Keywords

1 Introduction

The distribution system plays a crucial role among the other components of the electrical power system that is generation and transmission. The power system is becoming more and more complex with the increasing demand. In developing countries, power generation is usually insufficient to meet the increasing load demand. Therefore, it is necessary to reduce the total losses in the network of which the major part is contributed by the distribution system. The power industry is adopting deregulation to obtain the economic efficiency of power system operation. In the deregulated scenario, both generation and distribution companies are dedicated to their function [1], which avoids the monopoly and creates a competitive market environment between them. This forces the power utilities to fulfill the energy demand of the consumers at a reasonable cost.

Recently the electric vehicles (EVs) have gained importance due to the increasing air pollution, climate change, and increased oil prices. Distributed energy resources (DERs) such as EVs and distributed generation (DG) are growing as an opportunity to decarbonize the energy system. The necessity of EVs is very clear with their great potential to electrify the transportation sector [2]. The renewable-based DG sources create uncertainty in the power distribution system. At the same time, they also pose new technical challenges to the power system, which can be addressed with increased flexibility. For better utilization of electrical energy, the optimization of both distribution system operation and control becomes necessary. This can be achieved through the automation of the distribution system. One of the methods adopted is the remote control of the configuration by which losses in the branches of the entire system can be minimized. The distribution system reconfiguration is carried out by modifying the topological structure of the network by changing the status of the sectionalizing and tie-line switches [3, 4]. And also, the optimal switch operations may reduce losses in the system. Both of these are met by reconfiguration. Hence, both the ONR and less number of switching operations will reduce the power losses and they are met by the proposed ONR approach.

Most of the distribution systems generally operated in radial topology which enables suitable voltage and power flow control, reduced fault current, and easier protection coordination schemes over the meshed system. Typically, the radial distribution systems (RDSs) have two types of switches, namely, sectionalizing switches which are usually open, and tie-line switches which are usually closed [5]. When the fault occurs either on distributors or feeders, the tie-line switch allows some portion of the faulted part to be restored promptly, thereby enhancing the reliability of the system. According to Ref. [2], the power loss in the distribution network constitutes 70% of the total power loss. Therefore, the major cause of power interruption is due to problems in the distribution system.

1.1 Related Work

There has been considerable interest in the recent past to develop algorithms for feeder reconfiguration (FR) of the distribution system under various operating contingencies. Usually, distribution companies try to keep active power losses below the standard ones to gain profit rather than paying penalties. Thus the active power loss minimization is a major concern of the distribution system researchers, which has a significant impact on the maximum loadability of the network and hence, on the power system stability particularly in overburdened networks. Some of the established techniques to handle distribution systems under such a competitive scenario include network reconfiguration (NR), DG allocation, shunt capacitor placement, and simultaneous NR and DG allocation [6]. Hence, the existing distribution system requires to be optimized to satisfy the demand in the most reliable, economical, and environmentally friendlier way, while meeting the associated geographical or operational constraints.

A high-performance nonlinear sliding mode controller has been proposed in Ref. [7] for an EV charging system to improve the power factor (pf) to handle the unbalanced EV chargers and to compensate for voltage distortions. The super sense genetic algorithm (SSGA) is applied in Ref. [8] to solve the problem of complex combinatorial NR problem of RDSs. An approach for the optimal network rearrangement by incorporating the plugin electric vehicles (PEVs) proposed in Ref. [9] is based on the random programming model of the Monte Carlo simulation method. An approach for optimal placement and sizing of electric vehicle charging stations (EVCSs) on a distribution network is proposed in [10]. A fuzzy approach-based multi-objective heuristic technique for ONR in distribution systems considering the DGs is proposed in [11]. A single-phase (1-φ) EV charging coordination approach with the three-phase (3-φ) supply and chargers connected to the EVs with the less loaded phase of the feeder at the starting of charging has been proposed in [12]. The optimal planning approach of EVCSs and shunt capacitors is proposed in [13] and it is solved by using the dragonfly algorithm (DA).

