According to the characteristics of solar energy photovoltaic battery monomer equation according to of certain rules of series and parallel form photovoltaic array change different intensity of illumination and temperature, we can draw as Figure 2 and Figure 3 showed of photovoltaic array nonlinear output characteristic curve is:

Topology diagram of a three-phase photovoltaic system connected to an inverter. In Figure 4 :

In the three-phase stationary coordinate system, according to Kirchhoff’s current and voltage laws, the current and voltage equations of the grid-connected inverter can be obtained as [7] :

$C_1\frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}=i_{pv}-{\displaystyle \underset{k}{\sum }{i}_k{s}_k}$ (16)

$L\frac{\mathrm{d}{i}_k}{\mathrm{d}t}+R{i}_k=u_{kn}-u_{sk}-u_{on}$ (17)

$u_{kn}=u_{dc}\cdot s_k$ (18)

$u_{kn}=u_{ko}+u_{on}$ (19)

${\displaystyle \underset{k}{\sum }{i}_k={\displaystyle \underset{k}{\sum }{u}_{sk}=0}}$ (20)

Because the load of the photovoltaic inverter is symmetrical, the neutral point is 0. According to Equations (16) (20) it can be obtained:

$u_{on}=\frac{u_{dc}}{3}\cdot {\displaystyle \underset{k}{\sum }{s}_k}$ (21)

${\displaystyle \underset{k}{\sum }{u}_{ko}={\displaystyle \underset{k}{\sum }{u}_{kn}-3u_{on}=0}}$ (22)

where $k=a,b,c$ represents the switching state of an inverter bridge: When $s_k=1$ indicates the upper arm of the PV inverter is on and the lower arm is off, and $s_k=0$ indicates the upper arm is off and the lower arm is on. The inverter model in the three-phase stationary coordinate system is straightforward, but time-varying grid components complicate controller design. To simplify, this paper transforms the model from the three-phase stationary system to the synchronous rotating (d-q) system with the grid frequency. This conversion changes AC components to DC, easing control design.

First, Clarke transformation converts the three-phase system to a two-phase stationary system, aligned with phase A of the grid [8] . The mathematical model of the photovoltaic inverter synchronous coordinates, as shown in Figure 5 .

When the grid voltage is constant and inverter losses are neglected, the DC voltage of the grid-connected inverter is proportional to the d-axis component of the output current, and the active power p is also proportional to this current. Thus, the DC voltage can be controlled by regulating the active power [9] .

Figure 6 shows that the grid-connected inverter control system has an outer DC voltage loop and an inner loop for active and reactive currents. The DC voltage loop stabilizes or adjusts the DC side voltage. Due to the single-stage topology, the system often faces energy accumulation and DC voltage variations, which can lead to system collapse. DC voltage feedback can be achieved without static control using a PI controller.

The design of the voltage loop primarily aims to enhance the power supply’s ability to withstand interference with the load and is designed according to a type II system. To reduce overshoot, a filter link can be added to the reference voltage, or a ramp setting can be used to make the reference voltage change gradually. Assuming that the three-phase currents are symmetrical, the instantaneous power on the three-phase AC inductor is zero. We establish:

$C_1\frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}\cdot u_{dc}=u_{dc}{i}_{pv}-\frac{3}{2}{e}_d{i}_d$ (23)

It is further transformed to:

$C_1\frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}=i_{pv}-\frac{3e_d}{2u_{dc}}{i}_d$ (24)

Since the values of $e_d$ and $u_{dc}$ do not change much $-3e_d/2u_{dc}$, it can be represented by the constant K. Without considering the DC side disturbance, the transfer functions of active current and Dc voltage are expressed as

$\frac{U_{dc}\left(s\right)}{I_d\left(s\right)}=\frac{K}{s{C}_1}$ (25)

Let the PI regulator control equation be:

$i_q^{*}=\left(K_p+\frac{K_i}{s}\right)\left(u_{dc}^{*}-u_{dc}\right)=K_p\left(\frac{\tau s+1}{\tau s}\right)\left(u_{dc}^{*}-u_{dc}\right)$ (26)

Knowing the current closed-loop transfer function, $w\left(s\right)=1/\left(3T_{s}s+1\right)$ and considering the DC voltage sampling delay ( ${\text{e}}^{-T\tau }$), it can be approximated as $1/\left(Ts+1\right)=1/\left(4T_s+1\right)$. Combine equations (35) and (36) to get the voltage open-loop transfer function

${\omega }_o\left(s\right)=\frac{K_{p}K\left(\tau s+1\right)}{C_1\tau s^2\left(Ts+1\right)}$ (27)

According to a typical type II system design there are:

$\frac{K_{p}K}{C_1\tau }=\frac{h+1}{2{\tau }^2}$ (28)

Taking the intermediate bandwidth, $h=\frac{\tau }{4T_s}=5$, we get:

$K_p=\frac{3C_1}{20K{T}_s}$ (29)

$K_i=\frac{K_p}{20T_s}$ (30)

Design of the current loop PI, the sampling period of the current loop is the PWM switching period, $T_s$ the data acquisition ${\text{e}}^{-T_{s}s}$ and ${\text{e}}^{-0.5T_{s}s}$ control delays are respectively, the delay link can be equivalent to $1/\left(1.5T_{s}s+1\right)$.

