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Critical chain buffer management (CCBM) has been extensively studied in recent years. This paper investigates a new formulation of CCBM, the multimode chance-constrained CCBM problem. A flow-based mixed-integer linear programming model is described and the chance constraints are tackled using a scenario approach. A reinforcement learning (RL)-based algorithm is proposed to solve the problem. A factorial experiment is conducted and the results of this study indicate that solving the chance-constrained problem produces shorter project durations than the traditional approach that inserts time buffers into a baseline schedule generated by solving the deterministic problem. This paper also demonstrates that our RL method produces competitive schedules compared to established benchmarks. The importance of solving the chance-constrained problem and obtaining a project buffer tailored to the desired probability of completing the project on schedule directly from the solution is highlighted. Because of its potential for generating shorter schedules with the same on-time probabilities as the traditional approach, this research can be a useful aid for decision makers. [ABSTRACT FROM AUTHOR]

In this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of differentiability results for probability functions in infinite dimensions usable in this context. These results are subsequently combined with the optimal control setting to derive a novel Pontryagin’s optimality principle. [ABSTRACT FROM AUTHOR]

The existing distribution system planning methods do not fully consider improving power supply capacity and reliability through the coordination of multiple planning factors, and they are not comprehensive enough in quantifying planning risks. Therefore, this paper proposes a collaborative planning method for sources and networks that considers risk measurement. A multi-layer planning model is first constructed that includes a grid planning layer, a power planning layer, a switch planning layer, and an operation optimization layer. In the model, a risk measurement method combining opportunity constraints and conditional value-at-risk objectives is used to comprehensively assess the risk of the node voltage and branch current exceeding the limit caused by load uncertainty. Then, a solution strategy based on a genetic algorithm and a sparrow search algorithm is proposed to coordinate the contradiction between the solution time and the accuracy of the multi-layer model. Finally, taking a planned area to be expanded as an example, the results show that compared to the existing collaborative planning methods for sources and networks, the proposed method in this paper reduces the planning risks caused by load uncertainty by more than 50% and increases the annual net income of the power distribution company and the DG operators by RMB 1.5 million and RMB 1.1 million, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we tackle the resolution of chance-constrained problems reformulated via sample average approximation. The resulting data-driven deterministic reformulation takes the form of a large-scale mixed-integer program (MIP) cursed with Big-Ms. We introduce an exact resolution method for the MIP that combines the addition of a set of valid inequalities to tighten the linear relaxation bound with coefficient strengthening and constraint screening algorithms to improve its Big-Ms and considerably reduce its size. The proposed valid inequalities are based on the notion of k-envelopes and can be computed off-line using polynomial-time algorithms and added to the MIP program all at once. Furthermore, they are equally useful to boost the strengthening of the Big-Ms and the screening rate of superfluous constraints. We apply our procedures to a probabilistically constrained version of the DC optimal power flow problem with uncertain demand. The chance constraint requires that the probability of violating any of the power system's constraints be lower than some positive threshold. In a series of numerical experiments that involve five power systems of different size, we show the efficiency of the proposed methodology and compare it with some of the best performing convex inner approximations currently available in the literature. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported in part by the European Research Council under the EU Horizon 2020 research and innovation program [Grant 755705], in part by the Spanish Ministry of Science and Innovation [Grant AEI/10.13039/501100011033] through project PID2020-115460GB-I00, and in part by the Junta de Andalucía and the European Regional Development Fund through the research project P20_00153. Á. Porras is also financially supported by the Spanish Ministry of Science, Innovation and Universities through the University Teacher Training Program with fellowship number FPU19/03053. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.0302. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the maximizing of the probability P ζ ∣ ζ ∈ K (x) over a closed and convex set X , a special case of the chance-constrained optimization problem. Suppose K (x) ≜ ζ ∈ K ∣ c (x , ζ) ≥ 0 , and ζ is uniformly distributed on a convex and compact set K and c (x , ζ) is defined as either c (x , ζ) ≜ 1 - ζ T x m where m ≥ 0 (Setting A) or c (x , ζ) ≜ T x - ζ (Setting B). We show that in either setting, by leveraging recent findings in the context of non-Gaussian integrals of positively homogenous functions, P ζ ∣ ζ ∈ K (x) can be expressed as the expectation of a suitably defined continuous function F (∙ , ξ) with respect to an appropriately defined Gaussian density (or its variant), i.e. E p ~ F (x , ξ) . Aided by a recent observation in convex analysis, we then develop a convex representation of the original problem requiring the minimization of g E F (∙ , ξ) over X , where g is an appropriately defined smooth convex function. Traditional stochastic approximation schemes cannot contend with the minimization of g E F (∙ , ξ) over X , since conditionally unbiased sampled gradients are unavailable. We then develop a regularized variance-reduced stochastic approximation (r-VRSA) scheme that obviates the need for such unbiasedness by combining iterative regularization with variance-reduction. Notably, (r-VRSA) is characterized by almost-sure convergence guarantees, a convergence rate of O (1 / k 1 / 2 - a) in expected sub-optimality where a > 0 , and a sample complexity of O (1 / ϵ 6 + δ) where δ > 0 . To the best of our knowledge, this may be the first such scheme for probability maximization problems with convergence and rate guarantees. Preliminary numerics on a portfolio selection problem (Setting A) and a set-covering problem (Setting B) suggest that the scheme competes well with naive mini-batch SA schemes as well as integer programming approximation methods. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

