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electronics

Article

Game Theoretical Demand Response Management

and Short-Term Load Forecasting by Knowledge

Based Systems on the basis of Priority Index

Mahnoor Khan 1, Nadeem Javaid 1,*, Sajjad 1, Abdullah 2, Adnan Naseem 3, Salman Ahmed 4,

Muhammad Sajid Riaz 5, Mariam Akbar 1and Manzoor Ilahi 1

1Department of Computer Science, COMSATS University Islamabad, Islamabad 44000, Pakistan;

mahnoor.khan2794@gmail.com (M.K.); ciit.sajjad@gmail.com (S.); mariam.akbar@gmail.com (M.A.);

tamimy@comsats.edu.pk (M.I.)

2Department of Electrical Engineering, COMSATS University Islamabad, Islamabad 44000, Pakistan;

abdullahbjr@gmail.com

3Department of Computer Science and Information Technology, Alhamd Islamic University,

Islamabad 44000, Pakistan; adnan.naseem@alhamd.pk

4Department of Computer Science, Islamic International University, Islamabad 44000, Pakistan;

salmanresearchlab@gmail.com

5Department of Computer Science, Air University, Islamabad 44000, Pakistan; riaz.sajid@gmail.com

*Correspondence: nadeemjavaidqau@gmail.com

Received: 15 November 2018; Accepted: 7 December 2018; Published: 12 December 2018

Abstract:

Demand Response Management (DRM) is considered one of the crucial aspects of the

smart grid as it helps to lessen the production cost of electricity and utility bills. DRM becomes a

fascinating research area when numerous utility companies are involved and their announced prices

reﬂect consumer’s behavior. This paper discusses a Stackelberg game plan between consumers and

utility companies for efﬁcient energy management. For this purpose, analytical consequences (unique

solution) for the Stackelberg equilibrium are derived. Besides this, this paper presents a distributed

algorithm which converges for consumers and utilities. Moreover, different power consumption

activities on the basis of time series are becoming a basic need for load prediction in smart grid. Load

forecasting is taken as the signiﬁcant concerns in the power systems and energy management with

growing technology. The better precision of load forecasting minimizes the operational costs and

enhances the scheduling of the power system. The literature has discussed different techniques for

demand load forecasting like neural networks, fuzzy methods, Naïve Bayes, and regression based

techniques. This paper presents a novel knowledge based system for short-term load forecasting.

The algorithms of Afﬁnity Propagation and Binary Fireﬂy Algorithm are integrated in knowledge

based system. Besides, the proposed system has minimum operational time as compared to other

techniques used in the paper. Moreover, the precision of the proposed model is improved by a

different priority index to select similar days. The similarity in climate and date proximity are

considered all together in this index. Furthermore, the whole system is distributed in sub-systems

(regions) to measure the consequences of temperature. Additionally, the predicted load of the entire

system is evaluated by the combination of all predicted outcomes from all regions. The paper employs

the proposed knowledge based system on real time data. The proposed scheme is compared with

Deep Belief Network and Fuzzy Local Linear Model Tree in terms of accuracy and operational cost.

In addition, the presented system outperforms other techniques used in the paper and also decreases

the Mean Absolute Percentage Error (MAPE) on a yearly basis. Furthermore, the novel knowledge

based system gives more efﬁcient outcomes for demand load forecasting.

Electronics 2018,7, 431; doi:10.3390/electronics7120431 www.mdpi.com/journal/electronics

Electronics 2018,7, 431 2 of 34

Keywords:

behavioral analytics; Stackelberg game; demand response; knowledge based systems;

priority index; similar day; date proximity.

1. Introduction

In the modern day world, smart meters offer two way communication between the users and

the utilities. This communication leads towards a prevalent computing environment, which develops

large-scale data with high velocity and veracity [

1

]. The resultant data also give rise to a time series

concept. This phenomenon generally includes power consumption measurements of appliances over a

speciﬁc time interval [

2

]. The techniques of big data are proﬁcient enough to utilize resultant huge

volumes data of sequential time series. Moreover, these techniques also assist in data-driven decision

making. Besides, this big data can update utilities to learn power consumption patterns of consumers,

predicting demand and averting blackouts.

The utilities are keen on ﬁnding the optimal ways for cost reduction. Moreover, electricity

companies desire to increase their yields by acquainting their consumers with effective programs

like Demand Side Management (DSM) and demand response. Currently, marginal success has

been observed in achievement of goals for these programs. However, viable results still need to

be achieved [

3

]. Furthermore, implementation of DSM and demand response is a challenging task

for utilities. It is difﬁcult to comprehend and conclude the behavior of every individual consumer.

Moreover, it is also challenging to customize strategies that include proﬁt contrary to distress from

varying behavior of consumers on the basis of energy-saving policies introduced by utilities. Besides,

the association between consumer behavior and the constraints that affect power utilization patterns

are non-static, i.e., the activities of consumers keep on changing from time to time [4].

Usually, the behavior of consumers is reliant on weather and seasons, which has a capricious

effect on power utilization decisions. Thus, active participation of consumers in customized power

management is crucial for energy saving schemes. The companies should give timely response on

power consumption and associated costs [

5

]. Consequently, it is challenging to design such models

that are proﬁcient enough to evaluate energy time series from smart meters. Also, it is stimulating to

train the model that predicts power consumption.

The aforementioned discussion helps to study the inﬂuence of consumers’ behavior on power

consumption and to forecast the energy utilization patterns. This analysis can assist the utilities to

develop power saving strategies. Moreover, the utilities can design programs to stabilize the demand

and supply of energy ahead of time. For instance, short term forecast is related to daily and weekly

power usage. This type of prediction is best suitable when there is a need to enhance scheduling and

distribution. Alternatively, medium term forecasting is related to weekly and monthly forecasting.

Besides, long term forecasting is about yearly predictions of energy consumption. Medium and long

term predictions are capable of maintaining the equilibrium between the production of smart grid and

strategic scheduling [

6

]. However, such a task is very challenging as it is signiﬁcant to mine complex

interdependencies between appliance usages where numerous data streams are taking place.

Generally, DRM can be characterized in two extents, which are the utilities and consumers. There

has been substantial quantity of work done in power systems to maintain the balance between supply

and demand [

7

]. However, these studies have laid emphasis on the ﬁnancial aspects on the planning

and production levels. Moreover, these studies are unable to take both consumer and utility as a

substantial constituent. Contrariwise, the literature on consumer and utility has presented models to

increase user comfort, devoid of taking the cost of power or the proﬁts of the utilities [

8

]. This paper

takes motivation from this phenomenon. Moreover, this paper observes the increased proﬁts for

consumers and utilities.

This paper analyzes the collaborations between several utilities and consumers. Both entities

share mutual objectives, i.e., maximization of their payoffs. The utilities can increase their proﬁts by

Electronics 2018,7, 431 3 of 34

setting a suitable price per unit. Nonetheless, the users select a speciﬁed amount of power to purchase

from any utility on the basis of announced prices. Furthermore, the purchasing behavior of consumer

is dependent on the prices settled by the company. Likewise, the behavior of utilities is reliant for the

prices settled by other utilities. Thus, for solving these challenging collaborations between consumers

and utilities, this paper employs a game theoretical framework. This paper presents a Stackelberg

game plan between consumers and utilities. In this game, the utilities play a non-cooperative game

and the users look for their best optimum response.

The systematic and proﬁcient utilization of electrical power is a hot debate topic in today’s

world [

9

]. The optimal power management and maintaining balance between demand and supply are

considered as challenging tasks for modern power systems [

10

]. Moreover, the prediction of uncertain

production of renewable energy resources [

11

] and short-term load forecasting [

12

] are measured as

signiﬁcant components of the power grid for optimal power scheduling. Besides, short-term load

forecasting has wide applications in the energy market like load scheduling, unit commitment and

power production [

13

]. It has been observed in the literature that error maximization in short-term

load forecasting can result in substantial growth in the utility operating expenses. Thus, enhancing the

accuracy of predicted results is a challenging task and vital issue in power management.

The proximity of choosing a similar day to the target day is very crucial for selecting the similar

day along with temperature, according to previous studies. In this regard, this paper proposes a

different priority indexing technique for selection of similar days by analyzing the date proximity and

temperature similarity. Moreover, the date proximity used in this paper is the total number and nature

of days between selected and similar days. In contrast, the historic power load data is categorized

according to nature of days in demand prediction. Furthermore, this paper also presents four different

day types and two data-sets are presented for utilization of historical power load data. In addition,

the proposed knowledge based short-term load forecasting method employs monthly and weekly data

for two different data-sets. The best optimum results for short-term load forecasting will be achieved

by grouping of prediction results obtained from these two data-sets.

The consideration of exceptional temperature for any region is ineffectual because of variations

in temperatures in a vast topographical zone. A vast topographical zone is separated into three

climate types in [

14

]. Moreover, the temperature of three cities is labeled as cold, moderate, and

warm. The biased integration of these temperatures is presented as the temperature of the huge region.

The temperature is taken in [

15

] and the whole system is distributed in different regions. Besides,

the short-term load has been forecasted by some regression techniques. However, the precedence of

choosing similar days is also unnoticed in previous studies.

This paper divides the entire system in nine regions. Moreover, the climatic conditions of only

one city is chosen from every region. The knowledge based short-term load forecasting is employed to

every region after the consideration of temperature. In addition, the predicted power load of the entire

system is the aggregate of predicted load of particular regions. The impact of temperature is believed

to be much more efﬁcient and result improving when the system is divided.

The proposed system model is employed in Pakistan’s National Power Network (PNPN), which

is taken as a sample system in this paper. In the proposed system model, Afﬁnity Propagation

(AP) [

16

], and Binary Fireﬂy Algorithm (BFFA) are used as hybrid model. The proposed system

model shows a signiﬁcant decrease in MAPE in comparison with other traditional knowledge based

methods. This paper uses algorithms of Deep Belief Network (DBN) and Fuzzy Local Linear Model

Tree (F-LOLIMOT) for comparison purposes. The experimental results speciﬁes that the proposed

model requires minimum time for computation when associated with DBN and F-LOLIMOT.

The major research contributions of this paper include the proposition of the priority index for

selection of similar days by means of temperature of speciﬁed regions and date proximity. Moreover,

the historic power load is separated in two different data-sets in the paper. Subsequently, the data-sets

predict the short-term load and then the ﬁnal outcome is supposed to be more precise. The ﬁnal

Electronics 2018,7, 431 4 of 34

outcomes are achieved by the summation of predicted results from two data-sets. Furthermore, the

paper makes the impact of temperature effective by dividing the system in different regions.

The remaining paper is organized in following manner: Section II presents the previous work

done, Section III provides a brief discussion of a Stackelberg game and demonstrates the distinctiveness

and existence of the Stackelberg Equilibrium. Moreover, Section IV discusses the categorization of

knowledge based short-term load forecasting and Section V employs the proposed method on different

topographical regions. Moreover, results and their discussion are presented in Section VI and Section

VII concludes the paper.

2. Related Work

The challenges addressed in Section I are also discussed in the literature through methodologies of

big-data. A brief discussion of behavioral power consumption data to acquire better energy competence

are presented in [

6

]. Likewise, the inﬂuence of developmental ﬂuctuations for energy savings was

observed by [

17

]. The study also discussed the contribution of consumers to collaborate with the

utilities and better energy savings were highlighted.

The literature has proposed many novel methods for short-term load forecasting like fuzzy [

18

],

exponential smoothing [

19

], regression based [

20

], neural networks [

21

], and others. Moreover, every

proposed model has incorporated some techniques. For example, regression based processes are

usually comprised of Autoregressive Integrated Moving Average (ARIMA) [

22

], Auto-Regressive

Moving Average (ARMA) [

23

], Support Vector Regression (SVR) [

24

], and Auto-Regressive Moving

Average with Exogenous variable (ARMAX) [

25

]. Nevertheless, it is essential for aforementioned

techniques to learn the process by bulks of preceding data for tuning of various parameters.

Furthermore, the complexities of these techniques, minimum time of computation and memory

essentials of knowledge based model, can initiate a different perspective to knowledge based short-term

load forecasting.

In literature, there are some works cited in knowledge based systems that employ a similar day

method [

26

–

28

]. Although, there is a lot of room for enhancement in this scenario which can be studied.

The authors in [

29

] proposed a knowledge based system for short-term load demand forecasting.

However, the paper overlooked the consequences of temperature. The change in temperature can

cause ﬂuctuations in the load demand. Consequently, the effect of temperature must be included in

the short-term load forecasting. The different eight day categories are enumerated in [30].

Moreover, average stabilized loads of historic data for every day has been evaluated by means

of least and maximum load per hour. Furthermore, the least and maximum load for 11 days was

forecasted by means of regression techniques. The Mean Absolute Percentage Error (MAPE) of Irish

electrical power system attained was 2.52%. Moreover, the temperature was also incorporated in this

study and was associated with 3.86% by the statistical technique in [31].

The authors in [

32

] calculated the weighted mean load of every hour for three preceding and

similar days for short-term load forecasting. Moreover, the impact of temperature on prediction of

short-term load is also considered by means of exponential association between power demand and

temperature. Likewise, the mean prediction error for a daily peak load of France was attained 2.74%

in [

32

]. Besides, the consequences of temperature, wind pressure and humidity, was scrutinized in [

33

].

The MAPE calculated in this study was 1.43%. The study in [

23

] was almost equivalent to the proposed

model presented in [

22

]. Moreover, the MAPE achieved in this study was between 1.23% to 3.35% in

seven different states of America [34].

The mean prediction error for daily peak load in [

24

] was achieved 4.65% for weekdays and

7.08% for weekends of three different states of Turkey [

35

]. This mean prediction error was achieved

after smoothing the temperature discrepancies throughout the day. The precedence of similar days is

overlooked in previous studies. It is obvious that there are numerous days which are advantageous for

the knowledge based forecasting of load. Nevertheless, the best suitable preference of these same days

has a substantial effect on forecasting results.

Electronics 2018,7, 431 5 of 34

The consequences of temperature are neglected in [

36

] in terms of priority index. Moreover, in [

37

]

a priority index for medium term load prediction was presented. The proposed model was based on

the similarity of temperature for the selected day. The mean error achieved in [

37

] for Western States

of America was 3.25% for summer season. Besides, few values of error were attained that were more

than 6%. Though, the temperature was the only parameter, which was assessed in this study and the

proximity of chosen day to similar day was ignored. It is a well-known fact that same days do not

have alike temperature. Moreover, the similar days must be near to the target days in order to avert

the selection of similar days with similar temperature and different power load.

The work presented in [

38

] used the Bayesian network to forecast activities of different residents

by a particular appliance. However, the model was not efﬁcient enough to be functional towards real

world circumstances. The authors in [

39

] and [

40

] discussed a multi-label and time sequence based

classiﬁer model for a decision tree taking appliance association as a correlation. The basic purpose

of their model was to predict the power consumption of the appliance. Though, the authors merely

observed the past 24 h frame for future forecasting.

The work in [

41

] presented the association rule mining method to classify the interdependence

between power consumption and appliance usage to help power saving, anomaly detection,

and demand response. Nevertheless, this work lacked the proper rule mining process and

appliance-appliance association.

At present, Artiﬁcial Neural Network (ANN) and SVM are considered to work efﬁciently for

non-linear time series sequences. Karatasou et al. [

42

] demonstrated the practical implementation

of ANN in forecasting power expenditure of a building accompanied by statistical study. In [

43

,

44

],

a model is presented which hybrids the Support Vector Regression (SVR) and Immune Algorithm (IA)

to estimate local yearly report and power load in Taiwan.

Zhao et al. [

45

] presented a framework, which employed SVM to predict residential power

utilization in the humid area. Moreover, the study took meteorological conditions of that particular

area. Besides, Xuemei et al. [

46

] suggested Least Square Support Vector Machine (LS-SVM) for

chilling load prediction [

47

] for a residential zone in Singapore. The forecasting was done by hourly

weather information.

Wang et al. [

48

] discussed that the SVM based models have proven to be efﬁcient as compared to

ANN and ARIMA conﬁgurations. They employed Differential Evolution (DE) and SVM to predict

the conﬁgurations for yearly energy consumption. Conversely, the development of SVM model

is inﬂuenced by the category and constraints of the kernel function. Generally, it is discussed in

literature that the tuning constraints of SVM is a challenging task [

49

]. In addition, a number of models

are presented in literature to tune the parameters of SVM by techniques of machine learning and

artiﬁcial intelligence.

Ogliari et al. [

50

] proposed a hybrid model using Neural Network and Genetical Swarm

Optimization for energy prediction. The authors in [

51

] combined SVM with algorithms of Simulated

Snnealing to predict yearly load. On the subject of optimization techniques, Jaya Algorithm has

achieved attention in the last few years as a metaheuristic computing technique. The authors in [

52

]

and [

53

] observed that Jaya Algorithm outperforms other optimization techniques. Moreover, Jaya

Algorithm has also been employed for various real work applications.

There is a variety of literature available on the topic of game theory and DRM. In [

54

], the authors

have discussed power utilization and forecasting as a non-cooperative game plan. This basic aim

was to maximize the cost functions. Likewise, the authors in [

55

] have proposed a distributed set-up.

In this set-up, the cost function is demonstrated by its dependence on inclusive load. The consumers

adjusted their behavior for power consumption on the basis of cost function introduced by the utility.

The authors in [

56

] presented a theoretical framework for mutual optimization of investment and

functioning of a smart grid. Moreover, the aspects of power storing, renewable energy integration, and

demand response were taken into consideration. The paper signiﬁed the sharing of portfolio decisions,

Electronics 2018,7, 431 6 of 34

day-ahead pricing, and scheduling. They also presented the beneﬁts of integrated renewable energy

and demand response in terms of minimizing the sharing cost.

A robust optimization has been discussed in [

57

] in order to increase the utility of the end-user

by hourly prediction. The study presented in [

58

] laid emphasis on the knowledge and interest of

users to be aware of the announced electricity prices. The study proposed a technique to cope with

preferences of the consumers to increase power competence and consumer satisfaction. Moreover,

a dynamic cost price has been introduced to motivate users for attaining a cumulative load [

59

]. Also,

this load was handled by different utilities and DRM was scrutinized for bi-directional communication

between consumers in the micro-grid. The authors in [

60

] and [

61

] discussed the dynamic pricing in

detail for smart energy model of a smart grid. The discussed model was dependent on renewable

energy sources, which were further integrated with intelligent control that processed information from

a smart metering devices.

The studies discussed above are inadequate to meet the needs, i.e., the electricity ﬁrms considered

utility companies as a single ﬁrm. This study differs in this context as this incorporates numerous

utilities and consumers. Moreover, the basic aim of both entities is to increase their proﬁts

(remunerations) by game theoretic approach. Besides, there is a broad literature and ﬁndings available

on the Stackelberg game on the topics of proﬁts maximization, congestion control, and interactive

communication [62,63].

3. Game Theoretical Problem Formulation

This study takes nconsumers and

UC

utility companies in consideration. Besides, the energy

sources of the utilities include non-renewable and renewable resources. In literature, it is observed

that power generators, which are centered on the energy of fossils utilize a deﬁnite amount of energy.

Moreover, the energy of fossils is also supposed to be harmful for the environment. Contrarily,

renewable energy sources are considered environmentally friendly. However, renewable resources

have inherent natural stochastic behavior, which makes it difﬁcult to predict and control. The studies

show that uncertainties are common with renewable resources. Furthermore, Markov chain (discrete

time) has been extensively employed in literature for the generation of power from renewable

resources [64].

This study takes residential type consumers into account. In addition, all users have dissimilar

requirements for power consumption. The study also distinguishes the users based on their ﬁnancial

plans; i.e., purchasing power of electrical energy. Likewise, this study proposes a utility function for

every consumer. The function shows an increment using the total expanse of power that any consumer

is able to utilize. Moreover, this paper integrates cost parameters for every consumer.

The

UC

and nhave established a two way communication using the advanced metering

infrastructure for pricing swapping and information sharing. Conversely,

UC

can also communicate

with one another. The ncollect the value (cost) facts from the

UC

. In return, the

UC

then provide their

services to n.

Power initiation, dissemination, and expenditure can be divided in three ways [

65

]: Power

generators,

UC

, and n. This paper emphasizes the communication between nand

UC

. Moreover,

this paper assumes that

UC

show a ﬂuctuating behavior at the business level. Inspired from the game

theory models, the

UC

can play a vital part in an economical marketplace. No participant is capable

enough to affect the market price of electricity through his particular activities. Thus, the market price

is such constraint over which

UC

have no control. Moreover, the

UC

need to increase their production

up to the point where the minimal cost is equivalent to the cost of the market. This phenomenon occurs

once the total contributors increase and no contributor is authorized to govern an enormous power

generation quantity. Nonetheless, this study proposes a predetermined ﬁgure of

UC

(contributors).

This scenario depicts that every utility will announce its own price according to its generation capacity.

Table 1shows the list of symbols used.

Electronics 2018,7, 431 7 of 34

Table 1. List of Symbols Used.

