A Novel SLOPDM Solar Maximum Power Point Tracking Control Strategy for the Solar Photovoltaic Power System

Abstract

This study proposes a novel maximum power point tracking (MPPT) control strategy for the solar photovoltaic power system (SPPS). The proposed system adopts two solar photovoltaic modules of 430 W, which are connected to a boost converter and an MPPT controller, since the traditional MPPT algorithm (such as perturbation and observation [P&O] algorithm) can hardly reach maximum power point (MPP) under low irradiance level and partial shading conditions (PSC), which leads to the low efficiency of the SPPS. The speed of light optical path difference measurement (SLOPDM) MPPT control strategy has been developed in this study to overcome this problem. The estimation of the optical path angle difference is used as the basis for the proposed control strategy. This is done by determining the relationship between the optical path angle difference, solar photovoltaic power impedance R_spv and load R_o, and then calculating the duty cycle corresponding to the MPP, which then drives the boost converter to capture the MPP. The experimental results verify the proposed system, which shows the efficiency comparison between the SLOPDM MPPT algorithm, solar angle and horizon (SAH) algorithm, and P&O algorithm under PSC and uniform irradiance conditions (UIC) at irradiance levels of 700 W/m² and 65 W/m². It is evident from the comparison that the efficiency of the SLOPDM MPPT algorithm is 99% under both conditions, which is higher than the SAH and P&O algorithms. The SLOPDM MPPT algorithm can precisely, rapidly, and stably be operated at MPP. The contribution of this study is that the proposed MPPT control strategy can help achieve the high−performance of SPPS without changing the hardware circuit design and requiring any additional solar power meter. This reduces the cost and the complexity of the system significantly.

1. Introduction

Due to the global climate change issue [1], every country is concerned with reducing carbon emissions and improving the greenhouse effect. First, the consumption of coal, natural gas, and other fossil fuels can be reduced significantly for power generation [2] to achieve this goal. Second, people should reduce the use of fossil fuels in transportation [3], and walking or riding bicycles can be preferred for shorter distances instead of driving. Furthermore, the government has already been promoting the utilization of renewable energy, such as wind power [4], solar power [5], hydroelectric power [6], tidal power [7], geothermal power [8], and biomass power [9], etc. Renewable energy is connected in parallel with the mains through power conversion technology and provides power to the demand side, as shown in Figure 1. Electricity is supplied to the demand side, which includes residential buildings, industries, campuses, commercial applications, rail vehicles [10], electric vehicles [11], etc. It is obvious from the aforesaid uses that renewable energy has been integrated into people’s lives and has a wide range of applications.

Amongst the various renewable energy, the solar photovoltaic power system (SPPS) has drawn major attention of the researchers because SPPS does not generate noise during power generation and has a long service life [12]. The small solar photovoltaic module (SPVM) can be carried around conveniently and used in watches, computers, backpacks, blankets, etc. [13], which are more flexible in real life. Medium-sized solar photovoltaic arrays can be installed on rooftops to provide electricity for homes or farms [14]. A large-scale solar photovoltaic array parallel grid can improve users’ stable power quality [15].

