Mathematical modeling and extraction of parameters of solar photovoltaic module based on modified Newton–Raphson method
基于修正牛顿-拉斐逊法的太阳能光伏组件数学建模与参数提取
open access 开放存取
Highlights 亮点
- •The paper introduced a modified Newton-Raphson method.
论文介绍了一种改进的牛顿-拉斐森方法。 - •The method follows predictor–corrector rule.
该方法遵循预测-修正规则。 - •The method accuracy was measured by Root Mean Squared Error.
该方法的准确性是以均方根误差来衡量的。 - •Validated of method was done against experimental values, RTC cell and PWP210 module
根据实验值、RTC 电池和 PWP210 模块对该方法进行了验证 - •MATLAB simulation was done to study effect of temperature and solar intensity.
通过 MATLAB 仿真研究了温度和太阳强度的影响。
Abstract 摘要
This paper presents a numerical method for estimating four physical parameters of a single-diode circuit model based on manufacturer’s datasheet. A system of four non-linear equations are formed based on three key points of PV characteristics. The photocurrent, saturation current, ideality factor and the series resistance are solved iteratively using the proposed method. The suggested method is validated using RTC France solar cell, Chloride CHL285P and Photowatt PWP210 modules and the results are verified with respect to the in-field outdoor measurements. The proposed method shows a good agreement with the experimental data. Lastly, The model chosen is simulated under MATLB environment to assess the effects of external physical weather conditions, that is, temperature and solar irradiance. The advantage of the proposed method with respect of existing numerical techniques is that it converged faster than the widely used Newton method. Modeling of PV cell/module is essential in predicting performance of photovoltaic generators at any operating condition.
本文介绍了一种基于制造商数据表估算单二极管电路模型四个物理参数的数值方法。根据光伏特性的三个关键点,建立了由四个非线性方程组成的系统。光电流、饱和电流、ideality 因子和串联电阻的迭代求解采用了所建议的方法。使用 RTC France 太阳能电池、Chloride CHL285P 和 Photowatt PWP210 模块对所建议的方法进行了验证,结果与现场室外测量结果一致。建议的方法与实验数据显示出良好的一致性。最后,在 MATLB 环境下模拟了所选模型,以评估外部物理天气条件(即温度和太阳辐照度)的影响。与现有数值技术相比,所提出方法的优势在于它比广泛使用的牛顿方法收敛更快。光伏电池/模块建模对于预测光伏发电机在任何运行条件下的性能都至关重要。
本文介绍了一种基于制造商数据表估算单二极管电路模型四个物理参数的数值方法。根据光伏特性的三个关键点,建立了由四个非线性方程组成的系统。光电流、饱和电流、ideality 因子和串联电阻的迭代求解采用了所建议的方法。使用 RTC France 太阳能电池、Chloride CHL285P 和 Photowatt PWP210 模块对所建议的方法进行了验证,结果与现场室外测量结果一致。建议的方法与实验数据显示出良好的一致性。最后,在 MATLB 环境下模拟了所选模型,以评估外部物理天气条件(即温度和太阳辐照度)的影响。与现有数值技术相比,所提出方法的优势在于它比广泛使用的牛顿方法收敛更快。光伏电池/模块建模对于预测光伏发电机在任何运行条件下的性能都至关重要。
Keywords 关键词
Extraction of parameters
Mathematical modeling
Numerical method
Photovoltaic module
Single diode model
参数提取数学建模数值法光伏组件单二极管模型
Introduction 导言
Energy, the lifeblood of economic progress and a fundamental necessity for maintaining our quality of life and supporting all aspects of our economy, has witnessed a significant growth in demand over the past two decades [1], largely fueled by our heavy reliance on fossil fuels. However, the significant growth in energy demand has posed significant environmental challenges on a global level, emphasizing the need for immediate and decisive measures to address and mitigate its impact [2]. To address these