Probabilistic approach to investigate the impact of distributed generation on voltage deviation in distribution system

Abstract

The increasing penetration of renewable-based generation in the distribution network adds randomness to the bus voltage due to its stochastic characteristics. Consequently, the distribution network faces voltage limit violations at multiple buses. In this scenario, the traditional voltage control method fails to maintain the voltage profile of the network due to its slow response time. A fast and dynamic voltage control technique can be achieved by utilizing the voltage-controlling capability of available DGs in the network with the information about the impact of DG-integrated bus on change in voltage at any bus in the network. Evaluating this information using the traditional probabilistic load flow method is computationally inefficient for real-time applications. Therefore, in this paper, a principal component analysis (PCA)-based novel probabilistic voltage sensitivity index (PVSI) is proposed. It ranks the DG-integrated buses by evaluating the impact of power perturbation at DG-integrated buses on changes in voltage at any bus in the network. The PVSI is analytically derived to reduce the computation time for ranking. Further, the proposed PVSI is validated on the 69-bus and the 141-bus distribution systems by comparing it with the traditional Monte Carlo simulation (MCS) and joint differential entropy (JDE) in terms of accuracy and computation time. From the simulation, it is observed that PVSI gives a similar ranking as both methods in less computation time. Due to the advantage in computation time, the proposed method enables the distribution system operator (DSO) to use this information for fast and dynamic voltage control in a real-time scenario.

Formation of covariance matrix

$$\begin{aligned}{} & {} \Sigma _{(\Delta V_{oak},\Delta V_{o})}= \left[ \begin{array}{llll} \hbox {var}(\Delta V^r_{oak}) &{} \hbox {cov}(\Delta V^i_{oak},\Delta V^r_{oak}) &{} \hbox {cov}(\Delta V^r_{oak},\Delta V^i_{o})&{}\hbox {cov}(\Delta V^r_{oak},\Delta V^r_{o}) \\ \hbox {cov}(\Delta V^i_{oak},\Delta V^r_{oak}) &{} \hbox {var}(\Delta V^i_{oak}) &{} \hbox {cov}(\Delta V^i_{oak},\Delta V^r_{o}) &{} \hbox {cov}(\Delta V^i_{oak},\Delta V^i_{o}) \\ \hbox {cov}(\Delta V^r_{o},\Delta V^r_{oak}) &{} \hbox {cov}(\Delta V^i_{oak},\Delta V^r_{o}) &{} \hbox {var}(\Delta V^r_{o}) &{} \hbox {cov}(\Delta V^i_{o},\Delta V^r_{o})\\ \hbox {cov}(\Delta V^r_{oak},\Delta V^r_{o}) &{}\hbox {cov}(\Delta V^i_{oak},\Delta V^i_{o}) &{} \hbox {cov}(\Delta V^i_{o},\Delta V^r_{o}) &{} \hbox {var}(\Delta V^i_{o})\\ \end{array}\right] \end{aligned}$$

(A1)

$$\begin{aligned}{} & {} \Sigma _{(\Delta V^i_{oak},\Delta V^i_{o})}= \begin{bmatrix} \hbox {var}(\Delta V^i_{oak}) &{} \hbox {cov}(\Delta V^i_{oak},\Delta V^r_{o}) \\ \hbox {cov}(\Delta V^i_{oak},\Delta V^r_{o}) &{} \hbox {var}(\Delta V^i_{o}) \end{bmatrix} \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} \begin{aligned}&\hbox {Mean}(\Delta V^r_{o})=\sum _{\begin{array}{c} k=1 \end{array}}^nE(\Delta V^r_{oak})\\&\quad =-R_{oak}\sum _{\begin{array}{c} k=1 \end{array}}^nE(\Delta P_{oak}) +X_{oak}\sum _{\begin{array}{c} k=1 \end{array}}^nE(\Delta Q_{oak}) \end{aligned} \end{aligned}$$

(A3)

$$\begin{aligned}{} & {} \begin{aligned}&Mean(\Delta V^i_{o})=\sum _{\begin{array}{c} k=1 \end{array}}^nE(\Delta V^i_{oak})\\&\quad =-R_{oak}\sum _{\begin{array}{c} k=1 \end{array}}^nE(\Delta Q_{oak}) -X_{oak}\sum _{\begin{array}{c} k=1 \end{array}}^nE(\Delta P_{oak}) \end{aligned} \end{aligned}$$

