Deterministic Uncertainty Research Articles

A Robust Optimization Approach for Active and Reactive Power Management in Smart Distribution Networks Using Electric Vehicles

This paper presents a robust framework for active and reactive power management in distribution networks using electric vehicles (EVs). The method simultaneously minimizes the energy cost and the voltage deviation subject to network and EVs constraints. The uncertainties related to active and reactive loads, required energy to charge EV batteries, charge rate of batteries and charger capacity of EVs are modeled using deterministic uncertainty sets. First, based on duality theory, the max min form of the model is converted to a max form. Second, Benders decomposition is employed to solve the problem. The effectiveness of the proposed method is demonstrated with a 33-bus distribution network.

Coordinated Beamforming With Artificial Noise for Secure SWIPT Under Non-Linear EH Model: Centralized and Distributed Designs

This paper investigates the artificial noise (AN)-aided multi-cell coordinated beamforming (MCBF) for secure simultaneous wireless information and power transfer in both centralized and distributed manners. The proposed transmit design is formulated into a power-minimization problem to guarantee the authorized users’ information and energy harvesting (EH) requirements while avoiding the information interception by unauthorized users. Power splitting receiver architecture and the non-linear EH model are employed. Both perfect and imperfect channel state information (CSI) cases are considered. For the perfect CSI case, the non-robust design is presented by applying semi-definition relaxation (SDR). When no user harvests energy, the global optimum is guaranteed, and when some users harvest energy, approximate global optimum is achieved. For the imperfect CSI case, the worst-case robust design under the deterministic uncertainty channel model is studied, where a solving approach based on SDR and S-procedure is proposed, and the statistically robust design under the stochastic uncertainty channel model is also studied, where an upper bound to the global optimum is obtained by using SDR and Bernstein-type inequality. We further propose a distributed AN-aided MCBF design framework by using an alternating direction method of multipliers for the non-robust, worst-case robust, and statistically robust designs, with which each BS is able to optimize its own transmit design with the local CSI. Simulation results demonstrate our theoretical analysis, which show that our proposed distributed algorithm converges to the optimal results obtained by the centralized one. It also shows that employing the non-linear EH model is able to avoid false output power and save power consumption at the BSs.

Facility layout problem with QAP formulation under scenario-based uncertainty

ABSTRACTThe facility layout problem is usually treated as a deterministic problem and uncertainty regarding problem parameters has seldom been addressed. This study aims to investigate different ways of dealing with uncertainty to design a facility layout which attains robust and efficient performance under a finite number of possible scenarios. For this purpose, several mathematical models based on the quadratic assignment problem (QAP) formulation are developed. These formulations cover alternative approaches in stochastic programming and robust optimization literature such as: minimizing expected cost, maximum cost and maximum regret. Proposed models are solved using genetic algorithms incorporating operators and schemes that are specially selected and adapted for the models. Finally, a novel approach, where the optimization problem under scenario-based uncertainty is transformed into a multi-objective optimization problem by considering each scenario as a separate objective, is proposed. By solving the multi-objective counterpart of scenario-based QAP (mQAP), optimal solutions with respect to different robust performance measures can be obtained simultaneously in a Pareto optimal set. A multi-objective evolutionary algorithm is developed to solve the mQAP. Extensive numerical analysis enables comparison of the performance of these approaches and provides important insights about dealing with uncertainty in the facility layout problem.

Towards more resilient integrated power grid capacity expansion: A robust optimization approach with operational flexibility

This paper proposes a multi-period two-stage adaptive robust optimization model for long term power grid capacity expansion in an environmentally conscious way under generator output uncertainties due to natural variations and generator disruptions. In the first stage, the model plans generator addition and transmission line setup prior to, and robust to, uncertainty realization. In the second stage, it plans power generation and dispatch after, and flexible to, uncertainty realization. The model exploits the idea of strategic robustness and operational flexibility as a way to improve performance in the face of uncertainty. The robust optimization framework uses deterministic uncertainty sets, with parameters that permit control over the level of conservatism of the solution. Because the resulting two-stage model is semi-infinite and, therefore, intractable, it is reformulated as an affinely adjustable counterpart. The reformulation uses affine decision rules on second-stage variables and converts, after constraint processing, the semi-infinite model into a finite single-stage mixed integer programming model. The resulting model is tested on the IEEE 30-bus system and value gains are shown by comparing the performance of the model with a deterministic model and a stochastic programming model with varying risk aversions.