In ultra-dense networks (UDNs), the dense deployment of base stations (BSs) is facing challenges due to the pronounced unbalanced traffic loads, severe inter-cell interference, and uncertain traffic demands. In this paper, we tame traffic uncertainty for the joint optimization of BS activation and user association in UDNs to mitigate interference and balance traffic loads among BSs. Specifically, we address the traffic uncertainty by using chance constraint programming with the known first- and second-order statistics of the uncertain traffic. We formulate the joint BS activation and user association problem as a mixed integer non-linear programming problem, which is then decomposed into a set of user association sub-problems by modeling the BS states (active or idle) as a Markov chain. We solve the user association sub-problem at each BS state by transforming it into a convex problem over the positive orthant. In particular, at each BS state, the candidate serving BSs that lead to the optimal load balancing performance are identified for each user and parts of the user’s traffic are offloaded to the identified BSs. Based on the obtained solutions, we propose a distributed near-optimal BS activation and user association scheme. Numerical results demonstrate that our proposed scheme is more robust to traffic uncertainty and provides better load-balancing performance than the existing schemes. [ABSTRACT FROM AUTHOR]

This paper presents a scalable method for improving the solutions of ac optimal power flow (AC OPF) with respect to deviations in predicted power injections from wind and other uncertain generation resources. The aim of this paper is on providing solutions that are more robust to short-term deviations, and that optimize both the initial operating point and a parametrized response policy for control during fluctuations. We formulate this as a chance-constrained optimization problem. To obtain a tractable representation of the chance constraints, we introduce a number of modeling assumptions and leverage recent theoretical results to reformulate the problem as a convex, second-order cone program, which is efficiently solvable even for large instances. Our experiments demonstrate that the proposed procedure improves the feasibility and cost performance of the OPF solution, while the additional computation time is on the same magnitude as a single deterministic AC OPF calculation. [ABSTRACT FROM AUTHOR]

The path planning methods based on nonconvex constrained optimization, such as mixed-integer linear programming (MILP), have found various important applications, ranging from unmanned aerial vehicles (UAVs) and autonomous underwater vehicles (AUVs) to space vehicles. Moreover, their stochastic extensions have enabled risk-aware path planning, which explicitly limits the probability of failure to a user-specified bound. However, a major challenge of those path planning methods is constraint violation between discrete time steps. In the existing approach, a path is represented by a sequence of waypoints and the safety constraints (e.g., obstacle avoidance) are imposed on waypoints. Therefore, the trajectory between waypoints could violate the safety constraints. A naive continuous-time extension results in unrealistic computation cost. In this paper, we propose a novel approach to ensure constraint satisfaction between waypoints without employing a continuous-time formulation. The key idea is to enforce that the same inequality constraint is satisfied on any two adjacent time steps, under assumptions of polygonal obstacles and straight line trajectory between waypoints. The resulting problem encoding is MILP, which can be solved efficiently by commercial solvers. Thus, we also introduce novel extensions to risk-allocation path planners with improved scalability for real-world scenarios and run-time performance. While the proposed encoding approach is general, the particular emphasis of this paper is placed on the chance-constrained, nonconvex path-planning problem (CNPP). We provide extensive simulation results on CNPP to demonstrate the path safety and scalability of our encoding and related path planners. [ABSTRACT FROM AUTHOR]

Forecast uncertainties pose a considerable challenge to the success of model predictive control (MPC) in buildings. Numerous possibilities for considering forecast uncertainties in MPCs are available, but an in-depth comparison is lacking. This paper compares two main approaches to consider uncertainties: robust and stochastic MPC. They are benchmarked against a deterministic MPC and an MPC with perfect forecast. The MPCs utilize a holistic building model to reflect modern smart homes that include photovoltaic power generation and storage, thermally controlled loads, and smart appliances. Real-world data are used to identify the thermal building model. The performance of the various controllers is investigated under three levels of uncertainty for two building models with different envelope performance. For the highly insulated building, the deterministic MPC achieves satisfactory thermal comfort when the forecast error is medium or low, but the thermal comfort is compromised for high forecast errors. In the poorly insulated building, thermal comfort is compromised at medium and high forecast errors. Compared to the deterministic MPC, the robust MPC increases the electricity cost by up to 4.5% and provides complete temperature constraint satisfaction while the stochastic MPC increases the electricity cost by less than 1% and fulfills the thermal comfort requirements. [ABSTRACT FROM AUTHOR]

The rapid development of microgrids (MGs) with various prosumers promotes the accommodation of renewable distributed generation (RDG) and provides platforms for local energy sharing among prosumers. However, the operational uncertainties pose enormous challenges to the day-ahead bidding of MGs in the wholesale electricity market and there is an urgent need for a local market to facilitate the local energy sharing. Thus, this paper proposes a risk-averse day-ahead bidding strategy for MGs with full consideration of the multiple uncertainties originating from the wholesale electricity market, RDG and loads. Based on the transactive energy (TE) sharing concept, the local market is formulated as a Stackelberg game (SG) to effectively capture the strategic interaction among the MG and prosumers, where a distributed iterative algorithm with a bisection approach that only requires exchanging TE-related information is adopted to achieve the SG equilibrium without compromising privacy concerns. To handle the uncertainties of RDG and loads, the power balances are formulated as chance constraints and a data-driven quantile forecasting method is developed for achieving the computational tractability of chance constraints without any prior knowledge or probability distribution assumptions. Furthermore, a risk criterion of the conditional value-at-risk is incorporated in the day-ahead bidding model of MGs for risk aversion towards uncertainties of the wholesale electricity market. The effectiveness of the proposed solution is extensively demonstrated through numerical simulation. • A risk-averse day-ahead bidding strategy of TE sharing MGs is presented. • A SG-based local TE market with privacy protection is developed. • The wholesale market uncertainties are addressed in a risk-averse manner via CVaR. • The power balance chance constraints are built to handle RDG and load uncertainties. • A data-driven quantile forecasting method is proposed to solve chance constraints. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic lot sizing problem with probabilistic carbon emission constraints under the static uncertainty strategy and α service level constraints. We consider both the periodic and global carbon emission constraints and guarantee that the carbon emission constraints are met with the desired probabilities by using chance constraints. Though this problem can be observed in many real life production systems today, to the best of our knowledge, this is the first study on this problem. We formulate the problems as mixed integer linear programs and solve them by a commercial solver. In our computational experiments, we observe that the formulations can be solved for quite large problem instances in reasonable times. We compare the probabilistic global and periodic emission constraints according to their effects on the total cost and carbon emissions under different parameter settings, and present insights about the consequences of using these restrictions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, via chance constrained programming formulation and fuzzy membership, we give suggestions on a new fuzzy chance constrained least squares twin support vector machine, which can make data measurement noise efficiently. In this paper, we concentrate on least squares twin support vector machine classification when data distributions are uncertain statistically. The model's function is used to guarantee the small probability of misclassification for the uncertain data, with some known characters of the distribution. The fuzzy chance constrained least squares twin support vector machine model can be transformed into second-order cone programming (SOCP) through the properties of moment information of uncertain data and thus the dual problem of SOCP model is introduced. Besides, through the numerical experiments we also demonstrate the model's performance in real data and artificial data. [ABSTRACT FROM AUTHOR]

