A Chance Constrained Optimal Reserve Scheduling Approach for Economic Dispatch Considering Wind Penetration

The volatile wind power generation brings a full spectrum of problems to power system operation and management, ranging from transient system frequency fluctuation to steady state supply and demand balancing issue. In this paper, a novel wind integrated power system day-ahead economic dispatch model, with the consideration of generation and reserve cost is modelled and investigated. The proposed problem is first formulated as a chance constrained stochastic nonlinear programming U+0028 CCSNLP U+0029, and then transformed into a deterministic nonlinear programming U+0028 NLP U+0029. To tackle this NLP problem, a three-stage framework consists of particle swarm optimization U+0028 PSO U+0029, sequential quadratic programming U+0028 SQP U+0029 and Monte Carlo simulation U+0028 MCS U+0029 is proposed. The PSO is employed to heuristically search the line power flow limits, which are used by the SQP as constraints to solve the NLP problem. Then the solution from SQP is verified on benchmark system by using MCS. Finally, the verified results are feedback to the PSO as fitness value to update the particles. Simulation study on IEEE 30-bus system with wind power penetration is carried out, and the results demonstrate that the proposed dispatch model could be effectively solved by the proposed three-stage approach. © 2017 Chinese Association of Automation.

Fixed-Time Gradient Flows for Solving Constrained Optimization: A Unified Approach

The accelerated method in solving optimization problems has always been an absorbing topic.Based on the fixedtime(FxT)stability of nonlinear dynamical systems,we provide a unified approach for designing FxT gradient flows(FxTGFs).First,a general class of nonlinear functions in designing FxTGFs is provided.A unified method for designing first-order FxTGFs is shown under Polyak-Łjasiewicz inequality assumption,a weaker condition than strong convexity.When there exist both bounded and vanishing disturbances in the gradient flow,a specific class of nonsmooth robust FxTGFs with disturbance rejection is presented.Under the strict convexity assumption,Newton-based FxTGFs is given and further extended to solve time-varying optimization.Besides,the proposed FxTGFs are further used for solving equation-constrained optimization.Moreover,an FxT proximal gradient flow with a wide range of parameters is provided for solving nonsmooth composite optimization.To show the effectiveness of various FxTGFs,the static regret analyses for several typical FxTGFs are also provided in detail.Finally,the proposed FxTGFs are applied to solve two network problems,i.e.,the network consensus problem and solving a system linear equations,respectively,from the perspective of optimization.Particularly,by choosing component-wisely sign-preserving functions,these problems can be solved in a distributed way,which extends the existing results.The accelerated convergence and robustness of the proposed FxTGFs are validated in several numerical examples stemming from practical applications.

Even Search in a Promising Region for Constrained Multi-Objective Optimization

In recent years, a large number of approaches to constrained multi-objective optimization problems(CMOPs) have been proposed, focusing on developing tweaked strategies and techniques for handling constraints. However, an overly finetuned strategy or technique might overfit some problem types,resulting in a lack of versatility. In this article, we propose a generic search strategy that performs an even search in a promising region. The promising region, determined by obtained feasible non-dominated solutions, possesses two general properties.First, the constrained Pareto front(CPF) is included in the promising region. Second, as the number of feasible solutions increases or the convergence performance(i.e., approximation to the CPF) of these solutions improves, the promising region shrinks. Then we develop a new strategy named even search,which utilizes the non-dominated solutions to accelerate convergence and escape from local optima, and the feasible solutions under a constraint relaxation condition to exploit and detect feasible regions. Finally, a diversity measure is adopted to make sure that the individuals in the population evenly cover the valuable areas in the promising region. Experimental results on 45 instances from four benchmark test suites and 14 real-world CMOPs have demonstrated that searching evenly in the promising region can achieve competitive performance and excellent versatility compared to 11 most state-of-the-art methods tailored for CMOPs.

