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Optimization models for integrated biorefinery operations
Variations of physical and chemical characteristics of biomass lead to an uneven flow of biomass in a biorefinery, which reduces equipment utilization and increases operational costs. Uncertainty of biomass supply and high processing costs increase the risk of investing in the US’s cellulosic biofuel industry. We propose a stochastic programming model to streamline processes within a biorefinery. A chance constraint models system’s reliability requirement that the reactor is operating at a high utilization rate given uncertain biomass moisture content, particle size distribution, and equipment failure. The model identifies operating conditions of equipment and inventory level to maintain a continuous flow of biomass to the reactor. Furthermore, the sample average approximation method approximates the chance constraint and a bisection search-based heuristic solves this approximation. A case study is developed using real-life data collected at Idaho National Laboratory’s biomass processing facility. An extensive computational analysis indicates that sequencing of biomass bales based on moisture level, increasing storage capacity, and managing particle size distribution, increases utilization of the reactor and reduces operational costs.
An Empirical Quantile Estimation Approach for Chance-Constrained Nonlinear Optimization Problems
We investigate an empirical quantile estimation approach to solve chance-constrained nonlinear optimization problems. Our approach is based on the reformulation of the chance constraint as an equivalent quantile constraint to provide stronger signals on the gradient. In this approach, the value of the quantile function is estimated empirically from samples drawn from the random parameters, and the gradient of the quantile function is estimated via a finite-difference approximation on top of the quantile-function-value estimation. We establish a convergence theory of this approach within the framework of an augmented Lagrangian method for solving general nonlinear constrained optimization problems. The foundation of the convergence analysis is a concentration property of the empirical quantile process, and the analysis is divided based on whether or not the quantile function is differentiable. In contrast to the sampling-and-smoothing approach used in the literature, the method developed in this paper does not involve any smoothing function and hence the quantile-function gradient approximation is easier to implement and there are less accuracy-control parameters to tune. Furthermore, we demonstrate the effectiveness of this approach and compare it with a smoothing method for the quantile-gradient estimation. Numerical investigation shows that the two approaches are competitive for certain problem instances.
Optimization problems governed by systems of PDEs with uncertainties
This paper reviews current theoretical and numerical approaches to optimization problems governed by partial differential equations (PDEs) that depend on random variables or random fields. Such problems arise in many engineering, science, economics and societal decision-making tasks. This paper focuses on problems in which the governing PDEs are parametrized by the random variables/fields, and the decisions are made at the beginning and are not revised once uncertainty is revealed. Examples of such problems are presented to motivate the topic of this paper, and to illustrate the impact of different ways to model uncertainty in the formulations of the optimization problem and their impact on the solution. A linear–quadratic elliptic optimal control problem is used to provide a detailed discussion of the set-up for the risk-neutral optimization problem formulation, study the existence and characterization of its solution, and survey numerical methods for computing it. Different ways to model uncertainty in the PDE-constrained optimization problem are surveyed in an abstract setting, including risk measures, distributionally robust optimization formulations, probabilistic functions and chance constraints, and stochastic orders. Furthermore, approximation-based optimization approaches and stochastic methods for the solution of the large-scale PDE-constrained optimization problems under uncertainty are described. Some possible future research directions are outlined.
Convergence Analysis of Fixed Point Chance Constrained Optimal Power Flow Problems
For optimal power flow problems with chance constraints, a particularly effective method is based on a fixed point iteration applied to a sequence of deterministic power flow problems. However, a priori, the convergence of such an approach is not necessarily guaranteed. Here this article analyses the convergence conditions for this fixed point approach, and reports numerical experiments including for large IEEE networks.
Data-Driven Chance-Constrained Design of Voltage Droop Control for Distribution Networks: Preprint
This paper addresses the design of local control methods for voltage control in distribution networks with high level of distributed energy resources (DERs). The designed control methods adapt the active and reactive power output of distributed energy resources proportional to the deviation of the local measured voltage magnitudes from a reference voltage, which is referred to as droop control. Thus, the design focuses on determining the droop characteristics which satisfy network-wide voltage magnitude constraints. The uncertainty and variability of DERs renders the design of optimal droop controls very challenging. Hence, this paper proposes chance constraints to limit the risk from intermittent DERs, by designing droop control coefficients that guarantee the satisfaction of network operational constraints with a specific probability. In addition, the proposed approach relies entirely on historical data rather than assuming knowledge of the probability distributions that characterize the uncertainty of DERs. The efficacy of the proposed method is demonstrated on a 37-bus distribution feeder.
