Research on Dynamic Modeling and Parameter Identification of the Grid-Connected PV Power Generation System

Abstract

With the increasing proportion of renewable energy in the new power system, the grid-connected capacity of photovoltaic (PV) units shows an obvious upward trend, but its dynamic behavior under different penetration rates significantly affects the transient stability of the power system, so it is crucial to establish a dynamic model that meets the actual working conditions and select a suitable parameter identification method. Therefore, in this paper, based on the electromechanical transient characteristics of the grid-connected PV power generation system, the corresponding dynamic discrete equivalent model is established, and the simulation platform of the grid-connected PV power generation system is built in MATLAB/Simulink to study the adaptability of the dynamic discrete equivalent model of the grid-connected PV power generation system from the single and multiple scenarios using the ordinary least squares (OLS) and bat algorithm (BA) while comparing the generalization ability of the parameters identified by the two methods to the model. The simulation results show that the generalization ability of the parameters identified by the OLS and BA for the model in the single scenario is better, indicating that the model has good adaptability; the generalization ability of a set of general parameters identified by the BA for the model in the multiple scenarios is better than that of the OLS, indicating that the parameters identified by the BA have better adaptability. In conclusion, the dynamic discrete equivalent model of the grid-connected PV power generation system proposed in this paper can accurately reflect the dynamic characteristics of the grid-connected PV power generation system, and the parameters identified by the BA are more generalized than the OLS.

1. Introduction

With the increasingly harsh environment, the proportion of renewable energy in the new power system is gradually increasing; in particular, clean energy represented by PV units is connected to the grid on a large scale [1]. The high penetration rate of the grid-connected PV units has obviously changed the operating characteristics of the power grid, and the difficulty of fine-grained modeling and simulation of the power system has increased [2,3]. In addition, as the core part of the grid-connected PV inverters is flooding into the power system, the contradiction of “high proportion of inverter clusters, weak power grid system structure” is increasingly prominent. Under this background, the dynamic behavior of the grid-connected PV power generation system seriously affects the safe and stable operation of the power grid, and when a fault occurs in the power grid, the dynamic characteristics of the whole network will change rapidly. The existing dynamic model of the grid-connected PV power generation system has a complex structure and many parameters to be identified, so it is difficult to meet the needs of the power system simulation, and it lacks a certain degree of engineering practicality [4,5,6]. Therefore, in order to accurately describe the dynamic characteristics of the grid-connected PV power generation system, it is crucial to establish a dynamic model that meets the actual working conditions and select a suitable parameter identification method [7,8].

To study the dynamic characteristics of the grid-connected PV power generation system, a lot of research has been conducted on its dynamic modeling. The dynamic equivalent model of the PV modules was established for the electrical and thermal characteristics of the PV subsystem [9,10,11,12]. The equivalent description model of the grid-connected PV system was established according to the topology of the inverter to the power grid [13,14]. The single-phase grid-connected PV dynamic vector model was established based on the stability characteristics of the PV plants [15]. A discrete-time model was established by considering the PV power generation system as a dynamic nonlinear multimodal system [16,17,18]. The two-level PV dynamic mathematical model was established considering the dynamic response characteristics of the distribution network with a high PV penetration rate [19]. The linear model of the PV plants was established to describe the interactions between the various components of the grid-connected PV power generation system [20]. The three-level PV phasor model was established on the basis of realizing high-efficiency dynamic simulation of the large-scale power grid [21]. The single-phase and three-phase PV frequency-domain Norton equivalent model was established for simulating the PV system partition time-transient simulation [22]. The generalized discrete-time equivalent model uses the interface as the grid-connected PV bridge [23]. The data-driven hybrid equivalent model was established for the dynamic response process of multiple PV plants [24]. In summary, a lot of achievements have been made in the dynamic modeling of the grid-connected PV power generation system, but most of them focus on the modeling of the PV subsystems, and the proposed models have complex structures and difficult parameter identification. Furthermore, there are few dynamic models of the external fault characteristics of the grid-connected PV power generation system, and an accurate and unified equivalent model has not been formed.

Subsequently, after establishing the dynamic model of the grid-connected PV power generation system, it is necessary to select the suitable parameter identification method to ensure the accuracy of the model. The parameter estimation method based on nonlinear OLS was proposed to realize the optimal parameter identification of the PV modules; it was simple to operate and fast to solve [25]. The parameter identification strategy based on a simulated annealing particle swarm optimization (SAPSO) algorithm was proposed to determine the dynamic model parameters of the PV inverter and had a high identification accuracy [26]. The new hybrid seagull optimization algorithm (HSOA) was proposed to improve the efficiency of the PV model parameter identification and was superior in terms of accuracy and convergence [27]. The improved algorithm based on a novel meta-heuristic marine predator algorithm (MPA) was proposed to solve the problem of the large demand and slow solution speed of the PV plants model simulation, and shortened the solution time and avoided falling into local optimum [28]. The enhanced Lévy flying BA (ELBA) was proposed to reduce the dispersion of the grid-connected PV model parameter identification results, with a high solution quality and good stability of results [29]. In summary, the grid-connected PV power generation system model has made obvious breakthroughs in parameter identification, and more and more intelligent algorithms have been introduced, but it still faces many problems, such as the many parameters to be identified and the complicated solution processes. However, there are a few parameter identification methods proposed for the external response characteristics of the grid-connected PV power generation system, but the identification method that takes into account the solution speed and accuracy has not been found to obtain model parameters, and the model lacks certain universality.

Therefore, in this paper, based on the electromechanical transient characteristics of the grid-connected PV power generation system, the corresponding dynamic discrete equivalent model is established, and the simulation platform for the grid-connected PV power generation system is built in MATLAB/Simulink to study the adaptability of the dynamic discrete equivalent model of the grid-connected PV power generation system from the single and multiple scenarios using the OLS and BA, while comparing the generalization ability of the parameters identified by the two methods to the model. The contributions of the paper are as follows:

(1)

Deriving the dynamic discrete equivalent model of the grid-connected PV power generation system.

(2)

Analyzing the parameter identification process of the OLS and BA.

(3)

Comparing the generalization ability of the parameters identified by the OLS and BA to the model.

(4)

Verifying that the parameters identified by the BA are more generalized than the OLS.

The rest of this paper is organized as follows. The first section introduces the structure and control strategy of the grid-connected PV power generation system, the second section derives the dynamic discrete equivalent model of the grid-connected PV power generation system, the third section analyzes the OLS and BA parameter identification process, and the fourth section verifies the adaptability of the dynamic discrete equivalent model of the grid-connected PV power generation system from the single and multiple scenarios and compares the generalization ability of the parameters identified by the OLS and BA to the model.

2. Structure and Control Strategy of the Grid-Connected PV Power Generation System

2.1. Overall Structure

The grid-connected PV power generation system consists of the PV array, DC/DC booster converter, DC/AC three-phase inverter, LC filter, isolation transformer, and power grid. The PV array converts solar energy into low-voltage DC power through the photoelectric effect, this power is delivered to the DC/DC converter to generate stable high-voltage DC power through the boost circuit control, and the maximum power point tracking (MPPT) control strategy is introduced to ensure the maximum output power of the PV array. Then, the grid-connected inverter converts the DC power into AC power with the same frequency and phase as the power grid and feeds it into the power grid while meeting the grid-connected conditions. The overall structure of the grid-connected PV power generation system is shown in Figure 1.

The PV array uses the engineering practical model that can simulate any type of PV module, the light intensity and temperature changes of the PV array are assumed not to change in a very short period, and the MPPT adopts the disturbance observation method to adapt to the fast-changing atmospheric conditions. The DC/DC boost converter part ignores the switching action and adopts the dynamic mathematical model described by second-order differential equations. The DC/AC three-phase inverter adopts the dual-loop control structure with the voltage outer loop and current inner loop for the pulse width modulation (PWM), while d-q cross-coupling compensation is introduced to reduce the grid-connected circulating current.

2.2. Control System

The control structure of the grid-connected PV power generation system is shown in Figure 2. In the figure, L_f and C_f denote the filter inductance and capacitance of the low-pass filter, respectively; R_l and L_l denote the equivalent resistance and inductance of the transmission line from the PV power generation system to the public connection point (PCC), respectively; PLL denotes the phase-locked loop.

The main control parameters of the grid-connected PV inverter are shown in

Table 1. The main control parameters of the grid-connected PV inverter.

Parameter Name		Parameter Value
Voltage outer loop PI controller	Proportional link gain coefficient Kup	0.5
	Proportional link time constant Tup	0.003 s
	Integral link gain coefficient Kui	0.5
	Integral link time constant Tui	0.003 s
Current inner loop PI controller	Proportional link gain coefficient Kip	1
	Proportional link time constant Tip	0.002 s
	Integral link gain coefficient Kii	1
	Integral link time constant Tii	0.002 s

. The voltage regulator, as the outer-loop control, first controls the inverter DC-side output voltage value u_dc to track the given voltage value u_dcref, and then obtains the active current components reference value i_dref and the reactive current components reference value i_qref through the proportion integration (PI) controller. Usually, the grid-connected PV power generation system operates in the unit power factor state, so i_qref is taken as 0. The current regulator, as the inner-loop control, first takes the i_dref and i_qref of the voltage outer-loop output as input signals, and then makes the difference with the inverter AC-side output current values i_d and i_q and inputs them to the PI controller to realize the sinusoidal pulse width modulation (SPWM) modulated voltage wave by decoupling.

Through the dual-loop control and PWM, the inverter connected to the PV power plant and the regional power grid can realize the decoupling of the active power and reactive power. The power injected into the power grid by the PV power system can be expressed as Equation (1).

$$\left\{\begin{array}{l}P=\frac{3}{2}\left(U_{\mathrm{gd}}{I}_{\mathrm{d}}+U_{\mathrm{gq}}{I}_{\mathrm{q}}\right)\\ Q=-\frac{3}{2}\left(U_{\mathrm{gq}}{I}_{\mathrm{d}}-U_{\mathrm{gd}}{I}_{\mathrm{q}}\right)\end{array}\right.$$

(1)

Taking the d-axis as the direction of the AC-side voltage synthesis vector, the voltage component in the q-axis direction is 0, i.e., U_gq = 0. The grid-connected PV power generation system usually operates in the unit power factor state, so i_q = i_qref = 0. Therefore, the above equation can be expressed as Equation (2):

$$\left\{\begin{array}{l}P=\frac{3}{2}{U}_{\mathrm{gd}}{I}_{\mathrm{d}}\\ Q=0\end{array}\right.$$

(2)

where P and Q denote the active and reactive power emitted by the grid-connected PV power generation system, respectively; U_gd, U_gq and I_d, I_q denote the d-q axis components of the grid-connected voltage and the current of the PV power generation system, respectively.

According to the above equation, the control of the active and reactive power of the grid-connected PV power generation system can be achieved by controlling the d-q axis components of the grid-connected current of the PV power generation system. In the millisecond power system transient process, the process of the electromechanical transient characteristics containing the dynamic load of the PV power generation system is extremely short, and the external dynamic characteristics of the PV power generation system depend on the grid-connected point voltage variation.

