Two-stage Optimization Problem Research Articles

A two-stage polynomial approach to stochastic optimization of district heating networks

Abstract In this paper we use stochastic polynomial optimization to derive high-performance operating strategies for heating networks with uncertain or variable demand. The heat flow in district heating networks can be regulated by varying the supply temperature, the mass flow rate, or both simultaneously, leading to different operating strategies. The task of choosing the set-points within each strategy that minimize the network losses for a range of demand conditions can be cast as a two-stage stochastic optimization problem with polynomial objective and polynomial constraints. We derive a generalized moment problem (GMP) equivalent to such a two-stage stochastic optimization problem, and describe a hierarchy of moment relaxations approximating the optimal solution of the GMP. Under various network design parameters, we use the method to compute (approximately) optimal strategies when one of, or both, the mass flow rate and supply temperature are varied for a benchmark heat network. We report that the performance of an optimally-parameterized fixed-temperature variable-mass-flow strategy can approach that of a fully variable strategy.

Online Successive Convex Approximation for Two-Stage Stochastic Nonconvex Optimization

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.

Real-Time Pricing for Smart Grid with Multiple Companies and Multiple Users Using Two-Stage Optimization

AbstractIn this paper, we focus on the real-time interactions among multiple utility companies and multiple users and formulate real-time pricing (RTP) as a two-stage optimization problem. At the first stage, based on cost function, we propose a continuous supply function bidding mechanism to model the utility companies’ profit maximization problem, by which the analytic expression of electricity price is further derived. At the second stage, considering that individually optimal solution may not be socially optimal, we employ convex optimization with linear constraints to model the price anticipating users’ daily payoff maximum. Substitute the analytic expression of electricity price obtained at the first stage into the optimization problem at the second stage. Using customized proximal point algorithm (C-PPA), the optimization problem at the second stage is solved and electricity price is obtained accordingly. We also prove the existence and uniqueness of the Nash equilibrium in the mentioned two-stage optimization and the convergence of C-PPA. In addition, in order to make the algorithm more practical, a statistical approach is used to obtain the function of price only through online information exchange, instead of solving it directly. The proposed approach offers RTP, power production and load scheduling for multiple utility companies and multiple users in smart grid. Statistical approach helps to protect the company’s privacy and avoid the interference of random factors, and C-PPA has an advantage over Lagrangian algorithm because the former need not obtain the objection function of the dual optimization problem by solving an optimization problem with parameters. Simulation results show that the proposed framework can significantly reduce peak time loading and efficiently balance system energy distribution.

A Two-Stage Approach for Routing Multiple Unmanned Aerial Vehicles with Stochastic Fuel Consumption

The past decade has seen a substantial increase in the use of small unmanned aerial vehicles (UAVs) in both civil and military applications. This article addresses an important aspect of refueling in the context of routing multiple small UAVs to complete a surveillance or data collection mission. Specifically, this article formulates a multiple-UAV routing problem with the refueling constraint of minimizing the overall fuel consumption for all the vehicles as a two-stage stochastic optimization problem with uncertainty associated with the fuel consumption of each vehicle. The two-stage model allows for the application of sample average approximation (SAA). Although the SAA solution asymptotically converges to the optimal solution for the two-stage model, the SAA run time can be prohibitive for medium- and large-scale test instances. Hence, we develop a tabu search-based heuristic that exploits the model structure while considering the uncertainty in fuel consumption. Extensive computational experiments corroborate the benefits of the two-stage model compared to a deterministic model and the effectiveness of the heuristic for obtaining high-quality solutions.

Using Electrical Energy Storage to Mitigate Natural Gas-Supply Shortages

The recent leak of the Aliso Canyon natural gas storage facility calls for coordinated planning of natural gas and electric power systems with specific consideration of electrical energy storage. This paper proposes a coordinated planning model that fills this need. The model is formulated as a two-stage stochastic optimization problem, in which electricity- and natural gas-demand growth and natural gas-supply uncertainties are represented. We analyze the tradeoff between building electrical, natural gas, and electrical energy storage infrastructure. The sensitivity of electrical energy storage investment to the modeling of the operating conditions is also studied. The model is tested using a California-based case study.

