Stochastic Problem Research Articles

Nonlinear Effects in the Asymmetric Copula-Based Stochastic Frontier Model

Several stochastic frontier production analysis problems are not linear regression problems; two error components (inefficiency and noise) may not have symmetric dependence. In nonlinear problems, the data are divided into two or more regimes according to turning points or change points; the relationships between the output and all the inputs are allowed to differ between regimes. In the problem of two-error dependence, the inefficiency component generally plays a more dominant role than the noise component, thereby the output of each firm is mainly influenced by the inefficiency; in such cases, the symmetric dependence is not suitable. It is, therefore, desirable to come up with a new nonlinear asymmetric copula-based stochastic frontier model (SFM). Two nonlinear structures, namely kink and threshold and skew-normal copula, are suggested to SFM to deal with these problems. The Monte Carlo simulations and analysis of a real data set are employed to evaluate the accuracy and performance of proposed models. The results show that the proposed models show a higher performance when compared with the conventional linear SFM with symmetric two-error dependence.

Dynamic User-Scheduling and Power Allocation for SWIPT Aided Federated Learning: A Deep Learning Approach

Federated learning (FL) has been considered as a promising paradigm for enabling distributed machine learning (ML) in wireless networks. To address the limited energy capacity of wireless devices, we propose a simultaneous wireless information and power transfer (SWIPT) aided FL, in which one FL server (FLS) co-located at a cellular base station (BS) uses SWIPT to simultaneously broadcast the global model to wireless user-devices (UDs) and provide wireless power transfer to them. The UDs then use the harvested energy to train their local models and further transmit the local models to the FLS for aggregation. To improve the spectrum efficiency, we consider that the UDs form a non-orthogonal multiple access (NOMA) group for simultaneously sending their local models over the same spectrum channel. Taking the UDs' time-varying available energy and channel conditions into account, we propose a dynamic optimization of the UDs-scheduling, the BS's transmit-power allocation, and the UDs' power-splitting factors for SWIPT, with the objective of minimizing the long-term energy consumption while ensuring the FL convergence. The optimization problem, however, is challenging to solve since it is a finite-horizon dynamic programming problem but with an unknown stopping time, and moreover, the action space covers both discrete and continuous variables. To address these difficulties, we first execute a series of equivalent transformations to reduce the number of decision variables and then formulate the problem as a stochastic shortest path problem, based on which we propose an actor-critic deep reinforcement learning algorithm with the proximal policy optimization to efficiently learn the policy that dynamically adjusts the UDs-scheduling for FL as well as the BS's transmit-power for SWIPT. Numerical results validate the effectiveness and performance of our proposed algorithm. The results demonstrate that our proposed algorithm can effectively reduce the long-term energy consumption in comparison with two baseline algorithms.

Hopf bifurcation control of macroscopic traffic flow model considering vehicle braking effect.

Traffic congestion not only has a great impact on people's travel, but also increases energy consumption and air pollution. The control analysis of the macroscopic traffic flow model considering the vehicle braking effect is particularly important, reflecting the impact on the actual traffic flow density wave, so as to better solve the actual traffic problems. In this paper, based on a speed difference optimization speed model, the micro-macro-variables are transformed into a high-order continuous traffic flow model. Then, a random function considering the physical correlation of random components is added to the high-order continuous traffic flow model to establish a random traffic flow model that can reflect the uncertain behavior of traffic flow acceleration or deceleration. Based on this stochastic traffic model, the existence of Hopf bifurcation and bifurcation control of the traffic flow system model considering stochastic characteristics are derived by using Hopf bifurcation theorem. By Chebyshev polynomial approximation method, the stochastic problem of the system is transformed into the bifurcation control problem of its equivalent deterministic system. A feedback controller is designed to delay the occurrence of Hopf bifurcation and control the amplitude of the limit cycle. Without changing the equilibrium point of the system, the complete elimination of Hopf bifurcation can be achieved by controlling the amplitude of the limit cycle. That is, the feedback controller is used to modify the bifurcation characteristics of the system, such as the bifurcation appearing at the equilibrium point in the control system moves forward, moves backward or disappears, so as to achieve the effect of preventing or alleviating traffic congestion.

