Measurement Interval Effect on Photovoltaic Parameters Estimation

Abstract

Recently, the estimation of photovoltaic parameters has drawn the attention of researchers, and most of them propose new optimization methods to solve this problem. However, the process of photovoltaic parameters estimation can be affected by other aspects. In a real experimental setup, the I–V characteristic is obtained with IV tracers. Depending on their technical specifications, these instruments can influence the quality of the I–V characteristic, which in turn is inevitably linked to the estimation of photovoltaic parameters. Besides the uncertainties that accompany the measurement process, a major effect on parameters estimation is the size of the measurement interval of current and voltage, where some instruments are limited to measure a small portion of the characteristic or cannot reach their extremum regions. In this paper, three case studies are presented to analyse this phenomenon: different characteristic measurement starting points and different measurement intervals. In the simulation study the parameters are extracted from 1000 trial runs of the simulated I-V curve. The results are then validated using an experimental study where an IV tracer was built to measure the I–V characteristic. Both simulation and experimental studies concluded that starting the measurements at the open circuit voltage and having an interval spanning a minimum of half of the I–V curve results in an optimal estimation of photovoltaic parameters.

1. Introduction

Photovoltaic parameters extraction using new optimization methods is still a very active research topic. In recent years, a wide variety of optimization methods have emerged, especially algorithms that are inspired by natural phenomena. The majority of them are applied to extract photovoltaic parameters of different equivalent electrical circuits of photovoltaic cells [1,2,3,4,5,6].

The optimization process usually uses the ordinary least squares (OLS) as the cost function to be optimized [7], which is based on minimizing the difference between the measured and the estimated current in each iteration of the algorithm [1,2,4]. In [3,8], the authors suggested a new cost function to enhance the estimation of photovoltaic parameters. This function is based on the total least squares (TLS) which considers both the variations in current and in voltage. In [3], eleven metaheuristic algorithms were used to compare the performance of the suggested TLS cost function with the traditional OLS. That study concluded that using the total least squares function with a teaching learning-based optimization (TLBO) algorithm achieved the best results.

Photovoltaic parameters identification starts with current and voltage measurements to obtain the I–V experimental characteristic of the photovoltaic system. There are multiple high- and low-cost technologies used to build I–V tracers that can perform these measurements.

The first approach is the use of a load resistor that can be implemented through a variable resistor (rheostat) or a bank of resistors [9,10,11]. Although this method is straightforward and economical, it is time consuming, lacks precision and accuracy, the data resolution is limited, and it is hard to use for high-power photovoltaic modules.

The second technology is the capacitive load [11,12,13], which uses a bank of capacitors. It relies on the property that, when connected to a photovoltaic module, their charge rises, resulting in a current decrease while the voltage increases. In this circuit, a bank of resistors is also required to allow for the discharge of the capacitors. Many variants of this technology have been suggested in the literature [11,12,14]. However, to get measurements with minimum errors, expensive high-quality capacitors are need, and the required size of the capacitors can also present difficulties in the characterization of big photovoltaic modules.

Electronic loads such as power transistors can also be used [12,14,15,16]. A MOSFET or BJT can be connected to a programmable microcontroller to replace the variable load in the IV tracer circuit. This method has the disadvantage of inaccurate measurements, especially near the short circuit current area. This technology can only be used with low power photovoltaic panels, otherwise a heat sink and cooling system are needed.

The fourth method is the use of DC-DC converters [17,18]. In this approach the voltage generated by the photovoltaic module is controlled by the duty cycle of the converter. This measurement method cannot reach the short circuit current nor the open circuit voltage regions. It can also present some measurement errors because it cannot control duty cycles above 80%. A four-quadrant power supply can also be used, but it is highly expensive and requires specialized knowledge to operate [9,10,11].

The IV tracer technologies used to obtain the characteristic of photovoltaic systems affect the measurement interval of the current and voltage, since measurement errors are always present in the form of uncertainties, and in certain cases by not covering the whole voltage or current range [9,10,11].