This paper proposes a data-driven distributionally robust chance constrained real-time dispatch (DRCC-RTD) considering renewable generation forecasting errors. The proposed DRCC-RTD model minimizes the expected quadratic cost function and guarantees that the two-sided chance constraints are satisfied for any distribution in the ambiguity set. The Wasserstein-distance-based ambiguity set, which is a family of distributions centered at an empirical distribution, is employed to hedge against data perturbations. By applying the reformulation linearization technique (RLT) to relax the quadratic constraints of the worst-case costs and constructing linear reformulations of the DRCCs, the proposed DRCC-RTD model is cast into a deterministic linear programming (LP) problem, which can be solved efficiently by off-the-shelf solvers. Case studies are carried out on a 6-bus system and the IEEE 118-bus system to validate the effectiveness and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e. probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data. [ABSTRACT FROM AUTHOR]

Probability functions appearing in chance constraints are an ingredient of many practical applications. Understanding differentiability, and providing explicit formulae for gradients, allow us to build nonlinear programming methods for solving these optimization problems from practice. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by gas network applications, we investigate differentiability of probability functions acting on non-convex quadratic forms. We establish continuous differentiability for the broad class of elliptical random vectors under mild conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider chance constrained optimization of elliptic partial differential equation (CCPDE) systems with random parameters and constrained state variables. We demonstrate that, under standard assumptions, CCPDE is a convex optimization problem. Since chance constrained optimization problems are generally nonsmooth and difficult to solve directly, we propose a smoothing inner-outer approximation method to generate a sequence of smooth approximate problems for the CCPDE. Thus, the optimal solution of the convex CCPDE is approximable through optimal solutions of the inner-outer approximation problems. A numerical example demonstrates the viability of the proposed approach. [ABSTRACT FROM AUTHOR]

As the share of renewables in the grid increases, the operation of power systems becomes more challenging. The present paper proposes a method to formulate and solve chance-constrained optimal power flow while explicitly considering the full nonlinear ac power flow equations and stochastic uncertainties. We use polynomial chaos expansion to model the effects of arbitrary uncertainties of finite variance, which enables to predict and optimize the system state for a range of operating conditions. We apply chance constraints to limit the probability of violations of inequality constraints. Our method incorporates a more detailed and a more flexible description of both the controllable variables and the resulting system state than previous methods. Two case studies highlight the efficacy of the method, with a focus on satisfaction of the ac power flow equations and on the accurate computation of moments of all random variables. [ABSTRACT FROM AUTHOR]

Voltage control plays an important role in the operation of electricity distribution networks, especially with high penetration of distributed energy resources. These resources introduce significant and fast varying uncertainties. In this paper, we focus on reactive power compensation to control voltage in the presence of uncertainties. We adopt a chance constraint approach that accounts for arbitrary correlations between renewable resources at each of the buses. We show how the problem can be solved efficiently using historical samples analogously to the stochastic quasi-gradient methods. We also show that this optimization problem is convex for a wide variety of probabilistic distributions. Compared to conventional per-bus chance constraints, our formulation is more robust to uncertainty and more computationally tractable. We illustrate the results using standard IEEE distribution test feeders. [ABSTRACT FROM AUTHOR]

At present, electricity generation is responsible for more than a quarter of the greenhouse gas emissions in the US. Integrating significant amounts of renewable energy sources into a power grid is probably the most accessible way to reduce carbon emissions from power grids and slow down climate change. Unfortunately, the most accessible renewable power sources, such as wind and solar, are highly intermittent and thus bring a lot of uncertainty to power grid operation and challenge existing optimization and control policies. The chance-constrained (CC) alternating current optimal power flow (AC-OPF) framework finds the minimum cost of generation dispatch, maintaining the power grid operation within security limits with a prescribed probability. Unfortunately, the AC-OPF problem's chance-constrained extension is non-convex, computationally challenging, and requires knowledge of system parameters and also needs additional assumptions to be made about the behavior of renewable generation probability distribution. Known linear and convex approximations to the above problems, though tractable, are too conservative for operational practice and do not consider uncertainty in system parameters. This paper presents an alternative data-driven approach for solving the stochastic AC-OPF problem, based on Gaussian process regression (GPR) to close this gap. The Gaussian process (GP) approach learns a simple yet non-convex data-driven approximation to the AC power flow equations that can incorporate uncertain inputs. The latter is then used to determine the solution of CC-OPF efficiently, by accounting for both input and parameter uncertainty. The practical efficiency of the proposed approach using different approximations for GP-uncertainty propagation is validated and illustrated using a number of standard IEEE test cases. • Chance constrained optimal power flow based on data-driven approach. • Gaussian process regression for solving stochastic optimal power flow. • Gaussian processes for handling uncertainties in power systems. • Machine learning for power system optimization. [ABSTRACT FROM AUTHOR]