Symbol Meaning

UC Utility Companies

nAll consumers

n0Consumer

uc0Utility Company

dn0Demand of consumer

γn0Constant for user analysis

τn0Constant for user demand

ln Function for decision making

κuc0Price per unit

Bn0Total budget of consumer

Λn0,1,Λn0,2 ,Λn0,3 Lagrange multipliers

∇υcons Best condition of ﬁrst order

Euc0Available power of uc0

ξuc0+1Price of U C other than uc0

MInvertible matrix

|M| Determinant of M

=prod Strategy sets for M

=cons Strategy sets for n0

dGame plan for all n

dκ+Best feedback of all n

d+

n0Proposed best scheme for n

rIteration Number

δuc0Speed modiﬁcation constraint of uc0

IiInput Vector in SVM

OiTargeted Output in SVM

ETotal data in SVM

WWeight in SVM

tThreshold estimate in SVM

3.1. Analysis of User and Utility Company

The cost for every consumer shows ﬂuctuation when there are various utility ﬁrms having diverse

electricity costs. Moreover, the setting of cost is highly reliant on the rates of other

UC

. In this regard,

game theory offers an ordinary pattern to represent the activities of nand

UC

. Consequently, the

UC

settle the cost for each unit of energy and then publicize this to consumers. The users then respond

back to the cost by demanding an optimal amount of power from the

UC

. In this case,

UC

play ﬁrst.

The consumers then decide on the basis of announced prices. Moreover, both events are in sequence.

The events are that the utilities play primarily and at that time the consumers decide their verdict

based on the cost. Hence, this paper models the communication between the

UC

and nby a Stackelberg

game [

66

]. The proposed game model takes the

UC

as inﬂuential (leaders) and users as followers.

Moreover, the proposed model also considers the events as a multiple leaders and followers game.

3.1.1. Analysis of User Side

Assume that

dn0,uc0

is the request of consumer

n0

from a utility

uc0

. Hence, the value of a consumer

n0,Ccons,n0can be expressed as:

Ccons,n0=γn0∑

uc0∈U C

ln(dn,uc0+τdn0).∴∀uc0∈ UC (1)

Here

γn0

and

τn0

are constants. Also, the ln function is extensively employed in literature for user

making decisions [

67

]. The valuable function used for consumer

n0

in Equation (1) is interrelated to

the function γn0∑ln dn0,uc0.

Electronics 2018,7, 431 8 of 34

The consumer will recompense -

∞

when the valuable function

γn0∑ln dn0,uc0

is used regarding

uc0

, such that,

dn0,uc0

= 0. When

dn0,uc0

and

τn0

are equivalent to 0, then beneﬁt of

n0

regarding

uc0

begin to be ﬁnite. Generally, the representative cost of τn0= 1.

Suppose

κuc0

is the per unit cost given by any utility company

uc0

and

Bn0≥

0 is the total

expenditure of any consumer

n0

. Each

uc0

has given a distinct price rates of electrical energy

[κ0,κ1,......, κc] when n0∈n.

Subsequently, the

n0

computes the best demand response through

resolving best optimum solution (OS cons) given in Equation (2).

dn0=max(dn0,uc0)Ccons,n0∴∀uc0∈ U C, (2)

where ∑uc0∈UC κcdn0,uc0≤Bn0,

dn0,uc0≥0∴∀uc0∈ UC (3)

OS cons

is a convex optimization problem. Therefore, the obtained solution is distinctive

and optimal.

This paper considers the scrutiny accompanied by

UC

consumers and three

UC

s. Thus, they seek

for best optimum solution in this scenario for a speciﬁed uc0can be expressed as follows:

dn0=max(dn01,dn02)γn0

3

∑

uc0=1

ln(dn+τdn0,uc0), (4)

where κcdn0,1+κcdn0,2≤Bn0and dn0,1+dn0,2≥0.

The paper employs Lagrange multipliers (

Λn0,1

,

Λn0,2

,

Λn0,3

) for the respective

UC

and setting of

parameters as discussed above. Thus, the Equation (4) can be rewritten as:

υcons,n0=γn0

3

∑

uc0=1

ln(dn1 +τdn0,uc0)−Λn0,1(

3

∑

uc0=1

κcdn,uc0−B1) + Λn0,2dn0,1+Λn0,3 dn0,2. (5)

The values of the Lagrange multipliers are used as strategies for ﬁnding the local maximal and

minimal of the function subjected to inequality constraint. Thus, it improves the performance of

Equation (5).

Λn0,1(

3

∑

uc0=1

κcdn,uc0−Bn0) = 0. (6)

Moreover, setting

Λn0,2dn0,1

and

Λn0,3dn0,1

generates Equation (6) to 0. Whereas,

Λn0,1 >

0,

Λn0,2,Λn0,3 ,dn0,1, and dn0,2≥0.

The ﬁrst order optimality condition for linear, best optimum solution and maximization problem

is by setting

∇υcons =

0. Here,

υcons = (υcons ∀n0∈n

). All of the nare interconnected by

κc

. Also,

∇υcons =0 shows that,

(℘υcons,n0)(℘dn,uc0)−1=0∴∀n0∈n,uc0∈ UC. (7)

Also,

γn0−(τn0+dn0,1)(Λn0,1τ1+Λn0,2)(8)

and

γn0−(τn0+dn0,2)(Λn0,2τ1+Λn0,3). (9)

Next, this paper has considered four of the cases, which the n0can avail.

Electronics 2018,7, 431 9 of 34

Case 1

If

dn0,1

and

dn0,2

are greater than 0, then

Λn0,2 =Λn0,3 =

0. So, Equations (8) and (9) are

generalized as:

dn,uc0=γn0−(τn0κuc0Λn0,1), (10)

where n0∈nand uc0=1, 2, ..., n. Now, using Equation (6) in Equation (10),

3(γn0Λn0,1)−1=Bn0+τn0

3

∑

uc0=1

κuc0. (11)

Thus, Equation (11) becomes Equation (12) after simpliﬁcation.

3dn,uc0κuc0= (Bn0+τn0

3

∑

uc0=1

κuc0)−τn0(12)

Here value of uc0varies; i.e., 1, 2, or 3.

Case 2

If

dn0,1>

0 and

dn0,2

are equivalent to 0, then

τn0= (Bn0+τn0∑3

uc0=1κuc0)/

3

κuc0

. As discussed

above that

Λn0,2dn0,1

corresponds to 0. This paper derives Equation (13) by considering the cost of the

ﬁrst utility.

dn0,1=γn0−τn0Λn0,1κ1. (13)

This paper further expands Equation (6) to include extra parameter and ease simpliﬁcation. Thus,

Equation (14) is derived.

Λn0,1(γn0−κ1Λn0,1 τn0−Bn0) = 0. (14)

As Λn0,1 >0 and γn0−κ1Λn0,1τn0−Bn0=0, which refers to the point that Λn0,1 =γn0/(κ1τn0+

Bn0). Now, evaluating this in Equation (13),

dn0,1=κ1τn0+Bn0−τn0κ1. (15)

Equation (15) is now equivalent to Bn0/κ1. Moreover, Equation (15) can also be presented as:

dn0,1=τn0(κ1+κ2) + Bn0−3κ1τn0, (16)

where dn0,1= ((τn0(κ1+κ2) + Bn0)/3κ1) + ((τn0(κ1+κ2)−Bn0)/3κ1).

Case 3

If dn0,1is equivalent to 0 and dn0,2>0, then the identical scrutiny can be valuated as speciﬁed in

Case 2. This paper considered the cost of the second utility; thus, the demand of users with respect to

the second utility is given in Equation (17).

dn0,2=τn0(κ1+κ2) + Bn0−3κ2τn0. (17)

Subsequently, Equation (17) is now equivalent to Bn0/κ2.

Case 4

If

dn0,1

and

dn0,2

both are equivalent to 0, then

Λn0,1

,

Λn0,2

, and

Λn0,3

are real and positive values.

It is noted that Case 4 is assumed as best case which rarely occurs only when

κuc0=∞

or else

Bn0≥

0.

This paper has satisﬁed the power and cost parameters as equalities in Case 1, 2, and 3. However,

this scenario cannot be mapped on Case 4. This study further assumes that there are nconsumers

Electronics 2018,7, 431 10 of 34

in total and

UC

utilities that satisﬁes the equality conditions in previous cases for a given set of

κuc0

.

So, Equations (12), (16), and (17) can be combined in the above discussed scenario as:

dn0,uc0=τn0∑

uc0∈U C

κuc0+Bn0−κuc0U C τn0. (18)

In Equation (18), dn0,uc0≥0, n0∈nand uc0∈ UC. As dn0,uc0≥0. So,

τn0(∑

uc0∈U C

κ$) + Bn0>τn0κuc0(UC − 1). (19)

3.1.2. Analysis of Utility Companies

This study assumes that

Euc0

(

UC ∈ uc0

) depicts the available electrical energy of

UC

. The aim

of every

UC

is to vend the energy to gain maximum proﬁt. For instance, if there is only one

UC

then this ﬁrm will settle the price range according to its ease as there is no competition involved.

However, this study takes two basic strategies that decide the cost range of any

UC

. Firstly, it can be

the economical conditions of average consumers and secondly, it could be an aspect of competitiveness

among

UCs

. Furthermore, the

UCs

also take part in choosing the best optimum cost (game) with

another. Additionally, this study expresses the maximum proﬁt Eprod,uc0of any U C as:

Eprod,uc0(κuc0,ξuc0+1) = κuc0∑

n0∈n

dn0,uc0. (20)

Here,

ξuc0+1

is cost of

UC

apart from

uc0

. Thus, the best optimum solution for any

UC

can be

related in terms of OP pro d and can be expressed as:

ξ=max(κuc0)Eprod,uc0(ˇuc0,¸uc0+1),∴∀uc0∈ U C (21)

where

∑n∈n0dn0,uc0≤ Euc0

and

κuc0>

0,

∀U C ∈ uc0

. The maximum proﬁt of any

UC

is ﬂuctuating in

relation to energy for a constant

κuc0

. According to Equation (20), this phenomenon leads to parameters

of equality. Every

UC

proffers to vend all its energy to consumers. This paper assumes

υprod,uc0

to

resolve OP prod by:

υprod,uc0=κc∑

n0∈n

dn0,uc0−ζuc0(∑

n0∈n

dn0,uc0Euc0)(22)

The best optimal solution for the U C furthers presents ℘υprod,uc0/℘κuc0, which is equivalent to 0.

