The rapid evolution of high-precision synchronized time sources, such as GPS and 5G signals, has ushered in novel timing concepts for wide-area synchronized measurements. Additionally, these advancements have introduced an innovative signal-triggered sampling mechanism capable of dynamically adjusting the sampling rate based on specific requirements. This mechanism represents a pivotal development in the acquisition and processing of sinusoidal signals. By enabling the dynamic alteration of the sampling rate, applications can seamlessly adapt to diverse signal characteristics and meet varying requirements with enhanced flexibility. This progressive approach not only marks a significant leap forward in synchronized measurements but also underscores the adaptability and efficiency achievable through dynamic sampling rate adjustments.

The output frequency of the crystal oscillator plays a crucial role in determining the synchronized sampling interval. To mitigate the sampling time error outlined in the preceding section, a real-time monitoring approach is employed for the crystal oscillator’s output frequency. This real-time monitoring facilitates enhanced control over the accuracy of the sampling interval. Employing a timer, the real-time status of the crystal oscillator is continually monitored and transmitted to the data processing unit. This information is then utilized to dynamically adjust the sampling rate in real-time, thereby achieving adaptive sampling. This meticulous process ensures precise synchronization and contributes to the overall improvement in sampling interval control accuracy.

The frequency of a periodic signal is defined as the number of cycles contained in the periodic signal per unit of time as follow Eq. 1: f = n / t (1) where n is the number of cycles of a periodic signal in the time interval t.

The crystal oscillator exhibits superior short-term stability, ensuring that its output frequency remains constant within a synchronization signal period. Leveraging this period as a reference, the counting frequency of the timer can be determined by measuring the number of pulses occurring between adjacent pulse signals. Let f _osc denote the nominal crystal frequency of the v-bit master chip, and t _max is calculated as shown in Eq. 2. t max = 2 v / f osc × Z (2) where Z is octave factor.

When t _max is less than one synchronization signal cycle, the timer needs to be set to auto clear and restart mode. Use the rise of the pulse signal as the external interrupt signal, and read the values W ₁ and W ₂ of the counter when the adjacent pulse signal is interrupted. The actual output frequency f _t of the crystal can be expressed as follows Eq. 3: f t = 2 v × G − W 1 + W 2 Z f pulsar (3) where f _p ulsar is a synchronized signal frequency.

The actual pulse signal is subject to a zero-mean random error characterized by long-term stability. However, in the short term, jitter is inevitable, typically in the order of a few hundred nanoseconds. To mitigate the impact of short-term jitter on crystal frequency measurement, employing multiple pulses as a time reference proves advantageous. This not only extends the counting time of the counter but also provides a more robust time reference. Let the utilization of a continuous set of m pulses as a time reference be the approach to ascertain the crystal output frequency as follow Eq. 4: f t = 2 v × G m − W 1 + W 2 m Z f pulsar (4)

Using high-precision pulse signals, f _t can be updated in real time, which will effectively eliminate the influence of crystal frequency offset on synchronized sampling. The corrected sampling control parameter N _(t) at moment t as follows Eq. 5: N t = f t f s (5)

From the aforementioned analysis, it becomes evident that if the crystal frequency is not an integer multiple of the ideal sampling frequency, N(t) will incorporate a fractional part. Compounding this issue, the timer control parameter of the master chip can only be set as an integer, resulting in a residual part due to the non-integer division of N(t, leading to sampling interval errors. To address the challenges posed by the non-integer division inherent in utilizing the high transmission rate of 5G signals, the sampling rate f _s is dynamically adjusted based on the crystal frequency derived earlier. This dynamic adjustment effectively compensates for sampling time errors, bringing the equivalent sampling rate in proximity to the ideal sampling rate. The operational principle of variable interval sampling is depicted in Figure 1.

Adaptive synchronous sampling method mainly includes: real-time measurement of crystal output frequency, sampling control parameter adjustment, and variable sampling interval sampling. Using high-precision time pulse as a time reference to measure the crystal output frequency and trigger the first sampling command of synchronous sampling, using variable sampling interval control method, according to the real-time monitoring of the crystal output frequency to adaptively adjust the sampling control parameters, the specific steps are as follows in Figure 2.

