We address high-dimensional zero-one random parameters in two-stage convex conic optimization problems. Such parameters typically represent failures of network elements and constitute rare, high-impact random events in several applications. Given a sparse training dataset of the parameters, we motivate and study a distributionally robust formulation of the problem using a Wasserstein ambiguity set centered at the empirical distribution. We present a simple, tractable, and conservative approximation of this problem that can be efficiently computed and iteratively improved. Our method relies on a reformulation that optimizes over the convex hull of a mixed-integer conic programming representable set, followed by an approximation of this convex hull using lift-and-project techniques. We illustrate the practical viability and strong out-of-sample performance of our method on nonlinear optimal power flow and multi-commodity network design problems that are affected by random contingencies, and we report improvements of up to 20\% over existing sample average approximation and two-stage robust optimization methods.
In this paper we introduce a new parameterized Quadratic Decision Rule (QDR), a generalisation of the commonly employed Affine Decision Rule (ADR), for two-stage linear adjustable robust optimization problems with ellipsoidal uncertainty and show that (affinely parameterized) linear adjustable robust optimization problems with QDRs are numerically tractable by presenting exact semi-definite program (SDP) and second order cone program (SOCP) reformulations. Under these QDRs, we also establish that exact conic program reformulations also hold for two-stage linear ARO problems, containing also ad…
Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (…
Shunt FACTS devices, such as, a Static Var Compensator (SVC), are capable of providing local reactive power compensation. They are widely used in the network to reduce the real power loss and improve the voltage profile. This paper proposes a planning model based on mixed integer conic programming (MICP) to optimally allocate SVCs in the transmission network considering load uncertainty. The load uncertainties are represented by a number of scenarios. Reformulation and linearization techniques are utilized to transform the original non-convex model into a convex second order cone programming (…
This paper proposes a second-order conic programming (SOCP) approach to solve distributionally robust two-stage stochastic linear programs over 1-Wasserstein balls. We start from the case with distribution uncertainty only in the objective function and exactly reformulate it as an SOCP problem. Then, we study the case with distribution uncertainty only in constraints, and show that such a robust program is generally NP-hard as it involves a norm maximization problem over a polyhedron. However, it is reduced to an SOCP problem if the extreme points of the polyhedron are given as a prior. This m…
In this paper, we present and study a robust unit commitment model and some variants that consider complex temperature and demand uncertainties. Since there is a strong relationship among the efficiency of gas generators, demand, and temperature in practical systems, our robust models have both left- and right-hand-side (LHS and RHS, respectively) uncertainties. Unlike many existing robust models with RHS uncertainty only, the introduction of LHS uncertainty imposes a huge challenge in computing robust solutions. For those complex formulations, we analyze their structures, derive important pro…
This paper presents a new exact method to calculate worst-case parameter realizations in two-stage robust optimization problems with categorical or binary-valued uncertain data. Traditional exact algorithms for these problems, notably Benders decomposition and column-and-constraint generation, compute worst-case parameter realizations by solving mixed-integer bilinear optimization subproblems. However, their numerical solution can be computationally expensive not only due to their resulting large size after reformulating the bilinear terms, but also because decision-independent bounds on their…
We prove a general duality result for multi-stage portfolio optimization problems in markets with proportional transaction costs. The financial market is described by Kabanov's model of foreign exchange markets over a finite probability space and finite-horizon discrete time steps. This framework allows us to compare vector-valued portfolios under a partial ordering, so that our model does not require liquidation into some numeraire at terminal time. We embed the vector-valued portfolio problem into the set-optimization framework, and generate a problem dual to portfolio optimization. Using re…
Wayne State University（韦恩州立大学）；Virginia Tech（弗吉尼亚理工大学）；U.S. Army CCDC Ground Vehicle Systems Center（美国陆军CCDC地面车辆系统中心）；Aptiv（安波福）
Significant advances in sensing, robotics, and wireless networks have enabled the collaborative utilization of autonomous aerial, ground and underwater vehicles for various applications. However, to successfully harness the benefits of these unmanned ground vehicles (UGVs) in homeland security operations, it is critical to efficiently solve UGV path planning problem which lies at the heart of these operations. Furthermore, in the real-world applications of UGVs, these operations encounter uncertainties such as incomplete information about the target sites, travel times, and the availability of…
In this paper, we introduce a new class of decision rules, referred to as Constant Depth Decision Rules (CDDRs), for multistage optimization under linear constraints with uncertainty-affected right-hand sides. We consider two uncertainty classes: discrete uncertainties which can take at each stage at most a fixed number d of different values, and polytopic uncertainties which, at each stage, are elements of a convex hull of at most d points. Given the depth mu of the decision rule, the decision at stage t is expressed as the sum of t functions of mu consecutive values of the underlying uncerta…
This paper proposes a reformulation of the scenario-based two-stage unit commitment problem under uncertainty that allows finding unit-commitment plans that perform reasonably well both in expectation and for the worst case realization of the uncertainties. The proposed reformulation is based on partitioning the sample space of the uncertain factors by clustering the scenarios that approximate their probability distributions. It is, furthermore, very amenable to decomposition and parallelization using a column-and-constraint generation procedure.
