A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> The paper presents an integrated framework to analyze the issues of reactive power planning along with reactive power pricing. The planning problem involves optimal placement and sizing of capacitors in a network such that operating and investment costs are minimum. A simple bus-wise cost-benefit analysis (CBA) scheme is proposed which involves solving a modified optimal power flow problem (OPF) iteratively. The proposed CBA incorporates detailed hourly loading conditions at a bus and achieves a fairly accurate estimate of the benefits from capacitor placement. The formulation is directly handled by the well known MINOS code and is solved efficiently. It obviates the need to introduce integer variables and is thus suitable for large system applications. A two-part reactive power spot-pricing scheme is formulated, by which the investment and operational costs can be recovered by the utility. The proposed reactive power price has two parts-a fixed part to account for the investment costs of new capacitor at a bus and a variable spot price to account for the operating costs incurred in supplying the additional reactive power from generating units. <s> BIB001 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper presents a comparative study for three evolutionary algorithms (EAs) to the optimal reactive power planning (ORPP) problem: evolutionary programming, evolutionary strategy, and genetic algorithm. The ORPP problem is decomposed into P- and Q-optimization modules, and each module is optimized by the EAs in an iterative manner to obtain the global solution. The EA methods for the ORPP problem are evaluated against the IEEE 30-bus system as a common testbed, and the results are compared against each other and with those of linear programming. <s> BIB002 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> The paper proposes an application of hybrid evolutionary programming to reactive power planning. Reactive power planning is a nonsmooth and nondifferentiable optimization problem for a multiobjective function. The objective functions deals with the minimization of operating cost by reducing real power loss, improving the voltage profile and minimizing the allocation cost of additional reactive power sources. The proposed method is developed in such a way that a standard evolutionary programming (EP) method is acting as a base level search, which makes a quick decision to direct the search towards the optimal region, and local optimization by direct search and systematic reduction in size of search region method is next employed to do fine tuning. The reactive power planning using hybrid evolutionary programming is demonstrated with the IEEE 30-bus system. The comprehensive simulation results show that hybrid evolutionary programming is a suitable method to solve the reactive power planning problem. <s> BIB003 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Reactive power planning (RPP) involves optimal allocation and determination of the types and sizes of the installed capacitors. Traditionally, the locations for placing new VAr sources were either simply estimated or directly assumed. Essentially, it is a large-scale nonlinear optimization problem with a large number of variables and uncertain parameters. there are no known ways to solve such nonlinear programming problems (NLP) exactly in a reasonable time. This paper introduces mathematical formulations and the advantages and disadvantages of nine categories of methods, which are divided into two groups, conventional and advanced optimization methods. The first group of methods is often trapped by a local optimal solution; the second one can guarantee the global optimum but needs more computing time. The corresponding development history period is also split into a conventional period and an artificial intelligence period. In the second period, there is a tendency to integrate several approaches to solve the RPP problem. <s> BIB004 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper proposes an application of the Differential Evolution optimization algorithm to reactive power planning. This problem could be formulated mathematically as a nonlinear, non smooth, mixed-integer, multiobjective optimization problem. One of the objectives deals with the minimization of the real power losses that result in a reduction of the operational costs and improving the voltage profile across the system. The other objective minimizes the allocation of additional reactive power sources throughout the system. The proposed approach has been tested in a modified version of the IEEE 30-bus test system with satisfactory results. These results were compared with those obtained by another evolutionary computation technique (Evolutionary Programming). <s> BIB005 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Deregulation in the electricity supply industry has brought many new challenges to the problem of reactive power planning. Although the problem has been extensively studied, available standard optimization models and methods do not offer good solutions to this problem, especially in a competitive electricity market environment where many factors are uncertain. Given this background, a novel method for reactive power planning based on chance constrained programming is presented in this paper, with uncertain factors taken into account. A stochastic optimization model is first formulated under the presumption that the generator outputs and load demands can be modeled as