Numerical analysis of mathematical models of the factual contribution distribution in asymmetry and deviation of voltage at the common coupling points of energy supply systems.

UDC 621.317

Introduction. Non-compliance of power quality (PQ) to established standards are the reasons the marriage of products, equipment damage, and additional power losses both in consumers and electrical energy (EE) suppliers . According to some estimates  annual economical losses in several countries due to the low PQ arise USD 10-20 bln. For certain sectors of production decrease in PQ can cause damage to 3.800.000 EUR per event . Obviously, if this happens, it becomes a question of determining those responsible for lowering the PQ and compensation of economic damages to the injured party. The answer to it is to solve the problem of the distribution of factual contributions (FC) of sources of distortion (SD) in the distortion of the voltage at the point of common coupling (PCC) .

Problem definition. One of the new directions of development of the FC SD distribution methods in voltage distortion at PCC involves the use of mathematical models, drawn up in phase coordinates, given the distributed nature of SD in the power supply system (PSS), which are based on the principles of superposition  and exclusion .

A mathematical model of the distribution of FC of linear SD (undistorting the sinusoidal voltage waveform) in the voltage distortion is based on the principle of superposition, involves the expansion of distorting parts of voltages in each PCC from the activities of all SD according to the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where A is the incidence matrix; [Y.sub.undis] is the matrix of undistorted nodal conductivities of the PSS and EE consumers; [I.sub.dis i] is column matrix of distorted currents characterizing the i-th active or passive element with SD.

A mathematical model of the distribution of FC of linear SD in the voltage distortion is based on the principle of exclusion, involves determining the distorting of the voltage in each PCC, introduced by the i-th SD, by the following expression:

[U.sub.dis i] = [U.sub.dis] - [U.sup.ex SD 1.sub.dis], (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the matrix of distorted parts of voltages in the PCC from common action of all SD; [U.sup.ex SD i.sub.dis] is the matrix of distorted parts of voltages in the PCC with excluded distorted part of the i-th SD.

To check the adequacy and the comparison of the proposed new mathematical models of the distribution of FC of linear SD in voltage distortion at PCC it is necessary to perform the numerical analysis.

The goal of the investigation. To perform numerical analysis of mathematical models of distribution of FC of linear SD in voltage distortion in the PCC, based on the principles of superposition and exclusion.

Results of the investigation. We consider the PSS of seven EE power consumers (C) (see Fig. 1) consisting of a energy supply (ES), generalized electrical network (EN), a power transformer (T) and three overhead lines (OL).

The parameters of equivalent circuits of the elements of the considered PSS and EE consumers reduced to the voltage of 380 V are the following. The voltage on the buses of the ES: [[U.bar].sup.ES.sub.A] = 232 [angle] 0[degrees] V; [[U.bar].sup.ES.sub.B] = 232 [angle] 240[degrees] V; [[U.bar].sup.ES.sub.C] = 232 [angle] 120[degrees] V. Equivalent resistances of the generalized EN: [[Z.bar].sup.EN.sub.A] = 0.008 + j0.148 [ohm]; [[Z.bar].sup.EN.sub.B] = 0.008 + j0.04[ohm]; [[Z.bar].sup.EN.sub.C] = 0.008 + j0.056 [ohm]. Equivalent resistances of EE consumers:
i   [[Z.bar].sup.Ci.sub.A],   [[Z.bar].sup.Ci.sub.B],
[ohm]                     [ohm]

1         7.2 + j3.7                6.5 + j3.0
2         6.9 + j5.2                7.7 + j3.7
3         13.7+ j5.2                15.1 + j4.7
4         9.7 + j3.2                8.9 + j3.1
5         6.3 + j1.9                6.8 + j1.4
6         17.2 + j7.1               19.8 + j8.1
7         13.9 + j3.9               14.9 + j4.9

i   [[Z.bar].sup.Ci.sub.C],
[ohm]

1         6.74+ j3.5
2         6.87 + j3.9
3         14.2 + j4.6
4         10.5 + j3.5
5         7.2 + j1.9
6         15.6 + j6.5
7         15.1 + j4.5

Resistance of the power transformer:

[[Z.bar].sub.ph T] = 0.00105 + j0.0072 [ohm]. Conductivity of the power transformer: [[Y.bar].sub.pT] = 0.001375 + j0.0021 S. Specific resistance of the OL: [[Z.bar].sup.A25.sub.s] = 1.26 + j0.34 [ohm]/km; [[Z.bar].sup.A16.sub.s] = = 1.97 + j0.345 [ohm].