(A4)

$$\begin{aligned}{} & {} \begin{aligned}&\hbox {var}(\Delta V^r_{o})=\sum _{\begin{array}{c} k=1 \end{array}}^n \hbox {var}(\Delta V^r_{oak})\\&\quad =\sum _{\begin{array}{c} k=1 \end{array}}^n \hbox {var}(\Delta V^r_{oak})+\sum _{\begin{array}{c} j=1 \\ k\ne j \end{array}}^n(\Delta V^r_{oak}\Delta V^r_{oaj} ) \end{aligned} \end{aligned}$$

(A5)

$$\begin{aligned}{} & {} \begin{aligned}&\hbox {var}(\Delta V^i_{o})=\sum _{\begin{array}{c} k=1 \end{array}}^n \hbox {var}(\Delta V^i_{oak})\\&\quad =\sum _{\begin{array}{c} k=1 \end{array}}^n \hbox {var}(\Delta V^i_{oak})+\sum _{\begin{array}{c} j=1 \\ k\ne j \end{array}}^n(\Delta V^i_{oak}\Delta V^i_{oaj} ) \end{aligned} \end{aligned}$$

(A6)

$$\begin{aligned}{} & {} \hbox {cov}(\Delta V^r_{oak},\Delta V^r_{o})=E(\Delta V^r_{oak},\Delta V^r_{o}) =E(\Delta V^r_{oak}\sum _{\begin{array}{c} j=1 \end{array} }^n \Delta V^r_{oaj}) \end{aligned}$$

(A7)

$$\begin{aligned}{} & {} =E(\Delta V^r_{oak}\Delta V^r_{oak})+E(\sum _{\begin{array}{c} j=1 \\ k\ne j \end{array} }^n\Delta V^r_{oak}\Delta V^r_{oaj} ) \vspace{-.5cm}\nonumber \\{} & {} =\hbox {var}(\Delta V^r_{oak})+\sum _{\begin{array}{c} j=1 \\ k\ne j \end{array} }^nE(\Delta V^r_{oak}\Delta V^r_{oaj} ) \end{aligned}$$

(A8)

$$\begin{aligned}{} & {} \hbox {cov}(\Delta V^i_{oak},\Delta V^i_{o})=E(\Delta V^i_{oak},\Delta V^i_{o}) =E(\Delta V^i_{oak}\sum _{\begin{array}{c} j=1 \end{array} }^n \Delta V^i_{oaj})\nonumber \\{} & {} =\hbox {var}(\Delta V^i_{oak})+\sum _{\begin{array}{c} j=1 \\ k\ne j \end{array} }^nE(\Delta V^i_{oak}\Delta V^i_{oaj} ) \end{aligned}$$

(A9)

From (10)

$$\begin{aligned}{} & {} \hbox {var}(\Delta V^r_{oak})=\hbox {var}(\Delta Q_{ak}X_{oak}-\Delta P_{ak}R_{oak}) \end{aligned}$$

(A10)

$$\begin{aligned}{} & {} \begin{aligned}&=X^2_{oak}\hbox {var}(\Delta Q_{ak})+R^2_{oak}\hbox {var}(\Delta P_{ak}) \\&\quad -2R_{oak}X_{oak}\hbox {cov}(\Delta P_{ak}\Delta Q_{ak}) \\ \end{aligned} \end{aligned}$$

(A11)

$$\begin{aligned}{} & {} \begin{aligned}&E(\Delta V^r_{oak},\Delta V^r_{oaj}) =E((\Delta Q_{ak}X_{oak}-\Delta P_{ak}R_{oak})\\&\quad (\Delta Q_{aj}X_{oaj}-\Delta P_{aj}R_{oaj})) \end{aligned} \end{aligned}$$

