Stochastic Problem Formulation Research Articles

Improvement the resiliency of distribution network with routing, scheduling, and prepositioning mobile energy resources with limited fuel

Mobile energy resources (MERs) are equipment which are used to improve the resiliency of distribution networks after natural disasters. An important point about using these resources is optimal routing and scheduling them to supply essential loads. In this paper, to make the problem more actual, a new formulation considering limited fuel for the trunks that move these energy resources is presented. Then, refueling vehicle is used to lessen the effect of limited fuel. Therefore, an optimization algorithm is presented for optimal allocating refueling vehicle, simultaneously optimal routing and scheduling of limited-fuel MERs. In the next stage, to achieve a better result and reducing the restoration time, a new formulation is presented for prepositioning MERs considering the stochastic occurrence of damages in various points of the network. Latin hypercube sampling method is used as scenario reduction method to implement the stochastic problem formulation. To illustrate the efficiency of the proposed method considering limited fuel for MERs, refueling vehicles, and MERs prepositioning, simulations are carried out on IEEE 33-bus test system and the results are compared with the previous studies’ results.

АСИМПТОТИЧЕСКИЕ МЕТОДЫ В ЗАДАЧАХ СУПЕРВИЗОРНОГО УПРАВЛЕНИЯ АВТОНОМНЫХ ПОДВОДНЫХ РОБОТОВ

The work is devoted to the relevant problem of control synthesis for autonomous underwater vehicles(AUVs). Since AUVs must perform actions in accordance with a given program under conditionsof a volatile predictable environment, it is necessary to provide working tools, the use of which, togetherwith the AUV position and orientation control requires, along with continuous local stabilization means,implementation of supervising and coordinated control algorithms at the upper level. Such a two-levelcontrol scheme, which can be called supervisory, is proposed in this paper to be implemented using twotypes of asymptotic methods: to separate movements into fast and slow, an apparatus for analyzing singularly perturbed differential equations is used, and the upper level control is based on the principleof large deviations. The general synthesis task is to control slow movements and stabilize fast movements.At the same time, in the stochastic problem formulation, it is assumed that there is a randomperturbation in fast movements. Given that fast movements are stabilized, it is high probable that thenoise influence averaged and does not significantly affect the behavior of slow variables. However, withsufficiently long observation it is possible to find a situation where at some period the perturbation valuesare not only uncompensated, but, on the contrary, line up in a sequence as specially intended toform an apparent deviation of the slow sub-vector from equilibrium. It is known from the theory of largedeviations that such a trajectory is the only and most likely of all those leading to a certain critical event.At the same time, the phase of this process can be used to judge the critical event proximity. For thisreason, the large deviations theory usage makes it possible to organize the control of deviations from aiven object’ t jecto y, capable of giving estimates on the probability of the controlled deviationscritical values. As a result, it is shown that if accelerations are formed by fast subsystems, then in slowsubsystems it is possible not only to achieve acceptable quality and accuracy at a fixed interval, but alsoto ensure this regardless of the disturbances. The validity of the proposed approach to synthesis, basedon the separation of movements and the large deviations theory, is shown on the example of an autonomousunderwater vehicle with two rudders (fore and aft) in the task of controlling longitudinal motion ata given depth. Simulation results and their discussion are presented.

Inventory decisions for ameliorating products under consideration of stochastic demand

Product quality is a key factor in the food and drinks industry, allowing companies to distinguish themselves from their competitors and achieve higher profit margins. However, the quality of a product may be subject to change over time and thus complicate planning and decision-making processes. While most of the literature in this area focuses on quality change in the form of deterioration, there are also products that improve in quality over time, resulting in unique challenges with regard to production and inventory management. In this context, this paper introduces an inventory planning problem for ameliorating products under consideration of demand uncertainty and product loss due to evaporation over time. This problem is solved using a MILP model within a rolling horizon approach. The proposed model is illustrated on two case studies from the cheese and whisky industry and investigates different scenario settings. The general findings show that the differences in profit margins between the quality categories influence the optimal strategy of the manufacturer and that the optimal aging and storage time should be kept to a minimum. Moreover, the results show that the rolling horizon approach is well-suited to handle the stochastic setting and that high demand fulfilment can be achieved as long as an appropriate planning horizon for the different products and product categories is considered • Stochastic problem formulation to study inventory decisions for ameliorating products. • Illustration based on two case studies from the cheese and whisky industry. • Analysis of different scenario settings related to uncertainty and other aspects. • Findings show the impact of profit margins on the optimal strategy of the producer.

An Autonomous Task Assignment Paradigm for Autonomous Robotic In-Space Assembly.

The development of autonomous robotic systems is a key component in the expansion of space exploration and the development of infrastructures for in-space applications. An important capability for these robotic systems is the ability to maintain and repair structures in the absence of human input by autonomously generating valid task sequences and task to robot allocations. To this end, a novel stochastic problem formulation paired with a mixed integer programming assembly schedule generator has been developed to articulate the elements, constraints, and state of an assembly project and solve for an optimal assembly schedule. The developed formulations were tested with a set of hardware experiments that included generating an optimal schedule for an assembly and rescheduling during an assembly to plan a repair. This formulation and validation work provides a path forward for future research in the development of an autonomous system capable of building and maintaining in-space infrastructures.

