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REPORTING METHODOLOGY AND ALGORITHM OF MODES OF COMPLEX ENERGY SYSTEMS WITH PHASE
COORDINATES
Huseyngulu Guliyev1 and Famil Ibrahimov2
12Azerbaijan Technical University, Baku, Azerbaijan AZ1073, H. Javid avenue 25 1huseyngulu@mail.ru, 2amfanet@mail.ru
Abstract
A mathematical model, algorithm and program have been developed to study any types of complex asymmetric steady-state modes and transient processes of a multi-machine power system with a renewable energy source in phase coordinates, the results of which can be used in the operational control of power system operating modes with any type of emergency automation. The developed methodology and software package can also be used in industry to check the possibility of long-term operation in the considered asymmetrical mode from the point of view of the operating conditions of the system generators and electrical receivers, to determine the need to use baluns, to select their parameters and installation locations, to ensure the efficiency of asymmetrical modes , as well as for conducting various tests and analyzing accidents that have occurred.
Keywords: power system, asymmetrical steady-state modes, transient processes,
emergency automation, relay protection
I. Introduction
At present, methods and algorithms for calculating steady-state and transition modes of complex power systems are used in the replacement scheme of a symmetrical three-phase system. When modeling switching processes in non-symmetrical short-circuit and incomplete phase modes, instead of asymmetry, it is performed by adding a shunt or additional resistance. In modern conditions, where the integration of renewable energy sources and digital technologies into the energy system takes place, solving the mentioned problem with traditional methods and algorithms becomes significantly more complicated, and sometimes in complex asymmetric modes, when the transposition of electric transmission lines is not considered, when the parameters of the line and other elements of the system differ in phase, in substations when three-phase transformers are connected with special schemes, it is quite difficult and sometimes impossible to solve. At the same time, carrying out non-symmetric settled and transition modes in a fictitious two-axis coordinate system using transformation formulas significantly increases modeling and reporting errors and makes adequate decision-making difficult for mode control. Taking into account the above, the issue of expressing mode parameters in phase coordinates (a, b, c) during the calculation and study of steady and transition modes appears as an actual
Huseyngulu Guliyev and Famil Ibrahimov
REPORTING METHODOLOGY AND ALGORITHM OF MODES OF RT&A, No 1 (77)
COMPLEX ENERGY SYSTEMS WITH PHASE COORDINATES_Volume I9, March 2024
solution [1-4].
It should be noted that the advantage of the (a, b, c) coordinate system over other calculation systems, especially the d, q, 0 system, is that all mode quantities correspond to real-time values, and re-transformation and calculation of the results are not required to obtain the phase quantities.
The solution of the given problem in phase coordinates is quite universal, as the modeling of various types of non-symmetrical short-circuits and settled symmetric modes is relatively easy, and the simplicity of the reporting algorithm allows the use of modern high-performance computing systems. In this case, the main difficulty is to design a three-phase replacement scheme for the elements of the power system in non-symmetrical quasi-steady and transition modes. Therefore, the replacement schemes and mathematical models of the main elements of the power system (generator, transformer, power transmission line, load) for the calculation of non-symmetric modes were presented in [5-11], respectively.
In order to solve the problem posed in the conditions of integration of renewable energy sources, first, the calculation of the non-symmetrical settled mode is carried out in the phase coordinates, and here the pre-accident mode will differ significantly from the linear scheme of the mode due to the reasons we mentioned above. In the next stage, generators in a multi-machine complex energy system are combined with the algorithm of reporting transition processes in phase coordinates, taking into account the complete Park-Gorev equations [12-15].
In the work under review, the equations for the stator windings of conventional and wind generator machines are used in the (a, b, c) coordinate system, and for the rotor quantities, the d, q, 0 system is used. The periodic coefficients in the equations of synchronous and asynchronous machines are calculated as the angle between the stator and rotor axes in each interval of the mathematical solution of the equations. It should be noted that since their periodic coefficients are expressed as siny and cosy, their calculation does not cause any difficulties.
