Optimal planning of intra-city public charging stations

Intra-city Public Charging Stations (PCSs) play a crucial role in promoting the mass deployment of Electric Vehicles (EVs). To motivate the investment on PCSs, this work proposes a novel framework to find the optimal location and size of PCSs, which can maximize the benefit of the investment. The impacts of charging behaviors and urban land uses on the income of PCSs are taken into account. An agentbased trip chain model is used to represent the travel and charging patterns of EV owners. A cell-based geographic partition method based on Geographic Information System is employed to reflect the influence of land use on the dynamic and stochastic nature of EV charging behaviors. Based on the distributed charging demand, the optimal location and size of PCSs are determined by mixed-integer linear programming. V€ asterås, a Swedish city, is used as a case study to demonstrate the model's effectiveness. It is found that the charging demand served by a PCS is critical to its profitability, which is greatly affected by the charging behavior of drivers, the location and the service range of PCS. Moreover, charging price is another significant factor impacting profitability, and consequently the competitiveness of slow and fast PCSs. © 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
In recent years, the electrification of the transportation sector is thriving. Electric Vehicles (EVs) have shown clear advantages, such as environmental friendlyness, flexible operation, energy storage etc. However, it must be noted that the transfer from internal combustion engine vehicles to EVs should be accompanied by a highly renewable energy mix in order to maximize the benefit. For the fossil fuel dominated power production, indirect pollution still exists [1]. In countries with high penetration of renewable energy, different measures, such as tax reduction, subsidies, and extra local benefits, e.g., free parking, have been implemented to motivate the transition from petroleum-fueled vehicles to EVs [2]. Although EVs' penetration is rapidly increasing, reaching 7.2 million units in 2020 globally, EVs account for only a tiny market share [3,7].
Range anxiety is one of the most concerning issues for EV users and potential owners [5], and intra-city Public Charging Stations (PCSs) play a critical role in relieving range anxiety [6]. Recent years have seen a rapid development of public EV charging infrastructure. In 2019, the installed Publicly Accessible Chargers (PACs) reached 0.8 million in the world and accounted for an increasing share in the total installed light-duty vehicle chargers [7]. However, the number of installed PACs is far below the global EV stock. The underdevelopment of PCSs is regarded as the most significant barrier to EV penetration [8]. It is, therefore, essential to boost the intracity PCSs installation and motivate the associated investment.
Many efforts have been focused on optimizing the location and size of intra-city PCSs. For example, Cavadas et al. [9] and Sun et al. [10] conducted location optimization of PCSs based on residents' travel behaviors to maximize satisfied charging demand with constraints of a fixed budget. Xi et al. [11] and Wang et al. [12] employed the utility theory to maximize PACs' utilization with budget constraints. Zhu et al. [13] formulated a minimization problem for attaining minimal cost, including construction costs of PCSs and EVs' travel fuel costs. A method that can search for the shortest paths and a multiple domination model were adopted in studies [14,15] to minimize the number of PCSs while satisfying all the charging demand. In addition, the costs of electrification and energy transmission loss were also taken into account in PCS planning to reduce the impacts on grid reliability in Refs. [16,17]. Integration of solar power [18] and wind power [19] integration becomes a new principles for PCS development to realize a low carbon future. However, most of the previous studies focused on serviceability rather than the profitability of PCSs. It is also noted that high investment and low utilization of PCSs might lead to a low chance of profiting [20]. How intra-city PCSs should be deployed to gain maximal profits is vital for stakeholders and has not been comprehensively studied. There is a clear knowledge gap about quantitatively evaluating the economic performances of intra-city PCSs.
