Optimal Placement Of Facts Controllers Computer Science Essay
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THE drive towards deregulated environment may result in simultaneous installation of different FACTS controllers in power system. These multiple FACTS controllers have the potential to interact with each other. This interaction may either deteriorate or enhance system stability depending upon the chosen controls and placement of FACTS controllers. Hence there is a need to study the interaction between the FACTS controllers.
The various interactions can potentially occur between the different FACTS controllers, as well as, between FACTS controllers and Power System Stabilizers (PSS) in a multi-machine power system environment. These likely interactions have been classified into different frequency ranges and various interaction problems between FACTS controllers or FACTS to PSS's from voltage stability/ small signal stability viewpoint are presented in [1]-[2].
There are several methods proposed in literatures [3]-[86], [87]-[111], for placement of FACTS controllers in multi-machine power systems from different operating conditions viewpoint. References [3]-[5], classify three broad categories such as a sensitivity based methods, optimization based method, and artificial intelligence based techniques for placement of FACTS controllers from different operating conditions viewpoint in multi-machine power systems. The various sensitivity based methods have been proposed in literatures includes eigen-value analysis based methods [6]-[12], modal analysis techniques [13]-[15], index methods [16]-[23], residue-based methods [24]-[25],[43],[98]-[99], and other sensitivity based methods [26]-[37],[53],[87]-[97],[111]. The various optimization based methods have been proposed in literatures that includes non-linear optimization programming techniques [38],[39],[103], mixed integer-optimization programming techniques [40]-[42],[100]-[101], dynamic optimization programming algorithms [44], hybrid optimization programming algorithms [45], bellmann's optimization principle [46], decomposition coordination methods [47]-[48], curved space optimization techniques [104]. The various artificial intelligence (AI) based methods proposed in literature includes genetic algorithms (GA) [49]-[64],[105]-[106], [110], tabu search algorithms [65],[66], simulated annealing (SA) based approach [69]-[70],[107], particle swarm optimization (PSO) techniques [71]-[73],[80], artificial neural network (ANN) based algorithms [74]-[76], ant colony optimization (ACO) algorithms [77]-[78], graph search algorithms [79], fuzzy logic based approach [81]-[82], other techniques such as norm forms of diffeomorphism techniques [83], evolution strategies algorithms [84],[86], improved evolutionary programming [68], gravitational optimization techniques [85], benders decomposition techniques [42], augmented Lagrange multiplier approach [67], hybrid meta-heuristic approach [102], heuristic and algorithmic approach [108], energy approach [109].
This paper is organized as follows: Section 2 presents the review of various techniques/methods for placements of FACTS controllers in multi-machine power systems. Section 3 presents the summary of the paper. Section 4 presents the conclusions of the paper.
II. CLASSIFICATIONOF FACTS CONTROLLERS ALLOCATION TECHNIQUES
Three broad categories of allocation techniques for determining best suited location of FACTS controllers are sensitivity based methods, optimization based method, and artificial intelligence based techniques [3]-[5].
Sensitivity Based Methods
There are various sensitivity based methods such as a modal or eigen-value and index analysis. An eigen-value analysis based techniques has been proposed in [6] for the selection of parameters of stabilizers in multi-machine power system to enhance the damping of the power system oscillations. Reference [7], presents an eigen-value analysis based algorithms such as a participation factor method for identification of optimum location of power system stabilizers to enhance the damping of power system oscillations. Many literatures shows the existing methods of identifying the optimum sites for installing PSSs in multi-machine power systems are restricted to the sequential PSS application, which considers the enhancement of damping of just one critical electro-mechanical modes at a time and the eigen structure analysis of the open loop system, which does not take the control matrix (the B matrix in the linearized model dX/dt= AX+BU for power system) into considerations. An eigen-value analysis based techniques has been proposed for identifying the optimum sites for installing power system stabilizers (PSSs) in multi-machine power systems in [8]. The advantages of this proposed method is that it can identify the optimum sites for installing PSS so that several electro-mechanical modes are damped out simultaneously and it takes both the eigen structure of the open loop system and the control matrix into consideration. Reference [9], suggests an eigen-value analysis based approach for identifying the most effective FACTS controllers, locations, types and ratings that increase asset utilization of power systems. Reference [10], uses participation factor method has been suggested for the critical mode are used to determine the most suitable sites for SVC (Static Var Compensator) for system reinforcement. In [11]-[12], an eigen-value analysis based approach has been proposed for find the optimal location and rating of FACTS controllers (Static Var Compensator (SVC) and Thyristor Controlled Series Controller(TCSC)) and a continuation power flow is used to evaluate the effects of SVC and TCSC devices on power loadability). In [13] , a modal analysis algorithm has been suggested for allocation and control of FACTS devices for steady-state stability enhancement of large scale power system. A modal analysis algorithm has been suggested for placement of SVCs and selection of stability signals in power systems environments [14]. A new eigen solution free method of modal control