(1) The real-time crystal output frequency ft is obtained at moment t according to Equation (4);

Building upon the preceding section centered on high-precision pulse-sampled signals, the discrete-time signal for power system voltage or current can be expressed as follows Eq. 6: y k = A ⁡ cos k ω T i + ϕ + ε k = y ̂ k + ε k (6) where y _k is instantaneous signal value;A is amplitude;k is sampling instant;Ti is ideal sampling time;omegais radian frequency;ϕ is phase; ɛ _k is additive noise(assumed to be zero-mean Gaussian white noise with variance σ υ 2 ). Equation 1 can be written as Eq. 7 y k = y ̂ k + ε k (7) where y ̂ k is the estimated signal. It is known that the three consecutive samples of this single sinusoid satisfy the following relationship Equation 8. y ̂ k -   2 ⁡ cos ω T i y ̂ k − 1 + y ̂ k − 2 = 0 (8)

The system frequency can be found by accurately measuring the angular frequency of the voltage model in Eq. 8.

The presence of harmonics and DC attenuation distort the fundamental relationship of Equation 8 derived from the single-sine voltage signal model. n the presence of harmonics, the equation can be corrected as Eq. 9: y k = ∑ m = 1 M A m ⁡ cos m ω t + ϕ m + ε t (9) where m is harmonic order;M is highest frequency component of the signal; A _m is amplitude of the mth harmonic; ϕ _m is phase of the mth harmonic.

Relationship between three consecutive samples of the corrected signal as follow Eq. 10: y k = 2 ⁡ cos ω T i ⋅ y k − 1 − y k − 2 + ∑ m = 1 M [ A m 2 ⁡ cos m ω T i − 2 ⁡ cos ω T i sin m ω k − 1 T i + ϕ m ) ] (10)

If the measurement algorithm needs to be modified according to Eq. 10 in order to estimate the frequency very accurately, but in practice, the experimental results of the model given in Equation 8 are within reasonable accuracy even in the presence of harmonics.

In actual sampling, the actual sampling time Ts has time error with the ideal sampling time Ti due to the reasons such as crystal offset of the master control chipΔT.The sampling time is corrected as follows Eq. 11: T s = T i + Δ T (11)

In linear systems, Kalman filters serve as indispensable tools for tracking and estimation. However, the realm of engineering practice often encounters nonlinear systems, and for such cases, the extended Kalman filter emerges as a more effective approach compared to the conventional Kalman filter. Despite numerous algorithmic advancements proposed since the 1960s to enhance the Kalman filter’s performance, tackling the estimation of state equations for nonlinear state variables remains a challenging problem, crucial for improving algorithmic accuracy. In the context of power system synchronization measurements, various state estimation methods are employed within control systems. For nonlinear state estimation, the Kalman filter is utilized as a linearized model. While it exhibits a favorable response to weakly nonlinear systems, it may fall short in accurately representing highly robust nonlinear systems. Presently, the extended Kalman filter has gained widespread adoption for tracking and state estimation. Within the Kalman filtering algorithm, predictive modeling necessitates the incorporation of actual system and process noise. Meanwhile, updating modeling involves adjusting the predicted values. Consequently, the Kalman filtering algorithm operates as a pre-calibration algorithm, encompassing time update equations (prediction equation) and measurement update equations (calibration equation), as illustrated in Figure 3.

The preceding signal model satisfies the following Kalman filtered as follows Eq. 12 and as follows Eq. 13. x ̂ k + 1 = 1 0 0 0 2 ⁡ cos ω T s − 1 0 1 0 x ̂ k (12) y k = 0 2 ⁡ cos ω T s − 1 x ̂ k + ε k (13)

Based on the power system signal model, designing the ststa variable matrix shown in Eq. 14, x ̂ k = 2 ⁡ cos ω T s y ̂ k − 1 y ̂ k − 2 T = x ̂ k 1 x ̂ k 2 x ̂ k 3 T (14)

From the measurement equations, state equations and state variables of the system; the optimal nonlinear Kalman filtering algorithm as in Eqs 15, 16 is designed using the basic equations of Kalman filtering. x ̂ k | k = x ̂ k | k − 1 + K k y k − g x ̂ k | k − 1 (15) x ̂ k + 1 | k = f x ̂ k | k (16) where f x ̂ k = 2 ⁡ cos ω T s 2 ⁡ cos ω T s ⋅ y ̂ k − 1 − y ̂ k − 2 y ̂ k − 1 T ; g x ̂ k = 2 ⁡ cos ω T s ⋅ y ̂ k − 1 − y ̂ k − 2

