1 Introduction

Different water pumping systems use the electricity and diesel to run the pump. However, this kind of source has several drawbacks such as environmental pollution, low reliability, fossil fuel prices, low efficiency and high maintenance costs [1,2, 3]. Therefore, it is necessary to use a sustainable and appropriate alternative source to power the water pumping systems. The use of the renewable energy sources is the best solution to overcome the aforementioned drawbacks. Various renewable energy technologies such as solar, wind and biomass energies can be used for water pumping [4]. The solar photovoltaic technology is the most desired and suitable one because it is clean, naturally available and without noise [5].

As the output characteristics of the PV array are influenced by the meteorological conditions and the maximum efficiency is assured by one operating point, it’s necessary to introduce MPPT algorithms to function in maximum power point (MPP) under different irradiation levels. A variety of MPPT algorithms have been designed. Each algorithm has its own limitations, applications and specifications [6]. Perturb and observe and incremental conductance methods are the most used. Incremental conductance (INC) techniques provide good performance for most of the solar tracking applications [7]. The proposed PV water pumping system utilizes INC algorithm based on variable step size for better PV power optimization under fast changing atmospheric conditions [8].

Different types of motors are utilized for solar fed water pumping system with water pump. In [9], the authors use the synchronous reluctance motor. In [10], the authors introduce the induction motors [8]. Other researchers employ the permanent magnet synchronous motor to drive the pump [11]. In the PV water pumping applications, the induction motor provides better performance as compared to other commercial engines because it is a rugged machine, widely used in industries, cost effective and low cost [12].

However, the control of this type of motor is complex because of the nonlinear model. the scalar control technique is used to control the IM due to its lower cost and complexity but it offers unsatisfactory torque response on low frequencies [13]. In [14, 15], the authors propose the indirect field oriented control method. This technique provides good precision of speed and a satisfactory torque response on the full speed range. However, this control strategy depends on machine parameters. To achieve better control of the system, direct torque control (DTC) technique is introduced [16, 17].

This method offers better performance than scalar control and indirect field oriented control methods because it is simple to implement, less dependent on IM parameters and doesn’t require current regulation loops. Also, DTC has a fast response [18, 19]. However, the presence of the hysteresis controller generates large flux and torque ripples [20]. On the one hand, this is a result of using a fixed width of the hysteresis bands. Moreover, the use small values of bandwidths keeps the presence of ripples and increases inverter switching frequency, and on the other hand to the utilization of switching vector for complete sample time period. The insertion of fuzzy logic controller replaces the hysteresis controllers and preserves a constant switching frequency which allows to reduce the high flux and torque ripples. To overcome these drawbacks, the multilevel converter is utilized in [21, 22], however, this method increases the cost and the losses. Other researchers have proposed the space vector modulation to improve the performance of the conventional DTC [23], however, the flux and torque loop PI regulators need an accurate design. The intelligent techniques are also employed by many researchers to enhance the controller performances. In [24], the artificial neural network is utilized, but this technique requires powerful calculation processes and is very complex. Fuzzy logic controller ensures good performance, especially, for the complex system where the mathematical model does not exist or is severely nonlinear [25]. The behavior of a physical variable is represented by membership functions which are selected in a way that they describe the real behavior of the variable. The choice of the kind of membership functions depends on the application and the designer’s knowledge and experience, which are defined based on the experience of control systems. Sigmoidal, singleton, triangular, gaussian, trapezoidal and bell-shaped membership functions are Frequently employed. The triangular membership function turns out the most used and suitable membership function to control electric motors [26]. In [19], the hysteresis controllers and the look-up table are replaced by fuzzy logic blocks by keeping the conventional PI controller for speed regulation. As results, the flux and torque ripples are minimized. Although the advantages of the fuzzy logic controller, it can get more high performance by associating other techniques to drive the system. In [27], the authors propose an approach associated to fuzzy logic controller to select the optimal flux and variable step size Perturb and observe technique to extract the maximum power. This technique allows to reduce the losses and improve the efficiency. Moreover, the classical proportional-integral (PI) controller is used for speed regulation.

In [28], variable step size incremental conductance based maximum power tracking (MPPT) algorithm is utilized to control the duty ratio of DC-DC converter. The control of induction motor is achieved through fuzzy logic control based on direct torque method. However, the classical PI controller is adopted for speed regulation.

The conventional PI controller is the most widely employed for speed regulation due to its simple design and control structure. However, this controller provides larger overshoot, oscillation, lower response and the control performance is influenced by the external load disturbances and the variations of the parameters [29].

Moreover, the operating of the PV water pumping system using an optimal speed controller allows to start earlier, increase the pumped water and achieve a longer run time. In [30], the authors propose a genetic algorithm (GA) optimized PI and fuzzy sliding mode speed control for DTC technique. DTC using fuzzy-PI controllers is proposed in [31]. An improved Direct Torque Control using PI controller tuned using genetic algorithm is introduced in [32]. Therefore, an adaptive fuzzy controller for speed regulation is proposed in this paper to ensure optimal control performance during dynamic and steady state conditions.

However, to the authors’ best knowledge, the combining of adaptive fuzzy controller for the speed regulation and fuzzy logic DTC for a PV water pumping system has never been discussed. This allows us to propose a PV water pumping system based on the combining of the smart regulators (AFLC and FLC) for the first time in the literature.

Simulation studies of the proposed topology based on intelligent approaches will be investigated using Matlab/Simulink under various operating conditions to validate the suitability of the proposed PVWPS. Therefore, the use of intelligent techniques to control the PV water system provides better performance. The rest of the paper is organized as follows: the proposed system is designed in Sect. 2. Section 3 presents the conventional method. Section 4 describes the proposed control strategies. The results and discussion are given in Sect. 5. Finally, Sect. 6 provides the conclusions of this work and suggestions for future work.

2 Modeling of proposed PV system

The proposed PV water pumping system consists of an IM of 1,5 kW power rating and PV array of 1.88 kW peak power capacity under standard test conditions (Fig. 1) The characteristics of Csun235-60p PV panel and the used IM are listed in Tables 1 and 2 respectively.

Fig. 1
figure 1

Proposed PV water pumping system

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Table 1 Characteristics of PV panel
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Table 2 Motor parameters
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2.1 PV panel

Different types of PV panels are made to respond to the needs and demands of the users. To assure the necessary energy, the PV panels are associated in series or parallel. The equivalent circuit of the utilized PV panel is illustrated in Fig. 2 [7, 33, 34]

Fig. 2
figure 2

PV panel equivalent circuit

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The current produced by the PV panel can be expressed by:

$$I = I_{ph} - I_{0} \left( {\exp \frac{{q\left( {V + RI} \right)}}{{aKTN_{S} }} - 1} \right) - \frac{{\left( {V + IR} \right)}}{{R_{sh} }}$$
(1)

2.1.1 Boost converter

This type of converter has a simple structure (Fig. 3). Moreover, it’s characterized by higher efficiency of conversion. The outputs current and voltage of the converter can be written as follows [35]:

Fig. 3
figure 3

Boost converter

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Fig. 4
figure 4

Speed controller

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$${V}_{dc}= \frac{{V}_{pv}}{1- \alpha }$$
(2)
$${I}_{dc}= {I}_{pv}(1-\alpha )$$
(3)

The parameters of the of boost converter are calculated as follows:

Component

Expression

Used value

\({C}_{dc}\)

\({C}_{dc}\)= \(\frac{6a{V}_{LL}It}{[{V}_{dc}^{* 2}- {V}_{dc}^{2}]}\)

2000 μF

α

α = \(\frac{{V}_{dc}- {V}_{mp}}{{V}_{dc}}\)

0.26

\({L}_{pv}\)

\({L}_{pv}\)= \(\frac{D {V}_{mp}}{\Delta I {f}_{s}}\)

3 mH

where

\({V}_{dc}^{*}\): Estimated DC voltage

\({C}_{dc}\): DC link capacitor

\({V}_{dc}\): Selected DC voltage

\(I\): Phase current motor

\({V}_{LL}\): Phase voltage motor.

\(t\): Duration of transient

\(D\): Overloading factor

\({L}_{pv}\): Inductance of the Boost converter

\(\Delta I\): Amount of ripple content current

\({f}_{s}\): Frequency of switching,

α: Duty cycle.

