Modeling, Control and Validation of a Three-Phase Single-Stage Photovoltaic System

Abstract

The central inverter topology presents some advantages such as simplicity, low cost and high conversion efficiency, being the first option for interfacing photovoltaic mini-generation, whose shading and panel orientation studies are evaluated in the project planning phase. When it uses only one power converter, its control structures must ensure synchronization with the grid, tracking the maximum power generation point, appropriate power quality indices, and control of the active and reactive power injected into the grid. This work develops and contributes to mathematical models, the principles of formation of control structures, the decoupling process of the control loops, the treatment of nonlinearities, and the tuning of the controllers of a single-stage photovoltaic system that is integrated into the electrical grid through a three-phase voltage source inverter. Using the parameters and configurations of an actual inverter installed at the power plant CRESP (Reference Center for Solar Energy of Petrolina), mathematical modeling, implementation, and computational simulations were conducted in the time domain using MatLab^® software (R2021b). The results of the currents injected into the grid, voltages, active powers, and power factor at the connection point with the grid are presented, analyzed, and compared with real measurement data during one day of operation.

1. Introduction

Large photovoltaic (PV) energy conversion systems typically use a central inverter with a single-stage architecture, which presents, among other functionalities, the tracking of the maximum power point and the control of the current injected into the grid. In the literature, single-stage inverters are presented in voltage source inverter (VSI), current source inverter (CSI) or impedance source inverter (ZSI) topologies, which, in terms of output voltage, are classified as buck, boost and buck–boost, respectively [1]. Although the VSI requires a high voltage on the DC-link to establish the connection to the grid, compared to CSI and ZSI, their advantages are low in cost, simplicity, robustness and losses [2,3]. In PV systems with central inverter architecture, the VSI topology is the most widely used due to its high efficiency, simplicity of operation and attractive economic factors [4]. Therefore, this is the first option for large PV system applications, where shading and module orientation studies were assessed during the project planning.

The integration of renewable sources into the electrical system through power electronics is widely discussed in [5]. It is worth noting that VSI is a non-linear time-varying system, requiring detailed mathematical treatment to be represented by realistic simulations, such as those demanded in the performance analysis of control systems [6], and on power systems operation and planning for each study to be conducted [7]. To this end, different coordinate systems like $abc$, $\alpha \beta$, $qd$, and space phasors are concomitantly employed [8], which makes the modeling difficult to understand, given the different notations and concepts used. Using the $qd$ synchronous coordinate system, the modeling and current loop control of a three-phase VSI have been reported in [9,10,11]. However, the use of an ideal voltage source connected in parallel with the VSI input capacitor does not allow the characterization of the voltage dynamic behavior in this component. The integration modeling of PV systems to the power grid presents some deficiencies such as the absence of a systemic analysis of the control structures [12,13,14,15,16], the lack of design and treatment of control loop nonlinearities [12,13,14,15,17,18], the lack of a simple method for tuning controllers [19,20,21], and the lack of experimental validation in real systems [22,23]. Reference [24] presented the Digital Twin of a 17 kVA photovoltaic system to analyze the efficiency of a VSI under daily operating conditions. As they did not use control structures to generate PWM signals, it is impossible to assess whether the methodology can reproduce dynamic operating conditions for photovoltaic systems. In the industry, there is different software capable of estimating the efficiency of the components of a photovoltaic system under stationary operating conditions [25], which are essential in project phase planning. To analyze dynamic conditions, an EMTP (Electromagnetic Transients Program) is used, which, when coupled with dedicated hardware, makes it possible to evaluate power systems in real-time operation [26]. However, these tools are often not available to plant operators. Reference [27] proposed an elegant control system for a photovoltaic inverter, which can improve the transient stability of a synchronous generator connected to the grid when it is experiencing a fault. Although the methodology is promising, the plant models, the treatment of possible non-linearities, and the procedures for tuning the controllers were not described. Furthermore, based on the results presented, it is impossible to infer whether the models and the proposed control structure can reproduce the daily dynamic operating conditions of the photovoltaic plant. Reference [28] presents an iterative procedure considering power quality requirements for selecting the voltage loop bandwidth of grid-connected L-filter inverters. Although the results prove the methodology’s effectiveness, the absence of the specificities of the photovoltaic generator models and the maximum power point tracking (MPPT) algorithms does not allow inferences to be made about the system’s performance when dynamic irradiance and temperature conditions are considered. The VSI integration into single-stage photovoltaic energy conversion systems still presents challenges in terms of synthesizing the PWM signals of the switches, which must simultaneously meet the requirements of maintaining synchronism with the grid, tracking the point of maximum power, regulating the DC-link voltage and ensuring that the currents injected into the grid meet power quality criteria.

In this work, the modeling and treatment of the voltage loop nonlinearities for a three-phase VSI are developed with the objective of representing a commercial inverter in operation at CRESP, in which the configurations of the input PV arrangement, the structure and parameters of the output filter, and typical information present in the databook are known. However, given the industrial secrecy, the control structures and algorithms that define the operation of the inverter are unknown. To ensure power balance in the DC-link and thus ensure appropriate voltage levels, the VSI control system is formulated by a structure with two cascaded loops. The internal loop is characterized by decoupling the active and reactive power controls, as well as by the regulation of the current injected into the grid, while the external loop is characterized by the regulation of the voltage of the inverter DC-link and the tracking of the maximum power point of the PV generator. In order to circumvent the problem of steady-state error, usually observed in controllers described in stationary variables [29], this work employs control structures in the synchronous coordinate system, which allows the use of linear controllers that ensure zero error in the steady state. By applying a methodology for changing the time scale of irradiance and temperature measurements at the input of the PV system, commonly recorded minute-wise, the developed and implemented model allows simulations to be performed in a long duration time window, even when dealing with a dynamic model executed with an integration step on the order of μs. The dataset used for validation of the developed models was obtained in collaboration with CRESP, which has in its installations a 2.5 MW PV plant, a 1 MW floating solar plant, and two other concentrated solar power projects combined with green hydrogen production. As a means of verification, computational analyses are conducted using the MatLab^® software, and the results are validated through experimental measurements collected in the horizon of one day of operation. Assuming as input data the characteristics of the PV modules in the Standard Test Conditions (STC), the parameters of the VSI filters, the characteristics of the power grid, and the daily irradiance and temperature curves, the proposed model can be used to quantify the performance of different PV systems, since the synthesis of the control structures, the auto-tuning of the controllers and the methodology for simulating daily operation are independent of the PV arrangements.

