Contents Nomenclature
AbbreviationExpansioncost of all DG units | |
cost of total active power loss of the distribution system | |
cost of emission which targeted to minimize it | |
reflection of the voltage deviation on the cost | |
capital cost | |
variable fuel cost | |
operating and maintenance cost | |
interest rate (9%) | |
investment life time | |
total real power losses in all buses | |
active and reactive power injected at bus | |
generated active and reactive power of DG unit, respectively | |
maximum allowable active power generated from DG unit | |
active power injected from grid | |
demand of active and reactive power, respectively | |
reactive power losses | |
reactive power injected from grid | |
magnitudes of the voltage at buses | |
voltage deviation at load buses | |
maximum allowable voltage | |
voltage at load bus | |
maximum magnitude of voltage at load bus | |
minimum magnitude of voltage at load bus | |
angles of the voltage at buses | |
group of buses that connected to bus | |
susceptance and conductance of bus admittance matrix, respectively | |
weight factor that can be determined based on the | |
ratio of the feeder resistance to feeder reactance | |
resistance of branch | |
current flow of branch | |
total emission released from all DG units | |
emission products of DG units | |
factor of power loss | |
emission factor in | |
voltage deviation economic operator | |
transmission line loading | |
total available number of DG units | |
number of total branches | |
number of network buses | |
number of load buses | |
the | |
the | |
logarithmic spiral function | |
distance between the | |
constant indicates the shape of the logarithmic spiral | |
random number | |
the convergence constant | |
maximum number of flames | |
current number of iteration | |
maximum number of iterations |
Introduction
Recently, the distributed generators (DG) integrated into the distribution system have major positive impacts on the performance of the system, due to its ability to decrease the loss of transmission lines, improve the voltage stability, increasing the reliability and reducing the pollutant emission based on DG technology types [1]–[3].
The penetration of DG units in the distribution system is becoming more widespread because of the growth of demand load, reduction of pollutant emission and deregulated of the electrical power market. Several DG units technologies are used and categorized according to fuel energy used into dispatchable and non-dispatchable units. The former includes, for example, diesel generators, micro-turbine, and fuel cell. While, the later includes, renewable energy sources based on DG units such as solar photovoltaic systems, wind turbine generators, biomass and micro-hydro generators [4]–[7].
The effectiveness of DG performance is more related to the choice of their adequate location, types, and sizes, where the optimal selection will maximize the benefits of DG units used and avoid their drawbacks for the system such as increasing loss of the system, increasing the operating cost and voltage instability [8], [9]. Incorporating the DG units into the system has different impacts in the case of both steady-state and transient conditions. In the steady-state, there are some problems like reverse power flow, high power losses, voltage fluctuation, reactive power management, miscoordination of the protection scheme, poor power quality, regulation, and reliability of over-load tap changer (OLTC) [10]–[13]. On the other hand, the impacts in the transient state appear due to the islanding of DG units and the phenomena of the uncertainty of the output of DG units such as occurring from the variation of wind speed and shading effects in zones with PV [14]. The severity of these impacts is based on the locations of DG units with the amount of DG penetrations and the DG’s technology. Also, due to the nature of renewable DG units, the simultaneous variations of DG’s generations for supplying the demand load may cause under or over voltage. The effects of such phenomenon may affect by DG unit locations and weather conditions [15]. In addition, at a specific penetration level of DG units, the performance of the system is improved, but in contrast, beyond this level, the system was subject to degradation by substation and feeder loading, voltage deviation and increased power losses. Moreover, by increasing the penetration of DG units, the operation of the automatic voltage regulator (AVR) inside the OLTC of the transformer becomes more sophisticated and incapable because of occurring the phenomena of reverse power flow and accompanied with high voltage and current which can be controlled by employing different methods summarized in [9], [16].
Consequently, the problem of determining the optimal location and sizes of DG units has subject to great interest recently in order to achieve many objectives such as minimization of real power loss, improvement voltage profile, improvement power system quality and increasing both efficiency and reliability of the distribution system. So, various approaches are proposed in the literature to solve this problem [3], [17]–[21].
