CN108596475B - PMU data recovery method based on dynamic change of interpolation interval - Google Patents
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Abstract
The invention discloses a PMU data recovery method based on interpolation interval dynamic change, belonging to the technical field of PMU data recovery. The method comprises the following steps: according to the actual scene of PMU lost data, a single-point data loss scene and a continuous multi-point data loss scene are defined; determining the recovery sequence of the lost data, and providing a priority recovery method based on the dynamic change of an interpolation interval; and recovering the lost data, providing an improved cubic spline interpolation function, and providing constraint conditions for variables to enable a spline curve to be flatter. The method can effectively recover different types of lost data under different states of the system, solves the distortion problem of the traditional method under the continuous multipoint data loss scene, and has great significance and great value for ensuring the data quality of the PMU.
Description
Technical Field
The invention belongs to the technical field of PMU data recovery, and particularly relates to a PMU data recovery method based on dynamic change of interpolation intervals.
Background
With the development and utilization of large-scale renewable energy sources and the development of smart power grids, the China has currently built a super-large-scale complex interconnected power system which has the largest service population, the widest coverage range, the highest transmission voltage level and the largest renewable energy source accommodation in the world. The mechanism characteristics, the analysis method and the operation control mode of the power system all change fundamentally. The integrity of the power system is increasingly prominent, the systematic cascading failure of cross-region and cross-voltage grades gradually becomes a normal state, and the fine control requirement of a closed loop is obvious. The synchronous Phasor Measurement Units (PMUs) enable dynamic real-time monitoring of the power system due to their synchronicity, rapidity and accuracy, and can provide a data basis for system protection and closed-loop control. At present, 3000 left and right PMU devices are installed and operated in China, and all transformer substations of 220kV and above, main power plants and new energy grid-connected collection stations are covered. In addition, approximately 2000 commercial-grade PMUs have been installed in north america. However, due to the complex field environment, affected by the factors such as loss of synchronization signal, error of communication protocol, system overload, and transmission delay, the PMU inevitably has the problem of data loss, which seriously affects the application of the PMU in dynamic monitoring and closed-loop control, and even threatens the safety of the power grid. Under the background that the topology of a power grid is unknown and only PMU measurement information exists, the loss of PMU data can cause the power grid to be fragile and invisible, and the power grid is easy to be disturbed and attacked and even cause blackout accidents. Therefore, PMU data recovery has become a critical issue for ensuring the safety of the power system. In the prior art, a time series method, a matrix low rank method, a state estimation method, and the like are mostly adopted. The methods can effectively recover single-point lost data, but have no good recovery effect on continuous multi-point data loss.
Disclosure of Invention
In order to solve the above problems, the present invention provides a PMU data recovery method based on interpolation interval dynamic changes, which is characterized by comprising the following steps:
and 4, on the basis of determining the recovery sequence of the lost data, recovering the lost data under different scenes by using an improved standard cubic spline interpolation function.
The single-point data loss scene is a scene that only a single data loss exists in a group of PMU measurement data obtained within a period of time.
The continuous multipoint data loss scene is a scene that continuous multipoint data are lost in a group of PMU measurement data obtained within a period of time.
