Optimization of a PRT Fleet

Ahmed Saif 10/12/2009

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Abstract For personal rapid transit (PRT) systems to be able to compete with traditional transportation

means and achieve their objectives of cost-effectiveness, convenience and agility they need to

be operated efficiently and smartly. Rule-based operational strategies and simulation

techniques were used frequently to address the problem of vehicle routing in PRT systems.

Despite the apparent merits of these methods, there is still need for other approaches based on

optimization models and methods to provide more rigorous solutions of this routing problem.

The size, complexity and stochastic nature of these systems are the major obstacles for using

optimization techniques for this specific problem. A novel Mixed Integer Programming (MIP)

formulation of the PRT system operation in proposed to study and optimize the system. A

numerical example representing part of the PRT system at Masdar City was solved and

sensitivity analysis was conducted to study the effect of changing system parameters,

constraints, and objective functions. It has been shown that the proposed method is useful for

the system planner, not only during operations phase, but also in the capital expenditure

planning phase.

1- Introduction Personal Rapid Transit is considered one of the most promising urban transportation means.

This concept of personalized urban mass transit offering on-demand, non-stop transportation,

using small, automated vehicles on a network of specially-built guideways was initially

developed in the 1960’s. And since then, PRT related research, modeling and prototype testing

is undergoing continuously, yet slowly. The first real world PRT system was deployed by West

Virginia University in the early 1970’s to make connection between the downtown and the

university campus. However, commercial size PRT systems start to appear only very recently.

The PRT system at Masdar City, currently under construction, might be the first large scale

project of its kind, and is planned to form the backbone of the transportation system in the first

zero-carbon, car-free city.

PRT have several merits over traditional transit means, making it an attractive alternative for

many passengers and transportation system planners. The main characteristics of a PRT system

are [1]: a fleet of small automated vehicles traveling along a fixed guideway, offering private

transportation services to a small number of passengers, the voyage is uninterruptible because

of the offline stations and of the private transportation concept (the vehicle doesn't have to

stop for each passenger). Passengers enjoy the privacy, convenience, safety, and time saving

provided by a well-planned PRT system, while low polluting emissions level, low land usage,

automatic control, and low infrastructure cost are the PRT features most appreciated by

planners.

Despite the obvious advantages of PRT systems, several technical and systemic and regulatory

barriers have hindered a strong take-off of this technology. Nevertheless, recent improvements

in critical technologies for PRTs such as power storage, navigation, communication and material

technology have eased many of these technical challenges. Alongside that, developments in

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traffic systems concepts and tools enabled researchers and engineers to develop better

understanding of the best policies and practices for designing and managing successful PRT

systems. Last but not least, the evolving regulatory environment and enhanced public

awareness about traffic and environmental issues propelled the PRT proposition further.

Research in the area of PRT systems in moving in two main tracks: system design and

operational strategies. Important work has been done in the system design front to tackle

different problems such as: design of the guideway network (GN)[2,3], Station Design [3],

location of power supplies [4]. Deterministic, heuristic and evolutionary optimization techniques

have been used intensively to deal with these problems. These tools have been used also to

solve some operational planning problems such as the static vehicle routing (determining the

shortest path between two stations for the vehicle to follow) and the idle vehicle allocation

problems [3,5]. However, when it comes to dynamic routing, where the time/motion plan of

vehicles is to be determined, rule-based strategies and simulation techniques are preferred by

researchers. Reference [6] provides a review for the simulation models used for PRT systems.

Recently, K. Mueller has used discrete event simulation to study the PRT system at Masdar City

under two different dynamic routing rules: nearest station and longest queue [7]. Agent-based

techniques have been also proposed for studying PRT systems with autonomous vehicles

operating without central control [8].

Because of the complex, probabilistic and dynamic nature of the PRT fleet planning problem,

trying to deal with it using optimization tools is quite challenging. The problem is more complex

than the well-known Dynamic Vehicle Routing Problem with Time Window (DVRPTW) [9-10], an

NP-hard problem; even finding a feasible solution for a static VRPTW when the number of

vehicles is fixed is itself an NP-complete problem [11]. However, having a proper formulation

and solution strategy for this problem can be of great benefit to the transportation system

planner. The results obtained from running such a model can serve as the “reference strategy”

for all other rule-based operational strategies. By comparing the results obtained from

implementing these strategies with the optimum solution we can rank the efficiency of these

strategies and find if there is room for further improvement. Studying the optimization results

can also prove useful in developing rule-based strategies by observing the optimal strategy and

trying to infer certain rules out of it. Finally, through changing the parameters and bounds,

sensitivity and scenario analysis can be conducted more easily and reliably versus using

simulation tools.

This paper will be organized as follows: next section provides a concise description of the

problem with its underlying assumptions and conditions. In section 3, problem formulation will

be outlines, and a small numerical example to show the application of the proposed method will

be given in section 4. Sensitivity analysis and alternative formulations are presented in section 5.

Section 6 concludes the paper and identifies some possible extensions for this research.

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2- Problem Description A fleet of Personal Rapid Transit (RPT) is assigned to carry residents and visitors between a finite

number of stations at Masdar City under the following assumptions and conditions:

The demand pattern is deterministic and passengers’ arrival rate is known a priori. This

assumption might look unrealistic at first glance since the demand pattern for transit

services is stochastic and generally unpredictable. However, given the purpose of this

study outlined in the previous section, this assumption is acceptable.

Transportation service is not provided continuously throughout the day, but only for a

certain period of it (e.g. 20 hours). The remaining part of the day will be assigned for

checkout and maintenance of the PRTs. This assumption is here to define the

boundaries of the model in terms of time.

Trip time between two specific stations is constant and is not dependent on the traffic

condition. Again, this assumption, though idealistic, is used to make the model solvable

using deterministic optimization tools.

PRTs use a battery of certain capacity, which can be recharged, at a specific rate, from a

charging station located in a pivotal position within the network. This is taken as the

‘base-case’ which be altered when studying other alternatives in section 5.

Energy consumption depends on the state of the PRT, i.e. when in move it consumes

more power than when stationary.

When the PRT arrives to a station, it has to stay there for at least a minute for loading

and unloading of passengers.

All PRTs start their duty from the charging station and return there at the end of the

day. Also PRTs have their batteries fully charged at the beginning of the day.

Time is treated as discrete variable in the model and is counted in minutes. This

resolution is considered adequate for the purpose of this model, but smaller time units

can be used for more precise models.

The objective of this model is to answer the question: “How PRT vehicles shall be allocated to

carry passengers between PRT stations throughout the day at the least possible energy

consumption, while satisfying technical (battery capacity) and regulatory (service level)

constraints?”

3- Problem Formulation

3-1 Decision Variables

The following are the decision variables in this problem:

{

{

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{

{

( )

Where

{ }

{ }

{ } ( )

And when is used as a specific point of time it represents the beginning of the time interval.

3-2 Parameters / Inputs

The following parameters are given for the PRT system:

( )

3-3 Objective Function: Minimizing energy consumption

Here we will assume that the vehicle starts its duty with full battery charge and return to fully

recharge it at the end of the day, so the energy consumed throughout the day is equal to the

sum of energy charged in the batteries every time they return to the charging station. The only

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problem here is that is assumed binary, so might not be exactly realizable. A

new variable is defined, representing the residual charging capacity that cannot be fulfilled

according to the integer variable constraint mentioned earlier. So,

The objective function in this case is

Min ∑ ( ∑ )

3-5 Constraints

a. Routing constraints

The following set of constraints guarantees rational routing of the PRT fleet.

1- Each vehicle at any time period is either waiting at a station or moving toward a station.

∑∑

∑

2- If a vehicle is moving toward a station, either it was during the previous period moving

towards it or planning to depart from another station toward this station. Also, if a vehicle is

waiting in a station, either it was in the previous period moving toward it or waiting at the same

station.

∑

∑

3- Distance to target updating, if the vehicle is moving, this distance is decreasing gradually. And

when it starts a trip this distance gets step increase by the new trip time.

∑

∑∑

4- Any vehicle is occupied if it is travelling between two different stations:

∑

b. Passenger picking/waiting constraints

The following set of constraints deal with the relation between PRTs and passengers.

5- A vehicle can’t pick passengers between any two stations unless it is starting a trip between

them in the next period.

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6- A vehicle can’t pick passengers between two stations unless there are passengers waiting for

this trip

∑

7- People wait at PRT stations until a vehicle picks them to their destination. The queue length at

any moment is

∑

8- Maximum waiting time constraints

∑∑

9- No passengers are waiting at the very beginning

c. Boundary constraints

The following set of constraints defines the status of the PRT fleet at the beginning and end of

their duty.

10- All vehicles has to start their duty from the charging station and return to the same station

at the end of the day

∑

11- Batteries are fully charged at the beginning of the day.

12- Residual capacity to be charged at the end of the day

d. Charging constraints

The following set of constraints deal with the PRT battery charging state.

13- Battery charge at any moment is calculated as

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∑

( ∑

)

14- Each battery has the following charge limits

15- Vehicles can charge their batteries only when they are parking in the charging station

16- The charging station can charge only h vehicles at once.