2.2 Two level inverter

The inverter is introduced to ensure the DC/AC conversion and control the speed of the motor. It’s composed of 6 Insulated Gate Bipolar Transistors. The outputs voltages of the inverter in terms of Vdc can be written as follows [36]

$${V}_{a}= \frac{{V}_{dc}}{3} \left( {2S}_{a}{-S}_{b-}{S}_{c}\right)$$
(4)
$${V}_{b}= \frac{{V}_{dc}}{3}\left( {-S}_{a}{+2S}_{b-}{S}_{c}\right)$$
(5)
$${V}_{c}= \frac{{V}_{dc}}{3}\left( {-S}_{a}{-S}_{b}{+2S}_{c}\right)$$
(6)

where: \({S}_{a}{,S}_{b}{,S}_{c}\): logic control signals.

2.3 Induction motor

Based on the simplifying assumptions, the mathematical model of the induction machine can be expressed by the following equations [25]:

$$\left\{\begin{aligned}\frac{d{\phi }_{\alpha s}}{dt} &= {V}_{\alpha s} - {R}_{s }{i}_{\alpha s } \\ \frac{d{i}_{\alpha s }}{dt}&= -\left(\frac{{R}_{s }}{{\sigma l}_{s }}-\frac{{R}_{r }}{{\sigma l}_{r }} \right){i}_{\alpha s }-{w}_{r} {i}_{\beta s }+ \frac{{R}_{r }}{{\sigma l}_{r }{l}_{s }}{\phi }_{\alpha s}+ \frac{{w}_{r} }{{\sigma l}_{s }}{\phi }_{\beta s}+ \frac{1 }{{\sigma l}_{s }} {V}_{\alpha s} \\ \frac{d{\phi }_{\beta s}}{dt}& = {V}_{\beta s}- {R}_{r }{i}_{\beta s } \\ \frac{d{i}_{\beta s }}{dt}&= -\left(\frac{{R}_{s }}{{\sigma l}_{s }}-\frac{{R}_{r }}{{\sigma l}_{r }} \right){i}_{\beta s }-{w}_{r} {i}_{\alpha s }+ \frac{{R}_{r }}{{\sigma l}_{r }{l}_{s }}{\phi }_{\beta s}- \frac{{w}_{r} }{{\sigma l}_{s }}{\phi }_{\alpha s}+ \frac{1 }{{\sigma l}_{s }} {V}_{\beta s}\end{aligned}\right.$$
(7)

where

\({R}_{s }, {R}_{r}\): Stator and rotor resistances

\({i}_{\alpha s }, {i}_{\beta s}\): Stator current components

\({l}_{s }, {l}_{r }:\) Stator and rotor inductances

\({\phi }_{\alpha s},{\phi }_{\beta s}\): Stator flux components

\(\sigma\): leakage factor.

The expression of the electromagnetic torque \({T}_{em}\) can be written as follows:

$${T}_{em}= \frac{3}{2} P({\phi }_{\alpha s}{i}_{\beta s} - {\phi }_{\beta s}{i}_{\alpha s} )$$
(8)

2.4 Centrifugal pump

The load torque of the pump \({T}_{pump}\) can be expressed in terms of the proportionality constant \({K}_{pump}\) and the rotational speed as follows [37, 38]:

$$T_{pump} = K_{pump} w^{2}$$
(9)

3 Direct torque control strategy

To control the IM driven water pumping system, DTC method is introduced. It’s considered one of the most utilized ones. This classical technique uses the hysteresis controllers to control the stator flux and the torque. The two-level hysteresis controller is used for the stator flux in which the input is the flux error resulting from the comparison between the referential and estimated values while the output is a Boolean variable that determines the increase or the decrease in the flux. However, the three-level hysteresis controller is used for the torque to define the increase or the decrsease in the torque. The Boolean variables and the stator flux vector position allow to determine the adequate inverter state to attain an appropriate torque from the switching table (Table 3). Moreover, the reference torque is obtained from the PI speed controller. The correctors keep the torque and flux within defined limits [39, 40].