3. Dynamic Discrete Equivalent Model for the Grid-Connected PV Power Generation System

3.1. Single-Phase Equivalent Circuit Model

Because the inverter delivers power by injecting current into the power grid and controlling it, it can be equated to the current source. The power grid, on the other hand, is considered to maintain a constant voltage at a given moment, so it is equated to the voltage source. Based on this, the single-phase equivalent circuit model of the grid-connected PV power generation system is represented by the series combination of the controlled voltage source, equivalent resistance, and equivalent reactance; the specific structure is shown in Figure 3. In the figure, i_PV denotes the output current of the PV array; u_dc denotes the DC side voltage of the inverter; i_dc denotes the current injected into the inverter; U_i denotes the AC side voltage of the inverter; R and L denote the equivalent resistance and reactance from the inverter to the grid-connected point, respectively; I_L denotes the grid-connected current; U_g denotes the grid-connected voltage; and C denotes the voltage regulator capacitor on the DC side of the PV power generation system.

The dynamic differential equation of the grid-connected PV power generation system can be obtained from Kirchhoff’s voltage and current law (KVL, KCL), as shown in Equation (3).

$$\left\{\begin{array}{l}\frac{d{I}_{\mathrm{L}.\mathrm{abc}}}{\mathrm{d}t}=\frac{1}{L}\left(U_{\mathrm{i}.\mathrm{abc}}-U_{\mathrm{g}.\mathrm{abc}}-I_{\mathrm{L}.\mathrm{abc}}R\right)\\ \frac{d{u}_{\mathrm{dc}}}{dt}=\frac{1}{C}\left(i_{\mathrm{pv}}-i_{\mathrm{dc}}\right)\end{array}\right.$$

(3)

The third-order dynamic differential equation with the d-q axis components as state variables is obtained by subjecting Equation (3) to the Park transformation, as shown in Equation (4):

$$\left\{\begin{array}{l}\frac{d{I}_{\mathrm{L}.\mathrm{d}}}{dt}=\frac{1}{L}\left(U_{\mathrm{i}.\mathrm{d}}-U_{\mathrm{g}.\mathrm{d}}-I_{\mathrm{L}.\mathrm{d}}R\right)-\omega I_{\mathrm{L}.\mathrm{q}}\\ \frac{d{I}_{\mathrm{L}.\mathrm{q}}}{dt}=\frac{1}{L}\left(U_{\mathrm{i}.\mathrm{q}}-U_{\mathrm{g}.\mathrm{q}}-I_{\mathrm{L}.\mathrm{q}}R\right)+\omega I_{\mathrm{L}.\mathrm{d}}\\ \frac{d{u}_{\mathrm{dc}}}{dt}=\frac{1}{C}\left[i_{\mathrm{PV}}-1.5\left(S_{\mathrm{d}}{I}_{\mathrm{L}.\mathrm{d}}+S_{\mathrm{q}}{I}_{\mathrm{L}.\mathrm{q}}\right)\right]\end{array}\right.$$

(4)

where I_L.d, I_L.q and U_g.d, U_g.q denote the d-q axis components of the grid-connected current and the voltage of the PV power generation system, respectively; U_i.d and U_i.q denote the d-q axis components of the AC side voltage of the inverter, respectively; S_d and S_q denote the d-q axis components of the switching vector in the synchronous coordinate system, respectively; and ω denotes the angular frequency of the grid-connected voltage.

The expression of the output power of the grid-connected PV power generation system represented by the real and imaginary components of the grid-connected current can be obtained by performing the Parker inverse transformation on Equation (4), as shown in Equation (5).

$$\left\{\begin{array}{l}P=U_{\mathrm{g}.\mathrm{x}}{I}_{\mathrm{L}.\mathrm{x}}+U_{\mathrm{g}.\mathrm{y}}{I}_{\mathrm{L}.\mathrm{y}}\\ Q=U_{\mathrm{g}.\mathrm{y}}{I}_{\mathrm{L}.\mathrm{x}}-U_{\mathrm{g}.\mathrm{x}}{I}_{\mathrm{L}.\mathrm{y}}\end{array}\right.$$

(5)

3.2. Dynamic Discrete Equivalent Model

Based on the derivation of the mathematical equation of the single-phase equivalent circuit model, the output power of the grid-connected PV power generation system can be controlled by the real and imaginary components of the grid-connected current. To accurately and reasonably describe the dynamic characteristics of the grid-connected PV power generation system, the dynamic discrete equivalent model of the grid-connected PV power generation system is established based on the single-phase equivalent circuit model, and the corresponding mathematical equations are derived as follows [30]:

The incremental form of the d-q axis components of the grid-connected current of the PV power generation system can be obtained by applying the Laplace transform to Equation (4) and linearizing it, as shown in Equation (6).

$$\{\begin{array}{l}\Delta I_{\text{d}}\left(s\right)=\frac{B_{11}{s}^3\Delta U_{\text{gd}}\left(s\right)+s^2\left[B_{12}\Delta U_{\text{gd}}\left(s\right)+B_{13}\Delta U_{\text{gq}}\left(s\right)\right]}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}\text{+}\frac{s\left[B_{14}\Delta U_{\text{gd}}\left(s\right)+B_{15}\Delta U_{\text{gq}}\left(s\right)\right]}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}\\ \Delta I_{\text{q}}\left(s\right)=\frac{B_{21}{s}^3\Delta U_{\text{gq}}\left(s\right)+s^2\left[B_{22}\Delta U_{\text{gd}}\left(s\right)+B_{23}\Delta U_{\text{gq}}\left(s\right)\right]}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}\text{+}\frac{s\left[B_{24}\Delta U_{\text{gq}}\left(s\right)+B_{25}\Delta U_{\text{gd}}\left(s\right)\right]}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}\end{array}$$

