Levenberg-Marquardt algorithm for parameter identification of solar cell model Fayrouz DKHICHI, Benyounes OUKARFI Hassan II University, Faculty of Sciences and Techniques, Electronic, Electrotechnic, Automatic and Data Processing Laboratory, Electrical Engineering Department, BP 28800, Mohammedia - Morocco
[email protected],
[email protected] IPV
Abstract— The parameter identification issue of the single diode model of solar cell requires an efficient algorithm, due to the lack of information about the real values of the five parameters. For this purpose the Levenberg-Marquardt algorithm based on the minimization of the sum squared error function is proposed in this paper. The current - voltage (I - V) and power - voltage (P - V) curves of commercial silicon solar cell are calculated. The accuracy of the proposed method is compared with some related ones. The Levenberg-Marquardt algorithm provides firstly a good agreement between the calculated and the experimental curves of (I - V) and (P - V), and secondly the lower values of the power error. Keywords: Levenberg-Marquardt; identification; optimization; electrical parameters; power error; solar cell;
I.
INTRODUCTION
The optimization of photovoltaic (PV) power holds a great interest in many investigation works. The efficiency of this power is affected by the degradation of the solar cell’s internal electrical parameters. The optimal identification of these unknown parameters, leads to a good modeling of the solar cell to introduce an effective tool for evaluation, control and optimization of the solar cell system power. In the literature, the solar cell is presented by several equivalent models, where the single diode model is widely used. It reflects the high non-linearity of the PV current voltage (IPV–VPV) characteristic. The implicitness of this non linearity is not obvious to deal with, so to determine the model parameters, the identification issue requires an efficient optimization method. For this purpose, we propose in this paper the Levenberg-Marquardt algorithm which provides more precision in the parameters values identified, moreover, it allows a good agreement between the experimental and the calculated curves and subsequently a lower error power. II.
PROBLEM FORMULATION
A. Single diode model of the solar cell under illumination The single diode model of the solar cell is presented by the equivalent electrical circuit [1][2], schematized as follows:
Rs Rsh
Iph
VP V
Figure 1. Equivalent circuit of a solar cell
The mathematical model deducted from this circuit, shows the implicit relationship between the Ipv, Vpv and the five electrical parameters Rs, Rsh, Iph, Is, and n. I
=I
− I exp
− 1 +
(1)
Where: Rs: Series resistance representing the losses due to the various contacts and the connections. Rsh: Shunt resistance characterizing the leak currents of the junction. Iph: Photocurrent depending on both the illumination and the temperature. Is: Diode saturation current. n: Diode ideality factor. Vth: Thermal voltage[ A. T⁄q] With: A: Boltzmann’s constant [1.3806503.10-23 J/K]. q: Electrical charge of the electron [1.60217646.10-19 C]. T: Temperature of the cell by K. B. Optimization process In this study we determine the intrinsic parameters values of a solar cell from experimental measurements (IPV, VPV).To achieve this, we have adopted the principle of least squares in the parametric identification. This method consists in determining the values of the five parameters Rs, Rsh, Iph, Is and n, adjusting the model of Eq. 1 with the experimental values, through minimization of the error between the both. In the optimization process, the “Sum Squared Error” (SSE) is the function to be minimized, leading to an optimal solution.
The sum squared error function SSE is expressed as follows: SSE = F(θ) = ∑[I
(θ)– I
(θ)]² = ∑ ε(θ) ² (2)
IPVmes is the measured current, IPVmod is the model equation of the current, ε is the error between IPVmes and IPVmod, θ is the vector of the five parameters. To adjust the model to our measured data we have to find the minimum of the criterion F() with respect to the parameters. The minimization of F() cannot be made in an analytical intuitive way because of the strong non linearity of the characteristic IPV=f (VPV, ). Indeed, we note that the model of the solar cell presents a double non linearity. The first one is inherent to the equation itself (1). The second is related to some structural parameters: Rs and n. III.
LEVENBERG-MARQUARDT ALGORITHM
A. Levenberg-Marquardt method principle The Levenberg-Marquardt (LM) is a nonlinear optimization least squares method. It presents the combination of steepest descent and the Gauss-Newton methods, where it takes advantage of being less sensitive to initial values and fast when these values are near to the optimum. The vector of electrical parameters is modified according to the following expression: θ
= θ − [(F′′(θ) + λ I ) F (θ)]
(3)
The combination in the LM method is controlled by "damping factor" For large values of LM behaves as steepest descent, whereas when takes low values, LM becomes similar to the Gauss-Newton method. [3]
For the first area, the LSM algorithm is applied without any constraint; on the other hand for the nonlinear area, we can linearize it in a “logarithmic” way [4] by an accurate approximation.
IV.
RESULTS AND DISCUSSION
The experimental characteristic used in this study is of the commercial silicon solar cell (R.T.C France), with a diameter of 57 mm. The measured data of IPV and VPV are taken for an illumination of E = 1000W/m² and a temperature of T = 33 °C [5]. A. Identified parameters The LM algorithm converges to the optimal values of which are almost like those reported by Newton [5], Particle Swarm (PS) [6] and Genetic Algorithm (GA) [7]. TABLE II. FINAL VALUES OF THE FIVE PARAMETERS OBTAINED BY LM AND COMPARISON WITH OTHER METHODS
Parameters
LM
Newton
PS
GA
Rs (Ω
0.03644
0.0364
0.0313
0.0299
Rsh (Ω
53.8994
53.7634
64.1026
42.3729
Iph (A
0.7607
0.7608
0.7617
0.7619
Is (µA n
0.31934
0.3223
0.9980
0.8087
1.4800
1.4837
1.6000
1.5751
To study the accuracy of the parameters obtained and their impact on the power generated by the solar cell, the following statistical errors are used: Root Mean Squared Error (RMSE) :
B. Damping factor setting Due to the damping factor the switching between the two methods of steepest descent and Gauss-Newton is done automatically inside the LM algorithm. The setting value of this factor at each iteration is done according to the conventional method adopted in the literature and which comprises: When the sum squared error decreases, the parameters vector is updated. When the sum squared error increases, the parameters vector is still the same from the previous iteration. Where is a constant taken equal to 2 in our investigation.