Table 3 Switching Table
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Based on the model of the IM, the stator flux vector can be expressed as follows [41]:

$$\overrightarrow {{\Phi_{s} }} \left( t \right) = \mathop \int \limits_{0}^{t} ( \overrightarrow {{V_{S} }} \left( t \right) - R_{s} \overrightarrow {{i_{s} }} \left( t \right) ) dt$$
(10)

The magnitude and components of stator flux \(\Phi _{{s\alpha }}\) and \(\Phi _{{s\beta }}\) are estimated as follows [42]:

$$\Phi_{s\alpha } \left( t \right) = \mathop \int \limits_{0}^{t} ({ }V_{s\alpha } \left( t \right) - R_{s} i_{s\alpha } { }\left( t \right){ }){ }dt{ }$$
(11)
$$\Phi_{s\beta } \left( t \right) = \mathop \int \limits_{0}^{t} ({ }V_{s\beta } \left( t \right) - R_{s} i_{s\beta } { }\left( t \right){ }){ }dt{ }$$
(12)
$$\Phi _{s} = \sqrt {\Phi _{{s\alpha }}^{2} + \Phi _{{s\beta }}^{2} }$$
(13)

Moreover, the estimated electromagnetic torque can be expressed by:

$$T_{{em}} = \frac{3}{2}P(\Phi _{{s\alpha }} i _{{s\beta }} - \Phi _{{s\beta }} i _{{s\alpha }} )$$
(14)

In addition, the phase angle of the stator flux can be written as follows:

$$\Phi_{s} = { }\tan^{ - 1} \left( {\frac{{\Phi_{s\beta } }}{{\Phi_{s\alpha } }}} \right)$$
(15)

4 Speed PI controller synthesis

The PV water pumping system functions at variable speed which requires the use of speed controller for producing the reference torque (Fig. 4). The speed reference is determined by two components. The first one is obtained from DC link voltage controller as mentioned above. The second component is calculated by [37]:

$${\upomega }_{1}= \sqrt[3]{\frac{{\mathrm{P}}_{\mathrm{pv}}}{{\mathrm{K}}_{\mathrm{pump}}}}$$
(16)

The reference speed can be expressed by:

$${\upomega }_{\mathrm{ref}}= {\upomega }_{1}+ {\upomega }_{2}$$
(17)

The parameters of the speed controller are:

\({K}_{p}\)= 5.7, \({K}_{i}\)= 240.

5 DC-link voltage controller

The proportional-integral (PI) controller is introduced to maintain the DC bus voltage constant at its reference value (Fig. 5). Moreover, the second component of the reference speed (\({\omega }_{2}\)) is obtained from this controller. The voltage error (\({\Delta V}_{dc}\)) which results from the comparison between the reference bus voltage and the measured bus voltage (\({V}_{dc}\)) can be expressed as follows [43]:

$${\Delta V}_{{{\text{dc}}}} = {\text{ V}}_{{{\text{dc}}}}^{*} - {\text{V}}_{{{\text{dc}}}}$$
(18)
Fig. 5
figure 5

DC link voltage controller

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The output of the DC link voltage PI controller is expressed by:

$${{\omega }}_{1} = {{\omega }}_{{1\left( {{\text{K}} - 1} \right)}} + {\text{K}}_{{{\text{pdc}}}} \left\{ {\Delta {\text{V}}_{{{\text{dc}}\left( {\text{k}} \right)}} - \Delta {\text{V}}_{{{\text{dc}}\left( {{\text{k}} - 1} \right)}} } \right\} + {\text{K}}_{{{\text{idc}}}} \Delta {\text{V}}_{{{\text{dc}}({\text{k}})}}$$
(19)

The parameters of DC-link voltage controller are:

\({K}_{pdc}\) = 0.4, \({K}_{idc}\) = 1.7

6 Variable step size incremental conductance

Due to the low conversion efficiency of PV panel, the maximum power point tracking MPPT algorithm must be introduced to extract the maximum possible power from the PV array. Among the developed algorithms, the incremental conductance technique is often employed because of its high performance in terms of efficiency and tracking speed. It’s based on the power variation by calculating the incremental conductance and the conductance. The derivative of PV power by the voltage is zero means that the maximum power point is achieved. The equation of this algorithm can be expressed by:

$$\frac{{\mathrm{dP}}_{\mathrm{pv}}}{{\mathrm{dV}}_{\mathrm{pv}}}= \frac{\mathrm{d}({\mathrm{V}}_{\mathrm{pv}}{\mathrm{I}}_{\mathrm{pv}})}{{\mathrm{dV}}_{\mathrm{pv}}}= {\mathrm{I}}_{\mathrm{pv}}+ {\mathrm{V}}_{\mathrm{pv}}\frac{{\mathrm{dI}}_{\mathrm{pv}}}{{\mathrm{dV}}_{\mathrm{pv}}}=0$$
(20)

