**Figure 1.**
Proposed system model.

**Figure 1.**
Proposed system model.

**Figure 2.**
Pricing tariff. (**a**) Critical peak pricing.; (**b**) Real time pricing.

**Figure 2.**
Pricing tariff. (**a**) Critical peak pricing.; (**b**) Real time pricing.

**Figure 3.**
Hybrid bacterial flower pollination algorithm numerical example.

**Figure 3.**
Hybrid bacterial flower pollination algorithm numerical example.

**Figure 4.**
Load for single home using critical peak pricing. (**a**) load using operational time interval of 1 min; (**b**) load using operational time interval of 20 min; (**c**) load using operational time interval of 30 min; (**d**) load using operational time interval of 60 min.

**Figure 4.**
Load for single home using critical peak pricing. (**a**) load using operational time interval of 1 min; (**b**) load using operational time interval of 20 min; (**c**) load using operational time interval of 30 min; (**d**) load using operational time interval of 60 min.

**Figure 5.**
Load for single home using real time pricing. (**a**) load using operational time interval of 1 min; (**b**) load using operational time interval of 20 min; (**c**) load using operational time interval of 30 min; (**d**) load using operational time interval of 60 min.

**Figure 5.**
Load for single home using real time pricing. (**a**) load using operational time interval of 1 min; (**b**) load using operational time interval of 20 min; (**c**) load using operational time interval of 30 min; (**d**) load using operational time interval of 60 min.

**Figure 6.**
Total cost, waiting time and peak to average ratio for a single home using critical peak pricing. (**a**) Electricity cost using operational time interval of 1, 20, 30, 60 min; (**b**) Waiting time using operational time interval of 1, 20, 30, 60 min; (**c**) Peak to average ratio using operational time interval of 1, 20, 30, 60 min.

**Figure 6.**
Total cost, waiting time and peak to average ratio for a single home using critical peak pricing. (**a**) Electricity cost using operational time interval of 1, 20, 30, 60 min; (**b**) Waiting time using operational time interval of 1, 20, 30, 60 min; (**c**) Peak to average ratio using operational time interval of 1, 20, 30, 60 min.

**Figure 7.**
Total cost, waiting time and peak to average ratio for single home using real time pricing. (**a**) Electricity cost using operational time interval of 1, 20, 30, 60 min; (**b**) Waiting time using operational time interval of 1, 20, 30, 60 min; (**c**) Peak to average ratio using operational time interval of 1, 20, 30, 60 min.

**Figure 7.**
Total cost, waiting time and peak to average ratio for single home using real time pricing. (**a**) Electricity cost using operational time interval of 1, 20, 30, 60 min; (**b**) Waiting time using operational time interval of 1, 20, 30, 60 min; (**c**) Peak to average ratio using operational time interval of 1, 20, 30, 60 min.

**Figure 8.**
Load for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) load using operational time interval of 1 min for 10 homes; (**b**) load using operational time interval of 1 min for 30 homes; (**c**) load using operational time interval of 1 min for 50 homes.

**Figure 8.**
Load for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) load using operational time interval of 1 min for 10 homes; (**b**) load using operational time interval of 1 min for 30 homes; (**c**) load using operational time interval of 1 min for 50 homes.

**Figure 9.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) Electricity cost using operational time interval of 1 min; (**b**) Waiting time using operational time interval of 1 min; (**c**) Peak to average ratio using operational time interval of 1 min.

**Figure 9.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) Electricity cost using operational time interval of 1 min; (**b**) Waiting time using operational time interval of 1 min; (**c**) Peak to average ratio using operational time interval of 1 min.

**Figure 10.**
Load for multiple homes: (10, 30 and 50) using Real time pricing. (**a**) load using operational time interval of 1 min for 10 homes; (**b**) load using operational time interval of 1 min for 30 homes; (**c**) load using operational time interval of 1 min for 50 homes.

**Figure 10.**
Load for multiple homes: (10, 30 and 50) using Real time pricing. (**a**) load using operational time interval of 1 min for 10 homes; (**b**) load using operational time interval of 1 min for 30 homes; (**c**) load using operational time interval of 1 min for 50 homes.

**Figure 11.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using real time pricing. (**a**) Electricity cost using operational time interval of 1 min; (**b**) Waiting time using operational time interval of 1 min; (**c**) Peak to average ratio using operational time interval of 1 min.

**Figure 11.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using real time pricing. (**a**) Electricity cost using operational time interval of 1 min; (**b**) Waiting time using operational time interval of 1 min; (**c**) Peak to average ratio using operational time interval of 1 min.

**Figure 12.**
Load for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) load using operational time interval of 60 min for 10 homes; (**b**) load using operational time interval of 60 min for 30 homes; (**c**) load using operational time interval of 60 min for 50 homes.

**Figure 12.**
Load for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) load using operational time interval of 60 min for 10 homes; (**b**) load using operational time interval of 60 min for 30 homes; (**c**) load using operational time interval of 60 min for 50 homes.

**Figure 13.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) Electricity cost using operational time interval of 60 min; (**b**) Waiting time using operational time interval of 60 min; (**c**) Peak to average ratio using operational time interval of 60 min.

**Figure 13.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using critical peak pricing. (**a**) Electricity cost using operational time interval of 60 min; (**b**) Waiting time using operational time interval of 60 min; (**c**) Peak to average ratio using operational time interval of 60 min.

**Figure 14.**
Load for multiple homes: (10, 30 and 50) using real time pricing. (**a**) load using operational time interval of 60 min for 10 homes; (**b**) load using operational time interval of 60 min for 30 homes; (**c**) load using operational time interval of 60 min for 50 homes.

**Figure 14.**
Load for multiple homes: (10, 30 and 50) using real time pricing. (**a**) load using operational time interval of 60 min for 10 homes; (**b**) load using operational time interval of 60 min for 30 homes; (**c**) load using operational time interval of 60 min for 50 homes.

**Figure 15.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using real time pricing. (**a**) Electricity cost using operational time interval of 60 min; (**b**) Waiting time using operational time interval of 60 min; (**c**) Peak to average ratio using operational time interval of 60 min.

**Figure 15.**
Electricity cost, waiting time and peak to average ratio for multiple homes: (10, 30 and 50) using real time pricing. (**a**) Electricity cost using operational time interval of 60 min; (**b**) Waiting time using operational time interval of 60 min; (**c**) Peak to average ratio using operational time interval of 60 min.

**Figure 16.**
Feasible regions for a single home using critical peak pricing. (**a**) Operational time interval of 1 min; (**b**) Operational time interval of 20 min; (**c**) Operational time interval of 30 min; (**d**) Operational time interval of 60 min.

**Figure 16.**
Feasible regions for a single home using critical peak pricing. (**a**) Operational time interval of 1 min; (**b**) Operational time interval of 20 min; (**c**) Operational time interval of 30 min; (**d**) Operational time interval of 60 min.

**Figure 17.**
Feasible regions for a single home using real time pricing. (**a**) Operational time interval of 1 min; (**b**) Operational time interval of 20 min; (**c**) Operational time interval of 30 min; (**d**) Operational time interval of 60 min.

**Figure 17.**
Feasible regions for a single home using real time pricing. (**a**) Operational time interval of 1 min; (**b**) Operational time interval of 20 min; (**c**) Operational time interval of 30 min; (**d**) Operational time interval of 60 min.

**Table 1.**
Summarized literature review.

**Table 1.**
Summarized literature review.

Schemes | Achievement | Limitations |
---|

GA, BPSO, ACO [6] | Cost, PAR reduction and UC maximization | Computational complexity is not considered |

BFOA, BFOA, GA, BPSO, WDO, GBPSO [7] | Reduces the EC and limits PAR | Trade-off between EC and PAR is not considered |

GA, PSO, WDO, BFO, HGPSO [8] | Minimizes the electricity bill by | EC and PAR reduction are not considered |

scheduling household appliances | |

BFOA [9] | Reduces the EC with affordable UC | Trade-off among EC and |

UC is not considered |

BPSO [10] | Develops efficient scheme to minimize the EC | Privacy of user is not considered |

MOEA [11] | EC minimization and reduction in WT | Consumer’s threshold limit is not focused |

DR programs [13] | Minimize power consumption | Implements the DR program |

peak demand hours not considered |

BPSO [16] | Reduces peak hours demands | Peak demand is reduced |

Reduction in the bill | Electric cost is not considered |

HSA [17] | Reduces operational cost | UC is not considered |

DSM model is presented using GA [18] | Reduces operational cost, PAR | Time complexity is completely ignored |

In-place (PL) generalized algorithm [19] | EC and UC trade-off | Ignores the system complexity |

GA | EC and PAR reduction | System complexity is ignored |

current procedural terminology [20] | | |

GA [21] | EC and PAR is minimized by the proposed scheme | Installation cost is completely ignored |

GA [22] | EC is reduced by using GA | UC is ignored and PAR is also neglected |

GA [23] | EC is minimized with reduction in PAR | UC is neglected |

Hybrid algorithm using FPA and TS [24] | Hybrid version to optimize unconstrained problems | Ignores the optimization problem with multiple constrains |

FPA [25] | Side lobe level minimization and null placement | Ignores interferences in undesired direction |

**Table 2.**
Terms used in equations and their notations.

**Table 2.**
Terms used in equations and their notations.

Terms | Notations |
---|

Electric rate per slot (t) | $E{P}_{rate}^{t}$ |

Power rating per appliance (ap) | ${P}_{rate}^{ap}$ |

Maximum population size | ${N}_{p}$ |

Appliance load | ${L}_{oad}$ |

Scheduled load | ${L}_{oad}^{sch}$ |

Unscheduled L | ${L}_{oad}^{unsch}$ |

Domain of electric rate | ${E}_{rate}$ |

Fitness function | ${E}_{F}$ |

Load per slot (t) | ${L}_{oad}^{t}$ |

Appliances | $app$ |

**Table 3.**
Power rating and length of operational time for operational time interval 20 min.

**Table 3.**
Power rating and length of operational time for operational time interval 20 min.

Group | Appliances | PR (kWh) | LOTs |
---|

Controllable Appliances | Oven | 1.30 | 10.0 |

Kettle | 2.00 | 1.00 |

Coffee Maker | 0.80 | 4.00 |

Rice Cooker | 0.85 | 2.00 |

Blender | 0.30 | 2.00 |

Frying Pan | 1.10 | 3.00 |

Toaster | 0.90 | 1.00 |

Fan | 0.20 | 15.0 |

Shiftable appliances | Washing Machine | 0.50 | 6.00 |

Clothes Dryer | 1.20 | 6.00 |

Non-Shiftable Appliances | Dish Washer | 0.70 | 8.00 |

Vacuum Cleaner | 0.40 | 8.00 |

Hair Dryer | 1.50 | 2.00 |

Iron | 1.00 | 6.00 |

**Table 4.**
Terms used in hybrid bacterial flower pollination algorithm and their notations.

**Table 4.**
Terms used in hybrid bacterial flower pollination algorithm and their notations.

Terms | Notations |
---|

OTIs | t |

Total time in hours | T |

Upper bound | $\alpha $ |

Lower bound | $\beta $ |

Appliances | D |

Fitness | ${E}_{F}$ |

Maximum population size | ${N}_{p}$ |

Newly generated population | Xnew |

Old generated population | X |

**Table 5.**
Electricity cost for 24 h (for single home).

**Table 5.**
Electricity cost for 24 h (for single home).

Techniques | Cost (Cents) Using CPP | |
---|

1 min | 20 min | 30 min | 60 min |
---|

Unscheduled | 1.5323 × ${10}^{3}$ | 1.1210 × ${10}^{3}$ | 1.2912 × ${10}^{3}$ | 1.1319 × ${10}^{3}$ |

BFOA | 848.3800 | 829.7933 | 753.6100 | 952.7100 |

FPA | 785.6600 | 777.5267 | 804.0100 | 1.0423 × ${10}^{3}$ |

HBFPA | 785.6600 | 725.2600 | 608.0100 | 762.3100 |

**Table 6.**
Electricity cost for 24 h (For single home).

**Table 6.**
Electricity cost for 24 h (For single home).

Techniques | Cost (Cents) Using RTP | |
---|

1 min | 20 min | 30 min | 60 min |
---|

Unscheduled | 362.3626 | 269.1267 | 344.2463 | 333.1345 |

BFOA | 280.0945 | 265.0310 | 291.4513 | 275.6915 |

FPA | 286.3116 | 267.5580 | 300.4783 | 276.2255 |

HBFPA | 267.9894 | 235.0647 | 269.3313 | 275.1495 |

**Table 7.**
Peak to average ratio for 24 h (for single homes).

**Table 7.**
Peak to average ratio for 24 h (for single homes).

Techniques | PAR Using CPP | |
---|

1 min | 20 min | 30 min | 60 min |
---|

Unscheduled | 7.241 | 4.9 | 3.47 | 3.3784 |

BFOA | 3.4365 | 6.48 | 3.99 | 3.43 |

FPA | 3.5248 | 3.68 | 2.22 | 2.2289 |

HBFPA | 6.2047 | 4.4388 | 3.4657 | 3.2657 |

**Table 8.**
Peak to average ratio for 24 h (for single homes).

**Table 8.**
Peak to average ratio for 24 h (for single homes).

Techniques | PAR Using RTP | |
---|

1 min | 20 min | 30 min | 60 min |
---|

Unscheduled | 7.5619 | 6.3920 | 6.4850 | 3.6803 |

BFOA | 4.5242 | 3.6784 | 3.8245 | 3.3175 |

FPA | 5.4937 | 3.6784 | 3.4365 | 2.2289 |

HBFPA | 6.2047 | 4.3417 | 4.2125 | 3.2138 |

**Table 9.**
Waiting time for 24 h (for single homes).

**Table 9.**
Waiting time for 24 h (for single homes).

Techniques | Waiting Time Using CPP | |
---|

1 min | 20 min | 30 min | 60 min |
---|

BFOA | 137.3321 | 154.1667 | 86.7857 | 139.2857 |

FPA | 140.0812 | 153.3929 | 102.6786 | 147.8571 |

HBFPA | 214.3473 | 227.5595 | 149.8214 | 135.00 |

**Table 10.**
Waiting time for 24 h (for single homes).

**Table 10.**
Waiting time for 24 h (for single homes).

Techniques | Waiting Time Using RTP | |
---|

1 min | 20 min | 30 min | 60 min |
---|

BFOA | 54.9940 | 130.9762 | 80.2500 | 133.5714 |

FPA | 59.7152 | 147.9762 | 90.5357 | 165.00 |

HBFPA | 153.5777 | 228.5714 | 160.1786 | 130.00 |

**Table 11.**
Random power ratings in (kWh).

**Table 11.**
Random power ratings in (kWh).

Appliances | Power Rating 1 | Power Rating 2 | Power Rating 3 | Power Rating 4 |
---|

Washing-Machine | 0.50 | 0.70 | 0.90 | 0.40 |

Clothes Dryer | 10.0 | 1.20 | 1.40 | 1.60 |

Dish Washer | 0.38 | 0.50 | 0.70 | 0.80 |

Vacuum Cleaner | 0.80 | 1.00 | 0.20 | 0.50 |

Hair Dryer | 1.50 | 1.20 | 1.40 | 1.70 |

Iron | 1.00 | 1.30 | 1.50 | 1.20 |

Oven | 1.30 | 1.50 | 1.70 | 1.90 |

Kettle | 2.00 | 2.15 | 2.40 | 2.14 |

Coffee Maker | 0.80 | 0.40 | 0.50 | 0.20 |

Rice Cooker | 0.85 | 0.89 | 0.72 | 0.79 |

Blender | 0.30 | 0.47 | 0.40 | 0.70 |

Frying Pan | 1.10 | 1.50 | 1.90 | 2.00 |

Toaster | 0.90 | 1.00 | 0.50 | 0.70 |

Fan | 0.20 | 0.50 | 0.40 | 0.70 |

**Table 12.**
Electricity cost for 24 h (for multiple homes).

**Table 12.**
Electricity cost for 24 h (for multiple homes).

Techniques | Cost (Cents) Using CPP for OTI 1 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 4.2213 × ${10}^{5}$ | 1.2535 × ${10}^{6}$ | 2.1077 × ${10}^{6}$ |

BFOA | 3.8669 × ${10}^{5}$ | 1.1425 × ${10}^{6}$ | 1.9247 × ${10}^{6}$ |

FPA | 4.4563 × ${10}^{5}$ | 1.2881 × ${10}^{6}$ | 2.1096 × ${10}^{6}$ |

HBFPA | 0.2128 × ${10}^{5}$ | 0.7098 × ${10}^{5}$ | 1.2205 × ${10}^{6}$ |

**Table 13.**
Waiting time for 24 h (for multiple homes).

**Table 13.**
Waiting time for 24 h (for multiple homes).

Techniques | WT Using CPP for OTI 1 min |
---|

10 homes | 30 homes | 50 homes |
---|

BFOA | 544.0428 | 1.7259 × ${10}^{3}$ | 2.8516 × ${10}^{3}$ |

FPA | 722.7521 | 2.0134 × ${10}^{3}$ | 3.6380 × ${10}^{3}$ |

HBFPA | 1.5302 × ${10}^{3}$ | 4.5947 × ${10}^{3}$ | 7.6534 × ${10}^{3}$ |

**Table 14.**
Peak to average ratio for 24 h (for multiple homes).

**Table 14.**
Peak to average ratio for 24 h (for multiple homes).

Techniques | PAR Using CPP for OTI 1 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 75.9236 | 233.9259 | 381.1938 |

BFOA | 46.1947 | 137.6369 | 231.5394 |

FPA | 49.2012 | 150.6242 | 248.5587 |

HBFPA | 61.5829 | 180.9010 | 304.4469 |

**Table 15.**
Electricity cost for 24 h (for multiple homes).

**Table 15.**
Electricity cost for 24 h (for multiple homes).

Techniques | Cost (Cents) Using RTP for OTI 1 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 2.6912 × ${10}^{5}$ | 7.9680 × ${10}^{5}$ | 1.2990 × ${10}^{4}$ |

BFOA | 2.0976 × ${10}^{5}$ | 6.2206 × ${10}^{5}$ | 1.0124 × ${10}^{6}$ |

FPA | 2.1458 × ${10}^{5}$ | 6.3795 × ${10}^{5}$ | 1.0328 × ${10}^{6}$ |

HBFPA | 2.0176 × ${10}^{5}$ | 5.9593 × ${10}^{5}$ | 9.7160 × ${10}^{5}$ |

**Table 16.**
Waiting time for 24 h (for multiple homes).

**Table 16.**
Waiting time for 24 h (for multiple homes).

Techniques | WT Using RTP for OTI 1 min |
---|

10 homes | 30 homes | 50 homes |
---|

BFOA | 574.6065 | 1.6632 × ${10}^{3}$ | 6.7364 × ${10}^{3}$ |

FPA | 746.8147 | 2.1254 × ${10}^{3}$ | 4.5967 × ${10}^{3}$ |

HBFPA | 2.7175 × ${10}^{3}$ | 3.6963 × ${10}^{3}$ | 7.6552 × ${10}^{3}$ |

**Table 17.**
Peak to average ratio for 24 h (for multiple homes).

**Table 17.**
Peak to average ratio for 24 h (for multiple homes).

Techniques | PAR Using RTP for OTI 1 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 77.0644 | 232.1643 | 394.5380 |

BFOA | 45.6238 | 138.9392 | 234.6092 |

FPA | 50.2105 | 153.5715 | 253.3993 |

HBFPA | 57.1704 | 184.7282 | 302.1077 |

**Table 18.**
Electricity cost for 24 h (for multiple homes).

**Table 18.**
Electricity cost for 24 h (for multiple homes).

Techniques | Cost (Cents) Using CPP for OTI 60 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 1.5059 × ${10}^{4}$ | 4.2003 × ${10}^{4}$ | 7.1746 × ${10}^{4}$ |

BFOA | 1.0666 × ${10}^{4}$ | 2.9568 × ${10}^{4}$ | 4.9553 × ${10}^{4}$ |

FPA | 1.0836 × ${10}^{4}$ | 2.9822 × ${10}^{4}$ | 5.1076 × ${10}^{4}$ |

HBFPA | 0.83 × ${10}^{5}$ | 2.5075 × ${10}^{4}$ | 4.274 × ${10}^{4}$ |

**Table 19.**
Waiting time for 24 h (for multiple homes).

**Table 19.**
Waiting time for 24 h (for multiple homes).

Techniques | WT Using CPP for OTI 60 min |
---|

10 homes | 30 homes | 50 homes |
---|

BFOA | 1.0043 × ${10}^{3}$ | 3.2093 × ${10}^{3}$ | 5.1507 × ${10}^{3}$ |

FPA | 1.0757 × ${10}^{3}$ | 3.2429 × ${10}^{3}$ | 4.1714 × ${10}^{3}$ |

HBFPA | 1.3836 × ${10}^{3}$ | 4.1714 × ${10}^{3}$ | 6.8836 × ${10}^{3}$ |

**Table 20.**
Peak to average ratio for 24 h (for multiple homes).

**Table 20.**
Peak to average ratio for 24 h (for multiple homes).

Techniques | PAR Using CPP for OTI 60 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 38.7901 | 112.7401 | 189.7837 |

BFOA | 23.5727 | 78.1047 | 120.4385 |

FPA | 24.6797 | 75.6910 | 129.6416 |

HBFPA | 30.3457 | 98.0471 | 167.4858 |

**Table 21.**
Electricity cost for 24 h (for multiple homes).

**Table 21.**
Electricity cost for 24 h (for multiple homes).

Techniques | Cost (Cents) Using RTP for OTI 60 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 3.9954 × ${10}^{3}$ | 1.2170 × ${10}^{4}$ | 2.0437 × ${10}^{4}$ |

BFOA | 3.1541 × ${10}^{3}$ | 9.6811 × ${10}^{3}$ | 1.6242 × ${10}^{4}$ |

FPA | 3.1884 × ${10}^{3}$ | 9.8798 × ${10}^{3}$ | 1.6251 × ${10}^{4}$ |

HBFPA | 3.1877 × ${10}^{3}$ | 1.6242 × ${10}^{3}$ | 1.6039 × ${10}^{4}$ |

**Table 22.**
Waiting time for 24 h (for multiple homes).

**Table 22.**
Waiting time for 24 h (for multiple homes).

Techniques | WT Using RTP for OTI 60 min |
---|

10 homes | 30 homes | 50 homes |
---|

BFOA | 1.4314 × ${10}^{3}$ | 4.0729 × ${10}^{3}$ | 6.7364 × ${10}^{3}$ |

FPA | 1.4371 × ${10}^{3}$ | 3.8336 × ${10}^{3}$ | 6.869 × ${10}^{3}$ |

HBFPA | 1.3964 × ${10}^{3}$ | 4.1579 × ${10}^{3}$ | 6.9236 × ${10}^{3}$ |

**Table 23.**
Peak to average ratio for 24 h (for multiple homes).

**Table 23.**
Peak to average ratio for 24 h (for multiple homes).

Techniques | PAR Using RTP for OTI 60 min |
---|

10 homes | 30 homes | 50 homes |
---|

Unscheduled | 37.3187 | 115.6367 | 195.2858 |

BFOA | 25.9190 | 74.4312 | 122.2942 |

FPA | 25.8770 | 80.9041 | 129.3206 |

HBFPA | 33.8192 | 102.8952 | 166.9638 |