Multi-Objective Optimal Power Flow Problem Using Nelder–Mead based Prairie Dog Optimization Algorithm

doi:10.21203/rs.3.rs-3111489/v1

The Prairie Dog Optimization Algorithm (PDOA) is a novel metaheuristic algorithm that takes its inspiration from foraging behavior, burrow building activities and communication alarm of Prairie Dog. PDOA has gained extensive popularity among the research community and is now utilized to tackle a variety of optimization challenges. However, the algorithm does not have a healthy equilibrium condition between its exploration and exploitation process. To overcome this problem a novel hybrid algorithm namely Nelder–Mead based Prairie Dog Optimization Algorithm (NMPDOA) is proposed with the help of Nelder–Mead algorithm (NMA). In this paper, NMPDOA is used for solving the multi-objective Optimal Power Flow (MO-OPF) problems. In order to prove the applicability of the suggested technique, it has been applied to 20 different benchmark functions. The suggested technique is tested in the IEEE 30 and IEEE118 bus standard test system and applied in Indian 62 bus system. To determine the effectiveness of the algorithm, the results of PDOA and NMPDOA are compared with other previously published results discussed in the literature. In addition, PDOA and NMPDOA are also applied in two MO-OPF problems. The statistical analysis and t-Test analysis confirm the effectiveness and consistency of NMPDOA for solving real world highly nonlinear and mixed integer optimization problems.

MO-OPF

Metaheuristic algorithm

Prairie Dog Optimization Algorithm

Nelder–Mead Algorithm

Recently, Optimal Power Flow (OPF) has emerged as one of the most powerful optimization tools for improving the reliability and cost-effectiveness of power grids. In order to reduce real power loss (PLoss), improve Total Voltage deviation (TVD) and boost voltage stability index (LIndex), OPF provides the optimal settings for control variables such as Active power generation (P_Gen), generator bus voltage (V_Gen), transformer tap settings (TS), and reactive power compensators (RPC) while taking into consideration both the equality constraints (ECs) and inequality constraints (ICs). Due to the existence of continuous variables such as V_Gen and discrete variables such as TS and RPC, OPF problems are framed as mixed integer problems. The field of multi-objective optimization has come a long way and shown its ability to solve problems with several competing objectives in one attempt [1]. Multi-objective optimization is rapidly becoming an essential tool for engineers due to the growing complexity and difficulty of real-world problems [2].

Over the years, OPF problems have been solved by several mathematical models. Earlier stages of OPF issues are tackled using conventional techniques. In 1990 M. Bjelogrlic et al. implement Newton’s Method in OPF problems [3]. S. Granvile et al. implement Interior Point Method [4], and N. Deeb et al. impliment Decomposition Method [5] in OPF problem. The conventional strategy relies on starting with a partial answer and gradually refining it to the best solution as the number of iterations increases. Conventional methods are basically performing simple algebraic operations. Therefore, these methods are faster and have excellent convergence. However, conventional methods have some unavoidable issues such as local trapping, inability to properly manage constraints and error due to assumptions [6]. These unavoidable problems have been overcome to a large extent by evolutionary algorithm. Evolutionary algorithms are basically inspired by nature and based on the physical or psychological behaviors of living and non-living things [6].

In recent years, social communication and interaction-based evolutionary algorithms gain enormous attraction in solving OPF problems. In 2010 K. Mahadevan et al. implement Particle Swarm Optimization (PSO) algorithm [7] in OPF problem. In addition to this K. Iva et al. implemented Genetic Algorithm (GA) [8], D. Karaboga et al. implemented Artificial Bee colony (ABC) algorithm [9], and A. H. Khazali et al. implemented Harmony Search (HS) algorithm [10] in OPF problems. Furthermore, Duman et al. implemented gravitational search algorithm (GSA) to solve ORPD problems for IEEE standard system of 30,57 and 118 bus [11] in 2012. In 2016 Rajan et al. implemented exchange market algorithm (EMA) to minimize PLoss, and LIndex [12]. These algorithms may not assure an exact optimal solution but can provide good near-optimal solutions in a short time.

From a review of the relevant literature, it has been determined that, sometimes all the populations are trapped by a single solution which is called local trapping. This may cause premature convergence in search space. Therefore, many researchers developed hybrid algorithms for solving various nonlinear and mixed integer optimization problems [13]. In 2022 M. Ghasemi et al. implement Coulomb's and Franklin's laws algorithm (CFA) to solve OPF problems for large system [14]. B. Mandal et al. introduced a hybrid algorithm namely quasi-oppositional teaching learning-based optimization (QOTLBO) to solve OPF problems [15] in 2016. In 2017 G. Chen et al. developed an Improved GSA based conditional selection strategies (IGSA-CSS) algorithm to solve ORPD problems [16]. Rajan et al. implemented Nelder–Mead Algorithm (NMA) in firefly algorithm (FA) to solve ORPD problems for IEEE standard system of 30 and 118 bus [17] in 2015. In 2014 B. Shaw et al. developed opposition-based GSA (OGSA) and implemented to solve ORPD problems [18]. In addition to these evolutionary algorithms A. Shabanpour-Haghighi et al. developed a modified teaching–learning based optimization to solve MO-OPF problem for IEEE30 bus standard system in 2014 [19]. From the above literature it has been concluded that, the hybrid algorithms have shown superior problem-solving ability and overcome premature convergence problems. Algorithm hybridization is a process in which properties of two or more than two algorithms are combined to get the benefits from both the algorithm for better performance.

In 2022 F. A. Hashim et al. developed a nature inspired metaheuristic algorithm namely Prairie Dog Optimization Algorithm (PDOA) to solve complex optimization problems [20]. In [20] the superiority of PDOA is confirmed by outperforming 24 benchmark functions from all comparative algorithms out of 30 benchmark functions. Therefore, PDOA has gained enormous popularity among researchers and is often used in various disciplines for solving various nonlinear optimization problems. However, PDOA is a novel metaheuristic algorithm that hasn’t yet been used to solve any electrical engineering optimization problems.

In this research, the authors employ PDOA to address ORPD-related optimization challenges and tested on IEEE standard system of 30 and 118 bus and applied on 62 bus Indian utility systems. It has been observed that solutions obtained from PDOA are unable to outperform many well-known methods such as PSO [9], BFOA [17], MALO [21], HFA [17] and GSA [13] applied to ORPD problems.

The literature survey suggests that the effectiveness of any algorithms rely on substantially on its exploration and exploitation process. The method produces premature convergence if the exploitation phase dominates the exploration process. However, if the exploration phase dominates the exploitation process, the algorithm loses time searching over seamless areas, and sometimes the best answer may not be found. For better performance of any algorithm exploration and exploitation property must be properly balanced [22].

Looking at the vulnerabilities of the PDOA, an intelligent hybrid algorithm namely Nelder–Mead based Prairie Dog Optimization Algorithm (NMPDOA) is designed in this paper. In the proposed hybridization process, NMPDOA is designed by implementing the NMA [23] in PDOA that improves the convergence criteria of the PDOA. The suggested approach is implemented in 20 benchmark functions reported in Appendix A and compared to many well-known algorithms to determine its efficacy. NMPDOA is implemented in OPF problems such as minimization of PLoss, TVD and LIndex and tested on IEEE standard system of 30 and 118 bus and applied on 62 bus Indian utility systems. The NMPDOA is then applied in MO-OPF problem for IEEE standard system of 30 bus. The result obtained from NMPDOA are compared with PDOA and several state-of-art methods reported in the discussed literature.

In mathematical terms, the SO-OPF problem and MO-OPF problems may be represented as:

Minimize f(s,a)

Subject to $\left\{ {\begin{array}{*{20}{c}} {g(s,a)=0} \\ {h(s,a) \leqslant 0} \end{array}} \right.$ (1)

Minimize f_i(s,a)

Subject to $\left\{ {\begin{array}{*{20}{c}} {{g_i}(s,a)=0} \\ {{h_i}(s,a) \leqslant 0} \end{array}} \right.$ (2)

Where, f(s,a) represents the objective function. s represents dependent variables arranged in a vector, whereas a represents independent variables arranged in a vector. g(s,a) and h(s,a) represents the equality constraints and inequality constrains.

The variables of OPF objectives can be formulated mathematically as:

$$\begin{gathered} {a^T}=\left[ {{P_{Gen2}} \cdot \cdot \cdot {P_{GenNG}},{V_{Gen1}} \cdot \cdot \cdot {V_{GenNG}},{Q_{C1}} \cdot \cdot \cdot {Q_{CNC}},{T_1} \cdot \cdot \cdot {T_{NT}}} \right] \hfill \\ {s^T}=\left[ {{P_{Gen1}},{V_{D1}} \cdot \cdot \cdot {V_{DNPQ}},{Q_{Gen1}} \cdot \cdot \cdot {Q_{GenNG}},{S_{L1}} \cdot \cdot \cdot {S_{LNL}}} \right] \hfill \\ \end{gathered}$$

3

Where NG, NC, NT, NL and NPQ represent the number of generator bus, VAR compensator, tap changing transformer, transmission line (TL) and PQ bus respectively. Q_C and Q_Gen represents the VAR compensations and generator buses reactive power output respectively. S_L represents the line flow through the TL.

2.1. Objective functions:

The traditional OPF problem basically concerned with three major Single objective such as:

Case-1: Minimization of the active power loss in TL (PLoss)
Case-2: Minimization of the Voltage deviation at load busses (TVD)
Case-3: Enhance the Index of Voltage stability (L-Index)

The MO-OPF problems can be formalized by the combination of following objectives:

Case-4: Minimization of the combination of PLoss and Total fuel Cost (TFC).
Case-5: Minimization of the combination of TFC and Total Emission (TE).

