Direct Power Control of a Single Stage Current Source Inverter Grid-Tied PV System

: In this paper, a direct power predictive controller (DPPC) is derived for a current source inverter (CSI) based single stage photovoltaic (PV) system. The equations of the dynamics, including AC and DC ﬁlters, are formulated directly for the PV power and for the active and reactive power injected in the grid. Then, the prediction equations are synthesized straight for the power, and, at each time instant, the optimal switching vector that guarantees simultaneously the control of the power generated by the PV arrays, and the control of the reactive power in the connection to the grid is chosen. This approach aims to guarantee fast and accurate tracking of the power. The proposed system is then validated through simulation and experimental results, showing that the PV system is able to follow the power references, guaranteeing a fast response to a step in the power, and decoupled active and reactive power control, with minimum total harmonic distortion ( < 5%) of the currents injected in the grid.


Introduction
The requirements of decarbonization are leading to the massive integration of renewable energies in the grid [1,2]. In particular, photovoltaic (PV) systems are becoming increasingly popular, used both for domestic and industrial applications with an average annual growth rate of 60% [3].
Grid-connected PV inverters are required to deploy energy directly to the grid and they are usually connected to the low voltage (LV) or medium voltage (MV) distribution grid [4,5]. International standards regulate power quality in the connection of PV inverters to the grid, bounding the harmonic content of the injected currents and setting limits to power factor regulation. IEEE standard (Std) 1547-2018 [6] provides a set of requirements for interconnection and interoperability of distributed resources with power system interfaces; IEC 61000-3-2 [7] sets limits for current harmonic emissions; IEEE Std 519-2014 [8] recommends practices and requirements for harmonic control in electric power systems with a set of recommended limitations for voltage and current harmonics; and IEC 61727 [9] lays down a set of requirements for the interconnection of PV systems to the distribution grid. Two of the most important impositions of these standards is the bounding of the total harmonic distortion (THD) of the injected currents, up to the 50th harmonic, to around 5%, and setting the minimum state variables (inductor currents and capacitor voltages). Then, based on the equations of the power dynamics, the predictive controllers are synthesized to directly control the power in the PV and the reactive power in the connection to the grid. This is a novelty, as the usual approach is to control the AC inductor currents, which are state variables and are not exactly equal to the currents to be controlled (the grid currents). The dynamic response of the proposed DPPC is then experimentally evaluated under different operating conditions, and the power quality in the connection to the grid is assessed.
This paper is organized in five sections. In Section 2, the system is detailed and the model to characterize it is derived. In Section 3, the controller is synthesized, and in Section 4, the simulation and experimental results are obtained and compared. In Section 5, the main conclusions from the developed work are obtained and the main contributions of this work to the research field are summarized.

System Description
This section presents the model of the proposed system. The representation of the current source inverter (CSI), single stage three-phase grid-connected PV system is depicted in Figure 1. An approximate model of the PV array is used [39], being connected to the CSI through a filtering inductance L dc (with parasitic resistance r dc ). Then, the CSI is connected to the grid through a second order LC filter [27,40] designed and sized to guarantee compliance with the IEEE 1547 standard, ensuring a displacement power factor higher than 0.9 in the connection to the grid, and a total harmonic distortion of the injected AC grid currents that is lower than 5% [27,41].
Energies 2020, 13, x FOR PEER REVIEW 3 of 20 reactive power in the connection to the grid. This is a novelty, as the usual approach is to control the AC inductor currents, which are state variables and are not exactly equal to the currents to be controlled (the grid currents). The dynamic response of the proposed DPPC is then experimentally evaluated under different operating conditions, and the power quality in the connection to the grid is assessed. This paper is organized in five sections. In Section 2, the system is detailed and the model to characterize it is derived. In Section 3, the controller is synthesized, and in Section 4, the simulation and experimental results are obtained and compared. In Section 5, the main conclusions from the developed work are obtained and the main contributions of this work to the research field are summarized.

