Stochastic Linear Programming Problem Research Articles

SOLVING STOCHASTIC TRANSPORTATION ELECTRICITY PROBLEM WITH FUZZY INFORMATION ON PROBABILITY DISTRIBUTION USING MATLAB PROGRAM

This study focuses on MATLAB code programs of the entire stages of solving Stochastic Transportation Linear Programming Problems with Fuzzy Uncertainty Information on Probability Distribution Space (STLPPFI) with its algorithm outlines. A MATLAB code program of STLPPFI problem solver with algorithm outlines are proposed to solve STLPPFI model problems, and it utilizes many concepts as Alpha-Cut technique, Truth Degrees technique, Linear Fuzzy Membership Function (LFMF), Trapezoidal Fuzzy Number , Triangular Fuzzy Number , Linear Fuzzy Ranking Function (LFRF), Expectation Weighted Summation technique (EWS) and analyzing cases via second condition test of alpha-cut technique. The STLPPFI problem solver is utlized to convert STLPPFI into its corresponding equivalent Deterministic Transportation Linear Programming Problem (DTLPP) via defuzzifying from fuzziness on probability distribution space and derandomization randomness of problem formulation respectively. In addition, Dual-Simplex algorithm method with Vogel Approximation Algorithm Method (VAM) are used to obtain optimal solution from DTLPP. All MATLAB code programs with their proposed algorithm outlines are new except Dual-Simplex and VAM. The MATLAB code program of STLPPFI problem solver are more efficient along with a numerical example on electricity field illustrating practicability of this proposed MATLAB code program with its algorithm. Finally, the solution procedure illustrates the MATLAB code program of the proposed method is practical and applicable in the fields of energy and industry as it facilitates the method of transforming the energy at the lowest cost, least time running and is commercially applicable. Comparative comments are provided between Dual-Simplex and VAM in solution process.

Application of Stochastic Programming in Agricultural and Newsvendor Problems and It's Application in Real Life

The study of making the best decision under risk management in a variety of areas of our lives is known as Stochastic Programming. We will go through two-stage Stochastic Linear Programming approaches for a variety of real-world choice issues, as well as how to solve them. We will achieve this by constructing stochastic linear programming models based on real-world situations like the well-known Farmer's situation and News Vendors problems. The influence of pricing, Stochastic Integer Linear Programming problem, second stage Stochastic Integer Linear Programming problem, first stage Stochastic Binary Linear Programming problem, risk aversion problem, and continuous function for random variables based on two-stage SLP with the aid of Farmer's problem will all be examined. We will address the Newsvendor’s problem with Deterministic Equivalent Stochastic Linear Programming, an extension of Deterministic Stochastic Linear Programming for risk aversion with a high number of decision variables and restrictions, utilizing the two-stage Stochastic Linear Programming approach once more. Hand calculation is a challenging way to acquire the solution to the problems. As a result, we will use the programming language AMPL to design computer solutions for tackling both farmer and newsvendor difficulties. We will also utilize MATLAB to create graphs for the farmer's problem's continuous function. Dhaka Univ. J. Sci. 72(1): 30-45, 2024 (January)

Enhancing neutrosophic fuzzy goal programming approach for solving multi-objective probabilistic linear programming problems

This article considers a stochastic multi-objective linear programming problem in a neutrosophic framework. The right-hand sides of the constraints are random variables associated with their means and variances, and the single-valued trapezoidal neutrosophic numbers are included in the coefficients of the objective functions to investigate the problem. The problem is successfully converted into the related deterministic programming model based on the formulation of the scoring function and several distributions, such as the Uniform, Exponential, and Gama distributions. Applying fuzzy goal programming, the deterministic problem is solved. An example is then solved to demonstrate the solution methodology.

