Robust Model Predictive Control of a Renewable Energy Converter Under Parametric Uncertainty Conditions

The advanced control technique Model predictive control (MPC) is gaining popularity in power electronics for converters with distributed energy resources. It combines closed-loop control and minimizes errors and control effort. As a model-based control technique, MPC's performance can degrade due to plant disturbances, such as parametric errors or large perturbations in grid voltage or load current. Our research used an MPC with modulation on a converter connected to the grid with an inductive filter for integrating renewable energy sources. The margin of robust stability (RS), derived from the singular Value Decomposition (SVD), provides a theoretical investigation of the robustness of the MPC controller tuning in dependency on the cost function weight factors and the time horizons. In the experiments conducted on a 2 kW workbench, it was confirmed that the proposed controller is stable and robust in nominal and under severe parametric uncertainty conditions, addressing the power quality criteria defined in IEEE Std. 519-2014. The experimental results show that the proposed MPC controller tuning outperforms the classical PI current controller in nominal conditions and is more robust to uncertainty in the passive filter of the grid-connected converter.


I. INTRODUCTION
The renewable energy sources (RES) as solar PV and wind, into the electric power system is an essential option for reducing reliance on high carbon emission sources, such as oil derivatives and coal, which contribute significantly to global warming, according to [1].Both wind and PV have an inherent Direct Current (DC) nature, so power converters are necessary for integration into the local Alternating Current (AC) grid [2].
The most popular power converter topology used as a grid interface for RES [3] is the Voltage Source Converter (VSC) with a passive filter.According to [4], the passive filter is required to attenuate the harmonic components created by the switched voltage as well as nonlinearities such as the dead time of the VSC switches.In this work, we consider the inductive (L) filter topology due to its easy implementation as addressed in recent literature [5], [6], [7], [8].Although higher-order filters such as inductive-capacitive (LC) or inductive-capacitive-inductive (LCL) have advantages in terms of size, cost, and attenuation of harmonic distortions, the control system design and implementation complexity increases dramatically due to characteristic resonance, which necessitates either an active or passive technique of attenuation [9], [10].
The control system of the power converter for RES integration into the electric grid must address synchronization, power and power factor regulation, harmonic distortion, and disturbance in grid voltage or DC bus [4], [11], [12].The most common control technique includes the classical PI current controller [13], [14], [15], resonant proportional control (PR) [9], nonlinear controls by sliding modes or Fuzzy interface [4], state feedback [16], and MPC [17].
The MPC incorporates the characteristics of various sophisticated control approaches, such as closed-loop control, error and control effort minimization, straightforward implementation for multivariable systems, and the capacity to add physical restrictions [17], [18].Deadbeat, Finite Control Set (FCS-MPC), Continuous Control Set (CCS-MPC), and Generalized Predictive Control (GPC) are the primary types of MPC that are gaining traction in power converter applications [4], [17], [18], [19].The computing cost of MPC, particularly when limits are enforced, is a significant limitation [20].Furthermore, because the MPC is a modelbased technique, it may lose performance owing to parametric fluctuations and severe disturbances in load current or grid voltage, as cited by [4], [18].
The robustness is a desirable characteristic of the control techniques designed for grid-connected VSCs to reduce the effects of Total Harmonic Distortion (THD) in the grid current.Minimizing THD current waveform quality concentrates useful power in the fundamental component, indirectly improving energy conversion efficiency [21].As more renewable energy sources are integrated via power converters and non-linear electronic loads are used by local customers, it's common for grid-connected VSCs to operate under plant disturbances like parametric errors or distorted grid voltage [3], [12], [22], [23].The VSC control system must be capable of mitigating the impact of disturbances to improve power quality injected into the grid.
Several studies have been conducted to improve the robustness of VSC control systems.One such effort, presented in [24], developed a robust MPC with a Kalman filter observer to counter model mismatches.In another study [25], a robust Sliding Mode Controller (SMC) was designed, which performed well even in the presence of model parameter mismatch and disturbances in the grid voltage.Robust H ∞ controllers were developed in [26], [27], [28] using Linear Matrix Inequalities to solve the H ∞ norm optimization problem, considering different performance criteria.Additionally, an Active Disturbance Rejection Control (ADRC) was developed in [29] for the inner current control loop of a VSC.This method considers an extended state when accounting for disturbances, and it showed good performance in terms of disturbance rejection and fast response.
A recently published article in [6] aims to contribute to the existing literature on a specific problem.The article proposes the use of an MPC that employs a state-space model with PWM modulation for an L-filtered VSI connected to a distorted voltage grid.The authors demonstrate that the system can effectively attenuate harmonic components and meet the IEEE standard 519-2014, with THD less than 5%.However, the proposed method has some drawbacks, such as high computational cost and the need for empirical adjustment applied to the controller.These controllers are designed to eliminate the impact of harmonic elements on grid voltage.It is crucial to note that understanding harmonic components is essential for mitigating distortion.
This paper presents an explicit analysis of the stability and robustness of the CCS-MPC controller applied to gridconnected VSC with L-filter, in the synchronous rotation frame.This system serves as a grid interface for distributed RES, particularly during disturbances, contributing to applied control of power converters and energy transition.The RS margin, derived from SVD spectral gain for multi-variable systems, provides a theoretical investigation of the robustness of the CCS-MPC controller tuning under parametric uncertain and grid voltage disturbance [30], [31], [32].
The study of MPC type controller using the singular values began in our research group with the publication [33].The authors analyzed the MPC in conjunction with the repetitive controller (PRC, Predictive Repetitive Control) by varying the parameters of the Double Fed Induction Generator (DFIG) for wind generation.In the recent publication [34], the researchers noted the potential of using SVD for analyzing the stability and robustness of the MPC control system.The authors also applied the RPC technique to achieve zero stationary error in the current internal loop and improve the robustness of the grid-connected inverter under parametric uncertainty and disturbances in grid voltage.
In this study, the authors applied the expertise of SVD robustness analysis to the CCS-MPC controller.A detailed examination of the MPC parameters' time horizons and the cost function weighting matrix is conducted.The effect of time horizons and the cost function weighting matrix selection on the dynamic behavior of the closed-loop system for an MPC in direct power control was initially investigated in [35], but only in the nominal conditions of the plant.In Table 1, we highlight the differences between this paper and other studies conducted by the group.
The stiff voltage grid model is used for small-scale applications, resulting in minimal impact of grid impedance on system stability [36].The cost function's block diagonal matrices are reduced to identical diagonal matrices for simplicity.The main contributions of this article are presented below.
r Explicitly analysis of the stability and robustness of CCS-MPC controller applied to VSCs serving as grid interfaces for distributed RES.
r Experimental validation of the MPC controller's robust stability even under severe parametric uncertainty.
r In nominal conditions, the MPC controller tuning im- proves power quality.
r Experimental comparison with the classical PI cur- rent controller showed that the proposed controller had greater robustness.The structure of this paper is as follows: Section II presents the discrete model of the plant.In Section III, we describe the CCS-MPC control technique.Section IV presents the RS analysis of the CCS-MPC controller.The experimental robustness tests are provided in Section V.The main results are summarized in Section VI.

