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;
firstname.lastname@example.org (M.K.); email@example.com (S.); firstname.lastname@example.org (M.A.);
2Department of Electrical Engineering, COMSATS University Islamabad, Islamabad 44000, Pakistan;
3Department of Computer Science and Information Technology, Alhamd Islamic University,
Islamabad 44000, Pakistan; email@example.com
4Department of Computer Science, Islamic International University, Islamabad 44000, Pakistan;
5Department of Computer Science, Air University, Islamabad 44000, Pakistan; firstname.lastname@example.org
Received: 15 November 2018; Accepted: 7 December 2018; Published: 12 December 2018
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
behavioral analytics; Stackelberg game; demand response; knowledge based systems;
priority index; similar day; date proximity.
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 [
]. 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 [
]. 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 [
]. 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 .
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 [
]. 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 [
]. 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 [
]. 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 [
]. 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
]. The optimal power management and maintaining balance between demand and supply are
considered as challenging tasks for modern power systems [
]. Moreover, the prediction of uncertain
production of renewable energy resources [
] and short-term load forecasting [
] 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 [
]. 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 [
]. 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 [
] 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
], 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 [
]. Likewise, the inﬂuence of developmental ﬂuctuations for energy savings was
observed by [
]. 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 [
exponential smoothing [
], regression based [
], neural networks [
], and others. Moreover, every
proposed model has incorporated some techniques. For example, regression based processes are
usually comprised of Autoregressive Integrated Moving Average (ARIMA) [
Moving Average (ARMA) [
], Support Vector Regression (SVR) [
], and Auto-Regressive Moving
Average with Exogenous variable (ARMAX) [
]. 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
In literature, there are some works cited in knowledge based systems that employ a similar day
]. Although, there is a lot of room for enhancement in this scenario which can be studied.
The authors in [
] 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 .
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 .
The authors in [
] 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%
]. Besides, the consequences of temperature, wind pressure and humidity, was scrutinized in [
The MAPE calculated in this study was 1.43%. The study in [
] was almost equivalent to the proposed
model presented in [
]. Moreover, the MAPE achieved in this study was between 1.23% to 3.35% in
seven different states of America .
The mean prediction error for daily peak load in [
] was achieved 4.65% for weekdays and
7.08% for weekends of three different states of Turkey [
]. 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 [
] in terms of priority index. Moreover, in [
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 [
] 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 [
] 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 [
] and [
] 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 [
] 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
At present, Artiﬁcial Neural Network (ANN) and SVM are considered to work efﬁciently for
non-linear time series sequences. Karatasou et al. [
] demonstrated the practical implementation
of ANN in forecasting power expenditure of a building accompanied by statistical study. In [
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. [
] 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. [
] suggested Least Square Support Vector Machine (LS-SVM) for
chilling load prediction [
] for a residential zone in Singapore. The forecasting was done by hourly
Wang et al. [
] 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 [
]. In addition, a number of models
are presented in literature to tune the parameters of SVM by techniques of machine learning and
Ogliari et al. [
] proposed a hybrid model using Neural Network and Genetical Swarm
Optimization for energy prediction. The authors in [
] 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 [
] 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 [
], 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 [
] 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 [
] 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 [
] in order to increase the utility of the end-user
by hourly prediction. The study presented in [
] 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 [
this load was handled by different utilities and DRM was scrutinized for bi-directional communication
between consumers in the micro-grid. The authors in [
] and [
] 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
3. Game Theoretical Problem Formulation
This study takes nconsumers and
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
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.
and nhave established a two way communication using the advanced metering
infrastructure for pricing swapping and information sharing. Conversely,
can also communicate
with one another. The ncollect the value (cost) facts from the
. In return, the
then provide their
services to n.
Power initiation, dissemination, and expenditure can be divided in three ways [
, and n. This paper emphasizes the communication between nand
this paper assumes that
show a ﬂuctuating behavior at the business level. Inspired from the game
theory models, the
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
have no control. Moreover, the
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
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.
UC Utility Companies
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
|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
n0Proposed best scheme for n
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
. In this regard,
game theory offers an ordinary pattern to represent the activities of nand
. Consequently, the
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
. In this case,
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
and nby a Stackelberg
]. The proposed game model takes the
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
is the request of consumer
from a utility
. Hence, the value of a consumer
n0,Ccons,n0can be expressed as:
ln(dn,uc0+τdn0).∴∀uc0∈ UC (1)
are constants. Also, the ln function is extensively employed in literature for user
making decisions [
]. The valuable function used for consumer
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
is used regarding
, such that,
= 0. When
are equivalent to 0, then beneﬁt of
begin to be ﬁnite. Generally, the representative cost of τn0= 1.
is the per unit cost given by any utility company
0 is the total
expenditure of any consumer
has given a distinct price rates of electrical energy
[κ0,κ1,......, κc] when n0∈n.
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)
is a convex optimization problem. Therefore, the obtained solution is distinctive
This paper considers the scrutiny accompanied by
consumers and three
s. Thus, they seek
for best optimum solution in this scenario for a speciﬁed uc0can be expressed as follows:
where κcdn0,1+κcdn0,2≤Bn0and dn0,1+dn0,2≥0.
The paper employs Lagrange multipliers (
) for the respective
and setting of
parameters as discussed above. Thus, the Equation (4) can be rewritten as:
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
κcdn,uc0−Bn0) = 0. (6)
generates Equation (6) to 0. Whereas,
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 = (υcons ∀n0∈n
). All of the nare interconnected by
∇υcons =0 shows that,
(℘υcons,n0)(℘dn,uc0)−1=0∴∀n0∈n,uc0∈ UC. (7)
Next, this paper has considered four of the cases, which the n0can avail.
Electronics 2018,7, 431 9 of 34
are greater than 0, then
Λn0,2 =Λn0,3 =
0. So, Equations (8) and (9) are
where n0∈nand uc0=1, 2, ..., n. Now, using Equation (6) in Equation (10),
Thus, Equation (11) becomes Equation (12) after simpliﬁcation.
Here value of uc0varies; i.e., 1, 2, or 3.
are equivalent to 0, then
. As discussed
corresponds to 0. This paper derives Equation (13) by considering the cost of the
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),
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).
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.
both are equivalent to 0, then
are real and positive values.
It is noted that Case 4 is assumed as best case which rarely occurs only when
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
utilities that satisﬁes the equality conditions in previous cases for a given set of
So, Equations (12), (16), and (17) can be combined in the above discussed scenario as:
κuc0+Bn0−κuc0U C τn0. (18)
In Equation (18), dn0,uc0≥0, n0∈nand uc0∈ UC. As dn0,uc0≥0. So,
κ$) + Bn0>τn0κuc0(UC − 1). (19)
3.1.2. Analysis of Utility Companies
This study assumes that
UC ∈ uc0
) depicts the available electrical energy of
. The aim
is to vend the energy to gain maximum proﬁt. For instance, if there is only one
then this ﬁrm will settle the price range according to its ease as there is no competition involved.
However, this study takes two basic strategies that decide the cost range of any
. Firstly, it can be
the economical conditions of average consumers and secondly, it could be an aspect of competitiveness
. Furthermore, the
also take part in choosing the best optimum cost (game) with
another. Additionally, this study expresses the maximum proﬁt Eprod,uc0of any U C as:
Eprod,uc0(κuc0,ξuc0+1) = κuc0∑
is cost of
. Thus, the best optimum solution for any
related in terms of OP pro d and can be expressed as:
ξ=max(κuc0)Eprod,uc0(ˇuc0,¸uc0+1),∴∀uc0∈ U C (21)
∀U C ∈ uc0
. The maximum proﬁt of any
is ﬂuctuating in
relation to energy for a constant
. According to Equation (20), this phenomenon leads to parameters
of equality. Every
proffers to vend all its energy to consumers. This paper assumes
resolve OP prod by:
The best optimal solution for the U C furthers presents ℘υprod,uc0/℘κuc0, which is equivalent to 0.
uc0ρ(U C − 1)−ζuc0(ρ∑
+Bn0) = 0. (23)
. Moreover, the conditions used in Equations (21) and (22)
equations. Now, solving these three
, this study sets
U C ]
U C ]
can be evaluated by means of
employing Equation (18) for uc0,
κuc0=ρ(κ$∑$∈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
It can also be deduced from Equation (23) that
0. It refers
to the phenomenon that there is no essential need to play any game when
1. Therefore, the study
Electronics 2018,7, 431 11 of 34
merely focuses on the circumstances when
3. To handle the discussed scenario, Equation (23)
can now be computed as:
E1+J−H · · · −H
−H E2+J· · · −H
· · · · · · · · · · · ·
−H −H −H EU C +J
κU C ]
. From the above equations,
it can be concluded that Mis an invertible matrix. However, it could be expressed as:
This paper considers some cases to achieve closed-form solution of κ.