specified probability distributions. A method is then presented for solving the optimization problem using the Monte Carlo simulation method and genetic algorithm. Finally, a case study is used to illustrate the validity and essential features of the proposed model and methodology. <s> BIB006 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Increased load forecasts can severely deteriorate the performance of a power system. Reactive compensation devices are a common method to allow a power system to return to an acceptable performance level for an expected load. Reactive power planning (RPP) is used to determine the optimal placement of reactive devices for a set of objectives. RPP is a large scale multiple objectives highly constrained and partially discrete optimization problem that is very difficult to solve. Evolutionary algorithms have been used to solve RPP problems. However, new multi-objective evolutionary computational techniques have shown the ability to consider an optimization problem's objectives independently for the determination of Pareto Optimal solutions. This paper aims at applying the Non-Dominated Sorting Genetic Algorithm II (NSGAII) to a multi-objective RPP. The results from the case study presented show that there is great potential in the use of evolutionary computation for solving the multi-objective RPP. <s> BIB007 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> The key of reactive power planning (RPP), or Var planning, is the optimal allocation of reactive power sources considering location and size. Traditionally, the locations for placing new Var sources were either simply estimated or directly assumed. Recent research works have presented some rigorous optimization-based methods in RPP. This paper will first review various objectives of RPP. The objectives may consider many cost functions such as variable Var cost, fixed Var cost, real power losses, and fuel cost. Also considered may be the deviation of a given voltage schedule, voltage stability margin, or even a combination of different objectives as a multi-objective model. Secondly, different constraints in RPP are discussed. These different constraints are the key of various optimization models, identified as optimal power flow (OPF) model, security-constrained OPF (SCOPF) model, and SCOPF with voltage-stability consideration. Thirdly, the optimization-based models will be categorized as conventional algorithms, intelligent searches, and fuzzy set applications. The conventional algorithms include linear programming, nonlinear programming, mixed-integer nonlinear programming, etc. The intelligent searches include simulated annealing, evolutionary algorithms, and tabu search. The fuzzy set applications in RPP address the uncertainties in objectives and constraints. Finally, this paper will conclude the discussion with a summary matrix for different objectives, models, and algorithms. <s> BIB008 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper proposes an application of evolutionary programming (EP) to reactive power planning (RPP) considering the highest load buses, instead of considering the randomly selected load buses to install the reactive power sources. The proposed approach has been used in the IEEE 30-bus system. Simulation results, compared with those obtained by using the previous method of RPP using EP, considering the randomly selected load buses to install the reactive power sources are presented to show that, the present method is better for power system planning. <s> BIB009 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper presents a particle swarm optimization (PSO) approach for solving the contingency constrained optimal reactive power planning (CCORPP) problem. Generator bus voltage magnitudes, transformer tap settings and reactive power generation of capacitor bank are taken as the control variables. While solving the RPP problem using particle swarm optimization, the voltage magnitudes, the transformer tap setting and the reactive power generations of capacitor are taken as continuous variables. The proposed algorithm has been applied to find the optimal reactive power control variables in IEEE 30-bus system and is found to be more effective than the conventional methods and the Evolutionary Programming. <s> BIB010 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Optimal reactive power dispatch problem in power systems has thrown a growing influence on secure and economical operation of power systems. However, this issue is well known as a nonlinear, multimodal and mixed-variable problem. In the last decades, computation intelligence-based techniques, such as genetic algorithms (GAs), differential evolution (DE) algorithms and particle swarm optimization (PSO) algorithms, etc., have often been used for this aim. In this work, a seeker optimization algorithm (SOA)-based reactive power dispatch method is proposed. The SOA is based on the concept of simulating the act of human searching, where the search direction is based on the empirical gradient by evaluating the response to the position changes and the step length is based on uncertainty reasoning by using a simple Fuzzy rule. In this study, the algorithm's performance is evaluated on benchmark function optimization. Then, the SOA is applied to optimal reactive power dispatch on standard IEEE 57- and 118-bus power systems, and compared with conventional nonlinear programming method, two versions of GAs, three versions of DE algorithms and four versions of PSO algorithms. The simulation results show that the proposed approach is superior to the other listed algorithms and can be efficiently used for optimal reactive power dispatch. <s> BIB011 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Optimal reactive power dispatch (ORPD) has a growing impact on secure and economical operation of power systems. This issue is well known as a non-linear, multi-modal and multi-objective optimization problem where global optimization techniques are required in order to avoid local minima. In the last decades, computation intelligence-based techniques such as genetic algorithms (GAs), differential evolution (DE) algorithms and particle swarm optimization (PSO) algorithms, etc., have often been used for this aim. In this work, a seeker optimization algorithm (SOA) based method is proposed for ORPD considering static voltage stability and voltage deviation. The SOA is based on the concept of simulating the act of human searching where search direction is based on the empirical gradient by evaluating the response to the position changes and step length is based on uncertainty reasoning by using a simple Fuzzy rule. The algorithm's performance is studied with comparisons of two versions of GAs, three versions of DE algorithms and four versions of PSO algorithms on the IEEE 57 and 118-bus power systems. The simulation results show that the proposed approach performed better than the other listed algorithms and can be efficiently used for the ORPD problem. <s> BIB012 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> A metaheuristic technique for solving the short-term transmission network expansion and reactive power planning problems, at the same time, in regulated power systems using the AC model is presented. The problem is solved using a real genetic algorithm (RGA). For each topology proposed by RGA an indicator is employed to identify the weak buses for new reactive power sources allocation. The fitness function is calculated using the cost of each configuration as well as constraints deviation of an AC optimal power flow (OPF) in which the minimum reactive generation of new reactive sources and the active power losses are objectives. With allocation of reactive power sources at load buses, the circuit capacity increases and the cost of installation could be decreased. The method is tested in a well known test system, presenting good results when compared with other approaches. <s> BIB013 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Reactive Power Optimisation (RPO) is an important issue in deregulated electricity market. This paper presents a Hybrid Particle Swarm Optimisation (HPSO) approach for solving the RPO problem in electricity market. Particle Swarm Optimisation (PSO) is a population-based search algorithm for global optimisation and has a minimum number of control parameters. In the proposed HPSO, to avoid the possibility of trapping in local optimum, genetic operators namely crossover and mutation are applied to the PSO algorithm. The proposed algorithm has been applied to find the optimal reactive power control variables in IEEE 30-bus system and promising results have been obtained. <s> BIB014 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper addresses an optimal Reactive Power Planning (RPP) of power system. The Static Var Compensator (SVC) is introduced into power system in order to reactive power support and voltage control. The locations and the outputs of SVCs are determined using our proposed optimal reactive power planning model. The proposed method optimizes several objective functions at the same time within one general objective. The optimized objectives are minimization of total investment in reactive power support, average voltage deviation and minimization of total system loss. These objective functions are one of the most important objectives for every transmission and distribution systems. Harmony Search Algorithm (HS) is used to solve the optimization problem. The validity of the proposed method is tested on a typical power system. <s> BIB015 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Abstract This paper addresses an application of modified NSGA-II (MNSGA-II) by incorporating controlled elitism and dynamic crowding distance (DCD) strategies in NSGA-II to multiobjective optimal reactive power dispatch (ORPD) problem by minimizing real power loss and maximizing the system voltage stability. To validate the Pareto-front obtained using MNSGA-II, reference Pareto-front is generated using multiple runs of single objective optimization with weighted sum of objectives. For simulation purposes, IEEE 30 and IEEE 118 bus test systems are considered. The performance of MNSGA-II, NSGA-II and multiobjective particle swarm optimization (MOPSO) approaches are compared with respect to multiobjective performance measures. TOPSIS technique is applied on obtained non-dominated solutions to determine best compromise solution (BCS). Karush–Kuhn–Tucker (KKT) conditions are also applied on the obtained non-dominated solutions to substantiate a claim on optimality. Simulation results are quite promising and the MNSGA-II performs better than NSGA-II in maintaining diversity and authenticates its potential to solve multiobjective ORPD effectively. <s> BIB016 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Reactive power dispatch (RPD) is one of the important tasks in the operation and control of power system. This paper presents an efficient and reliable evolutionary-based approach to solve the RPD problem. The proposed approach employs differential