Research on photovoltaic parameters extraction from measured I–V characteristics has, so far, mainly focused on suggesting methodologies and comparing the obtained parameters to find the most efficient mathematical optimization algorithm for this task [1,2,3,8,19,20,21,22]. However, the effect of the measurement interval on the photovoltaic parameters’ extraction has not been addressed. Therefore, this paper’s key points are:

• The main contribution of this work is to present and analyse the influence of measurement intervals on photovoltaic parameters extraction.
• The optimization method used for photovoltaic parameters estimation is a hybrid method between teaching learning-based optimization and the Nelder–Mead algorithm (TLBONM) which is compared with the TLBO algorithm. A study comparing the two methods is presented.
• Three different cases are considered, with measurement starting points at the short circuit region; the maximum power point; and the open circuit region.
• The simulation results are validated using an experimental setup composed of the equivalent electric circuit of a single diode model.
• As the main result, the best starting point and the minimal measurement for parameters estimation are found.

2. Photovoltaic Parameters Extraction Methodology

2.1. Photovoltaic Cell Mathematical Model

The mathematical model of the single diode equivalent circuit of a photovoltaic solar cell is presented in Equation (1), where I and V are the output current and voltage. The photovoltaic parameters are the photocurrent I_ph; the saturation current I_s; the ideality factor of the diode n; the shunt resistance R_sh; and the series resistance R_s [23,24]. Additionally, q represents the electronic charge, k_B is Boltzmann constant, and T is the absolute temperature of the P-N junction. This equation can be solved using the Newton–Raphson method.

$$f(V,I)=I_{ph}-I_s\left[\exp \left(\frac{q\left(V+I{R}_s\right)}{n{k}_{B}T}\right)-1\right]-\frac{V+I{R}_s}{R_{sh}}-I=0.$$

(1)

To extract the photovoltaic parameters from the I–V measured characteristic, an objective function representing the deviations between the model and the measurements is minimized using an optimization algorithm in an iterative process.

2.2. TLS Cost Function used for Photovoltaic Parameters Extraction

The objective function used in this paper is based on the total least squares as expressed in Equation (2), where Î and $\widehat{V}$ are the estimated current and voltage and N is the number of data points. Since the mathematical model of the I–V characteristic of a photovoltaic cell is an implicit and continuous function (Equation (1)), the application of the objective function (Equation (2)) is a challenge, because it cannot be directly computed. In [8], the RMSE function based on Total Least Squares was analysed and an iterative method was proposed to calculate its values.

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum _{j=1}^N{(I_j-{\widehat{I}}_j)}^2+{(V_j-{\widehat{V}}_j)}^2}}$$

(2)

The performance of this function was also evaluated in [3] by comparing it with the usual cost function (i.e., ordinary least squares), when coupled with eleven different optimization algorithms with both single and double diode models.

2.3. Optimization Method

Following the work in [3], where multiple optimization algorithms are compared along with two cost functions, it was concluded that the teaching learning-based optimization (TLBO) algorithm had the best performance, and achieved the most accurate results when coupled with total least squares [TLBO-TLS].

However, it was also found that this algorithm is computationally more intensive than the others. To address this problem, this paper presents a hybrid method, which is computationally less intensive while having a similar performance as the TLBO. The hybrid method consists of combining the TLBO with the Nelder–Mead algorithm. The next sub-sections present a comparative study to specify which approach will be adopted for the analysis of the measurement interval effect on PV parameters estimation.

2.3.1. Teaching Learning-Based Optimization Algorithm (TLBO)

The TLBO is a genetic human behaviour-inspired algorithm that was first introduced by Rao et al. in [25]. It was inspired by the interaction between the teacher and the students and the interactions among students to improve the quality of learning. Starting from an initial population of students, the algorithm converges to the optimum solution. It is composed of two optimization steps. First, in the ‘Teacher phase’, the students try to reach the level of the teacher’s knowledge through

$$po{p}_{new}=po{p}_i+rand·\left(po{p}_T-TF·mean\right),$$

(3)

where $TF$ is the Teacher Factor coefficient, $mean$ is the average of all population, rand is a random number from the uniform distribution in the interval (0,1) and ${pop}_T$ is the best of the population, i.e., the $\mathrm{T}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{r}$. Then, the ‘Learner phase’ starts, where the students improve their knowledge by learning from each other. Each student (pop_i) is compared to a second, randomly chosen, student (pop_RS). If pop_i is better than pop_RS, then the student becomes

$$po{p}_{new}=po{p}_i+rand·\left(po{p}_i-po{p}_{RS}\right),$$

(4)

otherwise

$$po{p}_{new}=po{p}_i+rand·\left(po{p}_{RS}-po{p}_i\right).$$

(5)

The steps are repeated for each of the students (population) until the convergence criterion is reached. The solution corresponds to the student which has the minimum objective function value.

