# A New Nonlinear Controller for the Maximum Power Point Tracking of Photovoltaic Systems in Micro Grid Applications Based on Modified Anti-Disturbance Compensation

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## Abstract

**:**

## 1. Introduction

- i.
- A new nonlinear controller is proposed as a new nonlinear state error feedback (NLSEF). The new nonlinear controller consists of a proposed tracking differentiator (TD) combined with a new super twisting sliding mode controller (STC-SM).
- ii.
- A new nonlinear extended state observer (NLESO) is proposed to estimate the total disturbance of the PV system.
- iii.
- Last but not least, the modified ADRC is composed of the aforementioned proposed nonlinear controller (i.e., STC-SM) and the proposed NLESO to stabilize the system and track the MPP in the presence of disturbance and parameter variations.

## 2. PV System Characteristics and Modeling

#### 2.1. PV System Modeling

#### 2.2. PV System Characteristics

## 3. DC-DC Buck Converter

## 4. The Modified ADRC Design

#### 4.1. The Proposed Tracking Differentiator

#### 4.2. The Proposed Nonlinear Controllers

#### 4.2.1. The Proposed Super Twisting Sliding Mode Controller (STC-SM)

#### 4.2.2. The Proposed Nonlinear PID—Tracking Differentiator (NLPID-TD)

#### 4.3. The Proposed NLESO

## 5. Stability of the Closed-Loop System

**Assumption**

**1.**

**Theorem**

**1.**

**Proof.**

**Assumption**

**2**

- $\ell $and$\dot{\ell}$ are bounded, which, $\underset{0\le t\le \infty}{\mathit{sup}}\ell \le {c}_{1}\text{}$and $\underset{0\le t\le \infty}{\mathit{sup}}\dot{\ell}\le {c}_{2}$
- $\ell $and $\dot{\ell}$ are constant at the steady-state, which, $\underset{t\to \infty}{\mathrm{lim}}\ell ={c}_{3}$ and $\underset{t\to \infty}{\mathrm{lim}}\dot{\ell}\le 0$

**Assumption**

**3.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Assumption**

**3.**

## 6. Simulation Results

- Case study one: Irradiation changes with constant temperature at standard temperature conditions (STC).
- Case study two: Temperature changes with constant irradiation at standard temperature conditions (STC).
- Case study three: Load changes in both irradiation and temperature at standard temperature conditions (STC).

- i.
- Case study one. Irradiation changes with constant temperature at standard temperature conditions (STC).

- ii.
- Case study two. Temperature changes with constant irradiation at standard temperature conditions (STC).

- iii.
- Case study three: load change with both irradiation and temperature at standard temperature conditions (STC).

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. The Configuration of the ADRC Schemes

- 1.
- ADRC

- 2.
- LADRC

- 3.
- IADRC, STC-ADRC, and NLPD-ADRC

#### Appendix A.2. The Significance of the Different ADRC Schemes

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**Figure 2.**The PV characteristics (

**a**). V-I characteristics at different G and T = 25 °C. (

**b**) V-I characteristics at different T and $\mathrm{G}=1000\mathrm{W}/{\mathrm{m}}^{2}$. (

**c**) P-V characteristics at different G and T = 25 °C. (

**d**) P-V characteristics at different T and $\mathrm{G}=1000\text{}\mathrm{W}/{\mathrm{m}}^{2}$.

**Figure 4.**The MPPT–based proposed ADRC with the PV system with relative degree ($\rho =1)$, the numbers inside the parentheses in this figure indicate equation number in this paper.

**Figure 5.**Comparison between the proposed method (NLPD-ADRC) and LADRC, NLADRC, and IADRC under the irradiance change. (

**a**) PV Voltage. (

**b**) PV Current. (

**c**) PV Power. (

**d**) The converter output voltage (Load Voltage). (

**e**) The converter output current (Load Current). (

**f**) The converter output power (Load Power).

**Figure 6.**Comparison between the proposed method (STC-ADRC) and LADRC, NLADRC, and IADRC under irradiance changes. (

**a**) PV Voltage. (

**b**) PV Current. (

**c**) PV Power. (

**d**) The converter output voltage (Load Voltage). (

**e**) The converter output current (Load Current). (

**f**) The converter output power (Load Power).

**Figure 7.**Comparison between the proposed method (NLPD-ADRC) and LADRC, NLADRC, and IADRC under the temperature change. (

**a**) PV Voltage. (

**b**) PV Current. (

**c**) PV Power. (

**d**) The converter output voltage (Load Voltage). (

**e**) The converter output current (Load Current). (

**f**) The converter output power (Load Power).

**Figure 8.**Comparison between the proposed method (STC-ADRC) and LADRC, NLADRC, and IADRC under load changes by $\Delta {R}_{L}=\pm 20\%$. (

**a**) PV Voltage. (

**b**) PV Current. (

**c**) PV Power. (

**d**) The converter output voltage (Load Voltage). (

**e**) The converter output current (Load Current). (

**f**) The converter output power (Load Power).

**Table 1.**PV-cell sampled parameters [43].

Parameters | Description | Value | Unit |
---|---|---|---|

${K}_{0}$ | Boltzmann constant | $1.380\times {10}^{-23}$ | $\mathrm{J}/\mathrm{K}$ |

${q}_{0}$ | Electron charge | $1.602\times {10}^{-19}$ | $\mathrm{C}$ |

${I}_{SC}$ | Short circuit current at STC | $7.84\times {10}^{0}$ | $\mathrm{A}$ |

${V}_{OC}$ | Open circuit voltage at STC | $36.3\times {10}^{0}$ | $\mathrm{V}$ |

${K}_{v}$ | Temperature voltage constant | $18\times {10}^{-1}$ | $\mathrm{V}/\mathrm{K}\%$ |

${K}_{i}$ | Temperature current constant | $0.00175$ | $\mathrm{A}/\mathrm{K}\%$ |

${A}_{diode}$ | Diode ideality factor | $1.3$ | $\mathrm{Unitless}$ |

${T}_{ref}$ | Temperature at STC | $298$ | $\mathrm{K}$ |

${G}_{ref}$ | Irradiation at STC | $1000$ | $\mathrm{w}/{\mathrm{m}}^{2}$ |

${E}_{{g}_{0}}$ | Band-gap energy | $1.12$ | $\mathrm{Ev}$ |

${R}_{s}$ | Series resistance | $0.2656$ | $\mathsf{\Omega}$ |

${R}_{P}$ | Parallel resistance | $1000$ | $\mathsf{\Omega}$ |

${N}_{s}$ | Number of cells connected in series | $60$ | $\mathrm{Unitless}$ |

${N}_{P}$ | Number of cells connected in parallel | $1$ | $\mathrm{Unitless}$ |

${N}_{ms}$ | Number of modules connected in series | $10$ | $\mathrm{Unitless}$ |

Parameters | Description | Value | Unit |
---|---|---|---|

${C}_{1}$ | The input capacitor | $350$ | $mF$ |

${C}_{2}$ | The output capacitor | $35$ | $mF$ |

$L$ | Inductor | $270$ | $mH$ |

${R}_{L}$ | Load resistance | $1.6$ | $\mathsf{\Omega}$ |

${f}_{s}$ | Switching frequency | $50$ | $H\mathcal{z}$ |

Performance Index | Description | Mathematical Representation |
---|---|---|

$ITAE$ | Integral Time Absolute Error | ${{\displaystyle \int}}_{0}^{tf}t\left|e\left(t\right)\right|dt$ |

$IAU$ | Integral Absolute of the control signal | ${{\displaystyle \int}}_{0}^{tf}\left|u\left(t\right)\right|dt$ |

$ISU$ | Integral Square of the control signal | ${{\displaystyle \int}}_{0}^{tf}u{\left(t\right)}^{2}dt$ |

Scheme | ADRC Parts | ||
---|---|---|---|

TD | SEF | ESO | |

LADRC | $-$ | ${u}_{LPID}={k}_{p}\tilde{e}+{k}_{i}{\int}_{0}^{T}\tilde{e}dt+{k}_{d}\frac{d\tilde{e}}{dt};u={u}_{{0}_{LPID}}-{z}_{2}/{b}_{0}$ $\mathrm{Where}\text{}{\mathrm{k}}_{\mathrm{p}},{\text{}\mathrm{k}}_{\mathrm{i}},{\mathrm{k}}_{\mathrm{d}}$ are the proportional, integral, and derivative gains, respectively. | The LESO can be expressed as follows: $\{\begin{array}{l}{\dot{z}}_{1}={z}_{2}+{b}_{0}u+{\beta}_{1}\left({e}_{1}\right)\\ {\dot{z}}_{2}={\beta}_{2}\left({e}_{1}\right)\end{array}$ |

ADRC | $-$ | The NLSEF of [33] can be expressed as follows: $\{\begin{array}{l}fal\left({\tilde{e}}_{1},{\alpha}_{1},{\delta}_{1}\right)=\{\begin{array}{c}{\tilde{e}}_{1}/\left({\delta}_{1}{}^{1-{\alpha}_{1}}\right),x\le {\delta}_{1}\\ {\left|{\tilde{e}}_{1}\right|}^{{\alpha}_{1}}sign\left({\tilde{e}}_{1}\right),x{\delta}_{1}\end{array}\\ {u}_{0}=fal\left({\tilde{e}}_{1},{\alpha}_{1},{\delta}_{1}\right)\\ {u}_{1}={u}_{{0}_{NLSEF}}-{z}_{2}/{b}_{0}\end{array}$ | |

IADRC | $-$ | The INLSEF of [44] can be expressed as: $\{\begin{array}{c}{u}_{1}=\frac{{k}_{1}}{1+exp\left({\tilde{e}}^{2}\right)}{\left|\tilde{e}\right|}^{{\alpha}_{1}}sign\left(\tilde{e}\right)\\ {u}_{2}=\frac{{k}_{2}}{1+exp\left(\int {\tilde{e}}^{2}dt\right)}{\left|{\displaystyle \int}\tilde{e}dt\right|}^{{\alpha}_{2}}sign\left({\displaystyle \int}\tilde{e}dt\right)\\ {u}_{{0}_{INLSEF}}={u}_{1}+{u}_{2};u={u}_{INLSEF}-{z}_{2}/{b}_{0}\end{array}$ | The SMESO of [45] can be expressed as:$\{\begin{array}{c}{\dot{z}}_{1}\left(t\right)={z}_{2}\left(t\right)+{\beta}_{1}\left(k\left({e}_{1}\left(t\right)\right){e}_{1}\left(t\right)\right)\\ {\dot{z}}_{\rho +1}\left(t\right)={\beta}_{\rho +1}\left(k\left({e}_{1}\left(t\right)\right){e}_{1}\left(t\right)\right)\\ k\left({e}_{1}\left(t\right)\right)={k}_{\alpha}{\left|{e}_{1}\right|}^{\alpha -1}+{k}_{\beta}{\left|{e}_{1}\right|}^{\beta}\end{array}$ $\mathrm{where}\text{}k\left({e}_{1}\left(t\right)\right)$$\mathrm{is}\text{}\mathrm{a}\text{}\mathrm{nonlinear}\text{}\mathrm{function}\text{}\mathrm{and}\text{}\alpha \mathrm{and}\beta $$\mathrm{are}\text{}\mathrm{positive}\text{}\mathrm{tuning}\text{}\mathrm{parameters}\text{}\mathrm{that}\text{}\mathrm{must}\text{}\mathrm{be}\text{}\mathrm{less}\text{}\mathrm{than}\text{}1.\text{}{k}_{\alpha}\mathrm{and}{k}_{\beta}$ are the nonlinear function gains and are tuning parameters. |

STC-ADRC | Equation (12) | The proposed STC-SM of Equations (12) and (13). | Proposed NLSEO of Equations (16) and (17). |

NLPD-ADRC | Equation (12) | The proposed NLPD of Equation (33). | Proposed NLSEO of Equation (18). |

ADRC Unit | Parameter | Value | Parameter | Value |
---|---|---|---|---|

LPID | ${k}_{p}$ | $0.088$ | ${k}_{i}$ | 1 × 10^{−9} |

${k}_{d}$ | $0.088$ | $\delta $ | $10.326$ | |

LESO | ${\beta}_{1}$ | $1.78$ | ${\beta}_{2}$ | $0.792100$ |

ADRC Unit | Parameter | Value | Parameter | Value |
---|---|---|---|---|

NLSEF | ${\alpha}_{1}$ | $0.7729$ | ${\delta}_{1}$ | $0.3949$ |

LESO | ${\beta}_{1}$ | $102.3$ | ${\beta}_{2}$ | $2616.3225$ |

${b}_{0}$ | $7.375714$ | $-$ | $-$ |

ADRC Unit | Parameter | Value | Parameter | Value |
---|---|---|---|---|

INLSEF (NLPI) | ${k}_{11}$ | $1.524$ | ${k}_{2}$ | $0.6168$ |

${k}_{12}$ | $2.456$ | ${\mu}_{2}$ | $0.0222$ | |

${\mu}_{1}$ | $3.0164$ | ${\alpha}_{2}$ | $0.9593$ | |

${\alpha}_{1}$ | $0.9615$ | $\delta $ | $0.8865$ | |

SMESO | ${\beta}_{1}$ | $138.18$ | ${\beta}_{2}$ | $4773.4281$ |

${k}_{\alpha}$ | $0.9874$ | ${k}_{\beta}$ | $0.3604$ | |

$\alpha $ | $0.3506$ | $\beta $ | $0.2225$ | |

${b}_{0}$ | $11.176$ | $-$ | $-$ |

ADRC Unit | Parameter | Value | Parameter | Value |
---|---|---|---|---|

NLPD | ${k}_{1}$ | $6.1775$ | ${k}_{2}$ | $8.339$ |

${\alpha}_{1}$ | $0.8391$ | ${\alpha}_{2}$ | $0.8015$ | |

TD | $R$ | $55.38$ | ${a}_{2}$ | $7.842$ |

${a}_{1}$ | $0.142$ | $-$ | $-$ | |

NLESO | ${\beta}_{1}$ | $114.64$ | ${\beta}_{2}$ | $3285.582400$ |

${a}_{1}$ | $0.6129$ | ${b}_{0}$ | $6.305714$ |

ADRC Unit | Parameter | Value | Parameter | Value |
---|---|---|---|---|

STC-SM | $\kappa $ | $0.6704$ | $\xi $ | $0.74115$ |

$\U0001d4c5$ | $0.3035$ | $\delta $ | $0.6697$ | |

TD | $R$ | $55.38$ | ${a}_{2}$ | $7.842$ |

${a}_{1}$ | $0.142$ | $-$ | $-$ | |

NLESO | ${\beta}_{1}$ | $169.6$ | ${\beta}_{2}$ | $7191.04$ |

${a}_{1}$ | $0.5563$ | ${b}_{0}$ | $9.521429$ |

Performance Index | LADRC | ADRC | IADRC | NLPD-ADRC | STC-ADRC |
---|---|---|---|---|---|

$ITAE$ | $227.645291$ | $123.466037$ | $113.083615$ | $81.377283$ | $34.246021$ |

$IAU$ | $3124.721304$ | $1738.772900$ | $957.928495$ | $1275.646277$ | $1101.477163$ |

$ISU$ | $1365.077613$ | $112.698850$ | $3.401035$ | $9.959110$ | $7.666228$ |

$OPI$ | $3.742875$ | $0.721694$ | $0.402827$ | $0.351374$ | $0.199233$ |

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## Share and Cite

**MDPI and ACS Style**

Azar, A.T.; Abed, A.M.; Abdulmajeed, F.A.; Hameed, I.A.; Kamal, N.A.; Jawad, A.J.M.; Abbas, A.H.; Rashed, Z.A.; Hashim, Z.S.; Sahib, M.A.;
et al. A New Nonlinear Controller for the Maximum Power Point Tracking of Photovoltaic Systems in Micro Grid Applications Based on Modified Anti-Disturbance Compensation. *Sustainability* **2022**, *14*, 10511.
https://doi.org/10.3390/su141710511

**AMA Style**

Azar AT, Abed AM, Abdulmajeed FA, Hameed IA, Kamal NA, Jawad AJM, Abbas AH, Rashed ZA, Hashim ZS, Sahib MA,
et al. A New Nonlinear Controller for the Maximum Power Point Tracking of Photovoltaic Systems in Micro Grid Applications Based on Modified Anti-Disturbance Compensation. *Sustainability*. 2022; 14(17):10511.
https://doi.org/10.3390/su141710511

**Chicago/Turabian Style**

Azar, Ahmad Taher, Azher M. Abed, Farah Ayad Abdulmajeed, Ibrahim A. Hameed, Nashwa Ahmad Kamal, Anwar Jaafar Mohamad Jawad, Ali Hashim Abbas, Zainab Abdulateef Rashed, Zahraa Sabah Hashim, Mouayad A. Sahib,
and et al. 2022. "A New Nonlinear Controller for the Maximum Power Point Tracking of Photovoltaic Systems in Micro Grid Applications Based on Modified Anti-Disturbance Compensation" *Sustainability* 14, no. 17: 10511.
https://doi.org/10.3390/su141710511