A stochastic method to propagate uncertainties along large cores deterministic calculations

Deterministic uncertainty propagation methods are certainly powerful and time-sparing but their access to uncertainties related to the power map remains difficult due to a lack of numerical convergence. On the contrary, stochastic methods do not face such an issue and they enable a more rigorous access to uncertainty related to the PFNS. Our method combines an innovative transport calculation chain and a stochastic way of propagating uncertainties on nuclear data: first, our calculation scheme consists in the calculation of assembly self-shielded cross sections and a pin-by-pin flux calculation on the whole core. Validation was done and the required CPU time is suitable to allow numerous calculations. Then, we sample nuclear cross sections with consistent probability distribution functions with a correlated optimized Latin Hypercube Sampling. Finally, we deduce the power map uncertainties from the study of the output response functions. We performed our study on the system described in the framework of the OECD/NEA Expert Group in Uncertainty Analysis in Modelling. Results show the 238U inelastic scattering cross section, the 235U PFNS, the elastic scattering cross section of 1H and the 56Fe cross sections as major contributors to the total uncertainty on the power map: the power tilt between central and peripheral assemblies using COMAC-V2 covariance library amounts to 5.4% (1σ) (respectively 7.4% (1σ) using COMAC-V0).

Robust Consensus of Linear Feedback Protocols Over Uncertain Network Graphs

In this paper, we study the robust consensus problem for a group of linear discrete-time or continuous-time agents to coordinate over an uncertain communication network, which is to achieve consensus against transmission errors and noises. We model the network by communication links subject to deterministic uncertainties, which can be additive perturbations described by either some unknown transfer functions or norm bounded matrices. We show that the robust consensus problem can generally be solved by solving a simultaneous $H_\infty$ control problem for a set of low-dimensional subsystems. We also derive necessary conditions for the existence of a protocol achieving robust consensus. The results show that for discrete-time agents the uncertainty size must not exceed the inverse of the Mahler measure of the agents, while for continuous-time agents it must be less than the unity. Sufficient conditions in terms of linear matrix inequalities are further presented to design the robust consensus protocols.

Offset-Free Robust MPC of Systems with Mixed Stochastic and Deterministic Uncertainty

This paper presents a robust model predictive control (MPC) approach for offset-free tracking of piece-wise constant references in the presence of bounded deterministic and stochastic disturbances. The system is considered to be linear with two sources of additive bounded uncertainties on the states. The first uncertainty source accounts for unknown, deterministic structural/parametric plant-model mismatch. The second uncertainty source represents stochastic exogenous system disturbances. The proposed deterministic-stochastic robust MPC approach uses estimates of the deterministic model uncertainties to modify the nominal state and input targets. This allows for achieving offset-free tracking of the mean of the controlled variables. A non-conservative constraint tightening procedure is used to handle probabilistic state constraints and hard input constraints in the presence of stochastic uncertainties. The computational complexity of the proposed robust MPC approach is comparable to that of nominal MPC. The closed-loop performance of the proposed robust MPC approach is compared to that of robust tube-based MPC and stochastic MPC in a simulation study.

