Detection methods of unbalance factor and phase angle difference of positive and negative sequence

componet of three-phase voltage

Ma wenchuan1, Zhang shuo2, Shi nan2 Department of electrical engineering

Northeast Agriculture University, Harbin, China

mawenchuan@gmail.com

Wang shuwen1, Baiyu1, Li shumin1 Business and Management Department

Heilongjiang Technology and Business Vocational College Harbin, China

mawenchuan@gmail.com

Abstract—For achieving the compensation method which is used for balancing the three-phase unbalanced load by TCR, unbalance factor, phase angle difference of positive and negative sequence component of three-phase voltage should be detected. Accuracy and rapidity of detection is the key factors which affects three-phase unbalance compensation effect. This paper proposes a calculating method of unbalance factor, phase angle difference of positive and negative sequence component of three-phase voltage in the condition of supply voltage unbalance. The method can use dq transformation to achieve the relative parameters by delay sampling voltage to construct PLL signal. Simulation results show that the method can detect the parameters rapidly and accurately.

Keywords- asymmetrical supply voltage;dq transformation; unbalance factor; phase angle difference;

I. INTRODUCTION Usually detecting method on real part and imaginary part of

voltage vector is by constructing uα and uβ according to the actual voltage,

Where ( )sinu U tβ ω ϕ= + ( )cosu U tα ω ϕ= +

uα advanced uβ 90°, then it can get the equation

d

q

cos cos sinsin sin cos

u uU t tu uU t t

α

β

ϕ ω ωϕ ω ω

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

Transform uα and uβ to dq axis, the direct voltage component on dq axis is real part and imaginary part of voltage vector[1-4].

But due to uα is achieved by advancing uβ 90°, adopted data does not have Simultaneity[5-6], detecting time is about 5ms, it is difficulty to meat real-time request[7-8]. For solving the question the paper proposed an improved method on detecting real part and imaginary part of voltage vector.

II. IMPROVED DETECTING METHOD ON REAL AND IMAGINARY PART OF VOLTAGE VECTOR BASED ON DQ

TRANSFORMATION. In this section, by using sampling delay step and

trigonometric function orthogonality, it can achieve real part and imaginary part of voltage vector rapidly. The schematic diagram of algorithm is shown as Fig.1

Fig.1 Improved real and imaginary parts detection method of single-phase voltage based on dq transformation

A phase voltage is

( ) ( )0sinau t a tω θ= + (1)

where A is amplitude 0ω is angular frequency, sampling

( )u tα after a delay timeτ , phase shift isψ , and the delay signal is shown as:

( ) ( )s 0sinau t a tω θ ψ= + − (2)

Expansion formula ( )sdu t is shown as:

( ) ( ) ( ) ( ) ( )s 0 0sin cos cos sinau t a t a tω θ ψ ω θ ψ= + − + + − (3)

Where ( )cos ψ− and ( )sin ψ− is known.

Supported by Natural Science Foundation of Heilongjiang (LC201003)

978-1-4577-0547-2/12/$31.00 ©2012 IEEE

Orthogonal vectors ( )au tβ is

( ) ( ) ( ) ( ) ( )( )

sa0

coscos

sina

u t u tu t a t α

βψ

ω θψ

− −= + =

− (4)

A phase voltage Amplitude is

2 2( ( )) ( ( ))aa u t u tα β= + (5)

Constructing PLL by ( )u tα and ( )au tβ , and dq transform results of ( )u tα

and ( )au tβ is

( ) ( )( ) ( )

( )( )

d 0 0 0

q 0 0 0

cos sin sinsin cos cos0

u t t U tUu t t U t

ω θ ω θ ω θω θ ω θ ω θ

+ + +⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤= = ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ − + + +⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

(6)

Phase angel of a voltage is same to PLL, so phase voltage amplitude can achieve directly by dq transform.

And because system frequency will draft, phase draft ψ will produce error, for solving the question, ω can be achieved after derivation calculus according to ( ) ( )0sinu t a tα ω θ= + , calculation method is shown as:

( )( )

( )( ) ( )

2( sin ) sin( )sin sin

u t a t a tu t a t a tα

α

ω θ ω ω θωω θ ω θ

′′ ′′+ − += − = − = −+ +

(7)

Second derivative will be achieved by three point method.

During solving the question, delay time is determined byτ , and due to AD sampling rate restriction, τ is set integral multiple of sampling cycle.

III. METHOND ON UNBALANCE FACTOR, PHASE ANGLE DIFFERENCE OF POSITIVE AND NEGATIVE SEQUENCE COMPONET

OF THREE PHASE VOLTAGE Schematic diagram of calculating unbalance factor, phase

angle difference of positive and negative sequence component of three-phase voltage is shown as figure 2.

