Low Current Ripple Parameter-Free MPCC of Grid-Connected Inverters for PV Systems

Abstract

With an emphasis placed on a low-carbon economy, photovoltaic grid-connected inverters are moving toward the center of the stage. In order to address the problems related to the strong parameter dependence of the conventional model’s predictive control in grid-connected inverters, an improved parameter-free predictive current control is proposed. Relying on an extended state observer, an ultralocal model is employed to predict future currents without any parameters. The system can achieve a satisfactory performance in terms of dynamic response and robustness. Additionally, 30 virtual voltage vectors are extended for lower current ripples, which is followed by the use of a triangle candidate strategy to significantly ease the computing burden. In general, the proposed strategy omits parameter dependence, complex tuning work, and large tracking errors. The effectiveness of the proposed model is verified through the experimental results.

1. Introduction

Currently, the social economy and industrial production are developing rapidly, incurring an increasing amount of energy consumption. Nevertheless, fossil energy supplies are limited, such as oil, gas, and coal, the use of which can pollute the environment. In this manner, renewable sources of electricity, like solar energy, have been destined for a brighter future. For example, photovoltaic (PV) power-generation technology can effectively relieve the pressure of traditional thermal power generation; therefore, it has attracted significant attention recently [1,2,3]. As a bridge between circuits and the power grid, grid-connected inverters (GCIs) are an important component to realize the stability, reliability, and high efficiency of power-generation control systems [4,5].

There are two main conventional control strategies of GCI: pulse-width modulation and hysteresis-controlled pulse modulation [6,7]. The problems that are encountered by the two strategies are that the switching of frequency fails to maintain stability, and the total harmonic distortion is high. As the performance of digital processors improves, model predictive control (MPC) has been introduced to the control technology for grid-connected inverters [8,9,10,11]. MPC can omit the current inner loop control and pulse width modulation, yielding effortless application and strong adaptability. Specifically, Ref. [9] designed an MPC method based on the power prediction model, where the applied voltage vector (VV) was chosen by minimizing the power ripple. Therefore, flexible power regulation can be achieved. In addition to aiming for predictive power, predictive current control is another feasible technical route. Ref. [10] forecasts the possible behavior of the future grid current in accordance with the GCI model, namely model predictive current control (MPCC). Thereafter, the optimal VV can be selected so that the current error between the reference value and the predicted value is minimized in the cost function.

It is evident that the existing MPC strategies depend on accurate models of the grid-connected systems [12]. In practice, some uncertainties are inevitable in terms of inverter component parameters, considering the variations in climatic conditions. Therefore, satisfactory control performance can no longer be assured under model mismatches. So far, many researchers have posed a concern for the robustness of existing strategies [13,14,15].

On one hand, a variety of observers have been introduced in MPC to improve its performance and robustness. Particularly, disturbance observers are the principal force for this improvement due to their undisturbed grid current. In [13], a disturbance observer was built to estimate the actual uncertain disturbances in real time, and its output was employed to design a sliding mode controller so that the output voltage of the GCI could be controlled. In this manner, the robustness of the system was effectively improved, owing to the adaptively compensated disturbances. Similarly, Ref. [14] conceived of an extended Kalman filter (EKF) as the disturbance observer for estimating the grid voltage. Even though the grid voltages were seriously distorted, the current could still be controlled with a fast dynamic response, as expected. It should be noted that most observers are still determined by inverter mathematical models. In other words, these strategies fail to remove the influence of parameter mismatch. Furthermore, they require more effort for their tuning work.

Gradually, data-driven control has come to the attention of the PV system control [16,17]. These models introduce anomalous data at predetermined intensities and time slots, while the risk nature of false data injection fails to be reflected in real time. Additionally, a model-free predictive control based on the current gradient update was studied in [18], where no system parameters are required. When current gradient renewal is installed during one sampling period, the system control performance will deteriorate, and the grid-connected current will not meet national requirements. Simultaneously, Ref. [19] predicted the future current using the LaGrange interpolation algorithm rather than the slope change, avoiding the problem of stagnation. Gradually, diverse parameter-free MPC strategies have attracted significant attention in inverter control. These methods can be categorized into three types: ideal model-based, current gradient-based, and ultralocal model-based methods [20,21]. Ultralocal model-based predictive control is more reliable than the other two methods because it breaks away from the renewal stagnation of the look-up table [22].

