Optimal Power Flow Using the Jaya Algorithm

: This paper presents application of a new effective metaheuristic optimization method namely, the Jaya algorithm to deal with different optimum power ﬂow (OPF) problems. Unlike other population-based optimization methods, no algorithm-particular controlling parameters are required for this algorithm. In this work, three goal functions are considered for the OPF solution: generation cost minimization, real power loss reduction, and voltage stability improvement. In addition, the effect of distributed generation (DG) is incorporated into the OPF problem using a modiﬁed formulation. For best allocation of DG unit(s), a sensitivity-based procedure is introduced. Simulations are carried out on the modiﬁed IEEE 30-bus and IEEE 118-bus networks to determine the effectiveness of the Jaya algorithm. The single objective optimization cases are performed both with and without DG. For all considered cases, results demonstrate that Jaya algorithm can produce an optimum solution with rapid convergence. Statistical analysis is also carried out to check the reliability of the Jaya algorithm. The optimal solution obtained by the Jaya algorithm is compared with different stochastic algorithms, and demonstrably outperforms them in terms of solution optimality and solution feasibility, proving its effectiveness and potential. Notably, optimal placement of DGs results in even better solutions.


Introduction
Optimum power flow (OPF) solutions are crucial tools in electric power network operation [1,2].It is an astute power flow that utilizes optimization algorithms to regulate power grid control settings optimally amid diverse constraints [3,4].Many classical optimization algorithms have been utilized to deal with the OPF problem, like non-linear programming [5,6], the Newton algorithm [7], quadratic programming [8], and decomposition algorithms [9].A comprehensive review of the deterministic (conventional) optimization algorithms previously employed is presented in [10].Although these methods can achieve the globally optimal solution in some cases, they have certain shortcomings, such as getting trapped in local optima (i.e., insecure convergence properties), inability to tackle non-differentiable goal functions, and high sensitivity to initial search points.Furthermore, these algorithms cannot guarantee a global solution.Thus, proposing alternative methods to address the above-mentioned drawbacks is a necessity.
The significant development of computers in recent years has resulted in a trend of solving OPF problems using nature-inspired optimization techniques.Various stochastic optimization techniques

Problem Formulation
In this paper, three objectives are selected to deal with the OPF issue with and without incorporating DG effect: minimizing total fuel cost of power generation, real power loss minimization, and voltage stability improvement.The total fuel cost of a specified power network is represented by the following function: where N G is the number of generation units, whereas f i represent the fuel cost of the i-th generation unit that can be expressed by quadratic function as follows: where a i , b i , and c i represent the fuel cost coefficients of the i-th generation unit, whereas P Gi is the active power output of the ith generator.Meanwhile, the total system active power losses P loss can be defined as: where N is the number of network nodes; V i and V j are the voltage magnitudes for the i-th and j-th nodes, respectively; δ i and δ j are the node voltage angles of the i-th-j-th branch; and G ij refers to the conductivity between node i and node j.
The voltage stability of a particular power grid can be assessed using the widely utilized indicator, L-index, that is, L max [24]: where L k is the L-index of the kth load bus and N L is the number of load buses.Additionally, the optimization of the real power generation of DG units is another target toward achieving a superior solution to the OPF problem.Hence, DGs optimum penetration has been incorporated into the suggested formulation as an additional control variable.Mathematically, the proposed formulation of OPF problem that considers DG can be expressed as: subject to: a Equality constraints which stand for load flow equationsin polar form: where P Gi and Q Gi are the real and imaginary power generations at the i-th bus, respectively; P Di and Q Di are the real and imaginary power demands at the same i-th bus, respectively; and B ij is the susceptance through node i and node j. δ ij = δ i −δ j is the voltage angle variance.It is worth mentioning that other equivalent formulations for power flow equations are discussed in [25].
b Inequality constraints that represent the control variable constraints: The modified vector of control variables that considers the real power generation of DGs can be stated as: ., Q CN C , P DG1 , . . ., P DGN DG T (13) where P G refers to the active power generation outputs with the exception of swing bus, T i is the tapping ratio of the i-th transformer, V Gi is the voltage magnitude of the unit i, Q Ci is the imaginary power injection by the i-th shunt capacitor, P DGi is the real power generation of the i-th DG unit, N G is the number of generator buses, N T is the number of tap changing transformers, N C is the number of shunt capacitors, and N DG is the number of DG units.c Inequality constraints that comprise the dependent variable constraints.