have equivalent amount of energy available and capacity to produce, then
E3=· · · =EUC . Utilizing Equation (26),
E+J+H(1− UC)=κ. (28)
. Then the Equation (27) is used in Equation (19), so that the total demand to any
from n0is given as:
Bn0≥κτn0(UC − 1)−κτn0(UC − 1)(30)
Here, Equation (30) indicates that
0. This phenomenon indicates that now all
equivalent amount of power. Moreover, they have settled some pricing scheme that users have
Contrary to Case 1, this case considers that capacity of power generation is different for all
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| − ∑
|mi,j| ≥ 0, (31)
· · ·
. According to [
], a taut diagonal matrix is always non-singular and
positive. It is observed that
is taut and diagonal matrix as
− H(U C −
) = 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∑
is invertible; thus,
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
this paper assumes that
has increased the cost from
have same cost of
power generation. From Equation 19, suppose that any consumer ndemands power
any κuc0then the constraint in Equation (33) is satisﬁed.
κ$)τn0(UC − 1)−1). (33)
Now suppose that
fulﬁl the requirements of Equation (33). In this regard,
the necessities of consumers will show deviating behavior from dn0,uc0to d∗
uc0) + Bn0)
uc0U C −τn0. (34)
The differentiation among the necessities of any nfrom the ﬁrm uc0will now be expressed as:
∗(τn0(∑uc0∈UC κ$) + Bn0)
UC . (35)
From Equation (35), it is obvious that
0. Hence, the consumers are not capable
of demanding the total power generated by any
, i.e., the consumer will then demand for lesser
energy as required. Moreover, the proﬁt and cost of
will increase on the basis of consumer total
power demand. Thus, Equation (36) provides the balanced equation of demand and supply.
It is observed that in Equation (36),
. Thus, the
proﬁt gaining of
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
κuc0ma x ]
. As a matter of fact,
is owing to the
cost functions that is generated by
. Moreover, any
is not capable to lessen the price lower than
is the maximum range. According to
, the government has to settle
the cost, which consumers have to follow.
3.2. Proposed Stackelberg Game Modeling
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 .
This study assumes that
is the game-plan rectiﬁed for any
is the scheme
. Subsequently, the game-plan for
=prod ==proda× =prodb× · · · × =prod,U C
=cons ==consa× =consb× · · · × =cons,UC
is proposed Stackelberg
equilibrium for any uc0if,
Electronics 2018,7, 431 13 of 34
· · ·
is the game plan of all consumers n. Moreover, dand
best feedback of all consumers, i.e., d
. The best feedback of any consumer
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)
is supposed to be best optimum scheme for
d+and κ+is a Stackelberg equilibrium achieved for the game concerning the UC and n.
3.3. Distinctiveness of Stackelberg Equilibrium
has an exclusive maximum range (as discussed above) for
. Whenever the cost
planning game is played between the companies with a distinctive Nash Equilibrium, then the
Stackelberg game plan holds a special equilibrium.
An exclusive Nash equilibrium occurs in the cost selection game plan between
a distinctive Stackelberg equilibrium subsists as well.
Proof of Theorem 3.
There is equilibrium if
is a real value and
⊂ RU C
. On the topic of cost choosing of all
in Stackelberg game,
=prod = (=prod,1 × =prod,2 ×
· · · × =prod,U C )
κuc0⊂ =prod,U C
=prod = [κuc0,min
. Therefore, the game plan
is real value and ⊂ RU C .
is constant in
as discussed in Equation (20). Subsequently, the
f00(Eprod,uc0)according to κuc0is,
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,
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.
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.
acquire these demanding conditions from consumers. At that moment,
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).
In Equation (40), ris the repetition number and
is the rate modiﬁcation constraint of
updates its cost function, it sends this information to
. Furthermore, the
the demands and send this information back to
. Subsequently, the
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
Algorithm 1meets the best optimum solution for all
and nas the particular game plans are upgraded in a
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
will be negative in
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
11 Transmit the updated cost to n
12 Jump to 3
14 if κuc0,r+1≡κuc0then
17 Jump to 2
In Equation (40), if
0 then the signiﬁcant constraint for
not gaining a negative amount is
. 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
ampliﬁes only if
results and vice versa. However, in Equation (40), when
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
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 [
]. 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
]. 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
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 [
]. The hourly data
normalization of load demand is attained by distribution of load on hourly basis [
], which is shown
in Equation (42).