Meanwhile, the optimal voltage vector (VV) has been chosen from eight basic VVs in conventional MPCC. The number of candidate VVs is limited, and the switching state of the inverter stays the same during one period. Consequently, the current changes greatly, resulting in large current pulsation. In order to further improve the quality of grid-connected currents without increasing the hardware burden, virtual voltage vectors (VVVs) have been widely applied [23,24,25]. In essence, the active VVs take advantage of devised composition principle to generate several VVVs, which equips the grid with diverse amplitudes and angles compared to active VVs. Undoubtedly, the finite control set can be extensively expanded by synthesizing VVVs, and more candidate VVs can be prepared for rolling optimization in the cost function. The number of recursive operations in MPCs has also increased. Ref. [26] reduced the number of candidate vectors to three by pre-selecting them before performing the rolling optimization. Specifically, according to the position of the calculated reference VV in the space vector diagram, the three nearest VVs are selected in a small triangle for evaluation. Similarly, by introducing three auxiliary lines into the space vector diagram, three relatively suitable virtual voltage vectors are calculated in [27]. Although the number of iterations in the MPC is significantly reduced in [26,27], the sensitivity of the parameters is increased, owing to the dead-beat properties. The design logic is too complex in these methods.

Until now, such MPCC is still absent for GCI control, where both strong robustness and satisfactory quality of grid-connected current are taken into consideration. Therefore, for the sake of parameter-free control with reduced current ripples, this paper conceives an improved MPCC based on the ultralocal model. The main contributions are listed as follows: (1) A linear extended state observer (LESO) is designed to estimate the unknown part of system, so that the future grid current can be predicted accurately by building the ultralocal model. Only the sampling information is required instead of inverter parameters. This method successfully circumvents the complicated tuning work and the existence of abnormal data. (2) This paper establishes a synthesis mechanism of VVVs, based on which, a space distribution of 38 new vectors is built. Furthermore, the triangle candidate strategy is conducted to filter the optimal control set, followed the steps of hypotenuse selection and right-angle side selection. In this manner, the candidate VVs can be reduced to 7. It means that the quality of grid-connected current is greatly improved, and the calculation burden is significantly alleviated. (3) The problem of the delay due to the calculation time in the MPCC strategy has been solved by one-step delay compensation, so that the grid-connected current ripple is further reduced.

This paper expounds the research of the proposed parameter-free MPCC relying on the following parts. In Section 2, the model of GCI is constructed in detail. Then, in Section 3, the proposed parameter-free MPCC is elaborated, including the LESO, optimization strategy for VV selection, and delay compensation. Thereafter, experimental results are exhibited in Section 4. Finally, the research is concluded in Section 5.

2. Model of Grid-Connected Inverters for PV Systems

This paper concerns the typical two-stage grid-connected PV inverter system, where the front stage is a boost converter and the latter stage is a two-level voltage source inverter (2L-VSI), as shown in Figure 1.

Specifically, the equivalent circuit topology of 2L-VSI is presented in Figure 2, where the output terminal can be, respectively, linked to the positive-bus and negative-bus. Considering all possible combinations of switching states, eight voltage vectors can be produced, as shown in Figure 3.

Undoubtedly, the controlled object is a multi-order time-varying nonlinear system so it is ill-suited to design the controller in natural coordinates. Therefore, based on the Kirchhoff voltage law, dynamic equation of grid-connected current in synchronous rotating reference frame is given by

$$\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right]=\left[\begin{array}{cc}R& -{\omega }_e{L}_f\\ {\omega }_e{L}_f& R\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{cc}{L}_f& 0\\ 0& L_f\end{array}\right]\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{e}_d\\ e_q\end{array}\right]$$

(1)

where the subscripts d and q mean the components in the d and q axes, respectively. u represents the output voltage of the inverter; R denotes the line resistance; L_f represents the filter inductance; ω_e denotes the grid angular frequency; e represents grid voltage; i represents the grid-connected current.

