1 Introduction

Starting from the first reported method by Merlin and Back [1] in 1975, power distribution network reconfiguration (PDNR) methods have travelled a long path from the early single objective, computationally slow and mostly heuristic methods to the present day modern, multi-objective stochastic PDNR methods equipped with superfast simulators and latest visualizing tools. Looking at the vast number of published papers on this topic during the last three decades, one can always say that PDNR is undoubtedly one of the most discussed power system optimization problems for the researchers. The two fundamental challenges of PDNR methods for radial distribution networks (RDNs), as pointed out by Abadei and Kavasseri [2], are the extremely large combinatorial solution space and the requirement of a very fast loss estimation technique for repeated and continuous evaluation of each these configurations. Moreover, the gradual emergence of smart distribution systems with growing penetration of distributed generators (DGs) and modern FACTS devices, advance applications of information and communication techniques (ITC) for ensuring high-quality reliable power distribution, availability of new meta-heuristic optimization techniques and growing stochastic environment have, in fact, made the PDNR methods more challenging. Thus, initially only meant for overall active power loss reduction and feeder load balancing, the present day PDNR methods serve several other objectives (including even some conflicting objectives!!) with a clear focus on improving reliability and power quality indices in a stochastic environment.

Several review papers are available on PDNR methods in the literature. In 1994, Sarfi et al. [3] presented a survey of the state of the art in PDNR methods, which is quite old. Enacheanu et al. [4] presented a useful survey on encoding techniques for network topology meant for various meta-heuristic PDNR methods. Santos et al. [5] presented a very brief literature review of various PDNR approaches. Tsai and Hsu [6] briefly reviewed the various multi-objectives used in PDNR analysis. In 2011, Ferdavani et al. [7] presented a review of PDNR methods through heuristic approaches only. In [8], a thorough comparative analysis of population-based AI techniques for PDNR is presented. However, this population-based AI techniques include the genetic algorithm, particle swarm optimization and ant colony optimization only. Lavarato et al. [9] presented a brief literature review of the PDNR methods imposing radiality constraints. Guedes et al. [10] presented a short literature review of papers with a classification of PDNR methods as single or multi-objective. Kalambe and Agnihotri [11] presented a bibliographic review of the loss minimization methods in power distribution systems with PDNR as one among the various loss minimization options, so it could not throw lights on all related aspects of PDNR methods. Some of the patents on PDNR methods can also be seen [12–16].

In the light of above developments, this paper attempts to comprehensively review and classify the PDNR methods by tracing the evolution of the method with gradual automation of distribution system (DS) and the emergence of smart grid. This review has the following distinct features.

  • It traces the development of the modern multi-objective PDNR methods from the early single objective PDNR.

  • It classifies the PDNR approaches and presents a detailed chronological review.

  • It reviews the PDNR results with the most commonly referred test DSs.

  • It presents current trends in research and future research directions concerning PDNR.

2 PDNR methods: mono objective to multi-objectives

In 1975, Merlin and Back [1] took up the PDNR problem as finding the minimum spanning tree (MST) with least total active power loss and ever since; even today, minimization of power loss (PL) has remained the single most important objective. In fact, the main objective of PDNR, in normal conditions, is to minimize PL and in emergency, following a fault or during a planned outage, is to provide services to as many customers as possible, known as service restoration (SR). Energy loss minimization (ELM) is also considered instead of loss minimization (LM) in some papers. The early PDNR methods also considered feeder load balancing (FLB) or transformer load balancing (TLB), prevention of transformer and feeder overload (TFO), minimizing switching cost (SWC) and number of switching (NS), maximization of voltage stability index (VSI) as the other objectives but mostly achieved via LM.