κ2

uc0ρ(U C − 1)−ζuc0(ρ∑

$∈U C,$6=uc0

+Bn0) = 0. (23)

where

ρ=∑n0∈nτn0

and B

=∑n∈n0Bn0

. Moreover, the conditions used in Equations (21) and (22)

express

UC

equations. Now, solving these three

UC

, this study sets

κ+= [κ+

1

,

κ+

2

, .....,

κ+

U C ]

and

ζ+= [ζ+

1

,

ζ+

2

, .....,

ζ+

U C ]

. Furthermore,

D=d+

n0,uc0

can be evaluated by means of

κ+

. Consequently,

employing Equation (18) for uc0,

κuc0=ρ(κ$∑$∈U C,$6=uc0) + B

ρ(UC − 1) + Euc0UC . (24)

Now, using the current value of κuc0this study observes that,

ζuc0=ρ(U C − 1)( ρ(κ$∑$∈U C,$6=uc0) + B

UC(ρ+Euc0). (25)

It can also be deduced from Equation (23) that

ζuc0=ρκuc0(UC −

1

)

. Also,

ρ

and B

≥

0. It refers

to the phenomenon that there is no essential need to play any game when

UC =

1. Therefore, the study

Electronics 2018,7, 431 11 of 34

merely focuses on the circumstances when

UC ≥

3. To handle the discussed scenario, Equation (23)

can now be computed as:

Mκ=S. (26)

Here M=

E1+J−H · · · −H

−H E2+J· · · −H

· · · · · · · · · · · ·

−H −H −H EU C +J

,

κ= [κ1

,

κ2

, .....,

κU C ]

,J

=ρ(UC −

1

)UC−1

,

H=ρUC−1

, and

S=

B

UC−1

. From the above equations,

it can be concluded that Mis an invertible matrix. However, it could be expressed as:

κ=M−1S. (27)

This paper considers some cases to achieve closed-form solution of κ.

Case 1

All the

UC

have equivalent amount of energy available and capacity to produce, then

E1=E2=

E3=· · · =EUC . Utilizing Equation (26),

κuc0=S

E+J+H(1− UC)=κ. (28)

Likewise,

κ=B(UCE)−1(29)

Also,

E∝1

κ

. Then the Equation (27) is used in Equation (19), so that the total demand to any

UC

from n0is given as:

Bn0≥κτn0(UC − 1)−κτn0(UC − 1)(30)

Here, Equation (30) indicates that

Bn0≥

0. This phenomenon indicates that now all

UC

s produce

equivalent amount of power. Moreover, they have settled some pricing scheme that users have

to follow.

Case 2

Contrary to Case 1, this case considers that capacity of power generation is different for all

UC

s. The

M

in Equation (26) has some unique aspects, which relates that a real valued matrix

M= [mi,j,i,j=1, 2, · · · ,U C]∈ RU C is only considered diagonal as shown in Equation (31),

|mi,j| − ∑

j6=i

|mi,j| ≥ 0, (31)

where, i

=

1, 2,

· · ·

,

UC

. According to [

68

], a taut diagonal matrix is always non-singular and

|M|

is

positive. It is observed that

M

is taut and diagonal matrix as

Euc0+

J

− H(U C −

1

) = Euc0+ (ρ(U C −

1)) −(ρ(U C − 1)). Consequently, Euc0>0. Thus, Mis invertible.

Theorem 1. The distinctive solution achieved from Mis positive.

Proof of Theorem 1. The solution of Mis deduced by

κuc0=BU CUC−| |M|−1∑

$∈U C,$6=uc0

(E$+ρ). (32)

Since

|M|

is invertible; thus,

|M|

is positive if its eigenvalues are non zeros and show a symmetry

property. Also, the solution presented in Equation (32) depicts that κuc0>0.

Electronics 2018,7, 431 12 of 34

Theorem 2. The cost function discussed in Equation (27) is a best optimum solution for raising proﬁts.

Proof of Theorem 2.

Let the solution gained from Equation (27) be

κuc0

for any

uc0

. Moreover,

this paper assumes that

uc0

has increased the cost from

κuc0

to

κ∗

uc0

, while

UC

have same cost of

power generation. From Equation 19, suppose that any consumer ndemands power

dn0,uc0>

0 from

any κuc0then the constraint in Equation (33) is satisﬁed.

κuc0≤(Bn0+τn0(∑

uc0∈U C

κ$)τn0(UC − 1)−1). (33)

Now suppose that

κuc0

and

κ∗

uc0

fulﬁl the requirements of Equation (33). In this regard,

the necessities of consumers will show deviating behavior from dn0,uc0to d∗

n0,uc0as:

d∗

n0,uc0=(τn0(∑uc0∈UC κ$+κ∗

uc0) + Bn0)

κ∗

uc0U C −τn0. (34)

The differentiation among the necessities of any nfrom the ﬁrm uc0will now be expressed as:

dn0,uc0−d∗

n0,uc0=κ∗

uc0−κuc0

κ∗

uc0κuc0

∗(τn0(∑uc0∈UC κ$) + Bn0)

UC . (35)

From Equation (35), it is obvious that

dn0,uc0−d∗

n0,uc0>

0. Hence, the consumers are not capable

of demanding the total power generated by any

uc0

, i.e., the consumer will then demand for lesser

energy as required. Moreover, the proﬁt and cost of

uc0

will increase on the basis of consumer total

power demand. Thus, Equation (36) provides the balanced equation of demand and supply.

E∗

prod,uc0−Eprod,uc0=κ∗

uc0∑

n0∈n

d∗

n0,uc0−κuc0∑

n0∈n

dn0,uc0(36)

It is observed that in Equation (36),

E∗

prod,uc0(κ∗

uc0

,

ξuc0+1)<Eprod,uc0(κuc0

,

ξuc0+1)

. Thus, the

proﬁt gaining of

uc0

leads towards the loss and it is concluded that the price function presented in

Equation (27) is the best optimum function as it will result in ﬁnancial advantage.

On the subject of range of

κuc0

,

κuc0∈[κuc0mi n

,

κuc0ma x ]

. As a matter of fact,

κuc0min

is owing to the

cost functions that is generated by

UC

. Moreover, any

uc0

is not capable to lessen the price lower than

κuc0min

. Nonetheless,

κuc0ma x

is the maximum range. According to

κuc0ma x

, the government has to settle

the cost, which consumers have to follow.

3.2. Proposed Stackelberg Game Modeling

All the

UC

s partake to play the non-cooperative game with one another in order to settle the price

that will be further used by consumers. This is a critical point where Nash Equilibrium is required.

In a Stackelberg game, the equilibrium strategy for the followers is deﬁned as any strategy that is

compromised of the best response. The response is optimal as compared to the strategy that is adopted

or announced by the leaders [69].

This study assumes that

=prod,uc0

is the game-plan rectiﬁed for any

uc0

and

=cons,n0

is the scheme

planned for

n0

. Subsequently, the game-plan for

UC

will be

=prod ==proda× =prodb× · · · × =prod,U C

and

for

n0

will be

=cons ==consa× =consb× · · · × =cons,UC

. Thus,

κ+

uc0∈ =prod,uc0

is proposed Stackelberg

equilibrium for any uc0if,

Eprod,uc0(κ+,d(κ+)) ≥Eprod,uc0(κuc0,ξ+

uc0+1,d(κuc0,ξ+

uc0+1)). (37)

Electronics 2018,7, 431 13 of 34

Here,

κ+= [κ+

uc0]

,d

= [d1

,

d2

,

· · ·

,

dn]

is the game plan of all consumers n. Moreover, dand

κ+

are

best feedback of all consumers, i.e., d

∈ =cons

. The best feedback of any consumer

n0

for any particular

(κ1,κ2,· · · ,κuc0)∈(=prod,1 × =pro,2 × · ·· × =prod,U C )is:

κdn0=χcons,n0∈ =cons,n0∧Econs,n0(κ,χcons,n0)≥Econs,n0(κ,dn0). (38)

Here,

χcons,n0=d+

n0∈(κ+)dn0

. Thus,

d+

n0

is supposed to be best optimum scheme for

n0

. Besides,

d+and κ+is a Stackelberg equilibrium achieved for the game concerning the UC and n.

3.3. Distinctiveness of Stackelberg Equilibrium

The

OS cons

has an exclusive maximum range (as discussed above) for

κ

. Whenever the cost

planning game is played between the companies with a distinctive Nash Equilibrium, then the

Stackelberg game plan holds a special equilibrium.

Theorem 3.

An exclusive Nash equilibrium occurs in the cost selection game plan between

UC

. Likewise,

a distinctive Stackelberg equilibrium subsists as well.

Proof of Theorem 3.

There is equilibrium if

κ

is a real value and

⊂ RU C

. Moreover,

κE∗

prod,uc0

is

constant in

κ

. On the topic of cost choosing of all

UC

in Stackelberg game,

=prod = (=prod,1 × =prod,2 ×

· · · × =prod,U C )

. Here,

κuc0⊂ =prod,U C

. Moreover,

=prod = [κuc0,min

,

κuc0,max]

. Therefore, the game plan

is real value and ⊂ RU C .

Furthermore,

Eprod,uc0

is constant in

κuc0

as discussed in Equation (20). Subsequently, the

f00(Eprod,uc0)according to κuc0is,

℘2Eprod,uc0

℘κ2

uc0

=0. (39)

3.4. Distributed Algorithm

The users are now proﬁcient enough to compute their optimum demands on the basis of the cost

function provided by the utility companies as discussed in the preceding section. However, the different

utilities show the signiﬁcant response to policies announced by other companies. Moreover, it is

essential to calculate the price per unit. For this purpose,

uc0

should know the production capacity of

other utilities. Contrary to this, this paper proposes a distributed algorithm that further proves the

Stackelberg equilibrium of the game. The equilibrium is established in such a way that utilities are not

able to identify the constraints of each other.

The

uc0

establishes a subjective cost and then conduct their cost statistics to the users.

This communication is done efﬁciently by setting an interactive environment for utilities and

consumers. As a consequence, the consumers choose speciﬁc amount of electricity they need to

purchase from uc0.

All the

UC

acquire these demanding conditions from consumers. At that moment,

uc0

will analyze

the contrast among the available electrical energy and the entire energy needed by consumers from the

company. The U C will upgrade its price per unit with the help of Equation (40).

κuc0,r+1=dn0,uc0,r∑

n0∈n

−Euc0+δuc0κuc0,r. (40)

In Equation (40), ris the repetition number and

δuc0

is the rate modiﬁcation constraint of

uc0

.

Whenever a

uc0

updates its cost function, it sends this information to

n0

. Furthermore, the

n0

update

the demands and send this information back to

uc0

. Subsequently, the

UC

s will also update their

cost functions sequentially. Thus, the procedure lasts until the cost function shows convergence.

