The article deals with the issues of modeling static modes of autonomous generation systems with self-excited induction generator, taking into account the nonlinearity of the generator magnetization curve and with diferent load values. The simulation results, which have been verified experimentally, are presented and the threedimensional dependences of frequency and amplitude of generated voltage, obtained by mathematical modeling, are given.
The use of autonomous power supply systems has become of great importance according the current state of the Ukrainian energy sector [1, 2]. Such systems are used as backup power sources or to provide electricity in the absence of centralized power supply. In such systems, in particular, self-excited induction generators (SEIG) are used. Asynchronous induction machines with a squirrel-cage rotor are characterized by small weight-and-dimensions and high reliability due to the absence of sliding contacts. One of the significant disadvantages of such technical solution is the complexity of the self-excitation process, which depends on many parameters - the generator rotation speed, the capacitance value and the connection scheme of self-excitation capacitors, as well as the load value [3]. An important aspect of the operation of autonomous power supply systems with SEIG is the stability of their operation when the load value changes [4]. To analyze static operating modes, mathematical modeling of the autonomous power supply system is carried out at diferent load values [5].
In [6] it is shown that the mathematical model of an induction generator with self-excitation must necessarily take into account the nonlinearity of the magnetization curve of the machine. The approach of piecewise linear approximation of the magnetization curve [7, 8] is widespread. This approach makes it impossible to obtain analytical dependencies that describe the operation of an induction generator, which, in turn, complicates the study of the entire generation system. In [9] an analytical description of the magnetization curve is proposed, which allows obtaining analytical dependencies for studying the operation of induction generator
Since the load of SEIG can vary during operation, the capacitance of the self-excitation capacitors is adjusted to stabilize the output voltage [10, 11].
The purpose of this paper is to simulate the static operating modes of an autonomous generation system with a self-excited induction generator, taking into account the nonlinearity of magnetization curve depending on the magnitude of the applied load and to establish the relationship between the frequency and amplitude of the generated voltage and the generator load and the capacitor bank capacity, which will be useful for the development of generator voltage regulation systems.
Let us consider the circuit of an induction generator when connecting self-excitation capacitors in a “triangle”. We will consider the load to be active, because taking into account the inductive component leads to the need to solve sixth-order equations, which significantly complicates the obtaining of analytical dependencies [12]. Then the equivalent circuit will take the form (Figure. 1), where RL is the value of active resistance of the load in one phase: Ψ = − − Ψ, Ψ = − + ( − ) Ψ, = ︂[ 0 −1 1 0 ︂] where Ψ = [︀ Ψ Ψ ︀] , Ψ = [︀ Ψ
Ψ ]︀ are vectors of stator and rotor fluxes in d-q coordinate system, = [︀ ︀] , = [︀ ︀] are stator and rotor current vectors, = [︀
︀] is stator voltage vector in the coordinate system d–q, and are active resistances of the stator and rotor, is a number of pole pairs; is an angular velocity of the induction generator rotor; is an angular velocity of an arbitrary coordinate system d–q.
Let us consider the equivalent circuit of one phase of a stand-alone generation system (Figure 2)
+ + = 0,
(
As it is well known [11], [12] = , thus, equation (
The operation of a stand-alone generation system can be either stable or unstable. The stability region of steady-state modes depends on several factors [12]. In works [11, 12], an expression for the amplitude of the generated voltage at the equilibrium point has been derived.
| *| = √︂(︁
* * * 1 + − * 2 ︁) 2 + * 2 ︁( + ︁) 2 where is stator leakage inductance, * is magnetizing inductance, * is magnetizing current of induction generator. The symbol * will denote that the corresponding vector or variable is defined at the equilibrium point of the stand-alone generation system.
Another parameter of the steady-state operation mode of induction generator is generated voltage frequency. According to [11], its value is defined as (3) (4) (5) * =
− −
2 where * = * /2.
When the load of the generator or other parameters of the generation system change, the currents in the asynchronous machine, including the magnetizing current, also change [13]. This, in turn, leads to a change in the magnetizing inductance. Therefore, for each operating point in the steady-state mode, it is necessary to know the dependency = ( ). This dependency is not provided in the datasheets of asynchronous machines. The most common method for determining it is by experimentally obtaining the no-load characteristic of the asynchronous machine, and then deriving the dependency = ( ). based on it. However, this approach requires significant time and financial resources, as it necessitates a controllable voltage supply for the asynchronous machine. In [14], a method was shown for extrapolating an existing analytical dependency = ( ) from asynchronous machines of a similar type but diferent power ratings. This approach significantly simplifies the modeling process of the operation modes of stand-alone generation systems. Therefore, we will use this method for modeling the steady-state operation modes of the SEIG.
