A robust data clustering method for probabilistic load flow in wind integrated radial distribution networks

Electrical Power and Energy Systems 115 (2020) 105392

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

A robust data clustering method for probabilistic load flow in wind integrated radial distribution networks

T

⁎

Omid Sadeghiana, Arman Oshnoeib, Morteza Kheradmandib, , Rahmat Khezric, Behnam Mohammadi-Ivatlooa a

Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Department of Electrical Engineering, Shahid Beheshti University, Tehran, Iran c College of Science and Engineering, Flinders University, Adelaide, Australia b

A R T I C LE I N FO

A B S T R A C T

Keywords: K-means data clustering Monte Carlo simulation Probabilistic load flow Robust data clustering

Data clustering incorporated in Monte Carlo Simulation (MCS) proves efficient in Probabilistic Load Flow (PLF) of the power grids under uncertainty of renewable energy resources. Fixed cluster agents are assumed for the clusters in the investigations reported in literature. This assumption ignores the changeable characteristics of Normal Data Clustering (NDC). This implies that the agents may change in another execution of the NDC. Under such circumstances, providing precise results for the PLF during frequent executions is not practical and there is high error in the executions. This paper presents a robust data clustering (RDC) scheme to overcome the problem arising from varying results of the NDC, and provides closer solutions to MCS. The proposed RDC method obtains the average of solutions by performing numerous NDCs under a so-called normal to robust (N2R) factor so as to solve the PLF problem in wind-integrated radial distribution systems. The proposed method is applied to various IEEE test systems, and the results are discussed. The results demonstrate the efficacy of the proposed RDC method for probabilistic load flow.

1. Introduction The contribution of Renewable Energy Sources (RESs) is steadily increasing due to the carbon emission concerns. The Wind Turbines (WTs) comprise a significant proportion of renewable resources due to the progress in technology, economic advantage, and environmental friendliness [1]. The high proportion of generated power from the WTs, however, poses serious challenges to the planning and operation of power systems. The probabilistic load flow (PLF) analysis is among these challenges due to the stochastic nature of WTs. The traditional deterministic methods may no longer be appropriate since they are not able to consider the uncertainties in power system assessment. Application of probabilistic methods for such analysis is a key requirement to solve the load flow problems under uncertainties of RESs. Conventional Monte Carlo simulation (MCS) is a well-known probabilistic method for the purpose of PLF calculations by allowing to adopt the input uncertainties. However, the MCS approach relies on the high volume of data samples and its execution might be rather time-consuming [2]. In order to accelerate the MCS execution process and decreasing computational costs, data clustering approach has been widely used in the literature [3–7]. In data clustering, all input data are bunched into ⁎

specific categories, in which each category takes a representative and a probability. The clustered data as a limited version of data are directly utilized in the MCS calculations. In this regard, by increasing the number of clusters (NOC), the obtained results from the normal data clustering (NDC)-based MCS approach are closer to those of the simple MCS [8]. But, it should be noted that by increasing the NOC, the computational time may suffer. Additionally, it loses the accuracy when working for a lower NOC, which leads to deviations of corresponding security-constraints for the voltage magnitude of buses in distribution systems. The main point is that the stochastic nature of NDC has not been considered yet. The stochastic nature means that the agents may change for another execution of the NDC. Without taking this issue into consideration, data clustering approach is not capable of achieving the confidence and precise level of results during successive executions. Contributing to the outlined context, there is a vast body of literature investigating efficient approaches to solve the PLF problem. In [9], the Quasi-Monte Carlo method has been presented to assess the optimal PLF in the radial distribution network. An analytical method based on generalized polynomial chaos for the PLF evaluation has been introduced in [10]. This method firstly proposed in 2002 [11] as an extension of the polynomial chaos method and became an impressive

Corresponding author. E-mail address: [email protected] (M. Kheradmandi).

https://doi.org/10.1016/j.ijepes.2019.105392 Received 5 March 2019; Received in revised form 23 May 2019; Accepted 30 June 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

