太阳电池I-V方程显式求解原理研究及应用
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摘要
自1839年法国物理学家A．E．Becquerel发现光生伏特效应以来，由固体物理知识推导出的太阳电池Ⅰ-Ⅴ方程就成为研究太阳电池输出特性的基础。上述太阳电池Ⅰ-Ⅴ方程是一个隐式超越方程，其中的电流或电压均无法通过初等函数显式地用电压或电流来表示。因此在用该隐式表述解决太阳电池的实际问题时，常常采用各种近似方法来避免计算过程的迭代和推导过程中的繁琐。
鉴于初等函数无法给出太阳电池Ⅰ-Ⅴ方程关于电流或电压的显式表述，本文第二章引入超越函数Lambert W函数应用于太阳电池Ⅰ-Ⅴ方程，给出了太阳电池Ⅰ-Ⅴ方程关于电流和电压的显式表述，并以7种不同种类的太阳电池单体和组件为例，分别计算它们的特征点并绘制其Ⅰ-Ⅴ曲线，验证了显式表述可以准确、完整地表征太阳电池Ⅰ-Ⅴ特性。
其次，基于太阳电池电流和电压的显式表述，给出工程实际中几个关键问题的简便解决方法：
求解最佳负载。前人推导最佳负载表述都是基于隐式表达式，因而求导过程比较繁琐，需应用特别的数学技巧。由此求得的最佳负载表达式中包含最大功率下的电流和电压项，因而无法由模型参数值直接求得最佳负载值。而本文第三章运用了太阳电池关于电流和电压的显式表述后，仅依据最佳负载的定义式即可方便地给出太阳电池最佳负载的表达式，并且运用此表达式可由模型参数直接求得最佳负载值。基于此最佳负载的表达式，还可清晰地看出太阳电池既非电流源亦非电压源的特性，并研究各模型参数对最佳负载的影响。
预测组件性能。太阳电池组件的制造厂商仅提供标准测试条件下的一条Ⅰ-Ⅴ曲线，而实际使用时组件的工作状态通常并不在测试条件上，因此预测组件在各种工况下的输出特性就十分必要。本文第四章提出仅依靠一条标准测试条件下组件的Ⅰ-Ⅴ特性曲线就可准确预测组件在不同工况下输出特性的方法。该方法运用太阳电池电流和电压方程的显式表述，先巧妙地由一条标准测试条件下的Ⅰ-Ⅴ曲线提取出太阳电池模型参数，再结合模型参数随光照和温度的变化关系式，给出了不同光照和温度下组件性能特性的预测，并探讨了在极端工况下，组件输出特性的一些有意义的现象。
分析组件联接。在分析太阳电池组件联接特性时，由于在串联联接时需要单独地研究电压，并联联接时需要单独地分析电流，而Ⅰ-Ⅴ方程的隐式表述无法将电流和电压分开，因而给分析组件串、并联联接特性带来了困难。基于Lambert W函数的电流、电压的显式表述则为单独地分析电流和电压提供了可能。本文第五章依据太阳电池的显式表述依次讨论了组件联接应遵循的联接准则、阵列中被联接组件的工作状态分析以及模型参数对失配损失的影响等问题，最后还讨论了有旁通二极管和阻塞二极管的光伏阵列在受到不均匀光照时输出特性的变化。
Since the photovoltaic effect was found by the French physicist A. E. Becquerel in 1839, the solar cell I-V equation derived from the solid state physics has been to the foundation in solar cell output characteristics research. But this equation is an implicit transcendental equation, and may not be solved explicitly in general for I or V using common elementary functions. Therefore, when applied to solve some practical problems, various approximate methods are often used to avoid the iterative process of calculation and the cumbersome process of deviation.
As the explicit solution of solar cell I-V equation can not be given by elementary function, the transcendental function Lambert W function is applied to the solar cell I-V equation to express the explicit solution of current and voltage in the second chapter. After calculating characteristic points and plotting I-V curves of seven different types of solar cells and solar modules using explicit solution of solar cell I-V equation, respectively, the accuracy and validation of this expression are proofed.
Next, based on the explicit expression of solar cell current and voltage, simple solutions for several key issues in practical engineering are presented:
Solving the optimum load resistance. The derivations of optimum load resistance made by predecessor are based on the implicit expression of solar cell I-V equation, which leads to a cumbersome derivation process and a need for special math skills. Moreover, their optimum load expressions contain the maximum power current and voltage, and are not able to be used to calculate the value of optimum load resistance directly from the model parameters. But based on the explicit solution of solar cell current and voltage, the expression of optimum load resistance can be derived only using the definition of optimum load in the third chapter. And this expression of optimum load resistance can be used to calculate value of optimum load resistance directly from model parameters. This expression of optimum load resistance not only clearly indicates the property of solar cell as neither current source nor voltage source, but also can be used to study how the model parameters effect on the optimum load resistance.
Predicting the module performance. The solar module manufacturers provide only an I-V curve under the standard test condition, but the module actually does not work at the standard testing condition. Thus, the prediction of solar module output characteristics under various working conditions is very necessary. A method can accurately predict the module output characteristics under different working conditions which only needs a measured I-V curve under standard testing conditions is present in the fourth chapter. This method, firstly, extracts the solar module model parameters using an I-V curve under standard testing conditions. Secondly, together with the relationships of model parameters varying with illumination intensity and cell temperature, the prediction of solar module output characteristics under different work condition is made. At last, some significant phenomena of solar module output characteristics under the extreme working conditions are discussed.
Analyzing the connected module. When analyzing the characteristics of connected solar module, series connected module need study voltage separately and parallel connected module need study current separately. But the implicit I-V equation expression can not be used to study current and voltage separately, which make the analysis for characteristics of series and parallel connected module difficult. But the explicit expressions of current and voltage which based on the Lambert W function make analyzing current and voltage separately possible. In the fifth chapter, based on the explicit solution of solar cell I-V equation, the criterion of connecting module, the working state of module connected in array, and the effect of model parameters on mismatch loss are discussed. Finally, the output characteristics of photovoltaic array with bypass diodes and block diodes under non-uniform illumination are also studied.
鉴于初等函数无法给出太阳电池Ⅰ-Ⅴ方程关于电流或电压的显式表述，本文第二章引入超越函数Lambert W函数应用于太阳电池Ⅰ-Ⅴ方程，给出了太阳电池Ⅰ-Ⅴ方程关于电流和电压的显式表述，并以7种不同种类的太阳电池单体和组件为例，分别计算它们的特征点并绘制其Ⅰ-Ⅴ曲线，验证了显式表述可以准确、完整地表征太阳电池Ⅰ-Ⅴ特性。
其次，基于太阳电池电流和电压的显式表述，给出工程实际中几个关键问题的简便解决方法：
求解最佳负载。前人推导最佳负载表述都是基于隐式表达式，因而求导过程比较繁琐，需应用特别的数学技巧。由此求得的最佳负载表达式中包含最大功率下的电流和电压项，因而无法由模型参数值直接求得最佳负载值。而本文第三章运用了太阳电池关于电流和电压的显式表述后，仅依据最佳负载的定义式即可方便地给出太阳电池最佳负载的表达式，并且运用此表达式可由模型参数直接求得最佳负载值。基于此最佳负载的表达式，还可清晰地看出太阳电池既非电流源亦非电压源的特性，并研究各模型参数对最佳负载的影响。
预测组件性能。太阳电池组件的制造厂商仅提供标准测试条件下的一条Ⅰ-Ⅴ曲线，而实际使用时组件的工作状态通常并不在测试条件上，因此预测组件在各种工况下的输出特性就十分必要。本文第四章提出仅依靠一条标准测试条件下组件的Ⅰ-Ⅴ特性曲线就可准确预测组件在不同工况下输出特性的方法。该方法运用太阳电池电流和电压方程的显式表述，先巧妙地由一条标准测试条件下的Ⅰ-Ⅴ曲线提取出太阳电池模型参数，再结合模型参数随光照和温度的变化关系式，给出了不同光照和温度下组件性能特性的预测，并探讨了在极端工况下，组件输出特性的一些有意义的现象。
分析组件联接。在分析太阳电池组件联接特性时，由于在串联联接时需要单独地研究电压，并联联接时需要单独地分析电流，而Ⅰ-Ⅴ方程的隐式表述无法将电流和电压分开，因而给分析组件串、并联联接特性带来了困难。基于Lambert W函数的电流、电压的显式表述则为单独地分析电流和电压提供了可能。本文第五章依据太阳电池的显式表述依次讨论了组件联接应遵循的联接准则、阵列中被联接组件的工作状态分析以及模型参数对失配损失的影响等问题，最后还讨论了有旁通二极管和阻塞二极管的光伏阵列在受到不均匀光照时输出特性的变化。
Since the photovoltaic effect was found by the French physicist A. E. Becquerel in 1839, the solar cell I-V equation derived from the solid state physics has been to the foundation in solar cell output characteristics research. But this equation is an implicit transcendental equation, and may not be solved explicitly in general for I or V using common elementary functions. Therefore, when applied to solve some practical problems, various approximate methods are often used to avoid the iterative process of calculation and the cumbersome process of deviation.
As the explicit solution of solar cell I-V equation can not be given by elementary function, the transcendental function Lambert W function is applied to the solar cell I-V equation to express the explicit solution of current and voltage in the second chapter. After calculating characteristic points and plotting I-V curves of seven different types of solar cells and solar modules using explicit solution of solar cell I-V equation, respectively, the accuracy and validation of this expression are proofed.
Next, based on the explicit expression of solar cell current and voltage, simple solutions for several key issues in practical engineering are presented:
Solving the optimum load resistance. The derivations of optimum load resistance made by predecessor are based on the implicit expression of solar cell I-V equation, which leads to a cumbersome derivation process and a need for special math skills. Moreover, their optimum load expressions contain the maximum power current and voltage, and are not able to be used to calculate the value of optimum load resistance directly from the model parameters. But based on the explicit solution of solar cell current and voltage, the expression of optimum load resistance can be derived only using the definition of optimum load in the third chapter. And this expression of optimum load resistance can be used to calculate value of optimum load resistance directly from model parameters. This expression of optimum load resistance not only clearly indicates the property of solar cell as neither current source nor voltage source, but also can be used to study how the model parameters effect on the optimum load resistance.
Predicting the module performance. The solar module manufacturers provide only an I-V curve under the standard test condition, but the module actually does not work at the standard testing condition. Thus, the prediction of solar module output characteristics under various working conditions is very necessary. A method can accurately predict the module output characteristics under different working conditions which only needs a measured I-V curve under standard testing conditions is present in the fourth chapter. This method, firstly, extracts the solar module model parameters using an I-V curve under standard testing conditions. Secondly, together with the relationships of model parameters varying with illumination intensity and cell temperature, the prediction of solar module output characteristics under different work condition is made. At last, some significant phenomena of solar module output characteristics under the extreme working conditions are discussed.
Analyzing the connected module. When analyzing the characteristics of connected solar module, series connected module need study voltage separately and parallel connected module need study current separately. But the implicit I-V equation expression can not be used to study current and voltage separately, which make the analysis for characteristics of series and parallel connected module difficult. But the explicit expressions of current and voltage which based on the Lambert W function make analyzing current and voltage separately possible. In the fifth chapter, based on the explicit solution of solar cell I-V equation, the criterion of connecting module, the working state of module connected in array, and the effect of model parameters on mismatch loss are discussed. Finally, the output characteristics of photovoltaic array with bypass diodes and block diodes under non-uniform illumination are also studied.
引文
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