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Recently there has been a considerable increase in the penetration level of distributed energy resources (DERs) due to various factors, such as the increasing affordability of these resources, the global movement towards sustainable energy, and the energy democracy movement. However, the uncertainty and variability of DERs introduce new challenges for power system operations. Advanced techniques that account for the characteristics of DERs, i.e., their intermittency and human-in-the-loop factors, are essential to improving distribution system operations. This paper proposes a behavior-aware approach to analyze and aggregate prosumers’ participation in demand response (DR) programs. A convexified AC optimal power flow (ACOPF) via a second-order cone programming (SOCP) technique is used for system scheduling with DERs. A chance-constrained framework for the system operation is constructed as an iterative two-stage algorithm that can integrate loads, DERs’ uncertainty, and SOCP-based ACOPF into one framework to manage the violation probability of the distribution system’s security limits. The benefits of the analyzed prosumers’ behaviors are shown in this paper by comparing the optimal system scheduling with socially aware and non-socially aware approaches. The case study illustrates that the socially aware approach within the chance-constrained framework can utilize up to 43% more PV generation and improve the reliability and operation of distribution systems.

In this paper, we discuss optimality conditions for optimization problems subject to random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.

Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage.

On more electric aircraft (MEA), reducing fuel consumption and guaranteeing flight safety are pursued by efficient operational management of the electrical power system (EPS). Considering the growing number of onboard electric loads and the increasing complexity of EPS architecture due to the integration of multiple power converters and energy storage systems (ESSs), system-level operation control is required to manage power distribution, load scheduling, and ESSs. In this paper, a chance-constrained stochastic model predictive control (CC-SMPC) method is proposed to improve both the system operation in terms of the system's cost and reconfiguration activities as well as the ability to cope with uncertainties due to fluctuating load demands. Both normal and faulty operating conditions are investigated with multi-failure cases, resulting in different uncertainty propagation paths. The system's operational and technical requirements are formulated as a set of deterministic and probabilistic constraints in the CC-SMPC model. To verify the effectiveness of the proposed strategy, a comprehensive comparison study is conducted. Two uncertainty/failure cases are taken into account and simulations are performed for both offline and online control strategies while the Monte-Carlo algorithm is used for scenario generation. The results are evaluated using the proposed evaluation framework, showing that the CC-SMPC achieves better performance compared to deterministic MPC (DMPC) in both cases. In an offline testing framework, comparing the performance of DMPC and CC-SMPC strategies shows that CC-SMPC reduces the power constraint violations for batteries and generators in all cases following the selected confidence level. In addition, in an online testing framework with 1 % violation probability, the following results are observed in the two cases: In the EPS normal condition, CC-SMPC reduces the total cost by 31.4 % and the overall constraint violation cost by 93 %; while in the EPS faulty condition, CC-SMPC reduces the total cost by 4.37 %, and the overall constraint violation cost by 96 %.

In this paper, a sampling-based stochastic model predictive control (SMPC) algorithm is proposed for discrete- time linear systems subject to both parametric uncertainties and additive disturbances. One of the main drivers for the development of the proposed control strategy is the need for reliable and robust guidance and control strategies for automated rendezvous and proximity operations between spacecraft. To this end, the proposed control algorithm is validated on a floating spacecraft experimental testbed, proving that this solution is effectively implementable in real time. Parametric uncertainties due to the mass variations during operations, linearization errors, and disturbances due to external space environment are simulta- neously considered. The approach enables to suitably tighten the constraints to guarantee robust recursive feasibility when bounds on the uncertain variables are provided. Moreover, the offline sampling approach in the control design phase shifts all the intensive computations to the offline phase, thus greatly reducing the online computational cost, which usually constitutes the main limitation for the adoption of SMPC schemes, especially for low- cost on-board hardware. Numerical simulations and experiments show that the approach provides probabilistic guarantees on the success of the mission, even in rather uncertain and noisy situations, while improving the spacecraft performance in terms of fuel consumption.

The required time for surgical interventions in operating rooms (OR) may vary significantly from the predicted values depending on the type of operations being performed, the surgical team, and the patient. These deviations diminish the efficient utilization of OR resources and result in the disruption of projected surgery start times. This paper proposes a two-stage chance-constrained model to solve the OR scheduling problem under uncertainty. The goal is to minimize the costs associated with OR opening and overtime as well as reduce patient waiting times. The risk of OR overtime is controlled using chance constraints. Numerical experiments show that the proposed model provides a better trade-off between minimizing costs and reducing solution variability when compared to two existing models in the literature. It is also shown that the three models converge as the overtime probability threshold approaches one. Moreover, it is observed that the individual chance constraints result in the opening of fewer rooms, lower waiting times, and shorter solution times when compared to that of joint chance constraints. A decomposition algorithm is applied that solves large test instances of the OR scheduling problem, that of which is known to be NP-hard. Strong valid inequalities are derived in order to accelerate the convergence speed. The proposed approach outperformed both a commercial solver and a basic decomposition algorithm after solving all test instances with up to 89 surgeries and 20 ORs in less than 48 minutes.

