CURRENT FILTERING IN A THREE-PHASE THREE-WIRE POWER SYSTEM AT ASYMMETRIC SINUSOIDAL VOLTAGES.

UDC 621.314

Introduction. The unbalanced load of three-phase, three-wire power systems leads to a deterioration in the quality of electricity, which manifests in the emergence of negative sequence currents and pulsations of instantaneous power, which cause additional losses on the active supports of the transmission line and asymmetry of the supply voltages in the nodes of the overall connection of loads. Passive filters on reactive elements are efficiently used to balance the asymmetrical stationary linear load, calculation of which is based on two approaches: compensation of inactive components of input currents [1, 3-5] and elimination of the pulsating component of instantaneous power [2, 6]. However, in the case of asymmetry of voltages, the use of passive reactive compensators of both types leads to the emission into the transmission line of negative sequence currents and even greater voltage asymmetry, which contradicts the approaches outlined in [7]. Effective compensation of these currents is possible only by means of active filtration [8], and parallel active filters (PAF) are dominated successive ones by power of energy losses on their own elements. To further reduce the loss of PAF they are used in combination with reactive compensators [9]. Therefore, it is relevant to study the optimal distribution of asymmetric load currents between a three-phase source, PAF and a reactive compensator, which provides the minimum power loss when consuming sinusoidal symmetric source current.

Vectors of active currents in phase coordinates.

The periodic process in the section <A, B, C> of the three-wire power supply system is determined by three-coordinate vectors of instantaneous voltage and current values

[mathematical expression not reproducible], (1)

where [omega] is the circular frequency of the three-phase source; [u.sub.A], [u.sub.B], [u.sub.C] are the phase voltages deduced from the point of artificial grounding [5], [U.sub.A], [U.sub.B], [U.sub.C] and [[phi].sub.A], [[phi].sub.B], [[phi].sub.C] are their mean square and initial phases; the periodic current vector i(t) contains higher harmonic components in the case of nonlinear loading.

According to the Fryze [1] concept, active current

[i.sub.A](t) = P/[U.sup.2] u(t) (2)

provides the same active power P as the total current i(t).

The scalar coefficients of formula (2) can be found in the time domain as integrals

[mathematical expression not reproducible];

where [and] is the sign of transposition, T = 2 [pi]/[omega] is the period.

The current determined by the formula

[i.sub.N](t) = i(t) - [i.sub.A](t), (3)

is called inactive [1], it does not carry energy to load, but causes additional losses in the transmission line.

Since the vectors of active and inactive currents are mutually orthogonal, the correct ratio for the mean square values of these currents is:

[mathematical expression not reproducible].

In the case of compensation of inactive current by means of filtration, we will have a decrease in the power loss [DELTA]P in the transmission line, which can be characterized [10] by the coefficient of gain on the power loss:

[mathematical expression not reproducible], (4)

where r is the resistance of each wire of the transmission line; [lambda] = P/S = [I.sub.A]/I is the power factor; S = UI is the full power of the three-phase system.

In the sinusoidal mode of the three-phase voltage source, the active current vector also consists of sinusoidal temporal functions, so it is expedient [5] to introduce three-dimensional complex vectors (3Dphasors) of voltage and current in the same way [5].

[mathematical expression not reproducible]. (5)

The complex vector [[bar.i].sub.1] represents the harmonic component of the fundamental frequency of the vector i(t). In the time domain, it corresponds to a vector of instantaneous values i1(t) , which differs from the vector i(t) on the vector of higher harmonics

[mathematical expression not reproducible]. (6)

However, two complex vectors of (5) completely determine the active Fryze current in the frequency domain:

[mathematical expression not reproducible], (7)

where the sign * denotes complex conjugation, [[bar.i].sub.1N] is the complex vector of inactive current of the main frequency.

In the unbalanced mode of the three-phase source, the vectors [bar.u] and [[bar.i].sub.A] contain symmetric components of the negative sequence, which the modern Standard [7] refers to inactive components of the current to be compensated. To meet the requirements of the Standard, the active current must be formed as proportional to the reference voltage vector containing only the symmetric components of the direct sequence. We represent this vector in the frequency domain as proportional to the ort of a symmetric direct sequence [5]

[mathematical expression not reproducible], (8)

then the active current vector of the direct sequence is given by the expression

[mathematical expression not reproducible], (9)

where the coefficient of proportionality is determined from the condition of providing by this current under the action of the voltage vector [bar.u] of the same active power P, as the total current i(t) .

