An analytical solution to the economic dispatch problem (original) (raw)
Economic power dispatch for an interconnected power system based on reliability indices
Indonesian Journal of Electrical Engineering and Computer Science
Reliability indices are always one of the most important factors in the power systems. In this paper, the problem of the economic load dispatch (ELD) and the problem of economic emission load dispatch (CEELD) have been improved taking into account reliability indices. That is, the problem and reliability of ELD are proposed as combined economic load dispatch reliability (CELDR) and the problem CEELD is suggested as (CEELDR). In solving CELDR and CEELDR problems, tried to use power generators in a very reliable way to save system load, as well as minimum fuel and emission costs. In this effort, the ELD of power plants is successfully implemented in a single system containing 6 generating units, taking into account the reliability and emissions of the system with and without system power loss, inequality and inequality constraints, and valve point effects using the exchange market algorithm(EMA). The results suggest that reliability indicators in ELD can be used to create greater re...
Recent methodologies and approaches for the economic dispatch of generation in power systems
2012
ABSTRACT SUMMARY The new tendencies in the power system organization and the fast-changing technologies in the power industry dictate the need to keep track of the international experience and activities in the field of the modern economic dispatch problem. The goal of this paper was to provide a detailed account for papers published after 1990, the year that saw the beginning of major transformations in the power system organization. A comprehensive survey on mathematical formulations and a general background of methods, analyses, and developments in the field of economic dispatch is presented for the past 20 years based on more than 150 publications. The research literature in the field is classified into sections covering economic dispatch in both regulated and deregulated (reregulated) energy markets and where variable, partial predictable electricity generation is part of the generation portfolio. A database of the most common test systems used in the literature to test different economic dispatch methodologies is also provided. Copyright © 2012 John Wiley & Sons, Ltd.
Electric Power Systems Research, 2015
In this paper a probabilistic economic dispatch model considering thermal units (fuel generators), photovoltaic arrays and wind energy conversion systems is proposed. Wind speed, solar radiation and power demand are recognized as random variables. Unavailability of each type of power source is also considered. The solution strategy is based on the Monte Carlo method and non-linear constrained optimization. The optimal solution involves single and multidimensional probabilities, descriptive statistics, cluster and bimodal analysis. The proposed methodology yields the probability distributions of system marginal price, thermal (fuel based), solar and wind power generation and load shedding. The proposed model and methodology are applied to a case study of the Northern Chilean electrical system.
Optimal and reliable dispatch of supply and demand bids for competitive electricity markets
2000 Power Engineering Society Summer Meeting (Cat. No.00CH37134), 2000
This paper presents an optimal dispatch model for competitive electricity markets where independent system operators (ISOs) dispatch real power and ancillary services based on supply offers and demand bids. The model maximizes the "net" societal benefit while respecting various constraints. Zonal market clearing prices are determined and DC power flow equations are used to capture the physics of real power flows. In addition, integer decision variables are used to capture unit's onloff status and Automatic Generator Control (A M) / load following capability in a reliable way. Each producer is assigned a power flow on each inter-zonal connection and this will be used for accounting purpose. A six zone, 18-unit system is used as our case study with satisfactory results.
Optimal Power Dispatch via Multistage Stochastic Programming
Progress in Industrial Mathematics at ECMI 96, 1997
The short-term cost-optimal dispatch of electric power in a generation system under uncertain electricity demand is considered. The system comprises thermal and pumped-storage hydro units. An operation model is developed which represents a multistage mixed-integer stochastic program and a conceptual solution method using Lagrangian relaxation is sketched. For xed start-up and shutdown decisions an e cient algorithm for solving the multistage stochastic program is described and numerical results are reported.
Fast economic power dispatch method for power system planning studies
IET Generation, Transmission & Distribution, 2015
This study describes a method for solving the optimal economic power dispatch problem. The method proposed here uses linear programming, because linear programming-based formulations tend to be flexible, reliable and faster than their nonlinear counterparts. The proposed linear programming-based method is developed based on a linearised network model, in which voltage magnitudes and reactive power flows have both been accounted for, unlike traditional linearised power flow methods. A piecewise linear model is developed to handle the thermal capacities of transmission lines. Piecewise linear models are also developed to deal with the exponential loads, cost functions of generators and total power losses. The effectiveness of the proposed method is demonstrated on numerous standard test systems, including the IEEE 300-bus system. Further, a comparison with some other methods, which are available in the literature, is held to validate the consistency of the proposed method in finding the optimal or near-optimal global solution. The results show that the proposed method could lead to optimal or near-optimal global solution, and is appropriate to use for power system planning studies.
