*4.2. Incremental Conductance Modified (M-INC) Optimization Technique*

Under unstable irradiance and temperature conditions, a variation in voltage leads to a variation in the current. On the other hand, a voltage perturbation leads to a current perturbation when the photovoltaic field is subjected to a sudden change in atmospheric conditions [30]. Contrary to the classical incremental method [31], the M-INC algorithm distinguishes between these operating conditions and, thus, avoids the divergence caused by atmospheric disturbances. Indeed, the modified incremental conductance (M-INC) algorithm is supposed to act in the opposite way of the INC method when the system operates in the partial shading situation. This method is based on the power derivative being zero at MPP. It can be expressed as follows [32]:

$$\frac{dP}{dV} = \frac{d(IV)}{dV} = V\frac{dI}{dV} + I = 0\tag{6}$$

where *dP* is the power derivative, *dV* is the voltage derivative, and *dI* is the current derivative. Equation (6) can be expressed as follows:

$$\frac{\Delta I}{\Delta V} = \frac{dI}{dV} = -\frac{I}{V} \tag{7}$$

where Δ*V* and Δ*I* are the increments of voltage and current. The characteristics of the method can be obtained from the P–V curve and can be written as follows [32]:

$$\frac{dI}{dV} = -\frac{I}{V}\text{ at maximum power point}\tag{8}$$

$$-\frac{I}{V} \quad \prec \frac{dI}{dV} \text{ left to maximum power point} \tag{9}$$

$$-\frac{1}{V} \quad \succ \frac{dI}{dV}\text{ right to maximum power point}\tag{10}$$

The duty cycle of this technique is calculated by:

$$\mathfrak{a}\_{i}(n) = \mathfrak{a}\_{i}(n-1) \pm k\_{i} \left| \frac{P(n) - P(n-1)}{V(n) - V(n-1)} \right| = \mathfrak{a}\_{i}(n-1) \pm \delta \mathfrak{a}\_{i} \tag{11}$$

Figure 7 depicts a flowchart of the modified INC algorithm.

**Figure 7.** Flowchart of the M-INC algorithm.

#### *4.3. FL-INC Hybrid Optimization Technique*

The fuzzy logic of artificial intelligence is introduced in the incremental conductance technique to develop a new hybrid optimization method named FL-INC. The latter is applied to ensure a better optimization of the SWPS in case of very unstable weather conditions. Fuzzy logic allows the study and representation of imprecise knowledge and approximate reasoning [33]. Figure 8 shows the basic design of a fuzzy logic controller, which is composed of three stages. The first step is the fuzzification step, which consists in converting the input variables (physical variables) into linguistic variables (fuzzy variables), by establishing membership functions for the different input variables. The second step is the inference engine, which includes the inference block and the rule base to determine the output of the fuzzy controller from the inputs resulting from the fuzzification. Finally, defuzzification is the last step of the fuzzy controller; it allows for converting the fuzzy data provided by the inference mechanism into a physical or numerical quantity to define the decision process [34].

**Figure 8.** Fuzzy logic controller structure.

The optimization using the fuzzy logic technique can significantly reach the global power point. However, the performance of this control technique depends mainly on the human expertise. Indeed, the rules developed from the human operator's expertise are expressed in linguistic form. Moreover, these rules determine the dynamic performance of the fuzzy controller [35]. Therefore, applying this reasoning to the incremental algorithm improves the performance of the extracted power. Figure 9 shows a flowchart of the developed hybrid FL-INC technique.

**Figure 9.** Flowchart of the hybrid technique FL-INC.

The fuzzy controller proposed in this technique consists of two inputs, *E*(*k*) and Δ*E*(*k*), which are defined by the following equations:

$$E(k) = \frac{I(k) - I(k-1)}{V(k) - V(k-1)} + \frac{I(K)}{V(K)}\tag{12}$$

$$
\Delta E = E(k) - E(k-1) \tag{13}
$$

where *E*(*k*) is the derivative of the conductance calculated from the measured voltage and current. It is nullified when the operating point attains the MPP. Additionally, Δ*E*(*k*) is the error of the input *E*(*k*).

Additionally, the output of this fuzzy controller represents the change in the duty cycle (δα). Figure 10 shows the selected membership functions for E, ΔE, and δα.

**Figure 10.** Membership functions of the input (E, ΔE) and output (δα) variables.

The triangular membership function is chosen for all fuzzy sets because of its simplicity. The boundaries of the fuzzy variable range are usually normalized between −1 and +1. The control rules are used to make the decision and decide the action at the output of the fuzzy controller, which consists of 25 fuzzy rules, and are grouped in Table 4.


**Table 4.** Base of the fuzzy controller rules used for FL-INC.

Linguistic variables are expressed as: PB: positive big; PS: positive small; Z: zero; NS: negative small; and NB: negative big.

The control rules can be expressed as follows: rule: if (E is X) and (ΔE is Y), then (δα is W), where: X, Y, and W are the fuzzy sets of input and output variables.

The implementation of the different control rules is based on the instructions below:

