Passivity-Based Control Design Methodology for UPS Systems

This paper presents a passivity-based control (PBC) design methodology for three-phase voltage source inverters (VSI) for uninterruptable power supply (UPS) systems where reduced harmonic distortions for the nonlinear load, reduced output voltage overshoot, and a restricted settling time are required. The output filter design and modification for efficient control and existing challenges with the assignment of scaling coefficients of the output voltage, load, and inductor currents are addressed and analyzed. Notably, special attention is given to the modulator saturation issue through implementing an accurate converter model. Applications of the two versions of PBC in three-phase voltage source inverters using stationary αβ and rotating dq frames for a constant frequency of the output voltage are presented. Furthermore, the influence of the PBC parameters on the power converter performance is investigated. A comparative simulation and the experimental results validate the effectiveness of the presented passivity-based control design methodology.


Introduction
The control design of single-phase and three-phase voltage source inverters (VSI) for uninterruptable power supply (UPS) systems has been extensively discussed for the last four decades [1][2][3][4][5][6]. The PBC control strategies are implemented in renewable energy sources and energy storage systems [7]. Typically, two control types for the UPS systems are implemented in order to meet the standard requirements that are defined by IEC 62040-3 [8] for nonlinear rectifier resistive-capacitive (RC) load types. The first one is based on the single input single output (SISO) approach where only the output voltage is measured and controlled, while in the second approach, a multi input single output (MISO) controller is employed to regulate the output voltage based on the output current, inductor current, and the output voltage measurements. A proportional-integral-derivative controller (PID) is a standard SISO control (e.g., designed based on the quasi-continuous transfer function [9]). The other PID-like controller is based on the coefficient diagram method (CDM) [10]. The results of such a control can be satisfactory but treating the current as an unmeasured disturbance can lead to higher distortions of the output voltage than in the case of a MISO controller. Very good results can be obtained with a SISO controller using a plug-in repetitive controller (RPC) in the outer negative feedback loop, which works as a harmonics generator [3]. We designed an RPC using the zero phase error tracking compensation (ZPETC) technique [11]. The plug-in zero phase repetitive controller can be designed quite simply for a CDM controller in the inner control loop of an inverter [12]. However, an RPC has one serious disadvantage-it remembers the whole previous fundamental period (in some solutions or a three-wire delta balanced load (Figure 1), 2-level PWM: (2) Energies 2019, 12

Frames
The system with delta Δ or star Y loads presented in Figure 1 can be described by the initial (they will be modified further) space equations for small signal space variables in the abc frame. The input and output voltages for the star load Y are related to the zero-sequence vector (Equation (4)). In balanced systems, the neutral voltage v0 = vγ = 0. , The sinusoidal PWM modulation is preferred for UPS systems (there is a third harmonic of the line to neutral voltages in the space vector modulation (SVM)). The modulated space vector can be presented in the αβγ coordinates of the stationary frame after using the Clarke [5,33] transformation. For three-wire delta or star loads and for four-wire balanced systems, there is not a zero-sequence vector (γ coordinate) of the used voltage or current variables [34] and the αβ frame is used. The three-phase voltage or current components are defined as Equation (5)  For example, we assume R load = 43 Ω, f s = 12,800 Hz, L f = 1.1 mH, and C f > 1.8 µF for the delta load. These values are sufficient for reducing the maximum amplitude of the ripple harmonics to 3%. However, for 30 years, researchers of inverter controls have used approximately 50 µF [2,17,24]. The delay of every digital PWM control is at least one switching period. We store the calculated value in the PWM register and in the next switching period the width of the pulses is changed. Let us assume a step load decrease. The inductor works as the current source and most of the excessive inductor current flows through the filter capacitor C f , thereby increasing the output voltage (e.g., for a one phase for ∆I load = 5 A, switching period T s = 78 µs for switching frequency f s = 12,800 Hz, for C f = 2 µF, we will receive the unacceptable voltage pike ∆V out = 195 V, while for C f = 50 µF-only ∆V out = 7.8 V). Figure 2a shows the problem in an actual three-phase circuit. The delays in the inverter control loop will be longer than one switching period (the delay of the PWM modulator) while the overshoot will be the same as for the open loop circuit. However, the oversized filter capacitor has disadvantages-there is a much higher reactive power in this capacitor and much larger capacitor currents, which increase the power losses in the parasitic serial resistances and the output voltage has a much longer settling time (Figure 2b). The quality of the output voltage control is better for the low modulation index M because of the higher possible increase (1 − M)V DC of the first harmonic of the inverter bridge output voltage. Output voltage control quality is more efficient for lower output filter inductance values [32]. For a nonlinear rectifier RC load, the value of the voltage over the filter inductor should be steeply increased when the rectifier begins to conduct and a pulse current flows to the load capacitor. Hence, the maximum value of M [32] should be limited to approximately Equation (3):