challenges, it is imperative to adopt sustainable development strategies that offer long-term solutions. Among these strategies, renewable energy resources have emerged as highly efficient and effective alternatives [3]. In contrast to finite fossil fuel reserves, renewable energy sources like solar, wind, and hydropower have minimal adverse effects on the environment. Solar energy, in particular, has gained immense popularity worldwide due to its numerous benefits, including affordability, widespread availability, and a reduced ecological footprint [4]. This is primarily attributed to the capability of photovoltaic (PV) cell technology to directly convert sunlight into electricity [5], [6], making solar energy a promising and viable option for sustainable energy generation [7]. Despite the numerous advantages, the performance of photovoltaic (PV) modules is significantly impacted by external environmental conditions. These include solar intensity [5], ambient temperature [8], relative humidity [9], [10], wind speed [11], panel dust accumulation [12], and the tilt angle of the panel [13], [14], [15]. In light of this reality, the actual performance of PV module systems in real working environments often falls short of expectations [16]. To overcome this challenge, researchers have explored alternative methods for predicting the output characteristics and maximum power output of PV modules without relying on extensive experimental measurements. One such approach involves utilizing the manufacturer’s data, which provides information about the module’s electrical characteristics under standardized test conditions (STC) [17]. This data includes parameters like the maximum power point voltage, current, and temperature coefficients of the module. However, it is important to acknowledge that relying solely on manufacturer’s data for performance predictions may have limitations. Real-world operating conditions for PV modules can significantly differ from the standardized test conditions used to generate the manufacturer’s data [18]. Therefore, while predictive models based on manufacturer’s data can offer valuable insights, it is crucial to validate their accuracy through field measurements to ensure reliable performance predictions in actual working environments.
能源是经济进步的命脉,也是维持我们的生活质量和支持我们经济各方面发展的基本必需品,在过去二十年里,能源需求大幅增长[1],这主要是由于我们严重依赖化石燃料。然而,能源需求的大幅增长在全球范围内带来了重大的环境挑战,强调需要立即采取果断措施来应对和减轻其影响[2]。为应对这些挑战,当务之急是采取可提供长期解决方案的可持续发展战略。在这些战略中,可再生能源已成为高效和有效的替代能源[3]。与有限的化石燃料储备相比,太阳能、风能和水能等可再生能源对环境的不利影响微乎其微。特别是太阳能,由于其众多优点,包括经济实惠、广泛可用和减少生态足迹,已在全球范围内大受欢迎[4]。这主要归功于光伏(PV)电池技术能够直接将太阳光转化为电能[5]、[6],从而使太阳能成为可持续能源生产的一种前景广阔的可行选择[7]。尽管光伏(PV)组件具有诸多优势,但其性能受到外部环境条件的显著影响。这些因素包括太阳强度 [5]、环境温度 [8]、相对湿度 [9]、[10]、风速 [11]、面板积尘 [12] 和面板倾斜角度 [13]、[14]、[15]。鉴于这一现实,光伏组件系统在实际工作环境中的实际性能往往达不到预期[16]。 为了克服这一挑战,研究人员探索了其他方法,在不依赖大量实验测量的情况下预测光伏组件的输出特性和最大功率输出。其中一种方法是利用制造商的数据,这些数据提供了模块在标准化测试条件(STC)下的电气特性信息[17]。这些数据包括模块的最大功率点电压、电流和温度系数等参数。不过,必须承认,仅依靠制造商的数据进行性能预测可能存在局限性。光伏组件的实际运行条件可能与用于生成制造商数据的标准化测试条件大相径庭[18]。因此,虽然基于制造商数据的预测模型可以提供有价值的见解,但至关重要的是要通过现场测量来验证其准确性,以确保在实际工作环境中进行可靠的性能预测。
