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The concept of Urban Air Mobility (UAM) proposes to use revolutionary new electrical vertical takeoff and landing (eVTOL) aircraft to provide efficient and on-demand air transportation service between places previously underserved by the current aviation market. A key challenge for the success of UAM is how to manage large-scale autonomous flight operations with safety guarantee in high-density, dynamic and uncertain airspace environments. In this paper, a safety assured decentralized online guidance algorithm with airborne self-separation capability is proposed and analyzed for multi-aircraft autonomous flight operations under uncertainties. The problem is formulated as a multi-agent Markov Decision Process with continuous action space and is solved by a customized decentralized online algorithm based on Monte Carlo Tree Search (MCTS). To guarantee the safety of real-time autonomous flight operations in uncertain environments, the formulation of loss of chance constrained separation is introduced and integrated with the proposed MCTS algorithm. In addition, Gaussian process regression along with Bayesian optimization is employed to discretize the continuous action space, which helps shorten the flight time. A comprehensive numerical study shows that the proposed algorithm can provide safe onboard guidance with guaranteed low near mid-air collision probability in uncertain and high-density airspace environments. [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]

Wireless powered communication (WPC) has been considered as one of the key technologies in the Internet of Things (IoT) applications. In this paper, we study a wireless powered time-division duplex (TDD) multiuser multiple-input multiple-output (MU-MIMO) system, where the base station (BS) has its own power supply and all users can harvest radio frequency (RF) energy from the BS. We aim to maximize the users’ information rates by jointly optimizing the duration of users’ time slots and the signal covariance matrices of the BS and users. Different to the commonly used sum rate and max-min rate criteria, the proportional fairness of users’ rates is considered in the objective function. We first study the ideal case with the perfect channel state information (CSI), and show that the non-convex proportionally fair rate optimization problem can be transformed into an equivalent convex optimization problem. Then we consider practical systems with imperfect CSI, where the CSI mismatch follows a Gaussian distribution. A chance-constrained robust system design is proposed for this scenario, where the Bernstein inequality is applied to convert the chance constraints into the convex constraints. Finally, we consider a more general case where only partial knowledge of the CSI mismatch is available. In this case, the conditional value-at-risk (CVaR) method is applied to solve the distributionally robust system rate optimization problem. Simulation results are presented to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The enhanced index tracking (EIT) represents a popular investment strategy designed to create a portfolio of assets that outperforms a benchmark, while bearing a limited additional risk. This paper analyzes the EIT problem by the chance constraints (CC) paradigm and proposes a formulation where the return of the tracking portfolio is imposed to overcome the benchmark with a high probability value. Besides the CC-based formulation, where the eventual shortage is controlled in probabilistic terms, the paper introduces a model based on the Integrated version of the CC. Here the negative deviation of the portfolio performance from the benchmark is measured and the corresponding expected value is limited to be lower than a given threshold. Extensive computational experiments are carried out on different set of benchmark instances. Both the proposed formulations suggest investment strategies that track very closely the benchmark over the out-of-sample horizon and often achieve better performance. When compared with other existing strategies, the empirical analysis reveals that no optimization model clearly dominates the others, even though the formulation based on the traditional form of the CC seems to be very competitive. [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 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]