According to calculations carried out, the steady state operation of the considered PSS is characterized by parameters given in the Table 1. It follows from them that in the PCC No. 5, to which EE consumers C4 and C5 are connected, the coefficient of asymmetry of voltage by the zero sequence [K.sub.0U] and steady state deviation of the voltage [delta][U.sub.y] exceed the normal allowable values . On this basis, for the PCC No. 5 we determine FC of all SD in the distortion of its voltage.

According to the mathematical models (1) and (2) in the equivalent circuit of the individual elements of the PSS and EE customers distorted part of SD must be extracted and identified . If SD is a passive longitudinal element, its equivalent circuit will be determined by a series connection of two resistances, one of which describes the undistorted part ([[Z.bar].sup.undis.sub.el]), and the other--distorting part ([[Z.bar].sup.dis.sub.el]). If SD is a passive cross element, its equivalent circuit will be determined by the parallel connection of two conductivities [[Y.bar].sup.undis.sub.el] and [[Y.bar].sup.dis.sub.el]. For SD, which is an active element the equivalent circuit is provided as a serial connection of two EMF ([[E.bar].sup.undis.sub.ES] and [[E.bar].sup.dis.sub.ES]).

Determination of distorted part of any SD by voltage asymmetry is based on the deflection of its parameters from some symmetric state, for example for passive SD:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

The basis of determination of distorted parts of the SD by the deflection voltage are the principles of compliance with the required voltage levels on the ES buses and in control nodes of the PSS voltage as well as load of the individual elements of EN and EE consumers not exceeding permissible or maximum permissible values for them.

So, in the case of excess power of consumer above the maximum permitted its distorted part will be characterized by the following conductivity:

[[Y.bar].sup.dis.sub.ph Ci] = [DELTA][[S.bar].sup.*.sub.ph Ci]/[U.sup.2.sub.ph Ci], (4)

where [DELTA][[S.bar].sub.ph Ci] a is the part of the phase power of the i-th EE consumer, exceeding its maximum allowed value; [[U.bar].sub.ph Ci] is the phase voltage of the i-th EE consumer.

Distortion of the actual voltage on the ES buses ([[E.bar].sup.fact.sub.ph ES]) from the value required by the PSS operation mode ([[E.bar].sup.undis.sub.ph ES]) will characterize its distorted part:

[[E.bar].sup.dis.sub.ph ES] = [[E.bar].sup.undis.sub.ph ES] - [[E.bar].sup.fac.sub.ph ES] (5)

[[E.bar].sup.dis.sub.ph ES] = [[E.bar].sup.dis.sub.ph ES] - [[E.bar].sup.fact.sub.ph ES]. (5)

In our case, the maximum permitted power of electrical loads of each EE consumers are listed in Table 2. On operation mode conditions of the PSS the voltage on the ES buses must be maintained as 1,065 x [U.sub.nom]. Voltage regulation by the power transformer is not performed.

On the basis of the above expressions and additional information about the PSS operation, distorted and undistorted parameters of all its SD are determined (see Table 3 and Table 5). According to the mathematical models (1) and (2) the distribution of FC of linear SD in distortion of the voltage in the PCC No. 5 corresponds to the data given in Table 4 and Table 6. For a more visual representation these results are presented in Fig. 3 in graphical form.

We estimate the divergence of results for the distribution of FC of linear SD in distortion of voltages in the PCC No. 5, obtained on the basis of (1) and (2) mathematical models, by the relative root-mean-square deviation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where n is the total number of SD; symbols 1 and 2 correspond the mathematical model (1) and (2), respectively.

In our case, [delta] by the voltage asymmetry is 6.4 x [10.sup.-5]%, and by the voltage deviation - 8.1 x [10.sup.-4]%. These values lead to the conclusion of equivalence of (1) and (2) mathematical models and, consequently, their arbitrary choice for the solution of the problem of the distribution of FC of linear SD in the voltage distortion in the PCC.

