1 Introduction

In recent times, DGs have gained more consideration in power systems for handling the environmental and financial challenges instigated by fossil fuel based power plants. DGs are known as the electric power generations that can be directly connected to loads or DS [1]. The DGs such as wind turbines, solar photovoltaic (PV), full-cells, biomass can mitigate the emission of greenhouse gases (GHG) and climate changes. Moreover, the DS are facing several defies due to the increasing electricity consumption and operational constraints [2]. The DS have been enforced to deliver power to the customer continuity. Owing to low voltage level and high currents, DS have been suffered from severe power losses and voltage volatility. Consequently, the incorporation of DGs have been considered to overwhelm the aforesaid problems [3, 4]. So as to preserve the high efficiency and to enhance the performance of the DS, the placement of DGs should be optimal. Furthermore, the optimal allocation and sizing of DGs will minimize the power losses, energy cost, pollutant emission and THDv. Similarly, it increases the voltage profile and VSI. These concerns have stimulated the research effort towards the development of accurate and fast techniques for optimal DG placement and sizing to prevent voltage collapse and reduce the system’s power losses. In this context, several optimization techniques have been applied to solve optimal sizing and placement problem of DGs [5,6,7,8]. Detailed reviews on several optimal DG allocation algorithms and their implementation have been presented [9,10,11,12].

In recent years, several researches have been conducted to reduce the power loss and to increase the voltage profile in DS while optimizing optimal allocation problem of DGs and shunt capacitors [13]. Previously, analytical methods [14] and evolutionary algorithms like Generic Algorithm (GA) [15], Particle Swarm Optimization (PSO) [16, 17], hybrid PSO [18], hybrid Ant Colony Optimization (ACO) and Artificial Bee Colony (ABC) [19] have been applied for solving optimal allocation problem of single and multiple types of DGs. A logical methodology to optimally size and site both capacitors and DGs on DS has been presented [20] and PSO [21] has been implemented to improve the results. The meta-heuristic techniques for instance hybrid Harmony Search Algorithm (HSA) and Particle ABC (PABC) [22], Intersect Mutation Differential Evolution (IMDE) [23] and Backtracking Search Algorithm (BSA) [24] have all been employed to design and site both DGs and shunt capacitors. Most of these literatures have solved the problem with single objective namely real power loss minimization.

Most recently, Grey Wolf Optimizer (GWO) has been proposed to solve multiple objectives such as minimization of power loss and improvement of voltage profile [25]. A Multi Objective Shuffled Bat (Mo-SB) algorithm has been applied to select optimal placement of DGs by considering power loss and energy cost minimization [26]. Mo-SB algorithm has been proposed by hybrid Shuffled Frog Leaping Algorithm (SFLA) and Bat algorithm. Discrete Artificial Bee Colony (D-ABC) algorithm has been implemented to site DGs by considering the objectives such as maximization of system loadability and reduction of power losses [27]. Multiple objectives such as energy cost reduction, reliability enhancement, losses minimization and voltage deviation reduction have been considered while solving DGs optimization problem using Ant Lion Optimizer (ALO) [28] and Whale Optimization Algorithm (WOA) [29]. Lately, the improved chicken swarm optimization has been employed to place the charging stations optimally in the IEEE 33 bus system. The impact of Electric Vehicle load demand on the DS, in terms of per unit voltage profile, voltage stability index, average voltage deviation index and power loss has been discovered [30].

Firefly Algorithm (FA) has been proposed to increase the speed of exploration and exploitation, the sporadic patterns and activities of fireflies have been adapted [31]. FA outperforms PSO on continuous constrained optimization problems [32]. Owing to the benefits of the FA, it has been extensively employed in several engineering applications. The typical power system optimization problems such as economic dispatch and load forecasting problems have been solved using FA [33, 34]. This study contributes.

  • To solve the objectives such as minimization of power losses, minimization the voltage deviation, enhancement of VSI, reduction of pollutant emission and elimination of THDv simultaneously on a real-world power system.

  • To apply the new meta-heuristic optimization algorithm namely MOFA for DG placement in DS.

  • To test the proposed method on IEEE 33-bus and a practical 62 bus IUS by considering multiple objectives with different cases.

2 Mathematical Formulation

Six OFs for instance reduction of power losses, minimization of the voltage deviation, improvement of VSI, reduction of costs, reduction of pollutant emission, and elimination of average THDv for optimum DGs placement problem are selected.

