CN113268861B - Delay grid control model based on collaborative information transmission and verification method - Google Patents
Delay grid control model based on collaborative information transmission and verification method Download PDFInfo
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Abstract
The invention discloses a cooperative information transmission delay grid-based control model and a verification method, which belong to the technical field of traffic and comprise the following steps: s1: establishing a model; s2: carrying out linearization treatment; s3: analyzing linear stability; s4: carrying out nonlinear analysis; s5: simulating and outputting; the invention provides a grid control model, then obtains a new density equation through linearization treatment, analyzes linear stability in the equation to obtain a stable condition of a system in the environment, obtains a corresponding coexistence curve near a critical point of an unstable area through nonlinear analysis, performs numerical simulation, and finally performs system analysis on theoretical and simulation results to obtain a delay grid point control model based on cooperative information transmission and a verification method, so that system congestion can be effectively relieved, and energy consumption is reduced.
Description
Technical Field
The invention belongs to the technical field of traffic, and particularly relates to a cooperative information transmission delay grid-based control model and a verification method.
Background
With the continuous increase of the holding rate of the number of human automobiles, the system congestion problem shown on roads is increasingly severe, the increasingly severe system congestion problem is a major challenge facing people at present, and students also make intensive research on how to effectively relieve the system congestion problem all the time, and put forward models, such as micro models, macro models and grid models, for trying to solve the system congestion from different angles.
Until recently, nagatani proposed lattice hydrodynamic models by combining the ideas of microscopic models and macroscopic continuous medium kinetic models, and subsequently, a large number of lattice kinetic models appeared in succession, but these studies have not comprehensively considered the synergistic effects of flow delay and density delay, nor explored the impact of these factors on energy consumption control.
Disclosure of Invention
To solve the problems set forth in the background art described above. The invention provides a cooperative information transmission delay grid-based control model and a verification method, and has the characteristics of effectively relieving system congestion and reducing energy consumption.
In order to achieve the purpose, the invention provides the following technical scheme: a delay grid control model based on cooperative information transmission and a verification method thereof comprise the following steps:
s1: modeling
According to the lattice model proposed by Nagatani in 1998, a delay lattice point control model based on cooperative information transmission is proposed for the problem of traffic and density cooperative action which is not synthesized yet, namely:
wherein, λ a [ q ] j+1 -q j+1 (t-τ)]Represents the effect of the delay time of the system flow, lambda represents the induction coefficient of the flow delay, and kappa (rho) j+1 -ρ j+1 (t- τ)) represents the effect of the density delay time of the system, and κ represents the inductance of the density delay;
s2: linearization process
By eliminating the velocities in the equation, the density evolution is derived as follows:
and (3) carrying out linearization treatment on the density equation to obtain:
s3: linear stability analysis
The neutral stability condition of the system flow obtained by carrying out linear stability analysis by using a corresponding mathematical processing method is as follows:
the corresponding stabilization conditions were:
s4: non-linear analysis
Critical point (p) in the unstable region of the system c ,a c ) The nearby space defines a slow variable X and a time slow variable T:
X＝ε(j+bt)T＝ε 3 t 0＜ε＜＜1(7)，
where b is a constant to be determined, the density is expressed as:
ρ j (t)＝ρ c +εR(X,T) (8)，
by substituting formula (7) and formula (8) for formula (3) and expanding to the fifth order of ∈, it is possible to obtain:
the coefficients are obtained through mathematical processing:
to obtain the standard mKdV equation with high order minimums, the variables in equation (9) are transformed as follows:
thus, the mKdV equation containing the O (ε) term is obtained:
wherein:
if the O (epsilon) term in the formula is neglected, the formula
Normalized to the standard mKdV equation, its kink-anti-kink solution is:
kink-anti-kink wave propagation velocity is:
the kink-anti-kink solution of the mKdV equation is thus:
the density wave amplitude is:
accordingly, the coexistence curve can be expressed as
ρ＝ρ c ±A (22)；
S5: simulation and output
And performing numerical simulation according to theoretical analysis results obtained in the S3 and the S4, performing corresponding system analysis on the density and the flow provided by the model according to the results obtained by the numerical analysis, analyzing according to problems of energy, energy difference, energy loss, control and the like generated in actual vehicle running, and finally outputting a conclusion through system analysis and summary.
Further, in the present invention, in step S5, the conclusion is as follows:
s51: in the case of k =0, namely, without density delay, the traffic flow tends to be stable as the density delay inductance increases, namely, the traffic flow delay can alleviate traffic congestion;
s52: under the condition that the flow delay coefficient is not changed, the traffic flow tends to be in a stable state along with the increase of the density delay induction coefficient, namely, the density delay effect is also beneficial to improving the traffic stability;
s53: when the synergistic effect of the flow delay and the density delay is considered, the efficiency of the traffic system towards the steady state can be greatly improved, and thus the energy loss can be reduced macroscopically.
Compared with the prior art, the invention has the beneficial effects that:
the invention provides a grid control model, then obtains a new density equation through linearization treatment, analyzes linear stability in the equation to obtain a stable condition of a system in the environment, obtains a corresponding coexistence curve near a critical point of an unstable area through nonlinear analysis, performs numerical simulation, and finally performs system analysis on theoretical and simulation results to obtain a delay grid point control model based on cooperative information transmission and a verification method, so that system congestion can be effectively relieved, and energy consumption is reduced.
Drawings
FIG. 1 is a graph showing a combination of a stability curve and a coexistence curve according to the present invention;
FIG. 2 is a diagram of the spatial and temporal evolution of increased density with k =0 and λ according to the present invention;
FIG. 3 is a graph of the spatial and temporal evolution of increased density with λ =0.1 and κ according to the invention;
FIG. 4 is a density profile of the invention (κ =0, λ increasing) at 10300 time step;
FIG. 5 is a density profile of the invention (λ =0.1, κ increasing) at time 10300;
FIG. 6 is a flow chart of the present invention;
FIG. 7 is a plot of hysteresis (flow vs. density) of the present invention;
FIG. 8 is a graph of energy difference between adjacent times of the 25 th cell of the present invention;
fig. 9 is a graph of the energy difference between the 25 th cell and the front cell of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 1-9, the present invention provides the following technical solutions: a delay grid control model and a verification method based on collaborative information transmission comprise the following steps:
s1: modeling
According to the lattice model proposed by Nagatani in 1998, a coordinated information transmission delay lattice point control model is proposed for the problem of traffic and density synergy which is not integrated, namely:
wherein λ a [ q ] j+1 -q j+1 (t-τ)]Represents the effect of the delay time of the system flow, and lambda represents the inductance of the flow delay, kappa (rho) j+1 -ρ j+1 (t- τ)) represents the effect of the density delay time of the system, and κ represents the inductance of the density delay;
s2: linearization process
By eliminating the velocities in the equation, the density evolution is found as follows:
and (3) carrying out linearization treatment on the density equation to obtain:
s3: linear stability analysis
The neutral stability condition of the system flow obtained by linear stability analysis by using a corresponding mathematical processing method is as follows:
the corresponding stabilization conditions were:
as can be seen from equation (6), when κ and λ are both 0, the new model returns to the stable condition of Nagatani model, i.e. the neutral stability curve shown by the solid line in fig. 1, (a) is the case where κ =0 and λ increases, it is obvious that the neutral stability curve decreases with the increase of λ, i.e. the stable region is enlarged, and the system tends to be stable, which indicates that the flow delay induced effect is helpful to improve the system stability, and (b) is the case where λ =0.1 and λ increases, it is also clear that the neutral stability curve decreases with the increase of κ, i.e. the stable region is enlarged, which indicates that the density delay effect also has a positive effect on improving the system stability, and therefore, as is apparent from fig. 1, when κ and λ increase, the range of the delay time τ is enlarged, i.e. the system stability increases;
s4: non-linear analysis
Critical point (rho) in the unstable region of the system c ,a c ) The nearby space defines a slow variable X and a time slow variable T:
X＝ε(j+bt)T＝ε 3 t 0＜ε＜＜1 (7)，
where b is a constant to be determined, the density is expressed as:
ρ j (t)＝ρ c +εR(X,T) (8)，
by substituting formula (7) and formula (8) for formula (3) and expanding to the fifth order of ∈, we can obtain:
the coefficients are obtained through mathematical processing:
to obtain the standard mKdV equation with high order minimums, the variables in equation (9) are transformed as follows:
thus, an mKdV equation is obtained containing an O (ε) term:
wherein:
if the O (epsilon) term in the formula is neglected, the formula
Normalized to the standard mKdV equation, its kink-anti-kink solution is:
the propagation velocity of the kink-anti-kink wave is:
the kink-unkink solution from the mKdV equation is thus:
the density wave amplitude is:
accordingly, the coexistence curve can be expressed as
ρ＝ρ c ±A (22)，
The coexistence curve shown by the dotted line in fig. 1, (a) when κ and λ are both 0, the new model returns to the Nagatani original model, and matches the result of the linear analysis, it is obvious from (a) that when κ =0 and λ increases, the coexistence curve decreases and the stable region increases, which indicates that the flux delay effect has a positive effect on the system stability, while (b) when λ is a fixed value of 0.1 and λ increases, the coexistence curve also has a significant decrease trend and the stable region expands, which indicates that the density delay effect has a positive effect on improving the system stability, and the result obtained by combining fig. 1 indicates that both the flux delay response coefficient λ and the density delay response coefficient κ have a positive effect on improving the system stability, and when the flux delay and the density delay cooperate, the density delay effect plays an important role in improving the system stability;
s5: simulation and output
Carrying out numerical simulation on the formula (3), wherein a periodic boundary condition is adopted in the simulation process, and the initial parameters are set as follows:
N＝100,ρ c ＝ρ 0 ＝0.25,v max ＝2,a＝2.3，
the initial perturbation is set as:
ρ j (1)＝ρ j (0)＝ρ 0 -0.1,j＝50,
ρ j (1)＝ρ j (0)＝ρ 0 -0.1,j＝51,
ρ j (1)＝ρ j (0)＝ρ 0 ,others. (23)，
FIG. 2 is a graph of density spatio-temporal evolution after 10000 time steps when k =0 and the sensitivity coefficient λ of flow delay is increased, it can be seen that when a small perturbation is added to the system, kink-anti-kink density waves appear in all of (a), (b), (c) and (d), by contrast, both k and λ of (a) are 0, which is the simulation result of Nagatani model, and in (b), (c) and (d), k =0, only λ is increased, but it is still clear that the new model has a density spatio-temporal evolution graph in which the amplitude of density waves is gradually reduced with the increase of the density delay parameter λ, which indicates that the density delay effect can effectively suppress the system congestion,
further, when λ is set to be a fixed value, the density delay effect coefficient κ is increased to see whether the amplitude of the density wave can be reduced, and the purpose of suppressing the system congestion is also achieved, as is evident from (a), (b), (c) and (d) of fig. 3, when λ is set to be a fixed value of 0.1 and the density delay sensitivity coefficient κ is increased, the amplitude of the density wave is also continuously reduced, which indicates that the density delay effect also has a positive effect on the stability of the system, and particularly, (d), that is, when λ =0.1 and κ =0.3, the amplitude of the density wave tends to 0, and the system reaches a steady state, which indicates that the system can reach a steady state more quickly under the synergistic effect of the flow delay and the density delay,
further research shows that the density distribution diagram of 100 grids at the time step of t =10300 is simulated by adopting a mode of control variables, k =0 and lambda is increased, as is obvious from the attached figure 4, the fluctuation amplitude of the density distribution of 100 grids at the time step of t =10300 is reduced along with the increase of lambda, and consistent with the conclusion, the flow delay effect can improve the stability of the system,
further, λ is set as a fixed value to increase κ, and it is obvious from fig. 5 that λ is set as a fixed value at time step 10300, but the fluctuation amplitude of the density distribution of 100 grids still decreases with the increase of κ, which is consistent with the results of the previous analysis, the density delay effect has the effect of improving the stability of the system, and when κ =0.3 and λ =0.1, the density distribution amplitude tends to 0, the system reaches a steady state, which indicates that the system can more quickly approach the steady state under the synergistic effect of the flow delay and the density delay effect,