2.1.1. Minimization of the active power loss in TL: The formula for minimizing PLoss in TL is [21]:

Minimize F₁ = P_loss = $\sum\limits_{{k=1}}^{{NT}} {{g_k}} \left( {{V_a}^{2}+{V_b}^{2} - 2{V_a}{V_b}\cos \left( {{\delta _a} - {\delta _b}} \right)} \right)$ (4)

Where gk represents the k^th TL conductance. Va and Vb represents the voltage magnitude of a^th and b^th buses respectively. δ_a and δ_b represents the voltage angle.

2.1.2. Voltage deviation minimization at load busses: Sometimes in power system operation the load voltage may violate its limit. A problem formulation has been done in the OPF to minimize the violations from all the load buses. Minimization of TVD might enhance the voltage profile and allow power systems to operate more safely. TVD minimization at load busses can be formulated as [21]:

Minimize F₂ = VD =$\sum\limits_{{a=1}}^{{NPQ}} {\left| {{V_a} - {V_{re{f_a}}}} \right|}$ (5)

where Vrefa represents the reference voltage magnitude in a^th PQ bus (1pu).

2.1.3. Voltage stability enhancement: Currently, power systems are run near to their stability limitations for operational reasons that are both economical and ecologically benign. Therefore, Voltage stability is considered as an important issue. The ability of a power system to keep the voltage on any bus within a certain range is known as voltage stability. When there is a sudden disruption in the system, voltage instability occurs which may cause uncontrollably reducing in voltage. We can increase voltage stability by minimizing the maximum LIndex. The L-Index might have a range anywhere from 0 to 1, inclusive. The value of L-index near 0 would be considered the best solution, and a value near 1 would be considered the worst solution. L-index for j^th bus can be formulated as [21]:

L-index_j = $\left| {1 - \sum\limits_{{i=1}}^{{NPV}} {{F_i}_{j}\frac{{{V_i}}}{{{V_j}}}} } \right|$ (6)

${F_{ij}}={\text{ }} - {\left[ {{Y_1}} \right]^{ - 1}}\left[ {{Y_2}} \right]$

$$\left[ {\begin{array}{*{20}{c}} {{I_{PQ}}} \\ {{I_{PV}}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {{Y_p}}&{{Y_q}} \\ {{Y_r}}&{{Y_s}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V_{PQ}}} \\ {{V_{PV}}} \end{array}} \right]$$

7

where Yp and Yq represents the YBUS submatrices that were created by isolating the PQ bus and PV bus characteristics.

2.1.4. Minimization of simple fuel cost and Total Emission: F_cost^simple and F_emission^simple for n^th thermal generating units can be mathematically formulated as [24]:

$$F_{{TFC}}^{{simple}}=\sum\limits_{{n=1}}^{{NG}} {\left[ {{a_n}+{b_n}{P_{gn}}+{c_n}P_{{gn}}^{2}} \right]} \$ /h$$

8

$$F_{{TE}}^{{simple}}=\sum\limits_{{n=1}}^{{NG}} {\left[ {{s_n}+{t_n}{P_{gn}}+{u_n}P_{{gn}}^{2}+{v_n}\exp \left( {{w_n}{P_{gn}}} \right)} \right]} ton/h$$

9

Where a_n, b_n and c_n represents the cost coefficient of n^th generating unit. Sn, tn, un, vn and wn represents the emission coefficient of n^th generating unit. P_gn represents the real power generated in n^th generating units.

2.2. Multi-Objective Optimal power flow

In typical MO-OPF problem, multiple objectives are simultaneously optimized. The technique that solves the MO problem gets the Pareto-optimal front (POF), which is an array of solutions that are less significant than the others. Consider that the system has two alternative solutions, Xp and Xq. Xp is considered the non-dominant solution if the subsequent conditions are met [25].

$$\begin{gathered} \begin{array}{*{20}{c}} {\forall a \in \left\{ {1,2, \cdot \cdot \cdot ,{N_{obj}}} \right\}}&{{F_a}({X_p}) \leqslant {F_a}({X_q})} \end{array} \hfill \\ \begin{array}{*{20}{c}} {\exists b \in \left\{ {1,2, \cdot \cdot \cdot ,{N_{obj}}} \right\}}&{{F_b}({X_p})<{F_b}({X_q})} \end{array} \hfill \\ \end{gathered}$$

10

Where N_obj represents number of total objective functions used for formation of the multi objective optimization problem.

The objectives of a multi-objective problem often contradict with one another. Since there is not a single best option, the Pareto method instead seeks to provide an array of solutions. The best possible solution among them is named as the Pareto-optimal Solution (POS). After storing all of the POS, a fuzzy membership function (𝝻_a) is utilised to report the most preferred solution. The a^th objective function f_a may be stated in terms of a fuzzy membership function 𝝻_a as follows [25].

$${\mu _a}=\left\{ {\begin{array}{*{20}{c}} 1&{{f_a} \leqslant f_{a}^{{\hbox{max} }}} \\ {\frac{{f_{a}^{{\hbox{max} }} - {f_a}}}{{f_{a}^{{\hbox{max} }} - f_{a}^{{\hbox{min} }}}}}&{f_{a}^{{\hbox{min} }}<{f_a}<f_{a}^{{\hbox{max} }}} \\ 0&{{f_a} \geqslant f_{a}^{{\hbox{max} }}} \end{array}} \right.$$

11

Where, f_a^max and f_a^min represents the maximum and minimum value of a^th objective function.

The membership function 𝝻^k can be estimated for each Pareto front k can be mathematically formulated as,

$${\mu ^k}=\frac{{\sum\limits_{{a=1}}^{{Nobj}} {{W_a} \times \mu _{a}^{k}} }}{{\sum\limits_{{k=1}}^{M} {\sum\limits_{{a=1}}^{{Nobj}} {{W_a} \times \mu _{a}^{k}} } }}$$

12

Where, 𝝻^k represents the best optimized solution. W_a represents the weight coefficient. The significance of the objective function is used to establish the weight coefficient. M represents the number of non-dominated solution. Pareto analysis might be used to determine the nondominated solutions [19].

The current study takes a multi-objective optimisation approach by simultaneously minimizing the combination (PLoss – TFC), and (TFC - TE). The formulation for PLoss, TFC and TE and are given in equations (5, 8 and 9) respectively.

2.3. Constraints:

The solution to an optimization problem cannot be treated as a feasible solution if it doesn’t involve any sort of constraint violation. Several constraint management strategies, including Feasibility Rules (FR), ε -Constrained method, and the Penalty Factor (PF) method, have been created to handle impractical solutions and confine them to their minimum and maximum bounds [26]. The essential Equality Constraints (ECs) and Inequality Constraints (ICs) related to OPF problems and its handling process are discussed below.

2.3.1. Equality Constraints:

Power Flow Equation:

$$\begin{gathered} {P_{Ga}} - {P_{Lb}}=\sum\limits_{{b=1}}^{{NB}} {{V_a}{V_b}{G_{ab}}\cos \left( {{\delta _a} - {\delta _b}} \right)+} {V_a}{V_b}{B_{ab}}\sin \left( {{\delta _a} - {\delta _b}} \right)] \hfill \\ {Q_{Ga}} - {Q_{Lb}}=\sum\limits_{{b=1}}^{{NB}} {{V_a}{V_b}{G_{ab}}\cos \left( {{\delta _a} - {\delta _b}} \right) - } {V_a}{V_b}{B_{ab}}\sin \left( {{\delta _a} - {\delta _b}} \right)] \hfill \\ \end{gathered}$$

13

Where P_Ga-P_La and Q_Ga-Q_La represent total real and power injection in the system. NB represents the total bus number. G_ab and B_ab represent the conductance and susceptance between a^th and b^th bus respectively.

Active power balance constraint

The sum of the load demand (P_D) and PLoss must be equal to the amount of power produced, which can be mathematically formulated as

$$\sum\limits_{{n=1}}^{{NG}} {{P_{gn}} - {P_D} - {P_{Loss}}} =0$$

14

Where P_D represents the total load demand.

2.3.2. Inequality Constraints (ICs)

ICs on Independent Variables:

$$\left\{ {\begin{array}{*{20}{c}} {V_{{ge{n_a}}}^{{\hbox{min} }} \leqslant {V_{Ge{n_a}}} \leqslant V_{{ge{n_a}}}^{{\hbox{max} }}}&{a=1,2 \cdot \cdot \cdot NG} \\ {T_{a}^{{\hbox{min} }} \leqslant {T_a} \leqslant T_{a}^{{\hbox{max} }}}&{a=1,2 \cdot \cdot \cdot NT} \\ {Q_{{ca}}^{{\hbox{min} }} \leqslant {Q_{ca}} \leqslant Q_{{ca}}^{{\hbox{max} }}}&{a=1,2 \cdot \cdot \cdot NC} \end{array}} \right.$$

15

Where, V_Gena^min and V_Gena^max represents the boundary limit of V_Gen. T_a^min and T_a^max represents the boundary limit of of transformer taps. Q_Ca^min and Q_Ca^max represents boundary limit of VAR compensation.

ICs on dependent Variables:

Reactive power has a significant impact on load voltage. Small changes in control variables may cause large deviation in load voltage and line power flow. Therefore, load bus voltage should be bounded within their limits and line power flow through each TL should be less than their upper limits.

$$\left\{ {\begin{array}{*{20}{c}} {Q_{{Ge{n_a}}}^{{\hbox{min} }} \leqslant {Q_{Ge{n_a}}} \leqslant Q_{{Ge{n_a}}}^{{\hbox{max} }}}&{a=1,2 \cdot \cdot \cdot NG} \\ {V_{{La}}^{{\hbox{min} }} \leqslant {V_{La}} \leqslant V_{{La}}^{{\hbox{max} }}}&{a=1,2 \cdot \cdot \cdot NL} \\ {{S_{la}} \leqslant S_{{la}}^{{\hbox{max} }}}&{a=1,2 \cdot \cdot \cdot TL} \end{array}} \right.$$

16

Where Q_Gena^min and Q_Gena^max represents boundary limit of the reactive power of generator bus. V_La^min and V_La^max represents the boundary limit of PQ bus voltage.