2.1 Reduction of Power Losses

OF1 intends to minimalize the power losses and it is stated as [35]

$${\text{OF}}_{{1}} = \min \sum\limits_{i = 1}^{nL} {R_{i} } \left| {I_{i} } \right|^{2}$$
(1)

where, nLno. of branches, Rithe resistance of the i-th branch, Iicurrent magnitude of i-th branch.

2.2 Minimization the Voltage Deviation

OF2 intends to minimize the voltage deviation by following Eq. (2).

$${\text{OF}}_{{2}} = \min \sum\limits_{i = 0}^{N} {\left( {\frac{{v_{i} - v_{i}^{spec} }}{{v_{i}^{\max } - v_{i}^{\min } }}} \right)}^{2}$$
(2)

where. \({\text{v}}_{{\text{i}}}\) bus voltage magnitude, \(v_{i}^{spec}\) specified voltage magnitude (1.0 p.u.)

2.3 Improvement of VSI

OF3 intends to enhance VSI.

$${\text{OF}}_{{3}} = \min \left( {\frac{1}{{VSI\left( {m_{2} } \right)}}} \right)$$
(3)
$$\begin{gathered} {\text{VSI}}\left( {{\text{m}}_{{2}} } \right) = \left| {V(m_{1} )} \right|^{4} - 4\left[ {P(m_{2} )X_{ij} - Q(m_{2} )R_{ij} } \right]^{2} \hfill \\ - 4\left[ {P(m_{2} )R_{ij} + Q(m_{2} )X_{ij} } \right]\left| {V(m_{1} )} \right|^{2} \hfill \\ \end{gathered}$$
(4)

where, \(m_{1}\) slack bus, \(m_{2} ,m_{3} ,...,m_{n}\) bus no. (2, 3,..., nbus), Vbus voltage (p.u), P and Qactive power (MW) and reactive power (MVar), \(R_{ij}\) and \(X_{ij}\) resistance (Ω) and reactance (Ω) of branch ij,

2.4 Reduction of Cost

The cost is minimized by optimizing OF4.

$${\text{OF}}_{{4}} = \min \sum\limits_{i = 1}^{{N_{DG} }} {\left( {C_{{DG_{i} }} } \right)}$$
(5)

Where

$$C_{{DG_{i} }} = a + b \times PG_{i}$$
(6)
$$a = \frac{Capital\cos t(\$ /kW) \times Capacity(kW) \times Gr}{{Lifetime(year) \times 8760 \times LF}}$$
(7)
$$b = O\& M\cos t(\$ /kWh) + Fuel\cos t(\$ /kWh)$$
(8)

where, \(N_{DG}\) number of DGs, \(C_{{DG_{i} }}\) generation cost of i-th DG ($/kWh), a, bfixed and variable generation cost constants, \(PG_{i}\) active power at i-th DG unit (kW), Grthe annual rate of profit (5%), LFthe load factor of DGs (0.47), O & M costoperation and maintenance cost ($/kWh).

2.5 Reduction of Pollutant Emissions

OF5 has been modeled to reduce CO2, SO2, and NOx emissions [36].

$$OF_{5} = Min(f) = \sum\limits_{i = 1}^{{N_{DG} }} {E_{{DG_{i} }} + E_{Grid} }$$
(9)
$$E_{{DG_{i} }} = \left( {CO_{2}^{DG} + NO_{x}^{DG} + SO_{2}^{DG} } \right) \times PG_{i}$$
(10)
$$E_{Grid} = \left( {CO_{2}^{Grid} + NO_{x}^{Grid} + SO_{2}^{Grid} } \right) \times Pg_{Grid}$$
(11)

where, CO2, SO2, and NOxcarbon dioxide, sulfur dioxide, and nitrogen oxides; \(E_{Grid}\) and \(E_{{DG_{i} }}\) emissions generated by the grid and i-th DG unit.

2.6 Elimination of THDV

OF6 intends to eliminate the THDv by using Eq. (12).

$$OF_{6} = Min(THD_{V} ) = \frac{{\sum\nolimits_{i - 1}^{m} {THD_{{V_{i} }} } }}{m}$$
(12)

where m is the no. of buses.

and

$$THD_{vi} \left( \% \right) \le THD_{{v_{\max } }}$$
(13)

where, \(THD_{{v_{\max } }}\) is the maximum permissible bound at every bus (5%).

