Radial distribution network planning under uncertainty by applying different reliability cost models

Electrical Power and Energy Systems 117 (2020) 105655

Contents lists available at ScienceDirect

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

Radial distribution network planning under uncertainty by applying different reliability cost models

T

⁎

Jovan M. Nahmana, , Dragoslav M. Perićb a b

Academy of Engineering Sciences of Serbia, Belgrade, Serbia Tech. School of Professional Studies, Požarevac, Serbia

A R T I C LE I N FO

A B S T R A C T

Keywords: Distribution networks Planning Uncertain inputs Reliability costs

The combined steepest decent and artificial annealing optimization method for optimal planning of radial distribution networks is upgraded to take into account the uncertainty of inputs and various methods of determining the load curtailment costs. The optimization method considers the uncertainties of predicted power consumption, of line failure rates and of the power supplied by distributed generators. The search for the optimal network configuration is conducted with various cost models for the curtailed loads, for comparison. The calculations performed have shown that the reliability cost models considerably affect the optimal network configurations and associated total annual costs. Also, it is indicated that the distributed generators should be modeled in the planning of networks as they favorably affect the total costs by decreasing energy losses and by helping in maintaining network operation within the allowed current and voltage limits.

1. Introduction Planning of distribution networks is a complex task as it implies the consideration of costs of different origins, technical and ambient imposed constraints and customer requirements concerning the quality and reliability of power supply [1–4]. The goal of an optimization is to find the network solution that satisfies all requirements and constraints with minimum annual cost. The possible network configurations depend on the available routes, on the locations of source substation, load points and distributed generators, as well as on the adopted general concept of their structure. For typical rural networks with relatively small individual customers sparsely spread around the source substation, radial networks are preferred keeping low the capital cost. Such network structures are primarily considered in this paper. In searching for optimal network solutions various discrete search approaches were used in the past. The optimization of radial distribution networks was conducted using ant colony system algorithm [5]. The potential solutions are selected based on system experts’ suggestions. The optimization of radial distribution networks was also conducted by applying the dynamic programming technique and geographical information systems [6]. The network is formed stepwise by covering the loads closest to the source substation and then gradually the more distant loads. The same optimization technique was used to determine the optimal reinforcement and reconstruction steps for

⁎

typical network structures by accounting for capital, loss and undelivered energy costs [7]. The simulated annealing algorithm was applied [8] to find the best solution for further expansion of a radial distribution network through a number of alternative triangular substructures by considering the impact of the reliability associated costs. Branch exchange technique was applied [9] using the minimum spanning tree of the graph of available routes as the initial solution. This solution was then step wise reformed to meet the technical constraints with minimum capital and loss cost. Planning of distribution networks based upon urban maps has been also performed by applying evolutionary algorithms [10]. Effects of automation in network expansion planning have been analyzed by employing the genetic algorithm technique [11,12]. A multistage planning method for distribution networks incorporating the active management of distributed generators is proposed in [13]. Reference [14] presents a model for distribution network and renewable energy expansion planning by accounting for the demand reaction due to time-varying prices. A method for planning of new distribution networks with radial structure using the combination of the steepest descent and artificial annealing approaches has been developed in [15]. Methods for distribution system planning by considering the uncertainty of the power delivered by distributed generators [16] and of loads [17] weighted by adequate probabilities have been proposed. The uncertainty aspects in network expansion and reinforcement planning have been also considered by applying two-point

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

https://doi.org/10.1016/j.ijepes.2019.105655 Received 11 September 2019; Received in revised form 11 October 2019; Accepted 25 October 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

Electrical Power and Energy Systems 117 (2020) 105655

J.M. Nahman and D.M. Perić

The iteration process is terminated when the highest difference between two succeeding iterations of voltage drops is less than 0.1%. Currents flowing through network branches are calculated as

estimate method [18], stochastic programming based models [19], the information-gap decision theory [20], and fuzzy-sets [21]. The application of point estimate methods for modeling the uncertainties has been presented in [22] and [23]. The optimization method presented in this paper is designed for planning of new or further development or reconstruction of radial distribution networks. The approach suggested before in [15] is now upgraded by considering the contribution of wind generators (WGs) as well as by accounting for the uncertainty of various inputs. A new approach in assessing the reliability impacts is proposed based upon the energy providers costs combined with reliability constraints. The calculations performed have shown that the different reliability cost models can give network solutions that differ remarkably from one other in total cost and configuration. It is also shown that the WGs can substantially affect the network configuration and decrease the network cost. The performed comparative analyzes have shown that the considered uncertainties of input data give the same or nearly the same optimal network solutions and costs as in the case when the uncertain inputs are modeled only by their mean values.

[I ] = [Yb]·[A]T ·[U ]

In (5), [Yb] is the diagonal matrix of network branch admittances and [A] is the network incidence matrix. During the search for the optimal network radial configuration the aforementioned network structure matrices change. However, these changes can be easily made based upon the initial network edge matrix, as explained in [15]. 2.3. Supply interruption cost Usually, the reliability of a network is assessed by estimating the cost caused to customers by supply interruptions. In radial networks this cost is calculated by applying the following expression

Cr =

2. Modeling and optimization of networks

m−1

[AD]k (1)

k=1

In (1) m is the number of network nodes and [AD] is the m by m adjacency matrix of the network graph with elements AD(i,j) being equal to 1 if there is a branch connecting nodes i and j and equal to zero otherwise. The network is connected if all off - diagonal elements of [NC] are non zero. 2.2. Load flow analysis The load flow analysis is based upon the data on the predicted power consumed and produced at network nodes. In the calculation procedure the analysis of load flows and voltage drops are conducted by applying the general recursive approach based upon the network bus admittance matrix [15,25] that can be used both for meshed and radial network structures. The network source node is taken as slack bus and the calculated voltages at network nodes are voltage drops by reference to this node. The recursive expression for iteratively calculating the voltage drops are [15]

[U (k + 1) ] = [Yd]−1 ·[J ] + ([UN ] − [Yd]−1 ·[Y ])·[U (k ) ]

(2)

(k)

with [U ]. [J], [Y], [Yd], [UN] designating column vector of voltage drops at network nodes in iteration k, column vector of load currents, network bus admittance matrix, diagonal matrix built from the main diagonal of bus admittance matrix, and unit matrix, respectively. Load currents [J] which are taken as known input variables are determined as

[J ] = [S ∗ ]/ Vr

Cr =

(3)

ki

⎠

∑ λk α ⎛⎜∑ ci Pki D⎞⎟ k

with [S*] and Vr designating the vector of conjugated loads at network nodes and network base node voltage, respectively. The first iteration for nodes voltage drops is

[U (1) ] = [Yd]−1 ·[J ]

⎝

(6)

with λk , α , cki, and Pki being failure rate of branch k, load factor, cost per unit of interrupted load of the customers downstream from branch k if interruption lasts D hours, and peak load of these customers, respectively. Index k runs over all network branches. Index ki designates network branches downstream from branch k. Model (6) is simple and can be easily used in the assessment of network costs. However, the cki values are usually determined based upon rather uncertain assumptions about the circumstances at the fault occurrence [26]. It is usually presumed that the interruption will occure at a fixed day of the year and at the peak consumption. The main drawback of such an approach lies in the fact that the interuptions can happen at any instant during a day, with different cosequences causing different costs. A more complex approach in assessing the supply interruption costs for residental sector has been proposed in [27]. This approach considers each household activity and distribution network behaviour as Markov stochastic processes that overlap. Such a model makes it possible to determine the frequency and duration of the interruprtion of each household activity, being postponable or not, and to asses adequately the caused costs. The application of the mentioned approach in determining the costs for various types of customers requires good knowledge of cutomers’ activities during all caracteristc days of a year by making difference among year seasons and ordinary week days and hollydays. In engineering practice such data for customers are usually missing. Bearing the previous in mind, an alternative model for assessing the reliability in distribution system planning studies is also applied in this paper, for comparison. The alternative approach could be the following: a. Deterimine the amount lost of the incomes of the electrical energy providers in selling the electrical energy because of supply interruptions. The providers are, generally, the distribution network utility and the distributed generators owners. b. Prescribe the maximum allowable values of indices SAIDI and SAIFI. The mathematical model for calculating the reliability costs for a distribution network is now

In searching for the optimal radial network configuration within the set of available routes by stepwise excluding of most expensive branches the network connectivity check has to be performed after all exclusions of branches. As known [15,24], the connectivity of network configuration can be checked by the help of the matrix [NC] defined as

