Introduction

The current scenario confronts electrical distribution system operators with tremendous challenges, such as an exponential rise in load demand, low distribution system efficiency caused by large real power losses, and poor network voltage profiles. Distribution system I2R losses account for 13% of the total energy produced, according to recent studies [1]. Distribution system reconfiguration, connection of DTATCOMs & DGs in the electrical distribution system have all been used in literature to reduce I2R losses and improve system efficiency. Connecting DGs to the distribution system provides numerous additional advantages, including improved system loadability, reduced congestion in the T&D system, and enhanced EENS. These are just a few. As demand grows, it becomes imperative to increase the system's capacity in order to avoid the need for additional expansion of the infrastructure. These benefits can be harmed, however, by the incorrect positioning or size of DGs in the system. For this reason, in order to maximize the advantages while ensuring that various technical restrictions are met, it has been referred to as the "optimal connectivity of distributed generation (OCDG)" challenge.

The OCDG problem in radial distribution systems is discussed in depth in [2, 3].The OCDG problem in the distribution system was addressed by certain researchers in order to reduce I2R loss and improve the voltage profile of the network. For the OCDG problem, authors in [4,5,6] came up with a variety of analytical approaches. The radial distribution system's efficiency was hoped to be improved by a simulated annealing technique and a harmony search algorithm presented by authors in [7, 8]. DGs were located in the distribution system using loss-sensitivity indexes, and their sizes were employed as decision variables to optimize the intended outcomes. Mathematicians in [9] used a backtracking search optimization approach to reduce real power loss and enhance voltage distribution in a system. For DG placement, writers in [9] looked at the top (15–25%) of buses based on loss-sensitivity bus values, as opposed to the last two articles. Using the Firefly optimization technique and the Backtracking search optimization algorithm, authors in [10] tackled the OCDG and optimal capacitor problems at the same time. Authors in [11] used quasi-oppositional TLBO algorithm to solve a penalty-based multi-objective strategy to improve the OCDG problem's system efficiency, network voltage profile and stability index. Utilizing an optimization method to identify locations and DGs sizes is a better option than using loss-sensitivity indices and DGs sizes in an optimization algorithm to locate locations. DG locations and DG sizes are both treated as decision variables in this work, and an optimization approach is employed to obtain the best values for these variables.

The Non-dominant enhanced differential search multi-objective algorithm for OCDG problem was addressed in [12] by the authors to reduce the I2R loss, voltage deviation index, and network operation costs. To increase the real power loss reduction, voltage stability factor, and investment benefit of the distribution system, the authors of [13] used a Pareto-based multi-objective ant lion optimizer. A multi-objective approach has been found to be more effective in achieving all three objectives. With a multi-level load profile, the authors in [14] developed a novel objective function for loss mitigation and voltage profile improvement through the appropriate placement of DG and DSCs. To improve real power loss reduction and voltage stability margin in the distribution system, the authors in [15] looked at the OCDG problem under linear load changes from 0.5 to 1.5 times baseload. An algorithm for the best placement of dispatchable and non-dispatchable DGs to reduce energy loss and operating costs while also improving the distribution system's voltage profile under seasonal load variations was discussed in [16]. OCDG under varying distribution system loads was addressed in the last three articles, according to the authors. Authors in [17] used the HSA-PABC optimization algorithm to find the best location for Type-1 DGs and Type-3 DGs (0.85 leading power factor) under three different load levels of the distribution network to reduce power loss. The best percentage of loss reduction is obtained when Type-3 DGs are placed optimally in the distribution network, according to the results. A TLBO-GWO optimization technique was used by the authors of [18] to place Type-1 DG's and Type-3 DGs with optimum power factors in the distribution network in order to reduce I2R loss and improve reliability. For Type-3 DGs operating with an optimized power factor, the appropriate placement of these DGs minimizes network power loss to the lowest possible value, as evidenced by the above-mentioned articles. Therefore, in this work, we have chosen the best possible placement of Type-3 DGs with an optimized power factor in order to meet our goals.

In order to fulfil the goals of reducing real power loss, improving voltage profile, and balancing the load, an optimal distribution network reconfiguration issue is designed to determine the on/off status of tie and section switches positioned in the system by satisfying the technological restrictions. Bacterial foraging optimization techniques [19, 20] are used to construct an improved selective BPSO algorithm for the optimal network reconfiguration problem to reduce I2R loss. In [21, 22], a mathematical objective function is devised to reduce real power loss and improve the voltage profile of the network. According to [23], there is a Multi-objective Max–Min strategy for minimizing I2R loss, load balancing between branches and feeders, and the amount of switch operations. Type-1 DGs have been handled in [8, 24] to reduce the system's I2R loss using simultaneous ODNR and appropriate connection of Type-1 DGs. With the novel UVDA optimization technique, researchers in [25] attempted to minimize the real power loss in the distribution system by connecting Type-3 DGs optimally while also addressing optimal network reconfiguration issues.

