# Enhanced Kepler Optimization Method for Nonlinear Multi-Dimensional Optimal Power Flow

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Research Gap

#### 1.2. Novelties and Contributions of the Article

- ▪
- The traditional KOT was integrated with enhanced local escaping mechanisms to improve convergence speed and solution quality.
- ▪
- The enhancement of the KOT, called the proposed IKOT, was developed as an innovative approach to solving MDOPF problems.
- ▪
- Several objective functions of quadratic FC, FC considering the valve point, ENL, and EE were addressed.
- ▪
- Experimental validations of the IKOT algorithm on two standard IEEE 30- and 57-bus test systems demonstrate its superior performance compared to state-of-the-art algorithms in terms of solution quality, convergence speed, and robustness.

## 2. Problem Formulation

#### 2.1. Problem Objectives

#### 2.1.1. Fuel Generation Costs (FC)

_{1}, Pg

_{2}, …, Pg

_{Ngt}illustrates the generators output power in MW; c

_{v}, b

_{v}, and a

_{v}define the generator v cost coefficients.

_{v}and f

_{v}manifest the sinusoidal cost parameters of generator v. $Pg{t}_{v}^{\mathrm{min}}$ and $Pg{t}_{v}^{}$ clarify minimum limit and the output power of each generator v, respectively.

#### 2.1.2. Entire Network Losses (ENL)

_{vj}signifies conductance of each line between bus v and j; V and θ determine the Voltage and phase angle; N

_{bs}illustrates the buses’ number.

#### 2.1.3. Entire Emissions (EE)

_{v}, α

_{v}, and ξ

_{v}characterize the emission coefficients of generator v.

#### 2.2. Equality Constraints

_{1}, Qc

_{2}, …, Qc

_{Nq}illustrate MVAr of switched capacitors/reactors; B

_{vj}demonastrates mutual susceptance between buses v and j

#### 2.3. Inequality Constraints

- (i)
- Generator Constraints

_{1}, Qgt

_{2}, …, Qgt

_{Ngt}illustrate generators reactive power in MVAr; Vgt

_{1}, Vgt

_{2}, …, Vgt

_{Ngt}shows the generator voltages

- (ii)
- Transformer Constraints

_{1}, Tapc

_{2}, …, Tapc

_{Ntapc}Transformer tap settings

- (iii)
- Security Constraints

_{v}, …, VL

_{NPQ}manifest the Voltages at load buses; SF

_{1}, …, SF

_{NF}MVA characterize the loadings over transmission lines; Nq, denotes the number of VAr devices, NF illustrates Number of system lines.

## 3. Proposed IKOT for MDOPF Issue

#### 3.1. Standard KOT

_{p}planets is going to be assigned at random in Dim dimensions, reflecting the decision variables of an optimization issue, according to the underlying equation:

_{i,j}is a solution vector representing each planet containing the control variables with dimension Dim, while r is a randomized produced integer that ranges from 0 to 1. The initialization of the KOT is executed in a random way like other metaheuristics [55]. The constraints of the optimization process is defined through X

_{j,low}and X

_{j,up}, which are the lower and upper limits of each control variable (j), accordingly.

_{i}(t) describes the velocity of the i

^{th}object at time t; X

_{i}represents the i

^{th}object position; the symbol → that appears on the head of any variable indicates a vector form; U, U

_{1}, and U

_{2}tend to be integers that are selected at random from the set of numbers {0, 1}; F is an integer number randomly selected belongs to the set {−1, 1}; r

_{1}, r

_{2}, r

_{3}, r

_{4}, and r

_{5}are random uniformly distributed values within the bounds of [0, 1]; ε is a tiny number used for avoiding a divide-by-zero mistake; the masses of X

_{s}and X

_{i}are represented by Ms and m

_{i}, respectively; X

_{a}and X

_{b}belong to options drawn at random from the entire population; μ(t) signifies the universal gravitational constant; R

_{i}(t) reflects the distance at any time t between each object X

_{i}and the sun X

_{s}; and a

_{i}denotes the semimajor axis of the object i elliptical orbit at time t, and it is determined by Kepler’s third law, which is described in Equation (20), in the following manner:

_{i}illustrates the orbital interval of each i

^{th}object i, which is represented by an absolute value. R

_{i−norm}(t) denotes normalizing the Euclidian distance between X

_{s}and X

_{i}and can be described in the following way [51]:

_{i}(t + 1) represents the newly discovered location at time t + 1 of for any planet i, X

_{s}(t) provides the sun position regarding the determined best solution, and F acts as a flag employed to alter the searching directions. Fg

_{i}denotes the attracting gravitation force between the sun X

_{s}and any planet X

_{i}, as follows:

_{i}is a number between 0 and 1 and represents the eccentricity of an orbiting planet that was introduced to give KOT a stochastic quality; and Mn

_{s}and mn

_{i}signify the normalized values of Ms and m

_{i}, that describe the masses of Xs and Xi, respectively, and are provided by Equations (24) and (25), correspondingly [56]:

_{i}indicates the normalized value of R

_{i}that reflects the Euclidian distance as follows [56]:

_{2}is a number chosen at random from 0 to 1 in order to diverge the masses of different planets. μ(t) is a function that, in order to regulate searching precision, exponentially declines with time (t) and is described in the following manner [56]:

_{o}represents the starting value, t has become the present iteration number, and T

_{max}belongs to the total iterations’ number.

_{2}represents a cyclical control parameter which slowly decreases from one to two for T cycles over the course of the optimization procedure, as specified as follows:

- Initialization:
- ▪
- The algorithm begins by setting the initial population size (Np), which determines the number of candidate solutions to explore concurrently.
- ▪
- It also receives initial values for various parameters, including:
- ▪
- Xiou, Xop, Ho: These likely represent lower and upper bounds for the search space, along with a weighting factor (Ho) that might influence the exploration process.
- ▪
- y and T: The purpose of these variables is not explicitly defined in the figure.

- Fitness Evaluation:

- Identify the Best Solution:
- ▪
- An initial best solution is chosen, denoted by Xi(t = 1). This might be the solution with the fittest score from the initial population.
- ▪
- A variable i is set to 1, which serves as a counter for iterations.

- Selection and Random Vector Generation:
- ▪
- The algorithm selects two random solutions (Xa and Xb) from the current population.
- ▪
- It then generates a random vector using Equation (29), which is not provided in the image. This random vector might be injected into the population to introduce diversity and prevent stagnation.

- Evaluate Various Components:
- ▪
- Several calculations are performed to assess different aspects of the potential solutions:
- ▪
- Equations (17)–(19): These equations, not shown in the image, likely calculate terms related to the economic or environmental objectives being optimized. For instance, they might determine generation cost, emission levels, or a weighted combination of both.
- ▪
- Equation (23): This equation, also not shown, likely calculates a penalty function (Fg(t)) that might discourage solutions from violating certain constraints.
- ▪
- Equations (24) and (25): These equations, not shown, are likely used to compute quantities related to population diversity (Mn and mni, respectively).
- ▪
- Equation (26): This equation, not shown, calculates the normalized Euclidean distance (Rn(t)) of a solution, which might indicate how far it is from other solutions in the population.

- Update Using Weighted Mean:

- Boundary Check:

- Fitness Evaluation and Stopping Criteria:
- ▪
- After the update, the fitness of each solution in the population is re-evaluated using fit(X).
- ▪
- The iteration counter (t) is incremented by 1.
- ▪
- Two conditions are checked to determine if the algorithm should stop:
- ▪
- If i < Np (i is less than the population size), the algorithm has not evaluated all solutions in the current population and continues to the next iteration.
- ▪
- If an unspecified stopping criterion is met (likely a maximum number of iterations or a desired level of fitness improvement), the algorithm terminates.

- Update Best Solution:
- ▪
- If the stopping criteria are not met, the algorithm checks if the current best solution (Xi(t)) needs to be updated. It likely compares the fitness of the current best solution with the fittest solution found in the latest iteration.
- ▪
- If a new best solution is found, it is stored as Xi(t + 1).

- Update Xa and Repeat:
- ▪
- The algorithm updates the two selected solutions (Xa and Xb) for the next iteration. How these solutions are updated is not specified in the flowchart.
- ▪
- The process returns to step 4 and repeats until the stopping criteria are met.

- End:

_{1}}. This transfer mechanism represents approximately an equal way. Therefore, the exploitation strategy will start activation from the beginning of the iteration journey with approximately 50% of the solutions for each iteration. The equal probability of choosing the exploitation strategy from the start of the optimization process leads to premature exploitation. Premature exploitation can cause the algorithm to focus too early on local optima, hindering the thorough exploration of the search space. Effective optimization requires a balance between exploration (searching new areas) and exploitation (refining known good areas). An early and excessive focus on exploitation diminishes the algorithm’s ability to explore the search space adequately, potentially missing better solutions. The lack of a dynamic or adaptive mechanism to balance exploration and exploitation can result in suboptimal optimization performance. The algorithm may converge too quickly to suboptimal solutions due to insufficient exploration of the search space.