Optimal allocation of flexible AC transmission system (FACTS) for wind turbines integrated power system

AbstractThe importance of flexible alternating current transmission system (FACTS) devices is increasing day by day as they provide compensation for power systems. The installation of such devices solves many issues regarding the stability and power transferability of power systems. However, for optimizing the performance of FACTS devices, the location and sizes of FACTS devices must be selected very carefully. On the other side, the utilization of optimization techniques is increasing daily, as it is capable of solving nonlinear, random problems by its stochastic nature. As a result, it became a reliable tactic for solving stochastic problems almost in every aspect of our lives. Therefore, it is adopted for solving power system problems, like, FACTS devices allocation, distributed generators allocation, determining component's parameters, and more. In this paper, the optimal size and location of several FACTS devices are selected to achieve the following single objective functions separately: fuel cost minimization and power losses minimization, then combining the mentioned objective function into one multiobjective function for gross cost minimization. The power system optimized is the Institute of Electrical and Electronics Engineers 30‐bus standard system, and the FACTS devices installed are static VAR compensator, thyristor control series compensator, and thyristor‐controlled phase shifter, while the optimization is performed using different conventional and recent optimization algorithms, including a newly developed algorithm, typically Leader Walrus Optimization Algorithm “LWaOA,” making a comparison among such algorithms to investigate the algorithm that possesses the best performance. The results of the optimization processes show the superiority of LWaOA.

Discrete Chebyshev polynomials for the numerical solution of stochastic fractional two-dimensional Sobolev equation

In this work, the stochastic fractional two-dimensional Sobolev equation is introduced and a collocation method is proposed to solve it. The discrete Chebyshev polynomials are used as a proper family of basis functions to establish this collocation method. Some operational matrices related to conventional and stochastic integrals, as well as fractional and ordinary derivatives of these polynomials, are extracted and successfully used in making the expressed method. More precisely, by approximating the problem solution via a finite expansion of the expressed polynomials (where the expansion coefficients are unknown) and substituting it into the stochastic fractional problem, as well as by employing the above obtained matrices, a system of linear algebraic equations is obtained. Finally, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are found by solving this system. The correctness of the established method is examined by solving some numerical examples.

A loading contribution degree analysis-based strategy for time-variant reliability analysis of structures under multiple loading stochastic processes

This paper proposes a new single-loop method based on the loading contribution degree analysis (LCDA) and decoupling strategy for analyzing the time-variant reliability of structures considering multiple loading stochastic processes and degradation processes. In this method, the loading stochastic processes are first discretized using the expansion optimal linear estimation method. Then, the LCDA is carried out to obtain a total weighted loading stochastic process used to accurately reflects the structural response, thus constructing a single loading stochastic process analysis problem. Subsequently, a decoupling strategy is applied to decouple the degradation processes and the total weighted loading stochastic process to obtain the equivalent extreme-value limit state function (EEV-LSF) of the structure. Finally, an efficient probability distribution reconstruction tool is applied to derive the probability density function of EEV-LSF. Notably, the proposed LCDA can avoid the calculation of the response time series of a structure under the input loading stochastic processes, providing a novel perspective to simplify the calculation of structural extreme-value response. Additionally, the proposed method requires only one round of deterministic analysis to obtain the structural time-variant failure probability at the specified time, constructing an efficient single-loop analysis method. Three numerical examples are used to validate the proposed method.

