CN108596475B - PMU data recovery method based on dynamic change of interpolation interval - Google Patents

Abstract

The invention discloses a PMU data recovery method based on the dynamic change of interpolation interval, which belongs to the technical field of PMU data recovery. The method includes: defining a single-point data loss scenario and a continuous multi-point data loss scenario according to the actual scenario of the PMU losing data; determining the recovery sequence of the lost data, and proposing a priority recovery method based on the dynamic change of the interpolation interval; To recover the lost data, an improved cubic spline interpolation function is proposed, which makes the spline curve flatter by putting forward constraints on the variables. The present invention can effectively recover different types of lost data in different states of the system, solves the distortion problem of the traditional method in the scenario of continuous multi-point data loss, and has great significance and great value for ensuring the quality of PMU data.

Description

A PMU data recovery method based on dynamic change of interpolation interval

technical field

The invention belongs to the technical field of PMU data recovery, and in particular relates to a PMU data recovery method based on dynamic changes of interpolation intervals.

Background technique

With the development and utilization of large-scale renewable energy and the development of smart grid, my country has built a super-large-scale complex interconnected power system that serves the largest population, covers the widest range, has the highest transmission voltage level, and accommodates the most renewable energy in the world. The mechanism characteristics, analysis methods and operation control methods of power systems have undergone fundamental changes. The integrity of the power system has become increasingly prominent, and the systematic cascading failures across regions and voltage levels have gradually become a norm, and the demand for closed-loop refined control is obvious. Synchronized phasor measurement units (Phasor Measurement Units, PMUs) enable dynamic real-time monitoring of power systems because of their synchronization, rapidity and accuracy, and provide data foundation for system protection and closed-loop control. At present, about 3,000 PMU units have been installed and put into operation in China, covering all 220kV and above substations, main power plants and new energy grid-connected collection stations. In addition, approximately 2,000 commercial-grade PMUs have been installed in North America. However, due to the complex field environment and the influence of synchronization signal loss, communication protocol error, system overload, transmission delay and other factors, PMU inevitably has problems such as data loss, which seriously affects its application in dynamic monitoring and closed-loop control. Even threaten the security of the power grid. Under the background of unknown power grid topology and only PMU measurement information, the loss of PMU data will make the power grid fragile and unobservable, vulnerable to disturbance attacks and even cause blackouts. Therefore, PMU data recovery has become a key issue to ensure the security of the power system. In the prior art, time series method, matrix low-rank method, state estimation method, etc. are mostly used. All of these methods can effectively recover single-point data loss, but there is no good recovery effect for continuous multi-point data loss.

SUMMARY OF THE INVENTION

In view of the above problems, the present invention proposes a PMU data recovery method based on the dynamic change of the interpolation interval, which is characterized in that it includes the following steps:

Step 1. Establish two basic scenarios of PMU data loss, including single-point data loss scenarios and continuous multi-point data loss scenarios;

Step 2. Analyze the data loss type and determine the data loss scenario;

Step 3. Determine the recovery order of the lost data. In the scenario of continuous multi-point data loss, consider the priority allocation of data recovery, and calculate the recovery order of the lost data according to the parity of the number of lost data; if the data is lost at a single point In scenarios, priority allocation need not be considered;

Step 4. On the basis of determining the recovery sequence of the lost data, the improved standard cubic spline interpolation function is used to recover the lost data in different scenarios.

The single-point data loss scenario refers to a scenario in which only a single data loss exists in a set of PMU measurement data obtained within a period of time.

The continuous multi-point data loss scenario refers to a scenario in which there is continuous multi-point data loss in a set of obtained PMU measurement data within a period of time.