能源是经济进步的命脉,也是维持我们的生活质量和支持我们经济各方面发展的基本必需品,在过去二十年里,能源需求大幅增长[1],这主要是由于我们严重依赖化石燃料。然而,能源需求的大幅增长在全球范围内带来了重大的环境挑战,强调需要立即采取果断措施来应对和减轻其影响[2]。为应对这些挑战,当务之急是采取可提供长期解决方案的可持续发展战略。在这些战略中,可再生能源已成为高效和有效的替代能源[3]。与有限的化石燃料储备相比,太阳能、风能和水能等可再生能源对环境的不利影响微乎其微。特别是太阳能,由于其众多优点,包括经济实惠、广泛可用和减少生态足迹,已在全球范围内大受欢迎[4]。这主要归功于光伏(PV)电池技术能够直接将太阳光转化为电能[5]、[6],从而使太阳能成为可持续能源生产的一种前景广阔的可行选择[7]。尽管光伏(PV)组件具有诸多优势,但其性能受到外部环境条件的显著影响。这些因素包括太阳强度 [5]、环境温度 [8]、相对湿度 [9]、[10]、风速 [11]、面板积尘 [12] 和面板倾斜角度 [13]、[14]、[15]。鉴于这一现实,光伏组件系统在实际工作环境中的实际性能往往达不到预期[16]。 为了克服这一挑战,研究人员探索了其他方法,在不依赖大量实验测量的情况下预测光伏组件的输出特性和最大功率输出。其中一种方法是利用制造商的数据,这些数据提供了模块在标准化测试条件(STC)下的电气特性信息[17]。这些数据包括模块的最大功率点电压、电流和温度系数等参数。不过,必须承认,仅依靠制造商的数据进行性能预测可能存在局限性。光伏组件的实际运行条件可能与用于生成制造商数据的标准化测试条件大相径庭[18]。因此,虽然基于制造商数据的预测模型可以提供有价值的见解,但至关重要的是要通过现场测量来验证其准确性,以确保在实际工作环境中进行可靠的性能预测。
Efforts have been made by researchers to improve the performance of solar panels, leading to the development of various PV models. Based on equivalent circuit, the models are classified as single and double diodes. While single diode model consists of five, four or three unknown parameters, double diode model has seven unknown parameters. The five parameters of sinlge diode model are the photocurrent (), reverse saturation current (), series () and shunt () resistances, and the ideality factor (). The four parameter model of a single diode assumes the shunt resistance infinity and its effect is ignored [19], [20], [21]. The three parameter model proposes that the series resistance is zero and shunt resistance is infinity leading to their neglect. On the other hand the seven estimated parameters of double diode model includes the five parameters of the single parameter model with additional of second reverse saturation current and ideality factor for the second diode [22], [23].
研究人员一直在努力提高太阳能电池板的性能,从而开发出各种光伏模型。根据等效电路,模型可分为单二极管和双二极管。单二极管模型由五个、四个或三个未知参数组成,而双二极管模型则有七个未知参数。单二极管模型的五个参数分别是光电流( )、反向饱和电流( )、串联电阻( )和并联电阻( )以及表意系数( )。单二极管的四参数模型假定并联电阻为无穷大,其影响被忽略 [19]、[20]、[21]。三参数模型认为串联电阻为零,并联电阻为无穷大,因此忽略了它们的影响。另一方面,双二极管模型的七个估计参数包括单参数模型的五个参数以及第二个二极管的第二个反向饱和电流和理想系数 [22],[23]。
研究人员一直在努力提高太阳能电池板的性能,从而开发出各种光伏模型。根据等效电路,模型可分为单二极管和双二极管。单二极管模型由五个、四个或三个未知参数组成,而双二极管模型则有七个未知参数。单二极管模型的五个参数分别是光电流( )、反向饱和电流( )、串联电阻( )和并联电阻( )以及表意系数( )。单二极管的四参数模型假定并联电阻为无穷大,其影响被忽略 [19]、[20]、[21]。三参数模型认为串联电阻为零,并联电阻为无穷大,因此忽略了它们的影响。另一方面,双二极管模型的七个估计参数包括单参数模型的五个参数以及第二个二极管的第二个反向饱和电流和理想系数 [22],[23]。
There are various methods developed in literature for parameter extraction of PV circuit models. These approaches can be categorized into analytical, numerical (deterministic and stochastic) and hybrid methods [22]. For example, Jakhrani et al. [23] used numerical method to determine the output current and voltage of a five-parameter model for a single diode equivalent circuit. This method however, required extensive mathematical computations. Similarly, [24] proposed a numerical approach to extract the physical parameters based on single diode model. The ideality factor and series resistance were solved numerically using only two points, at short