An Adaptive Model with Joint Chance Constraints for a Hybrid Wind-Conventional Generator System.
Abstract not provided.
Singh, Bismark↗
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Results for “chance constraint”
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Joint Chance Constraints Reduction Through Learning in Active Distribution Networks
Due to an increase in distributed generation and controllable loads, distribution networks are frequently operating under high levels of uncertainty. Joint chance constraints, which seek to satisfy multiple constraints simultaneously with a prescribed probability, are one way to incorporate uncertainty across sets of constraints for optimization and control of these networks. Due to the complexity of evaluating these constraints directly, sampling approaches or approximations can be used to transform the joint chance constraint into deterministic constraints. However, sampling techniques may be extremely computationally expensive and not suitable for physical networks operating on fast timescales, and conservative approximations may needlessly result in a much higher cost of system operation. The proposed framework aims to provide a scalable, data-driven approach which learns operational trends in a power network, eliminates zero-probability events (e.g., inactive constraints), and uses this additional information to accurately and efficiently approximate the joint chance constraint directly.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Joint Chance Constraints in AC Optimal Power Flow: Improving Bounds through Learning
This paper considers distribution systems with a high penetration of distributed, renewable generation and addresses the problem of incorporating the associated uncertainty into the optimal operation of these networks. Joint chance constraints, which satisfy multiple constraints simultaneously with a prescribed probability, are one way to incorporate uncertainty across sets of constraints, leading to a chance-constrained optimal power flow problem. Departing from the computationally-heavy scenario-based approaches or approximations that transform the joint constraint into conservative deterministic constraints, this paper develops a scalable, data-driven approach which learns operational trends in a power network, eliminates zero-probability events (e.g., inactive constraints), and accurately and efficiently approximates bounds on the joint chance constraint iteratively. In particular, the proposed framework improves upon the classic methods based on the union bound (or Boole's inequality) by generating a much less conservative set of single chance constraints that also guarantees the satisfaction of the original joint constraint. The proposed framework is evaluated numerically using the IEEE 37-node test feeder, focusing on the problem of voltage regulation in distribution grids.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Distributionally robust polynomial chance-constraints under mixture ambiguity sets
Given $\mathrm{X}\subset\mathbb{R}^n, ε \in (0,1)$, a parametrized family of probability distributions $(μ_a)_{a\in \mathrm{A}}$ on $\mathrm{Ω}\subset \mathbb{R}^p$, we consider the feasible set $\mathrm{X}^*_ε \subset\mathrm{X}$ associated with the distributionally robust chance-constraint $\mathrm{X}^*_ε = \mathrm{x \in X : Prob}_μ[f(\mathrm{x}, ω) >0] > 1 - ε, \forall μ \in \mathscr{M}_{\mathrm{a}}$}, where $\mathscr{M}_{\mathrm{a}}$ is the set of all possibles mixtures of distributions $μ_{\mathrm{a}}, \mathrm{a \in A}$. For instance and typically, the family $\mathscr{M}_{\mathrm{a}}$ is the set of all mixtures of Gaussian distributions on $\mathbb{R}$ with mean and standard deviation $\mathrm{a}=(a,σ)$ in some compact set $\mathrm{A}\subset \mathbb{R}^2$. We provide a sequence of inner approximations $\mathrm{X}^d_ε = {\mathrm{x \in X} : w_d(\mathrm{x}) < ε}, d \in \mathbb{N}$, where $w_d$ is a polynomial of degree d whose vector of coefficients is an optimal solution of a semidefinite program. The size of the latter increases with the degree d. We also obtain the strong and highly desirable asymptotic guarantee that $λ(\mathrm{X}^*_ε$\$\mathrm{X}^d_ε$) → 0 as d increases, where λ is the Lebesgue measure on X. Same results are also obtained for the more intricated case of distributionally robust “joint” chance-constraints. Finally, there is a price to pay for this strong asymptotic guarantee which is the scalability of such a numerical scheme, and so far this important drawback makes it limited to problems of modest dimension.