Safe Reinforcement Learning-Based Transient Stability Control for Islanded Microgrids With Topology Reconfiguration
This paper proposes a safe reinforcement learning (RL)-based transient stability emergency control (TSEC) method for islanded microgrids. RL requires extensive interaction with the environment to learn control strategies, hence, a data-driven approach is used as a substitute for time-consuming time-domain simulation calculations. Deep sigma point processes (DSPP), which is a Gaussian process model, is utilized to predict the normal distribution of transient stability of microgrids and to construct a transient stability chance constraint. Reward-constrained policy optimization (RCPO) can simultaneously achieve objective prediction, policy learning, and constraint cost coefficient update across multiple timescales. RCPO interacts with the DSPP-based microgrid environment through a multi-process parallel manner, greatly increasing the training speed. Case studies on a real islanded microgrid demonstrate that the proposed method can efficiently and quickly obtain the optimal emergency control strategy while adhering to all hard constraints.
An Iterative Response-Surface-Based Approach for Chance-Constrained AC Optimal Power Flow Considering Dependent Uncertainty
A modern power system is characterized by a stochastic variation of the loads and an increasing penetration of renewable energy generation, which results in large uncertainties in its states. These uncertainties bring formidable challenges to the power system planning and operation process. To address these challenges, we propose a cost-effective, iterative response-surface-based approach for the chance-constrained AC optimal power-flow problem that aims to ensure the secure operation of the power systems considering dependent uncertainties. Starting from a stochastic-sampling-based framework, we first utilize the copula theory to simulate the dependence among multivariate uncertain inputs. Then, to reduce the prohibitive computational time required in the traditional Monte-Carlo method, we propose, instead of using the original complicated power-system model, to rely on a polynomial-chaos-based response surface. This response surface allows us to efficiently evaluate the time-consuming power-system model at arbitrary distributed sampled values with a negligible computational cost. This further enables us to efficiently conduct an online stochastic testing for the system states that not only screens out the statistical active constraints, but also assists in a better design of the tightened bounds without using any Gaussian or symmetric assumption. Finally, an iterative procedure is executed to fine-tune the optimal solution that better satisfies a predefined probability. The simulations conducted in multiple test systems demonstrate the excellent performance of the proposed method.
Thermal Overloading Risk Mitigation With a Semi-Analytical Probabilistic Model on Branch Current
A semi-analytical formulation is presented in this paper for the probability computation of branch current in multiphase systems. The developed formula is derived based on the linear power flow model in rectangular coordinates. The system uncertainty injections can be renewable energy resources or loads and are modeled using a Gaussian mixture model (GMM). The developed formula can be used to compute the line current violation probability as well as integrate into optimal power flow problem as chance-constraint relaxation. Here, the proposed formula is first compared with the Matlab embedded numerical integration function to show its performance. Besides, the semi-analytical formula is validated and compared with the Monte Carlo simulation method using the IEEE 123-bus system, EPRI Ckt5, and Ckt7 systems.
DNN-based policies for stochastic AC OPF
We report a prominent challenge to the safe and optimal operation of the modern power grid arises due to growing uncertainties in loads and renewables. Stochastic optimal power flow (SOPF) formulations provide a mechanism to handle these uncertainties by computing dispatch decisions and control policies that maintain feasibility under uncertainty. Most SOPF formulations consider simple control policies such as affine policies that are mathematically simple and resemble many policies used in current practice. Motivated by the efficacy of machine learning (ML) algorithms and the potential benefits of general control policies for cost and constraint enforcement, we put forth a deep neural network (DNN)-based policy that predicts the generator dispatch decisions in real time in response to uncertainty. The weights of the DNN are learnt using stochastic primal–dual updates that solve the SOPF without the need for prior generation of training labels and can explicitly account for the feasibility constraints in the SOPF. The advantages of the DNN policy over simpler policies and their efficacy in enforcing safety limits and producing near optimal solutions are demonstrated in the context of a chance constrained formulation on a number of test cases.
Learning Stochastic Parametric Differentiable Predictive Control Policies
We present a scalable unsupervised learning-based method for obtaining explicit control policies for model predictive control problems for stochastic linear systems with additive uncertainties subject to nonlinear chance constraints. We call the proposed method stochastic parametric differentiable predictive control (SP-DPC), which extends the recently proposed deterministic DPC policy optimization algorithm. We formulate the SP-DPC as a deterministic approximation to the stochastic parametric constrained optimal control problem via independent sampling of the problem's parameters and uncertainties. This formulation allows us to directly compute the policy gradients via automatic differentiation of the problem's value function, evaluated over sampled parameters and uncertainties. In particular, the computed expectation of the problem's value function is backpropagated through the finite-time closed-loop system rollouts parametrized by a known nominal system dynamics model and neural control policy. We also provide theoretical probabilistic guarantees on closed-loop stability and chance constraints satisfaction for systems controlled by learned neural policies. We demonstrate the computational efficiency and scalability of the proposed policy optimization algorithm in three numerical examples, including systems with a large number of states or subject to nonlinear constraints.