Entropy-metric estimation of the small data models with stochastic parameters

The formalization of dependencies between datasets, taking into account specific hypotheses about data properties, is a constantly relevant task, which is especially acute when it comes to small data. The aim of the study is to formalize the procedure for calculating optimal estimates of probability density functions of parameters of linear and nonlinear dynamic and static small data models, created taking into account specific hypotheses regarding the properties of the studied object. The research methodology includes probability theory and mathematical statistics, information theory, evaluation theory, and stochastic mathematical programming methods. The mathematical apparatus presented in the article is based on the principle of maximization of information entropy on sets determined as a result of a small number of censored measurements of “input" and “output" entities in the presence of noise. These data structures became the basis for the formalization of linear and nonlinear dynamic and static models of small data with stochastic parameters, which include both controlled and noise-oriented input and output measurement entities. For all variants of the above-mentioned small data models, the tasks of determining the optimal estimates of the probability density functions of the parameters were carried out. Formulated optimization problems are reduced to the forms canonical for the stochastic linear programming problem with probabilistic constraints.

Day-Ahead PV Generation Scheduling in Incentive Program for Accurate Renewable Forecasting

Photovoltaic (PV) power can be a reasonable alternative as a carbon-free power source in a global warming environment. However, when many PV generators are interconnected in power systems, inaccurate forecasting of PV generation leads to unstable power system operation. In order to help system operators maintain a reliable power balance, even when renewable capacity increases excessively, an incentive program has been introduced in Korea. The program is expected to improve the self-forecasting accuracy of distributed generators and enhance the reliability of power system operation by using the predicted output for day-ahead power system planning. In order to maximize the economic benefit of the incentive program, the PV site should offer a strategic schedule. This paper proposes a PV generation scheduling method that considers incentives for accurate renewable energy forecasting. The proposed method adjusts the predicted PV generation to the optimal generation schedule by considering the characteristics of PV energy deviation, energy storage system (ESS) operation, and PV curtailment. It then maximizes incentives by mitigating energy deviations using ESS and PV curtailment in real-time conditions. The PV scheduling problem is formulated as a stochastic mixed-integer linear programming (MILP) problem, considering energy deviation and daily revenue under expected PV operation scenarios. The numerical simulation results are presented to demonstrate the economic impact of the proposed method. The proposed method contributes to mitigating daily energy deviations and enhancing daily revenue.

Two-stage distributionally robust optimization for joint system design and maintenance scheduling in high-consequence systems

The failures of high-consequence systems can cause serious harm to humans, including loss of human health, life security, finance, and even social chaos. To protect high-consequence systems, both optimal system design and maintenance activities contribute to improving system reliability and social safety. The existing works generally optimize these two problems sequentially and assume that the degradation process of components is precisely known. However, sequential optimization often results in significant losses due to redundancies, and such a presumption usually cannot be guaranteed in practice, due to limited historical data or a lack of expert knowledge, referred to as epistemic uncertainty. To fill this gap, in this article, we consider an integrated optimization of system design and maintenance scheduling for multi-state high-consequence systems in which the component’s degradation is known with limited distributional information. To address this issue, we utilize the framework of distributionally robust optimization to provide a risk-averse decision to decision-makers even under the worst realizations of random parameters, and develop a two-stage integer distributionally robust model with moment-based ambiguity set to determine the system design and maintenance scheduling simultaneously. The proposed model can be converted to a tractable approximation as an integer linear stochastic programming problem. In order to solve large-scale problems, we develop a sample-based adaptive large neighborhood search algorithm to find the optimal system designs. In the numerical experiments, we present a case study on feedwater heating systems in nuclear power plants and demonstrate that an integrated optimization consideration creates significant benefits in profitability. We also present the out-of-sample performance of the distributionally robust design to avoid extreme risk.