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technique for probabilistic analysis of intricate systems. A synthetic approach based on Latin hypercube sampling and Cholesky decomposition for evaluation of the PLF has been accomplished in [12]. In [13], a novel solution for PLF analysis has been proposed for both balanced and unbalanced radial as well as weakly meshed networks. The method used discrete PDFs as input variables without assuming a predefined distribution. Analyzing the behavior of four Hongs point estimate schemes for the uncertainty of the PLF has been followed in [14] using the binominal and normal distributions to model the input random variables. The authors in [15] have concentrated on the PLF evaluation considering correlation between input random variables using uniform design sampling. Moreover, authors in [16] have proposed a PLF solution based on Latin hypercube sampling by adopting combined kernel density estimation with Nataf transformation to improve the calculation performance. To analyze the PLF, a cumulanttensor based method is accomplished in [17]. It determined the PDFs and reliability indices of the final outputs. Furthermore, general correlation among input random variables has been included in the analysis. A simplifications for the backward-forward load flow calculation that is able to analytically solve the PLF has been demonstrated in [18]. As well, a Bayesian inference perspective to solve the PLF problem has been developed in [19]. A likelihood-free method based on approximate Bayesian computation philosophy was generalized which increased the posterior distribution estimation of the state variables. In spite of these endeavors, none of them have paid attention to the probabilistic nature of RESs. In this regard, the authors in [20] have solved the PLF problem in a distribution network integrated with wind and photovoltaic systems. This paper makes use of a combinatory approach based on the MCS method and multi-linearized power flow equations. The same authors in [21] have presented an approach based on Taguchis orthogonal arrays to analyze the PLF of a wind-photovoltaic-integrated distribution system under load demand as well as wind and photovoltaic generations uncertainties. In [22], a new formulation for the PLF evaluation of distribution feeders is offered, in which the uncertainties of load demand and renewable resources are included. In [23], the impacts of high dimensional dependencies of wind speed among wind farms for the purpose of PLF have been examined. The kernel density estimate method was employed to evaluate the probability distribution of wind speed, and the pair copula method was used to estimate a joint PDF of wind speed among wind turbines. Also, the authors in [24] have provided a method based on the mean value first order saddle point approximation to assess the nonparametric PLF problem in the networks incorporating wind farms (WFs). Data clustering is well-known as an efficient method for decreasing the computational complexity of PLF studies. From standpoint of data clustering application, a particle swarm optimization (PSO) algorithm based clustering approach for the PLF considering wind uncertainty has been proposed in [3]. In [4], the direct PLF based on data clustering for radial distribution system including WF has been presented. In [5], the authors have proposed an approach based on data clustering for optimal capacitor allocation in wind integrated distribution systems. In a similar manner, application of data clustering approach has been propounded in [6] to carry out load flow problem considering distributed generation integration and load uncertainties. The total transfer capability evaluation of power systems in the presence of the wind farms (WFs) has been discussed in [7]. The research presented a hybrid approach based on data clustering and contingency enumeration to alleviate the intricacy of the total transfer capability problem by clustering the input data to a finite set. Nonetheless, these papers have not brought up the probabilistic nature of NDC in the PLF calculation. This paper proposes a robust data clustering (RDC) scheme to address the abovementioned drawback of the investigated studies for accurate analysis of PLF. The proposed method executes the NDC several times and then yields in the average value under a so-called normal to robust (N2R) factor. The proposed method is able to easily manipulate further analysis such as optimal allocation of capacitor and

Fig. 1. Relationship between MCS samples and convergence trend of MCS.

distributed generators in distribution networks facing any type of uncertainty and handling the correlation between them. The remarkable features of the proposed method can be itemized as follows:

• Presenting a simple model for RDS method which does not require further mathematical manipulation. • Accurate approximation of the PDF and CDF for the outputs. • Providing more precise results than the NDC and MCS methods while having a modest time burden • Successful application of the method on different test systems in-

cluding 10-bus, 33-bus, 69-bus, 85-bus, and 118-bus radial distribution systems.