An unmanned surface vehicles (USV) set point tracking problem is investigated in this paper. The stochastic model predictive control (SMPC) scheme is utilized to design the controller in order to reject the environment disturbances and meet the physical constraints. The design problem is formulated as a chance-constrained stochastic optimization problem, which is non-convex. Thus, the problem is computationally prohibitive. For this, the convex conditional value at risk (CVaR) approximation is introduced to convert the chance constraints into deterministic convex constraints. The converted constraints are then further transformed into the second order cone (SOC) constraints. Therefore, the proposed method is computationally tractable and hence can be implemented online. A numerical example is provided to illustrate the effectiveness of the proposed method.

We implement an exact mixed-integer programming algorithm for the chance-constrained multi-area reserve sizing problem in the presence of transmission network constraints. The problem can be cast as a two-stage stochastic mixed integer linear program using sample approximation. Due to the complicated structure of the problem, existing methods attempt to find a feasible solution based on heuristics. However, a recent development of integer programming techniques allow us to reformulate the problem into a form where we can solve it to optimality. In this paper, we apply this integer programming approach to solve our problem to optimality and compare the results with that of the existing heuristics.

Given the increased percentage of wind power in power systems, chance-constrained optimal power flow (CC-OPF) calculation, as a means to take wind power uncertainty into account with a guaranteed security level, is being promoted. Compared to the local CC-OPF within a regional grid, the global CC-OPF of a multi-regional interconnected grid is able to coordinate across different regions and therefore improve the economic efficiency when integrating high percentage of wind power generation. In this global problem, however, multiple regional independent system operators (ISOs) participate in the decision-making process, raising the need for distributed but coordinated approaches. Most notably, due to regulation restrictions, commercial interest, and data security, regional ISOs may refuse to share confidential information with others, including generation cost, load data, system topologies, and line parameters. But this information is needed to build and solve the global CC-OPF spanning multiple areas. To tackle these issues, this paper proposes a distributed CC-OPF method with confidentiality preservation, which enables regional ISOs to determine the optimal dispatchable generations within their regions without disclosing confidential data. This method does not require parameter tunings and will not suffer from convergence challenges. Results from IEEE test cases show that this method is highly accurate.

In this paper, we study the problem of linear optimization with probabilistic constraints. We suppose that the constraint row vectors are elliptically distributed. Further, the dependence of the rows is modeled by a family of Archimedean copulas, namely, the Gumbel-Hougaard family of copulas. Under mild assumptions, we prove the eventual convexity of the feasibility set.

The advent of renewable energy has huge implications for the design and control of power grids. Due to increasing supply-side uncertainty, traditional reliability constraints such as strict bounds on current, voltage and temperature in a transmission line have to be replaced by computationally demanding chance constraints. In this paper we use large deviations techniques to study the probability of current and temperature overloads in power grids with stochastic power injections, and develop corresponding safe capacity regions. In particular, we characterize the set of admissible power injections such that the probability of overloading of any line over a given time interval stays below a fixed target. We show how enforcing (stochastic) constraints on temperature, rather than on current, results in a less conservative approach and can thus lead to capacity gains., Comment: 12 pages (10 pages + 2 pages appendix), 2 figures. Revised version with extended numerical section

Medical drug shortages are an important issue in health care, since they can significantly affect patients’ health. Thus, selecting the appropriate distribution and inventory policies plays an important role in decreasing drug shortages. In this context, inventory routing models can be used to determine optimal policies in the context of medical drug distribution. However, in real-world conditions, some parameters in these models are subject to uncertainty. This paper examines the effects of uncertainty in the demand by relying on a two-stage stochastic programming approach to incorporate it into the optimization model. A two-stage model is then proposed and two different approaches based on chance constraints are used to assess the validity of the proposed model. In the first model, a scenario-based two-stage stochastic programming model without probabilistic constraint is proposed, while in the other two models, proposed for validation of the first model, probabilistic constraints are considered. A mathematical-programming based algorithm (a matheuristic) is proposed for solving the models. Moreover, the Latin hypercube sampling method is employed to generate scenarios for the scenario-based models. Numerical examples show the necessity of considering the stochastic nature of the problem and the accuracy of the proposed models and solution method.

