Enhanced Intelligent Closed Loop Direct Torque and Flux Control of Induction Motor for Standalone Photovoltaic Water Pumping System

Abstract

This paper aims to search for a high-performance low-cost standalone photovoltaic water pumping system (PVWPS) based on a three-phase induction motor (IM). In order to control the IM, a fuzzy direct torque control (FDTC) is proposed in this paper for overcoming the limitations of the conventional direct torque control (CDTC). In fact, the CDTC suffers from several problems such as torque ripples, current distortion, and switching frequency variations. These problems can be solved with the proposed FDTC. To ensure high performance of the PVWPS, the reference torque is generated using a fuzzy speed controller (FSC) instead of a conventional proportional integral speed controller. In order to extract the maximum amount of power, the proposed maximum power point tracking controller is based on variable step size perturb and observe to surmount the weakness of the conventional perturb and observe technique. The performance of the proposed FDTC based on the FSC under variable climatic conditions is demonstrated by digital simulation using Matlab/Simulink. The obtained results show the effectiveness of the suggested FDTC based on the FSC compared with the CDTC in terms of pumped water, reduction in flux and torque ripple, diminution of losses, and decrease in the stator current harmonic.

1. Introduction

In recent years, photovoltaic (PV) energy has become widely used in different applications, particularly in pumping systems [1]. Standalone PV pumping systems in rural areas are recommended since connecting to the grid where electricity is unavailable can be expensive and difficult [2,3]. In order to decrease the cost and increase the lifespan, our proposed system works without chemical energy storage, since the battery is the most expensive element in the PV system [4]. Major advantages encourage the use of PV pumping systems compared with diesel water pumping not only due to the exhaustion of the traditional energy and their effects on atmospheric pollution but also their need for less maintenance [5,6].

Despite the efficiency of the PV array, reaching 15–16% [7], the amount of power generated is continuously influenced by the atmospheric conditions such as solar radiation and temperature [1,8]. Because of nonlinear I-V and P-V curves, the generated PV power is also influenced by the nature of the load [8]. To surmount these weaknesses, maximum power point tracking (MPPT) techniques have been introduced to ensure optimal utilization of the PV array whatever the external conditions.

Recently, several MPPT techniques have been proposed and developed in the literature. Although those techniques have fast time response and high efficiency under varying weather conditions, each method has its principle, particularization, and application domain [9,10]. Three major groups of MPPT have been suggested. Conventional MPPT based on a fixed step, are known as perturb and observe (P&O) [11] or incremental conductance [12]. This kind of technical strategy is characterized by its simple structure. However, the oscillation around maximum power point (MPP) leads to an additional loss in the PV water pumping system (PVWPS) [8]. To surmount this drawback, non-conventional MPPT techniques based on artificial intelligence have been introduced. The most used are the fuzzy logic controller (FLC) [13], the artificial neural network (ANN) [14] and the particle swarm optimization (PSO) [15]. These techniques offer good performance compared with conventional algorithms, while they are characterized by their complex structures and their need for high-speed calculators for implementation [16]. To combine simple structure and good MPPT accuracy, a modified MPPT has been introduced. The most used in this group of MPPTs are variable step size P&O (VSS-P&O) and variable step size incremental conductance (VSS-INC) [9]. In the literature of artificial intelligence, VSS-INC and VSS-P&O are the most recommended in a PV water pumping system (PVWPS) [7,17]. Nevertheless, the simplicity and flexibility of VSS-P&O make this control strategy the most used [9].

Many types of motors have been used in PVWPSs [7]. Each one has its advantages and limitations. Due to its simple drive, decoupling between flux and torque DC motors has been used in PVWPSs [18]. However, the DC motor needs regular maintenance. To surmount this weakness, a permanent magnet brushless DC (BLDC) has been proposed [6]. However, these types of motors are recommended for low-power PV systems [19]. A sine-fed permanent magnet synchronous motor (PMSM) has been used in pumping systems [20]. Nevertheless, PMSMs are overshadowed by the induction motors (IMs) because of their rugged construction, low cost, no requirement for hall sensors, availability constraint, and are maintenance-free and high efficiency [7]. For this purpose, the IM is the most recommended in PV pumping applications [1,7].

Because of its nonlinear model, controlling speed, stator flux, and torque, the IM is relatively complex [16]. The field oriented control of induction motor drives represents a standard vector or rotor field-oriented control strategy. The main advantage of this control strategy compared with scalar-controlled drives is its fast dynamic response. The inherent coupling effect between the torque and flux in the machine is managed through decoupling (rotor flux orientation) control, which allows the torque and flux to be controlled independently [16,21]. This control strategy has the advantage of being highly performant in terms of time response of the torque during a wide speed control range [16]. However, this technique is influenced by the parameter variation in the IM and in the external load disturbances [21]. To surmount these drawbacks, direct torque control (DTC) has been proposed [22]. It offers good robustness under IM parameter variations. However, the use of hysteresis controllers gives rise to high ripples in the electromagnetic torque and in the stator flux, especially at low a speed, which causes mechanical vibration and produces additional noise. Moreover, total harmonic distortion (THD) of stator current will increase forward due to the high variation in the switching frequency [23]. Several solutions have been proposed in the literature to overcome the conventional DTC problems. In [24], a multilevel inverter was proposed to reduce the flux and torque ripples. However, the switching losses and the cost of the PVWPS will increase because of the important number of power devices. Furthermore, a multilevel inverter control strategy based on space vector modulation was introduced in [25,26,27]. The use of a high-level inverter helps to overcome torque and flux ripple. The weakness of this technique is the necessity of an accurate design of the flux and the torque [16]. To improve the DTC, a sliding mode control (SMC) was introduced in [28]. However, the main critical problem in this technique is the chattering phenomenon. Recently, many researchers have introduced artificial intelligence to improve DTC performance such as the ANN-based DTC. Nevertheless, the controller performance is influenced by the delays caused by the training process of the controller [29]. In our paper, the FLC is proposed to improve the performance of the DTC. The FLC has the advantage of working without a mathematical model, which makes it simple to design [30]. Moreover, intelligent control based on the FLC is characterized by its correspondence to a human logic and it allows fixing of the frequency commutation [8]. Consequently, the stator flux ripple and the THD current will be reduced.

Different configurations of the PVWPS have been suggested in the literature. The conventional control scheme of the PVWPS is composed of a fixed step size P&O (FSS-P&O) for the MPPT and a CDTC for the IM [9,23]. In [7], FSS-INC was put forward to obtain maximum power from the PV array and a scalar control for the IM. However, with a fixed step size, the performance is reduced. Since with a smaller step we obtain a slower response, the large step oscillation around MPPT will increase [31,32]. Despite its simple design and low cost, the scalar control strategy provides a weak torque response and the speed control accuracy is not good especially at low speed [33]. In [23], a sliding-mode control (SMC) is used to force the PV array working around the maximum power point (MPP). To drive the IM coupled with a centrifugal pump, the DTC is applied. The weakness of this control strategy is the instability of the DC link voltage [1], besides the DTC drawbacks. In [34], a standalone PV water pumping system with chemical storage was proposed. The proposed schemes have the advantage of working with reduced sensors. The control of the IM is based on DTC-SVM; the PV array operates at MPP using P&O algorithms. In the latter paper, the space vector modulation (SVM) has been associated with the DTC which reduces torque and flux ripples by an operation with a fixed switching frequency of the inverter. Nevertheless, the SVM-DTC based on PI controllers requires the knowledge of the exact model of the controlled system to determine its parameters. Moreover, the PI controllers have limited performances, especially during the presence of disturbances, uncertainties, and parameter variations. As a result, the dynamics and stability of the system will be affected [35]. The scalar control strategy has also been proposed for controlling pumping of the PV-system-based induction motor drive [36]. The scalar control is based on the steady-state model of the motor. The control is due to the magnitude variation of the control variables only and disregards the coupling effect in the machine. This method is simple and easy to implement; however, the inherent coupling effect between the torque and flux gives sluggish response and the system is prone to instability because of a high-order system effect. As a result, the scalar control technique has poor dynamic performance, especially at low speed range. In this work, a PVWPS based on IM with a fuzzy DTC (FDTC) is proposed. To track the maximum power from the PV array, VSS-P&O is used as well.

This paper deals with the optimization of a PVWPS, and a topology based on an intelligent control is suggested. VSS-P&O is proposed to attract the maximum power from the PV array. It consists in adjusting the optimal duty cycle of the boost converter. The first contribution consists in introducing the FDTC in order to improve the efficiency of the IM by reducing the torque ripples, current distortions, and commutation losses of the inverter. The DC link voltage is maintained at its reference using a PI controller. Secondly, a fuzzy speed controller (FSC) is introduced to generate the torque reference which improves the rotor speed control loop performances in terms of rapid response, tracking, and accuracy which consequently increases in pumped water.

This paper is organized as follows. The first section focuses on the presentation of the whole PVWPS and its specification. MPPT based on VSS-P&O is presented in the second section. Control strategies with a proportional–integral DC bus voltage controller, an FDTC, and a FSC are detailed in Section 3. The simulation results and the discussion of the whole PVWPS with two controllers to generate the torque reference are illustrated in Section 4. Finally, the conclusion of this work is presented in Section 5

2. Design of PV Pumping System

The synoptic diagram of the proposed system is depicted in Figure 1. It is composed of a PV array, a DC-DC boost converter to ensure working around MPP controlled by a VSS-P&O algorithm, a DC-DC buck converter to maintain the bus voltage at its reference value, a two-level voltage source inverter, and an IM coupled with a centrifugal pump. The inverter is controlled by an FDTC algorithm.

2.1. PV Generator Specification

2.1.1. PV Array

The number of panels is evaluated proportional to the rated power of the IM which is equal to 1.5 kW in this paper. Since the PVWPS presents some losses, the extracted power should be higher than the rated IM power. A PV array is selected using Equation (1):

$$P_{mpp}=\left(N_p\times I_{mpp}\right)·\left(N_s\times V_{mpp}\right)$$

(1)

where P_mpp represents the maximum power that can be generated at a given environmental radiation, V_mpp and I_mpp are the voltage and the current at the MPP, and N_s and N_p are serial and parallel panels to feed the induction motor coupled with a centrifugal pump. In this paper, the digital simulation will be carried out using eight series-connected panels to respect the required demand of energy.

In this work, the selected module to generate power is CSUN235-60P. The characteristic of the PV module is given in

Table 1. Characteristic of PV module at standard test condition (STC).

Parameter	Value
Module greatest power (Pmax)	235 W
Module voltage (Vmpp)	29.5 V
Module current (Impp)	7.97 A
Module open circuit voltage (Voc)	36.8 V
Module short circuit current (Isc)	8.59 A
Temperature coefficient at Voc (Kv)	−0.37%/°C
Temperature coefficient at Isc (Ki)	0.07%/°C
Number of cell	60
Array voltage at MPP (Vmpp)	236 V

.

2.1.2. PV Solar Module

The most used equivalent solar module with a single diode is illustrated by Figure 2 [2,13,17,23].

Since the association of the serial and parallel PV panels gives a PV array, its produced output current is calculated by the following expression [37]:

$$I_{pv}=N_p{I}_{ph}-N_p{I}_0(\exp (\frac{1}{V_t}(\frac{V_{pv}}{N_s}+\frac{R{I}_{pv}}{N_p})-1)-\frac{N_p}{R_{sh}}(\frac{V_{pv}}{N_s}+\frac{R{I}_{pv}}{N_p})$$

(2)

With

$$V_t=\frac{akT}{q}$$

(3)

2.2. DC-DC Boost-Buck Converter

Thanks to its high reliability, simple configuration, independent control, and important efficiency conversion, the boost-buck scheme chosen in this work is depicted in Figure 3 [8,38,39]. Moreover, the proposed topology has the advantage of flexibility in design since a DC link is inserted. It considered intermediate energy storage [38].

The mathematical model of the boost and the buck converters can be represented respectively by the following equations:

$$\{\begin{array}{l}\frac{d{V}_{dci}}{dt}=\frac{1}{C_{dci}}\left[\left(1-S\right)I_{L_{pv}}-I_b\right]\\ \frac{d{I}_{L_{pv}}}{dt}=\frac{1}{L_{pv}}\left[V_{pv}-\left(1-S\right)V_{dci}\right]\end{array}$$

(4)

$$\{\begin{array}{l}\frac{d{V}_{dc}}{dt}=\frac{1}{C_{dc}}\left[I_L-I_{inv}\right]\\ \frac{d{I}_L}{dt}=\frac{1}{L}\left[-V_{dc}+S^{\prime }{V}_{dci}\right]\end{array}$$

(5)

where S and S′ designs the logic control state of the power devices. Using the IM specifications, the DC bus voltage at the input of the inverter can be calculated as follows [7]:

$$V_{dc}=\frac{2\sqrt{2}}{\sqrt{3}}·\frac{V_{LL}}{m}$$

(6)

where m is the modulation index of a two-level three-phase inverter, and V_LL is the RMS line voltage across the motor chosen as 230 V.

The design of the boost-buck converters depend on the specifications of the load and the variation of power produced by the PV array [38]. The average intermediate voltage in this control strategy is considered V_dci = 560 V.

Duty cycles of both boost and buck ($\alpha$ et ${\alpha }^{\prime }$) converters are as follows [7,38]:

$$\alpha =\frac{V_{dci}-V_{mpp}pv}{V_{dci}},{\alpha }^{\prime }=\frac{V_{dc}}{V_{dci}}$$

(7)

where $V_{mpp}pv$ is the available voltage at the MPP. To provide energy in critical conditions such as the increase in the load and the variation in the environmental condition, the DC link voltage at capacitor is calculated by [7]:

$$C_{dc}=\frac{6\alpha V_{LL}·I_L·t}{\sqrt{3}(V_{dc}^{\ast 2}-V_{dc1}^2)}$$

(8)

where $\alpha$ is the duty ratio of the boost converter, t is the duration of the transient time, $V_{dc1}$ represents the DC bus voltage when $m=1$, and $I_L$ is the rated line IM current. The boost converter inductor is calculated by the following equation [7]:

$$L_{pv}=\frac{V_{mpp}pv·\alpha }{\Delta I·f_s}$$

(9)

where $f_s$ is the switching frequency, and $\Delta I$ is the ripple current.

The boost output capacitor C_dci depends on the output average current of the boost converter (I_b), switching frequency (fs), duty cycle ($\alpha$), and ripple content of the intermediate DC bus voltage (∆V_dci) [38]:

$$C_{dci}=\frac{I_b·\alpha }{\Delta V_{dci}·f_s}$$

(10)

where $I_b$ can be estimated by the following equation [38]:

$$I_b=\frac{P_{pv}}{V_{dci}}$$

(11)

where $P_{pv}$ is the PV power and $V_{dci}$ is the output voltage of the boost converter.

The design of the buck converter inductor is calculated by the following equation [38]:

$$L=\frac{V_{dci}·\left(1-{\alpha }^{\prime }\right){\alpha }^{\prime }}{\Delta I_L·f_s}$$

(12)

where $f_s$ is the switching frequency, and $\Delta I_L$ is the ripple content of the inductor current, $V_{dci}$ is the output voltage of the boost converter, and $\alpha \prime$ is the duty ratio of the buck converter.

Table 2. Parameters of boost-buck converter.

Component	Expression	Used Value
V *dc	$V_{dc}=2\sqrt{2}·V_{LL}/(\sqrt{3}·m)$	500 V
Cdc	$C_{dc}=\frac{6\alpha V_{LL}·I_L·t}{\sqrt{3}(V_{dc}^{\ast 2}-V_{dc1}^2)}$	2000 µF
$\alpha$/${\alpha }^{\prime }$	$\alpha =\frac{V_{dci}-V_{mpp}pv}{V_{dc}}$/${\alpha }^{\prime }=\frac{V_{dc}}{V_{dci}}$	0.57/0.89
Lpv	$L_{pv}=\frac{V_{mpp}pv·\alpha }{\Delta I·f_s}$	4 mH
$C_{dci}$	$C_{dci}=\frac{I_b·\alpha }{\Delta V_{dci}·f_s}$	200 µF
$L$	$L=\frac{V_{dci}·\left(1-{\alpha }^{\prime }\right){\alpha }^{\prime }}{\Delta I_L·f_s}$	3 mH

resumes the parameters of the boost-buck converter.

2.3. Induction Motor

The dynamics of the IM can be described in the $(\alpha -\beta )$ frame as follows:

-

Stator and rotor voltage in $(\alpha -\beta )$ frame:

$$\{\begin{array}{l}{V}_{s\alpha }=R_s{I}_{s\alpha }+\frac{d}{dt}{\varphi }_{s\alpha }\\ V_{s\beta }=R_s{I}_{s\beta }+\frac{d}{dt}{\varphi }_{s\beta }\\ 0=R_r{I}_{r\alpha }+\frac{d}{dt}{\varphi }_{r\alpha }+{\omega }_m{\varphi }_{r\beta }\\ 0=R_r{I}_{r\beta }+\frac{d}{dt}{\varphi }_{r\beta }+{\omega }_m{\varphi }_{r\alpha }\end{array}$$

(13)

-

Stator and rotor flux in $(\alpha -\beta )$ frame:

$$\{\begin{array}{l}{\varphi }_{s\alpha }=l_s{I}_{s\alpha }+M{I}_{r\alpha }\\ {\varphi }_{s\beta }=l_s{I}_{s\beta }+M{I}_{s\beta }\\ {\varphi }_{r\alpha }=l_r{I}_{r\alpha }+M{I}_{s\alpha }\\ {\varphi }_{r\beta }=l_r{I}_{r\beta }+M{I}_{s\beta }\end{array}$$

(14)

where $l_s,l_r$ and $M$ are the stator, the rotor, and the mutual inductance, $I_{s\alpha },I_{s\beta },I_{r\alpha }$, and $I_{r\beta }$ respectively denote the stator and rotor current in the $(\alpha -\beta )$ frame, and $R_s$ and $R_r$ are the stator and the rotor resistance.

-

Electromagnetic torque ‘’T_em’’ developed by the IM and its movement:

$$\{\begin{array}{l}{T}_{em}=\frac{3}{2}p\left({\varphi }_{s\alpha }{I}_{s\beta }-{\varphi }_{s\beta }{I}_{s\alpha }\right)\\ T_{em}-T_r=J\frac{d\mathsf{\Omega }}{dt}+f\mathsf{\Omega }\end{array}$$

(15)

where $p$ is the number of pair poles, $J$ is the inertia moment, and $f$ designs the viscous friction coefficient, T_r is the load torque in (Nm), and Ω is the IM rotor speed in (rad/s).

2.4. Design of Centrifugal Pump

As shown in Equation (16), the selected centrifugal pump load torque $T_l$ can be assumed to be proportional to the square of the IM rotor speed [23]:

$$T_l=k_{pump}·{\mathsf{\Omega }}^2$$

(16)

where $k_{pump}$ is the constant of the selected centrifugal pump. It is possible to estimate the pump constant using the rated value of the used IM. With 9.98 Nm and 1435 rpm:

$$k_{pump}=\frac{T}{{\mathsf{\Omega }}^2}=\frac{9.98}{{(2\ast \pi \ast 1435/60)}^2}=4.42\ast {10}^{-4} \mathrm{Nm}/{(\mathrm{rad}/\mathrm{s})}^2$$

(17)

Using similarity laws, the characteristic of a centrifugal pump $\left(P,H,Q\right)$ established for a rotation speed $N$, the new characteristic $\left(P^{\prime },H^{\prime },Q^{\prime }\right)$ can be obtained for any rotation speed $N^{\prime }$ as given by Equation (18) [17]:

$$Q^{\prime }=Q·\left(\frac{N^{\prime }}{N}\right);\hspace{0.17em}\hspace{0.17em}{H}^{\prime }=H·{\left(\frac{N^{\prime }}{N}\right)}^2;\hspace{0.17em}{P}^{\prime }=P·{\left(\frac{N^{\prime }}{N}\right)}^3$$

(18)

where $Q$ is the pumping flow in [L/s], $H$ is the pumping head in [m], and $N$ is the rotation speed in [tr/min].

3. Control Strategies

The proposed scheme is composed of two types of power converters:

▪

DC-DC boost-buck converters with a boost controlled by an MPPT algorithm to extract the maximum power from the PV array. To maintain the input voltage source inverter DC bus voltage at its reference, a buck converter is introduced.

▪

Two-level three-phase inverter to feed the IM coupled with a centrifugal pump. To improve the performance of the studied system, the FDTC is introduced.

Since the PVWPS functions at different speeds, producing the reference torque requires the use of a speed controller. The architecture for a standalone PVWPS is illustrated by Figure 4.

3.1. MPPT VSS P&O

Improving the efficiency of the PVWPS requires the use of a boost converter to operate the maximum point, as shown in Figure 5. Due to its nonlinear characteristic, the maximum power is generated to the load when there is an equivalence between the source and the load impedance [8].

Many algorithms have been processed in the literature to attack the MPP. Conventional MPPT based on FSS-P&O gives an acceptable performance [9]. However, during a sudden change in atmospheric conditions, the results present oscillations around the MPP. To overcome this drawback and to find a compromise between rapidity and precision of the studied system, VSS P&O is recommended [40]. To deduce the duty cycle of the power device, this algorithm is based on the measuring of the voltage and the current to calculate the power generated by the PV array. This algorithm functions as follows:

▪

If $\frac{d{P}_{pv}}{d{V}_{pv}}\rangle 0\to$The working point is on the left of the MPP.

▪

If $\frac{d{P}_{pv}}{d{V}_{pv}}\langle 0\to$The working point is on the right of the MPP.

▪

If $\frac{d{P}_{pv}}{d{V}_{pv}}\approx 0\to$The working point is at the MPP.

The step size is automatically adopted using the following equation:

$$Offset=Offset0\ast \left(\frac{\mathrm{\Delta }{P}_{pv}}{\mathrm{\Delta }{V}_{pv}}\right)$$

(19)

The flowchart of VSS P&O is depicted in Figure 6.

3.2. Resultant Speed Reference

The proposed studied system is designed to function at a variable speed. The reference speed can be calculated from the available power generated by the PV array, as mentioned in Equation (19). Therefore, the system operates in a closed loop.

If losses of the converters and the motor are neglected, the power generated by the PV array will be equal to the power available on the load. Using Equation (16), the reference speed can be obtained by the following equation:

$${\mathsf{\Omega }}_{ref}=\sqrt[3]{\frac{P_{pv}}{K_{pump}}}$$

(20)

Large numbers of techniques have been proposed to maintain the DC bus voltage at its reference. A PI controller was proposed by [41]. It consisted in converting the voltage error into speed. In [42], the MPPT control and the voltage regulation with a single stage were proposed. Due to its simplicity, the reduced voltage stress, and the decoupling control [38], boost-buck based on intelligent MPPT-PI DC bus voltage controllers are used in our paper.

3.3. DTC Based on Fuzzy Logic Controller for IM

To improve the performance of the suggested studied system, a DTC based on a fuzzy logic controller is introduced to control the IM. It consists in replacing the hysteresis controller and the switching table of the classical DTC by imposing the suitable vector state of the voltage source inverter lead in regulation of the stator flux and the torque of the IM.

The fuzzy switching table is elaborated using three inputs, which are the torque error $\epsilon (T_{em})$, the stator flux error $\epsilon (\varphi )$ and the stator flux angle $\left({\theta }_s\right)$, and one output $\left(V_i\right)$, which is one of the eight voltage source inverters. This control strategy is characterized by its high performance in terms of precision, stability, time response, and simplicity of implementation [16]. The basics of FDTC are illustrated in Figure 4.

3.3.1. Stator Flux and Electromagnetic Torque Estimation

In the $(\alpha -\beta )$ reference frame, the stator voltage can be computed using the V_dc bus voltage and the switches state (S_a, S_b, S_c) as follows:

$$\{\begin{array}{l}{v}_{s\alpha }=\sqrt{\frac{2}{3}}\left(S_a-\frac{1}{2}\left(S_b-S_c\right)\right)\\ v_{s\beta }=\frac{1}{\sqrt{2}}{V}_{dc}\left(S_b-S_c\right)\end{array}$$

(21)

Applying Concordia transformation, $i_{s\alpha }$ and $i_{s\beta }$ currents are expressed proportional to real currents $i_{sa}$, $i_{sb}$, and $i_{sc}$, according to the following equation:

$$\{\begin{array}{l}{i}_{s\alpha }=\sqrt{\frac{2}{3}}{i}_{sa}\\ i_{s\beta }=\frac{1}{\sqrt{2}}\left(i_{sb}-i_s{}_c\right)\end{array}$$

(22)

The module of the stator flux is calculated as follows:

$${\widehat{\varphi }}_s=\sqrt{{\widehat{\varphi }}_{s\alpha }^2+{\widehat{\varphi }}_{s\beta }^2}$$

(23)

The angle of the stator flux compared with the reference axes can be expressed by:

$${\theta }_s=arctg\left(\frac{{\widehat{\varphi }}_{s\beta }}{{\widehat{\varphi }}_{s\alpha }}\right)$$

(24)

where ${\varphi }_{s\left(\alpha ,\beta \right)}$ can be determined from Equation (13) as follows:

$${\widehat{\phi }}_{s(\alpha ,\beta )}={\displaystyle \int \left(v_{s(\alpha ,\beta )}-R_s{i}_{s(\alpha ,\beta )}\right)} dt$$

(25)

The electromagnetic torque is found by the estimated current, and the flux can be expressed as follows:

$$T_{em}=\frac{3}{2}p\left({\varphi }_{s\alpha }{i}_{s\beta }-{\varphi }_{s\beta }{i}_{s\alpha }\right)$$

(26)

3.3.2. Principle DTC Based on FLC

The FLC is introduced to determine the desired vector state of the inverter voltage source. The DTC based on the FLC for the IM is composed of three inputs and one output. The inputs are the torque error ${\epsilon }_T$, the stator flux error ${\epsilon }_{\varphi }$, and the stator flux angle $\theta$. The output is the desired vector state $v_i$. Three main parts can be distinguished in the FLC: the fuzzifier, the fuzzy control rules with fuzzy inferences, and the defuzzifier [16,43], as shown in Figure 7.

▪

Fuzzification:

It consists in converting analog inputs into fuzzy variables produced by membership functions. The inputs of the proposed studied system are the torque error ${\epsilon }_T$, the stator flux error ${\epsilon }_{\phi }$, and the flux position $\theta$, as depicted in Figure 7.

The first input is the torque error. To make its variation smaller, the torque error can be divided into five fuzzy sets: negative large (NL), negative small (NS), zero (Z), positive small (PS), and positive large (PL). The torque error membership function can be represented by three equidistant isosceles triangles and two symmetric trapezoidals. Variation is normalized in the interval [−1,1], as shown in Figure 8a.

In order to make the stator flux error variation medium, the stator flux error membership function can be represented by three linguistic sets: negative (N), zero (Z) and positive (P). Discourse is normalized at [−1,1], (P), and (N) variables which are represented by two trapezoidal membership functions, while (Z) is represented by an equidistant isosceles triangular membership function, as illustrated in Figure 8b.

The last input of the studied fuzzy controller is the stator flux angle. Its universe of discourse is equal to 360°. The stator flux angle can be divided into six sectors. To ameliorate precision, the universe of discourse of this fuzzy set is divided into 12 fuzzy sets. The stator flux angle membership function associated can be represented by twelve equidistant isosceles triangles of 60°, as depicted in Figure 8c.

▪

Fuzzy control rules:

Optimal switching states (S_a, S_b, S_c) are obtained by fuzzy rules. The latter are based on an expert knowledge and intuition to govern the behavior of the studied controller. The control algorithm contains 180 rules, which can be established from the three input membership functions (3 × 5 × 12).

Table 3. Fuzzy control rule table.

${\mathit{\epsilon }}_{\mathit{\varphi }}$	${\mathit{\epsilon }}_{\mathit{T}}$	θ1	θ2	θ3	θ4	θ5	θ6	θ7	θ8	θ9	θ10	θ11	θ12
P	PL	V2	V3	V3	V4	V4	V5	V5	V6	V6	V1	V1	V2
	PS	V2	V2	V3	V3	V4	V4	V5	V5	V6	V6	V1	V1
	Z	V0	V7	V7	V0	V0	V7	V7	V0	V0	V7	V7	V0
	NS	V1	V1	V2	V2	V3	V3	V4	V4	V5	V5	V6	V6
	NL	V6	V1	V1	V2	V2	V3	V3	V4	V4	V5	V5	V6
Z	PL	V2	V3	V3	V4	V4	V5	V5	V6	V6	V1	V1	V2
	PS	V2	V3	V3	V4	V4	V5	V5	V6	V6	V1	V1	V2
	Z	V7	V0	V0	V7	V7	V0	V0	V7	V7	V0	V0	V7
	NS	V7	V0	V0	V7	V7	V0	V0	V7	V7	V0	V0	V7
	NL	V6	V1	V1	V2	V2	V3	V3	V4	V4	V5	V5	V6
N	PL	V3	V4	V4	V5	V5	V6	V6	V1	V1	V2	V2	V3
	PS	V4	V4	V5	V5	V6	V6	V1	V1	V2	V2	V3	V3
	Z	V7	V7	V0	V0	V7	V7	V0	V0	V7	V7	V0	V0
	NS	V5	V5	V6	V6	V1	V1	V2	V2	V3	V3	V4	V4
	NL	V5	V6	V6	V1	V1	V2	V2	V3	V3	V4	V4	V5

presents the output vector depending on the input variables: the torque error, the flux error, and the stator flux angle.

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Fuzzy inferences:

The inference method adopted in this work is Mamdani’s method based on the min-max decision, which can be written by the following Equation (16).

$$\{\begin{array}{l}{\alpha }_i=\min \left({\mu }_{Ai}\left({\epsilon }_{\varphi }\right),{\mu }_{Bi}\left({\epsilon }_T\right),{\mu }_{Ci}\left(\theta \right)\right)\\ {\mu }_{Vi}^{\prime }\left(V\right)=\max \left({\alpha }_i,{\mu }_{Vi}\left(V\right)\right)\end{array}$$

(27)

where ${\mu }_{Ai}\left({\epsilon }_{\varphi }\right)$, ${\mu }_{Bi}\left({\epsilon }_T\right)$ and ${\mu }_{Ci}\left(\theta \right)$ respectively denote the membership value of the flux error, the torque error, and the stator flux angle, and ${\alpha }_i$ is the weighting factor for its rules.

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Defuzzification:

It consists of converting the fuzzy values back into a numerical value varying from V0–V7, which are one zero voltage (V0 or V7) and six non-zero voltage. The membership function of the output vector is illustrated by Figure 8c. The method used in this step is represented by Equation (28). This method is based on the maximum possibility distribution [16]:

$${\mu }^{\prime }{}_{Vout}\left(V\right)={\max }_{i=1}^{180}\max \left({\mu }_{Vout}^{\prime }\left(V\right)\right)$$

(28)

where ${\mu }^{\prime }{}_{Vout}$ designs the membership function degree of the output variable.

3.4. Fuzzy Speed Controller

The aim of this section is to generate the reference torque from the error speed. To surmount weaknesses based on the classical PI speed controller, such as the speed error and time response, a sensitivity fuzzy speed controller is proposed. The inputs of variables are the speed error and its variations [44]. The speed error can be written by Equation (29):

$$e\left(k\right)={\mathrm{\Omega }}_{ref}\left(k\right)-\mathrm{\Omega }\left(k\right)$$

(29)

The variation in the speed error is given by the following equation:

$$de\left(k\right)=e\left(k\right)-e\left(k-1\right)$$

(30)

The structure of the fuzzy speed controller with two variable inputs and one variable output designs the reference torque as illustrated by Figure 9:

The variable inputs (e), its variation (∆e), and the only variable output (∆u) can be divided into five linguistic variables: negative big (NB), negative (N), zero (ZE), positive (P), and positive big (PB). The membership function can be represented by three equidistant isosceles triangles and two symmetric trapezoidals as shown in Figure 10. Variables are normalized to −1,1.

4. Simulation Results and Discussion

In this section, a simulation study of the CDTC, the proposed FDTC based on a PI controller, and the FDTC based on a fuzzy speed controller for a pumping PV system is carried out using the Matlab/Simulink environment. The MPP is obtained using VSS-P&O. The parameters of the PV panel and the boost-buck converter are illustrated in Table 1 and Table 2. The simulation parameters of the IM, the centrifugal pump, and the controllers are illustrated in

Table 4. Parameters of IM and centrifugal pump.

Parameters of the IM	Value
Stator and rotor resistance (Rs/Rr) [Ω]	5.72/4.28
Stator, rotor, and mutual inductance (Ls/Lr/M) [H]	0.462/0.452/0.44
Nominal power (Pn) [W]	1500
Inertia moment (J) [kg·m2]	0.0049
Viscous friction (f) [N·m·s·rad−1]	1.5 × 10−4
Pair pole number (p)	2
Frequency (f) [Hz]	50
Rated voltage [V]	230
Parameters of the Centrifugal pump	Value
Pumping flow (Q) [L/s]	1.67 to 6.51
Pumping head (H) [m]	20 to 78
Parameters of controllers	Value
Integral and proportional gains of PI speed conytoller	Kpsc = 0.825, Kisc = 35
Fuzzy logic system parameters	G1T = −G2T = 0.52 NmG1f = −G2f = 0.024 Wb
Sampling period	50 µs
Torque hysteresis controler band for CDTC	HTe = ±0.1 Nm
Flux hysteresis controler band For CDTC	Hφ = ±0.01 Wb
Kpump	5.57 × 10−4

.