Some researchers took the OCDG and ODNR problems for enhancing the loadability of the network. Loadability (\(\lambda_{{{\text{max}}}}\)) of the system is termed as the maximum increase in network load level before the system voltage instability occurs. Figure 1 depicts loading curves A & B of a system without & with the connection of DGs, respectively. From Fig. 1, it is observed that curve B has a better system loadability than curve A, due to the connection of DGs optimally in the system and reconfigured network. And also from Fig. 1, it is noticed that enhancement in system loadability also improves the network voltage profile, i.e., at each loading level, curve B has a better voltage magnitude in comparison with curve A.

Fig. 1
figure 1

Impact of DGs placement and Network reconfiguration on system loadability

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Authors in [26,27,28] addressed the ODNR problem to enhance the system loadability make use of a fuzzy adaptation of the optimization algorithm, discrete ABC algorithm, enhanced HSO algorithm, respectively, and deduced that the ODNR enhances system loadability. In [29], the OCDG problem has used for the improvement of system loadabilty employing the hybrid PSO—k-matrix algorithm and drawn a conclusion that with 40% of real power injection by DGs into the system, real power loss mitigated to 65–70%, loadability improved to 15–40%. Researchers in [30], taken the OCDG and ODNR problems at a time for enhancement of system loadabilty and concluded that utmost enhancement of system loadabilty is noted in the case of DGs connected optimally in the optimal reconfigured network. From the latter two papers, it was observed that even though the system loadability is improved to utmost value but the percentage of real power loss mitigation is not up to the mark. And also, from the papers [29, 30], it was observed that improving the loadability of the system also improves the voltage profile of the system. Therefore, in this work authors considered the improvement of real power loss reduction and loadability of the system only which in turn also improves the voltage profile of the system. Since loadability of the system should be improved concerning the 100% load level of the system, in this work authors had not considered the load variations of the system. To extract the maximum number of benefits with less amount of power injected by DGs into the system, in this work authors have taken DGs penetration level as one of the objectives. To improve more than one objective at a time, researchers in [7, 10,11,12, 17] used either weighted-factor or Pareto-based or Max–Min based multi-objective methods. Among them, Max–Min based multi-objective method had advantages like no need to bother about weights or formation of the fronts. And also, since DGs penetration level is taken as one of the objectives, drive the authors of this work for selecting max–min multi-objective method rather than pareto-based multi-objective method.

Therefore, in this work, the multi-objective approach with the Max–Min method is used to mitigate the real power loss and maximize the system loadability \((\lambda_{{{\text{max}}}} )\). To improve the desired objectives two scenarios are considered, i.e., without and with DGs real power injection objective function. Under each scenario three cases are considered, i.e., optimization of single objectives is considered in case-1 & case-2, multi-objective optimization is considered in case-3. And each case having two sub-cases, the optimal connection of DGs in the initial configured network and the optimal connection of DGs in the optimal reconfigured network. BO algorithm is chosen to optimize the proposed objectives. The rest of the paper is organized as follows, section-2 introduces the mathematical formulation aspects of the work done in this work, section-3 will give brief insights of the BO optimization technique and thorough implementation aspects of it, Sect. 4 will illustrate the scenarios taken in this work and the associated results.

Problem formulation

Network real power loss

Real power loss \(\left( {P_{loss} } \right)\) have to be minimized for the enhancement of distribution system efficiency.

$$ f_{1} = {\text{minimize}} \left( {P_{{{\text{loss}}}} } \right) $$
(1)
$$ P_{{{\text{loss}}}} = J*R^{T} $$
(2)

where, J and R are branch current and branch resistance vectors of size nbr (number of branches). Backward/forward sweep-based load flow [31] is used to obtain \(P_{{{\text{loss}}}}\).

Loadability of the system

System loadability \(\left( {\lambda_{max} } \right)\) have to be maximized with a view for future load enhancement on the system.

$$ f_{2} = {\text{maximize}} \left( {\lambda_{{{\text{max}}}} } \right) $$
(3)

To obtain the \(\lambda_{{{\text{max}}}}\) of the system, authors had used the method developed in [32].

DGs penetration level

Placing of Distributed Generators in the distribution network changes the distribution system characteristics [33, 34] as bi-directional power flows, changing the passive distribution system network to active distribution network, change in fault current levels, etc. Therefore, to maintain the quality of the network, some of the authors in the literature limited the DG's real power injection into the distribution system. Authors in [9, 29, 35] limited the DGs real power penetration into the distribution system to 40% and 50%, respectively, and Authors in [36] taken DGs penetration level as one of the objectives and limited the DGs real power penetration into the system without violating stability margins. And also, from the literature, it was observed that at lower DGs penetration levels, a significant increase in DGs penetration level results in significant improvement in technical parameters. But at higher DGs penetration levels, a significant increase in DGs penetration level results in an insignificant improvement in benefits of the system.