evolution (DE) algorithm for optimal settings of RPD control variables. The proposed approach is examined and tested on the standard IEEE 30-bus test system with different objectives that reflect power losses minimization, voltage profile improvement, and voltage stability enhancement. The simulation results of the proposed approach are compared to those reported in the literature. The results demonstrate the potential of the proposed approach and show its effectiveness and robustness to solve the RPD problem. <s> BIB017 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Static Var compensator (SVC) is normally used in power system to improve voltage profile and reduce system power losses. In this paper, a relatively new optimization technique named as the improved harmony search algorithm (IHS) is applied to determine optimal location and size of SVC devices in a transmission network. A multi-criterion objective function comprising of both operational objectives and investment costs is considered. The results on the 57-bus test system showed that the IHS algorithm give lower power loss and better voltage improvement compared to the particle swarm optimization technique in solving the SVC placement and sizing problem. <s> BIB018 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper discusses the application of Modified Non-Dominated Sorting Genetic Algorithm-II (MNSGA-II) to multi-objective Reactive Power Planning (RPP) problem. The three objectives considered are minimization of combined operating and VAR allocation cost, bus voltage profile improvement and voltage stability enhancement. For maintaining good diversity in nondominated solutions, Dynamic Crowding Distance (DCD) procedure is implemented in NSGA-II and it is called as MNSGA-II. The standard IEEE 30-bus test system, practical 69-bus Indian system and IEEE 118-bus system are considered to analyze the performance of MNSGA-II. The results obtained using MNSGA-II are compared with NSGA-II and validated with reference pareto-front generated by conventional weighted sum method using Covariance Matrix Adapted Evolution Strategy (CMA-ES). The performance of NSGA-II and MNSGA-II are compared with respect to best, mean, worst and standard deviation of multi-objective performance measures namely gamma, spread, minimum spacing and Inverted Generational Distance (IGD) in 15 independent runs. The results show the effectiveness of MNSGA-II and confirm its potential to solve the multi-objective RPP problem. A decision-making procedure based on Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is used for finding best compromise solution from the set of pareto-solutions obtained through MNSGA-II. <s> BIB019 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> In deregulated electric power systems, one of the most important topics is to augment the total system social welfare. Many different methods have been carried out for this problem. One of the most viable methods to increasing system social welfare is to supply the reactive demands as locally. In this way, the reactive demands are supplied via locally reactive sources and the capacity of transmission lines is not used to transfer reactive power of demands. Thus, the congestion of lines is not reached and the system social welfare which is related to the congestion of lines is increased. In this scope, this study addresses an optimal Reactive Power Planning (RPP) in a deregulated power system. The proposed method optimizes two objective functions at the same time within one general objective. The optimized objectives are minimization of total investment in reactive power support and maximizing of total system social welfare. Genetic Algorithm (GA) is used to solve the optimization problem. The validity of the proposed method is verified on a typical power system. <s> BIB020 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper presents a multi-objective optimization methodology to solve the Optimal Reactive Power Flow (ORPF) problem. The e-constraint approach is implemented for the Multi-objective Mathematical Programming (MMP) formulation. The solution procedure uses Mixed Integer Non-Linear Programming (MINLP) model due to discrete variables, such as the tap settings of transformers and the reactive power output of capacitor banks. The optimum tap settings of transformers are directly determined in terms of the admittance matrix of the network since the admittance matrix is constructed in the optimization framework as additional equality constraints. The optimization problem is modeled in General Algebraic Modeling System (GAMS) software and solved using DICOPT solver. Simulation results are implemented on the IEEE 14-, 30-, and 118-bus test systems to simultaneously optimize the total fuel cost, power losses and the system loadability as objective functions. The simulation results show that the proposed algorithm is suitable and effective for the reactive power planning. <s> BIB021 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Abstract This work proposes a two-phase iterative simulation optimization approach to solve the optimal volt-ampere reactive (VAR) sources planning problem. In the upper phase, an Ordinal Optimization (OO)-based method is employed as the optimization methodology, which performs ranking and selection to choose better solutions from a candidate solution set based on the structural information of system extracted from the simulation methodology performed in the lower phase. The simulation methodology, which is based on the Dual Projected