A bi-level optimization framework for charging station design problem considering heterogeneous charging modes

Le Zhang; Ziling Zeng; Kun Gao

doi:10.1108/jicv-07-2021-0009

A bi-level optimization framework for charging station design problem considering heterogeneous charging modes

Zhang, Le; Zeng, Ziling; Gao, Kun 2022-02-17 00:00:00 Purpose – The purpose of this paper is to optimize the design of charging station deployed at the terminal station for electric transit, with explicit consideration of heterogenous charging modes. Design/methodology/approach – The authors proposed a bi-level model to optimize the decision-making at both tactical and operational levels simultaneously. Speciﬁcally, at the operational level (i.e. lower level), the service schedule and recharging plan of electric buses are optimized under speciﬁc design of charging station. The objective of lower-level model is to minimize total daily operational cost. This model is solved by a tailored column generation-based heuristic algorithm. At the tactical level (i.e. upper level), the design of charging station is optimized based upon the results obtained at the lower level. A tabu search algorithm is proposed subsequently to solve the upper-level model. Findings – This study conducted numerical cases to validate the applicability of the proposed model. Some managerial insights stemmed from numerical case studies are revealed and discussed, which can help transit agencies design charging station scientiﬁcally. Originality/value – The joint consideration of heterogeneous charging modes in charging station would further lower the operational cost of electric transit and speed up the market penetration of battery electric buses. Keywords Battery electric bus, Charging station design, Vehicle scheduling, Bi-level model, Heterogeneous charging modes Paper type Research paper powered by other energy sources, resulting in users’“range 1. Introduction anxiety” (Lebeau et al., 2016; Masmoudi et al.,2018; Qin et al., Electric transit is considered as the key to the world’s clean 2016; Li et al., 2019). To ensure normal operations, the transport future due to its high energy efﬁciency, zero emissions consumed electricity must be replenished by either battery (Lajunen, 2014; Jin et al.,2015; Xu et al., 2021; Qu et al., 2020; swapping or battery recharging (Li, 2013; Huang and Zhou, Zhang et al., 2020; Zhang et al., 2021) and shareability (Gao 2015; Wang et al.,2017; Tang et al., 2019; Liu and Ceder, et al.,2021; Bie et al.,2020; Meng and Qu, 2013; Wang et al., 2020; Rinaldi et al.,2020). Unfortunately, instead of mitigating 2018). Compared with diesel buses, battery electric buses this disadvantage, lack of sufﬁcient charging facilities further (BEBs) are able to improve energy efﬁciency by 50% and aggravates it (An, 2020). However, if sufﬁcient charging reduce greenhouse gas emissions by 98.36% (Mahmoud et al., facilities are deployed, it may cause severe budget burden for 2016). During the past decade, the public transit is electriﬁed transit system. Meanwhile, charging modes also affect the step by step. For example, in the USA, the share of BEBs in the charging efﬁciency as well as the infrastructure installation cost. Speciﬁcally, compared with normal charging, the ﬂeet size can bus market increased rapidly from 2% in 2007 to nearly 20% in be reduced signiﬁcantly through improved charging efﬁciency 2015 (Neff and Dickens, 2016); in Europe, the percentage of BEBs on the sales volumes of city buses is up to 10% by 2019, and this number rises up to around 20% in 2020. © Le Zhang, Ziling Zeng and Kun Gao. Published in Journal of Intelligent Undoubtedly, transit electriﬁcation is becoming an and Connected Vehicles. Published by Emerald Publishing Limited. This unstoppable trend. article is published under the Creative Commons Attribution (CC BY 4.0) Compared with diesel buses, the driving characteristics and licence. Anyone may reproduce, distribute, translate and create derivative refueling manner of BEBs are distinct. Speciﬁcally, BEBs works of this article (for both commercial and non-commercial purposes), generally have a much shorter operational range than buses subject to full attribution to the original publication and authors. The full terms of this licence maybe seen at http://creativecommons.org/licences/ by/4.0/legalcode The current issue and full text archive of this journal is available on Emerald This work is supported by National Natural Science Foundation of Insight at: https://www.emerald.com/insight/2399-9802.htm China (No. 72101115) and Natural Science Foundation of Jiangsu (No. BK20210316). Received 8 July 2021 Journal of Intelligent and Connected Vehicles Revised 28 September 2021 5/1 (2022) 8–16 5 December 2021 Emerald Publishing Limited [ISSN 2399-9802] [DOI 10.1108/JICV-07-2021-0009] Accepted 16 December 2021 8 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 by adaptation of fast charging mode, but it also causes higher Our key contributions from a theoretical and practical point infrastructure installation cost. Therefore, how to design the of view can be summarized as follows: charging station, trading off charging availability, charging We are, to our best knowledge, the ﬁrst to formulate and efﬁciency and limited budget become an important issue in solve charging station design problem with explicit transit electriﬁcation. To this end, we aim at studying the consideration of multiple charging modes and their optimal design of charging station deployed at the terminal corresponding effect on battery capacity fading. station for electric transit in this paper. To be speciﬁc, we A number of managerial insights stemmed from the propose a bi-level model, where the lower-level model numerical case study are outlined, which can serve as a optimizes the scheduling of BEBs given the design of charging solid theoretical foundation for more cost-efﬁcient station, including the number of charging facilities of different charging station design. charging modes (i.e. fast charging and normal charging); the The rest of this paper is organized as follows. Section 2 presents upper-level model optimizes the design of charging station with the problem formulation, i.e. a bi-level model. Section 3 explicit consideration of multiple charging modes. In the bi- elaborates the proposed solution approach for solving the level approach, the lower-level problem, i.e. optimal scheduling problem. The numerical cases are conducted in Section 4. of BEBs, is the key and difﬁcult part of the work. Therefore, we Conclusions are summarized in Section 5. next present the relevant studies in the realm of vehicle scheduling. 2. Mixed charging station design problem Bus scheduling problem consists of assigning buses to serve a series of timetabled trips with the objective of minimizing ﬂeet 2.1 Problem description size and/or operational costs. It is an extension of the well- In this section, a single-terminal transit network is considered known vehicle scheduling problem (VSP), which has been to deﬁne the optimization problem of charging station design, extensively studied in the literature (Markovic et al., 2015; as depicted in Figure 1. BEBs depart from the terminal station Schöbel, 2017). Generally speaking, VSP can be categorized to operate a sequence of scheduled round-trips, denoted as set into two groups: the single-depot VSP (SDVSP) (Paixão and V. For simplicity, we refer to the round-trip as trip for short Branco, 1987; Freling et al., 2001; Kang et al., 2019) and the from now. Charging facilities in mode q [ Q ={q , q } are 1 2 multiple-depot VSP (MDVSP) (Dell’Amico et al.,1993; deployed at terminal station with limited number C , where q q 1 Kliewer et al., 2006). Over the years, many varieties and indicates normal charging mode and q indicates fast charging extensions of VSP have been proposed to incorporate the real- mode. For each trip i([ V), the departure time s , travel time e i i world constraints and conditions, including VSP with multiple and the consumption of battery level relative to battery vehicle types (Ceder, 2011), VSP with route constraints (VSP- capacity, m are predeﬁned and deterministic. The objective of RC) (Bunte and Kliewer, 2009), the alternative fuel VSP (AF- this problem is to minimize the total cost of transit agency, VSP) (Li, 2013; Adler, 2014) and electric VSP (E-VSP) (Wen including bus acquisition cost, charging fee, maintenance cost et al.,2016). Among these varieties, VSP-RC, AF-VSP and E- of BEB ﬂeet and the cost incurred by the deployment of VSP are strongly motivated by electric vehicles. To accounting charging facilities. Therefore, the operators shall make for the speciﬁcs of electric vehicles, route duration or route decisions at both tactical and operational levels. To be speciﬁc, distance is constrained in VSP-RC (Haghani and Banihashemi, at the tactical level, the number of charging facilities in each 2002). If vehicles are allowed to be refueled at given recharging type deployed at the terminal station should be optimized and stations to prolong the total distance, that is AF-VSP. the vector of decision variables at this level is denoted by C ¼ However, traditional AF-VSP only considers full charging and {C jq [ Q}. At the operational level, the operators shall make the charging time is set as ﬁxed. Speciﬁcally, the vehicle’s fuel decisions on: how to assign BEBs to serve a series of trips level is set to be full after visiting any recharging stations. For satisfying the minimal battery level constraint and how to example, Li (2013) incorporated vehicle waiting time at optimize recharging schedule considering limited charging charging stations into the model, and the charging time was facilities (i.e. given speciﬁc charging station design). The simpliﬁed as ﬁxed by considering battery swapping. Later, E- VSP was proposed, where partial charging was allowed and the Figure 1 Single-terminal transit network charging time was usually assumed to be a linear function of the charged amount. Unfortunately, whereas much efforts have been made to deal with BEB scheduling, very little attention has been dedicated to explicitly modeling the design of charging station for electric transit, with full consideration of multiple charging modes. It will cause unforeseen operational cost when promoting transit electriﬁcation. In light of the above literature, this paper would employ the latest study in electric vehicle scheduling to study the optimal design of charging station, and the bi-level solution approach is adopted to ﬁx this problem. Numerical case studies were conducted to validate the applicability of the proposed model. It reveals that it is a cost-efﬁcient choice to deploy sufﬁcient charging facilities at the terminal station as the unit cost of charging facilities per day is much lower than that of BEBs. 9 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 X X X X corresponding decision variables at this level (i.e. service 1 m M1 m t s 1 e ; 8i 2 V i i itq itq t2T q2Q t2T q2Q sequence and charging strategy) are denoted by vector X. Notations used in this paper are summarized in Appendix. (1e) 2.2 Lower-level problem: optimal scheduling of battery f C ; 8t 2 T; q 2 Q (1f) itq q electric bus ﬂeet i2V The objective of the lower-level model is to minimize total 1 m M1 FðÞ SOC ; q i 1 itq operational cost, including bus acquisition cost, charging fee and t1 FðÞ SOCi;q11 maintenance cost within one day, where the maintenance cost is f 0 1 m M1 FðÞ SOC ; q ; itq i 1 it q1 1 t ¼t mainly incurred by battery degradation. It is worth to note that the 8t 2 T; i 2 V (1g) total charging fee is constant in our model, as it is related to the predeﬁned tripservice, whichis ﬁxed and independent of BEB schedule. Therefore, the objective function is simpliﬁed as the sum 1 m M1 FðÞ SOC ; q i 2 itq of bus acquisition cost and maintenance cost. The vector of decision variables at the operational level, X,can be deﬁned as X ={d , m , f 1 m M1 FðÞ SOC ; q ; 8t 2 T; i 2 V ij itq i 2 itq itq 2 2 f ji[ V, t[ T, q[ Q}, where: itq (1h) d [ {0, 1}: set to one if BEB serves trips i and j consecutively, ij where trip i begins earlier than trip j; and to zero otherwise, i[ V 1 m ðÞ 1 d M SOC 1 m 1ðÞ 1 d M; 8i 2 V i Oi i i Oi | O, j [ V | D, i = j.Here O denotes a virtual trip that every (1i) bus must serve before its ﬁrst real trip, and D denotes another virtual trip that each BEB serves after completing the last real trip of a day and being fully charged. The two notations are SOC lb; 8i 2 V (1j) deﬁned for the convenience of modeling work. The virtual X X trips’ travel times and electricity consumption are all set to zero: SOC SOC m 1ðÞ 1 d M1 M m ; 8i; j 2 V j i j ij itq m [ {0, 1}: set to one if BEB begins to charge with charging itq t2T q2Q mode q at time step t after ﬁnishing trip i and before serving the (1k) next trip, and to zero otherwise, i[ V, t[ T, q[ Q. f [ {0, 1}: set to one if BEB is under charging with itq X X SOC SOC m 1 d M M m ; 8i; j 2 V charging mode q at time step t after ﬁnishing trip i and before ðÞ j i j ij itq t2T q2Q serving the next trip, and to zero otherwise, i[ V, t[ T, q[ Q. In this model, we make the following assumptions: (1l) Assumption 1: The time is discretized with the unit time X X as 10 min. The time for full charge in mode q (i.e. normal SOC 1 m 1 1 d M1 1 m M; 8i; j 2 V ðÞ j j ij itq charging) and mode q (i.e. fast charging) are 2 h (i.e. 12 2 t2T q2Q time steps) and 10 min (i.e. 1 time step), respectively. (1m) Assumption 2: BEBs are fully charged when departing from the original depot, and are charged back to full when X X SOC 1 m ðÞ 1 d M 1 m M; 8i; j 2 V j j ij itq returning to destination depot. t2T q2Q Assumption 3: After ﬁnishing one trip, BEB can either be (1n) charged for one time and charged to full or serve the next trip consecutively without any charging activity. 2 jðÞ SOC ; 1ðÞ 1 SOC i i dðÞ SOC ; 1 ¼ W g q ; 8i 2 V; q 2 Q The lower-level model [P1] can be formulated as: q i ðÞ X XX X (1o) min J C jq 2 Q ¼ v ~ d 1 m dðÞ SOC ; 1 ½ P1 q Oi itq q i i2V i2V t2T q2Q X X s s 1 e ðÞ 1 d M M m ; 8i; j 2 V j i i ij itq (1a) t2T q2Q (1p) Subject to: ðÞ s m t1 F SOC ; q ðÞ 1 d M j i 1 ij itq d ¼ 1; 8j 2 V (1b) ij t2T i2V[O 1 m M; 8i; j 2 V (1q) itq X X t2T d d ¼ 0; 8i 2 V (1c) ij ji j2V[D j2V[O s m t1 FðÞ SOC ; q ðÞ 1 d M j i 2 ij itq X X t2T m 1; 8i 2 V (1d) 1 m M; 8i; j 2 V (1r) itq itq t2T q2Q t2T 10 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 X X 1 SoC ðÞ i m 11 M 1 d ; 8i 2 V (1s) iD itq SoC ¼ (2c) i;dev t2T q2Q X X Here the coefﬁcients g , g , g and g are constant model 1 2 3 4 m 1 MðÞ 1 d ; 8i 2 V (1t) iD itq parameters. t2T q2Q In this paper, we consider nonlinear charging proﬁle, where SOC increases nonlinearly with respect to the charging time, as In the above model, the objective function (1a) is to minimize presented in Figure 2. Speciﬁcally, the battery would undergo the total operational cost over the operation hours of one day, two phases, namely, CC phase and CV phase. In the ﬁrst phase including bus acquisition cost and maintenance cost, where v ~ (i.e. CC phase), the charging current is held constant and denotes the unit acquisition cost of BEB per day, and d hence the SOC increases linearly with time until the battery’s indicates the cost incurred by battery degradation with the state terminal voltage reaches the threshold. After that, the terminal of charge (SOC) from SOC (i.e. the SOC of BEB after just voltage keeps constant (i.e. CV phase), thus resulting in the ﬁnishing trip i) to 100% under charging mode q. Constraints current decreasing exponentially and the growth rate of SOC (1b) guarantee that each trip is served exactly once. decreasing with respect to the charging time. The pattern of Constraints (1c) represent covering and ﬂow conservation. SOC with respect to the charging time under normal charging Constraints (1d) state that after trip i, BEB may start charging mode can be approximated by piecewise linear function: in a certain time step with a certain charging mode. Constraints (1e) ensure that the starting time of charging activity after trip 0:8tt 2½Þ 0; 1 SOC t; q ¼ (3) ðÞ i should be no earlier than the end time of trip i, where M is a 1 ðÞ 0:81 0:2 t 1 t 2½ 1; 2 sufﬁciently large number. Constraints (1f) guarantee the number of charging facilities used in each time step cannot where t is in the unit of hour. Therefore, the number of time exceed its capacity. Constraints (1g) ((1h)) state that steps required to charge battery from SOC to full under normal F(SOC , q )(F(SOC , q )) time steps are occupied if normal i 1 i 2 charging mode can be calculated as follows: (fast) charging operation is applied. Here F(SOC , q) indicates the number of time steps required to charge battery from SOC 30ðÞ 1 SOC if SOC > 0:8 i i to full under charging mode q; F(SOC , q ) is always equal to 1 FðÞ SOC ; q ¼ 15 i 2 i 1 : 61 ðÞ 0:8 SOC if SOC 0:8 i i for all SOC [ (0, 1) due to assumption 1. Constraints (1i) i 2 indicate that BEB is fully charged when it departs from the (4) original depot O, where m means the consumption of battery level relative to battery capacity of trip i. Constraints (1j) Here function [a] returns the smallest integer that is no smaller guarantee that SOC should be no smaller than a predeﬁned than a. lower bound lb to reduce range anxiety. Constraints (1k-n) record the dynamic SOC of BEBs if d = 1. Constraints (1o) ij 2.3 Upper-level problem: optimal design of charging deﬁne the function d , where W indicates the battery station acquisition cost; x is the end-of-life related parameter. The The upper-level model can be formulated as follows: term j (SOC , 1) denotes the corresponding battery capacity fading rate, borrowed from Lam and Bauer (2012); g(q) refers min J ¼ A C 1 J ðÞ X; C (5a) q q ½ P2 ½ P1 to charging-mode related