We analyze the obtained FC distributions. Firstly, FC distribution of linear SD in the voltage distortion in the PCC is a vector (two-dimensional) quantity. It is obvious that in such a form the FC can not be used for the distribution of financial compensations for the reduction of PQ and it is necessary to provide a corresponding one-dimensional criterion. We set as the basis of the one-dimensional criterion of the FC distribution the FC scalar product in vector form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Such an approach means that this criterion assesses the FC by projections of vector FC [[U.bar].sup.PCC.sub.dis] on the total vector of the voltage distortion in the PCC [[U.bar].sup.PCC.sub/dis]. Omitting the module in the expression (7) it is possible to additionally take into account the effect of the voltage distortion compensation introduced by separate SD. In our case, this effect is most clearly demonstrated by the vectors [[U.bar].sup.FC C4.sub.0] and [[U.bar].sup.FC C6.sub.0] (Fig. 3,a).

Secondly, in the voltage distortion in the PCC No. 5 all SD PSS take part. Here, FC SD outside the PCC No. 5 may be comparable to or greater than FC SD connected directly to the PCC considered.

Third, the discrepancy of the FC ([[U.bar].sup.[DELTA].sub.dis] = = [[U.bar].sup.PCC No. 5.sub.dis] - [[U.bar].sup.[SIGMA]FC.sub.dis]) between all SD (Fig. 3,b) which is caused by not taking into account or the inaccuracy of the determination of distorted parts of some SD is possible. To eliminate it is enough to group the unknown or illdefined SD, belonging to the same subject of the energy market, for example, the PSS, and to determine their total FC by excluding from the total distortion level voltages in the PCC:

[[U.bar].sup.FC PSS.sub.dis] = [[U.bar].sup.PCC No. 5.sub.dis] - [[U.bar].sup.[SIGMA]FC Ci.sub.dis]. (8)

On the basis of the above, a one-dimensional distribution of the FC by the voltage asymmetry in the PCC No. 5 will be:
[FC.sub.DSi], %          EN            C1      C2      C3
8.6 x [10.sup.-6]    1.0    13.43   5.36
[PI]4         [PI]5   [PI]6   [PI]7
13.31          55.24   8.77     2.9

Assuming that distorted parts of the SD part from the side of the EE customers are identified accurately and distorted parts of the SD from the side of PSS elements are grouped, the one-dimensional distribution of the FC by the voltage deviation in the considered PCC will be:
[FC.sub.DSi], %    PSS     C1      C2      C3
80.64   4.46    10.56     0
[PI]4   [PI]5   [PI]6   [PI]7
4.34      0       0       0

The obtained results show that most part of the payments for compensation of economic losses for the subjects of the energy market in the PCC No. 5 from the voltage asymmetry falls on EE customers C5 (55.24%) and C2 (13.43%), and from the voltage deviations--on the PSS (80.64%) and EE C2 customer (10.56%).

Conclusions. Mathematical models of determination of FC of linear sources of SD in the voltage distortion in the PCC, based on the principles of superposition and exclusion, are equivalent. To assess the degree of participation of each SD in the voltage distortion in the PCC and the distribution of financial compensation to the injured party between all SD, a one-dimensional criterion of FC distribution based on the scalar product of vectors is developed. Not accounting the group of SD, belonging to one subject of the energy market, permits to determine their total FC as the discrepancy of the distribution of FC between all SD.

doi: 10.20998/2074-272X.2016.2.09

REFERENCES

[1.] Shidlovskiy A.K., Kuznetsov V.G., Nikolaenko V.G. Ekonomicheskaia otsenka posledstviy s nizheniia kachestva elektricheskoy energii v sovremennukh sistemakh elektrosnabzheniia [Economic evaluation of the effects of reducing the quality of electricity in modern power supply systems]. Kiev, IED AN USSR Publ., 1981.49 p. (Rus).

[2.] Zhezhelenko I.V., Saenko Yu.L. Kachestvo elektroenergii na promushlennukh predpriiatiiakh [Power quality in industrial plants]. Moscow, Energoatomizdat Publ., 2005. 261 p. (Rus).

[3.] Chepmen D. Price of low power quality. Energosberezhenie - Energy Saving, 2004, no.1, pp. 66-69. (Rus).

[4.] Sayenko Yu., Kalyuzhniy D. Analytical methods for determination of the factual contributions impact of the objects connected to power system on the distortion of symmetry and sinusoidal waveform of voltages. Przeglad Elektrotechniczny, 2015, vol.11, pp. 81-85. doi: 10.15199/48.2015.11.23.