In this paper, we propose the distributionally robust secure transmit schemes in multi-input single-output (MISO) downlink wireless networks which consist of a transmitter, a desired receiver, multiple eavesdroppers and an assisting jammer. The imperfect channel state information (CSI) is considered and the CSI errors are only captured by the mean and covariance. We first study the transmit power minimization by jointly designing the beamforming vector at transmitter and artificial noise(AN) covariance at jammer, while the lower bound of connection probability at desired receiver and lower bound of outage probability at eavesdroppers are guaranteed simultaneously. Since the chance-constrained problem is non-convex, we derive two safe convex approximations by exploiting the Conditional Value-at-risk (CVaR) based method and Bernstein-type inequality (BTiE) based method, respectively. Specifically, we extend the application of BTiE originally proposed with the Gaussianity assumption to other possible distributions which satisfy the proposed sufficient condition. Furthermore, the secrecy rate maximization is exploited under the constraint of total transmit power. The original problem is non-convex and fractional, hence we design the Bilevel Quick Search (BQS) method to make it tractable. Finally, the simulation results verify the effectiveness and the robustness of the proposed transmit schemes.

The increased penetration of plug-in electric vehicles asks for efficient algorithms to be adopted in parking lots equipped with charging units. In this paper, the peak power minimization problem for a plug-in charging station is addressed. A chance constraint approach is adopted in order to minimize the daily peak power, allowing for a tolerance on the charging service customer satisfaction expressing the probability that a vehicle leaves the station violating the agreed level of charge. Numerical simulations are provided to evaluate the performance of the proposed approach as well as to make a comparison with other techniques.

Freight distribution with electric vehicles (EVs) is a promising alternative to reduce the carbon footprint associated with city logistics. Algorithms for planning routes for EVs should take into account their relatively short driving range and the effects of traffic congestion on the battery consumption. This paper proposes new methodology and illustrates how it can be applied to solve an electric vehicle routing problem with stochastic and time-dependent travel times where battery recharging along routes is not allowed. First, a new method for generating network-consistent (correlated in time and space) and time-dependent speed scenarios is introduced. Second, a new technique for applying branch and price on instances defined on real street networks is developed. Computational experiments demonstrate the effectiveness of the approach for finding optimal or near-optimal solutions in instances with up to 133 customers and almost 1,500 road links. With a high probability, the routes in the obtained solutions can be performed by EVs without requiring intermediate recharging stops. An execution time control policy to further reduce the chances of stranded EVs is also presented. In addition, we measure the cost of independence, which is the impact on solution feasibility when travel times are assumed statistically independent. Last, we give directions on how to extend the proposed framework to handle recourse actions.

A comparative study of two state-of-the-art stochastic model predictive controllers for linear systems with parametric and additive uncertainties is presented. On the one hand, Stochastic Model Predictive Control (SMPC) is based on analytical methods and solves an optimal control problem (OCP) similar to a classic Model Predictive Control (MPC) with constraints. SMPC defines probabilistic constraints on the states, which are transformed into equivalent deterministic ones. On the other hand, Scenario-based Model Predictive Control (SCMPC) solves an OCP for a specified number of random realizations of uncertainties, also called scenarios. In this paper, Classic MPC, SMPC and SCMPC are compared through two numerical examples. Thanks to several Monte-Carlo simulations, performances of classic MPC, SMPC and SCMPC are compared using several criteria, such as number of successful runs, number of times the constraints are violated, integral absolute error and computational cost. Moreover, a Stochastic Model Predictive Control Toolbox was developed by the authors, available on MATLAB Central, in which it is possible to simulate a SMPC or a SCMPC to control multivariable linear systems with additive disturbances. This software was used to carry out part of the simulations of the numerical examples in this article and it can be used for results reproduction.

The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from $$L^2$$ L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.

This paper presents a comparison of two different models (Land et al (1993) and Olesen and Petersen (1995)), both designed to extend DEA to the case of stochastic inputs and outputs. The two models constitute two approaches within this area, that share certain characteristics. However, the two models behave very differently, and the choice between these two models can be confusing. This paper presents a systematic attempt to point out differences as well as similarities. It is demonstrated that the two models under some assumptions do have Lagrangian duals expressed in closed form. Similarities and differences are discussed based on a comparison of these dual structures. Weaknesses of the each of the two models are discussed and a merged model that combines attractive features of each of the two models is proposed.