P min
Slack ≤ P Slack ≤ P max Slack (14) The developed version of the vector of dependent variables that considers the reactive power generation of DG units can be defined as: where P Slack is the active power of the slack bus, V Li is the voltage of i-th load node, S line,i is the loading of the i-th transmission line, Q DGi is the imaginary power generation of the i-th DG unit, N L is the number of load buses, and N l is the number of transmission lines.
The literature shows several approaches to handling dependent variable constraints in population-based optimization algorithms.One efficient approach is to incorporate these constraints into the goal function as quadratic penalty expressions (i.e., the penalty factor method).Each term added to the goal function comprises a given penalty factor multiplied by the square of the violated value of a dependent variable.Consequently, any obtained infeasible candidate solution is denied.In this work, the modified goal function includes quadratic penalty terms of the extended set of dependent variables (i.e., P Slack , V L , Q G , S line , and Q DG ).Notably, a quadratic penalty term for the reactive power of DG is added.The mathematical formulation of the modified i-th goal function can be stated as follows: where F i is the i-th goal function to be optimized, F i is the i-th modified goal function, and λ P , λ V , λ O , λ S and λ DG are the penalty factors which can be assigned subjectively by a decision maker.Normally, using high penalties is desirable to insure that any infeasible solution is declined.x lim is the upper/lower bound value of the dependent variable x that can be stated in the following formulation: It is worth mentioning that the above mathematical formulation of the modified goal function is just used in case a dependent variable(s) violates the upper/lower limit.The main aim is to detect and avoid any discovered infeasible solution(s) during the optimization process.Notably, the choice of the penalty terms (λ P , λ V , λ O , λ S and λ DG ) play a crucial role.Based on application and designer experience, the penalty terms may be different for different boundaries.Using of different penalty factors lead to different solutions.To address this issue, a high unity penalty of 10000 has been considered in this paper for each dependent variables in case of violation the upper/lower limit.However, it would be desirable to update the penalty terms in the objective function dynamically in the algorithm.

The Jaya Algorithm
Jaya is a new population-based optimization algorithm introduced by Rao [22] to produce optimal solutions for constrained and unconstrained optimization problems.Unlike other population-based heuristic algorithms, Jaya has no algorithm-specific controlling parameter, and involves only the two ordinary controlling parameters of population size (m) (that is, the number of candidate solutions) and the number of generations (G n ) (that is, the total iterations).The optimization process of this technique is elicited on the basis of the idea that the solution determined for a specific problem have to shift toward the optimum solution and evade the inferior solution [22].The basic Jaya algorithm has only one phase according to the aforementioned concept, making it a simple optimization technique.Figure 1 illustrates the flowchart of Jaya algorithm.