then analyzing the flow distribution of the system, and giving flow fluctuation graphs of 200 time steps and 10100-10300 time steps at the beginning of a certain lattice point simulation, in fig. 6, simulating the flow of the 25 th lattice point, wherein (a) and (b) are respectively k =0, and the flow fluctuation conditions of the 200 time steps and 10100-10300 time steps at the beginning of the system under the condition that the lambda is increased, regardless of the evolution process of the 200 time steps or 10100-10300 time steps at the beginning of the simulation, when we set the k =0 and the flow delay induction coefficient lambda is increased, the fluctuation amplitude of the flow curve is gradually reduced, which shows that the flow delay effect is favorable for increasing the stability of the system, and (c) and (d) are respectively λ =0.1, and under the condition that the kappa is increased, the flow fluctuation conditions of the 200 time steps and 10100-10300 time steps at the beginning of the system under the condition that the lambda is constant 0.1, and the fluctuation amplitude of the flow curve is also reduced along with the increase of the density delay induction coefficient kappa, which shows that the congestion delay effect is also improved,
therefore, when the density delay effect and the flow delay effect act together, the efficiency of the system which tends to be stable can be greatly improved,
fig. 7 illustrates a hysteresis loop formed by a flow-density space, whose size can indicate the stability of the traffic system, and in order to verify the effects of the flow delay effect and the density delay effect, as in the above method, one of the parameters needs to be fixed and the other needs to be controlled,
as can be seen from fig. 7 (a), when the new model k =0 and λ is increased, the size of the hysteresis loop is gradually reduced, which indicates that the flow delay effect can reduce the hysteresis phenomenon of the system, and (b) shows that when the new model λ =0 and λ is increased, the area of the hysteresis loop is also contracted, and particularly when λ =0.1 and κ =0.3, the hysteresis loop is contracted to a point, and then the system tends to be in a stable state, which indicates that the flow delay and the density delay cooperate to effectively reduce the hysteresis phenomenon of the system, thereby improving the stability of the system,
according to the relation between the speed and the kinetic energy, if the system is stable, the variation of the speed along with the time is small, but if the system is unstable, the variation of the speed along with the time becomes large, so that the generated energy consumption and pollution are huge, the condition of traffic jam is represented by the energy variation between a certain grid point t time and the next time (t + 1), and the energy consumption equation is assumed as follows:
the energy consumption of the grid points varies as follows:
under the condition that other parameters are not changed, the 25 th grid is simulated, the energy change of the t moment and the next moment is explored, fig. 8 shows the system energy consumption change of the 200 time step before the early effect of the system and the 10100-10300 time step of the stable effect of the system, and (a) and (b) show that the system energy consumption change of the 200 time step before the early effect of the system and the 10100-10300 time step of the stable effect of the system are changed, when k =0 and λ is increased, the change situation of the energy difference curve amplitude of the adjacent moment is obviously seen, when λ is increased, the curve amplitude is reduced, which shows that the flow delay effect is smaller for reducing the system energy consumption change amplitude, and (c) and (d) show that the system energy consumption change of the 200 time step before the early effect of the system and the 10100-10300 time step of the stable effect of the system is reduced, when λ =0.1 is fixed, k is increased, the energy difference curve amplitude is also obviously reduced, which shows that the density delay effect is beneficial for reducing the energy consumption change of the system, and the stability of the system is increased,
in combination with the above two points, especially (d) when k =0.3 and λ =0.1, the amplitude of the energy consumption variation curve approaches 0, that is, the system is in a steady state, which indicates that when the system is under the synergistic effect of the dual effects of the density delay and the flow delay, the system can greatly improve the efficiency of reducing the energy consumption variation,
the energy consumption change of two adjacent grid points is further researched to represent the condition of system congestion of the whole system:
the energy change of the 25 th grid and the grid in front of the 25 th grid (namely, the 26 th grid) is selected for simulation, and fig. 9 shows that the energy difference curve fluctuates at about the 50 th time step, which indicates that the added small disturbance has an interference effect on the system, and the following conclusion is drawn from fig. 9:
(a) And (b) showing that the system is 200 time steps before the early effect and 10100-10300 time steps before the early effect, when k =0 and λ is increased, the amplitude of the change of the energy consumption change curve amplitude of the adjacent grids is reduced, which shows that the flow delay effect has a positive effect on reducing the energy loss of the system.
