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Power load forecasts based on hybrid PSO with Gaussian and adaptive mutation

and Wv-SVM

Qi Wu *

Key Laboratory of Measurement and Control of CSE (School of Automation, Southeast University), Ministry of Education, Nanjing, Jiangsu 210096, China

School of Mechanical Engineering, Southeast University, Nanjing, Jiangsu 210096, China

a r t i c l e i n f o

Keywords:

Load forecasts

Wv-SVM

Particle swarm optimization

Adaptive mutation

Gaussian mutation

a b s t r a c t

This paper presents a new load forecasting model based on hybrid particle swarm optimization with

Gaussian and adaptive mutation (HAGPSO) and wavelet v -support vector machine (Wv-SVM). Firstly, it

is proved that mother wavelet function can build a set of complete base through horizontal floating

and form the wavelet kernel function. And then, Wv-SVM with wavelet kernel function is proposed in this

paper. Secondly, aiming to the disadvantage of standard PSO, HAGPSO is proposed to seek the optimal

parameter of Wv-SVM. Finally, the load forecasting model based on HAGPSO and Wv-SVM is proposed

in this paper. The results of application in load forecasts show the proposed model is effective and

feasible.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

The theoretical study of load forecasting of power systems

started in the middle of last century, simultaneously with the

flourishing of system identification and modern control theories,

etc. Before that, because the scales of power systems were limited

and there were many uncertain factors, the study of load forecast-

ing had not taken shape. It was not until the 1980s that the theo-

retical study of mid-long term load forecasting began to occur, and

a series of forecasting methods, such as AR algorithm, MA algo-

rithm, General Exponential Smoothing algorithm, ARMA algorithm

and ARIMA algorithm, had been successively developed and are

widely accepted in the load forecasting of power systems at pres-

ent (Chenhui, 1987). With the improvement of the grey system,

manual neural network, expert system, genetic algorithm (Wu,

Yan, & Yang, 2008a) and other theories and methods, the method

of mid-long term load forecasting of power systems has continu-

ously improved (Benaouda, Murtagh, Starck, & Renaud, 2006;

Liang, 1997; Santos, Martins, & Pires, 2007; Topalli, Erkmen, &

Topalli, 2006; Ying & Pan, 2008). In general, most of the algorithms

above are based on the time series.

Recently, SVM which was developed by Vapnik (1995) is one of

the methods that receives increasing attention with remarkable re-

sults in the field of load forecasting (Hong, 2009; Pai & Hong, 2005;

Wu, Tzeng, & Lin, 2009). The main difference between NN and SVM

is the principleof riskminimization.ANN implements empiricalrisk

minimization (ERM) to minimize the error on the training data,

while SVM implements the principle of structural risk minimization

(SRM) by constructing an optimal separating hyper-plane in the hid-

den feature space, and using quadratic programming to find a un-

ique solution. SVM has yielded excellent generalization

performance that is significantly better than that of competingmethods in load forecasts(Hong, 2009; Pai& Hong, 2005; Wu,Tzeng

et al., 2009). However, for our used kernel functions so far, the SVM

cannot approach any curve in L2ðRnÞ space (quadratic continuous

integral space), because the kernel function which is used now is

not the complete orthonormal base. This character lead the SVM

cannot approach every curve in the L2ðRnÞ space. Similarly, the

regression SVMcannot approach every function. Therefore we need

find a new kernel function, and this function can build a set of com-

plete base through horizontal floating and flexing. As we know, this

kind of function has already existed, and it is the wavelet functions.

The SVM with wavelet kernel function is called by wavelet SVM

(WSVM). Reviewing the load forecasts literatures about support

vector machine technique (Hong, 2009; Pai & Hong, 2005; Wu,

Tzeng et al., 2009), little has been written about in the literature

on application of Wv-SVM to load forecast research field.

