Modeling and Stability Analysis of a Single-Phase Two-Stage Grid-Connected Photovoltaic System

Abstract

The stability issue of a single-phase two-stage grid-connected photovoltaic system is complicated due to the nonlinear v-i characteristic of the photovoltaic array as well as the interaction between power converters. Besides, even though linear system theory is widely used in stability analysis of balanced three-phase systems, the application of the same theory to single-phase systems meets serious challenges, since single-phase systems cannot be transformed into linear time-invariant systems simply using Park transformation as balanced three-phase systems. In this paper, (1) the integrated mathematical model of a single-phase two-stage grid-connected photovoltaic system is established, in which both DC-DC converter and DC-AC converter are included also the characteristic of the PV array is considered; (2) an observer-pattern modeling method is used to eliminate the time-varying variables; and (3) the stability of the system is studied using eigenvalue sensitivity and eigenvalue loci plots. Finally, simulation results are given to validate the proposed model and stability analysis.

1. Introduction

Under pressure from the energy crisis, photovoltaic (PV) energy has been more and more attractive for generating electricity. At the end of 2016, the total PV installation capacity around the world amounted to 305 GW [1]. The majority of PV installations are grid-connected PV systems, since they can deliver power to the grid directly and are more cost-effective than stand-alone systems [2]. Whereas large commercial PV systems are connected to the three-phase grid, single-phase topology is advantageous in small-scale PV systems such as residential systems due to its simplicity [3,4].

A typical grid-connected PV system is a two-stage system, where the first stage is normally a DC-DC converter for extracting power from the PV array and the second stage is a DC-AC converter for delivering power to the grid. However, the stability of such a system is a major concern. There are two main factors that make the stability analysis of a two-stage grid-connected PV system more difficult than other power electronic systems. First is the characteristic of the PV array. The v-i characteristic of a PV array is nonlinear and changes with the light intensity or temperature, thus the dynamics of a PV system can be vastly different from a traditional power electronic system fed from a constant voltage source. In some studies on stability analysis of PV systems, the PV array is replaced by a constant voltage source [5,6] or a constant current source [7]. These methods neglect the nonlinear characteristics of the PV array and may cause deviation between the theoretical analysis and the behavior of the real system [8]. Some studies take into account this characteristic of the PV array by using the v-i curve calculated from numerical techniques with the aid of a computer [8,9]. However, specific parameters of the PV array such as shunt resistance and series resistance are necessary for implementing the calculation. These parameters usually cannot be obtained from the datasheet. Second, a DC-DC converter and a DC-AC converter are connected in cascade. Even in a single power converter there exists complex behaviors such as bifurcation and chaos [10,11,12,13]. In this case, the behavior of the overall system may be more complicated than only one converter, since the interconnected converters will influence each other [14,15,16,17]. Hence, it is necessary to establish an integrated mathematical model for the entire single-phase two-stage grid-connected photovoltaic system that is able to describe the characteristic of the PV array as well as the interactions between two converters.

For a balanced three-phase system, application of Park transformation facilitates modeling of the system. The system can be first transformed into a Multiple-Input-Multiple-Output (MIMO) system in d-q reference frame and then be linearized around a fixed stead-state operating point [18]. Finally, the balanced three-phase system can be described using a linear time invariant (LTI) model. Thus, vast LTI theory tools can be applied to completing the controller design and stability analysis [19,20]. However, it is hard to put a single-phase system within the framework of an LTI model. The main difficulty for this is that linearization process must be performed around a fixed steady-state operating point rather than a steady-state time-periodic trajectory [18]. To deal with this problem, an observer-pattern modeling method [21,22] is proposed that eliminates the effect of time-variance.

In this paper, the stability analysis of the whole single-phase two-stage grid-connected PV system is presented. Both DC-DC converter and DC-AC converter will be included in the model. Also, the characteristic of the PV array will be considered. To avoid the lack of specific parameters of the PV array, the proposed model uses the basic parameters that are provided in all datasheets of PV arrays. The application of observer-pattern modeling method successfully transforms the system into time-invariant. With the proposed model, the stability of the system can be studied by calculating the eigenvalues of the Jacobian of the system.

2. System Description and Nonlinear Averaged Equations

Figure 1 presents the diagram of a single-phase two-stage grid-connected photovoltaic system. In this figure, C_in is the capacitance of the input filter, L_b and C_dc are the inductance and the capacitance of the boost converter, respectively, and L_f is the inductance of the output filter. The PV array generates electricity from solar radiation. A boost converter with an input filter connects the PV array to the DC bus in order to raise the output voltage of PV array to the voltage level of DC bus while implementing maximum power point tracking (MPPT). The boost converter is designed to operate in continuous conduction mode (CCM).

In this study, the Perturb & Observe (P&O) method is adopted for MPPT, since it is one of the most popular MPPT algorithms [23]. A full bridge inverter with an L filter supplies the power to the AC grid [24]. The full bridge inverter is controlled by a double loop controller, which is comprised of a voltage control loop and a current control loop. The voltage control loop regulates the DC bus voltage u_dc and generates the reference current i_ref for the current control loop. Then, the current control loop regulates the output current of full bridge inverter i_o. In order to facilitate the analysis, the boost converter and the full bridge inverter are assumed to have same switching frequency f_s.

In order to develop an integrated mathematical model, the system equations are derived for three parts: PV array part as given in Section 2.1, power stage part as described in Section 2.2, and controller part as detailed in Section 2.3.

2.1. PV Array

The output current i_pv of the PV array is related to the output voltage u_pv and also affected by other parameters of PV panel itself [25]. However, the manufacturer’s datasheet do not provide some of these parameters, such as equivalent series resistance and equivalent parallel resistance. Basic parameters provided in all datasheets of PV array are open circuit voltage U_OC, short circuit current I_SC, the current at maximum power point (MPP) I_M, and the voltage at maximum power point U_M. These parameters are all with the standard test condition (STC). To overcome the lack of detailed information of the PV array, a simplified model proposed in [26] is used in this study, which describes the terminal characteristic of the PV array with STC in the following equation:

$$i_{pv}=I_{SC}\left[1-A_1\left(e^{\frac{u_{pv}}{A_2{U}_{OC}}}-1\right)\right]$$

(1)

where $A_1=\left(1-\frac{I_M}{I_{SC}}\right)e^{-\frac{U_M}{A_2{U}_{OC}}}$, $A_2=\left(\frac{U_M}{U_{OC}}-1\right){\left[ln\left(1-\frac{I_M}{I_{SC}}\right)\right]}^{-1}$.

2.2. Power Stage Circuit

To describe the power stage circuit, four sate variables are used: the output voltage of PV array u_pv, the current of L_b which denoted as $i_{L_b}$, the DC bus voltage u_dc and the output current of full bridge inverter i_o. The averaged equations of the input filter and boost converter can be derived as follows [19]:

$$\{\begin{array}{l}\frac{d{u}_{pv}}{dt}=\frac{1}{C_{in}}\left(i_{pv}-i_{L_b}\right)\\ \frac{d{i}_{L_b}}{dt}=\frac{1}{L_b}\left[u_{pv}-\left(1-d_1\right)u_{dc}\right]\\ \frac{d{u}_{dc}}{dt}=\frac{1}{C_{dc}}\left[\left(1-d_1\right)i_{L_b}-i_{dc}\right]\end{array}$$

(2)

where d₁ is the duty cycle of the boost converter and i_dc is the input current of full bridge inverter.

According to the operation principle of the full bridge inverter, following equations can be derived [24]:

$$\{\begin{array}{l}{i}_{dc}=\left(2d_2-1\right)i_o\\ \frac{d{i}_o}{dt}=\frac{1}{L_f}\left[\left(2d_2-1\right)u_{dc}-u_g\right]\end{array}$$

(3)

where d₂ is the duty cycle of S₁ and S₄ in the full bridge inverter and u_g is the grid voltage.

Combine (2) and (3), then the averaged equations of the power stage circuit can be derived as presented in (4).

$$\{\begin{array}{l}\frac{d{u}_{pv}}{dt}=\frac{1}{C_{in}}\left(i_{pv}-i_{L_b}\right)\\ \frac{d{i}_{L_b}}{dt}=\frac{1}{L_b}\left[u_{pv}-\left(1-d_1\right)u_{dc}\right]\\ \frac{d{u}_{dc}}{dt}=\frac{1}{C_{dc}}\left[\left(1-d_1\right)i_{L_b}-\left(2d_2-1\right)i_o\right]\\ \frac{d{i}_o}{dt}=\frac{1}{L_f}\left[\left(2d_2-1\right)u_{dc}-u_g\right]\end{array}$$

(4)

2.3. Controller

The state equations corresponding to the Proportional-Integral (PI) controller can be expressed as follow:

$$\frac{dm\left(t\right)}{dt}=K_p\frac{de\left(t\right)}{dt}+\frac{K_p}{T_i}e(t)$$

(5)

where m(t) and e(t) are the output and the input signals of the PI controller, respectively, K_p is the gain of the PI controller, and T_i is the time constant of the PI controller.

Since the perturbation step size of P&O method is small, the reference voltage u_pvref given by the algorithm is assumed to be constant. Therefore, the state equation of PI controller 1 can be derived as

$$\frac{d{u}_{c1}}{dt}=K_{p1}\frac{d{u}_{pv}}{dt}+\frac{K_{p1}}{T_{i1}}\left(u_{pv}-u_{pvref}\right)$$

(6)

where u_c1 is the output signal of PI controller 1 and K_p1 and T_i1 are the gain and the time constant of the PI controller. u_c1 is compared to a ramp u_r in the Pulse Width Modulation (PWM) comparator to produce driving signal. To complete the model of MPPT controller, the duty cycle d₁ is expressed as following equation:

$$d_1=\frac{u_{c1}}{U_{M1}}$$

(7)

where U_M1 is the peak-to-peak amplitude of ramp u_r.

The state equation of PI controller 2 can be derived as

$$\frac{d{u}_e}{dt}=K_{p2}\frac{d{u}_{dc}}{dt}+\frac{K_{p2}}{T_{i2}}\left(u_{dc}-u_{dcref}\right)$$

(8)

where u_dcref is the reference voltage of DC bus, u_e is the output signal of PI controller 2, K_p2 and T_i2 is the gain and the time constant of the PI controller. Then the reference current i_ref can be expressed as:

$$i_{ref}=u_{e}sin\left(\omega t\right)$$

(9)

where ω is the angular frequency of power grid.

Thus the state equation of PI controller 3 can be derived as

$$\frac{d{u}_{c2}}{dt}=K_{p3}\left(u_e\omega cos(\omega t)+sin(\omega t)\frac{d{u}_e}{dt}-\frac{d{i}_o}{dt}\right)+\frac{K_{p3}}{T_{i3}}\left(u_{e}sin\left(\omega t\right)-i_o\right)$$

(10)

where u_c2 is the output signal of PI controller 3 and K_p3 and T_i3 are the gain and the time constant of the PI controller. u_c2 is compared to a triangular wave u_tri in the PWM comparator to generate driving signal. The duty cycle d₂ is expressed as following equation:

$$d_2=\frac{1}{2}\left(1+\frac{u_{c2}}{U_{M2}}\right)$$

(11)

where U_M2 is the peak-to-peak amplitude of triangular wave u_tri.

Combine (1), (4) and (6)–(11), the nonlinear averaged equations of the system can be written as

$$\{\begin{array}{l}\frac{d{u}_{pv}}{dt}=\frac{1}{C_{dc}}\left(I_{SC}\left[1-A_1\left(e^{\frac{u_{pv}}{A_2{U}_{OC}}}-1\right)\right]-i_{L_b}\right)\\ \frac{d{i}_{L_b}}{dt}=\frac{1}{L_b}\left[u_{pv}-\left(1-\frac{u_{c1}}{U_{M1}}\right)u_{dc}\right]\\ \frac{d{u}_{dc}}{dt}=\frac{1}{C_{dc}}\left[\left(1-\frac{u_{c1}}{U_{M1}}\right)i_{L_b}-\frac{u_{c2}{i}_o}{U_{M2}}\right]\\ \frac{d{i}_o}{dt}=\frac{1}{L_f}\left[\frac{u_{c2}}{U_{M2}}{u}_{dc}-U_{gm}sin(\omega t)\right]\\ \frac{d{u}_{c1}}{dt}=K_{p1}\frac{d{u}_{pv}}{dt}+\frac{K_{p1}}{T_{i1}}\left(u_{pv}-u_{pvref}\right)\\ \frac{d{u}_e}{dt}=K_{p2}\frac{d{u}_{dc}}{dt}+\frac{K_{p2}}{T_{i2}}\left(u_{dc}-u_{dcref}\right)\\ \frac{d{u}_{c2}}{dt}=K_{p3}\left(u_e\omega cos(\omega t)+sin(\omega t)\frac{d{u}_e}{dt}-\frac{d{i}_o}{dt}\right)+\frac{K_{p3}}{T_{i3}}\left(u_{e}sin(\omega t)-i_o\right)\end{array}$$

(12)

where U_gm is the amplitude of u_g.

3. Observer-Pattern Model

The equations shown in (12) contain sin(ωt) and cos(ωt), which means that the single-phase two-stage grid-connected photovoltaic system is a time-variant nonlinear system. This time-variance is the main difficulty in stability analysis of the system. To eliminating the effect of time-variance, the system is transformed into a time-invariant one using an observer-pattern modeling method [21,22]. Notice that sin(ωt) and cos(ωt) only exist in the expressions of di_o/dt and du_c2/dt. Thus, only i_o and u_c2 need to be processed.

First, the time-variance originated from fundamental frequency of the grid is removed by Park transformation. Implementation of Park transformation needs at least two orthogonal variables, so the concept of Imaginary Orthogonal Circuit is introduced [27]. Denote the corresponding Imaginary Orthogonal Circuit variables to i_o and u_c2 as i_oI and u_c2I. Since i_o and u_c2 are sinusoidal, i_oI and u_c2I maintain 90° phase shift with i_o and u_c2. The Park transformation can be expressed as

$$\{\begin{array}{l}{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}}=\left[\begin{array}{l}{i}_{od}\\ i_{oq}\end{array}\right]=\mathit{T}\left[\begin{array}{l}{i}_o\\ i_{oI}\end{array}\right]\\ {\mathit{u}}_{\mathit{c}2\mathit{d}\mathit{q}}=\left[\begin{array}{l}{u}_{c2d}\\ u_{c2q}\end{array}\right]=\mathit{T}\left[\begin{array}{l}{u}_{c2}\\ u_{c2I}\end{array}\right]\end{array}$$

(13)

where T is the transformation matrix given by (14).

$$\mathit{T}=\left[\begin{array}{cc}cos(\omega t)& sin(\omega t)\\ -sin\left(\omega t\right)& cos\left(\omega t\right)\end{array}\right]$$

(14)

Apply inverse transformation of (13) to the state equations of i_o and u_c2, resulting in the following equations:

$$\{\begin{array}{l}\frac{d}{dt}({\mathit{T}}^{-1}{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}})=\frac{u_{dc}}{L_f{U}_{M2}}{\mathit{T}}^{-1}{\mathit{u}}_{\mathit{c}2\mathit{d}\mathit{q}}-\frac{U_{gm}}{L_f}\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]\\ \frac{d}{dt}({\mathit{T}}^{-1}{\mathit{u}}_{\mathit{c}2\mathit{d}\mathit{q}})=K_{p3}{u}_e\omega \left[\begin{array}{c}cos\left(\omega t\right)\\ sin\left(\omega t\right)\end{array}\right]-\frac{K_{p3}}{L_f}\left(\frac{u_{dc}}{L_f{U}_{M2}}{\mathit{T}}^{-1}{\mathit{u}}_{\mathit{c}2\mathit{d}\mathit{q}}-\frac{U_{gm}}{L_f}\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]\right)\\ \hfill +K_{p3}\frac{d{u}_e}{dt}\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]+\frac{K_{p3}}{T_{i3}}{u}_e\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]-\frac{K_{p3}}{T_{i3}}{\mathit{T}}^{-1}{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}}\hfill \end{array}$$

(15)

Multiplying (15) by T gives

$$\{\begin{array}{l}\mathit{T}\frac{d}{dt}({\mathit{T}}^{-1}{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}})=\frac{u_{dc}}{L_f{U}_{M2}}{\mathit{u}}_{\mathit{c}2\mathit{d}\mathit{q}}-\frac{U_{gm}}{L_f}\mathit{T}\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]\\ \mathit{T}\frac{d}{dt}({\mathit{T}}^{-1}{\mathit{u}}_{\mathit{c}2\mathit{d}\mathit{q}})=\frac{K_{p3}}{T_{i3}}{u}_e\mathit{T}\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]-\frac{K_{p3}}{L_f}\left(\frac{u_{dc}}{L_f{U}_{M2}}{\mathit{u}}_{\mathit{c}2\mathit{d}\mathit{q}}-\frac{U_{gm}}{L_f}\mathit{T}\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]\right)\\ \hfill -\frac{K_{p3}}{T_{i3}}{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}}+K_{p3}\mathit{T}\left(u_e\omega \left[\begin{array}{c}cos\left(\omega t\right)\\ sin(\omega t)\end{array}\right]+\frac{d{u}_e}{dt}\left[\begin{array}{c}sin\left(\omega t\right)\\ -cos\left(\omega t\right)\end{array}\right]\right)\hfill \end{array}$$

(16)

Since $\mathit{T}\frac{d}{dt}({\mathit{T}}^{-1}{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}})=\mathit{T}\frac{d{\mathit{T}}^{-1}}{dt}{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}}+\frac{d{\mathit{i}}_{\mathit{o}\mathit{d}\mathit{q}}}{dt}$ and replacing i_odq with u_c2dq satisfies this equation as well, (16) can be expressed as

$$\{\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}{i}_{od}\\ i_{oq}\end{array}\right]=\frac{u_{dc}}{L_f{U}_{M2}}\left[\begin{array}{c}{u}_{c2d}\\ u_{c2q}\end{array}\right]-\frac{U_{gm}}{L_f}\left[\begin{array}{c}0\\ -1\end{array}\right]+\left[\begin{array}{cc}0& \omega \\ -\omega & 0\end{array}\right]\left[\begin{array}{c}{i}_{od}\\ i_{oq}\end{array}\right]\\ \frac{d}{dt}\left[\begin{array}{c}{u}_{c2d}\\ u_{c2q}\end{array}\right]=\left[\begin{array}{cc}0& \omega \\ -\omega & 0\end{array}\right]\left[\begin{array}{c}{u}_{c2d}\\ u_{c2q}\end{array}\right]+\frac{U_{gm}{K}_{p3}}{L_f}\left[\begin{array}{c}0\\ -1\end{array}\right]+\frac{K_{p3}}{T_{i3}}{u}_e\left[\begin{array}{c}0\\ -1\end{array}\right]\\ \hfill -\frac{K_{p3}}{T_{i3}}\left[\begin{array}{c}{i}_{od}\\ i_{oq}\end{array}\right]+K_{p3}\left(u_e\omega \left[\begin{array}{c}1\\ 0\end{array}\right]+\frac{d{u}_e}{dt}\left[\begin{array}{c}0\\ -1\end{array}\right]-\frac{u_{dc}}{L_f{U}_{M2}}\left[\begin{array}{c}{u}_{c2d}\\ u_{c2q}\end{array}\right]\right)\hfill \end{array}$$

(17)

Notice that

$$\{\begin{array}{l}{u}_{c2}=u_{c2d}cos\left(\omega t\right)-u_{c2q}sin\left(\omega t\right)\\ i_o=i_{od}cos\left(\omega t\right)-i_{oq}sin\left(\omega t\right)\end{array}$$

(18)

Thus, the product term u_c2i_o in the averaged Equation (12) can be substituted by

$$\begin{array}{ll}{u}_{c2}{i}_o=& \frac{1+cos\left(2\omega t\right)}{2}{u}_{c2d}{i}_{od}+\frac{1-cos\left(2\omega t\right)}{2}{u}_{c2q}{i}_{oq}\\ & -\frac{sin\left(2\omega t\right)}{2}(u_{c2d}{i}_{oq}+u_{c2q}{i}_{od})\end{array}$$

(19)

Since the Equation (19) contains cos(2ωt) and sin(2ωt), the equation is still time-variant and needs to be further processed. Assuming that g₁ = cos(2ωt), g₂ = sin(2ωt), the following equations can be constructed:

$$\{\begin{array}{l}\frac{d{g}_1}{dt}=-2\omega g_2\\ \frac{d{g}_2}{dt}=2\omega g_1\end{array}$$

(20)

Combining Equations (12), (17), (19) and (20), the observer-pattern model of the system can be written as

$$\{\begin{array}{l}\frac{d{u}_{pv}}{dt}=\frac{1}{C_{in}}\left(I_{SC}\left[1-A_1\left(e^{\frac{u_{pv}}{A_2{U}_{OC}}}-1\right)\right]-i_{L_b}\right)\\ \frac{d{i}_{L_b}}{dt}=\frac{1}{L_b}\left[u_{pv}-\left(1-\frac{u_{c1}}{U_{M1}}\right)u_{dc}\right]\\ \frac{d{u}_{dc}}{dt}=\frac{1}{C_{dc}}[\left(1-\frac{u_{c1}}{U_{M1}}\right)i_{L_b}-\frac{1+g_1}{2}{u}_{c2d}{i}_{od}-\frac{1-g_1}{2}{u}_{c2q}{i}_{oq}+\frac{g_2}{2}(u_{c2d}{i}_{oq}+u_{c2q}{i}_{od})]\\ \frac{d{i}_{od}}{dt}=\frac{u_{dc}{u}_{c2d}}{L_f{U}_{M2}}+\omega i_{oq}\\ \frac{d{i}_{oq}}{dt}=\frac{u_{dc}{u}_{c2q}}{L_f{U}_{M2}}+\frac{U_{gm}}{L_f}-\omega i_{od}\\ \frac{d{u}_{c1}}{dt}=K_{p1}\frac{d{u}_{pv}}{dt}+\frac{K_{p1}}{T_{i1}}\left(u_{pv}-u_{pvref}\right)\\ \frac{d{u}_e}{dt}=K_{p2}\frac{d{u}_{dc}}{dt}+\frac{K_{p2}}{T_{i2}}\left(u_{dc}-u_{dcref}\right)\\ \frac{d{u}_{c2d}}{dt}=K_{p3}{u}_e\omega -\frac{K_{p3}}{L_f{U}_{M2}}{u}_{dc}{u}_{c2d}-\frac{K_{p3}}{T_{i3}}{i}_{od}+\omega u_{c2q}\\ \frac{d{u}_{c2q}}{dt}=-K_{p3}\frac{d{u}_e}{dt}-\frac{K_{p3}}{L_f{U}_{M2}}{u}_{dc}{u}_{c2q}-\frac{U_{gm}{K}_{p3}}{L_f}-\frac{K_{p3}}{T_{i3}}{u}_e-\frac{K_{p3}}{T_{i3}}{i}_{oq}-\omega u_{c2d}\\ \frac{d{g}_1}{dt}=-2\omega g_2\\ \frac{d{g}_2}{dt}=2\omega g_1\end{array}$$

(21)

In order to verify the proposed model, simulation results obtained from PSIM are compared to the solutions of the model obtained from MATLAB. The simulation parameters, including PV array parameters and converter parameters, are given in

Table 1. Photovoltaic (PV) array parameters.

Parameter	Symbol	Quantity
Voltage at MPP	UM	119.6 V
Current at MPP	IM	8.36 A
Open circuit voltage	UOC	149.2 V
Short circuit current	ISC	8.81 A

and

Table 2. Converter parameters.

Parameter	Symbol	Quantity
Capacitance of input filter	Cin	1000 μF
Inductance of boost converter	Lb	10 mH
Capacitance of boost converter	Cdc	1500 μF
Inductance of output filter	Lf	25 mH
Amplitude of grid voltage	Ugm	220√2 V
Switching frequency	fs	10 kHz
Gain of PI controller 1	Kp1	0.05
Time constant of PI controller 1	Ti1	0.1
Gain of PI controller 2	Kp2	0.02
Time constant of PI controller 2	Ti2	0.01
Gain of PI controller 3	Kp3	1
Time constant of PI controller 3	Ti3	0.2

. Figure 2 and Figure 3 show the simulation waveforms of DC bus voltage u_dc and output current i_o, respectively. In both pictures, it can be seen obviously that observer-pattern model and simulation give almost the same results in steady state.

4. Stability Analysis

For simplicity, the observer-pattern model of the system can be written as

$$\dot{\mathit{X}}=\mathit{F}(\mathit{X})(\mathit{F}:{\mathit{R}}^{11}\to {\mathit{R}}^{11})$$

(22)

where X = [u_pv, $i_{L_b}$, u_dc, i_od, i_oq, u_c1, u_e, u_c2d, u_c2q, g₁, g₂]^T. By setting all the differential items in (21) to zero, the equilibrium point X^e = [u_pv^e, $i_{L_b}$^e, u_dc^e, i_od^e, i_oq^e, u_c1^e, u_e^e, u_c2d^e, u_c2q^e, g₁^e, g₂^e]^T is obtained. Then. the Jacobian A of the observer-pattern model at the equilibrium point can be derived as

$$\mathit{A}=\frac{\partial \mathit{F}}{\partial \mathit{X}}{|}_{\mathit{X}={\mathit{X}}^{\mathit{e}}}$$

(23)

Detailed description of Jacobian A is presented in Appendix A.

To analyze the stability of the system, the eigenvalues of Jacobian are used, which can be calculated by

$$det\left[\lambda \mathit{I}-\mathit{A}\right]=0$$

(24)

where I is unit matrix.

The gain and the time constant of three PI controllers are key parameters that affect the performance of the system. As these parameters change, the eigenvalues of the system also change. For the purpose of estimating the direction and size of the eigenvalue movement due to variations in system parameters, a sensitivity analysis is often used [28,29]. The eigenvalue sensitivity of λ_i with respect to an uncertain parameter μ can be calculated as follow:

$$\frac{\partial {\lambda }_i}{\partial \mu }=\frac{{\omega }_i{}^T(\partial \mathit{A}/\partial \mu ){\upsilon }_i}{{\omega }_i{}^T{\upsilon }_i}$$

(25)

where ω_i and υ_i are the left and right eigenvectors corresponding to the eigenvalue λ_i, respectively. If the real part of ∂λ_i/∂μ is positive, an increase in the parameter μ causes the eigenvalue λ_i to move towards right in horizontal direction. The size of the horizontal movement is decided by the magnitude of the real part of ∂λ_i/∂μ. Similarly, the imaginary part of ∂λ_i/∂μ is associated with the movement in vertical direction.

Table 3. Eigenvalue sensitivities.

	λ1,2	λ3,4	λ5	λ6,7	λ8,9	λ10,11
Kp1	−8.93 × 10-4 ± j7.47 × 10-5	5.57 ± j1.38 × 104	−9.31	−0.91 ± j0.145	−2.45 × 10-5 ± j2.42 × 10-4	0
Ti1	−2.78 × 10-7 ± j2.88 × 10-8	−47.5 ± j0.21	94.9	0.0286 ± j0.193	3.85 × 10-6 ± j3.28 × 10-7	0
Kp2	−937 ± j35.6	−0.605 ± j0.188	−0.664	−134 ± j553	0.00347 ± j0.00126	0
Ti2	−11.8 ± j0.68	−0.0211 ± j0.0846	1.47	11 ± j1144	1.22 × 10-4 ± j0.00224	0
Kp3	−1.6 × 104 ± j0.977	7.45 × 10-4 ± j0.00145	−1.35 × 10-4	0.236 ± j0.0336	0.00154 ± j0.00409	0
Ti3	−25 ± j7.68 × 10-4	2.62 × 10-5 ± j1.36 × 10-5	2.21 × 10-5	0.00141 ± j0.0192	25 ± j0.0208	0

lists the eigenvalue sensitivities with respect to PI controller parameters calculated by (25). For λ_1,2, the most sensitive parameter is K_p3 as the real part of the sensitivity of λ_1,2 with respect to K_p3 is the largest. The decrease in K_p3 makes λ_1,2 move toward right in the s-plane. For λ_3,4, the most sensitive parameters are T_i1 and K_p1. A negative perturbation in T_i1 makes λ_3,4 move towards right in the s-plane. In contrast, the decrease in K_p3 makes λ_3,4 move to left. For λ₅, the most critical parameter is T_i1. The increase in T_i1 leads to λ₅ moving towards right-half plane. For λ_6,7, the most sensitive parameter is K_p2. When K_p2 decreases, λ_6,7 moves towards right in the s-plane. For λ_8,9, the most sensitive parameters is T_i3. The increase in T_i3 leads to λ_8,9 moving towards right in the s-plane. λ_10,11 are insensitive to all these parameters listed in Table 3.

Figure 4 illustrates the loci of the eigenvalues with respect to various PI controller parameters. It is obviously that the eigenvalue loci in Figure 4 match the sensitivity analysis results above. Especially note that in Figure 4b, λ_3,4 move across the imaginary axis from the left-half plane to the right-half plane when T_i1 decreases to 0.01, which means the system becomes unstable.

The eigenvalues when T_i1 equals to 0.01 and 0.03 are listed in

Table 4. Eigenvalues when T_i1 = 0.01 and T_i1 = 0.03.

Ti1	λ1,2	λ3,4	λ5	λ6,7	λ8,9	λ10,11
0.01	−16016 ± j314	26.8 ± j1453	−94.7	−2.947 ± j22.55	−5 ± j314	±j628
0.03	−16016 ± j314	−4.743 ± j1451	−31.6	−2.927 ± j22.56	−5 ± j314	±j628

as a comparison. It can be seen clearly that λ_10,11, which originated from Equation (20), always equals to ±j628. Therefore, λ_10,11 is only associated with the angular frequency of power grid and has no influence on the stability of the system. Actually, λ_1,2 and λ_8,9 are different when T_i1 varies. However, the slight differences are disregarded in this paper. Neglecting λ_10,11, all eigenvalues are in the left half of the s-plane when T_i1 = 0.03, and the system operates in a stable state. However, when T_i1 decreases to 0.01, λ_3,4 becomes 26.8 ± j1453, which indicates the system is unstable and a low-frequency oscillation occurs. The frequency of the oscillation can be calculated according to the magnitude of the imaginary part of λ_3,4 as follows:

$$f_{osc}=\frac{1453}{2\pi }=231\mathrm{Hz}$$

(26)

Figure 5 presents the simulation results of u_dc obtained by PSIM when T_i1 equals to 0.03 and 0.01. In Figure 5a, the waveform of u_dc is sinusoidal and fluctuates around the nominal value. It can be seen clearly that u_dc contains a DC component and a ripple at 100 Hz. The ripple at 100 Hz is due to pulsating output power of the single-phase inverter [30]. In Figure 5c, the waveform of u_dc is non-sinusoidal. It can be seen from Figure 5d that the waveform contains a ripple at 100 Hz and a component at 230.5 Hz, which matches the theoretical analysis given by the observer-pattern model.

5. Conclusions

Modeling and stability analysis of a single-phase two-stage grid-connected photovoltaic system have been presented in this paper. (1) An integrated mathematical model including both DC-DC converter and DC-AC converter is developed to capture the dynamics of the system, also the nonlinear characteristic of the PV array is considered in the model; (2) An observer-pattern modeling method is applied to transform the system into time-invariant; (3) Critical controller parameters that influence the stability of the system are identified using eigenvalue sensitivity and eigenvalue loci plots. It is found that T_i1 is closely related to λ_3,4. The decrease in T_i1 makes λ_3,4 move to left in the s-plane and is disadvantageous to the stability of the system. The theoretical results have been validated by PSIM simulations.