(6)

where:

• $A_{11}=1$, $A_{12}=\frac{2R}{L}$, $A_{13}=\frac{4C{R}^2+3s_{\mathrm{q}}^{2}L+3s_{\mathrm{d}}^{2}L+4C{L}^2{\omega }^2}{4C{L}^2}$, $A_{14}=\frac{3s_{\mathrm{q}}^{2}R+3s_{\mathrm{d}}^{2}R}{4C{L}^2}$, $A_{15}=0$;
• $B_{11}=-\frac{1}{L}$, $B_{12}=-\frac{R}{L^2}$, $B_{13}=\frac{\omega }{L}$, $B_{14}=-\frac{3s_{\mathrm{q}}^2}{4C{L}^2}$, $B_{15}=\frac{3s_{\mathrm{d}}{s}_{\mathrm{q}}}{4C{L}^2}$;
• $B_{21}=-\frac{1}{L}$, $B_{22}=-\frac{R}{L^2}$, $B_{23}=-\frac{\omega }{L}$, $B_{24}=-\frac{3s_{\mathrm{d}}^2}{4C{L}^2}$, $B_{25}=\frac{3s_{\mathrm{d}}{s}_{\mathrm{q}}}{4C{L}^2}$.

The load model in the frequency domain can be obtained by deforming the incremental form of Equation (6), as shown in Equation (7).

$$\left[\begin{array}{l}\Delta I_{\mathrm{d}}\left(s\right)\\ \Delta I_{\mathrm{q}}\left(s\right)\end{array}\right]=\left[\begin{array}{l}{f}_1\left(s\right)f_2\left(s\right)\\ f_3\left(s\right)f_4\left(s\right)\end{array}\right]\left[\begin{array}{l}\Delta U_{\mathrm{gd}}\left(s\right)\\ \Delta U_{\mathrm{gq}}\left(s\right)\end{array}\right]$$

(7)

where:

$f_1\left(s\right)=\frac{B_{11}{s}^3+B_{12}{s}^2+B_{14}s}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}$, $f_2\left(s\right)=\frac{B_{13}{s}^2+B_{15}s}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}$,

$f_3\left(s\right)=\frac{B_{22}{s}^2+B_{25}s}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}$, $f_4\left(s\right)=\frac{B_{21}{s}^3+B_{23}{s}^2+B_{24}s}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}$.

The expression of the relationship between the grid-connected voltage and the real and imaginary parts of the grid-connected current of the PV power generation system can be obtained by coordinating the transformation of Equation (7), as shown in Equation (8).

$$\left[\begin{array}{l}\Delta I_{\mathrm{r}}\\ \Delta I_{\mathrm{j}}\end{array}\right]=\left[\begin{array}{l}G{F}_1{\mathrm{GF}}_2\\ B{H}_1{\mathrm{BH}}_2\end{array}\right]\left[\begin{array}{l}\Delta U_{\mathrm{gr}}\\ \Delta U_{\mathrm{gj}}\end{array}\right]$$

(8)

where:

$G{F}_1=\frac{\sin 2{\delta }_0}{2}\left[f_2\left(s\right)+f_3\left(s\right)\right]+{\sin }^2{\delta }_0{f}_1\left(s\right)+{\cos }^2{\delta }_0{f}_4\left(s\right)$,

$G{F}_2=\frac{\sin 2{\delta }_0}{2}\left[f_4\left(s\right)-f_1\left(s\right)\right]+{\sin }^2{\delta }_0{f}_2\left(s\right)-{\cos }^2{\delta }_0{f}_3\left(s\right)$,

$B{H}_1=\frac{\sin 2{\delta }_0}{2}\left[f_4\left(s\right)-f_1\left(s\right)\right]+{\sin }^2{\delta }_0{f}_3\left(s\right)-{\cos }^2{\delta }_0{f}_2\left(s\right)$,

$B{H}_2=-\frac{\sin 2{\delta }_0}{2}\left[f_2\left(s\right)+f_3\left(s\right)\right]+{\sin }^2{\delta }_0{f}_4\left(s\right)+{\cos }^2{\delta }_0{f}_1\left(s\right)$.

Then, convert Equation (8) into the transfer function of the real and imaginary parts of the grid-connected current concerning the grid-connected voltage, as shown in Equation (9).

$$\left\{\begin{array}{l}\frac{\Delta I_{\mathrm{r}}\left(s\right)}{\Delta U_{\mathrm{g}}\left(s\right)}=\frac{A_{\mathrm{r}}{s}^3+C_{\mathrm{r}}{s}^2+B_{\mathrm{r}}s}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}\\ \frac{\Delta I_{\mathrm{j}}\left(s\right)}{\Delta U_{\mathrm{g}}\left(s\right)}=\frac{A_{\mathrm{j}}{s}^3+C_{\mathrm{j}}{s}^2+B_{\mathrm{j}}s}{A_{11}{s}^4+A_{12}{s}^3+A_{13}{s}^2+A_{14}s+A_{15}}\end{array}\right.$$

(9)

where:

$A_{\mathrm{r}}=\frac{\sin 2{\delta }_0}{2}\left(B_{21}-B_{11}\right)+{\sin }^2{\delta }_0{B}_{11}+{\cos }^2{\delta }_0{B}_{21}$,

$A_{\mathrm{j}}=\frac{\sin 2{\delta }_0}{2}\left(B_{21}-B_{11}\right)+{\sin }^2{\delta }_0{B}_{21}+{\cos }^2{\delta }_0{B}_{11}$;

$B_{\mathrm{r}}=\frac{\sin 2{\delta }_0}{2}\left(B_{15}+B_{25}+B_{24}-B_{14}\right)+{\sin }^2{\delta }_0\left(B_{14}+B_{15}\right)+{\cos }^2{\delta }_0\left(B_{24}-B_{25}\right)$,

$B_{\mathrm{j}}=\frac{\sin 2{\delta }_0}{2}\left(B_{24}-B_{14}-B_{15}-B_{25}\right)+{\sin }^2{\delta }_0\left(B_{24}+B_{25}\right)+{\cos }^2{\delta }_0\left(B_{14}-B_{15}\right)$;

$C_{\mathrm{r}}=\frac{\sin 2{\delta }_0}{2}\left(B_{13}+B_{22}+B_{23}-B_{12}\right)+{\sin }^2{\delta }_0\left(B_{12}+B_{13}\right)+{\cos }^2{\delta }_0\left(B_{23}-B_{22}\right)$,

$C_{\mathrm{j}}=\frac{\sin 2{\delta }_0}{2}\left(B_{23}-B_{12}-B_{13}-B_{22}\right)+{\sin }^2{\delta }_0\left(B_{22}+B_{23}\right)+{\cos }^2{\delta }_0\left(B_{12}-B_{13}\right)$.

The difference equation model with the real and imaginary components of the grid-connected current as outputs can be obtained by performing bilinear transformation on Equation (9), as shown in Equation (10):

$$\left\{\begin{array}{l}\Delta I_{\mathrm{r}}\left(k+4\right)={\theta }_{\mathsf{\alpha }1}\Delta I_{\mathrm{r}}\left(k+3\right)+{\theta }_{\mathsf{\alpha }2}\Delta I_{\mathrm{r}}\left(k+2\right)+\\ {\theta }_{\mathsf{\alpha }3}\Delta I_{\mathrm{r}}\left(k+1\right)+{\theta }_{\mathsf{\alpha }4}\Delta I_{\mathrm{r}}\left(k\right)+{\theta }_{\mathsf{\alpha }5}\Delta U\left(k+4\right)+\\ {\theta }_{\mathsf{\alpha }6}\Delta U\left(k+3\right)+{\theta }_{\mathsf{\alpha }7}\Delta U\left(k+2\right)+{\theta }_{\mathsf{\alpha }8}\Delta U\left(k+1\right)+{\theta }_{\mathsf{\alpha }9}\Delta U\left(k\right)\\ \Delta I_{\mathrm{j}}\left(k+4\right)={\theta }_{\mathsf{\beta }1}\Delta I_{\mathrm{j}}\left(k+3\right)+{\theta }_{\mathsf{\beta }2}\Delta I_{\mathrm{j}}\left(k+2\right)+\\ {\theta }_{\mathsf{\beta }3}\Delta I_{\mathrm{j}}\left(k+1\right)+{\theta }_{\mathsf{\beta }4}\Delta I_{\mathrm{j}}\left(k\right)+{\theta }_{\mathsf{\beta }5}\Delta U\left(k+4\right)+\\ {\theta }_{\mathsf{\beta }6}\Delta U\left(k+3\right)+{\theta }_{\mathsf{\beta }7}\Delta U\left(k+2\right)+{\theta }_{\mathsf{\beta }8}\Delta U\left(k+1\right)+{\theta }_{\mathsf{\beta }9}\Delta U\left(k\right)\end{array}\right.$$

(10)

where k denotes the sampling moment of the grid-connected PV power generation system and θ_α1–θ_α9 and θ_β1–θ_β9 denote the real and imaginary part coefficients of the differential equation model, respectively. The expressions are as follows:

${\theta }_{\mathsf{\alpha }1}=\frac{-4A_{14}{h}^3+16A_{12}h+64A_{11}}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\alpha }2}=\frac{8A_{13}{h}^2-96A_{11}}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\alpha }3}=\frac{4A_{14}{h}^3-16A_{12}h+64A_{11}}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\alpha }4}=\frac{2A_{14}{h}^3-4A_{13}{h}^2+8A_{12}h-16A_{11}}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\alpha }5}=\frac{2B_{\mathrm{r}}{h}^3+4C_{\mathrm{r}}{h}^2+8A_{\mathrm{r}}h}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\alpha }6}=\frac{4B_{\mathrm{r}}{h}^3-16A_{\mathrm{r}}h}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\alpha }7}=\frac{-8C_{\mathrm{r}}{h}^2}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\alpha }8}=\frac{-4B_{\mathrm{r}}{h}^3+16A_{\mathrm{r}}h}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\alpha }9}=\frac{-2B_{\mathrm{r}}{h}^3+4C_{\mathrm{r}}{h}^2-8A_{\mathrm{r}}h}{2A_{14}{h}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$; ${\theta }_{\mathsf{\beta }1}=\frac{-4A_{14}{h}^3+16A_{12}h+64A_{11}}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\beta }2}=\frac{8A_{13}{h}^2-96A_{11}}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\beta }3}=\frac{4A_{14}{h}^3-16A_{12}h+64A_{11}}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\beta }4}=\frac{2A_{14}{h}^3-4A_{13}{h}^2+8A_{12}h-16A_{11}}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\beta }5}=\frac{2B_{\mathrm{j}}{h}^3+4C_{\mathrm{j}}{h}^2+8A_{\mathrm{j}}h}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\beta }6}=\frac{4B_{\mathrm{j}}{h}^3-16A_{\mathrm{j}}h}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\beta }7}=\frac{-8C_{\mathrm{j}}{h}^2}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$,

${\theta }_{\mathsf{\beta }8}=\frac{-4B_{\mathrm{j}}{h}^3+16A_{\mathrm{j}}h}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$, ${\theta }_{\mathsf{\beta }9}=\frac{-2B_{\mathrm{j}}{h}^3+4C_{\mathrm{j}}{h}^2-8A_{\mathrm{j}}h}{2A{}_{14}h{}^3+4A_{13}{h}^2+8A_{12}h+16A_{11}}$.

From Equation (10), it can be seen that the dynamic discrete equivalent model of the grid-connected PV power generation system needs to determine two sets of unknown parameters; namely, the current real and imaginary parts parameters, which are θ_α1, θ_α2, θ_α3, θ_α4, θ_α5, θ_α6, θ_α7, θ_α8, θ_α9 and θ_β1, θ_β2, θ_β3, θ_β4, θ_β5, θ_β6, θ_β7, θ_β8, θ_β9.

4. Dynamic Discrete Equivalent Model Parameter Identification Method

4.1. Parameter Identification Objective Function

This paper establishes the dynamic discrete equivalent model of the real and imaginary parts of the grid-connected current of the PV power generation system, and to accurately identify the two sets of parameters in this model, the reasonable objective function needs to be designed to minimize the error between the output of the dynamic discrete equivalent model and actual system of the grid-connected transmission and distribution. In this paper, the root-mean-square error (RMSE) is chosen as the objective function, as shown in Equation (11):

$$\left\{\begin{array}{l}E=\sqrt{\frac{1}{N}{\displaystyle \sum _{k=1}^N{\left(I_{\mathrm{r}k}-{\stackrel{\wedge }{I}}_{\mathrm{r}k}\right)}^2}}\\ \stackrel{\wedge }{E}=\sqrt{\frac{1}{N}{\displaystyle \sum _{k=1}^N{\left(I_{\mathrm{j}k}-{\stackrel{\wedge }{I}}_{\mathrm{j}k}\right)}^2}}\end{array}\right.$$

(11)

where N denotes the number of sampling points of the actual system; E and Ê denote the RMSE of the real and imaginary currents of the grid-connected PV power generation system, respectively; I_rk and I_jk denote the real and imaginary current values of the kth sampling point in the actual system, respectively; and Î_rk and Î_jk denote the real and imaginary current values of the k_th sampling point in the dynamic discrete equivalence model, respectively.

4.2. OLS Parameter Identification

The OLS parameter identification is mainly according to the known input and output data to identify model parameters, and the parameter identification process includes two parts, data acquisition and parameter identification, as shown in Figure 4. Among them, h(k) and y(k) denote the input and output variables of the grid-connected PV power generation system, respectively; f(k) and f*(k) denote the actual output and model output of the grid-connected PV power generation system, respectively; e(k) denotes the noise variable; θ*(k) denotes the parameter to be identified; J′(k) denotes the error between the actual output and model output of the grid-connected PV power generation system. First, the voltage U, active power P, and reactive power Q at the PCC of the grid-connected PV power generation system model are obtained as the known conditions of Equation (10). Then, the real part coefficients θ_α1–θ_α9 and imaginary part coefficients θ_β1–θ_β9 of the dynamic discrete equivalent model are identified. Finally, the dynamic data of the real current I_r, imaginary current I_j, and bus voltage U at the PCC are chosen, and the minimum variance J is used as the target correction parameter estimate.

The corresponding nonlinear model equation is analyzed with m sets of observations, as shown in Equation (12); the variance of the system is obtained, as shown in Equation (13); and there exists a set of optimal parameter estimates, as shown in Equation (14):

$$y_i=f\left(x_i,\theta \right),i=1,2,\dots ,m$$

(12)

where: x₁ = ∆I_r(k + 3), x₂ = ∆I_r(k + 2), x₃ = ∆I_r(k + 1), x₄ = ∆I_r(k), x₅ = ∆U(k + 4), x₆ = ∆U(k + 3), x₇ = ∆U(k + 2), x₈ = ∆U(k + 1), x₉ = ∆U(k); x₁₀ = ∆I_j(k + 3), x₁₁ = ∆I_j(k + 2), x₁₂ = ∆I_j(k + 1), x₁₃ = ∆I_j(k), x₁₄ = ∆U(k + 4), x₁₅ = ∆U(k + 3), x₁₆ = ∆U(k + 2), x₁₇ = ∆U(k + 1), x₁₈ = ∆U(k); y₁ = ∆I_r(k + 4), y₂ = ∆I_j(k + 4).

$$J={\displaystyle \sum _{i=1}^m{\epsilon }_i^2}={\epsilon }^T\epsilon ={\left[y_i-f\left(x_i,\theta \right)\right]}^T\left[y_i-f\left(x_i,\theta \right)\right]$$

(13)

$$\stackrel{\wedge }{\theta }={\left(x_i^T{x}_i\right)}^{-1}{x}_i^T{y}_i{}^{\prime }$$

(14)

4.3. BA Parameter Identification

The BA mainly uses ultrasound to locate prey, and the predation process includes global searches and local searches. The global search updates pulse frequency f_i, flight speed v_i, and position x_i by Equations (15)–(17) and pulse loudness A_i and pulse frequency R_i by Equations (18)–(19). The local search updates position x_i by Equation (20):

$$f_i=f_{\min }+\left(f_{\max }-f_{\min }\right)\beta$$

(15)

$$v_i^t=v_i^{t-1}+\left(x_i^{t-1}-x^{\ast }\right)f_i$$

(16)

$$x_i^t=x_i^{t-1}+v_i^t$$

(17)

$$A_i^{t+1}=\alpha A_i^t$$

(18)

$$R_i^{t+1}=R_i^0\left[1-e^{-\gamma t}\right]$$

(19)

$$x_{inew}=x_{iold}+\epsilon A_i^t$$

(20)

where β denotes the arbitrary value within [0, 1]; x* denotes the optimal value in the global search; f_min and f_max denote the minimum and maximum values of the pulse frequency, respectively; α denotes the decay coefficient of the pulse loudness, taking values in the range of [0, 1]; γ denotes the enhancement coefficient of the pulse frequency, taking values greater than 0; and ε denotes the random number within [−1, 1].

