Parameters identification of a photovoltaic module in a thermal system using meta-heuristic optimization methods

Introduction

The main applications of solar energy can be classified into two categories: thermal and photovoltaic systems. In the nature, only 20% of solar radiations incident on a PV module can increase the operating cell temperature, in which its performances are deteriorated [ ¹ ].

Consequently, the obtained energy conversion is reduced with order of 0.4–0.5% when environmental temperatures are progressively increased [ ¹ ]. To avoid this drawback, the overheating problem of the conventional PV cells is solved using the proposed cooling system which drops its cell temperatures to those neighboring the nominal temperature range.

The proposed solar system uses the water in the closed circuit in which its cells are cooled down in high temperatures. The advantages of this system are better heat absorption and lower production cost [ ² , ³ ]. Therefore, our study focuses on the comparison between the obtained electrical powers by both conventional PV and proposed PVT systems in different atmospheric conditions.

In the modeling step of actual PV/PVT systems, a good choice of the efficient model ensuring more accuracy of the actual system behavior is a key success factor for several analysis studies [ ⁴ ], such as diagnosis, synthesis and robustness of PV/PVT control law step against sensor noises, model parameter uncertainties and PV output power forecast [ ⁵ ]. Therefore, various electrical circuits’ oriented PV models have been proposed in the literature providing some optimal models where different intrinsic physical phenomena occurred in the electricity generation process. Among them, the equivalent circuit based upon a single diode is the most commonly adopted model for PV cells, accounting for the photon-generated current and the physics of the P–N junction of the PV cell.

In the design phase of single diode PV models, some unknown parameters should be well optimized such as the photo-generated current, the diode quality factor, the series and parallel resistors and others. An optimal set of these parameters is determined through solving an optimization problem which is previously formulated by the designer. Its fitness metric function (to be minimized) presents the mean square error given through discrepancy value between model prediction and actual measurement for each sampling time.

In the recent years, many researchers have been interested in designing efficient single diode PV models using some evolutionary computation algorithms such as GA or PSO algorithm or others [ ⁶ , ⁷ ]. Among them, Askarzadeh et al. identified the PV model parameters using the Bird Mating Optimizer BMO algorithm [ ⁸ ]. Fialho et al. determined these parameters through some analytical approaches where the PV system was linked to the electric grid [ ⁹ ]. Ogliari et al. estimated the model parameters by adopting the particle filter in the conventional PV power output forecast [ ¹⁰ ]. Soon and Low identified the single diode KC65T PV model given by three unknown electrical components, which were optimized by the PSO algorithm based upon log barrier constraint [ ¹¹ ]. These unknown electrical components have been identified by Qin and Kimball from field test data using PSO algorithm in which both total solar irradiance and environmental temperature variations are taken into account [ ¹² ]. These parameters have been identified from combining the GA by the Interior-Point Method IPM by Dizqah et al. [ ¹³ ]. Unfortunately, all proposed models are imprecisely described; the actual solar system behaviors when atmospheric conditions are changed in a wide range, particularly at high environmental temperature as well as the robustness of the developed models have not been considered.

This paper investigates the analysis of the above mentioned problem in which two following main contributions are proposed. The first one is to enhance the obtained electrical properties of the conventional PV system, regardless the effect of various atmospheric conditions. The second one is to decrease the obtained sensitivity of model parameters against environmental temperature variations. Therefore, the obtained electrical properties become depending only on the total solar irradiance variations. As a result, the validity of the proposed model will be extended in wide time range for different weathers such as hot and hazy weathers. The latter presents an important capital, especially, in synthesis control laws ensuring a good tracking of maximum power point MPPT.

The current paper starts in “ ^{Tools used for optimization} ” by introducing the mechanism of both GA and PSO algorithm. In “ ^{Circuit model of PV/PVT cells} ”, the design problem of single diode PV/PVT models is formulated. Its model parameters are then determined through experimental data recorded at different operating points in the “ ^{Experimental tests and study cases} ”. Robustness analysis of obtained PV/PVT models is established where other experimental data recorded at high temperatures and different total solar irradiances are taken into account. Finally, the current paper is ended by a conclusion given in “ ^Conclusion ”.

