Load Flow Equations Research Articles

A Security-constrained Economic Power Dispatch Technique Using Modified Subgradient Algorithm Based on Feasible Values and Pseudo Scaling Factor for a Power System Area Including Limited Energy Supply Thermal Units

A security-constrained power dispatch problem with non-convex cost function for a lossy electric power system area including limited energy supply thermal units fueled under a take-or-pay agreement is formulated. Then, an iterative solution method based on a modified subgradient algorithm based on feasible values and common pseudo scaling factor for limited energy supply thermal units is proposed and used to solve the problem. In the iterative proposed solution method, a modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the common pseudo scaling factor for limited energy supply thermal units is employed to adjust the amount of fuel consumed by the limited energy supply units during the considered operation period. Since all equality and inequality constraints in the non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles (the off-nominal tap settings), they are taken as independent variables except for those belonging to the reference bus. Load flow equations are added to the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, and off-nominal tap setting constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method given in this article before application of the modified subgradient algorithm based on feasible values to the optimization model. The proposed technique is tested on an IEEE 30-bus test system with a non-convex total cost rate curve. A portion of the dispatch problem considered in this article, where only a loading of the system in a subinterval is considered, was also solved by some other recently reported dispatch techniques. First, minimum total fuel cost rates and solution times obtained from the modified subgradient algorithm based on feasible values versus other techniques are compared, and the outperformance of the modified subgradient algorithm based on feasible values in terms of both total cost rate and solution time is demonstrated. After that, the optimal total fuel cost value and solution time of the system for the operation period for the cases where the fuel constraint is ignored and not ignored are presented, and it is shown that adding the fuel constraint into the dispatch problem further decreases the optimal total fuel cost for the considered dispatch problem. Finally, the modified subgradient algorithm based on feasible values was applied to the fuel constrained dispatch problem directly. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem.

Active Line Flow Control of Power System Network with FACTS Devices of choice using Soft Computing Technique

technology opens up new opportunity for operation and control of power system. Out of the various FACTS devices (viz-TCSC, TCPST, UPFC etc.), the right choices for the maximization of power flow in power system network demands attention which helps to achieve the active power flow up to their line limits without any constraint violation and with optimal investment on FACTS devices. In this paper, an algorithm has been developed for right choices of various combination of FACTS in the power network to enhance the power transfer capability of existing lines under normal condition very close to their line limits and has been applied for modified IEEE 14- bus system. This helps the optimal investment in FACTS devices and easy control. The proposed method is based on load flow and line flow equation In this paper Newton Raphson and modified simulated annealing technique are proposed

Secured loadability enhancement with TCSC in transmission system using computational intelligence techniques for pool and hybrid model

This paper proposes an algorithm for the optimal location and control of Flexible AC Transmission System (FACTS) devices for enhancing the loadability in transmission system using particle swarm optimization (PSO) and differential evolution (DE) for pool and hybrid model in deregulated electricity market. This approach uses AC load flow equations with the constraints on power generation, transmission line flow, magnitude of bus voltages and FACTS device settings. The bilateral transactions are modeled using secured bilateral transaction matrix utilizing AC distribution factor with the slack bus contribution. In this proposed approach, Thyristor Controlled Series Compensator (TCSC) is used. To validate the proposed approach, simulations are performed on 39 bus New England test system and IEEE 118 bus system. Comparisons are made in terms of maximum loadability, computation time and convergence characteristics. The simulation result obtained indicates that by optimal location and control of TCSC, using DE, enhances the loadability in transmission system with less computation time and faster convergence than using PSO. This comparative study concludes that by optimal location and control of TCSC, using DE will be more effective for secured loadability enhancement in transmission system of pool and hybrid model in deregulated electricity market.

On the determination of the solutions and maximum admissible power of the load flow equation via LMIS

It is well-known that the load flow equation plays a key role in power systems. This paper investigates the load flow equation addressing two fundamental problems, namely the determination of the solutions of this equation in a given region of interest, and the computation of the maximum admissible power for which this equation has a solution in such a region. For the first problem, it is shown that the sought solutions can be computed by solving an eigenvalue problem (EVP), i.e. a convex optimization problem with linear matrix inequality (LMI) constraints, and by looking for vectors with special structure into a linear subspace. For the second problem, it is shown that an upper bound of the sought maximum admissible power can be found by solving another EVP, moreover a sufficient and necessary condition for establishing the tightness of the found upper bound is provided. The proposed methodology is illustrated by some numerical examples. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

A New Formulation for Load Flow Solution of Power Systems With Series FACTS Devices

This paper addresses the solution of load flow equa-tions for a power system with series flexible ac -transmission systems(FACTS) devices. A novel formulation of equations using dual state variables (current magnitude and angle) and dual controlvariables (series injected real power and series voltage in quadra-ture with current) for series devices is proposed. These specifica-tions can be related to transmission line loading and device limits.Specifications like power flow through a series device can also be handled using this formulation. The load flow equations are solved using Newton-Raphson technique. A decoupled formulation is also proposed. Case studies are carried out on IEEE test systems withseveral types of specifications to validate the method.