However, the confirmation of unknown parameters of the Wv-

SVMis complicated process. In fact,it is a multivariableoptimization

problem in a continuous space. The appropriate parameter combi-

nation of models can enhance approximating degree of the original

series. Therefore, it is necessary to select an evolutionary algorithm

to seek theoptimal parameters of Wv-SVM. These unknown param-

eters have a great effect on the generalization performance of Wv-

SVM. An appropriate parameter combination corresponds to a high

generalization performance of Wv-SVM. Particle swarm optimiza-

tion (PSO), which is an evolutionary computation technique devel-

oped by Kennedy and Eberhart (1995), is considered as an

excellent technique to solve the combinatorial optimization prob-

0957-4174/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.eswa.2009.05.011

* Tel.: +86 25 51166581; fax: +86 25 511665260.

E-mail address: [email protected]

Expert Systems with Applications 37 (2010) 194–201

Contents lists available at ScienceDirect

Expert Systems with Applications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w a

mailto:[email protected]://www.sciencedirect.com/science/journal/09574174http://www.elsevier.com/locate/eswahttp://www.elsevier.com/locate/eswahttp://www.sciencedirect.com/science/journal/09574174mailto:[email protected]

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lems (Lin, Ying, Chen, & Lee, 2008; Shen, Shi, Kong, & Ye, 2007; Wu,

Liu, Xiong, & Liu, 2009; Wu, Yan, & Wang, 2009; Wu, 2009; Wu, Yan,

& Yang, 2008b; Wu, & Yan, 2009, in press; Yuan & Chu, 2007; Yang,

Yuan, Yuan, & Mao, 2007; Zhao & Yang, 2009).

PSO is based on the metaphor of social interaction and commu-

nication such as bird flocking. Original PSO is distinctly different

from other evolutionary-type methods in a way that it does not

use the filtering operation (such as crossover and mutation) andthe members of the entire population are maintained through the

search procedure so that information is socially shared among indi-

viduals to direct the search towards the best position in the search

space. Oneof themajor drawbacks of the standard PSOis its prema-

ture convergence. To overcome the shortage, there have been a lot

of reported works focused on the modification PSO such as in (Lin

et al., 2008; Shen et al., 2007; Wu et al., 2008b; Yuan & Chu,

2007; Zhao & Yang, 2009) to solve the parameter selection prob-

lemsof SVM,but little attention is given inWv-SVM. And then, a hy-

brid PSO with adaptive mutation and Gaussian mutation (HAGPSO)

is proposed to optimize the parameters of Wv-SVM in this paper.

Based on the above analysis, a new load forecasting model

based and Wv-SVM is proposed in this paper. Their superiority

over traditional model is verified in numerical simulation. The rest

of this paper is organized as follows. Section 2 introduces Wv-SVM.

HAGPSO is arranged in Section 3. In Section 4 the steps of HAGPSO

and forecasting method are described. Section 5 gives experimen-

tal simulation and results. Conclusions are drawn in the end.

2. Wavelet v -support vector machine (W v -SVM)

2.1. Wavelet kernel theory

Let us consider a set of data points ð x1; y1Þ; ð x2; y2Þ; . . . ; ð xl; ylÞ,

which are independently and randomly generated from an un-

known function. Specifically, xi is a column vector of attributes, y iis a scalar, which represents the dependent variable, and l denotes

the number of data points in the training set.The support vector’s kernel function can be described as not

only the product of point, such as K ð x; x0Þ ¼ K ðÞ, but also

the horizontal floating function, such as K ð x; x0Þ ¼ K ð x x0Þ. In fact,

if a function satisfied condition of Mercer, it is the allowable sup-

port vector kernel function.

Lemma 1 (Mercer, 1909). The symmetry function K ð x; x0Þ is the

kernel function of SVM if and only if: for all function g – 0 which

satisfies the condition of R

Rn g 2ðnÞdn < 1, we need satisfy the

condition as follows:Z Z K ð x; x0Þ g ð xÞ g ð xÞdxdx0 P 0; x; x0 2 R n ð1Þ

This theorem proposed a simple method to build kernel function.

