This paper proposes a novel data-driven adaptive robust optimization (ARO) framework for the unit commitment (UC) problem integrating wind power into smart grids. By leveraging a Dirichlet process mixture model, a data-driven uncertainty set for wind power forecast errors is constructed as a union of several basic uncertainty sets. Therefore, the proposed uncertainty set can flexibly capture a compact region of uncertainty in a nonparametric fashion. Based on this uncertainty set and wind power forecasts, a data-driven adaptive robust UC problem is then formulated as a four-level optimization problem. A decomposition-based algorithm is further developed. Compared to conventional robust UC models, the proposed approach does not presume single mode, symmetry, or independence in uncertainty. Moreover, it not only substantially withstands wind power forecast errors, but also significantly mitigates the conservatism issue by reducing operational costs. We also compare the proposed approach with the state-of-the-art data-driven ARO method based on principal component analysis and kernel smoothing to assess its performance. The effectiveness of the proposed approach is demonstrated with the six-bus and IEEE 118-bus systems. Computational results show that the proposed approach scales gracefully with problem size and generates solutions that are more cost effective than the existing data-driven ARO method.

Data-Driven Adaptive Robust Unit Commitment Under Wind Power Uncertainty: A Bayesian Nonparametric Approach

This paper studies a distributionally robust chance constrained program (DRCCP) with Wasserstein ambiguity set, where the uncertain constraints should be satisfied with a probability at least a given threshold for all the probability distributions of the uncertain parameters within a chosen Wasserstein distance from an empirical distribution. In this work, we investigate equivalent reformulations and approximations of such problems. We first show that a DRCCP can be reformulated as a conditional value-at-risk constrained optimization problem, and thus admits tight inner and outer approximations. We also show that a DRCCP of bounded feasible region is mixed integer representable by introducing big-M coefficients and additional binary variables. For a DRCCP with pure binary decision variables, by exploring the submodular structure, we show that it admits a big-M free formulation, which can be solved by a branch and cut algorithm. Finally, we present a numerical study to illustrate the effectiveness of the proposed formulations.

On distributionally robust chance constrained programs with Wasserstein distance

On Distributionally Robust Chance Constrained Programs with Wasserstein Distance

This paper proposed a Wasserstein metric-based distributionally robust approximate framework (WDRA), for unit commitment problem to manage the risk from uncertain wind power forecasted errors. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed framework minimizes the generating cost, start-up cost, shut-down cost, reserve cost, and the expected thermal generation adjusting cost under the worst-case distribution in the ambiguity set. The more historical available, the smaller the ambiguity set is and, hence, the less conservativeness the decision is. The size of the Wasserstein metric based robust counterpart (WDRC) model mainly depends on the size of sample set, which has a computation burden when more historical data are available. To overcome this drawback, this paper proposed an upper approximate of WDRC and verified the condition that makes the approximate model become exact. Comparisons with robust optimization and stochastic optimization illustrate that the proposed model can balance the economy and conservativeness effectively. Monte Carlo simulations on a modified IEEE-118 and a real 703-bus systems show that the proposed WDRA framework could reduce the computational time by several times when compared with WDRC.

Wasserstein Metric Based Distributionally Robust Approximate Framework for Unit Commitment

This article proposes a confidence interval based distributionally robust real-time economic dispatch (CI-DRED) approach, which considers the risk related to accommodating wind power. In this article, only the wind power curtailment and load shedding due to wind power disturbances are evaluated in the operational risk. The proposed approach can strike a balance between the operational costs and risk even when the wind power probability distribution cannot be precisely estimated. A novel ambiguity set is developed based on the imprecise probability theory, which can be constructed based on the point-wise or family-wise confidence intervals. The worst pair of distributions in the established ambiguity set is then identified, and the original CI-DRED problem is transformed into a determined nonlinear dispatch problem accordingly. By using the sequential convex optimization method and piecewise linear approximation method, the nonlinear dispatch model is reformulated as a mixed integer linear programming problem, for which off-the-shelf solvers are available. A fast inactive constraint filtration method is also applied to further relieve the computational burden. Numerical results on the IEEE 118-bus system and a real 445-bus system verify the effectiveness and efficiency of the proposed approach.

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Confidence Interval Based Distributionally Robust Real-Time Economic Dispatch Approach Considering Wind Power Accommodation Risk

Chance constrained optimal power flow (OPF) has been recognized as a promising framework to manage the risk from variable renewable energy (VRE). In the presence of VRE uncertainties, this paper discusses a distributionally robust chance constrained approximate ac-OPF. The power flow model employed in the proposed OPF formulation combines an exact ac power flow model at the nominal operation point and an approximate linear power flow model to reflect the system response under uncertainties. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed OPF model minimizes the expectation of the quadratic cost function w.r.t. the worst-case probability distribution and guarantees the chance constraints satisfied for any distribution in the ambiguity set. The whole method is data-driven in the sense that the ambiguity set is constructed from historical data without any presumption on the type of the probability distribution, and more data leads to smaller ambiguity set and less conservative strategy. Moreover, special problem structures of the proposed problem formulation are exploited to develop an efficient and scalable solution approach. Case studies are carried out on the IEEE 14 and 118 bus systems to show the accuracy and necessity of the approximate ac model and the attractive features of the distributionally robust optimization approach compared with other methods to deal with uncertainties.

Distributionally Robust Chance-Constrained Approximate AC-OPF With Wasserstein Metric

Chance constrained optimal power flow (OPF) has been recognized as a promising framework to manage the risk from variable renewable energy (VRE). In presence of VRE uncertainties, this paper discusses a distributionally robust chance constrained approximate AC-OPF. The power flow model employed in the proposed OPF formulation combines an exact AC power flow model at the nominal operation point and an approximate linear power flow model to reflect the system response under uncertainties. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed OPF model minimizes the expectation of the quadratic cost function w.r.t. the worst-case probability distribution and guarantees the chance constraints satisfied for any distribution in the ambiguity set. The whole method is data-driven in the sense that the ambiguity set is constructed from historical data without any presumption on the type of the probability distribution, and more data leads to smaller ambiguity set and less conservative strategy. Moreover, special problem structures of the proposed problem formulation are exploited to develop an efficient and scalable solution approach. Case studies are carried out on IEEE 14 and 118 bus systems to show the accuracy and necessity of the approximate AC model and the attractive features of the distributionally robust optimization approach compared with other methods to deal with uncertainties.

Li Yao

Papers

Distributionally Robust Chance-Constrained Approximate AC-OPF With Wasserstein Metric

Distributionally Robust Chance-Constrained Approximate AC-OPF with Wasserstein Metric

Distributionally Robust Chance-Constrained Voltage-Concerned DC-OPF with Wasserstein Metric