∑

∑ λk α ⎛⎜∑ cki Pki⎞⎟ k

2.1. Network connectivity check

[NC ] =

(5)

⎝

ki

⎠

(7)

Here ci designates providers income lost per unit of unsupplied energy. Other symbols have the same meaning as in (6). The reliability indices that should be determined for the distribution network are, as well known,

(4) 2

Electrical Power and Energy Systems 117 (2020) 105655

J.M. Nahman and D.M. Perić

SAIDI =

SAIFI =

∑k λk (∑ki DNki ) N

standard deviation σ1. [SL] is the vector of distribution network peak loads. The uncertainties associated with failure rates of lines are accounted for by multiplying the failure rate per unit of line length by random function f2 with mean 1 and adopted standard deviation σ2 . The inputs of wind generators (WGs) are modeled as strictly mutually correlated, which is a reasonably assumption as the generators are spread over a limited area occupied by the distribution network. The expected mean operating power of these generators is scaled by random function f3 with mean 1 and adopted standard deviation σ3 . In the analysis it was taken that the WGs connected to the considered faulted line do not operate during the repair duration as they, operating alone, cannot provide the energy of prescribed quality.

(8)

∑k λk (∑ki Nki ) (9)

N

with Nki designating the number of customers at the end of branch ki curtailed by the failure of branck k and N being the total number of network customers. These two indices provide a measure of the disturbances caused to the customers. Their maximum allowable values should be prescribed and applied as constraints in any reinforcement and further development activities for a network. According to Europian expirience [28] the preferable values that were achieved in 2016 by many countries of Europian Union are SAIDI < 400 min./ cust.yr. and SAIFI < 3 inter./cust.yr. For the customers that are very supply interruption sensitive the distribution utility can provide a higher reliability of supply using its available resources such as adequate modification of network configuration and/or placement of new switching devices. Let the annual cost of the modified solution and the solution when all customers are treated equally be Csc and Cnsc, respectively. Then, the annual payment of sensitive customers at node k to the distribution network utility for improved reliability of supply can be determined as

cpk =

Csc − Cnsc ·ΔWk ∑k ΔWk

2.6. Optimization In the optimization the combined calculation method presented in [15] is applied. The best radial configuration is searched for within the graph consisting of all line routes that are available considering the local circumstances. Thus, the widest possible basis for the search is used. The initial feasible minimum investment cost solution is determined by applying the steepest descent approach. The longest branches of the initial meshed network are eliminated one by one until a tree configuration is reached satisfying the prescribed constraints. This solution is further modified step by step using the simulated annealing technique to find the minimum total cost solution. The considered m = 3 uncertain inputs are modeled by applying the 2m + 1 point estimate method [Appendix] in load flow, undelivered power and SAIDI and SAIFI calculations. The possible violation of network constraints is checked using the expected values of these variables. In searching for the best locations for sectionalizers, the initial solution is taken to be that one halving the total load supplied by the feeder, which holds exactly if the loads are uniformly spread along the feeder. Two conductor cross sections, standardized for considered distribution networks, are used. Initially, the smaller of these cross sections is adopted for all branches. During the calculation flow higher cross section are used for some network branches in order to satisfy current and/or voltage constraints. The 2m + 1 point estimate method is applied as it gives very accurate results for composite power systems, as demonstrated in [22]. We have also checked the accuracy of the method for few considered network configurations using Monte Carlo simulation. It was found that, in all considered cases, the relative difference between the results obtained by these two approaches were less than 0.01% for calculated expected mean total cost and less than 0.15% for expected mean values of SAIDI and SAIFI indices. Maximum difference of standard declinations is found to be less than 1.5% for total cost and less than 1.6% for SAIDI and SAIFI indices.

