Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)

Shruthi Thangaraj; Vik Tor Goh; Timothy Tzen Vun Yap

doi:10.12688/f1000research.73182.1

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Research Article

Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)

[version 1; peer review: 1 approved with reservations, 1 not approved]

Shruthi Thangaraj¹, Vik Tor Goh ¹, Timothy Tzen Vun Yap²

PUBLISHED 28 Feb 2022

Author details Author details

¹ Faculty of Engineering, Multimedia University, Cyberjaya, Selangor, 63100, Malaysia
² Faculty of Computing and Informatics, Multimedia University, Cyberjaya, Selangor, 63100, Malaysia

Shruthi Thangaraj
Roles: Conceptualization, Data Curation, Formal Analysis, Investigation, Methodology, Software, Writing – Original Draft Preparation

Vik Tor Goh
Roles: Methodology, Supervision, Writing – Review & Editing

Timothy Tzen Vun Yap
Roles: Methodology, Supervision, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Research Synergy Foundation gateway.

This article is included in the Artificial Intelligence and Machine Learning gateway.

Abstract

Background: 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.

Keywords

phasor measurement unit, missing data, data recovery, smart grid, interpolation, cubic spline, data quality, data pre-processing

Corresponding author: Vik Tor Goh

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2022 Thangaraj S et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Thangaraj S, Goh VT and Yap TTV. Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU) [version 1; peer review: 1 approved with reservations, 1 not approved]. F1000Research 2022, 11:246 (https://doi.org/10.12688/f1000research.73182.1) First published: 28 Feb 2022, 11:246 (https://doi.org/10.12688/f1000research.73182.1) Latest published: 18 Dec 2023, 11:246 (https://doi.org/10.12688/f1000research.73182.3)

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.

In this paper, a modified recurrent equation-based method termed the Alpha Method (AM) for PMU missing data problem is proposed. The results are compared with the tri-diagonal matrix-based conventional cubic spline interpolation for the spline coefficients which is also termed the Linear Equations Method (LEM).

Literature review

The need to fill in the missing values in PMU and potential causes have been reviewed.⁵^–⁷ These works imply the need for missing data recovery techniques for PMU data to enhance the accuracy of the decision-making process and show the data quality and security risks associated with the missing data in PMU. One of the popular approaches is the matrix completion (MC) based on missing data recovery.⁸^–¹² The MC is the most exploited technique, however, a few of these were only theoretical approaches and a few approaches were only tested with simulated data.

Interpolation-based missing data recovery techniques¹³^–¹⁵ propose a reconstruction of missing values by a spatial interpolation or spatio-temporal interpolation of the values. Yet they require historical data of the same channel’s or time’s data for the interpolation. A few of the advanced/hybrid approaches¹⁶^,¹⁷ like k-nearest-neighbor and recurrent relation-based interpolations are not yet applied over the PMU data.

Missing data is a common problem in all fields of study; hence a variety of solutions are found to be effective based on the data pattern, data processing model, and data quality needs. However, adopting any conventional techniques available for treating missing values can get complex especially when solving the high precision and volume of PMU data.¹⁵ Therefore, there is a need for a missing data recovery method for PMU data. NASPI presents a variety of data requirements, attributes, and data quality problems for both static data and real-time data. There is a need for designing an effective data recovery method to work without the need for historical data processing and training.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 get complex and time-consuming when 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 $R to R$ . It is assumed that the value of $f$ is known only at $x_{1} \leq x_{2 .} \leq x_{i} \dots \leq x_{n} and let f (x_{i}) = a_{i} .$ Piecewise cubic spline interpolation is the problem of finding the $b_{i}$ , $c_{i}$ and $d_{i}$ coefficients of the cubic polynomials ${SF}_{i} for 0 \leq i \leq n - 1$ written in the form:

(1)

{SF}_{i} (x) = a_{i} + b_{i} (x - x_{i}) + c_{i} {(x - x_{i})}^{2} + d_{i} {(x - x_{i})}^{3}

Where $x$ can take any value between $x_{i}$ and $x_{i + 1}$ . That is,

(1a)

{SF}_{i} (x_{i}) = a_{i}

Let the first-order derivative of equation (1) be:

(2)

{SF}_{i}^{'} (x) = b_{i} + 2 c_{i} (x - x_{i}) + 3 d_{i} {(x - x_{i})}^{2}

The first-order derivative at $x_{i}$ for values of $1 \leq i \leq n - 1$ will be

(2a)

{SF}_{i}^{'} (x_{i}) = b_{i}

And the second-order derivative be:

(3)

{SF}_{i}^{''} (x) = 2 c_{i} + 6 d_{i} (x - x_{i})

The second-order derivative at $x_{i}$ for values of $1 \leq i \leq n - 1$ will be:

(3a)

{SF}_{i}^{''} (x_{i}) = 2 c_{i}

For a smooth fit between the adjacent pieces, the cubic spline interpolation requires that the following conditions hold:

1. The cubic functions should intersect at the points left and right, for $i = 0 to n - 1$
(4)
${SF}_{i} (x_{i + 1}) = {SF}_{i + 1} (x_{i}) = a_{i + 1}$
2. For each cubic function to join smoothly with its neighbors, the splines should have continuous first and second derivatives at the data points $i = 1, \dots, n - 1 :$
(5)
${SF}_{i}^{'} (x_{i + 1}) = {SF}_{i + 1}^{'} (x_{i}) = b_{i + 1}$
(6)
${SF}_{i}^{''} (x_{i + 1}) = {SF}_{i + 1}^{''} (x_{i}) = 2 . c_{i + 1}$

If $h_{i}$ = $x_{i + 1} - x_{i}$ and if $h_{i}$ is equal for all $i$ values, following Revesz,¹⁷ the relation between coefficients $a_{i}$ and $c_{i}$ can be resolved:

(7)

c_{i - 1} + 4 c_{i} + c_{i + 1} = \frac{3}{h^{2}} (a_{i - 1} - 2 a_{i} + a_{i + 1})

(8)

b_{i} = (a_{i + 1} - a_{i}) \frac{1}{h_{i}} - \frac{2 c_{i} + c_{i + 1}}{3} h_{i}

(9)

d_{i} = \frac{1}{3 . h_{i}} (c_{i + 1} - c_{i})

Equation (6) represents a system of linear equations for the unknowns $c_{i}$ for $0 \leq i \leq n$ . As the values of $a_{i}$ are known, the value of $c_{i}$ can be found by solving the tri-diagonal matrix-vector equation $Ax = B$ . While there are n+1 numbers of $c_{i}$ constants, equation (6) 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: $c_{0} = c_{n} = 0.0$ . That is, the second derivatives of the splines at the endpoints are assumed to be zero. Based on equation (4), a system of (N+1) linear equations of (N+1) variables can be formulated as:

(10)

\begin{matrix} A = [\begin{matrix} \begin{matrix} 1 & 0 & 0 0 \dots \\ 1 & 4 & \begin{matrix} 1 & 0 & \dots \end{matrix} \\ 0 & 1 & \begin{matrix} 4 & 1 & \dots \end{matrix} \end{matrix} & \begin{matrix} 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ & \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 & \dots \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} \dots \\ \dots \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} ⋮ & ⋮ & \begin{matrix} ⋮ & ⋮ \end{matrix} \\ 1 & 4 & \begin{matrix} 1 & 0 \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 1 \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} 4 \\ 0 \end{matrix} & \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix} \end{matrix}], x = [\begin{matrix} \begin{matrix} c_{0} \\ c_{1} \\ \begin{matrix} ⋮ \\ c_{n} \end{matrix} \end{matrix} \end{matrix}], and B = [\begin{matrix} \begin{matrix} 0 \\ \frac{3}{h^{1}} (a_{0} - 2 a_{1} + a_{2}) \end{matrix} \\ \begin{matrix} ⋮ \\ \frac{3}{h^{1}} (a_{n - 2} - 2 a_{n - 1} + a_{n}) \\ 0 \end{matrix} \end{matrix}] \end{matrix}

