Maximum Power Transfer of a Photovoltaic Microgeneration System Using PSO-Based Dynamic Modeling

Abstract

This research aims to implement an already developed algorithm to obtain the maximum power transfer of a solar generation field based on a dynamic approach. The study addresses the sizing of the load to be supplied, which is a residential building. On the other hand, it also considers the field sizing as a function of the load and the operating characteristics of the selected inverter. The irradiance data correspond to the hourly record of a station that is part of the network of meteorological stations in Quito. Quito was chosen as the location for this research due to the optimization algorithm’s practical application and the availability of experimental equipment. The demand sizing is based on the regulations of the distribution company with jurisdiction in the area, which makes it a suitable test bed for the algorithm. The optimization algorithm is developed using Python (version 3.9), and the analysis of the behavior of the solar panels is performed by dynamic modeling using the Vensim software (version 10.1.2). Finally, comparative results are presented between using and not using the investigated circuit and algorithm in the photovoltaic system, obtaining an improvement in the generation over a system without the use of these improvements, validating these results by implementing them in a test system, obtaining ranges higher than 10% of the initially generated power.

1. Introduction

In the current context, the optimization methods used to determine the point of maximum power transfer (MPPT) in photovoltaic systems have evolved significantly in recent years. This evolution has been driven by the need to maximize energy efficiency and reduce costs of electricity generation through solar energy [1,2]. Given the variability in the output power of photovoltaic solar panel arrays due to environmental conditions, various methods have been proposed to achieve the maximum power point (MPP) based on current and voltage measurements. These methods, known as maximum-power-point tracking (MPPT) techniques, use the panels’ current and voltage output data to determine a duty cycle for the inverters, maximizing the power output available at the connection to the inverter loads. These algorithms are based on the principle that the circuit’s output power is maximized when the source impedance matches the load impedance. The duty cycle D of the DC-DC booster is adjusted by the algorithm, using a mathematical expression that relates both impedances and the D cycle [3]. Among the most prominent methods are the perturbation and observation algorithm (P&O), the incremental conductance algorithm (InCond), and techniques based on artificial intelligence, such as genetic algorithms and neural networks. The first method, P&O, is widely used due to its ease of implementation and simplicity, but its performance is affected by irradiance conditions and changes in the ambient temperature [4]. The method InCond offers better accuracy in the detection of the point of maximum power under variable conditions, but it requires more computational resources. These limitations underscore the urgency for further research and development in the field of photovoltaic system optimization. However, recent advances in artificial intelligence have allowed the development of hybrid methods that combine traditional techniques with machine learning algorithms, achieving significant improvements in the efficiency and stability of the tracking [5]. Despite the advances above, MPPT optimization methods face several limitations. Traditional algorithms such as P&O and InCond can present stability and efficiency problems in partial irradiance conditions, known as partial shading. Based on these limitations, artificial intelligence, and heuristic optimization methods have shown great potential to overcome these limitations because they are better adapted to environmental variations and nonlinear characteristics of PV systems [6]. However, they require more processing power and are costly to implement. When mentioning the integration of these advanced algorithms in traditional photovoltaic systems, especially on a large scale, additional challenges appear in terms of stability and reliability as the technology advances, so it is essential to continue developing solutions that seek a balance between efficiency, cost, and complexity to optimize photovoltaic systems [7].

Grid-connected photovoltaic generation systems typically employ a two-stage conversion system to connect the panel to the grid, with both stages operated by power electronics [8,9,10,11,12]. The initial stage involves a DC/DC conversion that connects the panels to a coupling element, typically a capacitor; this is followed by a DC/AC inverter stage that delivers power to consumer loads [13,14,15]. Such systems empower users by facilitating the use of grid energy produced by the panels for self-consumption, maintaining a unidirectional power flow, and offering flexibility in energy management. According to [16], two types of PV systems are connected to the grid, one with battery backup and one without, providing users with options to suit their needs.

Ref. [14] proposes a source-impedance-type inverter (ZSI) capable of increasing or decreasing voltage based on its configuration for MPPT, operating via pulse width modulation (PWM). This method uses predictive control models (PCMs) for energy harvesting. It requires distinct control techniques, employing a predictive model based on MPPT with adaptive perturbations to maximize energy capture under various weather conditions. Ref. [17] uses a PSO algorithm for MPPT, first reviewing the model describing the PV array’s behavior based on numerical methods, then establishing generation references and control loops for both DC and AC, including a phase-locking detection algorithm. This approach effectively finds the global maximum power point (GMPP) under various conditions, improving overall system efficiency. Ref. [3] proposes an MPPT search algorithm based on the P&O algorithm, describing a DC-DC lifting-stage system that operates with PWM and a DC-AC inverter functioning through SPWM, including a phase-locked detection loop. This system models components through simulation, demonstrating adequate MPPT search as a function of climatic changes. Ref. [18] proposes an algorithm to improve the MPPT’s steady-state response and dynamic behavior using the incremental conductance method combined with PI control. This approach ensures no oscillations around the MPP. It reduces the response time to reach GMPP when operating conditions vary, requiring less computational time and fewer machine resources than traditional MPPT algorithms. For our research, we propose an MPPT scheme using PSO, as shown in Figure 1. It implements solar panels and PSO controllers in the inverter stage, ensuring maximum power transfer to the DC-DC converter, and then, to the inverter that feeds the load.