For the horizontal floating function, because hardly dividing this function into two same functions, we can give the condition of

horizontal floating kernel function.

Lemma 2 (Smola and Scholkopf, 1998). The horizontal floating func-

tion is allowable support vector’s kernel function if and only if the Fou-

rier transform of K ð xÞ need satisfy the condition follows:

F ½ xðxÞ ¼ ð2pÞn=2Z

Rnexpð jðx: xÞÞK ð xÞdx P 0; x 2 R n ð2Þ

If the wavelet function wð xÞ satisfied the conditions: wð xÞ 2

L2ðRÞ \ L1ðRÞ, and ŵð xÞ ¼ 0; ŵ is the Fourier transform of function

wð xÞ. The wavelet function group can be defined as:

wa;mð xÞ ¼ ðaÞ12w x ma

; x 2 R ð3Þ

where a is the so-called scaling parameter, m is the horizontal float-

ing coefficient, and wð xÞ is called the ‘‘mother wavelet”. The param-

eter of translation m 2 R and dilation a > 0, may be continuous or

discrete. For the function f ð xÞ; f ð xÞ 2 L2ðRÞ, The wavelet transform

f ð xÞ can be defined as:

W ða; mÞ ¼ ðaÞ12 Z þ1

1

f ð xÞw x ma dx ð4Þ

where wð xÞ stands for the complex conjugation of wð xÞ.

The wavelet transform W ða;mÞ can be considered as functions

of translation m with each scale a. Eq. (4) indicates the wavelet

analysis is a time-frequency analysis, or a time-scaled analysis. Dif-

ferent from the Short Time Fourier Transform (STFT), the wavelet

transform can be used for multi-scale analysis of a signal through

dilation and translation so it can extract time-frequency features

of a signal effectively.

Wavelet transform is also reversible, which provides the possi-

bility to reconstruct the original signal. A classical inversion for-

mula for f ð xÞ is:

f ð xÞ ¼ C 1wZ þ1

1

Z þ11

W ða; mÞwa;mð xÞda

a2 dm ð5Þ

where

C w ¼Z 1

1

jŵðwÞj2jwj dw 0. So far, because

the wavelet kernel function must satisfy the conditions of Lemma 2,

the number of wavelet kernel function which can be showed by

existent functions is few. Now, we give an existent wavelet kernel

function: Morlet wavelet kernel function, and we can prove that

this function can satisfy the condition of allowable support vector’s

kernel function. Morlet wavelet function is defined as follows:

w

ð x

Þ ¼ cos

ð1:75 x

Þexp

x2

2

ð10

Þ Theorem 1. Morlet wavelet kernel function is defined as:

K ð x; x0Þ ¼Yli¼1

cos 1:75 xi x0i

a

exp k xi x

0ik2

2a2

!;

x 2 R ld; xi 2 R d ð11Þand this kernel function is an allowable support vector kernel function.

Proof. According to Lemma 2, we only need to prove

F ½ xðxÞ ¼ ð2pÞl=2Z

Rldexpð jðx: xÞÞK ð xÞdx P 0 ð12Þ

where K ð xÞ ¼Ql

i¼1w

xi

a

¼Ql

i¼1 cos

1:75 xi

a

expðk xik

2=2a2Þ

; j denotesimaginary number unit. We have

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Z R

ldexpð jx xÞK ð xÞdx ¼

Z R

ldexpð jx xÞ

Yli¼1

cos 1:75 xia

exp k xik2

2a2

!!dx

¼ Yl

i¼1 Z 1

1exp

ð jxi xi

Þ expð j1:75 xi=aÞ þ expð j1:75 xi=aÞ

2

exp k xik2

2a2

!dxi ¼

Yli¼1

1

2

Z 11

exp k xik2

2 þ 1:75 j

a jxia

xi

!