[5.] Saenko Yu.L., Kalyuzhnyiy D.N. Superposition principle in mathematical models of the factual contribution distribution of linear sources of distortion in voltage distortion at the point of common coupling. Elektrifikatsiya transporta--Electrification of transport, 2015, no.10, pp. 123-133. (Rus).

[6.] Saenko Yu. L., Kalyuzhnyiy D. N. Exclusion principle in mathematical models of distribution of the factual contribution of the linear source of distortion in voltage distortion at the point of common coupling. Visnik Harkivskogo natsionalnogo tehnichnogo universitetu silskogo gospodarstva imeni Petra Vasilenka--Bulletin of Kharkiv Petro Vasylenko National Technical University of Agriculture, 2015, no.167, pp. 31-33. (Rus).

[7.] GOST 13109-97. Elektricheskaya energiya. Trebovaniya k kachestvu elektricheskoy energii v elektricheskih setyah obschego naznacheniya [State Standard 13109-97. Electric Energy. Requirements for the power quality in electric networks of general purpose]. Kiev, Gosstandart Ukrainy Publ., 1999. 33 p. (Rus).

[8.] Kalyuzhniy D.N. Presentation of linear sources of distortion in the mathematical models of their factual contribution distribution in voltage distortion at the point of common coupling. Energosberezhenie. Energetika. Energoaudit--Energy saving. Power engineering. Energy audit, 2015, no.11, pp. 19-25. (Rus).

Yu. L. Sayenko (1), Doctor of Technical Science, Professor, D.N. Kalyuzhniy (2), Candidate of Technical Science, Associate Professor,

(1) Pryazovskyi State Technical University, 7, Universytets'ka Str., Mariupol, 87500, Ukraine, phone +380 629 446551, e-mail: YuriSayenko@mail.ru.

(2) O.M. Beketov National University of Urban Economy in Kharkiv, 12, Revolution Str., Kharkiv, 61002, Ukraine, phone +380 50 5606835, e-mail: KalyuzhniyDN@mail.ru

Caption: Fig. 1. Power supply system of seven consumers of EE

Caption: Fig. 2. Equivalent circuit of the PSS and EE consumers in the single-phase version

Caption: Fig. 3. Graphical representation of the distribution of FC of linear SD in voltage distortion in the PCC No. 5 on the base of the mathematical model: (1): a) by the voltage asymmetry; b) by the voltage deviation
Table 1

Parameters of operation modes of the PSS and indicator of PQ in PCC

Parameters of the                      PCC PSS
operation mode of
the PSS                      1 *                    2 *

[[U.bar].sub.A]       401.84 [angle] 30    395.33 [angle] 28.67
([[U.bar].sub.AB]),
V

[[U.bar].sub.B]       401.84 [angle] -90   396.48 [angle] -91.57
([[U.bar].sub.BC]),
V

[[U.bar].sub.C]       401.84 [angle] 150   394.48 [angle] 148.4
([[U.bar].sub.BA]),
V

[delta][U.sub.y], %          5.75                  4.06

[K.sub.2U], %                 0                    0.293

[K.sub.OU], %                 0                      0

Parameters of the                           PCC PSS
operation mode of
the PSS                         3                        4

[[U.bar].sub.A]        224.09 [angle] -1.17     210.11 [angle] -0.88
([[U.bar].sub.AB]),
V

[[U.bar].sub.B]       228.26 [angle] -122.55   215.2 [angle] -122.42
([[U.bar].sub.BC]),
V

[[U.bar].sub.C]       231.05 [angle] 118.53    217.66 [angle] 11S.91
([[U.bar].sub.BA]),
V

[delta][U.sub.y], %            3 54                    -2.59

[K.sub.2U], %                  0.29                     0.35

[K.sub.OU], %                  1.6                      1.86

Parameters of the                            PCC PSS
operation mode of
the PSS                         5                        6

[[U.bar].sub.A]        202.01 [angle] -0.41     206.53 [angle] -0.85
([[U.bar].sub.AB]),
V

[[U.bar].sub.B]       206.45 [angle] -122.35   212.36 [angle] -122.23
([[U.bar].sub.BC]),
V

[[U.bar].sub.C]       210.76 [angle] 119.06     213.7 [angle] 119.14
([[U.bar].sub.BA]),
V

[delta][U.sub.y], %           -6.19                    -4,16

[K.sub.2U], %                  0.37                     0.39

[K.sub.OU], %                  2.21                     1.82

* Note. For the PCC No. 1, 2 the values of the
line voltages are indicated.

