Keywords

1 Introduction

Distributed generation (DG) has been more popular in recent years for usage in power grids. There are a number of ways in which DG units may affect system operating conditions, including the voltage profile, voltage stability, dependability, and the safety of the power market [1, 2]. Electricity is always a need, and distribution companies (DISCOs) work tirelessly to meet the needs of its consumers while keeping costs low. The design, operation, and maintenance of the DISCOs are based on the principle of maximizing return on investment at the lowest possible cost. Improving the voltage profile and minimizing power loss are crucial responsibilities for DISCOs to achieve maximum profit [3, 4]. To enhance the functioning of the system, the DISCOs are developing a great number of inventive technological concepts and programmes [5].

To maximize the DG owner’s return while minimizing the distribution company’s expenditure, this framework analyses and places DG’s units in the best possible location of the RDS [6, 7]. To maximize the best placement and appropriate value of DG units, a new parameter-free MFO approach is suggested [8,9,10,11]. The two searching operators of MFO, such as moths and flames, have the greatest capacity to find the ideal global solution with the least amount of computing time. Consequently, it effectively optimizes the size and ideal position of DG with conventional operational restrictions in order to boost the voltage profile and decrease DISCO network loss [12,13,14,15]. This paper considers the unpredictability of load demand, power production, electricity pricing, and dependability. The improved performance of the MFO algorithm is shown by testing the method on standard IEEE 33 node test system. The simulation result of voltage profile, power loss, DG location and size, and the cost–benefit analysis of DISCOs and DG owners are presented cutting-edge this research both numerically and graphically. Research based on comparisons was also carried out to prove the method’s efficacy.

2 Problem Formulation

2.1 Objective Functions of DISCOs

This study’s primary purpose is to maximize the return of DG owners and decrease the costs of DISCOs. The profit of DG owners is expressed numerically as,

$$\max \left( F \right) = {\text{MPF}}$$
(1)

Anywhere F remains the system’s impartial purpose and MPF remains the maximum profit of DG owners.

2.1.1 Maximum Profit Function

The ten-year change amid the benefits understood then the costs expended aimed at DG deployment is used to compute the maximum savings cost (Eq. 2).

$${\text{MPF}} = \sum {\text{Benefit}} - \sum {\text{Expenses}}$$
(2)

2.1.2 Net Present Value

Equation (3) demonstrates how to calculate the remaining contemporary worth issue (γ) using a 9% rise amount and a 12.5% attention amount to assess the worth of current costs over a planning period (N). The current cost value is calculated by multiplying this factor by each expense heading.

$${\text{NPV Factor}},\gamma = \mathop \sum \limits_{t = 1}^{N} \frac{{1 + {\text{IF}}}}{{1 + {\text{IR}}}}$$
(3)

NPV stands for remaining contemporary worth, IF for inflation, and IR for interest rate when t = 1, 2, 3,…, N.

2.1.3 Benefit Evaluation

The expected price of DG-generated power is US$ 300 per kWh, and the estimated cost of reducing energy loss is US$ 0.05 per kWh, for a total cost of US$ 300 per kWh.

2.1.4 Cost of Energy Loss Reduction

To simulate actual power losses, the load movement explanation for the test system is carried out deprived of DG, and then the procedure is repeated. The discrepancy in losses, which is indicated in Eq. (4), demonstrates a reduced net loss.

$${\text{NLR}} = P_{{{\text{loss}}}} - P_{{\text{loss,DG}}} ,$$
(4)

where NLR = net loss reduction, Ploss = system power loss without DG, and Ploss, DG = system power loss with DG. The DG loss reduction is transformed to a cost value using Eq. (5).

$$C_{{\left( {{\text{NLR}}} \right)}} = {\text{NLR}} \times \left( {{\text{cost of energy saving}}/{\text{kWh}}} \right) \times 8760$$
(5)

CNLR’s net present cost value is derived using Eq. (6).

$${\text{NPV}}\left( {C_{{{\text{NLR}}}} } \right) = C_{{{\text{NLR}}}} \mathop \sum \limits_{t = 1}^{N} \gamma t$$
(6)

wherever t = 1, 2,3,…,N; CNLR = cost of net loss reduction.

