Non-smooth Cost Functions Research Articles

Hybrid method for solving the non smooth cost function economic dispatch problem

This article is focused on hybrid method for solving the non-smooth cost function economic dispatch problem. The techniques were divided into two parts according to: the incremental cost rates are used to find the initial solution and bee colony optimization is used to find the optimal solution. The constraints of economic dispatch are power losses, load demand and practical operation constraints of generators. To verify the performance of the proposed algorithm, it is operated by the simulation on the MATLAB program and tests three case studies; three, six and thirteen generator units which compared to particle swarm optimization, cuckoo search algorithm, bat algorithm, firefly algorithm and bee colony optimization. The results show that the proposed algorithm is able to obtain higher quality solution efficiently than the others methods.

A Selection Hyper-Heuristic Algorithm for Multiobjective Dynamic Economic and Environmental Load Dispatch

Dynamic economic and environmental load dispatch (DEED) aims to determine the amount of electricity generated from power plants during the planning period to meet load demand while minimizing energy consumption costs and environmental pollution emission indicators subject to the operation requirements. This planning problem is usually expressed using a nonsmooth cost function, taking into account various equality and inequality constraints such as valve-point effects, operational limits, power balance, and ramp rate limits. This paper presents DEED models developed for a system consisting of thermal units, wind power generators, photovoltaic (PV) generators, and energy storage (ES). A selection hyper-heuristic algorithm is proposed to solve the problems. Three heuristic mutation operators formed a low-level operator pool to direct search the solution space of DEED. The high level of SHHA evaluates the performances of the low-level operators and dynamically adjusts the chosen probability of each operator. Simulation experiments were carried out on four systems of different types or sizes. The results verified the effectiveness of the proposed method.

Distributed optimal load frequency control considering nonsmooth cost functions

This work addresses the distributed frequency control problem in power systems considering controllable load with a nonsmooth cost. The nonsmoothness exists widely in power systems, such as tiered price, greatly challenging the design of distributed optimal controllers. In this paper, we first formulate an optimization problem that minimizes the nonsmooth regulation cost, where both capacity limits of controllable load and tie-line flow are considered. Then, a distributed controller is derived based on the primal–dual gradient method, where the Clark generalized gradient is utilized to address the nonsmoothness. We also prove the optimality of the equilibrium of the closed-loop system as well as its stability. Simulations carried out on the IEEE 68-bus system verifies the effectiveness of the proposed method.

Hybrid algorithm based on MBHS and PSO with Optimal Power Flow Problem for Non-Smooth Cost Functions

Optimal Power Flow (OPF) is the primary apparatus in power framework administrators for both working and arranging stages. It is made to define a couple of expectations dependent on power organizing factors with a couple of confinements. This paper examines the likelihood that utilizations couple of currently developed progressive ways to deal with anticipating answers for OPF issues based on ruler butterfly agreement search (MBHS) calculation which uses Particle swam streamlining (PSO) for ideal settings of OPF issue control factors. The standard IEEE 30-bus with IEEE 57-bus test framework is assessed and examined by the presentation of the proposed methodologies with different useful destinations and furthermore, the correlation is made to this strategy. At long last, the acquired outcomes that are recovered from the connected reproduction accommodate the MBHS and PSO with effective answers for the issue in OPF.

Noninferior Solution Grey Wolf Optimizer with an Independent Local Search Mechanism for Solving Economic Load Dispatch Problems

The economic load dispatch (ELD) problem is a complex optimization problem in power systems. The main task for this optimization problem is to minimize the total fuel cost of generators while also meeting the conditional constraints of valve-point loading effects, prohibited operating zones, and nonsmooth cost functions. In this paper, a novel grey wolf optimization (GWO), abbreviated as NGWO, is proposed to solve the ELD problem by introducing an independent local search strategy and a noninferior solution neighborhood independent local search technique to the original GWO algorithm to achieve the best problem solution. A local search strategy is added to the standard GWO algorithm in the NGWO, which is called GWOI, to search the local neighborhood of the global optimal point in depth and to guarantee a better candidate. In addition, a noninferior solution neighborhood independent local search method is introduced into the GWOI algorithm to find a better solution in the noninferior solution neighborhood and ensure the high probability of jumping out of the local optimum. The feasibility of the proposed NGWO method is verified on five different power systems, and it is compared with other selected methods in terms of the solution quality, convergence rate, and robustness. The compared experimental results indicate that the proposed NGWO method can efficiently solve ELD problems with higher-quality solutions.

