One of the most common control decisions faced by power system operators is the question of how to dispatch generation to meet demand for power. This is a complex optimization problem that includes many nonlinear, non convex constraints as well as inherent uncertainties about future demand for power and available generation. In this paper we develop convex formulations to appropriately model crucial classes of nonlinearities and stochastic effects. We focus on solving a nonlinear optimal power flow (OPF) problem that includes loss of synchrony constraints and models wind-farm caused fluctuations. In particular, we develop (a) a convex formulation of the deterministic phase-difference nonlinear Optimum Power Flow (OPF) problem; and (b) a probabilistic chance constrained OPF for angular stability, thermal overloads and generation limits that is computationally tractable.
Recently, chance-constrained stochastic electricity market designs have been proposed to address the shortcomings of scenario-based stochastic market designs. In particular, the use of chance-constrained market-clearing avoids trading off in-expectation and per-scenario characteristics and yields unique energy and reserves prices. However, current formulations rely on symmetric control policies based on the aggregated system imbalance, which restricts balancing reserve providers in their energy and reserve commitments. This paper extends existing chance-constrained market-clearing formulations…
Existing electricity market designs assume risk neutrality and lack risk-hedging instruments, which leads to suboptimal market outcomes and reduces the overall market efficiency. This paper enables risk-trading in the chance-constrained stochastic electricity market by introducing Arrow-Debreu Securities (ADS) and derives a risk-averse market-clearing model with risk trading. To enable risk trading, the probability space of underlying uncertainty is discretized in a finite number of outcomes, which makes it possible to design practical risk contracts and to produce energy, balancing reserve an…
The controller state and reference governor (CSRG) is an add-on scheme for nominal closed-loop systems with dynamic controllers which supervises the controller internal state and the reference input to the closed-loop system to enforce pointwise-in-time constraints. By admitting both controller state and reference modifications, the CSRG can achieve an enlarged constrained domain of attraction compared to conventional reference governor schemes where only reference modification is permitted. This paper studies the CSRG for systems subject to stochastic disturbances and chance constraints. We d…
While techniques have been developed for chance constrained stochastic optimal control using sample disturbance data that provide a probabilistic confidence bound for chance constraint satisfaction, far less is known about how to use sample data in a manner that can provide almost surely guarantees of chance constraint satisfaction. In this paper, we develop a method for solving chance constrained stochastic optimal control problems for linear time-invariant systems with unknown but sampled additive disturbances. We propose an open-loop control scheme for multi-vehicle planning, with both targ…
While many techniques have been developed for chance constrained stochastic optimal control with Gaussian disturbance processes, far less is known about computationally efficient methods to handle non-Gaussian processes. In this paper, we develop a method for solving chance constrained stochastic optimal control problems for linear time-invariant systems with general additive disturbances with finite moments and unimodal chance constraints. We propose an open-loop control scheme for multi-vehicle planning, with both target sets and collision avoidance constraints. Our method relies on the one-…
In this paper, a sample-based procedure for obtaining simple and computable approximations of chance-constrained sets is proposed. The procedure allows to control the complexity of the approximating set, by defining families of simple-approximating sets of given complexity. A probabilistic scaling procedure then allows to rescale these sets to obtain the desired probabilistic guarantees. The proposed approach is shown to be applicable in several problem in systems and control, such as the design of Stochastic Model Predictive Control schemes or the solution of probabilistic set membership esti…
Chance-constrained programming (CCP) is a promising approach to handle uncertainties in optimal power flow (OPF). However, conventional CCP usually assumes that uncertainties follow Gaussian distributions, which may not match reality. A few papers employed the Gaussian mixture model (GMM) to extend CCP to cases with non-Gaussian uncertainties, but they are only appropriate for cases with uncertainties on the right-hand side but not applicable to DC OPF that containing left-hand side uncertainties. To address this, we develop a tractable GMM-based chance-constrained DC OPF model. In this model,…
In this paper, we develop a chance-constrained decentralized electric vehicle (EV) charging control framework to achieve "valley-filling" meanwhile meeting individual charging requirements and satisfying distribution network constraints. The control design is formulated as an optimization problem with a stochastic non-separable objective function and globally coupled chance constraints. We propose a novel chance-constrained shrunken-primal-dual subgradient (CC-SPDS) algorithm to support the chance-constrained decentralized control scheme and verify its efficacy and convergence with a represent…
We study stochastic optimization problems with chance and risk constraints, where in the latter, risk is quantified in terms of the conditional value-at-risk (CVaR). We consider the distributionally robust versions of these problems, where the constraints are required to hold for a family of distributions constructed from the observed realizations of the uncertainty via the Wasserstein distance. Our main results establish that if the samples are drawn independently from an underlying distribution and the problems satisfy suitable technical assumptions, then the optimal value and optimizers of …