(A12)

$$\begin{aligned}{} & {} =E(\Delta Q_{aj}\Delta Q_{ak}X_{oak}X_{oaj})-E(\Delta Q_{aj}\Delta P_{ak}R_{oak}X_{oaj}) \nonumber \\{} & {} \quad -E(\Delta Q_{ak}\Delta P_{aj}R_{oaj}X_{oak})+E(\Delta P_{aj}\Delta P_{ak}R_{oak}X_{oak}) \nonumber \\{} & {} =X_{oak}X_{oaj}E(\Delta Q_{aj}\Delta Q_{ak})-R_{oak}X_{oaj}E(\Delta Q_{aj}\Delta P_{ak}) \nonumber \\{} & {} \quad -R_{oaj}X_{oai}E(\Delta Q_{ak}\Delta P_{aj})+R_{oai}X_{oak}E(\Delta P_{aj}\Delta P_{ak}) \nonumber \\= & {} X_{oak}X_{oaj}\hbox {cov}(\Delta Q_{aj}\Delta Q_{ak})-R_{oak}X_{oaj}\hbox {cov}(\Delta Q_{aj}\Delta P_{ak}) \nonumber \\{} & {} -R_{oaj}X_{oai}\hbox {cov}(\Delta Q_{ak}\Delta P_{aj})+R_{oai}X_{oak}\hbox {cov}(\Delta P_{aj}\Delta P_{ak}) \nonumber \\ \end{aligned}$$

(A13)

From (11)

$$\begin{aligned}{} & {} \begin{aligned} var(\Delta V^i_{oak})=\hbox {var}(-\Delta Q_{ak}R_{oak}-\Delta P_{ak}X_{oak}) \end{aligned} \end{aligned}$$

(A14)

$$\begin{aligned}{} & {} \begin{aligned}&\quad =R_{oak}\hbox {var}(\Delta Q_{ak})+X_{oak}\hbox {var}(\Delta P_{ak})\\&\quad +2X_{oak}R_{oak}\hbox {cov}(\Delta Q_{ak}\Delta P_{ak}) \end{aligned} \end{aligned}$$

(A15)

$$\begin{aligned}{} & {} \begin{aligned}&E(\Delta V^i_{oak},\Delta V^i_{oaj})=E((-\Delta Q_{ak}R_{oak}-\Delta P_{ak}X_{oak})\\&\quad (-\Delta Q_{aj}R_{oaj}-\Delta P_{aj}X_{oaj})) \end{aligned} \end{aligned}$$

(A16)

$$\begin{aligned} \nonumber \\{} & {} \begin{aligned}&R_{oai}R_{oaj}\hbox {cov}(\Delta Q_{ak}\Delta Q_{aj})+R_{oaj}X_{oak}\hbox {cov}(\Delta P_{ak}\Delta Q_{aj})\\&\quad X_{oaj}R_{oak}\hbox {cov}(\Delta P_{aj}\Delta Q_{ak})+X_{oaj}X_{oak}\hbox {cov}(\Delta P_{ak}\Delta P_{aj}) \end{aligned}\nonumber \\ \end{aligned}$$

(A17)

$$\begin{aligned}{} & {} \begin{aligned}&E(\Delta V^r_{oak},\Delta V^i_{o})=E((\Delta Q_{ak}R_{oak}-\Delta P_{ak}X_{oak})\\&\quad (-\Delta Q_{aj}R_{oaj}-\Delta P_{aj}X_{oaj})) \end{aligned} \end{aligned}$$

(A18)

$$\begin{aligned}{} & {} -X_{oak}R_{oaj}\hbox {cov}(\Delta Q_{ak}\Delta Q_{aj})+R_{oaj}R_{oak}\hbox {cov}(\Delta P_{ak}\Delta Q_{aj})\nonumber \\{} & {} \quad -X_{oak}X_{oak}\hbox {cov}(\Delta P_{aj}\Delta Q_{ak})+X_{oak}R_{oaj}\hbox {cov}(\Delta P_{ak}\Delta P_{aj}) \nonumber \\{} & {} E(\Delta V^i_{oak},\Delta V^r_{o})=-X_{oaj}R_{oak}\hbox {cov}(\Delta Q_{ak}\Delta Q_{aj})\nonumber \\{} & {} \quad +R_{oaj}R_{oak}\hbox {cov}(\Delta P_{aj}\Delta Q_{ak} +X_{oak}R_{oaj}\hbox {cov}(\Delta P_{ak}\Delta P_{aj})\nonumber \\{} & {} \quad -X_{oak}X_{oaj}\hbox {cov}(\Delta P_{ak}\Delta Q_{aj}) \end{aligned}$$

(A19)