Frames
The system with delta Δ or star Y loads presented in Figure 1 can be described by the initial (they will be modified further) space equations for small signal space variables in the abc frame. The input and output voltages for the star load Y are related to the zero-sequence vector (Equation (4)). In balanced systems, the neutral voltage v0 = vγ = 0. , The sinusoidal PWM modulation is preferred for UPS systems (there is a third harmonic of the line to neutral voltages in the space vector modulation (SVM)). The modulated space vector can be presented in the αβγ coordinates of the stationary frame after using the Clarke [5,33] transformation. For three-wire delta or star loads and for four-wire balanced systems, there is not a zero-sequence vector (γ coordinate) of the used voltage or current variables [34] and the αβ frame is used. The three-phase voltage or current components are defined as Equation (5) The state space model in the stationary αβ frame has the advantage of decoupling the models for both axes. Control in the stationary frame has two shifted π/2 sinusoidal waveforms of the M max < 0.65 (too low in practice) for f m = 50 Hz, L f = 3 mH, and R serial = 2 Ω is equal to the sum of all of the serial resistances in the load current path when the rectifier conducts including the equivalent series resistance (ESR) of the load capacitor. We used a lower value M = 0.3 in the simulation models and in the breadboard inverter to fully show the advantages of a PBC control. However, this value is too low for market solutions. Sometimes, it is better to permit higher distortions of the output voltage and use a modulation index M close to unity in order to better utilize the input DC voltage (modulation index close to unity leads to higher efficiency of a DC/AC inverter).

Initial State Space Models of Three-Phase Inverters in the Stationary αβ and Rotating dq Frames
The system with delta ∆ or star Y loads presented in Figure 1 can be described by the initial (they will be modified further) space equations for small signal space variables in the abc frame. The input and output voltages for the star load Y are related to the zero-sequence vector (Equation (4)). In balanced systems, the neutral voltage The sinusoidal PWM modulation is preferred for UPS systems (there is a third harmonic of the line to neutral voltages in the space vector modulation (SVM)). The modulated space vector can be presented in the αβγ coordinates of the stationary frame after using the Clarke [5,33] transformation. For three-wire delta or star loads and for four-wire balanced systems, there is not a zero-sequence vector (γ coordinate) of the used voltage or current variables [34] and the αβ frame is used. The three-phase voltage or current components are defined as Equation (5).
The state space model in the stationary αβ frame has the advantage of decoupling the models for both axes. Control in the stationary frame has two shifted π/2 sinusoidal waveforms of the reference voltages. This can be a disadvantage when the reference waveform has a variable frequency. We can use a Park transformation of the stationary αβ frame and present the space vector in the dq frame rotating with ω m (angular fundamental frequency of the output voltages). The models for the rotating dq frame are not decoupled (the state variables from one axis influence the other axis variables), which is a disadvantage of this transformation. The problem of decoupling can be solved in the control design [27]. The dq frame is used in the digital motor control of the induction machines. This paper will present the use of both stationary αβ and rotating dq frames in three-wire, three-phase VSI control systems. For a delta load (Figure 1), the Clarke transformation and reverse Clarke transformation are Equation (6) [5].
For a star load (Figure 1), the Clarke transformation and reverse Clarke transformation are Equation (7).
The vector d αβ (Equation (8)) of the output line currents is treated as a disturbance vector [27]. The index α, β or d, q means that the description is separate for the α or d and β or q orthogonal axes; the index αβ or dq means that the description concerns both axes simultaneously, e.g., the vector of disturbance variables d αβ and variables d α,β are Equation (8).
The initial type of x i state vector, and control and output vectors for the delta and star load (m α , m β are the control coefficients of α, β voltages) are Equation (9).
The initial state space model of the three-phase inverter ( Figure 1) in the stationary αβ frame is Equation (10). The only difference between the matrices for the delta and star loads is the value of the equivalent filter capacitor C fe : for the delta load C fe = 3C f and the star load C fe = C f .
The Park transformation and the reverse Park transformation for the balanced load are Equations (11) and (12).
x d x q = cos ω m t sin ω m t − sin ω m t cos ω m t x α x β (11) A balanced system (X 1m = X 2m = X 3m = X m ) in the steady state is described by Equation (5) after a Clarke transformation (Equation (6)), x α = X m cosω m t, x β = X m sinω m t, and after a Park transformation (Equation (11)), x d = X m , x q = 0-two constant values. This is the main advantage of using the dq frame in a power supply system [34]. The other definition of three-phase components in Equation (5) leads to different results. We use the description that concerns both axes simultaneously because the α or d and β or q variables should be included together in Hamiltonian (Equation (16))-they both participate in storing energy in the system. The final definition of the state variables x αβ,dq (Equations (13)-(15)) is defined in [27]. For all types of PBC control [21], we use the function H(x) (Hamiltonian) of the total energy stored in the system. For the αβ or dq frames, H(x) is equal to Equation (16).
We can notice that for the αβ or dq frames: The space Equations (18) are called the "perturbed Port-Hamiltonian model" of a physical system [13][14][15][16]21,29,35] for the αβ or dq frames because we can substitute (17) in (18). The vector of the input variables m αβ,dq and the vector of disturbance variables d αβ,dq is Equation (19).

