In order to make a robust and more reliable power grid, peak demand is taken into account rather than the average demand. As a consequence, natural resources are wasted, and the generation and distribution systems are under-utilized. Fast responding generators (e.g., coal and gas units), which are used to meet the peak demand, are not only expensive, but also have a high carbon emission rate. As a solution, different programs have been presented to shape the energy consumption profiles of users. Such programs aim to efficiently utilize the available generation so that new transmission and new generation infrastructures are minimally installed. These programs, known as demand side management (DSM) programs, aim either at scheduling consumption or reducing consumption [1
A DSM program provides support towards power grid functionalities in various areas, such as electricity market control, infrastructure maintenance and management of decentralized energy resources [2
]. In electricity markets, it informs the load controller about the latest load schedule and possible load reduction capabilities for each time step of the next day. Using this procedure, it schedules the load according to the objectives of interest associated with the power distribution systems [3
]. The load shapes indicate the daily or seasonal electricity demands of industrial or residential consumers between peak hours (PHs) and off-peak hours (OPHs). These shapes can be modified by six techniques [5
]: peak clipping, valley filling, load shifting, strategic conservation, strategic load growth and flexible load shape.
Peak clipping and valley filling are direct load control techniques. Peak clipping deals with the reduction of the peak loads, whereas valley filling considers the construction of loads for the off peak demands. Load shifting is the most effective and widely-used technique for load management in current power supply networks. It is concerned with shifting of the load from PHs to OPHs. Strategic conservation [5
] applies demand reduction methods at the customer side for achieving optimized load shapes. If there is a larger load demand, then the daily responses are optimized by load growth techniques (distributed energy resources) [5
The working of a generic DSM controller is shown in Figure 1
. The figure shows that DSM aims for: (i) electricity cost minimization; (ii) energy consumption minimization; (iii) peak to average ratio (PAR) minimization; and (iv) user comfort (UC) maximization. In the literature, many DSM techniques are proposed [8
] to achieve the aforementioned objectives. However, UC is not considered in most of these techniques, like [8
]. In these works, [11
] aim to reduce the electricity cost, and [20
] focus on minimizing the aggregated power consumption using integer linear programming and mixed integer linear programming. Similarly, electricity bills and aggregated power consumption are reduced in [22
] by using mixed integer non-linear programming. However, these techniques do not take into account the large number of different household appliances. Moreover, randomness in user load profiles makes the scheduling task more challenging.
In this paper, a heuristic algorithm-based DSM controller is designed for a residential area in a SG using the RTP scheme. In the designed DSM controller, five heuristic algorithms are implemented; GA, BPSO, wind-driven optimization (WDO), bacterial foraging optimization algorithm (BFOA) and our proposed hybrid genetic wind-driven (GWD) algorithm. These algorithms are chosen for implementation due to their flexibility for specified constraints and their low computational complexity [23
]. More distinctively, prioritized load shifting is carried out between PHs and OPHs using a large number of appliances in the residential area. For effective scheduling and ease of implementation, the appliances are divided into two classes: (i) Class A (non-shiftable appliances) and (ii) Class B (shiftable appliances). Simulations are conducted in MATLAB such that all of the selected heuristic algorithms are compared in terms of electricity cost, energy consumption, PAR and UC. Results show that our proposed hybrid GWD performs better than the other compared techniques in terms of the selected performance metrics. It is worth mentioning that the nomenclature and list of abbreviations are given in Table 1
and Table 2
The rest of the paper is organized as follows. Section 2
briefly describes the related work. Section 3
formulates the problem. The system model is given in Section 4
. Section 5
deals with the results and discussions. The paper is concluded in Section 6
2. Related Work
], the authors propose a technique for controlling the residential energy loads while maximizing UC and minimizing the electricity bill. A survey of home energy management for the residential customers is presented in [24
], where the authors focus on different techniques relating to shiftable, non-shiftable load and peak shaving. They use various pricing schemes, like RTP, TOU, CPP, IBR, etc. In [25
], a fully-automated EMSfor residential and commercial buildings is presented. They use the Q-learning algorithm for optimal DR mechanisms. Cristopher et al. [26
] design a new framework. They use SMs to decide the appliance schedules based on their load or power consumption. After scheduling, all of the data are transferred to the aggregator module, where the power consumption of all of the appliances is determined. The concept of load clustering is introduced in this approach, which comprises three clusters for scheduling purposes, as the first cluster is from 1 a.m. to 7 a.m., the second from 8 a.m. to 3 p.m. and the third from 3 p.m. to midnight. Two battery scheduling scenarios are used as: (i) the FCFS scheduling policy and (ii) appliance first scheduling policy. In FCFS, requests to consume electricity from clients are assigned priorities based on their arrival, whereas in the appliance first scenario, all electrical devices’ requests are given priority over battery charging.
Another methodology is proposed for minimizing the energy price under the dynamic pricing scheme to avoid PHs in [27
]. Its architecture comprises SM, CN, WSN and IHD. AMI controls bidirectional data flow between the utility and SM. The SM operates between MC and AMI. The MC organizes and controls the schedules of both controllable and uncontrollable electrical appliances, such that the optimal schedule is transmitted to each CN via the WSN. IHD invigilates the whole process. In [8
], GA is used to solve the scheduling problem under the RTP tariff in residential, commercial and industrial sectors. The authors present a novel approach known as the realistic scheduling mechanisms in [28
] for minimizing the customer inconvenience using the TOU pricing scheme. They organize three categories of appliances (ADA, ODA, OIA) and the algorithms relevant to their working times. They also use the BPSO algorithm for the scheduling of these appliances. In [9
], the researchers elaborate an efficient energy scheduling model and an algorithm based on artificial intelligence for residential area energy management in order to minimize the electricity cost. BPSO and GA are used for scheduling the optimal time of appliances and also for obtaining the best fitness values of the objective function.
For solving the numerically-constrained optimization problems, a review of BFOA is presented in [29
]. The authors discuss the taxonomy of constraint handling techniques, the main steps and adaptations to different schemes, including search space, step size, tumble-swim operator and the elimination-reproduction process. In [30
], a case study describes the electric demand model in rural households of Narino. Distributed privacy-friendly DSM is presented in [31
], which preserves users’ privacy by integrating data aggregation and perturbation. The authors describe that the users schedule their requests of appliances according to the aggregated energy consumption measurements as an additive white Gaussian process.
The authors in [32
] focus on cost and emission minimization approaches in data centers and corresponding cloud network infrastructures. They use renewable energy generation capability to enhance the reliability and energy efficiency in SG. They also improve the latency using the ICTs. The decentralized system framework presents DR mechanisms for the residential users to minimize electricity bills, maximize the UC and privacy in [33
]. In this framework, customers’ SMs integrate home load management modules for exchanging the load profiles’ information. Agents exchange information until they find an accurate load profile where the system does not get more improvement in the solution.
], an energy consumption management approach considers household users in which each house consists of two types of requests or demands: (i) essential and (ii) flexible, where flexible demands are further delay sensitive and delay tolerant. To optimize energy for both delay-sensitive and delay-tolerant demands, a new centralized algorithm is presented for scheduling. This approach also aims to minimize the total cost and delay of the flexible demands for obtaining optimal energy decisions. The authors design a cost-efficient demand side day-ahead bidding process and RTP mechanisms by using fractional programming methods in [35
], the authors present a survey of DSM optimization methods for the residential customers. They classify the DSM techniques into three dimensions as: (i) DSM for individual users and cooperative consumers; (ii) DSM as a deterministic model versus the stochastic method; and (iii) day-ahead DSM versus real-time DSM. The dynamic load priority method presents priorities to modify load priorities during the occurrence of demand response events in [37
]. A DR technique formulates the two-stage stochastic problem for energy resource scheduling; inciting the challenges of the renewable sources, electric vehicle and market price uncertainty. It reduces the overall operational cost of the energy aggregator by using stochastic programming [38
]. In [39
], global load balancing schemes describe the data center power management for minimizing the total electricity cost. They explain different components of the data centers as information technology equipment, the power delivery system and the cooling system in relationship with the SG’s features (power delivery, sustainability, peak shaving, etc.). A multi-objective optimization solution is designed using the market operator and the distributed network operator for a microgrid in [40
]. The generation of the price signal from the market operator and the power distribution system is specified using the Pareto-optimal solution. In [41
], a novel pricing strategy is proposed to investigate the robustness against renewable energy source power inputs. This scheme also focuses on the marginal befits and marginal cost of the power market using all existing information related to electricity demand, supply and energy imbalance.