analysis for the selection of the robust installing locations and feedback signals of FACTS based stabilizers in large-scale power systems is presented in [15]. An index based approach known as a Location Index for Effective Damping (LIED) method has been proposed for identifying the location of SVC and a Variable Series Capacitor (VSrC) in large-scale power systems for damping effectively in [16]. A structure preserving energy margin sensitivity based analysis has been addressed for determine the effectiveness of FACTS devices to improve transient stability of a power system in [17]. A controllability index method has been proposed for select the input signals for both single and multiple FACTS devices in small and large power systems for damping inter-area oscillations in [18]. Different input output controllability analysis are used to asses the most appropriate input signals (stability signals) for the SVC, the Static Synchronous Series Compensator (SSSC) and the Unified Power Flow Controller (UPFC) for achieving good damping of inter-area oscillations. A new method called the Extended Voltage Phasors Approach (EVPA) has been suggested for placement of FACTS controllers in power systems for identifying the most critical segments/bus in power system from the voltage stability view point in [19]. In [20]-[23], an index based method has been addressed for determine the suitable locations of FACTS devices in power system environments. A residues based approach has been proposed for allocation of FACTS controllers in power system to enhance the system stability [24]-[25]. An efficient algorithm for the solution of two important problems in the area of damping control of electro-mechanical oscillations in large scale power systems has been proposed in [26]. The proposed algorithms allow the determination the most suitable generators for installing power system stabilizers and the most suitable buses in the system for placing SVC in order to damp the critical modes of oscillation. A sensitivity based approach has been proposed for placement of FACTS controllers in open power markets to reduce the flows in heavily loaded lines, resulting in an increased loadability, low system loss, improved stability of the network, reduced cost of production and fulfilled contractual requirement by controlling the power flows in the network in [27]-[28] . A sensitivity based approach called Bus Static Participation Factor (BSPF) has been proposed for determine the optimal location of static VAR compensator (SVC) for voltage security enhancement in [29]. A sensitivity based approach has been proposed to determine the placement of TCSC and UPFC for enhancing the power system loadability [30]. In [31], a sensitivity analysis method has been proposed for determine the optimal placement of static VAR compensator (SVC) for voltage security enhancement in Algerian Distribution System. In [32], a sensitivity analysis and evolutionary programming techniques has been proposed for determine the optimal placement of UPFC in power system from the operational planning viewpoint. Sensitivity analysis is superior when compared to the others as sensitivity analysis gives the best possible installation location. References [33]-[37], presents a sensitivity based approach has been proposed for optimal placement of UPFC to enhance voltage stability margin under contingencies. Reference [87], suggests a normal form analysis approach based on sensitivity indices is used for sitting power systems stabilizers (PSS) in power systems network. Reference [88], suggested a Trajectory Sensitivity Analysis (TSA) technique for the evaluation of the effect of TCSC placement on transient stability. A sensitivity based technique is used for placement of Static Synchronous Series Compensator (SSSC) in power system network in [89]. Sensitivity based screening technique for greatly reducing the computation involved in determining the optimal location of a Unified Power Flow Controller (UPFC) in a power system [90]. A sensitivity analysis and evolution programming technique has been proposed for placement of UPFC in a power system in [91]. In [92], a sensitivity based technique used for determine the minimum amount of shunt reactive power (VAr) support which indirectly maximizes the real power transfer before voltage collapse is encountered. Sensitivity information that identifies weak buses is also available for locating effective VAr injection sites. A sensitivity based technique is used for determine optimal placement of Static Synchronous Compensator (STATCOM) and Unified Power Flow Controllers (UPFC) to enhancement of Dynamic Available Transfer Capability (ATC) under different contingencies in New England System [93]. A sensitivity factor based approach has been used in [94] for the optimal placement of the TCSC to minimize the congestion cost. In [95], a second-order sensitivity analysis technique used for optimal location of SVC and TCSC in power system. An eigen-value sensitivity based approach has been used for location and controller design of TCSC to enhance damping power system oscillations [96]. In [97], a new sensitivity factor, called System Loading Distribution Factor is used for determine the optimal location of UPFC in power system. A residues factor method has been used for determination of optimal location of the TCSC device to damp out the inter-area mode of oscillations [98]. In [99], a residues method based on sensitivity analysis technique is used for determine optimal location of the SVC to enhance the damping power system oscillations. A new approach based on sensitivity indices has been used for the optimal placement of various types of FACTS controllers such as TCSC, TCPAR and SVC in order to minimize total system reactive power loss and hence maximizing the static voltage stability in [111]. Magaji and Mustafa et al. [43] has been suggested a residue factor approach for optimal location of FACTS devices for damping oscillations of power systems. S. N. Singh and I. Erlich et al. [53] proposed for locating UPFC for enhancing power System loadability.
Optimization Based Techniques
This section reviews the optimal placement of FACTS controllers based on various optimization techniques such as a linear and quadratic programming, non-linear optimization programming, integer and mixed integer optimization programming, and dynamic optimization programming.
Non-Linear Optimization Programming (NLP) techniques
When the objective function and the constraints are non-linear, it forms non-linear programming (NLP). The difference between NLP and Linear Programming (LP) is analogous to the difference between a set of solving non-linear equations and a set of solving linear equations. In most of the NLP methods, the approach is to start from an initial guess and to determine a descent direction in which objective function decreases in case of minimization problems [5].
Reference [38], suggests a non-linear optimization programming techniques for assessing the placement of FACTS controllers in power system to damp electro-mechanical oscillations. A non-linear optimization programming techniques has been proposed for optimal network placement of SVC controller in [39] and a Benders Decomposition technique has been used for these solutions.
Integer and Mixed -Integer optimization Programming (IP & MIP) techniques
Reference [40], a mixed integer linear optimization programming algorithm has been proposed for the optimal placement of TCPST in multi-machine power systems. A mixed integer optimization programming algorithm has been proposed for optimal placement of Thyristor Controlled Phase Shifter Transformers (TCPSTs) in large scale power system for active flow and generation limits, and phase shifter constraints in [41]. A mixed integer optimization programming algorithm has been proposed for allocation of FACTS controllers in power system for security enhancement against voltage collapse and corrective controls, where the control effects by the devices to be installed are evaluated together with the other controls such as load shedding in contingencies to compute an optimal VAR planning [42]. In [100], a mixed integer non-linear optimization programming algorithm is used for determine the type, optimal number, optimal location of the TCSC for loadability enhancement in deregulated electricity markets. A mixed integer optimization programming algorithm has been used for optimal location of TCSC in a power system in [101].
Dynamic Programming (DP) techniques
Oliveira et al. suggested a dynamic optimization programming algorithm for allocation of FACTS devices in hydrothermal systems in order to minimize the expected thermal generation costs and the investments on FACTS devices in a pre-specified time interval [44]. Chang and Huang et al. showed that a hybrid optimization programming algorithm for optimal placement of SVC for voltage stability reinforcement [45]. In [46], a bellmann's optimality principle has been proposed for optimal sectionalizing switches allocation in distribution networks and also determines the optimal number of automatic sectionalizing switches devices. Lie and Deng et al. has addressed a decomposition coordination technique for optimal FACTS devices allocation in power system economic dispatch [47]. Zuwei and Lusan et al. presented review on the current status on the optimal placements of FACTS devices in power systems for enhances the damping of power system oscillations [48]. Orfanogianni and Bacher et al. suggested an optimization-based methodology is used for identify key locations of TCSC and UPFC include the nonlinear constraints of voltage limitation, zero megawatt active power exchange, voltage control, and reactive power exchange in the ac networks [103]. In [104], Curved Space Optimization (CSO) programming algorithm is used for allocation of SVC in a large power system.
Artificial Intelligence (AI) Based Techniques
This section reviews the optimal placement of FACTS controllers based on various Artificial Intelligence based techniques such as a Genetic Algorithm (GA), Expert System (ES), Artificial Neural Network (ANN), Tabu Search Optimization (TSO), Ant Colony Optimization (ACO) algorithm, Simulated Annealing (SA) approach, Particle Swarm Optimization (PSO) algorithm and Fuzzy Logic based approach.
