A Novel Adaptive Manta-Ray Foraging Optimization for Stochastic ORPD Considering Uncertainties of Wind Power and Load Demand
Abstract
:1. Introduction
- We propose a developed Adaptive Manta-Ray Foraging Optimization algorithm (AMRFO) for optimizing ORPD and SORPD.
- We solve the SORPD under the uncertainties of the loading and the output power of the WTs.
- We assess the performance of the system with and without the inclusion of the WTs for the total expected power loss and the total expected VDs.
- An extensive comparison is presented between the proposed algorithm with different algorithms, including SCSO, WOA, DO, AHA, and the conventional MRFO, to verify the effectiveness of the AMRFO on a standard benchmark function and IEEE 30-bus system.
2. Problem Formulation
2.1. Objective Functions
2.1.1. The Power Loss (PLoss)
2.1.2. The Voltage Deviations (VD)
2.1.3. The Total Expected Power Losses (TEPL)
2.1.4. The Total Expected Voltage Deviations (TEVD)
2.2. The System Constraints
2.2.1. Equality Constraints
2.2.2. Inequality Constraints
3. The Uncertainty Representation
4. Optimization Algorithm
4.1. Manta Ray Foraging Optimization (MRFO)
4.1.1. Chain Foraging
4.1.2. Cyclone Foraging
4.1.3. Somersault Foraging
4.2. The Adaptive Manta-Ray Foraging Optimization (AMRFO)
4.2.1. The Fitness Distance Balance
4.2.2. The Quasi-Oppositional Based Learning (QOBL)
4.2.3. Adaptive Levy Flight (ALF)
5. Results and Discussion
5.1. Application of the AMRFO on Standard Benchmark Functions
5.2. Application the AMRFO for Solving the ORPD Problems
5.2.1. Case 1: Solving the Conventional OPRD
5.2.2. Case 2: Solving the SORPD
- ▪
- An Adaptive Manta-Ray Foraging Optimization (AMRFO) is proposed based on the quasi-oppositional based learning fitness distance balance method, which was successfully applied to solve the conventional and stochastic ORPD problem.
- ▪
- The suggested AMRFO was validated on the standard benchmark functions compared to SCSO, WOA, GWO, AHA, DO, and the conventional MRFO in terms of the best, mean, worst and p-value.
- ▪
- In the case of solving the conventional ORPD, the minimum power loss and the VD that was obtained by the application of the AMRFO were 4.5279 MW and 0.0913 p.u., respectively.
- ▪
- The stochastic ORPD problem was solved by considering the uncertainties of the load demand and wind speed.
- ▪
- The TEPL decreased from 5.1025 MW to 4.5201 MW in the case of solving the SORPD without the inclusion of WTs, while this value was reduced from 5.1025 MW to 2.9011 MW compared to the base case with the inclusion of WTs. Likewise, the TEVD decreased from 0.7919 p.u. to 0.1349 p.u. without incorporating the WTs, and this decreased from 0.7919 p.u. to 0.1249 p.u. with the inclusion of WTs compared with the base case.
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
ORPD | Optimal reactive power dispatch |
AMRFO | Adaptive Manta-Ray Foraging Optimization |
MRFO | Manta-Ray Foraging Optimization |
SORPD | Stochastic ORPD |
FDB | Fitness distance balance selection |
ALF | An adaptive Levy Flight |
QOBL | Quasi Oppositional based learning |
MCS | The Monte Carlo simulation |
PLoss | The power loss |
TEPL | The total expected PLoss |
VD | The voltage deviations |
TEVD | The total expected VD |
Probability density function | |
SCSO | Sand Cat Swarm algorithm |
GWO | Grey Wolf Optimizer |
WOA | Whale Optimization Algorithm technique |
AHA | Artificial Hummingbird Algorithm |
