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energies

Article

An Optimized Home Energy Management System

with Integrated Renewable Energy and

Storage Resources

Adnan Ahmad 1, Asif Khan 1, Nadeem Javaid 1,*, Haﬁz Majid Hussain 2, Wadood Abdul 3,

Ahmad Almogren 3, Atif Alamri 3and Iftikhar Azim Niaz 1

1COMSATS Institute of Information Technology, Islamabad 44000, Pakistan;

adnan.a.yousafzai@gmail.com (A.A.); akbarech@gmail.com (A.K.); ianiaz@comsats.edu.pk (I.A.N.)

2Center for Advanced Studies in Engineering (CASE), Islamabad 44000, Pakistan;

majid_hussain47@yahoo.com

3Research Chair of Pervasive and Mobile Computing, College of Computer and Information Sciences,

King Saud University, Riyadh 11633, Saudi Arabia; aabdulwaheed@ksu.edu.sa (W.A.);

ahalmogren@ksu.edu.sa (A.A.); atif@ksu.edu.sa (A.A)

*Correspondence: nadeemjavaidqau@gmail.com; Tel.: +92-300-579-2728

Academic Editor: K. T. Chau

Received: 14 February 2017; Accepted: 11 April 2017; Published: 17 April 2017

Abstract:

Traditional power grid and its demand-side management (DSM) techniques are centralized

and mainly focus on industrial consumers. The ignorance of residential and commercial sectors

in DSM activities degrades the overall performance of a conventional grid. Therefore, the concept

of DSM and demand response (DR) via residential sector makes the smart grid (SG) superior over

the traditional grid. In this context, this paper proposes an optimized home energy management

system (OHEMS) that not only facilitates the integration of renewable energy source (RES) and energy

storage system (ESS) but also incorporates the residential sector into DSM activities. The proposed

OHEMS minimizes the electricity bill by scheduling the household appliances and ESS in response

to the dynamic pricing of electricity market. First, the constrained optimization problem is

mathematically formulated by using multiple knapsack problems, and then solved by using the

heuristic algorithms; genetic algorithm (GA), binary particle swarm optimization (BPSO), wind driven

optimization (WDO), bacterial foraging optimization (BFO) and hybrid GA-PSO (HGPO) algorithms.

The performance of the proposed scheme and heuristic algorithms is evaluated via MATLAB

simulations. Results illustrate that the integration of RES and ESS reduces the electricity bill and

peak-to-average ratio (PAR) by 19.94% and 21.55% respectively. Moreover, the HGPO algorithm

based home energy management system outperforms the other heuristic algorithms, and further

reduces the bill by 25.12% and PAR by 24.88%.

Keywords:

smart grid; demand side management; home energy management system; renewable

energy source; energy storage system; real time pricing; heuristic algorithms

1. Introduction

In recent decades, energy demand around the globe has shown the increasing trend. In past,

most of the power generation was being done from fossil fuels. However, to fulﬁl the increasing

electricity demand with minimal emissions of green house gases scientists have worked on

the new means of electricity generation: renewable and sustainable energy resources (RSERs).

But, the penetration of renewable energy sources (RESs) signiﬁcantly increased power system

complexity and dynamics [

1

], and the existing power system is not capable of maintaining its stability

if the integration of RESs and distributed generation (DG) is done at a large scale. In this context,

Energies 2017,10, 549; doi:10.3390/en10040549 www.mdpi.com/journal/energies

Energies 2017,10, 549 2 of 35

one of the present solutions is the transformation of the existing power grid into the smart grid (SG)

with cutting edge information and communication technologies (ICTs) [

2

]. These advanced ICTs not

only enable SG to incorporate the DG and RESs but also enhance the stability and reliability of power

system. European technology platform (European Commission, 2006) deﬁnes the SG as, “a smart

grid is an electricity network that can intelligently integrate the actions of all users connected to

it-generators, consumers and those that do both in order to efﬁciently deliver sustainable, economic

and secure electricity supplies”. SG has different kinds of operational and energy measures like smart

meters (SMs), smart appliances, renewable energy and electric energy storage resources. The vital

aspect of SG is the control of power production, transmission and distribution through advanced

ICTs. These ICTs enable SG to send control commands within the time limits deﬁned by numerous

international standards e.g., IEEE standards 1547 (i.e., the standards deﬁned for the control and

management of distributed energy resources) [

3

]. Moreover, SG makes possible the access of the power

system operator and end-users at the same time intelligently and efﬁciently.

The key factors that make SG superior over traditional grids are: two-way communication,

advanced metering infrastructure (AMI) and information management units (IMUs). They introduce

intelligence, automation and realtime control to power system. The two-way communication in SG not

only keeps the end-users well informed about the varying electricity prices, maintenance schedules of

the distribution network and events/failures that come either due to equipment failures or natural

disasters but also enables the operator to monitor and analyze the realtime data of energy consumption

and makes realtime decision about the operation activities and standby generators. A comprehensive

comparison of traditional grid and SG features is shown in Table 1.

The SG makes the integration of RESs and DGs practicable and involves the residential and

commercial users into demand-side management (DSM) and demand response (DR) activities [

4

].

DSM is the modiﬁcation of consumer demand for energy through various methods such as ﬁnancial

incentives and behavioral change through education. Usually, the goal of DSM is to encourage the

consumers to use less energy during peak hours, or to move the time of energy use to off-peak hours [

5

].

DSM techniques are used to optimize energy consumption pattern, to efﬁciently utilize the limited

energy resources and to enhance the overall efﬁciency of the power system. The term DR is used for

the programs designed to encourage end-users to make short-term reductions in energy demand in

response to a price signal from the electricity hourly market, or a trigger initiated by the electricity

grid operator. It is a change in the power consumption of an electric utility consumer to better match

the demand for power with the supply. DR seeks to adjust the demand for power instead of adjusting

the supply. However, it is totally impractical to ask the consumers to schedule their energy usage

by compromising their comfort level. Therefore, an automatic home energy management system

(HEMS) is required, however, a little awareness of consumers is required to know the beneﬁts of

various scheduling schemes. In this context, we present an OHEMS which not only integrates RES and

ESS into residential sector but also incorporate residential consumers into DSM activities.

The common objectives of different DSM and DR strategies in SG are the reduction of the electricity

cost and minimization of energy consumption in peak hours. To achieve these objectives numerous

algorithms for an efﬁcient HEMS have been proposed, such as integer linear programming (ILP) [

6

],

mixed integer linear programming (MILP) [

7

], multi-parametric programming [

8

], etc. However,

these techniques cannot tackle a large range of various household appliances having unpredictable,

non-linear and complicated energy usage patterns.

Energies 2017,10, 549 3 of 35

Table 1. A brief comparison of traditional grid and smart grid (SG).

Infrastructure Traditional Grid SG

Power system

Centralized generation.

Uni-directional ﬂow of energy

from utility to the consumers.

Decentralized generation. Two-way

ﬂow of energy between the utility

and the prosumers.

Power Losses

High power losses due to

centralized structure and inadequate

storage facilities.

Signiﬁcantly reduces the power

losses due to DG at distribution level

(i.e., the DG eliminates the losses of

transmission network).

Information system Aged metering and

monitoring system

Advanced metering and monitoring

system: AMI and supervisory

control and data acquisition system

(SCADA)

Communication system Wired technology Both wired and wireless

technologies

ESSs Main storage facility is pump-hydro

power plants.

Facilitate the distributed ESSs

integration

RSERs Mainly includes dispatchable RESs

(Hydro-power plants)

Provides decentralized control for

RSERs (solar, wind, tidal,

geo-thermal and biomass

energies etc.)

Self-healing

Reacts to stop further damage.

Emphasis is on protection of assets

following system faults.

Automatically senses and reacts to

actual and emerging contingencies.

Focus is on prevention.

Optimization of assets

Negligible incorporation of limited

operational data with assets

management processes and

technologies. Time based

maintenance.

Greatly expanded sensing and

measurement of grid conditions,

grid technologies deeply integrated

with assets management processes to

effectively manage assets and costs.

Condition based maintenance.

Consumer engagement

No proper involvement of the

consumers in DSM and DR activities

(i.e., no mechanism to send the

varying electricity prices to the

consumers in realtime and forced

load shedding is carried out to

maintain the balance between

demand and supply)

Provides dynamic pricing,

net metering and other incentive

based schemes.

Power Quality Focus only on the reduction of

failures and interruptions.

Ensure the quality of electricity for

the smooth operations of sensitive

electronics devices/equipments.

To overcome the deﬁciencies of the previously proposed HEMSs, this paper presents an optimized

home energy management system (OHEMS). The proposed scheme not only facilitates the integration

of RES and ESS into the residential sector but also reduces the prosumer’s electricity bill as well as the

PAR. In addition, the performance of the heuristic algorithms: genetic algorithm (GA), binary particle

swarm optimization (BPSO), wind driven optimization (WDO), bacterial foraging optimization (BFO)

and hybrid GA-PSO (HGPO) are also evaluated in terms of energy consumption pattern, and electricity

bill as well as PAR minimization. First, an energy management system (EMS) is designed using

exogenous grid signals; day-ahead pricing (DAP) signal, ambient temperature, and solar irradiance.

