Two Variable-Weather-Parameter Models and Linear Equivalent Models Expressed by Them for Photovoltaic Cell

For the conventional four-parameter model of photovoltaic (PV) cell, to simplify the number of its mathematical equations, in this paper, a variable-weather-parameter cell model (VWP model) is proposed. Furthermore, to decrease the number of its model parameters, a simplified variable-weather-parameter cell model (SVWP model) is also proposed. Based on them, the model parameters of linear equivalent models can be replaced by the conventional four-parameter model parameters, which finds the direct relationships between linear equivalent cell model and four-parameter cell model. Finally, some simulation experiments are done by comparing the proposed VWP and SVWP models with conventional four-parameter model of PV cell. The results show that, on the one hand, their output characteristics almost correspond with each other. On the other hand, these proposed cell models are very accurate under varying irradiance or temperature conditions. Meanwhile, some simulation experiments also verify that the linear equivalent models expressed by the VWP model and SVWP model are feasible, available and reasonable. By this work, both the conventional four-parameter model and linear equivalent models are greatly evolved and improved. On the one hand, the conventional four-parameter model can be greatly simplified, which makes its modeling and application easier. On the other hand, the bridge between it and linear equivalent models can be built successfully, which makes these linear models more standard and practical.

Hitherto, many research works on the mathematical models of PV cell, especially for its one-diode model and fourparameter model, have been done. On the one hand, lots of MPPT methods are proposed by using the one-diode cell model or four-parameter cell model. The different MPPT methods for PV cell were compared by Danandeh et al. [1].
To design a new MPPT strategy, the single-diode cell model was used by Moshksar and Ghanbari [2]. Based on the one-diode cell model, a tangent error MPPT method was proposed by Peng et al. [3], two perturb and observe methods (P&O method) were analyzed by Yang et al. [4] and Veerapen et al. [5], and a dual-tracking MPPT method under rapidly changing environmental conditions was presented by Jately and Arora [6]. To extract the solar cell parameters of the single-diode model, some approaches were presented and compared by Humada et al. [7], [8]. On the basis of the four-parameter model of PV cell, an improved P&O method was proposed by Li et al. [9]. To gain the fastest tracking speed, some variable-weather-parameter (VWP) methods based on four-parameter cell model were proposed by Li et al. [10]- [13]. In Refs. [10] and [11], three fundamental VWP methods were proposed. In Ref. [12], a VWP MPPT method was proposed by analyzing the input resistance. In Ref. [13], a VWP MPPT method for PV system was proposed by a defined characteristic resistance based on PV cell. However, by these works, the mathematical expressions of the fourparameter cell model is still not simplified, especially for the number of its mathematical equations. To address this issue, on the basis of these VWP methods, a VWP cell model is proposed in this paper. By it, not only the advantages of the VWP methods can be obtained, but also the number of the mathematical equations of the four-parameter cell model can be decreased from five to three. On the other hand, lots of works have been also done to study the model parameters of PV cell. A method to identify the cell model parameters of the one-diode circuit was proposed by Laudani et al. [14]. To predict the cell behaviour, an improved one-diode model was studied by Salmi et al. [15]. A matrix equation of PV cell was analyzed to describe the parallel-parallel topology by Kadri et al. [16]. A simple and complete cell model based on datasheet with varying environment was studied by Vergura [17]. An optimization method of the parameters and temperature dependence for the two-diode and one-diode models was presented by Barth et al. [18]. After simplifying the five-parameter cell model, the four-parameter model was presented by Nobuyoshi et al. [19]. Two linear cell models at the MPP were presented to linearize the V − I characteristic of the conventional four-parameter model of PV cell by Li [20]. It is obvious that the one-diode model which is regarded as the main model of PV cell has been analyzed and used widely. However, by these works, the number of model parameters of the four-parameter model is still not greatly simplified. To solve this problem, on the basis of the proposed VWP model of PV cell, a simplified variable-weather-parameter (SVWP) cell model is proposed in this paper. By it, not only the number of the model parameters of the four-parameter model can be decreased from four to two, but also the number of the mathematical equations of the four-parameter model can be decreased from five to one.
Finally, some works have been done to linearize the V − I curve of the conventional four-parameter model. For example, in Ref [20], two linear equivalent models (Norton equivalent model and Thevenin equivalent model) were proposed by Shaowu Li. The main aim of these proposed linear cell models is to use the nonlinear cell model in linear control theory more easily and conveniently. Meanwhile, some parameters which are the conclusions of three fundamental VWP methods were used as their model parameters. However, there exist some shortcomings for these two linear cell models. On the one hand, a piecewise function is used as their model parameters, which leads to the complex mathematical calculation. On the other hand, their model parameters are not expressed by the conventional cell models, so the fundamental characteristics of PV cell can not be reflected clearly. To make up for these demerits, in this paper, the model parameters of the proposed VWP model and SVWP model are used to express these two linear models. By this work, not only the trouble arising from piecewise function can be prevented, but also the direct relationships between four-parameter model and linear equivalent models can be built.
The main innovations and contributions of this work can be illustrated as follows: A VWP cell model, which can decrease the number of the equations of the conventional four-parameter model from five to three, is proposed. A SVWP cell model, which can decrease the number of the equations of the conventional four-parameter model from five to one, is proposed.
The proposed SVWP cell model can decrease the number of the model parameters of the conventional four-parameter model from four to two. The model parameters of two linear equivalent models are expressed by the VWP and SVWP cell models. The direct relationships between linear equivalent models and four-parameter cell model are firstly built by this work. This paper is arranged as follows: the VWP cell model and SVWP cell model are studied and proposed by the analysis of the four-parameter mathematical model in Section II. The model parameters of linear equivalent models are analyzed and expressed by the VWP model and SVWP model in Section III. The output characteristics and accuracy of the proposed VWP and SVWP models are analyzed, and the linear models with VWP and SVWP model parameters are 184886 VOLUME 8, 2020 tested in Section IV. There are some discussions and conclusions in Sections V and VI, respectively. Fig. 1 shows the equivalent circuit of the one-diode cell model [14], and Eq. (1) shows its mathematical expression [21], [22].