While many researchers and applications use the PSO algorithm in photovoltaic systems, this study’s contribution lies in dynamic modeling, which allows for the validation of the behavior of a power extractor circuit in conjunction with the PSO algorithm. However, most studies have focused on static approaches and have not considered the transient dynamics affecting system performance under natural conditions. Dynamic modeling in this context is not just a simulation but a comprehensive exploration of the interactions between the power extractor circuit and the PSO algorithm under various operating conditions. It meticulously considers abrupt changes in irradiance and temperature, which are common in PV applications. This thorough approach allows the stability and responsiveness of the system to be evaluated in real time, providing a more complete and accurate view of its performance. The integration of the PSO algorithm with the dynamic modeling of the circuit ensures that possible inefficiencies and energy losses are identified and mitigated, providing a robust solution applicable in rapidly changing real environments.

2. Solar Panel Model

Solar panels are components that take advantage of the photovoltaic effect to produce energy at voltage and current levels which depend on their construction characteristics and are reflected in their I-V curve. Solar cells are usually constructed of silicon or some other semiconductor material, with a typical generation voltage value of these devices being around 0.5 volts. The association of panels in series and parallel allows us to reach the required system voltage and current levels, respectively. The number of panels required is designed according to the maximum power needed for the loads to be supplied or by the power the chosen inverter can support. The power output of the array varies depending on the amount of solar radiation present at the installation site, the angles of the modules with regards to the location of the sun, the ambient and working temperature of the module, as well as the voltage at which the loads are absorbing the energy generated by this system [16].

2.1. Representation of a Photovoltaic Cell with Parallel Resistance

An accurate way to represent a PV solar cell is by a circuit containing series and parallel resistors that simulate the effects of irradiance and temperature on the panels, according to their nonlinear characteristics, which are then converted into DC. The series and parallel resistors allow voltage and current values to be obtained, respectively [19,20]. The mathematical expression that expresses the operation of this equivalence is as follows:

$$I_c=I_p-I_o[e^{{\displaystyle \frac{(q{V}_c+q{R}_s{I}_c)}{N_s{K}_S}}}-1]-{\displaystyle \frac{V_c+R_s{I}_c}{R_p}}$$

(1)

where:

• $I_o$ represents the saturation current;
• q represents the electron charge;
• $N_s$ represents the number of panels in series;
• $K_s$ represents Boltzmann’s constant;
• T represents the temperature;
• A represents the diode factor;
• $R_s$ represents the series resistance;
• $R_p$ represents the parallel resistance;
• $V_c$ is the output voltage;
• $I_c$ is the output current.

2.2. Representation of Photovoltaic Cell without Parallel Resistance

$$I_{rs}={\displaystyle \frac{I_{sc}}{e^{\left(\frac{q{V}_{oc}}{N_{s}AK{T}_{ref}}\right)}-1}}$$

(2)

$$I_s=I_{rs}{\left[{\displaystyle \frac{T_c}{T_{ref}}}\right]}^3\left[e^{\left({\displaystyle \frac{q{E}_q}{KA}}\right)\left({\displaystyle \frac{1}{T_{ref}}}-{\displaystyle \frac{1}{T_c}}\right)}\right]$$

(3)

$$I_{ph}=\left[I_{sc}+K_t(T_c-T_{ref})\right]{\displaystyle \frac{G}{1000}}$$

(4)

$$I_d=N_p{I}_s\left(e^{\left({\displaystyle \frac{q}{N_{s}KA{T}_c}}\right)(V_{pv}+I_{pv}{R}_s)}-1\right)$$

(5)

$$I_{pv}=N_p{I}_{pn}-I_d$$

(6)

where:

• $N_s:$ number of cells in series;
• $N_p:$ number of parallel cells;
• $k:$ Boltzmann constant, $1.38\times {10}^{-23}{\displaystyle \frac{m^{2}kg}{s^{2}K}}$;
• $q:$ electron charge, $1.6\times {10}^{-19}C$;
• $A:$ ideal diode constant, $1.5$;
• $I_{rs}:$ inverse saturation current;
• $I_s:$ module saturation current;
• $T_{ref}:$ module reference temperature;
• $T_c:$ module operating temperature;
• $I_{ph}:$ photo-current of a photovoltaic module;
• $I_{pv}:$ photovoltaic module output current;
• $V_{pv}:$ photovoltaic module output voltage;
• $K_i:$ temperature coefficient for short-circuit current;
• $R_s:$ series resistance, 0.2 $\Omega$;
• $G:$ solar irradiance incidence.

3. Maximum Power Transfer Point (MPPT)

Photovoltaic systems usually deliver 30% of their power received by solar radiation to the load. Therefore, methods have been developed to increase the power absorbed from the generation system. Hence, the maximum power transfer theorem states that the transfer is maximum when the instantaneous impedance of the load circuit is equal to the instantaneous impedance of the photovoltaic system; this means, therefore, that the source voltage must be increased to increase the power; this is possible thanks to using a power extraction circuit, which requires adjustment using a PWM signal to modify the source impedance with the load. However, performing the control using the voltage and current is also possible. In essence, the extraction circuit is a DC/DC converter to raise the voltage from the source, which requires a computerized control to search for the maximum power point, for which different methods have been developed, among which are slope rise (hill climbing), P&O, InCond, incremental resistance, fuzzy logic, neural networks, PSO, and sliding mode. The most widely used methods are reviewed in [21,22,23,24,25,26,27,28].

3.1. Methods to Obtain MPPT

To achieve maximum power transfer, a controller modifies the control input related to the DC/DC converter. Many methods exist currently, but this review focuses on the most widely used or studied ones.

3.1.1. P&O (Perturb and Observe)

The solar array’s power and output current variation varies non-linearly concerning the array voltage or the irradiance level present at that time. The maximum power point occurs when the array voltage is also located at a maximum point, and this happens at that time when the slope of the power–voltage curve tends to zero; to reach the optimum operating point of the system, requires the voltage to increase while the derivative of the power concerning the voltage is positive, and when this value is negative the voltage is made to decrease. The algorithm in question requires a disturbance itself that occurs in the duty cycle, so that a disturbance also occurs in the voltage; this technique uses the signal of the last disturbance and the last change in power to make a decision in which direction to produce the next disturbance to reach the point of maximum power, The simplicity of this method makes it one of the most widely used. However, the algorithm cannot reach the maximum power point and stay at that operating point because each disturbance causes the operating point to move around near the maximum power point.

3.1.2. Incremental Conductance (InCond)

This is a method accepted for its simplicity and good performance at low irradiance levels or when there is a rapid change in the irradiance level. The technique uses the voltage signal, current, and slope calculation, also used by the P&O method, knowing that the slope will be positive when the operating point is to the left of the end and the slope value will be negative when it is to the right of the point of maximum power. The main difference to the method above is that the changes are generated using the instantaneous impedance calculation of the PV system.

3.1.3. Comparison of Recent Research

Table 1. Comparison of metaheuristic algorithms for MPPT in photovoltaic systems.

Algorithm	Efficiency (%)	Convergence Time	MPPT Precision	Reference
PSO	98.5	2 s	High	[29]
INC	97.2	5 ms	Medium	[30]
P&O	96.8	1 s	Low	[30]
Genetic Algorithms	98.0	3 s	High	[31]
Harmonic Search Method	95.5	1.5 s	Medium	[32]

shows a comparison based on actual values from recent research evaluating the performance of PSO and other metaheuristic algorithms in PV systems.

3.1.4. PSO (Particle Swarm Optimization)

This method uses PSO to calculate or optimize the power extractor circuit’s duty cycle, using the present and previous values of the power delivered through the photovoltaic system. PSO is a metaheuristic optimization method that is based on the existence of particles with a given particle position and velocity, where each particle is a candidate for the solution. The best local solution and the best global solution then influences the positions of the particles. When PSO is used for MPPT control, the particle position corresponds to the current duty cycle while the particle velocity is the variation in the current duty cycle.

3.2. Power Extractor Circuit

Since the solar panel is considered a current source, it is necessary to calculate the existence of a capacitor that can act as a voltage source to work with the DC/DC boost. This circuit (Figure 2) is implemented in the proposed scheme’s MPPT controller. As seen in the implementation in the experimental generation system, we choose the capacitor calculated in the integrated circuit in this circuit.

$$C_a={\displaystyle \frac{D_y·V_{pv}}{4\Delta V_{pv}·{f_s}^2{I}_{dc}}}$$

(7)

$$D_y=1-{\displaystyle \frac{V_{pv}}{V_{dc}}}$$

(8)

$$L_a={\displaystyle \frac{V_{pv}(V_{dc}-V_{vp})}{\Delta I_{La}·f_s·V_{dc}}}$$

(9)

$$\Delta I_{La}=0.13I_{pv}{\displaystyle \frac{V_{dc}}{V_{pv}}}$$

(10)

$$C_1\ge {\displaystyle \frac{P_{pv}}{\Delta V_o·f_s·V_{dc}}}$$

(11)

where:

• $V_{pv}:$ input voltage to the converter from the PV array (V);
• $I_{pv}:$ maximum current that the photovoltaic array can deliver (A);
• $P_{pv}:$ rated power (W);
• $f_s:$ sampling frequency (Hz);
• $C_a:$ capacitance of the photovoltaic array (F);
• $C_1:$ capacitance of the DC link (F);
• $L_a:$ converter inductance (H);
• $V_{dc}:$ output converter voltage (V);
• $D_y:$ duty cycle of the converter;
• $\Delta V_{pv}:$ change in PV array voltage;
• $\Delta I_{La}:$ converter inductor ripple current (I);
• $\Delta V_o:$ output voltage ripple.

