Application of standard ZET-algorithm for gaps filling in wide area measurement system data assets

This article explores the application of Standard ZET-algorithm for gaps filling in Synchronized Phasor Measurement data assets received from Wide Area Measurement Systems. In spite of Synchronized Phasor Measurement data assets apparent advantages, they have gaps affecting the accuracy of solving power engineering tasks. Authors analysed the reasons of gap occurrence in data assets, provided experiments with voltage and current data assets and collected the key points about application of Standard ZET-algorithm.


Introduction
Synchronized Phasor Measurement (SPM) data assets received from Wide Area Measurement Systems (WAMS) are in widespread use for solving technological tasks in power systems [1]. These tasks cover the operational dispatch management and automatic emergency control: a) improving accuracy of power system state estimation [2]; b) on-line identifying a point of power system emergency separation into parts and then their synchronization; c) monitoring margin of power system stability; d) choosing volume of action adjustments for emergency response. Definitely, SPM data assets have unquestionable advantages: a) obtained from one node, they allow calculating the voltage of neighbouring nodes by load current values of a line, increased power system observability; b) they give an opportunity to control the phase imbalance, as the signal, when equipment tend to accident-caused failure [2]; c) high discreteness of regime parameters provides the possibility to analyse electromechanical transients for centralized and local automatics according to type of stability [3]. However, SPM data assets include noises and gaps inside, leading to errors in result data, especially, for technological tasks based on data analysis methods, and making a contribution to wrong conclusions and taking incorrect solutions [4,5]. This fact underlines the relevance of the research.

The reasons of gap occurrence in SPM data assets
In order to understand the nature of gaps, we analyzed the collecting process of SPM data assets received from WAMS ( Figure 1). PMU records SPM data assets received from current (CT) and voltage transformers (VT) set at a power station and/or substation. PMU assign the record time tag to every parameter in data assets. TSD generates time tags under synchronization of satellites [6]. After preprocessing SPM data assets in PDC, they are transmitted to the server of power system control center via communication net.
At the stage when data assets are recorded, gaps can be appeared due to PMU technical failures. It requires restarting PMU but it is not possible to restore PMU normal operation immediately since the additional adjustments are required. Moreover, the operating staff cannot fully complete these adjustments therefore required period of time for restoration is significantly increased.
Time synchronization malfunctions can also lead to gaps in data assets due to satellite communication channel failures. They cause by technical reasons or weather conditions in a location of power facilities. For example, low temperatures and high humidity provide icing transceiver facilities, consequently, the connection between a satellite and TSD temporary disappears. As a result the time tag shifts. Now this problem has short-term solution by integrating the special function to the program code of the system server. Unfortunately, this method of data synchronization cannot guarantee the absence of gaps in SPM data assets [4].
WAMS communication net needs to transmit SPM data assets from measurement and registration location to the server of power system control center. During transfer SPM data assets in on-line regime, they are often lost. These are because of connection loss between PDC and the server of power system control center; clients' reconnection; communication net reconfiguration; operation failures of net equipmentrouters, hubs, modems, servers; aggregations of SPM data assets received from several PMU to one protocol packet [5].
Specialists of System Operators and Joint-stock companies identified the dependence of gaps amount from the time of day ( Figure 2). They calculated the proportion of gaps in SPM data assets as percentage of total amount of received data in 10-minutes intervals per 24 hours period.  Figure 2 shows total amount of gaps in assets: grouped (gaps come one after another) and single gaps (red color); only grouped gaps (yellow color).
The most part of gaps appear at the beginning of a working shift, when clients' reconnections and communication net reconfiguration take frequently place. At the peak load period about 20% of data are lost. Therefore, these data assets can be used for technological tasks only after their processing by special gaps filling algorithms [7].