Then, the heuristic algorithms are applied to get an optimum solution for the formulated objective

function. The adequacy of proposed OHEMS is validated via simulations. The simulations are

conducted in two stages, in the ﬁrst case, the beneﬁts of RES and ESS integration are highlighted,

Energies 2017,10, 549 4 of 35

while in the second case the performance of GA, BPSO, WDO, BFO and HGPO algorithms based

HEMSs is compared in terms of electricity bill and PAR reduction, and uniform distribution of energy

consumption patterns.

The rest of the paper is organized as follows. Sections 2and 3brieﬂy describe the related work

and problem statement respectively. Section 4deals with problem formulation. Section 5and Section

6present the proposed system architecture and algorithms respectively. Results and discussions are

presented in Section 7, and the paper is concluded in Section 8.

2. Related Work

Recently, numerous DSM strategies have been proposed. Their common objectives are the

minimization of electricity cost, reduction of PAR, mitigation of carbon emissions, and enhancement

of power system efﬁciency. The appliances scheduling problem has been solved as an optimization

problem by numerous classical and heuristics algorithms. In this context, some of the recent studies

are given below, and a brief review of some state of the art trends is presented in Table 2.

In [

9

], the authors present a review of current trends in HEMS and DR in the residential

sector. The importance of HEMS for relocating and curtailment of the load is also discussed.

They give insights on existing optimization techniques: mathematical optimization, model predictive

control and heuristics algorithms. The impact of forecasting uncertainty, devices heterogeneity,

computational limitation and timing consideration in the design of optimization algorithms are

also discussed. However, the user comfort (UC) and appliances waiting time have not been discussed.

The authors in [

10

], present a compact survey of the current trends in HEMS. The challenges in the

implementation of HEMS are discussed and give the insights on current literature regarding DR,

DSM, appliance scheduling, and on single or multiple objective optimizations in HEMS. However,

the integration of RESs and ESSs into residential sector, their impacts on electricity bill and PAR have

not been addressed.

In [

11

], the authors discussed the importance of energy management and planning in smart cities

(SC). The paper presents a review on the planning and optimization of SC energy system. Four areas of

SC energy system: generation, storage, transportation and end user are addressed in detail. Although,

an ESS is integrated to supports residential load in grid events. But, the scheduling of appliances or

relocating of load and ESS in response to dynamic pricing of electricity market has not been discussed.

The authors in [

12

], give insights regarding the ways and manners that facilitate the integration of

RESs and DG in the SG and the concept of SCs have been comprehensively addressed. The obstacles

to the integration of DG into the existing distribution network have been highlighted. The effect

of DG on voltage stability and control at low and medium voltages are also discussed. Moreover,

the impact of DG on power quality, power system stability, and related events of voltage sag and

swell due to the failure of distributed sources are addressed in a comprehensive way. Although,

the authors discussed the impact of DGs and RESs integration on power quality and power system

stability. However, the integration of RESs and ESSs into residential sector and their role in DSM

programs have been ignored.

In [

13

], the authors present a detailed review of evaluating trends in smart homes (SHs) and SG.

They investigate the effectiveness of various communication technologies: Zig-Bee, Z-Wave, Wi-Fi and

wired protocols. The authors point out the merits and demerits of existing technologies, and products

available in the market. Moreover, the barriers, challenges, beneﬁts, and future trends regarding the

communication technologies and their role in the SG and SHs are also discussed. The authors in [14],

demonstrate a residential energy monitoring system (REMS). Three SHs powered by an in-house

RES (i.e., PV system) are considered for demonstration. The SH is equipped with data logger system

to measure and record the electricity production and demand patterns. The internal and external

temperature, as well as the humidity is recorded to accurately schedule the operation of heating

ventilation and air conditioning (HVAC) system. However, this paper did not address the integration

of ESS to utilize the RE more efﬁciently.

Energies 2017,10, 549 5 of 35

A. Mehmood et al. [

15

], present an in-depth review of load furcating (LF), current LF techniques

in existing power system, future trends, and its importance for the implementation of future SG is

discussed. They elaborate the two major types of LF: mathematical modelling and artiﬁcial intelligence

based computational models and their subcategories as well. The authors also present a comparative

study of various dynamic pricing schemes: realtime pricing (RTP), Time of Use pricing (ToUP) and

critical peak pricing(CPP).

Lee et al. [

16

], present linear programming (LP) based REMS for reduction of electricity

cost and PAR by charging the ESS from utility in off-peak hours and is discharged in peak hors.

The integration of RESs into residential sector is not considered and the charging of ESS from utility

is not an economical solution. In [

17

], the authors propose an ILP based HEMS with integrated

RES for appliances scheduling to shift the shiftable appliances from peak hours to off-peak hours.

However, the authors did not consider the UC and ESS integration. The authors in [

18

], discuss the

scheduling of home appliances for designed objective of electricity cost minimization or optimization

of electricity consumption pattern. MILP is used to schedule the household appliances and the RES.

An in-house RES not only reduces the electricity bills but the surplus energy is also sold to the utility to

generate revenue. Although, RES with HEMS is fruitful for both the utility and the prosumer, however,

the installation of RES may not be feasible for a single/small domestic consumer.

In [

19

], the authors proposed a hybrid technique of lighting search algorithm (LSA) and artiﬁcial

neural network (ANN) to generate an optimal operation patterns of the appliances. Four appliances

are modeled and developed in MATLAB/SIMULINK, with pre-deﬁned preferences of the consumer.

To optimize the appliances ON/OFF time, and to complete the assigned tasks with minimum cost,

a hybridization of LSA and ANN is used. The proposed technique signiﬁcantly reduces the electricity

cost and outperforms the hybrid of particle swarm optimization (PSO) based ANN. However, RES and

ESS has not been taken into account to more efﬁciently minimize the electricity bill. In addition,

the reduction of PAR and its impact on power system has not been discussed.

A comparative study of WDO and PSO is done in [

20

]. Household appliances are considered for

scheduling with the objectives of cost minimization and UC maximization. Moreover, Knapsack-WDO

(K-WDO) is also studied for the same objective functions. Results illustrate that K-WDO perform

better than WDO and PSO. However, in the proposed model the integration of RES is not considered

and an ESS is used to store energy from grid in off-peak hours which is not a viable solution.

Z. Weng et al. [

21

], demonstrate a fully automated energy management system (EMS) using

reinforcement learning (RL) techniques. The energy management and appliance scheduling problem

is solved by observe, learn and adapt (OLA) algorithm which adds more intelligence to EMS.

The proposed mechanism signiﬁcantly reduce the cost and PAR, but UC is compromised.

In [

22

], the authors discuss the problem of peak demand during certain hours. The concept

of clustering and smart charging has been introduced to maximize the beneﬁts in term of cost

reduction and UC. A GA-based EMS is designed which efﬁciently utilizes energy within the clusters by

scheduling the ESS and appliances. Results show that by appropriate scheduling of ESS and appliances

in RTP environment customer can get maximum saving over electricity bill. The authors in [

23

]

present the comparison of GA and PSO in terms of computational cost and computational efforts.

Results show that PSO needs less computational effort and computational cost to reach an optimal

solution as compared to GA. In [

24

], the authors demonstrate an efﬁcient HEMS architecture for

implementation of DSM in the residential sector. They combine RTP scheme with ToUP because the

use of only RTP signal may shift the peak demand to off-peak hours. To eliminate the creation of

new peaks, an objective function is properly formulated and solved by using GA. Results illustrate

that the hybridization of RTP and ToUP schemes is effective for reduction of bill and PAR. Moreover,

an acceptable trade-off between the UC and cost reduction is also achieved.

Energies 2017,10, 549 6 of 35

Table 2. Recent Trends: State of the art.

Technique Domain Desired Objective Findings Remarks

LP [16] REMS Reduction of electricity

bill and PAR

Objectives are achieved

via charging ESS from

grid in off-peak hours

and then discharging it

in peak hours

RES has not

been utilized

ILP [17]

Appliances

scheduling and

integration of RES

Reduction of electricity

bill and peak load

A signiﬁcant reduction

bill and peak loads is

achieved via RES

integration and

optimizing energy

consumption pattern

UC and ESS is

not considered

MILP [18]

HEMS,

and integration as

well as grid

interconnection of

RES

Reduction of cost and

PAR via RES utilization

Cost reduction is

achieved

Infeasible for

small scale

residential

consumers

LSA-ANN, and

PSO-ANN [19]

HEMS,

and appliances

scheduling

Cost reduction and

comparison of LSA-ANN

and PSO-ANN

LNA-ANN outperformed

the PSO-ANN, and

signiﬁcantly reduce the

electricity bill

UC and PAR

reduction is not

considered

PSO, WDO, and

K-WDO [20]

Appliances

scheduling

Minimization of

electricity cost and

maximization of UC

A favorable trade-off

between the cost and UC

is achieved, and K-WDO

performs better than

other algorithms

RES is not

exploited

RL [21]Fully-automated

EMS

Optimization of

appliances operating

time, and reduction

of PAR

Peaks formation is

avoided, and cost

reduction is achieved

UC and RES is

not considered

GA [22] REMS Electricity cost and

PAR reduction

New peaks formation is

avoided by dividing the

appliances into clusters

UC is

compromised,

and RES is not

considered

GA, and PSO [23]Appliances

scheduling

Optimization of

appliances operating time

to pay minimum

electricity bill

GA-based scheduling

achieves the desired

objectives with less

computational cost and

efforts

UC is

compromised

at achieving the

minimum

electricity bill

GA [24] HEMS Reduction of electricity

bill and PAR

Hybrid of RTP and ToUP

is ued to avoid the peaks

formation

UC is not

considered

Greedy iterative

algorithm [25]

EMS,

and optimization

of grid operation

Optimization of

consumers energy

consumption pattern for

grid station stability

RTP signal is used an

invisible hand to

optimized the energy

consumption pattern

UC is

compromised

IPSO [26]Grid station

stability

Reduction of load in

peak hours

The desired objective is

achieved by rejecting

extra load requests in

peak hours

Only passive

appliances are

considered,

and UC is

compromised.

P. Chavali et al. [

25

], present a distributed mechanism for REMS and grid optimization using

the greedy iterative algorithm. In the proposed technique, the electricity price is used as an invisible

hand to optimize the appliances scheduling and energy consumption. The proposed scheme did not

consider the user comfort in the problem formulation. In [

26

], the authors proposed an improved-PSO

(IPSO) for the solution of cost minimization problem. Results illustrate that the proposed IPSO brings

the user load curve near to the objective curve, where the objective curve and electricity price have

Energies 2017,10, 549 7 of 35

an inverse relationship. One of the objective functions is power system stability, and the proposed

scheme compromise on the UC by rejecting the load in peak hours.

3. Problem Statement

The rapid increase in household electronic appliances signiﬁcantly increase the electricity demand

of residential sector. Almost 40% of the generated electricity is consumed by the residential sector [

27

].

Currently, power generation is heavily dependent on fossil fuels and the hour of need is to fulﬁl the

inevitably increasing electricity demand with minimal emissions of greenhouse gases. Therefore,

scientists have worked to ﬁgure out new means of electricity generation i.e., RSERs. In this context,

the integration of RSERs and ESSs become lucrative for the researchers. Moreover, the conventional

power grid is already vulnerable to instability due to heavy loaded conditions and will be unable

to maintain its stability if the integration of RESs is done at a large scale. So, the research is

going on to implement the SG in a distributed manner, and HEMS is an integral part of SG to

optimize the energy consumption of household appliances. Keeping this objective in mind, we seek

to develop an OHEMS. Our objective is twofold: (i) integration of RES and ESS into residential

sector; (ii) energy management through appliances and resource scheduling. In order to achieve the

above-mentioned objectives, and to design an efﬁcient HEMS, various algorithms such as LP [

28

],

ILP [

29

], MILP [

30

], dynamic programming (DP) [

31

] and convex programming (CP) [

32

] have been

proposed. However, these techniques have very slow convergence rate, and in some cases, they are

unable to handle a large number of appliances. So, the heuristic algorithms such as GA [

33

], BPSO [

34

],

WDO [

35

] and BFO [

36

] are introduced to overcome these problems. The heuristic optimizations are

used where it is very difﬁcult to ﬁnd the exact optimal/feasible points. For example, in case of LP,

it is understood that the optimal solution must lies within the solutions points in that pool. However,

in heuristic optimization, there might be inﬁnite or more solution points in that solution space and

optimal solutions should be any one among those. Moreover, in the aforementioned techniques (LP, ILP,

MILP, DP and CP) based HEMSs, the integration of RESs, maximization of UC and adaptability with

dynamic pricing are ignored. Therefore, in this paper, not only the heuristic algorithms (GA, BPSO,

WDO, BFO, and HGPO) are used to design an OHEMS but their performance is also evaluated in

terms of energy consumption pattern, electricity bill and PAR reduction.

4. Problem Formulation

In this section, the mathematical models and constraints of photovoltaic (PV) system, ESS and

appliances are presented. Based on these descriptions, the optimization problem is formulated.

4.1. Energy Generation Model of PV System

The house of our smart prosumer is equipped with a rooftop PV system (i.e., RES). Although,

the RSERs mainly include solar, wind, tidal, geothermal, biomass, and biogass energies, however,

among them the most abundant and almost free of cost (i.e., having small operation and maintenance

costs) is solar energy and is available everywhere and to everyone. According to [

37

], the Earth receives

174,000 terawatts (TW) of incoming solar radiation at the upper atmosphere. Approximately 30% is

reﬂected back to space while the rest is absorbed by clouds, oceans and land masses. Most of the

world’s population live in areas with insolation levels of 150–300 watts/m

2

or 3.5–7.0 kWh/m

2

per day.

In this regard, the proposed OHEMS tries to maximize the beneﬁts from PV system and to minimize

the electricity bill, carbon emissions and PAR. The output power

EPV (t)

of PV system in kW at time

t

is calculated by Equation (1) [38],

EPV (t) = ηPV ·APV ·Ir(t)·1−0.005(Ta(t)−25)∀t(1)

where,

ηPV

is energy conversion efﬁciency of the PV system (%),

APV

is the area of the generator (m

2

),

Ir(t)

is the solar irradiance (kW/m

2

) at time

t

, 0.005 is temperature correction factor [

39

],

Ta(t)

is

Energies 2017,10, 549 8 of 35

the outdoor temperature (

◦

C) at time

t

and 25 is standard room temperature (

◦

C). The distribution of

hourly sun irradiation usually complies with a bimodal distribution that can be considered as a linear

blend of two unimodal distribution functions. The unimodal distribution functions could be modeled

by Weibull probability density function as shown in Equation (2) [38],

f(Ir(t)) = ζα1

β1Ir(t)

β1(α1−1)e−(Ir

β1)α1+ (1−ζ)α2

β2Ir(t)

β2(α2−1)e−(Ir

β2)α2, 0 <Ir(t)<∞(2)

where,

ζ

is a weighted factor,

α1

and

α2

are shape factors, together with

β1

and

β2

which are

scale factors.

4.2. Energy Storage Model

A small capacity ESS is used to store a portion of electricity generated by PV system, it is mainly

for exploiting the PV energy more efﬁciently. The ESS stores the energy only when its storage level is

lower then upper charge level. Electricity stored in the ESS at time

t

is presented by Equation (3) [

38

],

and the electricity charged, the electricity discharged and the self-discharging rate is taken into account.

The charging and discharging of ESS would lose some electrical energy, so turn-around efﬁciency of

ESS is considered.

ES(t) = ES(t−1) + κ·ηESS ·EPCh (t)−κ·EPDch (t)

ηESS ∀t(3)

where

ES

is stored energy (kWh) at time

t

,

κ

is time slot duration (hour),

ηESS

is ESS efﬁciency,

EPCh

is

the electric power (kW) supplied to ESS from RES at time and

EPDch

is is the electric power (kW)

supplied to the load from ESS at time t.

In order to maintain the storage and avoid overcharging/deep discharging, charge and discharge

rate of electricity, and energy stored in ESS should not exceed the limits deﬁned by the manufacturer.

EPCh(t)≤EPCh

UB (4)

EP(t)D ch ≤EPDch

LB (5)

ES(t)≤ESCh

UB (6)

where

EPCh

UB

is upper charge limit of ESS charge rate,

EPDch

LB

is lower limit of ESS discharge rate and

ESCh

UB is upper limit of ESS stored energy.

4.3. Energy Consumption Model

Let us suppose that the smart prosumer has two sets of appliances i.e.,

M

and

N

. The set

of shiftable (i.e., can be shifted to low price slots) appliances

M={a1

,

a2

,

a3

,

. . .

,

am}

and

the set of non-shiftable (i.e., will start operation on the time deﬁned by the user) appliances

N={b1

,

b2

,

b3

,

. . .

,

bn}

over a scheduling horizon of

t={

1, 2, 3, 4, 5,

. . .

, 24

}

. The daily energy

consumptions of shiftable and non-shiftable appliances are given by Equations (7) and (8) respectively,

Ea=

24

∑

t=1 m

∑

M=1

Ea

t,meM!={Ea

t1,meM+Ea

t2,meM+. . . +Ea

t24,meM}(7)

Eb=

24

∑

t=1 n

∑

N=1

Eb

t,neN!={Eb

t1,neN+Eb

t2,neN+. . . +Eb

t24,neN}(8)

Energies 2017,10, 549 9 of 35

where

Ea

t1,meM

,

Ea

t2,meM

,

. . .

,

Ea

t24,meM

denote the energy consumption of shiftable appliances and

Eb

t1,neN

,

Eb

t2,neN

,

. . .

,

Eb

t24,neN

represent the energy consumption of non-shiftable appliances at time

t

.

The total daily energy consumption Etotal of the prosumer load is calculated as,

Etotal =

24

∑

t=1 m

∑

M=1

Ea

t,meM+

n

∑

N=1

Eb

t,neN!(9)

4.4. PAR

PAR is a ratio of peak load consumed in a time slot

t

and the average of total load consumed

over the scheduling horizon, i.e., from

t=

1 to

t=

24. PAR tells us about the energy consumption

behaviour of the consumer and the operation of utility peak plants have a direct relationship with the

consumers PARs. So, it is beneﬁcial for the utility and consumer to reduce PAR so that power supply

and demand balance can be maintained. For single user it is calculated as follow,

PAR =maxEtotal (t)

1

T∑T

t=1Etotal(t)(10)

For multiple users “N” it can be calculated as,

PAR =maxEtotal (t,n)

1

T∑N

n=1∑T

t=1Etotal(t,n)(11)

4.5. Energy Pricing Model

Numerous electricity tariffs are available to deﬁne the energy pricing over a day. Such as ToUP,

DAP, peak pricing (PP), CPP and RTP [

40

]. In most of the appliances scheduling schemes, the pricing

of electricity is assumed to be DAP or ToUP, because the RTP increase the communication complexity.

