Search results for: optimal reactive power dispatch

Authors: Norziana Aminudin, Titik Khawa Abdul Rahman, Ismail Musirin

Abstract:

One of the factors to maintain system survivability is the adequate reactive power support to the system. Lack of reactive power support may cause undesirable voltage decay leading to total system instability. Thus, appropriate reactive power support scheme should be arranged in order to maintain system stability. The strength of a system capacity is normally denoted as system loadability. This paper presents the enhancement of system loadability through optimal reactive power planning technique using a newly developed optimization technique, termed as Multiagent Immune Evolutionary Programming (MAIEP). The concept of MAIEP is developed based on the combination of Multiagent System (MAS), Artificial Immune System (AIS) and Evolutionary Programming (EP). In realizing the effectiveness of the proposed technique, validation is conducted on the IEEE-26-Bus Reliability Test System. The results obtained from pre-optimization and post-optimization process were compared which eventually revealed the merit of MAIEP.

Keywords: Load margin, MAIEP, Maximum loading point, ORPP.

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Authors: Badr M. Alshammari, T. Guesmi

Abstract:

This study presents a modified version of the artificial bee colony (ABC) algorithm by including a local search technique for solving the non-convex economic power dispatch problem. The local search step is incorporated at the end of each iteration. Total system losses, valve-point loading effects and prohibited operating zones have been incorporated in the problem formulation. Thus, the problem becomes highly nonlinear and with discontinuous objective function. The proposed technique is validated using an IEEE benchmark system with ten thermal units. Simulation results demonstrate that the proposed optimization algorithm has better convergence characteristics in comparison with the original ABC algorithm.

Keywords: Economic power dispatch, artificial bee colony, valve-point loading effects, prohibited operating zones.

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Authors: Jagabondhu Hazra, Avinash Sinha

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Economic dispatch problem is an optimization problem where objective function is highly non linear, non-convex, non-differentiable and may have multiple local minima. Therefore, classical optimization methods may not converge or get trapped to any local minima. This paper presents a comparative study of four different evolutionary algorithms i.e. genetic algorithm, bacteria foraging optimization, ant colony optimization and particle swarm optimization for solving the economic dispatch problem. All the methods are tested on IEEE 30 bus test system. Simulation results are presented to show the comparative performance of these methods.

Keywords: Ant colony optimization, bacteria foraging optimization, economic dispatch, evolutionary algorithm, genetic algorithm, particle swarm optimization.

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Authors: Badr M. Alshammari, T. Guesmi

Abstract:

In recent years, the combined economic and emission power dispatch is one of the main problems of electrical power system. It aims to schedule the power generation of generators in order to minimize cost production and emission of harmful gases caused by fossil-fueled thermal units such as CO, CO₂, NO_x, and SO₂. To solve this complicated multi-objective problem, an improved version of the particle swarm optimization technique that includes non-dominated sorting concept has been proposed. Valve point loading effects and system losses have been considered. The three-unit and ten-unit benchmark systems have been used to show the effectiveness of the suggested optimization technique for solving this kind of nonconvex problem. The simulation results have been compared with those obtained using genetic algorithm based method. Comparison results show that the proposed approach can provide a higher quality solution with better performance.

Keywords: Power dispatch, valve point loading effects, multiobjective optimization, Pareto solutions.

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Authors: M. Tarafdar Haque, A.M. Kashtiban

Abstract:

The electric power industry is currently undergoing an unprecedented reform. One of the most exciting and potentially profitable recent developments is increasing usage of artificial intelligence techniques. The intention of this paper is to give an overview of using neural network (NN) techniques in power systems. According to the growth rate of NNs application in some power system subjects, this paper introduce a brief overview in fault diagnosis, security assessment, load forecasting, economic dispatch and harmonic analyzing. Advantages and disadvantages of using NNs in above mentioned subjects and the main challenges in these fields have been explained, too.

Keywords: Neural network, power system, security assessment, fault diagnosis, load forecasting, economic dispatch, harmonic analyzing.