4. Implementation of the Mathematical Model

PSO was selected because it is part of a set of intelligent optimization methods. One of its main advantages is its independence from the model structure; this means that it does not require a vast knowledge of the system to find optimal solutions. Likewise, this algorithm continues in local solutions. Random populations allow it to find different local solutions with simple equations determining optimal and global values. Finally, the ease of parameterization is mentioned since there are few parameters to modify before the start of the optimization process.

Table 2. PSO base algorithm.

Algorithm PSO
Step 1:	Start
Step 2:	Declaration of input variables
	$counter=0$→ variable in charge of posting income to the loop
	$dcurrent=0.5$$\to duty$ current cycle updated at each iteration
	$gbest=0.5$→ the best overall value.
	$p=\left[0000\right]$→ vector 1 × 4 fitness of each particle.
	$v=\left[0000\right]$→ vector 1 × 4 velocity of each particle.
	$pbest=\left[0000\right]$→ 1 × 4 vector best local value or each particle.
	$u=0$→ variation counter between neurons and data updating
	$dc=[00.30.60.9]$→ duty cicle of each particle.
Step 3:	For: each i-th particle
	Calculate the fitness value
	IF: current suitability value is the best, update
	IF: maximum value of neurons is reached, go to Step 4
	Otherwise: Return to Step 2
Step 4:	Calculate the best overall value
	YES: is the best overall value
	Update the best overall value
	Go to Step 5.
	Otherwise: Go to Step 4
Step 5:	Update the position and velocity of each particle.
	sentence 2
	sentence 3
	sentence 4
Step 6:	Yes: Meets convergence criterion
	go to Step 7
	Otherwise: Return to Step 3.
Step 7:	End

shows a description the proposed pseudocode.

4.1. PSO-Based Algorithm Method

The general equations for PSO are as follows:

$$V_i(K+1)=w{V}_i\left(K\right)+L_{1}ran{d}_1(P_{best,i}-X_i\left(K\right))+L_{2}ran{d}_2(G_{best}-X_i\left(K\right))$$

(12)

$$X_i(K+1)=X_i\left(K\right)+V_i(K+1)$$

(13)

where:

• w: weight of inertia;
• $L_1$: coefficient of cognitive acceleration;
• $L_2$: social acceleration coefficient;
• $ran{d}_1,ran{d}_2$: random numbers with uniform distribution between 0 and 1;
• $P_{best,i}$: best fitness value for each i-th particle;
• $G_{best}$: best global fitness value of all particles;
• K: iteration number;
  $V_i\left(K\right)$: velocity of each particle in the current iteration;
• $V_i(K+1)$: velocity of each particle in the next iteration;
• $X_i\left(K\right)$: position of each particle in the current iteration;
• $X_i(K+1)$: position of each particle in the next iteration.

4.2. Case Study

The demand estimation is carried out using the methodology suggested by Empresa Eléctrica Quito since the location is in the same city. A study of the most common elements in a type A residence and consumption stratum A was carried out, in addition to checking the factors related to the frequency of use (FFUn) and also the simultaneity factor (FS), since both factors are necessary to estimate the maximum unified demand (DMU) based on which we can finally obtain the power required by the transformation chamber if applicable, and which, at the same time, represents the power that the building could require at a given time.

Table 3. Power rating per device.

Device	Quantity	Pn (W)	CI (W)	FFU (%)	FS (%)
Illumination	18	20	360	98	40
Sound system	1	850	850	40	50
LED TV	2	115	230	70	60
PC	1	250	250	70	60
Printer	1	80	80	65	45
Blu-ray	1	50	50	60	20
Sound bar	1	150	150	45	15
Iron	1	1200	1200	75	40
Hair dryer	1	1800	1800	40	15
Table lamp	2	100	200	5	20
Router	1	22	22	75	65
Toaster	1	400	400	90	33
Coffee maker	1	400	400	90	33
Mixer	1	400	400	90	33

shows all projected electrical equipment and their power ratings. The total rated power for all the items listed corresponds to 21,042 W. However, the factors mentioned in the previous paragraphs still need to be added, so

Table 4. FFUn and FS factor.

Device	Quantity	Pn (W)	CI (W)	FFU (%)	FS (%)
Refrigerator	1	500	500	100	40
Fume hood	1	200	200	85	20
Microwave	1	1200	1200	80	30
Electric stove	1	5000	5000	60	35
Dryer	1	5000	5000	50	20
Washing machine	1	700	700	90	20
Dishwasher	1	1200	1200	65	35

shows the FFUn and FS factors for each listed device as a percentage.

Thus, each user’s (apartment’s) maximum unified demand corresponds to 3.89 kW, making the total demand for all users 31.09 kW. Considering the data shown in

Table 5. Parameters used to determine the apparent power of the building.