A phase is reference phase

( ) ( )( ) ( )( ) ( )

sin

sin

sin

a

b b b

c c c

u t U t

u t U t

u t U t

α ωω θω θ

=⎧⎪

= +⎨⎪ = +⎩

(8)

( ) ( )( ) ( )( ) ( )

cos

cos

cos

a

b b b

c c c

u t U t

u t U t

u t U t

βα

β

β

ω

ω θω θ

⎧ =⎪⎪ = +⎨⎪ = +⎪⎩

(9)

Fig.2 Detection method of unbalance factor and phase angle difference of positive and negative sequence of voltage

Constructing dq transform formula is

sin coscos sin

t tC

t tω ωω ω

−⎡ ⎤= ⎢ ⎥− −⎣ ⎦ (10)

It can get cos sinsin cos0

ad a

aq a

u uU t tu ut t

α

β

ω ωω ω

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

(11)

cos cos sinsin sin cos

bd bb b

bq bb b

u uU t tu uU t t β

θ ω ωθ ω ω

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

(12)

cos cos sinsin sin cos

cd cc c

cq cc c

u uU t tu uU t t β

θ ω ωθ ω ω

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

(13)

Real and imaginary part of aU•

、 bU•

and cU•

can be achieved according to above equation.

And the sequence component can be achieved by using the equation (14)

Where a1U ， a2U ， a0U is positive, negative and zero

sequence aptitude of aU•

、 bU•

、 cU•

,

1aθ ， 2aθ ， 0aθ is phase angle of a1U ， a2U ， a0U

2a1 1

22 2

0 0

1 cos +i sin1cos +i sin = 1 3

1 1 1cos +i sin

a1 a a1 a

a2 a a2 a b

a0 a a0 a c

UU UU U UU U U

α αθ θθ θ α αθ θ

⎡ ⎤⎡ ⎤×⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥× ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥×⎣ ⎦ ⎣ ⎦⎣ ⎦

(14)

And phase angle of positive, negative and zero sequence voltage can be achieved by using the equation (15)

11 2 2

1 1

22 2 2

2 2

cosarccoscos + sin

cosarccoscos + sin

a1 aa

a1 a a1 a

a2 aa

a2 a a2 a

UU U

UU U

θθθ θ

θθθ θ

⎧ ⎛ ⎞⎪ ⎜ ⎟=

⎜ ⎟⎪⎪ ⎝ ⎠⎨

⎛ ⎞⎪⎜ ⎟=⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

(15)

Because 1 1a a1 aU U θ= ∠ ， 2 2a a2 aU U θ= ∠ ， positive, negative and zero sequence of three-phase line voltage can be achieved as

( )( )

1 1 1 1 1

2 2 2 2 1

3 30

= 3 30

ab ab1 ab a1 a b1 b a1 a

ab ab2 ab a2 a b2 b a2 a

U U U U U

U U U U U

θ θ θ θ

θ θ θ θ

⎧ = ∠ = ∠ − ∠ = ∠ +⎪⎨

= ∠ ∠ − ∠ = ∠ −⎪⎩

(16)

Where ab1U and ab2U is positive, negative sequence amplitude of three-phase line phase.

a1θ ， a2θ is Positive, negative sequence phase angle of three-phase line voltage.

Unbalance factor can be calculated by equation (17)

2 2

2 2 22 2

1 1

3 cos + sincos + sin3

ab2 a a2 a a2 a

ab1 a1 a a1 aa1

U U U UkU U UU

θ θθ θ

= = = (17)

Difference of Phase angel is shown as

1 2 1 2 60ab ab a aθ θ θ θ θ= − = − + (18)

IV. SIMULATION ANALYSIS Algorithm is simulated by Matlab, System voltage is

5828.34 0au = ∠ ， 5998.94 -124.55bu = ∠ ，

5504.06 116.15cu = ∠ ，

Unbalance factor of voltage is 0.05k = ，

Positive sequence fundamental current of load is

1 3000 40i = ∠

Negative sequence fundamental current of load is

2 500 40i = ∠ − It contains 5、7、11 harmonic current.

Waveform of system voltage is shown as Fig.3

Fig.3 Waveform of system three-phase voltage

It will be seen from the Fig.3 that system voltage is asymmetrical and contains negative sequence.

Waveform of system load current is shown as Fig.4

Fig.4 Waveform of system three-phase load current

It will be seen from the figure that system current is asymmetrical and contains harmonic current.

Waveform of PLL is shown as Fig.5, It will be seen from the Fig.5 that the Waveform of PLL meets the request.

Fig.5 Waveform of PLL

Unbalance factor of system line-voltage is shown as Fig.6; it will be seen from the figure that the Unbalance factor of system line-voltage can be detected correctly and accuracy.

Fig.6 Unbalance factor of system line-voltage

Phase angel of line-voltage positive and negative sequence component, and its phase angle difference is shown as Fig.7

t/s0 0.04 0.08

-150

200

0.060.02

0

Phase angle of positive line voltage

Phase angle of positive line voltage

Difference of Phase angel

Fig.7 Phase angel of line-voltage positive and negative sequence component

and its phase angle difference

V. CONCLUSION Calculation process only contains few inverse cosine and

square root algorithm. DSP can realize the algorithm in a comparatively short time. Unbalance factor and phase angle difference can be achieved by this method correctly and accuracy.

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