2.7 Restraints

The equality and inequality restraints are deliberated while solving the problem.

The restraints for power balance requirements are pondered as

$$\sum\limits_{i = 1}^{NG} {PG_{i} } - P_{L} = P_{d}$$
(14)
$$\sum\limits_{i = 1}^{NG} {QG_{i} } - Q_{L} = Q_{d}$$
(15)

The extreme power produced by DGs must not exceed the allowable bounds of DS.

(a) Generator bounds

$$PG_{i}^{\min } \le PG_{i} \le PG_{i}^{\max }$$
(16)
$$QG_{i}^{\min } \le QG_{i} \le QG_{i}^{\max }$$
(17)

(b) Bus voltage bounds

$$0.95 \le V_{i} \le 1.05,i = 1,2,...,nbus$$
(18)

(c) DG Power factor bound

$$0.9 \le PF$$
(19)

where, \(P_{L}\) and \(Q_{L}\) active power losses (MW) and reactive power losses (MVar); \(P_{d}\) and \(Q_{d}\) active power load demand (MW) and reactive power load demand (MVar).

3 FA for Multi-Objective Optimization (MOO)

The description and the implementation procedure have been adopted from [8, 10, 35, 39]. Table 1 offers the data of the DGs.

Table 1 Data of the DGs [35]
Full size table

The MOO problem aims to optimize multiple and competing objective functions. It results with a non-dominated set of solutions namely Pareto solutions, rather than a single optimal solution. The formulation for a MOO problem can be expressed as follows

$$MinF(x) = \left[ {f_{1} (x),f_{2} (x),...,f_{p} (x)} \right]^{T}$$
(20)
$$h(x) = 0$$
$$g(x) \le 0$$

where f1, f2, …, fp are the objective functions and x is the set of decision variables and h(x) and g(x) are the equality and inequality constraints of the problem.

FA is an optimization algorithm that imitates the behavior of fireflies [37,38,39,40]. The initial population of fireflies has been expressed based on Eq. (21).

$$c_{i,d} = c_{d,\min } + rand \cdot \left( {c_{d,\max } - c_{d,\min } } \right)i \in \left[ {1,N_{F} } \right],d \in \left[ {1,D} \right]$$
(21)

where, NFNumber of fireflies; DDimension of the problem; randrandom number between 0 to 1.

The position vector \(c_{i} = \left( {c_{i1} ,c_{i2} ,...,c_{iD} } \right)\) of the firefly i, which has been deliberated as a Pareto optimal solution. The modernized tactic of FA has been provided in the Eq. (22).

$$c_{i}^{new} = c_{i}^{old} + \beta_{0} \exp \left( { - \gamma r_{ij}^{2} } \right)\left( {c_{j}^{old} - c_{i}^{old} } \right) + \alpha \xi_{i}$$
(22)

where, γlight absorption coefficient; β0attractiveness at rij = 0; αrandomization parameter; ξivector of random numbers with Gaussian or uniform distributions.

The distance between any two fireflies rij can be calculated by Eq. (23).

$$r_{ij} = \left\| {c_{i} - c_{j} } \right\| = \sqrt {\sum\limits_{k = 1}^{D} {\left( {c_{i,k} - c_{j,k} } \right)^{2} } }$$
(23)

where ci,k is the k-th dimension of independent variable ci of the firefly i, cj,k is the k-th component of the independent variable cj of the firefly j.

3.1 Evaluation of Pareto optimal solution

While optimizing multiple objectives, it is tough to choose a single optimal solution. So as to choose the “most appropriate” solution, the concept of optimality has been swapped by Pareto optimality. The Pareto optimal solution cannot be recognized with a single objective, and because of this, it has been determined without abating the enactment with more than one objective. A fuzzy approach has been employed to find the Pareto optimal solution. The fuzzy membership functions have been expressed by Eq. (24).

$$\mu_{Fi} (X) = \left\{ {\begin{array}{*{20}c} {1,} \\ {0,} \\ {\frac{{F_{i}^{\max } - F_{i} (X)}}{{F_{i}^{\max } - F_{i}^{\min } }},} \\ \end{array} } \right.\begin{array}{*{20}c} {F_{i} (X) \le F_{i}^{\min } } \\ {F_{i} (X) \ge F_{i}^{\max } } \\ {F_{i}^{\min } \le F_{i} (X) \le F_{i}^{\max } } \\ \end{array} i = 1,2$$
(24)