#### 3.2. Proposed IKOT

_{w}will be a probability factor which controls the LEO activation. In the range [0, 1], r

_{3}and r

_{4}signify randomized values; ϕ

_{1}and ϕ

_{2}imply two randomized values attained from a uniform distribution function within the set [−1; 1]; X

_{R1}and X

_{R2}are two picked solutions in a random way from the population.

_{1}and β

_{2}are two randomized number generated via Equations (32) and (33):

_{1}denotes an integer generated at random from the range [0, 1].

#### 3.3. Theoretical Explanation of IKOT

- Dynamic Balance between Exploration and Exploitation:
- ▪
- In the original KOT, the balance between exploration and exploitation is static, leading to early dominance of exploitation.
- ▪
- IKOT adjusts this balance dynamically by introducing LEO, which activates based on a probability factor Qw, thus providing a more adaptive approach to switching between exploration and exploitation.

- 2.
- Escaping Local Optima:
- ▪
- By incorporating LEO, IKOT can perturb the positions of planets more effectively, helping them escape local optima and explore new regions of the search space.

- 3.
- Key Steps of the Proposed IKOT
- Initialization: Randomly initialize the positions and velocities of planets.
- Exploration Phase: Utilize Equation (22) to update positions based on velocities.
- Exploitation Phase: Apply Equation (28) to refine positions near the best solution.
- Local Escaping Phase: Introduce LEO using Equation (31) to adjust positions and avoid local optima.
- Elitism: Ensure the best solutions are retained for the next iteration using Equation (30).

#### 3.4. The Methodology of IKOT for MDOPF

#### 3.4.1. Enhancement of IKOT for Encompassing Operational Constraints of Independent Variables

#### 3.4.2. Enhancement of IKOT for Encompassing Operational Constraints of Dependent Variables

## 4. Simulation Results

#### 4.1. Results of IEEE 30 Bus System

- ▪
- Scenario (Sc.) 1: Minimizing the FC
- ▪
- Sc. 2: Minimizing the FC considering Valve Point Effect (VpEFC)
- ▪
- Sc. 3: Minimizing the EE
- ▪
- Sc. 4: Minimizing the ENL

#### 4.1.1. Applications for Sc. 1

#### 4.1.2. VpEFC Minimizing (Sc. 2)

#### 4.1.3. EE Minimizing (Sc. 3)

_{2}/h was achieved by the proposed IKOT, while an average EE value of 0.2046853 tonCo

_{2}/h was achieved by the KOT. Similar to the previous analysis, the proposed IKOT algorithm showed slight improvements in the best EE, mean EE, and worst EE compared to the KOT algorithm. However, the most significant improvement was observed in the standard deviation, where the proposed IKOT algorithm achieved a substantial reduction of approximately 64.96%. The Wilcoxon signed rank test for this scenario is conducted in Table 9 to ensure that the IKOT is really significant compared to the KOT. It can be manifested from the table that the p-value was less than the significance level of 0.05 which means that the IKOT was really significant compared to the KOT.

#### 4.1.4. ENL Minimizing (Sc. 4)

#### 4.2. Results of IEEE 57 Bus System

- ▪
- Sc. 5: Minimizing the FC
- ▪
- Sc. 6: Minimizing the EE
- ▪
- Sc. 7: Minimizing the ENL

#### 4.3. Implications

- Reduced operational costs: By achieving lower fuel costs, the IKOT algorithm can potentially lead to significant cost savings for power generation companies. These savings can be passed on to consumers or reinvested in improving the power grid infrastructure.
- Improved environmental performance: By minimizing emissions, IKOT can contribute to reducing the environmental impact of power generation. This is becoming increasingly important as concerns about climate change grow.
- Reduced energy losses: Lower energy losses translate to a more efficient power grid. This can help to improve the overall reliability of the power system and reduce the need for additional power generation capacity.
- Faster computation times: The faster convergence characteristics of IKOT suggest that it may be computationally more efficient than conventional methods. This can be beneficial for real-time applications where quick solutions are needed.