2.4. Constraints Handling

Real-world optimization problems often include a number of constraints. Any optimization problem's solution can be considered practical if all of the restrictions are addressed. To keep the restrictions firmly within their boundaries, an effective approach to addressing constraints is necessary. As previously discussed, OPF is a real world highly nonlinear and mixed integer optimization problem that also requires an effective and reliable constrained handling approach for managing its ECs and ICs. Sections 2.3.1 and 2.3.2 of the current study define the methods used to deal with the constraints.

2.4.1 Handling Equality Constraints (ECs)

In an optimization problem, equality constraints are thought to be the hardest or most rigid constraints because they make the possible search space smaller. This slows down the optimization process in an actual way. So, the most common way to deal with equality constraints is to turn them into inequality constraints by using a small threshold value. Several additional methods for overcoming EC problems have also been presented in [26]. In the current OPF problem, the number of ECs increases as the bus size rises. For each bus, two ECs are associated, one for active power balance and the other for reactive power balance (described in equations (13). If there are "N" total vehicles in the system, "2N" ECs must be satisfied. In power system, these ECs are being addressed by load flow solution techniques. In the present work, the power flow problem is addressed by using the Newton-Raphson Load Flow (NRLF) solution technique. It should be emphasized that the ECs stated in equations (13 and 14) must be fully satisfied for NRLF to converge.

2.4.2 Handling inequality constraints (ICs)

Inequality limitations are lenient because they are simple to satisfy. Both the independent and dependent sides of the OPF problem include ICs, as shown by equations (15 and 16). To meet ICs on the side of the independent variable, a randomly generated value within that range is sufficient. The independent variable's upper or lower limit might be used to correct any violations that were spotted when updating the variable's value. However, ICs in the dependent variable may be addressed by punishing the solution that includes the violation [26]. The overall objective function taking into consideration power system operation and security limits is mathematically formulated using the penalty approach and is discussed in Eq. (17).

$$F=Fn+{\lambda _{QGena}}{\left( {{Q_{Gena}} - Q_{{Gena}}^{{\lim it}}} \right)^2}+{\lambda _{Ln}}{\left( {{V_{La}} - V_{{La}}^{{\lim it}}} \right)^2}+{\lambda _{TL}}{\left( {{S_{la}} - S_{{la}}^{{\lim it}}} \right)^2}$$

17

Where Fn is the minimization objective function. (Q_Gn-Q_Gn^limit), (V_Ln-V_Ln^limit) and (S_l-S_l^limit) represents the limit violation related to dependent variables. 𝜆_Gn, 𝜆_Ln and 𝜆_𝑇𝐿 are the penalty factors which can be chosen by trial and error method. Q_Gn^limit, V_Ln^limit and S_l^limit are the limiting values which can be mathematically formulated as:

$$Q_{{Gn}}^{{\lim it}}=\left\{ {\begin{array}{*{20}{l}} {Q_{{Gn}}^{{\hbox{min} }}}&{if,{Q_{Gn}} \leqslant Q_{{Gn}}^{{\hbox{min} }}} \\ {Q_{{Gn}}^{{\hbox{max} }}}&{if,{Q_{Gn}} \geqslant Q_{{Gn}}^{{\hbox{max} }}} \\ {{Q_{Gn}}}&{if,Q_{{Gn}}^{{\hbox{min} }} \leqslant {Q_{Gn}} \leqslant Q_{{Gn}}^{{\hbox{max} }}} \end{array}} \right.$$

18

$$V_{{Ln}}^{{\lim it}}=\left\{ {\begin{array}{*{20}{l}} {V_{{Ln}}^{{\hbox{min} }}}&{if,{V_{Ln}} \leqslant V_{{Ln}}^{{\hbox{min} }}} \\ {V_{{Ln}}^{{\hbox{max} }}}&{if,{V_{Ln}} \geqslant V_{{Ln}}^{{\hbox{max} }}} \\ {{V_{Ln}}}&{if,V_{{Ln}}^{{\hbox{min} }} \leqslant {V_{Ln}} \leqslant V_{{Ln}}^{{\hbox{max} }}} \end{array}} \right.$$

19

$$S_{l}^{{\lim it}}=\left\{ {\begin{array}{*{20}{c}} {{S_l}}&{if,{S_l} \leqslant S_{{_{l}}}^{{\hbox{max} }}} \\ {S_{l}^{{\hbox{max} }}}&{if,{S_l}>S_{{_{l}}}^{{\hbox{max} }}} \end{array}} \right.$$

20

Where Q_Gn, V_Ln, and S_l represent the reactive power at PV bus, voltage at load bus, and limit of line power flow respectively.

In 2022 A. E. Ezugwu et al. developed a novel nature-inspired metaheuristic algorithm namely Prairie Dog Optimization Algorithm (PDOA) to solve complex real-world optimization problems [20]. PDOA employs foraging behavior and burrow building activities to complete the exploration. The prairie dogs construct their burrows close to an abundance of food. When the food supply dwindles, they seek for a new one and construct new burrows around the new area. The prairie dogs employ two distinct communication alarm to communicate under various circumstances, such as predator threats or food availability. In the same manner as previous population-based algorithms, PDOA begins its search with an initial randomly generated population. In order to complete the process of exploration and exploitation, the suggested algorithm employs four prairie dog behaviours. The total iterations are divided into four parts. The exploration process is taken care by first two parts and the exploitation process is taken care by other two parts. The exploration process for food foraging and burrow building can be mathematically represented by Eqs. (21 and 22) respectively. The operational process of PDOA is given in Fig. 1.

$$S_{a}^{{t+1}}=GB_{a}^{t} - ECB_{a}^{t} \times \rho - \frac{{GB_{a}^{t} - A}}{{GB_{a}^{t}+\delta }} \times L(n)$$

21

$ECB_{a}^{t}=GB_{a}^{t} \times \delta +\frac{{S_{a}^{t} \times mean(S)}}{{GB_{a}^{t} \times \left( {UB - LB} \right)+\delta }}$

$$S_{a}^{{t+1}}=GB_{a}^{t} \times A - 1.5 \times p{\left( {1 - \frac{t}{{{t_{\hbox{max} }}}}} \right)^{\left( {\frac{{2t}}{{{t_{\hbox{max} }}}}} \right)}} \times L(n)$$

22

Where, S_a^t+1 and S_a^t represents the position of a^th prairie dog for current iteration previous iteration respectively. GB_a^t represents global best solution of the current iteration. ρ represents the communication sound. A represents position of a randomly chosen solution. δ represents the variation among the prairie dogs. L(n) represents the levy distribution. UB and LB represents the upper and lower bound of the control variables. t tmax represents the current iteration and maximum iteration set. p is chosen as 1 when t is even and is -1 when t is odd.

The exploitation process for food alarm and antipredation alarm can be mathematically represented by Eqs. (23 and 24) respectively.

$$S_{a}^{{t+1}}=GB_{a}^{t} - ECB_{a}^{t} \times \varepsilon - \frac{{GB_{a}^{t} - A}}{{GB_{a}^{t}+\delta }} \times R$$

23

$ECB_{a}^{t}=GB_{a}^{t} \times \delta +\frac{{S_{a}^{t} \times mean(S)}}{{GB_{a}^{t} \times \left( {UB - LB} \right)+\delta }}$

$$S_{a}^{{t+1}}=GB_{a}^{t} \times 1.5 \times {\left( {1 - \frac{t}{{{t_{\hbox{max} }}}}} \right)^{\left( {\frac{{2t}}{{{t_{\hbox{max} }}}}} \right)}} \times R$$

24

Where ϵ represents the quality of food source and R represents a random number generated between 0 and 1.

Pseudo code of PDOA implemented in OPF problem:

Initialization:

Set line date, bus data, maximum iteration, and PDOA parameters

Initialize the populations within the boundary limits

While t < tmax

For a = 1 to A

For b = 1 to B

Calculate the fitness (PLoss, TVD, TFC and LIndex)

Find Current best solution

Update Global best solution

If t < tmax/4 then

The solution is updated by food foraging activity (Eq. 21)

Else if tmax/4 < t < tmax/2

The solution is updated by burrowing activity (Eq. 22)

Else if tmax/2 < t < 3tmax/4

The solution is updated by food alarm (Eq. 23)

Else 3tmax/4 < t < tmax

The solution is updated by antipredation alarm (Eq. 24)

End

t = t + 1

End

Return Global Best

End

In non-convex optimization problems, the Nelder-Mead algorithm (NMA) is a simplex approach to find the local minima []. NMA creates a polyhedron with n + 1 vertices in an n-dimensional search space. The simplex technique is an iterative procedure that involves comparing the objective values at the n + 1 vertices of the common simplex and shifting the simplex closer to the optimal point. The steps of NMA's iterative process include "reflection" "expansion", "contraction", and “Shrinking”. The mathematical formulation of NMA is discussed below.

Reflection phase

The Reflection phase (X^R) can be mathematically formulated as

$${X^R}={X^0}+A({X^0} - {X^U})$$

25

${X^0}=\frac{1}{m}\sum\limits_{{\begin{array}{*{20}{c}} {n=1} \\ {n \ne h} \end{array}}}^{{m+1}} {{X^n}}$

If fitness associated with X^R is better than fitness of X^V, X^U is replaced with X^R.

Expansion Phase

The Expansion Phase (X^E) can be mathematically formulated as

$${X^E}={X^0}+B({X^R} - {X^0})$$

26

If fitness associated with X^E is better than fitness of X^B, X^U is replaced with X^E.

Contraction Phase

The Contraction Phase (X^R) can be mathematically formulated as

$${X^C}=\left\{ {\begin{array}{*{20}{c}} {{X^0}+C({X^R} - {X^0})}&{if,f({X^R})>f({X^U})} \\ {{X^0}+C({X^U} - {X^0})}&{else} \end{array}} \right.$$

27

If fitness associated with X^C is better than fitness of X^R, X^C is replaced with X^R.