2.1 Mono objective PDNR

Most of the papers reviewed consider the PDNR problem as a single objective of LM subject to current, voltage and radiality constraints (SOLMCVRC) as [4]:

$$\begin{aligned} { Min}\sum \limits _{b=1}^{N_{br}} {\left| {I_b} \right| } ^2{\cdot } R_b{\cdot }K_b\rightarrow \hbox {Minimization of PL}. \end{aligned}$$
(1)

Subject to constraints

$$\begin{aligned}&\displaystyle K_b\cdot \left| {I_b} \right| <\mathrm{I}_{\mathrm{m}\, ax} \rightarrow \hbox {Current constraints}\end{aligned}$$
(2)
$$\begin{aligned}&\displaystyle V_{i \min } <V_{i}<V_{i\,\max } \rightarrow \hbox {Voltage constraint}\end{aligned}$$
(3)
$$\begin{aligned}&\displaystyle g_I (I,k)=0\rightarrow \hbox {Kirchoff}' \hbox {s current law (KCL)}\end{aligned}$$
(4)
$$\begin{aligned}&\displaystyle g_V (V,k)=0\rightarrow \hbox {Kirchoff}' \hbox {s voltage law (KVL)}\end{aligned}$$
(5)
$$\begin{aligned}&\displaystyle \psi (k)=0\rightarrow \hbox {Radial topology constraints}. \end{aligned}$$
(6)

In above, \(N_{br}\) is the number of closed branches. \(R_{b}\), \(I_{b}\) and \(K_{b}\) are resistance, the current and topological status of the closed branch respectively. \(V_{i}\) is the voltage of the ith bus and \(V_{i,min}\) and \(V_{i,max}\) are the concerned minimum and maximum bus voltages respectively. The constraints (such as voltage or current as explained above) can be formulated as penalty functions in the objective function e.g Zhang et al. [17] considered LM as the main component whereas bus voltage deviation (BVD) and branch current limit (BCL) as the penalty functions given below:

$$\begin{aligned} M=PL+p_{1}.f_V+p_{2}.f_{C}. \end{aligned}$$
(7)

Here, \(f_{V}\) and \(f_{C}\) are the penalty functions for BVD and BCL respectively; \(p_{1}\) and \(p_{2}\) are the corresponding penalty coefficients. Gomes et al. [18] minimized the total cost of PDNR with two components such as cost for PL and branch utilization cost. Siti et al. [19] combined minimization of PL with phase and load balancing (PLB). Bahadoorsingh et al. [20] formulated an objective function for PDNR to minimize financial losses due to voltage sags without including cost due to power loss as

$$\begin{aligned} M=Min\left| {\sum \limits _{b=1}^{N_{b}} {f\cdot P_{c}\cdot P_{f}\cdot C} } \right| \end{aligned}$$
(8)

where f is the frequency of voltage sags at a particular site, \(P_{c}\) is the probability of a particular load composition at the site, \(P_{f}\) is the probability of equipment/process failure, C is the cost associated with the tripping of the equipment/process and \(N_{b}\) is the number of buses of interest. Prasad et al. [21] proposed an objective as combination of LM and maximization of system load balancing as:

$$\begin{aligned} LB_{sys}=\frac{1}{N_{br}}\sum \limits _{j=1}^{N_{br}} {\frac{S_{j}}{S_{jmax}}}. \end{aligned}$$
(9)

In above, \(S_{j}\) is the maximum apparent power flowing through branch-j and \(S_{jmax}\) is the maximum capacity of the branch-j. Carcamo-Gallardo et al. [22] defined energy not supplied (ENS) and set it as the PDNR objective. Cebrian and Kagan [23] considered to minimize the cost of energy loss (EL) with power quality costs like long duration interruptions (LDI) and customer process disruptions (CPD) due to voltage sags. Shariatkhah et al. [24] proposed a PDNR to minimize a cost function due to EL, customer interruption (CI) and SWC. Rosetti et al. [25] combined the minimization of EL with optimized DG allocation. Ghasemi and Moshtagh [26] combined several objectives in terms of minimizing cost as:

$$\begin{aligned} M=\min (LC+SWC-CIV). \end{aligned}$$
(10)

Here, LC, SWC and CIV represent the loss cost, switching cost and cost saved due to voltage profile improvement respectively. In norm-2 type, several objectives are considered as a vector and its distance between the vector containing the worst values of corresponding objectives are maximized. In PDNR case, these worst values refer to the values of the objective functions with the initial configuration. Niknam [27] and Niknam [28] used this approach to minimize PL, BVD, NS and FLB as below:

$$\begin{aligned}&Maximize\,\, J\nonumber \\&\quad =\sqrt{( {PL-PL_{0}})^2+( {BVD-BVD_{0}})^2+( {NS-NS_{0}})^2+( {FLB-FLB_{0}})^2} \end{aligned}$$
(11)

PL \(_{0}\), BVD \(_{0}\), NS \(_{0}\) and FLB \(_{0}\) are the values of the corresponding objective functions before reconfiguration. Thus, various objectives in PDNR can be classified [22] as a) minimization of PL or EL and b) minimization of restoration time c) optimization of reliability and power quality based cost functions.