A schematic representation of the dependency = ( ) obtained using the no-load characteristic of the machine, is shown in Figure 3 [14].
The magnetizing inductance as a function of the magnetizing current can be divided into three regions. In the first — ascending — section, the magnetizing inductance increases from the initial value 0 to . The inductance corresponds to the linear, unsaturated part of the magnetization curve of the asynchronous machine. In this region, only an unstable state is possible, meaning that both voltage generation and voltage collapse may occur [14]. In the second region — up to the saturation zone of the magnetic system — the inductance remains constant at . The third region corresponds to the saturation zone of the asynchronous machine, where the magnetizing inductance decreases — the descending part of the = ( ) curve.
In our study we will use on experimental dependence [12] of magnetizing inductance LM on the magnetizing current iM, obtained in the idle mode test of asynchronous machine 6343 with the following nameplate data: rated power of 0.37 kW; Nominal stator phase voltage 220 V mains rated frequency 50 Hz, rated speed 1450 rpm, which is presented on Figure.4.
An important characteristics of SEIG are its self-excitation borders [9]. Moreover, the border of spontaneous self-excitation is determined at = 0 and trigger self-excitation - in unsaturated linear section of the magnetization curve, which is characterized by the highest value LM [12]. Therefore, dependence = ( ) in Figure.3 have to be extrapolated, since the value = 0.
Figure.5 shows the results of magnetizing inductance curve and dynamical inductance approximation, described in [14]. Dependence = ( ) is presented in the form of four sections that are joined together without breaking the derivative.
In [14] was obtained that this approximation allows one to accurately describe the self-excitation border boundaries, which need to know primarily reliance = ( ) with small currents. However, as seen from Figure.5, in the steady-state operation modes a very significant deviation occurs between approximated and experimental dependencies, making this approach unacceptable to study these modes.
The descending part of the magnetization curve can be accurately described by polynomial equation of second order
Its factors may be determined using polyfit function in MATLAB. Figure.6 shows the results of magnetizing inductance and dynamical inductance approximation for values of approximation which were obtained using MATLAB.
= 312 + 22 + 3, 1 = 0.042433076890268; 2 = ˘0.383816481755985; 3 = 1.143841689235056, (7)
As seen from Figure.6, approximated magnetizing inductance accurately replicates the experimental curves on the descending part, and the dynamic inductance is negative and has growing part for large values of current, that is violating the physical nature of the phenomenon. Since in the study of steady-state of SEIG dynamic inductance is ignored, then this approach can be used in this case. But this approximation does not allow to consider the initial part of dependence = ( ) at low currents that will not allow its implementation for the study of boundary modes.
he inductance value is directly related to the dynamic inductance consider the magnetization curve approximation using dependence Ψ = ( ). Dependencies Ψ = ( ) that were obtained from experimental data of SEIG at mentioned above conditions for next values of the load 200 Ohm, 400 Ohm and no load case are depicted in Figure.7.
Approximation was performed by applying polyfit function in MATLAB to the saturated part of the flux linkage curve of SEIG for the case of no load. Points that are derived from the idle mode characteristics of SEIG are marked with *, Figure.8.
To account the unsaturated parts of magnetization curve the approximated dependence was amended with linear region in the left , that corresponds the inductance at unsaturated part of magnetizing curve (0, 105 < < 0, 33 ), and "sewn" to it part without breaking the original of the initial magnetization ( < 0, 105 ). For the magnetizing currents 0, 33 < < 4 flux linkage was approximated with fourth order polynomial equation:
Ψ = 14 + 23 + 32 + 4 + 5, 1 = ˘0.005214090677207; 2 = 0.082454101449568; 3 = ˘0.481133636330431; 4 = 1.225474520316153; 5 = ˘0.020348151810052. (8)
As shown in Figure.8, in the range of 0, 105 < < 0, 33 magnetization inductance is not constant, equal , and varies by a parabolic law. Therefore, in the area < 0, 33 magnetizing inductance approximation was adjusted as follows: at the site of 0, 105 < < 0, 33 and = = , it goes right in a downward part described with previously given fourth order polynomial equation. In addition, in the left it is described with a growing area of initial magnetization similarly as described in articles [6, 11].
To avoid the problems associated with negative values of dynamic inductance when extrapolating on the high values of current , the following correction was held. After the increase of magnetizing current, the dynamic inductance first adopts zero, it is assumed that it must continue zero, and the value of flux Ψ also remains constant Ψ = Ψ (at = 0). The results of this approximation are shown in Figure.9.
Therefore, taking all the aforementioned factors into account, the magnetizing inductance of the asynchronous machine can be expressed through the following dependencies: (9) (10) (11) (12) (13) • at < 1 where 1 – adjusted coeficient.
Dynamic inductance is defined as: • at 1 < < 2 • at 2 < < 3 • at > 3 = − 1( − 1)2,
2 = + − 11 − 1 (3 − 4 1) = = . 1 = 2 − 0 .
1 = 14 + 23 + 32 + 4 + 5;
= 413 + 322 + 23 + 4 If the value of 0 was obtained while = 0, then coeficient 1 will be determined by the ratio = Ψ , = 0. (14)
For the mentioned induction machine 6343 the next parameters of the approximation were accepted [12]: = 1.03115 ; 0 = 0.6345 ; 1 = 0.105 ; 2 = 0.2134; (15) 3 = 3.042 ; Ψ = 1.129833270853887 ; 1 = −0.005214090677207; 2 = 0.082454101449568; 3 = −0.481133636330431; 4 = 1.225474520316153;
5 = −0.020348151810052.
Thus, using equations (4) and (5), and taking into account the nonlinear dependence = ( ), the modeling of steady-state characteristics by using MatLab software’s numerical routines and packages was performed, and the dependencies of the amplitude and frequency of the asynchronous generator voltage were obtained for two values of active load (Figure 10 and Figure 11) — 400 Ohms and 200 Ohms — as well as for the no-load condition. The calculations were carried out assuming a self-excitation capacitor bank capacitance of 90 Ohms, with varying generator shaft rotational speed.
The results obtained from modeling the steady-state modes were compared with experimental data acquired for the induction generator A634U3. The rated parameters of the machine are: power — 0.37 kW; stator phase voltage — 220 V; supply frequency — 50 Hz; rotational speed — 1450 rpm. As can be seen from the presented graphs, the modeling results are consistent with the experimental characteristics.
The voltage and frequency values of the induction generator depend on three parameters: the load value, the capacitance of the self-excitation capacitor bank, and the generator shaft rotational speed [15]. To comprehensively assess the influence of these factors, we will model three-dimensional dependencies * (, 1/) and | *|(, 1/).
Figures 12 and 13 show the dependencies of the output voltage magnitude and frequency of the standalone generation system in steady-state operation mode, with varying load admittance 1/ within the self-excitation limits, at a generator shaft speed of 119.32 rad/s. Lines 1 indicate the boundaries of the self-excitation zone of the generation system, while lines 3 represent the maximum values of the function within this zone. Curves 2 on both graphs represent the projections of the three-dimensional graph onto the plane − 1/ .
Similarly, Figures 14 and 15 show the dependencies of voltage | * |(, 1/) and frequency * (, 1/) and with varying capacitance of the self-excitation capacitor bank within the self-excitation limits, at a generator shaft speed of 119.32 rad/s.
A detailed analysis of the obtained graphs reveals that the maximum values of both frequency and voltage occur under no-load conditions. This is consistent with the physical behavior of SEIG [16], where the absence of load minimizes voltage drop and reactive power consumption, allowing the system to reach its peak performance. Furthermore, in both cases examined, the three-dimensional plots of the output voltage magnitude demonstrate pronounced maxima located near the center of the self-excitation boundary region. This indicates that the optimal operating point, in terms of voltage stability, lies within this central zone rather than at the extreme boundaries, which may be associated with unstable excitation conditions. These findings underscore the importance of carefully selecting system parameters to ensure operation within this optimal excitation region.
In this paper, the steady-state operating modes of a stand-alone generation system with an induction generator were modeled. The investigation was carried out with consideration of an analytical representation of the magnetization curve of the induction generator, which made it possible to derive analytical dependencies for the generator’s voltage magnitude and frequency. A comparison between the modeling results and experimental data demonstrated a high level of agreement, thereby validating the accuracy of the proposed model.
Based on the developed models, three-dimensional plots were also obtained to analyze the combined influence of the self-excitation capacitor bank capacitance and load variation within the self-excitation limits of the induction generator, while maintaining a constant generator shaft speed. The analysis showed that the maximum values of the investigated quantities occurred under no-load conditions at the output of the stand-alone generation system.