THE FOLLOWING SPECIFICATION PARTICULARLY DESCRIBES THIS
INVENTION AND THE MANNER IN WHICH IT IS TO BE PERFORMED

This invention relates to a method of construction of unit vector using single grid voltage,

Power Quality is determined by variables, such as harmonics in current, power factor in the case of single phase systems. To maintain quality of electrical power to the consumers, many power quality solutions such as, Active Filter, STATic COMpensator (STATCOM) are available. In the case of power factor and harmonic compensation, grid tied three-phase or single-phase voltage source inverter based STATCOM is one of the versatile solutions. But in this system most of the popular control strategies demand generation of "unit vector" based on the grid voltage for vector orientation. For three phase system many method exists, including the one developed by Centre for Development of Advanced Computing (CDAC), for the construction of unit vector.

Construction of unit vectors for single phase system becomes complicated because only one grid voltage exists. Look up table is the existing and commonly used, to generate unit vectors for single phase system. This requires identification of zero crossing points. But presence of harmonics in grid voltages and the level of noise in the grid voltage sensing circuits will make estimation of zero crossing point difficult. In this invention a simple and efficient method to construct the unit vector for single-phase systems is proposed.
This unit vector always synchronizes with grid voltage irrespective of variation in grid frequency. This unit vector construction method can be implemented in digital platform, like Digital Signal Processor (DSP) or Field Programmable Gate Array (FPGA), effortlessly.

According to the prior art, the passive elements present in the electrical load are resistor, inductor and capacitor. The power flow from generating station to this passive load are mainly divided into two; active power and reactive power. Here active power will get an energy transformation from electrical to another form; light, heat, mechanical etc. But the reactive power is a circulating power, where it circulates between generating station and load. The aim of the STATCOM is to allow the circulation of reactive power locally, between STATCOM and load. Thus STATCOM brings the global circulation into local circulation. To do all this, it is important to distinguish between active and reactive power from the total power. To implement this, the in-phase and quadrature component of the current is separated and power computed. To separate out this from the total current we must know the angle information of the total current or where the angle 0° lies in the total current in real time. The unit vectors give this information. The unit vectors are two fundamental unity magnitude sinusoidal quantities, which are displaced by 90° from each other. Here one of the unit vectors, called "cosp", will be always in phase with grid voltage, i.e. in phase with active current. The other component is called "sinp". Therefore, the unit vector gives the phase information. If the unit vector consists of harmonics apart from fundamental then the active and reactive components extracted from the load will also contains the harmonics. Hence to avoid this, the unit vectors must contain only fundamental components. By using this unit vector, the active and reactive power due to fundamental only can be deduced even if grid voltage and load current are polluted by harmonics, since unit vector consists of only fundamental component.

The present known methods of estimating unit vectors in single phase systems are based on look up tables. For synchronizing look up table with the actual grid voltage, identification of zero crossing point of grid voltage is essential, presently Phase Locked Loop (PLL) does this job. The voltage harmonics and the noise restrict direct use of the voltage signals in the zero crossing detector. The harmonics are filtered using hardware band-pass filter. But, it is very difficult to design such a filter with precise choice of cut-off frequency that would separate out the fundamental without appreciable phase errors. Some work has been reported on the use of PLL to remove the effect of harmonics and noise in the grid voltage. But the design of a high performance PLL is not so easy when various non-idealities like multiple zero crossing in the grid voltage are occurring.

This invention is a simple and efficient method of unit vector construction without using the prior art PLL or band-pass filter, for single-phase systems. The novel feature of this method helps to minimize the effect of harmonics and noise present in the grid voltage feedback and gives automatic synchronisation. This unit vector construction method can be implemented on DSP or FPGA, effortlessly.

The description is divided into following parts:
> Construction of unit vectors
> Effect of grid frequency variation on unit vector
> Solution to mitigate the effect of frequency variation on unit vector

Let the grid phase voltages be,
VR = Vg sin(ωst) (1)

Sending VR through a low pass filter of corner frequency ωc1 as shown in figure 1, the output, FR, will be,

(2)
Equation (2) can be rewritten by substituting Laplace Transform of VR as,
(3)

Equation (3) is rewritten by replacing PART I by partial fraction as,
(4)

Taking Inverse Laplace Transform of equation (4),
(5)

In steady state the equation (5) can be rewritten as,

The above equation can be simplified as,
(6)
where
(7)

Sending the signal FR (output of low pass filter), again through a low pass filter of corner frequency ωc1, as shown in figure 2, the output, FR, from figure 2, will be,