To quantitatively assess the economy of intra-city PCSs, it is important to consider the key factors affecting the profit of PCSs, including location, charging demand, charging price, and the types of Electric Vehicle Supply Equipment (EVSE) in the PCS [8]. Planning intra-city PCSs is different from planning inter-city PCSs. Intercity PCSs are usually constructed in highway networks to serve 'onroute charging', and fast charging is the priority [21,22]. Intra-city PCSs mainly meet 'end-route charging' demands for shortdistance trips. Intra-city charging often happens at parking locations, varying with time of the day and type of land use [23,24]. Therefore, planning of intra-city PCSs needs to consider the spatial and temporal distribution of charging demand.
Modeling techniques of EV charging demand in the formulation of PCS planning problem have obtained extensive academic attentions. In the existing literature, the most popular methods are Flow-Capturing Location Models (FCLMs) and set-coverage location models (SCLMs). FCLM, which was firstly introduced by Hodgson [ [26] considered driving range limitations, and Chen et al. [27] investigated the impacts of drivers' time sensitivities on charging patterns. In contrast to the fixed and static shortest path principle, Yang [28] relaxed the assumptions on route choice and allowed small deviation from shortest path to simulate diverse user behaviors, and Li et al. [29] developed a multi-period optimization model to simulate the dynamics in the charging network. In other studies, FCLMs were extended to capture the stochastic nature of EV charging demand [21,30]. However, the applications of FCLMs are usually limited to the planning of refueling stations and inter-city PCSs that satisfies long distance travel demands, while SCLMs show better accuracy in intra-city PCS planning [31]. SCLM, proposed by Toregas [32], aggregates short distance charging demand in parking lots, which are generated using the method of node-based geographical segmentation. The disadvantage of SCLM is that charging demand is usually estimated based on the population density and vehicle ownership density, while ignores the dynamic and stochastic nature of EV travel and charging behaviors [33]. There have been a few studies that refined SCLMs to explicitly assess the mobility of EVs. Vazifeha et al. [34] conducted individual EV trajectory analysis based on call detail records. Bai et al. [35] formulated a data-driven optimization problem based on the GPS trajectory data of thousands of vehicles. It is noted that the charging patterns of EVs were usually oversimplified in the literature. For intra-city PCS planning, public charging demand should be distinguished from private ones. This is usually handled by assuming simplified travel and charging patterns that EVs are only charge at home or public areas [36,37]. However, in reality, EVs tend to be charged both at home and in public areas [38]. Some studies transformed the demand partition problem into choice criterion problems and associated demand levels with types of land use [39,40]. Luo et al. [41] investigated the temporal and spatial distribution of EV charging demand by acquiring EV parking distribution at different types of land uses. However, in these studies, the charging patterns of EVs and urban land uses are usually oversimplified, which can lead to significant deviations from real charging demand profiles.
Through reviewing the related works, the following gaps have been identified: (i) there is a lack of analysis about the economy of intra-city PCSs, which is crucial for motivating investment; (ii) there has not been a computationally effective method that can model both temporal and spatial distributions of charging demand and localize the demand corresponding to real-world urban areas; and (iii) there has not been a categorization method that can distinguish public charging demand from private charging demand.
To fill these gaps, this paper proposes a novel method that can consider the impacts of charging behaviors and urban land uses on optimizing the location and size of PCS in order to maximize the profits. An Agent-Based Trip Chain Model (ABTCM) is adopted to generate dynamic and stochastic charging demand; and a cellbased geographic partition method is conducted to localize the generated public charging demand to real-world urban areas. To optimize the siting and sizing of intra-city PCSs, a GIS-based Mixed Integer Linear Programming (MILP) optimization model is formulated. The proposed method is applied to V€ asterås, a Swedish city, to evaluate the impacts of service range, economic parameters and charging strategies on the profit of intra-city PCSs.
Following this introductory section, the remainder of the paper is structured as follows: In Section 2, the methodology for optimizing the planning of intra-city PCSs is presented. Section 3 applies the proposed methods to V€ asterås, Sweden, as a case study. Section 4 reports and discusses the results and Section 5 concludes the paper with a summary of key findings.