Uncertainty analysis of assembly and core-level calculations with application to CASMO-4E and SIMULATE-3

The topic of this paper is the development of sensitivity and uncertainty analysis capability to the CASMO-4/CASMO-4E – SIMULATE-3 code sequence in the context of the OECD/NEA benchmark ‘Uncertainty Analysis in Best-Estimate Modelling for Design, Operation and Safety Analysis of LWRs’ (UAM). The developed capability uses a two-step approach. In the first step, Uncertainties in nuclear data are propagated to two-group cross sections, diffusion coefficients, and assembly discontinuity factors. This is carried out using deterministic, perturbation-theory-based uncertainty analysis methodology. In the second step, a global covariance matrix, characterizing the uncertainties of the group constants, is formed, and the uncertainties are propagated through a full core SIMULATE calculation using a stochastic approach. This system enables the analysis of nuclear data related uncertainties in assembly-homogenized group constants, assembly discontinuity factors, and pin powers, as well as full core results such as multiplication factor and power distribution. The mathematical background of the deterministic uncertainty analysis methodology is reviewed and the main conclusions related to the implementation are summarized. Numerical results are presented for the full core Three Mile Island model in exercise I-3 of the UAM benchmark at hot zero power with all rods out and inserted. The computational efficiency of the calculations is discussed.

Covariance generation and uncertainty propagation for thermal and fast neutron induced fission yields

Fission product yields (FY) are fundamental nuclear data for several applications, including decay heat, shielding, dosimetry, burn-up calculations. To be safe and sustainable, modern and future nuclear systems require accurate knowledge on reactor parameters, with reduced margins of uncertainty. Present nuclear data libraries for FY do not provide consistent and complete uncertainty information which are limited, in many cases, to only variances. In the present work we propose a methodology to evaluate covariance matrices for thermal and fast neutron induced fission yields. The semi-empirical models adopted to evaluate the JEFF-3.1.1 FY library have been used in the Generalized Least Square Method available in CONRAD (COde for Nuclear Reaction Analysis and Data assimilation) to generate covariance matrices for several fissioning systems such as the thermal fission of U235, Pu239 and Pu241 and the fast fission of U238, Pu239 and Pu240. The impact of such covariances on nuclear applications has been estimated using deterministic and Monte Carlo uncertainty propagation techniques. We studied the effects on decay heat and reactivity loss uncertainty estimation for simplified test case geometries, such as PWR and SFR pin-cells. The impact on existing nuclear reactors, such as the Jules Horowitz Reactor under construction at CEA-Cadarache, has also been considered.

A Waveform Detector That Targets Template‐Decorrelated Signals and Achieves Its Predicted Performance, Part I: Demonstration with IMS Data

Abstract Waveform correlation detectors used in seismic monitoring scan multichannel data to test two competing hypotheses: that the data contain (1) a noisy, amplitude‐scaled version of a template waveform or (2) only noise. In reality, seismic wavefields include signals triggered by nontarget sources (background seismicity) and target signals that are only partially correlated with the waveform template. We reform the waveform correlation detector hypothesis test to accommodate deterministic uncertainty in template‐to‐target waveform similarity and thereby derive a new detector from convex set projections (the cone detector) for use in explosion monitoring. Our analyses give probability density functions that quantify the detectors’ degraded performance with decreasing waveform similarity. We then apply our results to three announced North Korean nuclear tests and use International Monitoring System (IMS) arrays to determine the probability that low magnitude, off‐site explosions can be reliably detected with a given waveform template. We demonstrate that cone detectors provide (1) an improved predictive capability over correlation detectors to identify such spatially separated explosive sources, (2) competitive detection rates, and (3) reduced false alarms on background seismicity. Online Material: Description and illustration of several theoretical and practical aspects of implementing the correlation and cone detector on geophysical waveform data.

Robust adaptive control for dynamic systems with mixed uncertainties

The control problem for single-input single-output (SISO) systems in the presence of mixed uncertainties, both stochastic and deterministic uncertainties, is considered. The stochastic uncertainties are modeled as exogenous noises, while the deterministic uncertainties are time invariant and appear as the unknown parameters which lie in a bounded interval. Based on a subdivision for the continuous interval, a robust adaptive controller is designed. The controller can not only realize the system output to track the desired output, but also learn a more accurate interval which contains the true value of the unknown parameter with a learning error given in advance. An example is given finally to demonstrate the effectiveness of the proposed method.

Active Disturbance Rejection Control Approach to Output-Feedback Stabilization of a Class of Uncertain Nonlinear Systems Subject to Stochastic Disturbance

The active disturbance rejection control (ADRC) is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control. However, all theoretical works up to present are limited to deterministic uncertainty. In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. We first design an extended state observer (ESO) that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance. It is shown that the resulting closed-loop system is practically stable in the mean-square topology. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.