The d and q-axis current control is symmetrical control, so taking the d-axis as an example, S-domain model of the PI controller is:

$k_p+\frac{k_i}{s}=\frac{k_p\left({\tau }_{i}s+1\right)}{{\tau }_{i}s}$ (31)

According to the typical ${\tau }_i=L/R$, Figure 5 as shown of the PI controller is

$W_o\left(s\right)=\frac{k_p}{R{\tau }_{i}s\left(1.5T_{s}s+1\right)}$ (32)

Then the PI controller parameters can be obtained:

$k_p=\frac{R{\tau }_i}{3T_s}=\frac{L}{3T_s}$ (33)

$k_i=\frac{k_p}{{\tau }_i}$ (34)

where $k_p$ and $k_i$ are the proportion coefficient and integral coefficient, respectively.

In this paper, we connected the photovoltaic system-inverter to the electricity grid of the Republic of Congo. The schematic representation of the electricity transmission network, shown in a single-line diagram (see Figure 7 ), provides a detailed overview of the elements of this complex network. This representation is crucial for understanding the layout and operation of the components that ensure electricity distribution across the country.

Firstly, the network consists of five power generation plants: The Congo Electric Power Plant (CEC) at node (1) located in the Pointe-Noire department, the Imboulou Power Plant at node (34) situated in the Pool department, the Djeno Power Plant at node (6) in the Pointe-Noire department, the Moukoukoulou Power Plant at node (19) in the Bouenza department, and finally, the Djoué Power Plant at node (23) in the Pool department, which is currently out of service. These plants play a crucial role in generating the energy necessary to supply the country. They are strategically located at various points across the Congolese territory to meet energy needs in a balanced manner. The geographical distribution of these plants aims to optimize production and minimize line losses.

Regarding consumption, the network includes 22 loads, representing the various locations where electricity is used. These loads are primarily localities situated across different departments throughout the network. Effective management of these loads is essential to ensure a stable and reliable supply to end users.

The network also features 24 transmission lines, which transport the electricity produced by the power plants to the loads. These lines often cover long distances, requiring rigorous planning and careful management to maintain distribution reliability. The transmission line infrastructure must be meticulously designed to minimize interruptions and ensure continuous delivery of electricity.

Finally, the network comprises 34 nodes, which are convergence points for the transmission lines. These nodes play a strategic role in the network by allowing the redistribution of electricity in various directions. Through these nodes, the network can be organized in a meshed manner, significantly improving supply security. In the event of a failure or malfunction in part of the network, the nodes enable the redirection of electricity through alternative routes, thereby ensuring service continuity.

The geographic and schematic representation of the electricity transmission network of the Republic of Congo offers a detailed and precise view of the entire system. It clearly visualizes the arrangement of power plants, loads, transmission lines, and nodes. This comprehensive view is essential for effective network management, facilitating its maintenance and future improvements

P&O is an advanced control technology for maximum power point tracking MPPT [10] [11] [12] [22] .

We design an intelligent algorithm to generate a reference voltage for the DC link. To overcome the limitations of the traditional P&O method, we have adopted a variable step size method, integrated into the MPPT control of a single-stage three-phase photovoltaic system. Simulations show improved performance [10] [11] [16] [20] - [23] with this method. Based on the DC reference voltage, the DC-AC converter adjusts the operating cycle. The behavior of solar panel indicating MPP and operating principle is shown in Figure 8 which indicates that the resulting change of PV power is observed as follows. When the PV module operating

point is on the left side of the curve ( $\frac{\Delta P}{\Delta V}$ is positive) which means the PV module output power increases, the perturbation of the PV module voltage should increase toward the MPP.

If the operating point of the module was on the right side of the curve $\frac{\Delta P}{\Delta V}$ is (negative), then the perturbation of the PV module voltage should be decreased toward the MPP.

A PI controller adjusts the DC voltage according to the measured error. This MPPT approach reacts more quickly to power variations caused by simultaneous changes in speed and irradiation, with irradiation estimated from the PI controller’s error. The DC voltage controller sets the reference current for the current controller, aiming to maintain a constant DC voltage despite rapid changes in atmospheric conditions.

