We investigate an approach to the approximation of ambiguous chance constrained programs (ACCP) in which the underlying distribution describing the random parameters is itself uncertain. We model this uncertainty with the assumption that the unknown distribution belongs to a closed ball centered around a fixed and known distribution. Using only samples drawn from the central distribution, we approximate ACCP with a robust sampled convex program (RSCP), and establish an upper bound on the probability that a solution to the RSCP violates the original ambiguous chance constraint, when the uncertainty set is defined in terms of the Prokhorov metric. Our bound on the constraint violation probability improves upon the existing bounds for RSCPs in the literature. We also consider another approach to approximating ACCP by means of a sampled convex program (SCP), which is built on samples drawn from the central distribution. Again, we provide upper bounds on the probability that a solution to the SCP violates the original ambiguous chance constraint for uncertainty sets defined according to a variety of metrics.

}, keywords = {RM14-002}, doi = {10.1109/CDC.2016.7799224}, author = {Tseng, Shih-Hao and Eilyan Bitar and Tang, Ao} } @article {1373, title = {Risk-limiting dispatch for integrating renewable power}, journal = {International Journal of Electrical Power \& Energy Systems}, volume = {44}, year = {2013}, month = {01/2013}, pages = {615 - 628}, abstract = {Risk-limiting dispatch or RLD is formulated as the optimal solution to a multi-stage, stochastic decision problem. At each stage, the system operator (SO) purchases forward energy and reserve capacity over a block or interval of time. The blocks get shorter as operations approach real time. Each decision is based on the most recent available information, including demand, renewable power, weather forecasts. The accumulated energy blocks must at each time t match the net demand D(t) = L(t) - W(t). The load L and renewable power W are both random processes. The expected cost of a dispatch is the sum of the costs of the energy and reserve capacity and the penalty or risk from mismatch between net demand and energy supply. The paper derives computable {\textquoteleft}closed-form{\textquoteright} formulas for RLD. Numerical examples demonstrate that the minimum expected cost can be substantially reduced by recognizing that risk from current decisions can be mitigated by future decisions; by additional intra-day energy and reserve capacity markets; and by better forecasts. These reductions are quantified and can be used to explore changes in the SO{\textquoteright}s decision structure, forecasting technology, and renewable penetration.}, keywords = {CERTS, reliability and markets, renewables integration, reserve markets, RM11-006}, issn = {01420615}, doi = {10.1016/j.ijepes.2012.07.048}, author = {Ram Rajagopal and Eilyan Bitar and Pravin Varaiya and Felix Wu} } @conference {1375, title = {Risk limiting dispatch of wind power}, booktitle = {2012 American Control Conference (ACC)}, year = {2012}, month = {06/2012}, pages = {4417 - 4422}, publisher = {IEEE}, organization = {IEEE}, address = {Montreal, QC}, abstract = {Integrating wind and solar power into the grid requires dispatching various types of reserve generation to compensate for the randomness of renewable power. The dispatch is usually determined by a system operator (SO) or an aggregator who {\textquoteleft}firms{\textquoteright} variable energy by bundling it with conventional power. The optimal dispatch is formulated as the solution to a stochastic control problem and shown to have a closed form that can be quickly computed. Different objectives and risk constraints can be included in the formulation and trade-offs can be evaluated. In particular one can quantify the influence of sequential forecasts on the total integration cost and the choice of dispatched generation. When the forecast error is Gaussian, the optimal dispatch policy can be precomputed.

}, keywords = {CERTS, reliability and markets, reserve generation, risk-limiting dispatch, RM11-006, wind power}, isbn = {978-1-4577-1095-7}, doi = {10.1109/ACC.2012.6315239}, author = {Ram Rajagopal and Eilyan Bitar and Felix Wu and Pravin Varaiya} }