Optimal Power Management of a DISCO with Integrations of Reliability Considerations and Wind Farm Based on Benders Decomposition

This paper presents a comprehensive framework model of a distribution company with security and reliability considerations. A probabilistic wind farm, which is a renewable energy resource, is modeled in this work. The requirement energy of distribution company can be either provided by distribution company′s own distributed generations or purchased from power market. Two reliability indices as well as DC load flow equations are also considered in order to satisfy reliability and security constraints, respectively. Since allocating proper spinning reserve improves reliability level, the amount of spinning reserve will be calculated iteratively. In this work, all equations are expressed in a linear fashion in which unit commitment formulation depends on binary variables associated with only on/off of units. The benders decomposition method is used to solve security‐based unit commitment.

Pseudo-loadflow formulation as a starting process for the Newton Raphson algorithm

This paper introduces new models which approximate the AC loadflow problem, but are able to converge (using the Newton Raphson algorithm) from a wider range of starting points. The solution of the pseudo-loadflow models can provide a robust starting process for the Newton Raphson solution of the conventional loadflow problem. It is also shown that pseudo-loadflow solutions exist in many cases where the AC loadflow equations do not appear to have any solution, and in such cases the pseudo-loadflow solution can provide useful information to assist in locating the cause of infeasibility of the AC loadflow model. Test results are presented for illustrative small network examples and also for larger test networks. The computational requirements of the proposed methods are similar to those of the conventional Newton Raphson loadflow algorithm.

Joint economic and emission dispatch in energy markets: A multiobjective mathematical programming approach

Abstract Economic load dispatch is the method of determining the most efficient, low-cost and reliable operation of a power system by dispatching the available electricity generation resources to supply the load on the system. Environmental concerns that arise due to the operation of fossil fuel fired electric generators, change the classical problem into multiobjective emission/economic dispatch (MEED) which is formulated as a constrained nonlinear multiobjective mathematical programming (MMP). The proposed MEED formulation includes emission minimization objective, AC load flow constraints and security constraints of the power system which usually are found simultaneously in real-world power systems. The proposed model has a more accurate evaluation of transmission losses obtained from the load flow equations. The MMP approach based on ɛ-constraint algorithm has been proposed for generating Pareto-optimal solutions of power systems MEED problem. Moreover, fuzzy decision making process is employed to extract one of the Pareto-optimal solutions as the best compromise nondominated solution. The proposed approach is simulated on the IEEE 30-bus six-generator test system and obtained results have been comprehensively compared with some of the most recently published research in the area (from the both aspects of precision and execution tome) which confirms the potential and effectiveness of the proposed approach.

An efficient coupled GA and load flow algorithm for optimal placement and sizing of distributed generators

A genetic algorithm is used in conjunction with an efficient load flow programme to determine the optimal locations and sizing of the predefined DGs within MATLAB software. The best location for the DGs and the sizing of the DGs are determined using the genetic algorithm. The branch electrical loss is considered as the objective function and the system total loss represent the fitness evaluation function for driving the GA. The load flow equations are considered as equality constraints and the equations of nodal voltage and branch capacity are considered as inequality constraints. The approach is tested on a 15 bus IEEE distribution feeder.

FACTS location and size for reactive power system compensation through the multi-objective optimization

FACTS location and size for reactive power system compensation through the multi-objective optimizationThe problem of the FACTS (Flexible Alternative Current Transmission System Devices) location and size for reactive power system compensation through the multi-objective optimization is presented in this paper. A new technique is proposed for the optimal setting, dimension and design of two kinds of FACTS namely: Static Volt Ampere reactive (VAR) Compensator (SVC) and Thyristor Controlled Series Compensator (TCSC) handling the minimization of transmission losses in electrical network. Using the proposed scheme, the type, the location and the rating of FACTS devices are optimized simultaneously. The problem to solve is multi criteria under constraints related to the load flow equations, the voltages, the transformer turn ratios, the active and reactive productions and the compensation devices. Its solution requires the the advanced algorithms to be applied. Thus, we propose an approach based on the evolutionary algorithms (EA) to solve multi-criterion problem. It is similar to the NSGA-II method (Ellitist Non Dominated Sorting Genetic Algorithm). The Pareto front is obtained for continuous, discrete and multiple of five MVArs (Mega Volt Ampere reactive) of compensator devices for the IEEE 57-bus test system (IEEE bus test is a standard network).

Power Flow Control with UPFC in Power Transmission System

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 . In this paper voltage source model of UPFC is incorporated in Newton Raphson algorithm in order to investigate the control of power flow. Normally there are two ways of solving the load flow equations with UPFC. The sequential method and the simultaneous method: In sequential method the equations of UPFC are separated from the system power balance equations. Both the set of equations are solved separately and sequentially. In simultaneous method all the equations are combined in to one set of non-linear algebraic equations. A jacobian matrix is then formed which is non symmetric in nature. The simultaneous method is used in this paper.

A Probability Method for Very Short-Term Steady State Analysis of a Distribution System with Wind Farms

In this paper, a probabilistic method is proposed to analyze the very short-term steady-state performance of an unbalanced distribution electrical system characterized by the presence of wind farms. This method, which can take into account the uncertainties of loads and wind productions, is based on a Monte Carlo simulation procedure applied to the non-linear three-phase load flow equations, including wind farm models. Bayesian time series models are used to predict the next hour's wind speed probability density functions, making possible a predictive evaluation of the very short-term system steady-state behavior. Numerical applications are presented and discussed with reference to the three-phase unbalanced IEEE 34-bus test distribution system in the presence of wind farms connected at different busbars.

Optimal Power Flow Using an Extended Conic Quadratic Formulation

Recent research has shown that the load flow equations describing the steady-state conditions in a meshed network can be placed in extended conic quadratic (ECQ) format. This paper presents a study of the implementation of the new load flow equations format in an optimal power flow (OPF) program which accounts for control devices such as tap-changing transformers, phase-shifting transformers, and unified power flow controllers. The proposed OPF representation retains the advantages of the ECQ format: 1) it can be easily integrated within optimization routines that require the evaluation of second-order derivatives, 2) it can be efficiently solved for using primal-dual interior-point methods, and 3) it can make use of linear programming scaling techniques for improving numerical conditioning. The ECQ-OPF program is employed to solve the economic dispatch and active power loss minimization problems. Numerical testing is used to validate the proposed approach by comparing against solution methods and results of standard test systems.

System network planning expansion using mathematical programming, genetic algorithms and tabu search

In this paper, system network planning expansion is formulated for mixed integer programming, a genetic algorithm (GA) and tabu search (TS). Compared with other optimization methods, GAs are suitable for traversing large search spaces, since they can do this relatively rapidly and because the use of mutation diverts the method away from local minima, which will tend to become more common as the search space increases in size. GA’s give an excellent trade off between solution quality and computing time and flexibility for taking into account specific constraints in real situations. TS has emerged as a new, highly efficient, search paradigm for finding quality solutions to combinatorial problems. It is characterized by gathering knowledge during the search and subsequently profiting from this knowledge. The attractiveness of the technique comes from its ability to escape local optimality. The cost function of this problem consists of the capital investment cost in discrete form, the cost of transmission losses and the power generation costs. The DC load flow equations for the network are embedded in the constraints of the mathematical model to avoid sub-optimal solutions that can arise if the enforcement of such constraints is done in an indirect way. The solution of the model gives the best line additions and also provides information regarding the optimal generation at each generation point. This method of solution is demonstrated on the expansion of a 10 bus bar system to 18 bus bars. Finally, a steady-state genetic algorithm is employed rather than generational replacement, also uniform crossover is used.

A Conic Quadratic Format for the Load Flow Equations of Meshed Networks

Recent research has shown that the load flow equations describing the steady-state conditions in a radial network can be formulated as a second-order cone program and solved using polynomial-time interior point methods. This paper extends the conic formulation to meshed networks by including a trigonometric functional constraint for the voltage angle spread on each line in the network. In particular, it is shown that the load flow solution of a meshed network can be obtained from a sequence of conic programming solutions. The proposed formulation and iterative procedure are validated by comparing with the Newton-Raphson solution of the standard power flow model. Testing is reported on seven networks. The proposed format of equations opens the door for new implementations of energy management system functions based on efficient and numerically robust conic quadratic optimization methods.