II. MODEL OF THE VSC IN ROTATION FRAME dq.
The grid-connected power converter (Fig. 1) is made up of a VSC and an L-filter [12], [18].By modulating the voltage V dc into a three-phase waveform v abc g timed with the grid common connection point (CCP), the VSC injects energy into the grid from the DC bus [37].The system's entire functionality (Fig. 1) is dependent on the VSC switches being properly driven by the space vector MPC control method in rotation frame dq [17].The MPC employs a predictive model, as specified in section III, as well as a cost function of the tracking error of i g,q and i g,d , as well as the control effort u dq [38].The crucial step is to minimize the cost function, which produces the reference signal for driving the VSC in the form v re f i (k) = u dq (k) + v g,dq (k) (Fig. 1).After converting to the αβ stationary frame, a space vector PWM modulator (SPWM) drives the VSC switch from the v re f i,dq signal [17].The design of the filter L g was based in (1), as proposed by [10].Here, I is the desired change of current in the inductor, f sw is the switching frequency, and V RMS is the effective value of the fundamental grid voltage.A PLL, as shown in Fig. 2, performs grid voltage synchronization, which is a simple and robust control loop that calculates the instantaneous phase θ g [39].The control loop (Fig. 2) requires the measurement of three-phase currents ( i abc g ) and voltages ( v abc g ) at the grid CCP, which are transformed to the synchronous frame dq first using the Clarke transform and then using θ g and the Parke transform [40].The grid voltage vector v g,dq and the d-axis of the rotating frame dq are forced into synchronism in the PLL (Fig. 2) at the same angle θ g as estimated by the PI control loop, so that the Parke transform results in v g,q = 0 and | v g,dq | = v g,d [39].Because it permits individual management of active power P and reactive power Q by employing only the components i g,d and i g,q , as established in (2) [17], this technique produces a weak coupling between the d and q components of the system variables.
(3) presents the continuous-time model for the VSC in the dq frame.The zero-order holder (ZOH) is used to approximate the derivative of the current in (3) in order to create the discrete-time model [41].In (4), the outcome of the formula manipulation is displayed, and in ( 5), the compact model terms are identified by square brackets.In (4) u dq = v i,dq − v g,dq is an auxiliary variable that represents the difference between VSC and network voltages [42].In ( 5)