Performance evaluation techniques are discussed on the basis of the dynamic parameters of the PV system. By comparing between two algorithms which are incremental conductance and P&O algorithm; improved incremental conductance algorithms, perturbations and observations techniques, one of them is better than others [10] - [22] ; the algorithm is the advanced control of technology one of the best, fastest and very easy to code. Due to the energy loss during the confusion and recovery periods when irrdiance changes, direct duty ratio control offers better energy utilization and better stability characteristics with the proposed of methods which are P&O algorithm. From Figures 8-10 , it can be seen that there is only one maximum power point under a certain light intensity, that is $\frac{\mathrm{d}P}{\mathrm{d}U}=0$. Taking the derivation of the voltage U on both sides of the instantaneous output power of the photovoltaic cell, the basic equation of this algorithms are follows as:

$P=U\cdot I$ then (35)

Taking the derivation of the voltage U on both sides of the instantaneous output power of the photovoltaic cell, $P=U\cdot I$ then is power of the cell since

$\frac{\mathrm{d}P}{\mathrm{d}U}=I+U\frac{\mathrm{d}I}{\mathrm{d}U}\cong I+U\frac{\Delta I}{\Delta U}$ (36)

Then $\frac{\mathrm{d}P}{\mathrm{d}U}=0$, is the output power of the photovoltaic cell reaches the maximum. Then it can be deduced that the following relationship must be satisfied when the operating point is at the maximum power point:

$\frac{\mathrm{d}I}{\mathrm{d}U}=-\frac{I}{U}$ (37)

Where $\frac{\Delta I}{\Delta U}$ is then the variation of current as follows as: Approximately $\frac{\mathrm{d}I}{\mathrm{d}U}$ is replacing, the criterion for maximum power tracking using the incremental conductance method as follows:

$\frac{\Delta I}{\Delta U}>-\frac{I}{U}$ Left side of MPPT (38)

$\frac{\Delta I}{\Delta U}=-\frac{I}{U}$ At the MPPT (39)

$\frac{\Delta I}{\Delta U}<-\frac{I}{U}$ right side of MPPT, (40)

The MPP can thus be tracked by comparing the instantaneous conductance $\frac{I}{U}$ to the incremental conductance $\frac{\Delta I}{\Delta U}$ and accordingly the voltage perturbation sign is determined still reaching the MPP [10] [11] [16] . The flow chart of the conventional Incremental conductance algorithm is shown in Figure 10 . If the irradiance increases or decreases, that’s, PV current increases or decreases, the MPP moves to right (left) with respect to PV voltage. When compared with this proposed method, low cost MPPTalgorithms as P&O Algorithm [22] , the main advantage of the incremental conductance algorithms is that it can determine the accurate direction to reach the MPP thus decreasing the steady–state oscillation and improving system response under rapidly changing conditions [10] - [22] . However, regarding the algorithm structure, conventional incremental conductance algorithm includes a number of division calculations and relatively complex decision-making process which in turn raises the need of a more powerful micro controller featuring higher clock frequency, larger memory and floating-point calculation capability decreasing the possibility of achieving a low cost. A solution to this conflicting situation is to have a variable step-size that gets smaller towards the MPP in order to balance the competing aims of convergence speed and tracking accuracy.

The scope $\frac{\Delta P}{\Delta U}$ can be calculated using the PV module voltage and current. The incremental conductance algorithm is derived by differentiating the PV module power with respect to voltage and setting the results equal to zero. For a direct control scheme which directly controls the converter switching without external control loops, the considered step is the change in converter duty ratio $\Delta D$ as shown in Equations (41)-(44).

$\Delta D=N_1\frac{\Delta P}{\Delta U}$ (41)

$\Delta P=P\left(k\right)-P\left(k-1\right)$ (42)

$\Delta U=U\left(k\right)-U\left(k-1\right)$ (43)

$\Delta D=D\left(k\right)-D\left(k-1\right)$ (44)

And $N_1$ is the scaling factor tuned at the design stage to adjust the conventional step-size $\Delta D$ to compromise between tracking accuracy [10] [18] and its convergence speed. The flow chart of the incremental conductance method is shown in Figure 10 . The main advantage of using the incremental conductance method [10] [11] is that the adaptability and control stability of the MPPT is high.

Stateflow is a MATLAB tool used to model and simulate decision logic systems and state machines, making it easier to design complex systems where states change based on events or conditions, as shown in Figure 11 .

Assessment of Proposed Incremental Conductance MPPT Algorithm Partial Shading, in conventional MPPT systems the energy produced by the shaded cell as complete power extraction from the PV array, can be preserved because it is inhibited by the cell [12] [23] . As shown in Figure 12 and also PV the charateristic of under uniform irradiance.