Non-smooth Cost Functions Research Articles

An Operator Splitting Scheme for Distributed Optimal Load-Side Frequency Control With Nonsmooth Cost Functions

An Operator Splitting Scheme for Distributed Optimal Load-Side Frequency Control With Nonsmooth Cost Functions

Correction to: Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions

Correction to: Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions

Nash Equilibrium Seeking Algorithm Design for Distributed Nonsmooth Multicluster Games Over Weight-Balanced Digraphs.

In this article, we study the multicluster games over weight-balanced digraphs, where the cost functions of all players are nonsmooth. Besides, in the problem, not only are the decisions of all players constrained by heterogeneous local constraints but also the decisions of players in the same cluster are constrained by coupling constraints. Due to the nonsmooth cost functions, the coupling constraints, the general local convex constraints, and the weight-balanced digraphs, existing Nash equilibrium seeking algorithms cannot solve our problem. In order to seek the Nash equilibrium of the game, we design a distributed algorithm based on subgradient descent, differential inclusions, and projection operations. In the algorithm, a distributed learning strategy is embedded for the players to estimate the decisions of other players. Moreover, we analyze the asymptotical convergence of the algorithm via set-valued LaSalle invariance principle. Finally, a numerical simulation about electricity market games is presented to illustrate the effectiveness of our result.

Neurodynamic approaches for multi-agent distributed optimization

This paper considers a class of multi-agent distributed convex optimization with a common set of constraints and provides several continuous-time neurodynamic approaches. In problem transformation, l1 and l2 penalty methods are used respectively to cast the linear consensus constraint into the objective function, which avoids introducing auxiliary variables and only involves information exchange among primal variables in the process of solving the problem. For nonsmooth cost functions, two differential inclusions with projection operator are proposed. Without convexity of the differential inclusions, the asymptotic behavior and convergence properties are explored. For smooth cost functions, by harnessing the smoothness of l2 penalty function, finite- and fixed-time convergent algorithms are provided via a specifically designed average consensus estimator. Finally, several numerical examples in the multi-agent simulation environment are conducted to illustrate the effectiveness of the proposed neurodynamic approaches.

Implementations of Golden Jackal Algorithm for Solving CCFELD Problems

This study offers the Golden Jackal Optimization (GJO) algorithm, an effective and trustworthy swarm optimization for tackling economic load dispatch (ELD) issues using cubic fuel cost functions. The presence of equal and unequal constraints of the non-smooth cost functions of a practical ELD has caused difficulties in finding an overall optimal result. The suggested GJO is tested first with quadratic cost functions as well as the cubic fuel cost functions to demonstrate its usefulness and efficiency. Three generator systems, five generator systems, six generating systems, 26 generators with quadratic and cubic fuel cost functions have all been used to assess the proposed GJO algorithm. Numerous case studies and evaluation with the other existing algorithms have substantiated that the suggested GJO technique yields outstanding outcomes.

Distributed neurodynamic algorithms for collaborative energy management in energy internet considering time-varying factors

This paper investigates the energy management problem of the energy Internet under time-varying conditions . In the context of coupled multi-energy networks, the energy Internet is considered to be composed of multiple energy bodies and requires collaborative planning of multiple energy networks. A model for distributed energy management with a non-smooth cost function and line congestion constraints is proposed, with the goal of reducing overall operating costs and improving customer benefits while considering load as a time-varying factor. Then, a neurodynamic time-varying algorithm for addressing the energy management problem executed in a fully distributed manner is proposed. On the one hand, the predictive effect of the differential feedback term is exploited and embedded in the implementation of the proposed algorithm, thus speeding up the convergence. On the other hand, the algorithm is executed in a distributed manner, and only limited information is exchanged among the agents to complete the optimal operation locally, thus reducing the communication burden and ensuring privacy and robustness. Finally, theoretical proofs guarantee the stability of the proposed algorithm, and simulation experiments illustrate the effectiveness and robustness of the proposed algorithm. • In this paper, a model for EI in a time-varying load scenario is proposed for optimal energy allocation among EBs integrating renewable energy, flexible loads, and multi-energy coupling, while trading energy. • A neurodynamic algorithm implemented in a fully distributed manner that enjoys hardware implementation and parallel computation is proposed in this paper. • The algorithm can effectively tackle EMPs with non-smooth objective functions and constraints under time-varying conditions.

A novel differential evolution algorithm for economic power dispatch problem

<p style='text-indent:20px;'>In power systems, Economic Power dispatch Problem (EPP) is an influential optimization problem which is a highly non-convex and non-linear optimization problem. In the current study, a novel version of Differential Evolution (NDE) is used to solve this particular problem. NDE algorithm enhances local and global search capability along with efficient utilization of time and space by making use of two elite features: selfadaptive control parameter and single population structure. The combined effect of these concepts improves the performance of Differential Evolution (DE) without compromising on quality of the solution and balances the exploitation and exploration capabilities of DE. The efficiency of NDE is validated by evaluating on three benchmark cases of the power system problem having constraints such as power balance and power generation along with nonsmooth cost function and is compared with other optimization algorithms. The Numerical outcomes uncovered that NDE performed well for all the benchmark cases and maintained a trade-off between convergence rate and efficiency.</p>

A Distributed Proximal-Gradient Based Algorithm for N-Coalition Games of Mixed-Order Players

This brief investigates the N-coalition games of mixed-order players, where first and second-order integrator-type players interact in the coalitions. Every player is associated with a nonsmooth cost function, and the objective of each coalition is to minimize its cost function which is the sum of local cost functions of the players. That is, the mixed-order players within the coalition collaboratively minimize their coalition’s cost function. To seek the Nash equilibrium of the N-coalition game, a distributed proximal-gradient based algorithm is proposed, where the proximal operator is used to deal with the nonsmoothness of the cost functions. The convergence of the proposed algorithm is analyzed by Lyapunov stability theorem. An example of mobile sensor networks is given to verify the effectiveness of the algorithm.

Optimal Generator Scheduling Considering Limit of Generator Operation Through Generator Capability Curve

Optimal generator scheduling is needed to save costs in an operation power system. In other hands, the operation of the generator in optimal condition must be considering the safety of the generator itself. This study proposes an optimal generator scheduling using a new modification of particle swarm optimization (PSO) algorithm considering the safety of the generator operation by visualized the operation of a generator through the generator capability curve. In term of scheduling, the MIW-PSO is applied with characteristics of power generator, such as smooth cost function and non-smooth cost function considering valve-point effects. The effectiveness and feasibility of the MIW-PSO were presented through four case studies and compared with previous literature in terms of power output and total operating cost. A case study is applied to show the performance of the optimal scheduling with MIW-PSO considering the safety of the generator operation by visualized the operation of a generator through the generator capability curve. The obtained results demonstrating the excellent performance of the proposed optimal scheduling both off to get optimal total cost and monitoring the safety of generator operation as well.

Solving a class of nonsmooth resource allocation problems with directed graphs through distributed Lipschitz continuous multi-proximal algorithms

In this paper, two distributed multi-proximal primal–dual algorithms are proposed to deal with a class of distributed nonsmooth resource allocation problems. In these problems, the global cost function is the summation of local convex and nonsmooth cost functions, each of which consists of one twice differentiable function and multiple nonsmooth functions. Communication graphs of underlying multi-agent systems are directed and strongly connected but not necessarily weighted-balanced. The multi-proximal splitting is introduced to deal with the difficulty caused by the unproximable property of the summation of those nonsmooth functions. Moreover, it can also guarantee the Lipschitz continuity of proposed algorithms. Auxiliary variables in the multi-proximal splitting are designed to estimate subgradients of nonsmooth functions. Theoretically, the convergence analysis is conducted by employing Lyapunov stability theory and integral input-to-state stability (iISS) theory with respect to sets. It shows that proposed algorithms converge to the optimal point while satisfying resource allocation constraints.

A New Hybrid Algorithm combining Ant Lion Optimization and Particle Swarm Optimization to Solve an Economic Dispatch Problem with non-smooth cost function

Wydawnictwo SIGMA-NOT wydaje czasopisma fachowe informujące swoich czytelników o najnowszych osiągnięciach naukowych i nowoczesnych rozwiązaniach technicznych w Polsce i na świecie, popularyzuje problemy techniczne oraz poszerza wiedzę i kulturę techniczną.

A multi-objective multi-verse optimization algorithm for dynamic load dispatch problems

In power system operation, optimal economic dispatch is imposed by the costs of increasing power generation, the increasing demand for electrical energy and the scarcity of energy resources. By satisfying all constraints, the most important thing is the economical load distribution in order to enable the generators used in the system to generate optimal power. The non-smooth cost function and emission with nonlinear constraints are the practical economic load dispatch issues that create a challenge that is effectively reduced. This paper presents a multi-objective multi-verse optimization scheme to minimize the dynamic economic load dispatch issue using valve-point effects. Along with all other necessary constraints, this algorithm preserves the ramp of unit required rate constraint. However, it maintains these limitations via the transaction duration to the next time horizon to eliminate the power system operation’s discontinuity, not only for its time horizon. The objective of this method is by satisfying various operational constraints and the power generator has the load requirement along with the minimization of cost. The proposed algorithm is tested on two test systems by varying the generating units as 40, 80, and 160. Simulation results are performed under the MATLAB environment and, the acquired results are compared with many existing algorithms in terms of fuel cost, emission cost, and robustness. The proposed scheme is very encouraging and proves the effectiveness of solving various dynamic economical load dispatch problems depending on the numerical outcomes.

Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions

This article deals with the calmness of a solution map for a Cournot Oligopoly Game with non-smooth cost functions. The fact that the cost functions are not supposed to be differentiable allows to consider cases where some firms have different units of production, with different marginal costs. In order to obtain results concerning calmness, we use a new technique based on an outer coderivative and on a mathematical induction on the number of players. It is concluded that the methodology used for the proofs can be replicated, in order to study the metric subregularity and calmness of multifunctions in a more general way.

Exponentially Convergent Algorithm Design for Constrained Distributed Optimization via Nonsmooth Approach

We develop an exponentially convergent distributed algorithm to minimize a sum of nonsmooth cost functions with a set constraint. The set constraint generally leads to the nonlinearity in distributed algorithms, and results in difficulties to derive an exponential rate. In this article, we remove the consensus constraints by an exact penalty method, and then propose a distributed projected subgradient algorithm by virtue of a differential inclusion and a differentiated projection operator. Resorting to nonsmooth approaches, we prove the convergence for this algorithm, and moreover, provide both the sublinear and exponential rates under some mild assumptions.

An Effective Algorithm for MAED Problems with a New Reliability Model at the Microgrid

This paper proposes a new framework for multi-area economic dispatch (MAED) in which the cost associated with the reliability consideration is taken into account together with the common operational and emission costs using expected energy not supplied (EENS) index. To improve the reliability level, the spinning reserve capacity is considered in the model as well. Furthermore, the MAED optimization problem and non-smooth cost functions are taken into account as well as other technical limitations such as tie-line capacity restriction, ramp rate limits, and prohibited operating zones at the microgrid. Considering all the above practical issues increases the complexity in terms of optimization, which, in turn, necessitates the use of a powerful optimization tool. A new successful algorithm inspired by phasor theory in mathematics, called phasor particle swarm optimization (PPSO), is used in this paper to address this problem. In PPSO, the particles’ update rules are driven by phase angles to essentially ensure a spread of variants across the population so that exploitation and exploration can be balanced. The optimal results obtained via simulations confirmed the capability of the proposed PPSO algorithm to find suitable optimal solutions for the proposed model.

A Multi-Restart Dynamic Harris Hawk Optimization Algorithm for the Economic Load Dispatch Problem

In this study, we propose an algorithm based on a recently proposed metaheuristic called the Harris Hawk Optimization (HHO). We utilized a dynamic control strategy to enhance the exploration capability. To further avoid trapping in local optima, we incorporate opposition-based learning (OLB) and a multi-restart strategy. The proposed algorithm is used to solve economic load dispatch (ELD) problems with non-smooth cost functions. The ELD problems have a very large search space and therefore it is difficult to find global optimum using analytical methods. The results of the proposed approach are very competitive compared with notable results from previous research.

