Modeling Method for Autonomous Current Inverters

1
and τ
2
already enter 
implicitly. This is fundamentally connected with the 
presence of three stages on the repeatability interval and, 
accordingly, two time moments τ
1
and τ
2
, associated with 
the transition from the first stage, to the second, and from 
the second to the third. 
Below are the results of mathematical modelling of a 
single-phase parallel ACI according to the above 
methodology, but taking into account the features of this 
scheme. In fig. 1 shows a single-phase circuit ACI and 
timing diagrams of the current and voltage of the load of the 
converter, which has a capacitive nature. 
The normal operating mode of the ACI is to operate at a 
constant input current i
d
= I
d
[22]. Obviously, the condition 
that provides an almost constant input current can be 
represented by a rough inequality that excludes noticeable 
fluctuations in the current i
d
during the interval between 
switching thyristors in the converter: 
,
(6)
where L
d
 - filter inductance in the current circuit i
d
; 
R
L
 - equivalent load resistance of the power supply; 
T - output frequency period; 
m - the number of switching power valves in the 
converter circuit. 
Fig. 1. Single-phase ACI circuit and timing diagrams of load current 
and voltage.
The switching capacitors C
k
are connected in parallel 
with the load and function only when the currents in the 
load phases are switched. The value of the capacitors does 
not depend on the reactive power of the load, which allows 
the autonomous current inverters to operate at any load and 
any frequency of the output voltage within the switching 
capacity of the capacitors. It follows from this that the mode 
of constant input current of the converter depends on both 
the magnitude L
d
and the output frequency of the ACI. 
From the description of the ACI operation modes, it 
follows that there is a need to have a load with a resultant 
capacitive 
impedance, 
which 
contributes 
to 
the 
instantaneous switching of the current in the load. For this, a 
compensating capacitor is connected in parallel to the L, R 
load. This combination in ACI is considered as its load. 
Corresponding algorithms for switching thyristors ACI with 
a given frequency contribute to the conversion of the 
continuous current of the power supply into an alternating 
current of the load and a compensating capacitor.
From the analysis of the processes on the interval δ it 
can be seen that the capacitive nature of the load provides a 
negative anode-cathode voltage of the thyristors conducting 
current for a time 
. (7) 
For successful switching of thyristors, it is necessary that 
this time is not less than the recovery time of thyristors’ t ≥ 
t
r
. 
The considered inverters are complex nonlinear circuits 
for which the calculation and analysis is performed at one or 
another level of accuracy. In modern methods of analysing 
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converting devices, the level of accuracy is determined by 
the following assumptions: 
a) valves are represented by equivalent circuits that
correspond to static volt-ampere characteristics; 
b) L, C - elements of converter circuits are represented
without losses and parasitic parameters and their 
characteristics are linearized;
c) transformers are represented by an equivalent circuit,
which corresponds to the linearized characteristic of the 
magnetization of the core without taking into account 
hysteresis phenomena and leakage fluxes, and most often - 
by an equivalent circuit of an ideal transformer. 
When calculating transient and steady-state processes of 
airspace, the analysis method based on the Laplace 
transform is often used. The way of applying the Laplace 
transform (operator method) is preferable for the following 
reasons. Firstly, the use of the classical theory of differential 
equations requires considerable preparatory work to 
compose a system of first-order differential equations. 
Secondly, the compilation of the system of differential 
equations of the n order with respect to the currents and 
voltages of interest is made purely formally and is not 
associated with the physics of processes in the substitution 
scheme. When implementing the operator method, the 
operator substitution diagram carries information about the 
development of the process at the stage under consideration 
and the initial conditions of this development. Besides, 
finding the image of currents and voltages of interest is 
reduced to a simple task of DC circuit calculation, which is 
the operator substitution diagram. In contrast to numerical 
integration 
methods, 
differentiation 
and 
integration 
procedures are excluded here. The analysis is reduced to 
working with algebraic equations. This eliminates the 
problem of choosing the integration step, convergence, etc. 
This justifies the choice of the operator method. In this case, 
for all possible working structures of the power circuit, 
equivalent operator equivalent circuits (OEC) are drawn up. 
Then, for these OECs, equations are written for the operator 
images of the variables. Solving these equations, images are 
found, and then the originals of the sought currents and 
voltages [22, 25]. 
The use of the operator method for the analysis of 
transient and steady-state processes in the above classical 
form leads, along with large preparatory procedures in the 
development of a mathematical model, to a deterioration in 
the quality of the resulting model: an increase in the amount 
of required memory, the complexity of the algorithm and a 
low speed of calculations. These disadvantages are primarily 
associated with a large number of equivalent OECs used in 
the development of a mathematical model. 