Chance-constrained programs (CCPs) constitute a difficult class of stochastic programs due to its possible nondifferentiability and nonconvexity even with simple linear random functionals. Existing approaches for solving the CCPs mainly deal with convex random functionals within the probability function. In the present paper, we consider two generalizations of the class of chance constraints commonly studied in the literature; one generalization involves probabilities of disjunctive nonconvex functional events and the other generalization involves mixed-signed affine combinations of the resulting probabilities; together, we coin the term affine chance constraint (ACC) system for these generalized chance constraints. Our proposed treatment of such an ACC system involves the fusion of several individually known ideas: (a) parameterized upper and lower approximations of the indicator function in the expectation formulation of probability; (b) external (i.e., fixed) versus internal (i.e., sequential) sampling-based approximation of the expectation operator; (c) constraint penalization as relaxations of feasibility; and (d) convexification of nonconvexity and nondifferentiability via surrogation. The integration of these techniques for solving the affine chance-constrained stochastic program (ACC-SP) is the main contribution of this paper. Indeed, combined together, these ideas lead to several algorithmic strategies with various degrees of practicality and computational efforts for the nonconvex ACC-SP. In an external sampling scheme, a given sample batch (presumably large) is applied to a penalty formulation of a fixed-accuracy approximation of the chance constraints of the problem via their expectation formulation. This results in a sample average approximation scheme, whose almost-sure convergence under a directional derivative condition to a Clarke stationary solution of the expectation constrained-SP as the sample sizes tend to infinity is established. In contrast, sequential sampling, along with surrogation leads to a sequential convex programming based algorithm whose asymptotic convergence for fixed- and diminishing-accuracy approximations of the indicator function can be established under prescribed increments of the sample sizes. [ABSTRACT FROM AUTHOR]

In recent years, with the continuous development of people's income and consumption level, consumers have higher and higher requirements for goods and services. The traditional warehousing-transportation method may lead to the decline of customer satisfaction level due to insufficient supply. Assuming that the demands of customers are unknown, we propose a two-stage distributionally robust optimization model with chance constraints, in which the ambiguity set contains all the probability distribution with the same first and second moments. For the sake of computation, the proposed model is equivalently transformed into a mixed-integer semi-definite programming problem. Since the existing optimization solver is challenging to solve the proposed model, this paper presents a modified primal-dual Benders' decomposition algorithm and proves the convergence of the algorithm. The validity of the proposed model is validated through the study of the storage and transportation problems of a perishable food supply chain in Shanghai. Compared with the non-robust optimization model, the traditional robust optimization model, and the distributionally robust optimization model based on Kullback-Leibler divergence, we show that the customer satisfaction level obtained by our method is improved by 9.1-13.4% on average in the out-of-sample datasets. [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]

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. [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]

To manage renewable generation and load consumption uncertainty, chance-constrained optimal power flow (OPF) formulations have been proposed. However, conventional solution approaches often rely on accurate estimates of uncertainty distributions, which are rarely available in reality. When the distributions are not known but can be limited to a set of plausible candidates, termed an ambiguity set, distributionally robust (DR) optimization can reduce out-of-sample violation of chance constraints. Nevertheless, a DR model may yield conservative solutions if the ambiguity set is too large. In view that most practical uncertainty distributions for renewable generation are unimodal, in this paper, we integrate unimodality into a moment-based ambiguity set to reduce the conservatism of a DR-OPF model. We review exact reformulations, approximations, and an online algorithm for solving this model. We extend these results to derive a new, offline solution algorithm. Specifically, this algorithm uses a parameter selection approach that searches for an optimal approximation of the DR-OPF model before solving it. This significantly improves the computational efficiency and solution quality. We evaluate the performance of the offline algorithm against existing solution approaches for DR-OPF using modified IEEE 118-bus and 300-bus systems with high penetrations of renewable generation. Results show that including unimodality reduces solution conservatism and cost without degrading reliability significantly. [ABSTRACT FROM AUTHOR]

This paper is concerned with the joint chance constraint for a system of linear inequalities. We discuss computationally tractble relaxations of this constraint based on various probability inequalities, including Chebyshev inequality, Petrov exponential inequalities, and others. Under the linear decision rule and additional assumptions about first and second order moments of the random vector, we establish several upper bounds for a single chance constraint. This approach is then extended to handle the joint linear constraint. It is shown that the relaxed constraints are second-order cone representable. Numerical test results are presented and the problem of how to choose proper probability inequalities is discussed. [ABSTRACT FROM AUTHOR]

