1 Introduction

The increase adulteration of the environment by contaminated wastes and emissions such as green house gases and CO\(_{2}\) is a serious problem that needs to be addressed [1]. Since the industrial revolution, the emission of CO\(_{2}\) has increased significantly based on a statistical study performed by NASA in 2016. The study showed that the atmospheric CO\(_{2}\) has never exceeded 300 parts per million (ppm) for 650,000 years, yet nowadays, the atmospheric CO\(_{2}\) has reached 400 ppm. The emissions have been increasing with a rate of 1.6% each year [2]. Hence, the greatest contribution of renewable energy technologies is generating nearly zero emissions of CO\(_{2}\) to the environment [3]. With this advantage, renewable energy has taken over many conventional power generation methods in variety of vital applications such as water pumping [4]. Water supply has always been a challenge in distant or isolated regions where infrastructure for water and energy distribution is missing [5]. Solar pumping technologies offer numerous benefits over diesel pumps, including enhanced efficiency and decreased operating and maintenance costs [6]. The PVPS has been used in countries such as India within households, animal farms, and different irrigation systems [7, 8]. The availability of pumping energy is well-matched to the demand for water, which is the biggest during the daytime. The use of the off-grid solar photovoltaic solution is considered a primary resource for tropical regions, where an electricity-grid source is not available [9]. Off-grid energy storage is required due to the highly intermittent nature of solar PV power generation. Typically, only physical storage solutions that tie electrical energy to something else are included in the concept of energy storage [10]. Other types of energy, such as thermal, chemical, kinetic, potential, or electrochemical energy [11]. These sorts of energy storage are designed to release energy when needed, typically in electrical or thermal form, while minimizing energy loss during the storage period. An electrochemical battery is a common demonstration of a physical storage option. Battery-free compared to battery-powered systems, PVPS is less costly and necessitates less maintenance. The storage batteries, however, offer the advantage of delivering steady performance throughout dark and cloudy hours. It is more cost-effective to install a water storage tank in the PVPS than to use batteries as a backup power source [12].

A performance investigation of PVPS with fixed tracking cost showed the superior performance of tracking system compared to fixed system [13] but with extra cost. A performance comparison between single stage and multistage centrifugal solar PV pumps showed high efficiency of single stage for low head application [14]. A study to establish the relation between the discharge rate and discharge increase with radiation and vice versa in same climatic conditions has been introduced by [15]. Different studies have proved the effect of MPPT on PVPS efficiency compared to conventional PVP systems [16,17,18]. The PVPS was categorized to have two main types; direct conversion PVPS and solar thermal system [19]. Focusing on direct conversion PVPS for the sake of this research, the system works by directly converting the solar energy into electricity powering the motors. PVPS was investigated in multiple countries due to the numerous advantages and suitability of such systems in critical applications. Configuration of PV array has to consider many factors such as the weather conditions and shading cases [20]. Power conditioning unit (PCU) is mainly responsible for regulating the transferred energy from the PV array to the motor pump. Based on the application and the motor used in the system, the PCU can be either DC-DC converter, or DC-AC converter [21]. Regarding the motors, some systems include DC motors benefiting from the DC motor high efficiency and simplicity of implementation. Although, DC motors are known to require often maintenance which can be costly in many cases [22]. One of the most reliable motors used in PVPS is induction motors. Unlike DC motors, induction motors are maintenance free, cheaper, and rugged in construction [23]. Water pumps come in two main types that are suitable to be used with PVPS, centrifugal pump which used rotation of an impeller to create centrifugal force while the impeller guide the water to the outlet [24]. The nonlinear system behaviour is caused by the stochastic variation in solar irradiance and time varying electrical characteristics of PV panels, furthermore the nonlinear characteristics of power converters, and the water pumps, [25]. The pump flow rate in PVPS is affected by the climatic conditions influencing the PV system, especially solar irradiance and air temperature variations. Measuring the performance of the pump depends on head (m) by which water is lifted, water to be pumped \(\left( \textrm{m}^{3}\right) \), system efficiency, diurnal variation in pump pressure due to change in irradiance, and pressure compensation [26]. These nonlinearities of the components of PVPS, i.e. PV panels, power converters, and induction motor pump, cause unexpected behaviour and unstable operation of PVPS, which significantly degrade the control performance over varying atmospheric conditions. This unexpected behaviour is related to the voltage-current characteristics of the PV panel, and the torque-speed characteristics of the induction motor. So, the main challenge is to extract the maximum power from the PV panel, while keeping the stable operation of the induction motor. Controlling the behaviour of the system around the MPP requires a Maximum Power Point Tracking (MPPT) controller. In [27, 28], a PVPS with a solar tracker and PV array was analysed to study the effect of applying an MPPT controller. The controller was proven to enhance the overall system performance regarding the generated power and water flow rate. After this first leading research, many approaches were followed to improve the application of MPPT controller on PV systems to ensure optimum performance. A study conducted by Bouchakour compared the application of P &O MPPT, Fuzzy MPPT, and the effect of using optimization algorithms such as GA and PSO [29]. Another study aimed to optimize the performance of MPPT controller by using metaheuristic algorithms such as GWO, grasshopper optimization (GHO) and rescue optimization algorithm (SRA) [30]. Ant lion optimization (ALO) also has been proposed for optimal allocation and sizing of distributed generation (DG) renewable sources for radial distribution system [31]. A recent approach to optimize the performance of a PV system is to use Fractional Order (FO) MPPT controller as proposed in [32]. Various studies have improved and applied FO-based MPPT controller with different approaches as in [33,34,35]. The improved fractional order variable step INC helps to attain stable output power under changing weather conditions [36].Two PI controllers for a PV-DC pump system were designed using the ABC optimization algorithm. The simulation results revealed that the designed ABC tuning PI controllers are reliable in their operation and provide excellent performance for changes in load torque [37]. Inspired by the conducted literature, MPPT controller has been applied on the proposed PVPS and optimized with metaheuristic algorithms as WOA and others. Also, a proposed topology of the solar water pumping system will be illustrated using boost converter in order to reduce the number of the solar panels as shown in Fig. 1.