An equilibrium optimizer algorithm has been applied to the ONR problem in Ref. [14] with loss reduction, voltage magnitude enhancement, and reliability indices improvement objectives. An efficient technique for balanced and unbalanced RDSs optimization by ONR and optimal capacitor placement has been proposed in [15]. The ONR allows better penetration of renewable energy sources (RESs) in the RDS and it is solved in Ref. [16] using the mixed particle swarm optimization (PSO) for loss minimization and voltage profile enhancement improvement. Reference [17] proposes optimal battery energy storage systems and allocation of PV-based DG have been solved by the PSO algorithm.

1.2 Scope and Contributions

From the literature on ONR with loss minimization objective in the distribution system, the research gap has been identified and explored the work area with current research performance and its limitations. From the literature, it has been identified that there is a requirement for solving the ONR problem by simultaneously installing the renewable-based DG units, shunt capacitors, and EVCSs. Renewable-based DG units, i.e., wind plants and solar PV farms have wind speed and solar insolation as input parameters and they are highly intermittent. The distribution load flow (DLF) used in this chapter is based on the iterative approach. The potential of this approach has made the ONR approach is very powerful and can be applied to any size of the distribution network. The ONR and DG allocation to strengthen the efficiency of distribution systems based on power loss minimization and voltage deviation minimization, as these are two major issues in the recent competitive power scenario, and they are considered as the objective functions with the presence of shunt capacitors and EVCSs. The simulation has occurred to both balanced as well as unbalanced radial distribution systems (RDSs).

This chapter is organized as follows: The description of RDS, ONR, and the summary of the literature work has been presented in Sect. 1. A brief description of distribution load flow (DLF) analysis has been presented in Sect. 2. Section 3 describes the modeling of shunt capacitors and EVCSs in the distribution system. Problem formulation is presented in Sect. 4. Section 5 describes the solution methodology. Section 6 describes the results and discussion on the 69 bus test system. The conclusions of this chapter have been summarized in Sect. 7.

2 Distribution Load Flow (DLF) Analysis

The analysis of DLF is basic but it is an essential mathematical tool for the analysis of distribution systems in both the planning and operational stages. The primary aim of the power flow analysis is to determine the magnitude and phase of steady-state voltage at all buses, active and reactive power flows in each line, for a specified loading. There are numerous power flow methods like Newton Raphson, Gauss-Seidel, fast decoupled methods, and many more methods with a modification in conventional ones. Due to the different properties of the distribution systems, these methods are not suitable for load flow analysis [18]. Certain applications including distribution automation and power system optimization require efficient and robust load flow solutions. Over the last few decades, these load flow methods have been evolved in several different dimensions to handle both static and dynamic power distribution system problems. Traditionally, the Backward Forward Sweep (BFS) load flow techniques are applied to distribution systems and it has two-step analyses. Other load flow methods include little modification into existing techniques for their advantage over the older ones. In literature, several conventional methods have been utilized to solve distribution system problems [19]. The open branches are electrically represented with very high impedance. Whenever a branch is connected, then its parameters are replaced with actual values (i.e., resistance and reactance values) and vice versa when a branch is removed or disconnected. In other words, when a branch exchange takes place, only those parameters will be modified accordingly for further processing. This saves a lot of computation burden.

The proposed load flow solution presented in this chapter depends on an iterative approach of the receiving end voltage of the RDS. It is successfully applied on ill-conditioned RDS with consideration of realistic load [20]. In the first step, the effective power at each bus is determined after forming adjacent branches and adjacent node matrices. A sparse technique is used to determine the branches and nodes beyond a particular node. The detailed mathematical formulation is given below considering the electrical equivalent of a branch connected between the nodes a and b of RDS, and it is shown in Fig. 1.

