Sell or Hold: A simple two-stage stochastic combinatorial optimization problem

Two-stage stochastic optimization, in which a long-term master problem is coupled with a family of short-term subproblems, plays a critical role in various application areas. However, most existing algorithms for two-stage stochastic optimization only work for special cases, and/or are based on the batch method, which requires huge memory and computational complexity. To the best of our knowledge, there still lack efficient and general two-stage online stochastic optimization algorithms. This paper proposes a two-stage online successive convex approximation (TOSCA) algorithm for general two-stage nonconvex stochastic optimization problems. At each iteration, the TOSCA algorithm first solves one short-term subproblem associated with the current realization of the system state. Then, it constructs a convex surrogate function for the objective of the long-term master problem. Finally, the long-term variables are updated by solving a convex approximation problem obtained by replacing the objective function in the long-term master problem with the convex surrogate function. We establish the almost sure convergence of the TOSCA algorithm and customize the algorithmic framework to solve three important application problems. Simulations show that the TOSCA algorithm can achieve superior performance over existing solutions.

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a standard stochastic programming formulation of these problems involves an exponential number of scenarios, therefore an exponential number of variables and constraints. We propose a new mixed-integer programming modeling approach where the number of variables and constraints is independent of the number of scenarios and scales at most pseudopolynomially with the problem size. The proposed modeling approach relies on state variables that track the system’s state as the uncertainty realizes sequentially. We demonstrate the advantages of the proposed approach in two applications arising in project scheduling and operating room allocation. Summary of Contribution: This paper proposes a new modeling approach for a class of two-stage stochastic optimization problems that is computationally more efficient than the traditional scenario-based stochastic integer programming models. The proposed modeling approach relies on state variables that track the system's state as the uncertainty realizes sequentially. We demonstrated the efficiency of the proposed approach by computational results on two applications in project scheduling and operating room allocation.

We consider barrier problems associated with two and multistage stochastic convex optimization problems. We show that the barrier recourse functions at any stage form a selfconcordant family with respect to the barrier parameter. We also show that the complexity value of the flrst stage problem increases additively with the number of stages and scenarios. We use these results to propose a prototype primal interior point decomposition algorithm for the two-stage and multistage stochastic convex optimization problems admitting self-concordant barriers.

Robust two-stage stochastic linear optimization with risk aversion

We consider distributionally robust two-stage stochastic linear optimization problems with higher-order (say \(p\ge 3\) and even possibly irrational) moment constraints in their ambiguity sets. We suggest to solve the dual form of the problem by a semi-infinite programming approach, which deals with a much simpler reformulation than the conic optimization approach. Some preliminary numerical results are reported.

We consider randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems containing mixed-integer decisions in the second stage. It is known that the second-stage optimal value function is piecewise linear-quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets. This structure is exploited to provide conditions implying that first and higher order terms of the integrand’s ANOVA decomposition (Math. Comp. 79 (2010), 953–966) have mixed weak first order partial derivatives. This leads to a good smooth approximation of the integrand and, hence, to good convergence rates of randomized QMC methods if the effective (superposition) dimension is low.

The time window assignment vehicle routing problem with product dependent deliveries

This paper models a virtual power plant (VPP) with high-penetration of distributed energy resources (DERs) to participate in the day ahead (DA) and futures markets and bilateral contracts with the aim of maximizing its profit. A two-stage stochastic optimization problem is developed that in the first stage, the VPP operator participates in the futures market and signs bilateral contracts. The VPP will participate in the DA market and supply its electrical loads in the second stage. The uncertainty parameters of the problem, including the DA market price, wind speed, and solar radiation, are first forecasted using the Long Short Term Memory (LSTM) neural network. Then the scenario generation and reduction method is used to cover the uncertainties in the predicted data. The problem has been simulated in three different cases, which indicate a significant increase in the profit of the VPP.

En-route charging navigation system is significant for electric vehicle (EV) owners to relieve the range anxiety. Considering the limited battery capacity of EVs, accurate battery energy consumption prediction is crucial for charging navigation in that it determines the maximum traveling distance. This article develops a comprehensive EV energy consumption prediction model and the impacts of the traffic flow on the motor efficiency and driving resistances have been thoroughly considered. Moreover, a stochastic traffic flow model is proposed to improve the energy consumption prediction precision, in which the related uncertainties, such as travel time, vehicle speed, etc., are captured. Then, a novel cost-effective en-route EV charging navigation problem is formulated as a two-stage stochastic optimization problem. In addition, the EV charging behavior and its interaction with the routing selection are characterized in the navigation problem. Finally, an improved Benders Algorithm is modified to obtain the global optimal solution of the proposed navigation model. The simulation verification is carried out in the actual road network of a certain city and the IEEE 33-node coupling system. Results demonstrate the feasibility and effectiveness of the proposed model and solution algorithm.