Algorithm 1supposes that n0=1 speciﬁes the ﬁrst consumer.

Electronics 2018,7, 431 14 of 34

Theorem 4. Given that ∀n0∈n, uc0∈ UC,r=1, 2, 3, · · · and

δuc0≥κuc0,r(Euc0U Cτn0)−τn0∑$∈UC∧$6=uc0κ$,r−Bn0

UCκ2

uc0,r

(41)

Algorithm 1meets the best optimum solution for all

UC

and nas the particular game plans are upgraded in a

speciﬁed order.

Proof of Theorem 4.

The feedback of a consumer as speciﬁed in Equation (18) is best optimum solution

for a particular κuc0.

Whenever, the cost per unit shows a converging behavior then the demand of every consumer

will coincides towards an established set. Therefore, it is necessary to discuss the converging behavior

of cost in order to demonstrate the changing performance of Algorithm 1.

Algorithm 1will only show the diverging behavior whenever the

κuc0,r

will be negative in

Equation (41).

Algorithm 1: Distributed Algorithm

1Randomly select κuc0,1 for r=1∀uc0∈ U C

2Consumer uc0=1, 2, 3, · · · ,n

3Calculate Equations (2)–(4) for κrby Equation (18)

4Send dn0,uc0,rto respective uc0

5Any uc0, which has not upgraded its value for r+1

6Evaluate κuc0,r+1by way of Equation (40)

7if κuc0,r+1−κuc0,1 =0then

8return Price value is not changed by uc0

9Jump to 8

10 else

11 Transmit the updated cost to n

12 Jump to 3

13 end

14 if κuc0,r+1≡κuc0then

15 Terminate

16 else

17 Jump to 2

18 end

In Equation (40), if

dn0,uc0,r∑n0∈n−Euc0≤

0 then the signiﬁcant constraint for

κuc0,r+1

for

not gaining a negative amount is

|dn0,uc0,r∑n∈n0−Euc0|(δuc0)−1

. Furthermore, the condition that is

discussed above can be revised as δuc0≥(Euc0−(dn0,uc0,r∑n0∈n)(κuc0,r)−1).

Equation (40) suggests that the cost

κuc0

ampliﬁes only if

dn0,uc0,r∑n0∈n−Euc0

gives positive

results and vice versa. However, in Equation (40), when

dn0,uc0,r∑n0∈n−Euc0=

0 the price value is not

changed. This particular condition is the established stage to which Algorithm 1shows converging

behavior. This stage is the Nash Equilibrium of the game plan (Stackelberg game between nand

UC

).

Afterwards, the UC will not show any ﬂuctuating behavior.

4. Knowledge Based Short-Term Load Forecasting

Knowledge based systems and computational intelligence are considered as major tools of artiﬁcial

intelligence. The knowledge based systems employs categorical representations of knowledge like

symbols and words [

70

]. The knowledge based systems are efﬁcient and simple as the categorical

Electronics 2018,7, 431 15 of 34

representation makes the knowledge readable and implicit for a human as compared to numerical

derived models in computational intelligence. The techniques of knowledge based systems incorporate

case based, model based, and rule based systems.

The major difference between a traditional program and knowledge based system is in their

structure [

71

]. The knowledge of the domain is closely associated with software for monitoring the

performance of that particular knowledge in a traditional program. However, the roles are clearly

divided in knowledge based systems. Moreover, there are two basic components of knowledge based

systems, which are knowledge base and inference engine. Nonetheless, some interface proﬁciencies

are also compulsory for a real-world system, as presented in Figure 1.

Knowledge Base Inference Engine

Interface

DataHumans and Experts Hardware and Software

Figure 1. Principle components of knowledge based system.

The paper categorizes knowledge based short-term load forecasting as classic and proposed.

The explanation of each is given below.

4.1. Classic Knowledge Based Short Term Load Forecasting

All categories of days are quantiﬁed initially in a classic knowledge based forecasting on the

basis of annual and weekly load curves. Moreover, this type of categorization of days is usually

associated with the user consumption behavior of a particular state. Besides, the annual growth rate in

load demand also plays a signiﬁcant role in typical knowledge based forecasting as historical load

data are also required. The annual growth in load demand is mostly reliant on different aspects like

growing economy or population. Consequently, normalization and stabilization of load data are

considered crucial in order to lessen the consequences of annual growth rate. Likewise, normalization

of data is also beneﬁcial to determine similarities in load curves more precisely [

72

]. The hourly data

normalization of load demand is attained by distribution of load on hourly basis [

73

], which is shown

in Equation (42).

Γ0

Sd,H=ΓSd,H

¯

X(ΓSd,H−1,ΓSd,H−2,· · · ,ΓSd,H−n). (42)

In Equation (42),

Γ0

Sd,H

is the load demand,

ΓSd,H

is the normalized value of data at any hour

H

of a similar day

Sd

, and

¯

X

is the mean of npreceding days. In addition,

H=

1, 2, 3,

· · ·

, 24. The load

Electronics 2018,7, 431 16 of 34

demand at any hour

H

of a target day can be obtained by normalization of load demand data of chosen

similar days and average load of npreceding hours, which is presented as:

Γtar,H=1

Dγ

×¯

X(Γtar,H−1,Γtar,H−2,· · · ,Γt ar,H−n). (43)

In Equation (42), tar indicates the target variable, which is predicted by the model for a speciﬁed

day. Moreover,

Γtar,H

is the predicted demand load for any hour

H

,

γ

is set of identical days, and

Dγ

is

total number of days chosen, which are similar. The minimum value of

Dγ

reduces the utilized historic

data and inadequate similar days, which are selected. Contrarily, the maximum value of

Dγ

indicate

that

γ

is comprised of vast historic data. Besides, a few number of days may have not sustainable

correlation with selected day according to this scenario.

4.2. Proposed Knowledge Based Short-Term Load Forecasting

The paper proposes a novel hybrid data mining technique in order to predict the load demand

by knowledge based systems. The proposed algorithm basically consists of two parts. The clustering

technique AP is used initially. The AP is employed in this scenario as it looks for noise in data and

then removes this noise from data, thus, decreases the instances of data. Subsequently, BFFA is used in

the next step for feature selection and classiﬁcation. Furthermore, Support Vector Regression (SVR) is

used as classiﬁer model in this proposed hybrid model. This proposed hybrid model chooses the most

relevant target variables and increases the accuracy of the system. Moreover, the proposed knowledge

based system is able to minimize the operational cost and maximizes the process of data mining for

selection of similar days.

The proposed knowledge based short-term load forecasting is categorized in three parts, which

are explained as follows.

4.2.1. Distribution of Historic Load Data

The selection of similar days from historic days is considered as crucial for knowledge based

forecasting. Moreover, the selection of similar months and days also have a signiﬁcant impact on the

results of short-term load forecasting. Therefore, this paper presents two historic data-sets, which are

well-deﬁned for every type of days. The ﬁrst data-set is comprised of similar days from preceding

month along with the selected date. Furthermore, the second data-set incorporates same days from

seven days earlier and subsequent to the target day of the week. The target year and similar days are

also chosen from all preceding years in both data-sets. Besides, the data-sets are speciﬁed by scrutiny

of annual load demand and meteorological conditions of Pakistan.

It is a well-known fact that temperature and load demand have a direct relationship with each

other. For example, usage of air conditioners and other cooling devices increases in summers especially.

This phenomenon shows variations in load curve and peak hour of the entire system. Moreover,

the impact of climatic conditions on the load demand in summers is usually more than other time of

year [74].

Figure 2illustrates the load curves for Thursday as an example. Moreover, this load curve is for

Pakistan and depicts all four seasons. It is obvious from Figure 2that the load level and hourly peaks

by day and nights shows a signiﬁcant ﬂuctuation in different spells. Therefore, it can be determined

that by maximization of the measured time, the range of both data-sets may affect the selection of

similar days with similar temperature. However, this phenomenon is not suitable for load curves

because changes in climate also affect load consumption behavior.

In the ﬁrst data-set, the same days are chosen from days that have equivalent month along with

the target day. Moreover, this paper has assumed that the selected day can also be similar to its month

or preceding month. Contrary to this, load curves from seven days earlier and subsequent to the

target day is more comparable to the target day when associated to load bends of the preceding month.

Consequently, the other data-set speciﬁes the consideration of these days in a data-set. Moreover,

Electronics 2018,7, 431 17 of 34

this paper assumes that this data-set must have a maximum weightage factor, in contrast to the ﬁrst

data-set. The priority index for both data-sets can be evaluated by Equation (43). The paper valuates

the ﬁnal results from the combination of results achieved from both data-sets as:

Γtar,H=W1×Γds1

tar,H+W2×Γds2

tar,H. (44)

In Equation (44),

Γds1

tar,H

and

Γds2

tar,H

are forecasted power load demand speciﬁed for each hour

H

and targeted day tar. Moreover,

W1

and

W2

are weights assigned to each data-set. Thus,

Γtar,H

is the

ﬁnal forecasting achieved by system for each hour Hand targeted day tar.

The proposed methodology for knowledge based forecasting is comprised of two main

constituents, which are

W1

and

Dγ

. Furthermore, the proposed method must also execute for training

data-set in order to choose the best optimum values of

W1

and

Dγ

. Subsequently, the proposed method

should be proﬁcient enough to select the execution, which gives the least prediction error. Besides,

the values of

W1

and

Dγ

are then selected as the optimal ones in order to predict the target day.

Moreover, this paper also assumes that the next 24 h are forecasted by preceding load demand data

and predicted loads of the day. This load demand data is achieved after prediction of the ﬁrst hour of

tar day by preceding load demand data.

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25

Hour (h)

System Load (MW)

Winter Autumn Spring Summer

Figure 2.

Variations in load behavior of sample Thursday during 2015 of Pakistan’s National Power

Network (PNPN).

4.2.2. Priority Index for Same Day

In knowledge based short-term load forecasting, temperature has a significant role. The fluctuating

behavior of climate and weather throughout a week or month shows a signiﬁcant effect on load curves.

Therefore, it is a vital part in choosing similar days for target year. Conversely, there can be different

motives that are the cause of divergence for load curves. For instance, the power evaluating strategies

and variations in utilization behaviors of Pakistan alter the levels of load demand. Thus, the selection

of similar days along with date proximity is effective to choose for knowledge based forecasting.

The paper determines a priority index of similar days as:

PISd

reg.= [℘temp,reg.×∑

C∈κr eg.

(tem pSd

C−tem ptar

C)2+℘1,reg.×(ηre g.)2]1

2. (45)

In Equation (45),

PISd

reg.

is the priority index of

Sd

in speciﬁc region,

tem pSd

C

and

tem ptar

C

are

the average temperatures of a speciﬁed city

C

on the daily basis for a similar day

Sd

and tar days,

correspondingly. Furthermore,

ηreg.

is total number of days between tar days and

Sd

days,

κreg.

are the

chosen cities from every region. This paper separates the system in seven different regions and from

every region only one city is selected.

Electronics 2018,7, 431 18 of 34

In this paper,

℘tem p,reg.

is considered as weighting factor of temperature, while

W1,reg.

is taken as

weighting factor of ηreg.. They are calculated as follows:

℘tem p,reg.=Dγ∑

Sd∈γreg.

×(tem pSd

C−tem ptar

C)2

Dcity

×∑

C∈κr eg.

(46)

and

W1,reg.=Dγ∑

Sd∈γreg.

×1

η2

reg

. (47)

In Equation (46),

Dcity

is total number of chosen cities from regions. Furthermore,

γreg.

are similar

days in a speciﬁed region in Equation (47).

This paper assumes that if variance of temperatures among tar and

Sd

is more than a determined

value

tem pds

, then this day is overlooked in

Sd

. Moreover, two days having huge differences in

temperature can depict different curve shapes of load demand. Likewise, this difference can cause

critical impact on knowledge based short-term load forecasting. In addition, this paper also employs

the priority index to the historical data and thus, speciﬁes similar days. Equations (44) and (45) have

signiﬁcant worth in this paper. The impact of temperature can be measured in an efﬁcient way from

these equations by dividing the PNPN. The next section speciﬁes this phenomenon.

4.2.3. Distribution of PNPN

The selection of exclusive temperature for huge topographical states usually affects the results

in short-term load forecasting. Therefore, an exclusive temperature could not be given to a huge

topographical state or zone in order to attain satisfactory forecasted outcomes. However, it is practical

to give an exclusive temperature to every region when the entire region is distributed. The distribution

of vast topographical zones has been observed in [

75

,

76

]. Nevertheless, these studies overlooked

priority index for similar day selection.

The paper distributes the region separately and then predicts the short-term load by consideration

of the proposed priority index for

Sd

selection. Furthermore, the forecasting of short-term load for

the entire system can be achieved by summation of predicted results from all regions. Besides, this

technique takes the temperature for

Sd

selection knowledge based load forecasting in an efﬁcient way.

4.2.4. Proposed Strategy

The similar days are computed by Equation (45) for every respective region. Subsequently,

Γds1

tar,H

and

Γds2

tar,H

are computed. Moreover,

Γtar,H

is attained as ultimate forecasting for every region by

selected similar days, according to Equation (44). The results obtained from all regions are combined

to achieve ﬁnal forecasted load for the entire system.

5. Application of Proposed Method on Vast Topographical Zone

This paper employs the knowledge based short-term load forecasting model on a vast

topographical region. Moreover, this paper has selected regions of Pakistan for implementation

of the proposed model. Pakistan has four seasons and different climates with signiﬁcant discrepancies

throughout the year. PNPN is a huge topographical system, which is distributed in nine regions that

are equivalent to regional electric utilities. The primary objective of PNPN in this study is to forecast

the demand load for every region. In addition, Figure 3presents the different colored portions along

with the mean of regions having high temperature throughout the year.

A city is selected from every region that is supposed to be the representative of the region.

Moreover, a city also speciﬁes the temperature of that particular region. There is no restriction on any

system to distribute into speciﬁed number of regions. However, the system can be divided according

Electronics 2018,7, 431 19 of 34

to the requirement of the system and ﬂuctuating behavior of weather. Figure 4depicts the changing

behavior of temperature for Lahore city as a sample.

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

2004

2005

1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31 1 10 20 31

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

Day

Hour Commencing

−10 0 10 20 30

Hourly Temperature (° C)

Figure 3. Heatmap and yearly weather conditions of sample region.

Date

0

5

10

15

2007−01 2007−07 2008−01 2008−07 2009−01 2009−07 2010−01

Wind speed (mph)

40

60

80

100 Humidity (%)

0

5

10

Rainfall (mm/day, averaged over a week)

960

980

1000

1020

1040 Air pressure (mb)

0

5

10

15

20

25 Outside temperature (°C)

Raw Data Smoothed Curve Median Value

Figure 4. Variations of temperature for average mid-day weather of Lahore.

Electronics 2018,7, 431 20 of 34

The investigation of PNPN demands more scrutiny of Pakistan’s user consumption behavioral

analytics. Monday is the ﬁrst working day of the week while Sunday is the last one. Moreover,

the seven days of the week are categorized into four types in Pakistan. The ﬁrst category of the day is

Monday, which is the ﬁrst working day in Pakistan. Monday has different power demand provisions,

especially in early morning (peak-hours). Furthermore, the days from Tuesday to Friday that are also

considered week-days in Pakistan, show the same load curve. The difference between Monday and

other days of the week is illustrated in Figure 5.

5000

6000

7000

8000

0 5 10 15 20 25

Hour (h)

P (MW)

Monday Tuesday

Figure 5.

Fluctuating Behavior of Load Curve in Pakistan and Difference of Monday and a

Sample Week-day.

Subsequently, another category of day is Friday and Saturday. In this category of days, the

operational hours of most workplaces and factories show a ﬂuctuating behavior in contradiction to

other week-days. Moreover, Sunday is supposed to be the rest day in Pakistan and is the last category

of day. The load curve and load demand depict an entire variating behavior from other categories of

day. Figure 6shows the ﬂuctuating behavior of load curve for a successive week.

−20 0 20 40 60

Hour (h)

P (MW)

Mon Tues Wed Thurs Fri Sat Sun

Figure 6. Fluctuating behavior of load curve in Pakistan of a particular week.

Electronics 2018,7, 431 21 of 34

The paper scrutinizes hourly load for nine regions of PNPN. In this regard, the data form the

duration of June 2015 to May 2017 is used as historic data for short-term load forecasting. Besides,

the paper predicts the load demand for the duration of June 2017 to May 2018. A city is chosen from

every region as a representative of that particular region. It is observed in the literature that there is no

concept of splitting the data-set into training and test data in knowledge based systems. Moreover,

the knowledge based systems use the entire historic data for choosing the best optimum results and

similar days as discussed in Section II. However, the data-sets are divided into training and test data

in DBN and F-LOLIMOT. This paper labels 77% of the data as training data and the remaining 23% of

the data as test data.

This paper performs sensitivity analysis on the PNPN and concludes that the optimal values

achieved for

Dγ

,

W1

, and

W2

are 8, 0.4, and 0.6, respectively. The sensitivity analysis is performed

by means of historic data for the duration of June 2015 to May 2018 in order to get the best optimum

parameter values. Moreover, the data for the duration of June 2017 to May 2018 is not utilized to get

the best optimum parameter values. The load demand for the speciﬁed time period of previous data

like from the duration of June 2016 to May 2017 is supposed to be the vital goal of prediction by the

load information and earlier than that period. This helps in selecting the best optimum parameter

values. The best optimal value is achieved when it has least prediction error for the speciﬁed period as

discussed above. The value of W1is changing from 0 to 1. Therefore, it is now obvious that the value

of W2will be calculated by W2=1−W1.

In addition, the best optimum values of

W1

and

W2

are evaluated by the scrutiny of the historic

data. Besides, data for the duration of June 2017 to May 2018 is not used in this analysis as this data is

for prediction purposes. Likewise, the value of

Dγ

is also attained from this method. This constraint

shows a ﬂuctuating behavior to achieve the least predicting error for a particular time spell. Table 2

presents the prediction error for every execution. In this table, the values of

W1

and

W2

show a variance

between 0 and 1. Nonetheless, the value of

Dγ

lies between 5 and 15. The best optimum values for

Dγ

,

W1and W2are 8, 0.4, and 0.6, respectively.

Table 2. Mean Absolute Percentage Error (MAPE) for every pair of Dγand W1for training data.

DγW1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

51.430 1.722 1.322 1.517 1.113 1.321 1.612 1.421 1.117 1.220 1.312

61.128 1.787 1.316 1.501 1.119 1.313 1.611 1.417 1.113 1.216 1.307

71.418 1.712 1.312 1.509 1.102 1.325 1.609 1.415 1.111 1.215 1.305

81.418 1.711 1.321 1.507 1.100 1.303 1.615 1.420 1.119 1.217 1.311

91.418 1.713 1.217 1.599 1.102 1.307 1.617 1.425 1.123 1.206 1.315

10 1.419 1.715 1.311 1.503 1.105 1.311 1.621 1.430 1.125 1.213 1.321

11 1.431 1.715 1.311 1.505 1.106 1.312 1.622 1.433 1.130 1.219 1.320

12 1.491 1.710 1.331 1.501 1.107 1.315 1.625 1.432 1.132 1.220 1.320

13 1.431 1.713 1.360 1.599 1.108 1.324 1.629 1.435 1.131 1.223 1.319

14 1.472 1.721 1.366 1.502 1.109 1.327 1.630 1.440 1.131 1.227 1.326

15 1.414 1.789 1.363 1.503 1.111 1.328 1.631 1.441 1.134 1.228 1.329

This paper further assumes that the proposed methodology employs the similar day load demand

data in the preceding years for the distinct days like public and religious holidays. This is done because

there is an inadequacy in the historic data. Therefore, the technique of priority index is not applicable

for distinct days. Consequently, it is one of the major reasons to observe the effect of temperature in

the priority index for normal days instead of distinct days.

The paper only lays emphasis on the short-term forecasting for normal days. Moreover, the

distinct days are overlooked from record for selection of similar day. Besides, the paper explains the

knowledge based short-term forecasting for Tuesday, 28 June 2016.

Electronics 2018,7, 431 22 of 34

1.

At ﬁrst, the days having a similar category of day are chosen on the basis of categorization of

target day. In this scenario, Tuesday is included in the second category of day classiﬁcation as

discussed above. Moreover, all the days between Tuesday to Friday are selected. However, all the

distinct days is overlooked for analytical purposes. Subsequently, these days are distributed in

two data-sets, as discussed in Section II.

2.

The priority index of every region is evaluated by Equation (45), for all chosen days. Table 3

presents the priority index of selected days for a sample region Islamabad as an example of

30 June 2015. Moreover, in this scenario the value of

℘tem p,reg.

is 0.03 and

℘1,reg.

is 1.5

×

10

−5

.

All the values and Table 3are associated with the second data-set of Islamabad for the speciﬁed

date. Every region and every data-set are different from one another.

Table 3. Selection of similar days on the basis of priority index values for 28 June 2016.

