Exact approach to the electric bus fleet scheduling

Available online at www.sciencedirect.com Available online at www.sciencedirect.com

ScienceDirect ScienceDirect

Transportation Research Procedia 00 (2019) 000–000 Available online at www.sciencedirect.com Transportation Research Procedia 00 (2019) 000–000

ScienceDirect

www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

Transportation Research Procedia 40 (2019) 1380–1387 www.elsevier.com/locate/procedia

13th International Scientific Conference on Sustainable, Modern and Safe Transport 13th International 2019), Scientific Conference on Sustainable, and Safe Transport (TRANSCOM High Tatras, Novy Smokovec –Modern Grand Hotel Bellevue, (TRANSCOM 2019),Slovak High Tatras, Novy Smokovec – Grand Hotel Bellevue, Republic, May 29-31, 2019 Slovak Republic, May 29-31, 2019

Exact approach to the electric bus fleet scheduling Exact approach to the electric bus fleet scheduling a a Maroš Janovec *, Michal Koháni Maroš Janoveca*, Michal Kohánia

Department of Mathematical Methods and Operations Research, Univerzitná 8215/1, Žilina, Research, Slovakia DepartmentofofŽilina, Mathematical Methods and01026 Operations University of Žilina, Univerzitná 8215/1, 01026 Žilina, Slovakia

a

aUniversity

Abstract Abstract In the recent years the research in the field of electromobility has been gaining a lot of interest. Research in the field of electromobility relates various problems connected to the limitedhas driving of vehicles, chargingininfrastructure In the recent years theto research in the field of electromobility been range gaining a lot of locating interest. of Research the field of and capacity of relates the electric network. Theseconnected limitationstoshould also be considered when solving the problem of transforming the electromobility to various problems the limited driving range of vehicles, locating of charging infrastructure public transport fleet of diesel engineThese busses into the should fleet ofalso electric vehicles. One the tasks should be solved is the and capacity of the electric network. limitations be considered whenofsolving the that problem of transforming problemtransport of scheduling. paper dealsbusses with the problem of of assigning to thethat set should of service public fleet ofThis diesel engine into the fleet electricavailable vehicles.electric One ofbuses the tasks be trips, solvedin isother the words scheduling of electric buses. Electric buses have of limited operation range and must charged operation. problem of scheduling. This paper deals with the problem assigning available electric buses be to the set of during servicethe trips, in other Charging process needs more time than refueling of diesel the number of and charging at charging station should be words scheduling of electric buses. Electric buses have busses limitedand operation range mustpoints be charged during the operation. considered. In ourneeds case itmore is assumed thatrefueling the charging is possible at the andofatcharging chosen locations. In the paper we propose Charging process time than of diesel busses and thedepot number points at charging station should bea linear mathematical model respects limitations. of the model willInbethe tested thepropose data seta considered. In our case it is that assumed that all thenecessary charging is possible atThe the performance depot and at chosen locations. paperonwe from the public transport in the all citynecessary of Žilinalimitations. using a standard IP solver to thewill quality of the linear mathematical modelsystem that respects The performance of evaluate the model be tested onsolution the dataand set computational time. from the public transport system in the city of Žilina using a standard IP solver to evaluate the quality of the solution and computational time. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility the scientific committee of the 13th International Scientific Conference on Sustainable, © 2019 The Authors. Published byof Elsevier B.V. committee Peer-review under responsibility of the scientific of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Peer-review under responsibility of the scientific Modern and Safe Transport (TRANSCOM 2019). committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Keywords: Electric vehicles; Scheduling problem; IP solver, Exact solution, Electric busses; Keywords: Electric vehicles; Scheduling problem; IP solver, Exact solution, Electric busses;

* Corresponding author. address:author. [email protected] * E-mail Corresponding E-mail address: [email protected] 2352-1465 © 2018 The Authors. Published by Elsevier B.V. Peer-review©under responsibility of the scientific committee 2352-1465 2018 The Authors. Published by Elsevier B.V. of the 13th International Scientific Conference on Sustainable, Moder n and Safe Transport (TRANSCOM 2019). Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Moder n and Safe Transport (TRANSCOM 2019). 2352-1465  2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). 10.1016/j.trpro.2019.07.191

2

Maroš Janovec et al. / Transportation Research Procedia 40 (2019) 1380–1387 Janovec and Koháni/ Transportation Research Procedia 00 (2019) 000–000