In this paper, the problem of active fault diagnosis (AFD) is investigated for a class of linear stochastic systems subject to chance constraints, where the input signals are specifically designed to enhance diagnosis performance by discriminating between multi-fault modes. As the misdiagnosis probability cannot be expressed in the closed form, a reachable upper bound on it is derived to construct a novel input design criterion for AFD based on the Cauchy–Schwarz divergence. A sufficient condition is provided to decompose the multivariate chance constraints into a set of univariate ones. A three-stage optimization approach is proposed to solve the input design problem incorporating chance constraints for AFD. In the upper stage, the best iteration parameter is determined for subsequent iterative optimization to obtain the least conservative risk allocation scheme. In the middle stage, the total risk of violating the multivariate constraints is optimally allocated to a set of univariate ones through iterative optimization, which can improve diagnosis performance. In the lower stage, a set of univariate chance constraints is equivalently converted into expectation constraints under the allocated risk, and then the input design problem under the constructed design criterion is solved to global optimality. The optimal input signals are injected, and the AFD decision is made based on real-time measurement information according to the minimum probability of misdiagnosis decision principle. Finally, simulation results on a four-tank system demonstrate the effectiveness and superiority of the proposed AFD method. [ABSTRACT FROM AUTHOR]

In this paper, integrated operation management of cooperative microgrids is formulated in the framework of stochastic predictive control. In the proposed scheme, a joint probabilistic constraint on the microgrids power exchange with the main grid couples operation of individual microgrids. In order to tackle the coupling constraint, a cooperative energy management strategy is proposed in which based on the statistical characteristics of uncertain parameters, the deterministic counterpart of the problem is derived and an efficient solution strategy is achieved. The proposed strategy is evaluated for an illustrative test case including two microgrids based on modified CIGRE benchmark. Moreover, statistical analysis is conducted to evaluate robustness characteristics of the solution strategy. [ABSTRACT FROM AUTHOR]

In this paper, a new finite-horizon H ∞ control problem is considered for a class of time-varying systems with stochastic nonlinearities, measurements degradation and chance constraints. The purpose of the addressed problem is to design the time-varying controller such that the closed-loop system satisfies the prespecified H ∞ disturbance attenuation requirement and certain chance constraints on the controlled output vector z k (i.e., the probability of the controlled output z k belonging to a given set is larger than a prescribed value). A modified maximum-volume-inscribed-ellipsoid (MVIE) method is employed to convert the chance constraint into some tractable inequalities, which could be conveniently handled by the recursive linear matrix inequality (RLMI) approach. Then, by using stochastic control analysis method, sufficient conditions are derived for the existence of the desired multi-objective controller and, furthermore, the gains of the desired controllers are characterized by means of RLMIs. Finally, two illustrative examples and a practical system are proposed to highlight the effectiveness and applicability of the presented controller design technology. [ABSTRACT FROM AUTHOR]

• Techno-economic analysis of battery storage systems providing frequency control. • A stochastic, data-driven optimisation algorithm to optimise the battery controller. • Use of frequency data and detailed, yet computationally efficient battery models. • Case study of Germany shows the highest margins for a battery rated at 1.6 MW/1.6 MWh. • Calendar ageing drives battery degradation, while cycle ageing has less impact. Optimal investment in battery energy storage systems, taking into account degradation, sizing and control, is crucial for the deployment of battery storage, of which providing frequency control is one of the major applications. In this paper, we present a holistic, data-driven framework to determine the optimal investment, size and controller of a battery storage system providing frequency control. We optimised the controller towards minimum degradation and electricity costs over its lifetime, while ensuring the delivery of frequency control services compliant with regulatory requirements. We adopted a detailed battery model, considering the dynamics and degradation when exposed to actual frequency data. Further, we used a stochastic optimisation objective while constraining the probability on unavailability to deliver the frequency control service. Through a thorough analysis, we were able to decrease the amount of data needed and thereby decrease the execution time while keeping the approximation error within limits. Using the proposed framework, we performed a techno-economic analysis of a battery providing 1 MW capacity in the German primary frequency control market. Results showed that a battery rated at 1.6 MW, 1.6 MWh has the highest net present value, yet this configuration is only profitable if costs are low enough or in case future frequency control prices do not decline too much. It transpires that calendar ageing drives battery degradation, whereas cycle ageing has less impact. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic variant of the traveling repairman problem with profits in which travel times are random. The introduction of the arrival time in the objective function instead of the travel time, which is common in most vehicle routing problems, poses compelling challenges, emphasized by the incorporation of the stochasticity in travel times and by the presence of profits. A risk-averse perspective is considered in the model, which is then formulated as a nonlinear integer model and heuristically solved by means of a beam search heuristic. Experimental results have been performed on instances adapted from the available deterministic datasets, to show the effectiveness of the solution approach. [ABSTRACT FROM AUTHOR]