II. PROPOSITION OF THE VWP MODELS A. THEORETICAL BASIS
Eqs.
In this work, Eqs. (3)-(7) will be used to analyze the characteristics of the parameters C 1 and C 2 , and to build the relationships between four parameters (I sc , V oc , I m and V m ) and S, T . Meanwhile, they are also used as the mathematical basis to propose two VWP models of PV cell.

B. PRINCIPLE OF THE VWP MODEL
When S and T are changing, the parameters I sc , V oc , I m and V m will also keep varying. Therefore, they can be all defined as the function of S, T , and represented by I sc (S, T ), V oc (S, T ), I m (S, T ) and V m (S, T ), respectively. It is obvious that these defined functions illustrate the relationships between four parameters (I sc , V oc , I m and V m ) and S, T .
According to Eqs. (8)- (10), it is obvious that the functions I sc (S, T ), V oc (S, T ), I m (S, T ) and V m (S, T ) are the key. On the one hand, according to Ref [12], I m (S, T ) and V m (S, T ) can be expressed by Eq. (11) and Eq. (12), respectively.
On the other hand, in order to obtain the expressions of I sc (S, T ) and V oc (S, T ), some simulation results shown by Meanwhile, according to Fig. 2(b), the approximation function shown in Eq. (14) can be given by fitting V oc − T curve when S keeps at 1000W/m 2 .
Take Eqs. (13) and (14) into account, the expression between V oc and S, T , is shown by Eq. (15).
According to Fig. 3(a), by fitting I sc −S curve, the approximation function shown in Eq. (16) can be given when T keeps at 25 • C.
Meanwhile, according to Fig. 3(b), the approximation function shown in Eq. (17) can be given by fitting I sc − T curve when S keeps at 1000W/m 2 .
Take Eqs. (16) and (17) into account, the expression between I sc and S, T , is shown by Eq. (18).
Finally, according to Eqs. (8)- (12), (15) and (18) , it is clear that, by the VWP model, the number of the mathematical equations of the conventional four-parameter cell model can be decreased from five to three. This simplification can be beneficial to its mathematical modeling and engineering application, which makes the use of the four-parameter cell model more convenient. Meanwhile, by this proposed VWP model, the direct relationships between S, T and PV cell can be clearly expressed, which reveals the regularity that the V − I characteristics of PV cell are influenced by the varying weather conditions.

C. SIMPLIFIED VWP MODEL
It is clear that the VWP model of PV cell expressed by Eqs. (8)-(10) is still complex, so it is inconvenient to use in practical application. Therefore some simulation experiments must be done to simplify it and the results can be shown by Figs. 4 and 5. Here, Fig.4(a) and Fig.4(b) show the C 1 − S and C 1 − T curves, respectively. Fig.5(a) and Fig.5(b) show the C 2 − S and C 2 − T curves, respectively.
It can be seen from Figs. 4(a) and 4(b) that C 1 approximately keeps constant all the time regardless of the changing irradiance or temperature, and its value is about 1.7416 × 10 −6 . Meanwhile, it can be also seen from Figs. 5(a) and 5(b), C 2 approximately keeps constant all the time regardless of the changing irradiance or temperature, and its value is about 7.5411 × 10 −2 .
According to Eq. (8) and characteristics of C 1 and C 2 , the VWP model of PV cell can be simplified and expressed by Eq. (19). This cell model can be named as ''simplified variable-weather-parameter model'' (SVWP model). It is obvious that, according to Eq. (19), the SVWP model of PV cell is only determined by I sc (S, T ), V oc (S, T ), C 1 and C 2 , which makes the VWP model expressed by Eqs. (8)-(10) greatly simplified when S and T keep varying.
Compare Eqs. (3)-(7) with Eq. (19), it is clear that, by the SVWP model, the conventional four-parameter cell model can be greatly simplified after C 1 and C 2 are regarded as two constants. This simplification can be illustrated from two aspects: on the one hand, the number of the model parameters of PV cell can decreased from four to two. On the other hand, the number of the mathematical equations of PV cell can be decreased from five to one. Therefore, by the proposition of the SVWP model, the conventional four-parameter cell model can be simplified more greatly than VWP model.
In addition, compare Eqs. (8)-(10) with Eq. (19), it is also clear that, by the SVWP model, not only the number of the model parameters of the VWP model can decreased from four to two, but also the number of the mathematical equations of the VWP model can be decreased from three to one.
In a word, both the VWP model and the SVWP model can greatly simplify the conventional four-parameter model of PV cell so that this conventional model can be used more easily and conveniently.

III. LINEAR EQUIVALENT MODELS BASED ON VWP MODEL PARAMETERS
After the VWP cell model and SVWP cell model have been proposed, their model parameters can be used to express the linear equivalent models. According to Ref [20], at the MPP, PV cell can be linearized as the Thevenin equivalent model (Fig.6(a)) and Norton equivalent model (Fig.6(b)). Their model parameters including R sM , V sM and I sM can be expressed by Eqs. (20)-(22), respectively. Where C can be According to Eqs. (20)-(23), for two linear cell models, their model parameters can be expressed by some state parameters of the DC/DC circuit (including C and P o max ). However, there exist some shortcomings arising from these parameters. On the one hand, the state parameters of the DC/DC circuit are changing with time, so the measured data are usually depended on to obtain the values of these parameters, which makes the use of these linear cell models inconvenient. On the other hand, for PV cell, these state parameters can not clearly illustrate the relationship between its linear models and its four-parameter model, so the cell characteristics of the linear models can not be shown. Therefore, in order to overcome these shortcomings, the question how the model parameters of the VWP model and SVWP model are used to express these linear equivalent models will be analyzed. and V m (S, T ), so the linear models expressed by I m (S, T ) and V m (S, T ) will be firstly analyzed.
On the one hand, for the model parameter R sM , it is well known that, at the MPP, Eqs. (24) and (25) must be satisfied according to Fig. 6.
According to Ref [12], at the MPP, Eqs. (26) and (27) are satisfied when the MPPT unit is the buck circuit.
On the other hand, for the model parameter V sM , according to Ref [12], Eq. (31) can be given.
Take the range of V m into account, Eq. (32) can be simplified as Eq. (33). Where C β ≈ 2.045.
Finally, for the model parameter I sM , submit Eqs.
In a word, according to Eqs.