System Description
This section presents the model of the proposed system. The representation of the current source inverter (CSI), single stage three-phase grid-connected PV system is depicted in Figure 1. An approximate model of the PV array is used [39], being connected to the CSI through a filtering inductance L dc (with parasitic resistance r dc ). Then, the CSI is connected to the grid through a second order LC filter [27,40] designed and sized to guarantee compliance with the IEEE 1547 standard, ensuring a displacement power factor higher than 0.9 in the connection to the grid, and a total harmonic distortion of the injected AC grid currents that is lower than 5% [27,41].
The model of the dynamics of the CSI based PV system is discussed in detail in the next sections.

PV Array Model
The PV array is modeled as shown in Figure 2, using the three parameters model representation [39], where iph is the photocurrent, dependent on the irradiation and temperature of the PV cell, and iD is the diode current. The model of the dynamics of the CSI based PV system is discussed in detail in the next sections.

PV Array Model
The PV array is modeled as shown in Figure 2, using the three parameters model representation [39], where i ph is the photocurrent, dependent on the irradiation and temperature of the PV cell, and i D is the diode current.
Energies 2020, 13, x FOR PEER REVIEW 3 of 20 reactive power in the connection to the grid. This is a novelty, as the usual approach is to control the AC inductor currents, which are state variables and are not exactly equal to the currents to be controlled (the grid currents). The dynamic response of the proposed DPPC is then experimentally evaluated under different operating conditions, and the power quality in the connection to the grid is assessed. This paper is organized in five sections. In Section 2, the system is detailed and the model to characterize it is derived. In Section 3, the controller is synthesized, and in Section 4, the simulation and experimental results are obtained and compared. In Section 5, the main conclusions from the developed work are obtained and the main contributions of this work to the research field are summarized.

System Description
This section presents the model of the proposed system. The representation of the current source inverter (CSI), single stage three-phase grid-connected PV system is depicted in Figure 1. An approximate model of the PV array is used [39], being connected to the CSI through a filtering inductance L dc (with parasitic resistance r dc ). Then, the CSI is connected to the grid through a second order LC filter [27,40] designed and sized to guarantee compliance with the IEEE 1547 standard, ensuring a displacement power factor higher than 0.9 in the connection to the grid, and a total harmonic distortion of the injected AC grid currents that is lower than 5% [27,41].
The model of the dynamics of the CSI based PV system is discussed in detail in the next sections.

PV Array Model
The PV array is modeled as shown in Figure 2, using the three parameters model representation [39], where iph is the photocurrent, dependent on the irradiation and temperature of the PV cell, and iD is the diode current.  Considering the equivalent model of the PV cell presented in Figure 2, the PV current can be obtained from Equation (1), The photocurrent (i ph ) of the PV cell changes with the irradiance level and cell temperature (T) according to Equation (2), where i sc is the short circuit current of the PV cell, G is irradiation level in kW/m 2 , G ref is the reference irradiation, 1 kW/m 2 , and µ T is the temperature coefficient of i sc . The diode current i D represented in Figure 2 can be obtained from Equation (3), where i o is the diode saturation current, q is the electric charge (1.6022 × 10 −19 C), k is the Boltzmann's constant (1.3806 × 10 −23 J/K), T is the cell temperature (K), and m is the diode quality factor (m = 1 for an ideal diode and m > 1 for a real diode). The diode's saturation current (i o ) changes with the cell temperature (T) and is expressed as in Equation (4): where i o r is the reference diode saturation current, T re f is the reference cell temperature and ε is the silica's characteristic. From Equation (1), the current of the PV cell, i PV (Equation (5)) can then be expressed as a function of the cell's voltage v PV , considering the diode current i D (Equation (3)), and an association of n p paralleled PV modules, and n s series-connected PV modules.
The PV data used in simulation, for the SunPower SPR305-WHT module, is presented in Table 1. The P-I characteristic of the PV is illustrated in Figure 3.