(10)

with ΔWk designating the increment of the annual amount of energy supplied to customer k achieved by the new, more reliable network solution. Index k runs over all sensitive customers. 2.4. Total network annual cost The annual capital cost equals

Cc =

∑

gk ck +

∑ gj cj

k∈M

j

(11)

with gk and ck being the capital recovery factor and the cost of line k, whereas gj and cj are the annual capital recovery and maintenance cost of switching device j and cost of it. M is the set of branches in the network configuration under consideration. The cost of the line going from the source substation includes both the line and corresponding substation costs. The annual cost of energy losses equals

Cl = 8760 cl

∑

r j |I j |2

j∈M

(12)

where cl, rj and M are cost per unit of energy lost, network branch j resistance and set of network branches, respectively. Load currents are determined from the load data by multiplying the maximum customer loads by loss factorβ

β = 0.15α + 0.85α 2

3. Application

(13)

3.1. Example I

The total network annual cost that should be minimized is

C = Cc + Cr + Cl

The graph of available network routes for a new rural 11 kV distribution network that should be planned is displayed in Fig. 1. Node 1 is source TS 35 kV/11 kV. There are 26 load points (TS 11 kV/0.4 kV) and 46 possible line routes. The planned network should be supplied by four WGs at nodes 7–10, as indicated by black square dots. Tables 1–4 quote the input data for the planning of the network under consideration. The data on installed capacity and assumed mean operating power of WGs are given in Table 5. The source substation equipment and building annual capital cost per outgoing line is taken to be 2.75 kUS$. The maximum allowed voltage declination from the rated voltage is 1 kV. In calculating the SAIDI and SAIFI indices it was adopted that the installed capacity of individual customers is equal 5 kVA and that the mean repair duration is D = 4 h for all k, i. The maximum allowable values for these two indices have been adopted to be

(14)

2.5. Uncertainty modeling The planning is always based upon some predictions associated with a degree of uncertainty. In the planning of new distribution networks the uncertain inputs are usually the customers’ peak loads, failure rates of lines, as well as the generation of distributed generators if there are any. The uncertainty of loads is modeled as

[SU ] = f1 ·[SL]

(15)

where f1 is an assumed scaling random function with mean 1 and 3

Electrical Power and Energy Systems 117 (2020) 105655

J.M. Nahman and D.M. Perić

Table 3 Complementary line data. Branch No.

Conductors Al./St. mm2 / mm2

Loading Capacity A

Impedance Ω/ km

Failure rate fl./(km yr)

1, 2, 3, 22, 30 other

35/6

145

0.8 + j 0.4

0.1

25/4

125

1.2 + j 0.4

0.1

Table 4 Cost and complementary load data. Power Factor

Load factor

ci US$/kW

cl US$/kWh

ck kUS$/km

g

0.9

0.6

5

0.1

1500

0.1

Table 5 Wind generators data.

Fig. 1. Graph of available network routes. Table 1 Consumption at load points. Load point No. Load, kVA

2 250

3 160

4 100

5 100

6 50

7 50

8 100

Load point No. Load, kVA Load point No. Load, kVA Load point No. Load, kVA

9 100 16 150 23 80

10 250 17 80 24 100

11 80 18 40 25 30

12 160 19 100 26 150

13 100 20 40 27 80

14 100 21 60

15 100 22 40

1

2

3

4

5

6

Length, km

4.10

3.65

4.20

2.10

1.50

1.85

Branch No.

7

8

9

10

11

12

Length, km Branch No.

1.90 13

1.80 14

0.85 15

1.00 16

1.25 17

2.09 18

Length, km Branch No.

2.00 19

2.20 20

2.30 21

1.70 22

1.32 23

1.60 24

Length, km Branch No.

1.45 25

2.75 26

1.35 27

2.80 28

0.95 29

0.55 30

Length, km Branch No.

1.30 31

0.93 32

1.95 33

0.68 34

1.31 35

2.00 36

Length, km Branch No.

1.10 37

0.60 38

0.75 39

0.50 40

0.49 41

1.05 42

Length, km Branch No.

0.63 43

0.65 44

0.80 45

0.45 46

0.43 –

0.36 –

Length, km

0.90

1.40

1.20

2.50

7

8

9

10

Rated power, kVA Mean power, kW

160 100

250 150

160 100

160 100

of lines are taken to be normally distributed with means equal to 1 and standard deviations being 0.15 and 0.25, respectively. The power produced by WGs is modeled by multiplying the corresponding mean power by the Weibull distributed scaling function with parameters α = 1.115, β = 5 and mean value 1.02. It is pertinent to mention that modeling the scaling functions by normal distributions allows for negative values of these functions. However, if the means of these distributions are four or more times greater than their standard deviations, as in our case, the probabilities of negative values of scaling functions are negligible small and can be ignored [29]. The choice was motivated by the assumption that the considered uncertain parameters can disperse from their mean values to higher or lower values with same probability. The optimal network solution has been searched for by applying two reliability cost models discussed before. The best solutions are determined by accounting for the uncertainty of inputs and by modeling them using only their mean values, for comparison. Eight cases have been considered:

Table 2 Lengths of network graf branches. Branch No.