2015
Optimal reactive power dispatch (ORPD) is a multi-variable problem with nonlinear constraints and continuous/discrete variables. Due to the stochastic behavior of loads, the ORPD requires a probabilistic mathematical model. In this paper, Monte Carlo simulation (MCS) is used for modeling of load uncertainties. Multi objective (MO) model is used for this aim. The objective functions of the proposed MO-ORPD are the real power losses and voltage stability (L-index). Since these two objectives are conflicted, the MO-ORPD problem is solved by weighted sum method. The problem is solved by considering bus voltage limits, the limits of branches power flow; transformers' tap changers and the amount of reactive power compensation at weak buses. The optimization models are implemented and solved in GAMS environment and the proposed method is examined on IEEE 14-bus test system. The obtained results illustrate the effectiveness of the proposed MO-ORPD problem for dealing with uncertainties ...
Microgrid dispatch and price of reliability using stochastic approximation
2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP), 2015
When properly operated, microgrids can facilitate the integration of stochastic renewable energy without compromising service reliability. However, in the context of multistage dispatching, finding the optimal day-ahead energy procurement that accounts for the variability of real-time operation is a computationally challenging task. This paper develops a computationally efficient two-stage economic dispatch scheme for a microgrid that exchanges energy with an external power system. The scheme is designed to minimize the generation and energy exchange costs, while setting limits on the microgridwide expected load not served. The day-ahead variables, which are the solution to the first stage, are found using a stochastic approximation saddle-point algorithm. The proposed algorithm is asymptotically convergent and can be efficiently implemented upon drawing samples from the distribution of the real-time state variables (wind energy, demand, and energy prices). Numerical tests using the IEEE 14-bus power system benchmark verify that the proposed scheme outperforms all other tested alternatives, even for very high wind power penetration.
Economic Dispatch: Applying Interval-Based Dependency Analysis to an Electric Power Problem
A common way to model uncertainty in the value of a quantity is to use a probability density function (PDF) or its integral, a probability distribution function (CDF). When two such values are combined to form a new value equal to their sum, product, max, etc., the new value is termed a derived distribution . It is well-known that derived distributions may be obtained by numerical convolution, Monte Carlo simulation, and analytically for specific classes of input distributions, under the assumption that the input distributions are independent. It is also possible to obtain derived distributions for specified dependency relationships other than independence. However, it is not always the case that the dependency relationship is known. Thus there is a need for obtaining solutions without assuming independence or any other specific dependency relationship. There are two numerical algorithms that have been implemented in software for this. Numerical approaches have the advantage of applicability to a very wide class of distributions. Probabilistic Arithmetic [6] is implemented in the commercially available software tool RiskCalc . Interval-Based Dependency Analysis (IBDA) [2], which extends our previous tool [1] by eliminating the independence assumption, is implemented in the software tool Statool and is available upon request from the authors. While the two tools have fundamental similarities [4], a significant difference with respect to the present problem is that IBDA supports, and Statool implements, excess width removal in the underlying interval calculations, from some expressions. In this paper we apply IBDA to generalize a solution to the well-known economic dispatch problem in electric power generation to the case where the dependency relationship between the fuel costs of two generators is unspecified.
Energies
In the context of the growing penetration of renewable power sources in power systems causing probabilistic contingency conditions, a suitable economic dispatch model is decisively needed. There is a lack of research in the field of probabilistic mathematical formulation considering the uncertainties due to the stochastic nature of renewables and contingency occurrence, as it is a very complex problem to be solved. The most appropriate model is the stochastic security-constrained economic dispatch (SSCED) model for optimized economic dispatch decisions during uncertainty. However, because of its complexity, it is rarely employed. This paper attempts to solve the complex SSCED problem in the presence of the uncertainty of resources and probabilistic contingency conditions, which is a novel effort in this regard. The SSCED is carried out over multiple periods to provide the load-following or contingency reserves. In the proposed SSCED, the uncertainty problem is addressed by modeling ...