Tools used for optimization

Circuit model of PV/PVT cells

Mathematical model of PV/PVT modules

The following equivalent electrical circuit based on a single diode is commonly used in modeling step of PV/PVT cells:

According to Fig.  ¹ , the electrical circuit model of PV/PVT cells consists of a current source assembled in parallel with a diode. A series resistor and a parallel resistor are added to describe the dissipation phenomena inside PV/PVT cells [ ¹⁷ , ¹⁸ – ¹⁹ ]. According to the equivalent circuit, the following expressions are established [ ²⁰ , ²¹ – ²² ]:

Ipv=Iph-ID-Vpv+Rs·IpvRp,

where Iph the photo-generated current, Ipv and Vpv are, respectively, the output current and the output voltage provided by the solar cell. ID the diode current given by:

ID=I0·-1+eq·Vpv+Rs·IpvK·Ta,

where Io is the reverse saturation current of the diode defined by:

I0=TaTn3n·Isc-1+eq·Vpv+Rs·IpvK·Ta·eq·Vgn·1Ta-1Tn,

Where n denotes the diode ideality factor. Moreover, the photo-generated current Iph is defined by:

Iph=GaGnIsc+KI(Ta-Tn),

where Ta is the absolute temperature, Tn is the nominal temperature given at Standard Test Conditions (STC), i.e., Tn=25∘C,KI is the constant weighting the temperature discrepancy ΔT=Ta-Tn,Ga is the total solar irradiance and Gn is the nominal total solar irradiance given at STC, i.e., Gn=1000W/m2 . According to Eq. ( ⁶ ), the maximum photocurrent Iphmax is determined when Ta and Ga reach its nominal values, i.e., Ta=Tn, and Ga=Gn . It yields in fact Iphmax=Isc . Moreover, the short resistor current Isc is given when the series resistance is low enough and the shunt resistance is high enough. Therefore, Iph should be limited by the upper bound Isc . Table  ¹ summarizes the meaning and the corresponding value of diverse electrical components.

Fig. 1

Table 1

Parameter	Quantity identification (unity)	Value
Isc	Short resistor current (A)	2.99
q	Elementary charge (c)	1.60 × 10−19
K	Boltzmann’s constant (J/K)	1.38 × 10−23
Voc	Open-circuit voltage (V)	20.80
Vg	Energy gap (eV)	1.20

Experimental tests and study cases

The comparative study has been presented here for two solar systems based upon ISOFOTON I-50 PV modules. The first one is the conventional PV cell operating without cooling. However, the second one is the proposed PVT cell that previously reinforced against high temperatures by means of the closed water circuit. These solar systems are positioned on the building roof of the applied research unit in renewable energy located in the south of Algeria.

In this study, both PV and PVT panels are inclined by an angle equals to the latitude of the area and each one has two sensors. The first sensor is a K-type thermocouple which measures the absolute temperature using the Campbell CS215 instrument. The second one is installed to measure the total solar irradiance using the Kipp and Zonen CMP21 pyranometer.

All recorded experimental data are carried out by the Agilent 34970 A. The experimental systems are shown in Fig.  ² .

Fig. 2

The typical electrical characteristics provided by both solar systems are summarized in Table  ² :

Note that, in severe weather conditions, absolute temperatures and total solar irradiances change, respectively, within 31.5∘C≤Ta≤44.5∘C and 600W/m2≤Ga≤1050W/m2 . The overheating problem of PVT cells is solved using the proposed cooling system which drops the PVT cell temperatures until those neighboring the nominal temperature range, i.e., 22.3∘C≤Ta≤26.9∘C . Consequently, the electrical properties provided by the PVT system depend only on a given total solar irradiance range. On the other side, the total solar irradiance range has been divided on two sub-ranges 650W/m2≤Ga≤750W/m2 and 800W/m2≤Ga≤1050W/m2 in which the same obtained electrical properties by the actual PVT system may be conserved. For that reason, two experimental measurements were performed in the modeling phase the actual PVT behavior. These perfect modeling requirements occur in the month of April, especially, from starting times 10h00 and 14h00. For both actual PV and PVT systems, experimental data were recorded every 30 s, in a clear day during the month of April 2015 from 10h00 to 12h30, yielding to 300 sampled measurements. In the same way, a second set of 300 other sampled measurements were recorded from 14h00 to 16h30. Therefore, a total set of N=600 sampled measurements were recorded. During the first N=300 measurements, the mean absolute temperatures and the mean total solar irradiances varied around Ta=22.3∘C and Ga=700.6544W/m2 , respectively. On the other hand, during the second N=300 measurements the mean absolute temperatures and the mean total solar irradiances varied around Ta=26.9∘C and Ga=900.3362W/m2 , respectively. The recorded experimental data are then stored in an on-board SD card for an off-line PV and PVT model parameters extraction. Its optimal sets are given by the GA and PSO algorithm using the following lower and upper boundary constraints:

Tables  ³ and ⁴ summarize the tuning parameters of the GA and PSO algorithm, which are given according to some guidelines proposed in [ ²³ , ²⁴ – ²⁵ ]:

Note that the GA and PSO algorithm are executed 20 times. After that, the best obtained fitness value is considered to design the single diode PV and PVT models.

Design of PV and PVT models

Design of first PV and PVT models

Note that one of most important factors that validate the GA and PSO algorithm is the best value of the fitness function which should be lower as much as possible. Therefore, Fig.  ³ shows the obtained fitness plots provided through GA and PSO algorithm during the extraction process of the first PV/PVT model parameters where the best minimization of the cost function is presented by the dashed blue line.

Fig. 3

According to Fig.  ³ , it is easy to observe that the GA converges within 50 generations whereas the PSO algorithm converges within 160 generations yielding also the best MSE minimization. The obtained first PV and PVT model parameters are summarized in Table  ⁵ in which the best parameters are mentioned in bold:

Table 5

		Model parameters				Jmin
		Iph	n	Rs	Rp
PVT	GA	2.0993	1.0000	0.1817	2.8967	5.95 × 10−4
	PSO	2.1626	1.0018	0.1936	2.1599	5.67 × 10−4
PV	GA	1.9728	1.5734	0.0343	4.7437	5.08 × 10−4
	PSO	2.0180	1.0000	0.1167	3.7548	4.94 × 10−4

To confirm these results, Fig.  ⁴ compares the actual current–voltage characteristics provided by the proposed PVT system with those determined through its corresponding first PVT models. In addition, Fig.  ⁵ compares the above mentioned characteristics given through the conventional actual PV system and its corresponding first PV models.

Fig. 4

Fig. 5

According to Figs.  ⁴ and ⁵ , the current–voltage characteristics, provided by actual PV and PVT systems, matched as close as possible with those given by the first PV and PVT models where the best results are ensured by the PSO algorithm. Now, the obtained actual and predicted power–voltage characteristics are compared in Fig.  ⁶ :

Fig. 6

According to Fig.  ⁶ , it is easy to observe that the obtained actual power–voltage characteristics are closely matching those determined through the corresponding models. This figure confirms also that the obtained power energy is enhanced by the actual PVT system with a maximal value of PPVT=26.19Watts given at the voltage V=14.66Volts . This maximal power is better than the one provided by the conventional PV system in which its maximal power reaches PPV=25.9Watts at the voltage V=14.50Volts . Note that, this comparison does not reduce the GA efficiency, as it will be shown in the next section.

Design of second PV and PVT models

In this section, the same tuning parameters summarized in Tables  ³ and ⁴ are used. Therefore, Fig.  ⁷ shows the obtained fitness plots provided through GA and PSO algorithm during the extraction process of the second PV and PVT model parameters where the best minimization of the cost function is presented by the dashed blue line.

Fig. 7

According to Fig.  ⁷ , it is easy to observe that the best fitness values obtained by GA and PSO algorithm are, respectively, provided within 50 and 175 generations, in which the best results are ensured by the GA. Note, the obtained second PV and PVT model parameters are summarized in Table  ⁶ in which the best parameters are mentioned in bold.

Table 6

		Parameters				Jmin
		Iph	n	Rs	Rp
PVT	GA	2.9822	1.4644	0.3237	2.1342	7.40 × 10−4
	PSO	2.9900	1.0000	1.2007	7.1147	31.70 × 10−4
PV	GA	2.8508	1.3110	0.4191	3.2142	8.54 × 10−4
	PSO	2.9900	1.0000	1.0825	5.5967	22.30 × 10−4

According to Table  ⁶ , it is easy to observe that the best minimization of the MSE criterion is performed by using the GA.

