Ellipsoidal Set Research Articles

Robust wind power capacity planning under fuel price uncertainty using conic duality theory and piecewise McCormick relaxation

Increasing the proportion of renewable energy sources (RESs) in power generation is crucial due to fossil fuel depletion and rising environmental pollution. In this regard, identifying the most lucrative sites and sizes for installing wind farms, as one of the fastest-growing types of RESs, is imperative. This paper presents a robust profit-oriented wind power capacity planning (WPCP) considering long- and short-term uncertainty. The model forms a two-stage min-max-min hierarchical structure. The first stage minimizes the investment cost plus maximum regret, while the second stage maximizes the profit under the worst-case uncertainty realization. Unlike the existing approaches where uncertainty in fossil fuel prices is neglected, we model fuel price uncertainty using both polyhedral and ellipsoidal uncertainty sets. In this respect, the third level is formulated as a bi-level program, with the upper level being the profit maximization and the lower level being the locational marginal price (LMP) calculation. In the case of the ellipsoidal set, the conic duality theory is employed to dualize the lower level. The piecewise McCormick relaxation (PMR) technique linearizes the bilinear terms. The nested column-and-constraint generation (NCCG) technique solves the formulated problem. A clarifying case study is employed to demonstrate the efficacy of the proposed model.

Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.

A Robust Optimization Model for Emergency Location Considering the Uncertainty and Correlation of Transportation Network Capacity

Emergencies often lead to the impairment of infrastructure systems, including transportation systems. It is necessary to analyze the uncertainty and correlation of transportation network capacity caused by emergencies, aiming at the problems of emergency facilities’ location and matching in emergency contexts. This study introduces novel concepts, such as flow distribution betweenness centrality (FD-BC) and the transport capacity effect coefficient (TC-EC). Furthermore, we introduce the ellipsoidal uncertainty set to characterize uncertainties in transport capacity. We construct a multi-criteria decision-making (MCDM) model and a multi-strength elitist genetic algorithm (multi-SEGA) to ensure the lower limit of transport capacity between demand and emergency points while minimizing decision-making costs. By designing an uncertain scenario example, we analyze the effect of the perturbation ratio and the uncertainty level on the robust location model. The following results were drawn: (1) Indicators FD-BC and TC-EC effectively indicated the importance of each section in the emergency transportation network. (2) The optimal value of the model’s objective function changed more significantly as the perturbation ratio and uncertainty level increased. (3) After reaching a certain uncertainty level, the robust model with an ellipsoidal uncertainty set became more conservative than the robust model with a box uncertainty set, which lacked practical significance. The research results guarantee the robustness of the emergency support system in uncertain conditions.

TWO APPROACHES TO THE FORMATION OF A QUANTITATIVE MEASURE OF STABILITY BASED ON MULTIPLE ESTIMATES OF THE PARAMETERS OF AN ENSEMBLE OF TRANSIENT PROCESSES

The article is devoted to the further development of the theory of stability of dynamic systems, namely of quantitative methods of stability assessment. A review and critical analysis of various approaches, which allow to introduce a quantitative measure of stability of dynamic systems to one degree or another, is given. The limitations of the existing methods, which are primarily related to the assessment of the behavior of the transient processes of individual trajectories, as well as the difficulty of obtaining an assessment of the behavior of the ensemble of transient processes when trying to apply the methods of N. D. Moiseyev. are substantiated. A method of quantitative assessment of a dynamic system stability based on the numerical estimates of the behavior of the area of initial deviations from the equilibrium position on the trajectories of the dynamic system is substantiated. Based on the Liouville formula, it is shown that changes in the volume of the area of the initial deviations on the trajectories of the system does not depend on the form of the latter one. This allowed to limit the area of initial deviations in the shape of a hypersphere and to obtain a simple expression for a quantitative measure of the stability of a linear stationary dynamic system, the geometric sense of which is to estimate the rate of change of the volume of the control surface. The article proposes and substantiates the criterion of uniformity of deformation of the area of initial deviations. The essence of the problem is that in the transient process, the values of some components of the phase vector may reach unacceptable deviations from the equilibrium position. A theoretical estimate of deformation non-uniformity for linear systems is obtained, which is taken to be the deviation of the trace of the ellipsoid matrix from the deviations of the trace of the hypersphere matrix of the corresponding volume. A method for a quantitative measure of the stability based on an integral quadratic functional calculated on a set of transient processes of initial deviations in the form of a set of ellipsoids with a normalized volume is proposed and substantiated. Diagonal positive normalized matrices are considered as a set of matrices of the integral quadratic criterion. A simple algorithm for calculation of the multiple integral quadratic criterion is proposed.