Substituting equation (3) for FR(s) and solving using partial fraction gives,
Equation

Taking inverse Laplace Transform of equation (8),
Equation

In steady state the equation (9) can be rewritten as,
Equation

The above equation can be simplified as,
Equation (10)

where,
Equation (11)

Now let us subtract FR'(t) given in equation (9) from FR(t) given in equation (5), as shown in Fig. 3 result in
Equation

In steady state the above equation can be simplified as,
Equation

The above equation can be simplified as,
Equation (12)

where, cosψ —(13)

Inference from the result:

The signal FR'(t) (equation (10)) and FR"(t) (equation (12)) are always 90° irrespective of change in grid frequency, ωs.

Construction of unit vectors from FR'(t) and FR"(t):

FIGURE 3 SHOWS THE BLOCK DIAGRAM OF CONSTRUCTION OF UNIT VECTORS.

Since FR'(t) and FR"(t) are displaced by 90°, FR'(t) reaches the peak when FR"(t) crosses the zero value. Similarly, FR"(t) reaches peak at the zero crossing of FR'(t). Therefore, by identifying the zero crossing point of FR'(t) and FR"(t), their peak can be measured. Since both signals are coming through the integrator the noise will be eliminated and clean zero crossing point will be obtained. The absolute value of peak of FR' is given by,
Equation (14)

Dividing FR'(t) with the absolute value of its peak gives,
Equation (15)

Similarly, the absolute value of peak of FR"(t) is given by,
Equation (16)

Dividing FR"(t) with the absolute value of its peak gives,
Equation (17)

From equation (11) and (13), it can be seen that when ωc = ωs (i.e. corner frequency of low pass filter is equal to grid frequency) the angle ψ= 0. Substituting this in equation (17) and (15), the unit vector is given by,
and (18)

Construction of unit vectors from the signal FR' and FR" are explained in figure 3. But the signal FR" is the arithmetic sum of FR and FR'. Therefore, the characteristics of unit vector generation block is decided by FR and FR'. Figure 4 shows that bode-plot of the signal FR and FR'. Figure shows the bandwidth of the system as, 200rad/sec (=31.8Hz).

Effect of grid frequency variation on unit vector

♦ Effect of variation of grid frequency on FR':

a) Change in magnitude of FR' due to change in grid frequency cos
Let the corner frequency of both low pass filter is kept equal to rated grid frequency, ωc=ωs, and let grid frequency varies as ωs=ωs+∆ω, then the variation in the magnitude of FR' is given by,
Equation (19)

where the following assumptions are made:

Compare to 1,1/2(Δω/ωs)= 0 and for binomial expansion — «1.

b) The phase error between grid voltage and unit vector, FR', due to change in grid frequency ωs

Let the corner frequency of both low pass filter is kept equal to rated grid frequency, ωc=ωs, and let grid frequency varies as ωs= ωs+Δ ω, then the variation in the phase of FR' is given by
Equation (20)

where the following assumption is made: Compare t

♦ Effect of variation of grid frequency on FR"

a) Change in magnitude of FR" due to change in grid frequency Δs
Let the corner frequency of both low pass filter is kept equal to rated grid frequency, ωc=ωs, and let grid frequency varies as ωs=ωc +Δω, then the variation in the magnitude of FR" is given by,
Equation (21)

where the following assumption is made: Compare to =0

b) The phase error between grid voltage and unit vector, FR", due to change in grid frequency ωs

Let the corner frequency of both low pass filter is kept equal to rated grid frequency, ωc=ωs, and let grid frequency varies as ωs=ωs+Δω then the variation in the phase of FR" is given by.
Equation (22)

where the followina assumption is made: Compare to =0

The phase angle error in FR' and FR' due to variation in grid frequency are identical and is given in equation (22) and it is drawn in figure 6 and figure 7. It shows that for a grid frequency error of 10% the phase angle error is 6°.

Inference from the result:

From equation (19) and equation (21) it can see that, the peak of FR' changes with variation in grid frequency whereas, the peak of FR" remains more or less constant for the change in grid frequency.

The signal FR'(t) (equation 10) and FR"(t) (equation 12) are always 90° irrespective of change in grid frequency, ωs. Therefore the peak of the signal can easily find out from the zero crossing of other signal and hence the two signal of unit magnitude can be deduced irrespective of change in grid frequency.

The angle ψ is dependent on grid frequency. Therefore, as grid frequency varies the phase angle between grid voltage and cosp (which is expected to be in phase with grid voltage) increases. This demand for a compensation scheme, which can keep the cosp in phase with grid voltage irrespective of frequency variation.