#### 4.4. Discussion

- Performance on larger systems: The study only investigates the performance of IKOT on the IEEE 30-bus and 57-bus test systems. It is important to evaluate how the algorithm scales to handle much larger power systems encountered in real-world applications.
- Integration with existing systems: Implementing a new optimization algorithm like IKOT may require modifications to existing power system control and management systems. Further research is needed to ensure smooth integration and compatibility.
- Comparison with other algorithms: While the study mentions comparisons with other optimization algorithms, a more comprehensive benchmark including popular methods would provide a clearer picture of IKOT’s relative strengths and weaknesses.
- Practical implementation challenges: The document focuses on the algorithmic aspects of IKOT. Further research is needed to explore the practical challenges of implementing IKOT in real-world power systems, such as data availability, handling uncertainties, and robustness to system disturbances.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**IEEE 30-bus schematic diagram [59].

**Figure 12.**Convergence characteristics of the proposed IKOT and KOT for Scs. 5, 6, and 7 of the IEEE 57-bus system.

Algorithm | Application | Advantage | Disadvantage |
---|---|---|---|

A modified Rao-2 algorithm [30] | IEEE 30 and IEEE 118 MDOPF | The FC function with and without the renewable energy sources | The Entire Network Losses (ENL) and the Emissions (EE) have not been taken into consideration |

krill herd algorithm (KHA) [31] | 26-bus and IEEE 57-bus MDOPF | FC, ENL, and voltage deviation functions | EE has not taken into consideration |

Opposition-based learning whale optimization algorithm [32] | IEEE 30 MDOPF | FC function | The large system and other objectives have not been considered |

A biogeography-based optimizer [33] | IEEE 30-bus and IEEE 57-bus MDOPF | The FC function has been treated as quadratic not sinusoidal | |

An adaptive multi-team perturbation-guiding JAYA [34] | The IEEE 30 and IEEE 118 of the MDOPF | FC and ENL functions | The EE have not been taken into consideration |

A differential-based harmony search algorithm [35] | The IEEE 57 and IEEE 118 of the MDOPF | FC and ENL functions | The EE have not been taken into consideration |

Salp swarm algorithm [36] | Different IEEE systems | FC function | The ENL and EE have not been taken into consideration |

Adaptive Differential Evolution [37] | A modified IEEE 30-bus test system | FC, ENL, and EE functions | The large systems have not been considered |

Variables | Base | Sc. 1 (FC (USD/h)) | Variables | Base | Sc. 1 (FC (USD/h)) | ||
---|---|---|---|---|---|---|---|

KOT | Proposed IKOT | KOT | Proposed IKOT | ||||

Vg _{1} | 1.050 | 1.0999842 | 1.0999925 | Qc _{20} | 0.000 | 4.3512481 | 4.6076627 |

Vg _{2} | 1.040 | 1.0875161 | 1.0874407 | Qc _{21} | 0.000 | 4.9641697 | 4.9763268 |

Vg _{5} | 1.010 | 1.060882 | 1.0609243 | Qc _{23} | 0.000 | 2.9795945 | 2.9102967 |

Vg _{8} | 1.010 | 1.0686429 | 1.0689624 | Qc _{24} | 0.000 | 4.9928187 | 4.9666918 |

Vg _{11} | 1.050 | 1.0995761 | 1.0999881 | Qc _{29} | 0.000 | 2.4323656 | 2.7008637 |

Vg _{13} | 1.050 | 1.0998924 | 1.0999718 | Pg _{1} | 99.240 | 177.0435773 | 176.9447176 |

Tap _{6–9} | 1.078 | 1.0525044 | 1.0529859 | Pg _{2} | 80.000 | 48.637559 | 48.607984 |

Tap _{6–10} | 1.069 | 0.9049149 | 0.901062 | Pg _{5} | 50.000 | 21.292769 | 21.391957 |

Tap _{4–12} | 1.032 | 0.9921532 | 0.9900401 | Pg _{8} | 20.000 | 21.061161 | 21.003969 |

Tap _{28–27} | 1.068 | 0.9654594 | 0.9656767 | Pg _{11} | 20.000 | 12.065562 | 11.975528 |

Qc _{10} | 0.000 | 4.9330193 | 4.999515 | Pg _{13} | 20.000 | 12.018054 | 12 |

Qc _{12} | 0.000 | 4.8984732 | 4.92006 | Cost_Pg | 901.9600 | 799.09219 | 799.08771 |