• Optimal Bayesian experiment design for parameter inference in constrained nonlinear systems. • Using surrogate models based on arbitrary polynomial chaos for efficient sample-based computation of the expected utility function. • Chance constrained handling in Bayesian experiment design. The optimal design of experiments is crucial for maximizing the information content of data across a wide-range of experimental goals. This paper presents a Bayesian approach to optimal experiment design (OED) for parameter inference in constrained, dynamic, and nonlinear systems under noisy, incomplete, and indirect measurements. Bayesian OED maximizes an expected utility objective, which accounts for prior and posterior uncertainty in the model parameters from an information-theoretic standpoint. Due to the complicated form of the expected utility, it must be estimated using sample-based methods and, in particular, a nested Monte Carlo estimator that is expensive to evaluate using the full dynamic model. We propose a novel surrogate model based on arbitrary polynomial chaos (aPC), which readily applies to any type of prior distribution. The aPC expansions are constructed locally at each design visited during the iterative optimization procedure. The main cost in aPC, which is the determination of the expansion coefficients, is minimized by estimating these coefficients from only a minimal set of dynamic model evaluations. Although sample-based estimators can also be applied to the chance constraints, this leads to a potentially large number of binary variables, such that a smooth moment-based approximation is preferred in this work. Numerical simulations indicate that the proposed surrogate can significantly lower the computational cost of the Bayesian OED, while guaranteeing the original chance constraints are satisfied without noticeably increasing the average time to find a solution. As such, this methodology appears to have the potential to pave the way for real-time or sequential dynamic experiment design in a fully Bayesian setting. [ABSTRACT FROM AUTHOR]

Highlights • A novel data-driven approach is proposed for stochastic model predictive control. • Support vector clustering is adopted to learn an uncertainty set from scenarios. • A calibration procedure is developed to provide desirable probabilistic guarantees. • Finally, an uncertainty set-induced robust optimization problem is solved. • The proposed method requires less scenarios and can reduce conservatism. Abstract Stochastic model predictive control (SMPC) has been a promising solution to complex control problems under uncertain disturbances. However, traditional SMPC approaches either require exact knowledge of probabilistic distributions, or rely on massive scenarios that are generated to represent uncertainties. In this paper, a novel scenario-based SMPC approach is proposed by actively learning a data-driven uncertainty set from available data with machine learning techniques. A systematical procedure is then proposed to further calibrate the uncertainty set, which gives appropriate probabilistic guarantee. The resulting data-driven uncertainty set is more compact than traditional norm-based sets, and can help reducing conservatism of control actions. Meanwhile, the proposed method requires less data samples than traditional scenario-based SMPC approaches, thereby enhancing the practicability of SMPC. Finally the optimal control problem is cast as a single-stage robust optimization problem, which can be solved efficiently by deriving the robust counterpart problem. The feasibility and stability issue is also discussed in detail. The efficacy of the proposed approach is demonstrated through a two-mass-spring system and a building energy control problem under uncertain disturbances. [ABSTRACT FROM AUTHOR]

Highlights • The study proposes an advanced control structure in 3 layers aiming the optimal efficiency energy management for Grid-Connected Hybrid Power Plant. • The control structure is based on a Stochastic Model Predictive Control and an optimal finite-state machine to take into account disturbances variations. • Disturbance estimation techniques based on of NAR Neural Networks are applied to have prediction of renewable sources. • The studied control method is compared, through simulation, to a deterministic MPC approach to illustrate an improved performance. Abstract This study presents a complete advanced control structure aimed at the optimal and most efficient energy management for a Grid-Connected Hybrid Power plant. This control scheme is composed of process supervision and process control layers, and it is a possible technology to enable improvements in the energy consumption of industrial systems subject to constraints and process demands. The proposed structure consists of the combination of a Model-Based Predictive Controller, formulated within the Chance Constraints framework to deal with stochastic disturbances (renewable sources, as solar irradiance), an optimal finite-state machine decision system and the use of disturbance estimation techniques for the prediction of renewable sources. The predictive controller uses feedforward compensation of estimated future disturbances, obtained by the use of Nonlinear Auto-Regressive Neural Networks with time delays. The proposed controller aims to perform the management of which energy system to use and to decide where to store energy between multiple storage options. This has to be done while always maximizing the use of renewable energy and optimizing energy generation due to contract rules (maintain maximal economic profit). The proposed method is applied to a case study of energy generation in a sugar cane power plant, with non-dispatchable renewable sources (such as photovoltaic and wind power generation), as well as dispatchable sources (as biomass and biogas). This hybrid power system is subject to operational constraints, as to produce steam in different pressures, sustain internal demands and, imperiously, produce and maintain an amount of electric power throughout each month, defined by strict contract rules with a local Distribution Network Operator (DNO). This paper aims to justify the use of this novel approach to optimal energy generation in hybrid microgrids through simulation, illustrating the performance improvement for different cases. [ABSTRACT FROM AUTHOR]

Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk constraints. This paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous. In particular, we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality. We find that the ambiguous risk constraints in this setting can be recast as a set of second-order cone (SOC) constraints. In order to facilitate the algorithmic implementation, we also derive efficient ways of finding violated SOC constraints. Finally, we demonstrate the theoretical results via computational case studies on power system operations. [ABSTRACT FROM AUTHOR]

The uncertainties originating from electricity markets, renewable distributed generations (RDG) and multi-energy loads pose huge challenges to the optimal energy management of the integrated energy system (IES) in competitive electricity markets. This paper develops an energy management framework (EMF) with a two-stage process for IES strategic participation in electricity markets, which can coordinate the operation of IES to comply with the two-settlement mechanism of the day-ahead and real-time markets. The former stage incorporates the conditional value-at-risk to make day-ahead bidding decisions for risk aversion towards uncertainties introduced by the electricity market. The latter coordinates the day-ahead bidding electricity to implement energy dispatch of IES for real-time market participation. Furthermore, supply-demand balances are formulated into chance constraints in the day-ahead bidding while real-time energy dispatch is implemented based on rolling model predictive control in response to uncertainties due to RDG and multi-energy loads. The proposed EMF is extensively assessed through two case studies in the PJM and Guangdong electricity markets. The numerical results demonstrate that the proposed EMF can effectively achieve day-ahead bidding of IES with risk aversion towards unexpected increases in operating costs caused by market uncertainties and determine the optimal operation scheme for its participation in the real-time market. • The energy management of IES for participating in electricity markets is explored. • A risk-averse day-ahead bidding strategy based on CVaR is developed for IES. • The rolling MPC is adopted for energy dispatch to participate in real-time market. • The uncertainties due to market, RDG and multi-energy loads are fully considered. • The proposed solution is assessed in the PJM and Guangdong electricity markets. [ABSTRACT FROM AUTHOR]