2.1.5 Cost of DG Power Generation

This inquiry has examined a PV system. The cost information aimed at this PV system is retrieved after [6] and computed using Eq. (7).

$$C_{\left( {{\text{DG,GEN}}} \right)} = \left( {{\text{DG Size}}} \right) \times \left( {\frac{{{\text{DG Generation cost}}}}{{{\text{kW}}}}} \right) - {\text{yr}}$$
(7)

The present-day charge worth of CDG, Gen is computed by Eq. (8).

$${\text{NPV}}\left( {C_{{\text{DG,Gen}}} } \right) = C_{{\text{DG,Gen}}} \mathop \sum \limits_{t = 1}^{N} \gamma t$$
(8)

wherever t = 1, 2,3,…,N; CDG, Gen = cost of DG power generation.

2.1.6 Expenses Cost

Amount comprises initial DG asset plus increasing O&M expenses throughout the forecasted time frame.

2.1.7 Operation and Maintenance Cost

This price takes into consideration the O&M expenses of grid-connected DG and is calculated as follows in Eq. (9). O&M data for DG implementation was gathered from

$$C_{\left( {{\text{DG,O}}\& {\text{M}}} \right)} = \left( {{\text{DG Size}}} \right) \times \left( {{\text{DGO}}\& {\text{Mcost}}/{\text{kW}} - {\text{yr}}} \right)$$
(9)

The present-day cost value of CDG, O&M is calculated using Eq. (10)

$${\text{NPV}}\;\left( {C_{{\text{DG,O}}\& {\text{M}}} } \right) = C_{{\text{DG,O}}\& {\text{M}}} \sum_{t = 1}^N {\gamma t}$$
(10)

wherever t = 1, 2,3,…,N; CDG, O&M = Cost of DG maintenance and operation.

2.1.8 Investment Cost

Investment cost of a solar PV-type DG is calculated at the best location using Eq. (11).

$$C_{{\left( {\text{DG,Inv}} \right)}} = \left( {\text{DG Size}} \right) \times \left( {{\text{DG Investment cost}}/{\text{kW}}} \right)$$
(11)

The economic justification of the aforementioned goal function is contingent upon the proper placement and rating of DG.

2.2 Limitations and Restrictions of DISCOs

The following restrictions apply to this optimization problem.

Bus voltages and branch currents limits during the planning process, locations and dimensions of DGs should be chosen so that branch currents and bus voltages remain within predetermined ranges. These restrictions are outlined below:

$$I_{b,t,j} \le I_{b}^{\max }$$
(12)
$$V^{\min } \le V_{n,t,j} \le V^{\max }$$
(13)

where the minimum and maximum voltage levels for each bus are and indicate the maximum allowable current in a given line due to thermal constraints.

2.2.1 DG Capacity Limit

Assume the following period defines the limits on the active and reactive capacity of each DG.

$$P_{DG,i}^{\min } \le P_{DG,i} \le P_{DG,i}^{\max }$$
(14)
$$Q_{DG,i}^{\min } \le Q_{DG,i} \le Q_{DG,i}^{\max }$$
(15)

In these inequalities, \(P_{DG,i}^{\min }\), \(P_{DG,i}^{\max }\), \(Q_{DG,i}^{\min }\), \(Q_{DG,i}^{\max }\) are the lowest and highest amounts of active and reactive power that the DG unit is capable of producing, respectively.

2.2.2 Contract Price Limit

Constraints imposed by the electricity market on the contract price between the DG owner and the DISCO may be represented as follows:

$$CP_{DG}^{\min } \le CP_{DG} \le CP_{DG}^{\max } ,$$
(16)

where \(CP_{DG}^{\min }\) and \(CP_{DG}^{\max }\) are the minimum and maximum contract price amounts that may be decided based on the market price of energy and other economic factors.