Shrinking Phase

The Shrinking Phase (Xⁿ) can be mathematically formulated as

$${X^n}={X^n}+\delta \left( {{X^B} - {X^n}} \right)$$

28

Where, A, B, and C represent the reflection coefficient, Expansion coefficient, and contraction coefficient respectively. Xⁿ, X^B, X^U, and X^V represent the current solution, best solution, worst solution, and second worst solution respectively.

The exploitation and exploration aspects of an algorithm need to be balanced for it to be effective. When exploitation phase dominates the exploration, the algorithm executes premature convergence. However, if the exploration phase dominates the exploitation process, the algorithm loses time searching over seamless areas, and sometimes the best answer may not be found [27]. Based on the observations on the characteristics of PDOA, it can be noticed that the exploitation phase and the exploration phase are not very well balanced in this algorithm. In view of these limitations, an intelligent hybridization approach has been designed to address complex optimization problems. In this paper PDOA is hybridized by merging the NMA which ultimately balanced the exploration and exploitation process and improves the convergence criteria of PDOA.

In this procedure, the objects are first organized in ascending order based on their fitness. Then divide all those objects into 2 groups. The first group of objects only contains the top 15 percent of objects, the second group contains the next 60 percent of objects, and the third group contains the remaining objects. The first group objects are considered as elite populations and are kept unchanged. The second group objects are iterated by PDOA and the third group objects are iterated by PDOA followed by the NMA. Afterward, if the maximum iteration is not complete, the populations of the three groups are joined together, and rearranged in ascending order and the process continues until the maximum number of repetitions is achieved. The process of NMPDOA is given in Fig. 2 and the Flowchart of proposed algorithm is given in Fig. 3.

5.1. Implementation of OPF in NMPDOA:

Step 1: Input data for NMPDOA

This step provides the information about the total number of Prairie Dog (n), the search space dimension (b), the minimum and maximum limits of each dimension and the termination criteria. The load flow data such as line data, and bus data are also provided in this step.

Step 2: Initialization of NMPDOA

The locations of objects represent the OPF problem's control variables. Initially all the control variables are chosen randomly within their predetermined limits and employed as a matrix having dimensions of the number of populations and number of control variables in the optimization process.

Step 3: Fitness Calculation

A fitness function (f_m) is used to assess each Prairie Dog fitness, and Eqs. (4,5 and 6) is used to determine the fitness values of all the objects.

Step 4: Ranking and grouping of all the Prairie Dogs

As a minimization problem, based on their fitness rating, arrange all of the objects in ascending order. Divide all objects into 3 groups. The top 15 percent objects are placed in the first group, the next 60 percent objects are placed in the second group, the rest are placed in the third group.

Step 5: Update the position of Prairie Dogs

The first group objects are considered as elite members among all the Prairie Dogs and are kept unchanged. The second group objects are iterated by PDOA and to improve the exploitation process the third group objects are iterated by PDOA as well as the NMA.

Step 6: Limit checking

After modifying the position of each Prairie Dogs, it should be checked that the position should not exceed its limit. If a variable violates its maximum limit, select it as the maximum value, and if a variable violates its minimum limit, select it as the minimum value.

Step 7: Termination criteria

Check whether the termination requirements are satisfied. if no then go to step 2 else proceed to step 7.

Step 8: Display the result

The object with the best fitness and its corresponding control variables are declared as the result.

In this paper, the performance of PDOA in dealing with the OPF problem are investigated by solving various objective functions, and then the same objective functions are solved by using NMPDOA. NMPDOA and PDOA are implemented in IEEE-30 bus [28] and IEEE-118 [29] bus power systems and applied to Indian 62 bus system [30]. MATLAB 2022b was used as the platform for the implementation of the suggested technique, and a personal computer equipped with (processor i7, CPU 2.50 GHz- 16GB RAM and 500GB SSD) was used for the simulation. The simulated result of NMPDOA is compared with PDOA and many other evolutionary algorithms. The statistical analysis and t-Test analysis in table (4, 9, 14 and 19) confirms the performance and reliability of the suggested algorithm over PDOA.

Table 1

System data description
	IEEE-30 bus [28]	IEEE-118 bus [29]	Indan utility-62 bus [30]
No. of Generator bus	6	54	19
Total no. of bus	30	118	62
No. of branches	41	186	89
No. of tap changing transformer	4	9	11
No. of Shunt capacitors	9	14	9
No. of load buses	24	64	43
Total Active power load (MW)	283.4	4242	2908
Total Reactive power load (MVAr)	126.2	1438	1270
Total active power generation (MW)	289.211	4375.36	2976.066
Total reactive power generation (MVAr)	108.922	881.92	770
Initial active power loss (MW)	5.811	133.36	68.066
Initial reactive power loss (MVAr)	32.417	-785.11	473.299

Table 2

Control variables for all the systems
Control Variable	IEEE-30 bus [21]		IEEE-118 bus [35–36]		Indian utility-62 bus [30]
Generator Voltage (p.u)	0.95	1.1	0.95	1.1	0.9	1.1
Transformer taps	0.9	1.1	0.9	1.1	0.9	1.1
Shunt Capacitor (MW)	0	5	-	-	0	5

6.1. IEEE 30 bus System:

Case-1: Minimization of the active power loss in TL

In this scenario, both PDOA and NMPDOA are used to determine the optimized control settings in order to reduce the amount of PLoss by using Eq. (4). Randomization is an inherent aspect of meta-heuristic algorithms. Therefore, it is important for taking a sufficient number of trial runs prior to claiming any solution to be the optimum one. Hence, 50 independent test runs are conducted for each algorithm and the best result is declared as the optimum solution. In addition, the results are compared to several recently developed algorithms described in the literature and presented in Fig. 7. Table 3 shows that, as compared to PDOA, NMPDOA offers a considerable reduction in power loss. The optimized result obtained from PDOA and NMPDOA are 4.6232 MW and 4.4891 MW respectively. i.e. a 20.45% and 22.76% decrease in power loss compared to the initial loss of 5.812 MW. The convergence curve of PDOA and NMPDOA are shown in Fig. 8. Furthermore, it has been observed that the convergence criteria of NMPDOA have shown a significant improvement over PDOA. Table 4 and Table 5 shows the t-Test analysis and statistical analysis for all the 3 cases.

Case-2: Minimization of the Voltage deviation at load busses

In this scenario, PDOA and NMPDOA are employed to minimize the TVD using Eq. (5). According to Table 3, the TVD is minimized by the PDOA to a value of 0.1162 pu which is 89.019% reduction as compared to the initial condition whereas it is only 0.0921 pu for the suggested approach which is 91.2965% reduction as compared to the initial condition. In addition, the results are compared to several recently developed algorithms described in the literature and presented in Fig. 8. Table 3 shows that, as compared to PDOA, NMPDOA offers a considerable reduction in TVD. Figure 5. illustrates the convergence characteristics of NMPDOA and the original PDOA. It can be noticed a substantial difference in between convergence criterion of NMPDOA and PDOA.

Case-3: Enhancement of the Voltage stability

In this scenario, PDOA and NMPDOA are implemented in the OPF problem to Enhance LIndex by using Eq. (6). Table 3 shows that, as compared to PDOA, NMPDOA offers a considerable improvement in LIndex value. From the 50 trial runs the best result obtained for PDOA and NMPDOA are 0.0979 pu and 0.0977 pu respectively. i.e. the optimized result of PDOA and NMPDOA shows a reduction of 41.9336% and 42.0522% with respect to the initial condition of Lindex value. Figure 9. shows that NMPDOA gives better results among all the compared results. Figure 6. depicts the convergence curve of PDOA and NMPDOA. Moreover, it has been shown that NMPDOA's convergence criteria have outperformed PDOA.

Table 3

Comparison result of IEEE 30 bus system, Case 1
Control Variables	Initial	PDOA	NMPDOA	PDOA	NMPDOA	PDOA	NMPDOA
Vg1	1.05	1.1	1.1	0.9995	1.0029	1.1	1.1
Vg2	1.04	1.054	1.0948	0.9662	0.9961	1.1	1.1
Vg5	1.01	1.0743	1.0749	1.0284	1.0208	1.1	1.1
Vg8	1.01	1.0755	1.098	1.0109	1.0045	1.1	1.0995
Vg11	1.05	1.0543	1.0988	1.0475	1.0001	1.1	1.1
Vg13	1.05	1.0801	1.0999	1.0322	1.0226	1.1	1.0976
T6-9	1.078	1.0147	0.9431	1.0293	1.0125	0.9093	0.9084
T6-10	1.069	1.0004	0.9	0.9494	0.9001	0.9079	0.9026
T4-12	1.032	1.0442	0.9058	1.0041	1.0115	0.9163	0.9014
T27-28	1.068	1.0142	0.9228	0.9738	0.9715	0.9088	0.9028
Qc10	0	3.4657	4.907	3.9165	5	5	5
Qc12	0	4.5865	5	0.0318	4.6471	5	5
Qc15	0	3.9529	5	5	5	5	5
Qc17	0	3.8365	5	0.062	0.2257	5	5
Qc20	0	4.8744	4.9881	3.4781	4.9613	5	5
Qc21	0	4.9285	5	4.0261	4.7302	5	5
Qc23	0	4.7556	5	3.0092	4.9984	5	5
Qc24	0	4.9076	5	5	4.7682	5	5
Qc29	0	1.9390	2.4919	3.4875	2.8922	5	5
Ploss (MW)	5.812	4.6232	4.4891	-	-	-	-
TVD (pu)	1.0582	-	-	0.1162	0.0921	-	-
LIndex (pu)	0.1686	-	-	-	-	0.0979	0.0977
% change	0	20.45	22.76	89.019	91.2965	41.9336	42.0522

