Stochastic Programming Optimization Research Articles

Pari-Mutuel Betting Markets: Racetracks and Lotteries Revisited

This survey discusses the state of the art in research in racetrack and lottery investment markets. Market efficiency and the pricing of various wagers are studied along with new developments since the Thaler & Ziemba (1988) review. The weak form inefficient market pricing approach using stochastic programming optimization models changed racetrack betting from handicapping to a financial market allowing professional syndicates to operate as hedge funds. Topics discussed include arbitrage and risk arbitrage, syndicates, betting exchange rebates, behavioral biases, and fundamental and mispricing information in racetrack and lottery markets. Similar models can be used to successfully trade stock market anomalies. Supplemental Materials are included online.

On Solving Stochastic Optimization Problems

Many optimization mathematical models, associated with the technical-economic processes of real-world problems, have elements of uncertainty in their structure, which places them in stochastic optimization programming. Their diversity and complexity, due to the large uncertainty space, require special methods of solving, because there is no general solution method. Within this context, in this paper we consider the category of optimization models that can contain random variable type coefficients and/or imposed probability levels on the constraints. The purpose of the paper is to propose a methodology dedicated to these studied models. Applying the methodology leads to developing a deterministic linear programming model, associated with the initial stochastic model. In fact, the proposed methodology reduces the stochastic formulation to a deterministic formulation. The methodology is illustrated with a numerical case study based on a manufacturing problem. Solving the obtained deterministic model is carried out in the version assisted by a specialized software product (WinQSB Version 2.0). It allows for the performing of a sensitivity analysis of the optimal solution, and/or a parametric analysis relative to certain model coefficients, both also presented in the paper. The main result of the study in this paper is the proposed methodology, which is applicable on a large scale, for any mathematical model of stochastic optimization of the mentioned type, regardless of complexity, dimensions and the domain of the process to which it is associated. The numerical results obtained when applying this methodology indicate its efficiency and effectiveness in finding the solution for the studied models. The approach to this issue in the present paper is determined by the wide range of stochastic optimization problems in the various studied real-life processes and by the imperative need to adopt the best decisions in conditions of uncertainty.

Incorporating catastrophe insurance in power distribution systems investment and planning for resilience enhancement

Severe damage and high economic losses caused by frequent extreme disasters are becoming unbearable for distribution system operators (DSOs) under climate change. In order to cope with extreme disasters, this paper proposes an investment portfolio and planning framework of multi-resource for resilience enhancement. In the investment portfolio and planning framework, insurance is incorporated to mitigate high economic losses as a risk sharing and resilience enhancement supplementary strategy, and co-optimized with security investment from the perspective of DSOs. Firstly, the insurance investment is modelled based on Gordon-Loeb model according to insurance industry principles and formulated as constraints by piecewise linearization techniques. Secondly, the investment portfolio and planning framework proposed in this paper is modeled as a scenario-based two-stage stochastic mixed integer linear optimization program (MILP). In the first stage, DSO makes investment and planning decisions including energy storage (ESS) allocation, line hardening strategy and insurance investment. Both operation strategies of normal operation scenarios and extreme disaster scenarios under climate change are optimized in the second stage. Finally, progressive hedging (PH) algorithm is applied to solve the two-stage stochastic MILP model. The proposed model and algorithm are tested on the modified IEEE 33-bus test system. The results verify the validity of investment portfolio and planning framework, and the results show that portfolio investment considering insurance can realize resilience enhancement in both load shedding and economic losses.

Economical revenue maximization in mobile edge caching and blockchain enabled space-air-ground integrated networks

In this paper, we study an edge caching and blockchain enabled space-air-ground integrated networking (SAGIN) network, where a low-earth-orbit (LEO) satellite serves as the content provider, and multiple edge caching enabled unmanned aerial vehicles (UAVs) will cache some contents to provide user equipments (UEs) with satisfactory content access services together with the satellite. Moreover, there’s a blockchain system that is deployed on UAVs, to provide the network with trust mechanism without requiring a centralized authority. From the standpoint of the operator, we intend to maximize the long-term averaged economical revenue by providing UEs with satisfactory and secure content access services. To achieve this purpose, we will jointly optimize the content placement of each UAV, content replacement when each UAV is full, the access control of each UE, and the blockchain deployment strategy about each UAV. the concept of queues in Lyapunov optimization is utilized to represent the backlog of edge equipment, ensuring the stability of virtual queues on UAVs and satellites, while satisfying the caching capacity constraints for content caching and blockchain deployment. Due to the tight coupling of optimization in each time slot and the variables within each time slot, our problem, which involves stochastic optimization and binary integer programming, is challenging to solve. To address this issue, we initially employ Lyapunov optimization theory to transform and decouple the problem into individual time-slot optimization problems. Subsequently, we utilize an effective heuristic algorithm called the fireworks algorithm to solve these individual optimization problems. However, the original fireworks algorithm cannot be directly applied to our problem due to its binary characteristics and inter-coupling constraints. Therefore, we have redesigned the explosion and mutation operations to adapt them to our specific problem. Simulation results demonstrate that our proposed algorithm outperforms other baseline algorithms.