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Energies 2020, 13, x FOR PEER REVIEW 5 of 20 The P-I characteristic of the PV is illustrated in Figure 3. The power supplied by the PV can be obtained from Equation (6) by multiplying both sides of Equation (5) by v PV : The power derivative can be calculated from Equation (7), and at maximum power 0 The derivative of the PV power can also be obtained from Equation (8) ( ) The PV power dynamics will be computed using this equation.

Current Source Inverter Model
To reduce the number of conversion stages and avoid the use of a step-up transformer in the connection to the grid, a CSI is used to perform the DC to AC conversion, as depicted in Figure 1. Ideal switches are used to model the converter, containing three switches for each phase, where the ON/OFF state of each switch S mn is defined as: The states of CSI switches can be represented by matrix S: The power derivative can be calculated from Equation (7), and at maximum power The derivative of the PV power can also be obtained from Equation (8), considering the time derivative of the PV voltage and current. At maximum power dP PV dt = 0 [42]: The PV power dynamics will be computed using this equation.

Current Source Inverter Model
To reduce the number of conversion stages and avoid the use of a step-up transformer in the connection to the grid, a CSI is used to perform the DC to AC conversion, as depicted in Figure 1. Ideal switches are used to model the converter, containing three switches for each phase, where the ON/OFF state of each switch S mn is defined as: The states of CSI switches can be represented by matrix S: To assure that the three-phase grid voltages are never short circuited and that the DC side inductive currents are never interrupted, the sum of all switch states connected to each output phase should be equal to 1 (Equation (10)). Considering this constraint, the switching states, output currents, and input voltage vectors of the CSI are presented in Table 2.
Using the switching matrix S, the converter voltages (v 1 , v 2 ) on the DC side and the grid side currents (i a , i b , i c ) can be obtained, respectively, from Equation (11): where (v sa , v sb , v sc ) are the converter grid side phase voltages and i PV is the DC current. This equation will be further used to predict the future DC voltage of the converter (v 12 ) and the grid side currents.

Equations of PV Power Dynamics
The equation of the DC current (i PV ) dynamics (Equation (12)) is obtained by applying Kirchhoff's laws to the circuit shown in Figure 1: where L dc and r dc represent the DC filtering inductance and its parasitic resistance, respectively. Using Equation (12) in Equation (8), the dynamics of PV power is obtained:

Equations of the Dynamics of Active and Reactive Power in the Connection to the Grid
Regarding the grid connection, the state variables, in abc coordinates, are the inductor currents (i la , i lb , i lc ) and the capacitor voltages (v sab , v sbc , v sca ).
The inductor currents (i la , i lb , i lc ) can be expressed as a function of the capacitors' voltages (v sab , v sbc , v sca ) and the grid voltages (v ga , v gb , v gc ), and are obtained from Equation (14), where L f represents the filter inductances [26,43].
Energies 2020, 13, 3165 The dynamics of the capacitor's voltages can be determined from Equation (15), where C f represents the filter capacitance.
The grid currents (i ga , i gb , i gc ) are then obtained from Equation (16), and depend on the inductor currents (i la , i lb , i lc ), on the filter inductance L f , and on the damping resistance R f .
Let us consider that X abc represents the system variables in abc coordinates. To obtain a decoupled system in αβ coordinates, the Concordia transformation (Equation (17)) is applied to Equations (14), and the variables X abc are transformed to αβ coordinates by computing X αβ = C T X abc .
Then, to obtain a stationary model of the system in dq coordinates, the Park transformation matrix D (Equation (18)) is used, where θ = ωt is the phase angle of the grid voltages (Equation (19)). Then, variables X αβ can be further transformed to X dq , by calculating X dq = D T X αβ .
In this reference frame, the grid voltages v gd , v gq are expressed by Equation (20): The full state space model of the CSI in dq coordinates can be obtained (Equation (21)) by applying the Concordia (Equation (17)) and Park (Equation (18)) transformation to Equations (14) and (15) [26,43]: Applying the Concordia (Equation (17)) and Park (Equation (18)) transformation to Equation (16), and considering the inductor's currents i ld , i lq dynamics (Equation (21)), the grid side currents are obtained in dq coordinates (Equation (22)), where ω is the grid angular frequency: The active and reactive power in the connection to the grid is calculated respectively by: Considering the grid currents (Equations (22) and (23)), the detailed equations of the active and reactive power are then calculated from Equation (24).
The active and reactive power (Equation (24)) do not depend directly on the control variables, CSI currents i d and i q . Consequently, to design the DPPC, it is necessary to calculate the first derivative of active and reactive power, considering that in dq coordinates the grid voltages v gd , v gq are constant, and their derivatives are zero: Using the derivatives of inductor currents and capacitor voltages (Equation (21)) in Equation (25), and neglecting cross terms, the dynamics of the active (P) and reactive (Q) power in the connection to the grid can be obtained as a function of the grid voltages v gd , v gq , the capacitor voltages v sd , v sq , and the CSI currents, i d , i q : Considering that in the chosen reference frame v gq = 0 (Equation (20)), the previous equation can be further simplified to: It is important to highlight that these equations represent the dynamics of the active and reactive power in the connection to the grid, and the predictive controller will be further synthesized using these equations.