1381

1. Introduction Electromobility is currently a highly-researched area. Great advantages of electric vehicles (EV) are very low CO2 emissions in the area of operation as well as potentially low operational costs. Together with the trend of decreasing purchasing price of electric vehicles and batteries, the electric vehicles are becoming interesting alternative to combustion engine vehicles. One of the places where the electric vehicles can be applied and tested is a public transport system in cities. This system is characteristic by high daily mileage per vehicle, therefore the transformation of the public transport fleet from conventional vehicles to electric vehicles can bring high reduction of operational costs. Main vehicle type used in cities is diesel engine bus. Consequently, the main stream of transformation is replacement of diesel buses by electric buses. Experiments with the application of electric buses on some lines were done in different cities, e.g. Soul (Kim and Song and Choi, 2015), Milton Keynes (Miles and Potter, 2014), Oporto (Perrotta et al., 2014), Putaraya (Teoh et al., 2017). These experiments were mainly conducted for data collection purposes. Therefore, the electric buses were applied only on specific lines, not as a complete transformation of the fleet. Before the transformation of the fleet several problems need to be addressed. One of the main ones are the line planning problem, the vehicle scheduling problem and crew scheduling problem. These problems are in their core the same as the problems which need to be solved when applying the fleet of diesel buses. However, the solution methods need to consider the specific nature of electric buses. The most problematic features are limited driving range of electric bus and the recharging time. Operational range of current electric buses is between 100 and 300 kilometers, which is not enough to drive a standard day workload without charging. Moreover, the charging time of the electric bus is far longer than the time of refueling a diesel bus and the driving range is influenced by weather conditions and outside temperature. This paper addresses the problem of assigning the available electric buses to the service trips, in other words the electric bus scheduling problem (EBSP). In the chapter 2 we will describe related works in the field. Chapter 3 is dedicated to the detailed problem description and formulation of the mathematical model for the solving of the electric bus fleet scheduling. In the last chapter we will describe solution method and numerical experiments on the selected test networks created from the real data provided by the public bus service operator in Žilina. 2. Related work The vehicle scheduling problem(VSP) is quite well researched and good solution methods are known. Our problem is branch of VSP, although it has more restrictions than the classical VSP. In order to different charging technologies the solution methodology differs significantly. It is caused mainly by the different charging times. The first technology is battery exchange system. It is quite perspective from the point of reduced time to refill the energy of electric bus. However, the charging of the battery is still needed, but it is done during the electric bus operation time. Kim, Song and Choi in their paper (Kim, Song and Choi, 2015) tried to show the usability of this technology. Their analysis and simulation were performed on the real data from experimental application of electric bus in Soul. The scheduling with battery exchange technology was researched by Chao and Xiaohong (Chao and Xiaohong, 2013), who adressed the single depot vehicle scheduling problem (SDVSP) with two independent objectives. The first objective represented the capital investment in electric bus purchase and the second one represented the charging demand. They used Non-Dominated Sorting Genetic Algorithm (NSGA-II) to solve the problem. A model of the scheduling problem with the exchange battery system was also proposed by Mirchandani et al. (Mirchandani et al., 2012). The scheduling was not solved on public transport system, but on service system car fleet. The next possible charging technology is opportunity charging. In this technology the charging is performed on chargers, which are situated at selected positions in the transport network. With application of this technology the extended charging time must be considered in the problem solving. Teoh et al. (Teoh et al., 2017) adressed the problem of designing transport system, which includes creation of timetables and scheduling of vehicles and crews.

1382

Maroš Janovec et al. / Transportation Research Procedia 40 (2019) 1380–1387 Janovec and Koháni/ Transportation Research Procedia 00 (2019) 000–000

3

They assume the use of fast charging technology. To solve the problem of scheduling the simulation was used. Although, the simulation approach is better at verification of the result than producing a solution. Other alternatives of solving the scheduling problem are mathematical programing methods. Niekerk et al. (Niekerk et al., 2017) proposed two mathematical models to solve the scheduling problem. These models focus on the scheduling with one depot and one vehicle type. The first of their models assumes that charging is linear process. In the second one they are able to describe also the non-linear charging process by discretizing the state of charge (SOC) of battery. In both models the time is discretized to intervals of the same length. The first model is solved to optimum by the mathematical programming methods for small and medium size problems. The column generation method is used to solve the second one and it is able to solve also bigger size problems. Linear model is also proposed by Sassi and Oularama (Sassi and Oularama, 2014). Their model also considers a combination of electric buses and diesel buses. Besides the basic constraints for battery state of charge, this model contains constraint for electric network capacity. The electric network constraint limits the maximal charging power at one-time interval, which reproduces decrease of charging power if more buses are connected. In addition, in the paper is proposed Sequential heuristic (SH) and Global heuristic (GH) to solve the problem. Also, they prove the NP-hardness of the Electric vehicle scheduling problem (E-VSP). The next model which describes the scheduling problem with the optimization of chargers is proposed by Rogge et al. (Rogge et al., 2018). The charging is performed only in depot and the electric bus is charged until the maximum battery capacity is reached. However, the proposed model is non-linear, their solution of the problem is based on heuristic and metaheuristic approach. In the paper a metaheuristic based on Grouping Genetic Algorithm (GGA) is also proposed and tested. 3. Problem description In this paper we propose a linear mathematical model for electric bus scheduling problem, which is based on Rogge’s model (Rogge et al., 2018). Unlike the Rogge’s model, our model is linear, and the focus is on application of partial charging (battery do not have to be fully charged) and charging in depot and also in charging stations on selected terminal stops, which was not considered in Rogge’s model. We also consider the continuity of time. Therefore, the model we present addresses the partial charging, charging at multiple locations and the continuous time, which was not researched by any of the above-mentioned authors. We assume that the charger’s locations and number of charging points are given. The main goal is to create schedules for each vehicle as well as schedules for each charger. We should use the minimal number of electric buses and respect the limitations of operational range of electric buses and limitations of the battery capacity. In the EBSP, we try to create schedule for electric bus. The schedule for one electric bus is a list of activities which consists of service trips and charging events. In the process of creating a schedule we have to take into account the standard constraints of vehicle scheduling as well as constraints that are specific for electric vehicles. The standard constraints include these limitations: • Each trip is assigned to exactly one vehicle. • The vehicle is assigned to a feasible sequence of service trips, in other words the vehicle cannot be assigned to two or more parallel trips. • The total cost is minimal. Other constraints which are different from the standard limitations of scheduling problem are the constraints considering the battery state of charge (SOC). They are: • The battery SOC of the bus has to be sufficient to drive during the whole bus service to the nearest depot or charger. • At chosen locations the battery can be charged. Although, during the charging the SOC of battery cannot exceed the maximal capacity of the battery. • At each charging point only one bus can charge in one time.

4

Maroš Janovec et al. / Transportation Research Procedia 40 (2019) 1380–1387 Janovec and Koháni/ Transportation Research Procedia 00 (2019) 000–000

1383

3.1. Formal problem formulation Let 𝑁𝑁 be a set of service trips, which needs to be served. Depot is represented by a source 𝐷𝐷0 and sink 𝐷𝐷n . We have a set 𝑅𝑅 of chargers and set of 𝑇𝑇 𝑟𝑟 , which represents charging events on charger 𝑟𝑟 ∈ 𝑅𝑅. Each service trip 𝑖𝑖 ∈ 𝑁𝑁 is characterized by a starting time 𝑠𝑠𝑖𝑖 , time 𝑡𝑡𝑖𝑖 which is spent to drive the service trip and amount of energy required to drive the service trip 𝑐𝑐𝑖𝑖 . Besides these characteristics a set of possible following service trips 𝐹𝐹(𝑖𝑖) is defined for each service trip. A service trip 𝑗𝑗 can be added as a follower of service trip 𝑖𝑖 only if the expression 𝑠𝑠𝑖𝑖 + 𝑡𝑡𝑖𝑖 + 𝑡𝑡𝑖𝑖𝑖𝑖 ≤ 𝑠𝑠𝑗𝑗 holds. The constant 𝑡𝑡𝑖𝑖𝑖𝑖 is time to drive from the end point of service trip 𝑖𝑖 to starting point of service trip 𝑗𝑗 and 𝑠𝑠𝑗𝑗 is starting time of service trip 𝑗𝑗. During the drive from 𝑖𝑖 to 𝑗𝑗 the energy is consumed, so we also need to know the amount of the consumed energy. It is expressed by the constant 𝑐𝑐𝑖𝑖𝑖𝑖 . We also define a set of possible predecessors 𝐵𝐵(𝑖𝑖) similarly to set of followers 𝐹𝐹(𝑖𝑖). The charger 𝑟𝑟 ∈ 𝑅𝑅 is defined by its location and charging speed 𝑞𝑞𝑟𝑟 . Constant 𝑡𝑡𝑖𝑖𝑖𝑖 expresses a time, which is needed to drive from the end of service trip 𝑖𝑖 to charger 𝑟𝑟. The constant 𝑡𝑡𝑟𝑟𝑟𝑟 is the opposite to 𝑡𝑡𝑖𝑖𝑖𝑖 and expresses the time to drive from charger 𝑟𝑟 to the start of service trip 𝑗𝑗. The energy consumption for these connections are defined by the constants 𝑐𝑐𝑖𝑖𝑖𝑖 and 𝑐𝑐𝑟𝑟𝑟𝑟 . In our model we have to work also with charging events. The number of the charging events at a charger 𝑟𝑟 is the same as the number of service trips. Then, the start of charging event 𝑠𝑠𝑟𝑟𝑟𝑟 for event 𝑡𝑡 is derived from service trip 𝑖𝑖 and is defined as 𝑠𝑠𝑟𝑟𝑟𝑟 = 𝑠𝑠𝑖𝑖 + 𝑡𝑡𝑖𝑖 + 𝑡𝑡𝑖𝑖𝑖𝑖 . The charging events on the same charger are ordered by increasing starting time. Next, we define a set of possible following charging events 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑖𝑖) and a set of possible previous charging events 𝐵𝐵𝐵𝐵(𝑟𝑟, 𝑖𝑖) for each service trip 𝑖𝑖 and each charger 𝑟𝑟. The charging event 𝑡𝑡 can be a follower of service trip 𝑖𝑖 if 𝑠𝑠𝑟𝑟𝑟𝑟 ≥ 𝑠𝑠𝑖𝑖 + 𝑡𝑡𝑖𝑖 + 𝑡𝑡𝑖𝑖𝑖𝑖 . Similarly, we define a set of possible following trips 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑡𝑡) and possible previous trips 𝐵𝐵𝐵𝐵(𝑟𝑟, 𝑡𝑡) of a charging event 𝑡𝑡 on charger 𝑟𝑟. Let the 𝐾𝐾 be a set of available vehicles. The battery in each vehicle 𝑘𝑘 ∈ 𝐾𝐾 is characterized by its minimal and 𝑘𝑘 𝑘𝑘 and 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 . These constants represent the interval of available energy. We define maximal state of charge 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 them to lower the degradation process of battery or to set a reserve energy for an electric vehicle (Rogge et al., 2015).

Fig. 1. Description of decision variables Let the set of binary decision variable 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 represents the decision, if the service trip 𝑗𝑗 is going to be served right 𝑘𝑘 will be after the service trip 𝑖𝑖 by vehicle 𝑘𝑘. The variable 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 exists only if 𝑗𝑗 ∈ 𝐹𝐹(𝑖𝑖). Binary decision variable 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 one, if the vehicle 𝑘𝑘 is going to charge at charging event 𝑡𝑡 on charger 𝑟𝑟 after finishing the service trip 𝑖𝑖. Also, the 𝑘𝑘 be one, if the vehicle 𝑘𝑘 is going charging event 𝑡𝑡 must be a follower of the service trip 𝑖𝑖. Let the binary decision 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 to serve the service trip 𝑗𝑗 after charging at the charging event 𝑡𝑡 on charger 𝑟𝑟. As before, for the service trip 𝑗𝑗 and 𝑘𝑘 , which represents charging event 𝑡𝑡 on charger 𝑟𝑟 must stand 𝑗𝑗 ∈ 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑡𝑡). The last binary decision variable is 𝑤𝑤𝑟𝑟𝑟𝑟 decision whether the vehicle 𝑘𝑘 continues charging at charging event 𝑡𝑡 + 1 on charger 𝑟𝑟 after charging at charging event 𝑡𝑡 on the same charger 𝑟𝑟. The visual description of the decision variables mentioned above is on Figure 1. To 𝑘𝑘 . Variable 𝑒𝑒𝑖𝑖𝑘𝑘 represents energy state of keep track of the energy state of battery we introduce variable 𝑒𝑒𝑖𝑖𝑘𝑘 and 𝜀𝜀𝑟𝑟𝑟𝑟 𝑘𝑘 vehicle 𝑘𝑘 just before the start of service trip 𝑖𝑖. Similarly, variable 𝜀𝜀𝑟𝑟𝑟𝑟 represents energy state of vehicle 𝑘𝑘 just before the start of charging event 𝑡𝑡 on charger 𝑟𝑟. The charged energy is limited not only by the maximum capacity of the battery, but also by available charging time at each event. The charging time is limited by the start of the next charging event at the same charger and by the starting time of the following service trip. In the specific example, if two events have the same starting time, then the first of them has a zero-length time interval for charging, but due to 𝑘𝑘 , it is possible to continue charging at the second event. the existence of the decision variable 𝑤𝑤𝑟𝑟𝑟𝑟

Maroš Janovec et al. / Transportation Research Procedia 40 (2019) 1380–1387 Janovec and Koháni/ Transportation Research Procedia 00 (2019) 000–000

1384

5

3.2. Mathematical model In this section we present our linear model formulation of the electric bus scheduling problem in the public transport system. 3.3. Objective function min ∑ ∑ 𝑥𝑥𝐷𝐷𝑘𝑘0𝑗𝑗

(1)

𝑘𝑘∈𝐾𝐾 𝑗𝑗∈𝑁𝑁

The objective (1) minimizes the number of used electric buses. 3.4. Vehicle scheduling constraints ∑ ∑ 𝑥𝑥ⅈ𝑗𝑗𝑘𝑘 + ∑ ∑

𝑘𝑘∈𝐾𝐾 𝑖𝑖∈𝐵𝐵(𝑗𝑗)

∑ 𝑥𝑥ⅈ𝑗𝑗𝑘𝑘 + ∑

𝑖𝑖∈𝐵𝐵(𝑗𝑗)

∑

𝑘𝑘∈𝐾𝐾 𝑟𝑟∈𝑅𝑅 𝑡𝑡∈𝐵𝐵𝐵𝐵(𝑟𝑟,𝑗𝑗)

𝑟𝑟∈𝑅𝑅 𝑡𝑡∈𝐵𝐵𝐵𝐵(𝑟𝑟,𝑗𝑗)

∑

𝑘𝑘 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 = 1 ; ∀𝑗𝑗 ∈ 𝑁𝑁

𝑘𝑘 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 = ∑ 𝑥𝑥𝑗𝑗𝑗𝑗𝑘𝑘 + ∑ 𝑙𝑙∈𝐹𝐹(𝑗𝑗)

∑

𝑟𝑟∈𝑅𝑅 𝑡𝑡∈𝐹𝐹𝐹𝐹(𝑟𝑟,𝑗𝑗)

𝑘𝑘 𝑦𝑦𝑗𝑗𝑗𝑗𝑗𝑗 ; ∀𝑗𝑗 ∈ 𝑁𝑁, ∀𝑘𝑘 ∈ 𝐾𝐾

𝑘𝑘 𝑘𝑘 + ∑ 𝑤𝑤𝑟𝑟𝑟𝑟−1 ≤ 1 ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 ∑ 𝑦𝑦𝑗𝑗𝑗𝑗𝑗𝑗

𝑘𝑘∈𝐾𝐾 𝑗𝑗∈𝐵𝐵𝐵𝐵(𝑟𝑟,𝑡𝑡)

𝑘𝑘 𝑘𝑘 ∑ 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑤𝑤𝑟𝑟𝑟𝑟−1 =

𝑖𝑖∈𝐵𝐵𝐵𝐵(𝑟𝑟,𝑡𝑡)

∑

𝑘𝑘∈𝐾𝐾

𝑘𝑘 𝑘𝑘 ∑ 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑤𝑤𝑟𝑟𝑟𝑟 ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾

𝑗𝑗∈𝐹𝐹𝐹𝐹(𝑟𝑟,𝑡𝑡)

(2) (3) (4) (5)

Constraints (2) ensure that each service trip is served by exactly one electric bus. As a flow constraint for each service trip, which means that if the bus come to serve the service trip it has to leave after serving, serves constraint (3). Constraints (4) define limitation, that at one moment at most one vehicle can charge at charger. Constraints (5) represent the flow constraint of electric bus during charging event, that means if the bus comes to charge it has to leave after charging. 3.5. Energy consumption constraints 𝑘𝑘 ; ∀𝑘𝑘 ∈ 𝐾𝐾 𝑒𝑒𝐷𝐷𝑘𝑘0 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚

(6)

𝑘𝑘 𝑘𝑘 + 𝑐𝑐𝑖𝑖 + ∑ 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 𝑐𝑐𝑖𝑖𝑖𝑖 + ∑ ∑ 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐𝑖𝑖𝑖𝑖 ; ∀𝑖𝑖 ∈ 𝑁𝑁 ; ∀𝑘𝑘 ∈ 𝐾𝐾 𝑒𝑒𝑖𝑖𝑘𝑘 ≥ 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚

(7)

𝑘𝑘 (1 − 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 ) ; ∀𝑗𝑗 ∈ 𝑁𝑁, ∀𝑖𝑖 ∈ 𝐵𝐵(𝑗𝑗), ∀𝑘𝑘 ∈ 𝐾𝐾 𝑒𝑒𝑗𝑗𝑘𝑘 ≤ 𝑒𝑒𝑖𝑖𝑘𝑘 − 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 (𝑐𝑐𝑖𝑖 + 𝑐𝑐𝑖𝑖𝑖𝑖 ) + 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚

(8)

𝑗𝑗∈𝐹𝐹(𝑖𝑖)

𝑟𝑟∈𝑅𝑅 𝑡𝑡∈𝐹𝐹𝐹𝐹(𝑟𝑟,𝑖𝑖)

𝑘𝑘 𝑒𝑒𝑗𝑗𝑘𝑘 ≥ 𝑒𝑒𝑖𝑖𝑘𝑘 − 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 (𝑐𝑐𝑖𝑖 + 𝑐𝑐𝑖𝑖𝑖𝑖 ) − 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 ) ; ∀𝑗𝑗 ∈ 𝑁𝑁, ∀𝑖𝑖 ∈ 𝐵𝐵(𝑗𝑗), ∀𝑘𝑘 ∈ 𝐾𝐾

𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑒𝑒𝑗𝑗𝑘𝑘 ≤ 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 + 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 ((𝑠𝑠𝑗𝑗 − 𝑡𝑡𝑟𝑟𝑟𝑟 − 𝑠𝑠𝑟𝑟𝑟𝑟 )𝑞𝑞𝑟𝑟 − 𝑐𝑐𝑟𝑟𝑟𝑟 ) + 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 ) ; ∀𝑗𝑗 ∈ 𝑁𝑁, ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝐵𝐵𝐵𝐵(𝑟𝑟, 𝑗𝑗), ∀𝑘𝑘 ∈ 𝐾𝐾 𝑘𝑘 𝑘𝑘 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 ≥ 𝑒𝑒𝑗𝑗𝑘𝑘 + 𝑐𝑐𝑟𝑟𝑟𝑟 − 𝑀𝑀𝑞𝑞𝑟𝑟 (1 − 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 ); ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾 , ∀𝑗𝑗 ∈ 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑡𝑡) 𝑘𝑘 𝑘𝑘 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 ≥ 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟+1 −𝑀𝑀𝑞𝑞𝑟𝑟 (1 − 𝑤𝑤𝑟𝑟𝑟𝑟 ) ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾

(9)

(10)

(11)

(12)

Janovec and Koháni/ Transportation Research Procedia 00 (2019) 000–000 Maroš Janovec et al. / Transportation Research Procedia 40 (2019) 1380–1387

6

1385

𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑒𝑒𝑗𝑗𝑘𝑘 + 𝑐𝑐𝑟𝑟𝑟𝑟 + 𝑀𝑀𝑞𝑞𝑟𝑟 (1 − 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 ) ≥ 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐𝑟𝑟𝑟𝑟 ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑗𝑗 ∈ 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑡𝑡)

(13)

𝑘𝑘 𝑘𝑘 𝑘𝑘 (𝑐𝑐𝑖𝑖 + 𝑐𝑐𝑖𝑖𝑖𝑖 ) − 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 ≥ 𝑒𝑒𝑖𝑖𝑘𝑘 − 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 (1 − 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 ) ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑖𝑖 ∈ 𝐵𝐵𝐵𝐵(𝑟𝑟, 𝑡𝑡)

(15)

𝑘𝑘 𝑘𝑘 𝑘𝑘 (𝑐𝑐𝑖𝑖 + 𝑐𝑐𝑖𝑖𝑖𝑖 ) + 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 ≤ 𝑒𝑒𝑖𝑖𝑘𝑘 − 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 (1 − 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 ) ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑖𝑖 ∈ 𝐵𝐵𝐵𝐵(𝑟𝑟, 𝑡𝑡)

(14)

𝑘𝑘 (𝑠𝑠 𝑘𝑘 𝑘𝑘 𝑟𝑟 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟+1 ≤ 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 + 𝑤𝑤𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟+1 − 𝑠𝑠𝑟𝑟𝑟𝑟 )𝑞𝑞𝑟𝑟 + 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝑤𝑤𝑟𝑟𝑟𝑟 ); ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 , ∀𝑘𝑘 ∈ 𝐾𝐾

(16)

𝑘𝑘 𝑘𝑘 𝑒𝑒𝑗𝑗𝑘𝑘 + 𝑐𝑐𝑟𝑟𝑟𝑟 − 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 + 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 ) ≥ 0 ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑗𝑗 ∈ 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑡𝑡)

(18)

𝑘𝑘 𝑘𝑘 (1 − 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 ) ≤ (𝑠𝑠𝑟𝑟𝑟𝑟+1 − 𝑠𝑠𝑟𝑟𝑟𝑟 )𝑞𝑞𝑟𝑟 ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑗𝑗 ∈ 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑡𝑡) 𝑒𝑒𝑗𝑗𝑘𝑘 + 𝑐𝑐𝑟𝑟𝑟𝑟 − 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 − 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 𝑘𝑘 𝑘𝑘 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟+1 − 𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 + 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝑤𝑤𝑟𝑟𝑟𝑟 ) ≥ 0 ; ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑘𝑘 ∈ 𝐾𝐾

(17)

(19)

Constraints (6) define that at the start of the day, the energy state is set to maximum level for each vehicle. Constraints (7) ensure, that each bus has enough energy to drive the service trip and to the next task or depot or charger. Constraints (8) and (9) define a preservation of energy between two service trips which follow one after another. Constraints (10) serves as limitation of energy state. The energy state before the service trip must be lower than the energy state of previous charging plus the maximum amount of energy that can be charged during the charging event minus energy consumption on the transfer from charger to service trip. Constraints (11) and (12) ensure that the energy state does not exceed maximum capacity during the charging process depending on the following trip respectively following charging event. The big constant M used in these constraints is a big number that guarantees relaxation of the limitation if the bus does not drive from the charger to the next trip or the bus does not continue at next charging event. The limitation that bus has enough energy after charging to travel from the charger to the next service trip represents constraints (13). Preservation of energy between the service trip and the charger, where the next charging happens, ensure constraints (14) and (15). Constraints (16) serve as an upper bound of charged energy at the charging event depending on the available time limited by the next charging event starting time. Specifically, if the charging events following one after another have the same starting time the amount of energy charged between them is zero. Constraints (17) limit the maximum charged energy during the charging event at the charger by the maximum energy that can be charged before the start of the next charging event on the same charger, if the bus continues after charging to the next the service trip. Constraints (18) and (19) define that the charged energy during the charging event must be non-negative depending on the following trip respectively following charging event. 3.6. Obligatory constraints 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 ∈ {0,1} ; ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑖𝑖 ∈ 𝑁𝑁 ∪ 𝐷𝐷0 ∪ 𝐷𝐷𝑛𝑛 , ∀𝑗𝑗 ∈ 𝐹𝐹(𝑖𝑖)

(20)

𝑘𝑘 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 ∈ {0,1} ; ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑖𝑖 ∈ 𝑁𝑁, ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑖𝑖)

(22)

𝑘𝑘 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 ∈ {0,1} ; ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 , ∀𝑗𝑗 ∈ 𝐹𝐹𝐹𝐹(𝑟𝑟, 𝑡𝑡)

(21)

𝑘𝑘 ∈ {0,1} ; ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟 𝑤𝑤𝑟𝑟𝑟𝑟

(23)

𝑒𝑒𝑖𝑖𝑘𝑘 ≥ 0 ; ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑖𝑖 ∈ 𝑁𝑁

(25)

𝜀𝜀𝑘𝑘𝑟𝑟𝑟𝑟 ≥ 0 ; ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑟𝑟 ∈ 𝑅𝑅, ∀𝑡𝑡 ∈ 𝑇𝑇 𝑟𝑟

(24)

1386

Maroš Janovec et al. / Transportation Research Procedia 40 (2019) 1380–1387 Janovec and Koháni/ Transportation Research Procedia 00 (2019) 000–000

7

𝑘𝑘 𝑘𝑘 𝑘𝑘 Constraints (20), (21), (22) and (23) define binary nature of the decision variables 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 , 𝑧𝑧𝑟𝑟𝑟𝑟𝑟𝑟 , 𝑦𝑦𝑖𝑖𝑖𝑖𝑖𝑖 and 𝑦𝑦𝑟𝑟𝑟𝑟 . Finally, 𝑘𝑘 𝑘𝑘 constraints (24) and (25) defines the range of variables 𝜀𝜀𝑟𝑟𝑟𝑟 and 𝑒𝑒𝑖𝑖 .

4. Numerical experiments

We performed number of experiments to test our model and see its limitations from the perspective of solution time and quality. The tests were performed on the machine equipped with Intel Core i5-7200U 2,5Ghz, 16GB of RAM. We solved our mathematical model exactly using the standard IP solver Xpress IVE. 4.1. Data and scenario description Data for numerical experiments were provided by the public transport operator DPMŽ in the city of Žilina. These data contain information about service trips performed by diesel buses during a working day. To test proposed model, we created four datasets. The first dataset (denoted as DS1) consists of 49 trips served on the bus line 26. The second dataset (DS2) covers the bus line 27 with 77 trips. The third dataset (DS3) contains 83 trips on the bus lines 26 and 29. The last dataset (DS4) is union of all trips on bus lines 26, 27 and 29 and contains 160 service trips. To be able to test proposed model, we need also the information about the charger stations location. We created three scenarios with different number of chargers and their locations. In the first scenario, denoted as scenario A, there are three chargers distributed at two locations, specifically one charger at the center of city and two chargers at the trolleybus depot. This situation represents the current location of the charging stations. In the second scenario, denoted as scenario B, we add two chargers at the bus depot, where all electric buses start their daily service. Though, we have five chargers in total in the second scenario. The last scenario, denoted as scenario C, we add one more charger at the train station to the total of six chargers. Each scenario is performed on the four datasets creating 12 experiments. In the experiments we test a homogenous fleet with one type of vehicle. For each experiment we define these common constants: • Maximum battery capacity - 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 = 140 kWh • Minimum battery capacity - 𝑆𝑆𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 = 0 kWh • Charging power of each charger 𝑞𝑞𝑟𝑟 = 1.33 kWh/min • Energy consumption 𝐶𝐶 = 0.8 𝑘𝑘𝑘𝑘ℎ/𝑘𝑘𝑘𝑘 • Average speed at deadhead trips 𝑣𝑣 = 35 𝑘𝑘𝑘𝑘/ℎ

4.2. Results

The results from the performed experiments are listed in Table 1. As we can see from the results the number of trips causes major increase in the complexity of the problem, which is shown by extended solution time. Second factor which increase the complexity is the number and location of available chargers. The changes in solution time can be seen in the Table 1. For reference, also the results of classical VSP are listed in the Table 1. The classic VSP which was used is showed below with the objective (26) and constraints (27) representing serving constraints, constraints (28) representing flow constraints and obligatory constraints (29). min ∑ ∑ 𝑥𝑥𝐷𝐷𝑘𝑘0𝑗𝑗

(26)

∑ ∑ 𝑥𝑥ⅈ𝑗𝑗𝑘𝑘 = 1 ; ∀𝑗𝑗 ∈ 𝑁𝑁

(27)

∑ 𝑥𝑥ⅈ𝑗𝑗𝑘𝑘 = ∑ 𝑥𝑥𝑗𝑗𝑗𝑗𝑘𝑘 ; ∀𝑗𝑗 ∈ 𝑁𝑁, ∀𝑘𝑘 ∈ 𝐾𝐾

(28)

𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 ∈ {0,1} ; ∀𝑘𝑘 ∈ 𝐾𝐾, ∀𝑖𝑖 ∈ 𝑁𝑁 ∪ 𝐷𝐷0 ∪ 𝐷𝐷𝑛𝑛 , ∀𝑗𝑗 ∈ 𝐹𝐹(𝑖𝑖)