Parameter	Value
Power factor	0.95
DMU (kVA)	4.09
Users	8.00
Diversity factor	2.54
Electricity demand
Due to technical losses	0.31
DD (kVA)	13.21
Transformer overload	10%
Power transformer	14.53

, we determine that the transformer should have a capacity of 14.53 kVA to supply the total load of the apartments (with communal lighting distributed). Therefore, the standard transformer that meets this need is 15 kVA. The demand curve is based on this power, representing the power supply required in maximum-, medium-, and low-demand scenarios. However, the maximum-demand point will be less than 100% of the planned demand [33].

Regarding the demand curve, it was considered based on the current measurements of a primary corresponding to a substation of the distributed company that governs the site area; this curve was taken to a value adjusted to the unit where the maximum current recorded was taken as the basis for the calculation of this mode, the point of maximum demand will have a unit value with which to place the maximum power in each scenario as a new base can quickly know the value of the power demand for each hour of the day, similar to what happens with the irradiance data. Figure 3 shows the demand curve for one specific day. Figure 4 shows the curve adjusted so that the maximum demand point corresponds to 75% of the capacity of the selected transformer. Although the demand is demonstrated for only one day, there are data for a total of 13 days, so it is feasible to place the system under multiple demand scenarios even though the hourly trend between the different days is similar, so that the performance on a typical day would be the same for most days within the period or month of data collection.

The irradiance and temperature data were extracted from the system available from the Secretary of Environment of the Municipality of Quito, which has a database of hourly data from its network of meteorological measurement stations. The Belisario station, strategically located in the center of Quito at coordinates −0.18 and −78.49 (latitude, longitude), was selected for its unique position. Figure 4 illustrates the irradiance measurement corresponding to the twelfth day of May of the current year. Figure 5 illustrates the temperature whose value is captured at the same measurement station on the same day.

The Municipality of Quito’s Secretary of Environment’s web page provides a reliable and precise source of information on irradiance, specifically in the section related to the atmospheric monitoring network. It can be seen that around noon, the highest irradiance value is obtained. In contrast, in the periods between 0:00 and 5:00 as well as between 18:00 and 23:59, the irradiance value is null, corresponding to the usual behavior of the solar resource in the city; it is also possible to notice that at noon of the study day there is a considerable decrease in the value of irradiance, which may be congruent with the existence of cloudiness in the area.

The following section studies the data proposed above to study the algorithm’s performance in improving the power transfer between the solar panels and the DC/AC converter, which is not part of this research. On the other hand, the solar panel has a 450 W panel from the manufacturer First Solar, whose characteristics in STC conditions (standard operating conditions) are shown in

Table 6. STC conditions.

Parameter	Value	Unit
Nominal Power	450	W
Efficiency	18.2	%
Vmpp	186.8	V
Impp	2.41	A
Voc	221.1	V
Isc	2.57	A
Vsys	1500	V
Ir	5	A
Icf	5	A

. In contrast, the operating characteristics in NOTC (nominal operating conditions) are shown in

Table 7. NOCT conditions.

Parameter	Value	Unit
Cell temperature	45	°C
Irradiance	800	W/m2
Ambient temperature	20	°C
Power rating	339.9	W
Vmp	175.2	V
Imp	1.94	A
Voc	208.8	V
Isc	2.07	A

.

The thermal coefficients of variation for the proposed panel are shown in

Table 8. Temperature coefficients.

Parameter	Value	Units
Tp	−0.32	%/°C
Tvoc	−0.28	%/°C
Tisc	0.04	%/°C

.

On the other hand, the design of the solar field requires information regarding the inverter to be used. For this, a 15 kVA inverter with a grid connection option, manufactured by Schneider Electric, whose model is PVSNVC15000, is proposed. The most relevant characteristics of the inverter are shown in

Table 9. Inverter characteristics.

Parameter	Value	Unit
Photovoltaic power	17	kW
Output voltage	230/400	V AC
Maximum output current	24	A AC
Frequency	50/60 ± 3	Hz
Power factor	0.8	-
Input voltage—open circuit	≤1000	V DC
Input voltage—starting	200	V DC
Input voltage at maximum power point	350…800	V DC
Input current at maximum power point	23	A
Efficiency	98%	-

.

By meticulously analyzing the operating data of the devices to be implemented, we can precisely determine the size of the solar field and the optimal arrangement of panels. These statements ensure that the inverter operates flawlessly under the conditions specified by the manufacturer. For the solar field under study, the suggestion requires between two and four panels in series and nine series strings. However, the maximum possible total would be 36 panels, with which there would be a potential power deficit of 800 W at the inverter input; however, increasing the number of panels in series or adding an additional parallel string would cause a violation of the limits indicated by the inverter manufacturer. Therefore, the solar array consists of 36 panels with nine strings, each with 4 panels connected in series while the nine strings are connected in parallel.

