A handbook for quickly querying different stability criteria.

Bases of Stability Criterion[1]

Middlebrook Criterion

Forbidden Region Criterion

AC Stability Criterion

Criterion of SUN Jian in balanced three-phase systems[2]

• Considering a single-phase system as an example
• Modeling the grid-connected inverters as a current source
• Can be extended to a balanced three-phase system

Generalized Nyquist Criterion

D-channel Criterion

• Using the D-channel element $Z_{Sdd}(s)$ and $Y_{Ldd}(s)$
• Not a sufficient condition for the stability judgement

Criterion based on the norm of the impedance [3] [4]

• Sufficient condition for the stability judgement

Singular-Value Criterion

Using the maximum singular-value of the matrix, which is also the 2-norm of the matrix:

$\max{\sigma}(Z_{Sdq}(j\omega))\times \max{\sigma}(Y_{Ldq}(j\omega))<1, \forall \omega \in (-\infin,+\infin)\\ \sigma(A)=\sqrt{\lambda(A^H\times A)}\\ A^H=\rm{conj}(\it A^T \rm)$

G-norm Criterion

$||Z_{Sdq}(j\omega)||_G \cdot ||Y_{Ldq}(j\omega)||_G<\frac{1}{4}, \forall\omega\in(-\infin, +\infin)\\ ||A_{m\times n}||_G=\max{|a_{ij}|}$

Infinity-One-Norm Criterion

$||Z_{Sdq}(j\omega)||_{\infin}\cdot ||Y_{Ldq}(j\omega)||_1 < \frac{1}{2}, \forall\omega\in(-\infin,+\infin)\\ ||A_{m\times n}||_1=\max_{1\le j\le n}{(\Sigma^{m}_{i=1}{|a_{ij}|})}\\ ||A_{m\times n}||_{\infin}=\max_{1\le i\le m}{(\Sigma^{n}_{j=1}{|a_{ij}|})}$

Infinity-Norm Criterion[3]

$||Z_{Sdq}(j\omega)||_{\infin}\cdot ||Y_{Ldq}(j\omega)||_{\infin}< 1, \forall\omega\in(-\infin,+\infin)\\ ||A_{m\times n}||_{\infin}=\max_{1\le i\le m}{(\Sigma^{n}_{j=1}{|a_{ij}|})}$

G-Sum-Norm Criterion[5]

G-Sum-Norm Criterion is the least conservative criterion, but the calculation is complicated.

$||Z_{Sdq}(j\omega)||_G \cdot ||Y_{Ldq}(j\omega)||_G<1, \forall\omega\in(-\infin, +\infin)\\ ||A_{m\times n}||_G=\max{|a_{ij}|}\\ ||A_{m\times n}||_{\rm sum}=\Sigma\Sigma{|a_{ij}|}$

Comparison of different AC stability criterions[3]

References

[1] 直流分布式电源系统稳定性研究, 张欣, PhD dissertation

[2] Impedance-Based Stability Criterion for Grid-Connected Inverters, SUN Jian, 2011, TPEL Letters

[3] Infinity-Norm of Impedance-Based Stability Criterion for Three-Phase AC Distributed Power Systems With Constant Power Loads, LIU Zeng, 2015, TPEL

[4] Stability criterion for AC power systems with regulated loads, M. Belkhayat, PhD dissertation, 1997

[5] 基于G-范数和sum-范数的三相交流级联系统稳定性判据, 刘方诚, 电机工程学报, 2014