A PMU data recovering method based on preferred selection strategy

1 Introduction

With the increasing developments of renewable sources and DC transmission lines, the real time monitoring and the close-loop control based on phasor measurement units(PMUs) become an ultimate key point in power grids.In recent years, there are nearly 2000 commercial grade PMUs installed across North America [1]. Moreover,about 3000 PMUs have been installed in China, including all 220kV and above substations, the main power plants and renewable energy sources grid-connected centralized stations [2].

The raw data collected from the power grid is the cornerstone of PMUs. However, due to the noise jamming,system configuration, signal transmission and other complicated factors, data quality of on-site PMUs is a big issue [3]. This affects the performance of PMU and makes it difficult to analyze the grid with missing data. Thus, this paper focuses on the imminent issue about how to recover missing PMU data accurately and effectively.

As for the issue above, three elements of data quality：accuracy, integrity and consistency have been defined in [4],and it also proposed some data cleaning methods in which the most popular strategy is to use the most likely value to fill the missing value. Analogously the data quality problem of PMU has been characterized by three qualities, including data accuracy, data availability and data timeliness [5].In order to recover missing data, time series methods based on the polynomial interpolation are presented in[7], [9]. In [6], the statistical results of different dataloss scenarios in North America were reviewed, based on which, the Lagrange interpolation polynomial method was proposed to estimate the incomplete and missing data [7].A quadratic prediction model was developed in [8]. And Based on the model, the algorithm combining of Kalman filter and smoothing techniques was adopted to condition the synchrophasor data [9]. These two methods are both simple and can be realized easily. However, the results are not good enough when continuous multi-point data were missing. Methods by exploiting the characteristics of PMU data are widely used in [10], [11]. In [10], the advanced data recovery algorithms exploiting the sparsity of the bad data was tested. And a new framework of recovering missing data by leveraging the approximate low-rank property of PMU data was presented in [11].Besides, the online algorithms of a state-space model and basic functions to predict power frequency was described in [12]. In spite that experiments are conducted to verify the effectiveness of the proposed methods, the continuous multi-point data loss is not taken into consideration.

In this paper, a new method based on the preferred selection strategy (PSS) is developed to recover the missing PMU data. First, two different scenarios of data loss are introduced. This is followed by a combination of preferred selection strategies and the function of cubic spline interpolation based on dichotomy. The main contribution of this paper is to show how the method is able to recover the missing PMU data which is artificial or obtained from the field. Also, the applicable scope of proposed method is discussed. And the interesting direction for future work is indicated at last.

2 The method in two different scenarios

The performance of detection and closed-loop control based on PMU may be affected by data loss issue.Incomplete data makes the grid unobservable and fragile,even causes large scale blackouts and leads to serious consequences [13]. The inherent relationship among data is studied to restore its integrity. Meanwhile, there is no standards on classification of data loss.

2.1 Two different scenarios of data loss

In order to simplify data loss issue, it was divided into two different scenarios： single-point data loss scenario and multi-point data loss scenario, as in Fig. 1.

Every square represents single PMU data in Fig. 1,the blue one represents known data, and the gray one represents missing data during a period of time. Fig. 1(a)shows single-point data loss scenario and Fig. 1(b) shows multi-point data loss scenario. Other data loss scenarios are equivalent to various combinations of them, such as discontinuous multi-point data loss. Then preferred selection strategies are presented.

Fig. 1 Two examples of scenarios. (a) is single-point data loss scenario and (b) is multi-point data loss scenario

2.2 Preferred selection strategies in two different scenarios

2.2.1 Single-point data loss scenario

Some widely used interpolation algorithms such as Lagrange interpolation polynomial method [7] are proven feasible in recovering missing data in single-point data loss scenario. However, the interpolation point on the curve only keeps continuous but not smooth [14]. From the mathematical point of view, the derivative may not be continuous. Due to the imperfection, a cubic spline interpolation method is used to keep the first and second derivative of sampling points continuous. It is obvious that the boundary condition of inflection point is met. The presented strategy avoids Runge’s phenomenon, in which oscillation can occur between points when interpolating using high degree polynomials.

Numerical experiments show that using eight foreand-spatial points to recover the single missing data has maximum accuracy. And it is demonstrated in Section 3.

2.2.2 Multi-point data loss scenario

When there is multi-point data loss, it is complicated to do lots of interpolation calculation. And results are not accurate enough. Therefore, a preferred selection strategy based on the dichotomy is proposed. According to the parity of multi-point number, the strategy has two cases.

On the one hand, if number of missing data is odd, detail process is in Algorithm 1. And represent missing data and the others represent known data,as is shown in Fig. 2.is recovered in the first stage,and are in the second stage,and are in the third stage.

On the other hand, if number is even, process is in Algorithm 2. And represent missing data and the others represent known data, as in Fig.3. Xn-1 and Xn are recovered in the first stage, Xn-3 and Xn+2 are in the second stage, Xn-2 and Xn+1 are in the third stage.

8： end if
9： All missing data have been recovered.

2.3 Function of cubic spline interpolation

A tabulated function y i = f( xi),M i =S′′(x i ),i = 1 ,2,… ,n, is given, focusing attention on one particular interval, between xi and xi+1 [15].The goal of cubic spline interpolation is to get an interpolation formula which is smooth in the first derivative, and continuous in the second derivative, both within an interval and at its boundaries.

So, the second derivative formula ofS( x) = Si( x)between xi and xi+1 is given：

where,

And the result of two times continuous integration of (1)

Because of the continuity S ′ ( x i- ) = S′( x i+), we can get

where,

Along with the additional boundary condition,

The equation (2) is calculated as：

The coefficient matrix is strictly diagonally dominant,so equations (3) has the unique solution. And the function is gotten. The combination of preferred selection strategies and the function of cubic spline interpolation base on dichotomy called PSS method is proposed.

The interpolation results of missing data not only depend on endpoints of missing interval, but also some points which have the same interval as missing data. The interpolation points calculated by PSS method do not have the obvious distortion and maintain the performance of curves. Compared with other methods, it does not damage the periodic signal of PMU data. The interpolation interval we selected meet the relationship that the interval between each selected points and corresponding interpolation points is equal, because it ensures the relationship between the sample interval and Nyquist frequency, according to the sampling theorem.

3 Case studies

Numerical experiments are tested on artificial modulation data and field PMU data with MATLAB in both static and transient stares. Compared with Lagrange interpolation polynomial method [7], the results of proposed method and Lagrange method are presented in this paper. While for the phasor measurement, its accuracy is evaluated with the total vector error (TVE) as：

where, X r(n) and X i (n) are the real and imaginary value of theoretical values of the input signal, X r ′( n)and X′i(n)are real and imaginary value of the estimates. The TVE of P class and M class PMUs is required to be less than 1% in static state in IEEE Standard C37.118 [8].

3.1 Simulation

Due to stability of PMU data, simulation of static state is not tested. We simulated the input signal for M class PMU when the power frequency has offset. The signal is followed as：

where, Xm means the phasor amplitude, f0means frequency offset, φ 0means the initial phases. And X m =57.73kV , f 0 =5Hz,φ 0=0.

3.1.1 Optimal number of data point for interpolation

The single randomly selected missing data is recovered by this method. We calculated many times, and results of TVE are the same owing to its periodicity. Results are as followed in Table 1.

Table 1 Optimal data points for interpolation

Data points 2 4 6 8 9 10 TVE 4.89% 0.16% 0.06% 0.07% 0.07% 0.07%

Table 1 shows that starting from eight points, TVE is constant and missing data is efficiently recovered. So, eight fore-and-spatial points are used to recover missing data.

3.1.2 Results in two different scenarios

We compared results of Lagrange method and PSS method using the given input signal of PMU in two different scenarios. And the corresponding results in the complex plane are presented in Fig. 4.

Fig. 4 Results of two methods in different scenarios. Two different scenarios are circled by curve. Blue circle is known data; red circle is missing data; in single-point data loss scenario, brown x symbol is resulted by Lagrange method and green diamond is resulted by PSS method; in multi-point data loss scenario, black star is resulted by Lagrange method and purple square is resulted by PSS method

Fig. 4. shows that in single-point data loss scenario the result of PSS method (TVE=0.07%) is better than Lagrange method (TVE=3.06%). While it is in multi-point data loss scenario, PSS method maintains the trend of known data,but Lagrange method distorts completely. TVE results are in Table2. Table2 shows that TVE of Lagrange method gets gradually large from the first point to the fifth point.The result is not only used to recover missing data. TVE of PSS method is less, it is useful for certain synchrophasor applications.