Decomposition of load currents in symmetric coordinates. For a detailed study of the difference between the vectors defined by formulas (7) and (9), we turn to the basis of symmetric coordinates [5]. Since the symmetric component of the zero sequence of each of these vectors is zero, we multiply them on the matrix containing the transposed orts of the symmetric components of the direct and negative sequences

[mathematical expression not reproducible].

As a result, we obtain the following expressions for complex voltage vectors in symmetric coordinates:

[mathematical expression not reproducible].

The transition from two-coordinate vectors in symmetric coordinates to three-coordinate complex vectors in phase coordinates is carried out by multiplying on the matrix [F.sub.0] - [parallel][[bar.e].sub.+] [[bar.e].sub.-][parallel]. Taking into account the relationship between the matrices

[mathematical expression not reproducible],

which stores the mean square values of the quantities in symmetric and phase coordinates, we obtain the following expressions for complex active current vectors in symmetric coordinates

[mathematical expression not reproducible]. (10)

Consumption from an asymmetric sinusoidal Fryze current source does not eliminate the pulsations of active power [5]. Its instantaneous value can be determined [2] using the vector

[mathematical expression not reproducible]

by the formula

[mathematical expression not reproducible],

where [mathematical expression not reproducible] is the complex parameter of asymmetry of a three-phase source.

If the input current of the three-phase system is determined by the vector [[??].sub.A], the instantaneous value of the pulsation of the active power is

[mathematical expression not reproducible].

Thus, the active current of a direct sequence creates a pulsation of instantaneous power, the amplitude of which is in 2/(1 + [[delta].sup.2]) times less than that one generated by the active Fryze current.

The difference between the active current vectors in symmetric coordinates obtained from (10), i.e.

[mathematical expression not reproducible] (11)

defines an additional current of compensation. The active power of this current is zero:

[mathematical expression not reproducible],

and it can be realized by PAF.

Thus, in order to realize the consumption from a three-phase current source of current active current of direct sequence, the main harmonic of the compensation current should contain the following components in symmetric coordinates:

[[??].sub.1C] = [i.sub.1] - [i.sub.A+] = [i.sub.1N] + [i.sub.[+ or -]]. (12)

In the time domain, the compensation current vector also includes a higher harmonic vector:

[i.sub.C+](t) = i(t) - [i.sub.A+](t) = [i.sub.1N](t) + [i.sub.[+ or -]](t) + [i.sub.H](t). (13)

The coefficient of gain on the power loss in the formation of active current of direct sequence

[mathematical expression not reproducible] (14)

exceeds the unit provided

[lambda][square root of (1 + [[delta].sup.2.sub.-])] < 1. (15)

Taking into account the orthogonality of the vectors [i.sub.C+](t), [i.sub.A+](t) and the limitation (15), the relative mean square value of the compensation current

[mathematical expression not reproducible].

At low values of the power factor, the rms value of the compensation current increases. To reduce the power loss of the active filter, it is advisable to use PAF in combination with a passive reactive compensator [9].

Current filtering for linear stationary load. If the load is linear and not changeable in time, then in formula (13) [i.sub.H](t) = 0, and all components of the currents are sinusoidal in the time domain, then the energy processes in the system are completely determined in the basis of symmetric coordinates. In this case, the compensation current vector corresponds to (12), and the sinusoidal inactive current can be completely generated by the reactive compensator in both the symmetric and asymmetric mode of the voltage source [5]. Therefore, in order to minimize the loss of PAF power, it is advisable to distribute the currents of the reactive compensator and the active filter in the integrated substitution circuit (Fig. 1) as follows:

[mathematical expression not reproducible].

This will reduce the mean square value of the PAF currents to the value

[mathematical expression not reproducible]

We obtain the formulas for the direct calculation of the parameters of the reactive compensator for the generation of the inactive Fryze current in the asymmetric mode of the three-phase source. Let the linear stationary load be characterized by a diagonal matrix of complex conductivities

[mathematical expression not reproducible].

In [5] it was shown that the Ohm law for input vectors of current and voltage is described by an expression in symmetric coordinates

[mathematical expression not reproducible], (16)

where

[mathematical expression not reproducible].