Many practical applications in control require that constraints on the inputs and states of the system are respected, while some performance criterion is optimized. In the presence of model uncertainties or disturbances, it is often sufficient to satisfy the state constraints for at least a prescribed share of the time, such as in building climate control or load mitigation for wind turbines. For such systems, this paper presents a new method of Scenario-Based Model Predictive Control (SCMPC). The basic idea is to optimize 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. Previous SCMPC approaches have suffered from a substantial gap between the rate of constraint violations specified in the optimal control problem and that actually observed in closed-loop operation of the controlled system. This paper identifies the two theoretical explanations for this gap. First, accounting for the special structure of the optimal control problem leads to a substantial reduction of the problem dimension. Second, the probabilistic constraints have to be interpreted as average-in-time, rather than pointwise-in-time. Based on these insights, a novel SCMPC method can be devised for general linear systems with additive and multiplicative disturbances, for which the number of scenarios is significantly reduced. The presented method retains the essential advantages of the general SCMPC approach, namely a low computational complexity and the ability to handle arbitrary probability distributions. Moreover, the computational complexity can be adjusted by a sample-and-remove strategy. [ABSTRACT FROM AUTHOR]

Chance constrained optimal power flow (OPF) has been recognized as a promising framework to manage the risk from variable renewable energy (VRE). In the presence of VRE uncertainties, this paper discusses a distributionally robust chance constrained approximate ac-OPF. The power flow model employed in the proposed OPF formulation combines an exact ac power flow model at the nominal operation point and an approximate linear power flow model to reflect the system response under uncertainties. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed OPF model minimizes the expectation of the quadratic cost function w.r.t. the worst-case probability distribution and guarantees the chance constraints satisfied for any distribution in the ambiguity set. The whole method is data-driven in the sense that the ambiguity set is constructed from historical data without any presumption on the type of the probability distribution, and more data leads to smaller ambiguity set and less conservative strategy. Moreover, special problem structures of the proposed problem formulation are exploited to develop an efficient and scalable solution approach. Case studies are carried out on the IEEE 14 and 118 bus systems to show the accuracy and necessity of the approximate ac model and the attractive features of the distributionally robust optimization approach compared with other methods to deal with uncertainties. [ABSTRACT FROM AUTHOR]

Stochastic model predictive control (SMPC) provides a probabilistic framework for MPC of systems with stochastic uncertainty. A key feature of SMPC is the inclusion of chance constraints, which enables a systematic trade-off between attainable control performance and probability of state constraint violations in a stochastic setting. This paper presents an overview of core concepts in SMPC in relation to MPC and stochastic optimal control, with numerical illustrations on a typical chemical process. Estimation of stochastic disturbances as well as the impact of estimation quality of stochastic disturbances on the SMPC performance are discussed. Some avenues for future research in SMPC are suggested. [ABSTRACT FROM AUTHOR]

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 nonlinearities. This paper proposes a semidefinite relaxation of a chance-constrained AC-OPF, which is able to provide guarantees for 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 24 and 118 bus system using realistic day-ahead forecast data and obtain tight near-global optimality guarantees. [ABSTRACT FROM PUBLISHER]

Higher levels of renewable electricity generation increase uncertainty in power system operation. To ensure secure system operation, new tools that account for this uncertainty are required. In this paper, we adopt a chance-constrained AC optimal power flow formulation, which guarantees that generation, power flows, and voltages remain within their bounds with a predefined probability. We then discuss different chance-constraint reformulations and solution approaches for the problem. We first describe an analytical reformulation based on partial linearization, which enables us to obtain a tractable representation of the optimization problem. We then provide an efficient algorithm based on an iterative solution scheme which alternates between solving a deterministic AC optimal power flow problem and assessing the impact of uncertainty. The flexibility of the iterative scheme enables not only scalable implementations, but also alternative chance-constraint reformulations. In particular, we suggest two sample-based reformulations that do not require any approximation or relaxation of the AC power flow equations. In a case study based on four different IEEE systems, we assess the performance of the method, and demonstrate scalability of the iterative scheme. We further show that the analytical reformulation accurately and efficiently enforces chance constraints in both in- and out-of-sample tests, and that the analytical reformulations outperforms the two alternative, sample-based chance constraint reformulations. [ABSTRACT FROM PUBLISHER]

This paper proposes a data-driven affinely adjustable distributionally robust method for unit commitment considering uncertain load and renewable generation forecasting errors. The proposed formulation minimizes expected total operation costs, including the costs of generation, reserve, wind curtailment, and load shedding, while guaranteeing the system security. Without any presumption about the probability distribution of the uncertainties, the proposed method constructs an ambiguity set of distributions using historical data and immunizes the operation strategies against the worst case distribution in the ambiguity set. The more historical data is available, the smaller the ambiguity set is and the less conservative the solution is. The formulation is finally cast into a mixed integer linear programming whose scale remains unchanged as the amount of historical data increases. Numerical results and Monte Carlo simulations on the 118- and 1888-bus systems demonstrate the favorable features of the proposed method. [ABSTRACT FROM AUTHOR]