2.2 Multi-objective PDNR methods

In literature several approaches are adopted when more than one objective are dealt in PDNR. The major approaches are discussed here.

2.2.1 Weighted sum multi-objective (WSMO) PDNR

Roytelman et al. [29] proposed a WSMO PDNR formulation to minimize PL, worst voltage drop (WVD), service interruption frequency (SIF), TLB and balanced service of important customers (BSIC).

$$\begin{aligned} M=w_{1} \cdot P_{Loss}+w_{2} \cdot { TLB}+w_{3} \cdot { WVD}+w_{4} \cdot { SIF}+w_{5} \cdot { BSIC} \end{aligned}$$
(12)

Santos et al. [5] formulated a WSMO based objective function considering PL, NS and number of out of service loads (NOSL). Macedo Braz and DeSouza [30] proposed a WSMO with objectives to minimize EL and NS. Zhang et al. [31] proposed reliability oriented WSMO PDNR with the following objectives

$$\begin{aligned} M=w_{1} \cdot { PL}+w_{2} \cdot { EENS}+w_{3} \cdot { SAIFI}+w_{4} \cdot { SAIDI}+w_{5} \cdot { ASAI} \end{aligned}$$
(13)

Where, EENS\(\rightarrow \) Expected Energy Not Supplied, SAIFI \(\rightarrow \)System Average Interruption Frequency Index, SAIDI\(\rightarrow \) System Average Interruption Duration Index and ASAI \(\rightarrow \) Average Service Availability Index.

However, in order to tackle the imprecision and ambiguity in reliability inputs, electrical parameters, and load data interval analysis is adopted in the paper [31]. Jazebi and Vahidi [32] considered the power quality based indices such as minimizing total harmonic distortion (THD) of the critical buses of the system and minimizing the voltage sag (VS) for a WSMO PDNR, where the individual objectives are normalized suitably. Ptfischer et al. [33] proposes analytic hierarchy analysis (AHA) for multi-criteria decision making considering objectives like minimizing EL, expected SAIFI (ESAIFI), expected ENS (EENS) subjected to constraints like over current limit, minimum and maximum bus voltage magnitudes. Benardon et al. [34] proposed a reliability-oriented WSMO as

$$\begin{aligned} M=w_{1} \cdot { EL}+w_{2} \cdot { EENS}+w_{3} \cdot { ESAIFI} \end{aligned}$$
(14)

Recently, Duan et al. [35] proposed a WSMO for objectives like minimizing PL, and reliability indices like SAIDI, SAIFI and EENS.

2.2.2 Fuzzy multi-objective (FMO) PDNR

When several objectives are simultaneously considered for optimization, it is preferred to go for optimization with soft limits (rather than hard limits) in terms of fuzzy membership values (MV), the approach popularly known as fuzzy multi-objective (FMO). Zhou et al. [36] proposed to minimize NS for RS and FLB by defining three fuzzy MVs. Another example as formulated by Huang [37] is

$$\begin{aligned} Max\,M = w_{1} \cdot \mu _{p}+w_{2} \cdot \sum \limits _{i=1}^{nb} {\mu _{vi}} +w_{3} \cdot \sum \limits _{i=1}^{nbr} {\mu _{ii}} +w_{4} \cdot \mu _{S} \end{aligned}$$
(15)