Fig. 1
figure 1

The proposed solar water pumping system topology

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Seeking a solution for the challenges met by fellow researchers in reaching the optimum step size, this research introduces a hybrid metaheuristics optimization with a fractional-order MPPT controller to enhance the PVPS’ performance. The conducted literature showed the main challenge of locating the MPP for complex PVPs with conventional algorithms. Thus, adopting a hybrid algorithm combining the online nature of fractional order controller tuned by metaheuristic algorithm to achieve robust MPP tracking. Due to the nonlinear behaviour of the proposed system, metaheuristic optimization algorithms are essential to extract system parameters with high accuracy. The comparative analysis conducted in [38] showed the impact of extracting PV system parameters with both analytical and metaheuristic approaches on system performance. The study showed the higher accuracy of the extracted parameters with metaheuristic algorithms such as Genetic Algorithm (GA). Besides the accuracy of such algorithms, metaheuristic techniques have higher convergence, which is required in such a complex system as the proposed PVPs.

In the presented study, three metaheuristic optimization algorithms will be used to tune the main controller of the system. The chosen algorithms were studied to confirm their suitability for such application. GWO is known for the effectiveness in finding the optimum solution compared to other algorithms such as PSO in PV sizing applications [39]. Besides GWO, ALO is heavily used in power generation applications such as optimal reactive power dispatch as the algorithm proved the ability to minimize the objective function while taking into consideration all system constraints. The third algorithm in the comparison is the WOA which has the ability to perform exploration and exploitation simultaneously. In this article, a novel approach is introduced to developing a fractional-order MPPT controller to control the induction motor. The controller is augmented with metaheuristic optimization algorithms as an additional contribution to this work. The contribution and novelty of this article can be summarized as follows:

  • Developing a batteries-free PVPS by using tanks for storage to reduce the cost.

  • Introducing a DC-DC converter-based optimal design of a PVPS using a three-phase induction pump.

  • Developing the optimal design steps and experimental implementation of a 1 KW three-phase PVPS.

  • Comparing two distinctive design techniques based on the available components in the local market.

  • Studying and experimentally analysing the characteristics of PV panels as constant (voltage /current) sources.

  • Experimentally analysing the unexpected behaviour of the induction motor-driven pumps powered by PV panels.

  • Developing a modified MPPT technique to overcome the potential unexpected behaviour of PVPS.

  • Studying how crucial the use of the DC-DC converter is, on the characteristics of PV panels and showing the nonlinearity of the PVPS using a family of curves.

  • Experimentally validating the performance of the modified FO-MPPT and applying several metaheuristics optimizations algorithms at various environmental conditions.

  • Studies are conducted to compare the performance of the modified FO-MPPT with the optimized conventional MPPT methods (\( P \& O\) and INC) at different climatic conditions and a dynamic load.

This article is organized as follows: Sect. 2 is dedicated for the design and implementation of the proposed system, Sect. 3 shows the mathematical model for the proposed system, Sect. 4 presents the experimental setup used for performance validation, Sect. 5 discusses the characteristics of the used induction motor. Sections 6 and 7 illustrate the proposed MPPT and its experimental implementation results.

2 Design and Implementation of Photo-Voltaic Pumping System (PVPS)

PVPS design demands a creative solution to face the challenge of maximizing the produced energy and the amount of pumped water while minimizing the total investment costs.

The design procedure of a suggested pilot PVPS is shown in Fig. 2 where, the water demanded per day \(\Phi _d = 205\) [\(\mathrm{m^3/day}\)] and the average clear sky period \(D_h = 7 \) [\(\textrm{hr}\)] at the proposed location, i.e. (Giza, Egypt). The demand water flow rate \(\Phi _{w}\) [\(\mathrm{m^3/hr}\)] could be calculated by [40]:

$$\begin{aligned} \Phi _{w} = \frac{\Phi _d}{D_h} \end{aligned}$$
(1)
Fig. 2
figure 2

PVPS Design process

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The hydraulic power \(P_{H}\) [watt] and the pump-motor electric power \(P_{e}\) [watt] can be calculated as follow:

$$\begin{aligned} P_{H}&= \frac{ \Phi \times \rho \times g \times H}{ 3.6 \times 10 ^6} \end{aligned}$$
(2)
$$\begin{aligned} P_{e}&=\frac{P_{H}}{\eta _{\textrm{Pump}}} \end{aligned}$$
(3)

where \(\rho \) is the water density [1000 \(\mathrm{kg/m^3}\)], g is the gravity constant [9.81 \(\mathrm{m^2/s}\)] and H is the dynamic head [m]. The solar panels array is designed to meet the electrical power demand \(P_{e}\) considering losses in the system. The maximum power of the PV array \(P_{\textrm{mpp}}\) can be calculated as:

$$\begin{aligned} P_{\textrm{mpp}} = N_{s} \times V_{\textrm{mpp}} \times N_{p} \times I_{\textrm{mpp}} \end{aligned}$$
(4)

Where \(N_{s}\) and \(N_{p}\) are the number of series and parallel connected PV modules.

Fig. 3
figure 3

Solar water pumping system cost

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Table 1 The Proposed PVPS Parameters
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Table 2 Average cost for the needed components
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A local market survey has been performed to investigate PVPS cost as shown in Fig. 3, which revealed that the cost of solar panels accounts for more than \(60\%\). Using the traditional design procedure may necessitate the production of custom components like PV panels and Power Conditioning Unit (PCU), which raises the cost. As a result, in the suggested design method, market-available components listed in Table 2 will be used to keep costs. The use of the market-available will affect the number of series PV panels \(N_{s}\) as in Eq. (5 ) to satisfy the 3\(\phi \) inverter operating voltage.

$$\begin{aligned} N_{s} = \frac{V_{\textrm{dc}}}{V_{\textrm{mpp}}} \end{aligned}$$
(5)

where the \(V_{\textrm{mpp}}\) is the value MPP voltage from the panels manual minus \(15\%\) of its value for redundancy factor and \(V_{\textrm{dc}}\) is dc voltage of (3\(\phi \) inverter) and is estimated from the following relation:

$$\begin{aligned} \lambda \times \frac{V_{\textrm{dc}}}{2 \sqrt{2}}=\frac{V_{L-L}}{\sqrt{3}} \end{aligned}$$

where \(\lambda \) is the modulation index and \(V_{L-L}\) is the line voltage across the motor terminals. Hence, \(V_{\textrm{dc}}=\frac{2\sqrt{2}}{\sqrt{3}} \times 230=375 \mathrm {~V}\), which is the required voltage when the modulation index is 1.

The pilot PVPS parameters using the traditional design procedure [40] are shown in Table. 1. Seven PV panels are needed to meet the 3\(\phi \) inverter operating voltage, although only three panels are required to meet electrical demand which considered wasted cost.

Considering the electrical power demand, only three panels are required; however, they do not provide the inverter with the needed voltage. This voltage insufficiency is inspired to be overcome by using a Boost chopper.

The usage of Boost chopper reduces the number of series PV panels while boosting PV voltage and saving the total PVPS cost. Boost converter with adjustable duty cycle (D) can be utilized to reduce the number of series solar as follows:

$$\begin{aligned} N_{\text{ s }}=\frac{V_{\textrm{dc}}}{V_{\text{ mpp }}\times (1-D)} \end{aligned}$$
(6)

As a commercial impact of the application of the Boost converter reduces the cost of PV array by \(56\%\); however, three Boost converters cost will be added, \(45\%\) of PVPS’s total cost has been saved. Also, the adjustable duty cycle enhances the redundancy performance and allow the proposed design to maintain the system performance under harsh conditions.

figure a

A Comparison between the traditional and the proposed PVSP design techniques using different sizes is shown in Table 3.

Table 3 Comparative analysis of optimal design of PVPS
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The analysis was performed using MATLAB platform to implement the proposed design algorithm 1. The results demonstrate the superiority of the proposed design in terms of cost reduction.

The average cost reduction ($ \(\downarrow \)) of the proposed is calculated as follows:

$$\begin{aligned} \$\downarrow =(N_{C}-N_{N} )* P_{C}- N_{N} * B_{C} [\$] \end{aligned}$$
(7)

and total cost reduction \(\%\) ($ \(\downarrow \%\) ):

$$\begin{aligned} \$ \downarrow \% = \frac{\$ \downarrow }{T_{C}} [\%] \end{aligned}$$
(8)

where \(N_{C}\), \(N_{N}\), \(P_{C}\), \(B_{C}\) and \(T_{C}\) are traditional method No. of the pv panels, proposed method No. of pv panels, cost of the PV panel, cost of the Boost converter and the traditional method total cost [$], respectively.

The proposed design reduced the total costs by 30 to 45 \(\%\). The proposed design promises an effective cost reduction in PVPS with electrical power range [0.4 to 2.2] kw, as shown in Table. 3.

Fig. 4
figure 4

Circuit diagram of PV with boost converter

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Table 4 Parameters of a single PV unit
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3 Modelling and Behaviour Analysis of the Proposed PVPS

This section illustrates the nonlinear mathematical model of the proposed PVPS and the effect of the Boost converter on the PV characteristics. Assuming that (Rs \(<<\) Rsh), the reduced equivalent circuit of the proposed PVPS can be used as shown in Fig. 4, [41]. Applying the parameters in table 4 and representing the output voltage and current of the PV cell by Vpv and Ipv, respectively, the electrical behaviour of a PV array with \(N_{p}\) parallel cells and \(N_{s}\) series cells could be characterized by Eq. (9).

Fig. 5
figure 5

Voltage - Current source behaviour of PV module

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$$\begin{aligned} I_{\textrm{p v}}{=}N_{p} \times I_{\textrm{p h}}{-}N_{p} \times I_{\textrm{Ds}}\left[ \exp \left( \frac{q\left( \frac{V_{\textrm{p v}}}{N_s}{+}\frac{R_s I_{\textrm{p v}}}{N_p}\right) }{\eta _{d} K_{b} T}{-}1\right) \right] \nonumber \\ \end{aligned}$$
(9)

where:

$$\begin{aligned} I_{\textrm{p h}}=\frac{G}{G_{o}}\left[ I_{\textrm{sh ,o}}+K_i\left( T-T_{o}\right) \right] \end{aligned}$$
(10)

As shown in Fig. 5, the behaviour of PV source could be divided into three regions constant voltage source (CVS), constant current source (CCS)and transition between (CVS \( \& \) CCS) sources.

The behaviour of PV source could be described as follow:

  • CCS (Region A): PV behaves as approximated constant current source type which is considered as PV current variation \({\delta I_{\textrm{pv}} \approx 0}\) and \(P_{\textrm{pv}} \propto V_{\textrm{pv}} \).

  • CVS (Region B): in which PV behaves as approximated constant voltage source as PV voltage variation \({\delta V_{\textrm{pv}} \approx 0}\) and \(P_{\textrm{pv}} \propto I_{\textrm{pv}} \).

  • Transition between (CVS \( \& \) CCS) (Region C): Indicates indeterminate source type that is not considered as constant current source or constant voltage source as \({\delta I_{\textrm{pv}}, \delta V_{\textrm{pv}} \ne 0}\) and \( {V_{\textrm{pv}}} \propto \frac{1}{I_{\textrm{pv}}}\).

A simplified PV behaviour, Converting the region (C) to borderer at the \(PV_{\mathrm{(mpp)}}\), has been proposed to reduce the transition between (CVS \( \& \) CCS). So the overall PV behaviour could be divided into two regions and consider \(PV_{\mathrm{(mpp)}}\) as a switching point between (CVS \( \& \) CCS) as follows:

  • \(V_{\textrm{pv}}<V_{\textrm{mpp}}\): considered as approximated constant current source type in which, \(P_{\textrm{pv}} \propto V_{\textrm{pv}}\) \( \& \) \(I_{\textrm{pv}} \approx I_{\textrm{mpp}}\).

  • \(V_{\textrm{pv}}> V_{\textrm{mpp}}\): considered as approximated constant voltage source type in which, \(P_{\textrm{pv}} \propto \frac{1}{V_{pv}}\) \( \& \) \(V_{\textrm{pv}} \approx V_{\textrm{mpp}}\).

The dynamical model of the boost converter shown in Fig. 4 is expressed as follows:

$$\begin{aligned} \begin{aligned} C_i\frac{dV_{\textrm{pv}}}{dt}=I_G-\frac{V_{\textrm{pv}}}{R_{\textrm{MPP}}}-I_{L} \\ L\frac{dI_{L}}{\textrm{dt}}=V_{\textrm{pv}}-(1-D)V_o \\ C_o\frac{dV_{o}}{dt}=I_L(1-D)-\frac{V_{o}}{R_{o}}-I_{o} \end{aligned} \end{aligned}$$
(11)

As illustrated in Fig. 6, the boost converter reduces the number of series panels while increasing PV voltage and decreasing PV current.

Fig. 6
figure 6

Boost converter effect on PV-characteristics at [G = 720, T = 24]

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Two sets of the PV panel connect to the boost chopper shown in Fig. 4 have been practically connected in series. A family characteristic curves of this connected PV panels are shown in Fig. 7. The nonlinear behaviour as well as transitions between (CVS \( \& \) CCS) is expected to adversely affect the load behaviour. This will be illustrated in details in Sect. 5.

Fig. 7
figure 7

PV characteristic curves at different climatic conditions

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4 Experimental Setup of the Proposed PVPS

The experimental setup shown in Fig. 8 has been implemented to evaluate the performance of the proposed PVPS at different climatic conditions. The proposed setup includes 2 "Suntech STP365S" PV panels, 2 Boost choppers, Variable Frequency Driver (VFD), Three phase induction motor, centrifugal pump, weather station (Temperature and irradiance sensors), Arduino (Uno) board as microcontroller and PC with MATLAB software. The experimental behaviour of the PVPS while changing the value of VFD frequency at different climatic conditions is illustrated in Fig. 10. The results show unexpected behaviour, as increasing the frequency value cause the PV power, pump power and water flow rate to increase until PV power reached the MPP and sharply decreased again. To overcome this unexpected behaviour, MPPT control is recommended to be applied. MPPT controller works to maintain the PV array at higher voltage under unexpected changes in conditions. The variations in surrounding weather conditions such as partial shading make it challenging for the PV system to reach the MPP due to the nonlinear behaviour of the system under these conditions. Thus, implementing an MPP tracking algorithm is essential to maximize the power output to drive the IM.

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Fig. 8
figure 8

Hardware implementation of the 1 KW prototype PVPS

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Hill-Climbing MPPT is primarily a closed loop algorithm that has been used in PV power applications [42]. Figure 9 shows the steps of applying MPPT algorithm to automatically adjust the operating point in order to achieve the maximum power consumption from PV panels [43]. The implementation process of MPPT consists of three main MPPT perturbation schemes, e.g. (Feed-forward Frequency control MPPT, Feed-back voltage control MPPT and Feed-back current control MPPT).

4.1 Scheme (A): Feed-forward Frequency Control MPPT

This scheme controls PVPS by using direct action frequency as MPPT perturbation parameter to allow the search for MPP. Figure 11 shows the block diagram of the feed-forward MPPT control using frequency f as perturbation parameter.

As shown in Fig. 12, this scheme did not overcome the phenomena of the unexpected behaviour. Also, the results show sudden drop of the hydraulic power after reaching the MPP. Thus, the feed-back MPPT control is proposed to be applied in scheme B.

4.2 Scheme (B): Feed-back Voltage Control MPPT

A closed loop PV voltage control was used as perturbation parameter of MPPT in order to overcome the unforeseen behaviour of PVPS.

The suggested feed-back voltage control MPPT scheme’s block design is shown in Fig. 13. The observed behaviour in Fig. 14 shows that the PVPS desired operation at MPP is still not accomplished. Despite of using voltage closed loop did not overcome the unexpected PVPS behaviour and power loses problem.

Fig. 9
figure 9

MPPT algorithm flowchart

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Fig. 10
figure 10

Experimental results of PVPS with variable f at different climatic conditions

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4.3 Scheme (C): Feed-back Current Control MPPT

A closed-loop PV current control has been used as perturbation parameter of MPPT. Figure 15 shows the block diagram of the proposed closed-loop current MPPT control. The obtained behaviour is shown in Fig. 16. The findings show that employing a feed-back current control MPPT technique to operate at MPP does not succeed. Furthermore, the sudden decrease in both PV and hydraulic power is still noticeable. The unpredictable behaviour that causes power losses was shown to be unrelated to the MPPT perturbation feed-forward or feed-back schemes. In order to control the induction motor efficiently using PV source, it is essential to examine the PV behaviour as fluctuating CVS - CCS feeding induction motor.

5 Dynamical Analysis of IM Driven by PV Source

The operation of the IM will be studied based on the observed nonlinear PV source dynamics and switching between CCS and CVS at Sect. 3. Figure 17 shows the equivalent circuit of IM with pump load.

Fig. 11
figure 11

Feed-forward Frequency control MPPT block diagram

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Fig. 12
figure 12

Experimental results Feed-forward frequency control MPPT

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Fig. 13
figure 13

Feed-back voltage control MPPT block diagram

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Fig. 14
figure 14

Experimental results of Feed-back voltage control MPPT

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Fig. 15
figure 15

Feed-back current control MPPT block diagram

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Fig. 16
figure 16

Experimental results of Feed-back current control MPPT

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Fig. 17
figure 17

\(3\phi \) IM+ pump circuit diagram

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The 3 \(\phi \) IM power can be calculated as follows:

$$\begin{aligned}&P_{\text{ m } }=3 V_{m} I_{m} \cos \phi =\omega _{m} \cdot T_{m} \end{aligned}$$
(12)

where \(\omega _{m}= \frac{120 \pi f}{p} [rpm]\) and \(\cos \phi =\frac{R_{m}}{Z_{m}}\). IM equivalent impedance \(Z_{m} \) determined as:

$$\begin{aligned} Z_{m}=\sqrt{R_{m}^{2}+\left( s X_{m}\right) ^{2}} \end{aligned}$$
(13)

The Torque-speed relation of the three phase IM is determined as follows:

$$\begin{aligned} \begin{aligned}&T_{m}=\frac{3}{\omega _{m}} \cdot V_{m}^{2} \cdot \frac{s R_{m}}{\left( R_{m}^{2}+\left( sx_{m}\right) ^{2}\right) } \\&T_{m}=\frac{3 \cdot p}{2 \pi f} \cdot V_{m}^{2} \cdot \frac{s R_{m}}{\left( R_{m}^{2}+\left( sx_{m}\right) ^{2}\right) } \end{aligned} \end{aligned}$$
(14)
Fig. 18
figure 18

Torque-speed characteristics of IM

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The Torque-speed relation of the three phase IM at CVS where \(V_{\textrm{dc}}= k*V_{m}\) at ideal PWM inverter where \(P_{dc} =P_{ac}\) is demonstrated by Eq.(15):

$$\begin{aligned} T_{m}= & {} \varepsilon \cdot V_{\textrm{dc}}^{2} \frac{s R_{m}}{R_{m}^{2}+(s x_{m}^{2})} \end{aligned}$$
(15)
$$\begin{aligned} \varepsilon= & {} \frac{3 \cdot p \cdot \eta {ac}}{2 \pi f \cdot k} \end{aligned}$$
(16)

