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Risk-Averse PDE-Constrained Optimization using the Conditional Value-At-Risk.
Abstract not provided.
Higher-moment buffered probability
In stochastic optimization, probabilities naturally arise as cost functionals and chance constraints. Unfortunately, these functions are difficult to handle both theoretically and computationally. The buffered probability of failure and its subsequent extensions were developed as numerically tractable, conservative surrogates for probabilistic computations. In this manuscript, we introduce the higher-moment buffered probability. Whereas the buffered probability is defined using the Conditional Value-at-Risk, the higher-moment buffered probability is defined using Higher-Moment Coherent Risk measures. In this way, the higher-moment buffered probability encodes information about the magnitude of tail moments, not simply the tail average. We prove that the higher-moment buffered probability is closed, monotonic, quasi-convex and can be computed by solving a smooth one-dimensional convex optimization problem. These properties enable smooth reformulations of both higher-moment buffered probability cost functionals and constraints.
Security Constrained Distributed Transaction Model for Multiple Prosumers
Massive access of renewable energy has prompted demand-side distributed resources to participate in regulation and improve flexibility of power systems. With large-scale access of massive, decentralized, and diverse distributed resources, demand-side market members have transformed from traditional “consumers” to “prosumers”. To explore the distributed transaction model of prosumers, in this paper, a multi-prosumer distributed transaction model is proposed, and the Conditional Value-at-Risk (CVaR) theory is applied to quantify potential risks caused by the stochastic characteristics inherited from renewable energy. First, a prosumer model under constraints of the distribution network including photovoltaic units, fuel cells, energy storage system, central air conditioning and flexible loads is established, and a multi-prosumer distributed transaction strategy is proposed to achieve power sharing among multiple prosumers. Second, a prosumer transaction model based on CVaR is constructed to measure risks inherited from the uncertainty of PV output within the prosumer and ensure safety of system operation in extreme PV output scenarios. Then, the alternating direction multiplier method (ADMM) is utilized to solve the constructed model efficiently. Finally, distributed transaction costs of prosumers are distributed fairly based on the generalized Nash equilibrium to maximize social benefits. Simulation results show the multi-prosumer distributed transaction mechanism established under the proposed generalized Nash equilibrium method can encourage power sharing among prosumers, increasing their own income and social benefits. Also, the CVaR can assist decision making of prosumers in weighting the risks and benefits, improving system resilience through energy management of prosumers.
Algorithms and analyses for stochastic optimization for turbofan noise reduction using parallel reduced-order modeling
Simulation-based optimization of acoustic liner design in a turbofan engine nacelle for noise reduction purposes can dramatically reduce the cost and time needed for experimental designs. Because uncertainties are inevitable in the design process, a stochastic optimization algorithm is posed based on the conditional value-at-risk measure so that an ideal acoustic liner impedance is determined that is robust in the presence of uncertainties. A parallel reduced-order modeling framework is developed that dramatically improves the computational efficiency of the stochastic optimization solver for a realistic nacelle geometry. The reduced stochastic optimization solver takes less than 500 s to execute. Additionally, well-posedness and finite element error analyses of the state system and optimization problem are provided.
Adversarial classification via distributional robustness with Wasserstein ambiguity
Abstract We study a model for adversarial classification based on distributionally robust chance constraints. We show that under Wasserstein ambiguity, the model aims to minimize the conditional value-at-risk of the distance to misclassification, and we explore links to adversarial classification models proposed earlier and to maximum-margin classifiers. We also provide a reformulation of the distributionally robust model for linear classification, and show it is equivalent to minimizing a regularized ramp loss objective. Numerical experiments show that, despite the nonconvexity of this formulation, standard descent methods appear to converge to the global minimizer for this problem. Inspired by this observation, we show that, for a certain class of distributions, the only stationary point of the regularized ramp loss minimization problem is the global minimizer.