þ exp k xik2

2 1:75i

a þ jxia

xi

!!

¼ Yl

i¼1

jaj ffiffiffiffiffiffiffi2pp

2

exp

ð1:75 xiaÞ2

2 !

þ exp ð1:75 þxiaÞ2

2

!! ð13Þ

Substituting formula (13) into Eq. (12), we can obtain Eq. (14).

F ½ xðxÞ ¼Yli¼1

jaj2

exp ð1:75 xiaÞ

2

2

!

þ exp ð1:75 þxiaÞ2

2

!! ð14Þ

where a – 0, we have

F ½ xðxÞ P 0 ð15Þ

This completes the proof of Theorem 1. h

2.2. Wavelet v -support vector machine

Combining the wavelet kernel function with v -SVM, we can

build a new SVM learning algorithm that is Wv-SVM. The structure

of Wv-SVM is shown in Fig. 1. For a set of data points ð x1; y1Þ;

ð x2; y2Þ; . . . ; ð xl; ylÞ, Wv-SVM can be described as:

minw;nðÞ ;e;b

sðw; nðÞ; eÞ ¼ 12

kwk2 þ C v eþ 1l

Xli¼1

ðni þ ni Þ !

ð16Þ

Subject to ðw xi þ bÞ yi 6 eþ ni ð17Þ yi ðw xi þ bÞ 6 eþ ni ð18Þn

ðÞi P 0; eP 0; b 2 R ð19Þ

where w and x i are a column vector with d dimensions, C > 0 is a

penalty factor, nðÞi ði ¼ 1; . . . ; lÞ are slack variables and m 2 (0,1] isan adjustable regularization parameter.

Problem (16) is a quadratic programming (QP) problem. By

means of the Wolfe principle, wavelet kernel function technique

and Karush–Kuhn–Tucker (KKT) conditions, we have the duality

problem (20) of the original optimal problem (16).

maxa;a

W ða; aÞ ¼ 12

Xli; j¼1

ðai aiÞða j a jÞK ð xi x jÞ

þXli¼1

ðai aiÞ yi ð20Þ

s:t: 0 6 ai;ai 6

C

l ð21ÞXli¼1

ðai aiÞ ¼ 0 ð22Þ

Xli¼1

ðai þai Þ 6 C v ð23Þ

Select the appropriate parameters C and v , and the optimal

mother wavelet function which can match well the original series

in some scope of scales as the kernel function of Wv-SVM model.

Then, W v -SVM output function is described as following:

f ð xÞ ¼X

l

i¼1ðai ai Þ

Yl

i¼1w

x j x jia

!þ b; b 2 R ð24Þ

where wð xÞ is wavelet transform function, a is the scaling parameter

of wavelet, a > 0. x j is the jth value of test vector x. x ji is the jth value

of sample vector x i.

Parameter b can be computed by Eq. (25), select the two scalars

a jða j 2 ð0; l=C ÞÞ and akðak 2 ð0; l=C ÞÞ, then we have

b ¼ 12

y j þ yk Xli¼1

ðai aiÞK ð xi; x jÞ þXli¼1

ðai aiÞK ð xi; xkÞ !" #

ð25Þ

3. Hybrid particle swarm optimization

The confirmation of unknown parameters of the Wv-SVM iscomplicated process. In fact, It is a multivariable optimization

problem in a continuous space. The appropriate parameter combi-

nation of models can enhance approximating degree of the original

series Therefore, it is necessary to select an intelligence algorithm

to get the optimal parameters of the proposed models. The

parameters of Wv-SVM have a great effect on the generalization

performance of Wv-SVM. An appropriate parameter combination

corresponds to a high generalization performance of the Wv-SVM.