3. Control Algorithm

3.1. Conventional Model Predictive Current Control

According to (1), the differential equation of dq-axes current can be obtained as

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]=\left[\begin{array}{cc}-R/L_f& {\omega }_e\\ -{\omega }_e& -R/L_f\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}(u_d-e_d)/L_f\\ (u_q-e_q)/L_f\end{array}\right]$$

(2)

Then, via the forward Euler discretization equation, the currents at (k + 1)th, namely [i_d(k + 1) and i_q(k + 1)] can be predicted as

$$\left[\begin{array}{c}{i}_d(k+1)\\ i_q(k+1)\end{array}\right]=\left[\begin{array}{cc}1-R{T}_s/L_f& {\omega }_e{T}_s\\ 1-R{T}_s/L_f& -{\omega }_e{T}_s\end{array}\right]\left[\begin{array}{c}{i}_d(k)\\ i_q(k)\end{array}\right]+\left[\begin{array}{cc}{T}_s/L_f& 0\\ 0& T_s/L_f\end{array}\right]\left[\begin{array}{c}{u}_d(k)-e_d(k)\\ u_q(k)-e_q(k)\end{array}\right]$$

(3)

where T_s denotes the sampling period; (k) and (k + 1) represent the components measured at (k)th and predicted at (k + 1)th, respectively. The stator voltages at (k)th used in (3) can be obtained as

$$\left[\begin{array}{c}{u}_d(k)\\ u_q(k)\end{array}\right]=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}{u}_{\alpha }(k)\\ u_{\beta }(k)\end{array}\right]$$

(4)

with

$$\left[\begin{array}{c}{u}_{\alpha }(k)\\ u_{\beta }(k)\end{array}\right]=\left[\begin{array}{c}\begin{array}{ccc}2/3\hfill & -1/2& -1/2\end{array}\\ \begin{array}{ccc}0& \sqrt{3}/3& -\sqrt{3}/3\hfill \end{array}\end{array}\right]\left[\begin{array}{c}{u}_a(k)\\ u_b(k)\\ u_c(k)\end{array}\right]$$

(5)

where the subscripts α and β mean the components in the α and β axes, respectively. θ is the power grid phase. u_x(k) = 0.5U_dc(S_x + 1), x∈{a, b, c}, S_x ∈ {0, 1} is the switch states of GCI as shown in Figure 3.

At last, all VVs will be evaluated by (6) after which the optimal VV can be chosen.

$$g=\left|i_d^{ref}(k+1)-i_d^{}(k+1)\right|+\left|i_q^{ref}(k+1)-i_q^{}(k+1)\right|$$

(6)

where the superscript ref means the ideal component.

Obviously, the prediction of future current is highly dependent on system parameters which fail to hold constant in the PV applications. So, it is necessary to develop an MPCC with better robustness against the parameter variation.

3.2. LESO Based on Ultralocal Model

In the modern control theory, the first-order ultralocal model of a single-variable system can be built as

$$\stackrel{·}{y}=F+\alpha u$$

(7)

where y and u represent the system output and input, respectively; α denotes a nonphysical scaling factor of system input; F is the sum of the known and unknown part of the system.

In conjunction with (2), the ultralocal mathematical model of grid-connected system can be constructed in a rotating dq frame as

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]=\left[\begin{array}{c}{F}_d\\ F_q\end{array}\right]+\left[\begin{array}{cc}{\alpha }_d& 0\\ 0& {\alpha }_q\end{array}\right]\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right]$$

(8)

with $\left[\begin{array}{c}{F}_d\\ F_q\end{array}\right]=\left[\begin{array}{cc}-R/L_f& {\omega }_e\\ -R/L_f& -{\omega }_e\end{array}\right]\left[\begin{array}{c}{i}_d(k)\\ i_q(k)\end{array}\right]-\left[\begin{array}{cc}1/L_f& 0\\ 0& 1/L_f\end{array}\right]\left[\begin{array}{c}{e}_d(k)\\ e_q(k)\end{array}\right]$; $\alpha =\frac{1}{L_f}$.

Obviously, α is constant so it can be confirmed by tuning in the practical system, while F is still much susceptible to motor parameters, and it decides the accuracy of the ultralocal model. In this way, according to (8), a LESO can be built as

$$\left\{\begin{array}{c}err=\stackrel{\wedge }{i_s}-i_s\\ \stackrel{·}{\stackrel{\wedge }{i_s}}=\stackrel{\wedge }{F}+\alpha u_s-{\beta }_{1}err\\ \stackrel{·}{\stackrel{\wedge }{F}}=-{\beta }_{2}err\end{array}\right.$$

(9)

where s ∈ {d, q}. The i_s and F are chosen as the state variables. The superscript “˄” represents the estimated value; β₁ and β₂ are the gain coefficients of LESO.

