Local and Global Optimization Methods for Power System Models: A Case Study on the Optimal Charging and Discharging Scheduling of Vehicle-to-Grid (V2G) Systems ^†

Abstract

Optimal scheduling of charging and discharging in V2G systems constitutes a highly complex optimization problem due to its nonlinear, mixed-integer, and multi-objective nature. This article proposes a hybrid local-global optimization methodology for scheduling charging/discharging processes in the context of Vehicle-to-Grid (V2G). The proposed methodology integrates a global convex optimization stage, which guarantees feasibility and optimization with respect to the constraints of the entire system (charging limits and energy cost minimization), with a local optimization stage, which adjusts the charging schedules of each vehicle based on personalized constraints (individual battery status and user preferences). The methodology explicitly incorporates scalability for short-term scheduling horizons relevant to daily and intraday operation. A comparative analysis is conducted across local and global methods under an operating scenario.

1. Introduction

In electrical power systems, optimization methods are an important tool for solving complex problems, planning, energy dispatch, and real-time operation, based on decision-making capabilities under multiple technical and economic constraints. Currently, the growing integration of electric vehicles (EVs) and the development of Vehicle-to-Grid (V2G) technology are leading to opportunities for optimization within the interaction between these transportation systems and the electrical distribution grid. In this context, when analyzing these new loads, it should be considered that electric vehicles no longer operate solely as mobile loads but rather become a distributed element or source capable of injecting energy, thus facilitating the balance of supply and demand, in addition to the possibility of integration with renewable generation systems and supporting ancillary services [1]. The possibility of different EVs being connected to the grid and their impact on the grid [2], regarding availability and energy needs, requires a coordinated process for optimal energy consumption by the grid [1]. In this sense, the optimal planning of the charging and discharging process under V2G technology arises, which becomes a high-dimensional optimization problem, considering the multiple objectives that can be set: minimizing costs, extending battery life, and ensuring user needs are met [3].

When considering the optimization of this V2G interaction process, different methods and schemes have been presented in the literature. For instance, the use of metaheuristic approaches to schedule optimal charging and discharging in V2G, applying algorithms such as Differential Evolution (DE), Particle Swarm Optimization (PSO), Whale Optimization Algorithm (WOA), and Grey Wolf Optimizer (GWO), based on the convergence of cost reduction [4,5,6,7,8]. In addition, solutions have been developed based on the use of multi-objective optimization methods, which focus on balancing revenue generation through the use of V2G and also consider aspects such as battery degradation in plug-in hybrid vehicles, for example [9].

Optimization problems focus on minimizing costs and battery life; however, the problem of how to optimally schedule the charging and discharging of EVs in a V2G scheme presents a challenge, considering the objectives related to minimizing operating costs, reducing the impact on battery life, and, as a complementary point, complying with the technical restrictions of the grid. Among the challenges related to solving this type of optimization problem is the high computational complexity, considering the presence of nonlinear models, mixed variables, and multiple objectives, as well as being designed under conditions of uncertainty in variables such as demand and generation. In this way, the idea of global and local optimization methods emerges. In the case of global methods, they are able to explore broad search spaces, as well as find solutions close to the global optimum given in multimodal problems. However, they require more computing time and are dependent on parameters [10,11,12].

In the case of local optimization, greater convergence speed and accuracy are obtained in well-conditioned environments, although there are limitations in terms of escaping local optima in non-convex problems [13,14].

Based on the limitations and advantages of each of these methods, the opportunity arises to define a coordinated hybrid method, combining global exploration with local refinement, in order to achieve a balance between solution quality and computational efficiency.

The idea of generating a hybrid model is essential in order to maximize the benefits of V2G systems without compromising technical and economic sustainability. The relevance of this contribution lies in the ability to develop a hybrid optimization scheme that can adapt to different grid configurations and time scales.

The integration of these optimization methods can be carried out through a sequential process: the global method provides aggregate-feasible allocations, which are adjusted locally using convex programming so that individual requirements, such as state of charge (SOC), can be guaranteed. Compared to heuristic approaches, convex optimization guarantees scalability, time convergence, and feasibility, making it suitable for the incorporation of large fleets of electric vehicles with strict operational constraints.

This article proposes the design, implementation, and evaluation of global and local optimization methods for the optimal scheduling of the charging and discharging process of V2G systems.

2. Vehicle-to-Grid (V2G) Schemes

V2G schemes offer the possibility of bi-directional energy transfer between an electric vehicle and the grid [15]. In this scheme, there is an interaction of energy and information between the electric vehicles and the distribution grid through the charging system (Figure 1). From V2G technology, electric vehicle batteries have the capacity not only to store energy but also to supply energy to the grid, storing surplus energy to maintain grid balance by connecting to the grid. Within the V2G scheme, electric vehicles can be considered as flexible loads, rather than a typical load that consumes only electrical energy [16].

3. Charging and Discharging Process of Electric Vehicles

When discussing strategies for charging/discharging electric vehicles, it is categorized into two main groups: Non-Coordinated Strategies and Coordinated Strategies.

• Uncoordinated Strategies: This strategy is very commonly used and involves a scheme that does not take scheduling into account. As it is a process that does not consider coordination, it does not make use of optimization techniques or price tracking mechanisms. It can be related to a purely random process, simply taking into account the connection of the electric vehicle to the grid [17].
• Coordinated Strategies: established for the case of a single EV or for a fleet of vehicles whose charging/discharging process occurs in a coordinated manner, using a scheduling scheme, based on the use of optimization techniques and the monitoring of pricing mechanisms [17,18].

This paper considers the use of the Coordinated Strategy, based on Figure 2.

4. Global and Local Optimization Methods

Referring to the global optimization problem, the focus is on finding a solution to a non-convex mathematical programming problem. We consider the definition of a “global” solution, as opposed to a local solution; that is, where a point is considered at which the objective function reaches an optimal value in relation to the entire search domain.

On the other hand, in the case of the solution of the local optimization problem, it is called optimal in relation to a given environment or range.

The relation with the convex-type solution is given by considering that the objective function and/or the feasible region are convex. In convex programming problems, every local optimum is also a global optimum, so that any method that solves a convex problem locally also solves it globally.

5. Convex Method Algorithm for Solving Global and Local Optimization

The definition of a convex function is based on the notion that it is generated if and only if it is convex when the restriction is given to any line that intersects its domain. An example of a convex function is shown in Figure 3.

Mathematically, a convex function is defined as a function that satisfies $f:R^n\to R$ and is a convex set for the domain of f or is a convex set for all x, $y\in domf$, and $\theta$ with a value between $0\le \theta \le 1$:

$$f(\theta x+(1-\theta \left)y\right)\le \theta f\left(x\right)+(1-\theta )f\left(y\right)$$

(1)

Performing a geometric analysis, the inequality behaves like a line segment between $(x,f(x\left)\right)$ y $(y,f(y\left)\right)$, which is a chord (Figure 3) from x to y that lies above the graph of f.

Convex optimization problems are based on the following formulation [20]:

$$minimize{f}_0\left(x\right)$$

(2)

$$subjectto{f}_i\left(x\right)\le b_i,i=1,...,m$$

(3)

where the functions $f_0,...,f_m:R^n\to R$ are convex, and satisfy the following linea- rity relation:

$$f_i(\alpha x+\beta y)\le \alpha f_i\left(x\right)+\beta f_i\left(y\right)$$

(4)

For all $x,y\in R^n$, and any $\alpha ,\beta \in R$ con $\alpha +\beta =1$, y $\alpha \ge 0$, $\beta \ge 0$.

6. Materials and Methods

The proposed methodology is based on a hybrid optimization model using local and global methods applied to the optimization of the charging and discharging process of electric vehicles connected to the grid or V2G systems. To better understand the proposal, Figure 4 provides a general overview of the optimization process to be developed, as well as a description of how each of the methods relates to the others.

From Figure 4, it can be understood that, in the case of global optimization, there will be a centralized process, where all electric vehicles, the electrical grid, and the impact on their variables will be considered within the model, in addition to having the capacity for total coordination. This will result in an optimal charging and discharging profile. On the other hand, in the case of local optimization, individual optimization of electric vehicles is performed. This results in an individual charging and discharging profile. Finally, considering the results obtained in each of the optimization models, a comparative analysis of results is performed for decision-making.

6.1. Global Optimization Process

The overall optimization process focuses on minimizing the total cost of electricity used in the charging process for a set of electric vehicles connected to the grid. It begins with the definition of the variables for each vehicle i and time t.

$$\begin{array}{cccc}\hfill & P_{i,t}^{ch}\ge 0\hfill & \hfill & \mathrm{Charging}\mathrm{power}\mathrm{of}\mathrm{vehicle}v\mathrm{at}\mathrm{time}t\left[\mathrm{kW}\right]\hfill \\ \hfill & P_{i,t}^{dis}\ge 0\hfill & \hfill & \mathrm{Discharge}\mathrm{power}\mathrm{of}\mathrm{vehicle}i\mathrm{at}\mathrm{time}t\left[\mathrm{kW}\right]\hfill \\ \hfill & SO{C}_{v,t}\hfill & \hfill & \mathrm{Charge}\mathrm{status}\mathrm{of}\mathrm{vehicle}i\mathrm{at}\mathrm{time}t\left[\mathrm{kWh}\right]\hfill \end{array}$$

The convention of net power is also defined as follows: $p_{i,t}=p_{i,t}^{ch}-p_{i,t}^{dis}$

For the SOC, it is defined for vehicle i at time t as follows:

$$SO{C}_{i,t}=SO{C}_{i,0}+{\displaystyle \frac{1}{E_i}}\sum _{\tau =1}^t({\eta }_i^{ch}{p}_{i,\tau }^{ch}-{\displaystyle \frac{1}{{\eta }_i^{dis}}}{p}_{i,\tau }^{dis})\ast \Delta t$$

(5)

where $\Delta t$ is defined as the duration of the interval in hours. An initial value of $\Delta t=1$ is considered.

The objective function (minimizing total energy cost), is calculated as follows:

$$min\sum _{t=1}^T{C}_{\mathrm{grid}}\left(t\right)·\left[D\left(t\right)-G\left(t\right)+\sum _{i=1}^{N_v}\left(P_{i,t}^{ch}-P_{i,t}^{dis}\right)\right]$$

(6)

The objective function only considers the cost of energy purchased from the grid. The following constraints are considered for the global optimization process:

• Active Power Limits per Electric Vehicle:

  $$P_i^{min}\le p_{i,t}^{ch}\le P_i^{max}$$

  (7)

  $$P_i^{min}\le p_{i,t}^{dis}\le P_i^{max}$$

  (8)

It can be appreciated in the case of charging and discharging power limits; it is required to define a minimum limit to which the electric vehicle will be discharged.

• Evolution and Dynamics of SOC:

  $$SO{C}_{i,t}=SO{C}_{i,0}+{\displaystyle \frac{1}{E_i}}\sum _{\tau =1}^t({\eta }_i^{ch}{p}_{i,\tau }^{ch}-{\displaystyle \frac{1}{{\eta }_i^{dis}}}{p}_{i,\tau }^{dis})$$

  (9)

• SOC Limits:

  $$SO{C}_{i,min}\le SO{C}_{i,t}\le SO{C}_{i,max}$$

  (10)

The limits of the SOC must be defined, taking into account the relationship between the minimum and maximum charging and discharging power of electric vehicles.

• Required SOC Final:

  $$SO{C}_{i,T}\ge SO{C}_i^{req}$$

  (11)

• Global Power Balance per Hour:

  $$\sum _{i=1}^N{p}_{i,t}^{ch}-\sum _{i=1}^N{p}_{i,t}^{dis}\le P_t^{grid}+G_t$$

  (12)

In the power balance equation, consumption from the electrical grid is considered to be the sum of the charging power minus the sum of the discharging power of electric vehicles. In addition, there is power from the grid ($P_t^{grid}$) in addition to renewable energy generation $G\left(t\right)$, if available (optional).

The proposed outline of the global optimization process is shown in Figure 5.

6.2. Local Optimization Process

In the local optimization process, each vehicle i is considered to solve its own individual problem. In this case, its impact on the grid is not taken into consideration. The general implementation of the optimization process is shown in Figure 6.

The formulation of the local optimization problem is detailed below:

• Objective Function:

  $$min\sum _{t=1}^T{c}_t(P_{i,t}^{ch}-P_{i,t}^{dis})$$

  (13)

The objective function considers the energy cost and the charging and charging power of the vehicle i at time t.

The constraints used in formulating the optimization problem are detailed below:

• Limits EV Charging and Discharging Power per Hour:

  $$0\le p_{i,t}^{ch}\le P_{i,t}$$

  (14)

  $$0\le p_{i,t}^{dis}\le P_{i,t}$$

  (15)

In the case where there is no prior assignment, it is considered: $P_{i,t}=P_i^{max}$

• Evolution and Dynamics of SOC:

  $$SO{C}_{i,t}=SO{C}_{i,0}+{\displaystyle \frac{1}{E_i}}\sum _{\tau =1}^t({\eta }_i^{ch}{p}_{i,\tau }^{ch}-{\displaystyle \frac{1}{{\eta }_i^{dis}}}{p}_{i,\tau }^{dis})$$

  (16)

• SOC Limits:

  $$SO{C}_{i,min}\le SO{C}_{i,t}\le SO{C}_{i,max}$$

  (17)

Similar to global optimization, it is also necessary to define the minimum and maximum SOC limits for each electric vehicle. i.

7. Results and Discussion

The proposed case study models the optimal charging schedule for a set of EVs over a 24-h period. Two approaches are compared:

• Local Optimization: A specific power limit per hour is defined for each vehicle. Planning is carried out independently for each EV.
• Global Optimization: A total power limit is defined, which is shared by all EVs every hour. Planning is carried out jointly for all EVs, and the minimum total cost of the system is sought.

In relation to the objective function, it focuses on minimizing the total energy cost required to charge the batteries so that the restrictions on capacity, minimum final state of charge (SOC), and power limits are met.

The parameters proposed for the case study are listed in the following

Table 1. Initial parameters.

Parameter	Value
Number of EV	30
Time Horizon	24 h
Battery Capacity	(40, 45, 50) [kWh] Average
Initial SOC	(0.2, 0.3, 0.4) Average
Final Min SOC	(0.9, 0.85, 0.8) Average
Electricity Prices	changing per hour [$/kWh]

.

The following constraints are additionally defined:

• Local: The maximum power allocated per vehicle = 2 kW per hour.
• Global: total power available per hour = 6 kW.

The proposed study scenario considers the following criteria:

• Local Optimization Scenario: each vehicle will be charged independently, and its maximum individual power (2 kW) will be limited, without sharing resources with other vehicles.
• Global Optimization Scenario: all vehicles will share a total power limit of 6 kW. In addition, the optimization process will decide how to distribute this energy to minimize the total cost.

Case of Study

In relation to Table 1, the initial conditions of the case study are parameterized and defined in order to solve the optimization problem.

Initially, a 24 h electricity cost profile was considered, as shown in Figure 7.

Then, the parameters for the initial and final SOC status of the batteries are defined, as shown in Figure 8.

The battery capacity is defined individually for each electric vehicle (Figure 9) taking into account the parameters of Table 1.

Once the conditions have been defined, the optimization problem is solved. At this point, the vehicle charging schedule is obtained as a result, considering both global and local optimization, as shown in Figure 10.

From Figure 10, it can be seen that each of the optimization processes meets different needs. In the case of global results, the impact on the grid and the charging needs of each EV are considered, without exceeding the established grid limit. For local results, restrictions are imposed based on the individual power limitations of each vehicle at each point in time.

Table 2. Summary of results.

Energy	Total
Local Energy [kWh]	839.35
Global Energy [kWh]	839.35

summarizes the results obtained in the optimization process in relation to the energy consumed by electric vehicles.

According to the vehicle fleet charging process, the total charge per hour shown in Figure 11 is obtained.

With regard to total energy consumption, the following general results have been obtained:

• Total energy charged (Local): 839.35 kWh;
• Total energy charged (global): 839.35 kWh.

As can be verified in the results obtained, the charging and discharging scheduling schemes comply with the restrictions established. However, it is necessary to consider the cost minimization obtained. In this regard, Figure 12 shows the total operating costs.

The results obtained in terms of total costs are:

• Total cost (Local): $106.23;
• Total cost (Global): $87.83.

The costs in terms of the results obtained in the overall optimization are minimally lower, representing a saving of 17.32% with the overall optimization.

8. Conclusions

The objective of minimizing the total cost of energy consumed to charge electric vehicles was successfully achieved. The results show that global optimization reduces the total cost compared to local optimization, meeting the goal of economic efficiency within the 24-h horizon. This highlights the importance of coordinated charging planning in electric transportation systems, especially when looking to take advantage of variable electricity prices.

Global optimization showed a more efficient use of available charging capacity, distributing total power among vehicles in such a way as to prioritize hours with lower electricity costs. In contrast, local optimization, limited by individual power, failed to take full advantage of periods of cheap energy, resulting in higher costs. This confirms that coordination in charging planning can generate significant savings and better use of energy resources.