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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]

Distribution system operators (DSOs) world-wide foresee a rapid roll-out of distributed energy resources. From the system perspective, their reliable and cost effective integration requires accounting for their physical properties in operating tools used by the DSO. This paper describes a decomposable approach to leverage the dispatch flexibility of thermostatically controlled loads (TCLs) for operating distribution systems with a high penetration level of photovoltaic resources. Each TCL ensemble is modeled using the Markov decision process (MDP). The MDP model is then integrated with a chance constrained optimal power flow that accounts for the uncertainty of photovoltaic resources. Since the integrated optimization model cannot be solved efficiently by existing dynamic programming methods or off-the-shelf solvers, this paper proposes an iterative spatio-temporal dual decomposition algorithm (ST-D2). We demonstrate the merits of the proposed integrated optimization and ST-D2 algorithm on the IEEE 33-bus test system. [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]

When multiple distributed energy resource (DER) aggregators exist in a non-cooperative power market, the calculation of individual aggregator’s bidding strategies could encounter significant uncertainties for considering DERs and competing market participants’ bidding strategies. In this paper, a bi-level bidding strategy optimization model is proposed for a DER aggregator which utilizes wind power, energy storage system (ESS), and curtailable load. At the upper level, the designated aggregator’s bidding strategy is optimized considering the wind power uncertainty. The wind forecast error is modeled by an ambiguity set using the data-driven approach. The information gap decision theory (IGDT) method is employed in this paper to maximize the risk level the designated aggregator can bear for a certain level of expected payoff. By detecting the worst case in wind power generation, the upper-level model is linearized as an MILP. The designated aggregator submits its bids to the market using the linear utility function acquired from linear regression. At the lower level, the market clearing is carried out using competing market participants’ bidding strategy scenarios. The scenarios and the corresponding probability are modeled through a data-driven approach. The market clearing problem is linearized using Taylor series. The price signal is iterated between the two levels as the proposed bi-level model is solved. Numerical results prove the validity and effectiveness of the proposed IGDT-based method. It is shown that the aggregator can adjust either the bidding quantities or coefficients to reach an expected payoff level. The bidding strategies are affected by uncertainties of wind power and competing bidding strategies. For an expected payoff level, when the designated aggregator is posed to consider a higher risk of wind power uncertainty, the aggregator can only bear a lower risk level from competing bidding strategies and vice versa. [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]

With the fast growth of wind power penetration, power systems need additional flexibility to cope with wind power ramping. Several electricity markets have established requirements for flexible ramping capacity (FRC) reserves. This paper addresses two crucial issues that have rarely been discussed in the literature: 1) how to characterize wind power ramping under different forecast values and 2) how to achieve a reasonable tradeoff between operational risks and FRC costs. Regarding the first issue, this paper proposes a concept of conditional distributions of wind power ramping, which is empirically verified by using simulation and real-world data. For the second issue, this paper develops an adjustable chance-constrained approach to optimally allocate FRC reserves. Equivalent tractable forms of the original problem are devised to improve computational efficiency. Tests carried out on a modified IEEE 118-bus system demonstrate the effectiveness and efficiency of the proposed method. [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]

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]

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]

The resource-constrained project scheduling problem (RCPSP) has been widely studied during the last few decades. In real-world projects, however, not all information is known in advance and uncertainty is an inevitable part of these projects. The chance-constrained resource-constrained project scheduling problem (CC-RCPSP) has been recently introduced to deal with uncertainty in the RCPSP. In this paper, we propose a branch-and-bound (B&B) algorithm and a mixed integer linear programming (MILP) formulation that solve a sample average approximation of the CC-RCPSP. We introduce two different branching schemes and eight different priority rules for the proposed B&B algorithm. The computational results suggest that the proposed B&B procedure clearly outperforms both a proposed MILP formulation and a branch-and-cut algorithm from the literature. [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]

An online optimal control strategy for power flow management in microgrids with on-site battery, renewable energy sources, and integrated electric vehicles (EVs) is presented in this paper. An optimization problem in the form of a mixed integer linear program is formulated. It is executed over a rolling time horizon using predicted values of the microgrid electricity demand, renewable energy generation, EV connection and disconnection times, and the EV state of charge at time of connection. The solution to this optimization problem provides the on-site storage and EV charge/discharge powers. Both bidirectional and unidirectional charging scenarios are considered for EVs. The proposed optimal controller maximizes economic benefits and ensures user-specified charge levels are reached at the time of EV disconnection from the microgrid. By formulating the problem as a stochastic chance constraints optimization, significant improvement is shown in the system robustness over conventional rolling horizon controller, while dealing with uncertainties in the predictions of demand/generation, and EV state of charge and connection/disconnection times. Results of Monte Carlo simulations show that the proposed chance constraints-based controller is highly effective in reducing cost and meeting the user desired EV charge level at time of disconnection from the microgrid, even in the presence of uncertainty. [ABSTRACT FROM PUBLISHER]

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 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 consider the robust secure beamformer design for multiple-input-single-output wiretap channels. Assuming that the eavesdroppers' channels are only partially available at the transmitter, we seek to maximize the secrecy rate under the transmit power and the secrecy rate outage probability constraint. The outage probability constraint requires that the secrecy rate exceed certain thresholds with high probability. Therefore, including such constraint in the design naturally ensures the desired robustness. Unfortunately, the presence of the probabilistic constraints makes the problem nonconvex and, hence, difficult to solve. In this paper, we investigate the outage probability constrained secrecy rate maximization problem using a novel two-step approach. Under a wide range of uncertainty models, our developed algorithms can obtain high-quality solutions, sometimes even exact global solutions, for the robust secure beamformer design problem. Simulation results are presented to verify the effectiveness and robustness of the proposed algorithms. [ABSTRACT FROM PUBLISHER]

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]

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

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]

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]

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]

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

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]

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]

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]

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