CEE 764 – Fall 2010 CEE 764 – Fall 2010 Topic 4 Topic 4 Bandwidth Optimization Bandwidth Optimization
CEE 764 – Fall 2010CEE 764 – Fall 2010
Bandwidth Optimization MethodologyBandwidth Optimization Methodology
Half-IntegerHalf-Integer Algorithms by Algorithms by Brooks Brooks and and LittleLittle Based on minimizing Based on minimizing interferencesinterferences Providing equal bandwidth solution when Providing equal bandwidth solution when
speeds are the same for both directionsspeeds are the same for both directions Calculating interference for each individual Calculating interference for each individual
intersection based on a referencing intersection based on a referencing intersection, intersection, mm
The referencing intersection The referencing intersection mm is the one is the one that has the minimum greenthat has the minimum green
CEE 764 – Fall 2010CEE 764 – Fall 2010
Half-Integer Offset (Center Red)Half-Integer Offset (Center Red)
Rm
Distance
m j
Ri
Gm
Gi
Tim
e
Rj
Gj
i
Sim
ult
aneo
us
Alt
ern
ate
CEE 764 – Fall 2010CEE 764 – Fall 2010
Simultaneous Offset - Forward Simultaneous Offset - Forward (Lower Interference)(Lower Interference)
Rm
Distance
m j
Rj
Gm Gj
Tim
e
IL
mjjLmjm NCRITR 5.05.0
Lmj CKNC 5.0
mjjmLL TRRCKI )(5.05.0
)(5.0 jmmjmj TTT 0
CEE 764 – Fall 2010CEE 764 – Fall 2010
Simultaneous Offset - Forward Simultaneous Offset - Forward (Upper Interference)(Upper Interference)
Rm
Distance
m j
Rj
Gm Gj
Tim
e
IU
mjjUmjm NCRITR 5.05.0
mjjmUU TRRCKI )(5.05.0
)(5.0 jmmjmj TTT
Umj CKNC 5.00
CEE 764 – Fall 2010CEE 764 – Fall 2010
Simultaneous Offset - Forward Simultaneous Offset - Forward (No Interference – Slack Time)(No Interference – Slack Time)
Rm
Distance
m j
Rj
Gm Gj
Tim
e
IS
0
CEE 764 – Fall 2010CEE 764 – Fall 2010
Simultaneous Offset - Forward Simultaneous Offset - Forward (No Interference – Slack Time)(No Interference – Slack Time)
Rm
Distance
m j
Rj
Gm Gj
Tim
e
IS
IS
CEE 764 – Fall 2010CEE 764 – Fall 2010
Alternate Offset - Forward Alternate Offset - Forward (Lower Interference)(Lower Interference)
Rm
Distance
m j
RjGm
Gj
Tim
e IL
mjjLmjm NCRITR 5.05.0
Lmj CKNC 5.0
mjjmLL TRRCKI )(5.05.0
)(5.0 jmmjmj TTT
0
CEE 764 – Fall 2010CEE 764 – Fall 2010
Alternate Offset - Forward Alternate Offset - Forward (Upper Interference)(Upper Interference)
Rm
Distance
m j
RjGm
Gj
Tim
e
IU
mjjUmjm NCRITR 5.05.0
mjjmUU TRRCKI )(5.05.0
)(5.0 jmmjmj TTT
0
Umj CKNC 5.0
CEE 764 – Fall 2010CEE 764 – Fall 2010
Alternate Offset - Forward Alternate Offset - Forward (No Interference)(No Interference)
Rm
Distance
m j
RjGm
Gj
Tim
e
IS
CEE 764 – Fall 2010CEE 764 – Fall 2010
Simultaneous Offset - Backward Simultaneous Offset - Backward (Lower Interference)(Lower Interference)
Rm
Distance
j m
Rj
GmGjTim
e
IL
mjjLmjm NCRITR 5.05.0
Lmj CKNC 5.0
)()(5.05.0 mjjmLL TRRCKI
)(5.0 jmmjmj TTT
0
CEE 764 – Fall 2010CEE 764 – Fall 2010
Summary of EquationsSummary of Equations
LjmmjjmL CKTTRRI 5.0)(5.0)(5.0
UjmmjjmU CKTTRRI 5.0)(5.0)(5.0
CCKTT jmmjmj 5.0)(5.00
jjmS GGGI