B. LINEAR MODELS EXPRESSED BY SVWP MODEL PARAMETERS
According to Section 2.3, the model parameters of the SVWP cell model include I sc (S, T ) and V oc (S, T ), so the linear models expressed by I sc (S, T ) and V oc (S, T ) will be analyzed.
On the one hand, for the model parameter R sM , Eq. (36) can be obtained by submitting Eq.(25) into Eq. (19).
Eq. (37) can be simplified as Eq. (38). where In the SVWP model, C 1 and C 2 are two constants, so C γ is also a constant according to Eq. (39).
On the other hand, for the model parameter V sM , according to Eqs. (21) and (23), Eq. (40) can be given. where It is clear that C δ is also a constant because C 1 and C 2 are two constants in the SVWP model.
Finally, for the model parameter I sM , submit Eqs. (38) and (40) into Eq. (22), Eq. (42) can be given. VOLUME 8, 2020 In order to obtain the values of C γ , C δ and C δ /C γ , some simulation results shown by Figs. 7 and 8 can be given.
Figs. 7 and 8 show that C γ and C δ can be regarded as the constants when S and T keep varying, and their values are 0.8893 and 1.6278, respectively. Meanwhile, the calculated value of C δ /C γ is 1.8304.
In a word, according to Eqs. (38), (40) and (42), the linear equivalent models shown by Fig. 6 can be expressed successfully by the SVWP model parameters I sc (S, T ) and V oc (S, T ). Here, I sc (S, T ) and V oc (S, T ) are the SVWP model parameters rather than VWP model parameters, although there are the same expressions, because Eqs. (38), (40) and (42) can be satisfied only by using these assumed constants C 1 , C 2 , C γ and C δ .      Therefore, a conclusion can be drawn that, when the solar irradiance or cell temperature keeps varying, the output characteristics of the VWP cell model can correspond with the conventional four-parameter model of PV cell to a great extent.

2) OUTPUT CHARACTERISTICS OF THE SVWP MODEL
To test the output characteristics of the proposed SVWP model of PV cell, some simulations are also done.  According to Figs. 13-16, a conclusion can be drawn that, when the weather parameters (irradiance and temperature) keep varying, the output characteristics of the SVWP cell model can correspond with the conventional four-parameter model of PV cell to a great extent.
In a word, for both VWP model and SVWP model, their output characteristics can always correspond well with conventional four-parameter model of PV cell regardless of varying irradiance or temperature.     Table 1. Figs. 17-22 show that, under varying irradiance conditions, there exist errors for both the VWP model and SVWP model, and their maximum values are usually around the maximum output voltage (V oc ). According to Table 1, the maximum value of the output current error is 130 mA and the maximum value of the output power error is 2.78 W under 800W/m 2 , 25 • C conditions. By contrast, their values at the MPP are 6.55 mA and 0.115 W for the VWP model, respectively, while they are 8.75 mA and 0.157 W for the SVWP model, respectively. Because PV system usually operates around the MPP, the error between VWP       Table 2.
Figs. 23-28 show that, under varying temperature conditions, there exist errors for both the VWP model and SVWP model and their maximum values are usually around the maximum output voltage (V oc ). Table 2 show that the maximum value of the output current error is 8.32 mA and the maximum value of the output power error is 0.172 W under 1000W/m 2 , 40 • C conditions. By contrast, their values at the  MPP are 2.95 mA and 0.05 W for the VWP model, respectively, while they are 0.56 mA and 0.0095 W for the SVWP model, respectively. Because PV system usually operates around the MPP, the error between VWP model or SVWP model and four-parameter model is very small, which also means the good accuracy of the VWP model and SVWP model. Therefore, a conclusion can be also drawn that, under different temperature conditions, both the VWP model and VOLUME 8, 2020     In a word, the proposed VWP and SVWP models of PV cell are very accurate, especially at the MPP, regardless of the varying irradiance or temperature, when the conventional four-parameter model is selected as the compared object.       approximately same as the PVLM regardless of the R sM − T , V sM − T or I sM − T curve. Clearly, these results reveal the feasibility, availability and rationality of the PVLM-VWP and PVLM-SVWP under varying temperature conditions. Therefore, a conclusion can be drawn that the characteristics of the PVLM-VWP and PVLM-SVWP can always correspond well with PVLM regardless of varying irradiance or temperature.