We formulate pure characteristics demand models under uncertainties of probability distributions as distributionally robust mathematical programs with stochastic complementarity constraints (DRMP-SCC). For any fixed first-stage variable and a random realization, the second-stage problem of DRMP-SCC is a monotone linear complementarity problem (LCP). To deal with uncertainties of probability distributions of the involved random variables in the stochastic LCP, we use the distributionally robust approach. Moreover, we propose an approximation problem with regularization and discretization to sol…
This paper studies the behavior of a strategic aggregator offering regulation capacity on behalf of a group of distributed energy resources (DERs, e.g. plug-in electric vehicles) in a power market. Our objective is to maximize the aggregator's revenue while controlling the risk of penalties due to poor service delivery. To achieve this goal, we propose data-driven risk-averse strategies to effectively handle uncertainties in: 1) The DER parameters (e.g., load demands and flexibilities) and 2) sub-hourly regulation signals (to the accuracy of every few seconds). We design both the day-ahead and…
Constraint based methods, such as the Flux Balance Analysis, are widely used to model cellular growth processes without relying on extensive information on the regulatory features. The regulation is instead substituted by an optimization problem usually aiming at maximal biomass accumulation. A recent extension to these methods called the dynamic enzyme-cost Flux Balance Analysis (deFBA) is a fully dynamic modeling method allowing for the prediction of necessary enzyme levels under changing environmental conditions. However, this method was designed for deterministic settings in which all dyna…
The increasing penetration of Renewable Energy Sources (RES) requires more Flexibility Resources (FR), generally thermal units and storages, must be kept in the system to accommodate the uncertainties from RES. The challenge is how the system can survive when the RES level is very high. In this paper, RESs are considered as full-role market participants. They can bid in the day-ahead market, and the powers they deliver to the market are controllable up to their maximum available powers. Therefore, RESs are effectively dispatchable and can function as FR providers. To integrate dispatchable ren…
This paper applies the N-block PCPM algorithm to solve multi-scale multi-stage stochastic programs, with the application to electricity capacity expansion models. Numerical results show that the proposed simplified N-block PCPM algorithm, along with the hybrid decomposition method, exhibits much better scalability for solving the resulting deterministic, large-scale block-separable optimization problem when compared with the ADMM algorithm and the PHA algorithm. The superiority of the algorithm's scalability is attributed to the two key features of the algorithm design: first, the proposed hyb…
Model predictive control is a control approach that minimizes a stage cost over a predicted system trajectory based on a model of the system and is capable of handling state and input constraints. For uncertain models, robust or adaptive methods can be used. Because the system model is used to calculate the control law, the closed-loop behavior of the system and thus its performance, measured by the sum of the stage costs, are related to the model used. If it is adapted online, a performance bound is difficult to obtain and thus the impact of model adaptation is mostly unknown. This work provi…
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
In this paper, we consider multi-stage stochastic optimization problems with convex objectives and conic constraints at each stage. We present a new stochastic first-order method, namely the dynamic stochastic approximation (DSA) algorithm, for solving these types of stochastic optimization problems. We show that DSA can achieve an optimal ${\cal O}(1/ε^4)$ rate of convergence in terms of the total number of required scenarios when applied to a three-stage stochastic optimization problem. We further show that this rate of convergence can be improved to ${\cal O}(1/ε^2)$ when the objective func…
This paper presents a chance constrained information gap decision model for multi-period microgrid expansion planning (MMEP) considering two categories of uncertainties, namely random and non-random uncertainties. The main task of MMEP is to determine the optimal sizing, type selection, and installation time of distributed energy resources (DER) in microgrid. In the proposed formulation, information gap decision theory (IGDT) is applied to hedge against non-random uncertainties of long-term demand growth. Then, chance constraints are imposed in the operational stage to address the random uncer…
The combination of boundary flow control and variable speed limit (VSL) is a widely used strategy for highway corridors to maintain safety and throughput when the capacity downstream is dropped. However, most proposed models assume fixed and known demand, which is not always true in reality. Following a recently proposed boundary flow and VSL control model that is based on the Lax-Hopf solution to the Lighthill-Whitham-Richards (LWR) model, this paper proposes a two-stage stochastic model to tackle the uncertainties in demand. Our model assumes the demand follows a discrete distribution and th…