New formulation and analysis of the system planning expansion model

AbstractIn this paper, an electrical power transmission network planning expansion is formulated for mixed‐integer programming, a genetic algorithm and tabu search (TS). Compared with other optimisation methods, GA's are suitable for traversing large search spaces since they can do this relatively rapidly and because the use of mutation diverts the method away from local minima, which will tend to become more common as the search space increases in size. GA's give an excellent trade‐off between solution quality and computing time and flexibility for taking into account specific constraints in real situations. TS has emerged as a new, highly efficient, search paradigm for finding quality solutions to combinatorial problems. It is characterised by gathering knowledge during the search, and subsequently profiting from this knowledge. The attractiveness of the technique comes from its ability to escape local optimality. The cost function of this problem consists of the capital investment cost in discrete form, the cost of transmission losses and the power generation costs. The DC load flow equations for the network are embedded in the constraints of the mathematical model to avoid sub‐optimal solutions that can arise if the enforcement of such constraints is done in an indirect way. The solution of the model gives the best line additions, and also provides information regarding the optimal generation at each generation point. This method of solution is demonstrated on the expansion of a 10 bus‐bar system to 18 bus‐bars. Finally, an empirical analysis of the effects of parameter values on genetic algorithm performance is tested. Copyright © 2007 John Wiley & Sons, Ltd.

Probabilistic three-phase load flow for unbalanced electrical distribution systems with wind farms

A probabilistic method is proposed to take into account the uncertainties of loads and wind production in electrical unbalanced distribution systems. The proposed method is based on Monte-Carlo simulation applied to the nonlinear three-phase load flow equations including wind farms, thereby taking into account all load and line unbalances that can characterise the distribution systems. The method allows the evaluation of phase-voltage and unbalance factor probability functions and in particular the maximum values and 95th percentiles, being the statistical measures of greatest interest in many international standards for power quality. Numerical applications are presented and discussed with reference to the three-phase unbalanced IEEE 34-bus test distribution system in presence of wind farms connected at different busbars.

Multi‐linear Monte Carlo simulation for probabilistic three‐phase load flow

AbstractSeveral transmission and distribution systems are known to have unbalanced lines and line sections carrying a mixture of single, double, and three‐phase loads; moreover, their steady‐state conditions are affected by unavoidable uncertainties that require to analyze these systems with probabilistic approaches. In this article, a probabilistic three‐phase load flow using a multi‐linear simulation algorithm is proposed. The method properly extends to the field of unbalanced systems a probabilistic method proposed for balanced systems. Numerical examples on the three‐phase unbalanced IEEE 13‐bus and 34‐bus distribution test systems are provided and discussed to illustrate the application of the proposed method. The results obtained with the proposed algorithm are compared with the ones obtained using a Monte Carlo simulation algorithm, which includes the non linear three‐phase load flow equations. Copyright © 2006 John Wiley & Sons, Ltd.

A Genetic Algorithm for the Corrective Control of Voltage and Reactive Power

This paper presents an algorithm for the selection of corrective control actions for bus voltage and generator reactive power in a power system. A genetic algorithm (GA) using linear approximation of load flow equations and a heuristic selection of participating controls were combined in a search method for the minimum number of control actions. The method was successfully applied on a 220-bus model for several cases of disturbed load flow, and calculation time was proven to be small enough to allow real-time application of the algorithm. The GA was compared with a integer programming-based solution method and showed a considerably reduced calculation time.

Continuation load flow using automatically determined branch megawatt losses as parameters

The conventional Newton's method is considered to be inadequate for the computation of the maximum loading point (MLP) of power systems since: (i) it encounters difficulties in the vicinity of the MLP: and (ii) the load flow Jacobian matrix becomes singular at the MLP. It is well known that continuation methods are powerful and useful tools that are able to trace the solution PV curve without experiencing such diffculties. However, continuation methods require a parameterisation so that a modified, well conditioned set of load flow equations is obtained. In particular, the Jacobian matrix associated with this modified set of equations should not be singular at the MLP. The authors propose that the actual power losses in transmission branches (lines and transformers) are used to parameterise the approach. Specific procedures for the automatic determination of the most appropriate parameter (branch) are proposed. Such procedures include the utilisation of fast voltage-stability indices. Simulation results are presented to show that the proposed method is able to trace the whole solution PV curve very efficiently.