Search results for: constrained optimization

Authors: Hyun-Chool Shin

Abstract:

In this paper we propose a simple adaptive algorithm iteratively solving the unit-norm constrained optimization problem. Instead of conventional parameter norm based normalization, the proposed algorithm incorporates scalar normalization which is computationally much simpler. The analysis of stationary point is presented to show that the proposed algorithm indeed solves the constrained optimization problem. The simulation results illustrate that the proposed algorithm performs as good as conventional ones while being computationally simpler.

Keywords: constrained optimization, unit-norm, LMS, principle component analysis.

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Authors: Tomoaki Hashimoto

Abstract:

In recent decades, probabilistic constrained optimal control problems have attracted much attention in many research fields. Although probabilistic constraints are generally intractable in an optimization problem, several tractable methods haven been proposed to handle probabilistic constraints. In most methods, probabilistic constraints are reduced to deterministic constraints that are tractable in an optimization problem. However, there is a gap between the transformed deterministic constraints in case of known and unknown probability distribution. This paper examines the conservativeness of probabilistic constrained optimization method for unknown probability distribution. The objective of this paper is to provide a quantitative assessment of the conservatism for tractable constraints in probabilistic constrained optimization with unknown probability distribution.

Keywords: Optimal control, stochastic systems, discrete-time systems, probabilistic constraints.

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

Abstract:

Direct search methods are evolutionary algorithms used to solve optimization problems. (DS) methods do not require any information about the gradient of the objective function at hand while searching for an optimum solution. One of such methods is Pattern Search (PS) algorithm. This paper presents a new approach based on a constrained pattern search algorithm to solve a security constrained power system economic dispatch problem (SCED). Operation of power systems demands a high degree of security to keep the system satisfactorily operating when subjected to disturbances, while and at the same time it is required to pay attention to the economic aspects. Pattern recognition technique is used first to assess dynamic security. Linear classifiers that determine the stability of electric power system are presented and added to other system stability and operational constraints. The problem is formulated as a constrained optimization problem in a way that insures a secure-economic system operation. Pattern search method is then applied to solve the constrained optimization formulation. In particular, the method is tested using one system. Simulation results of the proposed approach are compared with those reported in literature. The outcome is very encouraging and proves that pattern search (PS) is very applicable for solving security constrained power system economic dispatch problem (SCED).

Keywords: Security Constrained Economic Dispatch, Direct Search method, optimization.

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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: Fatemeh Azimi, Razeeh Sadat Aboutalebi, Amir Abbas Najafi

Abstract:

In this paper the multi-mode resource-constrained project scheduling problem with discounted cash flows is considered. Minimizing the makespan and maximization the net present value (NPV) are the two common objectives that have been investigated in the literature. We apply one evolutionary algorithm named multiobjective particle swarm optimization (MOPSO) to find Pareto front solutions. We used standard sets of instances from the project scheduling problem library (PSPLIB). The results are computationally compared respect to different metrics taken from the literature on evolutionary multi-objective optimization.

Keywords: Evolutionary multi-objective optimization makespan, multi-mode, resource constraint, net present value.

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Authors: Mohammad Taha, Dia abu al Nadi

Abstract:

In this paper, the phase control antenna array synthesis is presented. The problem is formulated as a constrained optimization problem that imposes nulls with prescribed level while maintaining the sidelobe at a prescribed level. For efficient use of the algorithm memory, compared to the well known Particle Swarm Optimization (PSO), the Accelerated Particle Swarm Optimization (APSO) is used to estimate the phase parameters of the synthesized array. The objective function is formed using a main objective and set of constraints with penalty factors that measure the violation of each feasible solution in the search space to each constraint. In this case the obtained feasible solution is guaranteed to satisfy all the constraints. Simulation results have shown significant performance increases and a decreased randomness in the parameter search space compared to a single objective conventional particle swarm optimization.

Keywords: Array synthesis, Sidelobe level control, Constrainedoptimization, Accelerated Particle Swarm Optimization.