97 MATHEMATICS AND COMPUTING↗
Approximating Joint Chance Constraints in Two-Stage Stochastic Programs
We present various approximations to joint chance constraints arising in two-stage stochastic programming models. Our approximations are derived from three classical inequalities: Markov's inequality, Chebysev's inequality, and Chernoff's bound. We provide preliminary computational results illustrating the quality of our approximation using a two-stage joint-chance-constrained stochastic program from the literature. We also briefly introduce other alternatives for constructing approximations for joint-chance-constrained two-stage programs.
97 MATHEMATICS AND COMPUTING↗
Chance Constraints for Improving the Security of AC Optimal Power Flow
This paper introduces a scalable method for improving the solutions of ac optimal power flow (AC OPF) with respect to deviations in predicted power injections from wind and other uncertain generation resources. The aim of this paper is on providing solutions that are more robust to short-term deviations, and that optimize both the initial operating point and a parametrized response policy for control during fluctuations. We formulate this as a chance-constrained optimization problem. To obtain a tractable representation of the chance constraints, we introduce a number of modeling assumptions and leverage recent theoretical results to reformulate the problem as a convex, second-order cone program, which is efficiently solvable even for large instances. Our experiments demonstrate that the proposed procedure improves the feasibility and cost performance of the OPF solution, while the additional computation time is on the same magnitude as a single deterministic AC OPF calculation.
24 POWER TRANSMISSION AND DISTRIBUTION↗
A Sequential Algorithm for Solving Nonlinear Optimization Problems with Chance Constraints
Not provided.
Mathematics↗
An Adaptive Model with Joint Chance Constraints for a Hybrid Wind-Conventional Generator System.
Abstract not provided.
Morton, David↗
Risk-Constrained Dynamic Programming for Optimal Mars Entry, Descent, and Landing
A chance-constrained dynamic programming algorithm was developed that is capable of making optimal sequential decisions within a user-specified risk bound. This work handles stochastic uncertainties over multiple stages in the CEMAT (Combined EDL-Mobility Analyses Tool) framework. It was demonstrated by a simulation of Mars entry, descent, and landing (EDL) using real landscape data obtained from the Mars Reconnaissance Orbiter. Although standard dynamic programming (DP) provides a general framework for optimal sequential decisionmaking under uncertainty, it typically achieves risk aversion by imposing an arbitrary penalty on failure states. Such a penalty-based approach cannot explicitly bound the probability of mission failure. A key idea behind the new approach is called risk allocation, which decomposes a joint chance constraint into a set of individual chance constraints and distributes risk over them. The joint chance constraint was reformulated into a constraint on an expectation over a sum of an indicator function, which can be incorporated into the cost function by dualizing the optimization problem. As a result, the chance-constraint optimization problem can be turned into an unconstrained optimization over a Lagrangian, which can be solved efficiently using a standard DP approach.
Ono, Masahiro↗
Efficient relaxations for joint chance constrained AC optimal power flow
Evolving power systems with increasing levels of stochasticity call for a need to solve optimal power flow problems with large quantities of random variables. Weather forecasts, electricity prices, and shifting load patterns introduce higher levels of uncertainty and can yield optimization problems that are difficult to solve in an efficient manner. Solution methods for single chance constraints in optimal power flow problems have been considered in the literature, ensuring single constraints are satisfied with a prescribed probability; however, joint chance constraints, ensuring multiple constraints are simultaneously satisfied, have predominantly been solved via scenario-based approaches or by utilizing Boole's inequality as an upper bound. In this paper, joint chance constraints are used to solve an AC optimal power flow problem while preventing overvoltages in distribution grids under high penetrations of photovoltaic systems. A tighter version of Boole's inequality is derived and used to provide a new upper bound on the joint chance constraint, and simulation results are shown demonstrating the benefit of the proposed upper bound. The new framework allows for a less conservative and more computationally efficient solution to considering joint chance constraints, specifically regarding preventing overvoltages.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Joint Chance-Constrained Dynamic Programming
This paper presents a novel dynamic programming algorithm with a joint chance constraint, which explicitly bounds the risk of failure in order to maintain the state within a specified feasible region. A joint chance constraint cannot be handled by existing constrained dynamic programming approaches since their application is limited to constraints in the same form as the cost function, that is, an expectation over a sum of one-stage costs. We overcome this challenge by reformulating the joint chance constraint into a constraint on an expectation over a sum of indicator functions, which can be incorporated into the cost function by dualizing the optimization problem. As a result, the primal variables can be optimized by a standard dynamic programming, while the dual variable is optimized by a root-finding algorithm that converges exponentially. Error bounds on the primal and dual objective values are rigorously derived. We demonstrate the algorithm on a path planning problem, as well as an optimal control problem for Mars entry, descent and landing. The simulations are conducted using a real terrain data of Mars, with four million discrete states at each time step.
