Optimal Allocation of Depots for Electric Bus Charging: Cost Minimization and Power System Impact Mitigation Using Mixed Integer Nonlinear Programming

Abstract

ABSTRACT

This paper proposes a novel approach for the optimal allocation of a depot for electric buses (EBs) charging in a specific transit service, considering the impact on the power system. The main objective is to minimize the total cost, achieved by minimizing the cost of the new cables connecting the depot station to the distribution system and the upgrade cost of existing lines to meet the additional loads. The outcomes are the optimal location of the depot, the optimal electric node bus in the distribution system to supply it, and the required system upgrades. The optimization problem is formulated as a Mixed Integer Nonlinear Programming (MINLP) model and solved using the General Algebraic Modeling System (GAMS). The methodology is tested on the H16 EB service line in Barcelona, Spain, and a typical electrical distribution system. Three case studies are presented in this paper. In the first case, the impact of a single service on the distribution network is analyzed, and in the second case, three H16 EB services are assumed to serve the network. The third case handles a multi‐route, multi‐terminal bus service to allocate and supply a depot capable of accommodating all the routes. This case will also include sensitivity analysis to test the robustness and reliability of the model. Results show that for a single H16 EB service, no line upgrades were needed, and the total cost per phase was $120,000. For three H16 EB services, three lines required upgrades, and the total cost per phase increased to $718,000. In the third case, the sensitivity analysis revealed that higher demand factors lead to increased costs due to more update requirements and voltage deviations. The results demonstrate that the minimum distance between the depot and node is not always the optimal or feasible solution that would prevent the depot load installation at a weak spot and meet the power flow constraints.

Introduction

In the past few years, human activities have been the main cause of two main environmental concerns: air pollution and global warming. The process of combusting fossil fuels is the main source of Green House Gas (GHG) emissions. Moreover, these emissions are the primary contributor to air pollution and global warming [1]. In the European Union (EU), the transportation sector is the largest sector that produces GHG emissions. Reports have shown that almost 20% of total CO₂ emissions, 39% of NO_x emissions, 11% of $${\mathrm{PM}}_{2.5}$$, and 20% of the CO emissions are from transportation in the EU and more specifically, diesel buses [2]. They are considered the most common means of transportation in and out of the cities. As a result, decarbonization solutions have been proposed to reverse the negative effects of climate change on the planet. This decarbonization can be achieved by replacing internal combustion engine vehicles (ICEV) with vehicles that run on fuel alternatives [3]. Over the past years, vehicles that run on low-carbon fossil fuels have been proposed and tested; however, the main issue of GHG emissions has not been completely resolved. Thus, vehicle electrification is the preferred approach for eliminating GHG emissions [4].

Buses contribute to 80% of the public transportation worldwide [5]. This indicates that if the electrification process was focused only on buses, the city's quality of life would improve drastically. As shown in Figure 1, the sales of electric buses (EBs) are expected to reach up to 80%, in 2040, from the overall sales of vehicles [6]. As a result, it is only logical to direct the research on EBs' resource and distribution power allocation. Unfortunately, the electrification of transportation buses will significantly impact the electric grid [7]. The uncoordinated times of EBs' charging result in overloading the grid since an EB might charge during peak hours and cause an overload on the power system. Thus, if the integration of the EBs into the grid is not properly planned, several problems may occur in the distribution system, such as voltage fluctuations, transformer overloading, and high energy losses [8–10].

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First, the voltage fluctuations are the result of the EBs demanding a large amount of power during charging and this causes voltage fluctuations in the distribution system between 90% and 110% of the nominal voltage [8]. Voltage fluctuations can lead to equipment damage, outages, and other problems in the distribution system. Second, the overload on the power system caused by the connection of EBs will overheat the transformer and might cause a power outage since the transformer exceeded its rated capacity [9]. Lastly, the energy losses are caused because the power lines and transformers used to distribute electricity have a limited capacity, and if the demand exceeds that capacity, it can result in energy losses [10]. Other power quality issues that must be addressed as well are voltage swells and sags, harmonics, and flicker. In [11], Sivaraman and Sharmeela highlight that electric vehicle (EV) charging stations introduce significant power quality issues, such as harmonics, DC injection, and voltage flicker, which can severely impact the grid. These issues arise because EV chargers, being power electronics-based equipment, draw nonlinear current from the supply voltage, leading to harmonic distortions. Additionally, the high switching frequency of chargers contributes to DC injection and rapid voltage changes, causing voltage flicker, especially with fast charging systems that cause increased harmonic distortions at higher states of charge. Srivastava et al. reviewed the power quality issues arising from the integration of EVs into distribution networks, highlighting the increased risk of voltage imbalance, transformer failures, and harmonic distortions due to both grid-to-vehicle (G2V) and vehicle-to-grid (V2G) operations, and discussed various mitigation measures such as active power filters and FACTS devices [12]. Similarly, Qin et al. thoroughly reviewed power quality issues, such as flicker, harmonics, and supraharmonics associated with EV charging, highlighting that these issues are primarily caused by the interaction between EV chargers and the grid, and proposing impedance-based stability criteria and active power filters as effective mitigation strategies [13]. Finally, Supponen et al. discussed the impact of EV charging on power quality through field tests and simulations, noting that high penetration of EV chargers can lead to significant harmonic distortions and voltage fluctuations, and emphasizing the importance of incorporating harmonic profiles into network design and planning [14].

Given the aforementioned problems, the integration of EBs with the distribution system without careful coordination or planning might be discouraged. This is due to the increased operational costs that resulted from fixing the problems that occurred to the distribution system. This redirected the focus of the researchers from focusing on replacing ICEVs with EVs to focusing on how to replace ICEVs with EVs and carefully integrate them into the distribution system without causing any critical problems.

Researchers have focused on three types of integrating solutions: charging infrastructure planning, bus scheduling, and charging scheduling [15]. Charging infrastructure planning focuses on allocating bus charging stations and determining the number of buses that will be charged at this station. Moreover, it focuses on optimizing the placement and scheduling of charging stations to ensure that EVs can travel between destinations without running out of charge. Planners will consider the costs of building and maintaining charging stations and their potential impact on the local communities [16]. A good example is the study of Mondal et al. [17], which utilizes the presence of EVs in a distribution network to optimally plan the allocation of charging and repair stations to improve the resilience of the distribution system. The article proposes a two-stage resilient restoration model that strategically pre-positions charging and repair stations using an integer nonlinear programming approach and optimizes the dispatch of EVs and Mobile Energy Resources (MERs) post-event with a mixed integer linear programming model. This method significantly reduces the energy not supplied (ENS) and improves the resilience index (RI) of the distribution system. Bus scheduling focuses on developing a detailed plan for the operation of the buses, the size of each bus, the route of the bus, and the number of times that a bus will complete the route within a given period [18]. On the other hand, charging scheduling focuses mainly on determining the charging behavior of the EBs in terms of the charging power, time, and site. Hu et al. [19] incorporate the latter two strategies by presenting a dispatching interval optimization model for electric buses under stochastic traffic conditions. The study considers the impact on the grid by modeling energy consumption in detail, accounting for travel speed, travel demand, and battery discharge rates. The incorporation of stochastic traffic conditions ensures realistic variations in energy consumption and charging needs. By optimizing dispatching intervals, the model indirectly reduces peak loads on the grid and distributes charging times more effectively. The use of an improved genetic algorithm with probabilistic evolution helps find robust solutions that handle uncertainties in electric bus operations. The model is validated with real-world data from Harbin, ensuring practical implementation without significant disruptions to the power supply.

It is suggested in the study [7] to include Battery Depot Operators (BDOs) in the planning process since the cost of building BDO will be low due to the advances in storage technologies. Furthermore, BDOs open the door for direct connection between the EBs and the grid, which facilitates the optimal allocation of the energy provided by the grid. Arif et al. [20] suggested an optimized solution where the profit from the BDO will be maximized while considering the grid stability. Much of the available research focuses on either minimizing the cost of building the BDOs or maximizing the profit obtained from them. However, they did not take into consideration minimizing the cost of upgrading the distribution system while benefiting from the BDOs. In this study [21], the authors present a novel planning method that employs a multi-objective optimization model to distribute photovoltaics (PVs) and fast charging stations (FCSs) in large-scale smart grids. The proposed optimization model in [22] allocates Renewable energy sources (RESs) and electric vehicle charging stations (EVCSs) while minimizing three objectives: voltage deviations, energy losses, and electric vehicles owners'dissatisfaction. The latter two papers considered the impact on the power network; nevertheless, the optimization models developed were solved using meta-heuristic techniques due to the complexity and multi-objectivity of the problems.

When focusing on cost minimization, a number of studies have explored various strategies to minimize the costs associated with the installation of new cables and the upgrading of existing lines, considering both technical and financial constraints. Hajimiragha et al. used a robust optimization approach to plan the transition to plug-in hybrid electric vehicles (PHEV), considering various uncertainties in vehicle usage patterns and aiming to minimize the overall costs associated with new infrastructure and necessary upgrades to handle increased loads [23]. Fernández et al. employed a large-scale distribution planning model, incorporating heuristic planning algorithms and a Geographic Information System (GIS), to assess the impact of different levels of plug-in electric vehicles (PEV) penetration on network investment and energy losses [24]. In [25], the authors developed reliability-based metrics using IEEE reliability standards and simulation models to quantify the maximum permissible load demand of electric vehicles, emphasizing cost-effective planning for new and upgraded infrastructure. A chargeable region optimization model, utilizing optimization software and probabilistic models to maximize the hosting capacity of distribution networks for electric vehicles while minimizing the costs of new cable installations and upgrades of existing lines was presented in [26]. Lastly, Sugihara and Funaki investigated the use of residential PVs to increase the hosting capacity and equality for fast charging stations, employing simulation tools and network analysis models to evaluate the impact of PV integration on the cost and capacity of the distribution network [27].

This article presents a novel optimization technique aimed at minimizing the costs associated with upgrading distribution lines and the installation of low-voltage cables connecting EB depots. Unlike previous studies, which primarily focused on the integration of EVs and the associated infrastructure costs in a more general sense, this work specifically addresses the economic impact of connecting EB depots to the distribution system. By strategically choosing the location of charging depots from three predetermined locations along a known route, the optimizer ensures efficient allocation of additional loads to the distribution system. The methodology includes a deterministic approach to decide on the optimal upgrading of distribution lines to handle the increased load, and this is applied to both single bus route and multiple bus route scenarios. This targeted approach not only aims to reduce the financial burden of infrastructure upgrades but also enhances the overall reliability and efficiency of the power distribution network when integrating electric buses. Tables 1 and 2 provide a comparative analysis between this work and the most relevant previous research, highlighting how this study differentiates itself by emphasizing specific methodologies, types of EVs, and unique objective functions, which demonstrates its distinct contributions to the field.

Table 1 Comparative analysis of this work and previous research (Part 1).

Aspect	This article	Hajimiragha et al. [23]	Fernández et al. [24]	Kamruzzaman and Benidris [25]
Solution method	Deterministic	Deterministic	Heuristic	Deterministic
Type of EV focused	EB	PHEV	PEV	EV
Objective function	Minimize costs of upgrading distribution lines and low-voltage cables supplying depots	Minimize costs associated with new infrastructure and necessary upgrades	Minimize investment costs, maintenance costs, and energy losses	Quantify maximum permissible load demand and plan cost-effective infrastructure upgrades
Tools used	GAMS	Not specified	Large-scale distribution planning model, GIS	IEEE reliability standards, simulation models
Power quality issues	Yes	No	Yes	Yes
Stochastic elements	No	No	No	No
Validation method	Simulating single and multiple bus routes based on real data	Simulation-based	Simulation-based	Simulation-based

Table 2 Comparative analysis of this work and previous research (Part 2).

Aspect	Zhao et al. [26]	Sugihara and Funaki [27]	Mondal et al. [17]	Hu et al. [19]
Solution method	Deterministic	Heuristic	Deterministic	Heuristic
Type of EV focused	EV	EV	EV	EB
Objective function	Maximize hosting capacity of distribution networks while minimizing costs of new cable installations and upgrades	Increase hosting capacity and equality for fast charging stations while minimizing costs	Improve resilience by reducing energy not supplied and improving the resilience index	Optimize dispatching intervals to reduce peak loads and distribute charging times
Tools used	Optimization software, probabilistic models	Simulation tools, network analysis models	Integer nonlinear programming, mixed integer linear programming	Static and stochastic model formulas, genetic algorithm
Power quality issues	Yes	Yes	No	Yes
Stochastic elements	Yes	Yes	Yes	Yes
Validation method	Simulation-based	Field tests and real data	Real-world data and case studies	Real-world data from bus operations in Harbin

The rest of the paper is organized as follows: Section 2 states the problem description, and Section 3 goes into detail on how the problem is formulated and the optimizing model is developed. The results obtained from the optimizing model are discussed in Section 4. Finally, Section 5 concludes the work of this paper.

Problem Description

The problem regarding EB services, which is the focus of this paper, is the allocation of a central depot station capable of charging up to 10 buses simultaneously. The depot is used only for the overnight charging of buses such that the buses have a fully charged battery at the beginning of each day to start the first cycle of the service. The rest of the EBs' energy requirements the bus's battery is not sized to supply the EB with enough energy for the whole service time-will be supplied by the opportunity chargers along the route stops.

The installation of a charging station with a power capacity of such a high magnitude will not only incur large costs but also have a substantial impact on the distribution network. Both sides will be considered in the paper when investigating the connection of the depot from one of its potential locations to the distribution network. The objective of the problem will be assigning the depot location to a specific stop on the route as well as deciding on the node to which the depot will connect to acquire the needed power from the grid's distribution network. The algorithm should not simply choose the node of the electrical system and depot location with the minimum distance between one another because when the effect on the distribution system is taken into account, the power flow constraints will most probably force the result to be otherwise. As a result of the large power drawn by the new load represented by the depot, some of the distribution system lines' active power capacity limit can be exceeded, hence the lines will need to be upgraded translating into extra costs. This occurrence can be observed in the distribution lines leading to the node connected to the depot; therefore, choosing the node and depot location based on the minimum distance for minimum low-voltage cable installation costs can sometimes result in very high line upgrade costs. Thus, not ending up as the optimal solution. It is worth mentioning that the minimum depot-to-node distance solution may not be feasible with regard to the power flow constraints. Figure 2 provides an illustrative description of the problem.

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The problem is further complicated by the presence of three identical EB service routes on the distribution system instead of one, where each route requires a depot to be installed at a certain stop and connected to a specific electrical system node. Lastly, the problem will address the depot allocation for a multi-route, multi-terminal single bus service. In this case, the candidate locations of the depot will not be specific stops along any of the routes. Instead, potential depot locations will be chosen such that each one is capable of servicing all routes and minimizes deadheaded kilometers. From these location options, the one that balances between cost and grid impact minimization is chosen.

EB Power Consumption Model

The EB's parameters will affect its energy consumption model, and its power requirement, $${P}_{\mathrm{EB}}$$, can be determined using Equation (1): 1$${P}_{\mathrm{EB}}={P}_{\mathrm{traction}}+{P}_{\mathrm{HVAC}}+{P}_{\mathrm{AUX}},$$ where $${P}_{\mathrm{traction}}$$ is the amount of power required by the bus to overcome the traction force and move at a specific speed. It is calculated according Equation (2). 2$${P}_{\mathrm{traction}}=\nu \times {f}_{\mathrm{traction}}={m}_{r}\alpha \nu +mg\nu {c}_{rr}cos(\theta )+\frac{1}{2}\rho {c}_{d}a\nu {(\nu -{\nu }_{{\rm{w}}})}^{2}+mg\nu sin(\theta )$$

$${m}_{r}$$: The rotating mass of the bus in $$\mathrm{kg}.{{\rm{m}}}^{2}$$.

$$\alpha $$: The bus's acceleration in $${\rm{m}}\unicode{x02215}{{\rm{s}}}^{2}$$.

$$\nu $$: Bus's speed in $${\rm{m}}\unicode{x02215}{\rm{s}}$$.

$$m$$: Bus's mass in $$\mathrm{kg}$$.

$${c}_{rr}$$: Rolling coefficient.

$$\theta $$: Angle due to the slope of the road in degrees.

$$\rho $$: Density of air in $$\mathrm{kg}\unicode{x02215}{{\rm{m}}}^{3}$$.

$${c}_{d}$$: Drag coefficient.

$$a$$: Frontal aerodynamic area of the EB.

$${\nu }_{{\rm{w}}}$$: Wind speed in $${\rm{m}}\unicode{x02215}{\rm{s}}$$.

$$g$$: Acceleration of free fall in $${\rm{m}}\unicode{x02215}{{\rm{s}}}^{2}$$.

It is worth mentioning that $${P}_{\mathrm{HVAC}}$$ varies according to the season as it represents the power consumed by the heating, ventilation, and air-conditioning systems. It is assumed to be between 29% and 49% of the total power consumed by the EB ($${P}_{\mathrm{EB}}$$) according to [28]. Finally, $${P}_{\text{AUX}}$$ is what is spent to power supplementary devices, such as lights, bus doors, radios, speakers, and so forth.

In-Depot Charger Model

Depending on the fleet size, a particular number of fast, three-phase DC chargers will be used for overnight charging of EBs. The buses are assumed to charge at the same time to study the maximum possible impact and peak demand on the distribution network, rather than staggered charging where buses are charged in batches.

To ensure an efficient charging process, the selection of DC chargers is based on the charging requirements of the electric buses. Factors such as the battery capacity, state of charge (SoC), and the required charging time are taken into account. For instance, if the fleet consists of buses with larger battery capacities, higher power chargers will be necessary to meet the overnight charging demands.

The power flow is solved per phase; hence, the single-phase power of the DC charger is considered. According to the maximum charging power capacity of the EB, the charger will be selected, and consequently, the new load on the network will be calculated.

The arrangement of the chargers within the depot is optimized to distribute the load evenly across the three phases, preventing any phase from being overloaded. This involves detailed load flow analysis and short-circuit studies to identify and mitigate potential bottlenecks in the distribution network.

EB Service Model

Detailed parameters and information should be known about the bus service to accurately apply the proposed optimization model. A bus service typically provides a specific number of buses per stop per hour ($$f$$), operates for a certain number of hours per day ($$T$$), and performs a number of daily cycles ($${N}_{\mathrm{Cycles}}$$). This reflects how many times the buses complete the route in a day, impacting the scheduling and maintenance planning. Route length ($$L$$) directly affects energy consumption and the required charging infrastructure, as longer routes will necessitate more frequent or higher capacity charging. The number of buses ($${N}_{\mathrm{Buses}}$$) indicates the fleet size, which needs to be managed efficiently to balance operational costs and service reliability. The buses will require charging, probably through both on-the-stop charging and in-depot/overnight charging. The service model also has a known number of stops along the route ($${N}_{\mathrm{Stops}}$$). The location, area, and available space surrounding each stop are studied to pick candidate stops that would accommodate a depot.

A multi-route service requires a more specific description. One depot is used for overnight parking and charging of the whole fleet, which then departs the next day to begin each bus's route from the main terminal. Figure 3 shows a three-route service with possible depot locations. Each route features two terminals: the main terminal, common to all routes, and a secondary terminal unique to each route.

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A bus cycle starts at the main terminal, traveling along its designated route and stopping at several bus stops until it reaches the route terminal. After a scheduled wait time at the route terminal, the bus returns to the main terminal, stopping at the designated bus stops along the way. Upon returning to the main terminal, the bus waits for another specific period before embarking on another roundtrip. The waiting periods at both terminals can be used for charging (in electric bus services) or adjusting the cycle time to prevent delays or early departures. The parameters of each route, such as the number of cycles per day, frequency, stops, and route length, differ, resulting in varying energy consumption for buses on different routes.

Technical Assumptions

In this study, the following technical assumptions are made to simplify and justify the problem description:

• All the buses charge at the same time: By considering simultaneous charging, we can assess the worst-case scenario for grid load and ensure the system's robustness under high demand conditions.

• A three-phase balanced system is considered: Assuming a balanced three-phase system simplifies the power flow analysis and ensures that the distribution network operates efficiently. This assumption helps in maintaining voltage stability and reduces the complexity of modeling unbalanced loads.

• The chargers used at the depot are DC fast chargers according to the charging capacity of each EB: DC fast chargers are capable of delivering high power, matching the charging capacity requirements of modern EBs, and ensuring they are ready for service the next day.

• The second case study assumes that all the routes of the three different services are identical while the third assumes a multi-route service having three distinct routes: This assumption allows for a comparative analysis between uniform route conditions and more complex, varied route scenarios. It helps in understanding the impact of route diversity on depot allocation.

• Both the first and the second case studies assign the depot to candidate stops along the route while the third one chooses off-route locations: This distinction is important to provide a planning model for services wanting to allocate the depot on-route as well as off-route.

• The presence of bus terminals is neglected for the single route service: For simplification, the single route service assumes buses operate continuously along a loop without designated terminals for extended stops. This assumption reduces the complexity of the model and focuses on the primary aspects of route length, stop frequency, and charging requirements.

Problem Formulation

The proposed optimization problem in this work is a mixed integer nonlinear programming (MINLP) problem and it is solved using General Algebraic Modeling System (GAMS). This part includes the detailed formulation of the optimization problem's objective function, constraints, and definitions of the variables and parameters used in the system model. The model is based on a three-phase balanced system; thus, it is developed per one phase and all results obtained will be that of one phase.

In the context of a three-phase unbalanced system, however, the problem formulation and power flow analysis become more complex compared to a balanced system. In an unbalanced system, the loads and impedances in each phase are not equal, leading to differences in voltage and current magnitudes and phase angles across the phases. This difference necessitates a phase-by-phase analysis to accurately model and analyze the power flow and the associated impacts on the distribution network.

For a three-phase unbalanced system, the power flow equations need to be reformulated to account for the asymmetry in phase impedances and loads. The use of sequence components (positive, negative, and zero sequences) becomes essential to decompose the unbalanced system into symmetrical components, facilitating the analysis. The voltage and current calculations will differ for each phase, and additional constraints will be required to ensure that the voltage and current limits are maintained individually for each phase [29, 30].

Furthermore, the incorporation of an unbalanced system into the planning model requires the inclusion of detailed phase-specific data for loads, line impedances, and other system parameters [31]. This ensures an accurate representation of the unbalanced conditions. The optimization process must then handle the increased computational complexity due to the larger number of variables and constraints.

Objective Function

The objective function is composed of four cost terms in dollars: a constant term $${C}_{\mathrm{Load}}$$, and three variable ones represented by $${C}_{\mathrm{Upgrade}}$$ and $${C}_{\mathrm{LVCC}}$$, where $${C}_{\mathrm{Load}}$$ is the cost of the total power consumed by the depot station at a certain instant, $${C}_{\mathrm{Upgrade}}$$ is the cost of upgrading the distribution network lines, if necessary, after the installation of the depot, and $${C}_{\mathrm{LVCC}}$$ is the low voltage cables cost (LVCC) used to connect the depot to one of the electrical nodes. Consequently, the objective function is as depicted in Equation (3). 3$$\mathop{\min }\limits_{\vartheta }Z={C}_{\mathrm{Load}}+{C}_{\mathrm{Upgrade}}+{C}_{\mathrm{LVCC}},$$ where $$Z$$ is the total cost and $$\vartheta $$ represents the vector of the control variables, which includes the optimal depot location, the optimal node bus to supply the depot, and the necessary distribution line upgrades. The constant term in the objective function ($${C}_{\mathrm{Load}}$$) can be calculated as follows: 4$${C}_{\mathrm{load}}={N}_{\mathrm{Buses}}\times {P}_{ch}\times {b}_{g},$$ where $${N}_{\mathrm{Buses}}$$ is the number of buses charging simultaneously at the depot, $${P}_{Ch}$$ is the charging power per phase for one EB in MW, and $${b}_{g}$$ is the cost coefficient of the grid generation in $/MW. The constraint is multiplied by 3 when three identical routes are considered. Second, $${C}_{\mathrm{Upgrade}}$$ is calculated according to Equation (5), where $$i$$ is the set of electrical nodes in the MV distribution system with $$n$$ number of buses and $$j$$ is an alias of $$i$$ to denote the line connected from bus $$i$$ to bus $$j$$ ($${L}_{ij}$$). Accordingly, $${a}_{ij}$$ is defined as a binary decision variable array such that each of its elements can take the value of 0 or 1, where 0 means that $${L}_{ij}$$ will not be carrying power exceeding its power capacity limit after the installation of the depot and ergo will not be upgraded to withstand double its capacity, while 1 suggests that line upgrade is necessary for the power flow constraints to uphold. In continuation, $${D}_{ij}$$ is the distance in km of $${L}_{ij}$$, and $${C}_{ij}$$ is the cost in $/km of the line upgrade. 5$${C}_{\mathrm{Upgrade}}=\sum ^{i}\sum ^{j}{a}_{ij}\times {D}_{ij}\times {C}_{ij}$$

Third, $${C}_{\mathrm{LVCC}}$$ is determined by Equation (6): 6$${C}_{\mathrm{LVCC}}=\sum ^{i}\sum ^{k}{X}_{i}\times {Y}_{k}\times {D}_{ik}\times {C}_{c},$$ where $$k$$ is the set of potential depot locations, $${X}_{i}$$ is a binary decision variable vector that can take a value of 1 or 0, indicating whether the depot will be connected to bus $$i$$ or not, respectively; $${Y}_{k}$$ is also a binary decision variable vector, which can either be 1 or 0 marking the allocation of the depot to one of the candidate stops, $${D}_{ik}$$ is the distance in $$km$$ between bus $$i$$ and candidate depot location $$k$$, and $${C}_{c}$$ is the cost in $/km of the low voltage cables that will be extended from bus $$i$$ to depot location $$k$$. Two additional $${C}_{LVCC}$$ components are added to the objective function when two more duplicate bus service routes that require a depot installation at a particular stop as well are included in the system model.

Constraints

The optimization technique must adhere to the constraints presented in this subsection while minimizing the combined costs.

• Power Balance Constraints

The active and reactive power balance constraints are shown in Equations (7) and (8), respectively. 7$${P}_{gi}-{P}_{di}-{X}_{i}\times {N}_{\mathrm{Buses}}\times {P}_{ch}=\sum ^{j}{V}_{i}\times {V}_{j}\times | {Y}_{ij}| \times cos({\gamma }_{ij}+{\delta }_{j}-{\delta }_{i}),$$ 8$${Q}_{{g}_{i}}-{Q}_{{d}_{i}}=-\sum ^{j}{V}_{i}\times {V}_{j}\times | {Y}_{ij}| \times sin({\gamma }_{ij}+{\delta }_{j}-{\delta }_{i}),$$ where $${P}_{{g}_{i}}$$ and $${Q}_{{g}_{i}}$$ are the active and reactive power generated at bus $$i$$ in MW and MVAR, respectively, and since the model is done on a distribution system, generation capability is limited only to bus 1, which represents the grid; henceforth $${P}_{{g}_{i}}$$ and $${Q}_{{g}_{i}}$$ for all electric nodes except for node 1 (grid node) are fixed to zero in the formulation. $${P}_{{d}_{i}}$$ and $${Q}_{{d}_{i}}$$ are the active and reactive power demand at $$i$$, $${V}_{i}$$ is the per unit (pu) voltage at bus $$i$$, $${V}_{j}$$ is the pu voltage at a bus $$j$$ connected to bus $$i$$, $${Y}_{ij}$$ is the admittance of $${L}_{ij}$$ - the $${Y}_{ij}$$ matrix is calculated from the system data line parameters: line resistance ($${R}_{ij}$$) and line reactance ($${X}_{ij}$$) - $${\gamma }_{ij}$$ is the admittance angle of $${L}_{ij}$$, $${\delta }_{i}$$ is the voltage phase angle at bus $$i$$, and $${\delta }_{j}$$ is the voltage phase angle at a bus $$j$$ connected to bus $$i$$. The term $${X}_{i}\times {N}_{Buses}\times {P}_{Ch}$$ subtracted from the left-hand side of Equation (7) represents the added demand on the system when a depot is installed, thence two similar terms are also subtracted when the two duplicate routes are considered.

• Power Flow Constraints

To know whether the power carried by $${L}_{ij}$$ ($${P}_{ij}$$) has exceeded the line limit when the depot load is added to the system, $${P}_{ij}$$ is calculated by Equation (9) [32]. 9$${P}_{ij}=\frac{{V}_{i}^{2}\times cos({\theta }_{ij})-{V}_{i}\times {V}_{j}\times cos({\delta }_{i}-{\delta }_{j}+{\theta }_{ij})}{{Z}_{ij}},$$ where $${Z}_{ij}$$ and $${\theta }_{ij}$$ are the impedance and the impedance angle of $${L}_{ij}$$ respectively. In addition, the constraint limiting the power flow in the distribution system lines to double their active power carrying capacity limit ($${P}_{\mathrm{limit}-ij}$$), i.e. when $${a}_{ij}$$ = 1 (if an upgrade is necessary), after the depot installation is represented in Equation (10). 10$${P}_{ij}\le {P}_{\mathrm{limit}-ij}\times (1+{a}_{ij})$$

• Electric Node Voltage Limits

Equation (11) ensures that the bus voltages remain within the pu voltage upper ($${V}_{\max }$$) and lower ($${V}_{\min }$$) bounds of the system. 11$${V}_{\min }\le {V}_{i}\le {V}_{\max }$$

• EB Service Constraints

To ensure that only one bus node is chosen to be connected to the depot and that the depot is allocated to only one of the candidate stops, constraints 12 and 13 are used respectively. $$P$$ is the index for bus service lines, such that $$P$$ = Service Line - 1, Service Line - 2, Service Line - 3. 12$$\sum ^{i}{X}_{i,P}=1$$ 13$$\sum ^{k}{Y}_{k,P}=1$$

To summarize, the optimization process is outlined in the pseudo-code presented in Figure 4, which details the steps involved in depot allocation and the minimization of upgrade and connection costs.

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Finally, to ensure the practicality and effectiveness of the proposed planning optimization framework, the feasibility will be assessed from the technical implementation, integration, and operation perspectives.

First, the use of GAMS to solve the MINLP problem is technically feasible. GAMS is a powerful tool for solving complex optimization problems, and its capability to handle large-scale, multi-variable scenarios aligns well with the requirements of the proposed model. Second, the assumption of a three-phase balanced system does not diminish the model's applicability to real-world systems, as many power distribution networks strive to maintain balanced loads to optimize performance and reliability. Third, the inclusion of detailed cost components, such as $${C}_{\mathrm{Load}}$$, $${C}_{\mathrm{Upgrade}}$$, and $${C}_{\mathrm{LVCC}}$$, ensures that the model comprehensively accounts for all relevant expenses, providing a realistic assessment of the financial implications. Moreover, the flexibility of the model to handle different scenarios, such as identical routes in the second case study and distinct routes in the third, demonstrates its adaptability. Lastly, the use of binary decision variables to determine depot locations and necessary upgrades reflects a clear and actionable strategy that can be readily implemented. This approach not only simplifies the decision-making process but also aligns well with practical infrastructure planning and development protocols.

Results and Discussions

The distribution system used in the case study is a typical MV 38-bus network. The system has a base rating of 100 MVA and operates at a voltage of 12.66 kV. The combined real and reactive power demands of this system are 3715 kW and 2300 kVAR, respectively. At nominal loading conditions, the real power losses amount to 202.67 kW, and the reactive power losses are 135.14 kVAR [33, 34]. In addition to these system specifications, the data of each line, $${L}_{ij}$$, ($${R}_{ij}$$, $${x}_{ij}$$, $${\gamma }_{ij}$$, $${\theta }_{ij}$$, etc.), as well as the active and reactive power loads, $${P}_{{d}_{i}}$$ and $${Q}_{{d}_{i}}$$, at each bus node are obtained from [35].

It is assumed that $${P}_{\mathrm{limit}-ij}$$ is around 1.2 times $${P}_{ij}$$ at peak conditions. Proper cables from [36] are selected based on the calculated limit. The price of each line cable in $/km, $${C}_{ij}$$, was estimated based on the El Sewedy Electric price list of 2022 [37] which was also used to approximate the price, $${C}_{c}$$, in $/km of the low voltage cables that will be extended from bus $$i$$ to location $$k$$. The distribution system is assumed to lie underground of the H16 bus line route in Barcelona, Spain as shown in Figure 5, and the parameters of the bus service are displayed in Table 3 [38]. Three potential locations for the depot are chosen from the route stops: Pg Taulat - Diagonal Mar ($$3\mathrm{rd}$$ stop), Parc de la Ciutad (15th stop), and Pg Zona Franca - Foc (33rd stop). These geographical locations were chosen based on their capability to accommodate a large depot and $${D}_{ik}$$ is calculated in km using the assumed lengths, $${D}_{ij}$$, of each line, $${L}_{ij}$$, as illustrated in Figure 5, similar to the approach used in [39] and [40].

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Table 3 Single route bus service parameters [38].

Parameter	Symbol	Value
Service frequency per stop	$$f$$	8 buses/h
Daily Service Time	$$T$$	17.5 h
Route cycle length	$$L$$	11.9 km
Number of stops per cycle	$${N}_{\mathrm{Stops}}$$	35 Stops

The EB used in the study is a BYD K9 series 40' Transit with a charging power per phase, $${P}_{Ch}$$, equal to 26.67 kW [41] in the first and second scenarios, and 100 kW in the third one. For the first and second cases, the service roughly requires 9 EBs based on the optimization algorithm proposed in [42]. However, in the third case, the multi-route service operates with 10 EBs. In all cases, EBs will charge to full capacity simultaneously in the depot overnight; however, the rest of their charging power requirements during the service time will be satisfied by the opportunity chargers along the route if needed. Therefore, the overnight charging of EBs is the primary load added by the depot to the distribution system. The cost coefficient of the grid generation, $${b}_{g}$$, is taken as 50 $/MW.

The impact on the distribution network is assessed through three case studies. The first case, as seen in Figure 5, has a single EB service supplied by the network. For the second case study, extra two bus service routes are put in the system as visible in Figure 6. In this case, the distribution network is lying under three EB service routes, which are assumed to be identical for simplicity. The last case study consists of a three-route service with the parameters shown in Table 4, which are assumed to be similar to real data. A single depot for the three routes will be allocated to either one of the three potential locations (A, B, or C) as seen in Figure 7. The distribution bus node to supply the depot will also be chosen as in the previous cases, and a sensitivity analysis will be performed on the model to test the impact of varying the demand at each node bus on the decision variables and bus voltage deviation.

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Table 4 Three-route bus service parameters.

Parameter	Route 1	Route 2	Route 3
$${f}_{s}$$ (buses/h)	3	2	3
$${T}_{\mathrm{cycle}}$$ (min)	60	75	40
$${t}_{\mathrm{terminal}}$$ (min)	5	5	5
$${T}_{\mathrm{service}}$$ (h)	16	16	16
$${N}_{\mathrm{stops}}$$	28	22	18
$${L}_{\mathrm{route}}$$ (km)	15	22	9
$${d}_{\mathrm{stops}}$$ (km)	0.5	1	0.5
$${\rm{\Delta }}{t}_{\mathrm{stops}}$$ (s)	30	30	30

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Case Study 1

As per the values mentioned above, for a single H16 bus line service, 9 buses will be charged in unison at the allocated depot. With each EB charging at 26.67 kW, the total extra load on the distribution system does not result in a detectable impact as showcased in Table 5. The optimizer connects the depot to node 14 and allocates it to the 15th stop (Location B). The optimization code cannot force the chosen $${D}_{ik}$$ to its minimum value (0.27 km) because the power flow constraints do not uphold, and the problem becomes infeasible. Instead, the second shortest $${D}_{ik}$$ (0.39 km) is chosen, which does not cause any value in the $${P}_{ij}$$ array to exceed the values in the $${P}_{\mathrm{limit}-ij}$$ matrix, thus the problem yields an all-zero $${a}_{ij}$$ matrix and no distribution line upgrades are necessary. This scenario amounts to a total cost of $120,000 per phase.

Table 5 Results of case study 1.

Extra load on the distribution System	Node connected to the depot	Depot allocation	Chosen $${D}_{ik}({\rm{k}}m)$$	Minimum $${D}_{ik}({\rm{k}}m)$$	Line upgrades	Cost per phase ($$Z$$)
9 Buses × 26.67 kW	14	Parc de la Ciutad-location B (15th stop)	0.39	0.27	None	$120,000

Case Study 2

In this case, three identical H16 EB service lines are considered on the same distribution power system. Hence, 27 buses will be charging in unison at the allocated depots, whereupon the results on the network will be more impactful. The results show that the Service line $$-1$$ depot is connected to node 13 and allocated to Location B, the service line $$-2$$ depot is connected to node 5 and allocated to Location B, whereas the service line $$-3$$ depot is connected to node 26 and allocated to Location C. To reduce the overall cost, the code prioritizes the minimization of $${C}_{\mathrm{Upgrade}}$$ over $${C}_{\mathrm{LVCC}}$$ for Service Line $$-1$$ by choosing the 6th shortest $${D}_{ik}$$ (1.16 km); however, the minimization of $${C}_{\mathrm{LVCC}}$$ is prioritized over $${C}_{\mathrm{Upgrade}}$$ for Service Line $$-2$$ and Service Line $$-3$$ by selecting the minimum $${D}_{ik}$$ in both routes: 0.49 km and 0.40 km, respectively. Following this specific approach in this scenario yields the minimum overall cost. Table 6 summarizes the aforementioned outcomes.

Table 6 Results of case study 2.

EB service	Node connected to the depot	Depot allocation	Chosen $${D}_{ik}({\rm{k}}m)$$	Minimum $${D}_{ik}({\rm{k}}m)$$
Service Line - 1	13	Parc de la Ciutad - Location B (15th Stop)	1.16	0.27
Service Line - 2	5	Parc de la Ciutad - Location B (15th Stop)	0.49	0.49
Service Line - 3	26	Pg Zona Franca - Foc - Location C (33rd stop)	0.40	0.40

The red lines in Figure 8 represent the lines, $${L}_{ij}$$, which have been upgraded to withstand double their power capacity. In the 38-bus electrical distribution System used, power is generated only from bus 1, as a result when the depot loads are added to the system, extra power flows starting from bus 1 up to buses 5, 13, and 26 where the new loads are connected. Most of the affected lines with the new power flow have sufficient power capacity limits such that upgrading is unnecessary; nevertheless, $${L}_{34}$$, $${L}_{45}$$, and $${L}_{56}$$ required an upgrade as shown in the figure. It is important to note that before bus number 5, the power flow was at its highest value since the bus was positioned before most of the initial system loads as well as the three new depot stations. Consequently, due to the consumption of $$\mathrm{Depot}\,\mathrm{load}$$ 2 at bus 5, the power flowing through the lines is reduced considerably such that only $${L}_{56}$$ requires an upgrade afterward. Table 7 displays the values of the individual cost terms, where $${C}_{\mathrm{Load}}$$ is $36, correctly triple its value from the previous case, the system upgrades cost $97,000 compared to null in case 1, and $${C}_{\mathrm{LVCC}}$$ is $621,000, which is approximately five times its value from the first case. These figures amount to a total cost of about $718,000 per phase.

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Table 7 Case study 2 analyzed cost components.

Cost term	Value
$${C}_{\mathrm{Load}}$$	$36
$${C}_{\mathrm{Upgrade}}$$	$97,000
$${C}_{\mathrm{LVCC}}$$	$621,000
Total cost per phase $$(Z)$$	$718,036

Case Study 3

The results shown in Table 8 highlight the advantage of connecting the depot to a node close to the slack bus (grid) within the distribution network. Specifically, the decision to connect to bus 2, which is near the slack bus, ensures that the additional load is connected to a robust part of the network. This strategic connection significantly minimizes the impact on the distribution system, thus eliminating the need for line upgrades.

Table 8 Results of case study 3.

Extra load on the distribution system	Node connected to the depot	Depot allocation	Chosen $${D}_{ik}({\rm{k}}m)$$	Minimum $${D}_{ik}({\rm{k}}m)$$	Line upgrades	Cost per phase ($$Z$$)
10 Buses $$\times 100$$ kW	2	Location B	0.80	0.80	None	$243,000

In this scenario, the minimum distance $${D}_{ik}$$ was selected, resulting in the lowest possible depot connection costs. As depicted in the table, the depot was allocated to Location B with a connection distance of 0.80 km. This minimal distance not only reduces the connection costs but also leverages the inherent strength of bus 2, ensuring that the network can handle the additional load without requiring further upgrades. The allocation problem in this case is straightforward, and the results are as expected, amounting to a total cost per phase of $243,000.

Sensitivity Analysis

The sensitivity analysis results, in this section, provide a comprehensive view of how varying the power demand factor (DF) affects the distribution network, especially in the presence of a new high load such as an EB depot.

At all demand factors, the optimizer consistently chooses to allocate the depot at Location B and connect it to node bus 2. This consistency highlights the optimality of this choice even at higher demands. As a result, the depot connection cost remains constant, as shown in Figure 9.

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From Table 9, we observe that the required line upgrades increase with higher load demand. Initially, the cost of upgrades rises at a steady rate, but after a DF of 1.2, the cost increases more sharply. This trend is evident in Figure 9, where the cost of system upgrades climbs significantly beyond DF 1.2. Consequently, the total cost per phase follows a similar pattern, primarily driven by the escalating cost of line upgrades, as this is the only variable cost component that changes with the demand factor.

Table 9 Line upgrades for different demand factors (DFs).

DF	1	1.15	1.2	1.25	1.3
Line upgrades	None	Line 1–2	Line 1–2	Line 1–2	Line 1–2
		Line 5–6	Line 5–6	Line 4–5	Line 2–3
				Line 5–6	Line 3–4
				Line 23–24	Line 4–5
					Line 5–6
					Line 23–24
					Line 6–26
					Line 27–28

Regarding the allowable voltage deviation, the results in Figure 10 indicate that it stays within the acceptable limits of the IEEE Standard 1547 (ś5%) only at the base case (DF 1). However, as the demand factor increases, the voltage deviation grows to 9.5% at DF 1.15 and reaches 9.9% at DF 1.3, aligning with the IEC 60038 standard (ś10%). It is important to note that while compliance with the IEC 60038 standard is not ideal, it may be permissible in regions with less stringent regulatory environments or where infrastructure limitations exist.

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In terms of proposed model efficiency, the deterministic nature of the MINLP approach allows for more accurate and faster convergence compared to heuristic methods. Heuristic models, such as those proposed by [27], often require multiple iterations to approximate a solution, which can be time-consuming and less efficient when dealing with large and complex systems. The proposed model significantly reduces computation time while maintaining accuracy, making it more suitable for practical implementation in large-scale networks, as demonstrated in the 38-bus distribution system case study. The results showed that the proposed model consistently identified the optimal depot locations while minimizing the required distribution line upgrades, offering an effective balance between system reliability and infrastructure investment. In terms of usability and implementation, GAMS does not require advanced technical expertise, making it accessible to grid operators and transit planners without the need for specialized coding skills. This practical ease of use ensures that the proposed model can be applied in real-world settings with minimal adjustments or additional development work. Moreover, the model's flexibility to handle both single-route and multi-route bus services, as demonstrated in the third case study, emphasizes its versatility. This adaptability to different operational scenarios is a clear advantage over heuristic-based models, which often struggle with multi-variable and multi-route configurations.

Conclusion

The optimization model developed in this research addresses the allocation of a central depot station capable of charging up to 10 electric buses simultaneously for an all-electric bus service route—the H16 bus service in Barcelona, Spain. This study considers the impact on a medium-voltage (MV) 38-bus distribution network to determine the optimal node that will supply the depot. The objective is to minimize the associated overall costs and identify necessary distribution line upgrades after introducing the new load. The approach is applied to three cases: a single EB service in the electrical network, three identical EB services in the same network, and a multi-route service with three distinct routes in the same network. The proposed technique provides the optimal solution that balances the cost of distribution network line upgrades and the cables connecting each depot to the node while ensuring that power flow constraints are met. For a single H16 EB service, none of the distribution lines required an upgrade, and the optimizer chose the second shortest $${D}_{ik}$$, resulting in a total cost per phase of $120,000. In the case of three identical H16 EB services, three lines needed an upgrade, and the shortest $${D}_{ik}$$ was chosen for two service routes, while the $${6}^{th}$$ shortest $${D}_{ik}$$ was chosen for the other. This scenario resulted in a total cost per phase of about $718,000. The third case study examined a multi-route service with three distinct routes, all served by a single depot. The optimizer consistently allocated the depot to Location B and connected it to node bus 2, highlighting the optimality of this choice even under higher demand factors. The total cost per phase in this scenario amounted to $243,000. Sensitivity analysis showed that required line upgrades increased with higher load demands, with a significant rise beyond a demand factor of 1.2. Additionally, while the voltage deviation was within the acceptable limits of the IEEE Standard 1547 at a base case, it increased to 9.5% at a 1.15 demand factor and reached 9.9% at a 1.3 demand factor, aligning with the IEC 60038 standard. This paper can be an optimal planning tool for grid operators as well as electric transit operators, as it provides valuable insights and strategies for efficient planning and management of the grid and electric transportation systems.

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