In order to express each three-phase element in phase coordinates, their description with a suitable three-phase replacement scheme was used in the calculation methodology. In this case, the operation of the complex transformation coefficient in different branching cases is taken into account for voltage regulation in power transformers and autotransformers. For the purpose of reporting, a matrix of nodal equations is established for separate elements of the energy system, and based on it, the results of the report are used not only for tuning relay protection and automation devices but also for more complex issues, in other words, mode symmetrization.
II. Solving the system of nodal equations in phase coordinates for the study of symmetric and non-symmetric regimes of the complex energy system
The steady-state mathematical model of a three-phase network is analogous to the model of a single-line network and is a system of nonlinear mathematical equations with complex coefficients and variables. All known methods can be used to solve it [16-18].
The well-known Gauss-Seidel method was used to solve the system of nodal voltage equations in the form of a current balance and the system of nodal equations written in the form of a matrix.
I Y II uu hi i\
Data for nodes is given as Pl + jQL load power for each phase, and for generators as Pg + jQc corresponding to each phase or PG , | UG | . As in a single-line circuit, a node is taken as a balancing node, for which the emf's are assumed to be 120° from each other in the three phases. The
voltage en
at the balancing node is determined according to the following expression:
Yi Y2 YI3 - ±o U s J U + Y T?6an Y\Ea
Y Y 22 Y Y 23 - ±o U 2 sj U 2 a YE?
Y y32 Y ±33 - ±o U3 SJ U3 a 2YiE0T
- Y - Y - ±0 3Y + Y 3±0 + yn 0 0 0
here a = 1Z120 = e] * ; Yl, Y2, Y0 - forward, reverse and zero sequence conductors; U1 ,U2,U3 -forward, reverse and zero sequence voltages; S, S2, S ~ the full powers of individual phases; Ebaal - is the EMF of the balancing node.
In this case, the following restrictions are taken into account according to voltage and power:
P < P < P
i mm — Gi — i max'
O < O < O
mm — x--Gi — max'
U- ■ <1 U- I < U- .
i mm I i I i max
i = 1,2,3,...., n
The allowable limits characterize the change of active power Pi, reactive power Qi and voltage modulus I JJi I at node i. The algorithm uses the procedure of accelerated accumulation of the iteration process, where the new acceleration coefficient wnew is recalculated depending on the given number of iterations of the old coefficient w start.
2
1 + , 1 -
(№start + A —1)2
(3)
here
x =
XI AU,
¡=1
/ (k+1)
X At/1
¡=1
/ (k)
(4)
new № —
start 0
№ —A
2
n
2
n
The value of the acceleration coefficient is taken in the range 1 <œ< 2. The iteration process ends after the given precision is met.
Considering the given expression, an algorithm and software were developed for the calculation of asymmetric modes in multi-machine complex systems with renewable energy sources, according to which the equations of the energy system elements are expressed in phase coordinates.
III. Determination of currents and voltages in phase coordinates in asymmetric
regimes of power systems
Calculation of short-circuit currents and single-phase modes can be performed based on the results of calculating the previous short-circuit pre-emergency mode. In this case, you can simultaneously simulate any type of short circuit, including short circuit through impedance. Short circuits and phase breaks are taken into account directly when drawing up nodal equations. At the
same time, all the necessary information is entered into the computer, taking into account the capacitive conductivities of the corresponding component lines, the resistance of the component transformers (autotransformers); resistances included in the neutral of transformers, generators, etc. The phase discontinuity of a branch can also be replaced by including an infinitely large resistance in it [19,20].
In the conductivity matrix | Y |, the short circuit is quite simply taken into account through transition resistance. To do this, it is enough to set the value of the contact resistance in the source data. When modeling a fault in the nodes of the system circuit, it is necessary to set a pre-provided code for the fault type. To calculate short-circuit currents and open-phase modes, the Gauss-Seidel method was used.