DO | Dandelion Optimizer |
SBR | The scenario-based reduction |
PSO | Particle Swarm Optimization |
OBL | The Opposition-based learning |
WT | Wind turbine |
TLBO | Teach learning-based optimization |
HPSO–TS | Hybrid PSO and Tabu search |
MPA | Marine predator Algorithm |
PSO-TVAC | PSO with time varying acceleration coefficients |
CLPSO | Comprehensive Learning PSO |
QOTLBO | Quasi Oppositional Teaching Learning based Optimization |
JA | Jaya Algorithm |
HSO | Harmony Search Algorithm |
SGA | Standard Genetic Algorithm |
DE | Differential Evolution |
SpGA | Specialized Genetic Algorithm |
FA | Firefly Algorithm |
ALO | Ant Lion Optimizer |
HPSO–TS | Hybrid PSO and Tabu search |
TS | Tabu search |
WOA | Whale Optimization Algorithm |
IDE | Improved Differential Evolution |
BBO | Biogeography-Based Optimization |
CLPSO | Comprehensive Learning PSO |
GSA | Gravitational Search Algorithm |
GSA-CSS | GSA With Conditional Selection Strategies |
IGSA-CSS | Improved GSA With Conditional Selection Strategies |
Appendix A. The Simulation Results for the Standard Objective Function
Fun | Algorithms | Average | Best | Worst | SD | p-Value | Mean Rank |
---|---|---|---|---|---|---|---|
F1 | SCSO | 1.7E−53 | 3.3E−61 | 3.8E−52 | 7.6E−53 | 9.73E−11 | 4 |
GWO | 6.4E−11 | 2.5E−12 | 3.31E−10 | 8.1E−11 | 9.73E−11 | 6 | |
WOA | 9.7E−33 | 4.3E−43 | 7.21E−32 | 2.1E−32 | 9.73E−11 | 5 | |
AHA | 8.1E−65 | 8.7E−82 | 1.81E−63 | 3.6E−64 | 9.73E−11 | 3 | |
DO | 8.2E−03 | 3.3E−03 | 1.51E−02 | 3.6E−03 | 9.73E−11 | 5 | |
MRFO | 1.7E−195 | 2.3E−215 | 3.41E−194 | 0.0E+00 | 9.73E−11 | 2 | |
AMRFO | 0.0 | 0.0 | 0.0 | 0.0 | - | 1 | |
F2 | SCSO | 5.51E−29 | 6.3E−34 | 4.5E−28 | 1.3E−28 | 1.37E−09 | 4 |
GWO | 3.61E−07 | 1.42E−07 | 7.71E−07 | 1.71E−07 | 1.38E−09 | 6 | |
WOA | 4.30E−23 | 2.01E−27 | 6.61E−22 | 1.41E−22 | 1.37E−09 | 5 | |
AHA | 1.50E−33 | 3.21E−40 | 2.9E−32 | 5.91E−33 | 1.37E−09 | 3 | |
DO | 3.51E−02 | 1.11E−02 | 7.91E−02 | 1.51E−02 | 1.37E−09 | 7 | |
MRFO | 2.51E−99 | 5.2E−108 | 6.21E−98 | 1.2E−98 | 1.37E−09 | 2 | |
AMRFO | 0.0 | 0.0 | 0.0 | 0.0 | - | 1 | |
F3 | SCSO | 2.91E−46 | 1.51E−55 | 6.91E−45 | 1.41E−45 | 9.73E−11 | 4 |
GWO | 1.51E+00 | 3.42E−02 | 2.11E+01 | 4.21E+0 | 9.73E−11 | 5 | |
WOA | 8.12E+04 | 2.81E+04 | 1.21E+05 | 2.1E+04 | 9.73E−11 | 7 | |
AHA | 3.01E−57 | 5.21E−72 | 7.42E−56 | 1.52E−56 | 9.73E−11 | 3 | |
DO | 6.81E+02 | 1.12E+02 | 2.02E+03 | 5.4E+02 | 9.73E−11 | 6 | |
MRFO | 6.72E−186 | 4.6E−200 | 1.71E−184 | 0 | 9.73E−11 | 2 | |
AMRFO | 0 | 0 | 0 | 0 | - | 1 | |
F4 | SCSO | 1.22E−24 | 4.71E−28 | 2.42E−23 | 4.81E−24 | 1.42E−09 | 4 |
GWO | 1.31E−02 | 4.52E−03 | 4.71E−02 | 1.12E−02 | 1.42E−09 | 5 | |
WOA | 5.81E+01 | 1.02E−05 | 9.02E+01 | 3.01E+01 | 1.42E−09 | 7 | |
AHA | 7.82E−31 | 4.12E−38 | 1.61E−29 | 3.12E−30 | 1.42E−09 | 3 | |
DO | 8.81E+00 | 2.31E+00 | 1.91E+01 | 4.92E+00 | 1.42E−09 | 6 | |
MRFO | 2.60E−98 | 4.2E−106 | 3.41E−97 | 7.40E−98 | 1.42E−09 | 2 | |
AMRFO | 0 | 0 | 0 | 0 | - | 1 | |
F5 | SCSO | 2.83E+01 | 2.73E+01 | 2.93E+01 | 5.63E−01 | 1.42E−09 | 3 |
GWO | 2.83E+01 | 2.64E+01 | 2.94E+01 | 8.23E−01 | 1.42E−09 | 4 | |
WOA | 2.92E+01 | 2.83E+01 | 2.93E+01 | 2.83E−01 | 1.42E−09 | 6 | |
AHA | 2.83E+01 | 2.73E+01 | 2.92E+01 | 5.231E−01 | 1.42E−09 | 5 | |
DO | 4.73E+01 | 2.53E+01 | 1.03E+02 | 2.73E+01 | 1.6E−09 | 7 | |
MRFO | 2.53E+01 | 2.43E+01 | 2.62E+01 | 4.32E−01 | 4.1E−07 | 2 | |
AMRFO | 2.41E+01 | 2.32E+01 | 2.60E+01 | 6.40E−01 | - | 1 | |
F6 | SCSO | 2.52E+00 | 1.41E+00 | 3.31E+00 | 4.41E−01 | 1.42E−09 | 7 |
GWO | 1.31E+00 | 2.61E−01 | 2.02E+00 | 4.61E−01 | 1.42E−09 | 6 | |
WOA | 1.3E+00 | 2.12E−01 | 1.91E+00 | 4.12E−01 | 1.42E−09 | 5 | |
AHA | 1.02E+00 | 2.91E−01 | 1.81E+00 | 3.71E−01 | 1.42E−09 | 4 | |
DO | 3.12E−03 | 6.61E−04 | 1.61E−02 | 3.71E−03 | 1.42E−09 | 3 | |
MRFO | 1.61E−04 | 7.52E−06 | 1.13E−03 | 2.41E−04 | 4.46E−08 | 2 | |
AMRFO | 9.52E−06 | 4.61E−07 | 4.91E−05 | 1.31E−05 | - | 1 | |
F7 | SCSO | 5.913E−04 | 1.92E−05 | 2.43E−03 | 6.91E−04 | 2.66E−06 | 4 |
GWO | 5.91E−03 | 3.13E−03 | 1.43E−02 | 2.72E−03 | 1.42E−09 | 5 | |
WOA | 7.81E−03 | 2.03E−04 | 4.22E−02 | 9.13E−03 | 1.42E−09 | 6 | |
AHA | 5.13E−04 | 5.31E−05 | 1.22E−03 | 3.13E−04 | 2.57E−08 | 3 | |
DO | 6.13E−02 | 2.03E−02 | 1.52E−01 | 3.61E−02 | 1.42E−09 | 7 | |
MRFO | 3.81E−04 | 1.43E−05 | 1.23E−03 | 2.72E−04 | 2.2E−06 | 2 | |
AMRFO | 7.61E−05 | 2.8E−06 | 1.71E−04 | 4.8E−05 | - | 1 | |
F8 | SCSO | −6.7E+03 | −7.7E+03 | −5.5E+03 | 6.4E+02 | 2.57E−09 | 4 |
GWO | −6.4E+03 | −7.6E+03 | −3.5E+03 | 9.6E+02 | 1.6E−09 | 5 | |
WOA | −9.7E+03 | −1.5E+04 | −7.1E+03 | 2.13E+03 | 0.077453 | 6 | |
AHA | −9.9E+03 | −1.4E+04 | −9.2E+03 | 4.5E+02 | 7.38E−09 | 3 | |
DO | −7.7E+03 | −8.5E+03 | −5.6E+03 | 7.31E+02 | 8.55E−08 | 7 | |
MRFO | −7.7E+03 | −9.2E+03 | −6.6E+03 | 7.41E+02 | 0.002991 | 2 | |
AMRFO | −8.4E+03 | −9.3E+03 | −7.3E+03 | 5.27E+02 | - | 1 | |
F9 | SCSO | 0 | 0 | 0 | 0 | N/A | 1 |
GWO | 1.22E+01 | 1.12E+00 | 2.61E+01 | 7.71E+00 | 9.73E−11 | 6 | |
WOA | 2.32E−15 | 0 | 5.72E−14 | 1.13E−14 | 0.337055 | 5 | |
AHA | 0 | 0 | 0 | 0 | N/A | 2 | |
DO | 5.61E+01 | 1.31E+01 | 1.42E+02 | 3.02E+01 | 9.73E−11 | 7 | |
MRFO | 0 | 0 | 0 | 0 | N/A | 3 | |
AMRFO | 0 | 0 | 0 | 0 | - | 4 | |
F10 | SCSO | 8.92E−16 | 8.91E−16 | 8.91E−16 | 0 | N/A | 2 |
GWO | 1.42E−06 | 5.1E−07 | 3.91E−06 | 7.61E−07 | 9.73E−11 | 4 | |
WOA | 1.02E−14 | 8.91E−16 | 2.22E−14 | 5.91E−15 | 2.74E−10 | 3 | |
AHA | 8.91E−16 | 8.91E−16 | 8.91E−16 | 0 | N/A | 1 | |
DO | 9.52E−02 | 1.32E−02 | 1.32E+00 | 2.61E−01 | 9.73E−11 | 5 | |
MRFO | 8.91E−16 | 8.91E−16 | 8.91E−16 | 0 | N/A | 1 | |
AMRFO | 8.91E−16 | 8.91E−16 | 8.91E−16 | 0 | - | 1 | |
F11 | SCSO | 0 | 0 | 0 | 0 | N/A | 1 |
GWO | 1.13E−02 | 8.11E−12 | 4.72E−02 | 1.42E−02 | 9.73E−11 | 3 | |
WOA | 4.62E−02 | 0 | 6.51E−01 | 1.63E−01 | 0.081168 | 5 | |
AHA | 0 | 0 | 0 | 0 | N/A | 2 | |
DO | 3.61E−02 | 6.02E−03 | 9.81E−02 | 2.21E−02 | 9.73E−11 | 4 | |
MRFO | 0 | 0 | 0 | 0 | N/A | 2 | |
AMRFO | 0 | 0 | 0 | 0 | N/A | 2 | |
F12 | SCSO | 1.32E−01 | 2.61E−02 | 3.33E−01 | 6.12E−02 | 4.13E−09 | 6 |
GWO | 8.31E−02 | 3.22E−02 | 1.81E−01 | 3.61E−02 | 1.46E−08 | 5 | |
WOA | 8.13E−02 | 1.33E−02 | 2.52E−01 | 5.72E−02 | 1.31E−08 | 4 | |
AHA | 2.63E−02 | 2.72E−03 | 8.22E−02 | 1.81E−02 | 2.57E−08 | 3 | |
DO | 4.92E−01 | 4.12E−05 | 2.71E+00 | 8.12E−01 | 7.38E−09 | 7 | |
MRFO | 6.61E−06 | 3.62E−07 | 2.63E−05 | 6.91E−06 | 8.86E−06 | 1 | |
AMRFO | 4.12E−03 | 1.33E−08 | 1.02E−01 | 2.13E−02 | - | 2 | |
F13 | SCSO | 2.41E+00 | 1.42E+00 | 2.91E+00 | 4.02E−01 | 0.006223 | 5 |
GWO | 1.12E+00 | 6.31E−01 | 1.61E+00 | 3.02E−01 | 3.7E−07 | 2 | |
WOA | 1.13E+00 | 3.02E−01 | 2.13E+00 | 4.41E−01 | 7.51E−07 | 3 | |
AHA | 2.31E+00 | 1.51E−01 | 2.91E+00 | 6.01E−01 | 0.00384 | 4 | |
DO | 4.41E−02 | 2.41E−04 | 5.31E−01 | 1.11E−01 | 1.8E−09 | 1 | |
MRFO | 2.51E+00 | 1.41E−02 | 3.01E+00 | 1.01E+00 | 0.004614 | 6 | |
AMRFO | 2.51E+00 | 3.01E−01 | 3.01E+00 | 7.81E−01 | - | 6 | |
F14 | SCSO | 4.23E+00 | 1.02E+00 | 1.31E+01 | 4.42E+00 | 1.38E−10 | 5 |