Fig. 1
figure 1

Electrical equivalent of a typical distribution system branch

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The amount of current flowing from node a to node b can be expressed as [20],

$$ I_{ab} = \frac{{\left| {V_{a} } \right|\angle \delta_{a} - \left| {V_{b} } \right|\angle \delta_{b} }}{{R_{ab} + jX_{ab} }} = \frac{{P_{b} - jQ_{b} }}{{\left( {\left| {V_{b} } \right|\angle \delta_{b} } \right)^{*} }} $$
(1)

where active power (\(P_{b}\)) can be expressed in terms of active power load at a bus/node i (\(P_{Di}\)) and active power loss of line k (\(P_{loss,k}\)). Mathematically, it can be expressed as,

$$ P_{b} = \mathop \sum \limits_{i = 1}^{{N_{b} }} P_{Di} + \mathop \sum \limits_{k = 1}^{{N_{br} }} P_{loss,k} $$
(2)

where \(N_{b}\) shows all the buses beyond the bus b. \(N_{br}\) shows all the branches beyond the bus b. From Eq. (1), \(P_{b}\) can be expressed as [21],

$$ P_{b} = \frac{{\left| {V_{a} } \right|\left| {V_{b} } \right|\sin \left( {\delta_{a} - \delta_{b} } \right) + R_{ab} Q_{b} }}{{X_{ab} }} $$
(3)

From the above equation, the voltage magnitude (\(\left| {V_{b} } \right|\)) and angle (\(\delta_{b}\)) at the end of receiving node can be calculated by using,

$$ \begin{aligned} \left| {V_{b} } \right| & = - \left[ {\left| {V_{a} } \right|\left( {\frac{{R_{ab} }}{{X_{ab} }}\sin \delta - \cos \delta } \right)} \right] \\ & \quad + \left[ {\left( {\left| {V_{a} } \right|\frac{{R_{ab} }}{{X_{ab} }}\sin \delta - \cos \delta } \right)^{2} - 4Q_{b} \left( {\frac{{R_{ab}^{2} }}{{X_{ab} }} + X_{ab} } \right)} \right]^{1/2} \\ \end{aligned} $$
(4)

where \(\delta = \delta_{a} - \delta_{b}\).

$$ \delta_{b} = \delta_{a} - \tan^{ - 1} \left[ {\frac{{P_{b} X_{ab} - Q_{b} R_{ab} }}{{\left| {V_{b} } \right|^{2} + P_{b} R_{ab} + Q_{b} X_{ab} }}} \right] $$
(5)

The active and reactive power losses are calculated by using Eq. (3), and they are expressed as,

$$ P_{loss,ab} = \frac{{\left( {P_{b}^{2} + Q_{b}^{2} } \right)R_{ab} }}{{\left| {V_{b} } \right|^{2} }} $$
(6)
$$ Q_{loss,ab} = \frac{{\left( {P_{b}^{2} + Q_{b}^{2} } \right)X_{ab} }}{{\left| {V_{b} } \right|^{2} }} $$
(7)

All such techniques work well with static systems where there is no change in the topology of the network [22]. Again under critical loading conditions, there is no guarantee of their convergence. Even in converged cases, these methods are very inefficient in respect of storage requirements and solution speed. Moreover, for dynamic systems, it is a challenge to arrange the line data as per the load flow requirement and to maintain the radiality, and ensure connectivity. This necessitates the utilization of improved data structure-based techniques.

3 Modeling of Shunt Capacitor and EVCS in the Distribution System

The ever-growing population leads to a significant increment in customer load demand. It leads to the placement of DGs in RDS being nearer to the load demand. Among the various renewable-based DGs, solar PV and wind energy are widely used as they are abundantly available. As there is a rapid growth in load demand, the line losses in the distribution network are quite high and need to be taken care of [23]. Various techniques have been implemented to RDSs apart from the DG penetration to optimize the power losses in the RDS. This section presents the modeling of shunt capacitors and EVCSs in the RDS.