PSO algorithm is considered as an excellent technique to solve the

combinatorial optimization problems. The proposed HAGPSO algo-

rithm is used to determine the parameters of Wv-SVM. The intelli-

gence system shown in Fig. 2 based on the HAGPSO algorithm and

Wv-SVM model can evaluate the performance of HAGPSO algo-

rithm by forecasting time series. The different Wv-SVMs in the dif-ferent Hilbert spaces are adopted to forecast the power load time

y

x

Σ

1( , )K x x

2( , )K x x

1w

iw

( , )n

K x x

nw

( , )i

K x x

2w

1 x

2 x

n x

Fig. 1. The architecture of Wv-SVM.

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series. For each particular region only the most adequate Wv-SVM

with the optimal parameters is used for the final forecasting.

To valuate forecasting capacity of the intelligence system, the

fitness function of AGPSO algorithm is designed as follows:

fitness ¼ 1l

Xli¼1

yi yi yi

2ð26Þ

where l is the size of the selected sample, yi denote the forecastingvalue of the selected sample, yi is original date of the selected

sample.

3.1. Standard particle swarm optimization

Similarly to evolutionary computation techniques, PSO (Yang

et al., 2007) uses a set of particles, representing potential solutions

to the problem under consideration. The swarm consists of n par-

ticles. Each particle has a position X i ¼ ð xi1; xi2; . . . ; xij; . . . xidÞ, a

velocity V i ¼ ðv i1;v i2; . . . ;v ij; . . . ;v idÞ, where i ¼ 1;2; . . . ;N ; j ¼

1;2; . . . ; d, and moves through an d-dimensional search space.

According to the global variant of the PSO, each particle moves to-

wards its best previous position and towards the best particle pg in

the swarm. Let us denote the best previously visited position of theith particle that gives the best fitness value as pi ¼ ð pi1; pi2; . . . ;

pij; . . . ; pidÞ, and the best previously visited position of the swarm

that gives best fitness as pg ¼ ð pg 1; pg 2; . . . ; pg j; . . . ; pg dÞ.

The change of position of each particle from one iteration to an-

other can be computed according to the distance between the cur-

rent position and its previous best position and the distance

between the current position and the best position of swarm. Then

the updating of velocity and particle position can be obtained by

using the following equations:

v kþ1ij ¼ w v kij þ c 1 r 1 pij xkij

þ c 2 r 2 pg j xkij

ð27Þ

xkþ1ij ¼ xkij þ v kþ1ij ð28Þ

where w is called inertia weight and is employed to control theimpact of the previous history of velocities on the current one. k

denotes the iteration number, c 1 is the cognition learning factor,

c 2 is the social learning factor, r 1 and r 2 are random numbers uni-

formly distributed in the range [0,1].

Thus, the particle flies through potential solutions towards pkiand pg k in a navigated way while still exploring new areas by the

stochastic mechanism to escape from local optima. Since there

was no actual mechanism for controlling the velocity of a particle,

it was necessary to impose a maximum value V max on it. If thevelocity exceeds the threshold, it is set equal to V max, which con-

trols the maximum travel distance at each iteration to avoid this

particle flying past good solutions. The PSO is terminated with a

maximal number of generations or the best particle position of

the entire swarm cannot be improved further after a sufficiently

large number of generations. The PSO has shown its robustness

and efficacy in solving function value optimization problems in

real number spaces.

3.2. Hybrid particle swarm optimization with Gaussian mutation and

adaptive mutation

Aiming to the disadvantage of the standard PSO, the adaptive

mutation operator is proposed to regulate the inertia weight of

velocity by means of the fitness value of object function and itera-

tive variable. The Gaussian mutation operator is considered to cor-

rect the direction of particle velocity. The aforementioned problem

is addressed by incorporating adaptive mutation and Gaussian

mutation for the previous velocity of the particle. Thus, the HAG-

PSO can update the velocity and particle position by using the fol-

lowing equations:2

v kþ1ij ¼ ð1 kÞwkijv kij þ kN 0;rki

þ c 1r 1 pij xkij þ c 2r 2 pg j xkij

ð29Þ

xkþ1ij ¼ xkij þ v kþ1ij ð30Þwkij

¼ b 1

f xki = f xkm þ ð

1

b

Þw0ij exp

ðak2

Þ ð31

Þrkþ1i ¼ rki expðN ið0;MrÞÞ ð32Þ

Fig. 2. The AGPSO optimizes the parameters of Wv-SVM.


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where i ¼ 1; 2; . . . ; N ; t ¼ 1;2. Mr is standard error of Gaussiandistribution, b is the adaptive coefficient, k is an increment coeffi-

cient, a is the coefficient of controlling particle velocity attenuation, f xki

is the fitness of the ith particle in the kth iterative process.

f xkm

is the optimal fitness of particle swarms in the k iterative

process.

The parameter w regulates the trade-off between the global and

local exploration abilities of the swarm. A large inertia weight facil-

itates global exploration, while a small one tends to facilitate local

exploration. A suitable value of the inertia weight w usually pro-

vides balance between global and local exploration abilities and

consequently results in a reduction of the number of iterations re-

quired to locate the optimum solution.

Adaptive mutation, which makes the quality of the solution de-

pend on mutation operator, is high effective mutation operator in

real code. The proposed adaptive mutation operator based on iter-

ative variable k and the fitness function value f ð xkÞ is described in

Eq. (31). Then, in first item of the right of Eq. (29), velocity inertia

weight wkij can provide balance between global and local explora-

tion abilities and consequently results in a reduction of the number

of iterations required to locate the optimum solution. In Eq. (31)

1 f xki = f xkm represents the particles with the bigger fitness

mutate in a smaller scope, while the ones with the smaller fitness

mutate in a big scope. w0ij expðak2

Þ represents the initial inertia

weight w0ij mutate in big scope and search the local optimal value

in bigger space in the start moment (smaller kÞ, while the param-

eter w0ij mutate in small scope and search the global optimal value

in small space and gradually reach the global optimal value in the

end moment (bigger kÞ.

The second item of Eq. (29) represents Gaussian mutation based

on the iterative variable k. The Gaussian mutation operator which

can correct the moving direction of particle velocity is represented

inEq. (32). In thestrategyof Gaussianmutation, theproposedveloc-

ity vector v kþ1 ¼ v kþ11 ;v kþ12 ; ;v

kþ1d

consists of last generation

velocity vector v k ¼ v k1;v k2; ;v

kd

and perturbation vector rk ¼

rk1;rk2; . . . ;r

kd . The perturbation vector mutates itself by Eq. (32)

on the each iterative process as a controlling vector of velocityvector.

The adaptive and Gaussian mutation operators can restore the

diversity loss of the population and improve the capacity of the

global search of the algorithm.

4. The procedures of HAGPSO and W v -SVM

The HAGPSO algorithm is described in steps as follows:

Algorithm 1

Step 1. Data preparation: Training, validation, and test sets are

represented as Tr, Va, and Te, respectively.

Step 2. Particle initialization and PSO parameters setting: Gener-ate initial particles. Set the PSO parameters including

number of particles ðnÞ, particle dimension ðmÞ, number

of maximal iterations ðkmaxÞ, error limitation of the fitness

function, velocity limitation ðV maxÞ, and inertia weight for

particle velocity ðw0Þ, Gaussian distribution ðN ð0;MrÞÞ,the perturbation momentum ðr0i Þ; the coefficient of controlling particle velocity attenuation ðaÞ, adaptivecoefficient ðbÞ, increment coefficient ðkÞ. Set iterative vari-

able: k ¼ 0. And perform the training process from Step

3–8.

Step 3. Set iterative variable: k ¼ k þ 1.

Step 4. Compute the fitness function value of each particle. Take

current particle as individual extremum point of every par-

ticle and do the particle with minimal fitness value as theglobal extremum point.

Step 5. Stop condition checking: if stopping criteria (maximum

iterations predefined or the error accuracy of the fitness

function) are met, go to Step 8. Otherwise, go to the next

step.