GWO | 5.51E+00 | 1.02E+00 | 1.31E+01 | 4.60E+00 | 1.38E−10 | 7 | |
WOA | 4.41E+00 | 1.02E+00 | 1.11E+01 | 3.61E+00 | 1.38E−10 | 6 | |
AHA | 1.23E+00 | 1.02E+00 | 3.02E+00 | 6.71E−01 | 1.83E−10 | 3 | |
DO | 1.43E+00 | 1.02E+00 | 3.02E+00 | 7.31E−01 | 1.37E−10 | 4 | |
MRFO | 1.01E+00 | 1.02E+00 | 1.02E+00 | 1.72E−16 | 7.52E−05 | 2 | |
AMRFO | 1.0E+00 | 1.02E+00 | 1.02E+00 | 4.51E−17 | - | 1 | |
F15 | SCSO | 5.21E−04 | 3.11E−04 | 1.21E−03 | 2.61E−04 | 1.42E−09 | 3 |
GWO | 3.01E−03 | 3.61E−04 | 2.01E−02 | 6.61E−03 | 1.42E−09 | 6 | |
WOA | 1.31E−03 | 3.31E−04 | 1.71E−02 | 3.31E−03 | 1.42E−09 | 5 | |
AHA | 3.12E−04 | 3.12E−04 | 3.41E−04 | 7.21E−06 | 1.42E−09 | 2 | |
DO | 4.71E−03 | 3.12E−04 | 2.02E−02 | 8.01E−03 | 1.42E−09 | 7 | |
MRFO | 1.22E−03 | 3.12E−04 | 2.02E−02 | 4.01E−03 | 1.42E−09 | 4 | |
AMRFO | 3.10E−04 | 3.10E−04 | 3.11E−04 | 9.34E−15 | - | 1 | |
F16 | SCSO | −1.0E+00 | −1.0E+00 | −1.0E+00 | 3.92E−09 | 1.87E−10 | 1 |
GWO | −1.0E+00 | −1.0E+00 | −1.0E+00 | 2.52E−07 | 1.87E−10 | 1 | |
WOA | −1.0E+00 | −1.0E+00 | −1.0E+00 | 9.22E−08 | 1.87E−10 | 1 | |
AHA | −1.0E+00 | −1.0E+00 | −1.0E+00 | 3.91E−12 | 8.8E−09 | 1 | |
DO | −1.0E+00 | −1.0E+00 | −1.0E+00 | 2.33E−11 | 1.87E−10 | 1 | |
MRFO | −1.0E+00 | −1.0E+00 | −1.0E+00 | 6.11E−16 | 0.018662 | 1 | |
AMRFO | −1.0E+00 | −1.0E+00 | −1.0E+00 | 6.61E−16 | - | 1 | |
F17 | SCSO | 4.0E−01 | 4.0E−01 | 4.0E−01 | 3.71E−07 | 9.73E−11 | 1 |
GWO | 4.0E−01 | 4.0E−01 | 4.0E−01 | 2.81E−06 | 9.73E−11 | 1 | |
WOA | 4.0E−01 | 4.0E−01 | 4.0E−01 | 1.41E−04 | 9.73E−11 | 1 | |
AHA | 4.0E−01 | 4.0E−01 | 4.0E−01 | 0 | N/A | 1 | |
DO | 4.0E−01 | 4.0E−01 | 4.0E−01 | 2.91E−10 | 9.73E−11 | 1 | |
MRFO | 4.0E−01 | 4.0E−01 | 4.0E−01 | 0 | N/A | 1 | |
AMRFO | 4.0E−01 | 4.0E−01 | 4.0E−01 | 0 | - | 1 | |
F18 | SCSO | 3.02E+00 | 3.02E+00 | 3.02E+00 | 3.62E−05 | 1.24E−09 | 1 |
GWO | 3.02E+00 | 3.02E+00 | 3.02E+00 | 3.42E−04 | 1.24E−09 | 1 | |
WOA | 3.02E+00 | 3.02E+00 | 3.02E+00 | 1.51E−03 | 1.24E−09 | 1 | |
AHA | 3.02E+00 | 3.02E+00 | 3.02E+00 | 2.02E−15 | 0.483014 | 1 | |
DO | 3.02E+00 | 3.02E+00 | 3.02E+00 | 1.13E−07 | 1.24E−09 | 1 | |
MRFO | 3.02E+00 | 3.02E+00 | 3.02E+00 | 1.51E−15 | 0.052141 | 1 | |
AMRFO | 3.02E+00 | 3.02E+00 | 3.02E+00 | 1.51E−15 | - | 1 | |
F19 | SCSO | −3.8E+00 | −3.8E+00 | −3.8E+00 | 3.1E−03 | 1.87E−10 | 1 |
GWO | −3.8E+00 | −3.9E+00 | −3.8E+00 | 2.5E−03 | 1.87E−10 | 1 | |
WOA | −3.7E+00 | −3.8E+00 | −3.6E+00 | 4.11E−02 | 1.87E−10 | 1 | |
AHA | −3.8E+00 | −3.8E+00 | −3.8E+00 | 5.82E−15 | 3.69E−08 | 1 | |
DO | −3.8E+00 | −3.8E+00 | −3.8E+00 | 1.1E−06 | 1.87E−10 | 1 | |
MRFO | −3.8E+00 | −3.8E+00 | −3.8E+00 | 2.3E−15 | 0.232368 | 1 | |
AMRFO | −3.9E+00 | −3.9E+00 | −3.9E+00 | 2.13E−15 | - | 1 | |
F20 | SCSO | −3.3E+00 | −3.3E+00 | −2.81E+00 | 1.12E−01 | 0.044224 | 1 |
GWO | −3.3E+00 | −3.3E+00 | −3.12E+00 | 8.11E−02 | 0.144383 | 1 | |
WOA | −3.2E+00 | −3.3E+00 | −3.1E+00 | 1.23E−01 | 0.010339 | 1 | |
AHA | −3.3E+00 | −3.3E+00 | −3.21E+00 | 3.92E−02 | 0.937112 | 1 | |
DO | −3.3E+00 | −3.3E+00 | −3.22E+00 | 3.92E−02 | 0.937112 | 1 | |
MRFO | −3.3E+00 | −3.3E+00 | −3.21E+00 | 6.1E−02 | 0.407218 | 1 | |
AMRFO | −3.3E+00 | −3.3E+00 | −3.20E+00 | 6.0E−02 | - | 1 | |
F21 | SCSO | −5.2E+00 | −1.0E+01 | −8.7E−01 | 1.31E+00 | 7.19E−10 | 7 |
GWO | −8.9E+00 | −1.0E+01 | −2.6E+00 | 2.51E+00 | 7.19E−10 | 3 | |
WOA | −7.4E+00 | −1.0E+01 | −2.5E+00 | 3.01E+00 | 7.19E−10 | 5 | |
AHA | −9.4E+00 | −1.0E+01 | −4.9E+00 | 1.91E+00 | 7.19E−10 | 2 | |
DO | −7.3E+00 | −1.0E+01 | −2.6E+00 | 3.51E+00 | 7.19E−10 | 6 | |