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Authors: Farhad Namdari, Reza Sedaghati

Abstract:

Economic dispatch (ED) has been considered to be one of the key functions in electric power system operation which can help to build up effective generating management plans. The practical ED problem has non-smooth cost function with nonlinear constraints which make it difficult to be effectively solved. This paper presents a novel heuristic and efficient optimization approach based on the new Bat algorithm (BA) to solve the practical non-smooth economic dispatch problem. The proposed algorithm easily takes care of different constraints. In addition, two newly introduced modifications method is developed to improve the variety of the bat population when increasing the convergence speed simultaneously. The simulation results obtained by the proposed algorithms are compared with the results obtained using other recently develop methods available in the literature.

Keywords: Non-smooth, economic dispatch, bat-inspired, nonlinear practical constraints, modified bat algorithm.

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Authors: N. Phanthuna V. Phupha N. Rugthaicharoencheep, S. Lerdwanittip

Abstract:

This paper presents an optimization technique to economic load dispatch (ELD) problems with considering the daily load patterns and generator constraints using a particle swarm optimization (PSO). The objective is to minimize the fuel cost. The optimization problem is subject to system constraints consisting of power balance and generation output of each units. The application of a constriction factor into PSO is a useful strategy to ensure convergence of the particle swarm algorithm. The proposed method is able to determine, the output power generation for all of the power generation units, so that the total constraint cost function is minimized. The performance of the developed methodology is demonstrated by case studies in test system of fifteen-generation units. The results show that the proposed algorithm scan give the minimum total cost of generation while satisfying all the constraints and benefiting greatly from saving in power loss reduction

Keywords: Particle Swarm Optimization, Economic Load Dispatch, Generator Constraints.

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Authors: P. Nammalvar, P. Meganathan, A. Balamuguran

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Among various active filters, shunt active filter is a viable solution for reactive power and harmonics compensation. In this paper, the SRF plan is used to generate current reference for compensation and conventional PI controllers were used as the controller to compensate the reactive power. The design of the closed loop controllers is reserved simple by modeling them as first order systems. Computationally uncomplicated and efficient SVM system is used in the present work for better utilization of dc bus voltage. The rating of shunt active filter has been finalized based on the reactive power demand of the selected reactive load. The proposed control and SVM technique are validated by simulating in MATLAB software.

Keywords: Shunt Active Filter, Space vector pulse width modulation, Voltage Source Converter, Reactive Power, Synchronous Reference Frame, Point of common coupling.

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Authors: Reza Sedaghati, Mojtaba Hakimzadeh, Mohammad Hasan Raouf, Mostafa Mirzadeh

Abstract:

Nowadays the growth of distributed generation within the bulk power system is feasible by using the optimal control of the transmission lines power flow. Static Synchronous Compensators (STATCOM) is effective for improving voltage stability but it can only exchange reactive power with the power grid. The integration of Superconducting Magnetic Energy Storage (SMES) with a STATCOM can extend the traditional STATCOM capabilities to four-quadrant bulk power system power flow control and providing exchange both the active and reactive power related to the STATCOM with the ac network. This paper shows how the SMES system can be connected to the ac system via the DC bus of a STATCOM and also analyzes how the integration of STATCOM and SMES allows the bus voltage regulation and power oscillation damping (POD) to be achieved simultaneously. The dynamic performance of the integrated STATCOM-SMES is evaluated through simulation by using PSCAD/EMTDC software and the compensation effectiveness of this integrated compensator is shown.

Keywords: STATCOM-SMES compensator, Power Oscillation Damping (POD), stabilizing, signal, voltage.

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Authors: K. Balamurugan, V. Dharmalingam, R. Muralisachithanandam, R. Sankaran

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This paper deals with the optimal choice and location of FACTS devices in deregulated power systems using Differential Evolution algorithm. The main objective of this paper is to achieve the power system economic generation allocation and dispatch in deregulated electricity market. Using the proposed method, the locations of the FACTS devices, their types and ratings are optimized simultaneously. Different kinds of FACTS devices such as TCSC and SVC are simulated in this study. Furthermore, their investment costs are also considered. Simulation results validate the capability of this new approach in minimizing the overall system cost function, which includes the investment costs of the FACTS devices and the bid offers of the market participants. The proposed algorithm is an effective and practical method for the choice and location of suitable FACTS devices in deregulated electricity market.

Keywords: FACTS Devices, Deregulated Electricity Market, Optimal Location, Differential Evolution, Mat Lab.

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Authors: B. Bahmani Firouzi, T. Niknam, M. Nayeripour

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With Power system movement toward restructuring along with factors such as life environment pollution, problems of transmission expansion and with advancement in construction technology of small generation units, it is expected that small units like wind turbines, fuel cells, photovoltaic, ... that most of the time connect to the distribution networks play a very essential role in electric power industry. With increase in developing usage of small generation units, management of distribution networks should be reviewed. The target of this paper is to present a new method for optimal management of active and reactive power in distribution networks with regard to costs pertaining to various types of dispersed generations, capacitors and cost of electric energy achieved from network. In other words, in this method it-s endeavored to select optimal sources of active and reactive power generation and controlling equipments such as dispersed generations, capacitors, under load tapchanger transformers and substations in a way that firstly costs in relation to them are minimized and secondly technical and physical constraints are regarded. Because the optimal management of distribution networks is an optimization problem with continuous and discrete variables, the new evolutionary method based on Ant Colony Algorithm has been applied. The simulation results of the method tested on two cases containing 23 and 34 buses exist and will be shown at later sections.

Keywords: Distributed Generation, Optimal Operation Management of distribution networks, Ant Colony Optimization(ACO).

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Authors: Bachir Bentouati, Lakhdar Chaib, Saliha Chettih, Gai-Ge Wang

Abstract:

The problem of economic dispatch (ED) is the basic problem of power framework, its main goal is to find the most favorable generation dispatch to generate each unit, reduce the whole power generation cost, and meet all system limitations. A heuristic algorithm, recently developed called Stud Krill Herd (SKH), has been employed in this paper to treat non-convex ED problems. The proposed KH has been modified using Stud selection and crossover (SSC) operator, to enhance the solution quality and avoid local optima. We are demonstrated SKH effects in two case study systems composed of 13-unit and 40-unit test systems to verify its performance and applicability in solving the ED problems. In the above systems, SKH can successfully obtain the best fuel generator and distribute the load requirements for the online generators. The results showed that the use of the proposed SKH method could reduce the total cost of generation and optimize the fulfillment of the load requirements.

Keywords: Stud Krill Herd, economic dispatch, crossover, stud selection, valve-point effect.

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Authors: Manisha Dubey, Gaurav Gupta, Anoop Arya

Abstract:

The demand for power supply increases day by day in developing countries like India henceforth demand of reactive power support in the form of ancillary services provider also has been increased. The multi-line and multi-type Flexible alternating current transmission system (FACTS) controllers are playing a vital role to regulate power flow through the transmission line. Unified power flow controller and interline power flow controller can be utilized to control reactive power flow through the transmission line. In a restructured power system, the demand of such controller is being popular due to their inherent capability. The transmission pricing by using reactive power cost allocation through modified matrix methodology has been proposed. The FACTS technologies have quite costly assembly, so it is very useful to apportion the expenses throughout the restructured electricity industry. Therefore, in this work, after embedding the FACTS devices into load flow, the impact on the costs allocated to users in fraction to the transmission framework utilization has been analyzed. From the obtained results, it is clear that the total cost recovery is enhanced towards the Reactive Power flow through the different transmission line for 5 bus test system. The fair pricing policy towards reactive power can be achieved by the proposed method incorporating FACTS controller towards cost recovery of the transmission network.

Keywords: Inter line power flow controller, Transmission Pricing, Unified power flow controller, cost allocation.

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Authors: J. Olamaei , T. Niknam, M. Nayeripour

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This paper presents an approach for daily optimal operation of distribution networks considering Distributed Generators (DGs). Due to private ownership of DGs, a cost based compensation method is used to encourage DGs in active and reactive power generation. The objective function is summation of electrical energy generated by DGs and substation bus (main bus) in the next day. A genetic algorithm is used to solve the optimal operation problem. The approach is tested on an IEEE34 buses distribution feeder.

Keywords: Distributed Generator, Daily Optimal Operation, Genetic Algorithm.

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Authors: Ali Dehghani, Mojtaba Hakimzadeh, Amir Habibi, Navid Mehdizadeh Afroozi

Abstract:

In this paper an isolated wind-diesel hybrid power system has been considered for reactive power control study having an induction generator for wind power conversion and synchronous alternator with automatic voltage regulator (AVR) for diesel unit is presented. The dynamic voltage stability evaluation is dependent on small signal analysis considering a Static VAR Compensator (SVC) and IEEE type -I excitation system. It's shown that the variable reactive power source like SVC is crucial to meet the varying demand of reactive power by induction generator and load and to acquire an excellent voltage regulation of the system with minimum fluctuations. Integral square error (ISE) criterion can be used to evaluate the optimum setting of gain parameters. Finally the dynamic responses of the power systems considered with optimum gain setting will also be presented.

Keywords: SVC, Small Signal Stability, Reactive Power, Control, Hybrid System.

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Authors: Shewit Tsegaye, Fekadu Shewarega

Abstract:

This paper presents a survey of articles, books and reports, which articulate the recent trends and aspects of Security Constrained Economic Dispatch (SCED). The period under consideration is 2008 through 2018. This is done to provide an up-to-date review of the recent major advancements in SCED, the state-of-the-art since 2008, identify further challenging developments needed in smarter grids, and indicate ways to address these challenges. This study consists of three areas of interest, which are very important and relevant for articulating the recent trends of SCED. These areas are: (i) SCED of power system with integrated renewable energy sources (IRES), (ii) SCED with post contingency corrective actions and (iii) Artificial intelligence based SCED.

Keywords: Security constrained economic dispatch, SCED of power system with IRES, SCED with post contingency corrective actions, artificial intelligence based SCED, IRES.

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Authors: A. K. Al-Othman, F. S. Al-Fares, K. M. EL-Nagger

Abstract:

This paper presents a new optimization technique based on quantum computing principles to solve a security constrained power system economic dispatch problem (SCED). The proposed technique is a population-based algorithm, which uses some quantum computing elements in coding and evolving groups of potential solutions to reach the optimum following a partially directed random approach. The SCED problem is formulated as a constrained optimization problem in a way that insures a secure-economic system operation. Real Coded Quantum-Inspired Evolution Algorithm (RQIEA) is then applied to solve the constrained optimization formulation. Simulation results of the proposed approach are compared with those reported in literature. The outcome is very encouraging and proves that RQIEA is very applicable for solving security constrained power system economic dispatch problem (SCED).

Keywords: State Estimation, Fuzzy Linear Regression, FuzzyLinear State Estimator (FLSE) and Measurements Uncertainty.

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Authors: Puneet Chawla, Balwinder Singh

Abstract:

Recently, increased attention has been devoted to the voltage instability phenomenon in power systems. Many techniques have been proposed in the literature for evaluating and predicting voltage stability using steady state analysis methods. In this paper P-V and Q-V curves have been generated for a 57 bus Patiala Rajpura circle of India. The power-flow program is developed in MATLAB using Newton Raphson method. Using Q-V curves the weakest bus of the power system and the maximum reactive power change permissible on that bus is calculated. STATCOMs are placed on the weakest bus to improve the voltage and hence voltage stability and also the power transmission capability of the line.

Keywords: Voltage stability, Reactive power, power flow, weakest bus, STATCOM.

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Authors: Navid Mehdizadeh Afroozi, Khodakhast Isapour, Mojtaba Hakimzadeh, Abdolmohammad Davodi

Abstract:

The objective of the Economic Dispatch(ED) Problems of electric power generation is to schedule the committed generating units outputs so as to meet the required load demand at minimum operating cost while satisfying all units and system equality and inequality constraints. This paper presents a new method of ED problems utilizing the Max-Min Ant System Optimization. Historically, traditional optimizations techniques have been used, such as linear and non-linear programming, but within the past decade the focus has shifted on the utilization of Evolutionary Algorithms, as an example Genetic Algorithms, Simulated Annealing and recently Ant Colony Optimization (ACO). In this paper we introduce the Max-Min Ant System based version of the Ant System. This algorithm encourages local searching around the best solution found in each iteration. To show its efficiency and effectiveness, the proposed Max-Min Ant System is applied to sample ED problems composed of 4 generators. Comparison to conventional genetic algorithms is presented.

Keywords: Economic Dispatch (ED), Ant Colony Optimization, Fuel Cost, Algorithm.

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Authors: S. Thongkeaw, M. Boonthienthong

Abstract:

This paper presents the techniques for voltage control in distribution system. It is integrated in the distribution management system. Voltage is an important parameter for the control of electrical power systems. The distribution network operators have the responsibility to regulate the voltage supplied to consumer within statutory limits. Traditionally, the On-Load Tap Changer (OLTC) transformer equipped with automatic voltage control (AVC) relays is the most popular and effective voltage control device. A static synchronous compensator (STATCOM) may be equipped with several controllers to perform multiple control functions. Static Var Compensation (SVC) is regulation slopes and available margins for var dispatch. The voltage control in distribution networks is established as a centralized analytical function in this paper. 

