a novel phasor estimation algorithm based on taylor … · a novel phasor estimation algorithm...

A NOVEL PHASOR ESTIMATION ALGORITHM BASED ON TAYLOR

EXPANSION AND PHASORLET

Mai R1, He Z

1, Kirby B

2, Edwards A

2, Bo Z

2, Fu L

1

1. College of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031 , China

2.Areva T&D UK, Automation & Information Systems

[email protected]

Abstract –The accurate measurement of real time syn-

chronous voltage and current phasors across the entire

network is crucial for the safe and efficient running of the

network. The Phasor Measurement Unit (PMU) is also a

prerequisite to further applications such as state estimation

and wide area control. A novel synchrophasor measure-

ment algorithm: the First Order Taylor expansion Phasor-

Let algorithm (FOPL), which is based on both Taylor

expansion and phasorlet algorithm, is proposed in this

paper. Firstly, slow varying signals in power systems are

modeled by Taylor expansion. The Phasorlet algorithm is

then employed to estimate the phasor of the signal. Finally

the estimation is revised via first derivative expressed by

two phasors of sequential data windows. Theoretical analy-

sis, simulation and analysis of real data demonstrate that

the performance of FOPL is superior to traditional DFT in

the presence of power swings and frequency deviations.

Keywords: Phasor Measurement Units (PMU) ，，，，

Phasorlet analysis, Taylor expansion，，，， discrete Fou-

rier transform (DFT) ，，，，Total Vector Error (TVE)

1 INTRODUCTION

Phasor Measurement Unit (PMU) measures the volt-

age and current phasors of power grid synchronously in

real time, allowing power grids to be operated in a safe

manner. PMU is also a prerequisite for other high-level

application such as state estimation, wide area control

and adaptive relaying. The accuracy of the synchronous

phasor estimation has a direct bearing on the effective-

ness of its application. Hence it is imperative that the

precision of the phasor estimation in a PMU conforms to

IEEE standard C37.118. The engineering application of

phasor measurements was first proposed by Phadke et

al. in the 1970s, and since then there have been a multi-

tude of publications on phasor estimation algorithms by

different scholars and research units with varying level

of success. These algorithms are mainly based either on

zero-crossing detection[1, 2], Discrete Fourier Trans-

form (DFT)[3-7] or Phasorlet analysis[8-11]. Those

based on zero-crossing detection are easy to understand

and to implement, but they are susceptible to errors in

the presence of harmonic distortion. Whilst DFT

based algorithms are widely adopted due to their inher-

ently superior harmonic rejection characteristic, the

accuracy of the algorithms under dynamic conditions is

not satisfactory for some applications. On the other

hand, phasorlet analysis is computationally intensive and

is still largely under research and development, although

their superior characteristic seems promising.

In a power system, signals from adjacent time inter-

vals are correlated. If the relationship between these

signals can be determined, it can be used to improve the

accuracy of phasor estimation. The derivatives in the

Taylor expansion represent the relationship between

adjacent phasors, so the use of Taylor expansion pro-

vides a means of characterizing this relationship.

After looking at phasorlet transformation as well as

the Taylor expressions for slow varying voltage and

current signals in a power system, this paper presents the

phasorlet expressions based on Taylor expansion and

compares its performance against that of DFT under

dynamic conditions. Theoretical analysis, simulations

and analysis of field data have shown that with minor

increase in computational requirements, the use of

FOPL can greatly improve the performance of phasor

estimation.

In this paper, we shall be examining the work carried

out. The paper proceeds as follows:

- Hypotheses adopted: signal modeling, definition of

instant value of signals, parameter variation;

- Rundown on existing calculation methods: zero-

crossing algorithm FFT and Phasorlet etc; and proposed

a simplified algorithm from Phasorlet.

- The search for a new type of algorithm that integrate

the advantages of Taylor expansion and simplified

phasorlet algorithm.

- Revised action: shift of phasor from calculation

point to report point, the expression of first derivative

and the flow chart of this algorithm is depicted in sec-

tion 2.

