 Original research
 Open access
 Published:
Impact of the penetration of distributed generation on optimal reactive power dispatch
Protection and Control of Modern Power Systems volumeÂ 5, ArticleÂ number:Â 31 (2020)
Abstract
Optimal reactive power dispatch (ORPD) is a complex and nonlinear problem, and is one of the subproblems of optimal power flow (OPF) in a power system. ORPD is formulated as a singleobjective problem to minimize the active power loss in a transmission system. In this work, power from distributed generation (DG) is integrated into a conventional power system and the ORPD problem is solved to minimize transmission line power loss. It proves that the application of DG not only contributes to power loss minimization and improvement of system stability but also reduces energy consumption from the conventional sources. A recently proposed metaheuristic algorithm known as the JAYA algorithm is applied to the standard IEEE 14, 30, 57 and 118 bus systems to solve the newly developed ORPD problem with the incorporation of DG. The simulation results prove the superiority of the JAYA algorithm over others. The respective optimal values of DG power that should be injected into the four IEEE test systems to obtain the minimum transmission line power losses are also provided.
1 Introduction
Minimizing power loss in transmission systems is a major area of research in power system engineering. Voltage collapse, as another major issue, is also attracting much research worldwide to find solutions to improve voltage stability and thus improve the security of the power system and make power transmission more economic. Optimal reactive power dispatch (ORPD) deals with not only the problem of increasing power loss with the expansion of power networks but also the increasing voltage instability problem. The ORPD problem is a subproblem of optimal power flow (OPF) whose solution helps determine the optimal values to the control variables such as the generator voltage, setting of the tapchanging transformer, and the optimal value of reactive power to be injected to compensate for the VAR demand, in order to simultaneously reduce the active power loss and improve voltage stability. Thus, the solution to the ORPD problem helps enhance the security of the power system and improve its economics. However, the ORPD problem is a complex, noncontinuous and nonlinear problem, and many conventional optimization techniques such as the Newton method, quadratic programming, linear programming, and interiorpoint methods, have failed to solve it since these methods have low accuracy, high complexity, and inability to find the local and global optima and thus result in insecure convergence [1,2,3,4,5,6].
Many modern stochastic and metaheuristic techniques have been applied to overcome these disadvantages, such as the genetic algorithm (GA) [7], improved GA [8], particle swarm optimization (PSO) [9], evolutionary programming (EP) [10], hybrid evolutionary strategy [11], the seeker optimization algorithm (SOA) [12], bacterialforaging optimization (BFO) [13], the gravitational search algorithm (GSA) [14], differential evolution (DE) [15], and the artificial bee colony algorithm (ABC) [16]. K. Medani et al. in [17] applied the whale optimization algorithm which was inspired by the bubblenet hunting technique of the humpback whales to solve the ORPD problem, while A. M. Shaheen et al. in [18] proposed a backtracking search optimizer (BSO) where five diversified generation strategies of mutation factor were applied. In [19], K. Lenin proposed an algorithm named Enhanced Red Wolf Optimization which is a hybrid of the wolf optimization (WO) and particle swarm optimization (PSO) algorithm, to solve the ORPD problem. In [20], an improved social spider optimization (ISSO) was used for determining the optimal solution of power loss in the ORPD problem. Zelan Li et al. [21] proposed an Antlion optimization algorithm (IALO) for a threebus system, whereas R. N. S Mei et al. [22] used two different algorithms, namely the MothFlame Optimizer and Ant Lion Optimizer, to optimize the ORPD problem.
This paper uses a novel algorithm, namely the JAYA algorithm developed by Rao [23], to solve the ORPD problem. Many other algorithms such as PSO and different variants of PSO, e.g., RPSO, LPSO, PSOCFA, Improved PSO Based on Success Rate (IPSOSR) [24], Fruit Fly optimization algorithm (FOA), and modified Fruit Fly optimization algorithm (MFOA) are also tested along with the JAYA algorithm. The results are compared to determine the best algorithm in terms of convergence, the ability to determine the optimal solution, and robustness.
The main contributions of the paper are as follows:

i)
Minimizing transmission line power loss by obtaining the optimal setting of the control variables within the system without violating the equality and inequality constraints.

ii)
Incorporating the concept of distributed generation (DG) into the ORPD problem to study its effect and analyze its contribution towards minimizing power loss and increasing system efficiency in the problem.

