Discontinuous Cost Functions Research Articles

Special issue on “Optimal design and operation of energy systems”

Special issue on “Optimal design and operation of energy systems”

Feature-Based MPPI Control with Applications to Maritime Systems

In this paper, a novel feature-based sampling strategy for nonlinear Model Predictive Path Integral (MPPI) control is presented. Using the MPPI approach, the optimal feedback control is calculated by solving a stochastic optimal control (OCP) problem online by evaluating the weighted inference of sampled stochastic trajectories. While the MPPI algorithm can be excellently parallelized, the closed-loop performance strongly depends on the information quality of the sampled trajectories. To draw samples, a proposal density is used. The solver’s and thus, the controller’s performance is of high quality if the sampled trajectories drawn from this proposal density are located in low-cost regions of state-space. In classical MPPI control, the explored state-space is strongly constrained by assumptions that refer to the control value’s covariance matrix, which are necessary for transforming the stochastic Hamilton–Jacobi–Bellman (HJB) equation into a linear second-order partial differential equation. To achieve excellent performance even with discontinuous cost functions, in this novel approach, knowledge-based features are introduced to constitute the proposal density and thus the low-cost region of state-space for exploration. This paper addresses the question of how the performance of the MPPI algorithm can be improved using a feature-based mixture of base densities. Furthermore, the developed algorithm is applied to an autonomous vessel that follows a track and concurrently avoids collisions using an emergency braking feature. Therefore, the presented feature-based MPPI algorithm is applied and analyzed in both simulation and full-scale experiments.

Economic nonlinear model predictive control of fatigue—Formulation and application to wind turbine control

AbstractIn this article, the estimation of fatigue is implemented in the cost function of a gradient‐based model predictive controller (MPC). This is a challenging problem, because calculating fatigue leads to a non‐standard and discontinuous cost function. Based on a brief previous publication, in the present work the method is derived, explained, and assessed in detail. The key enablers of the proposed method are a sequential implementation of MPC, the periodic substitution of discontinuous aspects of the cost function by linear functions, and the assumption of a sufficiently infrequent switching of this substitution. Fatigue cost gradients are obtained efficiently by automatic differentiation. The method is implemented in an economic nonlinear model predictive controller (ENMPC), which optimally balances revenue and fatigue cost. This novel formulation is applicable to a very wide range of domains, and it is demonstrated here on the control of a wind turbine. The proposed ENMPC is fully parameterized by physical and monetary variables, and outperforms a conventional ENMPC based on the literature. The method is assessed by considering various metrics, including frequency spectra, damage estimation, switching, and gradient dynamics, which together provide useful insight into its main characteristics and an initial assessment of its performance.

A heuristic method for the supplier selection and order quantity allocation problem

This paper addresses the so-called supplier-selection and order-quantity allocation problem. Because of the complexity of this of problem (e.g., non-linear, discontinuous cost function), a new heuristic method is proposed and tested. This heuristic method explores the search space in a short period of time to find cheaper solutions. In order to test the efficiency of the proposed heuristic, two metaheuristic algorithms are applied: particle swarm optimization (PSO) and differential evolution (DE). Additionally, two numerical examples were solved. In the first one, it is shown that the proposed heuristic performed best compared to other solutions previously published in the literature, both in terms of computational time and total cost. The complexity analysis found that in the worst case, the proposed heuristic executes on average 99.9% less numerical operations than PSO and DE. In the second numerical example, larger instances were studied. Our findings show that the proposed heuristic was able to find a feasible solution in 15 out 15 instances, while the PSO and the DE algorithms were unable to find a solution in 9 out of 15 instances. Therefore, the proposed heuristic does not just lead to lower total cost solutions, but it also performs a more exhaustive search in shorter computational times for larger instances of the problem. Finally, Wilcoxon and Kruskal-Wallis statistical tests demonstrate significant difference between the proposed heuristic and the PSO and DE; the proposed heuristic presented a lower median in most cases.

An Improved Grey Wolf Optimizer for a Supplier Selection and Order Quantity Allocation Problem

Supplier selection and order quantity allocation have a strong influence on a company’s profitability and the total cost of finished products. From an optimization perspective, the processes of selecting the right suppliers and allocating orders are modeled through a cost function that considers different elements, such as the price of raw materials, ordering costs, and holding costs. Obtaining the optimal solution for these models represents a complex problem due to their discontinuity, non-linearity, and high multi-modality. Under such conditions, it is not possible to use classical optimization methods. On the other hand, metaheuristic schemes have been extensively employed as alternative optimization techniques to solve difficult problems. Among the metaheuristic computation algorithms, the Grey Wolf Optimization (GWO) algorithm corresponds to a relatively new technique based on the hunting behavior of wolves. Even though GWO allows obtaining satisfying results, its limited exploration reduces its performance significantly when it faces high multi-modal and discontinuous cost functions. In this paper, a modified version of the GWO scheme is introduced to solve the complex optimization problems of supplier selection and order quantity allocation. The improved GWO method called iGWO includes weighted factors and a displacement vector to promote the exploration of the search strategy, avoiding the use of unfeasible solutions. In order to evaluate its performance, the proposed algorithm has been tested on a number of instances of a difficult problem found in the literature. The results show that the proposed algorithm not only obtains the optimal cost solutions, but also maintains a better search strategy, finding feasible solutions in all instances.