3.1 Modeling of Shunt Capacitor

Shunt capacitors supply the amount of reactive power to the RDS at the bus where they are connected. This in turn causes a reduction in reactive power flowing in the line. If the reactive power and the system voltage are assumed to be constant, then the losses are inversely proportional to the power factor, and hence improvement in the power factor causes a reduction in system losses. The other benefits of installing the shunt capacitors are voltage profile improvement, decrease in kVA loading, and reduces system improvement cost/kVA of load supplied [24]. And also, to overcome the compensation during the light load conditions, the automatic switching units can be provided but this switching equipment is costly and this, in turn, will limit the number of capacitors and thus the minimum capacity of the capacitor bank that has to be provided on the feeder.

The placement of shunt capacitors in the DS reduces the system losses, enhances the voltage profile, and also corrects the power factor. Figure 2 depicts the representation of the shunt capacitor in the DS. This capacitor injects reactive power (\(Q_{c}\)) into the system.

Fig. 2
figure 2

Representation of shunt capacitor in the distribution system

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Amount of reactive power injected at bus b (\(Q_{inj,b}\)) can be expressed by [25],

$$ Q_{inj,b} = Q_{Db} - Q_{c} $$
(8)

Now the active power loss with shunt capacitor (\(P_{loss,ab}^{c}\)) can be expressed as [25],

$$ P_{loss,ab}^{c} = \frac{{\left( {P_{b}^{2} + Q_{inj,b}^{2} } \right)R_{ab} }}{{\left| {V_{b} } \right|^{2} }} = \frac{{\left[ {P_{b}^{2} + \left( {Q_{Db} - Q_{c} } \right)^{2} } \right]R_{ab} }}{{\left| {V_{b} } \right|^{2} }} $$
(9)
$$ P_{loss,ab}^{c} = \frac{{\left( {P_{b}^{2} + Q_{b}^{2} } \right)R_{ab} }}{{\left| {V_{b} } \right|^{2} }} + \frac{{\left( {Q_{c}^{2} - 2Q_{Db} Q_{c} } \right)R_{ab} }}{{\left| {V_{b} } \right|^{2} }} = P_{loss,ab} + \Delta P_{loss,ab}^{c} $$
(10)

where \(\Delta P_{loss,ab}^{c}\) is the reduction in power loss, i.e., active power loss before and after placing the shunt capacitor [26], and it can be expressed from Eq. (10) as,

$$ \Delta P_{loss,ab}^{c} = \frac{{\left( {Q_{c}^{2} - 2Q_{Db} Q_{c} } \right)R_{ab} }}{{\left| {V_{b} } \right|^{2} }} $$
(11)

3.2 Modeling of EVCS in the Distribution System

Figure 3 depicts the representation of EVCS in the distribution system.

Fig. 3
figure 3

Representation of EVCS in the RDS

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The power demand of EVCS at bus b (\(P_{Db}^{EVCS}\)) can be calculated by using [27, 28],

$$ P_{Db}^{EVCS} = N_{EV}^{c} P_{G2V} \eta_{c} R_{c} - N_{EV}^{d} P_{V2G} \eta_{d} R_{d} $$
(12)

4 Problem Formulation

This section presents the general description of distribution systems and the mathematical modeling of optimal feeder reconfiguration (OFR) or optimal network reconfiguration (ONR). Factors that are affecting the increase in the power losses in the distribution network are feeder length, low voltage, low power factor, poor workmanship in fittings, and reduction of line losses. Various methods used for the reduction of distribution system losses are the construction of a new substation, reinforcement of feeder, reactive power compensation, HV distribution system, grading of conductors, and feeder reconfiguration [29]. In this work, two objectives, i.e., real power loss and voltage deviations are modeled with fuzzy sets [29, 30]. Some heuristics are developed to reduce the number of tie-line switching operations.