The BA parameter identification means solving for the parameter values corresponding to the minimum RMSE of the objective function based on the same inputs and outputs of the given model and actual system. In essence, it is also a function optimization problem, and its adaptability is shown in Figure 5. The general framework of the parameter identification of the dynamic discrete equivalent model of the grid-connected PV power generation system based on the BA is shown in

Table A1. The general framework for the parameter identification of the dynamic discrete equivalent model of the grid-connected PV power generation system based on the BA.

Steps	Project Content
1	Build the grid-connected PV power generation system model in MATLAB/Simulink.
2	Obtain the voltage U, active power P, and reactive power Q at the grid connection of the grid-connected PV power generation system model.
3	Initialize the population size n and the maximum number of iterations T.
4	Randomly initialize the position of each individual in the population.
5	Set t = 0.
6	WHILE t < T
7	Calculate the fitness of each individual according to Equations (10) and (11).
8	Determine the current optimal individual position of the population xbest(t).
9	Update individual pulse frequency fi, flight speed vi, and position xi according to Equations (15)–(17).
10	FOR1 i = 2, 3, …, n
11	IF1 rand > Ri
12	Update individual position xi according to Equation (17).
13	ELSE1
14	Update individual position xi according to Equation (20).
15	END IF1
16	END FOR1
17	FOR2 i = 2, 3, …, n
18	IF2 rand < Ai
19	Update individual position xi, pulse loudness Ai, and pulse frequency Ri according to Equations (17)–(19).
20	ELSE2
21	Update individual pulse frequency fi, flight speed vi, and position xi according to Equations (15)–(17).
22	END IF2
23	END FOR2
24	Calculate the fitness of each individual according to Equations (10) and (11).
25	Determine the current optimal individual position of the population xbest(t).
26	END WHILE
27	Output the optimal parameters of the dynamic discrete equivalent model of the grid-connected PV power generation system.

of Appendix A, where rand denotes the uniform random number within [0, 1].

5. Analysis of Simulation Results

5.1. Simulation System Overview

To study the adaptability of the dynamic discrete equivalent model of the grid-connected PV power generation system, the simulation platform of the grid-connected power PV generation system is built in MATLAB/Simulink, as shown in Figure 6.In the figure, the transmission system is the IEEE14 node system and the distribution system is the PV power generation system; the main parameters of the transmission and distribution system are given in

Table A2. The main parameters of the transmission and distribution system.

Serial Number	Parameter Name	Parameter Value
Transmission system (IEEE14 node system)	Reference voltage (Uref [kV])	23
	Standard frequency (fs [Hz])	50
	Series inductor (Ls [H])	0.618
	Series resistance (Rs [Ω])	0.4
Distribution system (PV power generation system)	Stand-alone capacity (Ps [kW])	500
	Rated voltage (Ur [V])	700
	Filter inductors (Lf [mH])	0.17
	Filter capacitors (Cf [F])	0.0018
	Step-up transformer ratio (k [kV])	35/230
	Open-circuit voltage (Uoc [V])	44.5
	Short-circuit current (Isc [A])	8.2
	Optimal operating voltage (Um [V])	35.5
	Optimal operating current (Im/A)	7.51
	Number of consistencies (Ns [piece])	300
	Number of parallels (Np [piece])	200
	Components conversion efficiency (ηc [%])	16.9

of Appendix A. Node 7 is set as the grid-connected point, and the PV power generation system is connected to the IEEE14 node system from node 7.

(1)

Simulation parameters setting

Simulation time t = 1 s, simulation step s = 0.001 s, sampling frequency f = 1000 Hz, three-phase short-circuit ground fault occurs at 0.5 s, and fault duration Δt = 0.05 s, using the discrete/ode45 solver. In addition, for the BA, fitness function f_Fitness = 1/E; population size sizepop = 20; the number of feature quantities sizeaim = 9; the maximum number of iterations maxiter = 1000; pulse frequency enhancement coefficient γ = 0.85; pulse frequency range [R_min, R_max] = [0, 1]; pulse loudness decay coefficient α = 0.1; pulse loudness range [A_min, A_max] = [0, 1]; and pulse frequency range [f_min, f_max] = [0, 2].

(2)

Simulation scenarios setting

In this study, the scenarios are formed by the combination of the PV penetration rate and the voltage dip depth. The PV penetration rate is controlled by changing the capacity of the PV connected to the IEEE14 node system, and the voltage dip depth is controlled by changing the fault resistance at the grid-connected bus. The single scenario is defined with the single PV penetration rate and voltage dip depth combination, and the multiple scenarios are defined with the multiple PV penetration rates and voltage dip depths combination.

Due to the page limitation, in the single scenario setting, the PV penetration rate is 48%, 56%, and 64%; the voltage dip depth is 5%, 15%, and 25%; the OLS and BA are used to identify the dynamic discrete equivalent model of the grid-connected PV power generation system, respectively; and the generalization ability of the parameters identified by the OLS and BA to the model is investigated. In the multiple scenarios setting, the PV penetration rate is increased from 10% to 80% (with a spacing of 8%); the voltage dip depth is increased from 5% to 50% (with a spacing of 5%); the OLS and BA are used to identify the dynamic discrete equivalent model of the grid-connected PV power generation system, respectively; a set of the general identification parameters is obtained; and the generalization ability of the general parameters identified by the OLS and BA to the model is investigated.