An adaptive local reduced basis method for solving PDEs with uncertain inputs and evaluating risk
Many physical systems are modeled using partial differential equations (PDEs) with uncertain or random inputs. For such systems, naively propagating a fixed number of samples of the input probability law (or an approximation thereof) through the PDE is often inadequate to accurately quantify the "risk" associated with critical system responses. In this paper, we develop a goal-oriented, adaptive sampling and local reduced basis approximation for PDEs with random inputs. Our method determines a set of samples and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. The samples are selected in an adaptive manner using an a posteriori error indicator. A notable advantage of the proposed approach is that the computational cost of the approximation during the adaptive process remains constant. We provide theoretical error bounds for our approximation and numerically demonstrate the performance of our method when compared to widely used adaptive sparse grid techniques. In addition, we tailor our approach to accurately quantify the risk of quantities of interest that depend on the PDE solution. We demonstrate our method on an advection-diffusion example and a Helmholtz example. Keywords: stochastic partial differential equation, adaptive sampling, reduced basis, a posteriori error estimation, coherent risk measure, conditional value-at-risk.
On the role of Battery Energy Storage Systems in the day-ahead Contingency-Constrained Unit Commitment problem under renewable penetration
The integration of variable Renewable Energy Sources (vRES) to alleviate greenhouse gas emissions has introduced significant challenges for power systems operations. These challenges include high levels of uncertainty due to the intermittence associated with vRES and therefore impose the need to devise a reliable and cost-effective day-ahead unit commitment and power and reserves scheduling for real-time operations. Also, this increasing penetration of vRES requires higher ramping capabilities from units originally designed for other purposes (e.g., base-load generation), which might be exacerbated during contingency states. Hence, in this work, we propose a methodology to address the day-ahead Contingency-Constrained Unit Commitment (CCUC) problem that leverages the participation of Battery Energy Storage Systems (BESSs) to address load-following and post-contingency management, therefore alleviating the ramping burden on conventional thermal generators. To do so, we formulate a three-level optimization problem that represents the decision-making process of obtaining the least-cost commitment, generation and reserves scheduling, while restricting the Conditional Value-at-Risk (CVaR) of the system imbalance at real-time operations to user-defined tolerance levels. In addition, we devise a computationally efficient solution approach for the proposed problem based on the Column-and Constraint Generation (CCG) algorithmic framework. Two numerical experiments are conducted to empirically illustrate the benefits of the proposed methodology. Key results indicate a reduction in real-time ramping needs and a better usage of the system resources, with a reduction in the overall system commitment levels and reserve scheduling costs when compared to a benchmark case in which storage is not available.
Stochastic scheduling of generating units with weekly energy storage: A hybrid decomposition approach
We propose a solution method for the large-scale stochastic unit commitment (SUC) problem with weekly-dispatched energy storage and significant weather-dependent stochastic generating capacity. Weekly storage facilities that mostly charge during weekends and discharge during weekdays require a weekly scheduling of generating units, which result in a large-scale optimization problem. This SUC problem is formulated as a two-stage stochastic model and we use the conditional value-at-risk as a risk measure. Using a Benders framework, the proposed solution method decomposes the problem into a mixed-integer linear master problem and linear and continuous subproblems. The master problem corresponds to the first-stage decisions throughout the week and includes all the commitment (binary) variables and their corresponding constraints. The subproblems correspond to the actual dispatch of the generating units on a weekly basis. Based on the success of column-and-constraint generation algorithms to solve robust optimization problems, we improve the low communication between the master problem and the subproblems in the standard Benders decomposition by adding primal variables and constraints from the subproblems to the master problem, which provides a better approximation of the recourse function. Furthermore, our computational experiments demonstrate the effectiveness of the proposed decomposition method using an instance of the South Carolina synthetic system with 90 generating units under 40 scenarios.
A data-driven network optimisation approach to coordinated control of distributed photovoltaic systems and smart buildings in distribution systems
The increasing integration of distributed energy resources, including demand-side resources and distributed photovoltaics (PVs), into distribution systems has resulted in more complicated power system operation. A data-driven network optimisation approach is proposed to coordinate the control of distributed PVs and smart buildings in distribution networks considering the uncertainties of solar power, outdoor temperature and heat gain associated with building thermal dynamics. These uncertain parameters have a significant impact on the operation and control of distributed PVs and smart buildings, bringing challenges to the distribution system operation. In the proposed data-driven distributionally robust optimisation (DRO) approach, the Wasserstein ball is used to construct an ambiguity set for the uncertain parameters, which does not require the probability distributions to be known. Furthermore, a conditional value-at-risk is incorporated into the Wasserstein-based DRO model and converted into a computationally tractable mixed-integer convex optimisation problem. Benchmarked with robust optimisation and chance-constrained programming, the proposed data-driven model can give a less conservative robust solution.