Genetic Algorithm(GA)
A genetic algorithm has been addressed for optimal location of phase shifters in the French network to reduce the flows in heavily loaded lines, resulting in an increased loadability of the network and a reduced cost of production [49]. A genetic algorithm has been addressed for optimal location of multiple type FACTS controllers in a power system. The optimization are performed on three parameters; the location of the devices, their types and their values. The system loadability is applied as measure of power system performance. Four different kinds of FACTS controllers are used as models for steady state studies: TCSC, TCPST, Thyristor Controlled Voltage Regulator (TCVR) and SVC in order to minimizing the overall system cost, which comprises of generation cost and investment cost of FACTS controllers [50]. Vijakumar and Kumudinidevi et al. presented a novel method for optimal location of FACTS controllers in a multi-machine power system. The location of FACTS controllers, their type and rated values are optimized simultaneously for the various FACTS controllers, TCSC and UPFC are considered in order to minimizing the overall system cost, which comprises of generation cost and investment cost of FACTS controllers [51]. A stochastic searching algorithm called as genetic algorithm has been proposed for optimal placement of static VAR compensator for enhancing voltage stability in [52]. In [54], a genetic algorithm (GA) based method has been proposed for determine the optimal placement of FACTS controllers in power system with the consideration of economics and cost effectiveness. In [55], a genetic algorithm (GA) based approach has been proposed for the optimal choice and allocation of FACTS devices in deregulated electricity power market is to achieve the power system economic generation allocation and dispatch in deregulated electricity market. The locations of the FACTS controllers, their type and ratings are optimized simultaneously. Reference [56], genetic algorithm (GA) and particle swarm optimization (PSO) has been proposed for optimal location and parameter setting of UPFC for enhancing power system security under single contingencies. A new genetic algorithm (GA) based approach [57] has been addressed for selection of the optimal number and location of UPFC devices in deregulated electric power systems. In [58], a novel method such as a genetic algorithm has been presented for optimal location of FACTS controllers in a multi-machine power system. The location of FACTS controllers, their type and rated values are optimized simultaneously for the various FACTS controllers such as a TCSC and UPFC are considered. Reference [59], a genetic algorithm (GA) has been proposed for location and parameters setting of UPFC for congestion management in pool market model. The planning method such as a micro-genetic algorithm (MGA) has been addressed for optimal type selection and placement of a FACTS device for power system stabilizing purpose in a multi-machine power system [60]. A heuristic approach using genetic algorithm has been addressed for optimal location of FACTS controllers in multi-machine power systems for enhancing the damping of power system oscillations in [61]-[64]. In [105], a non-traditional optimization technique, a Genetic Algorithm (GA) is conjunction with Fuzzy logic (FL) is used to optimize the various process parameters involved in introduction of FACTS devices such as a TCSC, Thyristor Controlled Phase Angle Regulator (TCPAR), SVC and UPFC in a power system. The various parameters taken into consideration were the location of the device, their type, and their rated value of the devices. A multi-objective optimal power flow and genetic algorithms used to determine the optimal number and location of UPFC devices in an assigned power system network for maximizing system capabilities, social welfare and to satisfy contractual requirements in an open market power [106]. An energy approach has been used for the optimal location of FACTS controllers/sensors in large-scale power systems in [109]. Reference [110], a genetic algorithm (GA) has been proposed for optimal choice and allocation of FACTS devices such as UPFC, TCSC, TCPST, and SVC in deregulated electricity market.
Tabu Search Algorithm (TS)
A TS algorithm has been addressed for optimal placement of FACTS controllers such as TCSC, TCPST, UPFC, and SVC in multi-machine power systems [65]-[66]. Reference [102], a hybrid-meta heuristic method based on tabu search in conjunction with evolutionary particle swarm optimization technique has been proposed for optimal location of UPFC in power system.
Simulated Annealing (SA) Algorithms
References [69], [70], a novel optimization based methodology such as a simulated annealing has been proposed for optimal location of FACTS devices such as TCSC and SVC in order to relive congestion in the transmission line while increasing static security margin and voltage profile of power system networks. In [107], the Goal Attainment (GA) method based on the SA approach is applied to solving general multi-objective VAR planning problems by assuming that the Decision Maker (DM) has goals for each of the objective functions. The VAR planning problem involves the determination of location and sizes of new compensators considering contingencies and voltage collapse problems in a power system.