Decentralized Control System Design for a Ternary Reactive Distillation with Inert Using Mixed Integer Optimization

AbstractInferential control of ternary reactive distillation with inert is controversially discussed in recent literature. For further clarification, a systematic study based on mixed integer dynamic optimization is presented. Special emphasis is placed on robustness with respect to various disturbances arising in practice. This gives rise to a stochastic problem formulation which is extended step by step. Through this, significant improvements can be achieved and it is concluded that inferential control of ternary reactive distillation with inert is feasible.

Minimax and risk averse multistage stochastic programming

In this paper we study relations between the minimax, risk averse and nested formulations of multistage stochastic programming problems. In particular, we discuss conditions for time consistency of such formulations of stochastic problems. We also describe a connection between law invariant coherent risk measures and the corresponding sets of probability measures in their dual representation. Finally, we discuss a minimax approach with moment constraints to the classical inventory model.

Implementation of Optimal Decisions in the Presence of Uncertainty

In this work, the implementation of optimal and robust decisions in the presence of various uncertainties comprising the model parameters, external conditions and the closed loop behavior of basic controllers is presented. In order to compute optimal and reliable decisions, a chance constrained optimization problem is formulated. The efficient solution approach is based on the relaxation of the original stochastic problem formulation to a standard NLP problem. By this means, nominal optimal solutions are relocated in order to guarantee both feasibility and process operation as close to the true optimum as possible. The solution implicates the minimization of additional costs which result from conservative strategies so as to compensate for uncertainty. The experimental verification of the developed approach is carried out on a distillation pilot plant for the separation of an azeotropic mixture.

Comparison of two methods for the stochastic least cost design of water distribution systems

The problem of stochastic (i.e. robust) water distribution system (WDS) design is formulated and solved here as an optimization problem under uncertainty. The objective is to minimize total design costs subject to a target level of system robustness. System robustness is defined as the probability of simultaneously satisfying minimum pressure head constraints at all nodes in the network. The sources of uncertainty analysed here are future water consumption and pipe roughnesses. All uncertain model input variables are assumed to be independent random variables following some pre-specified probability density function (PDF). Two new methods are developed to solve the aforementioned problem. In the Integration method, the stochastic problem formulation is replaced by a deterministic one. After some simplifications, a fast numerical integration method is used to quantify the uncertainties. The optimization problem is solved using a standard genetic algorithm (GA). The Sampling method solves the stochastic optimization problem directly by using the newly developed robust chance constrained GA. In this approach, a small number of Latin Hypercube (LH) samples are used to evaluate each solution’s fitness. The fitness values obtained this way are then averaged over the chromosome age. Both robust design methods are applied to a New York Tunnels rehabilitation case study. The results obtained lead to the following main conclusions: (i) neglecting demand uncertainty in WDS design may lead to serious under-design of such systems; (ii) both methods shown here are capable of identifying (near) optimal robust least cost designs achieving significant computational savings.

Two new approaches for the stochastic least cost design of water distribution systems

The problem of stochastic (i.e. robust) water distribution system (WDS) design is formulated and solved here as an optimisation problem under uncertainty. The objective is to minimise total design costs subject to a target level of system robustness. System robustness is defined as the probability of simultaneously satisfying minimum pressure head constraints at all nodes in the network. The decision variables are the alternative design options available for each pipe in the WDS. The only source of uncertainty analysed is the future water consumption uncertainty. Uncertain nodal demands are assumed to be independent random variables following some pre-specified probability density function (PDF). Two new methods are developed to solve the aforementioned problem. In the Integration method, the stochastic problem formulation is replaced with a deterministic one. After some simplifications, a fast numerical integration method is used to quantify the uncertainties. The optimisation problem is solved using the standard genetic algorithm (GA). The Sampling method solves the stochastic optimisation problem directly by using the newly developed robust chance constrained GA. In this approach, a small number of Latin Hypercube (LH) samples are used to evaluate each solution's fitness. The fitness values obtained this way are then averaged over the chromosome age. Both robust design methods are applied to a New York Tunnels rehabilitation case study. The optimal solutions are identified for different levels of robustness. The best solutions obtained are also compared to the previously identified optimal deterministic solution. The results obtained lead to the following conclusions: (1) Neglecting demand uncertainty in WDS design may lead to serious under-design of such systems; (2) Both methods shown here are capable of identifying (near) optimal robust least cost designs achieving significant computational savings.

Quantumlike formulation of stochastic problems

It is shown that a stochastic process can be viewed as a set of states (normalized positive linear functionals) over an Abelian C*-algebra. Alternatively, the stochastic process can be associated with a set of representations of the algebra as a subalgebra of the (noncommutative) C*-algebra of bounded operators in a Hilbert space. Then, an operator equation can be associated with every stochastic equation in some general conditions. The formalism is applied to Brownian motion. Then, we study the nonrelativistic motion of a single particle in stochastic electrodynamics, a theory which has been proposed as a possible alternative to quantum mechanics. The equations of motion, which are derived, coincide with the basic ones of quantum mechanics. The differences between this theory and quantum mechanics are summarized.