Step 6. Adopt the adaptive mutation operator by Eq. (31) and

Gaussian mutation operator by Eq. (32) to manipulate par-

ticle velocity.

Step 7. Update the particle position by Eqs. (29) and (30) and formnew particle swarms, go to step 3.

Step 8. End the training procedure, output the optimal particle.

On the basis of the Wv-SVM model, we can summarize an

estimation algorithm as the follows.

Algorithm 2

Step 1. Initialize the original data by normalization and fuzzifica-

tion, then form training patterns.

Step 2. Select the appropriate wavelet kernel function K , the con-

trol constant m and the penalty factor C . Construct the QPproblem (16) of the Wv-SVM.

Step 3. Solve the optimization problem and obtain the parame-

ters aðÞi . Compute the regression coefficient b by (25).Step 4. For a new forecasting task, extract load characteristics and

form a set of input variables x. Then compute the estima-

tion result ̂y by (24).

5. Experiment

To analyze the performance of the proposed HAGPSO algorithm,

the forecast of power load series by means of the intelligence sys-

tem based on HAGPSO and Wv-SVM is studied. To compare the

performance of HAGPSO algorithm, the standard PSO is also

adopted to optimize the parameters of Wv-SVM. The better algo-

rithm will give the better combinational parameters of Wv-SVM.

Therefore, there is a good forecasting capability provided by the

better combinational parameters in the regression estimation of

Wv-SVM. Better algorithm provides better forecasting capability.

To evaluate forecasting capacity of the intelligent system, some

evaluation indexes, such as mean absolute error (MAE), mean

absolute percentage error (MAPE) and mean square error (MSE),

are adopted to deal with the forecasting results of HAGPSOWv-

SVM and PSOWv-SVM.

In our experiments, power load series are selected from past

load record in a typical power company. The detailed characteristic

data and load series compose the corresponding training and test-

ing sample sets. During the process of the power load forecasting,

six influencing factors shown in Table 1, viz., sunlight, data, air

pressure, temperature, rainfall and humidity are taken into ac-

count. All linguistic information of gotten influencing factors is

dealt with fuzzy comprehensive evaluation (Feng & Xu, 1999)and form numerical information. Suppose the number of variables

is n, and n ¼ n1 þ n2, where n1 and n2, respectively denote the

number of fuzzy linguistic variables and crisp numerical variables.

The linguistic variables are evaluated in several description levels,

Table 1

Influencing factors of power load forecasts.

Load characteristics Unit Expression Weight

Sunlight Dimensionless Linguistic information 0.9

Data Dimensionless Linguistic information 0.7

Air pressure Dimensionless Linguistic information 0.68

Temperature Dimensionless Linguistic information 0.8

Rainfall Dimensionless Linguistic information 0.7

Humidity Dimensionless Linguistic information 0.4


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and a real number between 0 and 1 can be assigned to each

description level. Distinct numerical variables have different

dimensions and should be normalized firstly. The following nor-

malization is adopted:

xei ¼ xei min xei

li¼1

max xei

l

i¼1

min xei

l

i¼1

; e ¼ 1; 2; . . . ; n2 ð33Þ

where l is the number of samples, xe

i and xe

i denote the original valueand the normalized value, respectively. In fact, all the numerical

variables from (1) through (32) are the normalized values although

they are not marked by bars.

The proposed HAGPSO algorithm has been implemented in

Matlab 7.1 programming language. The experiments are made on

a 1.80 GHz Core(TM)2 CPU personal computer (PC) with 1.0 G

memory under Microsoft Windows XP professional. The initial

parameters of HAGPSO are given as follows: inertia weight:

w0 ¼ 0:9; positive acceleration constants: c 1; c 2 ¼ 2; the standard

error of Gaussian distribution: Mr ¼ 0:5; the adaptive coefficientb ¼ 0:8; increment coefficient: k ¼ 0:1; the fitness accuracy of the

Fig. 3. Mexican hat wavelet transform of load series in the scope of different scale.