A stochastic optimization approach for optimal Tail Assignment with knowledge-based predictive maintenance

AbstractRecent developments in the field of data analysis offer enormous potential to derive predictions of future aircraft maintenance necessities from observable aircraft attributes. This procedure is called Predictive Maintenance. The Predictive Maintenance generates considerable business benefits compared to alternatives like reactive or preventive maintenance for various reasons. Nevertheless, recognizing a maintenance necessity is only the first step, since the execution of the maintenance action has to fit into the airlines operational schedule. An essential step to realize financial benefits is to integrate Predictive Maintenance into the airlines optimization processes. To contribute to this task, we expand a well-known Tail Assignment model for the assignment of a potentially heterogeneous fleet of aircraft to a schedule of flights by a number of part failure scenarios. This results in a stochastic mixed integer linear optimization program, for which an optimization algorithm with solution guarantees is developed. This optimization algorithm is based on Benders Decomposition, which is concretized for the Tail Assignment problem and optimized for this task. Using this algorithm, a large proportion of the recovery costs for almost all instances tested is saved. The algorithm solves all instances in a reasonable amount of time. The algorithm uses a state-of-the-art mixed-integer linear solver, implementing a decomposition-based solution procedure for stochastic programs, called the L-shaped method. To demonstrate the potential of the approach, we benchmark our results using four strategies, motivated by commonly used preventive, reactive and passive approaches. Our approach leads to considerable cost savings when compared to each of the four benchmark approaches. The algorithms are tested on a set of instances with up to 80 flights based on a pre-pandemic schedule of a larger German airline. To the best of our knowledge, this article presents the first attempt to implement an exact optimization approach integrating malfunction predictions of the form presented into a Tail Assignment model.

A two-stage stochastic multi-objective resilience optimization model for network expansion of interdependent power–water networks under disruption

Network expansion of interdependent critical infrastructures under disruption uncertainty is modeled as a mixed-integer two-stage stochastic multi-objective optimization program. In this model, expected total cost and expected post-disaster resilience are considered competing objectives. Network resilience is quantified through network complexity and unmet demand. Functional relationships between critical infrastructures are modeled using a network-based approach with the physical interdependency enforced through demand constraints. Uncertainty is introduced as a set of random parameters corresponding to disruption scenarios. The proposed model is demonstrated in a case study of coupled power–water networks with the power flow in the grid modeled using linear DC power flow approximation equations. The deterministic-equivalent multi-objective model is solved using the augmented ϵ-constraint method. Solutions from stochastic and deterministic models are compared and the value of stochastic optimization is discussed.

Co-Optimization of Battery Storage Investment and Grid Expansion in Integrated Energy Systems

Battery storage is a flexible resource that can deliver a wide range of grid services quickly and efficiently. This article presents an investment planning model for battery storage, power transmission grid, and natural gas network in a stochastic gas–electric energy infrastructure. A bilevel stochastic optimization program is developed with an upper level investor and two interrelated lower level players. The investment decisions pertaining to the battery storage facilities and the expansion of power and gas systems are made by an independent investor anticipating the clearing results of gas and electricity markets, modeled as connected mixed-integer lower level programs. The nonconvexity of power generation and the randomness of power and gas demands, as well as renewable energy are considered in the formulation of the lower level problems. To compute this stochastic bilevel optimization with discrete decisions in both levels and interrelated lower level programs, we develop an exact solution methodology in a decomposed master-subproblem form. The application of the method is illustrated on two test systems. Experimental results show the modeling of gas and power grids’ interaction and the nonconvex nature of power production using the proposed methodology significantly affects the optimal costs and expansion plans.