However, the conventional incremental conductance technique uses a fixed step size to track the power which reduces the performance under rapidly changing atmospheric conditions. Therefore, The incremental conductance based on variable step size is introduced to track MPP using the derivative of the Ppv with respect Vpv and assure good functioning under rapidly changing atmospheric conditions (Fig. 6) [28]:

Fig. 6
figure 6

MPPT algorithm

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7 Proposed control strategies

The proposed control scheme of the PV water pumping system is composed of two steps: the first one consists of replacing the switching table by fuzzy logic controller while the second step consists of introducing the adaptive fuzzy logic controller for speed regulation.

7.1 Adaptive fuzzy logic controller

The bloc of the adaptive fuzzy logic controller consists of three principal steps (Fig. 7): fuzzification, bloc inference and defuzzification [44]. The inputs of the controller are speed error \({e}_{\omega }\)(k) and change of speed error \({\Delta e}_{\omega }\) while the outputs are \(\Delta {K}_{i}\) and \(\Delta {K}_{p}\) which allow to adapt the integral gain \({K}_{i}\) and proportional gain \({K}_{p}\).

In the fuzzification step, based on the appropriate gains values, the inputs and outputs normalized in the interval [−1, 1] and fuzzified using the following linguistic variables (Fig. 8):

  • NB is negative big;

  • PB is positive big.

  • NM is negative middle;

  • PM is positive middle

  • NS is negative small;

  • PS is positive small;

  • ZO is zero;

Fig. 7
figure 7

Structure of AFLC

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Fig. 8
figure 8

Membership functions

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Tables 4 and 5 resume the adaptive FLC fuzzy rules:

Table 4 \(\Delta {K}_{p}\) fuzzy rules
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Table 5 \(\Delta {K}_{i}\) fuzzy rules
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The fuzzy inference is achieved using the Mamdani’s Min–Max technique. The degree of membership for \(\Delta {K}_{p}\) and \(\Delta {K}_{i}\) can be determined as follows:

$$\mu _{{\text{D}}} \left( {\Delta {\text{K}}_{{\text{p}}} } \right) = \mathop {\text{V}}\limits_{{{\text{j}} = 1}}^{{49}} \left\{ {\left[ {\mu _{{{\text{Aj}}}} \left( {{\text{e}}_{\omega } } \right)} \right.} \right.\Lambda \mu _{{{\text{Bj}}}} \left( {\Delta {\text{ e}}_{\omega } } \right)]\Lambda \mu _{{{\text{Dj}}}} \left( {\Delta {\text{K}}_{{\text{p}}} } \right)\}$$
(21)
$$\mu _{\sigma } \left( {\Delta {\text{K}}_{{\text{p}}} } \right) = \mathop {\text{V}}\limits_{{{\text{j}} = 1}}^{{49}} \left\{ {\left[ {\mu _{{{\text{Aj}}}} \left( {{\text{e}}_{\omega } } \right)} \right.} \right.\Lambda \mu _{{{\text{Bj}}}} \left( {\Delta {\text{e}}_{\omega } } \right)]\Lambda \mu _{{\sigma j}} \left( {\Delta {\text{K}}_{{\text{p}}} } \right)\}$$
(22)

In the defuzzification step, the center of gravity technique is adopted to convert fuzzy variables to crisp values. Therefore, \(\Delta {K}_{p}\) and \(\Delta {K}_{i}\) can be determined by the following equations:

$$\Delta {\text{K}}_{{\text{p}}} \left( {{\text{e}}_{\omega } ,\Delta {\text{e}}_{\omega } } \right) = \frac{{\sum\nolimits_{{{\text{j}} = 1}}^{{49}} {{\text{P}}_{{{\text{Oj}}}} \mu _{{\sigma {\text{j}}}} (\Delta {\text{K}}_{{\text{p}}} )} }}{{\sum\nolimits_{{{\text{j}} = 1}}^{{49}} {\mu _{{\sigma {\text{j}}}} (\Delta {\text{K}}_{{\text{p}}} )} }}$$
(23)
$$\Delta {\text{K}}_{{\text{i}}} \left( {{\text{e}}_{\omega } ,\Delta {\text{e}}_{\omega } } \right) = \frac{{\sum\nolimits_{{{\text{j}} = 1}}^{{49}} {{\text{I}}_{{{\text{Oj}}}} \mu _{{\sigma {\text{j}}}} (\Delta {\text{K}}_{{\text{i}}} )} }}{{\sum\nolimits_{{{\text{j}} = 1}}^{{49}} {\mu _{{\sigma {\text{j}}}} (\Delta {\text{K}}_{{\text{i}}} )} }}$$
(24)