In the step 3, under the scenario of continuous multi-point data loss, the priority recovery method based on the dynamic change of the difference interval is used to calculate the recovery order of the lost data, and the specific calculation method is as follows:

(1) When the number of consecutive lost data is odd

Assume that there are 5 consecutive missing data in a set of PMU measurement data obtained in a period of time, namely X _n-2 , X _n-1 , X _n , X _n+1 , X _n+2 ;

Step 1. Input all PMU data X ₁ , X ₂ , . . . , X _m , the number of lost data N, the number of recovery times M, and the adjacent interval Z, where M=N;

Step 2. Calculate the adjacent interval Z ₁ of the selected points of the data to be restored in the first stage, and the calculation formula is as follows:

选择X_n前后各4个相邻间隔为Z₁的点，利用改进的标准三次样条插值函数恢复数据X_n；Step 3. Determine the data X _n to be restored in the first stage, where

Select 4 adjacent points with an interval of Z ₁ before and after X _n , and use the improved standard cubic spline interpolation function to restore the data X _n ;

Step 4. Calculate the adjacent interval Z ₂ of the selected points of the data to be restored in the second stage, and the calculation formula is as follows:

Z ₂ =Z ₁ -1

Step 5. Determine the data X _n-2 and X _n+2 to be restored in the second stage, use the existing data and the data X _n restored in the first stage, and use the improved standard cubic spline interpolation function to restore the data X _n-2 and X _n+2 ;

Step 6. Calculate the adjacent interval Z ₃ of the selected point of the data to be restored in the third stage, and the calculation formula is as follows:

Z ₃ =Z ₂ -1

Step 7. Determine the data Xn _-1 and Xn ₊₁ to be restored in the third stage, use the existing data and the data _Xn , Xn _-2 and Xn ₊₂ restored in the first two stages, and use the improved standard three times The spline interpolation function restores the data X _n-1 and X _n+1 ;

(2) When the number of consecutive lost data is even

Assume that there are 6 consecutive missing data in a set of PMU measurement data obtained in a period of time, namely X _n-3 , X _n-2 , X _n-1 , X _n , X _n+1 , X _{n +2} ;

Step a. Input all PMU data X ₁ , X ₂ , . . . , X _m , the number of lost data N, the number of recovery times M, and the adjacent interval Z, where M=N;

Step b. Calculate the adjacent interval Z ₁ of the selected points of the data to be restored in the first stage, and the calculation formula is as follows:

选择与待恢复数据前后各4个相邻间隔为Z₁的点，利用改进的标准三次样条插值函数恢复数据X_n-1和X_n；Step c, determine the data X _n-1 and X _n to be restored in the first stage, wherein

Select four adjacent points that are Z ₁ before and after the data to be restored, and use the improved standard cubic spline interpolation function to restore the data X _n-1 and X _n ;

Step d, calculate the adjacent interval Z ₂ of the selected points of the data to be restored in the second stage, and the calculation formula is as follows:

Z ₂ =Z ₁ -1

Step e. Determine the data X _n-3 and X _n+2 to be restored in the second stage, use the existing data and the data X _n-1 and X _n restored in the first stage, and use the improved standard cubic spline interpolation function to restore data Xn _-3 and Xn ₊₂ ;

Step f, calculate the adjacent interval Z ₃ of the selected point of the data to be restored in the third stage, and the calculation formula is as follows:

Z ₃ =Z ₂ -1

Step g, determine the data Xn _-2 and Xn ₊₁ to be restored in the third stage, use the existing data and the data Xn _-1 , _Xn , Xn _-3 and Xn ₊₂ restored in the first two stages, The data Xn _-2 and Xn ₊₁ are recovered using a modified standard cubic spline interpolation function.

The modeling method of the improved standard cubic spline interpolation function is:

(1) Construct a cubic spline interpolation function

given function

y _i =f(x _i ), i=1,2,...,n (1)

in,

a=x ₀ <x ₁ <x ₂ <...<x _n =b, [a, b]

If S(x)=y is on each subinterval [x _k , x _{k +1} ] (k=1, 2, . =y _i , i=1,2,...,n; and S(x), S'(x), S"(x) are continuous on [a, b], then S(x) is called f (x) cubic spline interpolation function on nodes x ₀ , x ₁ , x ₂ , ... x _n ;

(2) Let M _i =S″(x _i ), on the interval [x _i , x _i+1 ], the second derivative of S(x)=S _i (x) can be expressed as:

in,

h _i =x _i+1 -x _i

(3) Integrate formula (1) for two consecutive times to obtain:

From the continuity S'(x _i- )=S'(x _i+ ), we can get:

μ _i M _i-1 +2M _i +λ _i M _i =d _i (i=1,2,...,n-1) (5)

in,

(4) According to the boundary conditions S'(x ₀ )=y ₀ ', S'(x _n )=y _n ', formula (5) is expressed as the following matrix form:

The coefficient matrix of equation (7) is strictly diagonally dominant and has a unique solution,

S(x) S _i (x) on each interval [ _xi , x _i+1 ] is:

(5) Constructing an improved cubic spline interpolation function

Assuming that the maximum value and minimum value of S _i (x) on the interval [x _i-1 , x _i ] (i=1, 2, ..., n) are S _imax and S _{imin respectively} , then S( The extreme value difference of x) on the interval [a, b] is:

Assuming that S'(x ₀ )=y ₀ ', S'(x _n )=y _n ' is unknown, it can be obtained from the extreme value difference of S(x):

In the formula, f(x _i ) is an arbitrary function; a and b are the value intervals of the independent variable x; M _i is the second derivative of the cubic spline interpolation function.

The beneficial effects of the present invention are:

(1) The present invention can effectively recover data for different types of lost data when the system is in different states. In addition, the present invention is not constrained by the topology of the power grid, and can be realized only by inputting the real-time measurement data of the PMU. It solves the distortion problem of the traditional method in the scenario of continuous multi-point data loss, and provides an effective and feasible method for ensuring the quality of PMU data.

(2) A priority recovery strategy based on the dynamic change of the interpolation interval proposed by the present invention ensures the relationship between the sampling interval and the Nyquist frequency, avoids the damage of the periodic signal of the PMU data, and ensures the accuracy of recovery sex.

(3) The present invention improves the cubic spline interpolation function, provides constraints on variables M ₀ and _Mn , and ensures that the first and second derivatives of the spline interpolation function are continuous. At the same time, the Runge phenomenon is avoided and the spline curve is flatter.

Description of drawings

Accompanying drawing 1 is a kind of PMU data recovery method flow chart based on the dynamic change of interpolation interval;

Figure 2(a) is the basic scenario of PMU single-point data loss;

Accompanying drawing 2(b) is the basic scene of PMU continuous multi-point data loss;

Figure 3 (a) is a PMU discontinuous multi-point data loss scenario;

Figure 3(b) is a PMU complex data loss scenario;

Accompanying drawing 4 is the PMU data loss situation of odd number;

Accompanying drawing 5 is the PMU data loss situation of even number;

Accompanying drawing 6 compares two kinds of methods to different types of lost data recovery results;

Accompanying drawing 7 is the A-phase voltage amplitude and phase angle measurement data of a certain PMU;

Accompanying drawing 8 is the TVE comparison of single point loss data recovery result under steady state;

Accompanying drawing 9 is the TVE comparison of the data recovery result of single-point lost data under the transient state;

Accompanying drawing 10 (a) is the data recovery result of continuous multi-point loss under steady state;

Accompanying drawing 10(b) is the recovery result of continuous multi-point lost data under the transient state;

Accompanying drawing 11 is the influence of the number of lost data on the traditional method TVE under steady and transient state;

Accompanying drawing 12 (a) is the influence of the number of lost data on TVE of the inventive method under steady state;

Figure 12(b) is the influence of the number of lost data on the TVE of the method of the present invention under the transient state;

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

Accompanying drawing 1 is a kind of PMU data recovery method flow chart based on interpolation interval dynamic change, as shown in Fig. 1, described method comprises the following steps:

Step 1: Establish two basic scenarios of PMU data loss, including single-point data loss scenarios and continuous multi-point data loss scenarios;

Step 2: Analyze data loss types and determine data loss scenarios;

Step 3: Determine the recovery order of the lost data. In the scenario of continuous multi-point data loss, consider the priority allocation of data recovery, and calculate the recovery order of the lost data according to the parity of the number of lost data; if the data is lost at a single point In scenarios, priority allocation need not be considered;

Step 4: On the basis of determining the recovery sequence of lost data, recover lost data in different scenarios by improving the standard cubic spline interpolation function.