Therefore, in this paper, instead of limiting DGs real power injection to a fixed percentage say 40% or 50%, authors considered DGs penetration level as one of the objectives in scenario-2 along with the objectives considered in scenario-1, and a detailed analysis is presented in result section between scenario-1 and scenario-2 outcomes.

The mathematical modeling of the DGs real power injection into the system is taken as one of the objectives.

$$ f_{3} = {\text{minimize}} \left( {\mathop \sum \limits_{k = 1}^{ndg} P_{DG,k} = P_{T,DG} } \right) $$
(4)

where \(P_{DG,k}\) is the real power delivered by the kth DG unit, \(P_{T,DG}\) is the total real power delivered by the DGs units.

Max–Min method

In [23], the authors addressed the multi-objective Max–Min method for optimal network reconfiguration problem to select the comprised solutions between the objectives. The Max–Min method contains a Membership function for each objective function and has a value in the range [0–1]. The membership function for the minimization of the kth objective function is given as follows.

$$ M{F_k} = \begin{array}{*{20}{c}} {\;\;\;\;1,}\\ {\frac{{F_k^{max} - {F_k}}}{{F_k^{max} - F_k^{min}}},}\\ {\;\;\;\;\;0,} \end{array}\begin{array}{*{20}{c}} {{F_k} \le F_k^{min}}\\ {\,\,\, \quad F_k^{min} \le {F_k} \le F_k^{max}}\\ {0{F_k} \ge F_k^{max}} \end{array}$$
(5)

where \(F_{k}\),\( F_{k}^{max}\), and \(F_{k}^{min}\) are the kth objective function value, maximum and minimum values of the kth objective function, respectively. For maximization of the kth objective function, reciprocal of kth objective function value, minimum and maximum values of the kth objective function have to take to get \(F_{k}\), \(F_{k}^{min}\), and \(F_{k}^{max}\), respectively. The value of \(F_{k}^{min}\) is taken from the outcome of that single-objective optimization, the value of \(F_{k}^{max}\) is taken from the base load flow results.

Since the \(F_{k}^{max}\) is subtracted from the \(F_{k} \) in the numerator of the Membership value (\(MF_{k}\)) of an objective function, the objective function with the highest \(MF_{k}\) value is well improved and the objective function with the lowest \(MF_{k}\) value is less improved in terms of minimizing the objective function. Then a fuzzy decision for a comprised solution is defined as the choice of maximizing the lowest \(MF_{k} \) value. In other words, the multi-objective function is transformed into a single objective by maximizing the minimum value among all membership values as follows:

$$ {\text{Maximize of}} = \left\{ {min \left\{ {MF_{k} } \right\}} \right\} $$
(6)

The above maximization problem is converted into a minimization problem is as follows

$$ {\text{Minimize of}} = \left\{ {1 - \left\{ {min \left\{ {MF_{k} } \right\}} \right\}} \right\} $$
(7)

Constraints

The following constraints need to satisfy for the optimal network reconfiguration and connection of DGs to the distribution system.

  1. a.

    The voltage magnitude of the buses in the system should be within the permissible limits.

    $$ \left| {V_{min} } \right| < \left| {V_{i} } \right| < \left| {V_{max} } \right|\;\; i = 1,2 \ldots nb $$
    (8)

    where nb is the total number of buses are there in test system. In this paper, we have taken.

    \(\left| {V_{min} } \right|\)= 0.95 p.u. and \(\left| {V_{max} } \right|\)= 1.05 p.u.

  2. b.

    The magnitude of current in each branch should be less than the maximum current rating of the respective branch.

    $$ I_{J} \le I_{J}^{max} \;\;j = 1,2 \ldots nbr $$
    (9)

    where nbr is the total number of branches.

  3. c.

    Power injected by each DG (\(P_{DG,k}\)) must be less than the maximum power limit of DGs.

    $$ P_{DG,k} \le P_{DG,k}^{max} \;\;\;k = 1,2 \ldots ndg $$
    (10)

    where ndg is the number of DGs connected to the system. In this paper, the maximum real power injection by DGs (\(P_{DG,k}^{max}\)) limited to the total real power demand supplied by the DGs.

  4. d.

    Power factor of DGs must be between the minimum (\(pf_{k}^{min}\)) and unity power factor limits.

    $$ pf_{k}^{min} \le pf_{k} \le 1\;\; k = 1,2 \ldots ndg $$
    (11)

    In this paper, the minimum power factor of the DG unit is limited to 0.8.

