Low-Dimensional ODE Embedding to Convert Low-Resolution Meters Into "Virtual" PMUs

Power systems are integrating uncertain generations, demanding transient analyses using dynamic measurements. However, High-Resolution (HR) Phasor Measurement Units are few. The aim is to interpolate dynamic data for extensive but Low-Resolution (LR) meters. Existing interpolation methods capture data correlations but ignore the governing equations, i.e., Ordinary Differential Equations (ODEs) or Differential Algebraic Equations (DAEs). To solve DAEs, traditional solvers suffer from accumulative errors. The error can be reduced by fitting measurements in a recent gradient-based solver, i.e., Physics-Informed Neural Networks (PINNs). Nevertheless, PINN convergence is hard due to limited LR samples. To fill the missing information, it is noted that HR and LR data essentially lie in a low-dimensional embedding space governed by ODEs/DAEs. Hence, this paper proposes to smartly explore the embedding space through generating a good initial guess of LR data and enforcing the ODE/DAE constraints as refinement. For good initialization, the approach (1) captures the spatial-temporal correlations with another NN that maps from HR to LR variables, and (2) utilizes the rich HR data patterns to train the NN in Semi-Supervised Learning. Then, physical constraints are enforced to restrict the initial values, which leverage a PINN with a DAE function loss. For systems with unknown DAE parameters, a parameter estimation is introduced using measured and interpolated but erroneous data, where an error-corrected mechanism guarantees accuracy. The interpolation and estimation work coordinately, leading to the CoPIE: A Coordinate framework for Physics-informed Interpolation and Estimation. It is demonstrated that CoPIE has a much tighter error bound than other methods. Eventually, the high interpolation performance of CoPIE in different transmission and distribution systems is reported.