Critical buses identification for voltage stability assessment considering the application of modal analysis and a robust state estimation with bad data suppression

Abstract

This paper investigates the application of modal analysis in the results of a proposed robust state estimation technique to determine the most critical buses of power distribution systems within the context of voltage stability assessment. In the proposed approach, the Weighted Least Squares (WLS) technique is used to estimate state variables based on micro-PMUs (Phasor Measurement Units) and SMs (smart meters) measurements. The estimated voltage magnitudes and angles are used to form a reduced Jacobian matrix which provides the sensitivities of reactive powers and voltage magnitudes at each load bus. Based on the calculation of eigenvalues and eigenvectors, participation factors are estimated for each load bus. In order to minimize the influence of gross errors, a bad data detection procedure is performed using Chi-squared test. The proposed identification procedure is based on the Median Absolute Deviation (MAD) of the normalized measurement residuals. The corrupted measurements are re-weighted in the proposed methodology to suppress its influence on the results yielding high accurate estimates. Simulations are carried out using a 33-bus test system. The impact of distributed generation units, aleatory errors and bad data is evaluated by different case studies to prove the efficiency of the proposed method.

Appendix A: nonlinear functions

The nonlinear function ${\mathbf {h}}({\hat{\varvec{x}}})$ represents the estimated values calculated as function of the state variables in such a way that active/reactive powers, active/reactive flows and the branch currents can be determined from (28) to (30):

$$\begin{aligned} P_k&=V_k\sum \limits _{m\in K} V_m (G_{km}cos(\theta _{km})+B_{km}sen(\theta _{km})) \end{aligned}$$

(28)

$$\begin{aligned} Q_k&=V_k\sum \limits _{m\in K} V_m (G_{km}sen(\theta _{km})\nonumber \\&\quad -B_{km}cos(\theta _{km})) \end{aligned}$$

(29)

$$\begin{aligned}&\theta _{km}=\theta _k-\theta _m \end{aligned}$$

(30)

$$\begin{aligned} P_{km}&= V_k^2(g_{sk}+g_{km})-V_kV_m(g_{km}cos(\theta _{km})\nonumber \\&\quad +b_{km}sen(\theta _{km})) \end{aligned}$$

(31)

$$\begin{aligned} Q_{km}&= -V_i^2(b_{sk}+b_{km})-V_kV_m(g_{km}sen(\theta _{km})\nonumber \\&\quad -b_{km}cos(\theta _{km})) \end{aligned}$$

(32)

$$\begin{aligned} \dot{I}_{k}&=\frac{P_k-jQ_k}{\dot{V}_k} \end{aligned}$$

(33)

$$\begin{aligned} \dot{I}_{km}&=\frac{P_{km}-jQ_{km}}{\dot{V}_k} \end{aligned}$$

(34)

The elements of the Jacobian matrix related to active power measurements with respect to voltage angles and magnitudes are presented in (35).

$$\begin{aligned} \frac{\partial P_k}{\partial \theta _k}&=-V_k^2B_{kk} + \sum \limits _{m\in \Omega _k} V_kV_m(-G_{km}sin(\theta _{km}) +B_{km}cos(\theta _{km})) \nonumber \\ \frac{\partial P_k}{\partial \theta _m}&= V_k V_m (G_{km}sin(\theta _{km})-B_{km}cos(\theta _{km}))\nonumber \\ \frac{\partial P_k}{\partial V_k}&= -V_kG_{kk} + \sum \limits _{m\in \Omega _k} V_kV_m(G_{km}cos(\theta _{km}) +B_{km}sin(\theta _{km}))\nonumber \\ \frac{\partial P_k}{\partial V_m}&= V_k (G_{km}cos(\theta _{km})+B_{km}sin(\theta _{km})) \end{aligned}$$

(35)

The elements corresponding to the reactive power measurements are presented in the set of equations (36) with respect to voltage angles and magnitudes.

