Operational Scheduling of Household Appliances by Using Triple-Objective Optimization Algorithm Integrated with Multi-Criteria Decision Making

Abstract

Load scheduling is a key factor in demand side management (DSM), which manages available generation capacity with regard to the required demand. In this paper, a triple-objective load scheduling optimization problem (LSOP) is formulated for achieving optimal cost and peak demand as well as minimum customer inconvenience. A Henry gas solubility optimization (HGSO) algorithm that is based on multi-objective is used for solving LSOP. The proposed HGSO offers a set of compromise solutions that represent the tradeoff between the three objectives of the formulated problem. A set of all compromise solutions from the dominant Pareto front is achieved first, and then ranked by using MCDM so as to optimally sort these solutions. An entropy weighting method (EWM) is then used for computing the weights of various criteria that dominate the LSOP and is provided as a technique for ordering preferences by similarity to achieve the ideal solution (TOPSIS) so as to rank the sorted solutions. Two types of end-users are considered so as to show the effectiveness of the proposed LSOP: non-cooperative and cooperative users. The results of the proposed load scheduling method show the significance of the proposed method for both the cooperative and non-cooperative end-users. The proposed method achieves a cost of energy of R50.62 as a total cost of energy consumed by four non-cooperative end-users. The cost of energy for the cooperative end-users is found to be R47.39. Thus, saving in the energy cost unit is found to be around 5.5% by using the proposed method; moreover, the peak demand value is reduced by 9.7% when non-cooperative end-users becomes cooperative.

1. Introduction

The rapid growth and development of information and communication technologies (ICT) and pricing schemes (PS) have provided large opportunities for residential end-users to support energy saving [1]. The energy demands of the residential sector have increased globally with 30–40% of the total energy consumption for different reasons, namely the increasing of population, the wide range of urbanization, and the use of additional electrical appliances [2]. To meet these energy demands, more generation capacity is required. In order to tackle the increasing greenhouse gases and limit global warming crises, however, building traditional power plants is not recommended [3]; therefore, managing the available generation to meet the required demand is a critical aspect of demand side management (DSM). DSM tries to initiate the contribution of end-users by using ICT and PS to manage energy consumption and to mitigate the need for more generation capacity. Load scheduling (LS) is one of the most active tools for DSM to manage household appliances for a given price rate, which consequently decreases the energy bills of users and reduces the peak power load of the utility company [4].

A flexible pricing scheme is another key factor in demand side management, which encourages users to participate in demand response programs [5]. There are two types of pricing schemes. The first is price-based, such as real time pricing (RTP), Time-of-Use (ToU), critical peak pricing (CPP), and day-ahead pricing (DAP). When considering price-based types, the pricing rate changes based on the required demand, so that users can adjust their load according to a specific rate [6]. The second type of pricing scheme is incentive-based. When using the incentive-based type, users are motivated to minimize their energy consumption according to specific price offers or based on a contractual agreement. Accordingly, the program administrator gains a degree of authority to directly reduce, disconnect, and schedule in order to save cost. This can be achieved based on an agreement between the utility company and the user. Recently, such programs have included residential end-users as well as large commercial and industrial end-users. The centralized controllers are utilized to provide decision and control action to manage operating states and instantaneous demand of individual devices. Some program cases in this scheme offer interruptible tariffs, direct-load controls (DLCs), emergency programs, and demand-bidding programs. Depending on these DR techniques, the end-user load can be changed according to incentive offers and price changes, or in emergency cases that jeopardize grid operation. The outcomes of DR with regard to users and utility can be realized through reducing both user energy and cost, minimizing air pollution, dropping peak-energy prices, and reducing the need for new generation capacity [7].

Despite the available earlier studies, which have focused methods of controlling both power supply and the energy demand side, these earlier contributions lack the following points: First, the majority of the previous studies have relied only on a one price-based rate which, in turn, could be incompatible with other pricing rates. Second, commercial software packages have been used to solve most of the proposed multi-objective’s algorithms, which are not adaptable to modification and could not support global optimization. Moreover, combining multi-objective into single objective is widely dependent on the aggregated weighted sum model. Finally, it can be concluded that a multi-criteria decision-making method has rarely been used for sorting all possible solutions processes for such a problem.

According to the aforementioned research gaps, the contributions of this paper for the load scheduling optimization problem (LSOP) are listed as follows:

• A multi-objective LSOP that relies on dominant rank and utilizes the Henry gas solubility optimization (HGSO) approach to achieve optimal cost, optimal peak demand, and lowest end-user inconvenience. These objectives are devised according to HGSO for attaining a set of solutions for the specified criteria.
• A hybrid multi-criteria decision-making method is also utilized to identify compromise solutions. The entropy weighting method (EWM) is used to adapt the weights of various criteria that dominate the LSOP. According to these weights, TOPSIS provides a ranking for the given solutions. The multi-objective HGSO will produce sets of solutions for scheduling users’ loads to realize energy saving and optimal cost while ensuring lower inconvenience level. Merging EWM-TOPSIS with multi-objective HGSO aids the realization of the compromise solutions, which are based on predefined criteria among the entire optimal solution sets.
• Two types of pricing schemes, namely adaptive consumption level pricing scheme (ACLPS) and ToU, are utilized to validate the results by comparing their performances with other various pricing techniques.

The rest of the paper is organized as follows: Section 2 reviews the related works. Then, Section 3 describes the mathematical model of LSOP. After that, the proposed multi-objective HGSO algorithm and EWM-TOPSIS multicriteria decision-making methods are presented in Section 4. The performance evaluation of the proposed load scheduling method is presented in Section 5, and Section 6 discusses the validation of results. Finally, the conclusion of this work is posted in Section 7.