A safe reinforcement learning algorithm for supervisory control of power plants
Traditional control theory-based methods require tailored engineering for each system and constant fine-tuning. In power plant control, one often needs to obtain a precise representation of the system dynamics and carefully design the control scheme accordingly. Model-free Reinforcement learning (RL) has emerged as a promising solution for control tasks due to its ability to learn from trial-and-error interactions with the environment. It eliminates the need for explicitly modeling the environment’s dynamics, which is potentially inaccurate. However, the direct imposition of state constraints in power plant control raises challenges for standard RL methods. To address this, we propose a chance-constrained RL algorithm based on Proximal Policy Optimization for supervisory control. Our method employs Lagrangian relaxation to convert the constrained optimization problem into an unconstrained objective, where trainable Lagrange multipliers enforce the state constraints. In conclusion, our approach achieves the smallest distance of violation and violation rate in a load-follow maneuver for an advanced Nuclear Power Plant design.
A stochastic biomass blending problem in decentralized supply chains
Blending biomass materials of different physical or chemical properties provides an opportunity to adjust the quality of the feedstock to meet the specifications of the conversion platform. We propose a model which identifies the right mix of biomass to optimize the performance of the thermochemical conversion process at the mini-mum cost. This is a chance-constraint programming (CCP) model which takes into account the stochastic nature of biomass quality. The proposed CCP model ensures that process requirements, which are impacted by physical and chemical properties of biomass, are met most of the time. We consider two problem settings, a centralized and a decentralized supply chain. We propose a mixed-integer linear program to model the blending problem in the centralized setting and a bilevel program to model the blending problem in the decentralized setting. We use the sample average approximation method to approximate the chance constraints, and propose solution algorithms to solve this approximation. We develop a case study for South Carolina using data provided by the Billion Ton Study. Based on our results, the blends identified consist mainly of pine and softwood residues. The blends identified and the suppliers selected by both models are different. The cost of the centralized supply chain is 2%–6% lower. The implications of these results are twofold. First, these results could lead to improved collaborations in the supply chain. Second, these results provide an estimate of the approximation error from assuming centralized decision making in the supply chain.
Efficient Automated Driving Strategies Leveraging Anticipation and Optimal Control
Automated vehicles and advanced driver assistance systems bring computation, sensing, and communication technologies that exceed human abilities in some ways. For example, automated vehicles may sense a panorama all at once, do not suffer from human impairments and distractions, and could wirelessly communicate precise data with neighboring vehicles. Prototype and commercial deployments have demonstrated the capability to relieve human operators of some driving tasks up to and including fully autonomous taxi rides in some areas. The ultimate impact of this technology’s large-scale market penetration on energy efficiency remains unclear, with potential negative factors like road use by empty vehicles competing with positive ones like automatic eco-driving. Fundamentally enabled by historic and look-ahead data, this dissertation addresses the use of automated driving and driver assistance to optimize vehicle motion for energy efficiency. Facets of this problem include car following, co-optimized acceleration and lane change planning, and collaborative multi-agent guidance. Optimal control, especially model predictive control, is used extensively to improve energy efficiency while maintaining safe and timely driving via constraints. Techniques including chance constraints and mixed integer programming help overcome uncertainty and non-convexity challenges. Extensions of these techniques to tractor trailers on sloping roads are provided by making use of linear parameter-varying models. To approach the wheel-input energy eco-driving problem over generally shaped sloping roads with the computational potential for closed-loop implementation, a linear programming formulation is constructed. Distributed and collaborative techniques that enable connected and automated vehicles to accommodate their neighbors in traffic are also explored and compared to centralized control. Using simulations and vehicle-in-the-loop car following experiments, the proposed algorithms are benchmarked against others that do not make use of look-ahead information.