coefﬁcient, where the coefﬁcient of q2Q fast charging mode is larger than that of normal charging mode, i.e. g(q ) < g(q ). Constraints (1p-r) state the stating time of Subject to: 1 2 trip j should be no earlier than the ending time of trip i if d =1 ij X X and m ¼ 0; and the stating time of trip j should Figure 2 Illustration of nonlinear charging proﬁle itq t2T q2Q be no earlier than the ending time of charging operation applied X X after trip i if d = 1 and m ¼ 1. Constraints (1s) ij itq t2T q2Q and (1t) indicate that buses are charged back to full when returning to destination depot. We next present the exact mathematical form of function j (SOC , 1): g SoC g SoC i;avg i;dev 2 4 jðÞ SoC ; 1¼ g SoC e 1 g e i i;dev (2a) where 11 SoC SoC ¼ ; (2b) i;avg 11 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 X ¼ gCðÞ (5b) more details on the theory of CG, please refer to Merle et al. (1999). Brieﬂy, model [P4] is solved by repeatedly solving (i) a where A indicates the unit installation cost of charging facility restricted master problem with a subset of trip chains and (ii) a in mode q amortized to one day, measured in $/day; g(·) pricing subproblem to generate new trip chains with negative denotes the optimal lower-level solution for X under a given reduced costs. The restricted master problem is solved by design of charging station C, which is found by solving model commercial solvers directly (e.g. Cplex, Gurobi). The pricing [P1]. J (X, C) indicates the daily operational cost under problem is shortest path problem with resource constraint and [P1] solved by label-correcting algorithm with fully considering tactical decision C and operational decision X, which is special problem aspects: minimal battery level and battery consistent with the objective function of model [P1]. recharging. Each state is represented by a label, (k, b), where k is the last reached node and b represents the corresponding 3. Solution approach battery level. The cost of label (k, b)is ckðÞ ; b , representing the 3.1 Column generation-based heuristic algorithm for accumulative cost from original depot O. Now consider that solving lower-level model both label (k, b) and label (k; b), label (k, b) dominates (k; b)if To solve model [P1], we next reformulate it as an equivalent set (1)ckðÞ ; b ck; b , and (2) b b, where at least one of above partitioning model, which can be solved by column generation inequalities is strict. The label extension and dominance rule (CG). To this end, we introduce trip chain ﬁrstly. A trip chain are placed into a framework of the general label-correcting of BEB includes the service sequence, i.e. a series of trips sorted algorithm to solve the pricing problem. in an ascending order in terms of their departure time, and the To generate feasible integer solution, we next propose a charging strategy between any two consecutive trips. A trip heuristic algorithm based upon the CG procedure to obtain chain is feasible, if i) all the covered trips can be served on time; near-optimal integer solutions. In the proposed heuristic and ii) battery level is sufﬁcient to support all the covered trips algorithm, the inner procedure is the CG procedure; the outer given that BEB can be charged at terminal station if both time procedure is the selection strategies used to obtain an integer and charging facilities permit. Given the deﬁnition of trip chain, solution. the equivalent set partitioning model is presented as follows: Firstly, we modify constraint (6c) in model [P3] as: min J ¼ c l (6a) ½ r r P3 U l C ; 8t 2 T; q 2 Q (8) r tq r tq r2R r2R Subject to: Initially, C is constant and equal to C . Then the available tq q number of charging facilities at each time step may decrease as A l ¼ 1; 8i 2 V (6b) r2R the outer procedure proceeds in our heuristic algorithm. To be X speciﬁc, every time the inner CG procedure stops, we select the U l C ; 8t 2 T; q 2 Q (6c) r q tq trip chain (i.e. column) with the largest value of the decision r2R variables l . Then update C and V for a new iteration of the r tq above process: l 2fg 0; 1 ; 8r 2 R (6d) if BEB is charged in time step t under charging mode q according to the selected trip chain, update C / C – 1; tq tq In the above formulation, we denote R as the set of all feasible trip if trip i is served in the selected trip chain, update V / V/i; chains within one day, and c denotes the cost of trip chain r[ R, including the unit bus acquisition cost and maintenance cost The algorithm terminates when all the trips are served. along the trip chain. For each feasible trip chain r [ R,we deﬁne a binary variable l , which equals to one if and only if trip chain r is r 3.2. A tabu-search method for solving upper-level selected; and to zero otherwise. Objective function (6a) minimizes model the total cost over the operation hours of one day. Constraints The ﬁrst step of the tabu search is to initialize a feasible initial no (6 b) ensure that each trip i [ V is served exactly once in the 0 0 0 solution to model [P2], denoted by X C ; C . Then q q r 1 2 solution, where A is set to one if trip iiscoveredintripchain r and deﬁne a move as a change from a feasible solution X to a new to zero otherwise. Constraints (6c) guarantee that the charging feasible solution, where the change can be one of the following: facilities used in each time step are within the limited capacity C ; r (i) C ! C 1or C ! C 1 1; and (ii) C ! C 1or q q q q q q and U is set to one if BEB is under charging with charging mode 1 1 1 1 2 2 tq C ! C 1 1. At each move, the CG-based heuristic q q q at time step t in trip chain r and to zero otherwise. Constraints 2 2 algorithm presented in Section 3.1 is executed to ﬁnd the BEB (6d) deﬁne the domain of decision variable l , r[ R. utilization schedule and recharging plan. The total cost J is We next describe the CG procedure to solve the linear [P2] then calculated. Deﬁne the neighborhood of X,N(X), as the set relaxation of set partitioning model, deﬁned as model [P4], i.e. of feasible solutions that can be obtained by making one move the binary constraint of l is relaxed and replaced by constraints: from X. Further deﬁne the tabu list, TL, as the list of inverse l 0; 8r 2 R (7) moves of those most recent moves performed. The maximum length of tabu list is denoted as tabu_size. In each iteration, a We now describe CG procedure to solve model [P4]. The move is made according to one of the following two rules: description is kept for short in the interest of brevity because the If no move in N(X) can produce a lower total cost as algorithm is only a standard practice of the CG algorithm. For compared to the best solution so far, set the current move 12 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 to the one in N(X)\TL that produces the lowest total cost. price is 0.65 million RMB, and this cost covers the Following this rule, a move is made even if it produces a maintenance of vehicles over 8 years. The battery capacity is higher cost than the best solution so far. 162 kWh with unit price as 1130 RMB/kWh on average. If a move in N(X) \ TL produces a lower total cost than Therefore, the price of battery packs is about: the best solution so far, set the current move to the lowest- 1130 162 § ¼ 28008:2 $ (9) cost move in N(X). The tabu list TL is updated after each iteration. It is used to where § denotes the exchange rate of RMB to US dollar prevent the algorithm from returning to a solution attained in a ($/RMB) and equals 0.153 in this paper.[1] Hence, the price of previous iteration. The ﬁrst rule ﬁnds the best neighboring battery packs, W, is approximated as 28000 $. The price of solution that is generated not from any move in the tabu list. BEB without battery is: However, if a move in the tabu list can yield a better solution 650000 j 28000 ¼ 71450 $ (10) than the best one so far, that move is still selected according to the second rule. The algorithm ends when no better solution is The service life of bus (without batteries) is set as 12 years found after max_num_tb consecutive iterations. (Lajunen, 2014). Then, the unit acquisition cost of bus (without battery) per day is: 4. Numerical cases To validate our model, in this section we focus on a case study 365 ¼ 16:31 $ (11) with the terminal denoted as terminal station A. Departing from terminal station A, ﬁve yet-to-be-electriﬁed lines are Similarly, the unit acquisition cost of bus (without battery) per studied. The lengths of the ﬁve lines are 11 km, 11.5 km, day, v ~, is approximated as 16.5 $in this paper. 12.6 km, 19.3 km and 16.8 km respectively. Their travel We ﬁrstly examine the scenario where only one kind of durations (terminal to terminal) are 80 min, 90 min, 110 min, charging facilities is allowed to install at the terminal station. Without of loss of generality, we consider normal charging 150minand 130 min respectively. The timetables for these ﬁve lines are shown in Table 1. The technical parameters needed mode ﬁrstly and optimize the BEB service and charging for this paper are obtained from Yutong ZK6850BEVG53, as schedule under a range of charging station capacity: speciﬁed in Table 2. Speciﬁcally, Yutong ZK6850BEVG53 is a C 2½ 1; 21 ,as presented in Figure 3. Figure 3 showsthatthe kind of medium BEB, with vehicle length as 8.5 meters. Its optimal operational cost (the solid curve with triangle markers) decreases as C increases, until it reaches a threshold of 17. This threshold represents the maximum Table 1 Timetables for selected bus lines, departing from terminal station number of normal charging facilities needed for BEBs at terminal station A; i.e. any additional charging facilities Line 1 Line 2 Line 3 Line 4 Line 5 would be redundant, and the optimal total cost would stay the same. 07:00:00 06:10:00 07:00:00 06:20:00 06:40:00 The asterisk-marked solid curve represents the corresponding 07:20:00 Every 20 min 07:30:00 Every 30 min 07:10:00 total cost at the upper level. We note that, the optimal total cost 07:40:00 09:10:00 08:00:00 18:50:00 07:40:00 decreases as C increases, until it reaches to 17. After that, the Every 20 min Every 30 min Every 20 min Every 10 min q optimal total cost increases as C increases. This is because any 10:40:00 15:10:00 14:20:00 18:20:00 q more charging facilities would be redundant and the optimal Every 30 min Every 20 min 14:50:00 18:50:00 operational cost at the lower level would not reduce any more. 16:40:00 19:10:00 15:10:00 19:20:00 Therefore, for the studied transit network, it is optimal to Every 20 min Every 20 min 19:50:00 deploy 17 charging facilities at the terminal station A if the 19:20:00 20:10:00 operators only consider normal charging mode. Further Table 2 Parameter deﬁnitions and values Parameter Notation Value Unit Lower bound of battery level lb 20 % Unit acquisition cost of an electric bus (without battery) per day v 16.5 $/day Battery acquisition cost W 2.8e4 $ Unit cost of a normal charging facility per day A 5 $/day Unit cost of a fast charging facility per day A 30 $/day Coefﬁcient of battery capacity fading under normal charging mode g(q)1 – Coefﬁcient of battery capacity fading under fast charging mode g(q)2 – Coefﬁcient for battery degradation model g 4.09e-4 – Coefﬁcient for battery degradation model g 2.167 – Coefﬁcient for battery degradation model g 1.418e-5 – Coefﬁcient for battery degradation model g 6.13 – End-of-life related threshold x 0.2 – 13 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 When mixed design of charging station is considered, the model Figure 3 Effects of the number of normal chargers on the optimal cost reveals that the optimal charging station design is to install 2 fast 2.4 chargers and 2 normal chargers, where the optimal total cost is Operational cost optimized at the lower level $1.12 10 . Even though the daily saving brought by mixed design Total cost at the upper level Bus acquisition cost of charging station is minor, i.e. only $20. However, this beneﬁt Battery deterioration cost cannot be overlooked over 10years or more. 1.6 5. Conclusions 1.2 In this paper, we present a new mathematical formulation aimed at optimizing the design of charging station deployed at 0.8 the terminal station for electric transit. To this end, a bi-level model is built with full consideration of the decision-makings at 0.4 both tactical and operational levels. Speciﬁcally, the lower-level model optimizes the scheduling of BEBs given the design of 0 2 4 6 8 10121416182022 charging station, including the number of charging facilities Number of charging facilities under different charging mode (i.e. fast charging and normal charging), whereas the upper-level model optimizes the design of charging station given optimized BEB service sequence and investigation reveals that, as the unit cost of charging facilities charging plan at the lower level. per day is relatively low as compared with that of BEBs, it is In the future work we plan to explore more realistic scenarios cost-efﬁcient to deploy the charging facilities as more as needed where the “full-charging” assumption is relaxed, i.e. partial so that the total cost of transit agency can be saved by reducing charging among BEBs is allowed and the charging time ﬂeet size. depends on the amount of energy to be replenished following We also note that the SoC variation negatively affects battery more realistic non-linear charging proﬁle. Meanwhile, we aging rate: the larger the SoC variation is, the faster the battery would also plan to extend the transit network to the one with degrades. Therefore, to prolong battery life, the operators are multiple terminals, explore more efﬁcient solution approach encouraged to charge BEBs as frequently as possible, instead of with high quality, instead of using heuristic algorithm; consider as late as possible. As a trade-off, when the charging facilities agency budget for the installation of charging infrastructure and are relatively sufﬁcient, the electric buses tend to be charged as consider users’ psychological inertia (Gao et al., 2020). frequently as possible, to achieve more cost saving by extending batteries’ lifespan. We next examine the effect of the number of fast chargers on Note the optimal cost, as presented in Figure 4. Compared to normal 1 The result may deviate within a small range due to the charging, fast charging mode can reduce the ﬂeet size largely variation of exchange rate. due to its high charging efﬁciency, even though the installation cost of fast charging facilities is much higher than that of normal charging facilities. Therefore, the maximum number of fast References chargers required is only 3 with the minimal operational cost at the lower-level as $1.05 10 , as shown in Figure 4. The ﬁgure Adler, J.D. (2014), “Routing and scheduling of electric and also reveals that the minimal total cost at the upper level alternative-fuel vehicles”, Doctoral dissertation, Arizona ($1.14 10 ) occurs when the number of fast chargers is 3. State University. An, K. (2020), “Battery electric bus infrastructure planning Figure 4 Effects of the number of fast chargers on the optimal cost under demand uncertainty”, Transportation Research Part C: 2.4 Emerging Technologies, Vol. 111, pp. 572-587. 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List of notations Table A1 List of notations Notation Description Sets and Indices V Set of trips T Set of time steps R Set of all feasible trip chains R’ Subset of feasible trip chains Q Set of charging modes i, j Trip and node indices O Index of a virtual trip before each BEB’s ﬁrst trip of the day D Index of a virtual trip after each BEB’s last charging activity of the day T Time step index R Trip chain index Q Charging mode index Decision variables d [{0,1} Equals one if a BEB serves trips i and j in turn and consecutively, and zero otherwise, i [ V |{O}, j [ V | {D}, i= j ij l [ {0,1} Equals one if a BEB’s charging activity following trip i [ V starts at time step t [ T with charging mode q, and zero otherwise itq / [{0,1} Equals one if a BEB is being charged at time step t [ T between trip i [ V and the next trip it serves with charging mode q, and zero otherwise itq C Number of charging facilities in mode q deployed at terminal station k [ {0, 1} Equals one if trip chain r [ R is selected, and zero otherwise C Decision variables at the tactical level X Decision variables at the operational level Parameters and other variables s Departure time of trip i e Travel time of trip i Lb Minimum battery level to eliminate range anxiety m Battery consumption of trip i M A sufﬁciently large number ~ v Amortized acquisition and maintenance cost of a BEB per day q Normal charging mode q Fast charging mode F(SOC , q) Number of time steps required from SOC to full with charging mode q i i W Battery acquisition cost A Unit cost of a charging facility per day with charging mode q c Cost of trip train r c(q) Coefﬁcient of battery capacity fading under charging mode q r ,r ,r ,r Coefﬁcient for battery degradation model 1 2 3 4 X End-of-life related threshold n(SOC ,1) Battery capacity fading rate from SOC to full i i d (SOC ,1) Cost incurred by battery capacity fading from SOC to full under charging mode q q i i ckðÞ ; b Cost of label (k, b) A Equals one if trip i [ V is covered by trip chain r and zero otherwise U Equals one if a BEB in trip chain r is on a charger with charging mode q at time step t and zero otherwise tq N(X) Neighborhood of X TL Tabu list R Exchange rate of RMB to US dollar Corresponding author Le Zhang can be contacted at: le.zhang@njust.edu.cn For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: permissions@emeraldinsight.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Intelligent and Connected Vehicles Emerald Publishing http://www.deepdyve.com/lp/emerald-publishing/a-bi-level-optimization-framework-for-charging-station-design-problem-gv3L1AndaZ

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Publisher: Emerald Publishing
Copyright: © Le Zhang, Ziling Zeng and Kun Gao.
ISSN: 2399-9802
DOI: 10.1108/jicv-07-2021-0009
Publisher site: See Article on Publisher Site

Abstract

Purpose – The purpose of this paper is to optimize the design of charging station deployed at the terminal station for electric transit, with explicit consideration of heterogenous charging modes. Design/methodology/approach – The authors proposed a bi-level model to optimize the decision-making at both tactical and operational levels simultaneously. Speciﬁcally, at the operational level (i.e. lower level), the service schedule and recharging plan of electric buses are optimized under speciﬁc design of charging station. The objective of lower-level model is to minimize total daily operational cost. This model is solved by a tailored column generation-based heuristic algorithm. At the tactical level (i.e. upper level), the design of charging station is optimized based upon the results obtained at the lower level. A tabu search algorithm is proposed subsequently to solve the upper-level model. Findings – This study conducted numerical cases to validate the applicability of the proposed model. Some managerial insights stemmed from numerical case studies are revealed and discussed, which can help transit agencies design charging station scientiﬁcally. Originality/value – The joint consideration of heterogeneous charging modes in charging station would further lower the operational cost of electric transit and speed up the market penetration of battery electric buses. Keywords Battery electric bus, Charging station design, Vehicle scheduling, Bi-level model, Heterogeneous charging modes Paper type Research paper powered by other energy sources, resulting in users’“range 1. Introduction anxiety” (Lebeau et al., 2016; Masmoudi et al.,2018; Qin et al., Electric transit is considered as the key to the world’s clean 2016; Li et al., 2019). To ensure normal operations, the transport future due to its high energy efﬁciency, zero emissions consumed electricity must be replenished by either battery (Lajunen, 2014; Jin et al.,2015; Xu et al., 2021; Qu et al., 2020; swapping or battery recharging (Li, 2013; Huang and Zhou, Zhang et al., 2020; Zhang et al., 2021) and shareability (Gao 2015; Wang et al.,2017; Tang et al., 2019; Liu and Ceder, et al.,2021; Bie et al.,2020; Meng and Qu, 2013; Wang et al., 2020; Rinaldi et al.,2020). Unfortunately, instead of mitigating 2018). Compared with diesel buses, battery electric buses this disadvantage, lack of sufﬁcient charging facilities further (BEBs) are able to improve energy efﬁciency by 50% and aggravates it (An, 2020). However, if sufﬁcient charging reduce greenhouse gas emissions by 98.36% (Mahmoud et al., facilities are deployed, it may cause severe budget burden for 2016). During the past decade, the public transit is electriﬁed transit system. Meanwhile, charging modes also affect the step by step. For example, in the USA, the share of BEBs in the charging efﬁciency as well as the infrastructure installation cost. Speciﬁcally, compared with normal charging, the ﬂeet size can bus market increased rapidly from 2% in 2007 to nearly 20% in be reduced signiﬁcantly through improved charging efﬁciency 2015 (Neff and Dickens, 2016); in Europe, the percentage of BEBs on the sales volumes of city buses is up to 10% by 2019, and this number rises up to around 20% in 2020. © Le Zhang, Ziling Zeng and Kun Gao. Published in Journal of Intelligent Undoubtedly, transit electriﬁcation is becoming an and Connected Vehicles. Published by Emerald Publishing Limited. This unstoppable trend. article is published under the Creative Commons Attribution (CC BY 4.0) Compared with diesel buses, the driving characteristics and licence. Anyone may reproduce, distribute, translate and create derivative refueling manner of BEBs are distinct. Speciﬁcally, BEBs works of this article (for both commercial and non-commercial purposes), generally have a much shorter operational range than buses subject to full attribution to the original publication and authors. The full terms of this licence maybe seen at http://creativecommons.org/licences/ by/4.0/legalcode The current issue and full text archive of this journal is available on Emerald This work is supported by National Natural Science Foundation of Insight at: https://www.emerald.com/insight/2399-9802.htm China (No. 72101115) and Natural Science Foundation of Jiangsu (No. BK20210316). Received 8 July 2021 Journal of Intelligent and Connected Vehicles Revised 28 September 2021 5/1 (2022) 8–16 5 December 2021 Emerald Publishing Limited [ISSN 2399-9802] [DOI 10.1108/JICV-07-2021-0009] Accepted 16 December 2021 8 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 by adaptation of fast charging mode, but it also causes higher Our key contributions from a theoretical and practical point infrastructure installation cost. Therefore, how to design the of view can be summarized as follows: charging station, trading off charging availability, charging We are, to our best knowledge, the ﬁrst to formulate and efﬁciency and limited budget become an important issue in solve charging station design problem with explicit transit electriﬁcation. To this end, we aim at studying the consideration of multiple charging modes and their optimal design of charging station deployed at the terminal corresponding effect on battery capacity fading. station for electric transit in this paper. To be speciﬁc, we A number of managerial insights stemmed from the propose a bi-level model, where the lower-level model numerical case study are outlined, which can serve as a optimizes the scheduling of BEBs given the design of charging solid theoretical foundation for more cost-efﬁcient station, including the number of charging facilities of different charging station design. charging modes (i.e. fast charging and normal charging); the The rest of this paper is organized as follows. Section 2 presents upper-level model optimizes the design of charging station with the problem formulation, i.e. a bi-level model. Section 3 explicit consideration of multiple charging modes. In the bi- elaborates the proposed solution approach for solving the level approach, the lower-level problem, i.e. optimal scheduling problem. The numerical cases are conducted in Section 4. of BEBs, is the key and difﬁcult part of the work. Therefore, we Conclusions are summarized in Section 5. next present the relevant studies in the realm of vehicle scheduling. 2. Mixed charging station design problem Bus scheduling problem consists of assigning buses to serve a series of timetabled trips with the objective of minimizing ﬂeet 2.1 Problem description size and/or operational costs. It is an extension of the well- In this section, a single-terminal transit network is considered known vehicle scheduling problem (VSP), which has been to deﬁne the optimization problem of charging station design, extensively studied in the literature (Markovic et al., 2015; as depicted in Figure 1. BEBs depart from the terminal station Schöbel, 2017). Generally speaking, VSP can be categorized to operate a sequence of scheduled round-trips, denoted as set into two groups: the single-depot VSP (SDVSP) (Paixão and V. For simplicity, we refer to the round-trip as trip for short Branco, 1987; Freling et al., 2001; Kang et al., 2019) and the from now. Charging facilities in mode q [ Q ={q , q } are 1 2 multiple-depot VSP (MDVSP) (Dell’Amico et al.,1993; deployed at terminal station with limited number C , where q q 1 Kliewer et al., 2006). Over the years, many varieties and indicates normal charging mode and q indicates fast charging extensions of VSP have been proposed to incorporate the real- mode. For each trip i([ V), the departure time s , travel time e i i world constraints and conditions, including VSP with multiple and the consumption of battery level relative to battery vehicle types (Ceder, 2011), VSP with route constraints (VSP- capacity, m are predeﬁned and deterministic. The objective of RC) (Bunte and Kliewer, 2009), the alternative fuel VSP (AF- this problem is to minimize the total cost of transit agency, VSP) (Li, 2013; Adler, 2014) and electric VSP (E-VSP) (Wen including bus acquisition cost, charging fee, maintenance cost et al.,2016). Among these varieties, VSP-RC, AF-VSP and E- of BEB ﬂeet and the cost incurred by the deployment of VSP are strongly motivated by electric vehicles. To accounting charging facilities. Therefore, the operators shall make for the speciﬁcs of electric vehicles, route duration or route decisions at both tactical and operational levels. To be speciﬁc, distance is constrained in VSP-RC (Haghani and Banihashemi, at the tactical level, the number of charging facilities in each 2002). If vehicles are allowed to be refueled at given recharging type deployed at the terminal station should be optimized and stations to prolong the total distance, that is AF-VSP. the vector of decision variables at this level is denoted by C ¼ However, traditional AF-VSP only considers full charging and {C jq [ Q}. At the operational level, the operators shall make the charging time is set as ﬁxed. Speciﬁcally, the vehicle’s fuel decisions on: how to assign BEBs to serve a series of trips level is set to be full after visiting any recharging stations. For satisfying the minimal battery level constraint and how to example, Li (2013) incorporated vehicle waiting time at optimize recharging schedule considering limited charging charging stations into the model, and the charging time was facilities (i.e. given speciﬁc charging station design). The simpliﬁed as ﬁxed by considering battery swapping. Later, E- VSP was proposed, where partial charging was allowed and the Figure 1 Single-terminal transit network charging time was usually assumed to be a linear function of the charged amount. Unfortunately, whereas much efforts have been made to deal with BEB scheduling, very little attention has been dedicated to explicitly modeling the design of charging station for electric transit, with full consideration of multiple charging modes. It will cause unforeseen operational cost when promoting transit electriﬁcation. In light of the above literature, this paper would employ the latest study in electric vehicle scheduling to study the optimal design of charging station, and the bi-level solution approach is adopted to ﬁx this problem. Numerical case studies were conducted to validate the applicability of the proposed model. It reveals that it is a cost-efﬁcient choice to deploy sufﬁcient charging facilities at the terminal station as the unit cost of charging facilities per day is much lower than that of BEBs. 9 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 X X X X corresponding decision variables at this level (i.e. service 1 m M1 m t s 1 e ; 8i 2 V i i itq itq t2T q2Q t2T q2Q sequence and charging strategy) are denoted by vector X. Notations used in this paper are summarized in Appendix. (1e) 2.2 Lower-level problem: optimal scheduling of battery f C ; 8t 2 T; q 2 Q (1f) itq q electric bus ﬂeet i2V The objective of the lower-level model is to minimize total 1 m M1 FðÞ SOC ; q i 1 itq operational cost, including bus acquisition cost, charging fee and t1 FðÞ SOCi;q11 maintenance cost within one day, where the maintenance cost is f 0 1 m M1 FðÞ SOC ; q ; itq i 1 it q1 1 t ¼t mainly incurred by battery degradation. It is worth to note that the 8t 2 T; i 2 V (1g) total charging fee is constant in our model, as it is related to the predeﬁned tripservice, whichis ﬁxed and independent of BEB schedule. Therefore, the objective function is simpliﬁed as the sum 1 m M1 FðÞ SOC ; q i 2 itq of bus acquisition cost and maintenance cost. The vector of decision variables at the operational level, X,can be deﬁned as X ={d , m , f 1 m M1 FðÞ SOC ; q ; 8t 2 T; i 2 V ij itq i 2 itq itq 2 2 f ji[ V, t[ T, q[ Q}, where: itq (1h) d [ {0, 1}: set to one if BEB serves trips i and j consecutively, ij where trip i begins earlier than trip j; and to zero otherwise, i[ V 1 m ðÞ 1 d M SOC 1 m 1ðÞ 1 d M; 8i 2 V i Oi i i Oi | O, j [ V | D, i = j.Here O denotes a virtual trip that every (1i) bus must serve before its ﬁrst real trip, and D denotes another virtual trip that each BEB serves after completing the last real trip of a day and being fully charged. The two notations are SOC lb; 8i 2 V (1j) deﬁned for the convenience of modeling work. The virtual X X trips’ travel times and electricity consumption are all set to zero: SOC SOC m 1ðÞ 1 d M1 M m ; 8i; j 2 V j i j ij itq m [ {0, 1}: set to one if BEB begins to charge with charging itq t2T q2Q mode q at time step t after ﬁnishing trip i and before serving the (1k) next trip, and to zero otherwise, i[ V, t[ T, q[ Q. f [ {0, 1}: set to one if BEB is under charging with itq X X SOC SOC m 1 d M M m ; 8i; j 2 V charging mode q at time step t after ﬁnishing trip i and before ðÞ j i j ij itq t2T q2Q serving the next trip, and to zero otherwise, i[ V, t[ T, q[ Q. In this model, we make the following assumptions: (1l) Assumption 1: The time is discretized with the unit time X X as 10 min. The time for full charge in mode q (i.e. normal SOC 1 m 1 1 d M1 1 m M; 8i; j 2 V ðÞ j j ij itq charging) and mode q (i.e. fast charging) are 2 h (i.e. 12 2 t2T q2Q time steps) and 10 min (i.e. 1 time step), respectively. (1m) Assumption 2: BEBs are fully charged when departing from the original depot, and are charged back to full when X X SOC 1 m ðÞ 1 d M 1 m M; 8i; j 2 V j j ij itq returning to destination depot. t2T q2Q Assumption 3: After ﬁnishing one trip, BEB can either be (1n) charged for one time and charged to full or serve the next trip consecutively without any charging activity. 2 jðÞ SOC ; 1ðÞ 1 SOC i i dðÞ SOC ; 1 ¼ W g q ; 8i 2 V; q 2 Q The lower-level model [P1] can be formulated as: q i ðÞ X XX X (1o) min J C jq 2 Q ¼ v ~ d 1 m dðÞ SOC ; 1 ½ P1 q Oi itq q i i2V i2V t2T q2Q X X s s 1 e ðÞ 1 d M M m ; 8i; j 2 V j i i ij itq (1a) t2T q2Q (1p) Subject to: ðÞ s m t1 F SOC ; q ðÞ 1 d M j i 1 ij itq d ¼ 1; 8j 2 V (1b) ij t2T i2V[O 1 m M; 8i; j 2 V (1q) itq X X t2T d d ¼ 0; 8i 2 V (1c) ij ji j2V[D j2V[O s m t1 FðÞ SOC ; q ðÞ 1 d M j i 2 ij itq X X t2T m 1; 8i 2 V (1d) 1 m M; 8i; j 2 V (1r) itq itq t2T q2Q t2T 10 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 X X 1 SoC ðÞ i m 11 M 1 d ; 8i 2 V (1s) iD itq SoC ¼ (2c) i;dev t2T q2Q X X Here the coefﬁcients g , g , g and g are constant model 1 2 3 4 m 1 MðÞ 1 d ; 8i 2 V (1t) iD itq parameters. t2T q2Q In this paper, we consider nonlinear charging proﬁle, where SOC increases nonlinearly with respect to the charging time, as In the above model, the objective function (1a) is to minimize presented in Figure 2. Speciﬁcally, the battery would undergo the total operational cost over the operation hours of one day, two phases, namely, CC phase and CV phase. In the ﬁrst phase including bus acquisition cost and maintenance cost, where v ~ (i.e. CC phase), the charging current is held constant and denotes the unit acquisition cost of BEB per day, and d hence the SOC increases linearly with time until the battery’s indicates the cost incurred by battery degradation with the state terminal voltage reaches the threshold. After that, the terminal of charge (SOC) from SOC (i.e. the SOC of BEB after just voltage keeps constant (i.e. CV phase), thus resulting in the ﬁnishing trip i) to 100% under charging mode q. Constraints current decreasing exponentially and the growth rate of SOC (1b) guarantee that each trip is served exactly once. decreasing with respect to the charging time. The pattern of Constraints (1c) represent covering and ﬂow conservation. SOC with respect to the charging time under normal charging Constraints (1d) state that after trip i, BEB may start charging mode can be approximated by piecewise linear function: in a certain time step with a certain charging mode. Constraints (1e) ensure that the starting time of charging activity after trip 0:8tt 2½Þ 0; 1 SOC t; q ¼ (3) ðÞ i should be no earlier than the end time of trip i, where M is a 1 ðÞ 0:81 0:2 t 1 t 2½ 1; 2 sufﬁciently large number. Constraints (1f) guarantee the number of charging facilities used in each time step cannot where t is in the unit of hour. Therefore, the number of time exceed its capacity. Constraints (1g) ((1h)) state that steps required to charge battery from SOC to full under normal F(SOC , q )(F(SOC , q )) time steps are occupied if normal i 1 i 2 charging mode can be calculated as follows: (fast) charging operation is applied. Here F(SOC , q) indicates the number of time steps required to charge battery from SOC 30ðÞ 1 SOC if SOC > 0:8 i i to full under charging mode q; F(SOC , q ) is always equal to 1 FðÞ SOC ; q ¼ 15 i 2 i 1 : 61 ðÞ 0:8 SOC if SOC 0:8 i i for all SOC [ (0, 1) due to assumption 1. Constraints (1i) i 2 indicate that BEB is fully charged when it departs from the (4) original depot O, where m means the consumption of battery level relative to battery capacity of trip i. Constraints (1j) Here function [a] returns the smallest integer that is no smaller guarantee that SOC should be no smaller than a predeﬁned than a. lower bound lb to reduce range anxiety. Constraints (1k-n) record the dynamic SOC of BEBs if d = 1. Constraints (1o) ij 2.3 Upper-level problem: optimal design of charging deﬁne the function d , where W indicates the battery station acquisition cost; x is the end-of-life related parameter. The The upper-level model can be formulated as follows: term j (SOC , 1) denotes the corresponding battery capacity fading rate, borrowed from Lam and Bauer (2012); g(q) refers min J ¼ A C 1 J ðÞ X; C (5a) q q ½ P2 ½ P1 to charging-mode related coefﬁcient, where the coefﬁcient of q2Q fast charging mode is larger than that of normal charging mode, i.e. g(q ) < g(q ). Constraints (1p-r) state the stating time of Subject to: 1 2 trip j should be no earlier than the ending time of trip i if d =1 ij X X and m ¼ 0; and the stating time of trip j should Figure 2 Illustration of nonlinear charging proﬁle itq t2T q2Q be no earlier than the ending time of charging operation applied X X after trip i if d = 1 and m ¼ 1. Constraints (1s) ij itq t2T q2Q and (1t) indicate that buses are charged back to full when returning to