The values of Fmin and Fmax have been obtained through the payoff table. The singular optima of the objective functions have been estimated to make the payoff table for the competing objective functions p. Then the objective functions Fi, i = 1,..., p (\(F_{i}^{*}\) signifies the optimum value of Fi) has been resolved. The balance objective functions F1, …, Fi-1, Fi+1, …, Fp have been estimated and signified as \(F_{i}^{i} ,...,F_{i - 1}^{i} ,F_{i + 1}^{i} ,...,F_{p}^{i}\). The i-th row of the payoff table has \(F_{i}^{i} ,...,F_{i - 1}^{i} ,F_{i}^{*} ,F_{i + 1}^{i} ,...,F_{p}^{i}\). Similarly, all rows of the payoff table have been assessed. The j-th column of the payoff table has the estimated values for the objective function Fj among which the minimum and maximum values specify the bound of the objective function Fj for the implemented fuzzy method. The incessant and monotonic functions have been employed as membership functions. MOFA forms a population and searches in an objective space to select the optimal solution. While optimizing MOO, a set of non-dominated solutions has been saved in the origin. The solution X1 dominates X2 when the subsequent conditions have been satisfied:

$$\begin{gathered} \forall j \in \left\{ {1,2,...,n} \right\},F_{j} (X_{1} ) \le F_{j} (X_{2} ), \hfill \\ \exists m \in \left\{ {1,2,...,n} \right\},F_{m} (X_{1} ) < F_{m} (X_{2} ), \hfill \\ \end{gathered}$$
(25)

The normalized membership function has been calculated for each individual using Eq. (26).

$$N_{\mu } (j) = \frac{{\sum\nolimits_{i = 1}^{{N_{F} }} {w_{i} } \times \mu_{Fi} (X_{j} )}}{{\sum\nolimits_{j = 1}^{{N_{rep} }} {w_{i} \sum\nolimits_{i = 1}^{{N_{F} }} {w_{i} } \times \mu_{Fi} (X_{j} )} }}$$
(26)

This membership function has been executed to rank the non-dominated solutions consistent with the importance given over the objective functions and it has been signified as wi in Eq. (26).

4 Simulation Results

Two DS such as the IEEE 33-bus system [21] and the TANGEDCO 62-bus IUS [13] have been considered [25].

Three different cases are considered as follow.

Case 1

The position and size of DGs are estimated by considering three objectives (OF1, OF2, and OF3).

$${\text{MOF}}_{1} = {\text{ min}}\left( {{\text{OF}}_{1} ,{\text{ OF}}_{2} ,{\text{ OF}}_{3} } \right)$$
(27)

Case 2

The objectives for instance OF1, OF4, and OF5, have been optimized. MOF2 has been expressed as:

$${\text{MOF}}_{2} = {\text{ min}}\left( {{\text{OF}}_{1} ,{\text{ OF}}_{4} ,{\text{ OF}}_{5} } \right)$$
(28)

Case 3

Multiple DGs are placed by optimizing the objectives, such as OF2, OF4, and OF6. MOF3 has been framed as:

$${\text{MOF}}_{3} = {\text{ min}}\left( {{\text{OF}}_{2} ,{\text{ OF}}_{4} ,{\text{ OF}}_{6} } \right)$$
(29)

4.1 Results of 33-Bus System

The results of case 1 have been offered in Table 2. It evidently infers that the optimal location and capacity of DGs make an optimistic effect to minimalize energy loss, increase voltage profile and boost VSI. Table 3 provides the technical, financial, and environmental benefits by solving MOF2. Table 4 presents the minimized values of voltage deviation, energy cost and THDv by optimizing MOF3. Table 5 indicates that power loss, overall energy cost and pollutant emission have been minimized by 133.4388 kW, 2.117$/MWh, 4755.1 lb/h respectively after integrating DGs into the system.

