Fractional-Order Control of Grid-Connected Photovoltaic System Based on Synergetic and Sliding Mode Controllers

: Starting with the problem of connecting the photovoltaic (PV) system to the main grid, this article presents the control of a grid-connected PV system using fractional-order (FO) sliding mode control (SMC) and FO-synergetic controllers. The article presents the mathematical model of a PV system connected to the main grid together with the chain of intermediate elements and their control systems. To obtain a control system with superior performance, the robustness and superior performance of an SMC-type controller for the control of the u dc voltage in the DC intermediate circuit are combined with the advantages provided by the ﬂexibility of using synergetic control for the control of currents i d and i q . In addition, these control techniques are suitable for the control of nonlinear systems, and it is not necessary to linearize the controlled system around a static operating point; thus, the control system achieved is robust to parametric variations and provides the required static and dynamic performance. Further, by approaching the synthesis of these controllers using the fractional calculus for integration operators and differentiation operators, this article proposes a control system based on an FO-SMC controller combined with FO-synergetic controllers. The validation of the synthesis of the proposed control system is achieved through numerical simulations performed in Matlab/Simulink and by comparing it with a benchmark for the control of a grid-connected PV system implemented in Matlab/Simulink. Superior results of the proposed control system are obtained compared to other types of control algorithms.


Introduction
It is a well-known fact that it is important to use, to an increasing extent, renewable energy, characterized by the fact that it is generated from easily renewable sources which can thus be considered unlimited energy. Among these types of renewable energy sources, we refer to solar energy, wind energy, water energy, geothermal energy, etc. There is also a strong upward trend in the study and use of the PV system technology. Obviously, the study of its control systems was also developed in parallel with it [1].
Moreover, among the general approaches to the study of microgrid systems control, we can mention the study of the transient stability of a hybrid microgrid [2,3], the use of algorithms to offset lagging in the microgrid system [4], the optimization of the charging process of microgrid batteries [5,6], the study of the topologies of the converters used in the microgrid [7], the parallel coupling of the inverters in a microgrid [8], problems related to fault tolerance in the microgrid [9], and also problems related to the multi-grid dispatching in view of obtaining an economic optimum [10][11][12][13][14].
An inherent problem that arises is the control of the process of the PV system connection to a main grid. The components which ensure the connection of the PV system to the main grid are the DC-DC boost converter, the DC intermediate circuit, the DC-AC converter, the filtering block, and the transformer for the connection to the main grid, together with

•
We propose the cascade structure of the control system of the grid-connected PV system based on the robustness of an SMC-type controller for the control of u dc voltage in the DC intermediate circuit, combined with the flexibility of using synergetic control for the control of currents i d and i q . • The synthesis of the control laws by SMC-and synergetic-type controllers using the fractional calculus for integration operators and differentiation operators.

•
We realized the numerical simulations in Matlab/Simulink, and by comparing them with a benchmark for the control of a grid-connected PV system implemented in Matlab/Simulink, superior results of the proposed control system compared to other types of control algorithms are presented.
The other sections of the paper are structured as follows: Section 2 presents a mathematical model of the grid-connected PV system. Section 3 presents the control of the grid-connected PV system using the FO calculus for the SMC controller and synergetic controllers. Section 4 presents the numerical simulations in Matlab/Simulink for the control of the grid-connected PV system using FO-SMC and FO-synergetic controllers and the analysis thereof, and some conclusions are presented in Section 5.

