Output feedback nonlinear control of three-phase grid-connected PV generator

Received May 16, 2018 Revised Aug 18, 2018 Accepted Sep 3, 2018 This paper addresses the problem of controlling three-phase grid connected PV system involving a PV arrays, a voltage source inverter, a grid filter and an electric grid. This paper presents three main control objectives: i) ensuring the Maximum power point tracking (MPPT) in the side of PV panels, ii) guaranteeing a power factor unit in the side of the grid, iii) and ensuring the asymptotic stability of the closed loop system. Interestingly, the present study features the achievement of the above energetic goal without resorting to sensors of currents of the grid. To this end, an output-feedback control strategy combining a state observer and a nonlinear control laws is developed. The proposed output-feedback control strategy is backed by a formal analysis showing that all control objectives are actually achieved.


INTRODUCTION
It was very clear from recent studies and documentation that fossil fuels would last only a few more decades. The cost of fossil fuels has become a major challenge for all of humanity. Not only the economic value but also the environmental impacts of fossil fuels have clearly pushed us towards alternatives. The biggest alternatives that can really make a difference to sustainability, such as reducing greenhouse gas emissions and the long-term economy, are renewable energy sources (RES) such as wind and solar energy [1], [2]. PV system is increasing as a renewable source due to its advantages of little maintenance, absence of moving mechanical parts, no noise and no pollutant emission [3]. Furthermore, its cost is decreasing over the next ten years, while the deployment of PV systems continues to increase rapidly. With increasing PV penetration on the grid, eventually reaching hundreds of gigawatts (GW) of interconnected capacity, a variety of methods must be considered and implemented at different scales for a reliable and cost-effective connection into power grid.
Different control strategy for three-phase grid connected of PV modules has been largely dealt with in the specialist literature in the last few years [4]- [9]. Nevertheless, good integration of medium or large PV system in the grid may therefore require additional functionality from the inverter, such as control of reactive power. Moreover, the increase in the average size of a PV system may lead to new strategies such as eliminating the DC-DC converter, which is usually placed between the PV generator and inverter, and moving the MPPT to the inverter, which leads to increased simplicity, overall efficiency and a cost reduction. These two features are present in the three-phase inverter that is presented in this paper, with the addition of a Perturb and Observe (P&O) MPPT algorithm. The present paper is then focusing on the problem of controlling three phase grid-connected PV power generation systems. The control objectives are threefold: (i) global asymptotic stability of the whole closed-loop control system; (ii) achievement of the MPPT for the PV array; and (iii) ensuring a grid connection with unity Power Factor (PF). These objectives should be achieved in spite the climatic variables (temperature and radiation) changes. To this end, a nonlinear controller is developed using Lyapunov design technique. A theoretical analysis is developed to show that the controller actually meets its objectives a fact that is confirmed by simulation.
The paper is organized as follows: the three-phase grid connected PV system is described and modeled in Section 2. Section 3 is devoted to controller design and analysis. The controller tracking performances are illustrated by numerical simulation in Section 4.

RESEARCH METHOD 2.1. System overview
The main circuit of three-phase grid-connected photovoltaic system as shown in Figure 1. It consists of a PV arrays; a DC link capacitor C; a three phase inverter (including six power semiconductors) that is based upon to ensure a DC-AC power conversion and unity power factor; an inductor filter L with a series resistance r, and an electric grid. The control inputs of the system are a PWM signals , and taking values in the set {0,1}. The grid voltages , and constitute a three-phase balanced system.

PV Array model
An equivalent circuit for a PV cell as shown in Figure 2. Its current characteristic can be found in many places [10]- [13] and presents the following expression.
The meaning and typical values of the parameters given by (1) and (2) can be found in many places (see e.g. [14], [15], [16]). A is diode ideal factor, k is Boltzmann constant , T is temperature on absolute scale in K, q is electron charge and λ is the radiation in kW/m 2 , Iphr is the short-circuit current at 298 K and 1 kW/m 2 , KI = 0.0017A/K is the current temperature coefficient at Iphr, EG0 is the band gap for silicon, is reference temperature, Isatr is cell saturation current at Tr. PV array consists of Ns cells in series formed the panel and of Np panels in parallel according to the rated power required. The output voltage and current can be given by the following (3) Table 3. The data in Table 1 to Table 3 will be used for simulation. Table 2 shows the main characteristics of the PV array, designed using Sharp NU-183E1 modules connected in a proper series-parallel, making up a peak installed power of 71 kW. As there is no DC-DC converter between the PV generator and the inverter, the PV array configuration should be chosen such that the output voltage of the photovoltaic generator is adapted to the requirements of the inverter. In this case a 380V grid has been chosen, so the inverter would need at least 570V DC bus in order to be able to operate correctly. The minimum number of modules connected in series should be determined by the value of the minimum DC bus voltage and the worst-case climatic conditions. The PV array was found to require 28 series connected modules per string.    Table 2 : PV Array specifications using sharp NU-183E1 Table 3. Maximum power points (MPP): in Figure 3 and Figure 4 Parameter

Modeling of three-phase grid-connected PV system
The state-space model of a three-phase grid-connected photovoltaic system shown in Figure 1 can be obtained by the dynamic equations described in (5a)-(5d): Applying the Concordia transformation to (5a-d), the instantaneous model in stationary coordinates is givens as (6a)-(6b): is given by: According to the three-phase instantaneous active and reactive power theory, the instantaneous active power P and the instantaneous reactive power Q can be derived in three-phase abc coordinates for the three-phase photovoltaic grid-connected inverter: In the αβ coordinates, the instantaneous active power P and the instantaneous reactive power Q can be expressed as following:

OUTPUT FEEDBACK CONTROLLER DESIGN
The model (6a-c) is useful to build an accurate simulator for the studied system however it is not adequate to elaborate a continuous controller as it involves a binary input  u and  u . For control design purpose, it is more convenient to consider the following averaged model, obtained by averaging the model (6a-c) over one switching period [17]. On the basis of (10a-c), the next two subsections will be devoted to the observer design and the nonlinear controller design.

Observer design
Let us now consider in (10a-c) that is the only measurable variable. Then the following nonlinear observer is proposed: Where 1 x , 2 x , 3 x represent the estimates of the state variables and 0   being the observer design parameter.
Let us introduce the estimation errors ( 1 2 c ) From the error system (12a-c), one can state the following: Proposition: Consider the estimation error system (12a-c), obtained by combining the system (10a-c) and the nonlinear observer (11a-c). Then, , whatever the initial conditions, the state estimation error  converges exponentially to zero. Proof: Consider the following quadratic Lyapunov function: Its derivative, using (12a-c), can be obtained as follows: Using Young's Inequality (14a) becomes From (19), it is easy to show that: , which in turn shows that the estimates converge toward their true values. This ends the proof of the Proposition. In the next subsection, we focus on elaborating a controller that stabilizes the system.

Nonlinear controller design
With the aim of design an appropriate control for the model (10a-c) described in previous section, the control objectives, the control design and stability analysis will be investigated in this Section, taking into account the nonlinear feature and the multi-input multi-output (MIMO) aspect of the system. In order to define the control strategy, the first step is to establish control objectives, which can be summarized as follows: a. Maximum power point tracking (MPPT) of PV arrays, b. Unity power factor (PF) in the grid, c. Asymptotic stability of the whole system.
It is worth noting that all objectives must be achieved without sensing all variables. The PV voltage 3 x is the only measurable variable. The first control objective is to enforce the real power P to track the maximum power point PM. It's already point out; that this power can be controlled by the α-axis current iα and β-axis current iβ (see (9a)). In this paper the MPPT algorithm based on the Perturb and Observe (P&O) technique [18] is resorted to generate the coefficient  (involved in (22)) so that the active power P tracks its maximum value i.e. To achieve this objectives it suffices to enforce the α-axis current  i and β-axis current  i to track reference signals, say * 1 x and * 2 x , of the following forms: With  is any real positive parameter (although transient time-variations are allowed). Its generation will be seen later using the MPPT algorithm (see section 4). Once the control objectives are clearly defined, as the MIMO system is highly nonlinear, a Lyapunov based nonlinear control is proposed [17]. Given the observer guarantees that the errors converge to zero, the following controller design will be based on the estimate variables instead of their true values. Therefore, the following errors are introduced:  ( 2 5 ) In order to achieve the objectives: MPPT and power factor unit, one can seek that the errors 4 z and 5 z are vanishing. To these ends the dynamics of 4 z and 5 z have to be clearly defined. Deriving (24) and (25), it follows from (11a) and (11b) that: The goal, now, is to make 4 z and 5 z exponentially vanishing by enforcing its derivatives 4 z  and 5 z  to behave as follows: being design parameters and is a signum function. It is worth noting that the integral actions are introduced in (28) and (29) to allow a good robustness of the controller against unmodeled dynamics and perturbations. Combining (26) and (28) the first control law   is obtained: Finally, combining (27) and (29), the second control law   is also obtained Since the control laws are clearly defined, the concern now is to investigate the convergence of the errors 4 z and 5 z . To this end the following quadratic Lyapunov function is considered which, using (28) and (29), gives: As Vc is positive definite function and its derivative (34) is negative definite it follows that the equilibrium ) 0 , 0 ( ) , ( is globally asymptotically stable [18]. Which, in turn, gives )

