Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. II 181
Bartłomiej Bastian, Omelian Płachtyna Gdynia Maritime University
MATHEMATICAL MODEL OF SYNCHRONOUS GENERATOR FOR DIAGNOSTICS AND SETTINGS TESTS OF MARINE AVRS
Abstract: This article presents the idea of applying of the mathematical model of synchronous generator for diagnostics and settings tests of marine automatic voltage regulator (AVR). The algorithm of the authors numeric method named ‘the average step voltages’ is introduced. The model application and its physical realization in the laboratory are described. The proper online cooperation of the implemented model and the real AVR has been achieved. The obtained measuring results are shown. Keywords: synchronous generator, modeling, diagnostics, marine AVRS 1. Introduction
Designing and implementing new automatic voltage regulators (AVRs) for synchronous generators implies the need to perform tests for various static and dynamic, normal and malfunction states. Advancement in digital electronics and mathematic modeling methods give a way to perform AVR testing during the designing phase. That is possible by coupling the mathematic model of the electromechanical system with the real AVR. With the real-time operation simulation of the electromechanical system it’s possible to observe the AVR’s response in various operating modes. The above mentioned way can be also applied for testing marine AVRs in the exploitation stage i.e. during shipyard overhauls. In that case it would be possible to check the settings without connecting real loads. To realize the above idea it’s necessary to determine the mathematical model of the electromechanical system, where the major part is synchronous generator. Below, the mathematical model of synchronous generator is presented. The model is a part of the wider electromechanical system model [1] implemented in a programmable electronic device and connected to the real AVR.
2. The implemented mathematical model of synchronous generator
The implemented mathematical model fulfills the following requirements:
- being adequate for a real object,
- giving a possibility to access and modify parameters,
- operating in the real-time mode,
- being resistant to the numeric method errors in long time operation.
Equations parameters, which are provided by the manufacturer or can be easily determined by tests [2], give adequacy. The selected authors numeric method for equation solving, called “the average step voltages method”, gives calculation stability [3], [4], [5]. The implemented model includes electrical and mechanical balance equations [6], [7].
The electric balance equations in the form of matrix in the phase coordinates are:
0][ dt
diRu
, (1)
where:
TfCBA uuuuu )0,0,,,,(
- machine terminal
voltage vector;
],,,,,[][ QDfCBA RRRRRRdiagR - coil
resistance diagonal matrix (A, B, C - stator, f - excitation, D, Q – damper windings among d and q axes);
TQDfCBA iiiiiii ),,,,,(
- machine winding
current vector;
TQDfCBA ),,,,,(
- winding
linkage flux vector.
The winding linkage flux vector
is
calculated using the following equation:
iL
][ , (2)
where:
182 Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. II
][][
][][][
iiie
eiee
LL
LLL (3)
- machine self and mutual inductance matrix;
2
000
000
000
2
)(3)cos()cos()cos(
)cos(333636
)cos(363336
)cos(363633
0000
0)2cos()2cos()2cos(
0)2cos()2cos()2cos(
0)2cos()2cos()2cos(
3][
i
fad
i
ad
i
ad
i
ad
i
adqdqdqd
i
adqdqdqd
i
adqdqdqd
qdee
K
LL
K
L
K
L
K
LK
LLLLLLLLLLK
LLLLLLLLLLK
LLLLLLLLLL
LLL
(4)
equation (4) presents the stator and excitation self and mutual inductance matrix;
02
3)sin()cos(
)sin()cos(
)sin()cos(
][
i
ad
aqad
aqad
aqad
ei
K
LLL
LL
LL
L
(5)
- mutual inductance matrix between the stator and the excitation windings and the damper windings in d,q coordinates among d and q axis;
0)sin()sin()sin(2
3)cos()cos()cos(
3
2][
aqaqaq
i
adadadad
ie
LLLK
LLLL
L (6)
- mutual inductance matrix between the damper windings and the stator and the excitation dispersion inductance;
),( QaqDadii LLLLdiagL (7)
- diagonal self inductance matrix of the damper windings.