This paper presents a theoretical and methodological review of the literature relevant for the management of eutrophicated coastal ecosystems. First, we define a coastal ecosystem and argue that coastal ecosystem pollution is characterised by multiple, interacting, heterogeneous emissions at different, interdependent locations. We pay special attention to the fact that pollutant transports and transformations are stochastic due to variations in weather and other natural processes. In the second part of the paper we review various methodological aspects. First we characterise the basic approach which usually comes down to minimising total costs subject to a restriction on the loads of a single pollutant in a static setting. Next we consider various extensions, notably in terms of geographical scale, dynamics, multiple pollutants and uncertainty. Finally, we introduce the papers that make up the special issue. (C) 2003 Elsevier B.V All rights reserved.

5G New Radio (NR) is envisioned to provide three major services: enhanced Mobile Broadband (eMBB), Ultra Reliable Low Latency Communications (URLLC), and massive Machine Type Communication (mMTC). URLLC services (i.e., autonomous vehicles, industrial Internet of Things (IoT),..) require strict latency, on-way latency of 1 ms, with 99.999% reliability. eMBB applications aims extreme data rate while mMTC is designed to serve a large number of IoT devices that send a small data sporadically. In this paper, we address the resource scheduling problem of URLLC and eMBB traffics. First, the Resource Blocks (RBs) are allocated to eMBB users at the beginning of each time slot based on the channel state of each eMBB user and his previous average data rate up to current time slot. The RBs allocation problem modeled as as a 2-Dimensions Hopfield Neural Networks (2D-HNN) and the energy function of 2D-HNN is investigated to solve the RBs allocation problem. Then, the resource scheduling problem of URLLC and eMBB is formulated as an optimization problem with chance constraint. The chance constraint based problem aims to maximize the eMBB data rate while satisfying the URLLC critical constraints. The cumulative Distribution Function (CDF) of the stochastic URLLC traffic is investigated to relax the chance constraint into a linear constraint. The simulation results show efficiency of the proposed dynamic scheduling approach.

An industrial microgrid (IMG) consists in a microgrid involving manufacturer plants that are usually equipped with distributed generation facilities, industrial electric vehicles, energy storage systems, and so on. In this paper, the problem of IMG-efficient operation in the presence of plug-in electric vehicles is addressed. To this purpose, the schedule of the different device operations of IMGs has to be optimally computed, minimizing the operation cost while guaranteeing the electrical network stability and the production constraints. Such a problem is formulated in a receding horizon framework involving dynamic optimal power flow equations. Uncertainty affecting plug-in electric vehicles is handled by means of a chance constraint approach. The obtained nonconvex problem is then approximately solved by exploiting suitable convex relaxation techniques. The numerical simulations have been performed showing computational feasibility and robustness of the proposed approach against increased penetration of the electric vehicles.

The scenario-based optimization approach (`scenario approach') provides an intuitive way of approximating the solution to chance-constrained optimization programs, based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty (`scenarios'). A key merit of this approach is that it neither assumes knowledge of the uncertainty set, as it is common in robust optimization, nor of its probability distribution, as it is usually required in stochastic optimization. Moreover, the scenario approach is computationally efficient as its solution is based on a deterministic optimization program that is canonically convex, even when the original chance-constrained problem is not. Recently, researchers have obtained theoretical foundations for the scenario approach, providing a direct link between the number of scenarios and bounds on the constraint violation probability. These bounds are tight in the general case of an uncertain optimization problem with a single chance constraint. However, this paper shows that these bounds can be improved in situations where the constraints have a limited `support rank', a new concept that is introduced for the first time. This property is typically found in a large number of practical applications---most importantly, if the problem originally contains multiple chance constraints (e.g. multi-stage uncertain decision problems), or if a chance constraint belongs to a special class of constraints (e.g. linear or quadratic constraints). In these cases the quality of the scenario solution is improved while the same bound on the constraint violation probability is maintained, and also the computational complexity is reduced., This manuscript is the preprint of a paper submitted to the SIAM Journal on Optimization and it is subject to SIAM copyright. SIAM maintains the sole rights of distribution or publication of the work in all forms and media. If accepted, the copy of record will be available at http://www.siam.org

This paper proposes a reliability-based economic model predictive control (MPC) strategy for the management of generalised flow-based networks, integrating some ideas on network service reliability, dynamic safety stock planning, and degradation of equipment health. The proposed strategy is based on a single-layer economic optimisation problem with dynamic constraints, which includes two enhancements with respect to existing approaches. The first enhancement considers chance-constraint programming to compute an optimal inventory replenishment policy based on a desired risk acceptability level, leading to dynamical allocation of safety stocks in flow-based networks to satisfy non-stationary flow demands. The second enhancement computes a smart distribution of the control effort and maximises actuators’ availability by estimating their degradation and reliability. The proposed approach is illustrated with an application of water transport networks using the Barcelona network as the case study considered.