Differential Equations for Closed Sets in a Banach Space

Nonempty subsets of a vector space suggest themselves for describing shapes (without restrictions of regularity) or states (together with their deterministic uncertainty). Hence, there have been developed several suggestions how to extend differential and integral equations respectively to time-dependent subsets although subsets do not form a linear space in an obvious way. In this article, we summarize three central approaches which can handle possibly non-convex closed subsets: integral funnel solution (a.k.a. R-solutions), morphological primitives, and reachable sets (generalizing Aumann integrals). They are extended to a separable Banach space and characterized by means of semilinear evolution inclusions.We formulate conditions sufficient for the equivalence of these generalized concepts and, then this joint basis is used for specifying differential equations for closed (not necessarily convex or compact) subsets. Several further approaches in the literature prove to be special cases. In this purely metric setting, the counterpart of the Picard–Lindelof theorem (a.k.a. Cauchy–Lipschitz theorem) ensures the existence and uniqueness of set-valued solutions to initial value problems.

Robust Workload and Energy Management for Sustainable Data Centers

A large number of geo-distributed data centers begin to surge in the era of data deluge and information explosion. To meet the growing demand in massive data processing, the infrastructure of future data centers must be energy-efficient and sustainable. Facing this challenge, a systematic framework is put forth in this paper to integrate renewable energy sources (RES), distributed storage units, cooling facilities, as well as dynamic pricing into the workload and energy management tasks of a data center network. To cope with RES uncertainty, the resource allocation task is formulated as a robust optimization problem minimizing the worst-case net cost. Compared with existing stochastic optimization methods, the proposed approach entails a deterministic uncertainty set where generated RES reside, thus can be readily obtained in practice. It is further shown that the problem can be cast as a convex program, and then solved in a distributed fashion using the dual decomposition method. By exploiting the spatio-temporal diversity of local temperature, workload demand, energy prices, and renewable availability, the proposed approach outperforms existing alternatives, as corroborated by extensive numerical tests performed using real data.

Economic objectives, uncertainties and decision making in the energy sector

The German Energiewende has led to serious changes on the energy market. Technological innovations, growing technical and economical maturity and ongoing optimizations make renewable energies increasingly competitive. After describing appropriate economic objectives in the energy sector, the present paper highlights the difficulties arising in decision making in the area of renewable energies. Since decisions in the field of wind and solar energy depend on good weather forecasts, this paper focuses on the treatment of risks and uncertainties. For a wind power station convex combinations of Weibull distributions are presented as a general model for the description of the distributions of wind speeds. For the investigation of deterministic uncertainties a method is proposed, which has been recently developed for a specific case (Bischoff et al., Hard uncertainties in multiobjective layout optimization of photovoltaic power plants, 2015) and which is now formulated for the standard case of non-dynamic uncertainty sets. Finally, a new approach of error propagation is presented, which describes the propagation of errors in wind speeds to the resulting Weibull probability density functions.

Robust Power-Splitting SWIPT Beamforming for Broadcast Channels

This letter considers the multiple-input single-output (MISO) broadcast system for simultaneous wireless information and power transfer (SWIPT) using receiver power splitting and aims to optimize jointly the beamforming vectors and the power splitting ratios for minimizing the transmit power of the base station (BS) subject to the individual signal-to-interference-plus-noise ratio (SINR) and the energy-harvesting constraints at the mobile stations (MSs). However, the CSI is assumed imperfect but has a deterministic uncertainty region. Unlike existing attempts that resort to iterations guided by semidefinite relaxation (SDR), we propose a reverse convex nonsmooth optimization algorithm, which provides the near-optimal rank-one solution.