Constraints Separation Based Evolutionary Multitasking for Constrained Multi-Objective Optimization Problems

Constrained multi-objective optimization problems(CMOPs)generally contain multiple constraints,which not only form multiple discrete feasible regions but also reduce the size of optimal feasible regions,thus they propose serious challenges for solvers.Among all constraints,some constraints are highly correlated with optimal feasible regions;thus they can provide effective help to find feasible Pareto front.However,most of the existing constrained multi-objective evolutionary algorithms tackle constraints by regarding all constraints as a whole or directly ignoring all constraints,and do not consider judging the relations among constraints and do not utilize the information from promising single constraints.Therefore,this paper attempts to identify promising single constraints and utilize them to help solve CMOPs.To be specific,a CMOP is transformed into a multitasking optimization problem,where multiple auxiliary tasks are created to search for the Pareto fronts that only consider a single constraint respectively.Besides,an auxiliary task priority method is designed to identify and retain some high-related auxiliary tasks according to the information of relative positions and dominance relationships.Moreover,an improved tentative method is designed to find and transfer useful knowledge among tasks.Experimental results on three benchmark test suites and 11 realworld problems with different numbers of constraints show better or competitive performance of the proposed method when compared with eight state-of-the-art peer methods.

Constrained Multi-Objective Optimization With Deep Reinforcement Learning Assisted Operator Selection

Solving constrained multi-objective optimization problems with evolutionary algorithms has attracted considerable attention.Various constrained multi-objective optimization evolutionary algorithms(CMOEAs)have been developed with the use of different algorithmic strategies,evolutionary operators,and constraint-handling techniques.The performance of CMOEAs may be heavily dependent on the operators used,however,it is usually difficult to select suitable operators for the problem at hand.Hence,improving operator selection is promising and necessary for CMOEAs.This work proposes an online operator selection framework assisted by Deep Reinforcement Learning.The dynamics of the population,including convergence,diversity,and feasibility,are regarded as the state;the candidate operators are considered as actions;and the improvement of the population state is treated as the reward.By using a Q-network to learn a policy to estimate the Q-values of all actions,the proposed approach can adaptively select an operator that maximizes the improvement of the population according to the current state and thereby improve the algorithmic performance.The framework is embedded into four popular CMOEAs and assessed on 42 benchmark problems.The experimental results reveal that the proposed Deep Reinforcement Learning-assisted operator selection significantly improves the performance of these CMOEAs and the resulting algorithm obtains better versatility compared to nine state-of-the-art CMOEAs.

Improved Differential Evolution with Shrinking Space Technique for Constrained Optimization

Most of the current evolutionary algorithms for constrained optimization algorithm are low computational efficiency. In order to improve efficiency, an improved differential evolution with shrinking space technique and adaptive trade-off model, named ATMDE, is proposed to solve constrained optimization problems. The proposed ATMDE algorithm employs an improved differential evolution as the search optimizer to generate new offspring individuals into evolutionary population. For the con- straints, the adaptive trade-off model as one of the most important constraint-handling techniques is employed to select better individuals to retain into the next population, which could effectively handle multiple constraints. Then the shrinking space technique is designed to shrink the search region according to feedback information in order to improve computational efficiency without losing accuracy. The improved DE algorithm introduces three different mutant strategies to generate different offspring into evo- lutionary population. Moreover, a new mutant strategy called ＂DE/rand/best/l＂ is constructed to generate new individuals according to the feasibility proportion ofcurrent population. Finally, the effectiveness of the pro- posed method is verified by a suite of benchmark functions and practical engineering problems. This research presents a constrained evolutionary algorithm with high efficiency and accuracy for constrained optimization problems.

A hybrid cuckoo search algorithm with feasibility-based rule for constrained structural optimization

Constrained optimization problems are very important as they are encountered in many science and engineering applications.As a novel evolutionary computation technique,cuckoo search(CS) algorithm has attracted much attention and wide applications,owing to its easy implementation and quick convergence.A hybrid cuckoo pattern search algorithm(HCPS) with feasibility-based rule is proposed for solving constrained numerical and engineering design optimization problems.This algorithm can combine the stochastic exploration of the cuckoo search algorithm and the exploitation capability of the pattern search method.Simulation and comparisons based on several well-known benchmark test functions and structural design optimization problems demonstrate the effectiveness,efficiency and robustness of the proposed HCPS algorithm.