Comparison of metaheuristic optimisation methods for grid-edge technology that leverages heat pumps and thermal energy storage

Grid-edge technology can unlock flexibility from consumers to contribute to meeting the growing need for flexibility in European energy systems. Furthermore, power-to-heat technology such as heat pumps and thermal energy storage has been shown to both decarbonise heat and enable the cost-effective integration of more renewable electricity into the grid. The consumer's reaction to price signals in this context presents the opportunity to simultaneously unlock operational cost reductions for consumers and localised implicit demand-side flexibility to benefit grid operators.In this paper, the prediction accuracy, run-time, and reliability of several (metaheuristic) optimisation algorithms to derive optimal operation schedules for heat pump-based grid-edge technology are investigated. To compare effectiveness, an optimisation effectiveness indicator OEI is defined. Particle Swarm Optimisation (PSO) and Genetic Algorithm (GA) were found to be most effective and robust in yielding quasi-optimal minima for the non-linear, multi-modal, and discontinuous cost function. GA optimisation with binary variables is 5–15 times more effective than with continuous variables. Using continuous variables, PSO is more effective than GA due to smaller optimisation error, shorter run-time, and higher reliability (smaller standard deviation). Simulated Annealing and Direct (Pattern) Search were found to be not very effective.

Mixed Integer Programing for Multi-Tier Rebate with Discontinuous Cost Function

Mixed Integer Programing for Multi-Tier Rebate with Discontinuous Cost Function

Grandma Makes Granola

SummaryIn introductory calculus, most optimization problems begin with a continuous function. Departing from this trend, we describe a plausible micro-economic scenario with a discontinuous cost function. The case of multiple workers requires a more refined analysis than simply taking a derivative.

Design of memetic algorithms for the efficient optimization of chemical process synthesis problems with structural restrictions

In Urselmann et al. (2011a,b) we presented a memetic algorithm (MA) for the design optimization of reactive distillation columns. The MA is a combination of a problem-specific evolutionary algorithm (EA) that optimizes the design variables and a mathematical programming (MP) method that solves the continuous sub-problems with fixed discrete decisions which are proposed by the EA to local optimality. In comparison to the usual superstructure formulation, the search space of the MA is significantly reduced without excluding feasible solutions. The algorithm computes many different local optima and can handle structural restrictions and discontinuous cost functions. In this contribution, a systematic procedure to modify the MA to solve more complex design problems is described and demonstrated using the example of a reactive distillation column with an optional side- or pre-reactor with structural restrictions on the number of streams. New concepts to handle connected and optional unit operations are proposed.

A New modified particle swarm Optimization (PSO) Technique for Non-Convex Economic Dispatch

This paper gives a new modified technique for evaluating non-convex economic dispatch (NCED) problem having non-linear, discontinuous cost function. This new particle swarm optimization technique incorporates time varying acceleration coefficients (NPSOTVAC). The highly non-linear cost function due to ramp rate limit, prohibited operating zones and valve point loading effect is considered in the proposed method. The effectiveness of proposed NPSOTVAC algorithm is proved through test on 6 and 15 unit test systems.

Economic load dispatch with nonsmooth cost functions using evolutionary particle swarm optimization

AbstractThis paper presents an evolutionary particle swarm optimization (EPSO) method for solving the nonconvex economic load dispatch (ELD) problem. In practice, the nonconvex and the discontinuous cost function should be considered when optimizing ELD problem with constraints such as valve point effects, prohibited operating zones, ramp‐rate limits, and transmission loss of the system. In view of these constraints, the ELD problem is difficult to solve by any mathematical method. In EPSO, the evolutionary programming concept (combination, tournament competition, sorting, and selection) is employed in the classical PSO method in order to find the best individual and best group position based on the survival particle. The effectiveness of the EPSO is tested on 3‐, 6‐, 15‐, and 38‐unit systems. The results obtained by EPSO are also compared with classical PSO and other results reported in the literature. It is concluded that the EPSO method can produce lower generation cost compared to the existing methods. © 2013 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Solving the forward-reserve allocation problem in warehouse order pickingsystems

Many warehouses store at least some goods in two areas, a reserve areathat is efficient for storage and a forward area that is efficient fororder picking. The forward-reserve allocation problem determines the set ofStock-Keeping Units and their space allocations in the forward area to maximizethe forward area's benefit by trading off the relevant costs of orderpicking and internal replenishment. The mathematical model of this decisionresembles the classical knapsack problem with the additional complexity that ithas a discontinuous nonlinear cost function. A simple greedy heuristic has beenproposed in the literature to solve this problem. This paper proposes analternative branch-and-bound algorithm that can quickly solve the problem tooptimality. Heuristic and optimal solutions are numerically compared usingproblem instances based on real warehouse data. Results suggest that theheuristic solutions are very close to the optimal ones in terms of both theobjective value and the forward assignment.