Table 4

t-Test comparison of IEEE 30 bus system
t-Test	Case 1		Case 2		Case 3
t-Test	NMPDOA	PDOA	NMPDOA	PDOA	NMPDOA	PDOA
Mean	4.642826	5.153429	0.110102	0.180635	0.097862	0.109588
Variance	0.007682	0.076023	9.31E-05	0.002125	7.75E-09	4.45E-05
Observations	50	50	50	50	50	50
Hypothesized Mean Difference	0		0		0
Df	59		53		49
t Stat	-12.4793		-10.5893		-12.427
P(T < = t) one-tail	1.7E-18		5.47E-15		4.62E-17
t Critical one-tail	1.671093		1.674116		1.676551
P(T < = t) two-tail	3.4E-18		1.09E-14		9.23E-17
t Critical two-tail	2.000995		2.005746		2.009575

Table 5

Statistical Analysis of IEEE 30 bus system
	Case 1		Case 2		Case 3
	PDOA	NMPDOA	PDOA	NMPDOA	PDOA	NMPDOA
Best Value	4.6232063	4.489164	0.1162	0.0921	0.09794	0.0977
Worst Value	5.5164	4.783	0.2536	0.1256	0.1211	0.098
Mean Value	5.1534	4.6428	0.1806	0.1101	0.1096	0.0979
SD Value	0.2757	0.0876	0.0461	0.0097	0.0067	8.8025e-05

6.2. IEEE 118 bus System:

In order to check the efficiency and robustness of PDOA and NMPDOA for higher dimension optimization problems IEEE 118 bus system is taken. All three objectives such as PLoss, TVD and LIndex are implemented in PDOA and NMPDOA for IEEE 118 bus system. The system details are discussed in Table (6,7 and 8) respectively. For all the three cases 12 independent test runs are conducted for each algorithm and the best result is declared as the optimum solution. The acquired outcomes are compared with several recently developed algorithms described in the literature and presented in Fig. (13, 14 and 15) respectively. The total PLoss obtained from PDOA and NMPDOA are 123.1289 MW and 121.1289 MW respectively. i.e. a 20.45% and 22.76% decrease in power loss compared to the initial loss of 133.36MW. The TVD obtained from PDOA and NMPDOA are 0.369 pu and 0.2361 pu respectively. The maximum Lindex obtained from PDOA and NMPDOA are 0.0484 pu and 0.04781 pu respectively. The convergence curve of PDOA and NMPDOA are shown in Fig. (10, 11 and 12) respectively. Furthermore, it has been observed that the convergence criteria of NMPDOA have shown a significant improvement over PDOA. Table 9 and Table 10 shows the t-Test analysis and statistical analysis of all the 3 cases for IEEE 118 bus system.

Table 6

Comparison result of IEEE 118 bus system, Case 1
Control Variables	PDOA	NMPDOA	Vg62	1.0346	1.0548	Vg116	1.0277	1.0310
Vg1	0.9939	1.0029	Vg65	1.0381	0.9842	T8-5	0.9810	1.0080
Vg4	1.0281	1.0298	Vg66	1.0214	1.1000	T26-25	1.0114	0.9931
Vg6	1.0589	1.0345	Vg69	1.0742	1.0756	T30-17	0.9495	0.9921
Vg8	0.9831	0.9598	Vg70	1.0272	1.0382	T38-37	0.9720	0.9553
Vg10	0.9866	0.9750	Vg72	1.0417	1.0373	T63-59	0.9932	0.9407
Vg12	1.0007	0.9966	Vg73	1.0347	1.0415	T64-61	1.0037	0.9658
Vg15	1.0265	1.0593	Vg74	1.0340	1.0284	T65-66	0.9619	1.0093
Vg18	1.0407	1.0721	Vg76	1.0513	1.0759	T68-69	1.0073	0.9122
Vg19	1.0363	1.0077	Vg77	1.0115	1.0250	T81-80	0.9745	0.9866
Vg24	1.0490	1.0425	Vg80	1.0548	1.0457	Qc5	-4.4887	-23.6566
Vg25	1.0254	1.0787	Vg85	1.0008	0.9945	Qc34	8.8822	13.1915
Vg26	1.0998	1.1000	Vg87	1.0381	1.0320	Qc37	-4.0976	-16.2814
Vg27	1.0614	1.0574	Vg89	1.0734	1.0670	Qc44	6.2667	10.0000
Vg31	1.0187	1.0166	Vg90	1.0344	1.0529	Qc45	7.3327	9.5439
Vg32	1.0585	1.0435	Vg91	1.0466	1.0634	Qc46	9.3515	2.3079
Vg34	1.0219	1.0174	Vg92	1.0507	1.0879	Qc48	6.5447	3.6196
Vg36	1.0135	1.0318	Vg99	1.0378	1.0381	Qc74	4.5270	9.0057
Vg40	1.0068	1.0014	Vg100	1.0464	1.0486	Qc79	12.3636	5.3323
Vg42	1.0033	1.0063	Vg103	1.0666	1.0294	Qc82	11.7708	7.0913
Vg46	0.9869	0.9824	Vg104	1.0418	1.0442	Qc83	6.8131	6.7586
Vg49	1.0223	1.0131	Vg105	1.0096	1.0412	Qc105	10.1476	10.7384
Vg54	1.0054	1.0233	Vg107	1.0103	1.0251	Qc107	3.6533	0.9515
Vg55	1.0020	0.9821	Vg110	1.0490	1.0488	Qc110	5.4411	0.3553
Vg56	1.0442	1.0423	Vg111	1.0374	1.0317	PLoss	123.1289	121.1829
Vg59	1.0606	1.0659	Vg112	1.0231	0.9933	% Change	7.6718	9.131
Vg61	1.0012	1.0337	Vg113	1.0131	1.0158	% Change	7.6718	9.131

Table 7

Comparison result of IEEE 118 bus system, Case 2
Control Variables	PDOA	NMPDOA	Vg61	0.9917	0.9721	Vg112	0.9743	1.0695
Vg1	0.9992	0.9976	Vg62	0.9835	0.9521	Vg113	0.9803	0.9703
Vg4	0.9932	0.9830	Vg65	0.9796	0.9891	Vg116	0.9978	0.9944
Vg6	1.0153	1.0309	Vg66	1.0353	0.9963	T8-5	0.9256	0.9067
Vg8	0.9554	0.9561	Vg69	1.0332	1.0051	T26-25	0.9640	1.0534
Vg10	0.9709	0.9505	Vg70	1.0226	1.0560	T30-17	0.9868	0.9753
Vg12	1.0207	1.0049	Vg72	1.0338	1.0021	T38-37	0.9253	0.9543
Vg15	1.0050	0.9505	Vg73	0.9783	0.9705	T63-59	0.9964	0.9989
Vg18	1.0254	0.9541	Vg74	1.0177	1.0261	T64-61	0.9574	0.9755
Vg19	1.0927	1.0894	Vg76	1.0664	1.0636	T65-66	0.9972	0.9522
Vg24	1.0315	1.0261	Vg77	1.0572	1.0303	T68-69	0.9741	0.9808
Vg25	1.0059	1.0545	Vg80	0.9656	0.9816	T81-80	0.9654	0.9640
Vg26	0.9899	1.0365	Vg85	1.0260	1.0671	Qc5	-22.9725	-17.1521
Vg27	1.0073	1.0056	Vg87	1.0272	0.9891	Qc34	10.2192	6.4187
Vg31	1.0134	1.0014	Vg89	1.0162	1.0037	Qc37	-16.0771	-14.6454
Vg32	1.0688	1.0245	Vg90	1.0292	1.0163	Qc44	8.9067	8.2615
Vg34	0.9508	0.9553	Vg91	1.0109	1.0827	Qc45	1.3411	2.7222
Vg36	1.0099	1.0258	Vg92	1.0203	0.9519	Qc46	1.2034	1.5519
Vg40	1.0024	0.9960	Vg99	1.0241	1.0855	Qc48	9.3141	7.1370
Vg42	1.0136	1.0430	Vg100	0.9960	1.0535	Qc74	4.1044	8.5504
Vg46	1.0271	1.0319	Vg103	1.0241	0.9533	Qc79	9.5238	7.7613
Vg49	0.9548	0.9515	Vg104	1.0470	0.9674	Qc82	8.8372	14.0722
Vg54	1.0382	1.0015	Vg105	0.9752	0.9638	Qc83	5.8295	9.2405
Vg55	1.0428	1.0345	Vg107	1.0265	1.0443	Qc105	13.0319	4.6005
Vg56	1.0104	1.0572	Vg110	1.0266	1.0212	Qc107	1.9641	3.0490
Vg59	0.9975	1.0065	Vg111	1.0290	0.9840	Qc110	3.0247	2.2373
						TVD	0.3691	0.2361

