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A novel hybrid model based on neural network and multi-objectiveoptimization for effective load forecast | Tiếng Anh 12

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A novel hybrid model based on neural network and multi-objective

optimization for effective load forecast

Priyanka Singh

, Pragya Dwivedi

Computer Science and Engineering Department, MNNIT Allahabad, Allahabad, 211004, India

article info

Article history:

Received 17 January 2019

Received in revised form

21 May 2019

Accepted 10 June 2019

Available online 14 June 2019

Keywords:

Load forecasting

Artiﬁcial neural network

Multi-objective optimization

Hybrid forecasting model

Multi-objective follow the leader (MOFTL)

abstract

In recent years, increased attention has been paid by the researchers to predict accurate and stable load

due to its effect on the economy and need for proper management of power systems. However, most of

the previous research focused only on either reducing load forecast error or enhancing the stability, very

few studies focused on these two issues simultaneously. Introducing a forecasting model to solve both

independent objectives at the same time is a challenging task due to the complex behavior of the load

pattern.

Therefore, to achieve two objectives simultaneously, we propose a novel multi-objective algorithm

(MOFTL) based on Follow The Leader algorithm. The effectiveness of MOFTL has been shown by

comparing the results with three newly presented MOWCA, MOPSO and NSGA-II multi-objective algo-

rithms. Moreover, to validate the performance of MOFTL, we have combined MOFTL with neural network

termed as MOFTL-ANN to solve the problem of electrici ty load forecasting. The proposed hybrid model

outperforms baselin e models over two real-world electricity data sets namely England region and ERCOT

region. MOFTL-ANN shows improvement of 17.42%, 6.81%, 10.77% and 59.69% MAPE values for England

region and 4.20%, 4.16%, 1.14% and 21.85% MAPE values for ERCOT region over NSGA-II-ANN, FTL-ANN,

BPNN, and GRNN.

1. Introduction

Load forecasting is a central and integral process for periodic

planning, scheduling, controlling load generation, continuous dis-

tribution of electricity and expansion of facilities in the power

sectors [1,2]. Meanwhile, the study of load forecasting is a chal-

lenging task due to several uncertain factors such as demand

ﬂuctuation, complex historical load curve, calendar effects, fore-

casting models, random effects and many more [3].

In the past few years, load forecasting has been a very active and

popular research topic because of several reasons, deregulation of

the electricity market, increased ﬂuctuation in energy demands

and increased operating costs [4]. The poor forecasting methods

result in considerable losses to the economy, for example, a

research in 1985 presented that 1% increase in forecasting error

increased the associated operating costs of up to 10 million pounds

every year in the thermal British power system [5]. Therefore, the

development of an effective model with enhanced forecasting

abilities for accurate and stable load estimation is desired. So far,

several techniques and models have been proposed for load fore-

casting. In the initial stages of load forecasting, classical statistical

techniques like BoxeJenkins models and time series techniques

were widely used in load forecasting [6,7]. However, load data have

complex behavior [8] and are inﬂuenced by several factors [9], time

series techniques may exhibit poor performance. Thus, several

artiﬁcial intelligence techniques including artiﬁcial neural net-

works (ANNs) [10e13], fuzzy logic [14e16], expert systems, and

support vector machines (SVMs) [17,18] have been applied to time

series forecasting. Among these techniques, ANN has been widely

used for solving load forecasting problem [ 19e21].

Apart from these techniques, several hybrid models have been

presented in the literature to yield forecasting results with extreme

accuracy [33]. To ﬁ nd optimal hybrid model, several optimization

algorithms, such as particle swarm optimization (PSO) [34], genetic

algorithm (GA) [35], artiﬁcial bee colony (ABC) [36], differential

evolutionary (DE) [37 ,38], cuckoo search (CS) algorithm [39,40], ant

colony algorithm [

41] and follow the leader (FTL) [30] algorithm

have been applied for load prediction. In these studies [7,30,34,35],

* Corresponding author.

E-mail addresses: 2015rcs53@mnnit.ac.in, priyankamnnit925@gmail.com

(P. Singh), pragyadwi86@mnnit.ac.in (P. Dwivedi).

Contents lists available at ScienceDirect

Energy

journal homepage: www.elsevier.com/locate/energy

https://doi.org/10.1016/j.energy.2019.06.075

Energy 182 (2019) 606e622

researchers focused on improving the accuracy of forecasting

models by ﬁnding optimal weights of ANN, but stability also plays a

vital role in determining the effectiveness of a forecasting model.

Considering only one objective (accuracy or stability) at a time is

insufﬁcient for ﬁnding effective load forecasting model. Thus, a

hybrid model based on ANN should optimize the initial weights

while considering accuracy and overcome the instability of the ﬁnal

results simultaneously.

Researchers have developed an indeﬁnite number of learning

algorithm that are accurate and predicts outcomes with a high

degree of conﬁdence. But what happens when training the model

with a different/same subset of same training data set, will the

model perform with the same efﬁciency over repeated experi-

ments. ANN depends on random initial values, therefore, the ﬁnal

outcome varies over the same input data set. Every predictive

model has to be accurate and stable. A learning model is stable

when the predicted result does not change more than a certain

threshold for the different/same subset of data set over several

experiments. It is noteworthy that a hybrid model that achieves

high accuracy and strong stability at the same time is a multi-

objective problem (MOP) rather than a single-objective problem.

Over the past few decades, multi-objective problems (MOPs)

have attracted researchers in different real-world problems [42,43].

Many multi-objective algorithms such as binary coded elitist non-

dominated sorting GA (NSGA-II) [44e46], multi-objective ant lion

optimizer (MOALO) [ 47,48], multi-objective ﬂower pollination al-

gorithm (MOFPA) [49], multi-objective ﬁreﬂy algorithm (MOFA) [1],

multi-objective whale optimization algorithm (MOWOA) [50] etc.

have been introduced in last few years. Table 1 shows the research

contribution of different forecasting models. Findings from this

literature prove that hybrid forecasting models generate less fore-

casting error. But stability of the model is another important factor

in load forecasting that cannot be ignored. Based on the ﬁndings,

we can conclude that multi-objective optimization algorithms

overcome the shortcomings of single-objective optimization algo-

rithms and achieve improvement in accuracy and stability simul-

taneously. MOPs generate a series of optimal solutions, named

Pareto-optimal solutions (solutions are equally optimal) for solv-

ing different conﬂ icting objectives simultaneously. The ideal goal of

a MOP is to ﬁnd the set of solution termed as a Pareto-optimal set

which lies on the Pareto Front (PF) and, solutions have uniform

distribution along the true PF [44,51]. Among multiple trade-offs

and solutions found, choose only one solution using higher-level

information for practical purpose [46]. From the fact that multi-

objective algorithms can approximately ﬁnd the true Pareto-

optimal solutions of most MOPs, we have proposed a novel

multi-objective algorithm called Multi-Objective Follow The Leader

(MOFTL). This algorithm overcomes the problem of solving a single

objective function and enhance the accuracy and stability simul-

taneously for load prediction.

To obtain higher accuracy and better stability for load prediction

at the same time, a novel hybrid model involving the widely

acceptable neural network (NN), and our proposed multi-objective

algorithm (MOFTL) is presented in this paper. The validation of our

proposed multi-objective algorithm is done by testing on four

multi-objective benchmark problems and comparing with three

newly developed multi-objective algorithms: MOWCA, MOPSO,

and NSGA-II. For a comprehensive evaluation of MOFTL, hybrid

forecasting model MOFTL-ANN has been presented and applied for

load forecasting over two electricity load data sets collected from

England and Texas. Results obtained from different forecasting

models are tested by the Friedman test to show their statistical

signiﬁcance.

The major contributions of our work are as follows:

 A novel multi-objective follow the leader, MOFTL algorithm is

proposed in this paper.

 To validate the effectiveness of MOFTL, it is tested on four multi-

objective benchmark problems and compared with three multi-

objective algorithms: MOWCA, MOPSO, and NSGA-II.

 Further to obtain higher accuracy and excellent stability for load

forecasting, we combined MOFTL with a neural network called

MOFTL-ANN to optimize network weights.

 The comprehensive evaluation of MOFTL-ANN shows that it

overcomes the problem of solving single objective functions and

enhance the accuracy and stability for load forecasting.

The paper is organized as follows. Section 2 brie ﬂy describes

Table 1

Research contributions of forecasting algorithms in electricity load forecasting.

S.no. Forecasting

algorithm

Methodology Findings

1 Linear

regression

In Ref. [22], a regression-based daily peak load forecasting method with a

transformation technique is proposed.

Linear regression fails for nonlinear data due to poor non-linear ﬁtting

capability, lacks accuracy.

2 Multi

regression

In Ref. [23], for mid-term load forecast on an hourly basis of US utility,

multi-regression model has been used.

Multi regression technique has received limited success, it generates

higher forecasting error for abrupt changes in the environment.

3 Exponential

smoothing

In Ref. [24], ﬁve developed exponential smoothing models are compared

over one day ahead load forecast.

They are faster and efﬁcient but generates more error for a long-term load

forecast.

4 ARIMA In Ref. [25], ARIMA model and transfer function model are applied to STLF

by considering weather and load relationship.

ARIMA model generates great results for linear problems but is incapable

of solving nonlinear sensitive part of the load, it generates results only on

the basis of past and current data.

5 Fuzzy logic In Ref. [26], a fuzzy logic method is employed to forecast the gross

electricity demand of Turkey.

Fuzzy techniques generate better forecast results than statistical models. It

provides reliable result but shows a strong dependency on expert systems.

6 SVR In Ref. [9], a generic STLF strategy based on the support vector regression

machines is proposed. In Ref. [27], SVR is used to ﬁnd demand response for

an ofﬁce building.

SVR is a robust and accurate method. It is a highly effective model in

solving non-linear problems but fails due to higher running time and high

dependency between forecasting accuracy and selected SVM parameters.

7 ANN In Ref. [28], multilayer perceptron model (MLP) is used for the prediction of

long-term energy consumption. In Ref. [29], LSTM recurrent neural

network is used to solve the problem of load forecasting in a residential

area.

ANN has the capability to map the complex input and output relationship

but due to restricted generalization ability, the solution falls in a local

minimum. Also, for large neural network, high running time is required.

8 Hybrid

models

In Ref. [30], a newly proposed evolutionary algorithm is combined with

ANN to enhance forecasting accuracy. In Ref. [31], GRNN is combined with

a fruit ﬂy optimization algorithm to improve the capability of GRNN for

power load forecasting. In Ref. [32], modiﬁed ﬁreﬂy algorithm is used to

ﬁnd the optimal parameters of support vector regression.

The efﬁciency of the hybrid model is improved in terms of accuracy but

stability of model is another important factor that needs to be considered

during the load forecast.

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 607

multi-objective optimization and proposed multi-objective algo-

rithm. Section 3 presents the proposed hybrid model and section 4

demonstrates the performance of MOFTL over four multi-objective

benchmark problems and numerical results obtained from the

proposed hybrid model for electricity load forecast problem. In

Section 5, discussions based on experimental results are intro-

duced. Section 6 presents the conclusions of this paper.

2. Multi-objective follow the leader algorithm

In this section, the concept of multi-objective optimization and

some related description of the MOFTL are described.

2.1. Multi-objective optimization

Solving one objective function by an evolutionary algorithm is

single-objective optimization problem where an optimal solution is

chosen very conventionally. However, multi-objective optimization

problem solves more than one problem simultaneously. Unlike

single-objective optimization problem the output of MOP is series

of solutions called Pareto-optimal solutions, which are equally

optimal and their mapping in objective space generates Pareto

Front [47]. Selection of Pareto-optimal solutions that are uniformly

distributed along true Pareto Front is recommended. The deﬁnition

of multi-objective optimization problem, Pareto dominance, crowding

distance are given as follows.

Deﬁnition 1. Multi-objective optimization problem [52].

Minimize

Maximize : f

ðxÞ; m ¼ 1; 2; …::; M;

Subject to : g

ðxÞ0; j ¼ 1; 2; …::; J;

ðxÞ¼0; k ¼ 1; 2; …::; K;

ðLÞ

 x

ðUÞ

; i ¼ 1; 2; …::; n;

;

(1)

where M, J, K, n represents number of objective functions,

inequality constraints, equality constraints and number of vari-

ables, respectively. x

ðLÞ

and x

ðUÞ

are lower and upper boundaries of

for i

variable.

Deﬁnition 2. Pareto Dominance: [52] A solution x

ð1Þ

is said to

dominate the other solution x

ð2Þ

, if both the following conditions

are true:

 The solution x

ð1Þ

is no worse than x

ð2Þ

in all objectives. Thus, the

solutions are compared based on their objective function values

(or location of the corresponding points (z

ð1Þ

and z

ð2Þ

) on the

objective space).

 The solution x

ð1Þ

is strictly better than x

ð2Þ

in at least one

objective.

Based on comparison of dominance value, all individuals are

sorted into different front levels. Each front level has a rank, which

equals its non domination level. The individual in level 1 are

dominated by no other individuals; the individual in level 2 is

dominated by individuals of only level 1, and so on.

Deﬁnition 3. Crowding Distance: [44]

For MOPs, there is more than one objective function to be

minimized or maximized. To select the best solution in the popu-

lation that satisﬁes the condition of all objective function simul-

taneously, crowding-distance mechanism is used [53]. The

crowding distance of an individual is a measure of objective space

of individual that is not occupied by any other solution in the

population [52]. In other words, crowding distance is a measure of

how close an individual is to its neighboring candidates within

population [1]. It is always desired that location of solutions in

same front level have uniform distribution as, a large diversity of

the individuals can solve the stagnation problem of multi-objective

algorithms. Firstly, individuals are sorted on the basis of every m

objective function then boundary individuals of each level are

assigned inﬁnite value and crowding distance of other individuals

are calculated as follows.

¼ ∞

n1

i¼2

m¼1

iþ1

 f

i1

max

 f

min

(2)

where D

and D

are boundary individuals at each PF, n is number

of individuals at each PF, f

is ﬁtness value of m

objective function

of i

individual at PF f and m is number of objective functions and

the parameters f

max

and f

min

are the maximum and minimum

values of the m

objective function. Larger the average crowding

distance, better is the diversity in the population.

2.2. MOFTL (multi-objective follow the leader)

MOFTL is a multi-objective version of a novel meta-heuristic FTL

algorithm proposed by Singh and Dwivedi in 2018 inspired from

moving behavior of a sheep [30,54]. In ﬂock, sheep having highest

score is leader candidate and one with the lowest score is rear

candidate. When leader moves within the ﬂock, other candidates

follow them. Mathematical representation for updating the posi-

tion of every candidate is shown in eq. (2).

j;k;i

¼X

j;k;i

þ rand



j;leader;i

 X

j;rear;i



(3)

where, X

j;k;i

is updated position of sheep X

j;k;i

, X

j;leader;i

is position

value of the variable j for leader sheep in i

iteration, X

j;rear;i

position value of the variable j for rear sheep in i

iteration, rand is

a random value between 0 and 1 and

is inertia weight value lying

between 0 and 1 to control exploration and exploitation process

within the solution space. Fig. 1 shows sequential ﬂowchart of FTL.

Unlike single-objective FTL, multi-objective FTL has several

objective functions and should be evaluated simultaneously. A set

of non-dominated solutions are generated by checking domination

condition between two solutions. In MOFTL, all the solutions are

distributed within different front levels based on dominance con-

dition. Top front level ranks ﬁrst in which no individual is domi-

nated by other individuals, individuals in the second level are

dominated by some individuals on level 1, and so on. In the same

front level, there exist a number of solutions that are distributed

which is expected to be in a uniform manner, the crowding distance

operator helps in the selection process and measures closeness

between an individual and its neighbors. Larger average crowding

distance brings greater diversity in the population. For solving load

forecasting problem with MOFTL, we have considered two objec-

tive functions mean squared error (MSE) and standard deviation

(STD) to represent accuracy and stability respectively. The equation

of objective functions for load forecasting can be deﬁned as follows:

MSE ¼

i¼1

ðy

 y

(4)

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622608

STD ¼stdðy

 y

Þ; i ¼ 1; 2; …:N (5)

3. Proposed hybrid forecasting model

In this section, detailed study of artiﬁcial neural network and

hybrid MOFTL-ANN model is introduced.

3.1. Artiﬁcial neural network

An artiﬁcial neural network (ANN) is intelligence-based learning

system, inspired from functioning of human brain which consists of

several small processing elements called neurons [55,56]. These

neurons are interconnected and these connection have some

weight associated with themselves. Neurons receive information

through several input nodes, process them internally and generates

a response [57]. ANN being universal approximator can approxi-

mate large class of functions, generate higher accuracy over other

nonlinear model [58]. ANN architecture is composed of three

layers: input layer, hidden layer and output layer. In input layer,

signiﬁcant features which shows impact on prediction are consid-

ered and hidden layer/layers use these features to generate inter-

mediate solution and ﬁnal predicted value is produced at output

node of network [59]. Usually, feed forward neural network with

single hidden layer is preferred in time series forecasting. A simple

architecture of feed forward neural network is shown in Fig. 2.

Mathematical representation of relationship between inputs X ¼

; X

; …::X

and output Y

for logistic activation function in hidden

layer and linear function in output layer can be shown as follows

[30]:

j¼1

k;j

1 þ exp





i¼1

j;i



(6)

where, 1in, 1 j  mW

i;j

represents weight between nodes i

and j,

is activation function of j

node and n and m represents

number of nodes in input and hidden layer respectively.

So far several optimization algorithms have been combined

with neural network to ﬁnd optimal weights such as genetic al-

gorithm [35], PSO [60], fruit ﬂy optimization algorithm [31]to

predict accurate load forecasting. But ﬁnding a model that precisely

generate load along with greater stability is bigger problem of load

forecasting. Thus, a novel multi-objective algorithm has been pro-

posed in this paper which can not only predicts load with high

precision but also stable forecasting.

3.2. The MOFTL-ANN hybrid model

In this section, the weights of neural network are updated by a

multi-objective optimization algorithm, MOFTL to allow ANN to

achieve enhanced accuracy and stability simultaneously. For

effective load forecasting, it is recommended to consider both ac-

curacy and stability at the same time. Thus, the optimization of

initial weights of a neural network should satisfy both the objective

functions simultaneously. Mean squared error (MSE) f

in fore-

casting are highly considered for performance evaluation of load

forecasting models. In this study, standard deviation of forecasting

error is chosen as another indicator for representing the stability of

a forecasting model f

. Smaller the mean squared error and stan-

dard deviation value, higher is the accuracy and stability. The

pseudo-code of the hybrid MOFTL-ANN model is presented in Al-

gorithm 1.

4. Experiments and analysis

To verify the effectiveness of the proposed multi-objective al-

gorithm and hybrid model, two different experiments are per-

formed. In experiment 1, MOFTL is compared with three newly

developed multi-objective algorithms namely MOWCA [53,61],

MOPSO [62], and NSGA-II [46]. In Experiment II, the performance of

hybrid model (MOFTL-ANN)is compared with ﬁve state-of-art

forecasting models. Here, the obtained results are compared over

two load data sets namely, England region and ERCOT region. De-

tails of the experiment are described in the following subsections.

Fig. 1. Flowchart of FTL algorithm.

Fig. 2. Feed forward neural network.

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 609

4.1. Experiment I: tests of MOFTL

In this experiment, four multi-objective problem and three well

known multi-objective algorithms namely MOPSO, MOWCA and

NSGA-II have been used to test and compare the performance of

MOFTL algorithm. Mathematical representation of multi-objective

problems are as follows:

These four test functions and the performance metrics of

reverse generational distance (RGD) [63] along with three newly

developed multi-objective algorithms- MOWCA, MOPSO, and

NSGA-II verify the performance of our proposed algorithm MOFTL.

The mathematical formula for RGD is shown in eq. (7).

RGD ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j¼1

(7)

where, d

is Euclidean distance between i

true PF and nearest

members of obtained PF and N is number of true PF.

Each test function has been performed 100 times for 500 iter-

ations and 100 population to ﬁnd Pareto optimal solutions. The

results obtained from the experiment are shown in Table 2 and

Figs. 3e6. From Table 2, it can be easily demonstrated from all the

statistical values that MOFTL is superior to MOWCA, MOPSO, and

NSGA-II. Our proposed multi-objective algorithm yields the best

values of standard deviation for FON- 0.000156, SCH- 0.000366,

ZDT1- 0.010104 and ZDT3- 0.002242 multi-objective problems.

And these results prove that lower is standard deviation lesser in

scattering and better is the solution.

Remark. Results shown proves that MOFTL performs well and is

superior to MOWCA, MOPSO, NSGA-II. MOFTL achieves better

optimal solutions than others with higher accuracy and also ﬁnds

the best values of RGD. Thus, results show that our proposed al-

gorithm is effective and can be further used for solving real-world

multi-objective problems such as load forecasting problem of

predicting load with higher accuracy and stability.

4.2. Experiment II: tests of MOFTL-ANN

To deeply analyze the superior performance of the proposed

algorithm, MOFTL is implemented to solve the short-term load

forecasting problem. Implementation details are discussed in the

forthcoming subsections.

4.3. Data pre-processing

Before going for load prediction, data pre-processing is done to

eliminate spikes and outliers. The missing and extra data values are

managed and data normalization (MIN-MAX) is done between the

interval of ½1; 1 to reduce the training time. To perform different

experiments, the data set is divided into training and testing data.

4.4. Evaluation metrics

Since the past few decades, several evaluation metrics have

been developed to evaluate the performance of the load forecast.

However, there are no predeﬁned rules for selecting these metrics.

In this paper, we use the nine commonly used evaluation metrics

shown in Table 3 to evaluate the performance forecasting models.

Mean absolute error (MAE) indicate the overall level of errors [50].

Normalized mean squared error (NMSE) calculates the overall de-

viations between forecasted and measured load. The root of mean

squared error (RMSE) reﬂects the degree of differences between the

predicted and measured values and is more stable than MSE and

less sensitive to extreme errors. Mean absolute percent error

(MAPE) shows accuracy as a percentage of forecasting error.

Directional Change (DC) exhibits the movement directions or

turning points of prediction. Pearson's correlation coefﬁcient (r)

shows the correlation between the predicted value and the

observed value. Lastly, the index of agreement (IA) is a standard

metric of the degree of model prediction error [64]. In the following

subsections, we present two case studies on various data sets to

validate the effectiveness of MOFTL-ANN. ALL the experimental

parameters involved during the experiment is shown in Table 5.

The experiment has been performed 100 times with population

size of 50, 5,000 maximum iteration, 10 hidden neurons, one

output neuron and the number of input neurons is same as the

number of input parameters taken in the case study.

In order to validate the performance of our proposed hybrid

model MOFTL-ANN, we have compared our model to following

state-of-art models.

 NSGA-II-ANN (Neural network with NSGA-II)

 FTL-ANN (Neural network with Follow the leader algorithm)

 BPNN (Backpropagation neural network)

 GRNN (Generalized regression neural network)

4.5. Case study-1: ISO England

4.5.1. Data description

Hourly data set from England region from years 2004e2007 and

out-of-sample data from the years 2008 and 2009 are used for

training and testing purpose [65]. The statistical parameter of data

set is shown in Table 4 and the correlation coefﬁcient (r) between

independent variables given as input to a neural network and load

is shown in Table 6. Higher the value of r higher is the correlation

between the input parameter and load.

4.5.2. Result

The results obtained for the year 2008 and 2009 by all the

forecasting models are shown in Table 7 with their minimum,

average and maximum values. It clearly shows that MOFTL-ANN

brings higher accuracy in terms of all performance evaluation

metrics mentioned earlier. Moreover, from the table, it can be

observed that MOFTL enhances the prediction ability of ANN than

FON:

Minimize : f ð 1 Þ¼1  exp



i¼1



ﬃﬃﬃ



Minimize : f ð 2 Þ¼1  exp



i¼1



ﬃﬃﬃ



where;  4  xðiÞ4

SCH:

Minimize : f ð 1 Þ¼x

Minimize : f ð 2 Þ¼ðx  2Þ

where;  10

 xðiÞ10

ZDT1:

Minimize : f ð 1 Þ¼xð1Þ

Minimize : f ð 2 Þ¼g*

1 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

xð1Þ

where : g ¼ 1 þ



n  1



i¼2

xðiÞ



and 0  xðiÞ 1; 1  i  n

ZDT3:

Minimize : f ð 1 Þ¼xð1Þ

Minimize : f ð 2 Þ¼g*

1 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

xð1Þ



xð1Þ

sinð10

xð1ÞÞ



where : g ¼ 1 þ



n  1



i¼2

xðiÞ



and 0  xðiÞ 1; 1  i  n

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622610

NSGA-II does. For example, minimum values for MAPE of MOFTL-

ANN and NSGA-II-ANN are 3.2019% and 3.9615%, respectively. The

average MAE value of MOFTL-ANN is 504.1671 Mwh which is the

least by any of the compared forecasting models while GRNN has a

maximum average MAE value of 1199.392 Mwh. The average values

of MAE- 504.1671 Mwh, MEP- 15.26 862%, RMSE- 655.7201 Mwh,

NMSE- 0.05401 Mwh, MAPE- 3.4967%, Pearson correlation coefﬁ-

cient (r)- 0.9741 obtained by MOFTL-ANN makes it more reliable

forecasting model. However, MOFTL-ANN generates higher accu-

racy than FTL-ANN as the reason lies in the fact that FTL-ANN is

single-objective optimization i.e. enhancing the accuracy.

Table 8 and Table 9 shows monthly MAPE and MAE of all fore-

casting models for the years 2008 and 2009 respectively. MOFTL-

ANN signiﬁcantly outperforms all other forecasting models in

nearly every month due to their higher rate of directional predic-

tion while GRNN produced maximum values of MAE and MAPE for

Table 2

Results of the multi-objective algorithms (using RGD) on the four test functions adopted in this paper.

Algorithm FON Algorithm SCH

Min Avg Median Max Std. Min Avg Median Max Std.

MOFTL 0.00447282 0.00478 0.004756 0.005365 0.000156 MOFTL 0.01701711 0.017963 0.01793 0.018785 0.000366

MOWCA 0.01363076 0.117798 0.096826 0.429674 0.093677 MOWCA 0.01883266 0.020035 0.020002 0.021894 0.000675

MOPSO 0.0045397 0.00491 0.004892 0.005431 0.00019 MOPSO 0.01927343 0.021082 0.02084 0.026767 0.001125

NSGA-II 0.00437934 0.004857 0.004838 0.005422 0.000204 NSGA-II 0.01937874 0.021189 0.021199 0.023965 0.000993

Algorithm ZDT1 Algorithm ZDT3

Min Avg Median Max Std. Min Avg Median Max Std.

MOFTL 0.004333 0.008895 0.004785 0.340285 0.010104 MOFTL 0.190448 0.192423 0.191566 0.208109 0.002242

MOWCA 0.004412 0.026285 0.004876 0.319874 0.057746 MOWCA 0.191381 0.195551 0.19257 0.303958 0.01236

MOPSO 1.143838 1.457361 1.459175 1.615171 0.077166 MOPSO 0.804463 1.081175 1.092714 1.276132 0.09646

NSGA-II 0.111613 0.137112 0.13742 0.173442 0.033702 NSGA-II 0.195631 0.20023 0.199969 0.237757 0.00471

Algorithm 1: MOFTL-ANN.

Objective Functions:

Minimize

ðxÞ¼MSE

ðxÞ¼STD

;

Input:

train

- training samples

test

- testing samples

Output:

forecast

- forecasting data

Parameters:

Maxit - maximum number of iterations

Npop - number of sheep within a ﬂock

Dim - number of dimensions

- position of i

sheep

- rank value of i

sheep

- crowding distance values of i

sheep

- ﬁtness values of i

sheep

Begin

1 Set all the input parameters of MOFTL.

2 Initialize random positions of all sheep within the ﬂock of size Npop and Dim dimensions in the range of - 1 and þ 1.

3 For each sheep i within the ﬂock i ¼ 1:Npop

4 Compute corresponding ﬁtness function f

and f

using non-dominated sort ranking process.

5 Calculate crowding distance d

6 end for

7 =* Determine leader X

leader

and rear X

rear

sheep on the basis of their rank and crowding distance. =

8 While (i < Maxit)

9 =* Evaluate new solution and update position of sheep =

10 For each sheep i within the ﬂockj¼1:Npop

11 For k¼1:Dim

12 X

j;k;i

¼ X

j;k;i

þ rand

*ðX

j;leader;i

 X

j;rear;i

13 end for

14 end for

15 =* Check if any sheep goes beyond the boundary conditions.=

16 For each sheep i within the ﬂock i ¼ 1:Npop

17 Compute corresponding ﬁtness function f

and f

using non-dominated sort ranking process.

18 Calculate crowding distance d

19 end for

20 = Selection of sheep for next generation.=

21 Merge X

and X, sort them on the basis of rank and

crowding distance and update X with Npop sheep.

22 = Update leader and rear sheep.=

23 end while

24 Return X

25 Set the weights of neural network according to X.

26 Use X

train

to train ANN.

27 Use electricity load test data set X

test

to obtain the

predicted valve X

forecast

End

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 611

almost every month for both the years. In Tables 8 and 9, it can also

be noted that for a few months like Feb 2009, FTL-ANN generated

lower MAPE than MOFTL-ANN. The reason lies in the fact that FTL-

ANN is a single-objective optimization model while MOFTL-ANN

optimizes both accuracy and stability. But the overall result of

MOFTL-ANN cannot be ignored as it brings both stability and ac-

curacy simultaneously and overall MAPE and MAE of MOFTL-ANN

for the years 2008 and 2009 is lower than all compared fore-

casting models.

Fig. 7 shows the comparison of average MAE and MAPE value

Fig. 3. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for FON. (Note: PF represents Pareto Font).

Fig. 4. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for SCH. (Note: PF represents Pareto Font).

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622612

generated by all ﬁve forecasting model on a daily basis. The ﬁgure

clearly shows that MOFTL-ANN generated least MAE on each day of

the week while GRNN produced maximum MAE and MAPE. It can

also be noted that MAPE generated on Friday by FTL-ANN is less

than that of MOFTL-ANN due to the limitation of solving two

objective functions at the same time. Fig. 8-a shows a comparison of

average weekly load error found during load prediction for years

2008e2009. The ﬁgure reveals that MOFTL established to draw

least average weekly load error in almost all weeks for the years

2008 and 2009 while GRNN generated high peaks of average load

Fig. 5. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for ZDT1. (Note: PF represents Pareto Font).

Fig. 6. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for ZDT3. (Note: PF represents Pareto Font).

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 613

error during all the weeks of the testing period. Fig. 8-b shows the

comparison of load obtained after prediction by all ﬁve models with

the observed load. This comparison illustrates that MOFTL-ANN

achieves the most accurate prediction values and also load con-

sumption during weekends is less compared to weekdays. Fig. 9

shows the relationship between observed and predicted load by

all ﬁve forecasting models in which MOFTL-ANN show a strong

relation.

Remark. From the obtained experimental results, MOFTL-ANN

seems to be the most accurate hybrid model. These tables and

ﬁgures illustrate that our proposed hybrid model is the most su-

perior, accurate and stable forecasting model.

4.6. Case study-2: ERCOT

4.6.1. Data description

In this experiment, hourly load data set of ERCOT for the years

2009e2014 have been taken as training data and years 2015e2016

for testing purpose [66]. Table 6 shows the correlation between

input variables and load taken into consideration. ALL the statistical

parameters related to the ERCOT data set is shown in Table 4. Five

parameters namely, an hour of the day, the day of the week, 168-hr

lagged load, 24-hr lagged load and previous 24-hr average load has

been chosen for load prediction of ERCOT region.

4.6.2. Results

Table 10 shows the minimu, average and maximum values of all

the forecasting models namely, MOFTL-ANN, NSGA-II-ANN, FTL-

ANN, BPNN, and GRNN for the years 2015 and 2016. From the

obtained result, it can be clearly noticed that MOFTL-ANN ﬁnds a

better optimal solution than other forecasting models. Evaluation

metrics like average values of MAE- 1891.478 Mwh, MEP- 31.145%,

RMSE- 2586.809 Mwh, NMSE- 0.077475 Mwh, MAPE- 4.811885%,

Pearson correlation coefﬁcient (r)- 0.9606 makes MOFTL-ANN su-

perior than other forecasting models. GRNN generates a minimum

average directional change of 67.1173 while MOFTL-have maximum

- 71.8093.

Table 11 and Table 12 shows month wise MAE and MAPE error

values of all ﬁve forecasting models. The table compares the result

obtained from the experiment and found that MOFTL-ANN gener-

ates superior result. Very exceptionally, it can be noted that for the

month Dec 2015, FTL-ANN generated lesser MAPE of 4.44200%

while MAPE of MOFTL-ANN is 4.9073%. The reason for this

Table 3

Evaluation metrics.

S.No. Metric Equation

1 Mean absolute error

MAE ¼

j¼1



 y



2 Maximum error percentage

MEP ¼



ðy

 y



100

3 Root of mean squared error

RMSE ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j¼1

ðy

 y

4 Normalized mean squared error

NMSE ¼

j¼1

ðy

 y

N  1

j¼1

ðy

 yÞ

5 Mean absolute percent error

MAPE ¼

j¼1



 y



100

6 Daily peak mean absolute percent error

Daily Peak MAPE ¼



maxðTL

ÞmaxðFL

Þj

maxðTL

100

7 Pearson's correlation coefﬁcient

r ¼

j¼1

ðy

 yÞðy

 y

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j¼1

ðy

 yÞ

j¼1

ðy

 y

8 Directional change

DC ¼

100

N  1

N1

j¼1

, a

0; otherwise

1; if ðy

jþ1

 y

Þðy

jþ1

 y

Þ> 0

;

9 Index of agreement

IA ¼ 1 

j¼1

ðy

 y

j¼1

ðy

 yÞðy

 yÞ

Note: y

¼actual value of day j, y

¼predicted value of day j, y ¼ mean of actual value, y

¼mean of Predicted value, FL

¼forecast load value for every 24 h, TL

¼target

load for every 24 h, N ¼number of elements in training data.

Table 5

Experimental parameter values.

Forecasting model Experimental parameter Value

MOFTL-ANN MOFTL inertia weight 0.75

MOFTL maximum number of iterations 100

MOFTL population size 50

NSGA-II-ANN GA crossover ratio 0.7

GA mutation ratio 0.3

GA population size 50

FTL-ANN Tolerance value

8

FTL inertia weight 0.67

GA population size 50

BPNN Learning rate 0.1

GRNN Training precision

5

Table 4

Statistical parameters for the data used in this paper.

Region Samples # of elements Mean Max Median Min Std.

ISO New Pool England All 52608 15093.259 28130 15277 9040 2958.703

Training 35064 15270.068 28130 15473 9152 3007.685

Testing 17544 14739.883 26111 14982 9040 2825.413

ERCOT All 70128 37865.985 71092.609 35656.533 21378.730 9260.466

Training 52584 37208.379 68317.669 34917.492 21378.730 9155.398

Testing 17544 39837.002 71092.609 37625.894 24337.467 9294.055

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622614

phenomena is likely that FTL-ANN is a single-objective model

which can handle trends and randomness while MOFTL-ANN is a

multi-objective model and it has to handle both accuracy and sta-

bility simultaneously.

The comparison of average MAE and MAPE value generated by

all ﬁve forecasting model on a daily basis is shown in Fig. 10. The

ﬁgure clearly shows that MOFTL-ANN generated least MAE on each

day of the week except Sunday, BPNN produced minimum MAE.

Similarly, on Sunday and Tuesday, MAPE of BPNN is less than

MOFTL-ANN. The reason for such a situation lies in the nonlinear

behavior of data trends. GRNN produced maximum MAE and MAPE

on each day that proves its unsuitability for load prediction. Fig. 11-

a shows a comparison of average weekly load error for the year

Table 7

Forecasting results of the ﬁve forecasting models.

Forecasting Models MAE MEP RMSE NMSE MAPE Daily peak MAPE r DC IA

MOFTL-ANN Min 457.8729 11.49369 597.0174 0.044649 3.201904 2.504589 0.969801 72.55316 0.983564

Avg 504.1671 15.26862 655.7201 0.054012 3.496752 3.03031 0.974153 73.87733 0.988558

Max 547.7367 19.60747 711.1898 0.063359 3.828821 3.499827 0.979748 74.77056 0.986071

NSGA-II-ANN Min 582.4166 19.3054 779.6671 0.076147 3.92285 3.631563 0.952252 69.86832 0.96693

Avg 629.1621 20.92272 840.1871 0.088878 4.234364 4.094272 0.957317 70.9776 0.976367

Max 780.1065 24.75158 1001.752 0.125706 5.13332 6.045748 0.961517 72.08573 0.979975

FTL-ANN Min 523.3067 17.02067 692.2375 0.060027 3.543245 3.205244 0.959864 70.89437 0.979468

Avg 553.2902 19.94162 741.0733 0.068972 3.75242 3.675925 0.965272 72.49957 0.984357

Max 601.5597 22.40104 795.0949 0.079191 4.126287 3.940751 0.970286 73.98963 0.982063

BPNN Min 458.1641 16.82907 594.3669 0.044253 3.07581 2.648605 0.886737 62.68027 0.9298

Avg 579.5099 20.58969 775.7784 0.080456 3.919104 3.711811 0.959404 72.55772 0.977829

Max 1057.095 28.75839 1326.469 0.22041 7.345048 6.091889 0.979059 75.22659 0.988835

GRNN Min 624.5459 19.41066 859.5813 0.092557 4.21692 4.155094 0.911327 55.56632 0.559852

Avg 1199.392 29.77542 1542.116 0.328479 8.676287 7.147476 0.9415 62.11993 0.825696

Max 1812.585 36.30503 2285.68 0.654436 13.451 10.41273 0.953108 70.48965 0.974442

Table 8

Comparing monthly MAPE and MAE of the ﬁve forecasting models for years 2008.

Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN

MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE

Jan 2.755553 432.0074 3.772639 596.9734 3.582295 572.7084 5.525351 873.5551 7.229363 1178.512

Feb 2.708985 418.9039 3.138612 492.1177 2.980873 465.5721 5.680188 874.2147 5.448159 873.442

Mar 2.732399 393.22 3.79806 556.9457 3.05918 455.8828 4.905664 718.3088 3.640049 538.5853

Apr 2.727465 357.6796 3.464602 467.4888 3.19827 441.2072 4.056142 547.9548 7.78487 993.6697

May 2.628354 337.0947 3.29717 442.3028 3.233623 435.7069 3.474783 467.7393 8.615562 1065.285

Jun 3.369919 524.8216 5.553908 902.2284 5.28079 858.8203 7.778607 1262.292 8.850961 1459.154

July 2.98754 494.6581 4.945192 841.1846 4.74118 810.5707 7.668619 1318.876 7.755603 1381.83

Aug 2.497684 380.4252 4.308996 669.0496 4.199161 656.5596 6.241044 970.9933 5.466167 847.0691

Sept 3.082095 447.0101 4.532354 678.9162 4.434905 656.2067 6.373994 953.5084 8.372256 1206.877

Oct 2.583371 339.3431 2.866891 384.2469 3.053968 413.2177 3.693108 507.0441 6.57141 822.9664

Nov 3.219481 443.9945 3.580922 511.5356 3.926719 566.2431 6.0355 863.6141 6.522948 902.2965

Dec 3.968775 591.79 4.712281 715.2392 4.797472 732.7311 7.142112 1084.901 7.986139 1235.85

Table 9

Comparing monthly MAPE and MAE of the ﬁve forecasting models for years 2009.

Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN

MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE

Jan 3.551231 563.8089 3.286316 519.4472 3.287094 515.5938 5.696559 907.8121 8.251354 1354.426

Feb 3.573261 529.6609 3.559469 523.3133 3.342885 494.7484 5.496871 821.9467 5.333443 811.9557

Mar 4.195243 582.9211 3.915231 550.3764 3.338061 469.4018 5.03026 703.6413 6.178395 840.7755

Apr 3.742518 467.2049 3.965368 518.8956 3.509731 467.4801 4.864393 651.1306 8.555542 1049.714

May 3.52047 438.5759 3.532622 450.0189 3.220264 410.7452 5.18462 688.4336 9.541995 1133.21

Jun 3.055679 403.5158 3.156944 435.0509 3.493532 471.0829 4.262338 601.0097 6.238585 773.3182

July 2.746453 402.1409 4.059018 624.9799 4.089833 625.374 5.524043 858.5648 7.298816 1102.566

Aug 3.987541 637.3829 6.488447 1079.202 6.390219 1053.221 8.614247 1414.386 10.59996 1776.406

Sept 3.30444 442.0353 4.27097 587.1466 4.166079 568.5463 5.503848 761.8917 9.15875 1183.251

Oct 3.197969 410.5056 3.287985 439.7042 3.302124 447.0866 4.585281 632.3976 7.313592 902.4808

Nov 3.206459 411.9029 3.720922 506.2413 4.178728 566.2651 4.683497 641.6848 6.120391 747.6338

Dec 3.509934 537.4623 3.78914 588.6671 4.063178 626.7831 6.684146 1030.1 7.221532 1148.9

Table 6

Correlation coefﬁcient (r) between input parameters and the load data sets.

Input parameter r r

(ISO) (ERCOT)

Dry Bulb Temperature 0.19 e

Dew Point Temperature 0.07 e

Hour of the Day 0.51 0.3418

Day of the Week 0.03 0.0062

Holiday/Weekend indicator (0 or 1) 0.26 e

168-hr (previous week) Lagged Load 0.85 0.82

24-hr Lagged Load 0.90 0.94

Previous 24-hr Average Load 0.56 0.7477

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 615

Fig. 7. Comparison of MAE and MAPE of ﬁve forecasting models.

Fig. 8. Comparison of observed load with forecast load by ﬁve forecasting models.

Fig. 9. Relationship between observed load and forecasted load by ﬁve forecasting models for England region.

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622616

2015e2016. It can be clearly seen that average weekly load gener-

ated for the years 2015 and 2016 by MOFTL-ANN is minimum for

most of the weeks while GRNN seems to generate high peaks

during all the weeks of the testing period. Fig. 11-b shows the

comparison of the actual load with the load predicted by all ﬁve

forecasting models. This comparison illustrates that MOFTL-ANN

achieves the most accurate prediction values than any other

compared forecasting models. Fig. 12 shows a strong relationship

between observed and predicted load than other forecasting

models.

Remark. From all the experimental results shown in this case

study, it can be clearly stated that MOFTL-ANN can enhance accu-

racy and stability. Also, these results illustrate that our proposed

hybrid model is superior to other mentioned forecasting models.

5. Discussions

In this section, some insightful discussions for the experimental

results obtained are provided with more details.

5.1. The signiﬁcance of MOFTL-ANN: based on the Friedman

statistical test

This subsection further compares the forecasting abilities of the

proposed hybrid model with the compared forecasting models by

using the Friedman statistical test.

The Friedman test is a non-parametric, multiple comparisons

test that aims to detect signiﬁcant differences between the results

of two or more forecasting models [67]. The statistic F of the

Friedman test is shown as

F ¼

12N

qðq þ 1Þ

j¼1

Rank



qðq þ 1Þ

(8)

where N is the total number of forecasting results; q is a number of

forecasting models compared; Rank

is the average rank sum

received from each forecasting value for each model. The null hy-

pothesis for Friedman's test is all compared forecasting models

generate same forecasting errors and the alternative hypothesis is

the negation of the null hypothesis.

In this paper, the Friedman test is implemented under

¼ 0:05

signiﬁcance level. The results shown in Table 13 by the Friedman

test shows that forecasting results of all forecasting models are

different. At

¼ 0:05, the null hypothesis fails and the result shows

that every state-of-the-art model generates a unique forecasting

result.

5.2. Stability of the forecasting models

Error reduction in load forecasting is an important aspect of

showing the performance of forecasting models but the model

considering both accuracy and stability is preferred most. Thus, in

this paper, the standard deviation is utilized to evaluate the sta-

bility of the hybrid model and comparison models. Additionally, the

error metrics MAE and NMSE are the most important indicators

used to evaluate the performance of forecasting models. The

Table 10

Forecasting results of the ﬁve forecasting models for years 2015 and 2016.

Forecasting Models MAE MEP RMSE NMSE MAPE Daily peak MAPE r DC IA

MOFTL-ANN Min 1856.41 28.5497 2548.873 0.075212 4.735964 4.974963 0.959558 71.07108 0.979343

Avg 1891.478 31.1451 2586.809 0.077475 4.811885 5.09004 0.960604 71.80927 0.979783

Max 1918.213 32.58433 2624.876 0.079764 4.880519 5.231137 0.961706 72.37645 0.980342

NSGA-II-ANN Min 1921.28 32.07367 2627.326 0.079913 4.890396 5.122149 0.955678 68.43185 0.977111

Avg 1977.051 33.03538 2686.517 0.083565 5.022934 5.310469 0.957391 70.02856 0.977918

Max 2026.914 33.78354 2737.688 0.086768 5.165887 5.566259 0.959283 70.84877 0.979093

FTL-ANN Min 1973.962 32.93888 2681.217 0.083225 4.981819 5.324419 0.956565 69.51491 0.977315

Avg 1984.522 33.45472 2700.732 0.084443 5.020921 5.4492 0.956922 69.99658 0.977613

Max 1993.758 33.78354 2711.276 0.085101 5.066037 5.566259 0.957538 70.55806 0.978027

BPNN Min 1825.277 28.82691 2581.392 0.077143 4.630577 4.658941 0.953929 69.361 0.97624

Avg 1924.832 32.64605 2687.813 0.083703 4.867707 5.228421 0.957728 71.13037 0.978355

Max 2037.037 36.59138 2806.018 0.091153 5.146417 5.888341 0.961168 72.05723 0.980269

GRNN Min 2051.709 31.04214 2786.324 0.089878 5.294412 5.138398 0.91009 64.43026 0.95337

Avg 2405.141 32.18474 3156.274 0.116847 6.15749 6.838773 0.943329 67.11737 0.967091

Max 2948.473 36.05517 3958.327 0.18139 7.697825 8.62503 0.954116 69.89112 0.976004

Table 11

Comparing monthly MAPE and MAE of the ﬁve forecasting models for year 2015.

Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN

MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE

Jan 6.557777 2553.352 6.533426 2557.057 6.976802 2692.852 6.641719 2627.534 10.11026 3892.244

Feb 7.938058 3114.914 8.188373 3210.021 8.286305 3231.942 8.240336 3277.947 11.06183 4177.604

Mar 5.276459 1877.585 5.872915 2061.94 5.784634 2055.804 5.406229 1934.96 8.527302 2941.973

Apr 4.854562 1681.776 5.389833 1877.843 5.112522 1773.487 5.42361 1910.289 8.305066 2859.087

May 5.320253 2037.666 5.615279 2171.024 5.395839 2068.965 5.763074 2240.157 9.326754 3504.155

Jun 3.881549 1753.297 4.016854 1833.068 4.165233 1862.719 3.840786 1760.432 6.108553 2731.761

July 2.709052 1309.892 3.2211 1561.669 3.622653 1766.392 2.642116 1297.729 3.780322 1850.192

Aug 4.282922 2027.395 4.789138 2257.121 5.286263 2528.499 4.369791 2088.256 4.987809 2341.532

Sept 3.839865 1621.875 4.099053 1749.562 4.451325 1887.003 3.854501 1669.951 6.508854 2785.239

Oct 4.682193 1669.414 4.929145 1755.113 5.1547 1853.345 4.78041 1738.486 8.881864 3203.972

Nov 4.789588 1615.619 5.00434 1700.246 5.006814 1696.372 5.104504 1741.555 7.988978 2678.502

Dec 4.756019 1659.083 4.442005 1558.45 4.730951 1656.413 4.582497 1616.409 8.516669 2945.617

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 617

smaller the standard deviation is, the stronger the stability will be.

Table 14 shows the statistical values of MAE and NMSE for by

ﬁve forecasting models over 50 iterations. MOFTL-ANN yields best

statistical values such as a minimum MAE value of 457.8729,

maximum MAE of 547.7367, an average MAE value of 504.1671 and

standard deviation of 547.7367 while other forecasting models

yielded higher statistical values than our proposed hybrid model

for case study 1. Table also reveals that NMSE statistical values of

MOFTL-ANN have lower values than other forecasting models for

case study 1. The statistics show that MAE and NMSE of MOFTL-

ANN yields the least standard deviation of 23.0549 and 0.0 01633

respectively for case study 2.

Tabular results obtained show that MOFTL-ANN is superior to

compared forecasting models. Therefore, we can conclude that

MOFTL-ANN achieves the desirable forecasting stability, is more

stable than other compared models and can be an effective tech-

nique for electricity load forecasting in the real electricity market.

5.3. Run time

All the experiments are performed in the MATLAB R2017a

software on a Windows 10, 64-bit machine with Intel(R) Core(TM)

i5 CPU 760 @ 2.80 GHz with 1 processor.

Table 15 and Table 16 shows the computation time in seconds

obtained from experiment 1 and experiment 2. It can be observed

that the running time of MOFTL is 178.747056 s, 157.5818 s,

147.469428s and 165.879441 s for FON, SCH, ZDT1 and ZDT3 in

Experiment I and 1485.315609 s and 1698.172034 s in Experiment

II. This show that the computation time of MOFTL is shorter than

MOWCA and NSGA-II and demonstrates that it can ﬁnd Pareto

optimal solutions in less time. Similarly, the computation time of

MOFTL-ANN forecasting model is less that NSGA-II-ANN and GRNN.

However, the computation time of the proposed algorithm

(MOFTL) and proposed hybrid model (MOFTL-ANN) is little longer

than some of the comparison models, but this seems justiﬁed due

to its superior forecasting abilities. Further this computation time

can be minimized by using a high-performance systems.

5.4. Summary

Based on the analysis drawn from experiment- I, experiment- II

and the discussions above, the following conclusions can be drawn:

The proposed hybrid forecasting model based on MOFTL has very

strong forecasting ability and more stable forecasting results than

FTL-ANN, NSGA-II-ANN, BPNN, and GRNN.

 MOFTL optimization algorithm outperforms MOWCA, MOPSO,

and NSGA-II over multi-objective test functions. Also, MOFTL

has shorter computation time values of 178.747056 s, 157.5818 s,

147.469428 s, 165.879441 s for FON, SCH, ZDT1 and ZDT3 multi-

objective test functions over MOWCA and NSGA-II.

 MOFTL-ANN, a hybrid model enhanced the load forecasting

accuracy by improving the forecasting ability of ANN. The pro-

posed hybrid model shows improvement of 17.42%, 6.81%,

10.77% and 59.69% MAPE values for England region and 4.20%,

4.16%, 1.14% and 21.85% MAPE values for ERCOT region over

NSGA-II-ANN, FTL-ANN, BPNN, and GRNN forecasting models.

 Results from Friedman statistical test show that all compared

forecasting models generate different forecasting results.

Moreover, standard deviation values of 0.006027 and 0.001633

for England region and ERCOT region by the proposed hybrid

Table 12

Comparing monthly MAPE and MAE of the ﬁve forecasting models for the year 2016.

Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN

MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE

Jan 6.770209 2504.269 6.86443 2535.682 6.911179 2542.062 6.698792 2503.021 10.01666 3652.253

Feb 4.842234 1668.319 4.863737 1671.066 5.009048 1730.429 4.616692 1604.06 7.969788 2682.616

Mar 3.696451 1218.768 3.875041 1288.495 3.838115 1273.992 3.820356 1280.707 6.885492 2267.139

Apr 4.330271 1543.283 4.614292 1658.886 4.432312 1586.132 4.614475 1672.083 7.843856 2755.391

May 5.574658 2160.955 5.923943 2306.216 5.977383 2331.081 6.001755 2371.869 9.070539 3475.827

Jun 3.551048 1622.904 3.902762 1793.313 3.783588 1713.914 3.677367 1694.641 5.400061 2442.063

July 2.96804 1473.355 3.543854 1762.451 4.270406 2167.944 2.999465 1516.806 3.808573 1921.726

Aug 3.757903 1789.257 4.233035 2033.697 4.785633 2313.383 3.826978 1869.077 6.160521 2886.897

Sept 4.344463 1886.302 4.714515 2039.427 5.173031 2238.765 4.326536 1898.989 6.374026 2767.847

Oct 4.88653 1918.785 5.297524 2083.694 5.188784 2036.36 5.05911 2026.186 8.842943 3376.523

Nov 4.045829 1353.42 4.157906 1400.24 4.399363 1495.028 4.203166 1429.131 7.613245 2588.779

Dec 6.444455 2497.858 6.495171 2524.576 6.956649 2678.913 6.704293 2632.203 10.74334 4032.632

Fig. 10. Comparison of MAE and MAPE of ﬁve forecasting models.

P. Singh, P. Dwivedi / Energy 182 (2019) 606e62261 8

model is least compared to other forecasting models which

describe that MOFTL-ANN is the most stable one.

 A novel multi-objective optimization algorithm, namely MOFTL,

outperforms MOWCA, MOPSO, and NSGA-II and it adds a new

optimization algorithm for solving multi-objective optimization

problems. Due to high accuracy and stable load forecasting by

MOFTL-ANN, it can be considered as a new viable option for

time series forecasting abilities.

Remark. Compared to the state-of-the-art models, MOFTL-ANN

generates better results for experiment-I and experiment- II based

on evaluation criteria. On the basis of obtained results, we can

Fig. 11. Comparison of observed load with forecast load by ﬁve forecasting models.

Fig. 12. Relationship between observed load and forecasted load by ﬁve forecasting models for the ERCOT region.

Table 13

Results of Friedman test for Experiment II.

Compared Models Signiﬁcant level (

¼ 0:05) Compared Models Signiﬁcant level (

¼ 0:05)

Case Study I Case Study II

MOFTL-ANN vs. NSGA-II-ANN H

: e

¼ e

MOFTL-ANN vs. NSGA-II-ANN H

: e

¼ e

MOFTL-ANN vs. FTL-ANN F ¼8470.368 MOFTL-ANN vs. FTL-ANN F ¼3368.516

MOFTL-ANNR vs. BPNN P ¼0.00000 (Reject H

) MOFTL-ANNR vs. BPNN P ¼0.00000 (Reject H

)

MOFTL-ANN vs. GRNN MOFTL-ANN vs. GRNN

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 619

deduce that the MOFTL, can be used for solving multi-objective

optimization problems. Additionally, MOFTL-ANN is a robust,

highly accurate and stable hybrid model which can be used as a

practical forecasting model.

5.5. Real applications, limitations and future work

The proposed forecasting model can contribute to the electricity

market in dealing with future changes in load generation and

transmission. It can be used for many real applications such as wind

energy forecasting, hydro energy forecasting that are based on time

series forecasting as the proposed hybrid forecasting model shows

better ability to predict the future load with better accuracy and

stability. There are still some issues that must be resolved in future

studies: load affecting factors such as rainfall and wind speed can

be considered for more reliable and practical forecasting; advanced

data pre-processing method can improve the forecasting results.

6. Conclusions

Higher accuracy and excellent stability in electricity load fore-

casting play a very important role in the operation of power sys-

tems. This paper provides a great tool for energy providers to know

the amount of load demand in a particular region. The results

achieved from forecasting models show that our proposed hybrid

model results in outstanding forecasting performance. It is highly

desirable to develop a model that enhances both accuracy and

stability simultaneously.

To achieve such forecasting model, a novel multi-objective

optimization algorithm MOFTL have been proposed and com-

bined with the artiﬁcial neural network to optimize weights. To

verify the performance of the proposed hybrid model, nine per-

formance evaluation metrics including MAE, MEP, RMSE, NMSE,

MAPE, Pearson's correlation coefﬁcient, DC and IA have been used

in this paper. Based on the series of results drawn from experiment

II, the performance of our proposed hybrid model over two data

sets shows that MOFTL-ANN is capable of predicting load with

higher accuracy and effective stability. Experiment II shows the

effectiveness of the proposed hybrid models over compared models

for electricity load forecast. In both the case study, MOFTL generates

superior results than NSGA-II-ANN, FTL-ANN, BPNN, and GRNN for

all performance evaluation indexes with lesser computation time.

MOFTL-ANN shows improvement of 17.42%, 6.81%, 10.77% and

59.69% MAPE values over NSGA-II-ANN, FTL-ANN, BPNN and GRNN

for England region and 4.20%, 4.16%, 1.14% and 21.85% MAPE values

over NSGA-II-ANN, FTL-ANN, BPNN and GRNN for ERCOT region.

Abbreviations

ACO Ant Colony Optimization

ANN Artiﬁcial Neural Network

BPNN Backpropagation Neural Network

DC Directional Change

DE Differential Evolutionary

ERCOT Electric Reliability Council of Texas

FTL Follow The leader

GA Genetic Algorithm

GRNN Generalized Regression Neural Network

IA Index of Agreement

MAE Mean Absolute Error

MEP Maximum Error Percentage

MAPE Mean Absolute Percentage Error

MOALO Multi-Objective Ant Lion Optimizer

MOFA Multi-Objective Fireﬂy Algorithm

MOFPA Multi-Objective Flower Pollination Algorithm

MOFTL Multi-Objective Follow The Leader

MOP Multi-Objective Problem

MOPSO Multi-Objective Particle Swarm Optimization

MOWCA Multi-objective Water Cycle Algorithm

MOWOA Multi-Objective Whale Optimization Algorithm

NSGA-II Non-dominated Sorting GA

NMSE Normalized Mean Squared Error

PSO Particle Swarm Optimization

RMSE Root Mean Squared Error

SVM Support Vector Machine

Table 14

Statistical values of MAE and NMSE obtained from forecasting results of England and ERCOT region by forecasting models.

Experiment II

Case Study - 1

Forecasting model MAE NMSE

Min Max Avg Std Min Max Avg Std

MOFTL-ANN 457.8729 547.7367 504.1671 27.50089 0.044649 0.063359 0.054012 0.006027

NSGA-II-ANN 582.4166 780.1065 629.1621 57.21251 0.076147 0.125706 0.088878 0.014146

FTL-ANN 523.3067 601.5597 553.2902 27.89063 0.060027 0.079191 0.068972 0.007396

BPNN 458.1641 1057.095 579.5099 177.1325 0.044253 0.22041 0.080456 0.051533

GRNN 624.5459 1812.585 1199.392 437.0547 0.092557 0.654436 0.328479 0.206532

Experiment II

Case Study - 2

MOFTL-ANN 1856.41 1918.213 1891.478 23.05498 0.075212 0.079764 0.077475 0.001633

NSGA-II-ANN 1921.28 13876.34 3166.98 3762.978 0.079913 2.313992 0.306608 0.705325

FTL-ANN 2050.343 8963.795 6646.426 2081.043 0.090739 1.401217 0.85099 0.413967

BPNN 1825.277 2037.037 1924.832 72.61513 0.077143 0.091153 0.083703 0.005043

GRNN 2051.709 2948.473 2405.141 310.9689 0.089878 0.18139 0.116847 0.029389

Table 15

Computation time (seconds) of multi-objective algorithms over FON, SCH, ZDT1 and

ZDT2 test functions.

Models FON SCH ZDT1 ZDT3

MOFTL 178.747056 157.5818 147.469428 165.879441

MOWCA 191.321965 260.145963 229.378474 223.891435

MOPSO 58.09828 68.715269 67.32384 54.446691

NSGA-II 227.947349 207.794532 162.985784 162.985784

Table 16

Computation time (seconds) of multi-objective algorithms over case study - 1 and

case study - 2.