The rest of paper is organized as follows: WT modeling is illustrated in Section 2. The PLF configuration in the presence of wind turbine is described in Section 3. Section 4 explains the K-means clustering algorithm and its application in MCS and the proposed RDC scheme for the PLF problem. Simulation results are provided in Section 5, and the concluding remarks are mentioned in Section 6. 2. Wind farm modelling The output power of WT varies with wind speed which varies stochastically with time. The wind speed is commonly modeled by the Weibull probability distribution [25–27]. There are several methods to determine the Weibull parameters c and k [27]. In this study, the shape parameter k(dimensionless) and scale parameter c(m/s) are obtained as follows:

σ −1.086 k=⎛ ⎞ ⎝ u¯ ⎠

c=

(1)

u¯

(

Γ 1+

1 k

)

(2)

where u is the mean value of density function, σ represents the standard deviation, and Γ shows the Gamma function [27]. The relationship between the output power of the WT and inlet air speed is as follows [28]:

PWT

0 V ⩾ Vco or V ⩽ Vcin ⎧ ⎪ V − Vcin Vcin ⩽ V ⩽ Vr = V − V Pr ⎨ r cin ⎪ Pr Vr ⩽ V ⩽ Vco ⎩

(3)

where V , Vcin, Vco, Vr , and Pr represent the wind speed, down-cutting speed, up-cutting speed, nominal wind speed and its rated power, respectively. While PWT shows the active output power of WT. The active output power of WF (PWF ) is obtained by summation of output power of all individual WTs provided that all of them are available which can be mentioned as follows: 2

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Fig. 2. The obtained objective in 1000 execution for (a) 25 clusters (b) 100 clusters.

0.6

1 PDF/CDF

Objective

0.55 0.5 45 0.4 NDC

0

200

RDC

800

CDF of RDC PDF of NDC

0.6

PDF of RDC

0.4 0.2

MCS

400 600 Iteration

CDF of NDC

0.8

0 0.35

1000

0.4

0.45

0.5 0.55 Objective

(a)

0.6

0.65

(b)

Objective

0.6 X: 11 Y: 0.5024

0.55 0.5 0.45 0.4

RDC MCS

2

4

6

8 N2R

10

(c)

Fig. 3. Results of both approaches from the viewpoints of: (a) situation of NDC solutions (b) PDF and CDF, and (c) impact of N2R factor.

PWF =

∑ NWT PWT

drop in each branch will be calculated. Thus, the voltage at the end of b to b + 1 branch is obtained as follows:

(4)

where NWT shows the numbers of WTs.

Vb + 1 = Vb − ILb + 1 Zb + 1 3. Probabilistic load flow

where ILb + 1 and Zb + 1 are the current and impedance of the considered branch between b-th and b + 1-th bus, respectively. This stage of the load flow problem is called the forward step.

Considering the stochastic generation of the WF, the injected active power in a WF-connected bus (PbStoch ) is written as follows: stoch Pbstoch = PLb − PWF b

b = 1, 2, …, B

(5) 4. Methodology

Pbstoch

and PLb are the stochastic active generated power by the where WF, and the active power of the existent load in the WF-connected bus, respectively. Also b is index of buses. In this study, the backward-forward method is used for the load flow calculations. The operation of this method is based-on the direct application of the rules of Kirchhoff’s circuit law (KCL) and Kirchhoff’s voltage law (KVL) [29]. In this method, as the first step, initial values of one p.u. are assumed for the voltage magnitude over buses. The currents of each buses are then calculated as follows:

Vbk ,

Data clustering is the process of grouping a set of objects in such a way that objects in the same group (known as a cluster) are more similar to each other than to those in other groups (clusters). It is the main task of exploratory data mining, and a common technique for statistical data analysis used in many fields such as load flow patterns of renewable-integrated distribution systems. Data clustering is not a specific algorithm by itself; it is a general task that can be integrated via various algorithms for data mining. Different data clustering methods differ significantly in the notion of how the data must be clustered. Several methods have been applied by data clustering for grouping data based on similarities or differences of observations [30,31]. After categorizing, the volume of huge multi-dimensional data significantly decreases, which leads to a lower computational burden and less timeconsuming than the simple MCS. In this paper, the K-means algorithm as a method of data clustering is used to categorize the WFs power output. In the following, the K-means method and its application are discussed.