5. Analysis of Results

We conducted the first tests with a 275 W test panel under normal operating conditions with a constant irradiance of 1000 W/m² and a working temperature of 25 °C. During these tests, we made a significant finding. When we used a 1 $\mathrm{\mu }F$ capacitor we observed the output voltage signal of the panel oscillated, as shown in Figure 6. However, by adjusting the value of the output capacitor to 1000 $\mathrm{\mu }F$ we observed a remarkable change. As shown in Figure 7, the voltage signal stabilized, significantly improving the system’s performance. This adjustment also helped us avoid start-up oscillations, which is a crucial step toward enhancing the efficiency of the panel system.

It is well known that oscillations occur in the first milliseconds of simulation due to the change in irradiance values to which the panel is exposed. When the panel’s output voltage is stabilized, the voltage recorded at the output of the extractor circuit is more stable and has a higher peak. It is worth mentioning that in the iterative process the voltage usually stabilizes after a while.

When the voltage reaches its stable voltage under the conditions studied, there is a peak in the output voltage much higher than that recorded at the solar panel terminals; this validates the model’s functionality against changes in the irradiance value. Analyzing the panel’s output power, which was recorded at the output of the extractor circuit, as shown in Figure 8, also verified the existence of a higher peak power delivered to the change in the initial irradiance.

The peak power output, initially 150 W, soared to a remarkable 250 W with the introduction of our proposed algorithm. This substantial increase of 100 W, representing a variation in generation from 54.5% to 90.9% of nominal power at nominal operating conditions, clearly demonstrates the impressive impact of the algorithm and the power extractor circuit. In the specific case, the duty cycle was close to 0.9, corresponding to an impressive 90.9%. To meticulously study the initial performance of the algorithm, we proceeded to thoroughly examine the system with the inclusion of one single panel under working conditions with a constant irradiance of 1000 W/m², a working temperature of 25 °C, a continuous load of 100 ohms, the signal for pulse width modulation works at a frequency of 5 kHz, and the algorithm in charge of working with PSO starts from initial conditions with four particles. The simulation progressed during the iterative calculation process of the operating conditions of the panel where, once again, it could be verified that although, indeed, the output voltages of the panel and the extractor circuit do not show a considerable variation, the power is improved at all times due to the effect of the power extraction circuit, as shown in Figure 9.

The duty cycle for the specific case was stabilized during the process since the circuit’s operating conditions did not change at any time; at the beginning, with the initial conditions, the algorithm suffered oscillations typical of the method but soon stabilized to stable operating values, as shown in Figure 10.

The exact configuration was tested with an irradiance of 200 W/m², with which, consequently, we expect lower performance, as calculating the panel parameters involves an iterative process. Given the complexity of the equations, the variables related to the model will show a specific oscillatory behavior since the panel searches for an operating point before stabilizing its signal; even so, it is possible to note the validity of the model by checking that the power does not fluctuate with each iterative step of searching for the operating point but, on the contrary, remains relatively stable.

Figure 11 shows an increase in the performance of the energy generated during the short time of analysis since the existing valleys in the power output of the panel are reduced, the pulsating behavior in the power of the panel responds to the closing and opening of the switch that composes the power extractor circuit and the behavior of the existing inductor. As far as the voltage is concerned, the oscillatory behavior also demonstrates the search performed by the algorithm for a maximum transfer point since the output voltage behaves differently from that generated by the panel due to the search of the particles; the above is shown in Figure 12. It is worth mentioning that Figure 12 is displayed in seconds to show the detail of the response of the system; as has been explained, the calculations are iterative. This behavior is noticeable in the first 50 ms.

Once we validated the optimization method proposed in this research, we implemented a balance model using Vensim. This model ensured a balance between the power generated and the power demand. Figure 13 shows the implemented model.

Irradiance, temperature, and demand directly relate to the analysis time, allowing us to perform an hourly analysis. In contrast, Python (version 3.9) executes the algorithm to control the power extractor circuit and obtains the maximized PowerPoint, enabling us to achieve a balance and determine an economic benefit or detriment depending on the given conditions. We developed the described variables as lookup variables related to curves dependent on one variable. We designed the user interface with user friendliness, presenting these variables clearly and intuitively, as shown in Figure 14.

With the above, operating pairs for temperature and irradiance are found for each hour of the day. These pairs are inserted into the model generated in Python to obtain system operating points and the power to be generated to create the expected power balance. Figure 15 shows the environmental data used for 24 h.

As previously mentioned, there are time slots in which the irradiance values are zero so that the system will consistently not be able to generate power; these conditions were not tested to reduce the amount of processing time. Figure 16 shows the results obtained for the energy generated in each hour, taking into consideration the inverter data, since, for example, in hour 6 the irradiance level takes a value of 6.07 $\left[{\displaystyle \frac{W}{{\mathrm{m}}^2}}\right]$, which is an insufficient value to produce the necessary starting voltage, that for the inverter under study corresponds to 200 V. Hence, the inverter delivers power for 8 h. During this time, the operational needs are met until 16 h since the voltage is insufficient at 17 h.