This paper presents a stochastic predictive control algorithm for a number of microgrids connected to the same distribution system. Each microgrid includes a variety of distributed resources such as wind turbine, photo voltaic units, energy storage devices and loads. Considering the uncertainty of loads and renewable-based distributed resources, the power management problem is formulated in the framework of stochastic control. The microgrids operation are coupled through a joint probabilistic constraint which requires the power flow from utility to each microgrid limits to a pre-specified range. Utilizing probabilistic distribution function of uncertain variables, the deterministic counterpart of the problem is derived and a close-optimal solution of the problem is achieved. The Monte-Carlo simulation analysis is used to justify the robustness characteristics of the solution.

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the 1972 result of Ben-Tal and Hochman (BH) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. First, we use these results to treat ambiguous expected feasibility constraints to obtain exact reformulations for both functions that are convex and concave in the components of the random variable. This approach requires, however, the independence of the random variables and, moreover, may lead to an exponential number of terms in the resulting robust counterparts. We then show how upper bounds can be constructed that alleviate the independence restriction, and require only a linear number of terms, by exploiting models in which random variables are linearly aggregated. Moreover, using the BH bounds we derive three new safe tractable approximations of chance constraints of increasing computational complexity and quality. In a numerical study, we demonstrate the efficiency of our methods in solving stochastic optimization problems under mean-MAD ambiguity. The electronic companion is available at https://doi.org/10.1287/opre.2017.1688 .

In this paper, an adaptively constrained stochastic model predictive control (MPC) is proposed to achieve less-conservative coordination between energy storage units and uncertain renewable energy sources (RESs) in a microgrid (MG). Besides the economic objective of MG operation, the limits of state-of-charge (SOC) and discharging/charging power of the energy storage unit are formulated as chance constraints when accommodating uncertainties of RESs, considering mild violations of these constraints are allowed during long-term operation, and a closed-loop online update strategy is performed to adaptively tighten or relax constraints according to the actual deviation probability of violation level from the desired one as well as the current change rate of deviation probability. Numerical studies show that the proposed adaptively constrained stochastic MPC for MG optimal operation is much less conservative compared with the scenario optimization based robust MPC, and also presents a better convergence performance to the desired constraint violation level than other online update strategies.

The definition of the electric energy procurement plan represents a fundamental problem that any consumer has to deal with. Bilateral contracts, electricity market and self-production are the main supply sources that should be properly combined to satisfy the energy demand over a given time horizon at the minimum cost. The problem is made more complex by the presence of uncertainty, mainly related to the energy requirements and electricity market prices. Ignoring the uncertain nature of these elements can lead to the definition of procurement plans which are infeasible or overly expensive in a real setting. In this paper, we deal with the procurement problem under uncertainty by adopting the paradigm of joint chance constraints to define reliable plans that are feasible with a high probability level. Moreover, the proposed model includes in the objective function a risk measure to control undesirable effects caused by the random variations of the electricity market prices. The proposed model is applied to a real test case. The results show the benefit deriving from the stochastic optimization approach and the effect of considering different levels of risk aversion.

High penetration of renewable energy sources and the increasing share of stochastic loads require the explicit representation of uncertainty in tools such as the optimal power flow (OPF). Current approaches follow either a linearized approach or an iterative approximation of non-linearities. This paper proposes a semidefinite relaxation of a chance-constrained AC-OPF which is able to provide guarantees regarding global optimality. Using a piecewise affine policy, we can ensure tractability, accurately model large power deviations, and determine suitable corrective control policies for active power, reactive power, and voltage. We state a tractable formulation for two types of uncertainty sets. Using a scenario-based approach and making no prior assumptions about the probability distribution of the forecast errors, we obtain a robust formulation for a rectangular uncertainty set. Alternatively, assuming a Gaussian distribution of the forecast errors, we propose an analytical reformulation of the chance constraints suitable for semidefinite programming. We demonstrate the performance of our approach on the IEEE 9, 24 and 118 bus system using realistic day-ahead forecast data and obtain tight near-global optimality guarantees.