Coordinating the day-ahead operation scheduling for demand response and water desalination plants in smart grid

Integrating renewable energy resources (RES) is a challenge for power system operators due to their fluctuations and unpredictability. At the same time, the water shortage problem and the needs to desalinate more freshwater increase the prominence of sufficient energy resources for sustainable operation. Therefore, this paper presents a market-clearing mechanism in a co-optimization model that coordinates the operation of grid-connected reverse osmosis water desalination plants (RO-WDPs) and the operation of renewable-rich power systems. It is assumed that electric demands can participate in the provision of demand response (DR) to the market via multiple DR options. The DR options related to the general loads are demand shifting, load curtailment, distributed generation, and hybrid energy storage systems (HESS). These HESS includes battery storage and hydrogen storage systems. Pertaining to their special characteristics, a new DR option is proposed as a customized option for RO-WDPs. The market clearing mechanism is assumed to be based on the security-constrained unit commitment (SCUC) and formulated as a stochastic mixed-integer linear programming problem (MILP) to optimally schedule the day-ahead operation of the power system. The presented model is applied on 6-bus and IEEE 24-bus reliability test system (RTS) test power systems with significant penetration of RES to demonstrate its merits. The simulation results show that the system efficiency is enhanced by adding the energy flexibility of RO-WDP without endangering the water demand-supply when using the proposed coordinated model. Hence, the total operation cost is minimized, the RES integration is facilitated, and the hourly electricity prices are smoothened.

Profit sharing mechanisms in multi-owned cascaded hydrosystems

We consider the optimal management of hydropower generated by a cascade of three interconnected reservoirs owned by different agents. In this setting, water availability at the downhill reservoirs depends on decisions taken by the agents upstream. This creates an opportunity for the hydroplant at the top to withhold water and take advantage of situations with higher selling prices, which makes the overall decisions of the agents deviate from what can be considered best for the cascade as whole. In order to mitigate the market power of the hydroplant uphill, we propose a mechanism to enforce some collaborative behavior among the agents. This is achieved by agents transferring upstream fractions of their profit in exchange for water released from the top. The corresponding mathematical models are trilevel deterministic and trilevel stochastic linear programming problems, with uncertainty in prices and streamflows (exogenous inflows). For the stochastic variants, analyzed both in two- and multi-stage formulations, we propose new solution methods, extending to the trilevel nested setting the well-known L-Shaped and Stochastic Dual Dynamic Programming methods. A successful implementation of the later depends on certain floating cuts, that represent symbolically Benders-like linearizations. Convergence properties are discussed for some of the procedures. Numerical experiments confirm the interest of the approach, because with the proposed mechanism the top owner as well as the downhill hydroplants earn more money than when acting in an individualistic manner.

Multicarrier Microgrid Operation Model Using Stochastic Mixed Integer Linear Programming

The microgrid operation is addressed in this article based on a multicarrier energy hub. Natural gas, electricity, heating, cooling, hydrogen, carbon dioxide, and renewable energies are considered as the energy carriers. The designed microgrid optimizes and utilizes a wide range of resources at the same time including renewables, electrical storage, hybrid storage, heating-cooling storage, electric vehicles (EVs) charging station, power to gas unit, combined cooling-heating-power, and carbon capture-storage. The purpose is to reduce the environmental pollutions and operating costs. The resilience and flexibility of the energy hub is also improved. Vehicle to grid and fully-partial charge models are incorporated for EVs to improve the system resilience and supplying the critical loads following events. Different events are modeled to evaluate the system resilience. The model is expressed as a stochastic mixed integer linear programming problem. Both active and reactive powers are modeled. The microgrid is simulated under four different cases. The results show that the multitype energy storages reduce the annual cost of energy while the integrated charging station can decrease the load shedding.

AWS IoT analytics platform for microgrid operation management

Microgrid (MG) represents a promising opportunity for integrating renewable energy systems with the electric power grid. However, numerous complexities need to be addressed in the process. The electrical grid is complex, vulnerable, and centralized. Thus, the integration is challenging owing to the stochastic nature of renewable energy generation, which affects the possibility of reliable forecasting. The wastage due to poor estimation of clean energy generation discourages new investments in this area. However, recent advancements in big data technologies enable processing a large amount of data captured from multiple sources in real-time. It opens the possibility of improving the operational optimization of MGs and the performance of forecasting models. The overall MG problem is formulated using a two-stage stochastic mixed-integer linear programming problem with recourse. Amazon Web Services (AWS) IoT analytics platform inputs data in real-time and runs a sophisticated wind generation forecast analysis. The stochastic model is solved using the Sample Average Approximation (SAA) algorithm. The innovative methodology leads to significant improvements in the total average operating cost by integrating AWS IoT Analytics compared to traditional methods that use historical data. Computations are performed for different power-grid settings, including a power-outage, and different power generators units capacities with total operation average cost savings of 7.6% and 5.9%, respectively. Sensitivity analysis showed that the SSA algorithm could solve all the instances by providing high-quality solutions. The AWS IoT strategy outperformed at 7.7 % and 3.6% for optimality gap and CPU time, respectively. We examined an actual case in Peru for an agricultural application to assess the performance of a stochastic optimization model with a real-time IoT wind generation forecast strategy. The results revealed the following capabilities of our novel framework: (1) it can realize higher cost savings from the MG operating systems; (2) improve real-time renewable energy forecasting; (3) facilitate robust decision-making under conditions of uncertainty.

