There are different types of solar (photovoltaic) cells, and each type of cell has its own efficiency and cost [6] [7] . In this article, we study the single-diode model where the cell is presented as an electric current generator whose behavior is equivalent to a current source shunted by a single diode. To take into account physical phenomena at the cell level, the model is completed by a series resistor Rs shunt or parallel resistor Rp. The equivalent electrical diagram of the photovoltaic cell is given in ( Figure 2 ) below [7] .

The expression of the current characteristic of the photovoltaic panel (Ipv) is expressed as [7] follows:

$Ipv=Iph-\text{Is}\left[\exp \left(\frac{q\left(V+I.Rs\right)}{K.Tc.A}-1\right)\right]-\frac{V+I.Rs}{Rp}$ (1)

The equation giving the current characteristic of a GPV can be written as follows:

$I=Np.Iph-Np.Is\left[\exp \left(\frac{qVoc}{Ns.A.K.Tc}\right)-1\right]-\frac{\left(\frac{Np.V}{Np}+I.Rs\right)}{Rp}$ (2)

In this article, the work was done with the solar panel of type:

SOLTON Power SPI-215P whose electrical characteristics are shown in Figure 3 above.

The following figures represent respectively the current-voltage, power-voltage characteristics for a temperature T = 298 K˚ and a variable illumination (E) of [1,0.8, 0.6, 0.4, 0.2] * 1000 W/m² and also a variable temperature of [40 30 20]˚C.

The use of DC-DC converters allows the control of electrical power in circuits operating in direct current with great flexibility and high efficiency which in our case will allow us to pursue the optimum operating point [7] [8] . In our article, we have a Boost type converter whose electrical diagram is given in Figure 4 below.

The application of Kirchhoff's laws (Knot and Mesh) on the two equivalent circuits of the two operating phases gives [8] :

$0<\text{t}<\text{DTs}$

$\text{DTs}<\text{t}<\text{Ts}$

$\{\begin{array}{c}{i}_{C1}=C_1\frac{d{v}_i}{dt}=i_i-i_L\\ i_{C2}=C2\frac{d{v}_0}{dt}=-i_0\\ v_L=L\frac{d{i}_l}{dt}=v_i-R_L{i}_L\end{array}$ (3)

$\{\begin{array}{c}{i}_{C1}=C_1\frac{d{v}_i}{dt}=i_i-i_L\\ i_{C2}=C2\frac{d{v}_0}{dt}=i_L-i_0\\ v_L=L\frac{d{i}_L}{dt}=v_i-v_0-R_L{i}_L\end{array}$ (4)

D: The duty cycle of the pulses that control the switch.

Ts: Being the period of impulses

By applying to the systems of equations (3) and (4) we can obtain the following relations [9] [10] :

$\{\begin{array}{c}{i}_L=i_i-C_1\frac{d{v}_i}{dt}\\ i_0=\left(1-D\right)i_L-C2\frac{d{v}_0}{dt}\\ v_i=\left(1-D\right)v_0+R_L{i}_L+L\frac{d{i}_L}{dt}\end{array}$ (5)

Output current: $Io=\frac{P}{Vo}$

Percentage of voltage variation: $\Delta V=1\%$

Percentage of current variation: $\Delta I=5\%$

Inductance: $L=\frac{Vin\left(Vo-Vin\right)}{\Delta I*fs*Vo}=1.45mH$

Ability: $C=\frac{Io\left(Vo-Vin\right)}{ƒs*\Delta Vo*Vo}=3227uF$

In this part, we present a P&O control algorithm for tracking the maximum power delivered by the photovoltaic generator [11] [12] . The principle of this algorithm is to perform a disturbance on the voltage of the panel Vpv while acting on the duty cycle α. Indeed, following this disturbance, we calculate the power supplied by the PV panel at time k, then we compare it to the previous one at time (k − 1). If the power increases, we approach the maximum power point, “PMP” and the variation of the duty cycle is maintained in the same direction [13] . On the contrary, if the power decreases, we move away from the maximum power point, “PMP”. Then, we must reverse the direction of the variation of the duty cycle [14] . The general algorithm of this control is given by ( Figure 5 ) below.

The program corresponding to this algorithm has been implemented in the Matlab/Simulink environment.

The operation of the inverter is based on the switching state of the switches. Switching is provided by three logic functions (Sa, Sb, Sc) [14] [15] . These functions made it possible to establish the relationship (10) between the voltages in three phases (Va Vb Vc) of the millet mill actuator and the voltage of the direct current source (Vdc).