iii)
The superiority of the JAYA algorithm is established over other algorithms reported in the literature.
1.1 Distributed generation
Alternative sources of energy such as wind, solar, etc. are being used currently. In many cases, such sources of energy are used to generate power on a small scale in areas close to the end users. The end users consume power and any excess power is sent back to the grid. This approach is called distributed generation (DG) and it helps reduce coal consumption, the cost of generation, and transmission line power loss. Furthermore, the demand of consumers in remote areas can be fulfilled from the local generation and the risk of voltage collapse is also reduced. Much research has been carried out to increase the utilization of DG to enhance the security and economic growth of power systems [25,26,27,28,29,30].
In this work, DG power is supplied to the buses along with power from conventional sources to study the transmission line loss characteristic by solving the ORPD problem. The DG power is injected individually at each bus (except for the slack bus) within a specified limit and the ORPD problem is solved to determine the optimal values of the control variables for minimizing transmission line losses. The control variables chosen for the ORPD problem are the generator bus voltages, tap position of the tapchanging transformer, the VAR output of the compensating devices, and the injected DG active power. Thus for an nbus system, the ORPD problem is solved n1 times. The proposed algorithm is used to determine the optimal value of DG power for each bus in order to reduce transmission line loss for the ORPD problem. The power losses for the n1 buses are compared, and the bus with the minimum power loss and the corresponding injected DG power are selected.
2 Problem formulation
The objective of solving the ORPD problem is the minimization of power loss in transmission lines incorporating DG. The solution to this problem is to determine the optimal values of the control variables while simultaneously satisfying all the constraints in the system. First, the ORPD problem is solved without the incorporation of DG in the system, and power losses for the test cases are evaluated and compared using different optimization algorithms. The DG is then introduced and the algorithms again determine the power loss of the system with the penetration of DG. The objective function remains the same while the amount of DG power to be injected is considered as an additional control variable.
The objective function for the problem is expressed as [4]:
where Nl represents the total number of transmission lines, and the conductance of the k^{th} branch is G_{k}. V_{i} and V_{j} represent the magnitudes of the bus voltage for buses i and j, respectively, and Î´ _{ij} is the phase difference between V_{i} and V_{j}. The different constraint that need to be satisfied are discussed in the following subsections.
2.1 Constraints
The constraints are mainly categorized into equality constraints and inequality constraints as follows:
2.1.1 Equality constraints
These constraints depict the load flow equations as:
where the total number of buses is Nb, P_{gi} and Q_{gi} represent the active and reactive power generation, and P_{di} and Q_{di} are the active and reactive power load demands for the i^{th} bus, respectively. G_{ij} and B_{ij} represent the conductance and susceptance between the i^{th} and j^{th} buses, respectively.
2.1.2 Inequality constraints

Generator constraints:
The active and reactive power generation of the generator and its voltage magnitude are all set within their limits when solving the problem, as:
where, N_{g} represents the total number of generator buses, \( {V}_{gi}^{min} \), \( {P}_{gi}^{min} \) and \( {Q}_{gi}^{min} \) are the minimum limits and \( {V}_{gi}^{max} \), \( {P}_{gi}^{max} \) and \( {Q}_{gi}^{max} \) are the maximum limits of the generator bus voltages, active and reactive power, respectively. V_{gi}, P_{gi} and Q_{gi} are the voltage, active and reactive power generation at the i^{th} bus, respectively.

Transformer constraints:

VAR compensator constraints:

Operating constraints:
Equation 7 shows the maximum and minimum limits of the tap changing transformers, where N_{T} represents the number of tapchanging transformers in the system, T_{i} is the transformer tapsetting position at the i^{th} bus and \( {T}_i^{min} \) and \( {T}_i^{max} \) are its minimum and maximum limits. Equation 8 represents the limits of the reactive power to be injected by the VAR compensators, where N_{C} is the total number of shunt compensators at the buses, \( {Q}_{ci}^{min} \) and \( {Q}_{ci}^{max} \) are the minimum and maximum limits of the reactive power injection Q_{ci}, respectively. Equations 9 and 10 represent the operating constraints of load buses and the apparent power at the branches, where N_{PQ} depicts the total number of load buses, \( {S}_{Li}^{max} \) is the maximum apparent power flow at the i^{th} bus and S_{Li} is the apparent power at that branch. V_{Li} is the magnitude of the voltage at the i^{th} load bus and \( {V}_{Li}^{min} \) and \( {V}_{Li}^{max} \) are its minimum and maximum limits. The objective function in (1) is modified by considering the dependent variables as constraints using penalty coefficients as:
The limits of \( {V}_i^{lim} \) and \( {Q}_{gi}^{lim} \) are:
where, Î»_{V} and Î»_{Q} are the penalty coefficients, \( {N}_V^{lim} \) is the number of buses for which the voltages are outside limits and \( {N}_Q^{lim} \) is the number of buses for which the reactive power generations are outside limits.
3 JAYA algorithm
Many stochastic and metaheuristic techniques have been developed recently to solve this type of complex and nonlinear problem such as is the ORPD, including the JAYA algorithm proposed by R.V. Rao [23]. This algorithm has the ability to solve the optimization problem quickly to determine the optimal solution. It has a very high success and convergence rate compared with other algorithms as it has a tendency to move towards the best solution and move away from the worst in every iteration. This helps the algorithm to update new solutions by comparing it with the best without being stuck in local optima.
Let an objective function be f(x), where â€˜mâ€™ is the number of design variables (i.e. aâ€‰=â€‰1, 2, â€¦, m) and â€˜nâ€™ the number of populations (bâ€‰=â€‰1, 2, â€¦, n) for the i^{th} iteration. The population having the best solution of f(x) (i.e. f(x)_{best}) is called the best candidate and the population having the worst solution to the objective function (i.e. f(x)_{worst}) is called the worst. Assuming the value for the a^{th} variable of the b^{th} population in the i^{th} iteration is represented asÂ J_{a, b, i}, the value of the variable is updated as:
where J_{a, best, i} and J_{a, worst, i} are the best and worst solutions of the objective function of the a^{th} variable, respectively. r_{1} and r_{2} are two random numbers in the range of [0, 1]. Thus, this equation helps the variable to move closer to the best solution and away from the worst solution.
3.1 Implementation of JAYA algorithm in ORPD
The procedure for the implementation of the JAYA algorithm in solving the ORPD problem is shown in the flow chart in Fig.Â 1, and the detailed step by step descriptions are given below.