This method has been applied to multiple studies for photovoltaic parameter extraction [3,26,27,28]. In [3], the TLBO presented the best results when coupled with the TLS objective function. However, it was found that this method is time consuming, mostly because the TLBO is a stochastic algorithm and therefore requires a large population of candidate solutions.

2.3.2. Nelder–Mead Algorithm (NM)

Multiple studies have proposed different methods to extract the photovoltaic parameters from an I–V characteristic. In [29], the Nelder–Mead algorithm was used for photovoltaic parameters extraction where a new convergence condition was suggested to improve the convergence speed of the algorithm. The Nelder–Mead algorithm is a heuristic search method that can be used to minimise non-linear continuous functions. It starts with an initial simplex and, through the actions of reflection, expansion, contraction, and shrinking, can control the direction and the size of the simplex. This algorithm reaches the optimum point faster than the stochastic algorithms, however its convergence properties heavily depend on the initial simplex [8], and it is unsuitable for problems with multiple local minima. The convergence steps are detailed in [29] and the results of its application on the RTC France standard cell are presented in [3,8].

2.3.3. The Hybrid Methodology TLBO-NM-TLS

This paper suggests using a hybrid method (TLBO-NM-TLS) by coupling the teaching learning-based optimization with the Nelder–Mead and the total least squares cost function. The flowchart of this method is presented in Figure 1. A study was performed to compare the precision, accuracy and simulation time of three approaches: TLBO-TLS and two versions of the TLBO-NM-TLS method.

Table 1. Conditions for three different optimization methods.

Method	TLBO-TLS	TLBO-NM-TLS 1	TLBO-NM-TLS 2
Population size	25	25 for TLBO	25 for TLBO
		6 for NM	6 for NM
Maximum iterations	1000	150 for TLBO	150 for TLBO
		850 for NM	850 for NM
Number of trial runs	1000	1000	1000
Additional convergence conditions	---	---	An imposed tolerance on the maximum difference between the vertices and the centroid

presents the conditions of this comparative study. The additional convergence conditions used for the third method (TLBO-NM-TLS 2) consist of imposing a tolerance on the maximum distance between the centroid and the vertices and imposing another tolerance for the maximum difference between their cost functions values. This helps the algorithm converge with less iterations resulting in a shorter optimization time [29].

To compare the three methods, the mean and standard deviation of the RMSE, the mean values of single diode parameters, and the computation time were analysed. The results are presented in

Table 2. Comparison study of the three optimization methods.

	TLBO-TLS	TLBO-NM-TLS 1	TLBO-NM-TLS 2
Mean RMSE	5.473 × 10−4	5.473 × 10−4	5.473 × 10−4
Std RMSE	9.045 × 10−14	1.982 × 10−17	1.988 × 10−17
Mean $I_{ph}$ (A)	0.7607	0.7607	0.7607
Mean $I_s$ (A)	4.211 × 10−7	4.211 × 10−7	4.211 × 10−7
Mean $n$	1.507	1.507	1.507
Mean $R_{sh}$ (Ω)	55.21	55.21	55.21
Mean $R_s$ (Ω)	0.0345	0.0345	0.0345
Simulation time (s)	2.112 × 104	6.196 × 103	3.691 × 103

.

Table 2 shows that the method with the lowest RMSE was the TLBO-NM-TLS 1. However, its RMSE is very similar to the RMSE of TLBO-NM-TLS 2 while requiring almost double the computation time. Therefore, the TLBO-NM-TLS 2 method with additional convergence conditions was adopted in the analysis of the measurement interval effect on photovoltaic parameter estimation. The simulation time for this method is shown in bold in Table 2.