MRFO | −7.8E+00 | −1.0E+01 | E+00 | 2.61E+00 | 9.48E−05 | 4 | |
AMRFO | −1.0E+01 | −1.0E+01 | −1.0E+01 | 4.40E−15 | - | 1 | |
F22 | SCSO | −6.2E+00 | −1.0E+01 | −9.3E−01 | 2.62E+00 | 6.34E−09 | 6 |
GWO | −1.0E+01 | −1.0E+01 | −5.3E+00 | 1.12E+00 | 7.15E−09 | 1 | |
WOA | −6.6E+00 | −1.0E+01 | −2.1E+00 | 3.22E+00 | 3.91E−09 | 5 | |
AHA | −9.6E+00 | −1.0E+01 | −5.3E+00 | 1.92E+00 | 7.15E−09 | 3 | |
DO | −61+00 | −1.0E+01 | −1.9E+00 | 3.52E+00 | 1.65E−09 | 7 | |
MRFO | −7.2E+00 | −1.0E+01 | −2.9E+00 | 2.82E+00 | 3.66E−05 | 4 | |
AMRFO | −1.0E+01 | −1.0E+01 | −3.8E+00 | 1.31E+00 | - | 2 | |
F23 | SCSO | −6.1E+00 | −1.1E+01 | −2.91E+00 | 2.42E+00 | 7.54E−10 | 6 |
GWO | −1.0E+01 | −1.1E+01 | −2.51E+00 | 1.61E+00 | 7.54E−10 | 2 | |
WOA | −5.8E+00 | −1.1E+01 | −1.81E+00 | 2.91E+00 | 7.54E−10 | 7 | |
AHA | −9.9E+00 | −1.1E+01 | −5.31E+00 | 1.72E+00 | 7.54E−10 | 3 | |
DO | −7.6E+00 | −1.1E+01 | −2.51E+00 | 3.62E+00 | 7.54E−10 | 5 | |
MRFO | −8.3E+00 | −1.1E+01 | −3.91E+00 | 2.91E+00 | 1.05E−05 | 4 | |
AMRFO | −1.1E+01 | −1.1E+01 | −1.11E+01 | 2.81E−15 | - | 1 |
Appendix B. The Optimal Control Variables for Solving the SORPD
Without Incorporating WTs | ||||
Scenario No. | T11 | T12 | T15 | T36 |
1 | 1.02 | 0.99 | 0.97 | 1.03 |
2 | 1.02 | 1.05 | 1.05 | 1.02 |
3 | 0.98 | 1.04 | 1 | 0.96 |
4 | 1.06 | 0.97 | 1.04 | 1 |
5 | 1 | 1.07 | 1.04 | 1.02 |
6 | 0.97 | 1.03 | 1.04 | 0.99 |
7 | 0.95 | 1.01 | 1.02 | 0.99 |
8 | 0.99 | 1 | 1.05 | 1.05 |
9 | 1.01 | 1.01 | 1.03 | 1.01 |
10 | 1.01 | 1.04 | 1.04 | 1.08 |
Incorporating WTs | ||||
Scenario No. | T11 | T12 | T15 | T36 |
1 | 1.01 | 1.03 | 1.05 | 0.98 |
2 | 1 | 1.04 | 0.98 | 0.99 |
3 | 1.04 | 0.98 | 1.01 | 0.97 |
4 | 1.08 | 1.04 | 1.07 | 1.05 |
5 | 1.03 | 0.97 | 1.01 | 1.01 |
6 | 1.02 | 0.97 | 1.02 | 0.98 |
7 | 0.97 | 0.99 | 1.01 | 0.99 |
8 | 1.02 | 1.06 | 0.99 | 0.97 |
9 | 1.07 | 0.93 | 0.97 | 1.01 |
10 | 1.01 | 1.06 | 1.01 | 1 |
Without Incorporating WTs | |||||||||
Scenario No. | Q10 | Q12 | Q15 | Q17 | Q20 | Q21 | Q23 | Q24 | Q29 |
1 | 1.86 | 2.49 | 1.82 | 2.49 | 1.34 | 1.63 | 2.76 | 2.6 | 2.04 |
2 | 3.36 | 1.73 | 3.18 | 2.12 | 3.29 | 1.8 | 1.67 | 1.56 | 3.15 |
3 | 1.17 | 3.61 | 2.85 | 1.82 | 1.21 | 2.15 | 2.62 | 1.47 | 3.06 |
4 | 3.38 | 1.97 | 3 | 3.01 | 2.53 | 0.35 | 3.37 | 1.84 | 2.43 |
5 | 1.38 | 2.52 | 1.89 | 3.67 | 3.82 | 2.76 | 2.39 | 4 | 4.16 |
6 | 2.88 | 1.94 | 3.24 | 2.79 | 1.86 | 2.17 | 1.33 | 2.73 | 2.69 |
7 | 1.58 | 1.25 | 2.21 | 3.17 | 1.47 | 3.17 | 0.89 | 2.95 | 1.2 |
8 | 1.85 | 3.14 | 2.52 | 3.19 | 2.26 | 3.49 | 2.04 | 2.22 | 3.33 |
9 | 3.78 | 0.85 | 2.87 | 1.93 | 2.88 | 3.91 | 4.01 | 1.63 | 3.39 |
10 | 3.27 | 2.29 | 2.59 | 1.81 | 2.88 | 2.09 | 1.61 | 4.45 | 3.22 |
Incorporating WTs | |||||||||
Scenario No. | Q10 | Q12 | Q15 | Q17 | Q20 | Q21 | Q23 | Q24 | Q29 |
1 | 3.95 | 3.28 | 2.83 | 2.39 | 1.49 | 1.33 | 2.55 | 2.98 | 2.15 |
2 | 2.34 | 1.86 | 2.92 | 1.74 | 2.79 | 4.02 | 3.46 | 2.67 | 2.27 |
3 | 0.49 | 1.2 | 3.41 | 3.83 | 2.51 | 1.31 | 4.1 | 1.23 | 3.68 |
4 | 1.61 | 3.37 | 3.72 | 2.06 | 2.79 | 1.97 | 2.26 | 3.54 | 3.2 |
5 | 2.68 | 2.6 | 3.21 | 1.81 | 4.33 | 2.97 | 1.74 | 2.29 | 4.41 |
6 | 2.05 | 2.23 | 2.34 | 2.18 | 2.46 | 3.64 | 2.8 | 1.43 | 2.81 |
7 | 1.54 | 3.95 | 4.27 | 2.91 | 2.64 | 2.92 | 2.69 | 2.5 | 2.16 |
8 | 3.8 | 3.38 | 3.05 | 2.38 | 3.39 | 4.05 | 2.93 | 0.87 | 2.37 |
9 | 1.83 | 2.18 | 3.23 | 2.88 | 2.15 | 3.31 | 2.84 | 3.72 | 2.59 |
10 | 1.85 | 3.97 | 0.44 | 4.5 | 4.1 | 3.58 | 2.59 | 2.32 | 3.99 |
