Ramp Metering

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Introduction ​

ALINEA ALINEA is a simple but effective ramp metering strategy that utilizes a local traffic-responsive feedback loop to maintain maximum throughput at the downstream merge area of an on-ramp. The ALINEA algorithm calculates the metering rate at each time step by computing the difference between the downstream desired occupancy and the aggregated downstream occupancy at each time step, multiplying that difference by a regulator parameter, and adding that result to the metering rate of the previous time step. The following equation implicitly calculates the metering rate:

$$ r(k) = r(k-1) + {K_{R}}\times \big({O_{DES}}-{O_{OUT}}(k)\big) $$

where

Detectors are placed on every freeway lane 500 m downstream of the acceleration lane. The detector data for each lane is aggregated together at each time step to calculate the occupancy value that is compared to the desired occupancy. The on-ramp contains two detectors: a presence detector directly behind the ramp meter signal controller and a passage detector immediately after the ramp meter signal controller. The presence detector ensures that the signal rests on red when no vehicles are detected and ensures that the ramp metering algorithm is being applied when a vehicle is detected. The passage detector ensures that the system does not break down in case vehicles stay queued up behind the signal controller. The figure below illustrates the ALINEA setup used in SwashSim.

The ALINEA flow chart below reads as follows:



where

Fuzzy Logic ​​ Another method through which ramp metering can be integrated into SwashSim is fuzzy logic. Fuzzy logic is more like human language than traditional logical algorithms such as ALINEA. Rather than forcing a yes or a no, fuzzy logic utilizes the linguistic variables such as very low, low, medium, high, and very high. To process the approximate information for the real world, the fuzzy logic controller is actually a set of linguistic control rules related by the concepts of fuzzy implication and the rule of inference. In SwashSim, detectors are placed upstream, downstream, and on the on-ramp. The inputs for the upstream detectors are occupancy, flow, and speed. The inputs for the downstream detectors are speed and v/c ratio. The input for the ramp detectors is occupancy. A graphic representation of the fuzzy logic setup in SwashSim is shown in the figure below.



The graphic below is an example of a typical fuzzy logic system. The following is a description of the major components of the fuzzy logic system:


 * Fuzzifier: The fuzzifier translates each input into a set of fuzzy variables via membership functions.
 * Rules: The rules are the set of regulations that are based on expert opinions.
 * Inference: The inference stage involves applying fuzzy operators and implication methods to the rule base, allowing the fuzzy inputs to be converted into one fuzzy output.
 * Defuzzifier: The defuzzifier produces a crisp output based upon the fuzzy output.



Fuzzy Sets The fuzzification process involves translating each input into a fuzzy set with understandable terms such as "low" and "high". This is done using membership functions, which are different for each input type. Upstream mainline inputs use the Gaussian function, downstream mainline and ramp inputs use the Sigmoid function, and the output uses the triangular function. The full list of inputs along with their specific fuzzy sets and membership functions is shown below.

Inputs (Aggregated Performance Measures)

Aggregated Upstream Mainline Occupancy Considering the aggregated upstream mainline occupancy varies from 0 to 30% and is described as 3 Gaussian fuzzy sets: Low, medium, and high. The overlap is 50% and the parameters (center point and the sigma value) are found below.



$$ Guassian(x;c,σ) = e^{-\frac{1}{2}(\frac{x-c}{σ})^{2}} $$

Aggregated Upstream Mainline Flow Considering the upstream mainline flow varies from 0 to 2,000 veh/h/ln and is described as 3 Gaussian fuzzy sets: Low, medium, and high. The overlap is 50% and the parameters (center point and the sigma value) are found below. $$ Guassian(x;c,σ) = e^{-\frac{1}{2}(\frac{x-c}{σ})^{2}} $$

Aggregated Upstream Mainline Average Speed Considering the aggregated upstream mainline average speed varies from 0 to 110 mi/h and is described as 3 Gaussian fuzzy sets: Low, medium, and high. The overlap is 50% and the parameters (center point and the sigma value) are found using the plot shown below.



