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Page 409 Yang et al. Intell Robot 2024;4(4):406-21 I http://dx.doi.org/10.20517/ir.2024.24

Figure 1. PMU measurement in smart grids. PMU: Phasor measurement unit.

account for errors caused by noise or data packet loss. Since TSA can introduce time deviations that easily
bypass basic defense mechanisms such as smoothing filters and bad data detection (BDD), investigating the
impact of TSA on µPMU data collection and ensuring the reliability of grid monitoring data has become a
pressing issue that needs to be addressed.

2.1. Principle of system state estimation
In the power system network, µPMUs are strategically distributed across the buses, providing detailed mea-
surements of system dynamics. µPMUs capture both the complex voltage of the bus and the complex currents
flowing through all transmission lines connected to it, enabling precise monitoring of power flow dynamics.
Considering a local network, suppose there are buses, connected by transmission lines. The number
of transmission lines connected to bus n is . The set of other buses connected to bus is denoted as .

∈ . The bus voltage vector for all bus-es is denoted

The µPMU measurement vector is = 1 , · · · ,
×2
as = , ∈ , = 1, . . . , , where and represent the real and imaginary parts of ,
respectively. Similarly, for the branch connected to bus , the complex current vector of the bus transmission
× ×2
, , , ∈ , = 1, . . . , , where , and , represent the
line can be expressed as =
of the transmission line and bus . The measurement
real and imaginary parts of the complex current
vector is:
   
   | | cos 
   
   | | sin 
=   =   (1)

 ,   cos 
   
 ,   sin 

   
Where and are the corresponding phase angles of the voltage and current. In the classical state estima-
tion problem, the relationship between the µPMU measurements and the system state vector is established as
follows:

= ℎ ( ) + (2)

where ℎ( ) represents the measurement function of the system state vector = [ 1 , . . . , ] ∈ ; =

represents the random measurement errors introduced during the measurement process. These
1 , . . .
h i
2 2
2
errorsfollowaGaussiandistribution ∼ 0, ,andthecovariancematrix isdenotedas = , . . . , .
1
DuringthestateestimationprocessusingµPMUs, toimprovetheaccuracyoftheestimation, theweightedleast