III. MPC CONTROL FOR GRID-CONNECTED VSC
The MPC strategy's predictive model (Fig. 1) employs model from ( 5) to forecast the future behavior of power injected into the grid up to the sliding horizon n y [38], [43].The predictive model in ( 6) is formed by successive applications of ( 5), where: Y ∈ n y ×2 is the prediction vector of the system output defined in (7); x(k) are the actual measurements of the state variables; U ∈ 2n u ×1 is the prediction vector of the control signal up to the horizon n u defined in ( 9); is the predictive state matrix defined in (10).

A. COST FUNCTION
The quadratic cost function in (11) uses the predictions of the reference Y re f ∈ 2n y ×1 , the system output ( 6) and the control signal (8) [38].The terms y and u in (11) are diagonal weighting matrices of the tracking error and the control effort, respectively.
The vector Y re f is defined by the reference signal y re f (k) = i d,re f (k), i q,re f (k) , which is repeated n y times in (12) through multiplication by F = I I • • • I [18].In this sense, the grid voltage is regarded constant within the prediction horizon n y , which is compatible with DC values in the dq synchronous frame for any n y .This method is utilized in [43] to define the CCS-MPC in the αβ stationary frame, however it is limited to short time horizons [17].
According to [18], the weight factors y ∈ 2n y ×2n y and u ∈ 2n u ×2n u are positive definite block diagonal matrices.To decrease the degrees of freedom associated with the MPC controller, the cost function weighting block diagonal matrices are simplified to diagonal matrices with equal terms, as shown in ( 13) and ( 14).As a result, the important parameters for MPC-DPC performance are reduced to four: γ y , γ u , n y , and n u .

B. MINIMIZATION OF THE COST FUNCTION
We do not impose constraints for the system variables in the MPC method used in this study, allowing us to get an analytical solution for minimizing the cost function ( 11) by solving ∂J /∂U = 0 [44].The solution U presented in ( 15) is a control signal prediction vector, as stated in (8), with the first element being the control signal for the system's recent horizon u dq (k) [42].The control signal u dq (k) can be obtained from (15) as .
It should be noted that the control signal created by equation (15) provides continuous amplitude values that are applied to the SPWM, allowing the inverter to be driven with a wider range of voltage values rather than being limited to the eight voltage vectors used in standard FCS controllers [18].A more compressible way to define the control signal u dq (k + 1) is presented in (16), where K = W (M y M − u ) −1 M y is a constant matrix and independent of the measurements of the system variables, so can be calculated external to the real-time control loop [42].(16) computational load is similar to that of basic state feedback [44], [45], but it accomplishes error and control effort minimization at each time instant when the reference is updated and new measurements of the state variables are available.