In equations (4) to (7) the following values appear:
Ld - equivalent direct axis inductance
Lq - equivalent quadrature axis inductance
Lad - stator to direct axis mutual inductance
Laq - stator to quadrature axis mutual inductance
L0 – equivalent zero sequence inductance
LD - damper winding dispersion inductance among direct axis
LQ - damper winding dispersion inductance among quadrature axis
- stator phase A to pole axis angle
– 2/3.
The equation of motion:
SMMdt
dJ
(8)
where:
M - electromagnetic torque;
MS - mechanical torque (calculated from primer mover equations);
J - combined moment of inertia of rotating solids;
– angular speed.
3. The solving method
Equations (1) to (8) are solved using the average-step voltages method. For currents at the end of the integration step the following equation is used:
)(][ SS EURi
, (9)
where:
TQDfCBA iiiiiii ),,,,,( 1111111
- windings
currents at the end of the integration step;
TfCBA UUUUU )0,0,,,,( 1111
- average
terminal voltages at the integration step;
][1
][3
1][ 1L
tRRS
(10)
- equivalent resistance matrix;
][ 1L - self and mutual inductance matrix for
angle 1 (angle 1 corresponds to the end of the integration step, which can be calculated using the mechanical balance equation);
dt
idR
tiL
tREs
000 ][
6])[
1][
3
2(
(11)
Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. II 183
- equivalent machine windings EMF vector;
0i
- current vector at the start of the integration
step;
dt
id 0
- current derivative vector at the beginning
of the integration step, calculated from (1);
][ 0L - inductance matrix at the beginning of the
integration step;
t - integration step.
The algorithm of calculating variables after the integration step is as follows:
1. Inductances, linkage fluxes and electromagnetic torque are calculated using currents before the integration step
2. Derivations are calculated using equations (1) and (8).
3. The angle 1 is calculated according to the angular speed change at the and of the integration step.
4. Inductance matrix [L1] is calculated.
5. Equation (9) is calculated using (10) and (11)
according to the vector 1i
.
4. Program model implementation
Program model implementation is done in C++ language, which allows the user to achieve application portability between different operating systems. The application gives (using the presently available typical PC) hundreds calculating cycles per each network period. It’s enough to generate all necessary signals in the real time mode.
GENERATOR AND
NETWORK PARAMETERS
from fileEQUATIONS
according to circuit diagram
SOLVING
numeric method
SENDING
output card, screen
RECEIVING
input card, keyboard
SETUP AND STARTING
CONDITION
Fig 1. The structure of program model implementation
The structure of application is shown in fig. 1. The generator and network parameters are stored in an external file and can be easily changed. Starting conditions are included in the
program. The presented mathematical model of synchronous generator is embedded in the block named ‘equations’. That block contains also the mathematical model of the network load and configuration. The ‘Equations’ block acquires data (such as excitation current, load and power factor) from the ‘receiving’ block. The authors numeric method - the average step voltages method is included in the ‘solving’ block. The results are passed from the ‘solving’ to ‘sending’ block and next to the output card and display.
5. Physical realization
First trials of the implemented model were carried out in Marine Electrical Power Engineering laboratory of Gdynia Maritime University. That laboratory is mainly equipped with the main switchboard RG103A, 3 synchronous generators type GCf84a/4 manufactured by ELMOR driven by thyristor controlled DC machines (emulating 4-stroke diesel engines) and loads. Two types of AVR can be selected to control the generator excitation current:
- analog, TUR, manufactured by EFA
- digital, DECS 200, manufactured by BASLER
The digital AVR was chosen because:
- nowadays digital equipment is commonly used,
- it has separate terminals for measurement and power supply (there is no necessity to emulate high power signals)
- it uses standard signals (i.e. 4-20mA, 1A CT, RS 232).