- The evaluation of this algorithm under dynamic

conditions (frequency deviation and power swing) is

given in form of TVE.

- Evaluate algorithm from Field data off online: DFT

and FOTP are employed to estimate the phasor of a case

from field relay, which contains power swing, harmonic,

frequency deviation and their integration.

- A conclusion is given at the end.

2 BASIC THEORIES

2.1 Signal Representation[12]

In electrical power engineering, voltage and current

signals are often expressed in terms of a cosine function,

which is given mathematically by:

16th PSCC, Glasgow, Scotland, July 14-18, 2008

2 * 2

( ) =A cos(2 + )

2[ ( ) ]

2

j ft j ft

x t ft

X e X eπ π

π φ

−= ⋅ +

(1) In the above expression, A、 f 、φ are the mag-

nitude, frequency and initial phase angle of the signal

respectively. In phasor estimation, the signals used are

discretised form. To discretise (1), let cosf be the

estimated fundamental frequency and N be the number

of sampling points in one fundamental cycle. The sam-

pling interval is thus cos1/ ( )t f N∆ = ⋅ and the value of

the kth

sample is given by: ( ) =A cos( k+ ) k=0,1,...,N-1x k ω φ (2) where the angular frequency cos2 t fω π= ⋅ ∆ ⋅ repre-

sents the phase angle difference between two consecu-

tive sampling points. In (2), the cosine component can

be conveniently expressed in terms of phasors and their

complex conjugates as follows: cos cos2 2*2( ) = ( )

2

j k jl f j k jl fl lx k X e X e

ω π ω π− ⋅ ⋅ − + ⋅ ⋅⋅ + ⋅ (3)

where cos 20

jl flX X e

π⋅ ⋅= ⋅ represents the phasor of the

signal at t l= , whilst 0X represents the phasor at

0t = 。

2.2 Discrete Fourier Transform

Suppose that there is a complex exponential se-

quence as follow： 0( ) 0,1,..., 1j n

n e n Nωϕ = = − (4)

where 0 2 / Nω π= is the angular difference between

consecutive points of this sequence. ( )nϕ forms an

orthogonal basis which spans an N-dimensional or-

thogonal space. The DFT essentially pro-

jects { ( ),k=0,1,....,N-1}x k , which is the signal to be

analyzed, into this orthogonal space. This projection

process is realized by means of the convolution sum.

When the number of sampling points of the signal is M,

the phasor estimated by M point DFT at time l is

given by the following equation: 1

2( ) ( ) ( )

M

n

X l x n l nM

ϕ=

= ⋅ −∑% (5) M is the number of points in the data window. Equa-

tion (5) becomes the Full Cycle Discrete Fourier

Transform (FCDFT) when M N= , and becomes the

Half Cycle Discrete Transform (HCDFT)

when / 2M N= .

2.3 Phasorlet analysis[8]

In phasorlet analysis, the signal to be analyzed is

transformed by a family of functions generated by a

determining function of different resolutions (with vary-

ing levels of frequency resolution and time resolution).

Taking FCDFT for example, the summation in (5) can

be expressed in terms of the summation of 2j

jP =

2( 0,1, 2,..., log ( ))j N= partial sums, with each partial

sum consisting of /j jW N P= terms, as shown in (6)

below: 1

1 ( 1) 1

2 2( ) ( ) ( ) ( )

Pj pWj Pjjp

p n p Wj p

X l x n l n X lN N

ϕ−

= = − =

= ⋅ − =∑ ∑ ∑% % (6) where the thp partial sum ( ), 1,2,..., 2

j jpX k p =%

is referred as the phasorlet of the signal. More accu-

rately, it is the thp phasorlet of th

jP order and of reso-

lution j .

