# A Stochastic Optimization Algorithm to Enhance Controllers of Photovoltaic Systems

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## Abstract

**:**

## 1. Introduction

- (i)
- The FBL, used as a main control technique, is implemented to enable high local performance.
- (ii)
- The SMC technique is then combined with the FBL to attenuate the effects of random and matched disturbances. This combination is expected to improve the dynamic performance of the controlled system.
- (iii)
- Due to the uncertainties in the model, it is impossible to fully eliminate the disturbances utilizing only the conventional SMC. To overcome this problem, a method associated with an SMO technique is implemented to allow fine-tuning of the controller gains ensuring the efficiency of the PV system as a stand-alone power generator in a remote area.

## 2. Background and Problem Formulation

#### 2.1. Robust Nonlinear Control Strategy

**Definition**

**1**

**([30])**

**.**The system stated in (1) is assumed to have a local relative degree denoted by$r$at an operating point${\mathit{X}}_{0}$if:

- (i)
- ${\mathrm{L}}_{G}{\mathrm{L}}_{F}^{k}H\left(\mathit{X}\right)=0,$for all$\mathit{X}$around${\mathit{X}}_{0}$and for all$k<r-1$;
- (ii)
- ${\mathrm{L}}_{G}{\mathrm{L}}_{F}^{r-1}H\left({\mathit{X}}_{0}\right)\ne 0$,

**Definition**

**2**

**([30])**

**.**The relative degree${c}_{i}$ of the output $y\left(t\right)$, in relation to the disturbance, is expressed as the least integer${c}_{i}$ such that

- (i)
- When ${c}_{i}>r$, the impact of the disturbance is not as straightforward as the control input, with ${c}_{i}$ being given in (4). The entire information from the disturbance is accessible through the system states, and as such, the decoupling of $y\left(t\right)$ from the disturbance $\mathit{D}$ is rendered unnecessary.
- (ii)
- When ${c}_{i}=r$, the disturbance and control input have a similar influence on the system output, and feedforward performance is essential to perform decoupling.
- (iii)
- When ${c}_{i}<r$, the disturbance influences the output more straightforwardly than the control output. Some form of predictive activity is required to perform disturbance rejection.

**Theorem**

**1**

**([14])**

**.**Consider the nonlinear system stated in (1) and assume${c}_{i}$to be the relative degree specified in Definition 2. Then, the function$T\left(\mathit{X},\mathit{D}\right)$that facilitates the independence of$y\left(t\right)$from$\mathit{D}$is formulated as

#### 2.2. PV Model Description

^{−19}C), ${K}_{\mathrm{B}}$ is the Boltzmann constant (1.38 × 10

^{−23}J/K), $T$ is the cell temperature in Kelvin degrees (°K), and $n$ is the ideality factor of the diode. Replacing the expressions given in (9) and (10) in (8), we obtain

^{2}), ${K}_{\mathrm{I}}$ is the temperature coefficient of the cell short-circuiting current, ${T}_{\mathrm{ref}}$ is the reference temperature of the cell (in °K, with °K = 25 °C + 273), $E$ is the solar radiation in (W/m

^{2}), and ${E}_{\mathrm{ref}}$ is the reference insolation of the cell, which is equal to 1000 W/m

^{2}. In addition, the cell saturation current varies with the temperature of the cell, which is described as

#### 2.3. Nonlinear Control Design

- (i)
- Determining the PV system’s state space equations [38,39] as$$\{\begin{array}{c}{\dot{I}}_{\mathrm{A}}=\frac{{V}_{\mathrm{A}}}{L}-\frac{{K}_{\mathrm{B}}}{L}\mathsf{\Omega}-\frac{R}{L}{I}_{\mathrm{A}}+{d}_{1},\\ \dot{\Omega}=\frac{{K}_{\mathrm{B}}}{J}{I}_{\mathrm{A}}-\frac{{K}_{\mathrm{T}}+F}{J}\Omega -{d}_{2},\\ \begin{array}{c}{\dot{I}}_{\mathrm{L}}=-\frac{{V}_{\mathrm{A}}}{L}+\frac{{V}_{\mathrm{PV}}}{L}(U-\mathrm{sin}\left({V}_{\mathrm{A}}\right)),\\ {\dot{V}}_{\mathrm{A}}=\frac{{I}_{\mathrm{L}}}{C}-\frac{{V}_{\mathrm{A}}}{RC},\end{array}\end{array}$$
- (ii)
- Expressing the system in the conventional standard configuration given by$$\{\begin{array}{c}\dot{\mathit{X}}=F\left(\mathit{X}\right)+G\left(\mathit{X}\right)U+{\rm Y}\left(\mathit{X}\right)\mathit{D},\\ y=H\left(\mathit{X}\right),\end{array}$$$$\mathit{X}=\left[\begin{array}{c}{X}_{1}\\ {X}_{2}\\ \begin{array}{c}{X}_{3}\\ {X}_{4}\end{array}\end{array}\right]=\left[\begin{array}{c}{I}_{\mathrm{A}}\\ \mathsf{\Omega}\\ \begin{array}{c}{I}_{L}\\ {V}_{\mathrm{A}}\end{array}\end{array}\right],$$$$F\left(\mathit{X}\right)=\left[\begin{array}{c}\frac{{V}_{\mathrm{A}}}{L}-\frac{{K}_{\mathrm{B}}}{L}\mathsf{\Omega}-\frac{R}{L}{I}_{\mathrm{A}}\\ \frac{{K}_{\mathrm{B}}}{J}{I}_{\mathrm{A}}-\frac{{K}_{\mathrm{T}}+F}{J}\mathsf{\Omega}\text{}\\ \begin{array}{c}-\frac{{V}_{\mathrm{A}}}{L}-\frac{{V}_{\mathrm{PV}}}{L}\mathrm{sin}\left({V}_{\mathrm{A}}\right)\\ \frac{{I}_{\mathrm{L}}}{C}-\frac{{V}_{\mathrm{A}}}{RC}\end{array}\end{array}\right],\text{}G\left(\mathit{X}\right)=\left[\begin{array}{c}\begin{array}{c}0\\ 0\\ \frac{{V}_{\mathrm{PV}}}{L}\end{array}\\ 0\end{array}\right],$$$${\rm Y}\left(\mathit{X}\right)=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 0\end{array}\right],\mathit{D}=\left[\begin{array}{c}\begin{array}{c}{d}_{1}\\ {d}_{2}\\ {d}_{3}\end{array}\\ 0\end{array}\right],U=\rho ,\text{}y=H\left(\mathit{X}\right)={V}_{\mathrm{A}}.$$
- (iii)
- Stating $R=1.072\mathsf{\Omega}$ and $L=0.05H$, which denote the armature resistance and inductance, as mentioned, respectively; $J=476\times {10}^{-6}{\text{}\mathrm{kg}\text{}\mathrm{m}}^{2}$ and $F=88\times {10}^{-5}{\mathrm{m}}^{2}/\mathrm{s}$, which denote the DC machine’s inertia and viscous friction coefficients, as also mentioned, respectively; and C $=4000\times {10}^{-6}F,$ which represents the capacitor, as mentioned.
- (iv)
- Computing the relative degree $r$ by deriving y up to the point, when the control variable $U$ materializes to equations given by$$\{\begin{array}{c}y={V}_{\mathrm{A}},\\ \dot{y}={\dot{V}}_{\mathrm{A}}=\frac{{I}_{\mathrm{L}}}{C}-\frac{{V}_{\mathrm{A}}}{RC},\\ \begin{array}{c}\ddot{y}={\ddot{V}}_{\mathrm{A}}=-\frac{1}{LC}{V}_{\mathrm{A}}+\frac{{V}_{\mathrm{PV}}}{LC}U-\frac{{V}_{\mathrm{PV}}}{{L}^{2}}\mathrm{sin}\left({V}_{\mathrm{A}}\right)-\frac{1}{C}{d}_{3}-\frac{1}{RC}\left(\frac{{I}_{\mathrm{L}}}{C}-\frac{{V}_{\mathrm{A}}}{RC}\right),\end{array}\end{array}$$
- (v)
- Expressing the input controller as$$U=\frac{LCv+{V}_{\mathrm{A}}+\frac{L}{R}\left(\frac{{I}_{\mathrm{L}}}{C}-\frac{{V}_{\mathrm{A}}}{RC}\right)+{L}^{2}\mathrm{sin}\left({V}_{\mathrm{A}}\right)}{{V}_{\mathrm{PV}}}+T\left(\mathit{X},\mathit{D}\right),$$

## 3. Linearization, Sliding Modes, and Control Input Synthesis Methodology

#### 3.1. Input/Output Linearization and Second-Order Sliding Modes

#### 3.2. Implementation of the Disturbance-Free Control Model

#### 3.3. Control Implementation in Presence of Model Disturbances

#### 3.4. Analytic SOSM and Nonlinear Implementation

## 4. Control Design Based on the Slime Mould Algorithm

#### 4.1. Introduction to the Slime Mould Algorithm and Problem Expression

#### 4.2. Fundamentals of the Concept

#### 4.3. Algorithm Synthesis

Algorithm 1 SMO control strategy |

Step 1: Activate the PV system’s initial population as $\mathit{X}\left(0\right)={\left[{I}_{\mathrm{A}}\left(0\right),\text{}\mathsf{\Omega}\left(0\right),{I}_{\mathrm{L}}\left(0\right),{V}_{\mathrm{A}}\left(0\right)\right]}^{\mathrm{T}}$. Step 2: Fix the boundaries for all parameters. Step 3: Establish the fitness function expressed through $P={{\displaystyle \int}}_{0}^{\infty}e{\left(t\right)}^{2}\mathrm{d}t$. Step 4: Set the parameter population size of the SMO method and $\mathrm{max}\_\mathrm{iter}$. Step 5: Fix the locations for the slime mould ${\Gamma}_{i}$, for $i\left\{1,\dots ,n\right\}$, as $l\le \mathrm{max}\_\mathrm{iter}$: 5.1 Compute the fitness function for all the slime mould. 5.2 Ascertain the function T ( X,D) depicted by (7).5.3 Calculate the control law expressed by (24). 5.4 Re-assess for the finest fitness ${P}_{\mathrm{best}}=\mathrm{min}\left\{{{\displaystyle \int}}_{0}^{\infty}e{\left(t\right)}^{2}\mathrm{d}t\right\}$. 5.5 State the weight W by way of (36). 5.6 Re-evaluate $l,{\overrightarrow{V}}_{b},{\overrightarrow{V}}_{c}$ for the search portions. 5.7 Re-detect positions for each of the search portions through (37). 5.8 Do $i=i+1$. Step 6: Resume ${P}_{\mathrm{best}}=\mathrm{min}\left\{{{\displaystyle \int}}_{0}^{\infty}e{\left(t\right)}^{2}\mathrm{d}t\right\}$. Step 7: Confirm the control $U$ formulated in (31). |

#### 4.4. SMO Control Implementation and Simulation Breakdown

#### 4.5. New Scenario under Natural Irradiance

^{2}. The experimental results are reported next:

- (i)
- We state clearly that our designed scheme offers satisfactory performance as can be seen in Figure 16.
- (ii)
- The first study is performed by using the fundamental FBL technique without applying disturbing signals to the closed-loop system. This fact is shown in Figure 17. Notice that FBL provides high performance in stabilizing the controlled system.
- (iii)
- The second study is conducted by considering disturbed random and matched signals, as shown in Figure 18. Note that the dynamical performance is highly affected.
- (iv)
- Observe that, in this new scenario, as expected, the FBL controller totally loses its dynamical performance.
- (v)
- The designed SMO-SMC-FBL scheme developed in this paper is now compared to fundamental FBL (analytic technique ignoring disturbances), SOSM (analytic second-order sliding mode considering disturbance), and FBL (analytic technique considering disturbances). The results are shown in Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23, from where we detect that the superiority of the SMO technique is obvious. Indeed, the global behavior of the controlled system is stable and accurate.
- (vi)
- Therefore, the steady-state regime is attained in a quite satisfactory way.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Structure of the studied PV system, where ${I}_{\mathrm{PV}}$ and ${V}_{\mathrm{PV}}$ denote its current and voltage.

**Figure 3.**Electrical scheme of the studied system, where C is a capacitor, L is an inductor, ${I}_{\mathrm{L}}$ is the inductor current, and ${S}_{\mathrm{T}},{S}_{\mathrm{D}}$ are a transistor and a diode of the step-down converter, respectively.

**Figure 6.**System dynamics using I/O linearization controller for: (

**a**) Motor pump current, (

**b**) motor pump angular speed, (

**c**) motor pump voltage, (

**d**) DC converter inductance current, and (

**e**) control input stabilization.

**Figure 7.**I/O control performance considering unmeasured random and matched disturbances for: (

**a**) Motor pump current, (

**b**) motor pump angular speed, (

**c**) motor pump voltage, (

**d**) DC converter inductance current, (

**e**) control input stabilization.

**Figure 8.**Second-order sliding mode control performance considering unmeasured random and matched disturbances for: (

**a**) Motor pump current, (

**b**) motor pump angular speed, (

**c**) motor pump voltage, (

**d**) DC converter inductance current.

**Figure 16.**Simulation of a real profile of the natural irradiance around the value of $1000\text{}\mathrm{W}/{\mathrm{m}}^{2}$.

**Figure 17.**Dynamics of the motor pump with FBL control ignoring disturbances for: (

**a**) Motor pump current, (

**b**) motor pump angular speed. (

**c**) motor pump voltage, (

**d**) DC converter inductance current, (

**e**) control input stabilization.

**Figure 18.**Dynamics of the motor pump with FBL control including disturbances for: (

**a**) Motor pump current, (

**b**) motor pump angular speed, (

**c**) motor pump voltage, (

**d**) DC converter inductance current. (

**e**) control input stabilization.

Description | Parameter |
---|---|

Power at the maximum power point ($P\mathrm{mpp}$) | $190\mathrm{W}$ |

Voltage at the maximum power point ($V\mathrm{mpp}$) | $24.3\mathrm{V}$ |

Current at the maximum power point ($I\mathrm{mpp}$) | 7.82 A |

Open-circuit voltage (${V}_{\mathrm{OC}}$) | 30.6 V |

Short-circuit current (${I}_{\mathrm{SC}}$) | 8.5 A |

Number of cells per module | 50 |

Description | Values |
---|---|

PV generator | ${I}_{\mathrm{PH}}=4.4A;{I}_{\mathrm{S}}=52.75\times {10}^{-6}\mathrm{A};{V}_{\mathrm{T}}=6.73\mathrm{V}$ |

Capacitor | $C=4000\times {10}^{-6}\mathrm{Farad}$ |

The identified parameters of the DC motor | $R=1.07\mathsf{\Omega};L=0.05\mathrm{Henry};J=476\times {10}^{-6}\mathrm{kg}\text{}\times {\mathrm{m}}^{2};F=88\times {10}^{-5}\text{}\mathrm{per}\text{}\mathrm{unit},$${K}_{\mathrm{T}}=14\times {10}^{-4},{K}_{\mathrm{B}}=45\times {10}^{-3}\mathrm{per}\text{}\mathrm{unit}$ |

Parameters | Values |
---|---|

$q$ | $50$ |

$\overrightarrow{W}$ | $\mathrm{adaptive}$ |

${\overrightarrow{V}}_{b}$ | $-1$ to 1 |

${\overrightarrow{V}}_{c}$ | 1 to 0 |

**Table 4.**Controller gain tuning for several control inputs in the irradiance trajectory of Figure 4.

Time Slot | (8–9 h) | (9–10 h) | (10–11 h) | (11–14 h) | (14–15 h) | (15–16 h) |
---|---|---|---|---|---|---|

I/O FBL control | 10.50 | 10.15 | 9.34 | 8.00 | 10.01 | 11.00 |

Decoupling I/O FBL | 10.33 | 9.89 | 9.22 | 8.75 | 8.63 | 8.10 |

SOSM control | 8.00 | 7.74 | 7.10 | 6.68 | 7.00 | 7.50 |

SMO control | 4.35 | 3.56 | 3.00 | 2.88 | 3.45 | 3.72 |

Time Slot | (8–9 h) | (9–10 h) | (10–11 h) | (11–14 h) | (14–15 h) | (15–16 h) |
---|---|---|---|---|---|---|

I/O FBL control | 11.75 | 10.12 | 9.94 | 9.01 | 8.51 | 7.20 |

Decoupling I/O FBL | 20.23 | 19.91 | 19.45 | 18.17 | 17.263 | 16.51 |

SOSM control | 17.00 | 15.84 | 13.123 | 11.361 | 9.97 | 8.85 |

SMO control | 10 | 9.01 | 8.2 | 6.075 | 4.97 | 3.762 |

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**MDPI and ACS Style**

Charfeddine, S.; Alharbi, H.; Jerbi, H.; Kchaou, M.; Abbassi, R.; Leiva, V.
A Stochastic Optimization Algorithm to Enhance Controllers of Photovoltaic Systems. *Mathematics* **2022**, *10*, 2128.
https://doi.org/10.3390/math10122128

**AMA Style**

Charfeddine S, Alharbi H, Jerbi H, Kchaou M, Abbassi R, Leiva V.
A Stochastic Optimization Algorithm to Enhance Controllers of Photovoltaic Systems. *Mathematics*. 2022; 10(12):2128.
https://doi.org/10.3390/math10122128

**Chicago/Turabian Style**

Charfeddine, Samia, Hadeel Alharbi, Houssem Jerbi, Mourad Kchaou, Rabeh Abbassi, and Víctor Leiva.
2022. "A Stochastic Optimization Algorithm to Enhance Controllers of Photovoltaic Systems" *Mathematics* 10, no. 12: 2128.
https://doi.org/10.3390/math10122128