Where M is the objective function to be maximized and \(\mu _{P}\), \(\mu _{Vi}\), \(\mu _{Ii}\), \(\mu _{S }\)are the MVs to minimize the PL; sum of MVs to minimize the BVDs; sum of MVs to minimize the current violation and NS respectively. The corresponding weighting factors are \(w_{1}, w_{2}, w_{3}\) and \(w_{4}\) respectively. Hsiao [38] proposed an FMO for PL, BVD, NS and ensuring the reliability of services (ERS) in terms of minimizing the capacity margin of feeders and transformers. Venkatesh et al. [39] proposed an FMO PDNR for PL and VDI. Das [40] proposed an FMO formulation of PDNR method to minimize power loss (PL), bus voltage deviation (BVD), branch current loading (BCL) with FLB. Bernardon et al. [41] proposed a FMO PDNR to minimize PL and number of interrupted customers per year (NICY) subject to voltage, current and radiality constraints. Falagi et al. [42] formulated FMO PDNR with MFs defined for EENS and cost of sectionalizing switches (CSS) with DGs. Gupta et al. [43] proposed an FMO by formulating MVs for PL, BVD, BCL and NS. To take care of the changing degree of fuzzy MFs at different operating conditions, Grey Co-Relation Analysis (GCRA) based MO optimization is used by Tsai and Hsu [6]. Hooshmand and Soltani [44] implemented jointly PDNR and phase balancing as an FMO formulation defining three MFs for objectives LM, minimization of feeder neutral current (FNC) and phase balancing index (PBI). Malekpour et al. [45] proposed FMO with 4 objectives to optimize EL, the cost of electricity generation (CEG), emissions produced (CEP) and BVD.

2.2.3 Pareto optimal multi-objective (POMO) PDNR methods

As the DSs are continuously marching towards becoming smart grids, the reliability and power quality based objectives are becoming prominent, which are often conflicting in nature. The optimization problem gets complicated with such conflicting objectives. In such situations, when better values for all the individual objectives do not exist, non-dominant solutions having best possible trade-off among objectives are preferred, the set of such solutions is known as pareto front and are often preferred to single solutions because they can be very useful when considering problems of practical nature [46]. WSMO approach is known as convex pareto front approach, the \(\upvarepsilon \)-constrained MO is another approach where one of the objectives is made as the main target for minimization which sets a limit or \(\upvarepsilon \) for all other objectives. The \(\upvarepsilon \) can be decreased in a step wise manner so that various non-dominated solutions are obtained. Chiang and Jumeau [47, 48] proposed a two-stage \(\upvarepsilon \)-constrained PDNR with LM and FLB as the two objectives. Population-based meta heuristic optimization methods have the inherent advantages of obtaining a direct pareto front iteratively. Some examples of evolutionary methods capable of POMO are non-dominated sorting genetic algorithm (NSGA), pareto archived evolution strategy (PAES), micro-genetic algorithm (\(\mu \)GA), strong pareto genetic algorithm (SPGA), multi-objective particle swarm optimization (MOPSO), multi-objective tabu search optimization (MOTO) etc. [49, 50].

Mendoza et al. [51] considered two objectives, to minimize a cost function defined and ENS for pareto optimization. Mendoza et al. [49] considered four objectives: PL, SAIFI, system average interruption unavailability index (SAIUI), system average duration interruption index (SADII) and ENS for pareto optimization. Chandramohan et al. [46] formulated a POMO to minimize operating costs (OC) by minimizing real and reactive power loss along with maximizing operating reliability by minimizing total interruption cost (TIC) with usual voltage constraints as

(16)

Here, \(P_{Loss}\) and \(Q_{Loss}\) are the real and reactive power losses of the distribution system and \(K_{1}\) and \(K_{2}\) are respective cost coefficients. IC \(_{b}\) is the interruption cost of the bus-b. Niknam et al. [52] applied fuzzy clustering approach to select the best-compromised solution from a non-dominated set of solutions. Here, the objectives considered are LM, minimization of BVD, the cost of energy both by grid and renewable energy sources (CEG) and total emissions produced (CEP). Barbosa et al. [53] formulated a POMO PDNR for minimizing PL, VDI, current loading index (CLI) and NS. Guedes [10] proposed a pareto dominance based multi-objective heuristic PDNR to minimize total PL and maximum current (MC). Gupta et al. [54] formulated a POMO to minimize PL, BVD, average interruption frequency (AIF), average interruption unavailability (AIU) and ENS. Narimani et al. [55] proposed a POMO based PDNR with objectives like LM, minimizing operating cost (OC) of DGs and ENS.