For the calculation of the phasorlet ( )jpX k% , the

complex exponential sequence is redefined as follows: ( ), ( 1) 1( )

0, .

j jjp

n p W n pWn

ϕϕ

− ≤ ≤ −= others (7)

With this definition, the phasorlet of th

jP order can

be expressed as follows: 1

( 1)

2( ) ( ) ( )

W

pWjj jp p

j n p Wj

X l x n l nϕ−

= −

= ⋅ −∑% (8) In order to simplify the representation, only the low-

est frequency resolution phasorlet ( 2log ( )j N= ) is

considered. Equations (9) and (10) show the thp

and ( / 4)thp N+ estimate phasor respectively at the

time 0l = (phasorlets at different resolutions and time

can be easily derived using the same procedure). 0 02 ( 1)*(0) 2 ( )

jk j pjpX e X X e

ω ω− −= + ⋅% (9) 0 02 ( 1)*

/4 (0) 2 ( )jk j pj

p NX e X X eω ω− −

+ = − ⋅% (10) where 0,1,...,3 / 4p N= represent the different se-

quence numbers of partial sums. By adding equations

(9) , (10) and then rearranging the result, the phasor

estimation can be obtained: 0

0 /4( ) [ (0) (0)] / 2jk jj

p p NR p e X X Xω

+= = +% % (11) (11) shows that under ideal conditions, an estimation

exactly equal to the true value can be obtained from only

two points in a quarter of a full cycle. In a practical

system, the presence of system disturbances (e.g. fre-

quency deviation, harmonic distortion and power swing)

precludes the direct application of equation (10) due to

its limited accuracy as a result of its poor filtering per-

formance. This paper uses the mean value of multiple

phasorlet estimations of one cycle in order to improve

the accuracy of the result in the presence of system dis-

turbances. The phasorlet estimation of the thn data

window at time l is given by equation (12) by: 3 /4

1

2( ) ( )

3

N

l

p

W n R pN =

=⋅∑ (12)

where ( )W n is also known as the phasorlet estima-

tion of the Simplified PhasorLet Algorithm (SPLA). The

flow of data processing for SPLA and relationship be-

tween complex exponential sequences and data points is

shown in Figure 1. Only one cycle data are used in this


algorithm and in practical application, more data points

might be used to improve the filter performance. In

Figure 1, it can be shown that (12) is equal to the aver-

age value of a FCDFT and half of a HCDFT. So it just

needs to alter the coefficients of traditional DFT instead

of introducing more computation.

Figure 1: the flow of data processing for SPLA

2.4 Phasorlet algorithm based on Taylor Expansion

A real power system consists of multiple sources and

is open-ended. Its current and voltage values are influ-

enced by many factors. As such a static representation of

these dynamic signals will inevitably lead to significant

errors because of the signals’ dynamic nature. The errors

in estimation can be substantially reduced if the dynamic

behaviour of the signals can be somehow incorporated

in its representation. This paper introduces the Taylor

expansion to model current and voltage signals and

makes use of the first order derivative to capture the

dynamic behaviour of signals. The Taylor expansion can

be expressed as follows: 1 2 ( )0

1 1' '' ...

2! !

n nX X X t X t X tn

≈ + + + + (13) where 0X represents the true value of phasor in the

current data window and ( )nX is the nth order deriva-

tive of the phasor which represents the rate of change of

consecutive phasors. The higher the number of deriva-

tives considered, the closer is the estimation to the true

value, although more computation power is also re-

quired. Real power systems usually involve only slow

changing system conditions (e.g. power swing), which

means that the use of first order or second order deriva-

tive alone should provide sufficient accuracy. Substitut-

ing only up to the first order derivative term in (13)

into (1) and rearranging, the first order Taylor expan-

sion representation of the signal can be obtained: 2 * 20 0

2x(t) = [( ' ) ( ' ) ]

2

j ft j ftX X t e X X t e

π π−+ ⋅ + + ⋅ (14) When the system is in steady-state and operating at

the nominal frequency, 'X vanishes and X assumes a

constant value. Otherwise, 'X is non-zero and the esti-

mation varies with time as determined by 'X .

It is advantageous to choose the centre of data win-

dow as the reference point for the phasor, as this allows

certain types of errors contributed by odd derivatives to

cancel exactly. By substituting (14) into (12) and

rearranging, it is possible to derive (15), which is

known as the First Order PhasorLet Algorithm (FOPL).