Step 1:
â€ƒâ€ƒThe size of the population of the control variables and the total number of iterations for the problem are initialized.

Step 2:
â€ƒâ€ƒThe values of the control variables are randomly selected within their corresponding constraint limits.

Step 3:
â€ƒâ€ƒA standard IEEE bus system is chosen and the bus data and line data of the system are updated using the new values from the respective control variables. Then, the load flow operation using the NewtonRaphson method is executed.

Step 4:
â€ƒâ€ƒThe constraints are checked and if any constraint is violated, the control variables are reinitialized and steps 2 and 3 are repeated. If no constraint is violated, the power loss is then calculated using the results from the load flow.

Step 5:
â€ƒâ€ƒThe best and worst solutions are identified from the set of populations, i.e. the set resulting in the least power loss is declared as the â€˜best solutionâ€™ and the set with the highest power loss is declared as the â€˜worst solutionâ€™.

Step 6:
â€ƒâ€ƒThe iteration cycle commences.

Step 7:
â€ƒâ€ƒThe JAYA algorithm is initiated where the control variables forming the different populations are updated depending on the best and worst solutions using (14).

Step 8:
â€ƒâ€ƒAC load flow is reexecuted and the power loss is calculated for all different sets of population.

Step 9:
â€ƒâ€ƒThe results are compared to accept and reject the different sets of the control variables in each population depending on the best solution. The set of control variables having the best solution is accepted and ones with the worse solutions are updated with the previous best. Thus, a new best solution is determined after each iteration.

Step 10:
â€ƒâ€ƒThe process continues until the iteration reaches the maximum iteration.