SIMULATION RESULTS
The theoretical performances described by the theorem of an output feedback controller, including the control laws (30-31) and the nonlinear observer (11a-c), designed in Section 3, are now illustrated by simulation. The experimental setup, described by Figure 5, is simulated using MATLAB/SIMULINK. The characteristics of the controlled system are listed in Table 4. Note that the controlled system is simulated using. The instantaneous three phase model given by (5a-d).The model (10a-c) in α-β axis is only used in the controller design. The design parameters of the controller are given values of Table 5. These parameters have been selected using a 'trial-and-error' search method and proved to be suitable. It worth noting that the parameter  involved in (22)     Step value k 3 . 0 Figure 7 shows the perfect MPPT in the presence of radiation step changes meanwhile, the temperature is kept constant, equal to 298.15K(25°C). The simulated radiation profile is as follow: a first step change is performed between 500 and 1000W/m² at time t = 0.1 s and the second one between 1000 and 800 W/m² at time t = 0.2 s. The Figure 8 shows that the PV power captured varies between 35.1 kW and 71 kW and then returns to 57.2 kW.These values correspond is shows in Figure 3 to maximum points (M3, M1 and M2) of the curves associated to the considered radiation, respectively. The Figure 8 also shows that the voltage of the PV array Vpv varies between Vpv = 648.8 V and Vpv = 664 V and then returns to 661V, which correspond very well to the optimum voltages. Figure 9. shows the injected current and the grid voltage This Figure 9 clearly shows that the grid current is sinusoidal and in phase with the grid voltage , proving that the power factor unit is achieved. The alternating currents injected to the grid are illustrated by Figure 10.  Figure 11 shows the performances of the controller in presence of temperature step changes while the radiation λ is kept constant equal to 1000W/m 2 . The temperature step change is performed at time t=0.15s between T=25°C (298.15K) and T=55°C (328.15K). It is worth noting that these changes are very abrupt which is not realistic in practical case. Nevertheless, with this important change we want to show a good robustness of the proposed controller to achieving the MPPT objective. It can be seen from Figure 11 that the system tracks the new operating point very quickly. Indeed, the captured PV power P achieves the values 71.7kW et 61.7kW corresponding to the maximum points (M4 and M5) associated to the temperatures 25°C and 55°C, respectively as shown in Figure 4. 149 Figure 12 illustrates the grid current and the grid voltage vga, Figure 12 also shows that the current is sinusoidal and in phase with the voltage vga which proves the unity PF requirement. Finally, Figure 13 shows the zoomed three phase's grid currents and voltages.

CONCLUSION
We have addressed the problem of current sensorless control of the photovoltaic energy conversion system. The main feature of the system presented is it does not require an intermediate stage of DC-DC control, because the maximum power is set by the inverter itself by means of a Perturb & Observe algorithm.
The system dynamics has been described by the nonlinear state-space model (5a-b). Based on a transformed model in    axis, an output feedback controller, combining an observer and a state feedback control laws is designed and analyzed using a Lyapunov approach. The controller objectives are threefold: i) ensuring the MPPT in the side of PV generator; ii) guaranteeing a power factor unit in the side of the grid, iii) ensuring the global asymptotic stability of the closed loop system. Using both formal analysis and simulation, it has been proven that the obtained controller meets all the objectives. It is worth noting that all objectives are achieved without resorting to the measurement of all variables. The use of the estimator is motivated by the global cost and reliability considerations. Indeed, the fewer the number of sensors, the lower the global cost.