∗

∗

Pbstoch + jQb ⎞ S stoch Ibstoch (k ) = ⎜⎛ b k ⎟⎞ = ⎜⎛ ⎟ Vbk ⎝ Vb ⎠ ⎝ ⎠

(7)

(6)

Sbstoch

Qb , and represent the equivalent voltage, injected rewhere active power and the complex power of the b-th bus, respectively. Afterwards, starting from terminal buses, the current of each branch of radial distribution network is obtained. This phase of the load flow problem is well-known as the backward stage. After obtaining the current of each branch and based-on the given reference voltage, and by starting from the reference bus, the voltage 3

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h-th cluster members, respectively. Step 3: Calculate a new cluster agent using (9):

ai =

∑j ∈ Gh mj NG

i = 1, 2, …, k

(9)

where NG denotes the number of members of observation in cluster i. Step 4: Repeat steps 2 and 3 until the change in cluster representatives fall below a threshold. Step 5: After convergence, the probability of i-th agents ( pi ) is obtained by dividing the total number of member in this category to the total number of observations (N) as follows:

pi = ci/ N

(10)

In overall, the KM algorithm seeks to find the cluster agents (c1, c2, …, ck ) such that the sum of the squared distances of data points from the nearest cluster agents is minimized. Mathematically stated, k

min ∑

aj

∑

‖x i, j − cj ‖2

j=1 i=1

(11)

where k is the number of clusters; aj represents the number of data points in cluster j; and x i, j is the data point i in cluster j. The objective function denotes the Euclidean distance of data point x i, j from its associated agent cj . The KM algorithm is a local optimization method which is highly dependent on the choice of initial positions of cluster agents and different results may be obtained on different executions. Accordingly, the present work proposes a robust data clustering outline to overcome this drawback. 4.2. Monte Carlo simulation using K-means data clustering The MCS approach uses the wind speed samples. Since the output power of WF continuously varies with time, several initial conditions are produced. By increasing the number of samples in MCS, the computational time increases considerably. Therefore, if appropriate clusters are created by considering correlation among wind speed samples, it is expected to reduce the cost of MCS computation. To achieve this purpose, the KM algorithm is used to cluster the WFs power output. Since the values of WF power outputs have different magnitudes, these values are normalized separately in proportion to the maximum value to modify data units into a range of (0, 1). Then, the clustering technique is applied to the normalized values. Finally, resulted cluster agents are de-normalized (using multiplication by maximum vector (Max. power output of WF)) to convert to real data. Each cluster is specified by its probability and mean values of power outputs of WF in categories (cluster agent). The cluster agents are then used instead of all data sets.

Fig. 4. Conceptual diagram of the proposed RDC scheme.

4.1. K-means algorithm The K-means (KM) algorithm is a well-known clustering method which has been extensively used for classifying large volumes of data. It can be easily applied to practical problems for limitation of large-scale data [2]. The steps of the KM algorithm are as follows: Step 1: Specify the number of clusters (k) and select randomly initial observations from the whole set of observations as the cluster agents. Step 2: Assign the other remained observations to the clusters with the closest agent using (8):

mj ∈ Gh

if

i = 1, 2, …, K

|mj − ah | < |mj − ai|

(8)

4.3. Proposed robust data clustering approach

j = 1, 2, …, N

In this section, the proposed RDC scheme is introduced at first and then its utilization for the PLF evaluation in the radial distribution

where ai and ah are the agents of cluster i and h, N is the total number of observations, mj and Gh represent the j-th observation and the set of

Fig. 5. (a) Impact of connecting WF to different buses (b) Maximum impact of WF on the increased voltage of buses. 4

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1.05 1.03 1.01 0.99 0.97 0.95 0.93 0

50

100 150 Number of Clusters

200

1

0.98

0.9

0.96 CDF

Voltage magnitude (P.U.)

(a)

1

0.94 0.92 0.9

0.8 NDC

0.7

1

5

Base case

10

15 20 Buses No. NDC

25 MCS

30 33 RDC

0.6 0.91

MCS RDC

1 1.1 1.2 Voltage magnitude (P.U.)

(b)

1.3

(c)

Voltage magnitude (P.U.)

Fig. 6. Comparison of RDC and NDC from the viewpoints of: (a) situation of solutions corresponds to the voltage magnitude of bus 18, (b) voltage profile, and (c) CDF of voltage amplitude of bus 18.

