1. Introduction
Multi-objective optimization (MOO) is generally used to solve problems involving two or more conflicting objectives, and most of the problems in the real world are multi-objective in nature [
1]. In contrast to single-objective optimization problems, a single solution generally cannot minimize/maximize all the objective functions simultaneously in MOO problems [
2]. Instead, several solutions can be considered as optimal considering trade-offs among the different objectives. The set of all these possible optimal solutions form a Pareto front. A solution is known as Pareto optimal if none of the objective functions can be improved without deteriorating other objective functions. All Pareto optimal solutions are considered equally good if no preference information is provided [
3]. Due to the ability to provide acceptable solutions for various complex real-world problems, MOO is used in several fields, including power systems and microgrids [
4,
5,
6,
7,
8].
A multi-objective sizing problem is formulated [
4] considering three objectives, which include the loss of power supply, cost of energy, and dumped energy in microgrids. The developed model is solved using a multi-objective grey wolf optimizer. Similarly, goal-programming is used [
5] to solve a MOO problem where the degradation of batteries and curtailment of renewables in off-grid microgrids are the two conflicting objectives. The smoothness of the power output and total annual power amount are simultaneously considered [
6] for an integrated hydro/PV system. A non-dominated sorting genetic algorithm (NSGA-II) is utilized to solve the formulated problem. Similarly, the dimensioning of the hybrid power systems, including renewables, is considered [
7] for optimality and robustness. A multi-objective genetic algorithm is used to solve the planning problem. A MOO problem is formulated [
8] for a hybrid storage system for maximizing the utilization, reducing energy consumption, and preventing overcharging of the energy storage system.
Recently, due to the enhanced penetration of electric vehicles (EVs), several studies are being conducted on applying MOO for various aspects of EVs [
9,
10,
11,
12,
13]. A MOO problem is formulated [
9] for maximizing the profit while minimizing the deviation of profit for different EVs. The epsilon-constraint method is used to solve the formulated problem. Economic loss and peak-to-valley deviations of EVs are considered as the two competing objectives in one study [
10], and a modified weight aggregated particle swarm optimization algorithm is used. The power quality requirements of microgrids and user requirements of EVs are considered in another study [
11], and a dynamic programming-based model is developed to solve the MOO problem. Different uncertainties in the market price and demand response are considered for EVs in another study [
12], and a MOO problem is formulated. The two objectives considered are the cost and deviation of cost, and the weighted sum (WS) method, along with a fuzzy engine, is used to solve the problem. The fuel consumption cost and emissions are considered [
13] for a parameter matching optimization problem in EVs. To solve the problem, the MOO problem is transformed into a single-objective optimization problem using the weighted-sum method.
Due to the enhanced penetration of EVs, the interdependence of transport and power networks has increased. It could be challenging to provide power to EVs during major power outages. In addition, EVs are predicted to be the major source of transportation in the future; i.e., they will be used for diverse purposes ranging from personal use to emergency response [
14]. The allocation of energy to EVs during power outages is a challenging issue, and it can be modeled as a MOO problem. Recently, a few studies have been conducted on the energy allocation of EVs during major power outages. The prioritization of EVs is considered in particular studies [
15,
16] to allocate available energy to EVs during different events. Several factors are considered [
15] to determine the EV priority, such as battery energy level, distance to be traveled, and EV battery capacity. A game-theoretic approach is used [
16], and a priority factor is included in the utility function to prioritize different classes of EVs. The energy allocation to EVs is considered as a MOO problem in one study [
14], and social welfare, community wellbeing, and individual satisfaction are considered as the objectives.
However, there are several methods for solving MOO problems, such as methods with a prior articulation of the preferences, methods with a posterior articulation of the preferences, methods with no articulation of the preferences, and multi-objective heuristic methods [
2]. Some methods could be more beneficial for some cases than others. Therefore, it is required to analyze all these categories of MOO methods and determine the most suitable candidate for the problem at hand. Power allocation to EVs is a new yet critical issue for the future regarding transportation systems during outages. There is no such study on the comparison of different MOO methods for energy allocation to EVs during outages. In addition, indices are required to compare/analyze the performance of different methods on the same scale.
An attempt is made in this study to address the research gaps mentioned in the previous paragraph. Firstly, a MOO problem is formulated for allocating energy to EVs during outages. The developed MOO problem comprises three objective functions, one each for the societal level, community level, and individual level benefits of allocating energy to a particular EV. Then, one index is formulated for each of the three benefit levels mentioned above. Finally, EVs are ranked based on their value for society, the community, and individuals using the three indices. Five major MOO methods are selected, and their performance is analyzed for allocating energy to EVs during power outages. At least one method is selected from the three major categories of MOO, which includes methods with prior, posterior, and no articulation of preferences. In addition, the performance of the NSGA-II is also analyzed. To compare the performance of these methods, two indices are also proposed in this study: the demand fulfillment index and total demand fulfillment index. The demand fulfillment index considers the relative priority of different EVs while evaluating the number of EVs with the demand being fulfilled. However, the total demand fulfillment index compares the total number of EVs requiring a recharge to the total number of EVs with the demand being fulfilled. The performance of all five methods is compared based on the proposed indices. In addition, several other factors, such as the variance, computational complexity, and additional required constraints, are also compared for all the methods.
2. Multi-Objective Optimization Methods and Energy Allocation to EVs
MOO, also known as vector optimization, is the process of simultaneously optimizing a set of objective functions [
2]. Therefore, MOO techniques are known to be more realistic options for designers due to their ability to satisfy several objectives simultaneously [
1]. However, even for a trivial problem, it is unlikely to find a single solution that optimizes all the functions at the same time [
3]. In most cases, the objective functions are conflicting in nature, i.e., optimizing one can deteriorate the other and vice versa. Therefore, the goal of MOO techniques is to find a trade-off between these conflicting objectives. There could be several trade-off solutions for each MOO problem, and these solutions are known as Pareto optimal solutions. A solution is known to be Pareto optimal if none of the objective functions can be improved without deteriorating other objective functions. The set of all the Pareto optimal solutions is known as the Pareto front. All the solutions in the Pareto front are considered equally good and are considered optimal if there is no specific preference for any particular solution.
For the sake of visualization, an overview of a hypothetical bi-objective optimization problem is shown in
Figure 1. In
Figure 1, the anchor points refer to the optimal points of the individual objective functions. In addition, the anchor points refer to the worst value for the other objective function in the case of bi-objective functions [
17]. The utopia point is the ideal point where both the objective functions are optimized, which is generally unattainable. However, several methods try to minimize the distance from the utopia point. Finally, all the points on the Pareto front are attainable.
2.1. Categorization of Multi-Objective Optimization Techniques
The MOO techniques can be divided into three major groups based on the articulation of preferences. These groups comprise methods with a prior articulation of preferences, methods with a posterior articulation of preferences, and methods with no preferences. In the first case (prior articulation), the preference or priority of different objective functions is known in advance and this information is provided to the optimization algorithm. In the case of the second group (posterior articulation), the Pareto front is obtained first and then an optimal solution is selected from the Pareto front. In addition to these three categories, multi-objective genetic algorithms are also widely used for solving MOO problems. Therefore, an overview of all four of these categories and their representative examples is shown in
Figure 2. The performance of all the methods shown in
Figure 2 is compared for an energy allocation problem to EVs during system contingencies.
2.2. Energy Allocation to EVs during Contingencies
2.2.1. Power Contingencies and EVs
The penetration of EVs is increasing across the globe in order to reduce the carbon emissions from the transport sector. It is predicted in the Global EV Outlook 2021 report that the size of the EV fleet will reach between 145 and 230 million by 2030 [
18]. In the near future, EVs will not only be used for the personal commute but also emergency services, for example, by firefighters and hospitals. In addition, EVs will also be used for evacuation during major outages. During power outages, the locally available energy may not be sufficient to fulfill the needs of all the EVs. Therefore, different factors need to be considered to prioritize EVs and allocate the available energy. Recently, the allocation of power to EVs during major outages has been considered in several studies. For example, a non-cooperative game model is proposed in one case [
16], and the energy fulfillment of higher priority EVs is ensured first by including a priority factor in the utility function of the EVs. In another study [
14], three objectives are formulated for allocating energy to EVs. These objectives are social welfare, community wellbeing, and individual satisfaction. In another study, the EVs are prioritized and energy allocation is scheduled [
19] during both normal congestion and power outages.
Meanwhile, the integration of stationary energy storage and renewables in fast-charging stations is also being studied [
20,
21,
22]. These local resources can reduce the grid impacts of EVs during normal operation and can enhance the resilience of EVs during outages. However, due to the absence of connection with the grid and scarcity of locally available energy, additional factors need to be considered to prioritize EVs during outages. These factors could be the required energy, occupants of EVs, trip purpose, energy level, and time by which recharge is required. Based on these factors and the available energy, different objectives could be formulated. For example, the allocation of energy to all EVs provides individual satisfaction, but the community wellbeing and social welfare depend on the trip purpose and the occupants of the EVs since it is predicted that EVs will be the major source of transportation in the future and will be used for diverse purposes, ranging from personal use to public transport and emergency response. Therefore, these objectives need to be considered in allocating energy to EVs during power outages.
2.2.2. MOO for Power Allocation to EVs
To allocate energy to EVs during contingencies, a MOO problem is formulated in this section, similar to [
14]. The overall objective of the MOO problem is to maximize the satisfaction of EV owners/communities using three objective functions as given by (1). The problem is formulated in such a way that the first objective function (2) has the highest precedence; it is followed by the second objective function (3), and the third objective function (4) has the least precedence. The first objective refers to societal-level benefits that can be achieved by allocating energy to a certain EV. The second objective refers to community-level benefits, and, finally, the third objective refers to individual-level benefits, which can be achieved by allocating energy to a certain EV.
represent the rank of n
th EV corresponding to the first, second, and third objectives. These indices are formulated based on the trip purpose, occupants of EVs, departure time, and the difference between available and required energy. The vector
in Equations (2)–(4) is a binary vector that indicates whether the required energy of EVs is fulfilled or not. As indicated by (5), the binary variable takes a value of 1 if the energy allocated to the n
th EV (
) is greater than the required energy (
). Otherwise, the binary variable corresponding to the n
th EV takes a value of zero. Similarly,
represents the energy deficit for n
th EV and is computed as in (6). Equation (7) implies that energy allocated to each EV should be less than or equal to the amount of energy required by that EV. Finally, Equation (8) implies that the total amount of allocated energy to all EVs should be less than or equal to the available energy (
).
where,
subject to
5. Discussion and Analysis
In this section, the performance of the five MOO techniques discussed in the previous sections is compared. In the first part, the performance is analyzed using the proposed indices. In the second part, the complexity and the variance in the results are analyzed.
5.1. Evaluation via Performance Indices
The two indices proposed in this study are used in this section to analyze the performance of the five MOO methods. The first index (demand fulfillment index, DI) indicates the ratio of EVs with a particular rank to the total number of EVs having a non-zero value for that rank. Higher values imply more EVs with demand being fulfilled and vice versa. It can be observed from
Figure 7a that DI
1 is the highest in the case of LEX and NBI, while it is the lowest for the NSGA-II. The value on DI
2 is the highest for WS and lowest for MM, while the value of DI
3 is the highest for WS and lowest for LEX. Since the priority of F1 is the highest and that of F3 is the lowest, the results of LEX and NBI will be considered better. The WS is increasing the values of DI
2 and DI
3 at the cost of DI
1, which is generally not preferred.
Similarly, the second index (total fulfillment index, TI) shows the ratio of the total number of EVs with demand being fulfilled to the total number of EVs requiring a recharge. It can be observed from
Figure 7d that the value of TI is the highest for the WS method, while it is the lowest for the NSGA-II and LEX methods. The highest value in the WS is due to the consideration of the accumulated indices in the case of the WS, while, in the case of LEX, higher priority ranks are preferred irrespective of their accumulated indices. In the case of NSGA-II, there is no specific preference for any particular rank.
If other factors are not considered, the performance of LEX and NBI is suitable for objectives with clear priority/preferences due to their ability to maximize the allocation based on the priority. The performance of NSGA-II was the lowest in most of the cases, thus making it unsuitable for this problem. Finally, the WS method is better for maximizing the total number of EVs having their demands fulfilled at the cost of some degradation in the performance of the higher rank EVs. However, other factors also need to be considered to determine the best suitable candidate method, which is discussed in the following section.
5.2. Complexity and Variance Analysis
In the previous section, the best-performing case was selected for each method. However, in the case of the WS, LEX, and NBI methods, the outcome changes with a change in the priority or weights. Therefore, the maximum and minimum number of EVs survived by different methods and their variance is tabulated in
Table 5. It can be observed that the variance is maximum for the LEX while it is minimum for the NBI method (the MM and NSGA give single unique solutions). The higher variance in the LEX is due to a significant change in the outcome with the change in the priorities of the objective functions.
In terms of additional constraints, only one additional constraint is required for WS, while NBI and MM require I number of additional constraints, where I is the number of objective functions. Similarly, in the case of LEX, I-1 additional constraints are required, while NSGA-II does not require any additional constraints. An increase in the number of objective functions could increase the complexity of the problem where additional constraints are required.
The computational complexity of WS is the lowest due to the requirement of scalers only and a single additional constraint, while the complexity of NBI is higher due to the requirement of several runs and having I number of additional constraints. Similarly, the computational complexity of NSGA-II is also higher due to the requirement of more time in finding the optimal solution. MM and LEX have a medium level of complexity, where MM requires I additional constraints but provides a single solution. LEX requires I-1 additional constraints and provides different solutions based on the priority.
The analysis in this section and the previous section shows that NSGA-II is not suitable for this problem due to lower performance despite having higher computational complexity. LEX is the most suitable method if the priority of different EVs is known due to its ability to allocate energy based on the priority, similar to the findings of [
14]. In addition, it has a medium level of computational complexity. MM is suitable for cases where the priority is not known, and it can provide a single unique solution with a medium level of computational complexity.