Table 2 Results of MOF1 for IEEE 33 bus system
Full size table
Table 3 Results of MOF2 for IEEE 33 bus system
Full size table
Table 4 Results of MOF3 for IEEE 33 bus system
Full size table
Table 5 Comparison of results for IEEE 33 bus system
Full size table

4.2 Results of TANGEDCO 62-Bus IUS

The results of TANGEDCO 62-bus IUS indicate that a substantial minimization in power losses of 35.7842 MW is attained while optimizing MOF1. Table 6 specifies that the minimized voltage deviation of 0.0035 p.u has been achieved. The outcomes of MOF2and MOF3 have been presented in Tables 7 and 8. The PV and WT type DGs are vital in the minimization of emission, power loss and energy cost. A noticeable minimization in a power loss has been achieved from 40.4512 to 37.5452 MW. The energy cost has been minimized from 36.8121 $/MWh to 35.445 $/MWh. The pollutant emission has been decreased from 117,564.25 lb/h to 84,215.30 lb/h.

Table 6 Results of MOF1 for 62 bus IUS
Full size table
Table 7 Results of MOF2 for 62 bus IUS
Full size table
Table 8 Results of MOF3 for 62 bus IUS
Full size table

The voltage profile of the IEEE 33 bus and TANGEDCO 62 bus IUS are shown in Figs. 1 and 2 respectively. The voltage profile values have been plotted before and after the integration of DGs. The illustrative comparisons of the power losses, energy cost and pollution emission for before and after integrating DGs have been shown in Figs. 3 and 4 for IEEE 33 bus system and 62 bus IUS respectively.

Fig. 1
figure 1

Voltage profile for IEEE 33 bus system

Full size image
Fig. 2
figure 2

Voltage profile for 62 bus IUS

Full size image
Fig. 3
figure 3

Comparative results for IEEE 33 bus system on optimal placement problem

Full size image
Fig. 4
figure 4

Comparative results for 62 bus IUS on optimal placement problem

Full size image

The values of VSI are less than 1 for the system, hence the system is stable. The values shows VSI close to 1 infers that the system is impending its load limit and is nearly unstable. Else, the more VSI is close to zero the more the node is stable. In reality, the optimal DG placement is selected based on which node has the highest VSI value. Figures 5 and 6 show the Pareto front of IEEE 33 bus system and 62 bus IUS respectively using MOFA. The BCS for all the cases have been identified from the three-dimensional Pareto optimal set attained by optimizing the MOO. It has been noted that every Pareto optimal solution is an alternate choice for a decision-maker.

Fig. 5
figure 5

Three-dimensional Pareto optimal set attained by optimizing multiple objectives on IEEE 33 bus system

Full size image
Fig. 6
figure 6

Three-dimensional Pareto optimal set attained by optimizing multiple objectives on 62 bus IUS

Full size image

The power system planner can espouse one of them for the individual system depends on its preferences over the objective functions. After applying the MOFA to generate Pareto sets, the BCS among them has been selected using fuzzy decision-making according to the Eq. (24). For instance, the performance analysis for the results of MOF3 for 62 bus IUS is given in Tables 9. The best, worst, mean, variance and standard deviation (SD) [42] values are estimated and given in Table 10.

Table 9 Comparison of results before and after incorporating DGs for 62 bus IUS
Full size table
Table 10 Estimation of performance analysis the results of MOF3 for 62 bus IUS
Full size table

The consistency of the MOFA has been validated from the parameter variance and it has been estimated as 0.00245. The merits of MOFA is that it typically has good efficacy for certain problems and necessitate only a minimum number of iterations. Though, one of its limitations is the high possibility of being stuck in local optima since it is a local search algorithm.

5 Conclusion

In this study, the MOFA is executed to estimate the numbers, placements and capacities of the DGs for IEEE 33-bus system and 62-bus IUS. The formulations of the objectives for instance minimization of power loss, reduction of voltage deviation, improvement of VSI, reduction of cost, mitigation of emission and elimination of THDv are modeled. A systematic and exhaustive investigation has been performed. The implementation of MOFA for optimal placement of DGs on DS reveals its efficiency. The execution of MOFA on IEEE 33-bus system and 62-bus IUS demonstrates that with DGs penetration, the power losses, energy cost, and pollutant emission can be reduced remarkably. Moreover, the overall voltage quality has been improved considerably. It has been experienced that the DG units operating in voltage control mode and having reactive power regulation capability efficiently improve the voltage stability margin regardless of the system size. The MOFA offers a modest and precise solution and do not needed extensive computations. Lastly, the MOFA is based on naive algorithm through simple mathematical formulations. Hence, it can be simply protracted to deliberate the load variation and erratic characteristics of renewable based DGs. In future, the economic, environmental, social, and technical impacts of the renewable energy sources would be investigated. The power loss estimation has been performed under both leading and lagging power factors while sizing of DGs.