Stochastic model predictive controller for wind farm frequency regulation in waked conditions

As renewable energy penetrations increase, wind farms may be required to provide ancillary services such as frequency regulation. For wind farms, two issues complicate frequency regulation: turbine-wake interactions alter the available wind power, and the future power reference signal is uncertain. This paper presents a stochastic model predictive controller (MPC) for wind farms to track an uncertain frequency regulation signal while accounting for wakes. The controller uses a simplified Park model with wake propagation delay to predict turbine-wake interactions. The controller operates without any knowledge of the future power regulation signal or wind speed. The controller solves a nonconvex stochastic optimization program using scenario optimization. Our stochastic MPC tracks with 7%–10% less error than our deterministic MPC controllers.

Maximizing the utilization of existing grids for renewable energy integration

This paper presents a new model to maximize the utilization of existing transmission system infrastructure by optimally sizing and siting the future developments of variable renewable energy sources (VRES). The model tries to maximize the integration of VRES in power systems with minimum expected energy curtailment without relying on new investments in the transmission systems. The proposed model is formulated as a linear stochastic programming optimization problem where VRES output scenarios are generated such that their spatio-temporal correlations are maintained. The Progressive Hedging Algorithm (PHA) with bundled scenarios is utilized to solve the proposed model for large-scale cases. The proposed model is tested on the modified Garver 6-bus and IEEE 118-bus test systems, and its results are compared with the results of the conventional VRES integration model. These results and comparisons illustrate the effectiveness of the proposed approach in terms of maximizing VRES integration and enhancing computational performance.

Minimizing oracle-structured composite functions

We consider the problem of minimizing a composite convex function with two different access methods: an oracle, for which we can evaluate the value and gradient, and a structured function, which we access only by solving a convex optimization problem. We are motivated by two associated technological developments. For the oracle, systems like PyTorch or TensorFlow can automatically and efficiently compute gradients, given a computation graph description. For the structured function, systems like CVXPY accept a high level domain specific language description of the problem, and automatically translate it to a standard form for efficient solution. We develop a method that makes minimal assumptions about the two functions, does not require the tuning of algorithm parameters, and works well in practice across a variety of problems. Our algorithm combines a number of well-known ideas, including a low-rank quasi-Newton approximation of curvature, piecewise affine lower bounds from bundle-type methods, and two types of damping to ensure stability. We illustrate the method on stochastic optimization, utility maximization, and risk-averse programming problems, showing that our method is more efficient than standard solvers when the oracle function contains much data.

Shipper Cooperation in Stochastic Drone Delivery: A Dynamic Bayesian Game Approach

With the recent technological innovation, unmanned aerial vehicles, known as drones, have found numerous applications including package and parcel delivery for shippers. Drone delivery offers benefits over conventional ground-based vehicle delivery in terms of faster speed, lower cost, more environment-friendly, and less manpower needed. However, most of existing studies on drone delivery planning and scheduling focus on a single shipper and ignore uncertainty factors. As such, in this paper, we consider a scenario that multiple shippers can cooperate to minimize their drone delivery cost. We propose the Bayesian Shipper Cooperation in Stochastic Drone Delivery (BCoSDD) framework. The framework is composed of three functions, i.e., package assignment, shipper cooperation formation and cost management. The uncertainties of drone breakdown and misbehavior of cooperative shippers are taken into account by using multistage stochastic programming optimization and dynamic Bayesian coalition formation game. We conduct extensive performance evaluation of the BCoSDD framework by using customer locations from Solomon benchmark suite and a real Singapore logistics industry. As a result, the framework can help the shippers plan and schedule their drone delivery effectively.

Reliability‐based topology optimization for offshore wind farm collection system

AbstractAn optimization framework for global optimization of the cable layout topology for offshore wind farm (OWF) is presented. The framework designs and compares closed‐loop and radial layouts for the collection system of OWFs. For the former, a two‐stage stochastic optimization program based on a mixed integer linear programming (MILP) model is developed, while for the latter, a hop‐indexed full binary model is used. The purpose of the framework is to provide a common base for assessing both designs economically, using the same underlying contingency treatment. A discrete Markov model is implemented for calculating the cable failure probability, useful for estimating the time under contingency for multiple power generation scenarios. The objective function supports simultaneous optimization of (i) initial investment (network topology and cable sizing), (ii) total electrical power loss costs and (iii) operation costs due to energy curtailment from cable failures. Constraints are added accounting for common engineering aspects. The applicability of the full method is demonstrated by tackling three differently sized real‐world OWFs. Results show that (i) the profitability of either topology type depends strongly on the project size and wind turbine rating. Closed loop may be a competitive solution for large‐scale projects where large amounts of energy are potentially curtailed. (ii) The stochastic model presents low tractability to tackle large‐scale instances, increasing the required computing time and memory resources. (iii) Strategies must be adopted in order to apply stochastic optimization for modern OWFs, intending analytically or numerically simplification of mathematical models.