$$ Guassian(x;c,σ) = e^{-\frac{1}{2}(\frac{x-c}{σ})^{2}} $$

Aggregated Downstream Mainline Average Speed Aggregated downstream mainline average speed varies from 0 to 110 mi/h. It is described as only 1 Sigmoid fuzzy set which is very low, and the parameters (center point and the sigma value) are found using the plot shown below.



$$ Sig(x;a,c) = \frac{1}{1+e^{-a(x-c)}} $$

Aggregated Downstream Mainline v/c Aggregated Downstream mainline v/c ratio varies from 0 to 1. It is described as only 1 Sigmoid fuzzy set which is very high, and the parameters (center point and the sigma value) are found using the plot shown below.



$$ Sig(x;a,c) = \frac{1}{1+e^{-a(x-c)}} $$

Aggregated Ramp Demand/Queue Occupancy Both the aggregated ramp demand and queue occupancy vary from 0 to 50% described as only 1 Sigmoid fuzzy set which is very high, and the parameters (center point and the sigma value) are found using the plot shown below.



$$ Sig(x;a,c) = \frac{1}{1+e^{-a(x-c)}} $$

Inference Fuzzy logic relationship will be defined as a list of if-then pairs between the inputs condition and the outputs responses. The input conditions is a "premise" and the out responses is a "consequent". For example, as shown in the diagram below, the Rule Condition starts with a If and the Rule Outcome starts with a then. Within the rule condition, there is a AND or OR operation. The AND operation is analogous to the intersection of fuzzy sets, which takes the minimum value of given degree of membership which is between 0-1. The OR operation is analogous to the union of fuzzy sets, which takes the maximum value of given degree of membership which is between 0-1.

Rules 1 to Rule 3 The purpose of rule 1 through rule 3 is to form a complete rule base, which means at least one of the rules would work since the whole occupancy range is covered. Rule 4 to Rule 7 The rules couple speeds with either occupancy or flow to generate metering rates according to the fundamental diagram of traffic flow Rule 8 The purpose of this rule is to prevent the formation of downstream congestion rather just simply react to it. Volume/capacity-ratio (v/c-ratio) calculated with the historical measured maximum flow rate of downstream can be seen as a prediction of the downstream bottleneck behavior. Rule 9 Rule 9 is to prevent the excessive queue formation and to avoid a spillback onto the arterial street road on the basis of information collected from queue detector. Rule weights The rule weight is to stress the priority of each rule. Rule weighting scheme is flexible for different applications.

Output (Ramp Metering)

The metering rate varies from 240 veh/h to 900 veh/h, and is described as 3 triangular fuzzy sets: Low, medium, and high. The overlap is 50%.



$$ Triangle(x;a,b,c)=\left\{\begin{array}{l} 0,&\mbox{𝑥 ≤ 𝑎 }\\ \frac{x-a}{b-a},&\mbox{𝑎 ≤ 𝑥 ≤ b}\\ \frac{c-x}{c-b},&\mbox{𝑏 ≤ 𝑥 ≤ 𝑐}\\ 0,&\mbox{𝑥 ≥ 𝑐} \end{array}\right. $$

Defuzzification The defuzzication process is to convert a fuzzy output variable into a crisp value (metering rate). The centroid method is commonly used for the defuzzification process. In practice, a discrete fuzzy centroid equation is sued to replace the continuous centroid equation since it is easier to calculate. The equation is shown below.

$$ Metering Rate = \frac{\sum^3_{i=1}{w_{i}}{c_{i}}{I_{i}}}{\sum^3_{i=1}{w_{i}}{c_{i}}} $$

where

$$w(Low) = RuleOutcome[3] \times 2 + RuleOutcome[4] \times 2 + RuleOutcome[8] \times 3 $$ $$w(Medium) = RuleOutcome[2] \times 1.5 + RuleOutcome[5] \times 1 $$ $$w(High) = RuleOutcome[1] \times 21.5 + RuleOutcome[6] \times 1 + RuleOutcome[7] \times 1 + RuleOutcome[9] \times 3 $$ $$c(Low) = \frac{c-b}{3} + a $$ $$c(Medium) = b $$ $$c(High) = c - \frac{c-b}{3} $$ $$I(Low) = 165 $$ $$I(Medium) = 330 $$ $$I(High) = 165 $$

Test Networks ​ Diamond Interchange Loop Interchange

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