The optimal gain matrix is as follows Eq. 17: K k = P ̂ k | k − 1 ∂ g T ∂ x ̂ k , k − 1 . ∂ g ∂ x ̂ k , k − 1 P ̂ k | k − 1 ∂ g T ∂ x ̂ k , k − 1 + 1 (17)

The recursive equation for estimating the error variance matrix is as follows Eq. 18: P ̂ k | k = P ̂ k | k − 1 − K k ∂ g ∂ x ̂ k , k − 1 P ̂ k | k − 1 (18)

The prediction error variance can be expressed as follows Eq. 19: P ̂ k | k + 1 = F k P ̂ k | k F k T (19) where F k = ∂ f x ̂ k ∂ x ̂ k x k = x ̂ k | k = 1 0 0 x ̂ k | k 2 x ̂ k | k 1 − 1 0 1 0 ; ∂ g T ∂ x ̂ x ̂ k = x ̂ k | k = x ̂ k | k 2 x ̂ k | k 1 − 1 ; P ̂ k | k = E x k − x ̂ k | k x k − x ̂ k | k T / σ ν 2 ; P ̂ k + 1 | k = E x k + 1 − x ̂ k + 1 | k x k + 1 − x ̂ k + 1 | k T / σ ν 2 ; x ̂ 1 | 0 = x ̄ 1 ; P ̂ 1 | 0 = E x 1 − x ̄ 1 x 1 − x ̄ 1 T / σ ν 2 .

The filter is nonlinear and therefore, the gain K _k and the covariance matrix P ̂ k | k depend on the estimate x ̂ k | k of the state vector x _k .

The problem with all Kalman filtering algorithms is resetting the covariance matrix. After initial convergence, the gain K _k and the covariance matrix P _k|k stabilise at very small values. Subsequently, when certain parameters of the signal (amplitude, phase and frequency) change, the covariance matrix must be reset to obtain a higher gain in order to track the signal quickly.

The core idea of the covariance reconstruction extended Kalman filter algorithm is that the decision to set the covariance matrix to its initial value is based on a lag-type decision block. The hysteresis band is determined by the amount of noise and the nature of convergence. If the noise estimate is about 10% of the amplitude, then the lag band is chosen to be 20%–60% of the amplitude to avoid frequent resetting of the covariance matrix. A flag is set when the error exceeds a higher threshold and reset when the error falls below a lower threshold. If the flag is 1 and either of the Kalman gains is very small, then the covariance is reset and the flag is reset to 0 so that the covariance matrix is not immediately reset.

Calculation of lag band: The hysteresis band is calculated as the ratio of the residual’s paradigm to amplitude as shown in Eq. 20: lag = ‖ ϵ k ‖ Amplitude (20)

Decide whether to set the reset flag by comparing the value of the hysteresis band with the preset threshold as shown in Eq. 21: ResetFlag = 1 , if  lag > Threshold high 0 , otherwise (21)

If the flag is 1 and either of the Kalman gains is very small, then the covariance is reset and the flag is reset to 0 so that the covariance matrix is not immediately reset as shown in Eq. 22. if ResetFlag = 1  and  ‖ K k ‖  is very small, then  P k | k = P initial (22)

Generally, the frequency variation in power systems is limited to 5 Hz. Therefore, for faster tracking, the measured frequency by the filter is not allowed to vary beyond the 40–60 Hz band. This results in stable operation of the filter and does not lead the filter wayward.

Based on the above theory, this paper constructs the dynamic synchronous phase measurement algorithm based on EKF as follows in Figure 4.

1. k = 0;

The complexity analysis of the Covariance Reconstruction Extended Kalman Filter (CREKF) algorithm involves its primary steps, encompassing state prediction, state updating, covariance matrix updating, and the computation of the decision block.

The complexity analysis of the state prediction and update steps typically depends on the computational intricacy of the state and observation equations. In this paper, the equations have been linearized, resulting in a complexity denoted by O(N), where N represents the dimension of the state vector. The computation of the Kalman gain involves the Jacobian matrix of the observation equation, and thus its complexity is contingent upon the form of the observation equation, with a complexity denoted by O(N ²). The complexity of the covariance matrix update is O(N ³). The lagged decision block has a low complexity and consists mainly of the judgement and updating of the flags, the complexity of this step is of constant level with a complexity of O(1).