Design of the Direct Power Predictive Controller
A DPPC approach is used to control the generated PV power and the reactive power in the connection to the grid. This controller aims at choosing the optimal switching vector [34,35] from the possible converter switching combinations (Table 2).
Euler forward and backward methods are the approaches used the most in model predictive control for power converters and drives applications. The advantage of the Euler forward method is that it gives an explicit update equation, being easier to implement. On the other hand, the Euler backward method requires solving an implicit equation, but generally has better stability properties [31,32]. For this reason, the Euler backward (Equation (28)) integration method is chosen here to predict the PV power and the reactive power of the PV system in the connection to the grid.
The prediction of the state variables for future time t = k + 1, is obtained considering that the time interval h = T s is much lower than the period of the electric variables. Then the equations for the Euler backward (Equation (28)) integration method can be obtained.

Prediction of the PV Power
The predicted PV power at instant t = k + 1, can be determined from Equation (29), The PV current at the next time instant i PV (k + 1) can be obtained by discretizing Equation (12) at T s , where v 12 (k + 1) represents the future DC converter voltage and can be computed from Equation (11).
The predicted power P PV (k + 1) can also be obtained by discretizing Equation (13) at T s , with the same yield as Equation (29).
Regarding the future PV voltage v PV (k + 1), it can be considered that, at very small sampling, the predicted PV voltage is the same as the actual, v PV (k + 1) ≈ v PV (k). Another approach can be obtained using Equation (32), where the future value of PV voltage v PV (k + 1) will depend on the actual value v PV (k) and the expected changing in voltage ∆v PV (k + 1).
This change can be obtained from an MPPT algorithm, which defines the expected increase/decrease in voltage. The first approach is used in real-time simulation to reduce computation time and to decrease the sampling time as well.

Prediction of Active and Reactive Power in the Connection to the Grid
In the connection to the grid, the predicted active power P(k + 1) and reactive power Q(k + 1) can be calculated from Equation (27) by using the Euler backward method: The predicted power depends on the future capacitor voltages v sd (k + 1), v sq , (k + 1), which can be calculated by discretizing Equation (21): Substituting Equation (34) in Equation (33) and neglecting the cross terms, the future active and reactive power is obtained, considering that in the chosen reference frame, the grid voltage is nearly constant, thus v gd (k + 1) ≈ v gd (k).
From Equation (35), it is quite relevant to notice that the predicted values for the power injected in the grid only depend on the converter currents in the next time instant, i d (k + 1) and i q (k + 1), as expected. All the other variables, as the capacitor voltages (v sd , v sq ), are for the present sample t s = k and can be measured.