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Authors: G. Mesfin, M. Shuhaimi

Abstract:

This paper proposes a new optimization techniques for the optimization a gas processing plant uncertain feed and product flows. The problem is first formulated using a continuous linear deterministic approach. Subsequently, the single and joint chance constraint models for steady state process with timedependent uncertainties have been developed. The solution approach is based on converting the probabilistic problems into their equivalent deterministic form and solved at different confidence levels Case study for a real plant operation has been used to effectively implement the proposed model. The optimization results indicate that prior decision has to be made for in-operating plant under uncertain feed and product flows by satisfying all the constraints at 95% confidence level for single chance constrained and 85% confidence level for joint chance constrained optimizations cases.

Keywords: Butane, Feed composition, LPG, Productspecification, Propane.

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Authors: Tomoaki Hashimoto

Abstract:

Recently, optimal control problems subject to probabilistic constraints have attracted much attention in many research field. Although probabilistic constraints are generally intractable in optimization problems, several methods haven been proposed to deal with probabilistic constraints. In most methods, probabilistic constraints are transformed to deterministic constraints that are tractable in optimization problems. This paper examines a method for transforming probabilistic constraints into deterministic constraints for a class of probabilistic constrained optimal control problems.

Keywords: Optimal control, stochastic systems, discrete-time systems, probabilistic constraints.

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Authors: A. K. Sharma, N. P. Patidar, G. Agnihotri, D. K. Palwalia

Abstract:

This paper presents the performance state analysis of Self-Excited Induction Generator (SEIG) using Artificial Bee Colony (ABC) optimization technique. The total admittance of the induction machine is minimized to calculate the frequency and magnetizing reactance corresponding to any rotor speed, load impedance and excitation capacitance. The performance of SEIG is calculated using the optimized parameter found. The results obtained by ABC algorithm are compared with results from numerical method. The results obtained coincide with the numerical method results. This technique proves to be efficient in solving nonlinear constrained optimization problems and analyzing the performance of SEIG.

Keywords: Artificial bee colony, Steady state analysis, Selfexcited induction generator, Nonlinear constrained optimization.

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Authors: A.K.M. Khaled Ahsan Talukder, Taibun Nessa, Kaushik Roy

Abstract:

The weight constrained shortest path problem (WCSPP) is one of most several known basic problems in combinatorial optimization. Because of its importance in many areas of applications such as computer science, engineering and operations research, many researchers have extensively studied the WCSPP. This paper mainly concentrates on the reduction of total search space for finding WCSP using some existing Genetic Algorithm (GA). For this purpose, some controlled schemes of genetic operators are adopted on list chromosome representation. This approach gives a near optimum solution with smaller elapsed generation than classical GA technique. From further analysis on the matter, a new generalized schema theorem is also developed from the philosophy of Holland-s theorem.

Keywords: Genetic Algorithm, Evolutionary Optimization, Multi Objective Optimization, Non-linear Schema Theorem, WCSPP.

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Authors: P.S. Prakash, S. Selvan

Abstract:

Quality of Service (QoS) Routing aims to find path between source and destination satisfying the QoS requirements which efficiently using the network resources and underlying routing algorithm and to fmd low-cost paths that satisfy given QoS constraints. One of the key issues in providing end-to-end QoS guarantees in packet networks is determining feasible path that satisfies a number of QoS constraints. We present a Optimized Multi- Constrained Routing (OMCR) algorithm for the computation of constrained paths for QoS routing in computer networks. OMCR applies distance vector to construct a shortest path for each destination with reference to a given optimization metric, from which a set of feasible paths are derived at each node. OMCR is able to fmd feasible paths as well as optimize the utilization of network resources. OMCR operates with the hop-by-hop, connectionless routing model in IP Internet and does not create any loops while fmding the feasible paths. Nodes running OMCR not necessarily maintaining global view of network state such as topology, resource information and routing updates are sent only to neighboring nodes whereas its counterpart link-state routing method depend on complete network state for constrained path computation and that incurs excessive communication overhead.

Keywords: QoS Routing, Optimization, feasible path, multiple constraints.