Risk-Averse Generation Maintenance Scheduling With Microgrid Aggregators

This paper presents risk-averse long-term generation maintenance scheduling in the power systems with a considerable installed capacity of microgrids. Microgrid aggregators facilitate the participation of microgrids in the wholesale market. In this paper, the effect of microgrids as controllable demand entities on the generation maintenance scheduling practices in the power system is investigated. The uncertainties in the marginal cost of generation in microgrids, the generation capacity installed within the microgrids, and the system electricity demand are captured using respective nominal values and uncertainty intervals. Moreover, the contingencies in transmission network are addressed by introducing additional variables. A two-stage robust optimization problem is formulated to determine a trade-off among the performance and conservativeness of the procured solution in the long-term operation horizon. The problem is formulated as a mixed integer linear programming problem and column-and-constraint generation procedure is used to solve the problem. The master problem minimizes the maintenance cost of the generation units subjected to generation units’ constraints in the long-term operation horizon and the sub-problems determine the worst realization of the uncertainties and generate additional constraints in the master problem. The proposed methodology is applied to two case studies for a 6-bus and IEEE 118-bus power systems.

On the Sparse Beamformer Design

In designing acoustic broadband beamformers, the complexity can grow significantly when the number of microphones and the filter length increase. It is advantageous if many of the filter coefficients are zeroes so that the implementation can be executed with less computation. Moreover, the size of the array can also be pruned to reduce complexity. These problems are addressed in this paper. A suitable optimization model is proposed. Both array pruning and filter thinning can be solved together as a two-stage optimization problem to yield the final sparse designs. Numerical results show that the complexity of the designed beamformers can be reduced significantly with minimal effect on performance.

Decision making for sustainable aggregation of clean energy in day-ahead market: Uncertainty and risk

Abstract This paper addresses a strategy for decision-making of sustainable aggregation for clean energy participating in a day-ahead electricity market. The clean energy consists of wind turbines, photovoltaic arrays and energy storage, contributing to a better resilience of the system. The power delivered by a wind turbine or by a photovoltaic array is most certainly in deviation from the value associated with the bid accepted at a closing of a day-ahead electricity market. The deviation is due to the unpredicted variable nature of the respective sources of energy, leading to uncertainty and may imply a risk of loss of profit. So, uncertainty must be considered, and the addressed strategy considers uncertainty by scenarios of historical data and risk by the conditional value at risk. The strategy is a unified approach based on a risk-constrained and a two-stage stochastic optimization problem rewritten as a mixed-integer linear programming problem. A case study is presented to illustrate the main result and drive conclusions.

Coordination of Wind Farm and Pumped-Storage Hydro for a Self-Healing Power Grid

The increasing penetration of wind energy poses great challenges to the operation of power systems in normal and emergency states. However, energy storage technologies can help accommodate wind power uncertainty and variability due to their flexible characteristics. This paper focuses on the restoration phase, and provides a novel coordination strategy of wind and pumped-storage hydro (PSH) units for a faster and reliable self-healing process. The wind-PSH assisted power system restoration is formulated as a two-stage adaptive robust optimization problem. The first-stage problem determines the start-up sequence of generators and the energization times of transmission paths; and the second-stage problem decides load pickup sequences, wind power dispatch levels, and PSH units’ operating modes. The column-and-constraint generation decomposition algorithm is applied to solve the two-stage adaptive robust optimization problem, which has a mixed-integer optimization in the inner-level problem. The developed coordination strategy is tested on the modified IEEE 39-bus system. Numerical results demonstrate that the coordinated wind and PSH units can increase the total energy served, enhance wind power dispatchability, and reduce wind power curtailment.

A scenario decomposition algorithm for strategic time window assignment vehicle routing problems

We study the strategic decision-making problem of assigning time windows to customers in the context of vehicle routing applications that are affected by operational uncertainty. This problem, known as the Time Window Assignment Vehicle Routing Problem, can be viewed as a two-stage stochastic optimization problem, where time window assignments constitute first-stage decisions, vehicle routes adhering to the assigned time windows constitute second-stage decisions, and the objective is to minimize the expected routing costs. To that end, we develop in this paper a new scenario decomposition algorithm to solve the sampled deterministic equivalent of this stochastic model. From a modeling viewpoint, our approach can accommodate both continuous and discrete sets of feasible time window assignments as well as general scenario-based models of uncertainty for several routing-specific parameters, including customer demands and travel times, among others. From an algorithmic viewpoint, our approach can be easily parallelized, can utilize any available vehicle routing solver as a black box, and can be readily modified as a heuristic for large-scale instances. We perform a comprehensive computational study to demonstrate that our algorithm strongly outperforms all existing solution methods, as well as to quantify the trade-off between computational tractability and expected cost savings when considering a larger number of future scenarios during strategic time window assignment.