4.1 Fuzzy Membership Function for Active Power Loss Reduction (\(\mu L_{i}\))

The basic purpose for membership function, i.e., objective in the fuzzy domain is to minimize the active power loss of the system. The variable \(\alpha_{i}\) can be defined as

$$ \alpha_{i} = \frac{{P_{loss} \left( i \right)}}{{P_{loss}^{0} }}\quad for\,i = 1,2, \ldots ,N_{k} $$
(13)

where \(N_{k}\) is total number of lines in loop including the tie-line, when the kth tie-switch is closed, \(P_{loss} \left( i \right)\) is total active power loss when ith line in the loop is opened, and \(P_{loss}^{0}\) is total real power loss before the NR. Membership/objective function for active power loss reduction (\(\mu L_{i}\)) can be written as [30],

$$ \mu L_{i} = \left\{ {\begin{array}{*{20}l} {\frac{{\alpha_{\max } - \alpha_{i} }}{{\alpha_{\max } - \alpha_{\min } }}} \hfill & {for\,\alpha_{\min } < \alpha_{i} < \alpha_{\max } } \hfill \\ 1 \hfill & {for\,\alpha_{i} \le \alpha_{\min } } \hfill \\ 0 \hfill & {for\,\alpha_{i} \ge \alpha_{\max } } \hfill \\ \end{array} } \right. $$
(14)

4.2 Fuzzy Membership Function for Maximum Node Voltage Deviation (\(\mu V_{i}\))

The main aim of this function is to minimize the deviation of nodes’ voltage. The variable \(\beta_{i}\) can be expressed as

$$ \beta_{i} = \max \left( {\left| {V_{i,j} - V_{s} } \right|} \right)\quad for\,i = 1,2, \ldots ,N_{k} ;\;\;j = 1,2, \ldots ,N_{B} $$
(15)

where \(N_{B}\) is total number of buses in RDS, \(V_{s}\) is substation voltage, and \(V_{i,j}\) is jth bus voltage corresponding to the opening of the ith line [29, 30]. The fuzzy membership function for maximum bus voltage deviation (\(\mu V_{i}\)) can be expressed as,

$$ \mu V_{i} = \left\{ {\begin{array}{*{20}l} {\frac{{\beta_{\max } - \beta_{i} }}{{\beta_{\max } - \beta_{\min } }}} \hfill & {for\,\beta_{\min } < \beta_{i} < \beta_{\max } } \hfill \\ 1 \hfill & {for\,\beta_{i} \le \beta_{\min } } \hfill \\ 0 \hfill & {for\,\beta_{i} \ge \beta_{\max } } \hfill \\ \end{array} } \right. $$
(16)

4.3 Constraints

The active and reactive power balances of the RDS system including the DG units, shunt capacitors, and EVCSs are expressed as [31, 32],

$$ P_{D} = P_{G}^{Grid} + P_{Db}^{EVCS} + \mathop \sum \limits_{i = 1}^{{N_{DG} }} P_{DG,i} $$
(17)
$$ Q_{D} = Q_{G}^{Grid} + \mathop \sum \limits_{i = 1}^{{N_{DG} }} Q_{DG,i} + \mathop \sum \limits_{j = 1}^{{N_{c} }} Q_{c,j} $$
(18)

Voltages at each bus can be expressed as,

$$ 0.95 \le V_{i} \le 1.05 $$
(19)

Active and reactive powers of DG units can be expressed as [33, 34],

$$ P_{DG,i}^{\min } \le P_{DG,i} \le P_{DG,i}^{\max } $$
(20)
$$ Q_{DG,i}^{\min } \le Q_{DG,i} \le Q_{DG,i}^{\max } $$
(21)

4.4 Selection of Best-Compromised Solution

When optimizing two or more objectives simultaneously, a best-compromised solution needs to be determined [35]. The procedure for determining the best-compromised solution using the min-max principle is determined next:

The membership function values of the two objectives are determined. When the kth tie-line switch of RDS is closed, a loop is formed with number of lines in the loop \(N_{k}\). After opening the ith line in the loop, run the DLF to determine \(\mu L_{i}\) and \(\mu V_{i}\) for i = 1, 2, …, \(N_{k}\). Determine the fuzzy decision for overall satisfaction [36, 37] by using,

$$ D_{k,i} = \min \left( {\mu L_{i} , \mu V_{i} } \right)\quad for\,1,2, \ldots ,N_{k} $$
(22)