In the step 3, in a continuous multipoint data loss scene, a priority recovery method based on dynamic change of the difference interval is adopted to calculate a recovery sequence of lost data, and the specific calculation method is as follows:
(1) when the number of the continuous lost data is odd
Suppose there are 5 consecutive missing data, X respectively, in a set of PMU measurements taken over a period of timen-2，Xn-1，Xn，Xn+1，Xn+2；
Z2＝Z1-1
Z3＝Z2-1
(2) When the number of continuous lost data is even number
Suppose that it is in a period of timeIn the obtained PMU measurement data, there are 6 continuous lost data in the set of PMU measurement data, X respectivelyn-3，Xn-2，Xn-1，Xn，Xn+1，Xn+2；
Step a, inputting all PMU data X1，X2，...，XmThe method comprises the following steps of N, the number of lost data, M and an adjacent interval Z, wherein M is N;
step b, calculating the adjacent interval Z of the selected point of the data to be recovered in the first stage1The calculation formula is as follows:
step c, determining the data X to be recovered in the first stagen-1And XnWherein
Selecting the 4 adjacent intervals Z before and after the data to be recovered1Using an improved standard cubic spline interpolation function to recover the data Xn-1And Xn；
D, calculating the adjacent interval Z of the selected point of the data to be recovered in the second stage2The calculation formula is as follows:
Z2＝Z1-1
step e, determining the data X to be recovered in the second stagen-3And Xn+2Using existing data and first stage recovered data Xn-1And XnRecovering the data X using a modified standard cubic spline interpolation functionn-3And Xn+2；
Step f, calculating the adjacent interval Z of the selected point of the data to be recovered in the third stage3The calculation formula is as follows:
Z3＝Z2-1
step g, determining the data X to be recovered in the third stagen-2And Xn+1Using existing data and data X recovered from the first two stagesn-1、Xn、Xn-3And Xn+2Recovering the data X using a modified standard cubic spline interpolation functionn-2And Xn+1。
The improved modeling method of the standard cubic spline interpolation function comprises the following steps:
(1) constructing cubic spline interpolation function
Given function
yi＝f(xi)，i＝1，2，...，n (1)
Wherein,
a＝x0＜x1＜x2＜...＜xn＝b，[a，b]
if S (x) y is in each subinterval [ x ]k，xk+1](k-1, 2.., n-1) is a polynomial of not more than three degrees, and S (x)i)＝ y i1,2, ·, n; and S (x), S' (x) are in [ a, b ]]In the above sequence, S (x) is f (x) at node x0，x1，x2，…xnThe cubic spline interpolation function above;
(2) let Mi＝S″(xi) In the interval [ xi，xi+1]S (x) is Si(x) The second derivative of (d) can be expressed as:
wherein,
hi＝xi+1-xi
(3) integrating equation (1) twice in succession to obtain:
from the continuity S' (x)i-)＝S′(xi+) The following can be obtained:
μiMi-1+2Mi+λiMi＝di(i＝1，2，...，n-1) (5)
wherein,
(4) according to the boundary condition S' (x)0)＝y0′，S′(xn)＝yn', equation (5) is expressed in the form of the following matrix:
the coefficient matrix of equation (7) is strictly diagonal and has a unique solution,
s (x) in each interval [ x ]i，xi+1]Go to Si(x) Comprises the following steps:
(5) construction of improved cubic spline interpolation function
Suppose Si(x) In the interval [ xi-1，xi]The maximum and minimum values of (i ═ 1, 2.. times.n) are S, respectivelyimaxAnd SiminThen S (x) is in the interval [ a, b ]]The above extreme difference is:
suppose S' (x)0)＝y0′，S′(xn)＝yn' unknown, derived from the extreme difference of S (x):
in the formula, f (x)i) Is an arbitrary function; a. b is the value interval of the independent variable x; miThe second derivative of the cubic spline interpolation function.
The invention has the beneficial effects that:
(1) the invention can effectively recover data for different types of lost data when the system is in different states, and is not restricted by the topological structure of the power grid, and can be realized only by inputting PMU real-time measured data. The method solves the distortion problem of the traditional method under the continuous multipoint data loss scene, and provides an effective and feasible method for ensuring the data quality of the PMU.
(2) The priority recovery strategy based on the dynamic change of the interpolation interval ensures the relation between the sampling interval and the Nyquist frequency, avoids the damage of periodic signals of PMU data and ensures the accuracy of recovery.
(3) The invention improves the cubic spline interpolation function and provides a variable M0And MnThe constraint conditions ensure that the first derivative and the second derivative of the spline interpolation function are continuous. Meanwhile, the dragon lattice phenomenon is avoided, so that the spline curve is flatter.