Table 2 TVE results of two methods

Data Point point 1 Point 2 Point 3 Point 4 Point 5 TVE (Lagrange) 3.06% 12.14% 29.95% 58.92% 101.03%TVE (PSS) 4.03% 7.97% 6.09% 6.82% 4.41%

3.2 Testing of actual PMU data

PMUs are widely used in China, especially in places where there are abundant renewable energy sources. Over the past two years, sub-synchronous oscillation was usually detected in such areas and it spreaded far away, which causes voltage oscillation. Field data obtained from onsite PMUs in these areas is shown in Fig. 5 and the send frequency of PMU is 100Hz.

Fig. 5 Field PMU data of two states. (a) is static state and (b)is transient data

Fig. 5 shows that 6000 field PMU data points during one minute including voltage amplitude and angle. The voltage oscillates in transient state for 33s. Then the grid comes into static state.

3.2.1 Results of static and transient states

When the power grid is in static state, we select each data point as the missing one randomly and some continuous points like that. And When the power grid is in transient state, some data points the same as the previous are selected. The comparison between Lagrange method and PSS method is seen in Fig. 6.

Fig. 6 Results of two states. In (a), curve corresponds to the right coordinate, and the histogram corresponds to the right;TVE_Is and TVE_Ls mean that TVE of PSS and Lagrange method in static state; TVE_It and TVE_Lt mean that TVE of PSS and Lagrange method in transient state; In (b) and (c),blue circle is known data; red circle is missing data; black star is resulted by Lagrange method and purple square is resulted by PSS method of two different scenarios

Fig. 6 (a) shows that TVE of Lagrange and PSS method are both less than 1% among ten random points and average of the latter is better when there is single-point data loss in static state. In transient state, the curve shows that the single missing data cannot be recovered by Lagrange method and results are completely distorted, but PSS method has an accurate result as the same as that in static state.

Fig. 6 (b) shows that five continuous data points of static state are missing in multi-point loss scenario. Result of PSS method is obviously better than the other. By calculating TVE, it is demonstrated that PSS method is more accurate than Lagrange method and the proposed method works better.

Fig. 6 (c) shows that five continuous data points are missing in multi-point loss scenario when voltage oscillates.Proposed method can recover the missing data completely and accurately. The error of Lagrange method gets bigger from the first point to the fifth by analyzing TVE.

3.2.2 Results of numbers of missing data

In multi-data loss scenario, one data is randomly selected as the missing one. Then more and more continuous data are selected followed by it. The performance of both methods to recover these missing data is tested. Results of Lagrange and PSS method in two states are in Fig. 7 and Fig. 8.

As is illustrated in Fig. 7, TVE enlarges with the increase of missing data both in static and transient states,especially in the latter one, which is related to its own polynomial algorithm. It eventually leads to an invalid result due to the large departure of real data.

Fig. 7 Results of Lagrange method in two states. The number of missing data points is from 1 to 7. Curve corresponds to the left coordinate, and the histogram corresponds to the right.TVE_Ls means that TVE of Lagrange method in static state;TVE_Lt means that TVE of Lagrange method in transient state

Fig. 8 Results of Lagrange method in two states. The number of missing data points is from 1 to 7. Each rectangle represents TVE of each missing data

According to Fig. 8, recovered results are close to the real ones both in static and transient states, especially in former. Meanwhile, TVE results show that massive missing data have no effect on it and no clear relevant relationship to the number of missing data.

Through the above analysis, PSS method can replace the Lagrange method to recover missing data whether in single-point data loss scenario or multi-point data loss scenario. Also, this method has the good performance in both static and transient states. Meanwhile numbers of missing data do not have the clear relevant relationship to results.

4 Conclusions and future work

This paper addresses the PMU data loss issue. Data loss is classified into two different scenarios. We exploit the dichotomy to present a new method combined preferred selection strategies and function of cubic spline interpolation. Numerical experiments verified that the proposed method recover missing data of static and transient states accurately and effectively no matter in which scenario. Furthermore, it is much better than other methods when missing data are massive and continuous.

Without considering bad data, our future work is to develop a method which can identify and correct the bad data of PMU.

Acknowledgements

This work was supported in part by National Natural Science Foundation of China (NSFC) (51627811, 51707064);Project Supported by the National Key Research and Development Program of China (2017YFB090204) and Project of State Grid Corporation of China(SGTYHT/16-JS-198).