1

(Without applying a specific problem; it should be noted that in the PLF problem, the clusters’ agents inter within PLF formulation). The values of the objective are compared to the results of the MCS as the reference value. Fig. 2 shows the predicted objective for 1000 executions. As can be seen, the results for 100 clusters are closer to the MCS over 25 clusters, which illustrates the efficacy of the NDC. It should, however, be noted that the results obtained by NDC have a large oscillation around MCS over the iterations even for 100 clusters. To cope with this drawback, a robust model is proposed based on the oscillations of NDC results created around the MCS. In the proposed method, the NDC approach is executed several times instead of one time and the average of the executions under N2R factor is considered as the final solution, which is expected to have the smaller error than NDC. This method does not require extra equations and follows a simple scheme. Fig. 3(a) illustrates the results of both the NDC and the proposed RDC methods for 25 clusters in 1000 execution while the considered N2R for RDC is 10. As can be seen, the results of RDC are very closer to the simple MCS method. In addition, the PDF and CDF of solutions for both methods in 1000 execution are shown in Fig. 3(b). Fig. 3(c) illustrates the impact of the different N2R factors. The objective follows the reference value closely for N2R factors of 8 and 10 than in lower N2R factors, which verifies the effectiveness of the proposed RDC from the viewpoint of accuracy. In addition, the N2R factor has a linear relationship with time burden. This means that the time of calculation drastically increases for higher N2R factor. The computation burden for both the NDC and the proposed RDC can be written by (12) and (spseqn13), respectively. The conceptual diagram of the proposed RDC approach which is obtained from the NDC solutions is shown in Fig. 4.

0.99 0.98 0.97 0.96 0.95 0.94

1

2

3

4 5 6 7 Execution

8

9

10

Fig. 7. Comparison of both methods from viewpoints of position of voltage magnitude of bus 18 for different executions.

systems is evaluated. It can be concluded from the simulation results analysis of the previous research that the NDC-based MCS approach is an effective method for accurate analysis of the PLF. Accurate analysis means that the computational burden can be remarkably decreased, while keeping a high level of the accuracy. However, the notable drawback of the existing researches is the fixed clusters’ agents for each specific NOC. As the agents may change, these assumptions do not characterize the stochastic properties of the NDC and thus are not recognized as appropriate idea for the probabilistic problems. To further clarify this point, a random function with uniform distribution is adapted to generate the stochastic samples within the range (0–1). The convergence of the MCS is examined for various numbers of samples. Fig. 1 illustrates the convergence trend of MSC solutions over 1 to 7000 samples. The figure illustrates the decreasing dependency of MCS convergence to the number of samples. For around 7000 samples, the MCS solution converges to a final value, that is 0.5. A 7000-sample population is, therefore, used for Monte-Carlo simulation to deal with the uncertainty associated with the wind speed. The samples are categorized into 25 and 100 clusters, in which the optimal clusters’ agents are obtained using the KM algorithm. These agents are then multiplied by their probability, and their summation is directly considered as the output results to form the NDC solutions, which is called the objective

TNDC ≈ TMCS / NOC

(12)

TRDC ≈ N 2R (TMCS / NOC ) = N 2R TNDC

(13)

Let us focus on the application of both methods for PLF problem in the radial distribution networks. The voltage magnitude of the buses via the NDC approach is calculated by (14). In the proposed method, the average of the solutions (each solution is obtained from execution of NDC) is taken from (15), which represents the RDC solution. Equation 5

Electrical Power and Energy Systems 115 (2020) 105392

1 0.99 0.98 0.97 0.96 0.95 0.94

0.15

NDC RDC

0.1

PDF

Voltage magnitude (P.U.)

O. Sadeghian, et al.

0.05 0 0.94

100 Number of Clusters

CDF

(a)

0.95

0.96 0.97 0.98 0.99 Voltage magnitude (P.U.)

1

(b)

2

NDC

RDC

1.5

0.0440 pu

1

0.0180 pu

0.5 0 0.945

0.955 0.965 0.975 Voltage magnitude (pu)

0.985

0.995

(c)

Voltage magnitude (P.U.)

Fig. 8. (a) The position of 200 solutions associated with the voltage magnitude of bus 18 (b) PDF of the voltage magnitude of bus 18 for 200 solutions (c) CDF of the voltage magnitude of bus 18.