Pseudo Quasi-Newton (DPPQN) method, is employed in the lower phase. The proposed OO-based approach consists of four levels. The size and solution quality of the resulting candidate solution set is reduced and improved level by level, and the best solution obtained in the final level is the good enough capacitor placement pattern that we seek. Via numerous tests, this work demonstrates the efficiency of the proposed approach with respect to the genetic algorithm (GA) and tabu search (TS) on the IEEE 118-bus system. Test results show that the proposed approach is an excellent alternative for the optimal VAR sources planning problems. <s> BIB022 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Abstract In this paper a mathematical model for solving simultaneous transmission network expansion and reactive power planning problem (TEPRPP) via an AC model has been presented. A Real Genetic Algorithm (RGA) combined an Interior Point Method (IPM) aimed to obtaining a significant quality solution for such a problem has been employed. The proposed algorithm is tested on three systems; IEEE 24-bus system, 46-bus south Brazilian and the Southeast Network of Iran (SNI). The obtained results show the capability and the viability of the proposed methodology incorporating the AC model of the TEPRPP problem even in real world. <s> BIB023 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Abstract Reactive power dispatch (RPD) is one of the major issues of modern energy management system. This article presents an efficient genetic algorithm (GA) approach for modelling and solving RPD problem of power system in the framework of fuzzy goal programming (FGP) in uncertain environment. In the proposed approach, the objectives of the problem concerned with RPD problem are fuzzily described. In the solution process, the proposed GA method is used in the framework of FGP model in an iterative manner to reach a satisfactory decision. The proposed approach is tested on the standard IEEE 6-Generator 30-Bus System and compared with the solutions obtained in previous study. <s> BIB024 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper proposes an efficient differential evolution (DE) algorithm for the solution of the optimal reactive power dispatch (ORPD) problem. The main objective of ORPD is to minimize the total active power loss with optimal setting of control variables. The continuous control variables are generator bus voltage magnitudes. The discrete control variables are transformer tap settings and reactive power of shunt compensators. In DE algorithm the other form of differential mutation operator is used. It consists to add the global best individual in the differential mutation operator to improve the solution. The DE algorithm solution has been tested on the standard IEEE 30-Bus test system to minimize the total active power loss without and with voltage profile improvement. The results have been compared to the other heuristic methods such as standard genetic algorithm and particle swarm optimization method. Finally, simulation results show that this method converges to better solutions. <s> BIB025 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> Reactive power planning is vital for maintaining the voltage stability of power systems and evolutionary algorithms are highly useful for achieving this task. This paper compares the eectiveness of the dierential evolution (DE) and evolutionary programming (EP) algorithms in optimizing the reactive power planning of power systems under line outage contingency conditions. DE is ecient in exploration through the search space of the problem, while EP is simple and easy to implement. The low cost but fast response thyristor-controlled series capacitor (TCSC) exible alternating current transmission system (FACTS) device is incorporated to control the power ows. The optimal settings of the control variables of the generator voltages, transformer tap settings, and location and parameter settings of the TCSC are considered for reactive power planning and the resultant reactive power reserves. The eectiveness of the <s> BIB026 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> In this paper, a new methodology for Transmission Expansion Planning (TEP) in deregulated electricity market is presented. The proposed TEP is associated with Reactive Power Planning (RPP), reliability assessment and also consideration of wind and load uncertainties. The proposed planning aims at investment cost minimization, social welfare maximization and satisfying reliability constraint at the same time with taking into account wind and load uncertainties. Expected Energy Not Supplied (EENS) is used as an index for reliability evaluation. At first, Monte-Carlo simulation is used to obtain the Probability Density Function (PDF) of Wind Turbine Generator (WTG) output. Then, the WTG and load uncertainties are considered in TEP formulation. Particle Swarm Optimization (PSO) method is considered to solve the proposed planning problem which is a constrained nonlinear mixed integer optimization programming. Simulation results on two standard test systems (Garver and RTS systems) verify the effectiveness of the proposed planning for consideration of wind and load uncertainties in TEP problem under electricity market environment. Also, the proposed method leads to reduction of the total investment cost, the reliability improvement and the social welfare maximization. <s> BIB027 </s> A review of meta-heuristic algorithms for reactive power planning problem <s> General formulation of the RPP problem <s> This paper proposed an application of New Voltage Stability Index(NVSI) to Reactive Power Planning(RPP) using Soft Computing Technique based Differential Evolution(DE).NVSI is used to identify the weak buses for the Reactive Power Planning problem which involves process of experimental by voltage stability analysis based on the load variation. The Formulation of a New Voltage Stabilty Index(NVSI),which is originates from the equation of a two bus network,neglecting the resistance of (...) <s> BIB028