Mathematical Model of the Grid-Connected PV System
Following [15,27,31], which show the mathematical model of a grid-connected PV system, Figure 1 shows the general diagram of such a system. The PV array system is modeled in [15,27,31] and the input parameters are radiation and temperature. The list of component blocks additionally includes a DC boost converter along with the maximum power point tracking (MPPT) module and a three-phase DC-AC converter. The notations in Figure 1 are the usual ones. • We propose the cascade structure of the control system of the grid-connected PV system based on the robustness of an SMC-type controller for the control of udc voltage in the DC intermediate circuit, combined with the flexibility of using synergetic control for the control of currents id and iq. • The synthesis of the control laws by SMC-and synergetic-type controllers using the fractional calculus for integration operators and differentiation operators. • We realized the numerical simulations in Matlab/Simulink, and by comparing them with a benchmark for the control of a grid-connected PV system implemented in Matlab/Simulink, superior results of the proposed control system compared to other types of control algorithms are presented.
The other sections of the paper are structured as follows: Section 2 presents a mathematical model of the grid-connected PV system. Section 3 presents the control of the grid-connected PV system using the FO calculus for the SMC controller and synergetic controllers. Section 4 presents the numerical simulations in Matlab/Simulink for the control of the grid-connected PV system using FO-SMC and FO-synergetic controllers and the analysis thereof, and some conclusions are presented in Section 5.

Mathematical Model of the Grid-Connected PV System
Following [15,27,31], which show the mathematical model of a grid-connected PV system, Figure 1 shows the general diagram of such a system. The PV array system is modeled in [15,27,31] and the input parameters are radiation and temperature. The list of component blocks additionally includes a DC boost converter along with the maximum power point tracking (MPPT) module and a three-phase DC-AC converter. The notations in Figure 1 are the usual ones. Thus, the following equations can be written to describe the operation of the grid-connected PV system:  Thus, the following equations can be written to describe the operation of the gridconnected PV system: where u abc represents the output voltage of the DC-AC (voltage source converter-VSC) converter with u abc = u a u b u c T , e abc represents the grid voltage with e abc = e a e b e c T , and i abc represents the alternating current with i abc = i a i b i c T .
By applying this transformation to the abc reference system, we obtain the usual quantities in Figure 1 in the dq reference system (u dq0 = Pu abc , e dq0 = Pe abc , i dq0 = Pi abc ). Thus, Equation (4) becomes By components, this equation can be rewritten in the form of Equations (7) and (8): where u id and u iq represent the control variables for the command of the DC-AC converter (which is of voltage source converter-VSC-type). Equations (7) and (8) include the following notations: The MPPT algorithm is presented in [15,26,29]; it will provide the duty cycle signal (D) to control the DC boost converter. Thus, the following relations can be written:

Control of the Grid-Connected PV System
The control of a grid-connected PV system is presented at length in [15,27,31], both in normal operation and in low-voltage ride through (LVRT). The controllers used for the inner control loops of currents i d and i q are of the classic PI type or synergetic type [27], while the controller for the outer control loop of u dc is of the classic PI type [15,31]. Figure 2 shows the control scheme for the connection of a PV system to the power grid under normal operation. The control loops of currents i d and i q and of u dc are presented schematically in Figure 3. In this section, we will present several basic elements of the FO calculus to synthesize the fractional-type control laws in the case of using the design and synthesis procedures of the SMC and synergetic controllers.

Notions and Notations for Fractional-Order Calculus
To achieve a refinement of the differential and integral calculus, the non-integer order operator is added as aD α t , where the FO is noted with α, and the limits of the use of the operator are denoted a and t [28,29].

Notions and Notations for Fractional-Order Calculus
To achieve a refinement of the differential and integral calculus, the non-integer order operator is added as

Notions and Notations for Fractional-Order Calculus
To achieve a refinement of the differential and integral calculus, the non-integer order operator is added as  Further, a common alternative definition is given by the Riemann-Liouville differintegral [28,29]: where m − 1 < α < m, m ∈ N, and Γ(·) is Euler's gamma function. For the practical implementation by numerical calculation, the Grünwald-Letnikov definition is presented as follows [28,29]: where (·) is the integer part.
The Laplace transform can also be applied in the non-integer case similarly to the integer case (in terms of the power of the complex operator s). A special case is when the power α of operator s is of the commensurate order type q, (q ∈ R + , 0 < q < 1, α k = kq). For λ = s q , the transfer function H(λ) can be written as For the numerical implementations in embedded systems in real time, it is important to emphasize that the results of the fractional calculus cannot be implemented directly, but an integer-order approximation of these calculi is used on a specified frequency range (ω b , ω h ), by using Oustaloup recursive filters.
For s γ with 0 < γ < 1, an approximation can be used as follows [28,29]: where ω k , ω k , and K are given by A refined form of Oustaloup-type filters is given by the following relations [28]: ; where usually parameters b = 10 and d = 9.