IV. ROBUST STABILITY ANALYSIS
In this section, we analyze the CCS-MPC controller's robustness in the grid-connected VSC under parametric uncertainty using RS analysis.First, we verify nominal stability for various controller tuning by calculating the closed-loop eigenvalues.The details of the grid-connected VSC's nominal configuration can be found in Table 2.According to [46], the eigenvalues given by ( 17) are the closed-loop poles of the CCS-MPC controller, which can be deduced by substituting the control law from ( 16) into the discrete model from (5).
According to the research conducted by [23], the cost function weights are tuned by the ratio of γ u to γ y , which is the decisive factor in the minimum of the cost function, independent of the values of individual γ u and γ y .Here we adopt the tuning range of γ u /γ y from 10 −7 to 10 −1 and the time horizons n y = n u = [1,5,10].The value of γ y is set to be 10 5 arbitrarily.The CCS-MPC controller's coarse tuning involves adjusting the γ u /γ x ratio magnitude order.The blue curve in Fig. 3 represents the root locus of the closed-loop CCS-MPC current controller in the z-plane.It moves from the unit circle's edge to the center as the γ u /γ y tuning decreases.
The results in Fig. 3 show that the CCS-MPC control system is stable as the poles lie within the unit circle in the z-plane across the range of tunings examined.Additionally, it was observed that increasing the horizons n y and n u caused the poles to shift towards the center of the unit circle.The variations in tunings (3) and (4) when the horizons increased from n y = n u = 1 to n y = n u = 5 revealed this impact.To explicitly evaluate the stability, we apply the definitions of the RS margin proposed by [30] to calculate the tolerable parametric uncertainty in the L-filter ( L g and R g ).The symbol σ represents the singular values that indicate the spectral gain for multivariable systems [31].It helps to assess the sensitivity dynamics of the system towards changes in the model.This analysis provides valuable insights into the system's sensitivity to model variations.The RS margin is calculated from the structured singular value, which is an extension of the SVD concept [32].This value is obtained by applying it to the sensitivity function (S) [33].
The block diagram of the closed-loop CCS-MPC control is shown in Fig. 4. In this case, H (z) represents the dynamic model of the grid-connected power converter as described in section II and d in represents input disturbances, e.g.parameter uncertainty and grid voltage distortion [31].The sensitivity function S is the transfer function between the disturbance d in and the plant input signal u d , as defined in (18).
To conduct the RS analysis, we use the plant model represented by (5) with uncertainties in the L-filter parameters (L g and R g ) above and below the nominal values.We aim to

TABLE 3. Tolerable Parametric Uncertainty Equivalent to the Minimum RS Margin FIGURE 6. Worst disturbance analysis in the singular values σ(S) of the CCS-MPC controller tuning (4).
determine the minimum RS margin of the closed-loop CCS-MPC control system.By doing so, we can establish the range of parametric uncertainty within which stability is ensured.Fig. 5 displays the RS margin based on frequency for the five controller tunings highlighted in Fig. 3. Table 3 lists the allowable parametric uncertainty ( L g and R g ).
In Fig. 5, the minimum RS margin for tuning (4) is 0.55 at the critical frequency of 62.900 rd/s or 54% of the arbitrary uncertainty interval 0.5 mH < L g < 60 mH and 0.01 < R g < 4 ; that is, the close-loop stability is ensured in nominal condition and under parametric uncertainty limits [−62%, +117%] for L g and [−62%, +194%] for R g .It should be noted that the CCS-MPC controller has a high level of robustness, even in cases where the nominal poles are near the unit circle (Fig. 3).For example, tuning (1) has a tolerable range of parameter variation of [−84%, +236%] for L g .
To study how parametric uncertainty and the RS margin impact the closed-loop PRC control system, Fig. 6 displays the worst-case analysis of the spectral gain or singular value for tuning (4) [32].The figure reveals that even in the worst perturbation scenario (indicated by the blue dashed line), parametric uncertainty only leads to limited gain for any frequency.In Fig. 6, the singular value curves that indicate the worst-case scenario are represented by the blue line for the nominal system, blue dashed lines for parametric uncertainty samples, and red line for the worst-case gain.These curves show the envelope of the worst gain for each frequency, which indicates the robust stability barrier for the spectral gain of the PRC control system, as discussed in [47].
We conducted a more detailed analysis of the minimum RS margin for different CCS-MPC controller tuning within the range of 10 −7 < γ u /γ y < 10 −1 , building upon the RS analysis in Table 3.The resulting tolerable parametric uncertainty is presented in Fig. 7. Fig. 7 shows that the robust stability of the CCS-MPC controller is maximum for the tuning range 10 −3 < γ u /γ y < 10 −2 with n y = n u = 1, thus being able to maintain stability under parametric uncertainty up to the limits [−86%, +250%] for L g and [−82%, +892%] for R g .It is noteworthy that the maximum tolerance for parametric uncertainty remains relatively stable for longer time horizons (n y = n u = 5 and n y = n u = 10).However, it is crucial to consider that the γ u /γ y tuning for maximum robustness varies depending on the time horizon.The results in Fig. 7 show that the CCS-MPC control system maintains robust stability against parametric uncertainty ranging from a decrease of 44% to an increase of 67% in the L-filter inductance.