In short, the existing optimization techniques in [8
] are unable to handle the complexity of cost minimization and UC maximization problems due to their non-flexible nature. In fact, the solution of these non-linear problems lead to high computational complexity. Therefore, we use heuristic algorithms (GA, BPSO, WDO and BFOA) to solve these two problems. These algorithms support the multi-objective optimization problems and have flexible constraints and parameters, which are easy to handle. These algorithms are similar to population-based search methods [42
], which move from one population to another population in a number of iterations with improvement using a combination of deterministic and probabilistic rules. The comparison of the aforementioned techniques along with their achievements and drawbacks is listed in detail in Table 3
4. Proposed Solution
The proposed DSM techniques deal with the load management in a residential area for single and multiple homes. Its architecture consists of the number of homes, SMs, AMI and the utility companies. Let multiple homes be connected with a utility and SMs be installed in all of the homes as shown in Figure 2
. The AMI is used for bidirectional communication between SM and the utility. All homes have three types of appliances: (i) fixed; (ii) elastic; and (iii) shiftable. These appliances are also categorized into Class A and Class B based on their fixed or interruptible load profiles. Fixed load appliances are included in Class A, whereas elastic and shiftable are included in Class B. In other words, Class B contains interruptible appliances, which take part in the scheduling process.
The RTP tariff model is used for tracking the pattern of the total hourly costs of the consumed energy. Figure 3
shows that the appliances are scheduled by the appliances’ handler (EMC) during the specified time intervals using the given frame format. EMC schedules and checks appliances’ PB using the frame format. Each frame format consists of an eight-bit pattern, such that each appliance uses a specified bit pattern relating to its class ID, appliance ID, scheduling bit, interruptible or non-interruptible bit and priority bit. Based on the operational status of an appliance, its hourly cost schedule is tracked. In each class, every attribute uses a single bit, except class ID and appliance schedule, which use three- and two-bit patterns, respectively. This scenario is specific to these sets of the appliances using the given frame format for the proposed system’s test cases; however, it can be further extended to a larger set of appliances, and frame length can also be extended accordingly. Evolutionary algorithms are efficient in terms of computational complexity, however, at the cost of reduced accuracy. We prefer frame tracking over other evolutionary algorithms because it provides simple and efficient procedure in terms of relative accuracy and relative computational complexity. In the following subsections, the algorithms of GA, BPSO, WDO, BFOA and our proposed GWD algorithm are discussed in detail.
4.1. GA, BPSO, WDO and BFOA Algorithms
In this section, we modify the existing versions of GA, WDO, BPSO and BFAO to optimally schedule shiftable appliances. Firstly, the load is shifted to the OPHs subject to electricity cost minimization. In order to reduce peaks during the OPHs, each appliance is assigned a specific PB, which indicates the status (either ON or OFF) of the selected appliance. If an appliance is demanded to run in a specific time slot, its PB = 1; otherwise, its PB = 0. This status bit information is communicated via an RTP frame format.