Stochastic optimization for the scheduling of a grid-connected microgrid with a hybrid energy storage system considering multiple uncertainties

The deployment of a microgrid with renewable energy sources (RESs) generally considers two aspects i.e., the need for the energy storage and the presence of many uncertainties due to the intermittent nature of several RESs and the load demand variability. The latter justifies the use of stochastic optimization for the scheduling purpose. Therefore, this paper aims to investigate the optimal stochastic scheduling and evaluate the expected performance of a microgrid in grid-connected mode with a hybrid energy storage system consisting of a battery and a supercapacitor. The proposed optimization approach is formulated as a stochastic mixed-integer linear programming (MILP) problem and applied to a modified ORNL-DECC microgrid test system. The system uncertainties are associated with wind power, solar power, and load demand, which are represented by Gaussian distributed scenarios with standard deviations of 10%, 5%, and 3%, respectively. To investigate the effect of supercapacitor addition on the scheduling stochastic performance, the power rating of supercapacitor is varied between 0 and 20 kW. The analysis is performed based on the results of scheduling using the deterministic and the stochastic optimization schemes. In the deterministic scheme, the addition of supercapacitor is reviewed based on the achieved total cost. In the stochastic optimization scheme, we use the expected value of perfect information (EVPI) to observe the stochastic performance in addition to the expected total cost. The results are beneficial when assessing the advantage and disadvantage of supercapacitor deployment in a microgrid scheduling.

Two-stage stochastic energy procurement model for a large consumer in hydrothermal systems

The Energy Procurement (EP) problem faced by a large consumer is concerned with planning the energy procurement in the various energy markets available, such that its short- and medium-term demands are met, and the risks involved in such trading are mitigated. Although a number of EP models have been proposed for purely thermal systems, no specific model has been addressed for solving this problem for a large consumer located in a hydro-dominated system. In this paper we discuss the main specific features and issues involving EP problems for hydro-dominated markets. A central issue is the estimation of future energy prices in the pool market. In hydro-dominated systems, uncertainties in incremental water inflows into reservoirs affect directly such prices, as well as the estimated demands, and these are difficult correlations to be captured by a price estimation model. In this paper, we propose a Price Scenario Generation (PSG) model to estimate future pool prices, which is able to capture spatial and temporal correlation among uncertain prices, water inflows and demands. The estimated prices obtained by the PSG are introduced in the proposed Energy Procurement Model for Hydrothermal Systems (EPMHS), which calculates the optimal procurement decisions, involving the portions of energy traded in the pool, bilateral and futures markets, as well as the self-produced energy. The proposed EPMHS is formulated as a sequence of mixed-integer two-stage stochastic linear programming problems, where pool prices are handled as uncertain parameters. The EPMHS represents trading risks using the Conditional Value at Risk (CVaR) metric. We also propose a strategy for including yearly estimation of water inflows into the EPMHS, since prices in hydro-dominated markets are generally driven by water inflows forecasts. The model proposed is applied to the generation system of the northeast region of Brazil, and the results reveal coherent correlations between hydro and economic variables.