On the Convex Formulations of Robust Markov Decision Processes

Robust Markov decision processes (MDPs) are used for applications of dynamic optimization in uncertain environments and have been studied extensively. Many of the main properties and algorithms of MDPs, such as value iteration and policy iteration, extend directly to RMDPs. Surprisingly, there is no known analog of the MDP convex optimization formulation for solving RMDPs. This work describes the first convex optimization formulation of RMDPs under the classical sa-rectangularity and s-rectangularity assumptions. By using entropic regularization and exponential change of variables, we derive a convex formulation with a number of variables and constraints polynomial in the number of states and actions, but with large coefficients in the constraints. We further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or entropy-based uncertainty sets, showing that, in these cases, RMDPs can be reformulated as conic programs based on exponential cones, quadratic cones, and nonnegative orthants. Our work opens a new research direction for RMDPs and can serve as a first step toward obtaining a tractable convex formulation of RMDPs. Funding: The work in the paper was supported, in part, by NSF [Grants 2144601 and 1815275]; and Agence Nationale de la Recherche [Grant 11-LABX-0047].

Event-Based Distributed Set-Membership Estimation for Complex Networks Under Deception Attacks

This paper addresses the problem of event-based distributed set-membership estimation for complex networks with unknown but bounded (UBB) disturbances. To reflect the compromised data transmissions in cyber security, deception attacks are taken into consideration. Meanwhile, a novel estimation model is proposed against UBB disturbances. In order to schedule the signal transmissions between nodes and remote estimators, a novel decentralized dynamic periodic event-triggered mechanism (DPETM) with a time-varying threshold is developed for each node of the complex networks, which reduces the waste of communication resources and the complexity of computation. Thereafter, a series of distributed set-membership estimators are designed, whose parameters are explicitly determined in terms of the resolution of a particular linear matrix inequality (LMI) related to the information of the communication topology. An optimized ellipsoid estimation set is obtained by applying a recursive optimization algorithm. Finally, the simulation results are shown to demonstrate the viability of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by set-membership state estimation problem of complex networks in practical missions, such as military, environment, industry, etc. The set-membership estimation of complex networks provides a reliable confidence region for each system node. Event-triggered control is an effective method for the design of set-membership estimator. But the common results require the systems to monitor the measurements point-to-point, which leads to the huge consumption of calculation and communication resources. For this reason, this paper originally extends the DPETM to the discrete-time version from the field of continuous-time systems. Meanwhile, this paper considers the deception attacks in communication channels, and the generic framework established earlier can tackle simultaneously sector-bounded nonlinearity, UBB disturbances, and deception attacks. The main difficulty of this paper lies in the analysis for the sawtooth constraint of periodic samplings. For this difficulty, we introduce a piecewise auxiliary function, which is similar with the loop-function in the field of continuous-time systems. Together with recursive optimization algorithm, the detailed analysis method is proposed for the reliable confidence regions of each set-membership estimator.

The Possibilistic Mean-Variance Model with Uncertain Possibility Distributions

The possibilistic mean–variance (MV) model is the counterpart of Markowitz’s MV model in the possibility theory. This study aims to examine the possibilistic MV model when the possibility distributions of stock returns are uncertain triangular fuzzy numbers. We define an uncertainty vector and use its ellipsoidal uncertainty set in a minimax optimization problem to model this uncertainty. We also show that this minimax optimization problem reduces to a strictly convex minimization problem. Thus, unlike the possibilistic MV model, we get diversified optimal portfolios uniquely with our approach. After laying down the theoretical points of our approach, we illustrate it with a real-world example in the literature by using a software package for convex optimization. To the best of our knowledge, this is the first paper that considers uncertain possibility distributions in the possibilistic MV model.

A robust optimization method for new distribution systems based on adaptive data-driven polyhedral sets

In order to better describe the uncertainty of renewable energy output, this paper proposed a novel robust optimization method for new distribution systems based on adaptive data-driven polyhedral sets. First, an ellipsoidal uncertainty set was established using historical data on renewable energy output, and a data-driven convex hull polyhedral set was established by connecting high-dimensional ellipsoidal vertices; on this basis, an adaptive data-driven polyhedral set model was established to address the problem of high conservatism in the scaling process of convex hull polyhedral sets. Furthermore, a novel adaptive data-driven robust scheduling model for new distribution systems was established, and a column-and-constraint generation (C&amp;amp;CG) algorithm was used to solve the robust scheduling model. Finally, the improved IEEE-33 bus system simulation verification shows that the robust scheduling model for new distribution systems based on adaptive data-driven polyhedral sets can reduce conservatism and improve the robustness of optimization results.

A novel robust evaluation approach to improve systematic behavior of failure safety in water supply system under various ellipsoid uncertainties

This study proposes a novel robust optimization approach for an integrated water supply system, wherein the decision-makers attempt to improve failure safety of system under various uncertainty strategies. To cope with uncertainty, the ellipsoid uncertainty set is assumed to evaluate the best feasible solution in the direction of water supply under various strategies. We assessed the case of Hamoun watershed, a water-stressed watershed in southeastern of Iran, to evaluate the developed robust optimization model. In the following, the comparative feasibility under uncertainty levels is conducted to analyze the impacts of simulation strategies on the status of robust model. Based on the final results, the reliability of the model's objective functions experienced an increasing trend (58.3%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$58.3\\%$$\\end{document}), and the objective function values under the uncertainty strategies is greatly improved. The findings of the analysis show that the robust strategies in response to the failure safety achieve outstanding optimal objectives under uncertainty.

Synthesis of ℋ∞ Control for Descriptor Hybrid Systems with Actuator Saturation

This paper addresses a mode-dependent state-feedback H∞ control for stochastic descriptor hybrid systems, considering both the absence and presence of actuator saturation. Firstly, the necessary and sufficient conditions for the stochastic admissibility criterion with H∞ performance γ of the closed-loop system are proposed. Given the proposed non-convex condition, the author reformulates it into linear matrix inequalities (LMIs). Then, to extend the result to the systems with actuator saturation, the actuator-saturated control input is expressed as a linear combination of a given state-feedback control input and a virtual control input that always remains under the saturation level. To verify this expression, the set invariant condition is also suggested by using the singular mode-dependent Lyapunov function candidate. Therefore, the conditions for the existence of both the mode-dependent state-feedback H∞ control and the ellipsoidal shape invariant sets are successfully derived in terms of LMIs. Two numerical examples demonstrate the effectiveness of the proposed method by solving optimization problems subject to the proposed LMIs that minimize H∞ performance γ and maximize the invariant set, respectively.

A robust optimization approach to a repair shop network planning

Robust optimization model is a paradigm for decision-making under uncertainty, where parameters are given in the form of uncertainty sets. In this paper, we develop a robust optimization model for a repair shop network planning problem. The linear optimization model involves product of uncertain parameters in the constraints. We formulate its robust counterpart with the help of min-max regret and Lagrangian dual approach, considering the partial information of uncertain parameters is given in the form of ellipsoidal and polyhedral uncertainty sets. We also consider ellipsoidal+polyhedral uncertainty set, which is the intersection of ellipsoidal and polyhedral uncertainty sets. We apply the robust optimization model to a bi-objective multi-plant repair shop network planning problem where multiple pieces of equipment are repaired and overhauled using several resources over a multi-period planning horizon. We consider uncertainty in resource and demand parameters. Numerical examples are presented for illustrating the theoretical results.

Adaptive event-triggered reachable set control for Markov jump cyber-physical systems with time-varying delays

&lt;p&gt;In this paper, we proposed a reachable set control method for a class of Markov jump cyber-physical systems (MJCPSs) with time-varying delays, which addressed the challenges posed by false data injection (FDI) attacks to system security. The goal was to find the set of regions where all MJCPSs states were reachable from the origin in the presence of bounded disturbances. The adaptive event-triggered control strategy was introduced to save network resources. It also reduced the impact of FDI attacks and external disturbances on system security. The conservatism of the results were reduced by constructing the Lyapunov-Krasovskii (L-K) functional with time-varying delays. Difference terms were estimated by using the discrete Wirtinger inequality and the improved extended reciprocally convex matrix inequality, and the ellipsoid reachable set of the MJCPS was obtained. Then, the reachable set controller was obtained by linear matrix inequalities (LMIs) solving technique. Finally, an example simulation proved the validity of the results.&lt;/p&gt;

