Backstepping based power control of a three-phase single-stage grid-connected PV system

In order to reduce costs while maintaining superior performance, this paper presents a new control methodology of a three-phase grid connected photovoltaic system without using the intermediary DC/DC converter. Based on the synchronized nonlinear model of the whole photovoltaic system, two controllers have been proposed for the three-phase inverter in order to ensure the operation of the PV system at the maximum power point with unity power factor and minimum grid disturbance. Grid synchronization has been ensured by a three-phase 2 nd order PLL (Phase-Locked Loop). The stability of each controller is demonstrated by means of Lyapunov analysis and evaluated under changing atmospheric conditions using the Matlab/Simulink environment, the simulation results clearly demonstrate the performance provided by each controller.


INTRODUCTION
Nuclear energy is not likely to be a major source of world energy consumption because of the relative risks associated with unleashing the power of the atom and we have no choice but to invest heavily in renewable energy production. Solar energy is one of the most promising renewable electricity sources in the world. Moreover, the coordinated inter-connection of solar energy sources between global electricity markets can supply more flexibility and balancing to the grid. This is why the grid-connected photovoltaic systems have a great bright future.
Achieving the reduction of PV system manufacturing costs remains a major task. Among the trends to achieve such a reduction is the elimination of the DC/DC converter stage. Conventionally, the first converter stage, which is usually placed between the PV arrays and the inverter [1][2][3], achieves the MPPT whereas the inverter stage delivers and controls the energy injected into the grid. Therefore, to achieve this cost reduction, the three-phase inverter must also take care of the maximum power point tracking.
Maximum power point tracking is mandatory to maximize photovoltaic systems efficiency. To this end, several MPPT control strategies have been largely published in the last few years [1][2][3][4][5][6][7][8][9][10][11][12][13]. Perturb and observe (P&O) and incremental conductance (IC) algorithms are the most widely proposed in literature. They are the simplest algorithms to implement [3][4][5], but the dilemma -the choice between convergence speed and output fluctuations [5,6], has led in recent years to several research aimed at improving these two techniques [14][15][16][17]. The nonlinearity of photovoltaic characteristics makes the control of PV systems by conventional control strategies a complex task. However, the recent involvement of robust and nonlinear controls has enriched the field of research and has proved most appropriate for the control of nonlinear systems [7][8][9][10][11][12]. In particular, [6,18] are comparative studies between the backstepping control and other conventional controls which clearly showed the performance of the backstepping control. The photovoltaic array is a multiple associations, in series and parallel, of PV cells. The PV generator considered in this paper is composed by thirty-three SM55 Siemens panels connected in series. The electrical specifications for one panel are enlisted in Table 1. The modeling of the SM55 panel, using Matlab/Simulink, allowed tracing its characteristics for different values of irradiance and temperature which are shown in Figure 2 and Figure 3. Figure 4 shows the power-voltage characteristics of the PV generator considered in this paper under changing climatic conditions. The coordinates of the Maximum Power Points (MPP1, MPP2 and MPP3) summarized in Table 2 will be used for verification of the simulation results.    The basic power circuit of proposed single stage PV-system as shown in Figure 5 consists of a DClink capacitor which is directly connected to the three-phase inverter. Grid connection was performed through a low-pass filter used to reduce the ripple components due to the switching actions in PWM inverter. It is assumed that all three phase inverter switches are ideal, the low-pass filter phases are identical and the grid voltage is symmetric. The three-phase model is detailed in [7]. A simplified model can be obtained in the synchronous frame rotating at the angular frequency of the grid voltage. For this purpose, the powerinvariant dq-transformation, from balanced three phase electrical quantities to balanced two phase quadrature quantities, has been used: and T(θ) =√ where Ip; Vp : Are the PV array current and voltage

CONTROLLER DESIGN
The injected currents have to be synchronized with the grid voltages. To this end, the grid voltage phase angle is detected using a 2nd order phase locked loop (PLL). The structure of the PLL implemented in this work as shown in Figure 6 uses the grid voltage abc-dq transformation to track the grid voltages phase angle. A PI regulator was used to generate a corrective phase angle (est), that is fed back into the grid voltage abc-dq transformation module, from the quadrature voltage error [20]. The three-phase grid currents and voltages are transformed into direct and quadrature axis components, the controller outputs ( & ) are then transformed into three-phase components using the inverse dq-abc transformation, and then a PWM control is used to make them suitable for switching the inverter switches as shown in Figure 5. In the synchronous d-q reference frame (Eq=0), the injected powers are simplified , the active and reactive powers can be controlled by Id and Iq respectively: In order to achieve a minimum injection of reactive power the quadrature current reference must be set to zero. If we neglect ohmic loss, the power conservation principle gives: Finally, when the PV-generator power is at its maximum state, Figure 3, its derivative with respect to PV-voltage is zero. Table 3 summarizes the selected dynamic outputs and their references for each controller. where: = − is the direct current error, = − is the quadrature current error. The output chosen for the second controller is simpler than the first, but it is a challenging and time-consuming task to tune the appropriate backstepping parameters in this controller. This is largely due to the fact that establishing stability for switched systems is difficult. Therefore, in the first controller, we have chosen to also involve the direct current component Id in the control of the output power.

Formulation of MPPT control law (controller 1&2)
Let us define the first tracking error 1 and its LFC (Lyapunov Function Candidate) 1 , using (2), it is possible to deduce: The stabilizing controls and is chosen as follows: With the above choice ̇1 = − 1 1 2 becomes defined negative, ϵ 1 is proved to be asymptotically stable and converge to zero by the Lyapunov design. This means that 1 ⟶ 1 , so ⟶ ( ) .

Formulation of output power control law 3.2.1. First controller
According to Table 3 we define the error 2 and its derivative, deduced from (2), as follows: where: With the above choice ̇2 = − 1 1 2 − 2 2 2 < 0, 1 and 2 are proved to be stable and converge to zero by the Lyapunov design. This means that converge to . We are finally in a position to determine the control signals & . From (7) and (10), we deduce (11):

Second controller
The tracking error ϵ 2 is reduced to (12): 12) and the control low can be extracted directly from the second equation of mathematical model (2), by choosing K d such that: Using (7) and (13), we obtain the control signals (14):

SIMULATION RESULTS
The theoretical performances of the proposed controllers discussed in section three will be illustrated by simulation in this section. The PV system as shown in Figure 5 is simulated jointly with each controller, using the instantaneous three phase model, in Matlab/Simulink environment as shown in Figure 7. The model in d-q axis (2) is only used in the controllers design. Important simulation parameters are given in the Table 4. Controllers' parameters have been selected using a 'trial-and-error' search method. In order to prove the robustness of the control algorithms, the simulation is performed with the following scenario: a. A temperature increase from T=25°C (298.15K) to T=45°C (318.15K) after 1sec of start of simulation, then returns to T=25°C at 1.6sec, as shown in Figure 8. b. A solar irradiation drop from 1000W / m² to 800W / m² after 0.4 sec of start of simulation, then returns to 1000W / m² at 0.3 s as shown in Figure 9.  The first and second controllers are based respectively on (11) and (14), and use the same backstepping parameters as shown in Table 4. Simulation results of the PV array power and voltage under transient condition are shown in Figure 10, which correspond very well to the MPP coordinates summarized in Table 2.  Figure 11 illustrates the change of direct and quadrature current components. It can be clearly seen that the second controller can track the reference values with a fast transient response, but the first controller is the most accurate and the least disruptive of the grid. Figure 12 shows the grid voltage and the injected current obtained with each one of the proposed controllers (the grid current scale was multiplied by 10). It can easily be seen in the zoomed portions as shown in Figure 13 that the grid current is sinusoidal and in phase with the grid voltage which proving that the power factor unit is well achieved. Remark: It should be noted that it is very difficult to tune backstepping parameters of the second controller to establish the stability, unlike the first controller whose parameters influence precisely on the precision and settling time.   Figure 13. Zoomed portions of Figure 12

CONCLUSION
In this paper, two Backstepping controllers have been developed for a three phase single stage grid connected photovoltaic system, without using the conventional DC/DC converter for MPPT control, in order to ensure the operation of the PV system at maximum power point with unity power factor. From the simulation results, it has been proven that both algorithms, although they require more computation, are able to work under various levels of irradiation and temperature, it has also been deduced that the correct selection of system outputs helps to facilitate control tasks. Although the current of the first controller has fewer ripples compared to the second, the future work will deal with the mitigation of switching noise and the practical implementation.