* KL, KU, and K are integers** K is even - simultaneous; K is odd - alternate
maxIGII mUS
maxIGII mLS
CEE 764 – Fall 2010CEE 764 – Fall 2010
Summary of EquationsSummary of Equations
LjmmjjmL CKTTRRI 5.0)(5.0)(5.0
UjmmjjmU CKTTRRI 5.0)(5.0)(5.0
CCKTT jmmjmj 5.0)(5.00
0 jmSj GGIG
* KL, KU, and K are integers** K is even - simultaneous); K is odd - alternate
maxIGII mUS
maxIGII mLS
CEE 764 – Fall 2010CEE 764 – Fall 2010
Summary of Brooks’ AlgorithmSummary of Brooks’ Algorithm
a. Find intersection “a. Find intersection “mm” with smallest green” with smallest greenb. Travel times to right of “b. Travel times to right of “mm” (forward) are positive and to ” (forward) are positive and to the left (backward) are negativethe left (backward) are negativec. Calculate least allowable Ic. Calculate least allowable ILL and I and IUU for each intersection for each intersectiond. Perform total interference minimizationd. Perform total interference minimization
e. Identify the optimal progression band and offsetse. Identify the optimal progression band and offsets
f. Construct the time-space diagramf. Construct the time-space diagramg. Adjust split of directional bandwidth if desiredg. Adjust split of directional bandwidth if desired
][ ,, jLjUTotal IMaxIMaxMinIMin
Totalma IGB
CEE 764 – Fall 2010CEE 764 – Fall 2010
Graphical IllustrationGraphical Illustration
Distance
1 2
IU,1
Tim
e
m 3
IU,2IL,3
3,2,1, ),( LUUTotal IIIMaxI
CEE 764 – Fall 2010CEE 764 – Fall 2010
ExampleExample
LjmmjjmL CKTTRRI 5.0)(5.0)(5.0 UjmmjjmU CKTTRRI 5.0)(5.0)(5.0
CCKTT jmmjmj 5.0)(5.00 mjmS GGGI
Int. #Int. # GGjj RRjj DDmjmj -0.5(R-0.5(Rmm-R-Rjj)) TTmjmj 0.5(T0.5(Tmjmj+T+Tjmjm)) -I-ISS IImaxmax KKUU IIUU KKLL IILL θθmjmj
11 4545 3535 15001500 -2.5-2.5 -25.56-25.56 -25.56-25.56 -5-5 4040
2*2* 4040 4040 -- -- -- -- -- -- -- -- -- -- --
33 4545 3535 330330 -2.5-2.5 5.625.62 5.625.62 -5-5 00
11
-1-1
3.1203.120
-36.88-36.88
43.1243.12
00
11
-8.12-8.12
31.8731.87
44 5050 3030 18301830 -5.0-5.0 31.1931.19 31.1931.19
55 4545 3535 48304830 -2.5-2.5 82.3182.31 82.3182.31 -5-5
Speed is 40 mph
maxIGII mUS maxIGII mLS
CEE 764 – Fall 2010CEE 764 – Fall 2010
Illustration of Intersection #2 and #4 Illustration of Intersection #2 and #4 (Simultaneous Offset)(Simultaneous Offset)
Rm
Distance
m=2 #4
Rj
Gm
Gj
Tim
e
26.19
CEE 764 – Fall 2010CEE 764 – Fall 2010
Illustration of Intersection #2 and #4 Illustration of Intersection #2 and #4 (Alternate Offset)(Alternate Offset)
Rm
Distance
m=2 #4
RjGm
Gj
Tim
e 3.81
CEE 764 – Fall 2010CEE 764 – Fall 2010
Illustration of Intersection #2 and #5 Illustration of Intersection #2 and #5 (Simultaneous – Slack Times)(Simultaneous – Slack Times)
Rm
Distance
m=2 #5
Rj
Gm
Gj
Tim
e
4.81
0.19
CEE 764 – Fall 2010CEE 764 – Fall 2010
Example (continued)Example (continued)
Int. #Int. # GGjj IIUU IILL
11 4545 11.9411.94 23.0623.06
2*2* 4040 00 00
33 4545 3.123.12 31.8731.87
44 5050 26.1926.19 3.813.81
55 4545 -0.19-0.19 -4.81-4.81
Rank #Rank # Int. #Int. # IIUU IILL
11 44 26.1926.19 3.813.81