We use the current year’s tariff schedule and collect the data for a cost analysis. We assume each user is responsible for paying the demand. We take the design value of the transformer, 14.53 kVA, divide it by the total number of users, eight, and determine the consumption range, with the price set at USD 0.1285/kWH. Figure 17 shows the balance obtained for each hour, indicating that the benefit or detriment for the day in question amounts to USD $9.43$ to be paid to the distribution company for the total number of users. Without the photovoltaic system, the actual cost would have been USD $21.07$, saving USD $11.64$ for the analyzed day, that translates to a 55.2% saving with the installation of the photovoltaic system.

Negative values correspond to power deficits from the solar system, which users must pay to the distribution company, while positive values represent generation surpluses. We consider the surpluses to be paid to the user at the same cost as the demand. Figure 18 illustrates the economic benefit or detriment at each hour of the day.

To study the performance of the implemented model, the solar field is placed under the same environmental conditions as those tested with the power extraction algorithm. Figure 19 shows the results obtained without the power extraction circuit.

The results show that without using the extraction circuit, an average of 85% of the power obtained with the inclusion of the extractor circuit and the use of the PSO algorithm for the search for the maximum power transfer can be obtained. Figure 20 shows the surpluses or deficits presented for the first case using the extractor circuit and PSO.

Figure 21 shows the same balance, but for the second case neither the extractor circuit nor the PSO algorithm is considered, although the curve’s trend is similar to that obtained for the first case.

Figure 22 illustrates a comparison between both cases. It is possible to notice that during production hours, the deficit is more pronounced when the demand cannot be covered, and the surplus is lower when the generation exceeds the demand for the building; this will, therefore, alter the economic benefit or detriment.

Finally, Figure 23 compares the economic benefit or detriment with both cases studied; as with the power curves and their balances, the trend is similar, and only the values are altered since the economic benefit or detriment has a flat rate or constant value.

The sum of the values plotted above shows that the value to be paid to the distribution company for the total day in the second case is USD 10.97, which is slightly higher than the value obtained in the first case. From an economic perspective, Figure 24 illustrates the accumulated sum in each hour.

In the Figure 24 the accumulated sum always shows negative values; however, in the hours corresponding to those of higher generation, the decrease is more pronounced when the algorithm and circuit proposed in this research work are implemented.

5.1. Performance and Convergence Time

Finally, to compare the implemented model with other methods in the literature, the average performance obtained from the method in question is considered with values obtained for normal conditions and two additional methods: P&O (perturb and observe) and InCond (incremental conductance).

Table 10. Obtained results.

Irrad	Temp.	P [W]	P [W]	P [W]
$\mathbf{\left[}{\displaystyle \frac{\mathbf{W}}{{\mathbf{m}}^{\mathbf{2}}}}\mathbf{\right]}$	[°C]	P&O	InCond	PSO
		MPPT	MPPT	MPPT
600	25	3040	3040	3058.6
1000	25	5000	5000	5030.6
1000	40	4690	4685	4718.7
800	40	3785	3788	3808.2

shows the performance comparison, where it can be seen that although the variation is not significant, there is an improvement in performance without taking into account that PSO is a method with less variation in the operating point since it finds a single maximum point, while the additional methods base their performance on the constant variation of the duty cycle until it is oscillating near the operating point; the operating parameters of the methods mentioned above were obtained in [26], where a study of both methods was carried out.

Significantly, the data show an average improvement of 1% when implementing the control method under PSO, a notable advancement compared to the improvement that can be obtained by means of P&O and InCond, whose performances are similar. This underscores the progress in the field of algorithm performance. To determine the algorithm’s performance and the time required to find an optimal point, we conducted experiments, and the proposed system was tested under similar operating conditions in three different runs. Figure 25 shows that the algorithm consistently achieves a stable operating point around 0.35 s of simulation for the three tests performed. This stability, even with slight endpoint variations, is a reassuring aspect of the algorithm’s performance.

5.2. Implementation in an Experimental System

When implementing a PSO algorithm with a power extraction circuit in a PV system, it is observed that the results differ significantly, between 10% and 40%, from the global optimum when implemented in a real test system. This is due to several factors, such as changing environmental conditions, nonlinear characteristics of the solar panels, and inherent losses in the circuit hardware. In addition, in a real environment, variations in temperature, partial shading, and other external influences affect system performance, which is not always accurately reflected in simulations or theoretical models. These discrepancies between the ideal model and the practical implementation result in a noticeable difference in the efficiency and effectiveness of the PSO algorithm in maximizing the power extracted from the PV system. In the scheme shown in Figure 26, without considering the characteristics of the panels and load in the final implementation, as mentioned in previous sections, we implement the proposed algorithm on the test system and obtain the voltage characteristic curve, which we analyze with and without the algorithm.

The experimental system is located on the Universidad Politecnica Salesiana campus, which has an installed capacity of 1.2 kWH. We measured the generation conditions on a schedule of 7:00 h am to 19:00 h at standard conditions, then with the implementation of a PSO algorithm, and finally, implementing the extraction circuit and the PSO, giving the results shown in Figure 27.

