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Page 411 Yang et al. Intell Robot 2024;4(4):406-21 I http://dx.doi.org/10.20517/ir.2024.24

Table 1. Phase angle error caused by TSA
Reference Phase-angle error (degrees)
Jiang et al. [27] 52
Zhang et al. [26] ±60
Shepard et al. [28] 72
TSA: Time synchronization attack.

In system state estimation, the largest normalized residual (LNR) method is used for anomaly detection. This
method normalizes the residuals of each measurement and then compares the maximum normalized residual
to a predefined threshold. Specifically, if the LNR is below the threshold , it indicates that the measurement
does not contain significant bad data and can be accepted for further analysis and processing. Otherwise, the
measurement is considered abnormal, potentially containing erroneous data, and requires further inspection
and handling. This method effectively improves the reliability of data quality and enhances the overall stability
of the system.

2.2. TSA attack model
During a TSA, the attacker sends spoofed GPS signals, introducing incorrect time synchronization in µPMU
measurements. This disrupts the phase angles of synchronized data, causing a mismatch between measured
and actual phase values [25] . Assume that the attack occurs on bus . The power system operates at a frequency
[26] , causing the µPMU’s phase angle to change, leading to a
of 60 Hz. The attack induces a clock shift Δ

phase angle deviation Δ :

Δ = 2 Δ (11)

Table 1 presents the phase angle deviation errors caused by TSA. The phase angle deviation of the affected bus
can be expressed as:

exp Δ (12)
=

Through analysis, the relationship between the measurement vector under TSA and the normal measurement
vector can be derived. The phase angle deviation Δ falls within the interval [0, 2 ]or[− , ], depending on

the specific conditions. To further describe the measurement vector under TSA for all buses, the measurement
vector is:

Δ Δ
· 1 , . . . · , . . . , (13)
= 1 , . . . 1

represent the µPMUs affected by TSA. The mea-
Where ( = 1, . . . , ) is a diagonal matrix. 1 , . . . ,
surement vector for each affected bus can be expressed as:

= · (14)

The residual error vector is: