Robust non-linear control of a hybrid water pumping system based on induction motor

Received Feb 22, 2020 Revised Apr 26, 2020 Accepted May 19, 2020 This contribution presents a non-linear control of a hybrid pumping system supplied with a photovoltaic generator and a battery. This system is employed for delivering a continuous volume of water whatever the climatic conditions. In the DC side, a boost converter is controlled with the indirect double integral sliding mode controller (DISMC) for maximum power point tracking (MPPT). The DISMC is suitable for MPPT because it gives a fast response and reduces the amplitude of power oscillations. Then, a bidirectional buck-boost converter is adopted to ensure the energy management between the battery and the DC-bus, and this converter is controlled with integral sliding mode control (ISMC) theory. The non-linear predictive control (NPC) is chosen to drive an induction motor (IM), the NPC is known by its fast dynamic and high capacity to reject disturbances. The hybrid system is modelled in MATLAB/Simulink software. During simulations, the DISMC-MPPT is compared with other techniques such as sliding mode controller (SMC) MPPT and integral SMC MPPT, the DISMC provides the best tracking performances under different irradiances. Moreover, the designed controller for the bidirectional converter regulates the DC-link voltage with better performances than the classical PI controller. Lastly, the NPC regulates the speed of the IM with high robustness.


INTRODUCTION
The sliding mode control (SMC) is suitable for fast maximum power point tracking (MPPT) because it is known by stability, fast-response and non-sensitivity to parameter variation [1]. The conventional hysteresis-modulation (HM) based SMC suffers from variable switching frequency and from the chattering phenomenon due to the high switching frequency operation. These drawbacks motivated the researchers to design an indirect SMC based on the pulse width modulation (PWM) technique. The conventional SMC with PWM exhibits an unwanted steady-state error and slow response [2], to improve the response of indirect SMC an integral term is added to the existing sliding surface to constitute the integral sliding mode controller (ISMC) [3,4]. However, the construction of the indirect form of the ISMC derivates the state variables of the switching surface (=0), thus, the variable ∫ disappears from the equivalent control term u eq , and the correction of steady-state error is deteriorated [5,6]. Therefore, an integral term is added for the second time ∫(∫ )) to nullify the steady-state error introduced by the indirect ISMC, this new technique is called

CIRCUIT CONFIGURATION
A hybrid water pumping system is presented in Figure 1. The overall system comprises a PV array as a principal source of energy and a battery pack considered as a second power source. The system includes static DC-DC converters such as a unidirectional boost converter used for MPPT and a bidirectional buckboost converter which ensures the bidirectional flow of the energy between the DC-bus and the battery. Finally, a two-level inverter controls an induction motor with a centrifugal pump.

CONTROL STRATEGIES
Diverse controllers are proposed to control the hybrid system, and are listed as follows: • A DISMC based on PWM is designed to track the maximum power point (MPP) with high efficiency. • The bidirectional buck-boost converter is controlled with the ISMC theory to regulate the DC-bus voltage. • Control of the induction motor with the generalized predictive controller. Figure 2 shows the MPPT control scheme, in which a perturb and observe (P&O) algorithm generates the reference voltage V pv * . Then, a DISMC is used to force the state trajectory of the boost converter to follow the reference voltage given by the P&O algorithm.

Design of a double integral sliding mode controller for MPPT
Considering that the boost converter operates in continuous conduction mode (CCM), The average state-space model of the boost converter is presented in term of the switching signal u 1 as, Where, r pv is the dynamic resistance of the PV array. The general control law to control the voltage v pv is: Where S 1 is the switching surface, which is expressed as, S a e a e a e a e = + + + The terms a 1 -a 4 denote the sliding surface parameters, and e 1 -e 4 are the error signals.   [2,4]. The control law of the indirect sliding mode control is derived using PWM technique, comparing control signal v control1 with a ramp signal v ramp1 .

The control strategy of the bidirectional converter
The BDC is considered as the intermediate link between the battery and the DC bus. If the DC-link voltage is greater than the reference voltage v dc > V dc * , the BDC operates in buck mode to store the excess of energy in the battery. On the other hand, if v dc < V dc * the battery discharges to inject the required current into the DC-bus, in this case, the BDC works in boost mode [20,21]. The direction of the current is reversed from one mode to another. To control the BDC in both modes of operation two strategies of control are proposed as presented in Figure 3.

Design of a double integral sliding mode controller for the boost converter mode
Considering that the boost converter operates in CCM, and the chosen state variables are the inductor current i LB and the DC-link voltage v dc . The dynamic of the boost converter is described with the differential equations in (8), where u 2 refers to the state of the switch K B1 .
The sliding surface is formed with the state variables errors (9).
where e 1 -e 4 are the current and voltage errors, and B is the gain which amplifies the voltage error.
Substituting the dynamic of the step-up converter in the derivative of the sliding surface (2=0) gives, The equivalent control signal u 2eq of the boost converter is obtained by solving the following equation ̇2 =0, The control law is derived using the PWM technique, comparing the voltage control signal v control2 with a ramp signal v ramp2 .
Here, the factor G s =β ( 0<G s <1) is used to downscale voltage magnitude to a practical level. B 1 -B 3 are constant parameters determined according to reachability and stability conditions [5,22]. The duty ratio is multiplied with a signal of the pulse generator to ensure that the duty ratio is always less than 1. Figure 3 illustrates the control scheme of the implemented DISMC-PWM.