Therefore, we obtain:

$$\left\{\begin{array}{c}{\mathrm{K}}_{\mathrm{p }}= {\mathrm{K}}_{\mathrm{p }}\left(0\right)+\Delta {\mathrm{K}}_{\mathrm{p}} \\ {\mathrm{K}}_{\mathrm{i }}= {\mathrm{K}}_{\mathrm{i }}\left(0\right)+\Delta {\mathrm{K}}_{\mathrm{i}}\end{array}\right.$$
(25)

where: \({K}_{p }\left(0\right)\), \({K}_{i }\left(0\right)\): the initial parameters.

7.2 Fuzzy logic controller

In order to get more high performance, the fuzzy logic controller is introduced.

7.2.1 Fuzzification

Triangular and singleton MFs are utilized for the inputs and outputs respectively. The linguistic variables employed for membership functions are: negative (N) and positive (P) and zero (Z) for the flux error (Fig. 9a), negative large (NL), positive large (PL), negative small (NS), zero (Z) and positive small (PS) for the torque error (Fig. 9b), \({\theta }_{1}\) to \({\theta }_{12}\) for the sector angle determination (Fig. 9c) and “0”, “1” for the converter’s switches \({S}_{i}\) (Fig. 9d).

Fig. 9
figure 9

Membership functions of input and output variables

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7.2.2 Fuzzy rules

In this step, a suitable rule-base based on the input and the outputs must be elaborated. Therefore, it can be defined as follows:

IF(\(\varepsilon _{{\Phi _{s} }}\) is \({A}_{N,Z,P}\)) AND (\({\varepsilon }_{{T}_{em}}\) is \({B}_{NL,NS,Z,PS,PL}\)) AND ( \(\theta _{s}\) is \(\theta_{s \mathrm{1,12}}\)) THEN (\({S}_{a}\) is \({D}_{\mathrm{0,1}}\)) AND (\({S}_{b}\) is \({D}_{\mathrm{0,1}}\)) AND (\({S}_{c}\) is \({D}_{\mathrm{0,1}}\)).

where

\({A}_{N,Z,P}\), \({B}_{NL,NS,Z,PS,PL}\), \(\theta_{s \mathrm{1,12}}\): the inputs fuzzy sets.

\({D}_{\mathrm{0,1}}\): the outputs fuzzy sets.

The fuzzy inference system inputs contain three, five and twelve fuzzy sets respectively which involve a set of one hundred eighty rules (3 ∗ 5 ∗ 12). Thus, Table 6 illustrates the inference matrix which represents the rules.

Table 6 Fuzzy rules
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7.2.3 Defuzzification

This step consists of elaborating the switches ‘state which are obtained from the output of the fuzzy controller. Therefore, the max method is introduced and expressed by the following equation:

$$\mu {\text{Vout}}^{\prime } \left( {\text{V}} \right) = {\text{max}}_{{{\text{i}} = 1}}^{{180}} {\text{max}}(\mu {\text{Vout}}^{\prime } \left( {\text{V}} \right))$$
(26)

where:

\(\mu {\text{Vout}}^{{\prime }}\): membership degree.

8 Results and discussion

The response of the suggested PV water pumping system is analyzed with the help of simulations carried out in MATLAB/Simulink. Different modes of operation are analyzed under various solar insolation levels to evaluate the dynamic response of the PV system. A comparative study of the proposed PV system based on intelligence artificial techniques with the PV system based on the conventional DTC (C_DTC) is effectuated.