We study quasi-convex optimization problems, where only a subset of the constraints can be sampled, and yet one would like a probabilistic guarantee on the obtained solution with respect to the initial (unknown) optimization problem. Even though our results are partly applicable to general quasi-convex problems, in this work we introduce and study a particular subclass, which we call "quasi-linear problems". We provide optimality conditions for these problems. Thriving on this, we extend the approach of chance-constrained convex optimization to quasi-linear optimization problems. Finally, we s…
The problem of optimizing affine feedback laws that explicitly steer the mean and covariance of an uncertain system state in the presence of a Gaussian random field is considered. Spatially-dependent disturbances are successively approximated with respect to a nominal trajectory by a sequence of jointly Gaussian random vectors. Sequential updates to the nominal control inputs are computed via convex optimization that includes the effect of affine state feedback, the perturbing effects of spatial disturbances, and chance constraints on the closed-loop state and control. The developed method is …
We provide an exact deterministic reformulation for data-driven chance constrained programs over Wasserstein balls. For individual chance constraints as well as joint chance constraints with right-hand side uncertainty, our reformulation amounts to a mixed-integer conic program. In the special case of a Wasserstein ball with the $1$-norm or the $\infty$-norm, the cone is the nonnegative orthant, and the chance constrained program can be reformulated as a mixed-integer linear program. Our reformulation compares favourably to several state-of-the-art data-driven optimization schemes in our numer…
This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution u…
The uncertainty of multiple power loads and renewable energy generations (PLREG) in power systems increases the complexity of power flow analysis for decision-makers. The chance-constrained method can be applied to model the optimization problems of power flow under uncertainty. This paper develops a novel solution approach for chance-constrained AC optimal power flow (CCACOPF) problem based on the data-driven convexification of power flow and a fast algorithm for scenario technique (FAST). This method is computationally effective for mainly two reasons. First, the original nonconvex AC power …
The integration of distributed renewable generation (DRG) in distribution networks can be effectively promoted by scheduling flexible resources such as heating, ventilation, and air conditioning (HVAC) loads. However, finding the optimal scheduling for them is nontrivial because DRG outputs are highly uncertain. To address this issue, this paper proposes a learning-based joint chance-constrained approach to coordinate HVAC loads with DRG. Unlike cutting-edge works adopting individual chance constraints to manage uncertainties, this paper controls the violation probability of all critical const…
A method is developed to numerically solve chance constrained optimal control problems. The chance constraints are reformulated as nonlinear constraints that retain the probability properties of the original constraint. The reformulation transforms the chance constrained optimal control problem into a deterministic optimal control problem that can be solved numerically. The new method developed in this paper approximates the chance constraints using Markov Chain Monte Carlo (MCMC) sampling and kernel density estimators whose kernels have integral functions that bound the indicator function. Th…
This paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem via a Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). Through Lagrangian relaxation, we convert the chance-constrained (risk-constrained) SOC problem to a risk-minimizing SOC problem, the cost function of which possesses the time-additive Bellman structure. We show that the risk-minimizing control synthesis is equivalent to solving an HJB PDE whose boundary condition can be tuned appropriately to achieve a desired level of safety. Furthermore, it is shown that …
Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we establish an exact reformulation of chance constrained problems as a bilevel problems with convex lower-levels. We then derive a tractable penalty approach, where the penalized objective is a difference-of-convex function that we minimize with a suitable bundle algorithm. We release …
Agent behavior is arguably the greatest source of uncertainty in trajectory planning for autonomous vehicles. This problem has motivated significant amounts of work in the behavior prediction community on learning rich distributions of the future states and actions of agents. However, most current works on chance-constrained trajectory planning under agent or obstacle uncertainty either assume Gaussian uncertainty or linear constraints, which is limiting, or requires sampling, which can be computationally intractable to encode in an optimization problem. In this paper, we extend the state-of-t…
We propose a safe approximation to joint chance-constrained programming where the constraint functions are additively dependent on a normally-distributed random vector. The approximation is analytical, meaning that it requires neither numerical integrations nor sampling-based probability approximations. Under mild assumptions, the approximation is a standard nonlinear program. We compare this new safe approximation to another analytical safe approximation for joint chance-constrained programming based on Boole's inequality through two examples representing the constrained control of linear Gau…
We consider chance-constrained binary programs, where each row of the inequalities that involve uncertainty needs to be satisfied probabilistically. Only the information of the mean and covariance matrix is available, and we solve distributionally robust chance-constrained binary programs (DCBP). Using two different ambiguity sets, we equivalently reformulate the DCBPs as 0-1 second-order cone (SOC) programs. We further exploit the submodularity of 0-1 SOC constraints under special and general covariance matrices, and utilize the submodularity as well as lifting to derive extended polymatroid …
