Implementing Elephant Herding Optimization ...

Implementing Elephant Herding Optimization Algorithm with different Operation Time Intervals for Appliance Scheduling in Smart Grid Syed Muhammad Mohsin1 , Nadeem Javaid1,∗ , Sajjad Ahmad Madani1 , Syed Muhammad Abrar Akber2 , Sohaib Manzoor3 , Javed Ahmad4 1 COMSATS

Institute of Information Technology, Islamabad, 44000, Pakistan of CST, Huazhong University of Science and Technology, Wuhan, 430074, China 3 School of EIC, Huazhong University of Science and Technology, Wuhan, 430074, China 4 Department of Computer Science, Virtual Univeristy of Pakistan, Lahore ∗ Correspondence: www.njavaid.com, [email protected]

2 School

Abstract—Demand Side Management (DSM) is the strategy applied in smart grid domain for Home Energy Management (HEM) and balancing of electricity demand and supply. Numerous optimization techniques have been proposed by research community for HEM. In this study, we have implemented Elephant Herding Optimization (EHO) technique to achieve four objectives: cost, PAR and waiting time minimization with user comfort maximization. EHO has been implemented with three different Operation Time Intervals (OTIs) i.e., 05, 30 and 60 minutes. Simulation results showed that EHO performed much better than un-schedule case and EHO efficiency for shorter OTI is also better than longer OTIs. Index Terms—Smart Grid, Demand Side Management, Demand Response Program, Elephant Herding Optimization.

I. I NTRODUCTION Electric Power is produced by various sources like hydroelectric power plants, atomic power plants, wind energy, thermal power plants, solar power plants etc. Residential buildings are the important users of electric power and according to [1] 30% - 45% of around the globe generated electric power is required by residential buildings. Power demand of residential users is increasing with population which ultimately influences the grid stability. Research community has presented the smart grid concept to enhance effectiveness and reliability of entire electric power system [2], shown in figure 1. Two-way communication, Smart Appliances, Smart Meters (SMs), Renewable Energy Sources (RESs) are among essential parts of smart grid, shown in I. Demand Side Management (DSM) helps to avoid from unbalanced grid and has two main functions: load management and Demand Response (DR) program [3]. 15% of energy consumption of residential buildings can be decreased even without knowledge of electricity user [4]. According to the authors of [5] DSM is one the major factors to assist the

electricity users to take informed decisions about electricity usage. Load management, helps to avoid from load shedding, blackout and grids instability whereas, DR programs motivates the electricity users to alter their electricity demand according to the dynamic electricity rates announced by utility [6]. Peal Load Pricing (PLP), Time of Use (ToU), Day Ahead Real Time Pricing (DA-RTP), Critical Peak Pricing (CPP), Real Time Pricing (RTP), Inclined Block Rate (IBR) and Day Ahead Pricing (DAP) are prominent pricing schemes. RTP is considered as the most effective and the most efficient pricing scheme [7]. Key objectives of smart grid include cost minimization, peak to average ratio (PAR) minimization, power consumption minimization, user comfort maximization and RES integration. Mixed Integer Linear Programming (MILP) [8], Mixed Integer Non Linear Programming (MINLP) [9], Convex Programming (CP) [10], Genetic Algorithm (GA) [10], [12], Binary Particle Swarm Optimization (BPSO) [13], Ant Colony Optimization (ACO) algorithm [14], Differential Evolution (DE) algorithm [15] are few of the scheduling techniques proposed to achieve the smart grid objectives. State of the art related work reveals the vital contribution of the research community but non of the cited work has considered and investigated the effect of varying OTIs on the efficiency of EHO algorithm. In this study, we have considered three different operation time intervals (OTIs) and analyzed the efficiency of Elephant Herding Optimization (EHO) algorithm. Rest of the paper is organized as follows. Section II describes state of the art work. Elephant herding optimization technique has been discussed in section III and problem statement is given in section IV. System model is discussed in section V. Simulation results and their discussions are given in section VI whereas, section VII concludes this paper.