Drawings
FIG. 1 is a flow chart of a PMU data recovery method based on interpolation interval dynamic changes;
FIG. 2(a) is a basic scenario of PMU single point data loss;
FIG. 2(b) is a basic scenario of PMU continuous multipoint data loss;
FIG. 3(a) is a PMU discontinuous multipoint data loss scenario;
FIG. 3(b) is a PMU complex data loss scenario;
FIG. 4 shows an odd number of PMU data loss cases;
FIG. 5 shows an even number of PMU data loss cases;
FIG. 6 is a comparison of the recovery results for different types of lost data for the two methods;
FIG. 7 shows measured data of the amplitude and phase angle of the A-phase voltage of a PMU;
FIG. 8 is a TVE comparison of single point loss data recovery results in steady state;
fig. 9 is a TVE comparison of single-point missing data recovery results in a transient state;
FIG. 10(a) shows the recovery result of continuous multi-point missing data in steady state;
FIG. 10(b) shows the recovery result of the multi-point missing data under transient state;
FIG. 11 shows the influence of the number of lost data on TVE in steady and transient states;
FIG. 12(a) shows the effect of the number of lost data on the TVE of the present invention at steady state;
FIG. 12(b) is a diagram illustrating the influence of the number of missing data in transient on the TVE of the present invention;
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and examples.
Fig. 1 is a flowchart of a PMU data recovery method based on dynamic changes of interpolation intervals, as shown in fig. 1, the method includes the following steps:
step 1: establishing two basic scenes of PMU data loss, including a single-point data loss scene and a continuous multi-point data loss scene;
step 2: analyzing the loss type of the data and determining a data loss scene;
and step 3: determining the recovery sequence of the lost data, and if the recovery sequence of the lost data is in a continuous multipoint data loss scene, considering the priority distribution of data recovery and calculating the recovery sequence of the lost data according to the parity of the number of the lost data; if the single-point data loss scene exists, priority distribution does not need to be considered;
and 4, step 4: on the basis of determining the recovery sequence of the lost data, the lost data under different scenes are recovered by improving a standard cubic spline interpolation function.
Specifically, in step 1, in order to simplify the PMU data loss problem model and facilitate data recovery, two basic scenarios of data loss are defined according to accuracy, availability and real-time of PMU data and actual situation of field data loss, which are a single-point data loss scenario and a continuous multi-point data loss scenario respectively. Fig. 2(a) shows a single-point data loss scenario, fig. 2(b) shows a continuous multi-point data loss scenario, and in fig. 2(a) -2(b), squares represent PMU measurement data over a period of time, where white is lost data and black is known data. Other scenarios may be equivalent to different combinations of the two scenarios described above. For example, a discontinuous multipoint data loss scenario may be equivalent to a plurality of single point data loss scenarios, as shown in fig. 3 (a); a complex data loss scenario may be equivalent to a combination of single point data and continuous multipoint data loss, as shown in fig. 3 (b). According to fig. 2(a) and 3(a), a single-point data loss scenario may be defined as a scenario where only a single data loss exists in a set of PMU measurement data obtained over a period of time; according to fig. 2(b) and 3(b), a continuous multipoint data loss scenario may be defined as a scenario in which there is continuous multipoint data loss in a set of PMU measurement data obtained over a period of time.
Specifically, in step 3, the recovery sequence of the lost data is determined according to two basic scenarios of the PMU lost data established in step 1, and if in the single-point data loss scenario, priority allocation does not need to be considered. If in a scene of continuous multipoint data loss, because the existing method needs to carry out a large amount of calculation and the result error is large, the invention provides a priority recovery method based on dynamic change of an interpolation interval. The following two cases are discussed for the parity of the number of consecutive lost data:
(1) when the number of the continuously lost data is odd
FIG. 4 shows an odd number of PMU data loss cases, as shown in FIG. 4, assuming Xn-2，Xn-1，Xn，Xn+1，Xn+2For 5 consecutive lost data and the rest known data, in this case, the adopted priority recovery method specifically comprises the following steps:
Z2＝Z1-1
Z3＝Z2-1
(2) When the number of the continuous lost data is even