  5. e.

    Total real power (\(P_{T,DG}\)) and reactive power injected (\(Q_{T,DG}\)) by DGs must be less than the distribution system real (\(P_{load}\)) and reactive power (\(Q_{load}\)) demand.

    $$ \mathop \sum \limits_{k = 1}^{ndg} P_{DG,k} = P_{T,DG} \le P_{load} $$
    (12)
    $$ \mathop \sum \limits_{k = 1}^{ndg} Q_{DG,k} = Q_{T,DG} \le Q_{load} $$
    (13)
  6. f.

    Power balance constraints.

    $$ P_{sub} + P_{T,DG} = P_{load} + P_{loss} $$
    (14)
    $$ Q_{sub} + Q_{T,DG} = Q_{load} + Q_{loss} $$
    (15)

    where \(P_{sub}\), \(Q_{sub}\) are the real and reactive power demands at the substation.

  7. g.

    The ODNR problem requires checking the radiality status of the network. In this work spanning tree technique is used for checking the status of network radiality [37].

DG Placement performance indices

The following performance indices are considered to evaluate the impact of optimal DGs connection and optimal network configuration on the distribution system.

  1. a.

    Percentage Real power loss reduction

    $$ \% PLR = \frac{{P_{loss}^{b} - P_{loss}^{{\left( {DG + NR} \right)}} }}{{P_{loss}^{b} }} $$
    (16)

    where \(P_{loss}^{b}\) is the base case real power loss of the system, \(P_{loss}^{{\left( {DG + NR} \right)}}\) is the real power loss of the system after placement of DGs and network reconfiguration.

  2. b.

    Percentage Maximum Loadability improvement

    $$ \% MLI = \frac{{\lambda_{max}^{{\left( {DG + NR} \right)}} - \lambda_{max}^{b} }}{{\lambda_{max}^{b} }} $$
    (17)

Where \(\lambda_{max}^{b}\) is the base case maximum loadability of the system, \(\lambda_{max}^{{\left( {DG + NR} \right)}}\) is the maximum loadability of the system after placement of DGs and network reconfiguration.

Butterfly optimization algorithm

In the literature, various researchers have taken several optimization algorithms for the OCDG and ODNR problems. According to the “No Free lunch theorem,” no optimization algorithm gives exceptional results for all optimization problems. An optimization algorithm may give admirable results for some set of optimization problems and may give inferior results for another set of optimization problems. Performance-wise, all optimization algorithms are indistinguishable while solving a whole set of optimization problems. However, while choosing an optimization problem author of this paper have taken care of few things like since finding loadability of the distribution system is a very tedious process, authors try to avoid optimization algorithms with a two-stage evolutionary process like in cuckoo search algorithm, TLBO algorithm, etc., and algorithm should be easy in implementation. Since the Butterfly optimization (BO) algorithm is a new one and advantages like the ease in implementation have driven the authors to use this algorithm [38,39,40].

Sankalp A and S Singh developed the butterfly optimization (BO) method, a population-based meta-heuristic optimization strategy [41]. By drawing inspiration from butterfly mating and food seeking habits, the algorithm was created. They will rely on their sense of smell to find food and a partner for mating. In the process of searching for food, butterflies will release aromas with some force, and the potential of the scents/aromas is relative to the quantity of food source in the butterfly's neighborhood. It will emit a scent that will be picked up by others. If the other butterflies in the cluster are able to detect the aroma, they will move toward it. From one location to the next, butterflies will travel about in search of a good food source in this manner.

It is assumed that all butterflies are searching agents in the BO algorithm. Each agent will be assigned a specific location and a distinct fragrance. The scent of the agents is linked to the performance of the objective functions. In Eq. 18, the aroma's mathematical representation is provided.

$$ {\text{f}} = {\text{cI}}^{{\text{a}}} $$
(18)

where f, I, c & a are the magnitude of the aroma, stimulus intensity, sensor modality and power exponent. In the algorithm, I is taken as the fitness of the respective searching agent.

All agents will move to the new positions as per mathematical formulated Eqs. 19 & 20.

Update the position of ith agent using for global search Eq. 19 if \(random value\left[ {0,1} \right] < P\)

$$ x_{i}^{d} \left( {t + 1} \right) = x_{i}^{d} \left( t \right) + \left( {r^{2} {*}gbest - x_{i}^{d} \left( t \right)} \right) \quad {*} \,\, f_{i } i = 1 \ldots N $$
(19)

or Update the position of ith agent for local search using Eq. 20

$$ x_{i}^{d} \left( {t + 1} \right) = x_{i}^{d} \left( t \right) + \left( {r^{2} {*}x_{j}^{d} \left( t \right) - x_{k}^{d} \left( t \right)} \right) \quad {*}\,\, f_{i} i = 1 \ldots N $$
(20)

where \(x_{j}^{d} \left( t \right)\) and \(x_{k}^{d} \left( t \right)\) are random Jth and kth butterflies and N, d, r, p are Number of agents, Number of decision variables, arbitrary random number & switching probability lies in the range [0, 1].