2. Materials and Methods

The power processing system employed in this work is represented by the diagram shown in Figure 1, where the PLL (Phase-Locked Loop) block is the SRF-PLL (Synchronous Reference Frame–PLL) implementation responsible for tracking the phase angle used in the $\mathrm{abc}\to \mathrm{qd}$ transformation [17]; $C_{DC}$ is the DC-link capacitor or inverter input filter; $R_f$, $L_f$ and $C_f=3C_{\mathrm{\Delta }}$ (equivalent star connection capacitor) form the output LC filter; and $R_r$ and $L_r$ are the elements of the utility grid equivalent network.

2.1. Proposed Current Loop Description

The equations that describe the average model of the VSI output currents, the VSI pole voltages, and the PAC voltages can be expressed, respectively, by [30].

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L_f{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{inv-q}\\ i_{inv-d}\end{array}\right]& =\left[\begin{array}{cc}-R_f& -\omega L_f\\ \omega L_f& -R_f\end{array}\right]\left[\begin{array}{c}{i}_{inv-q}\\ i_{inv-d}\end{array}\right]+\left[\begin{array}{c}{v}_{inv-q}-v_{r-q}\\ v_{inv-d}-v_{r-d}\end{array}\right]\hfill \end{array}\end{array}$$

(1)

$$\left[\begin{array}{c}{v}_{inv-q}\hfill \\ v_{inv-d}\hfill \end{array}\right]=\left[\begin{array}{c}{d}_q\hfill \\ d_d\hfill \end{array}\right]v_{DC}$$

(2)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_f{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{v}_{r-q}\hfill \\ v_{r-d}\hfill \end{array}\right]& =C_f\left[\begin{array}{cc}0& -\omega \\ \omega & 0\end{array}\right]\left[\begin{array}{c}{v}_{r-q}\hfill \\ v_{r-d}\hfill \end{array}\right]+\left[\begin{array}{c}{i}_{inv-q}-i_{r-q}\hfill \\ i_{inv-d}-i_{r-d}\hfill \end{array}\right],\hfill \end{array}\end{array}$$

(3)

where $\omega$ is the grid’s operating angular frequency.

Although the synchronous $qd$ model provides a simplified representation of the power processing system, from (1), the control loops of the q and d components of the VSI output currents are coupled, which can degrade the control system performance, as evidenced in [29]. To compensate for the coupling effect, the voltage drop in the filter series impedance, $\mathrm{\Delta }{V}_f$, must be controlled as follows (4) [11,31]:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v_{inv-q}-v_{r-q}& =u_{cq}+\omega L_f{i}_{inv-d}\hfill \\ \hfill v_{inv-d}-v_{r-d}& =u_{cd}+\omega L_f{i}_{inv-d},\hfill \end{array}\end{array}$$

(4)

where $u_{cq}$ and $u_{cd}$ are new input variables of the system. Substituting (4) into (1), yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L_f{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{inv-q}\\ i_{inv-d}\end{array}\right]& =R_f\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{i}_{inv-q}\\ i_{inv-d}\end{array}\right]+\left[\begin{array}{c}{u}_{cq}\\ u_{cd}\end{array}\right],\hfill \end{array}\end{array}$$

(5)

which are first-order differential equations without coupling between the q and d axis state variables.

From (2) and (4), the quadrature-axis and direct-axis components of the inverter duty cycle, which allow the decoupling of the current loops to be performed, are, respectively, given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d_q& ={\displaystyle \frac{u_{cq}+\omega L_f{i}_{inv-d}+v_{r-q}}{v_{DC}}}\hfill \\ \hfill d_d& ={\displaystyle \frac{u_{cd}-\omega L_f{i}_{inv-q}+v_{r-d}}{v_{DC}}}.\hfill \end{array}\end{array}$$

(6)

As a conclusion, the duty cycle components are nonlinear functions on the inverter output currents, the control voltages and the DC-link voltage, given the quotient of the dynamic variables. An alternative to the linearization of (6) is the use of a control structure with two cascaded loops, one being an internal current loop and another an external voltage loop. Given the design methodology of this type of structure, the controllers must ensure that the loops operate independently, and it is a usual practice to make the internal loop much faster than the external loop [12]. Thus, regarding the time response of the current loop, the DC-link voltage can be assumed to be constant, which provides linearity to the duty cycle functions represented in (6).

Denoting by $I_{ref-q}\left(s\right)$ and $I_{ref-d}\left(s\right)$ the reference currents of the q and d axis loops, respectively, and similarly denoting by $C_{iq}\left(s\right)$ and $C_{id}\left(s\right)$ their respective current controllers, assuming zero initial conditions in (1) and (6), the system describing the current closed loop is shown in Figure 2.

After treating the meshes of the d and q axes [11], the results are as follows:

$$\begin{array}{cc}\hfill I_{inv-d}\left(s\right)& =I_{ref-d}\left(s\right)G_{ic}\left(s\right)\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill I_{inv-q}\left(s\right)& =I_{ref-q}\left(s\right)G_{ic}\left(s\right)\hfill \end{array}$$

(8)

$$G_{ic}\left(s\right)={\displaystyle \frac{\left(2\xi {\omega }_n-\frac{1}{T_i}\right)s+{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}},$$

(9)

where (9) is the current closed-loop transfer function, whose parameters are

$$\begin{array}{cc}\hfill {\omega }_n^2& ={\displaystyle \frac{K_i{K}_{PI}}{T_i{T}_{PI}}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \xi & ={\displaystyle \frac{1+K_i{K}_{PI}}{2{\omega }_n{T}_i}}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill K_i& ={\displaystyle \frac{1}{R_f}}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill T_i& ={\displaystyle \frac{L_f}{R_f}},\hfill \end{array}$$

(13)

where ${\omega }_n$ and $\xi$ are the natural frequency and damping fraction, respectively, that define the current closed-loop dynamics, while $K_i$ and $T_i$ correspond to the open-loop gain and time constant, respectively. The parameters $K_{PI}$ and $T_{PI}$ correspond to the gain and the time constant of the controller, respectively, whose determination is presented in section below.

Current Controller Tuning

In a standard second-order system, a good dynamic performance of control systems is usually achieved by adopting $\xi ={\displaystyle \frac{1}{\sqrt{2}}}$ [32]. However, when these systems have a zero added, the transients can be amplified depending on the position of the zero concerning the poles, which requires a new analysis of the $\xi$ value. Also, in these systems, the natural frequency, ${\omega }_n$, can be selected based on the premise that the higher the unitary gain crossover frequency of the open loop transfer function (OLTF), ${\omega }_{ci}$, the better the transient response of the closed loop system. However, since the theoretical limit is ${\omega }_{ci}\le {\displaystyle \frac{1}{2}}{\omega }_s$, where ${\omega }_s$ is the angular frequency of the pulse width modulation carrier [33], establishing a relationship between ${\omega }_n$, ${\omega }_s$, and $\xi$ becomes convenient.

Expressing the OLTF as a function of the design parameters, one can obtain

$$\mathrm{OLTF}\left(\mathrm{s}\right)={\displaystyle \frac{\left(2\xi {\omega }_n-\frac{1}{T_i}\right)s+{\omega }_n^2}{s^2+\frac{1}{T_i}s}}.$$

(14)

Evaluating the OLTF magnitude value at the unit gain crossover frequency, one has

$${|OLTF\left(\mathrm{s}\right)|}_{s=j{\omega }_{ci}}=\sqrt{{\displaystyle \frac{\left(2\xi {\omega }_n-\frac{1}{T_i}\right)j{\omega }_{ci}+{\omega }_n^2}{{\left(j{\omega }_{ci}\right)}^2+\frac{1}{T_i}j{\omega }_{ci}}}}=1,$$

(15)

where ${\omega }_n$ is given by

$${\omega }_n={\displaystyle \frac{{\omega }_{ci}^2\left(\sqrt{4T_i^2{\xi }^2{\omega }_{ci}^2+T_i^2+4{\xi }^2}+2\xi \right)}{T_i\left(4{\xi }^2{\omega }_{ci}^2+1\right)}}.$$

(16)

Substituting (16) into (9), it can be inferred that the closed-loop transfer function, $G_{ic}\left(s\right)$, can be represented as a function of $\xi$ and ${\omega }_{ci}=k_{ci}{\omega }_s$, for $0<k_{ci}\le {\displaystyle \frac{1}{2}}$. Since $f_s=5\mathrm{kHz}$ (VSI switching frequency), ${\omega }_s=2\pi f_s$, and the Figure 3 and Figure 4 show the step response of $G_{ic}\left(s\right)$ for different values of $\xi$ and $k_{ci}$. From the results, it can be inferred that adopting a $10\%$ overshoot and a settling time of $1.0$ ms as design criteria, the solution is in the neighborhood of $\xi =1.3$ and $k_{ci}\in \{0.5,0.4\}$. The lowest value of $k_{ci}$ was adopted in order to reduce possible oscillations in the output currents around the switching frequency. Substituting $\xi$ and ${\omega }_{ci}=0.4{\omega }_s$ in (16), ${\omega }_n=4.84·{10}^3\mathrm{rad}/\mathrm{s}$. Therefore, the parameters obtained for the proportional-integral (PI) controller were $T_{PI}=53.72$ ms and $K_{PI}=1.51$.

2.2. Proposed DC-Link Modeling

The inverters connected to the grid must operate synchronously with the voltages in the PCC, as well as control the injected active and reactive power flows. In other words, they must control the currents injected into the grid and the DC-link voltage. In PV configurations where no DC-DC stages are included, as in the central inverter or string topologies, the DC-link voltage control also performs the MPPT of the PV system [15]. In this work, for simplicity, the well-used conventional algorithm of perturb and observe (P&O) [34] was employed in the MPPT block.

The voltage at the VSI input capacitor depends on the input and output power balance [35]. Based on the control structure shown in Figure 1, the application of Kirchhoff’s law for the currents in the DC-link allows us to express

$$i_{pv}=i_C+i_{DC}.$$

(17)

Multiplying both sides of the equality by $v_{DC}$ gives the active power relations:

$$P_{pv}=i_C{v}_{DC}+P_{DC},$$

(18)

where, $P_{pv}$ and $P_{DC}=i_{DC}{v}_{DC}$ are the power converted by the photovoltaic modules and the power injected into the VSI DC-link, respectively.

Assuming that the switching losses are null, the instantaneous three-phase output power of the inverter is $P_{DC}=P_{\mathrm{inv}}$. Hence, the power balance equation in terms of the capacitor voltage holds:

$$P_{pv}=C{v}_{DC}{\displaystyle \frac{d}{dt}}{v}_{DC}+P_{inv}.$$

(19)

Expressing it in standard state space form yields

$${\displaystyle \frac{d}{dt}}{v}_{DC}^2={\displaystyle \frac{2}{C}}\left(P_{pv}-P_{inv}\right),$$

(20)

where $v_{DC}^2$ is the state variable.

Since the goal is to control the active and reactive power injected into the grid, it is convenient to represent $P_{inv}$ in terms of these components. From the system shown in Figure 1, one can write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L_f{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{inv-a}\left(t\right)\hfill \\ i_{inv-b}\left(t\right)\hfill \\ i_{inv-c}\left(t\right)\hfill \end{array}\right]& =\left[\begin{array}{ccc}-R_f& 0& 0\\ 0& -R_f& 0\\ 0& 0& -R_f\end{array}\right]\left[\begin{array}{c}{i}_{inv-a}\left(t\right)\\ i_{inv-b}\left(t\right)\\ i_{inv-c}\left(t\right)\end{array}\right]+\left[\begin{array}{c}{v}_{inv-a}\left(t\right)-v_{r-a}\left(t\right)\\ v_{inv-b}\left(t\right)-v_{r-b}\left(t\right)\\ v_{inv-c}\left(t\right)-v_{r-c}\left(t\right)\end{array}\right].\hfill \end{array}\end{array}$$

(21)

Pre-multiplying both sides of (21) by the matrix $\left[M\right]$ defined as

$$\left[M\right]=\left[\begin{array}{ccc}{i}_{inv-a}\left(t\right)& 0& 0\\ 0& i_{inv-b}\left(t\right)& 0\\ 0& 0& i_{inv-c}\left(t\right)\end{array}\right],$$

(22)

the instantaneous output power per inverter phase is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left[\begin{array}{c}{i}_{inv-a}\left(t\right)v_{inv-a}\left(t\right)\\ i_{inv-b}\left(t\right)v_{inv-b}\left(t\right)\\ i_{inv-c}\left(t\right)v_{inv-c}\left(t\right)\end{array}\right]& =\left[\begin{array}{ccc}{R}_f& 0& 0\\ 0& R_f& 0\\ 0& 0& R_f\end{array}\right]\left[\begin{array}{c}{i}_{inv-a}^2\left(t\right)\\ i_{inv-b}^2\left(t\right)\\ i_{inv-c}^2\left(t\right)\end{array}\right]\hfill \\ \hfill & +{\displaystyle \frac{L_f}{2}}{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{inv-a}^2\left(t\right)\hfill \\ i_{inv-b}^2\left(t\right)\hfill \\ i_{inv-c}^2\left(t\right)\hfill \end{array}\right]+\left[\begin{array}{c}{i}_{inv-a}\left(t\right)v_{r-a}\left(t\right)\hfill \\ i_{inv-b}\left(t\right)v_{r-b}\left(t\right)\hfill \\ i_{inv-c}\left(t\right)v_{r-c}\left(t\right)\hfill \end{array}\right].\hfill \end{array}\end{array}$$

(23)

Summing up the three-phase components, the instantaneous three-phase output power of the inverter can be represented by the superposition:

$$P_{inv}\left(t\right)=P_R\left(t\right)+P_L\left(t\right)+P_S\left(t\right),$$

(24)

where

$$P_R\left(t\right)=R_f\left(i_{inv-a}^2\left(t\right)+i_{inv-b}^2\left(t\right)+i_{inv-c}^2\left(t\right)\right)$$

(25)

$$P_L\left(t\right)={\displaystyle \frac{L_f}{2}}{\displaystyle \frac{d}{dt}}\left(i_{inv-a}^2\left(t\right)+i_{inv-b}^2\left(t\right)+i_{inv-c}^2\left(t\right)\right)$$

(26)

$$P_S\left(t\right)=v_{r-a}\left(t\right)i_{inv-a}\left(t\right)+v_{r-b}\left(t\right)i_{inv-b}\left(t\right)+v_{r-c}\left(t\right)i_{inv-c}\left(t\right),$$

(27)

where

• $P_R\left(t\right)$: component of the instantaneous three-phase power absorbed by the filter series resistors;
• $P_L\left(t\right)$: three-phase instantaneous power component stored by the filter inductors;
• $P_S\left(t\right)$: approximation of the three-phase instantaneous power component injected in the PCC.

Since this is a three-phase, three-wire system, the application of Park’s non-orthogonal transform to the components $P_L\left(t\right)$, $P_R\left(t\right)$, and $P_S\left(t\right)$ leads to

$$\begin{array}{cc}\hfill P_L\left(t\right)& ={\displaystyle \frac{L_f}{3}}{\displaystyle \frac{d}{dt}}\left(i_{inv-q}^2\left(t\right)+i_{inv-d}^2\left(t\right)\right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill P_R\left(t\right)& ={\displaystyle \frac{2R_f}{3}}\left(i_{inv-q}^2\left(t\right)+i_{inv-d}^2\left(t\right)\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill P_S\left(t\right)& ={\displaystyle \frac{3}{2}}\left(v_{r-d}\left(t\right)i_{inv-d}\left(t\right)+v_{r-q}\left(t\right)i_{inv-q}\left(t\right)\right).\hfill \end{array}$$

(30)

In turn, from instantaneous power theory [36], the reactive power injected into the PCC is given by

$$\begin{array}{c}\hfill Q_S\left(t\right)={\displaystyle \frac{3}{2}}\left(v_{r-q}\left(t\right)i_{inv-d}\left(t\right)-v_{r-d}\left(t\right)i_{inv-q}\left(t\right)\right).\end{array}$$

(31)

Therefore, from (28) to (31)

$$\begin{array}{cc}\hfill P_L\left(t\right)& ={\displaystyle \frac{4L_f}{27}}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{P_S^2\left(t\right)+Q_S^2\left(t\right)}{v_{r-d}^2\left(t\right)+v_{r-q}^2\left(t\right)}}\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill P_R\left(t\right)& ={\displaystyle \frac{8R_f}{27}}\left({\displaystyle \frac{P_S^2\left(t\right)+Q_S^2\left(t\right)}{v_{r-d}^2\left(t\right)+v_{r-q}^2\left(t\right)}}\right).\hfill \end{array}$$

(33)

Assuming that the PLL has zero phase error, the phase voltages when mapped in the synchronous reference have the following components:

$$\begin{array}{cc}\hfill v_{r-d}& =V_m\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill v_{r-q}& =0,\hfill \end{array}$$

(35)

where $V_m$ is the peak value of the phase voltage at PCC. Hence

$$\begin{array}{cc}\hfill P_L\left(t\right)& ={\displaystyle \frac{8L_f}{27V_m^2}}\left(P_S\left(t\right){\displaystyle \frac{d}{dt}}{P}_S\left(t\right)+Q_S\left(t\right){\displaystyle \frac{d}{dt}}{Q}_S\left(t\right)\right)\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill P_R\left(t\right)& ={\displaystyle \frac{8R_f}{27V_m^2}}\left(P_S^2\left(t\right)+Q_S^2\left(t\right)\right).\hfill \end{array}$$

(37)

Substituting (36) and (37) into (24), yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P_{inv}\left(t\right)& =P_S\left(t\right)+{\displaystyle \frac{8L_f}{27V_m^2}}\left(P_S\left(t\right){\displaystyle \frac{d}{dt}}{P}_S\left(t\right)+Q_S\left(t\right){\displaystyle \frac{d}{dt}}{Q}_S\left(t\right)\right)+{\displaystyle \frac{8R_f}{27V_m^2}}\left(P_S^2\left(t\right)+Q_S^2\left(t\right)\right).\hfill \end{array}\end{array}$$

(38)

Substituting (38) into (20), results in

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{DC}^2\left(t\right)}{dt}}& \approx {\displaystyle \frac{2}{C}}\left(P_{pv}\left(t\right)-P_S\left(t\right)\right)-{\displaystyle \frac{2}{C}}\left[{\displaystyle \frac{8L_f}{27V_m^2}}\left(P_S\left(t\right){\displaystyle \frac{d}{dt}}{P}_S\left(t\right)+Q_S\left(t\right){\displaystyle \frac{d}{dt}}{Q}_S\left(t\right)\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{2}{C}}\left[{\displaystyle \frac{8R_f}{27V_m^2}}\left(P_S^2\left(t\right)+Q_S^2\left(t\right)\right)\right],\hfill \end{array}\end{array}$$

(39)

which describes the dynamics of the DC-link, where $v_{DC}^2\left(t\right)$ is a system output variable and therefore a quantity to be controlled. Although $V_m$ may be assumed approximately constant, the presence of the terms $P_S\left(t\right){\displaystyle \frac{d}{dt}}{P}_S\left(t\right)$, $Q\left(t\right){\displaystyle \frac{d}{dt}}{Q}_S\left(t\right)$, $P_S^2\left(t\right)$, and $Q_S^2\left(t\right)$ makes the differential equation a non-linear one, which makes it difficult to obtain a control law to regulate the DC-link voltage. In this sense, a linearization process of the equation is usually employed [37].

2.3. Small-Signal Modeling

For the purpose of linearization of (39), in this work, $P_R\left(t\right)\ll P_S\left(t\right)$ was assumed. Therefore, the power dissipated in the series resistor of the LC filter is much smaller than the power injected into the PCC. Moreover, considering the dynamic system in an equilibrium state at the operating point, one has $P_S\left(t\right)=P_{Se},Q_S\left(t\right)=Q_{Se}$ and $P_{pv}\left(t\right)=P_{pve}$, and from (39)

$${\displaystyle \frac{2}{C}}\left(P_{pve}-P_{Se}\right)=0.$$

(40)

Thus, in the steady state, the power injected into the PCC should approximately match the power delivered by the PV generator.

By introducing a sinusoidal disturbance at the stationary operating point, the voltage at the DC-link can be described as

$${\displaystyle \frac{d}{dt}}\left(v_{CCe}^2+\mathrm{\Delta }{v}_{DC}^2\left(t\right)\right)\approx {\displaystyle \frac{2}{C}}\left(a_1+a_2+a_3\right),$$

(41)

where

$$\begin{array}{cc}\hfill a_1& =\left(P_{pve}+\mathrm{\Delta }{P}_{pv}\left(t\right)\right)-\left(P_{Se}+\mathrm{\Delta }{P}_S\left(t\right)\right)\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill a_2& =\left(1+{\displaystyle \frac{8L_f}{27V_m^2}}{\displaystyle \frac{d}{dt}}\left(P_{Se}+\mathrm{\Delta }{P}_S\left(t\right)\right)\right)\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill a_3& =-\left(Q_{Se}+\mathrm{\Delta }{Q}_S\left(t\right)\right)\left({\displaystyle \frac{8L_f}{27v_m^2}}\right){\displaystyle \frac{d}{dt}}\left(Q_{Se}+\mathrm{\Delta }{Q}_S\left(t\right)\right),\hfill \end{array}$$

(44)

where $\mathrm{\Delta }{v}_{DC}\left(t\right),\mathrm{\Delta }{P}_{pv}\left(t\right),\mathrm{\Delta }{P}_S\left(t\right)$, and $\mathrm{\Delta }{Q}_S\left(t\right)$ are disturbances or small sinusoidal signals.

By further extending the derivatives and adopting the equalities ${\displaystyle \frac{d}{dt}}{v}_{DCe}^2=0$, ${\displaystyle \frac{d}{dt}}{P}_{S_E}=0$, and ${\displaystyle \frac{d}{dt}}{Q}_{Se}=0$, the small-signal model of the DC-link voltage can be represented in the form

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\mathrm{\Delta }{v}_{DC}^2\left(t\right)\approx {\displaystyle \frac{2}{C}}\left({\tilde{a}}_1+{\tilde{a}}_2+{\tilde{a}}_3\right),\end{array}\end{array}$$

(45)

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tilde{a}}_1& =\mathrm{\Delta }{P}_{pv}\left(t\right)-P_{Se}\left({\displaystyle \frac{8L_f}{27V_m^2}}{\displaystyle \frac{d}{dt}}\mathrm{\Delta }{P}_S\left(t\right)\right)\hfill \\ \hfill {\tilde{a}}_2& =-\mathrm{\Delta }{P}_S\left(t\right)\left(1+{\displaystyle \frac{8L_f}{27V_m^2}}{\displaystyle \frac{d}{dt}}\mathrm{\Delta }{P}_S\left(t\right)\right)\hfill \\ \hfill {\tilde{a}}_3& =\left(-Q_{Se}-\mathrm{\Delta }{Q}_S\left(t\right)\right)\left({\displaystyle \frac{8L_f}{27V_m^2}}\right){\displaystyle \frac{d}{dt}}\mathrm{\Delta }{Q}_S\left(t\right).\hfill \end{array}\end{array}$$

(46)

By inspection of (46), the following terms make the small-signal model a nonlinear model:

$$\begin{array}{cc}\hfill {\varphi }_1& =\mathrm{\Delta }{P}_S\left(t\right)\left({\displaystyle \frac{8L_f}{27V_m^2}}\right){\displaystyle \frac{d}{dt}}\mathrm{\Delta }{P}_S\left(t\right)\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\varphi }_2& =\mathrm{\Delta }{Q}_S\left(t\right)\left({\displaystyle \frac{8L_f}{27V_m^2}}\right){\displaystyle \frac{d}{dt}}\mathrm{\Delta }{Q}_S\left(t\right).\hfill \end{array}$$

(48)

Thus, an alternative to make the small-signal model linear is to disregard these terms. In this condition, (45) is a linear system with multiple inputs, whose representation in the s domain becomes more convenient using the transfer matrix notation, namely

$$\mathrm{\Delta }{V}_{DC}^2\left(s\right)=\left[\begin{array}{ccc}{H}_1\left(s\right)\hfill & H_2\left(s\right)\hfill & H_3\left(s\right)\hfill \end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }{P}_{pv}\left(s\right)\\ \mathrm{\Delta }{P}_S\left(s\right)\\ \mathrm{\Delta }{Q}_S\left(s\right)\end{array}\right],$$

(49)

where

$$\begin{array}{cc}\hfill H_1\left(s\right)& ={\displaystyle \frac{2}{sC}}\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill H_2\left(s\right)& =-\left({\displaystyle \frac{2}{C}}\right){\displaystyle \frac{{\tau }_{P}s+1}{s}}\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill H_3\left(s\right)& =-{\displaystyle \frac{2}{C}}{\tau }_Q,\hfill \end{array}$$

(52)

with

$$\begin{array}{cc}\hfill {\tau }_P& =\left({\displaystyle \frac{8L_f{P}_{Se}}{27V_m^2}}\right)\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\tau }_Q& =\left({\displaystyle \frac{8L_f{Q}_{Se}}{27V_m^2}}\right).\hfill \end{array}$$

(54)

Figure 5 depicts (49) in a block diagram, whose inputs $\mathrm{\Delta }{Q}_S\left(s\right)$ and $\mathrm{\Delta }{P}_{pv}\left(s\right)$ will be assumed as disturbances. The disturbances in active power are associated with irradiance and temperature variations in the operation of the PV generators, while disturbances in reactive power are associated with the need for power factor regulation in the PCC. Both events may disturb the DC-link voltage, and a control action should be taken to ensure voltage stability.