This is the first study that integrates fleet assignment and robust schedule design.Nonlinear fuel and emission cost functions along with probabilistic chance constraints.Proposed conic reformulations could be solved by a commercial solver.Proposed two-stage algorithm is tested using a published schedule of a major airline. Assignment of aircraft types, each having different seat capacity, operational expenses and availabilities, critically affects airlines overall cost. In this paper, we assign fleet types to paths by considering not only flight timing and passenger demand, as commonly done in the literature, but also operational expenses, such as fuel burn and carbon emission costs associated with adjusting the cruise speed to ensure the passenger connections. In response to flight time uncertainty due to the airport congestions, we allow minor adjustments on the flight departure times in addition to cruise speed control, thereby satisfying the passenger connections at a desired service level. We model the uncertainty in flight duration via a random variable arising in chance constraints to ensure the passenger connections. Nonlinear fuel and carbon emission cost functions, chance constraints and binary aircraft assignment decisions make the problem significantly more difficult. To handle them, we use mixed-integer second order cone programming. We compare the performance of a schedule generated by the proposed model to the published schedule for a major U.S. airline. On the average, there exists a 20% overall operational cost saving compared to the published schedule. To solve the large scale problems in a reasonable time, we also develop a two-stage algorithm, which decomposes the problem into planning stages such as aircraft-path assignment and robust schedule generation, and then solves them sequentially.

Chance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties to a setting of control problems subject to (uniform) state chance constraints.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License., This paper proposes a reliability-based economic model predictive control (MPC) strategy for the management of generalised flow-based networks, integrating some ideas on network service reliability, dynamic safety stock planning, and degradation of equipment health. The proposed strategy is based on a single-layer economic optimisation problem with dynamic constraints, which includes two enhancements with respect to existing approaches. The first enhancement considers chance-constraint programming to compute an optimal inventory replenishment policy based on a desired risk acceptability level, leading to dynamically allocate safety stocks in flow-based networks to satisfy non-stationary flow demands. The second enhancement computes a smart distribution of the control effort and maximises actuators’ availability by estimating their degradation and reliability. The proposed approach is illustrated with an application of water transport networks using the Barcelona network as the considered case study., The authors wish to thank the projects that funded this research, i.e., the EU project EFFINET (FP7-ICT-2011-8-31855) and the Spanish projects ECOCIS (CICYT DPI2013-48243-C2-1-R) and SHERECS (CICYT DPI2011-26243) of the Spanish Ministry of Economy and Competitiveness.

In this paper we present a new combinatorial problem, called minmax multidimensional knapsack problem (MKP), motivated by a military logistics problem. The logistics problem is a two-period, two-level, chance-constrained problem with recourse. We show that the MKP is NP-hard and develop a practically efficient combinatorial algorithm for solving it. We also show that under some reasonable assumptions regarding the operational setting of the logistics problem, the chance-constrained optimization problem is decomposable into a series of MKPs that are solved separately. © 2007 Wiley Periodicals, Inc. Naval Research Logistics 54: 656-666, 2007 We present a new combinatorial model called minmax multidimensional knapsack problem (MKP). The MKP is motivated by a military logistics optimization problem typ- ical to tactical-level ground operations such as supply and resupply of ammunition to an artillery battalion in a 2-day operation. The problem is a two-period stochastic program- ming problem with recourse. However, under our reasonable operational assumptions, it is shown to have unique features that reduce it into two separate and sequential (albeit not inde- pendent) sets of MKPs. We show that the MKP is NP-hard and develop a new practically efficient algorithm for solving it. Stated simply, the military logistics problem is to find min- imum cost inventories that satisfy minimum responsiveness requirements. The requirements are expressed in terms of probabilities for satisfying demands in a two-period operation of a military unit that comprises several weapons. While some inventories must be determined before the operation, others can be set after the demands in the first period are realized—a situation that lends itself to a two-period, chance-constrained, stochastic programming model with recourse. Stochastic programming models with recourse have been applied in supply-chain and related problems, mostly in the context of optimizing expected values. Escudero et al. (9)

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the old result of Ben-Tal and Hochman (1972) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. We use these bounds to derive exact robust counterparts of expected feasibility of convex constraints and to construct new safe tractable approximations of chance constraints. Numerical examples show our method to be applicable to numerous applications of Robust Optimization, e.g., where implementation error or linear decision rules are present. Also, we show that the methodology can be used for optimization the average-case performance of worst-case optimal Robust Optimization solutions.

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the old result of Ben-Tal and Hochman (1972) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. We use these bounds to deriveexact robust counterparts of expected feasibility of convex constraints and to construct new safe tractable approximations of chance constraints. Numerical examples show our method to be applicable to numerous applications of Robust Optimization, e.g., where implementation error or linear decision rules are present.Also, we show that the methodology can be used for optimization the average-case performance of worst-caseoptimal Robust Optimization solutions.