2. Related Works

There are many studies that have addressed load scheduling for cost and energy saving. In [8], a mixed-integer nonlinear optimization technique within a ToU pricing program is presented to schedule actual user load profile. The outcomes of this research show an approximately 25% cost reduction. In [9], a dynamic residential load management scheme is presented. Based on a given allowance, users are motivated to manage their devices for optimal cost and energy saving. The performance of ToU demonstrates approximately 35% and 31% energy and cost saving, respectively. Similarly, a load scheduling system in a residential area is proposed in [10] in order to address end-users’ preferences. In addition, based on a ToU electricity tariff, multi-objective linear programming is utilized to reduce peak power load. This paper addresses the reduction of coordinated peak load within a multiple-household context. The presented approach seeks to reduce three objectives: scheduling end-user inconvenience, energy cost, and peak load demand. Based on these three objectives, both utility companies and end-users have the ability to adjust their priority in minimizing one over another. In this study, three multi-objective optimization approaches are applied, namely preemptive optimization, compromise optimization, and a normalized weighted-sum approach. The outcomes of the study reveal outstanding performance when analyzed with regard to three techniques in literature and end-users’ preferred schedules. An IoT-based home energy management scheme is provided in [11]. In order to improve the system’s efficiency in terms of energy consumption cost and end-users’ satisfaction, an enhanced butterfly optimization technique version is adapted. This approach is applied on a designed house; the outcomes are then compared with a normal consuming method. In [12], a multi-objective (cost and incontinence) algorithm is presented based on a multi-objective optimization differential evolution (MODE) scheme so as to achieve optimal energy and cost saving. In order to obtain all sets of solutions, the objectives are formulated by using MODE. After that, the compromise solutions are designated by using the analytic hierarchy process (AHP) and TOPSIS as a hybrid multi-criterion decision-making method. The analyses highlight how the proposed approach saves more than 32% of customer energy cost within ToU. On the other hand, the baseline operation, as well as customers’ peak loads, are minimized by 41%. In [13], an optimal scheduling method is proposed for household devices. The proposed approach addresses electricity cost, incentive, and end-users’ inconvenience under a ToU electricity charge. In addition, a study of the inconvenience weighting parameter effect on total costs is carried out. Then, the impact of incentive on optimization performance is evaluated. The simulation results show a 34.71% saving in the total costs of end-users. The authors in [14] propose a flexible demand management scheme to efficiently schedule residential load; the load operating time window is addressed as a constraint. The aim of this contribution is to emphasize constraints affecting the operation schedule of devices while contributing demand response events. Accordingly, an innovative crossover approach in genetic algorithms is adapted in this work. The presented method addresses dynamic pricing, distributed generation, and load shifting so as to reduce both energy cost and electricity bill. Real household workload data is presented as a case study by which four devices, scheduled for five days, and three various scenarios are examined. The adapted genetic model manages to realize up to 15% in bill reduction for multiple scenarios compared to business as usual. The operation management of residential appliances is provided in [15] through a demand-side management model. In order to reduce user dissatisfaction and electricity costs, a multi-objective optimization problem is presented, where residents’ tolerance to discomfort is addressed. The non-inferior solution sorting genetic algorithm (NSGA-II) and hybrid multi-objective particle swarm optimization (MOPSO) are presented to handle the load scheduling issue. After that, the selection of the most compromising solution within the Pareto solution set is achieved via TOPSIS method. It is concluded in this study that the hybrid algorithm can achieve higher efficiency, which validates the effectiveness of the optimization algorithm. Finally, determining the efficiency of the scheduling approach is demonstrated by providing a case study in this work. The authors in [16] propose an optimal schedule model for resident devices. An improved genetic algorithm (GA) is used to solve the optimization problem associated with load scheduling. The entropy method is adapted to perform a chromosome selection process in this improved GA. The mutation and blended crossover are carried though a correlation coefficient. The simulation is based on Python programming and is considered as a set of appliances regarding a single resident load profile. The outcomes of the simulations indicate a decrease in electricity cost as compared to the original operational case. The execution time of the proposed method is minimized by 2.76 s. The number of iterations that is considered by the enhanced GA is found to be comparatively lower than the standard GA. The authors in [17] proposes a DR scheduling model that targets four classes of residential building loads: non-interruptible and deferrable loads, interruptible and deferrable loads, air conditioning loads, and non-interruptible and nondeferrable loads. In order to minimize the inconvenience index and energy cost, a multi-objective optimization algorithm based on the nondominated sorting genetic algorithm II is utilized. The AC load flexibility is tested on the Dymola platform, where two kinds of DR strategies are developed for AC. About 25% of nocturnal peak load saving is achieved by applying this method, while 10% of the daily energy cost is reduced. Additional efforts are presented in [18] to explore the potential variations in peak-to-valley electricity consumption and electricity charges, which are considered under a smart home context. This is achieved by proposing a multi-objective smart home integrated management approach while considering the behavioral heterogeneity of appliances and household electricity consumption. The analysis indicates that smart homes involved in a power demand response can minimize peak load by 29.3–49.3%; moreover, the peak-to-valley difference can be reduced by 37.5–78.2%. Finally, in [19], the authors propose a non-cooperative Stackelberg model-based game theory that takes into account the impact of both users’ dissatisfaction with electricity consumption and load fluctuations in the power grid. In this approach, users are firstly categorized through induction and classification to gain insight into various electricity users’ categories. Next, two utility-based objective functions are set up. The power supply’s utility objective function is set to indicate the outcomes gained from the power supply process with regard to the power company. On the other hand, the utility objective function is regarding the power demand side and is set to show the degree of dissatisfaction for electricity users. The NSGA-II algorithm is used to solve the formulated objective functions in this work. Finally, the proposed algorithm is applied on an actual case while the sensitivity analysis of related parameters is considered to validate effectiveness.