A nonsmooth Stackelberg equilibrium problem via mixed variational inequalities

In this paper, we introduce and study a class of nonsmooth Stackelberg equilibrium problems with differential revenue functions and nonsmooth cost functions. First, we describe the best response set of the follower and the best strategy set of the leader for the considered nonsmooth Stackelberg equilibrium problem via mixed variational inequalities. Then, by using the resolvent operator method and the Banach fixed point theorem, we show the existence and uniqueness of its solution.

Optimisation of Classic Photometric Stereo by Non-convex Variational Minimisation

Estimating shape and appearance of a three-dimensional object from a given set of images is a classic research topic that is still actively pursued. Among the various techniques available, photometric stereo is distinguished by the assumption that the underlying input images are taken from the same point of view but under different lighting conditions. The most common techniques are conceptually close to the classic photometric stereo problem, meaning that the modelling encompasses a linearisation step and that the shape information is computed in terms of surface normals. In this work, instead of linearising we aim to stick to the original formulation of the photometric stereo problem, and we propose to minimise a much more natural objective function, namely the reprojection error in terms of depth. Minimising the resulting non-trivial variational model for photometric stereo allows to recover the depth of the photographed scene directly. As a solving strategy, we follow an approach based on a recently published optimisation scheme for non-convex and non-smooth cost functions. The main contributions of our paper are of theoretical nature. A technical novelty in our framework is the usage of matrix differential calculus. We supplement our approach by a detailed convergence analysis of the resulting optimisation algorithm and discuss possibilities to ease the computational complexity. At hand of an experimental evaluation we discuss important properties of the method. Overall, our strategy achieves more accurate results than other approaches that rely on the classic photometric stereo assumptions. The experiments also highlight some practical aspects of the underlying optimisation algorithm that may be of interest in a more general context.

Distributed Nonsmooth Optimization With Coupled Inequality Constraints via Modified Lagrangian Function

This technical note considers a distributed convex optimization problem with nonsmooth cost functions and coupled nonlinear inequality constraints. To solve the problem, we first propose a modified Lagrangian function containing local multipliers and a nonsmooth penalty function. Then we construct a distributed continuous-time algorithm by virtue of a projected primal-dual subgradient dynamics. Based on the nonsmooth analysis and Lyapunov function, we obtain the existence of the solution to the nonsmooth algorithm and its convergence.

Implementation of Smooth and Non-Smooth Fuel Cost Function for Economic Load Dispatch using Cuckoo Search Method

Economic Load Dispatch (ELD) is an important topic in the operation of power plants which can help to build up effecting generating management plans. The ELD problem has no smooth cost function with equality and inequality constraints which make it difficult to be effectively solved. The paper presents an application of Cuckoo Search Algorithm (CSA) to solve Economic Load Dispatch (ELD) problems with smooth and non-smooth fuel cost objective functions. Main objective of ELD is to determine the most economic generating dispatch required to satisfy the predicted load demands including line losses over a certain period of time while relaxing various equality and inequality constraints. The unit Min/Max operational constraints, effects of valve-point loading ripples and line losses are considered for the practical applications. This paper describes the implementation of smooth and non smooth fuel cost function by CSA Method and its comparison with BAT method. We have used 6 and 12 bus system for calculating their total fuel cost.

Robust Regression Estimation Based on Low-Dimensional Recurrent Neural Networks.

The robust Huber's M-estimator is widely used in signal and image processing, classification, and regression. From an optimization point of view, Huber's M-estimation problem is often formulated as a large-sized quadratic programming (QP) problem in view of its nonsmooth cost function. This paper presents a generalized regression estimator which minimizes a reduced-sized QP problem. The generalized regression estimator may be viewed as a significant generalization of several robust regression estimators including Huber's M-estimator. The performance of the generalized regression estimator is analyzed in terms of robustness and approximation accuracy. Furthermore, two low-dimensional recurrent neural networks (RNNs) are introduced for robust estimation. The two RNNs have low model complexity and enhanced computational efficiency. Finally, the experimental results of two examples and an application to image restoration are presented to substantiate superior performance of the proposed method over conventional algorithms for robust regression estimation in terms of approximation accuracy and convergence rate.

