Two-stage Adjustable Robust Optimization Research Articles

LP-based approximations for disjoint bilinear and two-stage adjustable robust optimization

LP-based approximations for disjoint bilinear and two-stage adjustable robust optimization

Pre-Perturbation Operational Strategy Scheduling in Microgrids by Two-Stage Adjustable Robust Optimization

A two-stage adaptive robust optimization is developed for pre-disturbance scheduling in microgrids (MGs) for handling uncertainties associated with electricity market prices, renewable generation, demand forecasts, and islanding events. The objective is to produce a reliable and optimal solution for MG operation that minimizes operational costs and the risk/failure in islanding events. In the literature, the uncertainty sets associated with islanding events cover a full scheduling period which results in a sub-optimal solution. In this paper, uncertainty sets corresponding to islanding events are modeled based on reliability/resiliency-oriented indexes of the MG/grid to achieve a more accurate/reliable solution. Besides, the Benders decomposition algorithm which is used to handle uncertainties in solving the optimization problem is time-consuming. Therefore, the column-and-constraint generation (C&CG) decomposition strategy is adopted to make the problem computationally tractable. Further, the uncertainty budget parameters are clarified to balance the conservatism and optimality (cost minimization) of the robust solution in uncertainty sets. The effectiveness of the proposed framework is evaluated and discussed by using a set of numerical studies with different scenarios in an MG. The simulations show that the proposed framework reduces operational costs by using the precise analysis of uncertainty budgets and a change in scheduling periods.

Data-driven adjustable robust Day-ahead economic dispatch strategy considering uncertainties of wind power generation and electric vehicles

Considering the intermittent features of wind power generation and electric vehicles, it is important for microgrid to formulate available dispatch strategy while ensuring the system economy. A two-stage data-driven adjustable robust optimization (ARO) model is presented in this paper to realize an optimal day-ahead economic dispatch strategy for the microgrid considering the uncertainties of wind power generation and electric vehicles. In the optimization model, an imprecise-Dirichlet-model-based nonparametric ambiguity set is constructed without any presumption about the probability distributions of the uncertainties. Further, a polyhedron uncertainty set obtained from the ambiguity set is driven by historical data, and this becomes narrower with an increase in the amount of the historical data used to construct the ambiguity set. The data-driven ARO problem is converted into a traditional two-stage ARO model with the obtained polyhedron uncertainty set. Further, this can be decomposed into a mixed integer linear programming (MILP) master problem and a sub problem with a max–min structure. Then, duality theory, Big-M method, and column and constraint generation (C&CG) algorithm are applied to achieve the optimized day-ahead economic dispatch strategy for the microgrid. Case studies illustrate the solution robustness, economic benefit, flexible adjustment capability and uncertainty depiction ability of the presented method.

On the Optimality of Affine Policies for Budgeted Uncertainty Sets

In this paper, we study the performance of affine policies for a two-stage, adjustable, robust optimization problem with a fixed recourse and an uncertain right-hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets, widely used in practice, in which we can specify a budget on the adversarial deviations of the uncertain parameters from the nominal values to adjust the level of conservatism. The two-stage adjustable robust optimization problem is hard to approximate within a factor better than [Formula: see text] even for budget of uncertainty sets in which [Formula: see text] is the number of decision variables. Affine policies, in which the second-stage decisions are constrained to be an affine function of the uncertain parameters provide a tractable approximation for the problem and have been observed to exhibit good empirical performance. We show that affine policies give an [Formula: see text]-approximation for the two-stage, adjustable, robust problem with fixed nonnegative recourse for budgeted uncertainty sets. This matches the hardness of approximation, and therefore, surprisingly, affine policies provide an optimal approximation for the problem (up to a constant factor). We also show strong theoretical performance bounds for affine policy for a significantly more general class of intersection of budgeted sets, including disjoint constrained budgeted sets, permutation invariant sets, and general intersection of budgeted sets. Our analysis relies on showing the existence of a near-optimal, feasible affine policy that satisfies certain nice structural properties. Based on these structural properties, we also present an alternate algorithm to compute a near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.

Adjustable robust optimization in enabling optimal day-ahead economic dispatch of CCHP-MG considering uncertainties of wind-solar power and electric vehicle

At present, electric vehicles (EVs), small-scale wind power, and solar power have been increasingly integrated into modern power system via the combined cooling heating and power based microgrid (CCHP-MG). However, inside the microgrid the uncertainties of EVs charging, wind power, and solar power significantly impact the economy of CCHP-MG operation. Therefore to improve the economy deteriorated by the uncertainties, this paper presents a two-stage adjustable robust optimization to achieve the minimal operational cost for CCHP-MG. Before the realizations of the uncertainties, the day-ahead stage as the first stage decides an operational strategy that can withstand the worst-case uncertainties. As long as the uncertainties are observed, the real-time stage as the second stage adjusts the operational units to compensate the errors caused by the day-ahead operational strategy. Due to the difficulties of the model solution, this paper further adopts the duality theory, Big-M method, and column-and-constraint generation (C & CG) decomposition to convert the model into two tractable mixed integer linear programming (MILP) problems. Further, C & CG iteration algorithm is also employed to solve the MILPs, which can ultimately provide an optimal economic day-ahead dispatch strategy capable of handling uncertainties. The experimental results demonstrate the effectiveness of the presented approach.