Three-Phase Voltage Source Inverter Controller Design
A passivity-based controller [13][14][15][16][17][18] was developed using an interconnection and damping assignment (IDA-PBC), which enables the control of a three-phase inverter to be decoupled in the dq frame [27][28][29]. For the presented inverter model cases, we assume that the coefficients of the matrixes J αβ,dq , R αβ,dq , G αβ,dq , and D αβ,dq do not depend on the state variables. In an actual inverter, the inductance of L f can be dependent on the inductor current and switching frequency [36][37][38].
The controller is designed to follow the reference state variables of the error vector (Equation (21)) in a closed loop system [27].
The closed loop dynamic (Equation (22)) of the tracking error is described in [27]. .
where: The J αβ,dq,a and R αβ,dq,a matrices are used to control a closed loop system. The closed loop energy function H(e αβ, dq ) (Equation (24)) is defined to ensure that the equilibrium is asymptotically stable [36] and will be achieved if H(e αβ,dq ) has the minimum in x αβ,dq,ref (Equation (25)).
dH(e αβ,dq ) The requirement (Equation (26)) is met [28] for a positively defined matrix (R αβ,dq + R αβ,dq,a ). The matrix R αβ,dq,a of injected damping R i (gain of the current error) and conductance K v (gain of the voltage error) is defined as Equation (27) and should enable requirement-Equation (26) to be met.
As the values of resistance R i and conductance K v increase, the speed of tracking error convergence increases but the oscillations of the output voltage can arise ( Figure 3).
, 0 0 1 0 In the αβ frame, Jαβ,a = 0 because it is not necessary to decouple the α, β variables. The roots λ1,2 (Equation (32)) of the characteristic polynomial (Equation (31)) of a closed loop system (Equation (22)) should be located in the left-half of the s-plane. Theoretically, this will always be fulfilled for the requirement of Equation (28). The location of roots λ1,2 on the complex plane theoretically ensures the stability of the PBC system. However, Figure 3 presents the simulations of the vo,uv line-to-line voltage, the io,u line current and the α axis control waveform for Rfe = 1 Ω, Cf = 50 μF, Lf = 3 mH and the inverter nonlinear rectifier RC2 load (Rload = 47 Ω, Cload = 470 μF) for PBC in the stationary αβ frame. The modulation index was From Equation (26), it is possible to initially define the range of the implemented gains (Equation (28)).
In the dq frame, J dq,a should decouple the voltage and current equations from the dq axes (Equations (29) and (30)).
In the αβ frame, J αβ,a = 0 because it is not necessary to decouple the α, β variables. The roots λ 1,2 (Equation (32)) of the characteristic polynomial (Equation (31)) of a closed loop system (Equation (22)) should be located in the left-half of the s-plane. Theoretically, this will always be fulfilled for the requirement of Equation (28).
The location of roots λ 1,2 on the complex plane theoretically ensures the stability of the PBC system. However, Figure 3 presents the simulations of the v o,uv line-to-line voltage, the i o,u line current and the α axis control waveform for R fe = 1 Ω, C f = 50 µF, L f = 3 mH and the inverter nonlinear rectifier RC 2 load (R load = 47 Ω, C load = 470 µF) for PBC in the stationary αβframe. The modulation index was set at M = 0.3. The lower value of M would give better results of control [32] but would be completely unrealistic. Figure 3 shows that for K v = 0 (the basic PBC), the output voltage error amplification is too low for R i = 10 Ω or 20 Ω and the THD v coefficient is high and for the K v > 2 Ω −1 for R i = 10 Ω or R i = 20 Ω, there are oscillations of the αaxis control waveform, which results in the oscillations of the output voltage. The saturation of the αaxis control waveform for K v R i > 40 inhibits the further reduction of output voltage distortions. For further simulations for both controllers, Improved PBC v.2 (IPBC2) and IDA-PBC, M = 0.3, R i = 10 Ω and K v = 2 Ω −1 were selected because of the relatively low THD v , the lack of control oscillations, and the lack of any αaxis control waveform saturation. The control law (Equation (33)) was determined in the αβframe (Equations (34) and (35)) and the dq frame (Equations (36)-(39)) by subtracting both sides of the Equations (16) for the αβframe or Equation (23) for the dq frame from both sides of Equation (28).
The difference control law for the stationary αβ frame (Equations (34) and (35)) for K v = 0 is the same as the control law for the conventional PBC [22]. Introducing K v (control of the output voltage error) makes the control law similar to the IPBC from [22]. However, the addition of the derivative of the output voltage error in the final control law should enable the dynamic changes in the output voltage to be reduced faster. This type of PBC will further be called IPBC2.
For a three-phase AC power supply in the αβ frame, the difference of reference voltage

