An improved NSGA‐II for the dynamic economic emission dispatch with the charging/discharging of plug‐in electric vehicles and home‐distributed photovoltaic generation

This paper investigates four energy utilization scenarios with or without home‐distributed photovoltaic generation (HDPG) to reduce the generation cost and pollutant emission of the dynamic economic emission dispatch with the charging/discharging of plug‐in electric vehicles (DEED‐PEV). The first scenario considers valley filling for the charging of PEVs. The second scenario combines valley filling and peak shaving for the charging and discharging of PEVs. The third scenario adds peak shaving of HDPG to the first scenario, followed by the peak shaving with the discharging of PEVs. The fourth scenario rearranges the distribution of photovoltaic (PV) power for the third scenario, and the PV power in the afternoon is stored by a photovoltaic energy storage system (PESS) and consumed in the evening. A universal procedure is designed for the valley filling and peak shaving of the four scenarios, which is beneficial for determining the filled and shaved loads with respect to certain time intervals. An NSGA‐II method based on a modified crossover and an elimination of individuals (NSGA‐II‐MCEI) is proposed for the multiobjective optimization of DEED‐PEV. The modified crossover can improve the convergence of NSGA‐II‐MCEI, and the elimination operator can maintain the evenness of the nondominated solutions. According to experimental results, scenario 4 achieves cost savings of 95386.62 $, 85636.87 $, and 6776.85 $, respectively, for Scenarios 1, 2, and 3, and it reaches emission reductions of 27617.64 kg, 17252.71 kg, and 220.98 kg, respectively, for scenarios 1, 2, and 3. Also, scenario 4 outperforms the other three scenarios for three weather conditions such as sunny day, cloudy day, and rainy day.


| INTRODUCTION
The global environment has encountered unprecedented challenges with the rapid development of global economy in the last decades, and people have reached a consensus that it is necessary and urgent to take a package of measures to reduce the effects of economic development on the environmental crisis.To overcome the effects of the pollutant emissions of traditional vehicles, many governments suggest cleaner travel modes and encourage the purchases of new energy vehicles (NEVs).Taking China, for example, the number of NEVs has increased from 0.91 million to 6.03 million during five-and-a-half years, and the number of plug-in electric vehicles (PEVs) has increased from 0.73 to 4.93 million during five-and-a-half years (Figure 1).
The goal of DEED-PEV is to adjust the power outputs of generation units to optimize generation cost and pollutant emission and handle various constraints. 1As depicted in Figure 2, the electric power customers come from buildings, hospitals, factories, schools, houses, charging stations, and so on.The power grid provides these customers with a large quantity of electric power that is produced by the generation units of power plants.So far, some studies have been conducted on PEVs.For example, an indepth analysis has been presented on electric vehicle charging station infrastructure, policy implications, and future trends in Mastoi et al., 2 a study of chargingdispatch strategies and vehicle-to-grid (V2G) technologies has been proposed for electric vehicles in distribution networks. 3There are also some related studies in Shaikh and colleagues, [4][5][6][7][8] and interested readers can refer to these references.
To satisfy the load demand of the growing number of PEVs, PV power and winder power [9][10][11] have been considered to be connected to the grid due to their merits of renewability and cleanliness.These energies have been essential supplements for traditional coal-fired power, and they play important roles in maintaining the efficient and stable operation of power systems.However, winder power has strong randomness and instability, and it is hard to accurately forecast and sufficiently utilize this kind of energy.On the other hand, solar energy, as an important clean and renewable energy, can alleviate the increasing pressure of power supply and improve the operation stability for power grid.Furthermore, the utilization of PV power is relatively convenient since it is mainly generated by solar panels in the daytime when the time periods around noon have the strongest sunlights (Figure 3). 12urthermore, the time period of traditional load peak is commonly located at 12 noon, and it is highly synchronous with the time period when PV power generation is the largest.To this end, PV power is preferred for the peak shaving of traditional load in comparison with wind power, and it will smooth the load distribution to some extent.
Demand side management (DSM) is an important means in decreasing the fluctuations of traditional loads, 13,14 and it has been widely used in smart grids.In Yang et al. 15 and Ma et al., 16 the authors added the charging loads of PEVs into DEED without considering the discharging power of PEVs, which imposes heavier pressure on the facility capacity and operation stability of traditional power systems.In Liang et al. 17 and Mei et al., 18 the authors studied the DEED3 PEV by the V2G 19,20 and grid-to-vehicle (G2V) 21,22 technologies.The F I G U R E 1 The numbers of new energy vehicle (NEVs) and plug-in electric vehicle (PEVs) during five-and-a-half years.valley of traditional loads was filled by the charging loads of PEVs with G2V, and the peak of traditional loads was shaved by the discharging power of PEVs with V2G.The DSM technology in Liang et al. 17 and Mei et al. 18 can smooth the load distribution by charging and discharging PEVs.Furthermore, the discharging power of PEVs can not only abate some burden for power systems but also increase the flexibility of power supply.
In addition, as a kind of common renewable energy, wind power was incorporated into DEED-PEV to satisfy a portion of the load demands from traditional consumers and PEVs. 23,24Nevertheless, winder power generation has strong randomicity, uncertainty, and intermittence, and it incurs the penalty cost associated with the overestimated and underestimated available wind power, which is harmful to the cost savings of power generation from the economic aspect.As another kind of potential renewable energy, PV power has been applied to various energy system, such as combined cooling, heating and power system 25 and combined heat and power system. 26,27PV power is more friendly to DEED-PEV than wind power, and it can be used for the peak shaving of the load demands because of the synchronization between PV power and traditional loads in daytime.The penetration of PV power in power grid will improve the flexibility of DSM and facilitate the minimizations of generation cost and pollutant emission for DEED-PEV.
Recently, a concept of home-distributed photovoltaic and vehicle-to-home (HDPV-V2H) has been presented in Chen et al., 12 and a household PEV can charge or discharge by its own bidirectional V2H device.The PV power is generated by the solar panels of the smart house with HDPV-V2H, and it can supply power for both household appliances and power grids.Accordingly, a large penetration of PV power of HDPV-V2H in power grid will significantly alleviate the pressure from the load demands of traditional users and PEV owners, which is helpful to the efficient energy utilization and reliable operation of the power system.In addition, the photovoltaic energy storage system (PESS) 28,29 can be utilized to store a part of photovoltaic (PV) power and release the stored PV power in the time intervals when load demands are badly needed.In case PESS is incorporated into HDPV-V2H, more economic and ecological profits will be obtained by rearranging the PV power.
In this paper, the HDPV-V2H with or without PESS is considered for DEED-PEV.In short, this paper has the following fourfold contributions.
(1) Four energy utilization scenarios are considered for DEED-PEV, and it is the first time that the fourth scenario involves the HDPV-V2H with PESS.Owning to the energy storage function of PESS, a part of PV power can be transferred from afternoon to evening, contributing to considerable cost saving and emission reduction.(2) A thorough valley filling approach (VFA) is developed for the valley filling of four energy utilization scenarios, and it applies to not only the valley in consecutive time intervals but also the one in both ends of a dispatching period.Unlike, Liang et al., 17 the charging inclinations of PEV drivers are considered in this paper, and the PEV loads have been assigned to all the two valleys of traditional loads by VFA, which is more humanized and flexible in practice.Unlike the literature work, the proposed work investigates more comprehensive and flexible energy utilization scenarios of DEED-PEV with HDPVV2H and PESS.The HDPV-V2H can provide considerable PV power for a house and reduce the dependency on the power grid.The PESS is able to store PV power and release it at the appropriate time, and thus it improves the energy utilization efficiency of the DEED-PEV significantly.It establishes a new formulation on DEED-PEV and presents an NSGA-II-MCEI for this issue.The fourth DEED-PEV scenario considered in this paper is more efficient than the ones in literature since it can achieve lower cost and emission, which contributes to the cost saving and emission reduction of the power system.
The remainder of this paper is organized as follows: Section 2 describes the formulation of DEED-PEV integrating HDPV-V2H with PESS.Section 3 presents the NSGA-IIMCEI for the multiobjective optimization related to the generation cost and pollutant emission of DEED-PEV.Section 4 investigates four energy utilization scenarios and uses three multiobjective evolutionary algorithms (MOEAs) to deal with eight DEED-PEV problems.Finally, conclusions are given in Section 5.

| RELATED WORK
According to Figure 4, a smart house with HDPV-V2H is composed of solar panels, PESS, household appliances, PEV and home energy management system (HEMS). 30,31Specifically, solar panels are responsible for HDPV, and they can supply sufficient power for household appliances and PEV.Furthermore, the surplus PV power will be either transferred to the power grid or stored by PESS, which improves the flexibility of energy utilization significantly.The bidirectional V2H system serves as a bridge between the PEV and the smart house, and the charging/ discharging of the PEV is coordinated by HEMS.Accordingly, a PEV can charge from the smart house that is not only equipped with solar panels but also connected to the power grid.In turn, it will discharge for the power grid and the household appliances if it has sufficient power in the battery.
Based on the above description of the smart house, the DEED-PEV integrating HDPVV2H with PESS can be formulated as follows.

| Problem formulation
DEED-PEV belongs to a kind of nonlinear constrained optimization problem, and it involves two objective functions and a series of constraints.Generally, the two objective functions are given by: where C P refers to the cost function and is associated with the power output of all conventional units in T time intervals.T means the number of time intervals in a dispatching period.N P stands for the number of units.P i t , denotes the power output of a conventional unit i i N ( = 1, …, ) P in the tth t T ( = 1, …, ) time interval, and P i,min represents the allowable minimum of P i t , .a b c e , , , i i i i , and f i denote the cost coefficients related to the ith unit.

 
where E P means the emission function and is related to the power outputs of N P units in T time intervals.μ k π σ , , , i i i i , and θ i represent the emission coefficients of the ith unit.
The functions C P and E P are subject to various practical constraints formulated in terms of the operating conditions of conventional units and the load demands from power customers.The constraints are given as follows: (1) Power generation limits where P i,min and P i,max represent, respectively, the allowable minimum and maximum for P i t , .They are associated with the unit index i but irrelevant to the time index t.
(2) Power balances  (5)   where B i j , , B i 0 , and B 00 stand for the transmission loss coefficients.To smooth loads as much as possible, the charging load P t ch, is commonly used for the valley filling of traditional loads, and the discharging power P t disch, is generally applied to the peak shaving of traditional loads.In addition, the PV generation P t pv, is applicable to the peak shaving of traditional loads due to the penetration of HDPV-V2H in power grid.Reasonable DSM will contribute to the minimizations of both C P and E P , and the valley filling and peak shaving in four scenarios will be investigated later.
(3) Up and down ramp rate limits UR i refers to the allowable maximum of the incre- ment between the power outputs in two adjacent time periods for unit i, and DR i means the allowable maximum of the reduction between the two power outputs.The constraints require that the difference between P i t , −1 and P i t , be in a reasonable range for the sake of machine maintaining.(4) Spinning reserve requirements 32 is taken as the net load based on valley filling and peak shaving.
(5) Prohibited operating zones (POZs) 34  ( ) where P i z L , and P i z U , mean, respectively, the lower and upper boundaries of the zth POZ for P i t , .Z i refers to the number of POZs for P i t , .Owning to the vibration in a shaft bearing or the faults in the machines themselves, P i t , falls outside a handful of POZs between P i,min and P i,max .Accordingly, the constraints require that the power output of each unit get rid of its POZs.

| Valley filling
In Liang et al., 17 all charging loads of PEVs are only assigned to the valley of traditional loads at night.In practice, a large number of public and private PEVs have a relatively long seven average driving time per day.Some of the PEVs have to charge in the daytime to ensure sufficient power in their batteries, while others will charge at night.In case the charging of the PEVs is only allowed in the night, the normal transportation of the PEVs will be affected significantly, especially in the daytime.To give consideration to both PEV owners' charging inclinations and economic power consumptions, the valleys of traditional loads in both daytime and night should be open to the charging of PEVs, and the total charging load in the daytime can be suitably smaller than the one in the night because of the lower electricity prices during the off-peak time intervals of the night.Furthermore, a thorough VFA is proposed to fill the daytime valley in consecutive time intervals and the night valley in both ends of a dispatching period, and it is described in Table 1.
where P ps,sum denotes the total load of peak shaving.t ¯s and t ¯e represent, respectively, the starting and ending time intervals of the loads to be shaved.P t d′, means the load to be shaved for the t th time interval.ξ stands for the threshold that determines the amount of the shaved load for P t d′, .The goal of peak shaving is to find the optimal threshold ξ to achieve d ¯ps = 0.That is, the above formulation can be treated as a simple unconstrained optimization problem that has one objective function d ¯ps related to the variable ξ .In this paper, the original differential evolution (DE) algorithm 35 is employed to deal with the optimization problem, and it can always find the optimal variable ξ that guarantees d ¯ps = 0. Accordingly, the shaved loads P t t t ( = ¯, …, ¯) t s e ps, can be determined.It should be noted that the other evolutionary algorithms can also be competent for the minimization of d ¯ps , such as genetic algorithm 36 and particle swarm optimization. 37The PSA based on DE is described in Table 2.
F and CR represent, respectively, the scale factor and crossover rate of DE.N pop and N gen denote, respectively, the population size and maximum generation number of DE. ξ min and ξ max refer to, respectively, the lower and upper boundaries of the variable ξ , and they are set as, respectively, 0 and the maximum of traditional loads in T time intervals.

| Constraint handling
Constraint handling is the foundation of multiobjective optimization, and it is responsible for satisfying all constraints of DEED-PEV.The constraints include power generation limits, power balances, up/down ramp rate limits, spinning reserve requirements, and POZs, and we modify a repair technique 38 to deal with the constraints.The modified repair technique (MRT) initially handles POZs.For any P i t , lying within a POZ P P ( , ) Like Zou et al., 38 MRT successively handles the power generation limits, up/down ramp rate limits, and power balances.Accordingly, the total constraint violation is calculated by: where VPB, VPOZ, and VSRR denote, respectively, the constraint violations of the power balances, POZs and spinning reserve requirements, and they are given by:   , , ,max , η i refers to a binary number, and η = 1 i indicates that unit i has one or more POZs and vice versa.
V total is taken as the primary evaluation criterion of individuals in terms of the constraint domination principle.39 In this case, a solution x (1) is considered to constraint dominate another solution x (2) , if any of the following conditions is true: (1) x (1) is feasible and x (2) is infeasible. (2)x (1) and x (2) are infeasible and x (1) has a smaller constraint violation value. (3)x (1) and x (2) are feasible and x (1) dominates x (2) according to the usual domination principle. 40,41 should be noted that x (1) and x (2) will be further evaluated by crowding distance if they have the same nondomination rank.Furthermore, x (1) is considered to be better than x (2) if the crowding distance of x (1) is larger than that of x (2) .

| Multiobjective optimization
The generation cost and pollutant emission need to be minimized for DEED-PEV, and multiobjective optimization is involved in this issue.Due to the low computation complexity and high comprehensive performance, NSGA-II has been one of the most widely used multiobjective evolutionary algorithms (MOEAs), especially for biobjective optimization. 42Recently, NSGA-II has been applied to various multiobjective optimization problems, such as nuclear reactor power control, 43 partition temperature of steel sheet, 44 and integrated process planning and scheduling in a battery packaging machinery workshop. 45esearchers have made efforts to further improve the comprehensive performance of NSGA-II for more complex multiobjective optimization problems.Liu et al. 46 boosted the computation efficiency of NSGA-II by introducing parent inheritance from Kumar and Guria, 47 and they carried out selections by using the roulette selection algorithm from Da Silva et al. 48Liu et al. 49 replaced the mutation of NSGA-II by the one from DE, which is beneficial for alleviating local convergence.Moreover, they divided all the individuals into several sequences, and the worst individual is deleted from each sequence in terms of crowding distance, which is helpful to the uniformity and stability of the individuals.Owning to the advantages, the two improved NSGA-II methods in Liu et al. 46,49 are considered for the biobjective optimization of DEED-PEV.In addition, we have proposed an NSGA-II variant to suit DEED-PEV, and it will be stated in Section 3.

| AN IMPROVED NSGA-II BASED ON A MODIFIED CROSSOVER AND AN ELIMINATION OF INDIVIDUALS
Due to the congenital defect of the simulated binary crossover (SBX), the searches of NSGA-II are mainly confined to the regions surrounding target vectors in the solution space.That is, SBX can only guarantee the local search of NSGA-II rather than global search.In addition, NSGA-II creates a mating pool that is composed of the parent and offspring individuals and selects the best N individuals according to their nondomination ranks and crowding distances.In case that N l select, individuals have to be selected from the N N N ( > ) select, ones with the same nondomination rank l, the N l select, individuals with the largest crowding distances will be selected.However, it is observed that this selection operator is likely to miss some potential individuals in some cases, which is harmful to the evenness of the nondominated solutions.Suppose the Pareto front of rank 1 in the mating pool includes N + 2 individuals and N individuals will be selected from the Pareto front.In other words, the other two individuals will not be considered for the next generation.In case the two individuals are adjacent to each other, the simultaneous elimination of the two individuals is likely to form a relatively large gap of this front, resulting in the unevenness of the nondominated solutions.
For NSGA-II, the convergence of individuals mainly depends on the genetic operators such as crossover and mutation, and the spread of individuals largely depends on the selection operator that is related to density estimation and crowded-comparison operator.To enhance the convergence and spread of individuals, this paper proposes an improved NSGA-II based on a modified crossover and an elimination of individuals (NSGA-II-MCEI).The details of NSGA-II-MCEI are stated as follows.

| Modified crossover
The modified crossover utilizes two strategies to update the two groups of target vectors in the mating pool.The first group includes the target vectors with nondomination rank l = 1, and the second group consists of the rest target vectors with nondomination rank l > 1.In case that the ith i N ( = 1, …, ) target vector belongs to nondomination rank l = 1, it will be updated by the first crossover strategy: ( ) where , ) denotes the jth (j = 1, …, D) variable of the ith target vector with nondomination rank l = 1 at the Gth ) r represents the jth variable of another randomly chosen target vector with nondomination rank l = 1 at the Gth generation.
means the jth variable of the ith offspring vector at the (G + 1)th generation.r stands for a random number with uniform distribution in [0, 1].Since the target vectors belong to the same nondomination rank l = 1, they are treated equally and any value between ) r will be a candidate for . In case that the ith target vector belongs to nondomination rank l > 1, it will be regulated by the second crossover strategy: ( ) where r′ refers to a random number with uniform distribution in [0, 1].x i j l G , ( , ) denotes the jth variable of the ith target vector with nondomination rank l > 1 at the Gth generation.
) b represents the jth variable of the best target vector with nondomination rank l = 1 at the Gth generation. in comparison with the other individuals of nondomination rank.l = 1 C P and E P stand for, respectively, the generation cost and pollutant emission of an individual with nondomination rank l = 1.C P,max and C P,min denote, respectively, the maximum and minimum of CP for all individuals with nondomination rank l = 1.E P,max and E P,min denote, respectively, the maximum and minimum of E P for all individuals with nondomination rank l = 1.In short, To improve the adaptability of the population for DEED-PEV, an adaptive formulation is designed for the individuals with different nondomination ranks, and it is given by: where p c i represents the crossover rate of the ith i N ( = 1, …, ) individual.p c,max and p c,min denote, respec- tively, the maximum and minimum crossover rates.l i and l max refer to, respectively, the nondomination rank of the ith individual and the highest nondomination rank for all individuals.In case l i is equal to or close to 1, p c i tends to be small and a minority of variables will be involved in Equation (17) for the ith individual.Accordingly, this individual is likely to carry out local search, suggesting that it just needs slight adjustments to maintain its advantage over the inferior individuals with high nondomination ranks.In case l i is equal to or close to l max , p c i tends to be large and a majority of variables will be adjusted by Equation (17)

| Elimination of individuals
In this paper, the selection operator of NSGA-II is replaced by the elimination of individuals (EI).The EI steps are elaborated as follows: . For any individual p in S G , determine its minimum Euclidean distance by G and find the associated The EI operator of NSGA-II-MCEI is quite different from the selection operator of NSGA-II.First, the EI operator uses Euclidean distance for density estimation in the objective space (As steps (7) and ( 8)), which is relatively simple and efficient from the aspects of formulation and calculation.Second, the selection operator is likely to miss some adjacent crowded individuals and form several large gaps of the front, which may lead to the unevenness of the front (Figure 5A).In contrast, the EI operator gradually eliminates the crowded individuals from F l when   S N > G and dynamically updates the minimum Euclidean distance and associated individual for every individual whose associated individual has been eliminated from F l (As step (9)).Therefore, the EI operator can efficiently avoid the large gaps of the front, which is helpful to the evenness of the front (Figure 5B).
Figure 6 shows the flowchart of NSGA-II-MCEI, and it is used to explain the methodology from an integral perspective.NSGA-II-MCEI makes improvements on the original NSGA-II by modifying the genetic operator and selection mechanism.
Accordingly, it has strong convergence and can obtain a number of good nondominated solutions for the DEED-PEV problems.

| EXPERIMENTAL RESULTS AND ANALYSIS
In this section, four energy utilization scenarios are presented for the DEED-PEV with or without HDPV-V2H.Especially, the fourth scenario rearranges the distribution of PV power by PESS for the peak shaving of traditional loads, which achieves lower levels of generation cost and pollutant emission than the other three scenarios.In addition, three improved NSGA-II approaches are applied to eight DEED-PEV problems, and NSGA-II-MCEI is able to obtain better nondominated solutions than the other two approaches for these problems.For each of the problems, decision maker can easily select one most desirable solution from the nondominated solutions of NSGA-II-MCEI according to the specific requirements on economy and environment.

| Four energy utilization scenarios
The Nissan Leaf with a battery capacity of 24 kWh 51 is selected as the PEV's type for four energy utilization scenarios.The state of charge (SOC) of a PEV is subject to the range [SOC , SOC ] min max for battery maintenance.SOC min and SOC max denote, respectively, the minimum and maximum SOCs, and they are 0.2 and 0.9, respectively. 52Suppose the consumed SOC of 0.25 is reserved for the daily traveling of each PEV, then the discharging SOC of 0.45 (SOC − SOC − 0.25) max min is available for V2G and bidirectional V2H.For each PEV, the charging lasts 1 h and the discharging also lasts 1 h.Suppose the SOC of a charging PEV equals to 0.35 that is slightly higher than SOC min , and it needs to increase to 0.75 for sufficient power utilization, which can be fulfilled by the G2V and bidirectional V2H technologies.The SOC of a discharging PEV equals to 0.8 which is close to SOC max , and it tends to decrease to 0.5 for tolerable power transferring and self-preservation, which can be realized by the V2G and bidirectional V2H technologies.Accordingly, the charging load of a charging PEV is equal to 9.6 kW, and the discharging power of a discharging PEV is equal to 7.2 kW.Let the total charging load P ch,sum be 2000 MW, then the number of charging PEVs is about 208,333 (2000 × 1000/9.6).Let the total discharging power P disch,sum be 200 MW, then the required number of discharging PEVs is about 27,778 (200 × 1000/7.2).
The first scenario considers the charging of PEVs for DEED-PEV and is shown in Figure 7A.The total charging load of 2000 MW is used for the valley filling of two valleys of traditional loads (Table 3).Specially, one fifth of the charging load is filled in the valley in daytime, and the rest is filled in the valley in night (Figure 8A).VFA is responsible for the valley filling, and details of VFA can be found in Section 2.2.The scenario involves a 10-unit system, 16 and its cost coefficients, emission coefficients, lower/upper power limits, ramp-up/rampdown rate limits are shown in Table 4.The transmission loss coefficients related to 10 conventional units are also from Ma et al. 16 and are provided in Equation (20).The POZs of all instances are from Arul et al. 50and are recorded in Table 5.
The third scenario incorporates HDPV-V2H into the second scenario and is shown in Figure 7C.The power outputs of the solar panels of a smart house in a sunny day are given in Table 6, 12 and the number of smart houses is equal to 20,000.As shown in Figure 9A,B, the PV power outputs are directly consumed and then the total discharging load of PEVs is used for the peak shaving.
The fourth scenario introduces PESS into the third scenario and is displayed in Figure 7D.To be more specific, the PV power between the time intervals 9:00 and 13:00 is directly consumed and then the PV power between the time intervals 14:00 and 18:00 is stored by the PESS.The storing efficiency of the PESS is 0.95, and the stored PV power is used to shave the peak surrounding the time interval 20:00 (Figure 9C).Finally, the discharging load of PEVs is applied to the peak shaving of the loads in the entire dispatching period (Figure 9D).The former peak shaving based on PV power belongs to a local one and the latter peak shaving based on the discharging load of PEVs belongs to a global one, and our proposed PSA is competent for both kinds of peak shaving.
The above four scenarios of DEED-PEV belong to multiobjective constrained optimization problems.Each DEED-PEV scenario has two objective functions, 240 real variables, 24 equality constraints, and 1540 inequality constraints.NSGA-II-MCEI is utilized to cope with the four DEED-PEV scenarios.Four sets of nondominated solutions are achieved by NSGA-II-MCEI for four scenarios, and they are visualized in Figure 10.The data of these solutions are recorded in the appendix due to space limitations.
As shown in Figure 10, the Pareto sets of Scenario 1, 2, 3 and 4 are denoted, respectively, by red square, green triangle, black diamond, and blue circle.The Pareto set of Scenario 4 is closer to the point of lower left corner than the Pareto sets of the other three scenarios, revealing the strong convergence of the nondominated solutions of Scenario 4. Furthermore, the non-dominated solutions of Scenario 4 can always dominate those of the other three scenarios, and thus Scenario 4 is able to provide a series of preferable solutions for the energy utilization of DEED-PEV.
The hypervolume values 53,54 of the four sets of nondominated solutions are equal to 0.0115, 0.1059, 0.8697, and 0.9782, respectively, for Scenarios 1, 2, 3, and 4. Therefore, the Pareto set of Scenario 4 has exhibits the best comprehensive performance for the simultaneous minimizations of generation cost and pollutant emission of DEED-PEV.In addition, a fuzzy-based method 55 is employed to select a compromise solution from the Pareto set of each scenario to equally treat both objectives of DEED-PEV.Four compromise solutions can be determined for the four energy utilization scenarios, and the objective function values of the solutions are shown in Table 7.
C* and E* represent, respectively, the generation cost and pollutant emission of a compromise solution.d C refers to the difference between the cost of Scenario i (i = 1, 2, 3) and that of Scenario 4. d E means the difference between the emission of Scenario i (i = 1, 2, 3) and that of Scenario 4. Scenarios 3 and 4 can obtain lower costs and emissions than Scenarios 1 and 2, suggesting that the DEED-PEV scenarios incorporating HDPV-V2H are more efficient than the ones without HDPV-V2H in energy utilization.Furthermore, Scenario 4 achieves cost saving and emission reduction with respect to Scenario 3,   Table 8 shows the hypervolume values of the Pareto sets obtained by five MOEAs for four scenarios.and the lowest level of H std for every scenario.In short, the high hypervolume levels and low standard deviation levels suggest the strong convergence and stability of NSGA-II-MCEI for four scenarios.

| Six complex energy utilization scenarios with different combinations of three weather conditions
Two or more weather conditions may exist in a day, thus six complex energy utilization scenarios with different combinations of three weather conditions are considered for DEED-PEV.All these scenarios are based on Scenario 4, and the combinations of three weather conditions are stated as follows: (1) Scenario 1* involves sunny day in time horizons [1,12] and cloudy day in time horizons [13,24].(2) Scenario 2* involves cloudy day in time horizons [1,12] and sunny day in time horizons [13,24].| 1713 (3) Scenario 3* involves sunny day in time horizons [1,12] and rainy day in time horizons [13,24].(4) Scenario 4* involves rainy day in time horizons [1,12]  and sunny day in time horizons [13,24].(5) Scenario 5* involves cloudy day in time horizons [1,12] and rainy day in time horizons [13,24].(6) Scenario 6* involves rainy day in time horizons [1,12]  and cloudy day in time horizons [13,24].
Five MOEAs, including i-NSGA-II-RWS, NSDE, MNSGA-II, INSGA-II, and NSGA-II-MCEI are used to solve these six scenarios.Each MOEA is executed 30 runs for a scenario, and the optimization results are shown in Table 9.
Table 9 shows the hypervolume values of the Pareto sets obtained by five MOEAs for six scenarios.The highest hypervolume levels and the lowest standard deviation levels are marked in boldface.NSGA-II-MCEI performs better than the other four MOEAs for six scenarios, and it obtains the greatest H H H , , max min mean , and the smallest H std for every scenario.Great hypervo- lumes reveal the desirable convergence of NSGA-II-MCEI, and small standard deviations indicate good stability of NSGA-II-MCEI.

| Three MOEAs for eight DEED-PEV problems with or without HDPV-V2H
Eight DEED-PEV problems involve the weather factors in the above four energy utilization scenarios.The first problem uses the first scenario that only considers the charging of PEVs.The second problem belongs to the second scenario that combines the charging and discharging of PEVs.The third problem comes from the third scenario which integrates the HDPV-V2H with the charging/discharging of PEVs.The fourth problem employs the fourth scenario, and it adds PESS to HDPV-V2H to rearrange the consumption of PV power.The fifth problem is similar to the third one, but it takes into account the cloudy day for the PV power generation of solar panels of smart houses instead of the sunny day (Table 6).The sixth problem is similar to the fourth one, but it rearranges the consumption of PV power under cloudy day conditions rather than sunny day.Like the third and fifth problems, the seventh problem belongs to the third energy utilization scenario, but it involves the rainy day condition for less PV power generation in comparison with the sunny day and the cloudy day (Table 6).Like the fourth and sixth problems, the eighth problem rely on Scenario 4 to smooth the load distribution, but the rainy day is involved in Scenario 4 instead of the other two weather conditions.
Three MOEAs are used to solve the eight DEED-PEV problems, and they are the NSGA-II with parent inheritance and roulette wheel selection (i-NSGA-II-RWS), 46 the NSGA-II based on DE operator (NSDE) 49 and our proposed NSGA-II-MCEI, respectively.The parameter settings of the first two MOEAs are the same as those of their original literature.For NSGA-II-MCEI, crossover rates p c,min and p c,max are set to 0.1 and 0.5, respectively.Mutation rate p m is set as 1/D, and D refers to the total number of variables related to the power outputs of N P conventional units in T time intervals.Additionally, the population size and maximum iteration number of each MOEA are set to 50 and 1500, respectively.All simulation files are programmed in MATLAB on a computer with Intel(R) Core(TM) i7-6700 processor (3.40 GHz) and 16 GB of memory.To avoid the arbitrary conclusion caused by the occasionality of individual experiments, each MOEA is executed 30 runs for a problem, and the experimental results of eight problems using three MOEAs are shown in Tables 10  and 11 and displayed in Figures 11, 12, and 13.For multiobjective optimization, hypervolume is an important and popular comprehensive indicator in measuring the convergence and diversity of the Pareto set of a MOEA.High hypervolume indicates strong convergence of a MOEA and vice versa.Therefore, it is used to evaluate the convergence of three MOEAs for eight problems.The hypervolume boxplots of i-NSGA-II-RWS, NSDE, and NSGA-II-MCEI are visualized in Figure 11 for eight DEED-PEV problems, and they help to observe the spread of hypervolumes of each MOA over 30 runs in a convenient and intuitive way.The boxplots of NSGA-II-MCEI are the highest for all problems, and thus NSGA-II-MCEI is able to acquire the nondominated solutions with large hypervolume for every problem.Also, the boxplot of NSGA-II-MCEI is the shortest for every problem, indicating the minor difference of 30 hypervolume values and favorable stability of NSGA-II-MCEI.The hypervolume boxplot of i-NSGA-II-RWS is the second highest for each problem, and its upper whisker is close to the lower whisker of NSGA-II-MCEI.On the contrary, the hypervolume boxplots of NSDE are the lowest, and there exist distinct height differences between the hypervolume boxplot of NSDE and that of another MOEA for every problem.In short, NSGA-II-MCEI has shown good comprehensive performance according to the high levels of hypervolume, and it has the ability to provide decision maker with desirable DEED-PEV solutions that give consideration to both generation cost and pollutant emission.
Coverage rate 53 is another key indicator of multiobjective optimization, and it is employed to verify the dominance of our proposed NSGA-II-MCEI over the other two MOEAs.Let X″ and X′ be, respectively, the Pareto sets of NSGA-II-MCEI and another MOEA, then a pair of coverage rates can be obtained by  C X X ( ″, ′) can be determined in the same way.Both   X″ and   X′ are equal to the population size N , and the coverage rates are shown in Table 11.

Problems
For each pair of C X X C X X ( ( ′, ″), ( ″, ′)), the larger coverage rate is stressed by boldface in Table 11.Every C X X ( ″, ′) is close to or equal to 1, and the smallest C X X ( ″, ′) is no less than 0.9607.In other words, most (or all) individuals of i-NSGA-II-RWS and NSDE are dominated by those of NSGA-II-MCEI.In the meantime, every C X X ( ′, ″) is equal to 0 that is much smaller than the value of the corresponding C X X ( ″, ′), and the individuals of i-NSGA-II-RWS and NSDE fail to dominate any of the individuals of NSGA-II-MCEI for every problem.The coverage rates in Table 11 are consistent with the hypervolume values in Table 10 and Figure 11, and they all reflect the relatively strong convergence and stability of NSGA-II-MCEI for eight DEED-PEV problems.
Figure 12 displays the Pareto-optimal points of i-NSGA-II-RWS, NSDE, and NSGA-IIMCEI for Problem 1.Since DEED-PEV aims to minimize both generation cost and pollutant emission, the point (2.5 × 10 , 2.75 × 10 ) 6 5 at the bottom-left corner of the displaying window can be considered as an ideal point.The Pareto-optimal points of NSGA-II-MCEI are close to the ideal point while the Paretooptimal points of NSDE are close to the top-right corner of the displaying window.Furthermore, owing to the slow convergence speed, the Pareto-optimal points of NSDE are far from those of the other two MOEAs, making it difficult to differentiate the distributions of the Pareto-optimal points.To visualize the distributions of the Pareto-optimal points in a clearer way, the Pareto-optimal points of the worst MOEA will be excluded from the displaying window for each problem.
Figure 13 displays the Pareto-optimal points of i-NSGA-II-RWS and NSGAII-MCEI for eight DEED-PEV problems.The Pareto-optimal points of NSGAII-MCEI are closer to the bottom-left corner of the displaying window in comparison with those of i-NSGA-II-RWS for every problem.Therefore, NSGAII-MCEI has a faster convergence speed than i-NSGA-II-RWS for the minimizations of both objective functions.Also, NSGA-II-MCEI has wider and more even distributions of Pareto-optimal points for the minimizations of both objective functions, and it demonstrates stronger exploration and exploitation abilities than i-NSGA-II-RWS in the two-dimensional objective space.Since NSGA-II-MCEI utilizes two kinds of modified crossover operators to regulate the solutions, it possesses relatively strong abilities of global and local searches in both solution and objective space, which is regarded as a major contribution to the desirable convergence and wide distribution of Pareto-optimal points.Furthermore, NSGA-II-MCEI applies an EI operator to the mating pool which is composed of parent and offspring individuals, and it dynamically updates the minimum Euclidean distance and associated individual for any individual whose associated individual has been excluded from the last front of S G .As a result, NSGA-II-MCEI is able to reserve half the mating pool in a more accurate way, which is beneficial for the even distribution of the Paretooptimal points.On the contrary, the distributions of the Pareto-optimal points of i-NSGA-II-RWS are narrow, crowded, and rough for most problems, and it is difficult for i-NSGA-II-RWS to gain a variety of potential dispatching strategies for eight DEED-PEV problems.
The spacing indicator 59 is employed to evaluate the evenness of the Pareto-optimal points obtained by three MOEAs for eight DEED-PEV problems.Table 12 records the average spacing of each MOEA in 30 independent runs and stresses the smallest spacing by boldface for each problem.NSGA-II-MCEI possesses the smallest spacing values for Problems 1, 2, 4, 6, and 7, and thus it can acquire the smoothest Pareto sets for most problems.Furthermore, the average spacing values in Table 12 are consistent with the distributions of Pareto sets in Figures 12 and 13, suggesting that the non-dominated solutions of NSGAII-MCEI are more refined and elastic than those of i-NSGA-II-RWS and NSDE for most problems.

| CONCLUSIONS
An NSGA-II-MCEI is proposed for minimizing the generation cost and pollutant emission of DEED-PEV, which aims to alleviate the conflicts between the two objects.For the crossover operator, the individuals with rank 1 exchange information with each other randomly, and the individuals with rank l  l ( 2) search in the neighborhood of the best one with rank 1.The modified crossover drives the individuals towards potential dispersed regions, which is beneficial for the wide distribution of the individuals.Additionally, the crowding distance is replaced by a more direct measuring criterion that calculates the Euclidean distance between an individual and its nearest neighbor in the objective space and records the neighbor as its associated individual.Furthermore, the selection of individuals is replaced by the elimination of individuals, and the latter updates the associated individual of every individual whose associated individual has been eliminated according to the Euclidean distance in the objective space.By combining the modified crossover and elimination of individuals, the NSGA-II-MCEI is able to produce a series of nondominated solutions with strong convergence and good evenness for DEED-PEV.Accordingly, the decision maker will finally select the most comprehensive solution from the non-dominated solutions in terms of the specific economic and environmental indicators.Four energy utilization scenarios with or without HDPG are considered to reduce the generation cost and pollutant emission of DEED-PEV as much as possible.By combining HDPG and PESS, scenario 4 is capable of storing a part of PV power and transferring it from afternoon to evening.The transferred PV power is used for the shaving of the second load peak, and the rest PV power is used for the shaving of the first load peak, contributing to a smoother distribution of loads.Experimental results suggest that the fourth scenario is more flexible and efficient than the other three ones in reducing the generation cost and pollutant emission of DEED-PEV, which is helpful to the economic development and environmental protection in the real world.APPENDIX A See Tables A1 and A2.

( 3 )
A universal peak shaving approach (PSA) is developed for the global or local peak shaving of three energy utilization scenarios.The global peak shaving can be used for scenarios 2, 3, and 4, and the local peak shaving is responsible for scenario 4. Accordingly, PSA enables each of the three scenarios to determine the shaved loads within a certain time four intervals according to the total power from PV power generation or discharging of PEVs.(4) An NSGA-II-MCEI is proposed for the multiobjective optimization of DEED-PEV.Two crossover strategies are designed to guide the individuals toward potentially dispersed regions, which is helpful to the convergence of NSGA-II-MCEI.Additionally, a comprehensive selection mechanism is established by Euclidean distance measurement, associating neighbors and elimination of crowded individuals.By combining the modified genetic operator and the comprehensive selection mechanism, the NSGA-II-MCEI is capable of obtaining a variety of nondominated solutions with strong convergence and good evenness for the DEED-PEV integrating HDPV-V2H with PESS.
, ) b plays the role of the leader in guiding inferior individuals towards the real Pareto front, and it can significantly improve their convergence speeds and avoid a lot of inefficient searches.

( 2 )− 1 ,( 7 )
Construct a mating pool R G by combining the parent population p G and ofspring population Q G , and calculate the number of individuals to be eliminated by.Normalize the objective function values of the individuals of S G by f ˆ= i objective function value for f i m , .f m max and f m min represent, respectively, the maximum and minimum of   S G objective function values for the mth objective dimension.(8) Calculate the Euclidean distance between each pair of individuals p and q by d = p q .

5
Comparison between two selection strategies of individuals.For example, six out of eight Pareto optimal points should be selected from the front in the 2D objective space.(A) Selection operator of NSGA-II.(B) EI operator of NSGA-II-MCEI.

F I G U R E 6
Flowchart of NSGA-II-MCEI.F I G U R E 7 The power flow of DEED-PEV in four scenarios.(A) Scenario 1. (B) Scenario 2. (C) Scenario 3. (D) Scenario 4. T A B L E 3 Traditional power loads (MW).
and thus the DEED-PEV scenario integrating HDPV-V2H with PESS can further optimize the energy utilization by suitably transferring a portion of the PV power.The pros and cons of a scenario depend largely on the smooth degree of the load distribution in a dispatching period.The smaller the standard deviation of consecutive T loads is, the smoother the load distribution exhibits.For the four scenarios, the standard deviations of loads are equal to 238.72, 224.09, 194.43, and 184.70 MW, respectively.It is clear that the standard deviation of loads of Scenario 4 is the smallest, and thus its load distribution is the smoothest.The data of the compromise solution of Scenario 4 is recorded in the appendix due to space limitations.Five MOEAs are used to solve the four scenarios, and they are the NSGA-II with parent inheritance and roulette wheel selection (i-NSGA-II-RWS),46 the NSGA-II based on DE operator (NSDE),49 modified nondominated sorting genetic algorithm II (MNSGA-II),56 improved nondominated sorting genetic algorithm-II (INSGA-II)57 and our proposed NSGA-II-MCEI, respectively.The parameter settings of the first two MOEAs are the same as those of their original literature.For NSGA-II-MCEI, crossover rates p c,min and p c,max are set to 0.1 and 0.5, respectively.Mutation rate p m is set as D 1/ , and D refers to the total number of variables related to the power outputs of N P conventional units in T time intervals.Additionally, the population size and maximum iteration number of each MOEA are set to 50 and 1500, respectively.All simulation files are programmed in MATLAB on a computer with Intel(R) Core(TM) i7-6700 processor (3.40 GHz) and 16 GB of memory.To avoid the arbitrary conclusion caused by the occasionality of individual experiment, each MOEA is executed 30 runs for a scenario, and the experimental results are given as follows.

8
, and H std refer to, respectively, the maximum, minimum, mean and standard deviation of 30 hypervolume values of the Pareto sets obtained by a MOEA for each scenario, and the highest hypervolume levels and the lowest standard deviation levels are marked in boldface.NSGA-II-MCEI outperforms the other four MOEAs for four scenarios, and it has the highest levels of H The bar charts with respect to the valley filling and peak shaving in the first two scenarios.(A) Valley filling by the charging of PEVs.(B) Peak shaving by the discharging of PEVs for (A).
L E 6 The PV power (kW) in a smart house with HDPV under three weather situations.t(h)P pv,t (kW)

F I G U R E 9
The bar charts with respect to the valley filling and peak shaving in the second two scenarios.(A) Peak shaving by the PV generation for Figure 8A.(B) Peak shaving by the discharging of PEVs for (A).(C) Peak shaving by the PV generation with ESS for Figure 8A.(D) Peak shaving by the discharging of PEVs for (C).

F
I G U R E 10 Pareto-optimal points obtained by NSGA-II-MCEI for four scenarios.T A B L E 7 The objective function values of the compromise solutions obtained by NSGA-II-MCEI for four scenarios.Scenarios C*($) d ($) ″), the denominator stands for the number of individuals in Pareto set X″, and the numerator represents the number of dominated individuals of set X″ by the individuals of set X′.The other coverage rate T A B L E 11 Coverage rates for eight DEED-PEV problems.

11
Hypervolume boxplots of three MOEAs for eight DEED-PEV cases.F I G U R E 12 Pareto-optimal points obtained by three MOEAs for Problem 1.F I G U R E 13 Pareto-optimal points of i-NSGA-II-RWS and NSGA-II-MCEI for eight DEED-PEV problems.

T
A B L E A1 Pareto sets of NSGA-II-MCEI for four scenarios.
universal PSA is developed for the peak shaving based on PV power and the discharging of PEVs.The formulation involved in PSA is given by: A s e time interval and is determined by: Algorithm description of the PSA based on DE.
P (14)T A B L E 214Stop the above steps and find the optimal solution ξ * 15 Calculate the shaved loads P t t t ( = , …, ) for the ith individual.As a result, this individual is likely to conduct global search, indicating that it seeks to make significant adjustments to pursue the superior individuals with low nondomination ranks.It is worth noting that p c ) Exclude N e individuals with the smallest Euclidean distance from F l in proper sequence.In case that two or more individuals have the same smallest Euclidean distance, the one whose second smallest Euclidean distance is the smallest among these individuals will be excluded from F l .Each time an individual p is excluded from F l , any individual whose associated individual is p has to update its minimum Euclidean distance and associated individual.(10)Add the rest of F l Basic data for the ten-unit system.POZs of the ten-unit system.
T A B L E 4

Table 10
and the lowest level of H std for every problem.Most H mean values of NSGA-II-MCEI are higher mean Hypervolume values using five MOEAs for four scenarios.between the H min of NSGA-II-MCEI and the H max of the other two algorithms.The high hypervolume levels suggest the preferable comprehensive performance of NSGA-II-MCEI, and it is mainly due to the modified crossover operator that drives individuals towards real Pareto front.As a result, it is able to find various good The Hmean of NSDE is lower than one-third of the H mean of NSGA-II-MCEI for every case.In the worst case, The H mean of NSDE is lower than one fifth of the H mean of NSGA-II-MCEI for Problem 1. Also, the H std values of NSDE are higher than 0.1, and these values are almost higher by one order of magnitude than those of NSGA-II-MCEI, revealing the instabilities T A B L E 9 Hypervolume values using five MOEAs for four scenarios.Bold values represent highest hypervolume levels and the lowest standard deviation levels. of 30 hypervolume values of NSDE for each problem.In addition, i-NSGA-II-RWS is the second-best MOEA for eight problems, and it always acquires the second-highest hypervolume values and the second-lowest H std values.A statistical analysis is performed to evaluate the performance difference among three MOEAs according to the Wilcoxon rank-sum test 58 at the significance level of 0.05.p Value is a significant indicator of the Wilcoxon ranksum test, and it is employed to judge whether NSGA-II-MCEI is statistically different from the other two MOEAs for eight DEED-PEV problems in this paper.Since the p values in Table 10 are much lower than 0.05, it can be concluded that NSGA-II-MCEI is statistically different from the other two MOEAs.More precisely, NSGA-II-MCEI is statistically better than the other two MOEAs based on the observation that it has higher H Note: T A B L E 10 Hypervolume values using three MOEAs for eight DEED-PEV cases.
Note: Bold values represent highest hypervolume levels and the lowest standard deviation levels.
T A B L E 12 Average spacing values acquired by three MOEAs for eight DEED-PEV problems.