Table 2
Maximal permitted power of the electrical
load of the EE consumers

Power             C1        C2         C3          C4

[S.sub.max],   S + j2.5   S + j2   3.5 + j1.5   4 + j1.5
kVA

Power             C5         C6          C7

[S.sub.max],   7 + j2.2   3 + j1.3   3.5 + j1.5
kVA

Table 3
Equivalent circuit parameters of the PSS elements and EE consumers
for the FC distribution by the voltage asymmetry

PSS element and power consumers

Equivalent circuit               EN           T *            T *
(winding)      (magnetic)

[[Z.bar.].sup.undis.sub.A],    0.008        0 00105       (0.001375
[ohm]                         +j0.048       +j0.0072      -j0.0021)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],         0             0              0
[ohm]
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.undis.sub.B],    0.008        0.00105       (0.001375
[ohm])                        +j-0.048      +j0.0072      -j0.0021)
([[Y.bar].sup.undis.sub.B],
S)

([[Z.bar].sup.dis.sub.B],     -j0.008          0              0
[ohm])
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],    0.008        0.00105       (0.001375
[ohm])                        +j0.048       +j0.0072      -j0.0021)
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],      j0.008          0              0
[ohm])
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]          --      [approximately      --
equal to] 0

PSS element and power consumers

Equivalent circuit             OL1 *      OL2 *      OL3 *

[[Z.bar.].sup.undis.sub.A],   0.1134     0.1576      0.1379
[ohm]                         +j0.031   +j0.0276    +j0.0242
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],         0          0          0
[ohm]
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.undis.sub.B],   0.1134     0.1576      0,1379
[ohm])                        +j0.031   +j0.02 76   +j0.0242
([[Y.bar].sup.undis.sub.B],
S)

([[Z.bar].sup.dis.sub.B],        0          0          0
[ohm])
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],   0.1134     0.1576      0.1379
[ohm])                        +j0.031   +j0.0276    +j0.0242
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],        0          0          0
[ohm])
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        0.1134     0.1576      0.1379
+j0.031   +j0.0276    +j0.0242

PSS element and power consumers

Equivalent circuit                  Cl               C2

[[Z.bar.].sup.undis.sub.A],       (0.118           (0.109
[ohm]                            -j0.059)         -j0.058)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],        (-0.00798         (0.00227
[ohm]                           +j0.0021)        -j0.00309)
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.undis.sub.B],       (0.118           (0.109
[ohm])                           -j0.059)         -j0.058)
([[Y.bar].sup.undis.sub.B],
S)

([[Z.bar].sup.dis.sub.B],       (0.008973        (-0.00342
[ohm])                          -j0.00029)      -j0.00 745)
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],       (0.118           (0.109
[ohm])                           -j0.059)         -j0,058)
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],       (-0.00099         (0.00153
[ohm])                          -j0.0021)        -j0.00435)
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately   [approximately
equal to] 0      equal to] 0

PSS element and power consumers

Equivalent circuit                  C3               C4

[[Z.bar.].sup.undis.sub.A],       (0.063           (0.093
[ohm]                            -j0.021)         -j0.031)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],        (0.001164        (0.000011
[ohm]                           -j0.00299)       -j0.00071)
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.undis.sub.B],       (0.063           (0.093
[ohm])                           -j0.021)         -j0.031)
([[Y.bar].sup.undis.sub.B],
S)

([[Z.bar].sup.dis.sub.B],       (-0.00261        (0.007239
[ohm])                          +j0.0024)        -j0.00352)
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],       (0.063           (0.093
[ohm])                           -j0.021)         -j0.031)
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],        (0.00109        (-0.00725
[ohm])                          +j0.0006)        +j0.0028)
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately   [approximately
equal to] 0      equal to] 0

PSS element and power consumers

Equivalent circuit                  C5               C6

[[Z.bar.].sup.undis.sub.A],       (0.139           (0.049
[ohm]                            -j0.036)         -j0.02)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],         (0.00669        (0.000488
[ohm]                           -j0.00815)       -j0.00018)
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.undis.sub.B],       (0.139           (0.049
[ohm])                           -j0.036)         -j0.02)
([[Y.bar].sup.undis.sub.B],
S)

