Nonlinear control for an optimized grid connection system of renewable energy resources

Received Oct 20, 2020 Revised Mar 2, 2021 Accepted Aug 9, 2021 This paper proposes an integral backstepping based nonlinear control strategy for a grid connected wind-photovoltaic hybrid system. The proposed control strategy aims at extracting the maximum power available while respecting the grid connection standards. The proposed system has a reduced number of power electronic converters, thereby ensuring lower costs and reduced energy losses, which improves the profitability and efficiency of the hybrid system. The effectiveness of the proposed topology and control methodology is validated using the MATLAB/Simulink software environment. The satisfactory results achieved under various atmospheric conditions and in different operating modes of the hybrid system, confirm the high efficiency of the proposed control strategy.


INTRODUCTION
Growing international concerns about climate change and the ongoing dangers associated with the exploitation of nuclear power generation technology are leading to new advances in electricity generation from renewable resources. Due to their environmental friendliness and sustainability, wind and solar energies have become widespread energy resources. But these resources are intermittent in nature, making it difficult to ensure a stable and continuous energy supply. The problem can be overcome by efficiently integrating local energy storage elements, but their limited life time adds an additional ongoing production cost.
Hybridization of renewable resources, especially if they are complementary in terms of availability, is an adequate solution to the problem of intermittency. Hybridization also reduces the number of power converters, which are typically dedicated to each resource, thus allowing efficient use of the installed converters. In addition, direct grid connection, whilst providing an intelligent energy management system to match production to use, is an appropriate approach to minimize or eliminate storage devices altogether.
A considerable academic literature on renewable power generation (RPG) is mainly devoted to their sizing, reliability, cost analysis, and energy management [1]- [6], whilst others contribute to their modelling and control techniques [7]- [9]. A maximum power point tracking (MPPT) controller is required to make the best use of the available energy. To this end, intensive work has been carried out on MPPT control of photovoltaic and wind energy systems [10]- [15], but most of proposed control methods are based on conventional methods, such as perturbation and observation or incremental conductance algorithms and proportional-integral (PI) controllers. In fact, these systems are nonlinear and the PI controller is designed for linear systems. The high performance of nonlinear controllers compared to conventional controllers has already been reported by several comparative studies [16], [17], and various nonlinear controllers have been proposed as efficient alternatives to conventional MPPT controls [18]- [22]. On another note, grid codes, such as the FERC standard interconnection agreements for wind and other alternative technologies (order No. 661-A), require that a power factor greater than 0.95 be maintained at the point of interconnection. For this purpose, power factor control for grid-connected RPG systems is of paramount importance [23]- [25].
Against this background, the present work proposes an efficient nonlinear control of a low-cost gridconnected PV-wind hybrid configuration. This paper assumes that the grid connection process, as detailed in [26], [27], has already been done. The proposed hybrid system components and their models are presented in the next section, and on the basis of these models nonlinear control laws are developed in the third section. The results of the simulations undertaken to validate and evaluate the proposed control strategy are presented in the fourth section, and a brief conclusion is given in the last section.

SYSTEM ELEMENTS AND THEIR MODELS
The proposed hybrid system configuration is illustrated in Figure 1. Using common electronic power converters, this configuration offers considerable savings in initial and operating costs (less energy loss). In fact, DC/DC and DC/AC converters dedicated to the conversion of photovoltaic (PV) energy are eliminated and the double fed induction generator (DFIG) converters will take care of the conversion of PV energy.

Irc
Lf GSC  (1) where: Cp is the wind turbine power coefficient; is the air density; R is the blade radius (in metres); is the wind speed (in m/s); Ω is the turbine angular speed (in rad/s); λ is the tip speed ratio and β is the blade pitch angle (in degrees). With cl=0.5176; c2=116; c3=0.4; c4=5; c5=21 and c6=0.0068. Figure 2 shows the characteristics of the wind turbine's power coefficient at different values of the blade pitch angle. The pitch control protects the turbine against turbulence and excessive overload, under normal conditions β=0.

Photovoltaic energy conversion
The mathematical expressions for the single-diode equivalent circuit of the PV cell are given by [19]: where: is the current crossing the PV cell; is the PV cell voltage; ℎ is the photocurrent; , and are the cell saturation current, the reverse saturation current and the short-circuits current; q is the electron charge; ℎ and are the intrinsic parallel and series resistors; K is the Boltzmann constant; is the diode ideality factor; E is the solar irradiance; is the reference irradiance (1 kW/m 2 ); T is the temperature on absolute scale; is the reference temperature (298,15K); is the short-circuit current temperature coefficient and 0 is the band-gap energy of the cell. Therefore, with a photovoltaic generator (PVG) consisting of Np strings in parallel, each string consists of Ns cells in series, the expression of the PVG current ( ) as a function of its voltage ( ) can be derived as follows: In this paper, a PVG made up of seventeen SM55 panels connected in series is considered. Electrical specifications for one panel are given in [21]. The Power-Voltage characteristics of the PVG under solar irradiance change are shown in Figure 3. The coordinates of the maximum power points shown in the zoomed-in parts of Figures 2 and 3 will be used to verify the simulation results.

State space representation
All measured 3-phase quantities are transformed into the stator-flux oriented dq-reference frame as shown in Figure 1. The DC-bus voltage, which is also the PVG voltage ( ), is governed by the following equation: where is the DC-bus capacitor; is the DC-current absorbed by the rotor side converter (RSC); and are the d-axis and q-axis components of the current absorbed by the grid side converter (GSC); and are the GSC control signal components in d-q frame.
In the synchronous dq-frame, the stator flux vector is aligned on the d-axis. Assuming that the stator flux magnitude is constant and the small drop in the stator resistance voltage is negligible, the stator voltage vector is also considered practically aligned on the q-axis of the dq-frame [27]- [29]. According to (4) and [18] an overall state space representation of the hybrid system under consideration is given by:

Tip speed ratio
Power coefficient where and are the rotor current components; and are the stator and rotor inductances; is the resistance of rotor windings; ( ) are the angular position of the d-q frame with respect to the reference frame attached to the stator (rotor); is the stator-rotor magnetizing inductance; is the leakage coefficient ( = 1 − 2 ⁄ ); and are the RSC control signal components; and are the resistance and the inductance of the L-filter; is the q-axis component of the stator voltage; is the quadrature component of the transformer (Tr) secondary voltage, see Figure 1, with = ; is the transformation ratio of Tr. The electromagnetic torque is expressed in the synchronous frame as: where is the number of pole pairs per phase in the DFIG stator windings; is the d-axis component of the stator flux.

DESIGN OF HYBRID SYSTEM CONTROLLERS
The grid voltage is assumed to be stable, and the stator flux is estimated as: The angular frequency ( ) and the phase angle ( ) of the synchronous d-q frame are generated using a three-phase phase-locked loop (PLL), as shown in Figure 4 [18], [21]. The RSC controller is designed to track the maximum power point (MPP) of the wind turbine and to inject the stator power with near unity power factor (UPF). The GSC controller is designed to track the MPP of the PVG and to maintain the reactive power injected by the GSC close to zero in Figure 1.

Designation of the hybrid system outputs and their references
The DFIG rotor dynamics obey Newton's second law for rotational dynamics, thus: where, and are the total inertia and total viscosity coefficient (turbine and DFIG), / is the aerodynamic torque applied to DFIG rotor, is the algebraic value of the DFIG electromagnetic torque and Ω its rotor speed. Operating the wind turbine at the maximum power point (i.e. λ=λopt and Cp=Cp_max) means that: Then, the optimal aerodynamic torque, / _ , is such that: 3 ; is the gearbox ratio and Ω is the DFIG rotor speed. The optimal electromagnetic torque ( _ ) and then the reference of the first output (q-axis rotor current; ) are derived as: (8) The reactive powers are expressed as a function of and (second and third outputs) as [18]: where is the reactive power injected by the DFIG stator; is the reactive powers injected by the GSC. In order to operate the hybrid system at or near UPF, the references of the direct currents are derived as: Since the PVG characteristics have a single extremum, as shown in Figure 3, the derivative of the PVG power with respect to its voltage ( ) was chosen as the fourth output of the system with a reference value of zero:

Elaboration of RSC control laws
Let us define the first error 1 and a first lyapunov function candidate (LFC) 1 as: where 1 = 1 ∫ 1 ( ) 0 , 1 is the design parameter of the integral action. 1 time-derivative is as: The q-axis component of RSC control signal, , is chosen as: where 1 is a strictly positive design parameter. 1 time derivative of the closed-loop system becomes: Similarly, the error between and its desired value, and a second LFC are defined as follows: where 2 = 2 ∫ 2 ( ) 0 , 2 is the design parameter of the integral action. 2 time-derivative is as: Then, the d-axis component of RSC control signal is chosen such that: where 2 is a strictly positive design parameter. With the above choice, 2 time derivative becomes:

Elaboration of GSC control laws
The error 3 , between and its desired value, and a third LFC are defined as: where 3 = 3 ∫ 3 ( ) 0 ; 3 is the design parameter of the integral action. So, 3 time-derivative is as: Then, the d-axis component of GSC control signal is chosen as: where: 1 = ; 2 = 0; 1 = 3 3 + 3 3 + − ; with 3 is a strictly positive design parameter.
The above choice guarantees the closed-loop negativity of the 3 time-derivative, which would become: The PV-MPPT error 4 and a fourth LFC, 4 , are defined as: Then, and are chosen in such a way that they satisfy:

Overall stability analysis
Let us define an overall Lyapunov function candidate such as: According to (19) , (23), (27) and (32), time-derivative of the closed-loop system is: Thus, is a positive definite function and has a negative definite derivative, therefore the tracking errors are asymptotically stable and converge to zero in Lyapunov's approach.

SIMULATION RESULTS
The closed-loop hybrid system is modeled in MATLAB-Simulink software in order to assess the performance of the designed controllers. The main parameters used for the simulation are summarized in Table 1. Figure 5 shows Matlab/Simulink diagram for the hybrid system. The solar radiation and wind velocity profiles shown in Figure 6 are used to perform this assessment.  Table 1. Parameters of controlled system A PI controller has been proposed in [29] for controlling the same hybrid system proposed in this paper and the MPPT algorithm "perturbe and observe" (including the PVG power limitation to avoid GSC overload) was used to adjust the PVG voltage. In order to provide a good idea of the proposed control performance, the simulation results obtained using the control strategy proposed in [29] are also presented (as the algorithm for tracking the maximum power of the turbine has not been specified in [29], the q-axis rotor current reference established in this paper has been used in order to provide MPPT control of the wind turbine).  The first simulation result, Figure 7, clearly shows that the PLL has successfully met its objective (q-axis stator flux is almost zero), and that its outputs are more accurate when used with the integral backstepping control. Figure 8 shows that the d-axis component of the stator voltage (which has been neglected in the design of the controllers) is indeed practically zero; its value does not exceed 0.2V with the proposed controller and ± 0.6V with the PI controller. Figure 9 shows the tracking efficiency of the q-axis rotor current reference provided by the proposed RSC controller, which results in very good tracking performance of the optimal electromagnetic torque, as shown in Figure 10. Figure 11 shows that the power coefficient of the wind turbine has been kept close to its maximum value regardless of variations in wind speed. Figure 12 shows that the d-axis rotor current was kept close to its reference value (with a tracking error of less than 0.5A) and hence the stator reactive power was kept close to zero, as shown in Figure 13. Moreover, the proposed GSC controller has been proven more efficient in controlling the reactive power injected by the GSC close to zero, as shown in Figure 14. . q-axis rotor current Figure 10. DFIG electromagnetic torque The total injected power, shown in Figure 15, reveals that the total reactive power is near zero and the active power injected using the proposed controller is greater than that generated using the PI controller. The wind speed profile in Figure 6 was used to evaluate the hybrid system during different operating modes of the DFIG, as can be seen in Figure 16. In the sub-synchronous mode, the power of the PVG is routed through both the GSC and the RSC. Therefore, the active power injected by the GSC becomes lower than that produced by the PVG, as can be seen in the zoomed-in portion of Figure 14.  18 show that the PVG voltage, and the PV power generated correspond very precisely to the MPP coordinates of the PVG shown in Figure 3. Figure 19 illustrates that the total current injected using the proposed controller exhibits less fluctuations than that obtained using the PI controller, and that the power factor has been maintained at unity independently of the rotor current behaviour shown in Figure 20.

CONCLUSION
A nonlinear control strategy for a grid-connected PV-wind hybrid system has been presented. The main control objectives are to guarantee the maximum extraction of locally available renewable energy and to operate the grid-connected hybrid system near unity power factor. The control laws have been developed in the stator-flux oriented reference frame on the basis of Lyapunov's stability theory. The soundness of the proposed control strategy has been confirmed by numerical simulations and its performance was compared to that obtained with a previously proposed one. The results obtained highlight the performance of the proposed nonlinear controller. Future work will concentrate on experimental testing of the proposed control strategy, as well as on improving the control of other renewable energy systems.