3. Simulation Study of Measurement Interval Effect on Photovoltaic Parameters Estimation

To study the effect of the measurement interval on the extraction of photovoltaic parameters, a MATLAB simulation following the block diagram shown in Figure 2 was performed. It starts by simulating measured data and then defining the interval used for parameter estimation. Then, the optimization process begins with the initialization of the TLBO population followed by the teachers’ and learners’ phases until the convergence conditions are met. The best result is then used as the initial simplex for the Nelder–Mead method, which will run until the optimum parameters are obtained.

A commercial silicon solar cell from Radiotechnique-Compelec (RTC) France [30] was adopted in this work for comparison between measured and simulated data. This analysis consists of extracting the parameters from 40 different intervals of the I–V characteristic. The algorithm starts with the smallest interval, which is equal to the whole voltage interval divided by 40. Three case studies, with different starting points, were considered: (0, I_sc), (V_mpp, I_mpp), and (V_oc, 0).

3.1. Simulated Data

For each case study, the parameters were estimated by computing the mean value of 1000 trial runs. To simulate measured data, additive white noises with RMS values noise_V and noise_I were, respectively, added to both voltage and current, following

$$V_{noisy}=V+nois{e}_V\cdot randn\left(length\left(V\right)\right),$$

(6)

$$I_{noisy}=I+nois{e}_I\cdot randn\left(length\left(I\right)\right).$$

(7)

3.2. Interval Selection

To study the estimation of photovoltaic parameters according to the interval of the current and voltage measurements, 40 different intervals of increasing size were studied. To choose the interval, the first step is to define a starting point: in an I–V characteristic, there are three important points that will correspond to each case study.

3.2.1. Case Study 1 (Starting at (0, I_sc))

In this case, the interval selection starts from the short circuit point (0, I_sc). Therefore, the voltage intervals are [0, V_h], where V_h is calculated using Equation (8) where h = 1, …, 40 represents the step to increase the interval size. Figure 3 shows the 40 intervals used in this case study.

$$V_h=V_{OC}\cdot h/40.$$

(8)

3.2.2. Case Study 2 (Starting at (V_mpp, I_mpp))

For this case, the interval selection starts from the maximum power point (V_mpp, I_mpp). The coordinates of the MPP (V_mpp, I_mpp) are found from the P–V characteristic and the limits of the voltage interval are [V_h1, V_h2], where the upper limit V_h2 is determined using Equation (9), with h = 1, …, 40 representing the step to increase the size of the interval, and the lower limit V_h1 being defined as the minimum value with the same power as V_h2 (Equation (10)), as shown in Figure 4.

$$V_{h_2}=V_{MPP}+\left(h\cdot \left(Voc-V_{MPP}\right)/40\right),$$

(9)

$$V_{h_1}=V\left(P\left(V_{h_2}\right)\right) \mathit{with} V_{h_1}\ne V_{h_2}.$$

(10)

3.2.3. Case Study 3 (Starting at (V_oc, 0))

In this case, the interval selection starts from the open voltage point (V_oc, 0). The selection process is simpler than in the previous case because the maximum value V_oc will be the same for all intervals and the minimum value is obtained from

$$V_h=Voc-\left(Voc\cdot h/40\right).$$

(11)

The voltage interval is therefore [V_h, V_oc] as shown in Figure 5.

3.3. Simulation Results

The results are compared using the mean values of the photovoltaic parameters (Figure 6), and their standard deviations over 1000 trial runs (Figure 7). Figure 8 presents the relative errors of these parameters when compared with the parameters obtained with the TLBO method in [3]. Figure 9 shows the mean of the absolute errors (MAE)

$$\mathrm{MAE}=\frac{1}{N}{\displaystyle \sum _{j=0}^{\mathrm{N}}\left|I_j-{\widehat{I}}_j\right|}$$

(12)

between the estimated current values Î_j and the measured current I_j of the commercial RTC solar cell, for the different intervals of each case study. The estimated current values Î_j were obtained by solving Equation (1) using the estimated PV parameters.

For the first case study (in red), with intervals starting from (0, I_sc), the convergence of saturation current, ideality factor, and series resistance start to stabilize just after the 37^th interval, as observed in the mean values of the parameters in Figure 6. This is also shown in Figure 7, where the standard deviations of these parameters decrease with the interval size. However, for the photocurrent and shunt resistance, the estimation process is optimized from the first interval, especially for the photocurrent. From Figure 6, the line representing I_ph is stable as the measurement’s interval increases and the standard deviation of the estimation of this parameter is very low in comparison with the others, showing that the estimation of this parameter is not influenced by interval size. This is also shown by the relative error of this parameter which, in this case, varies between 5.548 × 10⁻³ and 2.702 × 10⁻⁶.