Apply S-transform to (9), it can be rearranged as

$$\stackrel{\wedge }{i_s}\left(s\right)=\frac{\left({\beta }_{1}s+{\beta }_2\right)i_s+s\alpha u_s}{s^2+{\beta }_{1}s+{\beta }_2}$$

(10)

Therefore, the characteristic equation of the LESO can be obtained as

$$s^2+{\beta }_{1}s+{\beta }_2=0$$

(11)

To keep the system in a stable state, the roots of the characteristic equation should fall at −ω₀. On this premise, the gain coefficients β₁ and β₂ are set as

$$\left\{\begin{array}{l}{\beta }_1=2{\omega }_0\\ {\beta }_2={\omega }_0^2\end{array}\right.$$

(12)

where ω₀ is the bandwidth of the LESO. There is little doubt that an appropriate bandwidth is in favor of the satisfactory steady-state and dynamic performance of the LESO. For the further design of ω₀, (9) is transformed by forward Euler discretization equation and z-transform in turn, as

$$\left\{\begin{array}{c}err\left(z\right)=\stackrel{\wedge }{i_s}\left(z\right)-i_s\left(z\right)\\ z\stackrel{\wedge }{i_s}\left(z\right)=\stackrel{\wedge }{i_s}\left(z\right)+T_s\left[\stackrel{\wedge }{F}\left(z\right)+\alpha u_s\left(z\right)-{\beta }_{1}err\left(z\right)\right]\\ z\stackrel{\wedge }{F}\left(z\right)=\stackrel{\wedge }{F}\left(z\right)-T_s{\beta }_{2}err\left(z\right)\end{array}\right.$$

(13)

In fact, the sampling period is so small that the transfer function of the discrete system can be constructed as

$$G\left(z\right)=\frac{\stackrel{\wedge }{i_s}\left(z\right)}{i_s\left(z\right)}=\frac{{\beta }_{02}{T}_s+{\beta }_{01}z-{\beta }_{01}}{{\left(z-1\right)}^2+{\beta }_{02}{T}_s+{\beta }_{01}z-{\beta }_{01}}$$

(14)

where β₀₁ = β₁T_s, β₀₂ = β₂T_s.

To ensure the stability of the discrete system, the poles must be in the unit circle. The characteristic equation of the LESO is

$${\left(z-1\right)}^2+{\beta }_{02}{T}_s+{\beta }_{01}z-{\beta }_{01}=0$$

(15)

As mentioned above, we can find that β₀₁ = 2ω₀T_s, β₀₂ = ω₀²T_s. Therefore, the poles of G(z) can be solved from (15) as

$$z_1=z_2=1-{\omega }_0{T}_s$$

(16)

Gradually, the bandwidth ω₀ can be obtained as

$${\omega }_0=\frac{1-z_{1,2}}{T_s}$$

(17)

When z_1,2 is approaching 1, ω₀ becomes so small that the dynamic performance of LESO will be apt to deteriorate. Instead, when z_1,2 is approaching 0, ω₀ becomes so large that the system will diverge with weakened robustness. In this paper, z_1,2 is set as 0.1 and ω₀ is 9000 to keep quick dynamic response and system stability. Meanwhile, the nonphysical scaling factor α is set as 50 in the test.

According to the LESO based on the ultralocal model, the future currents at (k + 1)th can be predicted as

$$\left[\begin{array}{c}{i}_d(k+1)\\ i_q(k+1)\end{array}\right]=T_s\left[\begin{array}{c}\stackrel{\wedge }{F_d}(k)\\ \stackrel{\wedge }{F_q}(k)\end{array}\right]+T_s\left[\begin{array}{cc}\stackrel{\wedge }{{\alpha }_d}(k)& 0\\ 0& \stackrel{\wedge }{{\alpha }_q}(k)\end{array}\right]\left[\begin{array}{c}{u}_d(k)\\ u_q(k)\end{array}\right]+\left[\begin{array}{c}{i}_d(k)\\ i_q(k)\end{array}\right]$$

(18)

3.3. Optimization Strategy for Voltage Vector Selection

To improve the stable state performance of the control system, VVVs are applied to expand the finite control set so that more choices of candidate VVs can be evaluated in the cost function. Specifically, the eight voltage vectors in Figure 3 divide the control space into six sectors, and thus, the VVVs can be synthesized by three real VVs at most. When one sampling period is factitiously divided into three parts, three types of VVVs will be generated in Sector Ⅰ, as presented in Figure 4. The first type is synthesized by an active VV and a zero VV, like V₈. The second type is synthesized by two active VVs, like V₂₇. The last type is synthesized by two active VVs and a zero VV, like V₁₅. Via extending the vector synthesis of Sector Ⅰ to other sectors, a new VVs distribution can be obtained, where the added 30 VVVs can be found in

Table 1. The effects of different switching vectors on NPV.