$$\begin{aligned} \frac{\partial Q_k}{\partial \theta _k}= & {} -V_k^2G_{kk} + \sum \limits _{m\in \Omega _k} V_kV_m(G_{km}cos(\theta _{km}) +B_{km}sin(\theta _{km})) \nonumber \\ \frac{\partial Q_k}{\partial \theta _m}= & {} V_k V_m (-G_{km}cos(\theta _{km})-B_{km}sin(\theta _{km})) \frac{\partial Q_k}{\partial V_k}\nonumber \\= & {} -V_kB_{kk} + \sum \limits _{m\in \Omega _k} V_kV_m(-G_{km}sin(\theta _{km})\nonumber \\&\quad -B_{km}cos(\theta _{km})) \nonumber \\ \frac{\partial Q_k}{\partial V_m}&= V_k (G_{km}sin(\theta _{km})-B_{km}cos(\theta _{km})) \end{aligned}$$

(36)

The elements of ${\mathbf {H}}$ corresponding to the PMUs voltage magnitudes and angles are presented in (37) and (38), respectively.

$$\begin{aligned}&\frac{\partial V_k}{\partial \theta _k}= 0, \frac{\partial V_k}{\partial \theta _m}=0, \frac{\partial V_k}{\partial V_k}= 1, \frac{\partial V_k}{\partial V_m}= 0 \end{aligned}$$

(37)

$$\begin{aligned}&\frac{\partial \theta _k}{\partial \theta _k}= 1, \frac{\partial \theta _k}{\partial \theta _m}=0, \frac{\partial \theta _k}{\partial V_k}= 0, \frac{\partial \theta _k}{\partial V_m}= 0 \end{aligned}$$

(38)

The partial derivatives related to the branch currents measurements magnitudes provided by PMUs are given by equations (40). The elements corresponding to the branch currents angles are provided in (40).

$$\begin{aligned} \frac{\partial I_{km}}{\partial \theta _k}&= \frac{D_{km} V_k V_m}{\sqrt{E_{km}}} \nonumber \\ \frac{\partial I_{km}}{\partial \theta _m}&= - \frac{D_{km}V_k V_m}{\sqrt{E_{km}}}\nonumber \\ \frac{\partial I_{km}}{\partial V_k}&= \frac{A_{km}V_k+C_{km}V_m}{\sqrt{E_{km}}} \nonumber \\ \frac{\partial I_{km}}{\partial V_m}&= \frac{B_{km}V_k+C_{km}V_m}{\sqrt{E_{km}}} \nonumber \\ \end{aligned}$$

(39)

$$\begin{aligned} \frac{\partial \delta _{km}}{\partial \theta _k}&= \frac{A_{km} V_k^2+C_{km}V_k V_m}{{E_{km}}} \nonumber \\ \frac{\partial \delta _{km}}{\partial \theta _m}&= -\frac{B_{km} V_k^2+C_{km}V_k V_m}{{E_{km}}}\nonumber \\ \frac{\partial \delta _{km}}{\partial V_k}&= -\frac{D_{km} V_m}{{E_{km}}} \nonumber \\ \frac{\partial \delta _{km}}{\partial V_m}&= \frac{D_{km} V_k}{{E_{km}}} \nonumber \\ \end{aligned}$$

(40)

where $A_{km}, B_{km}, C_{km}, D_{km}$ are defined by (41):

$$\begin{aligned} A_{km}&=(G_{km}+G_{kk})^2+(B_{km}+B_{kk})^2\nonumber \\ B_{km}&=(G_{km}^2+B_{km}^2)\nonumber \\ C_{km}&=(B_{km}+B_{kk})(G_{km}sin(\theta _{km})-B_{km}cos(\theta _{km}))\nonumber \\&\quad -(G_{km}+G_{kk})(G_{km}cos(\theta _{km})+B_{km}sin(\theta _{km}))\nonumber \\ D_{km}&=(G_{km}+G_{kk})(G_{km}sin(\theta _{km})-B_{km}cos(\theta _{km})) \nonumber \\&\quad - (B_{km}+B_{kk})(G_{km}cos(\theta _{km})+B_{km}sin(\theta _{km}))\nonumber \\ E_{km}&=A_{km}V_k^2 + B_{km}V_m^2 + 2C_{km}V_k V_m \end{aligned}$$

(41)