In this regard, in the development of a mathematical 
model based on the operator method, it is proposed to use 
such equivalent circuits (hereinafter referred to as universal 
equivalent circuits), which would be equivalent to several 
possible structures of the power circuit. Such an approach to 
the development of a mathematical model, while retaining 
the advantages of the operator method, will make it possible 
to reduce the size of the model, reduce the required volume 
and time of calculation, and, thereby, will allow to more 
efficiently solve the problems of analysing the schemes 
under study. Therefore, in this work, to create a 
mathematical model of the investigated circuits, an operator 
method was adopted using universal OECs. When 
developing mathematical models, the following generally 
accepted assumptions were made: ideal valves, their 
switching is instantaneous, the magnetizing current of the 
power transformer is zero [22-24]. 
Thus, the development of mathematical models of the 
investigated VC circuits in the work is carried out in the 
following sequence: 
a) the set of possible types of equivalent circuits is
determined, which take place with the selected method of 
excitation and stabilization; 
b) universal equivalent circuits are drawn up, images and
originals necessary for calculating currents and voltages are 
displayed; 
c) the possible ways of the process development are
analysed and the sequence of changing the types of 
equivalent circuits at the timing intervals is identified, the 
boundary conditions for changing the stages of the process 
development are formed; 
d) a block diagram of the algorithm for calculating the
transient process is drawn up based on the search for the 
fulfil of the boundary conditions, taking into account the 
selected method of excitation and regulation. 
The analysis of possible OECs shows that the complete 
calculation of the transient process of a parallel ACI with a 
compensating device (CD) can be performed on the basis of 
two universal OECs. They differ from each other by the 
operating state of the CD unit. One of them corresponds to 
the disabled and the other to the enabled state of the CD. 
Their use for the analysis of different mode situations is 
determined by the corresponding initial conditions. For each 
of these universal OECs, calculation formulas for the 
instantaneous values of the currents and voltages sought are 
derived [22, 25]. 
III. RESULTS
Programming the obtained analytical expressions and 
making up the algorithm of the process paths, a 
mathematical model was obtained. On its basis, the 
calculation of transients was made, the timing diagrams of 
the currents and voltages sought, presented in Figs. 2 and 3, 
were constructed. 
Fig. 2. Timing diagrams of currents and voltages at start ACI. 
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Fig. 3. Timing diagrams of currents and voltages during load shedding: a) 
independent excitation; b) combined excitation. 
The timing diagrams (Figs. 2, 3) correspond to the 
power supply scheme based on a parallel inverter operating 
in the continuous inverter input current mode. It should be 
noted that the methodology described above can be used to 
develop mathematical models for calculating transient and 
steady-state processes not only for the stand-alone parallel 
inverter, but also for the parallel-series, series-parallel, 
serial-parallel, and in inverters with cut-off gates and with 
two-stage switching. The models are able to accommodate 
variations in the input voltage and load, as well as variations 
in both, taking into account the selected excitation method 
of the inverter, both at continuous and intermittent input 
currents to the inverter. An important aspect of the 
application of this simulation technique to engineering 
applications is the visualization of simulation results, which 
is of practical importance as it is difficult to imagine that an 
engineer could effectively use a simulation program that 
presents the output data as a file containing arrays of 
numerous elements.
In the article the methodology of mathematical 
modelling of valve converters with periodically changing 
structure of power circuit is considered by the example of 
AСI. The model takes into account the operation algorithm 
of the converter control system and load variation. This 
methodology can be applied to other types of power 
electronic valve converters with periodically changing 
structure of power circuit, such as rectifiers, autonomous 
voltage inverters, resonant inverters. 
IV. CONCLUSIONS
1.
Most of the methods have a certain relationship and
complement each other in this connection, in order to obtain 
reliable results that ensure the obtaining of technical 
parameters of the elements of valve converters based on 
ACI, ensuring the fulfilment of the requirements of the 
technical specifications for the design of the facility (taking 
into account the significant nonlinearity of valve elements 
with a variety of control systems) it is advisable to use a 
combined modelling technique. 
2.
For a variety of ACI circuits, the equivalent circuits
between the repetition periods are determined by the same 
differential ratios, differing from each other only by the 
coefficients of the sought variables. Therefore, considering 
the ACI as a linear system with some restrictions, in this 
work it is proposed to take for research the mathematical 
apparatus based on the Laplace transform and focus on the 
study of the coefficients of the operator equations of 
periodically changing structural schemes. The ratios of the 
coefficients of these equations can be considered as the 
main factors affecting the quality indicators of the operating 
modes of the considered ACI. 
3.
Taking into account the processes of energy
redistribution between the source and the load, as well as 
between the phases of the inverters in transient modes and at 
the moments of switching the valves, in the models and 
programs for calculating the ACI, requires their separate 
consideration from the processes that are established within 
the same repeat interval. When determining these dynamic 
processes, it becomes necessary to use the Laplace 
transform in conjunction with the method for determining 
the instantaneous values of currents and voltages of ACI at 
the boundaries of repeatability processes. 
4.
This methodology can be applied to other types of
power electronic valve converters with periodically 
changing structure of power circuit, such as rectifiers, 
autonomous voltage inverters, resonant inverters. 
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