Moreover, in ToUP the pricing horizon is divided into different blocks and a ﬁxed price is deﬁned for

each block. In this model, we use DAP in which the price of electricity changes on the hourly basis

and remains constant in an hour. The daily electricity bill of shiftable appliances

Ea

P

and non-shiftable

appliances Eb

Pis calculated by Equations (12) and (13) respectively,

Ea

P=

24

∑

t=1 m

∑

M=1Ea

meM(t)×Xa

meM(t)×PDA P(t)!(12)

Eb

P=

24

∑

t=1 n

∑

N=1Eb

neN(t)×Xb

neN(t)×PDA P(t)!(13)

Etotal

P=Ea

P+Eb

P=

24

∑

t=1 m

∑

M=1Ea

meM(t)×Xa

meM(t)×PDA P(t)(14)

+

n

∑

N=1Eb

neN(t)×Xb

neN(t)×PDA P(t)!

Xa

meM(t) = (1 if shiftable appliance is ON

0 if shiftable appliance is OFF (15)

Xb

neN(t) = (1 if non-shiftable appliance is ON

0 if non-shiftable appliance is OFF (16)

Energies 2017,10, 549 10 of 35

where

Xa

meM(t)

represent the

ON/OF F

state of a shiftable appliance

M

,

Xb

neN(t)

represent the

ON/OF F

state of an non-shiftable appliance

N

and

PDA P(t)

is DAP in the particular time slot

t

.

The electricity bill

EP(t)

an any time slot

t

after taking RES and ESS into consideration is calculated as,

EP(t) = Ea(t) + Eb(t)−EPV (t)−ES(τ)×PDAP(t)(17)

where,

τ

is a speciﬁc time slot between

t20

and

t24

that having highest bill. As shown in Figure 10 that

the RE is not available in those slots, so, the ESS is discharged to reduce the prosumer’s electricity bill.

4.6. Appliance Scheduling Problem

The main objectives of this work are: to minimize the electricity bill and PAR of the prosumer

by optimization the energy consumption pattern. The reduction in consumer PAR is beneﬁcial

for utility and all connected consumers, because it reduces the operations of utility peak plants:

standby generators. Let us suppose that a utility grid provides supply to

N

number of users and all

of them have HEMSs to optimize their energy consumption patterns and PARs which will deﬁnitely

results in an optimized grid operation. In this paper, the appliances scheduling problem of a single

prosumer is formulated as an optimization problem by using multiple knapsack problems (MKP).

MKP is a combinational optimization problem: to select the appliances for a particular hour from

a given set of appliances (

M

and

N

as deﬁned in Section 4.3). Each appliance has a value

X

which

represents the ON/OFF status of it, and a weight which shows the power rating of the appliance.

The number of the appliances to be selected (remain ON) for a particular hour depends on the objective

function and the constraints i.e., to pay as minimum electricity bill as possible and the total weight

(energy consumption) of the appliances must satisfy the constraints given by Equations (19) and (20).

The optimization problem is deﬁned as,

Objective function:

min Ea(t) + Eb(t)−EPV (t)−ES(τ)×PD AP (t)!(18)

Subject to:

Etotal (t)≤Egrid (t) + EPV (t) + ES(τ),∀1≤t≤24 (19)

Etotal (t)≥Emin

unsch (20)

τ0≤τsch ≤τmax (21)

where,

Egrid (t)

is the sanction load that an end-user can import from the utility grid at time

t

,

Emin

unsch

is

the minimum energy consumed in unscheduled scenario,

τ0

represents the lower limit of scheduling

horizon, τsch indicates the scheduling time and τmax shows the upper limit of scheduling horizon.

4.7. Feasible Region

An area enclosed by a speciﬁc set of points (constraints of optimization problem) is called feasible

region of the solution. Here, the objective function of our optimization problem is the minimization

of the electricity bill as given by Equation (18). It illustrates that the electricity bill depends on two

factors: amount of scheduled load and electricity price in a particular time slot. As the price signal is

deﬁned by the utility and we have no control over it, therefore, to minimize the electricity bill we can

only modify the shape of energy consumption pattern. The electricity price (DAP signal) we used has

range from 8.1 to 27.35 cents/kWh, and the four possible cases of the electricity bill using this DAP

signal and unscheduled load are as given in Table 3.

Based upon the values given in Table 3and maximum electricity bill in unscheduled scenario,

the constraints of the feasible region are,

Energies 2017,10, 549 11 of 35

C1:EMin(unschduled)≤Etotal(t)≤EMax(unschduled)

C2:EP(scheduled)(t)≤EP(Max,unscheduled)

C3:∑24

t=1Etotal

P(scheduled)≤∑24

t=1Etotal

P(unscheduled)

where the constraint

C1

shows that in any time slot

t

the scheduled load must be greater than

minimum unscheduled load and less than maximum unscheduled load,

C2

shows that in any time

slot

t

the electricity bill must be less than maximum electricity bill in unscheduled scenario and

C3

represents that in scheduled scenario the total electricity bill must be less than the total electricity bill

in unscheduled scenario.

Table 3. Possible cases.

Case Discerption

1Loadmin ,Pricemin

2Loadmin ,Pricema x

3Loadmax,Pricemin

4Loadmax,Pricem ax

5. Proposed System Architecture

In SG, DSM and DR ensure more stable and reliable grid operation. The main objective behind

the design and implementation of an HEMS is the reduction of electricity bill and PAR. From the utility

point of view, its two main beneﬁts are the management of energy resources and reduction of PAR.

While from the consumers point of view, it minimizes the electricity bill. These objectives are achieved

only with the modiﬁcation in energy consumption pattern of shiftable appliances in response to DAP

signal of electricity market deﬁned and broadcasted by the utility.

In the proposed OHEMS, it is assumed that in future SG each prosumer will have a HEMS.

To meet the energy demand with minimum electricity bill smart prosumer utilizes an in-house RES

and an ESS along with grid energy. HEMS of the grid friendly prosumer schedules the appliances and

the ESS to reduce the electricity bill and PAR in dynamic pricing environment of the electricity market.

In this context, the prosumer’s appliances are divided into two categories: shiftable (i.e., their start

time can be shifted to low price slots and would not be interrupted once they start operation) and

non-shiftable (i.e., their start time can not be deferred and would not be interrupted during operation)

appliances, as shown in Table 4.

Table 4. Load categorization.

Shiftable Loads Non-Shiftable Loads

Washing machine Personal computers

Air conditioner Security cameras

Clothes dryer Microwave oven

Water heater Refrigerator

Dish washer Television

ESS Lights

The proposed system architecture shown in Figure 1, mainly includes AMI, SM,

smart scheduler (SS), master controller (MC), PV system, direct current (DC)/alternating current

(AC) inverter, ESS, and appliances. In Figure 1the energy ﬂow is represented by solid lines while

information ﬂow is shown by dotted lines.

Energies 2017,10, 549 12 of 35

Figure 1. Proposed smart home (SH).

The integration of advanced ICTs into the conventional power grid for metering and

communication is known as AMI. It works as a backbone of the SG and enables the two-way

communication between the utility and the consumers. Furthermore, AMI is responsible for the

collection and transmission of energy consumption data from SM to the utility as well as for the

relaying of exogenous grid signals to the SM. The exogenous grid signals may be the price signal,

ambient temperature, solar irradiance or DR signal. The SM works as a communication gateway

between the SH and the utility. SM is typically installed between the AMI and EMC. The main

functions of SM are reading, processing and sending of energy consumption data to the utility via AMI

as well as the receiving and processing of DR and pricing signals from the utility. RSERs are considered

as a real alternative to fossil fuel power generation. RSERs mainly includes PV system, wind turbines,

small scale hydro-turbines and fuel cells. In the proposed scheme only PV system is utilized due

to its easy installation and small capital investment. A DC/AC inverter is used to convert the PV

system generated DC electricity into AC. An ESS works as a source and sink is regarded as a promising

solution for the integration of RSERs into distribution networks/residential sector. Therefore, in the

proposed OHEMS a small capacity ESS is used to efﬁciently exploit the PV system, and to reduce the

electricity bill in peak hours. An SS installed between SM and MC is programmed using heuristic

algorithms. The designed SS not only generates the optimal energy usage pattern for all appliances but

also sends it to the MC for execution. MC is the core of proposed OHEMS and controls the operation

of appliances and ESS according to the generated schedule by SS.

6. Scheduling Algorithms

The appliances scheduling problem formulated in Section 4is solved by using GA, BPSO, WDO,

BFO, and HGPO algorithms. Although, appliances scheduling problem has been solved by different

classical optimization techniques: LP, ILP, MILP and DP. However, these techniques can not handle

large number of appliances and face a lot of difﬁculties in convergence. Moreover, most of the classical

algorithms do not have the global perspective and often converge at the local optimum solution.

Energies 2017,10, 549 13 of 35

In contrast, the evolutionary algorithms: GA, BPSO, WDO, BFO, and HGPO algorithms give alternate

methods to solve complex problems, and outperform the classical optimization techniques.

6.1. GA

GA is an iterative optimization algorithm inspired by the natural genetic process of the living

organisms. Rather than working on a single solution, GA deals with different possible solutions in

each iteration [

41

]. GA begins its search with randomly initialized binary coded chromosomes.

