F1000ResearchF1000Research2046-1402F1000 Research LimitedLondon, UK10.12688/f1000research.73182.3Research ArticleArticlesModified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)
[version 3; peer review: 2 approved, 1 not approved]
ThangarajShruthiConceptualizationData CurationFormal AnalysisInvestigationMethodologySoftwareWriting – Original Draft Preparation1GohVik TorMethodologySupervisionWriting – Review & Editinghttps://orcid.org/0000-0003-4573-9493a1YapTimothy Tzen VunMethodologySupervisionWriting – Review & Editing2
Faculty of Engineering, Multimedia University, Cyberjaya, Selangor, 63100, Malaysia
Faculty of Computing and Informatics, Multimedia University, Cyberjaya, Selangor, 63100, Malaysiavtgoh@mmu.edu.my
Smart grid systems require high-quality Phasor Measurement Unit (PMU) data for proper operation, control, and decision-making. Missing PMU data may lead to improper actions or even blackouts. While the conventional cubic interpolation methods based on the solution of a set of linear equations to solve for the cubic spline coefficients have been applied by many researchers for interpolation of missing data, the computational complexity increases non-linearly with increasing data size.
Methods
In this work, a modified recurrent equation-based cubic spline interpolation procedure for recovering missing PMU data is proposed. The recurrent equation-based method makes the computations of spline constants simpler. Using PMU data from the State Load Despatch Center (SLDC) in Madhya Pradesh, India, a comparison of the root mean square error (RMSE) values and time of calculation (ToC) is calculated for both methods.
Results
The modified recurrent relation method could retrieve missing values 10 times faster when compared to the conventional cubic interpolation method based on the solution of a set of linear equations. The RMSE values have shown the proposed method is effective even for special cases of missing values (edges, continuous missing values).
Conclusions
The proposed method can retrieve any number of missing values at any location using observed data with a minimal number of calculations.
phasor measurement unitmissing datadata recoverysmart gridinterpolationcubic splinedata qualitydata pre-processingThe author(s) declared that no grants were involved in supporting this work.Amendments from Version 2
We have addressed the reviewer's concerns, namely the consistency in nomenclature and accuracy of equation numbers. Additionally, we included brief discussions about the results, characteristics of the dataset, and choice of evaluation metrics.
Introduction
The worldwide growing power systems highlight the need for better monitoring and control mechanisms to avoid major blackouts. Smart grids are intelligent systems that facilitate the development of communication, network, and computing technologies, protocols, and standards to integrate power system elements for two-way communication. This time-synchronized high-precision measurement device that is also known as a synchrophasor or Phasor Measurement Unit (PMU), gives clear information on the working of the entire grid. The PMU is used to monitor and control the power grid. It can help in providing real-time measurements by eliminating adverse conditions like blackouts. These combined characteristics of data availability, timeliness, and communication network contribute to the better performance of the PMU system. Although the role, impact,
1 architecture, technology,
2 applications, functionality, standards, and evolution of PMU (timing, measurement, communication, and data storage) have been released since 1995, the North American Synchro Phasor Initiative (NASPI) has highlighted the importance of data quality.
3 Data quality issues, their potential causes, and consequences are elaborated.
4–6 Generally, incomplete or missing data might affect the functionality of the entire system.
7 Hence, a way to handle missing values in PMU is mandatory for the effective functioning of the entire grid system.
With the advent of PMU systems, large datasets are generated and finding missing values using traditional cubic interpolation methods take larger computational time with the increase in data size. In this paper, a modified recurrent equation-based method termed the Alpha Method (AM) for PMU missing data problem is proposed. In this approach, a series of linear equations are solved using the modified recurrent equation to obtain a relationship between points on a spline, which is then used to estimate any missing values on the spline. We compare the proposed method to the more traditional method of solving linear equations, namely using tri-diagonal matrix or termed as the Linear Equations Method (LEM) in this paper. The proposed AM is computationally more efficient and takes less time to process than the LEM. Moreover, in real-time systems when the dataset grows progressively, we show that AM is better than LEM.
Literature review
The need to recover missing values in PMU data is vital to the proper operation of smart grids and the energy infrastructure. Literatures
5–7 indicate that missing data in PMU systems can negatively affect the accuracy of decision-making process and additionally, introduce security risks to the infrastructure. To address this problem, missing values have to be recovered and one of the more popular approaches is utilizing matrix completion.
8–12 Despite that, this approach is still largely theoretical and even so, viable methods utilizing this approach have only been tested on simulated data.
Alternatively, interpolation-based missing data recovery techniques
13–15 propose the reconstruction of missing values by a spatial interpolation or spatio-temporal interpolation of the values. Some work
16,17 even suggested advanced approaches utilizing
k-nearest neighbors and recurrent relation-based interpolations. However, in interpolation-based techniques, historical data such as channel or time data is needed for more accurate calculations. interpolation. As such, there is a need to design effective data recovery methods to work without the need for historical data processing.
3 So, a data-driven recovery technique capable of recovering missing entries with available or observed data is much needed. Moreover, the technique should not become overly complex or require high computational time as the size of the data grows.
Methods
Cubic spline interpolation is a widely used polynomial interpolation method for functions of one variable. Let
f be a function from
RtoR. It is assumed that the value of
fis known only at
x1≤x2.≤xi…≤xnand letfxi=ai. Piecewise cubic spline interpolation is the problem of finding the
bi,
ciand
di coefficients of the cubic polynomials
SFifor0≤i≤n−1 written in the form:
SFix=ai+bix−xi+cix−xi2+dix−xi3
Where
x can take any value between
xi and
xi+1. That is,
SFixi=ai
Let the first-order derivative of
equation (1) be:
SFi′x=bi+2cix−xi+3dix−xi2
The first-order derivative at
xi for values of
1≤i≤n−1 will be
SFi′xi=bi
And the second-order derivative be:
SFi′′x=2ci+6dix−xi
The second-order derivative at
xi for values of
1≤i≤n−1 will be:
SFi′′xi=2ci
For a smooth fit between the adjacent pieces, the cubic spline interpolation requires that the following conditions hold:
The cubic functions should intersect at the points left and right, for
i=0ton−1SFixi+1=SFi+1xi=ai+1
For each cubic function to join smoothly with its neighbors, the splines should have continuous first and second derivatives at the data points
i=1,…,n−1:SFi′xi+1=SFi+1′xi=bi+1SFi′′xi+1=SFi+1′′xi=2.ci+1
If
hi=
xi+1−xi and if
hi is equal for all
ivalues, following Revesz,
17 the relation between coefficients
ai and
ci can be resolved:
ci−1+4ci+ci+1=3h2ai−1−2ai+ai+1bi=ai+1−ai1hi−2ci+ci+13hidi=13.hici+1−ci
Equation (7) represents a system of linear equations for the unknowns
ci for
0≤i≤n. As the values of
aiare known, the value of
ci can be found by solving the tri-diagonal matrix-vector equation
Ax=B. While there are
n+1 numbers of
ci constants,
equation (7) yields only (
n-2) equations. Based on the nature or type of spline assumed two more equations representing the boundary conditions of the spline. In general, two types of splines may be considered: natural cubic spline and clamped cubic spline.
For natural cubic spline interpolation, the following boundary conditions are assumed:
c0=cn=0.0. That is, the second derivatives of the splines at the endpoints are assumed to be zero. Based on
equation (7), a system of (
N+1) linear equations of (
N+1) variables can be formulated as:
A=1000⋯00001410⋯00000141⋯0000⋮⋮⋮⋮⋮⋮⋮⋮⋮0000⋯14100000⋯01410000⋯0001,x=c0c1⋮cn,andB=03h2a0−2a1+a2⋮3h2an−2−2an−1+an0
For clamped cubic spline interpolation the following boundary conditions are assumed:
b0=f′(x0) and
bn=f′(xn), where the derivatives
f′(x0) and
f′(xn), are known constants. Thus, based on the boundary conditions assumed both natural and cubic splines result in
n+1 system of linear equations. The resulting system of
n+1 linear equations can be used to get unique solutions by any of the standard methods for solving a system of linear equations.
Once the values of
ci are found, the
bi and
di values can be obtained using
equations (8) and
(9) respectively. Similarly, under clamped spline interpolation,
A=2100⋯00001410⋯00000141⋯0000⋮⋮⋮⋮⋮⋮⋮⋮⋮0000⋯14100000⋯01410000⋯0012,x=c0c1⋮cnandB=3h2a1−a0−3hf′x03h2a0−2a1+a2⋮3h2an−2−2an−1+an3hf′x0−3h2an−an−1
Recurrent equation-based solution
Revesz,
17 chose boundary conditions that need to solve the tri-diagonal system given in
equation (7) where
xirational variables
ei rational constants,
r is a non-zero rational constant and
A is:
A=r100⋯00001410⋯00000141⋯0000⋮⋮⋮⋮⋮⋮⋮⋮⋮0000⋯14100000⋯01410000⋯0001,x=x1x2⋮xn−1xnandb=e1e2⋮en−1en
The first row of the new matrix in
equation (12) is shown to be equivalent to the first row of the clamped
b matrix
e1 is
e1=3r2ha1−a0h−f′x0+1r2c~1where,
c~1 is an estimate of
c1 and
r = 2+
√3≈3.732.
17
The chosen boundary conditions are such that the first row of the new matrix was the same as that of clamped cubic spline and while that of the last row was that of the natural cubic spline fixing the value of
cn as 0. Using
equation (12), the relationships between successive spline points can be obtained as:
xi+xi+1r=∑0≤k≤i−1−1kei−krk+1
Let
∝0,
∝ifor1<i≤n−1and∝n, respectively be:
∝0=0∝i=ei−∝i−1r=∑0≤k≤i−1−1kei−krk+1∝n=en
Based on the above, the closed form of solution for
xi can be given as:
xi=∑0≤k≤n−i−1r−k∝i+k
The above equation
(16) solves
xi no matter exactly what the initial values for
ei. This leads to a faster evaluation of the cubic spline than solving a tri-diagonal system. The major advantage of the method is when new measurements are added to the system. While conventional tri-diagonal matrix-based algorithm requires a complete redo of the entire computation,
equation (16) leads to a faster update for each
i ≤
n only with the addition of the term:
−1rn+1−i∝n+iand
xn+1=∝n+1. Similarly,
∝i constants can be updated by adding a single term
en+1
The system of linear equations given in
equation (7), in general, is solved by the standard solution of linear equations in the matrix form
Ax=b. Alternatively, it could be solved for
n variables by the recurrence relations given
equations (16) and
(17). The two methods, the first using the tri-diagonal matrix-based solution for the spline coefficients is termed the Linear Equations Method (LEM) and the second one using recurrence relations is termed the Alpha Method (AM). The algorithmic procedure for LEM and AM are given below.
Algorithmic procedure for regular tridiagonal matrix-based Linear Equation Method (LEM)
Step 1: Given the initial vector with missing values, separate them into two sets of vectors, the observed values vector
Robs and the missing values vector
RMiss, having sizes of
NO and
NM, respectively, such that
NO+
NM=
N.
Step 2:
Robs vector at
xi values of the (
NO-1) splines shall be the
ai coefficient vector.
Step 3: Using
ai, generate the RHS vector
B given in
equation (11).
Step 4: Generate a square coefficient matrix
A as given in
equation (11)
Step 5: Solve for the
civector is given in
(11), using the relation
Ax =
B
Step 6: Applying
ci in equations
(8) and
(9), compute the
bi and
di coefficient vectors for
n-2 points of the
Robs.
Step 7: Using the values of
ai,
bi,
cianddi, missing values can be found by the
equation (1) re-written as:
SFix=ai−bixi+cixi2−dixi3+bi−2cixi+3dixi2x+ci−3dixix2
Where
x represents the missing positions, between
xi and
xi+1 of spline
i.
Algorithmic procedure for recurrent equation-based Alpha Method (AM)
Step 1: Given the initial vector with missing values, separate them into two sets of vectors, the observed values vector
Robs and the missing values vector
RMiss, having sizes of
NO and
NM, respectively, such that
NO+
NM=
N.
Step 2: The
Robs vector at
xi values of the (
NO-1) splines is the
ai coefficient vector.
Step 3: Using
ai, generate the RHS vector
B given in
equation (11).
Step 4: Set
∝0=0and∝n=en, calculate the alpha vector using the relation.
∝i=ei−∝i−1r=∑0≤k≤i−1−1kei−krk+1 for
i values ranging from 1 to
NO-1
Step 5: Set
xn=∝n and solve for
ci values using the relation.
ci=∑0≤k≤n−i−1rk∝i+k
Step 6: Applying
ci in
equations (8) and
(9), compute the
bi and
di coefficient vectors for
n-2 points of the
Robs.
Step 7: Using the values of
ai,
bi,
cianddi, missing values can also be found using
equation (18), re-written here again for convenience:
SFix=ai−bixi+cixi2−dixi3+bi−2cixi+3dixi2x+ci−3dixix2
Where
x represents the missing positions, between
xi and
xi+1 of spline
i.
The modifications are as follows: In AM, rather than computing
E, alpha vectors and
ci coefficients for the full range of
NO-1 data points only the RHS,
E vector, was calculated for the full range of
NO-1 data points, while alpha vector and
ci were calculated only for
iandi+1 data elements, where
i is the missing data element. For the imputation of
i the element, only the
Ei vector for all
NO-1 data points,
∝i vector and
ci vectors for
iandi+1 and
bi and
di coefficients were essential for the calculation
ith missing element and its imputation.
In addition, using the AM, an effective procedure was demonstrated for the computation of the following cases: (i) missing first and the last element of the data vector, (ii) missing multiple data points at the beginning and the end, and (iii) missing multiple elements anywhere in the data vector. That is in
equation (18), when the current values of
A [
i] are replaced either with
A [
N-1] or
A [
i-1] based on the position of missing edge values or continuous values the Time of Calculation (ToC) and Root Mean Squared Error (RMSE) values have improved significantly.
The formula for RMSE is:
RMSE=∑i=1NPredicted−Actual2N
We have used RMSE and ToC as evaluation metrics to measure the effectiveness and efficiency of the proposed method because most literature used the same.
Results and discussion
A comparison between LEM and AM is shown here for the imputation of one-min real PMU system data having a size of 1490 data points for each of the 25 heterogeneous variables obtained from five different PMUs. Since our data does not have any missing values, we artificially introduced the missing values, of 10%, 20%, 30% in random.
A sample of one-minute PMU data for five PMUs’ was used in the study.
18 One minute of PMU data with 10%, 20%, and 30% missing data for five PMUs were evaluated.
When AM was employed, the average RMSE values were 0.83, 1.47, and 2.16 for 10%, 20%, and 30% of missing PMU data, respectively. This can be seen in
Figure 1. Moreover, for the same performance, AM showed significant improvements in its ToC as shown in
Figure 2. The average ToCs for AM were 1.35, 1.41, and 1.23s when recovering 10%, 20%, and 30% of its missing data.
By comparison, LEM had ToC values of 18.83, 16.02, 16.58s for 10%, 20%, and 30% of its missing data, respectively. The proposed method reduced the ToC by a factor of approximately 10 times. LEM had higher ToC values because it needed to solve the entire set of linear equations every time it needed to find the
b
i,
c
i, and
d
i coefficients. On the other hand, AM only needed to calculate these coefficients at two successive points of
i and
i+1.
Comparison of Root Mean Squared Error (RMSE) values.
Comparison of Time of Calculation (ToC).
Conclusions
In this study, AM was compared with LEM. However, because of the proliferation of the data, there is a need for customization of this technique to handle a high volume of data to reduce computational time and power. In the proposed method, the approaches demonstrated a reduced computational effort and time of calculation for solving the coefficient vectors. This study has made the following contributions: (i) the recurrent relation-based AM has been effectively employed in the imputation of PMU data and its advantages are demonstrated as an effective and efficient alternative to the conventional technique, and (ii) an effective procedure for handling missing values in special cases (edge, continuous values) is shown, which has not been addressed clearly in other methods. The proposed method has proven effective, and it only requires 10% effort in comparison to the LEM. Future research will focus on the application of the modified recurrent method in the analysis of real-time or stream PMU data.
Data availabilityUnderlying data
Harvard Dataverse: Underlying data for ‘Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)’, ‘PMU data’,
https://doi.org/10.7910/DVN/Y2LLJJ.
18
This project contains the following underlying data:
Data file: pmu1-1m-10.tab – One minute of data from PMU1 with 10% missing data
Data file: pmu1-1m-20.tab – One minute of data from PMU1 with 20% missing data
Data file: pmu1-1m-30.tab – One minute of data from PMU1 with 30% missing data
Data file: pmu2-1m-10.tab – One minute of data from PMU2 with 10% missing data
Data file: pmu2-1m-20.tab – One minute of data from PMU2 with 20% missing data
Data file: pmu2-1m-30.tab – One minute of data from PMU2 with 30% missing data
Data file: pmu3-1m-10.tab – One minute of data from PMU3 with 10% missing data
Data file: pmu3-1m-20.tab – One minute of data from PMU3 with 20% missing data
Data file: pmu3-1m-30.tab – One minute of data from PMU3 with 30% missing data
Data file: pmu4-1m-10.tab – One minute of data from PMU4 with 10% missing data
Data file: pmu4-1m-20.tab – One minute of data from PMU4 with 20% missing data
Data file: pmu4-1m-30.tab – One minute of data from PMU4 with 30% missing data
Data file: pmu5-1m-10.tab – One minute of data from PMU5 with 10% missing data
Data file: pmu5-1m-20.tab – One minute of data from PMU5 with 20% missing data
Data file: pmu5-1m-30.tab – One minute of data from PMU5 with 30% missing data
README.txt
Data are available under the terms of the
Creative Commons Zero “No rights reserved” data waiver (CC0 1.0 Public domain dedication).
The dataset presented in the work was obtained as real-world data from a regional Electricity authority in India. However, additional information such as the data source, the acquisition procedure, and the significance of the systemic variables are not detailed at this stage of algorithm development as the goal of this preliminary work is to demonstrate the efficacy of the proposed missing data recovery algorithm.
ReferencesPhadkeAGBiT:
Phasor measurement units, WAMS, and their applications in protection and control of power systems.J. Mod. Power Syst. Clean Energy.2018;6(4):619–629.
10.1007/s40565-018-0423-3UsmanMUFaruqueMO:
Applications of synchrophasor technologies in power systems.J. Mod. Power Syst. Clean Energy.2019;7(2):211–226.
10.1007/s40565-018-0455-8AmidanB:
Data Mining Techniques and Tools for Synchrophasor Data.North American SynchroPhasor Initiative (NASPI);2019, January;45.MillerLE:
PMU Data Quality: A Framework for the Attributes of PMU Data Quality and a Methodology for Examining Data Quality Impacts to Synchrophasor Applications.2017; no.March: pp.1–77.HuangC:
Data quality issues for synchrophasor applications Part I: a review.J. Mod. Power Syst. Clean Energy.2016;4(3):342–352.
10.1007/s40565-016-0217-4HuangC:
Data quality issues for synchrophasor applications Part II: problem formulation and potential solutions.J. Mod. Power Syst. Clean Energy.2016;4(3):353–361.
10.1007/s40565-016-0213-8FangX:
PMU Data Quality: A Framework for the Attributes of PMU Data Quality and a Methodology for Examining Data Quality Impacts to Synchrophasor Applications.IEEE Trans. Power Syst.2017;7(1):1–6.GenesCEsnaolaIPerlazaSM:
Recovering missing Data via matrix completion in electricity distribution systems. IEEE Workshop on Signal Processing Advances in Wireless Communications, SPAWC, 2016-Augus (July 2016).2016.GaoPWangMGhiocelSG:
Missing Data Recovery by Exploiting Low-Dimensionality in Power System Synchrophasor Measurements.IEEE Trans. Power Syst.2016;31(2):1006–1013.
10.1109/TPWRS.2015.2413935CaiJFCandèsEJShenZ:
A singular value thresholding algorithm for matrix completion.SIAM J. Optim.2010;20(4):1956–1982.
10.1137/080738970GenesCEsnaolaIIPerlazaSM:
Recovering missing data via matrix completion in electricity distribution systems. IEEE Workshop on Signal Processing Advances in Wireless Communications, SPAWC, 2016-August, 1–6.2016.HastieTMazu Missing DataerRLeeJD:
Matrix completion and low-rank SVD via fast alternating least squares.J. Mach. Learn. Res.2015;16:3367–3402.
31130828GrälerBPebesmaEHeuvelinkG:
Spatio-temporal interpolation using gstat.R Journal.2016;8(1):204–218.
10.32614/RJ-2016-014ChengSLuF:
A two-step method for missing spatio-temporal Data reconstruction.ISPRS Int. J. Geo Inf.2017;6(7).
10.3390/ijgi6070187DengMFanZLiuQ:
A Hybrid Method for Interpolating Missing Data in Heterogeneous Spatio-Temporal Datasets.ISPRS Int. J. Geo Inf.2016;5(2).
10.3390/ijgi5020013YangZLiuHBiT:
A PMU data recovering method based on preferred selection strategy.Glob. Energy Interconnect.2018;1(1):63–69.ReveszPZ:
A recurrence equation-based solution for the cubic spline interpolation problem.International Journal of Mathematical Models and Methods in Applied Sciences.2015;9(16):446–452.ThangarajSGohVTYapTTV:
PMU Data.2021.
Harvard Dataverse.
10.7910/DVN/Y2LLJJ10.5256/f1000research.160163.r231067Reviewer response for version 3YapWun She1Referee
Lee Kong Chian Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Kajang, Malaysia
Competing interests: No competing interests were disclosed.
The authors have addressed the concerns raised in my peer review report appropriately.
Is the work clearly and accurately presented and does it cite the current literature?
Partly
If applicable, is the statistical analysis and its interpretation appropriate?
Partly
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
Partly
Reviewer Expertise:
Artificial intelligence, cryptography, information security
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
10.5256/f1000research.142368.r220241Reviewer response for version 2YapWun She1Referee
Lee Kong Chian Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Kajang, Malaysia
Competing interests: No competing interests were disclosed.
The paper proposed a method to perform missing data recovery on PMU data. It uses a modified approach based on an established technique i.e. recurrent equation-based cubic spline interpolation. Although the paper can be difficult to follow in some sections especially the Methodology, overall, the method and its results do show some promise and is worth further investigation.
This paper can be considered for publication if the suggestions below are addressed.
Literature can be improved with a discussion about of some current missing data recovery techniques, especially the technique that the proposed method is based on, that is the recurrent equation-based method and tri-diagonal matrix-based conventional cubic spline interpolation. Those techniques are not discussed in the literature review despite their apparent importance to this paper.
Although not required in this paper due to the need for conciseness and perhaps page limits, it is recommended that the authors use some visual aids/figures/plots to help explain the notion of cubic spline in future manuscripts.
In Page 5, the section heading called "Recurrence equation-based solution" should be "Recurrent equation-based solution" for consistency with the rest of the text. Similarly, please ensure consistency throughout the paper. There are inconsistent terms such as LEM and LE (in Introduction), AM and AM Method (using method is redundant), etc.
Please check the numbering for the equations and the references/citations to them. For example, in Page 5, the authors state "The first row of the new matrix in (6) is shown..." would imply that Equation 6 is a matrix but Equation 6 is just a normal formula.
The dataset is lacking context but understandable due to the confidentiality of the data. It would help if more general characteristics of the dataset are given or an anonymised sampling is shown in a table. That would add depth to the discussion.
The discussion provides valuable insights into the comparison of the two methods, but additional details and context in certain areas could enhance the clarity and completeness of the findings. For example, the use of RMSE as an evaluation metric is common in imputation tasks, but it would be helpful to know if other metrics were considered or if there are specific reasons for choosing RMSE. Similarly, why was ToC used as a metric?
The comparison results that are provided in terms of RMSE and ToC seem to indicate that while Alpha Method has higher RMSE values compared to LE method, AM performs better in terms of time. Providing some insights in the discussion section into why this improvement occurred (e.g., the nature of the algorithm, computational complexity) would enhance the discussion.
Can the proposed method be used for other datasets with missing data or is it optimised specially for PMU data? A brief explanation to clarify the impact of PMU data would further improve the discussion.
Is the work clearly and accurately presented and does it cite the current literature?
Partly
If applicable, is the statistical analysis and its interpretation appropriate?
Partly
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
Partly
Reviewer Expertise:
Artificial intelligence, cryptography, information security
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.
GohVik TorMultimedia University, Malaysia
Competing interests: No competing interests were disclosed.
13122023
The literature review has been amended to improve its readability and conciseness.
We will include such visual aids in our future publications. Thank you for the suggestion.
The necessary changes have been made to ensure consistency in the nomenclatures and redundancies have been eliminated.
The numbering for the equations has been amended accordingly.
As pointed out, it will not be possible to share the full dataset due to confidentiality reasons. We briefly explained some characteristic in the Results and Discussion section and added some additional statements of the dataset in the Data Availability section. Furthermore, the dataset can be readily downloaded from the public repository and examined.
From our literature review, we determined that most literature use RMSE as an evaluation metric. As such, we adopt the same metric to ensure that our work can be fairly compared with other methods. We have added further clarification about this in the manuscript.
LEM had higher ToC values because it needed to solve the entire set of linear equations every time it needed to find the
b
i,
c
i, and
d
i coefficients. On the other hand, AM only needed to calculate these coefficients at two successive points of
i and
i+1. A brief discussion has been added in the Results and Discussion section explaining this.
The proposed data recovery method is dataset-neutral and can be used in any big data system. However, in our work, we utilized PMU data as a case study because PMU data was “big” and most importantly, it was readily available to us. For our future work, we will apply the proposed method on other datasets to examine its viability.
10.5256/f1000research.142368.r158516Reviewer response for version 2FooMathias1Refereehttps://orcid.org/0000-0003-1400-2659
School of Engineering, University of Warwick, Coventry, UK
Competing interests: No competing interests were disclosed.
The authors have addressed all my comments and the new manuscript reads better than before.
Is the work clearly and accurately presented and does it cite the current literature?
Partly
If applicable, is the statistical analysis and its interpretation appropriate?
Partly
Are all the source data underlying the results available to ensure full reproducibility?
Partly
Is the study design appropriate and is the work technically sound?
Partly
Are the conclusions drawn adequately supported by the results?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
No
Reviewer Expertise:
Dynamical system modelling
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
10.5256/f1000research.76818.r139819Reviewer response for version 1FarooqShaik Mullapathi1Refereehttps://orcid.org/0000-0003-3630-8974
Department of Computer Science and Engineering, K. S. R. M. College of Engineering (UGC-Autonomous), Kadapa, Andhra Pradesh, India
Competing interests: No competing interests were disclosed.
The manuscript proposes recurrent relation based alpha method to interpolate missing PMU data. Further, the authors try to prove that the proposed method reduces computational complexity.
However, the comments are as follows,
The Implementation of the proposed method is clearly missing (Hardware or software details used) in the manuscript which does not assist reproducing the results.
Most of the manuscript is dedicated for theoretical discussion about the proposed method. But a comparison between the existing methods with the proposed method is missing.
Add nomenclature that improves the readability of the manuscript.
Only data set of PMU values are presented (PMU Data Harvard Data verse) instead need to add discussion about the details of PMU Data.
Is the work clearly and accurately presented and does it cite the current literature?
Partly
If applicable, is the statistical analysis and its interpretation appropriate?
I cannot comment. A qualified statistician is required.
Are all the source data underlying the results available to ensure full reproducibility?
Partly
Is the study design appropriate and is the work technically sound?
Partly
Are the conclusions drawn adequately supported by the results?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
No
Reviewer Expertise:
Cyber security in smart grid communication network and VANET.
I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.
GohVik TorMultimedia University, Malaysia
Competing interests: No competing interests were disclosed.
16122022
The purpose of this preliminary paper is to introduce our work in missing data recovery using cubic spline interpolation, namely the mathematical foundation and algorithmic logic. These details have been presented and explained accordingly in the paper. Additionally, the data used is also available for download by interested parties. We aim to publish a more detailed paper soon which will contain more information such as those suggested by the reviewer. Thank you for the suggestion.
As stated earlier, the purpose of this preliminary paper is to introduce our proposed method, hence the emphasis on theoretical discussion. However, we have made an initial comparison with an existing method, namely the Linear Equation Method. This can be seen in the Results and discussion section.
The nomenclature is improved wherever possible to improve the readability of the manuscript.
The dataset presented in the work was obtained from a regional Electricity Authority in India. It was obtained for use as realistic data and brief details of the PMU data is now included. However, additional information such as the data source, the acquisition process, and the physical significance of the systemic variables are not detailed at this stage of algorithmic development as the main idea is only to demonstrate the efficacy of the missing data imputation algorithm. Nonetheless, we take note of this suggestion for our next submission. Thank you.
10.5256/f1000research.76818.r125681Reviewer response for version 1FooMathias1Refereehttps://orcid.org/0000-0003-1400-2659
School of Engineering, University of Warwick, Coventry, UK
Competing interests: No competing interests were disclosed.
In general, there is promising aspect of the proposed method but it has to be conveyed in a clearer manner. Here are my comments.
In Introduction section, the authors state that the comparison will be made with LEM. Can the author explain why specifically LEM is compared? Is that the current state-of-the-art method?
In Literature Review section, NASPI is mentioned but without proper definition of the acronym.
If my understanding is right, Equation (10) is a systems of linear equation of (7). Then, why does the h value in the B matrix have an exponent of 1 instead of 2 as of Equation (7)?
Statement above Equation (11): Unless I'm mistaken, there is no d coefficient to be solved from either equations (5) or (6).
Equation (11): Like Equation (10), can the authors clarify why the exponent of 1 is used for h?
Equation (13): Why is r taking this value? A bit more explanation would be helpful.
Equation (16): Why is there a 'for' in the equation?
Step 3 of LE method: There is no vector E in Equation (11).
Step 3 of AM method: There is no vector E in Equation (11).
Step 4 of AM method: There is no alpha term in Equation (11).
Results and Discussions section: Can the author explicitly write down the equation for RMSE?
Also, I am quite surprised with the huge difference in terms of RMSE between the two methods even for the case of 10% missing data considering the same equation (18) is used for both algorithms. The difference in ToC is understandable, but the vast difference in RMSE is a bit out of my expectation. Could the author briefly comment on the plausible reason for this huge difference in the RMSE value despite both algorithm using equation (18).
Overall comment: The mathematical derivation is not easy to follow and there are potential mistakes in citing the equations, which makes it even harder to follow. Thus, it is difficult to ascertain whether the results can be reproduced.
Is the work clearly and accurately presented and does it cite the current literature?
Partly
If applicable, is the statistical analysis and its interpretation appropriate?
Partly
Are all the source data underlying the results available to ensure full reproducibility?
Partly
Is the study design appropriate and is the work technically sound?
Partly
Are the conclusions drawn adequately supported by the results?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
No
Reviewer Expertise:
Dynamical system modelling
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.
GohVik TorMultimedia University, Malaysia
Competing interests: No competing interests were disclosed.
16122022
The idea of cubic spline is the development of a series of unique cubic polynomials that are fitted between the data points. Based on four continuity relations between points in the spline, the relationships between the spline coefficients shall result in a system of unique
n x n linear equations in the matrix form
Ax = B. The solution of this unique system of linear equations results in the values of constants at each spline point. Whenever some changes occur in any one of the splines, the system of linear equations must be solved for every specific change to fit the spline. The tri-diagonal method of solving a system of linear equations was employed in this study for comparison. Hence and from the reference, we made such a comparison to linear equation method.
The word is already introduced in the introduction section as North American Synchro Phasor Initiative (NASPI).
The exponent for
h in Equation (10) and (11) should be 2 instead of 1. We have made the corrections accordingly. Thank you.
As the reviewer correctly noted, the
d coefficient is not solved in Equations (5) or (6). It is instead solved using Equation (9). We have made the amendments accordingly.
The r-value in Equation (13) is a non-zero rational constant; the value is adapted from our reference work [17]. We have added this citation in the text.
The ‘
for’ should not be in Equation (16). It has been corrected.
Step 3 and Step 5 of the LE method has been corrected as vector
B instead of
E.
Reference to Equation (11) in Step 4 of the AM method has been removed.
The equation for RMSE has been added in the text.
Upon inspection, we found that the variables
b,
c, and
d of the cubic spline is found to be similar in both methods as the reviewer correctly predicted. We re-examined our results and have determined that due to an oversight, an error occurred in the final calculations of imputation values in the LE method. Instead of the coordinate numbers, the spline values at the coordinates were used for the calculation of missing values. As such, we have made the corrections to the results and discussions, as well as the plots in Figure 1 and Figure 2.
All corrections and suggestions given were incorporated and the revised version of the paper is written. We really appreciate the time spent by the reviewer for the useful suggestions and corrections made.