 - - 1.073 - TPEM 0.8655 8.706 0.016 9.532 - 2.18 103.51 FSDS 0.9462 0.203 0.019 6.4 - 1.48 37.74 UT 0.8655 8.7 0.005 73.78 × × Chargin g type 3 Node 15 MCS 0.9624 - 0.0172 - - 1.2151 - TPEM 0.9048 5.985 0.0152 11.84 - 2.5708 111.58 FSDS 0.9609 0.158 0.0182 5.63 - 1.4754 21.42 UT 0.9048 5.985 0.0027 84.57 × × 21 Table IV Average of err or indice s for statisti cal mo ments o f all v oltages i n the 34 - node IEEE test system Error indices (%) FSDS TPEM     0.0121 0.5302    0.4612 18.2353     0.5970 23.9296     0.0162 0.7064    0.6093 23.4291     0.7884 30.5604     0.0202 0.8822    0.7551 28.2373     0.9766 36.6145 Table V Statistic al cha racteri stics o f voltage at nod e 34 of IEEE 123 - node test system Method Mean ε μ ( %) STD ε STD ( %) Skewness ε Skew. (%) Chargin g type 1 Node 4 MCS 0.9927 - 0.0001 - - 0.0779 - TPE 0.9953 0.262 0.0001 14 - 0.148 89.8 FSDS 0.9927 0 0.0001 2.05 - 0.08 2.503 UT 0.9953 0.262 0.00004 59.5 × × Node 55 MCS 0.969 - 0.0006 - - 0.134 - TPEM 0.9804 1.09 0.0006 3.08 - 0.223 66.41 FSDS 0.969 0.002 0.0006 2.15 - 0.174 29.69 UT 0.9804 1.09 0.0003 48.72 × × Node 77 MCS 0.9334 - 0.001 - - 0.1754 - TPEM 0.957 2.574 0.0007 28.94 - 0.228 30.13 FSDS 0.9334 0.003 0.001 2.08 - 0.16 9.18 UT 0.957 2.574 0.0002 84.38 × × Node 116 MCS 0.9321 - 0.001 - - 0.2266 - TPEM 0.9563 2.6 0.0008 17.16 - 0.538 137.18 FSDS 0.9321 0.001 0.001 3.15 - 0.22 2.97 UT 0.9563 2.6 0.0001 85.49 × × Chargin g type 3 Node 33 MCS 0.968 - 0.001 - - 0.1784 - TPEM 0.979 1.155 0.0008 5.11 - 0.196 9.63 FSDS 0.967 0.006 0.001 3.1 - 0.173 2.85 UT 0.979 1.15 0.0005 41.01 × × Node 104 MCS 0.9302 - 0.001 - - 0.354 - TPEM 0.955 2.705 0.0008 35.11 - 0.297 15.97 FSDS 0.9302 0.141 0.001 1.32 - 0.317 10.43 UT 0.955 2.705 0.0001 87.23 × × 22 Table VI Average of error indices for statist ical moments of all voltages in the IEEE 34 - node test s ystem Error indices (%) FSDS TPEM     0. 0006 0.7291    0.0064 5.4555     0.0216 8.6908     0.0008 0.9734    0.0085 7.3523     0.0288 11.7522     0.001 1.2182    0.0107 9.2897     0.0359 14.8997 23 Figures Captions: Fig. 1: Steps of the proposed algorithm for PDF e stimati on of a rando m variab le Fig. 2: Flow chart f or prob abilis tic distr ibution power f low usi ng the propose d algori thm Fig. 3: Index M C with resp ect to diffe rent val ues of for t he volta ge at node 15 Fig. 4: STD sequenc e of M C for different number of random samples k n Fig. 5: PDF of the v oltage (p .u.) at no de 15, IEEE 34 - node test syste m Fig. 6: Expected value of voltages ( p.u.) in IEEE 34 - node test system Fig. 7: ST D of voltages (p.u.) of IEEE 34 - node test sy stem Fig. 8: PDF of volt age (p.u. ) at no de 34 of I EEE 12 3 - node tes t system Fig. 9: Expect ed value of voltages (p.u. ) in 123 - node IEEE test system Fig. 10: STD o f voltages (p.u .) in 123 - node IEEE test system 24  , a D - dime nsional hypercube.   =   for X    x                 = 󰇫 1, |     |    ,  = 1, 2, … ,  0,    =   (      )       (  ) = 1    1    (      )     Step 1 Step 2 Step 3 Step 4 Step 5 Fig. 1 Steps of the pr oposed al gorithm for PDF estim ation of a random variabl e 25 Fig. 2 Flowchar t for pr obabil istic di stribu tion pow er flow using the prop osed al gorithm 26 Fig. 3 Index M C with resp ect to diffe rent val ues of for t he volta ge at node 15 Fig. 4 STD sequence of M C for different number of random samples k n 27 Fig. 5 PDF of the voltage (p .u.) at n ode 15, I EEE 34 - node test syste m Fig. 6 Expecte d value of volt ages (p.u .) in I EEE 34 - node test system 28 Fig. 7 STD of voltages (p.u.) of IEEE 34 - node test syste m Fig. 8 PDF of volta ge (p.u. ) at node 34 of I EEE 123 - no de test s ystem 29 Fig. 9 Expecte d value of vol tages (p.u .) in 123 - node IEEE test system Fig. 10 STD of voltages (p. u.) in 123 - node IEEE test system 30