Least square and Kalman based methods for dynamic phasor estimation: a review
 Jalal Khodaparast^{1} and
 Mojtaba Khederzadeh^{1}Email author
https://doi.org/10.1186/s416010160032y
© The Author(s) 2017
Received: 25 June 2016
Accepted: 26 December 2016
Published: 26 January 2017
Abstract
The characterization of sinusoidal signals with time varying amplitude and phase is useful and applicable for many fields. Therefore several algorithms have been suggested to estimate main aspects of these signals. Within no standard approach to test the properties of these algorithms, it seems to be helpful to discuss a large class of algorithms according to their properties. In this paper, six methods of estimating dynamic phasor have been reviewed and discussed which three of them are based on least square and others are based on Kalman filter. Taylor expansion is used as a first step and continued with least square or Kalman filter in accordance with the proposal observer of each method. The theoretical processes of these methods are briefly clarified. The characterizations have been made by some tests in time and frequency domains. The tests include amplitude step, phase step, frequency step, frequency response, total vector error, transient monitor, noise, sample number, computation time, harmonic and DC offset which build a framework to compare the different methods.
Keywords
PMU Dynamic phasor Kalman filter Taylor series Least squareIntroduction
Due to the lack of recommended specific algorithms to estimate phasor in I E E E S t d.C37.118, phasor estimation has attracted lots of attentions recently [1]. Phasor estimation is a significant key of wide area monitoring and protecting in power systems. Fast and precise estimation is also necessary for accurate decision in power system control. Dynamic phasor application is not limited to P M U. For example, there are some utilizations in power system simulator programs [2]. Recent developments, particularly the emerging of power electronics based equipment like F A C T S devices, clarified an absence of suitable definition in the typical power system analysis methods which have considered the sinusoidal signal with constant amplitude and phase. For such components (power electronic based components) a full time domain simulation is needed due to incomplete concept of phasor. The concept of time varying phasor (dynamic phasor) has been proposed in [3] for the first time to overcome this problem. This concept has several advantages compared to timebased simulation. For example, it noticeably decreases the simulation time as advantage, but as a disadvantage, increases the number of variables and equations.
Several literatures discussed new algorithms of dynamic phasor estimation. In [4], a new method based on adaptive complex band pass filter was proposed to estimate phasor. Xianing et al. [5] proposed a method based on an angleshifted energy operated to extract the instantaneous amplitude. An integrated phasor and frequency estimation using a Fast Recursive Gauss Newton algorithm was proposed in [6]. A method based on modified Fourier transform to eliminate DC offset was suggested in [7]. A phasor estimation algorithm based on the least square curve fitting technique was presented in [8] for the distorted secondary current due to CT saturation. In [9], an innovative approach was proposed to estimate the phasor parameters including frequency, magnitude and angle in real time based on a newly constructed recursive wavelet transform. Reference [10] discussed phasor and frequency estimations under transient system conditions: electromagnetic and electromechanic. Maximally flat differentiators [11] and phasorlet [12] are other new methods for dynamic phasor estimation. Mai et al., [13]; Serna and Martin [14]; Serna [15] proposed modified forms of earlier methods.
Historically, Guass invented least square method and used it as estimator technique [16]. He suggested that the most appropriate value for the unknown parameter is the most probable one, which is the sum of the square of the observed and the computed values difference. Although Kalman filter is proposed fifty years ago, it is still one of the most important and common data fusion algorithms today. The great success of the Kalman filter is result of its low computational requirement, recursive property and its optimal estimation capability with Gaussian error [17]. The least square and Kalman filter based methods are discussed in this paper, as two general types of phasor estimation. Six specific methods based on these two types have been selected in this study which three of them are based on least square and others are based on Kalman filter.
Method1) Traditional method: This algorithm is based on zerothorder Taylor expansion and least square to estimate phasor [18].
Method2) Fourier Taylor method: This method is based on secondorder Taylor expansion and least square to approximate dynamic phasor [18].
Method3) Shank method: The idea of this method is based on consecutive delays of unit response (digital filter design theory) and least square method to estimate dynamic phasor [19].
Method4) Kalman Taylor method: The main concern of mentioned three methods is delay. In the next three methods, in contrast with the priors, Kalman filter is used as an alternative observer to address the delay challenge [20].
Method5) Fourier Kalman Taylor method: The main idea of this method is based on introducing augment state space which can overcome harmonic infiltration problem [21].
Method6) Modified Kalman Taylor method: The main contribution of this method is to modify modeling process of state space to decrease error bound [22].
The six concepts of algorithms are discussed as different common starting points in a unified manner. The main purpose of this paper is to review and provide a framework in order to compare past and future algorithms in this area.
Dynamic Phasor estimation
where the coefficients of the series (P _{0}, P _{1},P _{2},..., P _{ k },) are the derivatives of the dynamic phasor at the observation interval center. All six mentioned methods are similar until this step and differences come to show then.
Method1) traditional method
Method2) Fourier Taylor method
where P _{2}, P _{1}, P _{0}, \({p^{*}_{0}}\), \({P^{*}_{1}}\) and\({ P^{*}_{2}}\) are coefficients of secondorder Taylor series and their conjugated, respectively. N linear equations are created as (10):
It is clear from (12) that first and second derivatives of phasor can be calculated by this method.
Method3) Shank method
The mentioned three methods were based on least square. An important point about least square observer is its delay. It means that dynamic phasor is tracked with delay which will be shown in Section “Simulation results” later. To overcome this problem, next methods (Kalman filter based methods) have been proposed in literatures.
Method4) Kalman Taylor method
In (23), from left side, the first matrix is named S(n) and second one is named M S which is final measurement matrix. Finally (22) and (23) are considered as final state and final measurement equations. Kalman filter is applied to these two equations in two steps in order to estimate phasor. Predicting and updating steps are as:
where X(n−1) is rotated state vector at (n−1)^{ t h } sample and X ^{−}(n) is its prediction in n ^{ t h } sample. p ^{−}(n) is prior error covariance and \({\sigma ^{2}_{v}}\) is the variance of model error. Noise is assumed to affect only the rotated state vector despite of its derivatives thus considered as (h ^{ T }.h ^{ T }). In (24) H is the Hermitian transpose operator.
where:
X(n) is rotated state vector at sample n.
K(n), Kalman gain, reveals how much modification is needed for state variables based on measurement.
\({\sigma ^{2}_{w}}\) is measurement noise variance created by sensors.
p(n) is posterior error covariance. And
I is the unit matrix.
These Kalman equations make it possible to calculate X(n); Therefore dynamic phasor can be calculated based on estimated value of X(n).
Method5) Fourier Kalman Taylor method
where:
N is sample number in fundamental period.
a _{0}(t) and ϕ _{0}(t) are DC amplitude and phase.
a _{1}(t) and ϕ _{1}(t) are fundamental amplitude and phase.
a _{ N−1}(t) and ϕ _{ N−1}(t) are amplitude and phase of (N−1)^{ t h } harmonic.
f _{0} is zero (DC) frequency and f _{1} is the fundamental frequency of the signal.
Rest of dynamic phasor estimation in this method is the same as previous method.
Method6) Modified Kalman Taylor method
Kalman filter is used in method 6 as methods 4 and 5, so the rest of dynamic phasor process is similar to these methods.
Simulation results
According to Fig. 6, it is observed that phasor derivative estimations are not as accurate as the phasor estimation (Fig. 4) which indicates the elimination of higher terms in Taylor expansion. These derivatives have two important roles. First, they reduce error estimation as shown in simulation results; Second, they are able to calculate frequency and detect faults and power swings. It is the superiority of dynamic phasor compared to traditional concept of phasor.
In order to clarify this capability, consider a disturbance which occurs in a power system. It is important for us to be discovered immediately to take accurate actions. Power systems make use of distance relays in transmission lines to detect this condition. A distance relay is a device that measures the apparent impedance as an index of distance from the relay location. The power swing is a consequence of a severe disturbance like line fault, loss of generator unit and switching heavy load and creates large fluctuations (just like dynamic phasor condition) of active and reactive power between two areas of a power system. Power swing affects the distance relay behavior and causes its malfunction. Fast detection of power swing is interested in distance protection of transmission lines. Several methods have been proposed to solve this problem till now [24–29]. However the detection based on first and second derivatives of dynamic phasor can be a novel method and makes this aim accessible.