Step 11:
â€ƒâ€ƒThe optimal solution is obtained and the corresponding control variables are saved.
This whole process helps obtain the optimal values of the control variables for the best solution among all the sets of population.
4 Simulation results and discussions
To evaluate the performance of the JAYA algorithm, it is initially tested on 24 standard constrained benchmark functions (G01 â€“ G24) and the results are compared in TableÂ 1. It shows that the proposed algorithm is far superior and consistent in obtaining better results than the other wellestablished techniques. The results also depict the ability of the proposed technique in obtaining better results for all the functions under any constraints. The best and the mean values for each function using the JAYA algorithm are very close to each other, which implies that the algorithm is robust and produces results with minimum deviation compared to other techniques.
After obtaining this excellent performance of the proposed algorithm on the standard benchmark function with constraints, it is then implemented to solve the ORPD problem. In this paper, it is tested on the standard IEEE 14, 30, 57 and 118 bus systems along with other algorithms from the literature. It has also been tested to solve the minimum power loss of the ORPD problem with and without DG penetration, and the solutions are compared to those using different algorithms. The software used is MATLAB 2014b and the population size is 100 for all the cases in the paper.
The details of the number of individual parameters of the test systems are listed in TableÂ 2 and the system data of these test systems are obtained from [31].
4.1 Minimization of active power loss without DG injection
4.1.1 IEEE 14 bus system
The IEEE14 bus system has five generators at buses 1 (which is the slack bus), 2, 3, 6 and 8, respectively. There are 20 branches, and three tapchanging transformers between the lines 4â€“7, 4â€“9 and 5â€“6. Reactive power is injected at buses 9 and 14. The limits of the control variables (p.u. value) for the case study under the IEEE 14 bus system are as follows:
0.95â€‰â‰¤â€‰V_{g}â€‰â‰¤â€‰1.1;Â 0.95â€‰â‰¤â€‰V_{PQ}â€‰â‰¤â€‰1.05;Â 0.9â€‰â‰¤â€‰T_{i}â€‰â‰¤â€‰1.1 andÂ 0â€‰â‰¤â€‰Q_{c}â€‰â‰¤â€‰0.3.
The above control variables are used to solve the ORPD problem using the different algorithms and the simulation results are compared in TableÂ 3 to determine the best among them. Comparing the results in the table and the convergence characteristics shown in Fig.Â 2, it can be concluded that the JAYA algorithm has produced the best solution to the ORPD problem with a minimum loss of 12.227â€‰MW, and is superior to the other 37 algorithms.
4.1.2 IEEE 30 bus system
In the IEEE 30 bus system, there exist six generators situated at buses 1, 2, 5, 8, 11 and 13, respectively. Bus no. 1 is the slack bus, and there are 41 transmission lines with four branches having tapchanging transformers. Reactive power is injected by capacitor banks at bus no. 3, 10 and 24, respectively. The limits of the control variables (p.u. value) are as follows:
0.95â€‰â‰¤â€‰V_{g}â€‰â‰¤â€‰1.1;Â 0.95â€‰â‰¤â€‰V_{PQ}â€‰â‰¤â€‰1.1;Â 0.9â€‰â‰¤â€‰T_{i}â€‰â‰¤â€‰1.1 andÂ 0â€‰â‰¤â€‰Q_{c}â€‰â‰¤â€‰0.36.
The simulation results of the solution to the ORPD problem using the different algorithms for this test case are shown in TableÂ 4 along with the convergence characteristics in Fig.Â 3. It can be seen that the JAYA algorithm has produced the best results under the conditions of the control variables, resulting in the lowest line loss of 4.5983â€‰MW.
A statistical analysis of the algorithms is shown in TableÂ 5 for this particular test system. The best and worst values of the solutions of the ORPD problem along with the mean, standard deviation (std.), percentage of power saved, and the average computation time of the results for the different algorithms are compared. The results prove that the JAYA algorithm has obtained the best solution to the problem and is also the most consistent and robust with small std. and the maximum reduction of power loss of almost 18.7579% (4.5983â€‰MW). The time of convergence is modest and although the simulation speed is slower than a few others, the JAYA algorithm obtains the best solution and is much more favorable in terms of efficiency and economy than other methods.
In order to investigate how frequently the results from the different algorithms converge within a different range of solutions, the frequency of convergence for the IEEE 30 bus system under the inequality constraints of the control variables as mentioned earlier is compared in Table 6. It shows the number of times each algorithm has produced the solution within a specified range when the ORPD problem is run for 50 times for every single algorithm. The results show that the JAYA algorithm is undoubtedly the only one to produce all the results within the minimum range of 4.59â€“4.60â€‰MW. Although the MFOA technique is also consistent and has frequently obtained the solutions within the range of 5.11â€“5.20â€‰MW (49 times), the algorithm has failed to optimize the function to lower limits. Thus, the results prove that the JAYA algorithm has the capability of converging most frequently to the minimum solution.
4.1.3 IEEE 57 bus system
The standard IEEE 57 bus system has seven generators situated at buses 1, 2, 3, 6, 8, 9 and 12, respectively, where bus 1 is the slack bus. There are 15 branches out of a total of 80 having tapchanging transformers connected. The reactive power compensating devices are placed at buses 18, 25 and 53. The maximum and minimum limits of the control variables are given in TableÂ 7.
The simulation results for the ORPD problem using the different algorithms for the test case and the comparative convergence characteristics are shown in Table 8 and Fig.Â 4, respectively. It shows that the JAYA algorithm has reduced the power loss to 22.67%, a much lower level than the other algorithms. This is the best recorded solution for this particular test case under the mentioned inequality constraints.
4.1.4 IEEE 118 bus system
As the algorithm has successfully outperformed the other algorithms reported in the literature in optimizing the ORPD problem for the IEEE 57 bus system, it is now tested on the larger IEEE 118 bus system to observe its performance and ability. The IEEE 118 bus system has 54 generators, 14 shunt compensators, 9 tapchanging transformers, and a total of 186 transmission lines. The control variable limits are given in TableÂ 9. TableÂ 10 and Fig.Â 5 shows the simulation results and the convergence characteristics for the ORPD problem using the different algorithms, respectively.
The results show the superiority of the JAYA algorithm in determining the optimal solution, thus reducing the power loss to the lowest value of 105.4821â€‰MW (20.36%) for the test case compared to all the other algorithms without violating the limits of the constraints. This proves the JAYA algorithm to be the most efficient algorithms even for large scale power systems.
4.2 Minimization of power loss with DG injection
For the second part of the paper, DG power is injected individually at all the buses (except the slack bus) and the power losses are calculated using the same algorithms while keeping the constraints unchanged. The total number of control variables for each case without DG injection was listed in Table 2. When DG is penetrated into the system the number of control variables is increased by 1, and the DG power to be injected is taken as an additional control variable. The value of DG power is initially set at 100% of the maximum load demand for all the test cases considered in this paper. The algorithm then determines the optimal value of DG to be injected at each bus in order to produce the minimum power loss.
Moreover, the magnitude of the voltage of the bus at which the DG is injected is also considered as a control variable. Thus, when the DG is injected at any PV bus, the number of voltage control variables remains the same but for a PQ bus, it increases by 1.
The JAYA algorithm has been proved to be the best among all the reported algorithms in determining the minimum power loss without incorporating DG power. Thus, the proposed JAYA algorithm is used to determine the optimal value of DG to be injected at each bus to obtain the minimum power loss for all the test systems. The results are then compared and the optimal bus is located with the optimal value of DG to be injected for that particular bus. The study is repeated for all four test cases using different algorithms to determine which algorithm is able to determine the optimal value of DG at that optimal bus leading to minimum power loss. Such work on optimization of the ORPD problem with the concept of DG integration has not been discussed anywhere. This helps significantly reduce the power loss of the system that cannot be achieved using other methods of solving the ORPD problem. Moreover, it also encourages the use of nonconventional resources as the results obtained in this paper describes the details of the optimal amount of DG power to be integrated for a particular test case at the optimal bus location.
4.2.1 IEEE 14 bus system
The total generation of the test system is 272.6â€‰MW and the load demand is 259.11â€‰MW. The control variable limits are the same as in Section 4.1.1 for the case with no DG. TableÂ 11 shows the optimum solution of the ORPD problem for minimization of power loss when the optimal value of DG is injected at each bus, one at a time using the JAYA algorithm. The comparison of the results is also represented graphically in Fig. 6. From Table 11 and Fig. 6, it is observed that injecting the optimum DG power of 190.176833â€‰MW, which is 73.3962% of the demand, at bus no. 4 can achieve the minimum power loss of 2.9641â€‰MW.
Table 12 and Fig. 7 illustrate the significance of the DG power on power loss. The results show that the power loss would increase from 2.9641â€‰MW to 2.9645â€‰MW and 2.9651â€‰MW for the DG power increased from the optimal value of 190.176833â€‰MW to 191.18â€‰MW and 192â€‰MW respectively, and to 2.9643â€‰MW for the DG power decreased to 189â€‰MW. The other reported algorithms are now used to obtain the optimal value of DG at bus no. 4 and the ORPD problem is solved by optimizing the objective function f from (11). Here, the total number of control variables is 12 as the optimal power loss is obtained when the DG is inserted at bus no. 4, which is not a PV bus. The results and the convergence characteristics are shown in Table 13 and Fig. 8, respectively. These prove that the JAYA algorithm produces the bestoptimized value compared to all the other algorithms. The results from the two cases with and without DG penetration clearly show that the DG penetration has successfully reduced the power loss by 78.03% compared to a reduction of 9.36% under the similar condition without DG injection.
4.2.2 IEEE 30 bus system
The study on the IEEE 30 bus system has been performed on two different cases. In the first case, the optimal DG power is determined and the value is then fixed to obtain the power loss. However, in the second case, the DG power is considered variable, representing a probabilistic approach to observe the performance penetration of variable DG realistically in the ORPD problem. Here, the probabilistic approach of wind power is considered for the variable DG in the second case.
Without considering the variability of DG power
The total active power generation of the test system is 288.7â€‰MW and the load demand is 283.4â€‰MW. The control variable limits are the same as Section 4.1.2 with no DG. TableÂ 14 shows the optimal results of the ORPD problem when DG is individually placed on each bus. As shown, the minimum power loss for the IEEE 30 bus system is obtained when 104.34914â€‰MW DG, which is 36.8204% of the total demand, is placed at load bus no. 6. This reduces the power loss to 1.8574â€‰MW (67.18% reduction), whereas for the case without DG the loss was reduced by only 18.75%.
The significance of the optimum value of DG obtained by the JAYA algorithm is illustrated in TableÂ 15, which shows that when the DG value is increased or decreased by 1â€‰MW from the optimal value, there is an increase in power loss. Thus, the result obtained from the proposed algorithm is the optimal value of DG to be injected into the system for minimum power loss. Other reported algorithms are then used to optimize the ORPD problem by determining the optimal value of DG at bus no. 6 for minimum power loss. The results from TableÂ 16 and the convergence characteristics from Fig. 9 conclude that the minimum power loss is obtained by using the JAYA algorithm, indicating the superiority of the JAYA algorithm over other reported algorithms.
Considering the variability of DG power
In the work shown in Section 4.2.2.1, the uncertainty of DG power was not considered. In practical cases, the DG power is of a variable nature and thus needs to be considered to make the study more realistic. There are several reported cases in which the variability of renewable energy is integrated into the ORPD problem with a maximum capacity of DG of up to 110â€‰MW considered at bus 6. The Weibull probability distribution function [52], which considers the stochastic nature of wind power, is used and the variability of wind power is considered in two ways as follows.