Analytic Neural Network Gaussian Process Enabled Chance-Constrained Voltage Regulation for Active Distribution Systems with PVs, Batteries and EVs
This paper proposes an analytic neural network Gaussian process (NNGP)-based chance-constrained real-time voltage regulation method for active distribution systems with photovoltaics (PVs), batteries, and electric vehicles (EVs). NNGP can utilize historical measurement data to achieve real-time probabilistic node voltage estimation through Bayesian inference. Then, NNGP is fully analytically embedded into the optimal power flow model to perform voltage regulation and adapt to various topological changes. The uncertainties of voltage estimations are easily considered via the chance constraint, and it has been shown that the adoption of this chance constraint can significantly improve the reliability of voltage regulation under various scenarios. The comparison results with other methods, carried out on a real 759-node distribution system located in western Colorado, U.S., show that the proposed method can achieve accurate voltage estimation across different topologies and reliably perform voltage regulation considering PVs, batteries, and EVs.
Microgrid Tiered Circuits Effects for a Planned Housing Community in Puerto Rico
Puerto Rico faced a double strike from hurricanes Irma and Maria in 2017. The resulting damage required a comprehensive rebuild of electric infrastructure. There are plans and pilot projects to rebuild with microgrids to increase resilience. This paper provides a techno-economic analysis technique and case study of a potential future community in Puerto Rico that combines probabilistic microgrid design analysis with tiered circuits in building energy modeling. Tiered circuits in buildings allow electric load reduction via remote disconnection of non-critical circuits during an emergency. When coupled to a microgrid, tiered circuitry can reduce the chances of a microgrid's storage and generation resources being depleted. The analysis technique is applied to show 1) Approximate cost savings due to a tiered circuit structure and 2) Approximate cost savings gained by simultaneously considering resilience and sustainability constraints in the microgrid optimization. The analysis technique uses a resistive capacitive thermal model with load profiles for four tiers (tier 1-3 and non-critical loads). Three analyses were conducted using: 1) open-source software called Tiered Energy in Buildings and 2) the Microgrid Design Toolkit. For a fossil fuel based microgrid 30% of the total microgrid costs of 1.18 million USD were calculated where the non-tiered case keeps all loads 99.9% available and the tiered case keeps tier 1 at 99.9%, tier 2 at 95%, tier 3 at 80% availability, with no requirement on non-critical loads. The same comparison for a sustainable microgrid showed 8% cost savings on a 5.10 million USD microgrid due to tiered circuits. The results also showed 6-7% cost savings when our analysis technique optimizes sustainability and resilience simultaneously in comparison to doing microgrid resilience analysis and renewables net present value analysis independently. Though highly specific to our case study, similar assessments using our analysis technique can elucidate value of tiered circuits and simultaneous consideration of sustainability and resilience in other locations.
Optimal celestial bodies for dark matter detection
A wide variety of celestial bodies have been considered as dark matter detectors. Which stands the best chance of delivering the discovery of dark matter? Which is the most powerful dark matter detector? We investigate a range of objects, including the Sun, Earth, Jupiter, Brown Dwarfs, White Dwarfs, Neutron Stars, Stellar populations, and Exoplanets. We quantify how different objects are optimal dark matter detectors in different regimes by deconstructing some of the in-built assumptions in these search sensitivities, including observation potential and particle model assumptions. We find new constraints and future sensitivities across a range of dark matter annihilation final states. We quantify mediator properties leading to detectable celestial-body energy injection or Standard Model fluxes, and show how different objects can be expected to deliver corroborating signals. We discuss different search strategies, their opportunities and limitations, and the interplay of regimes where different celestial objects are optimal dark matter detectors. Deconstructing the assumptions of these searches leads us to point out a new search using the Galactic center stellar population that can provide greater sensitivity to the dark matter-nucleon scattering cross section than the Sun, despite being significantly further away in our Galaxy.
Secure Control Regions for Distributed Stochastic Systems with Application to Distributed Energy Resource Dispatch: Preprint
With the increasing connectedness and interdependence of systems that are stochastic in nature, the issue of how to manage and coordinate them for safe operation has evidently become more important. In many networked system architectures, the system-wide output has to be delicately managed; often within a prescribed set of bounds. In this paper, a novel control framework is proposed where the bounds on the outputs are translated into independent bounds on the controllable inputs of each subsystem. The main benefit of this framework is that respecting the individual control bounds suffices to guarantee that the system-wide outputs will remain within safe boundaries. Since the systems are assumed to be stochastic, the bounds on the output are introduced as probabilistic chance constraints. The benefits of this framework are demonstrated by applying it to the control of distributed energy resources in a distribution networks where main goal is to keep the voltage magnitudes with their prescribed bounds. The control bounds are evaluated using real data on an IEEE test system.