A. ROBUSTNESS TO GRID VOLTAGE DISTURBANCE
To ensure the stability of the proposed control system, it is important to consider the impact of disturbances in the grid voltage.This can be achieved by examining the input signal components of the plant described in (5), as shown in the frequency response depicted in Fig. 8.The disturbance we will be considering the uncertainty in the amplitude of the fundamental component, ranging from -100% to 100%,  as well as distortions caused by harmonic components up to the 19th order [6].This level of uncertainty in the grid voltage is significant and can be compared to contingencies, such as harmonic distortions or voltage sag.
The effect of the grid voltage disturbance in the minimum RS margin of the CCS-MPC controller is shown in Fig. 9.In this case, the time horizons were fixed at n y = n u = 10.In general, the minimum robust stability margin becomes narrower due to input signal disturbance, especially for tunings where the ratio γ u /γ y is greater than 3.5 × 10 −4 and the critical frequencies of σ (S) are within the input disturbance band, as shown in Fig. 5.The maximum stability margin for the L-filter inductance is reduced from [−86%, +250%] to [−80%, +203%] when the grid voltage distortion is considered.The RS margin in γ u /γ y ≤ 3.5 × 10 −4 is similar to that case where only the parametric uncertainty is analyzed.Curiously, in the worst case at γ u /γ y = 10 −6 , the distortion in the grid does not significantly impact the robustness margin of the controller.This suggests that with the correct tuning of the CCS-MPC controller, stability can be ensured in nominal conditions and theoretically accommodate a variation in the L-filter inductance of up to ±40%.

V. EXPERIMENTAL RESULTS OF THE CCS-MPC CONTROLLER
In this paper, we obtained our results using the experimental 3-phase bench presented in Fig. 10.The bench consists of several main components, including an electronic board for conditioning the three-phase voltage and current, a DSP Texas Instruments model TMS320F28335 to run control algorithms, a 2 kVA/800 Vdc/380 Vac three-phase/two-level power converter, a 13.5 mH inductive bank (L-filter), and an AC programmable power supply model SUPPLIER FCATHQ 4500 VA/380 V/500 Hz to simulate the power grid [41].The setup for the grid-connected VSC experiment is identical to the one outlined in Table 2.
We used the integral square error (ISE) to measure power regulation, which is defined in (19) [31].We also calculated the quality of energy injected into the grid using the total harmonic distortion (THD).Equation ( 19) defines e as the error vector for active and reactive power reference tracking, which is equal to . It is recommended by the IEEE 519-2014 standard [48] that the THD should not exceed 5%.

A. TUNING THE CCS-MPC UNDER NOMINAL CONDITIONS
Experimental tests in this section were conducted to compare the performance of four different CCS-MPC controller tunings numbered (2) through (5) with n y = n u = 1.Tuning (4) had a γ u /γ y value of 10 −5 and the resulting measurements for the i g,d and i g,q components of the grid current are shown in Fig. 11.shows the experimental results for the d-axis current component i g,d and the AC current and phase voltage in the grid.
For the tunings ( 3) and ( 4), the CCS-MPC controller guarantees the nominal stability and the power quality performance (THD below 5%) for at least 3 A current reference, equivalent to 20% of the converter's nominal power.
To better compare the performance of the CCS-MPC controller tuning experimentally, Fig. 14 displays the step response of the grid current i g,d , as well as the ISE and THD performances.Note that although tuning (2) shows nominal stability of the control system, it has a very large steady-state error (-30%).Although the ISE and THD performances of Tunings ( 4) and ( 5) are similar in Fig. 14(b), Tuning (5) exhibits a strong oscillation (± 6.5%) around the reference in the permanent regime as shown in Fig. 14(a).In addition, tuning (3) has a slight regime error and a slower transition compared to tuning (4).Moreover, tuning (4) has better power quality performance in terms of ISE and THD.Overall, CCS-MPC tuning with γ u /γ y = 10 −6 and n y = n u = 1 showed the best nominal performance.
The purpose of the results shown in Fig. 15 is to study how different time horizons affect the performance of the CCS-MPC controller tuning.It is worth noting that when the value of n y is increased from 1 to 10 for tuning (2) and (3), there is a significant improvement in both ISE and THD performance.However, the effect of the time horizon is less pronounced for tuning (4) and (5).Fig. 15 indicates that tuning (4) consistently produces the best ISE performance, regardless of the time horizon.Furthermore, when n y = n u = 10, tuning (4) achieves a THD performance of 0.91%, which is superior to tuning (5).Based on extensive experimentation, it has been determined that the optimal ISE and THD performance can