Small VVVs	Amplitude	Medium VVVs	Amplitude	Medium VVVs	Amplitude	Large VVVs	Amplitude	Large VVVs	Amplitude
V8	$\frac{2V_0 + V_1}{3}$	V14	$\frac{V_0 + 2V_1}{3}$	V20	$\frac{V_0 + 2V_4}{3}$	V26	$\frac{2V_1 + V_2}{3}$	V32	$\frac{2V_4 + V_5}{3}$
V9	$\frac{2V_0 + V_2}{3}$	V15	$\frac{V_0 + V_1 + V_2}{3}$	V21	$\frac{V_0 + V_4 + V_5}{3}$	V27	$\frac{2V_2 + V_1}{3}$	V33	$\frac{2V_5 + V_4}{3}$
V10	$\frac{2V_0 + V_3}{3}$	V16	$\frac{V_0 + 2V_2}{3}$	V22	$\frac{V_0 + 2V_5}{3}$	V28	$\frac{2V_2 + V_3}{3}$	V34	$\frac{2V_5 + V_6}{3}$
V11	$\frac{2V_0 + V_4}{3}$	V17	$\frac{V_0 + V_2 + V_3}{3}$	V23	$\frac{V_0 + V_5 + V_6}{3}$	V29	$\frac{2V_3 + V_2}{3}$	V35	$\frac{2V_6 + V_5}{3}$
V12	$\frac{2V_0 + V_5}{3}$	V18	$\frac{V_0 + 2V_3}{3}$	V24	$\frac{V_0 + 2V_6}{3}$	V30	$\frac{2V_3 + V_4}{3}$	V36	$\frac{2V_6 + V_1}{3}$
V13	$\frac{2V_0 + V_6}{3}$	V19	$\frac{V_0 + V_3 + V_4}{3}$	V25	$\frac{V_0 + V_6 + V_1}{3}$	V31	$\frac{2V_4 + V_3}{3}$	V37	$\frac{2V_1 + V_6}{3}$

. The whole voltage space is further divided into twelve right-angled triangle areas.

Undoubtedly, the increased VVs will bring a huge computational burden. Thus, an optimization selection of VVs is designed before the cost function evaluation. The diagram of the optimization selection is explained in Figure 5 and Figure 6.

First, on the principle of cost function minimization, the optimal vector from small VVVs is selected to locate the hypotenuse of the right triangle.

Then, as shown the example in Figure 6, if the hypotenuse is in the direction of V₉, the medium VVVs, V₁₅ and V₁₇, which are located on the corresponding right-angle sides, should be put into cost function to confirm a correct right-angled triangle.

Finally, all VVs located in this triangle are viewed as the candidate VVs, as V₀, V₂, V₇, V₉, V₁₅, V₁₆, and V₂₇. To sum up, only 7 candidate VVs should be considered in one sampling period rather than 38 VVs, and thus, the computation burden can be significantly reduced.

3.4. Delay Compensation

In the digital control system, a delay is hard to avoid due to the different moments between the current sampling and the new switching state application. As observed in Figure 7a, the former switching state is still applied during this delay interval, and thus the VV selected at (k − 1)th continues to be used at kth. In this case, the current deviates from the reference value. By analogy, the switching state chosen by sampling value at kth will be employed around (k + 1)th. No doubt the intrinsic delay makes the current oscillate around its reference, yielding the worsening current ripple.

To compensate for the delay caused by the sampling and calculation process, one-step delay compensation is conducted by building the prediction model. As seen in Figure 7b, at kth, the applied VV is the state chosen at (k − 1)th, and the future current can be predicted by (18). Furthermore, the current of (k + 2)th can be evaluated by all possible candidate VVs, as

$$\left[\begin{array}{c}{i}_d(k+2)\\ i_q(k+2)\end{array}\right]=T_s\left[\begin{array}{c}\stackrel{\wedge }{F_d}(k+1)\\ \stackrel{\wedge }{F_q}(k+1)\end{array}\right]+T_s\left[\begin{array}{cc}\stackrel{\wedge }{{\alpha }_d}(k+1)& 0\\ 0& \stackrel{\wedge }{{\alpha }_q}(k+1)\end{array}\right]\left[\begin{array}{c}{u}_d(k+1)\\ u_q(k+1)\end{array}\right]+\left[\begin{array}{c}{i}_d(k+1)\\ i_q(k+1)\end{array}\right]$$