Smart District Energy Management With Cooperative Microgrids

This paper faces the energy management problem of cooperative microgrids in a smart energy district. In particular, the aim of the research work is to propose an innovative optimization model to solve the problem of energy management in a district composed of several microgrids, taking into account uncertainties of key parameters. In this context, the objective of the paper is threefold: i) maximize the use of energy purchased at the day-ahead market; ii) minimize the need of additional and expensive energy in real-time iii) optimize the integration of renewable energy sources (RES), energy storage systems (ESS) and electric vehicle (EV) batteries in the microgrid. To these goals, the District Energy Management System (DEMS), i.e. the central controller of the district, must balance the microgrids energy demand with the optimal integration of RES, ESS and the batteries of EVs that are seen as movable storage devices shared among buildings. Moreover, the energy surplus can be sold back to the main power grid. The DEMS problem is solved by two approaches. In the first approach, the energy demand, the RES production and the costs are known and a linear programming problem is formalized and solved by the DEMS. In addition, a second approach is proposed in order to address the parameters uncertainty and is formalized as a stochastic linear programming problem. The optimization problems solutions provide the optimal strategy to schedule the charging and discharging operations of the storage systems and the electric vehicle batteries. A simulated case study demonstrates the benefits of the proposed approaches for the smart district.

Stochastic Multi-Objective Programming Problem: A Two-Phase Weighted Coefficient Approach

This paper deals with multi-objective stochastic linear programming problem. The problem is considered by introducing the coefficients of the decision variables and the right-hand-side parameters in the constraints as normal random variables. A method for converting the problem into its deterministic problem is proposed and hence two- phase approach with equal weights is proposed for finding an efficient solution. The advantages of the approach are: as weights which is positive, not necessarily equal and generate an efficient solution. A numerical example is given to illustrate the suggested methodology.

Stochastic optimization energy and reserve scheduling model application for alaçatı, Turkey

Stochastic programming is a popular method that is used to model systems that contain a high level of uncertainty. This paper presents a new model for energy and reserve scheduling for microgrid systems using stochastic programming to optimize the energy and reserve planning in order to minimize the operational costs of a microgrid structure with renewable energy resources. Purchased power from the main grid and demand response programs are used for both energy and reserve scheduling. Uncertainties related to renewable energy generation, day-ahead market prices as well as costs related to unbalanced energy are considered and integrated into the model via scenarios. The model is formulated as a two-stage stochastic linear programming problem, solved using the L-shaped method, subjected to a numerical analysis and compared with a deterministic scheduling and a no-reserve approach in order to emphasize its efficiency. The results of the optimization show that 9.448 kW's of the total 14.300 kW DSR bid is accepted by the operator and 4.254 kW (45% of total) is used as direct energy purchase while the rest 5.194 kW is used as stand by reserve. When the results of deterministic and stochastic models are considered, 70% of all realization outcomes of the stochastic model yielded lower costs when compared to both the deterministic and no-reserve approaches. Stochastic planning cost is %5 lower than deterministic planning across all realization outcomes. The proposed model has clear economic advantages, which can be translated into the energy sector.

A novel approach for solving stochastic problems with multiple objective functions

In this paper we suggest an approach for solving a multiobjective stochastic linear programming problem with normal multivariate distributions. Our approach is a combination between a multiobjective method and a nonconvex technique. The problem is first transformed into a deterministic multiobjective problem introducing the expected value criterion and an utility function that represents the decision makers preferences. The obtained problem is reduced to a mono-objective quadratic problem using a weighting method. This last problem is solved by DC (Difference of Convex) programming and DC algorithm. A numerical example is included for illustration.

Role of vanadium redox flow batteries in the energy management system of isolated microgrids