Fractional-Order Sliding Mode Control
Starting from Equation (3) and using S a , S b , and S c to denote the switching function for the DC-AC converter in Figure 1, the following equation is obtained in the abc frame: The switching functions Sd and S q can be obtained by using transformation (5): Based on these, Equation (19) becomes In Equation (21), i dc1 is given by the relations (9) and (10) which depend on the DC boost converter and the MPPT algorithm, which we will consider as an optimized form given by [15,31], so we will consider that it is necessary for the other terms of the right member to be calculated by the sliding mode control technique to maintain u dc at a prescribed value u dcref (which is considered constant or has slow variations relative to the variation of the other quantities in the control system). Additionally, in [26], it is demonstrated that, when the three-phase grid system is symmetrical, i d represents the direct current and reference i qref = 0 is selected, and thus Equation (21) becomes: Following the sliding mode control design procedure, the reference i dref for the inner control loop of currents i d and i q will be determined. Thus, we define the state variable x 1 as We define the switching surface S: where the state variable x 2 is defined by: To achieve convergence, the following is required: where ε and k are positive constants. From the calculation, we obtain: ..
and thus the following can be written: Following [32], to improve convergence and reduce high-frequency oscillations, the sgn function is replaced with the function below: For a = 4 and b = 0, h ∈ [−1 1], and a smoothed transition is achieved between −1 and 1. From this, the output of the SMC-type controller can be inferred: For the fractional case, the switching surface is defined as: where the fractional differential operator D is defined in relation (11). By calculating . S, we obtain: which can be rewritten using Equation (27): Using Equation (26), we obtain: By applying operator D 1−µ to relation (34), we obtain: The output of the FO-SMC type controller can thus be inferred from the following: Both in Equation (30) and in Equation (36), in order to avoid the uncontrolled increase in i dref due to the zero-crossings of the signal S d , it will be replaced in the practical implementation with S d = S d + c 3 , where c 3 > 0 becomes a new level of freedom in the design of the FO-SMC controller.

Fractional-Order Synergetic Control
It is well known that synergetic control can be considered as a generalization of sliding mode control. Thus, synergetic control can be applied to nonlinear systems described by the general form [27,32] .
where x represents the state vector, x ∈ n ; f (.) represents the continuous nonlinear function; u represents the control vector, u ∈ m , (m < n).
The synergetic control procedure includes the selection of a macrovariable ψ(x, t) which depends on the states of the system, for each control input. The system will be forced to evolve to manifolds ψ = 0, according to the following equation: where T > 0 is selected to obtain the desired convergence rate. By differentiating the macrovariable Ψ, we obtain: and by inserting the relation (39) into Equation (38), we obtain: The explicit forms of .
x states in the mathematical model of the controlled system are inserted into Equation (40). This results in the control law as follows: Next, we will apply the integer-order and fractional-order synergetic control procedures to replace the classic PI-type control loops of currents i d and i q . The outputs of the synergetic controller will be u d and u q (see Figure 2).
For the d-axis, for k d > 0, we choose the macrovariable Ψ d as follows: We define another state variable x 2 (in addition to the state variable x 1 in Equation (23)): Based on the relation (43) for slow variations of the reference quantities or quasistationary regime, the following relation can be written: Using these, we obtain the macrovariable derivative Ψ d defined in relation (42), of the following form: Based on these, for T = T 1 , Equation (40) becomes: Using Equation (7), Equation (46) becomes: After rearranging the terms in Equation (47), we can write: Based on this, we obtain the control law u d as follows: For axis d in the fractional case, the macrovariable is chosen: By deriving Equation (50), we obtain: .
Based on these, Equation (40) becomes Using Equation (7), Equation (52) becomes: After rearranging the terms in Equation (53), we can write: Based on this, we obtain the control law u d as follows: For the q-axis, for k q > 0, we choose the macrovariable Ψ q of the following form: We define another state variable x 3 (in addition to the state variables x 1 and x 2 in Equation (43)): Based on relation (57), because i qref is set to zero, we can write: Using these, we obtain the macrovariable derivative Ψ q defined in relation (56), of the following form: Based on these, for T = T 2 , Equation (40) becomes: Using Equation (8), Equation (60) becomes: After rearranging the terms in Equation (61), we can write: Based on this, we obtain the control law u q as follows: For the q-axis in the fractional case, the macrovariable is chosen: Based on these, Equation (40) becomes: By using Equation (8) and applying the fractional operator defined in relation (11) to both members of Equation (66), and considering that D −µ becomes I µ , we can write: After rearranging the terms in Equation (67), we can write: Based on this, we obtain the control law u q as follows: In the case of integer-order synergetic control, the control inputs u d and u q are provided by Equations (49) and (63), and in the case of the fractional synergetic control, the control inputs u d and u q are provided by Equations (55) and (69). By applying the inverse Park transform, the actual control inputs u abc (see Figure 2) are obtained as follows:

Numerical Simulations and Analysis for the Control of the Grid-Connected PV System Using FO-SMC and FO-Synergetic Controllers
In this section, starting from the controllers synthesized in the previous section, we will present the obtained results of the global system for the control of the grid-connected PV system, in which classic PI, synergetic, or FO-synergetic controllers are used for the inner control loops of currents i d and i q and classic PI, SMC, or FO-SMC controllers are used for the outer control loop of u dc . Owing to the levels of freedom and the refinement brought by the fractional calculus for the SMC and synergetic algorithms, it will be demonstrated, through numerical simulations, using the Matlab/Simulink environment, that superior performance is achieved using the proposed control system. The system described in Figures 2 and 3 is implemented in Matlab/Simulink and the block diagram is presented in Figure 4.
The presented implementation starts from an example in Matlab/Simulink [15], which presents the control system and the performances for a 100 kW model of the grid-connected PV array. The reference value of the voltage in the DC intermediate circuit is set to 500 V and the rated AC voltage supplied by the DC-AC converter is of 260 V. Further, the load is connected to the main grid through a distribution transformer with a rated voltage of 25 kV/260 V. The MPPT algorithm and its performances are presented and implemented in [15,31] and are used in the implementation presented in this article as a block function.
inner control loops of currents id and iq and classic PI, SMC, or FO-SMC controllers are used for the outer control loop of udc. Owing to the levels of freedom and the refinement brought by the fractional calculus for the SMC and synergetic algorithms, it will be demonstrated, through numerical simulations, using the Matlab/Simulink environment, that superior performance is achieved using the proposed control system. The system described in Figures 2 and 3 is implemented in Matlab/Simulink and the block diagram is presented in Figure 4. The presented implementation starts from an example in Matlab/Simulink [15], which presents the control system and the performances for a 100 kW model of the grid-connected PV array. The reference value of the voltage in the DC intermediate circuit is set to 500 V and the rated AC voltage supplied by the DC-AC converter is of 260 V. Further, the load is connected to the main grid through a distribution transformer with a rated voltage of 25 kV/260 V. The MPPT algorithm and its performances are presented and implemented in [15,31] and are used in the implementation presented in this article as a block function.
In order to have the smallest possible fluctuations when the DC-AC converter supplies a variable load, the importance of the precise control of the voltage level in the DC intermediate circuit-udc-is well known. For this, two cascade control loops are used, an outer one for the control of udc and two inner loops for the control of currents id and iq. The current reference and idref are supplied by the output of the outer loop controller, and iqref is set to zero [26]. Figures 5 and 6 show the Matlab/Simulink implementations of the control laws synthesized in Section 3 for the most complex case, where the controller of the outer control loop of voltage udc is of the FO-SMC type, and the inner control loops of currents id and iq are of the FO-synergetic type. Moreover, in Figure 4, the signal filtering at the In order to have the smallest possible fluctuations when the DC-AC converter supplies a variable load, the importance of the precise control of the voltage level in the DC intermediate circuit-u dc -is well known. For this, two cascade control loops are used, an outer one for the control of u dc and two inner loops for the control of currents i d and i q . The current reference and i dref are supplied by the output of the outer loop controller, and i qref is set to zero [26]. Figures 5 and 6 show the Matlab/Simulink implementations of the control laws synthesized in Section 3 for the most complex case, where the controller of the outer control loop of voltage u dc is of the FO-SMC type, and the inner control loops of currents i d and i q are of the FO-synergetic type. Moreover, in Figure 4, the signal filtering at the DC-AC converter output is achieved using a 10 kvar bank capacitor, which can be considered as the load for the control system.
The operation of the PV array is also implemented in [15], and the time variation of the irradiance and temperature input signals is shown in Figure 7. The PV array includes 330 SunPower-type modules which can supply a maximum of 100.7 kW (305.2 W/modules), and each module is characterized by an open-circuit voltage of V oc = 64.2 V and a shortcircuit current of I sc = 5.96 A. In the Matlab/Simulink implementation, the sample time used is of one microsecond for the Pulse Width Modulation (PWM) generator command signals for the DC-DC and DC-AC converters. For the control system of the voltage and currents, but also for the PLL-type synchronization loop, the sampling period is of 0.1 ms.