This paper proposes a methodology to model and solve the problem of stochastic economic dispatch incorporating renewable energies. In this context, demand and generation randomness (wind speed, solar radiation and rates of failure) are considered. Demand, wind speed, solar radiation and unavailability are modeled through Normal, Weibull, Beta and Uniform distributions respectively. The problem is therefore recognized as a stochastic process. Consequently, the cost of load shedding is considered. In order to define the optimal power allocation for each generator, the proposed methodology uses Group SO (3) orthogonal matrices (Lie's algebra), the marginal costs of the generators, the customer damage cost and Monte-Carlo trials. The result contains generation, marginal cost and load shedding statistics, among others.
Two-Timescale Stochastic Dispatch of Power Distribution Networks
2016
Smart distribution grids should optimally integrate stochastic renewable resources while effecting voltage regulation. Since some decisions have to be designed in advance, energy management is a multistage problem. For early stages, finding the optimal energy procurement accounting for the variability during real-time operation is a challenging task. The joint dispatch of slowand fast-timescale controls in a distribution grid is considered here. The substation voltage, the energy exchanged with a main grid, and the generation schedules for small diesel generators have to be decided on a slow timescale; whereas optimal photovoltaic inverter setpoints are found on a more frequent basis. While inverter and looser voltage regulation limits are imposed at all times, tighter bus voltage constraints are enforced on the average or in probability, thus enabling more efficient renewable integration. Upon reformulating the two-stage grid dispatch as a stochastic convex-concave problem, two dis...
Generation Contingency Constrained Economic Dispatch,” IEEE Transactions on Power Systems
IEEE Transactions on Power Systems
This paper presents a new model of the Generation Contingency Constrained Economic Dispatch problem and proposes a method for its solution. The operating policy of the Northern Ireland Electricity was the basis for the formulation, and software was implemented to support it. Since the Northern Ireland Electricity system operates in relative isolation the operating security criteria are rather stringent. In particular, it is required that loss of generation of any unit in the system must be covered by fast, 3 or 30second generation reserves on other units in the system. The fast response unit reserve capabilities are represented by concave curves. The solution method is based a nonlinear version of Danbig-Wolfe decomposition principle. Numerical results are presented.
International Journal of Smart Grid and Clean Energy
In order to alleviate the effects of greenhouse gas emissions, the environmental and economic dispatch (EED) is formulated as multiobjective optimization problem (MOP) solved by multiobjective immune algorithm (MOIA). Building on this model, the virtual power plant (VPP) is proposed involving distributed generation (DG), interruptible load (IL), and energy storage (ES) to participate in joint energy and reserve markets. The uncertainties of load prediction, DG, and IL are treated as an interval-based optimization in this study. The static and real-time simulations are conducted to demonstrate the validity of proposed stochastic EED model through the IEEE 30-bus test system.
A Reliable and Economic Power System Dispatch
Diyala Journal of Engineering Sciences, 2020
A line bus security index is added to the generation cost forming a new objective function a suitable for optimal power system dispatch. The value of this security index is directly proportional to the sum of the squares of voltages of both bus voltage and line flows. Normal as well as contingency states of a power system are successfully dispatched. Result obtaining during a normal state of both the 6-bus and modified 30-bus cost. Both systems are also dispatched when the outage contingencies are assumed. Successful dispatch results are obtained without any load shedding. Power system reliability is seen to be greatly improved through the present reliable optimum dispatch.
Derivation and Application of a New Transmission Loss Formula for Power System Economic Dispatch
Energies, 2018
The expression and calculation of transmission loss (TL) play key roles for solving the power system economic dispatch (ED) problem. ED including TL must compute the total TL and incremental transmission loss (ITL) by executing power flow equations. However, solving the power flow equations is time-consuming and may result in divergence by the iteration procedure. This approach is unsuitable for real-time ED in practical power systems. To avoid solving nonlinear power flow equations, most power companies continue to adopt the TL formula in ED. Traditional loss formulas are composed of network parameters and in terms of the generator's real power outputs. These formulas are derived by several assumptions, but these basic assumptions sacrifice accuracy. In this study, a new expression for the loss formula is proposed to improve the shortcomings of traditional loss formulas. The coefficients in the new loss formula can be obtained by recording the power losses according to varying real and reactive power outputs without any assumptions. The simultaneous equations of the second-order expansion of the Taylor series are then established. Finally, the corresponding coefficients can be calculated by solving the simultaneous equations. These new coefficients can be used in optimal real and reactive power dispatch problems. The proposed approach is tested by IEEE 14-bus and 30-bus systems, and the results are compared with those obtained from the traditional B coefficient method and the load flow method. The numerical results show that the proposed new loss formula for ED can hold high accuracy for different loading conditions and is very suitable for real-time applications.