The authors in [13
] have proposed a GA-based home energy management controller for a single home in a residential area using RTP tariffs. In this manuscript, a modified GA (an improved form of [13
]) is presented, which is shown in Algorithm 1. Objective functions (refer to Equations (1)–(3)) and their constraints are used by all of the selected optimization algorithms to find feasible solutions. Users input initial parameters for all appliances. GA creates a random population initially, which consists of a number of chromosomes represented by binary strings as the ON/OFF status of each appliance. Each chromosome is evaluated using Equations (1)–(3). RTP is used as the electricity pricing scheme. Key modifications that are implemented in GA (Algorithm 1 [13
]) to achieve the objectives in the proposed scheme and its expected outcomes are given in Table 4
|Algorithm 1: GA algorithm.|
], another energy management model is presented in which BPSO is used to meet the DSM challenges. The goal of this study is to minimize the electricity cost for residential area by scheduling shiftable loads. The authors use the TOU pricing model to calculate electricity bills of customers by investigating DR; however, they have ignored UC. Furthermore, in our proposed work, the objective function is formulated for cost minimization and UC maximization. BPSO is used to solve the designed optimization problem. RTP scheme is used for tracking the real-time behavior of the system. Thus, this proposed work gives a more significant solution for electricity bill minimization, PAR minimization and UC maximization. All steps of the proposed work are shown in Algorithm 2. Compared to [15
], BPSO is modified according to the customers’ requirements. Each particle in the generation is represented by a binary string denoted as states of an appliance. The proposed model is applicable for single and multiple homes in residential areas. In Table 5
, some suitable modifications and expected results in response to those modifications for the BPSO algorithm are given.
|Algorithm 2: BPSO algorithm.|
A WDO-based scheduling technique is presented in [10
] for comfort maximization of residential users. By considering appliance classes, user preferences and weather status, they model the UC and electricity cost. The WDO algorithm is used for minimizing electricity cost and maximizing UC. This work also analyses peak cost reduction in electricity bills by considering the TOU tariff. In this proposed work, household appliances are categorized on the basis of LOT and appliance power consumption. In order to make the scheduling process more efficient, delay and PB criteria (which are not considered in [10
]) are incorporated here for reducing electricity bills. In this study, WDO is enhanced in which LOT and the energy consumption of each appliance are calculated by evaluating the objective function (refer to Equations (1)–(3)) using constraints. Table 6
shows the enhancements made as per our proposed work and the expected results based on the enhancements. All steps of the implemented WDO algorithm are shown in Algorithm 3.
|Algorithm 3: WDO algorithm.|
], the authors propose a BFOA technique for grid resource scheduling. This technique is based on the hyper-heuristic resource scheduling algorithm, which has been designed to effectively schedule jobs on available resources in a grid environment. The authors evaluate the performance of the proposed BFOA algorithm by comparing it with the existing heuristic scheduling algorithms (GA and simulated annealing) using the makespan and cost performance metrics. Experimental results show that the proposed algorithm outperforms the existing algorithms in terms of cost minimization. In comparison to [17
], the proposed work introduces a new methodology of appliance scheduling for minimizing electricity cost, energy consumption and PAR, which benefits both customers and the utility. In this study, objective functions (refer to Equations (1)–(3)) and their constraints are modified according to the designed scenario. Table 7
contains the refinements made and their respective expected results. All steps of the proposed work are given in Algorithm 4.
|Algorithm 4: BFOA algorithm.|
4.2. Developing a Hybrid GWD Optimization Algorithm
In this algorithm, all of the stages of WDO are performed in a similar way as explained in Section 4.1
; however, the velocity updating steps for the global air pressure is replaced with GA’s crossover and mutation operations. In some cases, pressure values are very large, such that the updating velocities become too large, which degrade WDO’s performance. Thus, we replace these with GA’s crossover and mutation values. The scheduling procedure is followed as the same described in GA, BPSO, BFAO and WDO. It is evaluated with the help of the same objective functions (refer to Equations (1)–(3)). Detailed steps of this algorithm are shown in Algorithm 5. Modifications of the hybrid GWD and their respective expected outcomes are given in Table 8
|Algorithm 5: GWD algorithm.|
The metaheuristic algorithms do not guarantee exact reachability of the global optimum solution. The obtained solution is dependent on the set of random variables generated at the start of the metaheuristic optimization process. In our scenario, PSO, BFOA and WDO suffer from the global optima, and GA is a relatively better suited algorithm for the global optimal solution. In order to filter out the effects of random initializations, simulation runs of these algorithms are increased in number. However, this filtration is achieved at the cost of increased computational time. We have presented the statistical analysis of all of the algorithms with respect to cost and user comfort using the ANOVA in the Results Section after taking the average of the 10 runs.