Fig. 1. Smart Grid

II. R EVIEW OF E XISTING L ITERATURE Many optimization techniques have been proposed by research community to achieve the main objectives of DSM in Home Energy Management System (HEMS). Few of the prominent optimization techniques implemented for better DSM are given in the following. Summary of the cited literature review is shown in table I of this study. Authors of [13] have proposed an Energy Management Controller (EMC) for DSM in HEMS. Home appliances are categorized into three classes on the basis of their operating nature. Ant Colony Optimization (ACO), Binary Particle Swarm Optimization (BPSO) and Genetic Algorithm (GA) have been implemented in MATLAB for evaluation. ToU and IBR based pricing scheme have been used for electricity cost calculation. In terms of execution time, proposed energy management controller performed better with GA as compared to ACO and BPSO. Harmony Search algorithm (HSA) based optimization technique has been proposed by the authors of [16] to schedule RESs based storage system. ToU has been used as pricing scheme. Load and generation profiles of residential customers have been considered by authors during experiments. Simulation results show that HSA performed much better than GA. In [17], authors have proposed GA based algorithm for DSM of residential load. Cost minimization and user comfort maximization are the main objectives of this study. Household appliances have been categorized into five categories namely: thermostatically controlled appliances, elastic appliances, inelastic appliances, user aware appliances and regular appliances. Authors have implemented Intelligent Programmable Communication Thermostat (IPCT) and Conventional Pro-

grammable Communication Thermostat (CPCT) for scheduling of home appliances. Simulation results show 22.63% cost reduction and 22.77% minimization in PAR. Authors of [18] authors have implemented the proposed Load Satisfaction Algorithm (LSA) in three different budget scenarios of electricity user. Objective of this study are to minimize the electricity cost and maximize the user comfort. They have used ToU as pricing scheme for electric bill calculation. After implementation of proposed algorithm n three budget scenarios, authors have concluded that user’s budget directly affects user satisfaction. As user’s budget increase, user comfort also increases and vice-versa. Hybrid Differential Evolution with Harmony Search (DEHS) has been proposed by author of [19]. DE-HS has been proposed as scheduling algorithm for micro grid scheduling. Micro grid consists of different energy sources like Photo Voltaic (PV) system, Wind Energy, Electric Vehicles (EVs), battery storage and traditional generators. In the proposed technique EVs works as load and energy source. Proposed optimization technique has been implemented in two different scenarios: micro grid scheduling with EV and storage system and micro grid scheduling without EV and storage system. Authors concluded that DE-HS performed better in scenario 1 and electricity cost incurred during scenario 1 was 7.83% lesser than scenario 2. Authors of [20] have proposed Multi Objective Evolutionary Algorithm (MOEA) to minimize the cost and delay. Priorities have been assigned to home appliances to control their waiting time. Higher priority appliances can interrupt low priority appliances. Average delay calculation function has also been discussed by the authors. PAR is not considered in the pro-

posed technique and it has higher complexity. In [21], authors have proposed an ILP based optimization technique for consumers of residential area. Authors have implemented the proposed technique in three different scenarios. In first case electricity cost minimization is the primary concern of user and user does not care about his / her discomfort. In second case user only cares about his / discomfort whereas neglects electricity cost. In third case user cares about both parameters; discomfort and electricity cost. Significant trade-off is evident from implementation results of the proposed technique. A priority based scheduling algorithm has been proposed in [22] to minimize cost, PAR and energy consumption. Home appliances have been assigned priorities and all appliances switch ON/OFF on the basis of their assigned priority. Results show that proposed algorithm performed well in achieving the defined objectives but low priority appliances may face starvation in this scheme. Authors of [23] have used recursive formula approach to calculate peak demand. Proposed technique has been implemented in four different scenarios: non-scheduled, compressed, delay based and postponement request scenarios. They have considered infinite number of appliances to be scheduled in a smart home. RTP has been used as pricing scheme in the proposed scheme. RESs and user participation in HEM was also considered in the paper. Simulation results show that proposed technique consumed lesser computational time to calculate the peak demand. An opportunistic scheduling algorithm based on an optimal stopping rule has been proposed by the authors of [24]. Users have been categorized into three groups namely: active, passive and semi-active users. Modified First Come First Serve (MFCFS) algorithm has been implemented to minimize electricity cost and Priority Enable Early Deadline First (PEEDF) algorithm has been used in the proposed technique for user comfort maximization. Proposed scheme provided better results of user comfort maximization and cost reduction. In [25], a day-ahead scheduling model named as hybrid HSA with DE (HSDE) has been proposed for micro grid to minimize total generation and operation cost of micro grid. Power flow constraints have been taken into account by authors of this paper. Proposed scheme has been validated through comparison with other scheduling techniques like Hybrid Artificial Immune Algorithm with DE (AIADE), Hybrid Particle Swarm Optimization with DE (PSODE) and Hybrid GA with DE (GADE). In terms of cost reduction and execution time, HSDE performed better than AIADE, PSODE and GADE. MINLP has been implemented by the authors of [26]. Objective function of the proposed technique are electricity cost minimization and user comfort maximization. They have used RTP as pricing scheme for electricity cost calculations. Household appliances have been categorized into electrically controllable, thermally controllable and uncontrollable appliances. RESs integration and uncertainty of load has also been considered in the proposed scheme. Significant cost reduction has been recorded in the simulation results of the MINLP

based proposed technique. Generic DSM model with objective of cost, PAR and waiting time minimization has been proposed by the authors of [27]. Authors have implemented GA for scheduling of smart appliances using RTP pricing scheme for electricity bill calculation. Proposed technique has been implemented for two cases: single user and twenty users. Simulation results show that proposed algorithm performed well in the both cases. System complexity for twenty users is more than single user case. Two algorithms: Teaching and Learning Based Optimization (TLBO) and Shuffled Frog Learning (SFL) have been proposed by the authors of [28]. Authors have categorized the household appliances into shiftable, sheddable and nonsheddable loads. They have used ToU, RTP and CPP as pricing schemes. Cost reduction was the primary objective of this study. Simulation results show that TLBO provided better results than SFL and considerable decrease in consumer cost was also evident. III. O PTIMIZATION T ECHNIQUE Elephants are social creatures, and they live in social structures of females and calves. An elephant clan is composed of a few elephants under the command of Matriarch. A tribe comprises of one female with her calves or certain related females. Females like to live in family gatherings, while male elephants tend to live in disconnection, and they leave their family gathering when grown up. However, male elephants can remain in contact with elephants in their family. In EHO technique, grouping conduct of the elephants is considered for the formulation of an optimization technique. EHO technique was proposed by Wang et. al., in 2015 [29]. Authors have proposed this algorithm on the basis of herding or grouping behavior of elephants while considering following assumptions for elephants herding. • Subgroups of whole population of elephants is known as clans and each clan is composed of a fix number of elephants in it. • At one time, fixed number of elephants leave their clans and they live alone. • Matriarch in each clan is the head of clan which represents the best solution. A: Clan updating Operator This operator in EHO is used to update the next position of the elephants in the clan ci using equation number 1. x(new,ci,j) = xci,j + α(xbest,ci − xci,j )r

(1)

x(new,ci,j) and xci,j in the above equation show new updated position of elephant j and old position of elephant j respectively in clan ci. α ∈ [0, 1] denotes the scaling factor which determines the influence of matriarch in clan ci. xbest,ci shows matriarch ci, which is the fittest elephant in clan ci. r ∈ [0, 1] in equation number 1 denotes distribution, which is taken as uniform in this case.

TABLE I S UMMARY OF L ITERATURE R EVIEW Reference(s) [13]

Implemented Technique(s) ACO, BPSO and GA

[16] [17]

HSA GA, IPCT, CPCT

[18]

Load Satisfaction Algorithm

[19] [20]

DE-HS MOEA

[21]

ILP

[22]

Priority Based Scheduling Algorithm

[23]

Recursive Formula

[24]

MFCFS, PEEDF

[25]

HSDE

[26]

MINLP

[27]

GA

[28]

TLBO, SFL

Objective(s)

Limitation(s)

Cost and PAR minimization Cost minimization Cost and PAR minimization Cost minimization and user comfort maximization Cost minimization Cost and waiting time minimization Cost minimization and User comfort maximization Cost, PAR and energy consumption minimization Peak demand calculation under four scenarios Cost minimization and user comfort maximization Generation and electricity cost minimization Cost minimization and user comfort maximization Cost, PAR and waiting time minimization Cost minimization

Security and privacy are not considered. User comfort is not considered System complexity has increased PAR and cost saving is not considered

Emission of pollutant is not considered PAR, security and privacy are not considered. Complexity has increased. PAR and RES integration are not considered

Low priority appliances may face starvation

Higher power consumption due to infinite number of appliances RES installation cost is not considered

System complexity has increased Computational time has increased

System complexity has increased Delay, PAR and user comfort is not considered

B: Clan Separating Operator

TABLE II EHO PARAMETERS

This operator is used to separate the weak elephants out of the clan. Weak elephant is the male elephant who has matured and is about to leave the clan. Separating operator is explained in the equation number 2. x(worst,ci) = xmin + (xmax − xmin + 1)rand

Scenario

1

(2)

x(worst,ci) in the above equation represents the worst elephant. Whereas, xmax and xmin show upper bound and lower bound respectively, for elephant position in the clan ci. rand ∈ [0, 1] here denotes the type of distribution i.e., uniform in this case.

