  With the widespread use of power electronic devices, modern distribution
networks are turning into flexible distribution networks (FDNs), which have
enhanced active and reactive power flexibility at the
transmission-distribution-interface (TDI). However, owing to the stochastics
and volatility of distributed generation, the flexibility can change in real
time and can hardly be accurately captured using conventional discrete-time
(DT) assessment methods. This paper first proposes the notion of
continuous-time (CT) TDI active and reactive flexibility and establishes its
mathematical model. This model comprehensively considers the flexible devices
in the FDN and the impact of uncertainty of photovoltaic power generation and
load. In particular, a novel direction-factor-based metric is proposed to model
CT-TDI PQ flexibility. Moreover, an efficient solution method is designed to
address the difficulties in handling the infinite dimension of CT model and the
complexity of bi-objectivity from assessing both active and reactive
flexibility to be assessed. The solution successfully transforms the infinite
dimensional optimization into a finite dimensional problem and effectively
explores the PQ plane in a parallel pattern. Case studies show that the method
can more effectively assess the real-time TDI flexibility of an FDN relative to
conventional DT counterparts, and also reveals the impact of the relevant
factors, such as penetrations of flexible devices and levels of uncertainty.