IV. Modeling in phase coordinates of transient processes in complex regulated
power systems.
The equations of synchronous machines are modeled using the full Park-Gorev equations and simplified Lebedev-Zhdanov equations, taking into account electromagnetic transient processes in the rotor circuits [21-26]
The initial equations for calculating the modes of a synchronous machine in coordinates a, b, c are the following differential equations for the stator winding voltages:
PVa = ea " lara
PWb = % - Wb
PWc = ec - Wc
(5)
where \ya - flux linkage of the stator winding phases; ia, ib, ic - stator winding phase
currents; ra, rb, rc - active resistance of stator winding phases; ea, eb, ec - voltage at the terminals of
the generator stator phase windings; p = — - differentiation operator with respect to synchronous
dr
time t = 2nft.
To this system of equations one should add the stress equations for the rotor circuits and the rotor motion equations:
p¥f = ef -ifrf P ¥kd = -ikdrkd
p ¥kq = -ikqrkq
pS = -1 Mm + Me ) p0 = S
(6)
Vf »Vkd »Vkq
where lf » lkd » lkq
rf » rkd » rkq
- flux linkage of current and active resistance of the excitation winding and
damper circuits along the longitudinal and transverse axes; ef - voltage applied to the excitation winding; S - slip; H - inertial constant in el. rad; Mm - load torque on the shaft of a synchronous machine; Me - electromagnetic torque of synchronous machine; d - working angle (angle between
the transverse axis of the rotor and the representing vector of phase voltages.
To solve systems of equations (4) and (5) on a PC using any of the well-known numerical methods of Runge-Kutta, Adams Euler, etc. [3] it is necessary that the number of variables equals the number of equations. Experience shows that it is advisable to express all currents through the flux linkage of the circuits. For this purpose, well-known relationships obtained from calculations of symmetric modes using the Park-Gorev equations are used [3].
¿d = a¥d -b¥f -c¥kd
iq = S¥q - hWkq if = -b¥d + d¥f -e¥kd ikd = -c¥d -e¥f + f¥kd
ikq =-h¥q + k¥kq ■ _¥0
¿n =
(7)
where the coefficients a, b, c, d, e, f, g, h, k are expressed through the machine parameters as follows:
b =
c =
d =
e =
XfXkd Xad ; f XdXf - Xad
A A
XadXkd ' Xad ; g- Xkq
A XqXkq ~ ~Xaq
XadXf ~ ~Xad ■ h - _ Xaq
A ,h Xq Xkq X2 ; xaq
XdXkd ' ~ Xad ; k - _ Xq
A ; k - XqXkq xaq
XdXad ~ ~Xad .
A
(8)
A = X d X fXkd ~ xad ) _ xad (¡XadXkd ~ Xad ) _ Xad {XadXf ~ Xad ) .
The parameters included in these expressions represent the mutual or complete reactivity of the circuits.
X f = Xad + xf ; Xkd = xad + Xkd ; X kq = Xaq + Xkq ; Xd = Xad + Xe Xq = Xaq + Xl
To transition from stator currents id, iq, i0 to phase values ia, ib, ic we use the known relations.
ia = »o + jd cosr~ iq sinr;
ib = »0 + »d cos(r - p)~ »q sin(r - p); \ (9)
»c = »o + »d cos(r+p)- »q sin(r+p);
Yo = 1 {Ya + Yb + Yc );
Yd = 2 a cos y + Yb cos(/ -p)+{c cos(y + p)\ Yq = 2 {Ya sinr + Yb sin(y -p)+Yc sin(y + pH
x
u
a =
Huseyngulu Guliyev and Famil Ibrahimov
REPORTING METHODOLOGY AND ALGORITHM OF MODES OF RT&A, No 1 (77)
COMPLEX ENERGY SYSTEMS WITH PHASE COORDINATES_Volume I9, March 2024
2n
where P = ~ =120° for a machine with symmetrically arranged three-phase windings,
7T
y = t + 9 + — - the angle between the stationary axis of phase a and the rotating longitudinal axis of the rotor.