(c) And (d) showing that when λ =0.1 is fixed, κ increases, and the amplitude of the energy difference curve of adjacent lattices also shows a significantly decreasing trend, which indicates that the density delay effect has a positive effect on reducing the variation of the system energy consumption, increasing the stability of the system,
in combination with the above two points, especially (d) when k =0.3 and λ =0.1, the amplitude of the energy consumption variation curve of the adjacent grids approaches to 0, that is, the system is in a steady state, which shows that when the system is under the synergistic effect of the dual effects of density delay and flow delay, the system can greatly improve the efficiency of reducing energy consumption,
by two different energy loss analysis methods of adjacent time and adjacent grid, and by adding density delay and flow delay cooperative information transmission control, the energy consumption change trend can be effectively reduced, the energy consumption change control result is consistent with the analysis results of a density space-time diagram, a flow curve and a hysteresis loop diagram,
in conclusion, the density delay and the flow delay are both coordinated with the information transmission delay control, so that the efficiency of the system tending to the steady state can be greatly improved, the stability of the system is favorably improved, and the theoretical analysis result is consistent with the numerical simulation result.
Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.
Claims (1)
1. A delay grid control model and a verification method based on cooperative information transmission are characterized by comprising the following steps:
s1: modeling
According to the lattice model proposed by Nagatani in 1998, a coordinated information transmission delay lattice point control model is proposed for the problem of traffic and density synergy which is not integrated, namely:
wherein λ a [ q ] j+1 -q j+1 (t-τ)]Represents the effect of the delay time of the system flow, and lambda represents the inductance of the flow delay, kappa (rho) j+1 -ρ j+1 (t- τ)) represents the effect of the density delay time of the system, and κ represents the inductance of the density delay;
s2: linearization process
By eliminating the velocities in the equation, the density evolution is found as follows:
and (3) carrying out linearization treatment on the density equation to obtain:
s3: linear stability analysis
The neutral stability condition of the system flow obtained by linear stability analysis by using a corresponding mathematical processing method is as follows:
the corresponding stabilization conditions were:
s4: non-linear analysis
Critical point (p) in the unstable region of the system c ,a c ) The nearby space defines a slow variable X and a time slow variable T:
X＝ε(j+bt)T＝ε 3 t 0＜ε＜＜1 (7)，
where b is a constant to be determined, the density is expressed as:
ρ j (t)＝ρ c +εR(X,T) (8)，
by substituting formula (7) and formula (8) for formula (3) and expanding to the fifth order of ∈, it is possible to obtain:
the coefficients are obtained through mathematical processing:
to obtain the standard mKdV equation with high order minimums, the variables in equation (9) are transformed as follows:
thus, an mKdV equation is obtained containing an O (ε) term:
wherein:
if the O (epsilon) term in the formula is neglected, the formula
Normalized to the standard mKdV equation, its kink-anti-kink solution is:
the propagation velocity of the kink-anti-kink wave is:
the kink-anti-kink solution of the mKdV equation is thus:
the density wave amplitude is:
accordingly, the coexistence curve can be expressed as
ρ＝ρ c ±A (22)；
S5: simulation and output
Performing numerical simulation according to theoretical analysis results obtained in S3 and S4, performing corresponding system analysis on density and flow provided by the model according to the results obtained by the numerical analysis, analyzing according to energy, energy difference, energy loss and control problems generated in actual vehicle running, and finally outputting a conclusion through system analysis and summarization;
in step S5, the conclusion is as follows:
s51: in the case of k =0, namely, without density delay, the traffic flow tends to be stable as the density delay inductance increases, namely, the traffic flow delay can alleviate traffic congestion;
s52: under the condition that the flow delay coefficient is not changed, the traffic flow tends to be in a stable state along with the increase of the density delay induction coefficient, namely, the density delay effect is also beneficial to improving the traffic stability;
s53: when the synergistic effect of the flow delay and the density delay is considered, the efficiency of the traffic system towards the steady state can be greatly improved, and thus the energy loss can be reduced macroscopically.
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