Qc _{15} | 0.000 | 4.4012584 | 4.6575448 | Losses | 5.832400 | 8.6198222 | 8.623016 |

Qc _{17} | 0.000 | 4.7869863 | 4.8852291 |

Method | FCs (USD/h) | Method | FCs (USD/h) |
---|---|---|---|

Proposed IKOT | 799.0824 | IMFO [63] | 800.3848 |

KOT | 799.0835 | SOS [60] | 801.5733 |

MSA [66] | 800.5099 | ICA [72] | 801.843 |

NBO [74] | 799.7516 | CSO [61] | 799.8266 |

Developed GWO [64] | 800.433 | TLA [73] | 800.4212 |

GO [69] | 800.9728 | Adaptive GO [69] | 800.0212 |

JFS [76] | 799.1065 | ECHT-DE [67] | 800.4148 |

Improved EOA [65] | 799.688 | DHSA [40] | 802.2966 |

MCSO [62] | 799.3332 | GA [70] | 802.1962 |

BHBOA [40] | 799.9217 | Adaptive constraint DE [77] | 800.4113 |

pbest-DE [52] | 800.4115 | Self-adaptive penalty-DE [52] | 800.4293 |

Ensemble constraint handling-DE [52] | 800.4148 | Self-adaptive feasibility-DE [52] | 800.4131 |

Sc. 1 (FC (USD /h)) | ||
---|---|---|

KOT | Proposed IKOT | |

Best | 799.09219 | 799.08771 |

Mean | 799.09907 | 799.09261 |

Worst | 799.10668 | 799.09787 |

STD | 0.0043606 | 0.0029867 |

p-value | 2.7016 × 10^{−5} |

Variables | Base | Sc. 2 (VpEFC (USD/h)) | Variables | Base | Sc. 2 (VpEFC (USD/h)) | ||
---|---|---|---|---|---|---|---|

KOT | Proposed IKOT | KOT | Proposed IKOT | ||||

Vg _{1} | 1.050 | 1.099708 | 1.099944 | Qc _{20} | 0.000 | 4.687403 | 5 |

Vg _{2} | 1.040 | 1.07985 | 1.080144 | Qc _{21} | 0.000 | 4.997766 | 4.847008 |

Vg _{5} | 1.010 | 1.054655 | 1.055772 | Qc _{23} | 0.000 | 3.008146 | 2.069356 |

Vg _{8} | 1.010 | 1.06485 | 1.063306 | Qc _{24} | 0.000 | 4.325943 | 4.975328 |

Vg _{11} | 1.050 | 1.095728 | 1.1 | Qc _{29} | 0.000 | 2.873662 | 3.367333 |

Vg _{13} | 1.050 | 1.099848 | 1.098758 | Pg _{1} | 99.240 | 194.6822 | 194.6569 |

Tap _{6–9} | 1.078 | 1.075327 | 1.098192 | Pg _{2} | 80.000 | 48.09089 | 48.25121 |

Tap _{6–10} | 1.069 | 0.924231 | 0.900041 | Pg _{5} | 50.000 | 18.71153 | 18.61147 |

Tap _{4–12} | 1.032 | 1.031119 | 1.024952 | Pg _{8} | 20.000 | 10 | 10.00315 |

Tap _{28–27} | 1.068 | 0.983189 | 0.980161 | Pg _{11} | 20.000 | 10.02132 | 10 |

Qc _{10} | 0.000 | 4.238553 | 2.315208 | Pg _{13} | 20.000 | 12.02861 | 12.00017 |

Qc _{12} | 0.000 | 3.490664 | 5 | VpEFC | 901.9600 | 832.8738 | 832.8132 |

Qc _{15} | 0.000 | 3.288624 | 4.973036 | Losses | 5.832400 | 1.892896 | 1.979894 |