22 11 11.9411.94 23.0623.06
33 33 3.123.12 31.8731.87
44 2=m2=m 00 00
55 55 -0.19-0.19 -4.81-4.81
CEE 764 – Fall 2010CEE 764 – Fall 2010
Example (continued)Example (continued)
Intersection by RankIntersection by Rank IITotalTotal
11 22 33 44 55
UU UU UU UU UU 26.1926.19
LL UU UU UU UU 11.94 + 3.81 = 15.7511.94 + 3.81 = 15.75
LL LL UU UU UU
LL LL LL UU UU
LL LL LL LL UU
LL LL LL LL LL
InterferenceInterference RankRank Int. #Int. # K-valueK-value OffsetOffset θθmjmj
(Reference Start of Green)(Reference Start of Green)
LL 11 44 11 AlternateAlternate
UU 22 11 11 AlternateAlternate
UU 33 33 00 SimultaneousSimultaneous
UU 44 2=m2=m -- --
UU 55 55 22 SimultaneousSimultaneous
CEE 764 – Fall 2010CEE 764 – Fall 2010
Offset to Start of GreenOffset to Start of Green((AlternateAlternate))
Rm
Distance
m
Rj
Gm
θR,mj
Tim
e θG,mj
22,,mj
mjRmjG
RR
CEE 764 – Fall 2010CEE 764 – Fall 2010
Offset to Start of GreenOffset to Start of Green((SimultaneousSimultaneous))
Rm
Distance
m
Rj
Gm
θR,mj = 0
Tim
e θG,mj
22,,mj
mjRmjG
RC
R
CEE 764 – Fall 2010CEE 764 – Fall 2010
Bandwidth with LT PhasesBandwidth with LT Phases
Intersection 1 Intersection 2
Time
Space
CEE 764 – Fall 2010CEE 764 – Fall 2010
Major TermsMajor Terms
Cycle
Link length in-between
Phase B
OB through
IB left turn
IB through
OB left turn
CEE 764 – Fall 2010CEE 764 – Fall 2010
Bandwidth MaximizationBandwidth Maximization
Inbound Bandwidth
Outbound Bandwidth
Outbound
CEE 764 – Fall 2010CEE 764 – Fall 2010
Bandwidth MaximizationBandwidth Maximization
Inbound Bandwidth
Outbound Bandwidth
CEE 764 – Fall 2010CEE 764 – Fall 2010
Phasing SequencePhasing Sequence
Leading Lead-LagLag-LeadLagging
16 Combinations4 types of left turn sequence for each intersection
CEE 764 – Fall 2010CEE 764 – Fall 2010
MethodologyMethodology
Maximum Bandwidth = Bo + BiBo <= Gomin
Bi <= Gimin
Bmax = Go,min +Gi,min – Ii,min
X
j
CEE 764 – Fall 2010CEE 764 – Fall 2010
MethodologyMethodology
Bmax = Go,min +Gi,min – Ii,min
Exception:Bmax = Gi,min
Bmax = Constant
Interference
CEE 764 – Fall 2010CEE 764 – Fall 2010
MethodologyMethodology
Bmax = Go,min +Gi,min – Ii,min
Exception:Bmax = Gi,min
Bmax = Constant
Interference
CEE 764 – Fall 2010CEE 764 – Fall 2010
Upper InterferenceUpper Interference
Iup
X – Intersection that has the smallest inbound green
CEE 764 – Fall 2010CEE 764 – Fall 2010
Iujp = Gix - (-Rxn +Txj + Rjp + Gij + Tjx ) mod C
Iujp
0
-Rxn +Txj + Rjp + Gij
(-Rxn +Txj + Rjp + Gij + Tjx )
-Rxn +Txj
-Rxn +Txj + Rjp
-Rxn
Gix
Gix+n*CL
Intersection X Intersection j
-Rjp
Rxn
CEE 764 – Fall 2010CEE 764 – Fall 2010
Iujp = Gix - (-Rxn +Sox +Txj + Rjp + Gij + Tjx ) mod C
Iujp
0
-Rxn +Sox+Txj + Rjp + Gij
(-Rxn +Sox+Txj + Rjp + Gij + Tjx )
-Rxn +Sox+Txj
-Rxn +Sox+Txj + Rjp
-Rxn
Gix
Gix+n*CL
Intersection X Intersection j
-Rjp
Rxn-Rxn+Sox
CEE 764 – Fall 2010CEE 764 – Fall 2010
No valid upper interference
Iujp
(-Rxn +Sox+Txj + Rjp + Gij + Tjx )
Intersection X Intersection j
-Rjp
If Gix - (-Rxn +Sox+Txj + Rjp + Gij + Tjx )<-Sij, no valid upper interference.
