Passivity-Based Control Design Methodology for UPS Systems

Abstract

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.

1. 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 only half of it [3]). Therefore, when the harmonic disturbances vanish, the RPC has a fault response on the non-existent disturbance [12]. The MISO approach generally results in lower distortions of the output voltage for all types of disturbances as well as a higher robustness. Commonly, the load current is treated as an independent current source disturbance, thus making the inverter model independent of the load. However, since this method ignores the interaction of the inverter output voltage and the load current, it may not guarantee the required performance of a UPS system over a wide range of operating conditions that can change the location of the roots of the characteristic polynomial of a closed loop system [12]. In the recent years, the passivity-based control (PBC) approach that was introduced by Ortega [13] in 1989 and further extended in [14,15,16,17,18] has been employed for VSI. PBC is based on a MISO control and it is a suitable candidate for power conversion systems control (e.g., VSI) and has been widely used in induction motor drive systems [19,20]. When using PBC, a converter is treated as an “energy transformation multiport device” [21]. The idea of this controller is based on keeping the system behavior passive (i.e., the stored energy is always less than the supplied energy), which also requires “the injection” of proper damping [22]. The advantage of the PBC is the global stability of the controlled system. The choice of the acceptable gains of the controller is flexible and the control system is robust for the inverter parameter changes (parametric uncertainty) [22,23,24]. This paper proposes a PBC-based design methodology for three-phase inverters and presents a comparison of their control [25] using stationary αβ and rotating dq frames. In order to further improve the conventional PBC performance, proper output filter design as well as advanced PBC techniques that follow the IEC 62040-3 [8] and IEEE-519 [26] standard requirements will be considered. Notably, special attention is given to the modulator and controller saturation issues during dynamic loading. Thereby, the main aim of this paper is to provide guidelines and a design methodology on how to apply the simplest PBC approach for a three-phase VSI (simpler than e.g., the Interconnection and Damping Assignment Passivity-Based Control (IDA–PBC) idea presented in [27]) while maintaining a reasonable compromise between the ideal theory and the customer demands (with the more enhanced direct control of the output voltage than presented in [22]). The results of the modulator saturation effect, the problem of the restriction of the modulation index, the problem of the influence of the scaling factors in the measuring tracks on the PBC controller gains, and the final adjustment of the controller gains to reach the lowest distortions of the output voltage for the standard loads were not discussed yet [27,28,29]. The aim of the paper is to present a PBC version that can be easily implemented in a standard modern microcontroller delivering the pulse width modulation (PWM) signal with the acceptable high switching frequency. The presented difference control laws of PBC are devoted to this.

This paper is structured as follows. Section 2 describes the output filter sizing and selection. The state space modeling of an inverter in the stationary and rotating frames is presented in Section 3. Section 4 is dedicated to the design of different PBC. In Section 5, simulation and experimental results are provided to substantiate the effectiveness of the proposed design methodology for two versions of PBC. Finally, after the results and discussion, conclusions are drawn in Section 8.

2. The Output Filter Parameters

The basic approach to VSI design begins with the L_fC_f output filter design. The design should ensure a sufficiently low ripple voltage amplitude (IEC 62040-3 [8], IEEE519 [26]) and the total harmonic distortion of the output voltage (THD_V) level for static linear and nonlinear rectifier RC (R_load, C_load) loads. These calculations should permit the L_fC_f product value to be assigned. A cost function that is the sum of the absolute values of the reactive power in the filter inductor and the capacitor can then be created. The coefficients of both reactive powers that are assigned are equal [30,31]. For a three-wire star balanced load (Figure 1), a two-level PWM (six transistor h-bridge) [31] (n—phase line number, R_load—nominal load resistance, f_s—switching frequency, M—modulation index in actual inverters, M is set close to unity):

$$L_f=\sqrt{\frac{1}{M}}\frac{1}{f_s}{R}_{load},C_f>\frac{1}{f_s}\frac{1}{R_{load}},$$

(1)

or a three-wire delta balanced load (Figure 1), 2-level PWM:

$$L_f=\frac{1}{3}\frac{1}{f_s}{R}_{load},C_f>\frac{1}{f_s}\frac{1}{R_{load}}$$

(2)

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 ΔI_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 ΔV_out = 195 V, while for C_f = 50 μF—only ΔV_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):

$$M_{\max }<\frac{\sqrt{3}}{2}/(\frac{{\omega }_m{L}_f}{R_{serial}}+\sqrt{3}/2)$$

(3)

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).

3. Initial State Space Models of Three-Phase Inverters in the Stationary αβ and Rotating dq 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 = v_γ = 0.

$$v_0=v_{\gamma }=(v_{o,u}+v_{o,v}+v_{o,w})/3$$

(4)

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).

$${\left[\begin{array}{ccc}{x}_1& x_2& x_3\end{array}\right]}^T={\left[\begin{array}{ccc}{X}_{1m}\cos (\omega t)& X_{2m}\cos (\omega t-2\pi /3)& X_{3m}\cos (\omega t-4\pi /3)\end{array}\right]}^T$$

(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].

$$\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\frac{1}{2}& 0& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& 0\end{array}\right]\left[\begin{array}{c}{x}_{12}\\ x_{23}\\ x_{31}\end{array}\right],\left[\begin{array}{c}{x}_{12}\\ x_{23}\\ x_{31}\end{array}\right]=\left[\begin{array}{cc}\frac{3}{2}& -\frac{\sqrt{3}}{2}\\ 0& \sqrt{3}\\ -\frac{3}{2}& -\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]$$

(6)

For a star load (Figure 1), the Clarke transformation and reverse Clarke transformation are Equation (7).

$$\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right],\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]$$

(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).

$${\mathit{d}}_{\alpha \beta }={\left[\begin{array}{cc}{i}_{o,\alpha }& i_{o,\beta }\end{array}\right]}^T,d_{\alpha ,\beta }=i_{o,\alpha ,\beta }$$

(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).

$${\mathit{x}}_{i,\mathit{\alpha },\mathit{\beta }}={\left[\begin{array}{cc}{i}_{Lf,\alpha ,\beta }& v_{o,\alpha ,\beta }\end{array}\right]}^T,u_{\alpha ,\beta }=m_{\alpha ,\beta }{V}_{DC},{\mathit{y}}_{a,\beta }=v_{OUT\alpha ,\beta }$$

(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.

$${\dot{\mathit{x}}}_{i,\mathit{\alpha },\mathit{\beta }}=\left[\begin{array}{cc}-\frac{R_{fe}}{L_f}& -\frac{1}{L_f}\\ \frac{1}{C_{fe}}& 0\end{array}\right]{\mathit{x}}_{i,\alpha ,\beta }+\left[\begin{array}{c}0\\ -\frac{1}{C_{fe}}\end{array}\right]d_{\alpha ,\beta }+\left[\begin{array}{c}\frac{1}{L_f}\\ 0\end{array}\right]u_{\alpha ,\beta }$$

(10)

The Park transformation and the reverse Park transformation for the balanced load are Equations (11) and (12).

$$\left[\begin{array}{c}{x}_d\\ x_q\end{array}\right]=\left[\begin{array}{cc}\cos {\omega }_{m}t& \sin {\omega }_{m}t\\ -\sin {\omega }_{m}t& \cos {\omega }_{m}t\end{array}\right]\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]$$

(11)

$$\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]=\left[\begin{array}{cc}\cos {\omega }_{m}t& -\sin {\omega }_{m}t\\ \sin {\omega }_{m}t& \cos {\omega }_{m}t\end{array}\right]\left[\begin{array}{c}{x}_d\\ x_q\end{array}\right]$$

(12)

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_mcosω_mt, x_β = X_msinω_mt, 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).

$${\mathit{x}}_{\alpha \beta ,dq}={\left[\begin{array}{cccc}{L}_f{i}_{Lf,\alpha ,d}& L_f{i}_{Lf,\beta ,q}& C_{fe}{v}_{o,\alpha ,d}& C_{fe}{v}_{o,\beta ,q}\end{array}\right]}^T=\mathit{P}{\left[\begin{array}{cccc}{i}_{Lf,\alpha ,d}& i_{Lf,\beta ,q}& v_{o,\alpha ,d}& v_{o,\beta ,q}\end{array}\right]}^T$$

(13)

$${\left[\begin{array}{cccc}{i}_{Lf,\alpha ,d}& i_{Lf,\beta ,q}& v_{o,\alpha ,d}& v_{o,\beta ,q}\end{array}\right]}^T={\mathit{P}}^{-1}{\mathit{x}}_{\alpha \beta ,dq}$$

(14)

$$\mathit{P}=\left[\begin{array}{cccc}{L}_f& 0& 0& 0\\ 0& L_f& 0& 0\\ 0& 0& C_{fe}& 0\\ 0& 0& 0& C_{fe}\end{array}\right],{\mathit{P}}^{-1}=\left[\begin{array}{cccc}1/L_f& 0& 0& 0\\ 0& 1/L_f& 0& 0\\ 0& 0& 1/C_{fe}& 0\\ 0& 0& 0& 1/C_{fe}\end{array}\right]$$

(15)

$$H(x_{\alpha \beta ,dq})=\frac{1}{2}(L_f{i}_{Lf,\alpha ,d}{}^2+L_f{i}_{Lf,\beta ,q}{}^2+C_{fe}{v}_{o,\alpha ,d}{}^2+C_{fe}{v}_{o,\beta ,q}{}^2)=\frac{1}{2}{\mathit{x}}_{\alpha \beta ,dq}{}^T{\mathit{P}}^{-1}{\mathit{x}}_{\alpha \beta ,dq}$$

(16)

We can notice that for the αβ or dq frames:

$${\mathit{P}}^{-1}{\mathit{x}}_{\alpha \beta ,dq}=\partial H(x_{\alpha \beta ,dq})/\partial (x_{\alpha \beta ,dq})={\left[\begin{array}{cccc}{i}_{Lf,\alpha ,d}& i_{Lf,\beta ,q}& v_{o,\alpha ,d}& v_{o,\beta ,q}\end{array}\right]}^T$$

(17)

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).

$${\dot{x}}_{\alpha \beta ,dq}=[J_{\alpha \beta ,dq}-R_{\alpha \beta ,dq}]P^{-1}{x}_{\alpha \beta ,dq}+G_{\alpha \beta ,dq}{m}_{\alpha \beta ,dq}+D_{\alpha \beta ,dq}{d}_{\alpha \beta ,dq}$$

(18)

$$m_{\alpha \beta ,dq}={\left[\begin{array}{cc}{m}_{\alpha ,d}& m_{\beta ,q}\end{array}\right]}^T,d_{\alpha \beta ,dq}={\left[\begin{array}{cc}{i}_{o,\alpha ,d}& i_{o,\beta ,q}\end{array}\right]}^T$$

(19)

The input matrix G_αβ,dq, disturbance matrix D_αβ,dq, interconnection matrix J_αβ,dq (J_αβ is different from J_dq), and damping matrix R_αβ,dq are Equation (20).

$$\begin{gathered} G_{\alpha \beta ,dq}=\left[\begin{array}{cc}{V}_{DC}& 0\\ 0& V_{DC}\\ 0& 0\\ 0& 0\end{array}\right],D_{\alpha \beta ,dq}=\left[\begin{array}{cc}0& 0\\ 0& 0\\ -1& 0\\ 0& -1\end{array}\right],R_{\alpha \beta ,dq}=\left[\begin{array}{cccc}{R}_{fe}& 0& 0& 0\\ 0& R_{fe}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],J_{\alpha \beta }=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right], \\ {J}_{dq}=\left[\begin{array}{cccc}0& {\omega }_m& -1& 0\\ -{\omega }_m& 0& 0& -1\\ 1& 0& 0& {\omega }_m\\ 0& 1& -{\omega }_m& 0\end{array}\right] \end{gathered}$$

(20)

4. 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].

$$e_{\alpha \beta ,dq}=x_{\alpha \beta ,dq}-x_{\alpha \beta ,dq,ref}$$

(21)

The closed loop dynamic (Equation (22)) of the tracking error is described in [27].

$${\dot{e}}_{\alpha \beta ,dq}=[(J_{\alpha \beta ,dq}+J_{\alpha \beta ,dq,a})-(R_{\alpha \beta ,dq}+R_{\alpha \beta ,dq,a})]P^{-1}{e}_{\alpha \beta ,dq}$$

(22)

where:

$$P^{-1}{e}_{\alpha \beta ,dq}=\partial H(e_{\alpha \beta ,dq})/\partial e_{\alpha \beta ,dq}=\left[\begin{array}{c}{i}_{Lf\alpha ,d}-i_{Lf\alpha ,d,ref}\\ i_{Lf\beta ,q}-i_{Lf\beta ,q,ref}\\ v_{OUT\alpha ,d}-v_{OUT\alpha ,d,ref}\\ v_{OUT\beta ,q}-v_{OUT\beta ,q,ref}\end{array}\right]$$

(23)

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)).

$$H(e_{\alpha \beta ,dq})=\frac{1}{2}{e}_{\alpha \beta ,dq}{}^T{P}_{\alpha \beta ,dq}{}^{-1}{e}_{\alpha \beta ,dq}$$

(24)

$${\frac{\partial H(e_{\alpha \beta ,dq})}{\partial x_{\alpha \beta ,dq}}|}_{x_{\alpha \beta ,dq,ref}}=0\begin{array}{cc},& {\frac{{\partial }^{2}H(e_{\alpha \beta ,dq})}{\partial x_{\alpha \beta ,dq}{}^2}|}_{x_{\alpha \beta ,dq,ref}}>0\end{array}$$

(25)

The system is passive if the time derivative H(e_αβ,dq) is negative (Equation (26)).

$$\frac{\mathrm{d}H(e_{\alpha \beta ,dq})}{\mathrm{d}t}<0$$

(26)

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).

$$R_{\alpha \beta ,dq,a}=\left[\begin{array}{cccc}{R}_i& 0& 0& 0\\ 0& R_i& 0& 0\\ 0& 0& K_v& 0\\ 0& 0& 0& K_v\end{array}\right]$$

(27)

From Equation (26), it is possible to initially define the range of the implemented gains (Equation (28)).

$$R_{fe}+R_i>0\mathrm{and}{K}_v>0$$

(28)

In the dq frame, J_dq,a should decouple the voltage and current equations from the dq axes (Equations (29) and (30)).

$$J_{dq,a}=\left[\begin{array}{cccc}0& -{\omega }_m{L}_f& 0& 0\\ {\omega }_m{L}_f& 0& 0& 0\\ 0& 0& 0& -{\omega }_m{C}_{fe}\\ 0& 0& {\omega }_m{C}_{fe}& 0\end{array}\right]$$

(29)

$$J_{dq}-J_{dq,a}=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]=J_{\alpha \beta }$$

(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).

$$\det \{[(J_{\alpha \beta ,dq}+J_{\alpha \beta ,dq,a})-(R_{\alpha \beta ,dq}+R_{\alpha \beta ,dq,a})]-\lambda I\}=0$$

(31)

$${\lambda }_{1,2}=\frac{\left\{-[(R_{fe}+R_i)C_{fe}+L_f{K}_v]\pm \sqrt{{[(R_{fe}+R_i)C_{fe}+L_f{K}_v]}^2-4L_f{C}_{fe}[1+(R_{fe}+R_i)K_v]}\right\}}{2L_f{C}_{fe}}$$

(32)

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 RC₂ load (R_load = 47 Ω, C_load = 470 μF) for PBC in the stationary αβ frame. 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 α axis control waveform, which results in the oscillations of the output voltage. The saturation of the α axis control waveform for K_vR_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 α axis control waveform saturation. The control law (Equation (33)) was determined in the αβ frame (Equations (34) and (35)) and the dq frame (Equations (36)–(39)) by subtracting both sides of the Equations (16) for the αβ frame or Equation (23) for the dq frame from both sides of Equation (28).

$$\begin{array}{ll}{G}_{\alpha \beta ,dq}{m}_{\alpha \beta ,dq}=& -[J_{\alpha \beta ,dq}-R_{\alpha \beta ,dq}]P_{\alpha \beta ,dq}{}^{-1}{x}_{\alpha \beta ,dq,ref}+[J_{\alpha \beta ,dq,a}-R_{\alpha \beta ,dq,a}]P_{\alpha \beta ,dq}{}^{-1}(x_{\alpha \beta ,dq}-x_{\alpha \beta ,dq,ref})-\\ & -D_{\alpha \beta ,dq}{d}_{\alpha \beta ,dq}+{\dot{x}}_{\alpha \beta ,dq,ref}\end{array}$$

(33)

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.

$$\begin{array}{ll}{V}_{CTRL,\alpha ,\beta }(k)& =L_f[i_{Lf,\alpha ,\beta ,ref}(k)-i_{Lf,\alpha ,\beta ,ref}(k-1)]f_s+R_{fe}{i}_{Lf,\alpha ,\beta ,ref}(k)-\\ & -R_i[i_{Lf,\alpha ,\beta }(k)-i_{Lf,\alpha ,\beta ,ref}(k)]+v_{o,\alpha ,\beta ,ref}(k)\end{array}$$

(34)

here the reference inductor current i_Lf,α,β,ref is (35).

$$i_{Lf,\alpha ,\beta ,ref}(k)=C_{fe}[v_{o,\alpha ,\beta ,ref}(k)-v_{o,\alpha ,\beta ,ref}(k-1)]f_s-K_v[v_{o,\alpha ,\beta }(k)-v_{o,\alpha ,\beta ,ref}(k)]+i_{o,\alpha ,\beta }(k)$$