Methodology
The structure of the proposed method is depicted in Fig. 1. Charging revenues and costs, land rental costs and system investment costs are taken into consideration to assess the economic benefits of intra-city PCSs.
The charging demand generated from the Agent-Based Trip Chain Model (ABTCM) is localized to each geographic cell according to the type of land use and the volume of traffic flow. Then, the economic benefits of newly built PCSs are maximized with the MILP model considering charging revenues and costs, land rental costs and system investment costs. Charging revenues and costs are calculated based on electricity price, charging price of the selected EVSE and charging demand covered by the PCS. Land rental costs are determined by the locations of intra-city PCSs, which are selected from the potential intra-city parking locations identified by the GIS model. Investment costs are decided by the type and number of EVSE, which are dependent on the land use type and the magnitude of charging demand. The details of each sub model are introduced in the following sections.

The objective function
The siting and sizing of all PCSs are optimized simultaneously in maximizing the total profit of newly-built intra-city PCSs. The objective function is: where Pr is the profit of newly-built PCSs in the study area; p is the charging price per minute; x is a binary variable representing whether a PCS is built at a certain location, which is determined by Eqn (2); q represents the hourly charging demand covered by the PCS; n and pw denote the number and charging power of EVSE at the PCS, respectively; c is the total cost of a PCS; I represents the total number of potential locations; T represents the total number of hours during the simulation; and subscripts i and t denote the index of potential locations for intra-city PCSs and the index of the hour of the day. The total cost of a PCS consists of land rental cost (c rent ), system investment cost, (c inv ), fixed Operation and Maintenance (O&M) cost, (c O&M ), and electricity cost (c ele ), which are calculated as Eqns (3)e(7). The fixed O&M costs are assumed to be 3% of the investment costs [42].
where c park represents the parking fee for parking and installing charger during the period of use; and c EVSE and LT stand for the investment costs of an EVSE and its life span, respectively.

1) Constraints on charging demand
The hourly charging power covered by the ith PCS, q i;t , is constrained by the potential charging demand nearby (D i;t ) and the total installed charging capacity (n i *pw i ), as shown in Eqns (8e10). The potential charging demand at a PCS (D i;t ) is gathered from surrounding nodes according to Eqn (11), the demand nodes have to be in the PCS's service range (Eqn (12)) and cannot be assigned to another PCS (Eqn (13)).
where D denotes the potential charging demand that originates within the service range of a PCS and could be possibly covered by the PCS; j is the index of cells and corresponding demand nodes; J represents the total number of cells and corresponding demand nodes, which are further explained in Section 2.3; Dl is the charging demand at a demand node; rp is a binary variable denoting the coverage ability of a newly-built PCS; lp represents the Euclidean distance between a demand node and the PCS; and L represents the service range of the PCS.
2) The limits of the number of EVSE Eqns (14) and (15) respectively define the upper bound and lower bound of the number of EVSE.
where y i represents the maximum number of EVSE that can be installed at a PCS.
3) The limit of the total number of PCS Eqn (16) describes the maximum number of PCS that will be deployed.
where N is the maximum number of PCSs that can be deployed in the study area.

Dynamic EV charging demands at different land use types
To simulate the potential charging demand in urban areas, an ABTCM model is used, which can capture the dynamics of EV trips at three different land use types, corresponding to seven trip purposes (Fig. 2).
The trips of an EV agent in the ABTCM are characterized by six types of parameters: the place and time of departure, represented by p d and T d , the distance and time for driving, represented by d t and t d , and the destination and corresponding parking time, represented by p e and t p . The trip vector is represented by Eqns (17e19).
where m represents the index of trips. A trip chain is formulated for each EV agent, which starts from leaving home at time T d,1 . The parameters of the following trips are dependent on the previous trips. The ABTCM shows very good performances of modeling real charging demand, which has been validated in the authors' previous work [43]. The energy consumption of an EV trip is assumed to be proportionate to its driving distance. The energy discharge of EV battery after a trip is measured by State of Charge (SOC), which is calculated by Eqn (20).
where SOC 0 is the original SOC before the trip; E is the electricity consumption per kilometer; and C is the capacity of EV battery. In this study, two charging strategies, uncontrolled charging (UC) strategy and off-peak charging (OPC) strategy, are employed to simulate EV charging behaviors, and the detailed algorithm can be found in Ref. [43].
The accessibility to different types of EVSE is considered as an impact factor on EV charging behaviors. Based on the statistics in Refs. [44], EVSE is categorized into three types according to the range of power output, as shown in Table 1.
The simulation is implemented with a 1-min resolution to obtain the daily average charging demand as indicated by Eqn (21).
where Da represents the hourly average charging demand; the subscript lu denotes land use types; and R, W, P represent the residential, working and public land use type, respectively.