In the Matlab/Simulink implementation in [15], for the first 50 ms, the operation of the converters is bypassed during the period when the control system operates in the open loop. After the first 50 ms, the controllers come into operation both for the DC-DC converter and for the MPPT algorithm provided by [31], but also for the control of the DC-AC converter whose improved FO-SMC and FO-synergetic controllers proposed in this article provide superior performance compared to the classical PI controllers proposed in [15], both in stationary mode and in dynamic mode (see Figures 8-11).
330 SunPower-type modules which can supply a maximum of 100.7 kW (305.2 W/modules), and each module is characterized by an open-circuit voltage of Voc = 64.2 V and a short-circuit current of Isc = 5.96 A. In the Matlab/Simulink implementation, the sample time used is of one microsecond for the Pulse Width Modulation (PWM) generator command signals for the DC-DC and DC-AC converters. For the control system of the voltage and currents, but also for the PLL-type synchronization loop, the sampling period is of 0.1 ms.       In the Matlab/Simulink implementation in [15], for the first 50 ms, the operation of the converters is bypassed during the period when the control system operates in the open loop. After the first 50 ms, the controllers come into operation both for the DC-DC converter and for the MPPT algorithm provided by [31], but also for the control of the DC-AC converter whose improved FO-SMC and FO-synergetic controllers proposed in this article provide superior performance compared to the classical PI controllers proposed in [15], both in stationary mode and in dynamic mode (see . Figure 8a shows the response of the FO-SMC type control system for the control of the DC voltage udc combined with the FO-synergetic type control system for the control of currents id and iq (FO-SMC/FO-SYN controllers), if the DC voltage reference udcref = 500 V, and Figure 8b shows the response of the control system where the controllers used for both the control of the DC voltage udc and for the control of currents id and iq are of the PI type. After the validation of the control system start-up (after 50 ms), the MPPT algorithm start-up occurs (after 100 ms), and the end of the transitory regime (after 250 ms) is noted. In steady state, a steady-state error of 0.1 V, i.e., 0.02%, is noted for the FO-SMC/FO-SYN controllers, while the steady-state error for the PI controller is of 1 V, i.e., 0.2%. If the load is varied by a 30% increase or decrease, reaching 13 or 7 kvar, respectively, the superiority of the control of the grid-connected PV system based on FO-SMC/FO-SYN controllers is noted in Figures 9 and 10. (a) Regarding the dynamic regime, Figure 11 shows the response of the control of the grid-connected PV system based on FO-SMC/FO-SYN controllers as compared to PI controllers, where at time t = 1 s, the reference signal udcref undergoes a step variation at 550 V.
The parameters of the PI controller for udc control are Kp = 7 and Ki = 800 and the parameters of the PI controller for id and iq currents are Kp = 0.3 and Ki = 20 [15].
It is noted that the performances in the dynamic regime, as well as those in the stationary regime, are superior in the case of using FO-SMC/FO-SYN controllers, and in Figure 11, an override of 0.2% (1 V) and a response time of 20 ms are noted, compared to an override of 1% (5 V) and a response time of 50 ms in the case of PI controllers.  Figure 8a shows the response of the FO-SMC type control system for the control of the DC voltage u dc combined with the FO-synergetic type control system for the control of currents i d and i q (FO-SMC/FO-SYN controllers), if the DC voltage reference u dcref = 500 V, and Figure 8b shows the response of the control system where the controllers used for both the control of the DC voltage u dc and for the control of currents i d and i q are of the PI type. After the validation of the control system start-up (after 50 ms), the MPPT algorithm start-up occurs (after 100 ms), and the end of the transitory regime (after 250 ms) is noted. In steady state, a steady-state error of 0.1 V, i.e., 0.02%, is noted for the FO-SMC/FO-SYN controllers, while the steady-state error for the PI controller is of 1 V, i.e., 0.2%. If the load is varied by a 30% increase or decrease, reaching 13 or 7 kvar, respectively, the superiority of the control of the grid-connected PV system based on FO-SMC/FO-SYN controllers is noted in Figures 9 and 10.
Regarding the dynamic regime, Figure 11 shows the response of the control of the gridconnected PV system based on FO-SMC/FO-SYN controllers as compared to PI controllers, where at time t = 1 s, the reference signal u dcref undergoes a step variation at 550 V.
The parameters of the PI controller for u dc control are K p = 7 and K i = 800 and the parameters of the PI controller for i d and i q currents are K p = 0.3 and K i = 20 [15].