Risk-constrained probabilistic coordination in coupled transmission and distribution system

High penetration of renewable energy has instigated stochastic power injection at interconnection between transmission system (TS) and distribution system (DS). This paper delves into the intricate collaborative risk based stochastic dispatching challenges due to contingencies and the integration of renewable sources in the TS and DS. A framework of risk constrained probabilistic problem (RCPC) is presented, which is divided into Probabilistic coordination (PC) and risk-based stochastic security-constrained unit commitment (RSSCUC). The PC is a stochastic linear problem and RSSCUC is a stochastic mixed integer non-linear problem. Therefore, the probabilistic analytical target cascading along with polynomial chaos expansion (PCE) has been utilized for solving the PC problem for coupled TS and DS. Here, PCE has provided the proposed algorithm with the needed capability to present stochastic shared variables as coefficients for PC. Further, the benders decomposition algorithm has been utilized for solving RSSCUC, by dividing it into master problem and sub problem to generate feasibility cuts. The proposed technique reduces the computational burden on the system, as it requires single stochastic coordination problem for all scenarios as an alternative of each coordination problem for each scenario and polynomial coefficients instead of multiple scenarios for each shared variable between TS and DS. Different case studies have been performed utilizing the 6-bus system and IEEE 118-bus system as TSs, 7-bus, 9-bus, 85-bus and 69-bus systems as DSs. Results depict the efficacy of the proposed method.

Mathematical model of routing problem for hazardous biomedical waste: A multi-objective particle swarm optimization solution approach

Purpose: This model aims at solving a Green Heterogeneous and Stochastic Capacitated Vehicle Routing Problem that takes into account the risks and environmental hazards.
 Research Methodology: Regarding an NP-hard and complex problem, and after confirming the accuracy of the problem-solving in smaller dimensions by GAMS software, the problem is solved by the metaheuristic algorithm of multi-objective particle swarm optimization (MOPSO) and its coding in MATLAB software.
 Results: The results urge that using random sampling and probability distribution, non-deterministic parameters turned into deterministic ones, and high-quality solutions were obtained.
 Limitation: The proposed method is a routing problem and has been applied for the Green Heterogeneous and Stochastic Capacitated Vehicle Routing Problem. Future researchers may work on real data sets and hazardous biomedical waste data.
 Contribution: Based on the results presented, the model derived in this paper can support decisions such as routing, prioritization, time to reach each node, etc. so that the costs of routing, system reliability, environmental issues, and penalties for violation of the priority and maximum time elapsed for vehicles are taken into account.

Energy Efficiency Forecast as an Inverse Stochastic Problem: A Cross-Entropy Econometrics Approach

This paper forecasts the energy efficiency coefficients at the Polish province level (NUT-2), based on imperfect and contradictory knowledge. On the one hand, we have information on the aggregated national energy efficiency coefficients in the industrial, transport, household, and service sectors. On the other hand, we also have information on the energy intensity at the Polish province level. Since the two samples are of different natures and known with uncertainty, we are obviously dealing with an inverse stochastic problem whose solution requires particular statistical devices. The applied technique of non-extensive cross-entropy econometrics generalizes the Shannon-Kullback-Leibler approach based on the Gaussian assumptions. Its justification is explained throughout this paper from both methodological and empirical points of view. The model forecasts lead to the high-value energy efficiency estimates from quasi-unstructured sets of information. This constitutes the main contribution of this research. These outputs should provide energy policy units with valuable new devices for the optimization of the energy management processes on a disaggregated local level where, by contrast, different agents and households act decisively. On a global level, the proposed technique can be applied in different EU countries and elsewhere, in the context of experimental official statistics.

A stochastic LATIN method for stochastic and parameterized elastoplastic analysis

The LATIN method has been developed and successfully applied to a variety of deterministic problems, but few work has been developed for nonlinear stochastic problems. This paper presents a stochastic LATIN method to solve stochastic and/or parameterized elastoplastic problems. To this end, the stochastic solution is decoupled into spatial, temporal and stochastic spaces, and approximated by the sum of a set of products of triplets of spatial functions, temporal functions and random variables. Each triplet is then calculated in a greedy way using a stochastic LATIN iteration. The high efficiency of the proposed method relies on two aspects: The nonlinearity is efficiently handled by inheriting advantages of the classical LATIN method, and the randomness and/or parameters are effectively treated by a sample-based approximation of stochastic spaces. Further, the proposed method is not sensitive to the stochastic and/or parametric dimensions of inputs due to the sample description of stochastic spaces. It can thus be applied to high-dimensional stochastic and parameterized problems. Five numerical examples demonstrate the promising performance of the proposed stochastic LATIN method.