Specifically, in step 1, in order to simplify the PMU data loss problem model and facilitate data recovery, the present invention defines two data loss scenarios according to the accuracy, availability, real-time nature of PMU data, and the actual situation of data loss on site. There are two basic scenarios, namely single-point data loss scenario and continuous multi-point data loss scenario. Figure 2(a) is a single-point data loss scenario, and Figure 2(b) is a continuous multi-point data loss scenario. In Figures 2(a)-2(b), the squares represent the PMU measurement data within a period of time, Among them, white is missing data, black is known data. Other scenarios can be equivalent to different combinations of the above two scenarios. For example, a discontinuous multi-point data loss scenario can be equivalent to multiple single-point data loss scenarios, as shown in Figure 3(a); a complex data loss scenario can be equivalent to a combination of single-point data loss and continuous multi-point data loss, As shown in Figure 3(b). According to Figures 2(a) and 3(a), the single-point data loss scenario can be defined as a scenario in which only a single data loss exists in a set of PMU measurement data obtained within a period of time; according to Figure 2(b) , 3(b), the continuous multi-point data loss scenario can be defined as a scenario in which there is continuous multi-point data loss in a set of PMU measurement data obtained within a period of time.

Specifically, in the step 3, the recovery sequence of the lost data is determined according to the two basic scenarios in which the PMU loses data established in the step 1, and in the scenario of single-point data loss, priority assignment is not required. In the scenario of continuous multi-point data loss, since the existing method needs to perform a large amount of calculation and the result error is large, the present invention proposes a priority recovery method based on the dynamic change of the interpolation interval, which takes the data recovery into consideration. Priority allocation, according to the parity of the number of consecutive lost data, calculates the recovery sequence of lost data, which can effectively improve the calculation accuracy. Regarding the parity of the number of consecutively lost data, the following two cases are discussed:

(1) When the number of consecutive lost data is odd

Figure 4 shows the data loss situation of an odd number of PMUs. As shown in Figure 4, it is assumed that X _n-2 , X _n-1 , X _n , X _n+1 , and X _n+2 are 5 consecutive lost data, and the rest For known data, in this case, the specific steps of the priority recovery method adopted are as follows:

Step 1. Input all PMU data X ₁ , X ₂ , . . . , X _m , the number of lost data N, the number of recovery times M, and the adjacent interval Z, where M=N;

Step 2. Calculate the adjacent interval Z ₁ of the selected points of the data to be restored in the first stage, and the calculation formula is as follows:

选择X_n前后各4个相邻间隔为Z₁的点，利用改进的标准三次样条插值函数恢复数据X_n；Step 3. Determine the data X _n to be restored in the first stage, where

Select 4 adjacent points with an interval of Z ₁ before and after X _n , and use the improved standard cubic spline interpolation function to restore the data X _n ;

Step 4. Calculate the adjacent interval Z ₂ of the selected points of the data to be restored in the second stage, and the calculation formula is as follows:

Z ₂ =Z ₁ -1

Step 5. Determine the data X _n-2 and X _n+2 to be restored in the second stage, use the existing data and the data X _n restored in the first stage, and use the improved standard cubic spline interpolation function to restore the data X _n-2 and X _n+2 ;

Step 6. Calculate the adjacent interval Z ₃ of the selected point of the data to be restored in the third stage, and the calculation formula is as follows:

Z ₃ =Z ₂ -1

Step 7. Determine the data Xn _-1 and Xn ₊₁ to be restored in the third stage, use the existing data and the data _Xn , Xn _-2 and Xn ₊₂ restored in the first two stages, and use the improved standard three times The spline interpolation function restores the data Xn _-1 and Xn ₊₁ .

(2) When the number of consecutive lost data is even

Figure 5 shows the loss of even-numbered PMU data. As shown in Figure 5, it is assumed that X _n-3 , X _n-2 , X _n-1 , X _n , X _n+1 , and X _n+2 are 6 The data is lost continuously, and the rest are known data. In this case, the specific steps of the priority recovery method adopted are as follows:

Step a. Input all PMU data X ₁ , X ₂ , . . . , X _m , the number of lost data N, the number of recovery times M, and the adjacent interval Z, where M=N;

Step b. Calculate the adjacent interval Z ₁ of the selected points of the data to be restored in the first stage, and the calculation formula is as follows:

选择与待恢复数据前后各4个相邻间隔为Z₁的点，利用改进的标准三次样条插值函数恢复数据X_n-1和X_n；Step c, determine the data X _n-1 and X _n to be restored in the first stage, wherein