Application of Jaya Algorithm to OPF Problem
The following steps describe the proposed process of applying the Jaya algorithm to solve the OPF problem: Step 1 Define the branch data, active and reactive power load levels, and generation units data.
Specify the initial values of active power generation of PV buses, real power production of the DGs (in case of considering DG), reactive power injection for shunt compensators, voltage magnitudes of generation buses, and the tap setting of regulating transformers.
Step 2 Run the initial status load dispatch.Calculate the values of the goal functions for the initial status, that comprise generation cost reduction, real power loss reduction, and voltage stability improvement, using Equations ( 1), (3), and (4) , respectively.Step 3 Allocate the DG unit(s) to suitable site(s) on the basis of the sensitivity of active power loss and generation cost to both active and reactive power injection as stated in the following formulations: where   /  and   /  represent the sensitivities of active power losses to injected active power at the i-th bus and to injected reactive power at the k-th load bus, respectively.  /  is the sensitivity of active power losses to voltage angle of i-th bus, whereas   /  represents the sensitivity of active power losses to voltage magnitude of kth load bus.
[Jac] is the Jacobian matrix.
  where Cos/  and Cos/  represent the sensitivities of generation cost to each of active and reactive power injection at the i-th and k-th bus, respectively.Cos/  is the sensitivity of Flowchart demonstrating the optimization process of the basic Jaya algorithm [22].
where ∂P loss /∂P i and ∂P loss /∂Q K represent the sensitivities of active power losses to injected active power at the i-th bus and to injected reactive power at the k-th load bus, respectively.∂P loss /∂δ i is the sensitivity of active power losses to voltage angle of i-th bus, whereas ∂P loss /∂V K represents the sensitivity of active power losses to voltage magnitude of kth load bus.[Jac] is the Jacobian matrix.
where ∂Cost/∂P i and ∂Cost/∂Q K represent the sensitivities of generation cost to each of active and reactive power injection at the i-th and k-th bus, respectively.∂Cost/∂δ i is the sensitivity of generation cost to the voltage angle of the i-th bus, while the term ∂Cost/∂V K represents the sensitivity of generation cost to voltage magnitude of kth load bus.Note: This step 3 is used in case of considering DG effect.
Step 4 Define the i-th objective function F i (x,u) to be optimized (that is one of the objective functions described in Section 2).Initialize the number of control (design) variables (n), population size (m) (that is, the number of candidate solutions), number of iterations (G n ), and minimum and maximum limits of design variables (U L , L L ).
Step 5 Create an initial random population based on the defined controlling parameters within the pre-specified limits of design variables.This population is formulated as follows: where n is the number of control (design) variables, whereas m is the number of candidate solutions.
The value of the j-th control variable (X k,j ) in the k-th candidate solution can be written as follows: where rand(.) is a randomly produced number within the domain of (0-1); X max j and X min j are the upper and minimum boundaries of the j-th control variable.For more explanation, the physical elements of X k,j in the optimization work can be described as follows: Step 6 Run power flow program for each candidate solution and calculate the value of goal function that corresponds to each solution.
Step 7 Identify the best and worst solutions among the candidate solutions.
Step 8 Based on the best and worst solutions, modify all candidate solutions.The proposed modification is expressed as follows: Throughout the ith iteration (generation) and for each kth candidate solution, the value of the jth design variable (X j,k,i ) can be modified as follows [22]: where X j,k,i is the modified value of the j-th design variable, r 1,j,i and r 2,j,i are randomly generated numbers within the range of (0-1) for the j-th control variable.X j,best,i is the value of the j-th design variable for the top nominee solution.X j,worst,i is the value of the j-th design variable for the inferior nominee solution.The second term of the above equation stand for the propensity of the modified solution to proceed closer to the optimum solution.The third expression stands for the propensity of the solution to eschew the worst solution.
Step 9 For all updated solutions, if any control variable upper/lower limit is violated, replace the estimated value with the corresponding limit.
Step 10 Execute the power dispatch considering the modified vector of design variables.
Calculate the new values of the goal function and distributed generation size for each candidate solution.Add the assigned penalty(s) to the value of the goal function in case a dependent variable(s) violates the upper/lower limit, using Equation (20).
Step 11 For each candidate solution, compare the objective function F i (x,u) values for the previous and updated solution.Accept the updated solution if it is superior to the previous solution.Otherwise, keep the previous solution.
Step 12 Stop and report the optimal solution if the termination criterion is achieved.Otherwise, return to step 7.
For more explanation, the flowchart of the proposed application of Jaya algorithm to solve OPF problem is shown in Figure 2.

Results and Discussion
To examine the efficacy of the Jaya algorithm for single OPF problems considering DG, the algorithm is implemented in the IEEE 30-bus network and the IEEE 118-bus network.For the IEEE 30-bus network, the population size (m) and maximal number of generation (G n ) are set to 40 and 100, respectively.The case considered for the IEEE 118-bus network is carried out with a population size (m = 100) and a maximum of 300 iterations.The Jaya algorithm is performed in the computational environment of MATLAB R2015b [26] and implemented on a PC with a 2.7 GHz Intel ® Core™ (Intel Corporation, 2200 Mission College Blvd.: Santa Clara, CA, USA) i7 CPU and 16 GB RAM.

IEEE 30-Bus Network
The essential data of this network is shown in [27].The network has six generator units at buses 1, 2, 5, 8, 11, and 13.Load buses 10, 12, 15, 17, 20, 21, 23, 24, and 29 are equipped with switchable shunt capacitors.Four tap changing transformers are installed at lines 6-9, 6-10, 4-12, and 27-28.The voltage level bounds of all load buses are set to (0.95, 1.05) p.u.A prevalent DG unit which can produce active and reactive power is also utilized throughout the implementation of Jaya algorithm, with a 10 MW generation capability and 0.8 power factor.According to the results obtained from Equations ( 22) and ( 23), we found that bus number 30 is the nominee location for DGs accommodation with the top sensitivities of active power loss and generation cost to each of active and reactive power injection, that are (−0.1408),(−0.0516), (−0.0926), and (−0.0383), respectively.We noticed that bus 3 is the worst location, because it had the least sensitivities of active power loss and generation cost to both injected active and reactive powers, which are (−0.0362),(0.007), (−0.0208), and (−0.001).Accordingly, the aforementioned locations are used for DG accommodation throughout the implementation of the Jaya method for single optimization cases.The representative single-line digram of this network that contains all the required measures in the Jaya algorithm is illustrated in Figure 3. Accordingly, the aforementioned locations are used for DG accommodation throughout the implementation of the Jaya method for single optimization cases.The representative single-line digram of this network that contains all the required measures in the Jaya algorithm is illustrated in Figure 3.

Case 1: Fuel Cost Minimization
In this section, fuel cost minimization is considered the goal function during the implementation of Jaya algorithm with and without the influence of DG. Figure 4 illustrates the convergence graph of fuel cost minimization using the Jaya algorithm with and without incorporating the distributed generation unit at nodes 30 and 3. Without considering DG's effect, the method requires 49 iterations to obtain the optimum solution, which reveals the excellent convergence rate of the Jaya algorithm.The best adjustments of the design variables and optimal values of cost minimization are tabulated in Table 1.The results present a significant reduction in fuel cost from 902.0207 $/h to 800.4794 $/h, when the Jaya algorithm is executed without considering DG.In addition, the average computational time for single iteration for this case is 0.724 s.These results point out the effectiveness of the Jaya algorithm in terms of solution optimality and fast convergence.Fuel cost is compared with that obtained using various other heuristic optimization algorithms to validate the Jaya algorithm further.Table 2 demonstrates the supremacy of the Jaya algorithm over previous algorithms.Notably, the majority of obtained solutions using heuristic optimization algorithms (Table 2) are infeasible, which is principally due to voltage magnitude violations at one or more system load buses, as well as reactive power generation bound violations at one or more generation units.Notably, [23] also reported solution infeasibility for many previous methods, as compared in Table 2. Implementing Jaya method for fuel cost reduction when accommodating the DG at node 30 produces an even more significant reduction in fuel cost, reaching 768.0398 $/h, an attractive Lmax value (0.0969), and a great expansion of shunt compensators reactive power saving of up to 29.8391 MVAR.Most importantly, a considerable reduction in active power losses (8.4983 MW) is achieved.Additionally, as a worst site for DG accommodation, node 3 leads to a critical rise of 12.804% in the L-index that results in a slight decrease of 0.812% in losses as compared with the results gained for cost optimization without utilizing DG.These findings exhibit the superiority of the Jaya algorithm over several heuristics techniques for solving this type of problem, and reinforce the efficacy of the proposed DG placement technique.