The optimal solution is the maximum of overall degrees of satisfaction, and it is expressed as [38],

$$ OS_{k} = \max \left( {D_{k,i} } \right)\quad for\,1,2, \ldots ,N_{k} $$
(23)

5 Solution Methodology

This section presents heuristics for minimizing the number of operations of tie-line switches. Here, heuristic rules are developed to minimize the number of tie-line switch operations [39, 40]. The flow chart of the proposed solution methodology has been depicted in Fig. 4, and the step-by-step approach is presented next:

Fig. 4
figure 4

Flow chart of the proposed ONR/OFR algorithm

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  • Step 1: Read the RDS test system data.

  • Step 2: Execute the load flow solution as described in Sect. 2.

  • Step 3: Determine voltage difference (\(\Delta V_{tie}\)) across the open tie-line switches.

  • Step 4: Identify the open tie-line switch across which \(\Delta V_{tie}\) is maximum, and it can be represented as (\(\Delta {\text{V}}_{tie}^{\max }\)).

  • Step 5: If \(\Delta {\text{V}}_{tie}^{\max }\) > specified value (ε), then go to Step 6 else go to Step 11.

  • Step 6: When the selected tie-line switch is closed then identify the number of branches (\(N_{k}\)) in the loop.

  • Step 7: Open one line at a time in the loop, and determine the membership value for each objective function. Compute \(\mu L_{i}\) and \(\mu V_{i}\) using the Eqs. (14) and (16), respectively.

  • Step 8: Compute the overall degree of satisfaction using Eq. (22).

  • Step 9: Determine the optimal solution for the operation of the kth tie-line switch using Eq. (23).

  • Step 10: Make the number of tie-line switches (\(N_{tie}\)) equal to \(N_{tie} - 1\), and rearrange the coding of the rest of the tie-line switches, and go to Step 2.

  • Step 11: Display the output results.

6 Results and Discussion

The proposed ONR methodology has been implemented on IEEE 69 bus test system which has a single feeder with a single substation [41]. Figure 5 depicts the single-line diagram of 69 bus RDS. System load demand, line, and tie-line data have been taken from Ref. [42]. This system has 68 lines, i.e., sectionalizing switches, they are 1–68 and they are normally closed. Five tie-line switches (which form 5 loops) considered in this work are 69, 70, 71, 72, and 73, they open tie switches. The base voltage and kVA are 12.66 kV and 1000 kVA, respectively. The real and reactive power load of 3,802 kW and 2,694 kVAr, respectively. In this test system, the DG units are placed at buses 5, 28, 45, and 60; shunt capacitors are placed at buses 22, 36, and 64; EVCSs are placed at buses 18 and 59.

Fig. 5
figure 5

SLD of IEEE 69 bus RDS before ONR

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In the present work, the convergence criterion (\(\varepsilon\)) is considered as 0.01, and it has been assumed that \(\alpha_{\min }\) is 0.5, \(\alpha_{\max }\) is 1, \(\beta_{\min }\) is 0.05 and \(\beta_{\max }\) is 0.10. The active power loss obtained in the base case condition is 224.96 kW, and all the tie-line switches, i.e., 69, 70, 71, 72, and 73. The minimum voltage obtained in this base case is 0.9066 p.u. at bus 54.

6.1 Case 1: Tie-Line Switch Operation 1

In this case, the voltage difference across each tie-line switch is determined. The voltage differences across tie-line switches 69, 70, 71, 72 and 73 are 0.0031 p.u., 0.0008 p.u., 0.0416 p.u., 0.0742 p.u. and 0.0471 p.u., respectively. From these voltages, it can be observed that voltage difference across line number 72 is maximum, i.e., 0.0742 p.u. Table 1 presents the membership values of power loss and voltage deviation, and the overall satisfaction for tie-line switch operation 1. In this case, line number 72 is closed and the membership values for opened lines are presented in Table 1. The overall satisfaction has been determined by using Eq. (23), and they are presented in the table. From the results obtained, it is observed that by using the fuzzy set intersection, the fuzzy decision for overall satisfaction is obtained when line 46 is open and line 72 is closed. The obtained value of overall satisfaction is 0.7362, which is the maximum of \(D_{k,i}\).