Design of Integral sliding mode controller for the buck converter mode
Considering that the buck converter operates in CCM, and the chosen state variables are the inductor current I LB and the DC-link voltage V dc .The dynamic of the buck converter is described with the differential equations (14), where u 3 refers to the state of the switch K B2 .
where, c 1 and c 2 are the sliding surface parameters determined according to reachability condition [9,25], the errors are given by, the control discrete function of the buck converter is concluded from the transversality condition [25].
the derivative of the switching surface is given in equation (18), 3 3 solving the equation (3=0), the control signal would be: Figure 4 presents the non-linear predictive control of the induction motor. The mathematical model of the IM is presented in the two-dimensional stator reference frame (α-β) [26] is:

Non-linear predictive control of induction motor
where, 11 12 L h x operator is Lie derivative notation of the function h j with the respect to f(x).
The basic idea behind the predictive control consists of the construction of a control law u(t) able to force the system trajectory to follow the desired trajectory in a future horizon (t+τ r ). The control law is obtained through the optimization of the cost function expressed as, 0 1 ( ( ) y (t )) (y(t ) y (t )) d 2 using Taylor series expansion of the outputs and of the reference outputs the cost function can be rewritten as [15,18]:

SIMULATION RESULTS AND ANALYSIS
The hybrid pumping system has been tested in MATLAB / SimulinkTM software. The principal data of the hybrid pumping system are listed in Table 2. Figure 5 (a) shows that linear and sudden irradiances are applied to the system, and the temperature is fixed at 25 ° C value due to its little effect on power variation. Figure 5 (b) describes the evolution of PV voltage, the controller based on DISCM-MPPT tracks fastly the reference voltage V pv * provided by P&O algorithm. In addition, a zoomed view on the PV voltage curve illustrates that the voltage fluctuations are small when the MPP is reached. In order to make a fair comparison possible, the designed DISMC-MPPT is compared with both the conventional SMC-MPPT and the ISMC-MPPT under the same operating conditions of the system. Figure 5 (c) and Table 1 clearly show the improvement of the extracted power. From simulation data, it can be observed that the conventional SMC-MPPT presents the highest power fluctuations, the slower tracking speed and the largest steady-state error. The ISMC-MPPT increases the tracking speed, but the amplitude of power oscillation is not reduced significantly. In contrast, the DISM-MPPT reduces the amplitude of power oscillation and improves the other performances in comparison with ISMC-MPPT. On the other side, the sliding mode control scheme for the bidirectional converter is compared with the conventional PI control scheme. Figures 6 (a)  when the bidirectional converter switches from one mode (charging/discharging) to another. On the other hand, the DC-link voltage is accurately regulated with the sliding mode controllers, in this case, the system presents an insignificant drop of voltage and short setting time when the operating mode changes. Furthermore, the voltage ripples are reduced with the proposed controller. The state of charge (SOC) shown in Figure 6 (b) reflects the battery's charging/discharging modes. The SOC increases when the battery is charging and vice versa. Figure 6 (c) shows that the inductor L B operates in CCM mode, the battery receives the current from the DC-bus during charging (i LB negative) and delivers the current to DC-bus for the time of discharge (i LB positive). It can be observed from Figure 7 (a) that the non-linear predictive controller tracks the reference speed with a very fast dynamic. The electromagnetic torque curve presents reduced ripples because the curves of stator currents are quite close to the sinusoid form, which is depicted in Figure 7 (b) and Figure 7 (c). Figure 7 (d) shows that the decoupling between the flux and the torque is achieved accurately because the flux stays close to its reference. Since the NPC controller tracks the reference speed with high performance, the centrifugal pump parameters (flowrate, hydraulic power) are improved. Figure 7 (e) and Figure 7 (f) illustrate that these parameters follow exactly their references.

CONCLUSION
A robust control scheme of a hybrid water pumping system based on an induction motor was presented in this paper. First, a cascaded controller based on P&O and the indirect double integral SMC is designed to track the MPP. Seconde, the integral sliding mode control theory has been used to control the bidirectional buck-boost converter. The third stage of the system consists of a three-phase inverter which controls the speed and torque of an induction motor with the non-linear predictive control technique. Analyzing simulation results, it is found that the DISMC-MPPT tracks the MPP with better performances than both SMC-MPPT and ISMC-MPPT methods. The bidirectional converter controlled with the conventional PI controller is not able to preserve the desired response. However, the control scheme based on ISMC theory showed high robustness and operated properly in different irradiances and speeds. Moreover, the NPC exhibited high tracking performances of the speed, fast torque response and fewer ripples in the torque. Since the IM is controlled with high performances, the pump parameters (hydraulic power, flowrate) are improved.