([[Z.bar].sup.dis.sub.B],       (0.002271        (-0.00592
[ohm])                          +j0.0067)        +j0.0026)
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],       (0.139           (0.049
[ohm])                           -j0.036)         -j0.02)
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],       (-0.00896         (0.00543
[ohm])                          +j0.0015)        -j0.00243)
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately   [approximately
equal to] 0      equal to] 0

PSS element and
power consumers

Equivalent circuit                  C7

[[Z.bar.].sup.undis.sub.A],       (0.063
[ohm]                           -j0.019)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],         (0.00399
[ohm]                           -j0.00021)
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.undis.sub.B],       (0.063
[ohm])                           -j0,019)
([[Y.bar].sup.undis.sub.B],
S)

([[Z.bar].sup.dis.sub.B],       (-0.00213
[ohm])                          -j0.00099)
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],       (0.063
[ohm])                           -j0.019)
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],       (-0.00187
[ohm])                          +j0.0008)
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately
equal to] 0

* The power transformer and OL are accepted symmetrical elements.

Table 4
Distribution of FC of linear SD in voltage distortion by the
voltage asymmetry

Voltages of
symmetrical components

FC of the i-th SD in
voltage asymmetry

PCC   Indicator            Voltage of zero
of PQ                sequence, V/grad

Mathematical model based on the principle of SD superposition

5     [K.sub.OU],   2.21   [absolute value of     4.57
%                    ([[U.bar].sub.0])]
arg([[U.bar].sub.0])   142.3

Mathematical model based on the principle ofSD exclusion

5     [K.sub.OU],   2,21   [absolute value of     4.57
%                    ([[U.bar].sub.0])]
arg([[U.bar].sub.0])   142.3

Voltages of symmetrical components

FC of the i-th SD in voltage asymmetry

PCC          EN           Cl       C2        C3

Mathematical model based on the principle of SD superposition

5     4.9 x [10.sup.7]   1.125    1.336    0.363

2.68         49.42   -162.52   175.32

Mathematical model based on the principle ofSD exclusion

5      7 x [10.sup.4]    1,125    1.337    0.363

167,27        49.43   -162,49   175.35

Voltages of symmetrical components

FC of the i-th SD in voltage asymmetry

PCC    C4       C5       C6       C7       [SIGMA]

Mathematical model based on the principle of SD superposition

5     1.522   3.138    0.982     0,402       4.57

82.07   141.12   -97.23   -151.89      142.3

Mathematical model based on the principle of SD exclusion

5     1.524   3.138    0.983     0.402       4.569

82.02   141.1    -97.21   -151.8       142.3

Table 5
Equivalent circuit parameters of the PSS elements and EE consumers
for the FC distribution by the voltage deviation

PSS element and power consumers

Equivalent circuit              EN           T *            T *
(winding)      (magnetic)

[[Z.bar.].sup.undis.sub.A],    0.008       0.00105       (0.001175
[ohm]                         +j0.048      +j0.0072      -j0.0021)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],         0            0              0
[ohm]                          0.008       0.00105       (0.001)75
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.dis.sub.B],     +j0.04       +j0.0072      -j0.0021)
[ohm])                           0            0              0
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],    0.008       0.00105       (0.001)75
[ohm])                        +j0.056      +j0.0072      -j0.0021)
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],        0            0              0
[ohm])
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]         --      [approximately      --
equal to] 0

PSS element and power consumers

Equivalent circuit             OL1 *      OL2 *      OL3 *

[[Z.bar.].sup.undis.sub.A],    0.11)4     0.1576     0.1379
[ohm]                         +j0.031    +j0.0276   +j0.0242
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],         0          0          0
[ohm]                          0.1134     0.1576     0.1379
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.dis.sub.B],     +j0.031    +j0.0276   +j0.0242
[ohm])                           0          0          0
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],    0.1134     0.1576     0.1379
[ohm])                        +j0.031    +j0.0276   +j0.0242
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],        0          0          0
[ohm])
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]         0.1134     0.1576     0.1379
+j/0.031   +j0.0276   +j0.0242

PSS element and power consumers

Equivalent circuit                  C1               C2

[[Z.bar.].sup.undis.sub.A],       (0.11            (0.111
[ohm]                            -j0.049)         -j0.045)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],       (-j0.00697)       (-j0.016)
[ohm]                             0.127            0.106
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.dis.sub.B],        -j0.048          -j0.043
[ohm])                           (j/001)         (-j0.0073)
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],       0.117            0.106
[ohm])                           -j0.047          -j0.042
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],        (0.014)          (0,00453
[ohm])                                            -/0.002)
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately   [approximately
equal to] 0      equal to] 0