We can then model the inverter by a matrix $\left[T\right]$ensuring the continuous-alternating transition $\left[Vac\right]=\left[T\right]\left[Vdc\right]$with [15] [16] :

$\left[\begin{array}{c}Va\\ Vb\\ Vc\end{array}\right]=\left[\frac{Vdc}{3}\right]\left[\begin{array}{ccc}2& -1& -1\\ -1& 2& -1\\ -1& -1& 2\end{array}\right]\left[\begin{array}{c}Sa\\ Sb\\ Sc\end{array}\right]$ (6)

The cage asynchronous machine is the most used actuator in most motorized applications due to its simplicity, robustness and cost [17] . To simulate the mill with the machine it is necessary to proceed with its modeling.

The stator of the Asynchronous Machine (MAS) receives its power supply from a balanced three-phase voltage system. This power supply generates the formation of a rotating magnetic field (in accordance with the FERRARIS theorem) inside the space between the stator and the rotor of the machine called the air gap (as illustrated in Figure 6 ) [18] [19] .

The voltage equations for the three stator phases and the three rotor phases are as follows [19] - [21] :

$\{\begin{array}{c}\begin{array}{c}Va=VmaxSin\left(\text{θ}\right)\\ Vb=VmaxSin\left(\text{θ}-\frac{2\text{π}}{3}\right)\end{array}\\ Vc=VmaxSin\left(\text{θ}-\frac{4\text{π}}{3}\right)\end{array}$ (7)

Equation statoriques:

$\{\begin{array}{l}Vsa=Rs*isa+L\frac{d}{dt}isa\\ Vsb=Rs*isb+L\frac{d}{dt}isb\\ Vsc=Rs*isc+L\frac{d}{dt}isc\end{array}$ (8)

Equations Rotoriques:

$\{\begin{array}{l}Vra=Rr*ira+L\frac{d}{dt}ira\\ Vrb=Rr*irb+L\frac{d}{dt}irb\\ Vrc=Rr*irc+L\frac{d}{dt}irc\end{array}$ (9)

The transformation of the three-phase system into a two-phase system for the modeling of the synchronous machine (MAS) involves a change of reference that alters the physical quantities such as voltages, fluxes and currents [20] . This transformation results in equations independent of the angle θ and reduces the order of the machine equations. The Park transformation highlights the direct axis (d) in the Park frame, as well as the quadrature axis (q). Figure 7 below shows the principle of the Park transformation [20] [21] .

1) Tension equations

In a frame of reference linked to the rotating stator field, i.e. a frame of reference rotating at the stator pulsation speed ω = p. Ω = dș/dt, the voltage equations in the Park frame are written [21] :

$\begin{array}{c}\{\begin{array}{c}Vsd=Rs*isd+\frac{\text{d}\varnothing \text{sd}}{\text{dt}}-\omega s\varnothing sq\\ Vsq=Rs*isq+\frac{\text{d}\varnothing \text{sq}}{\text{dt}}+\omega s\varnothing sd\end{array}\\ \{\begin{array}{c}Vrd=Rr*ird+\frac{\text{d}\varnothing \text{rd}}{\text{dt}}-\omega r\varnothing rq=0\\ Vrq=Rr*irq+\frac{\text{d}\varnothing \text{rq}}{\text{dt}}+\omega r\varnothing rd=0\end{array}\end{array}$ (10)

2) Flow equations

By performing the Park transformation on the equations relating to fluxes and currents, we can derive the electromagnetic relations of the machine [21] .

$\{\begin{array}{c}\varnothing sd=Ls*isd+Lm*ird\\ \varnothing sq=Ls*isq+Lm*irq\end{array}$ (11)

$\{\begin{array}{c}\varnothing rd=Lr*ird+Lm*isd\\ \varnothing rq=Lr*irq+Lm*isq\end{array}$

3) Electromagnetic torque equations

It is necessary to enrich the electrical model by including the formulations of the electromagnetic torque as well as the speed, which allows to describe the mechanical behavior. The expression of the electromagnetic torque obtained is formulated as follows: [21]

$Cem=p*M\left(isq.ird-isd.irq\right)=\frac{p.M}{Lr}\left(\varnothing rd.isq-\varnothing rq.isd\right)$ (12)

4) Mechanical equations

$j.\frac{d\Omega }{dt}+f.\Omega =Cem-Cr$ (13)

So we can define the Park matrix as follows [6] [21] :