Table 8

Comparison result of IEEE 118 bus system, Case 3
Control Variables	PDOA	NMPDOA	Vg61	1.1	1.0174	Vg112	1.1	1.1000
Vg1	1.0195	1.0795	Vg62	1.1	1.1000	Vg113	1.1	1.1000
Vg4	1.0791	1.0990	Vg65	1.1	1.1000	Vg116	1.1	1.1000
Vg6	1.0289	1.0358	Vg66	1.1	1.1000	T8-5	1.1	1.0496
Vg8	0.9585	1.0928	Vg69	1.1	1.1000	T26-25	1.1	1.0586
Vg10	1.0231	1.1000	Vg70	1.1	1.1000	T30-17	1.1	1.0963
Vg12	1.0175	1.0818	Vg72	1.1	0.9864	T38-37	1.1	0.9882
Vg15	1.0745	1.1000	Vg73	1.1	1.0362	T63-59	1.1	1.1000
Vg18	1.0188	1.0985	Vg74	1.1	1.0922	T64-61	1.1	1.0336
Vg19	1.0604	1.0945	Vg76	1.1	1.1000	T65-66	1.1	0.9582
Vg24	1.0831	1.1000	Vg77	1.1	1.1000	T68-69	1.1	0.9099
Vg25	0.9523	1.0419	Vg80	1.1	1.1000	T81-80	1.1	1.0867
Vg26	0.9693	1.0310	Vg85	1.1	1.0140	Qc5	-34.1972	-4.3144
Vg27	0.9552	1.1000	Vg87	1.1	1.1000	Qc34	14	9.4737
Vg31	1.0710	1.0991	Vg89	1.1	1.0587	Qc37	-8.0497	-24.4000
Vg32	0.9713	1.1000	Vg90	1.1	1.0310	Qc44	10	10.0000
Vg34	1.0861	1.1000	Vg91	1.1	1.0997	Qc45	10	10.0000
Vg36	1.0942	1.1000	Vg92	1.1	1.1000	Qc46	10	9.4954
Vg40	1.0220	1.1000	Vg99	1.1	1.1000	Qc48	15	15.0000
Vg42	0.9514	1.1000	Vg100	1.1	1.1000	Qc74	9.3355	3.2511
Vg46	1.0928	1.1000	Vg103	1.1	1.0112	Qc79	18.0831	19.9917
Vg49	1.0516	1.0970	Vg104	1.1	1.0410	Qc82	20	20.0000
Vg54	0.9616	1.1000	Vg105	1.1	1.0818	Qc83	10	1.7882
Vg55	1.0883	1.1000	Vg107	1.1	1.1000	Qc105	14.2723	6.7551
Vg56	1.0669	1.1000	Vg110	1.1	0.9604	Qc107	5.1965	2.9134
Vg59	1.0899	1.0721	Vg111	1.1	1.0247	Qc110	5.2686	3.6124
						Lindex	0.0484	0.04766

Table 9

t-Test comparison of IEEE 118 bus system
t-Test	Case 1		Case 2		Case 3
t-Test	NMPDOA	PDOA	NMPDOA	PDOA	NMPDOA	PDOA
Mean	123.3456	125.2283	0.334828	0.540219	0.048893	0.050519
Variance	2.4557	1.724264	0.004015	0.004253	4.22E-07	1.24E-06
Observations	20	20	20	20	20	20
Hypothesized Mean Difference	0		0		0
Df	37		38		31
t Stat	-4.11811		-10.1021		-5.63734
P(T < = t) one-tail	0.000103		1.29E-12		1.73E-06
t Critical one-tail	1.687094		1.685954		1.695519
P(T < = t) two-tail	0.000206		2.57E-12		3.46E-06
t Critical two-tail	2.026192		2.024394		2.039513

Table 10

Statistical Analysis of IEEE 118 bus system
	Case 1		Case 2		Case 3
	PDOA	NMPDOA	PDOA	NMPDOA	PDOA	NMPDOA
Best Value	123.1289	121.1829	0.3690503	0.2361038	0.0484	0.04766
Worst Value	127.9664	125.7941	0.7031	0.4136	0.0523	0.0496
Mean Value	125.2283	123.3456	0.5402	0.3348	0.0505	0.0489
SD Value	1.5671	1.3131	0.0652	0.0634	0.0011	6.4988e-04

6.3. Indian 62 bus system:

In this paper we have developed an algorithm namely NMPDOA and implemented in all the three objectives of OPF problem for IEEE 30 bus and 118 bus test system. According on the prior result analysis, the suggested method outperforms the original PDOA and other previously published algorithms. We also implemented NMPDOA in 20 benchmark functions and compared with original PDOA and six previously published algorithm. Among them, NMPDOA outperforms all other algorithm for 16 benchmark functions. The Benchmark functions and their results are provided in Appendix A. Therefore, we have implemented PDOA and NMPDOA in Indian 62 bus system for all the three cases. The control variables are provided in Table (11, 12 and 13) respectively. For all the three cases 30 independent test runs are conducted for each algorithm and the best result is declared as the optimum solution. The total PLoss obtained from PDOA and NMPDOA are 66.94797 MW and 64.45826 MW respectively. i.e. a 1.6426% and 5.3004% decrease in power loss compared to the initial loss of 68.066 MW [39]. The TVD obtained from PDOA and NMPDOA are 0.5049 pu and 0.3883 pu respectively. The maximum Lindex obtained from PDOA and NMPDOA are 0.1632 pu and 0.1554 pu respectively. Figure (16, 17 and 18) illustrates the convergence characteristics of NMPDOA and the original PDOA for case 1, 2 and 3 respectively. It can be noticed a substantial difference in between convergence criterion of NMPDOA and PDOA. Table 14 and Table 15 shows the t-Test analysis and statistical analysis of all the 3 cases for Indian 62 bus system.

Table 11

Comparison result of Indian 62 bus system, Case 1
Control Variable	PDOA	NMPDOA	V50	1.1	1.1	T82	1.0210	1.0051
V1	1.1	1.1	V51	1.1	1.1	T83	1.0071	0.9996
V2	1.1	1.1	V52	1.1	1.1	T85	1.0031	0.9873
V5	1.1	1.1	V54	1.1	1.095473	Q12	2.4811	2.0397
V9	1.0909	1.0917	V57	1.1	1.1	Q13	3.2228	2.6347
V14	1.1	1.1	V58	1.1	1.096517	Q15	3.3156	0.2960
V17	1.0562	1.1	T3	0.93	0.90287	Q17	4.7314	4.0371
V23	1.0727	1.1	T11	1.0234	1.007386	Q20	4.5906	3.1390
V25	1.1	1.1	T12	0.9779	0.973991	Q21	1.2186	0.3757
V32	1.0743	1.0747	T13	1.1	1.1	Q23	2.2951	0.4248
V33	1.0791	1.0993	T14	1.0588	1.070763	Q24	1.2686	1.1440
V34	1.1	1.094	T37	0.9313	0.918953	Q29	2.5309	3.0862
V37	1.0822	1.1	T38	0.9929	0.974227	PLoss	66.94797	64.45826
V49	1.1	1.1	T39	0.9958	0.998306	% Change	1.6426	5.3004

Table 12

Comparison result of Indian 62 bus system, Case 2
Control Variable	PDOA	NMPDOA	V50	1.038	1.0175	T82	0.9616	1.0383
V1	1.0638	1.0893	V51	1.0866	1.0433	T83	0.9201	0.9286
V2	0.9947	0.9414	V52	1.0158	1.0768	T85	1.0038	1.0847
V5	0.9272	0.989	V54	1.0563	1.1	Q12	1.7742	2.9693
V9	0.9067	0.9635	V57	0.9435	1.0552	Q13	4.2975	3.2765
V14	1.0489	1.0409	V58	0.9781	0.9426	Q15	2.3251	0.507
V17	0.9471	0.9083	T3	0.9914	0.9329	Q17	1.9242	3.273
V23	1.0863	1.0531	T11	1.0236	1.0155	Q20	4.0324	4.4067
V25	1.0195	1.0423	T12	0.9877	0.903	Q21	2.6763	0.8991
V32	1.0564	1.0143	T13	1.0682	1.0917	Q23	1.8038	0.904
V33	0.9581	0.9066	T14	1.0276	1.0325	Q24	4.9143	4.4279
V34	1.0478	0.9944	T37	0.9136	0.9186	Q29	2.8123	2.0921
V37	1.0423	1.0778	T38	1.0087	1.0033	TVD	0.5049	0.3883
V49	1.0578	1.0457	T39	1.0114	1.0064

Table 13

Comparison result of Indian 62 bus system, Case 3
Control Variable	PDOA	NMPDOA	V50	1.1	1.087	T82	1.0993	1.0527
V1	1.1	1.1	V51	1.0965	0.9394	T83	1.0947	1.0562
V2	1.1	1.0962	V52	1.0948	1.0223	T85	1.0954	0.9099
V5	1.1	1.1	V54	1.0986	1.0015	Q12	3.3017	0
V9	1.0846	1.1	V57	1.1	0.9	Q13	2.4568	0
V14	1.1	1.1	V58	1.0996	1.0073	Q15	3.2009	3.4862
V17	1.1	1.0979	T3	0.9495	0.9017	Q17	0.9284	5
V23	1.0951	1.0168	T11	1.0988	1.0632	Q20	1.1867	1.7818
V25	1.1	0.9008	T12	0.9834	0.9001	Q21	3.3376	3.3934
V32	1.1	1.0603	T13	1.0999	0.9016	Q23	0.7002	0
V33	0.9832	1.0806	T14	1.0989	0.9743	Q24	2.671	3.8723
V34	1.0412	1.004	T37	0.9153	0.9052	Q29	4.2109	4.8462
V37	1.0459	1.1	T38	1.0887	0.9533	TVD	0.1632	0.1554
V49	1.0926	1.0475	T39	1.1	1.1

Table 14

t-Test comparison of Indian 62 bus system
t-Test	Case 1		Case 2		Case 3
t-Test	NMPDOA	PDOA	NMPDOA	PDOA	NMPDOA	PDOA
Mean	65.8659	70.4798	0.41787	0.582807	0.15861	0.17058
Variance	0.815126	5.257373	0.000348	0.002078	2.61E-06	2.24E-05
Observations	30	30	30	30	30	30
Hypothesized Mean Difference	0		0		0
Df	38		38		36
t Stat	-11.2709		-18.3413		-13.1157
P(T < = t) one-tail	5.54E-14		8.97E-21		1.4E-15
t Critical one-tail	1.685954		1.685954		1.688298
P(T < = t) two-tail	1.11E-13		1.79E-20		2.8E-15
t Critical two-tail	2.024394		2.024394		2.028094

Table 15

Statistical Analysis of Indian 62 bus system
	Case 1		Case 2		Case 3
	PDOA	NMPDOA	PDOA	NMPDOA	PDOA	NMPDOA
Best Value	66.94797	64.45826	0.5049	0.3883	0.1632	0.1554
Worst Value	76	67.4903	0.6437	0.438	0.1787	0.1617
Mean Value	70.4798	65.8659	0.5828	0.4178	0.1706	0.1586
SD Value	2.1837	0.9028	0.0456	0.0128	0.0047	0.0016

6.4 Multi-Objective Optimal Power Flow (MO-OPF) for IEEE 30 bus system

In this scenario, MO-OPF problems such as the combination of (PLoss – TFC) and (TFC – TE) for IEEE 30 bus standard system are implemented in PDOA and NMPDOA. For the case study 50 independent trial runs are conducted for both PDOA as well as NMPOA and the best result are declared as the optimum solution. Figures (21, 22, 23 and 24) displays the POS of the objective function. The optimized results and the control variables are discussed in Table (16 and 17). The comparison results of individual TFC and TE are presented in figure (19 and 20). The TFC obtained from PDOA and NMPDOA are 800.379 and 798.9149 respectively. The TE obtained from PDOA and NMPDOA are 0.2241 and 0.2219 respectively. The obtained results of PLoss – TFC and TFC – TE are compared with several previously published algorithm and presented in Table (17 and 18). It can be observed that the NMPDOA performs better than all the compared algorithm as well as the original PDOA. Table (19 and 20) shows the t-Test analysis and statistical analysis of TFC and TE for IEEE 30 bus system. For both PLoss- TFC and TFC- TE case 50 independent test runs are conducted for each algorithm and the best result is declared as the optimum solution.