Sustainable system design of electric powertrains—comparison of optimization methods

The transition within transportation towards battery electric vehicles can lead to a more sustainable future. To account for the development goal ‘climate action’ stated by the United Nations, it is mandatory, within the conceptual design phase, to derive energy-efficient system designs. One barrier is the uncertainty of the driving behaviour within the usage phase. This uncertainty is often addressed by using a stochastic synthesis process to derive representative driving cycles and by using cycle-based optimization. To deal with this uncertainty, a new approach based on a stochastic optimization program is presented. This leads to an optimization model that is solved with an exact solver. It is compared to a system design approach based on driving cycles and a genetic algorithm solver. Both approaches are applied to find efficient electric powertrains with fixed-speed and multi-speed transmissions. Hence, the similarities, differences and respective advantages of each optimization procedure are discussed.

Application of stochastic programming in groundwater pollution source identification

Groundwater pollution is a serious threat to the ecological environment and human life. It is necessary to determine the characteristics of pollution sources accurately and efficiently after the occurrence of pollution. The main basis for determining the characteristics of pollution sources is the pollutant concentration data of each observation well. However, due to the layout of the monitoring wells, the noise intensity of observation well concentration and unknown aquifer parameters, inversion results of groundwater pollution sources will be influenced significantly. Therefore, this article focuses on how to reduce its influence on the inversion results. In this article, a stochastic programming optimization model is introduced to explore its ability to control errors in complex situations. Results show that: in homogeneous aquifer, the normalized error (NE%) produced by simulation-optimization method is stable at about 2%, and the NE generated by stochastic programming method in confidence interval of 60%∼95% is between 0.20% and 5.42%. Moreover, stochastic programming model can effectively control the influence of noise. In the simulation-optimization model, when the noise intensity is 0.1 ∼ 0.5, the NE value is 2.01%∼12.68%, and the corresponding NE of stochastic programming model is 0.12%∼11.87%. Finally, this article considers the case that aquifer parameters are unknown (simultaneous identification of aquifer parameters and groundwater pollution sources). The results show that with the increase of the number of unknown aquifer parameters, the NE of the simulation-optimization model gradually increases from 1.16% to 8.75%. The NE value of the stochastic programming model decreases by 30% compared with the simulation-optimization model when the confidence level is 80%.

Joint RAN Slicing and Computation Offloading for Autonomous Vehicular Networks: A Learning-Assisted Hierarchical Approach

In this paper, a two-timescale radio access network (RAN) slicing and computing task offloading problem is investigated for a cloud-enabled autonomous vehicular network (C-AVN). We aim at jointly maximizing the communication and computing resource utilization with diverse quality-of-service (QoS) guarantee for autonomous driving tasks. Specifically, to capture the small-timescale network dynamics, a computing task scheduling problem is formulated as a stochastic optimization program, for maximizing the long-term network-wide computation load balancing with minimum task offloading variations. Due to the large problem size and unavailable network state transition probabilities, we employ cooperative multi-agent deep <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> -learning (MA-DQL) with fingerprint to solve the problem by learning the set of stationary task offloading policies with stabilized convergence. Given the task offloading decisions, we further study a RAN slicing problem in a large timescale, which is formulated as a convex optimization program. We focus on optimizing the radio resource slicing ratios among base stations, to maximize the aggregate network utility with statistical QoS provisioning for autonomous driving tasks. Based on the impact of radio resource slicing on computation load balancing, we propose a two-timescale hierarchical optimization framework to maximize both communication and computing resource utilization. Extensive simulation results are provided to demonstrate the effectiveness of the proposed framework in comparison with state-of-the-art schemes.