Qc _{17} | 0.000 | 2.893409 | 4.991811 | EE | 0.23909633 | 0.423181 | 0.423152 |

Sc. 2 (VpEFC (USD/h)) | ||
---|---|---|

KOT | Proposed IKOT | |

Best | 832.87384 | 832.81322 |

Mean | 832.94148 | 832.86706 |

Worst | 832.99096 | 832.915 |

STD | 0.032684 | 0.0331139 |

p-value | 1.8215 × 10^{−5} |

Variables | Base | Sc. 3 (EE (tonCo_{2}/h)) | |
---|---|---|---|

KOT | Proposed IKOT | ||

Vg _{1} | 1.050 | 1.0997405 | 1.0999696 |

Vg _{2} | 1.040 | 1.0949794 | 1.0964242 |

Vg _{5} | 1.010 | 1.0783116 | 1.0785521 |

Vg _{8} | 1.010 | 1.0849409 | 1.0859791 |

Vg _{11} | 1.050 | 1.1 | 1.0998134 |

Vg _{13} | 1.050 | 1.0998329 | 1.0998638 |

Tap _{6–9} | 1.078 | 1.0522669 | 1.0623459 |

Tap _{6–10} | 1.069 | 0.9084085 | 0.9034771 |

Tap _{4–12} | 1.032 | 0.989808 | 0.9860308 |

Tap _{28–27} | 1.068 | 0.9706455 | 0.9738471 |

Qc _{10} | 0.000 | 5 | 4.9752018 |

Qc _{12} | 0.000 | 5 | 4.9255699 |

Qc _{15} | 0.000 | 4.2479758 | 4.2423648 |

Qc _{17} | 0.000 | 5 | 4.990386 |

Qc _{20} | 0.000 | 3.9529735 | 4.3113485 |

Qc _{21} | 0.000 | 5 | 4.9913136 |

Qc _{23} | 0.000 | 2.474512 | 3.0899092 |

Qc _{24} | 0.000 | 4.975469 | 4.9414417 |

Qc _{29} | 0.000 | 2.2314195 | 2.3724372 |

Pg _{1} | 99.240 | 63.87452 | 63.93249 |

Pg _{2} | 80.000 | 67.509343 | 67.449144 |

Pg _{5} | 50.000 | 50 | 49.999916 |

Pg _{8} | 20.000 | 34.999381 | 34.999844 |

Pg _{11} | 20.000 | 30 | 29.999999 |

Pg _{13} | 20.000 | 39.999781 | 39.999944 |

Cost_Pg | 901.9600 | 943.6597 | 943.55787 |

Losses | 5.832400 | 2.9830268 | 2.9813334 |

EE | 0.23909633 | 0.2046835 | 0.2046825 |

Algorithm | EE | Algorithm | EE |
---|---|---|---|

Proposed IKOT | 0.204681894 | Adaptive GO [69] | 0.20484 |

KOT | 0.204682002 | MRFA [59] | 0.204754 |

NBO [62] | 0.2052063 | GO [69] | 0.20492 |

Stud KHA [78] | 0.2048 | modified TLA [79] | 0.20493 |

MCSO [62] | 0.2048911 | ECHT-DE [67] | 0.20482 |

KHA [78] | 0.2049 | ARCBT [33] | 0.2048 |

CSO [62] | 0.2051355 | JFS [76] | 0.204688 |

pbest-DE [52] | 0.2048 | Adaptive constraint DE [77] | 0.2048 |

ensemble constraint handling-DE [52] | 0.2048 | self-adaptive penalty-DE [52] | 0.2048 |