The Torque-speed relation of the three phase IM at fixed current source \(I_{\textrm{dc}}= I_{m}/3*k\) is presented by Eq. (17):

$$\begin{aligned} \begin{aligned}&T_{m}=\mu \cdot \frac{I_{\textrm{dc}}^{2} \cdot \left( R_{m}^{2}+\left( s x_{m}\right) ^{2}\right. }{s \cdot R_{m}} \\&\mu =\frac{p \cdot \eta {ac}}{6 \pi f} \cdot k^{2} \end{aligned} \end{aligned}$$
(17)

where \(\eta _{ac}\) : DC-AC inverter efficiency, \(T_{m}\): IM Torque ,f: frequency, \(V_{m}\) : IM Voltage ,s Slip factor ,\(R_{m}\) : IM Resistance, \(x_{m}\) : IM Inductance, k : = \( \lambda * f \) at constant flux operation and \(\lambda \) is the modulation index (0:1).

Figure 18 shows the torque-speed dynamical behaviour of constant voltage source, constant current, and PV source with different applied frequency. Increasing the frequency, according to dynamical behaviour, speeds up the motor and shifts the operating point to the right, resulting in more power at a specific torque. Whenever, the IM torque (\(T_{m})\) is greater than the pump torque (\(T_{L}\), the pump speed increases until reaching the Equilibrium Operating Point (EOP) of motor-pump set at which \(T_{L}\) = \(T_{m}\). The EOP analysis and torque-speed dynamical behaviour of CVS driven IM is demonstrated in Fig. 19 a with certain pump torque (\(T_{L}\)), frequency f and based on Eq. (15) could be illustrated as follow:

Fig. 19
figure 19

EOP of IM driven a CVS b CCS

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Table 5 EOP of IM at different feeding sources as Fig. 19
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  • Left side, at low speed, i.e. high slip value and \(T_{m} \propto \omega _{m}\) (lopsided operation region).

  • Right side, at high speed, i.e. low slip value and \(T_{m} \propto 1/\omega _{m}\) (equiponderant operation region).

Moreover, the EOP analysis torque-speed dynamical behaviour of CCS driven IM is demonstrated in Fig. 19 b with certain pump torque (\(T_{L}\)), frequency f and related to Eq. (17) can be examined as:

  • Left side, at low speed, i.e. high slip value, so and \(T_{m} \propto 1/\omega _{m}\) (equiponderant operation region).

  • Right side, at high speed, i.e. low slip value and \(T_{m} \propto \omega _{m}\) (lopsided operation region).

Table 5 illustrates EOP of IM at CVS and CSS Fed sources at different IM-pump operating points and load torque variation scenarios shown in Fig.19.

Finally, the EOP of IM driven by PV source can be examined using Fig. 20 which shows the EOP of IM driven by PV source. Changing the frequency f leads to change in the operating point and the relation between \(T_{L}\) and \(T_{m}\). EOP of motor-pump set moves from P1 to P4, then suddenly drop to P5 where P1 to P4 are equiponderant CVS and P5 is equiponderant CCS.

6 The Proposed MPPT Scheme of PVPS

According to the examined nature of PV generator, it has a nonlinear characteristics that are affected by the amount of irradiance and the temperature (\(G_{\textrm{pv}}, T_{\textrm{pv}})\). The open-circuit voltage of CVS is linearly dependent on PV temperature, on the other hand, the irradiance effect on the short-circuit current value of CCS. Furthermore, the switching point \(PV_{\textrm{mpp}}\) between CCS and CVS effects on the EOP of motor-pump as in Sect. 5, which moves due to the changes in irradiance or temperature. Moreover, employing the feed-back current and voltage control MPPT schemes illustrated in Sect. 4 was not effective to overcome the unexpected phenomena of IM driven by PV source.

It is necessary to have an effective perturbation parameter that reflects the variation in PV voltage and current.

Fig. 20
figure 20

EOP of IM with PV source fed at varying f

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Fig. 21
figure 21

Admittance and Resistance characteristics of PV panels

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The admittance (or impedance) of PV system is used as perturbation parameters instead of voltage and current, as shown in Fig.21. The admittance (\(\gamma _{pv}\)) was chosen rather than the impedance to start at point \(V_{pv}=V_{oc}, I_{pv}=0\) and \(\gamma _{pv} =0\). On the other hand, the impedance = \(\infty \). A closed-loop PV admittance control has been used as perturbation scheme of MPPT, where at MPP admittance Ipv/Vpv = \(\textrm{Impp} / \textrm{Vmpp}\). The obtained behaviour is shown in Fig. 23. The findings show that employing a feed-back admittance control MPPT technique, overcome the unexpected behaviour and provide a stable operation of IM.

Fig. 22
figure 22

Feed-back admittance MPPT control block diagram

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The technique for fractional order control systems has been widely used to enhance the performance of control systems. Motivated by the finding that a fractional order controller can outperform traditional integer order controllers. When compared to integer-order control, FO control has a more adaptable framework and extra degrees of freedom. As a result, it is possible to realize an accurate estimate of the system’s unknown uncertainties and increase the proposed system’s robustness and sensitivity by properly tuning the FO controller. Several scientists are currently using fractional order (FO) controllers to get the most stable performance of their systems. Fractional order INC shows a promising results in PV MPPT application with resistive load [25] and dynamic load [36].