The method described here for the mitigation of effect of frequency variation, does its job (keeping cosp in phase with grid voltage) for all range of frequencies.

From equation (11) and (13), it can be seen that when ωc = ωs the angle ψ = 0. But when grid frequency varies the angle ψ, no longer be zero. Therefore, the best way to keep the angle of unit vector to zero is that; subtract an angle ψ from ψ such that (ψ-ψ) is always zero. As a step towards to this consider the following equation.
Equation (23)
Equation (24)

For the implementation of the equations (23) and (24), Vg cosψ and Vg sinψ must be known. From equations (13) and (16), the following equation can be obtained.
Equation (25)

Adding equations (14) and (16) gives,
Equation (26)

Similarly subtracting equations (14) from equation (16) gives,
Equation (27)

Multiplying equations (26) and (27) and dividing by equation (14) gives,

Rewriting the above equation,
Equation (28)

Substituting equations (25) and (28) in equation (23) and simplifying using equation (12) and (16) gives,
Equation (29)

Similarly, substituting equations (25) and (28) in equation (24) gives,
Equation --(30)

Inference from the Result:

> The signal U1 is same as grid voltage both in magnitude as well as in phase
and U2 lag U1 by 90°. Therefore, the magnitude and phase of both U1 and U2 are independent of change in grid frequency, ωs.

> Since U1 and U2 are displaced by 90°, U1 reaches the peak when U2 crosses the zero value. Similarly, U2 reaches peak at the zero crossing of U1. Therefore, by identifying the zero crossing point of U1 and U2, their peak can be measured. The absolute value of peak of U1 is Vg

> Dividing U1 and U2 with its peak gives,
Equation

Figure 5 shows block diagram of construction of unit vectors with grid frequency compensation.

Fig. 6 shows comparison of % change in grid frequency vs. phase error with and without compensation for grid frequency variation.

Fig. 7 shows expanded view of figure 6

Experimental results

Test procedure

Since the grid frequency cannot be varied, it is simulated using the function generator. The block diagram given in figure 5 is build using assembly language in digital controller board.

The grid voltage (simulated using function generator) is read through Analog to Digital converter (ADC) into the processor. The block diagram (given in figure 5) implemented in DSP will read this grid voltage and generate unit vectors. To view the generated unit vector, it is send to the Digital to Analog Converter (DAC). The oscilloscope can be connected to the DAC to observe the unit vectors.

Next step is to simulate the grid frequency variation and observe its effect on unit vectors. For this, while running the program, using function generator the frequency is changed from 50Hz to 45Hz (10% variation) in step. Observe the effect on unit vectors.

Test result

The fundamental frequency is changed from 50Hz to 45Hz in step at time, t = t1. Figure 8 shows the waveform of unit vector implemented without any compensation scheme to eliminate the effect of grid frequency variation. Therefore, it can see that after time t = t1, the unit vector is not in phase with the grid voltage. The waveform clearly shows the zero phase error before t = t1 and an appreciable phase error after t = t1.

The figure 9 shows the waveform of unit vector implemented by including compensation for frequency variation. Here also the grid frequency is changed at t = t1. But from the waveform it is clear that it keeps the phase error as zero before t = t1 and after t = t1. This shows that the unit vector controller keeps the phase error equal to zero even if the grid frequency varies.

It is noreworthy that:

1. Unit vectors are generated using single grid voltage is a novel concept without using PLL or band-pass filter technique

2. The variation of the phase angle between grid voltage and unit vectors due to grid frequency variation is mitigated by using this innovative method. This method will work in wider frequency variation of input voltage. Hence it is useful for drives application with wider frequency variation.

3. This method can easily and accurately implemented in digital platform like, Digital Signal Processor or Field Programmable Gate Array

4. This method is free from multiple zero crossings present in input voltage, due to harmonics.

We Claim:

1. A method of construction of unit vector using single grid voltage without PLL or band pass filter technique where the grid voltage is VR = Vg sin(ωst); sending VR through a low pass filter of corner frequency ωc, as in the circuit in figure 1, the output, FR being sent again through a low pass filter of corner frequency ωc, as in the circuit in figure 2, to obtain the signals FR'(t) and FR'(t) which are always at 90° irrespective of change in grid frequency, cos. and constructing unit vectors therefrom as in the circuit in Fig. 3.

2. A method of construction of unit vector using single grid voltage substantially as herein described and illustrated with reference to the accompanying drawings.