Table 1 Membership values for tie-line switch operation 1
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6.2 Case 2: Tie-Line Switch Operation 2

The voltage differences across tie-line switches 69, 70, 71, 72 and 73 are 0.00312 p.u., 0.0008 p.u., 0.0416 p.u., 0.0742 p.u. and 0.0471 p.u., respectively. From these voltages, after the tie-line switch operation 1, it can be observed that voltage difference across line number 73 is maximum, i.e., 0.0471 p.u. Table 2 presents the membership values of power loss and voltage deviation, and the overall satisfaction for tie-line switch operation 2. In this case, line number 73 is closed and the membership values for opened lines are presented in Table 2. The overall satisfaction has been determined by using Eq. (23), and they are presented in the table. From the results obtained, it is observed that by using the fuzzy set intersection, the fuzzy decision for overall satisfaction is obtained when line 53 is open and line 73 is closed. The obtained value of overall satisfaction is 0.7575 which is the maximum of \(D_{k,i}\).

Table 2 Membership values for tie-line switch operation 2
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6.3 Case 3: Tie-Line Switch Operation 3

The voltage differences across tie-line switches 69, 70, 71, 72 and 73 are 0.00312 p.u., 0.0008 p.u., 0.0416 p.u., 0.0742 p.u. and 0.0471 p.u., respectively. From these voltages, after the tie-line switch operations 1 and 2, it can be observed that voltage difference across line number 71 is maximum, i.e., 0.0416 p.u. Table 3 presents the membership values of power loss and voltage deviation, and the overall satisfaction for tie-line switch operation 3. In this case, line number 71 is closed and the membership values for opened lines are presented in Table 3. The overall satisfaction has been determined by using Eq. (23), and they are presented in the table. From the results obtained, it is observed that by using the fuzzy set intersection, the fuzzy decision for overall satisfaction is obtained when line 13 is open and line 71 is closed. The obtained value of overall satisfaction is 0.8712 which is the maximum of \(D_{k,i}\).

Table 3 Membership values for tie-line switch operation 3
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6.4 Case 4: Tie-Line Switch Operation 4

The voltage differences across tie-line switches 69, 70, 71, 72 and 73 are 0.0031 p.u., 0.0008 p.u., 0.0416 p.u., 0.0742 p.u. and 0.0471 p.u., respectively. From these voltages, after the tie-line switch operations 1, 2, and 3, it can be observed that voltage difference across line number 69 is maximum, i.e., 0.0031 p.u. However, this voltage difference is less than ε (0.01). Therefore, there is no further network reconfiguration is required. Figure 6 depicts the final topology of IEEE 69 bus RDS after ONR. Bus voltages before and after the ONR are presented in Table 4. From this table, it can be observed the voltage profile has been improved after the proposed ONR approach.

Fig. 6
figure 6

SLD of IEEE 69 bus RDS after ONR

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Table 4 Bus voltages before and after the ONR
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7 Conclusions

This paper proposes an optimal network/feeder reconfiguration (ONR/OFR) problem of the radial distribution system (RDS), and it is solved by simultaneously allocating the distributed generation (DG), shunt capacitors, and electric vehicle charging stations (EVCSs). In the proposed ONR problem, the objectives, i.e., active power loss and voltage deviation minimizations are solved by using the fuzzy-based multi-objective methodology. An iterative approach-based distribution load flow (DLF) has been used in this work. The proposed algorithm identifies the ONR of feeders with the minimum number of tie-line switch operations. Simulation studies have been performed on 69 bus RDS.