Cell-based geographic partition for charging demand localization
The charging demand obtained from the ABTCM simulation is localized to the studied area according to the types of land uses. As shown in Fig. 3, six types of areas including: Residential Villa Area (RVA), Residential Apartment Area (RAA), Commercial Area (CA), Working Area (WA), Mixed-use Area (MA) and Natural Area (NA), are identified by GIS. The residential charging demand obtained from the ABTCM is localized to RVAs and RAAs, while, only the charging demands originated at RAAs can be satisfied by intra-city PCSs, because residential villas are usually equipped with private EV chargers. The working charging demand is localized to WAs, and the public charging demand is localized to CAs and MAs.
In this study, the parking lots are identified as potential locations for intra-city PCSs, owing to their accessibility and convenience. Each PCS is assumed to contain enough parking places.
As shown in Fig. 3, the area is partitioned into a number of equally sized square cells. Each cell contains a certain amount of charging demand, which is also called a demand node. The charging demand is localized based on the real distribution of traffic flow, which is measured by the traffic measurement points (TMPs) in the studied area. The traffic flow in each cell (f j ) is the average value of all TMPs in it (Eqn (22)). If there is no TMP in one cell, the traffic flow is assumed to be the average of its surrounding cells. The spatial distributions of traffic flows are determined by land use type and the function of areas, as given in Eqn (23). As shown in Eqn (24), the total charging demand, which is obtained from ABTCM (Eqn (21)) is localized to the corresponding areas based on the function of traffic flows.
The original charging demand in each cell is firstly satisfied by existing PCSs (Eqn (25)) and the remaining charging demand is the potential demand for newly-built PCSs. As shown in Fig. 3, the service ability, i.e., total installed capacity, of existing PCSs is distributed to each cell in proportion to the inversion of the distance between the PCS and the cell (Eqns (26) and (27)).
De z;j ¼ re z;j * 1 le z;j P J j¼1 À r ez;j * 1 le z;j Á ,P z (26) where f represents the amount of traffic flow; k and K are the index and the total number of TMPs in the cell, respectively; A represents the area covered by a type of land use; and Dc j is a 24-dimensional vector containing hourly charging demand at the node of cell j (Dc j;t ); the subscript z denotes the index of existing PCSs; De represents the charging demand on a demand node covered by an existing PCS; re is a binary variable denoting the coverage ability of an existing PCS; le represents the Euclidean distance between a demand node and an existing PCS; and P represents the total output power of an existing PCS.

Case study
V€ asterås, Sweden has an overall area of 67 km 2 . There were 44,192 private vehicles including 324 plug-in EVs in 2016. In the present study, the penetration rate of EV is assumed to be 5%, which implies a total of 2200 EVs.
Sweden had achieved almost carbon-free electricity supply by 2017, and set an ambitious objective to achieve 100% renewable electricity generation by 2040 [45]. The electricity supply by source in Sweden is depicted in Fig. 4.
The transport sector, consuming a large amount of oil, accounts for half of Swedish energy-related carbon emissions. Sweden aims to reduce 70% of greenhouse gas emissions of the transport sector from 2010 to 2030. Therefore, electrification of transport has been vigorously promoted, and public charging infrastructure has been greatly developed to strengthen the confidence of consumers. In 2017, Sweden had b the highest proportion of EV sales in the world, i.e., 6.3% in new car sales. Fig. 5 shows the distribution of different types of land uses, which is derived from OpenStreetMap and Google Earth. As shown in Fig.5(a), 8 PCSs with 40 chargers have already been installed in the city [46]. A total of 532 parking lots in this area are identified as potential locations for newly built intra-city PCSs. The map of the parking lots shown in Fig. 5(b), is obtained from OpenStreetMap [47]. As shown in Fig. 5(c), the traffic flow is monitored at 245 TMPs, and the data can be obtained from the traffic administration of V€ asterås [48]. The whole city is divided into 268 cells with the side length of 500 m as illustrated in Fig. 5(d).
The travel patterns are determined based the American National Household Travel Survey 2017 and some key parameters, i.e. departure time, travel distance and travel time, are calibrated by Swedish National Travel Survey (2015e2016) [49,50]. The charging patterns are extracted from an EV project [38].
The output power of EVSE and the cost for charging are taken from the existing PCSs in V€ asterås, Sweden [46], as shown in Table 2. In this paper, Level 2 and Level 3 EVSE are referred to as slow and fast chargers, respectively. According to the types of EVSE, the PCSs are referred to as Slow PCSs (SPCSs) and Fast PCSs (FPCSs), respectively. It is assumed that each PCS has at least 1 charger and at most 20 chargers. It is also assumed that FPCS will be constructed  only in CAs and SPCSs in WAs and RAAs. In addition, the parameters of Nissan Leaf EV are adopted to represent typical EV parameters. As shown in Fig. 6, the city is divided into three zones, corresponding to different land rental costs. Downtown in the inner ring (green star) has the highest land cost, and the hospital (red star) is the demarcation point of the second ring and the third ring. The information on EVSE is summarized in Table 2. The investment costs are derived from Refs. [42,51], which are commonly used by other literature.
To evaluate the impacts of charging strategies, service range and economic parameters on the profits of PCSs, four scenarios are generated, and the first scenario is a benchmarking scenario. Scenario 2 simulates the EV charging demand under the OPC strategy where EV owners are motivated to change their charging patterns to improve the grid reliability. Scenario 3 focuses on the impacts of    coverage ability of PCSs, where a larger service range, 1000 m, is considered. In Scenario 4, a sensitivity analysis on charging price is conducted. The details of the scenarios are tabulated in Table 3. In each scenario, three optimizations are conducted with the number of newly-built PCSs being no more than 5, 10 and 15. In addition, the number of EVs is assumed to remain the same.

Benchmarking scenario
The ABTCM is used to simulate the charging demand of 2200 EVs. To capture the comprehensive randomness of EV behaviors, the simulation was carried out for 77 days with a time step of 1 min. The simulation of the first seven days is removed to ensure that the battery SOC of the EV fleet have arrived in random states, and the following results start from the 8th day of simulation. Fig. 7 shows the typical one-day charging profiles at different land use types. The charging demand in a specific hour of day is averaged over the values of charging demand in the same hour of the 70 simulation days. The derived typical one-day charging profiles reflect the different travel and charging patterns on weekday and weekend. It is found that a large portion of daily charging demand is met at residential land use types, and 65% of the portion is satisfied in RVAs, which is not included in public charging demand. The peak of public charging demand appears at around 9 o'clock in the morning, which is mainly contributed by the charging demand in WAs and CAs.
To understand the impacts of the number of newly-built PCSs, it was assumed that 5, 10 and 15 PCSs would be built, and the optimized number of chargers, costs, and profits for all PCSs are listed in Table 4. The overall economic performances of the newly built PCSs are very good, mainly due to the high charging price in V€ asterås, which is 7.5e9.4 times of the electricity price. The charging demand covered by an average PCS increases with the total number of PCSs. The increase in covered charging demand simultaneously raises the profit and reduces the payback time. In contrast, the return on investments (ROI) decreases with the increase in the number of PCS, i.e., the ROI is 409.7%, 396.3%, and 375.3% for 5, 10 and 15 newly built PCSs, respectively, which indicates that the ROI is affected by the utilization rate of chargers.
The locations of the PCSs are illustrated in Fig. 8. Since most existing PCSs are in the city center, the newly-built PCSs are mainly located in the surrounding area. However, in general, the PCSs closer to the central tend to have a high level of profits due to a high level of charging demand.

The impacts of charging pattern
With the development of smart charging technologies, it is foreseen that off-peak charging will be greatly promoted, where EV owners are motivated to charge at home and reduce their reliance on PCSs [52]. As shown in Fig. 9, almost all the charging demand is shifted to evening time, and resultantly the charging demand in CAs and WAs is shifted to residential areas.
Although the total amount of charging demand remains unchanged, the public charging demand decreases rapidly. This causes a high peak of charging demand in the residential areas, of which the magnitude is twice larger than the peak in Scenario 1.
The detailed results are reported in Table 5. The change in charging patterns leads to significant decrease in profits (compared to Table 4). Yet, covered demand remains increasing, and the payback time decreases with the number of PCSs. However, the type, location and number of chargers of the newly-built PCSs are different from those in the benchmarking scenario. Compared to the benchmarking scenario, all the newly-built PCSs are SPCSs in residential areas and the total public charging demand in Scenario 2 is reduced by about 40%, resulting in fewer number of chargers on  average at every PCS.

The impacts of the coverage ability of PCSs
Service range is an important indicator for evaluating PCS service ability. Exiting studies assumed that service range should be slightly longer than people's acceptable walking distance from parking lots to destinations, i.e., 400 m. Some studies adopted the service range of refueling station. Regarding SPCS, EV owners usually park vehicles for a long period of time, and the service range   should be close to the acceptable walking distance to avoid impairing people's travel satisfaction. However, in anticipation of the advancement of fast charging and battery swap technology, the duration to serve a customer will become shorter and close to the service time of refueling station. Therefore, this study sets up service range of 500 m and 1000 m to investigate the impacts of cover ability.
For a certain type of PCS, a larger service range usually means a larger amount of covered charging demand, and thus means larger charging revenues. The relation between coverage ability and daily profits is illustrated in Fig. 10, based on the results for 5 and 10 PCSs in Scenario 1 and 3. Fig. 10 reflects that profit is linearly correlated to charging demand, which is under the limit that a PCS can serve. The profitability of FPCS is slightly lower than that of SPCS, because FPCS has higher investment. The comparison between FPCS and SPCS will be elaborated in Section 4.4.1. In addition to service range, larger penetration rates also mean larger amount of charging demand for each PCS. Given the lessons learned here, the profit of PCS will increase proportionally with the increase in EV penetration rate, while the return-on-investment (ROI) of PCS is similarly constant. Since the number of chargers has not reached the capacity limit, the number of chargers will increase with the increase in charging demands, while the locations of PCSs remain the same. If the penetration of EVs keeps increasing and reaches the capacity of potential locations, new PCSs will be built and the locations with high profitability have priority over others (Fig. 8).
The numerical results of Scenario 3 are presented in Table 6. Compared to the benchmarking scenario, the covered charging demand in Scenario 3 is 58.8% larger with 5 newly-built PCSs, 46.5% larger with 10 newly-built PCSs and 31.4% larger with 15 newlybuilt PCSs. Correspondingly, the number of chargers increases by 57.9%e89.5%, and the profits increase by 38.7%e57.5%. The covered demand of a charger on average is less than that in Scenario 1, which indicates a decrease in charger utilization as well as ROI. Fig. 11 depicts the cover ability of existing and newly-built PCSs in Scenario 1 and 3. The public charging demand covered by existing PCSs increases from 691.9 kWh in Scenario 1 to 1987.3 kWh in Scenario 3, accounting for about 23% of the total public charging demand. As shown in Fig. 11(d), the existing and newly-built PCSs with a service range of 10000 m can meet 95% of the total public charging demand. In addition, disproportionately huge investment is needed to meet the remaining 5% of the peak public charging demand. This will be discussed in detail in Section 4.4.2.

The impacts of charging price
The high charging price in V€ asterås is expected to decline in the future and this would encourage people to charge at PCSs. In this section, the impacts of charging price are evaluated by a sensitivity analysis.4.4.1 The impacts on the competition between FPCS and SPCS.
The cost for a FPCS is generally higher than that for a SPCS ( Table 2). One the one hand, the investment costs and O&M cost of a FPCS are higher than that of a SPCS. On the other, the fee charged per kWh at a FPCS is 80% of that at a SPCS. Therefore, SPCSs have higher profitability than FPCSs for serving the same amount of charging demand. This explains why the number of SPCS is usually much more than the number of FPCS.
To investigate the impacts of charging price on the competition between FPCS and SPCS, the charging price of FPCS is fixed, while the charging price of SPCS decreased by 0%e50%, with a step of 5%. The variation in the number of FPCS is shown in Fig. 12. The reduction in charging price of SPCS leads to a declining profitability as well as competitiveness, while the number of FPCSs increases. When the reduction in charging price of SPCS is smaller than 25%, the number of FPCS is fewer than the number of SPCS; when the charging price of SPCS reduces by 25%e30%, FPCS becomes competitive with SPCS; and the number of FPCS is more than the number of SPCS when the charging price of SPCS decrease over 35%. However, maximum 8 FPCSs could be constructed in a total of 15 PCSs, due to the limit of the total commercial land use area.
The geographical distribution of the 5 new PCSs with a 15% and a 45% reduction in SPCS charging price is shown in Fig. 13. Compared to the benchmarking scenario, SPCSs are replaced by FPCS, while the location of the PCSs remains nearly unchanged. This result reflects the fact that the PCS location mainly depends on the distribution of aggregated charging demand, and it certifies the robustness of the proposed method for optimizing PCS location.

The impacts on PCS payback time and sizing
To investigate the impacts of price variation on PCS payback time and sizing, the charging prices of both SPCS and FPCS decrease by 0%e80%, with a step of 10%. For SPCS, a 25% reduction in the charging price is taken as the reference for the sensitivity analysis.
The variation of ROI and payback time for 10 newly-built PCSs with different charging prices is illustrated in Fig. 14. ROI decreases and the payback time increases with the reduction in charging price. PCSs will break even when the charging price reduces by 80%, while the payback time prolonged to about ten years.
Variation of charging price also impacts on PCS sizing, which can further affect the coverage ability of PCSs. As depicted in Fig. 15, the charging capacity of PCSs decreases, and average working time increases along with the reduction of charging price. Compared to the benchmarking scenario, the peak charging power decreases by about 80% and the average daily working time of chargers increases from 8.6 h to 11.8 h, due to an 80% reduction in charging price.
To further evaluate the impacts of charging price on the coverage ability of PCS, a random SPCS is taken as an example. As shown in Fig.16Fig. 16, the peak demand can be satisfied by 3 chargers. When the charging price is reduced by 60%, indicated by the grey dashed curve, the optimization results show that only one      charger will be installed, and only the base demand will be met. The results with a high time resolution further prove the advantage of the proposed model in modeling the temporal heterogeneity of stochastic charging demand.

Conclusion
To optimize the location and size of Public Charging Stations (PCSs) and in order to maximize their profits, this paper proposes a framework, which combines an ABTCM, a cell-based geographic partition method and a GIS-based MILP optimization model. Such a framework provides a robust and adaptable approach to distinguish public charging demand from private charging demand and provides a real representation of the spatial and temporal distribution of public charging demand. The method has a good replicability. It can be applied to any city if the travel survey data, traffic flow measurement, and distribution of parking lot locations are available.
In this work, the proposed framework is verified by adopting it for a case study. The impacts of charging pattern, coverage ability of PCS and charging price on the planning of PCSs are specifically investigated. With the current charging price, the more charging demand a PCS serves, the higher profits it attains. Public charging demand can decrease by as much as 40% due to the variation of charging pattern, and the profits of a PCS would decrease significantly under the off-peak charging strategy. Due to the same reason, improving the coverage ability of a PCS can improve its profitability, and the location of PCSs gives priority to the areas with high charging demand. In addition, a decrease in charging price impairs the profitability of PCSs, and thus leads to smaller PCS sizes. By comparison, SPCS has competitiveness advantage over FPCS with the current charging price, while FPCS becomes competitive with SPCS when the charging price of SPCS reduces by 25%e30%.
This paper provides policy makers and PCS investors an effective tool for planning intra-city PCS investments and highlights the importance of charging prices for motivating EV penetration. This work focuses on the planning of PCS in reference to profitability and the capacity limit of the local low-voltage grid was not considered, due to lack of data. Future studies will consider the variation of charging speed and the impacts on low-voltage distribution network based on real-time distributed control algorithm for EV charging and vehicle to grid (V2G). In addition, in the current study, the demand simulation and PCS planning are relatively independent to avoid changing the existing travel and charging pattern of drivers. The impacts of newly-built PCSs on the travel and charging behaviors of EV owners can be evaluated in a more interactive way in future studies.