CEE 764 – Fall 2010CEE 764 – Fall 2010
T.T.=34 sec
61
38
45
16
55*
41
30
2216
55*
41
-30
0
30+34+22
30
30+34
30+45-34
Iu = 55-41=14
x
j
30+34+22+61
30+34+22+61+34=181=181-140=41
Iujp = Gix - (-Rxn + Txj + Rjp + Gij + Tjx ) mod C
CEE 764 – Fall 2010CEE 764 – Fall 2010
T.T.=34 sec
61
34
45
20
55*
41*
34
18
20
55*
41*
34
0
-(-34)=34
34-34
34-34+18
34-34+18+61-34=45
Iu = 55-45=10
x
j
34-34+18+61
Iujp = Gix - (-Rxn + Sox –Txj + Rjp + Gij -Tjx ) mod C
Intersection j has the smallest outbound green
Sox
CEE 764 – Fall 2010CEE 764 – Fall 2010
T.T.=34 sec
61
38
41*
20
55*
45
30
18
20
55*
45
30
0
-(-30)=30
30+4-34
30+4-34+18
30+4-34+18+61-34=45
Iu = 55-45=10
x
j
30+45
30+4-34+18+61
Iujp = Gix - (-Rxn + Sox –Txj + Rjp + Gij -Tjx ) mod C
Intersection j has the smallest outbound green
Sox30+4
CEE 764 – Fall 2010CEE 764 – Fall 2010
ILjp = (-Rxn +Txj – Sj+ Rjp + Tjx) mod C
ILjp
Slack Time, Sj
Sj = Goj – Go,minRxn
0
Go,min
x j
CEE 764 – Fall 2010CEE 764 – Fall 2010
T.T.=50 sec
60
30
45
24
55*
41
38
15
24
55*
41
-38
0
38+50-4+15
38
38+50
IL = 9
x
j
38+50-4+15+50=149
ILjp = (-Rxn +Txj – Sj+ Rjp + Tjx) mod C
38+50-4
CEE 764 – Fall 2010CEE 764 – Fall 2010
T.T.=34 sec
61
38
45
16
55*
41
30
22
16
55*
41
-30
0
30+34-4
30
Iu = 24+55-61=18
x
j
140-116=24>>6
No valid lower interference
30+34
30+34-4+22+61
30+34-4+22+34=116
30+34-4+22
Lower interference calculation causes upper interference occurring. Intersection j must move up by 4 sec to reduce upper interference. Inbound band = 55-14=41
24
CEE 764 – Fall 2010CEE 764 – Fall 2010
Lower InterferenceLower Interference
C - ILjp <= (Gij – Gix)=Sj
C ILjp
Gij
Gij – Gix
ILjp >= C – Sj
CEE 764 – Fall 2010CEE 764 – Fall 2010
T.T.=95 sec
30
41*
55*
16
18
45
38
0
2-sec slack
-16
-16+95-4-38+95=132
x
j
65
ILjp = (-Rxn + Txj – Soj+ Rjp + Tjx) mod C
-16+95
30
41*
55*
16
0
x
-16+95-4
-16+95-4-38
140-132=8<10No lower interference
CEE 764 – Fall 2010CEE 764 – Fall 2010
ExampleExample
There are three coordinated intersections A, B, and C with a cycle length of 140 sec. Travel times and phasing sequence and splits are shown below. Determine the maximum bandwidth and offsets.