Case 1:
â€ƒâ€ƒOverestimated wind power
This study shows the impact of overestimated power from the wind farm into the ORPD problem considering the uncertainty condition. The maximum power output from the wind farm is set as 110â€‰MW, as the optimal value of injected DG into the IEEE 30 bus system determined by the JAYA algorithm is 104.34914â€‰MW. The overestimated probabilistic approach of the Weibull probability distribution function determines the more realistic nature of the optimal wind power to be injected to minimize the power loss in the ORPD problem using the following equation:
where, P_{owi} represents the optimal value of injected wind power obtained using the overestimation of the Weibull probability distribution function, and k and c represent the shape and scale factors referred from [52]. w_{f} represents the forecasted wind power and W_{1} represents the actual power produced. The term f_{w}(w) is a probability density function of wind power output w, whereas v_{i} and v_{o} are the cutin and cutout wind velocities, respectively.
The analysis is carried out using the same number of techniques as discussed earlier and the details of the calculated results are shown in Table 17. The convergences of the different algorithms for this case are compared in Fig. 10. The results show that for the overestimation case, the optimal value of wind power is 59.2683â€‰MW (a reduction of 56.82% compared to the base case) as obtained by the JAYA algorithm, for which the power loss is 2.4442â€‰MW. This happens to be the lowest for this case among all the other algorithms. Thus, the analysis gives a realistic outcome of the penetration of wind power for the overestimated condition.

