We develop DeepOPF as a Deep Neural Network (DNN) approach for solving security-constrained direct current optimal power flow (SC-DCOPF) problems, which are critical for reliable and cost-effective power system operation. DeepOPF is inspired by the observation that solving the SC-DCOPF problem for a given power network is equivalent to depicting a highdimensional mapping between load inputs and generation and phase-angle outputs. We first construct and train a DNN to learn the mapping between the load inputs and the generations. We then directly compute the phase angles from the generations and loads by using the (linearized) power flow equations. Such a two-step procedure significantly reduces the dimension of the mapping to learn, subsequently cutting down the size of the DNN and the amount of training data/time needed. We further characterize a condition that allows us to tune the size of our neural network according to the desired approximation accuracy of the load-to-generation mapping. Simulation results of IEEE test cases show that DeepOPF always generates feasible solutions with negligible optimality loss, while speeding up the computing time by up to 400x as compared to a state-of-the-art solver.1 There are two types of SC-DCOPF problems, namely the preventive SC-DCOPF problem and the corrective SC-DCOPF problem. In the preventive SC-DCOPF problem, the system operating decisions cannot change once they are determined, thus they need to guarantee feasibility under both the preand post-contingency constraints. For the corrective SC-DCOPF problem, the system operator can have a short time (e.g., 5 minutes) [12] to adjust the operating points after the occurrence of each contingency. Our DeepOPF approach is applicable to both problems. We focus on the preventive SC-DCOPF problem in this paper for easy illustration.

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The AC-OPF problem is the key and challenging problem in the power system operation. When solving the AC-OPF problem, the feasibility issue is critical. In this paper, we develop an efficient Deep Neural Network (DNN) approach, DeepOPF, to ensure the feasibility of the generated solution. The idea is to train a DNN model to predict a set of independent operating variables, and then to directly compute the remaining dependable variables by solving the AC power flow equations. While this guarantees the power-flow balances, the principal difficulty lies in ensuring that the obtained solutions satisfy the operation limits of generations, voltages, and branch flow. We tackle this hurdle by employing a penalty approach in training the DNN. As the penalty gradients make the common first-order gradient-based algorithms prohibited due to the hardness of obtaining an explicit-form expression of the penalty gradients, we further apply a zero-order optimization technique to design the training algorithm to address the critical issue. The simulation results of the IEEE test case demonstrate the effectiveness of the penalty approach. Also, they show that DeepOPF can speed up the computing time by one order of magnitude compared to a state-of-the-art solver, at the expense of minor optimality loss.

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Deep Neural Networks (DNNs) approaches for the Optimal Power Flow (OPF) problem received considerable attention recently. A key challenge of these approaches lies in ensuring the feasibility of the predicted solutions to physical system constraints. Due to the inherent approximation errors, the solutions predicted by DNNs may violate the operating constraints, e.g., the transmission line capacities, limiting their applicability in practice. To address this challenge, we develop DeepOPF+ as a DNN approach based on the so-called "preventive" framework. Specifically, we calibrate the generation and transmission line limits used in the DNN training, thereby anticipating approximation errors and ensuring that the resulting predicted solutions remain feasible. We theoretically characterize the calibration magnitude necessary for ensuring universal feasibility. Our DeepOPF+ approach improves over existing DNN-based schemes in that it ensures feasibility and achieves a consistent speed up performance in both light-load and heavy-load regimes. Detailed simulation results on a range of test instances show that the proposed DeepOPF+ generates 100% feasible solutions with minor optimality loss. Meanwhile, it achieves a computational speedup of two orders of magnitude compared to state-of-the-art solvers. NOMENCLATUREVariable Definition N Set of buses, N |N |. G Set of generators. D Set of loads. E Set of branches. PG Power generation injection vector, [PG i , i ∈ N ]. P min G Minimum generator output vector, [P min G i , i ∈ N ]. P max G Maximum generator output vector, [P max G i , i ∈ N ]. PD Power load vector, [PD i , i ∈ N ]. Θ Voltage angle vector. θi Voltage angle for bus i. B Admittance matrix. xijLine reactance from bus i to j. P max T ij Line transmission limit from bus i to j. NhidThe number of hidden layers in the neural network.