B. VALIDATION OF THE ROBUSTNESS OF THE CCS-MPC CONTROLLER
This section presents the experimental validation of the CCS-MPC controller's robust stability under parametric variation in the L-filter.In the experiments, the nominal inductor of 13.2 mH was replaced by two others with values of 10 mH and 22 mH, corresponding to relative variations of -24% and +67%, respectively.These parametric deviations are within the stability margin for the controller tuning in γ u /γ y = 10 −5 and n y = n u = 10, estimated in [−58%, +105%] (Fig. 7).The d-axis step response under parametric variation is compared to the nominal case in Fig. 16(a), highlighting the performance metrics in transitory.Fig. 16(b) shows the ISE and THD power quality performance.
The results in Fig. 16(a) show that the CCS-MPC controller can track current references for parametric variations of −24% and +67%, which validate the robust stability of the proposed tuning.The blue line representing nominal transient performance (Fig. 16(a)) deteriorates significantly when the inductance of the L filter decreases, resulting in a strong current oscillation around the reference (± 10% in red line) and lower power quality or higher ISE and THD (Fig. 16(b)).Note that as the filter inductance increases, the current response in Fig. 16(a) becomes slower, resulting in a lower t s .The ISE degrades with an increase or decrease in L-filter inductance, while THD remains close to nominal during testing with the 22 mH inductor.The proposed PRC controller delivers a sinusoidal current with less than 5

C. PERFORMANCE COMPARISON WITH CLASSICAL CONTROL TECHNIQUES
In this section, we are comparing the robustness of the CCS-MPC controller tuning that is proposed in this paper with the classical PI controller in the synchronous rotation frame.To design the PI, we applied the well-established method proposed by [21] for the current control of a grid-connected VSC with L-filter.The method suggests using the gains k p = L g /τ i and k i = T s R g /τ i from the parameter τ i .The parameter τ i represents the time constant of the approximation of the closed-loop dynamics of the control system with a first-order model.It should be selected to be small for a fast response but large enough to ensure that the closed-loop bandwidth (1/τ i ) is smaller than the PWM switching frequency (20 kHz) [21].Through experimental testing, the PI controller's tuning was evaluated to determine its best performance in both transient and power quality ISE and THD.This is illustrated in Fig. 17.In Fig. 17(a), the measurements of the step response of the component current d injected into the network are displayed.Fig. 17(b) summarizes the ISE and THD performance for the tested tunings of the parameters τ i .The results indicate that the PI's ISE and THD performance improves when the τ i parameter is reduced to 100 μs, as even lower values tend to generate lower performance.Moreover, the PI with tuning at τ i = 100μs has the best settling time (t s = 1 ms) and the lowest overshoot (13%).It is worth noting that 14% is still too high when compared to the overshoot level of the best tuning of the CCS-MPC (blue line in Fig. 14).Nevertheless, the best tuning of the PI controller with τ i = 100μs generates the gains k p = 132 and k i = 0.25 and results in a phase margin of 75 • and gain margin of 12 dB [13].
The robustness tests of the designed PI were performed using the same parametric uncertain L g = [−24%, +67%] of the grid-connected VSC with L-filter described in section V-B.The step response for the PI controller is presented in Fig. 18   The grid current curves in Fig. 18 indicate that the PI's performance reduces significantly when the L filter's inductance is lowered.The PI's ability to follow the reference decreases more than that of the CCS-MPC in Fig. 16.In comparison to the CCS-MPC controller proposed in this article, the PI controller displayed inferior performance in ISE and THD during the robustness test.The CCS-MPC controller performs well under nominal conditions and is more robust to variations in the L-filter for grid-connected VSC.Additionally, the PI's performance deviation goes beyond the minimum THD performance, with values exceeding 5%, which violates the IEEE 519-2014 standard.
It is important to note that the PI controller is not designed to minimize the control effort as effectively as the MPC [15].The control effort of the MPC and PI methods are compared in Fig. 19.The MPC control effort (32.9 Vrms) is significantly lower than the PI controller (139.3Vrms), indicating the efficiency gains achieved through our proposed approach [18].In our efficiency comparison, our tests showed that CCS-MPC takes 31 microseconds to run, while PI takes 30 microseconds.Therefore, both controllers perform very similarly in terms of computational cost.