Particle Swarm Optimization (PSO) Algorithms
In [71], a Particle Swarm Optimization (PSO) algorithm has been addressed for the solution of the Optimal Power Flow (OPF) using controllable FACTS controllers to enhance economic solution. Rashed et al. suggested a Genetic Algorithm (GA) and PSO techniques for optimal location and parameter setting of TCSC to improve the power transfer capability, reduce active power losses, improve stabilities of the power network, and decrease the cost of power production and to fulfill the other control requirements by controlling the power flow in multi-machine power system network [72]. A Trinary Particle Swarm Optimization Technique has been proposed for optimal switch placement in distribution systems for achieving high distribution reliability levels and con-currently minimizing capital costs can be considered as the main issues. A novel three state approaches has been proposed for inspired from the discrete version of a powerful heuristic algorithm, PSO is developed and presented to determine the optimal number and locations of two types of switches (sectionalizes and breakers) in radial power systems automation is an important issue from the reliability and economical point of view [73]. In [74]-[76], an Artificial Intelligence Based Techniques has been addressed for optimal placement of FACTS controllers in large scale power system. In [80], a Particle Swarm Optimization (PSO) technique has been addressed for optimal location of FACTS controllers such as TCSC, SVC, and UPFC considering system loadability and cost of installation.
Ant Colony Optimization (ACO) algorithms
In [77], a methodology has been suggested for placement of sectionalizing switches in distribution networks in the presence of distributed generation sources for reliability improvement with consideration of economic aspects. In [78], an ACO algorithm has been addressed for the planning problem of electrical power distribution networks, stated as a mixed non-linear integer optimization problem, is solved using the Ant Colony System (ACS) algorithm. The ACS methodology is coupled with a conventional distribution system load flow algorithm and adapted to solve the primary distribution system planning problem. A Graph Search Algorithm has been addressed for optimal placement of fixed and switched capacitors on radial distribution systems to reduce power and energy losses, increases the available capacity of the feeders, and improves the feeder voltage profile [79].
Fuzzy Logic (FL) Algorithms
References [81]-[82], A fuzzy logic based approach has been addressed for optimal placement and sizing of FACTS controllers in power systems. In [83], the theory of the normal forms of diffeomorphism algorithm has been addressed for the SVC allocation in multi-machine power system for power system voltage stability enhancement. Luna and Maldonado et al. has been addressed a new methodology is based on the evolutionary strategies algorithm known as Evolution Strategies (ES) for optimally locating FACTS controllers in a power system for maximizes the system loadability while keeping the power system operating within appropriate security limits [84]. A Gravitational Optimization (GO) algorithm has been addressed for allocation of SVC in a power system in [85]. Kalyani et al. [86] has been suggested an Evolutionary algorithm for optimal location of UPFC and sequential quadratic programming (SQP) to optimize the UPFC control settings. In [108], a knowledge and algorithm based approach is used to VAR planning in a transmission system. The VAR planning problem involves the determination of location and sizes of new compensators considering contingencies and voltage collapse problems in a power system. Fang and Ngan et al. [67] suggested an augmented Lagrange Multipliers approach for optimal location of UPFC in power systems to enhances the steady state performance and significantly increase the loadability of the system. Hao et al. [68] has been proposed an improved evolutionary programming technique for optimal location and parameters settings of UPFCs to maximize the system laudability subject to the transmission line capability and specified voltage level.
III. SUMMARY OF THE PAPER
The following tables give summary of the paper as:
Methods/Techniques for Placement of FACTS controllers
Methods/Techniques point of view
Methods/Techniques
Total No. of Literatures Reviews out of 106 Literatures
% of Literatures Reviews out of 106 Literatures
Sensitivity based methods
48
45.28
Optimization based methods
14
13.20
AI-based techniques
44
41.51
Operating Parameters point of view
Operating Parameters of Power systems
Total No. of Literatures Reviews out of 106 Literatures
% of Literatures Reviews out of 106 Literatures
Damping of power system oscillations
16
15.09
Voltage Profile
20
18.87
Security of the power system
02
1.89
Small signal stability, transient stability
06
5.66
Power transfer capability through the lines
02
1.89
Dynamic performances of power systems
02
1.89
The loadability of the power system network
12
11.32
Others parameters point of view
46
43.39
From above tables it is concluded that the 45.28% of total literatures are reviews based on sensitivity methods, 13.20% of total literatures are reviews based on optimization programming and the 41.51% of total literatures are reviews on AI based techniques regarding with placement of FACTS controllers in multi-machine power systems. It is also concludes that the 15.09% of total literatures are reviews carryout from damping of power system oscillations, 18.87% of total literatures are reviews carryout from voltage stability, 1.89% of total literatures are reviews carryout from security of power system, 5.66% of total literatures are reviews carryout from small signal/transient/dynamic stability, 1.89% of total literatures are reviews carryout from power transfer capability through the lines, 1.89% of total literatures are reviews carryout from dynamic performance of power system, 11.32% of total literatures are reviews carryout from the loadability of power system, and 43.39% of total literatures are reviews carryout from other parameters viewpoints.
