Tightened voltage stability index approximation bound

Let a power distribution network have one slack bus and $n$ PQ loads, and let it be described by the relaxed branch flow model. In the voltage stability region, set

$$\rho=\rho(\varphi_{u,\operatorname{diag}}^{' -1}\varphi_{u,\operatorname{off}}^{'})$$

where $\rho(\cdot)$ denotes the spectral radius, and let $\operatorname{VSI}$ and $\operatorname{VSIA}$ denote the voltage stability index and its approximation, respectively. Tightened approximation bound. The bound in the main theorem can be tightened to

$$\operatorname{VSI}\leq\operatorname{VSIA}\leq\operatorname{VSI}-\frac{1}{n}\rho\,\ln(1-\rho).$$

The proposed tightening is motivated by the observation that the factor $n$ in the determinant approximation bound can be replaced by the number of eigenvalues whose magnitude is close to the spectral radius. The authors report that their simulations generally exhibit only one such eigenvalue, but the statement is presented as a conjectural implication rather than established as a theorem.