TREATMENT VERSUS PREVENTION OF NITROGEN FERTILIZER POLLUTION: AN INTER‐SECTORAL EXTERNALITY POLICY MODEL

Abstract This article presents an economic, hydrological, dynamic optimization model, which describes the negative external effects of nitrogen fertilizers on groundwater quality. The relative merits of treatment versus prevention of nitrogen pollution were analyzed. A dynamical water and nitrogen flow between land surface, the unsaturated zone, and groundwater was employed. A specific treatment technology, which gives rise to a discontinuous cost function, was also used. Applying the model to the coastal aquifer in Israel, our results showed that in a joint (agricultural and domestic) water source area that supplies a relatively small quantity of drinking water, it is more efficient to combine a policy that imposes restrictions on the use of nitrogen with a drinking‐water treatment process. However, when a relatively large quantity of drinking water is involved, imposing restrictions on the use of nitrogen only is more efficient. The paper, thus, is useful to planners of fast growing urban population centers with regard to regulation and can be used to calculate and evaluate specific policies.

Application of pattern search method to power system security constrained economic dispatch with non-smooth cost function

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) with non-smooth cost function. 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 three different test systems. 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). In addition, valve-point effect loading and total system losses are considered to further investigate the potential of the PS technique. Based on the results, it can be concluded that the PS has demonstrated ability in handling highly nonlinear discontinuous non-smooth cost function of the SCED.

Recent developments in exact strip analysis and optimum design of aerospace structures

The current paper outlines recent developments to algorithms and software for critical buckling and natural vibration analysis and optimum design of prismatic plate assemblies, based on the exact strip approach and the Wittrick—Williams algorithm. The current paper acts as a single source document discussing recent progress and planned future explorations in: initial local postbuckling of stiffened panels; discrete optimization of composite structures to satisfy manufacturing requirements; discontinuous cost functions; constraints on fundamental natural frequencies and frequency-free bands; a feasibility study of response surface optimization; and multi-level optimization of composite aircraft wings. The numerous references provide fuller technical details and illustrative examples.

Two-objective method for crisp and fuzzy interval comparison in optimization

In real optimization we always meet two main groups of criteria: requirements of useful outcomes increasing or expenses decreasing and demands of lower uncertainty or, in other words, risk minimization. Therefore, it seems advisable to formulate optimization problem under conditions of uncertainty, at least, two-objective on the basis of local criteria of outcomes increasing or expenses reduction and risk minimization. Generally, risk may be treated as the uncertainty of obtained result. In the considered situation, the degree of risk (uncertainty) may be defined in a natural way through the width of final interval objective function at the point of optimum achieved. To solve the given problem, the two-objective interval comparison technique has been developed taking into account the probability of supremacy of one interval over the other one and relation of compared widths of intervals. To illustrate the efficiency of the proposed method, the simple examples of minimization of interval double-extreme discontinuous cost function and fuzzy extension of Rosenbrock's test function are presented.

A modelling of route choice behaviour in transportation networks: an approach from reinforcement learning

This paper presents a simple yet powerful traffic equilibrium calculation method. The basic concept of the method is motivated from reinforcement learning with profit sharing. In the model, individual drivers are regarded as heterogeneous entities, being assumed to form their own values for each route through driving experiences and communications to the environment. The method realizes a disaggregate user equilibrium on a congested network so that it is useful to analyze the interrelationships between each driver's characteristics and the resultant network equilibrium. Further, this method not only covers from stochastic user equilibrium to deterministic user equilibrium, it also is applicable to a network with asymmetric cost functions or with discontinuous cost functions.

Head slider designs using approximation methods

This paper presents an approach to optimally design the air bearing surface (ABS) of the head slider by using the approximation methods. The reduced basis concept is used to reduce the number of design variables. In the numerical calculation, the progressive quadratic response surface modeling (PQRSM) is used to handle the non-smooth and discontinuous cost function. A multi-criteria optimization problem is formulated to enhance the flying performances over the entire recording band during the steady state and track seek operations. The optimal solutions of the sliders, whose target flying heights are 12 nm and 9 nm, are automatically obtained. The flying heights during the steady state operation become closer to the target values and the flying height variations during the track seek operation are smaller than those for the initial one. The pitch and roll angles are also kept within suitable ranges over the recording band.

Air-bearing design based on global approximation concepts

The purpose of this study is to automatically design the air-bearing surface shape to meet all desired flying performances using global approximation concepts. The reduced basis concept is used to reduce the number of design variables. The progressive quadratic response surface modeling is used to handle the nonsmooth and discontinuous cost and constraint functions. A multicriteria optimization problem is formulated to enhance the flying performance over the entire recording band during the steady-state and track-seek operations as well as the fast takeoff performance. Optimum solutions for the three target flying heights with four basis models are successfully obtained.

Minimization of Discontinuous Cost Functions by Smoothing

The minimization of functions that include discontinuous costs is considered. Sufficient conditions are introduced under which a smoothing scheme converges to the solution of the original problem. Some examples are presented and the numerical behavior of the proposed approach is commented upon.