WG at node No.

Case 1: the uncertainty of inputs is taken into account and the reliability costs are calculated according to (6). Case 2: the same as Case 1 if ignoring the uncertainties Case 3: the same as Case 1 but the reliability costs are calculated using (7) Case 4: the same as Case 3 if ignoring the uncertainties Case 5: the same as Case 3 if there are no WGs Case 6: the same as Case 5 if ignoring the uncertainties Case 7: the same as Case 3 after insertion of a sectionalizer at the sending end of branch 24 to increase the reliability of supply to interruptions sensitive customers at nodes 2 and 16 Case 8: the same as Case 7 if ignoring uncertainties. The optimal network solutions for the considered cases are displayed on Figs. 2 to 5. Triangles mark the locations of sectionalizers. The calculated costs for considered cases are presented in Table 4. The SAIDI and SAIFI indices for these solutions are given in Table 7. As can be observed, in all considered cases, by exclusion of Case 1 and Case 2, the uncertainty of inputs has generated network solutions that do not differ from these if this factor is neglected. For Case 1 and Case 2 very similar network solutions are obtained. It is noteworthy to stress that the difference between the network total costs if the

SAIDIm = 400 min./cust.yr. and SAIFI = 3 inter./cust.yr. It was also assumed that the circuit breakers in the source substation can operate as single time reclosers and in combination with a sectionalizer. All of the aforementioned assumptions are characteristic for the rural networks in the authors’ country. Random functions scaling the uncertainty of loads and failure rates 4

Electrical Power and Energy Systems 117 (2020) 105655

J.M. Nahman and D.M. Perić

Fig. 2. Optimal solution for Case 1.

Fig. 4. Optimal solutions for Case 3 and Case 4.

Fig. 3. Optimal solution for Case 2.

Fig. 5. Optimal solution for Case 5 and Case 6.

uncertainty of considered inputs is taken into account or neglected is comparatively small in all considered cases. Standard deviations of total costs are low when compared to their mean values. It can be explained by the fact that the considered uncertain inputs do not affect the cost of lines being the highest among all considered costs. SAIDI and SAIFI indices disperse considerably when uncertainty of inputs is accounted for but their mean values are the same as these determined by ignoring uncertainty. It can be observed that the proposed alternative reliability assessment model, applied in Case 3 to Case 8, yields the network solutions of considerably lower total costs when compared to the solutions obtained for Case 1 and Case 2. In Case 5 and Case 6 the voltage and current constraints dictated a network configuration that differs considerably from these obtained in other cases and has higher annual cost than solutions In Case 3 and Case 4. The main reason for the previous is the increase of magnitudes of currents flowing through the network causing higher losses and voltage drops that affect the network configuration.

Table 6 Annual cost in kUS$. Case

Total

σ

Lines

Switch. Devices.

Losses

Reliability.

1 2 3 4 5 6 7 8

75.46 74.44 61.04 60.31 66.93 66.09 61.65 60.91

4.02 0 2.55 0 3.88 0 2.63 0

41.82 41.52 42.19 42.19. 43.02 43.02 42.19 42.19

5.50 5.50 4.00 4.00 4.00 4.00 4.75 4.75

14.66 14.31 14.07 13.36 18.30 18.30 13.94 13.36

13.48 13.11 0.77 0.76 0.79 0.78 0.61 0.60

As can be seen from Table 6, the mean expected total cost in Case 7 is 610 US$ higher than that in Case 3. The increments of annual amounts of energy supplied to customers at nodes 2 and 16 due to the insertion of the sectionalizer in branch 24 are calculated to be 446 kWh and 267 kWh, respectively. By applying expression (10) we obtain that, 5

Electrical Power and Energy Systems 117 (2020) 105655

J.M. Nahman and D.M. Perić

Table 7 Saidi and saifi indices. Case

SAIDI min./cust. yr.