Multi-period, multi-timescale stochastic optimization model for simultaneous capacity investment and energy management decisions for hybrid Micro-Grids with green hydrogen production under uncertainty

Given the steep rises in renewable energy's proportion in the global energy mix expected over several decades, a systematic way to plan the long-term deployment is needed. The main challenges are how to handle the significant uncertainties in technologies and market dynamics over a large time horizon. The problem is further complicated by the fast-timescale volatility of renewable energy sources, potentially causing grid instability and unfulfilled demands. As a remedy, energy storage and power-to-hydrogen systems are considered in conjunction with energy management system but doing so raises the complexity of the planning problem further. In this work, the long-term capacity planning for a hybrid microgrid (HM) system is formulated as a multi-period stochastic decision problem that considers uncertainties occurring at multiple timescales. Long-term capacity decisions are inherently linked with energy dispatch and storage decisions occurring at fast-timescale and it is best to solve for them simultaneously. However, the computational demand for solving it becomes quickly intractable with problem size. To this end, we propose to develop a Markov decision process (MDP) formulation of the problem and use simulation-based reinforcement learning for multi-period capacity investments of the planning horizon. The MDP includes the policies used for dispatch and storage operation, which are represented by linear programming as a part of the simulation model. The effectiveness of our proposed method is demonstrated with a case study, where decisions over multiple decades are considered along with various uncertainties of multi-timescales. Economic and environmental assessments are performed, providing valuable guidelines for government's energy policy.

Stochastic flutter analysis of a three-degree-of-freedom airfoil under vertical turbulence disturbance

This paper investigates the nonlinear stochastic dynamic response of a three-degree-of-freedom(3-DOF) airfoil with high substructural nonlinearity under vertical turbulent disturbances. Considering a two-dimensional flow field, the Dryden turbulence model is used to describe the vertical turbulent disturbance, and a nonlinear stochastic aerodynamic model based on the Theodorsen theory is established for the stochastic flutter problem of the wing. An analysis is conducted to reduce the dimensionality of the airfoil system and solve the critical flutter velocity of the system. The Monte-Carlo method is applied to analyze the stochastic P-bifurcation problem of the system, and the effects of parameters such as the incoming velocity, turbulence scale and intensity on the stochastic dynamics behavior of the system are clarified. It is shown that the amplitude of the steady-state oscillation of the system increases with the incoming velocity and turbulence intensity, whereas the critical flutter speed decreases with increases in these parameters.

A simulation-based approximate dynamic programming approach to dynamic and stochastic resource-constrained multi-project scheduling problem

We consider the dynamic and stochastic resource-constrained multi-project scheduling problem which allows for the random arrival of projects and stochastic task durations. Completing projects generates rewards, which are reduced by a tardiness cost in the case of late completion. Multiple types of resource are available, and projects consume different amounts of these resources when under processing. The problem is modelled as an infinite-horizon discrete-time Markov decision process and seeks to maximise the expected discounted long-run profit. We use an approximate dynamic programming algorithm (ADP) with a linear approximation model which can be used for online decision making. Our approximation model uses project elements that are easily accessible by a decision-maker, with the model coefficients obtained offline via a combination of Monte Carlo simulation and least squares estimation. Our numerical study shows that ADP often statistically significantly outperforms the optimal reactive baseline algorithm (ORBA). In experiments on smaller problems however, both typically perform suboptimally compared to the optimal scheduler obtained by stochastic dynamic programming. ADP has an advantage over ORBA and dynamic programming in that ADP can be applied to larger problems. We also show that ADP generally produces statistically significantly higher profits than common algorithms used in practice, such as a rule-based algorithm and a reactive genetic algorithm.