Table 16

Comparison result of PLoss – TFC for IEEE 30 bus system
Control Variables	PDOA			NMPDOA
Control Variables	Min TFC	Comp. Sol	Min PLoss	Min TFC	Comp. Sol	Min PLoss
P_G1	159.6315	110.49021	92.81007	176.8334	109.6457	58.07628
P_G2	48.70484	52.00297	32.01948	48.83915	54.00486	73.65432
P_G5	20.91306	35.77005	50	21.30423	35.23194	50
P_G8	21.54655	35	35	21.17876	35	35
P_G11	11.45533	29.89968	29.99698	11.91499	30	30
P_G13	12.30077	25.04469	39.91807	12.01741	24.05598	40
V_g1	1.1	1.083926	1.1	1.1	1.1	1.1
V_g2	1.079851	1.028416	1.061807	1.087782	1.05221	1.057213
V_g5	1.040769	1.04007	1.075876	1.061569	1.071709	1.080123
V_g8	1.046992	1.045907	1.097443	1.069093	1.090522	1.087036
V_g11	1.058395	1.078722	1.085412	1.1	1.1	1.1
V_g13	1.06618	1.076594	1.1	1.1	1.1	1.1
T6-9	0.997092	0.991775	1.001038	1.034926	1.030642	1.035507
T6-10	1.014084	1.018777	0.955423	0.909295	0.914144	0.905001
T4-12	1.038481	1.029891	1.001483	0.989106	0.972323	0.968831
T27-28	0.998871	0.998218	1.000862	0.968934	0.96943	0.968669
Q10	4.171702	4.557924	4.264024	4.813152	4.518549	4.992269
Q12	3.589751	2.459172	3.172785	4.976918	4.93186	5
Q15	2.783272	3.338192	4.751528	5	4.97934	4.963206
Q17	4.000708	3.612191	2.710479	5	4.931062	4.94697
Q20	2.553074	2.651535	3.380364	4.443657	4.838273	4.657675
Q21	3.430683	4.135565	3.735755	4.969575	4.316941	4.960073
Q23	3.679326	4.70699	3.944272	4.971975	4.993176	4.997916
Q24	4.226332	4.173613	4.22924	5	4.932259	4.925023
Q29	2.295611	2.19335	2.330911	1.908023	1.967684	1.99097
Cost	800.379	848.2111	1247.362	798.9149	845.7689	954.2502
Ploss (MW)	8.848	4.8076	3.6554	8.6879	4.5385	3.3306

Table 17

Comparison result of TFC – TE for IEEE 30 bus system
Control Variables	PDOA			NMPDOA
Control Variables	Min TFC	Comp. Sol	Min TE	Min TFC	Comp. Sol	Min TE
P_G1	176.8869	92.2596	65.0484	176.487	93.3921	63.1972
P_G2	49.03512	64.3825	71.2688	48.9404	62.8952	68.0103
P_G5	21.49693	34.1902	49.8573	21.31156	31.223	49.9573
P_G8	20.70104	33.1297	34.9982	21.24269	34.1197	34.9999
P_G11	12.20367	29.1726	29.1398	11.97925	29.7921	31.3856
P_G13	12	36.098	37.3713	12.00046	36.0344	39.997
Cost	0.3665	0.2241	0.2063	0.3661	0.2219	0.2048
Ploss (MW)	800.1455	867.344	944.3754	798.9246	863.6201	949.964

Table 18

Comparison result of PLoss – TFC for IEEE 30 bus system
Algorithm	TFC	PLoss	Algorithm	TFC	TE
EGA–DQLF [40]	822.87	5.613	MOMICA [41]	865.066	0.2221
PSO [40]	847.011	5.6658	NKEA [41]	865.0815	0.2228
MOMICA [41]	848.0544	4.5603	BB-MOPSO [41]	865.0985	0.2227
NKEA [41]	848.68	4.5615	MNSGA-II [41]	865.5264	0.2228
BB-MOPSO [41]	849.7496	4.5607	MOICA [41]	865.5264	0.2246
MNSGA-II [41]	849.8742	4.5608	MA [44]	834	0.2499
MOICA [41]	850.9001	4.5625	SFLA [45]	872.8533	0.2249
HAS [42]	823.89	5.77	GA [45]	872.9601	0.227
DE [43]	820.85	6.13	PSO [45]	872.8731	0.2253
MA [44]	848.6486	5.1847	MSLFA [46]	867.713	0.2247
PDOA	848.2111	4.8076	PDOA	867.344	0.2241
NMPDOA	845.7689	4.5385	NMPDOA	863.6201	0.2219

Table 19

t-Test comparison of Indian 62 bus system
t-Test	TFC		TE
t-Test	NMPDOA	PDOA	NMPDOA	PDOA
Mean	802.5898	808.1063	800.689	807.3746
Variance	5.565939	17.96655	1.381069	14.90408
Observations	50	50	50	50
Hypothesized Mean Difference	0		0
Df	77		58
t Stat	-8.04107		-11.7147
P(T < = t) one-tail	4.24E-12		3.19E-17
t Critical one-tail	1.664885		1.671553
P(T < = t) two-tail	8.48E-12		6.39E-17
t Critical two-tail	1.991254		2.001717

Table 20

Statistical Analysis of Indian 62 bus system
	Case 1		Case 2
	PDOA	NMPDOA	PDOA	NMPDOA
Best Value	800.379	798.9149	800.3710	798.9104
Worst Value	815.7172	805.8621	813.4611	802.9371
Mean Value	808.1063	802.5898	807.3746	800.6890
SD Value	4.2387	2.3592	3.8606	1.1752

In this paper, the effectiveness of a recently developed metaheuristic algorithm namely PDOA in solving OPF problem is investigated. PDOA has superior convergence criteria in compared to a number of previously reported algorithms, but it has not been able to beat a number of recently developed algorithms. In this paper, the Nelder–Mead based Prairie Dog Optimization Algorithm (NMPDOA) is outlined. In the suggested approach, NMA for avoiding the premature convergence tendency of the original version of PDOA. The proposed algorithm is implemented in OPF problems such as real power loss minimization, voltage deviation minimization and improve voltage stability and tested on IEEE 30 bus and 118 bus test system and applied to Indian 62 bus system. The proposed algorithm is also applied to solve MO-OPF problems such as PLoss-TFC and TFC-TE for IEEE 30 bus system. The results of NMPDOA were compared with several previously published algorithms. It has been observed that the NMPDOA maintained lower mean and standard deviation across all the investigated scenarios. The t-test and statistical analysis confirms the efficacy and robustness of NMPDOA over PDOA. Furthermore, several widely-used benchmark functions are also implemented in NMPDOA to confirm its complex optimization problem handling capability over PDOA.

Funding Statement: The authors have not disclosed any funding.

Ethical Statement: Ethical approval is not applicable to this article.

Conflict of interest (COI) Statement: The authors declare that there is no conflict of interest regarding the publication of this paper.

Data availability Statement: Data will be made available on request.