Traffic control is at the core of research in transportation engineering because it is one of the best practices for reducing traffic congestion. It has been shown in recent years that the traffic control problem involving Lighthill-Whitham-Richards (LWR) model can be formulated as a Linear Programming (LP) problem given that the corresponding initial conditions and the model parameters in the fundamental diagram are fixed. However, the initial conditions can be uncertain when studying actual control problems. This paper presents a stochastic programming formulation of the boundary control problem involving chance constraints, to capture the uncertainty in the initial conditions. Different objective functions are explored using this framework, and the proposed model is validated by conducting case studies for both a single highway link and a highway network. In addition, the accuracy of relaxed optimal results is proved using Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

High penetration of renewable generation poses challenges to power system operation due to its uncertain nature. In droop-controlled microgrids, the voltage volatility induced by renewable uncertainties is aggravated by the high droop gains. This paper proposes a chance-constrained optimal power flow (CC-OPF) problem with power flow routers (PFRs) to better regulate the voltage profile in microgrids. PFR refers to a general type of network-side controller that brings more flexibility to the power network. Comparing with the normal CC-OPF that relies on node power flexibility only, the proposed model introduces a new dimension of control from power network to enhance system performance under renewable uncertainties. Adopting a partial linearization method and an iterative algorithm allows us to address the CC-OPF problem by iteratively solving a subproblem. Since the inclusion of PFRs complicates the subproblem and makes common solvers no longer applicable directly, a semidefinite programming relaxation is used to transform each subproblem into a convex form. The proposed method is verified on a modified IEEE 33-bus system and the results show that PFRs significantly reduce the standard deviations of voltage magnitudes and contribute to mitigating the voltage volatility, which makes the system operate in a more economic and secure way. [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]

This paper presents a chance constrained stochastic model predictive control (SMPC) approach for building climate control under combined parametric and additive uncertainties. The proposed SMPCap approach enables the quantification, and manipulation, of both the mean and covariance of the stochastic system states and inputs. Its enhanced uncertainty anticipation is shown to induce improved thermal comfort in closed-loop simulations compared to the conventional deterministic MPC (DMPC) and the state-of-the-art SMPCa only accounting for additive uncertainties, at the cost of a maximum relative increase in energy use of 21.6% and 4.2%, respectively. By incorporating the SMPCap strategy in an integrated optimal control and design (IOCD) approach, its additional added value for obtaining a more appropriate, yet robust, heat supply system sizing is illustrated. Via simulations, size reductions up to 33.3% are shown to be achievable for a terraced single-family dwelling without increasing thermal discomfort compared to an IOCD approach incorporating DMPC. [ABSTRACT FROM AUTHOR]

Considering the increased trend that power systems are closer to the operating bounds because of the increased demand and new challenges in consideration of gas systems and wind power, this paper investigates long-term voltage stability-constrained integrated electric and gas system optimal scheduling in consideration of wind energy integration. A sufficient condition, which is represented as an explicit function of voltage and injected power, is used to constrain power system long-term voltage stability. Due to bilinear terms in this condition, tightening piecewise McCormick envelope relaxation is used to convert it into convex constraints. The second-order cone programming (SOCP) formulation is employed to represent the operational constraints of the integrated electric and gas system. The loss of wind power probability, representing wind power uncertainties, is established by a chance-constrained programming model, which is transformed into a deterministic optimization model by means of the star-inequality-based extended formulation of sample average approximation. Two test systems, the 9-bus electric system with the 6-node gas system and the IEEE 118-bus electric system with the 40-node gas system, are used to validate the proposed model. [ABSTRACT FROM AUTHOR]