The detailed step-by-step implementation procedure of the proposed multi-objective Max–Min BO optimization technique for finding befitting DGs sizes, locations and power factors to obtain the desired objectives are given as follows.

  1. 1.

    Initialize optimization technique parameters N, d, p, c, a, Maximum number of iterations.

  2. 2.

    Read the test system line data and load data. Run the load flow algorithm for the base case to get \( F_{k}^{max}\) values for objectives power loss, system loadability & run single-objective optimization programs to get \(F_{k}^{min}\) values.

  3. 3.

    Initialize minimum and maximum limit values of decision variables. For optimal connection of DGs with optimized power factors problem, the total number of decision variables is equal to three times the number of DGs to be connected (DGs locations, DGs sizes, DGs power factors) in the system. Therefore, decision variables minimum and maximum variable limit vectors are shown in Eq. 21 and 22.

    $$ X_{min} = \left[ { L_{1min} L_{2min} ..L_{dmin} ,S_{1min} S_{2min} ..S_{dmin} ,pf_{1min} pf_{2min} ..pf_{dmin} } \right] $$
    (21)
    $$ X_{max} = \left[ { L_{1max} L_{2max} ..L_{dmax} ,S_{1max} S_{2max} ..S_{dmax} ,pf_{1max} pf_{2min} ..pf_{dmax} } \right] $$
    (22)

    where in the above vectors, L indicates DGs locations, S indicates DGs sizes and pf indicates DGs power factors.

  4. 4.

    Generate initial solutions using Eq. 23 as follows

    $$ x_{j}^{i} = \left[ {X_{min,j} + \left( {X_{max,j} - X_{min,j} } \right)*rand} \right] \,\,\, i = 1 \ldots .N, j = 1 \ldots \ldots length\left( {X_{min} } \right) $$
    (23)

A set of initial solutions generated using Eq. 23 is depicted in the matrix as follows

$$ X = \left[ {\begin{array}{*{20}c} {L_{1}^{1} \cdots L_{j}^{1} \ldots .L_{d}^{1} , S_{1}^{1} \cdots S_{j}^{1} \cdots S_{d}^{1} ,pf_{1}^{1} \cdots pf_{j}^{1} \cdots pf_{d}^{1} } \\ . \\ . \\ {L_{1}^{i} \cdots L_{j}^{i} \ldots .L_{d}^{i} , S_{1}^{i} \cdots S_{j}^{i} \cdots S_{d}^{i} ,pf_{1}^{i} \cdots pf_{j}^{i} \cdots pf_{d}^{i} } \\ . \\ . \\ {L_{1}^{N} \cdots L_{j}^{N} \ldots .L_{d}^{N} , S_{1}^{N} \cdots S_{j}^{N} \cdots S_{d}^{N} ,pf_{1}^{N} \cdots pf_{j}^{N} \cdots pf_{d}^{N} } \\ { } \\ \end{array} } \right] $$
(24)
  1. 5.

    Run the load flow program for each solution in matrix X and get power loss, system loadability and then calculate membership values for objectives in each solution set using Eq. 25 as follows

    $$M{F_{i,k}} = \begin{array}{*{20}{c}} {\;\;\;1,}\\ {\frac{{F_k^{max} - {F_{i,k}}}}{{F_k^{max} - F_k^{min}}},}\\ {\;\;\;0,} \end{array}\begin{array}{*{20}{c}} {~{F_{i,k}} \le F_k^{min}}\\ {\quad \, \, F_k^{min} \le {\;\;\;F_{i,k}} \le F_k^{max}}\\ {{F_{i,k}} \ge F_k^{max}} \end{array}~\;\;{\rm{for}}\;\;{\rm{i}} = {\rm{1}} \ldots {\rm{N}},{\rm{k}} = {\rm{1}} \ldots {\rm{M}}$$
    (25)

And then calculate objective function value or fitness value for each solution set in the matrix X using Eq. 26.

$$ of_{i} = 1 - \left\{ {min \left\{ {MF_{i,1} \ldots MF_{i,k} \ldots ..MF_{i,M} } \right\}} \right\} $$
(26)

As a whole, the whole fitness calculation method for all the agents is depicted in Eq. 27

$$ OF = \left[ {\begin{array}{*{20}c} {F_{1,1} \cdots F_{1,k} \cdots F_{1,M} \to MF_{1,1} \cdots MF_{1,k} \cdots MF_{1,M} \to of_{1} } \\ . \\ . \\ {F_{i,1} \cdots F_{i,k} \cdots F_{i,M} \to MF_{i,1} \cdots MF_{i,k} \cdots MF_{i,M} \to of_{i} } \\ . \\ . \\ {F_{N,1} \cdots F_{N,k} \cdots F_{N,M} \to MF_{N,1} \cdots MF_{N,k} \cdots MF_{N,M} \to of_{N} } \\ \end{array} } \right] $$
(27)

Find the solution with minimum objective function value (of) value and declare the corresponding solution set from matrix X as the global best solution.