Keywords: Voltage Control, Reactive Power, Distribution System.

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Authors: K. Balamurugan, R. Muralisachithanandam, V. Dharmalingam, R. Srikanth

Abstract:

This paper deals with the optimal choice and allocation of multi FACTS devices in Deregulated power system using Evolutionary Programming method. The objective is to achieve the power system economic generation allocation and dispatch in deregulated electricity market. Using the proposed method, the locations of the FACTS devices, their types and ratings are optimized simultaneously. Different kinds of FACTS devices are simulated in this study such as UPFC, TCSC, TCPST, and SVC. Simulation results validate the capability of this new approach in minimizing the overall system cost function, which includes the investment costs of the FACTS devices and the bid offers of the market participants. The proposed algorithm is an effective and practical method for the choice and allocation of FACTS devices in deregulated electricity market environment. The standard data of IEEE 14 Bus systems has been taken into account and simulated with aid of MAT-lab software and results were obtained.

Keywords: FACTS devices, Optimal allocation, Deregulated electricity market, Evolutionary programming, Mat Lab.

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Authors: Sidhartha Panda, N.P.Padhy

Abstract:

Wind power is among the most actively developing distributed generation (DG) technology. Majority of the wind power based DG technologies employ wind turbine induction generators (WTIG) instead of synchronous generators, for the technical advantages like: reduced size, increased robustness, lower cost, and increased electromechanical damping. However, dynamic changes of wind speed make the amount of active/reactive power injected/drawn to a WTIG embedded distribution network highly variable. This paper analyzes the effect of wind speed changes on the active and reactive power penetration to the wind energy embedded distribution network. Four types of wind speed changes namely; constant, linear change, gust change and random change of wind speed are considered in the analysis. The study is carried out by three-phase, non-linear, dynamic simulation of distribution system component models. Results obtained from the investigation are presented and discussed.

Keywords: Wind turbine induction generator, distribution network, active and reactive power, wind speed.

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Authors: Anant Oonsivilai, Kenedy A. Greyson

Abstract:

The demand of the energy management systems (EMS) set forth by modern power systems requires fast energy management systems. Contingency analysis is among the functions in EMS which is time consuming. In order to handle this limitation, this paper introduces agent based technology in the contingency analysis. The main function of agents is to speed up the performance. Negotiations process in decision making is explained and the issue set forth is the minimization of the operating costs. The IEEE 14 bus system and its line outage have been used in the research and simulation results are presented.

Keywords: Agents, model, negotiation, optimal dispatch, powersystems.

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Authors: Md. Sayeed Salam

Abstract:

This paper presents an efficient emission constrained economic dispatch algorithm that deals with nonlinear cost function and constraints. It is then incorporated into the dynamic programming based hydrothermal coordination program. The program has been tested on a practical utility system having 32 thermal and 12 hydro generating units. Test results show that a slight increase in production cost causes a substantial reduction in emission.

Keywords: Emission constraint, Hydrothermal coordination, and Economic dispatch algorithm.

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Authors: Samina Elyas Mubeen, R. K. Nema, Gayatri Agnihotri

Abstract:

In this paper the performance of unified power flow controller is investigated in controlling the flow of po wer over the transmission line. Voltage sources model is utilized to study the behaviour of the UPFC in regulating the active, reactive power and voltage profile. This model is incorporated in Newton Raphson algorithm for load flow studies. Simultaneous method is employed in which equations of UPFC and the power balance equations of network are combined in to one set of non-linear algebraic equations. It is solved according to the Newton raphson algorithm. Case studies are carried on standard 5 bus network. Simulation is done in Matlab. The result of network with and without using UPFC are compared in terms of active and reactive power flows in the line and active and reactive power flows at the bus to analyze the performance of UPFC.

Keywords: Newton-Raphson algorithm, Load flow, Unified power flow controller, Voltage source model.