Multistage stochastic dominance: an application to pension fund management

AbstractA pension fund manager typically decides the allocation of the pension fund assets taking into account a long-term sustainability goal. Many asset and liability management models, in the form of multistage stochastic programming problem, have been proposed to help the pension fund manager to define the optimal allocation given a multi-objective function. The recent literature proposes univariate stochastic dominance constraints to guarantee that the optimal strategy is able to stochastically dominate a benchmark portfolio. In this work we extend previous results (i) considering alternative types of multivariate stochastic dominance that appear more suitable in a multistage framework, (ii) proposing a way to measure the economic cost of introducing stochastic dominance constraints, (iii) proposing a sort of augmented stochastic dominance through a safety margin. Numerical results show the difference between the alternative ways to interpret and apply the multivariate stochastic dominance. These results are evaluated thanks to the proposed economic cost of the stochastic dominance constraints and either in presence or not of a safety margin.

A Two-Phase Pattern Generation and Production Planning Procedure for the Stochastic Skiving Process

The stochastic skiving stock problem (SSP), a relatively new combinatorial optimization problem, is considered in this paper. The conventional SSP seeks to determine the optimum structure that skives small pieces of different sizes side by side to form as many large items (products) as possible that meet a desired width. This study studies a multiproduct case for the SSP under uncertain demand and waste rate, including products of different widths. This stochastic version of the SSP considers a random demand for each product and a random waste rate during production. A two-stage stochastic programming approach with a recourse action is implemented to study this stochastic N P -hard problem on a large scale. Furthermore, the problem is solved in two phases. In the first phase, the dragonfly algorithm constructs minimal patterns that serve as an input for the next phase. The second phase performs sample-average approximation, solving the stochastic production problem. Results indicate that the two-phase heuristic approach is highly efficient regarding computational run time and provides robust solutions with an optimality gap of 0.3% for the worst-case scenario. In addition, we also compare the performance of the dragonfly algorithm (DA) to the particle swarm optimization (PSO) for pattern generation. Benchmarks indicate that the DA produces more robust minimal pattern sets as the tightness of the problem increases.

Application of high-credible statistical results calculation scheme based on least squares Quasi-Monte Carlo method in multimodal stochastic problems

Solving stochastic problems based on multimodal distributions with high accuracy and efficiency is the focus of current research in the stochastic field. However, since exact statistical results in practical engineering problems are unknown, how to efficiently obtain statistical results with high credibility is of great importance. Therefore, a high-credible statistical results calculation scheme is proposed in this paper. First, the multimodal model is transformed into a weighted sum form of multiple unimodal models by Gaussian mixture model. Then, a high-credible statistical results calculation scheme is designed based on the reusability of low-discrepancy sequences in the quasi-Monte Carlo (QMC) method. The scheme calculates the coefficient of variation for the random response statistical results of the unimodal model across various sample ranges. When the coefficient of variation is less than the tolerance error, which indicates low fluctuation, the statistical results are deemed to have converged to the exact solution. In order to further accelerate the calculation efficiency of the proposed scheme, this paper proposes a least squares quasi-Monte Carlo (LSQMC) method to yield high accuracy statistical moments. Compared with the averageness weights in the QMC method, the non-averageness weights obtained by LSQMC method can more effectively reflect the numerical Characteristics of the random variables and the location distribution of the sampling points in the probability space. Finally, numerical examples have demonstrated that the high-credible statistical results calculation scheme based on the LSQMC method can obtain the high accuracy statistical moments with high efficiency in multimodal stochastic problems.

An asynchronous parallel benders decomposition method for stochastic network design problems

Benders decomposition (BD) is one of the most popular solution algorithms for stochastic integer programs. The BD method decomposes stochastic problems into one master problem and multiple disjoint subproblems. It thus lends itself readily to parallelization. In almost all studies on the parallelization of this algorithm, the master problem remains idle until every subproblem is solved and vice versa. This can clearly result in an extremely inefficient parallel algorithm due to excessive idle times. On the other hand, relaxing the synchronization requirement can yield a nonconvergent or less efficient algorithm that may even underperform when compared to the sequential version. Addressing these issues, we introduce an asynchronous parallel BD method for stochastic network design problems. We show that the proposed algorithm converges to the global optimum and suggest various acceleration strategies to enhance its performance. We conduct an extensive numerical study on benchmark instances from a generic stochastic network design problem. The results indicate that using up to 20 processors, our asynchronous algorithm is on average 1.4 times faster than the conventional low-level parallel methods.

