Development of driver behavior model
5.3 Calibration and validation of model
5.3.2 Calibration of unknown model parameters
Various model parameters (C1, C2, β¦ C5) described earlier in Section 5.1 are calibrated in this subsection. The values of these parameters are tuned between certain ranges using genetic algorithms (GA), to obtain behavior of simulated stream similar to the field traffic stream by minimizing the error between two streams.
Table 5.1 Input parameters in simulation model from field data
Vehicle type 3W Car Bus 2W Truck LCV
Traffic composition (%) 8.11 43.39 3.08 40.90 1.89 2.63
Vehicle dimensions
Length (m)
Mean 2.92 3.75 9.21 1.83 6.74 5.97 Standard
deviation 0.08 0.87 1.12 0.17 1.49 1.67 Width
(m)
Mean 1.35 1.49 2.44 0.74 2.38 1.79 Standard
deviation 0.04 0.23 0.30 0.08 0.21 0.44
Lateral speed
Mean, m/s 1.08 1.04 1.03 1.23 0.94 1.03
85th percentile
(m/s) 2.85 2.83 2.73 3.25 2.47 2.72
Ξ» (negative
exponential) 0.62 0.68 0.68 0.57 0.75 0.68 Desired speed
distribution (km/h)
Mean, km/h 49.57 76.32 60.19 59.94 51.77 59.90 Standard deviation 6.41 17.66 11.63 14.08 12.94 14.98 Desired acceleration, ππππ₯ m/s2 1.32 2.56 1.21 2.31 1.00 1.86
Desired deceleration, m/s2 2.65 3.34 2.11 3.18 1.67 1.98
5.3.2.1 Description of calibrated parameters
The following parameters calibrated within their ranges, using genetic algorithms:
1. Parameter C1 for calculating interaction range: In Equation 5.1, the term πΆ1 π£π‘2 represents distance covered during braking (π£π‘2β2 Γ deceleration). The minimum deceleration may be assumed as 1 m/s2 whereas maximum deceleration as per Fambro et al. (2000) is 4.5 m/s2. Therefore upper and lower limits of πΆ1 (0.5 and 0.1, respectively) can be calculated from these values (1 (2 Γ deceleration)β ). Literature doesnβt yield any study related to estimation of interaction range of a driver in weak lane disciplined traffic, therefore, the constant C1 in Equation 5.1 is also considered for calibration from field data. The interaction range may or may not be site-specific.
2. Error in the perception of speed of LV by following driver (C2 and C3): It is basically the percentage incorrect speed perception of LV by the least risk-taking driver. Details about calculation of these errors are mentioned in Section 5.1.2. The perceived LV speed error limits maybe calculated based on the assumption that drivers would estimate negative error (positive relative speed or approaching the LV) with better accuracy than positive error.
The maximum error threshold is estimated from FHWAβs relative speed threshold with inter-vehicular spacing graph, for observation time 1 second (adopted from Evans and Rothery, 1973).
3. Ratio of normal braking to emergency braking (C4): The ratio of maximum deceleration (emergency breaking) to normal deceleration values are defined by a site-specific calibration constant C4. The maximum limit for emergency braking is based on the
assumption that drivers would not brake harder than g, the acceleration due to gravity, even in the worst possible scenario (Kudarauskas, 2007).
4. Ratio of speed offered by a gap to its proximity while calculating priority (Gap-priority):
It is the calibration constant C5 used in Equation 5.9.
These five parameters with their ranges are used as inputs to the objective function, described in Table 5.2. With various trial and errors (using GA), the parameters are assigned values by GA tool.
Objective function is described further.
Table 5.2 Model parameters used for calibration by GA
S. no. Parameter used for calculation of Parameter Lower limit Upper limit
1 Interaction range πΆ1 0.100 0.500
2 Perceived LV speed positive error πΆ2 0.000 0.100 3 Perceived LV speed negative error πΆ3 0.000 0.050
4 Emergency braking πΆ4 1.000 3.000
5 Gap-goodness (speed or proximity) πΆ5 0.200 5.000 5.3.2.2 Objective function
The optimization objective-function f used in this algorithm consists of error terms for deviation of model traffic from speed-flow relationship (macroscopic stream property) and speed distribution (microscopic property) relationships of the field; and is mentioned in Equation 5.12.