Models Case Study I Case Study II

MOFTL-ANN 1485.315609 1698.172034

NSGA-II-ANN 2477.819376 2866.610707

FTL-ANN 1137.724773 2978.105397

BPNN 524.8064923 399.106586

GRNN 3600.839397 3162.42325

P. Singh, P. Dwivedi / Energy 182 (2019) 606e622620

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Energy 182 (2019) 606e622
Contents lists available at ScienceDirect Energy
j o u r n a l h o me p a g e : w w w . e l s e v i e r . c o m/ l o ca t e / e n e r g y
A novel hybrid model based on neural network and multi-objective
optimization for effective load forecast
Priyanka Singh*, Pragya Dwivedi
Computer Science and Engineering Department, MNNIT Allahabad, Allahabad, 211004, India a r t i c l e i n f o a b s t r a c t Article history:
In recent years, increased attention has been paid by the researchers to predict accurate and stable load Received 17 January 2019
due to its effect on the economy and need for proper management of power systems. However, most of Received in revised form
the previous research focused only on either reducing load forecast error or enhancing the stability, very 21 May 2019
few studies focused on these two issues simultaneously. Introducing a forecasting model to solve both Accepted 10 June 2019
independent objectives at the same time is a challenging task due to the complex behavior of the load Available online 14 June 2019 pattern.
Therefore, to achieve two objectives simultaneously, we propose a novel multi-objective algorithm Keywords:
(MOFTL) based on Follow The Leader algorithm. The effectiveness of MOFTL has been shown by Load forecasting Artiﬁcial neural network
comparing the results with three newly presented MOWCA, MOPSO and NSGA-II multi-objective algo- Multi-objective optimization
rithms. Moreover, to validate the performance of MOFTL, we have combined MOFTL with neural network Hybrid forecasting model
termed as MOFTL-ANN to solve the problem of electricity load forecasting. The proposed hybrid model
Multi-objective follow the leader (MOFTL)
outperforms baseline models over two real-world electricity data sets namely England region and ERCOT
region. MOFTL-ANN shows improvement of 17.42%, 6.81%, 10.77% and 59.69% MAPE values for England
region and 4.20%, 4.16%, 1.14% and 21.85% MAPE values for ERCOT region over NSGA-II-ANN, FTL-ANN, BPNN, and GRNN.
© 2019 Elsevier Ltd. All rights reserved. 1. Introduction
development of an effective model with enhanced forecasting
abilities for accurate and stable load estimation is desired. So far,
Load forecasting is a central and integral process for periodic
several techniques and models have been proposed for load fore-
planning, scheduling, controlling load generation, continuous dis-
casting. In the initial stages of load forecasting, classical statistical
tribution of electricity and expansion of facilities in the power
techniques like BoxeJenkins models and time series techniques
sectors [1,2]. Meanwhile, the study of load forecasting is a chal-
were widely used in load forecasting [6,7]. However, load data have
lenging task due to several uncertain factors such as demand
complex behavior [8] and are inﬂuenced by several factors [9], time
ﬂuctuation, complex historical load curve, calendar effects, fore-
series techniques may exhibit poor performance. Thus, several
casting models, random effects and many more [3].
artiﬁcial intelligence techniques including artiﬁcial neural net-
In the past few years, load forecasting has been a very active and
works (ANNs) [10e13], fuzzy logic [14e16], expert systems, and
popular research topic because of several reasons, deregulation of
support vector machines (SVMs) [17,18] have been applied to time
the electricity market, increased ﬂuctuation in energy demands
series forecasting. Among these techniques, ANN has been widely
and increased operating costs [4]. The poor forecasting methods
used for solving load forecasting problem [19e21].
result in considerable losses to the economy, for example, a
Apart from these techniques, several hybrid models have been
research in 1985 presented that 1% increase in forecasting error
presented in the literature to yield forecasting results with extreme
increased the associated operating costs of up to 10 million pounds
accuracy [33]. To ﬁnd optimal hybrid model, several optimization
every year in the thermal British power system [5]. Therefore, the
algorithms, such as particle swarm optimization (PSO) [34], genetic
algorithm (GA) [35], artiﬁcial bee colony (ABC) [36], differential
evolutionary (DE) [37,38], cuckoo search (CS) algorithm [39,40], ant * Corresponding author.
colony algorithm [41] and follow the leader (FTL) [30] algorithm E-mail addresses: 2015rcs53@mnnit.ac.in, priyankamnnit925@gmail.com
have been applied for load prediction. In these studies [7,30,34,35],
(P. Singh), pragyadwi86@mnnit.ac.in (P. Dwivedi).
https://doi.org/10.1016/j.energy.2019.06.075
0360-5442/© 2019 Elsevier Ltd. All rights reserved.
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 607
researchers focused on improving the accuracy of forecasting
a MOP is to ﬁnd the set of solution termed as a Pareto-optimal set
models by ﬁnding optimal weights of ANN, but stability also plays a
which lies on the Pareto Front (PF) and, solutions have uniform
vital role in determining the effectiveness of a forecasting model.
distribution along the true PF [44,51]. Among multiple trade-offs
Considering only one objective (accuracy or stability) at a time is
and solutions found, choose only one solution using higher-level
insufﬁcient for ﬁnding effective load forecasting model. Thus, a
information for practical purpose [46]. From the fact that multi-
hybrid model based on ANN should optimize the initial weights
objective algorithms can approximately ﬁnd the true Pareto-
while considering accuracy and overcome the instability of the ﬁnal
optimal solutions of most MOPs, we have proposed a novel results simultaneously.
multi-objective algorithm called Multi-Objective Follow The Leader
Researchers have developed an indeﬁnite number of learning
(MOFTL). This algorithm overcomes the problem of solving a single
algorithm that are accurate and predicts outcomes with a high
objective function and enhance the accuracy and stability simul-
degree of conﬁdence. But what happens when training the model taneously for load prediction.
with a different/same subset of same training data set, will the
To obtain higher accuracy and better stability for load prediction
model perform with the same efﬁciency over repeated experi-
at the same time, a novel hybrid model involving the widely
ments. ANN depends on random initial values, therefore, the ﬁnal
acceptable neural network (NN), and our proposed multi-objective
outcome varies over the same input data set. Every predictive
algorithm (MOFTL) is presented in this paper. The validation of our
model has to be accurate and stable. A learning model is stable
proposed multi-objective algorithm is done by testing on four
when the predicted result does not change more than a certain
multi-objective benchmark problems and comparing with three
threshold for the different/same subset of data set over several
newly developed multi-objective algorithms: MOWCA, MOPSO,
experiments. It is noteworthy that a hybrid model that achieves
and NSGA-II. For a comprehensive evaluation of MOFTL, hybrid
high accuracy and strong stability at the same time is a multi-
forecasting model MOFTL-ANN has been presented and applied for
objective problem (MOP) rather than a single-objective problem.
load forecasting over two electricity load data sets collected from
Over the past few decades, multi-objective problems (MOPs)
England and Texas. Results obtained from different forecasting
have attracted researchers in different real-world problems [42,43].
models are tested by the Friedman test to show their statistical
Many multi-objective algorithms such as binary coded elitist non- signiﬁcance.
dominated sorting GA (NSGA-II) [44e46], multi-objective ant lion
The major contributions of our work are as follows:
optimizer (MOALO) [47,48], multi-objective ﬂower pollination al-
gorithm (MOFPA) [49], multi-objective ﬁreﬂy algorithm (MOFA) [1],
A novel multi-objective follow the leader, MOFTL algorithm is
multi-objective whale optimization algorithm (MOWOA) [50] etc. proposed in this paper.
have been introduced in last few years. Table 1 shows the research
To validate the effectiveness of MOFTL, it is tested on four multi-
contribution of different forecasting models. Findings from this
objective benchmark problems and compared with three multi-
literature prove that hybrid forecasting models generate less fore-
objective algorithms: MOWCA, MOPSO, and NSGA-II.
casting error. But stability of the model is another important factor
Further to obtain higher accuracy and excellent stability for load
in load forecasting that cannot be ignored. Based on the ﬁndings,
forecasting, we combined MOFTL with a neural network called
we can conclude that multi-objective optimization algorithms
MOFTL-ANN to optimize network weights.
overcome the shortcomings of single-objective optimization algo-
The comprehensive evaluation of MOFTL-ANN shows that it
rithms and achieve improvement in accuracy and stability simul-
overcomes the problem of solving single objective functions and
taneously. MOPs generate a series of optimal solutions, named
enhance the accuracy and stability for load forecasting.
Pareto-optimal solutions (solutions are equally optimal) for solv-
ing different conﬂicting objectives simultaneously. The ideal goal of
The paper is organized as follows. Section 2 brieﬂy describes Table 1
Research contributions of forecasting algorithms in electricity load forecasting. S.no. Forecasting Methodology Findings algorithm 1 Linear
In Ref. [22], a regression-based daily peak load forecasting method with a Linear regression fails for nonlinear data due to poor non-linear ﬁtting regression
transformation technique is proposed. capability, lacks accuracy. 2 Multi
In Ref. [23], for mid-term load forecast on an hourly basis of US utility,
Multi regression technique has received limited success, it generates regression
multi-regression model has been used.
higher forecasting error for abrupt changes in the environment. 3
Exponential In Ref. [24], ﬁve developed exponential smoothing models are compared They are faster and efﬁcient but generates more error for a long-term load smoothing
over one day ahead load forecast. forecast. 4 ARIMA
In Ref. [25], ARIMA model and transfer function model are applied to STLF ARIMA model generates great results for linear problems but is incapable
by considering weather and load relationship.
of solving nonlinear sensitive part of the load, it generates results only on
the basis of past and current data. 5 Fuzzy logic
In Ref. [26], a fuzzy logic method is employed to forecast the gross
Fuzzy techniques generate better forecast results than statistical models. It electricity demand of Turkey.
provides reliable result but shows a strong dependency on expert systems. 6 SVR
In Ref. [9], a generic STLF strategy based on the support vector regression SVR is a robust and accurate method. It is a highly effective model in
machines is proposed. In Ref. [27], SVR is used to ﬁnd demand response for solving non-linear problems but fails due to higher running time and high an ofﬁce building.
dependency between forecasting accuracy and selected SVM parameters. 7 ANN
In Ref. [28], multilayer perceptron model (MLP) is used for the prediction of ANN has the capability to map the complex input and output relationship
long-term energy consumption. In Ref. [29], LSTM recurrent neural
but due to restricted generalization ability, the solution falls in a local
network is used to solve the problem of load forecasting in a residential
minimum. Also, for large neural network, high running time is required. area. 8 Hybrid
In Ref. [30], a newly proposed evolutionary algorithm is combined with
The efﬁciency of the hybrid model is improved in terms of accuracy but models
ANN to enhance forecasting accuracy. In Ref. [31], GRNN is combined with stability of model is another important factor that needs to be considered
a fruit ﬂy optimization algorithm to improve the capability of GRNN for during the load forecast.
power load forecasting. In Ref. [32], modiﬁed ﬁreﬂy algorithm is used to
ﬁnd the optimal parameters of support vector regression. 608
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622
multi-objective optimization and proposed multi-objective algo-
population [52]. In other words, crowding distance is a measure of
rithm. Section 3 presents the proposed hybrid model and section 4
how close an individual is to its neighboring candidates within
demonstrates the performance of MOFTL over four multi-objective
population [1]. It is always desired that location of solutions in
benchmark problems and numerical results obtained from the
same front level have uniform distribution as, a large diversity of
proposed hybrid model for electricity load forecast problem. In
the individuals can solve the stagnation problem of multi-objective
Section 5, discussions based on experimental results are intro-
algorithms. Firstly, individuals are sorted on the basis of every mth
duced. Section 6 presents the conclusions of this paper.
objective function then boundary individuals of each level are
assigned inﬁnite value and crowding distance of other individuals
2. Multi-objective follow the leader algorithm are calculated as follows.
In this section, the concept of multi-objective optimization and D1 ¼ ∞
some related description of the MOFTL are described. Dn ¼ ∞ (2) Xn1XM f m
2.1. Multi-objective optimization D iþ1 f m i1 i ¼ i¼2 m¼1f max m f min m
Solving one objective function by an evolutionary algorithm is
single-objective optimization problem where an optimal solution is
where D1 and Dn are boundary individuals at each PF, n is number
chosen very conventionally. However, multi-objective optimization of individuals at each PF, f m
i is ﬁtness value of mth objective function
problem solves more than one problem simultaneously. Unlike
of ith individual at PF f and m is number of objective functions and
single-objective optimization problem the output of MOP is series the parameters f max m and f min m are the maximum and minimum
of solutions called Pareto-optimal solutions, which are equally
values of the mth objective function. Larger the average crowding
optimal and their mapping in objective space generates Pareto
distance, better is the diversity in the population.
Front [47]. Selection of Pareto-optimal solutions that are uniformly
distributed along true Pareto Front is recommended. The deﬁnition
2.2. MOFTL (multi-objective follow the leader)
of multi-objective optimization problem, Pareto dominance, crowding distance are given as follows.
MOFTL is a multi-objective version of a novel meta-heuristic FTL
Deﬁnition 1. Multi-objective optimization problem [52].
algorithm proposed by Singh and Dwivedi in 2018 inspired from 8 9
moving behavior of a sheep [30,54]. In ﬂock, sheep having highest > >
score is leader candidate and one with the lowest score is rear > > > Minimize=Maximize : f mðxÞ; m ¼ 1; 2; …::; M; > < > > Subject to : g =