In this paper we present power flow optimization of a residential DC microgrid that consists of photovoltaic array, batteries stack and fuel cells stack with electrolyser, and is connected to the grid via bidirectional power converter. The optimization problem aims to minimize microgrid operating costs and is formulated using a linear program that takes into account the storages charge and discharge efficiency. To account for power predictions uncertainty, optimization problem is defined in a stochastic framework by using chance constraints. Since we assume that the error in realization of power predictions will be compensated by utility grid, chance constraints are defined for power exchange between the microgrid and the utility grid. Finally, we investigate a stochastic model predictive control for the closed-loop power management in the microgrid. Performance verification of the proposed approach is performed on simulations for two-month period.

The number of renewable energy sources in electricity grids is growing and with it the associated uncertainty. For power system operation a good tradeoff between security and cost becomes increasingly important. We consider the problem of including uncertainty from renewable in-feed in a DC security constrained optimal power flow (SCOPF) by formulating chance constraints. The chance-constrained SCOPF is solved repeatedly and can be understood as a chance-constrained Model Predictive Control (MPC) problem, where the time coupling comes from storage in the system and the associated dynamics. The main focus of the paper is to reformulate the problem in a non-conservative way, i.e., to exploit the predefined violation level of the chance constraints based on their empirical closed-loop violations in order to achieve a good tradeoff between operational security and cost. We use and extend a recently proposed adaptive stochastic MPC scheme that starts with a standard conservative chance-constrained reformulation and then adapts the formulation of the constraints online based on the empirical violation level. The proposed approach is demonstrated on a 5-bus network.

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the old result of Ben-Tal and Hochman (1972) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. We use these bounds to derive exact robust counterparts of expected feasibility of convex constraints and to construct new safe tractable approximations of chance constraints. Numerical examples show our method to be applicable to numerous applications of Robust Optimization, e.g., where implementation error or linear decision rules are present. Also, we show that the methodology can be used for optimization the average-case performance of worst-case optimal Robust Optimization solutions.

International audience; The purpose of this work is to efficiently design disassembly lines taking into account the uncertainty of task processing times. The main contribution of the paper is the development of a decision tool that allows decision-makers to choose the best disassembly alternative (process), for an End of Life product (EOL), and assign the corresponding disassembly tasks to the workstations of the line under precedence and cycle time constraints. Task times are assumed to be random variables with known normal probability distributions. The case of presence of hazardous parts is studied and cycle time constraints are to be jointly satisfied with at least a certain probability level, or service level, fixed by the decision-maker. An AND/OR graph is used to model the precedence relationships among tasks. The objective is to minimise the line cost composed of the workstation operation costs and additional costs of workstations handling hazardous parts of the EOL product. To deal with task time uncertainties, lower and upper-bounding schemes using second-order cone programming and approximations with convex piecewise linear functions are developed. The applicability of the proposed solution approach is shown by solving to optimality a set of disassembly problem instances (EOL industrial products) from the literature.

Trabajo presentado a la 53rd IEEE Conference on Decision and Control (CDC 2014), celebrada del 15 al 17 de diciembre en Los Angeles, California (US).-- et al., Inventory management is one of the main tasks that the pharmacy department has to carry out in a hospital. It is a complex problem that requires to establish a tradeoff between different and contradictory optimization criteria. The complexity of the problem is increased due to the constraints that naturally arise in this type of applications. In this paper, which corresponds to preliminary works performed to implement advanced control techniques for pharmacy management in two Spanish hospitals, we propose and assess chance-constrained model predictive control (CC-MPC) as a mean to relieve this issue., The authors would like to acknowledge Junta de Andalucía (Pharmacon-trol Project, P12-TIC-2400), for funding this work.

This paper addresses a chance-constrained model predictive control (CC-MPC) strategy for the management of drinking water networks (DWNs) based on a finite horizon stochastic optimisation problem with joint probabilistic (chance) constraints. In this approach, water demands are considered additive stochastic disturbances with non-stationary uncertainty description, unbounded support and known (or approximated) quasi-concave probabilistic distribution. A deterministic equivalent of the stochastic problem is formulated using Boole's inequality to decompose joint chance constraints into single chance constraints and by considering a uniform allocation of risk to bound these later constraints. The resultant deterministic-equivalent optimisation problem is suitable to be solved with tractable quadratic programming (QP) or second order cone programming (SOCP) algorithms. The reformulation allows to explicitly and easily propagate uncertainty over the prediction horizon, and leads to a cost-efficient management of risk that consists in a dynamic back-off to avoid frequent violation of constraints. Results of applying the proposed approach to a real case study - the Barcelona DWN (Spain) - have shown that the network performance (in terms of operational costs) and the necessary back-off (to cope with stochastic disturbances) are optimised simultaneously within a single problem, keeping tractability of the solution, even in large-scale networks. The general formulation of the approach and the automatic computation of proper back-off within the MPC framework replace the need of experience-based heuristics or bi-level optimisation schemes that might compromise the trade-off between profits, reliability and computational burden., This work has been partially supported by the EU Project EFFINET (FP7-ICT-2011-8-31855) and the DGR of Generalitat de Catalunya (SAC group Ref. 2009/SGR/1491).