The chromosomes pattern of GA represent the

ON/OF F

state of appliances, and the length of

chromosomes shows the number of appliances.

Length of chromosomes = L (22)

where, Lis the number of household appliances.

Once the population (a set of solutions that shows the status of each appliance in a particular

time slot) is created, ﬁtness function of each possible solution is evaluated according to the objective

function of the optimization problem. Here, the ﬁtness of each population is evaluated using

Equation (18). Then, a new population is created by applying the natural genetic operators: crossover

and mutation. The GA parameters with their values on which it gives optimal results are listed

in Table 5.

Table 5. Genetic algorithm (GA) parameters.

Parameters Value

Number of iterations 500

Populationsize 200

Pm0.1

Pc0.9

n11

The working ﬂow of GA is shown in Figure 2. In each iteration, a new population is produced

through crossover and mutation. In the crossover step, two binary strings are crossover to create

a new

off spring. Crossover probability says how often crossover will be performed. If there is

no crossover, offspring is exact copy of parents. If there is a crossover, offspring is made from

parts of parents’ chromosome. If crossover probability is 100%, then all offspring is made by

crossover. If it is 0%, a whole new generation is made from exact copies of chromosomes from old

population [

42

]. Moreover,

a larger

crossover rate avoids premature convergence to the sub-optimal

solutions, so, the best crossover rate selected for optimization problems is 90% as given by,

Pc=0.9 (23)

To create randomness in the results so that the repetition of a population could be avoided we

use mutation process. It changes one or more principles gene in a chromosome from its initial state.

In natural genetic process, the probability of mutation is very low, so, an optimum mutation rate for

optimization problems is,

Pm=1−Pc(24)

Once crossover and mutation are done, again a population is generated and ﬁtness is evaluated

and compared with previous population.

Energies 2017,10, 549 14 of 35

Figure 2. Working ﬂow of GA.

6.2. BPSO

PSO is a nature-inspired optimization algorithm for ﬁnding an optimal solution within the search

space. The PSO algorithm inherently exists in the continuous domain. However, it can be modiﬁed

into the discrete domain, and its variants for the discrete domain is BPSO. The working ﬂow of BPSO

shown in Figure 3, mainly depends on four factors; initial position, initial velocity, particle own best

position and global best position among all the particles. In this algorithm, a population is randomly

initialized and scattered in the search space. The initial positions and velocities of the particles are

represented by

xi

=

x1+x2+· · · +xI

and

Vi

=

V1+V2+· · · +VI

respectively. The particles update

their velocities in each iteration by using Equation (25) [43],

24

∑

t=1

I

∑

i=1

Vt+1

i=

24

∑

t=1

I

∑

i=1

(wVt

j(j) + c1r1(Xlbest,i(j)) −xt

i(j)) + c2r2(Xgbest,i(j)−xt

i(j)) (25)

where

Vt+1

i

is the velocity of particle (appliance) in upcoming time slot,

w

is inertia factor,

Vt

j

is current

velocity,

r1

and

r2

are random numbers,

c1

and

c2

are local and global pulls respectively,

xt

i

is the

particle current position,

Xlbest

is local best position and

Xgbest

global best position. The velocities of

the particles are mapped between 0 and 1 by using sigmoid function given by Equation (26),

sig(Vt+1

i(j)) = 1

1+ex p(−Vt+1

i(j)) (26)

Energies 2017,10, 549 15 of 35

The random values assigned to each particle in the population are compared with the sigmoid

function to generate a binary coded population.

xt+1

i=(1sig(Vt+1

i(j)) <rij ,

0otherwise.(27)

In each iteration, particles record their positions with respect to neighbors. The local best positions

found by the particles are represented as Xlbest =Xlbest1+Xlbest2+· · · +Xlbest N . The local best values

are then compared with each other to ﬁnd the global best position. The

gth

particle among the particles

is said to be the global best position if it satisfy the objective function, and the global best positions

are represented as

Xgbest

=

Xgbest1+Xgbest2+· · · +XgbestN

. The main reason behind the use of global

best value rather than local best value is its faster convergence to the optimal solution. The global best

value is a binary coded string and represents the optimal

ON/OF F

status of the appliances. The ﬁtness

function of each particle is evaluated and compared with the corresponding cost of global best value,

and the string with minimum ﬁtness function is selected. The BPSO parameters and their values on

which BPSO gives optimal results are given in Table 6.

Figure 3. Steps involved in BPSO algorithm.

Energies 2017,10, 549 16 of 35

Table 6. Binary particle swarm optimization (BPSO) Parameters.

Parameters Values

Number of iterations 300

Swarmsize 200

vmax 4

vmin −4

wi2

wf0.4

c12

c22

n11

6.3. WDO

WDO is an another nature-inspired optimization algorithm based on the atmospheric motion

of air parcels. In this algorithm, inﬁnitely small air parcels move in N-dimensional search space.

The major difference between the WDO and other heuristic algorithms is the use of four different

forces to control the motion of air parcels in the atmosphere. These forces include pressure gradient

force, friction force, gravitational force, and Coriolis force. The pressure gradient force moves the air

parcels in the forward direction, while the frictional force resists their motion in the forward direction.

Furthermore, the gravitational force is a vertical force in three-dimensional search space that attracts

the air parcels towards the origin, and the deﬂection of air parcels in the atmosphere is due to the

Coriolis force. Mathematically, all four forces are represented as [44].

Fpg =−∆ρδν (28)

FC=−2Ω×µ(29)

FG=ρδν ×g(30)

FF=−ραµ (31)

where

Fpg

is pressure gradient force,

∆

is pressure gradient,

ρ

is air density,

δν

is ﬁnite volume of

the air,

FC

is Coriolis force,

Ω

represents the rotation of the earth,

µ

is velocity vector of the wind,

FG

vertical force directed toward the earth’s surface,

g

is acceleration of gravity,

FF

is friction force,

and αis friction coefﬁcient.

In each iteration, the position and velocity of the air parcels are updated by [44],

νnew = ((1−α)νold −gxold + [|Pmax

Po ld

|RT(xmax −xold )] −cνold

Po ld

(32)

and

xnew =xold + (νnew ×∆t)(33)

In Equation (32), the

((

1

−α)νold )

represents the opposite force continuously pushing the air

parcels towards their previous positions. Whereas the

gxold

shows the gravitational force which attracts

the air parcels towards the center of the earth. The motion of the air parcels towards the high-pressure

point/global best position is described by

[|Pmax

Pold |RT(xmax −xold )]

and

cνold

Pold

explains the Coriolis force

which controls the motion of the air parcels.

WDO creates random solutions from the

n

number of air parcels. After evaluating the ﬁtness

function, and updating velocities a new population is generated. The ﬁtness function of the new and

old air parcels are compared to obtain an optimal appliance scheduling pattern. The pressure term

used in WDO is just like ﬁtness term used in GA, BPSO and HGPO. The WDO parameters and their

Energies 2017,10, 549 17 of 35

values that give optimal results are given in Table 7and the main steps involved in WDO algorithm

are shown in Figure 4.

Table 7. Wind driven optimization (WDO) parameters.

Parameters Values

Number of iterations 500

Populationsize 200

dimMin −5

dimMax 5

vmin −0.3

vmax 0.3

RT 3

n 11

g 0.2

α0.4

Figure 4. Main steps of WDO algorithm.

Energies 2017,10, 549 18 of 35

6.4. BFO

BFO is a new addition to the family of nature-inspired optimization algorithms. BFO algorithm is

inspired by the social foraging behaviour of Escherichia coli. Since its inception, BFO has drawn the

attention of researchers from diverse ﬁelds of knowledge. In the BFO algorithm, the bacteria swim in

search of nutrients and select the best nutrients (solutions) to maximize its energy. The parameters

used in BFO and the values on which it gives optimal results are listed in Table 8.

Table 8. BFO parameters.

Parameter Value

Maximungeneration 500

Ne24

Nr5

Nc5

Np30

Ns2

Ci0.01

Ped 0.5

θ0.5

BFO algorithm consists of four steps; Chemotaxis, Swimming, Reproduction, and

Elimination-dispersal. BFO algorithm starts its search with parameters initialization, and after

initialization of parameters, under the chemotaxis step, initial states of appliances are evaluated

and then our system computes the new positions of bacteria (solution matrix). Mathematically the

chemotaxis steps of a bacterium can be represented by Equation (34) [45],

θi(j+1, k,l) = θi(j,k,l) + C(i)∆(i)

p∆T(i)∆(i)(34)

where

θi

(

j

,

k

,

l

) represents

i

-th bacterium at

j

-th chemotactic,

k

-th reproductive and

l

-th

elimination-dispersal step.

C(i)

is the size of step taken in the random direction speciﬁed by the

tumble, and ∆indicates a vector in the random direction whose elements lie in [−1, 1].

In next step, swimming loop is initialized to ﬁnd the current best state of appliances.

The swimming step is represented by Equation (35) [46],

Jcc(θ,P(j,k,l)) =

S

∑

i=1

Jcc(θi,θ(j,k,l)) =

S

∑

i=1

[−dattractant exp(−wattractant

n

∑

m=1

(θm−θi

m)2)] (35)

+

S

∑

i=1

[−hrepell ant exp(−wrepell ant

n

∑

m=1

(θm−θi

m)2)]

where

Jcc

(

θ

, P( j, k, l)) is the objective function,

S

is the total number of bacteria,

n

is the number of

appliances, and

θ

=[

θ1

,

θ1

,

· · ·

,

θn

]

n

is a n-dimensional search space. The

dattractant

,

wattractant

,

hrepell ant

and wrepell ant are different coefﬁcients that should be selected properly [47].

