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This paper extends the idea of the Thevenin equivalent into unbalanced 3&#x03C6; circuits and proposes a 3&#x03C6; longterm voltage stability indicator (VSI), that can identify critical loads in a 3&#x03C6; system. Furthermore, in order to identify whether the voltage stability limit is due to the transmission network or a distribution network, a transmission-distribution 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 3&#x03C6; Thevenin equivalent using PMU and &#x03BC;PMU 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 3&#x03C6;-VSI and TDDI over a wide range of scenarios and demonstrate the importance of &#x03BC;PMU measurements in identifying the regions causing long term voltage instability.

ieee transactions on smart grid

Monitoring Long Term Voltage Instability Due to Distribution and Transmission Interaction Using Unbalanced <inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math></inline-formula>PMU and PMU Measurements

Monitoring Long Term Voltage Instability Due to Distribution and Transmission Interaction Using Unbalanced $\mu$ PMU and PMU Measurements