A combination method using evolutionary algorithms in initial orbit determination for too short arc

With the combination of two evolutionary algorithms EDA and DE, a new method of initial orbit determination for satellites based on ground-based too-short-arc is established. Compared with other algorithms, the proposed method focuses on the most densely populated region in the solution space rather than the individual with best fitness value. Both the global information and local information are well fused in the search of optimum. In the method (a,e,M) are treated as variables of the optimization, and the optimization procedure is carried out as a two-stage hierarchical optimization problem which has three variables for each stage. Kernel density estimation is applied to build the probability distribution model without any assumptions of the specified distribution, accompanied by handling semi-major axis and eccentricity as a pair of dependent variables in the construction of the probability for the correlation between them in the practice. Numerical experiments with real ground-based observations show that the proposed method is applicable to too-short-arc with even 3 s, and the result of bias in several kilometers can be achieved with 5″ error added to angular measurements.

Robust Beamforming With Pilot Reuse Scheduling in a Heterogeneous Cloud Radio Access Network

This paper considers a downlink ultradense heterogeneous cloud radio access network, which guarantees seamless coverage and can provide high date rates. In order to reduce channel state information (CSI) feedback overhead, incomplete intercluster CSI is considered, i.e., each remote radio head or macro base station only measures the CSI from user equipments (UEs) in its serving cluster. To reduce pilot consumption, pilot reuse among UEs is assumed, resulting in imperfect intracluster CSI. A two-stage optimization problem is then formulated. In the first stage, a pilot scheduling algorithm is proposed to minimize the sum mean square error (MSE) of all channel estimates. Specifically, the minimum number of required pilots along with a feasible pilot allocation solution are first determined by applying the Dsatur algorithm, and adjustments based on the defined level of pilot contamination are then carried out for further improvement. Based on the pilot allocation result obtained in the first stage, the second stage aims at maximizing the sum spectral efficiency (SE) of the network by optimizing the beam vectors. Due to incomplete intercluster CSI and imperfect intracluster CSI, an explicit expression of each UE's achievable rate is unavailable. Hence, a lower bound on the achievable rate is derived based on Jensen's inequality, and an alternative robust transmission design algorithm along with its distributed realization are then proposed to maximize the derived tight lower bound. Simulation results show that compared with the existing algorithms, the system performance can be greatly improved by the proposed algorithms in terms of both sum MSE and sum SE.

Joint Energy Procurement and Demand Response Towards Optimal Deployment of Renewables

In this paper, joint energy procurement and demand response is studied from the perspective of the operator of a power system. The operator procures energy from both renewable energy sources (RESs) and the spot market. We observe the fact that the RESs may incur considerable infrastructure cost. This cost is taken into account and the optimal planning of renewables is examined by controlling the investment in RES infrastructures. Due to the uncertainty of renewables, the operator can also purchase energy directly from the spot market to compensate for the possible deficit incurred by the realization of the random renewable energy. By setting appropriate prices, the operator sells the collected energy to heterogeneous end users with different demand response characteristics. We model the decision making process of the operator as a two-stage optimization problem. The optimal decisions on the renewable deployment, energy purchase from the spot market, and pricing schemes are derived. Several solution structures are observed and a computationally efficient algorithm, requiring only closed-form calculation and simple bisection search, is proposed to compute the optimal decisions. Finally, numerical experiments are conducted to verify the optimality of the proposed algorithm and the solution structures observed theoretically. In particular, the impact of renewable penetration and the importance of its optimal design are highlighted.

Improving the Energy Efficiency of the ATAD System through Redesign using Integration Superstructure

The aim of this study is application of an approach for solution of two-stage stochastic optimization problem for design of heat-integrated Autothermal Thermophilic Aerobic Digestion (ATAD) bioreactors system for municipal wastewater treatment operating under uncertainties. It incorporates in a common superstructure of two models of heat integration with one heat storage tank and two heat storage tanks for storing the heat used for preheating the incoming into the ATAD system cold raw sludge. On the first stage of the approach heat exchanger networks areas and the volumes of the heat storage tanks are determined. On the second stage - the flows rates represented by respective heating and cooling times are determined. As an optimization criterion - the annual capital costs for needed for the purpose of redesign of the ATAD system equipment as heat exchangers, heat storage tanks and pumps is determined as well as operating costs related with energy consumption of used pumps. The optimization problem is solved using genetic algorithm. Implementation of the proposed approach results in reduction of the depth of the thermal shock and achievement of sustainability improvement of the ATAD system.

A robust basic cyclic scheduling problem

This paper addresses the Basic Cyclic Scheduling Problem where the processing times are affected by uncertainties. We formulate the problem as a two-stage robust optimization problem with polyhedral uncertainty set. We propose three exact algorithms for solving the problem. Two of them use a negative circuit detection algorithm as a subroutine and the last one is an Howard's algorithm adaptation. Results of numerical experiments on randomly generated instances show that the Howard's algorithm adaptation yields efficient results and opens perspectives on more difficult robust cyclic scheduling problems.

