Multiobjective Decision-Making Model for Power Scheduling Problem in Smart Homes

Abstract

The aim of this study was to solve power scheduling issues in smart homes to enable demand response in smart grids. The objective of demand response is to match demand with supply by reflecting supply expectations through consumer price signals, and especially to avoid peak demand during times of high prices and when supply is limited. Three objectives were considered: first, economic rationing by minimizing the total costs for consumers with the given hourly prices; second, to achieve better efficiency in terms of supply and greater stability in a power system by reducing peaks in usage or load, which is defined by minimizing the percentage of power rate; third, related to consumer comfort levels, by reducing variance in the schedule of appliances to actual usage periods requested. This multiobjective power scheduling problem for smart homes (PHPSH) was explored using a nondominated sorting genetic algorithm, called NSGA-II. The results showed that the Pareto-optimal solutions from NSGA-II are compatible with the weighted-sum-based model from the literature, and viable alternatives are available for end users with different weighted objectives.

1. Introduction

As human daily activities become more automated and digitalized, any power disruption can have dire consequences. Electricity crises can be caused by a high degree of uncertainty concerning both the demand for and supply of energy, especially when coupled with the utilization of sustainable and renewable energy. Sources of renewable energy, such as solar, wind, and hydropower, are intermittent and can make the power grid unstable.

According to the Energy Independence and Security Act of 2007, a smart grid is “a distribution system that allows for flow of information from a customer’s meter in two directions: both inside the house to thermostats, appliances, and other devices, and from the house back to the utility” [1]. To improve the reliability and efficiency of an electricity system, a smart grid uses digital technology to better balance supply and demand uncertainty with the capabilities of real-time monitoring and control. It integrates power and distribution systems, and implements advanced communication technology [2,3]. In terms of the integration of power generation and distribution systems,

Table 1. Comparisons in power system design features between smart grids and existing grids (adopted from [4]).

Design Feature	Smart Grid	Existing Grid
Device technology	Digital	Electromechanical
Direction of information	Two-way communication	One-way communication
Power production	Distributed generation	Centralized generation
Distribution structure	Network	Hierarchical
Monitoring method	Sensor throughout	Few sensors
Monitoring capability	Self-monitoring	Blind
Recovering capability	Self-healing	Manual restoration
Fault ride-through	Adaptive and islanding	Failures and blackouts
Fault detection	Remote check/test	Manual check/test
Control capability	Pervasive control	Limited control
Service selection	Many customer choices	Few customer choices

lists the major differences between smart grids and the existing model in terms of power system design features [4]. Issues identified in the smart grid design and implementation can be categorized into three system features: (1) a smart infrastructure system that focuses on the application of devices and communication technologies; (2) a smart management system that focuses on the process design in managing, monitoring, and control; (3) a smart protection system that focuses on failure protection, reliability, and the security of the entire grid with the use of storage and renewable energy [5].

However, the benefits and challenges of converting to a smart grid are not limited to supply. The Energy Independence and Security Act of 2007 defines a smart grid “to include a variety of operational and energy measures—including smart meters, smart appliances, renewable energy resources, and energy efficiency resources” [1]. Smart meters, referred to as advanced metering infrastructure (AMI), provide accurate billing information and allow customers to monitor their energy consumption and obtain real-time data, such as time-of-use pricing [6]. Due to the instability of power generation and high power generation costs, residential houses are encouraged to install AMIs to monitor electricity consumption and take advantage of electricity pricing policies. Ultimately, customers can reduce electricity costs by shifting energy usage from high-cost to low-cost periods [7]. This framework is often associated with demand response, a technique derived from the demand side of energy management. The purpose of demand-side management is to match the load shape to the power generation curve and to a higher level while maintaining grid reliability [7,8]. The most commonly considered objectives include: (1) minimization of electricity cost; (2) minimization of peak demand or peak to average ratio; (3) minimization of user discomfort; (4) maximization of utilization in local energy sources [9].

The demand response method is used to motivate customers to avoid electricity usage with time-sensitive tariffs, when the load might surpass the supply [8,10]. A report by the Environmental Change Institute at the University of Oxford in 2006 found that home users receiving immediate data from their meter made average savings ranging from 5% to 15%. Additionally, it showed that by implementing the time of use pricing technique, a reduction of up to 30% peak demand could be achieved [11]. The electricity consumed by smart appliances, such as washing machines, tumble dryers, and dishwashers, is lower by up to 50% compared with that of conventional appliances [12]. The promotion of time-of-use electricity prices allows users to improve control over their own electricity consumption, reduce electricity costs, and maintain the stability of the power grid by avoiding undesirable power peaks. The flexibility of smart appliances necessitates an optimal appliance schedule based on estimated time-of-use prices from power companies. This problem is referred to as the power scheduling problem for smart homes (PHPSH) [13,14].

Most scheduling problems can be solved using exact algorithms, heuristic algorithms, or approximation algorithms. Due to the complexity of the PHPSH, a genetic algorithm (GA), a metaheuristic and population-based algorithm, is the most adopted method in the literature [14]. However, despite the multiple objectives presented by the PSPSH, research has mainly focused on one objective and treated others as soft constraints. Some researchers, such as Makhademeh et al. [13], Rahim et al. [15], and Makhademeh et al. [16], solved the multiobjective optimization issue by aggregating objectives using the weighted sum method. These non-Pareto algorithms yielded only one final optimal solution based on the importance, or weight parameters, of each objective. The challenges of finding Pareto optimality are the possibly conflicting nature of these objectives and the inefficiency in Pareto approximation when a greater number of objectives are considered [17,18].

In this study, a multiobjective decision-making model was proposed for the PHPSH with the following three objectives: first, to minimize total electricity cost with real-time pricing provided by power companies at the beginning of day [9]; second, to minimize the peak-to-average ratio on that day [9]; third, to minimize user discomfort levels. A higher consumer comfort level means there is a smaller gap between the user’s expected time and the actual scheduled time in using appliances, which is almost as important as reducing the electricity cost for end users [13].

Furthermore, to provide consumers with schedule alternatives, the methodology of this study was to find Pareto-optimal solutions using a nondominated sorting genetic algorithm, called NSGA-II, which is a multiobjective evolutionary algorithm (MOEA). A Pareto-optimal solution is an optimal solution when not dominated by other feasible solutions in terms of at least one objective [18,19]. The major advantage of using MOEA is the ability to find multiple Pareto-optimal solutions in one run [19].

Proposed by Srinivas and Deb (1994), the nondominated sorting genetic algorithm (NSGA) is an extension of the genetic algorithm used to find Pareto-optimal solutions for multiobjective optimization problems. To avoid bias in objective weighting, NSGA abandons the use of fitness values as the basis for selecting chromosomes, and adopts a ranking selection method that calculates the crowding distance to rank chromosomes within subpopulations of good solutions [20].

Deb et al. (2002) tried to improve NSGA by restoring an elitism approach, called NSGA-II, to enhance convergence properties. In order to preserve and ensure that high-quality chromosomes can smoothly evolve, an elitism approach was adopted in the selection mechanism to retain the best chromosomes [21]. NSGA-II simultaneously deals with multiple objectives and yields a set of alternatives on the Pareto frontier as the optimum [22] and has been adopted in a variety of scheduling problems [23].

2. Mathematical Model and Methods

The power scheduling problem in a smart home (PSPSH) assumes that end users adopt a real-time pricing (RTP) strategy for power consumption. They are charged at hourly prices, which are quoted one day or less in advance to reflect the utility’s production costs. The notations used in this multiobjective decision making model are as follows:

• $n$: Number of time slots available for scheduling.
• $m$: Number of appliances.
• $l_{\mathrm{i}}$: Total number of operating time slots of appliance i,$i=1,2,\dots ,m$.
• $p{s}_i$: Electricity consumption of appliance i in a single time slot, $i=1,2,\dots ,m$.
• $E{S}_i$: Earliest time slot of preferred operating period for appliance i, $i=1,2,\dots ,m$.
• $L{F}_i$: Latest time slot of preferred operating period for appliance i, $i=1,2,\dots ,m$.
• $o_i$: The number of time slots in which appliance i is not operating during the preferred period, $i=1,2,\dots ,m$.
• $p{c}^j$: Electricity price at time period j,$j=1,2,\dots ,n$.
• $P_{max}$: The maximum power consumption in any time slot.
• $P_{avg}$: The average power consumption of all time slots.
• $Cost$: The total electricity cost.
• $C_{exp}$: The expected daily electricity cost set by the consumer.

The decision variable is a binary variable and defined as follows:

$$u_j^i=\left\{\begin{array}{l}1, \mathrm{if} \mathrm{appliance} i \mathrm{is} \mathrm{operating} \mathrm{at} \mathrm{time} j\\ 0, \mathrm{otherwise}\end{array}\right., i=1,2,\dots ,m;j=1,2,\dots ,n.$$

The mathematical model for our problem is as below:

$$\mathrm{Minimize} \mathrm{CR}=\frac{Cost}{C_{exp}}$$

(1)

$$\mathrm{Minimize} \mathrm{PAR}=\frac{P_{max}}{P_{avg}}$$

(2)

$$\mathrm{Minimize} \mathrm{WTR}={{\displaystyle \sum }}_{i=1}^m\frac{o_i}{(L{F}_i-E{S}_i)+1}$$

(3)

and subject to:

$${{\displaystyle \sum }}_{j=1}^n{u}_i^j=l_i, i=1,2,\dots ,m$$

(4)

$$Cost={{\displaystyle \sum }}_{j=1}^{n}p{c}^j\ast {{\displaystyle \sum }}_{i=1}^{m}p{s}_i\ast u_i^j.$$

(5)

$$P_{max}\ge {{\displaystyle \sum }}_{i=1}^{m}p{s}_i\ast u_i^j, j=1,2,\dots ,n$$

(6)

$$P_{avg}=\frac{{{\displaystyle \sum }}_{i=1}^{m}p{s}_i\ast u_i^j }{n}.$$

(7)

$$o_i={{\displaystyle \sum }}_{j=1}^{E{S}_i-1}{u}_i^j+{{\displaystyle \sum }}_{j=L{F}_i+1}^n{u}_i^j, i=1,2,\dots ,m$$

(8)

$$u_j^i\in \left\{0,1\right\}, i=1,2,\dots ,m; j=1,2,\dots ,n$$

(9)

There were three objectives for the PSPSH in this study. The first objective was to minimize the ratio of total electricity cost with respect to the expected daily electricity cost set by the consumer, which is called the cost rate (CR), and is defined in Equation (1). The second objective was to minimize the ratio of peak value of the power load to the average power load, which is referred to as percentage of power rate (PAR) and is defined in Equation (2). The third objective was to consider the user comfort level by minimizing the deviation in operating time from user preferences, which is referred to as waiting time rate (WTR), and is defined in Equation (3). Equation (4) ensures that the number of time slots for each appliance to be operational in meets its required time length. Equation (5) calculates the actual total cost. Equations (6) and (7) define the maximum and average power load. Equation (8) summarizes the number of time slots when any appliance is scheduled outside of the expected operating period. Equation (9) indicates that our decision variables are binary.

3. Numerical Example and Results

The numerical example used in this study was adopted from a study by Makhadmeh et al. [13] to simulate appliance scheduling based on the electricity consumption of a single household. The appliances for scheduling can be divided into shiftable appliances (SAs) and nonshiftable appliances (NSAs) [13]. SAs are appliances that can automatically run, such as dishwashers and washing machines, which can be scheduled in advance. NSAs are appliances that need to be manually operated, such as lamps and hair dryers, which cannot be scheduled in advance. Hence, SAs were used for power scheduling in our study.

3.1. Real-World Dataset

In the simulation, 10 types of appliances were considered and 39 appliances were scheduled on 7 June 2016. The time slot was set to 5 min due to appliances that had short operation times, and the total time slots were 288 in a day. All the appliances were SAs and could not be interrupted once they had start operating, and the start time of the appliances were within the expected operating period.

Table 2. Electricity consumption of the appliances (Adapted with permission from Ref. [13]. 2022, Springer Nature.)

No.	Appliance	Duration (min)	Power (kW/h)
1	Dishwasher	105	0.85
2	Air conditioner	60	1
3	Washing machine	55	0.38
4	Clothes dryer	60	0.8
5	Coffee maker	10	1.5
6	Electric water heater	70	4.5
7	Dehumidifier	120	0.05
8	Microwave	5	0.8
9	Electric vehicle	150	3
10	Refrigerator	1440	0.5

shows the electricity consumption of the appliances. It was assumed that each appliance had a fixed operating duration and power consumption rate. For instance, every time a consumer needed to use the dishwasher, it would run for 105 min and the power consumption rate was 0.85 kilowatts per hour (kW/h).

Table 3. Expected operation time slots of each appliance (adapted with permission from Ref. [13]. 2022, Springer Nature).

No. i	Type of Appliance	Operation Time ${\mathit{l}}_{\mathit{i}}$	Expected Operation Time Slots $\mathit{E}{\mathit{S}}_{\mathit{i}}$~ $\mathit{L}{\mathit{F}}_{\mathit{i}}$	No. i	Type of Appliance	Operation Time ${\mathit{l}}_{\mathit{i}}$	Expected Operation Time Slots $\mathit{E}{\mathit{S}}_{\mathit{i}}$~ $\mathit{L}{\mathit{F}}_{\mathit{i}}$
1	Dishwasher	21	108–156	21	Dehumidifier	6	1–24
2	Dishwasher	21	168–216	22	Dehumidifier	6	24–48
3	Dishwasher	21	240–288	23	Dehumidifier	6	48–72
4	Air conditioner	6	1–24	24	Dehumidifier	6	72–96
5	Air conditioner	6	24–48	25	Dehumidifier	6	96–120
6	Air conditioner	6	48–72	26	Dehumidifier	6	120–144
7	Air conditioner	6	72–96	27	Dehumidifier	6	144–168
8	Air conditioner	6	96–120	28	Dehumidifier	6	168–192
9	Air conditioner	6	120–144	29	Dehumidifier	6	192–216
10	Air conditioner	6	144–168	30	Dehumidifier	6	216–240
11	Air conditioner	6	168–192	31	Dehumidifier	6	240–264
12	Air conditioner	6	192–216	32	Dehumidifier	6	264–288
13	Air conditioner	6	216–240	33	Electric water heater	14	60–84
14	Air conditioner	6	240–264	34	Electric water heater	14	220–288
15	Air conditioner	6	264–288	35	Microwave	1	78–90
16	Washing machine	11	12–60	36	Microwave	1	156–168
17	Clothes dryer	12	60–96	37	Microwave	1	228–240
18	Coffee maker	2	60–90	38	Electric vehicle	30	1–90
19	Coffee maker	2	204–228	39	Electric vehicle	30	200–288
20	Refrigerator	288	1–288

shows the expected operating time slots for each appliance. For example, the first appliance operation was a dishwasher, and the operating time was converted into 21 time slots, where one time slot was equal to 5 min. The expected operation duration was also converted into time slots, from 9 a.m. to 1 p.m., and was represented by time slots 108 to 156. The hourly electricity price on 7 June 2016, taken from ComEd Hourly Pricing program [24], is shown in Figure 1.

3.2. Implementation of NAGS-II

The parameters used by NSGA-II were predetermined, including the population size, maximum number of generations, crossover rate, and mutation rate. The parameters used for the NSGA-II algorithm in this numerical example were as follows: the population size was set to 100, with a crossover rate of 0.9 and mutation rate of 0.02. The maximum generation was 200 [13]. It was assumed that all appliances needed to be scheduled and the expected daily electricity cost, $C_{exp}$, was set to USD 100. The flow chart of the NSGA-II algorithm is shown in Figure 2.

The design of the chromosome is illustrated in Figure 3. For each appliance, it was assumed that the earliest run time could not be scheduled before the earliest preferred time slot for that appliance. Once the appliance started operating, it needed to continue running until it reached the required operating time length. The chromosomes were sequenced according to the list of appliances, shown in Table 3, and each appliance had a fixed partial chromosome length equal to the number of time slots. Each gene had a binary value of 0 or 1, and each chromosome had 11,232 genes in total. The example in Figure 3 indicates that the operating period for Appliance 4 was from Time Slot 9 to 14, which meant that the air conditioner was scheduled to operate between 1:10 a.m. and 2:10 a.m. on 7 June 2016.

After the initial solution was generated, the elite strategy was used to select two chromosomes as parents sequentially for crossover, with the probability of crossover set by the crossover rate, or 0.9 in this instance. As shown in Figure 4, the crossover operator first generated crossover points by randomly selecting an appliance that excluded the first appliance. Two offspring were obtained by interchanging the section after the crossover point between the two parents. Then, the objective values of the offspring chromosomes were calculated when added to the offspring population.

Each chromosome in the parent population had a probability of mutation, 0.02 in this instance. To start a mutation, the operator first generated a mutation point by randomly selecting an appliance from all appliances, Appliance i as shown in Figure 5. Then, the partial chromosome for Appliance i was replaced by another that was randomly generated, similar to the steps taken to generate the initial population. The objective values of the mutated chromosome were then calculated when added to the offspring population.

After the offspring population was fully generated, a nondominated sorting was applied to sort the chromosomes in the combined population by the number of the chromosomes that were dominated. If the objective function of one chromosome value was better than that of all other chromosomes in all the objective functions, then zero chromosomes dominated it, meaning the chromosome was ranked first. When multiple chromosomes were in the same tank, their crowding distances (CDs) were calculated. The purpose of crowding distance is to ensure the spread of chromosomes. A larger crowding distance meant that one chromosome was far from other chromosomes in the same rank. The crowding distance in a given rank was calculated using Equations (10) and (11) as follows:

$$distanc{e}_j\left(i\right)=\left\{\begin{array}{c}\frac{F_j\left(i+1\right)-F_j\left(i-1\right)}{F_{j,max}-F_{j,min}},F_{j,min}<F_j\left(i\right)<F_{j,max}\\ \infty , \mathrm{otherwise}\end{array}\right., i\in F_j$$

(10)

$$\mathrm{CD}\left(i\right)={{\displaystyle \sum }}_{j=0}^{Z}distanc{e}_j\left(i\right), i\in F_j,$$

(11)

where j is the jth objective function among a set of Z objectives, i is the ith solution after sorting in the same rank and $F_j\left(i\right)$ is the ith sorted solution for jth objective function, and $F_{j,min}$ and $F_{j,max}$ are the minimum and maximum value for jth objective function in that rank, respectively [21].

After using nondominated sorting and calculating the crowding distance in each rank, the top-ranking chromosomes were selected for the next generation until it reached the population size. Figure 6 illustrates the procedure for selecting the next generation with nondominated sorting and crowding distance sorting. The algorithm stopped when it reached the maximum number of generations.

3.3. Experimental Results

When the NSGA-II stopped, a set of Pareto-optimal solutions were obtained from the chromosomes in the first rank. The final set of Pareto-optimal solutions, obtained by the NSGA-II, had a total of 90 solutions. The distribution of these Pareto-optimal solutions was approximated by the Pareto frontier, as shown in Figure 7.

For illustrative purposes, three solutions among the Pareto-optimal set are shown in

Table 4. Selected Pareto-optimal solutions obtained by NSGA-II.

Solution	CR	Cost(USD)	PAR	Pmax(kW/h)	WTR
(a)	0.9942	99.42	2.3331	0.75	0.4414
(b)	1.0278	102.78	1.8367	0.61	0.4414
(c)	1.3304	133.04	2.3331	0.75	0.4244

Bold values indicate the best value obtained by the comparison among all Pareto-optimal solutions obtained by NSGA-II in this trial run

. Solution (a) has the lowest electricity cost ratio (CR). Because the expected daily electricity cost was set to USD 100, the total electricity cost was included in Cost (USD). Solution (b) had the lowest percentage of power rate (PAR), and we also show the peak value of the power load in Pmax (kW/h). Solution (c) had the lowest waiting time rate (WTR). It can be observed that, as in Solution (a), in order to obtain the lowest electricity cost, the schedule would have a slightly higher peak power load and larger waiting time rate than Solutions (b) and (c). Our selected solutions are identified in Figure 7.

Bold values indicate the best value obtained by the comparison among all Pareto-optimal solutions obtained by NSGA-II in this trial run

Figure 8a–c show the power consumption curve for appliances operating within the expected time period, referred to as power scheduled, or outside the expected period, referred to as outside expected period, and the cumulated electricity cost in USD for Solutions (a), (b), and (c), respectively. It can be observed that Solution (a) tried to schedule operations between either 2:30 a.m. and 6 a.m. or between 9:30 p.m. and 12 a.m., when hourly prices were lower. To minimize PAR, Solution (b) moved operations to the last 75 min and spread across earlier time slots from as early as 8:30 p.m. and lowered peak power load from 0.75 kW/h to 0.61 kW/h, though electricity cost increased slightly by USD 3.36. Solution (c) had the least number of appliances running outside of the expected time slots, but the electricity cost was one-third more expensive that of than Solution (a).

These Pareto-optimal solutions provided alternatives, and final scheduling could be decided by user preference. It would be possible to demonstrate these three options to customers, as shown in Table 4, and they can decide based on personal preference. For example, customers may consider whether to improve their discomfort level, from 0.4414 to 0.4244, but it would then increase their electricity costs from USD 99.42 to 133.04. Alternatively, users can easily search for alternatives in the set of Pareto-optimal solutions with a better comfort level than Solution (a) and a lower electricity cost than Solution (c). Similar comparisons can be made concerning peak-to-average ratio and cost. For example, customers can have a lower power peak during the day, from 0.75 to 0.61 kW/h, but it would increase electricity costs from USD 99.42 to 102.78. This trade-off might be more appealing to end-users, if incentive programs were available to encourage low power peak from power companies.

4. Discussion

In order to study the robustness of the final Pareto-optimal solutions obtained by the NSGA-II, six experimental trials were conducted on the same numerical example, and we counted the number of Pareto-optimal solutions in each generation. In Figure 9, it shows that an initial, randomly generated population of size 100, conducted in all trial runs, had no Pareto-optimal points. The number of Pareto-optimal solutions quickly increased during the first 25 generations. The learning process significantly fluctuated before all trial runs ended, having more than 60 Pareto-optimal solutions after generation 175 or more. This demonstrated that NSGA-II is capable of maintaining a good population size in the nondominated region.

Another interesting feature is the design of objective functions on the convergence behavior of the NSGA-II algorithm. Instead of directly minimizing the actual cost, the first objective function adopts the concept of cost rate (CR), which is the ratio of total electricity cost with respect to the expected daily electricity cost, as in Equation (1). The reason is to be consistent with the other two objectives: PAR and WTR, as defined in Equations (2) and (3).

Based on the numerical results from the study by Makhadmeh et al. [13], the lowest cost was around USD 100 that day; therefore, the value of $C_{exp}$ was set to 100 and served as a scaling factor.

To investigate the effect of value ranges of objective functions on the convergence behavior of the NSGA-II algorithm, an additional experimental trial concerning the same power scheduling problem was conducted by replacing the definition of CR with Cost, or replacing the value of $C_{exp}$ to one. The number of Pareto-optimal solutions in each generation is demonstrated in Figure 10, and shows that increases in the number of Pareto-optimal solutions in early generation were much slower. The variation in convergence between different trial runs was much larger compared with Figure 9, as the value of $C_{exp}$ was 100. Moreover, not all trial runs had more than 60 Pareto-optimal solutions when the algorithm finished. Convergence behavior can be sensitive to the scaling design of objective functions.

5. Conclusions

In this study, we formulated the scheduling problem in a smart home (PSPSH) as a multiple-objective optimization problem. Then, the NSGA-II algorithm was applied to generate a set of Pareto-optimal solutions, as illustrated in the numerical example. One major benefit of using NSGA-II was avoiding any bias by scaling multiobjective functions into one objective. The final solution obtained by NSGA-II was a set of Pareto-optimal solutions, which provided good alternatives to accommodate user preferences without re-optimizing. Moreover, an extra effect in normalizing values among objective functions can be helpful in demonstrating the robustness of the convergence performance of NSGA-II algorithm. As multiple Pareto alternatives were made available with just one simulation run, it helps promote appliance scheduling in smart homes, being easily adjustable for different user preferences. Future work can focus on consumer perceptions in power scheduling and improve the presentation of alternatives during decision making. Different MOEA algorithms can also be investigated for robustness and efficacy in finding Pareto-optimal solutions with less computational effort.