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Performance analyses of proposed FDTC and VSS P&O

In this section, the performance of the PV system controlled by the proposed FDTC based on fuzzy speed controller is tested under variable irradiation profiles, as shown in Figure 11a.

The evolution of the produced power is illustrated by Figure 11b. This figure demonstrates that the produced power converges to the actual power, thanks to good performances of the VSS-P&O MPPT algorithm, which are archived in

Table 5. Efficiency obtained for VSS-P&O MPPT technique under different operating conditions.

Time (s)	0–1	2–3	4–5	6–7	8–9	10–11
Irradiation (W/m2)	200	400	600	800	1000	500
Average efficiency (Pout/Pmax) in (%)	98.29	98.36	98.51	99.12	99.52	98.41
Average efficiency (Pu/Pmax) for the overall system in (%)	80.12	80.18	80.30	80.80	81.12	80.22

. The average efficiency (P_out/P_max) is calculated utilizing the maximum power that can be produced by the PV generator, as depicted in Figure 11c.

Figure 12a presents the rotor speed evolution when the pumping system is controlled by the CDTC and the proposed FTDC. In this study, the reference speed is calculated through the produced power. In this case, the rotor speed control loop is based on the PI controller. This figure shows that the actual speed converges to its reference value with a negated fluctuation and good accuracy; but for the CDTC, the rotor speed presents a high fluctuation around its reference value. Relative to the CDTC, the water flow has very low fluctuation and good accuracy due to the suggested FDTC.

It can be seen that the proposed FDTC reduces the torque ripples and fluctuations, which consequently improves the pumping system performance and increases its service life. As illustrated by Figure 13b, the suggested FDTC also reduces the flux ripples. More comparison details are provided in

Table 6. Proposed FDTC performance: A comparative study.

	CDTC	FDTC
Torque ripples (Nm)	0.95	0.35
Flux ripples (Wb)	0.05	0.008
THD (%)	11.86	7.51
Stator current distortions	High	Low
Actual speed fluctuations	High	Neglected
Water flow fluctuations	High	Neglected

.

Figure 14 presents the three phases of the stator current. It can be noticed that the stator current features a perfect waveform without any distortions, as given by Figure 14b. However, for of the CDTC, the stator current presents important distortions.

Figure 15 illustrates the current total harmonics distortion (THD). It can be noticed that when the induction motor in controlled by the conventional DTC the stator phase current THD is evaluated by 11.86%, which is reduced to 7.51% in the case of the proposed fuzzy DTC.

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Performance analyses of the proposed fuzzy speed controller

The main objective in this section is to test the performance of the proposed fuzzy speed controller under sudden irradiation variation. As shown in Figure 16a, the pumping PV system starts by an irradiation equal to 500 W/m², which is increased to reach 1000 W/m² at t = 2.5 s. The produced power P_pv is provided by Figure 16b. It can be noticed that this figure shows that the generated power converges to the peak power for each irradiation value.

The rotor speed response is provided by Figure 17a. Relative to the PI controller, the proposed fuzzy speed controller offers good performances in terms of rapidity and convergence accuracy. The water flow evolution is illustrated by Figure 17b. It can be seen that for the proposed fuzzy speed controller, the water flow reaches the reference value quickly under sudden irradiation variations, which consequently increases the quantity of the pumped water. More comparison details are provided in

Table 7. Proposed fuzzy speed controller performances: A comparative study.

	PI Speed Controller	Fuzzy Speed Controller
Setting time	0.16	0.05
Dynamic	Medium	High
Robustness	Medium	Good
Complexity of implementation and tuning	Low	Meduim
Accuracy	Medium	Good

.

5. Conclusions

An improved performance of a standalone PVWPS based on intelligent techniques is investigated. In order to improve the efficiency of the PV system, a VSS-P&O is utilized to extract the maximum power from the PV generator. The IM is controlled through the conventional DTC, which is based on a two-hysteresis controller and a switching table. In order to overcome the conventional DTC limitations in terms of torque ripples and current distortion, an FDTC has been proposed. In order to improve the speed control loop, an FSC has been suggested to replace the conventional PI speed controller. A comparative study between the conventional DTC and the proposed FDTC is carried out. The proposed FSC is compared with the conventional PI speed controller. The obtained results demonstrated that the suggested FDTC-based FSC offers good performance in terms of torque ripples, current distortion reduction, and very fast speed response, which increases the pumped water quantity. The major improvements of this study are:

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Reduction in torque and flux ripple

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Improvement in stator current wave

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Increase in average efficiency

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Increase in water pumping

The next stage focuses on the experimental validation of the proposed control strategy. Other intelligent control will be developed to improve the PVWPS.