Many practical applications of control require that constraints on the inputs and states of the system be respected, while optimizing some performance criterion. In the presence of model uncertainties or disturbances, for many control applications it suffices to keep the state constraints at least for a prescribed share of the time, as e.g. in building climate control or load mitigation for wind turbines. For such systems, a new control method of Scenario-Based Model Predictive Control (SCMPC) is presented in this paper. It optimizes the control inputs over a finite horizon, subject to robust constraint satisfaction under a finite number of random scenarios of the uncertainty and/or disturbances. While previous approaches have shown to be conservative (i.e. to stay far below the specified rate of constraint violations), the new method is the first to account for the special structure of the MPC problem in order to significantly reduce the number of scenarios. In combination with a new framework for interpreting the probabilistic constraints as average-in-time, rather than pointwise-in-time, the conservatism is eliminated. The presented method retains the essential advantages of SCMPC, namely the reduced computational complexity and the handling of arbitrary probability distributions. It also allows for adopting sample-and-remove strategies, in order to trade performance against computational complexity., Pre-print submitted to Automatica

Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. In order to do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be successfully done by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz' code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, Deák's sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used in order to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. Later, the result is extended to alternative distributions with an emphasis on the multivariate Student (or T-) distribution.

Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario sets. The relevant distances of (multivariate) probability distributions for deriving quantitative stability results for such stochastic programs are B-discrepancies, where the class B of Borel sets depends on their structural properties. Hence, the optimal scenario reduction problem for such models is stated with respect to B-discrepancies. In this paper, upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or Kolmogorov metric) in some detail and provide some numerical experience.

This paper evaluates the consequences of considering environmental and economic risk in the analysis of cost-effective nitrogen abatement options in crop production. A farm-level mathematical programming model incorporating nitrogen leaching variability, field time variability, yield variability, and output price variability is developed. The empirical results reveal that requiring a high reliability with respect to a desired abatement target can be extremely costly, due to the high variability of nitrogen emissions. It appears to be sufficient to reduce average nitrogen load in order to reduce the environmental risk associated with nitrogen leaching variability, since a change to crops with lower average load also results in lower variability of nitrogen emissions. A farmer's degree of risk aversion has some effect on the economically optimal choice of crop mix. However, it is more important to consider the utilisation of machinery and labour resources and crop rotation effects, than considering risk aversion.

This paper addresses super-efficiency issue based on input relaxation model in stochastic data envelopment analysis. The proposed model is not limited to using the input amounts of evaluating DMU, and one can obtain a total ordering of units by using this method. The input relaxation super-efficiency model is developed in stochastic data envelopment analysis, and its deterministic equivalent, also, is derived which is a nonlinear program. Moreover, it is shown that the deterministic equivalent of the stochastic super-efficiency model can be converted to a quadratic program. As an empirical example, the proposed method is applied to the data of textile industry of China to rank efficient units. Finally, when allowable limits of data variations for evaluating DMU are permitted, the sensitivity analysis of the proposed model is discussed.

In this brief paper, a Stochastic Model Predictive Control formulation tractable for large-scale systems is developed. The proposed formulation combines the use of Affine Disturbance Feedback, a formulation successfully applied in robust control, with a deterministic reformulation of chance constraints. A novel approximation of the resulting stochastic finite horizon optimal control problem targeted at building climate control is introduced to ensure computational tractability. This work provides a systematic approach toward finding a control formulation which is shown to be useful for the application domain of building climate control. The analysis follows two steps: 1) a small-scale example reflecting the basic behavior of a building, but being simple enough for providing insight into the behavior of the considered approaches, is used to choose a suitable formulation; and 2) the chosen formulation is then further analyzed on a large-scale example from the project OptiControl, where people from industry and other research institutions worked together to create building models for realistic controller comparison. The proposed Stochastic Model Predictive Control formulation is compared with a theoretical benchmark and shown to outperform current control practice for buildings.