Bounded Uncertainty Set Research Articles

Smart charging and discharging of electric vehicles based on multi-objective robust optimization in smart cities

The combination of low-carbon electricity and electric vehicles brings considerable economic and environmental benefits but also introduces challenges due to the complexity and uncertainty of system synergies in smart cities. To fully exploit the advantages of photovoltaic power generation and electric vehicles and to release the potential of electric vehicles as distributed energy storage facilities, this paper develops a multi-objective robust optimization framework that accounts for the benefits of multiple parties of smart charging and discharging systems and depicts a bounded uncertainty set based on partial statistical information from real data. The original model is scalarized and linearized using efficient methods such as max-ordering scalarization and the robust augmented weighted Tchebycheff to facilitate the solution. Moreover, a smart charging and discharging scheduling strategy based on a convergent demand response strategy is proposed to achieve better demand-side management. A case study, based on real data from Car2go and the solar radiation intensity in Portland, Oregon, shows that (1) The methodological framework effectively addresses the smart scheduling issue that considers the interests of multiple parties in complex systems with uncertainty, in contrast to conventional methods. (2) The designed convergent demand response strategy promotes photovoltaic power generation and charging demand synergy and achieves regulation effects such as peak shaving and valley filling. (3) The proposed method and strategy yield good results in multiple aspects, such as charging costs, load regulation, and clean energy utilization, while enhancing the economic and environmental benefits.

A Robust Adaptive Objective Power Allocation in Cognitive NOMA Networks.

Cognitive radio (CR) is a candidate for opportunistic spectrum implementation in wireless communications, allowing secondary users (SUs) to share the spectrum with primary users (PUs). In this paper, a robust adaptive target power allocation strategy for cognitive nonorthogonal multiple access (NOMA) networks is proposed, which involves the maximum transmission power of each SU and interference power threshold under PU constraints. By introducing the signal-to-interference-plus-noise ratio (SINR) adjustment factor, the strategy enables single-station communication to achieve energy efficiency (EE) or high throughput (HT), thus making the target function more flexible. In the same communication scenario, different cognitive users can choose different communication targets that meet their needs. Different QoS can be selected by the same cognitive user at different times. In the case of imperfect channel state information (CSI), semi-infinite (SI) constraints with bounded uncertainty sets are transformed into an optimization problem under the worst case, which is solved by the dual decomposition method. Simulation results show that this strategy has good adaptive selectivity and robustness.

Robust Energy-Efficient Maximization for Cognitive NOMA Networks Under Channel Uncertainties

Resource allocation (RA) is a key technique to guarantee the quality of service of users and maximize the system capacity in cognitive nonorthogonal multiple access (NOMA) networks. However, traditional RA approaches have been proposed under perfect channel state information (CSI) which is too ideal in practical systems due to the impact of link delays, quantization errors, etc. In this article, with imperfect CSI, a downlink robust RA algorithm is proposed for robust transmission to maximize the sum energy efficiency (EE) of secondary users (SUs) under channel uncertainties. Meanwhile, it protects the minimum data rates of SUs, satisfying the maximum transmission power constraints of base stations and the maximum interference temperature constraint of each primary user (PU). The formulated problem is nonconvex, thus challenging to solve. For delay-tolerant services, to deal with the intractability of the problem caused by outage probability constraints, the closed-form expressions of outage probabilities of SUs and PUs are derived under Gaussian CSI error models. For delay-sensitive services, the robust constraints with bounded uncertainty sets are transformed into convex ones. Based on successive convex approximation and the parameter transformation approach, the original problem is converted into a closed-form geometric programming problem solved by dual decomposition methods and subgradient methods. Additionally, users’ outage probabilities and the minimum required transmit power under the two modeling approaches are analyzed. Computational complexity and sensitivity analysis are provided. The simulation results show the proposed algorithm serves good robustness and EE.

Research on VSC-MTDC Robust Optimal Power Flow Based on the Uncertainty of Wind Farm Output

Multi-terminal flexible direct current transmission based on voltage source converter (VSC-MTDC) is an effective measure to solve the problem of wind power consumption. In this paper, a method called robust optimization is proposed to calculate power flow in VSC-MTDC system with uncertain wind power. The model of uncertain output of wind farm is described by bounded uncertainty set. Besides, with differential evolution (DE) algorithm as a framework, reference value of VSC voltage and power control under uncertain wind power are solved by DE fitness evaluation function which comes from the combination of Newton-Raphson (NR) iteration power flow solution method and VSC-MTDC dynamic multi-objective power flow optimization model. At the end of this paper, the validity and superiority of this method is proved by comparison among a number of examples.

Robust Operation of Microgrids via Two-Stage Coordinated Energy Storage and Direct Load Control

This paper proposes a robust optimization approach for optimal operation of microgrids. The uncertain output variation of renewable energy sources (RESs) is addressed by collaboratively scheduling of energy storage (ES) and direct load control (DLC) through a two-stage complementary framework: an hour-ahead charging/discharging of ES and a quarter-hour-ahead activation of DLC. The objective is to maximize the total profit of the microgrid considering operation and maintenance costs of ES units, wind turbines and photovoltaics, and transaction with main grid and customer loads. Assuming the power output of RES randomly varies within a bounded uncertainty set, the problem is modeled to a two-stage robust optimization model and solved by a column-and-constraint generation algorithm. Compared with conventional operation methods, the ES and DLC are coordinated in different time-scales, and RES uncertainties are fully addressed during operation decision-making, ensuring the solutions to be optimal and robust for any realization of uncertainty. The proposed methodology is verified on the IEEE 33-bus distribution system through a wide range of different tests.

On the average performance of the adjustable RO and its use as an offline tool for multi-period production planning under uncertainty

Robust optimization (RO) is a distribution-free worst-case solution methodology designed for uncertain maximization problems via a max-min approach considering a bounded uncertainty set. It yields a feasible solution over this set with a guaranteed worst-case value. As opposed to a previous conception that RO is conservative based on optimal value analysis, we argue that in practice the uncertain parameters rarely take simultaneously the values of the worst-case scenario, and thus introduce a new performance measure based on simulated average values. To this end, we apply the adjustable RO (AARC) to a single new product multi-period production planning problem under an uncertain and bounded demand so as to maximize the total profit. The demand for the product is assumed to follow a typical life-cycle pattern, whose length is typically hard to anticipate. We suggest a novel approach to predict the production plan’s profitable cycle length, already at the outset of the planning horizon. The AARC is an offline method that is employed online and adjusted to past realizations of the demand by a linear decision rule (LDR). We compare it to an alternative offline method, aiming at maximum expected profit, applying the same LDR. Although the AARC maximizes the profit against a worst-case demand scenario, our empirical results show that the average performance of both methods is very similar. Further, AARC consistently guarantees a worst profit over the entire uncertainty set, and its model’s size is considerably smaller and thus exhibit superior performance.

Two-phase stochastic program for transit network design under demand uncertainty

This paper develops a reliability-based formulation for rapid transit network design under demand uncertainty. We use the notion of service reliability to confine the stochastic demand into a bounded uncertainty set that the rapid transit network is designed to cover. To evaluate the outcome of the service reliability chosen, flexible services are introduced to carry the demand overflow that exceeds the capacity of the rapid transit network such designed. A two-phase stochastic program is formulated, in which the transit line alignments and frequencies are determined in phase 1 for a specified level of service reliability; whereas in phase 2, flexible services are determined depending on the demand realization to capture the cost of demand overflow. Then the service reliability is optimized to minimize the combined rapid transit network cost obtained in phase 1, and the flexible services cost and passenger cost obtained in phase 2. The transit line alignments and passenger flows are studied under the principles of system optimal (SO) and user equilibrium (UE). We then develop a two-phase solution algorithm that combines the gradient method and neighborhood search and apply it to a series of networks. The results demonstrate the advantages of utilizing the two-phase formulation to determine the service reliability as compared with the traditional robust formulation that pre-specifies a robustness level.

Open vehicle routing problem with demand uncertainty and its robust strategies

We investigate the open vehicle routing problem with uncertain demands, where the vehicles do not necessarily return to their original locations after delivering goods to customers. We firstly describe the customer’s demand as specific bounded uncertainty sets with expected demand value and nominal value, and propose the robust optimization model that aim at minimizing transportation costs and unsatisfied demands in the specific bounded uncertainty sets. We propose four robust strategies to cope with the uncertain demand and an improved differential evolution algorithm (IDE) to solve the robust optimization model. Then we analyze the performance of four different robust strategies by considering the extra costs and unmet demand. Finally, the computational experiments indicate that the robust optimization greatly avoid unmet demand while incurring a small extra cost and the optimal return strategy is the best strategy by balancing the trade-off the cost and unmet demand among different robust strategies.

Worst-Case Violation of Sampled Convex Programs for Optimization with Uncertainty

A deterministic approach called robust optimization has been recently proposed to deal with optimization problems including inexact data, i.e., uncertainty. The basic idea of robust optimization is to seek a solution that is guaranteed to perform well in terms of feasibility and near-optimality for all possible realizations of the uncertain input data. To solve robust optimization problems, Calafiore and Campi have proposed a randomized approach based on sampling of constraints, where the number of samples is determined so that only a small portion of the original constraints is violated by the randomized solution. Our main concern is not only the probability of violation, but also the degree of violation, i.e., the worst-case violation. We derive an upper bound of the worst-case violation for the sampled convex programs and consider the relation between the probability of violation and the worst-case violation. The probability of violation and the degree of violation are simultaneously bounded by a prescribed value when the number of random samples is large enough. In addition, a confidence interval of the optimal value is obtained when the objective function includes uncertainty. Our method is applicable to not only a bounded uncertainty set but also an unbounded one. Hence, the scope of our method includes random sampling following an unbounded distribution such as the normal distribution.

Robust Channel Estimation and Detection for Single-Carrier and Multicarrier Block Transmission Systems

Block transmission techniques, including single-carrier (SC) and multicarrier (MC) communication techniques, have received much research interest recently for their better ability to handle the intersymbol interference problem than continuous transmission. Numerous detection schemes for SC and MC communication systems have been proposed. While these schemes may be derived from different principles, they usually rely on some initial estimate of the communication channel and/or the covariance matrix of the received signal. However, such estimates usually contain inherent estimation errors to which most existing detection schemes are known to be sensitive. In this paper, we develop robust estimation and detection schemes that explicitly account for channel and covariance matrix estimation errors by optimizing the worst-case performance over properly selected bounded uncertainty sets. Although the prior channel and covariance matrix-estimation errors are generally not bounded, we show that it is beneficial to refine the channel and covariance matrix estimates over properly chosen bounded uncertainty sets centered on the prior channel and covariance matrix estimates. Numerical results show that an improved performance is achieved by using the proposed robust approaches over the ones that ignore the prior estimation errors.

A robust optimisation approach for the milk run problem with time windows with inventory uncertainty: an auto industry supply chain case study

In this paper, we introduce a robust optimisation approach to solve the milk run system with time window with inventory uncertainty. This approach yields routes that minimise transportation costs while satisfying all inventory in a given bounded uncertainty set. The idea of the milk run problem has been used in the context of logistic and supply chain problems in order to manage the transportation of materials. Since the resulted problem formulation is NP-hard, and in order to solve the underlying problem, a novel algorithm entitled robust optimisation has been proposed. We apply the model to solve some numerical examples to show robust solution efficiency versus deterministic. Since the resulted problem illustrates that grows up time in this method is progressive, and in order to solve the large-scale problems, particle swarm optimisation has been proposed. We also observe that the robust solution amounts to a clever management of the remaining vehicle capacity compared to uniformly and non-uniformly distributing this slack over the vehicles.

