# An Evolutionary-Based MPPT Algorithm for Photovoltaic Systems under Dynamic Partial Shading

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Evolutionary-Based MPPT Algorithm for PV System Power Extraction

#### 2.1. MPPTs and PV System Issues Under Dynamic Partial Shading

#### 2.2. PSO-Based Algorithm Implementation

_{i}, and the position, x

_{i}, of the i-th agent at the k-th step of the searching process:

_{1}and c

_{2}are the acceleration coefficients, r

_{1}and r

_{2}are random numbers between 0 and 1, p

_{best,i}is the best position attained by the i-th particle and g

_{best}is the best position attained among all the particles.

_{1}and r

_{2}result in a very low velocity, thus a large number of iteration is required to reach the solution. On the other hand, too large changes in the velocity might cause the particles to move far from the neighborhood of the global maximum, opening up the possibility of converging to a local maximum instead of the global one. Moreover, in a partially shaded PV generator, the distance between the two consecutive peaks in the P-V curve is quite constant, about 80% of the open voltage of the string of PV cells connected in parallel with a bypass diode.

_{1}and c

_{2}. The search of the global maximum in the PSO-based MPPT algorithm is mathematically expressed by two equations that define the variation of the duty cycle, Δd

_{i}, and the duty cycle, d

_{i}, corresponding to the i-th agent at the k-th step of the searching process:

_{1}and c

_{2}are the acceleration coefficients, d

_{best,i}is the duty cycle corresponding to the maximum PV output power detected by the i-th particle and D

_{best}is the duty cycle corresponding to the maximum PV output power detected among all the particles.

- Step 1:
- Activation of the MPPT algorithm. First, the MPPT controller checks whether it is necessary to activate the search for a new GMPP by comparing the actual PV power, P
_{PV}(n), with the previously recorded maximum power, P_{MPP}. The absolute value of the difference between these powers over a threshold triggers the activation of the search for a new GMPP. - Step 2:
- PSO Initialization. This step is the beginning of each GMPP searching process. The initial positions of the particles, namely a first solution vector of duty cycles with N
_{p}elements, are calculated: they are linearly spaced between the minimum and maximum duty cycle to cover the whole search space. All of the variables that contain information related to a previous GMPP search are reset. Then, the algorithm transmits the first duty cycle to the power converter. The change of duty cycle causes a transient of the electrical system; the new steady state condition will be evaluated at the next time interval. - Step 3:
- Fitness Evaluation and Update Individual Best Data. The fitness value P
_{PV}(n) (actual PV output power) of the i-th particle, is calculated. Both the best individual position, d_{best,i}, and the best fitness, P_{PVbest,i}, are updated in case the fitness value is greater than the best fitness. If the particle that has just been evaluated is not the last one, the algorithm transmits the next duty cycle to the power converter, whose fitness function will be evaluated at the next time interval. Otherwise, global best data will be updated and operations after each k-th iteration will be performed. - Step 4:
- Update Global Best Data and End-of-Iteration checks. This step is the end of each k-th iteration. Both the global best position, D
_{best}, and the global best fitness, P_{PV,Gbest}, are updated in case the maximum of the best fitness values of particles is greater than the global best fitness. A counter, CounterG_{best}, is set to 1 at every global best position update, otherwise it is incremented. - Step 5:
- Convergence Determination and Reset Criterion. The convergence determination proposed in this work is based on the number of iterations without the update of D
_{best}: the convergence is reached in case a new D_{best}is not found in the last N_{Gbest}iterations. Moreover, a maximum number of iterations, N_{Iter}, is allowed to reach convergence. In case the convergence is not met within the number of allowed iterations, the algorithm has to update the velocity and position of each particle and has to perform another search iteration (move to Step 6). In case the convergence is met within the number of allowed iterations, the algorithm transmits D_{best}to the power converter to check the solution (move to Step 7). If the maximum number of allowed iterations is reached without convergence, the search for the GMPP has to be repeated (move to Step 2). - Step 6:
- Update Velocity and Position of Each Particle. After all the particles are evaluated and convergence is not achieved, the velocity and the position of each particle have to be updated by using Equations (3) and (4). Then, the algorithm transmits the first duty cycles to the power converter. At the next time interval, the algorithm will move to Step 3.
- Step 7:
- Check the GMPP. The duty cycle of the power converter is D
_{best}; the actual PV output power, P_{PV}(n), is compared with the global best fitness, P_{PV,Gbest}. The GMPP is reached if the absolute value of the difference between these powers is below a threshold. This check is necessary in case of dynamic partial shading, when the P-V curve changes significantly during the GMPP search process. Under these conditions, several fitness functions are sampled during the GMPP search process, making the information of d_{best,i}and D_{best}totally useless for tracking the actual GMPP. In case the result of the GMPP check is not successful, a new full scan is necessary and the algorithm immediately moves to Step 2. Otherwise, it is assumed that the GMPP has been reached and the power converter will be operated with the duty cycle D_{best}until a change in the environmental conditions, namely a change in the PV output power, triggers a new scan.

_{best}, P

_{PV,Gbest}occurs; N

_{Gbest}is 4 and N

_{Iter}is 8. Iterations from 3 to 6 are characterized by the same D

_{best}, thus the end of the sixth iteration triggers the check of the solution.

_{p}, the number of iterations required to meet the convergence criterion, and the sampling time T

_{s}. In order to avoid the measurement of the PV output power during transients, T

_{s}has to be larger than the power converter settling time. It is easy to prove that the search process requires a quite long time interval in which the PV generator works at different power values, even far from the GMPP.

## 3. Test Case Modeling

#### 3.1. Test Case PV System Model

_{PV}), the leakage or reverse saturation current (I

_{0}), the diode quality factor (n), the series resistance (R

_{s}) and the shunt resistance (R

_{sh}). Referring to the equivalent electric circuit in Figure 5, the I-V curve of a PV cell can be expressed based on Kirchhoff’s current law, Ohm’s law, and the Shockley diode equation:

_{C}, can be expressed as:

_{C}is the cell temperature and α

_{ISC}is the temperature coefficient for short-circuit current. In most cases, reference values are measured at standard test conditions (STC), that is with G

_{ref}equal to 1000 W/m

^{2}, cell temperature equal to 25 °C and Air Mass equal to 1.5. In case of partial shading on a cell, its light generated current is directly proportional to the ratio of the unshaded area of the cell, S

_{unshaded}, and its total surface, S

_{cell}. The unshaded area of each cell is calculated according to the shape of the shading object and its speed and position. The diode reverse saturation current can be expressed as:

^{−5}eV∙K

^{−1}) and E

_{g}is the bandgap energy of the silicon, that is temperature dependent and it is given, in eV, as:

_{out}(t) is the output voltage applied to the load resistor, and V

_{pv}(t) is the input voltage applied to the PV module. According to the MPPT test facility available in the SolarTechLab, the resistance of the load resistor, R

_{load}, is 37.5 Ω. At the PV module terminals, the load and the DC-DC converter can be replaced by an equivalent time-varying resistor, R

_{eq}(t).

#### 3.2. Test Case Shading Scenarios Model

## 4. Test Case Simulations and Results

- Perturb and Observe (P&O—A);
- Variable Step Perturb and Observe (P&O—B);
- Three Point Weight Comparison (P&O—C);
- Incremental Conductance (IC);
- Constant Voltage (CV);
- Open Voltage (OV);
- Short Current Pulse (SC);
- PSO-based MPPT algorithm with 3 particles (PSO—3p);
- PSO-based MPPT algorithm with 4 particles (PSO—4p);
- PSO-based MPPT algorithm with 5 particles (PSO—5p).

^{2}and 48 °C, respectively. Since MPPT algorithms regulate the PV generator both for irradiance changes and in case of dynamic partial shading, therefore constant irradiance is necessary to evaluate the behavior of the controller in the event of dynamic partial shading. Moreover, constant irradiance represents the environmental condition during a short time interval of a sunny day. The commonly used thermal model for predicting the temperature of PV cells in a PV module is based on the assumption that the difference between the cell and ambient temperature is proportional with the irradiance, thus constant irradiance corresponds to constant cell temperature. Partial shading on PV cells persists only for a few seconds: it is assumed that the cell temperature variation due to the passage of the moving object can be neglected.

- At the beginning of the simulation (conventional time stamp t = 0 s), the PV module is unshaded and the MPP algorithm is in steady state.
- The dynamical shading produced by the object moving in forward direction starts 10 s after the beginning of the simulation (conventional time stamp t = 10 s). The dynamical partial shading condition ends as soon as the moving object’s rear-end leaves the PV module. Dynamic partial shading lasts for 31.7 s in case of 2 × 1 shading objects, for 35.7 s in case of 3 × 1 shading objects and for 39.7 s in case of 4 × 1 shading objects;
- The dynamical shading produced by the object moving in backward direction starts at the conventional time stamp t = 70 s. As in the case of forward motion, the dynamical partial shading lasts from 31.7 s to 39.7 s, depending on the length of shading objects.
- The simulation ends at the conventional time stamp t = 120 s.

- the position of the shading object and the resulting unshaded area of each cell of the PV module are calculated;
- the I–V curves of each PV cell and the I–V curve of the PV module are calculated and the electrical circuit is solved, namely V
_{PV}(n) and I_{PV}(n) are calculated and P_{PV}(n) is derived; - one cycle of the MPPT algorithm is performed and the duty cycle is changed accordingly.

_{PV}(t) and I

_{PV}(t) are the actual voltage and current at the terminals of PV generator, V

_{MPP}(t) and I

_{MPP}(t) are the coordinates of the actual GMPP.

#### 4.1. Results: Rectangular Shape

#### 4.2. Results: Trapezoidal-A Shape

#### 4.3. Results: Trapezoidal-B Shape

#### 4.4. Results: Trapezoidal-C Shape

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

CI | Computational Intelligence |

CV | Constant Voltage |

DC/DC | Direct Current to Direct Current |

GMPP | Global Maximum Power Point |

IC | Incremental Conductance |

MPP | Maximum Power Point |

MPPT | Maximum Power Point Tracking |

OV | Open Voltage |

PSO | Particle Swarm Optimization |

PV | Photovoltaic |

P&O | Perturb and Observe |

SC | Short Current Pulse |

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**Figure 1.**Flowchart of Particle Swarm Optimization (PSO)-based Maximum Power Point (MPP) tracking method.

**Figure 6.**Diagram of the motion cycle: the shading object is moved forward and backward in one cycle.

**Figure 7.**Rectangular shape—2 × 1: (

**a**) P&O—B; (

**b**) PSO 3 particles; (

**c**) PSO 4 particles; (

**d**) PSO 5 particles.

Electrical Data | STC ^{(1)} | NOCT ^{(2)} | |
---|---|---|---|

Rated Power | P_{MPP} (W) | 285 | 208 |

Rated Voltage | V_{MPP} (V) | 31.3 | 28.4 |

Rated Current | I_{MPP} (A) | 9.10 | 7.33 |

Open-Circuit Voltage | V_{OC} (V) | 39.2 | 36.1 |

Short-Circuit Current | I_{SC} (A) | 9.73 | 7.87 |

^{(1)}Electrical values measured under Standard Test Conditions: 1000 W/m

^{2}, cell temperature 25 °C, AM 1.5;

^{(2)}Electrical values measured under Nominal Operating Conditions of cells: 800 W/m

^{2}, ambient temperature 20 °C, AM 1.5, wind speed 1 m/s, NOCT: 48 °C (nominal operating cell temperature).

Size | Rectangular | Trapezoidal-A | Trapezoidal-B | Trapezoidal-C |
---|---|---|---|---|

2 × 1 | ||||

3 × 1 | ||||

4 × 1 |

MPPT | Efficiency | |||||
---|---|---|---|---|---|---|

2 × 1 | 3 × 1 | 4 × 1 | ||||

Sim. | Exp. | Sim. | Exp. | Sim. | Exp. | |

Perturb & Observe—A | 95.34% | 93.66% | 96.19% | 97.93% | 95.37% | 95.41% |

Perturb & Observe—B | 95.85% | 95.54% | 93.01% | 93.29% | 94.23% | 92.76% |

Perturb & Observe—C | 93.40% | 91.04% | 94.34% | 91.94% | 93.48% | 94.52% |

Incremental Conductance | 94.92% | 92.74% | 95.88% | 94.97% | 95.28% | 95.68% |

Constant Voltage | 68.10% | 69.37% | 64.56% | 65.38% | 60.18% | 60.09% |

Open Voltage | 79.33% | 78.77% | 73.56% | 73.15% | 71.38% | 71.74% |

Short Current Pulse | 93.87% | 93.17% | 93.59% | 94.25% | 91.33% | 89.30% |

PSO—3 particles | 92.54% | 89.40% | 93.48% | 91.35% | 88.77% | 85.59% |

PSO—4 particles | 91.55% | 91.65% | 90.84% | 88.02% | 85.66% | 82.86% |

PSO—5 particles | 90.80% | 90.08% | 91.76% | 88.92% | 89.42% | 89.60% |

MPPT | Efficiency | ||
---|---|---|---|

2 × 1 | 3 × 1 | 4 × 1 | |

Perturb & Observe—A | 95.37% | 95.82% | 95.95% |

Perturb & Observe—B | 97.59% | 94.68% | 95.30% |

Perturb & Observe—C | 93.28% | 94.26% | 93.69% |

Incremental Conductance | 96.03% | 94.73% | 94.72% |

Constant Voltage | 69.90% | 66.41% | 62.19% |

Open Voltage | 69.22% | 73.29% | 72.17% |

Short Current Pulse | 93.75% | 93.47% | 91.42% |

PSO—3 particles | 92.77% | 93.44% | 89.71% |

PSO—4 particles | 87.89% | 90.91% | 90.64% |

PSO—5 particles | 90.34% | 91.57% | 91.36% |

MPPT | Efficiency | ||
---|---|---|---|

2 × 1 | 3 × 1 | 4 × 1 | |

Perturb & Observe—A | 97.39% | 94.52% | 95.41% |

Perturb & Observe—B | 97.91% | 95.81% | 91.14% |

Perturb & Observe—C | 95.62% | 92.58% | 93.47% |

Incremental Conductance | 97.06% | 92.94% | 92.83% |

Constant Voltage | 85.98% | 69.60% | 66.15% |

Open Voltage | 85.32% | 68.92% | 75.62% |

Short Current Pulse | 91.98% | 91.77% | 91.17% |

PSO—3 particles | 93.40% | 90.47% | 91.99% |

PSO—4 particles | 92.34% | 91.82% | 90.11% |

PSO—5 particles | 91.25% | 89.27% | 89.30% |

MPPT | Efficiency | ||
---|---|---|---|

2 × 1 | 3 × 1 | 4 × 1 | |

Perturb & Observe—A | 95.06% | 95.48% | 95.64% |

Perturb & Observe—B | 97.65% | 94.10% | 95.22% |

Perturb & Observe—C | 93.35% | 93.82% | 93.19% |

Incremental Conductance | 95.34% | 94.16% | 94.31% |

Constant Voltage | 71.59% | 67.37% | 63.15% |

Open Voltage | 70.92% | 73.81% | 72.42% |

Short Current Pulse | 93.01% | 92.81% | 91.77% |

PSO—3 particles | 91.07% | 92.62% | 88.54% |

PSO—4 particles | 91.78% | 92.02% | 90.40% |

PSO—5 particles | 92.47% | 91.61% | 91.10% |

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**MDPI and ACS Style**

Dolara, A.; Grimaccia, F.; Mussetta, M.; Ogliari, E.; Leva, S.
An Evolutionary-Based MPPT Algorithm for Photovoltaic Systems under Dynamic Partial Shading. *Appl. Sci.* **2018**, *8*, 558.
https://doi.org/10.3390/app8040558

**AMA Style**

Dolara A, Grimaccia F, Mussetta M, Ogliari E, Leva S.
An Evolutionary-Based MPPT Algorithm for Photovoltaic Systems under Dynamic Partial Shading. *Applied Sciences*. 2018; 8(4):558.
https://doi.org/10.3390/app8040558

**Chicago/Turabian Style**

Dolara, Alberto, Francesco Grimaccia, Marco Mussetta, Emanuele Ogliari, and Sonia Leva.
2018. "An Evolutionary-Based MPPT Algorithm for Photovoltaic Systems under Dynamic Partial Shading" *Applied Sciences* 8, no. 4: 558.
https://doi.org/10.3390/app8040558