2.4. Proposed DC-Link Voltage Control Loop

Assuming balanced three-phase voltages and a phase capture loop with zero error, the reference active and reactive power to be injected into the PCC when expressed in the synchronous referential are, respectively,

$$\begin{array}{cc}\hfill P_{S-ref}\left(t\right)& ={\displaystyle \frac{3}{2}}{V}_m{i}_{r-dref}\left(\mathrm{t}\right)\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill Q_{S-ref}\left(t\right)& =-{\displaystyle \frac{3}{2}}{V}_m{i}_{r-qref}\left(t\right).\hfill \end{array}$$

(56)

Otherwise, assuming that $V_m$ is held constant so that the power $P_{S-ref}\left(t\right)$ and $Q_{S-ref}\left(t\right)$ can be injected into the PCC, the direct and quadrature components of the currents injected into the grid have the following forms:

$$\begin{array}{cc}\hfill i_{r-dref}\left(\mathrm{t}\right)& ={\displaystyle \frac{2}{3V_m}}{P}_{S-ref}\left(t\right)\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill i_{r-qref}\left(\mathrm{t}\right)& =-{\displaystyle \frac{2}{3V_m}}{Q}_{S-ref}\left(t\right).\hfill \end{array}$$

(58)

Under the assumption that the average current in the LC filter capacitor is much smaller than the currents injected into the grid, one can adopt the approximations:

$$\begin{array}{cc}\hfill i_{ref-d}\left(t\right)& \approx i_{r-dref}\left(t\right)\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill i_{ref-q}\left(t\right)& \approx i_{r-qref}\left(t\right),\hfill \end{array}$$

(60)

where $i_{ref-d}\left(t\right)$ and $i_{ref-q}\left(t\right)$ are the inverter output currents of direct and quadrature axis references.

Substituting (57) into (7) and (58) into (8), the reference active and reactive power will be injected into the inverter output if the direct and quadrature components of the inverter output currents meet, respectively:

$$\begin{array}{cc}\hfill {\mathrm{I}}_{inv-d}\left(s\right)& \approx {\displaystyle \frac{2}{3V_m}}{P}_{S-ref}\left(s\right)G_{ic}\left(s\right)\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\mathrm{I}}_{inv-q}\left(s\right)& \approx -{\displaystyle \frac{2}{3V_m}}{Q}_{S-ref}\left(s\right)G_{ic}\left(s\right).\hfill \end{array}$$

(62)

Considering that the power stored/injected by the LC filter is much smaller than power injected into the grid, it leads to

$$\begin{array}{cc}\hfill {\mathrm{I}}_{inv-d}\left(s\right)& ={\displaystyle \frac{2}{3V_m}}{P}_{inv}\left(s\right)\approx {\displaystyle \frac{2}{3V_m}}{P}_S\left(s\right)\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill {\mathrm{I}}_{inv-q}\left(s\right)& =-{\displaystyle \frac{2}{3V_m}}{Q}_{inv}\left(s\right)\approx -{\displaystyle \frac{2}{3V_m}}{Q}_S\left(s\right).\hfill \end{array}$$

(64)

Then, (61) to (64) results in

$$\begin{array}{cc}\hfill {\displaystyle \frac{P_S\left(s\right)}{P_{S-ref}\left(s\right)}}& =G_{ic}\left(s\right)\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill {\displaystyle \frac{Q_S\left(s\right)}{Q_{S-ref}\left(s\right)}}& =G_{ic}\left(s\right),\hfill \end{array}$$

(66)

which indicates that the reference and output power are related according to the current closed-loop transfer functions.

From (65) and the block diagram shown in Figure 5, the DC-link regulation loop for the small-signal operating condition can be represented by Figure 6. The plant to be compensated, $G_v\left(s\right)$, is given by

$$G_v\left(s\right)=H_2\left(s\right)G_{ic}\left(s\right),$$

(67)

where $C_v\left(s\right)$ is the compensator plant, while $\mathrm{\Delta }{P}_{pv}\left(s\right)$ and $\mathrm{\Delta }{Q}_s\left(s\right)$ are the generated power and reactive power regulation disturbances injected into the PCC, respectively.

Voltage Controller Tuning

As the plant to be compensated is a third-order system with two zeros, we decided to use the frequency response technique to tune the controller. According to [32,38], open-loop frequency response information indirectly indicates the plant’s closed-loop performance. The phase margin of the OLTF gives an idea of the system’s damping, while the unity gain crossover frequency gives an estimate of the speed of the transient response.

The DC-link voltage control loop is dependent on the current loop. To ensure decoupling between both, voltage regulation should be slower than current regulation. In terms of frequency response, this suggests that the voltage OLTF should have a lower unit gain crossover frequency than the current OLTF. For the controller design specifications, it was considered that ${\omega }_{cv}=k_{cv}{\omega }_{ci}$ for $k_{cv}\le {\displaystyle \frac{1}{5}}$, and a phase margin ${30}^{°}\le {\phi }_m\le {90}^{°}$, whose values are in accordance with the recommendations reported in the literature [28,39].

According to [38], to meet the gain margin and phase margin specifications of a system, the proportional and integral gains of a PI controller are given, respectively, by

$$\begin{array}{cc}\hfill K_P& =-{\displaystyle \frac{1}{\rho \left({\omega }_{cv}\right)}}cos\left({\phi }_m-\psi \left({\omega }_{cv}\right)\right)\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill K_I& ={\displaystyle \frac{{\omega }_{cv}}{\rho \left({\omega }_{cv}\right)}}sen\left({\phi }_m-\psi \left({\omega }_{cv}\right)\right),\hfill \end{array}$$

(69)

where $\rho \left(\omega \right)$ and $\psi \left(\omega \right)$ are the gain and phase frequency responses of $G_v\left(s\right)$, respectively.