PSS element and power consumers

Equivalent circuit                  C3               C4

[[Z.bar.].sup.undis.sub.A],       (0.064           (0.093
[ohm]                            -j0.024)         -j0.031)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],            0                0
[ohm]                              0.06            0.094
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.dis.sub.B],        -j0.019          -j0.035
[ohm])                              0             (0.0059)
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],       0.064            0.086
[ohm])                           -j0.021          -j0.029
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],           0                0
[ohm])
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately   [approximately
equal to] 0      equal to] 0

PSS element and power consumers

Equivalent circuit                  C5               C6

[[Z.bar.].sup.undis.sub.A],       (0.145           (0.05
[ohm]                            -j0.044)         -j0-021)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],            0                0
[ohm]                             0.141            0.043
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.dis.sub.B],        -j0.029          -j0.018
[ohm])                              0                0
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],        0.13            0.055
[ohm])                           -j0.034          -j0.023
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],           0                0
[ohm])
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately   [approximately
equal to] 0      equal to] 0

PSS element and
power consumers

Equivalent circuit                  C7

[[Z.bar.].sup.undis.sub.A],       (0.067
[ohm]                            -j0.019)
([[Y.bar].sup.undis.sub.A],
S)

[[Z.bar].sup.dis.sub.A],            0
[ohm]                             0.061
([[Y.bar].sup.dis.sub.A],
S)

([[Z.bar].sup.dis.sub.B],         -j0.02
[ohm])                              0
([[Y.bar].sup.dis.sub.B],
S)

([[Z.bar].sup.undis.sub.C],       0.061
[ohm])                           -j0.018
([[Y.bar].sup.undis.sub.C],
S)

([[Z.bar].sup.dis.sub.C],           0
[ohm])
([[Y.bar].sup.dis.sub.C],
S)

[[Z.bar].sub.N], [ohm]        [approximately
equal to] 0

* The power transformer and OL are accepted symmetrical elements.

Table 6
Distribution of FC of linear SD in voltage distortion
by the voltage deviation

Voltage deviation by
direct sequence

PCC   Indicator                           Voltage deviation from
of PQ                               lower normal permitted

Mathematical model based on the principle of SD superposition

5     [delta][U.sub.y], %   -6.19   [U.sup.np.sub.min] -     2.615
[absolute value of
([[U.bar].sub.1])]

-arg([[U.bar].sub.1])    178.8

Mathematical model based on the principle of SD exclusion

5     [delta][U.sub.y], %   -6.19   [U.sup.np.sub.min] -     2.615
[absolute value of
([[U.bar].sub.1])]

-arg([[U.bar].sub.1])    178.8

Voltage deviation by direct sequence

FC of the i-th excluded SD in the voltage deviation

PCC   ES       C1       C2       C3   C4

Mathematical model based on the principle of SD superposition

5     2,051    012      0.423    0    0.122

178.68   166.26   129.12   0    -161.52

Mathematical model based on the principle ofSD exclusion

5     2.046    0.119    0.423    0    0.121

178.76   166.6    129.17   0    -161.68

Voltage deviation by
direct sequence

FC of the i-th excluded SD
in the voltage deviation

PCC   C5   C6   C7   [SIGMA]

Mathematical model based on the principle of SD superposition

5     0    0    0    2.575

0    0    0    171.S4

Mathematical model based on the principle of SD exclusion

5     0    0    0    2.57

0    0    0    171.9
COPYRIGHT 2016 Department of Electrical Apparatus of National Technical University, Kharkiv Polytechnic Institute
No portion of this article can be reproduced without the express written permission from the copyright holder.
Title Annotation: Printer friendly Cite/link Email Feedback Power Stations, Grids and Systems Sayenko, Yu.L.; Kalyuzhniy, D.N. Electrical Engineering & Electromechanics Report 1USA Mar 1, 2016 5172 Modifications of diode rectifier circuits for continuous insulation measurement in live AC IT networks. The relevance of determining responsibility for violation of power quality in terms of voltage fluctuations. Electric power distribution Electric power systems Mathematical models Numerical analysis