σ min./cust. yr

SAIFI int./cust. yr.

σ int./cust. yr

1 2 3 4 5 6 7 8

240 237 318 318 318 318 253 253

70 0 93 0 93 0 74 0

1.00 1.00 1.33 1.33 1.33 1.33 1.05 1.05

0.29 0 0.39 0 0.39 0 0.31 0

Fig. 7. Minimum annual cost network solution. Table 8 Optimal network costs, US$. Solution

Investment

Losses

Total

New Reference [1 5 ] Reference [2 6 ]

151,353 151,892 151,892

21,737 21,007 21,021

173,090 172,899 172,913

the total cost of the found optimal solution was calculated to be 3.7%. 4. Conclusions

Fig. 6. Available routes over oil production area.

The main idea of the paper was to investigate the effects the uncertainty of various inputs and different approaches for assessing the load curtailment costs could have in the planning of radial distribution networks. The optimization method applied in [15] has been upgraded by taking into account the uncertainty of various inputs affecting the costs. The results obtained for the network model, which is typical for rural networks in authors’ country, has indicated the following:

for increased reliability of supply, customers at nodes 2 and 16 should be charged annually by 381 US$ and 228 US$, respectively. The failure rate for nodes 2 and 16 in Case 7 is calculated to be 0.53 fl./yr, which is considerably lower than in Case 3 where this rate was 1.36 fl./yr. 3.2. Example II Fig. 6 displays a 34.5 kV network of available routes for feeding an oil production area with 21 load nodes spread over the site. The network is supplied by a 10 MVA substation. Capital cost per year for lines is 10 kUS$/km. Power factor of loads is 0.9, inductive. Maximum allowable voltage drops are 3%. The cost of energy losses is 0.05 kUS $/kWh, and loss factor equals 0.35. Other detailed data on lines and loads are given in [15] and [30]. The active source substation is connected at node 1 and the reserve one at node 2. The task was to find the least expensive radial network configuration concerning the investment and loss cost. Such analysis has been conducted already in [15] and [30] for fixed maximum loads by applying different optimization methods. In this paper we shall use the optimization approach presented before to analyze the effects of uncertainty of loads on the optimal network configuration and its total annual cost. The calculations are performed by modeling the uncertainty of network loads by multiplying them by the same normally distributed scaling function f1 applied in Example I. The minimum cost network structure obtained by optimization is presented in Fig. 7. It slightly differs from the optimal configuration determined in [15] and [30], which is due to the assumed uncertainty of loads. TABLE 8 quotes the mean expected cost of the found optimal solution as well as the costs obtained for the optimal solution found in afore mentioned two papers, for comparison. The standard deviation of

- The same or nearly the same optimal network solutions are obtained with and without taking into account the uncertainties of considered network data. - The alternative approaches in assessing the costs of load curtailment give network solutions that remarkably differ from one other in configuration and total cost. As the load curtailment costs of customers is not easy to assess with sufficient accuracy, the proposed alternative approach based upon the lost income of energy providers due to the undelivered energy, combined with constraints for SAIDI and SAIFI indices, could be considered as a simple and more transparent model for accounting for reliability issues in distribution networks planning. - A way for determining the cost for increasing the reliability of supply to load curtailment sensitive customers has been proposed and demonstrated. - The performed analyzes have shown that WGs can substantially affect the network configuration and decrease its cost and, therefore, should be taken into account in the planning procedure. Declaration of Competing Interest The authors declare that there is no conflict of interest.