The purpose of the RPP problem is to determine ''where" and ''how many" new VAR compensators must be added to a network for a predefined horizon of planning at minimum cost while satisfying an adequate voltage profile during normal conditions and contingencies. Fig. 1 illustrates the flowchart of the RPP problem. After defining the system data, the generation/load patterns are developed for a predefined horizon of planning. Then, the optimal locations of new reactive power sources are identified. They may be selected according to certain indices or all load buses may be considered as candidate buses BIB008 BIB004 . After that, the control variables (RPP variables) are optimized to achieve certain objective functions subject to set of equality and inequality constraints. Control variables include generator bus terminal voltages, reactive power generation of existing and new VAR sources and transformer tap ratio. The generator bus voltages are continuous in nature, while both reactive power generation of existing and new VAR sources and transformer tap ratio are discrete. The dependent variables include load bus voltage magnitude, active power generation at slack bus, the power flows through the transmission lines, and reactive power outputs of the generators. There are various objective functions that have been utilized in the RPP problem such as minimization of VAR investment cost and system operational cost of real power losses, improvement of voltage profile, and enhancement of voltage stability. However, the modeling of each objective has different shapes. Conventionally, the classical objective of the RPP problem is to achieve the minimum investment cost of additional reactive power supplies and minimize the system operational cost of power losses BIB013 BIB001   BIB006   BIB028  BIB002 BIB009 BIB003 BIB010 BIB014 as follows: where I C is the investment cost of new reactive power supplies and O C is the operational cost of power losses. The investment costs of VAR sources can be generally modeled with two components, a fixed installation cost at bus i (e i ) and a variable purchase cost of capacitive or inductive source at bus i (C ci | Q ci |),    BIB007 BIB006 BIB019 BIB005  BIB028  BIB002 BIB009 BIB003 BIB010 BIB014 as follows: where N c is the reactive compensator buses. This model requires considering the reactive power devices to be already installed before the optimization for its size. On the other hand, another general model of I C has been used as BIB013 BIB027 BIB023 BIB020 BIB015 : where N b is the total number of busses, and b C is the binary decision variables for installing capacitive source. Although the complexity of using binary variables to indicate whether the VAR source will be installed, this model will give a chance to consider all load buses to be candidates to install new reactive power sources. Traditionally, the annual cost of energy losses has been used as a direct measure to the operational costs (O C ) BIB013    BIB006 BIB019 BIB005   BIB028  BIB009 BIB003 BIB010 BIB014 as follows: where h is the per unit energy cost, d L is the duration of load level (h), N L is the number of load level duration, and P L loss s the real power loss during the period of load level L. On the other side, the minimization of network transmission power losses (P loss ) has been sometimes used directly instead of converting it to operational costs in the reactive power operation BIB021 BIB024 BIB016  BIB025 BIB017 BIB011 BIB012 and planning BIB022 BIB015 BIB018 BIB026 . Also, the power system has to satisfy equality and inequality constraints corresponding to the load flow model and operational variables as follows: 0 6 Q C e 6 Q max C e ; e ¼ 1; 2; . . . . . . N C ð11Þ where V i and V j are voltages at buses i and j, respectively; h ij is phase angle between buses i and j; G ij and B ij are mutual conductance and susceptance between buses i and j, respectively; (P gi À P Li ) and (Q gi À Q Li ) are the net real power injection at bus i, and the net reactive power injection at bus i, respectively; Q Ci is the capacitive or inductive power of existing VAR source installed at bus i. Q Ci n refers to the capacitive or inductive power of new VAR source installed at bus i. Q gi is the reactive power output of a generator i, and N pv refers to the total number of voltage-controlled buses. V i is the voltage magnitude of bus i. T k is the tapping change of a transformer k, and N t refers to the total number of on-load tap changing transformers. S flow refers to the apparent power flow, S max is the maximum MVA rating of the transmission lines and transformers, and N L refers to all transmission lines in the system. Q Ce is the reactive power output of existing VAR source at bus e, Q max C s its maximum capacity, and N C refers to the total number of existing VAR sources. n refers to the new installed VAR sources, and b C is always equal 1 for the investment cost of VAR sources modeled in Eq. (3). P s is the active power generation at the slack bus.