The SPPS is a diverse field and could have different research areas, including SPPS fault diagnosis [16], solar cell material upgrade [17], SPPS microgrid technology [18], and power electronics technology combined with the solar maximum power point tracking (MPPT) [19], etc. This study focuses on improving the efficiency of SPPS by developing a new MPPT technique that can help achieve higher efficiency under both uniform irradiance conditions (UIC) and partial shading conditions (PSC). Numerous studies focusing on MPPT have been presented. Kumar et al. discussed that the perturbation and observation (P&O) algorithm architecture is simple, easy to implement, and widely used in SPPS. However, the P&O algorithm uses the perturbation characteristic to track the maximum power point (MPP), which causes a power loss during the MPPT process. Additionally, the MPP cannot be achieved under PSC, resulting in low system performance [20]. Liu et al. examined the solar angle and horizon (SAH) algorithm to improve the performance of the hill−climbing algorithm. This SAH algorithm analyzes the solar angle and horizon with high efficiency under the UIC, but the efficiency decreases with environmental changes (as PSC) [21]. Lu et al. used the solar photovoltaic module output power and load (SPMOPL) MPPT control method to capture the MPP effectively. However, this SPMOPL MPPT control method needs huge data analysis (such as irradiance level, temperature, SPVM output voltage, etc.), which will cause a system burden [22]. Jagadeesan et al. proposed the right half−plane (RHP) MPPT control mechanism for tracking the MPP on the P_spv−V_spv curve of SPVM to improve the system efficiency. This control mechanism reduces the MPPT range and thus reducing the system burden. However, the system efficiency will be low if the irradiance level and temperature change are high [23]. Uno et al. discussed the dual MPPT control strategy, which analyzed the SPVM and power converter output voltage and current signals to estimate the best duty cycle and capturing MPP. The dual MPPT control strategy has high efficiency but requires multiple power MOSFETs and diodes, which increases the system cost [24]. Mobarak et al. introduced the parabolic MPPT control strategy, which can make calculations quickly and accurately to improve SPPS efficiency. However, this control strategy must estimate multiple peak power points with the parabolic equations many times under the PSC and track the MPP. This will increase the system burden and the MPPT time [25]. Zhu et al. discussed the finite−state−machine (FSM) MPPT that can improve the system efficiency under the UIC and PSC. However, the complicated control sequence increases the burden on the controller. Also, the system requires multiple power converters to be used in parallel, which increases the cost of the system [26]. Kiran et al. introduced the variable step size artificial neural network (VSSANN) MPPT method that captures MPP through artificial intelligence training to improve system efficiency. However, the system requires the installation of an additional solar power meter and a thermometer, which makes the system costly [27].

This study proposes a speed of light optical path difference measurement (SLOPDM) MPPT based on estimating optical path difference angle. The proposed method calculates the duty cycle for the boost converter to track the MPP by using the relationship between optical path difference angle, solar photovoltaic module impedance R_spv, and load R_o. The proposed SLOPDM control strategy helps achieve the high performance of the SPPS under both the UIC and PSC, verified by the experimental result. The proposed SLOPDM-based MPPT is fast, simple, low-cost, and does not cause a system burden.

Table 1. Comparison of various MPPT algorithms.

Algorithm	Complexity	Performance	MPPT Speed
P&O algorithm [20]	Low	Medium	Low
SAH algorithm [21]	Medium	High	Medium
SPMOPL algorithm [22]	Medium	High	Medium
RHP MPPT algorithm [23]	Low	Medium	High
Dual MPPT algorithm [24]	Medium	High	Medium
Parabolic MPPT algorithm [25]	Medium	High	High
FSM MPPT algorithm [26]	High	High	Medium
VSSANN MPPT algorithm [27]	Medium	High	High
Proposed SLOPDM algorithm	Low	High	High

shows a comparison of various MPPT algorithms. The following points can be concluded from this table. First, the proposed SLOPDM algorithm is the least complex among the aforementioned algorithms. Second, the proposed algorithm performs better than the P&O and RHP MPPT algorithms. Last, the MPP tracking speed of the proposed algorithm is superior to the P&O, SAH, SPMOPL, dual MPPT, and FSM MPPT algorithms.

2. The Proposed Solar Photovoltaic Power System

2.1. The Configuration of the Solar Photovoltaic Module

A solar photovoltaic cell (SPVC) is composed of multiple P−N junction semiconductors, which convert light energy into electrical energy. Figure 2 displays the equivalent circuit of a single SPVC, where I_ph is the current produced by the SPVC, R_pn is the non−linear resistance of the P−N junctions, D_pn is the P−N junction diode, R_sh and R_s represent the equivalent parallel resistance and the series resistance of the SPVC, respectively, V_spv and I_spv represent the output voltage and current of the SPVC [28].

The equivalent circuit of a single SPVC is shown in Figure 2. Using the P−N junction characteristic, the formula of its output current I_spv is expressed as follows:

$$I_{spv}=n_p{I}_{ph}-n_p{I}_r\left[\exp \left(\frac{q{V}_{spv}}{kTA{n}_s}\right)-1\right]$$

(1)

where n_p is the number of SPVC in parallel, n_s is the number of SPVC in series, q is the quantity of electric charge (1.6 × 10⁻¹⁹ C), k is the Boltzmann constant (1.38 × 10⁻²³ J/°K), T is the temperature of SPVC, and A is SPVC’s ideality factor.