5. Results and Discussion
In order to evaluate the proposed work, simulations are conducted in MATLAB using the RTP scheme. The 24-h time period is divided into PHs and OPHs for tracking the real-time behavior of the system. Four hours are taken as PHs (from 7 p.m.–10 p.m.) such that the PHs vary from season to season [43
]. From December–February, PHs are from 5 p.m.–9 p.m.; from March–May, PHs are 6 p.m.–10 p.m.; from June–August, PHs are 7 p.m.–10 p.m.; and from September-November, these vary accordingly. Four hours are used in this case (from 7 p.m.–10 p.m.) of one season, and the remaining all are included in OPHs.
There are two simulation scenarios that are discussed here: (i) single home and (ii) fifty homes. Each home has 12 appliances, and appliances are categorized into two classes: (i) Class A with fixed load appliances and (ii) Class B with shiftable and elastic load appliances, as shown in Table 9
. Figure 4
shows the RTP rates during each hour of the full day. The parameters of GA, BPSO, WDO, BFAO and GWD are given in Table 10
, Table 11
, Table 12
, Table 13
and Table 14
, respectively. To evaluate the performance of these algorithms, the following performance metrics are used.
Cost: Amount of electricity bills for the total number of units consumed per unit time in cents.
Energy Consumption: It is calculated as the total energy utilized per unit time in kilowatts per hour.
PAR: It is defined as the total peak load divided by average load during the whole day.
UC: It is calculated in terms of minimum cost and minimum appliance delay.
5.1. Single Home
The energy consumption of our proposed scheme hybrid GWD with respect to GA and WDO in unscheduled and scheduled cases is shown in Figure 5
. This figure shows that the maximum energy consumption values are 16.2 kWh, 11.8 kWh, 8.2 kWh and 4.1 kWh for the unscheduled case, scheduled GA, WDO and the hybrid GWD approach, respectively. The energy consumption of all algorithms is below their unscheduled cases. The energy consumption in GA, WDO and GWD is 56.89%, 67.18% and 65.87%; which is obtained by dividing the scheduled cost and unscheduled cost with percentage. It is important to note that the hybrid GWD algorithm is better than the simple WDO and GA in terms of energy consumption. GWD uses crossover and mutation operations from the GA, which helps with the faster convergence for achieving optimized results, and WDO uses explicit pressure values; however, when velocities are high, pressure values become extremely large, which leads to performance degradation.
The maximum amount of the electricity bill in the unscheduled case is 318.88 cents, as shown in Figure 6
. It is reduced to 78 cents in the case of GA, while it is reduced from 318 cents to 245 cents in WDO and up to 75 cents in GWD. The electricity cost in GA, WDO and GWD is 60%, 62% and 30%, respectively. During PHs, sufficient electricity cost reduction is achieved for all designed algorithms (GA, WDO and GWD). GWD performs better than the other algorithms in terms of the electricity cost reduction due to the amalgamation of crossover and mutation. The WDO’s cost is high due to its high pressure values.