Date Day Difference of Temperature Proximity of Date Index Value

4 June 2015 Thursday 1 371 0.1393

7 June 2015 Sunday 0 366 0.1282

10 June 2015 Wednesday 2 337 0.2859

11 June 2015 Thursday −1 332 0.2747

15 June 2015 Monday −2 266 0.1549

16 June 2015 Tuesday −5 265 0.3295

17 June 2015 Wednesday −3 264 0.3791

19 June 2015 Friday −4 263 0.3795

24 June 2016 Friday −1 4 0.0212

25 June 2016 Saturday −3 7 0.5701

26 June 2016 Sunday −2 6 0.1210

27 June 2016 Monday 0 5 0.0021

3.

The priority index and short-term load forecasting of every region is evaluated by Equations (42)

and (43) as discussed in Section II. In this scenario,

Dγ

and ﬁnal best suitable chosen similar days

are 25 June 2016, 26 June 2016, 27 June 2016, 4 June 2015, and 7 June 2015 in Islamabad. Moreover,

Table 3depicts that few same days show less difference in temperature rather than choosing same

days. However, they are overlooked in this paper as along with the difference in temperature,

the proximity of date has also signiﬁcant worth. For instance, 10 June 2015 and 11 June 2015

will have less difference in temperature as compared to 15 June 2015. However, such days are

neglected because they have maximum values of date proximity. Therefore, this paper can choose

a similar day that has maximum difference in temperature in the proposed methodology because

of proximities in date. Moreover, this phenomenon can produce more similar load curve shapes.

Besides, the same chosen days in Islamabad and other regions can cause a discrepancy in selecting

the same days from Islamabad for prediction of 28 June 2016.

4.

The predicted demand load of the entire system is combined load that is obtained from all regions

after short-term load forecasting is done for every respective region.

5.1. Deep Belief Network

In [

77

], the basis of DBN is presented brieﬂy. Moreover, the auto-correlation of load demand

data has been depicted in Figures 7–10 for the previous data. It is obvious from the auto-correlation

plots that the preceding data is more auto-correlated to experimental data, to some extent. This paper

performs Ljung Box [

78

] analysis of null supposition to check this assumption more quantitively.

The suppositions are as follows:

•S0

: The preceding data are disseminated autonomously, i.e., the correlation is 0 in the preceding

data from where the sample is chosen. Therefore, any experimental correlations in the preceding

data are the resultant from the unpredictability of the test group.

•S1

: The preceding data are not disseminated autonomously, i.e., the data show serial correlation.

Electronics 2018,7, 431 23 of 34

The auto-correlations tests are performed whose outcomes are shown in Table 4.

0 10 20 30 40

0.00 0.01 0.02 0.03 0.04 0.05

Original data: (0,1)

Figure 7.

Auto-correlation of preceding demand load data for day lags in deep belief network (DBN)

for original data (0, 1).

0 10 20 30 40

0.00 0.01 0.02 0.03 0.04 0.05

Resampled data: (0,1)

Figure 8.

Auto-correlation of preceding demand load data for day lags in DBN for original data (0, 1).

0 10 20 30 40

0.00 0.01 0.02 0.03 0.04 0.05

Original data: (1,2)

Figure 9.

Auto-correlation of preceding demand load data for day lags in DBN for original data (1, 2).

Electronics 2018,7, 431 24 of 34

0 10 20 30 40

0.00 0.01 0.02 0.03 0.04 0.05

Resampled data: (1,2)

Figure 10.

Auto-correlation of preceding demand load data for day lags in DBN for resampled

data (1,2).

Table 4. ρValues of the Ljung Box auto-correlation test with different region values.

Original Data Experimental Data Region Size

(0, 1) 1.00 ×10−70.5510981 8175

(0, 2) 6.75×10−40.6528330 14,798

(1, 1) 0.00×1000.4384530 16,856

(1, 2) 0.00×1000.7561250 15,087

The outcomes show that the preceding data is much more auto-correlated as compared to the

experimental data. It is often observed in literature that numerous testing process reject the

S0

for the

preceding data. However,

S0

is not rejected by experimental data. Therefore, there subsists a spatial

correlation in preceding data. Moreover, if sampling techniques are applied on the historic data then

this correlation can be disintegrated. The paper also performs sensitivity analysis and the structure of

DBN used for this paper includes one hidden layer with ﬁve neurons. Moreover, there are 25 neurons

are in input layer and 20 neurons in the output layer in the proposed architecture. These neurons

generate the prediction of load demand for the target day (24 h). On the topic of architecture of this

network, the input layer is comprised of two constraints for mean and maximum temperature for

selected day. Moreover, one constraint is for categorization of the forecasted day while the remaining

22 input constraints are associated with the preceding load demand data, which are as follows:

$τ=Γτ

m−Γτ

n

Γτ

n

. (48)

In Equation (48),

$τ

represents the total load demand data,

Γτ

m

and

Γτ

n

are demand load for

τth

hour (

τ=

1, 2, 3,

· · ·

, 24) preceding to selected day. This paper assumes that

Γτ

m

and

Γτ

n

represents the

τ

and

τ

-1 hours in Equation (48). Moreover, there are 20 neurons in the output (

OS τ

) layer of DBN

that signiﬁes the difference of load demand on the hourly basis for preceding and selected days,

OS τ=Γτ

tar −Γτ

n

Γτ

n

. (49)

The categorization of days in DBN are entirely divergent from knowledge based system.

According to Equations (48) and (49), Tuesday must be taken apart from days that range from

Wednesday to Friday. Therefore, in DBN ﬁve categories of days are taken for analysis.

Electronics 2018,7, 431 25 of 34

5.2. Fuzzy Local Linear Model Tree Algorithm

The paper employs F-LOLIMOT algorithm for training of the linear fuzzy model. The explanatory

analysis of F-LOLIMOT algorithm has been discussed in detail in [

79

]. Moreover, the F-LOLIMOT

algorithm is capable of predicting the hourly demand load, which is ahead than the current time

by means of climatic and load data. Figure 11 depicts that there are different inputs and outputs of

demand load and climatic data. This is done after sensitivity analysis on the system.

Furthermore, the lags of climate are the climatic condition of the preceding week and target

day. Likewise, the time lags of each hour load demand (inputs) are actually demand load data of

similar hour at preceding 9 and 10 days earlier than selected hour. It is obvious that the initial hour of

target day by utilizing preceding and recognized load data the upcoming hourly load is forecasted by

preceding data.

−1.0

−0.5

0.0

0.5

1.0

0 10 20 30

Day Lag

Autocorrelation

Figure 11.

Auto-correlation of preceding demand load data for day lags in Fuzzy Local Linear Model

Tree (F-LOLIMOT).

6. Results and Discussion

At ﬁrst, this section presents the evaluational measures that are used in this paper. Subsequently,

the results are discussed.

6.1. Evaluational Measures

In literature, Daily Maximum Error (DME), Maximum Distance Minimum Error (MDME),

and MAPE have been widely used in order to valuate the outcomes, which are achieved from

short-term forecasting. This paper has used MAPE, MDME and DME as:

MAPE =D−1

f

H=1

∑

Df

Γtar,H−Γν,H

Γν,H

(50)

and

DME =max(Γtar,H−Γν,H

Γν,H

). (51)

In Equations (50) and (51),

Df

are the hours that are forecasted and

Γν,H

is the real demand load at

speciﬁed hour

H

of tar day. This paper presents 4 implications to indicate the beneﬁts of the proposed

system. The implications are based on the forecasting of load demand for the duration of June 2015 to

June 2016. Moreover, these implications are made by climatic and load data, which lies in the range of

June 2015 to exactly one day before the target day. The paper takes this data as training data in this

scenario. The implications are:

Electronics 2018,7, 431 26 of 34

1. MAPE of short-term load forecasting throughout the year (Df= 9750)

2.

Average of DME throughout the year, which is referred as maximum distance and minimum error

3. Total number of days, which have MAPE higher than 3% (=3)

4. Total number of days, which have maximum error higher than 5% (=5)

The last two implications depict the division of errors, which are achieved from the results of

short-term load forecasting. In this paper, the proposed model minimizes the total number of exceeding

days from a certain limit and also enhances the performance of MAPE and DME.

6.2. Discussion of Results

The paper has evaluated the results on the basis of two assumed evaluations that are discussed

as follows.

6.2.1. Evaluation of Priority Index and Splitting Consequences on Knowledge Based Systems

This paper implements the proposed method on PNPN. In this regard, the following cases are

observed to discuss the consequences, which are associated with distribution of the forecasting results

and taking temperature in priority index.

1.

Case 1: Short-term load forecasting of PNPN without taking temperature and distribution of data

2.

Case 2: Short-term load forecasting of PNPN including consequences of data distribution without

taking the temperature

3.

Case 3: Short-term load forecasting of PNPN including including temperature without taking the

consequences of data distribution

4. Case 4: Short-term load forecasting of PNPN with temperature and distribution of data

The data distribution is overlooked in Case 1. Therefore, a distinctive temperature is not suitable

for the system. Moreover, the priority index is the center of attention in this case along with the date

proximity. Besides, the whole system is distributed in different sections in Case 2. Subsequently,

the prediction is performed for every respective section. The prediction of the entire system is a

combination of predicting outcomes in all sections. Case 2 differs from Case 1 as the data distribution

is carried out in this scenario. Nonetheless, the data distribution is also overlooked in Case 3. However,

the temperature is taken in consideration in terms of subjective average values in relation to the

demand from every region. The consequences of temperature are studied in Case 4. The paper

assumes Case 4 as a comprehensive case as it takes temperature in the priority index for selection of

similar days from every section.

Tables 5and 6presents the outcomes of the aforementioned cases for every category of the day.

It can be observed that MAPE of the entire system is minimum in Case 4 as compared to other cases.

The data distribution is done in Case 2 and Case 1 has overlooked this phenomenon. Thus, it is

proved that distributing the entire system can enhance the prediction outcomes. Moreover, the data

distribution also minimizes the MAPE and maximum distance and minimum error. Besides, the data

distribution among different regions minimizes the total number of days that go beyond acceptable

measures (=3and =5).

Table 5.

Consequences of priority index and data distribution on the forecasting results for

=5

and

=3

.

=5=3

Nature of Days Case 1 Case 2 Case 3 Case 4 Case 1 Case 2 Case 3 Case 4

Weekdays 10 8 7 65 4 6 8

Weekend 3 4 4 31 1 2 1

Yearly Mean 25 26 20 19 20 13 11 12

Electronics 2018,7, 431 27 of 34

Table 6.

Consequences of priority index and data distribution on the forecasting results for Maximum

Distance Minimum Error (MDME) and MAPE.