Without Incorporating WTs | ||||
Scenario No. | T11 | T12 | T15 | T36 |
1 | 0.96 | 0.96 | 1 | 0.97 |
2 | 1 | 1.04 | 1.02 | 0.95 |
3 | 1.03 | 0.93 | 0.99 | 0.97 |
4 | 1.01 | 0.94 | 1 | 0.94 |
5 | 1.03 | 0.91 | 0.96 | 0.97 |
6 | 0.99 | 1.06 | 0.98 | 0.96 |
7 | 1 | 0.96 | 1.01 | 0.96 |
8 | 0.94 | 0.94 | 0.98 | 1.01 |
9 | 1.02 | 1 | 0.99 | 0.97 |
10 | 0.93 | 1.01 | 1.02 | 0.98 |
Incorporating WTs | ||||
Scenario No. | T11 | T12 | T15 | T36 |
1 | 1.01 | 1.01 | 0.91 | 1 |
2 | 1.01 | 1.02 | 0.95 | 0.96 |
3 | 0.99 | 0.92 | 0.98 | 0.97 |
4 | 1 | 0.94 | 1 | 0.96 |
5 | 0.97 | 0.95 | 0.98 | 0.98 |
6 | 1.03 | 0.99 | 0.94 | 0.98 |
7 | 0.98 | 1.01 | 1.02 | 0.96 |
8 | 0.96 | 0.96 | 0.99 | 0.92 |
9 | 0.96 | 0.98 | 0.99 | 0.95 |
10 | 0.98 | 0.96 | 1 | 0.99 |
Without Incorporating WTs | |||||||||
Scenario No. | Q10 | Q12 | Q15 | Q17 | Q20 | Q21 | Q23 | Q24 | Q29 |
1 | 3.86 | 2.05 | 3.33 | 4.13 | 4.67 | 3.16 | 3.92 | 2.36 | 2.42 |
2 | 1.29 | 3.74 | 1.72 | 2.97 | 2.61 | 3.18 | 2.3 | 4.2 | 3.42 |
3 | 1.81 | 1.96 | 2.68 | 1.82 | 1.39 | 2.05 | 2.66 | 1.97 | 3.4 |
4 | 2.29 | 0.79 | 2.23 | 2.15 | 2.69 | 3.08 | 1.59 | 2.56 | 3.41 |
5 | 2.46 | 1.09 | 1.52 | 2.72 | 1.7 | 2.71 | 3.07 | 3.29 | 3.12 |
6 | 3.4 | 1.7 | 3.02 | 2.22 | 1.86 | 3.68 | 2.76 | 1.62 | 2.46 |
7 | 2.5 | 2.4 | 2.23 | 4.14 | 2.92 | 2.1 | 1.04 | 3.03 | 1.91 |
8 | 3.59 | 1.96 | 3.3 | 1.32 | 3.51 | 2.96 | 1.41 | 1.4 | 3.01 |
9 | 3.69 | 2.05 | 2.09 | 3.33 | 4.09 | 2.37 | 3.47 | 1.69 | 4.32 |
10 | 3.44 | 4.01 | 3.04 | 0.76 | 2.52 | 2.87 | 2.63 | 2.68 | 3.41 |
Incorporating WTs | |||||||||
Scenario No. | Q10 | Q12 | Q15 | Q17 | Q20 | Q21 | Q23 | Q24 | Q29 |
1 | 1.97 | 3.17 | 1.7 | 0.73 | 2.47 | 1.33 | 2.14 | 3 | 2.32 |
2 | 1.79 | 1.58 | 4.3 | 3.7 | 3.94 | 1.32 | 1.35 | 2.99 | 3.7 |
3 | 1.71 | 2.64 | 2.69 | 2.71 | 1.69 | 2.85 | 2.05 | 3.87 | 3.74 |
4 | 3.03 | 4.27 | 3.09 | 1.81 | 2.9 | 2.71 | 1.53 | 2.24 | 1.88 |
5 | 2.23 | 2.44 | 3.35 | 2.46 | 2.07 | 1.8 | 3.12 | 3.94 | 3.82 |
6 | 3.61 | 2.75 | 3.02 | 2.97 | 2.26 | 1.44 | 2.16 | 1.65 | 2.8 |
7 | 3.43 | 1.71 | 3.81 | 2.01 | 2.54 | 2.2 | 0.87 | 2.48 | 2.95 |
8 | 2.64 | 2.29 | 2.66 | 2.42 | 2.42 | 3.25 | 2.78 | 2.14 | 4.32 |
9 | 3 | 2.76 | 1.41 | 2 | 4.73 | 3.74 | 3.39 | 3.31 | 1.62 |
10 | 2.11 | 1.99 | 1.73 | 1.54 | 1.67 | 2.35 | 2.38 | 3.34 | 2.96 |
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Scenario Number | Loading % | Wind Speed (m/s) | |
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1 | 42.10 | 5.04 | 0.011 |
2 | 91.46 | 8.38 | 0.027 |
3 | 78.61 | 15.57 | 0.02 |
4 | 85.30 | 13.36 | 0.023 |
5 | 71.11 | 7.72 | 0.393 |
6 | 106.56 | 9.42 | 0.001 |
7 | 62.36 | 10.37 | 0.245 |
8 | 96.91 | 14.40 | 0.001 |
9 | 77.87 | 5.70 | 0.233 |
10 | 49.63 | 8.93 | 0.046 |
Algorithm | The Parameters |
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SCSO [56] | Max. iterations = 250, No. populations = 25, Phases control range (R) in range [−2rg, 2rg], Sensitivity range (rg) = [2, 0]. |
GWO [57] | Max. iterations = 250, No. populations = 25, a = [2, 0], A= [2, 0], C = 2. rand (0, 1) |
WOA [58] | Max. iterations = 250, No. populations = 25, a = [2, 0], A = [2, 0], , b = 1. |
AHA [59] | Max. iterations = 250, NO. populations = 25 |
DO [60] | Max. iterations = 250, No. populations = 25, = [0, 1], = [0, 1]. |
MRFO [37] | Max. iterations = 100, No. populations = 25, S = 2. |
AMRFO | Max. iterations = 100, No. populations = 25, S = 2. |
Control Variables | Min. | Max. | Minimization | Minimization |
---|---|---|---|---|