The PAR performance of all algorithms (GA, WDO and GWD) is shown in Figure 7
. This figure shows that PAR is significantly reduced in hybrid GWD as compared to the GA, WDO and unscheduled case. Results prove that our proposed algorithm effectively tackles the peak reduction problem. The PAR graph for GA, WDO and hybrid GWD displays that the power consumption of appliances is optimally distributed without creating peaks during the OPHs and PHs of the day. The PAR in GA, WDO and GWD is 60%, 75% and 40%. WDO has higher PAR than GA because it has higher pressure values of the particles, and GA is more effective in PAR reduction due to its ability to generate new populations of more feasible solutions using crossover and mutation. From these results, it is shown that the hybrid GWD approach outperforms all other schemes, because it uses the best features of both. Peak formation is a major drawback in the traditional electric power system, as it causes customers to pay high electricity bills, and the utility also suffers from high demand, which leads to blackouts or load shedding. The performance of these algorithms in this scenario is improved due to load shifting using appliances’ PBs, which causes utilities to fulfil the demands of customers and gives customers a chance to reduce their electricity bills.
In our proposed hybrid scheme, we have achieved the desired UC as shown in Figure 8
. It shows that UC is significantly reduced for GWD, GA and WDO as compared to the unscheduled case. By applying priority scheduling on the objective functions (refer to Equations (1)–(3)), this work enhanced the performance in terms of UC. UC of the unscheduled case is 98%, while in schedule WDO, GA and GWD, it is 60%. The maximum delay considered here is 4 h; otherwise, the utility has to pay a penalty for the users. There is a tradeoff in UC of all scheduled algorithms because only one scenario is considered here. However, the performance of this work is much better by considering the priority bits and minimum delay during scheduling.
All above simulations are performed for a single home; however, for testing the effects of the proposed scheme in multiple homes, multiple homes are taken in the next section. All of the modified algorithms (GA, BPSO, WDO and BFOA) are tested for 50 homes to investigate these in terms energy consumption minimization and electricity cost reduction. From Figure 13
, it is clear that the proposed work achieves significant results. As these algorithms are designed to satisfy the constraints of the objective function in 24 h, so that residential users get facilitated by reducing their electricity bills and that utilities get the benefit by keeping demand under the power capacity of the grid.
5.2. Fifty Homes
The energy consumption of GA, BPSO, WDO and BFOA is 15.00 kWh, 7.90 kWh, 11 kWh and 14.5 kWh, respectively, which is less than the unscheduled case as 16.5 kWh, approximately; as shown in Figure 9
, Figure 10
, Figure 11
and Figure 12
. The energy consumption in GA, BPSO, WDO and BFOA is 79%, 47%, 45% and 88%. GA is efficient among all of the others, though it considers a larger population size. It uses a natural selection operator, which reduces the convergence time towards the efficient solution during scheduling. BFOA is faster than BPSO and consumes less energy because BFOA is faster for a small population size. On the other hand, BPSO is suitable for a larger population size, and it also escapes from the local minima. WDO consumes more energy as compared to BPSO, BFOA and GA, because it has explicit pressure values of particles, causing performance degradation.
The electricity cost of the simulated algorithms is shown in Figure 13
, Figure 14
, Figure 15
and Figure 16
, which is obtained during the scheduling process. In each case, the scheduled costs of all four algorithms, GA, BPSO, WDO and BFOA, are 125.20, 175, 215 and 160 cents, respectively, which are lower than the unscheduled cost of 350. Furthermore, by using the PBs during appliance scheduling, the overall cost is reduced as compared to the unscheduled cases. After scheduling, the obtained electricity cost by using GA, BPSO, WDO and BFOA is 35%, 50%, 61% and 45%, respectively; whereas, in the unscheduled case, it is 100%. In this case, GA is the most effective algorithm even considering a larger population size than the other algorithms. GA uses the crossover and mutation operation, which is efficient in convergence and at finding the global optimal solution. BPSO uses linear and piecewise functions instead of natural selection operators, and it is mostly used for a large population size to avoid local minima. BFOA is suitable for a small population size, and it is more efficient than BPSO and GA in terms of convergence and energy efficiency. WDO suffers from pressure values, so it gives a higher cost than the others.