The second case study, starting from (V_mpp, I_mpp), presents better results than the first case for the saturation current, the ideality factor, and the series resistance. The curves of these parameters start to stabilize at the 30^th interval, and the standard deviation in Figure 7 shows that the estimation accuracy of these parameters increases with the interval size. The relative errors of these parameters are also smaller, as shown in Figure 8. However, the first case study remains better at estimating the photocurrent and the shunt resistance.

The third case study presents the best results when compared with the previous two cases for the extraction of the parameters saturation current, ideality factor, and series resistance. Especially for the series resistance, as shown in Figure 6 and Figure 7, the mean value remains stable, and the standard deviation has very low variation as the interval size increases. Its maximum relative error is 1.349 × 10⁻² while the minimum is 1.460 × 10⁻⁵. The estimated value of all parameters stabilized at around the 20^th interval; therefore, for the interval size, this case study presents faster convergence when compared to the previous cases. The shunt resistance is the parameter which presents the worst performance, but it is still better than the results of the second case study.

Overall, the results showed that the best method for parameter extraction from an I–V characteristic consists of starting the measurements at the open circuit point (V_oc, 0). This is shown in Figure 9 where the third case study exhibits a smaller mean of the absolute errors between estimated and measured currents for all the intervals studied.

4. Experimental Validation

4.1. Experimental Setup

Since the internal photovoltaic parameters are not provided by the manufacturers of photovoltaic panels, to validate these results experimentally, an equivalent electrical circuit was used instead of a photovoltaic cell. This circuit is presented in Figure 10. It was built so that the values of internal photovoltaic parameters, I_ph, R_sh and R_s, are known. The zoom in Figure 11 presents the circuit that was built according to the schematic in Figure 10. It includes a Keithley 228 A voltage/current source (100 V/1 A, 10 V/10 A) that has a four-quadrant operation and which was set up at 55 mA, a diode with the reference 1N5404, a 150 Ω shunt resistance, and a series resistance which was built with three 0.1 Ω resistances in parallel. To minimize the resistance of the connecting wires, the circuit was welded.

To obtain the I–V characteristic of the equivalent circuit, an IV tracer was set up as presented in Figure 10, where two instruments were used: one for the load variation and another for current and voltage measurements. A SONY Tektronix model AFG320 Arbitrary Function Generator creates a ramp signal that emulates a variable load. A NI USB 6009 data acquisition card (DAQ) from National Instruments with 8 analogue inputs, was used to measure both voltage and current. The current was measured through a 1 Ω reference resistance. Figure 11 shows the whole experimental set-up. The last component of this device is the computer where the NI-DAQmx drivers were installed. Matlab 9.13 was used to control the function generator using a general purpose interface bus (GPIB) and to read, save, and process the data from the DAQ.

4.2. Measurements Processing

The voltage and current measurements of the PV equivalent electric circuit are shown in Figure 12 with blue dots. To smooth the IV curve, a moving average was applied, resulting in the black curve in Figure 12. The interval limits for each case study were obtained with the method described in Section 3.2. The remaining data were chosen from the measurement by respecting the weighed Euclidian distance between each two successive points. The value of this distance was calculated by dividing the global weighed Euclidian distance of the interval on the desired size, which in this study was chosen to be 30.

4.3. Experimental Results Discussion

Figure 13 presents the estimated parameters according to the different measurement intervals (forty intervals) depending on the starting point as performed in the simulation results.

Figure 14 shows the relative error of the photocurrent and of the shunt and series resistances. The relative errors of the saturation current and the ideality factor were not calculated because their real values were not available.

Figure 15 shows the mean of the absolute errors between the measured and the estimated currents.

For the first case study, shown by red lines, I_ph and R_sh are more accurately estimated than in the other case studies, as shown by their relative error that is smaller than 0.362 × 10⁻² for I_ph and 0.1488 for R_sh. It can be concluded that the size of the measurement interval does not affect the estimation of these two parameters. However, for the remaining parameters (I_s, n, R_s), this case study presents lower performance, as seen in Figure 13.