For clamped cubic spline interpolation the following boundary conditions are assumed: $b_{0} = f^{'} (x_{0}$ ) and $b_{n} = f^{'} (x_{n}$ ), where the derivatives $f^{'} (x_{0}$ ) and $f^{'} (x_{n}$ ), 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 $c_{i}$ are obtained, using equations (5) and (6) respectively, the values of coefficients $b_{i}$ and $d_{i}$ can also be found. Similarly, under clamped spline interpolation,

(11)

\begin{matrix} A = [\begin{matrix} \begin{matrix} 2 & 1 & \begin{matrix} 0 & 0 & \dots \end{matrix} \\ 1 & 4 & \begin{matrix} 1 & 0 & \dots \end{matrix} \\ 0 & 1 & \begin{matrix} 4 & 1 & \dots \end{matrix} \end{matrix} & \begin{matrix} 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ & \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 & \dots \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} \dots \\ \dots \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} ⋮ & ⋮ & \begin{matrix} ⋮ & ⋮ \end{matrix} \\ 1 & 4 & \begin{matrix} 1 & 0 \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 1 \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} 4 \\ 1 \end{matrix} & \begin{matrix} 1 \\ 2 \end{matrix} \end{matrix} \end{matrix} \end{matrix}], x = [\begin{matrix} \begin{matrix} c_{0} \\ c_{1} \\ \begin{matrix} ⋮ \\ c_{n} \end{matrix} \end{matrix} \end{matrix}] and B = [\begin{matrix} \begin{matrix} \frac{3}{h^{2}} (a_{1} - a_{0}) - \frac{3}{h} f^{'} (x_{0}) \\ \frac{3}{h^{1}} (a_{0} - 2 a_{1} + a_{2}) \end{matrix} \\ \begin{matrix} ⋮ \\ \frac{3}{h^{1}} (a_{n - 2} - 2 a_{n - 1} + a_{n}) \\ \frac{3}{h} f^{'} (x_{0}) - \frac{3}{h^{2}} (a_{n} - a_{n - 1}) \end{matrix} \end{matrix}] \end{matrix}

Recurrence equation-based solution

Revesz,¹⁷ chose boundary conditions that need to solve the tri-diagonal system given in equation (6) where $x_{i}$ rational variables $e_{i}$ rational constants, r is a non-zero rational constant and A is:

(12)

\begin{matrix} A = [\begin{matrix} \begin{matrix} r & 1 & \begin{matrix} 0 & 0 & \dots \end{matrix} \\ 1 & 4 & \begin{matrix} 1 & 0 & \dots \end{matrix} \\ 0 & 1 & \begin{matrix} 4 & 1 & \dots \end{matrix} \end{matrix} & \begin{matrix} 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ & \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ 0 & 0 & \begin{matrix} 0 & 0 & \dots \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} \dots \\ \dots \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} ⋮ & ⋮ & \begin{matrix} ⋮ & ⋮ \end{matrix} \\ 1 & 4 & \begin{matrix} 1 & 0 \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 1 \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} 4 \\ 0 \end{matrix} & \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix} \end{matrix}], x = [\begin{matrix} \begin{matrix} \begin{array}{c} x_{1} \\ x_{2} \\ \begin{matrix} ⋮ \\ x_{\begin{matrix} n - 1 \\ x_{n} \end{matrix}} \end{matrix} \end{array} \end{matrix} \end{matrix}] and b = [\begin{matrix} \begin{array}{l} \begin{matrix} e_{1} \\ e_{2} \\ \begin{matrix} ⋮ \\ e_{\begin{matrix} n - 1 \\ e_{n} \end{matrix}} \end{matrix} \end{matrix} \end{array} \end{matrix}] \end{matrix}

The first row of the new matrix in (6) is shown to be equivalent to the first row of the clamped b matrix $e_{1}$ is

(13)

e_{1} = \frac{3 r}{2 h} (\frac{(a_{1} - a_{0})}{h} - f^{'} (x_{0})) + (1 \frac{r}{2}) {\tilde{c}}_{1}

where,

{\tilde{c}}_{1}

is an estimate of

c_{1}

and r = 2+

\sqrt 3 \approx 3.732 .

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 $c_{n}$ as 0.