candidate. When leader moves within the ﬂock, other candidates jðxÞ 0; j ¼ 1; 2; …::; J; >
follow them. Mathematical representation for updating the posi- > h > kðxÞ ¼ 0; k ¼ 1; 2; …::; K; > > > : >
tion of every candidate is shown in eq. (2). xðLÞ x ; i ¼ 1; 2; …::; n; > ; i i xðUÞ i 0 (1) X ¼ X X (3) j;k;i j;k;i þ randu* j;leader;i Xj;rear;i
where M, J, K, n represents number of objective functions,
where, X0j;k;i is updated position of sheep Xj;k;i, Xj;leader;i is position
inequality constraints, equality constraints and number of vari-
value of the variable j for leader sheep in ith iteration, Xj;rear;i is
ables, respectively. xðLÞ and xðUÞ are lower and upper boundaries of i i
position value of the variable j for rear sheep in ith iteration, rand is xi for ith variable.
a random value between 0 and 1 and u is inertia weight value lying
between 0 and 1 to control exploration and exploitation process
Deﬁnition 2. Pareto Dominance: [52] A solution xð1Þ is said to
within the solution space. Fig. 1 shows sequential ﬂowchart of FTL.
dominate the other solution xð2Þ, if both the following conditions
Unlike single-objective FTL, multi-objective FTL has several are true:
objective functions and should be evaluated simultaneously. A set
of non-dominated solutions are generated by checking domination
The solution xð1Þ is no worse than xð2Þ in all objectives. Thus, the
condition between two solutions. In MOFTL, all the solutions are
solutions are compared based on their objective function values
distributed within different front levels based on dominance con-
(or location of the corresponding points (zð1Þ and zð2Þ) on the
dition. Top front level ranks ﬁrst in which no individual is domi- objective space).
nated by other individuals, individuals in the second level are
The solution xð1Þ is strictly better than xð2Þ in at least one
dominated by some individuals on level 1, and so on. In the same objective.
front level, there exist a number of solutions that are distributed
which is expected to be in a uniform manner, the crowding distance
Based on comparison of dominance value, all individuals are
operator helps in the selection process and measures closeness
sorted into different front levels. Each front level has a rank, which
between an individual and its neighbors. Larger average crowding
equals its non domination level. The individual in level 1 are
distance brings greater diversity in the population. For solving load
dominated by no other individuals; the individual in level 2 is
forecasting problem with MOFTL, we have considered two objec-
dominated by individuals of only level 1, and so on.
tive functions mean squared error (MSE) and standard deviation
Deﬁnition 3. Crowding Distance: [44]
(STD) to represent accuracy and stability respectively. The equation
For MOPs, there is more than one objective function to be
of objective functions for load forecasting can be deﬁned as follows:
minimized or maximized. To select the best solution in the popu-
lation that satisﬁes the condition of all objective function simul- 1 X n MSE ¼ ðy0
taneously, crowding-distance mechanism is used [53]. The N i yiÞ2 (4) i¼1
crowding distance of an individual is a measure of objective space
of individual that is not occupied by any other solution in the
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 609
Fig. 2. Feed forward neural network. [30]: 0 1 X m 1 Y @ A k ¼ Wk;j: P þ Q n j (6) j¼1 1 þ exp i¼1Wj;iXi þ Qj
where, 1in, 1 j m Wi;j represents weight between nodes i
and j, Qj is activation function of jth node and n and m represents
number of nodes in input and hidden layer respectively.
So far several optimization algorithms have been combined
with neural network to ﬁnd optimal weights such as genetic al-
gorithm [35], PSO [60], fruit ﬂy optimization algorithm [31] to
predict accurate load forecasting. But ﬁnding a model that precisely
generate load along with greater stability is bigger problem of load
forecasting. Thus, a novel multi-objective algorithm has been pro-
Fig. 1. Flowchart of FTL algorithm.
posed in this paper which can not only predicts load with high
precision but also stable forecasting.
STD ¼ stdðy0i yiÞ; i ¼ 1; 2; …:N (5)
3.2. The MOFTL-ANN hybrid model
In this section, the weights of neural network are updated by a
3. Proposed hybrid forecasting model
multi-objective optimization algorithm, MOFTL to allow ANN to
achieve enhanced accuracy and stability simultaneously. For
In this section, detailed study of artiﬁcial neural network and
effective load forecasting, it is recommended to consider both ac-
hybrid MOFTL-ANN model is introduced.
curacy and stability at the same time. Thus, the optimization of
initial weights of a neural network should satisfy both the objective
3.1. Artiﬁcial neural network
functions simultaneously. Mean squared error (MSE) f1 in fore-
casting are highly considered for performance evaluation of load
An artiﬁcial neural network (ANN) is intelligence-based learning
forecasting models. In this study, standard deviation of forecasting
system, inspired from functioning of human brain which consists of
error is chosen as another indicator for representing the stability of
several small processing elements called neurons [55,56]. These
a forecasting model f2. Smaller the mean squared error and stan-
neurons are interconnected and these connection have some
dard deviation value, higher is the accuracy and stability. The
weight associated with themselves. Neurons receive information
pseudo-code of the hybrid MOFTL-ANN model is presented in Al-
through several input nodes, process them internally and generates gorithm 1.
a response [57]. ANN being universal approximator can approxi-
mate large class of functions, generate higher accuracy over other
nonlinear model [58]. ANN architecture is composed of three 4. Experiments and analysis
layers: input layer, hidden layer and output layer. In input layer,
signiﬁcant features which shows impact on prediction are consid-
To verify the effectiveness of the proposed multi-objective al-
ered and hidden layer/layers use these features to generate inter-
gorithm and hybrid model, two different experiments are per-
mediate solution and ﬁnal predicted value is produced at output
formed. In experiment 1, MOFTL is compared with three newly
node of network [59]. Usually, feed forward neural network with
developed multi-objective algorithms namely MOWCA [53,61],
single hidden layer is preferred in time series forecasting. A simple
MOPSO [62], and NSGA-II [46]. In Experiment II, the performance of
architecture of feed forward neural network is shown in Fig. 2.
hybrid model (MOFTL-ANN)is compared with ﬁve state-of-art
Mathematical representation of relationship between inputs X ¼
forecasting models. Here, the obtained results are compared over
X1; X2; …::Xn and output Yk for logistic activation function in hidden
two load data sets namely, England region and ERCOT region. De-
layer and linear function in output layer can be shown as follows
tails of the experiment are described in the following subsections. 610
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622
4.1. Experiment I: tests of MOFTL
forecasting problem. Implementation details are discussed in the forthcoming subsections.
In this experiment, four multi-objective problem and three well
known multi-objective algorithms namely MOPSO, MOWCA and 4.3. Data pre-processing
NSGA-II have been used to test and compare the performance of
MOFTL algorithm. Mathematical representation of multi-objective
Before going for load prediction, data pre-processing is done to problems are as follows:
eliminate spikes and outliers. The missing and extra data values are
managed and data normalization (MIN-MAX) is done between the
interval of ½ 1; 1 to reduce the training time. To perform different FON:
experiments, the data set is divided into training and testing data. ! P 1 2 Minimize : f ð1Þ ¼ 1 exp 3 pﬃﬃﬃ i¼1 xi 3 4.4. Evaluation metrics ! P 1 2 Minimize : f ð2Þ ¼ 1 exp 3 pﬃﬃﬃ i¼1 xi þ 3
Since the past few decades, several evaluation metrics have where; 4 xðiÞ 4
been developed to evaluate the performance of the load forecast. SCH:
However, there are no predeﬁned rules for selecting these metrics. Minimize : f ð1Þ ¼ x2
In this paper, we use the nine commonly used evaluation metrics Minimize : f ð2Þ ¼ ðx 2Þ2
shown in Table 3 to evaluate the performance forecasting models. where; 103 xðiÞ 103
Mean absolute error (MAE) indicate the overall level of errors [50]. ZDT1:
Normalized mean squared error (NMSE) calculates the overall de- Minimize : f ð1Þ ¼ xð1Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃ !
viations between forecasted and measured load. The root of mean xð1Þ Minimize : f ð2Þ ¼ g* 1 squared error (RMSE) re g
ﬂects the degree of differences between the X
predicted and measured values and is more stable than MSE and 9 N where : g ¼ 1 þ xðiÞ and 0 xðiÞ 1; 1 i n n 1 i¼2
less sensitive to extreme errors. Mean absolute percent error ZDT3:
(MAPE) shows accuracy as a percentage of forecasting error. Minimize : f ð1Þ ¼ xð1Þ ﬃﬃﬃﬃﬃﬃﬃﬃ s ﬃ
Directional Change (DC) exhibits the movement directions or ! xð1Þ xð1Þ
turning points of prediction. Pearson's correlation coef Minimize : f ð2Þ ¼ g* 1 sinð10pxð1ÞÞ ﬁcient (r) g g
shows the correlation between the predicted value and the 9 XN where : g ¼ 1 þ xðiÞ and 0 xðiÞ 1; 1 i n
observed value. Lastly, the index of agreement (IA) is a standard n 1 i¼2
metric of the degree of model prediction error [64]. In the following
subsections, we present two case studies on various data sets to
These four test functions and the performance metrics of
validate the effectiveness of MOFTL-ANN. ALL the experimental
reverse generational distance (RGD) [63] along with three newly
parameters involved during the experiment is shown in Table 5.
developed multi-objective algorithms- MOWCA, MOPSO, and
The experiment has been performed 100 times with population
NSGA-II verify the performance of our proposed algorithm MOFTL.
size of 50, 5,000 maximum iteration, 10 hidden neurons, one
The mathematical formula for RGD is shown in eq. (7).
output neuron and the number of input neurons is same as the
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v ﬃ
number of input parameters taken in the case study. u
In order to validate the performance of our proposed hybrid 1 X N u RGD ¼ t d2 (7)
model MOFTL-ANN, we have compared our model to following N j j¼1 state-of-art models.
where, dj is Euclidean distance between ith true PF and nearest
NSGA-II-ANN (Neural network with NSGA-II)
members of obtained PF and N is number of true PF.
FTL-ANN (Neural network with Follow the leader algorithm)
Each test function has been performed 100 times for 500 iter-
BPNN (Backpropagation neural network)
ations and 100 population to ﬁnd Pareto optimal solutions. The
GRNN (Generalized regression neural network)
results obtained from the experiment are shown in Table 2 and
Figs. 3e6. From Table 2, it can be easily demonstrated from all the
statistical values that MOFTL is superior to MOWCA, MOPSO, and 4.5. Case study-1: ISO England
NSGA-II. Our proposed multi-objective algorithm yields the best
values of standard deviation for FON- 0.000156, SCH- 0.000366, 4.5.1. Data description
ZDT1- 0.010104 and ZDT3- 0.002242 multi-objective problems.
Hourly data set from England region from years 2004e2007 and
And these results prove that lower is standard deviation lesser in
out-of-sample data from the years 2008 and 2009 are used for
scattering and better is the solution.
training and testing purpose [65]. The statistical parameter of data
Remark. Results shown proves that MOFTL performs well and is
set is shown in Table 4 and the correlation coefﬁcient (r) between
superior to MOWCA, MOPSO, NSGA-II. MOFTL achieves better
independent variables given as input to a neural network and load
optimal solutions than others with higher accuracy and also ﬁnds
is shown in Table 6. Higher the value of r higher is the correlation
the best values of RGD. Thus, results show that our proposed al-
between the input parameter and load.
gorithm is effective and can be further used for solving real-world
multi-objective problems such as load forecasting problem of 4.5.2. Result
predicting load with higher accuracy and stability.
The results obtained for the year 2008 and 2009 by all the
forecasting models are shown in Table 7 with their minimum,
4.2. Experiment II: tests of MOFTL-ANN
average and maximum values. It clearly shows that MOFTL-ANN
brings higher accuracy in terms of all performance evaluation
To deeply analyze the superior performance of the proposed
metrics mentioned earlier. Moreover, from the table, it can be
algorithm, MOFTL is implemented to solve the short-term load
observed that MOFTL enhances the prediction ability of ANN than
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 611 Table 2
Results of the multi-objective algorithms (using RGD) on the four test functions adopted in this paper. Algorithm FON Algorithm SCH Min Avg Median Max Std. Min Avg Median Max Std. MOFTL 0.00447282 0.00478 0.004756 0.005365 0.000156 MOFTL 0.01701711 0.017963 0.01793 0.018785 0.000366 MOWCA 0.01363076 0.117798 0.096826 0.429674 0.093677 MOWCA 0.01883266 0.020035 0.020002 0.021894 0.000675 MOPSO 0.0045397 0.00491 0.004892 0.005431 0.00019 MOPSO 0.01927343 0.021082 0.02084 0.026767 0.001125 NSGA-II 0.00437934 0.004857 0.004838 0.005422 0.000204 NSGA-II 0.01937874 0.021189 0.021199 0.023965 0.000993 Algorithm ZDT1 Algorithm ZDT3 Min Avg Median Max Std. Min Avg Median Max Std. MOFTL 0.004333 0.008895 0.004785 0.340285 0.010104 MOFTL 0.190448 0.192423 0.191566 0.208109 0.002242 MOWCA 0.004412 0.026285 0.004876 0.319874 0.057746 MOWCA 0.191381 0.195551 0.19257 0.303958 0.01236 MOPSO 1.143838 1.457361 1.459175 1.615171 0.077166 MOPSO 0.804463 1.081175 1.092714 1.276132 0.09646 NSGA-II 0.111613 0.137112 0.13742 0.173442 0.033702 NSGA-II 0.195631 0.20023 0.199969 0.237757 0.00471 Algorithm 1: MOFTL-ANN. Objective Functions: 8 9 < = f Minimize 1 ðxÞ ¼ MSE : f2ðxÞ ¼ STD ; Input: Xtrain - training samples Xtest - testing samples Output: Xforecast - forecasting data Parameters:
Maxit - maximum number of iterations
Npop - number of sheep within a ﬂock Dim - number of dimensions Xi - position of ith sheep Ri - rank value of ith sheep
di - crowding distance values of ith sheep
Fi - ﬁtness values of ith sheep Begin
1 Set all the input parameters of MOFTL.
2 Initialize random positions of all sheep within the ﬂock of size Npop and Dim dimensions in the range of - 1 and þ 1.
3 For each sheep i within the ﬂock i ¼ 1:Npop
4 Compute corresponding ﬁtness function f1 and f2 using non-dominated sort ranking process.
5 Calculate crowding distance di. 6 end for 7
=* Determine leader Xleader and rear Xrear sheep on the basis of their rank and crowding distance. = 8 While (i < Maxit) 9
=* Evaluate new solution and update position of sheep =
10 For each sheep i within the ﬂockj¼1:Npop 11 For k¼1:Dim 0 12 X ¼ X j;k;i
j;k;i þ randu*ðXj;leader;i Xj;rear;iÞ 13 end for 14 end for 15
=* Check if any sheep goes beyond the boundary conditions. =
16 For each sheep i within the ﬂock i ¼ 1:Npop
17 Compute corresponding ﬁtness function f1 and f2 using non-dominated sort ranking process.
18 Calculate crowding distance di. 19 end for