This is a formula for the corrected estimations of

phasors from phasorlet. *( ')

6 sin(2 / )s

j XP W

N f Nπ= −

⋅ ⋅ ⋅ (15)

where sf is the average frequency of the current win-

dow; P is the estimation of the true phasor value. In

order to calculate P , the first order derivative should be

expressed first. And it can be approximated based on the

current window phasor and last window phasor. This

represents the changing rate of phasors per second: ( ) ( ) ( -1)' ( )M M M sX W W f= − ⋅ (16) By substituting equation (16) into (15) and rear-

ranging, the phasor estimation at the centre of data win-

dow is given by: *( ) ( -1)

( ) ( )

( - )

6 sin(2 / )

M M

M M

j W WP W

N Nπ= −

⋅ ⋅ (17)

In practice, the centre time of the data window and

the reporting time do not always coincide. The phasor at

the centre of data windows needs to be shifted to the

phasor of reporting time, the maximum time shift being

equal to one half of a sampling period. The phase shift

and phasor at the reporting time are given as follows: s( ) 2

[ ( ) ']

r w

jr w

T T f

RP P T T X eθ

θ π= − ⋅ ⋅

= + − ⋅ ⋅ (18)

=

=

= e

r

w

shift angle

T report time

T time of centre of data window

P r vised Phasor of data window

RP phasor of report time

θ =

=

Figure 2: The flow chart of FOPL algorithm

Figure 2 show that the whole algorithm contains three

components: Phasorlet algorithm, revision and phasor

shifting. Firstly, the phasor of centre of data window is

estimated by SPLA via equation(12), providing the raw

phasor estimation. And then the raw phasor estimations

of current data window and last data window are em-


ployed to express the changing rate of the phasor-first

order derivative by equation (16). And the revised

phasor estimation will be got from the raw phasor esti-

mation via first order derivative by equation (17). Fi-

nally, not only the time of centre of data window and the

report time but also first order derivative will be in-

volved to shift the phasor from the centre of data win-

dow to report time by equation (18).

3 PERFORMANCE ANALYSIS

In order to provide a comprehensive comparison of

the performances of the various algorithms, this paper

computes the average, minimum, maximum and mean

square deviation of Total Vector Error (TVE) [12] for

the three different algorithms(FCDFT, SPLA and

FOPL) under two distinct operating conditions: fre-

quency deviation and power swing. The cases are evalu-

ated under a fixed sampling frequency of 1600Hz.

3.1 Evaluation criterion

TVE is commonly used to evaluate the performance

of phasor estimate algorithms. The TVE criterion meas-

ures errors in time synchronization, phasor magnitude

and angle. The definition of TVE is as follow:

2 2

2 2

( ( ) ) ( ( ) )r r i i

r r

R n R R n RTVE

R R

− + −=

+ (19)

real componenet of theoretical value

imaginary component of theoretical value

( ) real component of estimation

( ) imaginary component of estimation

r

i

r

i

R

R

R n

R n

=

=

=

=

( )R n RTVE

R

−=

Figure 3: Relationship between phasors and TVE

The relationship between the true value, the estima-

tion and the TVE is depicted in Figure 3. R is the true

phasor of the signal; ( )R n is the phasor estimation of

the signal; vector error ( E ) represents the difference

between R and ( )R n . And the TVE should be under

1% under steady state.

3.2 Frequency deviation

In practice, the instantaneous system frequency fluc-

tuates with time. If a fixed sampling frequency system is

employed, the sampling frequency of signal is not al-

ways an integer multiple of fundamental frequency,

leading to spectrum leakage and impacting on the accu-

racy of the phasor estimation. Although frequency track-

ing technology does help to reduce the effect of spec-

trum leakage, the delay, anti-alias filter delay and the

delay of the phase estimate algorithm, in estimating the

frequency means that spectrum leakage cannot be en-

tirely eliminated. Therefore, insensitivity to system

frequency deviations is an important characteristics of a

phasor estimate algorithm.