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Authors: Yanan Yang, Zhigang Wang, Xiang Chen

Abstract:

This paper presents the use of Legendre pseudospectral method for the optimization of finite-thrust orbital transfer for spacecrafts. In order to get an accurate solution, the System-s dynamics equations were normalized through a dimensionless method. The Legendre pseudospectral method is based on interpolating functions on Legendre-Gauss-Lobatto (LGL) quadrature nodes. This is used to transform the optimal control problem into a constrained parameter optimization problem. The developed novel optimization algorithm can be used to solve similar optimization problems of spacecraft finite-thrust orbital transfer. The results of a numerical simulation verified the validity of the proposed optimization method. The simulation results reveal that pseudospectral optimization method is a promising method for real-time trajectory optimization and provides good accuracy and fast convergence.

Keywords: Finite-thrust, Orbital transfer, Legendre pseudospectral method

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Authors: Vijaya K. Srivastava, Davide Spinello

Abstract:

This paper presents performance of two robust gradient-based heuristic optimization procedures based on 3ⁿ enumeration and tunneling approach to seek global optimum of constrained integer problems. Both these procedures consist of two distinct phases for locating the global optimum of integer problems with a linear or non-linear objective function subject to linear or non-linear constraints. In both procedures, in the first phase, a local minimum of the function is found using the gradient approach coupled with hemstitching moves when a constraint is violated in order to return the search to the feasible region. In the second phase, in one optimization procedure, the second sub-procedure examines 3ⁿ integer combinations on the boundary and within hypercube volume encompassing the result neighboring the result from the first phase and in the second optimization procedure a tunneling function is constructed at the local minimum of the first phase so as to find another point on the other side of the barrier where the function value is approximately the same. In the next cycle, the search for the global optimum commences in both optimization procedures again using this new-found point as the starting vector. The search continues and repeated for various step sizes along the function gradient as well as that along the vector normal to the violated constraints until no improvement in optimum value is found. The results from both these proposed optimization methods are presented and compared with one provided by popular MS Excel solver that is provided within MS Office suite and other published results.

Keywords: Constrained integer problems, enumerative search algorithm, Heuristic algorithm, tunneling algorithm.

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Authors: Akbar Banitalebi, Mohd Ismail Abd Aziz, Rohanin Ahmad

Abstract:

A new stochastic algorithm called Probabilistic Global Search Johor (PGSJ) has recently been established for global optimization of nonconvex real valued problems on finite dimensional Euclidean space. In this paper we present convergence guarantee for this algorithm in probabilistic sense without imposing any more condition. Then, we jointly utilize this algorithm along with control parameterization technique for the solution of constrained optimal control problem. The numerical simulations are also included to illustrate the efficiency and effectiveness of the PGSJ algorithm in the solution of control problems.

Keywords: Optimal Control Problem, Constraints, Direct Methods, Stochastic Algorithm

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Authors: Toshinori Nawata

Abstract:

In this paper we consider a nonlinear feedback control called augmented automatic choosing control (AACC) for nonlinear systems with constrained input using weighted gradient optimization automatic choosing functions. Constant term which arises from linearization of a given nonlinear system is treated as a coefficient of a stable zero dynamics. Parameters of the control are suboptimally selected by maximizing the stable region in the sense of Lyapunov with the aid of a genetic algorithm. This approach is applied to a field excitation control problem of power system to demonstrate the splendidness of the AACC. Simulation results show that the new controller can improve performance remarkably well.

Keywords: Augmented automatic choosing control, nonlinear control, genetic algorithm, zero dynamics.

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Authors: P.S.Prakash, S.Selvan

Abstract:

QoS Routing aims to find paths between senders and receivers satisfying the QoS requirements of the application which efficiently using the network resources and underlying routing algorithm to be able to find low-cost paths that satisfy given QoS constraints. The problem of finding least-cost routing is known to be NP hard or complete and some algorithms have been proposed to find a near optimal solution. But these heuristics or algorithms either impose relationships among the link metrics to reduce the complexity of the problem which may limit the general applicability of the heuristic, or are too costly in terms of execution time to be applicable to large networks. In this paper, we analyzed two algorithms namely Characterized Delay Constrained Routing (CDCR) and Optimized Delay Constrained Routing (ODCR). The CDCR algorithm dealt an approach for delay constrained routing that captures the trade-off between cost minimization and risk level regarding the delay constraint. The ODCR which uses an adaptive path weight function together with an additional constraint imposed on the path cost, to restrict search space and hence ODCR finds near optimal solution in much quicker time.

