Open-Loop Single-Phase Space State Model and Equivalent Circuit of a Non-Conventional Three-Phase Inverter

Vinícius Melo 1 , Alexandre Melo 2 , Walbermark Santos 3 , Jussara Fardin 3,* and Lucas Encarnação 3 1 Department of Automation and Control, Instituto Federal do Espírito Santo Campus Serra, Serra-ES 29173-087, Brazil; secchin@ifes.edu.br 2 Department of Electrical Engineering, Instituto Federal do Espírito Santo Campus Vitória, Vitória-ES 29040-780, Brazil; amelo@ifes.edu.br 3 Department of Electrical Engineering, Federal University of Espírito Santo Campus Goiabeiras, Vitória-ES 29075-910, Brazil; walbermark.santos@ufes.br (W.S.); lucas.encarnacao@ufes.br (L.E.) * Correspondence: jussara.fardin@ufes.br


Introduction
Alternating current (AC) voltages generated with amplitudes greater than the input source voltage are among the basic prerequisites in many applications such as three-phase electric drives, green energy sources, and uninterruptible power supply (UPS) systems, among others. The single-stage non-conventional inverters' topologies research for these applications have been increased lately [1][2][3][4]. In most of the cases, analyzing and solving the models are not simple due to the high system order and complex equations. Then, simplifying the inverter circuit in order to reduce its order model leads to a computational effort reduction. Consequently, it become more suitable in real-time control system applications [5].
Among the non-conventional inverters, the direct current-alternating current (DC-AC) topology based in DC-DC Buck-Boost converters has been noticed due to the possibility to obtain output voltage higher or lower than the input voltage. The Boost topology is not suitable for this application due to the fact that it yields an output voltage higher than or equal to the input voltage. In [6], the analysis of a single-phase inverter was presented based on two DC-DC bi-directional Buck-Boost converters, but all analysis was made using the full model of inverter. In [7], a single-phase differential DC-AC Buck-Boost converter is analyzed, and a novel control technique for variable frequency applications is proposed.
Usually, the DC-AC Buck-Boost application is focused in PV systems using single-phase inverters, such as [8][9][10][11][12], without using three-phase topologies due to its complexity. A three-phase Buck-Boost AC voltage regulator with twelve switches was presented in 2016 [13], and, in [14], simulation results of a six switch three-phase Buck-Boost inverter with resistive load. However, none of them approached a simplified model of the inverter for analysis and simulation purposes.
Recently, many topologies using Buck-Boost converters as a DC-AC converter has been increased such as presented in [15], where it was proposed an inverter with three Buck-Boost converters in bridge topology. In [16], a bidirectional twisted single-phase single-stage Buck-Boost DC-AC converter is described for applications with PV arrays or storage batteries, and, in [17], a single-stage isolated three-phase four-leg inverter has been proposed.
In this paper, the single-phase space state model and equivalent circuit of the non-conventional three-phase inverter with six IGBTs, based on Buck-Boost topology, is proposed. The full six order model for a resistive load is presented, and the simplification circuit obtained by reducing the full model to a simplified second order model, taking one leg of the inverter as reference. The simulations performed in the PLECS software to compare results of single-phase and three-phase circuit inverters for a resistive load confirm the advantages of using the simplified model. Finally, a three-phase 200 W buck-boost DC-AC converter prototype with resistive, inductive, and capacitive loads are implemented in order to obtain experimental results to support simulations and confirm that the theoretical proposed analyses are in accordance. The results show that the proposed circuit and model achieve a reliable and accurate solution that can be used to design and represent the dynamic behavior of a full model inverter in steady state.

The Three-Phase Line Voltages
The three-phase Buck-Boost inverter analyzed in this paper is shown in Figure 1. The converter is based in three bi-directional DC-DC Buck-Boost converters (legs 1, 2, and 3), which are able to generate three-phase AC voltages whose peak values can be greater or lower than the DC input voltage in a single stage. If each Buck-Boost converter is controlled to track DC-biased (unipolar) sinusoidal references that are phase shifted by 120 • from one another, the output line voltages will be a three-phase balanced sinusoidal system. The capacitors' voltage references can be written as shown in Equation (1): The terms V DC and v k = V peak sin(ωt + (1 − k)120 • ) represent the DC-bias and the sinusoidal voltage component of capacitors, respectively, where k = 1, 2, 3 represents the phase or inverter leg.
The line voltages are given by: Consequently: Based on Equations (2)-(4), these results give rise to the conclusion that v 12 , v 23 and v 31 are pure sinusoidal voltages with no DC-bias.

The Load Phase Voltages
Taking the float N reference, and considering that the three-phase load is balanced, it can be shown that: By replacing Equation (1) in (5), it can be obtained: Expanding the Equations (6) for all three-phases and adding them yields: Replacing (7) in (6), the load phase voltages can be written by:

The Three-Phase Model
The non-conventional three-phase inverter space state model shown in Figure 1 is given by Equation (9). In order to obtain the full model, each leg equation was derived through the determination of the respective line current in function of the load and capacitors' voltages. The modeling does not take into account the non-idealities of the inverter components. This space state representation was obtained taking into consideration the continuous conduction model of operation and the balanced three-phase load, even though this model can be used under unbalanced loads. This application will not be analyzed in this work since a single-phase model can only be useful for balanced systems.

The Equivalent Single-Phase Circuit Model
In order to obtain the equivalent single-phase circuit model, one of the inverter legs was taken as reference. For convenience, the first leg was chosen as shown in Figure 2, where Z represents the equivalent load of leg 1 equivalent circuit. Taking the Equation (5), the load voltage v Z can be obtained in accordance with the following steps:

Equivalent load
From Equation (10), it can be concluded that the equivalent load Z can be replaced by a load resistance in series with a DC source. By doing so, the equivalent single-phase circuit model of single stage Buck-Boost three-phase inverters can be represented by the electric circuit shown in Figure 3.  The DC source voltage level does not affect the AC output voltage, since its value is greater than the peak sinusoidal value of capacitor voltages, in order to avoid negative voltage at capacitors. In this work, the DC voltage level of capacitors is determined from Equation (11): The adopted value for v Ckmin in the simulation and prototype was 0.3V peak .

Steps of Operation
Due to the bi-directional current characteristics in each switch of equivalent circuit model presented in Figure 3, the converter will operate differently in each half cycle R load voltage, as shown in Figure 4. In the positive half cycle, only the IGBT T A can conduct when switched and diode T B can be forward biased. On the other hand, in the negative half cycle, only the IGBT T B can conduct when switched and the diode T A can be forward biased, as can be seen in Figures 5a and 6a.  Step 1: In this step, only the IBGT of switch T A is ON, and the switch B is OFF. During this step, the inductor is charged and the capacitor supplies energy to the resistive load and the V DC source, as shown in Figure 5a.
Step 2: In this case, only the diode of switch T B is ON, and the switch A is OFF. During this step, the inductor charges the capacitor and supply energy to the resistive load and the V DC source, as shown in Figure 5b.

2.
Negative half cycle: Step 3: In this step, only the diode of switch T A is ON, and the switch B is OFF. Now, the inductor supplies energy to the V g source. The V DC source supplies energy to the resistive load and charges the capacitor, as shown in Figure 6a.
Step 4: In this case, only the IGBT of switch T B is ON, and the switch A is OFF. Here, the inductor is charged from the V DC source and the capacitor. The resistive load is supplied by V DC source also. Figure 6b shows this step.

The Single-Phase Space State Model
As the power converter includes the duty cycle operation, the linear system analysis needs to be derived according to its operation steps described in the last section for each operation region.

Positive half cycle:
From Figure 5a, analysis of this circuit shows the follow equations: Considering i L and v C as state variables, Equations (12), (13) and (14) can be collected into state space representation given by From Figure 5b, it can be written: Gathering the Equations (17)-(19), the follow state space representation can be obtained: The average state equations for the positive half cycle can be constructed from Equations (15), (16), (20), and (21), which will lead to:

Negative half cycle:
From the circuit present in Figure 6a, the following equations can be written: In the same way, from Figure 6b, it can be obtained: Comparing the obtained equations from operation steps 1 and 2, it can be concluded that these steps have identical linear equations. The same is observed when comparing steps 3 and 4. This simplification is only possible because the semiconductors and inverter passive elements non-idealities were not taken into account.
Since then, the average state equations for the negative and positive half cycles will be the same. This conclusion rise up the average state equations that describe the equivalent single-phase space state model can be given by (30) and (31).
From Equation (30), the ideal duty cycle ratio function can be obtained, given by: Applying Equation (32) to a single-phase equivalent model, the duty cycle ratio can be expressed as: The same equation can be applied to each leg of three-phase Buck-Boost inverter. As a result: Equations (33) and (34) were used to obtain the simulation results presented in the next section.

Simulation Results
In order to validate the presented theory, simulations were performed in the PLECS software for three-phase and equivalent single-phase circuits shown in Figures 1 and 3, respectively, working in open-loop. The results presented here were obtained for the following specifications shown in Table 1. The schematic circuit simulating a three-phase inverter and its equivalent single-phase circuit on PLECS are shown in Figures 8 and 9, respectively. The simulation results were obtained for three load types: R load, RL load, and RC load. The RL case simulated a load transient condition in order to analyze the behavior of single-phase model under variable loads, in comparison with the full model.
The duty-cycle was calculated in accord with Equations (33) and (34) for single-phase and three-phase models, respectively. Their waveforms are presented in Figure 10.

R Load
In this case, a pure resistive 18Ω load is simulated. The main three-phase inverter circuit waveforms in steady state are presented in Figures 11-13. In Table 2, the RMS (root mean square), average, and peak-to-peak values for the currents and voltages are presented.   Figure 11a,b show, respectively, the three-phase model inductor currents and the comparison between the inductor currents of three-phase and single-phase models. Figure 12a,b show the three-phase model voltage capacitors, and the comparison of three-phase and single-phase models, respectively. Finally, Figure 13a,b show the three-phase model load phase voltage and the comparison for single and three-phase models, respectively. The small difference between both models and the reference is due to the inverter running in open-loop mode and there is no control of the output voltage inverter. On the other hand, these differences do not invalidate the proposed model. The calculus approximation can also yield some differences between the final values compared with the expected values. As it can be observed, the inductor currents and capacitor voltages for both models are quite similar as expected. The instantaneous average output load voltage for both models are very close to the expected reference waveform as shown in Figure 13b. This proves that the simulations performed have shown the single-phase model can represent with exactness the dynamic behavior of three-phase inverters in steady state.

RL Load
In this scenario, a 12 Ω resistor in series with a 22.1 mH inductor is simulated, whose impedance in 60 Hz is about 12 + j8.26Ω, with a power factor about 0.82. The main three-phase inverter circuit waveforms in steady state are presented in Figures 14-17. In Table 3, the RMS (root mean square), average, and peak-to-peak values for the currents and voltages are presented.     Figure 14a,b show, respectively, the three-phase model inductor currents and the comparison between the inductor currents of three-phase and single-phase models. Figure 15a,b show the three-phase model capacitor voltages and the comparison of three-phase and single-phase models, respectively. Figure 16a,b show the three-phase model load phase voltage and the comparison for single and three-phase models, respectively. As it can be observed, the inductors' currents and capacitors' voltages are quite similar as expected, as well as for the load phase voltage. Finally, Figure 17a shows the comparison between load current for both models, and Figure 17b shows the voltage and current at the RL load, where the shifting between them equal to 34.52 • can be observed, confirming that the power factor expected (cos 34.52 • = 0.82).

RC Load
In this scenario, a 6 Ω resistor in series with 235 µF capacitor is simulated, whose impedance in 60 Hz is about 6 + j11.29 Ω with a power factor about 0.43. The main three-phase inverter circuit waveforms in steady state are presented in Figures 18-21. In the Table 4, the RMS (root mean square), average, and peak-to-peak values for the currents and voltages are presented.     Figure 18a,b show, respectively, the three-phase model inductor currents and the comparison between the inductor currents of three-phase and single-phase models. Figure 19a,b show the three-phase model voltage capacitors, and the comparison of three-phase and single-phase models, respectively. Figure 20a,b show the three-phase model load phase voltage, and the comparison for single and three-phase models, respectively. As it can be observed, the inductors' currents and capacitors' voltages are quite similar as expected, as well as for the load phase voltage. Finally, Figure 21a shows the comparison between load current for both models, and Figure 21b shows the voltage and current at RC load. The shifting between them of 62.62 • can be observed that confirms the power factor expected (cos 62.62 • = 0.45).

RL Load with Transient
The series RL load in this case is initially made by a 24.7 Ω equivalent resistance with a 22.1 mH inductor, whose impedance in 60 Hz is about 24.7 + j8.26 Ω. After 100 ms, the resistive part is decreased by 10 Ω during 200 ms. Then, the resistive part goes back to its original value. Figure 22 shows the waveforms of RL load current and voltage waveforms, and state variables for single-phase and three-phase models.
In accordance with Figure 22, the capacitor and load phase voltages remain constant during this transition. The changing in the load and inductor currents is noticed when the load impedance changes its value. This behavior is expected, once the electrical current is inversely proportional to the impedance. The simulations show that with a little change in the load impedance, even the inverter running in open-loop, there is no disturbance in the output voltage inverter for both models. This allows for the conclusion that the single-phase proposed model is able to represent the dynamic behavior of the state variables even in transient loads.

Experimental Results
In order to support the simulation results of the proposed single-phase model, a three-phase Buck-Boost inverter prototype was performed as shown in Figure 23. The same values as indicated in Table 1 were implemented at the prototype for all cases analyzed. The duty cycle ratio is calculated at each step time of 50 µs in accordance with Equation (32).

R Load
The main experimental results are shown in Figures 24a,b and 25a,b in which the system operates under constant 18 Ω resistive load. Table 5 shows the electrical quantities for the simulated single-phase and three-phase prototype.  The practical results validate the proposed single-phase equivalent circuit and space state model. In accordance with Table 5, the simulated and experimental results are quite similar. Small differences are expected due to the model not taking into account some non idealities, such as resistances and parasitic inductances present in a real circuit, as well as switching noises that can cause some voltage/current sparks in the waveforms. The maximum static gain obtained ( v Cmax V g ) was 2.67.

RL Load
In this case, an identical RL used in simulations presented in Section 4.2 is fed by inverter. The main experimental results are shown in Figures 26-28 in which the system operates under constant load. Table 6 shows the electrical quantities for simulated single-phase and experimental three-phase prototype. These experimental results are in compliance with the simulations performed in Section 4.2.
In accordance with Table 6, the simulated and experimental results for RL load are quite similar. It leads to the conclusion that the single-phase model can represent the dynamic behavior of three-phase models under inductive balanced loads with precision.

RC Load
In this scenario, an identical RC used in simulations presented in Section 4.3 is fed by inverter. The main experimental results are shown in Figures 29-31 in which the system operates under a constant load. Table 7 shows the electrical quantities for the simulated single-phase and three-phase prototype.    In this case, taking the values shown in Table 7, small differences can be observed between the simulated single-phase model and experimental results of the inverter. These differences are due to the capacitors used in the load are electrolytic capacitors. These type of capacitors have large leaking currents, which can cause some variations in capacitance values. These facts do not invalidate the proposed model.

RL Transition Load
The series RL load used in this experimental case has the same parameters described in Section 4.4. Initially, it was made by a 24.7 Ω equivalent resistance with a 22.1 mH inductor. In the first transition, the resistive part of the RL load was decreased by 10 Ω. The waveforms of RL load phase voltages and currents in the first transition are shown in Figures 32a and 33a. The waveforms for inductor currents and capacitors voltages are shown in Figures 34a and 35a, respectively. In the second transition, the resistive part of the RL load backs the original value. The waveforms for load phase voltage and current, inductor currents, and capacitor voltages for the second transition are shown in Figures 32b,  33b, 34b and 35b. Several different situations were performed in order to compare the single-phase model simulations with the real three-phase inverter. The experimental results are in compliance with simulation results presented in Section 4.4, and validate the applicability of the single-phase model to represent the full model inverter in steady state. Furthermore, it can be useful to design and calculate the inductance and capacitance of inverter inductors and capacitors.

Conclusions
This work has presented an equivalent single-phase circuit of a three-phase Buck-Boost inverter with balanced load for analysis and modeling purposes. In order to confirm the study results, a 200 W three-phase inverter prototype with several types of output load was implemented. The behavior in steady state of the simulated single-phase proposed model demonstrated that it can be used with property in order to obtain previous simulated results of the three-phase Buck-Boost inverter. The simulated and experimental results show clearly that the single-phase equivalent circuit can represent each leg of the three-phase Buck-Boost inverter model with exactness. Furthermore, its simulation demands less computer effort than the three-phase model. Thus, it is useful for a design of the inverter, predictive control strategy, or other techniques that are crucial for solving the model.

Conflicts of Interest:
The authors declare no conflict of interest.