Robust QoS-Constrained Optimization of Downlink Multiuser MISO Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The knowledge of the channel at the transmit side of a communication system can be exploited by using precoding techniques, from which the overall transmission quality might benefit significantly. However, in practical wireless systems, the channel state information is prone to errors, which sometimes deteriorates the performance drastically. In this paper, we address the problem of robust transceiver design in a downlink multiuser system, with respect to the erroneous channel knowledge at the transmitter. The base station is equipped with an antenna array, while users have single antennas. The transceiver optimization is performed under a set of predefined users' quality-of-service constraints, defined as maximum mean square errors, or minimum signal-to-interference-plus-noise ratios (SINRs), which must be satisfied for all disturbances that belong to given, bounded uncertainty sets. Efficient numerical solutions are obtained using semidefinite programming methods from convex optimization theory. The proposed algorithms are found to outperform related approaches in the literature in terms of the achieved performance, while maintaining low computational complexity. The studied uncertainty models are applicable in mitigating typical errors that emerge as a result of quantization or channel estimation. </para>

On Inexact LPV Control Design of Continuous–Time Polytopic Systems

Robust dynamic output design most often relies on nonconvex problems. This is precisely the case with linear parameter-varying (LPV) control when the measured varying parameters do not exactly fit the real ones. This paper provides a solution in the convex programming framework with the use of linear matrix inequality (LMI) solvers in the case of polytopic parameter dependence. A bounded uncertainty set is defined from the difference between the measured and the true parameters from which LMI stabilization and performance conditions are obtained after bounding. As a desired property, the proposed conditions reduce to the classical LPV ones whenever the true parameters are available for control synthesis.

A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty

In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the Miller–Tucker–Zemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP. Our computational results on benchmark instances and on families of clustered instances show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. This is most pronounced for clustered instances under moderate uncertainty, where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We compare the robust optimization model with classic stochastic VRP models for this problem to illustrate the differences and similarities between them. We also observe that the robust solution amounts to a clever management of the remaining vehicle capacity compared to uniformly and non-uniformly distributing this slack over the vehicles.

Robust approach to channel estimation and detection for OFDM systems

A robust channel estimation and detection scheme has been developed that explicitly accounts for channel estimation error by optimising the worst-case performance over a properly selected bounded uncertainty set. Numerical results show improved performance using the proposed robust approach over the one that ignores the prior estimation errors.

A robust approach to channel estimation and detection for multi-carrier CDMA

We present a robust approach to channel estimation and multiuser detection (MUD) for multi-carrier CDMA by explicitly taking into account prior channel estimation errors. Specifically, robust channel estimates and MUD receivers are obtained by optimizing the worst-case performance over a properly selected bounded uncertainty set. Although the prior channel estimation error is not bounded, it is beneficial to refine the estimate over a properly chosen bounded uncertainty set. Numerical results show that linear MUD detectors resulting from our robust approach yield improved performance over those that ignore the prior estimation errors

Robust multiuser detection for multicarrier CDMA systems

Multiuser detection (MUD) for code-division multiple-access (CDMA) systems usually relies on some a priori channel estimates, which are obtained either blindly or by using training sequences, and the covariance matrix of the received signal, usually replaced by the sample covariance matrix. However, such prior estimates are often affected by errors that are typically ignored in subsequent detection. In this paper, we present robust channel estimation and MUD techniques for multicarrier (MC) CDMA by explicitly taking into account such estimation errors. The proposed techniques are obtained by optimizing the worst case performance over two bounded uncertainty sets pertaining to the two types of estimation errors. We show that although the estimation errors associated with the prior channel estimate and the sample covariance matrix are generally not bounded, it is beneficial to optimize the worst case performance over properly chosen bounded uncertainty sets determined by a parameter called bounding probability. At a slightly higher computational complexity, our proposed robust detectors are shown to yield improved performance over the standard detectors that ignore the prior estimation errors.

Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems

In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set.The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems.