Simulation-Based Rail Transit Optimization Model Myungseob (Edward) Kim, Paul Schonfeld, and Eungcheol Kim A proposed bilevel optimization process determines the depth of a dipped vertical alignment between rail transit stations as well as the cruising speed for each direction. This model also considers how regenerative braking may affect vertical alignment decisions. The optimized depth and directional cruising speeds were jointly obtained in a numerical example. A sensitivity analysis showed that regenerative braking reduced the total cost but did not significantly change the optimized depth of the dipped vertical alignment. The developed model is more realistic and useful than previous ones because it allows unequal station elevations, jointly optimizes decision variables, and considers regenerative braking.
stations to minimize the total cost (i.e., the supplier cost plus user cost). They found that the DVA reduces the cost of operations but without optimizing the DVA depth. Kim et al. also constrained the optimized cruising speed to be equal in both directions (12). These limitations are removed in the current study. Many electric transit vehicles, particularly ones recently built, use dynamic braking as their principal braking system (13). Dynamic braking can also recover braking energy and return it as electric energy to the power supply system. This braking variant is called regenerative braking. Dynamic braking is based on the fact that motors are inherently two-way electric-mechanical energy conversion devices. Normally, electric power is used by the motor to propel the vehicle. However, it is possible to reverse the motor into an electric generator, so that when the vehicle is in motion, the kinetic energy of the vehicle is used to generate electricity. The work involved in generating the electricity decelerates the vehicle (13). Thus, regenerative braking may complement and partially substitute for DVAs. In this study the depth of the DVA and the cruising speeds for each direction are jointly optimized. In rail transit systems the maximum authorized speed is dictated by the train control system, which is not analyzed here. The maximum reachable speed is analyzed here on the basis of kinematics, propulsion, braking, and passenger comfort considerations. However, the proposed models can impose speed constraints based on any criteria, including the limitations of control and signal systems. The effects of regenerative braking are further analyzed with the DVA profile. A numerical case study is used to verify the proposed framework for jointly optimizing the depth and the cruising speed in each direction. The following sensitivity analysis shows the impact on total cost with and without regenerative braking.
Rail transit can transport many passengers and can generally provide more reliable travel times than highway traffic. Rail transit operations may also reduce overall emissions. Many researchers have explored ways to improve rail transit through optimization of rail alignment (1, 2), joint optimization of rail and bus transit systems (3), optimized location of rail transit alignments and stations (4, 5), and phased development of rail transit systems (6). Train and crew scheduling have been optimized with genetic algorithms (7, 8). Several studies have considered the energy-saving design of rail transit alignments (9–12). Kim and Schonfeld introduced the dipped vertical alignment (DVA) profile (9) as a concept for exploiting gravity to reduce acceleration energy and brake wear. They found that the DVA profile reduces travel time as well as the required energy for train operation. Yeh extended the work of Kim and Schonfeld (9) by jointly optimizing vertical track alignment and operational characteristics (10). More recently, Kim and Chien analyzed various track alignments such as level, convex, and concave profiles for rail transit operations (11). Although the metro systems in New York City and elsewhere considered elevated lines more than a century ago (13), no evidence has been found that the DVA concept was applied. One major limitation of previous studies is their assumption that trains operate between stations with the same elevation. Kim et al. recently relaxed this constraint; this change allowed more realistic alignments (12). They also optimized the cruising speed between
DVA PROFILE Previous studies found that a DVA is useful in reducing travel time and energy costs (10, 12). However, they did not optimize the depth for applicable conditions. The value of depth is related to the length of SB (Figure 1); it is assumed that trains reach their maximum gradient along the vertical alignment. Along the alignment the rate of gradient change r should not exceed 0.05 on sag curves and 0.1 on crest curves (14). Thus sag sections should be twice as long as corresponding crest sections. For the DVA shown in Figure 1, Sections 1 and 7 are station platforms. Sections 2 to 6 are parabolic curves. Sections 2 and 6 are one-third of S1 and S2, respectively. Sections 3 and 5 are two-thirds of S1 and S2. SB is the section that has zero gradient. The total station spacing is the sum of one platform length plus five (parabolic or straight) sections.
M. E. Kim and P. Schonfeld, Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742. Current affiliation for M.E. Kim: Rail and Transit Division, Parsons Transportation Group, Parsons Corporation, 100 M Street, SE, Washington, DC, 20003. E. Kim, Department of Civil and Environmental Engineering, University of Incheon, Incheon 406-772, South Korea. Corresponding author: E. Kim,
[email protected]. Transportation Research Record: Journal of the Transportation Research Board, No. 2374, Transportation Research Board of the National Academies, Washington, D.C., 2013, pp. 143–153. DOI: 10.3141/2374-17 143
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FIGURE 1 DVA profile with depth and S B [depth and S B Þ 0 (S = section; E a, E b = elevation of Station A and Station B, respectively)].
The maximum value of vertical gradient allowed by the Washington (D.C.) Metropolitan Area Transit Authority (15) is 4% (i.e., ±4%). However, on the assumption of some adhesion and control improvements, gradients within ±4.5% were explored. A maximum gradient (Gmax) cannot be exceeded anywhere along the vertical alignment. In Figure 1, the points of the steepest gradient are at the boundaries between Sections 2 and 3 and between Sections 5 and 6. Thus,
Previous studies (10–13) also considered the percentage rate of the change in grade (r) per 30.48 m (100 ft), defined as r=
(G1 − G2 )
(1)
L
where G1 and G2 are the percentage of grade of the two intersecting tangents, respectively, and L is the curve length in 30.48-m (100-ft) stations. The generalized equations for the vertical alignment for Figure 1 are provided in Table 1.
Gmax ( = 4.5) ≤
2 ( d1 + d 2 ) S1
TABLE 1 Generalized DVA Equations Section Number
Equationa
Boundaryb
1
Y1 = Ea
2
3(d1 + d 2 ) P Y2 = Ea − X− 1 2 S12
3
Y3 = Ea +
P 3(d1 + d 2 ) X − S1 − 1 2S12 2
4
Y4 = Ea +
P 3(d1 + d 2 ) X − S1 − 1 2S12 2
5
Y5 = Ea +
3d 2 P X − S1 − S B − 1 2S22 2
6
Y6 = Ea −
3d 2 P X − S1 − S B − S2 − 1 2 S22
7
Y7 = Eb
( (
( ( (
)
0 ≤ X ≤ Pl /2 2
Pl /2 ≤ X ≤ 1/3S1 + Pl /2
)
) )
2
2
2
− ( d1 + d 2 )
1/3S1 + Pl /2 ≤ X ≤ S1 + Pl /2
− ( d1 + d 2 )
S1 + Pl /2 ≤ X ≤ S1 + SB + Pl /2
− ( d1 + d 2 )
)
2
− ( d1 )
Note: Ea = elevation of Station A; di = depth; X = horizontal coordinate of train. a Units: Yi = ft; X, S1, S2, SB, P1 = stations. b Units: stations.
S1 + SB + Pl /2 ≤ X ≤ S1 + SB + 2/3S2 + Pl /2 S1 + SB + 2/3S2 + Pl /2 ≤ X ≤ S1 + SB + S2 + Pl /2 S1 + SB + S2 + Pl /2 ≤ X ≤ S1 + SB + S2 + Pl
(2)
Kim, Schonfeld, and Kim
Gmax ( = 4.5) ≤
2 (d2 ) S2
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The DVA profiles with the minimum and maximum depth are shown in Figure 2. These depths are optimized in the following section.
(3)
where d1 = elevation distance, d2 = depth of DVA in Figure 1, and S1 and S2 = parabolic curves in Figure 1.
DETERMINISTIC SIMULATION MODEL A deterministic simulation model is used to obtain the travel times and energy consumption between stations. The simulation model is adapted from work by Kim et al. (12) and improved to consider regenerative braking.
From Equations 2 and 3, S1 and S2 can be rearranged as follows: S1 =
2 ( d1 + d 2 ) Gmax
(4)
S2 =
2 (d2 ) Gmax
(5) Simulation Assumptions
Total alignment length, S, is the summation of S1, S2, and SB: S = S1 + S2 + S B
The assumptions made in this model are as follows:
(6)
1. Parabolic vertical curves are used in which gradients are limited by the train’s climbing ability. 2. Horizontal curves are negligible for this analysis. 3. A concentrated mass is used to represent a train in motion; thus simulated trains start from the center of a platform and arrive at the center of the next platform. 4. The acceleration rate of trains is limited by whichever of the following three factors yields the lowest rate: available power, adhesion (friction), or passenger comfort; similarly, the braking rate is limited by power, adhesion, or passenger comfort. 5. The braking system can provide the maximum comfort-limited deceleration rate. 6. The cost of braking (i.e., brake maintenance and replacement) is proportional to the absorbed braking energy and can thus be represented with a cost per unit of braking energy and 7. Regenerative braking is available, which means that the energy can be recovered while the train is braking; the efficiency (percentage of recovered energy) of the regenerative braking is a given input; no additional cost for using regenerative braking is assumed.
where SB is the length of the section with 0% in the vertical alignment. Substitution of Equations 4 and 5 into Equation 6 gives the length of SB: 2 ( d1 + d 2 ) 2 ( d 2 ) 2d1 + 4 d 2 SB = S − S1 − S2 = S − + =S − Gmax Gmax Gmax
(7)
In Equation 7, S, d1, and Gmax are input values. Therefore, it can be noted that the value of SB and the value of d2 are interdependent. The maximum depth (d 2max) can be found when SB is zero; d 2max is obtained as follows: d 2max =
Gmax i S − 2d1 4
(8)
Also, if the minimum depth (d2 = 0) is applied, SB has the maximum length given by 2d S Bmax = S − 1 Gmax
(9)
(a) FIGURE 2 DVA profiles with (a) minimum depth (depth = 0). (continued on next page)
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(b) FIGURE 2 (continued) DVA profiles with (b) maximum depth (S B = 0).
Modeling Process During the simulation process, the train’s coordinates are updated along the vertical alignment profile at 0.05-s intervals. For each time interval, the simulation model computes the required tractive energy or braking energy. Maximum acceleration and deceleration rates of ±0.15 g are assumed for the passengers’ comfort and safety. For any given DVA profile, the simulation model finds the braking point at which a train must start decelerating so that it stops precisely at the next station. The overall process of the simulation model is shown in Figure 3. Train Motion and Energy Consumption
Fig m
F g m lbf ft s 2 lb
(10)
ρ i Wt
Ft =
375 i η i P V
Ft η P V lbf – hp mph
(13)
P = power, V = speed, η = transmission efficiency, and hp = horsepower.
308 i P V
Ft P V lbf hp mph
(14)
The adhesive force (Fa) is
The train’s acceleration is limited by net force, which is the difference between available tractive effort, Eavail, and train resistance, R. Thus, available acceleration aavail is aavail =
(12)
Tractive force Ft for a rail vehicle is
Ft =
a = acceleration, F = force (lbf), g = acceleration caused by gravity, and m = mass of train.
aavail Eavail R g ρ Wt 2 ft s 2 lbf lbf ft s – lb
Ea Ft Fa lbf lbf lbf
Since η is typically 0.82 (13), Equation 13 simplifies to
where
( Eavail − R) i g
Eavail = min {Ft , Fa }
where
Newton’s second law is used to determine vehicle acceleration at any point: a=
However, the available tractive effort Eavail is limited by both tractive force Ft and adhesive force Fa on the driving wheels. Therefore, it is the minimum of these two forces:
(11)
where ρ is the coefficient for rotating masses, usually 1.04 to 1.10 (13, 16), and Wt is the train mass.
Fa = ∝ i Fn
Fa ∝ Fn lbf – lbf
(15)
where Fn is the normal force and ∝ is the adhesive coefficient. Vuchic (13) provides relations between the adhesive coefficient ∝ and train speed V for dry conditions (in tunnels): ∝ = 0.3 − 0.0015 i 1.609V
∝ V – mph
(16)
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START Vehicle Characteristics Vertical Alignment Initial Accel, Cruising Speed
Vehicle Characteristics Vertical Alignment Initial Acceleration Rate
Run Simulation A to B Direction
Run Simulation A to B Direction
Run Simulation B to A Direction
Time = Time + Interval
Time = Time + Interval
aavail = (TE- Rt)g /
Wt, TE=min{ Ft, Fa}
aapplied = Min.{ aavail, amax, acruising} Update Information (X, Y, V) Use Kinematic Relations
Time = Time + Interval
a avail = (TE- Rt)g /
Apply Max. Acceleration Rate
aapplied = Min.{ aavail, amax, acruising}
Update Information (X, Y, V) Use Kinematic Relations
Update Information (X, Y, V) Use Kinematic Relations Calculate Energy, Save to TE(sum)
Save Distance and Speed Array DistanceBtoA(Time), SpeedBtoA(Time)
Save Result Time, Position(X, Y) Station, Speed, TE
Save Distance and Speed Array DistanceAtoB(Time), SpeedAtoB(Time) N N
Wt, TE=min{ Ft, Fa}
DistanceBtoA(Time) >= Station Spacing
DistanceAtoB(Time) >= Station Spacing
N
Braking Point ?
Y
Y
Y
Time = Time + Interval For i = 1: the number of array(SpeedAtoB) Apply Feasible Braking Rate For j = 1: the number of array(SpeedBtoA) Update Information (X, Y, V) Use Kinematic Relations
SpeedAtoB(i) >= SpeedBtoA(j)
Y Calculate Energy, Save to BE(sum) DistanceAtoB(i) >= DistanceBtoA(j) STOP ? N
Y Print Braking Point BrakingPoint = DistanceAtoB(i)
Print Result Time, Position(X, Y), Station, Speed, TE, BE
END
FIGURE 3 Simulation process adapted from work by Kim et al. [(12) TE = tractive energy; BE = braking energy.]
Fn = Wt cos θ
θ = tan −1
G 100
Fn Wt θ lbf lb radian
(17)
θ G radian %
(18)
where θ is the angle from level and G is the percentage of gradient. To avoid a negative value of adhesive force in Equation 15, it is assumed that the adhesive coefficient from Equation 16 applies for
speeds up to 62.15 mph (100 km/h), and beyond that speed the coefficient stays constant at 0.15. For the resistance computation, the modified Davis equation from Hay was applied (14). The unit resistance in pounds per ton is r = 0.6 +
20 KV 2 + 0.01V + + 20G + 0.8 D w wn w V n G ton mph axles %
(19)
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where
Papplied =
w = weight per axle (tons), K = air resistance coefficient, and n = number of axles per car.
Eapplied i V 308
∆eTE = Papplied Ri = r × Wt
l
R Rcar lbf lbf
i
(21)
For the comfort and safety of standing passengers in rail transit systems, the longitudinal acceleration and deceleration rates should be limited (usually to ±0.1 to 0.15). Therefore, the maximum allowable acceleration or deceleration rate a is −0.15 i g ≤ a ≤ 0.15 i g
g a ft s 2 ft s 2
(22)
When a constant cruise speed is sought, the acceleration limit acru is acru at i =
Vcruise − Vi ∆t
acruise at i Vcruise Vi ∆t ft s 2 ft s ft s s
(23)
aapplied = min {aavail , acru at i , amax }
aapplied aavail amax acru at i ft s 2 ft s 2 ft s 2 ft s 2
1 1.341
∆ e Papplied ∆ t kW-h hp s
(29)
Fb = min {Fbc , Fba }
(25)
where Fbc is the maximum comfort-limited braking force that is applicable with standing passengers. Fbc can be obtained by rearranging Equation 11: Fbc bmax ρ Wt R lbf ft s 2 – lb lbf
(26)
Fba is an adhesive-limited braking force, given in Equation 15. To compute energy consumption, the applied tractive effort, TEapplied, can be computed after the train acceleration (or deceleration) rate is determined: a applied i ρ i Wt g
TE applied a applied ρ Wt lbf ft s 2 – lb
On the basis of the increments of energy specified in Equation 29, the total energy required for tractive effort and the total braking energy that must be dissipated are separately accumulated.
JOINT OPTIMIZATION FRAMEWORK Objective Function The objective function minimizes total cost, which is the sum of both directional costs. The objective function is a function of the depth of the DVA and the cruising speed in each direction. For each directional cost between two stations, the user cost and supplier cost are added. The user cost, Cu, is determined by multiplying by the travel time between stations, the number of passengers per car, the number of cars in a train, and the time value for a passenger. The time value of a passenger is given as an input parameter. Supplier cost, Cs, includes the cost for tractive energy and for braking energy, the cost saving from regenerative braking, and the vehicle depreciation cost. The directional cost function is
(30)
where
For braking energy estimation, the available braking force (Fb) is the minimum of two forces:
TE applied = R +
i
+ Eb i vbraking − Erb i vrb
(24)
bmax i ρ i Wt g
∆t 3,600
Ct = Cu + Cs = t i Nc i Np i vuser + N c i t i vvehicle + Et i vtractive
Therefore, the applied acceleration rate aapplied is
Fbc = R +
i
(20)
The total resistance of a train R is the sum of the individual resistances of every vehicle i in that train: R = ∑ Ri
(28)
The energy consumption ΔeTE within time interval Δt is
The car resistance is r multiplied by the car weight in tons: r , Rcar Wt lbf ton
Papplied TE applied V hp lbf mph
(27)
If Eapplied is positive, some tractive force is needed and if it is negative, some braking force is needed. The power required for generating the applied tractive effort Papplied is
Ct = total cost for one-directional operation, Cu = user cost, Cs = supplier cost, t = travel time between stations, Nc = cars per train, Np = passengers per car, vuser = passenger time value ($/h), vvehicle = vehicle depreciation per revenue hour, Et = tractive effort required (kW-h), vtractive = unit tractive effort cost, Eb = braking energy required (kW-h), vbraking = unit braking energy cost, Erb = energy recovered by regenerative braking (kW-h), and vrb = unit cost of regenerative braking (= vtractive).
Optimization Framework A two-stage optimization framework is provided here for determining the values of the depth and cruising speeds for each direction (Figure 4). The depth of the DVA should be identical for both directions (assuming shared tunnels for the two directions), but the optimized cruising speeds may differ. Moreover, with given depth, finding the optimal cruising speed becomes a one-variable optimization problem. Therefore, cruising speed was determined through a constrained optimization process.
Kim, Schonfeld, and Kim
149
FIGURE 4 Optimization framework.
To optimize the cruising speed for each direction, golden section search and parabolic interpolation are used. To run the golden section search, lower and upper bounds of the variable (cruising speed here) must be assigned. The lower bound should exceed zero. For the upper bound, the simulation model was run without speed limitation and then the maximum reachable speed was obtained. This maximum possible speed is used as the upper bound for the cruising speed optimization. The objective function provided in Equation 30 is nonlinear. Thus, a nonlinear optimization algorithm is needed to minimize the total cost. Sequential quadratic programming (SQP) was applied
to optimize the DVA depth, which is the upper-level decision variable. SQP is a nonlinear optimization solver in MATLAB. Since SQP may get trapped in local optima, a global solver was created that utilizes multiple (k) SQPs. The k is the number of solvers used for finding solutions; k is an input value. With k, a starting point to run each solver is randomly selected between the lower and upper bounds of the depth. The lower bound is zero, with which SB has the maximum length shown in Equation 9. The upper bound is provided in Equation 8. Each solver starts to find the solutions from the randomly assigned starting point and then stops if the changes from the previous iteration are below 0.01 for the decision variable (depth)
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TABLE 2 Input Values for Numerical Example Parameter
TABLE 3 Optimized Results and Total Cost Breakdown
Default Value
Simulation Results
Spatial Inputs Station spacing [km (ft)] Platform length [m (ft)]
4.0 (13,200) 181.7 (600) × 2 stations
Elevation at Station A [m (ft)] Elevation at Station B [m (ft)] Elevation difference [m (ft)]
302.8 (1,000) 284.6 (940) 18.2 (60)
Train Inputs Car weight (tons/car) Car power (kw/car) Number of cars Number of axles per car
50 520 8 4
Cost and User Inputs Tractive effort unit cost ($/kW-h) Braking energy unit cost ($/kW-h) Vehicle depreciation cost ($/veh-h) Passengers per car Passenger time value ($/h)
0.3 0.2 50 50 10
Note: veh-h = vehicle hours.
and below 0.01 for the total cost. The global solution is found from the least-cost solution among the solutions obtained by the k solvers. For each solver, the active-set algorithm is used to find the solution because the nonlinear objective function in this problem cannot provide the hessian matrix. The detailed algorithm description can be obtained from Mathworks (17).
NUMERICAL EXAMPLE Input Values The input values used for the case study are shown in Table 2. Station spacing is about 13,200 ft, and the elevation difference between the two stations is 60 ft. A train has eight cars, with power and brakes on all axles. The passengers’ time value is assumed to be $10 per person hour. Vehicle depreciation cost is assumed to be $50 per vehicle hour.
Variable Travel time (s) Optimized speed (mph) Optimized depth (ft) Maximum gradient (%) Tractive effort required (kW-h) Tractive effort cost ($) Braking energy required (kW-h) Braking energy required cost ($) Regenerative energy saving ($) Vehicle depreciation cost ($) User cost ($) Total cost ($)
A to B (downhill)
B to A (uphill)
Sum
148.7 84.5 104.76 4.5 64.62 19.39 63.58 12.72 5.72 16.52 165.22 208.13
156.3 78.20 104.76 4.5 87.85 26.36 45.51 9.10 4.10 17.37 173.67 222.40
305.0 na na na 152.47 45.75 109.09 21.82 9.82 33.89 338.89 430.53
Note: na = not applicable.
Optimization Results Optimization results are shown in Table 3. The optimized depth is 104.76 ft, which is close to the maximum depth (105 ft) because the DVA can reduce travel time significantly, and reduced travel time decreases user cost. Optimized speeds for the downhill and uphill directions are 84.5 mph and 78.2 mph, respectively. Trains operating downhill reach higher speeds more quickly. Therefore, travel time is lower than it is uphill. Directional cost is $208.13 per train operating downhill between two stations, and $222.40 uphill. The total cost for both directions with jointly optimized depth and cruising speeds is $430.53 per train operating between two stations. Figures 5 through 8 show the downhill simulation profiles. Figure 5 shows the relationship between the downhill speed and the elevation profile. In Figure 6, the profiles of gradient versus elevation are shown, in which the maximum gradient is less than 4.5%. Figure 7 shows the profiles of speed versus acceleration rate. As expected, the acceleration rate is zero for the cruising section, in which the speed is constant. In Figure 7, the deceleration rate decreases at about 10,700 ft from the starting station. The reason is that the
FIGURE 5 Profile for speed versus elevation (downhill direction).
Kim, Schonfeld, and Kim
FIGURE 6 Profile for gradient versus elevation (downhill direction).
FIGURE 7 Profile for speed versus acceleration (downhill direction).
FIGURE 8 Profile for resistance versus gradient (downhill direction).
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TABLE 4 Sensitivity Analysis Results Regenerative Braking Efficiency (%) 0 30 (base) 45 100
Optimized Depth (ft)
Directional Cost (A to B) ($)
Directional Cost (B to A) ($)
Total Cost ($)
Cost Savings (%)
105.00 104.76 103.52 100.44
213.67 208.13 205.22 193.71
226.53 222.40 220.33 212.83
440.20 430.53 425.54 406.54
2.2 0.0 −1.2 −5.6
gradient at the 10,700-ft point changes from positive to negative. This change decreases the total resistance, so the available tractive force increases. Therefore, speed decreases slowly. Figure 8 shows that the total resistance is highly dependent on the gradient profile. Because of space limitations, the profiles for the opposite direction (uphill) are omitted here.
Sensitivity Analysis The following sensitivity analysis shows that regenerative braking is useful for reducing the energy cost, but the optimized depth changes are somewhat limited. An extreme case was explored in which regenerative braking efficiency is 100%. In this extreme case,
total cost decreases by 5.6% and 7.8% compared with the base case and the case with no regenerative braking, respectively. However, the optimized depths are not significantly reduced even if perfect regenerative braking is assumed. Therefore, although the regenerative braking recovers some fraction of the energy cost, the DVA alignment concept is still useful for saving energy in rail transit operations. The detailed sensitivity analysis results are shown in Table 4. The sensitivity analysis results are illustrated in Figures 9 and 10. Figure 9 shows that the cost reduction in the downhill direction (i.e., A to B) is greater than it is in the uphill direction. Figure 10 shows that the optimal depth change between no regenerative braking and perfect regenerative braking is about 5 ft. Moreover, according to Vuchic, the efficiency (percentage of energy recovery) of regenera-
FIGURE 9 Directional costs and total cost changes in sensitivity analysis.
FIGURE 10 Profiles for cost savings versus optimal depth by sensitivity analysis.
