Profit-Aware EV Utilisation Model for Sustainable Smart Cities: Joint Optimisation over EV System, Power Grid System, and City Road Grid System

Highlights

What are the main findings?

• For EVs that do not have the willingness or technical ability to discharge back to the grid, the proposed model successfully identifies the most financially profitable charging stations, routes, and schedules, all jointly optimised over the electric vehicle system, the power grid system, and the city road grid system.
• The performance has also been successfully proven for EVs that have the technical ability and willingness to discharge back to the grid.

What are the implications of the main findings?

• Traditional distance-based charging station suggestion models may put more stress on urban power grids. In contrast, the proposed profit-aware design can now be utilised to optimally manage this stress, thanks to the grid mismatch-driven nodal price allocation.
• This joint optimisation model can also be utilised to fairly compensate EV users for their additional inconveniences and to help the grid maintain its supply and demand mismatch, thereby ensuring stability.

Abstract

A sustainable city requires a sustainable means of transportation. This ambition is leading towards higher penetration of electric vehicles (EVs) in our cities, in both the private and commercial sectors, putting an ever greater burden on the existing power grid. Modern deregulated power grids vary electricity tariffs from location to location and from time to time to compensate for any additional burden. In this paper, we propose a profit-aware solution to strategically manage the movements of EVs in the city to support the grid while exploiting these locational, time-varying prices. This work is divided into three parts: (M1) profit-aware charging location and optimal route selection, (M2) profit-aware charging and discharging location and optimal route selection, and (M2b) profit-aware charging and discharging location and optimal route selection considering demand-side flexibility. This work is tested on the MATLAB programming platform using the Gurobi optimisation solver. From the extensive case studies, it is found that M1 can yield profits up to 2 times greater than those of its competitors, whereas M2 can achieve profits up to 2.5 times higher and simultaneously provide substantial grid support. Additionally, the M2b extension makes M2 more efficient in terms of grid support.

1. Introduction

The rapid adoption of electric vehicles (EVs) is a cornerstone in the evolution of sustainable smart cities. In 2024, global EV sales reached approximately 17.1 million units, marking a 25% increase from the previous year (https://rhomotion.com/news/over-17-million-evs-sold-in-2024-record-year/ (accessed on 1 November 2025).) EVs now account for more than 20% of new car sales worldwide, underscoring a significant transition toward cleaner urban mobility and energy systems. However, the benefits of EVs can only be fully realised if charging infrastructure and management strategies evolve in parallel. Today, most EV charging is uncoordinated and often concentrated during peak hours, which amplifies grid demand spikes. High-power consuming charging stations in particular draw substantial loads, creating localised imbalances. With millions of additional charging points projected worldwide, the implications for land use, grid reinforcement, and local reliability are significant. However, coordinated and smart charging can transform EVs from potential burdens into distributed storage assets that enhance renewable integration and grid resilience [1].

Smart charging and bidirectional vehicle-to-grid (V2G) operations are emerging as transformative solutions. Smart charging shifts demand to off-peak periods or aligns charging with renewable availability, while V2G enables EVs to discharge power back to the grid at times of high demand [2]. Together, these strategies allow EVs to integrate renewables, mitigate peak demand, and provide ancillary services. Yet, their widespread deployment introduces challenges such as voltage fluctuations, transformer stress, and rising system peaks [3]. Case studies highlight the complexity of EV integration. An energy management research study on microgrids demonstrates that EVs can reduce costs by nearly half while improving resilience [4]. Another work on a DC microgrid prototype confirms that optimal EV control can stabilise power flows and lessen dependence on the main grid [5]. At the city level, synergies between EVs and metro systems in Madrid illustrate how regenerative braking energy can be captured and stored in EV batteries [6]. Further examples from Valencia emphasise that effective policy, financial incentives, and infrastructure readiness are decisive in overcoming adoption barriers [7].

These studies also show that while technical and environmental considerations are crucial, economic incentives remain the key enabler of large-scale adoption. Without clear financial benefits, such as savings from time-variable tariffs or efficient use of demand response incentives, advanced charging schemes may see limited uptake. This paper proposes a profit-aware utilisation framework for EVs in sustainable smart cities. The framework integrates profitability for EV owners with grid stability and urban sustainability goals, leveraging a strategic optimisation approach that accounts for locational and time-varying electricity prices. With this approach, the proposed models position EVs not merely as clean transportation options but also as economically sustainable energy assets within the city.

1.1. Literature Review

The role of EVs in smart cities and sustainable energy systems has been studied extensively, with growing emphasis on smart charging, bidirectional V2G operations, and route optimisation. This section reviews recent contributions in three directions: smart charging, charging and discharging with grid impacts, and route optimisation.

EV Smart Charging: Uncoordinated charging, where EVs begin charging immediately upon plug-in, raises peak demand and stresses distribution assets. Smart charging coordinates charging with electricity prices, renewable generation, or grid constraints to enhance performance. On the behavioural side, Barman et al. [8] introduced a random utility model that jointly captures travel and charging choices, offering greater realism than simplified assumptions. Operational methods focus on cost, waiting time, and travel delay; ant colony optimisation reduced both waiting times and charging expenses [9], while centralised fleet scheduling improved peak demand reduction and PV utilisation compared with uncoordinated strategies [10]. Reinforcement learning has also gained traction; Sultanuddin et al. [11] showed that Double Deep Q-learning reduced load variance by 68% relative to uncontrolled charging. Renewable integration studies further demonstrate that workplace PV-EV systems improve renewable utilisation by more than 40% under the self-consumption–sufficiency balance (SCSB) metric [12].

Charging and Discharging with Grid Impact: While smart charging primarily addresses charging load management, recent work extends to the dual role of charging and discharging. Zheng and Yao [13] proposed a multi-objective allocation method for PV-based stations, showing that orderly V2G reduces both grid stress and costs. Tian et al. [14] developed a two-stage optimisation framework for distribution systems that improves stability and efficiency. Shaheen et al. [15] demonstrated that metaheuristic methods generate cost-effective V2G schedules that also deliver ancillary services. Reinforcement learning methods, such as recurrent proximal policy optimisation with LSTM forecasting, balance supply–demand under uncertainty [16]. These approaches highlight the resilience potential of V2G, though most remain system-level rather than mobility-aware.

At the distribution scale, Abiassaf et al. [17] showed that unmanaged large-scale EV adoption leads to voltage instability, transformer overloading, and congestion, particularly with fast charging. Optimised charging integrated with renewables mitigates these issues. Urban-level studies, such as Khalid et al.’s [18] on Stockholm’s grid, show that coordinated charging reduces congestion and losses and that PV and BESS integration further enhances outcomes. Microgrid studies add stochastic realism; Iqbal et al. [19] used Markov-based scheduling to account for travel variability, achieving cost-effective V2G while preserving mobility. Reliability-focused studies caution against unmanaged charging; Roy et al. [20] found accelerated transformer degradation under high EV penetration in rural Kentucky.

Route Optimisation with Charging and Discharging: As EV deployment accelerates, charging and discharging decisions along routes are increasingly embedded into routing models. Algafri et al. [21] proposed an optimisation model for station allocation while performing G2V/V2G transactions, achieving over 85% satisfaction for charging and nearly 99% for discharging requests. Jia et al. [22] proposed a deep reinforcement learning-based route optimisation framework for fuel cell EVs considering passenger numbers. Long-distance travel studies across Germany confirmed that charging requirements increase average travel time by 8%, with poor station distribution causing delays up to 30% [23], underscoring the importance of spatial planning. Advanced formulations explicitly capture nonlinear charging: Kim et al. [24] showed that ignoring state-of-charge dependencies yields infeasible routes, while Shahkamrani et al. [25] combined route mapping with day-ahead optimisation to reduce network losses by 25%.

Overall, the literature demonstrates strong advances in smart charging, V2G scheduling, and routing optimisation. Yet, most studies consider charging or routing in isolation, overlooking the strategic joint optimisation over EV charging and discharging opportunities, the power grid’s locational and temporal price variations, and city road connections and distances along a trip. Addressing this limitation is crucial to transforming EVs into profit-aware, mobility-integrated energy assets, supporting both user incentives and urban grid stability in sustainable smart cities.

1.2. Motivations

Smart cities––owing to their advanced ICT infrastructure––have better capability to support global sustainable goals. A sustainable city requires a sustainable means of transportation. This ambition is leading towards higher penetration of electric vehicles (EVs) in our cities, in both the private and commercial sectors, putting greater burden on the existing power grid. Furthermore, this burden is localised in some parts of the grid; hence EVs need to be mobilised by road in different locations of the city to get charged. In deregulated markets, electricity tariffs fluctuate across locations and time periods, adding further complexity and opportunity to EV charging and discharging decisions. While these dynamics help balance grid demand, they create economic and operational challenges for EV owners, fleet operators, and urban planners. Furthermore, diversity of these systems (which is interdisciplinary in nature), inherent nonlinearity of the constraints, and complexly co-related governing factors that need to interact with each other during the optimisation process appropriately make it tedious to develop comprehensive models that can reflect a real-world scenario.

Although prior studies have explored smart charging, V2G-enabled discharging, and routing optimisation, most approaches treat these aspects separately, which can be noticed in the summary of the literature presented in

Table 1. A comparative analysis of all studied works and their key contributions to three systems (Y: Yes).

	EV System		CG System	PG System		Temporal Pricings	Temporal and Locational Pricings
	Smart Charging	V2G (Charging + Discharging)	Route Optimisation	Grid Burden Management	Demand Flexibility
[8]	Y				Y	Y
[9]	Y			Y		Y
[10]	Y				Y	Y
[11]	Y			Y
[12]	Y				Y
[13]		Y		Y		Y
[14]		Y				Y
[15]		Y		Y		Y
[16]	Y	Y		Y		Y
[17]	Y			Y
[18]	Y			Y
[19]	Y			Y	Y	Y
[20]	Y				Y
[21]		Y	Y			Y
[23]	Y		Y
[24]	Y		Y
[25]	Y		Y			Y
PAUM-EV	Y	Y	Y	Y	Y	Y	Y

. Some have investigated two systems, but a through modelling of all three systems has rarely been presented. Few other articles have looked into the power network burdens but have chosen to impose their will on the EV users, instead of incentivising them with the nodal and temporal pricing models to achieve the profit goals of multiple system owners. As a result, opportunities to align profitability with grid stability and sustainability remain underutilised. This gap motivates the need for an integrated, profit-aware framework that strategically combines routing with charging and discharging decisions. By leveraging locational and time-varying prices, EVs can evolve from clean transport modes into active energy assets, supporting both urban mobility and resilient power systems.

1.3. Contributions

• A joint optimisation model is curated to combine the impacts and benefits of electric vehicles, power grids, and cities’ road grid systems.
• A profit-aware EV utilisation model (PAUM-EV-M1) is developed for charging location and optimal route selection for conventional EVs.
• A profit-aware EV utilisation model (PAUM-EV-M2) is developed for charging–discharging location and optimal route selection for V2G-enabled EVs.
• We extend the PAUM-EV-M2 model (PAUM-EV-M2b) to work under the flexibility and uncertainty of demand-side resources.

2. Profit-Aware Electric Vehicle Utilisation Model (PAUM-EV)

In this section, we propose three different models in which EVs are optimally utilised along with other flexible loads to attain higher profits without imposing infeasible burdens on the grid. Here, the decision-making process is managed by the utility-owned (of affiliated) aggregator that has real-time access to various operational parameters of the city’s power distribution network. EV users participate in this process through a mobile application and provide the necessary inputs; other fixed details are collected during the application registration. Depending on available time and technical feasibility, EV users may opt between smart charging (M1) or smart charging and discharging (M2) methods. Based on the selected options and surrounding possibilities, the aggregator chooses the objective and associated constraints to obtain the solutions.

Please note that in the introductory stage, the number of EVs and associated infrastructure is very low in the city to make any significant impact on the PG or CG system. During the mature EV adoption stage, a higher number of EVs can be observed in the city, and the required infrastructure is also well developed. However, in the intermediate stage, EVs grow faster, but the required infrastructure is not sufficient. Hence, this is a very appropriate scenario in which to deploy the proposed models.

2.1. Proposed PAUM Objective Function

The objective function in Equation (6) has three operational factors (Equations (1), (2) and (5)) and two profit-related factors (Equations (3) and (4)). The three operation factors include one optional DSM factor (Equation (5)). These factors are described mathematically as follows:

$$\begin{array}{cc}\hfill f_1& =N_1\times \sum _{\alpha \in A_{\tau }}{E}_{t,\alpha }\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill f_2& =N_2\times \sum _{l\in L}\sum _{\alpha \in A_{\tau }}(R_{l,\alpha }\times L_{l,\alpha })\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} f_3& =N_3\times \sum _{\tau \in T}\sum _{\alpha \in A_{\tau }}\left(\left(C_{\tau ,\alpha }-D_{\tau ,\alpha }\right)\times \mathrm{\Delta }t\times {\lambda }_{\tau ,l}^{\mathrm{grid}}\right)\times L_{l,\alpha }\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}{f}_4& =N_4\times \sum _{\tau \in T}\sum _{\alpha \in A_{\tau }}\left(D_{\tau ,\alpha }\times \mathrm{\Delta }t\times {\lambda }_{\alpha }^{\mathrm{deg}}\right)\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill f_5& =N_5\times \sum _{\tau \in T}\sum _{b\in B}{\beta }_{\tau ,b}\hspace{1em}\hfill \end{array}$$

(5)

All factors are normalised by dividing with the respective maximum feasible values, as denoted by $N_1$ to $N_5$. $N_5$ is also used as a binary switch to turn on or off the demand management ability. These five factors are arranged in objective functions as follows:

$$\underset{PGV,CGV,EVV}{\mathrm{Minimise}}(f_1+f_2+f_3+f_4+f_5)$$

(6)

Here, $f_1$ deals with the error between an EV’s demanded energy and the aggregator’s ability to cater the same; $f_2$ deals with selecting the charger location; $f_3$ and $f_4$ deal with the energy cost payable to the grid and the battery degradation cost on EV users; finally, $f_5$ deals with the amount of demand that needs to be curtailed to optimise the overall cost and feasibility. The entire proposed PAUM model, including Equation (6), is solved for three categories of decision variables, viz. power grid-related variables (PGVs), city grid-related variables (CGVs), and EV-related variables (EVVs), as discussed in the models in the following subsections.

2.2. PAUM System Modelling

The entire PAUM framework consists of three major systems: the EV system, the CG system, and the PG system. In the following subsections, the mathematical models for each system are presented, and the system interconnections are explained. In the later sections, their curated applications are elaborated. A conceptual diagram of coordination of all three systems is illustrated in Figure 1.

2.2.1. EV System Model

If $X_{\alpha }$ is the energy level of EV $\alpha$, and $C_{\alpha }$ and $D_{\alpha }$ are the associated charging and discharging powers in the EV, respectively, then decisions on all three variables for time t and for the multiple EV system $\alpha \in A_t$ can be evaluated by solving the following mathematical models:

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{X}_{\alpha ,t+1:\left|T\right|}& =X_{\alpha ,t:\left|T\right|-1}+\left(C_{\alpha ,t:\left|T\right|-1}\times {\eta }^C-D_{\alpha ,t:\left|T\right|-1}\times {\displaystyle \frac{1}{{\eta }^D}}\right)\times \mathrm{\Delta }t,\alpha \in A_t\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill X_{\alpha ,t+1:\left|T\right|}& \le X_{\alpha }^{max},\mathrm{and}{X}_{\alpha ,t+1:\left|T\right|}\ge X_{\alpha }^{min},\alpha \in A_t\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill C_{\alpha ,t+1:\left|T\right|}& \le C_{\alpha }^{max},\mathrm{and}{C}_{\alpha ,t+1:\left|T\right|}\ge C_{\alpha }^{min},\alpha \in A_t\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill D_{\alpha ,t+1:\left|T\right|}& \le D_{\alpha }^{max},\mathrm{and}{D}_{\alpha ,t+1:\left|T\right|}\ge D_{\alpha }^{min},\alpha \in A_t\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill X_{\alpha ,t^{\mathrm{arr}}}& =X_{\alpha }^{\mathrm{initial}},\alpha \in A_t\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill X_{\alpha ,t^{\mathrm{dep}}}& =X_{\alpha }^{\mathrm{desired}}-E_{\alpha },\alpha \in A_t\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill E_{{\alpha }^{\prime },t-1}& =0,{\alpha }^{\prime }\in A_{t-1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill X_{\alpha ,t+1:\left|T\right|}& \le X_{\alpha }^{\mathrm{desired},\mathrm{decision}},\alpha \in A_t\hspace{1em}\hspace{1em}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{X}_{\alpha ,1:t-1}& =X_{\alpha ,t-1}^{\mathrm{decision}},C_{\alpha ,1:t-1}=C_{\alpha ,t-1}^{\mathrm{decision}},\mathrm{and}{D}_{\alpha ,1:t-1}=D_{\alpha ,t-1}^{\mathrm{decision}},\alpha \in A_t\hfill \end{array}$$

(15)

Here, the battery capacity and SoC of the EVs are not explicitly modelled but are rather internalised here as energy levels of the EVs. If the model converges, we move to the next time instance $t:=t+1$. Then, the obtained decisions are stored for the next t as $X_{\alpha }^{\mathrm{desired},\mathrm{decision}}:=X_{\alpha }^{\mathrm{desired}}-E_{\alpha }$, $X_{\alpha ,t-1}^{\mathrm{decision}}:=X_{\alpha ,1:t-1}$, $C_{\alpha ,t-1}^{\mathrm{decision}}:=C_{\alpha ,1:t-1}$, and $D_{\alpha ,t-1}^{\mathrm{decision}}:=D_{\alpha ,1:t-1}$.

In this EV system model, Equation (7) establishes the relationship between the battery energy level and the charging and discharging powers. Equations (8)–(10) impose limitations on the energy level, as well as the charging and discharging powers, respectively. Equations (11) and (12) ensure the timely fulfilment of the energy needs. The energy fulfilment errors are evaluated by Equations (12)–(14). Finally, the constraints specified in Equation (15) are utilised for model predictive control (MPC) enforcement.

2.2.2. PG System Model

The power grid (PG) system is modelled using bus voltage (V) limits, angle ($\theta$) limits, real power generation ($P^G$) limits, load ($P^L$) limits, and lines flow ($P^{\mathrm{F}}$) limits. Here, the power flow equation, along with parameter limitations, is mathematically expressed in its polar form, as follows:

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{P}_b& =|V_b|\times \sum _{bb=1,bb\ne b}^{NB}|V_{bb}|\times |Y_{b,bb}|\times cos({\theta }_b-{\theta }_{bb}-{\varphi }_{b,bb}),\forall b\in B\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill P_{b,bb}^F& \le P_{b,bb}^{\max },\mathrm{and}{P}_{b,bb}^F\ge P_{b,bb}^{\min },\forall b,bb\in B\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill V_b& \le V_b^{\max },\mathrm{and}{V}_b\ge V_b^{\min },\forall b\in B\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\theta }_b& \le {\theta }_b^{\max },\mathrm{and}{\theta }_b\ge {\theta }_b^{\min },\forall b\in B\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}{P}_b^G& \le P_b^{\mathrm{G},\max },\mathrm{and}{P}_b^G\ge P_b^{\mathrm{G},\min },\forall b\in B\hfill \end{array}$$

(20)

Here, the power balance equations for fixed non-elastic PG bus loads and flexible EV loads are modelled as follows:

$$\sum _{b\in B}{P}_b^G-P_b^L-P_b^{Loss}-\sum _{\alpha \in A_t}(C_{\alpha ,b}-D_{\alpha ,b})\times \mathrm{\Delta }t\times L_{b,\alpha }=0$$

(21)

Along with the charging (C) and discharging (D) of EVs, this model also governs the decision of charging station location (L) on the grid. This $L_b$ can be made 0 for the buses without any charging station connection. The loss term $P_b^{Loss}$ for bus b can be evaluated by $P_{b,bb}^F+P_{bb,b}^F,\forall bb\in B$. Note that the impact of reactive power is not taken into account, as we are focusing on energy-related evaluations.

2.2.3. CG System Model

In this study, the city road infrastructure is modelled as a grid-like graph ${\mathcal{G}}^{\mathrm{CG}}=\{{\mathcal{V}}^{\mathrm{CG}},{\mathcal{E}}^{\mathrm{CG}}\}$, where each junction is equated with vertices and connecting roads are equated with edges. The distance of the road is presented on ${\mathcal{G}}^{\mathrm{CG}}$ as weights on the edges. Please refer to Section 3.1 for better understanding. In the graph, the minimum distance between source and destination nodes ${\mathcal{D}}_{v,vv}$ ($\forall v,vv\in {\mathcal{V}}^{\mathrm{CG}}$ and $v\ne vv$) is evaluated using the Dijkstra algorithm [26]. The decision vector L with two identical indices l and b is used to map the presence of EV charging stations on both the CG and the PG. Furthermore, the mapping of the PG onto the CG is also supported by the variables $\beta$, as discussed in Section 2.5.

The primary constraint in the CG system deals with $\mu$, i.e., the time required to travel over the shortest distance ${\mathcal{D}}_{v,vv}$ between two points v and $vv$. These are modelled as

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mu }_{\alpha ,l}& ={\mathcal{D}}_{v{0}_{\alpha },l}\times L_{l,\alpha }\times f^{DT}+f^{Trf}(v{0}_{\alpha },l),\alpha \in A_t,l\in \left\{\mathrm{EVCS}\right\}\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill tR& =t_{\alpha }^{\mathrm{arr}}+{\mu }_{\alpha ,l},\alpha \in A_t,l\in \left\{\mathrm{EVCS}\right\}\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill X_{\alpha ,tR}& =X_{\alpha }^{\mathrm{initial}},\alpha \in A_t,l\in \left\{\mathrm{EVCS}\right\}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\mathcal{D}}_{v{0}_{\alpha },l}& \times L_{l,\alpha }\le min\left({\mathcal{D}}_{v{0}_{\alpha },l}\right)+ϵ,\alpha \in A_t\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill L_{l,{\alpha }^{\prime }}& =L_{{\alpha }^{\prime }}^{\mathrm{decision}},{\alpha }^{\prime }\in A_{t-1}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(26)

Here, $v{0}_{\alpha }$ indicates the nearest node on the city grid from where EV user $\alpha$ has started the M.App. $f^{DT}$ is a factor that converts the distance into kilometres to approximate time in hours, neglecting traffic congestion. $f^{Trf}(v{0}_{\alpha },l)$ is a function that evaluates the traffic flow (in terms of time consumption) from starting location $v{0}_{\alpha }$ to destination location l, including the predicted temporal variations in traffic. In the CG system, Equation (22) defines the parameter $\mu$. Equations (23) and (24) implement the reach time $\mu$ feasibility onto the EV’s initial energy level and charging start time. Equation (25) imposes the minimum distance restriction on the decision-making process by using $ϵ$ to avoid assignment of a distant charger. Finally, Equation (26) enforces MPC and commitment continuity on the EVCS location selection variables.

2.3. PAUM-EV-M1: Profit-Aware Charging Location and Optimal Route Selection

In this model (M1), only the charging requirements of EVs are considered, making it suitable for a city without provisions for EV discharging or for EVs that are not designed to discharge. When an EV user starts the M.App, the aggregator collects the current battery level and current location of the EV in the city grid. Then, the EV user manually fills out the M.App with their final destination on the city grid and desired time to reach it. The aggregator has prior knowledge on the CG map, the PG map, CG—PG connecting maps, and the location of EV charging stations on those maps. Now, the aggregator solves the PAUM-EV optimisation models as per the process flow illustrated in Figure 2. Note that the governing constraint set $C1$—$C2$ is different for each model and is discussed in the sections below.

To consider only charging but not discharging, we slightly update Equations (7), (9), and (10) as follows:

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}{X}_{\alpha ,t+1:\left|T\right|}& =X_{\alpha ,t:\left|T\right|-1}+\left(C_{\alpha ,t:\left|T\right|-1}\times {\eta }^C\right)\times \mathrm{\Delta }t,\alpha \in A_t\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill C_{\alpha ,t+1:\left|T\right|}& \le C_{\alpha }^{max},\mathrm{and},C_{\alpha ,t+1:\left|T\right|}\ge 0,\alpha \in A_t\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill D_{\alpha ,t+1:\left|T\right|}& =0,\alpha \in A_t\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(29)

Here, the energy level inside the EV does not change due to discharging and does not go to a negative value; it only keeps increasing until departure. Hence, the key goal becomes to find the appropriate EVCS location and the charging schedule with other EV, CG, and PG system constraints being satisfied. To do so, we minimise the objective function in Equation (6) and solve the constraints in Equations (8), (11)–(15), (16)–(21), (22)–(26), and (27)–(29).

2.4. PAUM-EV-M2: Profit-Aware Charging–Discharging Location and Optimal Route Selection

In a favourable scenario where the city has EV discharging provision and EVs have V2G-enabled battery management systems, model M2 can be deployed. Here, the EV system model in Equations (7)–(15) is fully restored, the discharging of EVs can go from 0 to its minimum ($D^{\max }$) value, and similarly, charging can span freely between 0 and $C^{\max }$, as presented below:

$$\begin{array}{cc}\hfill X_{\alpha ,t+1:\left|T\right|}& =X_{\alpha ,t:\left|T\right|-1}+\left(C_{\alpha ,t:\left|T\right|-1}\times {\eta }^C-D_{\alpha ,t:\left|T\right|-1}\times {\displaystyle \frac{1}{{\eta }^D}}\right)\times \mathrm{\Delta }t,\alpha \in A_t\hfill \\ \hfill C_{\alpha ,t+1:\left|T\right|}& \le C_{\alpha }^{max},\mathrm{and},C_{\alpha ,t+1:\left|T\right|}\ge 0,\alpha \in A_t\hfill \\ \hfill D_{\alpha ,t+1:\left|T\right|}& \le D_{\alpha }^{max},\mathrm{and},D_{\alpha ,t+1:\left|T\right|}\ge 0,\alpha \in A_t\hfill \end{array}$$

Furthermore, unlike charging, the discharging efficiency ${\eta }^D$ gets divided, as EVs have to discharge more, i.e., $D_{\alpha ,t}/{\eta }^D$, for the grid to get a certain amount of energy $D_{\alpha ,t}$. To operate in model M2, the aggregator minimises Equation (6) while adhering to the EV, PG, and CG system models and limitations in Equations (7)–(15), (16)–(21) and (22)–(26).

2.5. PAUM-EV-M2b: Extension of PAUM-EV-M2 Considering the Demand Flexibility

Finally, having proposed a method to deal with EV charging and discharging in PAUM-EV-M2, we extended the model in PG systems to appropriately manage the flexible loads in the city, as follows:

$$\sum _{b\in B}{P}_{b,\tau }^G-P_{b,\tau }^L-P_{b,\tau }^{L,flex}\times {\beta }_{b,\tau }-\sum _{\alpha \in A_t}(C_{\alpha ,b,\tau }-D_{\alpha ,b,\tau })\times \mathrm{\Delta }t\times L_{b,\alpha ,\tau }=0,\forall \tau \in T$$

(30)

Here, the demand management variable $\beta$ is continuous and can be defined as

$${\beta }_{b,\tau }\le 1,\mathrm{and}{\beta }_{b,\tau }\ge 0$$

(31)

The electrical loads distributed across the city grid usually have the connections from the nearest electrical bus, and any deviation can be mapped on $P_b^{L,flex}$ and chosen with $\beta$.

2.6. Utility’s Pricing Strategy

If the urban area is subjected to ToU or flat tariffs, which is usually not responsive to the generation and demand mismatch, then an additional risk factor (say $\varphi$) must be included to make the pricing structure locationally variant.

$${\lambda }_b^{SG}={\lambda }^{SG}+\sum _{t\in T}\left({\varphi }_{\tau ,b}+{\psi }_{\tau ,b}\right)$$

(32)

where

$${\varphi }_{\tau ,b}=P_{\tau ,b}-{\displaystyle \frac{1}{\left|B\right|}}\times \sum _{b\in B}\sum _{t\in T}{P}_{\tau ,b}$$

(33)

Here, ${\lambda }^{SG}$ denotes the available prices. The factor $\psi$ is updated by the operator to compensate for long-term grid security and resilience challenges (exact evaluation is beyond the scope of this work). Further, if any nodal price is found to be negative, an offset value may also be added by deriving the absolute of minimum of ${\lambda }_b^{SG}$, $\forall b\in B$.

This mechanism prepares different ${\lambda }_b^{SG}$ for different buses $b\in B$, which now represent the true imbalance in the grid. In this paper, market clearing process is not the primary focus; in a real-world scenario, Equation (32) can be replaced with market-cleared LMP prices.

3. Results

3.1. Simulation Set-Up

The primary focus of this research study is to obtain a holistic and optimal solution through the joint optimisation of the EV system, the power grid system, and the city road grid system. Information on these three systems are collected as follows:

• The EV arrival and departure time and energy level, battery capacity, and efficiency information are obtained from the dataset referred in [27,28].
• Power grid system information, such as power generation, non-elastic load, and radial distribution line topology, is collected from the IEEE 33 Bus system data [28,29].
• City road grid information is derived from a check-board-pattern city, like Manhattan in New York, where each junction (graph vertices) is approximately equidistantly placed. In the present study, the graph edge weight is assumed to be 5km uniformly, as shown in Figure 3. It can easily be extended to non-uniform weights as shown in Figure 4.

The effectiveness of the proposed profit-aware EV utilisation model is tested on a workstation with 16GB of RAM and an Intel-i5 11th Gen 2.40 GHz processor by using the MATLAB 2024b programming platform and the Gurobi 10.0.1 optimisation solver.

The CG and PG interconnection mapping information is presented in a supplementary sheet in [28]. Figure 5 presents the equidistantly installed EV charging stations in the city, which are used in the present case studies. In Figure 6, the arrival and departure locations of the 30 EVs (mentioned as per the sequence in [27]) are depicted. Figure 7 illustrates the PG system and its mapping onto the CG system. It highlights the jurisdiction of the aggregator on the IEEE 33 bus system and shows how the lines and buses are spread across the CG system covering the load centres in the city. In this interconnected mapping, it is assumed that each bus in the PG system supplies all electrical loads in the CG squares surrounding it, extending to the road on the right and the road above. The colour bar indicates the flexible loads aggregated at each bus. Note that civil infrastructure issues are not considered in this research study, and the traffic characteristic is assumed to be static and sufficient.

In all the case studies, the real-time prices (RTPs) from Singapore’s wholesale market, based on the NEMS database’s October 2024 data [30], are used. Since the available sample rate in NEMS is 30 min, we consider two-day information to portray a 15 min interval for 24 h (96 samples). These RTPs (${\lambda }^{SG}$) do not reflect the generation and demand mismatch in our case study; rather they are cleared in the market for very different bids. Hence, to reflect the actual load imbalance in the nodes in the city, the risk factor pricing $\varphi$ in (32) is employed with ${\lambda }^{SG}$ for individual nodes in the PG. Note that the long-term factor $\psi$ is neglected here.

Utilising the simulation details presented above, the proposed model is validated on the following key points:

• Case-1: Charging schedule of EV, EVCS location, and route selection: It emphasises the performance of PAUM-EV-M1 with smart charging capability. It also shows the ability of the proposed model to appropriately optimise the route across the cities. Figure 8 of the revised manuscript illustrates the performance of the model in the city road grid system, and Figure 9 illustrates the EV users’ financial benefits.
• Case-2: Charging and discharging schedules of EVs: It emphasises the performance of the PAUM-EV-M2 model with smart charging and discharging capability. It is illustrated in Figure 10 of the revised manuscript along with the financial benefits of EV users. It also shows the ability of the model to support the power grid by reducing stress by appropriately mobilising EV loads.
• Case-3: Charging and discharging schedules along with demand management: It presents the performance of the PAUM-EV-M2b model with smart charging and discharging and appropriate management of the flexible demand loads. Table 1 presents how the flexible loads absorb the impact of other systems on the power grid system.

Furthermore, in these case studies, we study various parameters in the models to verify the effectiveness against (a) variations in the arrival and departure destinations of the EVs, (b) variations in initial and desired energy levels among the EVs, (c) different price profiles that vary across the locations and across the time horizon, (d) a state-of-the-art IEEE 33 bus benchmark system (for generation and demand parameters), and (e) two different types of road organisation, i.e., grid-like pattern and non-grid pattern.

3.2. Case-1: Charging Schedule of EVs, EVCS Location, and Route Selection

In this case study, we primarily assess the performance of PAUM-EV-M1 in the three different systems. In Figure 8 (left), the nodal and temporal price variation including the imbalance factor (as discussed in Equation (32)) is presented as a colour map. Its impact on driver route selection is illustrated in Figure 8 (right). As can be noticed, traditional drivers pick the nearest EVCS location and the route (the green line) that is not far from their envisaged travelling route (the blue line), despite their higher nodal prices. On the contrary, when we consider PAUM-EV-M1 at its full potential along with the restructured nodal prices, the profits are visible. EV optimally choose a slightly longer route (the brown line) and a distant EVCS, biased by the lower nodal prices, but manage to obtain a significant profit compared with the traditional strategy, as shown in Figure 9.

In Figure 9, four items are presented, namely, cost, distance, time, and schedule, and are compared between the traditional charging strategy (without the profit consideration) and the proposed PAUM-EV-M1 strategy. As can be noticed from Figure 9a,b, the financial benefits of EV owners are significantly higher with proposed model, approximately USD 260. On the contrary, the increases in travel distance (Figure 9c,d) and time (Figure 9e,f) are around 90 km and 17 min, respectively. These values grow as new EVs keep arriving to the charging station. The combined distance and combined travel time can be treated as a metric to compare the performance of the PAUM in CG system optimisation.

Other than the selection of the optimal EVCS nodes, another major reason behind this financial benefit can be understood by analysing the temporal variation in nodal prices and the charging schedules in Figure 9g,h. In contrast to the traditional charging strategy, where an EV gets the required charge continuously as soon as it arrives to the station, in the proposed model, charging is performed strategically when price is low, and the EV stays idle when prices are high. But, in this process of cost reduction, the model does not ignore the EV’s energy requirements. Furthermore, the initial flat-pricing periods indicate the moment when the EV initiate the M.App vs. the moment when EV charging initiated. Some EVs exhibit partial charging owing to their small stay duration.

3.3. Case-2: Charging and Discharging Schedules of EVs

In this case study, we assess the performance of the PAUM-EV-M2 strategy, which focuses on both strategic charging and discharging. It is also compared against the PAUM-EV-M1 strategy and traditional charging strategy capable of performing discharging to minimise the overall objective cost function. The obtained results are presented in Figure 10 and include four terms, i.e., cost, distance, arrival time, and schedules.

As can be noticed in Figure 10a,b, the financial benefit with the proposed PAUM-EV-M2 strategy is USD 390 higher than the traditional strategy. Furthermore, the benefit is better than just smartly charging with M1; in this case study, it is found to be around USD 80. The key reason behind this better performance is the strategic decision to discharge back to the grid (V2G process) when the price is high, which is visible in Figure 10g,h. In the traditional strategy, discharging is performed to achieve other system objectives but ignores the sole financial benefit of EV users. On the other hand, by analysing the charging and discharging schedules of EVs with temporal price variations, we can notice that energy discharging is performed during the high-price time, and charging is performed during low-price hours.

Moreover, similar to M1, the distance covered by EVs (Figure 10c,d) and arrival time of EVs (Figure 10e,f) are slightly higher in M2 compared with the traditional strategy; however, they remain nearly equal in M2 when compared with M1. Hence, looking at the financial benefits, it is advisable for the EV users to opt for the PAUM-EV-M2 model if the charging station and the EV has the technical feasibility to discharge back.

Impact on power grid: Furthermore, the effect of the proposed PAUM-EV-M2 model on the grid is studied and compared against the naive distance-focused EV charging model, as presented in Figure 11 and Figure 12, respectively. Here, the reduction in the demand burden on each PG bus is clearly visible in the proposed PAUM-EV-M2 model, compared with the naive one. It is also true when we divide the burden across the temporal horizon: the spike imposed on the grid is much lower in the proposed case compared to the naive one. The key reason for this deviation is peak demand-focused price calculation by the utility and the proposed profit-seeking EV charging and discharging model and profit-seeking charger allocation strategy. Therefore, from these case studies, it can be established that the proposed model is not only profitable for EV users but also beneficial for the grid operator.

3.4. Case-3: Charging and Discharging Schedules Along with Demand Management

Finally, in this case study, we analyse the performance of the extension of the PAUM-EV-M2 model (the M2b model) by incorporating the impact of flexible demand management. The impact is presented in

Table 2. Impact of flexible demand management (FDM) on aggregated non-EV loads on each bus.

Bus No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14
No FDM	0	9.6	8.64	11.52	5.76	5.76	19.2	19.2	5.76	5.76	4.32	5.76	5.76	11.52
FDM	0	9.3566	5.8518	9.1929	4.2996	4.4524	17.86	18.281	4.2345	4.2277	1.8671	3.3755	3.827	10.384
Bus No.	15	16	17	18	19	20	21	22	23	24	25	26	27	28
No FDM	5.76	5.76	5.76	8.64	8.64	8.64	8.64	8.64	8.64	40.32	40.32	5.76	5.76	5.76
FDM	3.3253	4.1615	4.7078	5.823	6.0122	6.9895	6.7726	6.8789	8.0168	39.416	38.907	5.0685	3.2271	5.1757
Bus No.	29	30	31	32	33
No FDM	11.52	19.2	14.4	20.16	5.76
FDM	10.842	18.688	14.4	20.16	5.76

(Note: FDM stands for flexible demand management).

. Since EV utilisation is also burdened by the power grid system constraints, sometimes, EV systems may experience additional adjustments in charging and discharging schedules to support this. These constraints are essential to grid stability and cannot be relaxed. Hence, to reduce the extra burden on the EV system, flexible demands in the power grid system are optimally reduced, as can be noticed in the table. Note that in this research study, very approximated power grid constraints are utilised, which can be extended in the future for more detailed modelling.

Hence, if it is technically feasible in the power grid system to initiate flexible load management, combining this with the EV utilisation model can be significantly beneficial for the power grid system as well as the EV system and will contribute towards the sustainability goals of a city.

3.5. Case-4: Non-Grid-Pattern City

The PAUM-EV-M2b framework is repeated again for a non-grid-pattern city (NPC), i.e., Singapore, with varying distances between the city road junctions. This additional information on the CG system is available in [28]. The new locations allocated to each arriving EV are presented in

Table 3. Charger locations allocated by PAUM-EV for grid-pattern city (GPC; e.g., Manhattan) and non-grid-pattern city (NPC; e.g., Singapore) against respective EV indices. Highlighted cells indicate the deviations in EVCS allocation due to changes in route distances.

EV index	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
GPC	15	15	9	9	15	2	2	9	8	8	15	9	15	8	8	15	13	15	15	9	8	15	15	15	15	15	12	9	2	3
NPC	15	15	9	9	15	2	9	9	8	9	15	9	15	2	8	9	13	9	15	9	8	15	15	9	15	15	15	9	2	2

.

As can be noticed, the locations allocated by the PAUM for a grid-pattern city (GPC) are different from an NPC in various instances. Since this is a multi-objective optimisation problem, and the outcome depends on various other factors, like grid prices and EV energy requirements; variation in distance in the CG system alone cannot affect the outcome entirely. However, allocation for eight EVs is found to be slightly changed, keeping the EV and PG system outcomes intact. This proves the effectiveness of the proposed PAUM-EV model for both GPC and NPC structures.

4. Discussion

• Computational complexity:
  In this model, five primary decision variables are used, as explained in Section 2.1. Here, the latter two are associated with charger location and demand management. However, the former three are associated with EVs, and their dimensions grow with the arrival of the new EVs. The increase in dimension of the decision variable matrix with the arrival of new EVs leads to an increase in computational complexity. This can be noticed in Figure 13, where execution time is plotted against each instance of EV arrival at any charging station in the city. Note that this rise in computation time is also due to PG constraint nonlinearity, the increase in the number of constraints with the arrival of new EVs, and the retention of solver cache. Without PG system constraints, each instance is solved within 10 s. To improve this time complexity, better computation hardware should be utilised by the utility.
• Step size/time interval:
  The impact of the step size $\mathrm{\Delta }t$ is quite critical. If we make $\mathrm{\Delta }t$ too large, the error between estimated SoC values and actual SoC values will be enormous; on the other hand, too small a $\mathrm{\Delta }t$ will increase the computational burden. Hence, a trade-off is always necessary. Depending on the available computational capacity at the utility and the requirement of fast decisions, the step size should be selected by the operator.
• EV penetration stages:
  The performance of the proposed joint optimisation model PAUM-EV may vary in accordance with various stages of EV penetration.
  At the introductory level, with few EVs on the road, this proposal can only serve EV users economically. At the maturity level, the presence of a large number of EVs can significantly impact grid stability; but the availability of power lines with sufficient carrying capacity may not affect the locational prices. However, performing joint optimisation at this level by providing an appropriate incentive will be financially beneficial for both the grid and EV users in critical scenarios.
  Based on our case studies, the following conclusions are drawn:
  • During the introductory stage, the optimisation model works well and caters for the requirement of all EVs ($30\times 50\mathrm{kWh}$).
  • At the intermediate level, if a 10% increase in EV load is taken, then the model requires a higher generation limit to converge without curtailing too much demand.
  • However, at the mature level, a 100% increase in the present EV load requires a significant change in the network parameters of the PG system to allow the joint optimisation models to converge.
• PV penetration and grid resilience:
  High penetration of intermittent renewables and peak load hours, if not handled timely, can deteriorate grid resilience. Dispatchable conventional generators and appropriate mobilisation of EVs with its flexible storage capacity can unlock a potential solution for this challenge (which is modelled in the PG system). Since the utility, with its enormous number of data, can forecast near-real-time events, it can easily evaluate the second term in Equation (32) (which is governed by power mismatch) and set the locational prices to lure the EVs accordingly. Furthermore, the factor ${\psi }_{tau,b}$ can also be updated to address any missing reserve issue if the LMP is beyond the utility’s immediate control.
• Policy requirements:
  This PAUM framework requires some policy-level support for its implementation. If the policy does not allow any variation in price or incentive, joint optimisation may not work as expected. Furthermore, for regulated market structure, the provision of additional incentives to EV users should be considered in the policy so that locational variations can be realised.
• Applications:
  Each of the three PAUM-EV models is designed for a specific scenario and task. The core goal is to develop a joint optimisation model for the EV, PG, and CG systems that can easily serve the interests of EV users as well as the electric utility. With traditional distance-based EV charging, power lines may experience higher stress at certain locations, which can lead to instability, and EV users may be forced to pay higher prices due to the LMP pricing structure. If the PAUM-EV models are deployed, they can suggest the least expensive location, which will indirectly help the grid with a balanced burden across the nodes.

5. Conclusions and Future Directions

In this paper, three profit-aware EC utilisation models are proposed to enhance the challenges in sustainability goals in cities. In this context, a joint optimisation strategy is developed for three different inter-related systems, namely, the EV system, the power grid system, and the city grid system. From several case studies on a combination of real-world and synthetic datasets, the following key conclusions are observed:

• Both M1 and M2 can yield more profit for EVs compared with their traditional counterpart, USD 260 and USD 390 in this case, while performing a minimum re-routing of the EVs. M2 is capable of obtaining better financial benefits than M1 (around USD 80), but it is subjected to technical feasibility at the EV and EVCS end.
• With the enhanced model, M2b, flexible demand management can further contribute towards reducing the 0–3 kW burden on the power grid across the city by optimally balancing the flexible loads subject to EV charging burden and price structure.
• In contrast to the traditional EV charging strategy, the proposed M1 and M2 models focus on both aspects of price structure in a city, namely, nodal variations and temporal variations. This makes the proposed models more profit-oriented compared with the traditional model, more grid requirement-aware, and hence, a better alternative to accomplish the sustainability goals of a city.
• The model is developed on the core assumption of market deregulation. If the market is regulated, policy must be designed so as to enable the operator to provide additional incentives to EV users. The availability of the charging infrastructure is another challenge; if enough options are not present, joint optimisation may not work appropriately for the city road grid system. If policy does not allow any variation in price as an incentive and enforces ToU-like pricing, then the PG system model may not behave as expected.

The proposed framework serves as a foundational tool for several research directions essential in present-day smart cities involving electric vehicle, power grid, and city road grid systems. A brief discussion on these directions is presented below:

• An extensive study will be carried out to understand the proposed model’s ability to lower carbon emissions in cities in various critical scenarios.
• The work will be extended to address macro-level grid challenges like energy security, baseload management, and infrastructure planning.
• Analyses will be carried out to understand societal constraints on flexible demand management.
• Extensive study will be conducted on different stages of PV and EV penetration in cities, equitable access, the impacts of traffic conditions, carbon caps, grid congestion forecasts, and generation mix conditions.
• The proposed model can be extended to work with fuel cell electric vehicles alongside battery electric vehicles for overall system-level enhancements.