Figure 7 and Figure 8 show the step response of the voltage loop for different values of $k_{cv}$ and ${\phi }_m$. From the curves, it can be inferred that, for a given unity gain crossover frequency, as the phase margin increases, the system’s response becomes less oscillatory and reduces the overshoot, while the time-based figures of merit increase. For different unity gain crossover frequencies, the higher the ${\omega }_{cv}$, the lower the time-based figures of merit. Therefore, the upper limit of ${\omega }_{cv}$ must be restricted in order to allow decoupling between the loops. As a heuristic decoupling criterion, we adopted the parameter that the peak time, $T_p$, of the voltage loop response should be at least ten times longer than that observed in the current loop, so $T_p\ge 3.5$ ms, but with a settling time of $T_s<3T_p$ and a maximum overshoot of 10%. These criteria are met for $k_{cv}={\displaystyle \frac{1}{15}}$ and ${\phi }_m={85}^{°}$, which result in controller parameters $K_P=-0.9415$ and $K_I=-193.40$.

3. Results and Discussions

The parameters shown in

Table 1. Parameters used in the simulation of the energy processing system.

Model Parameters	Value
$C_{\mathrm{\Delta }}$	$200\mathsf{\mu }\mathrm{F}$
$L_f$	$120\mathsf{\mu }\mathrm{H}$
$R_f$	$50\mathrm{m}\mathrm{\Omega }$
$L_r$	$150\mathsf{\mu }\mathrm{H}$
$R_r$	$1\mathrm{m}\mathrm{\Omega }$
$V_{int}$	$310\mathrm{V}$ peak
f	$60\mathrm{Hz}$
$f_c$	$5\mathrm{kHz}$
$C_{DC}$	$2525\mathsf{\mu }\mathrm{F}$

were used to perform the simulation of the system presented in Figure 1. Canadian Maxpower CS6U - 330P PV modules were employed in a PV arrangement formed by 96 strings with 19 modules per string, totaling 1824 modules or 602 kW in STC. Due to the intrinsic characteristics of mini-generation, all modules operate under homogeneous conditions, allowing the use of a discrete aggregate model [40], whose experimental validation is presented in [41]. As a reference, a WEG inverter in operation at CRESP was employed, with the following characteristics: model SIW700-T600-33, 655 kW, 700 V DC side, and 380 V AC side.

While conducting the simulations, the input signals $v_{DC-ref}$ and $I_{pv}$ were assumed to have the shapes shown in Figure 9, which are within the operating range recommended by the inverter manufacturer. For the aforementioned input signals, the dynamic behaviors of some quantities on the DC and AC sides are shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15.

From the results presented in Figure 10, the DC-link voltage has a good ability to track the reference voltage and, at the same time, has the ability to reject disturbances in the inverter input current. The largest overshoots observed occur during large variations in the input current, therefore, when there are significant changes in irradiance levels, such as cloud passes. Hence, depending on the operating point of the inverter, large changes in irradiance levels may lead to the activation of the inverter overvoltage protection.

Figure 11 depicts the behaviors of the reference active and reactive power as well as the powers injected into the PCC. The reference active power, $P_{S-ref}$, was taken as 0 ($V_{DC}{I}_{pv}$), thus corresponding to the power injected into the DC-link. The ratio $P_{S-ref}/P_{PAC}$ is defined as the inverter efficiency, which oscillates around the unit value, as displayed in Figure 12. To provide a unity power factor in PCC, the power $Q_{S-ref}$ was adjusted so as to compensate for the reactive power absorbed by the filter inductor, being necessary to adjust $Q_{S-ref}=-50\mathrm{kvar}$.

For the input signals $v_{DC-ref}$ and $I_{pv}$, the behavior of the direct and quadrature components of the currents injected into the grid and the PCC voltages are presented in Figure 13. As expected, the quadrature component of the current injected into the PCC is null, corroborating the value of the reactive power injected at this point. The direct component, $I_{r-d}$ presents the same dynamics experienced by this electrical quantity, once it is linked to the active power injected in the PCC, which can be verified by comparing the results shown in Figure 11 and Figure 13.

Based on the results shown in Figure 13, the quadrature component of the voltage at PCC presents a value close to zero, except for the presence of small disturbances, due to changes in the reference voltage of the DC-link and the currents injected into the grid. The aforementioned changes also affect the direct component of the voltage. These results illustrate the correct operation of the phase-capture loop and ensure good disturbance rejection capability of the control system.

The phase voltage signals present at the PCC, as well as of the currents injected into the grid, both in natural coordinates, are presented in Figure 14 and Figure 15, respectively. The higher the intensity of the current injected into the grid, the higher the voltage levels observed at the PCC. However, for the observed operating conditions, the voltages remain within acceptable limits.

Experimental Validation

In order to perform the validation of the models outlined in this work, a time series recorded by the meteorological station of CRESP was used for a partly cloudy day, whose irradiance and temperature dynamics are presented in Figure 16. The observation window extends from 6 a.m. to 5 p.m., totaling 11 h or 660 min of observation. The data acquisition system records data at a rate of one sample per minute, so the analyzed window has 660 samples.

Since the converter operates with a switching frequency $f_s=5$ kHz, the simulation of the system shown in Figure 2 requires an integration step $\mathrm{\Delta }t\le 20\mathsf{\mu }$s, which makes it unfeasible to conduct the simulations in an analysis window on the order of hours, as is the case. As a solution, the time scale of the time series was changed to the order of seconds, with the samples being distanced by an interval of 150 ms, which is the time required for the control system to stabilize the responses of the power processing system and, therefore, guarantee permanent steady-state operation. With the adopted solution, the time series is now represented by a window of approximately 99 s, enabling the simulations to be conducted for a daily-basis irradiance and temperature observation window. Once the simulations are completed in the reduced time scale, the time vector conversion to the natural scale can be obtained by

$$t_n=K_T·t_r,$$

(70)

where $t_r$ is the time vector in the reduced scale and $K_T={\displaystyle \frac{T_{\mathrm{natural}}}{T_{\mathrm{reduced}}}}$ is the scale factor, with $T_{\mathrm{natural}}$ corresponding to the time window in the natural scale and $T_{\mathrm{reduced}}$ to the time window in the reduced scale. For the present simulations,

$$t_n={\displaystyle \frac{660\min }{99\mathrm{s}}}·\left[00.150.30\cdots 99\right]\mathrm{s},$$

(71)

resulting in a time vector in the minute scale.