The authors in [3] proposed a novel method to determine the optimal size and location of DGs to not only reducing the power loss but ensuring the voltage stability of the system. The improved gravitational search algorithm is proposed in [19] to get the optimal placement and sizing of solar photovoltaic based DGs to minimize the total cost. A combined method of an intelligent water drop (IWD) and hybrid (GA) were proposed in [22] to determine the size and location of DGs in micro-grid for increasing voltage stability, reduce network losses and improve voltage profile. A hybrid fuzzy logic controller technique and ant-lion optimization algorithm’s with particle swarm optimization based combination is proposed in [23] to solve the optimal allocation of distributed generations in a radial distribution network to minimize the total cost of operation and deviation of voltage indexes. In [24] at different load levels, the objective function to find optimal location and sizing of DGs is reducing real and reactive power losses which solved by using biogeography-based optimization (BBO) algorithm. An efficient optimization algorithm to optimally allocate the multiple DG units in distribution systems based on sine cosine algorithm (SCA) and chaos map theory is proposed in [25] using three objective functions.
Nowadays, the DGs are planned optimally to achieve economic motive during the liberalized modern power market, so the optimally planning of distributed generators is very important for the operators in the distribution network [26]. In the real distribution network, there are different configurations with many huge buses. In addition to several load levels which may be taken into consideration at different periods, moreover, there are geographical and environmental constraints, this means a very large number of buses to be nominated for distributed generators. All of that may be making the choice of the optimal location and size is not easy and take a huge time. So it is better to use a technique reducing the nominated buses to save the time of searching according to network configuration. Therefore, the bus location index is employed in this paper to create a priority ranking list of candidate buses.
In 2015, the moth flame optimization (MFO) is proposed as a new technique to solve optimization problem in [27]. The MFO which is considered as one of the novel nature-inspired algorithms simulates the navigation method of moths for travelling for long distances. The MFO is appropriate for solving many practical optimization problems because of its brilliant characteristics [28]–[31].
As known, the balance between exploration and exploitation is the greatest significant features for any generalized approach. The exploration points to exploring the global search while the exploitation refers to the local search. According to the theory published in [32], no algorithm is the best appropriate for all the optimization problems. Therefore, there are different modifications are proposed by researchers to improve the characteristics of the MFO regarding the proper balance between exploitation and exploration capabilities.
The opposition based MFO method is proposed in [33] to overcome the disadvantages of the conventional MFO which are trapping in local optima and the slow convergence. In [34] the conventional MFO is combined with
This paper proposes a modified moth flame optimization (MMFO) algorithm. Two modifications are made in the original MFO to derive the proposed MMFO in order to improve the balance between the exploration-exploitation capabilities of the algorithm and speed up the convergence of the algorithm. Then, the proposed MMFO is employed to find the optimal location and sizing of DGs based on different dispatchable and non-dispatchable DGs units in order to minimize the total operating cost of the distribution system. The total objective function consists of the minimization of fuel cost, total real power loss, voltage deviation and pollution emission for some DGs is treated as weighted economic operators, where a multi-objective problem is converted to coefficient single objective function (CSOF) with considering some constraints of the system. The performance of the developed approach is tested using a standard test system and compared with other published methods to discover its notability for solving the problem described here. The contributions of this paper are to:
Propose the MMFO algorithm which improves the complementary features of the original MFO by improving the balance between the exploration and exploitation and avoiding the problems of the original MFO.
Introduce the problem formulation of finding the optimal location and sizing of DG units based on renewable energy sources to minimize the total operating cost considering four different objective functions.
Use the proposed MMFO algorithm to solve the above problem by converting the multi-objective function consists of four different functions into a coefficient single objective function (CSOF).
Enhance the solution of the above problem in comparison with the obtained results from published algorithms based on different cases and scenarios using the IEEE 69-bus test distribution system.
This paper is organized as follows: Section II introduces the bus location index (BLI) technique. The mathematical model of the objective problem is described in Section III. In Section IV the MMFO technique is discussed. The simulation results for the test system are presented with a discussion in Section V. Finally, Section VI presents the conclusions of the proposed work.