Coking coal is essential for the production of steel, and the quality of this coal significantly contributes to the quality of the produced steel. High quality coking coal has low ash content and a range of properties including volatile matter content and predicted coke strength. The coal is improved by processing after it has been mined. This processing varies and coal from multiple sources is blended. This paper introduces an original mixed integer programming model to maximise the profit of coal blending and processing. The model is computationally efficient and can be implemented at any coal mining and processing operation. The multi-period blending model incorporates stockpiling of raw material, and explicitly captures the geological variability of coal using chance constraints. A case study is evaluated and demonstrates that explicitly modelling geological variability can reduce the risk of breaching product specifications without any revenue loss. The improvement is achievable, without additional cost, by selecting the order that coal is fed into a processing plant. [ABSTRACT FROM AUTHOR]

This paper studies optimal day-ahead scheduling of grid-connected batteries that simultaneously provide three services: 1) load shifting, 2) real-time balancing, and 3) primary frequency control (PFC). The uncertainties of load and frequency are incorporated in the cost-minimizing scheduling problem via chance constraints. The resulting chance-constrained problem is then reformulated into a mixed-integer second-order cone program (MISOCP) that can be solved by commercial solvers. However, it is computationally formidable to obtain the globally optimal solution to the MISOCP due to the big problem size. To obtain a suboptimal solution quickly, a heuristic based on penalty alternating direction method (PADM) is developed to solve the MISOCP. Fixing the integer solution returned by the heuristic, we adopt the duality of the second-order cone program (SOCP) to price the three services in the local market. Theoretical analysis of the market equilibrium, individual rationality, and balanced budget is given. Real-world data of load, frequency, and price in the French grid is used in simulation. The results show that the proposed heuristic is computationally efficient, and the pricing results can guarantee a positive utility for each of the batteries, incentivizing them to provide services. [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 discusses distributionally robust geometric programs with individual or joint chance constraints. Several groups of uncertainty sets are considered: uncertainty sets with first two order moments information; uncertainty sets with known first order or first two order moments information under nonnegative support; uncertainty sets constrained by the Kullback–Leibler divergence with a normal or discrete reference distribution; uncertainty sets constrained by the Wasserstein distance under discrete, full, or nonnegative real-space support; and joint uncertainty sets for the product of random variables. Under each group of uncertainty sets, we find deterministic reformulations of the distributionally robust geometric programs with individual or joint chance constraints. Convexity, solution methods, and relationships of the reformulation programs are discussed. Finally, numerical tests are carried out on a shape optimization problem. [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.

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.

As more renewables are integrated into the power system, capacity expansion planners need more advanced long-term decision-making tools to properly model short-term stochastic production uncertainty and to explore its effects on expansion decisions. We develop a distributionally robust generation expansion planning model, accounting for a family of potential probability distributions of wind forecast error uncertainty. Aiming to include more realistic distributions, we construct more informed moment-based ambiguity sets by adding structural information of unimodality. We include operational-stage unit commitment constraints and model the risk of operational limit violations in two distinct forms: chance and conditional value-at-risk (CVaR) constraints. In both forms, the resulting expansion planning model is a mixed-integer second-order cone program. Using a thorough out-of-sample numerical analysis, we conclude: (i) the distributionally robust chance-constrained generation expansion planning model exhibits a better out-of-sample performance only if sufficiently accurate information about the first- and the second-order moments as well as the mode location of potential distributions is available; (ii) conversely, if such accurate information is unavailable, the distributionally robust CVaR-constrained generation expansion planning model outperforms; (iii) these two models have a similar performance when unimodality information is excluded. [ABSTRACT FROM AUTHOR]

This paper presents a fully risk-aware model predictive control (MPC) framework for chance-constrained discrete-time linear control systems with process noise. Conditional value-at-risk (CVaR) as a popular coherent risk measure is incorporated in both the constraints and the cost function of the MPC framework. This allows the system to navigate the entire spectrum of risk assessments, from worst-case to risk-neutral scenarios, ensuring both constraint satisfaction and performance optimization in stochastic environments. The recursive feasibility and risk-aware exponential stability of the resulting risk-aware MPC are demonstrated through rigorous theoretical analysis by considering the disturbance feedback policy parameterization. In the end, two numerical examples are given to elucidate the efficacy of the proposed method.