Uncertainties from deepening penetration of renewable energy resources have posed critical challenges to the secure and reliable operations of future electric grids. Among various approaches for decision making in uncertain environments, this paper focuses on chance-constrained optimization, which provides explicit probabilistic guarantees on the feasibility of optimal solutions. Although quite a few methods have been proposed to solve chance-constrained optimization problems, there is a lack of comprehensive review and comparative analysis of the proposed methods. Part I of this two-part pape…
This paper aims to provide a Dynamic Programming (DP) approach to solve the Mission-Wide Chance-Constrained Optimal Control Problems (MWCC-OCP). The mission-wide chance constraint guarantees that the probability that the entire state trajectory lies within a constraint/safe region is higher than a prescribed level, and is different from the stage-wise chance constraints imposed at individual time steps. The control objective is to find an optimal policy sequence that achieves both (i) satisfaction of a mission-wide chance constraint, and (ii) minimization of a cost function. By transforming th…
Chance constrained program is computationally intractable due to the existence of chance constraints, which are randomly disturbed and should be satisfied with a probability. This paper proposes a two-layer randomized algorithm to address chance constrained program. Randomized optimization is applied to search the optimizer which satisfies chance constraints in a framework of parallel algorithm. Firstly, multiple decision samples are extracted uniformly in the decision domain without considering the chance constraints. Then, in the second sampling layer, violation probabilities of all the extr…
University of New Mexico（新墨西哥大学）；Stanford University（斯坦福大学）
We present a data-driven algorithm for efficiently computing stochastic control policies for general joint chance constrained optimal control problems. Our approach leverages the theory of kernel distribution embeddings, which allows representing expectation operators as inner products in a reproducing kernel Hilbert space. This framework enables approximately reformulating the original problem using a dataset of observed trajectories from the system without imposing prior assumptions on the parameterization of the system dynamics or the structure of the uncertainty. By optimizing over a finit…
In recent years, there has been a huge trend to penetrate renewable energy sources into energy networks. However, these sources introduce uncertain power generation depending on environmental conditions. Therefore, finding 'optimal' and 'feasible' operation strategies is still a big challenge for network operators and thus, an appropriate optimization approach is of utmost importance. In this paper, we formulate the optimal power flow (OPF) with uncertainties as a chance constrained optimization problem. Since uncertainties in the network are usually 'non-Gaussian' distributed random variables…
A warm start method is developed for efficiently solving complex chance constrained optimal control problems. The warm start method addresses the computational challenges of solving chance constrained optimal control problems using biased kernel density estimators and Legendre-Gauss-Radau collocation with an $hp$ adaptive mesh refinement method. To address the computational challenges, the warm start method improves both the starting point for the chance constrained optimal control problem, as well as the efficiency of cycling through mesh refinement iterations. The improvement is accomplished…
In this paper, we propose a robust optimization-based heuristic algorithm for the chance-constrained binary knapsack problem (CKP). We assume that the weights of items are independent normally distributed. By utilizing the properties of the submodular function, the proposed method approximates the CKP to the robust knapsack problem with a cardinality constrained uncertainty set parameterized by a uncertainty budget parameter. The proposed approach obtains a heuristic solution by solving the approximated robust knapsack problem whose optimal solution can be obtained by solving the ordinary bina…
In this paper, we study unit commitment (UC) problems considering the uncertainty of load and wind power generation. UC problem is formulated as a chance-constrained two-stage stochastic programming problem where the chance constraint is used to restrict the probability of load imbalance. In addition to the conventional mixed integer linear programming formulation using Big-M, we present the bilinear mixed integer formulation of chance constraint, and then derive its linear counterpart using McCormick linearization method. Then, we develop a bilinear variant of Benders decomposition method, wh…
We consider chance-constrained problems with discrete random distribution. We aim for problems with a large number of scenarios. We propose a novel method based on the stochastic gradient descent method which performs updates of the decision variable based only on considering a few scenarios. We modify it to handle the non-separable objective. Complexity analysis and a comparison with the standard (batch) gradient descent method is provided. We give three examples with non-convex data and show that our method provides a good solution fast even when the number of scenarios is large.
We introduce a novel approach to reduce the computational effort of solving mixed-integer convex chance constrained programs through the scenario approach. Instead of reducing the number of required scenarios, we directly minimize the computational cost of the scenario program. We exploit the problem structure by efficiently partitioning the constraint function and considering a multiple chance constrained program that gives the same probabilistic guarantees as the original single chance constrained problem. We formulate the problem of finding the optimal partition, a partition achieving the l…
Chance-constrained programs (CCPs) constitute a difficult class of stochastic programs due to its possible nondifferentiability and nonconvexity even with simple linear random functionals. Existing approaches for solving the CCPs mainly deal with convex random functionals within the probability function. In the present paper, we consider two generalizations of the class of chance constraints commonly studied in the literature; one generalization involves probabilities of disjunctive nonconvex functional events and the other generalization involves mixed-signed affine combinations of the result…