0.99

5. Case studies

MCS RDC

0.98

To illustrate the performance of the proposed RDC method for PLF problem, five case studies including IEEE 10-bus, 33-bus, 69-bus, 85bus, and 118-bus radial distribution systems are examined. Detailed data for these systems are available in [32–36], respectively. The WF, which is formed by collecting several individual wind turbines, is added to the case studies. In this study, the WF comprises of 15 WTs. The following values are applied to calculate the wind power [37]:

0.97 0.96 0.95 2

4

6

8

10

Vc in = 4 m/s,

N2R

Table 1 Obtained values for objectives for various N2R factors. N2R

ϕ1k

ϕ2k

k ϕ11

k ϕ22

Min

1 2 3 4 5

2 4 6 8 10

0.0181 0.0124 0.0109 0.0101 0.0064

0.1241 0.2482 0.3723 0.4964 0.6205

1.0000 0.5107 0.3863 0.3176 0

0 0.2500 0.5000 0.7500 1.0000

0 0.2500 0.3863 0.3176 0

(15) can be written as (16).

VbNDC = ,x

∑

pi, x Vb, i, x

x = 1, 2, …, N 2R

b = 1, 2, …, B (14)

i∈K

VbRDC = VbRDC =

∑x ∈ N 2R ∑i ∈ K pi, x Vb, i, x N 2R ∑x ∈ N 2R VbNDC ,x N 2R

b = 1, 2, …, B

b = 1, 2, …, B

Vr = 10 m/s,

and

Pr = 500 kW

The IEEE 33-bus case study is the first investigated system. We have modified the network by installing the WF at bus 18. To show that why this bus has been selected, Fig. 5(a) shows the impact of WF on the voltage increase of buses based on the WF connection to different buses of the network. As seen, when the WF is connected to bus 18, the increase in the voltage of that bus has the highest value compared to the other buses. Fig. 5(b) shows the voltage profile of 33-bus system after the installation of the WF in bus 18. As can be observed, the maximum change in voltage magnitude is for bus 18 compared with the state that there is no WF in the distribution system. In order to illustrate the efficacy of the proposed RDC method, we compared the results with NDC and the MCS. It should be noted that the outputs of the method is intended for a planning method and thus is implemented in an off-line environment. Fig. 6(a) shows the position of the solutions corresponding to the voltage magnitude of bus 18 obtained through NDC and RDC methods for different NOCs including 25, 50, 75, 100, 125, 150, 175, 200 and N2R = 100. The voltage profile and voltage CDF are provided for NOC = 100 and N2R = 100 as shown in Fig. 6(b) and (c), respectively. The benchmark results determined by MCS are also represented in Figs. 6(a)–(c). As it is obviously observed from Figs. 6(a), by increasing the NOC, the position of NDC solutions tend to the result of the MCS and the results are associated with smaller errors than in lower NOCs. However, these solutions have an oscillation trend around the MCS, particularly for large NOC. Thus, increasing the NOC solely is not recognized sufficient for improving the accuracy level and it is reasonable to run data clustering several times (i.e. N2R factor times) and to calculate the average of solutions instead of a single execution obtained from NDC. As it is observed, the proposed method follows the MCS results more closely.

Fig. 9. Impact of different N2R factors on solutions corresponds to the voltage magnitude of bus 18.

k

Vc o = 22 m/s,

(15)

(16)

where x shows the index of execution of normal data clustering, and Vb, i, x represents the voltage of b-th bus obtained from i-th cluster in x-th RDC iteration. In addition, VbNDC are the predicted voltage of b-th bus , x , Vb in NDC for x-th execution and the predicted voltage of b-th bus in the RDC approach, respectively. 6

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Voltage magnitude (P.U.)

Voltage magnitude (P.U.)

O. Sadeghian, et al.

0.95 0.9 0.85

Base case With WF

0.8

maximum increaes

1

2

4

6 Buses No.

8

10

1.02 1 0.98 0.96 0.94

Base case

0.92 0.9

With WF maximum increaes

1

10

20

30 40 Buses No.

Base case With WF

0.98

Maximum increase

0.94 0.9 0.86

1

10

20

30

40 50 Buses No.

60

(b) Voltage magnitude (P.U.)

Voltage magnitude (P.U.)

(a) 1.02

50

60

70

1 0.95 0.9 0.85

Base case With WF Maximum increase

1

80 85

(c)

20

40

60 80 Buses No.

100

118

(d)

Fig. 10. Maximum increase in voltage magnitude due to presence of WF for (a) 10 bus (b) 69-bus (c) 85-bus, and (d) 118-bus systems.