Modeling and Measurement of a Three-Phase Inverter with IPBC2 and IDA-PBC Control for Standard Loads
The calculations of IPBC2 in the stationary αβ frame using only the Clarke transformation without any interactions and without the angular speed ω m as additional input are faster than the calculations of IDA-PBC [27] in the rotating dq frame. Both control systems were initially tested using the simulation models in MATLAB-Simulink and are implemented in the experimental model that was controlled with an STM32F407VG microprocessor. The switching frequency was f s = 12,800 Hz (256 switching periods in one fundamental period). For f s = 12,800 Hz, we have less than 78 µs for measuring three output voltages, three output currents, and three inductor currents (there are three independent Analog-to-Digital Converters (ADC)) in a three-phase inverter, calculating three Clarke transformations and executing the control laws for PBC in the stationary frame and the reverse Clarke transformations.
In IDA-PBC we should additionally calculate three Park transformations and three reverse Park transformations. The whole microprocessor program is based on PWM interrupts that call ADC conversions. The first and the second ADC interrupts after a conversion is finished call the next ADC conversion, the third ADC interrupt calls the subroutines with the Clark and Park transformations (only for IDA-PBC), calculates the control laws, reverses Clark and Park transformations, and finally the results of the calculations are stored in the three registers of PWM comparators, which will change the output pulse width in the next switching period. We can easily check that all of these activities do not exceed 78 µs-the lower priority interrupts and the main loop procedures should be available and executed.
The dynamic delta load that was tested was ∆470 Ω switched to ∆470||47 Ω and vice versa. The nonlinear rectifier RC 1 load (C load = 100 µF, R load = 47 Ω) or RC 2 load (C load = 470 µF, R load = 47 Ω) was used. The measured value [37] of the coil inductance with a Material Mix -26 iron-powder core [37] in the operating point was approximately L f = 3 mH and R fe = 1 Ω due to the power losses in the core [36,38]. For the delta load, C fe = 3C f = 150 µF. Modeling in MATLAB-Simulink does not solve some of the serious problems with separate scaling voltages and currents. In an actual inverter, the nominal values of the voltage and current have to be amplified to achieve approximately 2/3 of the maximum value of the analogue to digital converter (ADC) range (for 12-bit ADC the range is 0-4095) because we have to predict the instantaneous increase in the nominal value (e.g., for a step decrease of the load). The other problem is how to limit the maximum value of the control voltage in the input of the PWM modulator (the output value of the control law). In the presented actual inverter, the limit was ±3280 (for an input PWM modulator frequency of 84 MHz and a switching frequency f s = 12,800 Hz). In the MATLAB-Simulink model, it was ±1 (Figure 3). The actual delays and phase shifts of the filters that were used can only be measured in the device. During switching instants, there are spikes in the measured voltage waveforms and a voltage amplifier with the galvanic isolation can introduce a noise. The additional low pass filter (an anti-aliasing filter can be insufficient) can be an effective solution. These problems are absent in the MATLAB-Simulink model. The control signal should suppress the distortions of the output voltage. A similar analysis as for the simulation in Figure 3 was performed for the experimental model with IPBC2 and is partly presented in Figure 4a-d. Figure 4a-d presents the visualization-digital to analogue conversion