A Parameter Identification Scheme for Second-Order Highway Traffic Model Based on Differential Algebraic Methodology - Intelligent Computing for Sustainable Energy and Environment - page 88

λ i (5)
where T denotes the sampling time, k =0 ,...,K denotes the k -th time interval,
i =1 ,...,N denotes the i -thsectionofthehighway,and N is the number of
sections. The parameter meanings are shown in Table 1. According to previous
studies, the parameters τ , δ , κ and α are relatively deterministic and have a
small variation for a specific sections of highway, and the trac state estimation
and control results are known to be most sensitive to variations of the free speed
v f and critical density ρ cr . Therefore, this paper only treats the free speed and
critical density as the unknown model parameters that need to be identified. In
practice, other parameters can be determined by off-line model calibration and
regarded as known parameters in an on-line algorithm. The physical meanings of
the free speed and critical density are clearly illustrated through a fundamental
diagram, e.g. in the form of function (4). Fundamental diagram is a basic tool in
understanding the behavior of trac system through establishing the functional
relationship of the trac flow and the trac density of a given highway stretch.
One example of fundamental diagram is shown in Fig.2. Based on the fundamen-
tal diagram and trac measurements, it is indicated that highway segment has a
maximal flow rate, i.e. capacity. Once the trac flow reaches the regime behind
ρ cr , it becomes congested and throughput starts to decrease with the backward
propagation of shock waves. One of the major aims of the trac control is to
avoid the onset of congestions through changing the flow input at boundaries.

q i ( k )= ρ i ( k ) · v i ( k ) ·

Tabl e 1. Model parameter meanings

Parameter Unit Physical meaning
τ

hour Driver's reaction time

β % Turning rate of vehicles leaving off-ramps

κ

veh/km Additional tuning parameter

δ Ramp effect parameter

ρ cr veh/km Critical density

v f km/h free flow speed

r ( k ) veh/h On-ramp flow rate

s ( k ) veh/h Off-ramp flow rate

α Additional tuning parameter

3 Philosophy of Algebraic Identification Theory

Algebraic identification is based on elementary algebraic 4manipulations of the
following mathematical tools: differential algebra, operational calculus, and mod-
ule theory for a linear system [13][17][18][19].
Let us denote s as the differentiation operator and d/ds corresponds in the
time domain to the multiplication by t .

R ( s ) is the field of rational functions in

( s )[ ds ] is the set of linear differential the variable s with real coecients, and R