5.2. Simulation Verification in the Single Scenario

The PV penetration rate is set to 48%, 56%, and 64%; the voltage dip depth is 5%, 15%, and 25%; and the OLS and BA are used to identify the dynamic discrete equivalent model of the grid-connected PV power generation system, respectively; the fitting errors of the identified parameters to the model are shown in

Table 2. The fitting errors for the PV penetration rate of 48%.

Method	Voltage Dip 5%		Voltage Dip 15%		Voltage Dip 25%
	Current Real Part	Current Imaginary Part	Current Real Part	Current Imaginary Part	Current Real Part	Current Imaginary Part
OLS	9.391 × 10−5	2.558 × 10−4	1.722 × 10−4	3.897 × 10−4	4.917 × 10−4	4.256 × 10−4
BA	3.728 × 10−4	5.916 × 10−4	4.617 × 10−4	6.102 × 10−4	6.579 × 10−4	7.343 × 10−4

,

Table 3. The fitting errors for the PV penetration rate of 56%.

Method	Voltage Dip 5%		Voltage Dip 15%		Voltage Dip 25%
	Current Real Part	Current Imaginary Part	Current Real Part	Current Imaginary Part	Current Real Part	Current Imaginary Part
OLS	8.322 × 10−5	2.375 × 10−4	1.588 × 10−4	3.218 × 10−4	3.689 × 10−4	3.816 × 10−4
BA	2.688 × 10−4	6.441 × 10−4	3.505 × 10−4	6.986 × 10−4	5.326 × 10−4	7.894 × 10−4

and

Table 4. The fitting errors for the PV penetration rate of 64%.

Method	Voltage Dip 5%		Voltage Dip 15%		Voltage Dip 25%
	Current Real Part	Current Imaginary Part	Current Real Part	Current Imaginary Part	Current Real Part	Current Imaginary Part
OLS	8.094 × 10−5	2.569 × 10−4	1.241 × 10−4	3.913 × 10−4	2.997 × 10−4	4.565 × 10−4
BA	1.613 × 10−4	5.351 × 10−4	2.894 × 10−4	5.867 × 10−4	3.745 × 10−4	7.018 × 10−4

. The PV penetration rates of 48%, 56%, and 64% and the voltage dip depth of 25% are chosen to investigate the generalization ability of the parameters identified by the OLS and BA to the model, as shown in Figure 7, Figure 8 and Figure 9. In the figures, I_r and I_j denote the measured values of the grid-connected PV power generation system and I_r-OLS, I_j-OLS and I_r-BA, I_j-BA denote the model values of the grid-connected PV power generation system using the OLS and BA parameter identification.

From Table 2, Table 3 and Table 4, it can be seen that both the OLS and BA identify parameters with small fitting errors to the model in the single scenario, but the OLS is slightly better than the BA.

From Figure 5, Figure 6 and Figure 7, it can be seen that the parameters identified by the OLS and BA in the single scenario have better generalization ability to the model.

In summary, it is shown that the dynamic discrete equivalent model of the grid-connected PV power generation system proposed in this paper has good adaptability.

5.3. Simulation Verification in Multiple Scenarios

The PV penetration rate is increased from 10% to 80% (with spacing of 8%), the voltage dip depth is increased from 5% to 50% (with spacing of 5%), and the OLS and BA are used to identify the dynamic discrete equivalent model of the grid-connected PV power generation system, respectively; the generalization ability of a set of the general parameters to the model is shown in Figure 10 and Figure 11. In the figures, the X-axis denotes the voltage dip depth; the Y-axis denotes the PV penetration rate; the Z-axis denotes the fitting error; and the more yellow color indicates the larger fitting error, and the more blue color indicates the smaller fitting error.

From Figure 10 and Figure 11, it can be seen that when the PV penetration rate is in the range of 20–30% and the voltage dip depth is in the range of 15–40%, the BA has significantly better generalization ability than the OLS for the real part of the dynamic discrete equivalent model of the grid-connected PV power generation system; when the PV penetration rate is in the range of 30–40% and the voltage dip depth is in the range of 10–30%, the BA has significantly better generalization ability than the OLS for the imaginary part of the dynamic discrete equivalent model of the grid-connected PV power generation system.

In summary, the generalization ability of a set of the general parameters identified by the BA in the multiple scenarios is better than that of the OLS.

6. Conclusions

In this paper, based on the electromechanical transient characteristics of the grid-connected PV power generation system, the corresponding dynamic discrete equivalent model is established, and the simulation platform of the grid-connected PV power generation system is built in MATLAB/Simulink to study the adaptability of the dynamic discrete equivalent model of the grid-connected PV power generation system from the single and multiple scenarios using the OLS and BA while comparing the generalization ability of the parameters identified by the two methods to the model. Through the simulation experiments, it is found that the dynamic discrete equivalent model of the grid-connected PV power generation system proposed in this paper has good adaptability and can accurately reflect the dynamic characteristics of the grid-connected PV power generation system, and the parameters identified by the BA are more generalized than the OLS.

In this paper, the OLS and BA were selected for the parameter identification of the dynamic discrete equivalent model of the grid-connected PV power generation system without comparing the parameter identification effects of other intelligent algorithms. Therefore, in future research work, it is necessary to further analyze the parameter identification effects of the different intelligent algorithms based on the model proposed in this paper to ensure the generalizability of the model.