(14)

x_{i} + \frac{x_{i + 1}}{r} = \sum_{0 \leq k \leq (i - 1)} {(- 1)}^{k} \frac{e_{i - k}}{r^{k + 1}}

Let $\propto_{0}$ , $\propto_{i} for 1 < i \leq n - 1 and \propto_{n}$ , respectively be:

\propto_{0} = 0

(15)

\propto_{i} = \frac{e_{i - \propto_{i - 1}}}{r} = \sum_{0 \leq k \leq (i - 1)} {(- 1)}^{k} \frac{e_{i - k}}{r^{k + 1}}

\propto_{n} = e_{n}

Based on the above, the closed form of solution for $x_{i}$ can be given as:

(16)

for x_{i} = \sum_{0 \leq k \leq (n - i)} {(\frac{- 1}{r})}^{- k} \propto_{i + k}

The above equation solves $x_{i}$ no matter exactly what the initial values for $e_{i}$ . 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 (14) leads to a faster update for each i ≤ n only with the addition of the term:

(17)

{(\frac{- 1}{r})}^{n + 1 - i} \propto_{n + i}

and

x_{n + 1} = \propto_{n + 1} .

Similarly,

\propto_{i}

constants can be updated by adding a single term

e_{n + 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 (LE)

Step 1: Given the initial vector with missing values, separate them into two sets of vectors, the observed values vector $R_{obs}$ and the missing values vector $R_{Miss}$ , having sizes of NO and NM, respectively, such that NO+NM=N.

Step 2: $R_{obs}$ vector at $x_{i}$ values of the (NO-1) splines shall be the $a_{i}$ coefficient vector.

Step 3: Using $a_{i}$ , generate the RHS vector E given in equation (11).

Step 4: Generate a square coefficient matrix A as given in equation (11)

Step 5: Solve for the $c_{i}$ vector is given in (11), using the relation Ac=E

Step 6: Applying $c_{i}$ in equations compute the $b_{i}$ and $d_{i}$ coefficient vectors for n-2 points of the $R_{obs}$ ,

Step 7: Using the values of $a_{i}$ , $b_{i}$ , $c_{i} and d_{i}$ , missing values can be found by the equation (1) re-written as:

(18)

{SF}_{i} (x) = (a_{i} - b_{i} x_{i} + c_{i} x_{i}^{2} - d_{i} x_{i}^{3}) + (b_{i} - 2 c_{i} x_{i} + 3 d_{i} x_{i}^{2}) x + (c_{i} - 3 d_{i} x_{i}) x^{2}

Where x represents the missing positions, between $x_{i}$ and $x_{i + 1}$ of spline i.

Algorithmic procedure for recurrent equation-based Alpha Method (AM)

Step 2: The $R_{obs}$ vector at $x_{i}$ values of the (NO-1) splines are the $a_{i}$ coefficient vector.

Step 3: Using $a_{i}$ , generate the RHS vector E given in equation (11).

Step 4: Set $\propto_{0} = 0 and \propto_{n} = e_{n},$ given in equation (11) calculate the alpha vector using the relation.

$\propto_{i} = \frac{e_{i - \propto_{i - 1}}}{r} = \sum_{0 \leq k \leq (i - 1)} {(- 1)}^{k} \frac{e_{i - k}}{r^{k + 1}}$ for $i$ values ranging from 1 to NO-1

Step 5: Set $x_{n} = \propto_{n}$ and solve for $c_{i}$ values using the relation.

c_{i} = \sum_{0 \leq k \leq (n - i)} {(\frac{- 1}{r})}^{k} \propto_{i + k}

Step 6: Applying $c_{i}$ in equations compute the $b_{i}$ and $d_{i}$ coefficient vectors for n-2 points of the $R_{obs}$ ,

Step 7: Using the values of $a_{i}$ , $b_{i}$ , $c_{i} and d_{i}$ , missing values can also be found using equation (18), re-written here again for convenience:

(18)

{SF}_{i} (x) = (a_{i} - b_{i} x_{i} + c_{i} x_{i}^{2} - d_{i} x_{i}^{3}) + (b_{i} - 2 c_{i} x_{i} + 3 d_{i} x_{i}^{2}) x + (c_{i} - 3 d_{i} x_{i}) x^{2}

Where x represents the missing positions, between $x_{i}$ and $x_{i + 1}$ of spline i.