circuit point and at maximum power point. The method proposed showed good agreement with experimental values. In addition, El Achouby et al. [25] proposed numerical method to extract the five physical parameters at standard test conditions (STC). This ideality factor was made to vary and a system of four nonlinear equations was solved. The drawback of this approach is the suitable choice of initial values of the parameters. Moreover, Elkholy and Abou El-Ela [26] took an analytical approach by solving the parameters of a single diode model based on in-field outdoor data, demonstrating good agreement between simulated and experimental measurements. However, this method was complicated due to the requirement of extensive experimental data. [27] proposed a simple fitting method for estimating parameters of a single diode model, but its validity was limited to constant solar irradiance, making it impractical for real working environments. Rasheed et al. [28] presented the Chebyshev numerical method as an alternative to the Newton–Raphson method for solving the voltage of PV cells, but the increased computational burden from calculating second derivatives posed challenges. Currently, [29] introduced a new simplified method for the iterative estimation of the five PV module parameters. This method uses some approximations to determine the values of series resistance and ideality factor. The accuracy of this method was a bit challenge due to approximations while computing series resistance and ideality factor. [30] presented a numerical method for solving the series and shunt resistances using the Newton–Raphson approach and the Lambert W- function. However, this method needs a lot of assumptions to simplify the problem in order to extract the five parameters. Lastly, Manuel Godinho Rodrigues et al. [31] conducted a comparative study using numerical simulations to evaluate the five-parameter single diode model and the more advanced seven-parameter double diode model. The double diode model accounted for additional energy losses associated with charge carrier recombination in the depletion layers, but its high complexity level limited its practicality.
文献中有多种光伏电路模型参数提取方法。这些方法可分为分析方法、数值方法(确定性和随机性)和混合方法 [22]。例如,Jakhrani 等人 [23] 使用数值方法确定了单二极管等效电路五参数模型的输出电流和电压。然而,这种方法需要大量的数学计算。同样,[24] 提出了一种基于单二极管模型提取物理参数的数值方法。仅使用短路点和最大功率点两个点对理想化系数和串联电阻进行了数值求解。所提出的方法与实验值显示出良好的一致性。此外,El Achouby 等人[25] 提出了在标准测试条件 (STC) 下提取五个物理参数的数值方法。这种表意系数是变化的,并求解了一个由四个非线性方程组成的系统。这种方法的缺点是需要选择合适的参数初始值。此外,Elkholy 和 Abou El-Ela [26] 还采用了一种分析方法,即根据现场室外数据求解单个二极管模型的参数,结果表明模拟和实验测量结果之间的一致性很好。然而,由于需要大量实验数据,这种方法比较复杂。[27] 提出了一种简单的拟合方法来估算单二极管模型的参数,但其有效性仅限于恒定的太阳辐照度,因此在实际工作环境中并不实用。Rasheed 等人 [28] 提出了切比雪夫数值方法,作为牛顿-拉斐森方法的替代方法,用于求解光伏电池的电压,但计算二次导数所增加的计算负担带来了挑战。目前,[29] 提出了一种新的简化方法,用于迭代估计五个光伏组件参数。该方法使用一些近似值来确定串联电阻和表意系数的值。由于在计算串联电阻和意向系数时使用了近似值,该方法的准确性面临一定挑战。[30] 提出了一种利用牛顿-拉斐森方法和兰伯特 W- 函数求解串联和并联电阻的数值方法。然而,这种方法需要大量假设来简化问题,以提取五个参数。最后,Manuel Godinho Rodrigues 等人[31] 利用数值模拟进行了比较研究,评估了五参数单二极管模型和更先进的七参数双二极管模型。双二极管模型考虑到了与耗尽层中电荷载流子重组相关的额外能量损失,但其高复杂度限制了其实用性。