Ono, Masahiro↗
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.
active distribution systems↗
Modelling wind power spatial-temporal correlation in multi-interval optimal power flow: A sparse correlation matrix approach
The significantly increasing deployment of wind power necessitates that system operation considers the spatial-temporal correlation of power forecast from different wind power plants. How to model this spatial-temporal correlation in the actual system dispatch is challenging. In this paper, a sparse correlation matrix is utilized to efficiently model the spatial-temporal correlation of wind power forecast in the generation dispatch model. A novel, adjustable, and distributionally-robust chance-constrained multi-interval optimal power flow (ADRCC-MIOPF) model is proposed to obtain reliable economic dispatch (ED) solutions. The spatial-temporal correlation of wind power plants power forecasts is represented by the sparse correlation covariance matrix obtained from historical, time series wind power forecast error data. The chance constraints in the ADRCC-MIOPF model are transformed into a set of second-order-cone (SOC) constraints in which an adjustable coefficient in the chance constraints controls the robustness of the ADRCC-MIOPF model to the wind power forecast errors distribution. Case studies performed on the PJM 5-bus system and IEEE 118-bus system verify the proposed method to improve the system security and reduce the cost especially under the high wind penetration levels. All the cases can be solved within several minutes for both the small and large cases which validates the efficiency of the proposed sparse matrix model. In addition, considering the spatial-temporal correlation of wind power forecast and the distributional robustness of wind power forecast error leads to a more reliable economic dispatch with lower system violations.
17 WIND ENERGY↗
A scalable stochastic programming approach for the design of flexible systems
We study the problem of designing systems in order to minimize cost while meeting a given flexibility target. Flexibility is attained by enforcing a joint chance constraint, which ensures that the system will exhibit feasible operation with a given target probability level. Unfortunately, joint chance constraints are complicated mathematical objects that often need to be reformulated using mixed-integer programming (MIP) techniques. In this work, we cast the design problem as a conflict resolution problem that seeks to minimize cost while maximizing flexibility. We propose a purely continuous relaxation of this problem that provides a significantly more scalable approach relative to MIP methods and show that the formulation delivers solutions that closely approximate the Pareto set of the original joint chance-constrained problem.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
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.
09 BIOMASS FUELS↗
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.
Dollar, Robert Austin↗
Probabilistic feasibility assessment of sequestration reliance for climate targets
Countries worldwide are transforming their energy systems to achieve Carbon-Neutrality. Investing in renewable resources-based technologies and implementing Carbon Capture and Storage (CCS) are common strategies to achieve higher sequestration levels. Negative emissions through Bioenergy with CCS are expected to play an essential role in the transition to full decarbonization. On top of that, biomass is a limited resource that depends on environmental factors, which create uncertainties related to the amount that can be sustainably provided to the energy system. Here this study emphasizes the relevance of variability in carbon sequestration for achieving climate targets by 2050. Here this paper proposes a probabilistic approach that integrates the Global Change Analysis Model for Latin America (GCAM-LA) with a chance constraint approach. GCAM-LA is used to assess the impact on the energy sector of different limits of sustainable biomass and carbon budget scenarios. The risk associated with exceeding the sequestration capacity of a given region is modeled via Chance Constraint. Results show that electrification is an appropriate long-term decarbonization strategy. It smoothes the effects of uncertainty in sequestration capacity and responds to end-user demands. For this case study, higher levels of electrification are obtained at likelihood levels >66% for end-use sectors.
54 ENVIRONMENTAL SCIENCES↗