π = 2.0 β ππ πππΈ1β ππ πππΈ2 β¦ 5.12 (a)
ππ πππΈ2 = average(ππ πππΈπππ, ππ πππΈπ΅πππ, ππ πππΈπ΄π’π‘π) β¦ 5.12 (b) π€βπππ, ππ πππΈ1 is the normalized root mean square error between obtained speeds for particular per-minute flow level of model and field (for 1 hour duration); and ππ πππΈπππ, ππ πππΈπ΅πππ, ππ πππΈπ΄π’π‘π are the normalized root mean square errors between discretized frequency of speed distribution of car, two-wheeler and three-wheeler, respectively between field and model. The normalized MSE was used since the speed distribution and speed- flow relationships provide different values of errors and they need to be normalized to be combined to calculate a resultant error term.
5.3.2.3 Genetic Algorithms used to optimize model flow
The genetic algorithm consists of an objective function described by Equation 5.12 and proceeds as per following steps-
1. The algorithm begins by creating a random initial population of size 10, within the initial range mentioned in Table 5.2. One individual of the population consists of a set of values for respective parameters, within the initial limits.
2. The algorithm then creates a sequence of new populations. At each step, the algorithm uses the individuals in the current generation to create the next population. To create the new population, the algorithm performs the following steps:
ο· Score each individual of the current population by computing its fitness value by running the simulation program using the current values of the individual;
ο· Select the parents based on their fitness using Roulette wheel technique.
ο· The next generation is produced in three ways- (i) Directly pass some better fitness valued individuals to next generation (maximum 20% of the population); (ii) Produce children by crossover; (iii) Produce children by mutation. The next generation is produced only if there is an improvement to the objective function.
3. The algorithm is run for several generations and each generation produces an optimized value less than or equal to the earlier generationsβ value. Table 5.3 provides the optimized value by each generation and Fig. 5.7 highlights the optimization of objective function after each generation. In Table 5.3, best f(x) represents minimum value of optimization function amongst function values for all individuals in the population for the current generation, whereas mean f(x) represents their mean.
Fig. 5.7 Variation of objective function values after each generation of GA 1
1.1 1.2 1.3 1.4 1.5
0 5 10 15 20 25
Optimized value of objective function
No. of generations
Variation of objective function (model) with generations
4. From Fig. 5.7 and Table 5.3, it is observed that beyond 14th generation, the value of objective function remains consistent. Thus, the algorithm is terminated at the 25th generation. The obtained optimized values for πΆ1, πΆ2, πΆ3, πΆ4 and πΆ5 are 0.272, 0.085, 0.035, 2.57 and 0.556, respectively at the 25th generation.
5. The developed model is run with these optimized values and various parameters are calculated for further evaluation.
Table 5.3 Optimized value of objective function after each generation
Generation Best f(x)
Mean f(x)
No. of stalled generations
1 1.354 8.103 0
2 1.354 4.985 1
3 1.354 2.013 2
4 1.351 2.591 0
5 1.293 1.51 0
6 1.293 1.444 1
7 1.293 1.523 2
8 1.293 1.423 3
9 1.293 1.395 4
10 1.293 1.345 5
11 1.285 1.357 0
12 1.228 1.335 0
13 1.228 1.321 1
14 1.228 1.321 2
15 1.228 1.318 3
16 1.228 1.317 4
17 1.228 1.317 5
18 1.228 1.324 6
19 1.228 1.617 7
20 1.228 1.343 8
21 1.228 1.317 9
22 1.228 1.589 10
23 1.228 1.406 11
24 1.228 1.323 12
25 1.228 1.317 13