Kim, Schonfeld, and Kim
tive braking is generally up to 30%, which is the base case in this study (13). On the basis of the sensitivity analysis, regenerative braking slightly reduces the optimized DVA depth. However, regenerative braking can further reduce some of the construction cost because the optimized depth is reduced. The construction cost saving due to regenerative braking depends very much on local site conditions, which are beyond the scope of this paper.
CONCLUSIONS In this study, a joint optimization framework is provided in which the DVA depth and the cruising speeds for both directions are jointly obtained through two-stage optimization. Since the previous studies related to this work did not consider the optimization of the cruising speed in each direction and optimal depth, the model provided here can be used for more realistic operational analysis. Furthermore, this model also assesses regenerative braking, which is not considered in the relevant previous studies. For the numerical cases the analysis indicates that regenerative braking recovers some fraction of energy costs, and thus reduces the total cost. Although the optimized DVA depth change from regenerative braking is slight, the accumulated cost reduction from the long-term analysis may be significant. When travel time is less important, the optimized depth of the DVA may be decreased.
ACKNOWLEDGMENT This work was supported by a 2012 international cooperative research grant from Incheon National University in South Korea.
REFERENCES 1. Jha, M. K., P. Schonfeld, and S. Samanta. Optimizing Rail Transit Routes with Genetic Algorithms and Geographic Information System. Journal of Urban Planning and Development, Vol. 133, No. 3, 2007, pp. 161–171.
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2. Samanta, S., and M. K. Jha. Modeling a Rail Transit Alignment Considering Different Objectives. Transportation Research Part A, Vol. 45, No. 1, 2011, pp. 31–45. 3. Chien, S., and P. Schonfeld. Joint Optimization of a Rail Transit Line and Its Feeder Bus System. Journal of Advanced Transportation, Vol. 32, No. 3, 1998, pp. 253–284. 4. Lai, X. Optimization of Station Locations and Track Alignments for Rail Transit Lines. PhD dissertation. University of Maryland, College Park, 2012. 5. Lai, X., and P. Schonfeld. Optimizing Rail Transit Alignments Connecting Several Major Stations. Presented at 89th Annual Meeting of the Transportation Research Board, Washington, D.C., 2010. 6. Schonfeld, P., and W.-C. Cheng. Phased Development of Rail Transit Routes. Presented at 88th Annual Meeting of the Transportation Research Board, Washington, D.C., 2009. 7. Sahin, I., and S. Dundar. A Genetic Algorithm Solution for Train Scheduling. Presented at 90th Annual Meeting of the Transportation Research Board, Washington, D.C., 2011. 8. Liu, M., A. Haghani, and S. Toobaie. Genetic Algorithm-Based Column Generation Approach to Passenger Rail Crew Scheduling. In Transportation Research Record: Journal of the Transportation Research Board, No. 2159, Transportation Research Board of the National Academies, Washington, D.C., 2010, pp. 36–43. 9. Kim, D. N., and P. Schonfeld. Benefits of Dipped Vertical Alignment for Rail Transit Routes. Journal of Transportation Engineering, ASCE, Vol. 123, No. 1, 1997, pp. 20–27. 10. Yeh, S. Integrated Analysis of Vertical Alignment and Speed Profiles for Rail Transit Routes. Master’s thesis. University of Maryland, College Park, 2003. 11. Kim, K., and S. Chien. Simulation-Based Analysis of Train Controls Under Various Track Alignments. Journal of Transportation Engineering, Vol. 136, No. 11, 2010, pp. 937–948. 12. Kim, M., P. Schonfeld, and E. Kim. Comparison of Vertical Alignments for Rail Transit. Journal of Transportation Engineering, Vol. 139, No. 2, 2013, pp. 230–238. 13. Vuchic, V. R. Urban Transit: Systems and Technology. John Wiley & Sons, Inc., New York, 2007. 14. Hay, W. W. Railroad Engineering, 2nd ed. John Wiley & Sons, Inc., New York, 1982. 15. Washington Metropolitan Area Transit Authority. Manual of Design Criteria. Washington, D.C., 2002. 16. Vuchic, V. R. Urban Public Transportation. Prentice Hall, Inc., Englewood Cliffs, N.J., 1981. 17. MATLAB Product (Version 2011b) Documentation. Mathworks, Natick, Mass., 2012. The Rail Transit Infrastructure Committee peer-reviewed this paper.