self-adaptive feasibility-DE [52] | 0.2048 |

Sc. 3 (EE (USD/h)) | ||
---|---|---|

KOT | Proposed IKOT | |

Best | 0.2046835 | 0.2046825 |

Mean | 0.2046853 | 0.2046832 |

Worst | 0.2046878 | 0.2046838 |

STD | 1.175 × 10^{−6} | 4.115 × 10^{−7} |

p-value | 1.2290 × 10^{−5} |

Variables | Base | Sc. 4 (ENL (MW)) | |
---|---|---|---|

KOT | Proposed IKOT | ||

Vg _{1} | 1.050 | 1.099979 | 1.099997 |

Vg _{2} | 1.040 | 1.097268 | 1.097853 |

Vg _{5} | 1.010 | 1.07991 | 1.080156 |

Vg _{8} | 1.010 | 1.08744 | 1.087058 |

Vg _{11} | 1.050 | 1.099917 | 1.099964 |

Vg _{13} | 1.050 | 1.099971 | 1.099975 |

Tap _{6–9} | 1.078 | 1.067365 | 1.069314 |

Tap _{6–10} | 1.069 | 0.900134 | 0.900322 |

Tap _{4–12} | 1.032 | 0.987111 | 0.987483 |

Tap _{28–27} | 1.068 | 0.974319 | 0.974083 |

Qc _{10} | 0.000 | 5 | 4.994113 |

Qc _{12} | 0.000 | 4.941765 | 4.999269 |

Qc _{15} | 0.000 | 4.361995 | 4.690106 |

Qc _{17} | 0.000 | 4.91106 | 4.991473 |

Qc _{20} | 0.000 | 4.488789 | 4.314588 |

Qc _{21} | 0.000 | 4.998338 | 4.99429 |

Qc _{23} | 0.000 | 2.723986 | 2.655286 |

Qc _{24} | 0.000 | 4.962214 | 4.993665 |

Qc _{29} | 0.000 | 2.347776 | 2.305081 |

Pg _{1} | 99.240 | 51.27529 | 51.25339 |

Pg _{2} | 80.000 | 79.98773 | 79.99853 |

Pg _{5} | 50.000 | 49.99908 | 49.99977 |

Pg _{8} | 20.000 | 35 | 34.99976 |

Pg _{11} | 20.000 | 29.99779 | 29.99967 |

Pg _{13} | 20.000 | 39.99223 | 39.99993 |

FC | 901.9600 | 967.0152 | 967.0631 |

ENL | 5.832400 | 2.852124 | 2.851063 |

EE | 0.23909633 | 2.612035 | 2.608253 |

Sc. 4 (ENL (USD/h)) | ||
---|---|---|

KOT | Proposed IKOT | |

Best | 2.8521238 | 2.8510628 |

Mean | 2.8529505 | 2.8514778 |

Worst | 2.8539049 | 2.8519867 |

STD | 0.0004966 | 0.0002901 |

p-value | 8.2981 × 10^{−6} |

**Table 12.**Optimum outcomes obtained via the proposed IKOT and KOT for Scs. 5, 6, and 7 of the IEEE 57-bus system.

Variables | Base | Sc. 5 (FC (USD/h)) | Sc. 6 (EE (Ton/h)) | Sc. 7 (ENL (MW)) | |||
---|---|---|---|---|---|---|---|

KOT | IKOT | KOT | IKOT | KOT | IKOT | ||

Vg _{1} | 1.010 | 1.0528589 | 1.0490981 | 1.0512586 | 1.06 | 1.0512314 | 1.0599683 |

Vg _{2} | 1.010 | 1.0505598 | 1.0465119 | 1.0436187 | 1.056154 | 1.0446953 | 1.0566525 |

Vg _{3} | 1.010 | 1.0476581 | 1.0408582 | 1.0486027 | 1.053503 | 1.0443166 | 1.06 |

Vg _{6} | 1.010 | 1.057303 | 1.0526739 | 1.0415265 | 1.047782 | 1.0443103 | 1.06 |

Vg _{8} | 1.010 | 1.0594182 | 1.0598524 | 1.0495426 | 1.059955 | 1.0491184 | 1.06 |

Vg _{9} | 1.010 | 1.0359735 | 1.0334266 | 1.0336925 | 1.03923 | 1.0273124 | 1.0412399 |

Vg _{12} | 1.010 | 1.0401997 | 1.0352904 | 1.0462459 | 1.044963 | 1.0349537 | 1.04861 |

Tap _{4–18} | 0.970 | 1.071804 | 0.9660857 | 1.1 | 0.909915 | 0.9396409 | 1.0075575 |

Tap _{4–18} | 0.978 | 1.0341457 | 1.0019481 | 0.9792202 | 1.00174 | 1.0477643 | 0.9752831 |

Tap _{21–20} | 1.043 | 1.0415915 | 1.0145196 | 0.9813358 | 0.983893 | 0.9906537 | 0.981745 |

Tap _{24–25} | 1.000 | 0.9793877 | 1.0713834 | 1.0045759 | 1.087896 | 1.0067073 | 1.0542884 |

Tap _{24–25} | 1.000 | 1.0593901 | 0.9450148 | 1.0764719 | 1.023525 | 0.9733652 | 0.9612857 |

Tap _{24–26} | 1.043 | 1.0092304 | 1.0131516 | 0.9982919 | 1.05058 | 1.0479778 | 1.014219 |

Tap _{7–29} | 0.967 | 0.9564098 | 0.9497392 | 0.927779 | 1.048719 | 1.0423613 | 0.9752843 |

Tap _{34–32} | 0.975 | 1.0263921 | 0.9704916 | 0.9382754 | 1.048395 | 0.9909048 | 0.955498 |

Tap _{11–41} | 0.955 | 0.9400717 | 0.9 | 1.0131983 | 0.940488 | 0.931761 | 0.937831 |

