Online Successive Convex Approximation for Two-Stage Stochastic Nonconvex Optimization

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.

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.

With renewable energy sources (RESs) highly penetrating into the power system, new problems emerge for the independent system operator (ISO) to maintain and keep the power system safe and reliable in the day-ahead dispatching process under the fluctuation caused by renewable energy. In this paper, considering the small hydropower with no reservoir, different from the other hydro optimization research and wind power uncertain circumstances, a day-ahead scheduling model is proposed for a distributed power grid system which contains several distributed generators, such as small hydropower and wind power, and energy storage systems. To solve this model, a two-stage stochastic robust optimization approach is presented to smooth out hydro power and wind power output fluctuation with the aim of minimizing the total expected system operation cost under multiple cluster water inflow scenarios, and the worst case of wind power output uncertainty. More specifically, before dispatching and clearing, it is necessary to cluster the historical inflow scenarios of small hydropower into several typical scenarios via the Fuzzy C-means (FCM) clustering method, and then the clustering comprehensive quality (CCQ) method is also presented to evaluate whether these scenarios are representative, which has previously been ignored by cluster research. It can be found through numerical examples that FCM-CCQ can explain the classification more reasonably than the common clustering method. Then we optimize the two stage scheduling, which contain the pre-clearing stage and the rescheduling stage under each typical inflow scenario after clustering, and then calculate the final operating cost under the worst wind power output scenario. To conduct the proposed model, the day-ahead scheduling procedure on the Institute of Electrical and Electronics Engineers (IEEE) 30-bus test system is simulated with real hydropower and wind power data. Compared with traditional deterministic optimization, the results of two-stage stochastic robust optimization structured in this paper, increases the total cost of the system, but enhances the conservative scheduling strategy, improves the stability and reliability of the power system, and reduces the risk of decision-making simultaneously.

Robust two-stage stochastic linear optimization with risk aversion

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.

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

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.

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.

This study addresses the problem of creating single aircraft route within strict time constraints, taking into account the uncertainty of travel times. The approach involves a two-phase decision-making process: the initial phase focuses on selecting airports from a pool of candidates by optimizing for maximum customer satisfaction; the subsequent phase involves arranging the sequence of itinerary to these airports, factoring in variable travel times to enhance the overall experience. To tackle this complex issue, we have developed a two-stage stochastic optimization model that incorporates probability-based constraints. This model is then analyzed using the sample average approximation method. To validate the efficacy of our proposed model, we compared its performance with two established benchmark models. Additionally, case studies focused on the U.S. airline network suggest that operators can optimize airline route recommendations by enhancing both customer utility and their own profitability. This approach demonstrates that strategic adjustments in route planning can lead to mutual benefits, effectively balancing customer satisfaction with financial performance.

SUMMARY Wildfires are natural phenomena that play a crucial role in many forest and grassland ecosystems. Under favorable conditions, that is, extreme heat, availability of fuel, and high winds, they may spread beyond control and approach city limits. Power lines are vulnerable to wildfires in their vicinity, mainly because of increased conductor temperatures as a result of ambient temperature rising. This may lead to conductor annealing and/or increased sag. A violation of safety clearances of the lines could in turn lead to possible flashovers and onset of cascading failures. To address the situation, the operator may dynamically alter the ratings of the lines in the affected areas in order to reduce the flow of current through them, thereby assisting with conductor cooling. However, with reduced line capacities, the utility may not be able to supply all the loads from the main substation anymore, which may increase the risk of unserved load and load shedding. A solution is put forth in this paper that enables the system operator to dispatch distributed generators, demand responsive loads, and microgrids in order to supply loads under such emergency conditions. Because of the uncertain spread/severity of wildfires, the problem is stochastic in nature and is therefore modeled as two-stage stochastic optimization. First-stage variables determine the generation capacity to be purchased before the onset of the event, while second stage variables represent generation dispatch commands. The efficiency of the proposed solution is verified on a modified version of the Institute of Electrical and Electronics Engineers (IEEE) 123-bus test system. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, an optimal bidding strategy is proposed for microgrids that participate in both energy and spinning reserve markets for profit maximization. Various uncertain factors such as the day-ahead (DA) energy market prices, real-time (RT) energy prices, reserve prices, output power of the intermittent renewable resources and load demands are considered and modeled. The energy bids, reserve offers, unit commitment and power generation/consumption of the microgird are determined based on two-stage stochastic optimization. The first-stage is related to the DA market, while the second stage is related to the RT market. Comprehensive case studies, which are based on the codes and data of PJM market, have been carried out to verify the proposed strategy.

Urban integrated energy system stochastic robust optimization scheduling under multiple uncertainties

Designing an optimal multi-energy system with fast charging and hydrogen refueling station under uncertainties

Two-stage stochastic decision problems are characterized by the property that in stage one part of the decision has to be made before relevant data are observed and the rest of the decision can be made in stage two after data are available. It is generally assumed that while the data missing in stage one are not known, their probability distribution is known. In this paper, we consider a special form of such problems where the objective function is quadratic in the first and second stage decisions and the goal is to minimize the expected quadratic cost function. We assume that the decisions must lie in a standard simplex, but do not require convexity of the uncertain objective. It is well known that such quadratic optimization programs are NP-complete and thus belong to the most complex optimization problems. A viable method to attack such problems is to find bounds for the original complicated problem by solving easier approximate problems. We describe such approximations and show that by exploiting their properties, good but not necessarily optimal solutions can be found. We also present possible applications, such as selecting invited speakers for conferences with the aim to generate a community with high coherence. Finally, systematic numerical results are provided.

Virtual power plant (VPP) has gradually become a key technology to the increasing penetration of renewable energy. The uncertainty and variability of renewable energy and load demand pose significant challenges to the efficiency and stability of VPP's operation. In this paper, a data-driven two-stage stochastic robust optimization (SRO) scheduling model is proposed for a VPP considering wind power, solar power, load demand, and market price uncertainties. The objective is to maximize the profit of the VPP in the energy market and minimize the total system cost of the VPP in the reserve market under the worst-case realization of the uncertainties. The Dirichlet process mixture model (DPMM) and variational inference algorithm are employed for constructing the data-driven uncertainty ambiguity set considering the correlations among multiple uncertainties. The tailored column-and-constraint generation algorithm is developed to solve the SRO model iteratively by reformulating the second stage with the application of the Karush-Kuhn-Tucker conditions. Results from a case study illustrate the effectiveness and superiority of the proposed model.