Cost Function and Selection of Swtiching Vector
The selection of the optimal switching vector for the next sample is obtained by minimization of the cost function f (k + 1), that evaluates the error of the controlled variables. To ensure that the predicted active and reactive power reach the reference values, the cost function f (k + 1) is used to minimize the squared errors (Equation (36)), where P PV,re f and Q re f are considered as the set point references for the controller and w 1 , w 2 are the weights for the active and reactive power, respectively.
The procedure to select the optimal switching vectors is summarized in the flowchart of Figure 4. The proposed controller structure used in the simulations and to obtain the experimental results is presented in Figure 5.

Results
In this section, a 600 W, three-phase grid-connected CSI based PV system is simulated in a MATLAB/Simulink environment, and experimental results are obtained using the laboratory setup shown in Figure 6.
The experimental prototype is implemented using a controlled DC power supply (1.5 kW, 360 V, 15 A) to emulate the behavior of PV panels. The rapid prototyping software and hardware The proposed controller structure used in the simulations and to obtain the experimental results is presented in Figure 5. The proposed controller structure used in the simulations and to obtain the experimental results is presented in Figure 5.

Results
In this section, a 600 W, three-phase grid-connected CSI based PV system is simulated in a MATLAB/Simulink environment, and experimental results are obtained using the laboratory setup shown in Figure 6.
The experimental prototype is implemented using a controlled DC power supply (1.5 kW, 360 V, 15 A) to emulate the behavior of PV panels. The rapid prototyping software and hardware

Results
In this section, a 600 W, three-phase grid-connected CSI based PV system is simulated in a MATLAB/Simulink environment, and experimental results are obtained using the laboratory setup shown in Figure 6.
Energies 2020, 13, x FOR PEER REVIEW 13 of 20 DSPACE ds1103 and its Control Desk interface are used. A field-programmable gate array (FPGA) from Xilinx, model Virtex-6 FPGA ML605, is used to handle the timing requirements of the commutation process. The specification data of the experimental setup are listed in Table 3.

Response to a Step in Power
In this case, the proposed system is assessed for a PV power change from PPV = 400 W to PPV = 280 W at Q = 0 var. The main values used in the simulations and in the experiments are presented in Table 4. The system is tested at 110V phase grid voltage and around 70 V at the DC side (PV panel emulator).

70
Step: 400 to 280 0 Figure 7 depicts the simulation and experimental results for this case: Figure 7a,b shows one of the grid phase voltages vgan and the corresponding grid current iga, the power measured in the DC side Pdc, and the reactive power Q in the connection to the grid. When the PV power is reduced from PPV = 400 W to PPV = 280 W, the AC current reduces from iga = 1.4 A to iga = 1.07 A. At the PV power change, the DC current i dc reduces from i dc = 6 A to i dc = 4 A, as shown in Figure 7c,d. The experimental prototype is implemented using a controlled DC power supply (1.5 kW, 360 V, 15 A) to emulate the behavior of PV panels. The rapid prototyping software and hardware DSPACE ds1103 and its Control Desk interface are used. A field-programmable gate array (FPGA) from Xilinx, model Virtex-6 FPGA ML605, is used to handle the timing requirements of the commutation process. The specification data of the experimental setup are listed in Table 3. Table 3. System specification data.

Response to a Step in Power
In this case, the proposed system is assessed for a PV power change from P PV = 400 W to P PV = 280 W at Q = 0 var. The main values used in the simulations and in the experiments are presented in Table 4. The system is tested at 110 V phase grid voltage and around 70 V at the DC side (PV panel emulator). Table 4. Simulation data.