V. Modeling in phase coordinates of transient processes in complex regulated
power systems.
The block diagram of the algorithm for calculating asymmetric modes and transient processes in phase coordinates is shown in Figure 1.
As a result of calculating symmetrical or asymmetrical modes for each phase, the following are determined: modules and voltage angles in nodes, flows of active and reactive power along lines and transformers, losses in each element and in the system as a whole, generation of reactive power in those nodes where voltage modules are specified and other information if necessary.
When calculating short circuit (SC) modes, the output information is also displayed on the display screen in tabular form and includes: currents at the short circuit point, residual voltages in the circuit nodes and their phases, currents or power flows along the branches, including losses in each element and in the system generally.
In this work, it is possible to perform calculations during a short circuit at an intermediate point of a branch without introducing additional nodes into the design diagram.
To illustrate the performance of the developed methodology and program for calculating symmetrical and complex-asymmetrical modes in complex multi-machine power systems, let us consider several examples for a specific circuit shown in Figure 2. As can be seen, a 10 MW wind turbine is integrated into the system through a T3 transformer.
All necessary data for the system under study are presented in Tables 1 ^ 3. The values are presented in p. u. and reduced to Sb — 100MVA . Transformers T1-T3 have a connection diagram
Y0/A. Transformers T4 and T5 have three-phase-two-phase and star-zigzag connection schemes, respectively.
Tables 4 and 5 present the results of calculating symmetrical and asymmetrical modes in phase coordinates, where the load was represented by PL + jQL = const. Note that the voltage values in the secondary windings of a three-phase-two-phase transformer are distributed as follows:
Ua2-al = (U41 -Ua), Ue2_el = {UA2 -U4), where t/a2_al = 0,855Z83,15° - 0,197Z- 76,996° = 1,0424Z86,83°;
Us2_A = 0,836Z-5,011° -0,209Z-175,166° = 1,043Z-3,05°.
A comparison of the calculation results presented in Tables 4 and 5 indicates that the levels of voltage values in the nodes are different due to the different representations of loads in the nodes of the circuit.
H
Calculation of symmetrical or asymmetrical SSM
H
IE
Calculation of symmetry, or asymmetrical SSM followed by short circuit
Calculation only short circuit without calculating SSM
Entering information:
1. Number of short circuits
2. Types of short circuits
3. About the location. short circuit
4.Value about transient
resistance.
IL
Calculation of transient processes in phase coordinates
Entering initial data for:
1. Synchronous machines.
2. Excitation systems
3. Automatic speed regulator systems.
4. Load data.
5. Data on disturbances and actions of relay protection and automation
6. General information.
Figure 1: Block diagram of the algorithm for calculating power system modes in phase coordinates
0,05+j0,005 MVA
Gl
25,26,27
0,05+j0,005 MVA 42(b2), 44(b4)
13,14,15
3(<j
4,5,6
I T2 22,30,31
30+j19 MVA
19,20,21
i
45,46,47 (zig terminal) 48,49,50 (zag terminal)
Figure 2: Scheme of the studied multi-machine power system with renewable energy sources
Table 1: Initial data of power transmission lines of power systems (p.u.)
art node End node Positive sequence resistors Zero sequence resistances
Ri Xi Bi Ro Xo Bo
10, 11, 12 22, 23, 24 0,0145 0,0660 0,0108 0,0456 0,1944 0,0058
10, 11, 12 16, 17, 18 0,0110 0,0496 0,0080 0,0342 0,1458 0,0050
16, 17, 18 19, 20, 21 0,0091 0,0413 0,0068 0,0285 0,1215 0,0043
19, 20, 21 22, 23, 24 0,0056 0,0248 0,0041 0,0171 0,0729 0,0025
19, 20, 21 13, 14, 15 0,0035 0,0155 0,0021 0,0114 0,0486 0,0027
13, 14, 15 22, 23, 24 0,0013 0,0330 0,0054 0,0228 0,0972 0,0034
7, 8, 9 33, 34, 35 0,0073 0,0330 0,0054 0,0228 0,0972 0,0034
Table 2: Initial data of system generators (p.u.)
