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Abstract
This article applies the grey wolf optimizer and differential evolution (DE) algorithms to solve the optimal power flow (OPF) problem. Both algorithms are used to optimize single objective functions sequentially under the system constraints. Then, the DE algorithm is utilized to solve multi-objective OPF problems. The indicator of the static line stability index is incorporated into the OPF problem. The fuzzy-based Pareto front method is tested to find the best compromise point of multi-objective functions. The proposed algorithms are used to determine the optimal values of the continuous and discrete control variables. These algorithms are applied to the standard IEEE 30-bus and 118-bus systems with different scenarios. The simulation results are investigated and analyzed. The achieved results show the effectiveness of the proposed algorithms in comparison with the other recent heuristic algorithms in the literature.
NOMENCLATURE
| PGi | = | real power generation at bus i |
| ai, bi, ci | = | fuel cost coefficients of generating unit i |
| ei, fi | = | coefficients reflecting valve-point effects of generating unit i |
| Ng | = | number of generating units |
| |Vi|, |Vj| | = | voltage magnitude at bus i and bus j, respectively |
| δij | = | difference between δi and δj |
| |Zij| | = | impedance magnitude of line i–j |
| Rij | = | resistance between bus i and bus j |
| N | = | total number of buses |
| Xij | = | reactance between bus i and bus j |
| n | = | exponent factor |
| |Vmax|, |Vmin| | = | maximum and minimum bus voltage magnitudes, respectively |
| Gij | = | conductance between bus i and bus j |
| Bij | = | susceptance between bus i and bus j |
| QGi | = | reactive power generation at bus i |
| QCi | = | injected reactive power compensation at bus i |
| PDi | = | active power demand at bus i |
| QDi | = | reactive power demand at bus i |
| PminGi | = | lower real power generation limit of unit i |
| PmaxGi | = | upper real power generation limit of unit i |
| QminGi | = | lower reactive power generation limit of unit i |
| QmaxGi | = | upper reactive power generation limit of unit i |
| |Vmini| | = | lower limit of voltage magnitude at bus i |
| |Vmaxi| | = | upper limit of voltage magnitude at bus i |
| tmink | = | lower load tap setting of transformer k |
| tmaxk | = | upper load tap setting of transformer k |
| Nt | = | number of transformers |
| Sli | = | actual line flow |
| Sratedli | = | rated line transfer capacity |
| nbr | = | number of network lines |
| Qminci | = | lower limit of reactive power compensation at bus i |
| Qmaxci | = | maximum limit of reactive power compensation at bus i |
| Nc | = | number of nominated buses for reactive compensation |
| LSIij | = | line stability index of branch i–j |
| θ | = | impedance phase angle of line i–j |
| VPQ − i | = | voltage of load bus i |
| NPQ | = | number of load buses |
| X(t) | = | position vector of the grey wolf at an iteration t |
| XP(t) | = | position vector of the prey |
| A, C | = | coefficient vectors |
| Np | = | population size |
| D | = | problem dimension |
| Xminj, Xjmax | = | lower and upper boundaries of the jth decision variable |
| i | = | ith objective function |
| = | decision control vector | |
| Noj | = | number of objective functions |
| NEQ, NINQ | = | numbers of equality and inequality constraints, respectively |
| Fmini, Fmaxi | = | minimum and maximum values of the ith objective function among all Pareto solutions |
| M | = | number of Pareto solutions |
| μi | = | fuzzy membership function normalized in the range of 0–1 |
Additional information
Notes on contributors
Attia A. El-Fergany
Attia El-Fergany received the BSc degree (1994), MSc degree (1998), and PhD degree (2001), all in Electrical Power Engineering from Zagazig University in Zagazig, Egypt. He has been with the University of Zagazig since 1998, and presently as an Associate Professor of Electrical Power Engineering. Dr. El-Fergany has authored or co-authored numerous articles published in the international refereed journals and conferences. Attia has been given many awards for distinct international research publishing from Zagazig University, Egypt (2012–2014). In addition, he delivered successfully numerous short courses to worldwide graduated electrical engineers. He has participated in many field electrical technical studies. He is a Senior Member of the IEEE, a Member of the PES & the Education Society, and a Member of the IET. His research is concerned with the use of intelligent techniques to optimize operation, planning, and protection of the electric power systems.
Hany M. Hasanien
Hany M. Hasanien received his BSc, MSc, and PhD in electrical engineering from Ain Shams University, Faculty of Engineering, Cairo, Egypt, in 1999, 2004, and 2007, respectively. From 2008 to 2011, he was a Joint Researcher with Kitami Institute of Technology, Kitami, Japan. He is an Associate Professor at the Electrical Power and Machines Department, Faculty of Engineering, Ain Shams University. Currently, he is on leave as an Associate Professor at the Electrical Engineering Department, College of Engineering, King Saud University, Riyadh, KSA. He is a senior member of the IEEE and Power & Energy Society (PES). He is an Editorial Board Member of Electric Power Components and Systems. He was awarded the Encouraging Egypt Award for Engineering Sciences in 2012. His research interests include modern control techniques, power systems dynamics and control, renewable energy systems, and smart grids.