8.1 Starting state performances of PV the system

Figure 10a and b illustrate the solar radiation profile and the extracted PV power using MPPT algorithm for the insolation of 1000 W/m2 respectively. Figure 10c shows the extracted PV power without MPPT technique. It can be seen that the power reaches the maximun power (1880 W) while the power of the PV panel reaches 275 W without MPPT algorithm. Figure 10d and e show the rotor speed and the pumped water for both techniques. Using the proposed technique, the speed response is faster compared to conventional DTC. In addition, a speed overshoot appears and its value equal to 2% when the PV system is controlled by the classical technique.

Fig. 10
figure 10

Performance of PV system under starting state

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Figure 10f shows the electromagnetic torque provided by the IM controlled by the proposed technique and C_DTC. It’s clear that the torque overshoot of the conventional method is higher than this with artificial intelligence techniques. It reaches 15.5 Nm while the torque overshoot of the proposed techniques reaches 14.6 Nm. In addition, a high starting torque degrades the performance of the system. On the other hand, a significant reduction of torque ripples is obtained by using the proposed technique which leads to improve the performance and the efficiency of the PV water pumping system.

Figure 10g illustrates the response of the developed stator flux for both control strategies.it can be noticed that the proposed method based on artificial intelligence techniques provides less flux ripples.

Figure 10h and i show the trajectory of stator flux alpha and beta for the proposed method and C_DTC. In Fig. 10h, it can be observed that big oscillations are produced. From Fig. 10i, the aforementioned ripples and oscillations are reduced. Moreover, we obtain a perfect circular trajectory using the proposed method.

8.2 Sudden change of solar irradiance

A sudden change of solar irradiance is applied to PV water pumping from the solar insolation of 1000 W/m2 to 500 W/m2 at 3 s (Fig. 11a). Figure 11b and c illustrate the obtained PV power using VSS INC and without MPPT technique. We can notice that the maximum power is extracted from the PV panel using VSS INC for all solar irradiance levels.

Fig. 11
figure 11

Performance of PV system under variable insolation

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Figure 11d and e show the rotor speed and the pumped water for both techniques. The response of rotor speed obtained using the proposed method takes 0.9 s to reach the reference while the C_DTC technique takes 1.5 s. As consequence, the improvement of the response time is clear.

Figure 11f illustrates the developed electromagnetic torque of IM using both methods. At the change of the solar irradiance, the torque ripples of the PV water pumping system controlled by the proposed technique are minimized. Contrary to the PV system controlled by the C_DTC, the oscillations of the torque are higher.

Figure 11g shows the developed stator flux of the IM issued by C_DTC and the proposed method. It can be seen that the flux ripples based on artificial intelligence techniques are minimized during the simulation period.

Figure 11h and i show the trajectory of stator flux alpha and beta for the proposed method and C_DTC under sudden change of solar irradiance. From Fig . 11h, it can be observed high oscillations on stator flux using the C_DTC method. However, we observe that stator flux controlled by the proposed method assures perfect circular trajectory.

Table 7 resumes the main improvements of the proposed method based on artificial intelligence techniques compared to C_DTC.

Table 7 Performance of PV system
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From the presented data, the proposed method based on the combination of AFLC and FLC provides better performance in terms of reduction of overshoot, ripples and the improvement of response time.

A performance comparison of the proposed control strategy and other techniques used for PV water pumping system is summarized in Table 8. We mention that they do not refer to the same conditions because it is very difficult to find several studies effectuated under the same conditions. Hence, a comparison based on the response time, torque ripples and rise time is carried out. Therefore, the proposed PV water pumping system provides better performance.

Table 8 Comparison between the proposed technique and other methods used for PVWPS
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9 Conclusion

A solar PV water pumping system based on the combination of artificial intelligence techniques is presented. The proposed control consists of introducing the fuzzy logic controller to give the suitable switching keys to function the centrifugal pump and the adaptive fuzzy logic control for speed regulation. In addition, variable step size INC is introduced to ensure better operation of the proposed system. A comparative study between the proposed and conventional methods are investigated to prove the robustness of the proposed PV water pumping system. Under these comparisons, the suggested control strategies are more efficient and more likely to be utilized in PV water pumping applications compared to C_DTC. We can notice that the major improvements of this work are:

  • Minimization of flux and torque ripples

  • The increasing of pumped water flow

  • Reduction of overshoot and undershoot

  • Enhancement of the response time

The next stage of our research will be the experimental validation of the proposed control strategies. Besides, other artificial intelligence techniques will be proposed for PV water pumping applications.