Keywords: QoS, Delay, Routing, Optimization

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Authors: András Király, Tamás Varga, János Abonyi

Abstract:

Since supply chains highly impact the financial performance of companies, it is important to optimize and analyze their Key Performance Indicators (KPI). The synergistic combination of Particle Swarm Optimization (PSO) and Monte Carlo simulation is applied to determine the optimal reorder point of warehouses in supply chains. The goal of the optimization is the minimization of the objective function calculated as the linear combination of holding and order costs. The required values of service levels of the warehouses represent non-linear constraints in the PSO. The results illustrate that the developed stochastic simulator and optimization tool is flexible enough to handle complex situations.

Keywords: stochastic processes, empirical distributions, Monte Carlo simulation, PSO, supply chain management

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Authors: A. El Shahat, A. Keyhani, H. El Shewy

Abstract:

High Speed PM Generators driven by micro-turbines are widely used in Smart Grid System. So, this paper proposes comparative study among six classical, optimized and genetic analytical design cases for 400 kW output power at tip speed 200 m/s. These six design trials of High Speed Permanent Magnet Synchronous Generators (HSPMSGs) are: Classical Sizing; Unconstrained optimization for total losses and its minimization; Constrained optimized total mass with bounded constraints are introduced in the problem formulation. Then a genetic algorithm is formulated for obtaining maximum efficiency and minimizing machine size. In the second genetic problem formulation, we attempt to obtain minimum mass, the machine sizing that is constrained by the non-linear constraint function of machine losses. Finally, an optimum torque per ampere genetic sizing is predicted. All results are simulated with MATLAB, Optimization Toolbox and its Genetic Algorithm. Finally, six analytical design examples comparisons are introduced with study of machines waveforms, THD and rotor losses.

Keywords: High Speed, Micro - Turbines, Optimization, PM Generators, Smart Grid, MATLAB.

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Authors: Sangeeta Mondal, Vasundhara, Rajib Kar, Durbadal Mandal, S. P. Ghoshal

Abstract:

This paper presents an optimal design of linear phase digital high pass finite impulse response (FIR) filter using Improved Particle Swarm Optimization (IPSO). In the design process, the filter length, pass band and stop band frequencies, feasible pass band and stop band ripple sizes are specified. FIR filter design is a multi-modal optimization problem. An iterative method is introduced to find the optimal solution of FIR filter design problem. Evolutionary algorithms like real code genetic algorithm (RGA), particle swarm optimization (PSO), improved particle swarm optimization (IPSO) have been used in this work for the design of linear phase high pass FIR filter. IPSO is an improved PSO that proposes a new definition for the velocity vector and swarm updating and hence the solution quality is improved. A comparison of simulation results reveals the optimization efficacy of the algorithm over the prevailing optimization techniques for the solution of the multimodal, nondifferentiable, highly non-linear, and constrained FIR filter design problems.

Keywords: FIR Filter, IPSO, GA, PSO, Parks and McClellan Algorithm, Evolutionary Optimization, High Pass Filter

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Authors: Ivan C. Mustakerov, Daniela I. Borissova

Abstract:

The paper proposes an approach for design of modular systems based on original technique for modeling and formulation of combinatorial optimization problems. The proposed approach is described on the example of personal computer configuration design. It takes into account the existing compatibility restrictions between the modules and can be extended and modified to reflect different functional and users- requirements. The developed design modeling technique is used to formulate single objective nonlinear mixedinteger optimization tasks. The practical applicability of the developed approach is numerically tested on the basis of real modules data. Solutions of the formulated optimization tasks define the optimal configuration of the system that satisfies all compatibility restrictions and user requirements.

Keywords: Constrained discrete combinatorial choice, modular systems design, optimization problem, PC configuration.

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Authors: Prashant P. Bedekar, Sudhir R. Bhide, Vijay S. Kale

Abstract:

Overcurrent (OC) relays are the major protection devices in a distribution system. The operating time of the OC relays are to be coordinated properly to avoid the mal-operation of the backup relays. The OC relay time coordination in ring fed distribution networks is a highly constrained optimization problem which can be stated as a linear programming problem (LPP). The purpose is to find an optimum relay setting to minimize the time of operation of relays and at the same time, to keep the relays properly coordinated to avoid the mal-operation of relays. This paper presents two phase simplex method for optimum time coordination of OC relays. The method is based on the simplex algorithm which is used to find optimum solution of LPP. The method introduces artificial variables to get an initial basic feasible solution (IBFS). Artificial variables are removed using iterative process of first phase which minimizes the auxiliary objective function. The second phase minimizes the original objective function and gives the optimum time coordination of OC relays.