In Equation (1), I_r represents the SPVC’s reverse saturation current, which can be expressed as follows:

$$I_r=I_{rr}{\left(\frac{T}{T_r}\right)}^3\exp \left[\frac{q{E}_{Gap}}{kA}\left(\frac{1}{T_r}-\frac{1}{T}\right)\right]$$

(2)

where T_r is the reference temperature of SPVC, I_rr is the reverse saturation current of SPVC at temperature T_r, and E_Gap is the energy across the band gap of semiconductor material.

Figure 3 displays the schematic diagram of the SPVC, solar photovoltaic module (SPVM), and solar photovoltaic array [29]. Solar photovoltaic cells can be connected in parallel and series to form an SPVM, whereas a solar photovoltaic array is composed of multiple SPVMs with series and parallel connections.

2.2. Solar Photovoltaic Module’s Connection with Power Electronic Converter

This study adopts two SPVMs and implements the experiment with the solar photovoltaic simulator, where the specification of a single solar photovoltaic module is shown in

Table 2. Specifications of a single SPVM.

Parameters	Value
Voc	50 V
Isc	6 A
VMPP	40 V
IMPP	5.4 A
PMPP	215 W

. When the irradiance level (IL) = 1000 W/m² and the temperature = 25 °C, the single SPVM has an open−circuit voltage V_oc = 50 V, short−circuit current I_sc = 6 A, maximum power point voltage V_MPP = 40 V, maximum power point current I_MPP = 5.4 A, and maximum power point power P_MPP = 215 W.

In SPPS, the output voltage of SPVM is low. The power electronic converter can increase the voltage level up to a value required by the load and facilitate the stable voltage, so power electronics technology is widely used in SPPS. In addition, the MPPT controller can further enhance the SPVM output power (as shown in Figure 4) by sensing the output voltage or current of the SPVM. Accordingly, pulse−width modulation (PWM) signal is generated for the power switches which drive the converter. There are many kinds of power electronic converters employed in the MPPT controller for SPPS, such as single-ended primary inductor converter (SEPIC), buck-boost, Flyback, boost converters, etc. [30,31,32,33]. The boost converter [30] is most commonly used in MPPT systems mainly because of its merit of having a simple circuit structure, easy implementation, and high efficiency. Therefore, the boost converter has been selected as the power electronic converter in this study [30].

The relationship between output voltage V_o, SPVM voltage V_spv, and duty cycle D is shown as follows:

$$V_o=\frac{V_{spv}}{1-D}$$

(3)

The relationship between output current I_o, SPVM current I_spv, and duty cycle D is expressed as:

$$I_o=I_{spv}\left(1-D\right)$$

(4)

The relationship between the load impedance R_o, SPVM impedance R_spv, and duty cycle D is displayed as follows:

$$\frac{R_o}{R_{spv}}=\frac{1}{{\left(1-D\right)}^2}$$

(5)

where load impedance R_o = V_o/I_o and SPVM impedance R_spv = V_spv/I_spv.

Table 3. The SPVC efficiency range.

SPVC Material	Conversion Efficiency Range
Monocrystalline silicon	15~20%
Polycrystalline silicon	12~18%
Amorphous silicon (Sic, SiGe, and SiH)	6~9%
Monocrystalline compound (GaAs, InP)	18~30%
Polycrystalline compound (CdS, CdTe)	10~12%

shows the SPVC efficiency range [34]. The material included monocrystalline silicon, polycrystalline silicon, amorphous silicon, monocrystalline compound, and polycrystalline compound with a conversion efficiency range of 15~20%, 12~18%, 6~9%, 18~30%, and 10~12%, respectively. Different SPVC materials have different conversion efficiencies. Therefore, SPVM impedance and SPVM output power are also different.

When sunlight hits the SPVM connected to the load, it will generate V_spv and I_spv. However, the SPVM structure has a key effect on the SPVM impedance and the SPVM output power. An SPVM structure includes soda lime glass (SLG), ethylene vinyl acetate (EVA), SPVC, and back layer, etc. [34], as shown in Figure 5. If the SPVM structure is of poor quality, damaged, and has other abnormal problems, it will affect the SPVM output power. Although, many factors affect SPVM’s output power. However, this study proposed an MPPT that can be used to achieve the best system performance for various types of SPVM.