Using as input data the time series presented in Figure 16, Figure 17 presents the DC-link voltage behavior obtained using the proposed model, $v_{DC}$, as well as the equipment-monitored voltage in real operation. The computational model provides a voltage adherent to the experimental value for the different operating conditions present throughout the day, showing that the implemented control system operates assertively and robustly in the face of the variations of the PV generator parameters, which are dependent on the irradiance levels and the operating temperature of the equipment.

Figure 18 shows the behavior of the PV generator output current for one day of operation. The shape described by the current is similar to that observed for the irradiance throughout the day, corroborating the strong dependence between these quantities. With the output current of the experimentally validated PV generators, the mathematical model enables the development of new computational tools for generation performance monitoring and maintenance planning.

The behavior of the measured and simulated active power for the day under analysis is shown in Figure 19.

Although the string exhibits a power output of 602 kW at STC, based on the results, the generated power is less than the specified power output, even for irradiance levels greater than 1000 W/m². Such results are justified by the fact that the module has a temperature coefficient −0.40%/°C for maximum power. So each 1 °C increase in the module’s operating temperature, relative to STC, results in a reduction of the maximum string power by a factor of 0.40%.

Figure 20 depicts the dynamics of the effective value of the current injected into the grid for the analyzed day. Comparing the result obtained with that corresponding to the output current of the string (Figure 18), they present the same profile.

The behaviors of the active power and the power factor in the PCC for the analyzed day are shown in Figure 21 and Figure 22, respectively. As expected, the active power presents a similar profile to the one observed in the current injected into the grid, given that the reference active power injected in the PCC acts as input to the current control loop. From the presented results, the power factor assumes values close to unity during most of the day but operates with low values when irradiance levels are small, for example, at the beginning and the end of the day.

In order to quantify the adherence of the computational model response of the PV plant to the real operation measurements, Theil’s Inequality Coefficient (TIC) and the Root Mean Squared Error (RMSE) metrics were used, which are generally employed in the performance analysis of PV models [42,43]. The TIC and RMSE can be computed, respectively, by

$$\begin{array}{cc}\hfill \mathrm{TIC}& ={\displaystyle \frac{\sqrt{\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}{\sqrt{\frac{1}{n}{\sum }_{i=1}^n{y}_i^2}+\sqrt{\frac{1}{n}{\sum }_{i=1}^n{\widehat{y}}_i^2}}}\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill \mathrm{RMSE}& =\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2},\hfill \end{array}$$

(73)

where $y_i$ values are the experimental, measured values, whereas ${\widehat{y}}_i$ values are the values obtained through simulation and n the total number of observations. The TIC measurements provide values that lie between zero and one, with a value of zero indicating that the model is able to perfectly predict the measured data and a value of one indicating that the model is unable to perform prediction of the measurements. In general terms, a TIC value within the range of 0.2–0.3 suggests that the model has satisfactory predictive ability [44].

The results obtained for the curves displayed in Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 are presented in

Table 2. Conformity between simulations and measurements.

Variable	RMSE	TIC (pu)
$V_{DC}$ (V)	43.18	0.03
$I_{PV}$ (A)	34.59	0.04
$P_{DC}$ (kW)	27.64	0.05
$I_{r-RMS}$ (A)	78.07	0.10
$V_{r-RMS}$ (V)	48.02	0.06
$P_{CA}$ (kW)	52.01	0.10
Power factor	0.07	0.04

. According to the values obtained for the TIC, the plant model can be used to predict the actual behavior of the signals from the DC and AC sides of the plant.

4. Conclusions

From the concept of power balance, a differential equation that describes the DC-link voltage was presented in this paper. Since it is a non-linear equation, a linearization process was detailed, which allowed the presentation of a small-signal model for the DC-link, having as input the disturbances in the generated and injected power in the PCC. Assuming a phase capture loop with zero error, the currents injected into the grid were expressed in terms of active and reactive power injected into the PCC, establishing a control loop to regulate the voltage at the PCC. Since it is subordinated to the current loop, the PI controller was parameterized to ensure the decoupling of the loops and, at the same time, provide good performance rates during transient conditions.

Assuming a static grid equivalent model, the DC-link voltage and the current source were adopted as dynamic inputs of the power processing system, which were disturbed within the operating range recommended by the inverter manufacturer. Although the inverter allows for the power factor to be regulated at the PCC, the input reactive power of the voltage loop was kept constant and adjusted to ensure a unity power factor at the grid coupling point.

From the results obtained for the DC-link voltage, the control loop ensured traceability and disturbance rejection in the input signals. However, to avoid transients that could lead the voltage to assume values outside the recommendation range, which is likely to occur when there are large variations in the generated current, as happens in sudden changes in irradiance, it is suggested that the controllers be equipped with an anti-windup system.

The small discrepancy observed between the active power injected in the DC-link and the PCC confirmed the power balance and the good regulation of the voltage loop. Compensating for the reactive power absorbed by the output LC filter, the power factor at the PCC was kept close to the unity value for the different operating conditions. Nevertheless, depending on the parameterizations adopted for the inverter, the output reactive power can be regulated to assist the grid in maintaining voltage levels.

From the comparison of the experimental measurement results with those obtained through simulation, the aggregate model of the PV generator provided outstanding reproducibility of the DC-side voltage, current and power measurements, even though voltage drops in the DC cables, efficiency losses of the generator due to natural degradation and operation system faults were disregarded. Furthermore, the inverter and control system models proved to be representative of the plant in operation, given the reproduction of the DC-link voltage control, the MPPT, and the good capacity to reject the disturbances provided by the abrupt irradiance variation. Thus, the developed models present a good adherence to the measured data, suggesting that they can be used in dynamics studies for the adjustment of the control system and protection graduation, thus avoiding the untimely exit of the plant operation.

The model proposed can be interpreted as a Digital Twin of the real system in operation at the Reference Centre for Solar Energy of Petrolina, thus suggesting that it can be used as a tool, based on dynamic models, capable of assisting in the monitoring, diagnosis and operational studies of the PV system, such as the daily monitoring expected for energy production. Deviations from the experimental measurements about the computational results can be used as input data for properly calibrated intelligent systems to estimate the need to maintain the plant in the face of adverse factors, such as the loss of efficiency due to dirty photovoltaic modules.