Consensus Based Economic Dispatch including System Power Losses
International Journal of Engineering & Technology, 2018
Economic dispatch (ED) is an important class of optimization problem in Power System Operation. As both conventional and heuristic methods to solve EDP are centrally controlled, which may leads to some performance limitations, a Consensus based distributed algorithm is proposed in this paper to solve Economic Dispatch with inclusion of losses. Earlier, some papers dealt with the consensus based methods to solve Economic dispatch, but here in this paper the losses are included and the variation of losses at each iteration are also used to update the mismatch, which has some major prominence in the present day Power system environment. In this paper, the mismatch between load demand and total power generation is collectively learnt by the each generator, unlike the centralized approach, through the strongly connected communication network. MATLAB results in IEEE 6-bus system validate the potency and efficacy of the proposed technique
Probabilistic Economic Emission Dispatch Optimization of Multi-sources Power System
Energy Procedia, 2014
The interest on renewable energy resources is growing and the study of different integration aspects of these resources becomes very important to overcome problems caused by their variability or uncertainty. This paper treats the economic environmental power dispatch as a probabilistic multiobjective problem. The operation cost and green house gas emission functions are considered as the sum of deterministic part and probabilistic one. First, the problem is solved based on expected values of generated wind power then, using the cumulative density function (CDF) of each renewable energy source (RES), the CDF of the required reserve to compensate the RESs variability in order to keep the power balance. Then, respecting to the reserve contribution of each thermal generator, the probabilistic part of the global generation cost as well as its CDF are developed. Finally, the proposed approach is applied to solve the active power dispatch problem of IEEE 30-bus test system in two cases with and without RESs. The simulation results show that this method allows to get the complete information about the cumulative distribution function of the actual global cost of the system operation.
A simple recourse model for power dispatch under uncertain demand
Annals of Operations Research, 1995
Optimal power dispatch under uncertainty of power demand is tackled via a stochastic programming model with simple recourse. The decision variables correspond to generation policies of a system comprising thermal units, pumped storage plants and energy contracts. The paper is a case study to test the kernel estimation method in the context of stochastic programming. Kernel estimates are used to approximate the unknown probability distribution of power demand. General stability results from stochastic programming yield the asymptotic stability of optimal solutions. Kernel estimates lead to favourable numerical properties of the recourse model (no numerical integration, the optimization problem is smooth convex and of moderate dimension). Test runs based on real-life data are reported. We compute the value of the stochastic solution for di erent problem instances and compare the stochastic programming solution with deterministic solutions involving adjusted demand portions.
Solution of combined economic and emission dispatch problems of power systems without penalty
Applied Artificial Intelligence
Regular paper Solution of Combined Economic and Emission Dispatch problems using Galaxy-based Search Algorithm. JES Journal of Electrical Systems The Galaxy-based Search Algorithm (GbSA) is an optimization technique developed recently by Hamed Shah-Hosseini at Shahid Beheshti University-Iran [1, 2]. GbSA is a meta-heuristic that uses a modified Hill Climbing algorithm as a local search and resembles the spiral arms of some galaxies to search the optimum. In this paper, GbSA is proposed for solving the Combined Economic and Emission Dispatch (CEED) problem under some equality and inequality constraints. The equality constraints are the active power flow balance equations, while the inequality constraints are the minimum and maximum power output of each unit. The voltage levels and security are assumed to be constant. The CEED problem is obtained by considering both the economy and emission objectives. This bi-objective problem is converted into a single objective function using a price penalty factor. The validity of GbSA is tested on two sample systems and the results are compared to those reported in the recent literature. The study results are quite encouraging showing the good applicability of GbSA for CEED problem.