Keywords: Constrained optimization, LPP, Overcurrent relaycoordination, Two-phase simplex method.

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Authors: Kyoungwoo Park, Byeong Sam Kim, Hyo-Jae Lim, Ji Won Han, Park Kyoun Oh, Juhee Lee, Keun-Yeol Yu

Abstract:

Predictions of flow and heat transfer characteristics and shape optimization in internally finned circular tubes have been performed on three-dimensional periodically fully developed turbulent flow and thermal fields. For a trapezoidal fin profile, the effects of fin height h, upper fin widths d1, lower fin widths d2, and helix angle of fin ? on transport phenomena are investigated for the condition of fin number of N = 30. The CFD and mathematical optimization technique are coupled in order to optimize the shape of internally finned tube. The optimal solutions of the design variables (i.e., upper and lower fin widths, fin height and helix angle) are numerically obtained by minimizing the pressure loss and maximizing the heat transfer rate, simultaneously, for the limiting conditions of d1 = 0.5~1.5 mm, d2 = 0.5~1.5 mm, h= 0.5~1.5mm, ? = 10~30 degrees. The fully developed flow and thermal fields are predicted using the finite volume method and the optimization is carried out by means of the multi-objective genetic algorithm that is widely used in the constrained nonlinear optimization problem.

Keywords: Computational fluid dynamics, Genetic algorithm, Internally finned tube with helix angle, Optimization.

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Authors: Toshinori Nawata

Abstract:

In this paper a nonlinear feedback control called augmented automatic choosing control (AACC) for a class of nonlinear systems with constrained input is presented. When designed the control, a constant term which arises from linearization of a given nonlinear system is treated as a coefficient of a stable zero dynamics. Parameters of the control are suboptimally selected by maximizing the stable region in the sense of Lyapunov with the aid of a genetic algorithm. This approach is applied to a field excitation control problem of power system to demonstrate the splendidness of the AACC. Simulation results show that the new controller can improve performance remarkably well.

Keywords: Augmented automatic choosing control, nonlinear control, genetic algorithm, zero dynamics.

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Authors: Behrouz Afshar Nadjafi

Abstract:

Extensive research has been devoted to economic production quantity (EPQ) problem. However, no attention has been paid to problems where production period length is constrained. In this paper, we address the problem of deciding the optimal production quantity and the number of minor setups within each cycle, in which, production period length is constrained but a minor setup is possible for pass the constraint. A mathematical model is developed and Iterated Local Search (ILS) is proposed to solve this problem. Finally, solution procedure illustrated with a numerical example and results are analyzed.

Keywords: EPQ, Inventory control, minor setup, ILS.

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Authors: Songtao Chang, Yongji Wang, Lei Liu, Dangjun Zhao

Abstract:

Reentry trajectory optimization is a multi-constraints optimal control problem which is hard to solve. To tackle it, we proposed a new algorithm named CDEN(Constrained Differential Evolution Newton-Raphson Algorithm) based on Differential Evolution( DE) and Newton-Raphson.We transform the infinite dimensional optimal control problem to parameter optimization which is finite dimensional by discretize control parameter. In order to simplify the problem, we figure out the control parameter-s scope by process constraints. To handle constraints, we proposed a parameterless constraints handle process. Through comprehensive analyze the problem, we use a new algorithm integrated by DE and Newton-Raphson to solve it. It is validated by a reentry vehicle X-33, simulation results indicated that the algorithm is effective and robust.

Keywords: reentry vehicle, trajectory optimization, constraint optimal, differential evolution.