of the digital PWM α axis control signal for the IPBC2 control in the experimental model for the control parameters that were finally selected: Table 1). Figure 4d shows the control signal saturation that resulted in a decreased control quality when compared with Figure 4c. The different values of parameters R i and K v for the experimental model and the simulation were caused by the current and voltage scaling factors in the experimental model. The three-phase inverter MATLAB-Simulink simulation results were compared with the experimental model measurements for the open loop, IPBC2, and IDA-PBC controllers. The actual measured [36] values of the filter parameters were used. In the experimental model for IPBC2: Table 1). Further increasing R i and K v caused some oscillations in the output voltages.
values of the voltage and current have to be amplified to achieve approximately 2/3 of the maximum value of the analogue to digital converter (ADC) range (for 12-bit ADC the range is 0-4095) because we have to predict the instantaneous increase in the nominal value (e.g., for a step decrease of the load). The other problem is how to limit the maximum value of the control voltage in the input of the PWM modulator (the output value of the control law). In the presented actual inverter, the limit was ±3280 (for an input PWM modulator frequency of 84 MHz and a switching frequency fs = 12,800 Hz). In the MATLAB-Simulink model, it was ±1 (Figure 3). The actual delays and phase shifts of the filters that were used can only be measured in the device. During switching instants, there are spikes in the measured voltage waveforms and a voltage amplifier with the galvanic isolation can introduce a noise. The additional low pass filter (an anti-aliasing filter can be insufficient) can be an effective solution. These problems are absent in the MATLAB-Simulink model. The control signal should suppress the distortions of the output voltage. A similar analysis as for the simulation in Figure 3 was performed for the experimental model with IPBC2 and is partly presented in Figure 4ad. Figure 4a-d presents the visualization-digital to analogue conversion of the digital PWM α axis control signal for the IPBC2 control in the experimental model for the control parameters that were finally selected: Ri = 15 Ω, Kv = 0.8 Ω −1 for M = 0.3 (Table 1). Figure 4d shows the control signal saturation that resulted in a decreased control quality when compared with Figure 4c (Table 1). Further increasing Ri and Kv caused some oscillations in the output voltages. Table 1. Parameters of the simulation model and the inverter experimental models.    (Table 2) show that using different versions of PBC, the overshoot or undershoot for the step load was reduced in the experimental model from ±30% to ±5-±10% and the THDV was decreased from 10% to about 4-5% (in the simulations to 1%). The reasons for the different results of the simulations and measurements are discussed in the results. Both control systems, IPBC2 and IDA-PBC, gave similar results.   with the IDA-IPBC (dq frame), for the dynamic three-wire delta load, and the nonlinear rectifier RC 1 and RC 2 loads, respectively. The simulations and measurements gave very similar results for the open feedback loop, which verified the simulation model. Figure 5a-c, Figure 6a-c, Figure 7a-c, Figure 8a-c, Figure 9a-c, and Figure 10a-c are the simulations and measurements of the three-phase VSI with the open feedback loop, with the IPBC2 (αβ frame), with the IDA-IPBC (dq frame), for the dynamic three-wire delta load, and the nonlinear rectifier RC1 and RC2 loads, respectively. The simulations and measurements gave very similar results for the open feedback loop, which verified the simulation model.