V1 | 0.9 | 1.1 | 1.100 | 1.006 |
V2 | 0.9 | 1.1 | 1.094 | 1.007 |
V5 | 0.9 | 1.1 | 1.073 | 1.069 |
V8 | 0.9 | 1.1 | 1.075 | 1.000 |
V11 | 0.9 | 1.1 | 1.098 | 1.031 |
V13 | 0.9 | 1.1 | 1.100 | 1.024 |
T11 | 0.9 | 1.1 | 1.000 | 1.040 |
T12 | 0.9 | 1.1 | 0.930 | 0.910 |
T15 | 0.9 | 1.1 | 0.980 | 1.000 |
T36 | 0.9 | 1.1 | 0.970 | 0.970 |
Q10 | 0 | 0.05 | 2.500 | 4.570 |
Q12 | 0 | 0.05 | 4.840 | 0.380 |
Q15 | 0 | 0.05 | 4.770 | 4.810 |
Q17 | 0 | 0.05 | 4.760 | 0.870 |
Q20 | 0 | 0.05 | 4.140 | 4.740 |
Q21 | 0 | 0.05 | 4.930 | 4.980 |
Q23 | 0 | 0.05 | 3.950 | 4.970 |
Q24 | 0 | 0.05 | 4.920 | 4.850 |
Q29 | 0 | 0.05 | 2.430 | 2.850 |
4.5279 | 5.7852 | |||
1.9829 | 0.0913 | |||
0.1162 | 0.1367 |
Algorithm | Worst | Mean | Best |
---|---|---|---|
AMRFO | 4.6089 | 4.5488 | 4.5279 |
MRFO | 4.6532 | 4.5740 | 4.5308 |
SCSO | 4.7373 | 4.6537 | 4.5884 |
GWO | 4.7321 | 4.6795 | 4.6295 |
WOA | 4.9064 | 4.7969 | 4.6813 |
AHA | 5.7434 | 5.1461 | 4.9275 |
DO | 4.7657 | 4.6168 | 4.5526 |
Marine predator Algorithm(MPA) [68] | 4.6006 | 4.55389 | 4.5335 |
Jaya Algorithm (JA) [66] | NA | NA | 4.625 |
Ant Lion Optimizer (ALO) [18] | NA | NA | 4.5900 |
Harmony Search Algorithm (HSO) [24] | 4.9653 | 4.924 | 4.9059 |
PSO [24] | 5.0576 | 4.972 | 4.9239 |
Standard Genetic Algorithm (SGA) [24] | 5.1651 | 5.0378 | 4.9408 |
TLBO [26] | 4.57480 | 4.56950 | 4.5629 |
Quasi-oppositional TLBO [26] | 4.56170 | 4.56010 | 4.5594 |
Differential Evolution (DE) [8] | NA | NA | 4.5550 |
Specialized Genetic Algorithm (SpGA) [12] | NA | NA | 4.5692 |
Firefly Algorithm (FA) [67] | 4.59 | 4.578 | 4.5691 |
Hybrid PSO and Tabu search (HPSO–TS) [30] | NA | NA | 4.5213 |
Tabu search (TS) [30] | NA | NA | 4.9203 |
Whale Optimization Algorithm (WOA) [17] | NA | NA | 4.5943 |
PSO [17] | NA | NA | 4.6469 |
Improved Differential Evolution (IDE) [16] | NA | NA | 4.5521 |
Biogeography-Based Optimization (BBO) [22] | NA | NA | 4.5511 |
Comprehensive Learning PSO (CLPSO) [19] | NA | NA | 4.6282 |
Gravitational Search Algorithm (GSA) [27] | NA | NA | 5.00954 |
Improved GSA With Conditional Selection Strategies (IGSA-CSS) [27] | NA | NA | 4.76601 |
Algorithm | Worst | Mean | Best |
---|---|---|---|
AMRFO | 0.1218 | 0.1068 | 0.0913 |
MRFO | 0.1309 | 0.1158 | 0.1042 |
SCSO | 0.2174 | 0.1722 | 0.1338 |
GWO | 0.2021 | 0.1707 | 0.1356 |
WOA | 0.5001 | 0.2556 | 0.1749 |
AHA | 0.6492 | 0.3860 | 0.2076 |
DO | 0.1939 | 0.1460 | 0.1268 |
PSO based TVAC [69] | 0.5791 | 0.1597 | 0.1038 |
PSO with TVAC [69] | 0.5796 | 0.2376 | 0.2064 |
PG-PSO [69] | 0.2593 | 0.1440 | 0.1202 |
Social Spider Optimization [15] | 0.42681 | 0.2863 | 0.19304 |
Hybrid Salp Swarm Algorithm with Simulated Annealing [15] | 0.576439 | 0.308337 | 0.174701 |
Modified Salp Swarm Algorithm (MSSA) [15] | 1.860037 | 0.690254 | 0.230087 |
Salp Swarm Algorithm (SSA) [15] | 0.941759 | 0.374529 | 0.188411 |
Cuckoo search algorithm (CSA) [15] | 0.2076 | 0.16432 | 0.12692 |
Ant Lion Optimizer (ALO) [13] | NA | 0.1575 | 0.1192 |
Gravitational Search (GS) [27] | NA | NA | 0.17241 |
GSA and Conditional Selection Strategies (GSA-CSS) | NA | NA | 0.12394 |
Scenario No. | Loading % | |||||
---|---|---|---|---|---|---|
1 | 0.011 | 42.10 | 2.3387 | 0.0257 | 0.3412 | 0.0038 |
2 | 0.027 | 91.46 | 9.1826 | 0.2479 | 0.5218 | 0.0141 |
3 | 0.02 | 78.61 | 5.9414 | 0.1188 | 0.5981 | 0.0120 |
4 | 0.023 | 85.30 | 7.4740 | 0.1719 | 0.3789 | 0.0087 |
5 | 0.393 | 71.11 | 4.4702 | 1.7568 | 0.3293 | 0.1294 |