RMSE =
1 N
I
– I
Power Mean Absolute Error (PMAE): PMAE =
1 N
P
– P
Absolute Power Error (APE): APE = P
– P
TABLE III. COMPARISON BETWEEN THE ACCURACY OBTAINED BY LM AND THAT OF
C. Initialization method Most optimization algorithms require the initialization of the unknown parameters, the same principle also applies to the LM algorithm investigated in this paper. To estimate the initial values, we have used the Least Squares Method (LSM). This method is only applicable to linear systems parameters. This property is unchecked for our IPV = f(VPV) characteristic. However, we note that this characteristic presents two distinct areas; the first is linear while the second is not.
OTHER METHODS
Statistical error SEE
LM
RMSE PMAE
Newton
PS
GA
5.8005e-3
9.4632e-3
9.8680e- 4 9.6964e- 3 1.4936e- 2
1.9078e-2
3.3334e-4 3.0266e- 3 4.9768e- 3
5.7225e- 3
2.5625e-5 2.4445e-3
The table. III clearly shows the effectiveness of the LM algorithm, view it offers better accuracy explained by the lower values of RMSE and PMAE regarded to Newton, PS and GA. The lower value of RMSE obtained by LM reflects the compatibility between IPV and VPV calculated and measured, plotted in Fig. 2.
0.04 Absolute Power Error [W]
LM algorithm takes 115 iterations to converge to the optimal solution while references [5-7] that deal with Newton, GA and SA don’t specify the number of iteration required for the convergence of these algorithms.
LM Newton PS GA
0.03 0.02 0.01 0 0
0.1
0.2
0.3 Voltage [V]
0.4
0.5
0.6
Figure 4. Evolution of the APEi for LM, Newton, PS and GA algorithms
fitted curve mesured data
C. Convergence issue
0.6
2
10
0.4 0.2
Sum squared error
C u rren t [A ]
0.8
0
10
0
-2
10
-0.2 -0.2
-0.1
0
0.1
0.2 Voltage [V]
0.3
0.4
0.5
0.6
-4
10
Figure 2. IP V(VPV) characteristic of the silicon solar cell
-6
10
While the lowest value of PMAE obtained by LM reflects the compatibility between PPV and VPV calculated and measured in plotted in Fig. 3.
10
20
30
40
50
60 70 Iteration
80
90
100
110
120
Figure 5. Evolution of the sum squared error obtained by LM according to iterations
fitted curve mesured data
P o w er [W ]
0.4 0.3 0.2 0.1
Serie resistance Rs []
0.15 0.1 0.05 0 -0.05
0 -0.1
-0.1 -0.2 -0.2
-0.1
0
0.1
0.2 Voltage [V]
0.3
0.4
0.5
0.6
10
20
30
40
50
60 70 Iteration
80
90
100
110
120
Figure 6. Evolution of the serie resistance Rs according to iterations
B. Evolution of the power error To confirm the accuracy of the LM algorithm compared to Newton, PS and GA, we calculate APEi, which our algorithm has the lower values.
Shunt resistance Rsh []
64
Figure 3. PPV(VPV) characteristic of the silicon solar cell
62 60 58 56 54 52
10
20
30
40
50
60 70 Iteration
80
90
100
110
120
Figure 7. Evolution of the shunt resistance Rsh according to iterations
convergence process, the five parameters reflect different developments Fig. 6-10, which play an important role in the progressive reduction of sum squared error Fig.5.
Photocurrent Iph [A]
0.77 0.76 0.75
V.
0.74 0.73 0.72 10
20
30
40
50
60 70 Iteration
80
90
100
110
120
Figure 8. Evolution of the photocurrent Iph according to iterations -6
Saturation current Is [A]
x 10 8 6 4 2
This paper proposes LM algorithm for parameter identification of the solar cell single diode model. The identification process required the IPV (VPV) characteristic obtained from a 57 mm diameter commercial (R.T.C. France) silicon solar cell. The proposed method is more accurate than the compared methods, it provide lower values of power error, in addition, a high agreement between the experimental and calculated curves of IPV-VPV and PPV-VPV. Moreover the damping factor values are still given with a non-optimal way by the conventional method that not ensures convergence when the initial values of parameters change. Therefore, we will use in a future work an accurate seeker method for the values of the factor for more optimal convergence.
0 -2
REFERENCES 10
20
30
40
50
60 70 Iteration
80
90
100
110
120
Figure 9. Evolution of the saturation current Is according to iterations 1.9 Diode ideality factor n
CONCLUSION
[1]
[2]
1.8
[3]
1.7
[4]
1.6 1.5
[5] 1.4
10
20
30
40
50
60 70 Iteration
80
90
100
110
120
Figure 10. Evolution of the Diode ideality factor n according to iterations
As shown in Figs. 5-10, at the end of the convergence, the sum squared error function becomes constant and does not change where the five values of the parameters reach their optimums. Trough the 115 iterations required for the
[6]
[7]
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