2) ACCURACY OF THE MODEL PARAMETERS
Some simulation experiments are done to analyze the accuracy of the PVLM-VWP and PVLM-SVWP, and the results are shown in Table 3. Where R sM , V sM and I sM represent the parameter values of the PVLM calculated by Eqs. (20), (21) and (22) Table 3 shows that, firstly, R * sM , R sM and R sM are approximately equal to their corresponding R & sM and the differences among them are always less than 0.09 . Secondly, V * sM , V sM and V sM are approximately equal to their corresponding V & sM , the differences among them are always less than 0.33V while the average value is only 0.019V. Finally, I * sM , I sM and I sM are approximately equal to their corresponding I & sM , the differences among them are always less than 0.15A while the average value is only 0.021A.
In a word, when the model parameters of PVLM are expressed by the VWP model or SVWP model, their characteristics can be maintained very well and their good accuracy can be ensured, which illustrates the feasibility, availability and rationality of using the VWP and SVWP model to express linear equivalent model parameters.

V. DISCUSSIONS
There are some errors arising from the used curve fitting method to obtain Eqs. (11), (12), (15) and (18). Meanwhile, there are also some errors arising from these assumed constants including C 1 , C 2 , C α , C β , C γ and C δ . Therefore, one of the limitations on the VWP model, SVWP model PVLM-VWP and PVLM-SVWP is the existence of these errors. However, there are some arguments for these errors as follows: firstly, Fig. 4 and Fig. 5 show that C 1 and C 2 approximately keep constant all the time regardless of the changing irradiance or temperature, respectively. Meanwhile, Fig. 7 and Fig. 8 also show that C γ and C δ approximately keep constant all the time regardless of the varying irradiance or temperature, respectively. Therefore, it is reasonable that these parameters are assumed as the constants. Secondly, according to Section IV, these errors arising from not only curve fitting method but also those assumed constants are very small. Therefore, in practical application, these errors can be ignored to make the use of these cell models more convenient. Finally, it is well known that almost all theoretical researches to simplify the mathematical models inevitably lead to some errors. Therefore, these small errors in the theoretical analysis are acceptable.
In addition, all the proposed cell models (including PVLM-VWP and PVLM-SVWP) are studied on the basis of the conventional four-parameter model in this text. Therefore, their accuracy, rationality and applicability are constrained by the four-parameter cell model, which is the other limitation of these proposed models.
Although some drawbacks (or limitations) exist, the excellence and effectiveness of these cell models can still be guaranteed by the combination of the curve-fitting technique, MATLAB simulation analysis and VWP methods. The ways to overcome them include the error calibration, average value and so on.

VI. CONCLUSION AND FUTURE DIRECTIONS
In this paper, a VWP model of PV cell, which can simplify the number of the mathematical equations of the conventional four-parameter model from five to three, has been proposed. Meanwhile, to simplify this VWP model, a SVWP model, which can make the number of the model parameters decreased to two, has been also proposed. By the VWP model and SVWP model, the model parameters of the linear equivalent models have been expressed, which can build the direct relationship between four-parameter model and linear equivalent models. Finally, by some simulation experiments, the correspondence of the output characteristics between VWP model, SVWP model and conventional four-parameter model has been shown, the accuracy of the proposed VWP model and SVWP model compared with conventional four-parameter model has been illustrated, and the feasibility, availability and rationality of the linear equivalent models expressed by the VWP model and SVWP model has been verified.
Future work on the subject will be focused on the application of the proposed PVLM-VWP and PVLM-SVWP in the linear control theory. Here, their accuracy should be one of the research emphases when PV system cannot duly operate at the MPP under fast varying weather conditions.