The uncertainty associated with renewable energy sources introduces significant challenges in optimal power flow (OPF) analysis. A variety of new approaches have been proposed that use chance constraints to limit line or bus overload risk in OPF models. Most existing formulations assume that the probability distributions associated with the uncertainty are known a priori or can be estimated accurately from empirical data, and/or use separate chance constraints for upper and lower line/bus limits. In this paper, we propose a data driven distributionally robust chance constrained optimal power flow model (DRCC-OPF), which ensures that the worst-case probability of violating both the upper and lower limit of a line/bus capacity under a wide family of distributions is small. Assuming that we can estimate the first and second moments of the underlying distributions based on empirical data, we propose an exact reformulation of DRCC-OPF as a tractable convex program. The key theoretical result behind this reformulation is a second-order cone programming (SOCP) reformulation of a general two-sided distributionally robust chance constrained set by lifting the set to a higher dimensional space. Our numerical study shows that the proposed SOCP formulation can be solved efficiently and that the results of our model are quite robust. [ABSTRACT FROM PUBLISHER]

Electric energy generation from renewable energy sources is generally non-dispatchable due to its intrinsic volatility. Therefore, its integration into electricity markets and in power system operation is often based on volatility-compensating energy storage systems. Scheduling and control of this kind of coupled systems is usually based on hierarchical control and optimization. On the upper level, one solves an optimization problem to compute a dispatch schedule and a coherent allocation of energy reserves. On the lower level, one performs online adjustments of the dispatch schedule using, for example, model predictive control. In the present paper, we propose a formulation of the upper level optimization based on data-driven probabilistic forecasts of the power and energy output of the uncontrollable loads and generators dependent on renewable energy sources. Specifically, relying on probabilistic forecasts of both power and energy profiles of the uncertain demand/generation, we propose a novel framework to ensure the online feasibility of the dispatch schedule with a given security level. The efficacy of the proposed scheme is illustrated by simulations based on real household production and consumption data. [ABSTRACT FROM AUTHOR]

Uncertainties in oil demands have significant effects on the routing and scheduling of oil tankers and inevitably influence the pertinent fleet operational expenses, shortage and excess inventory penalties, and chartering costs. In this paper, we study a stochastic maritime scheduling transportation-inventory problem to transport crude oil from a source using a set of fully loaded heterogeneous vessels to be fully unloaded at a destination over a finite time horizon. Daily demands at the destination are normally distributed random variables and different penalties are imposed on the shortages/excesses in daily storage levels at the destination. The expected total cost of such a fleet operation, consists of the total vessels’ operational expenses, expected total penalties resulting from violating specified lower and upper bounds at the destination storage facility, and chartering expenses. We aim to simultaneously optimize the fleet schedules and maintain desirable daily storage levels to meet the stochastic demand requirements at the destination with acceptable reliability levels. We formulate the problem as a stochastic optimization model and then use chance constrained programming to convert it to an exact mixed-integer nonlinear program with a convex objective function and linear constraints. Solutions obtained via the proposed stochastic model and its deterministic counterpart are compared and analyzed to gain insights into the impact of the variations in demands and the probabilities of meeting demands on the overall operation and cost structure. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Most blind hyperspectral unmixing methods exploit convex geometry properties of hyperspectral data. The minimum volume simplex analysis (MVSA) is one of such methods, which, as many others, estimates the minimum volume (MV) simplex where the measured vectors live. MVSA was conceived to circumvent the matrix factorization step often implemented by MV-based algorithms and also to cope with outliers, which compromise the results produced by MV algorithms. Inspired by the recently proposed robust MV enclosing simplex (RMVES) algorithm, we herein introduce the robust MVSA (RMVSA), which is a version of MVSA robust to noise. As in RMVES, the robustness is achieved by employing chance constraints, which control the volume of the resulting simplex. RMVSA differs, however, substantially from RMVES in the way optimization is carried out. In this paper, we develop a linearization relaxation of the nonlinear chance constraints, which can greatly lighten the computational complex of chance constraint problems. The effectiveness of RMVSA is illustrated by comparing its performance with the state of the art. [ABSTRACT FROM PUBLISHER]

Under-Frequency Load Shedding (UFLS) schemes are the last resort to contain a frequency drop in the grid by disconnecting part of the demand. The allocation methods for selecting feeders that would contribute to the UFLS scheme have traditionally relied on the fact that electric demand followed fairly regular patterns, and could be forecast with high accuracy. However, recent integration of Distributed Generation (DG) increases the uncertainty in net consumption of feeders which, in turn, requires a reformulation of UFLS-allocation methods to account for this uncertainty. In this paper, a chance-constrained methodology for selecting feeders is proposed, with mathematical guarantees for the disconnection of the required amount of load with a certain pre-defined probability. The correlation in net-load forecasts among feeders is explicitly considered, given that uncertainty in DG power output is driven by meteorological conditions with high correlation across the network. Furthermore, this method is applicable either to systems with conventional UFLS schemes (where relays measure local frequency and trip if this magnitude falls below a certain threshold), or adaptive UFLS schemes (where relays are triggered by control signals sent in the few instants following a contingency). Relevant case studies demonstrate the applicability of the proposed method, and the need for explicit consideration of uncertainty in the UFLS-allocation process. • A chance-constrained methodology for selecting feeders contributing towards UFLS is proposed. • Uncertainty and correlation driven by Distributed Generation are explicitly considered. • A distributionally-robust option is given, requiring minimal information on uncertainty. • The method applies to several communication requirements for the operation of the power system. [ABSTRACT FROM AUTHOR]