20 = Selection of sheep for next generation. =
21 Merge X0 and X, sort them on the basis of rank and
crowding distance and update X with Npop sheep.
22 = Update leader and rear sheep. = 23 end while 24 Return X
25 Set the weights of neural network according to X. 26 Use Xtrain to train ANN.
27 Use electricity load test data set Xtest to obtain the predicted valve Xforecast. End
NSGA-II does. For example, minimum values for MAPE of MOFTL-
forecasting model. However, MOFTL-ANN generates higher accu-
ANN and NSGA-II-ANN are 3.2019% and 3.9615%, respectively. The
racy than FTL-ANN as the reason lies in the fact that FTL-ANN is
average MAE value of MOFTL-ANN is 504.1671 Mwh which is the
single-objective optimization i.e. enhancing the accuracy.
least by any of the compared forecasting models while GRNN has a
Table 8 and Table 9 shows monthly MAPE and MAE of all fore-
maximum average MAE value of 1199.392 Mwh. The average values
casting models for the years 2008 and 2009 respectively. MOFTL-
of MAE- 504.1671 Mwh, MEP- 15.26862%, RMSE- 655.7201 Mwh,
ANN signiﬁcantly outperforms all other forecasting models in
NMSE- 0.05401 Mwh, MAPE- 3.4967%, Pearson correlation coefﬁ-
nearly every month due to their higher rate of directional predic-
cient (r)- 0.9741 obtained by MOFTL-ANN makes it more reliable
tion while GRNN produced maximum values of MAE and MAPE for 612
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622
Fig. 3. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for FON. (Note: PF represents Pareto Font).
Fig. 4. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for SCH. (Note: PF represents Pareto Font).
almost every month for both the years. In Tables 8 and 9, it can also
MOFTL-ANN cannot be ignored as it brings both stability and ac-
be noted that for a few months like Feb 2009, FTL-ANN generated
curacy simultaneously and overall MAPE and MAE of MOFTL-ANN
lower MAPE than MOFTL-ANN. The reason lies in the fact that FTL-
for the years 2008 and 2009 is lower than all compared fore-
ANN is a single-objective optimization model while MOFTL-ANN casting models.
optimizes both accuracy and stability. But the overall result of
Fig. 7 shows the comparison of average MAE and MAPE value
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 613
Fig. 5. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for ZDT1. (Note: PF represents Pareto Font).
Fig. 6. Obtained Pareto optimal solutions by MOFTL, MOWCA, MOPSO and NSGA-II for ZDT3. (Note: PF represents Pareto Font).
generated by all ﬁve forecasting model on a daily basis. The ﬁgure
objective functions at the same time. Fig. 8-a shows a comparison of
clearly shows that MOFTL-ANN generated least MAE on each day of
average weekly load error found during load prediction for years
the week while GRNN produced maximum MAE and MAPE. It can
2008e2009. The ﬁgure reveals that MOFTL established to draw
also be noted that MAPE generated on Friday by FTL-ANN is less
least average weekly load error in almost all weeks for the years
than that of MOFTL-ANN due to the limitation of solving two
2008 and 2009 while GRNN generated high peaks of average load 614
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 Table 3 Evaluation metrics. S.No. Metric Equation 1 Mean absolute error 1X n MAE ¼ y N j¼1 j y0j 2 Maximum error percentage ð y0j yjÞ MEP ¼ 100 yj
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 Root of mean squared error 1Xn RMSE ¼ ðy N j¼1 j y0 jÞ2 4 Normalized mean squared error 1 X X n 1 n NMSE ¼ ðy ðy D2 j y0j Þ2 , D2 ¼ j yÞ2 N j¼1 N 1 j¼1 5 Mean absolute percent error 1X y n j y0 j MAPE ¼ 100 N j¼1 yj 6
Daily peak mean absolute percent error 1 jmaxðTL Daily Peak MAPE ¼ m Þ maxðFLm Þj 100 N maxðTLmÞ P 7
Pearson's correlation coefﬁcient n r ¼ j¼1ðyj yÞðy0j y0Þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P P n n j¼1ðyj yÞ2 j¼1ðy0j y0Þ2 8 9 8 Directional change 100 X < = N1 0; otherwise DC ¼ a N 1 j¼1
t , at ¼ : 1; if ðy0jþ1 yjÞðyjþ1 yjÞ > 0 ; P 9 Index of agreement N IA ¼ 1 j¼1ðyj yjÞ2 PNj¼1ðy0j yÞðyj yÞ2
Note: yj ¼ actual value of day j, y0j ¼ predicted value of day j, y ¼ mean of actual value, y0 ¼ mean of Predicted value, FLm ¼ forecast load value for every 24 h, TLm ¼ target
load for every 24 h, N ¼ number of elements in training data.
error during all the weeks of the testing period. Fig. 8-b shows the Table 5
comparison of load obtained after prediction by all ﬁve models with Experimental parameter values.
the observed load. This comparison illustrates that MOFTL-ANN Forecasting model Experimental parameter Value
achieves the most accurate prediction values and also load con- MOFTL-ANN MOFTL inertia weight 0.75
sumption during weekends is less compared to weekdays. Fig. 9
MOFTL maximum number of iterations 100
shows the relationship between observed and predicted load by MOFTL population size 50
all ﬁve forecasting models in which MOFTL-ANN show a strong NSGA-II-ANN GA crossover ratio 0.7 relation. GA mutation ratio 0.3 GA population size 50
Remark. From the obtained experimental results, MOFTL-ANN FTL-ANN Tolerance value 108
seems to be the most accurate hybrid model. These tables and FTL inertia weight 0.67
ﬁgures illustrate that our proposed hybrid model is the most su- GA population size 50
perior, accurate and stable forecasting model. BPNN Learning rate 0.1 GRNN Training precision 105 4.6. Case study-2: ERCOT
obtained result, it can be clearly noticed that MOFTL-ANN ﬁnds a 4.6.1. Data description
better optimal solution than other forecasting models. Evaluation
In this experiment, hourly load data set of ERCOT for the years
metrics like average values of MAE- 1891.478 Mwh, MEP- 31.145%,
2009e2014 have been taken as training data and years 2015e2016
RMSE- 2586.809 Mwh, NMSE- 0.077475 Mwh, MAPE- 4.811885%,
for testing purpose [66]. Table 6 shows the correlation between
Pearson correlation coefﬁcient (r)- 0.9606 makes MOFTL-ANN su-
input variables and load taken into consideration. ALL the statistical
perior than other forecasting models. GRNN generates a minimum
parameters related to the ERCOT data set is shown in Table 4. Five
average directional change of 67.1173 while MOFTL-have maximum
parameters namely, an hour of the day, the day of the week, 168-hr - 71.8093.
lagged load, 24-hr lagged load and previous 24-hr average load has
Table 11 and Table 12 shows month wise MAE and MAPE error
been chosen for load prediction of ERCOT region.
values of all ﬁve forecasting models. The table compares the result
obtained from the experiment and found that MOFTL-ANN gener- 4.6.2. Results
ates superior result. Very exceptionally, it can be noted that for the
Table 10 shows the minimu, average and maximum values of all
month Dec 2015, FTL-ANN generated lesser MAPE of 4.44200%
the forecasting models namely, MOFTL-ANN, NSGA-II-ANN, FTL-
while MAPE of MOFTL-ANN is 4.9073%. The reason for this
ANN, BPNN, and GRNN for the years 2015 and 2016. From the Table 4
Statistical parameters for the data used in this paper. Region Samples # of elements Mean Max Median Min Std. ISO New Pool England All 52608 15093.259 28130 15277 9040 2958.703 Training 35064 15270.068 28130 15473 9152 3007.685 Testing 17544 14739.883 26111 14982 9040 2825.413 ERCOT All 70128 37865.985 71092.609 35656.533 21378.730 9260.466 Training 52584 37208.379 68317.669 34917.492 21378.730 9155.398 Testing 17544 39837.002 71092.609 37625.894 24337.467 9294.055
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 615 Table 6
phenomena is likely that FTL-ANN is a single-objective model
Correlation coefﬁcient (r) between input parameters and the load data sets.
which can handle trends and randomness while MOFTL-ANN is a Input parameter r r
multi-objective model and it has to handle both accuracy and sta- bility simultaneously. (ISO) (ERCOT)
The comparison of average MAE and MAPE value generated by Dry Bulb Temperature 0.19 e
all ﬁve forecasting model on a daily basis is shown in Fig. 10. The Dew Point Temperature 0.07 e Hour of the Day 0.51 0.3418
ﬁgure clearly shows that MOFTL-ANN generated least MAE on each Day of the Week 0.03 0.0062
day of the week except Sunday, BPNN produced minimum MAE.
Holiday/Weekend indicator (0 or 1) 0.26 e
Similarly, on Sunday and Tuesday, MAPE of BPNN is less than
168-hr (previous week) Lagged Load 0.85 0.82
MOFTL-ANN. The reason for such a situation lies in the nonlinear 24-hr Lagged Load 0.90 0.94
behavior of data trends. GRNN produced maximum MAE and MAPE Previous 24-hr Average Load 0.56 0.7477
on each day that proves its unsuitability for load prediction. Fig. 11-
a shows a comparison of average weekly load error for the year Table 7
Forecasting results of the ﬁve forecasting models. Forecasting Models MAE MEP RMSE NMSE MAPE Daily peak MAPE r DC IA MOFTL-ANN Min 457.8729 11.49369 597.0174 0.044649 3.201904 2.504589 0.969801 72.55316 0.983564 Avg 504.1671 15.26862 655.7201 0.054012 3.496752 3.03031 0.974153 73.87733 0.988558 Max 547.7367 19.60747 711.1898 0.063359 3.828821 3.499827 0.979748 74.77056 0.986071 NSGA-II-ANN Min 582.4166 19.3054 779.6671 0.076147 3.92285 3.631563 0.952252 69.86832 0.96693 Avg 629.1621 20.92272 840.1871 0.088878 4.234364 4.094272 0.957317 70.9776 0.976367 Max 780.1065 24.75158 1001.752 0.125706 5.13332 6.045748 0.961517 72.08573 0.979975 FTL-ANN Min 523.3067 17.02067 692.2375 0.060027 3.543245 3.205244 0.959864 70.89437 0.979468 Avg 553.2902 19.94162 741.0733 0.068972 3.75242 3.675925 0.965272 72.49957 0.984357 Max 601.5597 22.40104 795.0949 0.079191 4.126287 3.940751 0.970286 73.98963 0.982063 BPNN Min 458.1641 16.82907 594.3669 0.044253 3.07581 2.648605 0.886737 62.68027 0.9298 Avg 579.5099 20.58969 775.7784 0.080456 3.919104 3.711811 0.959404 72.55772 0.977829 Max 1057.095 28.75839 1326.469 0.22041 7.345048 6.091889 0.979059 75.22659 0.988835 GRNN Min 624.5459 19.41066 859.5813 0.092557 4.21692 4.155094 0.911327 55.56632 0.559852 Avg 1199.392 29.77542 1542.116 0.328479 8.676287 7.147476 0.9415 62.11993 0.825696 Max 1812.585 36.30503 2285.68 0.654436 13.451 10.41273 0.953108 70.48965 0.974442 Table 8
Comparing monthly MAPE and MAE of the ﬁve forecasting models for years 2008. Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE Jan 2.755553 432.0074 3.772639 596.9734 3.582295 572.7084 5.525351 873.5551 7.229363 1178.512 Feb 2.708985 418.9039 3.138612 492.1177 2.980873 465.5721 5.680188 874.2147 5.448159 873.442 Mar 2.732399 393.22 3.79806 556.9457 3.05918 455.8828 4.905664 718.3088 3.640049 538.5853 Apr 2.727465 357.6796 3.464602 467.4888 3.19827 441.2072 4.056142 547.9548 7.78487 993.6697 May 2.628354 337.0947 3.29717 442.3028 3.233623 435.7069 3.474783 467.7393 8.615562 1065.285 Jun 3.369919 524.8216 5.553908 902.2284 5.28079 858.8203 7.778607 1262.292 8.850961 1459.154 July 2.98754 494.6581 4.945192 841.1846 4.74118 810.5707 7.668619 1318.876 7.755603 1381.83 Aug 2.497684 380.4252 4.308996 669.0496 4.199161 656.5596 6.241044 970.9933 5.466167 847.0691 Sept 3.082095 447.0101 4.532354 678.9162 4.434905 656.2067 6.373994 953.5084 8.372256 1206.877 Oct 2.583371 339.3431 2.866891 384.2469 3.053968 413.2177 3.693108 507.0441 6.57141 822.9664 Nov 3.219481 443.9945 3.580922 511.5356 3.926719 566.2431 6.0355 863.6141 6.522948 902.2965 Dec 3.968775 591.79 4.712281 715.2392 4.797472 732.7311 7.142112 1084.901 7.986139 1235.85 Table 9
Comparing monthly MAPE and MAE of the ﬁve forecasting models for years 2009. Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE Jan 3.551231 563.8089 3.286316 519.4472 3.287094 515.5938 5.696559 907.8121 8.251354 1354.426 Feb 3.573261 529.6609 3.559469 523.3133 3.342885 494.7484 5.496871 821.9467 5.333443 811.9557 Mar 4.195243 582.9211 3.915231 550.3764 3.338061 469.4018 5.03026 703.6413 6.178395 840.7755 Apr 3.742518 467.2049 3.965368 518.8956 3.509731 467.4801 4.864393 651.1306 8.555542 1049.714 May 3.52047 438.5759 3.532622 450.0189 3.220264 410.7452 5.18462 688.4336 9.541995 1133.21 Jun 3.055679 403.5158 3.156944 435.0509 3.493532 471.0829 4.262338 601.0097 6.238585 773.3182 July 2.746453 402.1409 4.059018 624.9799 4.089833 625.374 5.524043 858.5648 7.298816 1102.566 Aug 3.987541 637.3829 6.488447 1079.202 6.390219 1053.221 8.614247 1414.386 10.59996 1776.406 Sept 3.30444 442.0353 4.27097 587.1466 4.166079 568.5463 5.503848 761.8917 9.15875 1183.251 Oct 3.197969 410.5056 3.287985 439.7042 3.302124 447.0866 4.585281 632.3976 7.313592 902.4808 Nov 3.206459 411.9029 3.720922 506.2413 4.178728 566.2651 4.683497 641.6848 6.120391 747.6338 Dec 3.509934 537.4623 3.78914 588.6671 4.063178 626.7831 6.684146 1030.1 7.221532 1148.9 616
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622
Fig. 7. Comparison of MAE and MAPE of ﬁve forecasting models.
Fig. 8. Comparison of observed load with forecast load by ﬁve forecasting models.
Fig. 9. Relationship between observed load and forecasted load by ﬁve forecasting models for England region.
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 617 Table 10
Forecasting results of the ﬁve forecasting models for years 2015 and 2016. Forecasting Models MAE MEP RMSE NMSE MAPE Daily peak MAPE r DC IA MOFTL-ANN Min 1856.41 28.5497 2548.873 0.075212 4.735964 4.974963 0.959558 71.07108 0.979343 Avg 1891.478 31.1451 2586.809 0.077475 4.811885 5.09004 0.960604 71.80927 0.979783 Max 1918.213 32.58433 2624.876 0.079764 4.880519 5.231137 0.961706 72.37645 0.980342 NSGA-II-ANN Min 1921.28 32.07367 2627.326 0.079913 4.890396 5.122149 0.955678 68.43185 0.977111 Avg 1977.051 33.03538 2686.517 0.083565 5.022934 5.310469 0.957391 70.02856 0.977918 Max 2026.914 33.78354 2737.688 0.086768 5.165887 5.566259 0.959283 70.84877 0.979093 FTL-ANN Min 1973.962 32.93888 2681.217 0.083225 4.981819 5.324419 0.956565 69.51491 0.977315 Avg 1984.522 33.45472 2700.732 0.084443 5.020921 5.4492 0.956922 69.99658 0.977613 Max 1993.758 33.78354 2711.276 0.085101 5.066037 5.566259 0.957538 70.55806 0.978027 BPNN Min 1825.277 28.82691 2581.392 0.077143 4.630577 4.658941 0.953929 69.361 0.97624 Avg 1924.832 32.64605 2687.813 0.083703 4.867707 5.228421 0.957728 71.13037 0.978355 Max 2037.037 36.59138 2806.018 0.091153 5.146417 5.888341 0.961168 72.05723 0.980269 GRNN Min 2051.709 31.04214 2786.324 0.089878 5.294412 5.138398 0.91009 64.43026 0.95337 Avg 2405.141 32.18474 3156.274 0.116847 6.15749 6.838773 0.943329 67.11737 0.967091 Max 2948.473 36.05517 3958.327 0.18139 7.697825 8.62503 0.954116 69.89112 0.976004 Table 11
Comparing monthly MAPE and MAE of the ﬁve forecasting models for year 2015. Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE Jan 6.557777 2553.352 6.533426 2557.057 6.976802 2692.852 6.641719 2627.534 10.11026 3892.244 Feb 7.938058 3114.914 8.188373 3210.021 8.286305 3231.942 8.240336 3277.947 11.06183 4177.604 Mar 5.276459 1877.585 5.872915 2061.94 5.784634 2055.804 5.406229 1934.96 8.527302 2941.973 Apr 4.854562 1681.776 5.389833 1877.843 5.112522 1773.487 5.42361 1910.289 8.305066 2859.087 May 5.320253 2037.666 5.615279 2171.024 5.395839 2068.965 5.763074 2240.157 9.326754 3504.155 Jun 3.881549 1753.297 4.016854 1833.068 4.165233 1862.719 3.840786 1760.432 6.108553 2731.761 July 2.709052 1309.892 3.2211 1561.669 3.622653 1766.392 2.642116 1297.729 3.780322 1850.192 Aug 4.282922 2027.395 4.789138 2257.121 5.286263 2528.499 4.369791 2088.256 4.987809 2341.532 Sept 3.839865 1621.875 4.099053 1749.562 4.451325 1887.003 3.854501 1669.951 6.508854 2785.239 Oct 4.682193 1669.414 4.929145 1755.113 5.1547 1853.345 4.78041 1738.486 8.881864 3203.972 Nov 4.789588 1615.619 5.00434 1700.246 5.006814 1696.372 5.104504 1741.555 7.988978 2678.502 Dec 4.756019 1659.083 4.442005 1558.45 4.730951 1656.413 4.582497 1616.409 8.516669 2945.617