It is noted that the performances in the dynamic regime, as well as those in the stationary regime, are superior in the case of using FO-SMC/FO-SYN controllers, and in Figure 11, an override of 0.2% (1 V) and a response time of 20 ms are noted, compared to an override of 1% (5 V) and a response time of 50 ms in the case of PI controllers.  Figures 12-16 show a series of waveform graphs regarding the time evolution of the main inputs of interest in the control of the grid-connected PV system. Thus, Figure 12 shows the time evolution of id and iq currents, where it is noted that id follows the idref reference provided by the FO-SMC controller, while iq follows the set reference iqref = 0. Figure  13 shows the evolution of the average power and voltage of the PV array, Pmean and Umean.  Figures 12-16 show a series of waveform graphs regarding the time evolution of the main inputs of interest in the control of the grid-connected PV system. Thus, Figure 12 shows the time evolution of i d and i q currents, where it is noted that i d follows the i dref reference provided by the FO-SMC controller, while i q follows the set reference i qref = 0. Figure 13 shows the evolution of the average power and voltage of the PV array, P mean and U mean . Further, with regard to the DC-DC converter, Figure 13 presents the time evolution of the duty cycle signal provided by the MPPT algorithm, and with regard to the DC-AC converter, it presents the time evolution of the modulation index, a signal which is supplied by the control system of the VSC controller, which supplies control pulses. Figure 14 shows the evolution over time of the output voltage between two phases of the DC-AC converter. Figure 15 shows the time evolution of the voltage and current on a phase of the transformer for the connection to the main grid. Further, with regard to the DC-DC converter, Figure 13 presents the time evolution of the duty cycle signal provided by the MPPT algorithm, and with regard to the DC-AC converter, it presents the time evolution of the modulation index, a signal which is supplied by the control system of the VSC controller, which supplies control pulses. Figure 14 shows the evolution over time of the output voltage between two phases of the DC-AC converter. Figure 15 shows the time evolution of the voltage and current on a phase of the transformer for the connection to the main grid. Moreover, Figure 16 shows the time evolution of the active power which flows between the analyzed system and the main grid.   Further, with regard to the DC-DC converter, Figure 13 presents the time evolution of the duty cycle signal provided by the MPPT algorithm, and with regard to the DC-AC converter, it presents the time evolution of the modulation index, a signal which is supplied by the control system of the VSC controller, which supplies control pulses. Figure 14 shows the evolution over time of the output voltage between two phases of the DC-AC converter. Figure 15 shows the time evolution of the voltage and current on a phase of the transformer for the connection to the main grid. Moreover, Figure 16 shows the time evolution of the active power which flows between the analyzed system and the main grid.       Moreover, Figure 16 shows the time evolution of the active power which flows between the analyzed system and the main grid.
The control of the grid-connected PV system implemented in [15] is also discussed in [26], in which the control system of u dc voltage is of the SMC type, the control systems of currents i d and i q are of the classic PI type, and the time evolution of the irradiance and temperature signals is presented in Figure 17. Due to the fact that the basic model in Matlab/Simulink is complex and has all the aspects regarding the transformation chain from the PV array to the main grid connection and considering that it is used for comparison in other papers [15,26,27,31], this model can be considered as a benchmark for the control of the grid-connected PV system. Thus, the superior performance obtained by using the FO-SMC/FO-SYN controllers proposed in this article can be considered as a validation of the proposed control system. The control of the grid-connected PV system implemented in [15] is also discussed in [26], in which the control system of udc voltage is of the SMC type, the control systems of currents id and iq are of the classic PI type, and the time evolution of the irradiance and temperature signals is presented in Figure 17.    The control of the grid-connected PV system implemented in [15] is also discussed in [26], in which the control system of udc voltage is of the SMC type, the control systems of currents id and iq are of the classic PI type, and the time evolution of the irradiance and temperature signals is presented in Figure 17.        Due to the fact that the basic model in Matlab/Simulink is complex and has all the aspects regarding the transformation chain from the PV array to the main grid connection and considering that it is used for comparison in other papers [15,26,27,31], this model can be considered as a benchmark for the control of the grid-connected PV system. Thus, the superior performance obtained by using the FO-SMC/FO-SYN controllers proposed in this article can be considered as a validation of the proposed control system.