VI. CONCLUSION
In this paper, we propose a robust CCS-MPC controller for a grid-connected VSC with an L-filter to integrate renewable energy sources.We examined the robust stability of the CCS-MPC controller under different cost function weight factors and time horizons using the generalized singular values derived from SVD.We found that the controller was stable under all the different tunings we tested, even under severe grid voltage disturbances and uncertainties of up to ±40% in the L-filter inductance.
The effectiveness of the CCS-MPC controller tuning was tested on a 2 kW experimental bench.The best performance in terms of the quality of energy injected into the grid was achieved with a time horizon of 10 for the cost function weight factors γ u /γ y = 10 −5 .Parametric tests were conducted using inductances of 10 mH and 22 mH, which varied from -24% to +67% of the rated L-filter value.These tests verified the CCS-MPC controller's robust stability under parametric uncertainty.The CCS-MPC controller outperforms the classical PI current controller under nominal conditions and is more robust to variations in the L-filter for grid-connected VSC.

FIGURE 1 .
FIGURE 1. Space vector CCS-MPC strategy for the grid-connected VSC with L-filter using rotation frame dq.

FIGURE 2 .
FIGURE 2. Block diagram representing the operation of the PLL.

Algorithm 1 :
CCS-MPC algorithm.Result VSC switch drive.Load the system parameters: R, L, V ab , f g , V dc , f sw , γ y , γ u , n y , n u ; while CCS-MPC is active do Data acquisition: 3-phase i g,abc (k) and v g,abc (k); Apply Clark transform: i g,αβ (k) and v g,αβ (k); Active PLL: θ g ; Apply Park transform: i g,dq (k) and v g,dq (k); Apply (2): x(k) = [i g,d , i g,q ]; Build the system model (5): A d , B d , and C d ; Build the predictive model (6): , M and Y re f ; Build the cost function J (11); Minimize J using (16): u dq (k); v i,dq (k + 1) = u dq (k + 1) + v g,dq (k); Apply inverse Park transform: v i,αβ (k)); Apply v i,αβ (k) to the SPWM modulator; return VSC switches states; end TABLE 2. Nominal Parameters of the Grid-Connected VSC

FIGURE 4 .
FIGURE 4. Closed-loop block diagram of the CCS-MPC controller.

FIGURE 5 .
FIGURE 5. RS margin for five CCS-MPC controller tuning with n y = n u = 1.

FIGURE 7 .
FIGURE 7. Effect of the CCS-MPC controller tuning on robust stability or tolerable parametric uncertainty in the L-filter, relative to the nominal parameter L g = 13.2 mH and R g = 0.1 : (a) Inductance and (b) Internal Resistance.

FIGURE 8 .
FIGURE 8. Spectral profile of the grid voltage disturbance.

FIGURE 9 .
FIGURE 9. Minimum RS margin of the CCS-MPC controller under grid voltage disturbance.
Fig.12compares the AC current and voltage measured at the grid connection point.The experimental performance was evaluated with ISE = 10.36 dB and THD = 1.07%.Tuning (3) had a γ u /γ y value of 10 −4 and Fig.13

FIGURE 11 .FIGURE 12 .FIGURE 13 .
FIGURE 11.Measurements using the CCS-MPC tuning (4): components i g,d and i g,q of the grid current.

FIGURE 14 .
FIGURE 14.Comparison of the CCS-MPC controller performed using tunings (2) through (5): (a) Step response of d-axis grid current; (b) power quality ISE and THD.

FIGURE 15 .
FIGURE 15.Effect of the time horizons in the performance of the CCS-MPC controller tuning: (a) THD and (b) ISE.

FIGURE 16 .
FIGURE 16.Performance of the CCS-MPC controller under parametric uncertainty: (a) Step response of d-axis grid current; (b) power quality ISE and THD.
(a) and Fig. 18(b) compares the robustness under parametric uncertainty of the PI with the proposed CCS-MPC controller tuning.The graph shown in Fig. 18(b) displays the nominal performance of the controllers when L g = 13.2 mH

FIGURE 17 .
FIGURE 17. Experimental measurements of the PI controller tuning: (a) Step response of d-axis grid current; (b) power quality ISE and THD.

FIGURE 18 .
FIGURE 18.(a) Grid current response for the designed PI under parametric uncertainty; (b) Robustness comparison of the proposed CCS-MPC and PI.

FIGURE 19 .
FIGURE 19.Comparison of the control effort between the MPC and PI methods.