6 | 0.001 | 106.56 | 14.5927 | 0.0146 | 0.3957 | 0.0004 |
7 | 0.245 | 62.36 | 3.1513 | 0.7721 | 0.6169 | 0.1511 |
8 | 0.001 | 96.91 | 10.5326 | 0.0105 | 0.4962 | 0.0005 |
9 | 0.233 | 77.87 | 5.6409 | 1.3143 | 0.4812 | 0.1121 |
10 | 0.046 | 49.63 | 1.8994 | 0.0874 | 0.5166 | 0.0238 |
= 4.5201 | = 0.4558 |
Scenario No. | Loading % | Wind Speed (m/s) | (MW) | |||||
---|---|---|---|---|---|---|---|---|
1 | 0.011 | 42.10 | 5.04 | 11.747 | 1.3424 | 0.0148 | 0.3922 | 0.0043 |
2 | 0.027 | 91.46 | 8.38 | 31.032 | 5.9965 | 0.1619 | 0.4959 | 0.0134 |
3 | 0.02 | 78.61 | 15.57 | 72.496 | 2.5651 | 0.0513 | 0.4199 | 0.0084 |
4 | 0.023 | 85.30 | 13.36 | 59.762 | 3.7187 | 0.0855 | 0.7343 | 0.0169 |
5 | 0.393 | 71.11 | 7.72 | 27.238 | 2.8301 | 1.1122 | 0.4203 | 0.1652 |
6 | 0.001 | 106.56 | 9.42 | 37.034 | 12.4915 | 0.0125 | 0.2707 | 0.0003 |
7 | 0.245 | 62.36 | 10.37 | 42.543 | 1.5123 | 0.3705 | 0.4622 | 0.1132 |
8 | 0.001 | 96.91 | 14.40 | 65.779 | 4.9598 | 0.0050 | 0.4225 | 0.0004 |
9 | 0.233 | 77.87 | 5.70 | 15.581 | 4.4616 | 1.0396 | 0.5034 | 0.1173 |
10 | 0.046 | 49.63 | 8.93 | 34.184 | 1.0399 | 0.0478 | 0.3301 | 0.0152 |
TEPL = 2.9011 | TVED = 0.4546 |
Scenario No. | Loading % | |||||
---|---|---|---|---|---|---|
1 | 0.011 | 42.10 | 2.3753 | 0.0261 | 0.2164 | 0.0024 |
2 | 0.027 | 91.46 | 10.7310 | 0.2897 | 0.2510 | 0.0068 |
3 | 0.02 | 78.61 | 6.5879 | 0.1318 | 0.2130 | 0.0043 |
4 | 0.023 | 85.30 | 9.2064 | 0.2117 | 0.2145 | 0.0049 |
5 | 0.393 | 71.11 | 5.2047 | 2.0455 | 0.1282 | 0.0504 |
6 | 0.001 | 106.56 | 13.4473 | 0.0134 | 0.2799 | 0.0003 |
7 | 0.245 | 62.36 | 4.1113 | 1.0073 | 0.1292 | 0.0317 |
8 | 0.001 | 96.91 | 12.3517 | 0.0124 | 0.6952 | 0.0007 |
9 | 0.233 | 77.87 | 6.7513 | 1.5730 | 0.1130 | 0.0263 |
10 | 0.046 | 49.63 | 4.1757 | 0.1921 | 0.1559 | 0.0072 |
= 5.5030 | = 0.1349 |
Scenario NO. | Loading % | Wind Speed (m/s) | (MW) | |||||
---|---|---|---|---|---|---|---|---|
1 | 0.011 | 0.011 | 42.10 | 2.3753 | 3.5862 | 0.0394 | 0.1599 | 0.0018 |
2 | 0.027 | 0.027 | 91.46 | 10.7310 | 6.2666 | 0.1692 | 0.2493 | 0.0067 |
3 | 0.02 | 0.02 | 78.61 | 6.5879 | 2.6767 | 0.0535 | 0.2026 | 0.0041 |
4 | 0.023 | 0.023 | 85.30 | 9.2064 | 3.8184 | 0.0878 | 0.1718 | 0.0040 |
5 | 0.393 | 0.393 | 71.11 | 5.2047 | 3.7606 | 1.4779 | 0.1075 | 0.0422 |
6 | 0.001 | 0.001 | 106.56 | 13.4473 | 10.5488 | 0.0105 | 0.5717 | 0.0006 |
7 | 0.245 | 0.245 | 62.36 | 4.1113 | 4.9506 | 1.2129 | 0.1163 | 0.0285 |
8 | 0.001 | 0.001 | 96.91 | 12.3517 | 5.4928 | 0.0055 | 0.1931 | 0.0002 |
9 | 0.233 | 0.233 | 77.87 | 6.7513 | 6.3825 | 1.4871 | 0.1354 | 0.0315 |
10 | 0.046 | 0.046 | 49.63 | 4.1757 | 1.1932 | 0.0549 | 0.1178 | 0.0054 |
= 4.5989 | = 0.1249 |
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Almutairi, S.Z.; Mohamed, E.A.; El-Sousy, F.F.M. A Novel Adaptive Manta-Ray Foraging Optimization for Stochastic ORPD Considering Uncertainties of Wind Power and Load Demand. Mathematics 2023, 11, 2591. https://doi.org/10.3390/math11112591
Almutairi SZ, Mohamed EA, El-Sousy FFM. A Novel Adaptive Manta-Ray Foraging Optimization for Stochastic ORPD Considering Uncertainties of Wind Power and Load Demand. Mathematics. 2023; 11(11):2591. https://doi.org/10.3390/math11112591
Chicago/Turabian StyleAlmutairi, Sulaiman Z., Emad A. Mohamed, and Fayez F. M. El-Sousy. 2023. "A Novel Adaptive Manta-Ray Foraging Optimization for Stochastic ORPD Considering Uncertainties of Wind Power and Load Demand" Mathematics 11, no. 11: 2591. https://doi.org/10.3390/math11112591