When swimming steps complete, iterations of reproduction loop start in which only feasible

solutions are recorded to produce next generation. Then, the elimination-dispersion steps discard

the least feasible solutions, and new random samples are inserted with a low probability. This is

a very important process because infeasible solutions are eliminated, and the chances of repetition are

avoided. The working ﬂow of BFO algorithm is shown in Figure 5.

Energies 2017,10, 549 19 of 35

Figure 5. Steps involved in BFO algorithm.

6.5. HGPO

Our proposed HGPO algorithm combines the features of BPSO and GA to effectively reduce the

electricity bill and PAR. The GA and BPSO are chosen for hybridization because the GA is good in PAR

reduction, and BPSO is effective in bill reduction. Working procedure of HGPO consists of two stages,

ﬁrst, all steps of BPSO are followed, then, the crossover and mutation operators of GA are applied

to the current global best position

Xt

gbest

and the previous global best position

Xt−1

gbest

found by BPSO

algorithm. The application of crossover and mutation operators to the best positions give better results

than their application to the random population. The proposed HGPO algorithm is shown in Figure 6.

Energies 2017,10, 549 20 of 35

Figure 6. Steps involved in HGPO algorithm.

7. Results and Discussions

In this section, simulation results of the proposed OHEMS are presented. In the proposed scheme,

the integration of RES and ESS, as well as the performance of various heuristic algorithms (GA, BPSO,

WDO, BFO, and HGPO), is evaluated via two stages simulations. In the ﬁrst case, the integration

of RES and ESS into the residential sector are evaluated in terms of energy consumption pattern,

and electricity bill as well as PAR reduction. While in the second case, the same performance metrics

(energy consumption pattern, and electricity bill as well as PAR minimization) are used to evaluate the

effectiveness of GA, BPSO, WDO, BFO and HGPO algorithms for HEMSs. For simulations, we have

used MATLAB installed on Intel(R) Core(TM) i3-2370M CPU @ 2.4GHz and 2GB RAM with Windows 7.

The computational time of all heuristic algorithms is given in Table 9.

Table 9. Computational time.

Algorithm Computational Time (Sec)

GA 2.21

BPSO 1.86

WDO 2.84

BFO 3.59

HGPO 1.71

Energies 2017,10, 549 21 of 35

To demonstrate the proposed OHEMS, an end user with 11 passive appliances and an ESS

which works as a source and sink is considered. The descriptions of the appliances are shown in

Table 10, the column III shows the power rating of the appliances and column IV shows the length

of operation time of the corresponding appliance. In this paper, the length of operation time is

arbitrarily taken and remains the same in both scenarios (i.e., unscheduled and scheduled) to have

a fair comparison. It is assumed that the utility power supply is available around the clock to

support the prosumer’s load. Moreover, the utility has AMI to get the forecasted data of weather

conditions, ambient temperature, and solar irradiance from the metrological department, and to

broadcast it to the prosumers. The exogenous grid signals (DAP signal, forecasted ambient temperature,

and solar irradiance) used in the proposed OHEMS are shown in Figures 7–9respectively. The DAP

shown in Figure 7is deﬁned by the utility operator and the normalized form of solar irradiance and

temperature data obtain from METEONORM 6.1 for Islamabad region of Pakistan is presented in

Figures 8and 9.

Table 10. Description of the appliances.

Load Type Appliances Power Rating (kW) Daily Usage (Hours)

Shiftable

ESS 3 –

Washing machine 0.8 5

Air conditioner 1.3 10

Clothes dryer 0.7 4

Water heater 1 8

Dish washer 0.2 3

Un-shiftable

Personal computers 0.2 18

Security cameras 0.1 24

Microwave oven 0.5 7

Refrigerator 0.9 20

Television 0.2 8

Lights 0.1 6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

5

10

15

20

25

30

Time (hours)

Cost (Cents/kWh)

DAP

Figure 7. DAP signal.

Energies 2017,10, 549 22 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

5

10

15

20

25

30

35

Time (hours)

Temprature (°C)

Forecasted ambient temprature

Figure 8. Forecasted outdoor temperature.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Solar irradiance (kW/m2)

Solar irradiance

Figure 9. Solar irradiance.

The electricity generation by PV system modeled in Equation (1) mainly depends on energy

conversion efﬁciency of the solar generator, the effective area of the generator, solar irradiance,

and ambient temperature. Ninety percent of the estimated RE in any time slot of the scheduling

horizon is considered for consumption. This uncertainty of 10% is included to cater for the disparity

between the estimated and the actual generation. Moreover, 30% of the 90% of estimated RE in

each time slot is used for the charging of ESS as long as the charging level of ESS is between 10–90%.

The estimated RE, RE after taking 10% uncertainty, remaining RE after the charging of ESS and charging

level of ESS system are presented in Figures 10 and 11 respectively. The ESS is charged only from the

PV system in the day time. After the ESS charged fully, it is utilized in a high price time slot i.e.,

τ

as

given by Equation (17).

Energies 2017,10, 549 23 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

RES Generation (kW/h)

Estimated RE

90% of Estimated RE

Remaing RE after ESS charging

Figure 10. Estimated renewable energy.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.5

1

1.5

2

2.5

3

Time (hours)

Battery Storage Level (kW)

Battery charging level

Figure 11. Charging level of energy storage system (ESS).

7.1. Case 1: Integration of RES and ESS

In this case, the beneﬁts of RES and ESS integration into the residential sector are highlighted

in terms of energy consumption pattern, and electricity bill as well as PAR reduction. The load

distribution among the utility, RES and ESS is shown in Figure 12. It illustrates that in second scenario

a portion of the prosumer load shifted from utility to RES, and in the third scenario an ESS is also

integrated to support the prosumer’s load in peak hours.

Energies 2017,10, 549 24 of 35

Unscheduled Unscheduled+RES Unscheduled+RES+ESS

0

10

20

30

40

50

60

Load distribution(kW)

Load on ESS

Load on RES

Load on utility

Figure 12. Load distribution.

7.1.1. Energy Consumption

To explain the behavior of the energy consumption pattern, we deﬁne some arbitrary thresholds.

The descriptions of load and thresholds are given in Table 11.

Table 11. Thresholds of energy consumption.

Load Range (kW)

High peak 5 and above

Peak 4–4.9

Moderate 2–3.9

Minimum 1–1.9

Negligible 0.1–0.9

The energy consumption pattern of the prosumer load is shown in Figure 13. Results illustrate

that in the ﬁrst scenario (unscheduled load without RES and ESS), the energy consumption pattern has

a high peak load of 5.95 kW in time slot 1, and peak loads of 4.7 kW and 4.2 kW in time slots 21 and 9

respectively. Moreover, the energy consumption pattern shows a moderate behaviour in time slots 2–8,

10, 13–14, 17–20 and 22–24, and minimum energy is consumed in time slots 11–12 and 15–16.

The energy consumption pattern of the second scenario (unscheduled load with RES), shows

that the energy consumption remains the same in time slots 1–6 and 20–24. This is because the RE is

available from the PV system as shown in Figure 10. This illustrates that the in-house RES generates

electricity only in the time slots 7–19. Therefore, in time slots 7–19, the energy consumption is reduced

by the corresponding amount of RE available in that time slot.

In the third scenario (unscheduled load with RES and ESS), we also integrate an ESS to the

proposed SH. After the integration of ESS, energy consumption remains same in time slots 1–6 and

20–24; however, in time slots 7–8 and 10–19 the energy consumption becomes higher than the second

scenario. This increase occurs because 30% of the RE available in time slots 7–19 is used for the charging

of ESS. While, in time slot 9, the ESS is discharged and load on the utility is reduced by 73.60% and

67.16% as compared to ﬁrst and second scenarios respectively.

Energies 2017,10, 549 25 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

1

2

3

4

5

6

7

Time (hours)

Energy consumption (kWh)

Unsch hourly load

Unsch+RES hourly load

Unsch+RES+ESS hourly load

Figure 13. Energy consumption in case 1.

7.1.2. Electricity Cost

The corresponding electricity bill of the energy consumption is shown in Figure 14.

Results illustrate that in time slots 1–6 and 20–24 the electricity bill of all the three scenarios remains

the same. While, in time slots 7–19, the RE is available and signiﬁcantly reduces the electricity bill.

Particularly, in time slot 9, the RE reduces the electricity bill by 19.55%, and after the discharge of ESS,

the total reduction of 71.41% is achieved in the electricity bill.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

20

40

60

80

100

120

Time(hours)

Electricity Cost (Cents)

Unsch hourly cost

Unsch+RES hourly cost

Unsch+RES+ESS hourly cost

Figure 14. Electricity cost in case 1.

Energies 2017,10, 549 26 of 35

7.1.3. Total Cost

The total electricity bill of case 1 is shown in Figure 15. Results illustrate that the investment of a

small capital cost on the integration of RES reduces the electricity bill by 17.25%, and after adding EES

too, the bill is reduced by 19.94%. The reduction in electricity bill due to RES and ESS is summarized

in Table 12.