Basu, M., 2011. Economic environmental dispatch using multi-objective differential evolution. Applied soft computing, 11(2), pp.2845-2853.
Pulluri, H., Naresh, R. and Sharma, V., 2018. A solution network based on stud krill herd algorithm for optimal power flow problems. Soft Computing, 22, pp.159-176.
Bjelogrlic, M., Calovic, M.S., Ristanovic, P. and Babic, B.S., 1990. Application of Newton's optimal power flow in voltage/reactive power control. IEEE transactions on power systems, 5(4), pp.1447-1454.
Granville, S., 1994. Optimal reactive dispatch through interior point methods. IEEE Transactions on power systems, 9(1), pp.136-146.
Deeb, N. and Shahidehpour, S.M., 1990. Linear reactive power optimization in a large power network using the decomposition approach. IEEE Transactions on power systems, 5(2), pp.428-438.
Sarkheyli, A., Zain, A.M. and Sharif, S., 2015. The role of basic, modified and hybrid shuffled frog leaping algorithm on optimization problems: a review. Soft Computing, 19, pp.2011-2038.
Mahadevan, K. and Kannan, P.S., 2010. Comprehensive learning particle swarm optimization for reactive power dispatch. Applied soft computing, 10(2), pp.641-652.
Iba, K., 1994. Reactive power optimization by genetic algorithm. IEEE Transactions on power systems, 9(2), pp.685-692.
Karaboga, D. and Akay, B., 2009. A comparative study of artificial bee colony algorithm. Applied mathematics and computation, 214(1), pp.108-132.
Khazali, A.H. and Kalantar, M., 2011. Optimal reactive power dispatch based on harmony search algorithm. International Journal of Electrical Power & Energy Systems, 33(3), pp.684-692.
Duman, S., Sönmez, Y.U.S.U.F., Güvenç, U. and Yörükeren, N., 2012. Optimal reactive power dispatch using a gravitational search algorithm. IET generation, transmission & distribution, 6(6), pp.563-576.
Rajan, A. and Malakar, T., 2016. Exchange market algorithm based optimum reactive power dispatch. Applied Soft Computing, 43, pp.320-336.
Abd El-sattar, S., Kamel, S., Ebeed, M. and Jurado, F., 2021. An improved version of salp swarm algorithm for solving optimal power flow problem. Soft Computing, 25, pp.4027-4052.
Ghasemi, M., Akbari, E., Faraji Davoudkhani, I., Rahimnejad, A., Asadpoor, M.B. and Gadsden, S.A., 2022. Application of Coulomb’s and Franklin’s laws algorithm to solve large-scale optimal reactive power dispatch problems. Soft Computing, 26(24), pp.13899-13923.
Mandal, B. and Roy, P.K., 2013. Optimal reactive power dispatch using quasi-oppositional teaching learning based optimization. International Journal of Electrical Power & Energy Systems, 53, pp.123-134.
Chen, G., Liu, L., Zhang, Z. and Huang, S., 2017. Optimal reactive power dispatch by improved GSA-based algorithm with the novel strategies to handle constraints. Applied soft computing, 50, pp.58-70.
Rajan, A. and Malakar, T., 2015. Optimal reactive power dispatch using hybrid Nelder–Mead simplex based firefly algorithm. International Journal of Electrical Power & Energy Systems, 66, pp.9-24.
Shaw, B., Mukherjee, V. and Ghoshal, S.P., 2014. Solution of reactive power dispatch of power systems by an opposition-based gravitational search algorithm. International Journal of Electrical Power & Energy Systems, 55, pp.29-40.
Shabanpour-Haghighi, A., Seifi, A.R. and Niknam, T., 2014. A modified teaching–learning based optimization for multi-objective optimal power flow problem. Energy Conversion and Management, 77, pp.597-607.
Ezugwu, A.E., Agushaka, J.O., Abualigah, L., Mirjalili, S. and Gandomi, A.H., 2022. Prairie dog optimization algorithm. Neural Computing and Applications, 34(22), pp.20017-20065.
Rajan, A., Jeevan, K. and Malakar, T., 2017. Weighted elitism based Ant Lion Optimizer to solve optimum VAr planning problem. Applied soft computing, 55, pp.352-370.
Halder, S., Dora, B.K. and Bhat, S., 2022. An Enhanced Pathfinder Algorithm based MCSA for rotor breakage detection of induction motor. Journal of Computational Science, 64, p.101870.
Nelder, J.A. and Mead, R., 1965. A simplex method for function minimization. The computer journal, 7(4), pp.308-313.
Rajan, A. and Malakar, T., 2016. Optimum economic and emission dispatch using exchange market algorithm. International Journal of Electrical Power & Energy Systems, 82, pp.545-560.
Medina, M.A., Das, S., Coello, C.A.C. and Ramírez, J.M., 2014. Decomposition-based modern metaheuristic algorithms for multi-objective optimal power flow–A comparative study. Engineering Applications of Artificial Intelligence, 32, pp.10-20.
Biswas, P.P., Suganthan, P.N., Mallipeddi, R. and Amaratunga, G.A., 2020. Multi-objective optimal power flow solutions using a constraint handling technique of evolutionary algorithms. Soft Computing, 24, pp.2999-3023.
Nguyen, T.T. and Vo, D.N., 2020. Improved social spider optimization algorithm for optimal reactive power dispatch problem with different objectives. Neural Computing and Applications, 32(10), pp.5919-5950..
The IEEE 30-Bus Test System, http://labs.ece.uw.edu/pstca/pf30/pg_tca30bus.htm .
The IEEE 118-Bus Test System, https://labs.ece.uw.edu/pstca/pf118/pg_tca118bus.htm .
H. H. Kumar and R. Mageshvaran, “Load Flow Analysis and Optimal allocation of DG for Indian Utility 62 Bus Power System,” International Journal on Emerging Technologies, vol. 11, no. 2, pp. 874–886, 2020.
Dora, B.K., Bhat, S., Halder, S. and Sahoo, M., 2023, January. Solution of Reactive Power Dispatch problems using Enhanced Dwarf Mongoose Optimization Algorithm. In 2023 International Conference for Advancement in Technology (ICONAT) (pp. 1-6). IEEE.
Ghasemi, A., Valipour, K. and Tohidi, A., 2014. Multi objective optimal reactive power dispatch using a new multi objective strategy. International Journal of Electrical Power & Energy Systems, 57, pp.318-334.
Mouassa, S., Bouktir, T. and Salhi, A., 2017. Ant lion optimizer for solving optimal reactive power dispatch problem in power systems. Engineering science and technology, an international journal, 20(3), pp.885-895.
Sulaiman, M.H., Mustaffa, Z., Mohamed, M.R. and Aliman, O., 2015. Using the gray wolf optimizer for solving optimal reactive power dispatch problem. Applied Soft Computing, 32, pp.286-292.
A. Rajan, and T. Malakar, “ Optimal Active and Reactive power dispatch using hybrid Evolutionary Algorithm,” Master’s Thesis, National Institute of Technology Silchar, 2013.
Singh, R. P., Mukherjee, V., & Ghoshal, S. P. (2015). Optimal reactive power dispatch by particle swarm optimization with an aging leader and challengers. Applied Soft Computing, 29, 298-309.
Mugemanyi, S., Qu, Z., Rugema, F.X., Dong, Y., Bananeza, C. and Wang, L., 2020. Optimal reactive power dispatch using chaotic bat algorithm. IEEE Access, 8, pp.65830-65867.
Roy, P.K., Mandal, B. and Bhattacharya, K., 2012. Gravitational search algorithm based optimal reactive power dispatch for voltage stability enhancement. Electric Power Components and Systems, 40(9), pp.956-976.
Gnanadass, R., Padhy, N.P. and Manivannan, K., 2004. Assessment of available transfer capability for practical power systems with combined economic emission dispatch. Electric Power Systems Research, 69(2-3), pp.267-276.
Kumari, M.S. and Maheswarapu, S., 2010. Enhanced genetic algorithm based computation technique for multi-objective optimal power flow solution. International Journal of Electrical Power & Energy Systems, 32(6), pp.736-742.
Ghasemi, M., Ghavidel, S., Ghanbarian, M.M., Gharibzadeh, M. and Vahed, A.A., 2014. Multi-objective optimal power flow considering the cost, emission, voltage deviation and power losses using multi-objective modified imperialist competitive algorithm. Energy, 78, pp.276-289.
Sivasubramani, S. and Swarup, K.S., 2011. Multi-objective harmony search algorithm for optimal power flow problem. International Journal of Electrical Power & Energy Systems, 33(3), pp.745-752.
El-Fergany, A.A. and Hasanien, H.M., 2015. Single and multi-objective optimal power flow using grey wolf optimizer and differential evolution algorithms. Electric Power Components and Systems, 43(13), pp.1548-1559.
Khamees, A.K., Abdelaziz, A.Y., Eskaros, M.R., Alhelou, H.H. and Attia, M.A., 2021. Stochastic Modeling for Wind Energy and Multi-Objective Optimal Power Flow by Novel Meta-Heuristic Method. IEEE Access, 9, pp.158353-158366.
Yuan, X., Zhang, B., Wang, P., Liang, J., Yuan, Y., Huang, Y. and Lei, X., 2017. Multi-objective optimal power flow based on improved strength Pareto evolutionary algorithm. Energy, 122, pp.70-82.
Niknam, T., rasoul Narimani, M., Jabbari, M. and Malekpour, A.R., 2011. A modified shuffle frog leaping algorithm for multi-objective optimal power flow. Energy, 36(11), pp.6420-6432.
Naderi, E., Pourakbari-Kasmaei, M. and Abdi, H., 2019. An efficient particle swarm optimization algorithm to solve optimal power flow problem integrated with FACTS devices. Applied Soft Computing, 80, pp.243-262.
Adaryani, M.R. and Karami, A., 2013. Artificial bee colony algorithm for solving multi-objective optimal power flow problem. International Journal of Electrical Power & Energy Systems, 53, pp.219-230.
Niknam, T., Narimani, M.R., Aghaei, J., Tabatabaei, S. and Nayeripour, M., 2011. Modified honey bee mating optimisation to solve dynamic optimal power flow considering generator constraints. IET Generation, Transmission & Distribution, 5(10), pp.989-1002.
Shabanpour-Haghighi, A., Seifi, A.R. and Niknam, T., 2014. A modified teaching–learning based optimization for multi-objective optimal power flow problem. Energy Conversion and Management, 77, pp.597-607.
Daryani, N., Hagh, M.T. and Teimourzadeh, S., 2016. Adaptive group search optimization algorithm for multi-objective optimal power flow problem. Applied Soft Computing, 38, pp.1012-1024.

AppendixAB.docx

Multi-Objective Optimal Power Flow Problem Using Nelder–Mead based Prairie Dog Optimization Algorithm

Status:

Version 1

Abstract

Figures

1. Introduction

2. Problem Formulation

2.1. Objective functions:

2.2. Multi-Objective Optimal power flow

2.3. Constraints:

2.3.1. Equality Constraints:

2.3.2. Inequality Constraints (ICs)

2.4. Constraints Handling

2.4.1 Handling Equality Constraints (ECs)

2.4.2 Handling inequality constraints (ICs)

3. Prairie Dog Optimization Algorithm (PDOA)

4. Nelder–Mead Algorithm

5. Proposed algorithm

5.1. Implementation of OPF in NMPDOA:

6. Result Analysis

6.1. IEEE 30 bus System:

6.2. IEEE 118 bus System:

6.3. Indian 62 bus system:

6.4 Multi-Objective Optimal Power Flow (MO-OPF) for IEEE 30 bus system

7. Conclusion

Declarations

References

Supplementary Files

Status:

Version 1