Fig. 4. Morlet wavelet transform of load series in the scope of different scale.

Fig. 5. Gaussian wavelet transform of load series in the scope of different scale.


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normalized samples is equal to 0.0002; the coefficient of control-

ling particle velocity attenuation: a ¼ 2.The Morlet, Mexican hot and Gaussian wavelet are selected to

analyze the load series on the different scales shown in Figs. 3–5.Morlet wavelet transform is the best wavelet transform that can

inosculate the original load series on the scope of scale from 0.3

to 4 among all given wavelet transforms.

Therefore, Morlet wavelet can be ascertained as a kernel func-

tion of Wv-SVM model, three parameters also are determined as

follows:

v 2 ½0; 1; a 2 ½0:3; 2 and

C 2 maxð xi; jÞ minð xi; jÞl

103; maxð xi: jÞ minð xi; jÞl

103

The trend of fitness value of HAGPSO is shown in Fig. 6. It is

obvious that the HAGPSO is convergent. Therefore, HAGPSO is able

to be applied to seek the parameters of Wv-SVM.

The optimal combinational parameters are obtained byAlgorithm HAGPSO, viz., C ¼ 960:10;v ¼ 0:88 and a ¼ 0:89. Fig. 7

illuminates the load series forecasting results given by HAGPSO

and Wv-SVM.

For analyzing the parameter searching capacity of HAGPSO

algorithm, the standard PSO algorithm is used to optimize param-

eters of Wv-SVM by training the original load series, then give the

latest 12 weeks forecasting results of each model shown in Table 2.

The comparison between HAGPSO and PSO optimizing the

parameters of the same model (Wv-SVM) is shown in Table 3.

The Table 3 shows the error index distribution from two differ-

ent models. The MAE, MAPE and MSE of HAGPSOWv-SVM are bet-

ter than ones of PSOWv-SVM. It is obvious that adaptive and

Gaussian mutation operators can improve the global search ability

of particle swarm optimization algorithm. Experiment resultsshow that the forecast’s precision is improved by HAGPSO, com-

pared with PSO under the same conditions.

6. Conclusion

In this paper, a new load forecasting model based on HAGPSO

and Wv-SVM is proposed. A new version of PSO, viz., hybrid parti-

cle swarm optimization with adaptive mutation and Gaussian

mutation (HAGPSO), is also proposed to optimize the parameters

of Wv-SVM. The performance of the HAGPSOWv-SVM is evaluated

by means of forecasting the data of power loads, and the simula-

tion results demonstrate that the Wv-SVM is effective in dealing

with many dimensions, nonlinearity and finite samples. Moreover,

it is shown that the HAGPSO presented here is available for theWv-SVM to seek optimized parameters.

Fig. 6. The change trend of the fitness function.

Fig. 7. The load forecasting results based on HAGPSOWv-SVM model.

Table 2

Comparison of forecasting result from two different models.

The latest 12 weeks Real value Forecasting value

PSOWv-SVM HAGPSOWv-SVM

1 580 725 703

2 2046 2010 20183 908 858 880

4 1625 1585 1606

5 452 547 525

6 2937 2880 2920

7 1135 1046 1167

8 2580 2493 2499

9 2561 2508 2566

10 781 908 884

11 1489 1536 1516

12 1532 1519 1525

Table 3

Error statistic of two forecasting models.

Model MAE MAPE MSE

PSOWv-SVM 69.92 0.068 6292

HAGPSOWv-SVM 45.25 0.048 3473


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In our experiments, the fixed adaptive coefficients ðb; kÞ, the

second step control parameter Mr of normal mutation and theparameter a of control the velocity attenuation are adopted. How-

ever, how to choose an appropriate coefficient is not described in

this paper. The research on the velocity changes when different

above parameters are adopted is a meaningful problem for future

research.

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