Bus Location Index (BLI)
It is known that any change in the injected active and reactive power at any bus of the distribution system will lead to a change in total real power losses. According to this concept, the bus location index (BLI) is formulated as described in [36]. The real power losses can be written as in [37] as following.
\begin{equation*} P_{L}=\sum _{i=1}^{N}\sum _{j\in {k_{i}}}V_{i}V_{j}[G_{ij}\cos (\theta _{i}-\theta _{j})+\beta _{ij}\sin (\theta _{i}-\theta _{j})]\tag{1}\end{equation*}
The power balance equations are written as follow:\begin{align*} \Delta {P_{k}}=&P_{k}-V_{k}^{2}G_{kk}-V_{k}\sum _{j=1,j\neq {k}}^{N}V_{j}[G_{kj}\cos (\theta _{k}-\theta _{j}) \\&+\,\beta _{kj}\sin (\theta _{k}-\theta _{j})] \tag{2}\\ \Delta {Q_{k}}=&Q_{k}-V_{k}^{2}\beta _{kk}-V_{k}\sum _{j=1,j\neq {k}}^{N}V_{j}[G_{kj}\sin (\theta _{k}-\theta _{j}) \\&-\,\beta _{kj}\cos (\theta _{k}-\theta _{j})]\tag{3}\end{align*}
The variation in real power losses based on the previous concept may be described as follow in [37] \begin{align*} \begin{bmatrix} \dfrac {\partial {P_{L}}}{\partial {P}}\\[0.8pc] \dfrac {\partial {P_{L}}}{\partial {Q}} \end{bmatrix}=J^{-1} \begin{bmatrix} \dfrac {\partial {P_{L}}}{\partial {\theta }}\\[0.8pc] \dfrac {\partial {P_{L}}}{\partial {V}} \end{bmatrix}\tag{4}\end{align*}
\begin{align*} J=\begin{bmatrix} \dfrac {\partial {\Delta {P}}}{\partial {\Theta }} &\quad \dfrac {\partial {\Delta {P}}}{\partial {V}}\\[0.7pc] \dfrac {\partial {\Delta {Q}}}{\partial {\Theta }} &\quad \dfrac {\partial {\Delta {Q}}}{\partial {V}} \end{bmatrix}\tag{5}\end{align*}
The BLI can be expressed as follow in [38] for each bus.\begin{align*} BLI=&\omega \frac {\partial {P_{L}}}{\partial {P}}+(1-\omega)\frac {\partial {P_{L}}}{\partial {Q}} \tag{6}\\ \omega _{i}=&\frac {\frac {r_{i}}{x_{i}}} {\frac {r_{i}}{x_{i}}+1}\tag{7}\end{align*}
The weight factor may be a unique value and is calculated as the mean value of all weight factors of buses. This value is accepted particularly and can be used in BLI equation due to the
Objective Problem Formulation
The target of the objective problem proposed here is finding the optimal location and sizing of DGs based on renewable energy sources to minimize the total operating cost with considering equality and inequality constraints. Where some coefficients are utilized to integrate different objective functions for creating only CSOF which used to minimize the total cost of the system.
A. Cost Formulation of Dg Units
The total cost of DG units contains capital installation cost, fuel cost, operating and maintenance cost. It may be formulated as follow [39] \begin{align*} C_{DG}=&\sum _{i=1}^{N_{DG}}C(P_{DGi}) \tag{8}\\ C(P_{DGi})=&C_{c}P_{DGi_{max}}+\frac {(1+r)^{n}-1}{r(1+r)^{n}}(F_{c}+OM_{c})P_{DGi}\tag{9}\end{align*}
B. Total Power Loss Formulation
The active power losses of the distribution system can be described as follow [40] \begin{equation*} P_{LOSS}=\sum _{k=1}^{N_{b}}|{I_{k}}|^{2}R_{k}\tag{10}\end{equation*}
The cost of active power losses can be formulated as following:\begin{equation*} C_{P_{LOSS}}=P_{LOSS}W_{LOSS}\tag{11}\end{equation*}
C. Pollution Emission Formulation of Dg Units
According to DGs technologies, there are some types of DGs that generate \begin{align*} E_{DG}=&\sum _{i=1}^{N_{DG}}E(P_{DGi}) \tag{12}\\ E(P_{DGi})=&(CO_{2,DGi}+NO_{x,DGi}+SO_{2,DGi})P_{DGi}\tag{13}\end{align*}
The cost of emission released by DG units may be formulated as follow:\begin{equation*} C_{E}= E_{DG}W_{E}\tag{14}\end{equation*}
D. Voltage Deviation Formulation
The penetration of DG units may be cause variation in the distribution system voltage. Therefore, the voltage violation should be limited. The voltage deviation can be defined as follow [39] \begin{equation*} V_{D}=\sum _{i=1}^{N_{l}}|V_{i}-V_{M}|\tag{15}\end{equation*}
The reflection of the voltage deviation on the cost can be expressed as follow:\begin{equation*} C_{VD}= V_{D}W_{VD}\tag{16}\end{equation*}