FIG. 5 shows an even number of PMU data loss cases, as shown in FIG. 5, assuming Xn-3，Xn-2，Xn-1，Xn，Xn+1，Xn+2For 6 consecutive lost data and the rest known data, in this case, the adopted priority recovery method specifically comprises the following steps:
step a, inputting all PMU data X1，X2，...，XmThe method comprises the following steps of N, the number of lost data, M and an adjacent interval Z, wherein M is N;
step b, calculating the adjacent interval Z of the selected point of the data to be recovered in the first stage1The calculation formula is as follows:
step c, determining the data X to be recovered in the first stagen-1And XnWherein
Selecting the 4 adjacent intervals Z before and after the data to be recovered1Using an improved standard cubic spline interpolation function to recover the data Xn-1And Xn；
D, calculating the adjacent interval Z of the selected point of the data to be recovered in the second stage2The calculation formula is as follows:
Z2＝Z1-1
step e, determining the data X to be recovered in the second stagen-3And Xn+2Using existing data and first stage recovered data Xn-1And XnRecovering the data X using a modified standard cubic spline interpolation functionn-3And Xn+2；
Step f, calculating the adjacent interval Z of the selected point of the data to be recovered in the third stage3The calculation formula is as follows:
Z3＝Z2-1
step g, determining the data X to be recovered in the third stagen-2And Xn+1Using existing data and data X recovered from the first two stagesn-1、Xn、Xn-3And Xn+2By usingImproved standard cubic spline interpolation function recovery data Xn-2And Xn+1。
According to the sampling theorem, the adopted interval Z meets the condition that the interval between all selected points is equal to the interval of the point to be recovered so as to ensure the relation between the sampling interval and the Nyquist frequency. Compared with other algorithms, the recovery method does not damage the periodic signals of PMU data.
Specifically, in the step 4, based on the priority recovery strategy, the method of the present invention recovers the lost data by using an interpolation method. The cubic spline interpolation function is obtained by using the known data, so that the cubic spline interpolation function has higher fitting degree with the known data, and the lost data is obtained. In order to avoid the dragon phenomenon caused by using a high-order polynomial, the invention improves the cubic spline interpolation function, solves the improved cubic spline interpolation function meeting the extreme difference constraint condition and recovers the lost data, wherein the improved cubic spline interpolation function provides the constraint condition for variables on the basis of ensuring the continuity of the first derivative and the second derivative of the solved spline interpolation function, so that the curve has good smoothness, stronger linear approximation capability and can effectively represent the data change condition, and the modeling method comprises the following steps:
(1) constructing cubic spline interpolation function
Given function
yi＝f(xi)，i＝1，2，...，n (1)
Wherein,
a＝x0＜x1＜x2＜...＜xn＝b，[a，b]
if S (x) y is in each subinterval [ x ]k，xk+1](k-1, 2.., n-1) is a polynomial of not more than three degrees, and S (x)i)＝ y i1,2, ·, n; and S (x), S' (x) are in [ a, b ]]In the above sequence, S (x) is f (x) at node x0，x1，x2，...xnCubic spline interpolation function above.
(2) Let Mi＝S″(xi) Due to cubic spline interpolationThe interpolation formula may ensure that the first derivative is smooth at each segment interval and boundary, while the second derivative is continuous. Thus in [ x ]i，xi+1]S (x) is Si(x) The second derivative of (d) can be expressed as:
wherein,
hi＝xi+1-xi
(3) two successive integrations are performed on equation (1) to yield:
from the continuity S' (x)i-)＝S′(xi+) The following can be obtained:
μiMi-1+2Mi+λiMi＝di(i＝1，2，...，n-1) (5)
wherein,
(4) according to the boundary condition S' (x)0)＝y0′，S′(xn)＝yn', equation (5) is expressed in the form of the following matrix:
(5) construction of improved cubic spline interpolation function
Because the cubic spline interpolation function is a function related to boundary conditions, in order to enable the spline curve to be flatter, the extreme value difference is minimum according to the actual changing situation of the power system.
Suppose Si(x) In the interval [ xi-1，xi]The extrema at (i ═ 1, 2.. times., n) are each SimaxAnd SiminThen S (x) is in the interval [ a, b ]]The above extreme difference is:
suppose S' (x)0)＝y0′，S′(xn)＝ynIf unknown, the difference between the extreme values of S (x) is related to M0And MnTo obtain a function of M0And MnThe objective function of the unconstrained non-linear programming of (1) is:
formula (9) relates to M0And MnThe unconstrained nonlinear programming problem can be solved by a simplex substitution method because the objective function contains extremum operation and the parameters are determined by an equation system.
