Robust Convex Programming Research Articles

An Energy Management System for the Optimal Operation of BESS in DC Microgrids: A Robust Convex Programming Approach

This paper uses a convex reformulation to deal with the robust, optimal energy management of battery energy storage systems (BESS) and renewable energies in DC microgrids. This relaxed mathematical model guarantees the global optimum regarding energy management problems, even when including uncertainties in demand and renewable energy. The proposed robust model can reach the best scheduling for energy management in worst-case cost scenarios while satisfying all requirements. Furthermore, a model for power transfer losses in converter devices is added to the proposed model by using a binary polynomial representation, which is convexified. This convexification is performed by transforming the nonlinear non-convex equations of the optimal energy management model into second-order cone constraints. Four scenarios implemented in the modified IEEE 123-bus radial distribution feeder are proposed in order to analyze the robust optimal energy management strategy: the deterministic model, demand uncertainty, uncertainty in solar and wind generators, and uncertainty in demand and solar and wind generators. In all scenarios, the energy management model reduces the total energy costs. Simulation scenario 1 showed daily operating costs of about US$ 4376.82, while simulation scenarios 2, 3, and 4, including demand and generation uncertainties, increased the daily operating costs by about US$ 5243.23 (19.79%), US$ 5072.23 (15.88%), and US$ 5738.75 (31.12%), respectively. Scenario 4 showed the highest costs, as it involves more uncertainty. Hence, the robust optimal energy management strategy dispatches more energy from conventional generators, increasing the operating costs to satisfy those of the worst case.

Data-Driven Safety Verification of Discrete-Time Networks: A Compositional Approach

This letter proposes a compositional data-driven approach for safety verification of networks of discrete-time subsystems with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">formal guarantees</i> . Following a modular approach and for each subsystem, we search for a so-called sub-barrier candidate represented as a linear combination of a priori user-defined basis functions. We formulate the conditions on sub-barrier candidates as robust convex programs (RCPs) which are semi-infinite linear programs. Collecting sampled data from each subsystem, we approximate each RCP via a scenario convex program (SCP) which is a finite linear program. We provide an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">explicit</i> formula to compute the minimum number of sampled data guaranteeing a desired mismatch between the optimal value of each RCP and that of the corresponding SCP in a probabilistic sense. To ensure that the sum of the sub-barrier candidates is a barrier function for the whole network, we define a global dissipativity condition on top of the local SCPs. The local SCPs are thus related to each other via this global condition. This results in a large-scale optimization problem in a standard canonical form which is efficiently solved by the alternating direction method of multipliers (ADMM) algorithm. The effectiveness of our approach is illustrated by applying it to a room temperature control problem in a 100-room building.

Distributed Algorithms for Robust Convex Optimization via the Scenario Approach

This paper proposes distributed algorithms to solve robust convex optimization (RCO) when the constraints are affected by nonlinear uncertainty. We adopt a scenario approach by randomly sampling the uncertainty set. To facilitate the computational task, instead of using a single centralized processor to obtain a “global solution” of the scenario problem (SP), we resort to multiple interconnected processors that are distributed among different nodes of a network to simultaneously solve the SP. Then, we propose a primal-dual subgradient algorithm and a random projection algorithm to distributedly solve the SP over undirected and directed graphs, respectively. Both algorithms are given in an explicit recursive form with simple iterations, which are especially suited for processors with limited computational capability. We show that, if the underlying graph is strongly connected, each node asymptotically computes a common optimal solution to the SP with a convergence rate $O(1/(\sum _{t=1}^k\zeta ^t))$ , where $\lbrace \zeta ^t\rbrace$ is a sequence of appropriately decreasing stepsizes. That is, the RCO is effectively solved in a distributed way. The relations with the existing literature on robust convex programs are thoroughly discussed and an example of robust system identification is included to validate the effectiveness of our distributed algorithms.

Multivariate Predictive Analytics of Wind Power Data for Robust Control of Energy Storage

Short-term forecasting is frequently identified as an important tool for the effective management of wind generation. However, forecasting errors, inherent to the point forecasts, increase requirements for energy storage and can affect optimal system operation. Probabilistic forecasts can help tackle this issue by providing a proper characterization of forecasting errors in the optimization process. This paper proposes a multivariate model of forecasting data for wind generation. Predictive uncertainty intervals of wind power can be obtained by sampling from the proposed model. The main goal is to use empirical data models without linear or Gaussian approximations of the distributional or temporal variations. The predictive modeling is utilized within a case study of an energy storage system. A modified robust convex programming is used to maintain the practical robustness and feasibility of the solution based on the sampled scenarios from the model.

Performance Bounds for the Scenario Approach and an Extension to a Class of Non-Convex Programs

We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilistic bridge from the optimal value of SCP to the optimal values of RCP and CCP in which the uncertainty takes values in a general, possibly infinite dimensional, metric space. We then extend our results to a certain class of non-convex problems that includes, for example, binary decision variables. In the process, we also settle a measurability issue for a general class of scenario programs, which to date has been addressed by an assumption. Finally, we demonstrate the applicability of our results on a benchmark problem and a problem in fault detection and isolation.

Strong Duality in Robust Convex Programming: Complete Characterizations

Duality theory has played a key role in convex programming in the absence of data uncertainty. In this paper, we present a duality theory for convex programming problems in the face of data uncertainty via robust optimization. We characterize strong duality between the robust counterpart of an uncertain convex program and the optimistic counterpart of its uncertain Lagrangian dual. We provide a new robust characteristic cone constraint qualification which is necessary and sufficient for strong duality in the sense that the constraint qualification holds if and only if strong duality holds for every convex objective function of the program. We further show that this strong duality always holds for uncertain polyhedral convex programming problems by verifying our constraint qualification, where the uncertainty set is a polytope. We derive these results by way of first establishing a robust theorem of the alternative for parameterized convex inequality systems using conjugate analysis. We also give a convex characteristic cone constraint qualification that is necessary and sufficient for strong duality between the deterministic dual pair: the robust counterpart and its Lagrangian dual. Through simple numerical examples we also provide an insightful account of the development of our duality theory.

Robust convex quadratically constrained programs

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. In contrast to [4], where it is shown that such robust problems can be formulated as semidefinite programs, our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second-order cone programs (SOCP). We propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit SOCP and present examples where these classes of uncertainty sets are natural.