The system used in this study is structured as two SPVMs connected in series. The input of the boost converter is connected to SPVMs, whereas the output is connected to the load. Secondly, the MPPT control strategy is embedded in the controller (Figure 4). Finally, the controller generates the PWM signal for the boost converter to capture MPP depending on the proposed SLOPDM MPPT control strategy. The details about the proposed SLOPDM MPPT control strategy for the SPPS can be seen in Section 3.

3. Proposed Speed of Light Optical Path Difference Measurement MPPT Control Strategy

Figure 6 is the schematic diagram showing the relationship between the sun, earth, and planets, where the sunlight shines on the planet and the planet reflects the light to the earth. However, the earth revolves around the sun. It leads to the position of the planet seen on the earth being different from the actual position of the planet due to the speed combination of the earth’s revolution and light. It is similar to being on a running train when it’s raining outside, and the rain looks like it is falling diagonally when observed from inside the train car.

In the 18th century, Prof. Bradley proposed the speed of light optical path difference measurement (SLOPDM) method to calculate the actual planetary position. The planetary position could be calculated accurately using this method [35]. This study extends the use of the SLOPDM method for maximum power point tracking. So, the proposed MPPT method in this study is named SLOPDM MPPT. A detailed description of the proposed MPPT control strategy has been presented below.

In Figure 6, the earth’s revolution speed is $\stackrel{\rightharpoonup }{V}$, the speed of light reflected by the planet $\stackrel{\rightharpoonup }{C}$ is 3 × 10⁵ km/s, and the angle of the optical path difference is θ. Figure 7 demonstrates a schematic diagram of the relationship between the earth’s revolution speed $\stackrel{\rightharpoonup }{V}$, the light speed $\stackrel{\rightharpoonup }{C}$, and the angle of the optical path difference θ, which can also be represented by Equation (6).

$$\stackrel{\rightharpoonup }{C}=\frac{\stackrel{\rightharpoonup }{V}}{\tan \theta }$$

(6)

where $\stackrel{\rightharpoonup }{C}$ is constant as 3 × 10⁵ km/s, $\stackrel{\rightharpoonup }{V}$ changes with the distance between the sun and the earth. $\stackrel{\rightharpoonup }{V}$ becomes higher when the distance between the earth and the sun is reduced. Therefore, $\stackrel{\rightharpoonup }{V}$ is not constant.

Figure 8 is transformed from Figure 7, which shows the relationship between angle θ₁ and its opposite Y. In Figure 8, the radius of the circle is regarded as 1 depending on the light speed $\stackrel{\rightharpoonup }{C}$, which is a constant value; angle θ₁ corresponds to the angle of the optical path difference θ. Using Equation (5), Figure 7 and Figure 8 and assuming that Y is proportional to the $\stackrel{\rightharpoonup }{V}$ and R_spv, the relationship between $\stackrel{\rightharpoonup }{V}$, Y, R_spv, R_o, and duty cycle D are expressed as Equation (7).

$$\stackrel{\rightharpoonup }{V}\propto Y\propto R_{spv}=R_o{\left(1-D\right)}^2$$

(7)

When θ₁ = 0°, $\stackrel{\rightharpoonup }{C}$ is proportional to the radius (1). By substituting Equation (6) into (7), the relationship between tanθ₁, R_spv, R_o, and duty cycle D can be obtained as follows:

$$\begin{array}{c}\tan {\theta }_1·\stackrel{\rightharpoonup }{C}=\stackrel{\rightharpoonup }{V}\propto D^2-2D+\left(1-\frac{R_{spv}}{R_o}\right)=0\\ \tan {\theta }_1\propto D^2-2D+\left(1-\frac{R_{spv}}{R_o}\right)=0\end{array}$$

(8)

When θ₁ = 45°, Equation (8) will be transformed into Equation (9).

$$\begin{array}{c}\tan {\theta }_1·\stackrel{\rightharpoonup }{C}=\stackrel{\rightharpoonup }{V}\propto D^2-2D+\left(1-\frac{R_{spv}}{R_o}\right)=1\\ \tan {\theta }_1\propto -D^2+2D+\frac{R_{spv}}{R_o}=1\end{array}$$

(9)

If the value of R_spv and R_o are known, the duty cycle D at θ₁ = 0° and θ₁ = 45° can be calculated using Equations (8) and (9), respectively. Therefore, this study can estimate the MPPT’s duty cycle to achieve MPP using the relationship between tanθ₁, R_spv, R_o, and duty cycle D.