Disasters and disruptions such as the COVID-19 pandemic can significantly interrupt supply chains and industries. To control these disruptions, decision-makers must focus on supply chain resiliency. This paper proposes a multi-stage, multi-period green supply chain design model and six resilience strategies, with downstream and upstream disruptions taken into account to analyze both the ripple and bullwhip effect, respectively. To control the mentioned disruptions and handle the uncertainties of parameter estimations, a two-stage stochastic optimization approach is devised. The objectives are …
In this paper, a blood supply chain network, where the occurrence of disruption might interrupt the flow of Red Blood Cells, is dealt with. In principle, the probability of disruption is not the only property confiding the network, but unprecedented fluctuations in supplies and demands also contribute to the network shortages and outdated blood units. Although the consideration of parameter uncertainties is of paramount importance in the real-world circumstances for a decision-maker, she or he would be willing to monitor the network in a properly broader perspective. Therefore, one of the emin…
The type of decision dependent uncertainties (DDUs) imposes a great challenge in decision making, while existing methodologies are not sufficient to support many real practices. In this paper, we present a systematic study to handle this challenge in two-stage robust optimization~(RO). Our main contributions include three sophisticated variants of column-and-constraint generation method to exactly compute DDU-based two-stage RO. By a novel application of core concepts of linear programming, we provide rigorous analyses on their computational behaviors. Interestingly, in terms of the iteration …
This paper proposes a robust transient stability constrained optimal power flow problem that addresses renewable uncertainties by the coordination of generation re-dispatch and power flow router (PFR) tuning.PFR refers to a general type of network-side controller that enlarges the feasible region of the OPF problem. The coordination between network-side and generator-side control in the proposed model is more general than the traditional methods which focus on generation dispatch only. An offline-online solution framework is developed to solve the problem efficiently. Under this framework the …
The heating, ventilation and air condition (HVAC) system consumes the most energy in commercial buildings, consisting over 60% of total energy usage in the U.S. Flexible HVAC system setpoint scheduling could potentially save building energy costs. This paper first studies deterministic optimization, robust optimization, and stochastic optimization to minimize the daily operation cost with constraints of indoor air temperature comfort and mechanic operating requirement. Considering the uncertainties from ambient temperature, a Wasserstein metric-based distributionally robust optimization (DRO) …
Increasing integration of distributed energy resources (DERs) within distribution feeders provides unprecedented flexibility at the distribution-transmission interconnection. To exploit this flexibility and to use the capacity potential of aggregate DERs, feasible substation power injection trajectories need to be efficiently characterized. This paper provides an ellipsoidal inner approximation of the set of feasible power injection trajectories at the substation such that for any point in the set, there exists a feasible disaggregation strategy of DERs for any load uncertainty realization. Th…
We develop new optimization methodology for planning installation of Flexible Alternating Current Transmission System (FACTS) devices of the parallel and shunt types into large power transmission systems, which allows to delay or avoid installations of generally much more expensive power lines. Methodology takes as an input projected economic development, expressed through a paced growth of the system loads, as well as uncertainties, expressed through multiple scenarios of the growth. We price new devices according to their capacities. Installation cost contributes to the optimization objectiv…
Equilibrium models for energy markets under uncertain demand and supply have attracted considerable attentions. This paper focuses on modelling crude oil market share under the COVID-19 pandemic using two-stage stochastic equilibrium. We describe the uncertainties in the demand and supply by random variables and provide two types of production decisions (here-and-now and wait-and-see). The here-and-now decision in the first stage does not depend on the outcome of random events to be revealed in the future and the wait-and-see decision in the second stage is allowed to depend on the random even…
The H2 guaranteed cost decentralized control problem is investigated in this work. More specifically, on the basis of an appropriate H2 re-formulation that we put in place, the optimal control problem in the presence of parameter uncertainties is then suitably characterized by convex restriction and solved in parameter space. It is shown that a set of stabilizing decentralized controller gains for the uncertain system is parameterized in a convex set through appropriate convex restriction, and then an approximated conic optimization problem is constructed. This facilitates the use of the symme…