2

IV. P ROBLEM S TATEMENT Main focus of the smart grid research community has been on cost minimization, Peak to Average Ratio (PAR) minimization, load balancing and many optimization techniques have been proposed by the research community to achieve one or all of the mentioned optimization objectives. None of the cited work has considered and investigated the effect of varying OTIs on the efficiency of proposed optimization technique. In this study, we have considered the varying OTIs case. We have implemented elephant herding optimization algorithm with three different OTIs that are 05, 30 and 60 minutes, shown in table III.

3

Parameter Population size Maximum iterations Number of appliances Number of clans alpha beta Population size Maximum iterations Number of appliances Number of clans alpha beta Population size Maximum iterations Number of appliances Number of clans alpha beta

Value 24 100 8 2 0.9 0.1 48 100 8 2 0.9 0.1 50 100 8 2 0.9 0.1

Case of single home having multiple appliances has been considered for this study. These appliances are categorized into four classes on the basis of their operational characteristics, shown in table IV. Overall objectives of this study are stated below. • •

PAR minimization Cost minimization

4) Non-interruptible Class: Non-interruptible appliances are those appliances whose operation cannot be interrupted after startup of their operation e.g., once the laundry washer is scheduled then it will complete its OTIs. Its operation will not be interrupted during its duty cycle. B. Price Model

Fig. 2. System Model

• •

Load balancing User comfort maximization V. S YSTEM M ODEL

This section describes the architecture of the proposed system. In the proposed system, EMC has central role and manages the energy according to the implemented optimization technique. Utility provides price signal to the smart meter of smart home through EMC. User controls the EMC thorough internet. Proposed system model is shown in figure 2. In this study, we have considered the case of one home with multiple appliances; computer, television, refrigerator, light, electric vehicle charger, laundry washer, laundry dryer and dish washer. CPP is used as pricing scheme in the study. Three different scheduling horizons are taken in this study, which are 05, 30 and 60 minutes. Number of homes, number of appliances and pricing model remained same throughout the study. Electricity price has been taken in cents/KWh. EHO implementation parameters of three scenarios are shown in table II of this study. A. Load Categorization Considering power consumption pattern of home appliances, these are categorized into four classes namely: shiftable, non-shiftable, interruptible and non-interruptible class. Brief discussion about the mentioned categorization of the appliances is given in the following. 1) Shiftable Class: Shiftable appliances are those appliances whose complete operation can be shifted to any other time interval e.g., computer. 2) Non-shiftable Class: Non-shiftable appliances are those appliances whose operation cannot be shifted towards any other time interval e.g., refrigerator. 3) Interruptible Class: Interruptible appliances, also known as deferrable appliances, are those appliances whose operation can be interrupted e.g., electric vehicle charger.

Electricity consumers have to pay charges to utility in lieu of the electricity provision. Different pricing schemes are prevalent in the electricity market. These prices are usually variable according to time. During peak hours price is higher and during off peak hours electricity prices are comparatively low. Dynamic pricing model motivates the electricity users to shift their shiftable loads towards off peak hours which is beneficial for both users and utility. Customers get benefit in terms of lesser electricity bill whereas utility is benefited in terms of grid stability. ToU, CPP, PLP, RTP, DAP, DA-RTP, IBR are prominent pricing schemes used by utilities all over the world. In this study, we have used CPP as pricing model for all three scenarios. In this pricing model, price remains same during any time slot whereas, it can be different for different time slots. VI. R ESULTS AND D ISCUSSION Elephant herding optimization is an optimization technique which has been implemented through MATLAB in this study. Simulation results are discussed in detail in this section. EHO has been implemented in three different scenarios shown in table III. Household appliances have been categorized into four classes on the basis of their load nature. Names of appliance classes are shiftable class, non-shiftable class, interruptible class and non-interruptible class, shown in table IV of this study. TABLE III EHO I MPLEMENTATION S CENARIOS Scenario 1 2 3

