p i t = p i N × u i t u i t = 0 , t ∈ 1 , 24 ∪ t ∉ α i , β i ∑ t = α i t = β i u i t = H i α i < t < β i , α i − β i ≥ H i (1) As shown in Eq. 1, where i denotes power-adjustable load, p i t is the power consumed by power-adjustable load i at time t , and p i N is the nominal power of power-adjustable load i . The u i t is the status of load i at time t , where 0 is off and 1 is on; α i and β i are the working time range for power-adjustable load i ; and H i is the duration time of load i .

Air conditioners, water heaters, household batteries, and EVs are power-adjustable loads. Air conditioners have two states, cooling and heating, and its load models are as follows Eqs 2–6: u A C , C t = 1 , T A C , C , S + Δ T A C , C < T i n t u A C , C t − 1 , T A C , C , S < T i n t ≤ T A C , C , S + Δ T A C , C 0 , T i n t ≤ T A C , C , S (2) H C = ∑ t = 1 24 u A C , C t (3) u A C , H t = 1 , T A C , H , S + Δ T A C , H < T i n t u A C , H t − 1 , T A C , H , S − Δ T A C , H < T i n t ≤ T A C , H , S 0 , T i n t ≤ T A C , H , S (4) H H = ∑ t = 1 24 u A C , H t (5) p j t = P N C × u A C , C t refrigeration p j t = P N H × u A C , H t heating (6)

The relationship between the indoor temperature change and operating power of air condition can be expressed as Eq. 7: T in t = T in t − 1 + a T out t − T in t − 1 + b p t (7) where T A C , C , S and T A C , H , S are the temperature set in the cooling and heating state of the air conditioner at time t, respectively; Δ T A C , C and Δ T A C , H are the room temperature range set in the cooling and heating state of air conditioners, respectively; T in t is the indoor temperature at time t ; T out t is the outdoor temperature at time t ; a is the influence coefficient of outdoor temperature on indoor temperature; b is the operating coefficient of the air conditioner, where b < 0 means that the air conditioner operates in the cooling state and b > 0 means the air conditioner operates in the heating state; and u A C , C t and u A C , H t are the start-stop variable of the air conditioner at time t period, where 0 is stopping and 1 is starting.

The water heater load model is shown in Eq. 8: u W H t = 1 , T W H t < T W H , S − Δ T W H u W H t − 1 , T W H , S − Δ T W H ≤ T W H t < T W H , S 0 , T W H , S ≤ T W H t (8) where T W H , s is the water temperature set by the water heater; Δ T W H is the set range of water temperature; T W H t is the water temperature at time t ; and u W H t is the start-stop variable of the water heater at time t , where 0 is stopping and 1 is starting.

The output model of the BT is shown in Eq. 9: S O C B T t = C B T , n e t t / C B T , b c t S O C B T t + 1 = S O C B T t + p B T , c t × Δ t × θ B T , c / C B T , b a t S O C B T t + 1 = S O C B T t − p B T , d t × Δ t × θ B T , d / C B T , b a t 0 ≤ p B T , c t ≤ p B T , c , ⁡ max 0 ≤ p B T , d t ≤ p B T , d , ⁡ max S O C B T , ⁡ min ≤ S O C B T t ≤ S O C B T , ⁡ max ∑ t = 1 T   u B T t + 1 − u B T t ≤ λ B T p B T , c t × p B T , d t = 0 (9)

Output model of an EV shown in Eq. 10: S O C E V t = C E V , net   t C E V , b a t S O C E V t + 1 = S O C E V t + p E V , c t × Δ t × θ E V , c C E V , bat   S O C E V t + 1 = S O C E V t − p E V , d t × Δ t × θ E V , d C E V , b a t t 0 ≤ p E V , c t ≤ p E V , c , ⁡ max 0 ≤ p E V , d t ≤ p E V , d , ⁡ max S O C E V , ⁡ min ≤ S O C E V t ≤ S O C E V , ⁡ max ∑ t = 1 T   u E V t + 1 − u E V t ≤ λ E V p E V , c t × p E V , d t = 0 (10) where S O C B T t and S O C E V t are the BT and EV state of charging (SOC) at time t , respectively; C B T , n e t t and C E V , n e t t are the remaining battery capacity of the BT and EV at time t , respectively; p B T , C t and p B T , d t are the charging and discharging power of the BT at time t , respectively; p E V , c t and p E V , d t are the EV charging and discharging power at time t , respectively; θ B T , c and θ B T , d are the BT charging and discharging efficiency, respectively; θ E V , c and θ E V , d are the EV charging and discharging efficiency, respectively; P B T , c , ⁡ max and P BT , d , ⁡ max are the maximum charging and discharging power of the BT, respectively; P EV , c , ⁡ max and P EV , d , ⁡ max are the maximum charging and discharging power of the EV, respectively; S O C B T , ⁡ max and S O C B T , ⁡ min are the maximum and minimum charge state value of the BT, respectively; S O C E V , ⁡ max and S O C E V , ⁡ min are the maximum and minimum SOC value of the EV, respectively; u B T t and u E V t are the charging and discharging variables of the BT and EV at time t , respectively , where the value is 0 or 1; and λ BT and λ E V are limit of charging and discharging times of the BT and EV, respectively.

The time-transferable load has the delayed start function, which can transfer the working interval but cannot reduce the load. The startup and running time can be flexibly set according to the needs of users. p j t = p j N ∗ u j t u i t = 0 , t ∈ 1 , 24 ∪ t ∉ α j , β j ∑ t = t j start t j end u j t = ∑ t = α j β j u j t = H j t j end − t j start + 1 = H j α j ≤ t j start ≤ t ≤ t j e n d ≤ β j β j − α j ≥ H j (11) As shown in Eq. 11, where j denotes the time-transferable load; p j t is the electricity power of the time-transferable load j at time t ; p j N is the rated power of time-transferable load j ; u j t is start-variable 0–1, where 1 represents the operation of time-transferable load j and 0 indicates the time-transferable load is off; α j and β j are the time-transferable load j allowable start and stop time of operation, respectively; t j s t a r t and t j e n d are the time-transferable load j start and end time of actual operation, respectively; and H j is duration over which time-transferable load j needs to work.

The uncontrollable load scheduling model is shown in Eq. 12: u k t = 0 , t ∈ 1 , 24 ∪ t ∉ α k , β k 1 , t ∈ α k , β k (12) where u k t is the start-stop variable of uncontrollable load k at time t , where 0 indicates off; α k and β k are the allowable start and end time of uncontrollable load k , respectively.

For household users, the electricity purchase cost C 1 includes the electricity consumption cost of the uncontrollable load, time-transferable load, and power-adjustable load. A complete dispatching cycle can be divided into T periods, and the household user’s electricity purchase cost C 1 can be expressed as Eq. 13: C 1 = ∑ t = 1 T E t − p v , used t pr i b t (13) where pr i b t is the time-of-use electricity price at time t ; p v , used t is the photovoltaic power consumed by household appliances at time t ; and E t is the total energy consumption of all appliances at time t . E c c t = ∑ i = 1 m   p i t u i t Δ t (14) E c u t = ∑ j = 1 m   p j t u j t Δ t (15) E un   t = ∑ k = 1 m   p k t u k t Δ t (16) E t = E c c t + E c u t + E u n t (17) As shown in Eqs 14–17, where p i t , p j t , and p k t are the power of the power-adjustable load, time-transferable load, and uncontrollable load at time t , respectively; m , n , and l are the number of the corresponding load; E c c t , E c u t , and E u n t are the power consumption of power-adjustable load, time-transferable load and uncontrollable load at time t , respectively.

Household user’s profit C 2 from selling electricity is shown in Eqs 18–20: C 2 = ∑ t = 1 T   p g t pr i g t (18) p g t = p v , g t + p E V , d t (19) p v t = p v , used   t + p v , g t (20) where p v t is the photovoltaic supply power at time t and p v , g t is the photovoltaic power sold to the grid at time t . The p g t is the power that household users sell electricity to the grid at time t ; pr i g t is the price that users sell electricity at to the grid at time t ; and p E V , d t is the discharge power of EV at time t .

The total electricity cost C includes electricity purchase cost C 1 and profit from selling electricity, as shown in Eq. 21: C = C 1 − C 2 (21)

The carbon dioxide emission generated by the household user is shown in Eq. 22: Q c t = E t h E c c t + E c u t + E u n t − p v t (22) where Q c t is the carbon dioxide emission generated by the household user’s electricity usage at time t , and E t h is the carbon emission coefficient.

The amount of carbon emission quota M c t obtained by household users from external electricity purchase at time t is shown as Eq. 23: M c t = ε E c c t + E c u t + E u n t + p E V , c t (23) where ε is the carbon emissions quota allocation coefficient.

When the free carbon quota received by households is greater than their actual carbon emissions, household users can sell their excess carbon emissions to gain profits. If the free carbon quota received by household users is less than their actual carbon emissions, household users need to buy the required carbon emissions from the carbon trading market. Therefore, the carbon trading cost of household users at time t is shown as Eq. 24: R genbon t = q th Q c t − M c t (24) where R genbon t is the carbon trading cost of households at time t . When R genbon t is positive, it means households need to spend extra money to buy the required carbon quota; when R genbon t is negative, it means the income of households that can sell carbon quota. q t h is the carbon trading price.

The carbon quota M E V t obtained by the discharge of an EV at time t is as follows Eqs 25, 26: M E V t = p E V , c t − p E V , d t Δ t L E V E g a s − p E V , c t β t Δ t E t h + p E V , d t Δ t E t h (25) β t = E t − p v t E t (26) where Δ t is the time step; M E V t is the carbon quota owned by the EV at time t ; L E V is the distance that 1 kwh EV can travel; E g a s is the carbon emission of the fuel-using car driving 1 km; E t h is the carbon emission of per output power of thermal power; and β t is the proportion of thermal power capacity in EV charging quantity at time t , with the proportion of the total thermal power output in the total system output at time t used for calculation.

The carbon quota income R carbon t that the EV can sell at time t is shown in Eq. 27: R carbon t = q e v M E V t (27) where q e v is the EV carbon quota price.

Considering the total electricity cost of the household, household user’s carbon trading cost, battery degradation cost, and EV carbon quota income comprehensively, the objective of the HEMS is to minimize the total comprehensive operation cost: min ⁡ F = ⁡ min C + ∑ T t = 1 R genbon t − R carbon t + C battery t (28) As shown in Eq. 28, where C battery t represents the cost of battery degradation in both the EV and the BT. C battery t = c bat E bat + c L × p E V , d t L c × E bat DOD (29) As shown in Eq. 29, where c bat is the battery cost (which includes the EV battery and BT); c L is the labor cost for battery replacement; E bat represents the battery capacity; L c is the cycle life of batteries; and DOD is the discharge depth at L c .

The expected operation time represents the user’s comfort of energy use. Therefore, the energy consumption time of the time-transferable load is used to represent user satisfaction, and its satisfaction constraint is shown in Eq. 30: t j start ≤ t ≤ t j end (30)

During the scheduling process, users need to comply with the following power balance constraints: p b u y t + p V t + p E V , d t + p B T , d t = E c c t + E c u t + E u n t (31) As shown in Eq. 31, where the p b u y t is the user’s purchased power at time t .

In summary, the objective function is Eq. 28, and the power balance equality constraint is Eq. 31. Other relevant constraints have been given in the corresponding load models above.

During the parameter initialization phase, the standard PSO algorithm initializes particle positions and velocities with random numbers, resulting in suboptimal exploration of the solution space and limited global search capabilities, particularly in constraint optimization scenarios. Furthermore, the standard PSO algorithm is susceptible to premature convergence and loss of diversity, ultimately hindering its ability to attain highly accurate optimal solutions in HEMS optimization contexts.

The IPSO retains the population diversity, and the initial population is within the feasible domain of particles, which improves the quality of the initial population particle solution. The initial moment the particle swarm population is generated as shown in Eq. 32: x i , int = L b i + r n e w U b i − L b i (32) where x i , int is the initial particle population of the IPSO; r n e w are uniform random numbers for the IPSO; L b i is the lower limit of the i th particle solution in the IPSO; and U b i is the upper limit of the i th particle solution in the IPSO algorithm.

After the improvement, the PSO algorithm first compares the fitness value F i t i of each particle with the individual extreme p i d , and if F i t i < p i d , p i d is replaced with F i t i . F i t i is then compared with the global extremum p g d , and if F i t i < p g d , p g d is replaced with F i t i .

At the same time, in order to alleviate the shortcomings of premature convergence and diversity loss in the standard PSO and improve the solvability of the algorithm in constrained optimization problems, the IPSO improves the update of the standard PSO, and the update formula is shown in Eq. 33: x i , n + 1 = x i , n + b e t a × p i d − x i , n + a l p h a × r n e w U b i − L b i (33) where n is the current iteration number, x i , n is the position of the i th population in the IPSO; b e t a is adaptive coefficient; a l p h a is convergence factor p i d is Individual extremum.

The IPSO algorithm using Eq. 33 can update and be solved during the optimization process, effectively improving the algorithm’s constraint solving ability and enhancing the PSO algorithm’s ability to find the global optimal solution. However, after updating the position of particles in the population, there may be situations where some particles exceed the population constraint boundary, which greatly reduces the efficiency of the algorithm in searching for particles in the feasible domain. The IPSO algorithm improves the above situation by applying a boundary function to the updated particle population, thereby enhancing the efficiency of the algorithm in searching for feasible solutions. The boundary function of the IPSO algorithm is as follows Eq. 34: x i , n + 1 =   L b i , i f   x i , n + 1 ≤ L b i x i , n + 1 , i f   L b i < x i , n + 1 < U b i U b i , i f   x i , n + 1 ≥ U b i (34)

DE has fewer parameters and is relatively simple to calculate, making it widely used in power scheduling problems. The main process of DE can be shown as five parts (Initialization, Mutation, Crossing, Selection, and Termination):

(1) Initialization:

DE/current-to-rand/1 V i , G + 1 = X i , G + F i X best , G − X i , G + F i X r 1 , G − X r 2 , G (37)

DE/best/1 V i , G + 1 = X best , G + F i X r 1 , G − X r 2 , G (38) where r 0 , r 1 , r 2 ∈[1,N _P], NP is a random number that is not identical to each other. The X r 1 , G and X r 2 , G is the difference of randomly selecting two vectors. The X best , G is the optimal individual in the G generation population. The scaling factor is F i .

(3) Crossing:

This study assesses the performance of the algorithm by employing the Sphere, Ackley, Rastrigin, and Griewank functions, with an optimal value of 0 for these functions. The pertinent parameters of the test functions are presented in Table 1. Additionally, the DE, PSO, and IPSO methods are concurrently selected for comparison. Figure 3 illustrates the convergence trajectories of these algorithms, each independently solving the functions 500 times in a 100-dimensional space.

Figure 2 verifies the effectiveness of the IPSO. The IPSO has the best convergence effect compared to PSO and DE, while its final convergence value is closest to the optimal extreme value. The PSO optimization effect is the worst. By adjusting with the introduction of random learning factors, the convergence of the algorithm has been improved.

From Table 2, comparing the algorithms in solving complex functions, the IPSO has a faster convergence time than the DE. For the optimization process of the HEMS, a large amount of computing resources is necessary. After comparing the above algorithms, this study selects the IPSO for optimization.

In summary, the IPSO algorithm can enhance computational efficiency and global search capability. The flow chart of the HEMS can be seen in Figure 3. The steps and procedures of the HEMS optimization based on the IPSO algorithm are as follows:

Step 1:

Initialize the EV charging power, wind power generation, etc., as well as the initial parameters of the PSO algorithm such as attenuation coefficient a l p h a , b e t a , individual extremum p i d , global extremum p g d , etc.

Step 2:

Determine whether the time condition and the number of iterations conditions are met. If the conditions are met, proceed to Step 3, otherwise exit the program.

Step 3:

Calculate the medium and inequality constraints of the household energy management system and their penalty functions.

Step 4:

Obtain the objective function value.

Step 5:

If the individual extremum p i d is less than or equal to the global extremum p g d , the value of the global optimization solution is updated, otherwise Step 4 is returned at the same time.

Step 6:

Update particle swarm attenuation coefficient alpha, beta.

Step 7:

Update the position of the particle according to Eq. 33.

Step 8:

Determine whether the end condition is met, and if so, exit (error reaches set accuracy or reaches the maximum number of cycles), otherwise return to Step 3 to continue the calculation.

The following scenarios are set for the comparison of several different schemes:

Scenario 1:

Initial household electricity cost and carbon emissions do not consider optimal scheduling;

Scenario 2:

Optimal scheduling of household energy does not consider carbon trading and time satisfaction;

Scenario 3:

Optimal scheduling of household energy only considers time satisfaction and does not consider carbon trading;

Scenario 4:

Optimal scheduling of household energy only considers carbon trading and does not consider time satisfaction;

Scenario 5:

Optimal scheduling of household energy considering carbon trading and time satisfaction.

Scenario 2, Scenario 3 show that the carbon trading cost of household users and the carbon quota income of EVs are not considered. Scenario 4, Scenario 5 show that the carbon trading cost of household users and the carbon quota income of EVs are considered. Scenario 1 is the initial electricity cost and carbon emissions of households without considering optimal scheduling; in Scenario 2, time satisfaction constraints, user carbon trading costs, and the carbon quota income of EVs are not considered, and the goal is minimizing the total electricity cost; in Scenario 3, carbon trading is not considered, but the time satisfaction constraint is considered, and the total electricity cost is minimized; Scenario 4 considers the user’s carbon trading cost, EV carbon trading income, and the user’s total electricity cost, and the objective is minimizing the user’s comprehensive operation cost, but the time satisfaction constraint is ignored; Scenario 5 considers the user’s total electricity cost, carbon trading cost, and EV carbon trading income, with the objective being minimizing the household user’s comprehensive operating cost with time satisfaction constraints. During the optimization process, the user’s typical daily load in summer is selected as the optimization data.

Figure 4 shows the typical outdoor temperature, the optimized indoor temperature, and the water heater temperature. Figures 5–10 show the power in Scenario 1, Scenario 2, Scenario 3, Scenario 4, Scenario 5.

As shown in Figure 4, the air conditioner remains on, the indoor temperature decreases significantly, and the user’s room can reach the required temperature range. After the water heater is turned on, the water temperature rises to meet the user’s needs.

Figure 5 shows the load comparison between different scenarios. The load increased sharply from 3:00 to 6:00, and the load also increased between 22:00 and 24:00. From 8:00 to 16:00, the load of users decreased sharply. The reason for this was that some of the load was transferred to other periods to reduce the costs.

In Figure 6, the air conditioner has been turned on and the user does not consider the influence of the electricity price on the total household appliances’ cost. When the electricity price is high, the cost of electricity consumption increases accordingly. Discharge of the EV is not used and the charging/discharging behavior of the BT is relatively random. In Figure 7, all electrical appliances run at the lowest electricity price within the allowable time period. Meanwhile, EVs and batteries charge when at low electricity prices and discharge when at high electricity prices. When the photovoltaic output and BT discharge exceed the electricity used by electrical appliances, the user will sell the excess electricity at noon.

In Scenario 1, the electricity cost of the air conditioner is 13.8 CNY; in Scenario 2, the operation time of the air conditioner is significantly reduced, and its electricity cost is 3 CNY and 21.7% less than Scenario 1, and the indoor temperature meets the requirements of the user. The total electricity consumption of the water heater remains unchanged; in Scenario 1, its running time is 19:00 to 22:00 and the electricity cost is 2.7 CNY; in Scenario 2, the running time is 19:00 to 21:00 and 23:00 to 24:00 and the electricity cost is 2.25 CNY and 0.45 CNY, respectively, and 16.7% less than Scenario 1. The user’s water demand is met.

In Scenario 1, the EV is charged between 21:00 and 05:00, and the battery is charged. In Scenario 2, the EV and battery are charged between 23:00 and 07:00, and the electricity cost is 2.25 CNY less than in Scenario 1. In Scenario 1, the EV does not discharge and the BT discharges randomly. In Scenario 2, the discharge of the EV is at 19:00 to 23:00, the BT discharges at 10:00–15:00 and 20:00–23:00, and the electricity cost of the EV and BT is 3.91 CNY less than Scenario 1. The washing machine, rice cooker, dishwasher, vacuum cleaner, and electric kettle all operate during the valley period, when the electricity price is lowest, which significantly cuts the total household electricity cost compared with Scenario 1. However, in Scenario 2, the use of vacuum cleaners is advanced to 5:00, and the use of washing machine and dishwashers is delayed to 23:00, but the user’s usage habits are not considered, resulting in low satisfaction. Compared with Figure 6, considering the time satisfaction of the user with electricity consumption in Scenario 3 of Figure 8, the usage time distribution of the electrical appliances is more reasonable. The vacuum cleaner is transferred from 05:00 to 09:00, the dishwasher changes working time from 23:00 to 21:00, the washing machine changes working time from 23:00 to 17:00, and the electric kettle from 22:00 to 19:00. When the users’ time satisfaction constraint is met, the electricity cost of the time-transferable load is 1.32 CNY more than Scenario 2. In Figures 9, 10, the running conditions of most appliances do not change greatly when carbon trading is considered. In Scenario 4, Scenario 5, carbon trading is considered. The reduction in EV mileage is no less than the increase in the discharge of the EV. Figure 11 gives a comparison of the cost and emissions in each scenario, and the battery degradation cost in each scenario is shown in Table 6.

As shown in Figure 11, compared with Scenario 1, the total comprehensive operation cost and carbon emissions of Scenario 2, Scenario 3, Scenario 4, Scenario 5 have significantly decreased. After considering the time satisfaction constraint in Scenario 3, Scenario 5, compared with Scenario 2, Scenario 4, the electricity purchase cost increases by 1.13 CNY and 0.63 CNY, respectively. In Scenario 4, Scenario 5, when the carbon trading is considered, the system obtains carbon quota income, and the comprehensive total cost is reduced without carbon trading, and its carbon emissions are also reduced. As shown in Table 6, the battery degradation cost in Scenario 1 is much less than it in other scenarios, because EVs do not participate in scheduling, there is no battery degradation cost for EVs, and the overall battery degradation cost for households is lower than in other scenarios.

This paper discusses the influence of carbon trading price on dispatching and adjusts the carbon trading price to 0.39 CNY/kg, 0.49 CNY/kg, 0.59 CNY/kg, and 0.69 CNY/kg, respectively. Table 7 shows the impact of carbon trading prices on the HEMS.

With the increase in carbon trading price in Table 7, the negative carbon trading cost of users is increasing, the EV carbon quota income is also increasing, and the overall operating cost is decreasing. Therefore, the guidance and regulation of carbon trading prices can play a guiding role in the HEMS.