∑

4- Numerical Example To demonstrate the applicability of the proposed formulation, it has been tested on a trivial PRT

system consisting of three PRT vehicles, four PRT stations and operation period of thirty minutes

only. Although the system is very small, it was difficult to solve even using efficient Mixed

Integer Programming (MIP) solvers like BARON branch-and-reduce solver [12]. Nevertheless,

finding an efficient algorithm to solve such a complex problem is outside the scope of this paper.

System configurations and base case scenario (BCS) parameters are presented in figure (2) and

table (1), respectively. Most parameters were deducted or approximated from the real PRT

system of Masdar City, as presented in [7].

B

CD

A

32

4

32

1

Fig. (2): Trip time between PRT stations (in minutes)

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Table 1: base case scenario parameters

Parameter Value

Vehicle power consumption while moving 100 Wh/min

Vehicle power consumption while stationary 25 Wh/min

Battery recharging rate 400 Wh/min

Battery Capacity 1600 Wh

Minimum allowable battery charge 100 Wh

Maximum allowable passenger waiting time 2 min

Charging station capacity (max. number of vehicles that can charge their batteries simultaneously)

2

Minimum loading/unloading time 1 min

Table (2) depicts the demand pattern assumed for this problem, and the optimum solution of

this problem is portrayed in Table (3):

Table (2): Demand Pattern

time from station to station No. of PRTs requested

2 B D 1

3 D C 1

4 C D 1

5 D B 1

9 D B 1

15 C D 1

16 B C 1

19 D C 1

20 C D 1

21 D B 1

Table (3): Optimization results

Objective Function (minimum energy consumption) = 6,300 Wh

Optimal motion/charging plan

Minute PRT1 PRT2 PRT3

1 W(1) W(1) W(1)

2 W(1) W(1) W(1)

3 M(1-2) M(1-3) M(1-4)

4 W(2)-L M(1-3) M(1-4)

5 M(2-4) M(1-3) W(4)-L

6 M(2-4) W(3)-L M(4-3)

7 W(4)-LU M(3-4) M(4-3)

8 M(4-2) M(3-4) M(4-3)

9 M(4-2) M(3-4) M(4-3)

10 W(2)-U M(3-4) W(3)-U

11 M(2-1) W(4)-LU M(3-1)

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12 W(1)-C M(4-2) M(3-1)

13 W(1) M(4-2) M(3-1)

14 M(1-3) W(2)-U W(1)-C

15 M(1-3) M(2-1) W(1)-C

16 M(1-3) W(1)-C W(1)

17 W(3)-L W(1)-C M(1-2)

18 M(3-4) W(1) W(2)-L

19 M(3-4) M(1-4) M(2-3)

20 M(3-4) M(1-4) M(2-3)

21 M(3-4) W(4)-L M(2-3)

22 W(4)-U M(4-3) W(3)-LU

23 W(4)-L M(4-3) M(3-4)

24 M(4-2) M(4-3) M(3-4)

25 M(4-2) M(4-3) M(3-4)

26 W(2)-U W(3)-U M(3-4)

27 M(2-1) M(3-1) W(4)-U

28 W(1)-C M(3-1) M(4-1)

29 W(1) M(3-1) M(4-1)

30 W(1) W(1) W(1)

W(x): Waiting in station x M(x-y): Moving from station x to station y C: Charging battery L: Loading U: Unloading

5- Sensitivity analysis Our analysis of the PRT fleet routing can be expanded by changing the problem parameters,

constraints or objective functions and observe the impact on the planning strategy and results.

This sensitivity analysis is particularly useful in the planning phase when the planner has to

evaluate the cost of tightening/relaxing constraints and assess the benefits gained from adding

more resources. The following are only few examples to show how to use this model to conduct

useful sensitivity analysis and stress testing.

5-1 Changing system parameters

5-1-1 Battery Capacity

As have been seen in the given example, all vehicles had to return to the charging station in the

middle of the operation period to recharge their batteries. One alternative to reduce this

unnecessary movement is to increase the capacity of the PRT battery. If we add a second

battery to each PRT vehicle so the energy storage capacity is doubled from 160 Wh to 320 Wh,

the energy consumption of the PRT system will decrease to 5,625 Wh, a 10.7% reduction from

BCS. In larger system there can be even saving in the number of PRT vehicles needed in the

system. Of course, this saving has to be judged against the additional cost of the new batteries.

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5-1-2 Waiting time

If the service level (SL) of the PRT system is downgraded so longer waiting time is allowed, this

can have significant impact on the routing plan and consequently the system cost. Due to the

limited size of the test system, changing this parameter might slightly shift the vehicles

movement timing but the energy consumption will remain unchanged. However, this is not the

case in larger scale systems.

5-2 Adding/changing constraints

5-2-1 Stations capacity

PRT stations are usually designed to accommodate certain number of vehicles at the same time.

A constraint can be added to the problem formulation to characterize this fact as follows:

∑∑

{ }

Where is the PRT station capacity.

5-2-2 Battery charging in stations

Another way to tackle the energy storage limitations is to equip all the stations with power

chargers to be used by the vehicles while waiting to fill-up their batteries. If this option is to be

availed in the system, constraint 15 can be simply changed to:

∑∑

In our test system, applying this option will reduce the power consumption to 5,625 Wh, since

PRT vehicles don’t have to return to the charging station in the middle of the period.

5-3 Alternative objective functions

5-3-1 Minimizing waiting time

Waiting time was treated as a constraint in the original problem, but the operational planner

might want to recognize the highest service level achievable. In this case, the following objective

function can be used:

∑ ∑ ∑

5-3-2 Minimizing the number of PRT vehicles

PRT vehicles are very expensive compared to regular cars, so it is desirable to reduce this

sizeable capital expenditure through minimizing the number of PRT vehicles in the system, while

satisfying other constraints. For this purpose a new decision variable can be defined as:

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{

And a new constraint linking it to other decision variables is added. A vehicle can’t move if it

doesn’t exist.

∑ ∑

Finally, the new objective function becomes:

∑

In our simple problem the minimum number of vehicles that can satisfy the constraint is 3, so

there is no chance for cutting initial investment cost, however, in larger systems this objective

function can result in large saving. Moreover, an objective function combining both capital

expenditure and energy cost can be derived to minimize the total PRT system cost over its life

cycle.

Conclusion A new approach for planning the operations of PRT systems is proposed in this paper. The PRT

fleet routing problem was formulated as a mixed integer problem and solved using a standard

algorithm. The objective was to minimize the total energy consumption of the vehicles while

satisfying a set of service level and battery capacity constraints. A numerical example was given

to demonstrate the applicability of the formulation and solution algorithm. Moreover, some

examples of the application of the proposed model in sensitivity analysis were presented, which

can form a basis for further research in the area. Due to the complexity of the problem, an

efficient technique is required to optimize larger scale, real life systems.

References [1] Carnegie, J.A., AICPIPP, Voorhees, AM. (2007). Viability of Personal Rapid Transit In New Jersey. FINAL REPORT February 2007, Transportation Center, Rutgers, The State University of New Jersey, Hoffman P.S., Hamilton B.A., Inc. [2] J. Won and F. Karray, Personal Rapid Transit Network Design Using Genetic Algorithm and Ant Colony System Hybridization, Proc. of the International Conference on Control, Automation and Systems 2008, Soul, Korea. [3] J. Won, H. Choe, and F. Karray, Optimal Design of Personal Rapid Transit, Proceedings of the 2006 IEEE Intelligent Transportation Systems Conference, Toronto, Canada [4] N.X.H. Viet and H.S. Song, Locating Power Supplies on a Personal Rapid Transit System to Minimize System Losses, IEEE Transactions on industry applications, vol. 40, no. 6, Nov/Dec 2004.

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[5] I. Andréasson, Reallocation of empty personal rapid transit vehicles en route, Transportation Research Record 1838 (2003), pp. 36–41. [6] J. E. Anderson, Some History of PRT Simulation Programs, http://www.prtnz.com/component/option,com_docman/task,doc_download/gid,24/Itemid,37/ [7] K. Mueller, Simulation-based Analysis of Masdar City’s Personal Rapid Transit System, A Master Thesis. [8] F. Wang, M. Yang, and R. Yang, Simulation of Multi-Agent Based Cybernetic Transportation System, Simulation Modelling Practice and Theory, Volume 16, Issue 10, November 2008, Pages 1606-1614. [9] D.D. Bochtis, C.G. Sørensen, The vehicle routing problem in field logistics part I, Biosystems Engineering, Volume 104, Issue 4, December 2009, Pages 447-457 [10] M. M. Solomon. Algorithms for the Vehicle Routing and Scheduling Problems with Time Window Constraints. Operations Research, 35(2):254–265, 1987. [11] M. W. P. Savelsbergh. Local search in routing problems with time windows. Annals of Operations Research, 4(1):285–305, 1985. [12] GAMS Solver Descriptions, BARON, http://www.gams.com/dd/docs/solvers/baron.pdf