Conclusions
This article presents the control of a grid-connected PV system using FO-SMC and FO-synergetic controllers. The mathematical model of a PV system connected to the main grid is presented together with the chain of intermediate elements: the DC-DC boost converter, the DC intermediate circuit, the DC-AC converter, the filtering block, and the transformer for the connection to the main grid, together with their control systems. The robustness and superior performance of an SMC-type controller for the control of udc voltage in the DC intermediate circuit are combined with the advantages provided by the flexibility of using synergetic control for the control of currents id and iq. In addition, these control techniques are suitable for the control of nonlinear systems, and it is not necessary to linearize the controlled system around a static operating point; thus, the control system achieved is robust to parametric variations and provides the required static and dynamic performance. Further, by approaching the synthesis of these controllers using the fractional calculus for integration and differentiation operators, this article proposes a control system based on FO-SMC/FO-SYN controllers. The validation of the synthesis of the proposed control system is achieved by comparing it with a benchmark for the control of a grid-connected PV system implemented in Matlab/Simulink.

Conclusions
This article presents the control of a grid-connected PV system using FO-SMC and FOsynergetic controllers. The mathematical model of a PV system connected to the main grid is presented together with the chain of intermediate elements: the DC-DC boost converter, the DC intermediate circuit, the DC-AC converter, the filtering block, and the transformer for the connection to the main grid, together with their control systems. The robustness and superior performance of an SMC-type controller for the control of u dc voltage in the DC intermediate circuit are combined with the advantages provided by the flexibility of using synergetic control for the control of currents i d and i q . In addition, these control techniques are suitable for the control of nonlinear systems, and it is not necessary to linearize the controlled system around a static operating point; thus, the control system achieved is robust to parametric variations and provides the required static and dynamic performance. Further, by approaching the synthesis of these controllers using the fractional calculus for integration and differentiation operators, this article proposes a control system based on FO-SMC/FO-SYN controllers. The validation of the synthesis of the proposed control system is achieved by comparing it with a benchmark for the control of a grid-connected PV system implemented in Matlab/Simulink.