3 PDNR approaches

PDNR methods are broadly classified into three categories: (1) heuristics (2) meta-heuristic and (3) mathematical optimization based approaches. Of course, a fourth category is there which includes the hybrid methods based on the first three approaches. Heuristic-based approaches are the most popular ones, mainly due to the reason that these methods always give fast reconfiguration results and are based on distribution network operational experience, hence are simple to formulate. However, these methods not necessarily always give the global optimum values whereas meta-heuristic (the Greek prefix “meta” means “high level” [56]) PDNR methods take sufficiently longer time and are often complicated to formulate. These reconfiguration methods are based on genetic algorithm, simulated annealing, ant colony search, differential algorithm, harmony search, tabu search, gravitational search, particle swarm optimization, etc. Mathematical Optimization methods, otherwise, known as deterministic methods and the examples of this approach are linear programming, quadratic programming, nonlinear programming etc.

PDNR methods mathematically belong to the class of complex combinatorial non-differentiable optimization problem, where maintenance of radiality of the network, nonlinear nature of power flow constraints and necessity of exhaustive search for all possible configurations add to the complexity of the problem. It is for that reason; most of the initial PDNR approaches are heuristic search based on simple network operating principles with much-reduced solution space. Chiang and Jumeau [47, 48] termed these techniques otherwise as “the class of greedy search techniques which accepts only search movements that produce immediate improvement. As a result, these solution algorithms usually achieve local optimum solutions rather than global optimal solutions”. Sensitivity guided heuristics and several other inference mechanism based techniques are also included in this category.

The PDNR approaches can also be classified into two categories (a) branch exchange (BE) method (b) sequential switch opening method (SSO). In a branch exchange method, closure of any tie line switch must accompany with opening of any sectionalizing switch from the loop formed whereas in the second category, one or any number combination of the network tie line switches are closed first to form a meshed system and then sectionalizing switches are opened successively to regain radial configuration. Tables 1, 2 and 3 enlist a chronological order of various PDNR methods published in various journals of repute over last three decades for heuristic, mathematical optimization and meta heuristic-based approaches respectively. These tables also include brief information of the corresponding PDNR objectives and the type of load flow used. Figure 1 depicts the evolution of PDNR methods with the gradual emergence of smart grid.

Table 1 Heuristic search based PDNR methods
Full size table
Table 2 Mathematical optimization based PDNR methods
Full size table

4 Test systems and results

Looking at the huge number of papers available in the literature on PDNR methods, it is felt that there is a need for a review of the most commonly used test systems and reported reconfiguration results. Over the years, researchers have used several test DSs to explain their own approaches and compared it with other PDNR methods. Here, some of the most commonly referred test DSs are considered and the reconfiguration results of these systems as reported by the respective authors are presented. However, the reconfiguration results refer to LM objective only. The reported results (except for 16 bus system) include active power loss and minimum bus voltage magnitude before reconfiguration and active power loss, minimum bus voltage magnitude, open branches after reconfiguration. The reported results also include reported the number of LFs required and the execution time in second for the reconfiguration process.

Table 3 Meta heuristics based PDNR methods
Full size table
Fig. 1
figure 1

Evolution of PDNR methods

Full size image

4.1 16 bus test system (TS-1)

Civanlar et al. [60] introduced this three feeder test system (Fig. 2). The bus and branch numbering is kept the same as in the original case. The total active and reactive loads are 28.7 MW and 17.3 MVAr respectively. Capacitors are connected at 7 buses (5, 6, 9, 11, 12, 14 and 16) amounting a total of 11.4 MVAr. The base power and voltage are 100 MVA and 23 kV respectively. The line and load data of the system are available in [60]. This test system is mostly used by researchers to explain the corresponding proposed PDNR methods and for quick validation of results. There is unanimity for the power loss (511.4 kW) in the original network (Fig. 2) with open branches 15, 21 and 26 whereas the final power loss is 466.13 kW after reconfiguration with open branches 17, 19 and 26 (Fig. 3). The overall loss reduction is 8.85 %. The minimum voltage magnitude buses before and after reconfiguration are V\(_{12 }= 0.9693\) pu and V\(_{13 }= 0.9716\) pu respectively. Table 4 presents the number of load flows required by some PDNR methods (as reported in the concerned paper) for obtaining the minimum loss network from the original network (TS-1), which clearly shows that heuristic based PDNR methods are much faster than the meta-heuristic methods.