This paper proposes a day-ahead stochastic scheduling model in electricity markets. The model considers hourly forecast errors of system loads and variable renewable sources as well as random outages of power system components. A chance-constrained stochastic programming formulation with economic and reliability metrics is presented for the day-ahead scheduling. Reserve requirements and line flow limits are formulated as chance constraints in which power system reliability requirements are to be satisfied with a presumed level of high probability. The chance-constrained stochastic programming formulation is converted into a linear deterministic problem and a decomposition-based method is utilized to solve the day-ahead scheduling problem. Numerical tests are performed and the results are analyzed for a modified 31-bus system and an IEEE 118-bus system. The results show the viability of the proposed formulation for the day-ahead stochastic scheduling. Comparative evaluations of the proposed chance-constrained method and the Monte Carlo simulation (MCS) method are presented in the paper. [ABSTRACT FROM PUBLISHER]

• A distributed optimization model is proposed for the integrated electricity-natural gas networks. • The proposed model uses boundary consistency constraints to decouple the integrated networks. • P2G and gas turbines are used to deal with uncertainty risks. • P2G and gas storage tanks serve as distributed gas sources for the natural gas network. • The proposed model uses ATC to obtain a rapidly distributed solution. In recent years, the penetration of decentralized wind power generation in distribution networks has increased rapidly, and the uncertainties in wind power generation has posed great challenges on the operation of distribution networks. To this end, this paper establishes a distributed coordinated optimization model of integrated electricity-gas distribution networks considering power-to-gas (P2G) units and gas turbines to deal with the uncertainties of wind power output. First, this paper uses chance constraints to deal with the uncertainties of wind power output and converts it into a linear model that is easy to be solved. Secondly, considering the electricity distribution network and gas distribution network as different stakeholders, optimization models are established with the objectives of minimizing their respective operating costs, and the established models are solved in a distributed manner through the analytical target cascading (ATC). Finally, an integrated electricity-gas distribution system composed of the IEEE 33-node distribution network and 24-node natural gas network is used for simulation, which validate the proposed model and method. [ABSTRACT FROM AUTHOR]

Evolving power systems with increasing levels of stochasticity call for a need to solve optimal power flow problems with large quantities of random variables. Weather forecasts, electricity prices, and shifting load patterns introduce higher levels of uncertainty and can yield optimization problems that are difficult to solve in an efficient manner. Solution methods for single chance constraints in optimal power flow problems have been considered in the literature, ensuring single constraints are satisfied with a prescribed probability; however, joint chance constraints, ensuring multiple constraints are simultaneously satisfied, have predominantly been solved via scenario-based approaches or by utilizing Boole's inequality as an upper bound. In this paper, joint chance constraints are used to solve an AC optimal power flow problem while preventing overvoltages in distribution grids under high penetrations of photovoltaic systems. A tighter version of Boole's inequality is derived and used to provide a new upper bound on the joint chance constraint, and simulation results are shown demonstrating the benefit of the proposed upper bound. The new framework allows for a less conservative and more computationally efficient solution to considering joint chance constraints, specifically regarding preventing overvoltages. [ABSTRACT FROM AUTHOR]

To facilitate the integration of variable renewable generation and improve the resilience of electricity supply in a microgrid, this paper proposes an optimal scheduling strategy for microgrid operation considering constraints of islanding capability. A new concept, probability of successful islanding (PSI), indicating the probability that a microgrid maintains enough spinning reserve (both up and down) to meet local demand and accommodate local renewable generation after instantaneously islanding from the main grid, is developed. The PSI is formulated as mixed-integer linear program using multi-interval approximation taking into account the probability distributions of forecast errors of wind, PV and load. With the goal of minimizing the total operating cost while preserving user specified PSI, a chance-constrained optimization problem is formulated for the optimal scheduling of mirogrids and solved by mixed integer linear programming (MILP). Numerical simulations on a microgrid consisting of a wind turbine, a PV panel, a fuel cell, a micro-turbine, a diesel generator and a battery demonstrate the effectiveness of the proposed scheduling strategy. The relationship between PSI and various factors are verified. [ABSTRACT FROM AUTHOR]

In this paper, we study data-driven chance constrained stochastic programs, or more specifically, stochastic programs with distributionally robust chance constraints (DCCs) in a data-driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters. We consider a family of density-based confidence sets based on a general $$\phi $$ -divergence measure, and formulate DCC from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set. We derive an equivalent reformulation for DCC and show that it is equivalent to a classical chance constraint with a perturbed risk level. We also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general $$\phi $$ -divergence measures. In several special cases, our results can be strengthened such that we can derive closed-form expressions for the perturbed risk levels. In addition, we show that the conservatism of DCC vanishes as the size of historical data goes to infinity. Furthermore, we analyze the relationship between the conservatism of DCC and the size of historical data, which can help indicate the value of data. Finally, we conduct extensive computational experiments to test the performance of the proposed DCC model and compare various $$\phi $$ -divergence measures based on a capacitated lot-sizing problem with a quality-of-service requirement. [ABSTRACT FROM AUTHOR]

Due to rapid growth of population and development of economy, water resources allocation problems have aroused wide concern. Therefore, optimization of water resources systems is complex and uncertain, which is a severe challenge faced by water managers. In this paper, a factorial multi-stage stochastic programming with chance constraints approach is developed to deal with the issues of water-resources allocation under uncertainty and risk as well as their interactions. It can deal with uncertainties described as both interval numbers and probability distributions, and can also support the risk assessment within a multistage context. The solutions associated with different risk levels of constraint violation can be obtained, which can help characterize the relationship between the economic objective and the system risk. The inherent interactions between factors at different levels and their effects on total net benefits can be revealed through the analysis of multi-parameter interactions. [ABSTRACT FROM AUTHOR]