Integrating renewable energy and V2G uncertainty into optimal power flow: A gradient bald eagle search optimization algorithm with local escaping operator

AbstractHere, a new approach is proposed for solving the optimal power flow (OPF) problem in transmission networks using a Gradient Bald Eagle Search Algorithm (GBES) with a Local Escaping Operator (LEO). The method takes into account uncertainty of the renewable energy sources (wind energy and photovoltaic systems) and Vehicle‐to‐Grid (V2G) in the stochastic OPF problem. To improve the efficiency of the proposed technique and enhance its local exploitation capability, the LEO method's selection features are utilized. Monte Carlo methods are employed to estimate the generation costs of the renewable sources and PEVs and study their feasibility. The uncertainty of the renewable sources and PEVs is represented by Weibull, lognormal, and normal probability distribution functions (PDFs). The GBES approach is experimentally compared with well‐known meta‐heuristics using twenty‐three different test functions, and the results indicate its superiority over BES and other recently developed algorithms. Furthermore, the proposed method's effectiveness is evaluated using IEEE 30‐bus test system under various scenarios, and the simulation results demonstrate that it can effectively address OPF issues considering renewable energy sources and V2G, providing superior optimal solutions compared to other algorithms.

Three-Stage Stochastic Unit Commitment for Microgrids Toward Frequency Security via Renewable Energy Deloading

Renewable energy is boosting the deployment of microgrids (MGs) with stochastic and low inertia nature. To improve the operational efficiency of MGs while guaranteeing frequency security, a three-stage stochastic unit commitment problem is proposed, where renewable energy can be deloaded. In the first stage, the diesel generators (DGs) are scheduled, responding to uncertainties of loads and photovoltaic generator output. In the second stage, the outputs of DGs are optimized to reduce the operational cost under uncertain disturbances. In the third stage, a novel PV deloading strategy is proposed to test the frequency security of MGs. The three-stage optimization problem is formulated as a multi-stage stochastic optimization problem with recourse. This problem is then solved using a novel nested Benders decomposition algorithm with both dual cuts and partial primal cuts. Simulations are performed on an AC MG under different PV penetration levels and disturbances. The results verify the effectiveness of the proposed model in balancing operational efficiency, renewable energy utilization, and frequency security.

Optimising stochastic task allocation and scheduling plans for mission workers subject to learning-forgetting, fatigue-recovery, and stress-recovery effects

This study addresses the stochastic bi-objective task allocation and scheduling problems for mission workers subject to the complex (and joint) effect of learning-forgetting, fatigue-recovery, and stress-recovery processes. The mission consists of work-rest cycles with different types of tasks. Some task types are repetitive, not necessarily back-to-back, with some predecessors for other task types. The workers are multi-skilled, whose experience, learning, fatigue, and stress levels are updated, affecting their performance dynamically. A Markov Decision Process (MDP) is applied to formulate this stochastic problem, considering speed and accuracy as two measures of workforce performance. A decision could be “to repeat a task of a certain type” or “take a rest break till another worker becomes available”. The developed MDP model finds the optimal task allocation and work-break schedule for workers by minimising the sum of the tasks’ completion times and maximising their quality. Completion time is a continuous variable, and the quality index is a binary random variable, i.e., high or moderate, having a continuous probability of occurrence. The model is of a general form with a potential application in similar settings. The paper used the Sequential Greedy Assignment (SGA) and the Monte-Carlo Tree Search (MCTS) to solve the problem with their results compared. Numerical results with those from a sensitivity analysis are discussed.