Start End Positive sequence resistors Zero sequence resistances
node node Ri Xi Ro Xo
1, 2, 3 25, 26, 27 0,0 0,0967 0,0 0,0467
4, 5, 6 29, 30, 31 0,0 0,17 0,0 0,085
33, 34, 35 36, 37, 38 0,0 0,17 0,0 0,085
The obtained results of calculating the asymmetric mode indicate that the noted violations of the symmetric mode do not cause deep violations of the level of asymmetry of the mode parameters in the network circuit and such a mode is acceptable.
The calculation results for a single-phase short circuit at the generator terminals are presented in Figure 3, where a complete coincidence with the experimentally taken curves of the transition
process was obtained.
Table 3: Initial data of power transformers (p.u.)
Start node End node Connection type X
1, 2, 3 10, 11, 12 ^ Y 0,0533
4, 5, 6 13, 14, 15 aY 0,12
7, 8, 9 16, 17, 18 z±X 0,16
22, 23, 24 41, 42, 43, 44 I •—• 0,09(Xm) 0,16(Xr)
19, 20, 21 45, 46, 47 48, 49, 50 s T r^ 0,17
Table 4: Calculation results of the symmetrical mode in phase coordinates
Start End P Q Start End P Q
node node (MW) (MVAr) node node (MW) (MVAr)
1 25 -32,373 -15,122 25 1 32,373 16,230
2 26 -32,306 -14,879 26 2 32,306 15,976
3 27 -32,497 -14,971 27 3 32,497 16,081
4 29 -19,985 -9,405 29 4 19,985 10,151
5 30 -20,105 -9,534 30 5 20,105 10,292
6 31 -19,912 -9,582 31 6 19,912 10,329
10 16 12,079 3,889 16 10 -12,063 -4,690
11 17 12,106 3,965 17 11 -12,090 -4,768
12 18 11,943 4,030 18 12 -11,926 -4,833
10 22 12,317 5,404 22 10 -12,292 -6,470
11 23 12,230 5,382 23 11 -12,207 -6,452
12 24 12,103 5,536 24 12 -12,077 -6,610
13 19 6,511 1,803 19 13 -6,509 -2,090
14 20 6,437 1,662 20 14 -6,436 -1,950
15 21 6,466 1,749 21 15 -6,465 -2,037
13 22 4,002 0,986 22 13 -4,001 -1,568
14 23 3,912 0,880 23 14 -3,911 -1,463
15 24 3,974 0,972 24 15 -3,973 -1,555
16 19 4,617 3,748 19 16 -4,614 -4,470
17 20 4,489 3,649 20 17 -4,487 -4,373
18 21 4,444 3,785 21 18 -4,440 -4,508
19 22 0,996 0,186 22 19 -0,996 -0,626
20 23 0,924 0,139 23 20 -0,924 -0,579
21 24 0,988 0,203 24 21 -0,988 -0,643
7 33 -3,966 -6,288 33 7 3,969 5,701
8 34 -4,024 -6,332 34 8 4,028 5,745
9 35 -3,837 -6,442 35 9 3,841 5,856
33 36 -10,047 -8,021 36 33 10,047 8,272
34 37 -10,074 -8,038 37 34 10,075 8,292
35 38 -9,881 -8,178 38 35 9,880 8,428
Table 5: Calculation results for the asymmetric mode in phase coordinates
Start End P Q Start End P Q
node node (MW) (MVAR) node node (MW) (MVAR)
1 25 -38,839 -21,470 25 1 38,839 23,199
2 26 -32,511 -20,268 26 2 32,511 21,553
3 27 -37,071 -15,517 27 3 37,072 16,920
4 29 -24,543 -10,624 29 4 24,543 11,722
5 30 -16,229 -12,413 30 5 16,229 13,057
6 31 -19,228 -4,594 31 6 19,228 5,183
10 16 12,622 4,220 16 10 -12,605 -5,008
11 17 15,593 5,171 17 11 -15,566 -5,913
12 18 13,322 7,676 18 12 -13,295 -5,419
10 22 11,936 5,932 22 10 -11,913 -6,993
11 23 14,615 6,308 23 11 -14,580 -7,324
12 24 12,881 8,829 24 12 -12,844 -9,824