X(ΓSd,H−1,ΓSd,H−2,· · · ,ΓSd,H−n). (42)
In Equation (42),
is the load demand,
is the normalized value of data at any hour
of a similar day
is the mean of npreceding days. In addition,
1, 2, 3,
· · ·
, 24. The load
Electronics 2018,7, 431 16 of 34
demand at any hour
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:
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
is the predicted demand load for any hour
is set of identical days, and
total number of days chosen, which are similar. The minimum value of
reduces the utilized historic
data and inadequate similar days, which are selected. Contrarily, the maximum value of
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
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:
In Equation (44),
are forecasted power load demand speciﬁed for each hour
and targeted day tar. Moreover,
are weights assigned to each data-set. Thus,
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
. Furthermore, the proposed method must also execute for training
data-set in order to choose the best optimum values of
. Subsequently, the proposed method
should be proﬁcient enough to select the execution, which gives the least prediction error. Besides,
the values of
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.
0 5 10 15 20 25
System Load (MW)
Winter Autumn Spring Summer
Variations in load behavior of sample Thursday during 2015 of Pakistan’s National Power
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:
In Equation (45),
is the priority index of
in speciﬁc region,
the average temperatures of a speciﬁed city
on the daily basis for a similar day
and tar days,
is total number of days between tar days and
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,
is considered as weighting factor of temperature, while
is taken as
weighting factor of ηreg.. They are calculated as follows:
In Equation (46),
is total number of chosen cities from regions. Furthermore,
days in a speciﬁed region in Equation (47).
This paper assumes that if variance of temperatures among tar and
is more than a determined
, then this day is overlooked in
. 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 [
]. 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
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
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,
are computed. Moreover,
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.
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
−10 0 10 20 30
Hourly Temperature (° C)
Figure 3. Heatmap and yearly weather conditions of sample region.
2007−01 2007−07 2008−01 2008−07 2009−01 2009−07 2010−01
Wind speed (mph)
100 Humidity (%)
Rainfall (mm/day, averaged over a week)
1040 Air pressure (mb)
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.
0 5 10 15 20 25
Fluctuating Behavior of Load Curve in Pakistan and Difference of Monday and a
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
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
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
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
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
show a variance
between 0 and 1. Nonetheless, the value of
lies between 5 and 15. The best optimum values for
W1and W2are 8, 0.4, and 0.6, respectively.
Table 2. Mean Absolute Percentage Error (MAPE) for every pair of Dγand W1for training data.
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
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.
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
is 0.03 and
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
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,
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.
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
], 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 [
] analysis of null supposition to check this assumption more quantitively.
The suppositions are as follows:
: 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.
: 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)
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)
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)
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)
Auto-correlation of preceding demand load data for day lags in DBN for resampled
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
preceding data. However,
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:
In Equation (48),
represents the total load demand data,
are demand load for
1, 2, 3,
· · ·
, 24) preceding to selected day. This paper assumes that
-1 hours in Equation (48). Moreover, there are 20 neurons in the output (
) layer of DBN
that signiﬁes the difference of load demand on the hourly basis for preceding and selected days,
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 [
]. 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
0 10 20 30
Auto-correlation of preceding demand load data for day lags in Fuzzy Local Linear Model
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:
In Equations (50) and (51),
are the hours that are forecasted and
is the real demand load at
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)
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
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.
Case 1: Short-term load forecasting of PNPN without taking temperature and distribution of data
Case 2: Short-term load forecasting of PNPN including consequences of data distribution without
taking the temperature
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).
Consequences of priority index and data distribution on the forecasting results for
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
Consequences of priority index and data distribution on the forecasting results for Maximum
Distance Minimum Error (MDME) and 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
enhanced in Case 4 as compared to Case 3. Nonetheless,
has achieved enhanced results in Case 3 in
comparison with Case 4.
for target year. The optimum result achieved is for
= 7 and
= 0.3. The results achieved for
= 0.3 are approximately near to
= 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.
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
2. Case 2: Load forecasting by collected similar days in last data-set, Γds2
3. Case 3: Load forecasting by Γds1
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
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
. 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
. Nevertheless, integration of
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
0 5 10 15 20 25
Actual.Load Classic.Forecasting Modified.Forecasting
Comparative analysis and effect of proposed and traditional method for Monday,
19 September 2015.
0 5 10 15 20 25
Actual.Load Classic.Forecasting Modified.Forecasting
Comparative analysis and effect of proposed and traditional method for Wednesday,
13 June 2015.
Electronics 2018,7, 431 29 of 34
0 5 10 15 20 25
Actual.Load Classic.Forecasting Modified.Forecasting
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
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.
Comparison of Fuzzy Local Linear Model (F-LOLIMOT), deep belief network (DBN), and
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.
0 5 10 15 20 25
System Load (MW)
Actual.Load Similar.Day DBN F.LOLIMOT
Figure 15. Short-term load forecasting for a sample day.
0 5 10 15 20 25
Similar_Day DBN F_LOLIMOT
Figure 16. Error values for a sample day.
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
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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