Tap _{15–45} | 0.955 | 0.941064 | 0.9354061 | 0.9512305 | 0.967193 | 0.9923906 | 0.9491786 |

Tap _{14–46} | 0.900 | 0.952092 | 0.9248971 | 1.0823589 | 0.989219 | 0.9856068 | 0.9501329 |

Tap _{10–51} | 0.930 | 0.9427316 | 0.9281249 | 1.0101604 | 1.013262 | 0.9876213 | 0.9570339 |

Tap _{13–49} | 0.895 | 0.9145605 | 0.9108794 | 1.00109 | 0.984496 | 0.9557236 | 0.91698 |

Tap _{11–43} | 0.958 | 0.9241284 | 0.9261256 | 0.975155 | 0.990354 | 0.9948324 | 0.9799994 |

Tap _{40–56} | 0.958 | 1.0053302 | 1.0172206 | 0.9684819 | 1.06986 | 1.0445029 | 1.044269 |

Tap _{39–57} | 0.980 | 0.9809085 | 0.9625971 | 1.0140668 | 1.035583 | 1.017883 | 1.0478388 |

Tap _{9–55} | 0.940 | 0.969669 | 0.9431252 | 1.0408582 | 1.079417 | 1.0507726 | 0.9765349 |

Qc _{18} | 10.000 | 19.588645 | 16.833535 | 24.273151 | 4.044177 | 9.6662115 | 19.372384 |

Qc _{25} | 5.900 | 15.822386 | 13.469753 | 17.154082 | 16.24767 | 13.372862 | 9.937805 |

Qc _{53} | 6.300 | 14.669189 | 10.478507 | 2.9450366 | 14.62376 | 13.726467 | 12.958884 |

Pg _{1} | 478.635 | 142.3647278 | 144.2304109 | 328.6973467 | 336.0299972 | 185.9860937 | 165.2769025 |

Pg _{2} | 0.000 | 90.880229 | 89.614122 | 100 | 100 | 29.03113 | 35.605772 |

Pg _{3} | 40.000 | 44.99367 | 44.216258 | 140 | 139.9991 | 116.90941 | 140 |

Pg _{6} | 0.000 | 78.532235 | 70.349615 | 100 | 100 | 94.841309 | 90.058955 |

Pg _{8} | 450.000 | 455.1729 | 460.81591 | 249.12933 | 259.8651 | 326.41964 | 320.08809 |

Pg _{9} | 0.000 | 91.893133 | 94.588597 | 100 | 99.9803 | 98.460077 | 100 |

Pg _{12} | 310.000 | 361.90877 | 361.80614 | 257.74474 | 238.2888 | 410 | 409.75317 |

FC (USD/h) | 51345 | 41677.349 | 41666.963 | 48615.481 | 48775.07 | 43765.77 | 44634.796 |

EE (Ton/h) | 2.528 | 2.6117209 | 3.1226054 | 1.0484899 | 1.039368 | 1.5410877 | 2.683034 |

ENL (MW) | 27.8346 | 14.945665 | 14.821043 | 24.771408 | 23.36324 | 10.847654 | 9.9828921 |

Sc. 5 (FC (USD/h)) | Sc. 6 (ENL (MW)) | Sc. 7 (EE (Ton/h)) | |
---|---|---|---|

p-value | 3.7896 × 10^{−6} | 2.5631 × 10^{−6} | 1.7344 × 10^{−6} |

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## Share and Cite

**MDPI and ACS Style**

Alqahtani, M.H.; Almutairi, S.Z.; Shaheen, A.M.; Ginidi, A.R.
Enhanced Kepler Optimization Method for Nonlinear Multi-Dimensional Optimal Power Flow. *Axioms* **2024**, *13*, 419.
https://doi.org/10.3390/axioms13070419

**AMA Style**

Alqahtani MH, Almutairi SZ, Shaheen AM, Ginidi AR.
Enhanced Kepler Optimization Method for Nonlinear Multi-Dimensional Optimal Power Flow. *Axioms*. 2024; 13(7):419.
https://doi.org/10.3390/axioms13070419

**Chicago/Turabian Style**

Alqahtani, Mohammed H., Sulaiman Z. Almutairi, Abdullah M. Shaheen, and Ahmed R. Ginidi.
2024. "Enhanced Kepler Optimization Method for Nonlinear Multi-Dimensional Optimal Power Flow" *Axioms* 13, no. 7: 419.
https://doi.org/10.3390/axioms13070419