# Comparison of Traffic Flow Models with Real Traffic Data Based on a Quantitative Assessment
## 5 Discussion  
Considering the need to prepare the Polish Highway Capacity Manual, the objective of the article was to analyze existing models of the fundamental relationship, compare them, and select the best performing ones that could be used as a basis for the Polish method. The problem that we encountered when reviewing the literature was that there is no agreement of researchers on how to assess and compare traffic flow models. Many comparisons have an expert basis rather than a quantitative approach [4; 17; 29]. The most widely used in quantitative assessment is the model fitness to empirical data [25; 26; 27; 28]. The models' simplicity, compatibility with boundary conditions, and parameter values validity are rarely considered. On the other side, visual assessment of the models is hampered by the existence of the scatter in the real data explained by stochastic characteristics of traffic in real world [32; 33; 34; 35], which is revealed by many speed values corresponding to the same density. Taking these issues into account, we proposed a universal and quantitative method for assessing models of the fundamental relationship. Based on the literature [4; 9; 18; 24], we adopted four criteria for traffic flow models assessment and comparison: simplicity, empirical accuracy, correct estimation of model parameters, and meeting the boundary conditions; the criteria were quantified and assessed in a two-stage procedure. The final assessment is a weighted average of criteria assessment. A detailed analysis using the method confirms the conclusions from the literature review, which is that no model fully meets the requirements the researchers [4; 9; 18] expected of the fundamental relationship models. This suggests and comes as a confirmation of [PERSON] and [PERSON]'s conclusion [2] that the problem of finding the best model remains open. This is the most important conclusion from the study that encourages a search for a new model that could be applied to represent the empirical relations of traffic flow. Analysis results show that the simplest two and three-parameter models do not represent traffic flow characteristics well. While two-regime models offer a much better empirical accuracy, they do so at the cost of model simplicity. The best representation of empirical relations with a relatively simple mathematical function is offered by models that represent the relation \(\mathrm{v}=\mathrm{f}(\mathrm{k})\) with an S-shaped curve. This observation was picked by [PERSON] [4], who proposed a model that uses a sigmoidal curve to represent the relation. Similarly, [PERSON] et al. [14] proposed a model based on a bell-shaped curve. Thanks to the shape, it is possible to include the empirical data pattern where in the initial phase speed, it falls slowly until maximal flow is reached and starts falling much faster afterwards. The pace of the fall slows for low speeds. It also helps to take account of the break point where \(\mathrm{k}=\mathrm{k}_{\mathrm{opt}}\). However, most of these models fail to represent the area of maximum flow, and this is probably caused by the inevitable scatter in data that occurs at the highest flows and in congested traffic regime. We observed that some of the models (with the S-shaped curve representing the relation \(\mathrm{v}=\mathrm{f}(\mathrm{k})\)) that fail to meet boundary conditions ([PERSON] et al., [PERSON] and [PERSON]) seem to give a good representation of traffic flow, even in the intractable area of maximum flow. This raises some questions: Does the BC2 boundary condition have to be absolutely satisfied? Is failure to meet BC2 conditionally acceptable, and if so, under what condition? If our interest is solely in a free-flow regime before the maximum flow is reached (e.g., in methods for traffic condition assessment), is it crucial to meet BC in the congested regime? There are no obvious answers to these questions, and a lot will depend on the purpose of modeling.