Energy-Internet-oriented microgrid energy management system architecture and its application in China

In order to address the practical challenges posed by the increasing penetration of distributed energy resources and electric vehicles, the evolution from traditional power systems to Energy Internet and the rapidly changing market and policy environments in China, this paper proposes an Energy-Internet-oriented architecture of microgrid energy management system for China, considering specific technical requirements, markets and policies in China. Following the proposed architecture, the robust operation of a multi-energy microgrid with renewable energy and load uncertainties is formulated as a two-stage robust optimization problem and the column and constraints generation algorithm is used to solve it. A real-life case study in China, Beijing Goldwind Industrial Park, is conducted to demonstrate the proposed architecture and method. Simulation results show that by exporting surplus generation to the external customers and bulk power grid, as an important feature of the market dimension of the proposed architecture, the revenue of the microgrid is improved significantly compared to the traditional business mode. Moreover, as important features of technology and regulation dimensions of the proposed architecture, it is shown that smart charging of electric vehicles is able to improve the revenue of the microgrid as well. Finally, the results also show that the proposed robust operation method achieves robustness at the cost of lower revenue in non-worst scenarios.

Robust Constrained Operation of Integrated Electricity-Natural Gas System Considering Distributed Natural Gas Storage

A robust security-constrained unit commitment (robust SCUC) model is proposed in this paper to enhance the operational reliability of integrated electricity-natural gas system (IEGS) against possible transmission line outages. In this work, adjustable capability of natural gas contract is considered to avoid dramatic changes in real-time gas demand. Distributed natural gas storage is also included to smooth out power system demand curve and further improve operational efficiency at normal state and during contingencies. Nonlinear and nonconvex natural gas flow functions are relaxed into second-order cone (SOC) constraints, then the SOC-based column and constraint generation method is adopted to solve the proposed two-stage robust convex optimization problem. In this iterative method, the security-check subproblem feeds back worst case constraints to the first-stage master problem, until no violated scenarios can be detected. Simulations tested on 6-bus-6-node and 118-bus-10-node IEGS with natural gas storage show that the proposed storage model is effective in the robust SCUC solution for IEGS considering possible N - k contingencies with limited natural gas adjustments.

Comparing Spatial and Scenario Decomposition for Stochastic Hydrothermal Unit Commitment Problems

Solving very-large-scale optimization problems frequently require to decompose them in smaller subproblems, which are iteratively solved to produce useful information. One such approach is the Lagrangian relaxation (LR), a general technique that leads to many different decomposition schemes. The LR produces a lower bound of the objective function and useful information for heuristics aimed at constructing feasible primal solutions. In this paper, we compare the main LR strategies used so far for stochastic hydrothermal unit commitment problems, where uncertainty mainly concerns water availability in reservoirs and demand (weather conditions). The problem is customarily modeled as a two-stage mixed-integer optimization problem. We compare different decomposition strategies (unit and scenario schemes) in terms of quality of produced lower bound and running time. The schemes are assessed with various hydrothermal systems, considering different configuration of power plants, in terms of capacity and number of units.

The time window assignment vehicle routing problem with product dependent deliveries

This paper presents a new formulation for a time window assignment vehicle routing problem where time windows are defined for multiple product segments. This two-stage stochastic optimization problem is solved by means of a fix-and-optimize based matheuristic. The first stage assigns product dependent time windows while the second stage defines delivery schedules. Our approach outperforms a general-purpose solver and achieves an average cost decrease of 5.3% over expected value problem approaches. Furthermore, a sensitivity analysis on three operational models shows that it is possible to obtain significant savings compared to the solutions provided by a large European food retailer.

A Modified Penalty Parameter Approach for Optimal Estimation of UH with Simultaneous Estimation of Infiltration Parameters

The unit hydrograph and the infiltration parameters of a watershed can be obtained from observed rainfall–runoff data by using inverse optimization technique. This is a two-stage optimization problem. In the first stage, the infiltration parameters are obtained and the unit hydrograph ordinates are estimated in the second stage. In order to combine this two-stage method into a single stage one, a modified penalty parameter approach is proposed for converting the constrained optimization problem to an unconstrained one. The proposed approach is designed in such a way that the model initially obtains the infiltration parameters and then searches the optimal unit hydrograph ordinates. The optimization model is solved using Genetic Algorithms. A reduction factor is used in the penalty parameter approach so that the obtained optimal infiltration parameters are not destroyed during subsequent generation of genetic algorithms, required for searching optimal unit hydrograph ordinates. The performance of the proposed methodology is evaluated by using two example problems. The evaluation shows that the model is superior, simple in concept and also has the potential for field application.