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Authors: Ashish Saini, A.K. Saxena

Abstract:

In competitive electricity markets all over the world, an adoption of suitable transmission pricing model is a problem as transmission segment still operates as a monopoly. Transmission pricing is an important tool to promote investment for various transmission services in order to provide economic, secure and reliable electricity to bulk and retail customers. The nodal pricing based on SRMC (Short Run Marginal Cost) is found extremely useful by researchers for sending correct economic signals. The marginal prices must be determined as a part of solution to optimization problem i.e. to maximize the social welfare. The need to maximize the social welfare subject to number of system operational constraints is a major challenge from computation and societal point of views. The purpose of this paper is to present a nodal transmission pricing model based on SRMC by developing new mathematical expressions of real and reactive power marginal prices using GA-Fuzzy based optimal power flow framework. The impacts of selecting different social welfare functions on power marginal prices are analyzed and verified with results reported in literature. Network revenues for two different power systems are determined using expressions derived for real and reactive power marginal prices in this paper.

Keywords: Deregulation, electricity markets, nodal pricing, social welfare function, short run marginal cost.

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Authors: R. Oloomi, M. A. Sadrnia

Abstract:

This paper proposes a new optimal feedback controller for voltage source converters VSC's, for current regulated voltage source converters, which allows compensate the harmonics of current produced by nonlinear loads and load reactive power. The aim of the present paper is to describe a novel switching signal generation technique called optimal controller which guarantees that the injected currents follow the reference currents determined by the compensation strategy, with the smallest possible tracking error and fixed switching frequency. It is compared with well-known hysteresis current controller HCC. The validity of presented method and its comparison with HCC is studied through simulation results.

Keywords: Hysteresis Current Controller, Optimal Controller, Switching pattern, Voltage Source Converter.

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Authors: Baghdasaryan Marinka

Abstract:

Monitoring the distribution of electric power consumption in the technological process of ore grinding is conducted. As a result, the impacts of the mill filling rate, the productivity of the ore supply, the volumetric density of the grinding balls, the specific density of the ground ore, and the relative speed of the mill rotation on the specific consumption of electric power have been studied. The power and technological factors affecting the reactive power generated by the synchronous motors, operating within the technological scheme are studied. A block diagram for evaluating the power consumption modes of the technological process is presented, which includes the analysis of the technological scheme, the determination of the place and volumetric density of the ore-grinding mill, the evaluation of the technological and power factors affecting the energy saving process, as well as the assessment of the electric power standards.

Keywords: Electric power standard, factor, ore grinding, power consumption, reactive power, technological.

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Authors: Himanshu Shekhar Maharana, S. K .Dash

Abstract:

Economic Load Dispatch (ELD) proves to be a vital optimization process in electric power system for allocating generation amongst various units to compute the cost of generation, the cost of emission involving global warming gases like sulphur dioxide, nitrous oxide and carbon monoxide etc. In this dissertation, we emphasize ramp rate constriction factor based particle swarm optimization (RRCPSO) for analyzing various performance objectives, namely cost of generation, cost of emission, and a dual objective function involving both these objectives through the experimental simulated results. A 6-unit 30 bus IEEE test case system has been utilized for simulating the results involving improved weight factor advanced ramp rate limit constraints for optimizing total cost of generation and emission. This method increases the tendency of particles to venture into the solution space to ameliorate their convergence rates. Earlier works through dispersed PSO (DPSO) and constriction factor based PSO (CPSO) give rise to comparatively higher computational time and less good optimal solution at par with current dissertation. This paper deals with ramp rate and constriction factor based well defined ramp rate PSO to compute various objectives namely cost, emission and total objective etc. and compares the result with DPSO and weight improved PSO (WIPSO) techniques illustrating lesser computational time and better optimal solution. 

Keywords: Economic load dispatch, constriction factor based particle swarm optimization, dispersed particle swarm optimization, weight improved particle swarm optimization, ramp rate and constriction factor based particle swarm optimization.

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Authors: Saran Satsangi, Ashish Saini, Amit Saraswat

Abstract:

In this paper, a new K-means clustering based approach for identification of voltage control areas is developed. Voltage control areas are important for efficient reactive power management in power systems operating under deregulated environment. Although, voltage control areas are formed using conventional hierarchical clustering based method, but the present paper investigate the capability of K-means clustering for the purpose of forming voltage control areas. The proposed method is tested and compared for IEEE 14 bus and IEEE 30 bus systems. The results show that this K-means based method is competing with conventional hierarchical approach

Keywords: Voltage control areas, reactive power management, K-means clustering algorithm

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