Bee colony optimization with Taguchi method for solving the dynamic economic dispatch

Dynamic economic dispatch is to minimize the cost of power production of all the participating generators over a time horizon of 24 hours in one day. The dynamic economic dispatch with non-smooth cost functions, for which is formulated the optimal dispatch model of generations by considering the ramp up/down scheduling of power. This paper presents a Bee Colony Optimization (BCO) that applies the Taguchi Method (TM) to solve the Dynamic Economic Dispatch problem. The Taguchi method that involves the use of orthogonal arrays in estimating of the non-smooth cost function and Bee Colony Optimization is used to find the objective function under the operational of system constraints. The Taguchi method can global optimization for fast local convergence by minimizing the cost function in a few iterations. The effectiveness and efficiency of the TM-BCO is demonstrated by using a 10 unit of IEEE case with non-smooth fuel cost functions and is more effective than other previously developed algorithms. Moreover, the proposed approach presents significant computational benefits than traditional random search method especially for multi-unit systems with larger numbers of non-smooth cost functions and more complicated dynamic economic dispatch.

Imposing Class-Wise Feature Similarity in Stacked Autoencoders by Nuclear Norm Regularization

This work proposes a new formulation for supervised stacked autoencoder. We argue that features from the same class should be similar to each other and hence linearly dependent. This means that, when stacked as columns, the feature matrix for each class will be rank deficient (low-rank). We impose this constraint into the stacked autoencoder formulation in the form of nuclear norm penalties on class-wise feature matrices at each level. The nuclear norm penalty is the convex surrogate of rank, and promotes a low-rank solution as desired by our proposal. Owing to the nuclear norm penalties, our formulation is non-smooth; hence cannot be solved using gradient descent based techniques like backpropagation directly. Moreover we learn the stacked autoencoder in one go, without the usual pre-training followed by fine-tuning regime. Both the ends (non-smooth cost function and single stage training for all the layers simultaneously) are met by employing the variable splitting followed by augmented Lagrangian method of alternating directions. Two sets of experiments have been carried out. The first set is on a variety of benchmark datasets. Our method excels over other deep learning models compared against—class sparse stacked autoencoder, deep belief network and discriminative deep belief network. The second experiment is on the brain computer classification problem; we find that our method outperforms prior deep learning based solutions utilized for this task.

Hybrid PSO-SQP for solving combined heat and power dynamic economic emission dispatch problem with non-smooth cost function

In this paper we propose a hybrid particle swarm optimization (PSO) and sequential quadratic programming (SQP) for solving the combined heat and power dynamic economic emission dispatch (CHPDEED) problem. The primary objective of CHPDEED is to determine the optimal heat and power generation schedule of the online generating units over a fixed interval by simultaneously minimizing the generation cost and emission level and satisfying the dynamic constraints and other constraints. Taking into account the valve point effects, CHPDEED is considered as a multi-objective optimization problem with non-smooth characteristics. In the hybrid method, PSO is used as a global search to find near global optimal solution and this solution is used as initial for the SQP to find the global optimal solution at the end. The proposed method is verified using a test system consisting of eleven units and considering transmission line losses and valve point effects. The numerical results show the effectiveness and the superiority of the introduced method over other published methods.

Proximal Bundle Method for Contact Shape Optimization Problem

From the mathematical point of view, the contact shape optimization is a problem of nonlinear optimization with a specific structure, which can be exploited in its solution. In this paper, we show how to overcome the difficulties related to the nonsmooth cost function by using the proximal bundle methods. We describe all steps of the solution, including linearization, construction of a descent direction, line search, stopping criterion, etc. To illustrate the performance of the presented algorithm, we solve a shape optimization problem associated with the discretized two-dimensional contact problem with Coulomb's friction.

Reserve constrained dynamic economic dispatch with valve-point effect: A two-stage mixed integer linear programming approach

This paper proposes a deterministic two-stage mixed integer linear programming (TSMILP) approach to solve the reserve constrained dynamic economic dispatch (DED) problem considering valve-point effect (VPE). In stage one, the nonsmooth cost function and the transmission loss are piecewise linearized and consequently the DED problem is formulated as a mixed integer linear programming (MILP) problem, which can be solved by commercial solvers. In stage two, based on the solution obtained in stage one, a range compression technique is proposed to make a further exploitation in the subspace of the whole solution domain. Due to the linear approximation of the transmission loss, the solution obtained in stage two dose not strictly satisfies the power balance constraint. Hence, a forward procedure is employed to eliminate the error. The simulation results on four test systems show that TSMILP makes satisfactory performances, in comparison with the existing methods.