Overall, the scheduled peak formation rate is better than the unscheduled cases, and the desired results of the load shifting are achieved by the scheduling. The PAR obtained in GA, BPSO, WDO, BFOA and the unscheduled case is 26%, 25%, 12%, 2% and 46%, respectively. All of the high profile appliances are scheduled to low price rate hours. If the consumed energy in OPHs is high (creating peaks), then appliances are scheduled according to their PBs for reducing load and avoiding peak formation even during the low pricing rate hours. PAR in WDO, BPSO and BFOA is better than GA because GA is tested for a large set of populations, whereas all of the others are tested for a small population size, as shown in Figure 17
UC achieved by GA and BFAO is significantly greater than BPSO, WDO and the unscheduled case as shown in Figure 18
. The UC achieved in GA is nearly 0.9; BPSO is 0.5; WDO is 0.55; BFAO is 0.85; and it is 90%, 50%, 50% and 85%. Because during scheduling, all high power utilization appliances are shifted to OPHs, which facilitates the customers to pay less on the bill, so UC is maximized in BFOA and GA as compared to WDO and BPSO, which are the desired results obtained by the designed objective functions, and it is also beneficial for both customers and utilities.
In order to quantify the computational burden of the algorithms, we have chosen algorithm execution time (in s) as a performance metric. Figure 19
shows the execution time of the five simulated algorithms: GA, BPSO, WDO, BFOA and GWD. From the figure, it is evident that BPSO has the maximum computational burden (execution time = 88 s), and BFOA has the minimum computational burden (execution time = 8 s); a difference of 80 s. Similarly, GA, WDO and GWD take 13 s, 43 s and 32 s (to execute), respectively. The previous figures in the simulation Results Section show that GWD is relatively better than the compared algorithms in terms of the selected performance metrics, and Figure 19
shows the execution time of GWD as relatively moderate (better than WDO and worse than GA). To sum up, the GWD pays the cost of moderate execution time to achieve a considerable increase in UC and a decrease in both PAR and price.
5.3. Performance Trade-Offs in the Proposed Technique
After conducting the simulations, we have found some trade-offs and achievements. This approach is evaluated with the help of the following parameters: cost minimization, energy consumption minimization, UC maximization and PAR reduction. The achievements and trade-offs are mentioned in Table 15
5.4. Statistical Validation of GWD and Counter Part Algorithms Using ANOVA
In order to prove the metaheuristic algorithms’ stochastic nature, we have done the statistical analysis for checking their correctness and efficiency. Two algorithms are taken for comparison with our proposed algorithm in terms of the variance. The ANOVA is based on three assumptions [44
]: (i) all samples of the populations are normally distributed; (ii) all samples of the populations have equal variance; and (iii) all observations are mutually independent. In the table below, the analysis is described in detail for each sample population generated by the each individual algorithms.
Here, df indicates the degrees of freedom; MS represents the mean square test; and F represents the F test (taken by dividing the sum of squares and MS); and these are calculated using the equations from [44
]. We have done the ANOVA of three algorithms including our proposed algorithm. In this way, we have finally estimated that our proposed algorithm varies from them by a significant rate as shown in Table 16
In this work, a DSM controller is designed in which five heuristic algorithms (GA, BPSO, WDO, BFOA and our proposed hybrid GWD) are implemented. The hybrid GWD scheme reduced the electricity cost by approximately 10% in comparison to GA and 33% to WDO. On the other hand, GA provided the global optimal solution in scheduling and faster convergence, even when the population size is large. The GA outperformed BPSO, WDO and BFOA in terms of electricity cost and energy consumption. In contrast to the BPSO, BFOA is suitable for a small population, because it converges at a faster rate when the population size is small. Explicit particle pressure values make WDO the slowest to converge among all of the compared algorithms. The stochastic behavior of these algorithms is analyzed by statistical analysis. Assigning priority to appliances helped with efficient scheduling. Statistical analysis is performed by the ANOVA test, which is used to measure the variation in the algorithms’ performance metrics. In the future, we will focus on enhancing other heuristic algorithms to achieve the desired objectives.