Similar parameters of the jet compensator [mathematical expression not reproducible] for realization of the compensation current vector [mathematical expression not reproducible] are determined by the matrix-vector equation [5]:

[mathematical expression not reproducible].

We show that for a compensation currents vector in the form of an inactive Fryze current

[mathematical expression not reproducible] (17)

the parameter [b.sup.R.sub.+.] will always be a real number. For this we find the expression for the active power of the given load

[mathematical expression not reproducible],

where [mathematical expression not reproducible] and substitute this expression in formula (17) and further in (16).

After the transformation we have

[mathematical expression not reproducible]. (18)

The first coordinates of the vectors of formula (18) are real numbers, which proves the possibility of realizing the reactive compensator of the inactive Fryze current at an arbitrary combination of linear load parameters and an asymmetric source. The reactive conductivities of the compensator are determined from the system of equations (18) in the form

[mathematical expression not reproducible]. (19)

Example of simulation of currents filtration. We consider the hybrid filtration of input currents for a three-wire linear load determined by complex conductivities

[mathematical expression not reproducible],

and the asymmetry of the source is characterized by the parameter [[??].sub.-] - 0.2 j.

First of all, we define the parameters of the matrix of complex conductivities in symmetric coordinates

[mathematical expression not reproducible].

The values of the matrix of complex conductivities in accordance with (16)

[mathematical expression not reproducible].

[mathematical expression not reproducible].

We determine the parameter

[mathematical expression not reproducible]

and find the Fryze active current vector:

[mathematical expression not reproducible].

The value of the power factor

[mathematical expression not reproducible]

satisfies the condition (15):

[mathematical expression not reproducible].

Consequently, in accordance with (14), the formation of the active current of a direct sequence will bring savings in energy losses, which is estimated by the coefficient of gain

[k.sup.[DELTA]P.sub.A+] - 1/[0.553.sup.2] - 3.273.

Further by (18) we determine the parameters of the reactive compensator:

[mathematical expression not reproducible].

and form a matrix of complex conductivities with elements of reactive compensation:

[mathematical expression not reproducible].

The multiplication of this matrix on the vector of the input voltage gives the vector of the input current in the presence of a reactive compensator

[mathematical expression not reproducible],

which completely coincides with the previously determined vector of Fryze active current, which indicates the correctness of the calculation of the parameters of the compensator.

Generation of PAF of the vector of current determined by (11)

[mathematical expression not reproducible] (20)

provides the consumption from a three-phase source of active current vector of direct sequence:

[mathematical expression not reproducible].

The joint action of PAF and reactive compensator provides a total compensation current

[mathematical expression not reproducible],

thus the relative active value of PAF currents is

[mathematical expression not reproducible].

The reactive conductivities of the compensator are calculated by (19):

[b.sup.R.sub.AB] = 0.371G; [b.sup.RB.sub.C] = -0.227G;

Simulation of currents filtration in the time domain was carried out using the MATLAB model presented in Fig. 2

Parameters of the reactive elements of the compensator for G = 1 S and [omega] = 100[pi] rad/s are, respectively, [C.sub.AB] = 1.18 mF; [L.sub.BC] = 14.02 mH; [L.sub.CA] = 4.81 mH. Asymmetry of the voltages was realized by a sequential connection of sources of symmetric sinusoidal voltages of 100 V in direct sequence and 20 V in negative sequence with values of initial phases corresponding to parameters [U.sub.+] = 100[square root of (3)]B;[[??].sub.-] = 0.2 j. PAF has been simulated by dependent sources of currents controlled by source voltages. In order to generate filter currents in accordance with (11) and (20), the parameters of the dependent sources were taken equal

[G.sub.+] = -([g.sub.+] + [DELTA]g) [[delta].sub.2.sub.-] = 21.88 x [10.sup.-3] G; [G.sub.-] = [g.sub.+] + [DELTA]g = 0.547G.

The simulation results confirmed all calculated rms currents (Table 1) and the advantages of using a hybrid filter with the proposed currents distribution.

Conclusions. A principle of the distribution of compensating currents between the active filter and the reactive compensator of a three-phase three-wire power supply system with asymmetric sinusoidal voltages is proposed, which provides the consumption of symmetric sinusoidal source currents and minimizes the mean square value of the filter currents.