A new coding for solving large-scale non-convex economic dispatch problems without a penalty factor

In some optimization problems, it is necessary that a sum of variables converges to an expected value during the optimization process. In solving this type of problem, a penalty factor is usually employed in the objective function. Thus, some disadvantages such as increased search space is unavoidable. This research proposes a new coding based on particle swarm optimization in order to solve economic dispatch problems without a penalty factor, which, in turn, controls the sum of all variables. The suggested coding could effectively escape from infeasible solutions; thereby, search efficiency and solution quality would be dramatically improved. In order to verify the effectiveness of the proposed coding, it is implemented on various economic dispatch problems, i.e. test systems with 6, 15 and 40 generating units as well as a typical Korean system with 140 generating units, all with non-convex and non-smooth cost functions. In addition, the numerical results are compared with those of other well-established and valid techniques.

Interior search algorithm for optimal power flow with non-smooth cost functions

AbstractThe optimal power flow (OPF) problem considers as an optimization problem, in which the utility strives to reduce its global costs while pleasing all of its constraints. Therefore, Interior Search Algorithm (ISA) is applied to treat this problem. Where, ISA is a specific type of artificial intelligence and a mathematical programming, not a meta- heuristic algorithm. The effectiveness of this method in solving the OPF problem is evaluated on two test power systems, the IEEE 30-bus and the IEEE 57-bus test systems. For the first example, the ISA-OPF algorithm finds an answer that agrees with published results. For the 57-bus system, the ISA-OPF demonstrates its ability to transact with larger systems. Thus, the ISA-OPF algorithm is shown to be a robust tool to treat this optimization compared with other methods. Moreover, the advantage of ISA is that it has only one parameter of control which makes the simplicity in the main algorithm.

Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM

The quasistatic normal-compliance contact problem of isotropic homogeneous linear visco-elastic bodies with Coulomb friction at small strains in Kelvin–Voigt rheology is considered. The discretization is made by a semi-implicit formula in time and the Symmetric Galerkin Boundary Element Method (SGBEM) in space, assuming that the ratio of the viscosity and elasticity moduli is a given relaxation-time coefficient. The obtained recursive minimization problem, formulated only on the contact boundary, has a nonsmooth cost function. If the normal compliance responds linearly and the 2D problems are considered, then the cost function is piecewise-quadratic, which after a certain transformation gets the quadratic programming (QP) structure. However, it would lead to second-order cone programming in 3D problems. Finally, several computational tests are presented and analysed, with an additional discussion on numerical stability and convergence of the involved approximated Poincaré–Steklov operators.

An efficient hybrid MPSO-GA algorithm for solving non-smooth/non-convex economic dispatch problem with practical constraints

Abstract The main goal of Economic Dispatch (ED) is to determine the output of generating unit with least cost while satisfying the equality and inequality constraints. Valve-point effect, the ramp rate limits, prohibited operation zones (POZs), Multiple-fuel and transmission losses make the ED a complicated, non-linear constrained problem. Hence, in this paper, a new hybrid method based on modified particle swarm optimization and genetic algorithm (MPSO-GA) is proposed to solve such complicated problem. The feasibility of the proposed method is validated on six, ten and fifteen units test system. The results obtained by the MPSO-GA are compared with other techniques of the literature. These results substantiate the applicability of the proposed method for solving the constrained ED with non-smooth cost functions.

Optimum Dispatch of Hybrid Solar Thermal (HSTP) Electric Power Plant Using Non-Smooth Cost Function and Emission Function for IEEE-30 Bus System

The basic objective of economic load dispatch (ELD) is to optimize the total fuel cost of hybrid solar thermal electric power plant (HSTP). In ELD problems the cost function for each generator has been approximated by a single quadratic cost equation. As cost of coal increases, it becomes even more important have a good model for the production cost of each generator for the solar thermal hybrid system. A more accurate formulation is obtained for the ELD problem by expressing the generation cost function as a piece wise quadratic cost function. However, the solution methods for ELD problem with piece wise quadratic cost function requires much complicated algorithms such as the hierarchical structure approach along with evolutionary computations (ECs). A test system comprising of 10 units with 29 different fuel [7] cost equations is considered in this paper. The applied genetic algorithm method will provide optimal solution for the given load demand.