Type of the Model
The presented simulations and measurements (Table 2) show that using different versions of PBC, the overshoot or undershoot for the step load was reduced in the experimental model from ±30% to ±5-±10% and the THDV was decreased from 10% to about 4-5% (in the simulations to 1%). The reasons for the different results of the simulations and measurements are discussed in the results. Both control systems, IPBC2 and IDA-PBC, gave similar results.      The presented simulations and measurements (Table 2) show that using different versions of PBC, the overshoot or undershoot for the step load was reduced in the experimental model from ±30% to ±5-±10% and the THD V was decreased from 10% to about 4-5% (in the simulations to 1%). The reasons for the different results of the simulations and measurements are discussed in the results. Both control systems, IPBC2 and IDA-PBC, gave similar results.

Modulation Index Choice
The most important problem is the limitation of the input/output range of the modulator signals (Figures 3 and 4c,d ). This is one reason for the greater distortions of the output voltage in the experimental inverter than in the simulations. An excessive increase of the controller gains causes the saturation of the control signal and oscillations of the output voltage (the imaginary part of the roots of the characteristic equation of the closed loop system increases with the gain, Figure 3). Therefore, one of the most important problems in inverter control is maintaining a sufficient range for the possible changes in the input voltage of a PWM modulator. For a lower M modulation index, this range increases. However, using a very low value, e.g., M = 0.3, is not permissible for actual inverters. The compromise between the high (close to the unity) modulation index M that is used in actual inverters and the low M that enables a sufficient control dynamic depending on the value of the inductors in the output filter is required.

Controller Gains Adjustment
To design the control (IPBC2) of an inverter (Figure 11), we should test the output voltage for the selected nonlinear rectifier RC load for the assigned M modulation index and for the wide spectrum of gains R i > 0 and K v ≥ 0 and measure the THD V coefficient. For low values of these parameters, we will receive high values of THD V . We should increase R i and K v (the exemplary values are in Table 2). For K v > 0, at first the THD V coefficient decreases, it reaches the minimum, and then it increases. After crossing the minimum of THD V , the oscillations will appear in the output voltage and the control signal can become saturated. Close to the minimum of the THD V coefficient, we can designate the values of the parameters R i and K v (separately for the simulation and the experimental model) to avoid output voltage oscillations and the control signal waveform saturation (

Differences Between Results of the Simulation and the Experimental Inverter Measurements
The simulation results were better than the experimental inverter ( Figure 12) measurements (lower overshoot and undershoot, lower THD V of the output voltage for the rectifier RC load in IPBC2 and IDA-PBC controls), possibly because the selected scaling factors of the voltages and currents were imperfect. The saturation of the control signal had different levels in the simulations and in the experimental model; we always had a problem with the value of the modulation index versus the inductance of the filter coil (Equation (3)), which can inhibit the effective reduction of the output voltage distortions. The simulated voltages had higher amplitude than those in the experimental model and the ripple voltage in the simulations can cause a lower effect of THD V (THD V was lower for the simulations). The actual parameters of components of an inverter output filter in the operation point can be different from the nominal values [36,38], which can increase the differences between the simulations and theory.

Modulation Index Choice
The most important problem is the limitation of the input/output range of the modulator signals (Figures 3 and 4c-d). This is one reason for the greater distortions of the output voltage in the experimental inverter than in the simulations. An excessive increase of the controller gains causes the saturation of the control signal and oscillations of the output voltage (the imaginary part of the roots of the characteristic equation of the closed loop system increases with the gain, Figure 3). Therefore, one of the most important problems in inverter control is maintaining a sufficient range for the possible changes in the input voltage of a PWM modulator. For a lower M modulation index, this range increases. However, using a very low value, e.g., M = 0.3, is not permissible for actual inverters. The compromise between the high (close to the unity) modulation index M that is used in actual inverters and the low M that enables a sufficient control dynamic depending on the value of the inductors in the output filter is required.