Figure 4 shows the resulting angle error; magnitude

error and TVE when the different algorithms are applied

on a 48Hz signal for different initial angles from 0 to

360 degree when sampling frequency is 1600Hz. As

shown in Figure 4, angle error, magnitude error and

TVE vary periodically with the initial phase angle. The

performance of FOPL is much superior to that of the

other two algorithms. The maximum angle errors, mag-

nitude errors and TVEs are summarized in Table 1.

(a) Angle error between true value and estimate

-202Angle error/ °(b) Magnitude error between true value and estimate

202Mag error/pu%(c) TVE of the three algorithm

01TVE/% 230 90 180 270 angle/°

0 90 180 2700 90 180 270 angle/°

angle/°

Figure 4: Performance of three algorithms for 48Hz

Maximum Error

Angle(°) Mag(%) TVE (%)

FCDFT 1.286 2.323 2.389

SPLA 0.506 0.892 0.973

FOPL 0.159 0.305 0.377

Table 1: Maximum angle errors, magnitude errors and

TVEs for a 48Hz signal

The performance of the three algorithms under fre-

quency deviation can be further analyzed in terms of

statistics. The mean, minimum, maximum and variance

TVEs are plotted as a function of frequency deviation in

Figure 5(a, b, c, d) respectively. It is evident that the

mean TVE for FOPL is lower than that of FCDFT and

SPLA. The mean TVE for FOPL is only 1.3%, com-

pared to 5.33% and 1.975% respectively for FCDFT and

SPLA, when the signal frequency is 45Hz. To achieve a

1% maximum TVE, the maximum allowable system

frequency deviations for the different algorithms are:

FCDFT， ± 0.9Hz；FHCDFT， ± 2.1Hz；FOPL，± 3.6Hz. With the use of frequency tracking technol-

ogy, the deviation between estimated frequency and

fundamental frequency can be reduced significantly.

Under usual conditions, the frequency tracking can keep

within 2Hz± of the system frequency. Finally, the stabil-

ity of estimation performance as measured by the TVE


variance indicates that the FOPL gives the best results,

followed by SPLA and FCDFT.

Figure 5: Statistic performance during frequency deviation

3.3 Power swing

Let the power swing be described mathematically by: ( ) cos(2 ) 0.2ps psS t t f iπ= ⋅ + (20) Where 2psf Hz= is the frequency of power swing.

The frequency of the reference signal is 50Hz and data

window is taken at every sampling point. The perform-

ance of the three algorithms is shown in Figure 6.

It is seen that the angle error and magnitude error for

FCDFT exhibit larger oscillations during a power swing

than that of FOPL. The errors for FOPL are near zero.

The maximum angle errors, magnitude errors and TVEs

are summarized in Table 2, where it is evident that the

maximum TVE of FCDFT is 30 times higher than that

of FOPL in this condition. It is worth noting that at

125ms the TVE of FOPL remains close to zero, whilst

the TVEs for the other two algorithms exhibit maxima at

that time. The reason is that the first order derivative of

the power swing signal at that instant is -1, which corre-

sponds to the maximum absolute value for the first order

derivative and its second derivative is instantaneously

zero. Thus at that instant the errors in the FOPL arise

only from the third order derivative term and above,

giving a low TVE. On the other hand, because the first

order derivative is at its maximum and, at the same time

the absolute value of the signal phasor is a minimum,

maximum TVEs for FCDFT and SPLA occur at that

instant.

Maximum Error

Angle(°) Mag (%) TVE (%)

FCDFT 5.677 1.985 9.994

SPLA 1.892 0.665 3.325

FOPL 0.100 0.250 0.278

Table 2: Maximum errors during power swing

(a) Angle error during 2Hz power swing

(b) Magnitude error during 2Hz power swing

(c) TVE of three algorithms during 2Hz power swing

-505-20205100 50 100 t/ms150 2000 50 100 t/ms150 2000 50 100 t/ms150 200

Angle e

rror/ °

Mag e

rror/pu%

TVE/%

SPLAFCDFTFOPLFigure 6: Performance during a 2Hz power swing

TVE/%

Figure 7: Maximum of TVE under power swing condition

3.4 Harmonic rejection

According to the compliance criterion of IEEE stan-

dard c37.118, the harmonic from 2nd to 50th, whose

magnitude is 0.1pu, is selected to inject to the reference

signal at every 5 Hz. In order to make the fig clearer, the

mean TVEs from 2nd to 15th only are shown in Figure

8. The performance of harmonic rejection can be im-

proved via lengthening the data window.