Finally it is concluded that the maximum research work carryout from damping of power system oscillations and voltage stability point of view regarding with placement of FACTS controllers in multi-machine power systems.
IV. CONCLUSIONS
In this paper an attempt has been made to reviews, various AI based optimization methods used for the placement of FACTS controllers. Even through, excellent advancements have been made in classical methods i. e. sensitivity based method, they suffer with the following disadvantages: In most cases, mathematical formulations have to be simplified to get the solutions because of the extremely limited capability to solve real-word large-scale power system problems. They are weak in handling qualitative constraints. They have poor convergence, may get stuck at local optimum, they can find only a single optimized solution in a single simulation run, they become too slow if number of variables are large and they are computationally expensive for solution of a large power system.
Whereas, the major advantage of the AI based optimization methods is that they are relatively versatile for handling various qualitative constraints. AI methods can find multiple optimal solutions in single simulation run. So they are quite suitable in solving multi-objective optimization problems for placement FACTS controllers in multi-machine power system. In most cases, they can find the global optimum solution. The main advantages of ANN are: possesses learning ability, fast, appropriate for non-linear modeling, etc. whereas, large dimensionality and the choice of training methodology are some disadvantages of ANN. The advantages of Fuzzy method are: Accurately represents the operational constraints and fuzzified constraints are softer than traditional constraints. The advantages of GA methods are: It only uses the values of the objective function and less likely to get trapped at a local optimum. Higher computational time is its disadvantage. The advantages of EP are adaptability to change, ability to generate good enough solutions and rapid convergence. ACO and PSO are the latest entry in the field of optimization. The main advantages of ACO are positive feedback for recovery of good solutions, distributed computation, which avoids premature convergence. It has been mainly used in finding the shortest route in transmission network, short-term generation scheduling and optimal unit commitment. PSO can be used to solve complex optimization problems, which are non-linear, non-differentiable and multi-model. The main merits of PSO are its fast convergence speed and it can be realized simply for less parameters need adjusting. PSO has been mainly used to solve bi-objective generation scheduling, optimal reactive power dispatch and to minimize total cost of power generation. The applications of ACO and PSO for placement of FACTS controllers in multi-machine power system.
This paper has also addressed a survey of several technical literature concerned with various techniques/methods for placement FACTS controllers in multi-machine power system environments to show that the achieve significant improvements in operating parameters of the power systems such as small signal stability, transient stability, damping of power system oscillations, security of the power system, less active power loss, voltage profile, congestion management, quality of the power system, efficiency of power system operations, power transfer capability through the lines, dynamic performances of power systems, and the loadability of the power system network also increased. This review also shows that installing multiple controllers on the system may not improve the dynamic performance due to undesirable interactions. The tuning of one controller may affect other controllers and thus lead to unstable conditions. These issues should be taken into consideration when designing systems with multiple controllers. The implementation of a coordinated controller tuning procedure to avoid undesirable interactions in power systems, and thus improve overall dynamic performance is under this review.
Authors strongly believe that this survey article will be very much useful to the researchers for finding out the relevant references as well as the previous work done in the field of placement of FACTS Controllers for the various methods/techniques for placement of FACTS controllers in multi-machine power systems. So that further research work can be carried out.
ACKNOWLEDGMENT
The authors would like to thanks Dr. S. C. Srivastava, and Dr. S. N. Singh, Indian Institute of Technology, Kanpur, U.P., India, and Dr. K.S. Verma, and Dr. Deependra Singh, Kamla Nehru Institute of Technology, Sultanpur, U.P., India, for their valuables suggestions regarding placement and coordination techniques for FACTS controllers form voltage stability, and voltage security point of view in multi-machine power systems environments.