$\left[\text{P}\left(\text{θ}\right)\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\text{Cosθ}& \text{Cos}\left(\text{θ}-\frac{2\text{π}}{3}\right)& \text{Cos}\left(\text{θ}-\frac{4\text{π}}{3}\right)\\ -\text{Sinθ}& -\text{Sin}\left(\text{θ}-\frac{2\text{π}}{3}\right)& -\text{Sin}\left(\text{θ}-\frac{4\text{π}}{3}\right)\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]$ (14)

The inverse transformation of the Park Matrix exclaims:

$\left[\text{P}\left(\text{θ}\right)\right]{}^{-1}=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\text{Cosθ}& -\text{Sinθ}& \frac{1}{\sqrt{2}}\\ \text{Cos}\left(\text{θ}-\frac{2\text{π}}{3}\right)& -\text{Sin}\left(\text{θ}-\frac{2\text{π}}{3}\right)& \frac{1}{\sqrt{2}}\\ \text{Cos}\left(\text{θ}-\frac{4\text{π}}{3}\right)& -\text{Sin}(\text{θ}-\frac{4\text{π}}{3}& \frac{1}{\sqrt{2}}\end{array}\right]$ (15)

The millet mill can be considered as a mechanical load for the engine. The model of the artisanal mill is defined by the load torque profile. The characteristic observed in the operation is that the grinding is done in steady state. The mechanical parameters (speed and torque) of the mill are a function of the flow and the grain (D). According to their operating principle, the speed varies with the flow rate, so there is a relationship between speed and flow rate [6] [22] .

$\omega =\omega s-k*D$ (16)

With

$\omega$: the rotation speed under load (rd/s);

$\omega s$: the no-load rotation speed (rd/s);

K: the characteristic constant of the mill (rd/kg);

D: grain flow rate (Kg/s).

Using the fundamental operating concept of the artisanal mill as well as the interaction of the hammers (propellers) that reduce the grain size, it is possible to express the torque configuration as a fusion of the four pre-existing load couples. Equation (17) represents the formula of the mechanical characteristic as a function of the parameters (K, k1, k2, K3, K4) that define the specific properties of the mill [22]

$Cr=k_1+k_2\omega +k_3{\omega }^2+k_4{\omega }^{-1}$ (17)

With: Cr mill load torque (Nm);

K_1, K_2, K₃ K₄ and K: The characteristic constants of the Moulin.

The parameters of the mill used in this work are established by means of experimental tests. The corresponding values are presented in Table 1 below [22] .

Then the resisting torque (Cr) becomes:

$Cr=k_3{\left({\omega }_0-kxD\right)}^2+k_2\left({\omega }_0-kxD\right)+k_1$ (18)

V/f scalar control with SVM (support vector machines) strategy is used for the control of the grinding device.

The principle of scalar control is to keep the V/f ratio constant. Equation (19) gives the expression for the maximum torque [22] .

$C_{e\max }=\frac{3p{L}_m{}^2}{2L_s\left(L_s{L}_r-L_m\right)}{\left(\frac{V_S-R_S{I}_S}{{\omega }_s}\right)}^2$ (19)

The simplified expression, or the voltage drop across the stator resistance is neglected. However, at low speeds the voltage drop cannot be neglected. We propose to compensate the voltage drop by a threshold voltage (Vo) and to operate the motor at a standstill at a limit frequency (fo). Figure 8 shows the profile of the voltage amplitude as a function of frequency [22] [23] .

The profile made it possible to establish the relation (20) giving the amplitude of the voltage (Vs) and the frequency (fo) from the nominal values of the voltage (Vn) and the frequency (fn) [23] .

$V_s=\frac{\left(V_n-V_0\right)}{\left({ƒ}_n-{ƒ}_0\right)}\left(f_n-ƒ\right)+V_0$ (20)

The closed-loop strategy is used to control the system. This strategy uses a loop that, from the mill speed error (ω), increases the frequency of the stator voltages to correct the speed error due to slip (ω_g). Equation (21) gives the generated stator pulsation (ω_s) [23] .

${\text{ω}}_{\text{s}}={\text{ω}}_{\text{g}}-\text{ω}$ (21)

The three voltages (i = 1, 2, 3), generated from the frequency, are given by equation (22).

$V_i=\left(\frac{\left(V_n-V_0\right)}{\left({ƒ}_n-{ƒ}_0\right)}\left({ƒ}_n-ƒ\right)+V_0\right)\sin \left(2\pi ft-2\frac{\left(i-1\right)}{3}\pi \right)$ (22)

The generated voltages (Vi) and pulsations (ωs) are the input variables of the SVM (support vector machines) command.

The vector control technique is based on the orientation of the flux in the machine, at the stator, rotor or in the air gap, along one of the two axes (d) or (q) [24] . Therefore, the control of the currents along the d axis determines the magnetic state of the machine, while the control of the currents along the q axis provides us with the necessary torque [23] [24] .

The equation that shows the relationship between the flux ${\varnothing }_r$ and the current $i_{sd}$ is:

$\{\begin{array}{c}{C}_{em}=k*i_{sq}\\ \frac{M}{T_r}{i}_{sd}=\frac{d{\varnothing }_r}{dt}+\frac{1}{T_r}{\varnothing }_r\end{array}$ (23)

It is observed that the direct stator current component $i_{sd}$determines the rotor flux amplitude, while the quadratic stator current component $i_{sq}$ determines the torque.

1) Cruise control settings

The speed control chain can be schematically represented in ( Figure 9 ) below [25] :

$\{\begin{array}{c}\frac{1}{{\omega }_n{}^2}=\frac{J}{K_i}{K}_i=J.{\omega }_n{}^2\\ \frac{2\epsilon }{{\omega }_n}=\frac{f+K_p}{K_i}{K}_p=2\epsilon J{\omega }_n-f\end{array}$ (24)

T_r_w = 0.1; % Response time at 5% of speed

w_n_T_r_w = 3% The settings for speed

K_i_w = wn² * J

K_p_w = ((2*x_i * K_i_w)/wn) − f_r

2) Flow Regulator Settings

To ensure the proper operation of the machine, the total flux must be kept constant at its nominal value during speed variations or application of charge additives [24] [25] .

The block diagram is given by ( Figure 10 ):

$AG\left(s\right)=\frac{\frac{S{K}_{p+}{K}_i}{{\tau }_r}}{S^2.+S\left(\frac{L_m{K}_p+1}{{\tau }_r}\right)+\frac{L_m{K}_i}{{\tau }_r}}$ (25)

For the control of the closed-loop system, it is necessary to carefully choose the coefficients $K_{i}et{K}_p$. The transfer function of a closed-loop system of order 2 is given by [16] [25] .

$\{\begin{array}{c}\frac{{\text{L}}_{\text{m}}{K}_p+1}{{\text{τ}}_{\text{r}}}=2\rho K_p=\frac{1}{{\text{L}}_{\text{m}}}\left(\frac{2L_r\rho }{R_r}-1\right)\\ \frac{{\text{L}}_{\text{m}}{K}_i}{{\text{τ}}_{\text{r}}}=2{\rho }^2{K}_i=\frac{2L_r{\rho }^2}{R_r}\end{array}$ (26)

T_r_flux = 0.01; % Response time for the flow

% Parameters for the stream

K_i_phi = 1/(L_m * T_r_flux)

K_p_phi = (Lr/Rr) * K_i_phi

Figure 11 illustrates the configuration of the system integrating a solar panel (PV), a chopper, an inverter, a Solar Powered Motor (SPM) with its load (mill). This system is driven by a scalar control strategy [16] .

The modeling and implementation of the scalar control strategy was carried out using Matlab/Simulink. This approach allowed to simulate the system, and the results were obtained by running the simulation of the model with the parameters that will be provided in the Appendix.

Figure 14 illustrates the configuration of the system integrating a solar panel (PV), a chopper, an inverter, a Solar Powered Motor (SPM) with its load (mill). This system is driven by a FOC (field oriented control) control strategy.

The model creation and implementation of the FOC (Field Oriented Control) control strategy were performed using Matlab/Simulink. This method was used to simulate the system and the results were obtained by running the model simulation.

The rotation speed of the scalar control follows the reference in regulation and in pursuit with a rise time of 0.248 s and a response time of 0.8298 s after a transient regime observed at the time of start-up. The speed undergoes an overshoot of 9.341% for a time of 1.7 s then it returns to its setpoint and keeps this value in steady state until the load couples are applied, which it then regulates to return to the setpoint again.

The response of the vector control presents good results in pursuit (precision) of the setpoint and rejection of disturbances (Robustness) of load with an overshoot of 11.798%, a voltage peak of 352 V is observed then it quickly regains its setpoint with a rise time of 0.0119 s showing the speed of the system and a response time of 0.348 s.

To regulate the mill speed, we used both control methods: scalar control and vector control, which provide precise control and improve energy efficiency. A comparative study between the two methods shows that scalar control (Discussion 1), simpler and less expensive, is advantageous for a small-scale mill powered by an asynchronous motor and a photovoltaic system. Although vector control offers better performance (Discussion 2), its complexity and higher cost are not justified for rural applications where scalar control effectively meets the needs. After analysis, we chose scalar control as the optimal solution.