Decentralized optimal operation model for cooperative microgrids considering renewable energy uncertainties

An increasing number of microgrids (MGs) with high penetration of renewables are being integrated into the distribution system (DS). Multiple MGs as stakeholders in the future retail markets may sign a cooperation contract when faced with high uncertainties of renewable energies and complementary features among MGs. The optimal coordinated operation mechanism for the MGs would be essential. In this paper, we propose a decentralized optimal operation framework to coordinate the power exchange among MGs abiding by a cooperative contract based on the analytical target cascading (ATC) algorithm. Furthermore, to address the renewable energy uncertainties in the operation strategy of each MG, a two-stage adjustable robust optimization model is devised to coordinate the power exchange between the DS and MGs, the demand response, the output of controlled distributed generations (DGs), and the charging/discharging behavior of energy storage system (ESS). Then, the column and constraint generation method (CCG) is applied to solve the proposed robust optimization model. Comparative studies based on a DS with three typical MGs are performed to validate the effectiveness of the proposed methods.

A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization

We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right-hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large. A significant challenge in designing a practical piecewise affine policy is constructing good pieces of the uncertainty set. Here we address this challenge by introducing a new framework in which the uncertainty set is “approximated” by a “dominating” simplex. The corresponding policy is then based on a mapping from the uncertainty set to the simplex. Although our piecewise affine policy has exponentially many pieces, it can be computed efficiently by solving a compact linear program given the dominating simplex. Furthermore, we can find the dominating simplex in a closed form if the uncertainty set satisfies some symmetries and can be computed using a MIP in general. We would like to remark that our policy is an approximate piecewise-affine policy and is not necessarily a generalization of the class of affine policies. Nevertheless, the performance of our policy is significantly better than the affine policy for many important uncertainty sets, such as ellipsoids and norm-balls, both theoretically and numerically. For instance, for hypersphere uncertainty set, our piecewise affine policy can be computed by an LP and gives a $$O(m^{1/4})$$ -approximation whereas the affine policy requires us to solve a second order cone program and has a worst-case performance bound of $$O(\sqrt{m})$$ .

Robust Operation of Soft Open Points in Active Distribution Networks With High Penetration of Photovoltaic Integration

Distributed generators including photovoltaic (PV) panels have been integrated dramatically in active distribution networks (ADNs). Due to the strong volatility and uncertainty, the high penetration of PV generation immensely exacerbates the conditions of voltage violation in ADNs. However, the emerging flexible interconnection technology based on soft open points (SOPs) provides increased controllability and flexibility to the system operation. For fully exploiting the regulation ability of SOPs to address the problems caused by PV, this paper proposes a robust optimization method to achieve the robust optimal operation of SOPs in ADNs. A two-stage adjustable robust optimization model is built to tackle the uncertainties of PV outputs, in which robust operation strategies of SOPs are generated to eliminate the voltage violations and reduce the power losses of ADNs. A column-and-constraint generation algorithm is developed to solve the proposed robust optimization model, which are formulated as second-order cone program to facilitate the accuracy and computation efficiency. Case studies on the modified IEEE 33-node system and comparisons with the deterministic optimization approach are conducted to verify the effectiveness and robustness of the proposed method.

Robust Capacity Assessment of Distributed Generation in Unbalanced Distribution Networks Incorporating ANM Techniques

To settle a large-scale integration of renewable distributed generations (DGs), it requires to assess the maximal DG hosting capacity of active distribution networks (ADNs). For fully exploiting the ability of ADNs to accommodate DG, this paper proposes a robust comprehensive DG capacity assessment method considering three-phase power flow modelling and active network management (ANM) techniques. The two-stage adjustable robust optimization is employed to tackle the uncertainties of load demands and DG outputs. With our method, system planners can obtain the maximum penetration level of DGs with their optimal sizing and sitting decisions. Meanwhile, the robust optimal ANM schemes can be generated for each operation time period, including network reconfiguration, on-load-tap-changers regulation, and reactive power compensation. In addition, a three-step optimization algorithm is proposed to enhance the accuracy of DG capacity assessment results. The optimality and robustness of our method are validated via numerical tests on an unbalanced IEEE 33-bus distribution system.

A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides

We study two-stage adjustable robust linear programming in which the right-hand sides are uncertain and belong to a convex, compact uncertainty set. This problem is NP-hard, and the affine policy is a popular, tractable approximation. We prove that under standard and simple conditions, the two-stage problem can be reformulated as a copositive optimization problem, which in turn leads to a class of tractable, semidefinite-based approximations that are at least as strong as the affine policy. We investigate several examples from the literature demonstrating that our tractable approximations significantly improve the affine policy. In particular, our approach solves exactly in polynomial time a class of instances of increasing size for which the affine policy admits an arbitrarily large gap.