Using the transformation of Equations (5)−(9), the relationship between R_spv, angle θ₁, and duty cycle D can be drawn, as shown in Figure 9. In Figure 9, the right y−axis represents the impedance of SPVMs R_spv; the left y−axis represents the duty cycle D; the x−axis represents the angle; the straight blue line shows the results when R_spv changes from 1 Ω to 46 Ω; and the brown curve shows the results when D changes from 0 to 0.1. When load R_o = 200 Ω, R_spv = 34 Ω, angle = 34° using the Equations (5)–(9), 0.06 is the optimal duty cycle of the MPPT, because the proposed SLOPDM MPPT control strategy regards the optical path difference measurement of the light speed as the basis and considers the relationship between the load R_o and R_spv to calculate the MPPT duty cycle D. Therefore, the proposed SLOPDM control strategy can capture the MPP rapidly and accurately.

Figure 10 shows the flowchart of the proposed SLOPDM MPPT control strategy. First, the system measures V_spv, I_spv, V_o, and I_o. The R_spv and R_o will only be calculated when the boost converter’s output current I_o ≠ 0. Second, the dP_spv/dV_spv is the next parameter to be checked. If dP_spv/dV_spv = 0, the system is operating at MPP and the duty cycle D is fixed. By contrast, if dP_spv/dV_spv ≠ 0, the proposed SLOPDM MPPT will be performed. Next, the system calculates the angel with R_spv and R_o (Figure 9). Finally, the system substitutes R_spv, R_o, and angle θ₁ into Equation (8) to obtain the MPPT’s duty cycle D, and drives the boost converter to capture the MPP.

Figure 11 displays the architecture diagram of the solar photovoltaic simulator that connects the boost converter and embeds the proposed SLOPDM MPPT control strategy. The solar photovoltaic simulator simulates two SPVMs connected in series whose total rated power is 430 W, and the specification of a single SPVM is shown in Table 2. The specifications of the boost converter inductor L₁ and capacitor C₁, as well as the microcontroller unit (MCU), are shown in

Table 4. The specification of the boost converter and the MCU.

Parameter	Specification
MCU	Microchip, model number: 18F4520
L1 of the boost converter	0.5 mH
C1 of the boost converter	330 μF

. The control flow of the proposed SLOPDM MPPT strategy is explained as follows: First, the voltage and current sensors are employed at the solar photovoltaic simulator’s output V_spv and I_spv to capture the voltage V_spv,ref and current I_spv,ref signals to be transmitted to the MCU. Second, the boost converter’s output V_o and I_o uses voltage and current sensors to catch the voltage V_o,ref, and current I_o,ref signals transmitted to the MCU. Third, the V_spv,ref, I_spv,ref, V_o,ref, and I_o,ref signals are utilized to calculate the MPPT duty cycle using the proposed SLOPDM MPPT control strategy, whose operating frequency is 50 kHz. Finally, the MCU generates the MPPT duty cycle to drive the power MOSFET SW₁ of the boost converter, thus tracking the MPP.

4. Experiment Results

The prototype of the solar photovoltaic system used in this study is displayed in Figure 12. The specification of this experimental setup is shown in

Table 5. Specifications of the experimental setup.

Equipment/Components	Specification
Solar photovoltaic simulator	Chroma, model number: 62020H
Computer	ASUS, model number: X515E
Scope	Tektronix, model number: DPO 2014B
Power supply	Gwinstek, model number: GPS−4303Wanptek, model number: NPS306W
Load Ro	70 Ω and 450 Ω

. In this study, the proposed SLOPDM control strategy, SAH, and P&O algorithms have been experimentally implemented under both the UIC and PIC with the irradiance level (IL) of 700 W/m² and 65 W/m². A comparative analysis of these control strategies has also been performed to demonstrate the effectiveness of the proposed control strategy. The experiment results verify that the efficiency of the proposed SLOPDM algorithm is higher than that of the SAH and P&O algorithms.