(19)

Thereafter, the predictive currents from (19) are employed to (20) to evaluate the cost function of the candidate VVs in turn. The VV that minimizes cost function can be selected to drive the GCI for the following control period.

$$g=\left|i_d^{ref}(k+2)-i_d^{}(k+2)\right|+\left|i_q^{ref}(k+2)-i_q^{}(k+2)\right|$$

(20)

The overall diagram of the proposed MPCC control for grid-connected inverters is depicted in Figure 8. The reference voltage is obtained by maximum power point tracking. Only the sampling information is utilized in the ultralocal model to predict future current, which performs high robustness. Furthermore, VVVs have been reduced to seven via the triangle candidate strategy by which the calculation burden is successfully relieved. Finally, the applied switching states are selected after the delay compensation control.

3.5. Comparison between Proposed Method and Known Ones

A comparison between the proposed method and known ones is conducted, as summarized in

Table 2. Comparison between proposed strategy and known ones.

Reference	[10]	[11]	[14]	[26]	Proposed
Requires a model	Yes	Yes	Yes	Yes	Ultralocal data driven
Requires gain tuning	Yes	Yes	No	Yes	No
Current ripples	Low	High	High	Low	Low
Parameter robustness	Low	Low	High	Low	High
Relative Simplicity of algorithm	Middle	Low	Middle	High	Middle
Computational burden	High	Low	High	Middle	Low

. The future current is predicted by system parameters in [10,26], yielding weak robustness against the parameter variation. The former reduces the current ripples by multi-step prediction and the latter relies on virtual vectors instead. And Ref. [26] further reduces the computation by screening the candidate vectors. Ref. [11] controls grid-connected inverters by predicting power, which has extra tuning work and weak parameter robustness. The Kalman filter is utilized in [14] to predict the future current, which avoids the use of system parameters and has strong parameter robustness. However, the design logic of this strategy is complex, and only the basic voltage vectors are chosen for optimization, so the quality of grid-connected current should be improved.

4. Results

Aiming at validating the effectiveness of the proposed parameter-free MPCC for GCI, a 200W experimental prototype is constructed as shown in Figure 9. The three-phase inverter adopts FF300R12ME4 IGBT modules and the controller carries TMS320F28335 as the digital signal processor. It should be noted that the DC power is the input of the control platform, instead of traditional PV panel input, boost circuit and DC-link capacitance. The control platform consists of DSP, inverter, voltage sensors, and current sensors. The inductance and resistance are viewed as the grid filter. Three-phase grid-connected current can be produced through the inverter powered by 3 kW DC power. The current clamp (ETA 5301A) and the voltage probe (ETA 5010) are used for the real-time measurement of grid-connect current and voltage, respectively. All the parameters of the experimental machine are listed in

Table 3. Experimental prototype parameters.

Items	Specifications
DC-link voltage Udc	350 V
DC-link capacitance C	4700 μF
Filter inductance L	2 mH
Resistance R	0.5 Ω
Switching Frequency	10 kHz

. The switching and sampling frequency of the system are both set at 10 kHz.

4.1. Steady-State Performance

To highlight the advantage of steady-state performance in the proposed strategy, contrast experiments have been conducted with the condition of 4A current commanded. Figure 10 and Figure 11 illustrate the results of grid voltage and grid-connection current in the manner of conventional MPCC. It can be seen that the grid-connection currents are sinusoidal and in the same frequency and phase as the grid voltage. Nevertheless, the current prediction error is unavoidable because of the limitation of eight vectors, reflected in the low-order current harmonics. The THD of current is 6.8% and ill-suited to meet government standards. Instead, as shown in the results of the proposed MPCC in Figure 12 and Figure 13, the phase-A voltage of the grid is in line with the same phase of current, namely, the power factor is unity. Meanwhile, the expanded control vector sets are permissible when accompanied by the triangle candidate strategy, so that more accurate predictive currents and lower current ripples can be obtained. The THD value of the proposed MPCC is reduced to 1.59%, superior to the counterpart of conventional MPCC.