13 19 8,625 1,825 19 13 -8,523 -2,105
14 20 4,315 2,852 20 14 -4,314 -3,139
15 21 5,362 -1,541 21 15 -5,361 1,260
13 22 5,114 1,095 22 13 -5,111 -1,671
14 23 1,412 1,713 23 14 -1,411 -2,293
15 24 2,381 -2,027 24 15 -2,381 1,458
16 19 3,321 4,121 19 16 -3,319 -4,942
17 20 5,276 3,489 20 17 -5,272 -4,203
18 21 4,866 5,752 21 18 -4,861 -6,446
19 22 1,068 0,315 22 19 -1,067 -0,753
20 23 -0,993 0,455 23 20 0,994 -0,892
21 24 -0,400 -1,603 24 21 0,400 1,175
7 33 -3,408 -7,157 33 7 3,412 6,575
8 34 -2,302 -5,661 34 8 2,305 5,068
9 35 -4,268 -5,472 35 9 4,270 4,882
33 36 -10,082 -9,145 36 33 10,083 9,429
34 37 -8,970 -7,701 37 34 8,970 7,914
35 38 -10,947 -7,463 38 35 10,947 7,732
If, p.u.
2
1
J WW
0
t, c
-1-1-1-1-1—
0,04 0,08 0,12 0,16 0,20
Ia, p.u.
4
0
t, c
0,04
0,08
0,12
0,16
0,20
-4
Figure. 3: Current in the field winding (a) and in phase A (b) with a single-phase short circuit
VI. Conclusions
1. A mathematical model of a power system with a renewable energy source, an algorithm and a program have been developed for studying any types of longitudinal-transverse, complex asymmetric steady-state modes, short-circuit modes and transient processes in phase coordinates and a dialogue complex created on its basis, which can be used in operational dispatch control operating modes of power systems for any type of emergency automation.
2. The developed methodology and complex program can be used in industry to solve the following practical problems: to test the possibility of long-term operation in the considered asymmetrical mode from the point of view of the operating conditions of EPS generators and power receivers; to determine the need to use baluns, to select their parameters and installation locations, to ensure the efficiency of asymmetrical modes; for carrying out various tests, analyzing accidents that have occurred; for selecting response parameters and assessing the sensitivity of relay protection devices, parameters of automation devices.
3. The obtained comparative results of calculating the symmetrical and asymmetrical modes of power systems indicate that the levels of voltage values in the nodes are different due to the different representations of loads in the nodes of the circuit. The noted violations of the symmetrical mode do not cause deep violations of the level of asymmetry of the mode parameters in the network circuit and such a mode is acceptable.
[1] Merkuryev G. V., Shargin Yu. M. Formation of a mathematical model of the energy system for calculations of electromechanical transient processes // Electricity, 2008, No. 12, pp. 3-7
[2] Vainshtein R. A., Lozinsky K. S., Ivanov V. P., Kobytev M. I. Improvement of calculations of asymmetric modes in programs for calculating electromechanical transient processes // Electricity, 2008, No. 7, pp. 19-23
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