2.2.3 Power-Flow Constraints

Active and reactive power both need to be introduced into the system in order to satisfy the power-flow equations.

$$P_{n} = V_{n} \sum\limits_{m \in N} {V_{m} \left( {g_{mn} \cos \left( {\theta_{mn} } \right) + b_{mn} \sin \left( {\theta_{mn} } \right)} \right)}$$
(17)
$$Q_{n} = V_{n} \sum\limits_{m \in N} {V_{m} \left( {g_{mn} \sin \left( {\theta_{mn} } \right) + b_{mn} \cos \left( {\theta_{mn} } \right)} \right)}$$
(18)

where \(C_{{{\text{investment}}}}^{\max }\) reflects the maximum amount of funding the DG owner can afford Fig. 1.

Fig. 1
A 15-step flowchart. It includes start, read the line and load data such as that of kids and market parameters, initialize them, randomly generate the populations using a heuristic model, evaluate and replace with elites through maximum iterations, print the best, and stop.

Flowchart of DISCOs PROFIT maximization problem using proposed MFO algorithm

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3 Result and Discussion

In order to determine the location of the DG, its size, and conduct economic analysis in a competitive open market environment, this section shows how to apply the MFO algorithm to the DG planning problem. IEEE 33 node test system [16] is used with the MFO algorithm to examine performance. The test case takes into account the DG system that is based on solar PV [17]. The investment cost and operation and maintenance costs for DGs in the 100–1000 kW range are US$ 2493/kW and US$ 19/kW, respectively while they are US$ 2025/kW and US$ 16/kW for the 1000–2000 kW range. Table 1 lists the market price and duration details for DISCOs. Additionally, it is anticipated that the contract price will range between $35 and $50 per MWh. As far as actual power goes, DG has a value between 0.2 and 1 MW. The technical and financial details of the DGs are displayed in Table 2. Table 3 displays the variables that were used in this study. An Intel Core i3 computer running at 2.10 GHz and equipped with 4 GB of RAM was used to carry out the optimization method. MATLAB version R2014a was used. Table 1 documents and summarizes the test case’s findings for the IEEE 33 node RDS.

Table 1 Technical and commercial information of DISCOs
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Table 2 Commercial information of DGs
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Table 3 Details of the technical parameters
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3.1 33 Node Test System

Table 4 and Fig. 2 may be found in the supplementary materials [18, 19], and they provide line data, bus data, and a single-line schematic of the 33-bus test system, respectively. It has a reactive and real load of 3755 kW and 2330 kVAR, correspondingly, then operates at a radial voltage of 1266 kV with 33 buses and 32 branches.

Table 4 Base case voltage profile by installing DG in 33-bus node test system
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Fig. 2
An illustration of a 33-node test system. Systems 1 to 18 are in a row with rightward arrows, 19 to 22 and 26 to 33 connect vertically up from system 2, and 23 to 25 connect vertically down from system 3. D G is located at system 29.

Structure of 33 node test system with optimal location of DG

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By applying the MFO algorithm to the recommended RDS, the voltage contour and voltage stability directory are enhanced. The best sitting and sizing of DG is 29 as shown in Fig. 2 and 1.335924 MW each, respectively. The status of voltage profile and VSI for base case analysis with DG location is numerically reported in Tables 4 and 5. It is also graphically displayed in Figs. 3 and 4, respectively.

Table 5 Base case values of VSI with the placement of DG in 33-bus test system
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Fig. 3
A bar graph of voltage versus bus numbers from 1 to 33 for 2 categories. Base case voltage drops from 1 to 0.91 at bus number 17, peaks back to 1 at 19 and drops to 0.92 at 33. M F O voltage follows the same trend. Values are approximated.

Base case voltage profile by the placement of DG in 33-bus test system

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Fig. 4
A bar graph of V S I versus bus number for 2 categories. Base case V S I drops from 1 to 0.7 at bus number 17, peaks back to 1 at 19 and drops to 0.8 at 31. M F O V S I follows the same trend. Values are approximated.