destination depot. We next present the exact mathematical form of function j (SOC , 1): g SoC g SoC i;avg i;dev 2 4 jðÞ SoC ; 1¼ g SoC e 1 g e i i;dev (2a) where 11 SoC SoC ¼ ; (2b) i;avg 11 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 X ¼ gCðÞ (5b) more details on the theory of CG, please refer to Merle et al. (1999). Brieﬂy, model [P4] is solved by repeatedly solving (i) a where A indicates the unit installation cost of charging facility restricted master problem with a subset of trip chains and (ii) a in mode q amortized to one day, measured in $/day; g(·) pricing subproblem to generate new trip chains with negative denotes the optimal lower-level solution for X under a given reduced costs. The restricted master problem is solved by design of charging station C, which is found by solving model commercial solvers directly (e.g. Cplex, Gurobi). The pricing [P1]. J (X, C) indicates the daily operational cost under problem is shortest path problem with resource constraint and [P1] solved by label-correcting algorithm with fully considering tactical decision C and operational decision X, which is special problem aspects: minimal battery level and battery consistent with the objective function of model [P1]. recharging. Each state is represented by a label, (k, b), where k is the last reached node and b represents the corresponding 3. Solution approach battery level. The cost of label (k, b)is ckðÞ ; b , representing the 3.1 Column generation-based heuristic algorithm for accumulative cost from original depot O. Now consider that solving lower-level model both label (k, b) and label (k; b), label (k, b) dominates (k; b)if To solve model [P1], we next reformulate it as an equivalent set (1)ckðÞ ; b ck; b , and (2) b b, where at least one of above partitioning model, which can be solved by column generation inequalities is strict. The label extension and dominance rule (CG). To this end, we introduce trip chain ﬁrstly. A trip chain are placed into a framework of the general label-correcting of BEB includes the service sequence, i.e. a series of trips sorted algorithm to solve the pricing problem. in an ascending order in terms of their departure time, and the To generate feasible integer solution, we next propose a charging strategy between any two consecutive trips. A trip heuristic algorithm based upon the CG procedure to obtain chain is feasible, if i) all the covered trips can be served on time; near-optimal integer solutions. In the proposed heuristic and ii) battery level is sufﬁcient to support all the covered trips algorithm, the inner procedure is the CG procedure; the outer given that BEB can be charged at terminal station if both time procedure is the selection strategies used to obtain an integer and charging facilities permit. Given the deﬁnition of trip chain, solution. the equivalent set partitioning model is presented as follows: Firstly, we modify constraint (6c) in model [P3] as: min J ¼ c l (6a) ½ r r P3 U l C ; 8t 2 T; q 2 Q (8) r tq r tq r2R r2R Subject to: Initially, C is constant and equal to C . Then the available tq q number of charging facilities at each time step may decrease as A l ¼ 1; 8i 2 V (6b) r2R the outer procedure proceeds in our heuristic algorithm. To be X speciﬁc, every time the inner CG procedure stops, we select the U l C ; 8t 2 T; q 2 Q (6c) r q tq trip chain (i.e. column) with the largest value of the decision r2R variables l . Then update C and V for a new iteration of the r tq above process: l 2fg 0; 1 ; 8r 2 R (6d) if BEB is charged in time step t under charging mode q according to the selected trip chain, update C / C – 1; tq tq In the above formulation, we denote R as the set of all feasible trip if trip i is served in the selected trip chain, update V / V/i; chains within one day, and c denotes the cost of trip chain r[ R, including the unit bus acquisition cost and maintenance cost The algorithm terminates when all the trips are served. along the trip chain. For each feasible trip chain r [ R,we deﬁne a binary variable l , which equals to one if and only if trip chain r is r 3.2. A tabu-search method for solving upper-level selected; and to zero otherwise. Objective function (6a) minimizes model the total cost over the operation hours of one day. Constraints The ﬁrst step of the tabu search is to initialize a feasible initial no (6 b) ensure that each trip i [ V is served exactly once in the 0 0 0 solution to model [P2], denoted by X C ; C . Then q q r 1 2 solution, where A is set to one if trip iiscoveredintripchain r and deﬁne a move as a change from a feasible solution X to a new to zero otherwise. Constraints (6c) guarantee that the charging feasible solution, where the change can be one of the following: facilities used in each time step are within the limited capacity C ; r (i) C ! C 1or C ! C 1 1; and (ii) C ! C 1or q q q q q q and U is set to one if BEB is under charging with charging mode 1 1 1 1 2 2 tq C ! C 1 1. At each move, the CG-based heuristic q q q at time step t in trip chain r and to zero otherwise. Constraints 2 2 algorithm presented in Section 3.1 is executed to ﬁnd the BEB (6d) deﬁne the domain of decision variable l , r[ R. utilization schedule and recharging plan. The total cost J is We next describe the CG procedure to solve the linear [P2] then calculated. Deﬁne the neighborhood of X,N(X), as the set relaxation of set partitioning model, deﬁned as model [P4], i.e. of feasible solutions that can be obtained by making one move the binary constraint of l is relaxed and replaced by constraints: from X. Further deﬁne the tabu list, TL, as the list of inverse l 0; 8r 2 R (7) moves of those most recent moves performed. The maximum length of tabu list is denoted as tabu_size. In each iteration, a We now describe CG procedure to solve model [P4]. The move is made according to one of the following two rules: description is kept for short in the interest of brevity because the If no move in N(X) can produce a lower total cost as algorithm is only a standard practice of the CG algorithm. For compared to the best solution so far, set the current move 12 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 to the one in N(X)\TL that produces the lowest total cost. price is 0.65 million RMB, and this cost covers the Following this rule, a move is made even if it produces a maintenance of vehicles over 8 years. The battery capacity is higher cost than the best solution so far. 162 kWh with unit price as 1130 RMB/kWh on average. If a move in N(X) \ TL produces a lower total cost than Therefore, the price of battery packs is about: the best solution so far, set the current move to the lowest- 1130 162 § ¼ 28008:2 $ (9) cost move in N(X). The tabu list TL is updated after each iteration. It is used to where § denotes the exchange rate of RMB to US dollar prevent the algorithm from returning to a solution attained in a ($/RMB) and equals 0.153 in this paper.[1] Hence, the price of previous iteration. The ﬁrst rule ﬁnds the best neighboring battery packs, W, is approximated as 28000 $. The price of solution that is generated not from any move in the tabu list. BEB without battery is: However, if a move in the tabu list can yield a better solution 650000 j 28000 ¼ 71450 $ (10) than the best one so far, that move is still selected according to the second rule. The algorithm ends when no better solution is The service life of bus (without batteries) is set as 12 years found after max_num_tb consecutive iterations. (Lajunen, 2014). Then, the unit acquisition cost of bus (without battery) per day is: 4. Numerical cases To validate our model, in this section we focus on a case study 365 ¼ 16:31 $ (11) with the terminal denoted as terminal station A. Departing from terminal station A, ﬁve yet-to-be-electriﬁed lines are Similarly, the unit acquisition cost of bus (without battery) per studied. The lengths of the ﬁve lines are 11 km, 11.5 km, day, v ~, is approximated as 16.5 $in this paper. 