TVE

平均值/%

Figure 8: average TVEs during harmonic

4 ANALYSIS OF FIELD DATA

Practical sampling systems in a real power system

employ two main types of sampling methods: fixed rate

sampling and frequency tracking. Frequency tracking

technology can help to reduce the effect of spectrum

leakage, so it is widely used in the power system appli-

cations. Due to space constraint, only one set of field

data is presented and analyzed in this section. The cho-

sen case study, which incorporates elements of harmonic

distortion, frequency deviation and power swings em-

ploys frequency tracking.


0 100 200 300 400 500 600 t/ms0.61.01.6

Magnitue/p

u

700 8000.81.21.4

Figure(11)

Figure(12)

900 Figure 9: Amplitude estimations from FOPL and FCDFT of a power swing and fault voltage signal

Magnitude/p

u

Figure 10: Comparison of amplitude estimations un-

der fault power swing

Initially the power system was experiencing a power

swing and subsequently a fault occurred on a transmis-

sion line at 220ms. The line circuit breakers tripped as a

result to isolate the fault. As this is a double circuit

transmission line system, the power swing continued

through the remaining healthy transmission line and

persistent busbar voltage oscillation can be observed

during 230~450ms. At 460ms a fault occurred on the

remaining transmission line, causing its circuit breakers

to operate. Therefore, the system containing the busbar

where the voltage was measured transiently entered an

unstable state, before slowly settling back into a new

stable state. The fundamental frequency phasor magni-

tude responses obtained by three algorithms proposed

above are plotted in Figure 9. Frequency tracking tech-

nology was employed by the sampling system and there

are 32 data points in each fundamental cycle. It can be

seen that the performance of the FOPL is more stable

compared to FCDFT under dynamic conditions (which

consisted of a combination of frequency deviation, har-

monic distortion, power swing). As shown in Figure 10,

the magnitude estimation from FOPL is close to the

peak of the signal and it is very smooth, but there is a

large error and ripple in the magnitude response of

FCDFT because of dynamic nature of the signal. There

is always a finite delay in frequency tracking, so that the

estimated frequency will always lag the real frequency

during transient conditions. As such, the phasor estima-

tion produced by FCDFT has limited accuracy. In con-

trast, the estimation by FOPL is much closer to the real

phasor due to the inherent predictive effect offered by

the first order derivative term in the algorithm as shown

in Figure 11.

Magnitude/p

u

Figure 11: Comparison of Amplitude estimations under

frequency deviation

5 CONCLUSION

In this paper, a novel phasor estimate algorithm based

on Taylor expansion and Phasorlet analysis is proposed.

An improved performance can be obtained using the

simplified phasorlet algorithm (SPLA) compared to the

conventional FCDFT. Further improvement in perform-

ance is achieved with FOPL which, in addition to using

phasorlet transformation, also uses the Taylor expansion

to take advantage of the relationship between consecu-

tive data windows.


Under slow varying system conditions, FOPL can

dramatically improve phasor estimation accuracy at the

cost of slightly more computational overhead. Whether

it is in frequency deviation or power swing conditions,

FOPL demonstrates considerable advantages over other

methods, as its TVE can be maintained within the IEEE

standard C37.118 under most of dynamic conditions.

Nevertheless, limitations of FOPL are twofold. One is

that the accuracy of the current phasor estimation is

influenced by the previous phasor estimation, due to the

use of derivative. As a consequence, following a sudden

change in signal the TVE can be expected to increase

significantly. And the other one is that its performance

as a filter is not as good as FCDFT but it can be im-

proved, by involving more data point to the estimation

or employing pre-filtering to filter out harmonics.

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