Case 2:
â€ƒâ€ƒUnderestimated wind power
In this case, the underestimated approach of the Weibull probability distribution function is considered. The maximum wind power limit is set as 110â€‰MW and the optimal value of wind power is obtained for minimizing the power loss for the IEEE 30 bus system. The optimal value of the wind power using the Weibull probability distribution function is given as
where, P_{uwi} represents the optimal value of injected wind power obtained using the underestimation of the Weibull probability distribution function, The term v_{i} is the rated wind velocity and w_{r} the equivalent rated power of the wind farm.
The analysis is carried out using the different techniques and the results are displayed in Table 18, along with the convergence characteristics shown in Fig. 11. The results show that for the underestimated case of uncertain wind power, the optimal power output from the wind farm is 45.856â€‰MW for a power loss of 2.805â€‰MW (50.44% reduction compared to the base case). This optimal result is obtained using the JAYA algorithm and hence it is proved to be the best in optimizing this Weibull probability distribution functionbased ORPD problem. The uncertainty of DG power shows that the actual power under the realistic condition obtained from the wind farm is less than that obtained from Table 14. The overestimated output of wind power is higher than the underestimation and thus gives lower power loss. The difference in power loss is about 0.3608â€‰MW between the two estimations considering the best solutions from the JAYA algorithm.
4.2.3 EEE 57 bus system
The IEEE 57 bus system has a total active power generation of 1278.7â€‰MW and a load demand of 1250.8â€‰MW. The limits of the control variables are the same as Section 4.1.3 with no DG. TableÂ 19 shows the complete results of ORPD for minimizing power loss for 56 different cases where the DG is individually injected at each bus (except the slack bus no. 1). Table 19 shows that the power loss is minimum (i.e., 10.7774â€‰MW) when 271.898815â€‰MW DG is injected at the.
PQ bus 13, which is 21.738% of the total load demand of the system. These results are again obtained by the JAYA algorithm. The results from Table 20 show the significance of the optimal value of DG determined by the JAYA algorithm, as a small variation can lead to increased power loss. Other algorithms have also been used to optimize the ORPD problem by determining the optimum DG value at bus 13 and the results are illustrated in TableÂ 21 and Fig. 12.
From the results, it is concluded that the JAYA algorithm results in significantly lower power loss than the other algorithms.
The study shows that DG penetration reduces the power loss by 61.32% compared to 22.67% without DG penetration using the same algorithm.
4.2.4 IEEE 118 bus system
The active power generation and load demand of the IEEE 118 bus system are 4374.9â€‰MW and 4242.45â€‰MW, respectively. The limits of the control variables considered for this case are the same as Section 4.2.4 with no DG. Bus 69 is the slack bus, and the ORPD problems are solved using the JAYA algorithm with the individual injection of DG power at each of the other 117 buses. TableÂ 22 shows the details of the 117 solutions and indicates that the minimum power loss of 91.4174â€‰MW is obtained when a DG of power of 235.926829â€‰MW (5.5611% of the total load demand) is injected at bus 40. The penetration of DG reduces the power loss by almost 30.98% with the use of the JAYA algorithm, whereas for the system without DG injection, the proposed algorithm was only able to reduce the power loss by 20.36%.
The significance of the optimal value of DG obtained by the JAYA algorithm is illustrated in TableÂ 23. The data from TableÂ 24 and the convergence characteristics in Fig. 13 compare the results of the ORPD problem using different algorithms. It shows that the results obtained using the JAYA algorithm are the best of all the algorithms. In this case, the JAYA algorithm is not stuck in local optima and is able to optimize the problem to a much larger extent than the others. Thus, the JAYA algorithm is superior to other algorithms reported in the literature for all the test cases shown in this paper with the injection of DG in optimizing the ORPD problem.
5 Efficacy of JAYA algorithm
The efficacy of the JAYA algorithm can be explained as follows:

i.
Benchmark function â€“ The JAYA algorithm has been tested on 24 standard constrained benchmark functions (G01 â€“ G24) and the results were shown in Table 1. It has been proved to be the most robust and efficient algorithm by obtaining the best solution to all the different functions. Thus, this benchmark test has proved JAYA to be the best of all the algorithms reported on the optimization problem and thus can be tested on the nonlinear and highly constrained ORPD problem.

ii.
ORPD problem without DG â€“ The JAYA algorithm has proved to be the most efficient by consistently providing the optimal solutions to the problems for all the different bus systems. It has successfully obtained the best solution reported in the literature for the above four mentioned IEEE test bus systems for the ORPD problem.

iii.
ORPD problem with DG â€“ The JAYA algorithm has provided the best solution to the ORPD problem with DG compared to other algorithms discussed above. It has obtained the best value of DG to be penetrated to the system to minimize the power loss for all the four test bus systems.

iv.
Statistical analysis â€“ The statistical analysis of the JAYA algorithm and others was tested on the IEEE 30 bus system, and the results shown in Table 5 demonstrate that the JAYA algorithm is the most robust and has a minimum standard deviation compared to the others.

v.
Frequency of convergence â€“ This test, which is another method to judge the robustness of the algorithms, was performed for the different algorithms and the results were shown in Table 6 and its significance discussed in Section 4.1.2. Each algorithm was run 50 times and the results show that the JAYA algorithm has obtained the best solution for all 50 times within the range of 4.59â€“4.60â€‰MW, thus proving to be the most robust algorithm.

vi.
Convergence speed â€“ The convergence characteristics from the different test cases show that the JAYA algorithm may not be the fastest in terms of computation time, but it provides a good balance between convergence speed and obtaining the best solution.
Thus, these detailed comparisons show the ability of the JAYA algorithm to obtain the best solution to this critical optimization problem and to outperform many other well established techniques in respect of robustness, efficiency and, convergence speed.
6 Conclusion
This paper has shown the effect of the penetration of distributed generation (DG) into the ORPD problem for reducing the transmission line losses for the very first time and has provided a unique contribution in the study of the ORPD problem. A comprehensive study was carried out to locate the optimal bus and determine the corresponding optimal value of DG to be injected to minimize transmission line loss. The results show that power loss is minimized to a large extent when DG is injected into the system, establishing the advantages of the DG penetration in the optimization problem of ORPD. Using four different IEEE standard bus systems, it shows that if the optimal bus and value of DG are known, the power loss can be significantly reduced and system stability improved. This work reveals a new way of analyzing the ORPD problem and offers encouragement towards the utilization of renewable resources. The simulation results confirm that the JAYA algorithm is the best and efficient among the others reported in the literature, in terms of reliability, robustness, consistency, and rate of convergence in solving the ORPD problem for all the case studies. The JAYA algorithm gives consistent results under any condition without violating any equality and inequality constraint.
Availability of data and materials
The datasets used and analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 ABC:

Artificial Bee colony algorithm
 ACO_{R} :

Ant colony optimization algorithm extended to continuous domains
 ALCPSO:

Particle Swarm Optimization with Aging Leader and Challengers
 ALO:

Ant lion optimizer
 BA:

Bat algorithm
 BBDE:

Barebones DE
 BBO:

Biogeographybased Optimization
 BBPSO:

Barebones particle swarm optimization
 BDE:

Binary differential evolution
 BDE:

Best/1/bin DE
 BFO:

Bacterialforaging optimization
 BSO:

Backtracking search optimizer
 CIPSO:

Constant inertia weight conventional PSO
 CKHA:

Chaotic krill herd algorithm
 CLPSO:

Comprehensive Learning PSO
 CPSO:

Conventional PSO
 CPVEI HBMO:

Chaotic Parallel Vector Evaluated Interactive Honey Bee Mating Optimization
 CSA:

Common Scrambling Algorithm
 DDE:

Double differential evolution
 DE:

Differential evolution
 DG:

Distributed generation
 DSA:

Digital Signature Algorithm
 EMA:

Exchange market algorithm
 EP:

Evolutionary programming
 FOA:

Fruit Fly optimization algorithm
 FPA:

Flower Pollination Algorithm
 GA:

Genetic algorithm
 GBTLBO:

Gaussian barebones TLBO
 GBWCA:

GravityBase Objectsâ€™ Weight Clustering Algorithm
 GSA:

Gravitational search algorithm
 HBMO:

Honey Bee Mating Optimization
 HEP:

Hybrid evolutionary programming
 HSSSA:

HyperSpherical Search Algorithm
 HTS:

HIV testing services Algorithm
 IALO:

Antlion optimization algorithm
 ICA:

Imperialist competitive algorithm
 IPGSPSO:

Improved pseudo gradient searchparticle swarm optimization
 IPSOSR:

Improved PSO Based on Success Rate
 ISSO:

Improved social spider optimization algorithm
 IWO:

Invasive weed optimization
 LDIPSO:

Linearly decreasing inertia weight PSO
 LPSO:

LÃ©vy PSO
 MCBOA:

Modified colliding bodies optimization algorithm
 MDE:

Modified differential evolution
 MFOA:

Modified Fruit Fly optimization algorithm
 MGBTLBO:

Modified Gaussian barebones TLBO
 MICA:

Modified imperialist competitive algorithm
 MICAIWO:

Hybrid MICAIWO
 MOALO:

Multiobjective Ant Lion Optimizer
 MSFS:

Modified stochastic fractal search algorithm
 MSSA:

Multiobjective Salp Swarm Algorithm
 MTLA:

Modified teaching learning algorithm
 MTLADDE:

Hybrid MTLADDE
 NGBWCA:

Gaussian barebones WCA
 OGSA:

Oppositionbased gravitational search algorithm
 OPF:

Optimal power flow
 ORPD:

Optimal Reactive Power Dispatch
 PGPSO:

Pseudogradient PSO
 PGSWTPSO:

Pseudogradient Search Particle Swarm Optimization
 PSO:

Particle Swarm Optimization
 PSOCF:

Particle Swarm Optimization Collaborative filtering
 PSOCFA:

PSO with constriction factor
 PSOICA:

Particle Swarm Optimization Imperialist competitive algorithm hybrid
 PSOTVAC:

Particle swarm optimization with time varying acceleration coefficients
 PSOTVAC:

Particle Swarm Optimization with Time Varying Acceleration Coefficients
 PSOTVIW:

Timevarying inertia weighting strategy based on particle swarm optimization GSA
 QOTLBO:

Quasioppositional teaching learning based optimization
 RDE:

Rand/1/bin DE
 RPSO:

Rapid PSO
 RTS:

RealTime Scheduling
 SARCGA:

Selfadaptive real coded genetic algorithm
 SARGA:

Using selfadaptive real coded genetic algorithm
 SFLA:

Shuffled frog leaping algorithm
 SGA (F_{f1}):

Specialized genetic algorithm using (Ff1)
 SGA (F_{f2}):

Specialized genetic algorithm using (Ff2)
 SOA:

Seeker optimization algorithm
 SPSOTVAC:

Selforganizing hierarchical PSO with Time Varying Acceleration Coefficients
 SSA:

Salp Swarm Algorithm
 SSO:

Social spider optimization
 SWTPSO:

Stochastic weight tradeoff particle swarm optimization
 TLA:

Teaching learning algorithm
 TLBO:

Teaching Learning Based Optimization
 WCA:

Water Cycle Algorithm
 WOA:

Whale Optimization algorithm
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Mr. Tanmay Das carried out basic design, simulation work and prepared draft paper. Dr. Ranjit Roy and Dr. Kamal Krishna Mandal participated in checking simulation work, results & discussions, sequence of paper and helped to prepare the manuscript. All authors read and approved the final manuscript.
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Das, T., Roy, R. & Mandal, K.K. Impact of the penetration of distributed generation on optimal reactive power dispatch. Prot Control Mod Power Syst 5, 31 (2020). https://doi.org/10.1186/s41601020001775
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DOI: https://doi.org/10.1186/s41601020001775