Fig. 2
figure 2

TS-1 before reconfiguration

Full size image
Fig. 3
figure 3

TS-1 after reconfiguration

Full size image

4.2 33 Bus test system (TS-2)

This test system is extremely popular among the researchers for validation of reconfiguration methods and results. However, in literature two topologically same 33 bus systems are referred which lead to ambiguity. In fact, there is very little difference between the two as far as the line data are concerned. The load data and tie-line data are exactly the same. In this paper, the original 33 bus system (popularly known as Baran and Wu system) is named as TS-2A and the latter revised version is named as TS-2B. Baran and Wu [61] introduced the test system TS-2A (Fig. 4). The figure (as reported in [61]) has been redrawn and the bus numbering is slightly changed as followed by most of the researchers. The branches are indexed as 1 less than the corresponding receiving end bus and tie lines are numbered as shown in the Fig. 4. The total active and reactive loads are 3715 kW and 2300 kVAr respectively. The base power and voltage are 10 MVA and 12.66 kV respectively. The line and load data of the system are available in [61]. The total active power loss of the original network is 202.65 kW (by authors’ method) with open branches 33, 34, 35, 36 and 37.

Table 4 LF required by the PDNR methods for TS-1
Full size table

On the other hand, in literature, some researchers [35, 204] refer to a 33 bus test system as Baran and Wu test system but with slightly changed line data. For this system (named here as TS-2B), the line reactance of the line-1 (connecting bus 1 and 2 of Fig. 4) is 0.0477 \(\Omega \) instead of 0.0470 \(\Omega \). Similarly, the line resistance and reactance of line 7 (connecting bus 7 and 8 of Fig. 4) are 1.7114 \(\Omega \) and 1.2351 \(\Omega \) instead of 0.7111 \(\Omega \) and 0.2351 \(\Omega \) respectively. All other data including that of load and tie-line data are same as TS-2A. As a result, the total active power loss of TS-2B comes out to be 210.97 kW (by authors’ method) with open branches 33, 34, 35, 36 and 37. The various reconfiguration results for the TS-2A and TS-2B are presented in Table 5, which clearly shown how the two systems’ results have been mixed up leading to confusion. The reconfiguration results of TS-2B are shown as bold results in Table 5. The global minimum loss for TS-2A and TS-2B are reported to be 139.55 kW, however, many researchers report lesser values marked also as bold results in Table 5.

Fig. 4
figure 4

33 bus test system-TS-2 (A&B)

Full size image
Table 5 TS-2A and TS-2B reconfiguration results
Full size table

4.3 69 bus test system-TS-3

Like TS-2, in literature researchers have also considered two topologically same 69 bus networks, referred as TS-3A and TS-3B in this paper.

4.3.1 TS-3A

Baran and Wu [207] introduced this 12.66 kV test system (Fig. 5). The initial network has open branches 69 (11–43), 70 (13–21), 71 (15–46), 72 (50–59) and 73 (27–65). The total active and reactive loads are 3802.19 kW and 2964.6 kVAr respectively. The base power and voltage are 10 MVA and 12.66 kV respectively. The reconfiguration results for this test system are presented in Table 6. Although, the global minimum value is reported to be 99.6 kW, many researchers reported lesser values shown as bold results in Table 6. Here, again, heuristic methods are proven to be much faster.

4.3.2 TS-3B

Chiang and Jimeau [47, 48] introduced this test system (Fig. 6). The bus and branch numbering are maintained same as in the first paper. The initial network has open branches 69 (10–70), 70 (12–20), 71 (14–90), 72 (38–48) and 73 (26–54). The total active and reactive loads are 1107.91 kW and 897.93 kVAr respectively. For this test system, two sets of different reconfiguration results are reported. As pointed out by Ramos et al. [126], this is due to the value of base voltage considered. Chiang and Jimeau, considered the base voltage as 11/\(\surd \)3 kV resulting initial power loss as 69.76 kW, whereas, in papers where the base voltage is considered as 11 kV, the initial loss is 20.88 kW. The reported global minimum values are shown as bold results in Table 7.

Fig. 5
figure 5

69 bus test system-TS-3A

Full size image

4.4 70 Bus test system-TS-4

Two topologically same 70 bus test distribution systems are commonly considered by researchers, known as TS-4A [40] and TS-4B [97] in this paper.