(29)

𝑘𝑘∈𝐾𝐾 𝑗𝑗∈𝑁𝑁

𝑘𝑘∈𝐾𝐾 𝑖𝑖∈𝐵𝐵(𝑗𝑗)

𝑖𝑖∈𝐵𝐵(𝑗𝑗)

𝑙𝑙∈𝐹𝐹(𝑗𝑗)

8

Maroš Janovec et al. / Transportation Research Procedia 40 (2019) 1380–1387 Janovec and Koháni/ Transportation Research Procedia 00 (2019) 000–000

1387

The results of EBSP for the presented datasets are the same as the results for the classical VSP. Therefore, we can say, that for the specified lines the transformation from the classical buses to electric buses would be one classical bus for one electric bus exchange. Table 1. Results of experiments Dataset

DS1

DS2

DS3

DS4

Scenario

A

B

C

A

B

C

A

B

C

A

B

C

Num. of chargers

3

5

6

3

5

6

3

5

6

3

5

6

Num. of trips

49

49

49

77

77

77

83

83

83

160

160

160

Num. of buses in VSP

4

4

4

4

4

4

5

5

5

9

9

9

Solution of EBSP – num. of buses

4

4

4

4

4

4

5

5

5

9

9

9

Solution time of EBSP (s)

1.0

9.3

11.1

28.8

81.3

7.9

67.7

42.9

75.0

4182.5

6965.3

1194.6

5. Conclusions In this paper a model for electric bus scheduling problem was proposed. The model was tested on the datasets provided by the public transport operator DPMŽ in the city of Žilina. The experiments verified the model and proved the possibility of using this model to solve the scheduling problem in real situations. Also, during the experiments the bounds of problems which are possible to solve to optimality with the standard IP solver were found. In the reality we have to consider much larger scheduling problems than the experiments, which were performed. Due to these limitations another solution approach based on faster methods like heuristics and metaheuristics should be proposed in the future. Acknowledgement This work was supported by the research grants VEGA 1/0089/19 "Data analysis methods and decisions support tools for service systems supporting electric vehicles" and VEGA 1/0689/19 "Optimal design and economically efficient charging infrastructure deployment for electric buses in public transportation of smart cities". References Kim, Jeongyong & Song, Inho & Choi, W. (2015). An Electric Bus with a Battery Exchange System. Energies. 8. 6806-6819. 10.3390/en8076806. Miles, John & Potter, Stephen. (2014). Developing a viable electric bus service: The Milton Keynes demonstration project. Research in Transportation Economics. 48. 10.1016/j.retrec.2014.09.063. Perrotta, Deborah & Macedo, José & Rossetti, Rosaldo & Sousa, Jorge & Kokkinogenis, Zafeiris & Ribeiro, Bernardo & Afonso, J.L.. (2014). Route Planning for Electric Buses: A Case Study in Oporto. Procedia - Social and Behavioral Sciences. 111. 1004-1014. 10.1016/j.sbspro.2014.01.135. Teoh, Lay Eng & Khoo, Hooi Ling & Yoke Goh, Siew & Mun Chong, Lai. (2017). Scenario-based Electric Bus Operation: A Case Study of Putrajaya, Malaysia. International Journal of Transportation Science and Technology. 7. 10.1016/j.ijtst.2017.09.002. Chao, Zhu & Xiaohong, Chen. (2013). Optimizing Battery Electric Bus Transit Vehicle Scheduling with Battery Exchanging: Model and Case Study. Procedia - Social and Behavioral Sciences. 96. 2725-2736. 10.1016/j.sbspro.2013.08.306. Mirchandani, P., Madsen, O. B. G., & Adler, J. (2012). Scheduling and location issues in transforming service fleet vehicles to electric vehicles. Paper presented at 12th International Conference on Advanced Systems for Public Transport, Santiago, Chile. van Kooten Niekerk, Marcel & Akker, J.M. & Hoogeveen, J.A.. (2017). Scheduling electric vehicles. Public Transport. 9. 10.1007/s12469-0170164-0. Sassi, Ons & Oulamara, Ammar. (2014). Electric Vehicle Scheduling and Optimal Charging Problem: Complexity, Exact and Heuristic Approaches. International Journal of Production Research. 55. 10.1080/00207543.2016.1192695. Rogge, Matthias & van der Hurk, Evelien & Larsen, Allan & Sauer, Dirk. (2018). Electric bus fleet size and mix problem with optimization of charging infrastructure. Applied Energy. 211. 282-295. 10.1016/j.apenergy.2017.11.051. Rogge, M.& Wollny, S.& Sauer, D.U. Fast Charging Battery Buses for the Electrification of Urban Public Transport—A Feasibility Study Focusing on Charging Infrastructure and Energy Storage Requirements. Energies 2015, 8, 4587-4606.