The system frequency response (SFR) model describes the average network frequency response after a disturbance and has been applied to a wide variety of dynamic studies. However, the traditional literature does not provide a generic, analytical method for obtaining the SFR model parameters when the system contains multiple generators; instead, a numerical simulation-based approach or the operators' experience is the common practice to obtain an aggregated model. In this paper, an analytical method is proposed for aggregating the multi-machine SFR model into a single-machine model. The verification study indicates that the proposed aggregated SFR model can accurately represent the multi-machine SFR model. Furthermore, the detailed system simulation illustrates that the SFR model can also accurately represent the average frequency response of large systems for power system dynamic studies. Finally, three applications of the proposed method are explored, with system frequency control, frequency stability, and dynamic model reduction. The results show the method is promising with broad potential applications.

With the recent proliferation of open-source packages for computing, power system differential-algebraic equation (DAE) modeling and simulation are being revisited to reduce the programming efforts. Existing open-source tools require manual efforts to develop code for numerical equations, sparse Jacobians, and discontinuous components. This paper proposes a hybrid symbolic-numeric framework, exemplified by an open-source Python-based library ANDES, which consists of a symbolic layer for descriptive modeling and a numeric layer for vector-based numerical computation. This method enables the implementation of DAE models by mixing and matching modeling components, through which models are described. In the framework, a rich set of discontinuous components and standard transfer function blocks are provided besides essential modeling elements for rapid modeling. ANDES can automatically generate robust and fast numerical simulation code, as well as and high-quality documentation. Case studies present a) two implementations of turbine governor model TGOV1, b) power flow computation time break down for MATPOWER systems, c) validation of time-domain simulation with commercial software using three test systems with a variety of models, and d) the full eigenvalue analysis for Kundur's system. Validation shows that ANDES closely matches the commercial tool DSATools for power flow, time-domain simulation, and eigenvalue analysis.

With the increasing penetration of renewable energy, frequency response and its security are of significant concerns for reliable power system operations. Frequency-constrained unit commitment (FCUC) is proposed to address this challenge. Despite existing efforts in modeling frequency characteristics in unit commitment (UC), current strategies can only handle oversimplified low-order frequency response models and do not consider wide-range operating conditions. This paper presents a generic data-driven framework for FCUC under high renewable penetration. Deep neural networks (DNNs) are trained to predict the frequency response using real data or high-fidelity simulation data. Next, the DNN is reformulated as a set of mixed-integer linear constraints to be incorporated into the ordinary UC formulation. In the data generation phase, all possible power injections are considered, and a region-of-interest active sampling is proposed to include power injection samples with frequency nadirs closer to the UFLC threshold, which enhances the accuracy of frequency constraints in FCUC. The proposed FCUC is investigated on the IEEE 39-bus system. Then, a full-order dynamic model simulation using PSS/E verifies the effectiveness of FCUC in frequency-secure generator commitments.