70
Step: 400 to 280 0 Figure 7 depicts the simulation and experimental results for this case: Figure 7a,b shows one of the grid phase voltages v gan and the corresponding grid current i ga , the power measured in the DC side P dc , and the reactive power Q in the connection to the grid. When the PV power is reduced from P PV = 400 W to P PV = 280 W, the AC current reduces from i ga = 1.4 A to i ga = 1.07 A. At the PV power change, the DC current i dc reduces from i dc = 6 A to i dc = 4 A, as shown in Figure 7c,d. The results show that the power on the DC side is controlled and tracks the step in the reference. The phase current is nearly sinusoidal and is out of phase with the grid voltage (assuming power flow from grid side to DC side). Also, the reactive power in the connection to the grid is nearly zero and not affected by the step in the active power. The total harmonic distortion (THD) of the grid currents at maximum PV power (PPV = 540 W) is presented in Figure 8, calculated using a fast Fourier tool analysis (FFT) in PowerGUI block. It can be seen that the measured value, obtained with a Fluke 1735 (compliant with IEC 61000-4-7 Class II), is 4.1%, thus less than the 5% set as the maximum by international standards. The results show that the power on the DC side is controlled and tracks the step in the reference. The phase current is nearly sinusoidal and is out of phase with the grid voltage (assuming power flow from grid side to DC side). Also, the reactive power in the connection to the grid is nearly zero and not affected by the step in the active power.
The total harmonic distortion (THD) of the grid currents at maximum PV power (P PV = 540 W) is presented in Figure 8, calculated using a fast Fourier tool analysis (FFT) in PowerGUI block. It can be seen that the measured value, obtained with a Fluke 1735 (compliant with IEC 61000-4-7 Class II), is 4.1%, thus less than the 5% set as the maximum by international standards.
The total harmonic distortion (THD) of the grid currents at maximum PV power (PPV = 540 W) is presented in Figure 8, calculated using a fast Fourier tool analysis (FFT) in PowerGUI block. It can be seen that the measured value, obtained with a Fluke 1735 (compliant with IEC 61000-4-7 Class II), is 4.1%, thus less than the 5% set as the maximum by international standards.   can be seen that the current and voltage are out of phase, thus guaranteeing a nearly unitary power factor. The power supplied to the grid is slightly lower than 540 W and the reactive power is Q = 0 var.
These results show that the three-phase grid currents in Figure 9c,d are balanced and their ripple is low. Comparing the simulation and the experimental results, it can be seen that they are very similar, proving the effectiveness of the proposed DPPC.

Active Power and Reactive Power Control Using DPPC
In this section, the proposed system is tested for two operating scenarios: one for a step change in the reactive power and the other for the tracking of a PV power profile.
Step in the Reactive Power Q (Leading and Lagging) This case study presents the PV system performance for a step change in the reactive power. The These results show that the three-phase grid currents in Figure 9c,d are balanced and their ripple is low.
Comparing the simulation and the experimental results, it can be seen that they are very similar, proving the effectiveness of the proposed DPPC.

Active Power and Reactive Power Control Using DPPC
In this section, the proposed system is tested for two operating scenarios: one for a step change in the reactive power and the other for the tracking of a PV power profile.

4.2.1.
Step in the Reactive Power Q (Leading and Lagging) This case study presents the PV system performance for a step change in the reactive power. The injection of reactive power in the PV system has many different strategies that are summarized in [4]. Here, the set point for the reactive power is used as a fixed value, and the limitation of reactive power depends on the maximum power capacity of the CSI. Figure 10 shows the dynamic response of the grid-connected PV system when changing from lagging to leading mode. Figure 10a shows one of the grid phase voltages v gan and the current i ga , in the same phase, for reactive power changing from Q = −40 var to Q = 40 var at P = 350 W operation. Figure 10b shows the three-phase grid currents i gabc , (i gmax = 2.6 A) and the current in the DC link, i dc .
Energies 2020, 13, x FOR PEER REVIEW 16 of 20 the grid phase voltages vgan and the current iga, in the same phase, for reactive power changing from Q = -40 var to Q = 40 var at P = 350 W operation. Figure 10b shows the three-phase grid currents igabc, (igmax = 2.6 A) and the current in the DC link, idc.
The results depicted in Figure 10 show a transition from leading to lagging power factor, where the DPPC controller provides a fast, yet smooth response, tracking the step in the reactive power reference. The active power remains unaffected, showing the effectiveness of the active and reactive power decoupled control.
When the reactive power changes, then the phase of the grid currents also changes. However, there are no disturbances or significant distortion in the current waveforms.