Controller Gains Adjustment
To design the control (IPBC2) of an inverter (Figure 11), we should test the output voltage for the selected nonlinear rectifier RC load for the assigned M modulation index and for the wide spectrum of gains Ri > 0 and Kv ≥ 0 and measure the THDV coefficient. For low values of these parameters, we will receive high values of THDV. We should increase Ri and Kv (the exemplary values are in Table 2). For Kv > 0, at first the THDV coefficient decreases, it reaches the minimum, and then it increases. After crossing the minimum of THDV, the oscillations will appear in the output voltage and the control signal can become saturated. Close to the minimum of the THDV coefficient, we can designate the values of the parameters Ri and Kv (separately for the simulation and the experimental model) to avoid output voltage oscillations and the control signal waveform saturation (Figures 3, 4, and 11). A lower value of the gain Kv in the experimental inverter was selected (Figure 4) than in the simulation ( Figure 11) because there are output voltage and current scaling factors in the experimental inverter that can change the effective gain.

Differences Between Results of the Simulation and the Experimental Inverter Measurements
The simulation results were better than the experimental inverter ( Figure 12) measurements (lower overshoot and undershoot, lower THDV of the output voltage for the rectifier RC load in

Advantages of the Control in the Stationary Frame
Both IPBC2 and IDA-PBC that are described are based on the same state equations of the inverter. IPBC2 is described in the stationary αβ frame and the control laws in α and β axes are fully decoupled. This idea can easily be implemented in single-phase inverters, e.g., for efficiently decreasing the output voltage distortions of the Z-Source inverter [39]. The stationary αβ frame can be transformed to the rotating dq frame that implements the interconnections from one axis to the other. The idea of IDA-PBC control is simply to remove them. IDA-PBC requires many more calculations for the Park transformation but its constant reference values are its advantage. The results of using both control systems in a VSI are the same. Therefore, it is better to use the easier calculations in IPBC2 without the Park transformation.

The Steady-State Error Reduction
It seems that the control laws of IDA-PBC and IPBC2 do not include any direct integration of the output voltage error, which is a rather PD-like control of the output voltage. However, the control law considers the filter inductor current, which depends on integrating the voltage on the inductor, and thus indirectly on the inverter output voltage. In the classic PBC control versions that are presented in [22], there is no direct control of the output voltage at all. The output impedance of the inverter with the open control loop (without the transformer increasing voltage) is rather low. It can be calculated from the measurements that are presented in Figure 6a. In Figure 10a, the steady-state error in the inverter with IDA-PBC is close to zero.

Discussion
The presented PBC control has been widely presented in the literature [13][14][15][16][17][18]22,24,35]. The paper shows that simple calculations without decoupling in the three-phase control in the αβ frame result in similar distortions of the inverter output voltage as the control in the dq frame that requires more calculations and additional decoupling [27]. The paper focuses on the fact that quality of the inverter output voltage for the standard loads depends on parameters such as the modulation index [32], the saturation of the control signal, and taking in account the delay of the control system, the values of the output filter parameters, and their variability [36,38]. The paper presents the idea of initially adjusting the parameters (using simulations) of the improved PBC controller. Further research should enable the analytic calculations of the parameters of the PBC controller. This is important because the improved (with the direct control of the inverter output voltage) PBC seems be perfect for systems that convert energy such as e.g., inverters.

Conclusions
The output filter parameters can be initially calculated to achieve a low output voltage ripple in a steady state operation. However, the capacitance should be increased due to the output voltage increase during a switching period in cases where the load current decreases steeply and all of the inductor current flows through the filter capacitor. It was shown that both control systems, IPBC2 (using the stationary αβ frame) and IDA-PBC (using the rotating dq frame), resulted in a similar decrease in the distortions in the three-phase inverter output voltage. Therefore, for a constant frequency of the reference waveform, it is better to use IPBC2 without the Park and reverse Park transformations. The demands concerning the microprocessor that controls the system are lower when we use a stationary frame. The real quality of the output voltage depends on the maximum range of the PWM driver control signals. Lower M modulation indices result in better control results because it is possible to omit the controller saturation for the less restricted range of controller gains. However, this approach cannot be used in commercial designs where the modulation index is close to unity to fully utilize the input DC power source. The other restriction of the control results is that too large a value of the inductance in the output filter makes the current changes slower than required. It is clear that the product M max L fe should be limited. The simulations that were necessary before experimental tests have a significant disadvantage-they do not solve the problem of voltage and current measurement scaling. Moreover, the scaling coefficients are as important as the gains in the control law. The paper shows an easy way to adjust the controller gains (Figures 3 and 11). These gains in IPBC2 had a wide margin of tolerance.