3. Load Scheduling Mathematical Model

In this study, a multi-objective model is proposed for $M$ households to address cooperative and non-cooperative end-users’ load demand scheduling problems. The aim of this load scheduling model is to find the optimal operational time (on/off) for available household appliances within a given sampling time $\left(\tau \right)$ of 10 min of time index $\left(n\right)$ for a full one-day period ($N=24 \mathrm{h}$). The index of households is considered as $m$ for maximum number $M$. The non-cooperative model of individual user operation is summarized as follows [10]:

$$Cost={\sum }_n^N{\sum }_m^M{R}_n({\rho }_n{y}_{m,n}^{opt}-{\eta }_n{x}_{m,n})·∆\tau$$

(1)

$$Incon={\sum }_n^N{\sum }_m^M{z}_{m,n}$$

(2)

$$Peak=Max\left(R_n\right)$$

(3)

Subject to:

$$z_{m,n}=\sum _n^N\sum _m^M\left(y_{m,n}^{org}-y_{m,n}^{opt}\right)\forall m\in M,\forall n\in N$$

(4)

$$\sum _{B_n}^{S_n}{y}_{m,n}^{opt}\ge T_m\forall m\in M$$

(5)

$$y_{m,n}\in \left\{\mathrm{0,\; 1}\right\}$$

(6)

$$x_{m,n}\in \left\{\mathrm{0,\; 1}\right\}$$

(7)

$$∆\tau =\frac{60 \left(\mathrm{m}\mathrm{i}\mathrm{n}\right)}{\tau \left(\mathrm{m}\mathrm{i}\mathrm{n}\right)}$$

(8)

From the given model, there are three conflict objectives. These are energy cost ($Cost$), end-user inconvenience ($Incon$), and peak demand ($Peak$). Equation (1) represents the first objective, which is energy cost, and the aim of this objective is to minimize the end-user’s energy bill by optimally scheduling the operational time of given household appliances. The load scheduling is based on the provided price rate, which differs from the high rate during peak time and lower rate through non-peak time. Furthermore, to encourage users toward lowering their consumption, an incentive (${\eta }_n)$ is provided whenever users switch off some of their appliances ($y_{m,n}^{org}$) from peak time to be in a new set of operational appliances ${(y}_{m,n}^{opt}$) in non-peak time, which is denoted by $v_{m,n}$. Thus, the price rate (${\rho }_n)$ will be decreased by the amount of a given incentive to calculate the final cost according to energy consumption ($R_n$). The function value of ($z_{m,n}$) is calculated according to Equation (4).

Equation (2) refers to the second objective of the minimized end-user inconvenience ($z_{m,n})$. The main target of this objective is to minimize the difference between the operational time of the preferred user scheduling related to appliances ($y_{m,n}^{org})$ with respect to optimal operational time ($y_{m,n}^{opt}$), which may be late or postponed from the original operational case to minimize cost or peak demand. This objective function is formulated in Equation (4). As for the third objective, it is presented in Equation (3); it addresses the peak power consumption of the end-user load for a given time horizon. This objective seeks to minimize the peak power consumption of the end-user load by synchronizing the operation time of each load to flatten the peak edges of the consumption curve. This helps users to minimize energy cost while aiding the utility company by reducing consumption demand, especially during peak time with a high price rate. Equation (5) deals with the completion time of each appliance and is required to guarantee the optimal time given for each appliance $\left(y_{m,n}^{opt}\right)$ to meet the finishing time of each appliance’s process cycle $\left(T_m\right)$. Equations (6) and (7) refer to values $y_{m,n}$ and $x_{m,n}$, which are the logical statues of each appliance, indicating whether that appliance is ON (logic 1) or OFF (logic 0) during the current time slot. Equation (8) represents the sampling duration, which is one hour divided by the sampling time.

In the cooperative end-user model of the multiple-household model, it is assumed that all users work synchronically (concurrently) with optimal objectives (cost, inconvenience, and peak). Each equation in the above model is extended to include all end-users. This model can be described as follows [10]:

$$Cost=\sum _h^H\sum _n^N\sum _m^M{R}_n^h\left({\rho }_n{y}_{m,n}^{opt,h}-{{\eta }_n}_{}{x}_{m,n}^h\right)·∆\tau$$

(9)

$$Incon=\sum _h^H\sum _n^N\sum _m^M{z}_{m,n}^h$$

(10)

$$Peak=Max\left(R_n\right)$$

(11)

Subject to:

$${ z}_{m,n}^h=\sum _n^N\sum _m^M\left(y_{m,n}^{org,h}-y_{m,n}^{opt,h}\right)\forall m\in M,\forall n\in N$$

(12)

$$\sum _{B_n^h}^{S_n^h}{y}_{m,n}^{opt,h}\ge T_m^h\forall m\in M$$

(13)

$$y_{m,n}^h\in \left\{\mathrm{0,\; 1}\right\}$$

(14)

$$x_{m,n}^h\in \left\{\mathrm{0,\; 1}\right\}$$

(15)

$$∆\tau =\frac{60 \left(\mathrm{m}\mathrm{i}\mathrm{n}\right)}{\tau \left(\mathrm{m}\mathrm{i}\mathrm{n}\right)}$$

(16)

4. The Proposed Load Scheduling Method

The proposed method for scheduling household loads consists of two main stages. The first stage utilizes the HGSO as a multi-objective optimization algorithm to optimize the starting time of each device by minimizing the objective functions. After that, the second stage utilizes the MCDM method to rank the set of compromise solutions, which are obtained from stage one. It is worth mentioning that the second stage includes two parts: the first is EWM, which derives the weights of the criteria; the second is TOPSIS, which is used for ranking compromise solutions.

4.1. Multi-Objective Henry Gas Solubility Optimization (MHGSO) Algorithm

In 2019, the HGSO algorithm was proposed by Fatema et al. [20,21]. The algorithm depends on Henry’s law for the solubility of gas in fluid. The HGSO algorithm is a population-based metaheuristic evolutionary algorithm. Similar to other metaheuristic algorithms, HGSO could be defined in three basic steps: initializing a population that comprises a set of candidate solutions; a strategy for updating the solution; and evaluating the fitness function value. Accordingly, HGSO can be simplified through the next steps:

• Step 1: Initialization

A population with $N_P$ candidate solutions (individual vectors) is obtained in this step. The dimension of each individual vector is $D$ decision variables, and are distributed in a uniform basis and initialized based on the given formula:

$$X_{i,j}^G=X_{i,j}^L+rand\times \left(X_{i,j}^H-X_{i,j}^L\right)$$

(17)

$rand$ is a number randomly related to [0,1] interval, $X_{i,j}^L$ and $X_{i,j}^H$, which are both low and high boundaries of $i^{th}$ element within $j^{th}$ solution. $G$ represents the generation number ($G=1,2,\dots ,G_{max})$, which is the (iteration).

• Step 2: Clustering

The solution of the population can be defined as gases, and have been categorized into $N_c$ clusters based on gas type. Every cluster is assigned an equal value of gas types. Because each cluster includes the same kind of gases, the gases in $k^{th}$ cluster should have an equal Henry’s constant ($H_k$) value, which is calculated by Equation (18):

$$H_k^G=L_1\times rand \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} k=1,2,\dots ,N_c$$

(18)

where $L_1$ is a constant ($5\times {10}^{-2}$) [22] and $rand$ is a number that randomly belongs to (0, 1) period. The partial pressure regarding $j^{th}$ gas (individual vector) is related to $k^{th}$ cluster and is calculated according to the given equation:

$$P_{j,k}=L_2\times rand \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} k=1,2,\dots ,N \mathrm{a}\mathrm{n}\mathrm{d} k=1,2,\dots ,N_P$$

(19)

where $L_2$ is a constant with a value equal to 100 [22] and $rand$ is a random number which is selected in a random basis within (0, 1) interval. Based on Henry’s law, there will be another constant, $C_k$. This constant will be fixed for the gases in the cluster. $C_k$ is computed according to this formula:

$$C_k=L_3\times rand \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} k=1,2,\dots ,N_c$$

(20)

$L_3$ is a constant identical to $1\times {10}^{-2}$ [22], while $rand$ is a random value and is designated between (0, 1).

• Step 3: Objective function evaluation process

This step is initiated by evaluating the objective function related to the individual vector. Accordingly, the optimal solution ($X_{best})$ upon the population is located. Additionally, the optimal solution for every cluster ($X_{kbest})$ is determined. It is important to mention that minimizing the objective function is the goal of solving the current load scheduling optimization problem; therefore, the best solutions, $X_{best}$ and $X_{kbest}$, are those that correspond to the compromised objective function value regarding both population and cluster, respectively.

• Step 4: Henry’s coefficient updating

The Henry’s coefficient is updated for every iteration according to the following formula:

$$H_k^{G+1}=H_k^G\times exp\left(-C_k^G\left(\frac{1}{T^G}-\frac{1}{T^{\theta }}\right)\right)$$

(21)

where:

$$T^G=\exp \left(-\frac{G}{G_{max}}\right)$$

(22)

where $H_k^G$ and $H_k^{G+1}$ are the coefficients of Henry for $G$ and $G+1$ iterations, respectively. $T^{\theta }$ refers to a temperature constant (298.15); $G_{max}$ represents the total number of iterations.

• Step 5: Solubility computation

The solubility value of $j^{th}$ solution in $k^{th}$ cluster is derived according to the following equation:

$$S_{j,k}^G=Q\times H_k^G\times P_{j,k}$$

(23)

where $S_{j,k}^G$ is the solubility of $j^{th}$ gas in $k^{th}$ cluster for $G^{th}$ iteration. $Q$ is a constant with a value assigned to 1.

• Step 6: Solutions update process

The individual vectors of a population are updated for ${(G+1)}^{th}$ iteration using Equation (24):

$$X_{j,k}^{G+1}=X_{j,k}^G+F\times rand\times \gamma \times \left(X_{kbest}^G-X_{j,k}^G\right)+F\times rand\times \alpha \times \left(S_{j,k}^G\times X_{best}^G-X_{j,k}^G\right)$$

(24)

where:

$$\gamma =\beta \times \exp \left(-\frac{F_{best}^G+\epsilon }{F_{j,k}^G+\epsilon }\right)$$

(25)

where $\epsilon$ is a constant (0.05); $rand$ is a number that is randomly selected within [0,1] period; and $F$ is a flag set to obtain diversity when changing the direction of the search process. Both $\beta$ and $\alpha$ are constant values set to be 1. Where $\alpha$ indicates the impact of other solution vectors on $j^{th}$ gas in $k^{th}$ cluster. $\gamma$ shows the ability of $j^{th}$ individual vector in $k^{th}$ cluster to interrelate with other solution vectors that relate to its cluster. $F_{best}^G$ refers to the best fitness function value within the whole population in $G$ iteration. $F_{j,k}^G$ represents the fitness function value of $j^{th}$ solution that belongs to $k^{th}$ cluster in $G^{th}$ iteration.

• Step 7: Escaping of local optimum

The escaping strategy is set by reinitializing $N_w$ worst individual solutions within the population. It is worth mentioning that the worst solutions are the ones that have the biggest fitness function values as compared to the entire solutions of the population in $G^{th}$ iteration. The $N_w$ value is estimated by Equation (26):

$$N_w=N_P\times \left(rand\left(C_2-C_1\right)+C_1\right)$$

(26)

where $rand$ is a number randomly belonging to period [0,1]. $C_1$ and $C_2$ are constants with values equal to 0.1 and 0.2, respectively.

• Step 8: Update the worst solutions

In step 8, the $N_w$ worst solutions are substituted by new ones using Equation (17).

• Step 9: Dominance rank

In this step, an empirical population (${POP}_e$) is constructed by combining the solutions of $G^{th}$ iteration and the updated ones, which are obtained at the end of step 6. The size of ${POP}_e$ will be $D\times 2N_P$ elements. After that, the dominance rank value is computed for each solution of ${POP}_e$. The dominance rank value of a solution is equal to the number of solutions belonging to ${POP}_e$, which dominate the considered solution plus one. Then, the compromise solutions with the lowest dominance rank value are chosen to construct a new population with a $D\times N_P$ dimension for the next generation. Algorithm 1 illustrates the steps of the dominance ranking concept presented for initiating the new population of the next generation in the proposed HGSO algorithm.

Algorithm 1: Pseudo code of the proposed dominance rank in HGSO algorithm.

Combine the parent solutions and their updated ones to construct ${POP}_e$. Initiate 1 $\times 2N_P$ vector ($D_r$) to represent the dominance rank value of each solution in ${POP}_e$. Set the $2N_P$ elements of $D_r$ to zero. For $i=1$ to $2N_P$ do     For $j=1$ to ($2N_P-1$) do         If solution ($i$) is dominated by solution ($j$) where $i\ne j,$ then $D_r\left(i\right)=D_r\left(i\right)+1$.     End for End for Sort the elements of $D_r$ vector in ascending form. Sort the solutions in ${POP}_e$ based on $D_r$. Choose the first $N_P$ solutions from ${POP}_e$ to construct the new population for next generation.

The pseudo code of the proposed HGSO algorithm where it is used in the present study is illustrated by Algorithm 2.

Algorithm 2: Pseudo code of the proposed HGSO algorithm.

Initialization: Initialize $POP, H_k^0, C_k^0, P_{j, k}^0, L_1, L_2, L_3, \alpha , \beta , Q, f, \mathrm{and} \mathrm{compute}{S}_{j, k}^0$. Clustering the population: divide the $POP$ into $N_c$ clusters based on the type of gas. Evaluate the fitness function of each individual vector and find $X_{best}^0, X_{kbest}^0, \mathrm{and} F_{best}^0$. While $G\le G_{max}$ do      For $j=1$ to $N_P$ do          Compute $\gamma$ using Equation (25).          Update the solutions using Equation (24)     End for      Compute $T^G$ using Equation (22). Update Henry’s coefficient for each cluster using Equation (21). Update the solubility for each solution using Equation (23). Sort the solutions of each cluster in $POP$ based on fitness function. Find the worst solutions using Equation (26). Replace the worst solutions by new initialized solutions using Equation (17). Evaluate the fitness function for each solution of $POP$. Apply the dominance rank concept. Find $X_{best}^G, X_{kbest}^G, \mathrm{and} F_{best}^G$. Increment $G$ by one. End while Return $X_{best}$.

4.2. Multicriteria Decision Making

The results of the load scheduling optimization problem (LSOP) can be represented as an MCDM problem to identify the effective solution among a set of compromise solutions. The MCDM problem can be presented by a matrix that has $m$ solutions (alternatives) and $s$ criteria with weights ($w_j,$ where $j=1,2,\dots ,s$). In this section, the hybrid multicriteria decision-making (HMCDM) methods are presented in detail. The HMCDM is used for ranking superior solutions, which belong to the Pareto front of the load scheduling optimization problem (LSOP). The HMCDM includes the entropy weighting method (EWM) for computing the weights of various criteria, where it dominates the LSOP. Also, TOPSIS is considered for the ranking process.

4.2.1. Entropy

The various criteria which dominate the response of the problem have different degrees of significance. The significance of each criterion dominates the problem and can be reported by the value of weight. The sum of the weights of the criteria in the MCDM problem should be dominated by the following formula:

$$\sum _{j=1}^s{w}_j=1$$

(27)

where $w_j$ refers to $j^{th}$ criterion weight and $s$ is the total number of criteria, which dominate the problem. The EWM is proposed by Shannon and Weaver and developed further by Zeleny for establishing the weights of different criteria, which dominate the responses of the MCDM problem [23]. The EWM derives weight according to the decision matrix ($DM$), without using the preferences of experts or deciders around the criteria. In general, the EWM comprises three steps for obtaining the weights of each criterion: normalizing the decision matrix; computing the entropy index; and computing the weights [23]. These steps will be discussed in detail in the following subsections.

• Step 1: Normalizing

The cornerstone in the MCDM-problem is the $DM$, which includes the performance of various alternatives ($A_1,A_2,\dots ,A_q$) in terms of different criteria ($C_1,C_2,\dots ,C_s$). The $DM$ can be represented as follows:

$$DM=\begin{array}{cc}& \begin{array}{ccc}{C}_1& \dots \dots \dots \dots & C_s\\ w_1& \dots \dots \dots \dots & w_s\end{array}\\ \begin{array}{c}{A}_1\\ ⋮\\ A_q\end{array}& \left[\begin{array}{ccc}{a}_{11}& \dots \dots \dots & a_{1s}\\ ⋮& \dots \dots \dots & ⋮\\ a_{q1}& \dots \dots \dots & a_{qs}\end{array}\right]\end{array}$$

(28)

where $q$ is the total number of alternatives and $s$ is the total number of dominated criteria with various weights ($w_j$, where $j=\mathrm{1,2},\dots ,s$) that will be derived by EWM. $a_{ij}$ refers to the performance of $i^{th}$ alternative ($A_i$), which is dominated by $j^{th}$ criteria ($C_j$), where $j=1,2,\dots ,s$ and $i=1,2,\dots ,q$. In the first step of EWM, the elements of $DM$ matrix will be normalized by Equation (29):

$$r_{ij}=\frac{a_{ij}}{\sum _{i=1}^q{a}_{ij}},\mathrm{where}j=1,2,\dots ,s$$

(29)

where $r_{ij}$ refers to the normalized performance (element) of $i^{th}$ alternative ($A_i$) in terms of $j^{th}$ criteria ($C_j$).

• Step 2: Entropy Index

The entropy index ($e_j$) regarding $j^{th}$ criteria is computed in this step using the following formula. It should be noted that the value of the entropy index belongs to period [0,1] [23]:

$$e_j=-\frac{1}{\ln \left(q\right)}\times \sum _{i=1}^q{r}_{ij}\times \ln \left(r_{ij}\right),\mathrm{where}j=1,2,\dots ,s$$

(30)

• Step 3: Entropy Weights

The last step of EWM is computing the weights of each criterion using the degree of divergence ($1-e_j$) as presented below:

$$w_j=\frac{1-e_j}{\sum _{j=1}^s(1-e_j)}$$

(31)

4.2.2. TOPSIS

In 1980, Yoon and Hwang proposed the TOPSIS method to solve multi-dimensional MCDM problems [24]. In TOPSIS, sorting the alternatives (solutions) depends on determining the shortest distance from the ideal and negative ideal solutions. TOPSIS can be expressed by six steps as follows [25]:

• Step 1: Normalized Decision Matrix

In general, most MCDM problems include $n$-criteria measured by different units, which are called multi-dimensional criteria. Thus, normalizing the elements of $DM$ is inevitable and based on the following formula:

$$r_{ij}=\frac{a_{ij}}{\sqrt{\sum _{k=1}^q{\left(a_{kj}\right)}^2}}$$

(32)

The normalized decision matrix ($R$) is obtained when applying the first step as follows:

$$R=\left[\begin{array}{ccc}{r}_{11}& \cdots & r_{1s}\\ ⋮& \ddots & ⋮\\ r_{q1}& \cdots & r_{qs}\end{array}\right]$$

(33)

• Step 2: Weighted Normalized Decision Matrix

The normalized decision matrix ($R$) will be weighted by the weights, which were previously determined by EWM to construct the weighted normalized decision matrix ($V$). The elements of $V$-matrix are computed when multiplying the corresponding element of matrix $R$ by the convenient weight as presented in Equation (34):

$$v_{ij}=w_j{r}_{ij}, \mathrm{f}\mathrm{o}\mathrm{r} j=1,2,\dots ,s \mathrm{a}\mathrm{n}\mathrm{d} i=1,2,\dots ,q$$

(34)

Consequently, $V$ matrix, obtained by the current step, can be presented as:

$$V=\left[\begin{array}{ccc}{v}_{11}& \cdots & v_{1s}\\ ⋮& \ddots & ⋮\\ v_{q1}& \cdots & v_{qs}\end{array}\right],$$

(35)

• Step 3: Computation of Ideal and Negative Ideal Solutions

In step 3 of the TOPSIS method, an ideal solution ($A^{\mathrm{*}}$) and negative ideal solution ($A^{-}$) can be computed as follows:

$$\begin{array}{cc}\hfill A^{\mathrm{*}}& =\left\{\left({max}_i\left.v_{ij}\right|j\in J\right),\left({min}_i\left.v_{ij}\right|j\in J^{-}\right),i=1,2,\dots ,q\right\},\hfill \\ & =\left\{v_1^{\mathrm{*}},v_2^{\mathrm{*}},\dots ,v_s^{\mathrm{*}}\right\},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill A^{-}& =\left\{\left({min}_i\left.v_{ij}\right|j\in J\right),\left({max}_i\left.v_{ij}\right|j\in J^{-}\right),i=1,2,\dots ,q\right\},\hfill \\ & =\left\{v_1^{-},v_2^{-},\dots ,v_s^{-}\right\},\hfill \end{array}$$

(37)

where $J$ is the criterion of a benefit set with range [1, s]; and $J^{-}$ refers to the complement set of $J$ with [1, s] range, which presents the criterion of cost. It should be noted that the ideal solution is the preferable one; conversely, the negative ideal solution is the least preferable.

• Step 4: Separation Measure

The Euclidean distance with s-dimension is presented to compute the separation distance among each solution, which is a member of $V$ matrix, and the negative ideal/ideal solutions. The separation distance ($S^{\mathrm{*}}$) between the ideal solution ($v_j^{\mathrm{*}}$) and an alternative ($A_i$) can be presented by:

$${\displaystyle S_i^{\mathrm{*}}}=\sqrt{{\sum }_{j=1}^s{\left(v_{ij}-v_j^{\mathrm{*}}\right)}^2}, \mathrm{f}\mathrm{o}\mathrm{r} i=1,2,\dots ,q$$

(38)

Similarly, ($S^{-}$) refers to the separation distance between an alternative ($A_i$) and the negative ideal solution ($v_j^{-}$) over $s$ criteria that can be offered by:

$${\displaystyle S_i^{-}}=\sqrt{{\sum }_{j=1}^s{\left(v_{ij}-v_j^{-}\right)}^2}, \mathrm{f}\mathrm{o}\mathrm{r} i=1,2,\dots ,q$$

(39)

However, each solution of $V$ matrix has two separation distance values. The distance values are $S_i^{\mathrm{*}}$ and $S_i^{-}$, which are utilized to show the closeness of solutions from the ideal and negative ideal solutions, respectively.

• Step 5: Computing Relative Closeness

The relative closeness of a solution ($A_i$ for $i=1,2,\dots .,q$) with respect to the ideal solution can be shown by the following formula:

$$C_i^{\mathrm{*}}=\frac{S_i^{-}}{S_i^{-}+S_i^{\mathrm{*}}}, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} i=1,2,\dots ,q$$

(40)

It has to be indicated that the values of variable $C_i^{\mathrm{*}}$ lie within interval [0,1], where $C_i^{\mathrm{*}}=0$ only if $S_i^{-}=0\left(\mathrm{i}.\mathrm{e}., A_i=A_i^{-}\right)$, and $C_i^{\mathrm{*}}=1$ only if $S_i^{\mathrm{*}}=0\left(\mathrm{i}.\mathrm{e}., A_i=A_i^{\mathrm{*}}\right)$.

• Step 6: Sorting Alternatives Based on Ideal Solution’s Closeness

The $V$ matrix’s solutions are ranked in descending order based on the closet value to the ideal solution that is computed in the previous step. Consequently, the best solution is the one that has the highest value of closeness, i.e., it is the farthest to the negative ideal solution and the nearest to the ideal one.

The proposed HGSO optimization method and hybrid multicriteria decision-making method, which are used for solving the domestic load scheduling problem, are shown in Figure 1.

5. Experimental Results

The real household consumption profiles are adopted from [10] for four users ($H_1–H_4$) as tabulated in

Table 1. Rated consumption power and duration time for users’ profiles [10].

No.	Device Name	$\mathbf{Power} \mathbf{Rate} {\mathit{R}}_{\mathit{m}}$ (kW)				Duration (min/day)
		${\mathit{H}}_1$	${\mathit{H}}_2$	${\mathit{H}}_3$	${\mathit{H}}_4$	${\mathit{H}}_1$	${\mathit{H}}_2$	${\mathit{H}}_3$	${\mathit{H}}_4$
1	Washing machine	3	3	2	2	45	70	90	80
2	Drying machine	3.3	3.3	2	3	30	30	20	20
3	Elec. water heater 1	2.6	1.9	2	2.2	120	180	120	180
4	Elec. water heater 2	2.6	-	-	-	120	-	-	-
5	Kettle 1	1.9	-	-	-	10	-	-	-
6	Kettle 2	1.9	-	-	-	10	-	-	-
7	Toast machine	1.01	-	-	-	10	-	-	-
8	Hoover vacuum	1.2	1.2	1.8	1	30	30	10	20
9	Cooker 1	3	2.2	2	3	30	50	60	70
10	Cooker 2	3	-	-	-	50	-	-	-
11	Iron	1.235	-	-	-	48	-	-	-
12	Microwave	1.230	0.8	1.5	0.6	10	10	20	10
13	Dishwasher	2.5	1.5	1.2	1.5	150	80	120	100
14	DVD player	-	0.025	0.015	0.025	-	120	180	150

and

Table 2. The preferred operational time for users’ appliances [10].

No.	Device Name	$\mathbf{Starting} \mathbf{and} \mathbf{Ending} \mathbf{Time} ({\mathit{B}}_{\mathit{m}}-{\mathit{S}}_{\mathit{m}}$)
		${\mathit{H}}_1$	${\mathit{H}}_2$	${\mathit{H}}_3$	${\mathit{H}}_4$
1	Washing machine	16:00–22:00	10:00–15:00	18:00–22:00	13:00–19:00
2	Drying machine	16:00–20:20	11:00–15:00	19:00–22:00	14:00–19:00
3	Elec. water heater 1	04:00–08:10	05:00–09:00	05:00–09:00	05:00–09:00
4	Elec. water heater 2	16:00–22:00	-	-	-
5	Kettle 2	05:30–07:30	-	-	-
6	Kettle 2	17:40–20:00	-	-	-
7	Toast machine	05:00–07:00	-	-	-
8	Hoover vacuum	08:00–10:20	10:00–18:00	09:00–12:00	13:00–19:00
9	Cooker 1	05:00–07:00	10:30–15:30	10:30–15:30	10:30–15:30
10	Cooker 2	16:00–20:00	-	-	-
11	Iron	16:00–21:00	-	-	-
12	Microwave	16:00–19:00	17:00–21:00	16:00–18:00	13:00–19:00
13	Dishwasher	20:00–24:00	20:00–23:00	20:00–23:00	20:00–23:00
14	DVD player	-	10:00–23:00	08:00–23:00	13:00–19:00

. Table 1 shows the rated power consumption (kW) for each appliance and the duration time (min) for finishing its work. For some appliances, there are two operation times in a day, such as for the first house (electrical water heater, kettle, and cooker). In the case of the first house, the electrical water heater has two operation times for each one and needs about 120 min to complete its work. The preferred allowable operation time for each household appliance is given in Table 2. The defined duration periods are the preferred times that appliances should be functioning (switched ON), e.g., the washing machine’s preferred operation time is 16:00–22:00, 10:00–15:00, 18:00–22:00, and 13:00–19:00 for given households, respectively. The price and incentive rate are given in

Table 3. Pricing and incentive rate [10].

Time	Duration	Price Rate (R/kWh)	Incentives (R/kWh)
Peak	07:00–10:00 and 18:00–20:00	1.4400	0.2
Off-peak	Otherwise	0.4554	0.0

, which are adopted from [10], based on South Africa’s ToU tariff. From Table 3, the peak rate is about (1.44 R/kWh) for periods 07:00–10:00 and 18:00–22:00; otherwise, is the off-peak period, which is about (0.4554 R/kWh). R refers to South African currency (Rand or ZAR).

The proposed MHGSO is used to find the Pareto fronts and their Pareto optimal set of solutions for two different cases. The first case related to non-cooperative work for individual users, which means each user works independently (so that there are four scenarios relayed for numbers of households). The second case belongs to the cooperative work of users. This case covers four users together. The dimension of LSOP (number of appliances) for each case is tabulated in the third column of

Table 4. Parameter settings of MHGSO algorithm.

Cases	Scenarios	$\mathit{D}$	${\mathit{N}}_{\mathit{P}}$	${\mathit{G}}_{\mathit{m}\mathit{a}\mathit{x}}$
Non-Cooperative	First user	13	65	300
	Second user	8	40	300
	Third user	8	40	300
	Fourth user	8	40	300
Cooperative	Four users	37	185	300

. The number of individual solutions ($N_P$) in the population initiated for each scenario in the considered cases is $5D$, as illustrated in the fourth column of Table 4. The total number of generations ($G_{max}$) for MHGSO that is utilized for all cases is 300. Based on a series of experiments, it was found $G_{max}=300$ is adequate to obtain the optimal Pareto front for each case.

The proposed MHGSO algorithm is used to solve the LSOP by utilizing the data of the first user that has 13 appliances. The Pareto front of the first LSOP user is shown in Figure 2. Minimum cost, end-user inconvenience level, and peak values are R11, 29 slots, and 4500 W, respectively, as tabulated in

Table 5. Minimum criteria values of two cases.

Case	Scenarios	Cost (R)	Inconvenience (Slots)	Peak (W)
Non-Cooperative	First user	11.006	29	4500
	Second user	8.521	23	3300
	Third user	5.02	17	2000
	Fourth user	8.75	20	3000
Cooperative	Four users (All users)	35.045	156	7625

. The Pareto front of LSOP with the data of the second, third, and fourth users are illustrated in Figure 3, Figure 4 and Figure 5, respectively. It is worth noting that each user has 8 appliances. The minimum costs obtained for the second, third, and fourth users are R8.521, R5.02, and R8.75, respectively; the minimum peak values are 3300, 2000, and 3000 W, respectively.

For the second case of cooperative work for given users, Figure 6 expresses the Pareto front of LSOP using the data of four users, including the first, second, third, and fourth users simultaneously. In four users’ cases, LSOP consists of 37 appliances. The Pareto front of Figure 7 offers R35.045, 156 slots, and 7625 W, as minimum cost, inconvenience, and peak values, respectively.

At the end of the application of the MHGSO algorithm, a set of compromise solutions (Pareto front) is obtained. Therefore, a multicriteria decision-making method is used to sort the solutions from best to worst. An EWM is used to identify the weights of the criteria that dominate LSOP.

Table 6. Weights of criteria derived by EWM.

Case	Scenarios	${\mathit{w}}_1$	${\mathit{w}}_2$	${\mathit{w}}_3$
Non-Cooperative	First user	0.191	0.494	0.315
	Second user	0.014	0.392	0.594
	Third user	0.166	0.531	0.303
	Fourth user	0.085	0.531	0.384
Cooperative	Four users	0.142	0.488	0.370

illustrates the weights ($w_1,w_2, \mathrm{and} w_3$) for cost, inconvenience, and peak criterion, respectively, for both cases of the different users.

The TOPSIS multicriteria decision-making method is applied for ranking compromise solutions, where it corresponds to the Pareto front of each case. The $DM$ of each case is the Pareto front solutions with various numbers of alternatives, based on the number of solutions belonging to the Pareto front. As stated previously, the weights of the criteria that dominate the performance of LSOP are derived by using EWM.

Table 7. First three best ranking solutions for both cases.

Cases	Scenarios	Rank of Solutions	Cost (R)	Inconvenience (Slots)	Peak (W)
Non-Cooperative	First user	1	20.78	31	6135
		2	18.41	40	5600
		3	17.05	36	6835
	Second user	1	8.92	33	3400
		2	10.28	30	4200
		3	8.65	37	3300
	Third user	1	8.90	19	2015
		2	8.50	28	2015
		3	5.82	28	3215
	Fourth user	1	12.02	26	3025
		2	10.70	27	4025
		3	13.04	32	3025
Cooperative	Four users	1	47.39	203	9700
		2	48.91	200	10,110
		3	49.86	217	9100

shows the first three best ranking solutions for the two cases when applying the TOPSIS method.

According to Table 7, the user load consumption profile for the first non-cooperative rank solutions, based on the different number of users, are presented in Figure 7, Figure 8, Figure 9 and Figure 10. Figure 7 shows the consumption of the first user before and after applying the proposed load scheduling method. According to the load distribution of the first user, the total cost was reduced from R25.37 before load scheduling to R20.78 after applying the proposed load scheduling. The peak load reduced from 9.8 to 6.135 kW; the incontinence becomes 31 slots as shown in

Table 8. Results of first rank solution.

Cases	Scenarios	Before Scheduling			After Scheduling
		Total Energy (kWh)	Peak Load (W)	Total Cost (R)	Total Energy (kWh)	Peak Load (W)	Total Cost (R)	Incon (Slots)	Cost Reduction (%)	Peak Load Reduction (%)
Non-Cooperative	First user	27.18	9800	25.37	27.18	6135	20.78	31	18	37.3
	Second user	15.5	5500	9.05	15.5	3400	8.92	33	1.4	38.18
	Third user	12.9	2015	9.8	12.9	2015	8.89	19	9.2	0
	Fourth user	16.8	4030	12.6	16.8	3025	12.03	26	4.5	24.9
Cooperative	Four users	72.58	14,200	54.5	72.58	9700	47.3	203	13.2	31.6

. The saving thus becomes 18% for cost and about 37.3% for peak load. These savings of the first rank solution clearly reflect the importance of the given weights and are tabulated in Table 6 for the covered objectives. The results indicate high saving for peak and inconvenience because they offer high weights while the cost for this solution offers lower weight. The second rank of the first user provides significant reduction for peak to become 5600 kW. The third rank solutions of the first user show superior reduction in energy cost to be about R17.05, as shown in Table 7. These solutions show the importance of weights regarding the covered objectives. Figure 8, Figure 9 and Figure 10 show the consumption profiles of the other users. The cost and peak savings for the second, third, and fourth users are (1.4% and 38.18%), (9.2% and 0%), and (4.5% and 24.9), respectively.

In the case of cooperative end-users, Figure 11 shows the users’ consumption profiles before and after applying a scheduling method for the four users. Figure 11 provides the consumption disruption for the four users, which are 13.2% (R54 to R47.39) and 31.6% (14.2 to 9.7 kW) for cost and peak saving, respectively, as shown in Table 8.

Table 7 and Table 8 demonstrate the results of load scheduling after applying the proposed method for non-cooperative and cooperative work (four users). The total cost of non-cooperative is about R50.62 (user 1 to user 4); the total cost for cooperative is R47.39, which means about 5.5% of the cost is saving. The peak power level is also reduced from 10.3 kW to 9.3 kW as compared to the non-cooperative end-users’ case, which means approximately 9.7% peak saving.

6. Validation of the Proposed Load Scheduling Optimization Problem

To demonstrate the contribution of the proposed LSOP solving method, in contrast to the current published work, a comparison is conducted with different related contributions.

Table 9. Comparison results.

Before Scheduling				After Scheduling
References	Total Energy (kWh)	Peak Load (kW)	Total Cost (R)	Total Energy (kWh)	Peak Load (kW)	Total Cost (R)	$\mathit{I}\mathit{n}\mathit{c}\mathit{o}\mathit{n}$ (Slots)	Cost Reduction (%)	Peak Load Reduction (%)
Biased to minimized (cost) [8]	27.18	10.5	25.37	27.18	8.4	18.80	73	25	20
Biased to minimized (cost) [9]	27.18	10.5	25.37	27.18	6.8	17.38	-	31	35
multi-objective [12]	27.18	11.7	25.37	27.18	6.9	17.14	42	32	41
Proposed method for first rank solution	27.18	10.5	25.37	27.18	6.1	20.78	31	18	42

shows a comparison of the proposed method with results of [8,9,12] for the data of the first user. In [8,9], the authors focus on a single objective only, that of cost. Here, the cost saving is 25% and 31%, respectively. For peak load, the saving is 20% in [8] and 35% in [9]. In [12], the authors solve the LSOP based on two objectives, namely cost and inconvenience, while the savings are 32% and 41% for energy cost and peak demand, respectively. On the other hand, the proposed method deals with multi-objectives (cost, peak, and end-user inconvenience) concurrently for dominant solutions for all objectives. These Pareto front solutions are ranked according to the dominant rank method of MCDM to sort the predefined solutions-based weights by EWM. With regard to the first rank solution, the cost saving is found to be 18% while the peak saving is 42%. This solution provides more significance for peak and end-users’ inconvenience than cost.

7. Conclusions

Triple objectives-based load scheduling system was proposed to achieve optimal energy cost, minimum peak power demand, and lowest end-user inconvenience level. All sets of the Pareto front for domain solutions were presented, then these solutions were ranked by using MCDM according to a predetermined weight of the EWM method. The proposed system was applied for two cases of end-users: cooperative and non-cooperative. This assumption was made to support the effect of cooperative work on these objectives. The cost saving for cooperative work for all users was about 5.5%; the peak saving was 9.7%. These results confirmed that the cooperative work was more beneficial for users in saving costs. Accordingly, utility was also benefited by reducing the peak, and it minimizes the required generation capacity to support demand. The advantages of managing available generation capacity, which benefits users and utility companies, can lead to the reduction of GHG and aid in tackling the effects of weather warming and blackout crises.