It is shown that the direct sequence active current provides a gain on the power of losses in accordance with (14) and creates a pulsation of power with amplitude in 2/(1 + [[delta].sup.2.sub.-]) times less than the Fryze active current.

The possibility of compensation of inactive Friyze current by reactive elements at an arbitrary combination of linear load parameters and asymmetric sinusoidal source is proved, and direct formulas for calculating the parameters of a reactive compensator for generating Fryze inactive current are obtained.

An example of simulation of currents filtration of linear stationary load showed that the combined application of PAF and reactive compensator with the proposed distribution of currents of compensation ensured reduction of energy losses in the transmission line at 3.273 times, and the relative mean square value of PAF current was 12.9 % of the rms value of the total compensating current.

doi: 10.20998/2074-272X.2018.2.11

REFERENCES

[1.] Fryze S. Active, reactive and apparent power in circuits with nonsinusoidal voltage and current. Przeglqd Elektrotechniczny, 1931, no.7, 8, pp. 193-203.

[2.] Shidlovskii A.K., Kuznetsov V.G. Povysheniye kachestva elektroenergii v elektricheskikh setyakh [Improving of the power quality in electrical networks]. Kiev: Naukova Dumka Publ., 1985. 268 p.(Rus).

[3.] Hanzelka Z. Mitigation of voltage unbalance. Available at: http://www.leonardo-energy.org/chapter-5-mitigation-voltage-unbalance (accessed 22 May 2016).

[4.] Czarnecki L.S., Haley P.M. Unbalanced Power in Four-Wire Systems and Its Reactive Compensation. IEEE Transactions on Power Delivery, 2015, vol.30, no.1, pp. 53-63. doi: 10.1109/TPWRD.2014.2314599.

[5.] Sirotin Iu.A. Fryze's compensator and Fortescue transformation. Przeglqd Elektrotechniczny, 2011, no.1, pp. 101-106.

[6.] Sirotin Iu.A. Non-pulsed mode of supply in a three-phase system at asymmetrical voltage. Przeglqd Elektrotechniczny, 2013, no.7, pp. 54-58.

[7.] IEEE Std. 1459-2010. Definitions for the measurement of electric power quantities under sinusoidal non-sinusoidal, balanced or unbalanced conditions. doi: 10.1109/IEEESTD.2010.5439063.

[8.] Salmeron Revuelta P., Perez Litran S., Prieto Thomas J. Active power line conditioners design, simulation and implementation for improving power quality. Elsevier Inc.: Academic Press, 2016. 436 p.

[9.] Artemenko M.Y., Batrak L.M., Polishchuk S.Y., Mykhalskyi V.M., Shapoval I.A. Reactive compensation of non-active power in hybrid shunt filter of three-phase four-wire system at random load. Proceedings of 2016 2nd International conference on Intelligent Energy and Power Systems (IEPS). Kiev, 2016. doi: 10.1109/IEPS.2016.7521863.

[10.] Artemenko M.Y., Polishchuk S.Y., Mykhalskyi V.M., Shapoval I.A. Apparent power decompositions of the three phase power supply system to develop control algorithms of shunt active filter. Proceedings of the IEEE pirst Ukraine conference on Electrical and computer Engineering (UKRcON), 2017, pp. 495-499. doi: 10.1109/UKRCON.2017.8100537.

M.Yu. Artemenko (1), Doctor of Technical Science, Professor,

L.M. Batrak (1), candidate of Technical Science,

S.Y. Polishchuk (2), candidate of Technical Science,

(1) National Technical University of Ukraine <<Igor Sikorsky Kyiv Polytechnic Institute>>, 37, Prosp. Peremohy, Kyiv, Ukraine, 03056, e-mail: artemenko_m_ju@ukr.net, batrakln5@gmail.com

(2) The Institute of Electrodynamics of the NAS of Ukraine, 56, prospekt Peremogy, Kiev-57, 03680, Ukraine, e-mail: polischuk@ied.org.ua

Caption: Fig. 1. The substitution circuit

Caption: Fig. 2
```Table 1

[I.sup.2]   [I.sup.2.sub.SR]

Calculated    31751           9333
Measured       31920           9401

[I.sup.2.sub.SFR]   [I.sup.2.sub.F]

Calculated        9713                370
Measured           9712                371
```
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