Select four adjacent points that are Z ₁ before and after the data to be restored, and use the improved standard cubic spline interpolation function to restore the data X _n-1 and X _n ;

Step d, calculate the adjacent interval Z ₂ of the selected points of the data to be restored in the second stage, and the calculation formula is as follows:

Z ₂ =Z ₁ -1

Step e. Determine the data X _n-3 and X _n+2 to be restored in the second stage, use the existing data and the data X _n-1 and X _n restored in the first stage, and use the improved standard cubic spline interpolation function to restore data Xn _-3 and Xn ₊₂ ;

Step f, calculate the adjacent interval Z ₃ of the selected point of the data to be restored in the third stage, and the calculation formula is as follows:

Z ₃ =Z ₂ -1

Step g, determine the data Xn _-2 and Xn ₊₁ to be restored in the third stage, use the existing data and the data Xn _-1 , _Xn , Xn _-3 and Xn ₊₂ restored in the first two stages, The data Xn _-2 and Xn ₊₁ are recovered using a modified standard cubic spline interpolation function.

According to the sampling theorem, the adopted interval Z satisfies that the interval between each selected point and the interval to be restored are equal to ensure the relationship between the sampling interval and the Nyquist frequency. Compared with other algorithms, the recovery method does not damage the periodic signal of the PMU data.

Specifically, in the step 4, on the basis of the priority recovery strategy, the present invention adopts the interpolation method to recover the lost data. The cubic spline interpolation function is obtained by using the known data, so that it has a high degree of fit with the known data, and then the missing data is obtained. In order to avoid the Runge phenomenon caused by the use of high-order polynomials, the present invention improves the cubic spline interpolation function, obtains the improved cubic spline interpolation function under the constraint condition of extreme value difference, and restores the lost data. The cubic spline interpolation function is based on ensuring the continuity of the first and second derivatives of the obtained spline interpolation function, and puts forward constraints on the variables, so that the curve has good smoothness, strong linear approximation ability, and can effectively To characterize data changes, the modeling method is as follows:

(1) Construct a cubic spline interpolation function

given function

y _i =f(x _i ), i=1,2,...,n (1)

in,

a=x ₀ <x ₁ <x ₂ <...<x _n =b, [a, b]

If S(x)=y is on each subinterval [x _k , x _{k +1} ] (k=1, 2, . =y _i , i=1,2,...,n; and S(x), S'(x), S"(x) are continuous on [a, b], then S(x) is called f (x) Cubic spline interpolation function at nodes x ₀ , x ₁ , x ₂ , . . . x _n .

(2) Let M _i =S″( _xi ), because the interpolation formula obtained by the cubic spline interpolation method can ensure that the first-order derivative is smooth at each interval and boundary, and the second-order derivative is continuous at the same time. Therefore, in [xi _i , x _i+1 ], the second derivative of S(x)=S _i (x) can be expressed as:

in,

h _i =x _i+1 -x _i

(3) Perform two consecutive integrals on formula (1) to obtain:

From the continuity S'(x _i- )=S'(x _i+ ), we can get:

μ _i M _i-1 +2M _i +λ _i M _i =d _i (i=1,2,...,n-1) (5)

in,

(4) According to the boundary conditions S'(x ₀ )=y ₀ ', S'(x _n )=y _n ', formula (5) is expressed as the following matrix form:

(5) Constructing an improved cubic spline interpolation function

Since the cubic spline interpolation function is a function of boundary conditions, in order to make the spline curve flatter, the present invention obtains the minimum extreme value difference according to the actual situation of the power system change.

Assuming that the extreme values of S _i (x) in the interval [x _i-1 , x _i ] (i=1, 2, ..., n) are S _imax and S _{imin respectively} , then S(x) is in the interval [ The extreme value difference on a, b] is:

Assuming that S'(x ₀ )=y ₀ ' and S'(x _n )=y _n ' are unknown, the extreme value difference of S(x) is a function of M ₀ and _Mn , and we can obtain about M ₀ and M The objective function of the unconstrained nonlinear programming of _n is:

Formula (9) is an unconstrained nonlinear programming problem about M ₀ and _Mn . Since the objective function contains extreme value operations and the parameters are determined by the equation system, it can be solved by the simplex substitution method.

