# Dual Search Maximum Power Point (DSMPP) Algorithm Based on Mathematical Analysis under Shaded Conditions

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## Abstract

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## 1.Introduction

## 2. Photovoltaic (PV) System Model

#### 2.1. PV Array Model under Uniform Conditions

_{S}and R

_{SH}are the series and shunt resistances, respectively. The efficiency of a solar cell is quite sensitive to variations in the series resistance, but it is not as sensitive to variations in the shunt resistance. As a result, R

_{SH}can be assumed to be infinite; that is, the circuit is an open circuit. The current-voltage characteristics of a solar cell can be described by Equation (1):

_{PH}and I

_{S}are the photocurrent and saturation current, respectively; K

_{B}= 1.38 × 10

^{−23}J/K and q = 1.6 × 10

^{−19}C are Boltzmann’s constant and the elementary charge, respectively; A is an ideality factor with a value between 1 and 2.

#### 2.2. Mathematical Model of PV Array under Uniform and Partially Shaded Conditions

_{M}and current I

_{M}of a PV module formed by N

_{S}× N

_{P}solar cells are defined as follows:

_{S}is the number of cells connected in series and N

_{P}is the number of the cell’s columns in a module connected in parallel. In both equations, the subscript M denotes a module, and subscripts without M denote an individual solar cell. As shown in Figure 2, if N

_{SM}presents the number of modules connected in a string and N

_{PM}represents the number of strings connected in parallel in a PV array, the output voltage and the current of the PV array under identical conditions can be defined as follows [41,42]:

_{SM}and R

_{SA}are the total resistances of the PV module and the array, respectively. The output current of a PV array under uniform conditions can be derived as follows, whereas the subscripts A and M denote “Array” and “Module”, respectively:

_{DX}denotes the number of shaded modules in the Xth string. Under uniform conditions, N

_{DX}is zero. In this model, one bypass diode is connected in parallel with each module, and the voltage drop across the bypass diode of shaded module is assumed to be zero. In addition, the PV array is composed of N

_{SM}× N

_{PM}modules. Under partially shaded conditions, the related equations can be expressed as follows:

_{A}and I

_{SCA}are the output current and the short-circuit current of the PV array under PSCs, respectively. V

_{AX}and V

_{OCAX}denote the output voltage and open-circuit voltage in the X

_{TH}string, respectively, and R

_{SAX}denotes the resistance in the Xth string.

## 3. DC-DC Boost Converter

_{g}are the output and input voltages of the boost converter, respectively, and D is the duty cycle, which is defined as the ratio of the turn-on duration to the switching time period (T

_{S}).

## 4. Maximum Power Point Tracking (MPPT) Based on a Linear

#### Function under Partially Shaded Conditions (PSC)

**Figure 4.**Failure of the maximum power point tracking (MPPT) algorithm to find the global maximum power point (GMPP).

_{refnew}is the new reference voltage under PSCs; V

_{OCA}and I

_{SCA}are the open-circuit voltage and short-circuit current of the PV array, respectively; and I

_{PV}is the current of the PV array when the PS occurs.

**Figure 5.**Tracking of the GMPP based on the linear function method for the first case of the complicated scenario (

**a**) I-V curve and (

**b**) P-V curve.

**Figure 6.**Tracking of the GMPP based on the linear function method for the second case of the complicated scenario (

**a**) I-V curve and (

**b**) P-V curve.

- (1)
- The open-circuit voltage and short-circuit current methods are alternative techniques for obtaining the MPP. The open-circuit method is based on the relationship between the voltage of the PV array at the maximum power point (V
_{MPP}) and the open-circuit voltage of the PV array (V_{OCA}). The short-circuit current method is based on the relationship between the current of the PV array at the MPP (I_{MPP}) and the short-circuit current of the PV array (I_{SCA}). In these methods, the voltage and current at the MPP are approximately 80% of the open-circuit voltage and 92% of the short-circuit current, respectively [44,45,46]. - (2)
- In PSCs with multi-peak power points, the distance between peak powers are integral multiples of 80% of the open-circuit of the PV module (n × 0.8 × V
_{OC_Module}), where n is an integer. If the minimum number of different levels of the shaded modules in the strings is one, then the minimum value of n is one. In other words, the minimum distance between two consecutive peaks is 0.8 × V_{OC_Module}.

_{PM}× N

_{SM}configuration. For this investigation, five general stages that cover different scenarios are considered, and the results are presented in Table 1, where n

_{PS}is the number of strings that have shaded modules, V

_{ref}is the new reference voltage created under PSCs and calculated by Equation (14), n

_{MPP}is the number of peak powers in a related stage, and K is the calculated integer value (the number of shaded modules in the string) where the maximum power peaks are on the left side of V

_{ref}.

Stage | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|

Parameter | ||||||

n_{PS} | 1 | 2 | 2 | 3 | 3 | |

V_{reff} | 0.69V_{OCA} | 0.46V_{OCA} | 0.46V_{OCA} | 0.23V_{OCA} | 0.23V_{OCA} | |

n_{MPP} | 2 | 2 | 3 | 2 | 4 | |

K | 1, 2, 3, 4 | 3, 4 | 3, 4 | 4 | 4 | |

I_{MPP_}_{1} | 0.69I_{SCA} | 0.46I_{SCA} | 0.46I_{SCA} | 0.23I_{SCA} | 0.23I_{SCA} | |

V_{MPP_}_{1} | 0.8V_{OCA} | 0.8V_{OCA} | 0.8V_{OCA} | 0.8V_{OCA} | 0.8V_{OCA} | |

P_{MPP_}_{1} | 0.552V_{OCA}×I_{SCA} | 0.368V_{OCA}×I_{SCA} | 0.368V_{OCA}×I_{SCA} | 0.184V_{OCA}×I_{SCA} | 0.184V_{OCA}×I_{SCA} | |

I_{MPP_}_{2} | 0.92I_{SCA} | 0.92I_{SCA} | 0.69I_{SCA} | 0.92I_{SCA} | √ | |

V_{MPP_}_{2} | 0.48V_{OCA} | 0.32V_{OCA} | 0.32V_{OCA} | 0.16V_{OCA} | √ | |

P_{MPP_}_{2} | 0.44V_{OCA}×I_{SCA} | 0.294V_{OCA}×I_{SCA} | 0.22V_{OCA}×I_{SCA} | 0.147V_{OCA}×I_{SCA} | √ | |

I_{MPP_}_{3} | - | - | 0.92I_{SCA} | - | √ | |

V_{MPP_}_{3} | - | - | 0.16V_{OCA} | - | √ | |

P_{MPP_}_{3} | - | - | 0.147V_{OCA}×I_{SCA} | - | √ | |

I_{MPP_}_{4} | - | - | - | - | 0.92I_{SCA} | |

V_{MPP_}_{4} | - | - | - | - | 0.16V_{OCA} | |

P_{MPP_}_{4} | - | - | - | - | 0.147V_{OCA}×I_{SCA} |

_{OCA}. The value of the current at this point is calculated as follow:

_{PS}is the number of strings that have shaded modules. The new reference voltage for obtaining the new operating point under PSCs should be calculated by Equation (14), for which I

_{PV}is determined by Equation (15), where it is equal to I

_{MPP}

_{-PS}. Thus, this stage has only one string with shaded modules, and the reference voltage is acquired for N

_{PM}and n

_{PS}equal to 4 and 1, respectively, by using Equations (14) and (15):

_{MPP}; therefore, the analysis should begin by evaluating the possibility that the GMPP is located on the left side of the new reference voltage. Figure 7 shows the different zones for the voltage of the PV array in the P-V and I-V curves, where the MPP can be located in the zones according to the scenarios.

_{MPP_}

_{2}will be 0.64V

_{OCA}. The calculated values for V

_{ref}and V

_{MPP_}

_{2}are in the same zone (D), as shown in Figure 7, and thus, the second value of K must be 2. By substituting 2 for K in Equation (17), V

_{MPP_}

_{2}will be 0.48V

_{OCA}, and the corresponding value of I

_{MPP_}

_{2}is 0.92I

_{SCA}. The maximum power values for the two power peaks in this stage are obtained as follows:

_{MPP_}

_{2}is determined to decrease with increasing K and is located in zones A, B and C; consequently, P

_{MPP_}

_{2}is reduced. This analysis demonstrates that all MPPs on the left side of the reference voltage are not the GMPP.

_{OCA}, which is calculated using Equation (14). By solving Equation (18), the values obtained for K to locate the MPP on the left side of the reference voltage are 3 and 4. In Table 1, the current, voltage and power values of the second MPP for K equal to 3 are shown, and the power for the second MPP is less than that of the first MPP. Additionally, according to Equation (17), the voltage of the second MPP for K equal to 4 is reduced; consequently, the corresponding power is also reduced. This finding proves that the MPPs on the left side of the reference voltage are not the GMPP.

_{OCA}, which is in zone B, and the corresponding maximum power is 0.22V

_{OCA}×I

_{SCA}, which is lower than the first maximum power value in this stage. For K is equal to 4, the value of maximum power is 0.147V

_{OCA}×I

_{SCA}, which is also lower than the two other peak powers. In the fourth and fifth stages, the PV array with three shaded strings is considered, in which, for the fourth stage, the numbers of shaded modules in the strings are equal, but in the fifth stage, they are not equal; as a result, the numbers of peak powers in the fourth and fifth stages are 2 and 4, respectively. By solving Equation (18) for these two stages, the MPPs are determined to be located on the left side of the reference voltage when K is equal to 4; thus, as presented in Table 1, the peak power value is less than the peak powers located on the right side of the reference voltage. In Table 1, the values for the MPPs located on the right side of the reference voltage are indicated by check marks and can adopt different values for various values of K. According to the results presented in Table 1, the GMPP cannot be located on the left side of the reference voltage; consequently, the search zone is reduced, thereby minimizing the time required to obtain the GMPP.

## 5. Proposed Dual Search MPPT Algorithm

_{ref_mod}is the new operating point under PSCs. In this equation, the second term (α × (V

_{OCA}/N

_{SM})) has a significant effect in reducing the time required to achieve the MPP. Moreover, in Equation (19), the value of α should be adjusted carefully because an inappropriate value of α can lead to a shift in the new operating point to the wrong MPP zone and can miss the GMPP. As noted above, the minimum different level of the shaded modules in strings is one, thus the minimum value of n is one. In other words, the minimum distance between two consecutive peaks is 0.8 × V

_{OC_Module}. So, α should be adjusted to the value which avoids shifting the new reference voltage to the other peak powers. Thus, the maximum value for α is 0.8 and selecting the smaller value can lead to shift the reference voltage to wrong zone. For example, as shown in Figure 8, A or B is the location of the reference voltage that can shift to A1 and A2 or B1 by considering Equation (19), which leads to obtain the first MPPT in less time after transferring the operating point. In this work, according to investigations and analyses of P-V curves under PSC, the value of α is chosen as 0.75.

_{T}and T

_{C}are the irradiation and temperature under the new conditions, respectively [47,48]:

- Based on the above-described analyses, the GMPP is not on the left side of the new reference voltage created by the modified linear function.
- In P-V curves with multi-peak powers, when the GMPP is obtained, the magnitude of the subsequent MPPs decreases from either side.
- The minimum distance between two consecutive MPPs is 0.8 × V
_{OCM}. - When the duty cycle is the output of the P & O method, the PID controller is not needed, and consequently, the controller will be simplified.
- By carefully adjusting the step size of the duty cycle, the time required to reach the MPP and the overshoot and oscillations are significantly reduced, which can increase the efficiency of the system.
- In the modified linear function, the open-circuit voltage and the short-circuit current of the PV array are the most important parameters that should be updated by changing the irradiation to obtain the correct value of the new reference voltage.

_{Crit}), then partial shading occurs (B-4). The new reference voltage is calculated using the modified linear function, and the values of the power and voltage are stored (B-5). The existing operating point should be shifted to a new reference voltage (B-6), and thus, by considering the relation between the input and output voltages of a boost converter with a duty cycle, the controller should increase the duty cycle (B-7) to adjust the operating point to the neighbor of the new reference voltage. The value of ε

_{1}should be adjusted carefully because an inappropriate value can lead to miss the first MPPT after transferring the operating point (P

_{ref}). The value of ε

_{1}is set to 0.4 to ensure the modified P & O is able to start, for obtaining the MPP at neighbor of the specified point, and if using the bigger value for ε

_{1}may lead possibility of finding another MPP. During the time when the operating point is conducted near the reference voltage, simultaneously, the controller should recognize and store the possible MPPs on the right side of the new reference voltage. As shown in Figure 10, at the right side of the possible MPP, dp/dv is negative (B-8) and at the left side it is positive. When dp/dv becomes positive (B-9), the second MPP under PSCs is obtained, and the related values of power and voltage are stored (B-11). At the onset of the PSCs, the MPP drops as its voltage remains nearly constant; thus, the first MPP under PSCs is located just below the MPP under uniform conditions. The parameter j is used to count the number of possible MPPs obtained by scanning, and thus, when the first MPP is specified by scanning, j will be equal to 1 (B-10). The scanning and storing of possible MPPs are continued until the existing operating point reaches the new reference voltage (B-6); then, by using the modified P & O method, the power and voltage of the MPP at the new reference operating point are acquired and stored as P

_{ref}and V

_{ref}, respectively (B-13). In certain scenarios of shading, P

_{ref}is the same as the first MPP after shading (P

_{a}); thus, to determine whether P

_{ref}is the same as P

_{a}, the difference in voltage between these obtained points should be measured. If the difference is less than a predetermined value (ε

_{2}) (B-14), P

_{a}is the same point as P

_{re}

_{f}and P

_{re}

_{f}is the GMPP (B-15); otherwise, P

_{ref}should be compared with the last stored MPP during scanning (B-16). The value of ε

_{2}is considered 1 because at the start of the PSCs, it is possible for algorithm to store the voltage value (V

_{a}) during oscillation. According to the above-mentioned list of points to consider, if P

_{ref}is greater than the last stored value of power, P

_{ref}is the GMPP (B-17); otherwise, the duty cycle should be reduced (B-18) to ensure the next MPP can be reached (B-19); then, the values of power for j and j-1 are compared to find the GMPP (B-23). The value of j should be checked frequently to determine whether an operating point is the same as P

_{a}exists (B-25). If j is equal to 1, then the current MPP can be the first MPP after P

_{a}or the same as P

_{a}. In the case where the difference between the voltage of the existing point for the MPP and V

_{a}is less than a predetermined value (ε

_{2}) (B-26), P

_{a}is the GMPP (B-27); otherwise, the MPPs of P

_{(j)}and P

_{a}for j equals to 1 should be compared (B-28). If P

_{(j)}is greater than P

_{a}, then P

_{(j)}is the GMPP (B-29); otherwise, P

_{a}is the GMPP (B-30).

- (A)
- When PSCs do not occur, according to Equation (13), the PV voltage should increase rapidly to reach the MPP, and thus, a greater value of d is selected, which leads to a decrease in the time required to reach the MPP.
- (B)
- When the MPP under uniform condition or the GMPP under PSCs is obtained, the value of d should be adjusted to be lower so that the overshoot and oscillations can significantly be reduced.
- (C)
- Under PSCs, a large value of d should be selected to reach the operating point near the new reference voltage point, as calculated by Equation (19).
- (D)
- When the existing operating point is near the new reference voltage, such as in blocks 6 and 19, a small value of d should be selected to avoid missing the new operating point.

## 6. Simulation Results

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

Power in maximum point, MPP | 43 W |

Voltage in maximum point, V_{MPP} | 17.4 V |

Current in maximum point, I_{MPP} | 2.48 A |

Open circuit voltage, V_{OC} | 21.7 V |

Short circuit current, I_{SC} | 2.65 A |

Temperature coefficient of V_{OC} | −0.0821 V/°C |

Temperature coefficient of I_{SC} | 0.00106 A/°C |

Number of cells per module | 36 |

- SOP: Status of operation;
- T
_{GMPP}: The global maximum power point reaching time (s); - P
_{ave_uni}: The average maximum power point value in uniform condition (W); - P
_{ave_PSC}: The average maximum power point value under PSC (W); - P
_{ripp_uni}: The oscillation in power in uniform condition (W); - P
_{ripp_PSC}: The oscillation in power under PSC (W).

_{GMPP}for S1 and S3 are 0.07 and 0.074 s respectively. Moreover, the oscillation in power decreases significantly for S1 which the oscillation values for S1 and S3 are 1 and 4 W, respectively, when the GMPP is obtained (P

_{ripp_PSC}). In comparing the proposed method (S1) with S2 and S3 in terms of oscillation in power in uniform condition (P

_{ripp_uni}), S1 decreases the oscillation value significantly, and the values for S1, S2 and S3 are 0.3, 5 and 10 W, respectively. In terms of overshoot, S1 does not have any overshoot, which may be largely attributed to implementation of the modified P & O. By reducing the oscillation and overshoot while obtaining the GMPP, the system becomes more stable and the maximum available energy is reached.

Items | Scenario | SOP | T_{GMPP} | P_{ave_uni} | P_{ave_PSC} | P_{ripp_uni} | P_{ripp_PSC} | |
---|---|---|---|---|---|---|---|---|

System | ||||||||

S1 | 12a | Successful | 0.07 | 860 | 359.5 | 0.3 | 1 | |

S2 | 12a | Failed | - | 858 | - | 5 | - | |

S3 | 12a | Successful | 0.074 | 855 | 358 | 9 | 4 | |

S1 | 12b | Successful | 0.0545 | 860 | 398 | 0.3 | 0.1 | |

S2 | 12b | Failed | - | 858 | - | 5 | - | |

S3 | 12b | Successful | 0.0563 | 855 | 397 | 9 | 1.5 |

_{GMPP}) for S1 (0.0545 s) is less than S3 (0.0565 s). In addition, the oscillation in power (P

_{ripp_PSG}) for S1 is also less than S3 when the GMPP is obtained, with the values of 0.1 and 1.5 W, respectively.

**Figure 13.**PV output powers for the scenario of Figure 12a.

**Figure 14.**The PV output power for the scenario depicted in Figure 12b.

## 7. Hardware Implementation

_{GMPP}for S1 and S3 are 22.64 and 24.68 s, respectively. Moreover, the oscillation in power decreases significantly for S1 in which the oscillation values for S1 and S3 are 4.5 and 11.6 W, respectively, when the GMPP is obtained (P

_{ripp_PSC}). The performance of current and voltage of PV array for the scenario of Figure 16 for the proposed DSMPP algorithm is shown in Figure 18 which are from the time PS happens till when the GMPP is obtained.

**Figure 16.**P-V and I-V curves under PSC (

**a**) the detected GMPP by S1 and S3 and (

**b**) The failed detected GMPP by S2.

**Figure 17.**PV output power for the scenario of Figure 16.

**Figure 18.**The performance of voltage and current for the scenarios of Figure 16 by implementing DSMPP method.

_{GMPP}) for S1 (20.76 s) is less than S3 (24.08 s). In addition, the oscillation in power (P

_{ripp_PSG}) of S1 is also less than S3 when the GMPP is obtained, with values of 3.8 and 9.6 W, respectively. In Figure 21, the performance of voltage and current for the scenario of Figure 19 for the proposed DSMPP algorithm is shown.

**Figure 19.**P-V and I-V curves under PSC (

**a**) The detected GMPP by S1 and S3 and (

**b**) The failed detected GMPP by S2

**Figure 20.**PV output power for the scenario of Figure 19.

**Figure 21.**The performance of voltage and current for the scenarios of Figure 19 by implementing the DSMPP method.

_{GMPP}) are approximately 5.2, 13.88 and 21.8 s for S1, S2 and S3, respectively. By using the modified P & O method in S1, the oscillation in power is significantly reduced, as the values for S1, S2 and S3 at the GMPP (P

_{ripp_PSC}) are 4.5, 14.2 and 12.5 W, respectively.

_{GMPP}) are approximately 15.72, 23.48 and 27.24 s for S1, S2 and S3, respectively. By using the modified P & O method in S1, the oscillation in power is significantly reduced, as the values for S1, S2 and S3 at the GMPP (P

_{ripp_PSC}) are 3.2, 10.3 and 11.1 W, respectively. The performance of current and voltage of PV array for the scenarios of Figure 22a, and Figure 22b for the proposed DSMPP algorithm are shown in Figure 25 respectively, which are measured from the time PS happens till when the GMPP is obtained.

**Figure 22.**P-V and I-V curves for different scenarios of shading (

**a**) the detected GMPP by S1, S2, and S3 and (

**b**) the detected GMPP by S1, S2, and S3

**Figure 23.**PV output power for the scenario of Figure 22a.

**Figure 24.**PV output power for the scenario of Figure 22b.

_{GMPP}) in S1 is less than that in S2 and S3: the times to reach the GMPP are almost 13.48, 25.48 and 31.4 s for S1, S2 and S3, respectively. Moreover, the oscillation in power for S2 and S3 is greater than that for S1; the GMPP values (P

_{ripp_PSC}) are 2.8, 9.2 and 10.4 W for S1 S2 and S3, respectively. The performance of current and voltage of PV array for the scenarios of Figure 26 for the proposed DSMPP algorithm are shown in Figure 28 which are from the time PS happens till when the GMPP is obtained. In Table 4, the experimental results are summarized and the abbreviations are defined in Section 5.

**Figure 27.**PV output power for the scenario of Figure 26.

**Figure 28.**The performance of voltage and current for the scenarios of Figure 26 by implementing the DSMPP method.

Items | Scenario | SOP | T_{GMPP} | P_{ave_PSC} | P_{ripp_PSC} | |
---|---|---|---|---|---|---|

System | ||||||

S1 | 16 | Successful | 22.64 | 360 | 4.5 | |

S2 | 16 | Failed | - | 278.8 | - | |

S3 | 16 | Successful | 24.68 | 360 | 11.6 | |

S1 | 19 | Successful | 20.76 | 398.4 | 3.8 | |

S2 | 19 | Failed | - | 380.8 | - | |

S3 | 19 | Successful | 24.08 | 398.4 | 9.6 | |

S1 | 22.a | Successful | 5.2 | 429.9 | 4.5 | |

S2 | 22.a | Successful | 13.88 | 429.9 | 14.2 | |

S3 | 22.a | Successful | 21.8 | 429.9 | 12.5 | |

S1 | 22.b | Successful | 15.72 | 417.4 | 3.2 | |

S2 | 22.b | Successful | 23.48 | 417.4 | 10.3 | |

S3 | 22.b | Successful | 27.24 | 417.4 | 11.1 | |

S1 | 26 | Successful | 13.48 | 359.6 | 2.8 | |

S2 | 26 | Successful | 25.48 | 359.6 | 9.2 | |

S3 | 26 | Successful | 31.4 | 359.6 | 10.4 |

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Hajighorbani, S.; Radzi, M.A.M.; Kadir, M.Z.A.A.; Shafie, S.
Dual Search Maximum Power Point (DSMPP) Algorithm Based on Mathematical Analysis under Shaded Conditions. *Energies* **2015**, *8*, 12116-12146.
https://doi.org/10.3390/en81012116

**AMA Style**

Hajighorbani S, Radzi MAM, Kadir MZAA, Shafie S.
Dual Search Maximum Power Point (DSMPP) Algorithm Based on Mathematical Analysis under Shaded Conditions. *Energies*. 2015; 8(10):12116-12146.
https://doi.org/10.3390/en81012116

**Chicago/Turabian Style**

Hajighorbani, Shahrooz, Mohd Amran Mohd Radzi, Mohd Zainal Abidin Ab Kadir, and Suhaidi Shafie.
2015. "Dual Search Maximum Power Point (DSMPP) Algorithm Based on Mathematical Analysis under Shaded Conditions" *Energies* 8, no. 10: 12116-12146.
https://doi.org/10.3390/en81012116