MDME MAPE

Nature of Days Case 1 Case 2 Case 3 Case 4 Case 1 Case 2 Case 3 Case 4

Weekdays 3.70 3.65 2.95 2.58 1.23 1.33 1.25 1.29

Weekend 2.96 2.92 2.75 2.49 1.19 1.15 1.17 1.01

Yearly Mean 2.70 2.26 2.51 2.24 1.09 1.07 1.03 1.02

The consideration of temperature devoid of distributing the data in different regions is responsible

for reduction in valuation constraints when associated with Case 1 and Case 2. Nevertheless, the MAPE

of Case 4 is enhanced as compared to Case 3. Moreover, Case 4 has minimum days with maximum

error that is larger than 4%. Contrariwise, forecasting results are improved in Case 4 as it distributes

the data in different regions and takes temperature in priority index. The MAPE in Case 4 is 1.02 %

as depicted in Tables 5and 6. This achieved MAPE is approximately 8% improved than Case 2 and

almost 9% enhanced than Case 1. Besides, Case 4 has the total number of optimum days that exceeds

the acceptable criteria. The results achieved for minimum days with maximum error and

=5

are also

enhanced in Case 4 as compared to Case 3. Nonetheless,

=3

has achieved enhanced results in Case 3 in

comparison with Case 4.

Table 7presents

Dγ

and

W1

for target year. The optimum result achieved is for

Dγ

= 7 and

W1

= 0.3. The results achieved for

W1

= 0.3 are approximately near to

W1

= 0.4. Thus, the

achieved parameters from training data can give suitable outcomes and are proven appropriate

for the proposed method.

Table 7. MAPE for every pair of Dγand W1for target data.

DγW1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

51.115 1.161 1.089 1.075 1.096 1.049 1.078 1.088 1.117 1.121 1.125

61.015 1.029 1.021 1.017 1.019 1.022 1.026 1.031 1.040 1.045 1.050

71.043 1.012 1.011 1.009 1.016 1.025 1.043 1.045 1.052 1.105 1.106

81.301 1.318 1.313 1.314 1.321 1.325 1.329 1.342 1.378 1.389 1.391

91.208 1.219 1.217 1.216 1.223 1.234 1.249 1.265 1.290 1.301 1.315

10 1.305 1.315 1.321 1.311 1.326 1.336 1.349 1.367 1.387 1.403 1.421

11 1.308 1.329 1.325 1.326 1.331 1.341 1.352 1.353 1.376 1.391 1.415

12 1.309 1.301 1.327 1.328 1.345 1.347 1.358 1.367 1.395 1.412 1.428

13 1.309 1.302 1.324 1.331 1.337 1.348 1.362 1.381 1.413 1.426 1.443

14 1.403 1.436 1.431 1.435 1.443 1.453 1.466 1.487 1.503 1.529 1.525

15 1.404 1.414 1.416 1.423 1.439 1.465 1.494 1.511 1.534 1.529 1.549

6.2.2. Evaluation of Consequences on knowledge Based Systems from Preceding Data

The preceding data is categorized in two different sets as discussed in Section II. The paper studies

three cases in this subsection to depict the consequences of this type of categorization.

1. Case 1: Load forecasting by collected similar days in initial data-set, Γds1

tar,H

2. Case 2: Load forecasting by collected similar days in last data-set, Γds2

tar,H

3. Case 3: Load forecasting by Γds1

tar,Hand Γds2

tar,H, i.e., Γt ar,H=⇒Equation (44)

Table 8presents the outcomes of Case 1, Case 2, and Case 3 for every category of the day. Table 8

presents the outcomes of Case 1, Case 2, and Case 3 for every category of the day.

Electronics 2018,7, 431 28 of 34

Table 8. Consequences of taking Γds1

tar,Hand Γds2

tar,Hon forecasting.

=5=3MDME MAPE

Nature of Days Case 1 Case 2 Case 3 Case 1 Case 2 Case 3 Case 1 Case 2 Case 3 Case 1 Case 2 Case 3

Weekdays 10 13 78 7 73.21 2.73 2.71 1.81 1.52 1.26

Weekend 5 3 22 1 12.89 2.35 2.35 1.09 1.23 1.17

Yearly Mean 39 16 16 18 14 13 2.68 2.24 2.24 1.31 1.10 1.03

The MAPE of the entire system in Case 1 is maximum as compared to Case 2 and Case 3 in case

of

Γds1

tar,H

. Thus, it can be concluded that taking same days from selected or last month gives maximum

errors in forecasting results. Moreover, the total number of exceeding days from acceptable conditions

is not suitable, particularly

=5

. Nevertheless, integration of

Γds1

tar,H

and

Γds2

tar,H

gives enhanced results for

MAPE and minimum days with maximum error in Case 3. Furthermore, passed days from acceptable

conditions is lessened in Case 3.

Figures 12–14 depicts the comparative analysis of traditional and proposed forecasting method.

According to Figures 12–14, the days presented are four different days and these days belong to

dissimilar months. The predicted outcomes are then associated with real load demand. Moreover,

the results of the proposed system are much nearer to real load as compared to traditional

forecasting techniques.

2.4

2.7

3.0

3.3

0 5 10 15 20 25

Hour (h)

Load (MW)

Actual.Load Classic.Forecasting Modified.Forecasting

Figure 12.

Comparative analysis and effect of proposed and traditional method for Monday,

19 September 2015.

2.7

3.0

3.3

3.6

3.9

0 5 10 15 20 25

Hour (h)

Load (MW)

Actual.Load Classic.Forecasting Modified.Forecasting

Figure 13.

Comparative analysis and effect of proposed and traditional method for Wednesday,

13 June 2015.

Electronics 2018,7, 431 29 of 34

1.8

2.0

2.2

2.4

0 5 10 15 20 25

Hour (h)

Load (MW)

Actual.Load Classic.Forecasting Modified.Forecasting

Figure 14.

Comparative analysis and effect of proposed and traditional method for Sunday,

4 January 2015.

6.3. Comparative Analysis of Proposed Method, DBN, and F-LOLIMOT

The paper compares the results achieved from proposed knowledge based system with DBN and

F-LOLIMOT. The results are evaluated in terms of precision and operational time. The short-term load

predicting techniques is applied on PNPN to forecast the load demand for the duration of June 2017

to May 2018. Moreover, these predictions are based on temperature and load demand data, which

lies in the range of June 2015 to exactly one day before the target day. The results are presented in

Table 9, which shows that proposed knowledge based system has enhanced MAPE to 1.01. Besides,

the MAPE of

=5

and

=5

is also decreased. The DBN and F-LOLIMOT techniques show MAPE is

approximately higher than 3% for a month and approximately 5% greater in 47–50 days (maximum

error). Nonetheless, the proposed method has MAPE, which is greater than 3% in 15–18 days and 5%

with 23 days (maximum error). The variances discussed are notable enhancements in forecasting.

Table 9.

Comparison of Fuzzy Local Linear Model (F-LOLIMOT), deep belief network (DBN), and

proposed method.

Operational Time (s)

Technique =5=3MDME MAPE Training Time Executing Time

Proposed 17 10 2.83 1.10 15 0.41

DBN 50 42 2.89 1.21 29 0.52

F-LOLIMOT 42 35 3.43 1.50 215 0.81

On the topic of operational cost, the proposed knowledge based method takes minimum time

in training and executing in comparison with DBN and F-LOLIMOT. The proposed knowledge

based system, DBN, and F-LOLIMOT are executed to predict the days on a yearly basis. Besides,

the operational time is distributed to total number of predicted days in order to get the usual operational

time of prediction for a speciﬁed day. Moreover, the proposed system, DBN, and F-LOLIMOT are

executed with the same conditions. Besides, the parameters were tuned for every speciﬁed day

and forecasted demand load has been achieved for every technique. The paper distributes the day,

according to training and operational time in every technique. The proposed knowledge base systems

have less operational time as it does not require as much training as compared to DBN and F-LOLIMOT.

The proposed method lays emphasis on the selection of similar day and then predicts the load demand

as discussed above.

The forecasting of sample day is presented in Figures 15 and 16 by means of DBN, F-LOLIMOT,

and proposed knowledge based system. It is obvious that MAPE of the proposed method is 0.69

Electronics 2018,7, 431 30 of 34

for a sample day. This MAPE is lesser than MAPEs of DBN and F-LOLIMOT, which are 0.91 and

0.97 respectively. Moreover, the DME is minimized in the presented knowledge based system as

compared to others. The phenomenon of priority index is not suitable for special days (public holidays)

as discussed in earlier sections. Nevertheless, the special days can be forecasted by the presented

knowledge based system devoid of taking a priority index. Besides, the MAPE of the proposed system

is 1.30 for all days, together with special days. Nonetheless, the major aim of this paper is to study the

consequences of the priority index on the knowledge based system. Moreover, the scrutiny of special

days is beyond the scope of this paper.

2.50

2.75

3.00

3.25

3.50

0 5 10 15 20 25

Hour (h)

System Load (MW)

Actual.Load Similar.Day DBN F.LOLIMOT

Figure 15. Short-term load forecasting for a sample day.

0

1

2

3

4

0 5 10 15 20 25

Hour (h)

Error

Similar_Day DBN F_LOLIMOT

Figure 16. Error values for a sample day.

7. Conclusions

This paper presents a novel knowledge based short-term load forecasting method. The entire

system (region) is distributed in nine sub-systems (zones) by consideration of temperature to predict

the demand load more efﬁciently. The outcomes depict that distribution of huge topographical power

network improves the forecasting results. Moreover, this paper presents a novel priority index in which

climatic conditions and the date proximity of every particular region is observed. The algorithms of

AP and BFFA are hybridized in this paper to achieve better accuracy for a knowledge based system.

The proposed knowledge based system is veriﬁed on PNPN. The achieved outcomes depict that

proposed method minimizes the MAPE and other errors of forecasting in comparison with traditional

Electronics 2018,7, 431 31 of 34

forecasting techniques. Furthermore, the obtained results from proposed system are 15–20% improved

as compared to DBN and F-LOLIMOT techniques. Furthermore, this paper deﬁnes two standard

measures for error distribution. The outcomes verify that the total amount of exceeded days is reduced

through proposing knowledge based systems from acceptable criteria. This phenomenon speciﬁes

more efﬁcient forecasting results as compared to DBN, F-LOLIMOT, and traditional knowledge

based systems.

Author Contributions:

All authors have contributed to this paper with the same effort in ﬁnding available

literature, resources and writing the paper. Moreover, all authors have read and approved the ﬁnal manuscript.

Funding: This research received no external funding.

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

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