Precise photovoltaic (PV) behavior models are normally described by nonlinear analytical equations. To solve such equations, it is necessary to use iterative procedures. Aiming to make the computation easier, this paper proposes an approximate single-diode PV model that enables high-speed predictions for the electrical characteristics of commercial PV modules. Based on the experimental data, statistical analysis is conducted to validate the approximate model. Simulation results show that the calculated current-voltage (I-V) characteristics fit the measured data with high accuracy. Furthermore, compared with the existing modeling methods, the proposed model reduces the simulation time by approximately 30% in this work.
1. Introduction
Photovoltaic (PV) power market has grown rapidly in the last decade owing to the deterioration of the environmental quality and the escalation of fossil fuel price. Before installing a PV system, a good performance estimation of the adopted PV generators is necessary since the initial cost of the system is pretty high [1, 2]. Unfortunately, although PV generators always work under the operating environment far from the standard test conditions (STCs), PV manufacturers usually only list limited technical data measured at STCs, such as maximum power (Pmax), voltage at Pmax (Vmp), current at Pmax (Imp), short circuit current (Isc), and open-circuit voltage (Voc). For this reason, a reliable and flexible PV model that enables an accurate estimation of the PV generated electricity towards various operating conditions is of significance in the design phase.
Among numerous modeling approaches in the literature, the most widely used circuit-based PV model is the single-diode model (SDM), which consists of a series resistance (Rs), a shunt resistance (Rp), and a linear independent current source in parallel to a diode. The more accurate double-diode model (DDM) is available in [3]. It takes into consideration the recombination loss at the space depletion region of solar cells. In [4], the electrical characteristics of the multicrystalline solar cells are analyzed by a three-diode model (TDM), which further takes into account the influence of grain boundaries and leakage current through the peripheries. Although the DDM and TDM have certain advantages, the extra diodes increase the computational complexity. Accordingly, the SDM is considered to feature a good compromise between simplicity and accuracy. This may be the most likely reason why commercial simulation tools (e.g., PSIM [5] and PVsyst [6]) frequently apply the SDM in combination with the intricate dependence of the electric current on weather-related and environment factors, such as the ambient temperature (T) and the irradiance (G).
More recently, there is an increasing need for high-speed performance estimation as PV models are frequently used to aid real-time optimization of PV energy [7–16]. Ignoring the effect of the resistance is a typical approach to reduce the complexity of PV models. In [17], Mahmoud proposes the simplified single-diode model (SSDM) which removes the Rp from the general SDM. The further simplified single-diode model (FSSDM), also known as the ideal single-diode model (ISDM), neglects the Rs and Rp as well. Albeit their simplicity, accurate estimation of the electrical characteristics is not guaranteed [14]. Furthermore, tedious iterative root finding methods (e.g., Newton-Raphson method) are still needed in the SDM and SSDM to solve the implicit transcendental equations. In [18, 19], Jain et al. proposed Lambert-W function-based SDM which enables the solutions to be exact, explicit, and straightforward and is not necessary to ignore resistance effects. However, that model does not intrinsically reduce the complexity because the root of the Lambert W-function can only be calculated by using iterative approximations [20].
This paper proposes a simple yet accurate approximate single-diode model (ASDM) aiming to overcome the limitations in the existing simplified SDMs. The exponential diode behavior is approximated via continuous least squares approximation (CLSA), which permits designers or engineers to predict the current I by solving a closed-form expression. Only a simple numerical root-finding algorithm is required to determine the parameters of the ASDM. The accuracy and computational speed improvements are demonstrated by the simulation and experimental results.
The rest of the paper is organized as follows. In Section 2, we give a brief introduction to single-diode PV models. In Section 3, the proposed approximate model as well as its corresponding parameter estimation methods is presented. The accuracy and efficiency of the proposed method are validated in Section 4. This is followed by conclusions in Section 5.
2. Modeling of PV Modules
PV cells are made of a variety of semiconductor materials using different manufacturing processes. The working principle of PV cells is essentially based on the PV effect, which refers to the generation of a potential difference at the P-N junction in response to visible or other radiation. Figure 1 roughly demonstrates the basic structure of a silicon-based PV cell and its working mechanism. The PV cell consists of a thin layer of bulk Si or a thin Si film connected to electric terminals [21]. A thin metallic grid is connected on the top surface of the semiconductor. On the other side, the thin semiconductor layer is specially treated to form the P-N junction.
The basic structure of a silicon-based PV cell and its working mechanism.
PV module is a particular case of a series connected PV cells. When a PV module is exposed to light, the semiconductor materials absorb photons and accordingly charge carriers are generated. These carriers are separated by the P-N junction electric field and an electric current Ipv then flows through the external circuit. By eliminating the PV effect, a PV module behaves like a conventional diode that does not depend on any light parameters. The Shockley diode equation is generally used to describe the current flowing through the diode Id:
(1)Id=Io(eVD/nNsVt-1).
In (1), VD represents the electrical potential difference between the two ends of the diode. Io is the reverse saturation current, and n is the diode ideality factor. Vt denotes the thermal voltage of the PV module with Ns cells connected in series, and its value can be estimated as a function of T:
(2)Vt=kTq,
where k and q represent the Boltzmann constant (1.380650×10-23 J/K) and the electron charge (1.602176×10-19 C), respectively.
A simple approach, describing the value of I, is to assume that the superposition principle holds; that is, the total characteristic is the sum of the dark and illuminated characteristics [17, 21, 22]. Alternatively, the terminal current I is equal to the Ipv minus the current diverting through the diode:
(3)I=Ipv-Io(eV/nNsVt-1).
The modeling methods described so far consider the ideal behavior of PV modules based on a current source in parallel with an ideal diode. The SDM, whose circuit diagram is shown in Figure 2, improves the ideal model by recognizing Rs and Rp. Equation (4) mathematically describes the I-V characteristics of the SDM:
(4)I=Ipv-Io(e(V+IRs)/nNsVt-1)-V+IRsRp.
Circuit diagram of the SDM.
3. PV Model Approximation3.1. Function Approximation
Function approximation provides an approach to represent a complicated function f(x) (f(x)∈C[a,b]) by an easier form ϕ(x;a0,a1,…,an), where a0,a1,…,an are parameters to be determined so as to achieve the best approximation of f(x). The term least squares describes a frequently used means to solving overdetermined or inexactly specified equations (e.g., transcendental functions, integrals, and solutions of differential or algebraic equations) in an approximate sense [23]. Normally, least squares approximation (LSA) can be viewed as finding proper coefficients a0,a1,…,an so as to
(5)minimize∥f(x)-ϕ(x;a0,a1,…,an)∥2,
where ϕ(x;a0,a1,…,an) is usually a polynomial Pn(x) of degree at most n:
(6)Pn=a0+a1x+···+anxn=∑k=0nakxk.
The approximation problem might be regarded as a process of minimizing the error E, which is given in (7):
(7)E≡E(a0,a1,…,an)=∫ba(f(x)-Pn(x))2dx.
By applying the derivative to (7), we get
(8)∂E∂aj=-2∫abxjf(x)dx+2∑k=0nak∫abxj+kdx.
With the aim of finding real coefficients a0,a1,…,an, a necessary condition that should be considered is
(9)∂E∂aj=0,j=0,1,…,n.
After substituting (9) into (7), the linear normal equations, expressed by (10), can be derived to solve the unknown coefficients a0,a1,…,an. It has been proven that the normal equations always have a unique solution provided f(x)∈C[a,b] [24]. (10)∫abxjf(x)dx=∑k=0nak∫abxj+kdx,foreachj=0,1,…,n.
All the above approximation process is called continuous least square approximation (CLSA) in the field of applied mathematics.