Unscheduled Unscheduled+RES Unscheduled+RES+ESS

0

100

200

300

400

500

600

700

800

900

Total Cost (Cents)

Figure 15. Total electricity cost of the prosumer in case 1.

Table 12. Comparison of case 1 cost.

Scenario Cost (Cents) Difference (Cents) Reduction (%)

Unscheduled 862.66 - -

Unscheduled + RES 713.78 148.88 17.25%

Unscheduled + RES + ESS 690.63 172.03 19.94%

7.1.4. PAR

The PAR of the prosumer load is shown in Figure 16. Results illustrate that the integration

of RES reduces the PAR by 15.68% and after incorporating the ESS as well, the PAR is reduced by

21.55%. The reduction in PAR in all three scenarios is shown in Table 13. This reduction in PAR not

only enhances the stability and reliability of the power system but also reduces the electricity bill of

the prosumer.

Table 13. Comparison of case 1 PAR.

Scenario PAR Difference Reduction (%)

Unscheduled 3.0482 - -

Unscheduled + RES 2.5702 0.478 15.68%

Unscheduled + RES + ESS 2.3912 0.657 21.55%

Energies 2017,10, 549 27 of 35

Unscheduled Unscheduled+RES Unscheduled+RES+ESS

0

0.5

1

1.5

2

2.5

3

3.5

Case 1 PAR

Figure 16. Case 1 PAR.

7.2. Feasible Region of the Objective Function

The feasible region of the objective function (electricity bill) formulated in Equation (18) is shown

in Figure 17. In Figure 17, the trapezoid of

P1(

0.1, 0.81

)

,

P2(

0.1, 2.735

)

,

P4(

6, 48.6

)

and

P6(

6, 164.2

)

represents the overall region of electricity bill. Where, the point P1(0.1, 0.81)represents the electricity

bill when minimum possible load (0.1 kW) is scheduled in minimum price slot (21) of RTP signal,

and if the energy consumption is 0.1 kW in the maximum price slot (9) of RTP signal, the electricity

bill will be represented by the point

P2(

0.1, 2.735

)

. Moreover, the points

P4(

6, 164.2

)

and

P6(

6, 48.6

)

represent the bill when all appliances are in ON status in minimum and maximum price slots of RTP

signal respectively. Equation (18) shows that the objective function depends on two factors: amount of

load and electricity price in that time slot. The price signal is deﬁned by the utility and we have no

control over it; therefore, to minimize the electricity bill we can only modify the shape of the energy

consumption pattern. Furthermore, in scheduled load scenarios, the electricity cost in any time slot

must not exceed the maximum cost of the unscheduled scenario, i.e., 59.5 cents. After applying the

limit of 59.5 cents the feasible region of scheduled load electricity bill is shown by the pentagon of

P1(

0.1, 0.81

)

,

P2(

0.1, 2.735

)

,

P3(

2.13, 59.5

)

,

P5(

6, 59.5

)

and

P6(

6, 48.6

)

. Where, the point

P3(

2.13, 59.5

)

shows that in maximum price time slot of RTP signal the scheduled load must not exceed than 2.13 kW,

and point

P5(

6, 59.5

)

shows that the maximum possible load (6 kW) must not be scheduled in time

slots having price greater than 9.92 cents.

0123456

0

20

40

60

80

100

120

140

160

Energy consumption (kW)

Electricity Cost (Cents)

Feasible Region

P2 (0.1, 2.735)

P1 (0.1, 0.81)

P5 (6, 59.5)

P4 (6, 164)

P6 (6, 48.6)

P3 (2.13, 59.5)

Figure 17. Feasible region of objective function.

Energies 2017,10, 549 28 of 35

7.3. Case 2: OHEMS with Integrated RES and ESS

The distribution of the prosumer load among the utility, RES and ESS is illustrated in Figure 12.

It shows that in the second scenario of case 1, an RES is integrated to the SH while, in the third scenario,

an ESS is also utilized to support the prosumer’s load.

Here, in this section, the simulation results of the heuristic algorithms (GA, BPSO, WDO, BFO and

HGPO) based OHEMS are presented. In this case, the cost of unscheduled load with RES and

ESS is taken as a reference to validate the effectiveness of the heuristic algorithms for HEMSs.

The performance of heuristic algorithms (GA, BPSO, WDO, BFO and HGPO) is evaluated by using

the same performance metrics (energy usage pattern, and electricity bill as well as PAR reduction) of

case 1. The designed SS generates an optimal energy usage pattern for shiftable appliances in response

to grid signals and a large portion of load is shifted toward off-peak hours. This relocating of load

signiﬁcantly reduces the electricity bill as well as the PAR. Simulation results of the proposed scheme

are as follows:

7.3.1. Energy Consumption

The energy consumption patterns of scheduled and unscheduled loads with RES and ESS

are shown in Figure 18. In unscheduled load with RES and ESS scenario, the prosumer’s energy

consumption pattern has a high peak of 5.95 kW in time slot 1 and followed by a peak load of 4.8 kW

in time slot 21. A moderate behaviour of load is recorded in time slots 2–8, 13–14, 17–20, and 22–24.

Moreover, the energy consumption is found to be minimum in time slots 9–10, and negligible in time

slots 11–12 and 15–16.

GA-based scheduling creates no high peaks, and a peak load of 4.75 kW is consumed in time

slot 21; 4.12 kW in time slot 20; 4.07 kW in time slot 22; and 4.01 kW in time slot 24. The energy

consumption pattern is found to be moderate in time slots 1–6, 16–19 and 23. Furthermore, the energy

consumption is minimum in time slots 7–8, 10 and 13–15, and negligible in time slots 9 and 11–12.

The comparison of GA-based scheduled and unscheduled load shows that the maximum load in

GA-based scheduling is 20.16% less than the maximum load in the unscheduled scenario.

BPSO-based scheduling also does not create high peaks, and the energy consumption pattern

has peak load of 4.96 kW in time slots 2 and 22; 4.9 kW in time slot 21; 4.81 kW in time slot 20;

4.76 kW in time slot 19; 4.67 kW in time slots 17 and 19; 4.2 kW in time slot 3; and 4.12 kW in time

slot 16. The energy consumption is found to be moderate in time slots 5, 15 and 23. Additionally, a

minimum amount of energy is consumed in time slots 1, 4, 6 and 18, and negligible in time slots 7–14.

The comparison of BPSO-based scheduled and unscheduled load shows that the maximum load in

BPSO is 16.63% less than the maximum load in the unscheduled scenario.

In WDO-based scheduling, a high peak load of 5.75 kW is consumed in time slot 19 and 5.12 kW

in time slots 22–23. The energy consumption pattern has peak load of 4.95 kW in time slot 21; 4.08 kW

in time 20; and 4 kW in time slots 2 and 24. In addition, a moderate amount of energy is consumed

in time slots 3, 5, 16 and 17. The energy consumption is minimum in time slots 1, 4, 6, 15 and 18,

and negligible in time slots 7–14. The comparison of WDO-based scheduled and unscheduled load

shows that the maximum load in WDO is 3.36% less than the maximum in the unscheduled scenario.

The BFO algorithm neither creates high peaks nor peaks in energy consumption pattern.

The energy consumption is moderate in time slots 1–5, 17 and 19–24, and minimum in time slots

6–8, 11, 13–16 and 18. Moreover, the energy consumption is negligible in time slots 9–10 and 12.

The comparison of BFO-based scheduled and unscheduled load shows that the maximum load in BFO

is 36.13% less than the maximum load in the unscheduled scenario.

The proposed HGPO-based scheduling creates no high peaks, and a peak load of 4.7 kW is

consumed in time slot 19 and 4.61 kW in time slots 21–23. The energy consumption is moderate in time

slots 1–2, 4–6, 17 and 18. Moreover, minimum load is scheduled in time slots 3, 10, 13, 16, 20 and 24,

and negligible in time slots 7–9, 11–12 and 14–15. The comparison of HGPO-based scheduled and

Energies 2017,10, 549 29 of 35

unscheduled load shows that the maximum load in case of HGPO is 22.52% less than the maximum

load in the unscheduled scenario.

Figure 18 illustrates that the load shifting pattern of HGPO is balanced, and uniformly distribute

the load over the scheduling horizon. On the other hand, the other heuristic algorithms either

completely shift the load towards off-peak hours or consume higher energy in peak hours. The HGPO

outperforms the GA, BPSO, WDO and BFO algorithms by allowing the appliances to complete their

tasks with minimum delay and avoids peak formation as well.

Figure 18. Hourly energy consumption in case 2.

7.3.2. Electricity Cost

The corresponding electricity bill of unscheduled and scheduled load with RES and ESS is shown

in Figure 19. Results illustrate that the electricity bill of heuristic algorithms (GA, BPSO, WDO,

BFO and HGPO) based scheduled load remains within the feasible region. In GA-based scheduling,

the maximum electricity bill is 39.7 cents in time slot 7. The electricity bill in case of BPSO has

a maximum value of 43.6 cents in time slot 2 which is also less than the maximum bill (59.5) of

unscheduled scenario. In the case of WDO, the maximum electricity bill is 47.3 cents in time slot 21,

and the bill in all other slots also remains within the feasible region. The BFO-based scheduled load

has maximum bill of 49.8 cents in a time slot 8. In the proposed HGPO algorithm, the electricity

bill satisﬁes all the limits of the feasible region, and a maximum bill of 43.6 cents is recorded in time

slot 6. Moreover, in peak hours the electricity bill of scheduled load is comparatively less than the

unscheduled load, and the comparison of total electricity bills of the heuristic algorithms shows that

the performance of HGPO-based HEMS in term of bill reduction is better than GA, BPSO, WDO and

BFO based HEMS.