E. Equality and Inequality Constraints
According to the objective problem proposed, there are two types of constraints as following:
1) Equality Constraints
Power balance equation with considering DG units in the distribution system can be defined as follow [42] \begin{align*} P_{Grid}+\sum _{i=1}^{N_{DG}}P_{DGi}=&\sum _{j=1}^{N_{l}}P_{d}(j)+P_{LOSS} \tag{17}\\ Q_{Grid}+\sum _{i=1}^{N_{DG}}Q_{DGi}=&\sum _{j=1}^{N_{l}}Q_{d}(j)+Q_{LOSS}\tag{18}\end{align*}
2) Inequality Constraints
a: Voltage Limit Constraints
The voltage at each bus of the distribution system should be limited as following [42]:\begin{equation*} V_{Li}^{min}\leq V_{Li} \leq V_{Li}^{max}\tag{19}\end{equation*}
b: Dg Limit Constraints
The minimum and maximum allowable values of the active and reactive output power of DG units in the distribution system can be defined as follow:\begin{align*} P_{DGi}^{min}\leq P_{DGi}\leq&P_{DGi}^{max} \tag{20}\\ Q_{DGi}^{min}\leq Q_{DGi}\leq&Q_{DGi}^{max}\tag{21}\end{align*}
c: Feeder Constraints
The loading for each branch of distribution system should be limited using the following equation:\begin{equation*} S_{li} \leq S_{li}^{max}\tag{22}\end{equation*}
F. Total Objective Function
In this paper, the above four objective functions which are the minimization of fuel cost, total real power loss, voltage deviation and pollution emission for some DGs are combined and converted into CSOF based on some coeficients. Therefore, the total objective function can be expressed as follows:\begin{align*} \mathcal {F}=&C_{DG}+C_{P_{LOSS}}+ C_{E}+C_{VD} \\=&C_{DG}+P_{LOSS}W_{LOSS}+E_{DG}W_{E}+V_{D}W_{VD}\tag{23}\end{align*}
Proposed Optimization Algorithm
A. Moth Flame Optimization Overview
Moth-Flame Optimization (MFO) recently proposed in [27] is a population-based algorithm which mimics the moth’s navigation way in nature. It is based on the navigation way named transverse orientation of the moths. The main idea of the transverse orientation method is employing a fixed angle with respect to the moon by moths during flying. The moths attempt to keep the fixed angle when they see an artificial light that is very close in comparison with the moon, but they fail. So, they fly in a logarithmic spiral mechanism during convergence with the flame [27], [34].
The model of the MFO algorithm consists of two important components. They are moth and flame. The moths represent the members (solutions) which move around the search space, while the flames represent the best position (problem’s variables) found for these members. The MFO begins with an initial population of moths and flames which are randomly generated. The moth movement is oriented by the flames. The fitness value of each moth is then calculated based on the problem objective function. In the next iteration, the number of flames is decreased by removing the unfit flames which guide the moths to move to the fittest flame. These processes are repeated until only one flame remains which means that the best solution for the problem is obtained.
After the initial population of moths and flames are generated randomly, the mathematical model of transverse orientation behaviour can be expressed. The position of each moth which is guided by the flames can be updated as follows [27]:\begin{equation*} MO_{i}={g}(MO_{i},F_{k})\tag{24}\end{equation*}
The logarithmic spiral function can be expressed using the following equation [27]:\begin{equation*} {g}(MO_{i},F_{k})=D_{i}\cdot \exp (s\varepsilon)\cdot \cos (2\pi \varepsilon)+F_{k}\tag{25}\end{equation*}
\begin{equation*} D_{i}=|F_{k}-MO_{i}|\tag{26}\end{equation*}
The parameter
Also, to improve the exploitation property, the number of flames is decreased progressively with the iterations as follows [27]:\begin{equation*} \textrm {Flame number}=\textrm {round}\left ({N_{f}-it*\frac {N_{f}-it}{max_{it}}}\right)\tag{27}\end{equation*}
More details and the pseudo code of the original MFO algorithm can be found in [27].