12.6 km, 19.3 km and 16.8 km respectively. Their travel We ﬁrstly examine the scenario where only one kind of durations (terminal to terminal) are 80 min, 90 min, 110 min, charging facilities is allowed to install at the terminal station. Without of loss of generality, we consider normal charging 150minand 130 min respectively. The timetables for these ﬁve lines are shown in Table 1. The technical parameters needed mode ﬁrstly and optimize the BEB service and charging for this paper are obtained from Yutong ZK6850BEVG53, as schedule under a range of charging station capacity: speciﬁed in Table 2. Speciﬁcally, Yutong ZK6850BEVG53 is a C 2½ 1; 21 ,as presented in Figure 3. Figure 3 showsthatthe kind of medium BEB, with vehicle length as 8.5 meters. Its optimal operational cost (the solid curve with triangle markers) decreases as C increases, until it reaches a threshold of 17. This threshold represents the maximum Table 1 Timetables for selected bus lines, departing from terminal station number of normal charging facilities needed for BEBs at terminal station A; i.e. any additional charging facilities Line 1 Line 2 Line 3 Line 4 Line 5 would be redundant, and the optimal total cost would stay the same. 07:00:00 06:10:00 07:00:00 06:20:00 06:40:00 The asterisk-marked solid curve represents the corresponding 07:20:00 Every 20 min 07:30:00 Every 30 min 07:10:00 total cost at the upper level. We note that, the optimal total cost 07:40:00 09:10:00 08:00:00 18:50:00 07:40:00 decreases as C increases, until it reaches to 17. After that, the Every 20 min Every 30 min Every 20 min Every 10 min q optimal total cost increases as C increases. This is because any 10:40:00 15:10:00 14:20:00 18:20:00 q more charging facilities would be redundant and the optimal Every 30 min Every 20 min 14:50:00 18:50:00 operational cost at the lower level would not reduce any more. 16:40:00 19:10:00 15:10:00 19:20:00 Therefore, for the studied transit network, it is optimal to Every 20 min Every 20 min 19:50:00 deploy 17 charging facilities at the terminal station A if the 19:20:00 20:10:00 operators only consider normal charging mode. Further Table 2 Parameter deﬁnitions and values Parameter Notation Value Unit Lower bound of battery level lb 20 % Unit acquisition cost of an electric bus (without battery) per day v 16.5 $/day Battery acquisition cost W 2.8e4 $ Unit cost of a normal charging facility per day A 5 $/day Unit cost of a fast charging facility per day A 30 $/day Coefﬁcient of battery capacity fading under normal charging mode g(q)1 – Coefﬁcient of battery capacity fading under fast charging mode g(q)2 – Coefﬁcient for battery degradation model g 4.09e-4 – Coefﬁcient for battery degradation model g 2.167 – Coefﬁcient for battery degradation model g 1.418e-5 – Coefﬁcient for battery degradation model g 6.13 – End-of-life related threshold x 0.2 – 13 Heterogeneous charging modes Journal of Intelligent and Connected Vehicles Le Zhang, Ziling Zeng and Kun Gao Volume 5 · Number 1 · 2022 · 8–16 When mixed design of charging station is considered, the model Figure 3 Effects of the number of normal chargers on the optimal cost reveals that the optimal charging station design is to install 2 fast 2.4 chargers and 2 normal chargers, where the optimal total cost is Operational cost optimized at the lower level $1.12 10 . Even though the daily saving brought by mixed design Total cost at the upper level Bus acquisition cost of charging station is minor, i.e. only $20. However, this beneﬁt Battery deterioration cost cannot be overlooked over 10years or more. 1.6 5. Conclusions 1.2 In this paper, we present a new mathematical formulation aimed at optimizing the design of charging station deployed at 0.8 the terminal station for electric transit. To this end, a bi-level model is built with full consideration of the decision-makings at 0.4 both tactical and operational levels. Speciﬁcally, the lower-level model optimizes the scheduling of BEBs given the design of 0 2 4 6 8 10121416182022 charging station, including the number of charging facilities Number of charging facilities under different charging mode (i.e. fast charging and normal charging), whereas the upper-level model optimizes the design of charging station given optimized BEB service sequence and investigation reveals that, as the unit cost of charging facilities charging plan at the lower level. per day is relatively low as compared with that of BEBs, it is In the future work we plan to explore more realistic scenarios cost-efﬁcient to deploy the charging facilities as more as needed where the “full-charging” assumption is relaxed, i.e. partial so that the total cost of transit agency can be saved by reducing charging among BEBs is allowed and the charging time ﬂeet size. depends on the amount of energy to be replenished following We also note that the SoC variation negatively affects battery more realistic non-linear charging proﬁle. Meanwhile, we aging rate: the larger the SoC variation is, the faster the battery would also plan to extend the transit network to the one with degrades. Therefore, to prolong battery life, the operators are multiple terminals, explore more efﬁcient solution approach encouraged to charge BEBs as frequently as possible, instead of with high quality, instead of using heuristic algorithm; consider as late as possible. As a trade-off, when the charging facilities agency budget for the installation of charging infrastructure and are relatively sufﬁcient, the electric buses tend to be charged as consider users’ psychological inertia (Gao et al., 2020). frequently as possible, to achieve more cost saving by extending batteries’ lifespan. We next examine the effect of the number of fast chargers on Note the optimal cost, as presented in Figure 4. Compared to normal 1 The result may deviate within a small range due to the charging, fast charging mode can reduce the ﬂeet size largely variation of exchange rate. due to its high charging efﬁciency, even though the installation cost of fast charging facilities is much higher than that of normal charging facilities. Therefore, the maximum number of fast References chargers required is only 3 with the minimal operational cost at the lower-level as $1.05 10 , as shown in Figure 4. The ﬁgure Adler, J.D. (2014), “Routing and scheduling of electric and also reveals that the minimal total cost at the upper level alternative-fuel vehicles”, Doctoral dissertation, Arizona ($1.14 10 ) occurs when the number of fast chargers is 3. State University. An, K. (2020), “Battery electric bus infrastructure planning Figure 4 Effects of the number of fast chargers on the optimal cost under demand uncertainty”, Transportation Research Part C: 2.4 Emerging Technologies, Vol. 111, pp. 572-587. 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List of notations Table A1 List of notations Notation Description Sets and Indices V Set of trips T Set of time steps R Set of all feasible trip chains R’ Subset of feasible trip chains Q Set of charging modes i, j Trip and node indices O Index of a virtual trip before each BEB’s ﬁrst trip of the day D Index of a virtual trip after each BEB’s last charging activity of the day T Time step index R Trip chain index Q Charging mode index Decision variables d [{0,1} Equals one if a BEB serves trips i and j in turn and consecutively, and zero otherwise, i [ V |{O}, j [ V | {D}, i= j ij l [ {0,1} Equals one if a BEB’s charging activity following trip i [ V starts at time step t [ T with charging mode q, and zero otherwise itq / [{0,1} Equals one if a BEB is being charged at time step t [ T between trip i [ V and the next trip it serves with charging mode q, and zero otherwise itq C Number of charging facilities in mode q deployed at terminal station k [ {0, 1} Equals one if trip chain r [ R is selected, and zero otherwise C Decision variables at the tactical level X Decision variables at the operational level Parameters and other variables s Departure time of trip i e Travel time of trip i Lb Minimum battery level to eliminate range anxiety m Battery consumption of trip i M A sufﬁciently large number ~ v Amortized acquisition and maintenance cost of a BEB per day q Normal charging mode q Fast charging mode F(SOC , q) Number of time steps required from SOC to full with charging mode q i i W Battery acquisition cost A Unit cost of a charging facility per day with charging mode q c Cost of trip train r c(q) Coefﬁcient of battery capacity fading under charging mode q r ,r ,r ,r Coefﬁcient for battery degradation model 1 2 3 4 X End-of-life related threshold n(SOC ,1) Battery capacity fading rate from SOC to full i i d (SOC ,1) Cost incurred by battery capacity fading from SOC to full under charging mode q q i i ckðÞ ; b Cost of label (k, b) A Equals one if trip i [ V is covered by trip chain r and zero otherwise U Equals one if a BEB in trip chain r is on a charger with charging mode q at time step t and zero otherwise tq N(X) Neighborhood of X TL Tabu list R Exchange rate of RMB to US dollar Corresponding author Le Zhang can be contacted at: le.zhang@njust.edu.cn For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: permissions@emeraldinsight.com

Journal

Journal of Intelligent and Connected Vehicles – Emerald Publishing

Published: Feb 17, 2022

Keywords: Battery electric bus; Charging station design; Vehicle scheduling; Bi-level model; Heterogeneous charging modes

References

Routing and scheduling of electric and alternative-fuel vehicles

Adler, J.D.
Battery electric bus infrastructure planning under demand uncertainty

An, K.
Dynamic headway control for high‐frequency bus line based on speed guidance and intersection signal adjustment

Bie, Y.; Xiong, X.; Yan, Y.; Qu, X.
An overview on vehicle scheduling models

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A bi-level optimization framework for charging station design problem considering heterogeneous charging modes

A bi-level optimization framework for charging station design problem considering heterogeneous charging modes

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A bi-level optimization framework for charging station design problem considering heterogeneous charging modes

A bi-level optimization framework for charging station design problem considering heterogeneous charging modes

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