With the increasing penetration of renewable generation such as wind power in modern power systems, there are many new challenges arising in power system operation with respect to reliability and economy. In this work, we study a two-stage data-driven distributionally robust (DR) energy and reserve dispatch problem with uncertain wind power. Different from the general moment-based ambiguity set, we design a new distance-based ambiguity set to describe the uncertain probability distribution of wind power, which can be constructed in a data-driven manner from historical data. Base on this new ambiguity set, the second-stage worst-case expectation of the problem is reformulated to a combination of conditional value-at-risk (CVaR) and an expected cost with respect to a reference distribution. Thus, the proposed two-stage DR model becomes a two-stage stochastic optimization problem which can be readily solved. Case studies are carried out to verify the effectiveness of the proposed method based on the IEEE 6-bus test system and modified IEEE 118-bus test system. Simulation results show the value of data in controlling the conservatism of the problem, and the DR problem converges to the stochastic problem with fixed distribution as the data size goes to infinity.

This paper formalizes the two-stage recourse model and focuses on a stochastic generalization of the ordinary set cover problem to illustrate our technique for solving two-stage stochastic linear programs. The integrality constraints can be relaxed to yield a linear program. We give a compact formulation where only have the stage-1 variables and explain the basic idea by considering the two-stage stochastic set cover problem (SSC) and its formulation given by (SSC-P2). Furthermore, theorem forms for tacking various two-stage stochastic optimization problem be presented

The continuing liberalization and interconnection of energy markets worldwide has raised concerns about the inherent interdependency between primary energy supply and electric systems. With the growing interaction among energy carriers, limitations on the fuel delivery are increasingly relevant to the operation of power systems. One such issue is the impact of fuel supply in the Unit Commitment (UC). The contribution of this paper focuses on the development of a MIP UC that integrates the power and natural gas networks for the analysis of infrastructure outages. The problem is formulated as a two-stage stochastic optimization problem in which the first stage optimizes commitment and production decisions and the second computes deviations in levels of production and energy flows according to possible network contingencies. Case studies integrate the IEEE 24-bus system and Belgian high-calorific gas network to compute the effect of network uncertainty in the UC, showing the importance of interactions among energy systems.

Realizing the full potential of plug-in electric vehicle (PEVs) in power systems requires the development of business models for PEV owners and electric vehicle aggregators (EVAs). Most business models neglect the significant economic potential of PEV demand response. This paper addresses this challenge by proposing a novel business model to optimize the charging energy of PEVs for maximizing the owners’ profits. The proposed business model aims to overcome the opportunity cost neglect for PEV owners, whose charging energy and charging profiles are optimized with full consideration of the demand curves and market conditions. Lagrangian relaxation technology is used for the relaxation of the constraint of satisfying the charging demand, and as a result, the optimization potential becomes greater. The bidding/offering strategy is formulated as a two-stage stochastic optimization problem, considering the different market prices and initial and target state of energy (SOE) of the PEVs. By case studies and analyses, we demonstrate that the proposed business model can effectively overcome the opportunity cost neglect and increase the PEV owners’ profits. Furthermore, we demonstrate that the proposed business model is incentive-compatible. The PEV owners will be attracted by the proposed business model.

Decisions for a variable renewable resource generator’s commitment in the energy market are typically made in advance when little information is obtainable about wind availability and market prices. Much research has been published recommending various frameworks for addressing this issue. However, these frameworks are limited as they do not consider all markets a producer can participate in. Moreover, current stochastic programming models do not allow for uncertainty data to be updated as more accurate information becomes available. This work proposes two decision-making frameworks for a wind energy generator participating in day-ahead, intraday, reserve, and balancing markets. The first framework is a two-stage stochastic convex optimization approach, where both scenario-independent and scenario-dependent decisions are made concurrently. The second framework is a series of four two-stage stochastic optimization models wherein the results from each model feed into each subsequent model allowing for scenarios to be updated as more information becomes available to the decision-maker. In the simulation experiments, the multi-phase framework performs better than the single-phase in every run, and results in an average profit increase of 7%. The proposed optimization frameworks aid in better decision-making while addressing uncertainty related to variable resource generators and maximize the return on investment.

With the aggravation and evolution of global warming, natural disasters such as hurricanes occur more frequently, posing a great challenge to large-scale power systems. Therefore, the pre-position and reconfiguration of the microgrid defense resources by means of Mobile Energy Storage Vehicles (MEVs) and tie lines in damaged scenarios have attracted more and more attention. This paper proposes a novel two-stage optimization model with the consideration of MEVs and tie lines to minimize the shed loads and the outage duration of loads according to their proportional priorities. In the first stage, tie lines addition and MEVs pre-position are decided prior to a natural disaster; in the second stage, the switches of tie lines and original lines are operated and MEVs are allocated from staging locations to allocation nodes according to the specific damaged scenarios after the natural disaster strikes. The proposed load restoration method exploits the benefits of MEVs and ties lines by microgrid formation to pick up more critical loads. The progressive hedging algorithm is employed to solve the proposed scenario-based two-stage stochastic optimization problem. Finally, the effectiveness and superiority of the proposed model and applied algorithm are validated on an IEEE 33-bus test case.