A Risk-Driven Probabilistic Approach to Quantify Resilience in Power Distribution Systems
It is of growing concern to ensure resilience in power distribution systems to extreme weather events. However, there are no clear methodologies or metrics available for resilience assessment that allows system planners to assess the impact of appropriate planning measures and new operational procedures for resilience enhancement. In this paper, we propose a resilience metric using parameters that define system attributes and performance. To represent extreme events (tail probability), the conditional value-at-risk of each of the parameters are combined using Choquet Integral to evaluate the overall resilience. The effectiveness of the proposed resilience metric is studied within the simulation-based framework under extreme weather scenarios with the help of a modified IEEE 123-bus system. With the proposed framework, system operators will have additional flexibility to prioritize one investment over the others to enhance the resilience of the grid.
SU-E-T-368: Evaluating Dosimetric Outcome of Modulated Photon Radiotherapy (XMRT) Optimization for Head and Neck Patients
Purpose: The dosimetric outcome of optimized treatment plans obtained by modulating the photon beamlet energy and fluence on a small cohort of four Head and Neck (H and N) patients was investigated. This novel optimization technique is denoted XMRT for modulated photon radiotherapy. The dosimetric plans from XMRT for H and N treatment were compared to conventional, 6 MV intensity modulated radiotherapy (IMRT) optimization plans. Methods: An arrangement of two non-coplanar and five coplanar beams was used for all four H and N patients. Both XMRT and IMRT were subject to the same optimization algorithm, with XMRT optimization allowing both 6 and 18 MV beamlets while IMRT was restricted to 6 MV only. The optimization algorithm was based on a linear programming approach with partial-volume constraints implemented via the conditional value-at-risk method. H and N constraints were based off of those mentioned in the Radiation Therapy Oncology Group 1016 protocol. XMRT and IMRT solutions were assessed using metrics suggested by International Commission on Radiation Units and Measurements report 83. The Gurobi solver was used in conjunction with the CVX package to solve each optimization problem. Dose calculations and analysis were done in CERR using Monte Carlo dose calculation with VMC{sub ++}. Results: Both XMRT and IMRT solutions met all clinical criteria. Trade-offs were observed between improved dose uniformity to the primary target volume (PTV1) and increased dose to some of the surrounding healthy organs for XMRT compared to IMRT. On average, IMRT improved dose to the contralateral parotid gland and spinal cord while XMRT improved dose to the brainstem and mandible. Conclusion: Bi-energy XMRT optimization for H and N patients provides benefits in terms of improved dose uniformity to the primary target and reduced dose to some healthy structures, at the expense of increased dose to other healthy structures when compared with IMRT.
SU-G-BRC-02: A Novel Multi-Criteria Optimization Approach to Generate Deliverable Intensity-Modulated Radiation Therapy (IMRT) Treatment Plans
Purpose: To present a novel multi-criteria optimization (MCO) solution approach that generates treatment plans with deliverable apertures using column generation. Methods: We demonstrate our method with 10 locally advanced head-and-neck cancer cases retrospectively. In our MCO formulation, we defined an objective function for each structure in the treatment volume. This resulted in 9 objective functions, including 3 distinct objectives for primary target volume, high-risk and low-risk target volumes, 5 objectives for each of the organs-at-risk (OARs) (two parotid glands, spinal cord, brain stem and oral cavity), and one for the non-target non-OAR normal tissue. Conditional value-at-risk (CVaR) constraints were utilized to ensure at least certain fraction of the target volumes receiving the prescription doses. To directly generate deliverable plans, column generation algorithm was embedded within our MCO approach for aperture shape generation. Final dose distributions for all plans were generated using a Monte Carlo kernel-superposition dose calculation. We compared the MCO plans with the clinical plans, which were created by clinicians. Results: At least 95% target coverage was achieved by both MCO plans and clinical plans. However, the average conformity indices of clinical plans and the MCO plans were 1.95 and 1.35, respectively (31% reduction, p<0.01). Compared to the conventional clinical plan, the proposed MCO method achieved average reductions in left parotid mean dose of 5% (p=0.06), right parotid mean dose of 18% (p<0.01), oral cavity mean dose of 21% (p=0.03), spinal cord maximum dose of 20% (p<0.01), brain stem maximum dose of 61% (p<0.01), and normal tissue maximum dose of 5% (p<0.01), respectively. Conclusion: We demonstrated that the proposed MCO method was able to obtain deliverable IMRT treatment plans while achieving significant improvements in dosimetric plan quality.
Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization
Uncertainty is ubiquitous in virtually all engineering applications, and, for such problems, it is inadequate to simulate the underlying physics without quantifying the uncertainty in unknown or random inputs, boundary and initial conditions, and modeling assumptions. Here in this paper, we introduce a general framework for analyzing risk-averse optimization problems constrained by partial differential equations (PDEs). In particular, we postulate conditions on the random variable objective function as well as the PDE solution that guarantee existence of minimizers. Furthermore, we derive optimality conditions and apply our results to the control of an environmental contaminant. Lastly, we introduce a new risk measure, called the conditional entropic risk, that fuses desirable properties from both the conditional value-at-risk and the entropic risk measures.
Stochastic Optimal Power Flow Based on Data-Driven Distributionally Robust Optimization
We propose a data-driven method to solve a stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The objective is to determine power schedules for controllable devices in a power network to balance operational cost and conditional value-at-risk (CVaR) of device and network constraint violations. These decisions include scheduled power output adjustments and reserve policies, which specify planned reactions to forecast errors in order to accommodate fluctuating renewable energy sources. Instead of assuming the uncertainties across the networks follow prescribed probability distributions, we assume the distributions are only observable through a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real data-generating distribution, we formulate a distributionally robust optimization OPF problem to search for power schedules and reserve policies that are robust to sampling errors inherent in the dataset. A multi-stage closed-loop control strategy based on model predictive control (MPC) is also discussed. A simpIe numerical example illustrates inherent tradeoffs between operational cost and risk of constraint violation, and we show how our proposed method offers a data-driven framework to balance these objectives.
Data Driven UAM Flight Energy Consumption Prediction and Risk Assessment
With the current technological advancements revolutionizing the concept of Urban Air Mobility (UAM) and package delivery, there is also, a concurrent need to quantify the operational safety of these vehicles in terms of their associated risk. Conducting safe flight operations is critical for UAM vehicles which are electrically Vertical Takeoff and Landing (eVTOL) vehicles, to operate in current Air traffic control. In this paper, a data-driven method for UAM vehicle energy consumption prediction and risk quantification with conditional value-at-risk based on energy consumption distribution is presented. Significant factors affecting energy consumption, such as density altitude, aircraft design, airspeed, and collision avoidance algorithms, are considered in the data-driven based energy consumption prediction of different eVTOL
Power demand risk models on milling machines
The measurement of power demand risks in manufacturing power systems will benefit manufacturers and the wider society. If the risks can be characterized using manufacturing parameters, manufacturers can better control risks originating in those parameters, select less power-risky production plans, and reduce utility costs and resource consumption. The measurement of risk can also help manufacturers and power suppliers to protect their power systems from unexpected disturbances. Existing measures of risk, however, do not consider time duration, and thus cannot accurately quantify the risks in manufacturing power systems; the risks of a period of high power demand must be evaluated with the duration of the surge. Therefore, new methods of measuring power demand risks are proposed, adapting measures drawn from the field of finance. With a focus on milling operations, processing power is shown to be a function of processing amount and processing time, and a power demand distribution is directly derived as a joint distribution of and A and T. A bivariate random variable model with copulas is applied to examine the correlation in the joint distribution. Then, based on evaluation of a probability distribution of power demand from and, new risk measures are introduced. Herein illustrative examples are provided to show how the proposed measures can quantify the power demand risks from milling machines, based on manufacturing parameters. Certain manufacturing parameters are found to affect overall power demand risks, including i) raw material type, ii) variability in processing time, and iii) correlation between and. In the examples, these three factors increase power demand risks by up to 108%, 67%, and 1% respectively.