A New Strategy for Solving a Class of Constrained Nonlinear Optimization Problems Related to Weather and Climate Predictability

There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems： the lower bound of maximum predictable time, the upper bound of maximum prediction error, and the lower bound of maximum allowable initial error and parameter error. Highly effcient algorithms have been developed to solve the second optimization problem. And this optimization problem can be used in realistic models for weather and climate to study the upper bound of the maximum prediction error. Although a filtering strategy has been adopted to solve the other two problems, direct solutions are very time-consuming even for a very simple model, which therefore limits the applicability of these two predictability problems in realistic models. In this paper, a new strategy is designed to solve these problems, involving the use of the existing highly effcient algorithms for the second predictability problem in particular. Furthermore, a series of comparisons between the older filtering strategy and the new method are performed. It is demonstrated that the new strategy not only outputs the same results as the old one, but is also more computationally effcient. This would suggest that it is possible to study the predictability problems associated with these two nonlinear optimization problems in realistic forecast models of weather or climate.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

Evolutionary Multitasking With Global and Local Auxiliary Tasks for Constrained Multi-Objective Optimization

Constrained multi-objective optimization problems(CMOPs) include the optimization of objective functions and the satisfaction of constraint conditions, which challenge the solvers.To solve CMOPs, constrained multi-objective evolutionary algorithms(CMOEAs) have been developed. However, most of them tend to converge into local areas due to the loss of diversity. Evolutionary multitasking(EMT) is new model of solving complex optimization problems, through the knowledge transfer between the source task and other related tasks. Inspired by EMT, this paper develops a new EMT-based CMOEA to solve CMOPs, in which the main task, a global auxiliary task, and a local auxiliary task are created and optimized by one specific population respectively. The main task focuses on finding the feasible Pareto front(PF), and global and local auxiliary tasks are used to respectively enhance global and local diversity. Moreover, the global auxiliary task is used to implement the global search by ignoring constraints, so as to help the population of the main task pass through infeasible obstacles. The local auxiliary task is used to provide local diversity around the population of the main task, so as to exploit promising regions. Through the knowledge transfer among the three tasks, the search ability of the population of the main task will be significantly improved. Compared with other state-of-the-art CMOEAs, the experimental results on three benchmark test suites demonstrate the superior or competitive performance of the proposed CMOEA.

Hybridizing artificial bee colony with biogeography-based optimization for constrained mechanical design problems

A novel hybrid algorithm named ABC-BBO, which integrates artificial bee colony(ABC) algorithm with biogeography-based optimization(BBO) algorithm, is proposed to solve constrained mechanical design problems. ABC-BBO combined the exploration of ABC algorithm with the exploitation of BBO algorithm effectively, and hence it can generate the promising candidate individuals. The proposed hybrid algorithm speeds up the convergence and improves the algorithm's performance. Several benchmark test functions and mechanical design problems are applied to verifying the effects of these improvements and it is demonstrated that the performance of this proposed ABC-BBO is superior to or at least highly competitive with other population-based optimization approaches.

Multiobjective evolutionary algorithm for dynamic nonlinear constrained optimization problems

A new method to solve dynamic nonlinear constrained optimization problems （DNCOP） is proposed. First, the time （environment） variable period of DNCOP is divided into several equal subperiods. In each subperiod, the DNCOP is approximated by a static nonlinear constrained optimization problem （SNCOP）. Second, for each SNCOP, inspired by the idea of multiobjective optimization, it is transformed into a static bi-objective optimization problem. As a result, the original DNCOP is approximately transformed into several static bi-objective optimization problems. Third, a new multiobjective evolutionary algorithm is proposed based on a new selection operator and an improved nonuniformity mutation operator. The simulation results indicate that the proposed algorithm is effective for DNCOP.

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation （CNOP）, which is a nonlinear generalization of the linear singular vector （LSV）, is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming （SQP） algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） algorithm, and the spectral projected gradients （SPG2） algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

Feasible SQP Descent Method for Inequality Constrained Optimization Problems and Its Convergence

In this paper,the new SQP feasible descent algorithm for nonlinear constrained optimization problems presented,and under weaker conditions of relative,we proofed the new method still possesses global convergence and its strong convergence.The numerical results illustrate that the new methods are valid.

A TRUST REGION METHOD WITH A CONIC MODEL FOR NONLINEARLY CONSTRAINED OPTIMIZATION

Trust region methods are powerful and effective optimization methods. The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods. The advantages of the above two methods can be combined to form a more powerful method for constrained optimization. The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound. At the same time, the new algorithm still possesses robust global properties. The global convergence of the new algorithm under standard conditions is established.

Remarks on a benchmark nonlinear constrained optimization problem

Remarks on a benchmark nonlinear constrained optimization problem are made. Due to a citation error, two absolutely different results for the benchmark problem are obtained by independent researchers. Parallel simulated annealing using simplex method is employed in our study to solve the benchmark nonlinear constrained problem with mistaken formula and the best-known solution is obtained, whose optimality is testified by the Kuhn Tucker conditions.

MODIFIED INTEGRAL-LEVEL SET METHOD FOR THE CONSTRAINED SOLVING GLOBAL OPTIMIZATION

The constrained global optimization problem being considered, a modified integral_level set method was illustrated based on Chew_Zheng's paper on Integral Global Optimization and (Wu's) paper on Implementable Algorithm Convergence of Modified Integral_Level Set Method for Global Optimization Problem. It has two characters: 1) Each phase must construct a new function which has the same global optimal value as that of primitive objective function; 2) Comparing it with (Zheng's) method, solving level set procedure is avoided. An implementable algorithm also is given and it is proved that this algorithm is convergent.

A new primal-dual path-following interior-point algorithm for linearly constrained convex optimization

In this paper, a primal-dual path-following interior-point algorithm for linearly constrained convex optimization(LCCO) is presented.The algorithm is based on a new technique for finding a class of search directions and the strategy of the central path.At each iteration, only full-Newton steps are used.Finally, the favorable polynomial complexity bound for the algorithm with the small-update method is deserved, namely, O(√n log n /ε).

Novel constrained multi-objective biogeography-based optimization algorithm for robot path planning

A constrained multi-objective biogeography-based optimization algorithm （CMBOA） was proposed to solve robot path planning （RPP）. For RPP, the length and smoothness of path were taken as the optimization objectives, and the distance from the obstacles was constraint. In CMBOA, a new migration operator with disturbance factor was designed and applied to the feasible population to generate many more non-dominated feasible individuals; meanwhile, some infeasible individuals nearby feasible region were recombined with the nearest feasible ones to approach the feasibility. Compared with classical multi-objective evolutionary algorithms, the current study indicates that CM- BOA has better performance for RPP.

Feasibility-Guided Constraint-Handling Techniques for Engineering Optimization Problems

The particle swarm optimization(PSO)algorithm is an established nature-inspired population-based meta-heuristic that replicates the synchronizing movements of birds and sh.PSO is essentially an unconstrained algorithm and requires constraint handling techniques(CHTs)to solve constrained optimization problems(COPs).For this purpose,we integrate two CHTs,the superiority of feasibility(SF)and the violation constraint-handling(VCH),with a PSO.These CHTs distinguish feasible solutions from infeasible ones.Moreover,in SF,the selection of infeasible solutions is based on their degree of constraint violations,whereas in VCH,the number of constraint violations by an infeasible solution is of more importance.Therefore,a PSO is adapted for constrained optimization,yielding two constrained variants,denoted SF-PSO and VCH-PSO.Both SF-PSO and VCH-PSO are evaluated with respect to ve engineering problems:the Himmelblau’s nonlinear optimization,the welded beam design,the spring design,the pressure vessel design,and the three-bar truss design.The simulation results show that both algorithms are consistent in terms of their solutions to these problems,including their different available versions.Comparison of the SF-PSO and the VCHPSO with other existing algorithms on the tested problems shows that the proposed algorithms have lower computational cost in terms of the number of function evaluations used.We also report our disagreement with some unjust comparisons made by other researchers regarding the tested problems and their different variants.