文献中有多种光伏电路模型参数提取方法。这些方法可分为分析方法、数值方法(确定性和随机性)和混合方法 [22]。例如,Jakhrani 等人 [23] 使用数值方法确定了单二极管等效电路五参数模型的输出电流和电压。然而,这种方法需要大量的数学计算。同样,[24] 提出了一种基于单二极管模型提取物理参数的数值方法。仅使用短路点和最大功率点两个点对理想化系数和串联电阻进行了数值求解。所提出的方法与实验值显示出良好的一致性。此外,El Achouby 等人[25] 提出了在标准测试条件 (STC) 下提取五个物理参数的数值方法。这种表意系数是变化的,并求解了一个由四个非线性方程组成的系统。这种方法的缺点是需要选择合适的参数初始值。此外,Elkholy 和 Abou El-Ela [26] 还采用了一种分析方法,即根据现场室外数据求解单个二极管模型的参数,结果表明模拟和实验测量结果之间的一致性很好。然而,由于需要大量实验数据,这种方法比较复杂。[27] 提出了一种简单的拟合方法来估算单二极管模型的参数,但其有效性仅限于恒定的太阳辐照度,因此在实际工作环境中并不实用。Rasheed 等人 [28] 提出了切比雪夫数值方法,作为牛顿-拉斐森方法的替代方法,用于求解光伏电池的电压,但计算二次导数所增加的计算负担带来了挑战。目前,[29] 提出了一种新的简化方法,用于迭代估计五个光伏组件参数。该方法使用一些近似值来确定串联电阻和表意系数的值。由于在计算串联电阻和意向系数时使用了近似值,该方法的准确性面临一定挑战。[30] 提出了一种利用牛顿-拉斐森方法和兰伯特 W- 函数求解串联和并联电阻的数值方法。然而,这种方法需要大量假设来简化问题,以提取五个参数。最后,Manuel Godinho Rodrigues 等人[31] 利用数值模拟进行了比较研究,评估了五参数单二极管模型和更先进的七参数双二极管模型。双二极管模型考虑到了与耗尽层中电荷载流子重组相关的额外能量损失,但其高复杂度限制了其实用性。
As per literature review, numerical iterative methods have been widely applied in solving the non-linear, multi-parameter and transcendental equation of PV model. Of many iterative approaches, Newton–Raphson method is the most popular in this field due to its simplicity, accuracy and quadratic convergency rate [28], [32], [33]. Despite this dominance, Newton–Raphson has convergence shortcomings especially when the initial inputs are far from the actual roots [34], [35].
根据文献综述,数值迭代法已被广泛应用于求解光伏模型的非线性、多参数和超越方程。在众多迭代法中,牛顿-拉夫逊法因其简单、精确和二次收敛率而在该领域最受欢迎 [28]、[32]、[33]。尽管牛顿-拉夫逊法占主导地位,但它也存在收敛性缺陷,尤其是当初始输入远离实际根值时[34],[35]。
根据文献综述,数值迭代法已被广泛应用于求解光伏模型的非线性、多参数和超越方程。在众多迭代法中,牛顿-拉夫逊法因其简单、精确和二次收敛率而在该领域最受欢迎 [28]、[32]、[33]。尽管牛顿-拉夫逊法占主导地位,但它也存在收敛性缺陷,尤其是当初始输入远离实际根值时[34],[35]。
In this work, we present a modified Newton–Raphson Method proposed by [36] to estimate the parameters of single diode model. We first choose to use a simplified four parameter model of a single diode equivalent circuit. Next, we develop the corresponding mathematical equation of the model. Then, basing on three key points from manufacturer’s catalog, a system of four non-linear equations is formed and solved iteratively using the proposed method. This method advantageous than the standard Newton–Raphson one because its rate of convergence is enhanced to the order of . Thus, it offers a quick and accurate analysis of PV module performance. Circuit modeling of PV cells, as discussed by Batzelis and Papathanassiou, El Achouby et al. [37], [25], enables PV cell simulation and performance evaluation under varying conditions, aiding in monitoring system operation, forecasting power output, calculating power losses, and developing maximum power point tracking algorithms [38]. This paper is organized as follows: Section “Introduction” presents an introductory part of the study and the reviews of different approaches of solving mathematical models of PV module. Method part is found in Section “Method” in which the model formulation procedures and parameter extraction processes are discussed. Results and Discussion are presented in Section “Results and Discussion”. Lastly, conclusion is declared in Section “Conclusion”.
在这项工作中,我们采用 [36] 提出的修正牛顿-拉斐森方法来估计单二极管模型的参数。我们首先选择使用单二极管等效电路的简化四参数模型。接着,我们建立了模型的相应数学方程。然后,根据制造商目录中的三个关键点,形成四个非线性方程组,并使用所提出的方法进行迭代求解。与标准的 Newton-Raphson 方法相比,该方法的优势在于其收敛速度提高到了 的数量级。因此,它能快速准确地分析光伏组件的性能。正如 Batzelis 和 Papathanassiou、El Achouby 等人[37]、[25]所讨论的那样,光伏电池的电路建模可以在不同条件下对光伏电池进行模拟和性能评估,有助于监控系统运行、预测功率输出、计算功率损耗和开发最大功率点跟踪算法[38]。本文的结构如下:绪论 "部分介绍了本研究的前言部分,并回顾了求解光伏组件数学模型的不同方法。在 "方法 "部分,讨论了模型制定程序和参数提取过程。结果和讨论 "部分介绍了结果和讨论。最后,在 "结论 "部分给出结论。
在这项工作中,我们采用 [36] 提出的修正牛顿-拉斐森方法来估计单二极管模型的参数。我们首先选择使用单二极管等效电路的简化四参数模型。接着,我们建立了模型的相应数学方程。然后,根据制造商目录中的三个关键点,形成四个非线性方程组,并使用所提出的方法进行迭代求解。与标准的 Newton-Raphson 方法相比,该方法的优势在于其收敛速度提高到了 的数量级。因此,它能快速准确地分析光伏组件的性能。正如 Batzelis 和 Papathanassiou、El Achouby 等人[37]、[25]所讨论的那样,光伏电池的电路建模可以在不同条件下对光伏电池进行模拟和性能评估,有助于监控系统运行、预测功率输出、计算功率损耗和开发最大功率点跟踪算法[38]。本文的结构如下:绪论 "部分介绍了本研究的前言部分,并回顾了求解光伏组件数学模型的不同方法。在 "方法 "部分,讨论了模型制定程序和参数提取过程。结果和讨论 "部分介绍了结果和讨论。最后,在 "结论 "部分给出结论。
Method 方法
Model formulation 模型表述
PV cell is modeled as p–n junction diode because their working principle are similar. The equivalent circuit of the single diode model consists of the real diode connected in parallel with the ideal current source (), the series and the parallel resistors ( and ) as shown in Fig. 1. Referring Fig. 1, the Kirchhoff’s law is represented by the following equation: (1)Where; is the current output, is the photocurrent from the source, is the current through the diode and is the current through the shunt resistance. The photocurrent is the function of solar irradiation and temperature as shown in the equation [39] (2)
光伏电池的工作原理与 p-n 结二极管相似,因此将其建模为 p-n 结二极管。如图 1 所示,单二极管模型的等效电路由与理想电流源( )并联的实际二极管、串联电阻和并联电阻( 和 )组成。参照图 1,基尔霍夫定律用下式表示: (1) 其中, 为输出电流, 为来自光源的光电流, 为通过二极管的电流, 为通过并联电阻的电流。光电流 是太阳辐照度和温度的函数,如公式 [39] (2) 所示
光伏电池的工作原理与 p-n 结二极管相似,因此将其建模为 p-n 结二极管。如图 1 所示,单二极管模型的等效电路由与理想电流源( )并联的实际二极管、串联电阻和并联电阻( 和 )组成。参照图 1,基尔霍夫定律用下式表示: (1) 其中, 为输出电流, 为来自光源的光电流, 为通过二极管的电流, 为通过并联电阻的电流。光电流 是太阳辐照度和温度的函数,如公式 [39] (2) 所示
Where; is the solar irradiance, is the solar irradiance at reference point, is the photocurrent at reference point, is the temperature coefficient at short circuit, is an operating temperature and is the temperature at reference point The diode current is given by Shockley equation as follows [26] (3)Where:
式中: 为太阳辐照度, 为参考点的太阳辐照度, 为参考点的光电流, 为短路时的温度系数, 为工作温度, 为参考点的温度 二极管电流由肖克利方程给出,如下所示[26] (3) 式中
式中: 为太阳辐照度, 为参考点的太阳辐照度, 为参考点的光电流, 为短路时的温度系数, 为工作温度, 为参考点的温度 二极管电流由肖克利方程给出,如下所示[26] (3) 式中

Fig. 1. Equivalent Circuit Diagram for single diode five-parameter model [40].
图 1.单二极管五参数模型等效电路图 [40]。
– Charge of an electron
- 电子的电荷
- 电子的电荷
– Temperature of a cell
- 电池的温度
- 电池的温度
– Voltage output
- 电压输出
- 电压输出
– Diode reverse saturation current
- 二极管反向饱和电流
- 二极管反向饱和电流
The reverse saturation current depends on ambient temperature as shown in the equation [41] (4)Where (5)The series and shunt resistances can be estimated from curve as follows: (6)Where , and (7)The current through the shunt resistance is given by the following equation [42] (8)The ideality factor is given by [43], [44] as follows: (9)Under varying physical conditions, [26], [45] presents the analytical formulas to determine series and shunt resistances, and the ideality factor as follows: (10)Where, is the series resistance at reference conditions and is the corresponding parameter at the real experimental weather conditions. is the temperature coefficient at open circuit whose value is about . (11)Where, is the shunt resistance at reference conditions and is the parameter at the real outdoor conditions [26]. (12)Where, is the ideality factor of the diode at reference conditions and is the ideality factor at the real outdoor conditions [26]. Therefore, the final equation of a single diode five-parameter model is given as follows [25], [26], [46], [47], [48], [49] (13) [21] claims that the PV cell is not sensitive to shunt resistance and that it can be considered to be infinity without generating earth current leakage. In addition, a comparative study performed by [20] for different models of PV cell reveals that the effect of shunt resistance at low voltage and solar illumination is negligible. Moreover, a study by [19] shows that for higher power yield and fill factor, the shunt resistance should be large enough because low shunt resistance leads to high power loss. Therefore, the shunt resistance in this study is assumed to be infinity, hence the last term of the model equation is disregarded. By omitting this final component of the model, the influence of shunt resistance on PV module performance is ignored. The model equation remains only with four parameters . This helps to reduce the complexity of the model as given below: (14)For cells, Eq. (14) becomes: (15)
反向饱和电流取决于环境温度,如公式 [41] 所示 (4) 其中 (5) 串联电阻 和并联电阻 可根据 曲线估算如下: (6) 其中 和 (7) 通过并联电阻的电流由下式 [42] 得出 (8) 理想系数 由 [43], [44] 得出如下: (9) 在不同的物理条件下,[26]、[45] 提出了确定串联 和并联 电阻以及表意系数 的解析公式如下: (10) 其中, 为参考条件下的串联电阻, 为实际实验天气条件下的相应参数。 是开路时的温度系数,其值约为 。 (11) 其中, 是参考条件下的并联电阻,是实际室外条件下的参数 [26]。 (12) 其中, 是二极管在参考条件下的表意系数, 是实际室外条件下的表意系数[26]。因此,单二极管五参数模型的最终方程如下 [25]、[26]、[46]、[47]、[48]、[49] (13) [21] 称,光伏电池对并联电阻不敏感,可以认为其为无穷大,不会产生接地电流泄漏。此外,[20] 对不同型号的光伏电池进行的比较研究表明,在低电压和太阳光照下,并联电阻的影响可以忽略不计。 此外,[19] 的研究表明,为了获得更高的功率输出和填充因子,并联电阻应该足够大,因为并联电阻过小会导致功率损耗过大。因此,本研究假定并联电阻为无穷大,从而忽略了模型方程的最后一项。通过省略模型的最后一项,并联电阻对光伏组件性能的影响被忽略。模型方程只剩下四个参数 。这有助于降低模型的复杂性,如下所示: (14) 对于 电池,公式 (14) 变为: (15)
反向饱和电流取决于环境温度,如公式 [41] 所示 (4) 其中 (5) 串联电阻 和并联电阻 可根据 曲线估算如下: (6) 其中 和 (7) 通过并联电阻的电流由下式 [42] 得出 (8) 理想系数 由 [43], [44] 得出如下: (9) 在不同的物理条件下,[26]、[45] 提出了确定串联 和并联 电阻以及表意系数 的解析公式如下: (10) 其中, 为参考条件下的串联电阻, 为实际实验天气条件下的相应参数。 是开路时的温度系数,其值约为 。 (11) 其中, 是参考条件下的并联电阻,是实际室外条件下的参数 [26]。 (12) 其中, 是二极管在参考条件下的表意系数, 是实际室外条件下的表意系数[26]。因此,单二极管五参数模型的最终方程如下 [25]、[26]、[46]、[47]、[48]、[49] (13) [21] 称,光伏电池对并联电阻不敏感,可以认为其为无穷大,不会产生接地电流泄漏。此外,[20] 对不同型号的光伏电池进行的比较研究表明,在低电压和太阳光照下,并联电阻的影响可以忽略不计。 此外,[19] 的研究表明,为了获得更高的功率输出和填充因子,并联电阻应该足够大,因为并联电阻过小会导致功率损耗过大。因此,本研究假定并联电阻为无穷大,从而忽略了模型方程的最后一项。通过省略模型的最后一项,并联电阻对光伏组件性能的影响被忽略。模型方程只剩下四个参数 。这有助于降低模型的复杂性,如下所示: (14) 对于 电池,公式 (14) 变为: (15)
Determination of parameters of the model
确定模型参数
The model consists of four unknown parameters (, , , and ) whose values are required to be determined. To solve for these parameters, four equations are required. The three basic operating points on curve of the PV cell/module give the first three equations which are Eqs. (16), (17), (18)
该模型包括四个未知参数( 、 、 和 ),需要确定其值。要求解这些参数,需要四个方程。光伏电池/模块的 曲线上的三个基本工作点给出了前三个方程,即公式 (16)、(17) 和 (18)
该模型包括四个未知参数( 、 、 和 ),需要确定其值。要求解这些参数,需要四个方程。光伏电池/模块的 曲线上的三个基本工作点给出了前三个方程,即公式 (16)、(17) 和 (18)
- (a)At the open circuit, and (16)
开路时, 和 (16) - (b)At the short circuit, and (17)
短路时, 和 (17) - (c)
- (d)At maximum power point the slope, , where the power . Thus, we have the fourth equation as shown in Eq. (19) below (19)
在最大功率点 的斜率, ,其中功率 。因此,我们可以得到下式 (19) 所示的第四个方程 (19) .
If we let , , , and , then the system of equations can be written as follows (20)To obtain the values of model parameters, the system (20) is solved simultaneously by first plugging the values of , , and which are always given in manufacturer’s catalog. The proposed method, modified Newton–Raphson, is used to solve this system.
如果设 、 、 、 和 ,则方程组可写成 (20) 为获得模型参数值,首先插入制造商目录中通常给出的 、
如果设 、 、 、 和 ,则方程组可写成 (20) 为获得模型参数值,首先插入制造商目录中通常给出的 、