Fig. 23
figure 23

Experimental results of Feed-back admittance MPPT Control

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Fig. 24
figure 24

The proposed Control block diagram

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The proposed FO-MPPT main criteria using \(\gamma _{pv}\) as perturbation parameter can be expressed by Eq. (18):

$$\begin{aligned} \begin{aligned} \frac{d^{\alpha }}{dV_{pv}^{\alpha }}(\frac{-I_{pv}}{V_{pv}})&=(\frac{-1}{V_{pv}})(\frac{\Gamma (2)}{\Gamma (2-\alpha )})(I_{pv})^{1-\alpha } \\&\quad - I_{pv}\frac{\Gamma (0)}{\Gamma (-\alpha )}V_{pv}^{1-\alpha } \end{aligned} \end{aligned}$$
(18)
Table 6 Summary of MPPT implementation schemes
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The control procedure of FO-MPPT algorithm can be expressed by the flow chart in Fig.24. The procedure starts with measuring the PV voltage and current (\(V_{pv}, I_{pv})\) to determine the MPPT action according to the following conditions:

  • Condition 1: If (\( \Delta V^\alpha \ne 0 \& \frac{d^{\alpha }I}{dV^{\alpha }}=\frac{d^{\alpha }}{dV^{\alpha }}(\frac{-I_{0}}{V_{0}})\)) OR (\( \Delta V^\alpha =0 \& \Delta I =0 )\rightarrow \) keep current \(\gamma _{pv}\).

  • Condition 2: If (\( \Delta V^\alpha \ne 0 \& \frac{d^{\alpha }I}{dV^{\alpha }}>\frac{d^{\alpha }}{dV^{\alpha }}(\frac{-I_{0}}{V_{0}})\)) OR (\( \Delta V^\alpha =0 \& \Delta I > 0 )\rightarrow \) increase \(\gamma _{pv}(k)\) by \(\gamma _{pv}(k)=\gamma _{pv}(k-1)+\Delta \gamma _{pv}\) .

  • Condition 3: If (\( \Delta V^\alpha \ne 0 \& \frac{d^{\alpha }I}{dV^{\alpha }}<\frac{d^{\alpha }}{dV^{\alpha }}(\frac{-I_{0}}{V_{0}})\)) OR (\( \Delta V^\alpha =0 \& \Delta I < 0 )\rightarrow \) decrease \(\gamma _{pv}(k)\) by \(\gamma _{pv}(k)=\gamma _{pv}(k-1)-\Delta \gamma _{pv}\)

The effective parameters of MPPT performance are \(\alpha \) and \(\Delta \gamma _{pv}\) needs to be tuned using metaheuristics as recommended by [25, 36]. Table 6 illustrates the summary of the proposed MPPT schemes, the sensors used, the MPPT’s parameters to be tuned and the software tools used. Where k is the scaling factor mapping between the control signal and VFD’s frequency range (0:50) HZ. The experimental results also show that the settling time of the system with the different MPPT schemes is approximated [8:10] sec.

7 Results and Discussion of Real-time MPPT Implementation

This section demonstrates the implementation of the proposed MPPT results in real-time using PVPS hardware demonstrated in Sect. 4. Data gathering and manipulation of the proposed algorithm is illustrated using the block diagram of the process shown Fig. 24. The weather sensors collect the temperature and irradiance values (\(T_{pv}, G_{pv})\) to calculate the reference MPP value \(P_{mpp}(k)\) based on the mathematical model and PV in Sect. 3. The FO-MPPT input values, i.e. (\(V_{pv}\) and \(I_{pv}\)) are measured using voltage and current sensors to calculate the MPPT output reference control signal (\(\gamma \)) using MATLAB. The frequency f values are calculated using closed loop admittance algorithm implemented in Arduino and applied to the VFD as shown in Fig.24. Several metaheuristics algorithms namely Gray wolf Optimizer (GWO) [44], Ant-Lion Optimizer (ALO) [45], and whale optimizer algorithm (WOA) [46] have been used for MPPT parameters online tuning. The optimizer loop is running in parallel to the MPPT loop, where the cost function is calculated using Mean Square Error (MSE) between the reference \(P_{mpp}(k)\) and the measured \(P_{pv}(k)\). The optimizer generates the optimal MPPT parameters, i.e. ( \(\alpha \) and \(\Delta \gamma \)) as shown in algorithm 2 which are directly feed to MPPT controller loop. The results were obtained for the proposed MPPT controllers under the actual climatic conditions with the same optimization parameters (initial conditions and the number of population). The optimization cost function:

$$\begin{aligned} J\left( \theta _0,\theta _1\right) =\frac{1}{N} \sum _{K=1}^N(P_{PV}(K)-P_{\textrm{mpp}})^2 \end{aligned}$$
(19)

where \(\theta _0, \theta _1 \) are FO-MPPT Parameters, i.e. \(\alpha \) and \(\Delta \gamma _{\textrm{pv}}\).