  1. 6.

    Set iteration count = 0.

  2. 7.

    Update the aroma/fragrance of butterflies using Eq. 18.

  3. 8.

    Update the solutions of each agent using Eq. 19 & Eq. 20.

  4. 9.

    Calculate the objective function value or fitness value of each updated agent using the sequential process followed in Step 5.

  5. 10.

    Perform greedy selection between updated solutions and old solutions.

  6. 11.

    Update global best solution.

  7. 12.

    If the iteration count is less than the maximum number of iterations repeat steps 6–11 else print out the results such as global best solution, objective function values.

Results and discussion

In this section, the proposed BO technique for enhancement of the system loadability (\(\lambda_{{{\text{max}}}}\)) and real power loss mitigation is applied on 33 & 69 bus distribution test systems for the scenarios and cases shown in Table 1. In turn, each case is divided into two sub-cases a) optimal connection of DGs in the initial network without application of ODNR problem b) Optimal connection of DGs in the optimal reconfigured network which is obtained from the ODNR problem. The tuned BO algorithm parameters are shown in Table 2. All the simulations are implemented in MATLAB R2017a platform and carried out in computer having Core i7 7200U 3.10 GHz, 16 GB RAM.

Table 1 Scenarios and cases considered in this paper
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Table 2 BO algorithm parameters
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33 Bus radial distribution system

The line & load data of the system is taken from [29]. The system has 33 section switches and 5 tie switches. Normally tie switches are in open condition. The load on the system is 3.715 MW + j 2.3 MVAR. The base case real power loss is 210.98 kW, system loadability is 3.4, and the minimum voltage is 0.9038 p.u.

From the results of the ODNR problem, the points observed are.

  1. 1.

    In case of \(f_{1}\) optimization, the real power loss is reduced to 138.5513 kW. And also, in this case, system loadability is improved to 4.87. For this case, switches given by the algorithm are 7, 9, 14, 32, and 37.

  2. 2.

    In case of \(f_{2}\) optimization, the system loadability is enhanced to 5.23. And also, in this case, system network power loss is reduced to 139.9782 kW. For this case, switches given by the algorithm are 7, 9, 14, 28, and 32.

  3. 3.

    From the above observations, it is perceived that in the case of \(f_{2}\) maximization, both objectives is improved. Therefore, the optimal switches determined by the algorithm for enhancement of \(f_{2}\) are considered for case-3b.

Table 3 shows the outcomes of the OCDG problem for scenario 1. From the outcomes tabulated in Table 3, the succeeding points are observed. In case-1a & case-1b, the real power loss is reduced to 12.7458 kW & 18.7531 kW, respectively. It is observed that the real power loss of the system is reduced to the lowest value in the case of DGs placed in the initial configured network. In Case-2a & case-2b, system loadability is improved to 5.1 & 7.23 from 3.4 & 5.23, respectively. It is noticed that system loadability is improved to the utmost value in the case of DGs connected optimally in the optimal reconfigured network, i.e., in case-2b. From the outcomes of case-2a & case-2b, it is also noticed that real power loss is only reduced to 86.5804 kW and 98.8904 kW, respectively. To improve both loss reduction and system loadability, a multi-objective approach with the Max–Min method is taken in case-3. For case-3a, the minimum \( \left( {F_{k}^{max} } \right)\) and maximum \( (F_{k}^{min} ) \) objective function values taken for real power loss are 12 kW,210.98 kW and for maximum loadability are 1/5.1, 1/3.4. For case-3b, the minimum and maximum objective function values taken for real power loss are 18 kW, 139.9782 kW, for maximum loadability are 1/7.23,1/5.23. The convergence graphs for all cases of scenario-1 are shown in Fig. 2.

Table 3 Simulation results of 33 bus system for scenario-1
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Fig. 2
figure 2

Scenario-1 outcomes convergence graphs for 33 bus system

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From the results of case-3a & 3b, the points observed are as follows.

  1. 1.

    In case-3a, the system loadability is enhanced to 4.78 from 3.4, and loss is reduced to 39.1317 kW from 210.98 kW shows an improvement in both the objectives unlike in case-1a & case-2a.