B. Modified Moth Flam Optimization (MMFO)
To derive the proposed MMFO method, two modifications are made in the conventional MFO. In the conventional MFO, the convergence constant (\begin{equation*} c=\exp \left ({-\left ({\frac {it}{max_{it}/2}}\right)^{2}}\right)-2\tag{28}\end{equation*}
It is clear from (27) that the number of flames will reduce with iterations. This reduction in the flames’ number makes the balance between the exploitation and exploration. Therefore, equation (27) is modified in the proposed MMFO to enhance the balance between exploitation and exploration of the algorithm as follows:\begin{equation*} \textrm {Flame number}=\textrm {round}\left ({N_{f}*\exp \left ({-\frac {it}{max_{it}/4}}\right)}\right)\tag{29}\end{equation*}
Simulation Results and Discussions
The developed method presented in the previous section is applied to the IEEE-69 bus radial distribution test system which described in [43] and shown in Fig. 1. The system consists of the main root bus represent the utility grid at bus 69, 70 branches where the system voltage is 12.66 kV. The system load active and reactive power are 3.86 MW and 2.69 MVAR, respectively. While the maximum and minimum voltage limits are 0.95 and 1.05 p.u., respectively [41], [44].
The notability of the developed MMFO method in determining the optimal location and sizing of different DG units is proved in this paper compared with other published algorithms. These algorithms are ant lion optimizer (ALO) [45], grey wolf optimizer (GWO) [46], dragonfly algorithm (DA) [47], conventional MFO [27], modified JAYA (MJAYA) algorithm [48], and Salp swarm algorithm (SSA) [49].
In the implementation of the proposed MMFO and other meta-heuristic methods, many parameters should be chosen. In this paper, the appropriate values of these parameters are obtained based on empirical tests. All the numerical studies have been run on 2.9-GHz i7 PC with 8 GB of RAM using MATLAB 2014a.
Firstly, the sorting list of buses is obtained using the BLI method described in section II to create narrow candidate buses as shown in Fig. 2. Table 1 shows candidate buses sorting in descending order, accordingly, the first twenty buses from the table were chosen for primary locations of DG units. Then the MMFO method is used to find optimal placement and sizing of DG based on the proposed objective function.
The simulation results are executed with consideration of the system maximum load. Moreover, the maximum capacity of DG power is limited to 30 % of the total load demand. In addition, the DG units can deliver active power and reactive power where the DG units are represented as PQ model at power factor 0.9 [41], [44], [50].
To evaluate the MMFO method, different scenarios are carried out as follows:
Location and sizing for one DG unit
Location and sizing for two DG units
Location and sizing for three DG units
In all scenarios, there are six types of DG units (fuel cell, micro-turbine, photovoltaic, wind, hydro and biomass). The specifications of these types can be found in [39], [51]. Table 2 shows the capital costs, variable fuel cots, average operating and maintenance costs, and emission factors for
A. Location and Sizing for One DG Unit
According to different technologies of renewable energy sources as DG units, the simulation results for locating and sizing one DG unit using a developed MMFO in comparison with other techniques are shown in Table 3. In addition, the effect of using one DG unit on the voltage profile using the developed MMFO is shown in Fig. 3.
The results show that the minimum total cost of the system was obtained by using the proposed MMFO compared with other meta-heuristic optimization techniques. A great benefit is obvious of the proposed MMFO when using one micro-turbine at optimal bus 60 with optimal sizing of 0.6144 MVA where the total cost of the system is
B. Location and Sizing for Two DG Units
To confirm the efficiency of the developed MMFO to obtain the best location and size of DG, it is applied for two units of DG. The simulation results for locating and sizing two DG units using a developed MMFO in comparison with other recently published techniques are shown in Table 4. Also, the effect of using two DG units on voltage profile based on the developed MMFO is shown in Fig. 4.
These results prove that the minimum total cost of the system may be obtained using the proposed MMFO compared with other meta-heuristic techniques, based on employing two micro- turbines at optimal buses 60 and 63 with optimal sizing of 0.4423 and 0.6 MVA, respectively where the total cost of the system is equal to
C. Location and Sizing for Three DG Units
In this case, to prove the superiority of the developed MMFO for finding the optimal location and size of DG, it is applied for three units of DG. The simulation results for locating and sizing three DG units based on developed MMFO in comparison with other published techniques are shown in Tables 5 and 6. Furthermore, the effect of using three DG units on the voltage profile based on proposed MMFO is shown in Fig. 5.