Example 1

In order to further describe the present invention in detail, this embodiment uses Matlab to perform simulation tests in two states of the system steady state and transient state, and uses the PMU measured data to verify and compare with the recovery results of the existing algorithms. The measurement error of the synchrophasor is measured by the total vector error (TVE), and the calculation formula is as follows:

In the formula, X _r (n) and X _i (n) represent the real part and imaginary part of the theoretical value of the input signal, respectively, and X' _r (n) and X' _i (n) represent the real part and the estimated value of the input signal, respectively. imaginary part.

1. Simulation test

The steady state signal expression of the power system is as follows:

为初相角，且X_m＝57.73V，f₀＝50Hz，

where X _m is the phasor amplitude, f ₀ is the power frequency,

is the initial phase angle, and X _m =57.73V, f ₀ =50Hz,

Under ideal conditions, the phasor, frequency and frequency change rate of the above-mentioned signals will not change, and the output result will be constant, so it is easy to recover when data is lost. When the frequency offset occurs in the system, the transient signal expression of the power system is as follows:

In the formula, Δf is the frequency offset, and Δf=5Hz.

(1) Optimal number of interpolation points test

The invention uses the interpolation method to restore data, and the number of interpolation points directly affects the data restoration accuracy and calculation speed. Therefore, take the above frequency offset signal as an example to test the optimal number of interpolation points.

Randomly select a single point of lost data, use different interpolation points to restore the lost data, repeat the experiment 100 times, and record the results. The test results are shown in Table 1. In Table 1, TVE represents the mean value, and the results show that the adjacent 8 points of data before and after the lost data The recovery results are optimal.

Table 1 Optimal number of interpolation points

number of interpolation points 2 4 6 8 10 TVE() 4.89% 0.16 0.06 0.07 0.07

(2) Data recovery result when frequency offset

The data recovery method proposed by the present invention recovers different types of lost data in the frequency offset signal, and compares it with the existing method. The comparison result is shown in Fig. 6. The recovery effects of different types of lost data, in the figure, the axis of abscissa represents the real part and the imaginary part respectively, PS and LS are the recovery results of the method of the present invention and the existing method to single-point lost data, PM and LS are the method of the present invention And the recovery results of the existing methods for continuous multi-point lost data. It can be seen from FIG. 6 that the TVE of recovering single-point lost data using the method of the present invention is 0.07%, which is better than 3.06% of the existing method. When continuous multi-point data is lost, serious distortion occurs in the existing method, but the method proposed in the present invention can better restore the frequency offset signal and maintain the change trend of the data. The TVE comparison results of the two methods are shown in Table 2. It can be seen from Table 2 that the TVELM gradually increases with the increase of the lost data, and the TVEPM changes nonlinearly, which can effectively recover the lost data.

Table 2 TVE comparison of two methods

2. PMU measured data test

The invention analyzes the data obtained by the PMU monitoring during a certain synchronous oscillation in a new energy gathering area in western China, and performs data recovery verification. Figure 7 shows the measured data of the A-phase voltage amplitude and phase angle of the PMU during a certain synchronous oscillation.

(1) Comparison of single-point data loss recovery results under stable and transient conditions of the system

When the system is in a steady state, the method of the present invention and the existing method are respectively used to recover the randomly selected single point lost data. The TVE comparison of the lost data recovery results is shown in Figure 8. It can be seen from Figure 8 that 80% of the TVEPS in steady state are less than 1%, and only 30% of the TVELS are less than 1%, and the error of the recovery results varies greatly, which is difficult to use in practice. application in.

When the system is in a transient state, the same two methods are used to recover randomly selected single-point lost data. The TVE comparison of the lost data recovery results is shown in Figure 9. In Figure 9, PS corresponds to the primary vertical axis coordinate, and LS corresponds to the secondary vertical axis coordinate. It can be seen from this that when the subsynchronous oscillation occurs in the system, 90% of the TVEPS are less than 3%, and the recovery effect is good, and only a few TVELS are less than 5%, and the recovered data is seriously distorted. Due to the voltage oscillation, if the missing data is located at the peak or trough position, it is difficult to recover the change trend by this method only through the previous data.

