Extrapolating to Unseen Solar PV Levels: Physics-Informed Voltage Calculation in Low-Voltage Networks

Residential rooftop solar photovoltaic (PV) systems raise customer voltages in low voltage (LV) networks. Customer voltages must remain within statutory limits, set to ensure the adequate operation of the network and the equipment connected to it, so distribution companies need to know how much more PV a network can host. Answering that means calculating voltages at unseen PV levels, beyond any recorded to date. This calculation typically requires detailed LV electrical models, which are often unavailable or inaccurate for distribution companies, so no power flow can be run. Machine learning models such as neural networks (NNs), trained on smart-meter data (active and reactive power [$P,Q$], and voltage magnitude [$V$]), remove that requirement, but the unseen PV levels lie outside the range present in the training data, so the NN must extrapolate and calculates incorrect voltages. Two Physics-Informed Neural Network (PINN) formulations are compared, enforcing three relations among the NN's own voltage sensitivities, its derivatives with respect to $P$ and $Q$: a first-order Taylor approximation linking power and voltage changes, and the symmetry of both sensitivity matrices. None needs an electrical model. A weighted-loss (WL) PINN applies a fixed penalty weight, whereas an augmented Lagrangian (AL) PINN penalizes violations beyond a tolerance. Both are compared against a single-hidden-layer NN on a realistic Australian LV network of 126 customers, across low, medium and high impedance, using power-flow results as the benchmark. The models are trained at 25\% PV penetration and evaluated up to 100\%. There the NN misses a limit breach that both PINNs identify, while the AL-PINN locates the maximum voltage within 4.8 V against 10.9 V for the WL-PINN. Extrapolating to unseen PV levels is therefore possible without an electrical model, and tightening the AL-PINN's tolerance improves the calculation, whereas raising the penalty weight does not.