Number of Appliances 08 08 08

Algorithm EHO EHO EHO

Tariff CPP CPP CPP

OTIs 60 min 30 mim 05 min

Three different implementation scenarios have been shown in table III of this study. In scenario 1, EHO has been implemented on 08 appliances with CPP as pricing scheme and OTI is 60 minutes. In scenario 2, EHO has been implemented on 08 appliances with CPP as pricing scheme and OTI is 30 minutes. In scenario 3, EHO has been implemented on 08 appliances with CPP as pricing scheme and OTI is 05 minutes. Home Appliance categorization has been shown in table IV of this study. Shiftable appliances are those appliances whose complete operation can be shifted to any other time interval for e.g. computer. Non-shiftable appliances are those appliances whose operation cannot be shifted towards any other time interval for e.g. refrigerator. Interruptible appliances are those

TABLE IV A PPLIANCE C ATEGORIZATION 9

Appliance(s) Computer, Television Refrigerator, Light Electric Vehicle Laundry Washer, Laundry Dryer, Dish Washer

1400

8 7 6

Waiting Time

Category Shiftable Appliances Non-shiftable Appliances Interruptible Appliances Non-interruptible Appliances

5 4 3

1200 2

Total Cost (Cent)

1000

1 0

800

EHO

600

Fig. 5. Waiting Time in Scenario 1 400

200

0 Unscheduled

4.5

EHO

Unscheduled EHO

4

Fig. 3. Total Cost in Scenario 1

appliances whose operation can be interrupted for e.g. electric vehicle charger. Whereas, non-interruptible appliances are those appliances whose operation cannot be interrupted after startup of their operation for e.g. once the laundry washer is scheduled then it will complete its total OTIs. Simulation results of scenario 1 are given in table V whereas, its plots are shown in figures 3, 4, 5, 6 and 7. In this results table, unscheduled and EHO based scheduled cases are compared. After scheduling with EHO, 27.18% improvement in cost, 47.90% improvement in PAR and 27.85% improvement in peak load values have been recorded. Overall load in unscheduled case and scheduled case is equal.

Load (kWh)

3.5 3 2.5 2 1.5 1 0.5 0

5

10

15

20

Time (Hours)

Fig. 6. Load in Scenario 1

35 12 30 10

Load (kWh)

25

PAR

8

6

20

15

10 4 5 2 0 Unscheduled

EHO

0 Unscheduled

EHO

Fig. 7. Over All Load in Scenario 1 Fig. 4. PAR in Scenario 1

25

TABLE V S IMULATION R ESULTS OF S CENARIOS 1, 2 & 3 Scenario

1

2

3

Parameter Cost PAR Waiting Time Peak Load Over All Load Cost PAR Waiting Time Peak Load Over All Load Cost PAR Waiting Time Peak Load Over All Load

Unscheduled 1245.75 11.23 —– 4.20 30.09 589.89 5.83 —– 1.50 29.82 33.24 47.85 —– 0.4042 16.82

EHO 907.51 5.84 5.93 3.03 30.09 505.95 2.99 18.38 1.07 29.82 21.58 28.92 127.02 0.31 16.82

Achievement +27.18% +47.99% —– +27.85% Equal +14.23% +48.71% —– +28.67% Equal +35.07% +39.56% —– +19.10% Equal

600

6

5

400

4

300

PAR

Total Cost (Cent)

500

200

3

2

100

1

0 Unscheduled

EHO 0 Unscheduled

EHO

Fig. 8. Total Cost in Scenario 2 Fig. 9. PAR in Scenario 2

20 18 16 14

Waiting Time

In table V simulation results of scenario 2 are given whereas, its plots are shown in figure 8, 9, 10, 11 and 12. In this results table, unscheduled and EHO based scheduled cases are compared. After scheduling with EHO, total cost, PAR and peak load have been improved by 14.23%, 48.71% and 28.67% respectively. Overall load in unscheduled case and scheduled case is equal. Simulation results of scenario 3 are given in table V whereas, its plots are shown in figure 13, 14, 15, 16 and 17. In this results table, unscheduled and EHO based scheduled cases are compared. 35.07%, 34.67% and 19.10% improvements have been recorded in total cost, PAR and peak load. Overall load in unscheduled case and scheduled case is equal. Feasible region is the set of all possible points of any optimization problem in which objective function satisfies the results. Specific set points in the feasible region show constraints of the problem. In this study, we have found the feasible region of cost versus energy consumption (load) in three scenarios. Feasible region is sub set of total region, denoted by equation number 3. To find out the points for total region four

12 10 8 6 4 2 0 EHO

Fig. 10. Waiting Time in Scenario 2

1.5

50 Unscheduled EHO

45

1.3

40

1.2

35

1.1

30

PAR

Load (kWh)