Figure 19. Hourly electricity cost in case 2.

Energies 2017,10, 549 30 of 35

7.3.3. Total Cost

The comparison of overall electricity bill of the unscheduled and scheduled load with RES and

ESS is shown in Figure 20. The total electricity bill in unscheduled, and in scheduled load scenarios

using GA, BPSO, WDO, BFO and HGPO algorithms are 690.63, 622.97, 555.32, 584.24, 581.56 and

517.15 cents respectively. The comparison of total electricity bill shows that GA, BPSO, WDO, BFO

and HGPO algorithms based HEMS reduces the electricity bill by 9.80%, 19.60%, 15.40%, 15.80%,

and 25.12% respectively. The changes in electricity bill due to scheduling are listed in Table 14. In peak

hours the electricity charges of heuristic algorithms based scheduled load scenarios are signiﬁcantly

less than the unscheduled scenario. However, when the overall cost reduction is concerned the HGPO

gives better results than the other optimization algorithms. Moreover, the electricity consumption

pattern of HGPO is uniform, while the GA, BPSO, WDO and BFO algorithms create new peaks in

off-peak hours that degrade their overall performance.

Unsch+RES+ESS GA BPSO WDO BFO HGPO

0

100

200

300

400

500

600

700

Total Cost (Cents)

Figure 20. Case 2 total cost.

Table 14. Comparison of case 2 cost.

Scheduling Technique Cost (Cents) Difference (Cents) Reduction (%)

Unscheduled + RES + ESS 690.63 - -

GA scheduled + RES + ESS 622.97 67.66 9.80%

BPSO scheduled + RES + ESS 555.32 135.31 19.60%

WDO scheduled + RES + ESS 584.24 106.39 15.40%

BFO scheduled + RES + ESS 581.56 109.07 15.80%

HGPO scheduled + RES + ESS 517.15 173.48 25.12%

7.3.4. PAR

The PAR of unscheduled and scheduled load is shown in Figure 21. It illustrates that the proposed

HGPO algorithm minimizes the PAR by 24.88%. The GA, BPSO, WDO and BFO algorithms also

reduce the PAR by 14.09%, 3.30%, 22.10% and 33.54% respectively. The changes in PAR of the heuristic

algorithms based scheduled load are shown in Table 15. Although these algorithms are designed to

reduce the PAR and avoid peaks formation, the BPSO and WDO algorithms shift most of the load

to off-peak hours that creates new peaks. This new peaks formation disturbs the entire operation

schedule of the utility peak plants and utility impose a penalty on the consumer. The HGPO and BFO

based HEMS uniformly distributes the loads over the scheduling horizon and achieves the desired

Energies 2017,10, 549 31 of 35

objective. The performance of the GA, BPSO and WDO algorithms may be enhanced by setting the

proper thresholds of energy consumption in each slot; however, it will affect the UC and will increase

the waiting time of the appliances.

Unsch+RES+ESS GA BPSO WDO BFO HGPO

0

0.5

1

1.5

2

2.5

Case 2 PAR

Figure 21. Case 2 PAR.

Table 15. Comparison of case 2 PAR.

Scheduling Technique PAR Difference Reduction (%)

Unscheduled + RES + ESS 2.391 - -

GA scheduled + RES + ESS 2.054 0.337 14.09%

BPSO scheduled + RES + ESS 2.312 0.079 3.30%

WDO scheduled + RES + ESS 2.350 0.041 22.10%

BFO scheduled + RES + ESS 1.589 0.802 33.54%

HGPO scheduled + RES + ESS 1.796 0.595 24.88 %

8. Conclusions and Future Work

In this paper, an OHEMS is proposed, which incorporates the RES and ESS into the residential

sector. Results illustrate that the integration of RES and EES minimizes the electricity bill by 19.94%

and PAR by 21.55%. After the integration of RES and ESS, the constrained optimization problem is

mathematically formulated by using MKP, and solved by heuristic algorithms: GA, BPSO, WDO,

BFO and HGPO algorithms. The performance of the scheduling algorithms is evaluated in terms of

uniform distribution of load over the scheduling horizon, and reduction of electricity bill as well as

PAR. Simulation results show that, unlike GA, BPSO, WDO, and BFO, the HGPO algorithm uniformly

distributes the load over the scheduling horizon, and further reduces the electricity bill by 25.12% and

PAR by 24.88%. This reduction of PAR enhances the power system stability and ensures the stable

and reliable grid operation. From the above discussion, it is concluded that the HGPO-based HEMS

performs better than other heuristic algorithms and an overall reduction of 40.05% is achieved in cost

and 41.07% in PAR as compared to the unscheduled load without RES and ESS.

In future, we are interested in the coordination among the distributed micro sources, i.e., RES and

ESS to exploit the RSER utilization. This coordination of micro sources in a residential area will not

only enable the exchange of surplus RE among the prosumers but will also reduce the load on utility.

Acknowledgments:

This project was full ﬁnancially supported by the King Saud University, through Vice

Deanship of Research Chairs.

Energies 2017,10, 549 32 of 35

Author Contributions:

Adnan Ahmad, Asif Khan, and Nadeem Javaid proposed the system model and

classiﬁed the appliances. All authors performed extensive simulations for the manuscript. Haﬁz Majid Hussain,

and Iftikhar Azim Niaz wrote related work and simulation sections. Wadood Abdul, Ahmad Almogren,

and Atif Alamri wrote rest of the manuscript. Nadeem Javaid and Adnan Ahmad organized and reﬁned

the manuscript.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

Symbol Description Symbol Description

EPV Available energy from PV system Xgbest Global best value

ηPV Efﬁciency of PV system Xlbest Local best value

APV Area of PV generator Fpg Pressure gradient force

IrSolar radiation FCCoriolis force

TaAmbient temperature FGGravitational force

α1,α2Shape factors β1,β2Scale factors

ES Stored energy FFFriction force

κDuration of one time slot FGGravitational force

ηESS Efﬁciency of ESS Pold Pressure at current location

EPCh Charge rate of ESS at time Pmax High pressure point

EPD ch Discharge rate of ESS at time αConstant for update position

EPCh

UB Upper charge limit of ESS ωEarth rotation

EPD ch

LB Lower discharge limit of ESS ∆t Unit step time

ESUB Upper limit of energy storage νnew Updated velocity

MNumber of controllable appliances νold Current velocity

NNumber of un-shiftable appliances RUniversal gas constant

EaEnergy consumption of shiftable appliances Sig Sigmoid function

EbEnergy consumption of non-shiftable appliances nTotal number of appliances

Etotal Total energy consumption PDAP DAP signal

Ea

PElectricity cost of shiftable appliances energyconsumption CiStep size

Eb

PElectricity cost price of non-shiftable appliances energy consumption θPosition of bacteria

EPTotal bill of energy consumption ΩRotation of earth

Xa

meMON/OFF status of shiftable appliances δν Finite volume of air

Xb

nβeNON/OFF status of non-shiftable appliances wInertia factor

Egrid Available grid energy r1,r2Random numbers

Emin

unsch Minimum energy consumed in unscheduled senario c1Local pull

τ0Lower limit of scheduling horizon c2Global pull

τsch Scheduling time dimMax Upper limit of WDO dimensions

τmax Upper limit of scheduling horizon dimMin Lower limit of WDO dimensions

tTime slots ∆Pressure gradient

LLength of chromosomes µVelocity vector of wind

Vt+1

iParticle upcoming velocity Vt

jParticle current velocity

xt+1

iParticle upcoming position xt

iParticle current position

NeNumber of elimination steps PcProbability of crossover

NcNumber of chemotaxis steps PmProbability of mutation

NpNumber of population steps wiInitial weight constant

NsNumber of swimming steps wfFinal weight constant

NrNumber of reproduction steps vmax Upper limit of velocity

Ped Probability of elimination-dispersal vmin Lower limit of velocity

Energies 2017,10, 549 33 of 35

List of Acronyms

Acronym Description Acronym Description

SG Smart grid SHs Smart homes

SCs Smart cities REMS Residential energy management system

EMS Energy management system HE MS Home energy management system

DR Demand response DSM Demand side management

OH EMS Optimized home energy management system UC User comfort

RESs Renewable energy sources RSERs Renewable and sustainable energy resources

DGs Distributed generation ICTs Information and communication tecnologies

PV Photovoltaic ESS Energy storage system

SM Smart meter AM I Advance metering infrastructure

MC Master controller DC Direct current

AC Alternating current SS Smart scheduler

GA Genetic algorithm PSO Particle swarm optimization

BPSO Binary particle swarm optimization W DO Wind driven optimization

BFO Bacterial foraging optimization HGPO Hybrid GA-PSO

LSA Lighting search algorithm AN N Artiﬁcial neural networks

OL A Observe, learn and adopt LF Load forecasting

DP Dynamic programming IPSO Improved particle swarm optimization

CP Convex programming D AP Day ahead pricing

DA P Real time pricing ToUP Time of use pricing

PP Peak pricing CPP Critical peak pricing

LP Linear programming I LP Integer linear programming

MI LP Mixed integer linear programming RES Multiple knapsack problem

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