3.2. Approximate Single-Diode Model (ASDM)
In a typical SDM, the analytical expression of the forward I-V characteristics contains a transcendental function for predicting the value of Id, which is formulated as
(11)Id=Io(e(V+IRs)/nNsVt-1).
Assuming that the parameters are constant at a certain test condition, the value of I varies directly with the reference V. Let m=Rs/nNsVt; then Id can be rewritten as a function of I:
(12)Id(I)=IoemV/Rs·emI-Io.
CLSA provides a paradigm that simplifies the transcendental part of (12) into a polynomial of degree 1:
(13)emI≅a0+a1I.
By using the linear normal equations, namely, (10), the values of a0 and a1 can be solved. The detailed deduction process is given in the appendix, and their exact mathematical expressions are given in (14):
(14)a0=-2mImax2[(Imax-3m)emImax+(2Imax+3m)],a1=12mImax3[(Imax2-1m)emImax+(Imax2+1m)].
Accordingly, the ASDM can be formulated as a rational function:
(15)I≅Ipv-(IoeV/nNsVt)·a0-V/Rp1+(IoeV/nNsVt)·a1+Rs/Rp.
The methods of determining the parameters Ipv, Io, n, Rs, and Rp are presented in the next subsection.
3.3. Parameter Identification for the Proposed Model3.3.1. Analytical Methods for Predicting <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M74"><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mtext>pv</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M75"><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>
As a result of the PV effect, the photocurrent Ipv flows in a direction opposite to the forward dark current. Even when the PV module operates at short circuit, this current continues to flow and is measured as the short-circuit current Isc. From (3), it can be seen that the value of Ipv is approximately equal to the Isc in a high-quality PV module, and thus the assumption Isc≅Ipv is often used in PV modeling. Although the spectral short current density can be determined by analytical equations in [22], the required parameters are usually not given in the manufacturer’s tabular data. In view of the fact that the Isc depends linearly on the G and is also slightly influenced by the T, the Ipv can be given by (16) [21, 25, 26]:
(16)Ipv≅Isc=(Iscn+KiΔT)GGn,
where Iscn and Gn are the short current and irradiance at STCs, respectively. Ki, named short-circuit current coefficient, is a constant available in the datasheet. The difference between T and the standard test temperature Tn is denoted by ΔT.
The saturation current Io is the small current that flows when the P-N junction is reverse biased. The dependence of Io on the temperature was studied by Villalva et al. [21], in which the authors introduced (17) to predict the value of Io. In the expression, Kv is the open-circuit voltage coefficient and Vocn represents the open circuit voltage measured at the STCs:
(17)Io=(Iscn+KiΔT)e(Vocn+KvΔT)/nNsVt-1.
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The ideality factor n is an important parameter used to describe whether the P-N junction behaves close to or apart from the ideal case. As reported by [27], n and Rs significantly affect the shape of I-V curves around the maximum power point (MPP), whereas the Rp determines the slope of the I-V curve near the point Isc. With the aim of delivering a simplified calculation approach, the parameters of the ASDM are assumed to be constant and the variables x=(Rs,n) are solved by the equation system f(x) formed by the following.
The terminal current at the MPP:
(18)Imp≅Ipv-(IoeVmp/nNsVt)·a0-Vmp/Rp1+(IoeVmp/nNsVt)·a1+Rs/Rp.
The derivative of the terminal current with respect to the voltage at the MPP:
(19)∂I∂V|V=Vmp,I=Imp=-(a0+a1Imp)IoeVmp/nNsVt/nNsVt+1/Rp1+a1IoeVmp/nNsVt+Rs/Rp=-ImpVmp.
In the above equation system, a0, a1, Ipv, and Io are represented by (14), (16), and (17) with an STC environment set. By substituting the known operating points (0,Voc) and (Isc,0) into (15), 1/Rp and Rs/Rp are expressed as
(20)Rp=VocIpv-a0IoeVoc/nNsVt+Io,(21)RsRp=(1-a0-a1Isc)IoIsc.
Finally, the Newton method illustrated in [28] is capable of solving the unknowns n and Rs. In the numerical computing process, the kth generation of variable vector x gets the updated vector estimate:
(22)xk+1=xk-Jk-1f(xk),
where Jk is the Jacobian matrix of f(xk). Other parameters as well as I can be recovered by using (14)–(20).
4. Results and Discussion
The ASDM described in this paper is compared with the physical PV models in the commercial simulation tools, such as PSIM and PVsyst. The Villalva’s model [21], a famous comprehensive approach to modeling and simulation of PV arrays in the literature, is also used for comparison. These models are programmed in MATLAB, and their capability of predicting the electrical characteristics of PV modules is validated by the experimental I-V data extracted from the manufacturer’s datasheet. Four different PV modules produced with three various manufacturing techniques, namely, MSX60 (multicrystalline), KC200 GT (multi-crystalline), SQ150-PC (mono-crystalline), and HIT Power 180 (HIT) PV modules, are utilized for verification.
Aiming to evaluate the capability of the modeling methods to fit the characteristics of PV panels, statistical analysis is performed. In this paper, the fitness of PV models is described by the root mean square error (RMSE) and the mean absolute error (MAE) as well as the relative error (RE). They are mathematically expressed by the following equations:
(23)RMSE=1n∑i=1n(Ii-Ii^)2,MAE=1n∑i=1n|Ii-Ii^|,RE=|1-IiIi^|×100%,
where Ii and Ii^ present the simulated and measured current at the ith operating point among n measured I-V pairs, respectively. Table 1 lists the parameters of PV panels by using the methods described in Section 3.3, which deliver a convenient parameter estimation method that only requires the tabular information available in the datasheet. The obtained results are extracted under a set of STCs and are assumed to be constant in other operating conditions. The obtained RMSEs for the modules working under the STCs show a good agreement between the simulation results and experimental data.
Extracted ASDM parameters for different PV modules.
Module
n
Rs (Ω)
Rp (Ω)
a0
a1
RMSE
SQ150
1.6031
0.5334
808
0.9018
0.2877
2.10E-03
KC200GT
1.1266
0.2764
206
0.6939
0.3771
5.29E-02
MSX60
1.5390
0.1035
3140
0.9921
0.0843
6.23E-04
HIT180
1.6240
0.4929
781
0.9753
0.1583
2.24E-02
As soon as the model parameters are determined, the ASDM is able to predict the electrical characteristics of PV modules under varied atmospheric conditions. Figures 3, 4, and 5 show the I-V characteristics of MSX60 and KC200GT modules varying with different levels of irradiance and temperature. The simulation results of the PSIM and Villalva’s models are also plotted for reference. It is interesting to see that the ASDM obtains more accurate (I, V) above the 25°C, whereas the operating points of the Villalva’s model are closer to the measured data below the 25°C. Since the I-V curves of MSX60 at different irradiance levels are not issued in the datasheet, the related tests are not conducted in this work.
Current-voltage curves of a MSX60 PV module at various cell temperatures.
Current-voltage curves of a KC200GT PV module at various irradiance levels.
Current-voltage curves of a KC200GT PV module at various cell temperatures.
In order to further evaluate the estimation performance of the ASDM, more exhaustive tests have been conducted on the tested modules. Figures 6(a) and 6(b) show the MAEs of the simulated results subjected to irradiance variation, and all measurements are performed at a temperature of 25°C. On the other hand, Figures 6(c) and 6(d) demonstrate the MAEs of the ASDM model for MSX60 and HIT Power 180 modules working at the same irradiance of 1000 W/m^{2} but at different temperatures. In Figure 6, it is evident that the ASDM model outperforms the commercial tools (PSIM and PVsyst) in most cases and obtains better fitness quality than Villalva’s model at high irradiance and temperature levels.
Mean absolute errors of the PV models at different atmospheric conditions: (a) SQ150-PC; (b) MSX60; (c) KC200GT; (d) HIT 180.