Case 1: Fuel Cost Minimization
In this section, fuel cost minimization is considered the goal function during the implementation of Jaya algorithm with and without the influence of DG. Figure 4 illustrates the convergence graph of fuel cost minimization using the Jaya algorithm with and without incorporating the distributed generation unit at nodes 30 and 3. Without considering DG's effect, the method requires 49 iterations to obtain the optimum solution, which reveals the excellent convergence rate of the Jaya algorithm.The best adjustments of the design variables and optimal values of cost minimization are tabulated in Table 1.The results present a significant reduction in fuel cost from 902.0207 $/h to 800.4794 $/h, when the Jaya algorithm is executed without considering DG.In addition, the average computational time for single iteration for this case is 0.724 s.These results point out the effectiveness of the Jaya algorithm in terms of solution optimality and fast convergence.Fuel cost is compared with that obtained using various other heuristic optimization algorithms to validate the Jaya algorithm further.Table 2 demonstrates the supremacy of the Jaya algorithm over previous algorithms.Notably, the majority of obtained solutions using heuristic optimization algorithms (Table 2) are infeasible, which is principally due to voltage magnitude violations at one or more system load buses, as well as reactive power generation bound violations at one or more generation units.Notably, [23] also reported solution infeasibility for many previous methods, as compared in Table 2. Implementing Jaya method for fuel cost reduction when accommodating the DG at node 30 produces an even more significant reduction in fuel cost, reaching 768.0398 $/h, an attractive L max value (0.0969), and a great expansion of shunt compensators reactive power saving of up to 29.8391 MVAR.Most importantly, a considerable reduction in active power losses (8.4983 MW) is achieved.Additionally, as a worst site for DG accommodation, node 3 leads to a critical rise of 12.804% in the L-index that results in a slight decrease of 0.812% in losses as compared with the results gained for cost optimization without utilizing DG.These findings exhibit the superiority of the Jaya algorithm over several heuristics techniques for solving this type of problem, and reinforce the efficacy of the proposed DG placement technique.

Case 2: Real Power Losses Reduction
The goal considered in this case was active power losses minimization.The Jaya algorithm was utilized to attain the optimum solution, the results of which are given in Table 1.The Jaya algorithm is clearly efficient for determining the optimum settings of the control variable, which minimizes system losses.Consequently, real power losses is decreased significantly from 5.8482 MW to 3.1035 MW when Jaya algorithm is performed without considering DG. Figure 5 shows the steep convergence of real power losses using the Jaya algorithm while considering DG placement at buses 30 and 3.The goal considered in this case was active power losses minimization.The Jaya algorithm was utilized to attain the optimum solution, the results of which are given in Table 1.The Jaya algorithm is clearly efficient for determining the optimum settings of the control variable, which minimizes system losses.Consequently, real power losses is decreased significantly from 5.8482 MW to 3.1035 MW when Jaya algorithm is performed without considering DG. Figure 5 shows the steep convergence of real power losses using the Jaya algorithm while considering DG placement at buses 30 and 3.The algorithm fully reaches to the optimum solution with 65 iterations, demonstrating the rapid convergence of the Jaya algorithm.To test the effectiveness of the algorithm, estimated real power loss value is compared with that obtained using previously reported population-based optimization The algorithm fully reaches to the optimum solution with 65 iterations, demonstrating the rapid convergence of the Jaya algorithm.To test the effectiveness of the algorithm, estimated real power loss value is compared with that obtained using previously reported population-based optimization algorithms.Table 3 illustrates the superiority of the Jaya algorithm over these earlier algorithms.The solution obtained using HS [15] algorithm (Table 3) is infeasible because of the voltage magnitude violations at all load buses except bus 7 (Adaryani et al. have also reported this issue) [4].Implementing the Jaya method for loss minimization when placing the DG at the best site leads to further minimization (2.675040 MW), a considerable saving in generation cost (46.0839$/h) accompanied by an excellent voltage stability index value (0.0892).Furthermore, the shunt compensator reserve margin increases extensively by 110.787%.In contrast, placing a DG at node 3 (that is, the worst node for DG accommodation) while executing the OPF solution minimizes losses to 3.339 MW.This result is inferior when compared with loss minimization that does not utilize DG, which is 3.1035 MW.A critical increment of 11.871% L max is also obtained.The findings confirm that optimal placement of DG leads to a superior solution to the OPF problem.To sum up, these findings validate the supremacy of the Jaya algorithm over other techniques reported in the literature, both in terms of solution optimality and feasibility.

Case 3: Voltage Stability Improvement
In this section, enhancing voltage stability is selected as the goal function to be optimized using the Jaya algorithm while considering the effects of DG placement.The trend of enhancing system voltage stability is sketched in Figure 6.The results are given in Table 1.The findings point out that the voltage stability index is reinforced by 28.23%.Table 4 compares solutions achieved using Jaya method and other population-based optimization techniques, with the former yielding clearly superior results.
Remarkably, placing a DG unit at bus 30 while executing OPF significantly reduces L max index to 0.0851, whereas integrating a DG unit at the worst location (i.e., bus 3) increases L max index to 0.1371.