4.4.1 TS-4A

An 11 kV, 70 bus radial distribution system with 2 sub stations and 4 main feeders [40] is shown in Fig. 7. The base values for voltage and power are 11 kV and 100 MVA respectively. The substation buses are numbered as 1 and 70 respectively. The details of the test system with 11 tie lines are presented in [40]. The various reconfiguration results are presented in Table 8, where a minimum value of the loss is reported as 201.395 kW (marked bold in Table 8).

Table 6 TS-3A reconfiguration results
Full size table

4.4.2 TS-4B

TS-4B is topologically the same as TS-4A network with same line data. However, it has 8 tie lines as shown in Fig. 8 and the load data are also different. The details of the test system with 8 tie lines are available in [97]. The various reconfiguration results are presented in Table 9, where a minimum value of the loss is reported as 301.6 kW.

4.5 84-bus test system-TS-5

Su and Lee [158] introduced this 11.4 kV test system (Fig. 9), which has 11 feeders, 13 tie-lines, and 83 branches. All most all methods confirm the global minimum loss of 469.88 kW except in [26] and [178] which report a minimum loss of 463.29 kW shown as bold results in Table 10.

4.6 119 bus test system-TS-6

This system [17] is an 11 kV distribution system with 118 normally closed branches and 15 tie switches as shown in Fig. 10. The total active and reactive power loads are 22,709.7 kW and 17,041.1 kVAr. The reconfiguration results as reported in various papers are presented in Table 11.

Fig. 6
figure 6

69 bus test system-TS-3B

Full size image

4.7 136 bus test system-TS-7

This is a 13.8 kV real distribution system with 136 buses and 156 branches located in Brazil [142]. The total active and reactive power load for the system is 18,313.8 kW and 7932.5 kVAr respectively [111]. The reconfiguration results for this test system as reported by various researchers are presented in Table 12. Most of the authors report a minimum reconfiguration network with total active power loss of 280.1 kW with the exception of Swarnakar et al. [144, 173] reporting the lowest power loss of 279.75 kW marked as blue cell in Table 12. The fastest method is found to be that of Zin et al. [111], a heuristic method, which takes only 51 load flows.

Table 7 TS-3B reconfiguration results
Full size table
Fig. 7
figure 7

70 bus test system-TS-4A

Full size image

4.8 205 bus test system

The 205 bus test system is developed by [122] by triplicating the 69 bus TS-3B. The configuration and tie-line data are as presented in [122]. The reconfiguration results as reported in several papers are presented in Table 13.

4.9 Unbalanced test distribution systems (UTDS)

In literature, not many reconfiguration results with unbalanced radial systems have been reported. Here, some of the results are presented in Table 14.

5 Stochastic PDNR

With more uncertainties coming in smart distribution operations, PDNR with the stochastic environment has become the important current trend in research and perhaps has emerged as the most important future direction in PDNR research. With the growing trend of increased penetration of wind turbines (WTs), solar photo voltaic cells, fuel cells, plug-in hybrid electric vehicles(PHEVs) along with shift of focus to adopt probabilistic reliability models have brought in uncertainties into the system. These uncertainty effects are handled through the following methods: (1) Monte Carlo simulation (MCS), which is the most popular method but requires very high computational effort. (2) Analytical methods, which is computationally more efficient but require some mathematical assumptions to solve the problem. (3) Approximate methods: which overcome the shortcomings of the previous two, hence are more useful. Most well-known approximate methods are: First order second-moment method and point estimation method.

Table 8 TS-4A reconfiguration results
Full size table
Fig. 8
figure 8

70 bus test system-TS-4B

Full size image

Ammanulla et al. [188] used probabilistic reliability evaluation models for PDNR problem applying the minimal cutsets of components between the source and the load. A probabilistic power flow based on 2m PEM is employed to include uncertainty in the wind power generation output and demand concurrently[45, 199], wind speed variations, failure rate, repair rate forecast errors [206], and cost of power loss and customer interruption cost [211, 212]. In [213, 214] the 2m PEM is used to capture uncertainty associated with the load demand prediction error as well as the variation of price rise of natural gas for proton exchange membrane fuel cell power plants (PEM-FCPPs), tariff for buying electricity from PEM-FCPP and grid, tariff for selling electrical energy, operation and maintenance cost, hydrogen selling price and fuel cost for supplying residential loads. Niknam et al. [215] considered uncertainty due to wind power generation output and demand and adopted a scenario based two-stage methodology. Rostami et al. [216] and Kavousi-Fard et al. [217] used MCS solving probabilistic optimal power flow to model stochastic charging behavior of plug-in hybrid electric vehicles (PHEVs) under different charging strategies. In [218, 219], the 2m+1 PEM is used to capture the uncertainty of load and failure rate and repair rate forecast errors. Recently, Kavousi-Fard et al. [220] developed PDNR for smart grids with high penetration of PHEVs and wind power generation using a new stochastic framework based on unscented transformation (UT), which is an approximate method.