  2. 2.

    In case-3b, the system loadability is enhanced to 6.76 from 5.23 and loss is reduced to 42.7188 kW from 139.9782 kW shows an improvement in both the objectives unlike in case-1b & case-2b.

  3. 3.

    In scenario-1, the utmost percentage of improvement in both the objectives is observed in case-3b, i.e., in the case of DGs optimally connected in the optimal reconfigured network while optimizing \(f_{1}\) and \(f_{2}\) using the Max–Min method.

The minimum \(\left( {F_{k}^{min} } \right) \) and maximum \( \left( {F_{k}^{max} } \right)\) objective function values taken in scenario-2 for case-1a are 12 kW,210.98 kW, for case-1b are 1/5.1, 1/3.4, for case-2a are 18 kW, 139.9782 kW and for case-2b are 1/7.23,1/5.23 for system loadability. The minimum limit for DGs real power injection is taken as 50% of the system real power demand i.e., 3715*0.5 = 1857 kW, and the maximum real power injective power limit by DGs is taken as 100% injection level.

Table 4 shows the outcomes of the OCDG problem for scenario 2. Figure 3 depicts the comparison between the performance indices of scenario-1 & 2. From Fig. 3 it is observed that even though there is a significant difference between the % KVA injection by DGs into the distribution system in scenario-1 & scenario-2 cases, but the difference between the performance indices is very less. Therefore, it can be concluded that the optimal placement of DGs in scenario-2 gives a better improvement in objectives (% PLR & % MLI) with less amount of % KVA injection by the DGs into the system.

Table 4 Simulation results of 33 bus system for scenario-2
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Fig. 3
figure 3

Comparison between the performance indices of scenario 1 & 2 for 33 bus system

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From Table 4, the succeeding points are noticed. In the case of \(f_{1}\) and \(f_{3}\) optimization, loss is reduced to 23.715 kW & 23.446 kW in case-1a & case-1b, respectively. It is noticed that the amount of loss reduction is almost the same for both cases. In the case of \(f_{2}\) and \(f_{3}\) optimization, system loadability is improved to 4.73 & 6.69 in case-2a & case-2b, respectively, but the loss is reduced to 55.4613 kW and 56.2606 kW only. Therefore, to improve the real power loss reduction along with loadability, optimization of \(f_{1, } f_{2}\), and \(f_{3}\) are considered in case-3a & case-3b. The points observed from case-3a & case-3b are real power loss is reduced to 45.1702 kW, 46.3242 kW, respectively, system loadability is increased to 4.7, 6.64. From case-3a & 3b of scenario-2, it is concluded that the optimal connection of DGs in the reconfigured network shows better improvement in both the objectives, i.e., loss reduction and system loadability enhancement. The convergence graphs for all cases of scenario-2 are shown in Fig. 4. Based on the above discussions it can be concluded that among all the cases in scenario-1 & 2, the highest percentage of improvement in both the objectives is observed in case-3b of scenario-1, i.e., by the injection of 74.92% kVA into the system, real power loss is reduced to 79.75%, system lodability is increased by 98.92%. An almost equal percentage of improvement in both objectives with less amount of % kVA injection by DGs into the system is observed in case-3b of scenario-2, i.e., with 64.69% kVA injection into the system, the loss is reduced to 78.04%, system loadability is increased by 95.29%.

Fig. 4
figure 4

Scenario-2 outcomes convergence graphs for 33 bus system

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To access the capability of the BO optimization technique to the proposed methodology, the results obtained are contrasted with the befitting methods and algorithms that are accessible in the literature and shown in Table 5. From Table 5, it is observed that in case of power loss minimization by the optimal placing of DGs in the initial configured case & optimal reconfigured case, the proposed BO algorithm reduces the real power loss to 93.95% & 91.11, respectively, whereas HTLBO-GWO, HAS-PABC, UVDA reduces real power loss to 93.51%, 92.51%, and 87.98%, respectively. In the case of loadability maximization, the BO algorithm improves it to 50% whereas HPSO improves it to 48.23% only. In scenario-2, in the case of loss minimization, the loss is reduced to 88.76% with 53.01 kW injection by DGs into the system, whereas the BSOA algorithm reduces it to 85.94% with % 50 kW real power injection by DGs into the system. In [29], with 40% kW or 47.05 kVA injection by DGs into the system, real power loss reduced to 71.75%, system loadability increased to 26.76%. But with the proposed method in this paper, with 64.69% kVA injection by DGs into the system, real power loss reduced to 78.09%, maximum loadabilty increased to 95.29% that shows an improvement in both the objectives unlike the method in [29] which shows the efficacy of the proposed method.