6

Electrical Power and Energy Systems 117 (2020) 105655

J.M. Nahman and D.M. Perić

Appendix A. Optimization flow The optimization flow contains the following steps: (1) Store all input data figuring in the presented mathematical model. The already existing lines should be marked by the lowest indices for simple identification. These lines should not be removed in the optimization procedure. (2) Remove from the network the most expensive branch among the new available branches not considered before. (3) Check the connectivity of the created network. If the network is connected go to step 4. Else, do not exclude the selected branch from the network and return to step 2. (4) Check for the formed network configuration the violation of constraints. If any of these constraints is violated leave intact the selected branch and return to step 2 (5) Check if the formed network is radial. If it is not the case return to step 2. (6) Calculate the mean cost for network solution. Check for each feeder if the insertion of a sectionalizer decreases the feeder cost and, if it is the case, keep this solution. (7) Store the mean cost and the network configuration for further considerations. Define the initial “temperature ” Ti. Set i = 1 for the index of the simulated annealing procedure. (8) Include in the network by chance one of the branches removed in previous steps. Remove by chance a branch from the mesh formed by branch inclusion. Generate new ́ temperaturé Ti + 1 = Ti /(1 + b·Ti ) with b > 0 being a preselected number. Set i = i + 1. (9) Check for the violation of network constraints. If there is a violation leave the branch intact and return to step 8. (10) Calculate the network mean cost and associated standard deviation. Check for each feeder if the insertion of a sectionalizer decreases the mean feeder cost and, if it is the case, insert them. (11) Let ΔC be the difference between the cost of the best configuration found so far and the cost of the configuration generated in the previous step. Generate by chance a uniformly distributed random number R ranking from 0 to 1. If exp(ΔC / Ti ) > R , return to step 8 with the created network configuration. Otherwise, leave the branch intact and return to step 8 with the solution best so far. (12) Steps 8 to 11 are performed till i = G, with G being the preselected maximum number of iterations. B. The Hong’s 2m + 1 points estimate method [22,23] Let us consider function Z the arguments of which are m uncorrelated random variables vl , with probability distribution functions Fl having means μl and standard deviations σl , l = 1,…,m. In order to determine the moments of function Z, each function Fl is represented by three characteristic values pl,k

pl, k = μl + ξl, k σl

k = 1, 2

pl,3 = μl

(A1)

Parameters ξl, k and weights ωl, k , scaling the importance of pl,k values in representing function Fl, are determined to model exactly the first four standard moments about mean of this function. Their values are [22]

ξl, k =

λl,3 2

3

+ (−1)3 − k λl,4 − 4 λl2,3

ωl, k =

(−1)3 − k ξl, k (ξl,1 − ξl,2)

ωl,3 =

1 m

−

k = 1, 2

k = 1, 2

1 λl,4 − λl2,3

(A2)

In (A2) λl, j denotes the j-th coefficient of the moment about the mean of random variable vl. As known, λl,1 is zero, λl,2 equals to 1 whereas λl,3 and λl,4 are coefficients of skewness and kurtosis of Fl, respectively. For normal distribution λl,3 is zero and λl,4 equals 3. Characteristic values pl,k, modeling the impact of variable vl, give three estimates of function Z

Z (l, k ) = Z (μ1 , μ 2 , ...,pi, k , ...,μm − 1 , μm ),

k = 1, 2

Z (l, 3) = Z (μ1 , ...,μl , ...,μm )

(A3)

As all random variables generate Z(l,3), the j-th moment of Z can be determined using the following expressions 2

m

E (Z j ) = ∑l = 1 ∑k = 1 ωl, k Z (l, k ) j + ωZ (l, 3) j m

m

ω = ∑l = 1 ωl,3 = 1− ∑l = 1

1 λl,4 − λl2,3

(A4)

The mean value and standard deviation of Z can be calculated using (A4) as

μZ = E (Z ),

σZ =

E (Z 2) − μz2

(A5)

Determined characteristic parameters for applied scaling functions Characteristic values are: for customers’ loads: p1,1 = 1.260, p1,2 = 0.740, p1,3 = 1 for failure rates of lines: p2,1 = 1.433, p2,2 = 0.567, p2,3 = 1 for power generated by WGs: p3,1 = 1.402, p3,2 = 0.610 p3,3 = 1.02 The weights are: ωl, k = 1/6 for l, k = 1,2, ωl,3 = 0 , for l = 1,2,

7

Electrical Power and Energy Systems 117 (2020) 105655

J.M. Nahman and D.M. Perić

ω3,1 = 0.172,

ω3,2 = 0.177,

ω3,3 = −0.016

Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijepes.2019.105655.

simulated annealing technique. IEEE Trans Power Syst May 2008;31(2):790–5. [16] Martins VF, Borges LT. Active distribution network integrated planning incorporating distributed generation and load response uncertainties. IEEE Trans Power Syst 2011;26(4):2164–72. [17] Lotero RC, Contreras J. Distribution system planning with reliability. IEEE Trans Power Delivery 2011;26(4):2552–62. [18] Kavousi-Fard A, Niknam T. Optimal distribution feeder reconfiguration for reliability improvement considering uncertainty. IEEE Trans Power Delivery Juni. 2014.;29(3):1344–53. [19] Munoz-Delgado G, Contreras J, Arroyo JM. Multistage generation and network expansion plammimg in distribution systems considering uncertainty and reliability. IEEE Trans Power Syst 2016;31(5):3715–28. [20] Ahmadigorji M, Amjasy N, Dehghan S. A robust model for multiyear distribution network reinforcement planning based on information-gap decision theory. IEEE Trans Power Syst 2018;33(2):1339–51. [21] Ramirez-Rosado IJ, Dominguez-Navaro JA. Possibilistic model based on fuzzy sets for the multiobjective optimal planning of electric power distribution networks. IEEE Trans Power Syst 2004;19(4):1801–10. [22] Morales JM, Perez-Ruiz J. Point estimate schemes to solve the probabilistic power flow. IEEE Trans Power Syst 2007;22(4):1594–601. [23] Hong HP. An efficient point estimate method for probabilistic analysis. Reliab Eng Syst Safety 1998;59:261–7. [24] Christofides N. Graph Theory. New York: Academic Press; 1975. [25] Heydt GT. Computer Analysis Methods for Power Systems. New York: Macmillan P.C; 1986. [26] “Methods to consider customer interruption costs in power system analysis”, CIGRE Task Force report”, convener R. Billinton, Paris, 2002. [27] Nahman J, Vlajic-Naumovska I, Salamon D. Evaluation of supply interruption costs for residential sector in Serbia. EPSR 2014;110:39–44. [28] 6th CEER Benchmarking Report on the Quality of Electricity and Gas Supply, .2016. [29] Banks J, Carson JS. Discrete-event system simulation. New Jersey: Prentice-Hall; 1984. [30] Da Silva Y, Di Girolano J, Fereirra A. Optimizacion de circuitos aereos de distribucion et campos petroleros. Proc. IEEE Andean Region Conf. 1999;II:967–72.

References [1] Willis HL. Power Distribution Planning Reference Book. New York: Marcel-Decker; 1997. [2] Gönen T. Electric Power Distribution System Engineering. New York: McGraw-Hill; 1986. [3] Lakervi E, Holmes EJ. Electricity Distribution Networks Design. Stevenage: P. Peregrinus; 1995. [4] Brown RE. Electric Power Distribution Reliability. New York: CRC Press; 2009. [5] Gomez JF, et al. Ant colony system algorithm for the planning of primary distribution circuits. IEEE Trans Power Syst 2004;19(2):996–1004. [6] Boulaxis NG, Papadopoulos MP. Optimal feeder routing in distribution system planning using dynamic programming technicue and GIS facilities. IEEE Trans Power Del 2002;17(1):242–7. [7] Nahman J, Spirić J. Optimal planning of rural medium voltage distribution networks. El. Power Energy Syst 1997;19(8):549–56. [8] Jonnavithula S, Billinton R. Minimum cost analysis of feeder routing in distribution system planning. IEEE Trans Power Del Oct 2004;11(4):1935–40. [9] Goswami S. Distribution system planning using branch exchange technique. IEEE Trans Pow Syst 1997;12(2):718–23. [10] Diaz-Dorado E, Cidras J, Miguez E. Application of evolutionary algorithms for the planning of urban distribution networks of medium voltage. IEEE Trans Pow Sys 2002;17(3):879–84. [11] Heidari S, Fotuhi-Firuzabad M, Lehtonen M. Planning to equip the power distribution networks with automation system. IEEE Trans Power Syst 2017;32(5):3451–60. [12] Heidari S, Fotuhi-Firuzabad M, Kazemi S. Power distribution network expansion planning considering distribution automation. IEEE Trans Power Syst 2015;10(3):1261–9. [13] Koutsoukis NC, Georgilakis PS, Hatziargyriou ND. Multistage coordinated planning of active distribution networks. IEEE Trans Power Syst 2018;33(1):32–44. [14] Asensio M, Munoz-Delgado G, Contreras J. Bi-level approach to distribution network and renewable energy expansion planning considering demand response*. IEEE Trans Power Syst 2017;32(6):4298–309. [15] Nahman JM, Perić DM. Optimal planning of radial distribution networks by

8