RDC solutions (marked in red) are close to the MCS (marked in black) and have lower oscillations in comparison to NDC (marked in blue). As another analysis, Fig. 8(a) shows the positions of 200 solutions corresponding to the voltage magnitude of bus 18 obtained by NDC and RDC methods, which are compared for N2R = 10 and NOC = 100. The results of the proposed RDC are closer to the MCS results in comparison to the NDC method. The PDFs of the voltage magnitude of bus 18 for both the NDC and RDC approaches are shown in Fig. 8(b). As can be seen, the RDC approach has more appropriate PDF and follows the MCS more closely in comparison to NDC. The CDF of the voltage magnitude is depicted in Fig. 8(c) for bus 18. As it is observed, the CDF variations for the NDC is extended over an interval length of 0.0440 p.u. from 0.948 to 0.993, which is larger than the interval length of 0.0180 p.u. which is obtained for the RDC method. This suggests that the RDC leads

Table 2 Impact of WF on the increased voltage of buses in each test system. Test systems

Bus connected to the WF

Bus related to the highest value for increased voltage

Most increase in voltage (P.U.)

10-bus 69-bus 85-bus 118-bus

10 27 54 77

10 27 54 77

0.0374 0.0506 0.0752 0.0424

Fig. 7 shows the position of the solutions corresponds to the voltage magnitude of bus 18 obtained through NDC and RDC methods for NOC = 100 in 10 typical execution. It is clear from the Fig. 7 that the

Fig. 11. Comparison of RDC and NDC from the viewpoints of situation of solutions correspond to the voltage magnitude of wind connected buses for (a) 10 bus (b) 69-bus (c) 85-bus, and (d) 118-bus systems. 7

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Fig. 12. PDF of the voltage magnitude of WF connected buses for (a) 10 bus (b) 69-bus (c) 85-bus, and (d) 118-bus systems.

Fig. 13. CDF of the voltage magnitude of WF connected buses for (a) 10 bus (b) 69-bus (c) 85-bus, and (d) 118-bus systems. Table 3 Obtained optimal N2R factors in each test system. Systems

Optimal N2R

10-bus 69-bus 85-bus 118-bus

6 6 4 6

Table 4 Time burden of both approaches. Systems

TNDC (Sec) (NOC = 100)

TRDC (Sec) (NOC = 100)

TMCS (Sec)

10-bus 33-bus 69-bus 85-bus 118-bus

0.025271 0.062053 0.12885 0.16815 0.26509

0.025271∗N2R 0.062053∗N2R 0.12885∗N2R 0.16815∗N2R 0.26509∗N2R

1.6717 4.2292 9.0272 11.9810 17.2471

solutions to achieve the RDC solution. It should be emphasized that if the N2R factor decreases, the computation speed will improve but the accuracy may suffer and thus the desired value of N2R must be determined in various executions to achieve an accurate analysis of the PLF problem and improve the operation process of the proposed

to a smaller uncertainty than the NDC in the convergence trend for the voltage magnitude of bus 18. It is clear that the proposed RDC provides more accurate results for PLF problem compared to the NDC method. As mentioned earlier, N2R factor shows the number of NDC

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O. Sadeghian, et al.

Max. error

Mean

STD

Max. error

Mean

STD

10-bus

Neg: 0.0109 Pos: 0.0151

0.8740

0.00552

Neg: 0.0039 Pos: 0.0151

0.8743

0.00206

33-bus

Neg: 0.0178 Pos: 0.0272

0.9674

0.01001

Neg: 0.0078 Pos: 0.0102

0.9666

0.00368

69-bus

Neg: 0.0130 Pos: 0.0180

1.0055

0.00697

Neg: 0.0060 Pos: 0.0050

1.0056

0.00267

85-bus

Neg: 0.0239 Pos: 0.0151

0.9454

0.01165

Neg: 0.0079 Pos: 0.0151

0.9461

0.00382

accurate results compared to the NDC. It is concluded that the RDC method is also efficient for the PLF problem in large-scale test systems. In addition, the optimal N2R factors for 10-bus, 69-bus, 85-bus, and 118-bus systems are summarized in Table 3. Table 4 shows the approximate computational time for the proposed RDC method and the NDC and MCS methods. As can be observed, the higher values of N2R factor result in higher time burdens. Also, the proposed method is less time-consuming than the MCS, which is a worthwhile advantage of the RDC method. Finally, Table 5 shows the voltage error, mean and standard deviation of WF connected buses for all test systems. As can be seen, the calculated voltage error and the standard deviation of the proposed method are smaller than the values of NDC method, which indicates the higher accuracy level of the proposed method.