Handling the Constraints in Min-Max MPC

One of the major sources of conservativeness in min-max model predictive control (MPC) is the handling of constraints, where the ellipsoidal robust invariant set is utilized for the proposal of sufficient and conservative conditions for the satisfaction of the constraints. In this article, in order to reduce the conservativeness due to the constraint handling, we add additional relaxation variables to the physical constraints and propose a two-step approach through relaxing the constraints by the amount which is determined by the calculating of the maximal admissible set (MAS). The constraints are relaxed in an iterative manner to avoid the constraints violation, and the constraint relaxation variables are degrees of freedom for relaxing the constraints and improving the control performance. Moreover, we show that under certain circumstance, the physical constraints can be removed without the constraint violation. The proposed approach is shown to be recursively feasibility and its effectiveness is verified through an air conditioning control in a building energy system. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Buildings are account for large percentage of worldwide energy consumptions. One of the most applicable method for optimization of building energy system subject to multiple constraints is the model predictive control (MPC). However, the industrial MPC is usually not recursively feasible, which implies that the optimization problem can become infeasible and the software will be terminated at some time. In order to apply the MPC synthesis approach (MPC with recursive feasibility guarantee), we have to overcome the conservativeness problem due to the handling of constraints. We propose a useful approach in this work by introducing relaxation variables which act as degrees of freedom for improving the control performance, while the physical constraints are still satisfied. The proposed approach is verified through an example of a 24m <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^2$</tex-math> </inline-formula> office room located in Cyber-Physical Energy System (CPES) lab in Western China Science and Technology Innovation Harbour in Xianyang, China. The numerical results show the performance improve of the proposed approach.

Secure State Estimation for Artificial Neural Networks With Unknown-But-Bounded Noises: A Homomorphic Encryption Scheme.

This article is concerned with the secure state estimation problem for artificial neural networks (ANNs) subject to unknown-but-bounded noises, where sensors and the remote estimator are connected via open and bandwidth-limited communication networks. Using the encoding-decoding mechanism (EDM) and the Paillier encryption technique, a novel homomorphic encryption scheme (HES) is introduced, which aims to ensure the secure transmission of measurement information within communication networks that are constrained by bandwidth. Under this encoding-decoding-based HES, the data being transmitted can be encrypted into ciphertexts comprising finite bits. The emphasis of this research is placed on the development of a secure set-membership state estimation algorithm, which allows for the computation of estimates using encrypted data without the need for decryption, thereby ensuring data security throughout the entire estimation process. Taking into account the unknown-but-bounded noises, the underlying ANN, and the adopted HES, sufficient conditions are determined for the existence of the desired ellipsoidal set. The related secure state estimator gains are then derived by addressing optimization problems using the Lagrange multiplier method. Lastly, an example is presented to verify the effectiveness of the proposed secure state estimation approach.

Robustness analysis of uncertain time‐varying systems with unknown initial conditions

SummaryThis paper focuses on the robustness analysis of discrete‐time, linear time‐varying (LTV) systems subject to various uncertainties, such as static and dynamic, time‐invariant and time‐varying, linear perturbations, and unknown initial conditions. The proposed approach is based on integral quadratic constraint theory and allows for a potentially more accurate characterization of the set in which the initial state resides by imposing separate constraints on the initial values of the state variables as opposed to simply requiring the initial state to lie in some ellipsoid. The adopted problem formulation facilitates the analysis of uncertain LTV systems subject to disturbance inputs that are bounded pointwise in time, and the developed results enable determining useful pointwise bounds on the performance outputs given such inputs. The main analysis result is given for eventually periodic nominal systems, which include linear time‐invariant, finite horizon, and periodic systems as special cases. The analysis conditions are expressed as linear matrix inequalities. Two additional results stemming from the main analysis theorem are provided that can be used to determine overapproximated ellipsoidal reachable sets. Finally, the utility of the proposed approach is demonstrated in an illustrative example.