Table 7 shows the control parameters of the proposed metaheuristics Population size (N), Maximum iterations number, random number range and Number of variables [1,2] where in case integer MPPTs (\( P \& O\) and INC) is (\(\Delta \gamma \)) and FO-MPPT are (\(\Delta \gamma \) and \(\alpha \)). The optimization graphs (iteration vs Cost) are shown in Fig.28 where shows the high performance of WOA over ALO and GWO with FO-MPPT.

the experiments have been performed on the daily period from 8.30 AM to 4.30 PM. The overall experimental time period is divided into experiment/hour, during this experiment, each MPPT algorithm is been run for 6.5 minutes with sampling time = 25 sec for each optimizer iteration to validate the weather condition variations and evaluate the MPPT performance as shown in Fig.25. The PC configurations used for the experiments are a Core i7 processor, 64 GB Ram, GTX 1660 6GB graphics card.

Table 7 Control parameters used in ALO, GWO and WOA
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Fig. 25
figure 25

The proposed Real-Time Hardware Results Time Frame

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Table 8 Performance indices of comparative step response results of proposed MPPT techniques
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Fig. 26
figure 26

Comparative time response and controller action results of the proposed MPPT methods

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Fig. 27
figure 27

Statistical results of the proposed MPPT methods under the same climatic conditions

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A comparative study of the FO-MPPT is carried out different performance indices, i.e. the steady state power (\(P_{ss}\)), the rise time (\(T_r\)), and the settling time (\(T_s\)) to figure out the performance of the different MPPT techniques. Table 8 illustrates the obtained results performance indices of step time response. Figure 26 shows the obtained time response (\(P_{pv}\) vs Time) and control action (\(P_{pv}\) vs \(V_{pv}\)), also demonstrates the high time response performance and less controller action oscillation of FO-MPPT especially the FO-MPPT with WOA. In order to prove and compare the efficiency of different MPPT techniques to harvest maximum power from the PV modules, the output power of PV water pumping system was monitored under uncertain load and weather conditions. The efficiency (\(\eta \)) is shown in Eq. (20), the power steady state ripple (\(P_{rp}\)), and the minimum MPP tracking time \(\tau _r\) were also used as performance indices to compare the proposed FO techniques with the conventional MPPT methods. Figure 29 represents comparative results of the proposed FO-MPPT with conventional MPPT at different days with varying climatic conditions along with a the dynamic load. Figure 27 demonstrates the statistical results of the Mean Square Error (MSE) and the MPPT efficiency of the proposed MPPT with selected samples for four days over the year 2021-2022 where:

Fig. 28
figure 28

Optimization graphs (Cost vs Iterations)

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Fig. 29
figure 29

Comparative Average steady state results of the proposed MPPT methods using WOA for different days

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  • Day1: 25-November 2021 (Winter)

  • Day2: 1-January 2022 (Winter)

  • Day3: 18-March 2022 ( Summer)

  • Day4: 20-May 2022 (Summer)

The statistical results show the high performance of FO-MPPT over the integer MPPTs (P &O and INC) with the different metaheuristics optimization. The results in Table 9 present the high performance and efficiency of using the suggested FO-MPPT with WOA.

$$\begin{aligned} \begin{aligned}&P_{\textrm{avg}}(G,T)=\frac{1}{K} \sum _{0}^{K}(P_{PV}(K)) .\;\; and \\&\eta =\frac{P_{\textrm{mpp}}(G,T)}{P_{\textrm{avg}}(G,T)}*100 \% \end{aligned} \end{aligned}$$
(20)
Table 9 Performance of PVPS steady state results using the proposed MPPT with WOA for different days
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figure b

8 Conclusion and Future Work

In this article, an optimal design technique of PVPS has been introduced and compared with the traditional design. The proposed design technique saved between 30 to 45 \(\%\) of the PVPS total cost. The nonlinear behaviour of PV generator as CCS and CVS power source has been illustrated and shows the effect of this transition on the load behaviour. The EOP of IM driven by PV source has been analysed to recognize and investigate the unexpected behaviour of PVPS. In the presented study, a novel MPPT perturbation scheme using admittance has been proposed to achieve the sustainable operation of IM. A fractional-order MPPT control has been proposed and optimized using three metaheuristic techniques (GWO, ALO and WOA) to feed the IM with the maximum power. MATLAB is used to control the pilot PVPS and to study the suggested MPPT behaviour. Based on the conducted experiment, the obtained results can be summarized as follows:

  • The proposed FO-MPPT demonstrates better results compared to the traditional MPPT.

  • The FO-MPPT enhanced with metaheuristic optimization proves the ability to control IM-load, although the nonlinearity and the inconsistent behaviour of the PV generators.

  • The performance of FO-MPPT outgoes the other techniques since it provides an extra dynamical variable (\(\alpha \)) to the MPPT control.

  • WOA optimization technique shows higher MPPT tuning performance over the other metaheuristics techniques with fixed number of iterations and initial conditions.

For future consideration, this work could be extended in the future by studying the effect of the partial shading and apply more metaheuristics techniques. MATLAB R2020 has been used for the prototype and for research purposes. For commercial implementation, it is recommended to use an open-source software like python and windows applications like visual C.