Base case VSI and DG placement in 33-bus test system

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Table 6 shows the system’s minimum VSI, actual power loss, and reactive power loss with and without DG installation. The value of the indices is nearby zero (except VSI), and it is obvious that the performance of the system is enhanced. Since the optimal result of voltage stability index is nearby one, it indicates that the system voltage stability has been improved.

Table 6 Comparison of real and reactive power loss for 33 node system
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The simulated costs of various DISCO outlays over the course of a single year are shown in Table 7. The whole cost of DG ownership consists of the initial investment, as well as the running and maintenance expenses. The benefit, expenses and total profit of DISCOs for 10 year of planning period are clearly reported in Table 8 [20, 21]. The benefit, expenses and total profit of DISCOs are 21.59 M $/year, 3.81 M $/year and 17.78 M $/year, respectively.

Table 7 Simulation results of 33-bus system after 1 year
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Table 8 Simulation results of 33-bus system after 10 years of planning period
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Table 7 then Table 8 compare the consequences of the proposed multi-objective MFO approach to those of the fuzzy-DE algorithm to show which is more effective. From Table 8, it is established that the MFO algorithm provides higher benefit, total profits with less computational time by proper sitting and sizing of DG in RDS. The total profits of DISCOs are increased by 13.72% when compared with fuzzy-DE algorithm. Figure 5 depicts a comparison between the benefit, expenses and total profit of DISCOs and the present system. Figure 6 shows the convergence curve for the 33-bus test system. The data shows that it can be extrapolated that just 25 to 30 iterations are required for the strategy to provide the optimal global answer.

Fig. 5
A grouped bar graph of 2 categories of cost versus 3 categories of DISCOs. Benefits of DISCOs tops, followed by total profit, and expenses in order, for both M F O and fuzzy with D E.

Comparison of cost, benefit and profit of DISCOs for 33-bus test system

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Fig. 6
A line graph of objective function versus number of iteration. It plateaus at 1.63 from 0 to 10, rises sharply to 1.77 at 20, and plateaus after till 100.

Convergence curve of 33-bus test system

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The distribution load flow method may be used to freely calculate the base case voltage and VSI and to carry out the base case power flow. To enhance the voltage profile and increase DISCOs profits, the MFO algorithm is then utilized to find the sweet spot for the DG’s placement and size. It has more ability and well suitable for solving the problems of large system and to obtain the accurate solution by using it’s searching operators of moth and flame. The two operators are effectively tuning the optimal value of RDS for maximizing the profit of DISCOs. In this work, a single unit of DG is considered and rating of DG is 0 to 2 MW. In this algorithm, only limited number of parameters is used to obtain the solutions. The feasible control parameters obtained by the training process of MFO algorithm is as follows. Population size = 50; number of variables = 7; maximum number of iterations = 100.

4 Conclusion

The various voltage stability issues in both regulated and deregulated power systems are solved using the intelligent computational optimization algorithm moth-flame optimization (MFO) in this research work. The MFO meta-heuristic algorithm has been successfully used to increase the voltage stability of radial distribution systems, and it is regarded as one of the most promising of the meta-heuristic algorithms. Moths and flames are its two primary constituents. A solution has also been considered using both moths and flames. Through the proper placement and sizing of various capacitors, DG units, and network reconfiguration processes, the two searching operators of MFO have successfully improved the voltage profile, maximized net savings, and decreased network losses of RDS.

To enhance the advantages of DISCOs and DG owners in a deregulated environment, a devised MFO algorithm has also been used. The costs and income of the different DG owners and DISCOs are calculated here. It includes the costs of purchasing capacitors and DGs, as well as their costs of operation and maintenance. It also includes the costs of substations and customer interruptions, as well as the revenue and profits of DG owners. To demonstrate the effectiveness of the MFO, a numerical example using IEEE standard RDS test systems has been taken into consideration. The outcome demonstrates that, in comparison with other competing algorithms, the suggested algorithm offers an increase in profit with less computational time. Therefore, it can be said that the suggested MFO approach paves the best path for resolving power system optimization issues in a deregulated environment.