The algorithm for gaps filling
Nowadays the algorithms for gaps filling are constantly developing [7,8]. The simplest algorithms recommend deleting those rows and columns in data assets which have even one gap [9]. The complicated algorithms use methods of least squares and maximum likelihood [10], and EM-algorithm [11]. All of them are considered to be global ones. It means that the specified type of the dependence is relevant for the whole data assets.
Apart from global algorithms there are local algorithms which define the dependence type according to incomplete selection around a gap. These are standard ZET-algorithm and WANGAalgorithm [7,8].
This article illustrates application of standard ZET-algorithm. The choice of ZET-algorithm is due to hypotheses which lay in the core of the algorithm and the possibility of application for value forecasting in "time-property"-type time series.
Standard ZET-algorithm consists of three main hypotheses: a) the excessiveness: it visualizes that SPM data assets are redundant caused by conformity among rows and depending on each other columns; b) local competence: it states that in order to fill the gap, the analysis of a whole data asset is not needed, but only "competent" part including rows similar to i-row and columns similar to jcolumn are taken into account; c) linear dependency: a dependence among rows and columns must be linear. Standard ZET-algorithm has the preliminary stage, and then three main stages are realized l-times. At the preliminary stage, columns of SPM data asset are normalized according to dispersion for adjusting the properties to unified scale (1): Where a'ij is adjusted value; aij is initial value; j a is value of mathematical expectation; Aj is mean square deviation. Assumed that unknown gap has x and y coordinates (Figure 3). At the first main stage, from the parent SPM data asset where columns have been normalized according to dispersion by equation (1), for an unknown gap (Saxon blue color), a sub multitude of "competent" rows (blue color) is selected. Then, for "competent" rows, the "competent" columns (blue color) are selected to produce a "competent" matrix. The matrix order is Sij, when [1; ] ip  and [1; ] jq  . We must select (p-1) "competent" rows for y-row with an unknown gap and (q-1) "competent" columns for x-column with an unknown gap. The i-row "competence" Liy in relation to the y-row is defined by (2): where i is "competent" row number; tiy is completeness that is the number of values known for i-and y-rows; riy is Descartes' distance between rows. The "competent" y-row must not contain the gap on the x-position. The j-column "competence" Liy in relation to the x-column is defined as (3): where tjx is completeness that is the number of values known for the columns j and x; kjx is correlation coefficient for j-and x-columns. It should be emphasized that for kjx calculation we must use those column values which belong to competent rows. The "competent" column must not contain the gap on the y-position. At the second stage, reasonable coefficients σp and σq are defined for the goal of row and column forecasting gap. This procedure is similar as for rows as for columns. Given coefficient a (a=σp for rows and a=σq for columns within stated limits and with stated step size, we minimize the function (4) (6) and for "competent" columns by (7): 11 11 . qq qq ix The restored gap value is calculated by (8): The error of the gap value is defined by (9): where Bxy is initially known value.

The application of standard ZET-algorithm for gap filling in SPM data assets
We researched the application of standard ZET-algorithm by SPM data assets of current and voltage modules and phases. It should be noted that the data of current and voltage phases play a key role when the task of power system state estimation is decided. Meanwhile, it is impossible to understand, whether the current and voltage phase values are trusted or not, since a phase values can dynamically change from +π to -π during the time ( Figure 4). Two test assets are created based on the real SPM data assets. The first test asset consists of 25% of single gaps and 5% of grouped gaps, and the second one includes 5% of single gaps and 25% of grouped gaps (in following tables see asset I and asset II respectively).

Conclusion
Proposed researches and results of experiments demonstrate that: a) the excessiveness: it visualizes that SPM data assets are redundant caused by conformity among rows and depending on each other columns; b) the main advantage of standard ZET-algorithm is the robustness against occurrence of grouped gaps in SPM data assets; c) results of experiments from tables 1 and 2 illustrate that in assets when grouped gaps are dominant, they can be adequately filled by standard ZET-algorithm (Asset II); d) the average error for gap filling in assets of voltage and current phases is higher than for assets of voltage and current modules; e) results of experiments from tables 3 -6 prove the high calculation accuracy for both single and grouped gaps as in asset of phases as in asset of modules. The difference between real and calculated phase values is less than 2º; f) in case of phase value is changing near the boundary of [-180; +180] diapason, the gap value is calculated with low accuracy. Therefore, standard ZET-algorithm needs to be sophisticated by the algorithm for a "competent" matrix size correction.