The second case (blue lines) has the worst estimation of I_ph and R_sh. However, it is better than the previous case in estimating I_s, n, and R_s.

The third case study, presented by green lines, is more accurate and precise, as shown in Figure 14, at estimating I_s, n, and especially R_s, where its maximum relative error is 0.4274, corresponding to the first interval. The measurement interval size does not influence the estimation of this parameter. It is also better at extracting I_ph and R_sh than the second case.

Figure 15 shows that the best option for PV parameter estimation corresponds to the third case study (interval starting at V_oc), since its mean value of absolute errors is the smallest over the full measurements window. It can also be seen that its mean absolute error has a small fluctuation starting from the 20^th interval, which validated the simulation results presented in the previous section.

In this paper, a thorough investigation of parameter extraction methods using both experimental and simulation approaches was performed. The goal was to assess the consistency and reliability of results obtained from distinct starting points using different measurement interval sizes.

Three distinct case studies were analysed, as identified by the colour-coded lines in the results. These case studies aimed to extract the internal parameters (I_ph, R_sh, I_s, n, R_s) from I–V characteristics, providing a comprehensive overview of the performance of each case study.

Upon comparing the experimental and simulation results, several noteworthy trends emerged. The convergence behaviours of parameters in both approaches were similar in some instances, from which it can be concluded that each region of the IV characteristic depends more on certain parameters than on others. For instance, both experimental and simulation results indicated that I_ph and R_sh were estimated with high accuracy and precision in the short circuit region, while the I_s and n were better estimated at the maximum power region. Additionally, for better estimation of these two parameters along with R_s the open circuit region should be used. In both simulation and experimental studies, the open circuit region proved to be more efficient than the short circuit or the maximum power regions for photovoltaic parameters estimation. Relative to measurement interval size, both studies showed that measuring at least half of the I-V characteristic will lead to optimized PV parameters. These results are fundamental for building an efficient IV tracer with the capacity for photovoltaic parameters extraction.

5. Conclusions

This study falls within the framework of enhancing the process of photovoltaic parameters estimation. The current and voltage measurement procedure to obtain the I–V characteristic can heavily affect the parameters estimation. In this paper, the influence of the size of the window of the I–V measured characteristic was studied.

The work contains three case studies depending on the starting point of the intervals. These points are the known coordinates of any I–V characteristic which are defined by (0, I_sc), (V_mpp, I_mpp), and (V_oc, 0). The cost function that was used is based on total least squares to quantify the differences between the measured and estimated currents and voltages. A hybrid methodology based on combining the teaching learning-based optimization algorithm with the Nelder–Mead method was suggested for the optimization process to extract the photovoltaic parameters from the single diode model. This hybridization allows for good results with less simulation time as seen in the first section.

In the simulation study of the effect of the measurements’ interval size on parameter estimation, the I–V characteristic was simulated 1000 times by adding white noise to both current and voltage. For each case study, different interval lengths were analysed. Based on the results obtained, it was observed that the best approach to get photovoltaic parameters was by starting the I–V characteristic from the open circuit point (V_oc, 0), where it can obtain optimum parameters starting from the 20^th interval. This implies that the second half of any I–V photovoltaic characteristic can be sufficient for the estimation of its internal parameters.

This observation was validated by the experimental results, where an I–V tracer was built using a function generator and an NI data acquisition device. In this set-up, the current and voltage were measured from the electrical equivalent circuit of a photovoltaic cell (single diode) to compare the obtained values of I_ph, R_sh, and R_s with real ones.

Based on the results, it was observed that the most efficient approach to estimate the internal PV panel parameters is to start the measurements at the open circuit voltage and measure the second half of the I–V curve. These outcomes will help researchers build an optimized I–V tracer to enhance the photovoltaic parameters estimation process. Future work is needed to further improve the required time for the parameter estimation, as well as to analyse the impact of using the double diode model in such an instrument.

The work developed in this work will enable, through the monitorization of the PV internal parameters, the correlation of environmental conditions and ageing factors with those internal parameters. The objective is to be able to identify, using the internal parameters of a PV panel, the various effects that impact its performance.