Table 5 Comparison results of 33 bus system
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69 Bus radial distribution System

The line & load data of the system are taken from [29]. The system has 69 section switches and 5 tie switches. Normally tie switches are in open condition. The load on the system is 3.801 MW + j 2.693 MVAR. The base case real power loss is 224.9515 kW, loadability of the system is 3.21 and the minimum voltage is 0.9091 p.u.

From the results of the ODNR problem, the following points are observed. In the case of individual optimization of objective functions \(f_{1}\) & \(f_{2}\), switches given by the algorithm are the same, i.e., they are 14, 58, 61, 69, and 70. For these switch combinations real power loss is mitigated to 98.55 kW, lodability enhanced to 5.23. Therefore, the above-mentioned optimal switches are considered for the OCDG problem in the optimal reconfigured network case.

Table 6 shows the outcomes of the OCDG problem for scenario 1. In case-1a & 1b, the power loss is reduced to 4.487 kW & 5.3082 kW. It is observed that the power loss is reduced to the lowest value in the case of DGs connected optimally in the initial configured network. In case-2a & case-2b, the system loadability is improved to 4.91 &7.71, respectively, but the real power loss is only reduced to 89.8601 kW & 93.9651 kW. In case-3a & 3b, the system loadability is improved to 4.61 & 7.07 and real power loss is reduced to 30.2921 kW & 25.313, respectively. From scenario-1 outcomes, it can be deduced that both the loadability and real power loss reduction are well improved in case-3b. The convergence graphs for all cases of scenario-1 are shown in Fig. 5. Figure 6 depicts the comparison between the performance indices of scenario-1 & 2. From Fig. 6, it is noticed that the optimal connection of DGs in scenario-2 gives a better improvement in objectives (% PLR & % MLI) with less amount of % KVA injection by the DGs into the system.

Table 6 Simulation results of 69 bus system for scenario-1
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Fig. 5
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Scenario-1 outcomes convergence graphs for 69 bus system

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Fig. 6
figure 6

Comparison between the performance indices of scenario 1 & 2 for 69 bus system

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Table 7 shows the outcomes of the OCDG problem for scenario 2. In case-1a & 1b, the real power loss is reduced to 9.6078 & 7.0345 kW, respectively, but the system loadability is improved to 4.09 & 6.4 only. In case-2a & 2b, the system loadability is enhanced to 4.51 & 7.04, respectively, but the power loss is reduced to 35.096 kW & 46.448 kW only. Among case-3a & case-3b, better enhancement in both objectives is observed in optimal connection of DGs in optimal network reconfigured case, i.e., real power loss is reduced to 23.8112 kW and system loadability is enhanced to 6.94. The convergence graphs for all cases of scenario-2 are shown in Fig. 7. Based on the above discussions it can be concluded that among all the cases in scenario-1 & 2, better improvement in both objectives with less % KVA injection by DGs is observed in case-3b of scenario-2, i.e., real power loss is reduced to 89.414%, maximum loadability is increased to 116.19%.

Table 7 Simulation results of 69 bus system for scenario-2
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Fig. 7
figure 7

Scenario-2 outcomes convergence graphs for 69 bus system

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To access the capability of the BO optimization technique to the proposed methodology, the results obtained are contrasted with the befitting methods and algorithms that are accessible in the literature and shown in Table 8. The proposed algorithm yields to produce the same result produced by the HPSO algorithm in the literature concerning loadabilty of the system as an objective function and the proposed algorithm performs well in mitigating the real power with comparison to the HTLBO-GWO algorithm. In [29], with 40% KW or 47.06 KVA injection by DGs into the system, real power loss reduced to 87.206%, system loadability increased to 27.72%. But with the proposed method in this paper, with 63.98% KVA injection by DGs into the system, real power loss reduced to 89.414%, system loadabilty increased to 116.19% that shows an improvement in both the objectives unlike the method in [29] which shows the efficacy of the proposed method.

Table 8 Comparison results of 69 bus system
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Conclusion

In this work, OCDG and ODNR problems on radial distribution systems have been addressed to enhance the system efficiency and too apt upcoming load growth via I2R loss mitigation and system loadability enhancement. To achieve the objectives, two scenarios each consisting of three cases and each case having two sub-cases are considered. The concept of a spanning tree has been taken for confirming the radiality status of the system.BO optimization technique has been taken to optimize the proposed objective functions and implemented on 33 & 69 bus test systems. In both the test systems, the highest percentage of improvement in both the objectives with less amount of % KVA injection by DGs into the system is observed in case-3b of scenario-2. From the outcomes, it has observed that loss of the system is reduced to (75–89) %, loadability enhanced to (94–121) % with the injection of 64% KVA by DGs in 33 & 69 bus systems. BO algorithm has performed well in optimizing the proposed objectives when compared with the other algorithms in the literature.