The small size of isolated microgrids (IMGs), high intermittency and fluctuations of renewable generations, and low inertia of distributed energy resources make the frequency an important factor in the operation of IMGs that should be managed considering technical and economic constraints. The fast response time and long lifecycle of vanadium redox flow batteries (VRFBs) make them a promising candidate for energy and frequency management of IMGs. Accordingly, this paper proposes a novel energy management system (EMS) for IMGs which aims to optimize energy and reserve scheduling of VRFBs in coordination with droop controlled distributed generators. The objective of the EMS is to minimize the total operation cost of the IMG while dynamic and static frequency securities are preserved. In the proposed EMS, the hierarchical control structure of IMGs is precisely formulated and the VRFB is incorporated in the EMS to provide both energy and reserve in primary and secondary control levels. Moreover, to get accurate results, the nonlinear characteristics of VRFBs are considered in the developed model and linearized by an efficient approach. The proposed EMS is formulated as a two-stage stochastic mixed-integer linear programming problem (MILP) that guarantees the global optimal solution. Case studies on a test IMG indicate that optimal coordination of VRFBs in energy and reserve scheduling can reduce the operation cost more than 11.67%.

Efficient and sustainable closed-loop supply chain network design: A two-stage stochastic formulation with a hybrid solution methodology

In recent years, consumers and legislators have pushed companies to design their supply chain networks to consider environmental and social impacts as an important performance outcome. Due to the role of resource utilization as a key component of logistics network design, another primary goal of design is ensuring available scarce resources are used as efficiently as possible across all facilities. To address efficiency issues in a sustainable closed-loop supply chain network, a stochastic integrated multi-objective mixed integer nonlinear programming model is developed in this paper, in which sustainability outcomes as well as efficiency of facility resource utilization are considered in the design of a sustainable supply chain network. In doing so, efficiency is assessed using a bi-objective output-oriented data envelopment analysis model. A hybrid three-step solution methodology is presented that creates a linear form of the original mixed integer nonlinear programming problem using piecewise McCormick envelopes approach. In the second step, an aggregated single objective programming model is derived by exploiting the multi-choice goal programming. Finally, a Lagrangian relaxation algorithm is developed to effectively solve the latter stochastic single objective mixed integer linear programming problem. The application of the proposed approach is investigated with data drawn from a case study in the electronics industry. This case study illustrates how firms may balance sustainability and efficiency in the supply chain network design problem. Further, it demonstrates the integration of efficiency results in improving economic aspects of sustainability as well as social responsibility outcomes, but also highlights the trade-offs that exist between efficiency and environmental impacts.

Solving the multi-objective stochastic interval-valued linear fractional integer programming problem

In this paper, we consider a Multi-Objective Stochastic Interval-Valued Linear Fractional Integer Programming problem (MOSIVLFIP). We especially deal with a multi-objective stochastic fractional problem involving an inequality type of constraints, where all quantities on the right side are log-normal random variables, and the objective functions coefficients are fractional intervals. The proposed solving procedure is divided in three steps. In the first one, the probabilistic constraints are converted into deterministic ones by using the chance constrained programming technique. Then, the second step consists of transforming the studied problem objectives on an optimization problem with an interval-valued objective functions. Finally, by introducing the concept of weighted sum method, the equivalent converted problem obtained from the two first steps is transformed into a single objective deterministic fractional problem. The effectiveness of the proposed procedure is illustrated through a numerical example.

A Stochastic Framework for Optimal Island Formation During Two-Phase Natural Disasters

This article proposes a new three-stage stochastic framework for dealing with predictable two-phase natural disasters in distribution systems. This framework is a multiobjective optimization, in which the amount of curtailed energy, the number of switching actions, and the vulnerability of operational components are selected as the main criteria for decision-making process. The optimization problem is formulated in the form of a stochastic mixed-integer linear programming (MILP) problem. In this article, a windstorm event followed by flooding is analyzed as a two-phase natural disaster. In this regard, the uncertainties associated with gust-wind speed, floodwater depths, and load demands are taken into account by the proposed framework. The initial configurations of islands are formed just ahead of the storm event (first stage), and their borders are changed in the second stage, which is associated with the storm event and its aftermath. The final configurations of islands are determined by the third stage once the uncertainties of floodwater depths are revealed. In the proposed framework, the emergency generators (EGs) are assumed to be prone to flooding, and a novel approach is proposed for quantifying the flood-related failure probability of EGs. Likewise, overhead distribution structures are recognized as vulnerable components to storms. The proposed framework is implemented on a test system, and its effectiveness is investigated and verified through seven case studies.