(35)

The difference control law of IDA–PBC [27] for the rotating dq frame is Equations (36)–(39).

$$\begin{array}{ll}{V}_{CTRL,d}(k)& =L_f[i_{Lf,d,ref}(k)-i_{Lf,d,ref}(k-1)]f_s+R_{fe}{i}_{Lf,d,ref}(k)-{\omega }_{dq}{L}_f{i}_{Lf,q}-\\ & -R_i[i_{Lf,d}(k)-i_{Lf,d,ref}(k)]+v_{o,d,ref}(k)\end{array}$$

(36)

$$i_{Lf,d,ref}(k)=C_{fe}[v_{o,d,ref}(k)-v_{o,d,ref}(k-1)]f_s-{\omega }_{dq}{C}_{fe}{v}_{o,q}-K_v[v_{o,d}(k)-v_{o,d,ref}(k)]+i_{o,d}(k)$$

(37)

$$\begin{array}{ll}{V}_{CTRL,q}(k)& =L_f[i_{Lf,q,ref}(k)-i_{Lf,q,ref}(k-1)]f_s+R_{fe}{i}_{Lf,q,ref}(k)+{\omega }_{dq}{L}_f{i}_{Lf,d}-\\ & -R_i[i_{Lf,q}(k)-i_{Lf,q,ref}(k)]+v_{o,q,ref}(k)\end{array}$$

(38)

$$i_{Lf,q,ref}(k)=C_{fe}[v_{o,q,ref}(k)-v_{o,q,ref}(k-1)]f_s+{\omega }_{dq}{C}_{fe}{v}_{o,d}-K_v[v_{o,q}(k)-v_{o,q,ref}(k)]+i_{o,q}(k)$$

(39)

For a three-phase AC power supply in the αβ frame, the difference of reference voltage v_o,α,β,ref(k) − v_o,α,β,ref(k − 1) ≠ 0 because v_o,α,β,ref are two shifted values with π/2 sinusoidal waveforms. For the dq frame, v_o,d,q,ref(k) − v_o,d,q,ref(k − 1) = 0 because v_o,d,q,ref are two constant values.

5. 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 Δ470 Ω switched to Δ470||47 Ω and vice versa. The nonlinear rectifier RC₁ load (C_load = 100 µF, R_load = 47 Ω) or RC₂ 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: R_i = 15 Ω, K_v = 0.8 Ω⁻¹ for M = 0.3 (

Table 1. Parameters of the simulation model and the inverter experimental models.

Type of the Model	Lf (mH)	Rfe (Ω)	Cf (µF)	M	Vo,uv|max (V)	Ri (Ω)	Kv (Ω−1)
MATLAB Simulation	3	1	50	0.3	150	10	2
Experimental IPBC2	3	1	50	0.3	70	15	0.8
Experimental IDA–PBC	3	1	50	0.3	70	18	0.5

). 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: R_i = 15 Ω, K_v = 0.8 Ω⁻¹, M = 0.3, for IDA–PBC: R_i = 18 Ω, K_v = 0.5 Ω⁻¹, M = 0.3 (Table 1). Further increasing R_i and K_v caused some oscillations in the output voltages.

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.

The presented simulations and measurements (

Table 2. Results of the simulations and the measurements of the inverter experimental model.

	Ri (Ω)	Kv (Ω−1)	THDV RC1	THDV RC2	Overshoot Load Decrease	Undershoot Load Increase
No feedback. simulation	---	---	9.5%	9.1%	+23%	−20%
No feedback inverter	---	---	9.6%	10%	+32%	−28%
IPBC2 simulation	10	2	0.76%	1.2%	+4.5%	−5.5%
IPBC2 inverter	15	0.8	4.36%	5.46%	+7%	−4%
IDA–PBC simulation	10	2	0.75%	1.4%	+2.2%	−3%
IDA–PBC inverter	18	0.5	3.93%	4.96%	+10%	−7%

) 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.

6. Results

6.1. Modulation Index Choice

The most important problem is the limitation of the input/output range of the modulator signals (Figure 3 and Figure 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.

6.2. 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 (Figure 3, Figure 4, and Figure 11). A lower value of the gain K_v 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.

6.3. 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.

6.4. 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.

6.5. 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.

7. 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.

8. 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_maxL_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 (Figure 3 and Figure 11). These gains in IPBC2 had a wide margin of tolerance.