Fig 2. Test equipment
The application works on PC, equipped with I/O cards manufactured by ADVANTECH: PCI-1720 and PCI-1712. One of the cards leads out the calculated generator values of current
184 Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. II
and voltage. The other card measures the excitation current (converted to voltage by LEM) and forwards it into application. It is necessary to apply signal conditioning and to ensure galvanic separation, to make the implemented model cooperative with the real AVR. Therefore, APEX DC amplifiers are used: WN-21 for voltage and WP-05 for current. The block diagram of a test circuit is shown in fig 3.
Digital AVRDECS200
F-
INDUSTRIAL PC
Mathematical model
Card PCI-1720
Card PCI-1712
D/A Converter
A/D Converter
MAIN SUPPLYCONTROL SUPPLY
VOLTAGE MEASURMENT
CURRENT MEASURMENT
F+
LEM ConverterA/mV
APEX WN-21
APEX WP-05
Fig 3. Block diagram of a test circuit
Tests confirm the proper online cooperation of the implemented model and the real AVR. In order to check if the implemented model is adequate for real object measurements in two configurations are required:
- AVR with the real system
- AVR with the implemented model.
At the moment only the steady state measurements with active load have been carried out. Their results are shown in figure 4.
Fig 4. Test results.
We can observe that the excitation current, in a function of active load, in both configurations is similar. More tests, especially in dynamic states and with passive load, are planned for the future.
6. Summary
Diagnostics and settings tests of marine AVRs are reachable by using the application with the implemented real time model and signal conditioning equipment. The presented mathematical model of synchronous generator is principal and constitutes the most complex part of the network system model. The used authors numeric method provides stability and resistance to numeric method errors in long time operation. The obtained results are satisfactory and form grounds to carry out wider research.
7. Bibliography [1]. Płachtyna O, Bastian B., Kutsyk. A.: The mathematical model of ship's power station Electrotechnic and computer systems, No 03 (79), Technica, Kyiv 2011, Ukraine
[2]. Latek W.: Badanie maszyn elektrycznych w przemyśle WNT, 1979, Poland
[3]. Płachtyna O.: Tchislovyj odnokrokovyj metod analizu elektricznych kil i jogo zastosuvanija w zadaczach elektromechaniki Zeszyty Narodowego Uniwersytetu Technicznego w Charkowie No 30, 2008, Charków, Ukraine
[4]. Płachtyna O, Kłosowski Z. Żarnowski R.: Ocena skuteczności metody napięć średniokrokowych w porównaniu z klasycznymi metodami całkowania numerycznego w modelach matematycznych obwodów elektrycznych Kwartalnik Elektryka, Zeszyt No 1, 2011, Gliwice, Poland
[5]. Płachtyna O, Kłosowski Z. Żarnowski R.: Model matematyczny napędu prądu stałego w oparciu o metodę średnich napięć na długości kroku całkowania Przegląd Elektrotechniczny No 12a, 2011, Poland
[6]. Bajorek Z. Prokop J.: Elektromechaniczne przetworniki energii PR, Rzeszów, 1990, Poland
[7]. Paszek W.: Stany nieustalone maszyn elektrycznych prądu przemiennego WNT, Warszawa, 1986, Poland
Authors Bartłomiej Bastian, M.Sc B.Eng, Department of Marine Electrical Power Engineering, Gdynia Maritime University, Gdynia ul. Morska 81-87, Poland, e-mail: [email protected]
Omelian Płachtyna, Prof., Department of Marine Electronics, Gdynia Maritime University, Gdynia ul. Morska 81-87, Poland, e-mail: [email protected]
Reviewer
Prof. Mieczysław Ronkowski
Excitation current in a function of active load
0
0,5
1
1,5
2
2,5
3
3,5
4
0 5000 10000 15000 20000 25000
AVR - real system AVR - model
If[A]
P[W]
