A professional point of view suggests that photovoltaic systems should be installed at the optimum tilt angle and orientation. However, in photovoltaic systems integrated in buildings the flexibility of installation is common. This paper is organized in two different parts. In the first one, the energy losses caused by deviations from the tilt angle ($\beta$) and the orientation ($\gamma$) of the installation in relation to the ideal position are evaluated. This work considers the cloudy-sky conditions in each locality and theoretically calculates by applying the Cavaleri’s principle, the energy losses. Ten cities around the world, in the northern hemisphere, have been studied with a MATLAB code and the findings demonstrate that non-ideal tilt and azimuth angles can also lead to acceptable levels of electric energy generation. A photovoltaic system installed in South orientation ($\gamma =0°$) and $\beta$ deviations of up to 10 (°) in relation to the optimum tilt angle has a very small influence on the energy losses. The energy losses are: 5%, 10%, 15% and 20% when $\beta$ deviations are respectively: 21–23 (°), 31–33 (°), 37–40 (°) and 43–47 (°). Then, in the second part, an important application of this previous outcome comes out: the best distribution of the photovoltaic modules on a flat roof of irregular shape of an urban building is achieved.The aim of this work is to maximize the amount of energy get by a photovoltaic system. This engineering problem is highly complex as it involves 10 variables: the available flat roof area, the shape and the orientation of the available flat roof area, the dimensions (length and width) of the commercial photovoltaic modules, the orientation and the position of the photovoltaic modules, the number of the photovoltaic modules, the minimum distances (maintenance operations, to avoid shadowing effects) between rows of photovoltaic modules, and the minimum distance to the terrace boundary. In this context, this work aims to present a study to assist the decision-making.

This paper shows a packing algorithm (in Mathematica™) which maximizes the energy generation area of the solar photovoltaic system, considering shadings and distances required for maintenance. Eventually, using the initial study, it comes out the influence of $\beta$ on the potential capacity of the solar photovoltaic system and it is demonstrated that a decrease in the optimal tilt angle results in an increase up to 24% in the amount of obtained energy keeping invariable the available area. For example, in Almeria, with an optimum tilt angle of 30.3 (°) the amount of obtained energy is 149.8 (MWh) while with a tilt angle of 14 (°) the amount of obtained energy is 186.2 (MWh). This analysis enables to find the optimal answer to the following practical questions: what number of photovoltaic modules is required?, which is the right position for the photovoltaic modules?, and what orientation of photovoltaic modules is the right one?. There are many installers of photovoltaic systems who would benefit from studies about this issue.

专业观点建议光伏系统应以最佳倾斜角度和方向安装。然而，在集成在建筑物中的光伏系统中，安装的灵活性很常见。本文分为两个不同的部分。在第一个中，由于偏离倾斜角引起的能量损失（$\beta$) 和方向 ($\gamma$) 相对于理想位置的安装进行评估。这项工作考虑了每个地区的多云天空条件，并通过应用 Cavaleri 原理，从理论上计算了能量损失。世界上北半球的十个城市已经使用 MATLAB 代码进行了研究，结果表明，非理想的倾斜角和方位角也可以导致可接受的发电水平。安装在南向的光伏系统（$\gamma =0°$） 和 $\beta$与最佳倾斜角相关的高达 10 (°) 的偏差对能量损失的影响非常小。能量损失为：5%、10%、15%和20%，当$\beta$偏差分别为：21-23（°）、31-33（°）、37-40（°）和43-47（°）。然后，在第二部分，这个先前成果的一个重要应用出来了：在城市建筑的不规则形状的平屋顶上实现光伏组件的最佳分布。 这项工作的目的是最大限度地利用能量通过光伏系统获得。这个工程问题非常复杂，因为它涉及 10 个变量：可用平屋顶面积、可用平屋顶面积的形状和方向、商用光伏组件的尺寸（长度和宽度）、方向和位置光伏组件，光伏组件的数量，光伏组件行之间的最小距离（维护操作，以避免阴影效应），以及到平台边界的最小距离。在这种情况下，这项工作旨在提出一项研究来协助决策。

本文展示了一种打包算法（在 Mathematica™ 中），该算法可最大化太阳能光伏系统的发电面积，同时考虑到维护所需的阴影和距离。最终，利用最初的研究，得出了$\beta$关于太阳能光伏系统的潜在容量，并且证明最佳倾斜角的减小导致获得的能量增加高达 24%，保持可用面积不变。例如，在阿尔梅里亚，最佳倾角为 30.3 (°) 时获得的能量为 149.8 (MWh)，而倾角为 14 (°) 时获得的能量为 186.2 (MWh)。这种分析能够找到以下实际问题的最佳答案：需要多少个光伏组件？光伏组件的正确位置是什么？光伏组件的哪个方向是正确的？有许多光伏系统安装人员将从有关此问题的研究中受益。