To confirm these results, Fig.  ⁸ compares the actual current–voltage characteristics provided by the proposed PVT system with those determined through its corresponding second PVT models. In addition, Fig.  ⁹ compares the above mentioned characteristics given through the conventional actual PV system and its corresponding second PV models.

Fig. 8

Fig. 9

According to Figs.  ⁸ and ⁹ , the obtained current–voltage characteristics by the second PV and PVT models are matched as close as possible with those given through the actual PV and PVT systems where the GA gives the best models.

For this reason, only the second PV and PVT models based upon the GA are used to compare its power–voltage characteristics with those determined through the actual PV and PVT systems.

According to Fig.  ¹⁰ , it is clear to observe that the obtained power energy by the actual PVT system has the peak value PPVT=30.48Watts V=13.6Volts , which is better than the one provided by the actual PV system in which PPV=29.75Watts at V=13.30Volts .

Fig. 10

Validation of the obtained PV and PVT models

In this section, both first PV and PVT models based upon PSO algorithm and both second PV and PVT models based upon GA are validated in severe atmospheric conditions, which are recorded in July 2015.

Table  ⁷ summarizes the given absolute temperatures and the total solar irradiances at different times.

Table 7

Time	08H30	09H00	11H00	13H30	15H30
Ga(W/m2)	672.6580	686.1710	883.4676	1047.9080	936.4730
Ta(∘C)	34.09	36.01	37.88	40.92	39.09

According to Table  ⁷ the five power–voltage curves obtained by actual PV and PVT systems are compared with those provided by its corresponding models. The proposed comparisons are established according to the given total solar irradiance range. Figures  ¹¹ and ¹² compare the given power–voltage characteristics provided by actual PV and PVT systems and both first and second PV and PVT models.

Fig. 11

Fig. 12

According to Figs.  ¹¹ and ¹² the maximal powers provided by the actual PV and PVT systems can be arranged as the following histogrammes:

Table  ⁸ compares the given maximal powers in different weather conditions.

Table 8

		Pmax (PVT model1) = 26.19 Watt			Pmax (PV model1) = 25.90 Watt
Ga(W/m2)	Ta(∘C)	Actual PmaxPVT	Match ratio PmaxPVT\%	Match ratio error ξPVT%	Actual PmaxPV	Match ratio PmaxPV\%	Match ratio error ξPV%
672.658	34.09	24.54	93.700	6.3	17.58	67.87	32.124
686.171	36.10	25.16	96.067	3.93	19.43	75.019	24.981

		Pmax (PVT model2) = 30.48 Watt			Pmax (PV model2) = 29.75 Watt
883.4678	37.88	29.21	95.830	4.166	22.50	75.63	21.008
936.473	39.09	29.90	98.097	1.706	25.47	85.613	14.387
1047.908	40.92	29.68	97.375	2.6247	24.02	80.739	19.261

According to Figs.  ¹³ and ¹⁴ , it is obvious to confirm the following three main results:

Fig. 13

Fig. 14

• In high temperatures, the proposed PVT models ensure better robustness properties than those provided by the conventional PV models.

• The proposed PVT models have the ability to well model the actual PVT measurement regardless the severe atmospheric conditions.

• The proposed cooling system ensures the best electrical powers which become stationary in two different irradiation ranges and independently of temperature variations.

Conclusion

In this paper, the water-cooled PVT system is well modeled by two single diode PVT models according to the two total solar irradiance ranges and the absorbed temperature system. The optimal set of the PVT model parameters are identified through experimental data using both evolutionary optimization algorithms such as GA and PSO. The given current–voltage and power–voltage curves by the actual PV and PVT systems are compared to those given by the proposed PV and PVT models in nominal atmospheric conditions. The robustness of the best PV and PVT models are verified in severe atmospheric conditions in which the PVT model becomes more advantageous than the conventional PV one from an energetic point of view. So, the proposed PVT model becomes interesting for practical uses.