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]

We consider Reliability-based Robust Design Optimization (RRDO) where it is sought to optimize the mean of an objective function while satisfying constraints in probability. The high computational cost of the simulations underlying the objective and constraints strongly limits the number of evaluations and makes this type of problems particularly challenging. The numerical cost issue and the parametric uncertainties have been addressed with Bayesian optimization algorithms which leverage Gaussian processes of the objective and constraint functions. Current Bayesian optimization algorithms call the objective and constraint functions simultaneously at each iteration. This is often not necessary and overlooks calculation savings opportunity. This article proposes a new efficient RRDO Bayesian optimization algorithm that optimally selects for evaluation, not only the usual design variables, but also one or several constraints along with the uncertain parameters. The algorithm relies on a multi-output Gaussian model of the constraints. The coupling of constraints and their separated selection are gradually implemented in three algorithm variants which are compared to a reference Bayesian approach. The results are promising in terms of convergence speed, accuracy and stability as observed on a two, a four and a 27-dimensional problem. [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]

• 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]

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]

Industrial projects are plagued by uncertainties, often resulting in both time and cost overruns. This research introduces an innovative approach, employing Reinforcement Learning (RL), to address three distinct project management challenges within a setting of uncertain activity durations. The primary objective is to identify stable baseline schedules. The first challenge encompasses the multimode lean project management problem, wherein the goal is to maximize a project's value function while adhering to both due date and budget chance constraints. The second challenge involves the chance-constrained critical chain buffer management problem in a multimode context. Here, the aim is to minimize the project delivery date while considering resource constraints and duration-chance constraints. The third challenge revolves around striking a balance between the project value and its net present value (NPV) within a resource-constrained multimode environment. To tackle these three challenges, we devised mathematical programming models, some of which were solved optimally. Additionally, we developed competitive RL-based algorithms and verified their performance against established benchmarks. Our RL algorithms consistently generated schedules that compared favorably with the benchmarks, leading to higher project values and NPVs and shorter schedules while staying within the stakeholders' risk thresholds. The potential beneficiaries of this research are project managers and decision-makers who can use this approach to generate an efficient frontier of optimal project plans. [ABSTRACT FROM AUTHOR]

Uncertainty in classical stochastic programming models is often described solely by independent random parameters, ignoring their dependence on multidimensional features. We describe a novel contextual chance-constrained programming formulation that incorporates features, and argue that solutions that do not take them into account may not be implementable. Our formulation cannot be solved exactly in most cases, and we propose a tractable and fully data-driven approximate model that relies on weighted sums of random variables. We obtain a stochastic lower bound for the optimal value and feasibility results that include convergence to the true feasible set as the number of data points increases, as well as the minimal number of data points needed to obtain a feasible solution with high probability. We illustrate our findings in a vaccine allocation problem and compare the results with a naíve sample average approximation approach. [ABSTRACT FROM AUTHOR]

To avoid the problem of insufficient flexibility of the power grid along the cz railway due to source-load fluctuations, a distributionally robust optimization method considering flexibility is proposed in this paper. Firstly, Wasserstein distance is used to establish the uncertainty set of flexibility demand caused by source-load fluctuation, and a risk-cost model of insufficient flexibility is constructed to couple to the objective function. Combined with the flexibility chance constraint, the flexibility over-limit probability is limited to a certain confidence level in the fuzzy set, and a distributionally robust chance constraint optimization model based on Wasserstein distance is established. Then, the proposed model is transformed into a linear programming problem to be solved by dual theory and inner approximation transformation. Finally, a simulation study of the power grid along the CZ railway is carried out to verify the correctness and effectiveness of the model. The experimental results show that the proposed method can improve the flexibility capacity of the system without sacrificing economy.