Solving Power Economic Dispatch Problem with a Novel Quantum-Behaved Particle Swarm Optimization Algorithm

This paper proposes the shrink Gaussian distribution quantum-behaved optimization (SG-QPSO) algorithm to solve economic dispatch (ED) problems from the power systems area. By shrinking the Gaussian probability distribution near the learning inclination point of each particle iteratively, SG-QPSO maintains a strong global search capability at the beginning and strengthen its local search capability gradually. In this way, SG-QPSO improves the weak local search ability of QPSO and meets the needs of solving the ED optimization problem at different stages. The performance of the SG-QPSO algorithm was obtained by evaluating three different power systems containing many nonlinear features such as the ramp rate limits, prohibited operating zones, and nonsmooth cost functions and compared with other existing optimization algorithms in terms of solution quality, convergence, and robustness. Experimental results show that the SG-QPSO algorithm outperforms any other evaluated optimization algorithms in solving ED problems.

Bio-Inspired Dynamic Collective Choice in Large-Population Systems: A Robust Mean-Field Game Perspective.

Inspired by the collective decision making in biological systems, such as honeybee swarm searching for a new colony, we study a dynamic collective choice problem for large-population systems with the purpose of realizing certain advantageous features observed in biology. This problem focuses on the situation where a large number of heterogeneous agents subject to adversarial disturbances move from initial positions toward one of the destinations in a finite time while trying to remain close to the average trajectory of all agents. To overcome the complexity of this problem resulting from the large population and the heterogeneity of agents, and also to enforce some specific choices by individuals, we formulate the problem under consideration as a robust mean-field game with non-convex and non-smooth cost functions. Through Nash equivalence principle, we first deal with a single-player H∞ tracking problem by taking the population behavior as a fixed trajectory, and then establish a mean-field system to estimate the population behavior. Optimal control strategies and worst disturbances, independent of the population size, are designed, which give a way to realize the collective decision-making behavior emerged in biological systems. We further prove that the designed strategies constitute ϵN -Nash equilibrium, where ϵN goes toward zero as the number of agents increases to infinity. The effectiveness of the proposed results are illustrated through two simulation examples.

An Optimal Solution for Smooth and Non-Smooth Cost Functions-Based Economic Dispatch Problem

A modified particle swarm optimization and incorporated chaotic search to solve economic dispatch problems for smooth and non-smooth cost functions, considering prohibited operating zones and valve-point effects is proposed in this paper. An inertia weight modification of particle swarm optimization is introduced to enhance algorithm performance and generate optimal solutions with stable solution accuracy and offers faster convergence characteristic. Moreover, an incorporation of chaotic search, called logistic map, is used to increase the global searching capability. To demonstrate the effectiveness and feasibility of the proposed algorithm compared to the several existing methods in the literature, five systems with different criteria are verified. The results show the excellent performance of the proposed method to solve economic dispatch problems.

Robust nonlinear parameter estimation in tracer kinetic analysis using infinity norm regularization and particle swarm optimization

In positron emission tomography (PET) studies, the voxel-wise calculation of individual rate constants describing the tracer kinetics is quite challenging because of the nonlinear relationship between the rate constants and PET data and the high noise level in voxel data. Based on preliminary simulations using a standard two-tissue compartment model, we can hypothesize that it is possible to reduce errors in the rate constant estimates when constraining the overestimation of the larger of two exponents in the model equation. We thus propose a novel approach based on infinity-norm regularization for limiting this exponent. Owing to the non-smooth cost function of this regularization scheme, which prevents the use of conventional Jacobian-based optimization methods, we examined a proximal gradient algorithm and the particle swarm optimization (PSO) through a simulation study. Because it exploits multiple initial values, the PSO method shows much better convergence than the proximal gradient algorithm, which is susceptible to the initial values. In the implementation of PSO, the use of a Gamma distribution to govern random movements was shown to improve the convergence rate and stability compared to a uniform distribution. Consequently, Gamma-based PSO with regularization was shown to outperform all other methods tested, including the conventional basis function method and Levenberg–Marquardt algorithm, in terms of its statistical properties.