Adaptive backstepping control of induction motor powered by photovoltaic generator

Received Jul 20, 2020 Revised Oct 17, 2020 Accepted Dec 5, 2020 This paper is aimed at addressing the design of an effective adaptive nonlinear control of a photovoltaic (PV) water pumping system powering a submersible induction motor and a centrifugal water pump. Four objectives are achieved using an adaptive Backstepping controller. First, it is applied to ensure maximum power point tracking, and uses the latter as a reference in regulation of the rotor speed to convert the maximum electrical power into maximum mechanical power. Second, the adaptive controller is synthesized to control motor rotor flux and restrict the magnetic circuit to its linear interval. Third, it is used to online estimate the rotor time-constant and the load torque disturbance estimation. Finally, this controller is employed to limit the stator currents to protect induction motor windings. Mathematical modelling of the main elements of the system is presented. A sliding mode rotor flux estimator is employed in the output feedback control of the whole system. DC-AC converter is controlled by pulse width modulation. The feasibility, the robustness and the effectiveness of the proposed adaptive nonlinear controller are evaluated through simulations in MATLAB/Simulink environment.


INTRODUCTION
Renewable energies continue to gain popularity around the world, consolidating their dominance over fossil fuels in terms of new installed electricity production capacities. The use of coal, for its part, is continuing to decline worldwide. In 2019, renewable energies represented 72% of new installations in practically all regions, except in Africa and the Middle East, where their respective shares are 52% and 26%. These additions brought the share of renewable energy in total global electricity production capacity to 34.7%, compared to 33.3% at the end of 2018 [1].
The development of renewable energies is arousing a huge interest in the Moroccan national energy strategy. The latter is moving towards the diversification of energy supply sources by increasing the contribution of green energies to 42% of total installed electrical power by 2020. Energy efficiency, alongside the development of renewable energies, is a priority in the national energy strategy. The goal of this program is to save 12% of energy consumption in 2020 and 15% in 2030 [2].
The photovoltaic system is one of the most widely used renewable energy sources. One of its main applications is the direct solar photovoltaic water pumping system. This is due to its simplicity, maintenance free and its ability of operating without storage batteries. The system consists of a PV generator, a pumping

MODELLING OF THE SYSTEM
Solar photovoltaic water pumping system consists at least of the following elements: PV generator, induction motor, centrifugal water pump, controller, storage tank, DC link, rotor speed sensor.

Mathematical model of the PV generator
PV cells are generally associated in series and in parallel, and then encapsulated to form a photovoltaic module. Figure 1 illustrates the electrical equivalent circuit of a PV cell. PV generator is made up of interconnection of several modules to respond to the required power. PV modules are usually connected in series-parallel to increase the voltage and current at the generator output. Therefore, PV generator model is deduced from that the PV cell. Equations (1)-(3) describe the PV cell model.
where cell I , Vcell , ph I , Irs , Isc , , E, Tr, T, , q, K, Rss and Rsh are the current across the cell, the solar cell output voltage, photo-current, reverse saturation current, cell short-circuits current at 298 °K and 1 kW/m², reference solar irradiation in W/m 2 , solar irradiation in W/m 2 , cell's reference temperature, temperature on absolute scale, cell's short-circuit its current temperature coefficient, electron charge, Boltzmann's constant, and intrinsic series and parallel resistors of the cell respectively. Current and voltage outputs of the PV generator are deduced from those of the PV cell.
with Np and Ns being parallel and series wired PV solar panels respectively. With the aim to plot the nonlinear P-V characteristics of the PV generator and to define MPP under different atmospheric conditions, electrical specifications of the SM55 PV panel and MATLAB/Simulink environment are employed and given in Table 1.

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Adaptive nonlinear Backstepping controller must track these trajectories. MPP coordinates are recapitulated in Table 2 and used to validate the proposed controller. Figure 2 presents the P-V characteristic of PV generator under study. This generator is made up from 4 parallel chains, and each one is composed of 23 series SM55 PV panels.

Mathematical model of the DC-DC converter
This converter is used to boost PV generator voltage, to reduce the PV panels number used and to track the MPP. The boost converter state space representation is given as (5).
L being input voltage, output voltage, input capacitor, output capacitor, duty cycle, input inductance, output current, inductance current and output inductance respectively.

Mathematical model of the DC-AC converter
IM is supplied through a three-phase current controlled voltage source inverter (CC-VSI). This inverter is modeled using the relationship between the switching functions, the outputs voltages

Mathematical model of the induction motor
IM is supplied through a three-phase current controlled voltage source inverter (CC-VSI). This inverter is modeled using the relationship between the switching functions, the outputs and the DC-link voltages:

Mathematical model of the centrifugal pump
To extrapolate the "Head-Flow" characteristics of the pump corresponding to a different reference speed ' r  , the affinity laws can be used [10,19]. The current values of the flow ' q , the head ' h and the pressure r p can be expressed as a function of the reference values of the flow q , the head h and the pressure r p : Based on the centrifugal pump experimental data listed in the manufacturer's catalog, it is possible to choose the polynomial approximation, which is derived from the equation of the Euler pulse moment [26][27], to determine the "Head-Flow" characteristics of the pump. The pressure can be expressed by (10).
with  being the efficiency of the IM and the pump. Assuming that the efficiency remains constant when the speed varies, we can consider in this case that the centrifugal pump applies a load torque proportional to the square of the rotor speed. Therefore, the pump load torque is expressed as (12): where: The total dynamic head (TDH) is composed of the static and dynamic head and is given (14).
White [28] showed that the d h can be related to the flow and the pipeline system parameters (15). 2gdS flq h d = (15) with f, l, d, q, S and g being respectively the Darcy friction coefficient, the pipeline length, the pipeline diameter, the fluid flow, the section area and the gravity acceleration. There are other models of simplified hydraulic systems [19] adopted the (16).
With k being a constant to be designed using the centrifugal pump data. We will consider a centrifugal pump that delivers 20 m 3 /h to a water tank with a total dynamic head of 33.57 m and a power of 5.5 kW. The Grundfos NBG 65-40-315/334 pump, which uses the Grundfos performance curves, shown in Figure 3, is chosen. This pump is intended for drinking water supply and is equipped with a 5.5 kW 132SA type motor. At speeds other than the reference speed (1460 rpm) and using (10), the process of determining these four constants requires measurements of the pump head at a minimum of four flow rates. Based on the "Head-Flow" characteristics shown in Figure 3, these four parameters define the centrifugal pump model by the polynomial approximation in Simscape. The block diagram in Simscape is given in Figure 4. Figure 5 shows the pump and the pipeline "Head-Flow" characteristics. The use of the pump Simscape block requires the determination of the approximation coefficients c0, c1, c2 and c3. These coefficients are given (17).
The "Load torque-rotor speed" characteristic is reported in Figure 6. According to Figure 6, "Load torque-Rotor speed" characteristic of the centrifugal pump is highly nonlinear. This characteristic can be similar to that given in (12) in particular conditions. One of these conditions is to assume that the efficiency of the pump is constant.

ADAPTIVE CONTROLLER DESIGN
With the aim to achieve the MPP and to control the IM, the outputs to be controlled are expressed (18)- (21).
Rotor speed reference is given by: where MP  , mpp V and mpp I are the efficiency of the whole system, maximum voltage and maximum current of the PV generator respectively.

MPPT control law
Step 1: Let consider 1 as the MPP tracking error: The Lyapunov function and its derivative with respect to time can be defined as: The virtual control law 1  and its derivative 1  can be selected by forcing the Lyapunov function to be negative definite ( 0 [7,13]. with: (25) and 1 c being a strictly positive constant parameter to be chosen.
Step 2: Let consider the tracking error between the actual virtual control and its desired value: Let us define the second Lyapunov function and its derivative with respect to time: Duty cycle of DC-DC converter can be chosen in such way that the second Lyapunov function is negative definite: where 2 c is a constant parameter to be chosen.

Inverter control
Step 3: The tracking error between rotor speed and its reference and the error estimation of load torque can be expressed as: To guarantee stability and convergence, Lyapunov function is chosen positive and defined as: In this step, our objective is to select the control input which converges the origin 3  to 0. The control law is therefore given by (30): c being constant parameter to be designed.

Adaptive laws
Step 5: Let us consider a Lyapunov function and its derivative with repect to time:  + = The variation of rotor time-constant leads to the coupling between rotor flux and electromagnetic torque. Consequently, it is possible to deduce:

RESULTS AND DISCUSSION
To prove the excellent performance of the proposed controller and observers, we simulated the overall system in MATLAB/Simulink environment. The simulations are carried out at variable temperature and solar irradiance conditions. IM parameters are summarized in Table 3. The sudden variations in the solar irradiance from 1000 to 500 W/m 2 and the temperature from 25 °C to 50 °C as a function of time are considered. The reference value of the rotor flux is kept at its nominal value (1 Wb) while the mechanical speed reference is calculated referring to the optimal power. The rotor flux estimator is based on the sliding mode control [29]. This estimator requires knowledge of the stator resistance. To prove the robustness of the controller to the stator resistance variation, we programmed a slow variation of this resistance from 2.5 Ω (nominal value) to 3.3 Ω (32 percent variation).
Based on the simulation results, Figure 7 shows the controller reaches the MPP coordinates summarized in Table 1. In addition, the response of the mechanical speed obtained by the same controller tracks accurately its reference value. It is clearly obvious that the decoupling is achieved during the operation of the system and the quadrature component of the rotor flux varies only during the response time of the rotor time-constant observer.
The simulation results for these tests are shown in Figure 8 to Figure 15. It is quite remarkable from these figures that the adaptation of the rotor time-constant and of the load torque are accomplished whatever are the variations applied to the rotor and stator resistances of the machine and the variations in climatic conditions. In addition, these two observers are robust against the variations of stator resistance and optimal power. These responses do not exceed their desired values, require only measurable or estimated induction machine state variables and the obtained results are very satisfactory. Load torque observer is independent of the centrifugal pump variables and parameters. Unlike the controllers presented in reference [13], the use of anti-saturation function can eliminate the excessive currents at transient state of current controlled induction motor.

CONCLUSION
This paper presented the design and simulation of an adaptive vector control of the induction motor, driving a centrifugal pump, and supplied by a photovoltaic generator. This adaptive controller was introduced to online estimate rotor time constant variation in order to achieve the decoupling between rotor flux and electromagnetic torque and to avoid the use of mathematical model of the centrifugal pump in control laws. To optimize the control and adaptive laws, this controller is combined with indirect rotor field oriented control and the induction machine is fed by a current controlled voltage source inverter (CC-VSI). It can be seen from simulation results that the controller tracked accurately the MPP coordinates and the rotor speed reference. The proposed observers showed good results in the estimation of the rotor time-constant and the load torque disturbance and in the adaptation of these variables in an indirect vector control of the induction motor under abrupt climatic condition variations. In addition, these observers permitted achieving the decoupling between the rotor flux and the electromagnetic torque and their performances are very satisfactory. The control and adaptation laws were based only on the equations of the induction machine and had a slow dynamic response exhibiting the behavior of a first order system. The use of an anti-saturation function permitted to limit high current overshoot in transient state. Even if the adaptive laws were used in highly nonlinear applications, the robustness and the stability of the whole system were guaranteed and validated that these obtained performances will enable us to use them in real implementation.