118-bus

Neg: 0.0119 Pos: 0.0191

0.9114

0.00646

Neg: 0.0031 Pos: 0.0051

0.91025

0.00207

6. Conclusion

Table 5 Voltage error, mean and standard deviation of both approaches (pu). Systems

NDC

RDC

This paper introduces a robust data clustering scheme for the PLF problem considering the probabilistic character of the NDC method. The WF is considered as the source of uncertainty. The obtained results are compared with the NDC method and simple MCS for an efficient evaluation. The methods are examined in various radial distribution networks through adding WF in networks. The proposed method executes the NDC method and then takes the average of the obtained solutions under the N2R factor. This method does not require additional complicated equations and thus easy for implementation. The proposed method also deals with the correlation between uncertainties with no complexity. The obtained results demonstrate the high accuracy level of the proposed method in comparison to the NDC. In addition, it is less time-consuming than the MCS. The results illustrates the remarkable features of the proposed method. The research can be further extended for other probabilistic analysis such as optimal allocation of the capacitor and the distributed generators, and also the energy management problems.

standard deviation: STD, Neg: Negative, Pos: Positive.

method. Fig. 9 shows the voltage magnitude of bus 18 obtained from RDC method for NOC = 100 and different N2R factors. As it is shown, the higher value of N2R improves the accuracy level, but the computation burden is somewhat affected. Therefore, the accuracy level is a direct function of the taken N2R factor. An analysis is conducted to find an optimal value of N2R factor to keep the accuracy ϕ1 and the time burden ϕ2 under a desired value. In this analysis, at first, the values of these two objectives are obtained for different N2R factors. The best N2R factor is then selected by using the fuzzy satisfying approach [38]. In this method, a fuzzy membership number in the interval [0, 1] is assigned to the values of each objective function. The fuzzy membership numbers ϕ1 and ϕ2 are obtained as follows:

ϕ 11 =

ϕ1 − ϕ1min ϕ1max − ϕ1min

(17) Declaration of Competing Interest

ϕ22 =

ϕ2 − ϕ2min ϕ2max − ϕ2min

The authors declared that there is no conflict of interest.

(18)

At the first stage, the minimum value of ϕ11 and ϕ22 are obtained for all solutions. A solution which gives the maximum value of obtained minimum solutions is then selected as the best compromise solution. The obtained values for the maximum voltage error and time burden are given in Table 1 for different N2R factors for 33-bus system. Solution # 1 corresponds to minimization of ϕ2 whereas Solution#5 refers to minimization of ϕ1. Solution# 3 (i.e. N2R = 6) is the compromise solution obtained by fuzzy satisfying criterion In order to show the effectiveness of the RDC, the method is examined on the IEEE 10-bus, 69-bus, 85-bus, and 118-bus radial distribution systems, These systems are modified by installing a WF at buses of 10, 27, 54, and 77. As in the 33-bus test system, these buses are choosen due to the highest voltage increase that these buses show with connecting the WF. The impact of WF on the voltage change of buses is shown for all buses in Fig. 10. Table 2 provides more details about the most appropriate bus of the systems for WF connection. Similar to the 33-bus system, the position of the solutions corresponding to voltage magnitude of WF connected buses obtained from NDC (for different NOCs including 25, 50, 75, 100, 125, 150, 175, and 200) and RDC (with N2R = 100) methods are shown in Fig. 11. The PDFs of the voltage magnitude of the mentioned buses for both the NDC (with NOC = 100) and the RDC (with N2R = 10) approaches are shown in Fig. 12. In addition, the CDF of the voltage magnitude of the WF connected buses is shown in Fig. 13 for the test systems. As can be seen, as in 33-bus system, the RDC leads to a smaller uncertainty in the convergence trend of voltage magnitude than NDC for all test systems. It can, then, be verified that the proposed RDC method provides more

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