As we can see in the graph focused on the use of the PSO algorithm, the extractor circuit starts with a slight energy using the PSO algorithm. However, as the hours of the day increase, the chosen sensor heats up due to the Joule effect, and the delivered energy becomes similar to the one obtained when only the PSO algorithm is implemented, without using an extractor circuit, with a slight increase, which shows that our proposal has a slight advantage.

However, we still need to improve on the technology regarding hardware limitations.

Implementation Process

Implementing the proposed PSO algorithm requires a clear understanding of how the method works and how to integrate it into the inverter hardware and software; the general steps to implement this algorithm are described below:

i.

Understanding the algorithm: We must understand that the development of this MPPT algorithm works by perturbing the operating voltage of the solar panel and observing the change in the output power. If the power increases, the perturbation continues in the same direction; if it decreases, it decreases.

ii.

Data collection: The inverter, with its microcontroller, plays a pivotal role in the MPPT algorithm. It measures the solar panel’s voltage and current to calculate the output power. This measurement is achieved through voltage and current sensors connected to the inverter microcontroller.

iii.

Initialization of the swarm: A set of particles with random positions (solar panel operating voltages) and velocities is initialized. Each particle must have an initial position and store its best-known and global best positions.

iv.

Implementation of the algorithm: The algorithm is implemented in the firmware of the inverter microcontroller; this is achieved as long as the inverter allows it. Since we use experimental equipment, the firing code of the IGBT is open for modification.

v.

Hardware control: The microcontroller must be able to adjust the solar panel’s operating point. This involves controlling a DC-DC converter, such as a buck and boost converter, to change the input voltage to the inverter.

vi.

Optimization and testing: The PSO parameters, such as the wire size, inertia constants, and cognitive and social constants, are adjusted to improve the algorithm’s efficiency. Testing: Testing the algorithm under various lighting and temperature conditions is crucial. This ensures the algorithm can effectively find and track the maximum power point.

vii.

Implementation of protection: Protection mechanisms must be implemented to prevent damage to the inverter and solar panels, including protection against voltage, overcurrent, and overheating; usually, these systems already have these protections, for which we must only make adjustments to the system conditions.

viii.

Monitoring and updating: Monitoring and updating functions are integral to the system’s adaptability and performance maintenance. Monitoring functions are designed to oversee system performance and identify potential issues, while updating functions enable the microcontroller firmware to be enhanced, thereby improving the algorithm and introducing new features in the future.

6. Conclusions

The algorithm that searches for the maximum power transfer point can modify its results according to the system’s different operating conditions, climate, or related to the attached circuits.

The PSO optimization algorithm developed by Python is shown to be capable of performing under different operating conditions and sensitive to changes relative to the usual operation of a solar panel.

A sensitivity analysis of the PSO algorithm can be complicated due to the stochastic nature of the algorithm and the inherent variability of environmental conditions affecting PV systems. PSO parameters, such as particle number, acceleration coefficients, and inertia, can have nonlinear and complex impacts on the algorithm’s performance. In addition, changes in solar irradiance and temperature add another layer of uncertainty, making it difficult to isolate and measure the individual effect of each PSO parameter. For these reasons, obtaining conclusive and reproducible results is not an easy task.

Our findings show a stark contrast between the situation when the algorithm is implemented compared to when it is not. In the face of irradiance and load variations, the algorithm proves its worth by enabling the extraction of a staggering 85% more power than without its use.

The use of Vensim is justified, given its potential in causal analysis in conversion processes. Its versatility allows it to be interpreted in Python so that the parameters can be modified, read, and simulated from Python. For the present case, it has been possible to parameterize temporal variables with around 700 values, in addition to the facility to manipulate the iteration intervals by seconds, minutes, hours, days, and even years.

In conclusion, implementing an extraction circuit in a PV system using a particle swarm optimization (PSO) algorithm for maximum-power-point tracking (MPPT) can slightly increase the system’s energy delivery. However, its performance is limited by the Joule effect, which generates heating losses, and its constructional form, which can add complexity and cost to the system design.

Methods based on artificial intelligence and heuristic optimization should be further developed to improve and overcome the limitations of traditional techniques because they are better adapted to the changing environmental and nonlinear characteristics of PV systems. On the other hand, these new trends require greater data processing capacity, and their implementation costs are excessive. When integrating these advanced algorithms in traditional photovoltaic systems and, above all, on a large scale, there are limitations in efficiency and cost parameters, which must be overcome as technology advances.

Future Work

In the current context of MPPT optimization methods in PV systems, future work can focus on several areas to overcome existing limitations and improve overall performance. These include the development of hybrid algorithms that integrate artificial intelligence techniques with traditional methods, optimizing tracking efficiency and stability under varying environmental conditions. In addition, research could focus on reducing the cost and computational complexity of these advanced methods, facilitating their implementation in large-scale systems. Another field of interest is the improvement of the robustness and reliability of MPPT algorithms in partial shading situations through the use of distributed optimization techniques and collaborative algorithms among multiple PV modules. Also, the integration of IoT technologies and real-time machine learning could offer new opportunities for dynamic monitoring and adjustment of systems, further improving their efficiency and responsiveness to changing conditions.