diff --git "a/5dE4T4oBgHgl3EQfbwyA/content/tmp_files/load_file.txt" "b/5dE4T4oBgHgl3EQfbwyA/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/5dE4T4oBgHgl3EQfbwyA/content/tmp_files/load_file.txt" @@ -0,0 +1,1381 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf,len=1380 +page_content='Incorporating time-dependent demand patterns in the optimal location of capacitated charging stations Carlo Filippi(1) Gianfranco Guastaroba(1) Lorenzo Peirano(1) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Grazia Speranza(1) (1) University of Brescia, Department of Economics and Management, Brescia, Italy {carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='filippi, gianfranco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='guastaroba, lorenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='peirano, grazia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='speranza}@unibs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='it January 13, 2023 Abstract A massive use of electric vehicles is nowadays considered to be a key element of a sustainable transportation policy and the availability of charging stations is a crucial issue for their extensive use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Charging stations in an urban area have to be deployed in such a way that they can satisfy a demand that may dramatically vary in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In this paper we present an optimization model for the location of charging stations that takes into account the main specific features of the problem, in particular the different charging technologies, and their associated service time, and the fact that the demand depends on space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' To measure the importance of incorporating the time dependence in an optimization model, we also present a simpler model that extends a classical location model and does not include the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' A worst-case analysis and extensive computational experiments show that ignoring the temporal dimension of the problem may lead to a substantial amount of unsatisfied demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Keywords: Facility location, Charging stations, Electric vehicles, Demand patterns, Time-dependent optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 1 Introduction Sustainable transportation is one of the major challenges that modern countries are facing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Several sources indicate that the transportation sector generates the largest share of GreenHouse Gas (GHG) emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' According to the United States Environmental Protection Agency1, in 2020 the transportation sector produced 27% of the total GHG emissions in the US, mostly generated from burning fossil fuels by cars, trucks, ships, trains, and planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Domestic statistics issued by the UK government2 confirm that the transportation sector generated 27% of the total GHG emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The majority (91%) came from road transport vehicles, where the biggest contributors were cars and taxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Furthermore, data provided by the European Environment Agency3 highlight that in the EU more than 22% of the GHG emissions came from the transportation sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='epa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='gov/ghgemissions/sources-greenhouse-gas-emissions 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='uk/government/statistics/transport-and-environment-statistics-autumn-2021/tran sport-and-environment-statistics-autumn-2021 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='eea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='eu/data-and-maps/data/data-viewers/eea-greenhouse-gas-projections-data- viewer arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='05077v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='OC] 12 Jan 2023 Despite technical advances have made available a range of options for sustainable mobility, there are still important obstacles that must be overcome for their mass adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Among such options, Electric Vehicles (EVs) are considered one of the major directions to reduce the environmental impact of people mobility and make urban areas more sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In the 2021 edition of the Global EV Outlook 20214, the International Energy Agency pointed out that at the end of 2020 the global EVs stock hit 10 millions units, with 3 millions newly registered EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Europe was the fastest growing market, with a sales share equal to 10% and some leading countries, such as Norway, which registered a record high sales share of 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' This trend was accelerated by many countries of the European Union through substantial financial incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' However, the decision of potential EV buyers is still strongly affected by two major issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' On one hand, the purchase cost of an EV is still higher than that of a traditional internal combustion engine vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' On the other hand, the limited travel range of an EV and the long charging time are well-known to generate anxiety in the potential buyers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', Pevec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In fact, the willingness of drivers to purchase an EV strongly depends on the availability of charging stations nearby their points of interests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', home and work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' As the number of charging stations is growing, thanks to public and private investments, the location problem of such stations has attracted much attention (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' There are a number of factors that make the location of charging stations substantially different from other, more classical, location problems, in particular the choice of the charger to install (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', slow, quick, fast), and the characteristics of the charging demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The type of charger is a key factor to be taken into account, as it impacts the charging time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' As of the end of 2021, there exist three main types of charger (see Moloughney, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Level 1 chargers, also referred to as slow chargers, use common 120-volt outlets, and can take up to 40 hours to raise the level of a standard battery EV (with a 60 kWh sized battery) from 10% to 80% of the capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' These chargers are most suitable for private usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Level 2 chargers, sometimes called quick chargers, can charge up to 10 times faster than a level 1 charger, and are the most commonly used types for daily EV charging (see Moloughney, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Given the same battery characteristics mentioned above, the charging time is about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The level 3 or fast chargers can reduce the charging time to 40 minutes or even less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' For a comprehensive study regarding the state of the art on charging stations, the interested reader can refer to Pareek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The type of charger demanded by EVs is affected by the urban layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' For example, slow chargers will be demanded in residential areas so that EVs can be recharged over the night at low cost (an interesting study of the factors influencing the charging demand is provided in Wolbertus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In the classical location models a customer is characterized by the distance from any potential location and by a single quantity - a measure of the demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The models do not consider a temporal dimension of the problem which basically corresponds to assuming that the demand is uniformly distributed over the time period of interest of the location decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' On the contrary, the charging demand of EVs fluctuates over time, with peaks of demand in periods of time where the traffic volume is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Neglecting the demand dynamics may lead to solutions where the charging capacity deployed is not sufficient to satisfy the demand during the peak times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In this paper, we study the problem of determining an optimal deployment of charging stations for EVs within an urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Different types of chargers have to be located in pre-defined potential locations, modeled as nodes of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The urban area is partitioned in sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' A customer is associated with each section of the urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Its demand in a certain time interval is the number of EVs in that section that need to be recharged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The customer is located in the 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='iea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='org/reports/global-ev-outlook-2021 2 center of gravity of the section and is modeled as a node of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The urban area is also partitioned in zones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', commercial, industrial, or residential) which have different needs in terms of minimum number of each type of charger deployed in the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We have to determine, for each type of charger and each potential location, the number of chargers to be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Two criteria have a key role in this location problem: the cost of installing the chargers and the distance the customers have to travel to be recharged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We present, over a discretized time horizon, an optimization model that introduces a temporal dimension which, to the best of our knowledge, has never been introduced in the literature on location problems and captures the dynamics of the charging demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Assuming that a charger can take more than one period to fully recharge an EV, the proposed multi-period formulation includes constraints to keep track of the usage of chargers across consecutive time periods and to ensure that no other vehicles are assigned to any occupied charger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' This novel approach guarantees a correct sizing of the solution, in terms of number of stations opened and number of chargers installed, and ensures that the demand is completely satisfied in all time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In order to assess the value of introducing the temporal dimension in the location problem, which makes the optimization model more complex, we present a single-period optimization model that captures the same specificities of the problem but ignores the temporal aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In both models, the objective is the minimization of a convex combination of two terms: the total cost of deploying the charging stations and installing the chargers, and the average distance traveled by the customers to reach the assigned charging station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The two optimization models turn out to be Mixed Integer Linear Programming (MILP) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We compare the two models through a theoretical and a computational analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We show, through worst-case analysis, that a solution to the single-period model may fail to satisfy a large portion of the charging demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Extensive computational experiments are run on different classes of randomly generated instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The results confirm the importance of explicitly considering the dependence on time of the demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In fact, the single-period model is based on the common assumption that the charging demand is uniformly distributed across the planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In an application context such as the one at hand, where the demand fluctuates significantly during the day and across different zones of the same urban area, the single-period model produces solutions that are not capable of serving a large portion of the charging demand, especially in those time periods where the demand is prominently concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The computational experiments also include a parametric analysis of the relative weight assigned to the objective function components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In Section 2, the literature most closely related to our research is reviewed and the contribution of this paper is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In Section 3, after the presentation of the single-period extension of a classical location model, we provide the multi-period mathematical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In Section 4, we analyze the worst-case performance of the single-period model in terms of portion of unsatisfied charging demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Section 5 reports extensive computational experiments conducted on instances gener- ated to resemble demand dynamics frequently observed in different zones of a city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Finally, some concluding remarks are outlined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 2 Literature review The problem of determining an optimal location and size of charging stations for EVs has recently attracted an increasing academic attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Recent overviews of the main modeling and algorithmic approaches employed in this research area are available in Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2018), Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2019), and 3 Kchaou-Boujelben (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' For a general introduction on location problems the interested reader can refer to Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In the following, we focus on the papers that are most closely related to our research, and refer the interested reader to the above-mentioned surveys and the references cited therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' A first broad classification of the literature is based on the type of network considered (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' When only the distribution network is considered, the optimal location of charging stations must consider the potential adverse effects on the power grid, as an inappropriate placement of charging stations can be a threat to the power system security and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' On the other hand, when only the transportation network is taken into account, the main issue is to determine an optimal location of charging stations over a road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' This paper lies in the latter category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Within this category, the related literature can be further classified into two main streams of models called flow-based and node-based demand models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', see Kchaou-Boujelben, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In the literature, the majority of the research efforts are devoted to the flow-based demand models, whereas the number of papers adopting a node-based approach is still relatively limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' To the best of our knowledge, Anjos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2020) are the only authors that integrated, within the same optimization model, both a node-based and a flow-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The flow-based demand models are best suited for modeling long-haul (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', inter-urban) journeys where accounting for the limited driving range of EVs is important (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Anjos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Contributions to this line of research can be found, for example, in Kuby and Lim (2005), MirHassani and Ebrazi (2013), Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2016), and Hosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The present paper adopts a node-based demand model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In the class of node-based demand models, drivers demanding to charge their EVs are associated with one/few fixed locations, which represent, for instance, their residence, workplace or specific service facilities (such as commercial activities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' This approach is best suited for urban settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In fact, in such case EVs do not move much from the location where they need to be charged and their limited driving range can be neglected (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Anjos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The most common modeling approaches applied in the literature are based on the extension of classic discrete location models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', location-allocation as in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2016), set covering as in Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2016), and maximum coverage problems as in Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2019)) to incorporate technical constraints specific to EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Characteristics of the charging demand (such as the population size, the penetration rate of EVs, the type of zone, and the time of the day) are known to have a crucial impact on the optimal location of charging stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' To position the present paper within the literature, we classify the mathematical formulations into single-period and multi-period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In single-period optimization mod- els all the decision variables are time independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Although the spatial-temporal distribution of the charging demands is described by different authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', see Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2020, and the references cited therein), only few authors have proposed multi-period optimization models where the alloca- tion of the demand to the charging stations is time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The related stream of literature can be classified according to the length of the planning horizon considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' A long planning horizon is considered by some authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The basic rationale of these models is that locating charging stations is a long-term strategic decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' As a consequence, during these long periods of time the technology available, as well as the charging demand, may change significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Along this line of research, we mention the paper by Anjos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2020) where it is assumed that the locating decisions taken in a period have an impact on the charging demand in the subsequent periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In fact, potential EV buyers are influenced by the availability of charging opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Some papers have proposed multi-period optimization models that consider a short horizon, usually a day, divided in time periods, usually hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Our research belongs to this category of papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 4 To the best of our knowledge, Cavadas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2015) are the first authors to recognize the importance of incorporating into an optimization model the dynamics of the charging demand across the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The aim of the proposed multi-period model is the maximization of the total demand served, subject to a constraint on the budget available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The authors consider only one type of charger (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', a slow type) and the sizing of the charging stations is not part of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In the model we present in this paper, we address these shortcomings by considering multiple types of chargers and optimizing the quantities installed in each opened station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Rajabi-Ghahnavieh and Sadeghi-Barzani (2017) estimate the charging demand of EVs in different zones of a city and at different hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The authors consider the deployment of an unlimited number of fast chargers only and propose a non- linear optimization model that includes three cost components: the total opening cost, the total cost for the drivers to reach the assigned charging stations, and the cost of connecting the charging stations to the electric grid substations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The variability of the demand across the day is taken into consideration when determining the number of chargers to install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Nevertheless, the variables assigning EVs to stations are not time-dependent, and, hence, drivers demanding to charge their EVs at different hours are all assigned to the same station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In our paper, we allow the demand arising from the same location during the day to be assigned to different stations, depending on the evolution of the overall demand and the available cherging resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Moreover, we consider different types of chargers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Both short-term and long-term decisions are considered in Quddus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The main long-term decisions are related to the year, the location, and the type of charging stations to open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The short-term decisions are mainly related to the amount of power (provided by different sources, such as electric grid and renewable sources) to satisfy the hourly charging demand at a given location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Compared to our research, the drivers are, indirectly, pre-assigned to a charging station and, hence, the assignment is not part of the optimization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The authors cast the problem as a two-stage stochastic programming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Li and Jenn (2022) present an optimization model based on the concept of charging opportunities, which is measured through the time an individual stays at a given location within a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The authors separate the charging opportunities into home and non-home (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', public) categories, and allow the same individual to charge the EV multiple times at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The proposed optimization model determines the number of home and non-home chargers to install, as well as the times and locations for each individual to charge the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The model aims at minimizing the sum of the annual electricity cost for charging the EVs and the total cost of locating the home and non-home chargers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The number of chargers that can be installed in each location (called region by the authors) is unlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Finally, we mention the growing body of literature that addresses the problem of determining an optimal location of charging stations for EVs in car-sharing systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Brandst¨atter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2017, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Bekli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Although such problem has some characteristics in common with ours, it includes some operational characteristics that make it considerably different, for example the decisions about the number of EVs to acquire, the relocation of the EVs among stations, and the assumption that charging occurs only between two consecutive trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Contributions of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The contributions of this paper to the literature can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' ✓ We present a node-based multi-period optimization model for the location of charging stations that captures the dependence on time of the charging demand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' ✓ the multi-period model takes into account several characteristics of the real problem: multiple types of chargers (each with its own charging speed and installation cost), the capacitated nature of the charging stations (in terms of maximum number of chargers that can be in- 5 stalled), a minimum number of chargers to be installed in different zones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', commercial, residential, industrial);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' ✓ the multi-period model is compared to a single-period model through a worst-case analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' ✓ extensive computational experiments are presented that show, in particular, the importance of incorporating the dependence on time of the charging demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 3 Problem definition and mathematical formulations In this section, we first provide a general description of the location problem along with the notation that is common to the two optimization models that will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Then, the single-period MILP model is presented, together with the notation that is specific for the model, followed by the multi- period formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We consider the problem of determining, in an urban area, an optimal location of charging stations for EVs, along with the type and number of chargers to deploy in each station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' A maximum number of chargers, of each type and in total, can be deployed in each station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The location for any station can be selected from a pre-defined set of potential locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We introduce a complete bipartite network G = (I ∪ J , A), where I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' , I} is the set of demand nodes and J = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' , J} is the set of potential locations for the stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let cij be the travel distance from demand node i to station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' A fixed opening cost Fj is associated with each station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The opening cost does not include the cost of the chargers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We denote as K = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' , K} the set of types of chargers considered, and as fjk the cost of installing one charger of type k ∈ K in location j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let ujk be the maximum number of chargers of type k that can be installed in station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Similarly, uj denotes the maximum number of chargers that can be installed in total in station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The latter two parameters define, implicitly, the maximum charging capacity of station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Each node i is the center of gravity of a section of the urban area where the demand of the section is measured as the number of EVs that need to be recharged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We will introduce later, for each of the two optimization models, the planning horizon and the notation for the demand of a customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' For the sake of brevity, hereafter we refer to each potential location j simply as station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The demand must be entirely satisfied by the chargers that will be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' To take into account that different parts of the urban area have different needs in terms of type of charger desired, the urban area is partitioned in zones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', commercial, residential, industrial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We denote by L = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' , L} the set of zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We assume that, based on some preliminary analysis, in each zone ℓ ∈ L a minimum percentage ρℓk of chargers of type k must be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Each station j ∈ J belongs to a zone as well as each customer i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Thus, the zones imply a partition of both the stations and the demand points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' This partition does not restrict the allocation of demand to stations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', a demand point located in a zone can be assigned to a station located in a different zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Two criteria have a key role in this location problem: the cost of opening the stations and installing the chargers and the distance the customers have to travel to be recharged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The objective function we consider, to be minimized, is a convex combination of these two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The optimization problem is aimed at determining, for each type of charger and each station, the number of chargers to be deployed in such a way that the objective function is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 6 Both MILP models include the following decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let zj ∈ {0, 1}, with j ∈ J , be a binary variable that takes value 1 if station j is opened, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let yjk ∈ Z+, with j ∈ J and k ∈ K, be an integer variable that represents the number of chargers of type k installed in station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1 A single-period location model This section presents a single-period model for the location of the charging stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The MILP formulation, denoted as SP-CFL, is an extension of a classical CFL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Hereafter, we introduce the notation needed for the formulation, in addition to the one introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We consider a single planning period of length H and denote as di the total demand in i ∈ I, that is, the total number of EVs demanding to be recharged in i during H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let pk denote the average number of EVs fully recharged by one charger of type k during time period H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' For the sake of simplicity, we assume that pk does not depend on the type of EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The SP-CFL model also makes use of the following decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let xijk ∈ [0, 1], with i ∈ I, j ∈ J , and k ∈ K, be the fraction of the demand of node i assigned to a charger of type k in station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Then, the SP-CFL model can be stated as the following MILP: [SP-CFL] min λ · � � 1 � i∈I di � i∈I di � j∈J cij � k∈K xijk � � + (1 − λ) · � �� j∈J Fjzj + � j∈J � k∈K fjkyjk � � (1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' yjk ≤ ujkzj j ∈ J , k ∈ K (2) � k∈K yjk ≤ ujzj j ∈ J (3) � j∈J � k∈K xijk = 1 i ∈ I (4) � i∈I dixijk ≤ pkyjk j ∈ J , k ∈ K (5) xijk ≤ yjk i ∈ I, j ∈ J , k ∈ K (6) � j∈Aℓ yjk ≥ ρℓk � j∈Aℓ � k∈K yjk k ∈ K, ℓ ∈ L (7) zj ∈ {0, 1} j ∈ J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' yjk ∈ Z+ j ∈ J , k ∈ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' xijk ∈ [0, 1] i ∈ I, j ∈ J , k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (8) The objective function in (1) comprises two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The first one represents the average distance traveled by the EVs to reach the assigned station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The second term is the total cost of opening the 7 stations and installing the chargers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The two terms represent criteria of a substantially different nature: the first measures the quality of the service provided by the deployed stations and chargers to the drivers, whereas the second the cost of the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The two criteria are weighted by the trade-off parameter λ ∈ [0, 1], which is used to balance their importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Constraints (2) and (3) limit the number of chargers that can be installed in station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The former set bounds the number of chargers of type k to be lower than or equal to ujk, whereas the second set of constraints bounds the total number of chargers to be lower than or equal to uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Both sets of constraints (2) and (3) impose that no charger can be installed if station j is not open (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', zj = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Constraints (4) ensure that the demand of each node i ∈ I is entirely satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Constraints (5) guarantee that the number of EVs assigned to the chargers of type k deployed in station j is not greater than the charging capacity available (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', pkyjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' They also impose that no EV can be assigned to a type k of chargers in station j if no charger of that type is available (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', yjk = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Inequalities (6), which are redundant in this formulation, are well-known to yield a tighter Linear Programming (LP) relaxation than the equivalent formulation without them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', see Filippi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Constraints (7) guarantee that the number of chargers of type k installed in zone ℓ is at least equal to the minimum percentage ρℓk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Finally, constraints (8) define the domain of the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='2 A multi-period location model This section presents the MILP formulation for the multi-period model, henceforth denoted as the MP-CFL model, for the problem defined at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The planning period H of the single-period model is here partitioned into a number T of time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' For example, if H is a day, we may partition the day in hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let T = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' , T} denote the set of time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We denote as Rk the number of consecutive time periods needed to completely recharge a car using a charger of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Note that, similar to pk for the SP-CFL model, Rk does not depend on the type of EV but only on the type of charger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Furthermore, parameters pk and Rk are strictly related, as the latter is determined by dividing the length of the time horizon by pk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Rk = T pk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The demand of each node i ∈ I is no longer identified by a single value (di in the SP-CFL model) but by a time-dependent profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Let dt i denote the demand of node i ∈ I at the beginning of time period t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' A more detailed discussion about the demand profiles can be found in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We assume that the demand of a time period t must be served in that time period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', it cannot be postponed to a later time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' We say that a node is served by a charger of type k at time t if a charger is available at time t to start the charging which will occupy the charger for a total of Rk time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The capacity installed in each station must be sufficient to serve the charging demand assigned to that station in a time period and the demand assigned to the station in a previous time period that has not yet completed the charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Finally, let xt ijk ∈ [0, 1], with i ∈ I, j ∈ J , k ∈ K, and t ∈ T , be the fraction of the charging demand of node i to be served at time t that is assigned to a charger of type k in station j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The MP-CFL model is formulated as follows: [MP-CFL] 8 min λ · � � 1 � t∈T � i∈I dt i � t∈T � i∈I dt i � j∈J cij � k∈K xt ijk � � + (1 − λ) · � �� j∈J Fjzj + � j∈J � k∈K fjkyjk � � (9) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (2), (3), and (7) � j∈J � k∈K xt ijk = 1 i ∈ I, t ∈ T (10) xt ijk ≤ yjk i ∈ I, j ∈ J , k ∈ K, t ∈ T (11) � i∈I t−1 � τ=0 dt−τ i xt−τ ijk ≤ yjk j ∈ J , k ∈ K, t ∈ T : t < Rk (12) � i∈I Rk−1 � τ=0 dt−τ i xt−τ ijk ≤ yjk j ∈ J , k ∈ K, t ∈ T : t ≥ Rk (13) zj ∈ {0, 1} j ∈ J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' yjk ∈ Z+ j ∈ J , k ∈ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' xt ijk ∈ [0, 1] i ∈ I, j ∈ J , k ∈ K, t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (14) The objective function in (9) is the multi-period extension of function (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' For each node i ∈ I, constraints (10) ensure that the charging demand arising in each time period t is fully satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Akin to the objective function, also inequalities (11) are the multi-period extension of constraints (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Constraints (12) and (13) guarantee that the number of EVs that are charging in time period t at a charger of type k in station j is smaller than or equal to the number of available chargers of that type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', yjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Note that the second sum in (12) and (13) is used to keep track of the EVs that started to recharge in a previous time period but have not completed the charging in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Constraints (12) are defined for the first time periods in the planning horizon (such that t < Rk), whereas (13) are defined for the remaining time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Finally, constraints (14) define the domain of the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 4 Worst-case analysis In this section, we analyze the worst-case performance of the SP-CFL model in terms of the demand that cannot be satisfied if the optimal solution produced is implemented in a context where the demand fluctuates over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In fact, in this case if an optimal solution to the SP-CFL model is implemented, there is no guarantee that all the charging demand is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' As the SP-CFL model implicitly assumes that the charging demand is uniformly distributed across the planning horizon, when the demand fluctuates over time, there may be peak time periods where the chargers installed are not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 9 Theorem 1 When an optimal solution of the SP-CFL model is implemented, the fraction of the demand that does not find an available charger to be served may be up to 1 − 1 T , where T is the number of time periods of the planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' This bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Proof To prove the theorem, we build the following instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Recalling constraint (4) and summing up all constraints (5), the following chain of inequalities holds: � i∈I di (4) = � i∈I di � j∈J � k∈K xijk = � j∈J � k∈K � i∈I dixijk (5) ≤ � j∈J � k∈K pkyjk (15) for any feasible solution to the SP-CFL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Consider an instance where the travel distances are all negligible compared to the fixed opening and installing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' In this situation, the SP-CFL model would open the minimum number of charging stations and install the minimum number of chargers that are strictly necessary to satisfy the total demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' As a consequence, the value of the right-hand side of the rightmost inequality in (15) would be as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Additionally, suppose there is a single type of charger (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', K = 1) and that the total demand � i∈I di is a multiple of p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Recall that the latter parameter represents the number of EVs fully recharged by one charger during the planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Note that it can be determined by dividing the number of time periods T by the number of consecutive time periods needed to completely recharge an EV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Hence, at optimality, the inequality in (15) can be reformulated as follows: � i∈I di = � j∈J p1yj1 = T R1 � j∈J yj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' (16) Thus, the total number of chargers deployed is � j∈J yj1 = R1 � i∈I di T , which, assuming that R1 = 1, becomes � j∈J yj1 = � i∈I di T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Consider an extreme situation where the whole demand � i∈I di arises in one time period, say ˆt, whereas it is zero in the remaining periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The demand that can be satisfied in such time period is equal to the number of chargers installed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', � j∈J yj1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Given the assumptions above, this value is also equal to � i∈I di T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Thus, the amount of demand that does not find an available charger is equal to: � i∈I di − � i∈I di T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Figure 1 illustrates the construction for the special case where � i∈I di = T, which implies that � j∈J yj1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The whole demand, equal to T, arises in time period ˆt (green bar), whereas it is zero in the remaining periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The SP-CFL model assumes that such demand is uniformly distributed across the planning horizon (pink bars), and hence it opens one station equipped with one charger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' As a consequence, the charging demand that is not satisfied is T − 1 or, in percentage, T−1 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 10 Figure 1: An instance where the demand arises in time period ˆt (green bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The pink bars show a uniform distribution of the demand across the planning horizon, as implicitly assumed by the SP-CFL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 5 Experimental Analysis This section is devoted to the presentation and discussion of the computational experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' They were conducted on a Workstation HP Intel(R)-Xeon(R) at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='5GHz with 64 GB RAM (Win 10 Pro, 64 bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The processor is equipped with 6 physical cores, and all threads were used while solving each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The MILP models were implemented in Java, compiled within Apache NetBeans 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='3, and solved by means of CPLEX 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Each instance was solved with a CPU time limit of 3,600 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' All other CPLEX parameters were set at their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' First, we present the testing environment we used in our experiments, then we compare the optimal solutions for two illustrative examples generated ac- cording to two different urban structure models, and finally we provide detailed computational results comparing the solutions produced by the single-period and the multi-period models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1 Testing environment The generation of the charging demand and potential station locations follows the procedure de- scribed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' All the remaining parameters defining the testing environment are detailed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='1 Spatial and temporal charging demand generation As far as the urban structure is concerned, we considered two classic models, the concentric zone model and the sector model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The concentric zone model was proposed in 1925 by sociologist Ernest Burgess on the base of his human ecology theory, and was initially applied to the city of Chicago (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Burgess, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' It is, perhaps, the first theoretical model used to explain urban social structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The model depicts urban land usage as concentric rings: the business district is located in the center, whereas the remainder of the city is expanded in rings, each corresponding to a different land usage (such as industrial or residential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' The sector model was proposed in 1939 by land economist Homer Hoyt (see Hoyt, 1939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' It is a modification of the Burgess’ model where the city zones devoted to a specific land usage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=', business, residential, and productive) develop in sectors expanding from the original city center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfbwyA/content/2301.05077v1.pdf'} +page_content=' Though the actual structure of modern cities can hardly be captured by models as simple as Burgess’ and Hoyt’s, they are the basis of more 11 T : 1 0 t+1 t-1