4.1. Experimental Results under Uniform Irradiance Condition

Figure 13 displays the gate−source voltage of the power MOSFET (v_gs), output voltage (V_o), output current (I_o), and output power (P_o) waveforms of the boost converter under the test condition having IL = 700 W/m², the temperature = 25 °C obtained by implementing certain control strategies. Figure 14 shows the P_spv−V_spv characteristic curve of the SPVM at IL = 700 W/m² and the temperature = 25 °C obtained using certain control strategies.

Figure 13a displays the experimental results of the proposed SLOPDM MPPT control strategy. Figure 13b,c are the results of the SAH and P&O algorithms, respectively. First, R_spv and R_o were calculated using V_spv, I_spv, V_o, and I_o measurements of 21.2 Ω and 70 Ω, respectively. Second, angle θ₁ and MPPT duty cycle D were estimated using Equation (8) and Figure 9, which are 1.4° and 0.43, respectively. The proposed SLOPDM control strategy produced the duty cycle D = 0.43, V_o = 140 V, and V_MPP = 80.6 V (from Equation (3)) for the SPPS and the actuating point accurately operate at MPP P_mpp = 305 W at t = t_a as demonstrated in Figure 14a. The SAH algorithm’s actuating point operating at MPP P_mpp = 305 W is shown in Figure 14b, whereas the P&O algorithm’s actuating point operating at MPP P_mpp = 305 W is depicted in Figure 14c.

Figure 15 demonstrates the waveforms of the gate−source voltage of the power MOSFET (v_gs), output voltage (V_o), output current (I_o), and output power (P_o) of the boost converter at IL = 65 W/m² and temperature = 25 °C by implementing certain control strategies. Figure 16 shows the P_spv−V_spv characteristic curve of the SPVM at IL = 65 W/m² and temperature = 25 °C obtained using certain control strategies.

Figure 15a displays the experimental results of the proposed SLOPDM MPPT control strategy. Figure 15b,c are the results of the SAH and P&O algorithms, respectively. First, R_spv and R_o were calculated using V_spv, I_spv, V_o, and I_o measurements of 213.8 Ω and 450 Ω, respectively. Second, angle θ₁ and MPPT duty cycle D were estimated using Equation (8) and Figure 9, which are 2.47° and 0.28, respectively. The proposed SLOPDM control strategy produced the duty cycle D = 0.28, V_o = 105 V, and V_MPP = 75.7 V (from Equation (3)) for the SPPS and the actuating point accurately operate at MPP P_mpp = 26.6 W at t = t_a as demonstrated in Figure 16a. The SAH algorithm’s actuating point operates the same as the proposed method at MPP P_mpp = 26.6 W at t = t_a while the actuating point moves away from the MPP at t = t_b, and the power drops to P_mpp = 24.5 W, as shown in Figure 16b. This is because the SAH algorithm performs poorly under the low irradiance level condition (LILC). In Figure 16c, the P&O algorithm’s actuating point operated unstably until t = t_b. When t = t_c, the actuating point becomes unstable again. The MPP captured by the P&O algorithm P_mpp is 12.5 W, as depicted in Figure 16c. The reason is that the P&O algorithm cannot efficiently track MPP with perturbation characteristics under the LILC.

The above results are organized in

Table 6. The comparison of proposed SLOPDM, SAH, and P&O algorithms under UIC.

Algorithm	Efficiency
	IL = 700 W/m2	IL = 65 W/m2
SAH	99%	91%
P&O	99%	47%
SLOPDM	99%	99%

, which shows the comparison of proposed SLOPDM, SAH, and P&O algorithms under UIC. The proposed SLOPDM algorithm can reach 99% efficiency at the IL = 700 W/m² and 65 W/m², which is higher than the efficiency of SAH and P&O algorithms.

4.2. Experimental Results under the Partial Shading Conditions

Figure 17a shows the solar photovoltaic simulator simulating two SPVMs connected in series when the IL is 700 W/m^2, and the temperature is 25 °C. A single SPVM includes 32 SPVCs (8 × 4). Two SPVMs connected in series have 64 SPVCs (8 × 8), 15 of which are shaded. Under this condition, the SPVMs have I_sc = 4.7 A, V_oc = 90 V, V_mpp = 47.2 V, I_mpp = 4.25 A, and P_mpp = 200.6 W. Figure 17b shows that the solar photovoltaic simulator simulating two SPVMs connected in series when the IL is 65 W/m² and temperature = 25 °C. A single SPVM includes 32 SPVCs (8 × 4). Two SPVMs are connected in series, having 64 SPVCs (8 × 8), 30 of which are shaded. Under this condition, the SPVMs have I_sc = 0.68 A, V_oc = 57 V, V_mpp = 17.5 V, I_mpp = 0.628 A, and P_mpp = 11 W.

Figure 18 displays the P_spv−V_spv characteristic curve of the SPVM under PSC at IL = 700 W/m² and temperature = 25 °C. Figure 18a displays the experimental results of the proposed SLOPDM MPPT control strategy. The angle θ₁ and MPPT duty cycle D were estimated using Equation (8) and Figure 9. The actuating point is accurately operated at MPP P_mpp = 200 W. Figure 18b,c are the results of the SAH and the P&O algorithms, respectively. The SAH algorithm’s actuating point operated at P_mpp = 190 W due to its ability to operate around MPP under PSC. The P&O algorithm’s actuating point operated at P_mpp = 105 W, which shows that the P&O algorithm is unsuitable under PSC. This is because the actuating point of the P&O algorithm tracks the MPP with the perturbation characteristic, which is suitable to operate on a P_spv−V_spv characteristic curve with only one MPP (as in the case of UIC). However, P_spv−V_spv characteristic curve has multiple peak power points under PSC, and the P&O algorithm actuating point will not be able to judge the MPP immediately and produce low power [22].

Figure 19 demonstrate the P_spv−V_spv curve of the SPVM under PSC at IL = 65 W/m² and temperature = 25 °C. Figure 19a displays the experimental results of the proposed SLOPDM MPPT control strategy. First, uses the V_spv, I_spv, V_o, and I_o measurements. Second, angle θ₁ and MPPT duty cycle D were estimated using Equation (8) and Figure 9. The actuating point is accurately operated at MPP P_mpp = 10.9 W like Figure 18a. Figure 19b,c are the results of the SAH and P&O algorithms, respectively. The SAH algorithm’s actuating point operated at P_mpp = 10.3 W like Figure 18b. The P&O algorithm’s actuating point operated at P_mpp = 3.9 W like Figure 18c.

The above results are organized in

Table 7. The comparison of proposed SLOPDM, SAH, and P&O algorithms under PSC.

Algorithm	Efficiency
	IL = 700 W/m2	IL = 65 W/m2
SAH	95%	94%
P&O	52%	35%
SLOPDM	99%	99%

, which shows the comparison of proposed SLOPDM, SAH, and P&O algorithms under PSC. The proposed SLOPDM algorithm can reach 99% efficiency under the IL of 700 W/m² and 65 W/m², whose efficiency is higher than that of the SAH and P&O algorithms.

5. Conclusions

This research developed a novel SLOPDM MPPT control strategy for SPPS. The estimation of the optical path angle difference is used as the basis for the proposed control strategy. This is done by determining the relationship between the optical path angle difference, solar photovoltaic power impedance R_spv and load R_o, and then calculating the duty cycle corresponding to the MPP, which then drives the boost converter to capture the MPP. The proposed method can easily and rapidly achieve MPP. In this study, the experimental verification is carried out under both the UIC and PSC. The proposed SLOPDM algorithm performance is 99% under UIC, which is higher than that of the SAH and P&O algorithms. In addition, the proposed SLOPDM algorithm reached 99% under PSC with the irradiance level of 700 W/m² and 65 W/m², while the SAH algorithm efficiencies are 95% and 94%, and the P&O algorithm efficiencies are 52% and 35%, respectively. Under PSC, the proposed SLOPDM algorithm performed far better than the SAH and P&O algorithms. Finally, this novel control strategy does not need to change the hardware circuit design and requires any additional solar power meter. This reduces the cost and the complexity of the system significantly.

Future work can test and verify the proposed SLOPDM MPPT algorithm with multiple sets of SPPS. Furthermore, the related parameters can be modified to make it a faster MPPT control strategy, evaluate the period, and state that SPPS does not have to use MPPT, further improving SPPS efficiency.