Similar trend is observed in Tables 2 and 3, which show the REs of the calculated MPP locus at different operating conditions. In practical, predicting the locus of MPP is of importance in the improvement of power efficiency. For this reason, statistical analysis is conducted. Except for the tests on SQ150-PC module under high irradiance test condition, most REs of the ASDM are similar or even lower than those of others.
Relative errors of the calculated Imp at various irradiance levels.
Module
G (W/m^{2})
T (°C)
Relative error
ASDM
PSIM
PVsyst
Villalva et al. [21]
SQ150-PC
200
25
6.34%
0.51%
2.11%
1.51%
400
25
3.29%
2.81%
0.30%
2.99%
600
25
1.79%
3.60%
0.89%
2.68%
800
25
1.18%
3.33%
0.29%
1.18%
1000
25
0.59%
3.32%
0.00%
0.00%
KC200GT
200
25
1.93%
1.31%
0.39%
0.39%
400
25
0.39%
2.90%
1.93%
1.93%
600
25
0.00%
4.16%
1.15%
1.15%
800
25
0.00%
5.48%
0.38%
0.38%
1000
25
0.38%
6.81%
0.00%
0.00%
Relative errors of the calculated Imp at various temperature levels.
Module
G (W/m^{2})
T (°C)
Relative error
ASDM
PSIM
PVsyst
Villalva et al. [21]
MSX60
1000
0
1.59%
1.32%
1.59%
1.59%
1000
25
0.00%
2.16%
0.00%
0.00%
1000
50
1.31%
3.07%
1.31%
1.31%
1000
75
3.70%
4.00%
3.70%
3.70%
HIT180
1000
0
2.11%
0.33%
0.35%
2.11%
1000
25
0.00%
2.57%
2.59%
0.19%
1000
50
0.82%
1.51%
0.41%
0.82%
1000
75
0.22%
1.68%
1.12%
0.67%
The simulation results described so far verify the accuracy of the proposed ASDM. Besides its low-error estimation performance, the ASDM has the advantage of deriving the I-V characteristics in closed form, and thus it supports high-speed computing. Figure 7 makes a comparison among the efficiency of different PV models. In the tests, 10,000 operating points varied within the operating voltage range [0,Voc] are calculated in a general PC with a 2.40 GHZ Intel(R) Core (TM) 2 Duo CPU. It shows that the ASDM is able to reduce the simulation time by 30% compared to other tested models.
Simulation time of different PV models.
5. Conclusions
This paper has presented a simple approximate PV model which is capable of predicting the electrical characteristics of PV modules operating at a variety of atmospheric conditions. Continuous least squares approach is applied to fit the PV behaviors in a simple manner. The proposed mathematical modeling approach is easy and straightforward and doses not depend on iterative procedures to obtain solutions. The accuracy of the proposed model is evaluated through simulations. The results showed that the obtained current values are in good agreement with the experimental data. Future work integrating the real-time optimization of PV energy will highlight the value of this approximate modeling method.
AppendixThe Deduction Process of the Coefficients <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M152"><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M153"><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>
According to (10), the linear normal equation for emI can be rewritten in the following form:
(A.1)a0∫0ImaxdI+a1∫0ImaxIdI=∫0ImaxemIdI,a0∫0ImaxIdI+a1∫0ImaxI2dI=∫0ImaxIemIdI,
where Imax is the upper limit of the PV terminal current that is available at the I-V curves of the manufacturer’s datasheet. After performing the integration, it yields
(A.2)Imaxa0+Imax22a1=emImaxm-1m,Imax22a0+Imax33a1=ImaxmemImax-emImaxm2+1m2.
Equation (A.2) can be solved to obtain
(A.3)a0=-2mImax2[(Imax-3m)emImax+(2Imax+3m)],a1=12mImax3[(Imax2-1m)emImax+(Imax2+1m)].
Consequently, the least squares polynomial approximation of degree 1 for Id is
(A.4)Id(I)≅Io[emV/Rs·(a0+a1I)-1].
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