Loss Aversion Robust Optimization Model Under Distribution and Mean Return Ambiguity

From the aspect of behavioral finance, which is an emerging area integrating human behavior into finance, this work studies a robust portfolio problem for loss-averse investors under distribution and mean return ambiguity. A loss-aversion distributionally-robust optimization model is constructed if the return distribution of risky assets is unknown. Then, under the premise that the mean returns of risky assets belong to an ellipsoidal uncertainty set, a model under joint ambiguity in distribution and mean returns is constructed. This study solves both robust models and derives their analytical solutions, respectively. Moreover, the effect of ambiguity aversion and loss aversion on robust optimal portfolio returns is studied. The results show that ambiguity-neutral investors who do not know the return distribution obtain more robust optimal portfolio returns than ambiguity-averse investors who are unaware of both the distribution and mean return. The difference between them decreases with the increase of loss aversion coefficients and increases with ambiguity aversion coefficients. Both loss aversion and ambiguity aversion play important roles in investors&#x2019; behavioral portfolio selection.

Robust Radar Waveform Optimization Under Target Interpulse Fluctuation and Practical Constraints via Sequential Lagrange Dual Approximation

This correspondence considers the problem of robust radar waveform design against target interpulse fluctuation with constraints on the transmit energy, similarity, and signal dynamic range. We aim to make robust the transmit waveform and receive filter with respect to the Target Amplitude Vector (TAV) uncertainties by enhancing the worst-case Signal-to-Noise-Ratio (SNR) over the modeled ellipsoidal set. First of all, the filter is directly solved leveraging on the existence of saddle point condition, which leads to a suitable simplification of the original non-convex problem. Then, to handle the resultant max-min problem, the Sequential Lagrange Dual Approximation (SLDA) procedure is devised. Specifically, we update the transmit waveform via solving a sequence of max-min approximate problems, which are proven to be hidden convex relying on the theory of Lagrange duality. Finally, the filter is synthesized after solving a Quadratical Constraint Quadratic Programming (QCQP) problem based on the optimized waveform. Numerical results highlight the robustness of the proposed design which guarantees an enhanced worst-case SNR with respect to TAV uncertainties.

Existence of solutions to $$\Gamma $$-robust counterparts of gap function formulations of uncertain LCPs with ellipsoidal uncertainty sets

Existence of solutions to $$\Gamma $$-robust counterparts of gap function formulations of uncertain LCPs with ellipsoidal uncertainty sets

Adaptive Set-Membership State Estimation for Nonlinear Systems Under Bit Rate Allocation Mechanism: A Neural-Network-Based Approach.

In this article, the adaptive neural-network-based (NN-based) set-membership state estimation problem is studied for a class of nonlinear systems subject to bit rate constraints and unknown-but-bounded noises. The measurement output signals are transmitted from sensors to a remote estimator via a bit rate constrained communication channel. To relieve the communication burden and ameliorate the state estimation accuracy, a bit rate allocation mechanism is put forward for the sensor nodes by solving a constrained optimization problem. Subsequently, through the NN learning method, an NN-based set-membership estimator is designed to determine an ellipsoidal set that contains the system state, where the proposed estimator relies upon a prediction-correction structure. With the help of the mathematical induction technique and the set theory, sufficient conditions are obtained to ensure the existence of both the adaptive tuning parameters and the set-membership estimators, and then, the corresponding parameters and estimator gains are calculated by solving a set of optimization problems. In addition, the monotonicity of the upper bound on the squared estimation error with respect to the bit rate and the convergence of the NN weight are analyzed, respectively. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed state estimation algorithm.

Robust Bilevel Optimal Dispatch of Park Integrated Energy System Considering Renewable Energy Uncertainty

This paper focuses on optimizing the park integrated energy system (PIES) operation, and a robust bilevel optimal dispatch is proposed. Firstly, the robust uncertainty set is constructed based on the K-means++ algorithm to solve the uncertainty of renewable energy sources output in PIES. Then, the bi-level dispatch model is proposed, with the operator as the leader and consumers as the follower. The upper model establishes an electricity-heat-gas integrated energy network, and the lower model considers the demand response of consumers. Optimizing the pricing strategies of energy sources to determine the output of each energy conversion equipment and the demand response plan. Moreover, analyzing the decision-making process of the robust bi-level model and the solution method is given. Finally, case studies show that the proposed dispatch model can increase operator profits and reduce consumers’ energy costs. The in-sample and out-of-sample simulations demonstrate that the proposed ellipsoid uncertainty set possesses high compactness, good robustness, and low conservatism.