(2) Comparison of the recovery results of continuous multi-point data loss under the stable and transient state of the system

When the system is in steady state and transient state, the method of the present invention and the existing method are used to restore continuous multi-point lost data, and the results are shown in Figures 10(a)-10(b). 10(a), the method of the present invention has better ability to recover continuously lost data in steady state, and TVELM gradually increases with the increase of lost data, which is consistent with the simulation test results. Figure 10(b) shows the test results in the transient state, which can be obtained from Figure 10(b). When the system is in a transient state, the TVELM changes are the same as those in the steady state, and the recovery result has a large deviation. The TVEPM is below 1%, and the recovery The results are highly accurate.

(3) Analyze the impact of the number of missing data on TVE

By changing the number of lost data, compare the recovery effects of the two methods under steady and transient conditions, that is, the TVE changes. The comparison results are shown in Figures 11, 12(a), and 12(b). . The influence of the number of missing data in the transient state on the TVE of the existing method. In Figure 11, Lst represents the TVE change in the steady state, corresponding to the main vertical axis coordinate; Ltt represents the transient TVE change, corresponding to the secondary vertical axis coordinate. It can be seen from Figure 11 that both TVELst and TVELtt increase linearly with the increase of the number of lost data. However, this method only uses the data before the lost point, and cannot accurately correct each recovery result, so that the error gradually increases.

Figures 12(a)-12(b) are the effects of the number of lost data on the TVE of the method of the present invention in steady and transient states, which can be obtained from Figures 12(a)-12(b), in steady and transient states The following methods of the present invention can effectively restore data. The accuracy in steady state and transient state is much higher than that of existing methods. Lost data in steady state is easier to recover, and TVE in transient state is less than 1.1%. At the same time, with the increase of the number of lost data, the recovery order is also different according to the priority recovery method. Therefore, the TVE change has no obvious relationship with the number of lost data, and further research is needed.

Through the above tests, it is verified that the system can recover lost data under single-point and continuous multi-point loss scenarios in temporary and steady state. The test results show that this method has a good recovery effect.

This embodiment is only a preferred embodiment of the present invention, but the protection scope of the present invention is not limited to this. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention. , all should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be subject to the protection scope of the claims.

Claims (1)

1. a PMU data recovery method based on interpolation interval dynamic change, is characterized in that, comprises the following steps: Step 1. Establish two basic scenarios of PMU data loss, including single-point data loss scenarios and continuous multi-point data loss scenarios; The single-point data loss scenario refers to a scenario in which only a single data loss exists in a set of PMU measurement data obtained within a period of time; The continuous multi-point data loss scenario refers to a scenario in which continuous multi-point data loss exists in a set of PMU measurement data obtained within a period of time; Step 2. Analyze the data loss type and determine the data loss scenario; Step 3. Determine the recovery order of the lost data. In the scenario of continuous multi-point data loss, consider the priority allocation of data recovery, and calculate the recovery order of the lost data according to the parity of the number of lost data; if the data is lost at a single point In scenarios, priority allocation need not be considered; In the step 3, under the scenario of continuous multi-point data loss, the priority recovery method based on the dynamic change of the difference interval is used to calculate the recovery order of the lost data, and the specific calculation method is as follows: (1) When the number of consecutive lost data is odd Assume that in a period of time, there are 5 consecutive missing data in a set of PMU measurement data obtained, namely X _n-2 , X _n-1 , X _n , X _n+1 , X _n+2 ; Step 1. Input all PMU data X ₁ , X ₂ , . . . , X _m , the number of lost data N, the number of recovery times M, and the adjacent interval Z, where M=N; Step 2. Calculate the adjacent interval Z ₁ of the selected points of the data to be restored in the first stage, and the calculation formula is as follows:

Step 3. Determine the data X _n to be restored in the first stage, where

Select 4 adjacent points with an interval of Z ₁ before and after X _n , and use the improved standard cubic spline interpolation function to restore the data X _n ; Step 4. Calculate the adjacent interval Z ₂ of the selected points of the data to be restored in the second stage, and the calculation formula is as follows: Z ₂ =Z ₁ -1 Step 5. Determine the data X _n-2 and X _n+2 to be restored in the second stage, use the existing data and the data X _n restored in the first stage, and use the improved standard cubic spline interpolation function to restore the data X _n-2 and X _n+2 ; Step 6. Calculate the adjacent interval Z ₃ of the selected point of the data to be restored in the third stage, and the calculation formula is as follows: Z ₃ =Z ₂ -1 Step 7. Determine the data Xn _-1 and Xn ₊₁ to be restored in the third stage, use the existing data and the data _Xn , Xn _-2 and Xn ₊₂ restored in the first two stages, and use the improved standard three times The spline interpolation function restores the data X _n-1 and X _n+1 ; (2) When the number of consecutive lost data is even Assume that there are 6 consecutive missing data in a set of PMU measurement data obtained in a period of time, namely X _n-3 , X _n-2 , X _n-1 , X _n , X _n+1 , X _{n +2} ; Step a. Input all PMU data X ₁ , X ₂ , . . . , X _m , the number of lost data N, the number of recovery times M, and the adjacent interval Z, where M=N; Step b. Calculate the adjacent interval Z ₁ of the selected points of the data to be restored in the first stage, and the calculation formula is as follows:

Step c, determine the data X _n-1 and X _n to be restored in the first stage, wherein

Select four adjacent points that are Z ₁ before and after the data to be restored, and use the improved standard cubic spline interpolation function to restore the data X _n-1 and X _n ; Step d, calculate the adjacent interval Z ₂ of the selected points of the data to be restored in the second stage, and the calculation formula is as follows: Z ₂ =Z ₁ -1 Step e. Determine the data X _n-3 and X _n+2 to be restored in the second stage, use the existing data and the data X _n-1 and X _n restored in the first stage, and use the improved standard cubic spline interpolation function to restore data Xn _-3 and Xn ₊₂ ; Step f, calculate the adjacent interval Z ₃ of the selected point of the data to be restored in the third stage, and the calculation formula is as follows: Z ₃ =Z ₂ -1 Step g, determine the data Xn _-2 and Xn ₊₁ to be restored in the third stage, use the existing data and the data Xn _-1 , _Xn , Xn _-3 and Xn ₊₂ restored in the first two stages, Use the improved standard cubic spline interpolation function to restore the data Xn _-2 and Xn ₊₁ ; Step 4. On the basis of determining the recovery sequence of the lost data, use the improved standard cubic spline interpolation function to recover the lost data in different scenarios; The modeling method of the improved standard cubic spline interpolation function is: (1) Construct a cubic spline interpolation function given function y _i =f(x _i ), i=1,2,...,n (1) in, a=x ₀ <x ₁ <x ₂ <...<x _n =b, [a,b] If S(x)=y is on each subinterval [x _k ,x _k+1 ](k=1,2,...,n-1), is a polynomial of no more than three times, and S(x _i ) =y _i , i=1,2,...,n; and S(x), S′(x), S″(x) are continuous on [a, b], then S(x) is called f (x) cubic spline interpolation function on nodes x ₀ , x ₁ , x ₂ ,...x _n ; (2) Let M _i =S″(x _i ), on the interval [x _i ,x _i+1 ], the second derivative of S(x)=S _i (x) can be expressed as:

in, h _i =x _i+1 -x _i (3) Integrate formula (1) for two consecutive times to obtain:

From the continuity S'(x _i- )=S'(x _i+ ), we can get: μ _i M _i-1 +2M _i +λ _i M _i =d _i (i=1,2,...,n-1) (5) in,

(4) According to the boundary conditions S'(x ₀ )=y ₀ ', S'(x _n )=y _n ', formula (5) is expressed in the following matrix form:

The coefficient matrix of equation (7) is strictly diagonally dominant and has a unique solution, S(x) S _i (x) on each interval [x _i ,x _i+1 ] is:

(5) Constructing an improved cubic spline interpolation function Assuming that the maximum and minimum values of S _i (x) in the interval [x _i-1 , _xi ] (i=1,2,...,n) are S _imax and S _{imin respectively} , then S(x) The extreme value difference on the interval [a,b] is:

Assuming that S'(x ₀ )=y ₀ ', S'(x _n )=y _n ' is unknown, it can be obtained from the extreme value difference of S(x):

In the formula, f(x _i ) is an arbitrary function; a and b are the value intervals of the independent variable x; M _i is the second derivative of the cubic spline interpolation function.

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