Comparison of Statistical and Deterministic Frameworks of Uncertainty Quantification

Two different approaches to the prediction problem are compared employing a realistic example---combustion of natural gas---with 102 uncertain parameters and 76 quantities of interests. One approach, termed bound-to-bound data collaboration (abbreviated to B2B), deploys semidefinite programming algorithms where the initial bounds on unknowns are combined with initial bounds of experimental data to produce new uncertainty bounds for the unknowns that are consistent with the data and, finally, deterministic uncertainty bounds for prediction in new settings. The other approach is statistical and Bayesian, referred to as BCP (for Bayesian calibration and prediction). It places prior distributions on the unknown parameters and on the parameters of the measurement error distributions and produces posterior distributions for model parameters and posterior distributions for model predictions in new settings. The predictions from the two approaches are consistent; a very large degree of overlap exists between B2B bounds and the support of the BCP predictive distribution. Interpretation and comparison of the results is closely connected with assumptions made about the model and experimental data and how they are used in both settings. The principal conclusion is that use of both methods protects against possible violations of assumptions in the BCP approach and conservative specifications and predictions using B2B.

Aerodynamic design optimization of wind turbine rotors under geometric uncertainty

Presented is a robust optimization strategy for the aerodynamic design of horizontal axis wind turbine rotors including the variability of the annual energy production due to the uncertainty of the blade geometry caused by manufacturing and assembly errors. The energy production of a rotor designed with the proposed robust optimization approach features lower sensitivity to stochastic geometry errors with respect to that of a rotor designed with the conventional deterministic optimization approach that ignores these errors. The geometry uncertainty is represented by normal distributions of the blade pitch angle, and the twist angle and chord of the airfoils. The aerodynamic module is a blade-element momentum theory code. Both Monte Carlo sampling and the univariate reduced quadrature technique, a novel deterministic uncertainty analysis method, are used for uncertainty propagation. The performance of the two approaches is assessed in terms of accuracy and computational speed. A two-stage multi-objective evolution-based optimization strategy is used. Results highlight that, for the considered turbine type, the sensitivity of the annual energy production to rotor geometry errors can be reduced by reducing the rotational speed and increasing the blade loading. The primary objective of the paper is to highlight how to incorporate an efficient and accurate uncertainty propagation strategy in wind turbine design. The formulation of the considered design problem does not include all the engineering constraints adopted in real turbine design, but the proposed probabilistic design strategy is fairly independent of the problem definition and can be easily extended to turbine design systems of any complexity.

Robust Optimization Based Optimal DG Placement in Microgrids

This paper proposes a novel Microgrid (MG) planning methodology to decide optimal locations, sizes and mix of dispatchable and intermittent distributed generators (DGs). The long-term costs in the proposed planning model include investment, operation and maintenance (O&M), fuel and emission costs of DGs while the revenue includes payment by MG loads and utility grid. The problem is formulated as a mixed-integer program (MIP) considering the probabilistic nature of DG outputs and load consumption, wherein the costs are minimized and profits are maximized. The model is transformed to be a two-stage robust optimization problem. A column and constraint generation (CCG) framework is used to solve the problem. Compared with conventional MG planning approaches, the proposed model is more practical in that it fully considers the system uncertainties and only requires a deterministic uncertainty set, rather than a probability distribution of uncertain data which is difficult to obtain. Case studies of a MG with wind turbines, photovoltaic generators (PVs) and microturbines (MTs) demonstrate the effectiveness of the proposed methodology.

Two‐stage adjustable robust optimisation for unit commitment under uncertainty

A new adjustable robust optimisation formulation to solve unit commitment (UC) problems under uncertainty is proposed. Unlike the stochastic programming or chance-constrained approaches, the robust formulation does not require information on the exact probability distributions. This method protects power systems against all possible scenarios within a deterministic uncertainty set. The degree of conservatism can be conveniently controlled by the parameters of the uncertainty set. The formulation is divided into two stages, where here-and-now UC decisions are determined in the first stage, and the second stage makes wait-and-see dispatch decisions. The second-stage problems are reformulated into the equivalent form, so that a practical cutting-plane algorithm can be used to solve the proposed two-stage problems in an iterative manner. Numerical studies show that the robust formulation offers more consistent performance than the stochastic method when the underlying distribution of demand or generating unit reliability varies.