To analyze the current tracking performance of a proposed MPCC convincingly, the current prediction error is introduced by

$$error=(i_s{}^{ref}-i_s)/i_s{}^{ref}\times 100\%$$

(21)

Figure 14 presents dq-axes current prediction error results of the traditional MPCC and the proposed control. As seen in Figure 14a, the q-axis current prediction error is obviously less than 0 for a long time and the current prediction error ripple is large. As mentioned above, because the control vector sets are limited, sometimes the vector found by rolling optimization is not optimal or suboptimal, which may result in the deteriorating control performance, for example, the current quality decreases and even distorts. However, under the same operating condition, the proposed strategy can pull the dq-axes current prediction error back to 0, and the current prediction error ripple is reduced within 0.1. On the other hand, the strong robustness against the inverter parameter variation is successfully verified as displayed in Figure 15. When actual inverter parameters are inconsistent with ideal values according to the special operating environment, traditional MPCC suffers from large harmonics of grid-connected current and severe deterioration of steady-state performance. It is detrimental to the normal and safe operation of photovoltaic systems. Luckily, the grid-connected current is not affected by the model mismatch at all in the proposed MPCC, and the THD of the current is small enough to achieve grid-connected standards. This is because the future behaviors of the current are merely dependent on the sampling information, instead of inverter parameters. The strong robustness assists the stable operation of the system.

To be specific, the root mean square errors (RMSE) of the d-axis current obtained by (22) under different command currents, are summarized in Figure 16. Obviously, the RMSE of the d-axis current under the proposed MPCC stays below 1.0 at different given currents, lower than that under conventional MPCC. All thanks to the appropriate candidate VVs in the proposed strategy.

$$i_{\mathrm{RMSE}}=\sqrt{\frac{1}{M}{\displaystyle \sum _{k=1}^M{(i_d^{ref}(k)-i_d(k))}^2}}$$

(22)

where i_RMSE represents the RMSE of d-axis current, M denotes the number of sampling points.

In addition, the total harmonic distortion (THD) of all methods is presented in

Table 4. Comparison of quality of grid-connected current.

Strategy	id* = 1A, R′ = R	id* = 2A R′ = R	id* = 3A R′ = R	id* = 4A R′ = R	id* = 4A R′ = 2R
Conventional MPCC	THD = 7.22%	THD = 7.13%	THD = 6.57%	THD = 6.80%	THD = 23.60%
Proposed MPCC	THD = 2.28%	THD = 1.98%	THD = 1.94%	THD = 1.59%	THD = 1.73%

in the case of 1A, 2A, 3A, and 4A command currents. On the whole, the proposed MPCC performs better in view of the steady-state performance at the same sample frequency. Even when the system parameters mismatch like R′ = 2R, the grid-connected current of the proposed control strategy can also guarantee the grid-connected standard, with THD of 1.73%, much lower than THD of 23.60% in conventional MPCC.

4.2. Dynamic Performance

To show the dynamic performance of the proposed control strategy, contrast experiments are carried out between the traditional MPCC and the proposed MPCC. In Figure 17, the reference current is abruptly decreased from 4 A to 2 A. The grid-connected current and grid voltage remain in the same frequency and phase as expected in both strategies. The grid-connected current can quickly track the sudden change of the reference values. The proposed control strategy maintains the equal dynamic performance as the traditional strategy. Similarly, when the current command abruptly transforms from 2 A to 4 A as seen in Figure 18, the response time is short enough that the PV system can realize the maximum power point tracking in the proposed MPCC. On the whole, the proposed parameter-free MPCC can obtain acceptable dynamic performance.

5. Conclusions

This article proposes a parameter-free MPCC strategy extended from an LESO-based ultralocal model for PV GCI systems. Furthermore, the finite control set is expanded to reduce the grid-connection current harmonics and the triangle candidate strategy is utilized to relieve computing burden. The experimental results show that the proposed MPCC has better performance in the aspect of current ripples and steady-state performance when accurate parameters are used. The RMSE of the d-axis current stays below 1.0. Furthermore, even if the parameters of the PV inverter change in the poor operating environment, the system can still maintain a satisfactory grid-connection current with a THD of 1.59%. The proposed strategy presents strong robustness against parameter variation and satisfactory grid-connected current quality It is conducive to alleviating the shortage of resources in the west and promoting the rapid development of new energy industries, like solar cars, rooftop PV power stations and so on. In the future, we will give a further study on the effect of initial parameters and inverter nonlinearity on control performance.