The results clearly indicate that the superiority of the proposed MMFO over other methods, where it gives the minimum total cost. The minimum total cost obtained using the proposed MMFO with inserting three micro-turbines at buses 60, 61 and 63 with optimal sizing of 0.6477, 0.3185 and 0.2293 MVA, respectively and the total cost of the system equal to
D. Discussion
By employing the reduced sorting list of buses based on the BLI method, the computational time can be minimized. In addition, the efficiency of the investigation is improved whereas preserving the equilibrium through exploitative and exploratory.
From the simulation results, the optimal location and sizing of DG’s are different for all cases according to different technologies of DG’s based on the suggested objective function. However, the proposed MMFO technique offering great performance compared with other meta-heuristic optimization techniques. The main benefits of the proposed technique are achieved when using multiple DG units.
The outstanding performance of MMFO proposed is assigned through the locations of multiple DG units with different technologies simultaneously. The verification of the proposed method is achieved by considering the elaborate investigation for the location and sizing of two and three DG units with different technologies.
In case of using one DG with different technologies, the most optimal location bus has been 60 based on the proposed MMFO results with sizing varying between 0.6144 MW to 0.7975 MW.
It is seen that the locations of DGs are sequentially take placed by applying the proposed method for the PV, wind, and micro-turbine where the optimal location bus calculated in case of single DG is repeated in case of two DG units, finally the case of three DG units will contain the buses selected in previous cases.
When using three DG units, the proposed method gives better results in comparison with using one or two DG units referring to the reduction of power loss and voltage deviation. For example, Figures 6 and 7 show the relation between the size (number of DG units), power losses and voltage deviation, respectively when one, two or three micro-turbines are used. These figures indicate that the reduction of power loss is increased from 55.15 % and 72.08 % using one or two micro-turbine units, respectively to 74.4% using three micro-turbine units. Moreover, the voltage deviation is reduced from 1.3666 and 1.1205 using one or two micro-turbine units, respectively to 1.0607 % using three micro-turbine units.
The relation between the size and power losses when one, two or three micro-turbines are used based on MMFO.
The relation between the size and voltage deviation when one, two or three micro-turbines are used based on MMFO.
The multiple locations of DG units might be useful but should be inspected firstly economically as seen from results for different types of DG technology where the increasing number of DG units will increase the total system cost.
To show the convergence property of the developed MMFO in comparison with other methods, Figs. 8 and 9 illustrate the convergence curves of the developed MMFO and other methods in case of using three DG units. From these figures, one can observe that the objective function of the developed MMFO converges smoothly to the optimum solution without any unexpected oscillations which approves the convergence dependability of the developed MMFO. In addition, the developed MMFO needs fewer iterations in comparison with other methods to obtain the solution. This is due to the improved exploration and exploitation mechanism in the developed MMFO.
Comparison between the convergence characteristics of the developed MMFO and other methods in case of use three DG units: Part A.
Comparison between the convergence characteristics of the developed MMFO and other methods in case of use three DG units: Part B.
To further prove the effectiveness of the developed MMFO method over other methods, Figs. 10–12 show the computational time for the developed MMFO method and other algorithms for one, two, and three DG units, respectively. These results prove that the developed MMFO method is more effective than others when computational time is considered. The computational time of the developed MMFO is quite less and better than other optimization methods in most cases while it is comparable with other optimization methods in a few cases. As a whole, the developed MMFO is computationally efficient than the conventional MFO and other methods as a result of using the improved exploration and exploitation mechanism which accelerates the convergence of the MMFO method.
Conclusion
The MMFO algorithm is proposed in this paper to find the optimal location and sizing of DG units based on renewable energy sources to minimize the total operating cost considering four different objective functions. The MMFO is developed to overcome the disadvantages of the conventional MFO, by improving the balance between the exploration and exploitation and speed up the convergence of the algorithm. Also, the total objective function which consists of four different functions is converted to coefficient single objective function. The performance of the developed MMFO algorithm is verified using a standard test system and compared with some meta-heuristic methods to discover its notability based on different scenarios. It can be noticed from the results that the MMFO algorithm provided a better reduction of the objective function for all scenarios over other meta-heuristic methods used in the comparison. The comparisons’ results with other meta-heuristic methods evidently demonstrate that the MMFO algorithm outperformed these meta-heuristic methods whatever the number of DG units.