The modifications are as follows: In the AM method rather than computing E, alpha vectors and $c_{i}$ 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 $c_{i}$ were calculated only for $i and i + 1$ data elements, where $i$ is the missing data element. For the imputation of $i$ the element, only the $E_{i}$ vector for all NO-1 data points, $\propto_{i}$ vector and $c_{i}$ vectors for $i and i + 1$ and $b_{i}$ and $d_{i}$ coefficients were essential for the calculation $i^{th}$ 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 ToC and RMSE values have improved significantly.

Results and discussion

A comparison between LEM and AM methods 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%, 30% missing data respectively for five PMUs were evaluated.

When the AM method was employed, the average root mean squared error (RMSE) values were 0.5968, 0.9448, and 1.2445 for 10%, 20%, and 30% of missing PMU data respectively. This can be seen in Figure 1. Moreover, for the same performance, the AM method showed significant improvements in its time of calculation (ToC) as shown in Figure 2. The average ToCs for the proposed AM method were 2.132, 1.9634, and 1.738s when recovering 10%, 20%, and 30% of its missing data. By comparison, LEM had ToC values of 32.7679, 33.4482, and 36.7988s for 10%, 20%, and 30% of its missing data, respectively. The proposed method reduced the ToC by a factor of approximately 10 times.

Figure 1. Comparison of RMSE values: (A) alpha method; (B) linear equation method.

Figure 2. Comparison of Time of Calculation (ToC): (A) alpha method; (B) linear equation method.

Conclusions

In this study, the proposed AM method was compared with the LEM technique. 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 alpha method 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 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 availability

Underlying 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).

References

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4. Miller LE, et al.: 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.
5. Huang C, et al.: Data quality issues for synchrophasor applications Part I: a review. J. Mod. Power Syst. Clean Energy. 2016; 4(3): 342–352. Publisher Full Text
6. Huang C, et al.: Data quality issues for synchrophasor applications Part II: problem formulation and potential solutions. J. Mod. Power Syst. Clean Energy. 2016; 4(3): 353–361. Publisher Full Text
7. Fang X, et al.: 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.
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18. Thangaraj S, Goh VT, Yap TTV: PMU Data.2021. Harvard Dataverse. Publisher Full Text

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Version 3

VERSION 3 PUBLISHED 28 Feb 2022

Author details Author details

¹ Faculty of Engineering, Multimedia University, Cyberjaya, Selangor, 63100, Malaysia
² Faculty of Computing and Informatics, Multimedia University, Cyberjaya, Selangor, 63100, Malaysia

Shruthi Thangaraj
Roles: Conceptualization, Data Curation, Formal Analysis, Investigation, Methodology, Software, Writing – Original Draft Preparation

Vik Tor Goh
Roles: Methodology, Supervision, Writing – Review & Editing

Timothy Tzen Vun Yap
Roles: Methodology, Supervision, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

Article Versions (3)

version 3

Revised

Published: 18 Dec 2023, 11:246

https://doi.org/10.12688/f1000research.73182.3

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Published: 22 Dec 2022, 11:246

https://doi.org/10.12688/f1000research.73182.2

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Published: 28 Feb 2022, 11:246

https://doi.org/10.12688/f1000research.73182.1

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Thangaraj S, Goh VT and Yap TTV. Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU) [version 1; peer review: 1 approved with reservations, 1 not approved]. F1000Research 2022, 11:246 (https://doi.org/10.12688/f1000research.73182.1)

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Reviewer Report 12 Jul 2022

Shaik Mullapathi Farooq, Department of Computer Science and Engineering, K. S. R. M. College of Engineering (UGC-Autonomous), Kadapa, Andhra Pradesh, India

Not Approved

https://doi.org/10.5256/f1000research.76818.r139819

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
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
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
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

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.