2015e2016. It can be clearly seen that average weekly load gener-
of two or more forecasting models [67]. The statistic F of the
ated for the years 2015 and 2016 by MOFTL-ANN is minimum for Friedman test is shown as
most of the weeks while GRNN seems to generate high peaks " #
during all the weeks of the testing period. Fig. 11-b shows the 12N X q qðq þ 1Þ2 F ¼ Rank2 (8)
comparison of the actual load with the load predicted by all ﬁve qðq þ 1Þ j 4 j¼1
forecasting models. This comparison illustrates that MOFTL-ANN
achieves the most accurate prediction values than any other
where N is the total number of forecasting results; q is a number of
compared forecasting models. Fig. 12 shows a strong relationship
forecasting models compared; Rank
between observed and predicted load than other forecasting j is the average rank sum
received from each forecasting value for each model. The null hy- models.
pothesis for Friedman's test is all compared forecasting models
Remark. From all the experimental results shown in this case
generate same forecasting errors and the alternative hypothesis is
study, it can be clearly stated that MOFTL-ANN can enhance accu-
the negation of the null hypothesis.
racy and stability. Also, these results illustrate that our proposed
In this paper, the Friedman test is implemented under a ¼ 0:05
hybrid model is superior to other mentioned forecasting models.
signiﬁcance level. The results shown in Table 13 by the Friedman
test shows that forecasting results of all forecasting models are 5. Discussions
different. At a ¼ 0:05, the null hypothesis fails and the result shows
that every state-of-the-art model generates a unique forecasting
In this section, some insightful discussions for the experimental result.
results obtained are provided with more details.
5.2. Stability of the forecasting models
5.1. The signiﬁcance of MOFTL-ANN: based on the Friedman statistical test
Error reduction in load forecasting is an important aspect of
showing the performance of forecasting models but the model
This subsection further compares the forecasting abilities of the
considering both accuracy and stability is preferred most. Thus, in
proposed hybrid model with the compared forecasting models by
this paper, the standard deviation is utilized to evaluate the sta-
using the Friedman statistical test.
bility of the hybrid model and comparison models. Additionally, the
The Friedman test is a non-parametric, multiple comparisons
error metrics MAE and NMSE are the most important indicators
test that aims to detect signiﬁcant differences between the results
used to evaluate the performance of forecasting models. The 618
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 Table 12
Comparing monthly MAPE and MAE of the ﬁve forecasting models for the year 2016. Month MOFTL-ANN NSGA-II-ANN FTL-ANN BPNN GRNN MAPE MAE MAPE MAE MAPE MAE MAPE MAE MAPE MAE Jan 6.770209 2504.269 6.86443 2535.682 6.911179 2542.062 6.698792 2503.021 10.01666 3652.253 Feb 4.842234 1668.319 4.863737 1671.066 5.009048 1730.429 4.616692 1604.06 7.969788 2682.616 Mar 3.696451 1218.768 3.875041 1288.495 3.838115 1273.992 3.820356 1280.707 6.885492 2267.139 Apr 4.330271 1543.283 4.614292 1658.886 4.432312 1586.132 4.614475 1672.083 7.843856 2755.391 May 5.574658 2160.955 5.923943 2306.216 5.977383 2331.081 6.001755 2371.869 9.070539 3475.827 Jun 3.551048 1622.904 3.902762 1793.313 3.783588 1713.914 3.677367 1694.641 5.400061 2442.063 July 2.96804 1473.355 3.543854 1762.451 4.270406 2167.944 2.999465 1516.806 3.808573 1921.726 Aug 3.757903 1789.257 4.233035 2033.697 4.785633 2313.383 3.826978 1869.077 6.160521 2886.897 Sept 4.344463 1886.302 4.714515 2039.427 5.173031 2238.765 4.326536 1898.989 6.374026 2767.847 Oct 4.88653 1918.785 5.297524 2083.694 5.188784 2036.36 5.05911 2026.186 8.842943 3376.523 Nov 4.045829 1353.42 4.157906 1400.24 4.399363 1495.028 4.203166 1429.131 7.613245 2588.779 Dec 6.444455 2497.858 6.495171 2524.576 6.956649 2678.913 6.704293 2632.203 10.74334 4032.632
Fig. 10. Comparison of MAE and MAPE of ﬁve forecasting models.
smaller the standard deviation is, the stronger the stability will be.
MOFTL-ANN forecasting model is less that NSGA-II-ANN and GRNN.
Table 14 shows the statistical values of MAE and NMSE for by
However, the computation time of the proposed algorithm
ﬁve forecasting models over 50 iterations. MOFTL-ANN yields best
(MOFTL) and proposed hybrid model (MOFTL-ANN) is little longer
statistical values such as a minimum MAE value of 457.8729,
than some of the comparison models, but this seems justiﬁed due
maximum MAE of 547.7367, an average MAE value of 504.1671 and
to its superior forecasting abilities. Further this computation time
standard deviation of 547.7367 while other forecasting models
can be minimized by using a high-performance systems.
yielded higher statistical values than our proposed hybrid model
for case study 1. Table also reveals that NMSE statistical values of
MOFTL-ANN have lower values than other forecasting models for 5.4. Summary
case study 1. The statistics show that MAE and NMSE of MOFTL-
ANN yields the least standard deviation of 23.0549 and 0.001633
Based on the analysis drawn from experiment- I, experiment- II respectively for case study 2.
and the discussions above, the following conclusions can be drawn:
Tabular results obtained show that MOFTL-ANN is superior to
The proposed hybrid forecasting model based on MOFTL has very
compared forecasting models. Therefore, we can conclude that
strong forecasting ability and more stable forecasting results than
MOFTL-ANN achieves the desirable forecasting stability, is more
FTL-ANN, NSGA-II-ANN, BPNN, and GRNN.
stable than other compared models and can be an effective tech-
nique for electricity load forecasting in the real electricity market.
MOFTL optimization algorithm outperforms MOWCA, MOPSO,
and NSGA-II over multi-objective test functions. Also, MOFTL 5.3. Run time
has shorter computation time values of 178.747056 s, 157.5818 s,
147.469428 s, 165.879441 s for FON, SCH, ZDT1 and ZDT3 multi-
All the experiments are performed in the MATLAB R2017a
objective test functions over MOWCA and NSGA-II.
software on a Windows 10, 64-bit machine with Intel(R) Core(TM)
MOFTL-ANN, a hybrid model enhanced the load forecasting
i5 CPU 760 @ 2.80 GHz with 1 processor.
accuracy by improving the forecasting ability of ANN. The pro-
Table 15 and Table 16 shows the computation time in seconds
posed hybrid model shows improvement of 17.42%, 6.81%,
obtained from experiment 1 and experiment 2. It can be observed
10.77% and 59.69% MAPE values for England region and 4.20%,
that the running time of MOFTL is 178.747056 s, 157.5818 s,
4.16%, 1.14% and 21.85% MAPE values for ERCOT region over
147.469428s and 165.879441 s for FON, SCH, ZDT1 and ZDT3 in
NSGA-II-ANN, FTL-ANN, BPNN, and GRNN forecasting models.
Experiment I and 1485.315609 s and 1698.172034 s in Experiment
Results from Friedman statistical test show that all compared
II. This show that the computation time of MOFTL is shorter than
forecasting models generate different forecasting results.
MOWCA and NSGA-II and demonstrates that it can ﬁnd Pareto
Moreover, standard deviation values of 0.006027 and 0.001633
optimal solutions in less time. Similarly, the computation time of
for England region and ERCOT region by the proposed hybrid
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 619
Fig. 11. Comparison of observed load with forecast load by ﬁve forecasting models.
Fig. 12. Relationship between observed load and forecasted load by ﬁve forecasting models for the ERCOT region. Table 13
Results of Friedman test for Experiment II. Compared Models Signiﬁcant level (a ¼ 0:05) Compared Models Signiﬁcant level (a ¼ 0:05) Case Study I Case Study II MOFTL-ANN vs. NSGA-II-ANN
H0 : e1 ¼ e2 ¼ e3 ¼ e4 ¼ e5 MOFTL-ANN vs. NSGA-II-ANN
H0 : e1 ¼ e2 ¼ e3 ¼ e4 ¼ e5 MOFTL-ANN vs. FTL-ANN F ¼ 8470.368 MOFTL-ANN vs. FTL-ANN F ¼ 3368.516 MOFTL-ANNR vs. BPNN P ¼ 0.00000 (Reject H0) MOFTL-ANNR vs. BPNN P ¼ 0.00000 (Reject H0) MOFTL-ANN vs. GRNN MOFTL-ANN vs. GRNN
model is least compared to other forecasting models which
MOFTL-ANN, it can be considered as a new viable option for
describe that MOFTL-ANN is the most stable one.
time series forecasting abilities.
A novel multi-objective optimization algorithm, namely MOFTL,
outperforms MOWCA, MOPSO, and NSGA-II and it adds a new
Remark. Compared to the state-of-the-art models, MOFTL-ANN
optimization algorithm for solving multi-objective optimization
generates better results for experiment-I and experiment- II based
problems. Due to high accuracy and stable load forecasting by
on evaluation criteria. On the basis of obtained results, we can 620
P. Singh, P. Dwivedi / Energy 182 (2019) 606e622 Table 14
Statistical values of MAE and NMSE obtained from forecasting results of England and ERCOT region by forecasting models. Experiment II Forecasting model MAE NMSE Case Study - 1 Min Max Avg Std Min Max Avg Std MOFTL-ANN 457.8729 547.7367 504.1671 27.50089 0.044649 0.063359 0.054012 0.006027 NSGA-II-ANN 582.4166 780.1065 629.1621 57.21251 0.076147 0.125706 0.088878 0.014146 FTL-ANN 523.3067 601.5597 553.2902 27.89063 0.060027 0.079191 0.068972 0.007396 BPNN 458.1641 1057.095 579.5099 177.1325 0.044253 0.22041 0.080456 0.051533 GRNN 624.5459 1812.585 1199.392 437.0547 0.092557 0.654436 0.328479 0.206532 Experiment II MOFTL-ANN 1856.41 1918.213 1891.478 23.05498 0.075212 0.079764 0.077475 0.001633 Case Study - 2 NSGA-II-ANN 1921.28 13876.34 3166.98 3762.978 0.079913 2.313992 0.306608 0.705325 FTL-ANN 2050.343 8963.795 6646.426 2081.043 0.090739 1.401217 0.85099 0.413967 BPNN 1825.277 2037.037 1924.832 72.61513 0.077143 0.091153 0.083703 0.005043 GRNN 2051.709 2948.473 2405.141 310.9689 0.089878 0.18139 0.116847 0.029389 Table 15
To achieve such forecasting model, a novel multi-objective
Computation time (seconds) of multi-objective algorithms over FON, SCH, ZDT1 and
optimization algorithm MOFTL have been proposed and com- ZDT2 test functions.
bined with the artiﬁcial neural network to optimize weights. To Models FON SCH ZDT1 ZDT3
verify the performance of the proposed hybrid model, nine per- MOFTL 178.747056 157.5818 147.469428 165.879441
formance evaluation metrics including MAE, MEP, RMSE, NMSE, MOWCA 191.321965 260.145963 229.378474 223.891435
MAPE, Pearson's correlation coefﬁcient, DC and IA have been used MOPSO 58.09828 68.715269 67.32384 54.446691
in this paper. Based on the series of results drawn from experiment NSGA-II 227.947349 207.794532 162.985784 162.985784
II, the performance of our proposed hybrid model over two data
sets shows that MOFTL-ANN is capable of predicting load with
higher accuracy and effective stability. Experiment II shows the Table 16
effectiveness of the proposed hybrid models over compared models
Computation time (seconds) of multi-objective algorithms over case study - 1 and
for electricity load forecast. In both the case study, MOFTL generates case study - 2.
superior results than NSGA-II-ANN, FTL-ANN, BPNN, and GRNN for Models Case Study I Case Study II
all performance evaluation indexes with lesser computation time. MOFTL-ANN 1485.315609 1698.172034
MOFTL-ANN shows improvement of 17.42%, 6.81%, 10.77% and NSGA-II-ANN 2477.819376 2866.610707
59.69% MAPE values over NSGA-II-ANN, FTL-ANN, BPNN and GRNN FTL-ANN 1137.724773 2978.105397
for England region and 4.20%, 4.16%, 1.14% and 21.85% MAPE values BPNN 524.8064923 399.106586
over NSGA-II-ANN, FTL-ANN, BPNN and GRNN for ERCOT region. GRNN 3600.839397 3162.42325
deduce that the MOFTL, can be used for solving multi-objective
optimization problems. Additionally, MOFTL-ANN is a robust, Abbreviations
highly accurate and stable hybrid model which can be used as a practical forecasting model. ACO Ant Colony Optimization ANN Artiﬁcial Neural Network BPNN Backpropagation Neural Network
5.5. Real applications, limitations and future work DC Directional Change DE Differential Evolutionary
The proposed forecasting model can contribute to the electricity ERCOT
Electric Reliability Council of Texas
market in dealing with future changes in load generation and FTL Follow The leader
transmission. It can be used for many real applications such as wind GA Genetic Algorithm
energy forecasting, hydro energy forecasting that are based on time GRNN
Generalized Regression Neural Network
series forecasting as the proposed hybrid forecasting model shows IA Index of Agreement
better ability to predict the future load with better accuracy and MAE Mean Absolute Error
stability. There are still some issues that must be resolved in future MEP Maximum Error Percentage
studies: load affecting factors such as rainfall and wind speed can MAPE Mean Absolute Percentage Error
be considered for more reliable and practical forecasting; advanced MOALO
Multi-Objective Ant Lion Optimizer
data pre-processing method can improve the forecasting results. MOFA
Multi-Objective Fireﬂy Algorithm MOFPA
Multi-Objective Flower Pollination Algorithm 6. Conclusions MOFTL
Multi-Objective Follow The Leader MOP Multi-Objective Problem
Higher accuracy and excellent stability in electricity load fore- MOPSO
Multi-Objective Particle Swarm Optimization
casting play a very important role in the operation of power sys- MOWCA
Multi-objective Water Cycle Algorithm
tems. This paper provides a great tool for energy providers to know MOWOA
Multi-Objective Whale Optimization Algorithm
the amount of load demand in a particular region. The results NSGA-II Non-dominated Sorting GA
achieved from forecasting models show that our proposed hybrid NMSE Normalized Mean Squared Error
model results in outstanding forecasting performance. It is highly PSO Particle Swarm Optimization
desirable to develop a model that enhances both accuracy and RMSE Root Mean Squared Error stability simultaneously. SVM Support Vector Machine
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Document Outline

A novel hybrid model based on neural network and multi-objective optimization for effective load forecast
- 1. Introduction
- 2. Multi-objective follow the leader algorithm
  - 2.1. Multi-objective optimization
  - 2.2. MOFTL (multi-objective follow the leader)
- 3. Proposed hybrid forecasting model
  - 3.1. Artificial neural network
  - 3.2. The MOFTL-ANN hybrid model
- 4. Experiments and analysis
  - 4.1. Experiment I: tests of MOFTL
  - 4.2. Experiment II: tests of MOFTL-ANN
  - 4.3. Data pre-processing
  - 4.4. Evaluation metrics
  - 4.5. Case study-1: ISO England
    - 4.5.1. Data description
    - 4.5.2. Result
  - 4.6. Case study-2: ERCOT
    - 4.6.1. Data description
    - 4.6.2. Results
- 5. Discussions
  - 5.1. The significance of MOFTL-ANN: based on the Friedman statistical test
  - 5.2. Stability of the forecasting models
  - 5.3. Run time
  - 5.4. Summary
  - 5.5. Real applications, limitations and future work
- 6. Conclusions
- Abbreviations
- References

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