Table 9 TS-4B reconfiguration results
Full size table

6 Present trends and future directions in research

The review also identifies following research directions concerning PDNR methods.

Fig. 9
figure 9

84 bus test system-TS-5

Full size image

6.1 Application of new meta-heuristic approaches to the problem

This review reveals many new meta-heuristic methods, which are being proposed aiming to improve the PDNR approach such as GSA [55, 195], FWO [196, 197], TLBO [198, 199], QFA [200], ICA [201], DABC [204], BGSA [205], BA [206] which also confirms the scope of applying other new meta-heuristic optimization algorithms. Parameters of these heuristic approaches are selected by trial and error. These parameters can be tuned adaptively and automatically to improve the computational efficiency of the PDNR algorithms.

Table 10 TS-5 reconfiguration results
Full size table
Fig. 10
figure 10

119 bus test system-TS-6

Full size image

6.2 Inclusion of power quality and reliability indices in the objective

The review also reveals that for PDNR as far as the objectives are concerned, the focus has clearly shifted from traditional objectives like; minimization of PL, BVD, BCL, NS to maximization of reliability and power quality based indices. Most of the recently propose methods [20, 26, 31–35, 41, 49, 50, 54, 55, 188, 200, 203, 206] include these objectives. The review also reveals that researchers prefer meta-heuristic approaches while dealing with reliability and power quality indices. All these research work on PDNR has been carried out for a particular load level, or at light, medium and high load levels or based on a load profile data of a day. However, load growth, addition or expansion of lines need to be included in the PDNR algorithms. Many utilities use voltage regulators in the distribution networks. Therefore, it will be interesting to investigate the network reconfiguration considering voltage regulators.

6.3 Pareto optimality in multi-objectives

Clearly, the trend in MO-PDNR is for finding a set of non-dominated solutions or pareto fronts instead of going for a single optimized solution as this is best suited for practical operating conditions. Based on the requirement referring to a particular operating condition, a suitable solution can be selected out of the set of non-dominated pareto optimized solutions, e.g the fuzzy clustering approach used in [52, 55, 203, 206].

6.4 PDNR involving DGs and FACTS devices

With the growing penetration of DGs, many researchers have already proposed PDNR methods involving DGs [25, 34, 45, 55, 114, 115, 128, 132, 171, 193, 195, 197, 199, 202, 206].

Table 11 TS-6 reconfiguration results
Full size table
Table 12 TS-7 reconfiguration results
Full size table
Table 13 TS-8 reconfiguration results
Full size table
Table 14 Reconfiguration results of UTDSs
Full size table

Jazebi et al. [208] have proposed optimal placement of D-STATCOMs with PDNR in DS. Hence, further research is required to investigate PDNR with simultaneous placement of DGs, capacitors, FACTS devices and protection devices [209]. Issues such as maximizing the available delivery capability of UDSs with high penetration of DGs using PDNR [210] can be another interesting direction of research. It will be interesting to solve PDNR problem with the simultaneous placement of renewable DGs and energy storage devices considering future load growth.

7 Conclusions

In this paper, more than two hundred papers on PDNR methods have been reviewed. This review has clearly identified the various techniques adopted by researchers to solve PDNR problem. From this review it has been observed that heuristics methods mostly converge very fast to yield optimal configuration, however, the global optimal value cannot be guaranteed and in some cases these methods are not independent of initial configurations. Although, many meta heuristic methods have been proposed but out of all meta-heuristic PDNR methods, GA based PDNR methods are most popular and effective. Meta-heuristics methods although mostly give guaranteed global optimal results, but take long computational time to converge and hence, in some cases are blended with heuristics to increase the speed of convergence to make it suitable for real-time applications. Moreover, population-based meta-heuristic methods are preferred for POMO PDNRs and stochastic PDNRs.