Example 1
To further explain the present invention in detail, in this embodiment, Matlab is used to perform simulation test in both the steady state and the transient state of the system, and PMU measured data is used to verify and compare the data with the recovery result of the existing algorithm. The synchronous phasor measurement error is measured by a Total Vector Error (TVE), and the calculation formula is as follows:
in the formula, Xr(n) and Xi(n) denotes the real and imaginary parts, X ', respectively, of the theoretical value of the input signal'r(n) and X'iAnd (n) respectively represent the real part and the imaginary part of the estimated value of the input signal.
1. Simulation test
The power system steady-state signal expression is as follows:
in the formula, XmIs the phasor amplitude, f0Is the power frequency of the power line,
is a primary phase angle, and Xm＝57.73V，f0＝50Hz，
Under ideal conditions, the phasor, the frequency and the frequency change rate of the signal do not change at all, and the output result is constant, so that the signal is easy to recover when data is lost. When the frequency of the system is shifted, the power system transient signal expression is as follows:
in the formula, Δ f is a frequency offset, and Δ f is 5 Hz.
(1) Optimal interpolation point number test
The invention adopts an interpolation method to recover data, and the number of interpolation points directly influences the data recovery precision and the calculation speed. Therefore, taking the frequency offset signal as an example, the optimal number of interpolation points is tested.
Randomly selecting single-point lost data, recovering the lost data by adopting different interpolation points, repeating the experiment for 100 times, and recording the result, wherein the test result is shown in table 1, the TVE in table 1 represents the mean value, and the result shows that the recovery result of the adjacent 8-point data before and after the lost data is optimal.
TABLE 1 optimal number of interpolation points
| Number of |
2 | 4 | 6 | 8 | 10 |
| TVE() | 4.89％ | 0.16 | 0.06 | 0.07 | 0.07 |
(2) Data recovery results at frequency offset
The data recovery method provided by the invention is used for recovering different types of lost data in a frequency offset signal, and compared with the existing method, the comparison result is shown in figure 6, figure 6 shows the recovery effect of different types of lost data in a complex plane when frequency offset is carried out, in the figure, the horizontal and vertical axes respectively represent a real part and an imaginary part, PS and LS are the recovery results of the single-point lost data by the method and the existing method, and PM and LS are the recovery results of the continuous multi-point lost data by the method and the existing method. As can be seen from fig. 6, the TVE for recovering single-point loss data by the method of the present invention is 0.07%, which is better than 3.06% of the existing method. When continuous multipoint data is lost, the existing method has serious distortion, and the method provided by the invention can better recover the frequency offset signal and keep the variation trend of the data. The TVE comparison results of the two methods are shown in table 2, and it can be known from table 2 that the TVELM gradually increases with the increase of the lost data, and the TVEPM changes nonlinearly, so that the lost data can be effectively recovered.
TABLE 2 two methods TVE comparison
2. PMU measured data test
The method analyzes the data obtained by monitoring the PMU during the synchronous oscillation of a certain time in a certain new energy convergence area in the western China, and performs data recovery verification. The amplitude and phase angle measurement data of the A-phase voltage of the PMU at a certain synchronous oscillation are shown in FIG. 7.
(1) Comparing single-point data loss recovery results of system in stable and transient states
When the system is in a steady state, the randomly selected single-point lost data is recovered by respectively adopting the method and the existing method. Comparing the TVE of the lost data recovery result with that shown in fig. 8, it can be seen from fig. 8 that 80% of TVEPS is less than 1% and only 30% of TVELS is less than 1% in the steady state, and the error of the recovery result is greatly changed, which is difficult to be applied in practice.
When the system is in a transient state, the randomly selected single point of lost data is recovered by the same two methods. The comparison of the lost data recovery results TVE and LS is shown in fig. 9, where PS corresponds to the primary ordinate and LS corresponds to the secondary ordinate in fig. 9. Therefore, when the system generates subsynchronous oscillation, 90% of TVEPS is less than 3%, the recovery effect is better, only a few TVELS are less than 5%, and the recovery data are seriously distorted. Due to voltage oscillation, if the lost data is located at the position of a peak or a trough, the method is difficult to recover the variation trend only through the previous data.
(2) Continuous multipoint data loss recovery result comparison under stable and transient states of system
When the system is in a stable state and a transient state, the continuous multipoint lost data is recovered by respectively adopting the method of the invention and the existing method, the results are shown in fig. 10(a) -10(b), the test result in the stable state shown in fig. 10(a) can be obtained from fig. 10(a), the capability of recovering the continuous lost data by the method of the invention in the stable state is better, and the TVELM is gradually increased along with the increase of the lost data and is consistent with the simulation test result. FIG. 10(b) shows the test result in the transient state, and from FIG. 10(b), when the system is in the transient state, the TVELM variation condition is the same as that in the steady state, the recovery result has a larger deviation, the TVEPM is below 1%, and the recovery result has high accuracy.
(3) Analyzing the influence of the number of lost data on TVE
By changing the number of the lost data, the recovery effect of the two methods under stable and transient conditions, i.e. the TVE change conditions, are compared, and the comparison result is shown in fig. 11, 12(a) and 12(b), where fig. 11 shows the influence of the number of the lost data under stable and transient conditions on the TVE of the existing method, and in fig. 11, Lst represents the TVE change under stable conditions and corresponds to the principal vertical axis coordinate; ltt shows the TVE change at transient, corresponding to the secondary ordinate axis. As can be seen from fig. 11, both TVELst and TVELtt increase linearly with the increase in the number of lost data. However, this method only uses the data before the missing point, and cannot accurately correct each recovery result, so that the error gradually increases.
Fig. 12(a) -12(b) show the influence of the number of lost data on the TVE of the present invention under stable and transient conditions, and the TVE of the present invention can effectively recover data under both stable and transient conditions, as can be seen from fig. 12(a) -12 (b). The precision under steady state and transient state is much higher than that of the existing method, the lost data under steady state is easier to recover, and the TVE under transient state is not more than 1.1%. Meanwhile, as the number of the lost data increases, the recovery sequence is different according to the priority recovery method, so that the TVE change has no obvious relationship with the number of the lost data, and further research is still needed.
Through the tests, the lost data recovery under the single-point continuous multi-point loss scene is verified respectively when the system is in a temporary state and a steady state, and the test result shows that the method has a good recovery effect.
The present invention is not limited to the above embodiments, and any changes or substitutions that can be easily made by those skilled in the art within the technical scope of the present invention are also within the scope of the present invention. Therefore, the protection scope of the present invention shall 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 by comprising the following steps:
step 1, establishing two basic scenes of PMU data loss, including a single-point data loss scene and a continuous multipoint data loss scene;
the single-point data loss scene is a scene that only single data is lost in a group of PMU measured data obtained within a period of time;
the continuous multipoint data loss scene is a scene that continuous multipoint data are lost in a group of PMU measurement data obtained within a period of time;
step 2, analyzing the loss type of the data and determining a data loss scene;
step 3, determining the recovery sequence of the lost data, and if the recovery sequence of the lost data is in a continuous multipoint data loss scene, considering the priority distribution of data recovery and calculating the recovery sequence of the lost data according to the parity of the number of the lost data; if the single-point data loss scene exists, priority distribution does not need to be considered;
in the step 3, in a continuous multipoint data loss scene, a priority recovery method based on dynamic change of the difference interval is adopted to calculate a recovery sequence of lost data, and the specific calculation method is as follows:
(1) when the number of the continuous lost data is odd
Suppose there are 5 consecutive missing data, X respectively, in a set of PMU measurements taken over a period of timen-2,Xn-1,Xn,Xn+1,Xn+2；
Step 1, inputting all PMU data X1,X2,...,XmThe method comprises the following steps of N, the number of lost data, M and an adjacent interval Z, wherein M is N;
step 2, calculating the adjacent interval Z of the selected point of the data to be recovered in the first stage1The calculation formula is as follows:
step 3, determining the data X to be recovered in the first stagenWherein
Selection of XnThe front and the back are respectively adjacent to each other at an interval Z1Using an improved standard cubic spline interpolation function to recover the data Xn；
Step 4, calculating the adjacent interval Z of the selected point of the data to be recovered in the second stage2The calculation formula is as follows:
Z2＝Z1-1
step 5, determining the data X to be recovered at the second stagen-2And Xn+2Using existing data and first stage recovered data XnRecovering the data X using a modified standard cubic spline interpolation functionn-2And Xn+2；
Step 6, calculating the adjacent interval Z of the selected point of the data to be recovered in the third stage3The calculation formula is as follows:
Z3＝Z2-1
step 7, determining the data X to be recovered in the third stagen-1And Xn+1Using existing data and data X recovered from the first two stagesn、Xn-2And Xn+2Recovering the data X using a modified standard cubic spline interpolation functionn-1And Xn+1；
(2) When the number of continuous lost data is even number
Suppose there are 6 consecutive missing data, X respectively, in a set of PMU measurements taken over a period of timen-3,Xn-2,Xn-1,Xn,Xn+1,Xn+2；
Step a, inputting all PMU data X1,X2,...,XmThe method comprises the following steps of N, the number of lost data, M and an adjacent interval Z, wherein M is N;
step b, calculating the adjacent interval of the selected points of the data to be recovered in the first stageZ1The calculation formula is as follows:
step c, determining the data X to be recovered in the first stagen-1And XnWherein
Selecting the 4 adjacent intervals Z before and after the data to be recovered1Using an improved standard cubic spline interpolation function to recover the data Xn-1And Xn；
D, calculating the adjacent interval Z of the selected point of the data to be recovered in the second stage2The calculation formula is as follows:
Z2＝Z1-1
step e, determining the data X to be recovered in the second stagen-3And Xn+2Using existing data and first stage recovered data Xn-1And XnRecovering the data X using a modified standard cubic spline interpolation functionn-3And Xn+2；
Step f, calculating the adjacent interval Z of the selected point of the data to be recovered in the third stage3The calculation formula is as follows:
Z3＝Z2-1
step g, determining the data X to be recovered in the third stagen-2And Xn+1Using existing data and data X recovered from the first two stagesn-1、Xn、Xn-3And Xn+2Recovering the data X using a modified standard cubic spline interpolation functionn-2And Xn+1；
Step 4, on the basis of determining the recovery sequence of the lost data, recovering the lost data under different scenes by using an improved standard cubic spline interpolation function;
the improved modeling method of the standard cubic spline interpolation function comprises the following steps:
(1) constructing cubic spline interpolation function
Given function
yi＝f(xi)，i＝1,2,...,n (1)
Wherein,
a＝x0＜x1＜x2＜…＜xn＝b，[a,b]
if S (x) y is in each subinterval [ x ]k,xk+1](k-1, 2.., n-1) is a polynomial of not more than three degrees, and S (x)i)＝yi1,2, ·, n; and S (x), S' (x) are in [ a, b ]]In the above sequence, S (x) is f (x) at node x0,x1,x2,…xnThe cubic spline interpolation function above;
(2) let Mi＝S″(xi) In the interval [ xi,xi+1]S (x) is Si(x) The second derivative of (d) can be expressed as:
wherein,
hi＝xi+1-xi
(3) integrating equation (1) twice in succession to obtain:
from the continuity S' (x)i-)＝S′(xi+) The following can be obtained:
μiMi-1+2Mi+λiMi＝di(i＝1,2,...,n-1) (5)
wherein,
(4) according to the boundary condition S' (x)0)＝y0′,S′(xn)＝yn', equation (5) is expressed in the form of the following matrix:
the coefficient matrix of equation (7) is strictly diagonal and has a unique solution,
s (x) in each interval [ x ]i,xi+1]Go to Si(x) Comprises the following steps:
(5) construction of improved cubic spline interpolation function
Suppose Si(x) In the interval [ xi-1,xi]The maximum and minimum values of (i-1, 2, …, n) are SimaxAnd SiminThen S (x) is in the interval [ a, b ]]The above extreme difference is:
suppose S' (x)0)＝y0′,S′(xn)＝yn' unknown, derived from the extreme difference of S (x):
in the formula, f (x)i) Is an arbitrary function; a. b is the value interval of the independent variable x; miThe second derivative of the cubic spline interpolation function.
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