# Emerging Data-Driven Calibration Research on an Improved Link Performance Function in an Urban Area
## 2 Model  
### Problem Description  
In this paper, the calibration of a link performance function is studied. Congestion-related time consumption is a significant determinant of the route that a traveler chooses when using a road system. As a result, the connection between the journey time and loading for road transportation is a crucial component of the overall link performance function. Many academics have devoted their careers to related topics to clarify its function. Among these accomplishments, the BPR function is well known. The formula for it is as follows.  
\[T_{a}(v_{a})=T_{a}\Bigg{[}1+\alpha\bigg{(}\frac{v_{a}}{c_{a}}\bigg{)}^{\beta} \Bigg{]}. \tag{2}\]  
Here, \(T_{a}\) is the travel time under free-flow conditions on arc \(a\), \(a\) represents the volume of arc \(a\), and \(c_{a}\) denotes the capacity of arc \(a\). \(\alpha\) and \(\beta\) are regression parameters.  
According to Equation (2), as the volume of arc \(a\) grows, so does the travel time. The travel time will dramatically rise if the volume exceeds the capacity. This could be an indication of the volume-delay relationship due to congestion.  
Considering that the national conditions of China and the United States are different and that there are large differences in economic development and traffic conditions, the BPR function is not fully adapted to the road conditions in China. Therefore, the BPR function was improved and recalibrated in research on capacity in the National Key Transportation Technology Research and Development Program of China during the 9 th Five-Year Plan Period, and it was combined with data from large-scale actual traffic surveys in China [39]. The form of this new model (for convenience, it is termed BPR95) is as follows.  
\[T_{a}(v_{a})=\frac{T_{a}}{\alpha_{1}}\Bigg{[}1+\bigg{(}\frac{v_{a}}{c_{a}} \bigg{)}^{\beta}\Bigg{]}, \tag{3}\]\[\left\{\begin{array}{l}S_{a}=\frac{a_{1}S_{d}}{1+\left(\frac{a_{2}}{c_{a}}\right) ^{\beta}}\\ \beta=\alpha_{2}+\alpha_{3}\left(\frac{c_{a}}{c_{a}}\right)^{3}\end{array} \right.. \tag{4}\]  
Here, \(S_{a}\) is the average travel speed, \(S_{d}\) is the design speed, and \(\alpha_{1},\alpha_{2},\alpha_{3}\) are parameters to be calibrated.  
In addition to the time impediment, the monetary expenditure is also an important factor in the impediment of a road. Therefore, it is necessary to include money spent in the general cost function and to quantify time spent as money through the value of the time (VOT) parameter [40]. The general cost function for road transportation is shown as follows.  
\[F_{a}(v_{a})=r^{road}L_{a}+\theta T_{a}(v_{a}). \tag{5}\]  
Here, \(F_{a}(v_{a})\) represents the general cost of arc \(a\), \(r^{road}\) is the expenditure per kilometer on the road, \(L_{a}\) denotes the physical length of arc \(a\), and \(\theta\) is the VOT.  
The link performance function model proposed in this paper consists of two major components: \(T_{a}\), the time impedance, including the consumption of time and delays caused by tolls, road width, and other physical conditions on the section itself, as well as the characteristics of the traffic flow itself caused by time delays due to traffic congestion, is reflected in the BPR95 function; \(r^{road}L_{a}\) for the cost impedance, including transportation costs and road tolls. The scale of the link performance function's value \(F_{a}\) is RMB (yuan). Therefore, the scale of \(\theta\) is min/yuan and the scale of \(r^{road}\) is yuan/km.  
The parameter calibration problem is formulated as an optimization problem, and it is used to minimize the error between the real travel time patterns (empirical data) and the model-based travel time patterns. The error measurement is defined as follows:  
\[\min Z=\sqrt{\frac{\sum\limits_{i}\left(\frac{T_{i}^{data}-T_{i}^{model}}{T_{i} ^{data}}\right)^{2}}{N}}. \tag{6}\]  
Here, \(T_{i}^{model}\) denotes the model-based link travel time, \(T_{i}^{data}\) represents the real link travel time, and \(N\) is the number of samples.