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Authors: O. Castellanos Díaz, F. Schoeggl, H. W. Yarranton, M. A. Satyro, T. M. Lovestead, T. J. Bruno

Abstract:

The composition, vapour pressure, and heat capacity of nine biodiesel fuels from different sources were measured. The vapour pressure of the biodiesel fuels is modeled assuming an ideal liquid phase of the fatty acid methyl esters constituting the fuel. New methodologies to calculate the vapour pressure and ideal gas and liquid heat capacities of the biodiesel fuel constituents are proposed. Two alternative optimization scenarios are evaluated: 1) vapour pressure only; 2) vapour pressure constrained with liquid heat capacity. Without physical constraints, significant errors in liquid heat capacity predictions were found whereas the constrained correlation accurately fit both vapour pressure and liquid heat capacity.

Keywords: Biodiesel fuels, Fatty acid methyl ester, Heat capacity, Modeling, Vapour pressure

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Authors: P. S. Prakash, S. Selvan

Abstract:

QoS Routing aims to find paths between senders and receivers satisfying the QoS requirements of the application which efficiently using the network resources and underlying routing algorithm to be able to find low-cost paths that satisfy given QoS constraints. The problem of finding least-cost routing is known to be NP-hard or complete and some algorithms have been proposed to find a near optimal solution. But these heuristics or algorithms either impose relationships among the link metrics to reduce the complexity of the problem which may limit the general applicability of the heuristic, or are too costly in terms of execution time to be applicable to large networks. In this paper, we concentrate an algorithm that finds a near-optimal solution fast and we named this algorithm as optimized Delay Constrained Routing (ODCR), which uses an adaptive path weight function together with an additional constraint imposed on the path cost, to restrict search space and hence ODCR finds near optimal solution in much quicker time.

Keywords: QoS, Delay, Routing, Optimization.

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Authors: Phruksaphanrat B.

Abstract:

This research proposes a preemptive fuzzy goal programming model for multi-objective multi-mode resource constrained project scheduling problem. The objectives of the problem are minimization of the total time and the total cost of the project. Objective in a multi-mode resource-constrained project scheduling problem is often a minimization of makespan. However, both time and cost should be considered at the same time with different level of important priorities. Moreover, all elements of cost functions in a project are not included in the conventional cost objective function. Incomplete total project cost causes an error in finding the project scheduling time. In this research, preemptive fuzzy goal programming is presented to solve the multi-objective multi-mode resource constrained project scheduling problem. It can find the compromise solution of the problem. Moreover, it is also flexible in adjusting to find a variety of alternative solutions. 

Keywords: Multi-mode resource constrained project scheduling problem, Fuzzy set, Goal programming, Preemptive fuzzy goal programming.

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Authors: I. A. Farhat

Abstract:

The The dynamic economic dispatch (DED) problem is one of the complex constrained optimization problems that have nonlinear, con-convex and non-smooth objective functions. The purpose of the DED is to determine the optimal economic operation of the committed units while meeting the load demand. Associated to this constrained problem there exist highly nonlinear and non-convex practical constraints to be satisfied. Therefore, classical and derivative-based methods are likely not to converge to an optimal or near optimal solution to such a dynamic and large-scale problem. In this paper, an Artificial Immune System technique (AIS) is implemented and applied to solve the DED problem considering the transmission power losses and the valve-point effects in addition to the other operational constraints. To demonstrate the effectiveness of the proposed technique, two case studies are considered. The results obtained using the AIS are compared to those obtained by other methods reported in the literature and found better.

Keywords: Artificial Immune System (AIS), Dynamic Economic Dispatch (DED).

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Authors: I. A. Farhat

Abstract:

The dynamic economic dispatch (DED) problem is one of the complex constrained optimization problems that have nonlinear, con-convex and non-smooth objective functions. The purpose of the DED is to determine the optimal economic operation of the committed units while meeting the load demand.  Associated to this constrained problem there exist highly nonlinear and non-convex practical constraints to be satisfied. Therefore, classical and derivative-based methods are likely not to converge to an optimal or near optimal solution to such a dynamic and large-scale problem. In this paper, an Artificial Immune System technique (AIS) is implemented and applied to solve the DED problem considering the transmission power losses and the valve-point effects in addition to the other operational constraints. To demonstrate the effectiveness of the proposed technique, two case studies are considered. The results obtained using the AIS are compared to those obtained by other methods reported in the literature and found better.

Keywords: Artificial Immune System (AIS), Dynamic Economic Dispatch (DED).

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