Monitoring Long Term Voltage Instability due to Distribution & Transmission Interaction using Unbalanced μPMU & PMU Measurements

This paper extends the idea of the Thevenin equivalent into unbalanced 3φ circuits and proposes a 3φ long-term voltage stability indicator (VSI), that can identify critical loads in a system. Furthermore, in order to identify whether the voltage stability limit is due to the transmission network or a distribution network, a transmissiondistribution distinguishing index (TDDI) is proposed. The novelty in the proposed indices is that they can account for the unbalance in the lines and loads, enabling them to use unbalanced phasor measurements naturally. This is supported by mathematical derivations and numerical results. A convex optimization formulation to estimate the 3φ Thevenin equivalent using PMU & iPMU measurements is proposed, making it possible to calculate VSI and TDDI in an online model-free manner. Numerical simulations performed using co-simulation between Pypower and GridlabD are presented for the IEEE 9 bus and the 30 bus transmission networks combined with several modified IEEE 13 node and 37 node distribution networks. These case studies validate the proposed 3φ-VSI and TDDI over a wide range of scenarios and demonstrate the importance of iPMU measurements in identifying the regions causing long term voltage instability.