# A framework for smart charging strategies to further the adoption of electric buses in cities

Yacine Sehimi, Berk Celik, Fabrice Locment, Manuela Sechilariu

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Yacine Sehimi, Berk Celik, Fabrice Locment, Manuela Sechilariu. A framework for smart charging strategies to further the adoption of electric buses in cities. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0320. hal-05075204

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# A framework for smart charging strategies to further the adoption of electric buses in cities

Yacine Sehimi, Berk Celik, Fabrice Locment, Manuela Sechilariu

Avenues, Université de Technologie de Compiègne

Centre Pierre Guillaumat-CS90319

60203 Compiègne, France

E-Mail: {yacine.sehimi, berk.celik, fabrice.locment, manuela sechilariu}@utc.fr

**Acknowledgements**—This research is funded by ADEME France, project T-IPV grant number # 2308D0002.

**Index Terms**—Electric bus fleet, smart charging, aggregator, grid integration.

**Abstract**—The increasing deployment of electric buses in cities can create opportunities for bus operators to provide flexibility to the electrical grid and to reduce their operational costs. However, charging sessions must be properly monitored to maximize externalities while ensuring energy requirements of the fleet for their trips. This paper proposes a framework for analyzing the relevant charging strategies for a fleet of electric buses. After reviewing possible charging strategies in cooperation with grid ecosystem, a framework of interactions between actors is proposed. Modeling methods enabling to implement several charging strategies in a mathematical model are finally presented.

## I. INTRODUCTION

Despite their significant benefits in reducing the use of individual petroleum cars, internal combustion buses generate greenhouse gas emissions, pollution and noise. Therefore, they are progressively replaced by electric buses (EBs), which are a prominent solutions to mitigate these externalities. However, total cost of ownership of EBs is estimated 30% higher than their internal combustion counterparts [1]. It is still an obstacle on the road towards bus electrification. Moreover, the high power required to charge EBs can cause congestion and stress the reliability of the distribution grid, ultimately resulting in grid reinforcement costs for the Distribution System Operator (DSO).

Nevertheless, the total cost of ownership of EBs can be reduced if the charging costs are lowered. It can be achieved if charging sessions are wisely operated. Moreover, battery charging management offers flexibility to the Transmission System Operator (TSO) and an

help to face growing electrification of the transportation sector. Unlike private electric vehicles, EBs are owned by a company and the energy and availability of each individual vehicle is higher. Similarly to industrial consumers, the Public Transportation Operator (PTO) can decide to participate to ancillary services.

From these observations, scheduling charging operations appears to be a strong driver for the adoption of EBs. Charging scheduling of EBs has fostered attention of several studies. In [2], a charging optimization strategy aiming to maximize profits for the PTO and minimize their impact on LV feeder is implemented. Photovoltaic (PV) production and battery ESS are used and results show that peak power demand is highly reduced.

The purpose of this paper is to analyze how charging management strategies may help to further the adoption of EBs and mitigate their impacts on the electricity grid. The remainder of this article is organized as follows: Section II reviews the various methods of EB charging in term of charging mode, charging frequency and available services. Section III analyzes several frameworks of interactions between the PTO and grid actors. Finally Section IV proposes a model to assess the potential of a fleet of EBs to reduce its operational costs.

## II. CHARGING METHODS FOR EBs

### A. Charging mode: overnight, opportunity or terminal

EBs usually leave a depot (where all EBs are parked during the night) and achieve repeated trips between line terminals before going back to the depot. Charging sessions can therefore be achieved during the night when the bus are not in operation: this concept is called overnight charging. But EBs can also be charged at the stops of their line during operations or at the terminal stations before starting a service. In this case, the charging mode refers to opportunity (or terminal) charging.
Overnight charging has the lowest impact on buses operating conditions, because EBs are charged when not in operation. Moreover, Ebs are concentrated at a single location for a long period therefore the available flexibility is important. However, if EBs are only charged during the night at depot, their battery must be sized to ensure a full day of operation, resulting in a high battery size and a high initial investment.

Conversely, charging at terminals or at stops counters the previous limitation. As EBs are charged repeatedly along the route, the battery requirements are drastically reduced. For the terminal case, EBs' battery theoretically require the energy of a single terminal-to-terminal trip and in the case of charging at stops, the most energy intensive route between two stops may be the sizing criteria (assuming charging power at stops is sufficient).

Nevertheless, this benefit comes at the cost of a large charging infrastructure, especially for the opportunity charging case. Indeed, the time spent by EBs at stops or at terminal is generally short (a few seconds to a few minutes) so charging power requirements of the equipment must be sufficiently high to provide the required energy in this duration. Although some of the charging stations could be shared by several lines sharing the same stops or terminals, construction, hardware and grid connection cost could counteract the benefits of small battery sizes and costs.

To answer this question, the life cycle costs (LCCs) associated with the overnight, opportunity and terminal charging method have been compared [6]. Results show that charging at terminals could be the most cost effective option. Its LCC is even lower than diesel buses. Overnight charging seems to come at the highest cost, however smart charging algorithm have not been considered.

## B. EB charging ecosystem

### 1) Actors:

To provide electrical energy to EB's battery, several actors interact, directly or indirectly.

* The PTO who manages the fleet of EBs. It must ensure that EBs accomplish the required trips of the bus service, following a timetable. It is also responsible of the charging infrastructure, which can integrate renewable energy production and storage. Its objective is to minimize operational costs of EBs to ensure the lowest total cost of ownership throughout their lifetime.

* The DSO who operates medium and low voltage networks. DSOs must ensure voltage stability throughout their network and avoid grid congestion.

Their objective is to minimize grid reinforcement costs. To do so, they can benefit from flexible loads such as EB's charging power. Controlling these loads at appropriate time can help to avoid voltage imbalance and congestion and further delay grid upgrades (transformer and line power upgrade).

* The TSO who operates high and very high voltage networks. Its role is to transmit electricity between energy production facilities, grid of other countries (called interconnection) and distribution networks. They must balance production and consumption at any time. Imbalances cause frequency deviation which and power outages.

### 2) Markets and services:

To reduce charging cost, PTO can generate revenues by providing flexibility to DSO and TSO. Several services, called ancillary services, are available.

#### DSO services:

* Peak Power Shaving (DSO_PPS) : to avoid congestion on MV and LV grids, the load from EBs depot can be shifted from peak to off-peak hours. It allows to delay grid upgrades due to the apparition of new consumers.

* Voltage regulation (DSO_VR) : flexible EB charging can be used to ensure that the voltage deviation in MV and LV grids is limited. In France, 5% maximum deviation is required in the MV grid and 10 % in the LV grid.

* Losses minimization (DSO_LM) : Losses in LV and MV grid comprise transformer losses and line losses. Reducing EB power at appropriate times can limit these losses and costs for the DSO.

* Energy transmission cost minimization (DSO_ETCM) : DSO pays TSO for the energy transmission to its network. Using local PV production and flexible charging of EBs allows to reduce the bill and benefit from low price periods.

#### TSO services:

* Frequency containment reserve (TSO_FCR) : to limit frequency deviation, TSO can contract power capacities upward or downward. These capacities are activated automatically within 15 to 30 seconds, with a dynamic time between 1 and 10 seconds. This control is achieved at the European level (ENTSO-E members).

* Frequency Restoration Reserve (TSO_FRR) : this reserve allows FCR to recover its capacities for future activations. Part of this reserve is activated automatically (aFRR) within 15 minutes, an other is activated manually (mFRR).
* Replacement Reserves (TSO_RR) : this reserve progressively replace FRRs after their activation. The activation time is between 15 and 1 hour. The minimum power being 10 MW, it is unlikely that EBs are able to participate to this service

* Energy arbitrage (TSO_EA) : PTO can purchase and sell energy on a day ahead market monitored by the TSO. Using vehicle-to-grid (V2G) could ensure that its energy is sold at a competitive price.

### PTO services:

* Charging cost reduction (PTO_CCR) : PTO purchases electricity and its cost is comprised of grid tariffs (depending on the subscribed power, in €/MW), and energy cost (in €/MWh). Grid tariffs is fixed at subscription whereas energy price varies during the day. Charging scheduling strategy can be implemented to minimize this cost.

* Enhance renewable energy penetration (PTO_REP) : EBs can favor the introduction of renewable sources such as PV panels at the charging station location or at other locations. Charging management can ensure that a maximum energy production is absorbed when available.

* CO₂ emissions reduction (PTO_CO₂) : Charging sessions can be optimized in order to charge when the CO₂ footprint of the consumed energy is the lowest. It can help to further increase the decarbonation of PTO, especially in countries where the electricity mix is highly carbonated.

Table I gives a summary of several characteristics for the aforementioned services [10] [11]. Most of them require a power of 1 MW. For a fleet of EBs with a charging power of 150 kW at the depot, at least 7 EBs are necessary to participate to these services. However, terminal and opportunity charging schemes are more challenging for ancillary services as the temporal availability of EBs at the charging system is shorter.

It must be noted that EBs can participate to several services simultaneously. For instance, EB can reduce charging cost by providing services to the TSO such as FCR or FRR, as it has been studied in several previous works [3] [4] [5]. But DSO and TSO can also have competing interests. Indeed, activation of frequency reserves can occur at times where power demand is high and therefore when the need of peak shaving to limit congestion is the highest.

## III. FRAMEWORK FOR EB CHARGING MANAGEMENT

In this paper a framework enabling to capture the maximum benefits for a PTO while participating in the

previously mentioned grid services is proposed. First of all, previous works on this subject are reviewed.

### A. Aggregators

#### 1) Role of aggregator:

As seen in the previous part, by controlling its power flow in charging or discharging, an EB fleet can participate to various grid services and generate revenues. To do so, PTO must assess its available flexibility and determine which services are the most profitable while at the same times reducing the impact on its operating costs such as battery degradation. At the same time, the DSO aims to control the power flow from the depot in order to reduce grid congestion and voltage imbalance issues. It has been seen that both actors can have competing interests in controlling the charging of EBs.

An entity, called Electric Bus Aggregator (EBA), can play an intermediary role between these actors to face these challenges. The concept of aggregator has fostered attention in the context of private electric vehicles with V2G capability, but literature in the case study of EBAs is limited so far. Several frameworks of charging/discharging strategies for EBs have however been investigated.

#### 2) Type of aggregators:

* Independent aggregator

The EBA can be an independent entity from the PTO, DSO and TSO. Its objective is to generate profits by offering flexibility to the TSO while meeting grid constraints from the DSO and energy availability constraints from the PTO. At the same time, EBA sends price signals to the PTO, which determines a charging scheduling strategy to minimize operational costs. Operational costs include charging costs, revenues from selling energy to the EBA, and battery degradation costs [5].

This approach has several advantages. First, it allows the PTO to reach energy markets (day ahead

<table><caption>TABLE I
AVAILABLE SERVICES FOR CHARGING OPTIMIZATION</caption><thead><tr><td>Actor</td><td>Code</td><td>Min. power</td><td>Activation time</td><td>Frequency</td></tr></thead><tbody><tr><td>DSO</td><td>DSO_PPS</td><td>100 kW</td><td>10 min - 2 h</td><td>monthly</td></tr><tr><td>DSO</td><td>DSO_VR</td><td>10 kW</td><td>hours</td><td>daily</td></tr><tr><td>DSO</td><td>DSO_ETCM</td><td>10 kW</td><td>10 min - 10 h</td><td>continuous</td></tr><tr><td>TSO</td><td>TSO_FCR</td><td>1 MW</td><td>15 min - 30 min</td><td>daily-weekly</td></tr><tr><td>TSO</td><td>TSO_FRR</td><td>1 MW</td><td>no minimum</td><td>daily</td></tr><tr><td>TSO</td><td>TSO_RR</td><td>1 MW</td><td>≥ 15 min</td><td>daily</td></tr><tr><td>TSO</td><td>TSO_EA</td><td>1 MW</td><td>hours</td><td>daily</td></tr><tr><td>PTO</td><td>PTO_CCR</td><td>-</td><td>hours</td><td>daily</td></tr><tr><td>PTO / other</td><td>PTO_REP</td><td>100 kW</td><td>hours</td><td>daily</td></tr><tr><td>PTO</td><td>PTO_CO₂</td><td>-</td><td>hours</td><td>daily</td></tr></tbody></table>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Framework considered in this study</figcaption></figure>

and ancillary services) without having to interact
with a complex regulatory frameworks with respect
to its core sector. Indeed, from the PTO perspective,
a simple scheduling charging strategy based on
price signals must be implemented and availability
constraints must be sent to the EBA. The entry
barrier is therefore reduced.

On the other hand, a share of potential revenues
from energy flexibility are captured by the EBA
instead of PTO. EBA would only contract capacities
with TSO when it achieve profits. Therefore, charg-
ing cost reduction potential is reduced for PTO.

* DSO as aggregator

The DSO plays by nature an intermediary position
between PTO and TSO. DSO already trade energy
and provide flexibility to the TSO while meeting
power needs from the connected prosumers and
ensuring voltage stability on the network. A frame-
work with DSO as EBA is therefore the conven-
tional approach to date [12].

* PTO as aggregator

Lastly, the PTO itself can play the role of aggre-
gator. In this case, the main objective would be to
minimize charging costs. They would include energy
costs, grid tariffs, battery degradation costs but also
revenues from power flexibility provided to the TSO
and/or to the DSO. PTO can react to grid tariffs and
energy price provided by the DSO in order to reduce

congestion in the network. But it can also participate
directly to energy markets and frequency reserves
market without intermediary [3].

B. Proposed framework

The framework proposed in this paper is presented
in Fig. 1. The EBA is an entity that works solely for the
benefits of the actors and PTO's purpose is prioritized.
A hierarchical approach is therefore considered, where a
higher level corresponds to PTO's objective and a lower
level refers to DSO's and/or TSO's objective. Voltage
and power constraints are considered in order to limit
congestion and voltage imbalance in the distribution
network.

In the scenario proposed in Fig. 1, the higher objec-
tive is charging cost reduction and the lower objective
is to particpate to FCR and aFRR at the lowest cost for
the TSO. Other scenario can be considered to assess the
benefits of charging optimization for the DSO or TSO.
This hierchical approach would allow to capture the most
benefits for the PTO while at the same time ensuring that
this strategy is sufficiently relevant for grid actors.

IV. MODELING METHODS AND CHARGING ALGORITHMS

The proposed model is based on the framework
presented in Fig. 1 and previous works [3] [5]. Overnight
charging at depot is the selected charging mode because
<table><caption>TABLE II
DESCRIPTION OF VARIABLES USED IN THIS STUDY</caption><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td colspan="2"><strong>Sets</strong></td></tr><tr><td>i, I</td><td>EB index, and set</td></tr><tr><td>t, T</td><td>time index, and set</td></tr><tr><td>T<sup>FCR,con</sup></td><td>time periods of FCR contracts</td></tr><tr><td>T<sup>FCR,act</sup></td><td>time periods of FCR activation</td></tr><tr><td>T<sup>aFRRup,act</sup></td><td>time periods of aFRR upward</td></tr><tr><td>T<sup>aFRR,down</sup></td><td>time periods of aFRR downward</td></tr><tr><td colspan="2"><strong>Variables</strong></td></tr><tr><td>p<sub>t</sub><sup>depot</sup></td><td>depot power</td></tr><tr><td>p<sub>t</sub><sup>PV</sup></td><td>PV power</td></tr><tr><td>p<sub>t</sub><sup>EB</sup>, p<sub>t</sub><sup>ESS</sup></td><td>EB and ESS battery power</td></tr><tr><td>x<sub>t</sub><sup>FCR,act</sup></td><td>binary variable for FCR activation</td></tr><tr><td>x<sub>t</sub><sup>aFRR,up</sup></td><td>binary variable for upward aFRR activation</td></tr><tr><td>x<sub>t</sub><sup>aFRR,down</sup></td><td>binary variable for downward aFRR activation</td></tr><tr><td>π<sub>t</sub><sup>FCR</sup></td><td>capacity price for FCR provision</td></tr><tr><td>π<sub>t</sub><sup>aFRRup,P</sup></td><td>capacity price for upward aFRR provision</td></tr><tr><td>π<sub>t</sub><sup>aFRRdown,P</sup></td><td>capacity price for downward aFRR provision</td></tr><tr><td>π<sub>t</sub><sup>aFRRup,E</sup></td><td>binary variable for upward aFRR activation</td></tr><tr><td>π<sub>t</sub><sup>aFRRdown,E</sup></td><td>binary variable downward aFRR activation</td></tr><tr><td>e<sub>it</sub><sup>EB</sup>, e<sub>t</sub><sup>ESS</sup></td><td>EB and ESS battery energy</td></tr><tr><td colspan="2"><strong>Parameters</strong></td></tr><tr><td>Δt</td><td>time interval</td></tr><tr><td>t<sub>i</sub><sup>a</sup>, t<sub>i</sub><sup>d</sup></td><td>Arrival and departure time of EB i at the depot</td></tr><tr><td>p<sub>i</sub><sup>trip</sup></td><td>average power consumption of EB i during its trips with passengers</td></tr><tr><td>E<sup>EB</sup>, E<sup>ESS</sup></td><td>EB and ESS battery capacity</td></tr><tr><td>SOC<sup>min</sup></td><td>minimum SOC limit of EBs' battery</td></tr><tr><td>SOC<sup>max</sup></td><td>maximum SOC limit of EBs' battery</td></tr><tr><td>C<sup>rep,EB</sup>, C<sup>rep,ESS</sup></td><td>replacement cost of EB and ESS battery</td></tr><tr><td>Ah<sup>EB</sup>, Ah<sup>ESS</sup></td><td>Maximum ampere-hour throughput of EB and ESS battery before replacement</td></tr><tr><td>V<sup>EB</sup><sub>nom</sub>, V<sup>ESS</sup><sub>nom</sub></td><td>Nominal voltage of EB and ESS battery</td></tr><tr><td>P<sub>sub</sub></td><td>depot subscribed power</td></tr><tr><td>Π<sup>P</sup></td><td>grid tariff (€/kW)</td></tr><tr><td>Π<sup>E</sup><sub>t</sub></td><td>electricity price (€/kWh)</td></tr><tr><td>P<sup>ch,EB</sup>, P<sup>ch,ESS</sup></td><td>EB and ESS maximum charging power</td></tr><tr><td>P<sup>dis,EB</sup>, P<sup>dis,ESS</sup></td><td>EB and ESS maximum discharging power</td></tr><tr><td>P<sup>max,depot</sup></td><td>depot maximum power</td></tr><tr><td>η<sup>AC/DC</sup>, η<sup>DC/DC</sup></td><td>AC/DC and DC/DC converters efficiency</td></tr></tbody></table>

it has the most potential for ancillary services. Indeed, aggregating EBs at a defined location for a long period of time increases the power and energy available for these services. Variables and parameters are defined in Table II. A mathematical problem formulation allows to define the optimization problem with respect to the chosen charging strategy.

## A. Decision variables

The main decision variables considered are

* charging power of EBs and ESS $p_{i,t}^{EB}, p_{i,t}^{ESS} \in \mathbb{R}$

* selling capacity price for FCR provision $\pi_t^{FCR} \in \mathbb{R}^+$

* capacity and energy selling prices for aFRR upward provision $\pi^{aFRRup,P}, \pi^{aFRRup,E} \in \mathbb{R}^+$

* capacity and energy selling prices for aFRR downward provision $\pi^{aFRRdown,P}, \pi^{aFRRdown,E} \in \mathbb{R}^+$

## B. Objective function

The objective is to minimize the daily operating cost of the depot. The operating costs are comprised of electricity costs, battery degradation costs and revenues from ancillary services.

$$ \min \text{DOC} \qquad (1) $$

$$ \text{DOC} = C^{\text{elec}} + C^{\text{deg,EB}} + C^{\text{deg,ESS}} - R^{\text{services}} \quad (2) $$

$$ C^{\text{elec}} = \Pi^P P^{\text{sub}} + \sum_{t \in T} \Pi_t^E p_t^{\text{depot}} \Delta t \quad (3) $$

$$ C^{\text{deg,EB}} = \frac{C^{\text{rep,EB}}}{Ah^{EB} V_{\text{nom}}^{EB}} \sum_{t \in T} p_{i,t}^{EB} \Delta t \quad (4) $$

$$ C^{\text{deg,ESS}} = \frac{C^{\text{rep,ESS}}}{Ah^{ESS} V_{\text{nom}}^{ESS}} \sum_{t \in T} p_t^{ESS} \Delta t \quad (5) $$

$$ R^{\text{services}} = R^{\text{FCR}} + R^{\text{aFRR}} \quad (6) $$

$$ R^{\text{FCR}} = \sum_{t \in T} p_t^{\text{depot}} x_t^{\text{FCR,act}} \pi_t^{\text{FCR}} \quad (7) $$

$$ R^{\text{aFRR}} = R^{\text{aFRR,up}} + R^{\text{aFRR,down}} \quad (8) $$

$$ R^{\text{aFRR,up}} = \sum_{t \in T} p_t^{\text{depot}} x_t^{\text{aFRR,up}} (\pi_t^{\text{aFRRup,P}} + \pi_t^{\text{aFRRup,E}} \Delta t) \quad (9) $$

$$ R^{\text{aFRR,down}} = \sum_{t \in T} P_t^{\text{depot}} x_t^{\text{aFRR,down}} (\pi_t^{\text{aFRRdown,P}} + \pi_t^{\text{aFRRdown,E}} \Delta t) \quad (10) $$
Equation (3) shows that electricity costs are composed of a fixed cost based on subscribed power with the DSO, and a variable cost related to energy consumption. $P^{sub}$ can optionally be set as proportional to the maximum power consumed at depot. Equations (4)-(5) assume that battery degradation is proportional to the energy throughput of the battery.

Depot power is the sum of the power from the charging station, from the PV system and from ESS. Station power is the sum of the power of all the charge plugs.

$$p_t^{\text{depot}} = p_t^{\text{station}} + \eta^{\text{AC/DC}} \eta^{\text{DC/DC}} (p_t^{\text{ESS}} - p_t^{\text{PV}}) \quad (11)$$

$$p_t^{\text{station}} = \eta^{\text{AC/DC}} \eta^{\text{DC/DC}} \sum_{i \in \mathcal{I}} p_{i,t}^{\text{EB}}, \quad \forall t \in \mathcal{T} \quad (12)$$

The available energy in EBs and ESS' battery can also be defined with equations

$$e_{i,t}^{\text{EB}} = e_{i,t-1}^{\text{EB}} + p_{i,t}^{\text{EB}} \Delta t \quad (13)$$

$$e_t^{\text{ESS}} = e_{t-1}^{\text{ESS}} + p_t^{\text{ESS}} \Delta t \quad (14)$$

### C. Constraints

#### 1) Trips constraints:

$$p_{i,t}^{\text{EB}} = p_i^{\text{trip}} \quad \forall i \in \mathcal{I}, \forall t \in \mathcal{T} \setminus [t_i^a, t_i^d] \quad (15)$$

(15) ensures that EBs are discharged of the energy required to achieve their trips.

#### 2) Power constraints:

$$p_{i,t}^{\text{EB}} \le P^{\text{ch,EB}} \quad \forall i \in \mathcal{I}, \forall t \in \mathcal{T} \quad (16)$$

$$p_{i,t}^{\text{EB}} \ge P^{\text{dis,EB}} \quad \forall i \in \mathcal{I}, \forall t \in \mathcal{T} \quad (17)$$

$$p_t^{\text{ESS}} \le P^{\text{ch,ESS}}, \quad \forall t \in \mathcal{T} \quad (18)$$

$$p_t^{\text{ESS}} \ge P^{\text{dis,ESS}}, \quad \forall t \in \mathcal{T} \quad (19)$$

$$p_t^{\text{depot}} \le P^{\text{max,depot}}, \quad \forall t \in \mathcal{T} \quad (20)$$

Equations (16)-(19) ensures that EB and ESS are not charged and discharged more than the maximum power that the chargers can provide. Equation (20) limits the maximum power from the depot based on local distribution grid capability.

#### 3) Energy constraints:

$$e_{i,t}^{\text{EB}} \le E^{\text{EB}}.\text{SOC}^{\text{max}}, \quad \forall i \in \mathcal{I}, \forall t \in \mathcal{T} \quad (21)$$

$$e_{i,t}^{\text{EB}} \ge E^{\text{EB}}.\text{SOC}^{\text{min}}, \quad \forall i \in \mathcal{I}, \forall t \in \mathcal{T} \quad (22)$$

$$e_t^{\text{ESS}} \le E^{\text{EB}}.\text{SOC}^{\text{max}}, \quad \forall t \in \mathcal{T} \quad (23)$$

$$e_t^{\text{ESS}} \ge E^{\text{EB}}.\text{SOC}^{\text{min}}, \quad \forall t \in \mathcal{T} \quad (24)$$

$$e_{i,t}^{\text{EB}} = E^{\text{EB}}.\text{SOC}^{\text{max}}, \quad \forall i \in \mathcal{I} \quad (25)$$

Constraints (21)-(24) limits the state of charge of the batteries in a defined range as it is commonly done to maximize battery life. Equation (25) ensures that EBs leave the depot with a maximum energy capacity.

#### 4) FCR constraints:

To participate to FCR, the depot must provide symmetrical power for upward or downward frequency regulation [3]. In the case of upward regulation, power must be provided from the depot to the grid and the contrary for downward regulation. It is assumed that only EBs provide such service at the depot in order to limit the impact of PV production variability on the contracted reserves.

First, it is required to assess the maximum power that EBs are able to provide for upward and downward regulation.

$$p_{i,t}^{\text{up,max}} = \max\left(\frac{e_{i,t}^{\text{EB}} - e_{\min}}{\Delta t}, P^{\text{dis,EB}}\right) \quad (26)$$

$$p_{i,t}^{\text{down,max}} = \min\left(\frac{e_{\max} - e_{i,t}^{\text{EB}}}{\Delta t}, P^{\text{ch,EB}}\right) \quad (27)$$

$$p_t^{\text{up,max}} = \sum_{i \in \mathcal{I}} p_{i,t}^{\text{up,max}}, \quad \forall t \in \mathcal{T} \quad (28)$$

$$p_t^{\text{down,max}} = \sum_{i \in \mathcal{I}} p_{i,t}^{\text{up,max}}, \quad \forall t \in \mathcal{T} \quad (29)$$

$$p_t^{\text{FCR}} = \min(p_t^{\text{down,max}}, -p_t^{\text{up,max}}), \quad \forall t \in \mathcal{T} \quad (30)$$

Equation (26) (resp. 27) states that the maximum power provided by an EB for upward (resp. downward) regulation is limited by the charger discharging (resp. charging) power capability and the maximum discharge (resp. charge) power based on available energy. Equation (28) (resp. (29)) sums the available upward (resp. downward) power across the fleet to obtain the total available power. As symmetrical power must be provided for FCR, equation (30) allows to obtain the minimum power allowing symmetrical regulation.

Knowing the time when reserves are contracted for FCR (i.e. the set $\mathcal{T}^{\text{FCR,con}}$), it must be ensured that depot satisfies the minimum power requirement to provide this service. A condition variable $x_t^{\text{FCR}}$ is defined as follows.

$$x_t^{\text{FCR}} = \begin{cases} 1, & \text{if } p_t^{\text{FCR}} \ge P^{\text{FCR,min}} \text{ and } t \in \mathcal{T}^{\text{FCR,con}} \\ 0, & \text{else} \end{cases} \quad (31)$$

It can be noted that this condition variable can be transformed to linear inequality constraints using the big-M method [16]. Subsequently, an variable $x_t^{\text{FCR,act}}$ is defined to account of FCR's activation at the depot.

$$x_t^{\text{FCR,act}} = \begin{cases} x_t^{\text{FCR}}, & \text{if } t \in \mathcal{T}^{\text{FCR,act}} \\ 0, & \text{else} \end{cases} \quad (32)$$
Therefore, this variable equals 1 at activation times if depot has sufficient reserve available.

The activated power by TSO for FCR is proportional to frequency deviation. A variable $u_t^f$ is defined to account for this proportion

$$ u_t^f = \begin{cases} 1, & \text{if } f_t - f_{nom} \le -\Delta f_{max} \\ \frac{f_{nom} - f_t}{\Delta f_{max}}, & \text{if } -\Delta f_{max} \le f_t - f_{nom} \le -\Delta f_{min} \\ 0, & \text{if } -\Delta f_{min} < f_t - f_{nom} \le \Delta f_{min} \\ -\frac{f_{nom} - f_t}{\Delta f_{max}}, & \text{if } \Delta f_{min} \le f_t - f_{nom} \le \Delta f_{full} \\ -1, & \text{if } f_t - f_{nom} \ge \Delta f_{max} \end{cases} \quad (33) $$

Station power must equal activated power at activation times. It gives the equality constraint (36).

$$ p_t^{\text{depot}} = p_t^{\text{FCR}} u_t^f, \quad \forall t \in \mathcal{T}^{\text{FCR,act}}, x_t^{\text{FCR,act}} = 1 \quad (34) $$

5) aFRR constraints:

aFRR is contracted either upward or downward. Therefore the available reserves upward or downward are given by equations (26)-(29). Similarly to FCR, a condition variable $x_t^{aFRR}$ can be defined.

$$ x_t^{aFRR,up} = \begin{cases} 1, & \text{if } p_t^{up,max} \ge P^{aFRRup,min} \\ 0, & \text{else} \end{cases} \quad (35) $$

The process is equivalent for downward activation with $x_t^{aFRR,down}$. If this condition is met, the power reserve is fully activated at activation times.

$$ p_t^{\text{depot}} = p_t^{aFRR,up}, \quad \forall t \in \mathcal{T}^{aFRRup,act}, x_t^{aFRR,up} = 1 \quad (36) $$

and respectively for downward activation.

6) Price constraints:

Selling upward or downward reserve capacity must be beneficial for the PTO in terms of revenues but also in terms of competitive price for the TSO. Therefore, the selling price of capacity must be lower than the historical price from data. It gives the inequality constraints

$$ \pi_t^{\text{FCR}} \le \Pi_t^{\text{FCR}} \quad \forall t \in \mathcal{T}^{\text{FCR,con}} \quad (37) $$

$$ \pi_t^{aFRRup,P} \le \Pi_t^{aFRRup,P} \quad \forall t \in \mathcal{T}^{aFRRup,con} \quad (38) $$

$$ \pi_t^{aFRRup,E} \le \Pi_t^{aFRRup,E} \quad \forall t \in \mathcal{T}^{aFRRup,con} \quad (39) $$

7) Constraints from the DSO:

As mentioned earlier, DSO and TSO can have conflicting interests in managing the charging or discharging power from EBs. It must be ensured that voltage stability

is maintained throughout the MV distribution network. However the relation between the power from the depot and the node voltages in the MV grid is non-linear.

To remediate to this issue, the backward forward sweep method can be used [17]. The flowchart of a possible implementatbacion is presented in Fig. 2. First, a flat voltage profile is considered at all nodes. In the backward step, currents are computed at all branches of the network. In the forward sweep, voltage at every node is computed based on the current previously calculated. The operation is repeated until the node voltages converge.

If all voltage nodes are not in a tolerance region around the nominal value, depot maximum power is reduced and the process is repeated again. The process stops when the voltage tolerance criteria is met. Therefore the maximum power ensuring grid stability is determined.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Flowchart of the method used to comply with distribution grid constraints</figcaption></figure>

## V. PREVIOUS RESULTS

Aggregator-based charging strategies have been studied for buses in several previous works. In [5], a framework of independent aggregator is employed in a bilevel optimization approach. EBA objective is the upper-level and PTO objective is the lower level. Results show a 90% reduction in charging costs for the PTO and profits for the EBA when participating to ancillary services.
In [12], a bilevel optimization approach with DSO as aggregator is proposed. Energy is traded at the wholesale market. Energy price is determined at each feeder location with respect to grid congestion. At the upper-level, PTO aims to minimize charging costs and determines the optimal EB dispatch and charging schedule. This method proved effective in reducing both grid congestion and charging costs.

In [3], several charging optimization strategies have been compared: energy arbitrage, FCR, aFRR and peak shaving. In the frequency reserves scenario PTO provides bid prices and maximum up/down power capability directly to the TSO. Results showed that participation to aFRR could be the most effective option to reduce charging costs, rather than providing peak mitigation to the DSO or selling energy to energy markets.

## VI. CONCLUSION AND FUTURE WORKS

This paper introduced charging strategies for buses that can effectively be used to reduce operational costs for transport operators, while at the same time providing services to grid ecosystem. Services for the DSO and TSO have been reviewed and a framework involving the role of an EBA to manage various actors' interests has been highlighted. A mathematical model has been proposed to analyse the benefits of providing frequency control while avoiding congestion and high voltage deviations on the distribution network.

In future works, this framework will be implemented in a simulation model. The benefits and challenges for PTO, DSO and TSO will therefore be highlighted.

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