T.T.=34 sec T.T.=56 sec
61*
25
38
41*
16
64
18
64
50
30
57
18
A B C
CEE 764 – Fall 2010CEE 764 – Fall 2010
ExampleExampleT.T.=34 sec T.T.=56 sec
61*
25
38
41*
16
64
18
64
50
30
57
18
A B C
B: Iu =61-(-18+34-30+64+34)=61-84=-23, not valid (23>3); IL = (-18+34-9-30+34)=11*
C: Iu =61-(-18+90-25+64+90)=61-61=0*, IL = (-18+90-16-25+90)=121, not valid (140-121=19>3)
Therefore, it has a 11 lower interference caused by B and zero interference by C. Bandwidth = 41+(61-11)=91
CEE 764 – Fall 2010CEE 764 – Fall 2010
Homework (C=60)Homework (C=60)
T.T.=36 sec T.T.=51 sec
30
15
25
10
30
30
1010
10
25*
20
15
20
25
30
15
25
12
26
13
T.T.=27 sec T.T.=41 sec
CEE 764 – Fall 2010CEE 764 – Fall 2010
HomeworkHomework
There are three coordinated intersections A, B, and C with a cycle length of 140 sec. Travel times and phasing sequence and splits are shown below. Determine the maximum bandwidth and offsets.
T.T.=34 sec T.T.=56 sec
61
38
41*
20
55*
45
3018
25
18
64
57
CEE 764 – Fall 2010CEE 764 – Fall 2010
HomeworkHomework
There are two intersections A and B, running coordinated control for the major street, with the cycle length of 60 sec.
For intersection A, The outbound through time is 15 sec, and the inbound through time is 20 sec; the outbound left-turn interval is 10 sec, and the inbound left-turn interval is 15 sec.
For intersection B, The outbound through time is 20 sec, and the inbound through time is 25 sec; the outbound left-turn interval is 15 sec, and the inbound left-turn interval is 20 sec.
The distance between the two intersections is 1320 feet. Questions: Questions: Assuming intersection A is running leading left turns, and Intersection B is running leading left Assuming intersection A is running leading left turns, and Intersection B is running leading left
turn for outbound direction and lagging left turn for inbound direction. At the speed limit of 30 mph turn for outbound direction and lagging left turn for inbound direction. At the speed limit of 30 mph for both directions, what is the maximum total bandwidth (of both directions) between the two for both directions, what is the maximum total bandwidth (of both directions) between the two intersections?intersections?
If the speed limit is 20 mph, what is the maximum bandwidth when intersection A is running If the speed limit is 20 mph, what is the maximum bandwidth when intersection A is running leading left turn, and intersection B is running lagging left turn?leading left turn, and intersection B is running lagging left turn?
All red and yellow time can be ignored for the calculation. All red and yellow time can be ignored for the calculation. Please list all your calculations and adjustments if any (Tips: use Synchro to optimize bandwidth Please list all your calculations and adjustments if any (Tips: use Synchro to optimize bandwidth
manually to check your results)manually to check your results)
CEE 764 – Fall 2010CEE 764 – Fall 2010
Brooks’ Algorithm – A Special CaseBrooks’ Algorithm – A Special Case
CTGTGI jxjxjixU mod)(
Int. #Int. # GGjj RRjj DDmjmj -0.5(R-0.5(Rmm-R-Rjj)) TTmjmj 0.5(T0.5(Tmjmj+T+Tjmjm)) -I-ISS IImaxmax KKUU IIUU KKLL IILL θθmjmj
11 4545 3535 15001500 -2.5-2.5 -25.56-25.56 -25.56-25.56 -5-5 4040
2*2* 4040 4040 -- -- -- -- -- -- -- -- -- -- --
33 4545 3535 330330 -2.5-2.5 5.625.62 5.625.62 -5-5 00
11
-1-1
3.1203.120
-36.88-36.88
43.1243.12
00
11
-8.12-8.12
31.8731.87
44 5050 3030 18301830 -5.0-5.0 31.1931.19 31.1931.19 00
-1-1
26.1926.19
-13.81-13.81
00
11
-36.19-36.19
3.813.81
55 4545 3535 48304830 -2.5-2.5 82.3182.31 82.3182.31 -5-5 00
11
22
79.8179.81
35.8135.81
-0.19-0.19
11
22
-44.81-44.81
-4.81-4.81
Speed is 40 mph
maxIGII mUS maxIGII mLS
CTSTI jxjxjL mod)(