1.4

1

25

0.9

20

0.8

15

0.7

10

0.6

5

0.5

0 0

5

10

15

20

25

30

35

40

45

50

Unscheduled

EHO

Time (Hours)

Fig. 11. Load in Scenario 2

Fig. 14. PAR in Scenario 3

30

140

120

25

100

Waiting Time

Load (kWh)

20

15

80

60

10 40 5

20

0

0 Unscheduled

EHO

EHO

Fig. 12. Over All Load in Scenario 2

Fig. 15. Waiting Time in Scenario 3

35

0.45 Unscheduled EHO

0.4

30

0.35 0.3

Load (kWh)

Total Cost (Cent)

25

20

15

0.25 0.2 0.15

10 0.1 5

0.05

0

0 Unscheduled

EHO

0

50

100

150

200

Time (Hours)

Fig. 13. Total Cost in Scenario 3

Fig. 16. Load in Scenario 3

250

300

TABLE VII P OINTS FOR F EASIBLE R EGION

18 16

Scenarios

14

Load (kWh)

12

1

10 8 6

2

4 2 0 Unscheduled

EHO

3 Fig. 17. Over All Load in Scenario 3

Values (0.840, 8.0438) (0.840, 189.6905) (4.20, 40.2191) (4.20, 948.4524) (1, 225.822) (4.20, 225.822) (0.5000, 1.4250) (0.5000, 39.8428) (1.5000, 4.2750) (1.5000, 119.5283) (1, 79.6856) (1.5000, 79.6856) (0.0083, 0.00006557) (0.0083, 0.0150) (0.4042, 0.0032) (0.4042, 0.7271) (0.14, 0.2520) (0.4042, 0.2520)

1000

P 4(4.20, 948.4524) 800

Total Cost (Cents)

different cases are taken into account that are minimum load & minimum cost, minimum load & maximum cost, maximum load & minimum cost and maximum load & maximum cost. Values of the mentioned cases are given in table VI.

Points P1 P2 P3 P4 P5 P6 P1 P2 P3 P4 P5 P6 P1 P2 P3 P4 P5 P6

600

P 5(1, 225.822) P 6(4.20, 225.822)

P 2(0.840, 189.6905)

400

P 3(4.20, 40.2191)

200

F easibleRegion ⊂ T otalRegion

P 1(0.840, 8.0438)

(3) 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.4

0.45

Power Consumption

Fig. 18. Feasible Region in Scenario 1

0.8

TABLE VI D IFFERENT C ASES AND VALUES

P 4(0.4042, 0.7271)

0.7

Scenario 1

2

3

Point P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4

Cases Min. load & min. cost Min. load & max. cost Max. load & min. cost Max. load & max. cost Min. load & min. cost Min. load & max. cost Max. load & min. cost Max. load & max. cost Min. load & min. cost Min. load & max. cost Max. load & min. cost Max. load & max. cost

Load 0.840 0.840 4.20 4.20 0.5000 0.5000 1.5000 1.5000 0.0083 0.0083 0.4042 0.4042

Cost 8.0438 189.6905 40.2191 948.4524 1.4250 39.8428 4.2750 119.5283 0.00006557 0.0150 0.0032 0.7271

Total Cost (Cents)

0.6 0.5

P 5(0.14, 0.2520)

0.4

P 6(0.4042, 0.252)

0.3 0.2

P 2(0.0083, 0.0150)

P 3(0.4042 0.0032)

0.1

P 1(0.0083, 0.00006557)

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Power Consumption

Fig. 20. Feasible Region in Scenario 3

120

P 4(1.5000, 119.5283)

100

In table VI, values of the points are shown. By using these points total region is drawn and maximum of un-schedule cost is taken as taken as cut-off point to find out the feasible region from total region. Feasible regions of three scenario are shown in figures 18, 19 and 20. P1, P2, P3 and P4 constitute total region and after passing line through P5 and P6, we found feasible region, area shaded with light blue color. Feasible region is surrounded by P1, P2, P3, P5 and P6.

Total Cost (Cents)

P 5(1, 79.6856) 80

P 6(1.5000, 79.6856)

P 2(0.5000, 39.8428)

60 40

P 3(1.5000, 4.2750) 20

P 1(0.5000, 1.4250) 0 -20 0

0.5

1

Power Consumption

Fig. 19. Feasible Region in Scenario 2

1.5

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