T V E to examine error bound

Step amplitudephase benchmark tests to analyze dynamic response of the methods in amplitude and phase step condition

Step frequency benchmark tests to analyze dynamic response of the methods in frequency step condition

Frequency response to demonstrate the delay of the methods

Histogram tool to examine RMS error of amplitude estimation

Signal taken from a PMU to check presented methods in practical conditions

Harmonic and DC offset infiltration

Derivatives of amplitude and phase

Transient monitor index

Computation time

Sampling number

Noise infiltration
Magnitudephase step test
Frequency step test
Frequency measurement is an important parameter in power system operation because it indicates the dynamic balance between power generation and its consumption. To protect power system and detect islanding situation and its time, the frequency and its rate of change are utilized as indicators. Synchronized phasor measurements offer an opportunity to measure power system frequency. Dynamic phasor estimation, presented in this paper, as an inner part of P M U can be utilized in this field.
Frequency response
However method1 does not work appropriately when the input frequency deviates from nominal frequency even in very little amount. According to Fig. 11 (method1), magnitude response gain decreases sharply before and after fundamental frequency. This frequency domain analysis simply shows the drawback of method1 in oscillating condition. In contrast to method1, other methods show different behaviors around fundamental frequency. The main feature of an appropriate filter is to provide a gain by the value of two around positive and a zerogain in negative fundamental frequency. This is mainly because the signal model corresponds to two rotator components: first one rotates at positive and second one rotates at negative fundamental frequency. Therefore, complete elimination of negative one and complete pass of positive one is the main duty of the filters (presented methods).
This feature could be accessible by dynamic phasor concept as shown in Figs. 11 and 12. The flatter interval around fundamental frequency leads to lower distortion in phasor estimation. The more persistent flat gain of method 2 compared to method 1 validates the strength of dynamic phasor concept in oscillating conditions. These figures also help to explain the behavior of the estimations when other frequency components are present in the input signal. Methods 1, 2 and 5 provide zerogain in non fundamental component which demonstrates their ability to remove harmonic in the output.
However methods 3, 4 and 6 do not have this capability. So these methods have difficulties in harmonic conditions especially when subjected to DC component which is common in fault time. In addition, a useful result is attainable based on phase response of output. According to Fig. 11 (second row), all least square based methods (methods 1, 2 and 3) show nonzero around fundamental frequency point which indicates nonzero group delay when subjected to oscillation.
However Kalman filter based methods (methods 4, 5 and 6) show instantaneous estimation property because of their constant zero phase around nominal frequency. The great advantage of these methods is that they can be truly employed in P M U due to their synchronization in nanosecond scale.
It is worthy to note that good performances of all methods except method1, in off nominal frequency are limited around nominal frequency (small vicinity), but because frequency never deviates from the nominal value more than a few m H z in real conditions, the performances of these methods are acceptable. It is also worth noting that, noise effect in dynamic phasor concept is higher than traditional concept which is shown in noise simulation section. This fact can be explained by increase the flat gain length in the interval of fundamental frequency that makes methods more sensitive to noise.
Error bounds
where a(n) and \({\hat a(n)}\) are real and estimated amplitudes respectively. In statistics, histogram is a graphical representation of the data distribution. It is an estimate of the probability distribution of a continuous variable. Histogram tool creates disjoint categories (known as bins) and counts the number of observations that fall into each of these bins. Thus, by considering N as the total number of observations, K as the total number of bins, and m _{ k } as the number of observations in k t h bin, the histogram meets \({\sum _{k=1}^{k=K}}\).In this study, 20 bins and 150 observations have been considered.
Harmonic infiltration (power system test case)
In real conditions, power system signals may be polluted by harmonics or DC component which is necessary to be examined. In order to analyze oscillated signal along with harmonic, quite identical signal examined in [18] is used in this section. The data were taken from a P M U, installed in a substation. A threephase fault is created in one of power system lines at 0.1 s e c. The fault is cleared at 0.2 s e c by opening breakers at the ends of faulted line. Line removing causes a swing condition with frequency 5 H z. Measured signal consists of fundamental component (60 H z) and fifth harmonic component (300 H z,1.5 % of fundamental). This signal was sampled at 32 samples per cycle.
Noise infiltration
After this point, transient monitor increases steeply by increasing error variance. Generally Kalman based methods (methods 4, 5 and 6) show lower values compared to least square based methods according to Fig. 17. It is also observed that method 6 provides minimum T M index before critical point. However after critical point (Variance = 10^{−2}) method 1 shows minimum T M index because the dynamic phasor concept is more sensitive to noise due to its flat gain around fundamental frequency. This flat gain allows the noise to be transferred into the output (estimated dynamic phasor).
Simulation time
Conclusion
Phasor and its derivatives are very useful tools to improve the simulation programs of power system and its control. The constant amplitude and phase (traditional definition) impose serious restriction on monitor and control of power system, thus dynamic phasor attracted lots of attentions nowadays. Six dynamic estimation methods have been classified based on least square and Kalman filter; then they have been explained and compared in different test cases. All methods have both advantages and disadvantages. The defect of least square based methods is that they create a delay in estimation process but Kalman filter based methods provide instantaneous estimation. The advantage of all method except 1 is that the speed (first derivative) and acceleration (second derivative) of phasor could be calculated. Lack of comprehensive indices to explain the discrepancy of different method is motivating to establish a framework in order to compare presented methods. Twelve indices are utilized to form a complete benchmark in the paper including: TVE to examine error bound, amplitudephase step benchmark test to analyze dynamic response of the methods in amplitude and phase step condition, frequency step benchmark test to analyze dynamic response of the methods in frequency step condition, frequency response to demonstrate the delay of the methods, histogram tool to examine RMS error of amplitude estimation, signal taken from a PMU to check presented methods in practical conditions, harmonic and DC offset infiltration, derivatives of amplitude and phase to detect the fault occurrence time, transient monitor index, computation time, sampling number and noise infiltration. Simulation results show that the the dynamic phasor concept gives advantageous results in slow frequency oscillation.
Declarations
Authors’ contributions
JK, Ph.D. student, brings up the idea of this review, performed the primary simulations and drafted the manuscript. MK, Ph.D. supervisor, participated in revising the manuscript (response to the reviewer and editing grammatical lexical mistakes). Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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