CITE

Report a concern

Author Response 22 Dec 2022

Vik Tor Goh, Faculty of Engineering, Multimedia University, Cyberjaya, 63100, Malaysia

22 Dec 2022

Author Response

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 ... Continue reading 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.
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.
Competing Interests: No competing interests were disclosed. Close
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COMMENTS ON THIS REPORT

Author Response 22 Dec 2022

Vik Tor Goh, Faculty of Engineering, Multimedia University, Cyberjaya, 63100, Malaysia

22 Dec 2022

Author Response

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 ... Continue reading 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.
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.
Competing Interests: No competing interests were disclosed. Close
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Reviewer Report 10 Mar 2022

Mathias Foo, School of Engineering, University of Warwick, Coventry, UK

Approved with Reservations

https://doi.org/10.5256/f1000research.76818.r125681

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

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
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
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
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

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.

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Author Response 27 Dec 2023

Vik Tor Goh, Faculty of Engineering, Multimedia University, Cyberjaya, 63100, Malaysia

27 Dec 2023

Author Response

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 ... Continue reading 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.
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.
Competing Interests: No competing interests were disclosed. Close
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Author Response 27 Dec 2023

Vik Tor Goh, Faculty of Engineering, Multimedia University, Cyberjaya, 63100, Malaysia

27 Dec 2023

Author Response

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 ... Continue reading 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.
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.
Competing Interests: No competing interests were disclosed. Close
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Mathias Foo, University of Warwick, Coventry, UK
Shaik Mullapathi Farooq, K. S. R. M. College of Engineering (UGC-Autonomous), Kadapa, India
Wun She Yap, Universiti Tunku Abdul Rahman, Kajang, Malaysia

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2 Views

22 Dec 2023 | for Version 3

Wun She Yap, Lee Kong Chian Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Kajang, Malaysia

2 Views Cite this report Responses(0)

Approved

The authors have addressed the concerns raised in my peer review report appropriately.

Competing Interests

No competing interests were disclosed.

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.

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12 Views

27 Nov 2023 | for Version 2

Wun She Yap, Lee Kong Chian Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Kajang, Malaysia

12 Views Cite this report Responses(1)

Approved With Reservations

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
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

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
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Artificial intelligence, cryptography, information security

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Author Response

18 Jan 2024

Vik Tor Goh, Faculty of Engineering, Multimedia University, Cyberjaya, 63100, Malaysia

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.

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03 Jan 2023 | for Version 2

Mathias Foo, School of Engineering, University of Warwick, Coventry, UK

14 Views Cite this report Responses(0)

Approved

The authors have addressed all my comments and the new manuscript reads better than before.

Competing Interests

No competing interests were disclosed.

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.

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Reviewer Report

35 Views

12 Jul 2022 | for Version 1

Shaik Mullapathi Farooq, Department of Computer Science and Engineering, K. S. R. M. College of Engineering (UGC-Autonomous), Kadapa, Andhra Pradesh, India

35 Views Cite this report Responses(1)

Not Approved

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
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
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
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Cyber security in smart grid communication network and VANET.

Respond to this report

Responses (1)

Author Response

22 Dec 2022

Vik Tor Goh, Faculty of Engineering, Multimedia University, Cyberjaya, 63100, Malaysia

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.

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Competing Interests

No competing interests were disclosed.

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Reviewer Report

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10 Mar 2022 | for Version 1

Mathias Foo, School of Engineering, University of Warwick, Coventry, UK

31 Views Cite this report Responses(1)

Approved With Reservations

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
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

No
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
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Dynamical system modelling

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Author Response

27 Dec 2023

Vik Tor Goh, Faculty of Engineering, Multimedia University, Cyberjaya, 63100, Malaysia

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.

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Competing Interests

No competing interests were disclosed.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)

Abstract

Keywords

Introduction

Literature review

Methods

(1)

(1a)

(2)

(2a)

(3)

(3a)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

Recurrence equation-based solution

(12)

(13)

(14)

(15)

(16)

(17)

Algorithmic procedure for regular tridiagonal matrix-based Linear Equation Method (LE)

(18)

Algorithmic procedure for recurrent equation-based Alpha Method (AM)

(18)

Results and discussion

Figure 1. Comparison of RMSE values: (A) alpha method; (B) linear equation method.

Figure 2. Comparison of Time of Calculation (ToC): (A) alpha method; (B) linear equation method.

Conclusions

Data availability

Underlying data

References

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