PV Power Profile Control
In this case, the daily PV power profile and the injection of reactive power are controlled using the DPPC. The reactive power is controlled to be at its maximum value during the day, at maximum PV power, and to be zero at night. The grid-connected PV system acts as a STATCOM (Static Synchronous Compensator), as shown in Figure 11a, where it can be seen the maximum reactive power injection Q = 100 var in the grid during the day at the peak PV power profile. The experimental conditions for this case study are listed in Table 4. Step: 0 to 100 The results shown in Figure 11a indicate that the controller is able to track the maximum power. Also, the DPPC guarantees active and reactive power decoupling, as the step in the reactive power does not affect the active power.  The results depicted in Figure 10 show a transition from leading to lagging power factor, where the DPPC controller provides a fast, yet smooth response, tracking the step in the reactive power reference. The active power remains unaffected, showing the effectiveness of the active and reactive power decoupled control.
When the reactive power changes, then the phase of the grid currents also changes. However, there are no disturbances or significant distortion in the current waveforms.

PV Power Profile Control
In this case, the daily PV power profile and the injection of reactive power are controlled using the DPPC. The reactive power is controlled to be at its maximum value during the day, at maximum PV power, and to be zero at night. The grid-connected PV system acts as a STATCOM (Static Synchronous Compensator), as shown in Figure 11a, where it can be seen the maximum reactive power injection Q = 100 var in the grid during the day at the peak PV power profile. The experimental conditions for this case study are listed in Table 5.

Conclusions
The objective of this paper was to provide contributions to the increasing trend in using CSIs in PV systems, which result in increased power density, lower costs, and increased lifetime of the inverter.
A state of the art for CSI control for PV applications was provided while emphasizing the advantages of single state topologies against two stage topologies.
The equations of the dynamics of the PV power and the AC reactive and active power were derived and provided the basis on which the DPPC was designed.
The synthesized DPPC directly tracks the power references based on the prediction of the PV power and on the AC active and reactive power. A quadratic error cost function was proposed with separable parameterizable weights for the PV power and the grid reactive power.
Simulations and experimental validation were obtained for steps in both the active and reactive power, showing the ability of the controller to track both references separately while maintaining values of THD within standards regulations.
The tracking performance was tested experimentally by submitting the controller to a typical power profile where the converter was able to track the maximum power along the profile while showing the ability to control the reactive power fully decoupled from the active power reference.  Step: 0 to 100 The results shown in Figure 11a indicate that the controller is able to track the maximum power. Also, the DPPC guarantees active and reactive power decoupling, as the step in the reactive power does not affect the active power. Figure 11b,c present the grid phase voltage v ga and the corresponding grid phase current i ga for a step change in the reactive power Q, from Q = 0 var to Q = 100 var (Figure 11b), and then from Q = 100 var back to Q = 0 var (Figure 11c). The results obtained show that the grid current i ga is leading the grid voltage v ga at maximum Q and out of phase at Q = 0.

Conclusions
The objective of this paper was to provide contributions to the increasing trend in using CSIs in PV systems, which result in increased power density, lower costs, and increased lifetime of the inverter.
A state of the art for CSI control for PV applications was provided while emphasizing the advantages of single state topologies against two stage topologies.
The equations of the dynamics of the PV power and the AC reactive and active power were derived and provided the basis on which the DPPC was designed.
The synthesized DPPC directly tracks the power references based on the prediction of the PV power and on the AC active and reactive power. A quadratic error cost function was proposed with separable parameterizable weights for the PV power and the grid reactive power.
Simulations and experimental validation were obtained for steps in both the active and reactive power, showing the ability of the controller to track both references separately while maintaining values of THD within standards regulations.
The tracking performance was tested experimentally by submitting the controller to a typical power profile where the converter was able to track the maximum power along the profile while showing the ability to control the reactive power fully decoupled from the active power reference.