Case 3: Voltage Stability Improvement
In this section, enhancing voltage stability is selected as the goal function to be optimized using the Jaya algorithm while considering the effects of DG placement.The trend of enhancing system voltage stability is sketched in Figure 6.The results are given in Table 1.The findings point out that the voltage stability index is reinforced by 28.23%.Table 4 compares solutions achieved using Jaya method and other population-based optimization techniques, with the former yielding clearly superior results.Remarkably, placing a DG unit at bus 30 while executing OPF significantly reduces Lmax index to 0.0851, whereas integrating a DG unit at the worst location (i.e., bus 3) increases Lmax index to 0.1371.

Statistical Analysis
A statistical study is performed to examine the robustness and efficacy of the Jaya algorithm in solving OPF problems that considers DG effect.For each case, 50 independent trials of the Jaya algorithm with different initial populations (solutions) are carried out.As previously mentioned, for the modified IEEE 30-bus test network, population size (m) and utmost number of generation (G n ) are set to 40 and 100, respectively.In this paper, the statistical analysis indices employed are best value, worst value, average (mean) value, and standard variation (SD).Statistical analysis results are presented in Table 5.The table presents that the best, worst, and average values for all considered cases are very close to each other; hence the standard deviation values were small.These results affirm the robustness of the Jaya algorithm and its capability to explore the optimal solution in every run.optimization methods.Moreover, the optimal placement of DG units in terms of generation cost and power loss sensitivities, together with the Jaya algorithm approach, led to better solution for all single objective optimization cases.The erroneous placement of DG resulted in unattractive solutions.
In conclusion, the applicability, potential, and efficacy of the Jaya algorithm in solving OPF problems for small and large-scale power systems were confirmed.Jaya algorithm is a powerful tool for solving such problems and is a good candidate for solving OPF problems of practical power systems.In the future, the performance of the Jaya algorithm can be compared with other global solvers exist which can guarantee global optimality such as Lindoglobal, Antigone and Baron.

Figure 1 .
Figure 1.Flowchart demonstrating the optimization process of the basic Jaya algorithm [22].

Figure 1 .
Figure 1.Flowchart demonstrating the optimization process of the basic Jaya algorithm [22].

Figure 2 .
Figure 2. Flowchart of the application of Jaya algorithm for the OPF problem.

Figure 2 .
Figure 2. Flowchart of the application of Jaya algorithm for the OPF problem.

Figure 3 .
Figure 3.One line diagram of IEEE 30-bus network.

Figure 3 .
Figure 3.One line diagram of IEEE 30-bus network.

Figure 4 .
Figure 4. Convergence graph of fuel cost minimization using the Jaya algorithm with and without employing DG.Figure 4. Convergence graph of fuel cost minimization using the Jaya algorithm with and without employing DG.

Figure 4 .
Figure 4. Convergence graph of fuel cost minimization using the Jaya algorithm with and without employing DG.Figure 4. Convergence graph of fuel cost minimization using the Jaya algorithm with and without employing DG.

Figure 5 .
Figure 5. Convergence of Jaya method for case 2.

Figure 5 .
Figure 5. Convergence of Jaya method for case 2.

Figure 6 .
Figure 6.Convergence of Jaya method for case 3.

Figure 6 .
Figure 6.Convergence of Jaya method for case 3.

Table 1 .
Optimum setting of control variables for different cases using the Jaya algorithm without and with utilizing DG (modified IEEE 30-bus network).

Table 2 .
Comparison of the solutions obtained for cost reduction (modified IEEE 30-bus network).

Table 3 .
Comparison of the results obtained for real power losses (modified IEEE 30-bus network).
The values in bold type indicate optimum values; a Infeasible solution.

Table 4 .
Comparison of the results obtained for voltage stability enhancement (modified IEEE 30-bus network).
max Fuel Cost ($/h) Real Power Losses (MW)The values in bold type indicate optimum values; a Infeasible solution; NA, not available.

Table 5 .
Statistical results obtained over 50 independent trials of Jaya method without and with employing DG.