Parimutuel Betting Markets: Racetracks and Lotteries Revisited

This paper discusses the state of the art in research in racetrack and lottery investment markets. Market efficiency and the pricing of various wagers is studied along with new developments since the Thaler and Ziemba (1988) review. The weak form inefficient market approach using stochastic programming optimization models changed racetrack betting from handicapping to a financial market allowing professional syndicates to operate as successful hedge funds. The topics discussed include the role of arbitrage and risk arbitrage, syndicates, betting exchange rebates, behavioral biases, portfolio insurance and fundamental and mispricing information in racetrack and lottery markets as well as other sports betting markets. I co-authored the Beat the Racetrack books which provided a weak form winning system for place and show bets and analyzed racing as a financial market and Efficiency of Racetrack Betting Markets, the bible for Hong Kong racing betting syndicates. These made me known to practitioners and consultants and I have consulted for several racing syndicates

Optimization and validation of pumping system design and operation for water supply in high-rise buildings

The application of mathematical optimization methods for water supply system design and operation provides the capacity to increase the energy efficiency and to lower the investment costs considerably. We present a system approach for the optimal design and operation of pumping systems in real-world high-rise buildings that is based on the usage of mixed-integer nonlinear and mixed-integer linear modeling approaches. In addition, we consider different booster station topologies, i.e. parallel and series-parallel central booster stations as well as decentral booster stations. To confirm the validity of the underlying optimization models with real-world system behavior, we additionally present validation results based on experiments conducted on a modularly constructed pumping test rig. Within the models we consider layout and control decisions for different load scenarios, leading to a Deterministic Equivalent of a two-stage stochastic optimization program. We use a piecewise linearization as well as a piecewise relaxation of the pumps’ characteristics to derive mixed-integer linear models. Besides the solution with off-the-shelf solvers, we present a problem specific exact solving algorithm to improve the computation time. Focusing on the efficient exploration of the solution space, we divide the problem into smaller subproblems, which partly can be cut off in the solution process. Furthermore, we discuss the performance and applicability of the solution approaches for real buildings and analyze the technical aspects of the solutions from an engineer’s point of view, keeping in mind the economically important trade-off between investment and operation costs.

Challenges in the Management of Hydroelectric Generation in Power System Operations

The management of hydroelectric generation in the context of power system operations has been a difficult and important problem since the inception of power systems more than a century ago; however, various current developments are leading to important new associated challenges and opportunities: massive integration of variable renewable energy and other disruptive technologies, climate change effects on the availability of hydro inflows, and also new efficient techniques for optimization under uncertainty. Multistage stochastic optimization and stochastic dual dynamic programming are currently the dominant techniques for hydroelectric generation scheduling problems; however, there are many recent extensions and improvements on such techniques, and alternative approaches are being developed with significant potential for future concrete applications from power system operators and policy makers. In this context, this paper presents a literature review on hydroelectric generation scheduling models, and a discussion on the critical challenges, open research questions, and future lines of research associated to this problem.

Wind-hydrogen storage in distribution network expansion planning considering investment deferral and uncertainty

With respect to the recent developments of hydrogen storage system (HSS), it is relevant to model these storage units in the network expansion planning. Also, most of the available expansion planning tools consider constant locations and sizing for renewable resources and only study the impacts of renewables on the model. It seems that considering variable location and capacity for renewable energies and finding their optimal levels may result in more flexible model. With regard to these issues, this paper presents distribution network expansion planning incorporating wind power and hydrogen storage. The optimal site and size of wind and hydrogen systems are denoted. The stochastic optimization programming is addressed to minimize the plan budgets. The purpose is to defer the investment and operating budgets. The uncertainty modeling is developed to handle the load-wind errors. The achievements demonstrate that the model finds optimal location, sizing, operation pattern, and setting for wind turbines and HSSs while the planning cost is deferred and minimized.

Online Energy Management for Microgrids With CHP Co-Generation and Energy Storage

In this brief, the online energy management problem of a grid-connected microgrid is studied. The considered microgrid is a typical system that consists of renewable energy generations (RG), local co-generations with combined heat and power supply, the electricity and heat energy storage systems (ESS) (e.g., battery and thermal tank), and the centralized power grid (PG), as well as the external natural gas station. This brief aims to minimize the aggregate microgrid’s operation cost by formulating it as a stochastic nonconvex optimization programming, which is challenging to solve optimally due to the coupling feature of energy storage devices. First, by investigating the unique structure of the formulated optimization problem, we relax the constraints and transform the original nonconvex stochastic optimization problem into a convex optimization programming. Furthermore, we tackle the problem by developing a modified Lyapunov optimization approach and design an online energy management algorithm that does not require any statistic information of the random system inputs (e.g., the purchasing price from the PG, the harvesting electricity from the RG, and the charging levels of the ESS, and so on). Moreover, extensive empirical evaluations using real-world traces are performed to study the effectiveness of the proposed algorithm in practice. Our proposed algorithm can reduce 27.5% of the aggregate cost compared with the benchmark approach.